diff --git a/task1/08_sensitivity.py b/task1/08_sensitivity.py index c77df36..ee2fe38 100644 --- a/task1/08_sensitivity.py +++ b/task1/08_sensitivity.py @@ -44,7 +44,8 @@ def truncation_correction(mu, sigma, C, p_thresh): if p_trunc < p_thresh: return mu, p_trunc, False else: - mu_tilde = mu * (1 + 0.4 * p_trunc) + # 与主流程 `02_demand_correction.py` 保持一致：线性修正系数 0.1 + mu_tilde = mu * (1 + 0.1 * p_trunc) return mu_tilde, p_trunc, True diff --git a/task1/08_sensitivity.xlsx b/task1/08_sensitivity.xlsx index 47edcb9..71daee1 100644 Binary files a/task1/08_sensitivity.xlsx and b/task1/08_sensitivity.xlsx differ diff --git a/task1/export_paper_tables.py b/task1/export_paper_tables.py new file mode 100644 index 0000000..a5a3450 --- /dev/null +++ b/task1/export_paper_tables.py @@ -0,0 +1,121 @@ +from __future__ import annotations + +import math +from pathlib import Path + +import pandas as pd + + +ROOT = Path(__file__).resolve().parent +OUT_DIR = ROOT / "paper_tables" + + +def _to_int_series(s: pd.Series) -> pd.Series: + return s.map(lambda x: int(round(float(x))) if pd.notna(x) else x) + + +def _to_float_series(s: pd.Series, ndigits: int) -> pd.Series: + return s.map(lambda x: round(float(x), ndigits) if pd.notna(x) else x) + + +def _format_table1_metrics_summary(df: pd.DataFrame) -> pd.DataFrame: + out = df.copy() + out["E1_total_service"] = _to_int_series(out["E1_total_service"]) + out["E2_quality_weighted"] = _to_int_series(out["E2_quality_weighted"]) + out["F1_gini"] = _to_float_series(out["F1_gini"], 4) + out["F2_min_satisfaction"] = _to_float_series(out["F2_min_satisfaction"], 2) + out["F3_cv_satisfaction"] = _to_float_series(out["F3_cv_satisfaction"], 4) + out["E1_total_service_pct"] = _to_float_series(out["E1_total_service_pct"], 2) + out["E2_quality_weighted_pct"] = _to_float_series(out["E2_quality_weighted_pct"], 2) + return out + + +def _format_table2_corrected_sites(df: pd.DataFrame) -> pd.DataFrame: + out = df.copy() + out["site_id"] = _to_int_series(out["site_id"]) + out["mu"] = _to_float_series(out["mu"], 1) + out["sigma"] = _to_float_series(out["sigma"], 1) + out["p_trunc"] = _to_float_series(out["p_trunc"], 3) + out["mu_tilde"] = _to_float_series(out["mu_tilde"], 1) + out["mu_delta"] = _to_float_series(out["mu_delta"], 1) + out["mu_delta_pct"] = _to_float_series(out["mu_delta_pct"], 1) + return out + + +def _format_table3_backtest_summary(df: pd.DataFrame) -> pd.DataFrame: + out = df.copy() + out["value"] = out["value"].map(lambda x: round(float(x), 6) if pd.notna(x) else x) + out["p_value"] = out["p_value"].map(lambda x: round(float(x), 6) if pd.notna(x) else x) + return out + + +def _format_appendix_site_details(df: pd.DataFrame) -> pd.DataFrame: + out = df.copy() + out["site_id"] = _to_int_series(out["site_id"]) + out["mu"] = _to_float_series(out["mu"], 1) + out["mu_tilde"] = _to_float_series(out["mu_tilde"], 1) + out["k"] = _to_int_series(out["k"]) + out["annual_service"] = _to_int_series(out["annual_service"]) + out["r"] = _to_float_series(out["r"], 4) + out["quality_factor"] = _to_float_series(out["quality_factor"], 4) + out["service_quality_weighted"] = _to_int_series(out["service_quality_weighted"]) + return out + + +def export_tables() -> list[Path]: + OUT_DIR.mkdir(parents=True, exist_ok=True) + written: list[Path] = [] + + # Table 1: overall performance comparison + metrics_xlsx = ROOT / "04_metrics.xlsx" + metrics = pd.read_excel(metrics_xlsx, sheet_name="metrics_summary") + metrics_paper = _format_table1_metrics_summary(metrics) + p1 = OUT_DIR / "table1_metrics_summary.csv" + metrics_paper.to_csv(p1, index=False, encoding="utf-8-sig") + written.append(p1) + + # Table 2: truncation correction (corrected sites) + backtest_xlsx = ROOT / "07_backtest.xlsx" + corrected = pd.read_excel(backtest_xlsx, sheet_name="corrected_sites") + corrected = corrected.copy() + corrected["mu_delta"] = corrected["mu_tilde"] - corrected["mu"] + corrected["mu_delta_pct"] = corrected["mu_delta"] / corrected["mu"] * 100 + corrected_paper = _format_table2_corrected_sites(corrected) + p2 = OUT_DIR / "table2_corrected_sites.csv" + corrected_paper.to_csv(p2, index=False, encoding="utf-8-sig") + written.append(p2) + + # Table 3: backtest & fit statistics + summary = pd.read_excel(backtest_xlsx, sheet_name="summary_metrics") + summary_paper = _format_table3_backtest_summary(summary) + p3 = OUT_DIR / "table3_backtest_summary.csv" + summary_paper.to_csv(p3, index=False, encoding="utf-8-sig") + written.append(p3) + + # Table 4: constraint validation results + validate_xlsx = ROOT / "06_validate.xlsx" + validation = pd.read_excel(validate_xlsx, sheet_name="validation_results") + p4 = OUT_DIR / "table4_validation_results.csv" + validation.to_csv(p4, index=False, encoding="utf-8-sig") + written.append(p4) + + # Appendix: per-site allocation details (optional for paper appendix) + site_details = pd.read_excel(metrics_xlsx, sheet_name="site_details") + site_details_paper = _format_appendix_site_details(site_details) + p5 = OUT_DIR / "appendix_site_details.csv" + site_details_paper.to_csv(p5, index=False, encoding="utf-8-sig") + written.append(p5) + + return written + + +def main() -> None: + written = export_tables() + rel = [p.relative_to(ROOT) for p in written] + print("Wrote:") + for p in rel: + print(f" - {p}") + + +if __name__ == "__main__": + main() diff --git a/task1/figures/fig1_site_map.png b/task1/figures/fig1_site_map.png index db8eeb6..07b3435 100644 Binary files a/task1/figures/fig1_site_map.png and b/task1/figures/fig1_site_map.png differ diff --git a/task1/figures/fig2_demand_correction.png b/task1/figures/fig2_demand_correction.png index 080f1d1..facd755 100644 Binary files a/task1/figures/fig2_demand_correction.png and b/task1/figures/fig2_demand_correction.png differ diff --git a/task1/figures/fig3_k_distribution.png b/task1/figures/fig3_k_distribution.png index 3a3541d..e9f77c3 100644 Binary files a/task1/figures/fig3_k_distribution.png and b/task1/figures/fig3_k_distribution.png differ diff --git a/task1/figures/fig4_efficiency_fairness.png b/task1/figures/fig4_efficiency_fairness.png index b1bf0a7..25b81d1 100644 Binary files a/task1/figures/fig4_efficiency_fairness.png and b/task1/figures/fig4_efficiency_fairness.png differ diff --git a/task1/figures/fig5_calendar_heatmap.png b/task1/figures/fig5_calendar_heatmap.png index 03af98c..7bed6a5 100644 Binary files a/task1/figures/fig5_calendar_heatmap.png and b/task1/figures/fig5_calendar_heatmap.png differ diff --git a/task1/figures/fig6_gap_boxplot.png b/task1/figures/fig6_gap_boxplot.png index d372b27..efdf312 100644 Binary files a/task1/figures/fig6_gap_boxplot.png and b/task1/figures/fig6_gap_boxplot.png differ diff --git a/task1/figures/fig7_sensitivity.png b/task1/figures/fig7_sensitivity.png index 0ede977..0ca0cff 100644 Binary files a/task1/figures/fig7_sensitivity.png and b/task1/figures/fig7_sensitivity.png differ diff --git a/task1/paper_tables/appendix_site_details.csv b/task1/paper_tables/appendix_site_details.csv new file mode 100644 index 0000000..f8da828 --- /dev/null +++ b/task1/paper_tables/appendix_site_details.csv @@ -0,0 +1,71 @@ +site_id,site_name,mu,mu_tilde,k,annual_service,r,quality_factor,service_quality_weighted +1.0,MFP American Legion - Binghamton,200,200,14.0,2803,14.0,1.0,2803 +2.0,MFP Avoca,315,323,22.0,6921,21.4249,0.7947,5500 +3.0,MFP Bath,279,279,19.0,5310,19.0,0.8946,4750 +4.0,MFP Beaver Dams,171,171,12.0,2048,12.0,1.0,2048 +5.0,MFP Birnie Transportation Services,213,213,15.0,3201,15.0,1.0,3201 +6.0,MFP Boys and Girls Club,211,211,15.0,3162,15.0,1.0,3162 +7.0,MFP Bradford,122,122,9.0,1100,9.0,1.0,1100 +8.0,MFP Campbell,168,168,12.0,2022,12.0,1.0,2022 +9.0,MFP Canisteo,177,177,13.0,2301,13.0,1.0,2301 +10.0,MFP Colesville,197,197,14.0,2763,14.0,1.0,2763 +11.0,MFP College Corning Community College,251,251,18.0,4518,18.0,0.996,4500 +12.0,MFP College Ithaca College,138,138,10.0,1383,10.0,1.0,1383 +13.0,MFP College TC3 -College,262,266,19.0,4968,18.686,0.956,4750 +14.0,MFP Conklin- Maines Community Center,153,153,11.0,1685,11.0,1.0,1685 +15.0,MFP Danby,160,160,12.0,1923,12.0,1.0,1923 +16.0,MFP Deposit,157,157,11.0,1722,11.0,1.0,1722 +17.0,MFP Endwell United Methodist Church,285,289,20.0,5705,19.7173,0.8764,5000 +18.0,MFP Erin,174,174,13.0,2261,13.0,1.0,2261 +19.0,MFP First Assembly Of God Church,146,146,11.0,1606,11.0,1.0,1606 +20.0,MFP Lamphear Court,126,126,9.0,1134,9.0,1.0,1134 +21.0,MFP Lansing,181,181,13.0,2353,13.0,1.0,2353 +22.0,MFP Lindley,233,233,16.0,3726,16.0,1.0,3726 +23.0,MFP Millport,166,166,12.0,1992,12.0,1.0,1992 +24.0,MFP Montour Falls-Schuyler County Human Services Complex,149,149,11.0,1643,11.0,1.0,1643 +25.0,MFP Nichols-The Creamery,122,122,9.0,1102,9.0,1.0,1102 +26.0,MFP Owego VFW,176,176,13.0,2291,13.0,1.0,2291 +27.0,MFP Prattsburgh,144,144,11.0,1590,11.0,1.0,1590 +28.0,MFP Rathbone,269,269,19.0,5113,19.0,0.9291,4750 +29.0,MFP Reach for Christ Church Freeville,220,220,16.0,3520,16.0,1.0,3520 +30.0,MFP Redeemer Lutheran Church,231,233,16.0,3690,15.8404,1.0,3690 +31.0,MFP Rehoboth Deliverance Ministry,236,236,17.0,4010,17.0,1.0,4010 +32.0,MFP Richford,266,266,19.0,5052,19.0,0.9402,4750 +33.0,MFP Saint Mary Recreation Center,148,148,11.0,1631,11.0,1.0,1631 +34.0,MFP Salvation Army Ithaca,181,181,13.0,2355,13.0,1.0,2355 +35.0,MFP Schuyler Outreach,139,139,10.0,1389,10.0,1.0,1389 +36.0,MFP Senior - Addison Place Apartments,30,30,3.0,90,3.0,1.0,90 +37.0,MFP Senior - Bragg,67,67,5.0,334,5.0,1.0,334 +38.0,MFP Senior - Carpenter Apartments,31,31,3.0,93,3.0,1.0,93 +39.0,MFP Senior - Cayuga Meadows,26,26,3.0,78,3.0,1.0,78 +40.0,MFP Senior - CFS Lakeview,112,112,8.0,896,8.0,1.0,896 +41.0,MFP Senior - Conifer Village,34,34,3.0,101,3.0,1.0,101 +42.0,MFP Senior - Corning Senior Center,75,75,6.0,450,6.0,1.0,450 +43.0,MFP Senior - Dayspring,77,77,6.0,463,6.0,1.0,463 +44.0,MFP Senior - East Hill Senior Living,40,40,4.0,159,4.0,1.0,159 +45.0,"MFP Senior - Elizabeth Square, Waverly",29,29,3.0,87,3.0,1.0,87 +46.0,MFP Senior - Ellis Hollow,25,25,3.0,74,3.0,1.0,74 +47.0,MFP Senior - Flannery,62,62,5.0,309,5.0,1.0,309 +48.0,MFP Senior - Harry L Apartments,33,33,3.0,99,3.0,1.0,99 +49.0,MFP Senior - Jefferson Village,25,25,3.0,74,3.0,1.0,74 +50.0,MFP Senior - Lincoln Court,26,26,3.0,78,3.0,1.0,78 +51.0,MFP Senior - Long Meadow Senior Housing,35,35,3.0,104,3.0,1.0,104 +52.0,MFP Senior - Metro Plaza Apartments,56,56,5.0,282,5.0,1.0,282 +53.0,MFP Senior - North Shore Towers,58,58,5.0,292,5.0,1.0,292 +54.0,"MFP Senior - Northern Broome Senior Center, Whitney Point",51,51,4.0,204,4.0,1.0,204 +55.0,MFP Senior - Park Terrace Congregate Apartments,24,24,3.0,73,3.0,1.0,73 +56.0,MFP Senior - Springview Apartments,28,28,3.0,83,3.0,1.0,83 +57.0,MFP Senior - Titus Towers,73,73,6.0,437,6.0,1.0,437 +58.0,MFP Senior - Villa Serene,70,70,6.0,418,6.0,1.0,418 +59.0,MFP Senior - Village Square/Manor,34,34,3.0,103,3.0,1.0,103 +60.0,MFP Senior - Wells Apartments,24,24,3.0,70,3.0,1.0,70 +61.0,MFP Senior - Woodsedge Apartments,17,17,2.0,34,2.0,1.0,34 +62.0,MFP The Love Church,259,259,18.0,4668,18.0,0.964,4500 +63.0,MFP Troupsburg,149,149,11.0,1636,11.0,1.0,1636 +64.0,MFP Tuscarora,193,193,14.0,2697,14.0,1.0,2697 +65.0,MFP Van Etten,214,214,15.0,3206,15.0,1.0,3206 +66.0,MFP Waverly,397,429,29.0,11502,26.8129,0.6303,7250 +67.0,MFP Wayland,180,180,13.0,2345,13.0,1.0,2345 +68.0,MFP Whitney Point,203,203,14.0,2836,14.0,1.0,2836 +69.0,MFP Windsor,201,201,14.0,2813,14.0,1.0,2813 +70.0,MFP Woodhull,176,176,13.0,2288,13.0,1.0,2288 diff --git a/task1/paper_tables/table1_metrics_summary.csv b/task1/paper_tables/table1_metrics_summary.csv new file mode 100644 index 0000000..19c287a --- /dev/null +++ b/task1/paper_tables/table1_metrics_summary.csv @@ -0,0 +1,5 @@ +method,E1_total_service,E2_quality_weighted,F1_gini,F2_min_satisfaction,F3_cv_satisfaction,E1_total_service_pct,E2_quality_weighted_pct +Recommended (μ̃ proportional),139469,131462,0.3137,2.0,0.5603,0,0 +Baseline 1: Uniform,104797,101309,0.0244,9.2458,0.0486,-25,-23 +Baseline 2: 2019 Scaled,104071,100264,0.0915,5.0,0.1759,-25,-24 +Baseline 3: μ proportional (no correction),139129,131397,0.313,2.0,0.5561,0,0 diff --git a/task1/paper_tables/table2_corrected_sites.csv b/task1/paper_tables/table2_corrected_sites.csv new file mode 100644 index 0000000..7f76791 --- /dev/null +++ b/task1/paper_tables/table2_corrected_sites.csv @@ -0,0 +1,6 @@ +site_id,site_name,mu,sigma,p_trunc,mu_tilde,mu_delta,mu_delta_pct +2.0,MFP Avoca,315,57.3497,0.2684,323,8,3.0 +13.0,MFP College TC3 -College,262,91.9929,0.168,266,4,2.0 +17.0,MFP Endwell United Methodist Church,285,60.7815,0.1434,289,4,1.0 +30.0,MFP Redeemer Lutheran Church,231,93.4787,0.1007,233,2,1.0 +66.0,MFP Waverly,397,51.8754,0.8157,429,32,8.0 diff --git a/task1/paper_tables/table3_backtest_summary.csv b/task1/paper_tables/table3_backtest_summary.csv new file mode 100644 index 0000000..222d932 --- /dev/null +++ b/task1/paper_tables/table3_backtest_summary.csv @@ -0,0 +1,10 @@ +metric,value,p_value +"corr(visits_2019, μ)",0.0349,0.774 +"corr(visits_2019, μ̃)",0.0364,0.7645 +sites_increased,36.0, +sites_decreased,32.0, +sites_unchanged,2.0, +CV(k/μ̃),0.1974, +service_2019_actual,102473.0, +service_model,139469.0768, +improvement_pct,34.6117, diff --git a/task1/paper_tables/table4_validation_results.csv b/task1/paper_tables/table4_validation_results.csv new file mode 100644 index 0000000..fc1edf3 --- /dev/null +++ b/task1/paper_tables/table4_validation_results.csv @@ -0,0 +1,7 @@ +constraint,formula,expected,actual,passed,details +C1: 资源约束,Σk_i = 730,730,730,True,总访问次数 = 730 +C2: 覆盖约束,k_i >= 1,>= 1,min = 2,True,"最小访问次数 = 2, 违反站点数 = 0" +C3: 日容量约束,每日站点数 = 2,所有天 = 2,"min=2, max=2",True,不满足的天数 = 0 +C4: 无重复约束,site_1 ≠ site_2 (同一天),无重复,重复天数 = 0,True,无 +C5: 排程一致性,排程次数 = k_i,所有站点一致,不一致站点数 = 0,True,全部一致 +C6: 总天数,日历天数 = 365,365,365,True,日历包含 365 天 diff --git a/tu.JPG b/tu.JPG deleted file mode 100644 index 2a840bd..0000000 Binary files a/tu.JPG and /dev/null differ diff --git a/vscode/.Rhistory b/vscode/.Rhistory new file mode 100755 index 0000000..4d2e445 --- /dev/null +++ b/vscode/.Rhistory @@ -0,0 +1,512 @@ +df = tibble( +a = c(1,4,3,7), +b = a + rnorm(4) +) +df +df %>% +pivot_longer(1:2, names_to = "var", values_to = "val") +df %>% +pivot_longer(1:2, names_to = "var", values_to = "val") %>% +ggplot(aes(y = val, color = var)) + +geom_point() +df %>% +pivot_longer(1:2, names_to = "var", values_to = "val") %>% +ggplot(aes(x = 1:4, y = val, color = var)) + +geom_point() +df %>% +pivot_longer(1:2, names_to = "var", values_to = "val") %>% +mutate(x = 1:n()) %>% +ggplot(aes(x = x, y = val, color = var)) + +geom_point() +set.seed(1) +df = tibble( +x = 1:4, +a = c(1,4,3,7), +b = a + rnorm(4) +) +df %>% +pivot_longer(-x, names_to = "var", values_to = "val") %>% +mutate(x = 1:n()) %>% +ggplot(aes(x = x, y = val, color = var)) + +geom_point() +df %>% +pivot_longer(-x, names_to = "var", values_to = "val") %>% +ggplot(aes(x = x, y = val, color = var)) + +geom_point() +?geom_ribbon +df %>% +pivot_longer(-x, names_to = "var", values_to = "val") +df %>% +pivot_longer(-x, names_to = "var", values_to = "val") %>% +ggplot(aes(x = x, y = val, color = var)) + +geom_point() + +geom_ribbon(ymin = min(val), ymax = max(val)) +df %>% +pivot_longer(-x, names_to = "var", values_to = "val") %>% +ggplot(aes(x = x, y = val, color = var)) + +geom_point() + +geom_ribbon(aes(ymin = min(val), ymax = max(val))) +df %>% +pivot_longer(-x, names_to = "var", values_to = "val") %>% +ggplot(aes(x = x, y = val, color = var)) + +geom_point() + +geom_ribbon(aes(ymin = val, ymax = val)) +df +df %>% +ggplot(aes(x = x)) + +geom_ribbon(aes(ymin = a, ymax = b)) +df %>% +ggplot(aes(x = x)) + +geom_errorbar(aes(ymin = a, ymax = b)) +df %>% +pivot_longer(-x, names_to = "var", values_to = "val") +df %>% +pivot_longer(-x, names_to = "var", values_to = "val") %>% +ggplot(aes(x = x, y = val)) + +geom_line() +df %>% +pivot_longer(-x, names_to = "var", values_to = "val") %>% +ggplot(aes(x = x, y = val, color = var)) + +geom_line() +df %>% +pivot_longer(-x, names_to = "var", values_to = "val") %>% +ggplot(aes(x = x, y = val, color = var)) + +geom_vline() +?kmeans +36170/1228 +36170/1338 +library(tidyverse) +install.packages("D:/gurobi950/win64/R/gurobi_9.5-0.zip", repos = NULL, type = "win.binary") +install.packages("slam") +library(gurobi) +library(gurobi) +model = list() +model$A = matrix(c(1,1,0,0,1,1), nrow = 2, byrow = TRUE) +model$obj = c(1,2,3) +model$modelsense = "max" +model$rhs = c(1,1) +model$sense = c("<=", "<=") +result = gurobi(model) +result$objval +?c_across +library(tidyverse) +?c_across +library(mlr3verse) +library(mlr3proba) +library(survival) +task = tsk("rats") +learner = lrn("surv.coxh") +learner = lrn("surv.coxph") +measure = msr("surv.hung_auc") +install.packages("survAUC") +measure = msr("surv.hung_auc") +pred = learner$train(task)$predict(task) +pred$score(measure) +measure = msr("surv.cindex") +pred$score(measure) +measure = msr("surv.hung_auc", times = 60) +pred$score(measure, task) +task$data() +pred$score(measure, task, row_ids = 1:270) +pred = learner$train(task, row_ids = 1:270)$predict(task, row_ids = 271:300) +pred$score(measure, task) +pred$score(measure, task, train_set = 1:270) +pred = learner$train(task)$predict(task) +pred$score(measure, task, train_set = 1:270) +?survAUC::AUC.hc() +learner = lrn("surv.coxph") +measure = msr("surv.hung_auc", times = 60) +pred = learner$train(task, row_ids = 1:270)$predict(task, row_ids = 271:300) +pred$score(measure, task, train_set = 1:270) +library(tidyverse) +?mean +df = tibble( +x = c(2,NA,5,7), +y = LETTERS[1:4], +z = c(5,6,NA,NA) +) +df +df %>% +mutate(across(where(is.numeric), ~ .x[is.na(.x)] = mean(.x, na.rm = TRUE))) +?replace_na +starwars %>% +mutate(across(where(is.numeric)), ~ replace_na(.x, mean(.x, na.rm = TRUE))) +starwars %>% +mutate(across(where(is.numeric), ~ replace_na(.x, mean(.x, na.rm = TRUE)))) +library(tinytex) +pdflatex("mcmthesis.tex") +pdflatex("mcmthesis-demo.tex") +library(tinytex) +pdflatex(mcmthesis-demo.tex) +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +library(tinytex) +pdflatex("mcmthesis-demo.tex") +library(timetk) +holidays <- tk_make_holiday_sequence( +start_date = "2013-01-01", +end_date = "2013-12-31", +calendar = "NYSE") +holidays +idx <- tk_make_weekday_sequence("2012", +remove_weekends = TRUE, +remove_holidays = TRUE, calendar = "NYSE") +idx +tk_get_timeseries_summary(idx) +idx <- tk_make_weekday_sequence("2012", +remove_weekends = TRUE, +remove_holidays = TRUE, calendar = "NYSE") +tk_get_timeseries_summary(idx) +tk_make_holiday_sequence("2012", calendar = "NYSE") +?tk_make_weekday_sequence +?slidify +FB <- FANG %>% filter(symbol == "FB") +library(tidyverse) +library(tidyquant) +FB <- FANG %>% filter(symbol == "FB") +mean_roll_5 <- slidify(mean, .period = 5, .align = "right") +FB %>% +mutate(rolling_mean_5 = mean_roll_5(adjusted)) +as_mapper(~ .x ^ 2 + 1) +?roll_toslide +?roll_tos_lide +?roll_to_slide +?pslide +remove.packages("timetk") +install.packages("timetk") +install.packages("timetk") +idx <- tk_make_weekday_sequence("2012", +remove_weekends = TRUE, +remove_holidays = TRUE, calendar = "NYSE") +tk_get_timeseries_summary(idx) +library(timetk) +idx <- tk_make_weekday_sequence("2012", +remove_weekends = TRUE, +remove_holidays = TRUE, calendar = "NYSE") +tk_get_timeseries_summary(idx) +?tk_make_weekday_sequence +holidays <- tk_make_holiday_sequence( +start_date = "2016", +end_date = "2017", +calendar = "NYSE") +weekends <- tk_make_weekend_sequence( +start_date = "2016", +end_date = "2017") +holidays +weekends +weekends <- tk_make_weekend_sequence( +start_date = "2016-01-01", +end_date = "2017-12-31") +weekends +weekends <- tk_make_weekdays_sequence( +start_date = "2016-01-01", +end_date = "2017-12-31") +weekends <- tk_make_weekday_sequence( +start_date = "2016-01-01", +end_date = "2017-12-31") +weekends +library(tinytex) +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +library(tinytex) +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +library(lubridate) +begin = ymd_hm("2019-08-10 14:00") +end = ymd_hm("2020-03-05 18:15") +interval(begin, end) +begin %--% end +library(tinytex) +xelatex("mcmthesis-demo.tex") +library(tinytex) +pdflatex("mcmthesis-demo.tex") +library(tinytex) +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +library(tinytex) +pdflatex("mcmthesis-demo.tex") +library(tinytex) +pdflatex("MCM-ICM_Summary.tex") +pdflatex("MCM-ICM_Summary.tex") +library(tinytex) +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +library(tinytex) +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +library(tidyverse) +df = tribble( +~A, ~B +北京, 上海 +df = tribble( +~A, ~B, +北京, 上海 +南京, 西安 +df = tribble( +~A, ~B, +北京, 上海, +南京, 西安, +上海, 南京, +合肥, 沈阳, +上海, 北京, +上海, 南京, +西安, 南京, +天津, 广州) +?tribble +df = tribble( +~A, ~B, +北京, 上海, +南京, 西安, +上海, 南京, +合肥, 沈阳, +上海, 北京, +上海, 南京, +西安, 南京, +天津, 广州) +df = tribble( +~A, ~B, +"北京", "上海", +"南京", "西安", +"上海", "南京", +"合肥", "沈阳", +"上海", "北京", +"上海", "南京", +"西安", "南京", +"天津", "广州") +df +?distinct +library(sets) +install.packages("sets") +library(sets) +df %>% +mutate(C = set(A, B)) +df = tibble(x = c(1:3,3:1,4), y = c(2,4,4,2,1,2,1)) +library(tidyverse) +df = tibble(x = c(1:3,3:1,4), y = c(2,4,4,2,1,2,1)) +df +library(tidyverse) +df = tibble(x = c(1:3,3:1,4), y = c(2,4,4,2,1,2,1)) +df +f = function(...) { +dummy = rep(0, 4) +dummy[c(...)] = 1 +dummy +} +df = tibble(x = c(1:3,3:1,NA), y = c(2,4,4,2,1,2,1)) +df +f = function(...) { +dummy = rep(NA, 4) +dummy[c(...)] = 1 +dummy +} +df %>% +mutate(z = pmap(., f)) %>% +unnest_wider(z, names_sep = "") +f = function(...) { +v = c(...) +dummy = ifelse(any(is.na(v)), rep(NA, 4), rep(0,4)) +dummy[v] = 1 +dummy +} +df %>% +mutate(z = pmap(., f)) %>% +unnest_wider(z, names_sep = "") +df +f = function(...) { +v = c(...) +dummy = ifelse(any(is.na(v)), rep(NA,4), rep(0,4)) +dummy[v] = 1 +dummy +} +df %>% +mutate(z = pmap(., f)) %>% +unnest_wider(z, names_sep = "") +v = c(NA, 1) +any(is.na(v)) +dummy = rep(NA, 4) +dummy[v] +dummy[!is.na(v)] +vv +v +dummy[v] = 1 +dummy +v = c(2,4) +any(is.na(v)) +dummy = rep(0,4) +dummy +dummy[v] = 1 +dummy +df +f(1,2) +f(1,NA) +f(c(1,2)) +df +v = c(2,4) +dummy = ifelse(any(is.na(v)), rep(NA,4), rep(0,4)) +dummy +any(is.na(v)) +rep(0,4) +rep(NA,4) +is.na(v) +v = c(2.4) +v = c(2,4) +dummy = ifelse(any(is.na(v)), rep(NA,4), rep(0,4)) +any(is.na(v)) +ifelse(any(is.na(v)), rep(NA,4), rep(0,4)) +ifelse(any(is.na(v)), rep(NA,4), rep(0,4)) +if(any(is.na(v)) rep(NA,4) +if(any(is.na(v))) rep(NA,4) +else rep(0,4) +if(any(is.na(v))) dummy = rep(NA,4) +else dummy = rep(0,4) +if(any(is.na(v))) { +dummy = rep(NA,4) +} else { +dummy = rep(0,4) +} +f = function(...) { +v = c(...) +if(any(is.na(v))) { +dummy = rep(NA,4) +} else { +dummy = rep(0,4) +} +dummy[v] = 1 +dummy +} +f(2,4) +f(1,2) +f(1,NA) +f(NA,4) +f = function(...) { +v = c(...) +if(any(is.na(v))) { +dummy = rep(NA,4) +} else { +dummy = rep(0,4) +} +dummy[v] = 1 +dummy +} +df %>% +mutate(z = pmap(., f)) %>% +unnest_wider(z, names_sep = "") +f = function(...) { +v = c(...) +dummy = ifelse(any(is.na(v)), rep(NA,4), rep(0,4)) +dummy[v] = 1 +dummy +} +df %>% +mutate(z = pmap(., f)) %>% +unnest_wider(z, names_sep = "") +f = function(...) { +v = c(...) +dummy = ifelse(all(!is.na(v)), rep(0,4), rep(NA,4)) +dummy[v] = 1 +dummy +} +df %>% +mutate(z = pmap(., f)) %>% +unnest_wider(z, names_sep = "") +v = c(2,4) +ifelse(all(!is.na(v)), rep(0,4), rep(NA,4)) +rep(0,4) +3827.64 / 600 +(3827.64 - 800) * 0.2 * 0.7 +(3827.64 - 423.87) / 600 +5103.51 * 0.8 * 0.2 * 0.7 +(5103.51 - 571.59) / 800 +12758.78 * 0.8 * 0.2 * 0.7 +(12758.78 - 1428.98) / 2000 +df = tibble(x = c(1:3,3:1,NA), y = c(2,4,NA,2,1,2,1)) +df +f = function(...) { +v = c(...) +if(any(is.na(v))) { +dummy = rep(NA,4) +} else { +dummy = rep(0,4) +} +dummy[v] = 1 +dummy +} +df %>% +mutate(z = pmap(., f)) %>% +unnest_wider(z, names_sep = "") +df = tibble(x = c(1:3,3:1,NA), y = c(2,4,NA,2,1,2,NA)) +df +f = function(...) { +v = c(...) +if(any(is.na(v))) { +dummy = rep(NA,4) +} else { +dummy = rep(0,4) +} +dummy[v] = 1 +dummy +} +df %>% +mutate(z = pmap(., f)) %>% +unnest_wider(z, names_sep = "") +df +df %>% +mutate(z = pmap(., f)) %>% +unnest_wider(z, names_sep = "") +library(tinytex) +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +3827.63 / 600 +5.67 / 6.38 +80000 * 0.8887147 +library(tinytex) +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +library(tinytex) +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") +library(tinytex) +pdflatex("mcmthesis-demo.tex") +library(tinytex) +pdflatex("mcmthesis-demo.tex") +library(tinytex) +pdflatex("mcmthesis-demo.tex") +library(tinytex) +pdflatex("mcmthesis-demo.tex") +pdflatex("mcmthesis-demo.tex") 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a/vscode/.Rproj.user/07BFC725/sources/per/t/461C3D72-contents b/vscode/.Rproj.user/07BFC725/sources/per/t/461C3D72-contents new file mode 100755 index 0000000..8c1d3bf --- /dev/null +++ b/vscode/.Rproj.user/07BFC725/sources/per/t/461C3D72-contents @@ -0,0 +1,539 @@ +%% +%% This is file `mcmthesis-demo.tex', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% mcmthesis.dtx (with options: `demo') +%% +%% ----------------------------------- +%% +%% This is a generated file. +%% +%% Copyright (C) +%% 2010 -- 2015 by Zhaoli Wang +%% 2014 -- 2019 by Liam Huang +%% 2019 -- present by latexstudio.net +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public License, either version 1.3 +%% of this license or (at your option) any later version. +%% The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3 or later is part of all distributions of LaTeX +%% version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The Current Maintainer of this work is Liam Huang. +%% +%% +%% This is file `mcmthesis-demo.tex', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% mcmthesis.dtx (with options: `demo') +%% +%% ----------------------------------- +%% +%% This is a generated file. +%% +%% Copyright (C) +%% 2010 -- 2015 by Zhaoli Wang +%% 2014 -- 2019 by Liam Huang +%% 2019 -- present by latexstudio.net +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public License, either version 1.3 +%% of this license or (at your option) any later version. +%% The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3 or later is part of all distributions of LaTeX +%% version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The Current Maintainer of this work is Liam Huang. +%% +\documentclass{mcmthesis} +\mcmsetup{CTeX = false, % 使用 CTeX 套装时，设置为 true + tcn = {0000000}, problem = {A}, + sheet = true, titleinsheet = true, keywordsinsheet = true, + titlepage = false, abstract = false} + +\usepackage{newtxtext} % \usepackage{palatino} +\usepackage[backend=bibtex]{biblatex} % for RStudio Complie + +\usepackage{tocloft} +\setlength{\cftbeforesecskip}{6pt} +\renewcommand{\contentsname}{\hspace*{\fill}\Large\bfseries Contents \hspace*{\fill}} + +\title{Enjoy a Cozy and Green Bath} +% \author{\small \href{http://www.latexstudio.net/} +% {\includegraphics[width=7cm]{mcmthesis-logo}}} +\date{\today} + +\begin{document} + +\begin{abstract} + +A traditional bathtub cannot be reheated by itself, so users have to add hot water from time to time. Our goal is to establish a model of the temperature of bath water in space and time. Then we are expected to propose an optimal strategy for users to keep the temperature even and close to initial temperature and decrease water consumption. + +To simplify modeling process, we firstly assume there is no person in the bathtub. We regard the whole bathtub as a thermodynamic system and introduce heat transfer formulas. + +We establish two sub-models: adding water constantly and discontinuously. As for the former sub-model, we define the mean temperature of bath water. Introducing Newton cooling formula, we determine the heat transfer capacity. After deriving the value of parameters, we deduce formulas to derive results and simulate the change of temperature field via CFD. As for the second sub-model, we define an iteration consisting of two process: heating and standby. According to energy conservation law, we obtain the relationship of time and total heat dissipating capacity. Then we determine the mass flow and the time of adding hot water. We also use CFD to simulate the temperature field in second sub-model. + +In consideration of evaporation, we correct the results of sub-models referring to some scientists' studies. We define two evaluation criteria and compare the two sub-models. Adding water constantly is found to keep the temperature of bath water even and avoid wasting too much water, so it is recommended by us. + +Then we determine the influence of some factors: radiation heat transfer, the shape and volume of the tub, the shape/volume/temperature/motions of the person, the bubbles made from bubble bath additives. We focus on the influence of those factors to heat transfer and then conduct sensitivity analysis. The results indicate smaller bathtub with less surface area, lighter personal mass, less motions and more bubbles will decrease heat transfer and save water. + +Based on our model analysis and conclusions, we propose the optimal strategy for the user in a bathtub and explain the reason of uneven temperature throughout the bathtub. In addition, we make improvement for applying our model in real life. + +\begin{keywords} +Heat transfer, Thermodynamic system, CFD, Energy conservation +\end{keywords} + +\end{abstract} + +\maketitle + +%% Generate the Table of Contents, if it's needed. +% \renewcommand{\contentsname}{\centering Contents} +\tableofcontents % 若不想要目录, 注释掉该句 + +\newpage + + +\section{Introduction} + +\subsection{Background} + +Bathing in a tub is a perfect choice for those who have been worn out after a long day's working. A traditional bathtub is a simply water containment vessel without a secondary heating system or circulating jets. Thus the temperature of water in bathtub declines noticeably as time goes by, which will influent the experience of bathing. As a result, the bathing person needs to add a constant trickle of hot water from a faucet to reheat the bathing water. This way of bathing will result in waste of water because when the capacity of the bathtub is reached, excess water overflows the tub. + +An optimal bathing strategy is required for the person in a bathtub to get comfortable bathing experience while reducing the waste of water. + +\subsection{Literature Review} + +Korean physicist Gi-Beum Kim analyzed heat loss through free surface of water contained in bathtub due to conduction and evaporation \cite{1}. He derived a relational equation based on the basic theory of heat transfer to evaluate the performance of bath tubes. The major heat loss was found to be due to evaporation. Moreover, he found out that the speed of heat loss depends more on the humidity of the bathroom than the temperature of water contained in the bathtub. So, it is best to maintain the temperature of bathtub water to be between 41 to 45$^{\circ}$C and the humidity of bathroom to be 95\%. + +When it comes to convective heat transfer in bathtub, many studies can be referred to. Newton's law of cooling states that the rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings while under the effects of a breeze \cite{2}. Claude-Louis Navier and George Gabriel Stokes described the motion of viscous fluid substances with the Navier-Stokes equations. Those equations may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing \cite{3}. + +In addition, some numerical simulation software are applied in solving and analyzing problems that involve fluid flows. For example, Computational Fluid Dynamics (CFD) is a common one used to perform the calculations required to simulate the interaction of liquids and gases with surfaces defined by boundary conditions \cite{4}. + +\subsection{Restatement of the Problem} + +We are required to establish a model to determine the change of water temperature in space and time. Then we are expected to propose the best strategy for the person in the bathtub to keep the water temperature close to initial temperature and even throughout the tub. Reduction of waste of water is also needed. In addition, we have to consider the impact of different conditions on our model, such as different shapes and volumes of the bathtub, etc. + +In order to solve those problems, we will proceed as follows: + +\begin{itemize} +\item {\bf Stating assumptions}. By stating our assumptions, we will narrow the focus of our approach towards the problems and provide some insight into bathtub water temperature issues. + +\item {\bf Making notations}. We will give some notations which are important for us to clarify our models. + +\item {\bf Presenting our model}. In order to investigate the problem deeper, we divide our model into two sub-models. One is a steady convection heat transfer sub-model in which hot water is added constantly. The other one is an unsteady convection heat transfer sub-model where hot water is added discontinuously. + +\item {Defining evaluation criteria and comparing sub-models}. We define two main criteria to evaluate our model: the mean temperature of bath water and the amount of inflow water. + +\item {\bf Analysis of influencing factors}. In term of the impact of different factors on our model, we take those into consideration: the shape and volume of the tub, the shape/volume/temperature of the person in the bathtub, the motions made by the person in the bathtub and adding a bubble bath additive initially. + +\item {\bf Model testing and sensitivity analysis}. With the criteria defined before, we evaluate the reliability of our model and do the sensitivity analysis. + +\item {\bf Further discussion}. We discuss about different ways to arrange inflow faucets. Then we improve our model to apply them in reality. + +\item {\bf Evaluating the model}. We discuss about the strengths and weaknesses of our model: + +\begin{itemize} +\item[1)] ... +\item[2)] ... +\item[3)] ... +\item[4)] ... +\end{itemize} + +\end{itemize} + +\section{Assumptions and Justification} + +To simplify the problem and make it convenient for us to simulate real-life conditions, we make the following basic assumptions, each of which is properly justified. + +\begin{itemize} +\item {\bf The bath water is incompressible Non-Newtonian fluid}. The incompressible Non-Newtonian fluid is the basis of Navier–Stokes equations which are introduced to simulate the flow of bath water. + +\item {\bf All the physical properties of bath water, bathtub and air are assumed to be stable}. The change of those properties like specific heat, thermal conductivity and density is rather small according to some studies \cite{5}. It is complicated and unnecessary to consider these little change so we ignore them. + +\item {\bf There is no internal heat source in the system consisting of bathtub, hot water and air}. Before the person lies in the bathtub, no internal heat source exist except the system components. The circumstance where the person is in the bathtub will be investigated in our later discussion. + +\item {\bf We ignore radiative thermal exchange}. According to Stefan-Boltzmann’s law, the radiative thermal exchange can be ignored when the temperature is low. Refer to industrial standard \cite{6}, the temperature in bathroom is lower than 100 $^{\circ}$C, so it is reasonable for us to make this assumption. + +\item {\bf The temperature of the adding hot water from the faucet is stable}. This hypothesis can be easily achieved in reality and will simplify our process of solving the problem. +\end{itemize} + +\section{Notations} + +\begin{center} +\begin{tabular}{clc} +{\bf Symbols} & {\bf Description} & \quad {\bf Unit} \\[0.25cm] +$h$ & Convection heat transfer coefficient & \quad W/(m$^2 \cdot$ K) +\\[0.2cm] +$k$ & Thermal conductivity & \quad W/(m $\cdot$ K) \\[0.2cm] +$c_p$ & Specific heat & \quad J/(kg $\cdot$ K) \\[0.2cm] +$\rho$ & Density & \quad kg/m$^2$ \\[0.2cm] +$\delta$ & Thickness & \quad m \\[0.2cm] +$t$ & Temperature & \quad $^\circ$C, K \\[0.2cm] +$\tau$ & Time & \quad s, min, h \\[0.2cm] +$q_m$ & Mass flow & \quad kg/s \\[0.2cm] +$\Phi$ & Heat transfer power & \quad W \\[0.2cm] +$T$ & A period of time & \quad s, min, h \\[0.2cm] +$V$ & Volume & \quad m$^3$, L \\[0.2cm] +$M,\,m$ & Mass & \quad kg \\[0.2cm] +$A$ & Aera & \quad m$^2$ \\[0.2cm] +$a,\,b,\,c$ & The size of a bathtub & \quad m$^3$ +\end{tabular} +\end{center} + +\noindent where we define the main parameters while specific value of those parameters will be given later. + +\section{Model Overview} + +In our basic model, we aim at three goals: keeping the temperature as even as possible, making it close to the initial temperature and decreasing the water consumption. + +We start with the simple sub-model where hot water is added constantly. +At first we introduce convection heat transfer control equations in rectangular coordinate system. Then we define the mean temperature of bath water. + +Afterwards, we introduce Newton cooling formula to determine heat transfer +capacity. After deriving the value of parameters, we get calculating results via formula deduction and simulating results via CFD. + +Secondly, we present the complicated sub-model in which hot water is +added discontinuously. We define an iteration consisting of two process: +heating and standby. As for heating process, we derive control equations and boundary conditions. As for standby process, considering energy conservation law, we deduce the relationship of total heat dissipating capacity and time. + +Then we determine the time and amount of added hot water. After deriving the value of parameters, we get calculating results via formula deduction and simulating results via CFD. + +At last, we define two criteria to evaluate those two ways of adding hot water. Then we propose optimal strategy for the user in a bathtub. +The whole modeling process can be shown as follows. + +\begin{figure}[h] +\centering +\includegraphics[width=12cm]{fig1.jpg} +\caption{Modeling process} \label{fig1} +\end{figure} + +\section{Sub-model I : Adding Water Continuously} + +We first establish the sub-model based on the condition that a person add water continuously to reheat the bathing water. Then we use Computational Fluid Dynamics (CFD) to simulate the change of water temperature in the bathtub. At last, we evaluate the model with the criteria which have been defined before. + +\subsection{Model Establishment} + +Since we try to keep the temperature of the hot water in bathtub to be even, we have to derive the amount of inflow water and the energy dissipated by the hot water into the air. + +We derive the basic convection heat transfer control equations based on the former scientists’ achievement. Then, we define the mean temperature of bath water. Afterwards, we determine two types of heat transfer: the boundary heat transfer and the evaporation heat transfer. Combining thermodynamic formulas, we derive calculating results. Via Fluent software, we get simulation results. + +\subsubsection{Control Equations and Boundary Conditions} + +According to thermodynamics knowledge, we recall on basic convection +heat transfer control equations in rectangular coordinate system. Those +equations show the relationship of the temperature of the bathtub water in space. + +We assume the hot water in the bathtub as a cube. Then we put it into a +rectangular coordinate system. The length, width, and height of it is $a,\, b$ and $c$. + +\begin{figure}[h] +\centering +\includegraphics[width=8cm]{fig2.jpg} +\caption{Modeling process} \label{fig2} +\end{figure} + +In the basis of this, we introduce the following equations \cite{5}: + +\begin{itemize} +\item {\bf Continuity equation:} +\end{itemize} + +\begin{equation} \label{eq1} +\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} +\frac{\partial w}{\partial z} =0 +\end{equation} + +\noindent where the first component is the change of fluid mass along the $X$-ray. The second component is the change of fluid mass along the $Y$-ray. And the third component is the change of fluid mass along the $Z$-ray. The sum of the change in mass along those three directions is zero. + +\begin{itemize} +\item {\bf Moment differential equation (N-S equations):} +\end{itemize} + +\begin{equation} \label{eq2} +\left\{ +\begin{array}{l} \!\! +\rho \Big(u \dfrac{\partial u}{\partial x} + v \dfrac{\partial u}{\partial y} + w\dfrac{\partial u}{\partial z} \Big) = -\dfrac{\partial p}{\partial x} + \eta \Big(\dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial^2 u}{\partial y^2} + \dfrac{\partial^2 u}{\partial z^2} \Big) \\[0.3cm] +\rho \Big(u \dfrac{\partial v}{\partial x} + v \dfrac{\partial v}{\partial y} + w\dfrac{\partial v}{\partial z} \Big) = -\dfrac{\partial p}{\partial y} + \eta \Big(\dfrac{\partial^2 v}{\partial x^2} + \dfrac{\partial^2 v}{\partial y^2} + \dfrac{\partial^2 v}{\partial z^2} \Big) \\[0.3cm] +\rho \Big(u \dfrac{\partial w}{\partial x} + v \dfrac{\partial w}{\partial y} + w\dfrac{\partial w}{\partial z} \Big) = -g-\dfrac{\partial p}{\partial z} + \eta \Big(\dfrac{\partial^2 w}{\partial x^2} + \dfrac{\partial^2 w}{\partial y^2} + \dfrac{\partial^2 w}{\partial z^2} \Big) +\end{array} +\right. +\end{equation} + +\begin{itemize} +\item {\bf Energy differential equation:} +\end{itemize} + +\begin{equation} \label{eq3} +\rho c_p \Big( u\frac{\partial t}{\partial x} + v\frac{\partial t}{\partial y} + w\frac{\partial t}{\partial z} \Big) = \lambda \Big(\frac{\partial^2 t}{\partial x^2} + \frac{\partial^2 t}{\partial y^2} + \frac{\partial^2 t}{\partial z^2} \Big) +\end{equation} + +\noindent where the left three components are convection terms while the right three components are conduction terms. + +By Equation \eqref{eq3}, we have ...... + +...... + +On the right surface in Fig. \ref{fig2}, the water also transfers heat firstly with bathtub inner surfaces and then the heat comes into air. The boundary condition here is ...... + +\subsubsection{Definition of the Mean Temperature} + +...... + +\subsubsection{Determination of Heat Transfer Capacity} + +...... + +\section{Sub-model II: Adding Water Discontinuously} + +In order to establish the unsteady sub-model, we recall on the working principle of air conditioners. The heating performance of air conditions consist of two processes: heating and standby. After the user set a temperature, the air conditioner will begin to heat until the expected temperature is reached. Then it will go standby. When the temperature get below the expected temperature, the air conditioner begin to work again. As it works in this circle, the temperature remains the expected one. + +Inspired by this, we divide the bathtub working into two processes: adding +hot water until the expected temperature is reached, then keeping this +condition for a while unless the temperature is lower than a specific value. Iterating this circle ceaselessly will ensure the temperature kept relatively stable. + +\subsection{Heating Model} + +\subsubsection{Control Equations and Boundary Conditions} + +\subsubsection{Determination of Inflow Time and Amount} + +\subsection{Standby Model} + +\subsection{Results} + +\quad~ We first give the value of parameters based on others’ studies. Then we get the calculation results and simulating results via those data. + +\subsubsection{Determination of Parameters} + +After establishing the model, we have to determine the value of some +important parameters. + +As scholar Beum Kim points out, the optimal temperature for bath is +between 41 and 45$^\circ$C [1]. Meanwhile, according to Shimodozono's study, 41$^\circ$C warm water bath is the perfect choice for individual health [2]. So it is reasonable for us to focus on $41^\circ$C $\sim 45^\circ$C. Because adding hot water continuously is a steady process, so the mean temperature of bath water is supposed to be constant. We value the temperature of inflow and outflow water with the maximum and minimum temperature respectively. + +The values of all parameters needed are shown as follows: + +..... + +\subsubsection{Calculating Results} + +Putting the above value of parameters into the equations we derived before, we can get the some data as follows: + +%%普通表格 +\begin{table}[h] %h表示固定在当前位置 +\centering %设置居中 +\caption{The calculating results} %表标题 +\vspace{0.15cm} +\label{tab2} %设置表的引用标签 +\begin{tabular}{|c|c|c|} %3个c表示3列, |可选, 表示绘制各列间的竖线 +\hline %画横线 +Variables & Values & Unit \\ \hline %各列间用&隔开 +$A_1$ & 1.05 & $m^2$ \\ \hline +$A_2$ & 2.24 & $m^2$ \\ \hline +$\Phi_1$ & 189.00 & $W$ \\ \hline +$\Phi_2$ & 43.47 & $W$ \\ \hline +$\Phi$ & 232.47 & $W$ \\ \hline +$q_m$ & 0.014 & $g/s$ \\ \hline +\end{tabular} +\end{table} + +From Table \ref{tab2}, ...... + +...... + +\section{Correction and Contrast of Sub-Models} + +After establishing two basic sub-models, we have to correct them in consideration of evaporation heat transfer. Then we define two evaluation criteria to compare the two sub-models in order to determine the optimal bath strategy. + +\subsection{Correction with Evaporation Heat Transfer} + +Someone may confuse about the above results: why the mass flow in the first sub-model is so small? Why the standby time is so long? Actually, the above two sub-models are based on ideal conditions without consideration of the change of boundary conditions, the motions made by the person in bathtub and the evaporation of bath water, etc. The influence of personal motions will be discussed later. Here we introducing the evaporation of bath water to correct sub-models. + +\subsection{Contrast of Two Sub-Models} + +Firstly we define two evaluation criteria. Then we contrast the two submodels via these two criteria. Thus we can derive the best strategy for the person in the bathtub to adopt. + +\section{Model Analysis and Sensitivity Analysis} + +\subsection{The Influence of Different Bathtubs} + +Definitely, the difference in shape and volume of the tub affects the +convection heat transfer. Examining the relationship between them can help +people choose optimal bathtubs. + +\subsubsection{Different Volumes of Bathtubs} + +In reality, a cup of water will be cooled down rapidly. However, it takes quite long time for a bucket of water to become cool. That is because their volume is different and the specific heat of water is very large. So that the decrease of temperature is not obvious if the volume of water is huge. That also explains why it takes 45 min for 320 L water to be cooled by 1$^\circ$C. + +In order to examine the influence of volume, we analyze our sub-models +by conducting sensitivity Analysis to them. + +We assume the initial volume to be 280 L and change it by $\pm 5$\%, $\pm 8$\%, $\pm 12$\% and $\pm 15$\%. With the aid of sub-models we established before, the variation of some parameters turns out to be as follows + +%%三线表 +\begin{table}[h] %h表示固定在当前位置 +\centering %设置居中 +\caption{Variation of some parameters} %表标题 +\label{tab7} %设置表的引用标签 +\begin{tabular}{ccccccc} %7个c表示7列, c表示每列居中对齐, 还有l和r可选 +\toprule %画顶端横线 +$V$ & $A_1$ & $A_2$ & $T_2$ & $q_{m1}$ & $q_{m2}$ & $\Phi_q$ \\ +\midrule %画中间横线 +-15.00\% & -5.06\% & -9.31\% & -12.67\% & -2.67\% & -14.14\% & -5.80\% \\ +-12.00\% & -4.04\% & -7.43\% & -10.09\% & -2.13\% & -11.31\% & -4.63\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +\bottomrule %画底部横线 +\end{tabular} +\end{table} + +\section{Strength and Weakness} + +\subsection{Strength} + +\begin{itemize} +\item We analyze the problem based on thermodynamic formulas and laws, so that the model we established is of great validity. + +\item Our model is fairly robust due to our careful corrections in consideration of real-life situations and detailed sensitivity analysis. + +\item Via Fluent software, we simulate the time field of different areas throughout the bathtub. The outcome is vivid for us to understand the changing process. + +\item We come up with various criteria to compare different situations, like water consumption and the time of adding hot water. Hence an overall comparison can be made according to these criteria. + +\item Besides common factors, we still consider other factors, such as evaporation and radiation heat transfer. The evaporation turns out to be the main reason of heat loss, which corresponds with other scientist’s experimental outcome. +\end{itemize} + +\subsection{Weakness} + +\begin{itemize} +\item Having knowing the range of some parameters from others’ essays, we choose a value from them to apply in our model. Those values may not be reasonable in reality. + +\item Although we investigate a lot in the influence of personal motions, they are so complicated that need to be studied further. + +\item Limited to time, we do not conduct sensitivity analysis for the influence of personal surface area. +\end{itemize} + +\section{Further Discussion} + +In this part, we will focus on different distribution of inflow faucets. Then we discuss about the real-life application of our model. + +\begin{itemize} +\item Different Distribution of Inflow Faucets + +In our before discussion, we assume there being just one entrance of inflow. + +From the simulating outcome, we find the temperature of bath water is hardly even. So we come up with the idea of adding more entrances. + +The simulation turns out to be as follows + +\begin{figure}[h] +\centering +\includegraphics[width=12cm]{fig24.jpg} +\caption{The simulation results of different ways of arranging entrances} \label{fig24} +\end{figure} + +From the above figure, the more the entrances are, the evener the temperature will be. Recalling on the before simulation outcome, when there is only one entrance for inflow, the temperature of corners is quietly lower than the middle area. + +In conclusion, if we design more entrances, it will be easier to realize the goal to keep temperature even throughout the bathtub. + +\item Model Application + +Our before discussion is based on ideal assumptions. In reality, we have to make some corrections and improvement. + +\begin{itemize} +\item[1)] Adding hot water continually with the mass flow of 0.16 kg/s. This way can ensure even mean temperature throughout the bathtub and waste less water. + +\item[2)] The manufacturers can design an intelligent control system to monitor the temperature so that users can get more enjoyable bath experience. + +\item[3)] We recommend users to add bubble additives to slow down the water being cooler and help cleanse. The additives with lower thermal conductivity are optimal. + +\item[4)] The study method of our establishing model can be applied in other area relative to convection heat transfer, such as air conditioners. +\end{itemize} +\end{itemize} + +\begin{thebibliography}{99} +\addcontentsline{toc}{section}{Reference} + +\bibitem{1} Gi-Beum Kim. Change of the Warm Water Temperature for the Development of Smart Healthecare Bathing System. Hwahak konghak. 2006, 44(3): 270-276. +\bibitem{2} \url{https://en.wikipedia.org/wiki/Convective_heat_transfer#Newton.27s_law_of_cooling} +\bibitem{3} \url{https://en.wikipedia.org/wiki/Navier\%E2\%80\%93Stokes_equations} +\bibitem{4} \url{https://en.wikipedia.org/wiki/Computational_fluid_dynamics} +\bibitem{5} Holman J P. Heat Transfer (9th ed.), New York: McGraw-Hill, 2002. +\bibitem{6} Liu Weiguo, Chen Zhaoping, ZhangYin. Matlab program design and application. Beijing: Higher education press, 2002. (In Chinese) + +\end{thebibliography} + +\newpage + +\begin{letter}{Enjoy Your Bath Time!} + +From simulation results of real-life situations, we find it takes a period of time for the inflow hot water to spread throughout the bathtub. During this process, the bath water continues transferring heat into air, bathtub and the person in bathtub. The difference between heat transfer capacity makes the temperature of various areas to be different. So that it is difficult to get an evenly maintained temperature throughout the bath water. + +In order to enjoy a comfortable bath with even temperature of bath water and without wasting too much water, we propose the following suggestions. + +\begin{itemize} +\item Adding hot water consistently +\item Using smaller bathtub if possible +\item Decreasing motions during bath +\item Using bubble bath additives +\item Arranging more faucets of inflow +\end{itemize} + +\vspace{\parskip} + +Sincerely yours, + +Your friends + +\end{letter} + +\newpage + +\begin{appendices} + +\section{First appendix} + +In addition, your report must include a letter to the Chief Financial Officer (CFO) of the Goodgrant Foundation, Mr. Alpha Chiang, that describes the optimal investment strategy, your modeling approach and major results, and a brief discussion of your proposed concept of a return-on-investment (ROI). This letter should be no more than two pages in length. + +Here are simulation programmes we used in our model as follow.\\ + +\textbf{\textcolor[rgb]{0.98,0.00,0.00}{Input matlab source:}} +\lstinputlisting[language=Matlab]{./code/mcmthesis-matlab1.m} + +\section{Second appendix} + +some more text \textcolor[rgb]{0.98,0.00,0.00}{\textbf{Input C++ source:}} +\lstinputlisting[language=C++]{./code/mcmthesis-sudoku.cpp} + +\end{appendices} +\end{document} +%% +%% This work consists of these files mcmthesis.dtx, +%% figures/ and +%% code/, +%% and the derived files mcmthesis.cls, +%% mcmthesis-demo.tex, +%% README, +%% LICENSE, +%% mcmthesis.pdf and +%% mcmthesis-demo.pdf. +%% +%% End of file `mcmthesis-demo.tex'. diff --git a/vscode/.Rproj.user/07BFC725/sources/per/t/5BE5E0FB b/vscode/.Rproj.user/07BFC725/sources/per/t/5BE5E0FB new file mode 100755 index 0000000..17ccc47 --- /dev/null +++ b/vscode/.Rproj.user/07BFC725/sources/per/t/5BE5E0FB @@ -0,0 +1,26 @@ +{ + "id": "5BE5E0FB", + 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+\renewcommand{\figurename}{Figure} +\renewcommand{\tablename}{Table} +\graphicspath{{./}{./img/}{./fig/}{./image/}{./figure/}{./picture/} + {./imgs/}{./figs/}{./images/}{./figures/}{./pictures/}} +\def\maketitle{% + \let\saved@thepage\thepage + \let\thepage\relax + \ifMCM@opt@sheet + \makesheet + \fi + \newpage + \ifMCM@opt@titlepage + \MCM@maketitle + \fi + \newpage + \let\thepage\saved@thepage + \setcounter{page}{1} + \pagestyle{fancy} +} +\def\abstractname{Summary} +\ifMCM@opt@CTeX + \newbox\@abstract% + \setbox\@abstract\hbox{}% + \long\def\abstract{\bgroup\global\setbox\@abstract\vbox\bgroup\hsize\textwidth\leftskip1cm\rightskip1cm}% + \def\endabstract{\egroup\egroup} + \def\make@abstract{% + \begin{center} + \textbf{\abstractname} + \end{center} + \usebox\@abstract\par + } +\else + \RenewEnviron{abstract}{\xdef\@abstract{\expandonce\BODY}} + \def\make@abstract{% + \begin{center} + \textbf{\abstractname} + \end{center} + \@abstract\par + } +\fi +\newenvironment{letter}[1]{% + \par% + \bgroup\parindent0pt% + \begin{minipage}{5cm} + \flushleft #1% + \end{minipage}} + {\egroup\smallskip} + +\def\keywordsname{Keywords} +\ifMCM@opt@CTeX + \newbox\@keywords + \setbox\@keywords\hbox{} + \def\keywords{\global\setbox\@keywords\vbox\bgroup\noindent\leftskip0cm} + \def\endkeywords{\egroup}% + \def\make@keywords{% + \par\hskip.4cm\textbf{\keywordsname}: \usebox\@keywords\hfill\par + } +\else + \NewEnviron{keywords}{\xdef\@keywords{\expandonce\BODY}} + \def\make@keywords{% + \par\noindent\textbf{\keywordsname}: + \@keywords\par + } +\fi +\newcommand{\headset}{{\the\year}\\MCM/ICM\\Summary Sheet} +\newcommand{\problem}[1]{\mcmsetup{problem = #1}} +\def\makesheet{% + \pagestyle{empty}% + \null% + \vspace*{-5pc}% + \begin{center} + \begingroup + \setlength{\parindent}{0pt} + \begin{minipage}[t]{0.33\linewidth} + \bfseries\centering% + Problem Chosen\\[0.7pc] + {\Huge\textbf{\MCM@opt@problem}}\\[2.8pc] + \end{minipage}% + \begin{minipage}[t]{0.33\linewidth} + \centering% + \textbf{\headset}% + \end{minipage}% + \begin{minipage}[t]{0.33\linewidth} + \centering\bfseries% + Team Control Number\\[0.7pc] + {\Huge\textbf{\MCM@opt@tcn}}\\[2.8pc] + % {\Huge\textbf{\MCM@opt@tcn}}\\[2.8pc] + \end{minipage}\par + \rule{\linewidth}{0.8pt}\par + \endgroup + \vskip 10pt% + \ifMCM@opt@titleinsheet + \normalfont \LARGE \@title \par + \fi + \end{center} +\ifMCM@opt@keywordsinsheet + \make@abstract + \make@keywords +\else + \make@abstract +\fi} +\newcommand{\MCM@maketitle}{% + \begin{center}% + \let \footnote \thanks + {\LARGE \@title \par}% + \vskip 1.5em% + {\large + \lineskip .5em% + \begin{tabular}[t]{c}% + \@author + \end{tabular}\par}% + \vskip 1em% + {\large \@date}% + \end{center}% + \par + \vskip 1.5em% + \ifMCM@opt@abstract% + \make@abstract + \make@keywords + \fi% +} +\def\MCM@memoto{\relax} +\newcommand{\memoto}[1]{\gdef\MCM@memoto{#1}} +\def\MCM@memofrom{\relax} +\newcommand{\memofrom}[1]{\gdef\MCM@memofrom{#1}} +\def\MCM@memosubject{\relax} +\newcommand{\memosubject}[1]{\gdef\MCM@memosubject{#1}} +\def\MCM@memodate{\relax} +\newcommand{\memodate}[1]{\gdef\MCM@memodate{#1}} +\def\MCM@memologo{\relax} +\newcommand{\memologo}[1]{\gdef\MCM@memologo{\protect #1}} +\def\@letterheadaddress{\relax} +\newcommand{\lhaddress}[1]{\gdef\@letterheadaddress{#1}} +\newenvironment{memo}[1][Memorandum]{% + \pagestyle{plain}% + \ifthenelse{\equal{\MCM@memologo}{\relax}}{% + % without logo specified. + }{% + % with logo specified + \begin{minipage}[t]{\columnwidth}% + \begin{flushright} + \vspace{-0.6in} + \MCM@memologo + \vspace{0.5in} + \par\end{flushright}% + \end{minipage}% + } + \begin{center} + \LARGE\bfseries\scshape + #1 + \end{center} + \begin{description} + \ifthenelse{\equal{\MCM@memoto}{\relax}}{}{\item [{To:}] \MCM@memoto} + \ifthenelse{\equal{\MCM@memofrom}{\relax}}{}{\item [{From:}] \MCM@memofrom} + \ifthenelse{\equal{\MCM@memosubject}{\relax}}{}{\item [{Subject:}] \MCM@memosubject} + \ifthenelse{\equal{\MCM@memodate}{\relax}}{}{\item [{Date:}] \MCM@memodate} + \end{description} + \par\noindent + \rule[0.5ex]{\linewidth}{0.1pt}\par + \bigskip{} +}{% + \clearpage + \pagestyle{fancy}% +} +\newtheorem{Theorem}{Theorem}[section] +\newtheorem{Lemma}[Theorem]{Lemma} +\newtheorem{Corollary}[Theorem]{Corollary} +\newtheorem{Proposition}[Theorem]{Proposition} +\newtheorem{Definition}[Theorem]{Definition} +\newtheorem{Example}[Theorem]{Example} +\renewcommand\section{\@startsection{section}{1}{\z@}% + {-0pt\@plus -.2ex \@minus -.2ex}% + {1pt \@plus .2ex}% + {\rmfamily\Large\bfseries}} +\renewcommand\subsection{\@startsection{subsection}{2}{\z@}% + {-0pt\@plus -.2ex \@minus -.2ex}% + {1pt \@plus .2ex}% + {\rmfamily\large\bfseries}} +\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}% + {-.5ex\@plus -1ex \@minus -.2ex}% + {.25ex \@plus .2ex}% + {\rmfamily\normalsize\bfseries}} +\renewcommand\paragraph{\@startsection{paragraph}{4}{\z@}% + {1ex \@plus1ex 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+1,26 @@ +{ + "id": "82F97103", + "path": "C:/Users/zhang/Desktop/25美赛/MCM_Latex2025/reference.bib", + "project_path": "reference.bib", + "type": "text", + "hash": "721218634", + "contents": "", + "dirty": false, + "created": 1736701210568.0, + "source_on_save": false, + "relative_order": 3, + "properties": { + "source_window_id": "", + "Source": "Source", + "cursorPosition": "75,1", + "scrollLine": "50" + }, + "folds": "", + "lastKnownWriteTime": 1736716598, + "encoding": "UTF-8", + "collab_server": "", + "source_window": "", + "last_content_update": 1736716598157, + "read_only": false, + "read_only_alternatives": [] +} \ No newline at end of file diff --git a/vscode/.Rproj.user/1248C2B7/sources/per/t/82F97103-contents b/vscode/.Rproj.user/1248C2B7/sources/per/t/82F97103-contents new file mode 100755 index 0000000..2758623 --- /dev/null +++ b/vscode/.Rproj.user/1248C2B7/sources/per/t/82F97103-contents @@ -0,0 +1,76 @@ +@article{kim2006, + author = {Gi-Beum Kim}, + title = {Change of the Warm Water Temperature for the Development of Smart Healthecare Bathing System}, + journal = {Hwahak konghak}, + year = {2006}, + volume = {44}, + number = {3}, + pages = {270-276} +} + +@article{holman2002, + author = {Holman, J. P.}, + title = {Heat Transfer (9th ed.)}, + journal = {McGraw-Hill}, + year = {2002}, + address = {New York} +} + +@book{anderson2006, + author = {Anderson, D. A. and Tannehill, J. C. and Pletcher, R. H.}, + title = {Computational Fluid Dynamics and Heat Transfer}, + publisher = {Taylor \& Francis}, + year = {2006}, + edition = {2nd} +} + +@article{incropera2007, + author = {Incropera, F. P. and DeWitt, D. P. and Bergman, T. L. and Lavine, A. S.}, + title = {Fundamentals of Heat and Mass Transfer}, + journal = {John Wiley \& Sons}, + year = {2007}, + edition = {6th} +} + +@article{evaporation2018, + author = {Smith, J.}, + title = {Evaporation Effects in Bath Water Temperature Regulation}, + journal = {Journal of Thermal Science and Engineering}, + year = {2018}, + volume = {30}, + number = {4}, + pages = {456-467} +} + +@phdthesis{thesis2015, + author = {Brown, D.}, + title = {Thermal Modeling of Bath Water Systems}, + school = {University of Technology}, + year = {2015}, + address = {City} +} + +@misc{website2024, + author = {Thompson, H.}, + title = {Advanced CFD Techniques for Bath Water Simulation}, + howpublished = {\url{https://www.example.com/advanced-cfd}}, + note = {Accessed: 2024-10-01} +} + +@patent{patent2023, + author = {Wilson, G.}, + title = {Smart Bath Water Temperature Control System}, + number = {1234567}, + year = {2023}, + type = {US Patent}, + assignee = {SmartTech Inc.} +} + +@book{Liu02, + author = {Liu, Weiguo and Chen, Zhaoping and Zhang, Yin}, + title = {Matlab program design and application}, + publisher = {Higher education press}, + address = {Beijing}, + year = {2002}, + note = {In Chinese} +} diff --git a/vscode/.Rproj.user/1248C2B7/sources/per/t/A0D261B2 b/vscode/.Rproj.user/1248C2B7/sources/per/t/A0D261B2 new file mode 100755 index 0000000..7192d80 --- /dev/null +++ b/vscode/.Rproj.user/1248C2B7/sources/per/t/A0D261B2 @@ -0,0 +1,26 @@ +{ + "id": "A0D261B2", + "path": "C:/Users/zhang/Desktop/25美赛/MCM_Latex2025/mcmthesis-demo.tex", + "project_path": "mcmthesis-demo.tex", + "type": "tex", + "hash": "1436774386", + "contents": "", + "dirty": false, + "created": 1704816448925.0, + "source_on_save": false, + "relative_order": 1, + "properties": { + "source_window_id": "", + "Source": "Source", + "cursorPosition": "715,74", + "scrollLine": "701" + }, + "folds": "", + "lastKnownWriteTime": 1736716652, + "encoding": "UTF-8", + "collab_server": "", + "source_window": "", + "last_content_update": 1736716652098, + "read_only": false, + "read_only_alternatives": [] +} \ No newline at end of file diff --git a/vscode/.Rproj.user/1248C2B7/sources/per/t/A0D261B2-contents b/vscode/.Rproj.user/1248C2B7/sources/per/t/A0D261B2-contents new file mode 100755 index 0000000..8793e3a --- /dev/null +++ b/vscode/.Rproj.user/1248C2B7/sources/per/t/A0D261B2-contents @@ -0,0 +1,774 @@ +%% +%% This is file `mcmthesis-demo.tex', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% mcmthesis.dtx (with options: `demo') +%% +%% ----------------------------------- +%% +%% This is a generated file. +%% +%% Copyright (C) +%% 2010 -- 2015 by Zhaoli Wang +%% 2014 -- 2019 by Liam Huang +%% 2019 -- present by latexstudio.net +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public License, either version 1.3 +%% of this license or (at your option) any later version. +%% The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3 or later is part of all distributions of LaTeX +%% version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The Current Maintainer of this work is Liam Huang. +%% +%% +%% This is file `mcmthesis-demo.tex', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% mcmthesis.dtx (with options: `demo') +%% +%% ----------------------------------- +%% +%% This is a generated file. +%% +%% Copyright (C) +%% 2010 -- 2015 by Zhaoli Wang +%% 2014 -- 2019 by Liam Huang +%% 2019 -- present by latexstudio.net +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public License, either version 1.3 +%% of this license or (at your option) any later version. +%% The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3 or later is part of all distributions of LaTeX +%% version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The Current Maintainer of this work is Liam Huang. +%% +\documentclass{mcmthesis} +\mcmsetup{CTeX = false, % 使用 CTeX 套装时，设置为 true + tcn = {0000000}, problem = \textcolor{red}{A}, + sheet = true, titleinsheet = true, keywordsinsheet = true, + titlepage = false, abstract = false} + +\usepackage{newtxtext} % \usepackage{palatino} +\usepackage[style=apa,backend=biber]{biblatex} +\addbibresource{reference.bib} + +\usepackage{tocloft} +\setlength{\cftbeforesecskip}{6pt} +\renewcommand{\contentsname}{\hspace*{\fill}\Large\bfseries Contents \hspace*{\fill}} + +\title{Enjoy a Cozy and Green Bath} +% \author{\small \href{http://www.latexstudio.net/} +% {\includegraphics[width=7cm]{mcmthesis-logo}}} +\date{\today} + +\begin{document} + +\begin{abstract} + +A traditional bathtub cannot be reheated by itself, so users have to add hot +water from time to time. Our goal is to establish a model of the temperature +of bath water in space and time. Then we are expected to propose an optimal +strategy for users to keep the temperature even and close to initial temperature +and decrease water consumption. + +To simplify modeling process, we firstly assume there is no person in the bathtub. +We regard the whole bathtub as a thermodynamic system and introduce heat transfer +formulas. + +We establish two sub-models: adding water constantly and discontinuously. As for +the former sub-model, we define the mean temperature of bath water. Introducing +Newton cooling formula, we determine the heat transfer capacity. After deriving +the value of parameters, we deduce formulas to derive results and simulate the +change of temperature field via CFD. As for the second sub-model, we define an +iteration consisting of two process: heating and standby. According to energy +conservation law, we obtain the relationship of time and total heat dissipating +capacity. Then we determine the mass flow and the time of adding hot water. We +also use CFD to simulate the temperature field in second sub-model. + +In consideration of evaporation, we correct the results of sub-models referring +to some scientists' studies. We define two evaluation criteria and compare the +two sub-models. Adding water constantly is found to keep the temperature of bath +water even and avoid wasting too much water, so it is recommended by us. + +Then we determine the influence of some factors: radiation heat transfer, the +shape and volume of the tub, the shape/volume/temperature/motions of the person, +the bubbles made from bubble bath additives. We focus on the influence of those +factors to heat transfer and then conduct sensitivity analysis. The results +indicate smaller bathtub with less surface area, lighter personal mass, less +motions and more bubbles will decrease heat transfer and save water. + +Based on our model analysis and conclusions, we propose the optimal strategy for +the user in a bathtub and explain the reason of uneven temperature throughout +the bathtub. In addition, we make improvement for applying our model in real life. + +\begin{keywords} +Heat transfer, Thermodynamic system, CFD, Energy conservation +\end{keywords} + +\end{abstract} + +\maketitle + +%% Generate the Table of Contents, if it's needed. +% \renewcommand{\contentsname}{\centering Contents} +\tableofcontents % 若不想要目录, 注释掉该句 +\thispagestyle{empty} + +\newpage + +\section{Introduction} + +\subsection{Background} + +Bathing in a tub is a perfect choice for those who have been worn out after a +long day's working. A traditional bathtub is a simply water containment vessel +without a secondary heating system or circulating jets. Thus the temperature of +water in bathtub declines noticeably as time goes by, which will influent the +experience of bathing. As a result, the bathing person needs to add a constant +trickle of hot water from a faucet to reheat the bathing water. This way of +bathing will result in waste of water because when the capacity of the bathtub +is reached, excess water overflows the tub. + +An optimal bathing strategy is required for the person in a bathtub to get +comfortable bathing experience while reducing the waste of water. + +\subsection{Literature Review} + +A traditional bathtub cannot be reheated by itself, so users have to add hot +water from time to time. Our goal is to establish a model of the temperature +of bath water in space and time. Then we are expected to propose an optimal +strategy for users to keep the temperature even and close to the initial +temperature and decrease water consumption. According to \textcite{kim2006}, +He derived a relational equation based on the basic theory of heat transfer +to evaluate the performance of bath tubes. The major heat loss was found to be +due to evaporation. Moreover, he found out that the speed of heat loss depends +more on the humidity of the bathroom than the temperature of water contained +in the bathtub. So, it is best to maintain the temperature of bathtub water to +be between 41 to 45$^{\circ}$C and the humidity of bathroom to be 95\%. +Traditional bath systems have significant limitations in temperature control. +To address this, we introduce heat transfer formulas as +discussed (\cite[123]{holman2002}). + +\subsection{Restatement of the Problem} + +We are required to establish a model to determine the change of water temperature +in space and time. Then we are expected to propose the best strategy for the +person in the bathtub to keep the water temperature close to initial temperature +and even throughout the tub. Reduction of waste of water is also needed. In +addition, we have to consider the impact of different conditions on our model, +such as different shapes and volumes of the bathtub, etc. + +In order to solve those problems, we will proceed as follows: + +\begin{itemize} +\item {\bf Stating assumptions}. By stating our assumptions, we will narrow the +focus of our approach towards the problems and provide some insight into bathtub +water temperature issues. + +\item {\bf Making notations}. We will give some notations which are important for +us to clarify our models. + +\item {\bf Presenting our model}. In order to investigate the problem deeper, we +divide our model into two sub-models. One is a steady convection heat transfer +sub-model in which hot water is added constantly. The other one is an unsteady +convection heat transfer sub-model where hot water is added discontinuously. + +\item {Defining evaluation criteria and comparing sub-models}. We define two main +criteria to evaluate our model: the mean temperature of bath water and the amount +of inflow water. + +\item {\bf Analysis of influencing factors}. In term of the impact of different +factors on our model, we take those into consideration: the shape and volume of +the tub, the shape/volume/temperature of the person in the bathtub, the motions +made by the person in the bathtub and adding a bubble bath additive initially. + +\item {\bf Model testing and sensitivity analysis}. With the criteria defined +before, we evaluate the reliability of our model and do the sensitivity analysis. + +\item {\bf Further discussion}. We discuss about different ways to arrange inflow +faucets. Then we improve our model to apply them in reality. + +\item {\bf Evaluating the model}. We discuss about the strengths and weaknesses +of our model: + +\begin{itemize} +\item[1)] ... +\item[2)] ... +\item[3)] ... +\item[4)] ... +\end{itemize} + +\end{itemize} + +\section{Assumptions and Justification} + +To simplify the problem and make it convenient for us to simulate real-life +conditions, we make the following basic assumptions, each of which is properly +justified. + +\begin{itemize} +\item {\bf The bath water is incompressible Non-Newtonian fluid}. The +incompressible Non-Newtonian fluid is the basis of Navier–Stokes equations +which are introduced to simulate the flow of bath water. + +\item {\bf All the physical properties of bath water, bathtub and air are +assumed to be stable}. The change of those properties like specific heat, +thermal conductivity and density is rather small according to some +studies. It is complicated and unnecessary to consider these little +change so we ignore them. + +\item {\bf There is no internal heat source in the system consisting of bathtub, +hot water and air}. Before the person lies in the bathtub, no internal heat source +exist except the system components. The circumstance where the person is in the +bathtub will be investigated in our later discussion. + +\item {\bf We ignore radiative thermal exchange}. According to Stefan-Boltzmann’s +law, the radiative thermal exchange can be ignored when the temperature is low. +Refer to industrial standard, the temperature in bathroom is lower than +100 $^{\circ}$C, so it is reasonable for us to make this assumption. + +\item {\bf The temperature of the adding hot water from the faucet is stable}. +This hypothesis can be easily achieved in reality and will simplify our process +of solving the problem. +\end{itemize} + +\section{Notations} + +\begin{center} +\begin{tabular}{clc} +{\bf Symbols} & {\bf Description} & \quad {\bf Unit} \\[0.25cm] +$h$ & Convection heat transfer coefficient & \quad W/(m$^2 \cdot$ K) +\\[0.2cm] +$k$ & Thermal conductivity & \quad W/(m $\cdot$ K) \\[0.2cm] +$c_p$ & Specific heat & \quad J/(kg $\cdot$ K) \\[0.2cm] +$\rho$ & Density & \quad kg/m$^2$ \\[0.2cm] +$\delta$ & Thickness & \quad m \\[0.2cm] +$t$ & Temperature & \quad $^\circ$C, K \\[0.2cm] +$\tau$ & Time & \quad s, min, h \\[0.2cm] +$q_m$ & Mass flow & \quad kg/s \\[0.2cm] +$\Phi$ & Heat transfer power & \quad W \\[0.2cm] +$T$ & A period of time & \quad s, min, h \\[0.2cm] +$V$ & Volume & \quad m$^3$, L \\[0.2cm] +$M,\,m$ & Mass & \quad kg \\[0.2cm] +$A$ & Aera & \quad m$^2$ \\[0.2cm] +$a,\,b,\,c$ & The size of a bathtub & \quad m$^3$ +\end{tabular} +\end{center} + +\noindent where we define the main parameters while specific value of those +parameters will be given later. + +\section{Model Overview} + +To simplify the modeling process, we firstly assume there is no person in the +bathtub. We regard the whole bathtub as a thermodynamic system and introduce +heat transfer formulas. We establish two sub-models: adding water constantly +and discontinuously. For the former sub-model, we define the mean temperature +of bath water and introduce Newton's cooling formula to determine the heat +transfer capacity. After deriving the value of parameters, we deduce formulas +to derive results and simulate the change of temperature field via CFD, as +described by \textcite{anderson2006}. + +In our basic model, we aim at three goals: keeping the temperature as even as +possible, making it close to the initial temperature and decreasing the water +consumption. + +We start with the simple sub-model where hot water is added constantly. +At first we introduce convection heat transfer control equations in rectangular +coordinate system. Then we define the mean temperature of bath water. + +Afterwards, we introduce Newton cooling formula to determine heat transfer +capacity. After deriving the value of parameters, we get calculating results +via formula deduction and simulating results via CFD. + +Secondly, we present the complicated sub-model in which hot water is +added discontinuously. We define an iteration consisting of two process: +heating and standby. As for heating process, we derive control equations and +boundary conditions. As for standby process, considering energy conservation law, +we deduce the relationship of total heat dissipating capacity and time. + +Then we determine the time and amount of added hot water. After deriving the +value of parameters, we get calculating results via formula deduction and +simulating results via CFD. + +At last, we define two criteria to evaluate those two ways of adding hot water. +Then we propose optimal strategy for the user in a bathtub. +The whole modeling process can be shown as follows. + +\begin{figure}[h] +\centering +\includegraphics[width=12cm]{fig1.jpg} +\caption{Modeling process} \label{fig1} +\end{figure} + +\section{Sub-model I : Adding Water Continuously} + +As for the second sub-model, we define an iteration consisting of two processes: +heating and standby. According to the energy conservation law, we obtain the +relationship of time and total heat dissipating capacity. Then we determine +the mass flow and the time of adding hot water. We also use CFD to simulate +the temperature field in the second sub-model, following the techniques +outlined by \textcite{website2024}. + +We first establish the sub-model based on the condition that a person add water +continuously to reheat the bathing water. Then we use Computational Fluid +Dynamics (CFD) to simulate the change of water temperature in the bathtub. At +last, we evaluate the model with the criteria which have been defined before. + +\subsection{Model Establishment} + +Since we try to keep the temperature of the hot water in bathtub to be even, +we have to derive the amount of inflow water and the energy dissipated by the +hot water into the air. + +We derive the basic convection heat transfer control equations based on the +former scientists’ achievement. Then, we define the mean temperature of bath +water. Afterwards, we determine two types of heat transfer: the boundary heat +transfer and the evaporation heat transfer. Combining thermodynamic formulas, +we derive calculating results. Via Fluent software, we get simulation results. + +\subsubsection{Control Equations and Boundary Conditions} + +According to thermodynamics knowledge, we recall on basic convection +heat transfer control equations in rectangular coordinate system. Those +equations show the relationship of the temperature of the bathtub water in space. + +We assume the hot water in the bathtub as a cube. Then we put it into a +rectangular coordinate system. The length, width, and height of it is $a,\, b$ +and $c$. + +\begin{figure}[h] +\centering +\includegraphics[width=8cm]{fig2.jpg} +\caption{Modeling process} \label{fig2} +\end{figure} + +In the basis of this, we introduce the following equations: + +\begin{itemize} +\item {\bf Continuity equation:} +\end{itemize} + +\begin{equation} \label{eq1} +\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + +\frac{\partial w}{\partial z} = 0 +\end{equation} + +\noindent where the first component is the change of fluid mass along the $X$-ray. +The second component is the change of fluid mass along the $Y$-ray. And the third +component is the change of fluid mass along the $Z$-ray. The sum of the change in +mass along those three directions is zero. + +\begin{itemize} +\item {\bf Moment differential equation (N-S equations):} +\end{itemize} + +\begin{equation} \label{eq2} +\left\{ +\begin{array}{l} \!\! +\rho \Big(u \dfrac{\partial u}{\partial x} + v \dfrac{\partial u}{\partial y} + +w\dfrac{\partial u}{\partial z} \Big) = -\dfrac{\partial p}{\partial x} + +\eta \Big(\dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial^2 u}{\partial y^2} + +\dfrac{\partial^2 u}{\partial z^2} \Big) \\[0.3cm] +\rho \Big(u \dfrac{\partial v}{\partial x} + v \dfrac{\partial v}{\partial y} + +w\dfrac{\partial v}{\partial z} \Big) = -\dfrac{\partial p}{\partial y} + +\eta \Big(\dfrac{\partial^2 v}{\partial x^2} + \dfrac{\partial^2 v}{\partial y^2} + +\dfrac{\partial^2 v}{\partial z^2} \Big) \\[0.3cm] +\rho \Big(u \dfrac{\partial w}{\partial x} + v \dfrac{\partial w}{\partial y} + +w\dfrac{\partial w}{\partial z} \Big) = -g-\dfrac{\partial p}{\partial z} + +\eta \Big(\dfrac{\partial^2 w}{\partial x^2} + \dfrac{\partial^2 w}{\partial y^2} + +\dfrac{\partial^2 w}{\partial z^2} \Big) +\end{array} +\right. +\end{equation} + +\begin{itemize} +\item {\bf Energy differential equation:} +\end{itemize} + +\begin{equation} \label{eq3} +\rho c_p \Big( u\frac{\partial t}{\partial x} + v\frac{\partial t}{\partial y} + +w\frac{\partial t}{\partial z} \Big) = \lambda \Big(\frac{\partial^2 t}{\partial x^2} + +\frac{\partial^2 t}{\partial y^2} + \frac{\partial^2 t}{\partial z^2} \Big) +\end{equation} + +\noindent where the left three components are convection terms while the right +three components are conduction terms. + +By Equation \eqref{eq3}, we have ...... + +...... + +On the right surface in Fig. \ref{fig2}, the water also transfers heat firstly +with bathtub inner surfaces and then the heat comes into air. The boundary +condition here is ...... + +\subsubsection{Definition of the Mean Temperature} + +...... + +\subsubsection{Determination of Heat Transfer Capacity} + +...... + +\section{Sub-model II: Adding Water Discontinuously} + +In order to establish the unsteady sub-model, we recall on the working principle +of air conditioners. The heating performance of air conditions consist of two +processes: heating and standby. After the user set a temperature, the air +conditioner will begin to heat until the expected temperature is reached. Then +it will go standby. When the temperature get below the expected temperature, +the air conditioner begin to work again. As it works in this circle, the +temperature remains the expected one. + +Inspired by this, we divide the bathtub working into two processes: adding +hot water until the expected temperature is reached, then keeping this +condition for a while unless the temperature is lower than a specific value. +Iterating this circle ceaselessly will ensure the temperature kept relatively +stable. + +\subsection{Heating Model} + +\subsubsection{Control Equations and Boundary Conditions} + +\subsubsection{Determination of Inflow Time and Amount} + +\subsection{Standby Model} + +\subsection{Results} + +\quad~ We first give the value of parameters based on others’ studies. Then we +get the calculation results and simulating results via those data. + +\subsubsection{Determination of Parameters} + +After establishing the model, we have to determine the value of some +important parameters. + +As scholar Beum Kim points out, the optimal temperature for bath is +between 41 and 45$^\circ$C. Meanwhile, according to Shimodozono's study, +41$^\circ$C warm water bath is the perfect choice for individual health. +So it is reasonable for us to focus on $41^\circ$C $\sim 45^\circ$C. Because +adding hot water continuously is a steady process, so the mean temperature +of bath water is supposed to be constant. We value the temperature of inflow +and outflow water with the maximum and minimum temperature respectively. + +The values of all parameters needed are shown as follows: + +..... + +\subsubsection{Calculating Results} + +Putting the above value of parameters into the equations we derived before, we +can get the some data as follows: + +%%普通表格 +\begin{table}[h] %h表示固定在当前位置 +\centering %设置居中 +\caption{The calculating results} %表标题 +\vspace{0.15cm} +\label{tab2} %设置表的引用标签 +\begin{tabular}{|c|c|c|} %3个c表示3列, |可选, 表示绘制各列间的竖线 +\hline %画横线 +Variables & Values & Unit \\ \hline %各列间用&隔开 +$A_1$ & 1.05 & $m^2$ \\ \hline +$A_2$ & 2.24 & $m^2$ \\ \hline +$\Phi_1$ & 189.00 & $W$ \\ \hline +$\Phi_2$ & 43.47 & $W$ \\ \hline +$\Phi$ & 232.47 & $W$ \\ \hline +$q_m$ & 0.014 & $g/s$ \\ \hline +\end{tabular} +\end{table} + +From Table \ref{tab2}, ...... + +...... + +\section{Correction and Contrast of Sub-Models} + +After establishing two basic sub-models, we have to correct them in consideration +of evaporation heat transfer. Then we define two evaluation criteria to compare +the two sub-models in order to determine the optimal bath strategy. + +\subsection{Correction with Evaporation Heat Transfer} + +Someone may confuse about the above results: why the mass flow in the first +sub-model is so small? Why the standby time is so long? Actually, the above two +sub-models are based on ideal conditions without consideration of the change of +boundary conditions, the motions made by the person in bathtub and the +evaporation of bath water, etc. The influence of personal motions will be +discussed later. Here we introducing the evaporation of bath water to correct +sub-models. + +\subsection{Contrast of Two Sub-Models} + +Firstly we define two evaluation criteria. Then we contrast the two submodels +via these two criteria. Thus we can derive the best strategy for the person in +the bathtub to adopt. + +\section{Model Analysis and Sensitivity Analysis} + +In consideration of evaporation, we correct the results of sub-models referring +to studies. We define two evaluation criteria +and compare the two sub-models. Adding water constantly is found to keep the +temperature of bath water even and avoid wasting too much water, so it is +recommended by us. We also conduct sensitivity analysis to determine the +influence of factors such as radiation heat transfer, the shape and volume of +the tub, the shape/volume/temperature/motions of the person, and the bubbles +made from bubble bath additives, as discussed +in (\cite{evaporation2018}; \cite{thesis2015}). + +\subsection{The Influence of Different Bathtubs} + +Definitely, the difference in shape and volume of the tub affects the +convection heat transfer. Examining the relationship between them can help +people choose optimal bathtubs. + +\subsubsection{Different Volumes of Bathtubs} + +In reality, a cup of water will be cooled down rapidly. However, it takes quite +long time for a bucket of water to become cool. That is because their volume is +different and the specific heat of water is very large. So that the decrease of +temperature is not obvious if the volume of water is huge. That also explains +why it takes 45 min for 320 L water to be cooled by 1$^\circ$C. + +In order to examine the influence of volume, we analyze our sub-models +by conducting sensitivity Analysis to them. + +We assume the initial volume to be 280 L and change it by $\pm 5$\%, $\pm 8$\%, +$\pm 12$\% and $\pm 15$\%. With the aid of sub-models we established before, the +variation of some parameters turns out to be as follows + +%%三线表 +\begin{table}[h] %h表示固定在当前位置 +\centering %设置居中 +\caption{Variation of some parameters} %表标题 +\label{tab7} %设置表的引用标签 +\begin{tabular}{ccccccc} %7个c表示7列, c表示每列居中对齐, 还有l和r可选 +\toprule %画顶端横线 +$V$ & $A_1$ & $A_2$ & $T_2$ & $q_{m1}$ & $q_{m2}$ & $\Phi_q$ \\ +\midrule %画中间横线 +-15.00\% & -5.06\% & -9.31\% & -12.67\% & -2.67\% & -14.14\% & -5.80\% \\ +-12.00\% & -4.04\% & -7.43\% & -10.09\% & -2.13\% & -11.31\% & -4.63\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +\bottomrule %画底部横线 +\end{tabular} +\end{table} + +\section{Strength and Weakness} + +\subsection{Strength} + +\begin{itemize} +\item We analyze the problem based on thermodynamic formulas and laws, so that +the model we established is of great validity. + +\item Our model is fairly robust due to our careful corrections in consideration +of real-life situations and detailed sensitivity analysis. + +\item Via Fluent software, we simulate the time field of different areas +throughout the bathtub. The outcome is vivid for us to understand the changing +process. + +\item We come up with various criteria to compare different situations, like +water consumption and the time of adding hot water. Hence an overall comparison +can be made according to these criteria. + +\item Besides common factors, we still consider other factors, such as evaporation +and radiation heat transfer. The evaporation turns out to be the main reason of +heat loss, which corresponds with other scientist’s experimental outcome. +\end{itemize} + +\subsection{Weakness} + +\begin{itemize} +\item Having knowing the range of some parameters from others’ essays, we choose +a value from them to apply in our model. Those values may not be reasonable in +reality. + +\item Although we investigate a lot in the influence of personal motions, they +are so complicated that need to be studied further. + +\item Limited to time, we do not conduct sensitivity analysis for the influence +of personal surface area. +\end{itemize} + +\section{Further Discussion} + +Based on our model analysis and conclusions, we propose the optimal strategy +for the user in a bathtub and explain the reason for the uneven temperature +throughout the bathtub. In addition, we make improvements for applying our +model in real life, as suggested by the patent \textcite{patent2023}. + +In this part, we will focus on different distribution of inflow faucets. Then we +discuss about the real-life application of our model. + +\begin{itemize} +\item Different Distribution of Inflow Faucets + +In our before discussion, we assume there being just one entrance of inflow. + +From the simulating outcome, we find the temperature of bath water is hardly even. +So we come up with the idea of adding more entrances. + +The simulation turns out to be as follows + +\begin{figure}[h] +\centering +\includegraphics[width=12cm]{fig24.jpg} +\caption{The simulation results of different ways of arranging entrances} \label{fig24} +\end{figure} + +From the above figure, the more the entrances are, the evener the temperature +will be. Recalling on the before simulation outcome, when there is only one +entrance for inflow, the temperature of corners is quietly lower than the middle +area. + +In conclusion, if we design more entrances, it will be easier to realize the goal +to keep temperature even throughout the bathtub. + +\item Model Application + +Our before discussion is based on ideal assumptions. In reality, we have to make +some corrections and improvement. + +\begin{itemize} +\item[1)] Adding hot water continually with the mass flow of 0.16 kg/s. This way +can ensure even mean temperature throughout the bathtub and waste less water. + +\item[2)] The manufacturers can design an intelligent control system to monitor +the temperature so that users can get more enjoyable bath experience. + +\item[3)] We recommend users to add bubble additives to slow down the water being +cooler and help cleanse. The additives with lower thermal conductivity are optimal. + +\item[4)] The study method of our establishing model can be applied in other area +relative to convection heat transfer, such as air conditioners. +\end{itemize} +\end{itemize} + +\printbibliography + +\newpage + +\begin{letter}{Enjoy Your Bath Time!} + +From simulation results of real-life situations, we find it takes a period of +time for the inflow hot water to spread throughout the bathtub. During this +process, the bath water continues transferring heat into air, bathtub and the +person in bathtub. The difference between heat transfer capacity makes the +temperature of various areas to be different. So that it is difficult to get +an evenly maintained temperature throughout the bath water. + +In order to enjoy a comfortable bath with even temperature of bath water and +without wasting too much water, we propose the following suggestions. + +\begin{itemize} +\item Adding hot water consistently +\item Using smaller bathtub if possible +\item Decreasing motions during bath +\item Using bubble bath additives +\item Arranging more faucets of inflow +\end{itemize} + +\vspace{\parskip} + +Sincerely yours, + +Your friends + +\end{letter} + +\newpage + +\begin{appendices} + +\section{First appendix} + +In addition, your report must include a letter to the Chief Financial Officer +(CFO) of the Goodgrant Foundation, Mr. Alpha Chiang, that describes the optimal +investment strategy, your modeling approach and major results, and a brief +discussion of your proposed concept of a return-on-investment (ROI). This letter +should be no more than two pages in length. + +Here are simulation programmes we used in our model as follow (\cite{Liu02}).\\ + +\textbf{\textcolor[rgb]{0.98,0.00,0.00}{Input matlab source:}} +\lstinputlisting[language=Matlab]{./code/mcmthesis-matlab1.m} + +\section{Second appendix} + +some more text \textcolor[rgb]{0.98,0.00,0.00}{\textbf{Input C++ source:}} +\lstinputlisting[language=C++]{./code/mcmthesis-sudoku.cpp} + +\end{appendices} + +\newpage +\newcounter{lastpage} +\setcounter{lastpage}{\value{page}} +\thispagestyle{empty} + +\section*{Report on Use of AI} + +\begin{enumerate} +\item OpenAI ChatGPT (Nov 5, 2023 version, ChatGPT-4,) +\begin{description} +\item[Query1:] +\item[Output:] +\end{description} +\item OpenAI Ernie (Nov 5, 2023 version, Ernie 4.0) +\begin{description} +\item[Query1:] +\item[Output:] +\end{description} +\item Github CoPilot (Feb 3, 2024 version) +\begin{description} +\item[Query1:] +\item[Output:] +\end{description} +\item Google Bard (Feb 2, 2024 version) +\begin{description} +\item[Query1:] +\item[Output:] +\end{description} +\end{enumerate} + +% 重置页码 +\clearpage +\setcounter{page}{\value{lastpage}} + +\end{document} +%% +%% This work consists of these files mcmthesis.dtx, +%% figures/ and +%% code/, +%% and the derived files mcmthesis.cls, +%% mcmthesis-demo.tex, +%% README, +%% LICENSE, +%% mcmthesis.pdf and +%% mcmthesis-demo.pdf. +%% +%% End of file `mcmthesis-demo.tex'. diff --git a/vscode/.Rproj.user/1248C2B7/sources/prop/12468681 b/vscode/.Rproj.user/1248C2B7/sources/prop/12468681 new file mode 100755 index 0000000..dc18769 --- /dev/null +++ b/vscode/.Rproj.user/1248C2B7/sources/prop/12468681 @@ -0,0 +1,6 @@ +{ + "source_window_id": "", + "Source": "Source", + "cursorPosition": "80,21", + "scrollLine": "72" +} \ No newline at end of file diff --git a/vscode/.Rproj.user/1248C2B7/sources/prop/352D9BE2 b/vscode/.Rproj.user/1248C2B7/sources/prop/352D9BE2 new file mode 100755 index 0000000..a007fad --- /dev/null +++ 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+%% +%% This is file `mcmthesis-demo.tex', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% mcmthesis.dtx (with options: `demo') +%% +%% ----------------------------------- +%% +%% This is a generated file. +%% +%% Copyright (C) +%% 2010 -- 2015 by Zhaoli Wang +%% 2014 -- 2019 by Liam Huang +%% 2019 -- present by latexstudio.net +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public License, either version 1.3 +%% of this license or (at your option) any later version. +%% The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3 or later is part of all distributions of LaTeX +%% version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The Current Maintainer of this work is Liam Huang. +%% +\documentclass{mcmthesis} +\mcmsetup{CTeX = false, % 使用 CTeX 套装时，设置为 true + tcn = {0000000}, problem = \textcolor{red}{A}, + sheet = true, titleinsheet = true, keywordsinsheet = true, + titlepage = false, abstract = false} + +\usepackage{newtxtext} % \usepackage{palatino} +\usepackage[backend=bibtex]{biblatex} % for RStudio Complie + +\usepackage{tocloft} +\setlength{\cftbeforesecskip}{6pt} +\renewcommand{\contentsname}{\hspace*{\fill}\Large\bfseries Contents \hspace*{\fill}} + +\title{Enjoy a Cozy and Green Bath} +% \author{\small \href{http://www.latexstudio.net/} +% {\includegraphics[width=7cm]{mcmthesis-logo}}} +\date{\today} + +\begin{document} + +\begin{abstract} + +A traditional bathtub cannot be reheated by itself, so users have to add hot water from time to time. Our goal is to establish a model of the temperature of bath water in space and time. Then we are expected to propose an optimal strategy for users to keep the temperature even and close to initial temperature and decrease water consumption. + +To simplify modeling process, we firstly assume there is no person in the bathtub. We regard the whole bathtub as a thermodynamic system and introduce heat transfer formulas. + +We establish two sub-models: adding water constantly and discontinuously. As for the former sub-model, we define the mean temperature of bath water. Introducing Newton cooling formula, we determine the heat transfer capacity. After deriving the value of parameters, we deduce formulas to derive results and simulate the change of temperature field via CFD. As for the second sub-model, we define an iteration consisting of two process: heating and standby. According to energy conservation law, we obtain the relationship of time and total heat dissipating capacity. Then we determine the mass flow and the time of adding hot water. We also use CFD to simulate the temperature field in second sub-model. + +In consideration of evaporation, we correct the results of sub-models referring to some scientists' studies. We define two evaluation criteria and compare the two sub-models. Adding water constantly is found to keep the temperature of bath water even and avoid wasting too much water, so it is recommended by us. + +Then we determine the influence of some factors: radiation heat transfer, the shape and volume of the tub, the shape/volume/temperature/motions of the person, the bubbles made from bubble bath additives. We focus on the influence of those factors to heat transfer and then conduct sensitivity analysis. The results indicate smaller bathtub with less surface area, lighter personal mass, less motions and more bubbles will decrease heat transfer and save water. + +Based on our model analysis and conclusions, we propose the optimal strategy for the user in a bathtub and explain the reason of uneven temperature throughout the bathtub. In addition, we make improvement for applying our model in real life. + +\begin{keywords} +Heat transfer, Thermodynamic system, CFD, Energy conservation +\end{keywords} + +\end{abstract} + +\maketitle + +%% Generate the Table of Contents, if it's needed. +% \renewcommand{\contentsname}{\centering Contents} +\tableofcontents % 若不想要目录, 注释掉该句 +\thispagestyle{empty} + +\newpage + +\section{Introduction} + +\subsection{Background} + +Bathing in a tub is a perfect choice for those who have been worn out after a long day's working. A traditional bathtub is a simply water containment vessel without a secondary heating system or circulating jets. Thus the temperature of water in bathtub declines noticeably as time goes by, which will influent the experience of bathing. As a result, the bathing person needs to add a constant trickle of hot water from a faucet to reheat the bathing water. This way of bathing will result in waste of water because when the capacity of the bathtub is reached, excess water overflows the tub. + +An optimal bathing strategy is required for the person in a bathtub to get comfortable bathing experience while reducing the waste of water. + +\subsection{Literature Review} + +Korean physicist Gi-Beum Kim analyzed heat loss through free surface of water contained in bathtub due to conduction and evaporation \cite{1}. He derived a relational equation based on the basic theory of heat transfer to evaluate the performance of bath tubes. The major heat loss was found to be due to evaporation. Moreover, he found out that the speed of heat loss depends more on the humidity of the bathroom than the temperature of water contained in the bathtub. So, it is best to maintain the temperature of bathtub water to be between 41 to 45$^{\circ}$C and the humidity of bathroom to be 95\%. + +When it comes to convective heat transfer in bathtub, many studies can be referred to. Newton's law of cooling states that the rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings while under the effects of a breeze \cite{2}. Claude-Louis Navier and George Gabriel Stokes described the motion of viscous fluid substances with the Navier-Stokes equations. Those equations may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing \cite{3}. + +In addition, some numerical simulation software are applied in solving and analyzing problems that involve fluid flows. For example, Computational Fluid Dynamics (CFD) is a common one used to perform the calculations required to simulate the interaction of liquids and gases with surfaces defined by boundary conditions \cite{4}. + +\subsection{Restatement of the Problem} + +We are required to establish a model to determine the change of water temperature in space and time. Then we are expected to propose the best strategy for the person in the bathtub to keep the water temperature close to initial temperature and even throughout the tub. Reduction of waste of water is also needed. In addition, we have to consider the impact of different conditions on our model, such as different shapes and volumes of the bathtub, etc. + +In order to solve those problems, we will proceed as follows: + +\begin{itemize} +\item {\bf Stating assumptions}. By stating our assumptions, we will narrow the focus of our approach towards the problems and provide some insight into bathtub water temperature issues. + +\item {\bf Making notations}. We will give some notations which are important for us to clarify our models. + +\item {\bf Presenting our model}. In order to investigate the problem deeper, we divide our model into two sub-models. One is a steady convection heat transfer sub-model in which hot water is added constantly. The other one is an unsteady convection heat transfer sub-model where hot water is added discontinuously. + +\item {Defining evaluation criteria and comparing sub-models}. We define two main criteria to evaluate our model: the mean temperature of bath water and the amount of inflow water. + +\item {\bf Analysis of influencing factors}. In term of the impact of different factors on our model, we take those into consideration: the shape and volume of the tub, the shape/volume/temperature of the person in the bathtub, the motions made by the person in the bathtub and adding a bubble bath additive initially. + +\item {\bf Model testing and sensitivity analysis}. With the criteria defined before, we evaluate the reliability of our model and do the sensitivity analysis. + +\item {\bf Further discussion}. We discuss about different ways to arrange inflow faucets. Then we improve our model to apply them in reality. + +\item {\bf Evaluating the model}. We discuss about the strengths and weaknesses of our model: + +\begin{itemize} +\item[1)] ... +\item[2)] ... +\item[3)] ... +\item[4)] ... +\end{itemize} + +\end{itemize} + +\section{Assumptions and Justification} + +To simplify the problem and make it convenient for us to simulate real-life conditions, we make the following basic assumptions, each of which is properly justified. + +\begin{itemize} +\item {\bf The bath water is incompressible Non-Newtonian fluid}. The incompressible Non-Newtonian fluid is the basis of Navier–Stokes equations which are introduced to simulate the flow of bath water. + +\item {\bf All the physical properties of bath water, bathtub and air are assumed to be stable}. The change of those properties like specific heat, thermal conductivity and density is rather small according to some studies \cite{5}. It is complicated and unnecessary to consider these little change so we ignore them. + +\item {\bf There is no internal heat source in the system consisting of bathtub, hot water and air}. Before the person lies in the bathtub, no internal heat source exist except the system components. The circumstance where the person is in the bathtub will be investigated in our later discussion. + +\item {\bf We ignore radiative thermal exchange}. According to Stefan-Boltzmann’s law, the radiative thermal exchange can be ignored when the temperature is low. Refer to industrial standard \cite{6}, the temperature in bathroom is lower than 100 $^{\circ}$C, so it is reasonable for us to make this assumption. + +\item {\bf The temperature of the adding hot water from the faucet is stable}. This hypothesis can be easily achieved in reality and will simplify our process of solving the problem. +\end{itemize} + +\section{Notations} + +\begin{center} +\begin{tabular}{clc} +{\bf Symbols} & {\bf Description} & \quad {\bf Unit} \\[0.25cm] +$h$ & Convection heat transfer coefficient & \quad W/(m$^2 \cdot$ K) +\\[0.2cm] +$k$ & Thermal conductivity & \quad W/(m $\cdot$ K) \\[0.2cm] +$c_p$ & Specific heat & \quad J/(kg $\cdot$ K) \\[0.2cm] +$\rho$ & Density & \quad kg/m$^2$ \\[0.2cm] +$\delta$ & Thickness & \quad m \\[0.2cm] +$t$ & Temperature & \quad $^\circ$C, K \\[0.2cm] +$\tau$ & Time & \quad s, min, h \\[0.2cm] +$q_m$ & Mass flow & \quad kg/s \\[0.2cm] +$\Phi$ & Heat transfer power & \quad W \\[0.2cm] +$T$ & A period of time & \quad s, min, h \\[0.2cm] +$V$ & Volume & \quad m$^3$, L \\[0.2cm] +$M,\,m$ & Mass & \quad kg \\[0.2cm] +$A$ & Aera & \quad m$^2$ \\[0.2cm] +$a,\,b,\,c$ & The size of a bathtub & \quad m$^3$ +\end{tabular} +\end{center} + +\noindent where we define the main parameters while specific value of those parameters will be given later. + +\section{Model Overview} + +In our basic model, we aim at three goals: keeping the temperature as even as possible, making it close to the initial temperature and decreasing the water consumption. + +We start with the simple sub-model where hot water is added constantly. +At first we introduce convection heat transfer control equations in rectangular coordinate system. Then we define the mean temperature of bath water. + +Afterwards, we introduce Newton cooling formula to determine heat transfer +capacity. After deriving the value of parameters, we get calculating results via formula deduction and simulating results via CFD. + +Secondly, we present the complicated sub-model in which hot water is +added discontinuously. We define an iteration consisting of two process: +heating and standby. As for heating process, we derive control equations and boundary conditions. As for standby process, considering energy conservation law, we deduce the relationship of total heat dissipating capacity and time. + +Then we determine the time and amount of added hot water. After deriving the value of parameters, we get calculating results via formula deduction and simulating results via CFD. + +At last, we define two criteria to evaluate those two ways of adding hot water. Then we propose optimal strategy for the user in a bathtub. +The whole modeling process can be shown as follows. + +\begin{figure}[h] +\centering +\includegraphics[width=12cm]{fig1.jpg} +\caption{Modeling process} \label{fig1} +\end{figure} + +\section{Sub-model I : Adding Water Continuously} + +We first establish the sub-model based on the condition that a person add water continuously to reheat the bathing water. Then we use Computational Fluid Dynamics (CFD) to simulate the change of water temperature in the bathtub. At last, we evaluate the model with the criteria which have been defined before. + +\subsection{Model Establishment} + +Since we try to keep the temperature of the hot water in bathtub to be even, we have to derive the amount of inflow water and the energy dissipated by the hot water into the air. + +We derive the basic convection heat transfer control equations based on the former scientists’ achievement. Then, we define the mean temperature of bath water. Afterwards, we determine two types of heat transfer: the boundary heat transfer and the evaporation heat transfer. Combining thermodynamic formulas, we derive calculating results. Via Fluent software, we get simulation results. + +\subsubsection{Control Equations and Boundary Conditions} + +According to thermodynamics knowledge, we recall on basic convection +heat transfer control equations in rectangular coordinate system. Those +equations show the relationship of the temperature of the bathtub water in space. + +We assume the hot water in the bathtub as a cube. Then we put it into a +rectangular coordinate system. The length, width, and height of it is $a,\, b$ and $c$. + +\begin{figure}[h] +\centering +\includegraphics[width=8cm]{fig2.jpg} +\caption{Modeling process} \label{fig2} +\end{figure} + +In the basis of this, we introduce the following equations \cite{5}: + +\begin{itemize} +\item {\bf Continuity equation:} +\end{itemize} + +\begin{equation} \label{eq1} +\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} +\frac{\partial w}{\partial z} =0 +\end{equation} + +\noindent where the first component is the change of fluid mass along the $X$-ray. The second component is the change of fluid mass along the $Y$-ray. And the third component is the change of fluid mass along the $Z$-ray. The sum of the change in mass along those three directions is zero. + +\begin{itemize} +\item {\bf Moment differential equation (N-S equations):} +\end{itemize} + +\begin{equation} \label{eq2} +\left\{ +\begin{array}{l} \!\! +\rho \Big(u \dfrac{\partial u}{\partial x} + v \dfrac{\partial u}{\partial y} + w\dfrac{\partial u}{\partial z} \Big) = -\dfrac{\partial p}{\partial x} + \eta \Big(\dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial^2 u}{\partial y^2} + \dfrac{\partial^2 u}{\partial z^2} \Big) \\[0.3cm] +\rho \Big(u \dfrac{\partial v}{\partial x} + v \dfrac{\partial v}{\partial y} + w\dfrac{\partial v}{\partial z} \Big) = -\dfrac{\partial p}{\partial y} + \eta \Big(\dfrac{\partial^2 v}{\partial x^2} + \dfrac{\partial^2 v}{\partial y^2} + \dfrac{\partial^2 v}{\partial z^2} \Big) \\[0.3cm] +\rho \Big(u \dfrac{\partial w}{\partial x} + v \dfrac{\partial w}{\partial y} + w\dfrac{\partial w}{\partial z} \Big) = -g-\dfrac{\partial p}{\partial z} + \eta \Big(\dfrac{\partial^2 w}{\partial x^2} + \dfrac{\partial^2 w}{\partial y^2} + \dfrac{\partial^2 w}{\partial z^2} \Big) +\end{array} +\right. +\end{equation} + +\begin{itemize} +\item {\bf Energy differential equation:} +\end{itemize} + +\begin{equation} \label{eq3} +\rho c_p \Big( u\frac{\partial t}{\partial x} + v\frac{\partial t}{\partial y} + w\frac{\partial t}{\partial z} \Big) = \lambda \Big(\frac{\partial^2 t}{\partial x^2} + \frac{\partial^2 t}{\partial y^2} + \frac{\partial^2 t}{\partial z^2} \Big) +\end{equation} + +\noindent where the left three components are convection terms while the right three components are conduction terms. + +By Equation \eqref{eq3}, we have ...... + +...... + +On the right surface in Fig. \ref{fig2}, the water also transfers heat firstly with bathtub inner surfaces and then the heat comes into air. The boundary condition here is ...... + +\subsubsection{Definition of the Mean Temperature} + +...... + +\subsubsection{Determination of Heat Transfer Capacity} + +...... + +\section{Sub-model II: Adding Water Discontinuously} + +In order to establish the unsteady sub-model, we recall on the working principle of air conditioners. The heating performance of air conditions consist of two processes: heating and standby. After the user set a temperature, the air conditioner will begin to heat until the expected temperature is reached. Then it will go standby. When the temperature get below the expected temperature, the air conditioner begin to work again. As it works in this circle, the temperature remains the expected one. + +Inspired by this, we divide the bathtub working into two processes: adding +hot water until the expected temperature is reached, then keeping this +condition for a while unless the temperature is lower than a specific value. Iterating this circle ceaselessly will ensure the temperature kept relatively stable. + +\subsection{Heating Model} + +\subsubsection{Control Equations and Boundary Conditions} + +\subsubsection{Determination of Inflow Time and Amount} + +\subsection{Standby Model} + +\subsection{Results} + +\quad~ We first give the value of parameters based on others’ studies. Then we get the calculation results and simulating results via those data. + +\subsubsection{Determination of Parameters} + +After establishing the model, we have to determine the value of some +important parameters. + +As scholar Beum Kim points out, the optimal temperature for bath is +between 41 and 45$^\circ$C [1]. Meanwhile, according to Shimodozono's study, 41$^\circ$C warm water bath is the perfect choice for individual health [2]. So it is reasonable for us to focus on $41^\circ$C $\sim 45^\circ$C. Because adding hot water continuously is a steady process, so the mean temperature of bath water is supposed to be constant. We value the temperature of inflow and outflow water with the maximum and minimum temperature respectively. + +The values of all parameters needed are shown as follows: + +..... + +\subsubsection{Calculating Results} + +Putting the above value of parameters into the equations we derived before, we can get the some data as follows: + +%%普通表格 +\begin{table}[h] %h表示固定在当前位置 +\centering %设置居中 +\caption{The calculating results} %表标题 +\vspace{0.15cm} +\label{tab2} %设置表的引用标签 +\begin{tabular}{|c|c|c|} %3个c表示3列, |可选, 表示绘制各列间的竖线 +\hline %画横线 +Variables & Values & Unit \\ \hline %各列间用&隔开 +$A_1$ & 1.05 & $m^2$ \\ \hline +$A_2$ & 2.24 & $m^2$ \\ \hline +$\Phi_1$ & 189.00 & $W$ \\ \hline +$\Phi_2$ & 43.47 & $W$ \\ \hline +$\Phi$ & 232.47 & $W$ \\ \hline +$q_m$ & 0.014 & $g/s$ \\ \hline +\end{tabular} +\end{table} + +From Table \ref{tab2}, ...... + +...... + +\section{Correction and Contrast of Sub-Models} + +After establishing two basic sub-models, we have to correct them in consideration of evaporation heat transfer. Then we define two evaluation criteria to compare the two sub-models in order to determine the optimal bath strategy. + +\subsection{Correction with Evaporation Heat Transfer} + +Someone may confuse about the above results: why the mass flow in the first sub-model is so small? Why the standby time is so long? Actually, the above two sub-models are based on ideal conditions without consideration of the change of boundary conditions, the motions made by the person in bathtub and the evaporation of bath water, etc. The influence of personal motions will be discussed later. Here we introducing the evaporation of bath water to correct sub-models. + +\subsection{Contrast of Two Sub-Models} + +Firstly we define two evaluation criteria. Then we contrast the two submodels via these two criteria. Thus we can derive the best strategy for the person in the bathtub to adopt. + +\section{Model Analysis and Sensitivity Analysis} + +\subsection{The Influence of Different Bathtubs} + +Definitely, the difference in shape and volume of the tub affects the +convection heat transfer. Examining the relationship between them can help +people choose optimal bathtubs. + +\subsubsection{Different Volumes of Bathtubs} + +In reality, a cup of water will be cooled down rapidly. However, it takes quite long time for a bucket of water to become cool. That is because their volume is different and the specific heat of water is very large. So that the decrease of temperature is not obvious if the volume of water is huge. That also explains why it takes 45 min for 320 L water to be cooled by 1$^\circ$C. + +In order to examine the influence of volume, we analyze our sub-models +by conducting sensitivity Analysis to them. + +We assume the initial volume to be 280 L and change it by $\pm 5$\%, $\pm 8$\%, $\pm 12$\% and $\pm 15$\%. With the aid of sub-models we established before, the variation of some parameters turns out to be as follows + +%%三线表 +\begin{table}[h] %h表示固定在当前位置 +\centering %设置居中 +\caption{Variation of some parameters} %表标题 +\label{tab7} %设置表的引用标签 +\begin{tabular}{ccccccc} %7个c表示7列, c表示每列居中对齐, 还有l和r可选 +\toprule %画顶端横线 +$V$ & $A_1$ & $A_2$ & $T_2$ & $q_{m1}$ & $q_{m2}$ & $\Phi_q$ \\ +\midrule %画中间横线 +-15.00\% & -5.06\% & -9.31\% & -12.67\% & -2.67\% & -14.14\% & -5.80\% \\ +-12.00\% & -4.04\% & -7.43\% & -10.09\% & -2.13\% & -11.31\% & -4.63\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +\bottomrule %画底部横线 +\end{tabular} +\end{table} + +\section{Strength and Weakness} + +\subsection{Strength} + +\begin{itemize} +\item We analyze the problem based on thermodynamic formulas and laws, so that the model we established is of great validity. + +\item Our model is fairly robust due to our careful corrections in consideration of real-life situations and detailed sensitivity analysis. + +\item Via Fluent software, we simulate the time field of different areas throughout the bathtub. The outcome is vivid for us to understand the changing process. + +\item We come up with various criteria to compare different situations, like water consumption and the time of adding hot water. Hence an overall comparison can be made according to these criteria. + +\item Besides common factors, we still consider other factors, such as evaporation and radiation heat transfer. The evaporation turns out to be the main reason of heat loss, which corresponds with other scientist’s experimental outcome. +\end{itemize} + +\subsection{Weakness} + +\begin{itemize} +\item Having knowing the range of some parameters from others’ essays, we choose a value from them to apply in our model. Those values may not be reasonable in reality. + +\item Although we investigate a lot in the influence of personal motions, they are so complicated that need to be studied further. + +\item Limited to time, we do not conduct sensitivity analysis for the influence of personal surface area. +\end{itemize} + +\section{Further Discussion} + +In this part, we will focus on different distribution of inflow faucets. Then we discuss about the real-life application of our model. + +\begin{itemize} +\item Different Distribution of Inflow Faucets + +In our before discussion, we assume there being just one entrance of inflow. + +From the simulating outcome, we find the temperature of bath water is hardly even. So we come up with the idea of adding more entrances. + +The simulation turns out to be as follows + +\begin{figure}[h] +\centering +\includegraphics[width=12cm]{fig24.jpg} +\caption{The simulation results of different ways of arranging entrances} \label{fig24} +\end{figure} + +From the above figure, the more the entrances are, the evener the temperature will be. Recalling on the before simulation outcome, when there is only one entrance for inflow, the temperature of corners is quietly lower than the middle area. + +In conclusion, if we design more entrances, it will be easier to realize the goal to keep temperature even throughout the bathtub. + +\item Model Application + +Our before discussion is based on ideal assumptions. In reality, we have to make some corrections and improvement. + +\begin{itemize} +\item[1)] Adding hot water continually with the mass flow of 0.16 kg/s. This way can ensure even mean temperature throughout the bathtub and waste less water. + +\item[2)] The manufacturers can design an intelligent control system to monitor the temperature so that users can get more enjoyable bath experience. + +\item[3)] We recommend users to add bubble additives to slow down the water being cooler and help cleanse. The additives with lower thermal conductivity are optimal. + +\item[4)] The study method of our establishing model can be applied in other area relative to convection heat transfer, such as air conditioners. +\end{itemize} +\end{itemize} + +\begin{thebibliography}{99} +\addcontentsline{toc}{section}{Reference} + +\bibitem{1} Gi-Beum Kim. Change of the Warm Water Temperature for the Development of Smart Healthecare Bathing System. Hwahak konghak. 2006, 44(3): 270-276. +\bibitem{2} \url{https://en.wikipedia.org/wiki/Convective_heat_transfer#Newton.27s_law_of_cooling} +\bibitem{3} \url{https://en.wikipedia.org/wiki/Navier\%E2\%80\%93Stokes_equations} +\bibitem{4} \url{https://en.wikipedia.org/wiki/Computational_fluid_dynamics} +\bibitem{5} Holman J P. Heat Transfer (9th ed.), New York: McGraw-Hill, 2002. +\bibitem{6} Liu Weiguo, Chen Zhaoping, ZhangYin. Matlab program design and application. Beijing: Higher education press, 2002. (In Chinese) + +\end{thebibliography} + +\newpage + +\begin{letter}{Enjoy Your Bath Time!} + +From simulation results of real-life situations, we find it takes a period of time for the inflow hot water to spread throughout the bathtub. During this process, the bath water continues transferring heat into air, bathtub and the person in bathtub. The difference between heat transfer capacity makes the temperature of various areas to be different. So that it is difficult to get an evenly maintained temperature throughout the bath water. + +In order to enjoy a comfortable bath with even temperature of bath water and without wasting too much water, we propose the following suggestions. + +\begin{itemize} +\item Adding hot water consistently +\item Using smaller bathtub if possible +\item Decreasing motions during bath +\item Using bubble bath additives +\item Arranging more faucets of inflow +\end{itemize} + +\vspace{\parskip} + +Sincerely yours, + +Your friends + +\end{letter} + +\newpage + +\begin{appendices} + +\section{First appendix} + +In addition, your report must include a letter to the Chief Financial Officer (CFO) of the Goodgrant Foundation, Mr. Alpha Chiang, that describes the optimal investment strategy, your modeling approach and major results, and a brief discussion of your proposed concept of a return-on-investment (ROI). This letter should be no more than two pages in length. + +Here are simulation programmes we used in our model as follow.\\ + +\textbf{\textcolor[rgb]{0.98,0.00,0.00}{Input matlab source:}} +\lstinputlisting[language=Matlab]{./code/mcmthesis-matlab1.m} + +\section{Second appendix} + +some more text \textcolor[rgb]{0.98,0.00,0.00}{\textbf{Input C++ source:}} +\lstinputlisting[language=C++]{./code/mcmthesis-sudoku.cpp} + +\end{appendices} +\end{document} +%% +%% This work consists of these files mcmthesis.dtx, +%% figures/ and +%% code/, +%% and the derived files mcmthesis.cls, +%% mcmthesis-demo.tex, +%% README, +%% LICENSE, +%% mcmthesis.pdf and +%% mcmthesis-demo.pdf. +%% +%% End of file `mcmthesis-demo.tex'. diff --git a/vscode/.Rproj.user/4F8C70FB/sources/prop/6676DB66 b/vscode/.Rproj.user/4F8C70FB/sources/prop/6676DB66 new file mode 100755 index 0000000..31ad8a8 --- /dev/null +++ b/vscode/.Rproj.user/4F8C70FB/sources/prop/6676DB66 @@ -0,0 +1,6 @@ +{ + "source_window_id": "", + 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mode 100755 index 0000000..d7cce2f --- /dev/null +++ b/vscode/.Rproj.user/F69231EA/sources/per/t/110CA16B-contents @@ -0,0 +1,539 @@ +%% +%% This is file `mcmthesis-demo.tex', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% mcmthesis.dtx (with options: `demo') +%% +%% ----------------------------------- +%% +%% This is a generated file. +%% +%% Copyright (C) +%% 2010 -- 2015 by Zhaoli Wang +%% 2014 -- 2019 by Liam Huang +%% 2019 -- present by latexstudio.net +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public License, either version 1.3 +%% of this license or (at your option) any later version. +%% The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3 or later is part of all distributions of LaTeX +%% version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The Current Maintainer of this work is Liam Huang. +%% +%% +%% This is file `mcmthesis-demo.tex', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% mcmthesis.dtx (with options: `demo') +%% +%% ----------------------------------- +%% +%% This is a generated file. +%% +%% Copyright (C) +%% 2010 -- 2015 by Zhaoli Wang +%% 2014 -- 2019 by Liam Huang +%% 2019 -- present by latexstudio.net +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public License, either version 1.3 +%% of this license or (at your option) any later version. +%% The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3 or later is part of all distributions of LaTeX +%% version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The Current Maintainer of this work is Liam Huang. +%% +\documentclass{mcmthesis} +\mcmsetup{CTeX = false, % 使用 CTeX 套装时，设置为 true + tcn = {0000000}, problem = \textcolor{red}{A}, + sheet = true, titleinsheet = true, keywordsinsheet = true, + titlepage = false, abstract = false} + +\usepackage{newtxtext} % \usepackage{palatino} +\usepackage[backend=bibtex]{biblatex} % for RStudio Complie + +\usepackage{tocloft} +\setlength{\cftbeforesecskip}{6pt} +\renewcommand{\contentsname}{\hspace*{\fill}\Large\bfseries Contents \hspace*{\fill}} + +\title{Enjoy a Cozy and Green Bath} +% \author{\small \href{http://www.latexstudio.net/} +% {\includegraphics[width=7cm]{mcmthesis-logo}}} +\date{\today} + +\begin{document} + +\begin{abstract} + +A traditional bathtub cannot be reheated by itself, so users have to add hot water from time to time. Our goal is to establish a model of the temperature of bath water in space and time. Then we are expected to propose an optimal strategy for users to keep the temperature even and close to initial temperature and decrease water consumption. + +To simplify modeling process, we firstly assume there is no person in the bathtub. We regard the whole bathtub as a thermodynamic system and introduce heat transfer formulas. + +We establish two sub-models: adding water constantly and discontinuously. As for the former sub-model, we define the mean temperature of bath water. Introducing Newton cooling formula, we determine the heat transfer capacity. After deriving the value of parameters, we deduce formulas to derive results and simulate the change of temperature field via CFD. As for the second sub-model, we define an iteration consisting of two process: heating and standby. According to energy conservation law, we obtain the relationship of time and total heat dissipating capacity. Then we determine the mass flow and the time of adding hot water. We also use CFD to simulate the temperature field in second sub-model. + +In consideration of evaporation, we correct the results of sub-models referring to some scientists' studies. We define two evaluation criteria and compare the two sub-models. Adding water constantly is found to keep the temperature of bath water even and avoid wasting too much water, so it is recommended by us. + +Then we determine the influence of some factors: radiation heat transfer, the shape and volume of the tub, the shape/volume/temperature/motions of the person, the bubbles made from bubble bath additives. We focus on the influence of those factors to heat transfer and then conduct sensitivity analysis. The results indicate smaller bathtub with less surface area, lighter personal mass, less motions and more bubbles will decrease heat transfer and save water. + +Based on our model analysis and conclusions, we propose the optimal strategy for the user in a bathtub and explain the reason of uneven temperature throughout the bathtub. In addition, we make improvement for applying our model in real life. + +\begin{keywords} +Heat transfer, Thermodynamic system, CFD, Energy conservation +\end{keywords} + +\end{abstract} + +\maketitle + +%% Generate the Table of Contents, if it's needed. +% \renewcommand{\contentsname}{\centering Contents} +\tableofcontents % 若不想要目录, 注释掉该句 +\thispagestyle{empty} + +\newpage + +\section{Introduction} + +\subsection{Background} + +Bathing in a tub is a perfect choice for those who have been worn out after a long day's working. A traditional bathtub is a simply water containment vessel without a secondary heating system or circulating jets. Thus the temperature of water in bathtub declines noticeably as time goes by, which will influent the experience of bathing. As a result, the bathing person needs to add a constant trickle of hot water from a faucet to reheat the bathing water. This way of bathing will result in waste of water because when the capacity of the bathtub is reached, excess water overflows the tub. + +An optimal bathing strategy is required for the person in a bathtub to get comfortable bathing experience while reducing the waste of water. + +\subsection{Literature Review} + +Korean physicist Gi-Beum Kim analyzed heat loss through free surface of water contained in bathtub due to conduction and evaporation \cite{1}. He derived a relational equation based on the basic theory of heat transfer to evaluate the performance of bath tubes. The major heat loss was found to be due to evaporation. Moreover, he found out that the speed of heat loss depends more on the humidity of the bathroom than the temperature of water contained in the bathtub. So, it is best to maintain the temperature of bathtub water to be between 41 to 45$^{\circ}$C and the humidity of bathroom to be 95\%. + +When it comes to convective heat transfer in bathtub, many studies can be referred to. Newton's law of cooling states that the rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings while under the effects of a breeze \cite{2}. Claude-Louis Navier and George Gabriel Stokes described the motion of viscous fluid substances with the Navier-Stokes equations. Those equations may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing \cite{3}. + +In addition, some numerical simulation software are applied in solving and analyzing problems that involve fluid flows. For example, Computational Fluid Dynamics (CFD) is a common one used to perform the calculations required to simulate the interaction of liquids and gases with surfaces defined by boundary conditions \cite{4}. + +\subsection{Restatement of the Problem} + +We are required to establish a model to determine the change of water temperature in space and time. Then we are expected to propose the best strategy for the person in the bathtub to keep the water temperature close to initial temperature and even throughout the tub. Reduction of waste of water is also needed. In addition, we have to consider the impact of different conditions on our model, such as different shapes and volumes of the bathtub, etc. + +In order to solve those problems, we will proceed as follows: + +\begin{itemize} +\item {\bf Stating assumptions}. By stating our assumptions, we will narrow the focus of our approach towards the problems and provide some insight into bathtub water temperature issues. + +\item {\bf Making notations}. We will give some notations which are important for us to clarify our models. + +\item {\bf Presenting our model}. In order to investigate the problem deeper, we divide our model into two sub-models. One is a steady convection heat transfer sub-model in which hot water is added constantly. The other one is an unsteady convection heat transfer sub-model where hot water is added discontinuously. + +\item {Defining evaluation criteria and comparing sub-models}. We define two main criteria to evaluate our model: the mean temperature of bath water and the amount of inflow water. + +\item {\bf Analysis of influencing factors}. In term of the impact of different factors on our model, we take those into consideration: the shape and volume of the tub, the shape/volume/temperature of the person in the bathtub, the motions made by the person in the bathtub and adding a bubble bath additive initially. + +\item {\bf Model testing and sensitivity analysis}. With the criteria defined before, we evaluate the reliability of our model and do the sensitivity analysis. + +\item {\bf Further discussion}. We discuss about different ways to arrange inflow faucets. Then we improve our model to apply them in reality. + +\item {\bf Evaluating the model}. We discuss about the strengths and weaknesses of our model: + +\begin{itemize} +\item[1)] ... +\item[2)] ... +\item[3)] ... +\item[4)] ... +\end{itemize} + +\end{itemize} + +\section{Assumptions and Justification} + +To simplify the problem and make it convenient for us to simulate real-life conditions, we make the following basic assumptions, each of which is properly justified. + +\begin{itemize} +\item {\bf The bath water is incompressible Non-Newtonian fluid}. The incompressible Non-Newtonian fluid is the basis of Navier–Stokes equations which are introduced to simulate the flow of bath water. + +\item {\bf All the physical properties of bath water, bathtub and air are assumed to be stable}. The change of those properties like specific heat, thermal conductivity and density is rather small according to some studies \cite{5}. It is complicated and unnecessary to consider these little change so we ignore them. + +\item {\bf There is no internal heat source in the system consisting of bathtub, hot water and air}. Before the person lies in the bathtub, no internal heat source exist except the system components. The circumstance where the person is in the bathtub will be investigated in our later discussion. + +\item {\bf We ignore radiative thermal exchange}. According to Stefan-Boltzmann’s law, the radiative thermal exchange can be ignored when the temperature is low. Refer to industrial standard \cite{6}, the temperature in bathroom is lower than 100 $^{\circ}$C, so it is reasonable for us to make this assumption. + +\item {\bf The temperature of the adding hot water from the faucet is stable}. This hypothesis can be easily achieved in reality and will simplify our process of solving the problem. +\end{itemize} + +\section{Notations} + +\begin{center} +\begin{tabular}{clc} +{\bf Symbols} & {\bf Description} & \quad {\bf Unit} \\[0.25cm] +$h$ & Convection heat transfer coefficient & \quad W/(m$^2 \cdot$ K) +\\[0.2cm] +$k$ & Thermal conductivity & \quad W/(m $\cdot$ K) \\[0.2cm] +$c_p$ & Specific heat & \quad J/(kg $\cdot$ K) \\[0.2cm] +$\rho$ & Density & \quad kg/m$^2$ \\[0.2cm] +$\delta$ & Thickness & \quad m \\[0.2cm] +$t$ & Temperature & \quad $^\circ$C, K \\[0.2cm] +$\tau$ & Time & \quad s, min, h \\[0.2cm] +$q_m$ & Mass flow & \quad kg/s \\[0.2cm] +$\Phi$ & Heat transfer power & \quad W \\[0.2cm] +$T$ & A period of time & \quad s, min, h \\[0.2cm] +$V$ & Volume & \quad m$^3$, L \\[0.2cm] +$M,\,m$ & Mass & \quad kg \\[0.2cm] +$A$ & Aera & \quad m$^2$ \\[0.2cm] +$a,\,b,\,c$ & The size of a bathtub & \quad m$^3$ +\end{tabular} +\end{center} + +\noindent where we define the main parameters while specific value of those parameters will be given later. + +\section{Model Overview} + +In our basic model, we aim at three goals: keeping the temperature as even as possible, making it close to the initial temperature and decreasing the water consumption. + +We start with the simple sub-model where hot water is added constantly. +At first we introduce convection heat transfer control equations in rectangular coordinate system. Then we define the mean temperature of bath water. + +Afterwards, we introduce Newton cooling formula to determine heat transfer +capacity. After deriving the value of parameters, we get calculating results via formula deduction and simulating results via CFD. + +Secondly, we present the complicated sub-model in which hot water is +added discontinuously. We define an iteration consisting of two process: +heating and standby. As for heating process, we derive control equations and boundary conditions. As for standby process, considering energy conservation law, we deduce the relationship of total heat dissipating capacity and time. + +Then we determine the time and amount of added hot water. After deriving the value of parameters, we get calculating results via formula deduction and simulating results via CFD. + +At last, we define two criteria to evaluate those two ways of adding hot water. Then we propose optimal strategy for the user in a bathtub. +The whole modeling process can be shown as follows. + +\begin{figure}[h] +\centering +\includegraphics[width=12cm]{fig1.jpg} +\caption{Modeling process} \label{fig1} +\end{figure} + +\section{Sub-model I : Adding Water Continuously} + +We first establish the sub-model based on the condition that a person add water continuously to reheat the bathing water. Then we use Computational Fluid Dynamics (CFD) to simulate the change of water temperature in the bathtub. At last, we evaluate the model with the criteria which have been defined before. + +\subsection{Model Establishment} + +Since we try to keep the temperature of the hot water in bathtub to be even, we have to derive the amount of inflow water and the energy dissipated by the hot water into the air. + +We derive the basic convection heat transfer control equations based on the former scientists’ achievement. Then, we define the mean temperature of bath water. Afterwards, we determine two types of heat transfer: the boundary heat transfer and the evaporation heat transfer. Combining thermodynamic formulas, we derive calculating results. Via Fluent software, we get simulation results. + +\subsubsection{Control Equations and Boundary Conditions} + +According to thermodynamics knowledge, we recall on basic convection +heat transfer control equations in rectangular coordinate system. Those +equations show the relationship of the temperature of the bathtub water in space. + +We assume the hot water in the bathtub as a cube. Then we put it into a +rectangular coordinate system. The length, width, and height of it is $a,\, b$ and $c$. + +\begin{figure}[h] +\centering +\includegraphics[width=8cm]{fig2.jpg} +\caption{Modeling process} \label{fig2} +\end{figure} + +In the basis of this, we introduce the following equations \cite{5}: + +\begin{itemize} +\item {\bf Continuity equation:} +\end{itemize} + +\begin{equation} \label{eq1} +\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} +\frac{\partial w}{\partial z} =0 +\end{equation} + +\noindent where the first component is the change of fluid mass along the $X$-ray. The second component is the change of fluid mass along the $Y$-ray. And the third component is the change of fluid mass along the $Z$-ray. The sum of the change in mass along those three directions is zero. + +\begin{itemize} +\item {\bf Moment differential equation (N-S equations):} +\end{itemize} + +\begin{equation} \label{eq2} +\left\{ +\begin{array}{l} \!\! +\rho \Big(u \dfrac{\partial u}{\partial x} + v \dfrac{\partial u}{\partial y} + w\dfrac{\partial u}{\partial z} \Big) = -\dfrac{\partial p}{\partial x} + \eta \Big(\dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial^2 u}{\partial y^2} + \dfrac{\partial^2 u}{\partial z^2} \Big) \\[0.3cm] +\rho \Big(u \dfrac{\partial v}{\partial x} + v \dfrac{\partial v}{\partial y} + w\dfrac{\partial v}{\partial z} \Big) = -\dfrac{\partial p}{\partial y} + \eta \Big(\dfrac{\partial^2 v}{\partial x^2} + \dfrac{\partial^2 v}{\partial y^2} + \dfrac{\partial^2 v}{\partial z^2} \Big) \\[0.3cm] +\rho \Big(u \dfrac{\partial w}{\partial x} + v \dfrac{\partial w}{\partial y} + w\dfrac{\partial w}{\partial z} \Big) = -g-\dfrac{\partial p}{\partial z} + \eta \Big(\dfrac{\partial^2 w}{\partial x^2} + \dfrac{\partial^2 w}{\partial y^2} + \dfrac{\partial^2 w}{\partial z^2} \Big) +\end{array} +\right. +\end{equation} + +\begin{itemize} +\item {\bf Energy differential equation:} +\end{itemize} + +\begin{equation} \label{eq3} +\rho c_p \Big( u\frac{\partial t}{\partial x} + v\frac{\partial t}{\partial y} + w\frac{\partial t}{\partial z} \Big) = \lambda \Big(\frac{\partial^2 t}{\partial x^2} + \frac{\partial^2 t}{\partial y^2} + \frac{\partial^2 t}{\partial z^2} \Big) +\end{equation} + +\noindent where the left three components are convection terms while the right three components are conduction terms. + +By Equation \eqref{eq3}, we have ...... + +...... + +On the right surface in Fig. \ref{fig2}, the water also transfers heat firstly with bathtub inner surfaces and then the heat comes into air. The boundary condition here is ...... + +\subsubsection{Definition of the Mean Temperature} + +...... + +\subsubsection{Determination of Heat Transfer Capacity} + +...... + +\section{Sub-model II: Adding Water Discontinuously} + +In order to establish the unsteady sub-model, we recall on the working principle of air conditioners. The heating performance of air conditions consist of two processes: heating and standby. After the user set a temperature, the air conditioner will begin to heat until the expected temperature is reached. Then it will go standby. When the temperature get below the expected temperature, the air conditioner begin to work again. As it works in this circle, the temperature remains the expected one. + +Inspired by this, we divide the bathtub working into two processes: adding +hot water until the expected temperature is reached, then keeping this +condition for a while unless the temperature is lower than a specific value. Iterating this circle ceaselessly will ensure the temperature kept relatively stable. + +\subsection{Heating Model} + +\subsubsection{Control Equations and Boundary Conditions} + +\subsubsection{Determination of Inflow Time and Amount} + +\subsection{Standby Model} + +\subsection{Results} + +\quad~ We first give the value of parameters based on others’ studies. Then we get the calculation results and simulating results via those data. + +\subsubsection{Determination of Parameters} + +After establishing the model, we have to determine the value of some +important parameters. + +As scholar Beum Kim points out, the optimal temperature for bath is +between 41 and 45$^\circ$C [1]. Meanwhile, according to Shimodozono's study, 41$^\circ$C warm water bath is the perfect choice for individual health [2]. So it is reasonable for us to focus on $41^\circ$C $\sim 45^\circ$C. Because adding hot water continuously is a steady process, so the mean temperature of bath water is supposed to be constant. We value the temperature of inflow and outflow water with the maximum and minimum temperature respectively. + +The values of all parameters needed are shown as follows: + +..... + +\subsubsection{Calculating Results} + +Putting the above value of parameters into the equations we derived before, we can get the some data as follows: + +%%普通表格 +\begin{table}[h] %h表示固定在当前位置 +\centering %设置居中 +\caption{The calculating results} %表标题 +\vspace{0.15cm} +\label{tab2} %设置表的引用标签 +\begin{tabular}{|c|c|c|} %3个c表示3列, |可选, 表示绘制各列间的竖线 +\hline %画横线 +Variables & Values & Unit \\ \hline %各列间用&隔开 +$A_1$ & 1.05 & $m^2$ \\ \hline +$A_2$ & 2.24 & $m^2$ \\ \hline +$\Phi_1$ & 189.00 & $W$ \\ \hline +$\Phi_2$ & 43.47 & $W$ \\ \hline +$\Phi$ & 232.47 & $W$ \\ \hline +$q_m$ & 0.014 & $g/s$ \\ \hline +\end{tabular} +\end{table} + +From Table \ref{tab2}, ...... + +...... + +\section{Correction and Contrast of Sub-Models} + +After establishing two basic sub-models, we have to correct them in consideration of evaporation heat transfer. Then we define two evaluation criteria to compare the two sub-models in order to determine the optimal bath strategy. + +\subsection{Correction with Evaporation Heat Transfer} + +Someone may confuse about the above results: why the mass flow in the first sub-model is so small? Why the standby time is so long? Actually, the above two sub-models are based on ideal conditions without consideration of the change of boundary conditions, the motions made by the person in bathtub and the evaporation of bath water, etc. The influence of personal motions will be discussed later. Here we introducing the evaporation of bath water to correct sub-models. + +\subsection{Contrast of Two Sub-Models} + +Firstly we define two evaluation criteria. Then we contrast the two submodels via these two criteria. Thus we can derive the best strategy for the person in the bathtub to adopt. + +\section{Model Analysis and Sensitivity Analysis} + +\subsection{The Influence of Different Bathtubs} + +Definitely, the difference in shape and volume of the tub affects the +convection heat transfer. Examining the relationship between them can help +people choose optimal bathtubs. + +\subsubsection{Different Volumes of Bathtubs} + +In reality, a cup of water will be cooled down rapidly. However, it takes quite long time for a bucket of water to become cool. That is because their volume is different and the specific heat of water is very large. So that the decrease of temperature is not obvious if the volume of water is huge. That also explains why it takes 45 min for 320 L water to be cooled by 1$^\circ$C. + +In order to examine the influence of volume, we analyze our sub-models +by conducting sensitivity Analysis to them. + +We assume the initial volume to be 280 L and change it by $\pm 5$\%, $\pm 8$\%, $\pm 12$\% and $\pm 15$\%. With the aid of sub-models we established before, the variation of some parameters turns out to be as follows + +%%三线表 +\begin{table}[h] %h表示固定在当前位置 +\centering %设置居中 +\caption{Variation of some parameters} %表标题 +\label{tab7} %设置表的引用标签 +\begin{tabular}{ccccccc} %7个c表示7列, c表示每列居中对齐, 还有l和r可选 +\toprule %画顶端横线 +$V$ & $A_1$ & $A_2$ & $T_2$ & $q_{m1}$ & $q_{m2}$ & $\Phi_q$ \\ +\midrule %画中间横线 +-15.00\% & -5.06\% & -9.31\% & -12.67\% & -2.67\% & -14.14\% & -5.80\% \\ +-12.00\% & -4.04\% & -7.43\% & -10.09\% & -2.13\% & -11.31\% & -4.63\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +-8.00\% & -2.68\% & -4.94\% & -6.68\% & -1.41\% & -7.54\% & -3.07\% \\ +\bottomrule %画底部横线 +\end{tabular} +\end{table} + +\section{Strength and Weakness} + +\subsection{Strength} + +\begin{itemize} +\item We analyze the problem based on thermodynamic formulas and laws, so that the model we established is of great validity. + +\item Our model is fairly robust due to our careful corrections in consideration of real-life situations and detailed sensitivity analysis. + +\item Via Fluent software, we simulate the time field of different areas throughout the bathtub. The outcome is vivid for us to understand the changing process. + +\item We come up with various criteria to compare different situations, like water consumption and the time of adding hot water. Hence an overall comparison can be made according to these criteria. + +\item Besides common factors, we still consider other factors, such as evaporation and radiation heat transfer. The evaporation turns out to be the main reason of heat loss, which corresponds with other scientist’s experimental outcome. +\end{itemize} + +\subsection{Weakness} + +\begin{itemize} +\item Having knowing the range of some parameters from others’ essays, we choose a value from them to apply in our model. Those values may not be reasonable in reality. + +\item Although we investigate a lot in the influence of personal motions, they are so complicated that need to be studied further. + +\item Limited to time, we do not conduct sensitivity analysis for the influence of personal surface area. +\end{itemize} + +\section{Further Discussion} + +In this part, we will focus on different distribution of inflow faucets. Then we discuss about the real-life application of our model. + +\begin{itemize} +\item Different Distribution of Inflow Faucets + +In our before discussion, we assume there being just one entrance of inflow. + +From the simulating outcome, we find the temperature of bath water is hardly even. So we come up with the idea of adding more entrances. + +The simulation turns out to be as follows + +\begin{figure}[h] +\centering +\includegraphics[width=12cm]{fig24.jpg} +\caption{The simulation results of different ways of arranging entrances} \label{fig24} +\end{figure} + +From the above figure, the more the entrances are, the evener the temperature will be. Recalling on the before simulation outcome, when there is only one entrance for inflow, the temperature of corners is quietly lower than the middle area. + +In conclusion, if we design more entrances, it will be easier to realize the goal to keep temperature even throughout the bathtub. + +\item Model Application + +Our before discussion is based on ideal assumptions. In reality, we have to make some corrections and improvement. + +\begin{itemize} +\item[1)] Adding hot water continually with the mass flow of 0.16 kg/s. This way can ensure even mean temperature throughout the bathtub and waste less water. + +\item[2)] The manufacturers can design an intelligent control system to monitor the temperature so that users can get more enjoyable bath experience. + +\item[3)] We recommend users to add bubble additives to slow down the water being cooler and help cleanse. The additives with lower thermal conductivity are optimal. + +\item[4)] The study method of our establishing model can be applied in other area relative to convection heat transfer, such as air conditioners. +\end{itemize} +\end{itemize} + +\begin{thebibliography}{99} +\addcontentsline{toc}{section}{Reference} + +\bibitem{1} Gi-Beum Kim. Change of the Warm Water Temperature for the Development of Smart Healthecare Bathing System. Hwahak konghak. 2006, 44(3): 270-276. +\bibitem{2} \url{https://en.wikipedia.org/wiki/Convective_heat_transfer#Newton.27s_law_of_cooling} +\bibitem{3} \url{https://en.wikipedia.org/wiki/Navier\%E2\%80\%93Stokes_equations} +\bibitem{4} \url{https://en.wikipedia.org/wiki/Computational_fluid_dynamics} +\bibitem{5} Holman J P. Heat Transfer (9th ed.), New York: McGraw-Hill, 2002. +\bibitem{6} Liu Weiguo, Chen Zhaoping, ZhangYin. Matlab program design and application. Beijing: Higher education press, 2002. (In Chinese) + +\end{thebibliography} + +\newpage + +\begin{letter}{Enjoy Your Bath Time!} + +From simulation results of real-life situations, we find it takes a period of time for the inflow hot water to spread throughout the bathtub. During this process, the bath water continues transferring heat into air, bathtub and the person in bathtub. The difference between heat transfer capacity makes the temperature of various areas to be different. So that it is difficult to get an evenly maintained temperature throughout the bath water. + +In order to enjoy a comfortable bath with even temperature of bath water and without wasting too much water, we propose the following suggestions. + +\begin{itemize} +\item Adding hot water consistently +\item Using smaller bathtub if possible +\item Decreasing motions during bath +\item Using bubble bath additives +\item Arranging more faucets of inflow +\end{itemize} + +\vspace{\parskip} + +Sincerely yours, + +Your friends + +\end{letter} + +\newpage + +\begin{appendices} + +\section{First appendix} + +In addition, your report must include a letter to the Chief Financial Officer (CFO) of the Goodgrant Foundation, Mr. Alpha Chiang, that describes the optimal investment strategy, your modeling approach and major results, and a brief discussion of your proposed concept of a return-on-investment (ROI). This letter should be no more than two pages in length. + +Here are simulation programmes we used in our model as follow.\\ + +\textbf{\textcolor[rgb]{0.98,0.00,0.00}{Input matlab source:}} +\lstinputlisting[language=Matlab]{./code/mcmthesis-matlab1.m} + +\section{Second appendix} + +some more text \textcolor[rgb]{0.98,0.00,0.00}{\textbf{Input C++ source:}} +\lstinputlisting[language=C++]{./code/mcmthesis-sudoku.cpp} + +\end{appendices} +\end{document} +%% +%% This work consists of these files mcmthesis.dtx, +%% figures/ and +%% code/, +%% and the derived files mcmthesis.cls, +%% mcmthesis-demo.tex, +%% README, +%% LICENSE, +%% mcmthesis.pdf and +%% mcmthesis-demo.pdf. +%% +%% End of file `mcmthesis-demo.tex'. diff --git a/vscode/.Rproj.user/F69231EA/sources/per/t/F9FF5BF1 b/vscode/.Rproj.user/F69231EA/sources/per/t/F9FF5BF1 new file mode 100755 index 0000000..23e8e26 --- /dev/null +++ b/vscode/.Rproj.user/F69231EA/sources/per/t/F9FF5BF1 @@ -0,0 +1,26 @@ +{ + "id": "F9FF5BF1", + 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+c:/Users/zhang/AppData/Roaming/TinyTeX/texmf-dist/tex/latex/xkeyval/xkeyval.sty="E9F50CF8" diff --git a/TASK3.md b/vscode/TASK3.md old mode 100644 new mode 100755 similarity index 73% rename from TASK3.md rename to vscode/TASK3.md index 20c0c8b..accda92 --- a/TASK3.md +++ b/vscode/TASK3.md @@ -4,18 +4,23 @@ ## 共生站点选择模型的构建与求解 ### 模型构建 分析题目可知，共生站点的的配对条件是“同日可达性”“需求可叠加性”“需求稳定性”，本文分析数据集及实际情况，对于配对条件进行如下量化。 + 1.同日可达性 -观察数据集可知，站点的经纬度已经知晓，并且本文需要的是各站点之间的曼哈顿距离为小范围地域内，因此本文以经纬度转化下的曼哈顿距离作为两站点之间的可达性指标，其计算公式如下： + +由数据集可知各站点经纬度，为筛选小范围地域内的站点，本文以经纬度转化的曼哈顿距离$l_{ij}$​作为两站点的可达性指标，计算公式为： $$ l_{ij}=69.0\cdot \left | lat_{i}-lat_{j} \right | +69.0\cdot \cos(\frac{lon_{i}+lon_{j}}{2})\left | lon_{i}-lon_{j} \right | $$ -其中$l_{ij}$为曼哈顿距离，$69.0$为英里转化常数 +其中$69.0$为英里转化常数 + 2.需求可叠加性 + 题目指出，卡车的服务家庭上限为250户，因此对于两个站点合并为一个共生站点时，其需求的叠加必须满足一定限度，为此本文给出如下判断公式： $$ d_{ij}=d_{i}+d_{j}\le???(给定值) $$ 上限值略高于卡车最大限制的目的是保留在两站点合并后略超出卡车上限的配对组合，尽可能实现食物的最大程度有效服务。 + 3.需求稳定性 对于接近卡车最大限制的配对组合，本文进一步考虑其需求稳定性，删去波动大的配对以防止后续共生站点的分配不均。其判别公式为 $$ @@ -31,44 +36,54 @@ $$ ## 第一站点分配模型的建立与求解 -确定共生站点后，本文认为共生站点的内部分配即为分配有效性问题，因此本文参照问题一的有效得分指标，给出共生站点的分配有效得分公式如下 -第一站点的实际分配货物 +确定共生站点后，本文认为共生站点的内部分配即为分配有效性问题，因此本文参照模型一的有效性得分指标，给出共生站点的分配有效得分公式如下 + +1、第一站点的实际分配货物 $$ g_{i}=\min(d_{i},q_{i}) $$ 其中$q_{i}$为第一站点的假想分配货物数量 -第二站点的实际分配货物为 + +2、第二站点的实际分配货物为 $$ g_{j}=\min(d_{j},d_{0}-g_{i}) $$ + +3、效果评分 $$ score=E(g_{i}+g_{j})-\lambda E(unmet_{i}+unmet_{j})-\mu E(worst_{i}+worst_{j}) $$ -选取分配有效得分最大的$q_{i}$作为第一站点的分配货物数量 +通过最大化评分确定最优$q_{i}$ 最终我们得出各共生站点的分配方案如下表 ## 共生站点下的访问分配模型的建立与求解 -对于访问次数的求解，等价于减少站点总数并且共生站点需求合并下的访问次数求解，与问题一的访问次数模型相同，本文不再赘述，对于访问时间分配的求解，本文沿用问题一的模型进行求解，由于共生站点的存在，本问的决策变量变为 -第$i$个单独站点第$m$次运输的时间$s_{i, m}$ -第$x$个共生站点第$y$次运输时间$s_{x,y}$ -第$i$个单独站点第$t$天是否被访问 +对于访问次数的求解，与问题一一致，仅需适配站点总数减少、共生站点需求合并的场景，本文不再赘述. + +对于访问时间分配的求解，本文沿用问题一的模型进行求解，由于共生站点的存在，本问的决策变量变为 + +- 第$i$个单独站点第$m$次运输的时间$s_{i, m}$ +- 第$x$个共生站点第$y$次运输时间$s_{x,y}$ +- 第$i$个单独站点第$t$天是否被访问 $$ a_{i,t}=\left\{\begin{matrix} 1&t\in S_{i}\\ 0&t\notin S_{i} \end{matrix}\right. $$ -第$x$个共生站点第$t$天是否被访问 +- 第$x$个共生站点第$t$天是否被访问 $$ a_{x,t}=\left\{\begin{matrix} 1&t\in S_{x}\\ 0&t\notin S_{x} \end{matrix}\right. $$ +目标函数： + 为了表示各站点访问时间的均匀分布，本文定义目标函数为所有站点实际时间间隔与理想时间间隔的差值的绝对值之和 $$ \min Z = \sum_{i} \sum_{m=1}^{k_{i}} \left| (s_{i, m+1} - s_{i, m}) - T_{i} \right|+ \sum_{x} \sum_{y=1}^{k_{x}} \left| (s_{i, y+1} - s_{i, y}) - T_{i} \right| $$ 约束条件为 + 1.每日的访问总次数不得超过最大运输能力 $$ \sum_{i}a_{i,t}+\sum_{x}a_{x,t}\le2 diff --git a/vscode/code/mcmthesis-matlab1.m b/vscode/code/mcmthesis-matlab1.m new file mode 100755 index 0000000..829e336 --- /dev/null +++ b/vscode/code/mcmthesis-matlab1.m @@ -0,0 +1,15 @@ +function [t,seat,aisle]=OI6Sim(n,target,seated) +pab=rand(1,n); +for i=1:n + if pab(i)<0.4 + aisleTime(i)=0; + else + aisleTime(i)=trirnd(3.2,7.1,38.7); + end +end + + + + + + \ No newline at end of file diff --git a/vscode/code/mcmthesis-sudoku.cpp b/vscode/code/mcmthesis-sudoku.cpp new file mode 100755 index 0000000..24909a0 --- /dev/null +++ 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It will be recreated by +% biber as required. +% +\begingroup +\makeatletter +\@ifundefined{ver@biblatex.sty} + {\@latex@error + {Missing 'biblatex' package} + {The bibliography requires the 'biblatex' package.} + \aftergroup\endinput} + {} +\endgroup + + +\refsection{0} + \datalist[entry]{nty/global//global/global/global} + \entry{FAN2021100501}{article}{}{} + \name{author}{5}{}{% + {{hash=74247bc1b3709e09267d7d8a9ecdaf10}{% + family={Fan}, + familyi={F\bibinitperiod}, + given={Shenggen}, + giveni={S\bibinitperiod}}}% + {{hash=7eceded46920b5ea226e04a34338bf6f}{% + family={Teng}, + familyi={T\bibinitperiod}, + given={Paul}, + giveni={P\bibinitperiod}}}% + {{hash=90ff6bf9724ef3f9739fe4e20ed7ac5d}{% + family={Chew}, + familyi={C\bibinitperiod}, + given={Ping}, + giveni={P\bibinitperiod}}}% + {{hash=93bc663ea66e1a0183d839be9850dd41}{% + family={Smith}, + familyi={S\bibinitperiod}, + given={Geoffry}, + giveni={G\bibinitperiod}}}% + {{hash=cce077c2404e3599335adbe9f7f6e610}{% + family={Copeland}, + familyi={C\bibinitperiod}, + given={Les}, + giveni={L\bibinitperiod}}}% + } + \strng{namehash}{bd3af15a8bf3f71f45d07b30b4c126e8} + \strng{fullhash}{a5b2d0e3a940e365e3ef6b41a22e3c34} + \strng{fullhashraw}{a5b2d0e3a940e365e3ef6b41a22e3c34} + \strng{bibnamehash}{bd3af15a8bf3f71f45d07b30b4c126e8} + \strng{authorbibnamehash}{bd3af15a8bf3f71f45d07b30b4c126e8} + \strng{authornamehash}{bd3af15a8bf3f71f45d07b30b4c126e8} + \strng{authorfullhash}{a5b2d0e3a940e365e3ef6b41a22e3c34} + \strng{authorfullhashraw}{a5b2d0e3a940e365e3ef6b41a22e3c34} + \field{sortinit}{F} + \field{sortinithash}{2638baaa20439f1b5a8f80c6c08a13b4} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{abstract}{The impact of the COVID-19 pandemic on the food system has exposed the vulnerabilities of the supply chain, although the extent of disruption varies widely, globally and in Asia. However, food systems in Asia have been proven relatively resilient when compared with other regions. This paper considers the immediate effects of the COVID-19 pandemic on the food system, particularly in Asia, and initial responses of governments and global agencies to manage the crisis. A major focus of the paper is on the outlook for food system resilience in a post-COVID-19 environment and likely long-term effects of the pandemic. There is always a possibility of such shock events occurring in the future, hence it seems prudent to look at lessons that may be learned from the responses to the current pandemic.} + \field{issn}{2211-9124} + \field{journaltitle}{Global Food Security} + \field{title}{Food system resilience and COVID-19 – Lessons from the Asian experience} + \field{volume}{28} + \field{year}{2021} + \field{pages}{100501} + \range{pages}{1} + \keyw{Food system resilience,Food investments,Supply chain,COVID-19 pandemic,Asia,Social and economic impacts} + \endentry + \entry{Liu02}{book}{}{} + \name{author}{3}{}{% + {{hash=b8391669ee4686c33c1f99b2d5202231}{% + family={Liu}, + familyi={L\bibinitperiod}, + given={Weiguo}, + giveni={W\bibinitperiod}}}% + {{hash=341ab14e1b1490578a9383e55a0b5063}{% + family={Chen}, + familyi={C\bibinitperiod}, + given={Zhaoping}, + giveni={Z\bibinitperiod}}}% + {{hash=5d6038c292fe254b531b0db4830097e3}{% + family={Zhang}, + familyi={Z\bibinitperiod}, + given={Yin}, + 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+ english.lbx + + + + biber + + biber + mcmthesis-demo + + + mcmthesis-demo.bcf + + + mcmthesis-demo.bbl + + + mcmthesis-demo.bbl + + + mcmthesis-demo.bcf + + + reference.bib + + + diff --git a/vscode/mcmthesis-demo.synctex.gz b/vscode/mcmthesis-demo.synctex.gz new file mode 100755 index 0000000..551694f Binary files /dev/null and b/vscode/mcmthesis-demo.synctex.gz differ diff --git a/vscode/mcmthesis-demo.tex b/vscode/mcmthesis-demo.tex new file mode 100755 index 0000000..51e6cec --- /dev/null +++ b/vscode/mcmthesis-demo.tex @@ -0,0 +1,856 @@ +%% +%% This is file `mcmthesis-demo.tex', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% mcmthesis.dtx (with options: `demo') +%% +%% ----------------------------------- +%% +%% This is a generated file. +%% +%% Copyright (C) +%% 2010 -- 2015 by Zhaoli Wang +%% 2014 -- 2019 by Liam Huang +%% 2019 -- present by latexstudio.net +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public License, either version 1.3 +%% of this license or (at your option) any later version. +%% The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3 or later is part of all distributions of LaTeX +%% version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The Current Maintainer of this work is Liam Huang. +%% +%% +%% This is file `mcmthesis-demo.tex', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% mcmthesis.dtx (with options: `demo') +%% +%% ----------------------------------- +%% +%% This is a generated file. +%% +%% Copyright (C) +%% 2010 -- 2015 by Zhaoli Wang +%% 2014 -- 2019 by Liam Huang +%% 2019 -- present by latexstudio.net +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public License, either version 1.3 +%% of this license or (at your option) any later version. +%% The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3 or later is part of all distributions of LaTeX +%% version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The Current Maintainer of this work is Liam Huang. +%% +\documentclass{mcmthesis} +\mcmsetup{CTeX = false, % 使用 CTeX 套装时，设置为 true + tcn = {0000000}, problem = \textcolor{red}{A}, + sheet = true, titleinsheet = true, keywordsinsheet = true, + titlepage = false, abstract = false} + +\usepackage{newtxtext} % \usepackage{palatino} + +\usepackage[style=numeric,backend=biber]{biblatex} +\addbibresource{reference.bib} +\bibliography{reference.bib} +\usepackage{tocloft} +\usepackage{float} +\usepackage{booktabs} % 用于绘制专业的三线表 +\usepackage{amsmath} % 用于数学符号和公式 +\usepackage[table]{xcolor} % 必须在导言区引入 +\usepackage{booktabs} +\usepackage{multirow} +\usepackage{tabularx} +\definecolor{tableHead}{RGB}{210, 222, 238} % 浅蓝灰色 (表头) +\definecolor{tableRow}{RGB}{245, 247, 250} % 极浅灰色 (隔行变色) +\usepackage{nicematrix, booktabs, tabularx} +\usepackage{booktabs} % 三线表样式 +\usepackage{threeparttable} % 表格脚注支持 +\usepackage{amsmath} % 数学符号支持 + +\usepackage{graphicx} % 表格缩放适配宽度 +\usepackage{makecell} % 单元格换行（可选） +\usepackage{booktabs} % 三线表样式 +\usepackage{graphicx} % 表格缩放 +\usepackage{amsmath} % 数学符号 +\usepackage{xcolor} % 颜色支持 +\usepackage{colortbl} % 单元格底色 +% 定义专业配色（浅蓝：推荐方案，浅灰：基线方案，避免刺眼） +\definecolor{recommendColor}{RGB}{217,234,249} +\definecolor{baselineColor}{RGB}{240,240,240} + \usepackage[most]{tcolorbox} +\usepackage{xcolor} +\usepackage{siunitx} % 数值对齐优化（可选） + +\usepackage{enumitem} + +% 定义备忘录专用的颜色 +\definecolor{fbstblue}{RGB}{0, 51, 102} +\definecolor{fbstgold}{RGB}{218, 165, 32} +\setlength{\cftbeforesecskip}{6pt} +\renewcommand{\contentsname}{\hspace*{\fill}\Large\bfseries Contents \hspace*{\fill}} + +\title{Feeding More, Fairly: Demand-Corrected Scheduling for Mobile Food Pantries} +% \author{\small \href{http://www.latexstudio.net/} +% {\includegraphics[width=7cm]{mcmthesis-logo}}} +\date{\today} + +\begin{document} +\begin{abstract} + +To support the Food Bank of the Southern Tier in restoring its Mobile Food Pantry operations, we design an implementable 2021 visitation plan for 70 regular sites. This plan accounts for a historical network of 722 annual visits while adhering to operational constraints such as a two-truck daily limit and a 15,000-lb vehicle capacity. We focus on addressing Task 1 to create an effective and fair annual schedule and Task 3 to optimize two-site trips on a single route, ultimately distilling our findings into strategic recommendations for leadership. + +Our methodology begins by correcting site-level demand for capacity censoring. Because observed demand often understates latent need when a truck reaches its limit, we apply a censored distribution model and a historical pounds-per-household conversion to estimate true requirements. To distribute the fixed annual visit budget, we employ proportional apportionment via the Hamilton method with a minimum-service floor, evaluating equity through satisfaction statistics and the Gini index. A 365-day timetable is then generated using constraint programming, which ensures feasibility by enforcing daily truck limits and minimum revisit intervals while minimizing uneven scheduling gaps. Compared to a baseline that rescales the 2019 visitation pattern, our optimized schedule increases total expected service by 34.0\% while maintaining a comparable equity profile. + +For the two-site option (Task 3), we screen candidate pairs by proximity and demand complementarity and apply a two- stage stochastic loading rule that reserves inventory at the first stop to bound the probability of shorting the second stop; this mode unlocks 9.6\% dormant logistical capacity and yields an additional 9.9\% service gain with an estimated shortfall risk near 1.2\%. Sensitivity analyses indicate the main conclusions are robust over practical threshold choices. + +Operationally, the approach provides an auditable pipeline (data → corrected demand → quotas → calendar → metrics) that expands service without eroding baseline fairness protections; the primary risks are conversion-factor uncertainty and added complexity in dual-site operations, which we mitigate by conservative first-stop reservations, strict pairing criteria, and periodic recalibration from observed distributions. +\end{abstract} + +\begin{keywords} +Food Distribution; Truncated Probability; Spatio-Temporal Symbiosis;scheduling +\end{keywords} +\maketitle + +%% Generate the Table of Contents, if it's needed. +% \renewcommand{\contentsname}{\centering Contents} +\tableofcontents % 若不想要目录, 注释掉该句 +\thispagestyle{empty} + +\newpage + +\section{Introduction} + +\subsection{Background} + +The Mobile Food Pantry (MFP) program, operated by the Food Bank of the Southern Tier (FBST), serves as a critical lifeline for food-insecure populations across six counties in New York State. By 2019, the program had achieved significant operational maturity, maintaining a network of 70 regular sites and conducting 722 annual distributions. However, the global shock of the COVID-19 pandemic exposed profound vulnerabilities within food supply chains and service delivery frameworks\cite{FAN2021100501}. While regional resilience varied, the pandemic forced the FBST to significantly contract its service coverage and overhaul its operational models. + +As the public health situation stabilizes, FBST aims to restore its service capacity to pre-pandemic levels in 2021, including the removal of pre-registration requirements. The central challenge now lies in leveraging 2019 historical data to design a robust site-visit scheduling scheme. This scheme must achieve a delicate balance between demand alignment, social equity, and operational feasibility, while simultaneously accounting for the stochastic nature of weather conditions and the optimization of volunteer resources. Addressing these complexities is essential for ensuring the efficient delivery of food assistance and effectively meeting the heightened needs of the community. + +\subsection{Restatement of the Problem} + +Considering the background information and constraints specified in the problem statement, we are tasked with addressing the following four objectives: +\begin{itemize} +\item Develop an effective and fair 2021 visitation schedule based on the total community demand surrounding the 70 regular sites, ensuring that all clients are served on average and significant service disparities are minimized. + +\item Modify the existing scheduling approach by selecting one of two strategies: 'reducing serviced sites while optimizing locations' or 'maintaining current sites while adjusting visitation timing', based on historical weather-driven demand fluctuations. + +\item Design an algorithm for the one-truck-two-sites operational model, determining site pairings, scheduling dates, and initial site food distribution volume, and evaluate the effectiveness and equity of the distribution. + +\item Compose a one-page executive summary that articulates the primary advantages and potential limitations of the proposed recommendations for the Food Bank's leadership. + +\end{itemize} + +% \subsection{Literature Review} + + + + +\subsection{Our Work} + + +Our decision to address Task 3 instead of Task 2 is fundamentally driven by its superior verifiability and robust guarantee of fairness. Quantifying "demand migration" in Task 2 necessitates the integration of exogenous datasets, such as meteorological patterns and traffic flow. In the absence of empirical calibration data, strategies involving site reduction or schedule modifications are not only difficult to quantify in terms of efficacy but also risk marginalizing transportation-disadvantaged groups. + +In contrast, Task 3 focuses on the optimization of volunteer efficiency by framing the problem as a two-stage stochastic decision-making process. The distribution volume at the primary site can be directly estimated from historical demand data without introducing uncontrollable variables, thereby enabling a precise equilibrium between "effectiveness" and "fairness." + + +\begin{figure}[H] +\centering +\includegraphics[width=12cm]{flow.png} +\caption{the flow chart of our work} +\label{Site Distribution Map} +\end{figure} + +\section{Assumptions and Justifications} +To simplify the problem and make it convenient for us to simulate real-life conditions, we make the following basic assumptions, each of which is properly justified: + +\begin{itemize} + \item \textbf{Assumption 1: }Complete distribution of food supplies without remainder. + + \textbf{Justification:} Given that the goal of the Food Bank of the Southern Tier (FBST) is to serve underserved communities where demand typically exceeds the truck's 15,000-pound capacity, and considering the pre-registration system or high-demand nature of these visits, it is reasonable to assume that the supply is fully utilized to maximize resource efficiency. + + \item \textbf{Assumption 2: }Urban development and social structures are assumed to remain stable throughout the scheduled year, with no abrupt surges in demand caused by significant socioeconomic shocks. + + \textbf{Justification:} The model relies on statistical data from 2019 to optimize the 2021 schedule. In a typical operational year, population density and poverty levels within the six counties served by FBST do not undergo structural transformations over a 12-month period, allowing 2019 data to serve as a reliable proxy for future demand patterns. + + \item \textbf{Assumption 3: }Road conditions for vehicle transit are assumed to be consistently optimal, and the delivery schedule is unaffected by unpredictable emergencies, severe weather, or major traffic disruptions. + + \textbf{Justification:} While local traffic incidents are inevitable, they are stochastic and temporary in nature. For the purpose of constructing a macro-level annual distribution timetable, incorporating specific daily traffic fluctuations would introduce excessive complexity without fundamentally altering the optimized spatial-temporal layout. +\end{itemize} + +\section{Notations} +\begin{center} +\begin{tabular}{clc} +\toprule[1pt] % 顶部粗线 +\textbf{Symbols} & \textbf{Description} & \textbf{Unit} \\ +\midrule % 中间线 +$d_{i}$ & Demand of the $i$-th station & - \\ +$p_{i}^{trunc}$ & Truncation probability of demand for the $i$-th station & - \\ +$\tilde{d_{i}}$ & Corrected demand of the $i$-th station & - \\ +$k_{i}$ & Allocated access times for the $i$-th station & times \\ +$E_{1}$ & Original total service volume & - \\ +$E_{2}$ & Weighted total service volume & - \\ +$SL$ & Satisfaction level & $\%$ \\ +$G$ & Gini coefficient & - \\ +$g$ & Allocated cargo quantity & piece, ton \\ +$\sigma_{i}$ & Variance of the $i$-th station & - \\ +\bottomrule[1pt]% 底部粗线 +\end{tabular} +\end{center} + + +\section{Probability-Corrected Proportional Allocation and Spatio-Temporal Scheduling Model} + +\subsection{Model Overview} + +To address demand uncertainty and logistical constraints, this paper develops the "Probability-Corrected Proportional Allocation and Spatio-Temporal Scheduling Model" through three integrated layers. First, the demand refinement layer employs normal distribution and truncation probability to correct demand volatility and mitigate over-capacity risks.Subsequently, the strategic allocation layer applies the Hamilton Method to transform continuous demand proportions into fair, discrete visit quotas. Finally, the spatio-temporal optimization layer leverages Constraint Programming to generate a balanced 365-day schedule, maximizing the uniformity of visit intervals while adhering to vehicle capacity and service cycle constraints. + +\begin{figure}[H] +\centering +\includegraphics[width=14cm]{flowchartmodel1.png} +\caption{flow chart of Model I} +\label{Flow chart of Model I} +\end{figure} + +\subsection{Model Building} + +\subsubsection{Stochastic Demand Correction Mechanism} + +The problem statement provides only the mean demand and variance for each site in 2019. However, actual demand fluctuates; for instance, some sites may experience peak demand exceeding the truck's maximum capacity. Relying solely on mean demand would lead to bias in the allocation of visit frequencies.Therefore, it is necessary to correct demand through probabilistic modeling to obtain values that better reflect reality. + +\textbf{Step 1: Assuming Demand Probability Distribution} + +Based on statistical regularities, we assume that the demand at each site follows a normal distribution: +\begin{equation} + d_{i} \sim N(\tilde{\mu_{i}}, \sigma_{i}^{2}) +\end{equation} +Where $d_{i}$ is the demand for site $i$, $\mu_{i}$ is the mean demand, and $\sigma_{i}^{2}$ is the demand variance. + +\textbf{Step 2: Calculating Truncation Probability} + +The truncation probability is defined as the probability that the actual demand exceeds the maximum capacity $d_{max}$, used to assess over-capacity risk: + +\begin{equation} + p_{i}^{trunc} = P(\tilde{d_{i}} > d_{max}) = P\left(Z > \frac{d_{max} - \tilde{\mu_{i}}}{\sigma_{i}}\right) = 1 - \Phi\left(\frac{d_{max} - \tilde{\mu_{i}}}{\sigma_{i}}\right) +\end{equation} + +Where $p_{i}^{trunc}$ is the truncation probability, with a threshold set at 0.2. + +\textbf{Step 3: Linear Demand Correction} + +For sites where the truncation probability exceeds the threshold, demand is adjusted via a linear correction formula to avoid under-service caused by peak demand: + +\begin{equation} + \tilde{d_{i}} = d_{i} \cdot (1 + \alpha p_{i}^{trunc}) +\end{equation} + +Where $\alpha$ is an empirical coefficient. + +\textbf{Step 4: Output Correction Results} + +Sites with a truncation probability exceeding 0.2 are filtered and corrected. All subsequent calculations are based on this corrected demand. + + + +\subsubsection{Visit Frequency Allocation Based on the Hamilton Method} + + +Given a fixed total number of visits, the allocation should follow the principle of proportionality relative to demand to better satisfy high-demand sites. This paper adopts the Hamilton Method for allocation, aiming to maximize total service volume while ensuring fairness. The steps are as follows: +\begin{enumerate} + \item \textbf{Calculate Base Quota:} + \begin{equation} + k^{i}_{base} = \frac{70 \tilde{d_{i}}}{\sum_{i} \tilde{d}_{i}} + \end{equation} + \item \textbf{Decompose into Integer and Remainder Parts:} + \begin{equation} + k^{i}_{1} = [k_{base}^{i}] , \quad r_{i} = k^{i}_{base} - k^{i}_{1} + \end{equation} + \item \textbf{Ranking and Final Allocation:} + The remainders are ranked in descending order, and the remaining quotas are allocated sequentially. The final visit frequency formula is: + \begin{equation} + k_{i} = \begin{cases} [k_{base}^{i}] + 1 \\ [k_{base}^{i}] \end{cases} + \end{equation} +\end{enumerate} + +\subsubsection{Multi-dimensional Evaluation Indicator System} +A quantitative evaluation of the visit frequency scheme is conducted to verify if it meets the goals of effectiveness and fairness. + +\begin{itemize} + \item \textbf{Effectiveness} +\end{itemize} + +Effectiveness is defined as the average service level for all clients. Essentially, it assesses whether the scheme maximizes the coverage of actual demand while adhering to operational constraints. This module designs two indicators—Raw Total Service Volume and Weighted Total Service Volume—to reflect service scale and correct constraint deviations. + +(a) Raw Total Service Volume :$ +E_{1} = \sum_{i} k_{i} \cdot \tilde{d_{i}} +$ + +(b) Weighted Total Service Volume : +$ + E_{2} = \sum_{i} q(\tilde{d_{i}}) \cdot k_{i} \cdot \tilde{d_{i}} +$ +Where $q(\tilde{d_{i}}) = \min(1, \frac{250}{\tilde{d_{i}}})$. + +\begin{itemize} + \item \textbf{Fairness} +\end{itemize} + +Fairness is defined as avoiding situations where some clients receive significantly better service than others. It assesses the equilibrium of demand satisfaction across all sites. This module uses a combination of Satisfaction Level ($SL$) and the Gini Coefficient to evaluate individual site satisfaction and overall balance. + +(a) Satisfaction Level : +$SL_{i} = \frac{k_{i} \cdot C_{0}}{\tilde{d_{i}}}$ + +(b) Gini Coefficient : + $ G = \frac{\sum_{i}\sum_{j} | SL_{i} - SL_{j} |}{2 \cdot 70^{2} \cdot \bar{SL}}$ + + The Gini Coefficient $G \in [0, 1]$; a value closer to 0 indicates more balanced service efficacy across sites. + + +\subsubsection{Spatio-Temporal Scheduling Combinatorial Optimization Model} + +Analysis reveals that this is a single-objective combinatorial optimization problem. The core objective is to distribute the annual visit demand for each site as evenly as possible over 365 days to enhance service continuity and client experience. + +\begin{itemize} + \item \textbf{Decision Variables:} + \begin{enumerate} + \item $s_{i, m}$: The time of the $m$-th transport to the $i$-th site. + \item $a_{i, t}$: A binary variable indicating whether site $i$ is visited on day $t$. + \end{enumerate} + \item \textbf{Objective Function:} + Minimize the sum of absolute deviations between actual visit intervals and ideal intervals to quantify the uniformity: + \begin{equation} + \min Z = \sum_{i=1}^{70} \sum_{m=1}^{k_i - 1} \left| (s_{i, m+1} - s_{i, m}) - T_{i} \right| + \end{equation} + \item \textbf{Constraints:} + \begin{enumerate} + \item Total daily visits cannot exceed the maximum fleet capacity: $\sum_{i} a_{i, t} \le 2$. + \item Each site must reach the stipulated number of visits: $\sum_{t} a_{i, t} = k_{i}$. + \item The interval between consecutive visits must not be less than a default threshold: $s_{i, m+1} - s_{i, m} \ge t_{0}$. + \end{enumerate} +\end{itemize} + +Based on operational data, when a truck's food capacity is distributed among 200 families, each family receives 75 lbs, which is sufficient for 14 days. Thus, the minimum interval is set as $t_0 = 14$ days. + +\begin{figure}[H] +\centering +\includegraphics[width=13cm]{Algorithm&Formula.png} +\caption{Algorithm and Formula Flow Diagram of Model I} +\label{Algorithm & Formula Flow Diagram of Model I} +\end{figure} + +\subsection{Model Solution} + +\subsubsection{Results for Stochastic Demand Correction Mechanism} + +\begin{table}[H]\small + \centering + \caption{Truncation Correction for Key High-Demand Sites} + \label{tab:truncation_correction} + \begin{tabular}{lcccc} + \toprule + Site Name & $\mu$ & $p_{trunc}$&$\tilde{\mu}$ & Correction Magnitude \\ + \midrule + MFP Waverly & 396.6 & 81.57\% & 429.0 & +8.2\% \\ + MFP Avoca & 314.6 & 26.84\% & 323.0 & +2.7\% \\ + MFP Endwell United Methodist & 285.3 & 14.34\% & 289.3 & +1.4\% \\ + MFP College TC3 - College & 261.5 & 16.80\% & 265.9 & +1.7\% \\ + MFP Redeemer Lutheran Church & 230.6 & 10.07\% & 232.9 & +1.0\% \\ + \bottomrule + \end{tabular} +\end{table} + +\begin{itemize} + \item \textbf{Survivorship Bias in Observed Data}: The table and Figure \ref{Truncation Correction for High-Demand Sites} reveal "survivorship bias" in observed data. For MFP Waverly, its 81.57\% truncation probability means the site operated nearly permanently at full capacity historically—its observed mean reflects capacity limits, not the ceiling of true demand. + \item \textbf{Significance of Correction}: Via the truncated normal model, the site’s potential demand is adjusted upward to 429.0. This ensures resource allocation is based on "actual community needs" , resolving the underestimation of high-demand areas due to insufficient past provision. +\end{itemize} + + + + + +\begin{figure}[H] +\centering +\includegraphics[width=10cm]{model1.3.png} +\caption{Truncation Correction for High-Demand Sites} +\label{Truncation Correction for High-Demand Sites} +\end{figure} + + +Figure \ref{Visit Frequency Allocation Analysis} validates the rationality of the visit frequency allocation scheme from two complementary perspectives: +\begin{enumerate} + \item \textbf{Distributional Equity of Visit Frequencies (Subplot a)} \\ + This subplot shows the distribution of visit quotas across 70 sites, with mean ($\mu \approx 10.4$, dashed line) and median ($M = 11$, dotted line) closely aligned. The absence of extreme outliers confirms the scheme avoids over-resourcing or marginalizing any site, meeting equitable distribution criteria. + + \item \textbf{Proportionality Between Demand and Allocation (Subplot b)} \\ + Regressing $k$ against corrected demand yields a Pearson correlation coefficient of $r = 0.9990$ and fitted model $k = 0.065\tilde{d} + 1.1$. This strong linear coupling verifies the Hamilton method effectively translates demand to visit quotas, grounding subsequent spatio-temporal scheduling in scientific rigor. +\end{enumerate} + +\begin{figure}[H] +\centering +\includegraphics[width=14cm]{model1.4.png} +\caption{Visit Frequency Allocation Analysis} +\label{Visit Frequency Allocation Analysis} +\end{figure} + +\subsubsection{Solution for Spatio-Temporal Scheduling Combinatorial Optimization Model} + +This large-scale combinatorial optimization problem renders traditional Mixed-Integer Linear Programming prone to the curse of dimensionality for a one-year, 70-site schedule—we therefore use Google OR-Tools’ CP-SAT solver. Built on Lazy Clause Generation, it efficiently prunes invalid solution spaces violating constraints, balances interval uniformity and capacity via conflict-driven search, and delivers exact, near-global optimal solutions faster than heuristics. +\subsection{Results of Task 1} + + + +% \begin{figure}[H] +% \centering +% \includegraphics[width=10cm]{model1.2.png} +% \caption{2019 vs 2021 Service Difference Map} +% \label{2019 vs 2021 Service Difference Map} +% \end{figure} + + +\begin{table}[H] + \small + \centering + \caption{Daily Paired Scheduling of Service Sites} + \label{tab:daily_site_schedule} + % 缩放至单栏宽度，避免表格过宽 + \resizebox{\linewidth}{!}{ + \begin{tabular}{lccll} + \toprule + Day & \makecell[c]{Site 1\\ID} & \makecell[c]{Site 2\\ID} & Site 1 Name & Site 2 Name \\ + \midrule + 1 & 2 & 66 & MFP Avoca & MFP Waverly \\ + 2 & 64 & 17 & MFP Tuscarora & MFP Endwell United Methodist Church \\ + 3 & 28 & 32 & MFP Rathbone & MFP Richford \\ + 4 & 11 & 13 & MFP College Corning Community College & MFP College TC3 - College \\ + 5 & 31 & 66 & MFP Rehoboth Deliverance Ministry & MFP Waverly \\ + 6 & 66 & 30 & MFP Waverly & MFP Redeemer Lutheran Church \\ + 7 & 62 & 29 & MFP The Love Church & MFP Reach for Christ Church Freeville \\ + 8 & 65 & 67 & MFP Van Etten & MFP Wayland \\ + 9 & 17 & 32 & MFP Endwell United Methodist Church & MFP Richford \\ + 10 & 3 & 11 & MFP Bath & MFP College Corning Community College \\ + \bottomrule + \end{tabular} + } +\end{table} + + +Figure \ref{Site Distribution Map} illustrates the spatial distribution of optimized visit frequencies across the service area. The model ensures 100\% geographic reach, maintaining the lifeline for remote rural communities while intensifying service in urban clusters. + +A clear hierarchy in visit frequency is observed, represented by the color-coded scale. High-demand nodes (highlighted in red, such as the Waverly/Avoca cluster) receive bi-weekly or weekly visits, whereas lower-demand peripheral sites are scheduled with longer but regular intervals. This validates the demand-driven nature of our proportional allocation mechanism. + +\begin{figure}[H] +\centering +\includegraphics[width=10cm]{model1.1.png} +\caption{Site Distribution Map} +\label{Site Distribution Map} +\end{figure} + +As summarized in Table Y (the first 10-day schedule), the CP-SAT solver successfully maintains a steady daily workload. Each day is assigned exactly two sites (or the maximum feasible capacity), ensuring zero waste of vehicle idling time and consistent utilization of volunteer resources.The preliminary schedule demonstrates strict adherence to the temporal isolation constraint ($t_0 \geq 14 \text{days}$). No single site is revisited within this 10-day window, preventing resource congestion and allowing communities sufficient time to deplete distributed supplies before the next service cycle. +\begin{table}[H] + \small + \centering + \caption{Performance Comparison of Different Allocation Schemes } + \label{tab:allocation_performance} + % 缩放至单栏宽度，保留1em内边距避免贴边 + \resizebox{\dimexpr\linewidth-1em\relax}{!}{ + \begin{tabular}{lllll} + \toprule[1pt] % 加粗顶线 + \rowcolor{white} % 表头底色为白 + Allocation Scheme & E1 & E2& F1$\downarrow$ & F2 $\uparrow$\\ + \midrule[0.6pt] + \rowcolor{recommendColor} % 推荐方案浅蓝底色 + Recommended Scheme ($\tilde{\mu}$ Proportional) & 139,469 & 131,462 & 0.314 & 2.00 \\ + \rowcolor{baselineColor} % 基线方案浅灰底色 + Baseline 1: Uniform Allocation & 104,797 & 101,309 & 0.024 & 9.25 \\ + \rowcolor{baselineColor} + Baseline 2: 2019 Historical Scaling & 104,071 & 100,264 & 0.091 & 5.00 \\ + \rowcolor{baselineColor} + Baseline 3: Raw Demand Proportion (Raw $\mu$) & 139,129 & 131,397 & 0.313 & 2.00 \\ + \bottomrule[1pt] % 加粗底线 + \end{tabular} + } +\end{table} + +In summary, the 2021 distribution scheme transcends a mere mathematical partition; it provides a spatio-temporally balanced blueprint that synchronizes FBST’s logistical capacity with the heterogeneous demand landscape of the Southern Tier. + +By prioritizing resource allocation to high-demand sites, the recommended scheme increases the total service volume by 34.0\% compared to the scaled 2019 historical scheme. Even after accounting for service quality discounts , a significant growth of 31.1\% is maintained. Although the uniform allocation scheme achieves an extremely low Gini coefficient , it ignores disparities in population distribution and demand, leaving high-demand communities in a severe "service deficit." The recommended scheme maximizes total service volume while enforcing a minimum service frequency threshold . The value characteristics of its Satisfaction Gini Coefficient precisely reflect the normal distribution pattern resulting from the accurate matching between resources and demand. + + + +\begin{figure}[H] +\centering +\includegraphics[width=10cm]{model1.5.png} +\caption{Efficiency Fairness Plot} +\label{efficiency fairness plot} +\end{figure} + +\section{Dual-Site Collaborative Scheduling Optimization Model Based on the Spatio-Temporal Symbiosis Mechanism} + +\subsection{Model Overview} + +Building upon the foundation of Model I, this chapter introduces a tripartite architectural framework centered on the "Symbiotic Sites" strategy to transcend the traditional one-truck-one-site constraint and unlock dormant logistical capacity. + +The model's logic initiates with a rigorous screening phase, where candidate site pairs are evaluated through a multi-dimensional filter encompassing spatial proximity, Demand Accumulation Moderation, and Demand Stability. Once viable pairs are identified, an internal resource allocation mechanism employs a multi-objective utility function to determine the optimal cargo distribution between the paired sites, effectively reconciling the intrinsic trade-off between service effectiveness and cross-site fairness. Ultimately, these symbiotic pairs are integrated as composite decision units into an enhanced global scheduling algorithm, which synchronizes their requirements with independent sites to generate a unified and optimized distribution timetable for the entire year. + +\begin{figure}[H] +\centering +\includegraphics[width=10cm]{flowchartmodel2.png} +\caption{flow chart of Model II} +\label{flow chart of Model II} +\end{figure} + +\subsection{Model Building} + +\subsubsection{Selection Criteria for Symbiotic Sites} + +To ensure the operational feasibility and scientific rigor of the symbiotic strategy, we construct a three-dimensional quantitative screening system to strictly define the entry conditions for symbiotic site pairs. + +\textbf{(1) Spatial Proximity} + +The latitudes and longitudes of all sites are known from the dataset.To identify sites within a small geographic range, a Modified Manhattan Distance($l_{ij}$) is used to quantify the geographic correlation between sites $i$ and $j$. This formula is adapted to the spherical projection characteristics of geographic coordinates: +\begin{equation} + l_{ij} = 69.0 \cdot |lat_{i} - lat_{j}| + 69.0 \cdot \cos\left(\frac{lon_{i} + lon_{j}}{2}\right) |lon_{i} - lon_{j}| +\end{equation} + +Where the coefficient $69.0$ is the mileage conversion constant transforming coordinate differences into surface distance in miles. + +\begin{figure}[H] +\centering +\includegraphics[width=11cm]{distance.png} +\caption{Schematic Diagram of Manhattan Distance Calculation} +\label{Schematic Diagram of Manhattan Distance Calculation} +\end{figure} + +\textbf{(2) Demand Accumulation Moderation} + +Let the service demands of sites $i$ and $j$ be $d_i$ and $d_j$. The maximum truck load is $Q_{max} = 250$ families. To maintain a buffer for scheduling optimization, the upper limit for joint demand is set at 450. Symbiotic pairs must satisfy: + +\begin{equation} + d_{ij} = d_{i} + d_{j} \le 450 +\end{equation} + +\textbf{(3) Demand Stability} + +For candidate pairs nearing the truck's capacity limit, we further consider demand stability. Pairs with high volatility are excluded to prevent subsequent imbalance in symbiotic site allocation. The criterion is: +\begin{equation} + \frac{\sigma_{i}}{d_{i}} \le 0.5, \quad \frac{\sigma_{j}}{d_{j}} \le 0.5 +\end{equation} + +\subsubsection{Internal Resource Allocation Model for Symbiotic Pairs} +For the screened symbiotic pairs $\{i, j\}$, the cargo allocation strategy must be optimized. We define a virtual allocation $q_i$ as the pre-allocated cargo volume for site $i$. Based on Model I, a comprehensive utility scoring function is constructed to solve for the optimal quota: + +\begin{enumerate} + \item \textbf{Actual cargo distribution for the first site:} $g_{i} = \min(d_{i}, q_{i})$ + \item \textbf{Actual cargo distribution for the second site:} $g_{j} = \min(d_{j}, d_{0} - g_{i})$ + \item \textbf{Utility Maximization Objective:} + \begin{itemize} + \item \textbf{Objective 1:} Maximize expected total service volume: $\max E(g_{i} + g_{j})$ + \item \textbf{Objective 2:} Minimize the service gap (Fairness): + $\min U_{i} + U_{j}$ + \end{itemize} +\end{enumerate} + +To balance service effectiveness and allocation fairness, the dual objectives are fused into a unified Allocation Effectiveness Score: +\begin{equation} + score_{ij}^{A} = E(g_{i} + g_{j}) - \lambda E(U_{i} + U_{j}) - \mu E(\max(U_{i}, U_{j})) +\end{equation} + +The $q_i$ that yields the maximum score is selected as the allocation quantity for the first site. + + + +\subsubsection{Global Scheduling Model Adapted for Symbiotic Sites} +The calculation of visit frequencies follows the same logic as Model I, adapted for the scenario where the total number of decision units is reduced and symbiotic demands are merged. + +For visit time allocation, we reconstruct the decision variables, objective function, and constraints to achieve coordinated scheduling of independent and symbiotic sites. +\begin{itemize} + \item \textbf{Decision Variables:} + \begin{enumerate} + \item $s_{i, m}$: The time of the $m$-th transport to independent site $i$. + \item $s_{x, y}$: The time of the $y$-th transport to symbiotic pair $x$. + \item $a_{i, t}$: A binary variable indicating whether independent site $i$ is visited on day $t$. + \item $a_{x, t}$: A binary variable indicating whether symbiotic pair $x$ is visited on day $t$. + \end{enumerate} + \item \textbf{Objective Function:} + Minimize the sum of absolute deviations between actual visit intervals and ideal intervals to ensure uniform distribution of visits across all units: + \begin{equation} + \min Z = \sum_{i} \sum_{m=1}^{k_{i}-1} \left| (s_{i, m+1} - s_{i, m}) - T_{i} \right| + \sum_{x} \sum_{y=1}^{k_{x}-1} \left| (s_{x, y+1} - s_{x, y}) - T_{x} \right| + \end{equation} + \item \textbf{Constraints:} + \begin{enumerate} + \item Total daily visits cannot exceed the maximum fleet capacity: $\sum_{i} a_{i, t} + \sum_{x} a_{x, t} \le 2$. + \item Each unit must reach the stipulated number of visits: $\sum_{t} a_{i, t} = k_{i}, \quad \sum_{t} a_{x, t} = k_{x}$. + \item The interval between consecutive visits must not be less than a default threshold: $s_{i, m+1} - s_{i, m} \ge t_{0}, \quad s_{x, y+1} - s_{x, y} \ge t_{0}$. + \end{enumerate} +\end{itemize} + +\subsubsection{Evaluation Indicator System for Symbiotic Strategy} +In light of the symbiotic site mechanism, the evaluation metrics from Model I are updated as follows: + +\begin{table}[H]\small +\centering +\caption{Comparison of Evaluation Indicators between Model I and Model II} +\label{tab:comparison} +\renewcommand{\arraystretch}{1.5} % 进一步增加行高，让公式更舒展 + +% 自定义列定义：L为左对齐，C为居中对齐，均带有防止拉伸间隙的命令 +\begin{NiceTabularX}{\textwidth}{l >{\raggedright\arraybackslash}p{3.5cm} >{\centering\arraybackslash}X >{\centering\arraybackslash}X}[ + code-before = { + \rowcolor{tableHead}{1} + \rowcolors{2}{white}{tableRow} + } +] +\toprule +\textbf{Dimension} & \textbf{Metric (Symbol)} & \textbf{Model I (Baseline)} & \textbf{Model II (Symbiotic)} \\ \midrule + +% 第一组：Effectiveness +\Block{2-1}{\textbf{Effectiveness}} + & Service Volume ($E_1$) & $\displaystyle \sum k_i \tilde{d}_i$ & $\displaystyle \sum k_i \tilde{d}_i + \sum k_x \tilde{d}_x \uparrow$ \\ + & Weighted Volume ($E_2$) & $\displaystyle \sum q(\tilde{d}_i) k_i \tilde{d}_i$ & \small Optimized for joint capacity \\ \midrule + +% 第二组：Fairness +\Block{2-1}{\textbf{Fairness}} + & Satisfaction ($SL$) & $k_i C_0 / \tilde{d}_i$ & \small Balanced within pairs \\ + & Gini Coefficient ($G$) & \small Overall dispersion $G$ & \small Lowered $G'$ (Better balance) \\ \midrule + +% 第三组：Logistics +\Block{2-1}{\textbf{Logistics}} + & Decision Units ($N$) & \small 70 independent sites & \small $N_{indep} + N_{pairs} < 70$ \\ + & Risk Control & \small Truncation Prob. ($p_{trunc}$) & \small Service Gap Risk ($P_x$) \\ +\bottomrule +\end{NiceTabularX} +\end{table} + + + + +\subsection{Model Solution} + +\subsubsection{Solving for Symbiotic Site Screening} +Using a greedy approach, sites are prioritized for pairing based on a Pairing Score($score_{ij}^{P}$): +\begin{equation} + score_{ij}^{P} = \frac{\tilde{d_{i}} + \tilde{d_{j}}}{d_{0}} - \beta \frac{l_{ij}}{l_{0}} - \gamma \frac{\sigma^{2}_{i} + \sigma^{2}_{j}}{(\tilde{d_{i}} + \tilde{d_{j}})^{2}} +\end{equation} +Candidate combinations with higher scores are paired first, ensuring each site is paired at most once. + +\begin{figure}[H] +\centering +\includegraphics[width=10cm]{model2.1.png} +\caption{Allocation Strategy Scatter} +\label{Allocation Strategy Scatter} +\end{figure} + +This set of scatter plots aims to verify the physical meaning of the analytical solution for the optimal allocation quantity , i.e., $q^* = \frac{\sigma_j \mu_i + \sigma_i (Q - \mu_j)}{\sigma_i + \sigma_j}$. The left plot shows a strong positive correlation ($R^2 \approx 1$) between the proportion of site $i$'s average demand in the paired total demand and its allocation ratio , confirming that average demand is the basis for allocation—sites with higher demand receive more reserves. The middle plot reveals a negative correlation between the proportion of site $i$'s own volatility and its allocation ratio, while the right plot indicates a positive correlation between the volatility proportion of partner site $j$ and site $i$'s allocation ratio . These results validate the model's "Risk Buffering Mechanism": contrary to intuition, the model does not blindly "fill" high-volatility sites but retains more flexibility. This nonlinear allocation strategy is the core advantage that makes the model superior to simple proportional allocation. + + +\subsection{Results of Task 3} +\begin{table}[H] + \footnotesize + \centering + \caption{Key Parameters of Paired Service Sites} + \label{tab:paired_site_params} + + \begin{tabular}{cccccccccccc} + \toprule[1pt] + {ID$_\text{i}$} & {ID$_\text{j}$} & {Dist} & {$\mu_\text{i}$} & {$\mu_\text{j}$} & {$\mu_\text{sum}$} & {$\sigma_\text{i}$} & {$\sigma_\text{j}$} & {CV$_\text{i}$} & {CV$_\text{j}$} & {$k_\text{i}$} & {$k_\text{j}$} \\ + \midrule[0.6pt] + 21 & 61 & 1.1681 & 181.0000 & 17.2000 & 198.2000 & 23.2938 & 4.2374 & 0.1287 & 0.2464 & 13 & 2 \\ + 29 & 39 & 10.1829 & 220.0000 & 25.8889 & 245.8889 & 22.6176 & 3.3333 & 0.1028 & 0.1288 & 16 & 3 \\ + 38 & 55 & 3.3738 & 31.0909 & 24.3636 & 55.4545 & 6.7595 & 4.4107 & 0.2174 & 0.1810 & 3 & 3 \\ + 48 & 68 & 14.9932 & 32.8333 & 202.5455 & 235.3788 & 6.2423 & 30.4479 & 0.1901 & 0.1503 & 3 & 14 \\ + 50 & 69 & 13.7204 & 26.0000 & 200.9091 & 226.9091 & 4.8990 & 34.8954 & 0.1884 & 0.1737 & 3 & 14 \\ + \bottomrule[1pt] + \end{tabular} + +\end{table} +This table reports five representative dual-site pairings and shows how the model balances feasibility, efficiency, and risk. First, all paired sites are geographically close (about 1.17--14.99 miles), which supports operational feasibility and limits extra travel. Second, the demand structure is largely complementary: several pairs match a higher-demand site with a lower-demand site , keeping the combined expected demand $\mu_{\text{sum}}$ within a manageable range (about 55.45--245.89) so that residual truck capacity can be utilized rather than wasted. Third, uncertainty is controlled through the coefficient of variation : the paired sites generally exhibit moderate variability (roughly 0.10--0.25), and higher-variability small sites are coupled with more stable main sites to reduce the probability of shortfall at the second stop. Finally, the allocated visit frequencies reflect need: the primary (higher-$\mu$) sites receive more annual visits , while the smaller sites retain fewer but necessary visits , improving per-trip productivity while maintaining basic coverage equity. + +\begin{figure}[H] +\centering +\includegraphics[width=11cm]{model2.2.png} +\caption{Paired Service Site Map} +\label{Paired Service Site Map} +\end{figure} + +\begin{itemize} + \item \textbf{Effectiveness analysis} +\end{itemize} +\begin{table}[H] + \footnotesize + \centering + \caption{Performance Comparison} + \label{tab:task3_effectiveness} + + \begin{tabular}{lccrr} + \toprule[1pt] + Performance Indicator & Task 1 & Task 3 & Change & Change (\%) \\ + \midrule[0.6pt] + E1 (Total Service Volume) & 139,469 & 153,307 & +13,838 & +9.9 \\ + E2 (Quality-Weighted Volume) & 131,462 & 139,701 & +8,240 & +6.3 \\ + F1 (Gini Coefficient) & 0.314 & 0.322 & +0.008 & +2.5 \\ + F2 (Minimum Satisfaction Rate) & 2.000 & 2.000 & 0.000 & 0.0 \\ + R1 (Shortfall Risk) & 0 & 0.012 & +0.012 & New \\ + RS (Resource Savings) & 0\% & 9.6\% & +9.6\% & New \\ + \bottomrule[1pt] + \end{tabular} +\end{table} + +This section compares core performance metrics between Task 1 (traditional single-site mode) and Task 3 (dual-site optimized mode), verifying the significant improvements of the new model: + +1. \textbf{Efficiency Gains}: The total service volume increased by 9.9\% , directly driven by the reallocation of 70 released scheduling slots (9.6\% of total capacity) to the entire network. The quality-weighted service volume still achieved a 6.3\% growth, despite a slight reduction in average allocation per household under the dual-site mode. + +2. \textbf{Risk Control}: The shortfall risk rose from 0 to 1.2\%, which is far below the industry-accepted 5\% threshold—proving the model trades minimal risk for substantial efficiency improvements. Meanwhile, the 9.6\% resource savings enable FBST to serve nearly 10\% more communities with existing trucks/volunteers, or cut operational costs by 9.6\% while maintaining current service levels. + +3. \textbf{Equity Maintenance}: The minimum satisfaction rate remained unchanged at 2.0, ensuring the "baseline protection" of service equity. The Gini coefficient slightly increased by 2.5\% , indicating minor inequity introduced by efficiency gains, which can be adjusted via pairing admission constraints or merging ratio parameter tuning. + +\begin{itemize} + \item \textbf{Fairness analysis} +\end{itemize} + +Fig. \ref{Fairness diagnostics} presents the fairness diagnostics for 70 service sites: the left histogram (satisfaction rate distribution) checks for "long-tail sites" with suppressed satisfaction rates ($r_i$), and no excessive concentration at extremely low values is observed (mean $r_i=11.34$); the right Lorenz curve reflects fairness—closer proximity to the diagonal equality line indicates higher equity, and the Gini coefficient here aligns with the F1 index in Table \ref{tab:task3_effectiveness}. Overall, the fairness metrics stay comparable to the baseline, and the minimum satisfaction rate remains unchanged, confirming that efficiency gains do not come at the cost of service to the most vulnerable sites. + +\begin{figure}[H] +\centering +\includegraphics[width=12cm]{model2.3.png} +\caption{Fairness diagnostics} +\label{Fairness diagnostics} +\end{figure} + +\section{Sensitivity Analysis} + +\subsection{Sensitivity Analysis for Task 1} + +\begin{figure}[H] +\centering +\includegraphics[width=11cm]{task1.png} +\caption{Sensitivity Analysis for Task 1} + +\end{figure} +To verify the robustness and reliability of the model parameters, we perform a sensitivity analysis on the critical parameters in Task 1: the capacity threshold $C_0$ and the truncation probability threshold $p_0^{trunc}$. By testing twenty different parameter combinations, we evaluate the performance of various evaluation metrics and their relative deviations from the baseline values provided in the paper. + +% 待插图 +The results indicate that the overall fluctuations of all metrics remain stable. Specifically, $C_0$ is identified as the primary factor influencing the results; its variation significantly affects how many sites are categorized as high-demand sites requiring correction. We observe that fluctuations in evaluation metrics caused by different $p_0^{trunc}$ values (under the same $C_0$) are substantially smaller than those caused by changes in $C_0$ itself. Furthermore, the magnitude of these changes remains relatively low, demonstrating that parameter variations do not alter the core model philosophy of "demand-based allocation, fairness, and feasibility." Consequently, our selection of $C_0 = 350$ and $p_0^{trunc} = 0.1$ is justified as both reasonable and robust. + +\subsection{Sensitivity Analysis for Task 3} +To further validate the reliability of the dual-site collaborative scheduling model, we conduct a sensitivity analysis on the key parameters of Task 3. Performance metrics and their respective deviations were recorded across multiple parameter combinations. + +\begin{figure}[H] +\centering +\includegraphics[width=11cm]{task3.png} +\caption{Sensitivity Analysis for Task 3} + +\end{figure} +Our analysis yields the following observations: +\begin{itemize} + \item \textbf{Distance Threshold ($l_{max}$):} As the distance threshold $l_{max}$ increases from 10 to approximately 22 miles, the number of possible symbiotic site pairs increases from 27 to 34. Beyond this point, further loosening the distance constraint results in negligible marginal gains in the total service volume. To ensure a sufficient number of pairs for operational flexibility, we set $l_{max} = 50$. + \item \textbf{Demand Threshold ($d_{max}$):} Variations in the demand threshold have a minimal impact on the total service volume, with a fluctuation of only approximately $2\%$. However, increasing this value significantly elevates the \textit{Service Gap Risk}. To prioritize distribution effectiveness and maximize food utilization, we adopt $d_{max} = 450$, accepting a higher localized risk to ensure no capacity is wasted. + \item \textbf{Volatility Threshold ($CV_{min}$):} At lower threshold values, the $CV_{min}$ (Coefficient of Variation) shows a distinct negative correlation with the total service volume. As the threshold increases, the system enters a plateau phase and reaches its peak around $0.56$. To balance the maximization of total service volume with the mitigation of service gap risks, we select a volatility threshold of $0.5$. +\end{itemize} + +In summary, the parameter selections in our model ensure both high total service volume and superior food distribution effectiveness, demonstrating that the model parameters are highly reliable and robust under varying conditions. + + +\section{Strengths and Weaknesses} + +\subsection{Strengths} +\begin{itemize} + \item The concept of "true demand" is innovatively proposed by utilizing truncation probability to assess the credibility of mean demand. This approach aligns more closely with real-world demand fluctuations and significantly enhances the practical applicability of the model. + \item The selection of model parameters is highly consistent with the core objectives of effectiveness and fairness. This ensures that the evaluation metrics can accurately reflect the scheduling performance and the overall distribution efficiency of the generated timetable. + \item The model transforms a complex large-scale scheduling problem into a structured three-layer logic, which simplifies the computational complexity while maintaining high solution quality and robustness. +\end{itemize} + +\subsection{Weaknesses and Possible Improvement} +\begin{itemize} + \item Instead of integrating actual road network data, the model employs Manhattan distance based on latitude and longitude coordinates to estimate accessibility. This simplification may introduce certain spatial deviations compared to real-world driving routes. + \item Due to the lack of real-time demand quantification, the temporal scheduling resolution is limited to a daily basis. This lack of granularity means the model may not fully address intra-day demand surges or the needs of clients in specific time-slots. + \item The evaluation indicator system is not sufficiently comprehensive. In future work, more diverse dimensions such as logistical costs, fuel consumption, and volunteer satisfaction could be incorporated to provide a more holistic assessment of the distribution scheme. +\end{itemize} + + +\section{Conclusion} + +This paper develops a demand-responsive and equity-aware scheduling framework for restoring FBST's Mobile Food Pantry services across 70 sites under daily fleet capacity constraints. In Model I (PPA-STS), we correct observed demand using a truncation-based probabilistic mechanism to infer latent community need at capacity-constrained sites, allocate annual visit quotas via the Hamilton method, and generate a 365-day timetable using CP-SAT while enforcing the minimum 14-day interval constraint. Compared with scaled 2019 scheduling, the recommended scheme increases total service volume by 34.0\% (31.1\% quality-weighted) while maintaining baseline equity protection through a minimum satisfaction rate of 2.0. + +To further improve volunteer and vehicle utilization, Model II extends the framework with a dual-site collaborative strategy. By screening geographically proximate and demand-complementary site pairs and optimizing the first-stop load-out through a two-stage stochastic allocation score, the approach unlocks 9.6\% dormant capacity and yields an additional 9.9\% increase in total service volume (6.3\% quality-weighted), with limited shortfall risk (1.2\%) and only minor changes in overall dispersion. Sensitivity analyses confirm that these conclusions remain stable under reasonable variations of key thresholds. + +Overall, the proposed models transform annual scheduling into a robust decision process that jointly advances effectiveness, fairness, and operational feasibility. Future work should incorporate road-network travel times, collect 2021 ``unserved headcount'' and attendance data to calibrate demand and weather effects, and extend the objective system to include costs, volunteer availability, and finer temporal resolution. + + + +\newpage +% 备忘录头部 +\begin{center} + \begin{tcolorbox}[colback=fbstblue!5, colframe=fbstblue, arc=0mm, width=\textwidth, boxrule=1.5pt] + \begin{flushleft} + {\LARGE \textbf{\color{fbstblue}MEMORANDUM}} \\ + + \begin{tabular}{ll} + \textbf{TO:} & Board of Directors, Food Bank of the Southern Tier (FBST) \\ + \textbf{FROM:} & Modeling Team \\ + \textbf{SUBJECT:} & \textbf{2021 Service Restoration: Unlocking Operational Capacity} + \end{tabular} + \end{flushleft} + \end{tcolorbox} +\end{center} + +\noindent \textbf{\large Executive Overview} \\ +As FBST restores capacity in 2021, our audit revealed that static scheduling masked true community demand. The proposed framework utilizes statistical correction and route optimization to serve an estimated 153,307 families annually—a 48\% increase over 2019 levels—within existing operational feasibility and resource constraints . + +\noindent {\color{fbstblue}\rule{\textwidth}{1.5pt}} + +\noindent \textbf{1. Addressing Under-Estimated Demand} \\ +Analysis identified a “Capacity Ceiling” where 2019 demand in key hubs was under-estimated by up to 19\%. +\begin{itemize}[leftmargin=1.5em, nosep] + \item \textbf{The Solution:} We implemented a \textit{Demand-Proportional Allocation System}. By reallocating visits to high-need hubs, service volume increases by 34.6\% while ensuring all sites meet a minimum service frequency ($k \ge 2$). +\end{itemize} + +\noindent \textbf{2. Expanding Coverage via Shared-Truck Strategies} \\ +To break logistical bottlenecks, we introduced a \textit{Paired-Site Routing Protocol} based on symbiotic site selection. +\begin{itemize}[leftmargin=1.5em, nosep] + \item \textbf{Mechanism:} Trucks visit two proximal sites ($<$50 mi), merging demands to reduce total decision units . By pairing volatile sites with stable ones, we mitigate food shortfall risks to 1.2\%. + \item \textbf{Impact:} This strategy frees 70 scheduling slots, expanding coverage by 9.9\% while maximizing food utilization with a $d_{max}$ of 450 . +\end{itemize} + +\noindent \textbf{Strategic Recommendations} +\begin{enumerate}[leftmargin=1.5em, nosep] + \item \textbf{Adopt Dynamic Load-Out Sheets:} Operationalize generated manifestos to ensure uniform food distribution and efficient pruning of schedules that violate the 14-day interval . + \item \textbf{Data Reform:} Require drivers to log “unserved headcount” in 2021 to replace statistical estimates with empirical demand data for the 2022 cycle. +\end{enumerate} + +\begin{tcolorbox}[colback=fbstgold!10, colframe=fbstgold, boxrule=1pt, arc=2mm] +\textbf{Conclusion:} Our solution transforms scheduling into a dynamic, demand-responsive system using a global optimization model . This delivers $\sim$49,000 more meals than 2019 by acknowledging hidden demand and optimizing route density. +\end{tcolorbox} +\newpage +\printbibliography[title={References}, heading=bibintoc] + +% 重置页码 +\clearpage +% \setcounter{page}{\value{lastpage}} + +\end{document} +%% +%% This work consists of these files mcmthesis.dtx, +%% figures/ and +%% code/, +%% and the derived files mcmthesis.cls, +%% mcmthesis-demo.tex, +%% README, +%% LICENSE, +%% mcmthesis.pdf and +%% mcmthesis-demo.pdf. +%% +%% End of file `mcmthesis-demo.tex'. diff --git a/vscode/mcmthesis-demo.toc b/vscode/mcmthesis-demo.toc new file mode 100755 index 0000000..8b036e2 --- /dev/null +++ b/vscode/mcmthesis-demo.toc @@ -0,0 +1,35 @@ +\contentsline {section}{\numberline {1}Introduction}{3}{section.1}% +\contentsline {subsection}{\numberline {1.1}Background}{3}{subsection.1.1}% +\contentsline {subsection}{\numberline {1.2}Restatement of the Problem}{3}{subsection.1.2}% +\contentsline {subsection}{\numberline {1.3}Our Work}{3}{subsection.1.3}% +\contentsline {section}{\numberline {2}Assumptions and Justifications}{4}{section.2}% +\contentsline {section}{\numberline {3}Notations}{5}{section.3}% +\contentsline {section}{\numberline {4}Probability-Corrected Proportional Allocation and Spatio-Temporal Scheduling Model}{5}{section.4}% +\contentsline {subsection}{\numberline {4.1}Model Overview}{5}{subsection.4.1}% +\contentsline {subsection}{\numberline {4.2}Model Building}{6}{subsection.4.2}% +\contentsline {subsubsection}{\numberline {4.2.1}Stochastic Demand Correction Mechanism}{6}{subsubsection.4.2.1}% +\contentsline {subsubsection}{\numberline {4.2.2}Visit Frequency Allocation Based on the Hamilton Method}{6}{subsubsection.4.2.2}% +\contentsline {subsubsection}{\numberline {4.2.3}Multi-dimensional Evaluation Indicator System}{7}{subsubsection.4.2.3}% +\contentsline {subsubsection}{\numberline {4.2.4}Spatio-Temporal Scheduling Combinatorial Optimization Model}{7}{subsubsection.4.2.4}% +\contentsline {subsection}{\numberline {4.3}Model Solution}{9}{subsection.4.3}% +\contentsline {subsubsection}{\numberline {4.3.1}Results for Stochastic Demand Correction Mechanism}{9}{subsubsection.4.3.1}% +\contentsline {subsubsection}{\numberline {4.3.2}Solution for Spatio-Temporal Scheduling Combinatorial Optimization Model}{10}{subsubsection.4.3.2}% +\contentsline {subsection}{\numberline {4.4}Results of Task 1}{10}{subsection.4.4}% +\contentsline {section}{\numberline {5}Dual-Site Collaborative Scheduling Optimization Model Based on the Spatio-Temporal Symbiosis Mechanism}{12}{section.5}% +\contentsline {subsection}{\numberline {5.1}Model Overview}{12}{subsection.5.1}% +\contentsline {subsection}{\numberline {5.2}Model Building}{13}{subsection.5.2}% +\contentsline {subsubsection}{\numberline {5.2.1}Selection Criteria for Symbiotic Sites}{13}{subsubsection.5.2.1}% +\contentsline {subsubsection}{\numberline {5.2.2}Internal Resource Allocation Model for Symbiotic Pairs}{14}{subsubsection.5.2.2}% +\contentsline {subsubsection}{\numberline {5.2.3}Global Scheduling Model Adapted for Symbiotic Sites}{15}{subsubsection.5.2.3}% +\contentsline {subsubsection}{\numberline {5.2.4}Evaluation Indicator System for Symbiotic Strategy}{15}{subsubsection.5.2.4}% +\contentsline {subsection}{\numberline {5.3}Model Solution}{16}{subsection.5.3}% +\contentsline {subsubsection}{\numberline {5.3.1}Solving for Symbiotic Site Screening}{16}{subsubsection.5.3.1}% +\contentsline {subsection}{\numberline {5.4}Results of Task 3}{16}{subsection.5.4}% +\contentsline {section}{\numberline {6}Sensitivity Analysis}{18}{section.6}% +\contentsline {subsection}{\numberline {6.1}Sensitivity Analysis for Task 1}{18}{subsection.6.1}% +\contentsline {subsection}{\numberline {6.2}Sensitivity Analysis for Task 3}{19}{subsection.6.2}% +\contentsline {section}{\numberline {7}Strengths and Weaknesses}{20}{section.7}% +\contentsline {subsection}{\numberline {7.1}Strengths}{20}{subsection.7.1}% +\contentsline {subsection}{\numberline {7.2}Weaknesses and Possible Improvement}{20}{subsection.7.2}% +\contentsline {section}{\numberline {8}Conclusion}{20}{section.8}% +\contentsline {section}{References}{23}{section*.1}% diff --git a/vscode/mcmthesis.cls b/vscode/mcmthesis.cls new file mode 100755 index 0000000..b03640c --- /dev/null +++ b/vscode/mcmthesis.cls @@ -0,0 +1,352 @@ +%% +%% This is file `mcmthesis.cls', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% mcmthesis.dtx (with options: `class') +%% +%% ----------------------------------- +%% +%% This is a generated file. +%% +%% Copyright (C) +%% 2010 -- 2015 by Zhaoli Wang +%% 2014 -- 2019 by Liam Huang +%% 2019 -- present by latexstudio.net +%% +%% This work may be distributed and/or modified under the +%% conditions of the LaTeX Project Public License, either version 1.3 +%% of this license or (at your option) any later version. +%% The latest version of this license is in +%% http://www.latex-project.org/lppl.txt +%% and version 1.3 or later is part of all distributions of LaTeX +%% version 2005/12/01 or later. +%% +%% This work has the LPPL maintenance status `maintained'. +%% +%% The Current Maintainer of this work is Liam Huang. +%% +\NeedsTeXFormat{LaTeX2e}[1999/12/01] +\ProvidesClass{mcmthesis} + [2020/01/18 v6.3 The Thesis Template Designed For MCM/ICM] +\typeout{The Thesis Template Designed For MCM/ICM} +\def\MCMversion{v6.3} +\RequirePackage{xkeyval} +\RequirePackage{etoolbox} +\define@boolkey{MCM}[MCM@opt@]{CTeX}[false]{} +\define@boolkey{MCM}[MCM@opt@]{titlepage}[true]{} +\define@boolkey{MCM}[MCM@opt@]{abstract}[true]{} +\define@boolkey{MCM}[MCM@opt@]{sheet}[true]{} +\define@boolkey{MCM}[MCM@opt@]{titleinsheet}[false]{} +\define@boolkey{MCM}[MCM@opt@]{keywordsinsheet}[false]{} +\define@cmdkeys{MCM}[MCM@opt@]{tcn,problem} +\define@key{MCM}{tcn}[0000]{\gdef\MCM@opt@tcn{#1}} +\define@key{MCM}{problem}[A]{\gdef\MCM@opt@problem{#1}} +\setkeys{MCM}{tcn=0000,problem=B} + +\define@key{mcmthesis.cls}{tcn}[0000]{\gdef\MCM@opt@tcn{#1}} +\define@key{mcmthesis.cls}{problem}[A]{\gdef\MCM@opt@problem{#1}} +\define@boolkey{mcmthesis.cls}[MCM@opt@]{titlepage}{} +\define@boolkey{mcmthesis.cls}[MCM@opt@]{abstract}{} +\define@boolkey{mcmthesis.cls}[MCM@opt@]{sheet}{} +\define@boolkey{mcmthesis.cls}[MCM@opt@]{titleinsheet}{} +\define@boolkey{mcmthesis.cls}[MCM@opt@]{keywordsinsheet}{} +\MCM@opt@sheettrue +\MCM@opt@titlepagetrue +\MCM@opt@titleinsheetfalse +\MCM@opt@keywordsinsheetfalse +\MCM@opt@abstracttrue +\newcommand{\mcmsetup}[1]{\setkeys{MCM}{#1}} +\ProcessOptionsX\relax +\LoadClass[a4paper, 12pt]{article} +\newcommand{\team}{Team \#\ \MCM@opt@tcn} +\RequirePackage{fancyhdr, fancybox} +\RequirePackage{ifthen} +\RequirePackage{lastpage} +\RequirePackage{listings} +\RequirePackage[toc, page, title, titletoc, header]{appendix} +\RequirePackage{paralist} +\RequirePackage{amsthm, amsfonts} +\RequirePackage{amsmath, bm} +\RequirePackage{amssymb, mathrsfs} +\RequirePackage{latexsym} +\RequirePackage{longtable, multirow, hhline, tabularx, array} +\RequirePackage{flafter} +\RequirePackage{pifont, calc} +\RequirePackage{colortbl, booktabs} +\RequirePackage{geometry} +\RequirePackage[T1]{fontenc} +\RequirePackage[scaled]{berasans} +\RequirePackage{hyperref} +\RequirePackage{ifpdf, ifxetex} +\ifMCM@opt@CTeX +\else + \RequirePackage{environ} +\fi +\ifpdf + \RequirePackage{graphicx} + \RequirePackage{epstopdf} +\else + \ifxetex + \RequirePackage{graphicx} + \else + \RequirePackage[dvipdfmx]{graphicx} + \RequirePackage{bmpsize} + \fi +\fi +\RequirePackage[svgnames]{xcolor} +\ifpdf + \hypersetup{hidelinks} +\else + \ifxetex + \hypersetup{hidelinks} + \else + \hypersetup{dvipdfm, hidelinks} + \fi +\fi +\geometry{a4paper, margin = 1in} +\pagestyle{fancy} +\fancyhf{} +\lhead{\small\sffamily \team} +\rhead{\small\sffamily Page \thepage\ of \pageref{LastPage}} +% \rhead{\small\sffamily Page \thepage} +\setlength\parskip{.5\baselineskip} +\renewcommand\tableofcontents{% + \centerline{\normalfont\Large\bfseries\sffamily\contentsname + \@mkboth{% + \MakeUppercase\contentsname}{\MakeUppercase\contentsname}}% + \vskip 5ex% + \@starttoc{toc}% + } +\setcounter{totalnumber}{4} +\setcounter{topnumber}{2} +\setcounter{bottomnumber}{2} +\renewcommand{\textfraction}{0.15} +\renewcommand{\topfraction}{0.85} +\renewcommand{\bottomfraction}{0.65} +\renewcommand{\floatpagefraction}{0.60} +\renewcommand{\figurename}{Figure} +\renewcommand{\tablename}{Table} +\graphicspath{{./}{./img/}{./fig/}{./image/}{./figure/}{./picture/} + {./imgs/}{./figs/}{./images/}{./figures/}{./pictures/}} +\def\maketitle{% + \let\saved@thepage\thepage + \let\thepage\relax + \ifMCM@opt@sheet + \makesheet + \fi + \newpage + \ifMCM@opt@titlepage + \MCM@maketitle + \fi + \newpage + \let\thepage\saved@thepage + \setcounter{page}{2} + \pagestyle{fancy} +} +\def\abstractname{\large{Summary}} +\ifMCM@opt@CTeX + \newbox\@abstract% + \setbox\@abstract\hbox{}% + \long\def\abstract{\bgroup\global\setbox\@abstract\vbox\bgroup\hsize\textwidth\leftskip1cm\rightskip1cm}% + \def\endabstract{\egroup\egroup} + \def\make@abstract{% + \begin{center} + \textbf{\abstractname} + \end{center} + \usebox\@abstract\par + } +\else + \RenewEnviron{abstract}{\xdef\@abstract{\expandonce\BODY}} + \def\make@abstract{% + \begin{center} + \vspace*{-0.4cm}% + \textbf{\abstractname} + \end{center} + \vspace*{-0.4cm}% + \@abstract\par + } +\fi +\newenvironment{letter}[1]{% + \par% + \bgroup\parindent0pt% + \begin{minipage}{5cm} + \flushleft #1% + \end{minipage}} + {\egroup\smallskip} + +\def\keywordsname{Keywords} +\ifMCM@opt@CTeX + \newbox\@keywords + \setbox\@keywords\hbox{} + \def\keywords{\global\setbox\@keywords\vbox\bgroup\noindent\leftskip0cm} + \def\endkeywords{\egroup}% + \def\make@keywords{% + \par\hskip.4cm\textbf{\keywordsname}: \usebox\@keywords\hfill\par + } +\else + \NewEnviron{keywords}{\xdef\@keywords{\expandonce\BODY}} + \def\make@keywords{% + \par\noindent\textbf{\keywordsname}: + \@keywords\par + } +\fi +\newcommand{\headset}{{\the\year}\\MCM/ICM\\Summary Sheet} +\newcommand{\problem}[1]{\mcmsetup{problem = #1}} +\def\makesheet{% + \pagestyle{empty}% + \null% + \vspace*{-5pc}% + \begin{center} + \begingroup + \setlength{\parindent}{0pt} + \begin{minipage}[t]{0.33\linewidth} + \bfseries\centering% + Problem Chosen\\[0.7pc] + {\Huge\textbf{\MCM@opt@problem}}\\[2.8pc] + \end{minipage}% + \begin{minipage}[t]{0.33\linewidth} + \centering% + \textbf{\headset}% + \end{minipage}% + \begin{minipage}[t]{0.33\linewidth} + \centering\bfseries% + Team Control Number\\[0.7pc] + {\Huge\textbf{\textcolor{red}{\MCM@opt@tcn}}}\\[2.8pc] + % {\Huge\textbf{\MCM@opt@tcn}}\\[2.8pc] + \end{minipage}\par + \rule{\linewidth}{1.0pt}\par + \endgroup + % \vskip 10pt% + \ifMCM@opt@titleinsheet + \normalfont \LARGE \@title \par + \fi + \end{center} +\ifMCM@opt@keywordsinsheet + \make@abstract + \make@keywords +\else + \make@abstract +\fi} +\newcommand{\MCM@maketitle}{% + \begin{center}% + \let \footnote \thanks + {\LARGE \@title \par}% + \vskip 1.5em% + {\large + \lineskip .5em% + \begin{tabular}[t]{c}% + \@author + \end{tabular}\par}% + \vskip 1em% + {\large \@date}% + \end{center}% + \par + \vskip 1.5em% + \ifMCM@opt@abstract% + \make@abstract + \make@keywords + \fi% +} +\def\MCM@memoto{\relax} +\newcommand{\memoto}[1]{\gdef\MCM@memoto{#1}} +\def\MCM@memofrom{\relax} +\newcommand{\memofrom}[1]{\gdef\MCM@memofrom{#1}} +\def\MCM@memosubject{\relax} +\newcommand{\memosubject}[1]{\gdef\MCM@memosubject{#1}} +\def\MCM@memodate{\relax} +\newcommand{\memodate}[1]{\gdef\MCM@memodate{#1}} +\def\MCM@memologo{\relax} +\newcommand{\memologo}[1]{\gdef\MCM@memologo{\protect #1}} +\def\@letterheadaddress{\relax} +\newcommand{\lhaddress}[1]{\gdef\@letterheadaddress{#1}} +\newenvironment{memo}[1][Memorandum]{% + \pagestyle{plain}% + \ifthenelse{\equal{\MCM@memologo}{\relax}}{% + % without logo specified. + }{% + % with logo specified + \begin{minipage}[t]{\columnwidth}% + \begin{flushright} + \vspace{-0.6in} + \MCM@memologo + \vspace{0.5in} + \par\end{flushright}% + \end{minipage}% + } + \begin{center} + \LARGE\bfseries\scshape + #1 + \end{center} + \begin{description} + \ifthenelse{\equal{\MCM@memoto}{\relax}}{}{\item [{To:}] \MCM@memoto} + \ifthenelse{\equal{\MCM@memofrom}{\relax}}{}{\item [{From:}] \MCM@memofrom} + \ifthenelse{\equal{\MCM@memosubject}{\relax}}{}{\item [{Subject:}] \MCM@memosubject} + \ifthenelse{\equal{\MCM@memodate}{\relax}}{}{\item [{Date:}] \MCM@memodate} + \end{description} + \par\noindent + \rule[0.5ex]{\linewidth}{0.1pt}\par + \bigskip{} +}{% + \clearpage + \pagestyle{fancy}% +} +\newtheorem{Theorem}{Theorem}[section] +\newtheorem{Lemma}[Theorem]{Lemma} +\newtheorem{Corollary}[Theorem]{Corollary} +\newtheorem{Proposition}[Theorem]{Proposition} +\newtheorem{Definition}[Theorem]{Definition} +\newtheorem{Example}[Theorem]{Example} +\renewcommand\section{\@startsection{section}{1}{\z@}% + {-0pt\@plus -.2ex \@minus -.2ex}% + {1pt \@plus .2ex}% + {\rmfamily\Large\bfseries}} +\renewcommand\subsection{\@startsection{subsection}{2}{\z@}% + {-0pt\@plus -.2ex \@minus -.2ex}% + {1pt \@plus .2ex}% + {\rmfamily\large\bfseries}} +\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}% + {-.5ex\@plus -1ex \@minus -.2ex}% + {.25ex \@plus .2ex}% + {\rmfamily\normalsize\bfseries}} +\renewcommand\paragraph{\@startsection{paragraph}{4}{\z@}% + {1ex \@plus1ex \@minus.2ex}% + {-1em}% + {\rmfamily\normalsize}} + +\providecommand{\dif}{\mathop{}\!\mathrm{d}} +\providecommand{\me}{\mathrm{e}} +\providecommand{\mi}{\mathrm{i}} + +\definecolor{grey}{rgb}{0.8,0.8,0.8} +\definecolor{darkgreen}{rgb}{0,0.3,0} +\definecolor{darkblue}{rgb}{0,0,0.3} +\def\lstbasicfont{\fontfamily{pcr}\selectfont\footnotesize} +\lstset{% + % numbers=left, + % numberstyle=\small,% + showstringspaces=false, + showspaces=false,% + tabsize=4,% + frame=lines,% + basicstyle={\footnotesize\lstbasicfont},% + keywordstyle=\color{darkblue}\bfseries,% + identifierstyle=,% + commentstyle=\color{darkgreen},%\itshape,% + stringstyle=\color{black}% +} +\lstloadlanguages{C,C++,Java,Matlab,Mathematica} +\endinput +%% +%% This work consists of these files mcmthesis.dtx, +%% figures/ and +%% code/, +%% and the derived files mcmthesis.cls, +%% mcmthesis-demo.tex, +%% README, +%% LICENSE, +%% mcmthesis.pdf and +%% mcmthesis-demo.pdf. +%% +%% End of file `mcmthesis.cls'. diff --git a/vscode/mcmthesis.synctex.gz b/vscode/mcmthesis.synctex.gz new file mode 100755 index 0000000..189c605 Binary files /dev/null and b/vscode/mcmthesis.synctex.gz differ diff --git a/vscode/picture.aux b/vscode/picture.aux new file mode 100755 index 0000000..b640121 --- /dev/null +++ b/vscode/picture.aux @@ -0,0 +1,2 @@ +\relax +\gdef \@abspage@last{1} diff --git a/vscode/picture.fdb_latexmk b/vscode/picture.fdb_latexmk new file mode 100755 index 0000000..cf464aa --- 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I am going to ignore this key. + +See the pgfkeys package documentation for explanation. +Type H for immediate help. + ... + +l.20 \node[variable, below=of ntotal] + (kbase) {$k_{base} \in [5, 10]$}; +(That was another \errmessage.) + +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <9> on input line 23. +LaTeX Font Info: External font `cmex10' loaded for size +(Font) <6> on input line 23. + +d:/XHJ/mathmo/MCM2026/20260116/vscode/picture.tex:26: Package pgfkeys Error: The key '/tikz/variable' requires a value. I am going to ignore this key. + +See the pgfkeys package documentation for explanation. +Type H for immediate help. + ... + +l.26 \node[variable, right=of nfree] + (kie) { +(That was another \errmessage.) + +d:/XHJ/mathmo/MCM2026/20260116/vscode/picture.tex:27: Undefined control sequence. +l.27 $k_{i_e} = k_{base} + \text + {round}\left(N_{free} \cdot \frac{D_... +The control sequence at the end of the top line +of your error message was never \def'ed. 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Perhaps you misspelled it. + +See the pgfkeys package documentation for explanation. +Type H for immediate help. + ... + +l.67 ...effbox) [fit=(eff) (score) (gini) (judge)] + {}; +(That was another \errmessage.) + + + +[1 + + +Non-PDF special ignored! + papersize=538.80806pt,247.52835pt +{c:/texlive/texlive/2025/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] (./picture.aux) + *********** +LaTeX2e <2024-11-01> patch level 2 +L3 programming layer <2025-01-18> + *********** + ) +Here is how much of TeX's memory you used: + 13523 strings out of 473190 + 288281 string characters out of 5715811 + 666049 words of memory out of 5000000 + 36561 multiletter control sequences out of 15000+600000 + 560968 words of font info for 43 fonts, out of 8000000 for 9000 + 1141 hyphenation exceptions out of 8191 + 117i,8n,121p,508b,855s stack positions out of 10000i,1000n,20000p,200000b,200000s + +Output written on picture.pdf (1 page, 152892 bytes). +PDF statistics: + 86 PDF objects out of 1000 (max. 8388607) + 52 compressed objects within 1 object stream + 0 named destinations out of 1000 (max. 500000) + 13 words of extra memory for PDF output out of 10000 (max. 10000000) + diff --git a/vscode/picture.pdf b/vscode/picture.pdf new file mode 100755 index 0000000..00c048f Binary files /dev/null and b/vscode/picture.pdf differ diff --git a/vscode/picture.synctex.gz b/vscode/picture.synctex.gz new file mode 100755 index 0000000..51a3713 Binary files /dev/null and b/vscode/picture.synctex.gz differ diff --git a/vscode/picture.tex b/vscode/picture.tex new file mode 100755 index 0000000..52c4dd8 --- /dev/null +++ b/vscode/picture.tex @@ -0,0 +1,72 @@ +\documentclass[tikz,border=10pt]{standalone} +\usetikzlibrary{shapes.geometric, arrows.meta, positioning, calc, backgrounds} + +\begin{document} + +\begin{tikzpicture}[ + node distance=1.5cm and 2cm, + % 设置节点样式 + base/.style={rectangle, rounded corners, draw=black, minimum width=2.5cm, minimum height=1cm, text centered, font=\small, fill=white}, + variable/.style={base, fill=blue!10, draw=blue!50!black}, + process/.style={base, fill=orange!10, draw=orange!50!black}, + decision/.style={diamond, aspect=2, draw=red!50!black, fill=red!5!white, font=\scriptsize, inner sep=0pt}, + formula/.style={font=\scriptsize, text=black!80}, + % 设置区域样式 + container/.style={draw, dashed, inner sep=15pt, rounded corners} +] + + % --- 1. Fairness Allocation Part --- + \node[variable] (ntotal) {$N_{total} = 730$}; + \node[variable, below=of ntotal] (kbase) {$k_{base} \in [5, 10]$}; + + \node[process, right=of $(ntotal.south)!0.5!(kbase.north)$] (nfree) { + $N_{free} = N_{total} - 70 \cdot k_{base}$ + }; + + \node[variable, right=of nfree] (kie) { + $k_{i_e} = k_{base} + \text{round}\left(N_{free} \cdot \frac{D_i}{\sum D_i}\right)$ + }; + + % --- 2. Effectiveness Scoring Part --- + \node[process, below=2cm of kie] (eff) { + $annual\_eff_i = k_{i_e} \cdot \min(d_i, d_0)$ + }; + + \node[process, below=1cm of eff] (score) { + $score_i = \frac{eff_i}{D_i} - \alpha \cdot unmet_i - \beta \cdot waste_i$ + }; + + % --- 3. Validation Part --- + \node[variable, left=of score] (gini) { + $G = \frac{\sum \sum |score_i - score_j|}{2n^2 \overline{score}}$ + }; + + \node[decision, left=of gini] (judge) {$G < 0.2$?}; + + % --- 连线 --- + \draw[-Stealth, thick] (ntotal) -- (nfree); + \draw[-Stealth, thick] (kbase) -- (nfree); + \draw[-Stealth, thick] (nfree) -- node[above, formula] {Demand Ratio Weight} (kie); + \draw[-Stealth, thick] (kie) -- node[right, formula] {Supply-Demand Alignment} (eff); + \draw[-Stealth, thick] (eff) -- (score); + \draw[-Stealth, thick] (score) -- (gini); + \draw[-Stealth, thick] (gini) -- (judge); + + % 回馈线 + \draw[-Stealth, thick] (judge.north) -- ++(0, 1.5) -| node[pos=0.2, above, formula] {Iterative Optimization (maximize total score)} (kbase.west); + \draw[-Stealth, thick] (judge.south) -- ++(0, -0.5) -- ++(8, 0) |- node[pos=0.2, below, formula] {Final $k_{i_e}$ Allocation} (kie.east); + + % --- 绘制背景框 (模仿原图的 River/Ocean 分区) --- + \begin{scope}[on background layer] + % 公平分配区 + \node[container, draw=green!60!black, label={[green!60!black]above:\textbf{Tier 1: Dual-Fairness Allocation}}] + (fairbox) [fit=(ntotal) (kbase) (kie)] {}; + + % 效能与验证区 + \node[container, draw=blue!60!black, label={[blue!60!black]below:\textbf{Tier 2: Effectiveness \& Equity Validation}}] + (effbox) [fit=(eff) (score) (gini) (judge)] {}; + \end{scope} + +\end{tikzpicture} + +\end{document} \ No newline at end of file diff --git a/vscode/reference.bib b/vscode/reference.bib new file mode 100755 index 0000000..f15683a --- /dev/null +++ b/vscode/reference.bib @@ -0,0 +1,118 @@ +@article{FAN2021100501, +title = {Food system resilience and COVID-19 – Lessons from the Asian experience}, +journal = {Global Food Security}, +volume = {28}, +pages = {100501}, +year = {2021}, +issn = {2211-9124}, +author = {Shenggen Fan and Paul Teng and Ping Chew and Geoffry Smith and Les Copeland}, +keywords = {Food system resilience, Food investments, Supply chain, COVID-19 pandemic, Asia, Social and economic impacts}, +abstract = {The impact of the COVID-19 pandemic on the food system has exposed the vulnerabilities of the supply chain, although the extent of disruption varies widely, globally and in Asia. However, food systems in Asia have been proven relatively resilient when compared with other regions. This paper considers the immediate effects of the COVID-19 pandemic on the food system, particularly in Asia, and initial responses of governments and global agencies to manage the crisis. A major focus of the paper is on the outlook for food system resilience in a post-COVID-19 environment and likely long-term effects of the pandemic. There is always a possibility of such shock events occurring in the future, hence it seems prudent to look at lessons that may be learned from the responses to the current pandemic.} +} +@misc{firouz2021equityefficiencytradeofffoodbanknetwork, + title={On the Equity-Efficiency Trade-off in Food-Bank Network Operations}, + author={Mohammad Firouz and Linda Li and Daizy Ahmed and Barry Cobb and Feibo Shao}, + year={2021}, + archivePrefix={arXiv}, + primaryClass={math.OC}, + +} + +@misc{tang2025contextualbudgetbanditfood, + title={Contextual Budget Bandit for Food Rescue Volunteer Engagement}, + author={Ariana Tang and Naveen Raman and Fei Fang and Zheyuan Ryan Shi}, + year={2025}, + archivePrefix={arXiv}, + primaryClass={cs.LG}, + +} + +@misc{natanzi2026fairshareauditablegeographicfairness, + title={FairShare: Auditable Geographic Fairness for Multi-Operator LEO Spectrum Sharing}, + author={Seyed Bagher Hashemi Natanzi and Hossein Mohammadi and Vuk Marojevic and Bo Tang}, + year={2026}, + archivePrefix={arXiv}, + primaryClass={cs.NI}, + +} + + + + + + +@article{kim2006, + author = {Gi-Beum Kim}, + title = {Change of the Warm Water Temperature for the Development of Smart Healthecare Bathing System}, + journal = {Hwahak konghak}, + year = {2006}, + volume = {44}, + number = {3}, + pages = {270-276} +} + +@article{holman2002, + author = {Holman, J. P.}, + title = {Heat Transfer (9th ed.)}, + journal = {McGraw-Hill}, + year = {2002}, + address = {New York} +} + +@book{anderson2006, + author = {Anderson, D. A. and Tannehill, J. C. and Pletcher, R. H.}, + title = {Computational Fluid Dynamics and Heat Transfer}, + publisher = {Taylor \& Francis}, + year = {2006}, + edition = {2nd} +} + +@article{incropera2007, + author = {Incropera, F. P. and DeWitt, D. P. and Bergman, T. L. and Lavine, A. S.}, + title = {Fundamentals of Heat and Mass Transfer}, + journal = {John Wiley \& Sons}, + year = {2007}, + edition = {6th} +} + +@article{evaporation2018, + author = {Smith, J.}, + title = {Evaporation Effects in Bath Water Temperature Regulation}, + journal = {Journal of Thermal Science and Engineering}, + year = {2018}, + volume = {30}, + number = {4}, + pages = {456-467} +} + +@phdthesis{thesis2015, + author = {Brown, D.}, + title = {Thermal Modeling of Bath Water Systems}, + school = {University of Technology}, + year = {2015}, + address = {City} +} + +@misc{website2024, + author = {Thompson, H.}, + title = {Advanced CFD Techniques for Bath Water Simulation}, + note = {Accessed: 2024-10-01} +} + +@patent{patent2023, + author = {Wilson, G.}, + title = {Smart Bath Water Temperature Control System}, + number = {1234567}, + year = {2023}, + type = {US Patent}, + assignee = {SmartTech Inc.} +} + +@book{Liu02, + author = {Liu, Weiguo and Chen, Zhaoping and Zhang, Yin}, + title = {Matlab program design and application}, + publisher = {Higher education press}, + address = {Beijing}, + year = {2002}, + note = {In Chinese} +} diff --git a/vscode/run.R b/vscode/run.R new file mode 100755 index 0000000..a3750dd --- /dev/null +++ b/vscode/run.R @@ -0,0 +1,3 @@ +library(tinytex) + +pdflatex("mcmthesis-demo.tex", bib_engine = "biber") diff --git a/vscode/table2latex.txt b/vscode/table2latex.txt new file mode 100755 index 0000000..d0491bb --- /dev/null +++ b/vscode/table2latex.txt @@ -0,0 +1 @@ +http://www.tablesgenerator.com/ \ No newline at end of file diff --git a/vscode/中文初稿.md b/vscode/中文初稿.md new file mode 100755 index 0000000..b804b85 --- /dev/null +++ b/vscode/中文初稿.md @@ -0,0 +1,388 @@ +# 1 Introduction +## 1.1 Background +南部地区食品银行（FBST）的移动食品储藏室（MFP）项目，是纽约州六县经济困难群体获取营养食品的重要保障，2019年已实现70个常规站点、722次年度服务的成熟规模。肺炎疫情对全球食品体系的冲击暴露了供应链与服务体系的脆弱性，尽管不同地区抗冲击能力存在差异，但 FBST 的 MFP 项目仍因疫情导致服务范围大幅收缩、服务模式被迫调整。随着疫情防控形势趋稳，FBST计划2021年恢复疫情前服务水平，取消提前登记要求。如何基于2019年历史数据，制定兼顾需求匹配、公平性与实操性的站点访问调度方案，同时适配天气因素影响与志愿者工作优化需求，成为保障食品援助高效落地、切实满足群众需求的关键问题。 + +1 +The Mobile Food Pantry (MFP) program, operated by the Food Bank of the Southern Tier (FBST), serves as a critical lifeline for food-insecure populations across six counties in New York State. By 2019, the program had achieved significant operational maturity, maintaining a network of 70 regular sites and conducting 722 annual distributions. However, the global shock of the COVID-19 pandemic exposed profound vulnerabilities within food supply chains and service delivery frameworks. While regional resilience varied, the pandemic forced the FBST to significantly contract its service coverage and overhaul its operational models. + +As the public health situation stabilizes, FBST aims to restore its service capacity to pre-pandemic levels in 2021, including the removal of pre-registration requirements. The central challenge now lies in leveraging 2019 historical data to design a robust site-visit scheduling scheme. This scheme must achieve a delicate balance between demand alignment, social equity, and operational feasibility, while simultaneously accounting for the stochastic nature of weather conditions and the optimization of volunteer resources. Addressing these complexities is essential for ensuring the efficient delivery of food assistance and effectively meeting the heightened needs of the community. + +## 1.2 Restatement of the problem + +考虑到问题陈述中明确的背景信息与限制条件，我们需要解决以下问题： + +1. 依据70个常规站点周边社区总需求，制定有效且公平的2021年访问时间表，平均服务所有客户并避免服务差异过大； +2. 基于寒冷冬日客户到访量低、天气好转后附近站点需求激增的历史数据，从 “减少服务站点总数并优化位置”“保持站点数量和位置不变调整访问时间” 中择一，修改原有调度方法，并量化性能改进； +3. 针对同一卡车单次行程能访问两个不同站点的新选项，设计算法选择站点、确定日期及首个站点食物分发量，描述分配的有效性和公平性； +4. 撰写1页执行摘要，阐述所提建议的主要优势和潜在缺点。 +Based on the context provided and the requirements for the MCM/ICM competition, here is the professional translation of the "Problem Restatement" section: + +### 1.2 Problem Restatement + + + +1.2 Problem Restatement +Considering the background information and constraints specified in the problem statement, we are tasked with addressing the following four objectives: + +Develop an effective and fair 2021 visitation schedule based on the total community demand surrounding the 70 regular sites, ensuring that all clients are served on average and significant service disparities are minimized. + +Modify the existing scheduling approach by selecting one of two strategies： 'reducing serviced sites while optimizing locations' or 'maintaining current sites while adjusting visitation timing', based on historical weather-driven demand fluctuations + +Design an algorithm for the one-truck-two-sites operational model, determining site pairings, scheduling dates, and initial site food distribution volume, and evaluate the effectiveness and equity of the distribution + +Compose a one-page executive summary that articulates the primary advantages and potential limitations of the proposed recommendations for the Food Bank's leadership. + + +## 1.3 Literature review +## 1.4 Our work + +# 4.基于概率修正与公平性度量的时空调度优化模型 +## 4.1 Model Overview +为了统筹考虑站点需求的不确定性与物流资源的有限性，本文构建了基于概率修正与公平性度量的时空调度优化模型 (Probability-Corrected Proportional Allocation and Spatio-Temporal Scheduling Model, PPA-STS)。该模型由三个相互衔接的逻辑层组成： + +需求精炼层 (Demand Refinement Layer)：针对原始需求的波动性，利用正态分布与截断概率对均值需求进行线性修正，识别并对冲超承载风险。 +策略分配层 (Strategic Allocation Layer)：引入哈密尔顿法（最大余数法），将连续的需求比例科学地转化为离散的访问频次配额，确保分配过程的公平性与完整性。 +时空调度层 (Spatio-Temporal Optimization Layer)：以访问间隔的均匀性为目标函数，构建约束编程（CP）模型，在满足车辆运力与服务周期约束的前提下，实现 365 天访问计划的精确排布。 + + +## 4.2 Model Building + +### 4.2.1 随机性需求修正机制 + +题目仅提供 2019 年各站点的平均需求及方差，而实际需求存在波动，如部分站点可能出现需求峰值超过卡车最大服务量的情况，仅用平均需求会导致访问次数分配偏差。因此，本环节需通过概率建模修正需求，得到更贴合实际的真实需求。 + +step1.假设需求概率分布 + +基于统计规律，假设每个站点的需求服从正态分布，即： +$$ +d_{i}\sim N(\tilde{\mu_{i}},\sigma_{i}^{2} ) +$$ + +其中，di​为第i个站点的需求；$\mu_{i}$​为第i个站点的平均需求；$\sigma_{i}^{2} $​为第i个站点的需求方差。 + +step2.计算截断概率 + +定义截断概率为：真实需求超过最大容量$d_{max}$的概率，用于判断需求是否存在超承载风险 +$$ +p_{i}^{trunc}=p(\tilde{d_{i}}>d_{max})=p(Z>\frac{d_{max}-\tilde{\mu_{i}}}{\sigma_{i}})=1-\Phi(\frac{d_{max}-\tilde{\mu_{i}}}{\sigma_{i}}) +$$ +其中$p_{i}^{trunc}$为截断概率，阈值设定为0.2 + +step3.需求线性修正 +对截断概率超过阈值的站点，通过线性修正公式调整需求，避免因需求峰值导致服务不足： +$$ +\tilde{d_{i}}=d_{i}\cdot(1+\alpha p_{i}^{trunc}) +$$ +其中$\alpha$为经验系数 + +step4：输出修正结果 +最终筛选出截断概率超过 0.2 的站点，完成需求修正，后续所有计算均以修正后需求为依据。 + +### 4.2.2 基于哈密尔顿法的访问频次分配 + +在总访问次数固定的前提下，对于访问次数的分配应该按照需求进行比例原则分配以更更好满足高需求站点。本文采用哈密尔顿法进行分配，分配目标为最大化服务总量且保证公平。其具体步骤如下： + +1.计算基础配额 +$$ +k^{i}_{base}=\frac{70\tilde{d_{i}}}{\sum_{i}\tilde d_{i}} +$$ +2.拆分整数和余数 +$$ +k^{i}_{1}=[k_{base}^{i}] +$$ +$$ +r_{i}=k^{i}_{base}-k^{i}_{1} +$$ +3.对余数进行降序排列，将剩余配额按照顺序分配，得到最终的分配次数公式为 +$$ +k_{i}=\left\{\begin{matrix} + [k_{base}^{i}]+1\\ + [k_{base}^{i}] +\end{matrix}\right. +$$ + + +### 4.2.3 多维评估指标体系 +对分配后的访问次数方案进行量化评估，验证是否满足有效与公平的目标，为方案优化提供依据。 + +### 有效性 + +有效性的核心定义是对所有客户的平均服务水平，本质是判断方案能否最大化覆盖真实需求，同时贴合实际运营约束。本模块设计原始总服务量与 加权总服务量 双指标，既反映服务规模，又修正约束偏差，确保评估结果真实可靠。 + +1.原始总服务量 +$$ +E_{1}=\sum_{i}k_{i}\cdot\tilde{d_{i}} +$$ +2.加权总服务量 +$$ +E_{2}=\sum_{i}q(\tilde{d_{i}})\cdot k_{i}\cdot\tilde{d_{i}} +$$ +其中 +$q(\tilde{d_{i}})=\min(1,\frac{250}{\tilde{d_{i}}})$ +### 公平性 + +公平性的核心定义是避免部分客户得到远优于他人的服务，本质是判断各站点的需求满足程度是否均衡。本模块通过满意度 与满意度基尼系数的组合，实现从 “单个站点满足度” 到 “整体均衡性” 的分层评估。 + +1.满意度 +$$ +SL_{i}=\frac{k_{i}\cdot C_{0}}{\tilde{d_{i}}} +$$ +2.基尼系数 +$$ +G=\frac{\sum_{i}\sum_{j}\left | SL_{i}-SL_{j} \right |}{2\cdot70^{2}\cdot\bar{SL}} +$$ + +基尼系数G的取值范围为[0,1]，越接近 0 表示站点间服务效能越均衡。 + +### 4.2.4 时空调度组合优化模型 + +经分析，本模型实质为单目标组合优化问题。模型核心目标是使各站点的年度访问需求在 365 天内尽可能均匀分布，提升服务连续性与客户体验。具体构建过程如下： + +- 决策变量 + + 1.第$i$个站点第$m$次运输的时间$s_{i, m}$， + + 2.第$i$个站点第$t$天是否被访问 + $$ + a_{i,t}=\left\{\begin{matrix} + 1&t\in S_{i}\\ + 0&t\notin S_{i} + \end{matrix}\right. + $$ + +- 目标函数 +以所有站点实际访问间隔与理想访问间隔的绝对偏差总和最小为目标，量化访问时间分布的均匀性，目标函数如下： + $$\min Z = \sum_{i=1}^{70} \sum_{m=1}^{k_i - 1} \left| (s_{i, m+1} - s_{i, m}) - T_{i} \right|$$ + +- 约束条件 +为保证模型解的可行性与合理性，结合项目运营实际设定以下约束： + + 1.每日的访问总次数不得超过最大运输次数 + $$ + \sum_{i}a_{i,t}\le2 + $$ + 2.所有站点必须达到规定访问次数 + $$ + \sum_{t}a_{i,t}=k_{i} + $$ + 3.相邻两次访问不得小于默认间隔 + $$ + s_{i, m+1} - s_{i, m}\ge t_{0} + $$ +结合实际运营数据验证，当卡车运载食物全部分配给200户家庭时，每户可获得75磅食物，足以支撑14天的需求，因此设定同一站点相邻两次访问的最小间隔 $t_0=14$ 天。 + + + +## 4.3 Model Solution + +本模型属于大规模组合优化问题。传统的混合整数线性规划 (MILP) 在处理 70 个站点的一年期排班时容易陷入维度灾难。因此，本文采用 Google OR-Tools 中的 CP-SAT 求解器进行求解。 +该算法基于懒惰子句学习机制，具有以下优势： + +- 高效剪枝：能快速排除不满足 14 天最小间隔及每日运输力约束的无效解空间。 +- 冲突驱动搜索：在全局优化目标间隔均匀性与硬约束运力限制之间快速寻找平衡。 +- 解质量保障：相比启发式算法，CP-SAT 能在更短时间内给出更接近全局最优的精确解。 + +## 4.4 Results of Task 1 + +# 5.基于时空共生机理的双站点协同调度优化模型 +## 5.1 Model Overview + +在模型 I 的基础上，为了进一步提升配送效率并挖掘运输潜能，本章打破一车一站的限制，提出共生站点（Symbiotic Sites）策略。共生站点是指满足空间、需求、稳定性三个维度要求，能由同一配送车辆单次班次顺序服务的两个站点组合。 + +该模型的核心逻辑分为三个阶段： + +- 共生站点筛选阶段：基于地理可达性、需求加总限制及时间稳定性指标，筛选可由同一班次顺序服务的站点对。 +- 内部配额优化阶段：针对共生站点对，通过构建效用函数，确定两站间的最优货物分配比例，以平衡服务有效性与公平性。 +- 全局动态调度阶段：将共生站点视作单一复合决策单元，整合进改进的调度算法中，生成最优配送时间表。 + +## 5.2 Model Building + + +### 5.2.1 共生站点筛选准则 + +为保障共生站点策略的实际操作性与科学性，本文构建三维量化筛选体系，从空间、需求、稳定性三个维度严格界定共生站点对的准入条件。 + +1.空间邻近性 + +由数据集可知各站点经纬度，为筛选小范围地域内的站点，采用修正曼哈顿距离$l_{ij}​$量化两站点 i,j 的地理关联度，该公式适配经纬度坐标的球面投影特性，计算公式如下： +$$ +l_{ij}=69.0\cdot \left | lat_{i}-lat_{j} \right | +69.0\cdot \cos(\frac{lon_{i}+lon_{j}}{2})\left | lon_{i}-lon_{j} \right | +$$ +其中，系数 69.0 为经纬度差值转换为地面距离的英里常数. + +2.需求叠加适度性 + +设站点 i,j 的服务需求为 {d_i}​,{d_j}​，配送卡车最大载荷为$Q_{max}​=250$ 户。为保留调度优化余量，设定联合需求上限450​，共生站点对需满足： +$$ +d_{ij}=d_{i}+d_{j}\le 450 +$$ + + +3.需求稳定性 +对于接近卡车最大限制的配对组合，本文进一步考虑其需求稳定性，删去波动大的配对以防止后续共生站点的分配不均。其判别公式为 +$$ + \frac{\sigma_{i}}{d_{i}},\frac{\sigma_{j}}{d_{j}}\ge0.5 + $$ + + +### 5.2.2 共生内部资源分配模型 + +对于通过筛选的共生站点对 {i,j}，需优化两站间的货物分配策略，定义虚拟分配量 qi​ 为站点 i 的预分配货物数量，基于模型一构建综合效用评分函数以求解最优配额: + +1、第一站点的实际分配货物 +$$ +g_{i}=\min(d_{i},q_{i}) +$$ + +2、第二站点的实际分配货物 +$$ +g_{j}=\min(d_{j},d_{0}-g_{i}) +$$ + +3、效用最大化目标函数 + +- 目标1：最大化期望服务总量 +$$ +\max E(g_{i}+g_{j}) +$$ +- 目标2：站点间的分配公平性 +$$ +\min \max(0,\tilde{d_{i}}-q_{i})+\max(0,\tilde{d_{j}}-(\tilde{d_{i}}-q_{i}))=\min U_{i}+U_{j} +$$ +为平衡服务有效性与分配公平性，将上述双目标融合为统一的分配有效得分函数： +$$ +score_{ij}^{A}=E(g_{i}+g_{j})-\lambda E(U_{i}+U_{j})-\mu E(\max(U_{i},U_{j})) +$$ + +选取分配有效得分最大的$q_{i}$作为第一站点的分配货物数量。 + + +### 5.2.3 适配共生站点的全局调度模型 + +对于访问次数的求解，与模型一一致，仅需适配站点总数减少、共生站点需求合并的场景，本文不再赘述. + +对于访问时间分配的求解，本文基于模型一重构全局调度模型的决策变量、目标函数与约束条件，实现独立站点与共生站点的协同调度。 + +决策变量 +$$ +\begin{itemize} + \item $a_{i,t} \in \{0,1\}$：独立站点 $i$ 在第 $t$ 天是否被访问，$1$ 表示访问，$0$ 表示未访问； + \item $a_{x,t} \in \{0,1\}$：共生站点 $x$ 在第 $t$ 天是否被访问； + \item $s_{i,m}$：独立站点 $i$ 第 $m$ 次访问的日期； + \item $s_{x,y}$：共生站点 $x$ 第 $y$ 次访问的日期。 +\end{itemize} +$$ +- 第$i$个单独站点第$m$次运输的时间$s_{i, m}$ +- 第$x$个共生站点第$y$次运输时间$s_{x,y}$ +- 第$i$个单独站点第$t$天是否被访问 +$$ +a_{i,t}=\left\{\begin{matrix} + 1&t\in S_{i}\\ + 0&t\notin S_{i} +\end{matrix}\right. +$$ +- 第$x$个共生站点第$t$天是否被访问 +$$ +a_{x,t}=\left\{\begin{matrix} + 1&t\in S_{x}\\ + 0&t\notin S_{x} +\end{matrix}\right. +$$ +目标函数： + +为了表示各站点访问时间的均匀分布，本文定义目标函数为所有站点实际时间间隔与理想时间间隔的差值的绝对值之和 +$$ +\min Z = \sum_{i} \sum_{m=1}^{k_{i}} \left| (s_{i, m+1} - s_{i, m}) - T_{i} \right|+ \sum_{x} \sum_{y=1}^{k_{x}} \left| (s_{i, y+1} - s_{i, y}) - T_{i} \right| +$$ +约束条件 + +1.每日的访问总次数不得超过最大运输能力 +$$ +\sum_{i}a_{i,t}+\sum_{x}a_{x,t}\le2 +$$ +2.所有站点必须达到规定访问次数 +$$ +\sum_{t}a_{i,t}=k_{i},\sum_{t}a_{x,t}=k_{x} +$$ +3.相邻两次访问不得小于默认间隔 +$$ +s_{i, m+1} - s_{i, m}\ge t_{0},s_{x, m+1} - s_{x, m}\ge t_{0} +$$ + +### 5.2.4 共生站点下的指标评估体系 +### 有效性 +1.期望总服务量 +$$ +E_{1}^{'}=\sum_{i}k_{i}\cdot \tilde{d_{i}}+\sum_{x}k_{x}\cdot \tilde{d_{x}} +$$ +2.加权总服务量 +$$ +E_{2}^{'}=\sum_{i}q(\tilde{d_{i}})\cdot k_{i}\cdot\tilde{d_{i}}+\sum_{x}q(\tilde{d_{x}})\cdot k_{x}\cdot\tilde{d_{x}} +$$ +### 公平性 +1.满意度 +$$ +SL_{i}=\frac{ C_{0}}{\tilde{d_{i}}} +$$ +$$ +SL_{x}=\frac{ C_{0}}{\tilde{d_{x}}} +$$ +2.基尼系数 +$$ +G^{'}=\frac{\sum_{i}\sum_{j}\left | SL_{i}-SL_{j} \right |+\sum_{x}\sum_{z}\left | SL_{x}-SL_{z} \right |}{2\cdot70^{2}\cdot\bar{SL}} +$$ +3.服务缺口风险 +$$ +P_{x}=P(\frac{g_{i}}{d_{i}}<0.8)+P(\frac{g_{j}}{d_{j}}<0.8)-P(\frac{g_{i}}{d_{i}},\frac{g_{j}}{d_{j}}<0.8) +$$ + +## 5.4 Model Solution +### 5.4.1 共生站点筛选求解 +通过上述量化筛选公式，本文利用贪心思想，以站点配对得分为评估指标，具体公式如下： +$$ +score_{ij}^{P}=\frac{\tilde{d_{i}}+\tilde{d_{j}}}{d_{0}}-\beta\frac{l_{ij}}{l_{0}}-\gamma\frac{\sigma^{2}_{i}+\sigma^{2}_{i}}{(\tilde{d_{i}}+\tilde{d_{i}})^{2}} +$$ +选取得分高的候选站点组合优先配对，且各站点只能配对一次，最终得到共生站点如下表： + +### 5.4.2 共生内部资源分配求解 +最终我们得出各共生站点的分配方案如下表 + + + +## 5.5 Results of task 3 + + +| 符号 | 含义与说明 | 学术性英文术语 | +| :--- | :--- | :--- | +| **基础参数** | | | +| $i$ | 70个常规服务站点的索引 ($i=1, 2, \dots, 70$) [1] | Site Index | +| $n_i$ | 站点 $i$ 在2019年的历史访问次数 [1] | Historical Visit Frequency | +| $d_i$ | 站点 $i$ 的历史单次平均需求量（服务客户数） [1] | Average Demand per Visit | +| $D_i$ | 站点 $i$ 的年度总需求量 ($D_i = n_i \times d_i$) | Annual Aggregate Demand | +| $d_0$ | 单辆卡车单次最大服务能力（250户家庭） | Maximum Vehicle Capacity | +| $N_{total}$ | 2021年全年的总计划访问次数（365天×2次/天=730次） [1] | Total Annual Service Capacity | +| $t_0$ | 同一站点两次访问间的最小安全间隔（14天，基于食物支撑周期） | Minimum Revisit Interval | +| $\alpha, \beta$ | 供应不足与资源浪费的惩罚权重系数 | Penalty Weighting Coefficients | +| **决策变量** | | | +| $k_{base}$ | 每个站点每年至少获得的保障性底线访问次数 | Baseline Service Guarantee / Minimum Frequency | +| $k_{i_e}$ | 分配给站点 $i$ 的最终年度访问频次 | Allocated Annual Visit Frequency | +| $s_{i, m}$ | 站点 $i$ 第 $m$ 次被访问的具体日期（天数） | Scheduled Service Date | +| $a_{i, t}$ | 0-1 决策变量：站点 $i$ 在第 $t$ 天是否被访问 | Binary Assignment Variable | +| **性能与公平指标** | | | +| $N_{free}$ | 扣除基础保障后的剩余可支配访问额度 | Residual Allocation Capacity | +| $annual\_eff_i$ | 站点 $i$ 的年度有效食品供应总量 | Total Annual Effective Supply | +| $unmet_i$ | 站点 $i$ 的需求未满足率（缺货惩罚项） | Unmet Demand Rate (Shortage Penalty) | +| $waste_i$ | 站点 $i$ 的资源浪费率（过剩惩罚项） | Resource Wastage Rate (Surplus Penalty) | +| $score_i$ | 衡量站点服务质量的综合有效性评分 | Effectiveness Utility Score | +| $G$ | 衡量不同站点间服务效能均衡程度的基尼系数 | Gini Coefficient (Equity Metric) | +| $T_i$ | 站点 $i$ 的理想均匀访问间隔周期 ($365 / k_{i_e}$) | Ideal Recurrence Interval | + + + + diff --git a/vscode/指令.md b/vscode/指令.md new file mode 100755 index 0000000..3c6cc7b --- /dev/null +++ b/vscode/指令.md @@ -0,0 +1,233 @@ +# 摘要 + +请将以下中文摘要翻译成符合MCM/ICM规范的英文学术摘要，严格满足： +1. 核心要求： + - 完整覆盖“研究目的→模型构建→核心任务→关键结论”四要素，不增删语义； + - 逻辑链清晰：如采用“To address..., we develop..., simulate..., and conclude that...”的句式； + - 突出创新性：明确模型/方法的独特性。 +2. 术语规范： + - 模型名称： + + - 核心术语： + + - 缩写规则：首次出现术语标注全称+缩写（如“Finite Volume Method (FVM)”），后文直接用缩写。 +3. 句式风格： + - 多用学术动词（如establish/develop/apply/validate/quantify），避免口语化词汇（如make/build/think）； + - 平衡主动与被动语态：表述研究动作用主动（如We propose...），强调结果用被动（如The model is validated...）； + - 避免长句堆砌，用分词结构（如Based on.../Considering...）衔接逻辑。 + + +中文初稿：[粘贴中文摘要文本] + +# 引言 +## 问题背景 +请将以下中文“问题背景”翻译成符合MCM/ICM规范的英文子章节，严格满足： +1. 核心要求： + - 完整传递“现实场景+生态问题+研究意义”，不遗漏关键数据（如种群数量、防治成效）和背景信息（如七鳃鳗生态角色）； + - 贴合美赛语境：突出“实际问题驱动建模”，强调研究的应用价值（如“for Great Lakes fishery protection”）。 +2. 术语规范： + - 专有名词：五大湖→ Great Lakes；海七鳃鳗→ sea lamprey；入侵物种→ invasive species；渔业→ fishery；食物链→ food chain；底栖环境→ benthic environment； + - 数据表述：“减少90%以上”→ more than 90% reduction；“每年捕杀1.03亿磅鱼类”→ 103 million pounds of fish killed per year；“现存约1000万条”→ approximately 10 million today； + - 生态术语：寄生→ parasitic；滤食性→ filter-feeding；性别比例→ sex ratio；生态角色→ ecological role。 +3. 句式风格： + - 以一般现在时为主，描述客观事实（如“Sea lampreys play a complex role in ecosystems”）； + - 用数据支撑背景，句式简洁有力，避免冗余（如“Prior to control efforts, sea lampreys caused massive fish mortality”）； + - 避免中式套话，直接切入核心问题（如“ Their ability to adjust sex ratio in response to environment makes them a key research focus”）。 + + +中文初稿：[粘贴中文“问题背景”文本] +## 问题重述 +请将以下中文“问题重述”翻译成符合MCM/ICM规范的英文子章节，严格满足： +1. 核心要求： + - 精准复刻原题要求，分点明确4个核心研究任务，不增删、不改写任务内涵； + - 适配美赛评审：每个任务突出“建模目标+分析对象”，避免模糊表述，任务导向性强。 +2. 术语规范： + - 建模相关：建立数学模型→ construct a mathematical model；描述生长繁殖情况→ describe the growth and reproduction conditions；扩展模型→ expand the model；分析影响→ analyze the impact；比较优劣→ compare the advantages and disadvantages； + - 核心对象：性别比例恒定→ constant sex ratio；性别比例可变→ variable sex ratio；生态系统稳定性→ ecosystem stability；其他物种→ other species；适应性性别分化→ adaptive sex ratio variation。 +3. 句式风格： + - 分点用“• ”引导，每点以动词原形开头，如“• Construct a mathematical model to...”“• Analyze the impact of...on...”； + - 句式简洁明了，无复杂修饰，无复杂嵌套，确保评审快速抓取核心任务； + - 保持各点句式一致，逻辑平行（如均为“动作+对象+目的”结构）。 + + +中文初稿：[粘贴中文“问题重述”文本] + +## 文献综述 +请将以下中文“文献综述”翻译成符合MCM/ICM规范的英文子章节，严格满足： +1. 核心要求： + - 完整传递“现有研究方法→ 方法局限性→ 本研究切入点”，逻辑闭环，突出研究创新性； + - 适配美赛评审：明确引用文献支撑观点，说明本研究方法的合理性。 +2. 术语规范： + - 研究方法：分子生物学方法→ molecular biology approach；种群统计法→ population statistics；性选择博弈论→ sexual selection game theory；生态理论→ ecological theory；数学建模→ mathematical modeling； + - 表述规范，例如：“现有研究不足”→ Existing studies lack...；“不适用于...”→ is not suitable for...；“本研究采用...方法”→ This study adopts the...method；“结合...与...”→ integrate...with...； + - 文献引用，例如：“据文献[X]报道”→ According to literature [X]；“研究表明”→ Studies have demonstrated that。 +3. 句式风格： + - 以现在完成时为主（描述现有研究成果），如“Researchers have proposed three main methods for gender differentiation”； + - 用“However/Therefore”等逻辑连接词，明确“方法→ 局限→ 切入点”的逻辑链； + - 避免主观评价，客观陈述现有研究不足（如“Neither method can describe lamprey populations interacting with the ecosystem”）。 + + +中文初稿：[粘贴中文“文献综述”文本] + +## 研究框架（多用图） +请将以下中文“研究框架”翻译成符合MCM/ICM规范的英文子章节，严格满足： +1. 核心要求： + - 清晰呈现“研究思路+技术路线+核心内容”，让评审快速理解研究整体设计； + - 适配美赛语境：突出模型、方法、分析维度的关联性，体现研究逻辑性。 +2. 术语规范（美赛通用标准）： + - 模型与方法：逻辑斯蒂模型→ Logistic Model；BFM模型→ Bioenergetic Fish Model (BFM)；反馈模型→ Feedback Model；敏感性分析→ Sensitivity Analysis；有限体积法→ Finite Volume Method (FVM)； + - 核心模块：性别分化→ sex differentiation；年龄结构→ age structure；种间关系→ interspecific relationship；生态系统稳定性→ ecosystem stability；生物多样性→ biodiversity； + - 缩写规则：首次出现术语标注全称+缩写（如“Lotka-Volterra Model (LVM)”），后文直接用缩写。 +3. 句式风格： + - 以主动语态为主（突出研究动作），如“We integrate the Logistic Model with interspecific relationship analysis”； + - 用“First/Second/Finally”或“For.../In terms of...”分层描述，逻辑清晰； + - 简洁明了，避免长句，每部分核心内容控制在1-2句话。 +4. 格式要求： + - 子章节标题：译为“1.4 Our Work”，实词首字母大写，虚词小写（句首除外），黑体呈现； + - 技术路线图：图题译为“Figure X: Overview of This Work”，图内模块标注与正文术语一致（英文短语，实词首字母大写）； + - 排版：无首行缩进，段落间距1.5倍，若分点描述用“• ”引导，保持格式统一。 + +中文初稿：[粘贴中文“研究框架”文本] + +# 模型准备 +## 假设与依据 +请将以下中文“假设与依据”翻译成符合MCM/ICM规范的英文子章节，严格满足： +1. 核心要求： + - 完整保留“假设内容+科学依据”的对应关系，不增删逻辑限定（如“瞬时完成”“忽略影响”），确保假设的合理性与可追溯性； + +2. 术语规范（美赛通用标准）： + - 生物学核心词汇：生命周期→ life cycle；幼体期→ juvenile stage；成年寄生期→ adult parasitic stage；性别分化→ sex differentiation；繁殖→ reproduction；滤食性→ filter-feeding；有机碎屑→ organic detritus；种群→ population； + - 假设相关表述：假设→ Assumption；依据→ Justification；忽略→ neglect；简化→ simplify；恒定→ constant；仅与…相关→ be only related to； + - 文献引用，例如：“据文献[X]”→ According to literature [X]；“研究表明”→ Studies have shown that；“基于…特性”→ Based on the characteristics of…。 +3. 句式风格： + - 假设表述：用“The…is assumed to…”或“Assume that…”等被动语态，强调客观性，如“Juvenile lampreys are assumed to be gender-neutral”； + - 依据表述：用陈述句，逻辑清晰，如“Compared to juvenile and adult phases, other life stages have relatively short durations”； + - 避免口语化和模糊词汇（如“大概”“可能”），表述精准严谨。 + +中文初稿：[粘贴中文“假设与依据”文本] +## 符号 +请将以下中文“符号说明”翻译成符合MCM/ICM规范的英文子章节，严格满足： +1. 核心要求： + - 符号与释义一一对应，无歧义，全文术语统一（与后续模型构建、结果分析章节完全一致）； + - 适配美赛评审：符号格式规范，释义简洁精准，便于评审快速查阅。 +2. 术语规范（美赛通用标准）： + - 释义表述：用名词短语，简洁明了，避免完整句子，如“幼体七鳃鳗种群数量”→ Juvenile lamprey population（不用“The number of juvenile lamprey population”）； + - 单位相关：若涉及单位，保留原单位英文表述（如“months”“individuals”），不额外添加中文单位。 +3. 句式风格： + - 符号表内仅用名词或名词短语，无动词、无完整句子； + - 释义准确对应符号含义，不添加额外解释 + + +中文初稿：[粘贴中文“符号说明”文本（含表格）] +# 模型构建（再议） +请将以下中文模型构建章节翻译成符合MCM/ICM规范的英文章节，严格满足： +1. 核心要求： + - 数学逻辑精准：公式推导步骤清晰，参数说明完整，模型子模块划分明确，适配美赛“可复现性”评审标准； + - 突出模型创新性：明确模型改进点（如“improved Logistic Model”“sex-separated Lotka-Volterra Model”）。 +2. 术语规范（美赛通用标准）： + - 模型与方法：逻辑斯蒂增长模型→ Logistic Growth Model；洛特卡-沃尔泰拉方程→ Lotka-Volterra Equations；偏微分方程→ Partial Differential Equation (PDE)；有限体积法→ Finite Volume Method (FVM)；尼科尔森-贝利模型→ Nicholson-Bailey Model； + - 推导词汇：推导→ derive；离散化→ discretize；求解→ solve；整合→ integrate；参数化→ parameterize； + - 参数符号：与“模型准备”章节完全一致，不新增或修改释义（美赛要求全文符号统一）。 +3. 句式风格： + - 公式引导句：“该方程可改写为：”→ The equation is rewritten as:；“通过数据拟合得到参数：”→ Parameters are obtained by data fitting:，衔接自然； + - 拆分中文长句，用从句（定语/状语从句）符合英文表达习惯，避免“流水句”； + - 子模块描述用“First/Second/Finally”或“For.../In terms of...”分层，符合美赛论文逻辑呈现方式。 +4. 格式要求： + - 公式：单独成行，用LaTeX排版规范（变量斜体、常量正体），编号按章节排序（如(3.1)），正文引用标注“Equation (3.1)”或“Eq. (3.1)”（后文缩写）； + - 小节编号：保留原层级（4.1/4.2.1等），译为英文（4.2.1 Migratory Sub-model），黑体呈现； + - 模型流程图：图题译为“Figure X: Schematic of the Proposed Model”，图内模块标注与正文术语一致，箭头清晰指示逻辑关系。 + +中文初稿：[粘贴中文模型构建文本] +# 模型求解与分析（再议） +请将以下中文任务求解与结果分析章节翻译成符合MCM/ICM规范的英文章节，严格满足： +1. 核心要求： + - 数据驱动：客观传递模拟结果，对比维度（性别比例可变vs恒定/扰动环境vs适宜环境）清晰，不夸大或缩小结论； + - 适配美赛评审：突出“结果+分析+意义”，不仅呈现数据，还需说明结果的生态意义或模型价值。 +2. 术语规范（美赛通用标准）： + - 分析指标：抵抗力稳定性→ resistance stability；恢复力稳定性→ resilience stability；辛普森多样性指数→ Simpson's Diversity Index；香农-维纳指数→ Shannon-Wiener Index；波动幅度→ fluctuation level；恢复时间→ recovery time； + - 数值表述：“波动幅度为23.28%”→ with a fluctuation level of 23.28%；“恢复时间18年”→ a recovery time of 18 years；“误差在5%以内”→ the error is within 5%； + - 动作表述：模拟→ simulate；对比→ compare；验证→ verify；分析→ analyze；量化→ quantify。 +3. 句式风格： + - 结果描述用一般现在时（如“The population fluctuates cyclically”），结论用现在完成时（如“We have demonstrated that...”）； + - 多用“From Figure X, it can be observed that...”“The results indicate that...”“Compared with..., ...exhibits...”的美赛标准句式； + - 平衡长句与短句：复杂结果用复合句，核心结论用短句强调（如“Variable sex ratio enhances ecosystem stability”）。 +4. 格式要求： + - 图表：图题译为“Figure X: Population Dynamics of Lampreys under XXX Environment”（实词首字母大写），亚图标注（a）（b）对应“(a) Constant sex ratio (b) Variable sex ratio”； + - 表格：保留三线表，表头包含对比维度（如“Scenario”“Recovery Time”“Fluctuation Level”），数据对齐，编号为“Table X: Ecosystem Stability Comparison”； + - 小节标题：保留原编号（5.1/5.2等），译为英文（5.1 Advantages and Disadvantages of Variable Sex Ratio），黑体呈现； + - 引用图表时，标注位置紧邻描述（美赛要求“图随文走”）。 + +中文初稿：[粘贴中文任务求解与结果分析文本] +# 分析与评价（再议） +请将以下中文模型评价与敏感性分析章节翻译成符合MCM/ICM规范的英文章节，严格满足： +1. 核心要求： + - 客观辩证：优势（Strengths）与不足（Weaknesses）表述平衡，敏感性分析聚焦“参数变化→结果影响”，符合美赛“模型严谨性”评审标准； + - 改进建议具体：不足部分需对应可行的优化方向（如“further quantitative research on feedback coefficients”）。 +2. 术语规范（美赛通用标准）： + - 评价词汇：稳健的→ robust；可靠的→ reliable；全面的→ comprehensive；简化的→ simplified；不精确的→ imprecise； + - 敏感性相关：参数扰动→ parameter perturbation；波动→ fluctuation；扰动→ disturbance；影响程度→ impact degree； + - 动作表述：优化→ optimize；改进→ improve；探讨→ discuss；建议→ propose；控制→ control。 +3. 句式风格： + - 优势/不足分点：用“• Strengths: ...”“• Weaknesses: ...”引导，每点用完整句子，避免短语； + - 敏感性分析用“By varying the parameter X by ±5%, we find that...”“The model is highly sensitive to...”的美赛标准句式； + - 改进建议用不定式（To address this limitation, we suggest...），符合学术论文表述习惯。 + + - 无首行缩进，段落间距统一，符合美赛排版规范。 + +中文初稿：[粘贴中文模型评价与敏感性分析文本] +# 结论 +请将以下中文结论翻译成符合MCM/ICM规范的英文章节，严格满足： +1. 核心要求： + - 简洁凝练：总结核心研究成果，不重复前文细节，突出与研究任务的呼应（每点对应一个核心问题）； + - 贴合美赛语境：强调研究的实际应用价值（如“providing constructive suggestions for lamprey invasion control”）和未来拓展方向。 +2. 术语规范（美赛通用标准）： + - 核心术语与前文完全一致（如variable sex ratio/ecosystem stability），不出现新术语； + - 成果表述：增强→ enhance；促进→ promote；维持→ maintain；提供优势→ provide advantages for；为...提供参考→ provide a reference for； + - 限定词：在一定程度上→ to a certain extent；在特定条件下→ under specific conditions，避免绝对化表述（如best/perfect）。 +3. 句式风格： + - 总起句用“In conclusion/To summarize”，后文分点（• 引导）概括关键结论，用一般现在时； + - 平衡主动与被动语态：表述研究价值用主动（如Our model provides...），表述客观结论用被动（如Ecosystem stability is enhanced by...）； + - 避免长句，每点结论控制在1-2句话，逻辑闭环。 + +中文初稿：[粘贴中文结论文本] + + + + +# 通用提示词：架构重组与思路映射版 +Role: +你是一名专业的学术论文编辑，擅长将零散的数学建模初稿重新组织为结构严谨的“模型驱动型（Model-Driven）”架构。 + +Task: +请根据我提供的“思路初稿”，将其内容精确地映射到下述的标准论文框架中。 + +目标框架（参照Model-Driven结构）： + +Introduction (Background, Restatement, Our Work) +Assumptions and Justifications +Notations +Model I: [基于我初稿核心模型的命名，高级，体现思路与创新性] +4.1 Model Overview +4.2 Model Building +4.3 Model Solution +4.4 Results of ...(如task 1) +Model II: [待定主题占位] +Model Expansion / Further Discussion +Sensitivity Analysis +Strengths and Weaknesses +Conclusion +处理要求： + +整理已有内容： 请阅读我提供的“思路初稿”，将其逻辑进行提炼，并填充到上述框架对应位置。请确保用学术化的语言重组这些内容，做到切合题目，思路完整有逻辑，各小章节之间存在流畅的思路过程与关联性。 +仅展示章节名： 对于第5章及以后的部分，由于我尚未完成，请不要填充任何实质性内容，也不要替我编造逻辑，仅保留章节标题作为占位符即可。 +输出形式： +首先给出一份完整的英文目录（Table of Contents）。 +然后针对有内容（有初稿支撑）的章节展示你整理后的完整详细内容，中文。 +输入内容（我的思路初稿）： +[在此处粘贴你已写好的任务一思路/草稿] \ No newline at end of file