Dalhousie university | Usemore Soap Company | Supply Chain 5-Year Plan | | Eric Noel | 12/5/2010 |

Abstract
The Usemore soap company currently has 4 plants and 10 warehouses in its distribution network. In response to forecasted demands in 5 years, Usemore is considering building 2 new plants and 6 new warehouses. The problem is modeled as a linear program with objective to minimize cost. However, the model must be formulated as piecewise linear in order to account for the non-linear warehousing costs. The recommended course of action for Usemore is to build 1 new plant, shut down 5 of the existing public warehouses, and open 5 of the new warehouses.

Table of Contents Abstract ii 1. Introduction 1 2. Current Situation 2 2.1 Problem Statement 2 2.2 Problem Data 2 3. Proposed Model 2 3.1 Explanation 2 3.2 Formulation 2 4. Solution 4 4.1 Plants 4 4.2 Warehouses 4 4.3 Average Inbound and Outbound Distances 5 5. Discussion 5 6. Conclusion 5 7. Bibliography 5

1. Introduction
The Usemore soap company currently has 4 plants and 10 warehouses within its distribution network. In response to forecasted demands in 5 years, Usemore is considering building 2 new plants and 6 new warehouses.
The problem was solved using a linear programming approach utilizing a GLPK (GNU MathProg) formulation. In addition to this report, the following files were required to solve the problem: * Usemore.mod : GLPK model * Plants.csv : Model plants data in CSV format * Potential_plants.csv: Model potential plants data in CSV format * Warehouses.csv: Model warehouses plants data in CSV format * Potential_warehouses.csv: Model potential warehouses data in CSV format * Costing_info.csv: Model warehouse costing data in CSV format * Demand.csv: Model customer demand data in CSV format
Using glpsol or gusek, the model is then solved and generates output files: * optimal_warehouse_flows.csv: Optimal warehouse to customer flows in CSV format * optimal_plant_flows.csv: Optimal plant to warehouse flows in CSV format
The purpose of this report is to discuss the formulation of the MIP and answer the following questions: 1. How many plants and warehouses should be operated? 2. Where should they be located? 3. Which customers should be assigned to which warehouse? 4. Which warehouses should be assigned to which plants? 5. What is the average length of the inbound warehouse flows? 6. What is the average length of the outbound warehouse flows?

2. Current Situation
2.1 Problem Statement
A detailed problem statement is included in Appendix A: Usemore.doc
2.2 Problem Data
Plant and warehouse locations, various costs, customer demands and forecasts are included in Appendix B: Usemore.xls
3. Proposed Model
This section will discuss the model adapted to the problem as well as its formulation.
3.1 Explanation
The problem is modeled as a linear program with objective to minimize the cost of the distribution network. However, the model must be formulated as piecewise linear in order to account for the non-linear warehousing costs.

Figure [ 1 ] - Non-linear warehousing cost (Warehouse 1)
The function was converted into 10 linear pieces. Additionally, since there are toggle variables (to build a plant or not to build), the problem changes from an LP to an MIP (Mixed integer program).
3.2 Formulation
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The following is the actual GLPK model formulation:
# Usemore piecewise GLPK implementation (GNU MathProg)
# Written by Eric Noel
# December 5, 2010
# Dalhousie University

set P; # Set of plants (includes potentials) set PP; # Subset of potential plants set EP, default P diff PP; # Subset of existing plants set W; # Set of warehouses (including potentials) set PW; # Subset of potential warehouses set EW, default W diff PW; # Subset of existing warehouses set D; # Demand Zones set n, default 0..9; # Piecewise pieces

# Parameters param XP{i in P}; # Plant X Coordinate param YP{i in P}; # Plany Y Coordinate param XW{i in W}; # Warehouse X Coordinate param YW{i in W}; # Warehouse Y Coordinate param PC{i in P}; # Plant capacity param PT{i in P}; # Plant throughput param PV{i in P}; # Plant variable cost param GF{i in D}; # 5-yr growth factor param XD{i in D}; # Demand Zone X Coordinate param YD{i in D}; # Demand Zone Y Coordinate param DD{i in D}; # Demand Zone Actual Demand param SR{i in W}; # Storage rate param HR{i in W}; # Handling rate param order_cost{i in W}; # Order Cost param order_size{i in W}; # Order Size param customer_cost{i in W}; # Customer Cost param customer_size{i in W}; # Customer Size

# Piecewise X values param a{i in n}, default (350000/10)*i;

# Piecewise Y values param b{i in n, j in W}, default (a[i])^0.58 * (SR[j] * 293.8 + HR[j] + 35.3);

# Variables var UseWarehouse{i in W}, binary; # Build warehouse i (No cost) var BuildPlant{i in P}, binary; # Build or upgrade (potential) plant i var x1{i in P, j in W}, >=0; # Flow from factory to warehouses and demand zones var x2{i in W, j in D}, >=0; # Flow from warehouse to demand zone var lambda{i in n, j in W}, >=0, <= 1; # Piecewise lambdas var z{i in n, j in W}, binary; # Adjency variables

# Input parameters from CSV files table dataset1 IN "CSV" "plants.csv" : P <- [Plant], XP ~ X, YP ~ Y, PC ~ capacity, PT ~ throughput, PV ~ variable_cost; table dataset2 IN "CSV" "potential_plants.csv" : PP <- [Plant]; table dataset3 IN "CSV" "warehouses.csv" : W <- [Warehouse], XW ~ X, YW ~ Y; table dataset4 IN "CSV" "potential_warehouses.csv" : PW <- [Warehouse]; table dataset6 IN "CSV" "demand.csv" : D <- [center], XD ~ x, YD ~ y, DD ~ demand, GF ~ GF; table dataset7 IN "CSV" "costing_info.csv" : [Warehouse], SR ~ storage, HR ~ handling, order_cost ~ order_cost, order_size ~ order_size, customer_cost ~ customer_cost, customer_size ~ customer_size;

# Objective minimize cost: sum{i in P, j in W} (0.92 + x1[i,j] * (0.0034*(0.36*((XP[i]-XW[j])^2 + (YP[i]-YW[j])^2)^0.5) + PV[i] + order_cost[j]/order_size[j])) + sum{i in W, j in D} (5.45 + x2[i,j] * (0.0037*(0.36*((XW[i]-XD[j])^2 + (YW[i]-YD[j])^2)^0.5))+customer_cost[i]/customer_size[i]) + sum{i in P} 4000000*BuildPlant[i] + sum{i in n, j in W} lambda[i,j]*b[i,j];

# Demand must be met
s.t. Demand{j in D}: sum{i in W} x2[i,j] >= DD[j]*GF[j];

# Flow from potential plant must be 0 if warehouse is not built.
s.t. Plants{i in PP}: sum{j in W} x1[i,j] <= BuildPlant[i]*999999999;

# Flow from potential warehouses must be 0 if warehouse is not built.
s.t. Warehouses1{i in W}: sum{j in D} x2[i,j] <= UseWarehouse[i]*350000;
s.t. Warehouses2{i in W}: sum{j in D} x2[i,j] >= UseWarehouse[i]*10400;

# Flow from warehouse to customers must equal flow from plants to warehouses
s.t. Flow{i in W}: sum{j in P} x1[j,i] - sum{j in D} x2[i,j] = 0;

# Plant capacity constraint
s.t. PlantCapacity{i in P}: sum{j in W} x1[i,j] <= PC[i] + 1000000*BuildPlant[i];

# Piecewise constraints
s.t. lambda1{j in W}: sum{i in n} lambda[i,j] = 1;
s.t. lambda2{i in n, j in W}: z[i,j] - lambda[i,j] >= 0;
s.t. WarehouseAsFnLambda{i in W}: sum{j in P} x1[j,i] - sum{k in n} a[k] * lambda[k,i] <= 0;
s.t. adjency{i in W, j in 0 .. 7, k in (j+2) .. 9}: z[j,i] + z[k,i] <= 1;

# Display Solution solve; display cost, UseWarehouse, BuildPlant;

# Create output file for plant flows table tab_result1{i in P, j in W} OUT "CSV" "optimal_plant_flows.csv" : i ~ plant, j ~ market, x1[i,j] ~ flow;

# Create output file for warehouse flows table tab_result2{i in W, j in D} OUT "CSV" "optimal_warehouse_flows.csv" : i ~ warehouse, j ~ market, x2[i,j] ~ flow;

------------------------------------------------- end; 4. Solution
The recommended course of action for Usemore is to build 1 new plant, shut down 5 of the existing public warehouses, and open 5 of the new warehouses for a total cost of $46,691,314. Plants and warehouse configurations are summarized in Table 1 and Table 2.
4.1 Plants
The following is a summary of the GLPK output file optimal_plant_flows.csv:
Table [ 1 ]: Warehouses to Plants Plant | Warehouses | COVINGTON,KY | COVINGTON,KY, CHICAGO,IL, CLEVELAND,OH | NEW YORK,NY | NEW YORK,NY | ARLINGTON,TX | ARLINGTON,TX | LONG BEACH,CA | LONG BEACH,CA | MEMPHIS,TN | MEMPHIS,TN, DAVENPORT,IA, NEW ORLEANS,LA |

4.2 Warehouses
The following is a summary of the GLPK output file optimal_warehouse_flows.csv:
Table [ 2 ]: Customers to Warehouses Warehouse | Customers | COVINGTON,KY | 1, 2, 5, 6, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 80, 82, 83, 85, 86, 87, 91, 115, 116, 117, 118, 139, 140 | BOSTON,MA | 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 60, 61 | MEMPHIS,TN | 3, 4, 15, 29, 30, 31, 32, 33, 34, 35, 36, 39, 89, 90, 92, 93, 100, 101, 104, 105, 107, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 140, 141, 142, 143, 144, 145, 146, 147, 148, 151, 153, 154, 155, 156 | CLEVELAND,OH | 7, 8, 17, 66, 68, 69, 70, 71 | DAVENPORT,IA | 92, 94 | NEW ORLEANS,LA | 149 | NEW YORK,NY | 57, 58, 59, 60, 62, 63, 64, 65, 67, 72, 73, 74, 75, 76, 77, 78, 79, 81, 83, 84 | ARLINGTON,TX | 37, 38, 40, 41, 102, 106, 108, 109, 110, 111, 112, 113, 114, 150, 152, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 190 | LONG BEACH,CA | 173, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 191 | CHICAGO,IL | 22, 23, 24, 25, 26, 27, 28, 88, 95, 96, 97, 98, 99, 103 |
4.3 Average Inbound and Outbound Distances
The following summarizes the average distances covered after having applying the K = 0.36 distance conversion factor: * Average inbound distance covered: 363.60 (unit distance) * Average outbound distance covered: 387.06 (unit distance)
These distances were computed within the model, outputted to optimal_plant_flows.csv and optimal_warehouse_flows.csv, and then averaged.
5. Discussion
The implementation of the GLPK MIP was successful. However, the 10-piece piecewise approximation has likely altered the optimal solution, since the LP relaxation suggested that plant flows be anywhere from 100-100,000 – and the MIP is only accounting for a few specific cases. It is in fact possible to break the cost function into more pieces, but the processing time required to solve the model becomes too lengthy (several hours).
However, the implementation is successful and flexible. Parameters of the CSV files may easily be altered and the model may solve different cases. For example, if there are an additional 20 warehouse locations to consider, simply add them to the data set and re-run the model. Additionally, if time is available, the cost function may be broken into several more pieces for a more optimal solution.
6. Conclusion
As stated, the recommended course of action for Usemore is to build 1 new plant, shut down 5 of the existing public warehouses, and open 5 of the new warehouses for a total cost of $46,691,314. Should Usemore want to consider even more plants and warehouses, the MIP formulation provided is flexible enough to easily accommodate new data sets.
7. Bibliography
Uday Venkatadri, IENG 4579: Supply Chain Managenent The Supply Chain Design Problem, Department of Industrial Engineering, Dalhousie University, 2008