Determine the number of refrigerators to be shipped from the plants to the warehouses and then from the warehouses to retailers to minimize the cost. Formulate the problem as a linear program with an objective function and all constraints. Determine the optimal solution for the linear program using any software you want. What are the optimal shipping routes and minimum cost?

This is an extension of the transportation model (lecture notes: slides 52-55). There are now intermediate transshipment points added between the sources (plants) and destinations (retailers). Items being shipped from a Plant (pi) must be shipped to a Warehouse (wj) before being shipped to the Retailer (rk). Each Plant will have an associated supply (si) and each Retailer will have a demand (dk). The number of plants is n, number of warehouses is q and the number of retailers is m. The edges (i,j) from plant (pi) to warehouse (wj) have costs associated denoted cp(i,j). The edges (j,k) from a warehouse (wj) to a retailer (rk) have costs associated denoted cw(j,k). The graph below shows the transshipment map for a manufacturer of refrigerators. Refrigerators are produced at four plants and then shipped to a warehouse (weekly) before going to the retailer.

R1 R2 P1 W1 R3 P2 W2 R4 P3 R5 W3 P4 R6 R7 Below are the costs of shipping from a plant to a warehouse and then a warehouse to a retailer. If it is impossible to ship between the two locations an X is placed in the table. cost W1 W2 W3 P1 $10 $15 X P2 $11 $8 X P3 $13 $8 $9 P4 X $14 $8 cost R1 R2 R3 R4 R5 R6 R7 W1 $5 $6 $7 $10 X X W2 X X $12 $8 $10 $14 X W3 X X X $14 $12 $12 $6 The tables below give the capacity of each plant (supply) and the demand for each retailer (per week). P1 P2 P3 P4 Supply 150 450 250 150 R1 R2 R3 R4 R5 R6 R7 Demand 100 150 100 200 200 150 100