2- A supply chain system consists of 4 layers, suppliers, warehouses, retailers

Question

2- A supply chain system consists of 4 layers, suppliers, warehouses, retailers and customers. The shipping costs between each pair of layers are listed in Tables 1 to 3. The supply capacity of suppliers 1, 2 and 3 are 14000, 8000 and 6000 respectively. The demand for customer 1, 2, 3 and 4 is 3500, 6000, 8500, and 7000 respectively. (8 points) a) Define sets and decision variables clearly. b) Write the balance capacity constraint for the second retailer. c) Write the balance capacity constraint for the third warehouse. d) Write the supply capacity constraint for third supplier e) Write the demand constraint for the fourth customer f) Write the objective function. g) What is the number of decision variables in this problem? h) What is the number of constraints in this problem? Table 1: Shipping fee per unit of item from supplier to warehouse From To Supplier 1 Supplier 2 Supplier 3 Warehouse1 Warehouse 2 Warehouse 3 S8 $2 $4 $5 $3 $1 $7 $8 Table 2: Shipping fee per unit of item from warehouse to retailer. From ToRetailer 1 Warehouse 1 Retailer 2 Retailer 3 $10 Retailer 4 $11 $9 S5 S8 S6 $8 $9 $8 $7 Warehouse 2 $7 Warehouse 3 Table 3: Shipping fee per unit of item from retailer to customer Customer 1 Customer 2 Customer 3 Customer 4 Retailer 1 Retailer 2 Retailer 3 Retailer4 S5 $7 $4 $6 S6 $8 $4 $7 $4 $6 $3 $8 $8 $9 $5 $6

Explanation / Answer

(a) Decision variables:

SiWj = Quantity to be shipped from Supplier i to Warehouse j , where i=1,2,3 and j=1,2,3

WjRk = Quantity to be shipped from Warehouse j to Retailer k, where j=1,2,3 and k=1,2,3,4

RkCm = Quantity to be shipped from Retailer k to Customer m , where k=1,2,3,4 and m=1,2,3,4

(b) Balance constraint equation for second retailer: W1R2+W2R2+W3R2 = R2C1+R2C2+R2C3+R2C4

(c) Balance constraint equation for third warehouse: S1W3+S2W3+S3S3 = W3R1+W3R2+W3R3+W3R4

(d) Supply capacity constraint for the third supplier: S3W1+S3W2+S3W3 <= 6000

(e) Demand constraint for the fourth customer: R1C4+R2C4+R3C4+R4C4 = 7000

(f) Objective: Minimize Xij*Cij + Xjk*Cjk + Xkm*Ckm, where Cij, Cjk, Ckm are shipping fee(cost) per unit from respective source node to destination node.

(g) Number of decision variables: 3 suppliers * 3 warehouses + 3 warehouses * 4 retailer + 4 retailers * 4 customers = 3*3+3*4+4*4 = 37

(h) Number of constraints: 3 supplier capacity constraints + 3 warehouse balanace capacity constraints + 4 retailer capacity constraints + 4 customer demand constraints = 14

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