9. An automotive manufacturer has four plants to annually manufacture 500,000 ca
ID: 371750 • Letter: 9
Question
9. An automotive manufacturer has four plants to annually manufacture 500,000 cars of each of its three models: X, Y, and Z. Each plant can only manufacture a single model, and each car can only be made at a single plant. If plants 3 and 4 are used, then labour union contracts dictate that plant 1 must also be used. The fixed costs of operating each plant and the variable costs for producing one car of each model are: Plant Fixed Cost Model X Model Y Model Z Var. Cost Var. Cost Var. Cost $12,000 $16,000 $9,000 $6 billion $15,000 $18,000 $11,000 $4 billion $17,000 $19,000 $12,000 $2 billion $19,000$22000 $14,000 $7 billion 2 4 Formulate an Integer Program to minimize the annual production cost. Page 4 of 4Explanation / Answer
Formulate ILP model as below:
Decision variable: Pi = 1 indicates that plant i is used.
Xij = 1 indicates that plant i is selected for manufacturing j model, where i=1,2,3,4 and j=X,Y,Z
Pi and Xij are binary variables
Objective: Minimize 7P1 + 6P2 + 4P3 + 2P4 + (12000*X1x + 16000*X1y + 9000*X1z + 15000*X2x + 18000*X2y + 11000*X2z + 17000*X3x + 19000*X3y + 12000*X3z + 19000*X4x + 22000*X4y + 14000*X4z )*500000/109
objective function is minimization of production cost including fixed and variable. Total cost is expressed in $ billions.
Constraints:
P1 - (X1x + X1y + X1z) = 0
P2 - (X2x + X2y + X2z) = 0
P3 - (X3x + X3y + X3z) = 0
P4 - (X4x + X4y + X4z) = 0
X1x + X2x + X3x + X4x = 1
X1y + X2y + X3y + X4y = 1
X1z + X2z + X3z + X4z = 1
(P3 + P4) - P1 <= 1
Pi, Xij = 0,1
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