Answer (a): Let i denotes the plant and j denotes the warehouse. Let X ij be the

Question

Answer (a):

Let i denotes the plant and j denotes the warehouse.

Let Xij be the number of units to be transported form ith plant to jth warehouse.

Let yi be the cost fixed cost of ith plant

The objective of this problem is to minimize the total cost of transportation and operating cost. Therefore, the problem may be stated as follows:

Minimize Z = 7x11 + 6x12 + 6x13 + 6x14 + 8x15 + 4x21 + 3x22 + 5x23 + 8x24 + 4x25 + 8x31 + 7x32 + 8x33 + 9x34 + 8x35 + 9x41 + 2x42 + 4x43 + 7x44 + 9x45 + 3500 y1 + 4000 y2 + 2000 y3 + 3000 y4

(The above equation includes total transportation as well as fixed cost of operation)

Subject to

X11 + x12 + x13 + x14 +x15 400 (capacity constraint of plant 1)

X21 + x22 + x23 + x24 + x25 350 (capacity constraint of plant 2)

X31 + x32 + x33 + x34 + x35 300 (capacity constraint of plant 3)

X41 + x42 + x43 + x44 + x45 500 (capacity constraint of plant 4)

X11 + x21 + x31 + x41 = 100 (demand constraint of warehouse 1)

X12 + x22 + x32 + x42 = 200 (demand constraint of warehouse 2)

X13 + x23 + x33 + x43 = 160 (demand constraint of warehouse 3)

X14 + x24 + x34 + x44 = 220 (demand constraint of warehouse 4)

X15 + x25 + x35 + x45 = 50 (demand constraint of warehouse 5)

Xij 0                     i = 1,2,3,4                             j= 1,2,3,4,5

Yi = 0,1                  i = 1,2,3,4            

In this problem, yi would be equal to zero, if a particular plant is not run, and equal to one if it is used. If a plant is not operated, its capacity would be reduced to zero, as would be its fixed cost.

Answer (b): Condition 1:

Let i denotes the plant

If plant i is operated, it will take the value of 1, otherwise it will take the value of 0

If plant 1 is operated, it will take the value of 1, else 0. Similarly, if plant 2 is operated, it will take the value of 1, else 0.

The first constraint, i.e. either Plant 1 or plant 2 (or both) must be operated can be stated as:

P1 + P2 1 (where P1 and P2 denotes plant 1 and plant 2)

In above equation, any one plant which will be operated, will take the value of 1. The other plant will take the value of 0. Hence the constraint condition of either plant 1 or plant 2 must be operated gets satisfied. And if both the plants must be operated, both plants will take the value of 1. Hence the RHS of the above equation will be satisfied because, if both plants are operated, the LHS will be equal to 2 which is greater than 1.

Condition 2:

If plant 1 (P1) is operated, plant 2 (P2) must be operated but plant 4 (P4) cannot be operated.

This can be stated as:

P1 + P2 + P4 0

The explanation of the above stated constraint is as given below:

If P1 is not operated (i.e. when it takes the value 0), then P2 can be operated or cannot be operated (i.e. P2 can take the value of 0 or 1). At the same time, plant 4 (P4) can be operated or cannot be operated (i.e. P4 can also take the value of 0 or 1).

Therefore, if P1 = 0, and P2 =0, then P4 can be 0 or 1 satisfying the above constraint equation.

If P1 = 0, P2 = 1, then P4 can be 0 or 1 satisfying the above constraint equation.

If P1 = 1, then P2 must be equal to 1 and P4 must be equal to 0. Therefore, if P1 = 1, P2 = 1, and P4 = 0, then still it satisfies the above constraint equation.

Explanation / Answer

Part b is separate from part a

Exercise There are four plants and five warehouses in a distribution network. The per unit variable cost of serving each warehouse (WH) from each plant (P), the demand at each warehouse, the capacity of each plant, and the fixed cost of opening an operating each plant are provided in the tables below: Variable Cost/Unit Pl P2 P3 P4 Demand Fixed Cost 3500 4000 2000 3000 WH 74 89 WH26 372 WH3 6 5 84 WH4 689 7 WH584 100 200 160 220 50 PlantCapacitv 400 350 300 500 a) Formulate an integer programming model that can be used to determine the least costly distribution strategy. (Define the decision variables, state the objective function, and specify the constraints clearly.) b) Formulate the following restriction:s " Either Plant 1 or Plant 2 (or both) must be operated " If Plant 1 is operated, Plant 2 must also be operated but Plant 4 cannot be operated.

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