Question

A fashion & textile manufacturer currently has four manufacturing plants (Plant A, B, C, D) and two retail outlets (Outlet 1 and 2). Due to expected reduction in demand, the company is considering whether closing some of the plants would be possible. The details of the weekly forecasted demand at outlets, weekly supply at plants, and the unit cost to transport between plants and outlets are shown as follows.

	Outlet 1	Outlet 2	Supply	Fixed Annual Operating Cost
Plant A	$5	$7	3,560	$750,000
Plant B	4.5	6	3,200	830,000
Plant C	6	8	2,800	680,000
Plant D	9	7	4,200	825,000
Forecasted Demand	4,500	6,000

The plant can produce only if it is operated, and the quantity should not exceed the weekly supply. Plant B can be used only if either plant A or plant C or both plants are used. Both Plant C and D should be operated at the same time. The manufacturer does not need to pay the fixed annual operating cost if the plant is not operated.

Questions:

1. Formulate an integer programming model for this problem to determine which plants should be operated and the number of products that should be shipped from each plant to each outlet to minimize the total cost, which includes the operating cost and transportation cost.                                                                                    

2. Solve the problem using solver (you need to attach the Excel worksheet as evidence).      

Answer 1

a

Formulating integer programming model.

Decision variables:

Yi = 1 if plant i should produce; else 0 if plant i should not produce and shut down

Y1=1 if plant 1 should produce; else 0 if plant A should not produce and shut down

Y2=1 if plant 2 should produce; else 0 if plant B should not produce and shut down

Y1=3 if plant 3 should produce; else 0 if plant C should not produce and shut down

Y4=1 if plant 4 should produce; else 0 if plant D should not produce and shut down

Xij = Number of units to be produced and shipped from plant i to Outlet j

X11 = Number of units to be produced and shipped from plant A to Outlet 1

X12 = Number of units to be produced and shipped from plant A to Outlet 2

X21 = Number of units to be produced and shipped from plant B to Outlet 1

X22 = Number of units to be produced and shipped from plant B to Outlet 2

X31 = Number of units to be produced and shipped from plant C to Outlet 1

X32 = Number of units to be produced and shipped from plant C to Outlet 2

X41 = Number of units to be produced and shipped from plant D to Outlet 1

X42 = Number of units to be produced and shipped from plant D to Outlet 2

Objective function

The objective is to minimize total cost. Total cost is the sum of fixed operating cost and variable transport cost from each plant to each outlet.

MIN Z= (750000Y1+830000Y2+680000Y3+825000Y4)+ (5Y1X11+7Y1X12+4.5Y2X21+6Y2X22+6Y3X31+8Y3X32+9Y4X41+7Y4X42)

Constraints:

Supply constraints:

X11+X12<= 3560

X21+X22<=3200

X31+X32<= 2800

X41+X42<= 4200

Demand constraints are:

Y1X11+Y2X21+Y3X31+Y4X41>=4500

Y1X12+Y2X22+Y3X32+Y4X42 >=6000

Y1 + Y3 >= Y2    (Plant B can be used only if either plant A or plant C or both plants are used)

Y3 = Y4    (Both Plant C and D should be operated at the same time)

Y1+Y2+Y3+Y4 <= 3

Y1, Y2, Y3, Y4 = {0,1}; binary variable

X11, X12, X21, X22, X31, X32, X41, X42 = {0, 1, 2....}; integer variable

b.

We will solve this integer programming problem using Excel Solver. Below Excel screen print show the approach.

Go to File-> Options-> Excel Add ins -> Solver Add in
Now go to Data -> Solver and setup Solver with decision variables, objective function and constraints.

Click on Solve to get the optimal solution as below:

Hence, only plant A, C and D should be operated and following units should be transported as optimal schedule.

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