The Terraco Motor Company has produced a lightweight, all-terrain vehicle code-named “J99 Terra” for the military. The company is now planning to sell the Terra to the public. It has five plants that manufacture the vehicle and four regional distribution centers. The company is unsure of public demand for the Terra, so it is considering reducing its fixed operating costs by closing one or more plants, even though it would incur an increase in transportation costs. The relevant costs for the problem are provided in the following table. The transportation costs are per thousand vehicles shipped; for example, the cost of shipping 1,000 vehicles from plant 1 to warehouse C is $32,000.                                              Transportation Costs ($1,000s) to Warehousefrom plantABCDAnnual Production CapacityAnnual Fixed Operation Costs15621326512,0002,100,0002184673518,000850,00031271415214,0001,800,00043024612810,0001,100,00054550263116,000900,000annual demand6,00014,0008,00010,000   Formulate an integer programming model for this problem and solve it by using Excel to assist the company in determining which plants should remain open and which should be closed and the number of vehicles that should be shipped from each plant to each warehouse to minimize total cost.

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Asked Apr 3, 2019
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The Terraco Motor Company has produced a lightweight, all-terrain vehicle code-named “J99 Terra” for the military. The company is now planning to sell the Terra to the public. It has five plants that manufacture the vehicle and four regional distribution centers. The company is unsure of public demand for the Terra, so it is considering reducing its fixed operating costs by closing one or more plants, even though it would incur an increase in transportation costs. The relevant costs for the problem are provided in the following table. The transportation costs are per thousand vehicles shipped; for example, the cost of shipping 1,000 vehicles from plant 1 to warehouse C is $32,000.

 

                                             Transportation Costs ($1,000s) to Warehouse
from plant A B C D Annual Production Capacity Annual Fixed Operation Costs
1 56 21 32 65 12,000 2,100,000
2 18 46 7 35 18,000 850,000
3 12 71 41 52 14,000 1,800,000
4 30 24 61 28 10,000 1,100,000
5 45 50 26 31 16,000 900,000
annual demand 6,000 14,000 8,000 10,000    

 

Formulate an integer programming model for this problem and solve it by using Excel to assist the company in determining which plants should remain open and which should be closed and the number of vehicles that should be shipped from each plant to each warehouse to minimize total cost.

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Expert Answer

Step 1

We model the problem by using the following decision variables:

  • 20 variables to represent the number of vehicles transported from plant to warehouse such as x1A, x1B,…
  • 5 variables to represent the binary decision variable to operate a plant or not represented as y1, y2,…

The objective function would be represented as the sum of the fixed costs for operating the plants and the cost of transportation between the plants and warehouses. The objective is to minimize the cost.

The demand and supply constraints would be as below:

Where ai represent the capacity at each plant, bj represent the demand at each warehouse.

Also Xij are integers and Yi are 0 or 1 (binary).

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Step 2

The Microsoft Excel ® model is given below (formula sheet, the result sheet as well as the solver settings) :

Formula sheet and Solver Sheet shown below:

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