A snack food producer runs four different plants that supply product to four different regional distribution centers

Question

The division operations manager is focused on one product, so he creates a table showing each plant's monthly capacity and each distribution center's monthly demand (both amounts in cases) for the product. The division manager supplements this table with the cost data to ship one case from each plant to each distribution center. Formulate an objective function and constraints that will solve this problem using linear programming.

Center 1 Center 2 Center 3 Center 4 Monthly Capacity
Plant A $2 $7 $5 $4 8000
Plant B $9 $4 $7 $6 12000
Plant C $7 $6 $4 $3 7500
Plant D $4 $8 $3 $5 5000
Monthly Demand 9000 8500 8000 7000

Answer 1

This is a cost minimization problem with 16 decision variables, one for each combination of plant and center; there are 8 constraints, one for each plant's capacity and one for each center's demand.

Objective Min Z = $2xA1 + $7xA2 + $5xA3 + $4xA4 + $9xB1 + $4xB2 + $7xB3 + $6xB4 + $7xC1 + $6xC2 + $4xC3 + $3xC4 + $4xD1 + $8xD2 + $3xD3 + $5xD4

Subject to
Plant A: xA1 + xA2 + xA3 + xA4 = 8000

Plant B: xB1 + xB2 + xB3 + xB4 = 12000

Plant C: xC1 + xC2 + xC3 + xC4 = 7500

Plant D: xD1 + xD2 + xD3 + xD4 = 5000

Center 1: xA1 + xB1 + xC1 + xD1 = 9000

Center 2: xA2 + xB2 + xC2 + xD2 = 8500

Center 3: xA3 + xB3 + xC3 + xD3 = 8000

Center 4: xA4 + xB4 + xC4 + xD4 = 7000

The optimal solution is:

Center 1 Center 2 Center 3 Center 4 Monthly Capacity
Plant A 8,000 8,000
Plant B 8,500 3,500 12,000
Plant C 500 7,000 7,500
Plant D 1,000 4,000 5,000
Monthly Demand 9,000 8,500 8,000 7,000 $113,500