A potato chip manufacturing company is contemplating the purchase of a potato farm to meet the increased demand of potato chips. The available capacity at the three plants A, B and C are shown in the first table below. The second table shows the fixed cost and the projected annual harvest at the six farms (1, 2, 3, 4, 5,6) being considered. The unit cost of transporting a ton of potato from each of the six farms to each of the three plants are shown in the third Table.

Develop and solve a binary integer linear program (BILP) to help the manufacturing company decide which of six farms should be purchased.

Plant		Available Capacity (Tons)
A		12
B		10
C		14
Farm		Fixed Cost ($ 1000)		Projected Annual Harvest (Tons)
1		405		11.2
2		390		10.5
3		450		12.8
4		368		9.3
5		520		10.8
6		465		9.6
From Farm	To Plant A ($/Ton)		To Plant B ($/Ton)		To Plant C $/Ton)
1	18		15		12
2	13		10		17
3	16		14		18
4	19		15		16
5	17		19		12
6	14		16		12

Decision Variables

y_i = 0 if farm i is not selected, and 1 if farm i is selected; i = 1,2,3,4,5,6

x_ij = potatoes (1000 tons) shipped from farm i to plant j; j = A,B,C.

Objective Function

Minimize Z = 18x_1A+ 15x_1B+ 12x_1C+ 13x_2A+ 10x_2B+ 17x_2C+ 16x₃+ 14x_3B

+18x_3C+ 19x_4A+ 15x_4b+ 16x_4C+ 17x_5A+ 19x_5B+12x_5C+ 14x_6A

+ 16x_6B+ 12x_6C+ 405y₁+ 390y₂+ 450y₃+ 368y₄+ 520y₅+ 465y₆

Constraints

x_1A + x_1B + x_1C - 11.2y₁ 0 x_2A + x_2B + x_2C -10.5y₂ 0

x_3A + x_3A + x_3C - 12.8y₃ 0 x_4A + x_4b + x_4C - 9.3y₄ 0

x_5A + x_5B + x_5B - 10.8y₅ 0 x_6A + x_6B + X_6C - 9.6y₆ 0

x_1A + x_2A + x_3A + x_4A + x_5A + x_6A = 12

x_1B + x_2B + x_3B + x_4B + x_5B + x_6B = 10

x_1C + x_2C + x_3C + x_4C + x_5C + x_6C = 14

x_ij 0 y_i = 0 or 1

AB I Plant J K L D Plant M Farm Qty Shipped 18 $/Ton 15 10 13 Fixed Cost 405 390 450 368 520 465 Annual Harvest 11.2 10.5 12.8 9.3 10.8 9.6 Purchased or not Harvest Shipped Excess Capacity 11.2 10.5 12.8 0.161290323 1.5 0 2.77556E-17 -2.66454E-16 19 14 12 Demand Received Unmet Demand 12 Total Cost 1776.654839 AB I Plant J K L D Plant M Farm Qty Shipped 18 $/Ton 15 10 13 Fixed Cost 405 390 450 368 520 465 Annual Harvest 11.2 10.5 12.8 9.3 10.8 9.6 Purchased or not Harvest Shipped Excess Capacity 11.2 10.5 12.8 0.161290323 1.5 0 2.77556E-17 -2.66454E-16 19 14 12 Demand Received Unmet Demand 12 Total Cost 1776.654839