Question

5. A company operates two fruit processing plants. The growers are willing to supply fresh fruit in the following amounts: Apples: 200 tonnes at Kiwifruit: 310 tonnes at $1000/tonne Pears: 420 tonnes at $900/tonne $1100/tonne Shipping costs in $ per tonne are To Plant A To Plant B Apples Kiwifruit 200 300 350 250 Pears 600 400 Plant capacities and labour costs are: Plant A Plant B Capacity 460 tonnes 560 tonnes $26/tonne $21/tonne Labour cost The canned fruits are sold at $5000/tonne to the distributors. The company can sell at this price all they can produce. The objective is to find the best mixture of the quantities supplied by the three growers to the two plants so that the company maximises its profits (a) Formulate the LP. You will need 6 decision variables, the amounts of each fruit sent to each plant. (b) Solve the LP using Excel. What is the optimal solution and the optimal value of the profit?

Answer 1

5.

a)Let xij = weight of i fruit units shipped to plant j

where i = {1 for Apples,2 for Kiwifruit,3 for Pears} and j = {A for Plant A,B for Plant B}

Profit = Revenue - (cost of fresh fruit+shipping cost+labour cost)

Profit on x1A per ton = 5000 - (1100+300+26) = 3574

Profit on x2A per ton = 5000 - (1000+200+26) = 3774

Profit on x3A per ton = 5000 - (900+600+26) = 3474

Profit on x1B per ton = 5000 - (1100+350+21) = 3529

Profit on x2B per ton = 5000 - (1000+250+21) = 3729

Profit on x3B per ton = 5000 - (900+400+21) = 3679

Objective is to maximize profit so objective function = Max 3574x1A+3774x2A+3474x3A+3529x1B+3729x2B+3679x3B

Subject to,

Supply constraints

x1A+x1B <= 200

x2A+x2B <= 310

x3A+x3B <= 420

Plant capacity constraint

x1A+x2A+x3A <= 460

x1B+x2B+x3B <= 560

xij >= 0 (non-negativity constraint)

b)
solving in excel we get following optimal solution:

	To Plant A	To Plant B
Apples	150	50
Kiwifruit	310	0
Pears	0	420

Optimal value of profit = 3427670

	x1A	X2A	x3A	x1B	x2B	x3B
	150	310	0	50	0	420				Objective function	3427670
Supply constraints	1			1			200	<=	200
		1			1		310	<=	310
			1			1	420	<=	420
Capacity constraints	1	1	1				460	<=	460
				1	1	1	470	<=	560