A firm produces tractors and lawn mowers and make a profit of $30 on each tracto

Question

A firm produces tractors and lawn mowers and make a profit of $30 on each tractors and $30 on each lawn mower. And they sell all they can produce. The time requirement in the machine shop, fabrication, and tractor assembly as well as the maximum time available for each process are given below. Machine Shop Fabrication Assembly Tractor 2 hours 2 hours 1 hour Lawn Mower 1 hour 3 hours 0 hour Hrs. Available 60 hours 120 hours 45 hours a) Formulate a linear program to maximize the profit. Clearly define your decision variables, constraints and objective function b) Solve the formulated linear graphically and find the optimal solution and the profit company will make c) Determine manually the sensitivity range for the profit per tractors. Show your manual calculations d) By inspecting your graph, determine the shadow price for assembly, Explain the rationale to your answer e) Calculate manually the maximum amount a manager should be willing to pay for one additional hour of machine time. Show your calculations f) Determine manually and by inspecting your graph the sensitivity range assembly? Explain rationale your answer

Explanation / Answer

Decision Variables are the number of Tractors (say X) and number of Lawn mowers (say Y) to be produced.

Objective function is to maximize Profit = Maximize 30X + 30Y

Constraints are in terms of availability of hours of Machine shop, Fabrication and Assembly for Tractors & Lawn mowers as follows:

2X + Y <= 60                2X + 3Y <= 120 and X <= 45 and X,Y >= 0

The solution is as follows:

X Y RHS RHS Maximize 30 30 Max 30X + 30Y Machine shop 2 1 <= 60 2X + Y <= 60 Fabrication 2 3 <= 120 2X + 3Y <= 120 Assembly 1 0 <= 45 X <= 45

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