A company is considering launching two new products that have large sales potent

Question

A company is considering launching two new products that have large sales potential. Product 1 requires some of the production capacity in Plants 1 and 3, but none in Plant 2. Product 2 needs only Plants 2 and 3. The marketing division has concluded that the company could sell as much of either product as could be produced by these plants. However, because both products would be competing for the same production capacity in Plant 3, it is not clear which mix of the two products would be most profitable. Each product will be produced in batches of 20, so the production rate is defined as the number of batches produced per week. Any combination of production rates that satisfies these restrictions is permitted, including producing none of one product and as much as possible of the other. The number of hours of production time available per week in each plant for both products as well as the number of hours of production time required for each batch produced of each new product in each plant is given in the table below. The profit per batch of product 1 is $3000 and of product 2 is $2000. Answer the following

a) Formulate a linear mathematical program to determine what the production rates should be for the two products in order to maximize their total profit, subject to the restrictions imposed by the limited production capacities available in the three plants. Clearly define your decision variables, constraints and objective function

b) Use the graphical method to solve the model and find the optimal amounts of batches of each product and the optimal amount of profit

c) Using manual calculations, determine by how much the profit per batch of each product can increase or decrease before the optimal solution changes

d) By inspecting your graphical solution, how much would an additional hour of available hour in Plant 2 should be worth. Justify your answer based on your graph

e) By inspecting your graphical solution, by how many hours, if any, can you reduce the time available in Plant 1 without affecting the optimal amount of profit

f) If the new products had been required to return a net profit of at least $50,000 per week to justify discontinuing part of the current product line. What would be the additional constraint and how does this change the solution. Demonstrate this change on your graphical solution

Production Time per Batch, Hours Product Production Time Available per Week, Hours Plant 12 18

Explanation / Answer

A.

Objective function:

Maximize

Z=X11*3000+X13*3000+ X22*2000+X23*2000

Such that

X11*1 <= 4

X22*2 <= 12

X13*3+ X23*2 <= 18

X12,13,22,23 >= 0

Product 1 Product 1 Plant 1 1 0 Plant 2 0 2 Plant 3 3 2 X1 X2 Product 1 Product 1 Constraints Plant 1 4 4 <= 4 Plant 2 12 12 <= 12 Plant 3 18 0 18 <= 18 22 12 Profit 3000 2000 Total profit 90000 Maximization

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