Recall the following problem from Homework 2 Consider a wind

Recall the following problem from Homework 2: Consider a window company with only three

employees which makes two dierent kinds of hand-crafted windows: a wood-framed and an aluminum-

framed window. They earn $60 prot for each wood-framed window and $30 prot for each aluminum-framed

window. Employee 1 makes the wood frames, and can make 6 per day. Employee 2 makes the aluminum

frames, and can make 4 per day. Employee 3 forms and cuts the glass, and can make 48 square feet of glass

per day. Each wood-framed window uses 6 square feet of glass and each aluminum-framed window uses 8

square feet of glass.

Let x1 and x2 be the number of wood-framed and aluminum-framed windows, respectively. The linear

programming model can be formulated as:

maximize 60x1 + 30x2

subject to 6x1 + 8x2 <=48

x1 <=6

x2 <= 4

x1; x2 >= 0

The optimal solution to this problem is (x1; x2 ) = (6; 1: 5), with an optimal cost of 405.

a) Suppose that one day, Employee 1 oers to work overtime to produce up to 9 wood frames instead of 6,

but asks that he be paid an extra xed amount of $90. Can his oer be evaluated without re-solving the

model? Explain why or why not.

b) Suppose that on a dierent day, Employee 1 oers to work overtime to produce up to 7 wood frames

instead of 6, but asks that he be paid an extra xed amount of $30. Is this a good deal from the company\'s

point of view?

c) What would be the change in total prot if the company agreed to Employee 1\'s oer from part b)?

Solution

Answer:

By given optimum solution,

Sale Price each Unit of X1 & X2

Total Cost for 7.5 Units = $405/7.5 Units = $ 54 Each unit for X1

                                                                   = $54/2 = $ 27 for X2

a) offer of employee 1 cannot be evaluated without re-solving the model as limiting factors have changed by offer of production of 9 Units with extra payment of $90.

b) Yes, it is good deal from company\'s point of view as profit on product X1 is $60 and demand by Emloyee 1 is $ 30 only provided  Employee 3 is ready to produce extra 6 squire feet of Glass for less than $30 for this 1 unit of X1 as he has already  reached his limit of 48 Squire feet of Glass (6x 6 for X1 + 8x 1.5 for X2).

c) Total Profit of the company will increase by $30 if Employee 3 is ready to produce extra 6 squire feet of glass without any charges.