Question

Sasha, a doctor based in Toronto had to stop working temporarily due to the health and safety restrictions imposed by the government due to the outbreak of the swine flu. She started to make two types of children's wooden toys in her basement, cars (C) and dolls (D). Cars can yield a contribution margin of $9 each and dolls have a contribution margin of $8 each. Since her electric saw overheats, Sasha can make no more than 7 cars or 14 dolls each day. Since she doesn't have equipment for drying the lacquer finish that she puts on the toys, the drying operation limits her to 16 cars or 8 dolls per day. Sasha wants to know what is the combination of cars and dolls that she should make each day to maximize her profits, subject to her constraints. a) Formulate a linear programming problem to depict Sasha's objective algebraically. b) What are the corner points in Sasha's feasible region? [Note: You do not need to graph the feasible region in your answer file, but you may want to draw it in a scrap paper to understand the problem better and solve it c) Solve this problem using the corner point method. Find the optimal combination of cars and dolls. What is the optimal profit at that combination?

Answer 1

Let she makes C number of cars and D number of dolls.

It is given that Cars yield $9 and dolls yield $8

Hence the total profit from C cars and D dolls=9C+8D

The objective is to maximisethis profit,

Hence the objective function is: Max Z = 9C+8D

It is given that due to the over hear of electric saw, she cannot make more than 7 cars and 14 dolls each day

Hence, C<= 7, D<=14..........(1)

Due to the drying optionlimitation she cannot make more than 16 cars and 8 dolls each day

Hence, C<=16, D<=8..............(2)

From (1) and (2), we can derive the constraints as:

C<=7 and D<=8

(a) Hence the LPP is:

Max Z = 9C+8D

C<=7

D<=8

C,D >= 0

(b) The feasible region is unbounded, but to satisfy the non negative restrictions, we can consider the region only in the first quadrant

Here the region is OPQR with corner points

O(0,0), P(7,0), Q(7,8) and R(8,0)

(c) Now let's find Z at each of the corner points (Corner point method)

Z = 9C+8D

At O(0,0), Z = 0

At P(7,0), Z = 63

At Q(7,8), Z = 63+64 = 127

At R (8,0), Z =72

Max Z = 127 at Q(7,8)

Hence the optimal solution is :

Maximum Profit,Z = 127 when no. of cars, C =7 and no.of dolls, D = 8