QUESTION 5 IS JUST NEEDED TO ANSWER 8 and 9 5. Small Motors Company manufactures

Question

QUESTION 5 IS JUST NEEDED TO ANSWER 8 and 9
5. Small Motors Company manufactures compact and subcompact cars. The production of each
car requires a certain amount of raw material and labor, as specified in the following table:
Raw Material (pounds) Labor (hours)
Compact 200 18
Subcompact 150 20
Unit Cost ($) 10 70
Total Available 80,000 9,000
The marketing division has estimated that at most 1500 compacts can be sold at $10,000 each
and that at most 200 subcompacts can be sold at $8000 each. As Vice-President of Scheduling
you formulated the following model, where C is the number of compact cars and S is the number
of subcompact cars to produce.
Maximize 6740 C + 5100 S
Subject to
200 C + 150 S < 80000
18 C + 20 S < 9000
C < 1500
S < 200
C , S > 0

8. For the case of Small Motors Company in Problem 5, use your results from Problem 6 to
determine which profit margin is most sensitive in the sense that a smaller change up or down
affects the optimal production plan. Explain. (4 points)

9. For the case of Small Motors Company in Problem 5, the union contract is up for renewal.
Small Motors now has the choice to agree with one of two new provisions with the following
requirements:
Provision 1
Labor (hours)
Provision 2
Labor (hours)
Compact 18 18
Subcompact 20 20
Unit Cost ($) 70 70
Total Available 10,000 8,000
Use your results from Problem 7 to determine which provision to sign and why. Show your
work.

7. For each constraint that holds with equality at the optimal solution of the linear program in
Problem 5, perform the following:
a) Graphically and algebraically determine the sensitivity range of the right-hand-side value.

Explanation / Answer

I have already answered 5th problem previously when u asked

the optimal solution is C = 250 S = 200

objective fun f= 6740C + 5100S = 2705000 at optimal solution

200 C + 150 S < 80000

s<200

these are the 2 constraints that hold with equality at optimal solution

Q 8

for this we need to check the range of optimality of the coefficients of objective fun

f= k1 C + k2 S

slope of obj fun = -k1/k2 this should lie between the slopes of 2 constraints mentioned above

when k2 = 5100

-4/3 < -k1/5100 < 0

-6800 < -k1 < 0

0< k1 < 6800

its now at 6740 , so it can rise by 60 $ with out affecting optimal solution

when k1 = 6740

-4/3 < -6740/k2 < 0

0 < 6740/k2 < 4/3

k2 > 5055

its already at 5100 , so it can only decrease by $45

by this analysis I can say that profit margin for subcompact is more sensitive than for compact

Q 7

decrease RHS of 18C + 20 S < 9000

now 18C +20S <8000 now the new optimal solution is C = 307 S = 123 where f = 2696480

if 18 C + 20S < 10000 now the optimal solution does not change so f = 2705000

Q 9

from Q 7 I like to prefer provision 1

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