Solution Key
Average = 22.00
Range 13 – 25
BOSTON COLLEGE
GRADUATE SCHOOL OF MANAGEMENT
Operations and Strategic Management Department
MD716 Modeling and Decision Analysis
Quiz 1
Spring 2005
This quiz is printed on both sides of the page. There are a total of 6 questions
(labeled a-f).
The Monet Company produces four types of picture frames, which we label 1, 2, 3 and 4.
The four types of frames differ with respect to size, shape and materials used. Each type
requires a certain amount of skilled labor, metal and glass, as shown in the table below.
This table also lists the unit selling price Monet charges for each type of frame. During
the coming week, Monet can purchase up to 4000 hours of skilled labor, 6000 ounces of
metal and 10,000 ounces of glass. The unit costs are $8.00 per labor hour, $0.50 per
ounce of metal and $0.75 per ounce of glass. Also, market constraints are such that it is
impossible to sell more than 1000 type 1 frames, 2000 type 2 frames, 500 type 3 frames
and 1000 type 4 frames, and Monet does not want to keep any frames in inventory at the
end of the week.
Skilled labor
Metal
Glass
Selling price
Frame #1
2
4
6
$28.50
Frame #2
1
2
2
$12.50
Frame #3
3
1
1
$29.25
Frame #4
2
2
2
$21.50
The production manager at Monet has set up the following linear program to determine
the mix of frames to produce that maximizes profit and stays within the resource
availability and maximum sales constraints.
Maximize 6x1 + 2x2 + 4x3 + 3x4
Subject to
2x1 + x2 + 3x3 + 2x4 ≤ 4000
4x1 + 2x2 + x3 + 2x4 ≤ 6000
6x1 + 2x2 + x3 + 2x4 ≤ 10,000
x1 ≤ 1000
x2 ≤ 2000
x3 ≤ 500
x4 ≤ 1000
x1, x2, x3, x4 ≥ 0
a. (4 points) The coefficient 6 in the objective function is the contribution to profit for
each type 1 frame, show (and explain) how this number is computed.
Contribution to profit = selling price – labor cost – metal cost – glass cost
= 28.50 – 2(8) – 4(.5) – 6(.75)
= $6
b. (4 points) Two weeks ago Monet produced 500 of each of the four types of frames.
Last week they produced 1000 of frame types 1, 2 and 4, and they produced none of
frame type 3. Is it feasible to produce either of these product mixes this week? Explain.
Would either of these solutions be optimal this week? Explain without referring to the
Excel spreadsheet or Solver reports.
By substituting x1=x2=x3=x4=500 into the seven constraints you obtain:
4000≤4000 binding
4500≤6000 non-binding
5500≤6000 non-binding
500≤1000 non-binding
500≤2000 non-binding
500≤500 binding
500≤1000 non-binding
Therefore, the solution is feasible. It is NOT optimal because the objective function is not
parallel to the binding constraints. And if you reduce the production of frame #3 by 1
unit you can increase the production of the more profitable frame #1 by I unit, thus
improving on the current solution.
By substituting x1=x2= x4=1000, x3=0 into the constraints you obtain:
5000>4000 not feasible
Therefore, the solution is NOT feasible and NOT optimal.
The production manager at Monet has used Excel and Solver to optimize the linear
programming model. Attached are an Excel spreadsheet, a Solver Answer report and a
Solver Sensitivity report for this problem.
c. (4 points) What is the optimal solution? How many of each frame type should Monet
produce? How much profit will this product mix yield? How many labor hours will they
use? How many ounces of metal and glass will they use? You may use any of the
information provided to answer this question.
Produce 1000 of frame #1, 800 of frame #2, 400 of frame #3 and 0 of frame #4
Profit for this mix is $9200
This mix uses 4000 hours of labor, 6000 ounces of metal and 8000 ounces of glass
d. (4 points) Monet’s sales department has been distributing a $1 discount coupon for
type 1 frames. What impact will this have on the optimal product mix? What impact will
this have on total profit? You may use any of the information provided to answer this
question.
A $1 discount will reduce the profit contribution of frame #1 by $1. This is less than the
allowable decrease so the product mix will not change. Total profit will decrease by
$1(1000) = $1000 (assuming that all customers redeem the coupon).
e. (4 points) Monet is considering purchasing additional metal from a new supplier.
What is the maximum they should pay and what is the maximum amount of metal they
should purchase at this price? Explain. Should Monet consider looking for an additional
supplier of glass? Why or why not? You may use any of the information provided to
answer this question.
Monet should pay up to $.90 (shadow price of $.40 + cost of metal$.50) for additional
metal up to an additional 2000 ounces (the allowable increase).
Monet already has excess glass so there is no need to seek an additional supplier.
f. (5 points) Monet is concerned about becoming too reliant on the production of a single
product. To address this concern they would like to make sure that no one frame type is
produced in a quantity that exceeds 40% of the total product mix. What changes should
they make to the linear programming model to implement this policy? Without resolving
the model, what impact would you expect these changes to have on the current optimal
product mix and profit?
The following constraints should be added:
x1 ≤ .4(x1+x2+x3+x4)
x2 ≤ .4(x1+x2+x3+x4)
x3 ≤ .4(x1+x2+x3+x4)
x4 ≤ .4(x1+x2+x3+x4)
The current solution violates the first of these constraints. Therefore, production of
frame #1 will be reduced, production of the other frame types will likely change and
profit will decrease.
Name____________________________________________________ Section_______