Sensitivity Analysis Practice Problems
Problem #1: The Judson Cooperation Problem (Given on previous Midterm exam)
The Judson Corporation has acquired 100 lots on which it is about to build a home on each lot. Two styles
of homes are to be built, the “Cape Cod” (X1) and the “Ranch Home” (X2). Judson wishes to build these
homes over the next nine months. During this time, Judson will have available 13,000 man-hours of
bricklayer labor and 12,000 hours of carpenter labor. Each Cape Cod house requires 200 man-hours of
carpentry labor and 50 man-hours of bricklayer labor. Each Ranch Home requires 120 hours of bricklayer
labor and 100 man-hours of carpentry. The profit contribution of Cape Cod is projected to be $5,100 while
that of a Ranch Home is projected at $5,000. The problem when formulated as an LP and solved is as follows:
Max. Z = $5,100 X1 + $5,000 X2
Subject to
X1 +
X2 <= 100
200X1 + 100X2 <= 12000
50 X1 + 120X2 <= 13000
a) What is the optimal planned mix of homes that Judson will build? What will the total profit be?
b) How much Judson Corporation is willing to pay for an additional hour of bricklayer labor? What is the
allowable increase of hours for which Judson is willing to pay that price?
c) One of Judson’s salesmen who is native of Massachusetts feels certain that he could sell the Cape Cods
for $2000 more each than Judson is currently projecting. Should Judson change its planned mix of homes?
Justify.
d) A gentleman who owns 10 vacant lots adjacent to Judson’s 100 lots needs some money quickly and
offers to sell his 10 lots for $60,000. Should Judson buy? Justify
e) Suppose that Judson’s has 2000 additional hours of bricklayer labor and 1,000 hours less of Carpenter
labor. What effect will this have on the shadow prices? Justify.
Problem 2 (Given on a previous midterm exam)
A company produces tools at two plants and sells them to three customers. The cost of producing 1000
tools at a plant and shipping them to a customer is given in Table below:
Customer1
Customer2
Customer3
Plant 1
$60
$30
$160
Plant 2
$130
$70
$170
Customers 1 and 3 pay $200 per thousand tools; customer 2 pays $150 per thousand tools. To produce 1000
tools at plant 1, 200 hours of labor are needed, while 300 hours are needed at plant 2. A total of 5500 hours
of labor are available for use at the two plants. Additional labor hours can be purchased at $20 per labor
hour. Plant 1 can produce up to 10,000 tools and plant 2, up to 12,000 tools. Demand by each customer is
assumed unlimited.
If we let Xij = number of tools (in thousands) produced at plant i and shipped to customer j, and
L = number of additional hour purchased.
The problem when formulated as an LP and solved is as follows:
Max. Z = 140 X11 + 120 X12 + 40 X13 + 70 X21 + 80 X22 + 30 X23 - 20 L
Subject to
C1
1 X11 + 1 X12 + 1 X13 <= 10
C2
1 X21 + 1 X22 + 1 X23 <= 12
C3
200 X11 + 200 X12 + 200 X13 + 300 X21 + 300 X22
+ 300 X23 - 1 L <= 5500
a) If it costs $70 to produce 1000 tools at plant 1 and ship them to customer 1, what would be the new
solution to the problem and the profit?
b) If the price of an additional hour of labor were reduced to $4, would the company purchase any
additional labor?
c) A consultant offers to increase plant 1’s production capacity by 5000 tools for a cost of $400.
Should the company take the offer?
d) If the company were given 5 extra hours of labor, what would the profit become?