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# MIDTERM EXAM

## INTRODUCTIION TO MANAGEMENT SCIENCE

Question # 1 (6 points):
Find the complete optimal solution to this Linear Programming problem.

Subject to
12x1 + 4x2 >= 48
10x1 + 5x2 >= 50
4x1 + 8x2 >= 32
x1, x2 >= 0

## Question # 2 (12 points):

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:

## 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 labour are needed, while 300 hours
are needed at plant 2. A total of 5500 hours of labour are available for use at the two
plants. Additional labour hours can be purchased at \$20 per labour 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.

## 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

2
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? (6 points)

b) If the price of an additional hour of labor were reduced to \$4, would the company
purchase any additional labor? (6 points)

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? (6 points)

d) If the company were given 5 extra hours of labor, what would the profit become? (6
points)

## Question # 3 (12 points):

Margaret Young’s family owns five panels of farmland broken into a southwest
sector, north sector, west sector, and southwest sector. Young is involved primarily
in growing wheat, alfalfa, and barley crops and is currently preparing her
production plan for next year. The Pennsylvania Water Authority has just
announced its yearly water allotment, with the Young farm receiving 7,400 acre-
feet. Each parcel can only tolerate a specified amount of irrigation per growing
season, as specified below:

## Southeast 2000 3200

North 2300 3400
Northwest 600 800
West 1100 500
Southwest 500 600

Each of Young’s crops needs a minimum amount of water per acre and there is a
projected limit of each crop. Crop data follow:

feet)

## Wheat 110,000 bushels 1.6

Alfalfa 1,800 tons 2.9
Barley 2,200 tons 3.5

Young’s best estimate is that she can sell wheat at a net profit of \$2 per bushel, alfalfa at
\$40 per ton, and barley at \$50 per ton. One acre of land yields an average of 1.5 tons of
alfalfa and 2.2 tons of barley. The wheat yield is approximately 50 bushels per acre.

Formulate Young’s production plan. (Define the decision variables, objective function
and the constraints).
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