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ADM 2302 Midterm Exam (sample 2)

Problem 1 (12 points)

Consider the following linear program:

Max Z = X₁ – 2X₂

Subject to

– 4X₁ + 3X₂ <= 3

X₁ – X₂ <= 3

X₁, X₂ >= 0 

a) Graph the feasible region for the problem. (4 points)

b) Is the feasible region unbounded? Explain. (2 points)

c) Find the optimal solution. (4 points)

d) Does an unbounded feasible region imply that the optimal solution to the linear program will be unbounded? (2 points)

Problem 2 (20 points)

A large sporting goods store is placing an order for bicycles with its supplier. Four models can be ordered: the adult Open Trail, the adult Citscape, the girl’s Sea Sprite, and the boy’s Trail Blazer.  It is assumed that every bike ordered will be sold, and their profits, respectively, are 30, 25, 22, and 20.  The LP model should maximize the profit.  There are several conditions that the store needs to worry about.  One of these is space to hold the inventory.  The adult bikes need two feet, but each children’s bike needs only one foot.  The store has 500 feet of space.  There are 1200 hours of assembly time available.  The children’s bikes need 4 hours each; the Open Trail needs 5 hours and the Cityscape needs 6 hours.  The store would like to place an order for at least 275 bikes.

The problem when formulated as an LP and solved is as follows:

Max. Z = 30 X1 +25 X2 + 22 X3 + 20 X4

Subject to

C1 2 X1 + 2 X2 + X3 + X4 <= 500

C2 5 X1 + 6 X2 + 4 X3 + 4 X4 <= 1200

C3 X1 + X2 + X3 + X4 >= 275

X1, X2, X3, X4 >= 0

a) What would be the optimal order of each kind of bike and the optimal profit if the profit on the Cityscape increases to $35? (4 points)

b) Suppose you work in the store and you are offered 200 additional hours of assembly time for a total cost of $1200. What would you say yes or no? Justify. (3 points)

c) How much storage space will be left unused? Justify. (3 points)

d) What would the profit be if we require 5 more bikes in inventory? Justify. (3 points)

e) Which resource should the company work to increase, inventory space or assembly time? Justify. (2 points)

f) By how much the profit of the Trail Blazer bike has to increase in order to become profitable to order it? Justify. (2 points)

g) What would happen if the profit of Open trail drops to $28 and the profit of Citscape increases to $27? Justify. (3 points)

Problem 3 (10 points)

Sound Electronics, Inc., produces a battery-operated tape recorder at plants located in Martinsville, North Carolina; Plymouth, New York; and Franklin, Missouri.  The unit transportation cost for shipments from the three plants to distribution centers in Chicago, Dallas, and New York are as follows:

To

From Chicago Dallas New York 

Martinsville 1.45 1.60 1.40

Plymouth 1.10 2.25 0.60

Franklin 1.20 1.20 1.80 

After considering transportation costs, management had decided that under no circumstances will it use the Plymouth-Dallas route.

The plant capacities and distributor orders for the next month are as follows: 

Plant Capacity (units) Distributor Orders (units) 

Martinsville 400 Chicago 400

Plymouth 600 Dallas 400

Franklin 300 New York 400

Because of different wage scales at the three plants, the unit production cost varies from plant to plant.  Assuming the costs are $29.50 per unit at Martinsville, $31.20 per unit at Plymouth, and $30.35 per unit at Franklin.

Formulate the corresponding Linear Programming model that minimizes production and transportation costs (define decision variables the objective function and the constraints).

Problem 4:(18 points)

CSL is a chain of computer service stores.  The number of hours of skilled repair time that CSL requires during next five month is as follow:

Month 1 (January): 6,000 hours

Month 2 (February): 7,000 hours

Month 3 (March): 8,000 hours

Month 4 (April): 9,500 hours

Month 5 (May): 11,000 hours

At the beginning of January, 50 skilled technicians work for CSL.  Each skilled technician can work up to 160 hour per month.

In order to meet future demands, new technicians must be trained.  It takes one month to train a new technician.  During the month of training a trainee must be supervised for 50 hours by an experienced technician.

Each experienced technician is paid $2,000 a month (even if he or she does not work the full 160 hours).  During the month of training, a trainee is paid $1,000 a month.  At the end of each month, 5% of CSL’s experienced technicians quit to join Plum Computers.

Formulate the corresponding linear programming model, whose solution will enable CSL to minimize the total labor cost incurred in meeting the service requirements for the next five month (define decision variables the objective function and the constraints).

Hint:

Total labor cost = cost of paying the experienced technicians + cost of paying the trainees