Practice Exam
DS 523
Spring 2008
1. Let A, B, and C be the amounts invested in companies A, B, and C. If no more than 50% of the total
investment can be in company B, then
a. B  5
b. A  .5B + C  0
c. .5A  B  .5C  0
d. .5A + .5B  .5C  0
ANS: D
2. The problem which deals with the distribution of goods from several sources to several destinations is the
a. maximal flow problem
b. transportation problem
c. assignment problem
d. shortest-route problem
ANS: B
3. The objective of the transportation problem is to
a. identify one origin that can satisfy total demand at the destinations and at the same time
minimize total shipping cost.
b. minimize the number of origins used to satisfy total demand at the destinations.
c. minimize the number of shipments necessary to satisfy total demand at the destinations.
d. minimize the cost of shipping products from several origins to several destinations.
ANS: D
4. The assignment problem constraint x31 + x32 + x33 + x34  2 means
a. agent 3 can be assigned to 2 tasks.
b. agent 2 can be assigned to 3 tasks.
c. a mixture of agents 1, 2, 3, and 4 will be assigned to tasks.
d. there is no feasible solution.
ANS: A
5. Constraints in a transshipment problem
a. correspond to arcs.
b. include a variable for every arc.
c. require the sum of the shipments out of an origin node to equal supply.
d. All of the alternatives are correct.
ANS: B
6. A constraint with a positive slack value
a. will have a positive dual price.
.b. will have a negative dual price.
c. will have a dual price of zero.
d. has no restrictions for its dual price.
ANS: C
7. The amount by which an objective function coefficient can change before a different set of values for the
decision variables becomes optimal is the
a. optimal solution.
b. dual solution.
c. range of optimality.
d. range of feasibility.
ANS: C
8. The 100% Rule compares
a. proposed changes to allowed changes.
b. new values to original values.
c. objective function changes to right-hand side changes.
d. dual prices to reduced costs.
ANS: A
9. A section of output from The Management Scientist is shown here.
Variable
Lower Limit
Current Value
Upper Limit
1
60
100
120
What will happen to the solution if the objective function coefficient for variable 1 decreases by 20?
a. Nothing. The values of the decision variables, the dual prices, and the objective function
will all remain the same.
b. The value of the objective function will change, but the values of the decision variables
and the dual prices will remain the same.
c. The same decision variables will be positive, but their values, the objective function value,
and the dual prices will change.
d. The problem will need to be resolved to find the new optimal solution and dual price.
ANS: B
10. The amount that the objective function coefficient of a decision variable would have to improve before
that variable would have a positive value in the solution is the
a. dual price.
b. surplus variable.
c. reduced cost.
d. upper limit.
ANS: C
1.Tots Toys makes a plastic tricycle that is composed of three major components: a handlebar-front wheel-pedal
assembly, a seat and frame unit, and rear wheels. The company has orders for 12,000 of these tricycles.
Current schedules yield the following information.
Component
Front
Seat/Frame
Rear wheel (each)
Available
Requirements
Plastic
3
4
.5
50000
Time
10
6
2
160000
Space
2
2
.1
30000
Cost to
Manufacture
8
6
1
Cost to
Purchase
12
9
3
The company obviously does not have the resources available to manufacture everything needed for the
completion of 12000 tricycles so has gathered purchase information for each component. Develop a linear
programming model to tell the company how many of each component should be manufactured and how
many should be purchased in order to provide 12000 fully completed tricycles at the minimum cost.
ANS:
Let
FM = number of fronts made
SM = number of seats made
WM = number of wheels made
FP = number of fronts purchased
SP = number of seats purchased
WP = number of wheels purchased
Min
8FM + 6SM + 1WM + 12FP + 9SP + 3WP
3FM + 4SM + .5WM  50000
10FM + 6SM + 2WM  160000
2FM + 2SM + .1WM  30000
FM + FP  12000
SM + SP  12000
WM + WP  24000
FM, SM, WM, FP, SP, WP  0
2.Canning Transport is to move goods from three factories to three distribution centers. Information about the
move is given below. Give the network model and the linear programming model for this problem.
s.t.
Source
A
B
C
Supply
200
100
150
Destination
X
Y
Z
Demand
50
125
125
Shipping costs are:
Source
A
B
C
Destination
X
Y
Z
3
2
5
9
10
-5
6
4
(Source B cannot ship to destination Z)
ANS:
Min
3XAX + 2XAY + 5XAZ + 9XBX + 10XBY + 5XCX + 6XCY + 4XCZ
XAX + XAY + XAZ  200
XBX + XBY  100
XCX + XCY + XCZ  150
XDX + XDY + XDZ  50
XAX + XBX + XCX + XDX = 250
X AY + XBY + XCY + XDY = 125
XAZ + XBZ + XCZ + XDZ = 125
Xij  0
s.t.
PTS: 1
TOP: Transportation problem
3.The binding constraints for this problem are the first and the horizontal axis.
Min
x1 + 2x2
s.t.
x1 + x2  300
2x1 + x2  400
2x1 + 5x2  750
x1 , x2  0
a.
b.
c.
d.
e.
Keeping c2 fixed at 2, over what range can c1 vary before there is a change in the optimal
solution point?
Keeping c1 fixed at 1, over what range can c2 vary before there is a change in the optimal
solution point?
If the objective function becomes Min 1.5x1 + 2x2, what will be the optimal values of x1,
x2, and the objective function?
If the objective function becomes Min 7x1 + 6x2, what constraints will be binding?
Find the dual price for each constraint in the original problem.
ANS:
a.
b.
c.
d.
e.
.8  c1  2
1  c2  2.5
x1 = 300, x2 = 0, z = 450
Constraints 1 and 3 will be binding.
Dual prices are 1, 0, 0.