Answers – April 22, 2002
Correct choices shown in bold, underlined

IFSM 300

Introduction to Management Science

Sections 0101 and 0201

Dr. R. Robinson

Exam 2

April 15, 2002

This is an open-book, in-class examination containing 25 questions, each worth 4 points, totaling 100 points.

  1. In an LP model, a lower limit (that is, smallest amount permitted) usually is described by a ______constraint:

(circle one)

  2. =
  4. None of the above.
  1. A slack variable in an LP model represents:

(circle one)

  1. The amount by which the left side of a  constraint is larger than the right side.
  2. The amount by which the left side of a  constraint is larger than the right side.
  3. The amount by which the left side of a  constraint is smaller than the right side.
  4. The amount by which the left side of a  constraint is smaller than the right side.
  5. None of the above.
  1. An LP model solution that satisfies all of the constraints is called:

(circle one)

  1. An optimum solution.
  2. A feasible solution.
  3. A nonnegative solution.
  4. None of the above.

Table 1 Multiproduct Manufacturing Company plans to produce x1 units of product 1 and x2 units of product 2 in the next month. MMC’s prices and costs are:

Product / Selling Price Per Unit / Production Cost Per Unit
1 / $45 / $28
2 / $75 / $52
  1. See table 1. Management wants to improve profit. Therefore their objective in an LP model would be to select x1 and x2 so as to:

(circle one)

  1. MIN 28 x1 + 52 x2
  2. MAX 45 x1 + 75 x2
  3. MIN 45 x1 + 75 x2
  4. MAX 17 x1 + 23 x2
  5. None of the above.
  1. A  constraint in an LP model is binding only if, when evaluated at optimality, the constraint’s:

(circle one)

  1. Left side is  the right side.
  2. Left side is = the right side.
  3. Left side is  the right side.
  4. None of the above.

Table 2 You have formulated this LP model:

MAX 20 x + 10 y

ST

12 x + 15 y  180

15 x + 10 y  150

3 x – 8 y  0

(x, y  0)

In the following graph, x is plotted on the horizontal axis and y on the vertical axis. The solid lines are boundaries of the constraints. The dotted line is an objective-function line.

6. See table 2. On the graph, the feasible region is:

(circle one)

  1. I
  2. II
  3. III
  4. IV
  5. V
  6. None of the above.

7. See table 2. The equation of the objective-function line shown on the graph is:

(circle one)

  1. 20 x + 10 y = 7
  2. 20 x + 10 y = 15
  3. 20 x + 10 y = 30
  4. 20 x + 10 y = 140
  5. None of the above.
  1. Without doing any calculations, you can tell by looking at the graph that the optimum corner (extreme point) is:

(circle one)

  1. Point A
  2. Point B
  3. Point C
  4. Point D
  5. Point E
  6. None of the above.
  1. In Lindo output, a negative dual price (shadow price) shows, per unit increase in the right side of a constraint, the:

(circle one)

  1. Increase in the optimum objective-function value.
  2. Decrease in the optimum objective-function value.
  3. Improvement in the optimum objective-function value.
  4. Worsening in the optimum objective-function value.
  5. None of the above.
  1. The Lindo sensitivity report shows the range over which a constraint’s right side may

be revised without changing that constraint’s dual price. This range, in the

“Righthand Side Ranges” section of the report, is:

(circle one)

  1. The interval between 0 and the ALLOWABLE INCREASE.
  2. The difference between the ALLOWABLE INCREASE and the ALLOWABLE DECREASE.
  3. The interval starting at the current righthand side minus the ALLOWABLE DECREASE and going up to the current righthand side plus the ALLOWABLE INCREASE.
  4. None of the above.

Table 3 You want to examine the dual of an LP model. Your original model is:

MAX 3 x1 + 2 x2 + 6 x3

ST

4 x1 + 2 x2 + 3 x3 100

2 x1 + x2 – 2 x3 200

4 x2 + x3 200

(x1, x2, x3 0)

11. See table 3. The number of constraints in the dual model is:

(circle one)

  1. 1
  2. 2
  3. 3 (excluding nonnegativity conditions)
  4. 4 (including nonnegativity conditions)

12. See table 3. The first dual-model constraint is:

(circle one)

  1. 4 u1 + 2 u2 + 4 u3 3
  2. – 4 u1 + 2 u2 – 4 u3  3
  3. 4 u1 + 2 u2 3
  4. – 4 u1 + 2 u2 3
  5. None of the above.

13. Suppose a dummy source must be added to a transportation model. Then the cost per

unit of shipping to any destination from this dummy source is entered in the

transportation table as:

(circle one)

  1. $M
  2. $0
  3. The minimum of any unit shipping cost to that destination.
  4. None of the above.

Table 4 Longhaul Logistics is trucking cargo from three factories to three distribution centers. Management wants to minimize total shipping cost. Various data are:

Source / Supply / Destination / Demand
A / 200 / X / 50
B / 100 / Y / 125
C / 150 / Z / 125

Unit shipping costs:

Destination
Source / X / Y / Z
A / 3 / 2 / 5
B / 9 / 10 / --
C / 5 / 6 / 4
(Source B cannot ship to destination Z)

14. See table 4. You decide to set this up as a simpler, natural LP model. Then, in Lindo,

theobjective function contains this term for shipment from source B to destination Z:

(circle one)

  1. 9999 x23
  2. x23
  3. No term
  4. None of the above.

Table 5 The following diagram shows a transshipment model. The numbers on the left are supplies, the numbers on the arcs are unit shipping costs, and the numbers on the right are demands.

15. See table 5. Your model for this problem is a simpler, natural LP model in network-

flow form. The flow at each node is described by a constraint. In Lindo style, the

constraint for node 2 is:

(circle one)

  1. x24 + x25 = 400
  2. x24 + x25 – x32 = 0
  3. x24 + x25 – x32 = 400
  4. 5 x24 + 6 x25 – 2 x32 = 0
  5. None of the above.

16. An assignment model with four tasks to be accomplished and four individuals

available to do those tasks, as entered into Lindo, will have _____ decision variables.

(circle one)

  1. 4
  2. 8
  3. 16
  4. None of the above.

17. In an assignment model, you prepare the assignment table that is ready for application

of the Hungarian method by applying the idea of penalties (or “opportunity costs”)

when:

(circle one)

  1. supply of to-be-assigned entities is greater than demand from places-to-be-

assigned entities.

B.demand from places-to-be-assigned entities is greater than the supply of

to-be-assigned entities.

  1. The objective function is to be minimized.
  2. The objective function is to be maximized.
  3. None of the above.

Table 6 The following diagram shows times to distribute a message in a network.

18. See table 6. You want to determine how messages should be passed within the

network (how nodes should be connected) in order to minimize the total of all times

between connected nodes. This may be accomplished by formulating a ______

model:

(circle one)

  1. Traveling salesman.
  2. Maximum flow.
  3. Shortest route.
  4. Minimum spanning tree.
  5. None of the above.

19. See table 6 and question 18. The minimized total time is:

(circle one)

  1. 33
  2. 31
  3. 30
  4. 28
  5. 27

Table 7 The times to make a round of pizza deliveries are shown in this diagram:


20. See table 7. The Pisa Pizza delivery person wants to make the rounds and come back to the shop in the shortest time. This can be set up as a ______model:

(circle one)

  1. Shortest route.
  2. Traveling salesman.
  3. Minimum spanning tree.
  4. Maximum flow.
  5. None of the above.

21. See table 7 and question 20. The number of arcs in the solution will be:

(circle one)

  1. 3
  2. 4
  3. 5
  4. None of the above.

Table 8 A beverage distributor makes deliveries from its warehouse to each of its six

other locations. The permitted paths and associated distances are shown in this diagram:

22. See table 8. The distributor wants to know the shortest route to each location.

Applying the shortest-route algorithm, and starting at the warehouse, the first

connection selected is W-2, and the second W-5. The third connection should be:

(circle one)

  1. W-6
  2. 2-3
  3. 5-6
  4. None of the above.

23. See table 8 and question 22. The label at node 7 will be:

(circle one)

  1. [14,6]
  2. [15,6]
  3. [19,4]
  4. None of the above.

Table 9 Rollingstock Railway Company operates a network of lines entering and leaving a city. Speed limits, track reconstruction, and train-length restrictions lead to the flow diagram below, where the numbers represent how many train cars can pass per hour.

24. See table 9. RRC wants to find the maximum number of train cars per hour than may

move through the city. Applying the appropriate algorithm, and starting at the

beginning, the largest number of cars per hour that may be planned in the first step

(without splitting the flow partway though the network) is:

(circle one)

  1. 300
  2. 400
  3. 500
  4. 600

25. See table 9 and question 24. When RRC completes its analysis, the planned

maximum total number of train cars per hour is:

  1. 1,100
  2. 1,200
  3. 1,300
  4. 1,400
  5. 1,500

End of Exam

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