1. EMGT 501
HW #1
3.1-4
3.1-11
3.4-16
Due Day: Sep 12

2. 3.1-4 Use the graphical method to solve the problem:

3. (Machine Hours per Week)
The Omega Manufacturing Manufacturing Company has discontinued the production of a certain unprofitable product line. This act created considerable excess production capacity. Management is considering devoting this excess capacity to one ore more of three products; call them products 1, 2, and 3. The available capacity on the machines that might limit output is summarized in the following table:
Available Time
(Machine Hours per Week)
500
350
150
Machine Type
Milling machine
Lathe
Grinder

4. The number of machine hours required for each unit of the respective products is
Productivity coefficient (in machine hours per unit)
Machine Type
Milling machine
Lathe
Grinder
Product 1 9 5 3
Product 2 3 4
Product 3 5 2
The sales department indicates that the sales potential for products 1 and 2 exceeds the maximum production rate and that the sales potential for product 3 is 20 units per week. The unit profit would be $50, $20, and $25, respectively, on products 1, 2, and 3. The objective is to determine how much of each product Omega should produce to maximize profit.
(a) Formulate a linear programming model for this problem.
(b) Use a computer to solve this model by the simplex method.

5. A cargo plane has three compartments for storing cargo: front, center, and back. These compartments have capacity limits on both weight and space, as summarized below:
Weight
Capacity
(Tons)
Space
Capacity
(Cubic Feet)
Compartment
Front
Center
Back 12 18 10
7,000
9,000
5,000
Further more, the weight of the cargo in the respective compartment must be the same proportion of that compartment’s weight capacity to maintain the balance of the airplane.

6. The following four cargoes have been offered for shipment on an upcoming flight as space is available:
Weight
(Tons)
Volume
(Cubic Feet/Ton)
Profit
($/Ton)
Cargo 1 2 3 4 20 16 25 13
500
700
600
400
320
400
360
290
Any portion of these cargoes can be accepted. The objective is to determine how much (if any) of each cargo should be accepted and how to distribute each among the compartments to maximize the total profit for the flight.
(a) Formulate a linear programming model for this problem.
(b) Solve this model by the simplex method to find one of its
multiple optimal solutions.

7. EMGT 501
HW #1
Answer
3-1-4
3-1-11
3-4-16

8. 3-1-4 Max s.t. (1) (2) (3) (4) (3) (2) and (1) 10 (4) 8 6 (13,5)
Optimal Solution
with Z=31 4 2

9. 3-1-11
(a) Let

10. (b)

11. 3-4-16

13. (b)

14. EMGT 501
HW #2
4.4-6(b) (c)
Due Day: Sep. 19

15. 4.4-6 Consider the following problem.
(b) Work through the simplex method step by step in tabular form.
(c) Use a software package based on the simplex method to solve
the problem.

16. EMGT 501
HW #2
Answer
4.4-6
(b), (c)

17. . 3 / 230 with ) 80 , 50 20 ( *) *, solution Optimal = Z x 4.4-6 (b)
Basis Z X 1 2 3 4 5 6
RHS -2 -4 -3 60 40 80 -1
3/4
1/2
1/4 15
5/4
3/2
-1/4 25
-5/4
-3/4 35
11/6
5/6
2/3
230/3
1/3
-1/3
20/3 X
5/6 1
-1/6
2/3
50/3 3 X
-5/3
-2/3
-1/3 1
80/3 6
(c) . 3 /
230
with ) 80 , 50 20 ( *) *,
solution
Optimal 6 5 4 2 1 = Z x

18. Dual Formulation of DEA
EMGT 501
HW #3
6.1-4
6.1-5
Dual Formulation of DEA

19. 6.1-4
For each of the following linear programming models, give your recommendation on which is the more efficient way (probably) to obtain an optimal solution: by applying the simplex method directly to this primal problem or by applying the simplex method directly to the dual problem instead. Explain.
(a) Maximize
subject to
(b) Maximize
subject to
and
and

20. 6.1-5
Consider the following problem.
Maximize
subject to
and
(a) Construct the dual problem.
(b) Use duality theory to show that the optimal solution
for the primal problem has

21. Formulate the Dual Form of the DEA Problem
Primal Form of DEA
Max
s.t.

22. Dual Formulation of DEA
EMGT 501
HW#3
Answer
6.1-4
6.1-5
Dual Formulation of DEA

23. 6.1-4 (a) Dual formulation becomes
Min
s.t.
# of constraints of Dual = 3
# of constraints of Primal = 5
So, Dual is better than Primal because the size of B-1 in Dual is smaller than that of Primal.

24. (b) Dual formulation becomes
Min
s.t.
# of constraints of Dual = 5
# of constraints of Primal = 3
So, Primal is better than Dual because the size of B-1 in Primal is smaller than that of Dual.

25. 6.1-5
(a)
Min
s.t.
(b)
It is clear that Z*=0, y1*=0, y2*=0.

26. Primal Form
Max
s.t.
Dual Form
Min
s.t.

27. EMGT 501
HW#4
10.3-1
10.4-5

28. 10.3-1
You and several friends are about to prepare a lasagna dinner. The tasks to be performed, their immediate predecessors, and their estimated durations are as follows:
Tasks that
Task Task Description Must Precede Time
A Buy the mozzarella cheese* minutes
B Slice the mozzarella A minutes
C Beat 2 eggs minutes
D Mix eggs and ricotta cheese C minutes
E Cut up onions and mushrooms minutes
F Cook the tomato sauce E minutes
G Boil large quantity of water minutes
H Boil the lasagna noodles G minutes
I Drain the lasagna noodles H minutes
J Assemble all the ingredients I, F, D, B minutes
K Preheat the oven minutes
L Bake the lasagna J, K minutes

29. * There is none in the refrigerator.
(a) Construct the project network for preparing this dinner.
(b) Find all the paths and path lengths through this project
network. Which of these paths is a critical path?
(c) Find the earliest start time and earliest finish time for each
activity.
(d) Find the latest start time and latest finish time for each activity.
(e) Find the slack for each activity. Which of the paths is a
critical path?
(f) Because of a phone call, you were interrupted for 6 minutes when you should have been cutting the onions and mushrooms. By how much will the dinner be delayed? If you use your food processor, which reduces the cutting time from 7 to 2 minutes, will the dinner still be delayed?

30. 10.4-5
Sharon Lowe, vice president for marketing for the Electronic Toys Company, is about to begin a project to design an advertising campaign for a new line of toys. She wants the project completed within 57 days in time to launch the advertising campaign at the beginning of the Christmas season.
Sharon has identified the six activities (labeled A, B, …, F) needed to execute this project. Considering the order in which these activities need to occur, she also has constructed the following project network. A C E F
START
FINISH B D

31. Using the PERT three-estimate approach, Sharon has obtained the following estimates of the duration of each activity.
Optimistic Most Likely Pessimistic
Activity Estimate Estimate Estimate
A days days days
B days days days
C days days days
D days days days
E days days days
F days days days
(a) Find the estimate of the mean and variance of the duration of
each activity.
(b) Find the mean critical path.

32. (c) Use the mean critical path to find the approximate probability
that the advertising campaign will be ready to launch within
57 days.
(d) Now consider the other path through the project network.
Find the approximate probability that this path will be
completed within 57 days.
(e) Since these paths do not overlap, a better estimate of the
probability that the project will finish within 57 days can be
obtained as follows. The project will finish within 57 days if
both paths are completed within 57 days. Therefore, the
approximate probability that the project will finish within 57
days is the product of the probabilities found in parts (c) and
(d). Perform this calculation. What does this answer say
about the accuracy of the standard procedure used in part (c)?

33. HW #4
Answer
10.3-1
10.4-5

34. Start
10.3-1 a) 2 7 30 15 15 A C E G K 10 5 3 25 B D F H 10 J I 2 L 30
Finish

36. c, d & e)
Activity
Start A B C D E F G H I J K L
Finish ES 30 2 7 15 25 35 45 75 EF 30 35 2 5 7 32 15 25 27 45 75 LS 30 32 3 10 8 23 33 35 45 75 LF 30 35 32 10 23 33 45 75
Slack 30 3 8
Critical Path
Yes No
Critical Path: Start A B J L Finish

37. f )
Dinner will be delayed 3 minutes because of the phone call. If the food processor is used, then dinner will not be delayed because there was 3 minutes of slack and 5 minutes of cutting time saved.

38. 10.4 – 5 a) 4 6 F 18 E 9 27 D 1 15 C 16 23 B 12 A
Activity b)

40. HW #5
14.5-2

41. 14.5-2
Consider the game having the following payoff table.
(a) Use the approach described in Sec to formulate the problem of finding optimal mixed strategies according to the minimax criterion as a linear programming problem.
(b) Use the simplex method to find these optimal mixed strategies.

42. Answer
HW #5
14.5-2

43. 14.5-2 a) b)
Solve Automatically by the Simplex Method
Optimal Solution
Value of the Objective Function: Z = 2.368

44. HW #6
Due Day: Nov. 7

45.
The leading brewery on the West Coast (labeled A) has hired an OR analyst to analyze its market position. It is particularly concerned about its major competitor (labeled B). The analyst believes that brand switching can be modeled as a Markov chain using three states, with states A and B representing customers drinking beer produced from the aforementioned breweries and the analyst has constructed the following (one-step) transition matrix from past data.
What are the steady-state market shares for the two major breweries?

46.
A soap company specializes in a luxury type of bath soap. The sales of this soap fluctuate between two levels - “Low” and “High” - depending upon two factors: (1) whether they advertise, and (2) the advertising and marketing of new products being done by competitors. The second factor is out of the company’s control, but it is trying to determine what its own advertising policy should be. For example, the marketing manager’s proposal is to advertise when sales are low but not to advertise when sales are high. Advertising in any quarter of a year has its primary impact on sales in the following quarter. Therefore, at the beginning of each quarter, the needed information is available to forecast accurately whether sales will be low or high that quarter and to decide whether to advertise that quarter.

47. The cost of advertising is $1 million for each quarter of a year in which it is done. When advertising is done during a quarter, the probability of having high sales the next quarter is 1/2 or 3/4, depending upon whether the current quarter’s sales are low or high. These probabilities go down to 1/4 or 1/2 when advertising is not done during the current quarter. The company’s quarterly profits (excluding advertising costs) are $4 million when sales are high but only $2 million when sales are low. (Hereafter, use units of millions of dollars.)
(a) Construct the (one-step) transition matrix for each of the following advertising strategies: (i) never advertise, (ii) always advertise, (iii) follow the marketing manager’s proposal.
(b) Determine the steady-state probabilities manually for each of the three cases in part (a).
(c) Find the long-run expected average profit (including a deduction for advertising costs) per quarter for each of the three advertising strategies in part (a). Which of these strategies is best according to this measure of performance?

48. HW #6
ANSWER
16.5-4
16.5-6

49. 16.5-4

50. 16.5-6 1 1 1

52. HW #7
17.2-3
17.5-1
17.5-9
Due Day: Dec. 2

53. 17.2-3
Mom-and-Pop’s Grocery Store has a small adjacent parking lot with three parking spaces reserved for the store’s customers. During store hours, cars enter the lot and use one of the spaces at a mean rate of 2 per hour. For n = 0, 1, 2, 3, the probability Pn that exactly n spaces currently are being used is P0 = 0.2, P1 = 0.3, P2 = 0.3, P3 = 0.2.
(a) Describe how this parking lot can be interpreted as being a queueing system. In particular, identify the customers and the servers. What is the service being provided? What constitutes a service time? What is the queue capacity?
(b) Determine the basic measures of performance - L, Lq, W, and Wq - for this queueing system.
(c) Use the results from part (b) to determine the average length of time that a car remains in a parking space.

54. 17.5-1
Consider the birth-and-death process with all
and for n = 3, 4, …
(a) Display the rate diagram.
(b) Calculate P0, P1, P2, P3, and Pn for n = 4, 5, ...
(c) Calculate L, Lq, W, and Wq.

55. 17.5-9
A certain small grocery store has a single checkout stand with a full-time cashier. Customers arrive at the stand “randomly” (i.e., a Poisson input process) at a mean rate of 30 per hour. When there is only one customer at the stand, she is processed by the cashier alone, with an expected service time of 1.5 minutes. However, the stock boy has been given standard instructions that whenever there is more than one customer at the stand, he is to help the cashier by bagging the groceries. This help reduces the expected time required to process a customer to 1 minute. In both cases, the service-time distribution is exponential.
(a) Construct the rate diagram for this queueing system.
(b) What is the steady-state probability distribution of the number of customers at the checkout stand?
(c) Derive L for this system. (Hint: Refer to the derivation of L for the M/M/1 model at the beginning of Sec ) Use this information to determine Lq, W, and Wq.

56. HW #7
ANSWER
17.2-3
17.5-1
17.5-9

57. (a)
A parking lot is a queueing system for providing cars with parking opportunities.
The parking spaces are servers.
The service time is the amount of time a car spends in a system.
The queue capacity is 0.

58. (b)
(c)

59. 17.5-1
(a) 3 2 1
. . . 1 2 3 4 2 2 2 2

60. (b)

61. (c)

62. 17.5-9
(a)
. . . 1 2 3

63. (b)

64. (c)

65. HW #8
19.3-2
Due Day: Dec. 12

66. 19.3-2
The demand for a product is 600 units per week, and the items are with drawn at a constant rate. The setup cost for placing an order to replenish inventory is $25. The unit cost of each item is $3, and the inventory holding cost is $0.05 per item per week.
(a) Assuming shortages are not allowed, determine how often
to order and what size the order should be.
(b) If shortages are allowed but cost $2 per item per week,
determine how often to order and what size the order should be.

67.
In the basic EOQ model, suppose the stock is replenished uniformly (rather than instantaneously) at the rate of b items per unit time until the order quantity Q is fulfilled. Withdrawals from the inventory are made at the rate of a items per unit time, where a < b. Replenishments and withdrawals of the inventory are made simultaneously. For example, if Q is 60, b is 3 per day, and a is 2 per day, then 3 units of stock arrive each day for days 1 to 20, 31 to 50, and so on, whereas units are withdrawn at the rate of 2 per day every day. The diagram of inventory level versus time is given below for this example.

68. HW #8
Answer
19.3-2

69. 19.3-2
(a)
(b)

70.
(a)
(b)