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3.1-11 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

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

0

Product 3

5

0

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.

3.4-16 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.

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

20

16

25

13

1

2

3

4

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.

(a) Let

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.

Basis

Z

X

X

X

X

X

X

RHS

1

2

3

4

5

6

Z

1

-2

-4

-3

0

0

0

0

X

0

3

4

2

1

0

0

60

4

X

0

2

1

2

0

1

0

40

5

X

0

1

3

2

0

0

1

80

6

Z

1

1

0

-1

1

0

0

60

X

0

3/4

1

1/2

1/4

0

0

15

2

X

0

5/4

0

3/2

-1/4

1

0

25

5

X

0

-5/4

0

1/2

-3/4

0

1

35

6

Z

1

11/6

0

0

5/6

2/3

0

230/3

X

0

1/3

1

0

1/3

-1/3

2

=

Optimal

solution

(

x

*,

x

*,

x

*,

x

*,

x

*,

x

*)

(

0

,

20

/

3

,

50

/

3

,

0

,

0

,

80

/

3

)

1

2

3

4

5

6

=

with

Z

230

/

3

.

0

20/3

X

0

5/6

0

1

-1/6

2/3

0

50/3

3

X

0

-5/3

0

0

-2/3

-1/3

1

80/3

6

(c)

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

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

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.

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.

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* 30 minutes

B Slice the mozzarella A 5 minutes

C Beat 2 eggs 2 minutes

D Mix eggs and ricotta cheese C 3 minutes

E Cut up onions and mushrooms 7 minutes

F Cook the tomato sauce E 25 minutes

G Boil large quantity of water 15 minutes

H Boil the lasagna noodles G 10 minutes

I Drain the lasagna noodles H 2 minutes

J Assemble all the ingredients I, F, D, B 10 minutes

K Preheat the oven 15 minutes

L Bake the lasagna J, K 30 minutes

* 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?

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

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 12 days 12 days 12 days

B 15 days 21 days 39 days

C 12 days 15 days 18 days

D 18 days 27 days 36 days

E 12 days 18 days 24 days

F 2 days 5 days 14 days

(a) Find the estimate of the mean and variance of the duration of

each activity.

(b) Find the mean critical path.

(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)?

ES

0

0

30

0

2

0

7

0

15

25

35

0

45

75

EF

0

30

35

2

5

7

32

15

25

27

45

15

75

75

LS

0

0

30

30

32

3

10

8

23

33

35

30

45

75

LF

0

30

35

32

35

10

35

23

33

35

45

45

75

75

Slack

0

0

0

30

30

3

3

8

8

8

0

30

0

0

Activity

Start

A

B

C

D

E

F

G

H

I

J

K

L

Finish

Critical Path

Yes

Yes

Yes

No

No

No

No

No

No

No

Yes

No

Yes

Yes

Critical Path: Start A B J L Finish

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.

14.5-2

Consider the game having the following payoff table.

(a) Use the approach described in Sec. 14.5 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.

a)

b)

Solve Automatically by the Simplex Method

Optimal Solution

Value of the Objective Function: Z = 2.368

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?

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.

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?

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 Pnthat 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.

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 Pnfor n = 4, 5, ...

(c) Calculate L, Lq, W, and Wq.

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. 17.6.) Use this information to determine Lq, W, and Wq.

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.

(c)

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.

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.

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