Problems


[Page 422 ( continued )]
1.

A manufacturing company produces products 1, 2, and 3. The three products have the following resource requirements and produce the following profit:

Product

Labor (hr./unit)

Material (lb./unit)

Profit ($/unit)

1

5

4

$3

2

2

6

5

3

4

3

2


At present the firm has a daily labor capacity of 240 available hours and a daily supply of 400 pounds of material. The general linear programming formulation for this problem is as follows :

Management has developed the following set of goals, arranged in order of their importance to the firm:

  1. Because of recent labor relations difficulties, management wants to avoid underutilization of normal production capacity.

  2. Management has established a satisfactory profit level of $500 per day.

  3. Overtime is to be minimized as much as possible.

  4. Management wants to minimize the purchase of additional materials to avoid handling and storage problems.

Formulate a goal programming model to determine the number of each product to produce to best satisfy the goals.

2.

In Problem 15 in Chapter 6, Computers Unlimited sells microcomputers and distributes them from three warehouses to four universities. The available supply at the three warehouses, demand at the four universities, and shipping costs are shown in the following table:

 

University

 

Warehouse

Tech

A&M

State

Central

Supply

Richmond

$22

17

30

18

420

Atlanta

15

35

20

25

610

Washington

28

21

16

14

340

Demand

520

250

400

380

 


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Instead of its original objective of cost minimization, Computers Unlimited has indicated the following goals, arranged in order of their importance:

  1. A&M has been one of its better long- term customers, so Computers Unlimited wants to meet all of A&M's demands.

  2. Because of recent problems with a trucking union, it wants to ship at least 80 units from the Washington warehouse to Central University.

  3. To maintain the best possible relations with all its customers, Computers Unlimited would like to meet no less than 80% of each customer's demand.

  4. It would like to keep total transportation costs to no more than 110% of the $22,470 total cost achieved with the optimal allocation, using the transportation solution method.

  5. Because of dissatisfaction with the trucking firm it uses for the Atlanta-to-State deliveries, it would like to minimize the number of units shipped over this route.

  1. Formulate a goal programming model for this problem to determine the number of microcomputers to ship on each route to achieve the goals.

  2. Solve this model by using the computer.

3.

The Bay City Parks and Recreation Department has received a federal grant of $600,000 to expand its public recreation facilities. City council representatives have demanded four different types of facilitiesgymnasiums, athletic fields, tennis courts, and swimming pools. In fact, the demand by various communities in the city has been for 7 gyms, 10 athletic fields, 8 tennis courts, and 12 swimming pools. Each facility costs a certain amount, requires a certain number of acres, and is expected to be used a certain amount, as follows:

Facility

Cost

Required Acres

Expected Usage (people/week)

Gymnasium

$80,000

4

1,500

Athletic field

24,000

8

3,000

Tennis court

15,000

3

500

Swimming pool

40,000

5

1,000


The Parks and Recreation Department has located 50 acres of land for construction (although more land could be located, if necessary). The department has established the following goals, listed in order of their priority:

  1. The department wants to spend the total grant because any amount not spent must be returned to the government.

  2. The department wants the facilities to be used by a total of at least 20,000 people each week.

  3. The department wants to avoid having to secure more than the 50 acres of land already located.

  4. The department would like to meet the demands of the city council for new facilities. However, this goal should be weighted according to the number of people expected to use each facility.

  1. Formulate a goal programming model to determine how many of each type of facility should be constructed to best achieve the city's goals.

  2. Solve this model by using the computer so that the solution values are integers.

4.

A farmer in the Midwest has 1,000 acres of land on which she intends to plant corn, wheat, and soybeans. Each acre of corn costs $100 for preparation, requires 7 worker-days of labor, and yields a profit of $30. An acre of wheat costs $120 to prepare, requires 10 worker-days of labor, and yields $40 profit. An acre of soybeans costs $70 to prepare, requires 8 worker-days, and yields $20 profit. The farmer has taken out a loan of $80,000 for crop preparation and has contracted with a union for 6,000 worker-days of labor. A midwestern granary has agreed to purchase 200 acres of corn, 500 acres of wheat, and 300 acres of soybeans. The farmer has established the following goals, in order of their importance:


[Page 424]
  1. To maintain good relations with the union, the labor contract must be honored; that is, the full 6,000 worker-days of labor contracted for must be used.

  2. Preparation costs should not exceed the loan amount so that additional loans will not have to be secured.

  3. The farmer desires a profit of at least $105,000 to remain in good financial condition.

  4. Contracting for excess labor should be avoided.

  5. The farmer would like to use as much of the available acreage as possible.

  6. The farmer would like to meet the sales agreement with the granary. However, the goal should be weighted according to the profit returned by each crop.

  1. Formulate a goal programming model to determine the number of acres of each crop the farmer should plant to satisfy the goals in the best possible way.

  2. Solve this model by using the computer.

5.

The Growall Fertilizer Company produces three types of fertilizerSupergro, Dynaplant, and Soilsaver. The company has the capacity to produce a maximum of 2,000 tons of fertilizer in a week. It costs $800 to produce a ton of Supergro, $1,500 for Dynaplant, and $500 for Soilsaver. The production process requires 10 hours of labor for a ton of Supergro, 12 hours for a ton of Dynaplant, and 18 hours for a ton of Soilsaver. The company has 800 hours of normal production labor available each week. Each week the company can expect a demand for 800 tons of Supergro, 900 tons of Dynaplant, and 1,100 tons of Soilsaver. The company has established the following goals, in order of their priority:

  1. The company does not want to spend over $20,000 per week on production, if possible.

  2. The company would like to limit overtime to 100 hours per week.

  3. The company wants to meet demand for all three fertilizers; however, it is twice as important to meet the demand for Supergro as it is to meet the demand for Dynaplant, and it is twice as important to meet the demand for Dynaplant as it is to meet the demand for Soilsaver.

  4. It is desirable to avoid producing under capacity, if possible.

  5. Because of union agreements, the company wants to avoid underutilization of labor.

  1. Formulate a goal programming model to determine the number of tons of each brand of fertilizer to produce to satisfy the goals.

  2. Solve this model by using the computer.

6.

The Barrett Textile Mill was checked by inspectors enforcing Occupational Safety and Health Administration (OSHA) codes. The inspectors found violations in four categories: hazardous materials, fire protection, hand- powered tools, and machine guarding. In each category the mill was not in 100% compliance. Each percentage point of increase in the compliance level in each category will reduce the frequency of accidents, decrease the accident cost per worker, and constitute progress toward satisfying the OSHA compliance level. However, achieving compliance does cost the mill money. The following table shows the benefits (in accident frequency and accident cost per worker) and the costs of a percentage point increase in compliance in each category:

Category

Accident Frequency Reduction (accidents/10 5 hr. of exposure)

Accident Cost/Worker Reduction

Cost/Percentage Point Compliance

1. Hazardous materials

0.18

$1.21

$135

2. Fire protection

0.11

0.48

87

3. Hand-powered tools

0.17

0.54

58

4. Machine guarding

0.21

1.04

160



[Page 425]

To achieve 100% compliance in all four categories, the mill would have to increase compliance in hazardous materials by 60 percentage points (i.e., it is now at 40% compliance), in fire protection by 28 percentage points, in hand-powered tools by 35 percentage points, and in machine guarding by 17 percentage points. However, the management of the mill faces a dilemma, in that only $52,000 is available to spend on safety. Any larger expenditure could jeopardize the financial standing of the mill. Thus, management hopes to achieve a level of accident reduction and compliance that is within the company's budget limitation and that will satisfy OSHA authorities enough to temporarily delay punitive action. Therefore, management has established four goals, listed here in order of importance:

  1. Do not exceed the budget constraint of $52,000.

  2. Achieve the percentage increases in compliance necessary to achieve 100% compliance in each category.

  3. Achieve total accident frequency reduction of 20 accidents/105 hours of exposure. (This goal denotes management's desire to minimize the frequency of accidents even if 100% compliance cannot be achieved in all categories.)

  4. Reduce the total accident cost per worker by $115.

  1. Formulate a goal programming model to determine the percentage points of compliance needed in each category to satisfy the goals.

  2. Solve this model by using the computer.

7.

Solve the following goal programming model graphically and by using the computer:

8.

Solve the following goal programming model graphically and by using the computer:

9.

Solve the following goal programming model by using the computer:

10.

The Wearever Carpet Company manufactures two brands of carpetshag and sculpturedin 100-yard lots. It requires 8 hours to produce one lot of shag carpet and 6 hours to produce one lot of sculptured carpet. The company has the following production goals, in prioritized order:


[Page 426]
  1. Do not underutilize production capacity, which is 480 hours.

  2. Achieve product demand of 40 (100-yard) lots for shag and 50 (100-yard) lots for sculptured carpet. Meeting demand for shag is more important than meeting demand for sculptured, by a ratio of 5 to 2.

  3. Limit production overtime to 20 hours.

  1. Formulate a goal programming model to determine the amount of shag and sculptured carpet to produce to best meet the company's goals.

  2. Solve this model by using the computer.

11.

The East Midvale Textile Company produces denim and brushed-cotton cloth. The average production rate for both types of cloth is 1,000 yards per hour , and the normal weekly production capacity (running two shifts) is 80 hours. The marketing department estimates that the maximum weekly demand is for 60,000 yards of denim and 35,000 yards of brushed cotton. The profit is $3.00 per yard for denim and $2.00 per yard for brushed cotton. The company has established the following four goals, listed in order of importance:

  1. Eliminate underutilization of production capacity to maintain stable employment levels.

  2. Limit overtime to 10 hours.

  3. Meet demand for denim and brushed cotton weighted according to profit for each.

  4. Minimize overtime as much as possible.

  1. Formulate a goal programming model to determine the number of yards (in 1,000-yard lots) to produce to satisfy the goals.

  2. Solve this model by using the computer.

12.

The Oregon Atlantic Company produces two kinds of papernewsprint and white wrapping paper (butcher paper). It requires 5 minutes to produce a yard of newsprint and 8 minutes to produce a yard of wrapping paper. The company has 4,800 minutes of normal production capacity available each week. The profit is $0.20 for a yard of newsprint and $0.25 for a yard of wrapping paper. The weekly demand is for 500 yards of newsprint and 400 yards of wrapping paper. The company has established the following goals, in order of priority:

  1. Limit overtime to 480 minutes.

  2. Achieve a profit of $300 each week.

  3. Fulfill the demand for the products in order of magnitude of their profits.

  4. Avoid underutilization of production capacity.

  1. Formulate a goal programming model to determine the number of yards of each type of paper to produce weekly to satisfy the various goals.

  2. Solve the goal programming model by using the computer.

13.

A rural clinic hires its staff from nearby cities and towns on a part-time basis. The clinic attempts to have a general practitioner (GP), a nurse, and an internist on duty during at least a portion of each week. The clinic has a weekly budget of $1,200. A GP charges the clinic $40 per hour, a nurse charges $20 per hour, and an internist charges $150 per hour. The clinic has established the following goals, in order of priority:

  1. A nurse should be available at least 30 hours per week.

  2. The weekly budget of $1,200 should not be exceeded.

  3. A GP or an internist should be available at least 20 hours per week.

  4. An internist should be available at least 6 hours per week.

  1. Formulate a goal programming model to determine the number of hours to hire each staff member to satisfy the various goals.

  2. Solve the model by using the computer.


[Page 427]
14.

The Eaststate Manufacturing Company produces four different airplane parts from fabricated sheet metal for several major aircraft companies. The manufacturing process consists of four operationsstamping, assembly, finishing, and packaging. The processing times per unit for each operation and total available hours per year to produce these parts are as follows:

 

Part (hr./unit)

 

Operation

1

2

3

4

Total Hours/Year

Stamping

0.06

0.17

0.10

0.14

700

Assembly

0.18

0.20

0.14

700

Finishing

0.07

0.20

0.08

0.12

800

Packaging

0.09

0.12

0.07

0.15

600


The sheet metal required for each part, the estimated annual demand, and the profit per part are as follows:

Part

Sheet Metal (ft. 2 )

Estimated Annual Demand

Profits

1

2.6

2,600

$ 90

2

1.4

1,800

100

3

2.5

4,100

80

4

3.2

1,200

120


The company has 15,000 square feet of fabricated metal delivered each month. The company has the following prioritized production goals:

  1. Avoid overtime, which would erode profit levels.

  2. Meet parts demand.

  3. Achieve an annual profit of $700,000.

  4. Avoid ordering more material because a surcharge is required by the supplier for changing the standard monthly order.

  1. Formulate a goal programming model to determine the amount of each part to produce to achieve the company's objectives.

  2. Solve this model by using the computer.

  3. How would the solution be affected if the first two priorities were reversed ?

15.

Mac's Warehouse is a large discount store that operates 7 days per week. The store needs the following number of full-time employees working each day of the week:

Day

Number of Employees

Day

Number of Employees

Sunday

47

Thursday

34

Monday

22

Friday

43

Tuesday

28

Saturday

53

Wednesday

35

   

Each employee must work 5 consecutive days each week and then have 2 days off. For example, any employee who works Sunday through Thursday has Friday and Saturday off. The store currently has a total of 60 employees available to work. Mac's has developed the following set of prioritized goals for employee scheduling:


[Page 428]
  1. The store would like to avoid hiring any additional employees.

  2. The most important days for the store to be fully staffed are Saturday and Sunday.

  3. The next most important day to be fully staffed is Friday.

  4. The store would like to be fully staffed the remaining 4 days in the week.

  1. Formulate a goal programming model to determine the number of employees who should begin their 5-day workweek each day of the week to achieve the store's objectives.

  2. Solve this model by using the computer.

16.

Infocomp Systems Lab is a research and development (R&D) company that develops computer systems and software primarily for the medical industry. The lab has proposals from its own researchers for eight new projects. Each of the proposed research projects requires limited resources, and it is not possible to undertake all of them. The following table shows the developmental budget, the number of researchers, and the expected annual sales from each project if successfully developed and implemented:

Project

Developmental Budget ($1,000,000s)

Number of Research Personnel

Expected Annual Sales ($1,000,000s)

1

$0.675

6

$0.82

2

1.050

5

1.75

3

0.725

7

1.60

4

0.430

8

1.90

5

1.240

10

0.93

6

0.890

6

1.70

7

1.620

7

1.30

8

1.200

6

1.80


The lab has developed the following set of prioritized goals for selecting which projects to initiate:

  1. The company would like to remain within a total developmental budget of $5,000,000.

  2. The number of available research personnel is 27, and Infocomp would like to avoid obtaining extra researchers.

  3. The company would like the expected future annual sales from the implemented projects to be at least $6,500,000.

  4. Projects 1, 3, 4, and 6 are considered offensive in that they represent new product initiatives, while projects 2, 5, 7, and 8 are existing product upgrades and thus defensive in nature. The lab would like to select at least two projects from each group .

  5. Projects 2, 3, 5, 6, and 7 are considered the most risky of the projects, and the company would prefer not to select any more than three of these projects.

  6. The lab's owner has indicated that she would like to see projects 5 and 6 initiated if doing so does not interfere with the achievement of any of the more important goals determined by the lab's top management.

  1. Formulate a goal programming model to determine which projects Infocomp Systems Lab should select to best achieve its goals.

  2. Solve this model by using the computer. (Note that the solution requires 01 integer values for the variables in the model.)

17.

Dampier Associates is a holding company that specializes in buying out small to medium- sized textile companies. The company is currently considering purchasing three companies in the Carolinas: Alton Textiles, Bonham Mills, and Core Textiles. The main criteria the company uses to determine which companies it will purchase are current profitability and growth potential. Dampier moderately prefers growth potential over profitability in making a purchase decision. Dampier's pairwise comparisons for the three target textile companies it is considering are as follows:


[Page 429]
 

Profitability

Company

A

B

C

A

1

1/3

7

B

3

1

9

C

1/7

1/9

1


 

Growth Potential

Company

A

B

C

A

1

1/2

1/5

B

2

1

1/3

C

5

3

1


Develop an overall ranking of the three companies for Dampier Associates by using AHP.

18.

Check the pairwise comparisons for Dampier Associates in Problem 17 for consistency.

19.

Bernard Mee, the head of the department of management science at Tech, is evaluating faculty for raises at the end of the academic year. He is considering three faculty members for raises: John Abbott, Megan Bates, and Debbie Cook. Faculty evaluations are based on three criteriateaching, research, and service. Professor Mee's pairwise comparisons for each of the three faculty members for each criterion and his pairwise comparison matrix for the three criteria are as follows:

 

Teaching

Faculty Member

A

B

C

A

1

2

1/3

B

1/2

1

1/5

C

3

5

1


 

Research

Faculty Member

A

B

C

A

1

3

1/2

B

1/3

1

1

C

2

1

1


 

Service

Faculty Member

A

B

C

A

1

3

6

B

1/3

1

2

C

1/6

1/2

1



[Page 430]

Criterion

Teaching

Research

Service

Teaching

1

3

5

Research

1/3

1

2

Service

1/5

1/2

1


Determine an overall ranking of the three faculty members by using AHP.

20.

Check the pairwise comparisons for the criteria in Problem 19 for consistency and indicate whether the level of consistency is acceptable.

21.

Megan Moppett is a sales representative for Technical Software Systems (TSS), and she receives a commission for every new system installation she sells to a client. Her earnings during the past few years have been very high, and she wants to invest in a mutual fund. She is considering three funds: the Temple Global Fund, the Alliance Blue Chip Fund, and the Madison Bond Fund. She has three criteria for selectionpotential return (based on historical trends and forecasts), risk, and the fund's load factor. Megan's pairwise comparisons for the funds for each of their criteria and her pairwise comparison of the three criteria are as follows:

 

Return

Fund

Global

Blue Chip

Bond

Global

1

1/4

2

Blue Chip

4

1

6

Bond

1/2

1/6

1


 

Risk

Fund

Global

Blue Chip

Bond

Global

1

2

1/3

Blue Chip

1/2

1

1/5

Bond

3

5

1


 

Load

Fund

Global

Blue Chip

Bond

Global

1

1

1/3

Blue Chip

1

1

1/3

Bond

3

3

1


Criterion

Return

Risk

Load

Return

1

3

5

Risk

1/3

1

2

Load

1/5

1/2

1


Determine the fund in which Megan should invest.

22.

In Problem 21, if Megan Moppett has $85,000 to invest and she wants to diversify by investing in all three funds, how much should she invest in each?


[Page 431]
23.

Alex Wall is shopping for a new four-wheel-drive utility vehicle and has identified three models from which she will choosean Explorer, a Trooper, and a Passport. She will make her selection based on Consumer Digest ratings, price, and each vehicle's appearance. Following are Alex's pairwise comparisons for the vehicles for each of her criteria and her criteria preferences:

 

Consumer Digest Rating

Vehicle

Explorer

Trooper

Passport

Explorer

1

4

3

Trooper

1/4

1

1/2

Passport

1/3

2

1


 

Price

Vehicle

Explorer

Trooper

Passport

Explorer

1

1/4

1/6

Trooper

4

1

2

Passport

6

1/2

1


 

Appearance

Vehicle

Explorer

Trooper

Passport

Explorer

1

4

3

Trooper

1/4

1

1/2

Passport

1/3

2

1


Criterion

Consumer Digest Rating

Price

Appearance

Consumer Digest rating

1

2

4

Price

1/2

1

3

Appearance

1/4

1/3

1


Using AHP, determine which vehicle Alex should purchase.

24.

Station WRCH in Richmond, Virginia, is interviewing candidates for the job of news anchor on its 6:00 P.M. Eyewitness News show. There are three final candidates for the jobJune Pawlie, Kellie Cooric, and Tim Brokenaw. The criteria the station manager will use to make the selection are camera appearance, intelligence, and speaking ability (or speech). The station manager's pairwise comparisons for the job for each of these candidates and for the three criteria are as follows:

 

Appearance

Anchor

Pawlie

Cooric

Brokenaw

Pawlie

1

2

7

Cooric

1/2

1

5

Brokenaw

1/7

1/5

1



[Page 432]
 

Intelligence

Anchor

Pawlie

Cooric

Brokenaw

Pawlie

1

1/3

1/4

Cooric

3

1

1/2

Brokenaw

4

2

1


 

Speech

Anchor

Pawlie

Cooric

Brokenaw

Pawlie

1

1/3

2

Cooric

3

1

6

Brokenaw

1/2

1/6

1


Criterion

Appearance

Intelligence

Speech

Appearance

1

8

3

Intelligence

1/8

1

1/5

Speech

1/3

5

1


Using AHP, determine which candidate the station manager should hire as news anchor.

25.

Carol Latta is visiting hotels in Los Angeles to decide where to hold a convention for a national organization of college business school teachers she represents. There are three hotels from which to choosethe Cheraton, the Milton, and the Harriott. The criteria she is to use to make her selection are ambiance, location (based on safety and walking distance to attractions and restaurants ), and cost to the organization. Following are the pairwise comparisons she has developed that indicate her preference for each hotel for each criterion and her pairwise comparisons for the criteria:

 

Ambiance

Hotel

Cheraton

Milton

Harriott

Cheraton

1

1/2

1/5

Milton

2

1

1/3

Harriott

5

3

1


 

Location

Hotel

Cheraton

Milton

Harriott

Cheraton

1

5

3

Milton

1/5

1

1/4

Harriott

1/3

4

1


 

Cost

Hotel

Cheraton

Milton

Harriott

Cheraton

1

2

5

Milton

1/2

1

2

Harriott

1/5

1/2

1



[Page 433]

Criterion

Ambiance

Location

Cost

Ambiance

1

2

4

Location

1/2

1

3

Cost

1/4

1/3

1


Develop an overall ranking of the three hotels, using AHP, to help Carol Latta decide where to hold the meeting.

26.

Aaron Zeitel is a high school senior deciding which college to attend in the fall. He has narrowed his choices to three liberal arts schools: Arrington, Barton, and Claiborne. His criteria for selection are the school's academic reputation, location (and especially proximity to his home), the cost of tuition and room and board, and the social and cultural opportunities available. Following are Aaron's pairwise comparisons of the schools for each of the four criteria and his pairwise comparisons for the criteria:

 

Academic

College

A

B

C

A

1

1/2

3

B

2

1

4

C

1/3

1/4

1


 

Location

College

A

B

C

A

1

1

3

B

1

1

5

C

1/3

1/5

1


 

Cost

College

A

B

C

A

1

1/2

1/4

B

2

1

1/2

C

4

2

1


 

Social

College

A

B

C

A

1

3

6

B

1/3

1

3

C

1/6

1/3

1


Criterion

Academic

Location

Cost

Social

Academic

1

4

1/2

3

Location

1/4

1

1/5

1/2

Cost

2

5

1

3

Social

1/3

2

1/3

1



[Page 434]

Using AHP, determine which college Aaron should select and check the consistency of the pairwise comparison matrix for the criteria.

27.

Whitney Eggleston operates a computerized dating service for students at Tech. She uses AHP to help match her clients . Whitney is attempting to match Chris with either Robin, Terry, or Kelly. She evaluates her clients according to three criteriaphysical attractiveness, intelligence, and personality, and she had Chris do pairwise comparisons on this set of criteria, as follows:

Criterion

Attractiveness

Intelligence

Personality

Attractiveness

1

1

3

Intelligence

1

1

2

Personality

1/3

1/2

1


Whitney herself did the pairwise comparisons for Robin, Terry, and Kelly, based on their data sheets and a personal interview with each:

 

Attractiveness

Client

Robin

Terry

Kelly

Robin

1

3

5

Terry

1/3

1

2

Kelly

1/5

1/2

1


 

Intelligence

Client

Robin

Terry

Kelly

Robin

1

2

1/2

Terry

1/2

1

1/4

Kelly

2

4

1


 

Personality

Client

Robin

Terry

Kelly

Robin

1

2

1/3

Terry

1/2

1

1/2

Kelly

3

2

1


Who is the best match for Chris, according to Whitney's AHP analysis?

28.

Rockingham Systems is considering three R&D projects it has identified as A, B, and C. Rockingham is not sure it will undertake all three projects, so it wants to rank them in terms of preferability. Rockingham will use three criteria to rank the projectsprofit potential, probability of success, and cost. Following are Rockingham's pairwise comparisons for the projects for each of the three criteria and for the criteria:

 

Profit

Project

A

B

C

A

1

4

6

B

1/4

1

2

C

1/6

1/2

1



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P (Success)

Project

A

B

C

A

1

2

1/3

B

1/2

1

1/6

C

3

6

1


 

Cost

Project

A

B

C

A

1

1/3

1/4

B

3

1

1/2

C

4

2

1


Criteria

Profit

P (Success)

Cost

Profit

1

2

6

P (Success)

1/2

1

4

Cost

1/6

1/4

1


Rank the projects for Rockingham Systems by using AHP.

29.

Professor Rakes is selecting a new graduate assistant from a pool of second-year MBA students. He will make his selection based on the student's grade point average (GPA) to date, the overall GMAT ( entrance exam) score, and the undergraduate degree discipline. Professor Rakes has developed the following pairwise comparisons for these criteria:

Criteria

GPA

GMAT

Degree

GPA

1

5

1/3

GMAT

1/5

1

1/6

Degree

3

6

1


The files of the three students, Adrian, Bon, and Corey, provide the following data:

Student

GPA

GMAT

Degree

Adrian

3.6

560

Business

Bon

3.8

670

English

Corey

3.0

610

Engineering


Use AHP to help Professor Rakes select a graduate assistant.

30.

The Bay City Parks and Recreation Department is considering building several new facilities, including a gym, an athletic field, a tennis pavilion, and a pool. It will base its decision on which facilities to build depending on projected usage (from surveys) and construction and operating costs. The department strongly prefers usage to cost as a criterion for selection. Following are the department's pairwise comparisons, reflecting its preferences for each facility for the two criteria:


[Page 436]
 

Usage

Facility

Gym

Field

Tennis

Pool

Gym

1

1/3

3

2

Field

3

1

5

4

Tennis

1/3

1/5

1

1/3

Pool

1/2

1/4

3

1


 

Cost

Facility

Gym

Field

Tennis

Pool

Gym

1

1/4

1/2

3

Field

4

1

3

7

Tennis

2

1/3

1

4

Pool

1/3

1/7

1/4

1


Rank the facilities, using AHP, and check the pairwise comparisons for consistency.

31.

Students at a college in Ohio are planning a spring break vacation to one of three locations: Myrtle Beach (MB), Daytona Beach (DB), or Ft. Lauderdale (FL). They are to base their decision on three criteriaweather, cost, and potential fun (based on an Internet survey of friends and acquaintances at other colleges). The students have developed the following pairwise comparisons for the locations for each criterion and for the three criteria:

 

Weather

Location

MB

DB

FL

MB

1

1/3

1/3

DB

3

1

1

FL

3

1

1


 

Cost

Location

MB

DB

FL

MB

1

3

5

DB

1/3

1

2

FL

1/5

1/2

1


 

Fun

Location

MB

DB

FL

MB

1

1/2

5

DB

2

1

3

FL

1/5

1/3

1



[Page 437]

Criteria

Weather

Cost

Fun

Weather

1

4

1/4

Cost

1/4

1

1/5

Fun

4

5

1


If the students use AHP to help make a decision, which location will they select for their spring break vacation?

32.

Check the pairwise comparisons in Problem 31 for consistency.

33.

The management science and information technology majors at Tech select one of two available options within the majordecision support systems (DSS) or operations management (OM). Student advisers use AHP with the students to determine which option they should select. The criteria used by the advisers are student aptitude and interests, faculty who teach in the options, and potential job availability. An adviser has helped one major develop the following pairwise comparisons:

 

Aptitude

Option

DSS

OM

DSS

1

3

OM

1/3

1


 

Faculty

Option

DSS

OM

DSS

1

1/5

OM

5

1


 

Jobs

Option

DSS

OM

DSS

1

4

OM

1/4

1


Criterion

Aptitude

Faculty

Jobs

Aptitude

1

1/2

1/4

Faculty

2

1

1/3

Jobs

4

3

1


Which option should the student select?

34.

The Des Moines Twisters, a women's professional basketball team, needs a new point guard and is considering signing one of three college playersKeisha Jones, Natasha Franklin, and Kathleen Taylor. The team will base its decision of which player to sign on four criteria: the player's shooting ability, rebounding ability, ball handling skills, and attitude. Following is a survey of the three players' college game statistics for their senior year:


[Page 438]
 

Per-Game Statistics

Player

Points

Field Goal (%)

Free Throw (%)

Rebounds

Assists

Turnovers

Minutes

Keisha Jones

18.6

47.5

75.7

10.3

4.1

3.2

33.9

Natasha Franklin

22.9

38.5

64.2

12.4

3.6

1.8

32.5

Kathleen Taylor

16.5

62.1

88.3

7.6

6.7

0.6

37.4


Although the coach can base some of her pairwise comparisons on these statistics, it is more difficult to assess attitude. She has interviewed each player, and she moderately prefers Keisha Jones's attitude to Kathleen Taylor's and strongly prefers Keisha Jones's attitude to Natasha Franklin's.

Assume the coach's role and develop pairwise comparisons for all four of the criteria and between the criteria. Use AHP to determine which player you would select.

35.

The town of Blacksburg needs a larger modern middle school. The current middle school is in the center of town and is over 40 years old. There are two proposals for a new schoolrenovate and expand the current facility and keep it in town or build a new school on the outskirts of town. Different groups in town have strong feelings about the proposals. Some citizens want to retain the sense of tradition of the old school and like it in town, where it helps engender a sense of community. Others view the old school as antiquated and beyond saving and believe keeping the school in town near bars, traffic, and college students to be negative. The county school board will make the final decision. The school board has asked several management science professors from the local college to use AHP to help evaluate the proposals. The school board has identified four groups from which it wants to solicit input regarding their preferences: the middle school PTA, the middle school teachers, current and former middle school students, and the town council. The management science professors have developed the following pairwise comparison matrices for each of these groups:

 

PTA

Proposal

Renovate

New

Renovate

1

1/3

New

3

1


 

Teachers

Proposal

Renovate

New

Renovate

1

1/9

New

9

1


 

Students

Proposal

Renovate

New

Renovate

1

2

New

1/2

1


 

Town Council

Proposal

Renovate

New

Renovate

1

5

New

1/5

1



[Page 439]

The school board's pairwise comparison of the four groups from which it is soliciting preferences is as follows:

Group

PTA

Teachers

Students

Town Council

PTA

1

5

2

1/4

Teachers

1/5

1

1/4

1/7

Students

1/2

4

1

1/5

Town council

4

7

5

1


  1. Based on the AHP analysis conducted by the management science professors, which proposal should the school board select?

  2. Check the school board's pairwise comparison of the criteria for consistency.

36.

Given the following pairwise comparisons, indicate your preferences according to the preference scale in Table 9.1.

  1. Steak to chicken

  2. Hot dogs to hamburgers

  3. Republicans to Democrats

  4. Soccer to football

  5. College basketball to professional basketball

  6. Management science to management

  7. Domino's to Pizza Hut

  8. McDonald's to Wendy's

  9. Ford to Honda

  10. Dickens to Faulkner

  11. Beatles to Beethoven

  12. New York to Los Angeles

  13. Chicago to Atlanta

  14. American League to National League

37.

Federated Health Care has contracted to be Tech's primary health care provider for faculty and staff. There are three major hospitals in the area (within 35 miles)County, Memorial, and Generalthat have full-service emergency rooms. Federated wants to designate one of the hospitals as its primary care emergency rooms for its members. The company's criteria for selection are quality of medical care, as determined by a patient survey; distance to the emergency room by the majority of its members; speed of medical attention at the emergency room; and cost. Following are the pairwise comparisons of the emergency rooms for each of the four criteria and the pairwise comparisons for the criteria:

 

Medical Care

Hospital

County

Memorial

General

County

1

1/6

1/3

Memorial

6

1

3

General

3

1/3

1


 

Distance

Hospital

County

Memorial

General

County

1

7

4

Memorial

1/7

1

2

General

1/4

1/2

1



[Page 440]
 

Speed of Attention

Hospital

County

Memorial

General

County

1

1/2

3

Memorial

2

1

4

General

1/3

1/4

1


 

Cost

Hospital

County

Memorial

General

County

1

6

4

Memorial

1/6

1

1/2

General

1/4

2

1


Criterion

Medical Care

Distance

Speed of Attention

Cost

Medical care

1

8

6

3

Distance

1/8

1

1/2

1/6

Speed of attention

1/6

2

1

1/4

Cost

1/3

6

4

1


Using AHP, determine which hospital emergency room Federated Health Care should designate as its primary care provider.

38.

The department of management science at Tech offers four sections of introductory quantitative methods in the fall semester, each taught by a different teacher. A group of students have developed the following criteria to rank the sections: time and day, grading history of the instructor, classroom atmosphere (i.e., relaxed or formal), amount of homework, and the teacher's sense of humor. Following are the students' preferences for these criteria:

Criterion

Time/Day

Grading

Atmosphere

Homework

Jokes

Time/day

1

2

7

3

8

Grading

1/2

1

6

3

9

Atmosphere

1/7

1/6

1

1/4

2

Homework

1/3

1/3

4

1

7

Jokes

1/8

1/9

1/2

1/7

1


Following are the students' pairwise comparisons of the sections for the criteria:

 

Time/Day

Section

1

2

3

4

1

1

3

5

7

2

1/3

1

2

5

3

1/5

1/2

1

3

4

1/7

1/5

1/3

1



[Page 441]
 

Grading

Section

1

2

3

4

1

1

1/7

1/8

2

2

7

1

2

6

3

8

1/2

1

5

4

1/2

1/6

1/5

1


 

Atmosphere

Section

1

2

3

4

1

1

6

3

3

2

1/6

1

3

2

3

1/3

1/3

1

1

4

1/3

1/2

1

1


 

Homework

Section

1

2

3

4

1

1

1/4

1/8

1/2

2

4

1

1/3

4

3

8

3

1

5

4

2

1/4

1/5

1


 

Jokes

Section

1

2

3

4

1

1

1/5

3

4

2

5

1

6

7

3

1/3

1/6

1

3

4

1/4

1/7

1/3

1


  1. Using AHP, rank the courses for the students.

  2. Using this framework and these criteria, but with your own preferences for a course with multiple sections that you are familiar with at your school, develop a ranking of the sections.

39.

A faculty committee in the department of management science at Tech is evaluating three new textbooks for its introductory management science course, which all business students are required to take. The texts , identified by the authors, are Adams/Jones, Barnes, and Cook/Smith. The committee's selection criteria are topical coverage, readability, cost, and the available supplements. Following are the committee's pairwise comparisons of the three textbooks for each of the four criteria and the committee's pairwise comparisons for the criteria:

 

Coverage

Textbook

A

B

C

A

1

1/5

1/4

B

5

1

3

C

4

1/3

1



[Page 442]
 

Readability

Textbook

A

B

C

A

1

2

3

B

1/2

1

3

C

1/3

1/3

1


 

Cost

Textbook

A

B

C

A

1

1/2

1/5

B

2

1

1/3

C

5

3

1


 

Supplements

Textbook

A

B

C

A

1

4

7

B

1/4

1

3

C

1/7

1/3

1


Criterion

Coverage

Readability

Cost

Supplements

Coverage

1

1/2

1/4

2

Readability

2

1

1/3

5

Cost

4

3

1

3

Supplements

1/2

1/5

1/3

1


Using AHP, determine which textbook the committee should select. Check the consistency of the pairwise comparison matrix for the criteria.

40.

On the day of the professional football draft, the owner, general manager, and coaching staff of the New York Gladiators are attempting to decide which of the available players they should select as their pick approaches. They have three players on their big board whom they would like: Al Stonecrusher, a defensive lineman, Bruce Kowslaski, a tight end, and Charlie Speedman, a running back. They evaluate players according to four criteriathe player's projected salary demands, speed, and size (i.e., height and weight) and the team's position needs. Following are the group's pairwise comparisons for the players for each of the four criteria and the pairwise comparisons for the criteria:

 

Salary

Player

A

B

C

A

1

5

3

B

1/5

1

1/2

C

1/3

2

1



[Page 443]
 

Speed

Player

A

B

C

A

1

1/5

3

B

5

1

6

C

1/3

1/6

1


 

Position

Player

A

B

C

A

1

1/2

1/5

B

2

1

1/3

C

5

3

1


 

Size

Player

A

B

C

A

1

4

2

B

1/4

1

1/3

C

1/2

3

1


Criterion

Salary

Speed

Position

Size

Salary

1

1/5

1/3

1/2

Speed

5

1

3

2

Position

3

1/3

1

2

Size

2

1/2

1/2

1


Using AHP, determine which player the Gladiators should select. Check the consistency of the pair-wise comparison matrix for the criteria.

41.

Visit one or more local car dealers and select four new models of cars you might like to purchase. Using AHP and your own preferences, rank your selections according to the following criteria: price, style/appearance, reliability/maintenance, engine size, gas mileage, safety, and features/options. (You may need to access some additional references, such as Consumer Reports , to facilitate your judgments .) After you have ranked your car selections using AHP, develop a scoring model to perform the same analysis. Then compare the results of the two models and discuss which method your prefer.

42.

Arsenal Electronics is to construct a new $1.2 billion semiconductor plant and has selected four small towns in the Midwest as potential sites. The important decision criteria and grades for each town are as follows:


[Page 444]
   

Town

Decision Criterion

Weight

Abbeton

Bayside

Cane Creek

Dunnville

Work ethic

0.18

80

90

70

75

Quality of life

0.16

75

85

95

90

Labor laws/unionization

0.12

90

90

60

70

Infrastructure

0.10

60

50

60

70

Education

0.08

75

90

85

95

Labor skill and education

0.07

75

65

70

75

Cost of living

0.06

70

75

85

75

Taxes

0.05

65

70

55

60

Incentive package

0.05

90

95

70

75

Government regulations

0.03

40

50

65

55

Environmental regulations

0.03

65

60

70

80

Transportation

0.03

90

80

95

80

Space for expansion

0.02

90

95

90

90

Urban proximity

0.02

60

90

70

80


Develop a scoring model to determine in which town the plant should be built.

43.

The Dynaco Manufacturing Company is to build a new plant to make ring bearings (used in automobiles and trucks ). The site selection team is evaluating three sites, and it has graded the important weighted criteria for each as follows:

   

Building Site

Decision Criterion

Weight

1

2

3

Labor pool and climate

0.30

80

65

90

Proximity to suppliers

0.20

100

91

75

Wage rates

0.15

60

95

72

Community environment

0.15

75

80

80

Proximity to customers

0.10

65

90

95

Shipping modes

0.05

85

92

65

Air service

0.05

50

65

90


Develop a scoring model to determine which site the selection team should recommend.

44.

Exotech Computers manufactures computer components such as chips, circuit boards , motherboards, keyboards, LCD panels, and the like and sells them around the world. It wants to construct a new warehouse and distribution center in Asia to serve emerging markets there. It has identified sites in Shanghai, Hong Kong, and Singapore and has graded the important weighted decision criteria for each site as follows:


[Page 445]
   

Asian City

Decision Criterion

Weight

Shanghai

Hong Kong

Singapore

Political stability

0.25

50

60

90

Economic growth

0.18

90

70

75

Port facilities

0.15

60

95

90

Container support

0.10

50

80

90

Land construction cost

0.08

90

20

30

Transportation/distribution

0.08

50

80

70

Duties and tariffs

0.07

70

90

90

Trade regulations

0.05

70

95

95

Airline service

0.02

60

80

70

Area roads

0.02

60

70

80


Recommend a site based on these decision criteria and grades.

45.

State University is to construct a new student center and athletic complex that will include a bookstore, a post office, theaters, a market, a minimall, meeting rooms, a swimming pool, and weight and exercise rooms. The university administration has hired a site selection specialist to identify the best potential sites on campus for the new facility. The site specialist has identified four sites on campus and has graded the important weighted decision criteria for each site as follows:

   

Campus Site

Decision Criterion

Weight

South

West A

West B

East

Proximity to housing

0.23

70

90

65

85

Student traffic

0.22

75

80

60

85

Parking availability

0.16

90

60

80

70

Plot size, terrain

0.12

80

70

90

75

Infrastructure

0.10

50

60

40

60

Off-campus accessibility

0.06

90

70

70

70

Proximity to dining facilities

0.05

60

80

70

90

Visitor traffic

0.04

70

80

65

55

Landscape and aesthetics

0.02

50

40

60

70


Which site should the specialist recommend?

46.

Following an all-star school soccer career, Kelly Williams has been offered scholarships to five universities. She has had a difficult time making a decision and has decided to use a scoring model to help evaluate the different offers. The following table includes a weighted list of criteria she has developed and a grade showing how well each school satisfies the criteria:


[Page 446]
   

University

Decision Criterion

Weight

Tech

State

A&M

Central

Western

Playing time

0.29

61

52

93

83

77

Coach

0.20

87

92

66

55

80

Conference affiliation

0.17

42

57

80

92

63

School prestige

0.13

98

72

65

81

83

Program status

0.08

78

82

59

62

75

Degree program/major

0.08

72

80

75

93

89

Dollar value of scholarship

0.05

93

88

75

82

74


Rank the universities according to their scores to assist Kelly in making her decision.

47.

The Carter family wants to purchase a time-share condominium for 1 week during the summer in Hilton Head, South Carolina. There are many resort complexes from which to choose, and the Carters narrowed their list to five. They have graded their choices according to the following weighted criteria:

   

Resort Location

Decision Criterion

Weight

Albermarle

Beachfront

Calypso

Dafuskie

Edenisle

Specific summer week

.40

80

70

70

90

60

Cost

.20

50

70

90

60

90

Proximity to beach

.15

70

60

70

50

80

Condominium quality

.05

90

80

90

60

80

Pool size

.05

40

60

70

80

100

Cleanliness

.05

70

90

90

80

90

Crowdedness of beach

.05

30

80

50

70

60

Condominium size

.05

100

70

90

80

90


Use a scoring model to determine a recommended resort for the Carters' time-share purchase.

48.

Robin Dillon has recently accepted a new job in the Washington, DC, area and has been hunting for a condominium to purchase. From friends and coworkers she has compiled a list of five possible condominium complexes that she might move into. The following table indicates the weighted criteria that Robin intends to use in her decision-making process and a grade indicating how well each complex satisfies each criterion:

   

Condominium Complex

Decision Criterion

Weight

Fairfax Forrest

Dupont Gardens

Tysons Terrace

Alexandria Commons

Manassas Farms

Purchase price

.30

92

85

75

62

79

Neighborhood location

.18

76

63

95

90

80

Proximity to Metro train

.12

78

75

76

85

60

Shopping

.10

65

80

98

92

75

Security

.10

75

78

90

95

83

Recreational facilities

.05

96

90

82

81

93

Distance to job

.05

85

67

95

75

65

Condo floor plan

.05

80

78

86

92

90

Complex size

.05

65

60

92

89

70



[Page 447]

Use a scoring model to help Robin determine which condominium she should purchase.

49.

The New River Rapids Under-15 Girls' Travel Soccer Team is determining which tournments it should enter during the fall season . The manager has identified six possible tournaments, and the team wants to enter three of them. The team manager first surveyed the parents and players and determined a list of decision criteria and their relative weights. The manager then asked the parents and players to grade the tournaments on how well they satisfied the criteria. The criteria, weights, and average grades are summarized as follows:

   

Tournament

Decision Criterion

Weight

Roanoke

Greensboro

Bristol

Richmond

Knoxville

Charlotte

Distance

0.24

81

73

85

92

66

84

Tournament dates

0.21

91

88

79

83

75

69

Hotel cost

0.16

86

75

92

67

73

71

Level of competition

0.14

88

93

63

79

89

94

Field quality

0.10

84

91

70

87

77

81

Shopping

0.07

81

83

77

92

90

97

Attractions

0.06

73

82

80

91

96

87

Restaurants

0.02

83

88

65

95

72

91


To what three tournaments should the manager recommend the team apply, given the preferences indicated by the parents and players?

50.

Select four fast-food restaurants (e.g., McDonald's, Burger King, Wendy's, Domino's) in your area and develop a scoring model that includes decision criteria, weights, and grades to rank the restaurants from best to worst.




Introduction to Management Science
Introduction to Management Science (10th Edition)
ISBN: 0136064361
EAN: 2147483647
Year: 2006
Pages: 358

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