Example Problems Solutions

[Page 418 ( continued )]

As a prelude to the problems section, the following examples demonstrate the formulations and solutions of a goal programming problem, and an AHP problem.

Problem Statement

Rucklehouse Public Relations has contracted to do a survey following an election primary in New Hampshire. The firm must assign interviewers to carry out the survey. The interviews are conducted by telephone and in person. One person can conduct 80 telephone interviews or 40 personal interviews in a day. It costs $50 per day for a telephone interviewer and $70 per day for a personal interviewer. The following three goals, which are listed in order of their priority, have been established by the firm to ensure a representative survey:

1. At least 3,000 total interviews should be conducted.

2. An interviewer should conduct only one type of interview each day. The firm wants to maintain its daily budget of $2,500.

3. At least 1,000 interviews should be by telephone.

Formulate a goal programming model to determine the number of interviewers to hire to satisfy these goals and then solve the model.

---

[Page 419]

Solution

Step  1.	Model Formulation where x 1 = number of telephone interviewers x 2 = number of personal interviewers
Step  2.	The QM for Windows Solution

Problem Statement

Grace LeMans wants to purchase a new mountain bike, and she is considering three modelsthe Xandu Mark III, the Yellow Hawk Z9, and the Zodiak MB5. Grace has identified three criteria for selection on which she will base her decision: purchase price, gear action, and weight/durability. Grace has developed the following pairwise comparison matrices for the three criteria:

	Price
Bike	X	Y	Z
X	1	3	6
Y	1/3	1	2
Z	1/6	1/2	1

---

[Page 420]

	Gear Action
Bike	X	Y	Z
X	1	1/3	1/7
Y	3	1	1/4
Z	7	4	1

	Weight/Durability
Bike	X	Y	Z
X	1	3	1
Y	1/3	1	1/2
Z	1	2	1

Grace has prioritized her decision criteria according to the following pairwise comparisons:

Criteria	Price	Gears	Weight
Price	1	3	5
Gears	1/3	1	2
Weight	1/5	1/2	1

Using AHP, develop an overall ranking of the three bikes Grace is considering.

Solution

Step  1.

Develop Normalized Matrices and Preference Vectors for All the Pairwise Comparison Matrices for Criteria

	Price
Bike	X	Y	Z	Row Averages
X	0.6667	0.6667	0.6667	0.6667
Y	0.2222	0.2222	0.2222	0.2222
Z	0.1111	0.1111	0.1111	0.1111
				1.0000

	Gear Action
Bike	X	Y	Z	Row Averages
X	0.0909	0.0625	0.1026	0.0853
Y	0.2727	0.1875	0.1795	0.2132
Z	0.6364	0.7500	0.7179	0.7014
				1.0000

---

[Page 421]

	Weight/Durability
Bike	X	Y	Z	Row Averages
X	0.4286	0.5000	0.4000	0.4429
Y	0.1429	0.1667	0.2000	0.1698
Z	0.4286	0.3333	0.4000	0.3873
				1.0000

The preference vectors are summarized in the following matrix:

	Criteria
Bike	Price	Gears	Weight
X	0.6667	0.0853	0.4429
Y	0.2222	0.2132	0.1698
Z	0.1111	0.7014	0.3873

Step  2.

Rank the Criteria

Criteria	Price	Gears	Weight	Row Averages
Price	0.6522	0.6667	0.6250	0.6479
Gears	0.2174	0.2222	0.2500	0.2299
Weight	0.1304	0.1111	0.1250	0.1222
				1.0000

The preference vector for the criteria is

Step  3.

Develop an Overall Ranking

---

[Page 422]

The ranking of the three bikes, in order of the magnitude of their scores, is

Bike	Score
Xandu	0.5057
Zodiak	0.2806
Yellow Hawk	0.2138
	1.0000