As a prelude to the problems section, the following examples demonstrate the formulations and solutions of a goal programming problem, and an AHP problem.
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:
At least 3,000 total interviews should be conducted.
An interviewer should conduct only one type of interview each day. The firm wants to maintain its daily budget of $2,500.
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.
Step 1.  Model Formulation where x _{ 1 } = number of telephone interviewers x _{ 2 } = number of personal interviewers 
Step 2.  The QM for Windows Solution 
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 
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.
Step 1.  Develop Normalized Matrices and Preference Vectors for All the Pairwise Comparison Matrices for Criteria
The preference vectors are summarized in the following matrix:
 
Step 2.  Rank the Criteria
The preference vector for the criteria is
 
Step 3.  Develop an Overall Ranking

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 