Analytic Hierarchy Process (AHP) By : BasmahAlQadheeb- FatimahAlOtaibi

Analytic Hierarchy Process (AHP)  Is one of Multi Criteria decision making method that was originally developed by Prof. Thomas L. Saaty.  Is an excellent modeling structure for representing multicriteria (multiple goals, multiple objectives) problems—with sets of criteria and alternatives (choices)-commonly found in business environments.  In short, it is a method to derive ratio scales from paired comparisons

Level 0 Level 1 Level2

 Level 0 is the goal of the analysis. Level 1 is multi criteria that consist of several factors . Level 2 in is the alternative choices.  The input of AHP can be obtained from actual measurement such as price, weight etc., or from subjective opinion such as satisfaction feelings and preference.  AHP allow some small inconsistency in judgment because human is not always consistent.

Standard Preference Table PREFERENCE LEVEL NUMERICAL VALUE Equally preferred Equally to moderately preferred Moderately preferred Moderately to strongly preferred Strongly preferred Strongly to very strongly preferred Very strongly preferred Very strongly to extremely preferred Extremely preferred 1 2 3 4 5 6 7 8 9

Analytic Hierarchy Process  Step 1: Structure a hierarchy. Define the problem, determine the criteria and identify the alternatives.  For example:  Suppose that John has three fruits Apple ,banana and Cherry . We wanted to ask him, Which fruit you like better than the others, and how much you like it in comparison with the others

 Step 2: Make pairwise comparisons. Rate the relative importance between each pair of decision alternatives and criteria.

 You may observe that the number of comparisons is a combination of the number of things to be compared. Since we have 3 objects (Apple, Banana and Cheery), we have 3 comparisons.  Table below shows the number of comparisons:

Making Comparison Matrix  We have 3 by 3 matrix  The diagonal elements of the matrix are always 1 and we only need to fill up the upper triangular matrix.  How to fill up the upper triangular matrix is using the following rules:  If the judgment value is on the left side of 1, we put the actual judgment value.  If the judgment value is on the right side of 1, we put the reciprocal value .

 John made subjective judgment on which fruit he likes best, like the following

 Comparing apple and banana, John slightly favor banana, thus we put 1/3 in the row 1 column 2 of the matrix.  Comparing Apple and Cherry, John stronglylikes apple, thus we put actual judgment 5 on the first row, last column of the matrix.  Comparing banana and cherry, banana is dominant. Thus we put his actual judgment on the second row, last column of the matrix. Then based on his preference values above, we have a reciprocal matrix like this:

 Suppose we have 3 by 3 reciprocal matrix from paired comparison

 We sum each column of the reciprocal matrix to get

 Then we divide each element of the matrix with the sum of its column, we have normalized relative weight. The sum of each column is 1.

► Step 3: Synthesize the results to determine the best alternative. Obtain the final results.

 The normalized principal Eigen vector can be obtained by averaging across the rows  The normalized principal Eigen vector is also called priority vector

 In our example above,  Apple is 28.28%, Banana is 64.34% and Cherry is 7.38%.  John most preferable fruit is Banana, followed by Apple and Cheery

Our sample problem  Jilley Bean Co. is selecting a new location to expand its operations. The company want to use AHP to help it decide which location to build its new plant. Jilley Bean Co. has four criteria they will base their decision on these are the following: property price, distance from suppliers, the quality of the labor pool, and the cost of labor. They have three locations to decide from.

Matrices given criteria and preferences B 9 7 5 3 1 3 5 7 9 PRICE DISTANCE A B C 1 3 2 6 1/3 1/5 1/6 1/9 1/2 5 9 LABOR WAGES 7 4 1/7 1/4 A C 9 7 5 3 1 3 5 7 9 B C 9 7 5 3 1 3 5 7 9

How it is done ~ STEP ONE & TWO PRICE A B C 1 3 2 1- First sum (add up) all the values in each column. 1/3 1/5 ½ 5 SUM 11/6 9 16/5 2- Next the values in each column are divided by the corresponding column sums 6/11 3/9 5/8 2/11 1/9 1/16 3/11 5/9 5/16 NOTICE: the values in each column sum to 1.

STEP THREE PRICE A B C R w Average = 1/3 * 6/11 + 3/9 + 5/8 0.0512 2/11 + 1/9 + 1/16 0.1185 3/11 + 5/9 + 5/16 0.3803 SUM 1 o Next convert fractions to decimals and find the average of each row.

STEP FOUR  Find the average for all the criterion by doing steps 1-3 on all the criteria. Arriving at the following Price Distance Labor Wages A 0.0512 0.2819 0.1790 0.1561 B 0.1185 0.0598 0.6850 0.6196 C 0.3803 0.6583 0.1360 0.2243

STEP FIVE  Rank the criteria in order of importance ~use the same method used in ranking each criterion. Price Distance Labor Wages 1 1/5 3 4 5 9 7 1/3 1/9 2 ¼ 1/7 1/2 SUM 6.583 1.454 13.5 14

STEP 6-9  Repeat steps 1-4 with the new matrices. You should arrive at the following : Price Distance Labor Wages 0.1519 0.1376 0.2222 0.285 7 0.7595 0.6878 0.6667 0.5 0.0506 0.0764 0.0741 0.142 9 0.0380 0.0983 0.0370 0.071 4 SUM 1

= ¼ * Row Average Price Distance Labor Wages 0.1519 0.1376+ + 0.2222+ 0.1519 0.1376+ + 0.2222+ 0.2857 0.1994 = Distance ¼ * 0.7595+ 0.6878+ 0.6667+ 0.5 0.6535 = Labor 0.0506+ 0.0764+ 0.0741+ 0.1429 0.086 Wages 0.0380+ 0.0983+ 0.0370+ 0.0714 0.0612 SUM 1

Take the criteria matrix and multiple it by the preference vector Price Distance Labor Wages 0.1994 A 0.0512 0.2819 0.1790 0.1561 X 0.6535 B 0.1185 0.0598 0.6850 0.6196 0.086 C 0.3803 0.6583 0.1360 0.2243 0.0612 Location A score =0.1994(0.0512)+0.6535(0.2819)+0.086(0.1790)+0.0612(0.1561)=0.3091 Location B score =0.1994(0.1185)+0.6535(0.2819)+0.086(0.1790)+0.0612(0.1561)=0.1595 Location C score =0.1994(0.3803)+0.6535(0.2819)+0.086(0.1790)+0.0612(0.1561)=0.5314

 In our example above,  A is 30.91%, B is 15.95% and C is 53.14%.  Location C should be chosen for Jilley Bean Co. to built a plant.