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Pairwise Comparison Matrix

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By now you know how to make paired comparisons. In this section you will learn how to make a reciprocal matrix from pair wise comparisons.
For example John has 3 kinds of fruits to be compared and he made subjective judgment on which fruit he likes best, like the following

 

 
 


We can make a matrix from the 3 comparisons above. Because we have three comparisons, thus 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:
  1. If the judgment value is on the left side of 1, we put the actual judgment value.
  2. If the judgment value is on the right side of 1, we put the reciprocal value .
Comparing apple and banana, John slightly favor banana, thus we put in the row 1 column 2 of the matrix. Comparing Apple and Cherry, John strongly likes 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
 

To fill the lower triangular matrix, we use the reciprocal values of the upper diagonal. If is the element of row column of the matrix, then the lower diagonal is filled using this formula 
 
 Thus now we have complete comparison matrix
 
 Notice that all the element in the comparison matrix are positive, or


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Analytic Hierarchy Process (AHP) Example

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One Monday morning Peter, instead of attending class, was mulling over his four job offers.
His offers came from Acme Manufacturing (A), Bankers Bank (B), Creative Consulting (C) and Dynamic Decision Making (D). He knew that factors such as location, salary, job content, and long-term prospects were important to him, but he wanted some way to formalize the relative importance, and some way to evaluate each job offer. Luckily, he attended the following Tuesday class of MSTC, who showed him one way to think about these problems. This technique is called the Analytic Hierarchy Process.

The first step in AHP is to ignore the jobs and just decide the relative importance of the objectives. Peter does this by comparing each pair of objectives and ranking them on the following scale: Comparing objective i and objective j (where i is assumed to be at least as important as j), give a value aij as follows:


 Value aij   Comparison description 
 1    Objectives i and jare of equal importance  
 3    Objectives i is weakly more important that j 
 5    Objectives i is strongly more important that j 
 7    Objectives i is very strongly more important that j 
 9    Objectives i is absolutely more important that j 
Pairwise comparison values

Of course, we set aii = 1. Furthermore, if we set aij = k , then we set aji = 1 k . Peter, thinking hard about his preferences, comes up with the following table:

 Location   Salary   Content   Long 
 Location   1    15   13   12 
 Salary   5    1    2    4  
 Content   3    12   1    3  
 Long   2    14   13   1  
Preferences on Objectives

Now, the AHP is going to make some simple calculations to determine the overall weight
that Peter is assigning to each objective: this weight will be between 0 and 1, and the total
weights will add up to 1. We do that by taking each entry and dividing by the sum of the
column it appears in. For instance the (Location,Location) entry would end up as
 1  / 1+ 5+ 3+ 2       =       0.091.

The other entries become:
ahp example
Weights on Objectives

This suggests that about half of the objective weight is on salary, 30% on amount of job
content, 13% on long term prospects, and 9% on location.
Now, why does this magical transformation make sense? If we read down the first column in the original matrix, we have the values of each of the objectives, normalized by setting the value of location to be 1. Similarly, the second column are the values, normalizing with salary equals 1. For a perfectly consistent decision maker, each column should be identical, except for the normalization. By dividing by the total in each column, therefore, we would expect identical columns, with each entry giving the relative weight of the row’s objective. By averaging across each row, we correct for any small inconsistencies in the decision making process.

Our next step is to evaluate all the jobs on each objective. For instance, if we take Location, if we prefer to be in the northeast (and preferably Boston), and the jobs are located in Pittsburgh, New York, Boston, and San Francisco respectively, then we might get the following matrix:

 A   B   C   D 
 Acme(A)   1    12   13   5  
 Bankers(B)   2    1    12   7  
 Creative(C)   3    2    1    9  
 Dynamic(D)   15   17   19   1  
Location Scores

Again we can normalize (divide by the sums of the columns, and average across rows to get
the relative weights of each job with regards to location.) In this case, we get the following:

 A   B   C   D   Avg. 
 Acme(A)   0.161    0.137    0.171    0.227    0.174  
 Bankers(B)   0.322    0.275    0.257    0.312    0.293  
 Creative(C)   0.484    0.549    0.514    0.409    0.489  
 Dynamic(D)   0.032    0.040    0.057    0.045    0.044  
 P= 1  
Relative Location Scores

“Location Value” is about 49% for C, 29% for B, 17% for A and D has about 4%. We can go through a similar process with Salary, Content, and Long-term prospects. Suppose the relative values for the objectives can be given as follows:
Recalling our overall weights, we can now get a value for each job. The value for Acme Manufacturing is
(0.174)(0.086) + (0.050)(0.496) + (0.210)(0.289) + (0.510)(0.130) = 0.164

   (A)cme   (B)ankers   (C)reative   (D)ynamic 
 Location   0.174    0.293    0.489    0.044  
 Salary  0.050   0.444   0.312   0.194 
Content  0.210   0.038   0.354   0.398  
 Long   0.510    0.012    0.290    0.188  
Relative scores for each objective

Similarly, the value Bankers Bank is
(0.293)(0.086) + (0.444)(0.496) + (0.038)(0.289) + (0.012)(0.130) = 0.256
The value for Creative Consultants is 0.335, and that for Dynamic Decision is 0.238. Creative Consultants it is! Peter immediately makes his decision.

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Procedure AHP

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AHP is one method to help prepare a priority of the various options using several criteria (multi-criteria). Because of its multi-criteria, AHP enough widely used in the preparation of priority. For example, for set priorities for research, management research institute often using several criteria such as the impact of research, costs, human resource capacity, and also possible execution time.

In addition to the multi-criteria, AHP is also based on a structured and logical process. Selection or preparation of priority done with a logical and structured procedure. Activity was conducted by experts associated with the representative alternatives will be drawn up priorities (Bougeois, 2005).

Broadly speaking, there are three stages in the preparation AHP priorities, namely:

1. Decomposition of the problem
In setting priorities, then the problem of prioritization should be able to be decomposed into the goal (goal) of an activity, identification of options, and the formulation of criteria to choose a priority (Figure 1). The first step is formulate the goal of drafting a priority. in OPEC strategy formulation case, the goal of OPEC is to improve the welfare of members of OPEC and increase OPEC's social and political role in the international forum. For the case of election supplier, the purpose of the activities is to choose the best supplier.

In the case of the selection of research proposals, the purpose of the activities may look for topic / best research proposal. Once the destination can be determined, the next step is determining the criteria of that goal. For the case of OPEC, criteria objectives are (i) the stabilization of revenues; (ii) conservation of oil deposits; and (iii) an increase in OPEC's political role in the international forum. For selection of suppliers, the indicators used include (i) logistics capabilities; (ii) production capability; and (iii) the ability commercial / finance.

Based on the objectives and criteria, several options need to be identified. Such options should be the choices potential, so the number is not too much choice. For case OPEC, the choice of strategy is (i) the stability of production and price; (ii) production and export quotas; (iii) fluctuations (shock) production;and (iv) maintain the current policy is applied. For supplier selection case, the options available are 3suppliers namely (i) Kirsehir; (ii) Bastas; and (iii) AKYUS. For preparation of research priorities, the possible choices is the title / topic research proposed by researchers.

2. Assessment for comparing elements results decomposition
Once the problem is decomposed, then there are two stages of assessment or comparing between elements of comparison between criteria and the comparison between options for each criterion. Comparison among the criteria intended to determine weights for each criteria. On the other hand, the comparison between options for each criterion. In other words, this judgment is intended to see how important a choice seen from certain criteria.

In conducting the assessment / comparison, experts who developed AHP uses a scale of 1/9 to 9. If option A and B being equal (indifferent), then A and B were each given If for example the value 1. A better / more preferably from B, then A given grades 3 and B rated 1/3. If A is much preferred to B, then A eg rated 7 and B rated 1/7. This assessment will not be used in this article because of the lack of a logical manner.As an example, if A and B are worth 7 1/7, then the difference between A and B is almost close to 700%.

An alternate assessment that is used by Bourgeois (2005) who wore a scale between 0.1 to 1.9 is considered more logical as presented in Table 1. If A slightly better / preferred from B, then A and B rated 1.3 rated 0.7, indicating a distance of about 30% of the value of 1. If A is much preferred by B, then the value of A becomes B 1.6 to 0.4. The way such assessments will be used in this paper.


Assessment  Value A  Value B
A very much preferred than B  1.9  0.1
A far more preferable than B 1.6 0.4
A slightly more preferred than B 1.3  0.7
A equals B 1.0  1.0
A slightly less preferred than B  0.7 1.3
A far less favored than B 0.4  1.6
A very much less favored than B  0.1 1.9
Scale Ratings

By using assessments such as Table 1, then comparison between the criteria would result in Table 2 below. For facilitate, in the table assumed there were only four criteria. From The table can be summarized as follows:

  • Cij is the result of assessment / comparison between criteria i to j
  • Ci. is owned criteria sale value to i
  • C is the sum of all values Ci.
  • Weight  criteria to Iobtained by dividing the value of Ci. with C.

Criteria   CR1    CR2    CR3    CR4  Total   Weight 
 CR1    -   c12    c13    c14    c1.    bc1= c1./c  
 CR2    c21    -   c23    c24    c2.    bc2=c2./c  
 CR3    c31    c32    -   c34    c3.    bc3=c3./c  
 CR4    c41    c42    c43    -   c4.    bc4=c4./c  
Total           C    
Comparison between Criteria

 
Using the same procedure, it is performed comparison between options (OP) for each criterion. table 3 The following illustrates the comparison between options (4 choices) to criterion 1 (C1) with the following explanation:
  • Oij an assessment / comparison between the choice i with k for criteria to j
  • Oi. is the sum value of the owned options to i
  • O is the sum of all values oi.
  • BOij an option value to i for criteria to j
The choice between the assessment process continues for all criteria. For the record, the assessment should be done by experts and major stakeholders. Typically, the number of experts vary depending the availability of resources. Assessment can be done with distributing questionnaires to each expert or by conduct a meeting of experts to conduct an assessment The. For this case study, the assessment carried out by gather experts.


 C1    OP1    OP2    OP3    OP4  Total Weight
 OP1    -   o12    o13    o14    o1.    bo11=o1./o  
 OP2    o21    -   o23    o24    o2.    bo21=o2./o  
 OP3    o31    o32    -   o34    o3.    bo31=o3./o  
 OP4    o41    o42    o43    -   o4.    bo41=o4./o  
Total          O    
Comparisons between options for Criterion C1
 

3. Synthesis of priorities
Synthesis of the assessment results is the final stage of AHP. on Essentially, this synthesis is the sum of the weights obtained by each option on each criterion after a given weight of these criteria. In general, the value of an option is as follows :
 
 ibop   =   ijboΣ   jbc*    ............................................. (1)  
     i=1      
 bopi = value / weights for selection to i

The formula can also be presented in tabular form. For simplicity, it is assumed there are four criteria with four options as Table 4 below. As an example of the value of the priority / weight of option 1 (OP1) obtained by multiplying the weight value on the criteria the value associated with the criteria for option 1 as following:

bopi = bo11* bc1+ bo12* bc2 + bo13* bc3+ bo14* bc4 ................. (2)

 It is synonymous done to option 2, 3 and 4. With  comparing the values obtained by each option, the priority  can be prepared based on the magnitude of that value. The higher the value  an option, the higher the priority, and vice versa.

 
   CR1    CR2    CR3    CR4    Prioritas  
   bc1    bc2    bc3    bc4    bopi  
 OP1    bo11    bo12    bo13    bo14    bop1  
 OP2    bo21    bo22    bo23    bo23    bop2  
 OP3    bo31    bo32    bo33    bo34    bop3  
 OP4    bo41    bo42    bo43    bo44    bop4  
Synthesis Assessment

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