Pages: 9 (886 words) Published: November 9, 2013
Chapter 6

Analytic Hierarchy Process (AHP) Example
One Monday morning1 , 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
1
3
5
7
9

Comparison
Objectives i
Objectives i
Objectives i
Objectives i
Objectives i

description
and j are of equal importance
is weakly more important that j
is strongly more important that j
is very strongly more important that j
is absolutely more important that j

Table 6.1: Pairwise comparison values
Of course, we set aii = 1. Furthermore, if we set aij = k , then we set aji = thinking hard about his preferences, comes up with the following table: Location
Salary
Content
Long

Location
1
5
3
2

Salary

Content
1
3

1
2

1

2
1

Peter,

Long

1
5

1
k.

4
3
1

1
2
1
4

1
3

Table 6.2: 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

= 0.091.
1+5+3+2
1

Thanks to Michael Trick for the details of this example

1

2

6. Analytic Hierarchy Process (AHP) Example

The other entries become:
Location
Salary
Content
Long

Location
0.091
0.455
0.273
0.182

Salary
0.102
0.513
0.256
0.128

Content
0.091
0.545
0.273
0.091

Long
0.059
0.471
0.353
0.118

Avg.
0.086
0.496
0.289
0.130
=1

Table 6.3: 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:

Acme (A)
Bankers (B)
Creative (C)
Dynamic (D)

A
1
2
3

B

C

1
2

1
2

1
3
1
2

1

1
5

1
7

1
9

D
5
7
9
1

Table 6.4: 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: Acme (A)
Bankers (B)
Creative (C)
Dynamic (D)

A
0.161
0.322
0.484
0.032

B...

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