Analytic Hierarchy Process (AHP) Example

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:

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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