*Sensitivity analysis is a fundamental concept in the effective use and implementation of quantitative decision models, whose purpose is to assess the stability of an optimal solution under changes in the parameters.* (Dantzig)

#### Weighted sum model (Alternative Evaluation)

In AHP the preference *P _{i}* of alternative

*A*is calculated using the following formula (weighted sum model):

_{i}(1)

*w*ith

*W*the weight of criterion

_{j}*C*, and

_{j}*a*the performance measure of alternative

_{ij}*A*with respect to criterion

_{i}*C*. Performance values are normalized.

_{j}(2)

**Example**

Sensitivity analysis will answer two questions:

- Which is the most critical criterion, and
- Which is the most critical performance measure,

changing the ranking between two alternatives?

#### The most critical criterion

The *most critical criterion* is defined as the criterion *C _{k}*, with the smallest change of the current weight

*W*by the amount of

_{k}*δ*changing the ranking between the alternatives

_{kij}*A*and

_{i}*A*.

_{j}The *Absolute-Top* (or AT) *critical criterion* is the most critical criterion with the smallest change *δ _{kij}* changing the ranking of the best (top) alternative.

The *Absolute-Any* (or AA) *critical criterion *is the most critical criterion with the smallest change *δ _{kij}* changing any ranking of alternatives.

For each pair of alternatives *A _{i}*,

*A*, with

_{j}*i*= 1 to

*n*and

*i*<

*j*we calculate

(3)with .

**Example**

Table 2

- The absolute-top critical criterion is Neighbourhood: a change from 18.8% by -8% will change the ranking between the top alternative A1 (House A) and alternative A2 (House B).
- The absolute-any critical criterion is the same as above, as -8% is the smallest value in the table.

As the weight uncertainty for the criterion *Neighbourhood* is +1.4% and -1.3%, the solution is stable.

** ****The most critical measure of performance**

The *most critical measure of performance* is defined as the minimum change of the current value of *a _{ij}* such that the current ranking between alternative

*A*and

_{i}*A*will change.

_{j}For all alternatives *A _{i}* and

*A*with

_{j}*i*≠

*j*and and each criterion we calculate

(4)

with .

**Example**

Table 3

- The
*absolute-any**critical performance measure*is found for alternative(House C) under the criterion*A*_{3}*Financing*. A change from 27.9% by 20.4% will change its ranking with alternative(House B), i.e. only a (drastic) change from 27.9% to 48.3% of the evaluation of House C with respect to Financing would change the ranking of House C and House B.*A*_{2}

#### Implementation in AHP-OS

For alternative evaluation the method described above is implemented in AHP-OS. On the group result page in the *Group Result Menu* tick the checkbox *var* and then click *Refresh*.

Under the headline Sensitivity Analysis TA and AA critical criterion as well as AA critical performance measure will be displayed. You can download the complete tables as csv files with a click on *Download*.

#### References

Triantaphyllou, E., Sánchez, A., *A sensitivity analysis approach for some deterministic multi-criteria decision making methods*, Decision Sciences, Vol. 28, No. 1, pp. 151-194, (1997).

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