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## Cross Tabulation

The simplest multi criteria decision making is to put into a cross table of criteria and alternatives. Then we put subjective score value on each cell of the table. The sum (or normalized sum) of and compute the sum of all factors for each alternatives.

For example, we have 3 alternative choices X, Y and Z and four criteria to decide the alternatives A, B, C and D. You can input any name for alternatives and criteria. The values on the table 2 below are any number certain range for each factor. The only similarity between these numbers is that they have the same interpretation that higher values are preferable than smaller values.

Table 2: Evaluation based on scores of each factor

 Criteria | Alternatives Choice X Choice Y Choice Z Range Factor A 1 4 5 0-5 Factor B 20 70 50 1-100 Factor C -2 0 1 -2 to +2 Factor D 0.4 0.75 0.4 0 to 1 Sum 19.4 74.75 56.4 Normalized Score 12.9% 49.7% 37.5%

If you have many alternatives, sometimes it is easier to compare the sum value of each choice by normalizing them. Total sums is 150.55 (=19.4+74.75+56.4). The sum of each choice is normalized by division of each sum with the total sums. For instance, choice X is normalized into 19.4/150.55*100%= 12.9%. Clearly choice Y is preferable than choice Z while choice Z is better than X.

However, you will notice that the range of value for each factors are not the same. It is quite unfair to sum all the values of multiple criteria and compare the result. Clearly factor B is dominant because the range has higher value. To be fair, we can propose two solutions:

1. Instead of using arbitrary values for each factor, we just rank the choice for each factor. Smaller rank value is more preferable than higher rank.
2. We transform the score value of each factor according to the range value such that each factor will have the same range.

In the next sections, let us try the two solutions one by one.

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Preferable reference for this tutorial is

Teknomo, Kardi. (2006) Analytic Hierarchy Process (AHP) Tutorial .
http://people.revoledu.com/kardi/tutorial/AHP/