# Full details of Analytic Hierarchy Process AHP

AHP is analytic hierarchy process. This method can be used to calculate weights of criteria this is a simple decision matrix with four criteria that is price, storage space, camera and looks with five alternatives. Each alternative has their own value of criteria associated with them. The first-and-foremost step in AHP is creating the hierarchical structure in which goal is kept in the first level.

In this example, the goal is to buy the best mobile-phone criteria is kept in the second level and alternative is kept in level three. Each alternative has their own value of criteria associated with them. Example, each mobile phone will have their own price or cost associated with them. Similarly, each mobile phone will have their own value of storage space.

Second step is to create a pairwise comparison matrix. This pairwise matrix gives the relative importance of various attribute with respect to the goal. If we take this example, how important is price while buying a mobile or what is the importance of storage space when we buy mobile phone.

This pairwise comparison matrix is created with the help of scale of relative importance this is the scale of relative importance in which one is for equal importance. 3 is given for moderate importance five for strong importance 7 for very strong 9 and extremely important values. The length of pairwise matrix is equivalent to the number of criteria used in decision making process.

Here we have a 4×4 matrix. As we have four criteria i.e. price, storage space, camera and looks, we will have a 4×4 matrix the value in the pair wise matrix depend upon the decision maker or the person who want to buy the mobile phone.

What will be the value of this cell for that some question should be asked to the person who is buying the mobile phone how important is Big Data Analytics price or cost with respect to storage space for a person like me. Price of cost is of a strong importance than storage space. If storage space is given X value, then price or cost will be given 5X value. We can see here that for strong importance a value of 5 is given now. Next what we have to do is we have to divide the row element by the column element now.

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Here price is the row element and storage space is a column element. The storage space has become an x value and price as 5x. So, here the value will be 5X divided by X which is equivalent to 5. Here storage space is given X value and price 5X. So, it will give 1/5. For this particular cell, the question asked should be how important is price or cost with respect to camera. Price or cost is of moderate to strong importance than camera.

So, if camera is given X value, price and cost will be given 4X value so we can also take anything in intermediate between 3 & 5, that is of 4 importance. That is moderate to strong importance will be assigned a 4 value. Similarly, here camera to price will be given 1/4 value. Now how important is camera with respect to storage space? Camera is of equal to moderate importance than storage space. So if camera is given 2X value storage space will be given X value. So, here we will get a value of 2 well vice versa storage to camera will be given 1/2 value. Similarly, we can assign values to each cell. The fractional value has been converted to decimal value and the sum of each value is calculated, that is 1 + 0.2 + 0.25 + 0.14 will give 1.59.

Normalized pair wise matrix is calculated. All the elements of the column is divided by the sum of the column. Here we can see that 1 (one) is divided by 1.5 (one point five), 9.2 (nine point two) is divided by 1.59 (one point five nine) and so on.

This is the normalized pairwise matrix. Here I have calculated the criteria weights, the weights are calculated by averaging all the elements in the row we have just added all these elements and divided it with the number of criteria which will give the criteria of weight. Next step is calculating the consistency that is to check whether the calculated value is correct or not.

For this I have taken the same pairwise comparison matrix which is not normalized. I have multiplied each value in the column with the criteria value. So, here 1(one) has been multiplied by the criteria weight that is 0.6038. Similarly, 0.2 is multiplied with the criteria 0.6038. Next, we calculate this ratio of weighted sum value and criteria weight.

Now, we calculate it for each row. On solving, we get this value now lambda max is calculated by taking the average of all these values. Next, we calculate the consistency index CI which is given by the formula lambda max minus n upon n-1 (n minus 1). In this example, n is 4 as we have four criteria.

Finally, we calculate the consistency ratio which is given by dividing the consistency index with random index (RI), random index is the consistency index of randomly generated pairwise matrix. I have shown the random index table for up to 10 criteria in our example the random index for n equal to 4 is 0.90 so have just calculated the consistency ratio since the value of consistency ratio CR is 0.037311 for the proportion of inconsistency CR is less than 0.10 which is the standard we can assume that our metrics is reasonably consistent so we may continue with the process of decision making using AHP based on the requirement of buyer.

These criteria weights can be used by the decision maker for further calculation. So, here the price has been given 60 percent weighted storage space 13.65 percent weightage camera 19.58 percent and looks a weightage of 6.46 percent.