In wise by Matrix relationship. In this way,

Inour day to day life we take various decisions. Most of these decisions arebased on our judgment which is based on some criterion that we consider beforemaking decision. Let’s take basic example- For buying a new shirt, when weenter into showroom we consider various alternatives, evaluate them and at lastchoose them. We generally oversee the real procedure behind choosing. Weconsider various criterions before buying that shirt for example- Color of theshirt, price of the shirt, comfortable, design, brand etc. Generally one shirtcannot maximize these entire criterion hence we prioritize our criterion.

Inour mind we rank our criterion, we give different weightage to them and basedon those weightage, the alternative which maximize the fulfillment of criterionwe select that alternative. Thiswas the very basic example given to understand the basic process of decisionmaking. In real life, we encounter various complex situations where decisionmaking becomes very difficult. Moreover, various people has various priorities,which make group decision even more complex.Itis basically a structured technique for dealing with complex decisions that wasdeveloped by Thomas L.

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Saaty in the 1980.  Thebasic concept of this model is comparing variables by using pair wise by Matrixrelationship. In this way, variables are compared and weights are given to eachvariable in respect to the other.      Multi Criteria Decision Making The analytic hierarchy process is applied at variousareas where we need to evaluate best alternative for decision making. Let’s seesome examples- GOAL CRITERIA ALTERNATIVES Selection of College Location A Reputation B Placement C Fees Selection of Apartment Price A Area Location B Colony C Selection of Leader Experience A Education B Benefit C Leadership quality D Selection of new car Price A Availability of spares B service C Discount  The analytic hierarchy process can be used in allthese decision making process. Thedetermination of criteria and alternatives are subjective in nature.

The listof criteria and alternatives that is mentioned above is not exhausted. They donotcover all possible criteria or all possible alternatives but basically showpossible options. Different people may prepare different lists. Criteria’s maybe broken down into more detail sub-criteria and many alternatives can be addedaccordingly.Mostof the decisions making are based on individual judgment but sometime groupdecisions are also made.Theanalytic hierarchy process solves this problem in various steps. Steps in analytic hierarchy process: 1.

      Startwith Defining the problem statement.2.      Developa model: Breakdown the model into a hierarchy of goals, criteria andalternatives.3.      Constructa pair wise comparison matrix for criteria by –a)      Definethe comparison scale.

b)      Comparethe criteria pair wise.c)      Giveweightage to one criterion in comparison to other.4.      Preparenormalized matrix.5.      Findout weights for criteria’s using normalized matrix.6.

      Findout inconsistency using normalized matrix.7.      Constructlocal priorities matrix for the alternatives. (Derive priorities for eachpriorities separately)8.      Repeatthe steps- 4-6 for alternative comparison matrix.9.      Findout global weights for alternative.10.

  Performsensitivity analysis to know how changes in the weights of the criterion affectthe choice of alternatives.11.  Takethe managerial decision based on results and sensitivity analysis. Wecan better understand all these steps by solving a problem.

Let’sstart with a problem where NTPC Ltd. Has to decide which is the best locationfor the project out of two options A and B. Decision of project is based on 3basic criteria’s. 1.      Availabilityof coal2.      Availabilityof water3.

      Availabilityof market  Example 1: NTPC Project selection Step: 1Define the problem statement– Selection of NTPC ProjectStep: 2Develop a Model: Goals , Criteria’s and Alternatives.  Step: 3 Comparison matrix for criteria’s. Notall the criteria’s are having same priorities. We have to find out relativepriorities (weightage) of criteria’s. It is called relative because theobtained criteria priorities are measured with respect to each other. We needto give priority of one over other using some fixed scale.Wecan use Saaty’s pair wise comparison scale for this purpose.Thepair wise comparison scale can be used to rank one criterion over others.

Total number of comparisonrequired for the pair-wise comparison matrix is given by  n(n-1)/2          wheren = number of elements Heretotal number of criteria’s are three; Therefore total number of comparisonrequired will be (3*2)/2 = 3.Pair wise comparisonmatrix:Based on the criteria’s we have to form pair wise comparison matrix comparingtwo at a time. For example we have to give weight of coal over water or weightof water over market.Wecan form a matrix like A: Selection of Project Coal Water Market Coal 1 7 3 Water 0.1429 1 0.3333 Market 0.

3333 3 1  Thiscomparison matrix shows the weight of one criterion over other. Diagonal of thematrix is always 1 as the criteria’s are being compared by them.Value 7 shows theweightage of coal over water (Very strongly more important). We can write (Weight of Coal/ Weightof water) = 7Similarly (Weight ofwater/ Weight of market) = 3 (Moderately more important)Values under diagonalis nothing but the inverse of values above the diagonal as we know from matrixproperty that                                     Aij= (1/Aji)After preparation ofcomparison matrix the next step is to prepare the normalized matrix.Step-4 Normalized matrix To prepare the normalize matrix A’,the first step is to add the column ratio weightage given in comparison matrixand divide this sum to each ratio weightage of comparison matrix A. 1.47619 11.

00000 4.33333 Selection of Project Coal Water Market Coal 0.6774 0.6364 0.6923 Water 0.0968 0.0909 0.0769 Market 0.

2258 0.2727 0.2308  Step-5 Weights of criterion Average of rows of Matrix A gives the weightage matrix w orNormalized Principal Eigen vector.In this case, W here is given by  W coal 0.

6687 W water 0.0882 W market 0.2431  The normalized principal Eigen vector is also calledpriority vector.Weightagematrix W clearly shows that weightage given to Coal > Market > WaterWecan interpretate that for the decision of project selection of NTPC, we givemore weightage to availability of coal followed by market and water.Step-6 Consistency check Itis important to check for the consistency of the judgment given for criteria’s.Theidea of consistency is very simple.

If one prefer availability of coal 7 timesthan availability of water that is (Wcoal/Wwater)= 7 and if we preferavailability of coal 3 times than availability of market that is(Wcoal/Wmarket)=3 then its very easy to understand that preference availabilityof water over market should be 3/7 = 0.43. But, when we ask the same questionfor preference of availability of water over availability of market,respondent’s answer is 0.333.

We can generalize thisby saying that any comparison matrix is consistence if –A12*A23= A13Ithappens because the respondent is not consistence with the answer in the matrixof judgment. Some inconsistency is always expected in Analytical HierarchyProcess as the comparing value is totally depends on individual subjectivediscretion. It also varies with person to person and group to group. We canfind out the inconsistency present in the analysis. Moreover; we can get theacceptable range of consistency. In Analytical Hierarchy Process analysisacceptable range of inconsistency is given by consistency ratio value of 0.1 Consistencyratio is the ratio of consistency index (CI) and random consistency index (RI).Random consistencyindex (RI) can be obtained by the saaty table or can be directly found out byusing the formula-RI= 1.98* (N-2)/NWhereN is the size of matrix A.Tofind the consistency index (CI) we have to first find out Principal Eigen Value– ? maxItcan be obtained by from the summation of products between each element of Eigenvector and the sum of columns of the reciprocal matrix.After getting value ofPrincipal Eigen Value – ? max, ConsistencyIndex can be found out using formula CI= (? max-n)/ (n-1)Letsunderstand the concept by the above example.Reciprocal Matrix/ ComparisonMatrix A – Selection of Project Coal Water Market Coal 1 7 3 Water 0.1429 1 0.3333 Market 0.3333 3 1  NormalizedEigen Vector Matrix- W: W coal 0.6687 W water 0.0882 W market 0.2431  Inexcel multiplication of these two matrix can easily be done with the help offormula = mmult() 2.0154 0.2648 0.7306  Summationof all these three values gives Principal Eigen value– ? max.If a matrix is consistent then A*W=N*WIf a matrix is not consistent thenAW’ = ? max*W’ = ? max*W’