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Decision making by AHP and ANPPradeep S & Nihar Barik NITIE, Mumbai

Multi Criteria Decision makingy Multi Criteria Decision Making(MCDM) is a systematic

procedure for transforming complex decision problems by a sequence of transparent steps assists the decision maker in arriving at a rational decision.

Why should we use it? MCDM enables multiple stakeholder preferences to be modeled MCDM offers improved coordination and collaboration MCDM can be implemented to integrate spatial information

How does MCDM works

AHP???y A Multi Criteria

Decision Making method for complicated and unstructured problems.y An approach that uses

a hierarchical model having levels of goal, criteria, possible subcriteria, and alternatives.

A powerful and understandable methodology that allows groups or individuals to combine qualitative and quantitative factors in decision making process

Analytic hierarchy process (AHP) was proposed by Saaty (1977, 1980) to model subjective decision-making processes in a hierarchical system.

The Theoretical Foundation of the AHPBased on pairwise comparison Analyzes the criteria + alternatives in a structured manner and ranks alternatives Assumes decision maker can provide paired comparisons based on knowledge, intuition and decision maker never judges one alternative infinitely over other. All decision problems are considered as a hierarchical structure in the AHP The applications of AHP can refer to corporate planning, portfolio selection, and benefit/cost analysis by government agencies for resource allocation purposes.

The general form of the AHP

The main four steps of the AHPStep 1. Set up the hierarchical system by decomposing the problem into a hierarchy of interrelated elements. Step 2. Compare the comparative weights between the attributes of the decision elements to form the reciprocal matrix. Step 3. Synthesize the individual subjective judgments and estimate the relative weights. Step 4. Aggregate the relative weights of the decision elements to determine the best alternatives/strategies.

Primary Scales of MeasurementScale NominalNumbers Assigned to Runners Rank Order of WinnersThird place Second place First place Finish7 8 3

Ordinal

Finish

Interval

Performance

Rating on a 0 to 10 Scale

8.2

9.1

9.6

Ratio

Time to Finish, in Seconds

15.2

14.1

13.4

A Classification of Scaling TechniquesScaling Techniques

Comparative Scales

Noncomparative Scales

Paired Comparison

Rank Order

Constant Sum

Q-Sort and Other Procedures

Continuous Itemized Rating Scales Rating Scales

Likert

Semantic Differential

Stapel

Paired ComparisonTo make tradeoffs among the many objectives and many criteria, the judgments that are usually made in qualitative terms are expressed numerically. To do this, rather than simply assigning a score out of a persons memory that appears reasonable, one must make reciprocal pair wise comparisons in a carefully designed scientific way.

Fundamental Scale1 3 5 7 9 2,4,6,8 Equal Importance Moderate importance of one over another Strong or essential importance Very strong or demonstrated importance Extreme importance Intermediate values

Use reciprocals for inverse comparisons

Comparison MatrixGiven: Three apples of different sizes.The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may h

Apple A

Apple B

Apple C

We Assess Their Relative Sizes By Forming RatiosSize Comparison Apple A Apple B Apple C Apple A S1/S1 S2 / S1 S3 / S1 Apple B S1/S2 S2 / S2 S3 / S2 Apple C S1/S3 S2 / S3 S3 / S3

Pairwise ComparisonsSizeApple ASize ComparisonThe image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again.

Apple BApple BThe image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again.

Apple CApple CResulting Relative Size Priority of Apple Eigenvector

Apple AThe image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again.

The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may h

Apple AThe image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again.

1

2

6

6/10

A

Apple BThe image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may h

1/2

1

3

3/10

B

Apple C

1/6

1/3

1

1/10

C

When the judgments are consistent, as they are here, any normalized column gives the priorities.

ConsistencyIn this example Apple B is 3 times larger than Apple C. We can obtain this value directly from the comparisons of Apple A with Apples B & C as 6/2 = 3. But if we were to use judgment we may have guessed it as 4. In that case we would have been inconsistent. Now guessing it as 4 is not as bad as guessing it as 5 or more. The farther we are from the true value the more inconsistent we are. The AHP provides a theory for checking the inconsistency throughout the matrix and allowing a certain level of overall inconsistency but not more.

If we wish to compare a set of n attributes pairwise according to their relative importance weights, where the attributes are denoted by a1 , a2 ,..., an and the weights are denoted by w , w ,..., wn , then the pairwise comparisons can be 1 2 represented as: a11 L M A ! ai1 L M an1 L

a1 j L M aij L M anj L

ain M ain , where a ! 1/ a , a ! a / a , a ! ij ij ik jk ij ji M ann

i

/

j

.

By multiplying A by w yield w1 w1 w1 L L w1 wj wn w w1 M M M 1 M M wi wi wi , or L L Aw ! w j ! n w j ! nw w wj wn 1 M M M M M w wn n wn wn wk w L w L w n j 1

( A nI ) w = 0.

Since solving the above equation is the eigenvalue problem, we can derive the comparative weights by find the eigenvector w with respec to Pmax Which satisfies Aw ! Pmax w , where Pmax is the largest eigenvalue of the matrix A. Furthermore, in order to ensure the consistency of the subjective perception and the accuracy of the comparative weights, two indices, including the consistency indexes (C.I.) and the consistency ratio (C.R.), are suggested. The equation of the C.I. can be expressed as:

C.I. = (Pmax n) / (n-1), where Pmax is the largest eigenvalue, and n denotes the numbers of the attributes.

Saaty (1980) suggested that the value of the C.I. should not exceed 0.1 for a confident result. On the other hand, the C.R. can be calculated as:C .I . , R .I .

C .R . !

where R.I. refer to an random consistency index which is derived from a large sample of randomly generated reciprocal matrices using the scale1/ 9,1/ 8,K ,1,K ,8,9.

The R.I. with respect to different size matrix is shown as Number of elements R.I. 3 4 5 6 7 8 9 10 11 12 13

0.52 0.89 1.11 1.25 1.35 1.40 1.45 1.49 1.51 1.54 1.56

The C.R. should be under 0.1 for a reliable result, and 0.2 is the maximum tolerated level.

Typical AHP Problem

Linear HierarchyGoal Criteria component, cluster (Level) element Alternatives

Subcriteria

Learning AHP through illustration

Solving Problem By AHPGoal

PriceCriterias

Mileage

Prestige

Comfort

Alternatives

Car 1

Car 2

Car 3

Level I Comparison Comparing the Cluster node with respect to Goal nodePrice Price Mileage Prestige Comfort 1 1/3 1/4 1/3 Mileage Prestige comfort 3 1 1/2 3 4 2 1 4 3 1/3 1/4 1

Totalling

1.92

7.50

11.00

4.58

Normalizing the matrixPrice Price Mileage Prestige Comfort Mileage Prestige comfort 0.522 0.400 0.364 0.655 0.174 0.133 0.182 0.073 max = 1.92 (0.48525) + (7.50 )(0.1405) + 0.130 0.067 0.091 0.055 (11)(0.08575) + 0.289 * 4.58 = 4.249 0.174 0.400 0.364 0.218NEV Price Mileage Prestige Comfort 0.48525 0.1405 0.08575 0.289 Calculations =(0.522+0.4+0.364+0.655)/4 =(0.174+0.133+0.182+0.073)/4 =(0.13+0.067+0.091+0.055)/4 =(0.174+0.4+0.364+0.218)/4

Z1

max = 1.92 (0.48525) + (7.50 )(0.1405) + (11)(0.08575) + 0.289 * 4.58 = 4.249 Consistency Index = ( max n)/(n-1) = (4.249-4)/3= 0.08307 R.I for (n=4) = 0.89 C.R = C.I/R.I = 0.09 < 10% So consistent

Comparing wrt Price node in Alternative clusterCar 1 Car 1 Car 2 Car 3 1.00 1/7 1/4 Car 2 7.00 1.00 5 Car 3 NEV 4.00 1/5 1.00 Car 1 Car 2 Car 3 0.670 0.073 0.203

Normalizing and Equating the above table we get

A1

Comparing wrt Mileage node in Alternative clusterCar 1 Car 1 Car 2 Car 3 1.00 1.00 1.00 Car 2 1.00 1.00 1/2 Car 3 NEV 1.00 2.00 1.00 Car 1 Car 2 Car 3 0.328 0.411 0.261

Normalizing and Equating the above table we get

Mileage Car 1 Car 2 Car 3

Car 1 0.33 0.33 0.33

Car 2 0.4 0.4 0.2

Car 3 0.25 0.5 0.25

A2

= 3.0556 , Consistency Index = 0.02778 . Relative index for N= 3 matrix is 0.52 . So consistency ratio is 0. 0534

Comparing wrt Prestige node in Alternative clusterCar 1 Car 1 Car 2 Car 3 1.00 1/3 1/7 Car 2 3.00 1.00 1/5 Car 3 NEV 7.00 5.00 1.00 Car 1 Car 2 Car 3

0.643 0.283 0.074

Normalizing and Equating the above table we get

Mileage Car 1 Car 2 Car 3

Car 10.675676 0.225225 0.096525

Car 20.714286 0.238095 0.047619

Car 30.538462 0.384615 0.076923

A3

= 3.093 , Consistency Index = 0.0469 . Relative index for N= 3 matrix is 0.52 . So consistency ratio is 0. 09

Comparing wrt Comfort node in Alternative clusterCar 1 Car 1 Car 2 Car 3 1.00 1/3 5.00 Car 2 3.00 1.00 7.00 Car 3 NEV 1/5 1/7 1.00 Car 1 Car 2 Car 3 0.193 0.083 0.724

Normalizing and Equating the above table we get

Mileage Car 1 Car 2 Car 3

Car 1

Car 2

Car 3

A4

0.157978 0.272727 0.149254 0.052659 0.090909 0.10661 0.789889 0.636364 0.746269

= 3.114 , Consistency Index = 0.057 . Relative index for N= 3 matrix is 0.52 . So consistency ratio is 0. 109

Final Matrix table for AHP prioritiesPrice 0.485 Car 1 0.670 Mileage 0.140 0.328 Prestige 0.086 0.643 Comfort 0.289 0.193

Car2 Car3

0.073 0.203

0.411 0.261

0.283 0.074

0.083 0.724

Car 1 priority = 0.485*0.670+0.140*0.328+0.086*0.643+0.289*0.193 = 0.482 Car 2 priority = 0.485*0.073+0.140*0.411+0.086*0.283+0.289*0.083 = 0.141 Car 3 priority = 0.485*0.203+0.140*0.261+0.086*0.074+0.289*0.724 = 0.351 So Select Car 1 as it has high priority Value . (PleaseNote we have not considered the dependency between various criterias while evaluating)

So Why The Analytic Network Process (ANP ) ????

The Analytic Network Process (ANP) involves Dependence and Feedback Real life problems involve dependence and feedback. Such phenomena can not be dealt with in the framework of a hierarchy but we can by using a network with priorities. With feedback the alternatives can depend on the criteria as in a hierarchy but may also depend on each other. The criteria themselves can depend on the alternatives and on each other as well. Feedback improves the priorities derived from judgments and makes prediction more accurate.

Analytical Hierarchy ProcessA methodology that allows groups or individuals to deal with the interconnections(dependence and feedback) between factors of complex structure in decision making process

A Multi Criteria Decision Making method for complicated and unstructured problems

An approach that uses a network model having clusters of elements(criteria and alternatives)

NETWORKS, DEPENDENCE AND FEEDBACKy A network has clusters of elements, with the elements in one

cluster being connected to elements in another cluster (outer dependence) or the same cluster (inner dependence).y A network or feedback has sources and sinks. A source node is

an origin of paths of influence and never a destination of such paths. A sink node is a destination of paths of influence and never an origin of such paths. A full network can include source nodes; intermediate nodes that fall on paths from source nodes, lie on cycles, or fall on paths to sink nodes; and finally sink nodes. Some networks can contain only source and sink nodes. Still others can include only source and cycle nodes or cycle and sink nodes or only cycle nodes. A decision problem involving feedback arises often in practice.

Feedback Network with components having Inner and Outer Dependence among Their Elements

C4 C1Feedback

Arc from component C4 to C2 indicates the outer dependence of the elements in C2 on the elements in C4 with respect to a common property.

C2No arrow enters are source components such as C1 No arrow leaves are known as sink components arrows both enter and exit leave are known as transient components such asC2, C3 and C4 C2and C3 forms a cycle

C3

Loop in a component indicates inner dependence of the elements in that component with respect to a common property.

Steps Of Analytical Network Process (ANP)Step 1-Deconstructing a problem into a complete set of hierarchical or network models Step 2-Generating pairwise comparisons to estimate the relative importance of various elements at each level and Derive the local weights using the AHP.

Steps Of Analytical Network Process (ANP) (continued)Step 3- Formulate the supermatrix according to the results of the local weights and the network structure. Step 4- Raise the supermatrix to limiting powers for obtaining the final results .

Super Matrixy The influence of elements in the network on other elements

in that network can be represented in supermatrix:C1 e11 W11 C1 M e1n1 e21 e22 W21 W ! C2 M e2 n2 M M M em1 em 2 Wm1 Cm M emnm e12 W12 L W1m C2 L Cm e11 L e1n1 e21 L e2 n2 L em1 L emnm

W22

L

W2 m

Where Cm denotes the mth cluster, emn denotes the nth element in the mth cluster, and Wij is the local priority matrix of the influence of the elements compared in the jth cluster to the ith cluster. In addition, if the jth cluster has no influence to the ith cluster, then Wij=0.

M Wm 2

M M M L

M Wmm

Super Matrix : ExampleNetwork structure

Supermatrix

Twelve ANP Steps

Step 1: Define the decision problemy . Describe the decision problem in detail including

its objectives, criteria and subcriteria, actors and their objectives and the possible outcomes of that decision. y Give details of influences that determine how that decision may come out

Step 2: Determine Control and ubCriteriay Determine the control criteria and subcriteria in the

four control hierarchies one each for the benefits, opportunities, costs and risks of that decision y obtain their priorities from paired comparisons matrices. y If a control criterion or subcriterion has a global priority of 3% or less, you may consider carefully eliminating it from further consideration.

Step 3. Determine the most general network of clustersy Determine the most general network of clusters or components)

and their elements that applies to all the control criteria. y Number and arrange the clusters and their elements in a convenient way (perhaps in a column). y Use the identical label to represent the same cluster and the same elements for all the control criteria. y Determine the most general network of clusters

Step 4. Determine Clusters and Elementsy For each control criterion or subcriterion, determine the

clusters of the general feedback system with their elements y Connect them according to their outer and inner dependence influences. y An arrow is drawn from a cluster to any cluster whose elements influence it. y Describe the decision problem in detail including its objectives, criteria and subcriteria, actors and their objectives and the possible outcomes of that decision.

Step 5. Determine the approachy Determine the approach you want to follow in the

analysis of each cluster or element, influencing (the preferred approach) other clusters and elements with respect to a criterion, or being influenced by other clusters and elements. y The sense (being influenced or influencing) must apply to all the criteria for the four control hierarchies for the entire decision.

Step 6. Supermatrix Constructiony For each control criterion, construct the supermatrix by

laying out the clusters in the order they are numbered and all the elements in each cluster both vertically on the left and horizontally at the top. y Enter in the appropriate position the priorities derived from the paired comparisons as subcolumns of the corresponding column of the supermatrix.

Step 7. Perform Paired Comparisonsy Perform paired comparisons on the elements within the

clusters themselves according to their influence on each element in another cluster they are connected to (outer dependence) or on elements in their own cluster (inner dependence). y Comparisons of elements according to which element influences a given element more and how strongly more than another element it is compared with are made with a control criterion or subcriterion of the control hierarchy in mind.

Step 8. Paired Comparisons on the Clustersy Perform paired comparisons on the clusters as they

influence each cluster to which they are connected with respect to the given control criterion. y The derived weights are used to weight the elements of the corresponding column blocks of the supermatrix. Assign a zero when there is no influence. Thus obtain the weighted column stochastic supermatrix.

Step 9. Compute Limit Priorities of the Stochastic SupermatrixCompute the limit priorities of the stochastic supermatrix according to whether it is y irreducible (primitive or imprimitive [cyclic]) or y reducible with one being a simple or a multiple root and whether the system is cyclic or not. Two kinds of outcomes are possible. y In the first all the columns of the matrix are identical and each gives the relative priorities of the elements from which the priorities of the elements in each cluster are normalized to one. y In the second the limit cycles in blocks and the different limits are summed and averaged and again normalized to one for each cluster.

Step 10. Synthesize the Limiting Prioritiesy 10. Synthesize the limiting priorities by weighting each idealized

limit vector by the weight of its control criterion and adding the resulting vectors for each of the four merits: Benefits (B), Opportunities (O), Costs (C) and Risks (R). y There are now four vectors, one for each of the four merits. An answer involving marginal values of the merits is obtained by forming the ratio BO/CR for each alternative from the four vectors. The alternative with the largest ratio is chosen for some decisions. y Companies and individuals with limited resources often prefer this type of synthesis.

Step 11. Determine the strategic criteria and their prioritiesy Determine strategic criteria and their priorities to rate

the four merits one at a time. Normalize the four ratings thus obtained. y For each alternative, subtract the costs and risks from the sum of the benefits and opportunities. y At other times one may add the weighted reciprocals of the costs and risks. y Still at other times one may subtract the costs from one and risks from one and then weight and add them to the weighted benefits and opportunities. y In all, we have four different formulas for synthesis.

Step 12. Sensitivity Analysisy Perform sensitivity analysis on the final outcome and

interpret the results of sensitivity observing how large or small these ratios are. y Can another outcome that is close also serve as a best outcome? Why? y By noting how stable this outcome is. Compare it with the other outcomes by taking ratios. Can another outcome that is close also serve as a best outcome? Why?

Learning ANP through illustration

Method II . Now evaluating the same model of buying a car using Using Analaytic Network Process. We have taken into consideration the effect of Alternative on CriteriasGoal

Criterias

Price

Mileage

Prestige

Comfort

Outer Dependency Exist Alternat ives Car 1 Car 2 Car 3

Comparing the Alternative node with respect to Criteria node Comparison with respect to Car 1 Price Price Mileage Prestige Comfort 1 1/5 1/7 2 Mileage Prestige comfort 5 1.00 2.00 5.00 7 1/2 1.00 7.00 2 1/5 0.14 Price 1.00 Mileage Normalizing and Equating the above table we getPrice Price Mileage Prestige Comfort 0.543478 0.108696 0.07764 0.271739 Mileage 0.384615 0.076923 0.153846 0.384615 Prestige 0.451613 0.032258 0.064516 0.451613 comfort 0.598802 0.05988 0.042772 0.299401

NEV 0.495 0.069 0.085 0.352

Prestige Comfort

B1Consistency ratio is less than 10 %

Comparing the Alternative node with respect to Criteria node Comparison with respect to Car 2 Price Price Mileage Prestige Comfort 1 1/2 1/3 2.00 Mileage Prestige comfort 2 1.00 1.00 3.00 3 1.00 1.00 3.00 1/2 1/3 1/3 1.00 Mileage Normalizing and Equating the above table we getPrice Price Mileage Prestige Comfort Mileage Prestige comfort

NEV Price 0.288 0.138 0.127 0.447

Prestige Comfort

0.261097 0.285714 0.130548 0.142857 0.087032 0.142857 0.522193 0.428571

0.375 0.230415 0.125 0.125 0.15361 0.15361

B2Consistency ratio is less than 10 %

0.375 0.460829

Comparing the Alternative node with respect to Criteria node Comparison with respect to Car 3 Price Price Mileage Prestige Comfort 1 6.00 5.00 2.00 Mileage Prestige comfort 1/6 1.00 1.00 3.00 1/5 3 1 0.20 1/2 5 2 1 Mileage Normalizing and Equating the above table we getPrice Price Mileage Prestige Comfort Mileage Prestige comfort

NEV Price 0.068 0.561 0.250 0.121

Prestige Comfort

0.071429 0.098039 0.042553 0.058824 0.428571 0.588235 0.638298 0.588235 0.357143 0.196078 0.212766 0.235294 0.142857 0.117647 0.106383 0.117647

B3Consistency ratio is less than 10 %

Now we have to form the un weighted Super matrix as indicated with the values

Car 1 Car 1 Car 2 Car 3 Price Mileage Prestige Comfort B1 0 0 0

Car 2 0 0 0

Car 3 0 0 0

Price

Mileage Prestige comfort

A1

A2

A3

A4

0 B2 B3 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

Car 1 Car 1 Car 2 Car 3 Price Mileage Prestige Comfort 0 0 0 0.495 0.069 0.085 0.352

Car 2 0 0 0 0.288 0.138 0.127 0.447

Car 3 0 0 0 0.068 0.561 0.250 0.121

Price 0.670 0.073 0.203 0 0 0 0

Mileage 0.328 0.411 0.261 0 0 0 0

Prestige 0.643 0.283 0.074 0 0 0 0

comfort 0.193 0.083 0.724 0 0 0 0

After Normalizing we need to find the Limiting value of the Super Matrix. This is found / reached by multiplying the matrix to nth Value .

Car 1 Car1 Car2 Car3 Price Mileage

Car 2

Car 3

Price

Mileage Prestige Comfort

0.22192 0.22192 0.22192 0.22192 0.22192 0.22192 0.22192 0.09679 0.09679 0.09679 0.09679 0.09679 0.09679 0.09679 0.18129 0.18129 0.18129 0.18129 0.18129 0.18129 0.18129 0.15072 0.15072 0.15072 0.15072 0.15072 0.15072 0.15072 0.13059 0.13059 0.13059 0.13059 0.13059 0.13059 0.13059

Prestige 0.07465 0.07465 0.07465 0.07465 0.07465 0.07465 0.07465 Comfort 0.14404 0.14404 0.14404 0.14404 0.14404 0.14404 0.14404

Now for finding the ANP priorities of the Alternatives we need to Normalize the respective Cluster Car 1 Car 2 Car 3 Calculations : Car 1 = (0.22192) / (0.22192+0.09679+0.1 8129)

0.44384 0.19358 0.36258 ANP (With Outer dependies) 0.44384 0.19358 0.36258

AHP Car 1 Car 2 Car 3 0.482 0.141 0.351

Conclusion after using ANP (Outer dependance) : Model to be selected still remains the same but the priorities have changed The weight age for Car 1 has decreased by 8 % and Car 2 weightage has increased drastically

Method III .. Evaluating the same model using ANP (with Inner dependancies and outer dependancies ) `Inner Dependency and Outer dependency exists in this case Price Goal

Criterias

Mileage

Prestige

Comfort

Alternatives

Car 1

Car 2

Car 3

So Comparison between clusters for (inner Dependance) Say In Criterias Node Price is influenced by Prestige & Comfort & Prestige is influenced by Price & Comfort 1. Comparison of Prestige and comfort with respect to Price Prestige Comfort Prestige Comfort 1 1/3 3 1 Prestige Comfort Normalized matrix Prestige 0.75 0.25 Comfort 0.75 0.25

2. Comparison of Price and Comfort with respect to Prestige

Normalizing we get Price Comfort 0.833 0.1667

Price Price Comfort 1 1/5

Comfort 5 1 Price Comfort

0.833 0.1667

Then forming the Super Matrix.

Car 1 Car 1 Car 2 Car 3 Price Mileage Prestige Comfort B1 0 0 0

Car 2 0 0 0

Car 3 0 0 0

Price A1 0 0

Prestige comfort A3 0.8333 0 0 0.1677 0 0 0 A4

B2

B3 0.75 0.25

Values have come up in this cluster due to inner dependency between criterias

After inserting the values the Unweighted Super matrix will look like

Car 1 Car 1 Car 2 Car 3 Price Mileage Prestige Comfort 0 0 0 0.495 0.069 0.085 0.352

Car 2 0 0 0 0.288 0.138 0.127 0.447

Car 3 0 0 0 0.068 0.561 0.250 0.121

Price 0.670 0.073 0.203 0 0 0.75 0.25

Mileage Prestige comfort 0.328 0.411 0.261 0 0 0 0 0.643 0.283 0.074 0.8333 0 0 0.1677 0 0 0 0.193 0.083 0.724

Normalizing the Super Matrix Car 1 Car 1 Car 2 Car 3 Price Mileage Prestige Comfort 0 0 0 0.49942 0.06591 0.07945 0.35522 Car 2 0 0 0 0.28795 0.13766 0.1258 0.44859 Car 3 0 0 0 0.06632 0.56614 0.24737 0.12018

Price 0.34354 0.03459 0.12187 0 0 0.375 0.125

Mileage 0.32748 0.4126 0.25992 0 0 0 0

Prestige 0.32456 0.13948 0.03596 0.41667 0 0 0.08333

comfort 0.18839 0.08096 0.73064 0 0 0 0

Finding the Limiting power of the matrix by Matrix Multiplication .. (This is an important step in getting the priorities. ) Car 1 Car 1 Car 2 Car 3 Price Mileage 0.1702 0.08535 0.16768 0.17551 0.1179 Car 2 0.1702 0.08535 0.16768 0.17551 0.1179 0.13156 0.1518 Car 3 0.1702 0.08535 0.16768 0.17551 0.1179 0.13156 0.1518 Price 0.1702 0.08535 0.16768 0.17551 0.1179 0.13156 0.1518 Mileage Prestige comfort 0.1702 0.08535 0.16768 0.17551 0.1179 0.13156 0.1518 0.1702 0.08535 0.16768 0.17551 0.1179 0.13156 0.1518 0.1702 0.08535 0.16768 0.17551 0.1179 0.13156 0.1518

Prestige 0.13156 Comfort 0.1518

Now for finding the ANP priorities of the Alternatives we need to Normalize the respective Cluster Car 1 Car 2 Car 3 0.1702 0.08535 0.16768 0.1702 0.08535 0.16768 0.1702 0.08535 0.16768

Normalizing the above matrix we get the ANP priorities as ..

Car 1 Car 2 Car 3

0.392145 0.211663 0.396191 So Select Car 3

A comparisonANP (With Outer dependies) 0.44384 0.19358 0.36258 ANP (With Inner decencies 0.392145 0.201663 0.396191

AHP Car 1 Car 2 Car 3 0.482 0.141 0.351

Car 3 is now having more weightage than Car 1 .. . So the person will choose Car 3

Advantages of the Analytic Hierarchy Process and the Analytic Network Process over other Multi Criteria Decision Making methods As compared to other MCDM approaches, AHP/ANP is not

proportionately complicated, and this helps improve management understanding and transparency of the modeling technique. They have the supplemental power of being able to mix

quantitative and qualitative factors into a decision. This approach can be fit together with other solution approach

such as optimization, and goal programming. AHP/ANP may use a hierarchical structuring of the factors

involved. The hierarchical structuring is universal to the composition of virtually all complex systems, and is a natural problem-solving paradigm in the face of complexity.

In AHP/ANP, judgment elicitations are completed using

a decompositional approach, which has been shown in experimental studies to reduce decision-making errors. AHP/ANP is a technique that can prove valuable in

helping multiple parties (stakeholders) arrive at an agreeable solution due to its structure, and if implemented appropriately can be used as a consensus-building tool AHP has also been validated from the decision makers

perspective as well in recent empirical studies

Limitations of ANP:y ANP is more complex than the AHP and it increases the

effort This can be overcome by using software Super Decisions, Expert Choicey 2. In case, if there are several alternatives in the decision

making a number of pair wise comparisons would be quite demanding

References : Saaty, T. L. (2005). Theory and Applications of the Analytic Network Process. Pittsburgh, PA: RWS Publications, 4922 Ellsworth Avenue, Pittsburgh, PA 15213. Decision making with dependence and feedback. The Analytic Network Process Thomas L.Satty The Analytic Hierachy process An Exposition by Ernest h Forman , School of Business and Public Management , George Washington University , Washington D C 20052

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