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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