Comparing alternatives using multiple criteria

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1 Comparig alteratives usig multiple criteria Des L. Bricker Dept of Mechaical & Idustrial Egieerig The Uiversity of Ioa AHP 9/4/00 page of 9 AHP 9/4/00 page 3 of 9 Whe a decisio-maker has multiple objectives, it is difficult to choose betee alteratives, e.g. Choosig hich of several job offers to accept o Salary o Locatio o Opportuity for advacemet o Persoal iterests Selectig hich automobile to purchase o Price o Safety o Fuel ecoomy o Poer of egie Selectig hich uiversity to atted Selectig plat site EXAMPLE You are tryig to decide hether to live i Chicago or Ne York City, based upo four criteria:. Housig cost. Cultural opportuities 3. Quality of schools 4. Crime level Which city should you choose? AHP 9/4/00 page of 9 AHP 9/4/00 page 4 of 9

2 Suppose that you have determied the relative importace of the criteria: Weights for criteria: Criterio Housig cost Culture Schools Crime Weight AHP is a method for systematically determiig the eights of the criteria & of the cities ith respect to the criteria. You have rated the to cities o each criterio: Housig cost Culture Schools Crime Ne York Chicago (The scores i all cases have bee chose so as to sum to 00%, but eed ot have bee!) AHP 9/4/00 page 5 of 9 AHP 9/4/00 page 7 of 9 Based upo these values, you ca o compute a score for each city: Ne York: = 0.4 Chicago: = 0.6 Accordigly, your better choice ould seem to be Chicago! (developed by Thomas Saaty). Develop the eights for each criterio by a. Developig a pairise compariso matrix b. Computig eigevector c. Checkig cosistecy. Develop the eights for each alterative ith respect to each criterio by a. Developig a pairise compariso matrix b. Computig eigevector c. Checkig cosistecy 3. Calculate the eighted average for each alterative 4. Choose the alterative yieldig the highest score. AHP 9/4/00 page 6 of 9 AHP 9/4/00 page 8 of 9

3 Pairise Comparisos i Create a pairise compariso matrix A here aij = > j idicates the preferece of alterative i over alterative j. j (The for sake of cosistecy, a = ji = a <.) ij a a a i a a a A = = a a a Cosistecy: If alterative #i is a ij times as desirable as alterative #j, & alterative #j is a jk times as desirable as alterative #k, the cosistecy ould require that alterative #i is a ij a jk times as desirable as alterative #k! AHP 9/4/00 page 9 of 9 AHP 9/4/00 page of 9 The stadard AHP approach assumes that criteria i & j are compared, ad the folloig ratig is assiged to the more preferred oe: Ratig Descriptio Equally preferred 3 Moderately preferred 5 Strogly preferred 7 Very strogly preferred 9 Extremely strogly preferred Ratigs, 4, 6 & 8 may be used as ell, ith obvious iterpretatios. Simple matrix multiplicatio demostrates that (if perfectly cosistet) : A = = = e.g., first ro of A times is = 3 3 ad so A= AHP 9/4/00 page 0 of 9 AHP 9/4/00 page of 9

4 This meas that is a eigevector of the matrix A, ad λ = the correspodig eigevalue! (It ca be sho that if A is cosistet, is the largest eigevalue of A, ad if ot perfectly cosistet, the largest eigevalue is larger tha. Saaty proposed that the largest eigevalue λ max > be selected, ad the correspodig eigevector used as the eights. It is stadard procedure (but ot ecessary) to ormalize, e.g., either i = or max{ i } = i= i=, The deviatio of λ max from (the perfectly cosistet situatio) is a idicator of the icosistecy of the compariso matrix A. λmax Cosistecy idex: CI = CI Cosistecy Ratio: CR = ACI here ACI is the average cosistecy idex for a large umber of radom A matrices: ACI AHP 9/4/00 page 3 of 9 AHP 9/4/00 page 5 of 9 Alterative determiatio of eights A simpler averagig scheme yields a set of eights hich are approximately that obtaied by the eigeaalysis: Normalize the colums of A: ( j) a j a j = a ij i= aj If cosistet, the CR (cosistecy ratio) ill be 0. Saaty suggests that if CR > 0., oe should go back ad revise the comparisos! Compute the average of the ormalized colums: ( j) j = = AHP 9/4/00 page 4 of 9 AHP 9/4/00 page 6 of 9

5 Example: Suppose that the pairise compariso matrix is A = A eigeaalysis yields the largest eigevalue λ max =4.438 ith its eigevector = [ , 0.06, 0.038, 0.75 ] Note: Usig the averagig method to approximate the true eigevector = [ , 0.06, 0.038, 0.75 ] yields the eights =[0.544, 0.83, 0.040, ] hich are close eough that they ould probably lead to the same coclusio i AHP. AHP 9/4/00 page 7 of 9 AHP 9/4/00 page 9 of 9 The cosistecy idex is therefore λmax CI = = = CI ad the cosistecy ratio is CR = = = 0.63 ACI 0.90 This suggests that there is too much icosistecy i the pairise comparisos! AHP 9/4/00 page 8 of 9

Comparing alternatives using multiple criteria

Comparing alternatives using multiple criteria Denns L. Bricker Dept of Mechanical & Industrial Engineering The University of Iowa AHP 2/4/2003 page 1 of 22 When a decision-maker has multiple objectives,