The Analytic Hierarchy Process

Transcription

1 The Anlytic Hierrchy Process The Anlytic Hierrchy Process (AHP), introduced by Thoms Sty (980), is n effective tool for deling with complex decision mking, nd my id the decision mker to set priorities nd mke the best decision. By reducing complex decisions to series of pirwise comprisons, nd then synthesizing the results, the AHP helps to cpture both subective nd obective spects of decision. n ddition, the AHP incorportes useful technique for checking the consistency of the decision mker s evlutions, thus reducing the bis in the decision mking process. How the AHP works The AHP considers set of evlution criteri, nd set of lterntive options mong which the best decision is to be mde. t is importnt to note tht, since some of the criteri could be contrsting, it is not true in generl tht the best option is the one which optimizes ech single criterion, rther the one which chieves the most suitble trde-off mong the different criteri. The AHP genertes weight for ech evlution criterion ccording to the decision mker s pirwise comprisons of the criteri. The higher the weight, the more importnt the corresponding criterion. Next, for fixed criterion, the AHP ssigns score to ech option ccording to the decision mker s pirwise comprisons of the options bsed on tht criterion. The higher the score, the better the performnce of the option with respect to the considered criterion. Finlly, the AHP combines the criteri weights nd the options scores, thus determining globl score for ech option, nd consequent rnking. The globl score for given option is weighted sum of the scores it obtined with respect to ll the criteri. 2 Fetures of the AHP The AHP is very flexible nd powerful tool becuse the scores, nd therefore the finl rnking, re obtined on the bsis of the pirwise reltive evlutions of both the criteri nd the options provided by the user. The computtions mde by the AHP re lwys guided by the decision mker s experience, nd the AHP cn thus be considered s tool tht is ble to trnslte the evlutions (both qulittive nd quntittive) mde by the decision mker into multicriteri rnking. n ddition, the AHP is simple becuse there is no need of building complex expert system with the decision mker s knowledge embedded in it. On the other hnd, the AHP my require lrge number of evlutions by the user, especilly for problems with mny criteri nd options. Although every single evlution is very simple, since it only requires the decision mker to express how two options or criteri compre to ech other, the lod of the evlution tsk my become unresonble. n fct the number of pirwise comprisons grows qudrticlly with the number of criteri nd options. For instnce, when compring 0 lterntives on 4 criteri, 4 3/2=6 comprisons re requested to build the weight vector, nd 4 (0 9/2)=80 pirwise comprisons re needed to build the score mtrix. However, in order to reduce the decision mker s worklod the AHP cn be completely or prtilly utomted by specifying suitble thresholds for utomticlly deciding some pirwise comprisons. 3 mplementtion of the AHP The AHP cn be implemented in three simple consecutive steps: ) Computing the vector of criteri weights.

2 2) Computing the mtrix of option scores. 3) Rnking the options. Ech step will be described in detil in the following. t is ssumed tht m evlution criteri re considered, nd n options re to be evluted. A useful technique for checking the relibility of the results will be lso introduced. 3. Computing the vector of criteri weights n order to compute the weights for the different criteri, the AHP strts creting pirwise comprison mtrix A. The mtrix A is m m rel mtrix, where m is the number of evlution criteri considered. Ech entry k of the mtrix A represents the importnce of the th criterion reltive to the kth criterion. f k >, then the th criterion is more importnt thn the kth criterion, while if k <, then the th criterion is less importnt thn the kth criterion. f two criteri hve the sme importnce, then the entry k is. The entries k nd k stisfy the following constrint: =. () k k Obviously, = for ll. The reltive importnce between two criteri is mesured ccording to numericl scle from to 9, s shown in Tble, where it is ssumed tht the th criterion is eqully or more importnt thn the kth criterion. The phrses in the nterprettion column of Tble re only suggestive, nd my be used to trnslte the decision mker s qulittive evlutions of the reltive importnce between two criteri into numbers. t is lso possible to ssign intermedite vlues which do not correspond to precise interprettion. The vlues in the mtrix A re by construction pirwise consistent, see (). On the other hnd, the rtings my in generl show slight inconsistencies. However these do not cuse serious difficulties for the AHP. Vlue of k nterprettion nd k re eqully importnt 3 is slightly more importnt thn k 5 is more importnt thn k 7 is strongly more importnt thn k 9 is bsolutely more importnt thn k Tble. Tble of reltive scores. Once the mtrix A is built, it is possible to derive from A the normlized pirwise comprison mtrix A norm by mking equl to the sum of the entries on ech column, i.e. ech entry of the mtrix A norm is computed s k = m l= Finlly, the criteri weight vector w (tht is n m-dimensionl column vector) is built by verging the entries on ech row of A norm, i.e. w = m l= For mtrix A, i denotes the entry in the ith row nd the th column of A. For vector v, v i denotes the ith element of v. k m lk l.. k (2)

3 3.2 Computing the mtrix of option scores The mtrix of option scores is n m rel mtrix S. Ech entry s i of S represents the score of the ith option with respect to the th criterion. n order to derive such scores, pirwise comprison mtrix B is first built for ech of the m criteri, =,...,m. The mtrix B is n n rel mtrix, where n is the number of options evluted. Ech entry of the mtrix B represents the evlution of the ith option compred to the hth option with respect to the th criterion. f option is better thn the hth option, while if >, then the ith <, then the ith option is worse thn the hth option. f two options re evluted s equivlent with respect to the th criterion, then the entry is. The entries nd stisfy the following constrint: b hi ( ) ( ) bhi nd b ii = for ll i. An evlution scle similr to the one introduced in Tble my be used to trnslte the decision mker s pirwise evlutions into numbers. Second, the AHP pplies to ech mtrix B the sme two-step procedure described for the pirwise comprison mtrix A, i.e. it divides ech entry by the sum of the entries in the sme column, nd then it verges the entries on ech row, thus obtining the score vectors s, =,...,m. The vector s contins the scores of the evluted options with respect to the th criterion. Finlly, the score mtrix S is obtined s i.e. the th column of S corresponds to s. = (4) S = [ s ()... s (m) ] (5) Remrk. n the considered DSS structure, the pirwise option evlutions re performed by compring the vlues of the performnce indictors corresponding to the decision criteri. Hence, this step of the AHP cn be considered s trnsformtion of the indictor mtrix into the score mtrix S. 3.3 Rnking the options Once the weight vector w nd the score mtrix S hve been computed, the AHP obtins vector v of globl scores by multiplying S nd w, i.e. v = S w (6) The ith entry v i of v represents the globl score ssigned by the AHP to the ith option. As the finl step, the option rnking is ccomplished by ordering the globl scores in decresing order. 4 Checking the consistency When mny pirwise comprisons re performed, some inconsistencies my typiclly rise. One exmple is the following. Assume tht 3 criteri re considered, nd the decision mker evlutes tht the first criterion is slightly more importnt thn the second criterion, while the second criterion is slightly more importnt thn the third criterion. An evident inconsistency rises if the decision mker evlutes by mistke tht the third criterion is eqully or more importnt thn the first criterion. On the other hnd, slight inconsistency rises if the decision mker evlutes tht the

4 first criterion is lso slightly more importnt thn the third criterion. A consistent evlution would be, for instnce, tht the first criterion is more importnt thn the third criterion. The AHP incorportes n effective technique for checking the consistency of the evlutions mde by the decision mker when building ech of the pirwise comprison mtrices involved in the process, nmely the mtrix A nd the mtrices B. The technique relies on the computtion of suitble consistency index, nd will be described only for the mtrix A. t is strightforwrd to dpt it to the cse of the mtrices B by replcing A with B, w with s (), nd m with n. The Consistency ndex (C) is obtined by first computing the sclr x s the verge of the elements of the vector whose th element is the rtio of the th element of the vector A w to the corresponding element of the vector w. Then, x m C =. (7) m A perfectly consistent decision mker should lwys obtin C=0, but smll vlues of inconsistency my be tolerted. n prticulr, if C < 0. (8) R the inconsistencies re tolerble, nd relible result my be expected from the AHP. n (8) R is the Rndom ndex, i.e. the consistency index when the entries of A re completely rndom. The vlues of R for smll problems (m 0) re shown in Tble 2. m R Tble 2. Vlues of the Rndom ndex (R) for smll problems. The mtrices A corresponding to the cses considered in the bove exmple re shown below, together with their consistency evlution bsed on the computtion of the consistency index. Note tht the conclusions re s expected. 3 /3 /3 3 C/R =.50 inconsistent 3 /3 3 3 /3 3 C/R = 0.8 slightly inconsistent /3 /3 3 5 /3 3 C/R = consistent /5 /3 5 Automting the pirwise comprisons Although every single AHP evlution is very simple (the decision mker is only required to express how two criteri or lterntives compre to ech other), the lod of the evlution tsk my become unresonble nd tedious for the decision mker when mny criteri nd lterntives re

5 considered. However, in order to llevite the decision mker's worklod, some pirwise comprisons cn be completely or prtilly utomted. A simple method is suggested in the following. Let the th criterion be expressed by n ttribute which ssumes vlues in the intervl [,min,,mx ], (i) (h) nd let nd be the instnces of the ttribute under the ith nd hth control options, respectively. Assume tht the lrger the vlue of the ttribute, the better the system performnce ccording to the th criterion. f, the element of B cn be computed s b ( ) ih = 8,mx,min A similr expression holds if the smller the vlue of the ttribute, the better the system performnce ccording to the th criterion. f, the element of B cn be computed s b ( ) ih = 8,mx,min Note tht (0) nd () re liner functions of the difference functions cn be designed by exploiting specific knowledge nd/or experience. 6 An illustrtive exmple i h (0) (). Of course, More sophisticted An exmple will be here described in order to clrify the mechnism of the AHP. m=3 evlution criteri re considered, nd n=3 lterntives re evluted. Ech criterion is expressed by n ttribute. The lrger the vlue of the ttribute, the better the performnce of the option with respect to the corresponding criterion. The decision mker first builds the following pirwise comprison mtrix for the three criteri: /3 /5 to which corresponds the weight vector w = [ ] T. Then, bsed on the vlues ssumed by the ttributes for the three options (see Figure ), the decision mker builds the following pirwise comprison mtrices: 3 / / 5 () = (2) B / 3 5, B = 5 5, / 7 / 5 / 5 0 A 3 A 2 A B Criterion = / 5 / 9 5 / A A 3 A 2 Criterion 2 0 A 3 A 2 A Criterion 3 Figure. Vlues of the ttributes for the lterntives A, A 2 nd A 3 (the scle on ech xis is not relevnt).

6 () to which correspond the score vectors = [ ] T (2) s, s = [ ] T, nd s = [ ] T. Hence, the score mtrix S is S = [ s () s (2) s ] = nd the globl score vector is v = S w = [ ] T. Note tht the first option turns out to be the most preferble, though it is the worst of the three with respect to the second criterion. References Sty, T.L., 980. The Anlytic Hierrchy Process. McGrw-Hill, New York.