A sensitivity analysis algorithm for hierarchical decision models

Transcription

1 European Journal of Operational Research 85 (28) Decision Support A sensitivity analysis algorith for hierarchical decision odels Hongyi hen *, Dundar F. Kocaoglu Departent of Engineering and Technology Manageent, 9 SW 4th Avenue, LL Suite 5, ortland, OR 972, United States Received 5 June 26; accepted 3 Deceer 26 Availale online 4 January 27 Astract In this paper, a coprehensive algorith is developed to analyze the sensitivity of hierarchical decision odels (HDM), including the analytic hierarchy process and its variants, to single and ultiple changes in the local contriution atrices at any level of the decision hierarchy. The algorith is applicale to all HDM that use an additive function to derive the overall contriution vector. It is independent of pairwise coparison scales, judgent quantification techniques and group opinion coining ethods. The allowale range/region of perturations, contriution tolerance, operating point sensitivity coefficient, total sensitivity coefficient and the ost critical decision eleent at a certain level are identified in the HDM SA algorith. An exaple is given to deonstrate the application of the algorith and show that HDM SA can reveal inforation ore significant and useful than siply nowing the ran order of the decision alternatives. Ó 27 Elsevier.V. All rights reserved. Keywords: Roustness and sensitivity analysis; Multiple criteria analysis; Decision analysis. Introduction As the world has ecoe ore coplex, decision proles have followed suit and ust contend with increasingly coplex relationships and interactions aong the decision eleents. To assist decision aers and analysts, different ethods have een developed to decopose proles into hierarchical levels and forulate hierarchical decision odels (HDM). In addition to the well-nown analytic hierarchy process (AH) developed y Saaty (98), several other odels ased on the sae asic concept of dealing with ultiple decision levels ut using different pairwise coparison scales and judgent quantification techniques were developed concurrent with or shortly after the introduction of AH (i.e., Kocaoglu, 976, 983; hu et al., 979; Johnson et al., 98; Hihn, 98; elton and Gear, 983, 985; Jensen, 984; Ra, 988; Lootsa, 999). In HDM, the local contriutions of decision eleents at one level to decision eleents on the next higher level, derived fro different judgent quantification ethods, are supplied as interediate input to the hierarchical odel. Decisions otained y evaluating the final raning of alternatives are ased on the local contriutions. However, values of the local contriutions are seldo nown at a % confidence level and * orresponding author. Tel.: ; fax: E-ail address: hongyipdx.edu (H. hen) /$ - see front atter Ó 27 Elsevier.V. All rights reserved. doi:.6/j.ejor

2 H. hen, D.F. Kocaoglu / European Journal of Operational Research 85 (28) are suject to variations as the environent changes. esides, the various pairwise coparison scales and judgent quantification techniques eployed in HDM usually yield different local contriution values, and thus different results for the sae prole (Triantaphyllou, 2), and various group-opinion coining ethods (i.e., Ferrell, 985; arzilai and Lootsa, 997; Saaty, 2; Hastie and Kaeda, 25) ay change the current decision. Therefore, the solution of a prole is not coplete with the ere deterination of a ran order of decision alternatives. In order to develop an overall strategy to eet the various contingencies, one needs to conduct a sensitivity analysis (SA) for the HDM results. SA has een regarded as a fundaental concept in the effective use and ipleentation of quantitative decision odels (Dantzig, 963; Evans, 984). It has several iportant roles and serves different purposes in the decision-aing process (e.g., Dantzig, 963; Howard, 968; Alexander, 989; Kelton et al., 998; Harrell et al., 2; Saltelli, 24; Reilly, 2). SA can even provide inforation ore significant and useful than siply nowing the odel solution (hillips et al., 976). onducting a SA for HDM can: (i) help visualize the ipact of changes at the policy and strategy levels on decisions at the operational level; (ii) test the roustness of the recoended decision (Ho, 24); (iii) identify the critical eleents of the decision (Aracost and Hosseini, 994; Triantaphyllou and Sanchez, 997); (iv) generate scenarios of possile ranings of decision alternatives under different conditions (Winerae and reswic, 23); (v) help judgent providers (the experts) reach consensus (Yeh et al., 2); and (vi) offer answers to what if questions. There is considerale literature on the developent of SA for various operations research and anageent science odels (Triantaphyllou and Sanchez, 997). However, literature on the SA for HDM is liited. Most literature in the field of HDM has een focused on the applications side (Golden et al., 989; Foran and Gass, 2). Theoretical studies have een geared toward analyzing and solving the raning irregularity proles (elton and Gear, 983, 985; Triantaphyllou, 2) and coparing and evaluating different HDM (Triantaphyllou, 2). Soe studies in this group relevant to SA tried to identify the ranges in which the values in the AH pairwise coparison atrix can vary without causing the ran reversal prole (Arel and Vargas, 99; Moreno-Jienez and Vargas, 993; Sugihara and Tanaa, 2; Faras et al., 24). In uch of the literature where HDM, especially AH, were applied to help solve proles, a asic SA was conducted y increentally changing the nuerical values of specific proles and graphically showing the corresponding trend of changes in the odel result (artovi, 994; orthic and Scheiner, 998; Yeh et al., 2; Winerae and reswic, 23; Ho, 24). Such literature constitutes the first group of sensitivity studies of HDM, naely the nuerical increental analysis, which is an iteration-ased and data-dependant process. Expert hoice (99), software ased on AH, elongs to this group since it offers a asic SA function with which users can alter one of the criteria weights and see graphically how the gloal priorities change. However, the function is very liited: it does not allow users to change values at levels other than the first level of the decision hierarchy, nor does it allow ore than one change at a tie. Other researchers eploy a siulations approach to study the sensitivity of HDM. They replace values in the local contriution atrix with proaility distriutions and calculate the expected value of rans after hundreds of siulation runs (Hauser and Tadiaalla, 996; utler et al., 997). The proailistic input introduces stochasticity to the output, thus aing the odel non-deterinistic. The algorith proposed in this paper focuses on deterinistic additive HDM ut is also applicale to non-deterinistic additive HDM: If the proaility distriutions of the contriution values are nown in an interval, our algorith can deterine the proaility of ran changes and generate scenarios of ranings while the contriution value varies within the interval. This is significantly ore inforation than that provided y the expected rans otained in the siulations approach. The third group of sensitivity studies of HDM is through atheatical deduction, which is usually used when siple closed-for expressions can descrie the relationship etween inputs and outputs. opared to nuerical increental analysis and the siulations approach, atheatical deduction has etter perforance (rich with inforation, precisely defined threshold value to any decial place, % accurate once the deduction process has een verified), less coputational coplexity (fast, does not depend on repetitive iterations or large replications) and equal generality (sae assuptions). Since perforance, coputational coplexity and generality are the three characteristics to e copared while evaluating systes ethods (Klir, 2), atheatical deduction is identified to e the approach preferred overall to study the sensitivity of HDM. Major studies that eployed atheatical deduction are discussed elow.

3 268 H. hen, D.F. Kocaoglu / European Journal of Operational Research 85 (28) Aracost and Hosseini (994) defined the deterinant attriute in an AH decision hierarchy as the one that ost differentiates the final raning of the alternatives. Masuda (99) and Huang (22) investigated the situation when the entire AH contriution atrix is pertured and proposed two different sensitivity coefficients, oth as easureents of the lielihood of ran changes: The closer to zero the coefficient is, the less liely the ran changes will occur. Triantaphyllou and Sanchez (997) studied the threshold of a single change in the first-level contriution vector of a weighted su odel (including AH) and a weighted product odel, and proposed a sensitivity coefficient without noralizing the perturation threshold (see Appendix A). For ultiplicative AH, Aguaron and Moreno-Jienez (2) proposed a local staility index as the reciprocal of a local staility interval in which a judgent, alternative, or atrix can vary without changing the raning of alternatives local priorities ðx i Þ. To suarize, although researchers frequently understand the iportance of SA for HDM, no study has developed a coprehensive algorith to exaine the sensitivity of HDM results in a fast, accurate and precise way. To close the research gap, we propose an HDM SA algorith to study the odel s roustness to changes in every local contriution atrix at different levels. The algorith is independent fro the pairwise coparison scales and judgent quantification techniques and is applicale to all HDM ased on an additive relationship. A coprehensive SA for HDM using the ultiplicative function proposed y arzilai and Lootsa (997), which is less widely used and is invalid in certain situations (Vargas, 998), will e studied in future research. The next section egins y introducing the odel structure of additive HDM and clarifying the notations used in this paper. Five groups of propositions that constitute the HDM SA algorith are then presented to define several sensitivity indicators of HDM in different situations. The SA of adding new decision alternatives to an existing odel is also addressed. Data fro a recent h.d. dissertation (Ho, 24) is eployed as a siple exaple to illustrate the application of the algorith and show the significant insight gained through HDM SA. ontriutions and future wor conclude the paper. 2. The HDM SA algorith 2.. HDM odel structure and notations Since atheatical deduction in syolic for is eployed to study the sensitivity of HDM results to variations at different levels of the decision hierarchy, instead of the typical three-level odel used in ost of the literature (e.g., Aracost and Hosseini, 994; Triantaphyllou, 2), a MOGSA (ission-ojective-goalstrategy-action) odel (leland and Kocaoglu, 98), which consists of five decision levels, is used to represent the general HDM odel structure in this paper. In applications, the levels of the hierarchy can e extended or reduced according to specific needs. The notations used in this paper are as follows (see Fig. ): O G the th ojective, ¼ ; 2;...; L the th goal, ¼ ; 2;...; K Fig.. HDM odel structure.

4 S j A i the jth strategy, j ¼ ; 2;...; J the ith action, i ¼ ; 2;...; I L nuer of ojectives K nuer of goals I nuer of actions J nuer of strategies A i overall contriution of the ith action A i to the ission r the ran of i. A r rans efore A rþn, which indicates A r > A rþn i A O contriution of the ith action A i to the th ojective O contriution of the ith action A i to the th goal G i ij A S S j S G j O H. hen, D.F. Kocaoglu / European Journal of Operational Research 85 (28) contriution of the ith action A i to the jth strategy S j overall contriution of the jth strategy to the ission contriution of the jth strategy to the th goal contriution of the th goal to the th ojective contriution of the th ojective to the ission The ters criteria weights (Ra, 988; Triantaphyllou and Sanchez, 997; Yeh et al., 2; Winerae and reswic, 23), priority (Saaty, 98, 2; Masuda, 99; Aracost and Hosseini, 994; Aguaron and Moreno-Jienez, 2; Huang, 22) and perforance values (Triantaphyllou and Sanchez, 997) used in the literature are called contriutions in this study ecause they are actually easureents of the contriution of a decision eleent to another eleent on a higher level. At the otto level of the decision hierarchy, actions are the decision alternatives under evaluation. They are raned according to their overall contriution to the ission, denoted as A i, which is calculated y taing the su-product of all the local contriution atrices etween M and A levels: A i ¼ XL ¼ A O i O ¼ XL ¼ X K i O ¼ XL ¼ X K X J j¼ ij S G j O : ðþ All the values in the atrices are noralized so that the contriutions to each decision eleent add up to : X L ¼ O ¼ ; X K ¼ ; X J j¼ S G j ¼ ; X I i¼ ij ¼ : ð2þ (Note that the ajor difference etween additive HDM and ultiplicative HDM lies in this aggregation step.) 2.2. Assuptions All the assuptions that apply to additive HDM are applicale in this study. In addition, it is assued that when perturations are induced on any of the contriutions, the values of other related contriutions will e changed according to their original ratio scale relationships, so the contriutions of different decision eleents to a higher-level decision eleent still add up to. For exaple, if M perturations induced on contriutions of M goals, G s, to a specific ojective, O, the new values of G O ðnewþ ¼G O þ : The new values of other s will e ðnewþ ¼G O þ ; with G O ¼XM K 6¼... M ð ¼ ; 2;...; MÞ are s will e ð3aþ : ð3þ (* indicates that perturation(s) are induced on contriution(s) related to that specific decision eleent.)

5 27 H. hen, D.F. Kocaoglu / European Journal of Operational Research 85 (28) Tolerance analysis Tolerance is defined as the allowale range in which a contriution value can vary without changing the ran order of decision alternatives. To deterine the tolerance of each contriution, the allowale range of perturations on the contriution is calculated first. The allowale range of perturations corresponds to the slac or allowale increase and decrease, as used in the sensitivity analysis of linear prograing (Murty, 976; hillips et al., 976). The logic ehind deducting the allowale range of perturations is: Suppose originally A r rans efore A t, indicating ð A r > A t Þ; the ran order of A r and A t will e preserved if the new value of A r is still greater than or equal to the new value of A t. Therefore, the relationships etween the perturation(s) and the contriutions can e found y representing the new values of A r and A t with an expression containing the original contriutions and the induced perturation(s). For details of the atheatical deductions, please refer to Appendix. As noted in the literature (Triantaphyllou and Sanchez, 997; Aguaron and Moreno-Jienez, 2; arron and Schidt, 988), decision aers ay e interested in either the raning of all decision alternatives or only the top choice in different cases. In this paper three situations are considered to preserve the current ran order of: (i) a pair of decision alternatives, (ii) all decision alternatives, and (iii) the est alternative. In an effort to offer a coprehensive algorith, we present three groups of propositions in the following susections, covering situations when ultiple and single perturations are induced on local contriution atrices fro the top to the otto level of the decision hierarchy. Tolerance of the local contriutions at each level is also defined First level contriution vector Theore. Let O O 6 O 6 O ; M O 6 M O ; ¼ ; 2;...; M denote M perturations induced on M of the O s, which are O ; the original raning of A r and A rþn will not reverse if: O O þ O O 2 þþ O O 2 þþ O O M M ; ð4aþ where ¼ A r A rþn ; ð4þ O ¼ XL A O rþn; A O r A O rþn; O L ¼ ¼ O 6¼... 6¼ M... M þ XL ¼ 6¼... M A O O r L ¼ O 6¼... M : ð4cþ The top choice will reain at the top ran if the aove condition is satisfied for all r ¼ and n ¼ ; 2;...; I. The ran order of all A i s will reain unchanged if the aove condition is satisfied for all r ¼ ; 2;...; I, and n ¼. Theore defines an M diensional allowale region for M perturations induced in the first level contriution vector O. As long as the values of the perturations fall into this allowale region, current ran orders will reain unchanged. When ðm ¼ Þ, which eans only one O value is pertured, the threshold of the perturation can e deterined y orollary.. orollary.. Let O O 6 O 6 O denote the perturation induced on one of the O s, which is O ; the original raning of A r and A rþn will not reverse if: O O ; where ¼ A r A rþn ; O ¼ A O rþn; A O r A O ¼; 6¼ XL L ¼; 6¼ O rþn; O þ XL A O O r L ¼; 6¼ ¼; 6¼ O ð5aþ ð5þ : ð5cþ The top choice will reain at the top ran if the aove condition is satisfied for all r ¼ and n ¼ ; 2;...; I. The ran order of all A i s will reain unchanged if the aove condition is satisfied for all r ¼ ; 2;...; I, and n ¼.

6 Thresholds of the single perturation O, denoted as eo (negative) and eo þ (positive), to preserve current raning of interested A i s can e calculated fro (5a) (5c). oining the feasiility constraint O 6 O 6 O, which protects any O value fro going elow zero or aove one, the allowale range of perturations on O, denoted as do ; do þ, can e derived as Maxf O ; eo g; Minf O ; eo þ g. Then, the tolerance of the corresponding contriution O is do þ O ; do þ þ O. As long as the value of O is within this tolerance range, the final raning of A i s under consideration will reain unchanged. To derive the allowale range of perturations or the tolerance of a O, I inequalities need to e satisfied in oth cases: to either preserve the top-raned alternative only or to preserve the ran order for all A i s. I is the nuer of decision alternatives Middle levels of the decision hierarchy Theore 2 and its corollaries are applicale to perturation(s) induced in iddle-level contriution atrices, such as GO and SG j in the MOGSA odel. Notations used in this group of propositions are fro the GO atrix. Theore 2. Let a 6 a 6 a ; M a 6 M a G O ; ¼ ; 2;...; M denote M a perturations induced on M of the a s (contriutions of M goals G to the ath changing ojective O a Þ, t 6 t 6 t ; T t 6 T t G O ; t ¼ ; 2;...; T denote T perturations induced t on T of the s (contriutions of T goals G t to the th changing ojective O ), q c 6 q c 6 q c ; Q q c 6 Q q c G O ; q ¼ ; 2;...; Q denote Q perturations q c induced on Q of the c s (contriutions of Q goals G q to the cth changing ojective O c ); the original raning of A r and A rþn will not reverse if: XM G O a þ XT þ XQ a ; ð6aþ q c where ¼ A r A rþn 2 ; G O ¼ O a 6 A G a 4 G O ¼ O t G O q c ¼ O 6 c 4 H. hen, D.F. Kocaoglu / European Journal of Operational Research 85 (28) rþn; rþn; t A G rþn; q A G r r t A G r q t þ XK 6¼ þ XK 6¼ t þ XK 6¼ q G O t O O c r rþn; r rþn; G O t a K 6¼ a K 6¼ t r rþn; G O c K 6¼ q 3 ð6þ 7 5 ; ð6cþ c ; ð6dþ : ð6eþ The top choice will reain at the top ran if the aove condition is satisfied for all r ¼ and n ¼ ; 2;...; I. The ran order of all A i s will reain unchanged if the aove condition is satisfied for all r ¼ ; 2;...; I, and n ¼. Theore 2 deals with a general situation when different nuers (M,T,Q) of the local contriutions to three ojectives O a ; O and O are pertured (see Fig. 2). It defines a ðm þ T þ QÞ diensional allowale c region for the ðm þ T þ QÞ perturations induced in the local contriution atrix GO. When contriutions to ore than three ojectives need to e changed, (6a) can e extended y adding ore X x¼ G O x s h x h following the sae pattern, using x to represent the nuer of perturations induced for each and h to h differentiate the new O h to which the x contriutions will e pertured. When there is only one eing changed, the threshold of such change can e deterined y orollary 2..

7 272 H. hen, D.F. Kocaoglu / European Journal of Operational Research 85 (28) Fig. 2. ontriutions of ultiple goals to ultiple ojectives. orollary 2.. Let G O < G O < G O denote a perturation induced on one of the G O s, which is (contriution of a specific goal G to a specific ojective O ); the original raning of A r and A rþn will not reverse if: G O ; where ¼ A r A rþn 2 ; GO ¼ O 6 4 A G rþn; A G r þ XK 6¼ A G r XK A G rþn; 6¼ A G O 3 K 6¼ ð7aþ ð7þ 7 5 : ð7cþ The top choice will reain at the top ran if the aove condition is satisfied for all r ¼ and n ¼ ; 2;...; I. The ran order of all A i s will reain unchanged if the aove condition is satisfied for all r ¼ ; 2;...; I, and n ¼. The thresholds of in oth directions, denoted as eg O and eg O þ, can e derived fro (6a) (6c). Then, the allowale range of perturations on is dg O, where d G O ¼ MaxfG O and d G O þ ¼ Min G O orollary 2.2. Let G O < < G O ; K ;6¼ G O < M < K ;6¼ G O ; ¼ ; 2;...; MÞ denote M perturations induced on M of the s, which are (contriutions of specific goals G s to a specific ojective O, see Fig. 3); the original raning of A r and A rþn will not reverse if: ; dg O þ ; eg O g ; eg O þ. The tolerance of contriution G O is dg O þ G O ; dg O þ þ G O. G O þ 2 G O 2 where ¼ A r A rþn 2 ; G O ¼ O 6 4 A G rþn; A G r þ XK þþ G O 6¼... M r rþn; þþ G O M M ; K 6¼... M ð8aþ 3 ð8þ 7 5 : ð8cþ The top choice will reain at the top ran if the aove condition is satisfied for all r ¼ and n ¼ ; 2;...; I. The ran order of all A i s will reain unchanged if the aove condition is satisfied for all r ¼ ; 2;...; I, and n ¼ otto level of the decision hierarchy Theore 3 and its corollaries deal with perturations induced in atrix AS ij, which is the otto level of the decision hierarchy. Since the decision alternatives level is involved in the analysis, situations are Fig. 3. ontriutions of ultiple goals G to a specific ojective O.

8 differentiated when the perturation(s) are induced on the decision alternatives eing copared or not. onsequently, ore coplex propositions are developed to address various situations. Theore 3. Let i A S j a i 6 j a i 6 j A S a i ; M j a i 6 M j a A S i ; ¼ ; 2;...; M denote M j a perturations induced in M of the ij A S a s (contriutions of M actions A i to the ath changing strategy S j a Þ, i t j i 6 t j i 6 t j i ; T t j i 6 T t j A S i ; t ¼ ; 2;...; T denote T perturations induced in t j T of the ij s (contriutions of T actions A i t to the th changing strategy S j ), i q j c i 6 q j c i 6 q j c i ; Q q j c i A S 6 Q q j c A S i ; q ¼ ; 2;...; Q denote Q perturations induced q j c in Q of the c s (contriutions of Q actions A i q to the cth changing strategy S j c ); the original raning of A r and A rþn will not reverse if: A r A rþn S j r j S j a XM A r A rþn S j c XQ i j a rþn;j c i¼;i6¼i A S q ij c i q j c rþn;j a i¼;i6¼i A S ij a S j XT i t j ðwhen soe perturations are induced on s ut not on rþn;j sþ; S j rþn j þ S j a XM þ S j c XQ A r A rþn S j a XM H. hen, D.F. Kocaoglu / European Journal of Operational Research 85 (28) i j a i c q j c i¼;i6¼i A S q ij c a i¼;i6¼i A S ij a þ S j XT i t j rþn;j i¼;i6¼i t ij i¼;i6¼i t ij ðwhen soe perturations are induced on rþn;j s ut not on A S sþ; ð9þ þ S j c XQ i A S a j a i q j c rþn;j a i¼;i6¼i A S ij a A S c rþn;j c i¼;i6¼i A S q ij c þ S j XT i t j ðwhen soe perturations are induced on oth A r A rþn S j A S rþn j r j rþn;j i¼;i6¼i t A S ij s and rþn;j sþ; ðwhen no perturation is induced on nor rþn;j Þ: The top choice will reain at the top ran if all the aove conditions, (9a) (9d), are satisfied for all r ¼ and n ¼ ; 2;...; I. The original raning for all A i s will reain unchanged if all the aove conditions, (9a) (9d), are satisfied for all r ¼ ; 2;...; I, and n ¼. Theore 3 deals with a general situation when different nuers (M,T,Q) of the local contriutions to three strategies (S j a, S j and S j ) are pertured (see Fig. 4). When contriutions to ore than three strategies c ð9aþ ð9cþ ð9dþ Fig. 4. ontriutions of ultiple actions to ultiple strategies.

9 274 H. hen, D.F. Kocaoglu / European Journal of Operational Research 85 (28) need to e changed, (9a) (9c) can e extended y adding ore S j h X x¼ ij i h x j I h i¼;i6¼i x sae pattern, using x to represent the nuer of perturations induced for each ij h new S j to which the x contriutions will e pertured. When only one A S h ij such a perturation can e deterined ased on orollary 3.. orollary 3.. Let i j is i A S j reverse if: ij h! following the and h to differentiate the value is pertured, the threshold of A S i j 6 i j 6 A S i j (contriution of a specific action A i to a specific strategy S j ); the original raning of A r and A rþn will not denote the perturation induced on one of the ij s, which i j A S ij ; ðaþ where ¼ A r A rþn ; ðþ A S ij ¼ S j A S A S rþn;j i¼;i6¼i A S ij or A S ij ¼ S j þ if i j! i¼;i6¼rþn A S ij rþn;j I i¼;i6¼r A S ij or A S ij ¼ S j þ A S! is induced on neither A S nor rþn;j if i j if i j is induced on A S rþn;j is induced on A S ; ðcþ ; ðdþ : ðeþ The top choice will reain at the top ran if all the aove conditions are satisfied for all r ¼ and n ¼ ; 2;...; I. The original raning for all A i s will reain unchanged if all the aove conditions, (a) (e), are satisfied for all r ¼ ; 2;...; I, and n ¼. The thresholds of i j in oth directions, denoted as ea S ij and ea S ijþ, can e derived fro (a) (e). The h i allowale range of perturations on ij is d A S ij ; da S ijþ, where h d A S ij ¼ MaxfA S i Minf i j ; ea S ijþ gþ. The tolerance of contriution A S i j is da S ij þ A S i j ; da S ijþ þ A S i j. orollary 3.2. Let i j A S i < j i < j A S i j; I i¼;i6¼i A S ij < M i < j i j ; ea S ij g and i¼;i6¼i A S ij d A S ijþ ¼ ; 2;...; MÞ denote M perturations induced on M of the ij A S s, which are i (contriutions of M specific j actions A i s to a specific strategy S j, see Fig. 5); the original raning of A r and A rþn will not reverse if: A r A rþn XM i j S j A S or A r A ðrþnþ A S ðrþnþ j r j or A r A ðrþnþ XM i6¼rþn i j S j if one of the A S i j s; which is A S rþn;j I i¼;i6¼i A S ij S j I i¼;i6¼i A S ij ðrþnþ j if i j s are induced on neither A S nor rþn;j A S if i j s are induced on oth A S and rþn;j! þ ðrþnþ j S j þ in this case; is induced on A S rþn;j i¼;i6¼i A S ij ; ¼ ðaþ ðþ ðcþ Fig. 5. ontriutions of ultiple actions A i to a specific strategy S j.

10 H. hen, D.F. Kocaoglu / European Journal of Operational Research 85 (28) or A r XM A rþn ;i6¼r i j S j rþn;j I i¼;i6¼i A S ij if one of the A S i j s; which is r j rþn;j I i¼;i6¼i A S ij r j S j þ in this case; is induced on A S! : ðdþ The top choice will reain at the top ran if (a) (d), are satisfied for all r ¼ and n ¼ ; 2;...; I. The original raning for all A i s will reain unchanged if all the aove conditions, (a) (d), are satisfied for all r ¼ ; 2;...; I, and n ¼. orollary 3.3. Let i j i j < i j < A S i j ; ¼ ; 2;...; M denote M perturations induced on M of the ij s (contriutions of a specific action A i to M specific strategies S j s, see Fig. 6); the original raning of A r and A rþn will not reverse if: A r A rþn XM S j i j A S A S rþn;j I i¼ A S ij when perturations are induced on neither s nor rþn;j s ð2aþ! or A r A rþn XM S j rþn j þ when perturations are induced on rþn;j s Fig. 6. ontriutions of a specific action A i to specific strategies S j. i¼;i6¼t A S ij rþn;j i¼;i6¼r A S ij or A r XM A rþn S j r j þ A S! ð2þ when perturations are induced on : The top choice will reain at the top ran if (2a) (2c) are satisfied for all r ¼ and t ¼ r þ ; r þ 2;...; rþ I. The original raning for all A i s will reain unchanged if all the aove conditions, (2a) (2c), are satisfied for all r ¼ ; 2;...; I and t ¼ r þ. s ð2cþ Suary The aove three groups of propositions define the allowale region of perturations and tolerance of contriutions at any level of an additive decision hierarchy. Tale suarizes the level(s) of the contriution vector/atrix and the nuer of induced perturations that each proposition deals with. The nuer of inequalities that have to e satisfied in each situation is also specified. When the perturation nuer equals two, a two-diensional allowale region for the two perturations is defined y the inequalities. When it increases to three, the allowale region for the three perturations is a three-diensional polyhedron, as shown in Fig. 7, with its hyperplanes defined y the inequalities. The origin, where the values of the three perturations are all zero, represents no changes Sensitivity coefficients Different sensitivity coefficients (S) for HDM have een proposed in the literature (Masuda, 99; Triantaphyllou and Sanchez, 997; Huang, 22). Masuda (99) defined the S as the standard deviation of the

11 276 H. hen, D.F. Kocaoglu / European Journal of Operational Research 85 (28) Tale Suary of ropositions to 3.3 Theores (T) and orollaries () Level (s) in HDM Nuer of perturations Nuer of inequalities * ondition onditions 2 and 3 T Top M 2+M I+ M. Top 2 I T2(Fig. 2) Middles M + T + Q M+ T+ Q +4 I + M + T + Q Middles 2 I 2.2 (Fig. 3) Middles M 2 + M I+ M T3(Fig. 4) otto M + T + Q M+ T + Q +4 I + M + T + Q otto 2 I 3.2 (Fig. 5) otto M 2 + M I+ M 3.3 (Fig. 6) otto M +M I+ M * ondition : Ran order of a pair of decision alternative is of concern. ondition 2: Ran order of all the decision alternatives is of concern. ondition 3: Ran order of the top choice is of concern. Fig. 7. The allowale region for perturations. extree vector of an AH odel. Huang (22) showed that Masuda s definition was invalid in certain situations and defined another S ased on Masuda s wor, also as a easureent of the lielihood of range changes. The S proposed y Triantaphyllou and Sanchez (997) is the reciprocal of the sallest percentage y which the contriution ust change to reverse the alternatives raning. Siilar to the sensitivity coefficient concept, a local staility index is defined y Aguaron and Moreno-Jienez (2) as the reciprocal of the local staility interval in ultiplicative AH. In this paper, to give as coplete inforation as possile, two sensitivity coefficients are proposed: the operating point sensitivity coefficient (OS) and the total sensitivity coefficient (TS). The OS is defined as the shortest distance fro the current contriution value to the edges of its tolerance. It is dependent on the contriution s current value (the operating point) and directions of the change (increasing or decreasing). TS specifies that the shorter the tolerances of a decision eleent s contriutions are, the ore sensitive the final decision is to variations of that decision eleent. Evans (984) noted that if the current paraetric value is located near the center of * (allowale region), then the decision is roust. The OS defined in this paper indicates the roustness of the current decision, while the TS reveals ore aout how flexile the input values can e without changing the decision. They give different ut equally iportant inforation and thus should e used together. Theore 4.. If the allowale range of perturations on O is do ; do þ to preserve the final raning of Ai s, the OS and TS of O are OSðO Þ¼Minfjd O j; jdo þ jg; TSðO Þ¼jd O þ do j: ð3aþ ð3þ

12 Theore 4.2. If the allowale range of perturations on is d G O ; dg O þ to preserve the final raning of Ai s, the OS and TS of G are OSðG Þ¼Min H. hen, D.F. Kocaoglu / European Journal of Operational Research 85 (28) L fjdg O j; jdg O þ jg; TSðG Þ¼Min 6 6L fjdg O þ dg O jg: h i Theore 4.3. If the allowale range of perturations on ij is d A S ij ; da S ijþ the OS and TS of A i are: ð4aþ ð4þ to preserve the final raning of A i s, OSðA i Þ¼Min 6j6J fjda S ij j; jda S ijþ jg; ð5aþ TSðA i Þ¼Min 6j6J fjda S ijþ da S ij jg: ð5þ The saller the sensitivity coefficients of a decision eleent are, the ore sensitive the decision is to variations of that eleent. If the TS of a decision eleent is one, eaning the tolerance is fro zero to one, the decision is not sensitive at all to changes that occur to the contriutions of this eleent. In addition, the TS of a contriution is also the proaility of varying that contriution value etween zero and one without changing the current ranings of A i s. The aove theores are ased on one-way SA in which the influence of an input to the decision is analyzed while eeping other inputs at their ase values (leen, 996; Reilly, 2). Extending the analysis to ultiple siultaneous changes, we can study the sensitivity of a certain decision level in the hierarchy. Recall that in the tolerance analysis section, an M-diensional allowale region is defined for M perturations induced on any local contriution vectors to preserve the final raning of A i s. ased on the sae logic, the shortest distance fro the origin to all hyperplanes of the M-diensional polyhedron and the polyhedron s volue deterine the roustness of the current odel regarding changes to the M contriution values. As to what TS reveals in the one-diensional analysis, since the volue of the M perturations feasile region is one, the volue of the M-diensional polyhedron is also the proaility of eeping A i s ran orders unchanged when the M contriutions vary fro zero to one ritical decision eleents In several previous studies, researchers tried to identify the ost influential variales with respect to the ran ordering of the alternatives (Howard, 968) or deterinant attriute that strongly contriutes to the choice aong alternatives (Aracost and Hosseini, 994). In this paper, the ost critical decision eleent is defined as the one whose influence on the final decision is ost sensitive to perturations, as defined y Triantaphyllou and Sanchez (997). Extending their definition to ultiple levels of the decision hierarchy, we get: Theore 5. The ost critical decision eleent at a given level of the decision hierarchy for current raning of A i s is the decision eleent corresponding to the sallest TS and OS at that level. In situations when the sallest TS and OS do not occur on the sae decision eleent, there can e two different decision eleents, and each one can e considered the ost critical in different situations. Additional analysis can also e carried out to deterine which one is ore critical Adding new decision alternatives There are situations where new decision eleents need to e added after a hierarchical decision odel has een uilt. Adding new decision eleents to the iddle levels of the decision hierarchy will change all the contriution atrices. In this case, it is suggested that a new decision hierarchy e constructed and the overall contriution vector e recalculated. However, introducing new decision alternatives only changes the otto level of the decision hierarchy; and SA can e applied to that special case.

13 278 H. hen, D.F. Kocaoglu / European Journal of Operational Research 85 (28) With the assistance of orollary 3.3, the ipact of adding a new decision alternative can e studied y assuing that the current contriutions of the new decision alternatives are zero and the new contriutions are i j, where ði ¼ I þ Þ and ð ¼ ; 2;...; JÞ. The currently top-raned decision alternative will reain unchanged as long as inequality (2) is satisfied for ðr ¼ Þ and ðn ¼ IÞ. The current raning of all decision alternatives will reain unchanged if (2) is satisfied for ðr ¼ ; 2;...; IÞ and ðr þ n ¼ I þ Þ, with the new decision alternative raned last. ased on the sae logic, adding ultiple new decision alternatives can e analyzed using Theore 3. The entire decision hierarchy does not have to e re-calculated. 3. An exaple All the propositions in the tolerance analysis section are verified using data fro a recent h.d. dissertation y Ho (24). The verification shows that whenever the perturations induced to the local contriution atrices go eyond their allowale region, the raning of the interested decision alternatives will e changed. Due to liited space, the detailed verification process will not e shown in this paper. The purpose of the exaple here is to deonstrate the use of HDM SA and show insights that are not availale or intuitively recognizale without conducting an HDM SA. Ho s odel evaluated five eerging technologies in Taiwan s seiconductor foundry industry y using a hierarchical decision odel containing four levels: overall copetitive success, copetitive goals, technology strategies, and technology alternatives. Applying orollary., Theores 4. and 5, the sensitivity of the copetitive-goals level is studied in a one-way SA (leen, 996). Local contriutions of copetitive goals to, aggregated contriutions of technology alternatives to the copetitive the overall copetitive success O goals AO i, and overall contriutions of technology alternatives to the copetitive success A i are suarized in Tales 2 4. First, the ran order of all the technology alternatives is considered. When r ¼, n ¼ : ased on (5), weget¼ A r A rþn ¼ A ðþ A ð2þ ¼ A 2 A 5 ¼ :235 :224 ¼ :46: (Note that r and r þ n are the rans of the alternatives. In Tale 4 we can see that the second and the fifth technology raned first and second. Therefore, we have A 2 A 5 in the aove expression.) Fro (5c), we get O ¼ A O 5 A O 5 O 4 A O ¼2 O 2 O 4 ¼2 ¼2 O ¼ :24 :24 ð:22 :25=:64 þ :24 :2=:64 þ :24 :8=:64Þ X4 A O 2 ¼2 þ X4 þ ð:2 :25=:64 þ :22 :22=:64 þ :2 :8=:64Þ ¼ :228: Fro (5a), we get O Tale 2 First level contriution vector O 6 O ¼ :46 :228 ¼ :64: O opetitive goals O ost leadership roduct leadership ustoer leadership Maret leadership Overall copetitive success Tale 3 Aggregated contriution atrix AO i A O i Technology alternatives A i 3 9 n Hi Lo Factory integration ost leadership roduct leadership ustoer leadership Maret leadership

14 H. hen, D.F. Kocaoglu / European Journal of Operational Research 85 (28) Tale 4 Overall contriution vector A i A i 3 9 n Hi Lo Factory integration Overall copetitive success urrent raning (3) () (5) (4) (2) Repeating the sae steps for n ¼, and r ¼ 2; 3; 4, we get all the inequalities that need to e satisfied, which 8 9 O 6 :64 ðwhen r ¼ ; n ¼ Þ >< >= are O : ðwhen r ¼ 2; n ¼ Þ O. oining the with the feasiility constraint [.36,.64], the 6 :466 ðwhen r ¼ 3; n ¼ Þ >: >; O 4:23 ðwhen r ¼ 4; n ¼ Þ allowale range of O is [.,.466]. Fro (3a) and (3), OSðO Þ¼Minf:; :466g ¼:, TSðO Þ¼j:466 þ :j ¼:476. Repeating the sae steps for O 2, O 3, and O 4, the sensitivity of a single change to contriutions of copetitive goals to overall success is deterined, as suarized in Tale 5. In this case, the rans of all technology alternatives are considered. Fro Tale 5 and Fig. 8, we can see that OSs and TSs give the sae inforation aout the criticality order of the decision eleents on the copetitive-goals level. The sallest OS and TS oth occur on O 2, aing product leadership the ost critical copetitive goal to preserve the current raning of all technology alternatives. Since TS(O 2 ) is.258, there is a 74.2% chance that the current ran order of the technology alternatives will change when O 2, the contriution of product leadership to overall success, varies fro zero to one. If we are only concerned with the current top-raned technology alternative, orollary. is applied y taing r ¼, n ¼ ; 2; 3; 4 to calculate the sensitivity indicators for O ð ¼ ; 2; 3; 4Þ. In this case, oth OS Tale 5 HDM SA at O level to preserve the raning of all A i O O 2 O 3 O 4 ase values Allowale ranges of perturations [.,.466] [.25,.8] [.69,.79] [.8,.82] Tolerance [.35,.8256] [,.258] [.43, ] [, ] OS ðo Þ TS ðo Þ Fig. 8. OS and TS as indicators of the criticality of O.

15 28 H. hen, D.F. Kocaoglu / European Journal of Operational Research 85 (28) Tale 6 HDM SA at O level to preserve the raning of the top A i O O 2 O 3 O 4 ase values Allowale ranges of perturations [.285,.64] [.25,.77] [.2,.79] [.8,.82] Tolerance [.75, ] [,.427] [, ] [, ] OS (O ) TS (O ) and TS indicate that O 2 is also the ost critical copetitive goal for 9 n linewidth to reain as the top choice. The HDM SA result is suarized in Tale 6. The HDM SA result shows that the ran of the top technology, 9 n linewidth, is not sensitive to changes on O 3 (contriution of custoer leadership to overall success) and O 4 (contriution of aret leadership to overall success). There is zero chance that the technology alternatives ran will change when these two values vary fro zero to one. However, it is sensitive to value increases on O 2 (contriution of product leadership to overall success). If the value of O 2 increases ore than.77, the inequality defined y (4a) in Definition. is not satisfied when r ¼ and n ¼ 2, which indicates that the current ran order of the firstand third-raned technologies will reverse. Interpreting this fro the perspective of how changes at the policy level will affect decisions at the operational level, the sensitivity analysis indicates that if the Taiwan seiconductor foundry industry shifts the ephasis of its copetitive goals to product leadership ore than 7.7%, then 3 wafer technology should e the top technology to e developed. In addition, HDM SA also indicates that the current second-raned technology, factory integration, is doinated y 9 n linewidth technology, which currently rans first, regardless of how the contriutions of copetitive goals change. However, the third-raned technology, 3 wafer, is sensitive to changes in the copetitive goals: it will ecoe the top choice when product leadership is ephasized or cost leadership is deephasized. This result ay draw the attention of decision aers and cause the to reconsider the resource allocation for these top three technologies, depending on how certain they are of the current contriution value assigned to each copetitive goal and how liely the ephasis on the copetitive goals will shift. 4. oncluding rears In this paper, we propose a coprehensive HDM SA algorith to analyze the ipacts of single and ultiple changes to the local contriution vector/atrices at any level of a decision hierarchy. In four groups of propositions, the allowale range/region of perturations and contriution tolerance are defined to eep the raning of interested decision alternatives unchanged; two sensitivity coefficients, operating point sensitivity coefficient and total sensitivity coefficient, are proposed to evaluate the roustness of a hierarchical odel; and the ost critical decision eleent at a given level to aintain the current decision is identified. The algorith is independent of the various pairwise coparison scales, judgent quantification techniques and group opinion coining ethods used y different researchers. Even though the tolerance analysis section is ased on the additive relationship to aggregate local contriution atrices into an overall contriution vector, the deductive logic can e easily applied to ultiplicative HDM. The tolerance analysis eployed atheatical deduction in syolic for in defining the allowale range/ region of perturations and contriution tolerance. opared to other ethods eployed in the literature, it has etter perforance (rich with inforation, precisely defined threshold value to any decial place, % accurate once the deduction process is verified), less coputational coplexity (fast, does not depend on repetitive iterations or large replications) and equal generality (sae assuptions). These propositions are tested and verified y data fro a dissertation y Ho (24) that used an additive hierarchical decision odel to evaluate eerging technologies. An exaple in which three HDM SA propositions are applied to Ho s odel is presented to deonstrate the practical application of the algorith. While the HDM SA algorith deals with changes to the local contriution atrices, which are the interediate input to HDM and thus ae the algorith independent of the different pairwise coparison scales

16 H. hen, D.F. Kocaoglu / European Journal of Operational Research 85 (28) and judgent quantification techniques, the disadvantage of such an approach is that it does not reveal the direct ipact of judgent changes to the odel results. A future extension of this study would e to lin the current algorith to different judgent quantification ethods and analyze odel s sensitivity to varying judgents. Future wor for HDM SA will also address ultiple siultaneous changes at different levels of the decision hierarchy. In addition, ased on the sae deductive logic, SA for ultiplicative HDM can e developed to define the allowale range/region of perturations at any level of the decision hierarchy. Future wor also includes applying the whole set of HDM SA algoriths to a large nuer of proles eing addressed y HDM. Acnowledgeent We gratefully acnowledge the financial support fro Maseeh Fellowship awarded y Fariorz Maseeh ollege of Engineering and oputer Science at ortland State University. The valuale suggestions y arry Anderson, Ti Anderson, Tugrul Dai, Hua Tang, Wayne Waeland, EJOR anonyous reviewers, and the editorial tea are highly appreciated. Appendix A. The flaw in Triantaphyllou and Sanchez s SA ethod To deduct the threshold value of perturations on W which will alter the ran order of A and A 2, Tri- and Sanchez (997) defined the perturation as d ;;2, and the new value of W as antaphyllou W ¼ W d ;;2. To preserve the property that all weights add up to, weights are noralized as follows, with W i denoting the noralized value: W ¼ W W þ W 2 þþw n W 2 ¼ W 2 W þ W 2 þþw n : : : W n ¼ W n W þ W : 2 þþw n ðaþ ða2þ ða3þ If we use d ;;2 to represent the actual threshold instead of the un-noralized threshold d ;;2, we have: W d ;;2 ¼ W ¼ W W d ;;2 W þ W ¼ ¼ W d ;;2 n 2 þþw n W d ;;2 þ W 2 þþw n i¼ W ; ða4þ i d ;;2! W d ;;2 Xn W i d ;;2 ¼ W d ;;2 ; ða5þ i¼!! W Xn W i d ;;2 d ;;2 Xn W i d ;;2 ¼ W d ;;2 ; ða6þ i¼ i¼ d ;;2 ¼ W n i¼ W i d ;;2 W þ d ;;2 n i¼ W ¼ W W d ;;2 n i d ;;2 i¼ W : ða7þ i d ;;2 The actual threshold, d ;i;j (shown as d ;;2 in the aove expression), is a value different fro d ;i;j (shown as d ;;2 in the aove expression) as was assued in the Triantaphyllou and Sanchez study; d ;i;j is a function of d ;i;j ut not equal to it. For exaple, if the contriution values, W i s, are.4,.3,.2 and., and the d ;i;j defined y Triantaphyllou and Sanchez is., they conclude that efore noralization, W can go down to

17 282 H. hen, D.F. Kocaoglu / European Journal of Operational Research 85 (28) without altering the ran order of the decision alternatives. However, after noralization, W can only go down to.33, not to.3, and the other contriution values are changed to.33,.22, and.. The actual threshold of the change on W is :7 ¼ :4 :4: : instead of.. Appendix. Matheatical deduction for ropositions. to Matheatical deduction for Theore When M perturations O O < O < O ; L ¼; 6¼ O < M O on M of the O s, which are O, the new values of O are O ðnewþ ¼O þ O : < L ¼; 6¼ O are induced ased on the assuption, the other O s will e changed according to their original ratio scales. Therefore, new values of other O s are O ðnewþ ¼O þ O ; with O ¼ XM O O L : ¼; 6¼ O Therefore, the new values of A i can e represented as A i ðnewþ ¼XM ¼ XM ð O þ O ÞA O i þ XL O A O i þ XL ¼; 6¼ ¼; 6¼ Since M O A O i þ L ¼; 6¼ O... M A O A i ðnewþ ¼A i þ XM O A O i XL ð O þ O ÞA O i O A O i þ XM ¼; 6¼ i ¼ A i then A O i O A O i XL ¼; 6¼ A O i M O O L : ¼; 6¼ O M O O L : ð:þ ¼; 6¼ O The raning of A r and A rþn will not e reversed if A r ðnewþ A rþnðnewþ. y sustituting Eq. (.) in the inequality, we get: A r þ XM O A O r A rþn þ XM A r A rþn XM A r A rþn XM XM XL ¼; 6¼ O A O rþn; O A O rþn; 2 A O r XL ¼; 6¼ XL ¼; 6¼ O A O r þ XL 4 O A O rþn; ¼; 6¼ M O O L ¼; 6¼ O A O r M A O rþn; O O L ; ¼; 6¼ O A O rþn; M O O L ¼; 6¼ O A O r XL ¼; 6¼ M O O L ; ¼; 6¼ O A O rþn; O L ¼; 6¼ O þ XL ¼; 6¼ A O r O L ¼; 6¼ O ð:2þ The top choice will reain at the top ran if the aove condition is satisfied for all r ¼ and n ¼ ; 2;...; I, which eans A ðnewþ A 2 ðnewþ; A ðnewþ A 3 ðnewþ;...; A ðnewþ A I ðnewþ. 3 A5:

18 H. hen, D.F. Kocaoglu / European Journal of Operational Research 85 (28) The ran order for all A i s will reain the sae if the aove condition is satisfied for all r ¼ ; 2;...; I, and n ¼, which eans A ðnewþ A 2 ðnewþ A r ðnewþ A I ðnewþ..2. Matheatical deduction for Theore 2 When M perturations a ð ¼ ; 2;...; MÞ are induced in M of the G O a s, denoted as, T pertur- a s, denoted as ations t ðq ¼ ; 2;...; QÞ are induced in Q of the of a ðt ¼ ; 2;...; T Þ are induced in T of the s and other G O a s will e ðnewþ ¼G O a þ a ; a ðnewþ ¼G O a a c s, denoted as þ G O ; with a ¼XM The new values of s and of other t s will e ðnewþ ¼ t t ðnewþ ¼ The new values of q c þ ; t ðnewþ q ¼G O c þ q c ; q c ðnewþ ¼ c c þ ; with ¼ XT s and of other G O s will e c þ ; with c ¼ XQ c Therefore, the new values of A i s can e represented as AðnewÞ i ¼ XL X K O G O i þ XK O a a ¼ 6¼ q c a K, Q perturations t q c, ased on the assuptions, the new values a 6¼ t K 6¼ t G O c q c K 6¼ q a c XM a K 6¼ : : : a a A i 6¼ a 6¼ 6¼ 6¼ t 6¼ c 6¼ q þ XM O a þ XT þ XQ O O c þ a a i þ XK 6¼ t t q c þ t þ q c i þ XK t 6¼ q i q ; O O c XT t K 6¼ t XQ c G O c q c 6¼ q K c A i A i

19 284 H. hen, D.F. Kocaoglu / European Journal of Operational Research 85 (28) Since A i A i ðnewþ ¼XL ¼ XK X K 6¼ t XK 6¼ q O G O i XK 6¼ O O c ¼ L K ¼ O G O A i ðnewþ ¼A i XK XK O a 6¼ 6¼ t XK 6¼ q O O c O a X T t K 6¼ t X M a K 6¼ a A i þ XT O X Q G O c q c K A i þ XQ 6¼ c q i, then X M a K 6¼ X T t K 6¼ t a a A i þ XM A i þ XT O X Q G O c q c K A i þ XQ 6¼ c q a A i þ XM t i t O c q c A G i q : O a a A G i t i t O a a A G i O c q c A G i q : ð2:þ The raning of A r and A rþn will not e reversed if A r ðnewþ A rþnðnewþ. y sustituting Eq. (2.) in the inequality, we get: A XK r þ XT 6¼ A rþn XK O a A G r O 6¼ t XK 6¼ q X M a K 6¼ t r XK t 6¼ q XK 6¼ O rþn; O A G c rþn; O A G a rþn; O c A G r a a X M a K 6¼ X T t K 6¼ t A þ XM O a a A G r X Q G O c q c K 6¼ c q a X Q G O c q c K 6¼ c q a A þ XT A þ XM A þ XQ O XK A þ XQ 6¼ t O O a a A G rþn; t rþn; t O c q c A G rþn; q ; r O c q c A G r q X T t K 6¼ t A