Sensitivity Analysis in a Generic Multi-Attribute Decision Support System

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1 Senstvty Analyss n a Generc Mult-Attrbute Decson Support System Sxto Ríos-Insua, Antono Jménez and Alfonso Mateos Department of Artfcal Intellgence, Madrd Techncal Unversty Campus de Montegancedo s/n, Boadlla de Monte, Madrd, SPAIN Abstract. Ths paper descrbes several possble senstvty analyses assocated wth a generc mult-attrbute Decson Support System, whch s capable of consderng all the steps n the Decson Analyss cycle and s amed at adng decson-makers n the choce of the most preferred alternatve. The system evaluates the set of alternatves through an addtve multattrbute utlty functon that allows for mprecse assgnments concernng utltes and weghts and uncertanty about consequences n terms of ntervals nstead of sngle values. The frst senstvty analyss s related to the tradtonal approach that can be used to answer what f questons. A seres of graphcal dsplays are an excellent ad n the decson-makng process, because they gve useful and mportant nsght nto the fnal rankng. Then, other complementary procedures are consdered that explot the mprecse nputs, generatng results that provde nsghts nto the model recommendatons. Ths leads to the computaton of non-domnated and potentally optmal alternatves, a weght stablty nterval assessment and smulaton technues that utlze Monte Carlo smulaton methods allowng smultaneous weght changes. Key words: Decson Support System, Imprecson, Multattrbute Utlty, Senstvty Analyss.. Introducton The dffculty n solvng complex real decson-makng problems has recently led to the development of better decson support tools to deal wth the ncreasngly nvolved dffcultes. The tools to deal wth such problems should be able to consder some of the basc aspects that arse n real problems, lke the presence of multple conflctng objectves, as well as mprecson concernng assgnments. To deal wth such complex stuatons, we have developed a decson support system (DSS) that allows to construct an objectve tree wth up to 200 objectves and 0 objectve levels. The DSS s based on an addtve mult-attrbute utlty model (Keeney and Raffa, 993), whch allows for mprecson concernng the nputs. Thus, the process of assessng ndvdual utlty functons and constant scales s not very demandng and s, hence, less stressful for decson makers (DMs). Furthermore, we consder the stuaton where the consequences of the alternatves can be entered as ranges or ntervals, modelled through contnuous unform dstrbutons, nstead of sngle values as an approach under certanty would demand, see (Mateos et al, 200). The startng pont wll be to establsh a set of n attrbutes assocated wth the lowest-level objectves and denoted by X,...,X n. Thus, the consequence of each alternatve S q S, where S s L U L U L U the avalable alternatves set, s a vector of ntervals ([ x, x ],...,[ x, x ],...,[ x, x ]), where L U x and x are the lower (L) and upper (U) endponts of the mprecse or uncertan consequence for attrbute X, respectvely. We also consder the possblty of substtutng each nterval by a q q q nq nq

2 P L U sngle value gven by the average x ( x + x )/ 2 = (P means precson), havng then a precse consequence, f deemed approprate by the DM. Next, an analyss focused on judgemental nputs must be conducted to assess mprecse utlty functons on attrbutes, whch leads to a famly of utlty functons for each one wth assessed extreme functons denoted by u L and u U, respectvely (Jménez et al, 2003). The evaluaton P process calls for precse utlty functons n the problem-solvng process, where u s the precse functon, obtaned by fttng natural cubc splnes through the md-ponts of the famly of utlty functons. Smlarly, mprecse scalng factors or weghts for each objectve and attrbute n the P, where w s the respectve average normalzed value. For the aggregaton nto a global utlty, we assume the addtve ndependence condton (Keeney and Raffa, 993) or an approxmaton (Raffa, 982; q n P P P p Stewart, 996), whch leads to the utlty functon of the form u( S ) = = k ( ) u x, where k s the precse weght obtaned by multplyng the weghts w P n the path from the global objectve of the herarchy to attrbute X. However, due to the possble mprecson n our approach concernng the consequences, the utlty for each alternatve S q can gven by a utlty n L L L n U U U k u x, k u x (provded that the utlty functons are ncreasng. If any L U objectve tree are assessed, obtanng the respectve normalzed ntervals [ w, w ] [ ] nterval ( ) ( ) = = utlty functon s decreasng, the change would be obvous). Fnally, we have recently developed several senstvty analyses n the system, whch wll be explaned n the followng sectons, because, as s well known, senstvty analyss s an essental ad n any quanttatve model to study the robustness of the fnal rankng of the alternatves. Ths methodology has been mplemented on a PC-based DSS, where all the process-relevant nformaton can be entered to help DMs arrve at the best or a satsfactory alternatve (Jménez et al, 2003). 2. Graphcs-Supported Evaluaton and Senstvty Analyss Once all the mprecse (or precse) nformaton related to utltes, weghts and consequences, denoted by u U, k K and x X, respectvely, has been entered n the DSS, t computes an Alternatves Classfcaton, whch provdes the rankng based on ther precse values, as well as the respectve assocated utlty ntervals, see Fgure. Fgure. The utlty ntervals and the strateges ranked accordng to ther precse utltes 2

3 Ths graph can be helpful for the DMs, because, besdes provdng the rank based on the precse utlty, t gves the utlty ntervals that reflect the mprecson concernng the preferences and the uncertanty about the consequences. If the DM changes a utlty, weght or consequence value, the system takes care of how these changes are propagated through the objectve herarchy and, automatcally, recalculates the alternatves classfcaton. Another graphcal ad s provded by Weght Stablty Intervals for each one of the objectves or attrbutes n the herarchy. Fgure 2 shows the stablty nterval, where ts weght can vary wthout affectng the overall rankng of the alternatves for a second level objectve named, Health mpact. The present value s 0.689, and we fnd that f ths value s changed to another one outsde the nterval [0.68, 0.8], then the above rankng would change. Fgure 2. The stablty weght nterval for a gven objectve Another dsplay that s potentally useful for DMs s the Stacked Bar Rankng, whch provdes the average utlty assgned to each alternatve by breakng down the contrbuton for the dfferent attrbutes n the utlty bar, see Fgure 3. Fgure 3 also shows another graph named Measure Utltes for Alternatves, whch dsplays the utlty of the dfferent attrbutes for a gven alternatve through bars whose wdth corresponds to the weght of the attrbute n queston. Fgure 3. Dsplays of the Stacked Bar Rankng and Measure Utltes for Alternatves Another dsplay, Fgure 4, shows the normalzed weght ntervals and normalzed average weghts assocated wth each attrbute. Ths fgure also shows the correlaton between pars of selected attrbutes for all the alternatves. Fnally, there s another possble dsplay, named Compare Alternatves Graph, whch compares two selected alternatves wth respect to the utltes for each attrbute n the model, thus provdng how much more utlty each alternatve has wth respect to the other for each attrbute. 3

4 Fgure 4. Dsplay of the normalzed nterval and average weghts, and correlaton between alternatves 3. Nondomnated and Potentally Optmal Alternatves The ranges provded for utltes, weghts and consequences are now used for smultaneous varaton of all the data, takng advantage of advances n optmzaton theory to make t easer for the DM to choose an optmal soluton. Thus, although t has been possble to output the rankng of alternatves based on ther precse values, another possblty offered by the DSS s to explot all the nformaton about such ranges to better choose the most preferred alternatve, as well as to reduce the set of alternatves of nterest (Mateos et al, 2003). The man thrust wll be, therefore, to order the alternatves n a Pareto sense. The DSS computes the Non-Domnated and Potentally Optmal alternatves by solvng the optmzaton problems (for all the possble pars of alternatves), respectvely, q p mn f qp = u( S ) u( S ) mn f p p q s.t. u U, k K, x X s.t. u( S ) u( S ) + f 0, p q u U, k K, x X, Fgure 5 shows the non-domnated and the potentally optmal alternatves, remndng us that the optmal one s Lake Lmng. Note that alternatve No Acton s not potentally optmal, and the set of alternatves to be consdered by the DM has been reduced accordngly. p Fgure 5. Vew of the non-domnated and the potentally optmal alternatves 4

5 4. Weght Smulaton The system also performs smulaton technues, whch allows for smultaneous changes of the weghts and generates results that can be easly analysed statstcally to provde more nsghts nto the multattrbute model recommendatons (Butler et al, 997). We propose selectng weghts at random usng a computer smulaton program so that the results of many combnatons of weghts, ncludng a complete rankng, can be explored effcently. Three general classes of smulaton are consdered: Random Weghts. As an extreme case, weghts for attrbutes are generated completely at random. Ths approach mples no knowledge whatsoever of the relatve mportance of the attrbutes. Rank Order Weghts. Randomly generatng weghts whle preservng ther attrbutes rank order places substantal restrctons on the doman of possble weghts that are consstent wth the DM s judgement of attrbute mportance. Therefore, the results from the rank order smulaton may provde more meanngful results. Response Dstrbuton Weghts. The thrd type of senstvty analyss usng smulaton recognzes that the weght assessment procedure s subject to varaton. Now, attrbute weghts are randomly assgned values takng nto account the nterval weghts provded by the DMs n the weghts assgnment. As an ad, the system dsplays for each of the above cases a multple boxplot for the alternatves and computes several statstcs (mode, mn, 25 th percentle,..., mean, st. devaton) about the rankngs of each alternatve, see Fgure 6. All ths nformaton can be useful for dscardng some alternatves. Fgure 6. Multple boxplot for the result of a smulaton and some statstcs about the rankng Acknowledgements: Ths paper was supported by the Mnstry of Scence and Technology project DPI References Butler, J., Ja, J. and J. Dyer (997); Smulaton Technues for the Senstvty Analyss of Multcrtera Decson Models; European Journal of Operatonal Research, Vol. 03 (pp ) 5

6 Jménez, A., Ríos-Insua, S. and A. Mateos (2003); A Decson Support System for Multattrbute Utlty Evaluaton based on Imprecse Assgnments; Decson Support Systems (to appear) Keeney, R.L. and H. Raffa (993); Decson wth Multple Objectves; Cambrdge U.P. Mateos, A., Jménez, A. and S. Ríos-Insua (2003); Solvng Domnance and Potental Optmalty n Imprecse Mult-Attrbute Addtve Problems; Relablty Engneerng and System Safety, Vol. 79, No. 2 (pp ) Mateos, A., Ríos-Insua, S. and E. Gallego (200); Postoptmal Analyss n a Mult-Attrbute Decson Model for Restorng Contamnated Aquatc Ecosystems; Journal of the Operatonal Research Socety, Vol. 52 (pp. -2) Raffa, H. (982); The Art and Scence of Negotaton; Harvard Unversty Press; Cambrdge. Stewart, Th.J. (996); Robustness of Addtve Value Functon Method n MCDM; Journal of Mult- Crtera Decson Analyss, Vol. 5 (pp ) 6

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