Normative and descriptive approaches to multiattribute decision making

Transcription

1 De. 009, Volume 8, No. (Serial No.78) China-USA Business Review, ISSN 57-54, USA Normative and desriptive approahes to multiattribute deision making Milan Terek (Department of Statistis, University of Eonomis in Bratislava, Bratislava 855, Slovakia) Abstrat: The main problem in multiattribute deision making is deiding how to trade off inreased value on one attribute for lower value on another attribute. Generally, there are two approahes to multiattribute deision making: Normative approah assuming that the rational behavior of deision maker is based on a set of aioms but desriptive approah based on pairwise omparisons. Firstly, the paper deals with the normative approah. It illustrates the utilization of normative approah on the eample from the area of management of a purhase in a firm, in whih the hoie from the finite set of alternatives is neessary. Then, the same deision problem is solved with aid of analyti hierarhy proess developed by Saaty. Finally, the omparison of the two approahes is made. Key words: multiattribute deision making; normative approah; desriptive approah; utility funtion; deision tree; analyti hierarhy proess; omparison. Introdution The main problem in multiattribute deision making is deiding how to trade off inreased value on one attribute for lower value on another attribute. This deision making requires the deision maker s judgment. The attribute is the means to measure aomplishment of the fundamental objetive. The term attribute will be used to refer to the quantity measured on an attribute sale. For eample, if an objetive is to maimize profit, then the attribute sale might be defined in terms of EUR.. Approahes to multiattribute deision making Generally there are two approahes: () The normative approah. To multiattribute deision making assumes that rational behavior of deision maker is based on a set of aioms: Ordering and transitivity, redution of ompound unertain events, ontinuity, substitutability, monotoniity, invariane, finiteness. If deision maker aepts the aioms: It is possible to find a utility funtion for evaluation the onsequenes (the deision maker s subjetive preferene struture an be aptured by an m-dimensional preferene funtion); The deision making is onsistent with maimizing of epeted utility. () The desriptive approah. It is based on pairwise omparisons of the alternatives for all attributes. The results are represented in a matri. Different sales an be used. The weights for the attributes epress their mutual importane. The weighted preferenes have to be aggregated for eah alternative. There are two different methods * This paper was elaborated with the support of the grant ageny VEGA in the framework of the projet number /047/08: Quantitative Methods in Si Sigma Strategy. Milan Terek, Ph.D., professor, Department of Statistis, University of Eonomis in Bratislava; researh fields: deision analysis, statistial quality ontrol, data mining. More about two approahes see Beroggi (999). More in detail see Clemen and Reilly, 00, pp

2 to it. The first one is to add up all the values in a row for eah alternative and attribute preferential aggregation within attribute. Another method is to put the matries on top of eah other and to add up the entries for eah ell preferential aggregation aross attributes. The analyti hierarhy proess (AHP) assumes that deision maker omplies with reiproal symmetry but not neessarily with intensity transitivity. To redue the number of pairwise omparisons, an aggregation of attributes into hierarhial lusters may be done. Then, the relative importane of the attributes an be omputed. Preferential aggregation leads to a preferential intensity matri. Preferential aggregation is often assumed to be additive. On the basis of the preferential values, preferential orders for the feasible alternatives an be determined. The desriptive approah an lead to inonsistenies in the assessments. The different methods of the resolution of inonsistenies are known (Beroggi, 999, pp.69-7). Firstly, the normative approah and an additive utility funtion will be desribed.. Multiattribute deision making using utility funtion Firstly, suppose there is only one attribute. The risk attitude of deision maker an be aptured by a subjetive preferene funtion, named utility funtion u(). The utility funtion represents a way to translate for eample EUROs or other measurement units of the attribute into utility units. If the utility funtion is onave, the deision maker is risk-averse; If it is onve, the deision maker is risk-seeking; If it is a straight line, the deision maker is risk-neutral (Raiffa, 968; Brown, 005). If more attributes are supposed, deision maker s subjetive preferene struture an be aptured by an m-dimensional preferene funtion, ( ) u,,..., m, where,,, m are attributes. This funtion will be alled m-dimensional utility funtion, and an additive utility funtion will be used. Eample: A new ar is needed in the ompany. This ar will be used mainly to drive in the ity. The deision maker has a basi idea of the brand and type of a ar. There are different types that have remained in his loser seletion. Let us mark them A, A, A. Assume that there are only two parameters of the ars whih are important to deision maker the purhase prie and the average fuel onsumption for 00km in the ity. The deision maker has the values of the two parameters (see Table ). Table The values of the parameters of the ars to A A A Purhase prie (EUR) Fuel onsumption (l/00km) One of the simple ways of reahing a reasonable solution will be demonstrated.. The additive utility funtion Suppose that there are individual utility funtions, ( ) u, u ( ),, ( ) u for m different attributes from m. Assume that eah individual utility funtion assigns values from 0 to to the worst and best levels on that partiular attribute. The utility funtion is said to be additive if it an be epressed as the sum of the individual utility funtions: u (,,..., m ) = k u ( ) + k u ( ) + + k u m m ( m ) () where k, k,, k m are the weights, and all weights are positive and they add up to.. Assessing individual utility funtions m m

3 The so alled proportional soring will be used. The most desired value of the attribute will be assigned the utility, the least desired, the utility 0 (from the values of the attribute of all relevant alternatives). The utilities of the other alternatives will be alulated aording to the relation: Utility (the value of the attribute of alternative) = (the value of the attribute of alternative the least desired attribute s value)/(the most desired attribute s value the least desired attribute s value) Let us alulate the utility for the prie u (800) and for the fuel onsumption u s ( 8.5), of the alternative A. u (800) = = u s ( 8.5) = = u 900 = 0, Table are the utilities for the prize and the fuel onsumption of all alternatives, apparently ( ) ( 8000) u = 0, u ( 8.) =. u =, (9.0) s s Table The utilities for the prie and the fuel onsumption A A A u ( A i ) ( ) s A i u Assessing weights All that left now is, to assign weight to the attribute aording to the judgment of deision maker and the alternatives (final utilities of the alternatives) an be evaluated by: u( A i ) = k u ( A i ) + k s u s ( A i ) () Generally, there are three methods of assessing the weights: priing out, swing weighting and lottery weights (Clemen & Reilly, 00, pp.64-60; Clemen, 99). Suppose that aording to one of their, the author obtained k = 0.65, ks = Now the values of the additive utility funtion u( Ai ) u( A ) = = 0.75 of the alternatives an be alulated u( A ) = = 0.70 u( A ) = = 0.65 So, the best alternative is A..4 Certainty versus unertainty Really, the deision problem in the eample onerns deision making under ertainty. When making deisions under ertainty, we an use ordinal utility funtions (value funtions). Ordinal utility funtions are only required to rank-order sure outomes in a way whih is onsistent with the deision maker s preferenes for those outomes. There is no onern with lotteries or unertain outomes (Clemen, 00, p.60). If deision maker has to make a deision under unertainty, he/she should use ardinal utility funtion. A ardinal utility funtion appropriately inorporates risk attitude of deision maker, so that lotteries are rank-ordered in a way that is onsistent with deision maker s risk attitudes. All of the weight-assessment methods an be used regardless of the presene or absene of unertainty. If a speifi multiattribute preferene model was established, the weight-assessment methods amount to various ways 4

4 of establishing indifferene among speifi lotteries or onsequenes. The weights an be derived on the basis of these indifferene judgements. The proportional-sores tehnique is a speial ase, and it may be used under onditions of unertainty by a deision maker who is risk-neutral for the speified attributes. The distintion between ordinal and ardinal utility models an be important in theoretial models of eonomi behavior, pratial deision analysis appliations rarely distinguish between the two (Clemen, 00, p.6). Eample ontinuing : Let s etend our eample to one more attribute. Add the attribute: the osts of maintaining of the ar during its life span. Suppose, the osts of maintaining are unertain events and deision maker formulated the prior probability distribution for the outomes of these unertain events. The onsequenes of the outomes in EUROs are also known. The struture of the deision problem with all needed data is in the deision tree in Fig.. A A A (0.85) (0.60) (0.85) (0.85) (0.60) (0.85) (0.85) (0.60) (0.85) Fig. Deision tree In Fig. : is ost of maintaining; is purhase prie; is fuel onsumption. Fig. is the deision tree of the deision problem with the utilities alulated by the proportional soring method (Terek, 007). We will onsider: The prie the weight k = 0.6, the fuel onsumption the weight ks = 0. and the osts of maintenane the weight kn = 0.. The -dimensional additive utility funtion is alulated aording to: u(,, ) = 0.un( ) + 0.6u( ) + 0.us( ) () Finally, the epeted utilities EU are alulated: EU ( A ) = EU ( A ) = EU ( A ) = So the highest utility value has the alternative A..5 Interations of attributes Until now, suppose that we are able to model preferenes aurately with additive utility funtion. Really, we need it for the additive independene. Generally, the situations in whih attributes interat are more subtle. For eample, two attributes may be 5

5 substitutes for eah other. On the other hand, attributes ould be omplementary (for eample, the suess of various phases of a projet). Suh interations annot be modeled with aid of the additive utility funtion. The additive model is simply an additive ombination of preferenes for individual attributes. To apture the interations as well as risk attitudes, we must think more generally. It is possible to think about many attributes at one. The multiattribute utility theory onepts using only two attributes will be presented. The ideas an be etended to more attributes. Generally, there are two approahes to assessment a multiattribute utility funtion. The first approah onsists in the diret assessment of the multiattribute utility funtion, and the other approah onsists in the thinking about a multiattribute utility funtion that is omposed of the individual utility funtion: u(, y) = + u ( ) + u ( y) u ( ) u ( y) (4) X Y + 4 The importane of any suh formulation is that it eases the assessment. This way of modeling multiattribute utility funtion sometimes is alled separability. The separability requires some onditions. These onditions onern how the preferenes interat among the attributes. X Y A A A (0.85) (0.60) (0.85) (0.85) (0.60) (0.85) (0.85) (0.60) (0.85) Fig. u n ( i ) u ( ) u s ( ) u(,, ) Deision tree with utilities.6 Independene onditions.6. Preferential independene The mutual preferential independene is needed for separability. An attribute Y is said to preferentially independent of X if preferenes for speifi outomes of Y do not depend on the level of attribute X (Clemen & Reilly, 00, p.647). Mutual preferential independene holds for many people and many situations, or that at least it is a reasonable approimation. It is like deomposability property for an objetive s hierarhy. The mutual preferential independene is the ondition that is needed for the ordinal additive utility funtion. For the deision making under unertainty, mutual preferential independene is not strong enough. It is neessary, but not suffiient ondition for obtaining separability of a ardinal multiattribute utility funtion (Clemen & Reilly, 00, p.647)..6. Utility independene 6

6 An attribute Y is onsidered utility independent of attribute X if preferenes for unertain hoies involving different levels of Y are independent of the value of X. Imagine assessing a ertainty equivalent for a lottery involving only outomes in Y. If our ertainty equivalent amount for the Y lottery is the same no matter what the level of X, then Y is utility independent of X; If X also is utility independent of Y, then the two attributes are mutually utility independent (Clemen & Reilly, 00, p.648). Utility independene is analogous to preferential independene, eept that the assessments are made under unertainty. Almost all reported multiattribute appliations assume utility independene and thus are able to use a deomposable utility funtion (Clemen & Reilly, 00, p.648)..6. Determining if independene eists We an imagine the series of pairwise omparisons that involve one of the attributes. With the other attribute fied at its lowest level, the deision maker should deide whih outome in eah pair does he/she prefer, and then, hange the level of the fied attribute. If the omparisons will be the same regardless of the fied level of the other attribute, then preferential independene holds. If the preferential independene is studied, the dialogue onerns the omparisons between sure outomes of Y for fied values of X. The study on the utility independene is the same, eept that the pairwise omparisons would be omparisons between lotteries involving attribute Y. To establish mutual preferential or utility independene, the roles of X and Y would have to be reversed to determine whether pairwise omparisons of outomes or lotteries in X depended on fied values for Y. If eah attribute turns out to be independent of the other, then mutual utility or preferential independene holds (Clemen & Reilly, 00, p.650)..6.4 Additive independene If we want to model preferenes with additive utility funtion, the additive independene is needed. Suppose X and Y are mutually utility independent and deision maker is indifferent between lotteries A and B: 4, with probability 0.5; A: ( ) y ( ) + y +, with probability 0.5; B: (, y ) + with probability 0.5; (, y ) + with probability 0.5; If this is the ase, then EU(A) = EU(B) and the additive utility funtion an be used. 5 In this ase, hanges in lotteries in one attribute do not affet preferenes in lotteries in other attribute. For utility independene, hanges in sure levels of one attribute do not affet preferenes for lotteries in the other attribute..6.5 Three or more attributes When a deision problem involves three or more attributes, modeling preferenes is more diffiult. Under ertain onditions, the multipliative utility funtion an be used..6.6 When independene fails Suppose we want assess a two-attribute utility funtion over attributes X and Y. The author has found that The eample of a dialog between analyst and deision maker an be seen in Keeney and Raiffa (976). 4 The worst level of the attribute is assigned by, and the best level, by +. 5 The proof see in Clemen and Reilly (00, p.65). 7

7 neither X nor Y is independent of eah other. A reasonable utility funtion an be obtained by two ways. The first possibility is to perform a diret assessment. Pik the best and worst (, y) pairs, assign then utility values of and 0, and then use the referene gambles to assess the utility values for other points. The seond approah is to transform the attributes and proeed to analyze the problem with the new set. The new set of attributes must apture the ritial aspets of the problem, and they must be measurable (Clemen & Reilly, 00, p.660). Now the desriptive approah to multiattribute deision making will be studied. 4. Analyti hierarhy proess There are a lot of desriptive preferene aggregation methods. The methods of the lassi ELECTRE, PROMETHEE and a lot of others are well-known and applied. 6 From this large set of methods, the author will desribe more in details with the analyti hierarhy proess (AHP), developed by Thomas L. Saaty, desribed in Anderson, Sweeney, Williams, Martin (008, pp ). The method is designed to solve omple multiattribute deision problems. AHP requires the deision maker to provide assessments about the relative importane of eah attribute and then speify a preferene for eah deision alternative using eah attribute. The output of AHP is a ranking of alternatives refleting the deision maker preferenes. Eample ontinuing. The problem given in Eample ontinuing will be solved with the aid of the Saaty method. Firstly, the preferenes for the attributes have to be determined. We need to onstrut a matri of the pairwise omparison ratings. Suppose we will use the omparison sale in Table (Anderson, Sweeney, Williams & Martin, 008, p.747). Intermediate judgments in Table, suh as moderately to strongly more important are also possible and will reeive a numerial rating 4. Table Comparison sale for the importane of attributes Verbal judgment Numerial rating Etremely more important 9 8 Very strongly more important 7 6 Strongly more important 5 4 Moderately more important Equally important Then, the deision maker has to realize the pairwise omparisons. In eah omparison he/she must selet more important attribute and then epress a judgment of how important the seleted attribute is. Suppose the judgments of deision maker are showed in Table 4. The numbers in Table 4 epress the preferenes of attributes. For eample, the number in the third row and first olumn means that for the deision maker, attribute is moderately more important to attribute (numerial 6 A onise overview of desriptive preferene aggregation methods an be found in Vinke (99). 8

8 rating of preferene is equal to ). Logially, attribute is for him less important than (numerial rating is equal to /). Table 4 Judgments of deision maker onerning weights /6 / 6 / Sum 0 0/6 0/ The pairwise omparison matri for attributes is showed in Table 4. On the basis of omparison matri, the alulation of the weight (priority) of eah attribute is possible. This proedure of AHP is referred to as synthesization. The following simple proedure was used. () Divide eah element in the pairwise omparison matri by its olumn total. () Compute the average of the elements in eah row. These averages provide the weights for the attributes (see Table 5). Table 5 Weights of attributes Weight In the net step of AHP, the onsisteny of the judgments has to be verified. For eample, if attribute ompared to attribute has a numerial rating, and attribute ompared to attribute has numerial rating, then perfet onsisteny of attribute ompared to attribute would have a numerial rating of 6 ( ). Some degrees of inonsisteny an be epeted. In Anderson, Sweeney, Williams and Martin (008, pp ), a simple method of verifying onsisteny is desribed. If the degree of inonsisteny is unaeptable, the deision maker should review and revise the pairwise omparisons before ontinuing with the AHP analysis. In this eample, the ideal onsisteny of weights was met. Then, the use omparison proedure to determine the priorities for the ars, eah of the attributes is needed. And the orresponding sale has to be reated. In the eample, the author will use the sale in Table 6 (Anderson, Sweeney, Williams & Martin, 008, p.75). The pairwise omparisons of the alternatives aording to the prie will be realised. Suppose the judgments of deision maker are showed in Table 7. In Table 8, the synthesization is done. In the ase of the osts of maintaining, we an not work diretly with the probability distributions, but only with some their harateristis. The author will use the epeted values. So, firstly the epeted values of the osts of maintaining for eah alternative will be omputed: E( A ) = 500 E( A ) = 800 E( A ) = 00 9

9 Then, we an proeed as in the ase of the prie. Suppose the judgments of deision maker are in Table 9. Table 6 Comparison sale for the preferene of alternatives Verbal judgment Numerial rating Etremely preferred 9 8 Very strongly preferred 7 6 Strongly preferred 5 4 Moderately preferred Equally preferred Table 7 Judgments of deision maker onerning alternatives aording to prie A A A A /4 /8 A 4 / A 8 Sum /4 /8 Table 8 The priorities of alternatives aording to prie A A A Priorities A / / / / A 4/ 4/ 4/ 4/ A 8/ 8/ 8/ 8/ Table 9 Judgments of deision maker onerning alternatives aording to osts of maintaining A A A A 8 A / 4 A /8 /4 Sum /8 /4 In Table 0, the synthesization is done. By the same proedure also the priorities of alternatives aording to the fuel onsumption were obtained. The results are in Table. Now, an overall priority ranking an be omputed. For eah alternative, the weighted sum of priorities will be omputed: () For alternative A : 0. 8/ / / 0.54 () For alternative A : 0. 4/ / + 0. /

10 () For alternative A : 0. / / + 0. / So, the best alternative is A. A different result as before was obtained. Table 0 The priorities of alternatives aording to osts of maintaining A A A Priorities A 8/ 8/ 8/ 8/ A 4/ 4/ 4/ 4/ A / / / / Table The priorities of alternatives aording to fuel onsumption A A A Priorities A 9/ 9/ 9/ 9/ A / / / / A / / / / 5. The omparison of the normative and desriptive approahes The desriptive approah an lead to inonsistenies in the assessments. In desriptive approah, the alternatives near replias an lead to strutural instability. Funtional stability is affeted by dependenies among attributes and alternatives. Numerial instability an our due to hanges in parameters (Beroggi, 999, p.). The advantage of the methods of desriptive approah is their relative simpliity in omparison with the proedures of normative approah. In normative approah, a probability distribution of the outomes of the unertain events an be epliitly inluded to the deision making proedure. In desriptive approah, only some harateristis of the probability distributions, alulated before the deision making proedure is applied an be epliitly used. In the appliation of Saaty method, in the eample, the means of the distributions alulated before the method was applied. In the presented eample, the author used the linear individual utility funtions. Generally, the individual utility funtions in normative approah an be linear or non-linear. If a ardinal utility funtion is applied, the normative approah will be able to determine the attitude of deision maker toward the risk. The individual utility funtion is linear when the deision maker is risk neutral and non-linear when he/she is risk-averse or risk-seeking. When the ordinal utility funtion is applied, the individual utility funtions an also be linear or non-linear. This information is epliitly inluded to deision making proess. In the author s opinion it is a great advantage of the normative approah. The normative approah is sometimes ritiized beause deision makers do not always behave rationally. Some behavioral paradoes are known. 7 These are situations in whih intelligent people make deisions that violate one or more of the aioms, and make deisions whih are inonsistent with the epeted utility. These paradoes do not invalidate the idea that we should still make deisions aording to the epeted utility (Clemen & Reilly, 00, p.588). (to be ontinued on Page 4) 7 See for eample Clemen and Reilly, 00, pp