Redesigning Decision Matrix Method with an indeterminacy-based inference process

Transcription

1 Redesgnng Decson Matrx Method wth an ndetermnacy-based nference process Jose L. Salmeron a* and Florentn Smarandache b a Pablo de Olavde Unversty at Sevlle (Span) b Unversty of New Mexco, Gallup (USA) ABSTRACT For academcs and practtoners concerned wth computers, busness and mathematcs, one central ssue s supportng decson makers. In ths paper, we propose a generalzaton of Decson Matrx Method (DMM), usng Neutrosophc logc. It emerges as an alternatve to the exstng logcs and t represents a mathematcal model of uncertanty and ndetermnacy. Ths paper proposes the Neutrosophc Decson Matrx Method as a more realstc tool for decson makng. In addton, a de-neutrosophcaton process s ncluded. Keywords: Decson Matrx Method, Neutrosophc Decson Matrx Method, Neutrosophc Logc, Decson Makng. Mathematcs Subject Classfcaton: Neutrosophc Logc. * Correspondng author e-mal address: salmeron@upo.es

2 1. INTRODUCTION For academcs and practtoners concerned wth computers, busness and mathematcs, one central ssue s supportng decson makers. In that sense, makng coherent decsons requres knowledge about the current or future state of the world and the path to formulatng a ft response (Zack, 2007). The authors propose a generalzaton of Decson Matrx Method (DMM), or Pugh Method as sometmes s called, usng Neutrosophc logc (Smarandache, 1999). The man strengths of ths paper are two-folds: t provdes a more realstc method that supports group decsons wth several alternatves and t presents a de-neutrosophcaton process. We thnk ths s an useful endeavour. The remander of ths paper s structured as follows: Secton 3 revews Decson Matrx Method; Secton 3 shows a bref overvew of Neutrosophc Logc and proposes Neutrosophc Decson Matrx Method and de-neutrosophcaton process; the fnal secton shows the paper s conclusons. 2. DECISION MATRIX METHOD BACKGROUND Decson Matrx Method (DMM) was developed by Stuart Pugh (1996) as an approach for selectng concept alternatves. DMM s a method (Murphy, 1979) that allows decson makers to systematcally dentfy and analyze the strength of relatonshps between sets of nformaton. Ths technque s especally nterestng for lookng at large numbers of factors and assessng each relatve mportance. Furthermore, DMM s a method for alternatve selecton usng a scorng matrx. DMM s often used throughout plannng actvtes to select product/servce features and goals and to develop process stages and weght optons.

3 DMM s brefly exposed. At the frst tme an evaluaton team s establshed. Frstly, the team selects a lst of weghted crtera and then evaluates each alternatve aganst the prevous crtera. That electon could be done usng any technque or mx of them (dscusson meetngs, branstormng, and so on). Ths one must be refned n an teratve process. The next step s to assgn a relatve weght to each crteron. Usually, ten ponts are dstrbuted among the crtera. Ths assgnment must be done by team consensus. In addton, each team member can assgn weghts by hmself, then the numbers for each crteron are added for a composte crteron weghtng. Follow that, L-shaped matrx s drawn. Ths knd of matrx relates two groups of tems to each other (or one group to tself). In the last step, the alternatves are scored relatve to crtera. Fgure 1. Buldng a Decson Matrx

4 Some optons are showed n Table 1. Table 1. Assessng alternatves Method Values range 1 Ratng scale for each alternatve. For example {1=low, 2=medum, 3=hgh} 2 For each crteron, rank-order all alternatves accordng to each fts the crteron. Order them wth 1 beng the opton that s least ft to crteron. 3 Establsh a reference. It may be one of the alternatves or any current product/servce. For each crteron, rate each other alternatve n comparson to the baselne. For example: Scores of {-1=worse, 0=same, +1=better} Wder scales could be used. At the end, multply each alternatve s ratng by ts weght. Add the ponts for each alternatve. The alternatve wth the hghest score wll be the team s proposal. Let C be the crtera vector of a DMM. C = c, c,..., c ) where c j belongs to the ( 1 2 n crtera domnon of the problem and n s the total number of crtera. Let W be the weghts crtera vector of a DMM. W = w, w,..., w ) where [, N ) N w j 0. ( 1 2 n Let A be the ratng vector of alternatve. A = a, a,..., a ) where a { 1,0,1 }. ( 1 2 n m Consder the matrx D be defned by D = ( a j ) where a j s the ratng of alternatve to the crteron j, a { 1,0,1 }. D s called the ratng matrx of the DMM. j Consder the vector S be defned by S = W D, beng D = ( s1, s2,..., sm) where s k s the product of weght by alternatve j and m s the number of alternatves.

5 ( s s... s ) = ( w w... w ) 1 2 m 1 2 n a11 a12... a1n am 1 am2... a mn The hghest s k wll be the team s proposal for the problem analyzed. Addtonally, alternatves have been ranked by the team. It s mportant to note that s k measures only rate of alternatve j respect to weght, tll now any scholar has not contemplated the ndetermnacy of any relaton between alternatves and crtera. When we deal wth unsupervsed data, there are stuatons when team can not to determne any rate. Our proposal ncludes ndetermnacy n DMM generatng more realstc results. In our opnon, ncludng ndetermnacy n DMM s an useful endeavour. 3. NEUTROSOPHIC LOGIC FUNDAMENTALS Neutrosophc Logc (Smarandache, 1999) emerges as an alternatve to the exstng logcs and t represents a mathematcal model of uncertanty, and ndetermnacy. A logc n whch each proposton s estmated to have the percentage of truth n a subset T, the percentage of ndetermnacy n a subset I, and the percentage of falsty n a subset F, s called Neutrosophc Logc. It uses a subset of truth (or ndetermnacy, or falsty), nstead of usng a number, because n many cases, humans are not able to exactly determne the percentages of truth and of falsty but to approxmate them: for example a proposton s between 30-40% true. The subsets are not necessarly ntervals, but any sets (dscrete, contnuous, open or closed or half-open/ half-closed nterval, ntersectons or unons of the prevous sets, etc.) n accordance wth the gven proposton. A subset may have one element only n specal cases of ths logc. It s mperatve to menton

6 here that the Neutrosophc logc s a strat generalzaton of the theory of Intutonst Fuzzy Logc. Accordng to Ashbacher (2002), Neutrosophc Logc s an extenson of Fuzzy Logc (Zadeh, 1965) n whch ndetermnacy s ncluded. It has become very essental that the noton of neutrosophc logc play a vtal role n several of the real world problems lke law, medcne, ndustry, fnance, IT, stocks and share, and so on. Statc context of Neutrosophc logc s showed n Fgure 2. Fgure 2. Statc context of Neutrosophc logc Fuzzy theory measures the grade of membershp or the non-exstence of a membershp n the revolutonary way but fuzzy theory has faled to attrbute the concept when the relatons between notons or nodes or concepts n problems are ndetermnate. In fact one can say the ncluson of the concept of ndetermnate stuaton wth fuzzy concepts wll form the neutrosophc concepts. In NL each proposton s estmated to have the percentage of truth n a subset T, the percentage of ndetermnacy n a subset I, and the percentage of falsty n a subset F. We use a subset of truth (or ndetermnacy, or falsty), nstead of a number only, because n many cases we are not able to exactly determne the percentages of truth and of falsty but to approxmate them: for example a proposton s between 30-40% true and between 60-70% false, even worst: between 30-40%

7 or 45-50% true (accordng to varous analyzers), and 60% or between 66-70% false. The subsets are not necessary ntervals, but any sets (dscrete, contnuous, open or closed or half-open/half-closed nterval, ntersectons or unons of the prevous sets, etc.) n accordance wth the gven proposton. A subset may have one element only n specal cases of ths logc. Statcally T, I, F are subsets, but dynamcally they are functons/operators dependng on many known or unknown parameters. Constants (T, I, F) truth-values, where T, I, F are standard or non-standard subsets of the non-standard nterval ],1 [ + 0, where n nf = nf T + nf I + nf F - 0, and n sup = sup T + sup I + sup F 3 +. Statcally T, I, F are subsets, but dynamcally T, I, F are functons/operators dependng on many known or unknown parameters. The NL s a formal frame tryng to measure the truth, ndetermnacy, and falsehood. The hypothess s that no theory s exempted from paradoxes, because of the language mprecson, metaphorc expresson, varous levels or meta-levels of understandng/nterpretaton whch mght overlap Usng ndetermnacy n Decson Matrx Method We propose a redesgn of the DMM called Neutrosophc Decson Matrx Method (NDMM). Ths proposal ncludes ndetermnacy n alternatves ratng and not s used to weghts. It s because weghts are the quantfed value of crtera. They are selected by the team. Therefore, an ndetermnacy weght has no sense. On the other hand, t s possble to consder ndetermnacy to alternatves ratng. A Neutrosophc Decson Matrx s a neutrosophc matrx wth neutrosophc values (alternatves ratngs or ndetermnaces as elements). Consder the matrx D be defned by D = ( a j ) where a j s the neutrosophc value of alternatve to the crteron j. D s called the ratng matrx of the NDMM. In that

8 sense, [, 1] I a j 1. We would nterpret ths expresson as representng the total group of numbers as the unon of two other groups. The frst nterval would start at -1 and proceed toward +1. The second would be an ndetermnacy value. The total set of numbers would be all those n the frst group along wth the ndetermnacy value. Note that I [ 1,1 ] that nterval. In fact, we have that = { x 1 x 1}, snce t s an ndetermnate value n a j. In addton, we propose a de-neutrosophcaton process n NDMM. Ths one s based on max-mn values of I. A neutrosophc value s transformed n an nterval wth two values, the maxmum and the mnmum value for I. In that sense, the neutrosophc scores wll be an area, where the upper lmt has I = 1 and the lower lmt has I = 1. The soluton set s U n χ = s j j=1, where j s the alternatves number and s s the score of each one. Any s k k j belongs to the complement of c χ. Alternatve selected s the global maxmum n χ. It s an alternatve A m where s * m s ;, m χ. De-neutrosophcaton process wll be appled wthn the followng applcaton. * s m s a lne (y axs value fxed) represented the score of alternatve A m. It s possble that s s, snce s s a lne and j s j s an area after deneutrosophcaton process. We select accordng to s s j f s max s > otherwse j mn s 2 j + k k ( max s s ) j 3.2. An applcaton Ths example llustrates the mprovements of NDMM versus DMM. NDMM proposal allows to represent ndetermnacy n a decsonal framework. Let C be

9 the crtera vector of a decson problem. C = c, c, c, ) where c j belongs to the crtera domnon of the problem. ( c4 Let W be the weghts crtera vector of a DMM. W = w, w, w, ). We have ( w4 used a three-valued scale from 1 (less mportance) to 3 (more mportance). w w w w = 3 = 3 = 2 = 1 Three dfferent alternatves are been consderng. We call t order of each one. A, where s the Consder the followng Neutrosophc Decson Matrx where alternatves and ratngs are showed. Each column represents the ratngs for an alternatve and each row gves the crteron ratngs for all the alternatves. Neutrosoph c 5 2 Decson Matrx = I I 7 We have used a scale from 1 (less ft) to 10 (more ft). Indetermnacy s ntroduced n the second alternatve (second crteron) and the thrd alternatve (thrd crteron). We show the S vector wth the product of weght by alternatve j as a result. ( I I ) = ( ) I I 7 The neutrosophc score of each alternatve s showed.

10 A1 = 44 scores n = A2 = I A3 = I If we consder scores got for second and thrd alternatves as equatons the representaton would be the showed n Fgure 3. Obvously, A 1 s the best opton. Fgure 3. Alternatves neutrosophc scores The next step s the de-neutrosophcaton process. We replace I [ 0,1] both maxmum and mnmum values. A1 = 44 scores d = A2 = [ 28,31] A3 = [ 43,45] Fgure 4 shows the de-neutrosophc results. The results show alternatves 2 and 3 as areas. In ths case 0 k 1. A3 wll be selected f and only f k > 0. It s more realstc vew from DMM.

11 Fgure 4. Alternatves de-neutrosophc scores 4. CONCLUSIONS Numerous scentfc publcatons address the ssue of decson makng n every felds. But, lttle efforts have been done for processng ndetermnacy n ths context. Ths paper shows a formal method for processng ndetermnacy n Decson Matrx Method and nclude a de-neutrosophcaton process. The man outputs of ths paper are two-folds: t provdes a neutrosophc tool for decson makng and t also ncludes ndetermnacy n a decson tool. In ths paper a renewed Decson Matrx Method has been proposed. As a methodologcal support, we have used Neutrosophc Logc. Ths emergng logc extends the lmts of nformaton for supportng decson makng and so on. Usng NDMM decson makers are not forced to select ratngs when ther knowledge s not enough for t. In that sense, NDMM s a more realstc tool snce experts judgements are focused on ther expertse. Anyway, more research s needed about Neutrosophc logc lmt and applcatons. Incorporatng the analyss of NDMM, the study proposes an nnovatve way for decson makng.

12 5. REFERENCES Ashbacher, C. Introducton to Neutrosophc Logc. Amercan Research Press. Rehoboth, Murphy, K.R. Comment on Pugh's method and model for assessng envronmental effects. Organzatonal Behavor and Human Performance, 23(1), 1979, pp Pugh, S. Creatng Innovatve Products Usng Total Desgn. Addson Wesley Longman, Smarandache, F. A Unfyng Feld n Logcs: Neutrosophc Logc, Amercan Research Press, Rehoboth, Avalable at Zack, M.H. The role of decson support systems n an ndetermnate world. Decson Support Systems Forthcomng. Zadeh, L.A. Fuzzy Sets, Informaton and Control 8(3), 1965, pp