Possibility/Necessity-Based Probabilistic Expectation Models for Linear Programming Problems with Discrete Fuzzy Random Variables

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1 S S symmetry Article Possibility/Necessity-Based Probabilistic Expectatio Models for Liear Programmig Problems with Discrete Fuzzy Radom Variables Hideki Katagiri 1, *, Kosuke Kato 2 ad Takeshi Uo 3 1 Departmet of Idustrial Egieerig, Faculty of Egieerig, Kaagawa Uiversity, Rokkakubashi, Yokohama-shi, Kaagawa , Japa 2 Departmet of Computer Sciece, Hiroshima Istitute of Techology, Miyake, Saeki-ku, Hiroshima , Japa; k.katoh.me@it-hiroshima.ac.jp 3 Departmet of Mathematical Sciece,Graduate School of Techology, Idustrial ad Social Sciece, Tokushima Uiversity, 2-1, Miamijosajima-cho, Tokushima-shi, Tokushima , Japa; uo.takeshi@tokushima-u.ac.jp * Correspodece: katagiri@kaagawa.ac.jp; Tel.: Received: 7 September 217; Accepted: 6 October 217; Published: 3 October 217 Abstract: This paper cosiders liear programmig problems LPPs) where the objective fuctios ivolve discrete fuzzy radom variables fuzzy set-valued discrete radom variables). New decisio makig models, which are useful i fuzzy stochastic eviromets, are proposed based o both possibility theory ad probability theory. I multi-objective cases, Pareto optimal solutios of the proposed models are ewly defied. Computatioal algorithms for obtaiig the Pareto optimal solutios of the proposed models are provided. It is show that problems ivolvig discrete fuzzy radom variables ca be trasformed ito determiistic oliear mathematical programmig problems which ca be solved through a covetioal mathematical programmig solver uder practically reasoable assumptios. A umerical example of agriculture productio problems is give to demostrate the applicability of the proposed models to real-world problems i fuzzy stochastic eviromets. Keywords: discrete fuzzy radom variable; liear programmig; possibility measure; ecessity measure; expectatio model; Pareto optimal solutio 1. Itroductio Oe of the traditioal tools for takig ito cosideratio ucertaity of parameters ivolved i mathematical programmig problems is stochastic programmig 1,2]. Stochastic programmig approaches implicitly assume that ucertai parameters ivolved i problems ca be expressed as radom variables. For example, demadig amouts of products are ofte mathematically modeled as radom variables. I this case, realized values of radom parameters uder evet occurrece are assumed to be represeted with determiistic values such as real values. O the other had, radom variables are ot always suitable to estimate parameters of problems, whe huma judgmets ad/or kowledge have to be mathematically hadled. It is worth utilizig ot oly historical or past data but also experts kowledge or judgmets ivolvig ambiguity or vagueess which are ofte represeted as fuzzy sets. Simultaeous cosideratio of fuzziess ad radomess is highly importat i modelig decisio makig problems, because decisio makig by humas i stochastic eviromets is itrisically based ot oly o radomess but also o fuzziess. I the last decade, mathematical models which take ito cosideratio both fuzziess ad radomess have cosiderably draw Symmetry 217, 9, 254; doi:1.339/sym

2 Symmetry 217, 9, of 34 attetios i the research field of decisio makig such as liear programmig 3 12], iteger programmig 13], ivetory 14,15], trasportatio 16], facility layout 17], flood maagemet 18] ad etwork optimizatio 19,2]. I this paper, we focus o mathematical optimizatio models i fuzzy stochastic decisio makig situatios where possible realized values of radom parameters i liear programmig problems LPPs) are ambiguously estimated by experts as fuzzy sets or fuzzy umbers. Such fuzzy set-valued radom variables, amely, radom parameters whose realized values are represeted with fuzzy sets, ca be expressed as fuzzy radom variables 21 26]. Previous studies o fuzzy radom LPPs have maily focused o the case where the coefficiets of the objective fuctio ad the costraits are expressed by cotiuous fuzzy radom variables, which is a exteded cocept of cotiuous radom variables. Fuzzy radom optimizatio models were firstly developed by Luhadjula ad his colleagues 27,28] as LPPs with fuzzy radom variable coefficiets, ad further studied by Liu 29,3], Katagiri et al. 4,6] ad Yao 11] ad so o. A brief survey of major fuzzy stochastic programmig models icludig mathematical programmig models usig fuzzy radom variables was foud i the paper by Luhadjula 8]. O the other had, there are a few studies 13,31,32] o LPPs with discrete fuzzy radom variables. As will be discussed later i more details, it is quite importat to propose more geeral fuzzy radom LPP models i order to wide the rage of applicatio of fuzzy stochastic programmig, which motivates this article to provide ew geeralized mathematical programmig models with discrete fuzzy radom variables. This paper is orgaized as follows: I Sectio 2, the defiitios of fuzzy radom variables are itroduced. Some types of fuzzy radom variables are ewly defied. Sectio 3 focuses o discrete fuzzy radom variables ad defies some types of discrete fuzzy radom variables. Sectio 4 formulates a sigle/multiple objective LPP where the coefficiets of the objective fuctios) are discrete fuzzy radom variables. I Sectio 5, we costruct ew optimizatio criteria for optimizatio problems with discrete fuzzy radom variables, which are based both o possibility theory ad o probability theory. Sectio 6 proposes decisio makig models usig optimizatio criteria itroduced i Sectio 5, ad defies weak) Pareto optimal solutios of the proposed models i the multi-objective case. Sectio 7 discusses how the proposed model ca be solved ad costruct a algorithm for obtaiig a Pareto optimal solutio of the proposed models. I Sectio 8, we execute a umerical experimet with a example of agriculture productio problems i order to demostrate the applicability of the proposed model to real-world decisio makig problems. It is show that the R 33] laguage ca be used to solve the problems with hudreds of decisio variables i a practical computatioal time. Fially, Sectio 9 summarizes this paper ad discusses future research works. 2. Prelimiaries I this sectio, we review some mathematical cocepts related to discrete fuzzy radom variables, such as covex fuzzy sets ad fuzzy umbers. Defiitios of fuzzy radom variables are also provided Fuzzy Set ad Fuzzy Number As a preparatio for the itroductio of fuzzy radom variables, we firstly itroduce the defiitio of fuzzy sets. Defiitio 1. Normal covex fuzzy set) A ormal covex fuzzy set is characterized by a membership fuctio µã : R, 1], that is, µãx), 1], for all x R, such that A α is a oempty compact iterval A α x R µãx) α if α, 1] clsupp µã) if α,

3 Symmetry 217, 9, of 34 where clsupp µã) deotes the closure of set supp µã, ad supp µã deotes a support of membership fuctio µã. A L-R fuzzy umber was itroduced by Dubois ad Prade 34] ad is defied based o a ormal covex fuzzy set. Defiitio 2. L-R fuzzy umber) A ormal covex fuzzy set F is said to be a L-R fuzzy umber, deoted by d, β, γ) LR, if its membership fuctio µ F is defied as follows: ) d τ L if τ d µ F τ) β ) 1) τ d R if τ > d, γ where L ad R are referece fuctios satisfyig the followig coditios: 1. Lt) ad Rt) are oicreasig for ay t >. 2. L) R) Lt) L t) ad Rt) R t) for ay t R. 4. There exists a t L > such that Lt) holds for ay t larger tha tl. Similarly, there exists a tr > such that Rt) holds for ay t larger tha t R. Fuzzy umbers are regarded as exteded cocepts of real umbers because F is reduced to a real umber d if β γ i Defiitio 2. Fuzzy umbers are a useful tool for represetig huma kowledge ad/or estimatio. Figure 1 shows a typical membership fuctio of a L-R fuzzy umber. µ ~ τ) F 1 L R β d γ τ Figure 1. L-R fuzzy umber. I Defiitio 2, if L R ad β γ, we call such a L-R fuzzy umber a L fuzzy umber. I other words, L fuzzy umbers are symmetric fuzzy umbers defied as follows: Defiitio 3. L fuzzy umber) A ormal covex fuzzy set F is said to be a L fuzzy umber if its membership fuctio µ F is defied as follows: ) d τ µ F τ) L, 2) β where L is a referece fuctio satisfyig the followig coditios: 1. Lt) is oicreasig for ay t >. 2. L) Lt) L t) for ay t R. 4. There exists a t L > such that Lt) holds for ay t larger tha tl.

4 Symmetry 217, 9, of 34 Figure 2 shows a typical membership fuctio of a L fuzzy umber. µ ~ τ) F 1 L L β d β τ Figure 2. L fuzzy umber I Defiitio 2, if Lt) Rt) max, 1 t, we call such a L-R fuzzy umber a triagular fuzzy umber. Defiitio 4. Triagular fuzzy umber) A L-R fuzzy umber F is said to be a triagular fuzzy umber, deoted by d, β, γ) tri, if the referece fuctios L ad R of a L-R fuzzy umber are give as Lt) Rt) max1 t, ). I other words, a triagular fuzzy umber F is characterized by the followig piece-wise liear membership fuctio: max µ F τ) 1 max 1 d τ, β τ d, γ if τ d if τ > d. Figure 3 shows a typical membership fuctio of a triagular fuzzy umber. µ ~ τ) F 1 3) L R β d γ τ Figure 3. Triagular fuzzy umber Fuzzy Radom Variable I this sectio, we review ad defie some importat cocepts uderlyig fuzzy radom programmig problems. There are maily two defiitios of fuzzy radom variables. A fuzzy radom variable was firstly defied by Kwakeraak 24] as a exteded cocept of radom variables i the sese that the realized values for give evets or scearios are ot real but fuzzy umbers. Kruse ad Meyer 35] provided some cocepts similar to the model by Kwakeraak. Puri ad Ralescu 25] defied fuzzy radom variables as radom fuzzy sets ad developed a mathematical basis of fuzzy radom variables with Klemmet 23]. Overviews of fuzzy radom variables have bee provided by Gil et al. 22] ad Shapiro 26]. We itroduce a geeral defiitio of fuzzy radom variables, which is based o the works of Kwakeraak 24], Kruse ad Meyer 35] ad Gil et al. 21]:

5 Symmetry 217, 9, of 34 Defiitio 5. Fuzzy radom variable) Let Ω, F, P) be a probability space ad FR) deote the set of all fuzzy umbers i R, where FR) deotes a class of ormal covex fuzzy subsets of R havig compact α level set for α, 1]. A fuzzy radom variable is a mappig Ā : Ω FR) such that for ay α, 1] ad all ω Ω, the real-valued mappig if Ā α : Ω R, satisfyig if à α ω) ifãω)) α ad sup Ā α : Ω R, satisfyig sup à α ω) supãω)) α are real-valued radom variables, that is, Borel measurable real-valued fuctios. Ãω)) α is a oempty compact iterval defied by Ãω)) α x R µãω) x) α if α, 1] clsupp µãω) ) if α, where µãω) is the membership fuctio of a fuzzy set Ãω), clsupp µãω) ) deotes the closure of set supp µãω), ad supp µãω) deotes a support of fuctio µãω) Special Types of Fuzzy Radom Variables Used i Decisio Makig For the purpose of applyig fuzzy radom variables to decisio makig problems, Katagiri et al. 4,6,2,36,37] itroduced some special types of fuzzy radom variables where the realized values of radom variables for give evets or scearios are L-R fuzzy umbers or triagular fuzzy umbers. Sice these fuzzy radom variables are useful for modelig various decisio makig problems, we categorize these fuzzy radom variables ito several types such as L-R fuzzy radom variable, L fuzzy radom variable ad triagular fuzzy radom variable, together with their examples which were origially itroduced i the previous papers. First, we defie L-R fuzzy radom variable as follows: Defiitio 6. L-R fuzzy radom variable) Let d, β ad γ be radom variables whose realizatio for a give evet ω Ω are dω), βω) ad γω), respectively, where Ω is a sample space, ad βω) ad γω) are positive costats for ay ω Ω. The, a fuzzy radom variable F is said to be a L-R fuzzy radom variable, deoted by d, β, γ) LR, if its realized values Fω) dω), βω), γω)) LR for ay evet ω Ω are L-R fuzzy umbers defied as ) dω) τ L µ Fω) τ) βω) ) τ dω) R γω) if τ dω) if τ > dω). 4) L-R fuzzy radom variables were itroduced to decisio makig problems such as a portfolio selectio problem 38], a LPP 36] ad a multi-objective programmig problem 4]. I these studies, the coefficiets of objective fuctios are represeted as a special type of L-R fuzzy radom variables i which the spread parameters β ad γ are costats, ot radom variables, as show i the followig example:

6 Symmetry 217, 9, of 34 Example 1. I Defiitio 6, let ḡ be a Gaussia ormal) radom variable Nm, σ 2 ) where m is the mea ad σ is the stadard deviatio. Also, let β ad γ be positive costats, ot radom variables. The, F is a kid of L-R fuzzy radom variables if the membership fuctio of the realizatio of F is defied as ) gω) τ L µ Fω) τ) β ) τ gω) R γ if τ gω) if τ > gω), where gω) is a realized value of ḡ for a give evet ω Ω, ad Ω is a sample space. Aother example of L-R fuzzy radom variables is show i the study o a multi-objective LPP 6] as follows: Example 2. Let d, β ad γ be radom variables expressed as d d 1 t + d 2, β β 1 t + β 2, γ γ 1 t + γ 2, where t is a radom variable whose mea ad variace are m ad σ 2, respectively, ad d 1, d 2, β 1, β 2, γ 1 ad γ 2 are costat values. The, Ā is a kid of L-R fuzzy radom variables if the membership fuctio of the realizatio of Ā is defied as ) d1 tω) + d 2 τ L β 1 tω) + β 2 µãω) τ) ) τ d1 tω) + d 2 R γ 1 tω) + γ 2 if τ d 1 tω) + d 2 if τ > d 1 tω) + d 2, where tω) is a realized value of t for a give evet ω Ω, ad Ω is a sample space. Whe the referece fuctios of left-had ad right-had sides are the same i Defiitio 6, amely, if it holds L R, we call such a L-R fuzzy radom variable a L fuzzy radom variable defied as follows: Defiitio 7. L fuzzy radom variable) Let d ad β be radom variables whose realizatio for a give evet ω Ω are dω) ad βω), respectively, where Ω is a sample space, ad βω) is a positive costat for ay ω Ω. The, a fuzzy radom variable F is said to be a L fuzzy radom variable if its realized values for ay evet ω Ω are L-R fuzzy umbers defied as ) dω) τ µ Fω) τ) L. 7) βω) 5) 6) L fuzzy radom variables were itroduced i etwork optimizatio problems such as bottleeck miimum spaig tree problems 2,39]. I these studies, the cost for costructig each edge i a optimal etwork costructio problem was expressed as a L fuzzy radom variable show i the followig example: Example 3. I Defiitio 7, let ḡ be a Gaussia ormal) radom variable Nm, σ 2 ) where m is the mea ad σ is the stadard deviatio. Also, let β be a positive costat, ot a radom variable. The, F is a kid of L fuzzy radom variables if the membership fuctio of the realizatio of F is defied as ) gω) τ µ Fω) τ) L, 8) β

7 Symmetry 217, 9, of 34 where gω) is a realized value of ḡ for a give evet ω Ω, ad Ω is a sample space. The L fuzzy radom variable show i Example 3 ca be iterpreted as a hybrid umber. The hybrid umber, which was origially itroduced by Kaufma ad Gupta 4], is composed of a series of fuzzy umbers, ad is obtaied by shiftig fuzzy umbers i a radom way alog the abscissa. Especially if Lt) Rt) max, 1 t i Defiitio 6, we call such a L-R fuzzy radom variable a triagular fuzzy radom variable. Defiitio 8. Triagular fuzzy radom variable) A L-R fuzzy radom variable F is said to be a triagular fuzzy radom variable, deoted by d, β, γ) tri, if the realizatio Fω) for each ω k Ω is represeted by a triagular fuzzy umber dω), βω), γω)) tri, where Ω is a sample space. I other words, a discrete triagular fuzzy radom variable F is a discrete fuzzy radom variable whose realizatio for each evet ω is a triagular fuzzy umber characterized by the followig membership fuctio: max µ Fω) τ) 1 max 1 dω) τ, βω) τ dω), γω) if τ dω) if τ > dω). Triagular fuzzy radom variables were itroduced i the study o a multi-objective LPP 37]. I this study, spread parameters β ad γ are ot radom variables but costat values as show i the followig example: Example 4. I Defiitio 8, let β ad γ be positive costats, ot radom variables. The, F is a kid of triagular fuzzy radom variables if the membership fuctio of the realizatio of F is defied as max µ Fω) τ) 1 max 1 dω) τ, β τ dω), γ if τ dω) if τ > dω), where dω) is a realized value of d for a give evet ω Ω, ad Ω is a sample space. 3. Discrete Fuzzy Radom Variable I this sectio, we discuss discrete fuzzy radom variable as for a preparatio for proposig a ew framework of LPPs with discrete fuzzy radom variables. Firstly, we review the defiitio of discrete fuzzy radom variable give by Kawakeraak 41]. Secodly, we provide the defiitio of discrete L-R fuzzy radom variable ad that of discrete triagular fuzzy radom variable which was applied to a etwork optimizatio problem 31], a LPP 32] ad a multi-objective -1 programmig problem 13]. Defiitios of Discrete Fuzzy Radom Variables I the 197s, Kwakeraak 41] origially proposed a cocept of discrete fuzzy radom variable. I this paper, we provide the defiitio of discrete fuzzy radom variable as follows: 9) 1) Defiitio 9. Discrete fuzzy radom variable) Let Ω be a set of evets such that the occurrece probability of each evet ω k Ω is p k ad that k p k 1. Let F k be a fuzzy set characterized by a membership fuctio µ F k, ad let F be a set of F k, k K, where K is

8 Symmetry 217, 9, of 34 a idex set of k. Let F be a mappig from Ω to F such that Fω k ) F k. The, a mappig F is said to be a discrete fuzzy radom variable. Cosiderig the applicability of the discrete fuzzy radom variables i real-world decisio makig, we defie discrete L-R fuzzy radom variables as a special type of discrete fuzzy radom variables. Defiitio 1. Discrete L-R fuzzy radom variable) A discrete fuzzy radom variable F is said to be a discrete L-R fuzzy radom variable, deoted by d, β, γ) LR, if the realizatio of F d, β, γ) LR for ay evet ω k Ω is a L-R fuzzy umber F k d k, β k, γ k ) LR, where d k, β k ad γ k are the realized values of d, β, ad γ for a give evet ω k Ω, respectively, ad Ω is a sample space. The, F is a L-R fuzzy radom variable i which the membership fuctio of the realizatio F k for each evet ω k Ω is defied as ) dk τ L if τ d β k k µ F k τ) ) 11) τ dk R if τ > d k. γ k The followig is a example of discrete L-R fuzzy radom variables: Example 5. Cosider β k ad γ k vary depedet o evets or scearios. The, F is a discrete L-R fuzzy radom variable i which the membership fuctios of the realized fuzzy umbers of F k, k 1, 2, 3 are defied as follows: ) 3 τ L 35 µ F 1 τ) ) τ 3 R 2 if τ 3 if τ > 3, ) 2 τ L 25 µ F 2 τ) ) τ 2 R 1 ) 1 τ L 3 µ F 3 τ) ) τ 1 R 15 if τ 2 if τ > 2, if τ 1 if τ > 1. 12) Figure 4 shows a typical membership fuctio of a discrete L-R fuzzy radom variable. Figure 4. Discrete L-R fuzzy radom variable.

9 Symmetry 217, 9, of 34 I particular, if Lt) Rt) max, 1 t i Defiitio 1, we call such a discrete L-R fuzzy radom variable a discrete triagular fuzzy radom variable. Defiitio 11. Discrete triagular fuzzy radom variable) A discrete L-R fuzzy radom variable F is said to be a discrete triagular fuzzy radom variable, deoted by d, β, γ) tri, if the realizatio F k for each ω k Ω is represeted by a triagular fuzzy umber d k, β k, γ k ) tri, where Ω is a sample space. I other words, a discrete triagular fuzzy radom variable F is a discrete fuzzy radom variable whose realizatio for each evet ω k is a triagular fuzzy umber characterized by the followig membership fuctio: 1 d k τ max µ F k τ), β k max 1 τ d k, γ k if τ d k if τ > d k. Example 6. Whe it holds that Lt) Rt) max, 1 t i Example 5, F is a discrete triagular fuzzy radom variable whose realized values F k, k 1, 2, 3 are characterized by the followig membership fuctio: 13) max µ F 1 τ) max µ F 2 τ) max µ F 3 τ) 1 max 1 1 max 1 1 max 1 3 τ, 35 τ 3, 2 2 τ, 25 τ 2, 1 1 τ, 3 τ 1, 15 if τ 3 if τ > 3, if τ 2 if τ > 2, if τ 1 if τ > 1. 14) Figure 5 shows a typical membership fuctio of a discrete triagular fuzzy radom variable. Figure 5. Discrete triagular fuzzy radom variable. Discrete triagular fuzzy radom variables were firstly itroduced i some of previous studies o a etwork optimizatio problem 31], a LPP 32] ad a multi-objective -1 programmig problem 13]. I these studies, the spread parameters β k ad γ k do ot vary with evets ω k but they are fixed as costats for ay evets. To the author s best kowledge, there has bee o study o liear

10 Symmetry 217, 9, of 34 programmig model where the spread parameters β k ad γ k of discrete fuzzy radom variables vary with differet evets ω k Ω. I the ext sectio we shall propose ew liear programmig models with discrete fuzzy radom variables i which spread parameters vary with stochastic evets. 4. Problem Formulatio Assumig that the coefficiets of the objective fuctios are give as discrete fuzzy radom variables, we cosider the followig fuzzy radom programmig problem: miimize C l x, l 1, 2,..., q subject to Ax b, x, where C l C l1,..., C l ), l 1, 2,..., q are dimesioal coefficiet row vectors whose elemets are discrete fuzzy radom variables, x is a dimesioal decisio variable colum vector, A is a m coefficiet matrix, ad b is a m dimesioal colum vector. Whe the umber of objective fuctios is equal to 1 q 1), the problem 15) becomes a sigle-objective fuzzy radom programmig problem; otherwise, whe q 2, 15) is a multi-objective fuzzy radom programmig problem. I problem 15), all the objective fuctios are to be miimized. Without loss of geerality, this paper cosiders miimizatio problems, because ay maximizatio problems ca be trasformed ito miimizatio problems by multiplyig the origial objective fuctio i the maximizatio problem by Model Usig Discrete L-R Fuzzy Radom Variables I problem 15), we firstly cosider the case where each elemet C lj of the coefficiet vectors C l C l1,..., C l ), l 1, 2,..., q i 15) is a discrete L-R fuzzy radom variable d lj, β lj, γ lj ) LR whose realizatio for a give evet ω lk Ω l is a L-R fuzzy umber C ljk dljk, β ljk, γ ljk ) LR, l 1, 2,..., q, j 1, 2,...,, k 1, 2,..., with the membership fuctio defied as ) dljk τ L β ljk µ C ljk τ) ) τ dljk R γ ljk if τ d ljk if τ > d ljk, 15) 16) where Ω l ωl1, ω l2,..., ω lrl deotes a set of evets related to the lth objective fuctio. I 16), the values of d ljk, β ljk ad γ ljk are costat, ad β ljk ad γ ljk are positive. The probability that each evet ω lk occurs is give as p lk, where p lk 1, l 1, 2,..., q. Figure 6 shows that a typical membership fuctio of a L-R fuzzy umber defied by 16). Figure 6. Realized values C ljk for the kth evet of a discrete L-R fuzzy radom variable C lj.

11 Symmetry 217, 9, of 34 Through the exteded sum of fuzzy umbers 42] based o the Zadeh s extesio priciple 43], the objective fuctio C l x is represeted by a sigle fuzzy radom variable whose realized value for a evet or sceario ω lk is a L-R fuzzy umber C lk x d lk x, β lk x, γ lk x) LR characterized by the membership fuctio ) dlk x υ L if υ d µ C lk x υ) β lk x lk x ) 17) υ dlk x R if υ > d γ lk x lk x, where d lk, β lk ad γ lk are dimesioal colum vectors whose values vary depedet o evets ω lk Ω l, l 1, 2,..., q. Figure 7 shows that the membership fuctio of a L-R fuzzy umber defied by 17). Figure 7. Realized values C ljk for the kth evet of a discrete L-R fuzzy radom variable C lj Model Usig Discrete Triagular Fuzzy Radom Variables As a special type of discrete L-R fuzzy radom variable defied i 17), we also cosider the case where C lj is a discrete triagular fuzzy radom variable i which its realized values for evets or scearios are triagular fuzzy umbers C ljk d ljk, β ljk, γ ljk ) tri for ω kl Ω l, l 1, 2,..., q, j 1, 2,...,, k 1, 2,..., with the followig membership fuctio: max 1 d ljk τ, β ljk µ C ljk τ) max 1 τ d ljk, γ ljk if τ d ljk if τ > d ljk. 18) The, through the Zadeh s extesio priciple, the realized value of each objective fuctio C l x for a give evet ω lk is represeted by a sigle triagular fuzzy umber d lk x, β lk x, γ lk x) tri which is characterized by max 1 d lkx υ, if υ d lk x µ C lk x υ) max β lk x 1 υ d lkx, γ lk x Figures 8 ad 9 show the membership fuctios of C ljk ad C lk x. if υ > d lk x. 19)

12 Symmetry 217, 9, of 34 Figure 8. Realized value C ljk for the kth evet of a discrete triagular fuzzy radom variable C lj. Figure 9. Realized value C lk x for the kth evet of a discrete triagular fuzzy radom variable C l x. 5. Possibility/Necessity-Based Probabilistic Expectatio This sectio is devoted to discussig optimizatio criteria to solve problem 15) with discrete fuzzy radom variables whose realized values are give as L-R fuzzy umbers defied by 17) or triagular fuzzy umbers defied by 19). It should be oted here that problem 15) is ot a well-defied mathematical programmig problem because, eve whe a decisio vector x is determied, the objective fuctio value C lk x is ot determied as a costat due to both radomess ad fuzziess of C lk. I other words, a certai optimizatio criterio is eeded to compare the value of fuzzy radom objective fuctio. I this sectio, we propose some useful optimizatio criteria based o both possibility ad probability measures, called a possibility/ecessity-based probabilistic expectatio Prelimiary: Possibility ad Necessity Measures As a preparatio for optimizatio criteria i fuzzy stochastic decisio makig eviromets, we review the defiitio of possibility ad ecessity measures, ad discuss how the measures are applied to our problems with discrete fuzzy radom variables Possibility Measure Cosiderig that membership fuctios of fuzzy sets ca be regarded as possibilistic distributios of possibilistic variables 44], a defiitio of possibility measure is give 34,44] as follows: Defiitio 12. Possibility measure) Let à ad B be fuzzy sets characterized by membership fuctios µã ad µ B, respectively. The, uder a possibilistic distributio of µã of a possibilistic variable α, possibility measure of the evet that α is i a fuzzy set B is defied as follows: Πà B) sup mi µãv), µ B v)). 2) v

13 Symmetry 217, 9, of 34 I decisio makig situatios where the objective fuctio is to be miimized, decisio makers DMs) ofte have a fuzzy goal such as the objective fuctio value C lk x is substatially less tha or equal to a certai value f l, which is expressed by C lk x < f l, where < deotes substatially less tha or equal to defied i 12). Let µ G l be a membership fuctio of fuzzy set G l such that the degree of y beig substatially less tha or equal to a certai value f l is represeted with µ G l y). Assume that a certai evet ω lk has occurred, o the basis of possibility theory ad otatios 2). The, the degree of possibility that C lk x satisfies fuzzy goal G amely, the degree of possibility that the objective fuctio value C lk x for ay evet ω lk Ω l is substatially less tha or equal to a certai aspiratio level f l ) is defied as Π C lk x < ) f l Π C lk x G l ) sup mi µ y C lk x y), µ G l y), l 1, 2,..., q, k 1, 2,...,. 21) Figure 1 illustrates the degree of possibility defied by 21) for a fixed evet ω lk, which is the ordiate of the crossig poit betwee the membership fuctios of fuzzy goal G l ad the objective fuctio C lk x. Figure 1. Degree of possibility Π C lk x < ) f l Necessity Measure For DMs who make decisios from pessimistic view poits, a ecessity measure is recommeded. The ecessity measure defied by Zadeh 44] ad Dubois ad Prade 34] is as follows: Defiitio 13. Necessity measure) Let à ad B be fuzzy sets characterized by membership fuctios µã ad µ B, respectively. The, uder a possibilistic distributio of µã of a possibilistic variable α, the ecessity measure of the evet that α is i a fuzzy set B is defied as follows: Nà B) if max 1 µ v à v), µ B v)). 22) The, i view of 22), the degree of ecessity that the objective fuctio value C lk x for ay evet ω lk Ω l satisfies the fuzzy goal G l is defied as N C lk x < ) f l N C lk x G l ) if max 1 µ y C lk x y), µ G l y), l 1, 2,..., q. 23) Figure 11 illustrates the degree of ecessity defied by 23), which is the ordiate of the crossig poit betwee the membership fuctios of fuzzy goal G l ad the upside-dow of the membership fuctio of the objective fuctio C lk x.

14 Symmetry 217, 9, of 34 Figure 11. Degree of ecessity N C lk x < ) f l. Possibilistic programmig 45,46] is oe of the most promisig tools for hadlig mathematical optimizatio problems with ambiguous parameters Optimizatio Criteria i Fuzzy Radom Eviromets Possibilistic programmig approaches caot directly be applied to solvig problems with discrete fuzzy radom variables. This is because the degrees of possibility or ecessity defied i 21) or 23) are ot costats but vary depedet o evets ω lk. I this sectio, takig ito cosideratio both fuzziess ad radomess ivolved i the coefficiets of the problems, we ewly propose some useful optimizatio criteria for problems with discrete fuzzy radom variables. As ovel optimizatio criteria, we provide possibility-based probabilistic expectatio PPE) ad ecessity-based probabilistic expectatio NPE) as follows: Defiitio 14. Possibility-based probabilistic expectatio PPE)) Let C l C l1,..., C l ), l 1, 2,..., q be dimesioal coefficiet row vectors of fuzzy radom variables i multi-objective) LPP 15). Suppose that the realized value of C lj is a fuzzy set or fuzzy umber as a special case) C ljk. Let Π C lk x < ) f l be the degree of possibility for a fixed evet ω lk defied i 21). By usig p lk which is the probability that a evet or sceario w lk occurs, the optimizatio criterio called a possibility-based probabilistic expectatio PPE) is defied ad calculated as follows: E Π C l x < )] f l p lk Π C lk x < ) f l p lk Π C lk x G l ) 24) p lk sup mi µ y C lk x y), µ G l y), l 1, 2,..., q, where E ] deotes a probabilistic expectatio. Possibility measures are recommeded to optimistic DMs. O the other had, sice DMs are ot always optimistic i geeral, we itroduce the followig ew optimizatio criterio based o ecessity measures i order to costruct a optimizatio criterio for pessimistic DMs: Defiitio 15. Necessity-based probabilistic expectatio NPE)) Let C l C l1,..., C l ), l 1, 2,..., q be dimesioal coefficiet row vectors of fuzzy radom variables i multi-objective) LPP 15). Suppose that the realized value of C lj is a fuzzy set or fuzzy umber as a special

15 Symmetry 217, 9, of 34 case) C ljk. Let N C lk x < ) f l be the degree of ecessity for a fixed evet ω lk defied i 21). The, the followig optimizatio criterio is said to be ecessity-based probabilistic expectatio NPE): E N C l x < )] f l p lk N C lk x < ) f l p lk N C lk x G l ) 25) p lk if max 1 µ y C lk x y), µ G l y), l 1, 2,..., q. 6. Discrete Fuzzy Radom Liear Programmig Models Usig Possibility/Necessity-Based Probabilistic Expectatio O the basis of the ew optimizatio criteria defied as 24) or 25) i the previous sectio, we propose ew liear programmig-based decisio makig models i fuzzy stochastic eviromets Possibility-Based Probabilistic Expectatio PPE) Model Whe the DM is optimistic, it is reasoable to use the model based o PPE. The, we cosider the followig problem to maximize the probabilistic expectatio of the degree of possibility: Possibility-based probabilistic expectatio model PPE model)] maximize E Π C l x < )] f l, l 1, 2,..., q subject to x X, 26) where the objective fuctios of problem 26) are give as 24). I geeral, problem 26) is a multi-objective programmig problem. Especially i the case of q 1, 26) becomes a sigle-objective programmig problem, ad the optimal solutio is a feasible solutio which maximizes the objective fuctio. O the other had, whe q 1, the problem to be solved has multiple objective fuctios, which meas there does ot geerally exist a complete solutio that simultaeously maximizes all the objective fuctios. I such multi-objective cases, oe of reasoable solutio approaches to 26) is to seek a solutio satisfyig Pareto optimality, called a Pareto optimal solutio. We defie Pareto optimal solutios of 26). Firstly, we itroduce the cocepts of weak Pareto optimal solutio as follows: Defiitio 16. Weak Pareto optimal solutio of PPE model) x X is said to be a weak Pareto optimal solutio of the possibility-based probabilistic expectatio model if ad oly if there is o x X such that E Π C l x < )] f l > E Π C l x < )] f l for all l 1, 2,..., q. As a stroger cocept tha a weak Pareto optimal solutio, a strog) Pareto optimal solutio of 26) is defied as follows: Defiitio 17. Strog) Pareto optimal solutio of PPE model) x X is said to be a strog) Pareto optimal solutio of the possibility-based probabilistic expectatio model if ad oly if there is o x X such that E Π C l x < )] f l E Π C l x < )] f l for all l 1, 2,..., q, ad that E Π C l x < )] f l > E Π C l x < )] f l for at least oe l 1, 2,..., q.

16 Symmetry 217, 9, of 34 I order to obtai a weak/strog) Pareto optimal solutio of PPE model, we cosider the followig maximi problem, which is oe of scalarizatio methods for obtaiig a weak/strog) Pareto optimal solutio of multi-objective programmig problems 47]: Maximi problem for PPE model] maximize subject to x X. mi E Π C l x < )] f l l 1,2,...,q 27) I the theory of multi-objective optimizatio, it is kow that a optimal solutio of the maximi problem assures at least weak Pareto optimality. The, we show the followig propositio: Propositio 1. Weak Pareto optimality of the maximi problem for PPE model) Let x be a optimal solutio of problem 27). The, x is a weak Pareto optimal solutio of problem 26), amely, a weak Pareto optimal solutio for PPE model. Proof. Assume that a optimal solutio x of 27) is ot a weak Pareto optimal solutio of PPE model defied i Defiitio 16. The, there exists a feasible solutio ˆx X of 27) such that E Π C l ˆx < )] f l > E Π C l x < )] f l for all l 1, 2,..., q. The, it follows mi E Π C l ˆx < )] f l l )] > mi E Π C l x < fl. l This cotradicts the fact that x is a optimal solutio of 27). Sice a optimal solutio of 27) is ot always a strog) Pareto optimal solutio but oly a weak Pareto optimal solutio i geeral, we cosider the followig augmeted maximi problems i order to fid a solutio satisfyig strog Pareto optimality istead of weak Pareto optimality. Augmeted maximi problem for PPE model] maximize z Π x) subject to x X, mi E Π C l x < )] f l l 1,2,...,q + ρ q l1 E Π C l x < )] f l 28) where ρ is a sufficietly small positive costat, say 1 6. I the theory of multi-objective optimizatio 47], it is kow that a optimal solutio of the augmeted maximi problem assures strog) Pareto optimality. The, we obtai the followig propositio: Propositio 2. Strog) Pareto optimality of augmeted maximi problem for PPE model) Let x be a optimal solutio of problem 28). The, x is a strog) Pareto optimal solutio of 26), amely, a strog) Pareto optimal solutio for PPE model. Proof. Assume that a optimal solutio of 28), deoted by x, is ot strog) Pareto optimal solutio of PPE model. The, there exists ˆx such that

17 Symmetry 217, 9, of 34 E Π C l x < )] f l E Π C l x < )] f l E Π C l x < )] f l for all l for at least oe l 1, 2,..., q. The, it follows mi E Π C l ˆx < )] f l l 1,2,...,q ρ q l1 E Π C l ˆx < )] f l > ρ 1, 2,..., q, ad that E Π C l x < )] f l )] mi E Π C l x < fl l 1,2,...,q q l1 )] E Π C l x < fl. > Therefore, it holds that mi E Π C l ˆx < )] f l l 1,2,...,q )] > mi E Π C l x < fl l 1,2,...,q + ρ + ρ q l1 q l1 E Π C l ˆx < )] f l )] E Π C l x < fl. This cotradicts the fact that x is a optimal solutio of the augmeted miimax problem Necessity-Based Probabilistic Expectatio Model NPE Model) Ulike the case discussed i the previous sectio, whe the DM is pessimistic, the NPE model is recommeded, istead of the PPE model. This sectio is devoted to addressig how the ecessity-based probabilistic expectatio NPE) model based o 25) ca be solved i the case of liear membership fuctios. Usig the ecessity-based probabilistic mea defied i 25), we cosider aother ew decisio makig model called NPE model ad formulate the mathematical programmig problem as follows: Necessity-based probabilistic expectatio model NPE model)] maximize E N C l x < )] f l, l 1, 2,..., q subject to x X. 29) Whe q 1, 29) is a sigle-objective problem. Otherwise, amely, whe q 2, 29) is a multi-objective problem i which a solutio satisfyig strog) Pareto optimality, called a strog) Pareto optimal solutio, is cosidered to be a reasoable optimal solutio. We defie strog) Pareto optimal solutios of 29). The cocept of weak Pareto optimal solutio for NPE model is defied as follows: Defiitio 18. Weak Pareto optimal solutio of NPE model) x X is said to be a weak Pareto optimal solutio of the ecessity-based probabilistic expectatio model if ad oly if there is o x X such that E N C l x < )] f l > E N C l x < )] f l for all l 1, 2,..., q. As a stroger cocept tha weak Pareto optimal solutios, strog) Pareto optimal solutios of 29) is defied as follows: Defiitio 19. Strog) Pareto optimal solutio of NPE model) x X is said to be a strog) Pareto optimal solutio of the ecessity-based probabilistic expectatio model if ad oly if there is o x X such that E N C l x < )] f l E N C l x < )] f l for all l 1, 2,..., q, ad that E N C l x < )] f l > E N C l x < )] f l for at least oe l 1, 2,..., q.

18 Symmetry 217, 9, of 34 Scalarizatio-Based Problems for Obtaiig a Pareto Optimal Solutio I order to obtai a weak) Pareto optimal solutio of NPE model, we cosider the followig maximi problem, which is oe of well-kow scalarizatio methods for solvig multi-objective optimizatio problems: Maximi problem for NPE model] maximize subject to x X. mi E N l 1,2,...,q C l x < )] f l 3) Similar to the case of PPE model discussed i the previous sectio, we obtai the followig propositio: Propositio 3. Weak Pareto optimality of the maximi problem for NPE model) Let x be a optimal solutio of problem 3). The, x is a weak Pareto optimal solutio of 29), amely, a weak Pareto optimal solutio for NPE model. Sice the proof of Propositio 3 is very similar to that of Propositio 1, we omit its proof. Similar to the property of the optimal solutio of problem 29), a optimal solutio of 3) is ot always a strog) Pareto optimal solutio but oly a weak Pareto optimal solutio i geeral. To fid a solutio satisfyig strog) Pareto optimality istead of weak Pareto optimality, we cosider the followig augmeted maximi problem. Augmeted maximi problem for NPE model] maximize subject to x X, mi E N C l x < )] f l l 1,2,...,q + ρ q l1 E N C l x < )] f l 31) where ρ is a sufficietly small positive costat, say 1 6. The, we obtai the followig propositio: Propositio 4. Strog) Pareto optimality of augmeted maximi problem for NPE model) Let x be a optimal solutio of problem 31). The, x is a strog) Pareto optimal solutio of 29), amely, a strog) Pareto optimal solutio for NPE model. We omit the proof of Propositio 4 because it is similar to that of Propositio Solutio Algorithms 7.1. Solutio Algorithm for the PPE Model Now we discuss how to solve problem 28) i order to obtai a strog) Pareto optimal solutio for the PPE model. Here, we focus o the case where all the membership fuctios of fuzzy umbers ad fuzzy goals are represeted by liear membership fuctios. To be more specific, we restrict ourselves to cosiderig the case that the coefficiets of objective fuctio i 15) are triagular fuzzy radom

19 Symmetry 217, 9, of 34 variables defied i 11), ad that the membership fuctio of the fuzzy goal G l for the lth objective fuctio is the followig piecewise liear membership fuctio, called a liear membership fuctio: µ G l y) 1 if y < fl 1, l 1, 2,..., q y fl fl 1 fl if fl 1 y fl if y > fl, where f l ad f 1 l are parameters whose values are determied by a DM. Figure 12 shows the membership of fuzzy goal G l which is expressed by a liear membership fuctio. 32) Figure 12. Liear membership fuctio µ G l of a fuzzy goal G l. From a practical aspect, it is importat to show how to determie the values of parameters fl ad fl 1 i the liear membership fuctios 32) of fuzzy goals G l, l 1, 2,..., q. Whe a DM ca easily set the values of fl ad fl 1, l 1, 2,..., q, these values should be determied by the DM s ow idea or choice. O the other had, whe it is difficult for a DM to determie the parameter values of fuzzy goals, we recommed that the values of fl 1 ad fl are determied as follows: f 1 l f l max p lk d ljk ˆx l j r 1,2,...,q miimize p lk d ljk ˆx r j fo 1, 2,..., q, 33) where ˆx l deotes a optimal solutio of the followig lth optimizatio problem which has a sigle objective fuctio: subject to x X p lk d ljk x j fo 1, 2,..., q. 34) The above calculatio method is similar to the Zimmerma s method 48] which was origially itroduced for fuzzy o-stochastic) liear programmig. We cosider the case where the coefficiets of the objective fuctio are give as discrete triagular fuzzy radom variables. Assumig that µ C ljk ad µ G l are give by 18) ad 32), respectively, we ca show that the followig theorem holds:

20 Symmetry 217, 9, of 34 Theorem 1. Assume that C lj is a discrete triagular fuzzy radom variable whose realized values for evets are triagular fuzzy umbers characterized by 18), ad that the membership fuctio of each fuzzy goal G l is characterized by 32) ad 33). The, the possibility-based probabilistic expectatio PPE) is calculated as E Π C l x < )] f l ] p lk mi 1, max, glk Πx), l 1, 2,..., q, 35) where g Π lk x) β ljk d ljk )x j + f l β ljk x j fl 1 + fl, l 1, 2,..., q, k 1, 2,...,. 36) Proof. The calculatio of Π C lk x < ) f l is doe by dividig ito three cases, amely, 1) Case 1: If d lk x < fl 1, 2) Case 2: If f l 1 d lk x fl + γ lk x, 3) Case 3: If d lk x > fl + γ lk x. 1. Case 1: If d lk x < fl 1 the value of Π C lk x < ) f l is equal to 1, as show i Figure Case 2: If fl 1 d lk x fl + γ lk x, the value of Π C lk x < ) f l is calculated as the ordiate of the crossig poit betwee the membership fuctio of fuzzy goal G l ad the objective fuctio C lk x, as show i Figure 14. The abscissa of the crossig poit of two fuctios µ C lk x ad µ G l ) is obtaied by solvig the equatio 1 d lkx y β lk x y f l fl 1 fl, l 1, 2,..., q, k 1, 2,...,. 37) The, the solutio y Π lk y Π lk f 1 l of 37) is β ljk x j + fl f 1 l ) d ljk x j, l 1, 2,..., q, k 1, 2,...,. β ljk x j fl 1 + fl Cosequetly, the ordiate of the crossig poit is calculated as µ G l y Π lk ) µ C lk x yπ lk ) β ljk d ljk )x j + f l β ljk x j fl 1 + fl g Π lk x) ) fo 1, 2,..., q, k 1, 2,...,. 3. Case 3: If d lk x > fl + γ lk, the value of Π C lk x < ) f l is equal to, as show i Figure 15.

21 Symmetry 217, 9, of 34 Therefore, the computatioal results of the above three cases ca be itegrated ad represeted as a sigle form Π C lk x < f l ) Π C lk x G l ) 1 if d lk x < f 1 l g Π lk x) if f 1 l d lk x f l + γ lk x if d lk x > f l + γ lk x mi 1, max, glk Πx)]. Cosequetly, E Π C l x < )] f l is calculated based o Defiitio 14 as follows: E Π C l x < )] f l p lk Π C lk x < ) f l p lk Π C lk x G l ) ] p lk mi 1, max, glk Πx), l 1, 2,..., q. Figures illustrate the degrees of possibility that the fuzzy goal G l is fulfilled uder the possibility distributio µ C lk x, each of which is correspodig to Case 1, Case 2 ad Case 3, respectively. I each figure, the bold lie expresses the value of mi µ C lk x y), µ G l y). I Figure 13, the maximum of the bold lie is 1. I Figure 14, the maximum of the bold lie is betwee ad 1. I Figure 15, the maximum of the bold lie is. Figure 13. Case 1 i the proof of Theorem 1. Figure 14. Case 2 i the proof of Theorem 1.

22 Symmetry 217, 9, of 34 Figure 15. Case 3 i the proof of Theorem 1. From Propositio 2, the optimal solutio of augmeted maximi problem 28) is a strog) Pareto optimal solutio. I the case of liear membership fuctios, the augmeted maximi problem 28) for the PPE model is formulated usig 35) ad 36) as follows: Augmeted maximi problem for PPE model liear membership fuctio case)] maximize z Π x) +ρ q l1 subject to x X, mi l 1,2,...,q p lk mi p lk mi 1, max 1, max, g Π lk x) ] ], glk Πx) 38) where glk Π is give by 36), amely, g Π lk x) β ljk d ljk )x j + f l β ljk x j fl 1 + fl, l 1, 2,..., q, k 1, 2,...,, ad ρ is a sufficietly small positive costat. Now we summarize a algorithm for obtaiig a strog) Pareto optimal solutio of possibility-based probabilistic expectatio model PPE model) i the multi-objective case. A algorithm for obtaiig a strog) Pareto optimal solutio of PPE model liear membership fuctio case)] Step 1: Calculatio of possible objective fuctio values) Usig a liear programmig techique, solve idividual miimizatio problems 34) for l 1, 2,..., q, amely miimize subject to x X p lk d ljk x j fo 1, 2,..., q, ad obtai optimal solutios x l of the lth miimizatio problems fo 1, 2,..., q. Step 2: Settig of membership fuctios of fuzzy goals) Ask the DM to specify the values of fl ad fl 1, l 1, 2,..., q. If the DM has o idea of how f l

23 Symmetry 217, 9, of 34 ad fl 1, l 1, 2,..., q are determied, the the DM ca set the followig values calculated by 33) as f 1 l f l max p lk d ljk ˆx l j r 1,2,...,q mi l 1,2,...,q p lk d ljk ˆx r j fo 1, 2,..., q, usig the optimal solutios x l obtaied i Step 1. Step 3: Derivatio of a strog Pareto optimal solutio of PPE model) Usig a oliear programmig techique, solve the followig augmeted maximi problem 38): ] maximize p lk mi 1, max, glk Πx) where g Π lk x) +ρ q l1 subject to x X, p lk mi β ljk d ljk )x j + f l β ljk x j fl 1 + fl ad ρ is a sufficietly small positive costat. ] 1, max, glk Πx), l 1, 2,..., q, k 1, 2,...,, It should be oted here that 38) is a oliear programmig problem NLPP) with liear costraits i which the objective fuctio has poits at which the gradiet is ot calculated. I such a case, a certai heuristic or metaheuristic algorithm ca be used to solve the problem. Aother applicable solutio method is the Nelder-Mead method 49] which ca solve a liear-costraied NLPPs without ay iformatio o the derivative of the objective fuctio ad costraits Solutio Algorithm for the NPE Model Whe a DM is pessimistic for the attaied objective fuctio values, a ecessity-based probabilistic expectatio NPE) model is recommeded. I a maer similar to Theorem 1 which holds for the PPE model, we obtai the followig theorem with respect to the NPE model: Theorem 2. Assume that C lj is a discrete triagular fuzzy radom variable whose realized values for evets are triagular fuzzy umbers characterized by 18), ad that the membership fuctio of each fuzzy goal G l is characterized by 32) ad 33). The, the ecessity-based probabilistic expectatio defied i 25) is calculated as E N C l x < )] f l ] p lk mi 1, max, glk Nx), l 1, 2,..., q, 39) where g N lk x) d ljk x j + f l γ ljk x j fl 1 + fl, l 1, 2,..., q, k 1, 2,...,. 4)

24 Symmetry 217, 9, of 34 Proof. From the defiitio of ecessity measure, the calculatio of N C lk x < ) f l is doe by dividig by three cases, amely, 1) Case 1: If d lk x < fl 1, 2) Case 2: If f l 1 d lk x fl + γ lk x, 3) Case 3: If d lk x > fl + γ lk x. 1. Case 1: If d lk + γ lk )x < fl 1, the value of N C lk x < ) f l is equal to 1, as show i Figure Case 2: If fl 1 γ lk x d lk x fl + γ lk x, the value of N C lk x < ) f l is calculated as the ordiate of the crossig poit betwee the membership fuctios of fuzzy goal G l ad the objective fuctio C lk x, as show i Figure 17. The abscissa of the crossig poit of two fuctios µ C lk x ad µ G l ) is obtaied by solvig the equatio y d lk x γ lk x y f l fl 1 fl, l 1, 2,..., q, k 1, 2,...,. 41) The, the solutio of 41) is y N lk f l γ ljk x j + fl f 1 l ) d ljk x j, l 1, 2,..., q, k 1, 2,...,. γ ljk x j fl 1 + fl Cosequetly, the ordiate of the crossig poit is calculated as µ G l ylk N ) 1 µ C lk x yn lk ) d ljk x j + f l γ ljk x j fl 1 + fl g N lk x) ) fo 1, 2,..., q, k 1, 2,...,. 3. Case 3: If d lk x > f l, the value of N C lk x < f l ) is equal to, as show i Figure 18. The computatioal results of the three cases above ca be itegrated ad expressed as a sigle form N C lk x < f l ) N C lk x G l ) 1 if d lk x < f 1 l γ lk x g N lk x) if f 1 l γ lk x d lk x f l + γ lk x if d lk x > f l mi 1, max, g N lk x)]. Cosequetly, the ecessity-based probabilistic expectatio defied i 25) is calculated as E N C l x < )] f l p lk N C lk x < ) f l p lk N C lk x G l ) p lk mi ] 1, max, glk Nx), l 1, 2,..., q.

25 Symmetry 217, 9, of 34 Figures illustrate the degrees of ecessity that the fuzzy goal G l is fulfilled uder the possibility distributio µ C lk x, each of which is correspodig to Case 1, Case 2 ad Case 3, respectively. I each figure, the bold lie expresses the values of max1 µ C lk x y l), µ G l y l ). I Figure 16, the miimum of the bold lie is 1. I Figure 17, the miimum of the bold lie is betwee ad 1. I Figure 18, the miimum of the bold lie is. Figure 16. Case 1 i the proof of Theorem 2. Figure 17. Case 2 i the proof of Theorem 2. Figure 18. Case 3 i the proof of Theorem 2. From Propositio 4, the optimal solutio of augmeted maximi problem 31) is a strog) Pareto optimal solutio of NPE model. I the case of liear membership fuctios, the augmeted maximi problem 42) for NPE model is formulated usig 39) ad 4) as follows:

26 Symmetry 217, 9, of 34 Augmeted maximi problem for the NPE model liear membership fuctio case)] maximize z N x) +ρ q l1 subject to x X, mi l 1,2,...,q p lk mi p lk mi 1, max 1, max, g N lk x) ] ], glk Nx) 42) where glk N x) is give by 4), amely, g N lk x) d ljk x j + f l γ ljk x j fl 1 + fl, l 1, 2,..., q, k 1, 2,...,, ad ρ is a sufficietly small positive costat. Now we summarize a algorithm for obtaiig a strog) Pareto optimal solutio of the ecessity-based probabilistic expectatio model NPE model) i the multi-objective case. A algorithm for obtaiig a strog) Pareto optimal solutio of NPE model liear membership fuctio case)] Step 1: Calculatio of possible objective fuctio values) By usig a liear programmig techique, solve idividual miimizatio problems 34) for l 1, 2,..., q, amely miimize subject to x X p lk d ljk x j fo 1, 2,..., q, ad obtai optimal solutios x l of the lth miimizatio problems fo 1, 2,..., q. Step 2: Settig of membership fuctios of fuzzy goals) Ask the DM to specify the values of fl ad fl 1, l 1, 2,..., q. If the decisio has o idea of how fl ad fl 1, l 1, 2,..., q are determied, the DM could set the values calculated by 33) as f 1 l f l max p lk d ljk ˆx l j r 1,2,...,q p lk d ljk ˆx r j fo 1, 2,..., q, usig the optimal solutios x l obtaied i Step 1. Step 3: Derivatio of a strog) Pareto optimal solutio of the NPE model) Solve the followig augmeted maximi problem 42) usig a oliear programmig techique: maximize mi l 1,2,...,q +ρ q l1 subject to x X, ] p lk mi 1, max, glk Nx) p lk mi ] 1, max, glk Nx)

27 Symmetry 217, 9, of 34 where glk N x) is give by 4), amely, g N lk x) d ljk x j + f l γ ljk x j fl 1 + fl ad ρ is a sufficietly small positive costat., l 1, 2,..., q, k 1, 2,...,, Similar to the case of PPE model i the previous sectio, 42) is a oliear programmig problem with liear costraits, which ca be solved by a certai oliear programmig techique. 8. Numerical Experimets I order to demostrate feasibility ad efficiecy of the proposed model, we cosider a example of agriculture productio problems. Oe of classical approaches to crop plaig problems is stochastic programmig 1] usig several stochastic evets or scearios related to climate ad/or ecoomic coditios. However, it is sometimes difficult to defiitely estimate the exact values of the profit ad the workig time i crop plaig problems because of lack of data ad/or some factors such as huma skills. Zeg et al. 5] cosidered a fuzzy multi-objective programmig approach to a crop plaig problem. I this sectio, we apply the proposed model to solve a crop plaig problem i a fuzzy stochastic eviromet where the profit ad the workig times are give as discrete fuzzy radom variables. I order to solve the problem, we employ costroptim fuctio which is prepared as a stadard fuctio i the R laguage 33] ad is ofte used as a solver for NLPPs with liear costraits. It should be stressed here that some state-of-the-art algorithms based o heuristics or metaheuristics 51] may solve problems more efficietly. Noetheless, we do ot propose a specific solutio algorithm for the proposed model i this article, because the proposal of a specific solutio algorithm is ot a purpose of this paper. The R laguage is easy to use for may researchers ad practical persos, eve if they are ot good at writig their ow programmig codes Crop Area Plaig Problem Uder a Fuzzy Radom Eviromet Assume that a agricultural compay DM) produces 5 kids of summer vegetables bell pepper, cucumber, eggplat, tomato ad watermelo). We cosider the followig fuzzy radom LPP with bi-objective fuctios q 2), 5 decisio variables 5) ad 5 costraits m 5): maximize C 11 x 1 + C 12 x 2 + C 13 x 3 + C 14 x 4 + C 15 x 5 miimize C 21 x 1 + C 22 x 2 + C 23 x 3 + C 24 x 4 + C 25 x 5 subject to a 11 x 1 + a 12 x 2 + a 13 x 3 + a 14 x 4 + a 15 x 5 b 1 a 21 x 1 + a 22 x 2 + a 23 x 3 + a 24 x 4 + a 25 x 5 b 2 a 31 x 1 + a 32 x 2 + a 33 x 3 + a 34 x 4 + a 35 x 5 b 3 a 41 x 1 + a 42 x 2 + a 43 x 3 + a 44 x 4 + a 45 x 5 b 4 a 51 x 1 + a 52 x 2 + a 53 x 3 + a 54 x 4 + a 55 x 5 b 5 x j, j 1, 2,..., 5, where the first objective fuctio represets the profit 1 thousad ye) eared by producig ad sellig vegetables, ad the secod oe expresses the total workig time 8 hours). Let x j, j 1, 2,..., 5 deote the growig area 1 3 m 2 ) of vegetables j 1 bell pepper), j 2 cucumber), j 3 eggplat), j 4 tomato) ad j 5 watermelo), respectively. I the objective fuctio, let C 1j ad C 2j be the profit ad the workig time per uit of vegetables j 1, 2,..., 5, respectively. Assume that C 1j ad C 2j, j 1, 2,..., 5 are estimated as discrete triagular fuzzy radom variables. O the basis of the research results o relatioships betwee vegetable 43)

28 Symmetry 217, 9, of 34 diseases ad humidity 52], we assume that the umber of evets scearios) related to the 1st objective fuctio ad the secod oe are 5 r 1 r 2 5). To be more specific, the set of evets are give as Ω 1 ω 11, ω 12,..., ω 15 ad Ω 2 ω 21, ω 22,..., ω 25 as show i Table 1. Table 1. Evets related to the 1st ad 2d objective fuctios. Evet Probability Situatio ω 11 p 11.5 average aual temperature is ormal. ω 12 p average aual temperature is high. ω 13 p average aual temperature is low. ω 14 p 14.6 it happes a epidemic disease for cucurbitaceous vegetables such as cucumber ad watermelo, due to a very high-temperature. ω 15 p 15.4 it happes a epidemic disease for solaaceae vegetables such as bell pepper, eggplat ad tomato, due to a very low-temperature. ω 21 p 21.5 average aual humidity is ormal. ω 22 p 22.2 average aual humidity is high. ω 23 p average aual humidity is low. ω 24 p 24.8 it happes a epidemic disease for cucurbitaceous vegetables such as cucumber ad watermelo, caused by very low-humidity. ω 25 p 25.6 it happes a epidemic disease for solaaceae vegetables such as bell pepper, eggplat ad tomato, due to a very low-temperature. Tables 2 ad 3 show the parameter values of the realized fuzzy umbers C 1jk d 1jk, β 1jk, γ 1jk ) tri ad C 2jk d 2jk, β 2jk, γ 2jk ) tri, j 1, 2,..., 5, k 1, 2,..., 5, which characterize fuzzy radom variables C 1j ad C 2j, j 1, 2,..., 5, respectively. The values of d 1jk are give i Table 2, each of which is based o the statistical data i 27 by the Japaese Miistry of Agriculture, Forestry ad Fisheries JMAFF) 53]. The values of d 2jk are give i Table 3, each of which is based o the report of Mekoe et al. 54]. By takig ito cosideratio the degree of risk of producig differet vegetables, it is assumed that the parameter values of β ljk ad γ ljk for j 1, 2, 5 are larger tha those for j 3, 4. Table 2. Parameters of C 1jk i the 1st objective fuctio. Parameter k 1 k 2 k 3 k 4 k 5 d 11k d 12k d 13k d 14k d 15k γ 11k γ 12k γ 13k γ 14k γ 15k β 11k β 12k β 13k β 14k β 15k

29 Symmetry 217, 9, of 34 Table 3. Parameters of C 2jk i the 2d objective fuctio. Parameter k 1 k 2 k 3 k 4 k 5 d 21k d 22k d 23k d 24k d 25k β 21k β 22k β 23k β 24k β 25k γ 21k γ 22k γ 23k γ 24k γ 25k As show i problem 43), there are five costraits i the crop plaig problem. Table 4 shows the coefficiets of these costraits. The 1st costrait reflects that there is the uppeimit of the total cost of croppig, sales, etc. The uit of a 1j ad b 1 i the 1st costrait is coverted from a uit area to 1 thousad ye, based o the statistical data i 27 by JMAFF 53]. The 2d costrait ad the 3rd oe represets the uppeimit ad the loweimit of the total growig area of vegetables, respectively. The 4th ad 5th costraits represet that the agricultural compay sigs cotracts with two major customers for sellig certai amouts of specific vegetables. I these two costraits, the uit for these costraits is coverted from area to kilo gram, based o the statistical data i 27 by JMAFF 53]. Table 4. Left-had side coefficiets i costraits. LHS Value j 1 j 2 j 3 j 4 j 5 a 1j a 2j a 3j a 4j a 5j Table 5. Right-had side values i costraits. RHS Value i 1 i 2 i 3 i 4 i 5 b i 3, ,5. 7. I problem 43), the 1st objective fuctio is the profit to be maximized. Sice the algorithm proposed i Sectio 5 is valid for miimizatio problems, we trasform the maximizatio problem ito a miimizatio problem by multiplyig the origial 1st objective fuctio by 1 as follows: miimize C 11 x 1 C 12 x 2 C 13 x 3 C 14 x 4 C 15 x 5 miimize C 21 x 1 + C 22 x 2 + C 23 x 3 + C 24 x 4 + C 25 x 5 subject to a 11 x 1 + a 12 x 2 + a 13 x 3 + a 14 x 4 + a 15 x 5 b 1 a 21 x 1 + a 22 x 2 + a 23 x 3 + a 24 x 4 + a 25 x 5 b 2 a 31 x 1 + a 32 x 2 + a 33 x 3 + a 34 x 4 + a 35 x 5 b 3 a 41 x 1 + a 42 x 2 + a 43 x 3 + a 44 x 4 + a 45 x 5 b 4 a 51 x 1 + a 52 x 2 + a 53 x 3 + a 54 x 4 + a 55 x 5 b 5 x j, j 1, 2,..., 5. 44)

30 Symmetry 217, 9, of 34 I order to utilize the results obtaied i the previous sectios, we trasform maximizatio ito miimizatio of the 1st objective fuctio by settig C ij C ij as follows: miimize C 11 x 1 + C 12 x 2 + C 13 x 3 + C 14 x 4 + C 15 x 5 miimize C 21 x 1 + C 22 x 2 + C 23 x 3 + C 24 x 4 + C 25 x 5 subject to a 11 x 1 + a 12 x 2 + a 13 x 3 + a 14 x 4 + a 15 x 5 b 1 a 21 x 1 + a 22 x 2 + a 23 x 3 + a 24 x 4 + a 25 x 5 b 2 45) a 31 x 1 + a 32 x 2 + a 33 x 3 + a 34 x 4 + a 35 x 5 b 3 a 41 x 1 + a 42 x 2 + a 43 x 3 + a 44 x 4 + a 45 x 5 b 4 a 51 x 1 + a 52 x 2 + a 53 x 3 + a 54 x 4 + a 55 x 5 b 5 x j, j 1, 2,..., 5, where C ij are discrete triagular fuzzy radom variables expressed as C ijk d ijk, β ijk, γ ijk ) tri. The, the followig remark should be oted: Remark 1. Let C ad C ij be L-R fuzzy radom variables expressed as C ijk d ijk, β ijk, γ ijk ) tri ad C ijk d ijk, β ijk, γ ijk ) tri. If C ij C ij, it holds that d ijk d ijk, β ijk γ ijk, γ ijk β ijk. The, the values of parameters i the 1st objective fuctio as show i Table 2 ca be replaced by Table 6, where we use the property of triagular fuzzy radom variables described i Remark 1. Based o the algorithm proposed i the previous sectio, a Pareto optimal solutio i the crop plaig problem is obtaied. Firstly, the fuzzy goals for each objective fuctio are give, by solvig LPPs i Step 1 ad computig fl 1 ad fl fo 1, 2 i Step 2, as f1 1, f 1 ) , ) ad f2 1, f 2 ) , ). I Step 3, based o the DM s preferece, the augmeted maximi problems 38) ad/or 42) are solved, which correspods to the possibility-based probabilistic expectatio model PPE model) ad the ecessity-based probabilistic expectatio model NPE model), respectively. Sice the obtaied solutios through the fuctio i the R laguage do ot always satisfy global optimality but local optimality, we apply this fuctio to 1 iitial solutios that are radomly geerated, ad select the best solutio amog 1 local optimal solutios. Thus, we obtai the followig optimal solutios for PPE model ad NPE model: x Π 65.74, 24.25,., 4.87, 189.1), z Π x Π ).5693, x N.13, 163.8, 5.35, , ), z N x N ).4668, where x Π ad x N are optimal solutios of 38) ad 42), respectively, ad z Π x Π ) ad z N x N ) are their objective fuctio values, respectively. From the computatioal results, the possibility-based probabilistic expectatio model PPE model) teds to crop high-risk high-retur vegetables such as bell pepper, cucumber ad watermelo, ad few areas are assiged to other vegetables. O the other had, the ecessity-based probabilistic expectatio model NPE model) has a tedecy to icrease the croppig areas of low-risk low-retur vegetables such as tomato ad to decrease those of high-risk low-retur vegetables such as bell pepper.

31 Symmetry 217, 9, of 34 Table 6. Parameters of C 1jk C 1jk ) i the 1st objective fuctio. Parameter k 1 k 2 k 3 k 4 k 5 d 11k d 12k d 13k d 14k d 15k β 11k β 12k β 13k β 14k β 15k γ 11k γ 12k γ 13k γ 14k γ 15k Computatioal Times for Differet Size Problems As we metioed before, we do ot propose a specific solutio algorithm for the proposed model, because it may take much time ad efforts for researchers or practical persos to write programmig codes eve if we propose ew solutio algorithms. Istead of proposig a specific solutio algorithm, we use the R laguage ad show the R laguage ca solve problems with hudreds of decisio variables i a practical computatioal time. We expect that the use of the R laguage ca promote the use of our model for solvig real-world problems i fuzzy stochastic eviromets. I order to show the applicability of the R laguage to our model i terms of computatioal time, we coduct additioal experimets usig 5 umerical examples i which the umber of decisio variables ad that of costrait are differet. To be more precise, the umbers of decisio variables i 7 examples are 1, 3, 6, 1, 15, 2, 25, respectively. The umber of costraits i each example is set to be the half umber of decisio variables. To focus o the effect of the umber of decisio variables ad that of costraits, the umber of the objective fuctios ad that of evets scearios) are fixed. To be more specific, we fix the umber of the objective fuctios ad that of evets scearios) as 5 q 5) ad 1 1, l 1, 2,..., 5), respectively. The values of parameters d ljk are radomly chose i 5, 4,..., 5, β ljk ad γ ljk are the absolute values of the products of d ljk ad values radomly chose i.1,.2] fo 1, 2,..., 5, j 1, 2,...,, k 1, 2,..., 1. As for the costraits, the values of a ij i matrix A are radomly chose i 1, 2,..., 1, ad the values of b i are give as the sum of elemets i a i for ay i 1, 2,..., m, j 1, 2,...,. Similar to the experimet i Sectio 8.1, we used costroptim fuctio i the R laguage ad coducted 3 rus i which iitial solutios are radomly geerated. We coducted this umerical experimet usig R versio 3.2. o imac OS X Yosemite versio 1.1.3, CPU: 3.4 GHz Itel Core i7, RAM: 32 GB 16 MHz DDR3). Table 7 shows computatioal times for solvig 7 problem istaces. Table 7. Computatioal times for differet size problems. No. of Decisio Variable CPU Times s) Figure 19 shows the relatioship betwee the umber of decisio variables ad computatioal times obtaied i Table 7. It is show from this graph that the computatioal time liearly icreases as the umbers of decisio variables ad costraits icrease.

32 Symmetry 217, 9, of 34 Figure 19. Relatioship betwee the umber of decisio variables ad computatioal times. 9. Coclusios I this paper, we have cosidered LPPs i which the coefficiets of the objective fuctios are discrete fuzzy radom variables. Icorporatig possibility ad ecessity measures ito a probability measure, we have proposed ew decisio makig models i fuzzy stochastic eviromets, called possibility/ecessity-based probabilistic expectatio model PPE/NPE model), which is to maximize the expectatio of the degree of possibility or ecessity that the objective fuctio values satisfy the give fuzzy goals. It has bee show that the formulated problems based o the proposed models ca be trasformed ito determiistic oliear multi-objective) programmig problems, especially, ito more simple problems whe liear membership fuctios are used. I additio, we have defied strog) Pareto optimal solutios of the proposed models i the multi-objective case, ad proposed a algorithm for obtaiig a solutio satisfyig strog) Pareto optimality. I order to show how the proposed model ca be applied to real-world problems, we have coducted a umerical experimet with a agriculture productio problem. We also have demostrated that a stadard fuctio i the R laguage is applicable to solve the problems with hudreds of decisio variables i a practical computatioal time. I the ear future, we will show a geeralized variace miimizatio model which is a exteded versio discussed i the previous study 32]. Furthermore, some applicatios of the proposed models to real-world problems will be discussed elsewhere. Ackowledgmets: This work was supported by JSPS KAKENHI Grat Number 17K1276. Author Cotributios: Hideki Katagiri proposed the method ad wrote the paper. Kosuke Kato ad Takeshi Uo aalyzed the validity of the method ad performed the umerical experimets. Coflicts of Iterest: The authors declare o coflict of iterest. Refereces 1. Birge, J.R.; Louveaux, F. Itroductio to Stochastic Programmig, 2d ed.; Spriger: Berli/Heidelberg, Germay, Datzig, G.B. Liear programmig uder ucertaity. Maag. Sci. 1955, 1, Aiche, F.; Abbas, M.; Dubois, D. Chace-costraied programmig with fuzzy stochastic coefficiets. Fuzzy Optim. Decis. Mak. 213, 12, Katagiri, H.; Sakawa, M. Iteractive multiobjective fuzzy radom programmig through the level set-based probability model. If. Sci. 211, 181,

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