Available online at www.sciencedirect.com Chaos, Solitons and Fractals 36 (2008) 217–225 www.elsevier.com/locate/chaos Fuzzy upper bounds and their applications M. Soleimani-damaneh * Department of Mathematics, Faculty of Mathematical Science and Computer Engineering, Teacher Training University, 599 Taleghani Avenue, Tehran 15618, Iran Accepted 7 June 2006 Communicated by Prof. B.G. Sidharth Abstract This paper considers the concept of fuzzy upper bounds and provides some relevant applications. Considering a fuzzy DEA model, the existence of a fuzzy upper bound for the objective function of the model is shown and an effective approach to solve that model is introduced. Some dual interpretations are provided, which are useful for practical purposes. Applications of the concept of fuzzy upper bounds in two physical problems are pointed out. Ó 2006 Elsevier Ltd. All rights reserved. 1. Introduction In recent years, fuzzy systems have been used in a variety of problems ranging from quantum optics and gravity [18], particle systems [19–21,26], economical systems [13,25], medicine [1,3], to bioinformatics and computational biology [5,8]. A fuzzy function is a generalization of a classical function, and different features of the classical concept of a function can be considered to be fuzzy rather than crisp. Different degrees of fuzzification of the classical concept of a function were reviewed by Zimmermann [31], while we consider a hybrid concept of these types as a fuzzifying function. Also the concept of the upper bound of a fuzzy function is a generalization of the classical concept of upper bound in crisp analysis. Analogously, a fuzzy upper bound notion for fuzzy functions can be introduced, after proposing a signed distance. Nowadays evaluation of decision making units (DMUs), by using the mathematical programming-based techniques, has allocated to itself a wide variety of research in operations research (OR) field. Data envelopment analysis (DEA) is one of those techniques and was first proposed by Charnes et al. [6]. One can find several fuzzy mathematical programming-based approaches to evaluate DMUs in the DEA literature. Kao and Liu [14,15] use the notion of fuzziness and they transform a fuzzy DEA model to a family of crisp DEA models by applying the a-cut approach. Sengupta [23] analyzes the results of fuzzy DEA models by Zimmermann’s method [30]. Hougaard [12] suggests the application of efficiency score aggregated by value judgments or manager opinions. Guo and Tanaka [11] extend the CCR efficiency score in the crisp case to the fuzzy number and they consider the relationship between DEA and regression analysis. * Tel.: +98 21 2093947; fax: +98 21 7507772. E-mail addresses: soleimani_d@yahoo.com, m_soleimani@tmu.ac.ir. 0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.06.042 218 M. Soleimani-damaneh / Chaos, Solitons and Fractals 36 (2008) 217–225 Jahanshahloo et al. [13], Soleimani-damaneh et al. [25], Léon et al. [16], and Lertworasirikul et al. [17] develop some models using possibilistic programming. In the present paper, we consider the additive model as a free coordinate, orientation-less, and slack-based model in crisp DEA and extend this model to be a fuzzy DEA model using a fuzzy signed distance and fuzzy upper bound concepts. We use both primal envelopment model and dual multiplier model. Some results are obtained with respect to both computational and interpretation aspects. The rest of this paper unfolds as follows: Section 2 contains some basic notions and results about fuzzy numbers and fuzzy upper bounds. Section 3 extends the additive model to be a fuzzy model, using the fuzzy input-output data. In this section the existence of fuzzy upper bound for the objective function of the related fuzzy model is established. Section 4 shows that the primal and dual form of the respective model can be interpreted from a standard DEA view. Section 5 exhibits the abilities of the proposed approach. Finally, in Section 6 some applications of the concept of fuzzy upper bounds are provided in two physical problems. 2. Fuzzy numbers and fuzzy upper bounds e : X ! ½0; 1. The support of A; e supp A, e is the closure of the set A fuzzy set on a set X is a classical function A e e : R ! ½0; 1 on R, satisfying three conditions: (a) A e is an upper semi fx 2 X j AðxÞ > 0g. A fuzzy number is a fuzzy set A e is a compact interval, and (c) if supp A e ¼ ½a; b, then there exist c, d, such that continuous function on R, (b) supp A e is non-decreasing on the interval [a, c], equal to 1 on the interval [c, d], and non-increasing on a 6 c 6 d 6 b and A e denoted by ½ A e , is the interval [d, b]. The a-cut set of A, a n o e e ½ Aa ¼ x 2 Rj AðxÞ P a ð1Þ e ¼ supp A. e The lower and upper endpoints of any a-cut set, ½ A e , are represented by ½ A eL for each a 2 (0, 1], while ½ A 0 a a U e and ½ Aa , respectively. Also let F ðRÞ be the family of fuzzy numbers on R. e and B e as follows: In this paper we define the weighted signed distance of A Z 1   ~ A; e U  ½ B e BÞ e L þ ½ A e U da; e L  ½ B e ¼ ð2Þ SðaÞ ½ A dð a a a a 0 where S(a) is a reducing function that allows the decision maker to decrease the influence of the lower a-cuts of fuzzy numbers and to adjust the influence of the upper levels as is deemed appropriate, see [27, p. 354]. Now, we define the ranking system on F ðRÞ as ~ A; ~ A; ~ A; e if dð e BÞ e if dð e BÞ eB e BÞ eA e > 0; eA e < 0; e if dð e ¼ 0: B B and A In fact, d~ is an extension of a distance introduced in [28]. Now, we extend the mean value concept, introduced by Carlsson and Fuller [4], to a weighted crisp possibilistic mean value, using the reducing function, as follows: Z 1   e L þ ½ A e U da: e ¼ ð3Þ SðaÞ ½ A Mð AÞ a a 0 Thus the following relation holds: ~ A; e  BÞ e B e BÞ e 8 A; e 2 F ðRÞ: e ¼ Mð A dð ð4Þ e ðB eA e Therefore the reason why we use d~ ordering for ranking the fuzzy numbers can be explained as B e or B e if the weighted crisp possibilistic mean value of A eB e  AÞ e is positive (negative or zero), which is an extension A of the crisp notion. We consider a fuzzy function f~ : D ! S where f~ is a fuzzy function from a fuzzy domain, D, into a fuzzy range, S. The upper bound (lower bound) concept of a fuzzy function (of the range of a fuzzy function) can be described as a generalization of that in the crisp case. Grzegorzewski [10] has defined a concept as ‘‘upper (lower) horizon’’ of a given set of fuzzy numbers. He has used this concept to provide a metric space and an order on the fuzzy numbers. Nieto [22] used the upper bound concept for solving fuzzy differential equations. Unlike Grzegorzewski [10] we define the fuzzy upper bound notion using the distance introduced in the previous section. g is a distanceDefinition 1. Let f~ : X ! F ðRÞ be a fuzzy function from X  F ðRÞ into F ðRÞ. The fuzzy number UB ~ based upper bound for f if ~ UB; g AÞ e P 0; dð e 2 f~ ðX Þ; 8A where d~ is as defined in (2). ð5Þ M. Soleimani-damaneh / Chaos, Solitons and Fractals 36 (2008) 217–225 219 For a more plausible use, we can also define the upper bound concept from a possibilistic point of view. Possibility e and B e are two theory was first formulated by Zadeh [29] and has been developed by many scholars. Suppose that A e<B e is a fuzzy event (see [17,29]) with the following possibility measure: fuzzy numbers (variables), then A e < BÞ e e ¼ supt;z2R fminð AðtÞ; e possð A BðzÞÞjt < zg: e6B e¼B e and A e can be defined in the same way. After this brief introduction The possibilities of the fuzzy events A about the possibility theory, we define the possibilistic upper bound concept as follows: g is a possibilistic Definition 2. Let f~ : X ! F ðRÞ be a fuzzy function from X  F ðRÞ into F ðRÞ. The fuzzy number UB upper bound for f~ if g < AÞ e ¼ 0; possð UB e 2 f~ ðX Þ: 8A The following theorem shows a relationship between the above definitions. g is a distance-based upper bound if it is a possibilistic upper bound. Theorem 1. UB g as a possibilistic upper bound, then Proof. Consider UB n o g e e 2 f~ ðX Þ: AðzÞÞjt < z ¼ 0; 8 A sup minð UBðtÞ; t;z2R Hence for all a 2 (0, 1], we get e AðzÞ ¼ 0; this results in g L; 8A e 2 f~ ðX Þ 8z > ½ UB a g L 6 ½ UB g U; e U 6 ½ UB e L 6 ½ A ½ A a a a a and this completes the proof. h e 2 f~ ðX Þ 8a 2 ð0; 1; 8 A The following example shows that the reverse of the above theorem is not necessarily correct. Example 1. Suppose that f~ : F ðRÞ ! F ðRÞ has been defined by f~  ð2; 1; 1Þ, as a constant fuzzy function, where (2, 1, 1) is a triangular fuzzy number. It is evident that (3, 2, 1) is a distance-based upper bound for f~ , while it is not a possibilistic upper bound for f~ . In this paper we focus on the distance-based notion as an extension of the crisp upper bound notion, and we solve a fuzzy DEA model using it. 3. A fuzzy DEA model Let us assume that we have a set of DMUs consisting of DMUj, j = 1, . . . , n, with fuzzy input–output vectors ð~xj ; ~y j Þ, in which ~xj 2 ðF ðRÞP0 Þm and ~y j 2 ðF ðRÞP0 Þs where F ðRÞP0 is the family of all non-negative fuzzy numbers. A fuzzy e is named a non-negative fuzzy number if ½ A e L P 0. Now we consider the additive model as a free coordinate, number A 0 orientation-less, and slack-based model in crisp DEA and extend this model to be a fuzzy DEA model. There are several types of additive models, from which we select the following: m s X X ~s ~sþ max þ ðM1Þ i r i¼1 s:t: n X r¼1 kj~xij þ ~s xio ; i ¼~ i ¼ 1; . . . ; m; ð6Þ kj ~y rj  ~sþ y ro ; r ¼ ~ r ¼ 1; . . . ; s; ð7Þ j¼1 n X j¼1 n X kj ¼ 1; j¼1 kj P 0; ~s sþ i ;~ r 2 F ðRÞP0 for all indices: 220 M. Soleimani-damaneh / Chaos, Solitons and Fractals 36 (2008) 217–225 DMUo is DMU under assessment. The ideal purpose of model (M1) is to make the value of the objective function g as much as possible. Therefore, we should find the values of k; ~s , and ~sþ to make approach an upper bound, UB,   Pm  Ps þ g as much as possible. To this end, we must minimize d~ UB; g P ~s þ P ~sþ , under feasi þ r¼1~sr approach UB i¼1~ i i r r sibility. Hence we consider the following crisp model. Z 1 m s X   U X  þ U g U þ ½ UB gL ~si a  ~sr a Z o ¼ min SðaÞ ½ UB a a 0 Z s:t: ! i¼1 m s X   L X  þ L ~si a  ~sr a da  i¼1 1 SðaÞ 0 (  X  U U L kj ½~xij a þ ½~xij La þ ½~s s xio U i a þ ½~ i a  ½~ a j ) 1 SðaÞ 0 ( ½~y ro La n X ðM2Þ r¼1 ½~xio La da ¼ 0; Z r¼1 X ) i ¼ 1; . . . ; m; U L kj ð½~y rj U y rj La Þ  ½~sþ sþ y ro U a þ ½~ r a  ½~ r a  ½~ a j da ¼ 0; r ¼ 1; . . . ; s; kj ¼ 1; j¼1  L  þ L sr 0 P 0 for all indices: kj P 0; ~s i 0; ~ Definition 3. DMUo is an inefficient DMU if Zo = 0; otherwise this DMU is named -inefficient, if 0 < Zo 6 , where  is a user-specified constant, exhibiting the acceptable tolerance value of inefficiency. g The following theorem demonstrates the existence of What remains to wrap up the discussion is determining the UB. that. e ¼ Theorem 2. M Pm Pn i¼1 þ xij j¼1 ~ f~ : Po  Rn m Ps Pn is a distance-based upper bound for y rj j¼1 ~ r¼1 s ðF ðRÞÞ ðF ðRÞÞ ! F ðRÞ m s X X ~s ~sþ ðk; ~s ; ~sþ Þ7! i þ r i¼1 r¼1 for each o 2 {1, 2, . . . , n}, where Po is the feasible set of (M1), corresponding to DMUo. P P e ¼ m ~s þ s ~sþ ¼ f~ ðk; ~s ; ~sþ Þ 2 f~ ðPo Þ, then Proof. Consider an arbitrary o 2 {1, 2, . . . , n} and suppose that A i¼1 i r¼1 r ( ) Z 1 n X  U   L kj ð½~xij U xij La Þ þ ~s si a  ½~xio U SðaÞ xio La da ¼ 0; 8i a þ ½~ i a þ ~ a  ½~ 0 j¼1 and this results in ( ) Z 1 Z m m X   U X   L ~si a þ ~si a da 6 SðaÞ 0 i¼1 1 0 i¼1 6 Z 1 ( ) m m X X U L SðaÞ ½~xio a þ ½~xio a da SðaÞ 0 ( i¼1 m X i¼1 i¼1 n X ½~xij U a j¼1 ) n m X X L þ ½~xij a da: i¼1 j¼1 Also Z 1 SðaÞ 0 ( n X j¼1 kj ð½~y rj U a þ ½~y rj La Þ  U ½~sþ r a  L ½~sþ r a  ½~y ro U a  ½~y ro La ) da ¼ 0; 8r ð8Þ 221 M. Soleimani-damaneh / Chaos, Solitons and Fractals 36 (2008) 217–225 and this results in ( ) Z 1 Z s X n s X X L þ U þ L SðaÞ kj ð½~y rj U þ ½~ y  Þ  ½~ s   ½~ s  da ¼ rj a a r a r a 0 r¼1 j¼1 1 SðaÞ 0 r¼1 ( P ) s s X X L ½~y ro U þ ½~ y  da: ro a a r¼1 ð9Þ r¼1 L L ¼ 1; kj P 0, and ½~s sþ i 0 ; ½~ r 0 P 0; 8i; r; j. Now we get (   Z 1 X U X L XX L XX XX U XX e ¼ e;A ½~s ½~s ½~y rj La  ½~xij a þ ½~y rj U ½~xij a þ SðaÞ d~ M i a i a  a þ Furthermore j kj 0 i r j j ) Z X U X L þ ½~ s  ½~sþ   da P  r a r a r ¼ Z 1 0 ( 1 0 r ) ( i r j i i j X U XX XX L ½~sþ sþ ½~y rj La  ½~y rj U SðaÞ r a  ½~ r a a þ r r j r j ) X X ½~y ro La da P 0; ½~y ro U SðaÞ a þ r r using (8), (9), and the constraint pletes the proof. h P ~ M e P 0 for each A e 2 f~ ðPo Þ and this come ; AÞ kj ¼ 1 and kj P 0; 8j. Therefore dð 4. Equivalent standard DEA model and its dual interpretation Using some simple changes of variables, model (M2) reduces to the following model: ! Z 1 Z 1   X U X L X U X L þ þ   g U þ ½ UB g L ¼ max ½~ s  ½~ s  þ ½~ s  þ ½~ s  þ SðaÞ SðaÞ ½ UB da  Zo þ r a r a i a i a a a 0 0 s:t: i r i Constraints of model ðM3Þ r ðM2Þ: Considering Z 1   L U s da; SðaÞ ½~s s i a þ ½~ i a i ¼ 0 sþ r ¼ xij ¼ Z Z 1 0 1 0 y rj ¼ p¼ Z Z 0 1 0 1   L U da; SðaÞ ½~sþ sþ r a þ ½~ r a   da; SðaÞ ½~xij La þ ½~xij U a   da; SðaÞ ½~y rj La þ ½~y rj U a   g U þ ½ UB g L da; SðaÞ ½ UB a a we can rewrite model (M3) as, X X s sþ Z o þ p ¼ max i þ r i s:t: X ðM4Þ r kj xij þ s xio ; i ¼ i ¼ 1; . . . ; m; kj y rj  sþ y ro ; r ¼  r ¼ 1; . . . ; s; j X j X kj ¼ 1; kj ; s sþ i ; r P 0; for all indices: j It is evident that this model is a crisp standard additive DEA model and can be solved by standard DEA solvers. Also we can classify DMUs using this model. P  P þ si þ rsr ¼ 0 in model (M4); otherwise DMUo is an -efficient DMU if Definition o is efficient if i P  4.PDMU 0 < isi þ rsþ r 6  in this model. 222 M. Soleimani-damaneh / Chaos, Solitons and Fractals 36 (2008) 217–225 This definition is similar to the crisp case and regarding a theorem in standard crisp DEA (see [7, Chapter 4]), it is obvious that there exists at least one efficient DMU. Also for each DMUo under assessment, there exist some efficient
P P projection points: ^~xo ; ^~y o ¼ xj ; j kj ~y j . The proof of this assertion is similar to that of Theorem 4.5 in [7]. See j kj ~ Section 5 for a numerical illustration of these matters. Considering vi, ur, and u as dual variables of model (M4), the dual of this model is as follows: X X ur y ro  u min vi xio  s:t: X ðDM4Þ r i vi xij  X ur y rj  u P 0; j ¼ 1; . . . ; n; r i vi P 1; ur P 1; u is free: Using the definitions of y rj ; xij and (2), this model is equivalent to ! X X vi~xio ur ~y ro þ u; max d~ s:t: d~ r i X X r ur ~y rj þ u; i vi~xij ! 6 0; ðM5Þ j ¼ 1; . . . ; n; vi P 1; ur P 1; u is free: ~ a; ~ As a generalization of real numbers we can consider dð~ bÞ 6 0 and ~ b 2 F ðRÞ>0 equivalent to ~a~b 61. Hence considering ~a ~ ~ the maximization of dð~a; bÞ as equivalent to the maximization of ~b is reasonable note that ~a~b 6 1 . Therefore (M5) can be interpreted as P y þu r ur ~ P ro ðM6Þ max xio i vi ~ P y þu r ur ~ P rj s:t: 6 1; j ¼ 1; . . . ; n; xij i vi ~ ur P 1; vi P 1; u is free: This model is similar to the ratio model in the basic literature of DEA and considers the efficiency (inefficiency) as a ratio of virtual output to virtual input. u refers to the variable returns to scale assumption of technology. Indeed, attending to (DM4), (M5), and their other versions as multiplier fuzzy DEA models can be very useful. These models are useful to treat the weights restrictions (WRs) in fuzzy DEA models and related discussions, see [7, Chapter 6] for details about WRs in DEA. These models can also be effective to determine returns-to-scale (RTS) in fuzzy DEA, see [2,7, Chapter 5] for further details about RTS in DEA. In brief, these models and the subjects mentioned (WRs and RTS) can be considered for further research. 5. Properties of the provided approach and illustrative example In this section we address the abilities of the proposed method as well as its advantages. Also, to illustrate how the proposed method is applied to find the efficiency position and the efficiency pattern, a numerical example is considered. As mentioned in the previous section, the present approach can be used to capture almost all types of input–output data such that the related defining integrals of y rj s, xij s, and p are well-known, while, it cannot be seen in some existing approaches, e.g., [11,13,16,17]. Note that for LR-fuzzy number these integrals are well-known. The proposed method determines the efficiency position of DMUs, using a standard DEA model, (M4), and hence it can be utilized by available DEA solvers. In fact, determination of the efficiency position of DMUs by the present approach requires solving only one LP, while some customary approaches need to solve many LPs [11,13,14,17], and some others increase the number of the constraints of models largely [16]. The introduced approach preserves the duality property, an important property of optimization models, which helps us in some respects: obtaining a projection point as a suggested benchmark (see [7]); incorporating weights restrictions in multiplier model (DM4); determination of super-efficiency, returns-to-scale; etc. But almost none of the references in fuzzy DEA literature has the concept of duality been addressed. 223 M. Soleimani-damaneh / Chaos, Solitons and Fractals 36 (2008) 217–225 Table 1 DMUs Input-1 Input-2 Output-1 Output-2 A B C D E F G H I (16, 17, 1, 2) (8, 11, 4, 1) (14, 16, 7, 1) (11, 13.5, 5.5, 1) (16, 18, 7, 1) (13.5, 15.75, 6.25, 1) (8, 18, 5, 1) (8, 12, 3, 1) (6, 9, 4, 1) (40, 42, 2, 2) (14, 18, 2, 4) (11, 13, 1, 1) (12.5, 15.5, 1.5, 2.5) (46, 48, 2, 4) (29.25, 31.75, 1.75, 3.25) (15, 16, 2, 2) (16, 16, 2, 4) (12.5, 20, 2, 5) (30, 31, 4, 1) (40, 44, 2, 2) (10, 12, 4, 1) (25, 28, 3, 1.5) (42, 44, 2, 2) (33.5, 36, 2.5, 1.75) (40, 42, 2, 4) (40, 46, 4, 2) (40, 44, 1, 1) (16, 18, 2, 1) (75.5, 79.5, 5.5, 4) (86, 88, 7, 1) (81, 82, 3, 5.5) (14.5, 16, 1.5, 4.5) (47.6, 49.87, 3.87, 3.5) (76, 79.5, 5, 3) (78, 79.5, 6, 4) (74, 79.5, 5.5, 4.5) For a numerical illustration, consider nine DMUs with data shown in Table 1. The input–output data are trapezoidal fuzzy numbers and S(a) = a is assumed. In this example the upperbound (anti-ideal) is e ¼ ð1145:85; 1249:37; 122:87; 85Þ M and hence p  1191:30: For instance, for DMUD model (M4) is as follows:  þ   Z D þ p ¼ max s 1 þ s2 þ s1 þ s2 s:t: 16:7k1 þ 9k2 þ 14k3 þ 11:5k4 þ 16k5 þ 13:75k6 þ 12:3k7 þ 9:7k8 þ 7k9 þ s 1 ¼ 11:5; 41k1 þ 16:3k2 þ 12k3 þ 14:2k4 þ 47:3k5 þ 30:7k6 þ 15:5k7 þ 16:3k8 þ 16:75k9 þ s 2 ¼ 14:2; 30k1 þ 42k2 þ 10:5k3 þ 26:25k4 þ 43k5 þ 34:6k6 þ 41:3k7 þ 42:7k8 þ 42k9  sþ 1 ¼ 26:25; 16:8k1 þ 77:25k2 þ 86k3 þ 81:9k4 þ 15:75k5 þ 48:7k6 þ 77:4k7 þ 78:4k8 þ 76:7k9  sþ 2 ¼ 81:9;  kj P 0; j ¼ 1; . . . ; 9; ~si ; ~sþ i P 0; i ¼ 1; 2: In this model the technological coefficients are calculated from the formulae of xij s and y rj s, introduced in Section 5. One of the optimal solutions to this model is ðk1 ¼ k2 ¼ k4 ¼ k5 ¼ k6 ¼ k7 ¼ 0; k3 ¼ 0:5023; k8 ¼ 0:3643; sþ s zD þ p ¼ 0:2595Þ: s 2 ¼ 0; 1 ¼ 2 ¼ 0:0777; k9 ¼ 0:1333; sþ 1 ¼ 0:1818; This means that DMUD is not efficient, and one of its efficiency patterns (projection points) is as follows: n X ^~x1D ¼ kj ~x1j ¼ ð7:7326; 11:097; 3:635; 1Þ; j¼1 ^~x2D ¼ n X kj ~x2j ¼ ð14:5273; 17:5362; 2; 4:139Þ; j¼1 ^~y 1D ¼ n X kj ~y 1j ¼ ð40; 44:7242; 2:5951; 0:9999Þ; j¼1 ^~y 2D ¼ n X kj ~y 2j ¼ ð75:1104; 79:5; 3:4213; 4:6625Þ; j¼1 which is an efficient point. Corresponding to DMUD, the distance between the objective function value of (M2) and the anti-ideal is 224 M. Soleimani-damaneh / Chaos, Solitons and Fractals 36 (2008) 217–225 Z D ¼ p  ðthe optimal value of ðM4Þ ¼ p  0:2595 ¼ 1191:04: Furthermore, DMUs B, C, E, G, H, and I are efficient units, i.e., the optimal value of the objective function of (M4), related to them, is zero. And the optimal value of that related to DMUs A and F are 106 and 56.25, respectively. The projection points, related to both these DMUs, are DMUH. Note that in the optimal solution of (M4) related to both DMUs A and F we have k8 ¼ 1 and kj ¼ 0; j 6¼ 8. 6. Applications in physics and engineering Sections 3–5 of this paper discuss a DEA model which has many applications for interpreting the productivity of the complex economical and engineering systems. In this section some other applications of the concept of fuzzy upper bounds are provided concisely. In fact two physical applications are dealt with. (i) Determining the maximal flow in a network is a main and known problem in some applied fields of research including physics, optimization, graph theory, etc. As can be seen in [24], this results in a time-continuous linear programming problem. Some authors have surveyed the solvability of this problem in a fuzzy time-continuous framework [24]. Similar to the manner used in Sections 3–5 one can use the possibilistic upper bound concept provided in this paper to solve this problem. Since the manner is similar to that used in the previous section, we do not peruse it here. (ii) The second application discusses the Ritz variational method [9] during the calculation of the ground-state energy in a fuzzy framework. Consider a Hamilton H, and an arbitrary square integrable function W, so that hWjWi = 1. Considering W as a fuzzy function and the ranking system as defined in Section 2, similar to [9] it can be shown that hWjHjWi is a fuzzy upper bound on E0 (ground-state energy). Now hWjHjWi should be minimized with respect to a number of parameters (a1, a2, . . .). This can be done by minimizing the distance between E0 and hWjHjWi. The rest of the discussion is the same as that provided in Sections 3–5 and hence is not pursued here. 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