Fuzzy linear programming under interval uncertainty based on IFS representation

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Fuzzy Sets and Systems 188 (2012) 68 – 87 www.elsevier.com/locate/fss

Fuzzy linear programming under interval uncertainty based on IFS representation Dipti Dubey∗ , Suresh Chandra, Aparna Mehra Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi 110016, India Received 11 June 2010; received in revised form 19 September 2011; accepted 20 September 2011 Available online 24 September 2011

Abstract The equivalence between the interval-valued fuzzy set (IVFS) and the intuitionistic fuzzy set (IFS) is exploited to study linear programming problems involving interval uncertainty modeled using IFS. The non-membership of IFS is constructed with three different viewpoints viz., optimistic, pessimistic, and mixed. These constructions along with their indeterminacy factors result in S-shaped membership functions in the fuzzy counterparts of the intuitionistic fuzzy linear programming models. The solution methodology of Yang et al. [45], and its subsequent generalization by Lin and Chen [33] are used to compute the optimal solutions of the three fuzzy linear programming models. © 2011 Elsevier B.V. All rights reserved. Keywords: Fuzzy linear programming; Interval uncertainty; Intuitionistic fuzzy sets; Intuitionistic fuzzy goals; S-shaped membership function

1. Introduction Fuzzy set (FS) theory [46] has been extensively used to capture linguistic uncertainty in decision making problems particularly in optimization problems. In fuzzy sets, the membership degree of an element in [0, 1] expresses the degree of belongingness of an element to a fuzzy set. However, no objective procedure is available for the experts to assign the crisp membership degrees to the elements in a FS. In such a case, it is more reasonable to represent the membership degree of each element to the FS by means of an interval. It was suggested by Zadeh [47] to alleviate the fuzzy set theory to the interval-valued fuzzy set theory. An IVFS is a fuzzy set in which the membership degree is assumed to belong to an interval. On the other hand, in early 1980s, Atanassov [2] introduced another extension of Zadeh’s FS namely the intuitionistic fuzzy set. IFS assigns to each element of the universe both a degree of membership and the degree of non-membership, which are more or less independent, related only by the constraint that the sum of two degrees must not exceed one. Subsequently, in [2,3,5,6] Atanassov developed the IFS theory. Although the two extensions, IVFS and IFS, were introduced independently yet surprisingly the two have been shown to be equivalent first in [7] and later in [12,15,18]. This framework forms the background of our present study wherein we shall be viewing the interval uncertainty (in the IVFS) using IFS representation.

∗ Corresponding author.

E-mail addresses: [email protected] (D. Dubey), [email protected] (S. Chandra), [email protected] (A. Mehra). 0165-0114/$ - see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2011.09.008

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Last two decades had seen the IFS theory crossing some milestones with its own share of controversy regarding its nomenclature [4,6,18,22]. The conflict has been mainly due to the already established field of intuitionistic logic (IL). Atanassov’s IFS theory and the IL differ in their mathematical structure and treatment thereby making it confusing to use the same terminology for two different concepts. But seeing the statistical data of research publications and defended theses in the IFS theory, Atanassov [4] pointed out that the change of name would lead to a terminological chaos. However, Dubois et al. [18] disagreed with his arguments stating that it would be crucial and helpful for further development of Atanassov’s IFS theory to refrain from this name. They also suggested to call intuitionistic fuzzy sets as Atanassov’s intuitionistic fuzzy sets or just the bipolar fuzzy sets. A bipolar fuzzy set is a pair of fuzzy sets, one of which represent positive and the other represents the negative aspects of the given information, see [19,20]. Some other extensions of fuzzy set theory have also been reported in literature, for example, vague sets, gray sets, to name a few, but all these concepts turned out to be equivalent to Atanassov’s intuitionistic fuzzy sets or bipolar sets. For detailed links between these objects and adequate theorems which present the isomorphisms between them, one may refer to [15,16]. However, till now there is no consensus among scientists on final name for IFS. The researchers continue to use the name IFS, and some of the recent publications [5,25,26,28,31,42–44] are strong evidence for it. In the same spirit, we shall henceforth be using the same name IFS to be understood in the sense of Atanassov’s IFS. Despite certain difficulties, the IF sets have already been used for (in alphabetic order of applications) clustering [43], medical diagnosis [13], multicriteria decision making [26–31,35,42], pattern recognition [25,41], to name a few. Besides these, there is a rich theory associated with IFS, for instance, refer to [3,10,12,14,17] and the references there in. The first serious attempt to use IFS in optimization problems was made by Angelov [1] who proposed an optimization model by considering degrees of rejection of objective(s) and constraints together with their degrees of acceptance. He formulated an intuitionistic fuzzy optimization (IFO) model by adopting the approach of maximizing the degree of acceptance of intuitionistic fuzzy (IF) objective(s) and of constraints and minimizing the degree of rejection of IF objective(s) and constraints. Subsequently he formulated a crisp optimization problem using the IF aggregation operators in conjunction with the Bellman and Zadeh’s [9] extension principle. Recently, in [44], Yager pointed out the difficulty in using a straight away extension of the Bellman and Zadeh’s extension principle for aggregating IF decisions. He also suggested an alternative approach to choose the optimal decision while preserving the spirit of the Bellman and Zadeh’s extension principle. We shall be describing these issues in brief in the section to follow. The corrective measure suggested by Yager [44] inspired us to modify the IFO model of [1] to propose new models for optimization problems in setup in intuitionistic fuzzy scenario. The paper is planned as follows. In Section 2, we present a set of concepts from IFS theory which facilitate further discussion. In Section 3, a general framework for fuzzy linear programming problems under interval uncertainty is explained. We provide three different interpretations for IF inequality, namely IF essentially greater than equal to, by constructing the non-membership function in three different ways viz., optimistic, pessimistic, and mixed. In Section 4, we develop the corresponding three IFO models and formulated their fuzzy counterparts. We recognize that the membership functions of the resultant fuzzy optimization problems are S-shaped, thereby making us to apply the methods for solving fuzzy programming problems with piecewise linear S-shaped membership functions. In the present work we have used the methods proposed by Yang et al. [45] and Lin and Chen [33] to solve fuzzy programming problems with S-shaped membership functions by modeling their corresponding crisp optimization problems. Section 5 highlights some important issues needing attention, while Section 6 concludes the discussion. 2. Preliminaries Let X denotes the universe set. An IVFS A is defined by a function A from X to the set of closed subintervals in [0, 1], say, A (x) = [1 (x), 2 (x)], x ∈ X. An IFS A assigns to each element x of the universe X a membership degree A (x) ∈ [0, 1] and a non-membership degree A (x) ∈ [0, 1] such that A (x) + A (x) ≤ 1. Obviously, when A (x) + A (x) = 1, for all elements x ∈ X , the traditional fuzzy set concept is recovered. An IFS A is mathematically represented as {x, A (x), A (x)|x ∈ X }. Clearly, the aforementioned notions are mathematically isomorphic by taking 1 (x) = A (x), 2 (x) = 1 − A (x), x ∈ X. The standard intersection of two IFS A and B is an IFS C whose membership and non-membership functions are respectively defined as C (x) = min{A (x), B (x)} and C (x) = max{A (x), B (x)}, while the standard union of

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two IFS A and B is an IFS D whose membership and non-membership functions are respectively defined as D (x) = max{A (x), B (x)} and D (x) = min{A (x), B (x)}. The standard negation of an IFS A = {x, A (x), A (x)|x ∈ X } is an IFS defined by A = {x, A (x), A (x)|x ∈ X }. It is worth to remark here that this involutive negation acting on the pair of membership and non-membership functions makes IFS theory formally collapse to IVFS theory. This collapse was noticed in [7,12,18,44]. Furthermore, since the negation of a membership–non-membership pair comes down to swapping the membership and the non-membership functions, the IFS can also be viewed in the spirit of type II bipolarity [19]. The value A (x) = 1 − A (x) − A (x), x ∈ X , is regarded as a measure of indeterminacy [2] (or non-determinacy or undecidedness [12]). Further, A (x) = (1 − A (x)) − A (x) can also be viewed as length of the membership interval [A (x), 1 −  A (x)], i.e., interval uncertainty. This framework enables us to use IFS as a tool to represent the interval uncertainty in this paper. Analogous to fuzzy optimization, an IFO problem can be described as a two-stage process which includes aggregation of goals and constraints and then defuzzification to form crisp optimization problem. In the symmetric model, proposed by Bellman and Zadeh [9], there is no difference between objectives and constraints. If G i , i = 1, . . . , p, denote the p IF goals and C j , j = 1, . . . , q, denote the q IF constraints in a space of alternatives X , then by the Bellman and Zadeh’s extension principle [9], an IF decision D can be viewed as an IFS given by D = {x,  D (x),  D (x)|x ∈ X }, where  D (x) = min(G i (x), C j (x)), and  D (x) = max(G i (x), C j (x)). i, j

i, j

(1)

With this principle at the background, Angelov [1] formulated an optimization problem in which the degrees of rejection of objective(s) and of constraints are considered together with the degrees of their acceptance. He associated a value function with D as VD (x) =  D (x) −  D (x), x ∈ X , and the optimal solution is taken in the sense of finding an x ∗ ∈ X such that VD (x ∗ ) = maxx∈X VD (x), or equivalently to solve   max min(G i (x), C j (x)) − max(G i (x), C j (x)) . x∈X

i, j

i, j

The problem has been transformed by Angelov [1] to the following crisp optimization problem: max  −  s.t. G i (x) ≥ , i = 1, . . . , p, G i (x) ≤ , i = 1, . . . , p, C j (x) ≥ , C j (x) ≤ ,

j = 1, . . . , q, j = 1, . . . , q,

 +  ≤ 1,  ≥ ,  ≥ 0, x ∈ X. The foregoing discussion shows that the approach in [1] is a straightforward application of the Bellman and Zadeh’s extension principle. However, as we have remarked earlier, Yager [44] pointed out certain demerits of this approach. For instance, consider two alternatives x and y with  D (x) = 0.49,  D (x) = 0.51 and  D (y) = 0,  D (y) = 0. Since  D (x) −  D (x) = −0.02 and  D (y) −  D (y) = 0, the obvious optimal decision among the two is y. This seems rather strange that despite the membership of acceptance of x being clearly much more than that of y, still y persists to be the decision maker (DM) optimal choice. In [44], Yager suggested an alternative way to view the process involved in using VD (x). He transformed the value function VD (x) to the function FD (x) = 21 VD (x) + 21 =  D (x) + 21  D (x) instead. Note that we are using the same order comparison on this transformation, that is, if VD (x) > VD (y) then FD (x) > FD (y). This motivated Yager [44] to suggest a more general possibility of using the function FD (x) =  D (x) +  D (x),  ∈ (0, 1],

(2)

where  D (x) and  D (x) are as defined in (1). In fact this approach is not altogether new. If instead we view D as IVFS [ D (x), 1 −  D (x)], and rewrite (2) as follows: FD (x) = (1 − ) D (x) + (1 −  D (x)),  ∈ (0, 1]

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then (2) is equivalent to the Hurwicz pessimism–optimism criterion [24] in decision theory, viz., first choose a real number  ∈ [0, 1], and then find an alternative x for which FD takes the largest possible value. The distinction between using VD (x) and FD (x) is essentially in how much of indeterminacy we are able to resolve. The larger value of  indicates that we favor alternatives with imprecise membership more, in other words, indeterminacy gets more resolved in favor of membership. On the other hand, lower values of  show that indeterminacy gets more resolved in favor of non-membership. It is important to note that on using FD (x) to compare the alternatives, the IFS D gets essentially transformed into a standard FS described by {x, FD (x), 1 − FD (x)|x ∈ X }. The IF decision D can thus be interpreted as first aggregating all IF goals and IF constraints as in (1) followed by converting the resultant IFS to the FS by resolving the indeterminacy factor in the aggregated IF decision D. In this way one can associate an equivalent fuzzy optimization problem max x∈X FD (x) with IFO problem. However, in this paper, we propose to first resolve the indeterminacy factors individually in each IF goal and IF constraint, and thereafter aggregate the resultant fuzzy sets by using the Bellman and Zadeh’s extension principle. Consequently, the decision, denoted by D˜ (to distinguish it from [44]) is a FS whose membership function is defined by  D˜  (x) = min(G i (x) + G i (x), C j (x) + C j (x)). i, j

Since for all i, j, i = 1, . . . , p, j = 1, . . . , q, and x ∈ X , G i (x), C j (x) ≥ min{G i (x), C j (x)} and i, j

− G i (x), −C j (x) ≥ − max{G i (x), C j (x)} i, j

it can easily be shown that for any  ∈ (0, 1], max  D˜  (x) ≥ max FD (x). x∈X

x∈X

This observation clearly indicates that our approach to develop a new decision making model, motivated by the approach of Yager [44], in fact yield a better optimal value. Now, with no additional information in place, we have taken  = 21 in our subsequent discussion. Nevertheless it is important to realize that this choice does not stand in present study for the results can be worked out with any other value of  in [0, 1]. 3. Fuzzy linear programming problem under interval uncertainty We shall first be describing as to how we interpret the IF linear inequality. Let us consider a general IF linear inequality a T x I F b, where a ∈ Rn , x ∈ Rn , and b ∈ R. This inequality can be characterized as an IFS given by {a T x, (a T x), (a T x)|a T x ∈ R}, where the membership and the non-membership functions are to be understood in the sense described below. Let p > 0 be the tolerance for membership. The membership function (·) is then defined as ⎧ a T x ≤ b − p, ⎪ ⎨ 0, (a T x) = h 1 (a T x), b − p ≤ a T x ≤ b, ⎪ ⎩ 1, a T x ≥ b, where h 1 : R → [0, 1] is a continuous non-decreasing function with h 1 (b − p) = 0 and h 1 (b) = 1. For constructing the non-membership function (·), we realize that the only relation to be satisfied is (a T x) ≤ 1 − (a T x). Now for a T x ≥ b, (a T x) = 1, and hence (a T x) = 0. For a T x < b we have considered three cases in the sequel. One is when (a T x) does not become one when a T x < b − p, that is, there exists q > 0 such that (a T x) < 1 for b − p − q ≤ a T x ≤ b − p. The second is when (a T x) becomes zero before b. In other words, there exists r ∈ (0, p) such that (a T x) = 0 for a T x ≥ b − p + r . The third one is a combination of these two cases. In first case, we define the non-membership function (·) as follows (Fig. 1): ⎧ a T x ≤ b − p − q, ⎪ ⎨ 1, (a T x) = h 2 (a T x), b − p − q ≤ a T x ≤ b, ⎪ ⎩ 0, a T x ≥ b,

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1



µ

b−p−qb−p

b

aTx

Fig. 1. Membership and non-membership functions in first case.

1



b−p

µ

b − p + rb

aTx

Fig. 2. Membership and non-membership functions in second case.

where q > 0, h 2 : R → [0, 1] is a continuous non-increasing function such that h 2 (b − p − q) = 1, h 2 (b) = 0, h 1 (a T x) + h 2 (a T x) ≤ 1. We define the non-membership function for the second case as follows (Fig. 2): ⎧ ⎪ a T x ≤ b − p, ⎨ 1, T (a x) = h 3 (a T x), b − p ≤ a T x ≤ b − p + r, ⎪ ⎩ 0, a T x ≥ b − p + r, where r ∈ (0, p), h 3 : R → [0, 1] is a continuous non-increasing function such that h 3 (b − p) = 1, h 3 (b − p +r ) = 0, h 1 (a T x) + h 3 (a T x) ≤ 1. The non-membership function in the third case is defined next (Fig. 3). ⎧ ⎪ a T x ≤ b − p − s, ⎨ 1, (a T x) = h 4 (a T x), b − p − s ≤ a T x ≤ b − p − s + w, ⎪ ⎩ 0, a T x ≥ b − p − s + w, where s > 0, w ∈ (s, p + s), and h 4 : R → [0, 1] is a continuous non-increasing function such that h 4 (b − p − s) = 1, h 4 (b − p − s + w) = 0, h 1 (a T x) + h 4 (a T x) ≤ 1. It is important to note here that in the first case there is an interval [b − p − q, b − p] in which the membership degree is zero but the non-membership degree is not one. While in the second case there is an interval [b − p + r, b] in which the non-membership degree is zero but the membership degree is not one. Thus, depending on the construction of the non-membership function, in the present work, the IF inequality a T x I F b has been interpreted in three different ways namely: 1. the optimistic approach, 2. the pessimistic approach, 3. the mixed approach.

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1



b−p−s b−p

73

µ

b − p − s + wb

aTx

Fig. 3. Membership and non-membership functions in third case.

aTx

aTx = b aTx = b − p aTx = b − p − q

x

Fig. 4. Interpretation of tolerances in optimistic approach.

We shall call the IF inequality a T x I F b as IF essentially greater than equal to in the spirit of Zimmerman’s interpretation [48] of the analogous fuzzy linear inequality. Furthermore, we say a T x I F b if and only if (−a)T x I F (−b). Also, the positive numbers p, q, r, s and w, that appeared in describing the membership and the non-membership functions in the three approaches, shall now onwards be called tolerances and they are to be taken in an appropriate sense. We shall be associating p with membership function; q and r with the non-membership function in optimistic and pessimistic approaches respectively; while s and w shall be associated with the mixed approach construction of the non-membership function. It is important to remark here that the three viewpoints have been associated only with the construction of the non-membership function and not to be given any other meaning. In the context of IFO problem, as we shall see in the next section, the IF inequalities act as elastic constraints that allow us to discriminate between more or less possible values of a variable x. The membership and the non-membership functions describing IF inequality can be interpreted respectively as strong feasible solutions and the strong infeasible solutions. For the purpose of understanding, consider a fuzzy optimization problem in which the feasible set is described by only one fuzzy inequality a T xb. Recall that in a classical fuzzy optimization problem with  as fuzzy essentially greater than equal to, the membership function value (a T x) represents the degree of accepting an element x in the feasible set of an optimization problem. Thus any element x for which a T x ≥ b is surely a feasible solution (and hence preferred one) while an element x for which a T x < b − p, p > 0, is certainly an infeasible solution, see [8] for more details. On the other hand, in analogous IFO problem with say only one constraint of the form a T x I F b, similar interpretations can be provided to the tolerance parameters. In the optimistic approach of the IF scenario, we continue to interpret the (strong) feasible solution in the same sense while the upper bound of infeasibility is relaxed to b − p − q, q > 0, such that any x for which b − p − q ≤ a T x < b − p, q > 0, is surely not feasible (but it is not to be understood as an infeasible solution). The DM is not ready to accept all such x but also not rejecting them out rightly due to indeterminacy. However, an x with a T x < b − p − q, is certainly (strong) infeasible. The discussion is summarized in Fig. 4. In the pessimistic approach of the IF scenario, we can view any element x which yields b − p + r ≤ a T x < b, 0 < r < p, as surely not an infeasible solution (again it is not to be taken as a feasible solution). Here, the DM is

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not rejecting such solutions but at the same time not ready to accept them completely. The mixed case can also be interpreted in a similar manner. Any element x satisfying b − p − s + w ≤ a T x < b can be viewed as not an infeasible solution, while an element x satisfying b − p − s ≤ a T x < b − p is treated as not a feasible solution. The same also clarifies our position in Section 2 that an IFS or IF relation can be taken in the spirit of type II bipolarity. We are now ready to describe the general model of a linear programming problem with IF inequality and IF objective function. Consider the IF linear programming problem (IFLPP) max ˜

cT x AiT x I F bi , i = 1, . . . , m,

s.t.

x ∈ S, where x ∈ Rn , c ∈ Rn , Ai ∈ Rn , bi ∈ R, i = 1, . . . , m, and the set S ⊆ Rn can be described by crisp linear inequalities (if any). Moreover, the objective function max ˜ is to be understood in the IF sense of satisfaction of an aspiration level Z 0 of the DM. Then the problem (IFLPP) is equivalent to finding x ∈ Rn such that cT x  I F Z 0 , AiT x  I F bi , i = 1, . . . , m, x ∈ S. Note that in the above scheme there is no distinction between the IF objective function and the IF constraints. Thus the model is fully symmetric with respect to the objective function and the constraints. Now, let 0 (·) and 0 (·) denote respectively the membership function and the non-membership function for the objective IF linear inequality, and i (·) and i (·), i = 1, . . . , m, denote respectively the membership functions and the non-membership functions for the IF linear inequalities corresponding to the original IF constraints in (IFLPP). All the functions are taken in the afore described sense. Further, on account of our discussion in Section 2 leading to (2), we associate new membership functions with the objective and the constraints IF inequalities respectively by resolving half of their indeterminacy in favor of their membership as follows: f 0 (c T x) = 0 (c T x) + 21 0 (c T x),

f i (AiT x) = i (AiT x) + 21 i (AiT x), i = 1, . . . , m.

Exploiting the Bellman and Zadeh’s extension principle, problem (IFLPP) can be re-casted as follows: T x)}, max min{ f 0 (c T x), f 1 (A1T x), . . . , f m (Am x∈S

which is equivalent to the following crisp optimization problem: (ECP) max s.t.

 f 0 (c T x) ≥ , f i (AiT x) ≥ , i = 1, . . . , m, x ∈ S,  ∈ [0, 1].

Definition 1. If (x ∗ , ∗ ) is an optimal solution of (ECP) then x ∗ is an optimal solution of (IFLPP) and ∗ is the degree up to which the aspiration level Z 0 of the DM is met. If (ECP) is infeasible then (IFLPP) is said to possess no solution. Remark 1. It may be noted that if there is no indeterminacy factor in the IFO problem (IFLPP), that is, all the nonmembership functions are standard complements of the corresponding membership functions, then we are in a fuzzy scenario, and problem (ECP) reduces to max  s.t. 0 (c T x) ≥ , i (AiT x) ≥ , i = 1, . . . , m, x ∈ S,

 ∈ [0, 1],

which is same as the one formulated by Zimmermann [48] in fuzzy setup.

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4. The three models Since the non-membership function of the IF inequality can be described in three ways, we propose three models for the IF linear programming problem (IFLPP) to capture the same. Moreover, the functions h i , i = 1, . . . , 4, used in describing the membership and the non-membership functions can be nonlinear functions. However, in present context, for easy understanding and obvious computational advantage all are taken as linear functions. 4.1. Optimistic approach Consider the IF inequality c T x I F Z 0 . Let p0 > 0, q0 > 0 be the tolerances for the objective function. Then we define the membership and the non-membership functions as continuous linear functions

0 (c T x) =

⎧ 0, ⎪ ⎪ ⎨ 1+ ⎪ ⎪ ⎩ 1,

c T x ≤ Z 0 − p0 , cT x − Z 0 , Z 0 − p0 ≤ c T x ≤ Z 0 , p0 cT x ≥ Z 0

and

0 (c T x) =

⎧ 1, ⎪ ⎪ ⎨ 1− ⎪ ⎪ ⎩ 0,

c T x ≤ Z 0 − p0 − q 0 , c T x − (Z 0 − p0 − q0 ) , Z 0 − p0 − q 0 ≤ c T x ≤ Z 0 , p0 + q 0 cT x ≥ Z 0 .

Observe that in the interval [Z 0 − p0 − q0 , Z 0 − p0 ], the membership degree of achieving the aspiration level Z 0 is zero while the non-membership degree is not one. Thereby indicating that the DM is not keen to accept any value below Z 0 − p0 but at the same time he is not out rightly rejecting the values between Z 0 − p0 − q0 and Z 0 − p0 . For this reason we call such an approach optimistic. Let pi > 0, qi > 0, be the tolerances for the IF inequality in the ith constraint. We define the membership and the non-membership functions as follows: ⎧ ⎪ 1, AiT x ≤ bi , ⎪ ⎪ ⎨ T i (AiT x) = 1 + bi − Ai x , bi ≤ A T x ≤ bi + pi , i ⎪ ⎪ pi ⎪ ⎩ 0, AiT x ≥ bi + pi and

i (AiT x) =

⎧ ⎪ 0, ⎪ ⎪ ⎨ 1+ ⎪ ⎪ ⎪ ⎩ 1,

AiT x ≤ bi , AiT x − (bi + pi + qi ) , bi ≤ AiT x ≤ bi + pi + qi , pi + qi AiT x ≥ bi + pi + qi .

Consequently, the functions f 0 (·), f i (·), i = 1, . . . , m, become ⎧ 0, c T x ≤ Z 0 − p0 − q 0 , ⎪ ⎪ ⎪ ⎪ ⎪ T ⎪ ⎪ ⎨ c x − (Z 0 − p0 − q0 ) , Z 0 − p0 − q 0 ≤ c T x ≤ Z 0 − p0 , T 2( p + q ) 0 0 f 0 (c x) =   ⎪ 2 p0 + q 0 ⎪ ⎪ 1 + (c T x − Z 0 ) , Z 0 − p0 ≤ c T x ≤ Z 0 , ⎪ ⎪ ⎪ 2 p ( p + q ) 0 0 0 ⎪ ⎩ 1, cT x ≥ Z 0

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f 0 (cT x )

f i (A Ti x ) 1

1

f i1

f 02 )

f i2

f 01 Z 0 − p0 − q0

Z 0 − p0

Z0

cT x

bi

bi + pi

bi + pi + qi

A Ti x

Fig. 5. Membership functions in optimistic approach for (IFLPP).

and

⎧ 1, ⎪ ⎪   ⎪ ⎪ 2 pi + qi ⎪ T ⎪ ⎪ ⎨ 1 + (bi − Ai x) 2 p ( p + q ) , i i i f i (AiT x) = Tx ⎪ + p + q − A b i i i ⎪ i ⎪ , ⎪ ⎪ ⎪ 2( pi + qi ) ⎪ ⎩ 0,

AiT x ≤ bi , bi ≤ AiT x ≤ bi + pi , bi + pi ≤ AiT x ≤ bi + pi + qi , AiT x ≥ bi + pi + qi .

Here f 0 (·) and f i (·) are piecewise linear S-shaped functions and f i (·), i = 1, . . . , m, are inverted S-shaped as depicted in Fig. 5. So problem (IFLPP) reduces to a fuzzy optimization problem with piecewise linear membership functions. Since the S-shaped membership functions have convex type break points, the problem (IFLPP) can be equivalently formulated as a mixed integer linear program. Following the approach of Yang et al. [45], the equivalent crisp problem is given by max  s.t. f 01 (c T x) + M0 ≥ , f 02 (c T x) + M(1 − 0 ) ≥ , f i1 (AiT x) + Mi ≥ , i = 1, . . . , m, f i2 (AiT x) + M(1 − i ) ≥ , i = 1, . . . , m, x ∈ S,  ∈ [0, 1], i ∈ {0, 1}, i = 0, 1, . . . , m, where M is a large positive real number. On simplification, we finally have the following crisp mixed integer linear programming problem to solve: (COPO) max  s.t. c T x + 2M( p0 + q0 )0 − 2( p0 + q0 ) ≥ Z 0 − p0 − q0 , (2 p0 + q0 )c T x − 2M p0 ( p0 + q0 )0 − 2 p0 ( p0 + q0 ), ≥ Z 0 (2 p0 + q0 ) − 2(1 + M) p0 ( p0 + q0 ), (2 pi + qi )AiT x − 2M pi ( pi + qi )i + 2 pi ( pi + qi ), ≤ (2 pi + bi )( pi + qi ) + bi pi , i = 1, . . . , m, AiT x + 2M( pi + qi )i + 2( pi + qi ), ≤ (2M + 1)( pi + qi ) + bi , i = 1, . . . , m, x ∈ S,  ∈ [0, 1], i ∈ {0, 1}, i = 0, 1, . . . , m. To illustrate the working of this IFO model, we present the following example. Example 1. Consider max ˜ 2x1 + x2 s.t. x1  I F 3, 2x1 + x2  I F 7, x1 + x2 ≤ 4, x1 , x2 ≥ 0.

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Take Z 0 = 8, p0 = 3, q0 = 2, p1 = 2, q1 = 21 , p2 = 2, q2 = 1. Using the above method, the equivalent crisp model is given by max  s.t. 0.2x1 + 0.1x2 + M0 −  ≥ 0.3, −0.53x1 − 0.26x2 + M0 +  ≤ M − 1.13, 0.45x1 − M1 +  ≤ 2.35, 0.2x1 + M1 +  ≤ M + 1.1, 10x1 + 5x2 − 12M2 + 12 ≤ 47, 2x1 + x2 + 6M2 + 6 ≤ 6M + 10, x1 + x2 ≤ 4, x1 , x2 ≥ 0,  ∈ [0, 1], 0 , 1 , 2 ∈ {0, 1} and M is large positive real number. The optimal solution is x1∗ = 3.3888, x2∗ = 0.6111,  = 0.8250, 0 = 1, 1 = 0 and 2 = 0. Thus the optimal solution of the given IF linear program is x1∗ = 3.3888, x2∗ = 0.6111, and the DM aspiration level of 8 is met with 82.50%. We next present a real-life example of a transportation problem inspired by a similar example in [1]. We used the IF theory to model the problem under available information. Consider the following transportation problem. Example 2. Loads from three ports have to be divided between four markets. Cost of a delivery from ith port to the jth market (in thousand of unit) are given in the respective cells of the following table. The demands of loads in every market and the capacity of loads in every port are given (in tons) respectively in last row and column of the table. An optimal transportation plan x(x ∈ R3×4 ) which minimizes the cost is to be determined.

Market1 Port1 Port2 Port3 Demand

Market2

Market3

Market4

10 2 8

7 7 5

4 10 3

1 6 2

200

200

100

350

Capacity 400 200 350

Practically, the demands of markets are determined on the basis of sales forecasting. If the prognosis for market 2 is about 200 tons and for market 4 is about 350 tons, the corresponding IF linear program is given by ˜ 10x11 + 7x12 + 4x13 + x14 + 2x21 + 7x22 + 10x23 + 6x24 + 8x31 + 5x32 + 3x33 + 2x34 min s.t. x11 + x12 + x13 + x14 ≤ 400, x21 + x22 + x23 + x24 ≤ 200, x31 + x32 + x33 + x34 ≤ 350, x11 + x21 + x31 ≥ 200, x12 + x22 + x32  I F 200, x13 + x23 + x33 ≥ 100, x14 + x24 + x34  I F 350, xi j ≥ 0, i = 1, . . . , 3, j = 1, . . . , 4. Suppose Z 0 = 2000, p0 = 500, q0 = 200, p5 = 30, q5 = 10, p7 = 40, q7 = 20. In this problem we assume that the minimum demands of market 2 and market 4 are not precisely known priori. For instance, the demand information of market 2 is that a supply above 200 tons is (strongly) preferred, while a supply below 160 tons is completely rejected, i.e., the DM at market 2 has no reason to accept supply below 160 tons. A supply between 160 tons and 170 tons is not

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out rightly rejected, i.e., the DM has no reason to reject the same. The demand information of market 4 can similarly be interpreted. Using the optimistic approach, the equivalent crisp model is given as follows: max  s.t. 1200x11 + 8400x12 + 4800x13, +1200x14 + 2400x21 + 8400x22, +12, 000x23 + 7200x24 + 9600x31, +6000x32 + 3600x33 + 2400x34, −70, 000M0 + 70, 0000 ≤ 3, 100, 000, 10x11 + 7x12 + 4x13 + x14 + 2x21 + 7x22, +10x23 + 6x24 + 8x31 + 5x32 +, 3x33 + 2x34 + 1400M0 + 1400 ≤ 1400M + 2700, x12 + x22 + x32 + 80M1 − 80 ≥ 160, 70x12 + 70x22 + 70x32 − 2400M1 − 2400 ≥ −1200 − 1200M, x14 + x24 + x34 + 120M2 − 120 ≥ 290, 100x14 + 100x24 + 100x34 − 4800M2 − 4800 ≥ −1300 − 4800M, x11 + x12 + x13 + x14 ≤ 400, x21 + x22 + x23 + x24 ≤ 200, x31 + x32 + x33 + x34 ≤ 350, x13 + x23 + x33 ≥ 100, x11 + x21 + x31 ≥ 200, xi j ≥ 0, i = 1, . . . , 3, j = 1, . . . , 4,  ∈ [0, 1], 0 , 1 , 2 ∈ {0, 1} ∗ = 400, x ∗ = 200, x and M is large positive real number. The optimal solution is x14 32 = 186.85, x 33 = 163.14, 21  = 0.6165, 0 = 0, 1 = 1, 2 = 1. Consequently, the optimal solution of the original transportation problem is as follows with 61.65% satisfaction.

Market1 Port1 Port2 Port3

200

Demand

200

Market2

Market3

186.85

163.12

200

100

Market4

Capacity

400

400 200 350

350

There are plenty of real life applications of IFS or equivalently IVFS to quote all here. We urge the interested readers to refer to [38] and many other relevant references therein as well as heretofore in the present paper. 4.2. Pessimistic approach Let p0 , r0 , with 0 < r0 < p0 , be the tolerances for the IF linear inequality c T x I F Z 0 . Define the membership and the non-membership functions as continuous linear functions as follows: ⎧ 0, c T x ≤ Z 0 − p0 , ⎪ ⎪ ⎨ T c x − Z0 0 (c T x) = 1 + , Z 0 − p0 ≤ c T x ≤ Z 0 , ⎪ p0 ⎪ ⎩ 1, cT x ≥ Z 0

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and

0 (c T x) =

⎧ 1, ⎪ ⎪ ⎪ ⎨ 1− ⎪ ⎪ ⎪ ⎩ 0,

c T x ≤ Z 0 − p0 , c T x − (Z 0 − p0 ) , Z 0 − p0 ≤ c T x ≤ Z 0 − p0 + r 0 , r0 c T x ≥ Z 0 − p0 + r 0 .

Observe that in the interval [Z 0 − p0 + r0 , Z 0 ], the membership degree of achieving the aspiration level Z 0 is not one despite the fact that its non-membership degree is zero there in. In other words, the DM is not willing to reject the values between Z 0 − p0 + r0 and Z 0 but at the same time not out rightly accepting them. In this spirit we call such an approach as pessimistic. Let pi , ri , with 0 < ri < pi , be the tolerances for the IF linear inequality in the ith constraint, i = 1, . . . , m. We define the membership and the non-membership functions as follows:

i (AiT x) =

⎧ 1, ⎪ ⎪ ⎪ ⎨ 1+ ⎪ ⎪ ⎪ ⎩ 0,

AiT x ≤ bi , bi − AiT x , bi ≤ AiT x ≤ bi + pi , pi AiT x ≥ bi + pi

and

i (AiT x) =

⎧ 0, ⎪ ⎪ ⎪ ⎨ 1+ ⎪ ⎪ ⎪ ⎩ 1,

AiT x ≤ bi + pi − ri , AiT x − (bi + pi ) , bi + pi − ri ≤ AiT x ≤ bi + pi , ri AiT x ≥ bi + pi .

Consequently, the functions f 0 (·), f i (·), i = 1, . . . , m, become ⎧ 0, ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ p0 + r 0 cT x − Z 0 ⎪ ⎨ 1+ , 2r0 p0 f 0 (c T x) = ⎪ ⎪ cT x − Z 0 ⎪ ⎪ ⎪ 1+ , ⎪ ⎪ 2 p0 ⎪ ⎩ 1,

c T x ≤ Z 0 − p0 , Z 0 − p0 ≤ c T x ≤ Z 0 − p0 + r 0 , Z 0 − p0 + r 0 ≤ c T x ≤ Z 0 , cT x ≥ Z 0

and ⎧ 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ bi − AiT x ⎪ ⎪ , ⎨1+ 2pi f i (AiT x) = ⎪ bi − AiT x pi + ri ⎪ ⎪ ⎪ 1+ , ⎪ ⎪ 2ri pi ⎪ ⎪ ⎩ 0,

AiT x ≤ bi, bi ≤ AiT x ≤ bi + pi − ri , bi + pi − ri ≤ AiT x ≤ bi + pi , AiT x ≥ bi + pi .

Here, f 0 (·) and f i (·) are piecewise linear S-shaped functions and f i (·), i = 1, . . . , m, are inverted S-shaped as shown in Fig. 6. Note that this time the membership functions have concave break points unlike the previous case where they have convex break points. Again adopting the method of Yang et al. [45] the resultant optimization model

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D. Dubey et al. / Fuzzy Sets and Systems 188 (2012) 68 – 87 f 0 (cT x )

f i (A iT x ) 1

1

fi1 pi + r i

f 02

2pi

p0 + r 0 2p0

fi2

f 01 Z 0 − p0

Z 0 − p0 + r 0

Z0

cT x

bi

bi + pi − r i

bi + pi

A Ti x

Fig. 6. Membership functions in pessimistic approach for (IFLPP).

is given by max  s.t. f 01 (c T x) ≥ , f 02 (c T x) ≥ , f i1 (AiT x) ≥ , f i2 (AiT x) ≥ , i = 1, . . . , m, x ∈ S,  ∈ [0, 1], which on simplification yields the following crisp linear programming problem: (COPP) max  s.t. ( p0 + r0 )c T x − 2 p0 r0  ≥ ( p0 + r0 )(Z 0 − p0 ), c T x − 2 p 0  ≥ Z 0 − 2 p0 , AiT x + 2 pi  ≤ bi + 2 pi , ( pi + ri )AiT x + 2 pi ri  ≤ ( pi + ri )(bi + pi ), i = 1, . . . , m, x ∈ S,  ∈ [0, 1]. Example 3. Recall Example 1, same as taken to illustrate the optimistic case. max ˜ 2x1 + x2 s.t. x1  I F 3, 2x1 + x2  I F 7, x1 + x2 ≤ 4, x1 , x2 ≥ 0. Take Z 0 = 8, p0 = 3, r0 = 2, p1 = 2, r1 = 1, p2 = 2, r2 = 1. Using the aforementioned model, the equivalent crisp problem becomes max  s.t. 0.8333x1 + 0.4166x2 −  ≥ 2.0833, 0.3333x1 + 0.1666x2 −  ≥ 0.3333, 0.25x1 +  ≤ 1.75, 0.75x1 +  ≤ 3.75, 2x1 + x2 + 4 ≤ 11, 6x1 + 3x2 + 4 ≤ 27, x1 + x2 ≤ 4, x1 , x2 ≥ 0,  ∈ [0, 1].

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The optimal solution is x1∗ = 3.4002, x2∗ = 0.5997,  = 0.8999. Thus the aspiration level 8 of DM is met with 89.99%. What we observe in the above example is that the aspiration level Z 0 = 8 of the DM is met with higher degree in the pessimistic scenario than what it was in the optimistic case. This may appear unconventional. On the other hand, consider the following IFO problem with IF objective and crisp constraints. Example 4. We need to solve max ˜ 2x1 + x2 s.t. x1 ≤ 3, x1 + x2 ≤ 4, x1 , x2 ≥ 0. Take Z 0 = 9 and p0 = 3 in both optimistic and pessimistic approaches, and q0 = 2, r0 = 2, in the optimistic and the pessimistic approaches respectively. Then the optimal solutions of the associated crisp programming problems are, x1∗ = 3, x2∗ = 1,  = 0.4667 for the optimistic case, and x1∗ = 3, x2∗ = 1,  = 0.4166 for the pessimistic case. Here the degree with which the aspiration level of the DM is met in the optimistic case is more than the one in the pessimistic case. In view of above Examples 2–4, it is irresistible to have a deeper look at the two optimization models (optimistic and pessimistic) from their optimal values perspective and identify the reasons as to why the optimistic case sometimes yield a lower optimal value than its pessimistic counterpart. In this regard, we have made some observations which are summarized below. Remark 2. Consider the following IF linear programming problem with crisp constraints, where the intuitionistic fuzziness appears only in the objective function: (IFOCC) max ˜ s.t.

cT x AiT x ≤ bi , i = 1, . . . , m, x ∈ S.

Let Z 0 be the aspiration level of DM for the IF objective function, p0 , q0 be the tolerances for the model with optimistic approach, and p0 , r0 be the tolerances for the model with pessimistic approach. Let (xopt , opt , 0opt ) and (x pes ,  pes ) denote the optimal solutions for the equivalent crisp problems (COPO) for the optimistic and (COPP) for the pessimistic cases respectively. Then, ⎧ T T ⎪ ⎨ opt >  pes if c xopt , c x pes ∈ (Z 0 − p0 − q0 , Z 0 − ), opt <  pes if c T xopt , c T x pes ∈ (Z 0 − , Z 0 ), ⎪ ⎩ opt =  pes otherwise, where = ( p0 − r0 )( p0 + q0 )/( p0 + q0 − r0 ) ∈ ( p0 − r0 , p0 ). This has also been depicted in Fig. 7. However, for a general (IFLPP), such observations are difficult to make. 4.3. Mixed approach Let p0 > 0, s0 > 0, w0 > 0, with s0 < w0 < p0 + s0 , be the tolerances for the IF linear inequality c T x I F Z 0 . In analogy to the foregoing discussion, define the membership function and the non-membership function as continuous linear functions as follows: ⎧ 0, c T x ≤ Z 0 − p0, ⎪ ⎪ ⎨ T 0 (c T x) = 1 + c x − Z 0 , Z 0 − p0 ≤ c T x ≤ Z 0 , ⎪ p0 ⎪ ⎩ 1, cT x ≥ Z 0

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D. Dubey et al. / Fuzzy Sets and Systems 188 (2012) 68 – 87 f 0 (cT x ) 1

Z 0 − p0 − q0

Z 0 − p0 Z 0 − ξ Z 0 − p0 + r 0

Z0

cTx

Fig. 7. Optimistic (solid) and pessimistic (dotted) memberships for (IFOCC).

and

0 (c T x) =

⎧ 1, ⎪ ⎪ ⎪ ⎨ 1− ⎪ ⎪ ⎪ ⎩ 0,

c T x ≤ Z 0 − p0 − s0 , c T x − (Z 0 − p0 − s0 ) , Z 0 − p0 − s0 ≤ c T x ≤ Z 0 − p0 − s0 + w0 , w0 c T x ≥ Z 0 − p0 − s0 + w0 .

Observe that in the interval [Z 0 − p0 − s0 , Z 0 − p0 ], the membership degree of achieving the aspiration level Z 0 is zero while its non-membership degree is not one. On other hand, in the interval [Z 0 − p0 − s0 + w0 , Z 0 ], the non-membership degree of achieving the aspiration level Z 0 is zero while its membership degree is not one. Thereby we call it a mixed approach. Let pi , si , wi , with si < wi < pi + si , be the tolerances for the IF linear inequality in the ith constraint. The membership and the non-membership functions are defined as follows:

i (AiT x) =

⎧ 1, ⎪ ⎪ ⎪ ⎨ 1+ ⎪ ⎪ ⎪ ⎩ 0,

AiT x ≤ bi , bi − AiT x , bi ≤ AiT x ≤ bi + pi , pi AiT x ≥ bi + pi

and

i (AiT x) =

⎧ 0, ⎪ ⎪ ⎪ ⎨ 1+ ⎪ ⎪ ⎪ ⎩ 1,

AiT x ≤ bi + pi + si − wi , AiT x − (bi + pi + si ) , bi + pi + si − wi ≤ AiT x ≤ bi + pi + si , wi AiT x ≥ bi + pi + si .

Therefore, ⎧ 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ c T x − (Z 0 − p0 − s0 ) ⎪ ⎪ , ⎪ ⎪ 2w0 ⎪ ⎪ ⎨   f 0 (c T x) = (c T x − Z 0 ) p0 + w0 + p0 + s0 + w0 , ⎪ ⎪ 2 p0 w0 2w0 ⎪ ⎪ ⎪ T ⎪ ⎪ c x − Z0 ⎪ ⎪ 1+ , ⎪ ⎪ 2 p0 ⎪ ⎩ 1,

c T x ≤ Z 0 − p0 − s0 , Z 0 − p0 − s0 ≤ c T x ≤ Z 0 − p0 , Z 0 − p0 ≤ c T x ≤ Z 0 − p0 − s0 + w0 , Z 0 − p0 − s0 + w0 ≤ c T x ≤ Z 0 , cT x ≥ Z 0

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f 0 (cT x ) 1

f 03

p +w −s 2p

f 02 s 2w

f 01 Z 0 − p0 − s 0

Z 0 − p0

Z 0 − p0 − s 0 + w 0 Z 0

cT x

Fig. 8. Membership function f 0 for mixed (IFLPP). f i (A Ti x ) 1

f i1

s 2w

f i2 p +w −s 2p

f i3 bi

bi + pi + s i

bi + pi + s i − w i bi + pi

A Ti x

Fig. 9. Membership function f i for mixed (IFLPP).

and

⎧ 1, ⎪ ⎪ ⎪ ⎪ ⎪ bi − AiT x ⎪ ⎪ , 1 + ⎪ ⎪ ⎪ 2 pi  ⎪    ⎪ ⎪ pi + wi pi + si + wi ⎨ T + , (bi − Ai x) f i (AiT x) = 2 pi wi 2wi ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ b + pi + si − AiT x ⎪ ⎪ i , ⎪ ⎪ 2si ⎪ ⎩ 0,

AiT x ≤ bi , bi ≤ AiT x ≤ bi + pi + si − wi ,

bi + pi + si − wi ≤ AiT x ≤ bi + pi , bi + pi ≤ AiT x ≤ bi + pi + si , AiT x ≥ bi + pi + si .

f 0 (·) and f i (·) are piecewise linear S-shaped functions and f i (·), i = 1, . . . , m, are inverted S-shaped. Note that, unlike the previous two cases, the functions f 0 (·) and f i (·) have both convex type and concave type break points as shown in Figs. 8 and 9. Again using the generalized model of Lin [33], the mixed approach model of (IFLPP) can be transformed to the following equivalent crisp mixed integer linear programming problem: (COPM) max  s.t. f 01 (c T x) + M0 ≥ , f 02 (c T x) + M(1 − 0 ) ≥ , f 03 (c T x) + M(1 − 0 ) ≥ , f i1 (AiT x) + Mi ≥ , i = 1, . . . , m, f i2 (AiT x) + Mi ≥ , i = 1, . . . , m, f i3 (AiT x) + M(1 − i ) ≥ , i = 1, . . . , m, x ∈ S,  ∈ [0, 1], i ∈ {0, 1}, i = 0, 1, . . . , m, where M is a large positive real number.

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We present the following example to illustrate the working of this case. Example 5. Consider the IFLPP max ˜ 2x1 + x2 s.t. x1  I F 3, 2x1 + x2  I F 7, x1 + x2 ≤ 4, x1 , x2 ≥ 0. Take Z 0 = 8, p0 = 3, s0 = 2, w0 = 3, p1 = 2, s1 = 1, w1 = 2, p2 = 3, s2 = 1, w2 = 2. Using the mixed approach, the equivalent crisp optimization problem is given by (COPM)1 max  s.t. 2x1 + x2 + 6M0 − 6 ≥ 3, −12x1 − 6x2 + 18M0 + 18 ≤ 18M − 16, −2x1 − x2 + 6M0 + 6 ≤ 6M − 2, x1 − 4M1 + 4 ≤ 7, 2x1 − 4M1 + 4 ≤ 11, x1 + 4M1 + 4 ≤ 4M + 5, 2x1 + x2 − 6M2 + 6 ≤ −13, 0.8333x1 + 0.4166x2 − M2 +  ≤ 4.4166, x1 + 0.5x2 + M2 +  ≤ M + 5.5, x1 + x2 ≤ 4, x1 , x2 ≥ 0,  ∈ [0, 1], 0 , 1 , 2 ∈ {0, 1} and M is large positive real number. The optimal solution is x1∗ = 0.34, x2∗ = 0.6,  = 0.9, 0 = 1, 1 = 0, 2 = 1. Hence the optimal solution of the given IF linear program is x1∗ = 0.34, x2∗ = 0.6, and the DM aspiration level of 8 is met with 90%. 5. Additional comments There are some important observations for the proposed models that we would like to share with our readers. These observations, summarized below in the form of Remarks, would also bring forth some questions for future investigations. Note 1. Membership function holds the key in designing any model based on fuzzy set theory. Generating a suitable membership function for a fuzzy variable has long been recognized as a challenging issues in fuzzy systems design. Several efforts have been made for automatic generation of membership function using fuzzy rules, optimization techniques and expert knowledge. The pioneer works (in order of their appearance in literature) of Homaifar and McCormick [23], Medasani et al. [37], Makrehchi et al. [36], Liao et al. [32], to name a few, can be referred for details. Recently, some attempts have also been made to construct type-2 fuzzy membership function, see [11]. A natural query is that can these or similar other techniques be used to generate non-membership functions? The initial answer appears to be positive but the same needs to be supported with numerical evidences. Note 2. Another critical issue is the choice of the aspiration level Z 0 and the tolerance parameters pi , qi , ri , si , wi , i = 0, 1, . . . , m,. The parameters values are assumed to be provided by the DM, whose judgement and expertise we rely on. An extensive dialogue with the DM can throw more light and provide more information to adjust and tune

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in these parameters to their best judged values during the course of modeling. Furthermore, at the end stages, we can also perform post optimality analysis to examine the effect of small perturbations in the values of these parameters on the optimal objective values of the three models. Since the crisp optimization problems in all three cases are either mixed integer linear programming problems or linear programming problems, performing a post optimality analysis is a straightforward task. To support Note 2, we perform the post optimality analysis on Z 0 in Example 5. The corresponding crisp optimization problem (C O P M)1 is solved using software LINGO 12.0 [34] on a 32 bits Window platform. The solution report also generates dual price figures for each constraint. Note any change in the value of Z 0 causes change in the right hand side values of the first three constraints in (C O P M)1 . The dual prices for the first, second and third constraint are 0, 0, and 0.1 respectively. Consequently, if we take Z 0 = 9, the changed optimal objective value is 0.9 − 0.1 × 1 = 0.8. If instead we take Z 0 = 7, the changed optimal objective value is 0.9 + 0.1 × 1 = 1. It is observed that the optimal basis remains same even after these minor changes in (C O M P)1 . Post optimality analysis for Z 0 can easily be done for other models also. A post optimality analysis for the other parameters can also be attempted in similar way, although the latter appears to be difficult as these parameters contribute toward both the coefficient matrices and the right hand side vectors of the crisp models simultaneously. Can the theory of post optimality analysis or sensitivity analysis be utilized to design a mechanism to help DM finds reasonable values of the parameters, is an interesting exercise. Note 3. A natural instinct that the optimistic approach would yield a higher degree of satisfaction in the optimal solution vis a´ vis its pessimistic version or mixed version, is not always met. The preliminary reason appears to be in the way we resolve the indeterminacy factor. By allowing more sophisticated rule-based variation in the choice of  ∈ (0, 1], we may be in a position to design better and more robust models for IFO problems. An open research issue is how to estimate the appropriate value of ? This can be taken up in future research. Note 4. The equivalent crisp optimization problems proposed in the paper may be infeasible (in the traditional sense). However if these problems are feasible then they possess optimal solutions. A natural question therefore is as to how to model an equivalent crisp optimization problem that is necessarily feasible? Note 5. During the course of revision of this paper, it has been suggested to have a look at the concept of bipolar fuzzy sets with reference to the cumulative prospect theory [40], and apply the latter to model the optimization problem. Though the idea is interesting and well taken, but it is beyond the scope of the present paper. We would surely work along the suggested lines in near future. In this context, we would also like to take note of a very recent research article by Freson et al. [21]. In nutshell, the hidden treasure in the concept of bipolarity needs to be explored in optimization theory in near future. 6. Conclusions In this paper, we studied the symmetric model for linear programming problems (wherein there is no distinction between the objective function and constraints) set up in the intuitionistic fuzzy scenario. We examined three ways to interpret the IF inequality namely, IF essentially greater than equal to, by taking optimistic, pessimistic, and mixed approaches in constructing the non-membership function. It is important to remark here that the approach adopted in [39] and other related research articles to model linear programming problem using IVFS differs from the one taken by us in the present work. Herein, we have equivalently represented the interval uncertainty by the corresponding indeterminacy factor in IFS representation. Besides the Bellman and Zadeh’s extension principle, we also took inspiration mainly from the Hurwicz criterion [24] and a recent research article by Yager [44] wherein he had suggested to resolve the indeterminacy factor before applying the extension principle to decision making problems. We have resolved half the indeterminacy part in favor of membership to avoid some undesirable scenarios discussed in Section 2. This is a reasonable starting point, since, in general, we do not have any additional information about the level by which the DM would like to resolve indeterminacy, something that is required to resolve it completely. Still, through this approach, we are able to overcome certain demerits of the IF linear programming model studied by Angelov [1].

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Even though the membership and the non-membership functions in the initial IFO problem are taken to be linear, the equivalent fuzzy optimization models involve S-shaped membership functions. Depending on the shape of the S-shaped membership functions and the nature of their break points, the crisp models either result in mixed integer linear programs involving binary variables or linear programs with all real variables. These problems can easily be solved to get the optimal solution of the original IFO problem. Acknowledgments The first author is thankful to the National Board of Higher Mathematics, India, for providing financial grant. The authors deeply thank the anonymous referees and the Editors-in Chiefs, B. De Baets, D. Dubois and E. Hüllermeier, for their enriching suggestions which considerably improve the overall presentation as well as our understanding of the concepts. Glad Deschrijver deserves our special thanks for valuable friendly inputs on this work. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

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