Pareto-optimal solutions for multi-objective flexible linear programming

1 Introduction

A linear programming (LP) problem traditionally involves hard constraints; but specifying strict satisfaction of constraints can lead to inconsistencies resulting in vacuousness of feasible set. A flexible linear programming (FLP) involving flexible version of constraints can be of much help in such a situation. Also, if there are more than one objective functions to be optimized simultaneously in an FLP problem then it is termed as a multi-objective FLP (MOFLP) problem. The MOFLP problems are often investigated under the name multi-objective fuzzy linear programming problems [12, 29 32] because fuzzy sets are used as a principal tool to mathematically represent the flexible constraints. Engineering design [13, 17], machine scheduling problem [19], pattern classification [15], water supply planning and resource management [16], production planning [8] and project management [26] are some potential real life applications where the optimization problems are MOFLP problems. For a recent survey of fuzzy set based optimization problems, one can look at [14, 25] and references therein.

An important aspect of a MOFLP problem is its solution concept. The notion of Pareto-optimality is a widely accepted solution concept for classical multi-objective linear programming (MOLP). But surprisingly there is no comprehensive and analogous study on a solution concept for MOFLP problems. Werners [22] extended the notion of Pareto-optimality for MOFLP and called it fuzzy-efficiency. The concept of fuzzy-efficiency for MOFLP is partially inconsistent with the concept of Pareto-optimality for MOLP. In MOLP, constraints are used to describe the set of feasible solutions and objective functions are used to characterize the Pareto-optimal solutions among the feasible solutions. This spirit is somewhat missing in the notion of fuzzy-efficiency where both objective functions and constraints participate equally in defining it. In other words, it is not clear as to what is the meaning of a feasible solution of MOFLP problem in terms of flexible constraints. And indeed it makes sense to talk about feasibility ahead of any kind of optimality/Pareto-optimality for an optimization problem inhand.

With this motivation, in this paper, a solution concept of Pareto-optimality for MOFLP problem is defined and an approach is proposed to extract one such solution from the set of optimal solutions of an equivalent optimization problem formulated by Dubey andMehra [5].

Note that all the approaches to model MOFLP are based on the assumption that preferences can be expressed using numerical values in [0,1]. But often the experts suffer from a lack of knowledge about the exact preference values that they can attach with their decisions. In such a case, intervals have been used by decision makers to express their preferences. One can refer to [1, 9] to get more insight on the significance of using interval-valued preferences in flexible constraints. If the preference degree of a vector to be feasible with respect to a flexible constraint of MOFLP is an interval then, such a constraint is a flexible constraint with interval uncertainty. Thus, flexible constraint with interval uncertainty generalizes the notion of flexible constraint. If at least one constraint in a MOFLP is a flexible constraint with interval uncertainty, then a problem is termed as MOFLP with interval uncertainty (MOFLPIU).

In [5], MOFLP is studied from a perspective of bipolar view in preference modeling [2, 6] and a scheme is proposed to solve MOFLP problem. Our second aim in this paper is to extend this bipolar approach to MOFLPIU problem.

Since the degree to which a solution of MOFLPIU satisfies a constraint is an interval, the obvious question is: how to compare interval satisfaction degrees of two solutions? There are two possible ways to answer this question: one is to fix a linear order on the set L = {[a, b]  | a, b ∈ [0, 1] , a ≤ b}, namely an order that allows to compare any two intervals, thereafter aggregate the interval-valued preferences using some aggregation operator on L. Another way could be to first resolve the interval uncertainty and then aggregate the resultant preferences which take the values in numeral scale [0, 1] . The latter approach is opted in this paper. The prime reason for taking this route is that the standard order ‘≤’ is a total order on [0, 1], thereby making ([0, 1] , ≤) a complete lattice; thus any two acceptable solutions in ([0, 1] , ≤) are comparable. Another reason is that this approach results in an optimization problem which is rather simple to solve.

In [4, 8], authors have studied flexible linear programming with interval uncertainty (FLPIU) and proposed different schemes to solve FLPIU problems. In [4], a flexible constraint with interval uncertainty is mathematically represented using Atanassov’s intutionistic (I) fuzzy set. The crux of this approach is that intervals can be compared via a selection of precise substitute to intervals and hence these I-fuzzy sets are transformed to fuzzy sets by resolving λ times indeterminacy in I-fuzzy sets in favor of the membership function. By doing so, FLPIU is essentially transformed to FLP. In [7, 8], an interval type-2 fuzzy LP problem is studied and two approaches are proposed to solve it. One of these two approaches transforms the problem into an equivalent fuzzy LP problem and the other one transforms it into an interval-valued LPproblem.

In this paper, the approach proposed in [4] is extended to MOFLPIU problem by integrating the concept of bipolarity in it. An interval-valued (IV) fuzzy set representation is used to represent a flexible constraint with interval uncertainty. The interval uncertainty is resolved using the Hurwicz optimism-pessimism criterion [10] and then the resultant MOFLP problem is studied in the heterogeneous bipolar framework [2, 4]. It is important to remark here that our approach can not be compared with approaches presented in [7, 8] for the reason that the modelling frameworks are different while we use bipolar approach, which is more reasonable [5], to model MOFLPIU whereas in [7, 8] the model proposed is derived from the traditional Ballman and Zadeh’s principle. Secondly, we have worked with Pareto-optimality and a method to extract the same in a multi-objective framework, while no such idea is involved in [7, 8].

The paper is organized as follows. In Section 2, a solution concept of Pareto-optimality is defined for MOFLP problems and an approach is proposed to extract a Pareto-optimal solution of a MOFLP problem. In Section 3, a mathematical representation of a flexible constraint with interval uncertainty as an IV-fuzzy set and the Hurwicz optimism-pessimism criterion are explained. An equivalent optimization problem of MOFLPIU problem is formulated in Section 4. In Section 5, an optimistic-pessimistic approach is proposed to formulate a deterministic equivalent of this optimization problem. The approach is illustrated through a numerical example. The paper ends with some concluding remarks in Section 6.

2 Multi-objective flexible linear programming

In this section, the solution concept of Pareto-optimality is extended to MOFLP problem and a method is proposed to extract a Pareto-optimal solution of it.

The general framework of MOFLP problem is

( MOFLP ) max ( c 1 T x , … , c p T x ) subject to a i T x ⪯ b i i = 1 , … , m x ∈ S ,

where c k ∈ ℝ n , k = 1 , … , p , a i ∈ ℝ n , b i ∈ ℝ , i = 1 , … , m , and S ⊆ ℝ n comprises of all linear inequalities in x. The constraints a i T x ⪯ b i are flexible resource constraints in which the resources availability b _i can be relaxed to positive tolerances r _i ,  i = 1, …, m. Here the symbol “⪯” is essentially less than or equal to and to be understood in the sense of a chosen membership function μ _{R i} for the fuzzy set R _i corresponding to the ith flexible constraint. For i = 1, …, m,  μ _{R i} is defined as follows: μ R i ( a i T x ) = { 1 , a i T x < b i , h R i ( a i T x ) , b i ≤ a i T x ≤ b i + r i , 0 , a i T x > b i + r i , where r _i is a positive admissible tolerance in the flexible constraint R _i and h R i : ℝ → [ 0 , 1 ] is a continuous monotonically decreasing function such that h _{R i}  (b _i  + r _i ) =0,  h _{R i}  (b _i ) =1 .

3 An IV-linear inequality and Hurwicz criterion

From now onwards the paper discusses the MOFLPIU problem. This section aims to build a brief background to facilitate further discussion.

A general model of MOFLPIU can be stated as follows:

( MOFLPIU ) max ( c 1 T x , … , c p T x ) subject to a i T x ⪯ IV b i i = 1 , … , m x ∈ S ,

where c k ∈ ℝ n , k = 1 , … , p , a i ∈ ℝ n , b i ∈ ℝ , i = 1 , … , m , and S ⊆ ℝ n comprises of all linear inequalities in x. The symbol ‘’ represents that preference values for each instantiation of the variables of a constraint are interval and call it as ‘interval-valued (IV)- linear inequality’.

In [4], an IV-linear inequality is mathematically represented using I-fuzzy set. In this paper an IV-fuzzy set is used as a mathematical tool to represent an IV-linear inequality. An IV-fuzzy set A is characterized by its membership function μ A : X → L , where L = {[a, b]  | a, b ∈ [0, 1] , a ≤ b}, which assigns to each x ∈ X, an interval-valued membership degree, i.e. μ A ( x ) = [ μ _ A ( x ) , μ ¯ A ( x ) ], where μ _ A , μ ¯ A : X → [ 0 , 1 ], are respectively lower and upper membership functions. Though motivation behind the two extensions viz. I-fuzzy set and IV-fuzzy set of a fuzzy set is different but they turned out to be mathematically equivalent; to convince one can refer to [3].

For a ∈ ℝ n and b ∈ ℝ , a general IV-linear inequality, aTx⪰IVb, can be characterized by an IV-fuzzy set A defined by the following lower and upper membership functions.

The lower membership function μ _ A is defined as μ _ A ( a T x ) = { 0 , a T x < b - g , h 1 ( a T x ) , b - g ≤ a T x ≤ b , 1 , a T x > b where g > 0 is the tolerance in μ _ A of a ^T x and h 1 : ℝ → [ 0 , 1 ] is a continuous non-decreasing function with h₁ (b - g) =0 and h₁ (b) =1.

Now, on the lines of [4], the upper membership function can be constructed in three ways namely, optimistic, pessimistic, and mixed.

In optimistic case, the upper membership function is defined as follows:

μ ¯ A ( a T x ) = { 0 , a T x < b - g - q , h 2 ( a T x ) , b - g - q ≤ a T x ≤ b , 1 , a T x > b where q > 0 , h 2 : ℝ → [ 0 , 1 ] is a continuous non-decreasing function such that h₂ (b - g - q) =0,  h₂ (b) =1, and h₁ (a ^T x) ≤ h₂ (a ^T x) (see Fig. 1).

The upper membership function for the pessimistic case is defined as follows: μ ¯ A ( a T x ) = { 0 , a T x < b - g , h 3 ( a T x ) , b - g ≤ a T x ≤ b - g + d , 1 , a T x > b - g + d where d ∈ ( 0 , g ) , h 3 : ℝ → [ 0 , 1 ] is a continuous non-decreasing function such that h₃ (b - g) =0,  h₃ (b - g + d) =1,  h₁ (a ^T x) ≤ h₃ (a ^T x) (see Fig. 2).

The upper membership function in the mixed case is defined next. μ ¯ A ( a T x ) = { 0 , a T x ≤ b - g - q ˇ , h 4 ( a T x ) , b - g - q ˇ ≤ a T x ≤ b - g - q ˇ + d ˇ , 1 , a T x ≥ b - g - q ˇ + d ˇ where q ˇ > 0 , d ˇ ∈ ( q ˇ , g + q ˇ ) , and h 4 : ℝ → [ 0 , 1 ] is a continuous non-decreasing function such that h 4 ( b - g - q ˇ ) = 0 , h 4 ( b - g - q ˇ + d ˇ ) = 1 , h 1 ( a T x )≤h₄ (a ^T x) (see Fig. 3).

Note that IV-linear inequality aTx⪯IVb, is interpreted as -aTx⪰IV - b .

As discussed earlier, the degree to which a solution of (MOFLPIU) satisfies a constraint is an interval. Therefore our main concern is; how to compare interval satisfaction degrees of two solutions? There are many methods available to compare intervals in the literature but herein the Hurwicz optimism-pessimism criterion [10] is used as it allows comparison of intervals via a selection of precise substitute to intervals.

The Hurwicz optimism-pessimism criterion, proposed in 1951, ascribes to a decision a value which is a weighted sum of its worst and best possible values, ( 1 - λ ) a _ + λ a ¯; parameter λ ∈ [0, 1] being interpreted as a degree of optimism.

Let [a₁, b₁] ,   [a₂, b₂]   ∈  L  =   {[a, b] |a, b  ∈   [0, 1] , a  ≤  b} and λ ∈ [0, 1]; Define [ a 1 , b 1 ] ≤ λ [ a 2 , b 2 ] ⇔ ( 1 - λ ) a 1 + λ b 1 ≤ ( 1 - λ ) a 2 + λ b 2 . Then ≤ _λ is a pre-order on L and hence any two intervals in L are comparable. In [11], the authors have elaborated that why it is reasonable to use the Hurwicz optimism-pessimism criterion for comparing two intervals.

4 The Equivalent Model for MOFLPIU

In this section, the solution concept of Pareto-optimality is extended to (MOFLPIU) and then an equivalent optimization problem is formulated for computationally extracting a Pareto-optimal solution of (MOFLPIU).

Let μ _ R i and μ ¯ R i , i = 1 , … , m , denote respectively the lower membership function and the upper membership function of the IV-fuzzy set R i representing an IV-linear inequality corresponding to ith flexible constraint with interval uncertainty of (MOFLPIU).

Definition 4. For λ ∈ [0, 1], the feasibility degree of x ∈ S of (MOFLPIU) is defined as β x λ = min { ( 1 - λ ) μ _ R 1 ( a 1 T x ) + λ μ ¯ R 1 ( a 1 T x ) , … , ( 1 - λ ) μ _ R m ( a m T x ) + λ μ ¯ R m ( a m T x ) } . Then, x is called as β x λ - feasible solution of MOFLPIU.

Definition 5. For λ ∈ [0, 1], a β x λ - feasible solution x ∈ S is Pareto-optimal for (MOFLPIU) if there does not exist another β y λ - feasible solution y ∈ S such that β y λ ≥ β x λ and c k T y ≥ c k T x ,  k = 1, … p, and either β y λ > β x λ or c k T y > c k T x for at least one k.

The maximization in (MOFLPIU) is interpreted in the sense of satisfaction of aspiration levels Z _k ,  k = 1, …, p, which are given by DM for the kth objective c k T x. Then problem (MOFLPIU) is equivalent to the following flexible constraint satisfaction problem with interval uncertainty (FCSPIU) which requires to find an x ∈ ℝ n such that ( FCSPIU ) c k T x ⪰ IV Z k k = 1 , … , p a i T x ⪯ IV b i i = 1 , … , m x ∈ S .

Let μ _ G k and μ ¯ G k , k = 1 , … , p , denote respectively the lower membership function and the upper membership function of the IV-fuzzy set G k representing an IV-linear inequality corresponding to kth objective function.

Using the Hurwicz criterion, with degree of optimism λ ∈ [0, 1], IV-fuzzy sets G k and R i are transformed into fuzzy sets G _k and R _i respectively with the following membership functions: f G k ( c k T x ) = ( 1 - λ ) μ _ G k ( c k T x ) + λ μ ¯ G k ( c k T x ) k = 1 , … , p f R i ( a i T x ) = ( 1 - λ ) μ _ R i ( a i T x ) + λ μ ¯ R i ( a i T x ) i = 1 , … , m . Consequently, problem (FCSPIU) transforms to the following FCSP which requires to find x ∈ ℝ n such that

( FCSP - 2 ) c k T x ⪰ Z k k = 1 , … , p a i T x ⪯ b i i = 1 , … , m x ∈ S .

The bipolar approach proposed in [5] is followed next and hence the resultant fuzzy sets G _k ,  k = 1, …, p, are aggregated using a strictly monotone OWA operator and the fuzzy sets R _i ,  i = 1, …, m, using the ‘min’ operator.

μ D P ( x ) = F ( f G 1 ( c 1 T x ) , , … , f G p ( c p T x ) ) ,and

μ D N ( x ) = min ( f R 1 ( a 1 T x ) , … , f R m ( a m T x ) ) .

The problem (MOFLPIU) is equivalent to

max x ∈ S min { μ D P ( x ) , μ D N ( x ) } , which can be re-casted into the following problem:

( OP-4 ) max α subject to F ( f G 1 ( c 1 T x ) , … , f G p ( c p T x ) ) ≥ α f R i ( a i T x ) ≥ α i = 1 , … , m x ∈ S 0 ≤ α ≤ 1.

A Pareto-optimal solution of (MOFLPIU) can be extracted from the set of optimal solutions of (OP-4) on the same lines as explained in Subsection 1 in the context of (MOFLP).

5 The optimistic-pessimistic approach

In this section, an optimistic-pessimistic approach is proposed where IV-linear inequalities corresponding to objective functions are interpreted in optimistic sense while the IV-linear inequalities corresponding to the original flexible constraints with interval uncertainty are taken in the pessimistic sense. The reason for doing so is not far fetched and will shortly be revealed.

Consider the kth IV-linear inequality c k T x ⪰ IV Z k. Let g _k  > 0, q _k  > 0 be the tolerances for the kth objective function. Then the lower membership and the upper-membership functions in optimistic case are defined as continuous linear functions

μ _ G k ( c k T x ) = { 0 , c k T x < Z k - g k , 1 + c k T x - Z k g k , Z k - g k ≤ c k T x ≤ Z k , 1 , c k T x > Z k and μ ¯ G k ( c k T x ) = { 0 , c k T x < Z k - g k - q k , c k T x - ( Z k - g k - q k ) g k + q k , Z k - g k - q k ≤ c k T x ≤ Z k , 1 , c k T x > Z k .

Let r _i ,  t _i , with 0 < t _i  < r _i , be the tolerances for the IV linear inequality in the ith constraint, i = 1, …, m. The lower membership and the upper membership functions in pessimistic case are defined as follows: μ _ R i ( a i T x ) = { 1 , a i T x < b i , 1 + b i - a i T x r i , b i ≤ a i T x ≤ b i + r i , 0 , a i T x > b i + r i and μ ¯ R i ( a i T x ) = { 1 , a i T x < b i + r i - t i , ( b i + r i ) - a i T x t i , b i + r i - t i ≤ a i T x ≤ b i + r i , 0 , a i T x > b i + r i .

For λ = 1 2, the functions f _{G k} ,  k = 1, …, p, in optimistic case and  f _{R i} ,  i = 1, …, m, in pessimistic case become:

∥ f G k ( c k T x ) = { 0 , c k T x < Z k - g k - q k , f G k 1 ( c k T x ) , Z k - g k - q k ≤ c k T x ≤ Z k - g k , f G k 2 ( c k T x ) , Z k - g k ≤ c k T x ≤ Z k , 1 , c k T x > Z k f R i ( a i T x ) = { 1 , a i T x < b i , f R i 1 ( a i T x ) , b i ≤ a i T x ≤ b i + r i - t i , f R i 2 ( a i T x ) , b i + r i - t i ≤ a i T x ≤ b i + r i , 0 , a i T x > b i + r i . where f G k 1 ( c k T x ) = c k T x - ( Z k - g k - q k ) 2 ( g k + q k ) ,

f G k 2 ( c k T x ) = 1 + ( c k T x - Z k ) ( 2 g k + q k 2 g k ( g k + q k ) ) ,

f R i 1 ( a i T x ) = 1 + b i - a i T x 2 r i ,

and f R i 2 ( a i T x ) = r i + t i 2 t i ( 1 + b i - a i T x r i ) .

The resultant f _{G k} ,  k = 1, …, p, and f _{R i} ,  i = 1, …, m are S-shaped. Note that on the same lines as above f _{G k} ,  k = 1, …, p, and f _{R i} ,  i = 1, …, m can be computed respectively in pessimistic and optimistic cases.

We can observe from Fig. 4 that taking optimistic approach to interpret IV-inequalities c k T x ⪰ IV Z k , k = 1 , … , p , allows us to relax the lower bounds for aspiration levels in the resultant membership functions f _{G k} ,  k = 1, …, p. As objective functions correspond to positive preferences of DM, it makes sense to use the optimistic approach. Further, as constraints correspond to negative preferences of DM, in view of Fig. 5, it makes sense to take the pessimistic approach to interpret IV-inequalities

When F is a strictly monotone OWA operator, following the method of [27] problem (OP-4) can be reformulated as the following mixed integer linear programming problem: ( MILP - 1 ) max α subject to GY ≤ 0 y j e - O 1 - M Δ k 1 - M Δ j 2 ≤ 0 j = 1 , … , p y j e - O 2 - ( 1 - M ) Δ k 1 - M Δ j 2 ≤ 0 j = 1 , … , p e T Δ j ≤ p - j j = 1 , … , p w 1 y 1 + … + w p y p ≥ α f R i 1 ( a i T x ) ≥ α i = 1 , … , m f R i 2 ( a i T x ) ≥ α i = 1 , … , m δ G k , δ J k ∈ { 0 , 1 } j , k = 1 , … , p 0 ≤ y j ≤ 1 , j = 1 , … , p 0 ≤ α ≤ 1 , x ∈ S , ∥ whereG = [ - 1 1 0 … 0 0 0 - 1 1 … 0 0 ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 0 … - 1 1 ]: a (p - 1) × p matrix; Y: a p column vector whose components are y _j ; e: a p column vector with all entries 1; O₁: a p column vector whose components are

f G k 1 ( c k T x ); O₂: a p column vector whose components are

f G k 2 ( c k T x ); Δ k 1: a p column vector whose components are _{dt Gk}  ; Δ j 2: a p column vector whose components are _dtj k ; M: a large positive real number.

In particular, when F is an ‘average’ operator, following the approach of Yang et al. [28], problem (OP-4) can be reformulated as the following mixed integer linear programming problem:

(MILP - 2) max α

subject to

f G k 1 ( c k T x ) + M δ G k ≥ α k k = 1 , … , p f G k 2 ( c k T x ) + M ( 1 - δ G k ) ≥ α k k = 1 , … , p f R i 1 ( a i T x ) ≥ α i = 1 , … , m f R i 2 ( a i T x ) ≥ α i = 1 , … , m 1 p ∑ k = 1 p α k ≥ α x ∈ S , α k ∈ [ 0 , 1 ] k = 1 , … , p δ G k ∈ { 0 , 1 } k = 1 , … , p .

where M is large positive real number.

On the same lines as Proposition 1 in [5] one can easily prove the following result.

Proposition 1. Let F be an average operator. Then problem (OP-4) has an optimal solution if and only if problem (MILP-1) (or (MILP-2)) has an optimal solution. Further, their optimal objective values are equal.

Remark 1. Although, in this discussion value of λ is 1 2 , yet for any choice λ ∈ (0, 1) the resultant membership functions f _{G k} ,  k = 1, …, p, and f _{R i} ,  i = 1, …, m, will be S-shaped functions. The proposed approach still works for any choice of λ ∈ (0, 1) .

In order to illustrate this approach, consider the following example of MOFLPIU.

max 2 x 1 + x 2 with Z 1 = 8 , g 1 = 3 , q 1 = 2 max 3 x 1 + 4 x 2 with Z 2 = 13 , g 2 = 3 , q 2 = 2 max x 1 + 5 x 2 with Z 3 = 14 , g 3 = 2 , q 3 = 1 subject to x 1 ≼ I V 3 with r 1 = 2 , s 1 = 1 2 x 1 + x 2 ≼ I V 7 with r 2 = 3 , s 2 = 1 x 1 + x 2 ≤ 4 x 1 , x 2 ≥ 0.

For λ = 1 2, the equivalent optimization problem analogous to (MILP-2) is given by

max α s u b j e c t t o 2 x 1 + x 2 + 10 M δ G 1 − 10 α 1 ≥ 3 2 x 1 + x 2 − 3.75 M δ G 1 − 3.75 α 1 ≥ − 3.75 M + 4.25 3 x 1 + 4 x 2 + 10 M δ G 2 − 10 α 2 ≥ 8 3 x 1 + 4 x 2 − 3.75 M δ G 3 − 3.75 α 2 ≥ − 3.75 M + 9.25 x 1 + 5 x 2 + 6 M δ G 3 − 6 α 3 ≥ 11 x 1 + 5 x 2 − 2.4 M δ G 3 − 2.4 α 3 ≥ − 2.4 M + 8.6 x 1 + 4 α ≤ 7 x 1 + 2 α ≤ 5 2 x 1 + x 2 + 6 α ≤ 13 2 x 1 + x 2 + 2.4 α ≤ 10 x 1 + x 2 ≤ 4 x 1 , x 2 ≥ 0 1 3 ∑ k = 1 3 α k ≥ α α , α k ∈ [ 0 , 1 ] k = 1 , 2 , 3 δ G k ∈ { 0 , 1 } k = 1 , 2 , 3 ,

and M is large positive real number.

The optimal solution is x 1 * = 1.5 , x 2 * = 2.5 , α 1 * = 0.3333 , α 2 * = 1 , α 3 * = 1 with an overall satisfaction 0.7779. This problem has a unique optimal solution and therefore x 1 * = 1.5 , x 2 * = 2.5 is a Pareto-optimal solution of given MOFLPIU.

6 Conclusions

In this paper, an attempt is made to build a theoretical framework for multi-objective linear programming subject to hard and flexible constraints by defining the solution concept of Pareto-optimality. The bipolar approach [5] is extended to solve MOFLP for Pareto-optimal solution. This study is extended to MOFLPIU problems. A bipolar approach is proposed to formulate equivalent optimization problem of MOFLPIU by extending the approach in [4] to solve FLPIU. An optimistic-pessimistic framework is proposed to formulate the deterministic equivalent optimization problem. In MOFLPIU, bipolarity is integrated in two stages. First in the equivalent optimization problem formulation and secondly in its deterministic equivalent formulation by taking an optimistic approach to represent the flexible constraints with interval uncertainty corresponding to objective functions and a pessimistic approach for the remaining flexible constraints with interval uncertainty.

The other way to model (MOFLPIU) could be by considering some total order on the set L = {[a, b] | [a, b] ⊆ [0, 1]} , in order to appropriately define interval-valued preference relation, and then aggregate the interval-valued preferences directly without resolving the interval uncertainty first (i.e., the Hurwicz criterion). Though apparently this approach will lead to rather hard mathematics and complex optimization models but nevertheless it can be taken up in future research.

7 Appendix

Proposition 2. Let (x^**, θ^**, β^**) be an optimal solution of problem (OP-2). If either (x^**, θ^**, β^**) is unique or μ G k ( c k T x * * ) < 1 , ∀ k = 1 , … , p , then x^** is a Pareto-optimal solution of (MOFLP).

Proof. Suppose on the contrary that x^** is not a Pareto-optimal solution of (MOFLP). Then there exists a β _y - feasible solution y of (MOFLP) such that either

β y > β * * , c k T y ≥ c k T x * * , k = 1 , … , p , or

As μ _{G k} ,  k = 1, …, p, are monotonically increasing and F is strictly increasing, F ( μ G 1 ( c 1 T y ) , … , μ G p ( c p T y ) ) ≥ F ( μ G 1 ( c 1 T x * * ) , … , μ G p ( c p T x * * ) ) ≥ θ * * . Therefore, (y, θ^**, β^**) is feasible to (OP-2).

In case (i), as μ _{G k} ,  k = 1, …, p, are monotonically increasing and F is strictly increasing, F ( μ G 1 ( c 1 T y ) , … , μ G p ( c p T y ) ) + β y > F ( μ G 1 ( c 1 T x * * ) , … , μ G p ( c p T x * * ) ) + β * * ≥ θ * * + β * * , which contradicts optimality of (x^**, θ^**, β^**) for (OP-2). Hence this case is ruled out. The following discussion pertains to case (ii).

If (x^**, θ^**, β^**) is the unique optimal solution of (OP-2), then θ^** + β^** is the optimal objective value and hence (y, θ^**, β^**) is also an optimal solution of (OP-2) contradicting uniqueness of (x^**, θ^**, β^**).

Next consider the case when μ G k ( c k T x * * ) < 1 , k = 1 , … , p .

Then, for k = 1, …, p,   c k T x * * ∈ [ Z k - g k , Z k ) .Hence, μ G k ( c k T y ) ≥ μ G k ( c k T x * * ) , k = 1 , … , p , k ≠ k 0 ,and μ G k 0 ( c k 0 T x * * ) > μ G k 0 ( c k 0 T y ) since μ _{G k} is strictly increasing function over [Z _k  - g _k , Z _k ) . Again, F is strictly increasing so, F ( μ G 1 ( c 1 T y ) , … , μ G p ( c p T y ) ) + β y > F ( μ G 1 ( c 1 T x * * ) , … , μ G p ( c p T x * * ) ) + β * * ≥ θ * * + β * * , which contradicts optimality of (x^**, θ^**, β^**) for(OP-2). Hence the requisite result follows.□

Proposition 3. Let ( x ˆ , p s x ˆ ( s = 1 , … , l ) ) be an optimal solution of problem (OP-3). Then, x ˆ is a Pareto-optimal solution of problem (MOFLP).

Proof. It is evident that ( x ˆ , θ * * , β * * ) is an optimal solution of (OP-2). Suppose x ˆ is not a Pareto-optimal solution of (MOFLP). Then there exists a β _y - feasible solution y of the (MOFLP) such that either

β y > β * * , c k T y ≥ c k T x ˆ , k = 1 , … , p , or