New efficiency conditions for multiobjective interval-valued programming problems

Information Sciences 420 (2017) 235–248 Contents lists available at ScienceDirect Information Sciences journal homepage: www.elsevier.com/locate/ins New efficiency conditions for multiobjective interval-valued programming problems R R. Osuna-Gómez a , ∗, B. Hernández-Jiménez b , Y. Chalco-Cano c , G. Ruiz-Garzón d a Departamento de Estadística e I.O., Fac. Matemáticas, Universidad de Sevilla, c/Tarfia s/n 41012 Sevilla, Spain b Departamento de Economía, Métodos Cuantitativos e H. Económica. Área de Estadística e I.O., Universidad Pablo de Olavide, Sevilla, Spain c Instituto de Alta Investigación, Universidad de Tarapacá, Casilla, 7D, Arica, Chile d Departamento de Estadística e I.O., Universidad de Cádiz, Campus de Jerez, 11405, Jerez, Spain a r t i c l e i n f o Article history: Received 4 February 2016 Revised 11 July 2017 Accepted 6 August 2017 Available online 7 August 2017 Keywords: Multiobjective programming Interval-valued functions Generalized Hukuhara differentiability Fritz–John and Karush–Kuhn–Tucker efficiency conditions a b s t r a c t In this paper, we focus on necessary and sufficient efficiency conditions for optimization problems with multiple objectives and a feasible set defined by interval-valued functions. A new concept of Fritz-John and Karush–Kuhn–Tucker-type points is introduced for this mathematical programming problem based on the gH-derivative concept. The innovation and importance of these concepts are presented from a practical and computational point of view. The problem is approached directly, without transforming it into a real-valued programming problem, thereby attaining theoretical results that are more powerful and computationally more efficient under weaker hypotheses. We also provide necessary conditions for efficiency, which have been inexistent in the relevant literature to date. The identification of necessary conditions is important for the development of future computational optimization techniques in an interval-valued environment. We introduce new generalized convexity notions for gH-differentiable interval-valued problems which are a generalization of previous concepts and we prove a sufficient efficiency condition based on these concepts. Finally, the efficiency conditions for deterministic programming problems are shown to be particular instances of the results proved in this paper. The theoretical developments are illustrated and justified through several numerical examples. © 2017 Elsevier Inc. All rights reserved. 1. Introduction In many real-life engineering and economic problems, the occurrence of vagueness and/or imprecision in the data is inevitable due to the increasing complexity of the environment. Deterministic or classic optimization models therefore remain inadequate or insufficient to model real-life optimization problems. The inaccuracy lies in the objectives pursued and/or in the definition of possible alternatives. Due to the representation and utilization of imprecise, vague and uncertain information that abounds in real-world situations, various probabilistic and non-probabilistic approaches have been developed in recent years, such as interval analysis, fuzzy set theory and stochastic theory, respectively, whose advantages and disadvantages differ greatly. If the coefficients R The research in this paper has been supported by Fondecyt-Chile 1151154 project and by MINECO /FEDER(UE) through grant MTM2015-66185-P ∗Corresponding author. E-mail addresses: r[email protected] (R. Osuna-Gómez), [email protected] (B. Hernández-Jiménez), [email protected] (Y. Chalco-Cano), [email protected] (G. Ruiz-Garzón). http://dx.doi.org/10.1016/j.ins.2017.08.022 0020-0255/© 2017 Elsevier Inc. All rights reserved. 236 R. Osuna-Gómez et al. / Information Sciences 420 (2017) 235–248 are assumed as random variables with known distributions then we have a stochastic optimization problem. In fuzzy optimization, the coefficients are assumed to be fuzzy sets, and if it is possible to establish a range of variation in the data, then an interval-valued optimization problem is suitable. Several authors are of the opinion that imprecision cannot be equated with randomness, [40] , in fact, there is a qualitatively different type of imprecision and vagueness which cannot be tackled with probabilistic tools. On the other hand, stochastic optimization and fuzzy optimization involve several drawbacks. Firstly, the distribution and membership specification functions are very subjective. Secondly, the stochastic and fuzzy optimization approaches require the evaluation of the solution on the whole uncertainty set in order to determine its expected cost, which is, in general, computationally demanding. In addition, for a decision maker it is not always easy to specify the membership function or probability distribution in an inexact environment. Therefore, interval-valued optimization problems may provide an alternative choice for the consideration of uncertainty in optimization problems. Several of the most important factors that show the importance of investigating the uncertainty interval include simplicity, measurement, privacy, and round-off errors [28] . The modelling of a system under interval uncertainty requires no additional information such as membership function and probability distribution. It is often possible to establish a variation range in the data, and therefore modelling the problem using interval-valued functions for objectives and constraints becomes an alternative. Consequently, interval analysis was introduced as an attempt to handle interval (non-probabilistic) uncertainty and/or imprecision that appears in many mathematical and/or computer models of deterministic real-world phenomena. In the literature, numerous examples can be found where imprecision in real problems is formulated using interval analysis as a mathematical tool. In [8,9,13,16,19] , real examples of problems modelled as data envelopment analysis, goal programming, inexact linear programming problems, optimal real-time control, and minimax regret solutions are shown, where the vagueness is expressed by closed intervals. In [44] , a multi-period portfolio selection problem is discussed in which the returns and the risk of risky assets are characterized by intervals. The interval analysis approach is applied for the handling of imprecise data in financial markets and an interval optimization model is proposed. In [20] , an uncertain multiobjective optimization method is suggested to deal with the crashworthiness design problem of vehicles, in which the uncertainties of the parameters are described by intervals. In the aforementioned examples, and others such as, those found in [3] and [31] , the interval-valued programming problem is converted via any appropriate strategy into a deterministic programming problem, with the aim of applying conventional optimization techniques. In [29] , an overview is provided of multiobjective linear programming problems with interval coefficients. In [38] , weak and strong duality theorems are obtained for a multiobjective problem defined by interval-valued linear objectives and constraints. In [37] , multiobjective programming problems with interval-valued objective functions and real-valued constraints are considered and sufficient Karush–Kuhn–Tucker (hereinafter KKT) optimality conditions are achieved, by supposing differentiable convex or pseudoconvex real-valued endpoint functions. In [33] , the authors consider two order relations on the set of all closed intervals and propose many Pareto optimal-solution concepts for multiobjective programming problems with interval-valued objective functions. These KKT-type sufficient optimality conditions are derived under convexity and/or pseudo-convexity assumptions for endpoint functions. In this paper, efficiency conditions are derived for a multiobjective optimization problem where the objectives and constraints are defined by interval-valued functions. Optimality conditions proposed in previous papers published on this subject are real-valued type (i.e., equations), and not interval-valued type. We define vector interval-valued Karush–Kuhn– Tucker and Fritz-John solutions (hereinafter referred to as vector interval KKT and FJ solutions). These definitions are not equations: they imply that zero belongs to an interval, but they coincide with the classic definitions when the problem is defined by real-valued functions, that is, they generalize the classic definitions to interval-valued enviroment. This approach enables advantage to be taken of all information provided by the interval-valued functions as models of imprecision. Moreover, previous papers suppose very restrictive hypotheses of differentiability for interval-valued functions involved in the problem, see [6] . The convexity or pseudo-convexity for the endpoint functions are also assumed, although very simple functions (such that F (x ) = Cx with C a bounded closed interval, which is a generalization of a linear function) do not verify these hypotheses. Futhermore we provide the necessary efficiency conditions for the interval-valued problems, which have not been proved in previous papers published on this subject. These conditions are crucial for the future development of computational optimization techniques in interval-valued environments. This paper is organized as follows: in Section 2 , basic properties and arithmetic for intervals are introduced. The generalized Hukuhara difference is applied to define the difference between any two closed intervals. Using this gH-difference and the limit concept for interval-valued functions, we consider the gH-derivative of an interval-valued function. In Section 3 , a brief example is given to illustrate our study. We formulate the multiobjective programming problem with interval-valued objective and constraint functions, and we give several solution concepts for this problem. In Section 4 , the necessary KKT and FJ-type efficiency conditions are derived for the solution concepts given in Section 3 . In Section 5 , new generalized convexity concepts are introduced for the multiobjective interval-valued programming problem in order to prove sufficient KKT and FJ-type efficiency conditions. In Section 6 , the theoretical developments are illustrated through numerical examples. R. Osuna-Gómez et al. / Information Sciences 420 (2017) 235–248 237 2. Preliminaries We denote K C as the family of all bounded closed intervals in R , i.e., K C = { [ a , a ] / a , a ∈ R and a ≤a } . For A = [ a , a ] , B = b , b ∈ K C and λ∈ R , we consider the following operations A + B = [ a , a ] + b , b = a + b , a + b , (1) λA = λ[ a , a ] = [ λa , λa ] if λ≥0 , [ λa , λa ] if λ< 0 . (2) From (1) and (2) , then we have −A = [ −a , −a ] and B −A = B + (−A ) = b −a , b −a (the Minkowski difference). The space K C with operations (1) and (2) is not a linear space since an interval has no inverse element and therefore subtraction has no suitable properties. For example, if A = [ −1 , 1] , then A −A = A + ( −1 ) A = A + [ −1 , 1 ] = [ −1 , 1 ] + [ −1 , 1 ] = [ −2 , 2 ]  = { 0 } (3) One crucial concept for optimization theory involves the adequate definition of derivative. To this end it is necessary to evaluate the difference between two intervals. To overcome (3) , Hukuhara defined what has been named the Hukuhara difference. If A = B + Cthen the Hukuhara difference (H-difference) of A and B , denoted as A −H B, is equal to C . Although, A −H A = { 0 } , the H-difference of two intervals does not always exist. For example, let A = [ −2 , −1] and B = [ −4 , 0] , therefore A −H B exists if there exists C such that: [ −2 , −1] = [ −4 , 0] + [ c , c ] Therefore, c = 2 and c = −1 , but C is not an interval. On this topic, Stefanini and Bede in [34] introduced the following definition. Definition 1. The generalized Hukuhara difference ( gH -difference) of two intervals A and B is defined as follows: A gH B = C ⇔ (a ) A = B + C or (b) B = A + (−1) C. In case (a), the gH -difference coincides with the H-difference. Note that the gH -difference and the difference defined in [23] are the same concept. The gH -difference exists for any two compact intervals, it is unique, A gH A = { 0 } and it is equal to (see [5,34] ): A gH B = min a −b , a −b , max a −b , a −b . For example, if A = [ −1 , 1] , A gH A = [ min { (−1) −(−1) , 1 −1 } , max { (−1) −(−1) , 1 −1 } ] =[0 , 0] . Let us give an ordering relation between any two closed intervals. Definition 2. Let A = [ a , a ] and B = [ b , b ] be two closed intervals in R . It is said that • A B ⇔ a ≤b and ¯ a ≤¯ b . • A B ⇔ A B and A  = B, i.e., a ≤b , and ¯ a ≤¯ b , with a strict inequality. • A ≺B ⇔ a < b and ¯ a < ¯ b . It is clear that A ≺B ⇒ A B ⇒ A B . Lemma 1. For gH -difference it is verified that A B if and only if A gH B [0 , 0] . Analogously, this holds for and ≺. Proof. A B ⇔ a ≤b and a ≤b ⇔ a −b ≤0 and a −b ≤0 ⇔ min a −b , a −b ≤0 and max a −b , a −b ≤0 ⇔ A gH B [0 , 0] . A B ⇔ a ≤b , a ≤b and a  = b or a  = b ⇔ min a −b , a −b ≤0 and max a −b , a −b ≤0 with a strict inequality ⇔ A gH B [0 , 0] . Similarly, this holds for ≺.  Remark 1. We introduce the following notation. 1. As a generalization of (2) , let us consider A = [ a , a ] ∈ K C and λ= (λ1 , λ2 ) ∈ R 2 and we denote the linear combination as λ×A , λ×A = λ1 a + λ2 a ∈ R . It is immediate that 238 R. Osuna-Gómez et al. / Information Sciences 420 (2017) 235–248 • If λࣛ 0 ( λ1 , λ2 ≥0) then A B ⇒ λ×A ≤λ×B. • If λ≥0 ( λ1 , λ2 ≥0 and λ = 0) then A B ⇒ λ×A ≤λ×B, A ≺B ⇒ λ×A < λ×B. • If λ> 0 ( λ1 , λ2 > 0) then A B ⇒ λ×A ≤λ×B, A B ⇒ λ×A < λ×B. 2. If A = (A 1 , A 2 , . . . , A p ) is a vector of p closed intervals, A i ∈ K C , i = 1 , . . . , pand λ= (λ1 , λ2 , . . . , λp ) with λi ∈ R 2 then we can denote λ×A = p  i =1 λi ×A i ∈ R . If λ×A = 0 with λij ≥0 but λ = 0 then 0 ∈ u T a , v T a , where a = ( a 1 , . . . , a p ) , a = ( a 1 , . . . , a p ) and u, v ∈ R p , u, v ≥0, but not both zero. Since, if we consider λ1 to be the greatest common divisor of { λ11 , . . . λp1 } and λ2 to be the greatest common divisor of { λ12 , . . . λp2 } , then since λ = 0 we obtain λ1 + λ2 > 0 , and we can suppose that λ1 + λ2 = 1 , therefore λi 1 = λ1 u i , λi 2 = λ2 v i , λ×A = λ1 ( p  i =1 u i a i ) + λ2 ( p  i =1 v i a i ) ⇔ 0 ∈ u T a , v T a . (4) Henceforth, let T be an open and non-empty subset of R . Let F : T → K C be an interval-valued function. We will denote F (t) = [ f (t ) , f (t )] , where f (t) ≤f (t) , ∀ t ∈ T . The functions f and f are called endpoint functions of F . Based on the gH -difference, the following differentiability concept for interval-valued functions can be introduced. Definition 3. [23,34] Let t 0 ∈ T and F : T → K C be. The generalized Hukuhara derivative ( gH -derivative) of F at t 0 is defined as F  (t 0 ) = lim h → 0 F (t 0 + h ) gH F (t 0 ) h . (5) If F  (t 0 ) ∈ K C that satisfies (5) exists, then we say that F is generalized Hukuhara differentiable ( gH -differentiable) at t 0 . In (5) , the limit is taken in the metric space (K C , H) , where H is defined by H(A, B ) = max | a −b | , a −b . The following result relates the gH -differentiability of F to the differentiability of its endpoint functions. Theorem 1. [4] , [23] Let F : T → K C be an interval-valued function. Then F is gH-differentiable at t 0 ∈ T if and only if one of the following cases holds: (a) f and f are differentiable at t 0 and F  (t 0 ) = min ( f )  (t 0 ) , ( f )  (t 0 ) , max ( f )  (t 0 ) , ( f )  (t 0 ) . (b) The lateral derivatives ( f )  −(t 0 ) , ( f )  + (t 0 ) , ( f )  −(t 0 ) and ( f )  + (t 0 ) exist and satisfy ( f )  −(t 0 ) = ( f )  + (t 0 ) and ( f )  + (t 0 ) = ( f )  −(t 0 ) . Moreover F  (t 0 ) = min ( f )  −(t 0 ) , ( f )  −(t 0 ) , max ( f )  −(t 0 ) , ( f )  −(t 0 )  = min ( f )  + (t 0 ) , ( f )  + (t 0 ) , max ( f )  + (t 0 ) , ( f )  + (t 0 ) . R. Osuna-Gómez et al. / Information Sciences 420 (2017) 235–248 239 Example 1. Consider the interval-valued function defined by F (t) = [ −t 2 , t 2 ] , whereby f (t) = −t 2 and f (t) = t 2 are differentiable functions on R and F  (t 0 ) = [ −2 t 0 , 2 t 0 ] if t 0 ≥0 [2 t 0 , −2 t 0 ] if t 0 < 0 = [ −2 , 2] t 0 . Regarding F (t) = [ −| t| , | t| ] , we have that f (t) = −| t| and f (t) = | t| are not differentiable functions at t 0 = 0 . However ( f )  + (0) = ( f )  −(0) = −1 and ( f )  −(0) = ( f )  + (0) = 1 . Therefore, F (0 + h ) gH F (0) h = 1 h ([ −| h | , | h | ] gH [0 , 0]) = | h | h [ −1 , 1] , and F  (0) = [ −1 , 1] . Remark 2. There are other derivative concepts for interval-valued functions which have been used in interval-valued optimization: the Hukuhara derivative ( H -derivative [12] ) and the endpoint derivative ( e -derivative) or weakly derivative [36] . It is well known that if F is H -differentiable, then it has the property that the diameter len ( F ( t )) (length of F ( t )) is nondecreasing when t increases [4,5,34] . On the other hand, F is e -differentiable or weakly-differentiable if and only if the endpoint functions f and f are differentiable. Hence, we can clearly see that the gH -derivative is a more general concept than the Hukuhara and endpoint derivatives. Futhermore, both the Hukuhara and endpoint derivatives are very restrictive, as shown in [6] . 3. Multiobjective constrained interval-valued optimization: motivation and definitions Various approaches, in addition to the traditional Markowitz model, have been proposed in the literature for the analysis of portfolio selection problems. Among these approaches, we can cite the possibilistic portfolio models, which treat the expected return rates of the securities as fuzzy or possibilistic variables, instead of as random variables. In fact, another way to treat uncertainty in decision-making problems consists of assuming that the data is not exact, and that it varies in given intervals. Interval analysis is thus appropriate to handle imprecise input data. Interval programming models of portfolio selection are constructed in [10,17,18,39,41,44] . We can consider a simple example to illustrate our study [3] . Example 2. Let us consider a portfolio management problem and assume that there are 2 investment types, the return of the j th investment type is denoted as R j (j = 1 , 2) and the proportion of total investment funds devoted to this investment type is denoted as x j , i.e., x 1 + x 2 = 1 . In the real setting, since R j s vary due to uncertainties, these are assumed to be random variables which can be represented by the vector of averages γ= (γ1 , γ2 ) ( γj : average rate for R j ) and the covariance matrix { σjk } ( σjk : covariance between R j and R k with σjk = σkj ). The average return associated with the portfolio x = (x 1 , x 2 ) is given by γ1 x 1 + γ2 x 2 and the risk of investment can be formulated as f(x ) = 1 2  2 j=1  2 k =1 σjk x j x k . From a practical point of view, due to various kinds of uncertainties, it is usually difficult to specify the coefficients, i.e., γj and σjk . However, there are some cases where one can estimate the upper and lower bound of the coefficients, [25–27] . It is therefore necessary to solve a nonlinear multiobjective interval-valued optimization model. Minimize  −([ γ1 , γ1 ] x 1 + [ γ2 , γ2 ] x 2 ) , 2  j=1 2  k =1 [ σjk , σjk ] x j x k  , s.t.: x 1 + x 2 = 1 , x 1 , x 2 ≥0 , which can be rewritten as Minimize F (t) = (F 1 (t) , F 2 (t)) , s.t.: −t ≤0 , t −1 ≤0 , t ∈ R , (6) where F 1 , F 2 are interval-valued functions such that F 1 (t) = [ f 1 (t ) , f 1 (t )] and F 2 (t) = [ f 2 (t ) , f 2 (t )] where t = x 1 , 1 −t = x 2 , f 1 (t) = ( γ2 −γ1 ) t −γ2 , f 1 (t) = ( γ2 −γ1 ) t −γ2 , f 2 (t) = ( σ11 −2 σ12 + σ22 ) t 2 + 2( σ12 −σ22 ) t + σ22 , 240 R. Osuna-Gómez et al. / Information Sciences 420 (2017) 235–248 f 2 (t) = ( σ11 −2 σ12 + σ22 ) t 2 + 2( σ12 −σ22 ) t + σ22 . Henceforth, let us consider a general multiobjective problem with objectives and constraints represented by intervalvalued functions. (MIVOP )  Min F (t) , subject to: G j (t) A j , j = 1 , ... , m, t ∈ T , where T is an open set of R , F and G j are interval-valued functions, F : T → ( K C ) p , F (t) = (F 1 (t ) , . . . , F p (t )) , G j : T → K C and A j ∈ K C for j = 1 , . . . , m . The feasible set is denoted as X = { t : G j (t) A j ∀ j = 1 , . . . , m } . By Lemma 1 , G j (t) A j ⇔ H j (t) = G j (t) gH A j [0 , 0] . This is therefore equivalent to considering the following multiobjective optimization problem: (MIVOP ) Min F (t) , s.t.: H j (t) [0 , 0] , j = 1 , . . . , m, t ∈ T . Obviously, H j ( t ) [0, 0] if and only if h j (t) ≤0 , therefore the previous problem is equivalent to the following problem with real-valued constraints. (MIVOP ) Min F (t) , subject to: h j (t) ≤0 , j = 1 , . . . , m, t ∈ T . Example 3. Consider the interval-valued constraint G (t) = [ −1 , 1] t [0 , 1] , and hence g (t) = −t if t ≥0 t if t ≤0 , g (t) = t if t ≥0 −t if t ≤0 , G (t) [0 , 1] ⇔ −t ≤0 and t ≤1 if t ≥0 ⇔ 0 ≤t ≤1 t ≤0 and −t ≤1 if t ≤0 ⇔ −1 ≤t ≤0 . This is equivalent to g (t) −1 ≤0 . Lemma 2. The feasible sets of (MIVOP) and (MIVOP)’ are the same. Proof. By definition, h j (t) = max { g j (t) −a j , g j (t) −a j } , therefore if t is a feasible solution for (MIVOP), then g j (t) ≤a j , and g j (t) ≤a j ⇔ g j (t) −a j ≤0 , and g j (t) −a j ≤0 and hence h j (t) ≤0 and t is a feasible solution for (MIVOP). Reciprocally, if h j (t) ≤0 then H j ( t ) [0, 0], leading to g j (t) −a j ≤0 and g j (t) −a j ≤0 and therefore t is a feasible solution for (MIVOP)’.  Since (MIVOP) has several interval-valued conflicting objective functions, an exact solution of (MIVOP) which minimizes all objective functions simultaneously, may not exist. Therefore, like the classic multiobjetive problem, the solution of (MIVOP) is a compromise or Pareto solution. Definition 4. Let F be a vector interval-valued function defined on T . It is said that t ∗∈ X is a 1. (local) strictly efficient solution of (MIVOP) if there does not exist t ∈ X ( ∃ δ> 0, t ∈ B ( t ∗, δ) ∩ X ) such that F i ( t ) F i ( t ∗), ∀ i and ∃ k such that F k ( t ) F k ( t ∗) (i.e F ( t )  = F ( t ∗)), 2. (local) efficient solution of (MIVOP) if there does not exist t ∈ X ( ∃ δ> 0, t ∈ B ( t ∗, δ) ∩ X ) such that F i ( t ) F i ( t ∗), ∀ i and ∃ k such that F k ( t ) ≺F k ( t ∗), 3. (local) strictly weakly efficient solution of (MIVOP) if there does not exist t ∈ X ( ∃ δ> 0, t ∈ B ( t ∗, δ) ∩ X ) such that F i ( t ) F i ( t ∗), ∀ i , 4. (local) weakly efficient of (MIVOP) if there does not exist t ∈ X ( ∃ δ> 0, t ∈ B ( t ∗, δ) ∩ X ) such that F i ( t ) ≺F i ( t ∗), ∀ i , R. Osuna-Gómez et al. / Information Sciences 420 (2017) 235–248 241 where B ( t ∗, δ) is a neighbourhood of t ∗with radius δ. In [37] , strictly efficient solutions are defined as type-I Pareto solutions, and strictly weakly efficient solutions are defined as weakly type-I Pareto solutions. If we consider a multiobjective optimization problem defined by real-valued functions we have the following equivalences for Pareto-type solutions: • strictly efficient ≡efficient ≡efficient Pareto, • strictly weakly efficient ≡weakly efficient ≡weakly efficient Pareto. Remark 3. The following relations are immediate: efficient ⇐ strictly efficient ⇓ ⇓ weakly efficient ⇐ strictly weakly efficient Example 4. Let us consider the vector interval-valued function F (t) = (F 1 (t) , F 2 ( t )) where F 1 (t) = [0 , 1] and F 2 (t) = [ −2 , −1] t are defined on t ≥0. Then t = 0 is not an efficient solution and it is not a strictly efficient solution for F because, there exists t = 1 such that F 1 (1) = F 1 (0) and F 2 (1) ≺F 2 (0). However t = 0 is a strictly weakly efficient solution and therefore t = 0 is a weakly efficient solution, because no other t exits such that F 1 ( t ) F 1 (0). Let us now consider the vector interval-valued function F (t) = (F 1 (t) , F 2 (t)) defined on t ≥0, where F 1 (t) = [ −1 , 0] tand F 2 (t) = [ −2 , 0] t. Then t = 0 is a weakly efficient solution but not a strictly weakly efficient solution, because there exists t = 1 such that F 1 (1) F 1 (0) and F 2 (1) F 2 (0). In several papers, techniques directly derived from real-valued multiobjective optimization are used to solve intervalvalued multiobjective optimization problems. These papers consider a real-valued function ˆ F (t) = ( f 1 (t ) , f 1 (t ) , . . . , f p (t) , f p (t)) with 2 ×p objective functions to which conventional optimization techniques are directly applied. However this represents a loss of implicit contained information in the interval-valued functions. Let us consider the previous example with F (t) = (F 1 (t) , F 2 (t)) defined on t ≥0, where F 1 (t) = [ −1 , 0] tand F 2 (t) = [ −2 , 0] t. t = 0 is therefore a weakly efficient solution for ˆ F , but is not a strictly weakly efficient solution for F . This illustrates that the solution concepts for interval-valued multiobjective problems are more general than the solution concepts for real-valued multiobjective problems. Modelling of a real-life problem using interval-valued functions is therefore more advantageous than using real-valued modelling since, among other reasons, it allows better reflecting reality and offers more possibilities for solutions to the decision-maker. However, if an interval-valued problem is transformed into a real-valued problem then such advantages are lost. Therefore, it would be interesting to obtain theoretical results that would allow direct resolution techniques to be developed for multiobjective interval-valued problems. 4. Necessary efficiency conditions for multiobjective interval-valued problem In this section, necessary efficiency conditions are provided for interval-valued multiobjective programming problems. It is important to emphasize the innovation of the results proved in this section. In order to obtain the necessary efficiency conditions it is essential to prove the following geometric type result. Theorem 2. Let t ∗∈ X be a local weakly efficient solution for (MIVOP), where F i are gH-differentiable functions at t ∗, for all i = 1 , . . . , p. Let I(t ∗) = { j = 1 , . . . , m : h j (t ∗) = 0 } , whereby the functions h j with j ∈ I ( t ∗) are differentiable functions at t ∗, and h j with j ∈ I ( t ∗) are continuous functions at t ∗. Then the following system has no solution at y ∈ R . yF  i (t ∗) ≺[0 , 0] i = 1 , . . . , p y h  j (t ∗) ≤0 j ∈ I(t ∗) . (7) Proof. Suppose that (7) has a solution at y ∈ R . From Theorem 1 on the characterization of gH-differentiable functions from endpoint functions, we know that for each objective function, F i , one of the following cases holds: (a) f i and f i are differentiable functions and F  i (t ∗) = [ min { f  i (t ∗) , f  i (t ∗) } , max { f  i (t ∗) , f  i (t ∗) } ] ≺[0 , 0] . Therefore, if y is a solution for (7) , then f  i (t ∗) y < 0 and f  i (t ∗) y < 0 . 242 R. Osuna-Gómez et al. / Information Sciences 420 (2017) 235–248 Since f and f are real-valued functions, the classic directional derivative definition yields, y f  i (t ∗) = lim h → 0 f i (t ∗+ yh ) −f i (t ∗) h < 0 , y f  i (t ∗) = lim h → 0 f i (t ∗+ yh ) −f i (t ∗) h < 0 . Therefore, ∀ ν> 0, ∃ i < 0 such that if | h | < i , then  f i (t ∗+ yh ) −f i (t ∗) h −y f  i (t ∗)  < ν If we consider ν= −1 2 y f  i (t ∗) > 0 , and if | h | < i , then −ν< f i (t ∗+ yh ) −f i (t ∗) h −y f  i (t ∗) < ν f i (t ∗+ yh ) −f i (t ∗) h < y f  i (t ∗) + ν= 1 2 y f  i (t ∗) < 0 . If h ∈ (0, i ), then f i (t ∗+ yh ) −f i (t ∗) < 0 . (8) Analogously ∃ i > 0 such that ∀ h ∈ (0 , i ) it holds f i (t ∗+ yh ) −f i (t ∗) < 0 . (9) (b) There exist ( f i )  + (t ∗) , ( f i )  −(t ∗) , ( f i )  + (t ∗) and ( f i )  −(t ∗) such that, from Theorem 1 , F  i (t ∗) = min ( f i )  + (t ∗) , ( f i )  + (t ∗) , max ( f i )  + (t ∗) , ( f i )  + (t ∗) . Therefore yF  i (t ∗) ≺[0 , 0] ⇒ y ( f i )  + (t ∗) < 0 and y ( f i )  + (t ∗) < 0 . Hence, there exists + i > 0 , such that for all h , with h ∈ (0 , + i ) f i (t ∗+ yh ) < f i (t ∗) . (10) And there exists + i > 0 , such that for all h , with h ∈ (0 , + i ) f i (t ∗+ yh ) < f i (t ∗) . (11) By taking = min { i , i , + i , + i : i = 1 , . . . , n } , h ∈ (0, ), from (8) –(11) , we obtain that f i (t ∗+ yh ) < f i (t ∗) and f i (t ∗+ yh ) < f i (t ∗) . (12) On the other hand, if there exists a y ∈ R solution of (7) , then by arguing in a similar way, for all j ∈ I ( t ∗), ∃ ρj > 0 such that when h ∈ (0, ρj ) h j (t ∗+ yh ) ≤h j (t ∗) = 0 . Moreover, for all j ∈ I ( t ∗), lim h → 0 h j (t ∗+ yh ) = h j (t ∗) < 0 , since h j is continuous at t ∗. Therefore there exists ρj , j ∈ I ( t ∗) such that, if h ∈ (0, ρj ) then h j (t ∗+ hy ) ≤0 . We assume ρ= min { ρj : j = 1 , . . . , m } . And finally, by taking 0 < h < min { , ρ, δ}, it shown that t ∗+ hy ∈ X ∩ B (t ∗, δ) and F i (t ∗+ yh ) ≺F i (t ∗) , i = 1 , . . . , p, which contradicts the hypothesis.  Along these same lines the following theorem can be proved Theorem 3. Let t ∗∈ X be a local strictly weakly efficient solution for (MIVOP), where F i are gH-differentiable functions at t ∗, for all i = 1 , . . . , p. Let I(t ∗) = { j = 1 , . . . , m : h j (t ∗) = 0 } , whereby the functions h j with j ∈ I ( t ∗) are differentiable functions at t ∗, and h j with j ∈ I ( t ∗) are continuous functions at t ∗. Then the following system has no solution at y ∈ R . yF  i (t ∗) [0 , 0] i = 1 , . . . , p y h j (t ∗) ≤0 j ∈ I(t ∗) . (13) R. Osuna-Gómez et al. / Information Sciences 420 (2017) 235–248 243 We can now transform the previous geometric conditions into necessary interval-valued Fritz-John and Karush–Kuhn– Tucker-type efficiency conditions. Definition 5. Let F i be gH-differentiable functions on T for all i = 1 , . . . , p, and h j are differentiable on T for all j = 1 , . . . m . A feasible solution t ∗∈ X is said to be a vector interval Fritz-John solution for (MIVOP) if there exist λ1 , . . . , λp ∈ R 2 , λi ࣛ 0, μ1 , . . . , μm ∈ R , μj ≥0 but ( λ, μ)  = 0 with λ= (λ1 , . . . λp ) ∈ M 2 ×p a 2 ×p matrix and μ= (μ1 , . . . μm ) ∈ R m such that λ×F  (t ∗) + μT h  (t ∗) = 0 , (14) μj h j (t ∗) = 0 , j = 1 , . . . , m. (15) Definition 6. Let F i be gH-differentiable functions on T for all i = 1 , . . . , pand h j differentiable on T for all j = 1 , . . . m . A feasible solution t ∗∈ X is said to be a vector interval Karush–Kuhn–Tucker solution for (MIVOP) if there exist λ1 , . . . , λp ∈ R 2 , λi ࣛ 0 and μ1 , . . . , μm ∈ R , μj ≥0 with λ= (λ1 , . . . λp ) ∈ M 2 ×p , λ = 0 and μ= (μ1 , . . . μm ) ∈ R m such that (14) and (15) are verified. Remark 4. If (MIVOP) is an optimization problem defined by real-valued functions, then F  (t) ∈ R p and F  i (t) = min [ F  i (t ∗)] = max [ F  i (t ∗)] and constraints h j = h j = h j , and therefore (14) remains as p  i =1 (λi 1 + λi 2 ) F  i (t ∗) + m  j=1 μj h  j (t ∗) = 0 , by considering that λ= (λ11 + λ12 , . . . , λp1 + λp2 ) , (14) can be rewritten as λT F  (t ∗) + μT h  (t ∗) = 0 , which is the classic definition of vector FJ and KKT points, [30] . Remark 5. Notice that condition (15) implies that μj = 0 for j ∈ I ( t ∗), so from Remark 1 , vector interval FJ and KKT definitions are equivalent to 0 ∈  p  i =1 u i min [ F  i (t ∗)] , p  i =1 v i max [ F  i (t ∗)]  +  j∈ I(t ∗) μj ¯ h  j (t ∗) . This means that the 0 is in the transported interval determined by positive linear combinations of derivatives of F at t ∗. Theorem 4. Let t ∗∈ X be a local weakly efficient solution for (MIVOP), where F i are gH-differentiable functions at t ∗for all i = 1 , . . . , pand h j with j = 1 , . . . m are differentiable functions at t ∗. Therefore t ∗is a vector interval Fritz-John solution for (MIVOP). Proof. Suppose that the following lineal system has a solution at y ∈ R , yA < 0 yB < 0 yC < 0  (16) where A, B ∈ M p×1 and C ∈ M k ×1 with k = card(I(t ∗)) A = ⎛ ⎝ min [ F  1 (t ∗)] . . . min [ F  p (t ∗)] ⎞ ⎠ , B = ⎛ ⎝ max [ F  1 (t ∗)] . . . max [ F  p (t ∗)] ⎞ ⎠ , C = ⎛ ⎜ ⎝ h  1 (t ∗) . . . h  k (t ∗) ⎞ ⎟ ⎠ .