Study on multi-objective nonlinear programming problem with rough parameters

Abstract

Rough set theory, introduced by Pawlak in 1981, is one of the important theories to express the vagueness not by means of membership but employing a boundary region of a set, i.e., an object is approximately determined based on some knowledge. In our real-life, there exists several parameters which impact simultaneously on each other and hence dealing with such different parameters and their conflictness create a multi-objective nonlinear programming problem (MONLPP). The objective of the paper is to deal with a MONLPP with rough parameters in the constraint set. The considered MONLPP with rough parameters are converted into the two-single objective problems namely, lower and upper approximate problems by using the weighted averaging and the ɛ- constraints methods and hence discussed their efficient solutions. The Karush-Kuhn-Tucker’s optimality conditions are applied to solve these two lower and upper approximate problems. In addition, the rough weights and the rough parameter ɛ are determined by the lower and upper the approximations corresponding each efficient solution. Finally, two numerical examples are considered to demonstrate the stated approach and discuss their advantages over the existing ones.

Keywords

Multiobjective nonlinear programming rough set lower approximation programming problem upper approximation programming problem weighting method parametric analysis

1 Introduction

In rough set theory, it is simply assumed that the knowledge is based on the ability to classify objects and is formally represented by an equivalence relation called the indiscernibility relation. The indiscernibility relation induces an approximation space on the universe, which is made by equivalence classes of indiscernible objects. The concept of a rough set is introduced by Pawlak et al. [22] and Pawlak [23] is one of the important theories to express the vagueness not by means of membership but employing a boundary region of a set. A rough set concept is different from ordinary as well as fuzzy sets. For instance, in an ordinary set, an object is identified with a characteristic function while in a fuzzy set, a partial degree of membership between 0 and 1 is considered to express the uncertainty in the data. On the other hand, in a rough set, an object is approximately determined based on some knowledge. In the literature, several authors have applied the concept of rough set to the diverse application such as artificial intelligence, expert systems, civil engineering [4], medical data analysis [8], data mining [21 , 36], Pattern recognition [20, 25], and decision theory [16, 29].

According to the decision maker (DM) influence in the optimization process, multiobjective optimization (MO) methods can be classified [15] into the following four categories:

Methods where DM does not provide information (no-preference methods),

Methods where a posteriori information is used (posteriori methods),

Methods where a priori information is used (priori methods),

Methods where progressive information is used (interactive methods).

In the light of the multi-objective programming problems, Tarabia et al. [30] developed a modified approach based on the ranking functions for solving fuzzy multiobjective programming problems. Sasaki and Gen [27] proposed a hybridized genetic algorithm for solving multiple-objective nonlinear programming having fuzzy multiple objective functions and constraints with generalized upper bounding structure. Khalifa [17] studied multi-objective nonlinear programming with rough interval parameters in the constraints set. Ammar and Emsimir [2] proposed an algorithm to solve fully rough integer linear programming problems. Ammar and Al-Asfar [3] proposed two approaches for solving fully fuzzy rough multiobjective nonlinear programming problems. Wang and Chaing [34] applied user preference enabling method to solve general constrained nonlinear MO problems. Kundu and Islam [18] introduced an interactive method to design a high reliable and productivity system with minimum cost to solve multi-objective reliability optimization problem. Walia et al. [33] studied the effect of capital investment and warehouses space on profits as well as shortage cost through sensitivity analysis and compared the efficiency of fuzzy nonlinear programming and intuitionistic fuzzy optimization techniques to obtain the solution. Ahmed [1] proposed a method to solve MO problems with intuitionistic fuzzy parameters. Liu et al. [19] introduced a new systematic method for determining an optimal operation scheme for minimizing octane number loss and operational risks. Garg [10] presented an interactive MO model to solve the reliability problems by using soft-computing techniques. To find the optimal solution of MO problem, several others have converted them into the single-objective optimization problems by using the transformation techniques such as weighted method etc. However, the necessary and the sufficient conditions to find an optimum solution is provided by Karush-Kuhn-Tucker (KKT) conditions. To solve the non-linear problem that arises due to the KKT conditions, Bialas and Karwan [5] proposed a parametric complementary pivot algorithm. Fortuny-Amat and McCarl’s [9] approached enforces the complementary slackness conditions by transforming the formulation into a much larger mixed integer programming problem. Apart from these, recently, several authors introduced enormous researches by using the features of the rough set in the optimization problems to solve the problems. For more details, we refer to read [7 , 35] and their corresponding references.

The rough set approach seems to be of fundamental importance to cognitive sciences, especially in the areas of machine learning, knowledge discovery from data bases, expert systems, inductive reasoning and pattern recognition. Khalifa [17] studied the multiobjective nonlinear programming in the constraints and introduced the stability set of the first kind. To keep the existing literature on an inexact rough interval for the normalized pentagonal fuzzy numbers in the line with such a scenario, the literature demands carving out a conceptual framework for solving such problems under a more generalized version, that is, the inexact interval for the normalized pentagonal fuzzy numbers. Therefore, in this study, multiobjective nonlinear programming problem with roughness in the parameters of the constraints set is conceptualized. Moreover, the weighting method is applied for solving the lower and upper approximations problems, and the stability set of the first kind is investigated and determined depending on the ɛ - constraintsmethod. Along with some important results, numerical examples are presented for illustration. Based on the study conducted above, the main objectives of the proposed study are:

To study the multi-objective nonlinear programming problem with rough parameters.

To present a rough multiobjective nonlinear programming (R-MONLP) problem to address the problems and characterized it with rough parameters in the constraints.

To specify the concept of rough efficient solution in the R-MONLP problems.

To validate the proposed study with the support of illustrative numerical examples.

The rest of the paper is outlined as follows: Section 2 briefly outline the basic concept related to rough set. In Section 3, we introduced the concept of R-MONLP problems along the lower and upper approximation with rough parameters. The solution to determine the rough weights and roughness of the parameters for each of the considered problem is presented in Section 4 and 5 respectively. In section 6, we discuss the method to determine the roughness parameters in the constraints corresponding to the considered R-MONLP problems. Section 7 gives the illustrative examples to demonstrate the approach. Finally, the concluding remark is summarized in Section 8.

2 Preliminaries

In this section, we present some of the basic concepts and results based on heptagonal fuzzy number, an inexact rough interval approximation and their arithmetic operations,

Definition 1 [28] A fuzzy number ${\tilde{A}}_{H}^{N} (a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7})$ is a heptagonal fuzzy number (HFN), given in Fig. 1, whereas $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7} \in ℝ$ and its membership function is defined by

Fig. 1

Heptagonal fuzzy number.

$\begin{matrix} μ_{{\tilde{A}}_{H}^{N}} = {\begin{matrix} \frac{1}{3} (\frac{{x - a}_{1}}{a_{2} - a_{1}}) for a_{1} ⩽ x ⩽ a_{2}, \\ \frac{1}{3} + \frac{1}{3} (\frac{{x - a}_{2}}{a_{3} - a_{2}}) for a_{2} ⩽ x ⩽ a_{3}, \\ \frac{2}{3} + \frac{1}{3} (\frac{{x - a}_{3}}{a_{4} - a_{3}}) for a_{3} ⩽ x ⩽ a_{4}, \\ 1 - \frac{1}{3} (\frac{{x - a}_{4}}{a_{5} - a_{4}}) for a_{4} ⩽ x ⩽ a_{5}, \\ \frac{2}{3} - \frac{1}{3} (\frac{{x - a}_{5}}{a_{6} - a_{5}}) for a_{5} ⩽ x ⩽ a_{6}, \\ \frac{1}{3} (\frac{{x - a}_{2}}{a_{3} - a_{2}}) for a_{6} ⩽ x ⩽ a_{7}, \\ 0, for x < a_{1} and x > a_{7} . \end{matrix} \end{matrix}$

Definition 2. [28] A HFN can be characterized, by the so-called interval of confidence at level α, as $\begin{matrix} {\tilde{A}}_{H_{α}}^{N} (x) & = {x \in X : μ_{{\tilde{A}}_{H}^{N}} ⩾ α} \\ = {\begin{matrix} [P^{-} (α), P^{+} (u)] for α \in [0, \frac{1}{3}], \\ [Q^{-} (α), Q^{+} (u)] for α \in [\frac{1}{3}, \frac{2}{3}], \\ [R^{-} (α), R^{+} (u)] for α \in [\frac{2}{3}, 1] . \end{matrix} \end{matrix}$

Definition 3. Let ${\tilde{A}}_{H}^{N} = (a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7})$ and ${\tilde{B}}_{H}^{N} = (b_{1}, b_{2}, b_{3}, b_{4}, b_{5}, b_{6}, b_{7})$ be two HFNs. Then, the basic operations are defined as $\begin{matrix} Addition : {\tilde{A}}_{H}^{N} \oplus {\tilde{B}}_{H}^{N} = (a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}) \\ \oplus (b_{1}, b_{2}, b_{3}, b_{4}, b_{5}, b_{6}, b_{7}) \\ = (a_{1} + b_{1}, a_{2} + b_{2}, a_{3} + b_{3}, \\ a_{4} + b_{4}, a_{5} + b_{5}, a_{6} + b_{6}, a_{7} + b_{7}), \\ Subtraction : {\tilde{A}}_{H}^{N} ⊖ {\tilde{B}}_{H}^{N} = (a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}) \\ ⊖ (b_{1}, b_{2}, b_{3}, b_{4}, b_{5}, b_{6}, b_{7}) \\ = (a_{1} - b_{7}, a_{2} - b_{6}, a_{3} - b_{5}, a_{4} \\ - b_{4}, a_{5} - b_{3}, a_{6} - b_{2}, a_{7} - b_{1}), \\ Scalar multiplication : k {\tilde{A}}_{H}^{N} \\ = {\begin{matrix} k (a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}), k ⩾ 0, \\ k (a_{7}, a_{6}, a_{5}, a_{4}, a_{3}, a_{2}, a_{1}), k < 0 . \end{matrix} \end{matrix}$

Definition 4. A rough interval approximation x^R of normalized heptagonal fuzzy number ${\tilde{A}}_{H}^{N} = (a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7})$ is defined as an interval with known lower and upper bounds while the distribution information for x is unknown: $\begin{matrix} {\tilde{A}}^{R} = [A_{α}^{(UAI)} : A_{α}^{(LAI)}], \end{matrix}$ where $A_{α}^{(LAI)} = inf {x \in ℝ : μ_{{\tilde{A}}_{H}^{N}} ⩾ \frac{1}{3}}, {and A}_{α}^{(UAI)} = sup {x \in ℝ : μ_{{\tilde{A}}_{H}^{N}} ⩾ \frac{1}{3}}$ , are the upper and lower approximation intervals of ${\tilde{A}}^{R}$ , respectively.

If ${\tilde{A}}_{H}^{N} (a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7})$ , the rough interval of ${\tilde{A}}_{H}^{N}$ is ${\tilde{A}}^{R} = [[a_{2}, a_{6}] : [a_{3,} a_{5}]]$ .

Definition 5. Let ${\tilde{A}}^{R} = [A_{α}^{(UAI)} : A_{α}^{(LAI)}]$ , and ${\tilde{B}}^{R} = [B_{α}^{(UAI)} : B_{α}^{(LAI)}]$ be two rough intervals, ${\tilde{A}}^{R} > 0$ and ${\tilde{B}}^{R} > 0$ . The arithmetic operations are defined as

${\tilde{A}}^{R} \oplus {\tilde{B}}^{R} = [[A_{α}^{(UAI)} + B_{α}^{(UAI)}] : [A_{α}^{(LAI)} + B_{α}^{(LAI)}]]$

${\tilde{A}}^{R} ⊖ {\tilde{B}}^{R} = [[A_{α}^{(UAI)} - B_{α}^{(UAI)}] : [A_{α}^{(LAI)} - B_{α}^{(LAI)}]]$

${\tilde{A}}^{R} \otimes {\tilde{B}}^{R} = [[A_{α}^{(UAI)} \times B_{α}^{(UAI)}] : [A_{α}^{(LAI)} \times B_{α}^{(LAI)}]]$

${\tilde{A}}^{R} ø {\tilde{B}}^{R} = [[A_{α}^{(UAI)} / B_{α}^{(UAI)}] : [A_{α}^{(LAI)} / B_{α}^{(LAI)}]]$

If $A_{α}^{(UAI)} = [A_{α}^{- (UAI)}, A_{α}^{+ (UAI)}], A_{α}^{(LAI)} = [A_{α}^{- (LAI)}, A_{α}^{+ (LAI)}], B_{α}^{(UAI)} = [B_{α}^{- (UAI)}, B_{α}^{+ (UAI)}]$ , and $B_{α}^{(LAI)} = [B_{α}^{- (LAI)}, B_{α}^{+ (LAI)}]$ , where $A_{α}^{- (UAI)}, A_{α}^{+ (UAI)}, A_{α}^{- (LAI)}, A_{α}^{+ (LAI)}, B_{α}^{- (UAI)}, B_{α}^{+ (UAI)}, B_{α}^{- (LAI)}, B_{α}^{+ (LAI)}$ are the deterministic numbers and also are the lower and upper bounds of $A_{α}^{(UAI)}, A_{α}^{(LAI)}, B_{α}^{(UAI)}$ , and $B_{α}^{(LAI)}$ , respectively. Then, Definition 5 becomes

${\tilde{A}}^{R} \oplus {\tilde{B}}^{R} = [[A_{α}^{- (UAI)} + B_{α}^{- (UAI)}, A_{α}^{+ (UAI)} + B_{α}^{+ (UAI)}] : [A_{α}^{- (LAI)} + B_{α}^{- (LAI)}, A_{α}^{+ (LAI)} + B_{α}^{+ (LAI)}]]$

${\tilde{A}}^{R} ⊖ {\tilde{B}}^{R} = [[A_{α}^{- (UAI)} - B_{α}^{+ (UAI)}, A_{α}^{+ (UAI)} - B_{α}^{- (UAI)}] : [A_{α}^{- (LAI)} - B_{α}^{+ (LAI)}, A_{α}^{+ (LAI)} - B_{α}^{- (LAI)}]]$

${\tilde{A}}^{R} \otimes {\tilde{B}}^{R} = [[A_{α}^{- (UAI)} \times B_{α}^{- (UAI)}, A_{α}^{+ (UAI)} \times B_{α}^{+ (UAI)}] : [A_{α}^{- (LAI)} \times B_{α}^{- (LAI)}, A_{α}^{+ (LAI)} \times B_{α}^{+ (LAI)}]]$

${\tilde{A}}^{R} ø {\tilde{B}}^{R} = [[A_{α}^{- (UAI)} / B_{α}^{+ (UAI)}, A_{α}^{+ (UAI)} / B_{α}^{- (UAI)}] : [A_{α}^{- (LAI)} / B_{α}^{+ (LAI)}, A_{α}^{+ (LAI)} / B_{α}^{- (LAI)}]]$

Definition 6. For ${\tilde{A}}^{R} = [A_{α}^{(UAI)} : A_{α}^{(LAI)}]$ , and ${\tilde{B}}^{R} = [B_{α}^{(UAI)} : B_{α}^{(LAI)}]$ , their order relations are as follows

${\tilde{A}}^{R} ≲ {\tilde{B}}^{R} if A_{α}^{+ (UAI)} ⩽ B_{α}^{+ (UAI)} and A_{α}^{- (UAI)} ⩽ B_{α}^{- (UAI)}$

${\tilde{A}}^{R} ≺ {\tilde{B}}^{R} \Leftrightarrow A^{R} ⩽ B^{R} {andA}^{R} \neq B^{R}$

3 Problem formulation and solution concepts

In this section, we define the concept of R-MONLP problem with rough parameters.

Consider the following multiobjective nonlinear programming with roughness in the constraints as

$\begin{matrix} (R - MONLP) F (x) = min f_{j} (x), j = \bar{1, m} \\ Subject to \\ x \in X (b^{R}) = {x \in ℝ^{n} : H (x) ⩽ {\tilde{b}}^{R}} . \end{matrix}$ (1) where, $F : ℝ^{n} \to ℝ^{m}, H : ℝ^{n} \to ℝ^{k}$ are the convex functions of class C⁽¹⁾ on $ℝ^{n}, F = (f_{1}, f_{2}, {\dots, f_{m})}^{T}$ , $H = {(h_{1}, h_{2},, \dots, h_{r})}^{T}, r = \bar{1, k}$ and ${\tilde{b}}^{R} = {({\tilde{b}}_{1}^{R}, {\tilde{b}}_{2}^{R}, \dots, {\tilde{b}}_{r}^{R})}^{T}, r = \bar{1, k}$ represents the vector of rough parameters in the constraints $h_{r} (x), r = \bar{1, k}$ . Here, $r = \bar{1, k}$ represent r = 1, 2, 3, …, k

The rough parameters are represented by rough numbers, which can be characterized by lower and upper approximations, $b_{r}^{LA}$ , and $b_{r}^{UP}$ ; respectively. For the convince, we may write, ${\tilde{b}}_{r}^{R} \in [b_{r}^{LAI} : b_{r}^{UAI}], r = \bar{1, k}$ . It clear that the feasible set is characterized by two sets namely, the lower and upper approximation sets; respectively and denoted by X^LAI, and X^UAI and defined by:

$X^{LAI} = {x \in ℝ^{n} : h_{r} (x) ⩾ {\tilde{b}}_{r}^{LAI}, r = \bar{1, k}},$ (2)

$X^{UAI} = {x \in ℝ^{n} : h_{r} (x) ⩽ {\tilde{b}}_{r}^{UAI}, r = \bar{1, k}}$ (3)

According to Equations (2) and (3), the R - MONLP is divided into two problems, namely the lower and upper approximation multiobjective objective programming problems.

$\begin{matrix} (LAP) min f_{j} (x), j = \bar{1, m} \\ Subject to \\ x \in X^{LAI} = {x \in ℝ^{n} : h_{r} (x) ⩾ {\tilde{b}}_{r}^{LAI}, r = \bar{1, k}} \end{matrix}$ (4)

$\begin{matrix} (UAP) min f_{j} (x), j = \bar{1, m} \\ Subject to \\ x \in X^{UAI} = {{x \in ℝ^{n} : h_{r} (x) ⩽ {\tilde{b}}_{r}^{UAI}, r = \bar{1, k}}} . \end{matrix}$ (5)

Definition 7. [6] The solutions x^*LAI ∈ X^LAI, andx^*UAI ∈ X^UAI are said to be efficient solutions for problems (4), and (5) if there is no x^LAI ∈ X^LAI (x^UAI ∈ X^UAI) such that f_j (x^LAI) ⩽ f_j (x^*LAI) (f_j (x^UAI) ⩾ f_j (x^*UAI)) and f_j (x^LAI) ≠ f_j (x^*LAI) (f_j (x^UAI) ≠ f_j (x^*UAI)) with strict inequality holding for at least one j.

Definition 8. [6] The solution x^R ∈ X is said to be a rough efficient solution for the R - MONLP if x^R ∈ [x^*LAI, x^*UAI], where x^*LAI and x^*UAI are the lower and upper efficient solutions for problems (4) and (5); respectively.

Next, we state a result for the rough efficient solution for the R-MONLP problems.

Proposition 1. Let $f_{j} (x), j = \bar{1, m}$ be convex functions on a convex set $X ({\tilde{b}}^{R})$ , and x^*LAI is an efficient solution for problem (4). If x^*UAIdominates x^*LAI over X, then x^R* ∈ [x^*LAI, x^*UAI] is a rough efficient solution for the R - MONLP.

Proof. Since x^R* ∈ [x^*LAI, x^*UAI], then there is $\bar{θ} \in [0, 1]$ with ${\tilde{x}}^{R *} = (1 - \bar{θ}) x^{* LAI} + {\bar{θ x}}^{* UAI}$ . Suppose that x^R* is not an efficient solution for R - MONLP, therefore, there is an $\bar{x} \in X$ such that: $f_{j} (\bar{x}) ⩽ f_{j} (x^{R *}), j = \bar{1, m}, i \neq e, f_{e} (\bar{x}) < f_{e} (x^{R *}) .$

Let $\bar{x} \in X^{LAI}$ , and from the convexity assumption of f_j, we have

$f_{j} (\bar{x}) ⩽ (1 - \bar{θ}) f_{j} (x^{* LAI}) + {\bar{θ f}}_{j} (x^{* UAI}) ⩽ f_{j} (x^{* LAI}) + \bar{θ} (f_{j} (x^{* UAI}) - f_{j} (x^{* LAI}))$ ,

$f_{e} (\bar{x}) ⩽ f_{e} (x^{* LAI}) + \bar{θ} (f_{j} (x^{* UAI}) - f_{j} (x^{* LAI})) .$ Since x^*UAI dominates x^*LAI, so

$f_{j} (x^{* UAI}) - f_{j} (x^{* LAI}) ⩽ 0, j = \bar{1, m}$ , and $f_{j} (x^{* UAI}) - f_{j} (x^{* LAI}) < 0, j = \bar{1, m}$ .

Therefore

$f_{j} (\bar{x}) ⩽ f_{j} (x^{* LAI}), f_{e} (\bar{x}) < f_{e} (x^{* LAI}) .$

This is a contradiction that x^*LAI is an efficient solution for problem (4).

Let $\bar{x} \notin X^{LAI}$ , then from the convexity of f_j on X, we have for any x^LAI ∈ X^LAI

$\begin{matrix} \begin{matrix} f_{j} (θ x^{LAI} + (1 - θ) \bar{x}) ⩽ θ f_{j} (x^{LAI}) + (1 - θ) f_{j} (\bar{x}) \\ ⩽ θ f_{j} (x^{LAI}) + (1 - θ) f_{j} (x^{R *}) ⩽ θ f_{j} (x^{LAI}) + (1 - θ) \\ [(1 - \bar{θ}) f_{j} (x^{* LAI}) + {\bar{θ f}}_{j} (x^{* UAI})] ⩽ θ f_{j} (x^{LAI}) + (1 - θ) \\ (f_{j} (x^{* LAI}) + \bar{θ} (f_{j} (x^{* UAI}) - f_{j} (x^{* LAI}))), θ \in [0, 1], \end{matrix} \end{matrix}$

with strict inequality holds for at least one j. Put x^LAI = x^*LAI, we obtain $\begin{matrix} \begin{matrix} f_{j} (θ x^{* LAI} + (1 - θ) \bar{x}) ⩽ f_{j} (x^{* LAI}) \\ + \bar{θ} (1 - θ) (f_{j} (x^{* UAI}) - f_{j} (x^{* LAI})), θ \in [0, 1] . \end{matrix} \end{matrix}$

Since x^*UAI dominates x^*LAI, so $f_{j} (θ x^{* LAI} + (1 - θ) \bar{x}) ⩽ f_{i} (x^{* LAI})$ with at least one inequality holds for at least one j, i.e., there is $\hat{x} \in X^{LAI}$ corresponds to $\hat{θ}$ such that $f_{j} (\hat{x}) ⩽ f_{j} (x^{* LAI}), j = \bar{1, m}, f_{j} (\hat{x}) < f_{j} (x^{* LAI})$ . This is a contradiction that x^*LAI is an efficient solution to problem (4). Hence, the result.

Proposition 2. Let $f_{j} (x), j = \bar{1, m}$ be convex function on a convex set X. Let x^*UAI is an efficient solution for problem (5). For x^R* ∈ [x^*LAI, x^*UAI], if x^*LA dominates x^*UAI on X, then x^R* is an efficient solution for R - MONLP.

Proof. Since x^R* ∈ [x^*LAI, x^*UAI], then there is an $\bar{θ} \in [0, 1]$ such that $x^{R *} = (1 - \bar{θ}) x^{* LAI} + \bar{θ} - x^{* UAI} .$ Suppose that x^R* is not an efficient solution for R - MONLP then there is an $\bar{x} \in X$ such that $f_{j} (\bar{x}) ⩽ f_{j} (x^{R *}), j = \bar{1, m}$ , and $f_{j} (\bar{x}) < f_{j} (x^{R *}) .$

Let $\bar{x} \in X^{UAI}$ , then from the convexity of f_j, we have $\begin{matrix} f_{j} (\bar{x}) ⩽ (1 - \bar{θ}) f_{j} (x^{* LAI}) + {\bar{θ f}}_{i} (x^{* UAI}) \end{matrix}$ with strict inequality holds for at least one j. Since x^*LAI dominates x^*UAI, then $f_{j} (x^{* LAI}) ⩽ f_{j} (x^{* UAI}), j = \bar{1, m}$ , f_j (x^*LAI) < f_j (x^*UAI), and $f_{j} (\bar{x}) ⩽ (1 - \bar{θ}) f_{j} (x^{* UAI}) + {\bar{θ f}}_{j} (x^{* UAI}), j = \bar{1, m}$ , $f_{i} (\bar{x}) < f_{j} (x^{* UAI})$ . This is a contradiction that x^*UAI is an efficient solution for problem (5).

Let $\bar{x} \notin X^{UAI}$ , then for any x^UAI ∈ X^UAI, and from the convexity of f_j, we have

$f_{j} (θ x^{UAI} + (1 - θ) \bar{x}) ⩽ θ f_{j} (x^{UAI}) + (1 - θ) f_{j} (\bar{x}) ⩽ θ f_{j} (x^{UAI}) + (1 - θ) ((1 - \bar{θ}) f_{j} (x^{* LAI} + {\bar{θ f}}_{i} (x^{* UAI})))$ , θ ∈ [0, 1] , with strict inequality holds for at least one j.

Since x^*LAI dominates x^*UAI, then f_j (x^*LAI) ⩽ f_j (x^*UAI) , and f_j (x^*LAI) < f_j (x^*UAI), and $f_{j} (θ x^{UAI} + (1 - θ) \bar{x})) ⩽ θ f_{j} (x^{UAI}) + (1 - θ) f_{j} (x^{* UAI})$ . Put x^UAI = x^*UAI, we have $f_{j} (θ x^{* UAI} + (1 - θ) \bar{x}) ⩽ f_{j} (x^{* UAI})$ , with strict inequality holds for at least one j, then there is an $\hat{x} = θ x^{* UAI} + (1 - θ) \bar{x}$ corresponds to $\hat{θ}$ such that $f_{j} (\hat{x}) ⩽ f_{j} (x^{* UAI}), j = \bar{1, m}$ ; and that $f_{j} (\hat{x}) < f_{j} (x^{* UAI})$ , which contradicts the assumption that x^*UAI is an efficient solution of (5).

Hence, the result.

4 Roughness weights corresponding to the solution of the R - MONLP

The lower approximation-programming problem (4) can be treated using the weighting method as

$\begin{matrix} min \sum_{i = 1}^{m} γ_{j}^{LAI} f_{j}^{LA} (x) \\ Subject to \end{matrix}$ (6) $\begin{matrix} X^{LAI} = {x \in ℝ^{n} : h_{r} (x) - {\tilde{B}}_{r}^{LAI} ⩾ 0, r = \bar{1, k}}; \end{matrix}$ $\begin{matrix} γ_{j}^{LAI} ⩾ 0, j = \bar{1, m}, and \sum_{i = 1}^{m} γ_{j}^{LAI} = 1; \end{matrix}$ and the upper approximation-programming problem (5) can be treated using the weighting method as

$min \sum_{i = 1}^{m} γ_{j}^{UAI} f_{j}^{UA} (x)$ (7)

${x \in ℝ^{n} : h_{r} (x) - b_{r}^{UAI} ⩽ 0, r = \bar{1, k}}; γ_{j}^{UAI} ⩾ 0, j = \bar{1, m}$ , and $\sum_{i = 1}^{m} γ_{j}^{UAI} = 1;$ and $0 \neq γ_{j}^{UAI} = \frac{\bar{f_{j}} - \underline{f_{j}}}{\sum_{j = 1}^{m} (\bar{f_{j}} - \underline{f_{j}})}$ , where $\bar{f_{j}},$ and $\underline{f_{j}}$ are the individual maximum and minimum of f_j (x); respectively.

Theorem 1. Let $f_{j} (x), j = \bar{1, m}$ and $h_{r} (x), r = \bar{1, k}$ are differentiable and continuous on a neighborhood of x^*LAI and x^*UAI. If x^*LA is an efficient solution of problem (4) and x^*UAI is an efficient solution of problem (5), then the weights corresponding to the R - MONLP solution are the rough with lower and upper approximations; respectively $\begin{matrix} γ_{m}^{LAI} = & \frac{{\bar{a}}_{m}}{{\bar{b}}_{mm}}, γ_{j}^{LAI} = \frac{1}{{\bar{b}}_{jj}} ({\bar{a}}_{j} - \sum_{j = i + 1}^{m} {\bar{b}}_{ji} γ_{j}^{LAI}), \\ j = \bar{1, m - 1} \end{matrix}$ $\begin{matrix} γ_{m}^{UAI} = & \frac{{\hat{a}}_{m}}{{\hat{b}}_{mm}}, γ_{j}^{UAI} = \frac{1}{{\hat{b}}_{jj}} ({\hat{a}}_{j} - \sum_{j = i + 1}^{m} {\hat{b}}_{ji} γ_{j}^{UAI}), \\ j = \bar{1, m - 1} . \end{matrix}$

Where, $\begin{matrix} {\bar{a}}_{m} = \sum_{r = 1}^{k} γ_{r} \frac{δ (b_{r}^{LAI} - h_{r} (x^{* LAI}))}{δ x_{s}}, s = \bar{1, m,} \end{matrix}$ $\begin{matrix} {\bar{b}}_{mm} = \sum_{j = 1}^{m} \frac{δ f_{j} (x^{* LAI})}{δ x_{s}}, s = \bar{1, m}, \end{matrix}$ $\begin{matrix} {\hat{a}}_{m} = \sum_{r = 1}^{k} γ_{r} \frac{δ (h_{r} (x^{* UAI}) - b_{r}^{UAI})}{δ x_{s}}, s = \bar{1, m,} and \end{matrix}$ $\begin{matrix} {\hat{b}}_{mm} = \sum_{j = 1}^{m} \frac{δ f_{j} (x^{* UAI})}{δ x_{s}}, s = \bar{1, m} . \end{matrix}$

Proof. Since $f_{j} (x), j = \bar{1, m}$ and $h_{r} (x), r = \bar{1, k}$ are differentiable and continuous functions at a neighborhood of x^*LA, and x^*LA is an efficient solution for problem (4), then it satisfies the following Karush-Kuhn - Tuckers’ optimality conditions as: $\begin{matrix} \sum_{j = 1}^{m} γ_{j}^{LA} \frac{δ f_{j} (x^{* LA})}{δ x_{s}}, + \sum_{r = 1}^{k} γ_{r} \frac{δ (b_{r}^{LA} - h_{r} (x^{* LA}))}{δ x_{s}} \\ = 0, s = \bar{1, n}, \end{matrix}$ $\begin{matrix} γ_{r} (b_{r}^{LA} - h_{r} (x^{* LA}) = 0, r = \bar{1, k}, \end{matrix}$ $\begin{matrix} γ_{r} ⩾ 0, r = \bar{1, k} . \end{matrix}$

Let us consider the following two cases:

Case 1: If $b_{r}^{LAI} - h_{r} (x^{* LAI}) > 0$ , then γ_r = 0 and the Lagrangian function can be formulated as follows: $\begin{matrix} \sum_{j = 1}^{m} γ_{j}^{LA} \frac{δ f_{j} (x^{* LAI})}{δ x_{s}} = 0, s = \bar{1, n} . \end{matrix}$

Let $b_{sj} = \frac{δ f_{j}^{LA} (x^{* LAI})}{δ x_{s}}, s = \bar{1, n}$ . Then, we have the following system:

$\begin{matrix} b_{11} γ_{1}^{LAI} + b_{12} γ_{2}^{LAI} + \dots + b_{1 m} γ_{m}^{LAI} = 0, \\ b_{21} γ_{1}^{LAI} + b_{22} γ_{2}^{LAI} + \dots + b_{2 m} γ_{m}^{LAI} = 0, \\ \cdot \cdot \dots \cdot \\ \cdot \cdot \dots \cdot \\ \cdot \cdot \dots \cdot \\ b_{n 1} γ_{1}^{LAI} + b_{n 2} γ_{2}^{LAI} + \dots + b_{nm} γ_{m}^{LAI} = 0 . \end{matrix}$ (8)

Remark 1. It is clear that system (8) is a linear homogeneous system.

Case 2: If $b_{r}^{LAI} - h_{r} (x^{* LAI}) = 0$ then γ_rj ⩾ 0 and the Lagrangian function takes the form $\begin{matrix} \sum_{j = 1}^{m} γ_{j}^{LA} \frac{δ f_{j}^{LA} (x^{* LAI})}{δ x_{s}} \\ = \sum_{r = 1}^{k} γ_{r} \frac{δ (b_{r}^{LAI} - h_{r} (x^{* LAI}))}{δ x_{s}} = 0, s = \bar{1, n}, \end{matrix}$

and we get the following nonhomogeneous linear system

$\begin{matrix} b_{11} γ_{1}^{LAI} + b_{12} γ_{2}^{LAI} + \dots + b_{1 m} γ_{m}^{LAI} = a_{1}, \\ b_{21} γ_{1}^{LAI} + b_{22} γ_{2}^{LAI} + \dots + b_{2 m} γ_{m}^{LAI} = a_{2}, \\ \cdot \cdot \dots \cdot \\ \cdot \cdot \dots \cdot \\ \cdot \cdot \dots \cdot \\ b_{n 1} γ_{1}^{LAI} + b_{n 2} γ_{2}^{LAI} + \dots + b_{nm} γ_{m}^{LA} = a_{n} . \end{matrix}$ (9)

For system (9), we obtain the following subcases:

If n < m, this leads to infinite number of solutions,

If n = m, this leads to a unique solution which can be obtained by applying any methods for solving linear systems, say Gauss elimination method.

Let rewrite system (9) as

$U γ^{LAI} = V,$ (10)

Where, U = (b_ij) is a m × n matrix defined as

$\begin{matrix} U = (\begin{matrix} \frac{δ f_{1} (x^{* LAI})}{δ x_{1}} \frac{δ f_{2} (x^{* LAI})}{δ x_{1}} & \dots & \frac{δ f_{m} (x^{* LAI})}{δ x_{1}} \\ ⋮ & ⋱ & ⋮ \\ \frac{δ f_{1} (x^{* LAI})}{δ x_{m}} \dots \frac{δ f_{2} (x^{* LAI})}{δ x_{m}} & \dots & \frac{δ f_{m} (x^{* LAI})}{δ x_{m}} \end{matrix}), and \end{matrix}$ $\begin{matrix} V = (\begin{matrix} γ_{1} \frac{δ (h_{1} (x^{* LAI}) - b_{1}^{LAI})}{δ x_{1}} + γ_{2} \frac{δ (h_{2} (x^{* LAI}) - b_{2}^{LAI})}{δ x_{1}} + & \dots & γ_{k} \frac{δ (h_{k} (x^{* LAI}) - b_{k}^{LAI})}{δ x_{1}} \\ ⋮ & ⋱ & ⋮ \\ γ_{1} \frac{δ (h_{1} (x^{* LAI}) - b_{1}^{LAI})}{δ x_{m}} + γ_{2} \frac{δ (h_{2} (x^{* LAI}) - b_{2}^{LAI})}{δ x_{m}} & \dots & γ_{k} \frac{δ (h_{k} (x^{* LAI}) - b_{k}^{LAI})}{δ x_{m}} \end{matrix}) \end{matrix}$

Then, we can rewrite the system (10) as $\begin{matrix} (\begin{matrix} γ_{1}^{LAI} \\ γ_{2}^{LAI} \\ . \\ . \\ . \\ γ_{m}^{LAI} \end{matrix}) = (\begin{matrix} b_{11}^{LAI} b_{12}^{LAI} \dots b_{1 m}^{LAI} \\ b_{21}^{LAI} b_{22}^{LAI} \dots b_{2 m}^{LAI} \\ . \\ . \\ . \\ b_{m 1}^{LAI} b_{m 2}^{LAI} \dots b_{mm}^{LAI} \end{matrix}) (\begin{matrix} a_{1}^{LAI} \\ a_{2}^{LAI} \\ . \\ . \\ . \\ a_{m}^{LAI} \end{matrix}), \end{matrix}$ which can be converted into $\begin{matrix} (\begin{matrix} γ_{1}^{LAI} \\ γ_{2}^{LAI} \\ . \\ . \\ . \\ γ_{m}^{LAI} \end{matrix}) = (\begin{matrix} {\bar{b}}_{11}^{LAI} b_{12}^{LAI} \dots b_{1 m}^{LAI} \\ 0 {\bar{b}}_{22}^{LAI} \dots b_{2 m}^{LAI} \\ . \\ . \\ . \\ 00 \dots 0 {\bar{b}}_{mm}^{LAI} \end{matrix}) (\begin{matrix} a_{1}^{LAI} \\ a_{2}^{LAI} \\ . \\ . \\ . \\ a_{m}^{LAI} \end{matrix}) \end{matrix}$

If ${\bar{b}}_{mm}^{LAI} = 0$ , then there is no solution.

If ${\bar{b}}_{mm}^{LAI} \neq 0$ , then by using backward substitution, the solution is

$\begin{matrix} γ_{m}^{LAI} = & \frac{{\bar{a}}_{m}^{LAI}}{{\bar{b}}_{mm}^{LA}}, γ_{j}^{LAI} = \frac{1}{{\bar{b}}_{jj}^{LAI}} ({\bar{a}}_{j}^{LAI} - \sum_{i = j + 1}^{n} {\bar{b}}_{ji}^{LA} γ_{i}^{LAI}), \\ j = \bar{1, m - 1,} \end{matrix}$

(iii) If n > m, this case is similar to (ii).

The weights of problem (7) can be determined as in the same of lower approximation problem (6) as

$\begin{matrix} γ_{m}^{UAI} = & \frac{{\hat{a}}_{m}^{UAI}}{{\hat{b}}_{mm}^{UAI}}, γ_{j}^{UAI} = \frac{1}{{\hat{b}}_{jj}^{UAI}} ({\hat{a}}_{j}^{UAI} - \sum_{i = j + 1}^{n} {\hat{b}}_{ji}^{UAI} γ_{i}^{UAI}), \\ j = \bar{1, m - 1,} \end{matrix}$

Thus, the weigh corresponding to the solution of R - MONLP problem (i.e., x^R ∈ [x^*LAI, x^*UAI]) is rough with γ^R ∈ [γ^LAI, γ^UAI], where $\begin{matrix} γ^{LAI *} & = {γ_{m}^{LAI}, γ_{j}^{LAI}} \\ = {\frac{{\bar{a}}_{m}}{{\bar{b}}_{mm}}, \frac{1}{\bar{b_{ji}}} ({\bar{a}}_{j}^{LAI} - \sum_{i = j + 1}^{n} {\bar{b}}_{ji}^{LAI} γ_{i}^{LAI})} \end{matrix}$ $\begin{matrix} γ^{UAI *} = & {γ_{m}^{UAI}, γ_{j}^{UAI}} \\ = {\frac{{\hat{a}}_{m}}{{\hat{b}}_{mm}}, \frac{1}{{\hat{a}}_{ji}} ({\hat{a}}_{j}^{UAI} - \sum_{i = j + 1}^{n} b_{ji}^{UAI} γ_{i}^{UAI})} . \end{matrix}$

If x^*LAI = x^*UAI, then $b_{ij}^{LAI} = b_{ij}^{UAI}$ and $a_{i}^{LAI} = a_{i}^{UAI}$ and hence γ^LAI = γ^UAI.

If x^*LAI ≠ x^*UAI, then γ^LAI ≠ γ^UAI.

5 Roughness of parameters corresponding to the solution of the R - MONLP

The lower approximation problem (4) is converted into a single objective linear programming by using the ɛ- constraints method as $\begin{matrix} \begin{matrix} {(LP)}_{ɛ} min f_{g} (x) \\ Subject to \\ f_{j} (x) ⩽ ɛ_{j}^{R}, j = \bar{1, q}, j \neq g, \\ x \in X (\tilde{B}) = {x \in ℝ^{n} : h_{r} (x) ⩽ {\tilde{B}}_{r}^{R}, r = \bar{1, k}} . \end{matrix} \end{matrix}$

Where, ${\tilde{B}}_{r}^{R} \in [b_{r}^{LAI}, b_{r}^{UAI}]$ , and $ɛ_{j}^{R} \in [ɛ_{j}^{LA} I, ɛ_{j}^{UAI}]$ .

Clearly, the (R - MONLP) problem can convert into the following two problems, namely, Lower and upper approximation problems each of them can be formulated as

Lower approximation problem:

$\begin{matrix} min f_{g} (x) \\ Subject to \\ x \in X^{LAI} = {x \in ℝ^{n} : h_{r} (x) ⩾ b_{r}^{LAI}, r = \bar{1, k}}, \\ f_{j} (x) ⩾ ɛ_{j}^{LAI}, j = \bar{1, q}, j \neq g, and \end{matrix}$ (11)

Upper approximation problem:

$\begin{matrix} min f_{g} (x) \\ Subject to \\ x \in X^{UAI} = {x \in ℝ^{n} : h_{r} (x) ⩽ b_{r}^{UAI}, r = \bar{1, k}}, \\ f_{j} (x) ⩽ ɛ_{j}^{UAI}, j = \bar{1, q}, j \neq g . \end{matrix}$ (12)

Lemma 1. x^*LAI ∈ IntX^LAI is an efficient solution for the R - MONLP, if it is a unique optimal solution for problem (11) for some g with $ɛ_{j}^{* LAI} = f_{j} (x^{* LAI}), j = \bar{1, q}, j \neq g$ .

Proof. Assume that x^*LAI ∈ X^LAI is a unique solution for problem (4). Therefore, for any $\hat{x} \in X^{LAI}, f_{g} (\hat{x}) > f_{g} (x^{* LAI})$ , $and f_{j} (\hat{x}) ⩾ f_{j} (x^{* LAI}) j = \bar{1, q}, j \neq g$ ,which leads to x^*LAI is an efficient solution on X^LAI .

Let $\hat{x} \in X - X^{LAI}$ dominated x^*LA (i. e., $f_{j} (\hat{x}) ⩽ f_{j} (x^{* LAI})$ , $, j = \bar{1, q}, j \neq g$ , and $f_{j} (\hat{x}) < f_{j} (x^{* LAI})$ ). From the convexity assumption of $f_{j} (x), j = \bar{1, q}$ , we get

$f_{j} (θ \hat{x} + (1 - θ) x^{* LAI}) ⩽ θ f_{i} (\hat{x}) + (1 - θ) f_{j} (x^{* LAI}), 0 ⩽ θ ⩽ 1$ ,and

$f_{j} (θ \hat{x} + (1 - θ) x^{* LAI}) ⩽ f_{j} (x^{* LAI}), j = \bar{1, q}$ , and $f_{j} (θ \hat{x} + (1 - θ) x^{* LAI}) < f_{j} (x^{* LAI}) .$

Since, x^*LAI ∈ IntX^LAI, then for a certain θ and $\bar{x} \in θ \hat{x} + (1 - θ) x^{* LAI}, \bar{x} \in X^{LAI}$ and $f_{j} (\bar{x}) ⩽ f_{i} (x^{* LAI}), j = \bar{1, q}$ ; $f_{j} (\bar{x}) < f_{j} (x^{* LAI})$ , which contradicts that x^*LAI is an efficient solution on X^LAI . Hence the result.

Lemma 2. x^*UAI ∈ IntX^UAI is an efficient solution for the R - MONLP problem if it is a unique optimal solution for problem (12) for some k with $ɛ_{i}^{* UAI} = f_{i} (x^{* UAI}), i = \bar{1, m}$ , i ≠ k.

Proof. Similar to the proof of Lemma 1.

6 Determination of rough parameters (ɛ^LAI, ɛ^{UA
I}) in the constraints corresponding to the R - MONLP problem solution

Let x^*LAI ∈ X^LAI be an optimal solution for problem (11) corresponding to $ɛ_{j}^{* LAI}$ and f_j (x) are differentiable in a neighborhood of x^*LAI, then by applying the Karush-Kuhn - Tucker’s optimality conditions for problem (11), we have $\begin{matrix} \frac{δ f_{g} (x^{* LAI})}{δ x} + γ_{r} \frac{δ (b_{r}^{LAI} - h_{r} (x^{* LAI}))}{δ x} \\ + ϑ_{jr} \frac{δ (ɛ_{j}^{LAI} - f_{j} (x^{* LAI}))}{δ x} = 0, \end{matrix}$ $\begin{matrix} γ_{r} (b_{r}^{LAI} - h_{r} (x^{* LAI})) = 0, \\ r = \bar{1, k}; ϑ_{j} (ɛ_{j}^{LAI} - f_{j} (x^{* LAI})) = 0, \end{matrix}$ $\begin{matrix} h_{r} (x^{* LAI}) ⩾ b_{r}^{LAI}, f_{j} (x^{* LAI}) ⩾ ɛ_{j}^{LAI} \end{matrix}$ $\begin{matrix} γ_{r} ⩾ 0, r = \bar{1, k}, ϑ_{j} ⩾ 0, j = \bar{1, p} . \end{matrix}$

Consider the following four cases:

Case 1: For γ_r = 0, ϑ_j = 0. Then, $b_{r}^{LAI} ⩾ h_{r} (x^{* LAI}), ɛ_{j}^{LAI} > f_{j} (x^{* LAI})$ ,

Case 2: For γ_r ≠ 0, ϑ_j = 0. Then, $b_{r}^{LAI} ⩾ h_{r} (x^{* LAI}), ɛ_{j}^{LAI} > f_{j} (x^{* LAI})$ ,

Case 3: For γ_r = 0, ϑ_j ≠ 0. Then, $b_{r}^{LAI} ⩾ h_{r} (x^{* LAI}), ɛ_{j}^{LAI} = f_{j} (x^{* LAI})$ ,

Case 4: For γ_r ≠ 0, ϑ_j ≠ 0. Then, $b_{r}^{LAI} = h_{r} (x^{* LAI}), ɛ_{j}^{LAI} = f_{j} (x^{* LAI})$ .

Let x^*UAI ∈ X^UAI be an optimal solution for problem (12) corresponding to $ɛ_{j}^{* UAI}$ and f_j (x) are differentiable in a neighborhood of x^*UAI, then by applying the KKT optimality conditions for problem (12), we have $\begin{matrix} \frac{δ f_{g} (x^{* UAI})}{δ x} + γ_{r} \frac{δ (b_{r}^{UAI} - h_{r} (x^{* LAI}))}{δ x} \\ + ϑ_{jr} \frac{δ (ɛ_{j}^{UAI} - f_{j} (x^{* UAI}))}{δ x} = 0, \end{matrix}$ $\begin{matrix} γ_{r} (b_{r}^{UAI} - h_{r} (x^{* UAI})) = 0, \\ r = \bar{1, k}; ϑ_{j} (ɛ_{j}^{UAI} - f_{j} (x^{* UAI})) = 0, \end{matrix}$ $\begin{matrix} h_{r} (x^{* UAI}) ⩾ b_{r}^{UAI}, f_{j} (x^{* UAI}) ⩾ ɛ_{j}^{UAI} \end{matrix}$ $\begin{matrix} γ_{r} ⩾ 0, r = \bar{1, k}, ϑ_{j} ⩾ 0, j = \bar{1, p} . \end{matrix}$

Consider the following four cases:

Case i: For γ_r = 0, ϑ_j = 0. Then, $b_{r}^{UAI} ⩾ h_{r} (x^{* UAI}), ɛ_{j}^{UAI} > f_{j} (x^{* UAI})$ ,

Case ii: For γ_r ≠ 0, ϑ_j = 0. Then, $b_{r}^{UAI} ⩾ h_{r} (x^{* UAI}), ɛ_{j}^{UAI} > f_{j} (x^{* UAI})$ ,

Case iii: For γ_r = 0, ϑ_j ≠ 0. Then, $b_{r}^{UA} ⩾ h_{r} (x^{* UAI}), ɛ_{j}^{UAI} = f_{j} (x^{* UAI})$ ,

Case iv: For γ_r ≠ 0, ϑ_j ≠ 0. Then, $b_{r}^{UAI} = h_{r} (x^{* UAI}), ɛ_{j}^{LAI} = f_{j} (x^{* UAI})$ .

According to Lemma 1 and Lemma 2, it is clear that x^R ∈ [x^*LAI, x^*UAI] is a solution for the R - MONLP problem corresponding to $ɛ_{J}^{R} \in [ɛ_{j}^{* LAI}, ɛ_{u}^{* UAI}]$ .

7 Numerical examples

In this section, we give two numerical examples to illustrate the working of the above stated theory.

Example 1. Consider the following R - MONLP problem

$\begin{matrix} minF (x) = {(f_{1} (x), f_{2} (x), f_{3} (x))}^{T} \\ Subject to \\ h (x) ⩽ {\tilde{B}}^{R} \end{matrix}$ (13) where, f₁ (x) = x - 1, f₂ (x) = 5 - x, f₃ (x) = (x - 3) ² + 1 and h (x) = 1 - x, $\begin{matrix} {\tilde{b}}_{H}^{N} = (- 6, - 5, - 4, 1, 3, 5, 7), {\tilde{B}}^{R} \in [- 5, 5] . \end{matrix}$

The lower approximation problem corresponding to problem (13) is formulated as:

$\begin{matrix} minF (x) = {(f_{1} (x), f_{2} (x), f_{3} (x))}^{T} \\ Subject to \\ x ⩽ 6 . \end{matrix}$ (14)

The upper approximation problem corresponding to problem (13) is formulated as:

$\begin{matrix} minF (x) = {(f_{1} (x), f_{2} (x), f_{3} (x))}^{T} \\ Subject to \\ x ⩾ - 4 . \end{matrix}$ (15)

The efficient solution set of problem (14) is $\begin{matrix} X^{* LAI} = {x \in ℝ : x ⩽ 3} . \end{matrix}$

The efficient solution set of problem (14) is $\begin{matrix} X^{* UAI} = {x \in ℝ : - 4 ⩽ x ⩽ 3} \end{matrix}$

By applying the weighting method for the lower and upper approximation problems (14) and (15); respectively, we have

$\begin{matrix} {(LAP)}_{γ} \min (γ_{1}^{LAI} (1 - x) \\ + γ_{2}^{LAI} (5 - x) + γ_{2}^{LAI} ({(x - 3)}^{2} + 1)) \\ Subject to \\ x ⩽ 6, and \end{matrix}$ (16)

$\begin{matrix} {(UAP)}_{γ} \min (γ_{1}^{UAI} (1 - x) \\ + γ_{2}^{UAI} (5 - x) + γ_{2}^{UAI} ({(x - 3)}^{2} + 1)) \\ Subject to \\ x ⩾ - 4 . \end{matrix}$ (17)

Using the KKT optimality conditions for problems (16) and (17); respectively, we obtain $\begin{matrix} γ_{1}^{LAI} - γ_{2}^{LAI} + 2 γ_{3}^{LAI} (x^{* LAI} - 3) + δ_{1} = 0, \end{matrix}$ $\begin{matrix} δ_{1} (x^{* LAI} - 6) = 0, \end{matrix}$ $\begin{matrix} x - 6 ⩽ 0, \end{matrix}$

$δ_{1} ⩾ 0,$ (18) $\begin{matrix} γ_{1}^{LAI} = γ_{2}^{LAI} - 2 γ_{2}^{LAI} (x^{* LAI} - 3) - δ_{1}, \end{matrix}$ $\begin{matrix} γ_{2}^{LAI} = γ_{1}^{LAI} + 2 γ_{2}^{LAI} (x^{* LAI} - 3) + δ_{1}, \end{matrix}$ $\begin{matrix} γ_{3}^{LAI} = \frac{γ_{2}^{LAI} + γ_{1}^{LAI} - δ_{1}}{2 (x^{* LAI} - 3)}, \end{matrix}$ and $\begin{matrix} γ_{1}^{UAI} - γ_{2}^{UAI} + 2 γ_{3}^{UAI} (x^{* UAI} - 3) - ϑ_{1} = 0, \end{matrix}$ $\begin{matrix} ϑ_{1} (- x^{* UAI} - 4) = 0, \end{matrix}$ $\begin{matrix} - x - 4 ⩽ 0, \end{matrix}$

$ϑ_{1} ⩾ 0,$ (19) $\begin{matrix} γ_{1}^{UAI} = γ_{2}^{UAI} - 2 γ_{3}^{UAI} (x^{* UAI} - 3) + ϑ_{1}, \end{matrix}$ $\begin{matrix} γ_{2}^{UAI} = γ_{1}^{UAI} + 2 γ_{3}^{UAI} (x^{* UAI} - 3) - ϑ_{1}, \end{matrix}$ $\begin{matrix} γ_{3}^{UAI} = \frac{γ_{2}^{UAI} - γ_{1}^{UAI} + ϑ_{1}}{2 (x^{* UAI} - 3)} . \end{matrix}$

Thus, the set of weights corresponding to the solution x^* of the R - MONLP takes the form

$\begin{matrix} (γ_{1}^{LAI}, γ_{2}^{LAI}, γ_{3}^{LAI}) ⩽ ({\tilde{γ}}_{1}^{R}, {\tilde{γ}}_{2}^{R}, {\tilde{γ}}_{3}^{R}) ⩽ (γ_{1}^{UAI}, γ_{2}^{UAI}, γ_{3}^{UAI}) \end{matrix}$

For x^*LAI = x^*UAI = 2 ∈ X^*LAI ∩ X^*UAI, we have $\begin{matrix} γ_{1}^{LAI} = γ_{2}^{LAI} - 2 γ_{2}^{LAI} (x^{* LAI} - 3) - δ_{1}, \end{matrix}$ $\begin{matrix} γ_{1}^{UAI} = γ_{2}^{UAI} - 2 γ_{3}^{UAI} (x^{* UAI} - 3) + ϑ_{1} \end{matrix}$ $\begin{matrix} γ_{2}^{LAI} = γ_{1}^{LAI} + 2 γ_{2}^{LAI} (x^{* LAI} - 3) + δ_{1}, \end{matrix}$ $\begin{matrix} γ_{2}^{UAI} = γ_{1}^{UAI} + 2 γ_{3}^{UAI} (x^{* UAI} - 3) - ϑ_{1}, \end{matrix}$ $\begin{matrix} γ_{3}^{LAI} = \frac{γ_{2}^{LAI} + γ_{1}^{LAI} - δ_{1}}{2 (x^{* LAI} - 3)}, \end{matrix}$ $\begin{matrix} γ_{3}^{UAI} = \frac{γ_{2}^{UAI} - γ_{1}^{UAI} + ϑ_{1}}{2 (x^{* UAI} - 3)} . \end{matrix}$

Since, $\sum_{j = 1}^{3} γ_{j}^{LAI} = 1$ , and $\sum_{j = 1}^{3} γ_{j}^{UAI} = 1$ , we have $\begin{matrix} \begin{matrix} γ_{1}^{LAI} = \frac{1}{2} - \frac{1}{3} δ_{1}, γ_{1}^{UAI} = \frac{1}{3} (2 + ϑ_{1}), \\ γ_{2}^{LAI} = \frac{1}{2}, γ_{2}^{UAI} = 0, \\ γ_{3}^{LAI} = \frac{1}{3} δ_{1} . γ_{3}^{UAI} = \frac{1}{3} (1 - ϑ_{1}) . \end{matrix} \end{matrix}$

Thus,

$(\frac{1}{2} - \frac{1}{3} δ_{1}, \frac{1}{2}, \frac{1}{3} δ_{1}) ⩽ ({\tilde{γ}}_{1}^{R}, {\tilde{γ}}_{2}^{R}, {\tilde{γ}}_{3}^{R}) ⩽ (\frac{1}{3} (2 + ϑ_{1}), 0, \frac{1}{3} (1 - ϑ_{1}))$ , where $0 ⩽ δ_{1} ⩽ \frac{3}{2}, 0 ⩽ ϑ_{1} ⩽ 1$ .

Example 2. Consider the following R - MONLP

$\begin{matrix} minF (x) = {(f_{1} (x), f_{2} (x), f_{3} (x))}^{T} \\ Subject to \\ h (x) ⩽ {\tilde{B}}^{R} . \end{matrix}$ (20)

Where, f₁ (x) = x - 1, f₂ (x) = 5 - x, f₃ (x) = (x - 3) ² + 1 and h (x) = 1 - x, $\begin{matrix} {\tilde{b}}_{H}^{N} = (- 6, - 5, - 4, 1, 3, 5, 7), {\tilde{B}}^{R} \in [- 5, 5] . \end{matrix}$

Using the ɛ- constraints method, problem (20) becomes

$\begin{matrix} min f_{3} (x) = {(x - 3)}^{2} + 1 \\ Subject to \\ 1 - x ⩽ {\tilde{B}}^{R}, {\tilde{B}}^{R} \in [- 5, 5], \\ x - 1 ⩽ {\tilde{ɛ}}_{1}^{R}, {\tilde{ɛ}}_{1}^{R} \in [ɛ_{1}^{LAI}, ɛ_{1}^{UAI}], \\ 5 - x ⩽ {\tilde{ɛ}}_{2}^{R}, [ɛ_{2}^{LAI}, ɛ_{2}^{UAI}], \end{matrix}$ (21)

The lower approximation problem corresponding to problem (21) is formulated as:

$\begin{matrix} {(LAP)}_{ɛ} min {(x - 3)}^{2} + 1 \\ Subject to \\ x ⩽ 6, 1 - x + ɛ_{1}^{LAI} ⩽ 0, 5 - x + ɛ_{2}^{LAI} ⩽ 0 \end{matrix}$ (22)

The upper approximation problem corresponding to problem (21) is formulated as:

$\begin{matrix} {(UAP)}_{ɛ} min {(x - 3)}^{2} + 1 \\ Subject to \\ - x - 4 ⩽ 0, x - 1 - ɛ_{1}^{UAI} ⩽ 0, 5 - x - ɛ_{2}^{LAI} ⩽ 0 . \end{matrix}$ (23)

The KKT optimality conditions for problem (22), we have $\begin{matrix} 2 (x^{* LAI} - 3) + γ_{1} + ϑ_{1} - ϑ_{2} = 0, \end{matrix}$ $\begin{matrix} γ_{1} (x^{* LAI} - 6) = 0, \end{matrix}$ $\begin{matrix} x - 6 ⩽ 0, \end{matrix}$ $\begin{matrix} ϑ_{1} (1 - x^{* LAI} - ɛ_{1}^{LAI}) = 0, \end{matrix}$ $\begin{matrix} ϑ_{2} (x^{* LAI} - 5 + ɛ_{2}^{LAI}) = 0 \end{matrix}$ $\begin{matrix} γ_{1}, ϑ_{2}, ϑ_{2} ⩾ 0 . \end{matrix}$

For γ₁ = ϑ₁ = ϑ₂ = 0. Then x^*LAI = 3.

For γ₁ ≠ 0, ϑ₁ = ϑ₂ = 0. Then $x^{* LAI} = 3 - \frac{γ_{1}}{2}, x^{* LAI} ⩽ 3$ .

For γ₁ = 0, ϑ₁, ϑ₂ ≠ 0. Then $x^{* LAI} = \frac{1}{2} (ϑ_{2} - ϑ_{1}) + 3, x^{* LAI} ⩽ 3$ .

For γ₁, ϑ₁, ϑ₂ ≠ 0. Then $x^{* LAI} = \frac{1}{2} (ϑ_{2} - ϑ_{1} - γ_{1}) + 3, x^{* LAI} ⩽ 3,$ and $ɛ_{1}^{LAI} ⩽ x^{* LAI} - 1,$ $ɛ_{2}^{LAI} ⩽ 5 - x^{* LAI}$

Table 1

Comparisons of different researcher’s contributions

	Weighting method	ɛ-constraints method	KKT optimality	Efficient solution	Parametric study	Environment
Khalifa [17]	√	×	×	√	×	Rough set
Osman et al., [21]	×	×	×	√	×	Fuzzy set
Ammar, and Emsimir, [2]	√	×	√	√	×	Fuzzy set
Ahmed [1]	×	×	√	√	×	Fuzzy set
Ammar and Al-Asfar, [3]	√	×	√	√	×	Fuzzy set
This study	√	√	√	√	√	Rough set

The KKT optimality conditions for problem (23), we have $\begin{matrix} 2 (x^{* UAI} - 3) - γ_{1} + ϑ_{1} - ϑ_{2} = 0, \end{matrix}$ $\begin{matrix} text γ_{1} (- x^{* UAI} - 4) = 0, \end{matrix}$ $\begin{matrix} - x - 4 ⩽ 0, \end{matrix}$ $\begin{matrix} ϑ_{1} (- 1 + x^{* UAI} - ɛ_{1}^{UAI}) = 0, \end{matrix}$ $\begin{matrix} ϑ_{2} (- x^{* UAI} + 5 - ɛ_{2}^{LAI}) = 0 \end{matrix}$ $\begin{matrix} γ_{1}, ϑ_{2}, ϑ_{2} ⩾ 0 . \end{matrix}$

For γ₁ = ϑ₁ = ϑ₂ = 0. Then x^*UAI = 3.

For γ₁ ≠ 0, ϑ₁ = ϑ₂ = 0. Then $x^{* UAI} = 3 + \frac{γ_{1}}{2}, - 4 ⩽ x^{* UAI} ⩽ 3$ .

For γ₁ = 0, ϑ₁, ϑ₂ ≠ 0. Then $x^{* UAI} = \frac{1}{2} (ϑ_{2} - ϑ_{1}) + 3, - 4 ⩽ x^{* UAI} ⩽ 3$ .

For γ₁, ϑ₁, ϑ₂ ≠ 0. Then $x^{* UAI} = \frac{1}{2} (ϑ_{2} - ϑ_{1} + γ_{1}) + 3, - 4 ⩽ x^{* UAI} ⩽ 3$ , and $ɛ_{1}^{UAI} ⩾ x^{* UAI} - 1,$ $ɛ_{2}^{UAI} ⩽ 5 - x^{* UAI}$

Then the solution x^* ∈ [x^*LAI, x^*UAI] for the R - MONLP corresponding to $ɛ^{* R} \in [ɛ_{j}^{LAI}, ɛ_{j}^{* UAI}]$ where $ɛ_{1}^{LAI} ⩽ x^{* LAI} - 1$ , $ɛ_{2}^{LAI} ⩽ 5 - x^{* LAI}$ , and $ɛ_{1}^{UAI} ⩾ x^{* UAI} - 1$ , $ɛ_{2}^{UAI} ⩽ 5 - x^{* UAI}$ . It is obvious that the results obtained by the proposed method is more generalized than it is obtained by Khalifa (2018). In addition to this, we summarize the characteristic comparison of the proposed study with the existing recent studies [1–3 , 21] in Table 1. The symbol “√” in the table represent that it obeys the corresponding characteristics while “×” represent it does not. From this comparative study, we can analyze that the proposed study is well equipped with the features of the efficient, parametric and other using the rough parameters features while others fail to satisfy certain characteristics.

8 Conclusions and future work

In this paper, a concept of multiobjective nonlinear programming problem with rough parameters (R-MONLP) in the constraint set has introduced. In our real-life, there exists several parameters which impact simultaneously on each other and hence dealing with such different parameters and their conflictness create a MONLPP. The objective of the paper is to deal with a MONLPP with rough parameters in the constraint set. The considered R-MONLP problem has converted into the corresponding two approximation problems, namely lower and upper approximation problems. From these sub-problems, we apply the weighting method with the help of the KKT optimality conditions to obtain their solutions. Further, based on the ɛ- constraints method, rough weights and rough parameters ɛ in the constraints corresponding to the solution are determined related to lower and upper problems. Some propositions and the results are derived for their efficient solution of the problems. The present study has been illustrated with two numerical examples and demonstrate their functionally. Finally, a characteristic comparison of the stated work with the existing study are examined in the Table 1, which reveals that the proposed results have more beneficial to solve the optimization problems. In the future work, we may include the further extension of this study to other fuzzy-like structure (i.e., interval-valued fuzzy set, Neutrosophic set, Pythagorean fuzzy set, Spherical fuzzy set etc. with more discussion and suggestive comments. Some points for the future research scope are stated below:

Determination the relationship between the rough weights and rough parameter ɛ for rough multiobjective programming.

A study of duality in rough multiobjective programming problem.

A parametric analysis for rough programming problem with roughness in the objective function.

A parametric study for rough programming problem with roughness in the constraints.

Footnotes

Acknowledgments

The authors are thankful to the Editor-in-Chief and the anonymous referees for their helpful and suggestive comments. The authors would like to thank the Deanship of Scientific Research, Qassim University for funding the publication of this project.

Conflicts of interest

The authors declare no conflicts of interest.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

The authors would like to thank the Deanship of Scientific Research, Qassim University for funding the publication of this project.

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