A study of uncertainty multi-objective nonlinear programming problems for rough intervals

Abstract

In real conditions, the parameters of multi-objective nonlinear programming (MONLP) problem models can’t be determined exactly. Hence in this paper, we concerned with studying the uncertainty of MONLP problems. We propose algorithms to solve rough and fully-rough-interval multi-objective nonlinear programming (RIMONLP and FRIMONLP) problems, to determine optimal rough solutions value and rough decision variables, where all coefficients and decision variables in the objective functions and constraints are rough intervals (RIs). For the RIMONLP and FRIMONLP problems solving methodology are presented using the weighting method and slice-sum method with Kuhn-Tucker conditions, We will structure two nonlinear programming (NLP) problems. In the first one of this NLP problem, all of its variables and coefficients are the lower approximation (LAI) it’s RIs. The second NLP problems are upper approximation intervals (UAI) of RIs. Subsequently, both NLP problems are sliced into two crisp nonlinear problems. NLP is utilized because numerous real systems are inherently nonlinear. Also, rough intervals are so important for dealing with uncertainty and inaccurate data in decision-making (DM) problems. The suggested algorithms enable us to the optimal solutions in the largest range of possible solution. Finally, Illustrative examples of the results are given.

Keywords

Multi-objective nonlinear programming (MONLP)rough-interval (RI)Fully-rough-interval Slice-sum method and Kuhn-Tucker (KT) conditions

1 Introduction

In fact, many of our real-life phenomena are nonlinear in nature; that is why we need nonlinear programming tools capable of dealing with many conflicting goals. In this situation, the conventional single-objective methods and linear programming are not sufficient; we need other ways of thinking, other concepts and other approaches-multi-objective nonlinear optimization see [16].

The rough set theory (RST) suggested by [24], may be regarded as a successful mathematics tool for dealing with analyses of inaccurate and ambiguous data, which can be used in machine learning, knowledge discovery and pattern recognition [1 , 19] and [31]. The main notion in Pawlak’s rough set model is an equivalence relation. All equivalence classes form a partition of a specific universe. An arbitrary subset enables approximated by utilizing the equivalence classes into two subsets named the lower and upper approximation. The equivalence relationship is a strict 2 condition that may reduce RS’s applications in real and practical problems. Hence, different extensions of the initial Pawlak’s rough set from the equivalence relation to more general mathematical concepts were developed, for example, binary relations by [30].

The RST transacts with an approximation of an arbitrary subset of a universe through a pair of observable or definable subsets named lower and upper approximation. In other words, the lower approximation of the concept is the set of all elementary concepts which are embedded in it, while the upper approximation is the set of all elementary concepts that have a non-empty intersection with the concepts see [26]. In [32] defined a rough programming problem and characterized a rough optimal solution. A mathematical model for solving integer linear programming problems was given in [2 –5] and [6]. and the fuzzy rough solution set of multi-objective integer linear fractional programming problem was introduced by [6]. [29] have dealt with rough multi-objective transportation problems. The concept of solving traditional interval programming combined with fuzzy programming is used to construct the solution approach see [21]. [17] proposed a fuzzy interval goal programming to release the restrictions of FGP with single coefficient modelling. [23] proposed a level-bound method for solving fully fuzzy interval integer transportation problem. [20] introduced an analysis of interval programming utilizing rough interval.

In this paper, we introduce rough and fully–rough-interval MONLP problem such that all coefficients and variables in both the objective functions and constraints are rough intervals. Basic notions of rough set and rough intervals are given in section 2. We give solution proceedings to characterize the rough solution set of the RIMONLP problem with rough intervals in both the objective and constraints functions in section 3. An algorithm to deduce the fully rough solution set of the fully rough MONLP problem with a fully-rough-interval of parameters and variables are given in section 4. For the above two problems, numerical examples are given.

2 Preliminaries

In this section, we introduce some notions and definition related to the FRMONLP problem. The following concepts can be found in in [6 , 22] and [24].

2.1 Rough sets and rough intervals

In this section, we introduce the definition of rough sets, rough interval and basic operations for rough intervals.

1.2.1 Approximation space

Let U be denoted a finite and non-empty set called the universe, and let R ∈ U × U denoted an equivalence relation on U. The pair (U, R) is called an approximation space (Pawlak approximation space). If two elements x, y ∈ U are in the same equivalence class (i. e., x R y). The quotient set of U by the relation R is denoted by U/R and U/R = {X₁, X₂, X₃, . . . , X_n} where X_i is an equivalence class of R, i = 1, 2, . . . , n [24 , 27].

Definition 1. (Rough Sets) [24 –26]: For an approximation space (U, R) by a rough approximation in (U, R) we mean a mapping Apr (X) : P (U) → P (U) × P (U) defined by for every X ∈ P (U), Apr (X) = (X^LA, X^UA), where ${\begin{matrix} X^{L A} = {x \in U | [x]_{R} \subseteq X} \\ X^{U A} = {x \in U | [x]_{R} \cap X \neq φ} . \end{matrix}$

The pair (X^L A, X^U A) is called a rough set with a reference set X in approximation space (U, R). Here [x] _R denoted the equivalence class of R containing x, X^L A is called a lower approximation of X in (U, R) and X^U A is called an upper approximation of X in (U, R). The set B (X) = X^U A - X^L A is called the boundary region of X

Remark 1.

Set X is crisp if the boundary region of X is empty.

X is a rough set if the boundary region is not empty.

For a rough set X^R=(X^LA, X^UA)) the following properties hold:

X^L A ⊆ X ⊆ X^U A.

φ^L A = φ^U A = φ, U^L A = U^U A = U.

(X ∪ Y) ^U A = X^U A ∪ Y^U A, (X ∩ Y) ^L A = X^L A ∩ Y^L A.

X ⊆ Y ⇒ X^L A ⊆ Y^L A And X^U A ⊆ Y^U A.

Definition 2. (Rough Interval) [19, 22]: Suppose X is denoting a compact set of real numbers. A rough interval X^R is defined as an interval with known lower and upper bounds, but unknown distribution information for X denoted by X^R = [X^(LAI) ; X^(UAI)] where X^(LAI) and X^(UAI) are lower and upper approximation intervals of X^R, respectively with X^(LAI) ⊆ X^(UAI).

Proposition 1. For the rough interval X ^R , the following holds:

X^R ⩾ _R 0^R iff X^(LAI) ⩾ 0 and X^(UAI) ⩾ 0.

X^R ⩽ _R 0^R iff X^(LAI) ⩽ 0 and X^(UAI) ⩽ 0.

Let I^R denote the set of all rough intervals in R. Suppose A^R, B^R ∈ I^R we can write A^R = [A^LAI : A^UAI] and B^R = [B^LAI : B^UAI] such that A^LAI = [a^L L, a^U L] and A^UAI = [a^L U, a^U U] where a^L L, a^U L, a^L U and a^U U ∈ R. Where L L, U L , L U, U U, to represent lower lower, upper lower, lower upper, upper upper, respectively. Similarly, we can define B^L A I, B^U A I.

Proposition 2. Let A^R = [A^LAI : A^UAI] and B^R = [B^LAI : B^UAI] be two rough intervals in R, the following holds:

A^R = ^R B^R iff A^LAI = B^LAI and A^UAI = B^UAI.

A^R ⩽ _R B^R iff A^LAI ⩽ B^LAI and A^UAI ⩽ B^UAI.

A^R ⩾ _R B^R iff A^LAI ⩾ B^LAI and A^UAI ⩾ B^UAI.

2.2 Basic operations for rough interval

For any rough interval A^R, B^R where A^R ⩾ 0^R and B^R ⩾ 0 we can define the operation on rough intervals as follows [19 , 23]:

Addition: A^R ⊕ B^R = [[A^LAI + B^LAI] : [A^UAI + B^UAI]]

such that

${\begin{matrix} [A^{LAI} + B^{LAI}] = [a^{L L} + b^{L L}, a^{U L} + b^{U L}], \\ [A^{UAI} + B^{UAI}] = [a^{L U} + b^{L U}, a^{U U} + b^{U U}] . \end{matrix}$

Subtraction: A^R _ B^R = [[A^LAI - B^LAI] : [A^UAI - B^UAI]]

such that

${\begin{matrix} [A^{LAI} - B^{LAI}] = [a^{L L} - b^{U L}, a^{L L} - b^{U L}], \\ [A^{UAI} - B^{UAI}] = [a^{L U} - b^{U U}, a^{U U} - b^{L U}] . \end{matrix}$

Multiplication: A^R ⊗ B^R = [[A^LAI × B^LAI] : [A^UAI × B^UAI]]

such that

${\begin{matrix} [A^{LAI} \times B^{LAI}] = [a^{L L} \times b^{L L}, a^{U L} \times b^{U L}], \\ [A^{UAI} \times B^{UAI}] = [a^{L U} \times b^{L U}, a^{U U} \times b^{U U}] . \end{matrix}$

Division: A^R/B^R = [[A^LAI/B^LAI] : [A^UAI/B^UAI]]

such that ${\begin{matrix} [A^{LAI} / B^{LAI}] = [a^{L L} / b^{U L}, a^{U L} / b^{L L}], \\ [A^{UAI} / B^{UAI}] = [a^{L U} / b^{U U}, a^{U U} / b^{L U}] . \end{matrix}$

3 Rough-interval multi-objective nonlinear problem (RIMONLP)

Multi-objective nonlinear programming problem with rough intervals for coefficient and variables (RIMONLP) may be formulated as: $\begin{matrix} (RIMONLP) : min f_{r}^{r} (x^{R}) = & \sum_{j = 1}^{m} c_{r j}^{R} x_{j}^{δ_{j}}, r = 1, 2, . . ., k; δ_{j} \in Z^{+} \\ Subject to \\ \sum_{j = 1}^{m} a_{j}^{R} x_{j}^{δ_{j}} ⩽ b_{i}^{R}, \\ x_{j}^{δ_{j}} ⩾ 0, i \in I = {1, 2, . . ., n}; j \in J = {1, 2, . . ., m} \end{matrix}}$ (1)

The weighting single objective function as RNLP problem Kasia (1998) of (RIMONLP) problem (1) is: $\begin{matrix} (RNLP (w)) : & minimize \sum_{r = 1}^{k} w_{r} (\sum_{j = 1}^{m} c_{r j}^{R} x_{j}^{δ_{j}}), δ_{j} \in Z^{+} \\ subject to \\ \sum_{j = 1}^{m} a_{i j}^{R} x_{j}^{δ_{j}} ⩽ b_{i}^{R}, i \in I, j \in J \\ x_{j} ⩾ 0 for r ⩾ 0, r = 1, 2, . . ., k, \\ w_{r} \in W = {w : 0 ⩽ w ⩽ 1, \sum_{r = 1}^{m} w_{r} = 1} . \end{matrix}}$ (2)

The relationship between the optimal solution x^* of the weighting problem (2) and the efficient solution of problem (1) can be characterized by the following theorems [10].

Theorem 1. If x^* (w^*) is an optimal solution to the weighting problem (2) for some w^* > 0, then x^* is an efficient solution to the problem (1).

Theorem 2. If x^* is an efficient solution to the problem (1) then there exists w^* ∈ W such that x^* solves RNLP^R (w) ^* problem (2) and if either one of the following two conditions holds:

$w_{r}^{*} > 0, for all r = 1, 2, \dots k, or$

x^* is theunique optimal solution for a given (w^*).

The weighting problem determines the complete set of the optimal solution of problem (2) if the problem is convex. The general RINLP(w) problem (3) can be written as follows problem:

$\begin{matrix} RINLP (w) : minimize \sum_{r = 1}^{k} w_{r} (\sum_{j = 1}^{m} ([c_{r j}^{L L}, c_{r j}^{U L}] : [c_{r j}^{L U}, c_{r j}^{U U}])) x_{j}^{δ_{j}}, \\ subject to \\ \sum_{i = 1}^{m} ([a_{ij}^{LL}, a_{ij}^{UL}], [a_{ij}^{LU}, a_{ij}^{UU}]) x_{j}^{δ_{j}} ⩽ ([b_{i}^{LL}, b_{i}^{UL}], [b_{i}^{LU}, b_{i}^{UU}]), i \in I, j \in J, δ_{j} \in Z^{+} \\ x_{j} ⩾ 0, j \in J, w_{r} \in W = {w : 0 ⩽ w ⩽ 1, \sum_{r = 1}^{m} w_{r} = 1} . \end{matrix}}$ (3)

It can be written as:

$\begin{matrix} \begin{matrix} ([{NLP}^{LAI}, {NLP}^{UAI}]) = Min \sum_{r = 1}^{k} w_{r} \sum_{j = 1}^{m} ([c_{rj}^{L L}, c_{rj}^{U L}] : [c_{rj}^{L U}, c_{rj}^{U U}]) x_{j}^{δ_{j}} \\ subject to \\ \sum_{j = 1}^{m} [a_{i j}^{L L}, a_{i j}^{U L}] : [a_{i j}^{L U}, a_{i j}^{U U}]) x_{j}^{δ_{j}} ⩽ [b_{i}^{L L}, b_{i}^{U L}] : [b_{i}^{L U}, b_{i}^{U U}], i \in I, j \in J \\ x_{j} ⩾ 0, j \in J, w_{r} \in W = {w : 0 ⩽ w ⩽ 1, \sum_{r = 1}^{m} w_{r} = 1}, x^{δ_{j}} \in R^{n}, δ_{j} \in Z^{+} \end{matrix}} \end{matrix}$ (4)

Where $([c_{j}^{L L}, c_{j}^{U L}] : [c_{j}^{L U}, c_{j}^{U U}]), ([a_{i j}^{L L}, a_{i j}^{U L}] : [a_{i j}^{L U}, a_{i j}^{U U}])$ and $([b_{i}^{L L}, b_{i}^{L U}] :$ $[b_{i}^{L U}, b_{i}^{U U}])$ are rough intervals. The RINLP(w) problem (4) can be sliced into four multi-objective nonlinear programming problems namely, upper upper approximation (NLP^UU), lower upper approximation (NLP^UL), lower approximation (NLP^LL), and lower upper approximation (NLP^LU) problems as in the following: The possibly optimal range of the upper approximation interval [UAI] as

[NLP^LU, NLP^UU] by solving interval nonlinear programming as:

$\begin{matrix} {NLP}^{UAI} : f^{U} = & min \sum_{r = 1}^{k} w_{r} (\sum_{j = 1}^{m} [c_{j}^{L U}, c_{j}^{U U}]) x_{j}^{δ_{j}}, \\ subject to \\ \sum_{j = 1}^{m} [a_{i j}^{L U}, a_{i j}^{U U}] x_{j}^{δ_{j}} ⩽ [b_{i}^{L U}, b_{i}^{U U}] \\ x_{j} ⩾ 0, i \in I, j \in J; δ_{j} \in Z^{+}, w_{r} \in W \end{matrix}}$ (5)

The above possibly optimal range of problem NLP^UAL by solving in two classical NLPs as follows:

$\begin{matrix} {NLP}^{U U} : f^{UU} = & min \sum_{r = 1}^{k} w_{r} (\sum_{j = 1}^{m} c_{j}^{U U} x_{j}^{δ_{j}}), \\ subject to \\ \sum_{j = 1}^{m} a_{i j}^{U U} x_{j}^{δ_{j}} ⩽ b_{i}^{U U}, \\ x_{j} ⩾ 0, i \in I, j \in J; δ_{j} \in Z^{+}, w_{r} \in W \end{matrix}}$ (6) $\begin{matrix} {NLP}^{L U} : f^{LU} = & min \sum_{r = 1}^{k} w_{r} (\sum_{j = 1}^{m} c_{j}^{U U} x_{j}^{δ_{j}}), \\ subject to \\ \sum_{j = 1}^{m} a_{i j}^{L U} x_{j}^{δ_{j}} ⩽ b_{i}^{L U}, \\ x_{j} ⩾ 0, i \in I, j \in J; δ_{j} \in Z^{+}, w_{r} \in W \end{matrix}}$ (7)

The surely optimal range or the lower approximation interval [LAI] as [NLP^L L, NLP^U L] follows by solving interval nonlinear programming as:

$\begin{matrix} \begin{matrix} {NLP}^{LAI} : & f^{L} = min \sum_{r = 1}^{k} w_{r} (\sum_{j = 1}^{m} [c_{j}^{L L}, c_{j}^{U L}]) x_{j}^{δ_{j}} \\ subject to \\ \sum_{j = 1}^{m} [a_{i j}^{L L}, a_{i j}^{U L}] x_{j}^{δ_{j}} ⩽ [b_{i}^{L L}, b_{i}^{L U}], \\ x_{j} ⩾ 0, i \in I, j \in J; δ_{j} \in Z^{+}, w_{r} \in W \end{matrix}}, \end{matrix}$ (8)

The above surely optimal range of problem NLP^L A I follows by solving in the two classical NLPs:

$\begin{matrix} {NLP}^{L L} : f^{LL} = & min \sum_{r = 1}^{k} w_{r} (\sum_{j = 1}^{m} c_{j}^{L L} x_{j}^{δ_{j}}), \\ subject to \\ \sum_{j = 1}^{m} a_{i j}^{L L} x_{j}^{δ_{j}} ⩽ b_{i}^{L L}, \\ x_{j} ⩾ 0, i \in I, j \in J; δ_{j} \in Z^{+}, w_{r} \in W \end{matrix}}$ (9)

and

$\begin{matrix} {NLP}^{U L} : f^{UL} = & min \sum_{r = 1}^{k} w_{r} (\sum_{j = 1}^{m} c_{j}^{U L} x_{j}^{δ_{j}}), \\ subject to \\ \sum_{j = 1}^{m} a_{i j}^{U L} x_{j}^{δ_{j}} ⩽ b_{i}^{U L}, \\ x_{j} ⩾ 0, i \in I, j \in J; δ_{j} \in Z^{+}, w_{r} \in W \end{matrix}}$ (10)

We denote solutions to NLP^UU, NLP^UL, NLP^LL and NLP^LU as $x_{j}^{* uu}, x_{j}^{* u ℓ}$ , x^*ℓℓ and $x_{j}^{* ℓ u}$ , respectively.

Definition 4. If RIMONLP problems are NLP of RINLP(w) problem (4):

The interval [NLP^L A I] = [NLP^L L, NLP^U L] ([NLP^U A I] = [NLP^L U, NLP^U U]) is surely (possibly) optimal range of problem RINLP (4), if the optimal range of each RINLP is a subset of [NLP^L L, NLP^U L] ([NLP^L U, NLP^U U]).

Let [NLP^L L, NLP^L U] ([NLP^L U, NLP^U L]) be surely (possibly) optimal range of the problem RINLP (4). Then the rough interval: [[NLP^L L, NLP^U L] : [NLP^L U, NLP^U U]] is called the rough optimal range of RINLP(w) problem (4).

Any point optimal values belong to [NLP^LL, NLP^UL] ([NLP^LU, NLP^UU]), is called a completely (rather) satisfactory solution of RINLP(w) problem (4).

x^*R is the surely-feasible solution of the RINLP problem, iff it belongs to the lower approximation of the feasible set of NLP^UAI problem.

x^*R is the possibly feasible solution of the RINLP problem, iff it belongs to the upper approximation of the feasible set of NLP^LAI problem.

The relation between the rough optimal solution set of RINLP(w) problem (4) and four NLP^UU, NLP^UL, NLP^LL and NLP^LU problems, respectively as in the next theorem.

Theorem 3. If the sets ${x_{j}^{* u u}}_{j \in J}, {x_{j}^{* u ℓ}}_{j \in J}, {x_{j}^{* ℓ ℓ}}_{j \in J} and {x_{j}^{* ℓ u}}_{j \in J}$ , are the optimal solution of NLP^UU, NLP^ULNLP^LL and NLP^LU of the RINLP(w) (4) problem, with the minimum optimal value f^*UU, f^*UL, f^*LL and f^*LU, respectively, then the set of rough intervals. ${([x_{j}^{* ℓ ℓ}, x_{j}^{* u ℓ}] : [x_{j}^{* ℓ u}, x_{j}^{* u u}])}_{j \in J}$ is an optimal solution for the problem RINLP(w) (4) with minimum optimal values ([f^*L L, f^*U L] : [f^*L U, f^*U U]) provided $x_{j}^{* ℓ u} ⩽ x_{j}^{* ℓ ℓ} ⩽ x_{j}^{* u ℓ} ⩽ x_{j}^{* u u}$ , for j ∈ J.

Proof. Since solutions ${x_{j}^{* u u}}_{j \in J}, {x_{j}^{* u ℓ}}_{j \in J}, {x_{j}^{* ℓ ℓ}}_{j \in J} and {x_{j}^{* ℓ u}}_{j \in J}$ are optimal for the problems NLP^U U, NLP^U L, NLP^LL and NLP^LU, respectively and $x_{j}^{* ℓ u} ⩽ x_{j}^{* ℓ ℓ} ⩽ x_{j}^{* u ℓ} ⩽ x_{j}^{* u u}$ , for j ∈ J, then we can conclude that the set of solutions ${([x_{j}^{* ℓ ℓ}, x_{j}^{* u ℓ}] : [x_{j}^{* ℓ u}, x_{j}^{* u u}])}_{j \in J}$ are a feasible solution to the problem (4). Let ${([x_{j}^{* ℓ ℓ}, x_{j}^{* u ℓ}] : [x_{j}^{* ℓ u}, x_{j}^{* u u}]), j \in J}$ are a feasible solution to RINLP(w) (4) problem. Therefore, ${x_{j}^{* u u}, for all j \in J}, {x_{j}^{* u ℓ}}, {x_{j}^{* ℓ ℓ}} and {x_{j}^{* ℓ u}}$ for j ∈ J are feasible solutions for NLP^U U, NLP^U L, NLP^LL and NLP^LU problems, respectively. Follows that: $\begin{matrix} f^{* U U} & = \sum_{r = 1}^{k} w_{r}^{*} \sum_{j = 1}^{m} c_{rj}^{* U U} x_{j}^{* δ_{j} u u} \\ ⩽ \sum_{r = 1}^{k} w_{r}^{*} \sum_{j = 1}^{m} c_{rj}^{* U U} x_{j}^{δ_{j} u u} \end{matrix}$ (11) $\begin{matrix} f^{* U L} & = \sum_{r = 1}^{k} w_{r}^{*} \sum_{j = 1}^{m} c_{j}^{* U L} x_{j}^{* δ_{j} u ℓ} \\ ⩽ \sum_{r = 1}^{k} w_{r}^{*} \sum_{j = 1}^{m} c_{j}^{* U L} x_{j}^{δ_{j} u ℓ} \end{matrix}$ (12) $\begin{matrix} f^{* L L} & = \sum_{r = 1}^{k} w_{r}^{*} \sum_{j = 1}^{m} c_{j}^{* L L} x_{j}^{* δ_{j} ℓ ℓ} \\ ⩽ \sum_{r = 1}^{k} w_{r}^{*} \sum_{j = 1}^{m} c_{j}^{* L L} x_{j}^{δ_{j} ℓ ℓ} \end{matrix}$ (13) $\begin{matrix} f^{* L U} & = \sum_{r = 1}^{k} w_{r}^{*} \sum_{j = 1}^{m} c_{j}^{* L U} x_{j}^{* δ_{j} ℓ u} \\ ⩽ \sum_{r = 1}^{k} w_{r}^{*} \sum_{j = 1}^{m} c_{j}^{* L U} x_{j}^{δ_{j} ℓ u} \end{matrix}$ (14)

This implies that $\begin{matrix} ([f^{* L L}, f^{* U L}], [f^{* L U}, f^{* U U}] \\ = \sum_{j = 1}^{m} ([c_{j}^{L L}, c_{j}^{U L}], [c_{j}^{L U}, c_{j}^{U U}]) (x_{j}^{* δ_{j}}), \end{matrix}$ where $\begin{matrix} \sum_{j = 1}^{m} ([c_{j}^{L L}, c_{j}^{U L}], [c_{j}^{L U}, c_{j}^{U U}]) (x_{j}^{δ_{j}}) \\ ⩽ \sum_{j = 1}^{m} ([c_{j}^{L L}, c_{j}^{U L}], [c_{j}^{L U}, c_{j}^{U U}]) (x_{j}^{δ_{j}}) . \end{matrix}$

Therefore, the set of solutions ${([x_{j}^{* ℓ ℓ}, x_{j}^{* u ℓ}] : [x_{j}^{* ℓ u}, x_{j}^{* u u}])}_{j \in J}$ is a rough optimal solution set of RINLP(w) (4) problem with minimum optimal rough values ([f^*L L, f^*U L] : [f^*L U, f^*U U]). Hence, the theorem is provided.

3.1 Solution procedures for RIMONLP problem

Formulate the real-life problem as a RIMONLP problem (1),

Construct the weighting sum nonlinear RINLP(w) problem (4) of the RIMONLP problem.

Find the possibly optimal range of the NLP^UAI (5) problem.

According to Step 3, the possibly optimal range of problem NLP^UAL follows by solving two classical NLP^UU and NLP^LU problems (6), (7).

Find the surely optimal range of problem NLP^LAI problem (8).

According to Step 5, the surely optimal range of problem NLP^UAL follows by solving two classical NLP^LL and NLP^UL problems (9), (10).

We denote solutions to NLP^U U, NLP^L U, NLP^U L, NLP^L L as $x_{j}^{u u}, x_{j}^{ℓ u}$ , $x_{j}^{u ℓ}, x_{j}^{ℓ ℓ}$ , respectively, where the rough optimal value $\begin{matrix} f (x_{j}^{* R}) = ([f (x^{* ℓ ℓ}), f (x^{* u ℓ})], \\ [f (x^{* ℓ u}, f (x^{* u u})]) . \end{matrix}$

In addition, it follows that:

The possible values range for the problem NLP^U A I are [f (x^*ℓu) , f (x^*u u)].

The surely optimal values range for problem NLP^L A I are [f (x^*ℓℓ) , f (x^*u ℓ)].

The set ${[x_{j}^{* ℓ ℓ}, x_{j}^{* u ℓ}], j \in J}$ are completely satisfactory solutions.

The set ${[x_{j}^{* ℓ u}, x_{j}^{* u u}], j \in J}$ are rather satisfactory solutions.

Example 1. Consider the following MONLP RI $\begin{matrix} Min f^{R} = (c_{11}^{R} x_{1}^{2} + c_{12}^{R} x_{2}^{2}, c_{21}^{R} x_{1} + c_{22}^{R} x_{2}) \\ s . t . \\ a_{11}^{R} x_{1} + a_{12}^{R} x_{2} ⩽ b_{1}^{R}, \\ a_{21}^{R} x_{1} + a_{22}^{R} x_{2} ⩽ b_{2}^{R}, \\ x_{1}, x_{2} ⩾ 0, \end{matrix}$ where $\begin{matrix} c_{11}^{R} = [2, 3] : [1, 4], & c_{12}^{R} = [3, 4] : [2, 5], \\ c_{21}^{R} = [4, 5] : [3, 6], & c_{22}^{R} = [5, 6] : [4, 7], \\ a_{11}^{R} = [3, 4] : [2, 5], & a_{12}^{R} = [2, 3] : [1, 4], \\ a_{21}^{R} = [1, 1.5] : [0.5, 2], & a_{22}^{R} = [2, 4] : [1, 4], \\ b_{1}^{R} = [20, 30] : [10, 40], & b_{2}^{R} = [15, 20] : [5, 30] . \end{matrix}$

Then $\begin{matrix} min (([2, 3] : [1, 4]) x_{1}^{2} + ([3, 4] : [2, 5]) x_{2}^{2}, \\ ([5, 6] : [4, 8]) x_{1} + ([5, 6] : [4, 7]) x_{2}) \\ s . t . \\ ([3, 4] : [2, 5]) x_{1} + ([2, 3] : [1, 4]) x_{2} \\ ⩽ ([20, 30] : [10, 40]), \\ ([1, 1.5] : [0.5, 2]) x_{1} + ([2, 4] : [1, 4]) x_{2} \\ ⩽ ([15, 20] : [5, 30]), \\ x_{1}, x_{2} ⩾ 0 \end{matrix}$

Convert the RIMONLP to the single objective by weighting method $f (x) = w_{1} f_{1} (x) + w_{2} f_{2} (x),$

For w₁ = w₂ = 0.5, then f (x) take the form: $\begin{matrix} Min ([1, 1.5] : [0.5, 2]) x_{1}^{2} + ([1.5, 2] : [1, 2.5]) x_{2}^{2} \\ + ([2, 2.5] : [1.5, 3]) x_{1} + ([2.5, 3] : [2, 3.5]) x_{2} \end{matrix}$

To solve above problem, we must solve two-interval nonlinear programming problems with NLP^U and NLP^Las follows: $\begin{matrix} {NLP}^{U} = [{NLP}^{L U}, {NLP}^{U U}] = Min [0.5, 2] x_{1}^{2} \\ + [1, 2 . 5] x_{2}^{2} + [1.5, 3] x_{1} + [2, 3.5] x_{2} \\ s . t . \\ [2, 5] x_{1} + [1, 4] x_{2} ⩽ [10, 40], \\ [0.5, 2] x_{1} + [1, 4] x_{2} ⩽ [5, 30], \\ x_{1}, x_{2} ⩾ 0, \end{matrix}$ and $\begin{matrix} {NLP}^{L} = [{NLP}^{L L}, {NLP}^{U L}] = Min [1, 1.5] x_{1}^{2} \\ + [1.5, 2] x_{2}^{2} + [2, 2.5] x_{1} + [2.5, 3] x_{2} \\ s . t . \\ [3, 4] x_{1} + [2, 3] x_{2} ⩽ [20, 30], \\ [1, 1.5] x_{1} + [2, 4] x_{2} ⩽ [15, 20], \\ x_{1}, x_{2} ⩾ 0, \end{matrix}$

In interval nonlinear problem NLP^U is transformed into two problems NLP^UU and NLP^UL as the following: also. $\begin{matrix} {NLP}^{U U} = min 2 x_{1}^{2} + 2.5 x_{2}^{2} + 3 x_{1} + 3.5 x_{2} \\ s . t . \\ 5 x_{1} + 4 x_{2} ⩽ 40, \\ 2 x_{1} + 4 x_{2} ⩽ 30, \\ x_{j} ⩾ 0, j = 1, 2, \end{matrix}$ and $\begin{matrix} {NLP}^{U L} = min 1.5 x_{1}^{2} + 2 x_{2}^{2} + 2.5 x_{1} + 3 x_{2} \\ s . t . \\ 4 x_{1} + 3 x_{2} ⩽ 30, \\ 1.5 x_{1} + 4 x_{2} ⩽ 20, \\ x_{j}^{u ℓ} ⩽ x_{j}^{u u}, x_{j} ⩾ 0, j = 1, 2, \end{matrix}$

Also, problem NLP^L is transformed into two problems NLP^LL and NLP^LU as the following: $\begin{matrix} {NLP}^{L L} = min 1 x_{1}^{2} + 1.5 x_{2}^{2} + 2 x_{1} + 2.5 x_{2} \\ s . t . \\ 3 x_{1} + 2 x_{2} ⩽ 20, \\ 1 x_{1} + 2 x_{2} ⩽ 15, \\ x_{j}^{ℓ ℓ} ⩽ x_{j}^{u ℓ}; x_{j} ⩾ 0, j = 1, 2, \end{matrix}$ and $\begin{matrix} {NLP}^{L U} = min 0.5 x_{1}^{2} + 1 x_{2}^{2} + 1.5 x_{2} + 2 x_{1} \\ s . t . \\ 2 x_{1} + 4 x_{2} ⩽ 10, \\ 2 x_{1} + 4 x_{2} ⩽ 10, \\ x_{j}^{ℓ u} ⩽ x_{j}^{ℓ ℓ}; x_{j} ⩾ 0, j = 1, 2 . \end{matrix}$

We used KT conditions to find efficient values and efficient solution for UAI and LAI.

Table 1

Problem f ^R $x_{1}^{}$ $x_{2}^{*}$

NLP^U U 99.68 2.17 5.13

NLP^U L 60.17 2.17 4.19

NLP^L L 45.86 2.17 4.19

NLP^L U 28.82 2.17 3.92

Problem	f ^*R	$x_{1}^{*}$	$x_{2}^{*}$
NLP^U U	99.68	2.17	5.13
NLP^U L	60.17	2.17	4.19
NLP^L L	45.86	2.17	4.19
NLP^L U	28.82	2.17	3.92

4 Fully–rough-interval multi-objective nonlinear programming problem

A fully-rough-interval multi-objective nonlinear programming problem is: defined as:

$\begin{matrix} FRIMONLP : & min f_{r}^{R} (c^{R}, x^{R}) = \sum_{j = 1}^{m} c_{r j}^{R} x_{j}^{δ_{j} R}, r = 1, 2 . . ., k; δ_{j} \in Z^{+} \\ subject to \\ \sum_{j = 1}^{m} a_{i j}^{R} x_{j}^{δ_{j} R} ⩽ b_{i}^{R}, \\ x_{j}^{R} ⩾ 0, i \in I; j \in J \end{matrix}}$ (15)

Where $f_{r}^{R}$ , r = 1,...,k denote rough objective functions, $A^{R} = [a_{i j}^{R}]_{m \times n}$ , $B^{R} = b_{i}^{R}, i \in I$ and $c_{r}^{R} = (c_{1}^{R}, c_{2}^{R}, . . ., c_{m}^{R})^{T}$ ; r = 1, . . ., k, i ∈ I, j ∈ J are rough intervals parameters, $x_{j}^{δ R}$ nonlinear rough-interval decision variables as:

$c_{r j}^{R} = ([c_{r j}^{L L}, c_{r j}^{U L}] : [c_{r j}^{L U}, c_{r j}^{U U}])$ ,

$a_{i j}^{R} = ([a_{i j}^{L L}, a_{i j}^{U L}] : [a_{i j}^{L U}, a_{i j}^{U U}])$

$b_{i}^{R} = ([b_{i}^{L L}, b_{i}^{U L}] : [b_{i}^{L U}, b_{i}^{U U}])$ ,

$x_{j}^{δ_{j} R} = ([x_{j}^{L L δ_{j}}, (x_{j}^{U L δ_{j}}] : [x_{j}^{L U δ_{j}}, x_{j}^{U U δ_{j}}])$ ,

r = 1, 2, . . . , k ; i = 1, 2, . . . , n ; j = 1, 2, . . . , m.

Definition 5. (Rough efficient solution): The rough vector x^*R ∈ M^Rwhich satisfies the conditions in problem (16) is called a rough efficient solution of problem (15), iff there does not exist another x^R ∈ M^R such that $f_{r}^{R} (c^{* R}, x^{R}) ⩽ f_{r}^{* R} (c^{* R}, x^{* R}),$

for all r and $f_{r}^{R} (c^{* R}, x^{R}) < f_{r}^{* R} (c^{* R}, x^{* R})$ for at least one r = 1, 2, . . . , k.

4.1 The socialization problem

The multi-objective optimization FRIMONLP problem is modified into the following problem to be called a weighting that problem FNLP^R(w):

$\begin{matrix} {FNLP}^{R} (w) & F^{R} = min \sum_{r = 1}^{k} w_{r} (\sum_{j = 1}^{m} c_{r j}^{R} x_{j}^{δ_{j} R}), \\ subject to \\ M^{R} = {\begin{matrix} \sum_{j = 1}^{m} a_{i j}^{R} x_{j}^{δ_{j} R} ⩽ b_{i}^{R}, \\ x^{R} ⩾ 0, i \in I; j \in J, r ⩾ 2, δ_{j} \in Z^{+}, w_{r} \in W \end{matrix}} \end{matrix}}$ (16)

Theorem 4. If x^*R (w^*) ∈ M^R is an optimal solution of the weighing FNLP^R(w^*) problem (16) for some w^* > 0, then x^*R is an efficient solution of problem (15).

Proof. Follows as in [10]

Theorem 5. If x^*R (w^*) ∈ M^R is an optimal solution of FNLP^R(w) problem (16), then there exists w^* ∈ W such that x^*R solves FNLP^R (w^*) and if either one of the following two conditions holds:

$w_{r}^{*} > 0$ , for all r = 1, 2, . . . , k, or

x^*R is the unique optimal solution for a given (w^*).

Proof. Follows as in [10]

The weighting problem determines the complete set of the optimal solution of problem (16) if the problem is convex. The idea was associating each objective function with a weighting coefficient and minimized the weighted sum of the objectives. The rough interval of the problem FNLP^R (w^*) can be written as in the following:

$\begin{matrix} {FNLP}^{R} : & F^{R} = \min \sum_{r = 1}^{k} w_{r} (\sum_{j = 1}^{m} ([c_{r j}^{L L}, c_{r j}^{U L}] : [c_{r j}^{L U}, c_{r j}^{U U}]) \otimes \\ ([x_{j}^{δ_{j} L L}, x_{j}^{δ_{j} U L}] : [x_{j}^{δ_{j} L U}, x_{j}^{δ_{j} U U}])) \\ subject to \\ M^{R} = {\begin{matrix} \sum_{j = 1}^{m} ([a_{i j}^{L L}, a_{i j}^{U L}] : [a_{i j}^{L U}, a_{i j}^{U U}]) \otimes ([x_{j}^{δ_{j} LL}, x_{j}^{δ_{j} U L}] : \\ [x_{j}^{δ_{j} L U}, x_{j}^{δ_{j} U U}]) ⩽ ([b_{i}^{L L}, b_{i}^{U L}] : [b_{i}^{L U}, b_{i}^{U U}]) \\ x_{j}^{L L}, x_{j}^{U L}, x_{j}^{L U}, x_{j}^{U U} ⩾ 0, \\ i \in I; j \in J, r ⩾ 2, δ_{j} \in Z^{+}, w_{r} \in W \end{matrix}} \end{matrix}}$ (17)

where

$([x_{j}^{δ_{j} L L}, x_{j}^{δ_{j} U L}] : [x_{j}^{δ_{j} L U}, x_{j}^{δ_{j} U U}])$ ,

$([c_{r j}^{L L}, c_{r j}^{U L}] : [c_{r j}^{L U}, c_{r j}^{U U}])$ ,

$([a_{i j}^{L L}, a_{i j}^{U L}] : [a_{i j}^{L U}, a_{i j}^{U U}])$ ,

$([b_{i}^{L L}, b_{i}^{U L}] : [b_{i}^{L U}, b_{i}^{U U}])$

are rough intervals coefficient and variables in the objective function and the constraints. The fully rough nonlinear programming problem (16) -(17) can be written as two NLP problem with rough interval as in the following:

$\begin{matrix} {FNLP}^{U A I} : & F^{U} = \min \sum_{r = 1}^{k} w_{r} (\sum_{j = 1}^{m} [c_{r j}^{L U}, c_{r j}^{U U}]) \otimes [x_{j}^{δ_{j} L U}, x_{j}^{δ_{j} U U}]) \\ subject to \\ \sum_{j = 1}^{m} [a_{i j}^{L U}, a_{i j}^{U U}] \otimes [x_{j}^{δ_{j} L U}, x_{j}^{δ_{j} U U}] ⩽ [b_{i}^{L U}, b_{i}^{U U}], \\ x_{j}^{L U}, x_{j}^{U U} ⩾ 0, i \in I; j \in J, r ⩾ 2, δ_{j} \in Z^{+}, w_{r} \in W \end{matrix}}$ (18) $\begin{matrix} {FNLP}^{L A I} : & F^{L} = \min \sum_{r = 1}^{k} w_{r} (\sum_{j = 1}^{m} [c_{j}^{L L}, c_{j}^{U L}]) \otimes [x_{j}^{δ_{j} L U}, x_{j}^{δ_{j} U U}]) \\ subject to \\ \sum_{j = 1}^{m} [a_{i j}^{L L}, a_{i j}^{U L}] \otimes [x_{j}^{δ_{j} L U}, x_{j}^{δ_{j} U U}] ⩽ [b_{i}^{L L}, b_{i}^{U L}], \\ x_{j}^{L L}, x_{j}^{U L} ⩾ 0, i \in I; j \in J, r ⩾ 2, δ_{j} \in Z^{+}, w_{r} \in W \end{matrix}}$ (19)

The equation (18) can be rewrite as in the following NLP problems:

$\begin{matrix} {FNLP}^{U U} : & F^{UU} = \min \sum_{r = 1}^{k} w_{r} (\sum_{j = 1}^{m} c_{j}^{U U} x_{j}^{δ_{j} U U}) \\ subject to \\ \sum_{j = 1}^{m} a_{i j}^{U U} x_{j}^{δ_{j} U U} ⩽ b_{i}^{U U}, \\ x_{j}^{U U} ⩾ 0, i \in I; j \in J, r ⩾ 2, δ_{j} \in Z^{+}, w_{r} \in W \end{matrix}}$ (20) and

$\begin{matrix} {FNLP}^{L U} : & F^{LU} = m in \sum_{r = 1}^{k} w_{r} (\sum_{j = 1}^{m} c_{j}^{L U} x_{j}^{δ_{j} L U}) \\ subject to \\ \sum_{j = 1}^{m} a_{i j}^{L U} x_{j}^{δ_{j} L U} ⩽ b_{i}^{L U}, \\ x_{j}^{L U} ⩾ 0, i \in I; j \in J, r ⩾ 2, δ_{j} \in Z^{+}, w_{r} \in W \end{matrix}}$ (21)

Also, the Equation (19) can be rewritten as in the following NLP problems:

$\begin{matrix} {FNLP}^{L L} : & F^{LL} = \min \sum_{r = 1}^{k} w_{r} (\sum_{j = 1}^{m} c_{j}^{L L} x_{j}^{δ_{j} L L}) \\ subject to \\ \sum_{j = 1}^{m} a_{i j}^{L L} x_{j}^{δ_{j} L L} ⩽ b_{i}^{L L}, \\ x_{j}^{L L} ⩾ 0, i \in I; j \in J, r ⩾ 2, δ_{j} \in Z^{+}, w_{r} \in W \end{matrix}}$ (22)

$\begin{matrix} {FNLP}^{U L} : & F^{UL} = \min \sum_{r = 1}^{k} w_{r} (\sum_{j = 1}^{m} c_{j}^{U L} x_{j}^{δ_{j} U L}) \\ subject to \\ \sum_{j = 1}^{m} a_{i j}^{U L} x_{j}^{δ_{j} U L} ⩽ b_{i}^{U L}, \\ x_{j}^{U L} ⩾ 0, i \in I; j \in J, r ⩾ 2, δ_{j} \in Z^{+}, w_{r} \in W \end{matrix}}$ (23)

The possibly optimal range of the upper approximation interval [UAI] as [FNLP^LU: FNLP^UU] follows by solving interval nonlinear programming NLP^UU problem (20) and

$\begin{matrix} {FNLP}^{L U} : & F^{LU} = m in \sum_{r = 1}^{k} w_{r} (\sum_{j = 1}^{m} c_{j}^{L U} x_{j}^{δ_{j} L U}) \\ subject to \\ \sum_{j = 1}^{m} a_{i j}^{L U} x_{j}^{δ_{j} L U} ⩽ b_{i}^{L U}, \\ x_{j}^{L U} ⩽ x_{j}^{* L L}, x_{j}^{L U} ⩾ 0, \\ i \in I; j \in J, r ⩾ 2, δ_{j} \in Z^{+}, w_{r} \in W \end{matrix}}$ (24)

Where x^*LL is the solution of NLP^LL Problem.

The surely optimal range of problem FNLP^L A I follows by solving the two classical FNLPs:

$\begin{matrix} {FNLP}^{L L} : & F^{LL} = \min \sum_{r = 1}^{k} w_{r} (\sum_{j = 1}^{m} c_{j}^{L L} x_{j}^{δ_{j} L L}) \\ subject to \\ \sum_{j = 1}^{m} a_{i j}^{L L} x_{j}^{δ_{j} L L} ⩽ b_{i}^{L L}, \\ x_{j}^{L L} ⩽ x_{j}^{* U L}, x_{j}^{L L} ⩾ 0, \\ i \in I; j \in J, r ⩾ 2, δ_{j} \in Z^{+}, w_{r} \in W \end{matrix}}$ (25)

Where x^*UL is the solution of FNLP^UL Problem.

$\begin{matrix} {FNLP}^{U L} : & F^{UL} = \min \sum_{r = 1}^{k} w_{r} (\sum_{j = 1}^{m} c_{j}^{U L} x_{j}^{δ_{j} U L}) \\ subject to \\ \sum_{j = 1}^{m} a_{i j}^{U L} x_{j}^{δ_{j} U L} ⩽ b_{i}^{U L}, \\ x_{j}^{U L} ⩽ x_{j}^{* U U}, x_{j}^{U L} ⩾ 0, \\ i \in I; j \in J, r ⩾ 2, δ_{j} \in Z^{+}, w_{r} \in W \end{matrix}}$ (26)

Where x^*UU is the solution of NLP ^UU Problem. The rough optimal values (F^*R) and the rough efficient solutions $(x_{j}^{* R})$ will be as: $F^{* R} = ([F^{* L L}, F^{* U L}] : [F^{* L U}, F^{* U U}]),,$ $x_{j}^{* R} = ([x_{j}^{* L L}, x_{j}^{* U L}] : [x_{j}^{* L U}, x_{j}^{* U U}]), j \in J . .$

In addition: The possible values range for a problem NLP^U A I (18) are [F^*L U, F^*U U].

The surely optimal values range for a problem NLP^L A I (19) are [F^*L L, F^*U L].

Also, the interval $[x_{j}^{* L L}, x_{j}^{* U L}], j \in J$ are completely satisfactory solutions.

The interval $3 x_{1}^{U L} + x_{2}^{U L} ⩽ 110$ are rather satisfactory solutions.

4.2 Solution procedures for FRIMONLP problem

Formulate the real-life problem as a FRIMONLP problem (15),

Construct the weighting sum nonlinear FNLP(w) problem (16-17) of FRIMONLP problem.

Find the possibly optimal range of the upper approximation interval FNLP^UAI (18) problem.

According to Step 3, the possibly optimal range of problem NLP^UAL follows by solving two classical FNLP^UU(w) and FNLP^LU(w) problems (20) and (21).

Find the surely optimal range of the lower approximation interval problem FNLP^LAI problem (19).

According to Step 5, the surely optimal range of problem FNLP^UAL follows by solving two classical FNLP^LL and FNLP^UL problems (22) and (23).

We denote solutions to FNLP^UU, FNLP^LU, FNLP^UL and FNLP^LL as, $[x_{j}^{* L U}, x_{j}^{* U U}], [x_{j}^{* L L}, x_{j}^{* U L}]$ , respectively, where the rough optimal value $\begin{matrix} F (x_{j}^{* R}) = ([F (x^{* LL}), F (x^{* UL})], \\ [F (x^{* LU}), F (x^{* UU})]) . \end{matrix}$

Theorem 6. If the sets ${x_{j}^{* u u}}_{j \in J}, {x_{j}^{* u ℓ}}_{j \in J}, {x_{j}^{* ℓ ℓ}}_{j \in J} and {x_{j}^{* ℓ u}}_{j \in J}$ are the optimal solution of FNLP^UU, FNLP^UL, FNLP^LL and FNLP^LU of the FNLP^R (17) problem, with the minimum optimal value F^*UU, F^*UL, F^*LL and F^*LU, respectively, then the set of rough intervals. ${([x_{j}^{* ℓ ℓ}, x_{j}^{* u ℓ}] : [x_{j}^{* ℓ u}, x_{j}^{* u u}])$ forall j ∈ J}, is an optimal solution for the FNLP^R (17) problem with minimum optimal values ([F^*L L, F^*U L] : [F^*L U, F^*U U]) provided $x_{j}^{* ℓ u} ⩽ x_{j}^{* ℓ ℓ} ⩽ x_{j}^{* u ℓ} ⩽ x_{j}^{* u u}$ , forj ∈ J.

Proof. Follows as Theorem 3.

Example 2. Consider the following problem: $\begin{matrix} f^{R} = Min (c_{11}^{R} (x_{1}^{2})^{R} + c_{12}^{R} (x_{2}^{2})^{R} \\ + c_{31}^{R} x_{1}^{R}, c_{21}^{R} x_{2}^{R} + c_{22}^{R} x_{2}^{R}) \\ s . t . \\ a_{11}^{R} x_{1}^{R} + a_{12}^{R} x_{2}^{R} ⩽ b_{1}^{R}, \\ a_{21}^{R} x_{1}^{R} + a_{22}^{R} x_{2}^{R} ⩽ b_{2}^{R}, \\ x_{1}^{R}, x_{2}^{R} ⩾ 0 and rough interval, \end{matrix}$ where

$\begin{matrix} c_{11}^{R} = [3, 3.5] : [2.5, 5], & c_{12}^{R} = [2.5, 3] : [2, 4], \\ c_{13}^{R} = [2.5, 3] : [2, 4], & c_{21}^{R} = [3, 4] : [2, 5], \\ c_{22}^{R} = [5, 6] : [4, 8], & a_{11}^{R} = [2.5, 3] : [2, 3.5], \\ a_{12}^{R} = [1, 1] : [0.5, 2], & a_{21}^{R} = [3, 3.5] : [2.5, 4], \\ a_{22}^{R} = [0.5, 1.5] : [0.25, 2], & b_{1}^{R} = [90, 110] : [50, 136], \\ b_{2}^{R} = [55, 70] : [35, 150] \end{matrix}$ $\begin{matrix} Min f^{R} = (\begin{matrix} (([3, 3.5] : [2.5, 5]) \otimes ([(x_{1}^{2})^{L L}, (x_{1}^{2})^{U L}] : [(x_{1}^{2})^{L U}, (x_{1}^{2})^{U U}]) \\ \oplus ([2.5, 3] : [2, 4]) \otimes [[(x_{2}^{2})^{L L}, (x_{2}^{2})^{U L}] : [(x_{2}^{2})^{L U}, (x_{2}^{2})^{U U}] \\ \oplus ([2.5, 3] : [2, 4]) \otimes [x_{1}^{L L}, x_{1}^{U L}] : [x_{1}^{L U}, x_{1}^{U U}]) \\ ([3, 4] : [2, 5]) \otimes [(x_{1}^{L L})^{2}, (x_{1}^{2})^{U L}] : [(x_{1}^{2})^{L U}, (x_{1}^{2})^{U U}] \\ \oplus ([5, 6] : [4, 8]) \otimes [x_{2}^{L L}, x_{2}^{U L}] : [x_{2}^{L U}, x_{2}^{U U}]) \end{matrix}) \\ s . t . \\ ([2.5, 3] : [2, 3.5]) \otimes [x_{1}^{L L}, x_{1}^{U L}] : [x_{1}^{L U}, x_{1}^{U U}] \\ \oplus ([1, 1] : [0.5, 2]) \otimes [x_{2}^{LL}, x_{2}^{U L}] : [x_{2}^{LU}, x_{2}^{UU}] ⩽ ([90, 110] : [50, 136]), \\ ([3, 3.5] : [2.5, 4]) \otimes ([x_{1}^{L L}, x_{1}^{U L}] : [x_{1}^{L U}, x_{1}^{U U}]) \oplus ([0.5, 1.5] : [0.25, 2]) \\ \otimes ([x_{2}^{L L}, x_{2}^{U L}] : [x_{2}^{L U}, x_{2}^{U U}] ⩽ ([55, 70] : [35, 150]) \\ x_{j}^{L L}, x_{j}^{L U}, x_{j}^{U L}, x_{j}^{U U} ⩾ 0, j = 1, 2 \end{matrix}$

Convert the MONLPFRI to single objective by weighting method for w₁ = w₂ =0.5 the objective function is: $Min f^{R} = (\begin{matrix} ([3, 3.75] : [2.25, 5]) \otimes [(x_{1}^{2})^{L L}, (x_{1}^{2})^{U L}] : [(x_{1}^{2})^{L U}, (x_{1}^{2})^{U U}] \\ \oplus ([3.75, 4.5] : [3.25, 6]) \otimes ([(x_{2}^{2})^{L L}, (x_{2}^{2})^{U L}] : [(x_{2}^{2})^{L U}, (x_{2}^{2})^{U U}]) \\ \oplus ([1.25, 1.5] : [1, 2]) \otimes ([x_{1}^{L L}, x_{1}^{U L}] : [x_{1}^{L U}, x_{1}^{U U}]) \\ \oplus ([2.5, 3] : [2, 4]) \otimes ([x_{2}^{L L}, x_{2}^{U L}] : ([x_{2}^{L U}, x_{2}^{U U}]) \end{matrix})$

To solve above problem, we have to solve two interval nonlinear programming problems with upper approximation interval (UAI) and lower approximation interval (LAI),

$\begin{matrix} {NLP}^{U} = [{NLP}^{L U}, {NLP}^{U U}] = Min \\ (\begin{matrix} [2.25, 5] \otimes [(x_{1}^{2})^{L U}, (x_{1}^{2})^{U U}] \oplus [3.25, 6] \otimes [(x_{2}^{2})^{L U}, (x_{2}^{2})^{U U}] \\ \oplus [1, 2] \otimes [x_{1}^{L U}, x_{1}^{U U}] \oplus [2, 4] \otimes [x_{2}^{L U}, x_{2}^{U U}] \end{matrix}) \\ s . t . \\ [2, 3.5] \otimes [x_{1}^{L U}, x_{1}^{U U}] \oplus [0.5, 2] \otimes [x_{2}^{L U}, x_{2}^{U U}] ⩽ [50, 136], \\ [2.5, 4] \otimes [x_{1}^{L U}, x_{1}^{U U}] \oplus [0.25, 2] \otimes [x_{2}^{L U}, x_{2}^{U U}] ⩽ [35, 150], \\ x_{j}^{L U}, x_{j}^{U U} ⩾ 0, j = 1, 2 . \end{matrix}$ $\begin{matrix} {NLP}^{L} = [{NLP}^{L L}, {NLP}^{U L}] \\ = Min [3, 3.75] \otimes [(x_{1}^{2})^{L L}, (x_{1}^{2})^{U L}] \oplus [3.75, 4.5] \\ \otimes [(x_{2}^{2})^{L L}, (x_{2}^{2})^{U L}] \oplus [1.25, 1.5] \otimes [x_{1}^{L L}, x_{1}^{U L}] \\ \oplus [2.5, 3] \otimes ([x_{2}^{L L}, x_{2}^{U L}] \\ s . t . \\ [2.5, 3] \otimes [x_{1}^{L L}, x_{1}^{U L}] \oplus [1, 1] \otimes [x_{2}^{L L}, x_{2}^{U L}] ⩽ [90, 110], \\ [3, 3.5] \otimes [x_{1}^{L L}, x_{1}^{U L}] \oplus [0.5, 1.5] \otimes [x_{2}^{L L}, x_{2}^{U L}] ⩽ [55, 70], \\ x_{j}^{L L}, x_{j}^{U L} ⩾ 0, j = 1, 2 . \end{matrix}$

In the FNLP^R problem NLP^U is transformed into two crisp NLP problems NLP^UU and NLP^LU, also the NLP^L is transformed into two crisp NLP problems NLP^LL and NLP^UL as the following:

${NLP}^{U U} = min 5 (x_{1}^{2})^{U U} + 2 (x_{2}^{2})^{U U} + 2 x_{1}^{U U} + 4 x_{2}^{U U}$

subject to $3.5 x_{1}^{U U} + 2 x_{2}^{U U} ⩽ 136,$ $4 x_{1}^{U U} + 2 x_{2}^{U U} ⩽ 150,$ $x_{j}^{U U} ⩾ 0, j = 1, 2 .$ and $\begin{matrix} {NLP}^{U L} = & min 3.75 (x_{1}^{2})^{U L} + 4.5 (x_{2}^{2})^{U L} \\ + 1.5 x_{1}^{U L} + 3 x_{2}^{U L} \end{matrix}$

subject to $3 x_{1}^{U L} + x_{2}^{U L} ⩽ 110,$ $3.5 x_{1}^{U L} + 1.5 x_{2}^{U L} ⩽ 70,$ $x_{j}^{U L} ⩽ x_{j}^{U U}, x_{j}^{U L} ⩾ 0, j = 1, 2 .$

Also $\begin{matrix} {NLP}^{L L} = & min 3 (x_{1}^{2})^{L L} + 3.75 (x_{2}^{2})^{L L} \\ + 1.25 x_{1}^{L L} + 2.5 x_{2}^{L L} \end{matrix}$

subject to $2.5 x_{1}^{L L} + x_{2}^{L L} ⩽ 90,$ $3 x_{1}^{L L} + 0.5 x_{2}^{L L} ⩽ 55,$ $x_{j}^{L L} ⩽ x_{j}^{U L}, x_{j}^{L L} ⩾ 0, j = 1, 2 .$ and $\begin{matrix} {NLP}^{L U} = & min 2.25 (x_{1}^{2})^{L U} + 3.25 (x_{2}^{2})^{L U} + x_{1}^{L U} \\ + 2 x_{2}^{L U} \end{matrix}$

subject to

2 x_{1}^{L U} + 0.5 x_{2}^{L U} ⩽ 50,

2.5 x_{1}^{L U} + 0.25 x_{2}^{L U} ⩽ 35,

x_{j}^{L U} ⩽ x_{j}^{L L}, x_{j}^{L U} ⩾ 0, j = 1, 2 .

Problems	f ^*R	$x_{1}^{* R}$	$x_{2}^{* R}$
NLP^U U	9138	28	29
NLP^U L	4097.75	7.57	29
NLP^L L	3407	7.57	29
NLP^L U	1158.73	7.57	17.43

where the rough efficient values range solutions for

\begin{matrix} F^{* R} & = ([{NLP}^{L A I}, {NLP}^{U A I}]) \\ = ([3407, 4097.75] : [1158.73, 9138]) . \end{matrix}

The rough optimal solutions are: $\begin{matrix} x_{1}^{* R} = ([7.57, 7.57] : [7.57, 28]), \\ x_{2}^{* R} = ([29, 29] : [17.43, 29]), \end{matrix}$

And the possibly optimal values range solution for NLP^UAI is $[{NLP}^{L U}, {NLP}^{U U}] = [1158.73, 9138] .$

The surely optimal values range solutions for NLP^LAI is: $[{NLP}^{L L}, {NLP}^{U L}] = [3407, 4097.75] .$

In addition, the completely satisfactory solution for $\begin{matrix} = [7.57, 7.57], [x_{2}^{* L L}, x_{2}^{* U L}] \\ = [29, 29] . \end{matrix}$

And the rather satisfactory solution for $\begin{matrix} [x_{1}^{* L U}, x_{1}^{* U U}] & = [7.57, 28], [x_{2}^{* L U}, x_{2}^{* U U}] \\ = [17.43, 29] . \end{matrix}$

5 Conclusion

In this paper, we presented a methodology for solving two types of uncertainty multi-objective nonlinear programming problems. The first one is the multi-objective nonlinear programming (RIMONLP) problem with rough coefficient in both objectives and constraints functions. We characterize the rough efficient solution set and the rough set of optimal value for (RIMONLP) problem by a given procedure. The second one is the fully-rough-interval multi-objective nonlinear programming (FRIMONLP) problem. In this problem, all coefficients and variables in the objectives and the constraints functions are rough numbers. An algorithm to deduce the rough efficient solution set, and it does corresponding the rough optimal value was introduced. The algorithms of the two problems depend on the Slice-sum method and the weighting sum method. For each algorithm, we give illustrative examples.

Footnotes

Acknowledgments

The authors would like to express their gratitude to the anonymous reviewers for the comments and suggestions, which greatly helped them to improve the paper.

References

Abu-Donia

M.H.

, Multi knowledge based rough approximation and applications, Knowledge-Based System 26 (2012), 20–29.

Ammar

and Emsimir

, A mathematical model for solving fuzzy integer linear programming problems with fully rough intervals, Granular Computing, (2020). Dor: 10.1007/s41066-020-00216-4.

Ammar

and Emsimir

, A mathematical model for solving integer linear programming problems, African Journal of Mathematics and Computer Science Research 13(1) (2020). DOI: 105897/AJMCSR2019.0804.

Ammar

E.E.

and El-Wahed

, A study of probabilistic multi-objective linear fractional programming problems under fuzziness, International Journal of Industrial Engineering and Production Research 31(1) (2020), 1–12.

Ammar

E.E.

and El-Wahed

, An Interactive Approach for Solving the Multi-objective Minimum Cost Flow Problem in the Fuzzy Environment, Journal of Mathematics 2020 (2020), 6247423.

Ammar

E.E.

and Eljabi

, On solving fuzzy rough multi-objective integer linear fractional programming problem, Journal of Intelligent & Fuzzy Systems 37(5) (2019).

Atteya

T.E.M.

, Rough multiple objective programming, European Journal of the Operational Research 248(1) (2016), 204–210.

Bazaraa

M.S.

, Jarvis

J.J.

and Sherali

H.D.

, Linear programming and network flows, John Wiely and Sons, New York, (2010).

Bronson

, Theory and problems of operations research, Copyright by McGraw Hill Book Company Singapore, (1983).

10.

Chankong

and Himes

Y.Y.

, Multiobjective Decision Making theory and methodology, Elsevier Science Publishing Co. Ine, (1983).

11.

Chen

S.Y.

, Classifying credit ratings for Asian banks using integrating feature selection and the CPDA-based rough set approach, Knowledge-Based Systems 26 (2012), 259–270.

12.

Formica

, Semantic web search based on rough sets and fuzzy formal concept analysis, Knowledge-Based Systems 26 (2012), 40–47.

13.

George

and Yuan

, Fuzzy sets and fuzzy logic: Theory ad Applications, Prentice-Hall, (2008).

14.

Gupta

P.K.

and Mohan

, Problems in Operations Research, Sultan Chand and Sons, (2006).

15.

Hamzehee

, Yaghoobi

M.A.

and Mashinchi

, Linear Programming with Rough Interval Coefficients, Journal of Intelligent and Fuzzy Systems (26) (2014), 1179–1189.

16.

Kasia

, Nonlinear multi-objective optimization, Kluwer Academic, Publishers (1998).

17.

Kouaissah

and Hocine

, Optimizing sustainable and renewable energy portfolios using a fuzzy interval goal programming approach, Computers & Industrial Engineering 144 (2020), 10448.

18.

Kuhn

H.W.

and Tucker

A.W.

, Nonlinear programming proccedings of the second Berkeiey Smposium on Mathematical statistics and Probability (ed. j. Neyman), University of California (1951) Press, PP. 481–492.

19.

, Huang

and He

, An inexact rough-interval fuzzy linear programming method for generating conjunctive water-allocation strategies to agricultural irrigation systems, App Math Mod (35) (2011), 4330–4340.

20.

Midya

and Roy

S.K.

, Analysis of interval programming in different environments and its application to fixed-charge transportation problem, Discret Math Algorithms Appl 9(3) (2017), 1750040.

21.

Osman

M.S.

, El-Sherbiny

M.M.

, Khalifa

and Farag

H.H.

, A fuzzy technique for solving rough interval multi-objective transportation problem, Int J Comput Appl 147(10) (2016), 49–57.

22.

Pandian

, Natarajan

and Akilbasha

, Fully rough integer interval transportation problems, International Journal of Pharmacy & Technology 8(2) (2016), 13866–13876.

23.

Pandian

, Natarajan

and Akilbasha

, Fuzzy interval integer transportation problems, Int J of Pure and Applied Mathematics 119(9) (2018), 133–142.

24.

Pawlak

, Rough sets, International Journal of Information and Computer Sciences 11(5) (1982), 341–356.

25.

Pawlak

, Rough sets: Theoretical aspects of reasoning about data, system theory, Knowledge Engineering and Problem Solving, Kluwer Academic Publishers, Dordrecht, the Netherlands, vol. 9, (1991).

26.

Pawlak

and Skowron

, Rudiment of rough sets, Information Sciences (177) (2007), 3–27.

27.

Rebollédo

, Rough intervals-enhancing intervals for qualitative modeling of technical systems, Artificial Intelligence 170 (2006), 667–685.

28.

Roy

S.K.

and Mula

, Rough set approach to bi-matrix game, International Journal of Operational Research 23(2) (2015), 229–244.

29.

Roy

S.K.

, Midya

and Vincent

F.Y.

, Multi-objective fixed-charge transportation problem with random rough variables, Int J Uncertain Fuzziness Knowl Based Syst 26(6) (2018), 971–996.

30.

Slowinski

, A generalized definition of rough approximations based on similarity, IEEE Transactions on Knowledge and Data Engineering 12(2) (2000), 331–336.

31.

Wang.

, Huang

and Hand

B.T.

, Yang, An interval-valued fuzzy-stochastic programming approach and its application to municipal solid waste management, Environmental Modelling & Software 29 (2016), 24–36.

32.

Youness

, Characterising solutions of rough programming problems, European Journal of Operational Research (168) (2006), 1019–1029.

A study of uncertainty multi-objective nonlinear programming problems for rough intervals

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Rough sets and rough intervals

1.2.1 Approximation space

2.2 Basic operations for rough interval

3 Rough-interval multi-objective nonlinear problem (RIMONLP)

Table 1 Problem f *R x 1 * x 2 * NLPU U 99.68 2.17 5.13 NLPU L 60.17 2.17 4.19 NLPL L 45.86 2.17 4.19 NLPL U 28.82 2.17 3.92

5 Conclusion

Footnotes

Acknowledgments

References

Table 1

Problem f ^R $x_{1}^{}$ $x_{2}^{*}$

NLP^U U 99.68 2.17 5.13

NLP^U L 60.17 2.17 4.19

NLP^L L 45.86 2.17 4.19

NLP^L U 28.82 2.17 3.92