Robust linear programming with IT2 rough coefficients

Abstract

In this paper, a robust linear programming is considered, where all of its coefficients in the objective function and constraints are rough intervals or IT2 rough interval coefficients. First, we allow the IT2 rough intervals to transform into rough intervals using [α₁, α₂] level. Then, a robust two-step solution method (RTSM) is developed to solve the robust linear programming problem with IT2 rough interval coefficients (LPIT2RIC). Finally, an example is presented to demonstrate the results.

Keywords

Linear programming uncertainty T2 fuzzy set theory rough set theory IT2 rough coefficients

1 Introduction

Linear programming was one of the most popular methods used in optimization problems [1, 2]. A linear programming problem aims to minimize (or maximize) a linear objective function subject to a set of linear constraints. However, in many real-world management problems, almost all decision making and optimization problems are confronted with uncertainty, mostly including interval, stochastic, fuzzy and roughness. The existence of complex uncertainties makes it difficult to obtain optimal strategies. For example, waste generation rate within a city is related to many socio-economic and environmental factors, and exhibits uncertain and dynamic features; the efficiency of a municipal wastewater treatment plant is affected by wastewater flow rate, and is uncertain in nature; regional air quality is mainly influenced by air pollutant emissions within this area, which also present uncertain characteristics. Such uncertainties can lead to increased complexities in the related optimization efforts.

In the past decade, many researchers have developed interest in optimization problems with interval, stochastic, and fuzzy [3 –6]. Among these efforts, the T2 fuzzy set theory and rough set theory were particularly considered in modeling parameters for tackling subjective information derived from decision-makers. The concept of rough set theory was proposed by Pawlak [7] as a method for the joint management of uncertainty and vagueness. The rough interval (RI), proposed by Robolledo [8], is used to deal with partially ill-defined or unknown variables and parameters. RI was introduced to adapt rough set principles to model continuous variables. Note that, rough sets were used only to handle discrete objects, and could not represent continuous values. RI is a particular case of rough sets. It fulfills all rough sets’ properties and core concepts, including the upper and lower approximation definitions [8]. Type-2(T2) fuzzy set (FS), the most attractive method to measure the dispersion off formal fuzzy sets, was proposed by L.A. Zadeh [9] as an extension of ordinary fuzzy sets. It has also been seen as one of the methods to increase the fuzziness of a relation of membership. However, the most common reason that the T2 FS method has not been widely used is that it is more difficult to understand than the type-1(T1) FS method. Presently, there has been a logical progression from T1 FS to the interval T2 fuzzy sets (IT2FS) method or T2 fuzzy intervals. The IT2FS is an accessible type of the general T2 FS method because it is easier to compute, especially when there are large uncertain variables, and it can yield more promising solutions than T1 FS [10].

In practice, dual and even multiple uncertainties may exist particularly in systems with high vagueness and ambiguity. Thus, the development of a method that can reflect multiple-uncertain characteristics in both modeling formulation and resulting solutions is necessary. Zhang [11] presented a general framework for the study of IT2 rough fuzzy set-the upper and lower approximations of which are IT2 fuzzy sets.

The objective of this study is to propose a robust linear programming with rough interval and IT2 rough interval coefficients to address above challenges. In addition, a robust two-step method (RTSM) is developed for solving this problem. The RTSM can help to at least guarantee feasibility of the generated solutions under the best-case constraints, while the dimensions of the decision space are not significantly compromised. The rest of the paper is organized as follows. In Section 2, IT2 fuzzy set is presented. In Section 3, IT2 rough interval is discussed. Section 4 proposes a solution method for a robust linear programming with IT2 rough interval coefficients. In Section 5, an example of this problem is presented. Finally, concluding remarks are given in Section 6.

2 IT2 fuzzy set

Zadeh introduced the concept of type-2 fuzzy sets in 1975 [9] as follow:

Definition 2.1. A type-2 (T2) fuzzy set, denoted by $\tilde{A}$ , is characterized by a type-2 membership function $μ_{\tilde{A}} (x, u)$ , where x ∈ U and u ∈ J_x ⊆ I, i.e., $\tilde{A} = {((x, u), μ_{\tilde{A}} (x, u)) | \forall x \in U, \forall u \in J_{x} \subseteq I}$ in which $0 \leq μ_{\tilde{A}} (x, u) \leq 1$ , $\tilde{A}$ can also be expressed as follows: $\tilde{A} = \int_{x \in U} \int_{u \in J_{x}} μ_{\tilde{A}} (x, u) / (x, u), J_{x} \subseteq I$ where ∫ denotes the union among all the admissible x and u. For a discrete universe of discourse, ∫ is replaced by Σ.

The class of all type-2 fuzzy sets of the universe U is denoted by $\tilde{F} (U)$ .

Definition 2.2. Interval type-2 (IT2) fuzzy set is defined as a T2 fuzzy set where all its secondary grades are unity ∀f_x (u) =1. An IT2 fuzzy set can be completely determined using its FOU as follow [12, 13]: $FOU (\tilde{A}) = \int_{x \in X} [\underline{μ_{\tilde{A}} (x)}, \bar{μ_{\tilde{A}} (x)}] .$

3 IT2 rough interval

Definition 3.1. The qualitative value A is called a rough interval when one can assign two closed intervals A^* and A_* on R to it, where A_* ⊆ A^*. Moreover,

If x ∈ A_* then A surely takes x (denoted by x ∈ A).

If x ∈ A^* then A possibly takes x.

If x ∉ A^* then A surely does not take x(denoted by x ∉ A).

A^* and A_* are called the lower approximation interval (LAI) and the upper approximation interval (UAI) of A, respectively. Further, A is denoted by A = [A_*, A^*].

Definition 3.2. Let U and W be two nonempty universes, and let $\tilde{R} \in {\tilde{F}}_{IT 2} (U \times W)$ . The triple $(U, W, \tilde{R})$ is called an IT2 fuzzy approximation space. For any $\tilde{A} \in {\tilde{F}}_{IT 2} (W)$ , the upper and lower approximations of $\tilde{A}$ with respect to $(U, W, \tilde{R})$ , denoted by $\bar{\tilde{R}} (\tilde{A})$ and $\tilde{\underline{R}} (\tilde{A})$ , are two IT2 fuzzy sets and are, respectively defined as follows: $\begin{matrix} \bar{\tilde{R}} (\tilde{A}) & = & \int_{x \in U} \int_{u \in D \bar{\tilde{R}} (\tilde{A}) (x)} 1 / (x, u), \\ \tilde{\underline{R}} (\tilde{A}) & = & \int_{x \in U} \int_{u \in D \tilde{\underline{R}} (\tilde{A}) (x)} 1 / (x, u), \end{matrix}$ where for any x ∈ U, $D \bar{\tilde{R}} (\tilde{A}) (x) = [μ_{\underline{D} \bar{\tilde{R}} (\tilde{A})} (x), μ_{\bar{D} \bar{\tilde{R}} (\tilde{A})} (x)]$ , $D \tilde{\underline{R}} (\tilde{A}) (x) = [μ_{\underline{D} \tilde{\underline{R}} (\tilde{A})} (x), μ_{\bar{D} \tilde{\underline{R}} (\tilde{A})} (x)]$ .

If for any x ∈ U, $\bar{\tilde{R}} (\tilde{A}) = \tilde{\underline{R}} (\tilde{A})$ , then the IT2 fuzzy set $\tilde{A}$ is definable with respect to the IT2 fuzzy approximation space $(U, W, \tilde{R})$ . Otherwise, the IT2 fuzzy set $\tilde{A}$ is rough with respect to the IT2 fuzzy approximation space, and $[\bar{\tilde{R}} (\tilde{A}), \tilde{\underline{R}} (\tilde{A})]$ is called an IT2 fuzzy rough set. Meanwhile, the mappings $\bar{\tilde{R}} : {\bar{\tilde{F}}}_{IT 2} (W) \to {\bar{\tilde{F}}}_{IT 2} (U)$ and $\tilde{\underline{R}} : {\tilde{\underline{F}}}_{IT 2} (W) \to {\tilde{\underline{F}}}_{IT 2} (U)$ are referred as the upper IT2 fuzzy rough approximation operator and the lower IT2 fuzzy rough approximation operator.

Definition 3.3. [11, 14] Let $\tilde{A} \in {\tilde{F}}_{IT 2} (W)$ and [α₁, α₂] ∈ [I]; then, we define ${\tilde{A}}_{[α_{1}, α_{2}]} = {x \in W | μ_{\underline{D} \tilde{A}} (x) \geq α_{1}, μ_{\underline{D} \tilde{A}} (x) \geq α_{2}}$ . Furthermore, we define ${\tilde{A}}_{α_{1}} = {x \in W | μ_{\underline{D} \tilde{A}} (x) \geq α_{1}} = [{\tilde{A}}_{α_{1}}^{l}, {\tilde{A}}_{α_{1}}^{u}]$ , and ${\tilde{A}}_{α_{2}} = {x \in W | μ_{\bar{D} \tilde{A}} (x) \geq α_{2}} = [{\tilde{A}}_{α_{2}}^{l}, {\tilde{A}}_{α_{2}}^{u}]$ .

Example. Figure 1 illustrates an IT2 fuzzy set $\tilde{A}$ . For [α₁, α₂] = [0.3, 0.5], we can calculate the set of $\tilde{A}$ under the lever [0.3, 0.5], ${\tilde{A}}_{[0.3, 0.5]} = [[0 . 28, 0 . 63], [0 . 3, 0 . 5]]$ .

Fig.1

Set of $\tilde{A}$ under the lever [0.3, 0.5].

The arithmetic operations on the IT2 RIs are based on Moore’s interval arithmetic [15]. The IT2 RIs arithmetic separates an arithmetic operation, such as addition or subtraction, into two identical interval operations: one operates over the UAIs, while the other uses the corresponding LAI as operands. We state these arithmetic operations as follows:

Let ${\tilde{A}}_{[α_{1}, α_{2}]} = [[{\tilde{A}}_{α_{1}}^{l}, {\tilde{A}}_{α_{1}}^{u}], [{\tilde{A}}_{α_{2}}^{l}, {\tilde{A}}_{α_{2}}^{u}]]$ ,

Addition: ${\tilde{A}}_{[α_{1}, α_{2}]} + {\tilde{B}}_{[α_{1}, α_{2}]} = [[{\tilde{A}}_{α_{1}}^{l} + {\tilde{B}}_{α_{1}}^{l}, {\tilde{A}}_{α_{1}}^{u} + {\tilde{B}}_{α_{1}}^{u}], [{\tilde{A}}_{α_{2}}^{l} + {\tilde{B}}_{α_{2}}^{l}, {\tilde{A}}_{α_{2}}^{u} + {\tilde{B}}_{α_{2}}^{u}]]$ ,

Negation: $- {\tilde{A}}_{[α_{1}, α_{2}]} = [[- {\tilde{A}}_{α_{1}}^{u}, - {\tilde{A}}_{α_{1}}^{l}], [- {\tilde{A}}_{α_{2}}^{u}, - {\tilde{A}}_{α_{2}}^{l}]]$ ,

Subtraction: ${\tilde{A}}_{[α_{1}, α_{2}]} - {\tilde{B}}_{[α_{1}, α_{2}]} = [[{\tilde{A}}_{α_{1}}^{l} - {\tilde{B}}_{α_{1}}^{u}, {\tilde{A}}_{α_{1}}^{u} - {\tilde{B}}_{α_{1}}^{l}], [{\tilde{A}}_{α_{2}}^{l} - {\tilde{B}}_{α_{2}}^{u}, {\tilde{A}}_{α_{2}}^{u} - {\tilde{B}}_{α_{2}}^{l}]]$ .

4 Robust linear programming with IT2 rough interval coefficients

Chinneck and Ramadan transformed the original interval LP into two LPs with crisp coefficients [16]. In this section, we consider linear programming problems with IT2 rough interval coefficients (LPIT2RIC) for right hand side parameters (RHS). Let us consider a robust linear programming problem with IT2 rough interval coefficients for RHS as:

$\begin{matrix} Max z = \sum_{i = 1}^{n} [[{\underline{c}}_{i}^{l}, {\underline{c}}_{i}^{u}], [{\bar{c}}_{i}^{l}, {\bar{c}}_{i}^{u}]] x_{i} \\ s . t . \\ \sum_{i = 1}^{n} [[{\underline{a}}_{ij}^{l}, {\underline{a}}_{ij}^{u}], [{\bar{a}}_{ij}^{l}, {\bar{a}}_{ij}^{u}]] x_{i} \leq [\tilde{\underline{R}} ({\tilde{A}}_{j}), \bar{\tilde{R}} ({\tilde{A}}_{j})], \\ j = 1, 2, \dots, m, \\ x_{i} \geq 0, i = 1, 2, \dots, n, \end{matrix}$ (1) where $[c_{i}^{l}, c_{i}^{u}]$ and $[a_{ij}^{l}, a_{ij}^{u}]$ (i = 1, 2, …, n, j = 1, 2, …, m) are rough interval coefficients and $[\tilde{\underline{R}} ({\tilde{A}}_{j}), \bar{\tilde{R}} ({\tilde{A}}_{j})]$ (j = 1, 2, …, m) are IT2 rough interval coefficients.

According to the arithmetic operations on the IT2 RIs introduced in Section 3, we select a pre-defuzzification level named [α₁, α₂] for all IT2 rough interval coefficients. Model (1) can be rewritten as:

$\begin{matrix} Max z = \sum_{i = 1}^{n} [[{\underline{c}}_{i}^{l}, {\underline{c}}_{i}^{u}], [{\bar{c}}_{i}^{l}, {\bar{c}}_{i}^{u}]] x_{i} \\ s . t . \\ \sum_{i = 1}^{n} [[{\underline{a}}_{ij}^{l}, {\underline{a}}_{ij}^{u}], [{\bar{a}}_{ij}^{l}, {\bar{a}}_{ij}^{u}]] x_{i} \\ \leq [[{({\tilde{A}}_{j})}_{α_{1}}^{l}, {({\tilde{A}}_{j})}_{α_{1}}^{u}], [{({\tilde{A}}_{j})}_{α_{2}}^{l}, {({\tilde{A}}_{j})}_{α_{2}}^{u}]], \\ j = 1, 2, \dots, m, \\ x_{i} \geq 0, i = 1, 2, \dots, n . \end{matrix}$ (2)

Definition 4.1. Consider all the corresponding LPIT2RIC problems and ILP problems of Problem (2). Let $[{\underline{z}}^{l}, {\underline{z}}^{u}]$ ( $[{\bar{z}}^{l}, {\bar{z}}^{u}]$ ) be surely (possibly) optimal range of Problem (2). Then the rough interval $[[{\underline{z}}^{l}, {\underline{z}}^{u}], [{\bar{z}}^{l}, {\bar{z}}^{u}]]$ is called the rough optimal range of Problem (2). In order to solve model (2), we transfer the proposed model to two boundary models corresponding to

$\begin{matrix} Max [{\underline{z}}^{l}, {\underline{z}}^{u}] = \sum_{i = 1}^{n} [{\underline{c}}_{i}^{l}, {\underline{c}}_{i}^{u}] x_{i} \\ s . t . \\ \sum_{i = 1}^{n} [{\underline{a}}_{ij}^{l}, {\underline{a}}_{ij}^{u}] x_{i} \leq [{({\tilde{A}}_{j})}_{α_{1}}^{l}, {({\tilde{A}}_{j})}_{α_{1}}^{u}], \\ j = 1, 2, \dots, m, \\ x_{i} \geq 0, i = 1, 2, \dots, n, \end{matrix}$ (3) and

$\begin{matrix} Max [{\bar{z}}^{l}, {\bar{z}}^{u}] = \sum_{i = 1}^{n} [{\bar{c}}_{i}^{l}, {\bar{c}}_{i}^{u}] x_{i} \\ s . t . \\ \sum_{i = 1}^{n} [{\bar{a}}_{ij}^{l}, {\bar{a}}_{ij}^{u}] x_{i} \leq [{({\tilde{A}}_{j})}_{α_{2}}^{l}, {({\tilde{A}}_{j})}_{α_{2}}^{u}], \\ j = 1, 2, \dots, m, \\ x_{i} \geq 0, i = 1, 2, \dots, n, \end{matrix}$ (4)

According to [16], we convert each boundary model to two LP models corresponding to upper and lower approximation intervals. $\begin{matrix} LP 1 : \\ Max {\underline{z}}^{l} = \sum_{i = 1}^{n} {\underline{c}}_{i}^{l} x_{i} \\ s . t . \\ \sum_{i = 1}^{n} {\underline{a}}_{ij}^{l} x_{i} \leq {({\tilde{A}}_{j})}_{α_{1}}^{l}, j = 1, 2, \dots, m, \\ x_{i} \geq 0, i = 1, 2, \dots, n . \end{matrix}$ (5) $\begin{matrix} LP 2 : \\ Max {\underline{z}}^{u} = \sum_{i = 1}^{n} {\underline{c}}_{i}^{u} x_{i} \\ s . t . \\ \sum_{i = 1}^{n} {\underline{a}}_{ij}^{u} x_{i} \leq {({\tilde{A}}_{j})}_{α_{1}}^{u}, j = 1, 2, \dots, m, \\ x_{i} \geq 0, i = 1, 2, \dots, n . \end{matrix}$ (6) $\begin{matrix} LP 3 : \\ Max {\bar{z}}^{l} = \sum_{i = 1}^{n} {\bar{c}}_{i}^{l} x_{i} \\ s . t . \\ \sum_{i = 1}^{n} {\bar{a}}_{ij}^{l} x_{i} \leq {({\tilde{A}}_{j})}_{α_{2}}^{l}, j = 1, 2, \dots, m, \\ x_{i} \geq 0, i = 1, 2, \dots, n . \end{matrix}$ (7) $\begin{matrix} LP 4 : \\ Max {\bar{z}}^{u} = \sum_{i = 1}^{n} {\bar{c}}_{i}^{u} x_{i} \\ s . t . \\ \sum_{i = 1}^{n} {\bar{a}}_{ij}^{u} x_{i} \leq {({\tilde{A}}_{j})}_{α_{2}}^{u}, j = 1, 2, \dots, m, \\ x_{i} \geq 0, i = 1, 2, \dots, n . \end{matrix}$ (8)

Dealing with rough interval inequality constraints and variables, similar to the ILP problems, are more difficult. In details, for n interval coefficients $[c_{i}^{l}, c_{i}^{u}]$ (i = 1, 2, …, n) in the objective function, the former k coefficients are assumed to be positive (i.e., $c_{i}^{l}, c_{i}^{u} \geq 0$ , for i = 1, 2, …, k), and the latter (n - k) coefficients are negative (i.e., $c_{i}^{l}, c_{i}^{u} \leq 0$ , for i = k + 1, … , n). Fan et al. [17] proposed a robust two-step solution method (RTSM) to solve the fuzzy interval programming.

Theorem 4.1. If and only if: $\sum_{i = 1}^{k} a_{ij}^{l} x_{i} + \sum_{i = k + 1}^{n} a_{ij}^{l} x_{i} \leq {({\tilde{A}}_{j})}_{α_{1}}^{u},$ solutions obtained from submodel (3) will not lead to violation of the best-case constraint ( $\sum_{j = 1}^{n} a_{ij}^{l} x_{j} \leq b_{i}^{u}$ ).

Proof. Consider the first constraint of submodel (3): $\begin{matrix} \sum_{i = 1}^{k} [{\underline{a}}_{ij}^{l}, {\underline{a}}_{ij}^{u}] x_{i} + \sum_{i = k + 1}^{n} [{\underline{a}}_{ij}^{l}, {\underline{a}}_{ij}^{u}] x_{i} \\ \leq [{({\tilde{A}}_{j})}_{α_{1}}^{l}, {({\tilde{A}}_{j})}_{α_{1}}^{u}], \end{matrix}$

According to [11], we have $\begin{matrix} \sum_{i = 1}^{k} {| a_{ij} |}^{l} Sign (a_{ij}) x_{i}^{u} + \sum_{i = k + 1}^{n} {| a_{ij} |}^{u} Sign (a_{ij}) x_{i}^{l} \\ \leq {({\tilde{A}}_{j})}_{α_{1}}^{u} \end{matrix}$ and $\begin{matrix} \sum_{i = 1}^{k} {| a_{ij} |}^{u} Sign (a_{ij}) x_{j}^{l} + \sum_{i = k + 1}^{n} {| a_{ij} |}^{l} Sign (a_{ij}) x_{i}^{u} \\ \leq {({\tilde{A}}_{j})}_{α_{1}}^{l} . \end{matrix}$

Two criteria exist for determining whether additional constraints are needed to avoid violation of the best-case constraint:

If a_ij × x_i ≥ 0, ∀ i, then we have a_ij ≥ 0 (i = 1, 2, …, k) and a_ij ≤ 0 (i = k + 1, …, n). Thus, ${| a_{ij} |}^{l} Sign (a_{ij}) x_{i}^{u} = a_{ij}^{l} x_{i}^{u}$ , and ${| a_{ij} |}^{u} Sign (a_{ij}) x_{i}^{l} = a_{ij}^{l} x_{i}^{l}$ hold. Therefore, we have $\sum_{i = 1}^{k} {| a_{ij} |}^{l} Sign (a_{ij}) x_{i}^{u} + \sum_{i = k + 1}^{n} {| a_{ij} |}^{u} Sign (a_{ij}) x_{i}^{l} = \sum_{i = 1}^{k} a_{ij}^{l} x_{i}^{u} + \sum_{i = k + 1}^{n} a_{ij}^{l} x_{i}^{u} \leq {({\tilde{A}}_{j})}_{α_{1}}^{u}$ . For any $x_{i} \in [x_{i}^{l}, x_{i}^{u}]$ (i = 1, 2, …, n), $a_{ij}^{l} x_{i} \leq a_{ij}^{l} x_{i}^{u}$ (j = 1, 2, …, k) and $a_{ij}^{l} x_{i} \leq a_{ij}^{l} x_{i}^{l}$ (i = k + 1, …, n) hold. Con- sequently, we have $\sum_{i = 1}^{k} a_{ij}^{l} x_{i} + \sum_{i = k + 1}^{n} a_{ij}^{l} x_{i} \leq \sum_{i = 1}^{k} a_{ij}^{l} x_{i}^{u} + \sum_{i = k + 1}^{n} a_{ij}^{l} x_{i}^{u} \leq {({\tilde{A}}_{j})}_{α_{1}}^{u}$ .

If a_ij × x_i ≤ 0, ∀ i, then we have a_ij ≤ 0 (i = 1, 2, …, k) and a_ij ≥ 0 (i = k + 1, …, n). Thus, ${| a_{ij} |}^{u} Sign (a_{ij}^{u}) x_{i}^{l} = a_{ij}^{l} x_{i}^{l}$ and ${| a_{ij} |}^{l} Sign (a_{ij}) x_{i}^{u} = a_{ij}^{l} x_{i}^{u}$ hold. Consequently, we have $\sum_{i = 1}^{k} {| a_{ij} |}^{u} Sign (a_{ij}) x_{i}^{l} + \sum_{i = k + 1}^{n} {| a_{ij} |}^{l} Sign (a_{ij}) x_{i}^{u} = \sum_{i = 1}^{k} a_{ij}^{l} x_{i}^{l} + \sum_{i = k + 1}^{n} a_{ij}^{l} x_{i}^{u} \leq {({\tilde{A}}_{j})}_{α_{1}}^{l} \leq {({\tilde{A}}_{j})}_{α_{1}}^{u}$ . For any $x_{i} \in [x_{i}^{l}, x_{i}^{u}]$ (i = 1, 2, …, n), $a_{ij}^{l} x_{i} \leq a_{ij}^{l} x_{i}^{l}$ (i = 1, 2, …, k) and $a_{ij}^{l} x_{i} \leq a_{ij}^{l} x_{i}^{u}$ (i = k + 1, …, n) hold. Therefore, we have $\sum_{i = 1}^{k} a_{ij}^{l} x_{i} + \sum_{i = k + 1}^{n} a_{ij}^{l} x_{i} \leq \sum_{i = 1}^{k} a_{ij}^{l} x_{i}^{l} + \sum_{i = k + 1}^{n} a_{ij}^{l} x_{i}^{u} \leq {({\tilde{A}}_{j})}_{α_{1}}^{l} \leq {({\tilde{A}}_{j})}_{α_{1}}^{u}$ .

We can obtain the same conclusion to submodel (4).

Remark. To solve model (3), the submodel corresponding to ${\underline{z}}^{l}$ would be solved first because of the following reasons:

The solutions of the submodel corresponding to ${\underline{z}}^{l}$ [i.e. $x_{iopt}^{l}$ (i = 1, 2, …, k) and $x_{iopt}^{u}$ (i = k + 1, …, n)] are identified from set Q^-, since Q^- ⊆ Q⁺, ∃ x_i ∈ Q⁺, such that $x_{i}^{u} \geq x_{iopt}^{l}$ (i = 1, 2, …, k) and $x_{i}^{l} \leq x_{iopt}^{u}$ (i = k + 1, …, n).

For model (3), the submodel corresponding to ${\underline{z}}^{l}$ would use relatively strict constraints (i.e., lower-bound values for the constraints’ right-hand sides), and the submodel corresponding to ${\underline{z}}^{u}$ would use relatively relaxed constraints (i.e., upper-bound values for the right-hand sides). From Theorem 4.1, additional constraints based on the solutions corresponding to ${\underline{z}}^{l}$ should be interposed for the submodel corresponding to ${\underline{z}}^{u}$ . This will help avoid violation of the best-case constraints as the decision variables fluctuate within the generated decision space.

Based on the above analysis, the first boundary model (3) can be presented as follows

$\begin{matrix} Max {\underline{z}}^{l} = \sum_{i = 1}^{k} {\underline{c}}_{i}^{l} x_{i}^{l} + \sum_{i = k + 1}^{n} {\underline{c}}_{i}^{l} x_{i}^{u} \\ s . t . \\ \sum_{i = 1}^{k} {| {\underline{a}}_{ij} |}^{u} Sign ({\underline{a}}_{ij}) x_{i}^{l} + \sum_{i = k + 1}^{n} {| {\underline{a}}_{ij} |}^{l} Sign ({\underline{a}}_{ij}) x_{i}^{u} \\ \leq {({\tilde{A}}_{j})}_{α_{2}}^{l}, j = 1, 2, \dots, m, \\ x_{i}^{l} \geq 0, i = 1, 2, \dots, k, \\ x_{i}^{l} \geq 0, i = k + 1, k + 2, \dots, n \end{matrix}$ (9) solutions of $x_{iopt}^{l}$ (i = 1, 2, …, k) and $x_{iopt}^{u}$ (i = k + 1, k + 2, …, n) can be obtained for submodel (9). The optimistic submodel which corresponds to ${\underline{z}}^{u}$ can be formulated as follows

$\begin{matrix} Max {\underline{z}}^{u} = \sum_{i = 1}^{k} {\underline{c}}_{i}^{u} x_{i}^{u} + \sum_{i = k + 1}^{n} {\underline{c}}_{i}^{u} x_{i}^{l} \\ s . t . \\ \sum_{i = 1}^{k} {| {\underline{a}}_{ij} |}^{l} Sign ({\underline{a}}_{ij}) x_{i}^{u} + \sum_{i = k + 1}^{n} {| {\underline{a}}_{ij} |}^{u} Sign ({\underline{a}}_{ij}) x_{i}^{l} \\ \leq {({\tilde{A}}_{j})}_{α_{2}}^{u}, j = 1, 2, \dots, m, \\ Max {x_{i} \geq 0, i = 1, 2, \dots, n | \sum_{i = 1}^{n} {\underline{a}}_{ij}^{l} x_{i}} \\ \leq {({\tilde{A}}_{j})}_{α_{2}}^{u} \\ x_{i}^{u} \geq x_{iopt}^{l}, i = 1, 2, \dots, k, \\ x_{i}^{l} \leq x_{iopt}^{u}, i = k + 1, k + 2, \dots, n, \\ x_{i}^{l} \geq 0, i = 1, 2, \dots, k, \\ x_{i}^{l} \geq 0, i = k + 1, k + 2, \dots, n, \end{matrix}$ (10)

For constraint $Max {x_{i} \geq 0, i = 1, 2, \dots, n | \sum_{i = 1}^{n} {\underline{a}}_{ij}^{l} x_{i}} \leq {({\tilde{A}}_{j})}_{α_{2}}^{u}$ , if ${\underline{a}}_{ij}^{l} > 0$ , then $x_{i} = x_{i}^{u}$ ; otherwise, $x_{i} = x_{i}^{l}$ . Therefore, the ultimate surely optimal range for model (2) is $[{\underline{z}}_{opt}^{l}, {\underline{z}}_{opt}^{u}]$ . Similarly, the ultimate possible optimal range for model (2) is $[{\bar{z}}_{opt}^{l}, {\bar{z}}_{opt}^{u}]$ .

The following numerical example is given to illustrate the results.

5 Illustrative case study of environmental management

An illustrative case of a robust linear programming with rough interval coefficients and IT2 rough interval coefficients will be analyzed to demonstrate the solution process of RTSM.

$\begin{matrix} Max f = [[25, 30], [26, 28]] x_{1} \\ - [[5.5, 6.0], [5.7, 5.8]] x_{2} \\ s . t . \\ [[7, 10], [8, 9]] x_{1} - [[11, 14], [12, 13]] x_{2} \\ \leq [\tilde{\underline{R}} (\tilde{4}), \bar{\tilde{R}} (\tilde{4})], \\ [[0.9, 1.2], [1.0, 1.1]] x_{1} + [[0.18, 0.21], \\ [0.19, 0.2]] x_{2} \leq [\tilde{\underline{R}} (\tilde{6.8}), \bar{\tilde{R}} (\tilde{6.8})] \\ x_{1} \geq 0, x_{2} \geq 0 . \end{matrix}$ (11)

In the above problem, it is assumed that the objective is to maximize economic return, while constraints denote respectively limitations in resource availability and pollutant-emission allowance.

According to the arithmetic operations on the IT2 RIs introduced in Section 3, we select a pre-defuzzification [α₁, α₂] = [0.5, 0.7] for all IT2 rough interval coefficients, then $[[{(\tilde{4})}_{α_{1}}^{l}, {(\tilde{4})}_{α_{1}}^{u}], [{(\tilde{4})}_{α_{2}}^{l}, {(\tilde{4})}_{α_{2}}^{u}]] = [[3.8, 4.2], [3.9, 4.0]]$ and $[[{(\tilde{6.8})}_{α_{1}}^{l}, {(\tilde{6.8})}_{α_{1}}^{u}], [{(\tilde{6.8})}_{α_{2}}^{l}, {(\tilde{6.8})}_{α_{2}}^{u}]] = [[6.5, 7], [6.6, 6.9]]$ .

Model (11) can be rewritten as:

$\begin{matrix} Max f = [[25, 30], [26, 28]] x_{1} \\ - [[5.5, 6.0], [5.7, 5.8]] x_{2} \\ s . t . \\ [[7, 10], [8, 9]] x_{1} - [[11, 14], [12, 13]] x_{2} \\ \leq [[3.8, 4.2], [3.9, 4.0]], \\ [[0.9, 1.2], [1.0, 1.1]] x_{1} + [[0.18, 0.21], \\ [0.19, 0.2]] x_{2} \leq [[6.5, 7], [6.6, 6.9]] \\ x_{1} \geq 0, x_{2} \geq 0 \end{matrix}$ (12)

Based on RTSM, the rough optimal ranges of model (11) are: $\begin{matrix} x_{1 opt} & = & [[4.81, 6.97], [5.36, 6.15]], \\ x_{2 opt} & = & [[3.19, 4.03], [3.47, 3.70]], \\ f_{opt} & = & [[96.13, 191.63], [117.96, 152.28]] \end{matrix}$

With the above RTSM, an additional constraint [i.e. constraints $0.9 {\underline{x}}_{1}^{u} + 0.18 {\underline{x}}_{2 opt}^{u} \leq 7,$ and $1 . 0 {\bar{x}}_{1}^{u} + 0.1 9 {\bar{x}}_{2 opt}^{u} \leq 6 . 9,$ presented as dotted lines in Fig. 2(a) and (b), respectively] is proposed to prevent the violation of the best-case constraint.

Fig.2

(a) Solution area of the first boundary model. (b) Solution area of the second boundary model.

6 Conclusions

The concept of IT2 rough interval was presented as an extended form for expressing uncertainties in LP model. First, by introducing the fuzzy tolerance measures [α₁, α₂] level, the IT2 rough interval can be transformed into a rough interval. Then, a robust two-step method (RTSM) was applied to solve the linear programming problems with IT2 rough interval coefficients. Through the integrating of the IT2 rough interval and the RTSM, the model can simultaneously handle parameter uncertainties and can generate reasonable ranges for decision variables and objective function. The developed method was applied to a numerical example to demonstrate that solutions obtained cannot violate the model’s best-case constraint when decision variables fluctuate within their solution intervals. Moreover, the results also suggest that this inexact optimization technique is effective and can be applied to many practical environmental and engineering problems that involve uncertainties.

Footnotes

Acknowledgments

This research was supported Fujian Young and Middle-aged Teacher Education Research Foundation (JAT170424), Xiamen University of Technology Foreign Science and Technology Cooperation and Communication Foundation (E201400200), Xiamen University of Technology High-level personnel Foundation (YKJ14038), Fujian Class A Foundation (JA14242). The writers are very grateful to the editor and the anonymous reviewers for their insightful comments and suggestions.

References

Bazaraa

M.S.

, Jarvis

J.J.

and Sherali

H.D.

, Linear programming and network flows, John Wiley and Sons, New York, 2010.

Dantaig

G.B.

and Thapa

M.N.

, Linear programming: Theory and extensions, Springer Verlag, Berlin, 2003.

Jin

, Huang

G.H.

, Fan

Y.R.

, Wang

and Wu

, A pseudo-optimal inexact stochastic interval T2 fuzzy sets approach for energy and environmental systems planning under uncertainty: A case study for Xiamen City of China, Applied Energy 138 (2015), 71–90.

H.W.

, Huang

G.H.

and He

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

Nie

X.H.

, Huang

G.H.

, Li

Y.P.

and Liu

, IFRP: A hybrid interval-parameter fuzzy robust programming approach for waste management planning under uncertainty, Journal of Environmental Management 84 (2007), 1–11.

Chang

N.B.

and Wang

S.F.

, A fuzzy goal programming approach for the optimal planning of metropolitan solid waste management systems, Journal of Operation Research 32(4) (1997), 303–321.

Pawlak

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

Rebolledo

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

Zadeh

L.A.

, Electronics Research Laboratory, University of California, Calculus of fuzzy restrictions, 1975.

10.

Mendez

J.M.

, Modelling and control of coiling entry temperature using interval type-2 fuzzy logic systems, Ironmaking Steelmaking 37(2) (2010), 126–134.

11.

Zhou

, Guo

H.C.

, Huang

and Huang

G.H.

, The interval linear programming: A revisit, Journal of Environmental Informatics 11(1) (2008), 1–10.

12.

Mendel

J.M.

, John

R.I.

and Liu

, Interval type-2 fuzzy logic systems made simple, IEEE Transactions on Fuzzy Systems 14(6) (2006), 808–821.

13.

John

R.I.

and Mendel

J.M.

, Type-2 fuzzy sets made simple, IEEE Transaction on Fuzzy Systems 10(2) (2002), 117–127.

14.

Zhang

, On characterization of generalized interval type-2 fuzzy rough sets, Information Sciences 219 (2013), 124–150.

15.

Moore

R.E.

, Introduction to interval analysis, Siam, 2009.

16.

Chinneck

J.W.

and Ramadan

, Linear programming with interval coefficients, Journal of the Operational Research Society 51(2) (2000), 209–220.

17.

Fan

Y.R.

, Huang

G.H.

and Li

Y.P.

, Robust interval linear programming for environmental decision making under uncertainty, Engineering Optimization 44(11) (2012), 1321–1336.