Result: Three different optimization techniques for solving the fully rough interval multi-level linear programming problem

Title:

Three different optimization techniques for solving the fully rough interval multi-level linear programming problem

Authors:

E. Fathy, E. Ammar, M.A. Helmy

Source:

Journal of Intelligent & Fuzzy Systems. 45:2593-2610

Publisher Information:

SAGE Publications, 2023.

Publication Year:

2023

Subject Terms:

0211 other engineering and technologies, 0202 electrical engineering, electronic engineering, information engineering, 02 engineering and technology

Document Type:

Academic journal Article

ISSN:

1875-8967
1064-1246

DOI:

10.3233/jifs-230057

Accession Number:

edsair.doi...........9d38b3076a6550e5073be6093e4c788f

Database:

OpenAIRE

Further Information

Due to the importance of the multi-level fully rough interval linear programming (MLFRILP) problem to address a wide range of management and optimization challenges in practical applications, such as policymaking, supply chain management, energy management, and so on, few researchers have specifically discussed this point. This paper presents an easy and systematic roadmap of studies of the currently available literature on rough multi-level programming problems and improvements related to group procedures in seven basic categories for future researchers and also introduces the concept of multi-level fully rough interval optimization. We start remodeling the problem into its sixteen crisp linear programming LP problems using the interval method and slice sum method. All crisp LPs can be reduced to four crisp LPs. In addition, three different optimization techniques were used to solve the complex multi-level linear programming issues. A numerical example is also provided to further clarify each strategy. Finally, we have a comparison of the methods used for solving the MLFRILP problem.

AN0170719011;flu01aug.23;2023Aug29.06:04;v2.2.500

Three different optimization techniques for solving the fully rough interval multi-level linear programming problem

Keywords: Constraint method; interval arithmetic; interactive approach; fuzzy approach; rough interval programming; slice sum method

1 Introduction

The effectiveness of rough set (RS) theory as a useful mathematical technique for dealing with incomplete data analysis and knowledge gaps has been proven. Jerbi et al. [[20]] proposed the innovative malware detection technique known as "variable precision rough set malware detection," which offers powerful detection rules and can reveal the new nature of specific software. It is predicated on such hybridization. The adaptive multi-granulation decision-theoretic rough sets model is what Zhang et al. [[34]] suggest as a generalized MG-DTRS model. This model has the capacity to adjust a compensation coefficient in order to adaptively acquire a pair of probabilistic thresholds. The "shifting strategy" was used in Osman et al.'s [[24]] study to divide the difficult task into four manageable challenges that will be worked on simultaneously. For each problem, a membership function was constructed to improve a model of fuzzy goal programming for obtaining a satisfactory solution to the multilevel, multi-objective fractional programming problem. An extension of the interval method was presented in Fathy [[15]]. A modified version of the fuzzy approach that was created in the fully rough environment to solve the linear model was used to deal with the fully rough problem's roughness. In many fields' real-life application problems, the coefficients of a model of a linear programming problem may not be exactly defined because of market globalization in the current time and some other uncontrollable factors; therefore, Ammar et al. [[4]] and Abohany et al. [[2]] used the slice-sum method to solve this problem. Fathy et al. [[16]] provided an application that was used to determine the optimality for the cost of the solid MLLP transportation problem in a rough interval environment.

The multi-level programming problem is a series of optimization issues where the outcome of one depends on the decisions made by higher decision-makers (DMs). The decision of the upper-level restrictions affects the decision of the lower-level restrictions in this crucial phase. It was applied in many fields of real-life application problems, such as engineering, environment, medicine, banking, economic systems, management sciences, and transportation problems, as in Zhang et al. [[32]]. Emam et al. [[12]] demonstrated a mathematical approach that used the fuzzy decision approach and bound and decomposition strategy to find a fuzzy optimal solution for its problem, and Emam et al. [[13]] deduced an interactive approach to solving the same problem based on the multi-objective linear (MOL) programming technique, By comparing the result found in [[12]] and the algorithm proposed in [[13]], we found that the result of the algorithm in [[13]] is better than the result that was found in [[12]]. In Fathy [[14]], the decomposition technique was used to break down the fuzzy problem into three crisp problems, namely, middle-multi-objective integer quadratic problems (MOIQP), upper-MOIQP, and lower-MOIQP.

In order to address a wide range of management and optimization challenges in practical applications, researchers have underlined the significance of creating a number of different multilevel decision-making (MLDM) techniques. These applications generally fall into the following areas: policymaking [[5], [28]]; supply chain management [[26], [31]]; energy management [[21], [29]]; safety and accident management [[3], [6]]; traffic and transportation network design [[7], [30]]; network interdiction [[22]]; incapacitated lot-sizing problems [[17], [27]]; and so on. It is suggested in Cui et al.'s [[8]] that a parameter optimization method based on multi-level pattern matching be used to adapt the best operation parameters in the current superior operational pattern library in order to enhance the transfer bar product quality. This method is applied to the furnace's rough rolling link.

In many sectors, including engineering, finance, economics, and other disciplines, MLFRILP has grown to be a commonly utilized technique. The following factors make the MLFRILP problem difficult to answer using traditional techniques:

• a hierarchical decision structure with independent and frequently conflicting objectives.

• Rough intervals for the coefficients in the goal functions and limitations as well as the decision variables.

This work proposes approaches to tackle the MLFRILP problem based on the interval method and slice sum method in order to get around these issues. The following are the primary contributions of the methodologies we've presented:

1. The MLFRILP problem, where all decision variables and parameters are described by rough intervals, is addressed, and various proposed solutions are shown.

2. Using the interval approach, the MLFRILP issue is split into four crisp linear problems at each level, with the crisp problems having an additional bounded variable constraint imposed. The optimization variables for the crisp problems are then taken from all lower problems.

3. A conceptual analysis of three optimization algorithms—the constraint method, interactive approach, and fuzzy approach—is provided for solving crisp linear problems based on rough interval data.

4. Using a numerical example, the efficacy of three approaches is shown.

5. A comparison of the suggested methods is conducted.

6. The improved approaches offer the best answer for a variety of MLFRILP models with ambiguous interval parameters.

The scope of this research is to address the MLFRILP problem. Section 2, "Survey of Research on Problems Associated with Rough Multi-Level Programming Problems in the Period 2017–2022," Section 3 states some important definitions and arithmetic operations that will be used throughout this paper. Section 4 formulates the model of the considered problem. The transformation of the fully rough interval is obtained in Section 5. Subsection 5.1 describes the rough interval coefficient transformation. The transformation of rough interval variables is described in Section 5.2. In Section 6, the methods to solve the MLFRILP problem are explained. Subsection 6.1 discusses the constraint method for the MLFRILP problem. Subsection 6.2 presents an interactive model for the MLFRILP problem. Subsection 6.3 presents a fuzzy approach to the MLFRILP problem. A numerical example is also provided to further clarify each strategy in Section 7. Section 8 gave a comparison of the methods used for solving MLFRILP. Section 9: Conclusion.

2 A survey of research on problems associated with rough multi-level programming problems in...

Rough multilevel decision-making (RMLDM) procedures point to ways to deal with decentralised administration problems that highlight decision-making entities dispersed on different levels of the chain. Noteworthy endeavours have been given to understanding the basic concepts and developing differing solution algorithms related to RMLDM by researchers in both the mathematics/computer science and business ranges. The importance of creating a variety of fully rough interval multilevel decision-making (FRMLDM) processes has been stressed by researchers in order to address a range of management and optimization issues in real-world applications. and have effectively picked up experience in this range. As a result, a high-quality audit of current patterns is required, not only for theoretical research but also for real-world changes in RMLDM in business.

This section efficiently surveys up-to-date RMLDM procedures and clusters related procedure improvements into seven fundamental categories: rough bi-level decision problems, Rough three-level programming problems, rough multilevel decision problems, fully rough multilevel programming problems, fuzzy rough bi-level decision-making problems, fuzzy rough three-level decision-making problems, and the applications of these methods in several domains are given in Table 1.

Table 1 Summary of surveys up-to-date rough multilevel decision-making procedures and clusters related procedure improvements into seven fundamental categories

<table><colgroup><col align="left" /><col align="center" /><col align="center" /></colgroup><thead valign="top"><tr><td>Main categories of RMLDM techniques</td><td>Reference</td><td>No. of listed references</td></tr></thead><tbody><tr><td>Rough bi-level decision problems</td><td>[<xref ref-type="bibr" rid="bibr20">20</xref>]</td><td>1</td></tr><tr><td>Rough three-level programming problems</td><td>[<xref ref-type="bibr" rid="bibr1">1</xref>, <xref ref-type="bibr" rid="bibr11">11</xref>, <xref ref-type="bibr" rid="bibr13">13</xref>, <xref ref-type="bibr" rid="bibr23">23</xref>, <xref ref-type="bibr" rid="bibr35">35</xref>]</td><td>5</td></tr><tr><td>Rough multilevel decision problems</td><td>[<xref ref-type="bibr" rid="bibr8">8</xref>, <xref ref-type="bibr" rid="bibr16">16</xref>, <xref ref-type="bibr" rid="bibr34">34</xref>]</td><td>3</td></tr><tr><td>Fully rough multilevel programming problems</td><td>[<xref ref-type="bibr" rid="bibr2">2</xref>, <xref ref-type="bibr" rid="bibr14">14</xref>, <xref ref-type="bibr" rid="bibr24">24</xref>]</td><td>3</td></tr><tr><td>Fuzzy rough bi-level decision-making problems</td><td>[<xref ref-type="bibr" rid="bibr9">9</xref>]</td><td>1</td></tr><tr><td>Fuzzy rough three-level decision-making problems</td><td>[<xref ref-type="bibr" rid="bibr33">33</xref>]</td><td>1</td></tr><tr><td>Applications of RMLDM techniques</td><td>[<xref ref-type="bibr" rid="bibr2">2</xref>, <xref ref-type="bibr" rid="bibr8">8</xref>, <xref ref-type="bibr" rid="bibr16">16</xref>, <xref ref-type="bibr" rid="bibr20">20</xref>, <xref ref-type="bibr" rid="bibr33">33-34</xref>, <xref ref-type="bibr" rid="bibr35">35</xref>]</td><td>7</td></tr></tbody></table>

By giving state-of-the-art information, this study will specifically bolster researchers and viable experts in their understanding of how advancements in theoretical research come about and their applications in connection with multilevel decision-making methods. Relevant common concepts were briefly described, while relevant references were included in the pre-examinations.

3 Basic preliminaries

In this section, the authors provide some important definitions of a rough interval (RI) and arithmetic operations on all RIs that will be used throughout this paper. [[19], [25]] contain the definitions that follow.

<bold>Definition 1.</bold> RI can be thought of as a qualitative value from an ill-defined concept based on a variable x ∈ R.

<bold>Definition 2.</bold> The qualitative value A is called a RI when one can assign two closed intervals <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>A</mi><mo>-</mo></munder></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mi>A</mi><mo>¯</mo></mover></math> on R to it where <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>A</mi><mo>-</mo></munder><mo>⊆</mo><mover accent="true"><mi>A</mi><mo>¯</mo></mover><mo>.</mo></math>

<bold>Definition 3.</bold> <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>A</mi><mo>-</mo></munder></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mi>A</mi><mo>¯</mo></mover></math> are called the lower approximation interval and the upper approximation interval of A, respectively. Further, A is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>A</mi><mi>r</mi><mo>*</mo></msubsup><mo>=</mo><mo>(</mo><munder><mi>A</mi><mo>-</mo></munder></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mi>A</mi><mo>¯</mo></mover><mo>)</mo><mo>.</mo></math>

<bold>Definition 4.</bold> <sups>.</sups>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><msup><munder><mi>a</mi><mo>-</mo></munder><mi>L</mi></msup><mo>,</mo><msup><munder><mi>a</mi><mo>-</mo></munder><mi>U</mi></msup></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi>L</mi></msup><mo>,</mo><msup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi>U</mi></msup></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><msup><munder><mi>b</mi><mo>-</mo></munder><mi>L</mi></msup><mo>,</mo><msup><munder><mi>b</mi><mo>-</mo></munder><mi>U</mi></msup></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>L</mi></msup><mo>,</mo><msup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>U</mi></msup></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow></math> be two RIs, then the basic arithmetic operations will be defined as follows:

(i) Addition: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>⊕</mo><mi>B</mi><mo>=</mo><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><msup><munder><mi>a</mi><mo>-</mo></munder><mi>L</mi></msup><mo>+</mo><msup><munder><mi>b</mi><mo>-</mo></munder><mi>L</mi></msup><mo>,</mo><msup><munder><mi>a</mi><mo>-</mo></munder><mi>U</mi></msup><mo>+</mo><msup><munder><mi>b</mi><mo>-</mo></munder><mi>U</mi></msup></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi>L</mi></msup><mo>+</mo><msup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>L</mi></msup><mo>,</mo><msup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi>U</mi></msup><mo>+</mo><msup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>U</mi></msup></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow><mo>.</mo></math>

(ii) Subtraction: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>-</mo><mi>B</mi><mo>=</mo><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><msup><munder><mi>a</mi><mo>-</mo></munder><mi>L</mi></msup><mo>-</mo><msup><munder><mi>b</mi><mo>-</mo></munder><mi>U</mi></msup><mo>,</mo><msup><munder><mi>a</mi><mo>-</mo></munder><mi>U</mi></msup><mo>-</mo><msup><munder><mi>b</mi><mo>-</mo></munder><mi>L</mi></msup></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi>L</mi></msup><mo>-</mo><msup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>L</mi></msup><mo>,</mo><msup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi>U</mi></msup><mo>-</mo><msup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>U</mi></msup></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow><mo>.</mo></math>

(iii) Negation: <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>A</mi><mo>=</mo><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><msup><munder><mi>a</mi><mo>-</mo></munder><mi>U</mi></msup><mo>,</mo><mo>-</mo><msup><munder><mi>a</mi><mo>-</mo></munder><mi>L</mi></msup></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><mo>-</mo><msup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi>U</mi></msup><mo>,</mo><mo>-</mo><msup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi>L</mi></msup></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow></math> .

(iv) Scalar Multiplication: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="italic">KA</mi><mo>=</mo><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><mi>K</mi><msup><munder><mi>a</mi><mo>-</mo></munder><mi>L</mi></msup><mo>,</mo><mi>K</mi><msup><munder><mi>a</mi><mo>-</mo></munder><mi>U</mi></msup></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><mi>K</mi><msup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi>L</mi></msup><mo>,</mo><mi>K</mi><msup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi>U</mi></msup></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow></math> , if k is a positive real number.

(v) Multiplication: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>⊗</mo><mi>B</mi><mo>=</mo><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><msup><munder><mi>a</mi><mo>-</mo></munder><mi>L</mi></msup><msup><munder><mi>b</mi><mo>-</mo></munder><mi>L</mi></msup><mo>,</mo><msup><munder><mi>a</mi><mo>-</mo></munder><mi>U</mi></msup><msup><munder><mi>b</mi><mo>-</mo></munder><mi>U</mi></msup></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi>L</mi></msup><msup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>L</mi></msup><mo>,</mo><msup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi>U</mi></msup><msup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>U</mi></msup></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow><mo>,</mo><mi mathvariant="normal">if</mi><mi>A</mi><mo>,</mo><mi>B</mi><mo>⩾</mo><mn>0</mn><mo>.</mo></math>

<bold>Definition 5.</bold> A RI <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><msup><munder><mi>a</mi><mo>-</mo></munder><mi>L</mi></msup><mo>,</mo><msup><munder><mi>a</mi><mo>-</mo></munder><mi>U</mi></msup></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi>L</mi></msup><mo>,</mo><msup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi>U</mi></msup></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow></math> is said to be a rough positive integer, if <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><munder><mi>a</mi><mo>-</mo></munder><mi>L</mi></msup><mo>,</mo><msup><munder><mi>a</mi><mo>-</mo></munder><mi>U</mi></msup><mo>,</mo><msup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi>L</mi></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi>U</mi></msup></math> are positive integers.

<bold>Definition 6.</bold> Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><msup><munder><mi>a</mi><mo>-</mo></munder><mi>L</mi></msup><mo>,</mo><msup><munder><mi>a</mi><mo>-</mo></munder><mi>U</mi></msup></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi>L</mi></msup><mo>,</mo><msup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi>U</mi></msup></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><msup><munder><mi>b</mi><mo>-</mo></munder><mi>L</mi></msup><mo>,</mo><msup><munder><mi>b</mi><mo>-</mo></munder><mi>U</mi></msup></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>L</mi></msup><mo>,</mo><msup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>U</mi></msup></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow></math> . be two RIs, then

(i) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>⩾</mo><mi>B</mi><mo>,</mo><mi mathvariant="normal">if</mi><msup><munder><mi>a</mi><mo>-</mo></munder><mi>L</mi></msup><mo>⩾</mo><msup><munder><mi>b</mi><mo>-</mo></munder><mi>L</mi></msup><mo>,</mo><msup><munder><mi>a</mi><mo>-</mo></munder><mi>U</mi></msup><mo>⩾</mo><msup><munder><mi>b</mi><mo>-</mo></munder><mi>U</mi></msup><mo>,</mo><msup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi>L</mi></msup><mo>⩾</mo><msup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>L</mi></msup><mo>,</mo><msup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi>U</mi></msup><mo>⩾</mo><msup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>U</mi></msup><mo>.</mo></math>

(ii) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>⩽</mo><mi>B</mi><mo>,</mo><mi mathvariant="normal">if</mi><msup><munder><mi>a</mi><mo>-</mo></munder><mi>L</mi></msup><mo>⩽</mo><msup><munder><mi>b</mi><mo>-</mo></munder><mi>L</mi></msup><mo>,</mo><msup><munder><mi>a</mi><mo>-</mo></munder><mi>U</mi></msup><mo>⩽</mo><msup><munder><mi>b</mi><mo>-</mo></munder><mi>U</mi></msup><mo>,</mo><msup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi>L</mi></msup><mo>⩽</mo><msup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>L</mi></msup><mo>,</mo><msup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi>U</mi></msup><mo>⩾</mo><msup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>U</mi></msup><mo>.</mo></math>

(iii) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>B</mi><mo>,</mo><mi mathvariant="normal">if</mi><msup><munder><mi>a</mi><mo>-</mo></munder><mi>L</mi></msup><mo>=</mo><msup><munder><mi>b</mi><mo>-</mo></munder><mi>L</mi></msup><mo>,</mo><msup><munder><mi>a</mi><mo>-</mo></munder><mi>U</mi></msup><mo>=</mo><msup><munder><mi>b</mi><mo>-</mo></munder><mi>U</mi></msup><mo>,</mo><msup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi>L</mi></msup><mo>=</mo><msup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>L</mi></msup><mo>,</mo><msup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi>U</mi></msup><mo>=</mo><msup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>U</mi></msup><mo>.</mo></math>

4 Problem formulation and solution concept

Uncertainty-based optimization is useful in solving many real-world problems. In many practical applications, uncertainty optimization is crucial. Our goal in this research is to demonstrate how to resolve an MLFRILP problem.

The following formulation of the MLFRILP problem is possible:

(1.a) <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>[</mo><mrow><msup><mn>1</mn><mi mathvariant="normal">st</mi></msup><mo>-</mo><mi mathvariant="normal">LevelDM</mi></mrow><mo>]</mo></mrow><mo>:</mo><munder><mo>max</mo><mrow><msubsup><mi>x</mi><mn>1</mn><mi>R</mi></msubsup></mrow></munder><msubsup><mi>F</mi><mn>1</mn><mi>R</mi></msubsup><mrow><mo>(</mo><mrow><msup><mi>x</mi><mi>R</mi></msup></mrow><mo>)</mo></mrow><mo>=</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>c</mi><mrow><mn>1</mn><mi>j</mi></mrow><mi>R</mi></msubsup><msubsup><mi>x</mi><mi>j</mi><mi>R</mi></msubsup><mo>,</mo></math>

where <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mn>2</mn><mi>R</mi></msubsup><mo>,</mo><mo>...</mo><mo>,</mo><msubsup><mi>x</mi><mi>n</mi><mi>R</mi></msubsup></math> solves

(1.b) <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><msup><mn>2</mn><mi mathvariant="normal">nd</mi></msup><mo>-</mo><mi mathvariant="normal">LevelDM</mi><mo>]</mo><mo>:</mo><munder><mo>max</mo><mrow><msubsup><mi>x</mi><mn>2</mn><mi>R</mi></msubsup></mrow></munder><msubsup><mi>F</mi><mn>2</mn><mi>R</mi></msubsup><mrow><mo>(</mo><mrow><msup><mi>x</mi><mi>R</mi></msup></mrow><mo>)</mo></mrow><mo>=</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>c</mi><mrow><mn>2</mn><mi>j</mi></mrow><mi>R</mi></msubsup><msubsup><mi>x</mi><mi>j</mi><mi>R</mi></msubsup><mo>,</mo></math>

where <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mn>3</mn><mi>R</mi></msubsup><mo>,</mo><mo>...</mo><mo>,</mo><msubsup><mi>x</mi><mi>n</mi><mi>R</mi></msubsup></math> solves

⋮

where <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mi>k</mi><mi>R</mi></msubsup><mo>,</mo><mo>...</mo><mo>,</mo><msubsup><mi>x</mi><mi>n</mi><mi>R</mi></msubsup></math> solves

(1.c) <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mo>[</mo><mi mathvariant="normal">k</mi></mrow><mi mathvariant="normal">th</mi></msup><mo>-</mo><mi mathvariant="normal">LevelDM</mi><mo>]</mo><mo>:</mo><munder><mo>max</mo><mrow><msubsup><mi>x</mi><mi>k</mi><mi>R</mi></msubsup></mrow></munder><msubsup><mi>F</mi><mi>k</mi><mi>R</mi></msubsup><mrow><mo>(</mo><mrow><msup><mi>x</mi><mi>R</mi></msup></mrow><mo>)</mo></mrow><mo>=</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>c</mi><mi mathvariant="italic">kj</mi><mi>R</mi></msubsup><msubsup><mi>x</mi><mi>j</mi><mi>R</mi></msubsup><mo>,</mo></math>

where <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mi>R</mi></msubsup><mo>,</mo><mo>...</mo><mo>,</mo><msubsup><mi>x</mi><mi>n</mi><mi>R</mi></msubsup></math> solves

subject to

(1.d) <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>G</mi><mi>R</mi></msup><mo>=</mo><mrow><mo>{</mo><mrow><mrow><mtable><mtr><mtd columnalign="center"><mrow><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>a</mi><mi mathvariant="italic">ij</mi><mi>R</mi></msubsup><msubsup><mi>x</mi><mi>j</mi><mi>R</mi></msubsup><mo>⩽</mo><msubsup><mi>b</mi><mi>i</mi><mi>R</mi></msubsup><mo>,</mo><mrow><mo>(</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>m</mi></mrow><mo>)</mo></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="center"><mrow><msubsup><mi>x</mi><mi>j</mi><mi>R</mi></msubsup><mo>⩾</mo><msup><mn>0</mn><mi>R</mi></msup><mo>,</mo><mrow><mo>(</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></mrow></mrow><mo /></mrow></math>

where <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mi>j</mi><mi>R</mi></msubsup><mo>=</mo><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mi>j</mi><mi>L</mi><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mi>j</mi><mi>U</mi></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>j</mi><mi>L</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>j</mi><mi>U</mi></msubsup></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow></math> , (j = 1, ..., n) are RI decision variables, <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>c</mi><mi mathvariant="italic">rj</mi><mi>R</mi></msubsup><mo>=</mo><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><msubsup><munder><mi>c</mi><mo>-</mo></munder><mi mathvariant="italic">rj</mi><mi>L</mi></msubsup><mo>,</mo><msubsup><munder><mi>c</mi><mo>-</mo></munder><mi mathvariant="italic">rj</mi><mi>U</mi></msubsup></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mover accent="true"><mi>c</mi><mo>¯</mo></mover><mi mathvariant="italic">rj</mi><mi>L</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>c</mi><mo>¯</mo></mover><mi mathvariant="italic">rj</mi><mi>U</mi></msubsup></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>a</mi><mi mathvariant="italic">ij</mi><mi>R</mi></msubsup><mo>=</mo><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><munder><mi>a</mi><mo>-</mo></munder><mi mathvariant="italic">ij</mi><mi>L</mi><mo>,</mo><munder><mi>a</mi><mo>-</mo></munder><mi mathvariant="italic">ij</mi><mi>U</mi></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi mathvariant="italic">ij</mi><mi>L</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi mathvariant="italic">ij</mi><mi>U</mi></msubsup></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>b</mi><mi>i</mi><mi>R</mi></msubsup><mo>=</mo><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><munder><mi>b</mi><mo>-</mo></munder><mi>i</mi><mi>L</mi><mo>,</mo><munder><mi>b</mi><mo>-</mo></munder><mi>i</mi><mi>U</mi></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>i</mi><mi>L</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>i</mi><mi>U</mi></msubsup></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow></math> , (r = 1, 2, ..., k ; j = 1, ..., n ; j = 1, ..., n) Are RI parameters.

Then the Problem (1.a) –(1.d), can be rewritten for r<sups>th</sups> -level (r = 1, 2, ..., k) in the following form:

(2.a) <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>[</mo><mrow><msup><mi mathvariant="normal">r</mi><mi mathvariant="normal">th</mi></msup><mi mathvariant="normal">Level</mi></mrow><mo>]</mo></mrow><mo>:</mo><munder><mo>max</mo><mrow><msubsup><mi>x</mi><mi>r</mi><mi>R</mi></msubsup></mrow></munder><msubsup><mi>F</mi><mi>r</mi><mi>R</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><msubsup><munder><mi>c</mi><mo>-</mo></munder><mi mathvariant="italic">rj</mi><mi>L</mi></msubsup><mo>,</mo><msubsup><munder><mi>c</mi><mo>-</mo></munder><mi mathvariant="italic">rj</mi><mi>U</mi></msubsup></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mover accent="true"><mi>c</mi><mo>¯</mo></mover><mi mathvariant="italic">rj</mi><mi>L</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>c</mi><mo>¯</mo></mover><mi mathvariant="italic">rj</mi><mi>U</mi></msubsup></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow><msubsup><mi>x</mi><mi>j</mi><mi>R</mi></msubsup><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></math>

where <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mi>R</mi></msubsup><mo>,</mo><mo>...</mo><mo>,</mo><msubsup><mi>x</mi><mi>n</mi><mi>R</mi></msubsup></math> solves

subject to

(2.b) <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="center"><msup><mi>G</mi><mi>R</mi></msup><mo>=</mo></mtd></mtr><mtr><mtd columnalign="center"><mrow><mo>{</mo><mrow><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><munder><mi>a</mi><mo>-</mo></munder><mi mathvariant="italic">ij</mi><mi>L</mi><mo>,</mo><munder><mi>a</mi><mo>-</mo></munder><mi mathvariant="italic">ij</mi><mi>U</mi></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi mathvariant="italic">ij</mi><mi>L</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi mathvariant="italic">ij</mi><mi>U</mi></msubsup></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo /></mrow></mtd></mtr><mtr><mtd columnalign="center"><mtext> </mtext><mtext> </mtext><mtext> </mtext><msubsup><mi>x</mi><mi>j</mi><mi>R</mi></msubsup><mo>⩽</mo><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><munder><mi>b</mi><mo>-</mo></munder><mi>i</mi><mi>L</mi><mo>,</mo><munder><mi>b</mi><mo>-</mo></munder><mi>i</mi><mi>U</mi></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>i</mi><mi>L</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>i</mi><mi>U</mi></msubsup></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><mtext> </mtext><mtext> </mtext><mtext> </mtext><msubsup><mi>x</mi><mi>j</mi><mi>R</mi></msubsup><mo>⩾</mo><msup><mn>0</mn><mi>R</mi></msup><mo>,</mo><mo>(</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mrow><mo /><mrow><mo>,</mo><mi>n</mi><mo>)</mo></mrow><mo>}</mo></mrow></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></math>

In the aforementioned Problems (2.a) and (2.b), the variables <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mi>j</mi><mi>R</mi></msubsup></math> are RI variables for the objective function and constraints, the coefficients of the objective function are RI coefficients <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><msubsup><munder><mi>c</mi><mo>-</mo></munder><mi mathvariant="italic">rj</mi><mi>L</mi></msubsup><mo>,</mo><msubsup><munder><mi>c</mi><mo>-</mo></munder><mi mathvariant="italic">rj</mi><mi>U</mi></msubsup></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mover accent="true"><mi>c</mi><mo>¯</mo></mover><mi mathvariant="italic">rj</mi><mi>L</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>c</mi><mo>¯</mo></mover><mi mathvariant="italic">rj</mi><mi>U</mi></msubsup></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow><mo>,</mo><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>k</mi><mo>.</mo></math> , the coefficients of the constraint are RI coefficients <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mrow><mrow><mo>[</mo><mrow><munder><mi>a</mi><mo>-</mo></munder><mi mathvariant="italic">ij</mi><mi>L</mi><mo>,</mo><munder><mi>a</mi><mo>-</mo></munder><mi mathvariant="italic">ij</mi><mi>U</mi></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi mathvariant="italic">ij</mi><mi>L</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi mathvariant="italic">ij</mi><mi>U</mi></msubsup></mrow><mo>]</mo></mrow></mrow><mo>)</mo></math> , and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mrow><mrow><mo>[</mo><mrow><munder><mi>b</mi><mo>-</mo></munder><mi>i</mi><mi>L</mi><mo>,</mo><munder><mi>b</mi><mo>-</mo></munder><mi>i</mi><mi>U</mi></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>i</mi><mi>L</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>i</mi><mi>U</mi></msubsup></mrow><mo>]</mo></mrow></mrow><mo>)</mo></math> are RIs of constants.

5 The transformation of MLFRILP problem into a crisp model

To split the MLFRILP problem into two LP problems with an interval coefficient, apply the interval method [[19]]. One of these issues is linear programming (LP), whose coefficients are all upper approximations of FRIs, namely P<sups>U</sups>, and whose coefficients are all lower approximations of FRIs, namely P<sups>L</sups> respectively. Then each (MLFRILP) problems transform into two LPs with RI decision variables, <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>P</mi><mn>1</mn><mi>U</mi></msubsup></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>P</mi><mn>2</mn><mi>U</mi></msubsup></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>P</mi><mn>1</mn><mi>L</mi></msubsup></math> , and <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>P</mi><mn>2</mn><mi>L</mi></msubsup></math> . Secondly, use the slice sum method [[25]] to convert each LPs with RI decision variables into four crisp LPs, <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>P</mi><mrow><mo>ℓ</mo><mi>h</mi></mrow><mi>U</mi></msubsup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>P</mi><mrow><mo>ℓ</mo><mi>h</mi></mrow><mi>U</mi></msubsup></math> , (ℓ =1, 2 ; h = 1, 2, 3, 4). All crisp LPs can be simplified to only four crisp LPs, P<subs>1</subs>, P<subs>2</subs>, P<subs>3</subs> and P<subs>4</subs>.

5.1 The transformation of RI coefficient

P <sups>L</sups> of the r<sups>th</sups>Level DM:

subject to

(3.a) <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="center"><munderover><mi>G</mi><mo>-</mo><mi>L</mi></munderover><mo>=</mo><mo>{</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><munder><mi>a</mi><mo>-</mo></munder><mi mathvariant="italic">ij</mi><mi>U</mi><msubsup><mi>x</mi><mi>j</mi><mi>R</mi></msubsup><mo>⩽</mo><munder><mi>b</mi><mo>-</mo></munder><mi>i</mi><mi>L</mi><mo>,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>m</mi><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><msubsup><mi>x</mi><mi>j</mi><mi>R</mi></msubsup><mo>⩾</mo><mn>0</mn><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi><mo>}</mo></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></mrow><mo>(</mo><mn>3</mn><mo>.</mo><mi mathvariant="normal">a</mi><mo>)</mo></math>

subject to

(3.b) <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="center"><munderover><mi>G</mi><mo>-</mo><mi>U</mi></munderover><mo>=</mo><mo>{</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><munder><mi>a</mi><mo>-</mo></munder><mi mathvariant="italic">ij</mi><mi>L</mi><msubsup><mi>x</mi><mi>j</mi><mi>R</mi></msubsup><mo>⩽</mo><munder><mi>b</mi><mo>-</mo></munder><mi>i</mi><mi>U</mi><mo>,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>m</mi><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><msubsup><mi>x</mi><mi>j</mi><mi>R</mi></msubsup><mo>⩾</mo><mn>0</mn><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi><mo>}</mo></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></mrow></math>

The following theorems are required and helpful after dividing the RI coefficient in the goal functions and constraints into upper and lower intervals to design a crisp equivalent model.

Theorem 1 [[19]].Assume that the solution to problems (3.a) and (3.b) is inside the optimal range. The issue range (2.a) is then the unquestionably optimal range (2.b).

While finding the possible optimal range of problems (2.a) and (2.b), which led to the next two LP problems, will give you the possibly optimal (PO) range of the r<sups>th</sups> - L DM utilizing interval method [[19]].

P <sups>U</sups> of the r<sups>th</sups> - LDM:

subject to

(4.a) <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="center"><msup><mover accent="true"><mi>G</mi><mo>¯</mo></mover><mi>L</mi></msup><mo>=</mo><mo>{</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi mathvariant="italic">ij</mi><mi>U</mi></msubsup><msubsup><mi>x</mi><mi>j</mi><mi>R</mi></msubsup><mo>⩽</mo><msubsup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>i</mi><mi>L</mi></msubsup><mo>,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>m</mi><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><msubsup><mi>x</mi><mi>j</mi><mi>R</mi></msubsup><mo>⩾</mo><mn>0</mn><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi><mo>}</mo></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></mrow></math>

subject to

(4.b) <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="center"><msup><mover accent="true"><mi>G</mi><mo>¯</mo></mover><mi>U</mi></msup><mo>=</mo><mo>{</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi mathvariant="italic">ij</mi><mi>L</mi></msubsup><msubsup><mi>x</mi><mi>j</mi><mi>R</mi></msubsup><mo>⩽</mo><msubsup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>i</mi><mi>U</mi></msubsup><mo>,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>m</mi><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><msubsup><mi>x</mi><mi>j</mi><mi>R</mi></msubsup><mo>⩾</mo><mn>0</mn><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi><mo>}</mo></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></mrow></math>

Theorem 2 [[19]].Assume that the solution to Problems 4.a and 4.b is within its optimal range. The range of the problem (2.a) is then identical to the unquestionably optimal range (2.b).

5.2 The transformation of RI variables

Use the slice sum method [[25]] that is a method for solving FRIs problems.

Let <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>F</mi><mi>r</mi><mi>R</mi></msubsup><mo>=</mo><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><msubsup><munder><mi>f</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi></msubsup><mo>,</mo><msubsup><munder><mi>f</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi></msubsup></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>k</mi></mrow><mo>)</mo></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mi>j</mi><mi>R</mi></msubsup><mo>=</mo><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mi>L</mi></msubsup><mo>,</mo><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mi>U</mi></msubsup></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mi>x</mi><mi>j</mi><mrow><mo>*</mo><mi>L</mi></mrow></msubsup><mo>,</mo><msubsup><mi>x</mi><mi>j</mi><mrow><mo>*</mo><mi>U</mi></mrow></msubsup></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>(</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi></mrow><mo>)</mo></mrow></math> .

(5.a) <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="center"><mrow><mo>[</mo><mrow><msup><mi mathvariant="normal">r</mi><mi mathvariant="normal">th</mi></msup><mi mathvariant="normal">Level</mi><mo>,</mo><mrow><mo>(</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mo>:</mo><mspace width="thinmathspace" /><munder><mo>max</mo><mrow><msubsup><mi>x</mi><mi>r</mi><mi>R</mi></msubsup></mrow></munder><msubsup><mi>F</mi><mi>r</mi><mi>R</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo></mtd></mtr><mtr><mtd columnalign="center"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><msubsup><munder><mi>c</mi><mo>-</mo></munder><mi mathvariant="italic">rj</mi><mi>L</mi></msubsup><mo>,</mo><msubsup><munder><mi>c</mi><mo>-</mo></munder><mi mathvariant="italic">rj</mi><mi>U</mi></msubsup></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mover accent="true"><mi>c</mi><mo>¯</mo></mover><mi mathvariant="italic">rj</mi><mi>L</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>c</mi><mo>¯</mo></mover><mi mathvariant="italic">rj</mi><mi>U</mi></msubsup></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mi>L</mi></msubsup><mo>,</mo><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mi>U</mi></msubsup></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mi>x</mi><mi>j</mi><mrow><mo>*</mo><mi>L</mi></mrow></msubsup><mo>,</mo><msubsup><mi>x</mi><mi>j</mi><mrow><mo>*</mo><mi>U</mi></mrow></msubsup></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext></math>

subject to

(5.b) <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="center"><msup><mi>G</mi><mi>R</mi></msup><mo>=</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><munder><mi>a</mi><mo>-</mo></munder><mi mathvariant="italic">ij</mi><mi>L</mi><mo>,</mo><munder><mi>a</mi><mo>-</mo></munder><mi mathvariant="italic">ij</mi><mi>U</mi></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi mathvariant="italic">ij</mi><mi>L</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi mathvariant="italic">ij</mi><mi>U</mi></msubsup></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="center"><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mi>L</mi></msubsup><mo>,</mo><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mi>U</mi></msubsup></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mi>x</mi><mi>j</mi><mrow><mo>*</mo><mi>L</mi></mrow></msubsup><mo>,</mo><msubsup><mi>x</mi><mi>j</mi><mrow><mo>*</mo><mi>U</mi></mrow></msubsup></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow></mtd></mtr><mtr><mtd columnalign="center"><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>⩽</mo><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><munder><mi>b</mi><mo>-</mo></munder><mi>i</mi><mi>L</mi><mo>,</mo><munder><mi>b</mi><mo>-</mo></munder><mi>i</mi><mi>U</mi></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>i</mi><mi>L</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>i</mi><mi>U</mi></msubsup></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></mrow></math>

<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mi>j</mi><mi>R</mi></msubsup><mo>⩾</mo><mn>0</mn><mo>,</mo><mrow><mo>(</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi></mrow><mo>)</mo></mrow><mo>,</mo></math> are RI variables of the constraints and the objective function. In (3.a) problem with RIs variables can be sliced into the following four LP problems using slice sum method [[25]].

subject to

(6) <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd columnalign="center"><munderover><mi>G</mi><mo>-</mo><mi>L</mi></munderover><mo>=</mo><mo>{</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><munder><mi>a</mi><mo>-</mo></munder><mi mathvariant="italic">ij</mi><mi>U</mi><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mi>L</mi></msubsup><mo>⩽</mo><munder><mi>b</mi><mo>-</mo></munder><mi>i</mi><mi>L</mi><mo>,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>m</mi><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mi>L</mi></msubsup><mo>⩾</mo><mn>0</mn><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi><mo>}</mo></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></math>

subject to

(7) <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd columnalign="center"><munderover><mi>G</mi><mo>-</mo><mi>L</mi></munderover><mo>=</mo><mo>{</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><munder><mi>a</mi><mo>-</mo></munder><mi mathvariant="italic">ij</mi><mi>U</mi><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mi>U</mi></msubsup><mo>⩽</mo><munder><mi>b</mi><mo>-</mo></munder><mi>i</mi><mi>L</mi><mo>,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>m</mi><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mi>U</mi></msubsup><mo>⩾</mo><mn>0</mn><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi><mo>}</mo></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></math>

subject to

(8) <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd columnalign="center"><munderover><mi>G</mi><mo>-</mo><mi>L</mi></munderover><mo>=</mo><mo>{</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><munder><mi>a</mi><mo>-</mo></munder><mi mathvariant="italic">ij</mi><mi>U</mi><msubsup><mi>x</mi><mi>j</mi><mrow><mo>*</mo><mi>L</mi></mrow></msubsup><mo>⩽</mo><munder><mi>b</mi><mo>-</mo></munder><mi>i</mi><mi>L</mi><mo>,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>m</mi><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><msubsup><mi>x</mi><mi>j</mi><mrow><mo>*</mo><mi>L</mi></mrow></msubsup><mo>⩾</mo><mn>0</mn><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi><mo>}</mo></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></math>

subject to

(9) <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd columnalign="center"><munderover><mi>G</mi><mo>-</mo><mi>L</mi></munderover><mo>=</mo><mo>{</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><munder><mi>a</mi><mo>-</mo></munder><mi mathvariant="italic">ij</mi><mi>U</mi><msubsup><mi>x</mi><mi>j</mi><mrow><mo>*</mo><mi>U</mi></mrow></msubsup><mo>⩽</mo><munder><mi>b</mi><mo>-</mo></munder><mi>i</mi><mi>L</mi><mo>,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>m</mi><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><msubsup><mi>x</mi><mi>j</mi><mrow><mo>*</mo><mi>U</mi></mrow></msubsup><mo>⩾</mo><mn>0</mn><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi><mo>}</mo></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></math>

Solving problems (6)–(9), leads to <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mi>L</mi></msubsup><mo>=</mo><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mi>U</mi></msubsup><mo>=</mo><msubsup><mi>x</mi><mi>j</mi><mrow><mo>*</mo><mi>L</mi></mrow></msubsup><mo>=</mo><msubsup><mi>x</mi><mi>j</mi><mrow><mo>*</mo><mi>U</mi></mrow></msubsup><mo>=</mo><munder><mi>x</mi><mo>-</mo></munder><mi>j</mi><mi>L</mi><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi><mo>.</mo></math> so, the four LP problems can be simplified to the following LP problem.

subject to

(10) <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd columnalign="center"><munderover><mi>G</mi><mo>-</mo><mi>L</mi></munderover><mo>=</mo><mo>{</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><munder><mi>a</mi><mo>-</mo></munder><mi mathvariant="italic">ij</mi><mi>U</mi><munder><mi>x</mi><mo>-</mo></munder><mi>j</mi><mi>L</mi><mo>⩽</mo><munder><mi>b</mi><mo>-</mo></munder><mi>i</mi><mi>L</mi><mo>,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>m</mi><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><munder><mi>x</mi><mo>-</mo></munder><mi>j</mi><mi>L</mi><mo>⩾</mo><mn>0</mn><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi><mo>}</mo></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></math>

The problems (3.a) through (4.b) can also be rewritten as the following four LP problems:

Lower approximation lower bound (LALB):

subject to

(11.a) <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="center"><munderover><mi>G</mi><mo>-</mo><mi>L</mi></munderover><mo>=</mo><mo>{</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><munder><mi>a</mi><mo>-</mo></munder><mi mathvariant="italic">ij</mi><mi>U</mi><munder><mi>x</mi><mo>-</mo></munder><mi>j</mi><mi>L</mi><mo>⩽</mo><munder><mi>b</mi><mo>-</mo></munder><mi>i</mi><mi>L</mi><mo>,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>m</mi><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><munder><mi>c</mi><mo>-</mo></munder><mi mathvariant="italic">rj</mi><mi>L</mi></msubsup><munder><mi>x</mi><mo>-</mo></munder><mi>j</mi><mi>L</mi><mo>⩽</mo><mi mathvariant="italic">Max</mi><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><munder><mi>c</mi><mo>-</mo></munder><mi mathvariant="italic">rj</mi><mi>U</mi></msubsup><munder><mi>x</mi><mo>-</mo></munder><mi>j</mi><mi>U</mi><mo>,</mo><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>k</mi></mtd></mtr><mtr><mtd columnalign="center"><munder><mi>x</mi><mo>-</mo></munder><mi>j</mi><mi>L</mi><mo>⩾</mo><mn>0</mn><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi><mo>}</mo></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></mrow><mo>(</mo><mn>11</mn><mo>.</mo><mi mathvariant="normal">a</mi><mo>)</mo></math>

Upper Approximation Lower Bound (UALB):

subject to

(11.b) <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="center"><munderover><mi>G</mi><mo>-</mo><mi>U</mi></munderover><mo>=</mo><mo>{</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><munder><mi>a</mi><mo>-</mo></munder><mi mathvariant="italic">ij</mi><mi>L</mi><munder><mi>x</mi><mo>-</mo></munder><mi>j</mi><mi>U</mi><mo>⩽</mo><munder><mi>b</mi><mo>-</mo></munder><mi>i</mi><mi>U</mi><mo>,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>m</mi><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><munder><mi>c</mi><mo>-</mo></munder><mi mathvariant="italic">rj</mi><mi>L</mi></msubsup><munder><mi>x</mi><mo>-</mo></munder><mi>j</mi><mi>L</mi><mo>⩽</mo><mi mathvariant="italic">Max</mi><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><munder><mi>c</mi><mo>-</mo></munder><mi mathvariant="italic">rj</mi><mi>U</mi></msubsup><munder><mi>x</mi><mo>-</mo></munder><mi>j</mi><mi>U</mi><mo>,</mo><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>k</mi></mtd></mtr><mtr><mtd columnalign="center"><munder><mi>x</mi><mo>-</mo></munder><mi>j</mi><mi>U</mi><mo>⩾</mo><mn>0</mn><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi><mo>}</mo></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></mrow></math>

Lower approximation upper bound (LAUB):

subject to

(12.a) <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="center"><msup><mover accent="true"><mi>G</mi><mo>¯</mo></mover><mi>L</mi></msup><mo>=</mo><mo>{</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi mathvariant="italic">ij</mi><mi>U</mi></msubsup><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>j</mi><mi>L</mi></msubsup><mo>⩽</mo><msubsup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>i</mi><mi>L</mi></msubsup><mo>,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>m</mi><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mover accent="true"><mi>c</mi><mo>¯</mo></mover><mi mathvariant="italic">rj</mi><mi>L</mi></msubsup><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>j</mi><mi>L</mi></msubsup><mo>⩽</mo><mi mathvariant="italic">Max</mi><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mover accent="true"><mi>c</mi><mo>¯</mo></mover><mi mathvariant="italic">rj</mi><mi>U</mi></msubsup><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>j</mi><mi>U</mi></msubsup><mo>,</mo><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>k</mi><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>j</mi><mi>L</mi></msubsup><mo>⩾</mo><mn>0</mn><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi><mo>}</mo></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></mrow></math>

Upper approximation upper bound (UAUB):

subject to

(12.b) <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="center"><msup><mover accent="true"><mi>G</mi><mo>¯</mo></mover><mi>U</mi></msup><mo>=</mo><mo>{</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mover accent="true"><mi>a</mi><mo>¯</mo></mover><mi mathvariant="italic">ij</mi><mi>L</mi></msubsup><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>j</mi><mi>U</mi></msubsup><mo>⩽</mo><msubsup><mover accent="true"><mi>b</mi><mo>¯</mo></mover><mi>i</mi><mi>U</mi></msubsup><mo>,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>m</mi><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mover accent="true"><mi>c</mi><mo>¯</mo></mover><mi mathvariant="italic">rj</mi><mi>L</mi></msubsup><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>j</mi><mi>L</mi></msubsup><mo>⩽</mo><mi mathvariant="italic">Max</mi><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mover accent="true"><mi>c</mi><mo>¯</mo></mover><mi mathvariant="italic">rj</mi><mi>U</mi></msubsup><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>j</mi><mi>U</mi></msubsup><mo>,</mo><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>k</mi><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>j</mi><mi>U</mi></msubsup><mo>⩾</mo><mn>0</mn><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi><mo>}</mo></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></mrow></math>

Theorem 3 [[25]]. If <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>j</mi><mrow><mo>*</mo><mi>U</mi></mrow></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>j</mi><mrow><mo>*</mo><mi>L</mi></mrow></msubsup><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mi>j</mi><mrow><mo>*</mo><mi>L</mi></mrow><mo>,</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>x</mi><mo>-</mo></munder><mi>j</mi><mrow><mo>*</mo><mi>U</mi></mrow><mo>,</mo><mrow><mo>(</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mi>n</mi></mrow><mo>)</mo></mrow></math> is an optimal solution for the UAUB, LAUB, LALB, and UALB problems, respectively, then the set of RIs <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mrow><mrow><mo>[</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mi>j</mi><mrow><mo>*</mo><mi>L</mi></mrow><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mi>j</mi><mrow><mo>*</mo><mi>U</mi></mrow></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>j</mi><mrow><mo>*</mo><mi>L</mi></mrow></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>j</mi><mrow><mo>*</mo><mi>U</mi></mrow></msubsup></mrow><mo>]</mo></mrow></mrow><mo>)</mo></math> is an optimal solution for the MLFRILP problem such that <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>j</mi><mrow><mo>*</mo><mi>L</mi></mrow></msubsup><mo>⩽</mo><munder><mi>x</mi><mo>-</mo></munder><mi>j</mi><mrow><mo>*</mo><mi>L</mi></mrow><mo>⩽</mo><munder><mi>x</mi><mo>-</mo></munder><mi>j</mi><mrow><mo>*</mo><mi>U</mi></mrow><mo>⩽</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>j</mi><mrow><mo>*</mo><mi>U</mi></mrow></msubsup><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mi>n</mi><mo>.</mo></math>

6 The optimization approaches on MLFRILP Problem

6.1 The constraint method for MLFRILP problem

In multi-level optimization, the constraint approach [[11]] is employed, where upper levels supply lower levels with sufficient solutions that are acceptable in rank order. In order to get the possibly and SO-range solutions for the rth- L DM problem, the constraint technique first solves the sharp LP problems. <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mi mathvariant="italic">rth</mi></msubsup><mo>=</mo><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>j</mi></mrow><mi mathvariant="italic">Lrth</mi><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>j</mi></mrow><mi mathvariant="italic">Urth</mi></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mrow><mo>*</mo><mi>j</mi></mrow><mi mathvariant="italic">Lrth</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mrow><mo>*</mo><mi>j</mi></mrow><mi mathvariant="italic">Urth</mi></msubsup></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>k</mi></mrow><mo>)</mo></mrow><mo>,</mo></math> (j = 1, 2, ..., n) Describes his or her problem from the perspective of the rth-L DM by setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>{</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Lrth</mi><mo>=</mo><munder><mi>x</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Urth</mi><mo>=</mo><munder><mi>x</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Lrth</mi></msubsup><mo>=</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Urth</mi></msubsup><mo>=</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup></mrow><mo>}</mo></math> , and doing this twice more till r = k. The <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mn>1</mn></mrow><mi mathvariant="italic">rth</mi></msubsup><mo>,</mo><msubsup><mi>x</mi><mrow><mo>*</mo><mn>2</mn></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup><mo>,</mo><msubsup><mi>x</mi><mrow><mo>*</mo><mn>3</mn></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup><mo>,</mo><mo>...</mo><mo>,</mo><msubsup><mi>x</mi><mrow><mo>*</mo><mi>n</mi></mrow><mi>k</mi></msubsup></mrow><mo>)</mo></math> namely a preferred RI solution to the rth-L DM. Finally, the MLFRILP problem's best solution is eventually discovered.

6.1.1 A flowchart for solving MLFRIL

The decision-making procedure of the constraint method is presented in Fig. 1.

Graph: Fig. 1 The decision-making procedure of the constraint method.

6.2 An interactive model for MLFRILP problem

The MLFRILP problem is resolved using an interactive model [[13], [10]]. The 1<sups>st</sups>Level DM initially provides the 2<sups>nd</sups>Level DM with the preferred solutions that are acceptable in rank order, and the 2<sups>nd</sups>Level DM then adopts the 1<sups>st</sups>Level DM's preferred solutions to acquire the solutions and eventually arrive at the 1<sups>st</sups>Level DM's satisfying solution. Finally, the 1<sups>st</sups>Level DM, the 2<sups>nd</sups>Level DM, etc., and the (k - 1) <sups>th</sups>L DM decide the preferred solution of the MLFRILP according to the satisfactoriness test functions as follows:

Let <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mi mathvariant="italic">rth</mi></msubsup><mo>=</mo><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>j</mi></mrow><mi mathvariant="italic">Lrth</mi><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>j</mi></mrow><mi mathvariant="italic">Urth</mi></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mrow><mo>*</mo><mi>j</mi></mrow><mi mathvariant="italic">Lrth</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mrow><mo>*</mo><mi>j</mi></mrow><mi mathvariant="italic">Urth</mi></msubsup></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow></math> (r = 1, 2, ..., k) , (j = 1, 2, ..., n) is the optimal rough solution of the r<sups>th</sups>Level DM. The (r + 1)th-Level DM defines his/her problem in the point of view of the rthLevel DM by setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>{</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Lrth</mi><mo>=</mo><munder><mi>x</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Urth</mi><mo>=</mo><munder><mi>x</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Lrth</mi></msubsup><mo>=</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Urth</mi></msubsup><mo>=</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup></mrow><mo>}</mo></math> .

Now the satisfactoriness test functions of the rth-Level DM:

(13) <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><munder><mi>f</mi><mo>-</mo></munder><mrow><mi>r</mi><mo>-</mo><mn>1</mn></mrow><mi>L</mi><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mn>1</mn><mi mathvariant="italic">Lr</mi></msubsup><mo>,</mo><mo>...</mo><mo>,</mo><msubsup><mi>x</mi><mi>n</mi><mrow><mi>L</mi><mrow><mo>(</mo><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></msubsup></mrow><mo>)</mo></mrow><mo>-</mo><munder><mi>f</mi><mo>-</mo></munder><mrow><mi>r</mi><mo>-</mo><mn>1</mn></mrow><mi>L</mi><msub><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mn>1</mn><mi mathvariant="italic">Lr</mi></msubsup><mo>,</mo><mo>...</mo><mo>,</mo><msubsup><mi>x</mi><mi>n</mi><mi mathvariant="italic">Lk</mi></msubsup></mrow><mo>)</mo></mrow><mn>2</mn></msub></mrow><mrow><munder><mi>f</mi><mo>-</mo></munder><mrow><mi>r</mi><mo>-</mo><mn>1</mn></mrow><mi>L</mi><msub><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mn>1</mn><mi mathvariant="italic">Lr</mi></msubsup><mo>,</mo><msubsup><mi>x</mi><mn>2</mn><mrow><mi>L</mi><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></msubsup><mo>,</mo><mo>...</mo><mo>,</mo><msubsup><mi>x</mi><mi>n</mi><mi mathvariant="italic">Lk</mi></msubsup></mrow><mo>)</mo></mrow><mn>2</mn></msub></mrow></mfrac><mo>⩽</mo><munderover><mi>δ</mi><mrow><mo>-</mo></mrow><mi mathvariant="italic">rL</mi></munderover><mo>.</mo></math>

So, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mn>1</mn></mrow><mi mathvariant="italic">rth</mi></msubsup><mo>,</mo><msubsup><mi>x</mi><mrow><mo>*</mo><mn>2</mn></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup><mo>,</mo><msubsup><mi>x</mi><mrow><mo>*</mo><mn>3</mn></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup><mo>,</mo><mo>...</mo><mo>,</mo><msubsup><mi>x</mi><mrow><mo>*</mo><mi>n</mi></mrow><mi>k</mi></msubsup></mrow><mo>)</mo></math> is a preferred RI solution to the r<sups>th</sups>Level DM, where the r<sups>th</sups>Level DM assigns a value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>δ</mi><mi>r</mi></msup><mo>=</mo><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><msubsup><mi>δ</mi><mo>-</mo><mi mathvariant="italic">Lr</mi></msubsup><mo>,</mo><msubsup><mi>δ</mi><mo>-</mo><mi mathvariant="italic">Ur</mi></msubsup></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msup><mover accent="true"><mi>δ</mi><mo>¯</mo></mover><mi mathvariant="italic">Lr</mi></msup><mo>,</mo><msup><mover accent="true"><mi>δ</mi><mo>¯</mo></mover><mi mathvariant="italic">Ur</mi></msup></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow></math> , which means the <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mn>1</mn></mrow><mi mathvariant="italic">rth</mi></msubsup><mo>,</mo><msubsup><mi>x</mi><mrow><mo>*</mo><mn>2</mn></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup><mo>,</mo><msubsup><mi>x</mi><mrow><mo>*</mo><mn>3</mn></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup><mo>,</mo><mo>...</mo><mo>,</mo><msubsup><mi>x</mi><mrow><mo>*</mo><mi>n</mi></mrow><mi>k</mi></msubsup></mrow><mo>)</mo></math> is a preferred RI solution the MLFRILP problem

6.2.1 A flowchart for solving MLFRILP

The decision-making procedure of the interactive method is presented in Fig. 2.

Graph: Fig. 2 The decision-making procedure of the interactive method.

6.3 Optimization method of the fuzzy approach

The MLFRILP problem is resolved in this part using a fuzzy technique. The MLFRILP issue is solved using this approach. The r<sups>th</sups>Level DM initially obtains the optimal RI solution <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>F</mi><mi>r</mi><mo>*</mo></msubsup><mo>=</mo><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><msubsup><mi>f</mi><mrow><mi>r</mi><mo>*</mo></mrow><mi>L</mi></msubsup><mo>,</mo><msubsup><mi>f</mi><mrow><mi>r</mi><mo>*</mo></mrow><mi>U</mi></msubsup></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mi>f</mi><mi>r</mi><mrow><mo>*</mo><mi>L</mi></mrow></msubsup><mo>,</mo><msubsup><mi>f</mi><mi>r</mi><mrow><mo>*</mo><mi>U</mi></mrow></msubsup></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow></math> with optimal decision variables <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mi mathvariant="italic">rth</mi></msubsup><mo>=</mo><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>j</mi></mrow><mi mathvariant="italic">Lrth</mi><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>j</mi></mrow><mi mathvariant="italic">Urth</mi></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mrow><mo>*</mo><mi>j</mi></mrow><mi mathvariant="italic">Lrth</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mrow><mo>*</mo><mi>j</mi></mrow><mi mathvariant="italic">Urth</mi></msubsup></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow></math> (j = 1, 2, ..., n ; r = 1, 2, ..., k).

Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><munder><mi>t</mi><mo>-</mo></munder><mn>1</mn><mi>L</mi><mo>,</mo><munder><mi>t</mi><mo>-</mo></munder><mn>1</mn><mi>U</mi></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mn>1</mn><mi>L</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mn>1</mn><mi>U</mi></msubsup></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow><mo>,</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mrow><mrow><mo>[</mo><mrow><munder><mi>t</mi><mo>-</mo></munder><mn>2</mn><mi>L</mi><mo>,</mo><munder><mi>t</mi><mo>-</mo></munder><mn>2</mn><mi>U</mi></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mn>2</mn><mi>L</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mn>2</mn><mi>U</mi></msubsup></mrow><mo>]</mo></mrow></mrow><mo>)</mo></math> , and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mrow><mrow><mo>[</mo><mrow><munder><mi>t</mi><mo>-</mo></munder><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mi>L</mi><mo>,</mo><munder><mi>t</mi><mo>-</mo></munder><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mi>U</mi></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mi>L</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mi>U</mi></msubsup></mrow><mo>]</mo></mrow></mrow><mo>)</mo></math> be maximum tolerance values on the RI decision variables <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mi>j</mi><mi mathvariant="italic">rth</mi></msubsup><mo>=</mo><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mi>j</mi><mi mathvariant="italic">Lrth</mi><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mi>j</mi><mi mathvariant="italic">Urth</mi></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>j</mi><mi mathvariant="italic">Lrth</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>j</mi><mi mathvariant="italic">Urth</mi></msubsup></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow></math> , (j = 1, 2, ... , n ; r = 1, 2, ..., k - 1) recognized by the r<sups>th</sups>Level DM.

Then, the membership functions for the RI decision variables <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mi>j</mi><mi mathvariant="italic">rth</mi></msubsup><mrow><mo>(</mo><mrow><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>k</mi><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow></math> can be formulated as follows:

(14) <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd columnalign="center"><mi>μ</mi><mrow><mo>(</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi></mrow><mo>)</mo></mrow><mo>=</mo></mtd></mtr><mtr><mtd columnalign="center"><mrow><mo>{</mo><mrow><mrow><mtable><mtr><mtd columnalign="center"><mrow><mfrac><mrow><munder><mi>x</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi><mo>-</mo><mrow><mo>(</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Lrth</mi><mo>-</mo><munder><mi>t</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi></mrow><mo>)</mo></mrow></mrow><mrow><munder><mi>t</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi></mrow></mfrac><mo>,</mo><mi mathvariant="italic">if</mi><munder><mi>x</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Lrth</mi><mo>-</mo><munder><mi>t</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi><mo>⩽</mo><munder><mi>x</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi><mo>⩽</mo><munder><mi>x</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Lrth</mi><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="center"><mrow><mfrac><mrow><mo>-</mo><munder><mi>x</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi><mo>+</mo><mrow><mo>(</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Lrth</mi><mo>+</mo><munder><mi>t</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi></mrow><mo>)</mo></mrow></mrow><mrow><munder><mi>t</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi></mrow></mfrac><mo>,</mo><mi mathvariant="italic">if</mi><munder><mi>x</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Lrth</mi><mo>⩽</mo><munder><mi>x</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi><mo>⩽</mo><munder><mi>x</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Lrth</mi><mo>-</mo><munder><mi>t</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi><mo>.</mo></mrow></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></mrow></mrow><mo /></mrow></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></math>

(15) <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd columnalign="center"><mi>μ</mi><mrow><mo>(</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi></mrow><mo>)</mo></mrow><mo>=</mo></mtd></mtr><mtr><mtd columnalign="center"><mrow><mo>{</mo><mrow><mrow><mtable><mtr><mtd columnalign="center"><mrow><mfrac><mrow><munder><mi>x</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi><mo>-</mo><mrow><mo>(</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Urth</mi><mo>-</mo><munder><mi>t</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi></mrow><mo>)</mo></mrow></mrow><mrow><munder><mi>t</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi></mrow></mfrac><mo>,</mo><mi mathvariant="italic">if</mi><munder><mi>x</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Urth</mi><mo>-</mo><munder><mi>t</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi><mo>⩽</mo><munder><mi>x</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi><mo>⩽</mo><munder><mi>x</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Urth</mi><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="center"><mrow><mfrac><mrow><mo>-</mo><munder><mi>x</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi><mo>+</mo><mrow><mo>(</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Urth</mi><mo>+</mo><munder><mi>t</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi></mrow><mo>)</mo></mrow></mrow><mrow><munder><mi>t</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi></mrow></mfrac><mo>,</mo><mi mathvariant="italic">if</mi><munder><mi>x</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Urth</mi><mo>⩽</mo><munder><mi>x</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi><mo>⩽</mo><munder><mi>x</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Urth</mi><mo>-</mo><munder><mi>t</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi><mo>.</mo></mrow></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></mrow></mrow><mo /></mrow></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></math>

(16) <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd columnalign="center"><mi>μ</mi><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup></mrow><mo>)</mo></mrow><mo>=</mo></mtd></mtr><mtr><mtd columnalign="center"><mrow><mo>{</mo><mrow><mrow><mtable><mtr><mtd columnalign="center"><mrow><mfrac><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup><mo>-</mo><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Lrth</mi></msubsup><mo>-</mo><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup></mrow><mo>)</mo></mrow></mrow><mrow><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup></mrow></mfrac><mo>,</mo><mi mathvariant="italic">if</mi><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Lrth</mi></msubsup><mo>-</mo><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup><mo>⩽</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup><mo>⩽</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Lrth</mi></msubsup><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="center"><mrow><mfrac><mrow><mo>-</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup><mo>+</mo><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Lrth</mi></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup></mrow><mo>)</mo></mrow></mrow><mrow><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup></mrow></mfrac><mo>,</mo><mi mathvariant="italic">if</mi><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Lrth</mi></msubsup><mo>⩽</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup><mo>⩽</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Lrth</mi></msubsup><mo>-</mo><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup><mo>.</mo></mrow></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></mrow></mrow><mo /></mrow></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></math>

(17) <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd columnalign="center"><mi>μ</mi><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup></mrow><mo>)</mo></mrow><mo>=</mo></mtd></mtr><mtr><mtd columnalign="center"><mrow><mo>{</mo><mrow><mrow><mtable><mtr><mtd columnalign="center"><mrow><mfrac><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup><mo>-</mo><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Urth</mi></msubsup><mo>-</mo><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup></mrow><mo>)</mo></mrow></mrow><mrow><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup></mrow></mfrac><mo>,</mo><mi mathvariant="italic">if</mi><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Urth</mi></msubsup><mo>-</mo><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup><mo>⩽</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup><mo>⩽</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Urth</mi></msubsup><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="center"><mrow><mfrac><mrow><mo>-</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup><mo>+</mo><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Urth</mi></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup></mrow><mo>)</mo></mrow></mrow><mrow><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup></mrow></mfrac><mo>,</mo><mi mathvariant="italic">if</mi><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Urth</mi></msubsup><mo>⩽</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup><mo>⩽</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Urth</mi></msubsup><mo>-</mo><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup><mo>.</mo></mrow></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></mrow></mrow><mo /></mrow></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></math>

The following can be assumed for the membership functions of the r<sups>th</sups>Level DM (r = 1, 2, ..., k):

(18) <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd columnalign="center"><msub><mi>μ</mi><mrow><munder><mi>L</mi><mo>-</mo></munder></mrow></msub><mrow><mo>(</mo><mrow><msubsup><munder><mi>f</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>=</mo></mtd></mtr><mtr><mtd columnalign="center"><mrow><mo>{</mo><mrow><mtable><mtr><mtd columnalign="center"><mn>1</mn><mo>,</mo><msubsup><munder><mi>f</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>⩾</mo><munder><mi>f</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>r</mi></mrow><mi>L</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><mfrac><mrow><msubsup><munder><mi>f</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>-</mo><msubsup><munder><mi>f</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi></msubsup><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup></mrow><mo>)</mo></mrow></mrow><mrow><munder><mi>f</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>r</mi></mrow><mi>L</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>-</mo><msubsup><munder><mi>f</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi></msubsup><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup></mrow><mo>)</mo></mrow></mrow></mfrac><mo>,</mo><msubsup><munder><mi>f</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi></msubsup><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup></mrow><mo>)</mo></mrow><mo><</mo><msubsup><munder><mi>f</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo><</mo><munder><mi>f</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>r</mi></mrow><mi>L</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><mn>0</mn><mo>,</mo><msubsup><munder><mi>f</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>⩽</mo><msubsup><munder><mi>f</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi></msubsup><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></mrow><mo /></mrow></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></math>

(19) <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd columnalign="center"><msub><mi>μ</mi><mrow><munder><mi>U</mi><mo>-</mo></munder></mrow></msub><mrow><mo>(</mo><mrow><msubsup><munder><mi>f</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>=</mo></mtd></mtr><mtr><mtd columnalign="center"><mrow><mo>{</mo><mrow><mtable><mtr><mtd columnalign="center"><mn>1</mn><mo>,</mo><msubsup><munder><mi>f</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>⩾</mo><munder><mi>f</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>r</mi></mrow><mi>U</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><mfrac><mrow><msubsup><munder><mi>f</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>-</mo><msubsup><munder><mi>f</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi></msubsup><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup></mrow><mo>)</mo></mrow></mrow><mrow><munder><mi>f</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>r</mi></mrow><mi>U</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>-</mo><msubsup><munder><mi>f</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi></msubsup><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup></mrow><mo>)</mo></mrow></mrow></mfrac><mo>,</mo><msubsup><munder><mi>f</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi></msubsup><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup></mrow><mo>)</mo></mrow><mo><</mo><msubsup><munder><mi>f</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo><</mo><munder><mi>f</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>r</mi></mrow><mi>U</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><mn>0</mn><mo>,</mo><msubsup><munder><mi>f</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>⩽</mo><msubsup><munder><mi>f</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi></msubsup><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></mrow><mo /></mrow></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></math>

(20) <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd columnalign="center"><msub><mi>μ</mi><mrow><mover accent="true"><mi>L</mi><mo>¯</mo></mover></mrow></msub><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>=</mo></mtd></mtr><mtr><mtd columnalign="center"><mrow><mo>{</mo><mrow><mtable><mtr><mtd columnalign="center"><mn>1</mn><mo>,</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>⩾</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mrow><mo>*</mo><mi>L</mi></mrow></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><mfrac><mrow><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>-</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup></mrow><mo>)</mo></mrow></mrow><mrow><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mrow><mo>*</mo><mi>L</mi></mrow></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>-</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup></mrow><mo>)</mo></mrow></mrow></mfrac><mo>,</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup></mrow><mo>)</mo></mrow><mo><</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo><</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mrow><mo>*</mo><mi>L</mi></mrow></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><mn>0</mn><mo>,</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>⩽</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></mrow><mo /></mrow></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></math>

(21) <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd columnalign="center"><msub><mi>μ</mi><mrow><mover accent="true"><mi>U</mi><mo>¯</mo></mover></mrow></msub><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>=</mo></mtd></mtr><mtr><mtd columnalign="center"><mrow><mo>{</mo><mrow><mtable><mtr><mtd columnalign="center"><mn>1</mn><mo>,</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>⩾</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mrow><mo>*</mo><mi>U</mi></mrow></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><mfrac><mrow><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>-</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup></mrow><mo>)</mo></mrow></mrow><mrow><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mrow><mo>*</mo><mi>U</mi></mrow></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>-</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup></mrow><mo>)</mo></mrow></mrow></mfrac><mo>,</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup></mrow><mo>)</mo></mrow><mo><</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo><</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mrow><mo>*</mo><mi>U</mi></mrow></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><mn>0</mn><mo>,</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>⩽</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></mrow><mo /></mrow></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></math>

Such that at r = k, then <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><munder><mi>f</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi></msubsup><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup></mrow><mo>)</mo></mrow><mo>=</mo><msubsup><munder><mi>f</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi></msubsup><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup></mrow><mo>)</mo></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><munder><mi>f</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi></msubsup><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup></mrow><mo>)</mo></mrow><mo>=</mo><msubsup><munder><mi>f</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi></msubsup><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup></mrow><mo>)</mo></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup></mrow><mo>)</mo></mrow><mo>=</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup></mrow><mo>)</mo></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup></mrow><mo>)</mo></mrow><mo>=</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup><mrow><mo>(</mo><mrow><msubsup><mi>x</mi><mrow><mo>*</mo><mi>j</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mo>-</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi mathvariant="italic">th</mi></mrow></msubsup></mrow><mo>)</mo></mrow></math> .

The following Tchebycheff problems will be resolved in order to produce a suitable result and ensure the satisfaction of all decision-makers:

subject to

(22) <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd columnalign="center"><mi>x</mi><mo>∈</mo><msup><mover accent="true"><mi>G</mi><mo>¯</mo></mover><mi>L</mi></msup><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><msub><mi>μ</mi><mrow><mover accent="true"><mi>L</mi><mo>¯</mo></mover></mrow></msub><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>⩾</mo><msup><mi>ω</mi><mn>1</mn></msup><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><mfrac><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup><mo>-</mo><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Lrth</mi></msubsup><mo>-</mo><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup></mrow><mo>)</mo></mrow></mrow><mrow><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup></mrow></mfrac><mo>⩾</mo><msup><mi>ω</mi><mn>1</mn></msup><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><mfrac><mrow><mo>-</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup><mo>+</mo><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Lrth</mi></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup></mrow><mo>)</mo></mrow></mrow><mrow><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup></mrow></mfrac><mo>⩾</mo><msup><mi>ω</mi><mn>1</mn></msup><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mi>r</mi><mi>L</mi></msubsup><mo>></mo><mn>0</mn><mo>,</mo><mspace width="thinmathspace" /><msup><mi>ω</mi><mn>1</mn></msup><mo>∈</mo><mrow><mo>[</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow><mo>]</mo></mrow><mo>.</mo></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></math>

subject to

(23) <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd columnalign="center"><mi>x</mi><mo>∈</mo><msup><mover accent="true"><mi>G</mi><mo>¯</mo></mover><mi>U</mi></msup><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><msub><mi>μ</mi><mrow><mover accent="true"><mi>U</mi><mo>¯</mo></mover></mrow></msub><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>⩾</mo><msup><mi>ω</mi><mn>4</mn></msup><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><mfrac><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup><mo>-</mo><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Urth</mi></msubsup><mo>-</mo><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup></mrow><mo>)</mo></mrow></mrow><mrow><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup></mrow></mfrac><mo>⩾</mo><msup><mi>ω</mi><mn>4</mn></msup><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><mfrac><mrow><mo>-</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup><mo>+</mo><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Urth</mi></msubsup><mo>+</mo><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup></mrow><mo>)</mo></mrow></mrow><mrow><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup></mrow></mfrac><mo>⩾</mo><msup><mi>ω</mi><mn>4</mn></msup><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mi>r</mi><mi>U</mi></msubsup><mo>></mo><mn>0</mn><mo>,</mo><mspace width="thinmathspace" /><msup><mi>ω</mi><mn>4</mn></msup><mo>∈</mo><mrow><mo>[</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow><mo>]</mo></mrow><mo>.</mo></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></math>

subject to

(24) <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd columnalign="center"><mi>x</mi><mo>∈</mo><munderover><mi>G</mi><mo>-</mo><mi>L</mi></munderover><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><msub><mi>μ</mi><mrow><munder><mi>L</mi><mo>-</mo></munder></mrow></msub><mrow><mo>(</mo><mrow><msubsup><munder><mi>f</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>⩾</mo><msup><mi>ω</mi><mn>2</mn></msup><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><mfrac><mrow><munder><mi>x</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi><mo>-</mo><mrow><mo>(</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Lrth</mi><mo>-</mo><munder><mi>t</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi></mrow><mo>)</mo></mrow></mrow><mrow><munder><mi>t</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi></mrow></mfrac><mo>⩾</mo><msup><mi>ω</mi><mn>2</mn></msup><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><mfrac><mrow><mo>-</mo><munder><mi>x</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi><mo>+</mo><mrow><mo>(</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Lrth</mi><mo>+</mo><munder><mi>t</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi></mrow><mo>)</mo></mrow></mrow><mrow><munder><mi>t</mi><mo>-</mo></munder><mi>r</mi><mi>L</mi></mrow></mfrac><mo>⩾</mo><msup><mi>ω</mi><mn>2</mn></msup><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><msubsup><munder accentunder="true"><mi>t</mi><mo stretchy="true">_</mo></munder><mi>r</mi><mi>L</mi></msubsup><mo>></mo><mn>0</mn><mo>,</mo><mspace width="thinmathspace" /><msup><mi>ω</mi><mn>2</mn></msup><mo>∈</mo><mrow><mo>[</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow><mo>]</mo></mrow><mo>.</mo></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></math>

<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><munder accentunder="true"><mi mathvariant="italic">Tch</mi><mo stretchy="true">_</mo></munder><mn>3</mn></msup><mo>:</mo><mo>max</mo><msup><mi>ω</mi><mn>3</mn></msup></math> ,

subject to

(25) <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd columnalign="center"><mi>x</mi><mo>∈</mo><munderover><mi>G</mi><mo>-</mo><mi>U</mi></munderover><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><msub><mi>μ</mi><mrow><munder><mi>U</mi><mo>-</mo></munder></mrow></msub><mrow><mo>(</mo><mrow><msubsup><munder><mi>f</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi></msubsup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>⩾</mo><msup><mi>ω</mi><mn>3</mn></msup><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><mfrac><mrow><munder><mi>x</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi><mo>-</mo><mrow><mo>(</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Urth</mi><mo>-</mo><munder><mi>t</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi></mrow><mo>)</mo></mrow></mrow><mrow><munder><mi>t</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi></mrow></mfrac><mo>⩾</mo><msup><mi>ω</mi><mn>3</mn></msup><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><mfrac><mrow><mo>-</mo><munder><mi>x</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi><mo>+</mo><mrow><mo>(</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mrow><mo>*</mo><mi>r</mi></mrow><mi mathvariant="italic">Urth</mi><mo>+</mo><munder><mi>t</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi></mrow><mo>)</mo></mrow></mrow><mrow><munder><mi>t</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi></mrow></mfrac><mo>⩾</mo><msup><mi>ω</mi><mn>3</mn></msup><mo>,</mo></mtd></mtr><mtr><mtd columnalign="center"><munder><mi>t</mi><mo>-</mo></munder><mi>r</mi><mi>U</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace width="thinmathspace" /><msup><mi>ω</mi><mn>3</mn></msup><mo>∈</mo><mrow><mo>[</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow><mo>]</mo></mrow><mo>.</mo></mtd></mtr><mtr><mtd columnalign="center" /></mtr></mtable></math>

6.3.1 A flowchart for solving MLFRILP

The decision-making procedure of the fuzzy approach is presented in Fig. 3.

Graph: Fig. 3 The decision-making procedure of the fuzzy method.

7 Numerical example

A numerical example is resolved utilizing the three computational approaches—(a) constraint method, (b) interactive model, and (c) fuzzy approach—for the solution of MLP problems with fully rough parameters and fully rough decision-making variables. Think about the following illustration of a three-level programming issue using RI parameters and RI decision variables in the objective functions and constraints:

[1<sups>st</sups> - LDM]: <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi mathvariant="italic">max</mi><mrow><msubsup><mi>x</mi><mn>1</mn><mi>R</mi></msubsup></mrow></munder><msubsup><mi>F</mi><mn>1</mn><mi>R</mi></msubsup><mo>=</mo><munder><mi mathvariant="italic">max</mi><mrow><msubsup><mi>x</mi><mn>1</mn><mi>R</mi></msubsup></mrow></munder><mrow><mo>{</mo><mrow><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow><mrow><mo>]</mo><mo>,</mo><mo>[</mo></mrow><mrow><mn>1</mn><mo>,</mo><mn>4</mn></mrow></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow><mo>⊗</mo><msubsup><mi>x</mi><mn>1</mn><mi>R</mi></msubsup><mo>⊕</mo><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><mrow><mn>1</mn><mo>,</mo><mn>3</mn></mrow><mrow><mo>]</mo><mo>,</mo><mo>[</mo></mrow><mrow><mn>1</mn><mo>,</mo><mn>6</mn></mrow></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow><mo>⊗</mo><msubsup><mi>x</mi><mn>2</mn><mi>R</mi></msubsup><mo>⊕</mo><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><mrow><mn>2</mn><mo>,</mo><mn>2</mn></mrow><mrow><mo>]</mo><mo>,</mo><mo>[</mo></mrow><mrow><mn>2</mn><mo>,</mo><mn>5</mn></mrow></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow><mo>⊗</mo><msubsup><mi>x</mi><mn>3</mn><mi>R</mi></msubsup><mo>⊕</mo><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><mrow><mn>2</mn><mo>,</mo><mn>5</mn></mrow><mrow><mo>]</mo><mo>,</mo><mo>[</mo></mrow><mrow><mn>1</mn><mo>,</mo><mn>7</mn></mrow></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow></mrow><mo>}</mo></mrow><mo>,</mo></math>

Where <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mn>2</mn><mi>R</mi></msubsup><mo>,</mo><msubsup><mi>x</mi><mn>3</mn><mi>R</mi></msubsup></math> solve

[2<sups>nd</sups>LDM]: <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi mathvariant="italic">max</mi><mrow><msubsup><mi>x</mi><mn>2</mn><mi>R</mi></msubsup></mrow></munder><msubsup><mi>F</mi><mn>2</mn><mi>R</mi></msubsup><mo>=</mo><munder><mi mathvariant="italic">max</mi><mrow><msubsup><mi>x</mi><mn>2</mn><mi>R</mi></msubsup></mrow></munder><mrow><mo>{</mo><mrow><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow><mrow><mo>]</mo><mo>,</mo><mo>[</mo></mrow><mrow><mn>1</mn><mo>,</mo><mn>4</mn></mrow><mo>]</mo></mrow><mo>]</mo></mrow><mo>⊗</mo><msubsup><mi>x</mi><mn>1</mn><mi>R</mi></msubsup><mo>⊕</mo><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><mrow><mn>5</mn><mo>,</mo><mn>7</mn></mrow><mrow><mo>]</mo><mo>,</mo><mo>[</mo></mrow><mrow><mn>3</mn><mo>,</mo><mn>8</mn></mrow></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow><mo>⊗</mo><msubsup><mi>x</mi><mn>2</mn><mi>R</mi></msubsup><mo>⊕</mo><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><mrow><mn>3</mn><mo>,</mo><mn>5</mn></mrow><mrow><mo>]</mo><mo>,</mo><mo>[</mo></mrow><mrow><mn>2</mn><mo>,</mo><mn>8</mn></mrow></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow><mo>⊗</mo><msubsup><mi>x</mi><mn>3</mn><mi>R</mi></msubsup><mo>⊕</mo><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><mrow><mn>5</mn><mo>,</mo><mn>7</mn></mrow><mrow><mo>]</mo><mo>,</mo><mo>[</mo></mrow><mrow><mn>2</mn><mo>,</mo><mn>10</mn></mrow></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow></mrow><mo>}</mo></mrow><mo>,</mo></math>

Where <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mn>3</mn><mi>R</mi></msubsup></math> solve

[3<sups>rd</sups>LDM]: <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi mathvariant="italic">max</mi><mrow><msubsup><mi>x</mi><mn>3</mn><mi>R</mi></msubsup></mrow></munder><msubsup><mi>F</mi><mn>3</mn><mi>R</mi></msubsup><mo>=</mo><munder><mi mathvariant="italic">max</mi><mrow><msubsup><mi>x</mi><mn>3</mn><mi>R</mi></msubsup></mrow></munder><mrow><mo>{</mo><mrow><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><mrow><mn>3</mn><mo>,</mo><mn>5</mn></mrow><mrow><mo>]</mo><mo>,</mo><mo>[</mo></mrow><mrow><mn>2</mn><mo>,</mo><mn>7</mn></mrow></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow><mo>⊗</mo><msubsup><mi>x</mi><mn>1</mn><mi>R</mi></msubsup><mo>⊕</mo><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><mrow><mn>4</mn><mo>,</mo><mn>5</mn></mrow><mrow><mo>]</mo><mo>,</mo><mo>[</mo></mrow><mrow><mn>3</mn><mo>,</mo><mn>6</mn></mrow></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow><mo>⊗</mo><msubsup><mi>x</mi><mn>2</mn><mi>R</mi></msubsup><mo>⊕</mo><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><mrow><mn>5</mn><mo>,</mo><mn>8</mn></mrow><mrow><mo>]</mo><mo>,</mo><mo>[</mo></mrow><mrow><mn>3</mn><mo>,</mo><mn>9</mn></mrow></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow><mo>⊗</mo><msubsup><mi>x</mi><mn>3</mn><mi>R</mi></msubsup><mo>⊕</mo><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><mrow><mn>3</mn><mo>,</mo><mn>6</mn></mrow><mrow><mo>]</mo><mo>,</mo><mo>[</mo></mrow><mrow><mn>1</mn><mo>,</mo><mn>8</mn></mrow></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow></mrow><mo>}</mo></mrow><mo>,</mo></math>

subject to

<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>{</mo><mrow><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><mrow><mn>5</mn><mo>,</mo><mn>7</mn></mrow><mrow><mo>]</mo><mo>,</mo><mo>[</mo></mrow><mrow><mn>5</mn><mo>,</mo><mn>9</mn></mrow></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow><mo>⊗</mo><msubsup><mi>x</mi><mn>1</mn><mi>R</mi></msubsup><mo>⊕</mo><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><mrow><mn>3</mn><mo>,</mo><mn>5</mn></mrow><mrow><mo>]</mo><mo>,</mo><mo>[</mo></mrow><mrow><mn>3</mn><mo>,</mo><mn>8</mn></mrow></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow><mo>⊗</mo><msubsup><mi>x</mi><mn>2</mn><mi>R</mi></msubsup></mrow><mo /></mrow><mo>⩾</mo><mrow><mo>[</mo><mrow><mrow><mo>[</mo><mrow><mrow><mn>4</mn><mo>,</mo><mn>8</mn></mrow><mrow><mo>]</mo><mo>,</mo><mo>[</mo></mrow><mrow><mn>3</mn><mo>,</mo><mn>16</mn></mrow></mrow><mo>]</mo></mrow></mrow><mo>]</mo></mrow><mo>,</mo><msubsup><mi>x</mi><mn>1</mn><mi>R</mi></msubsup><mo>,</mo><msubsup><mi>x</mi><mn>2</mn><mi>R</mi></msubsup><mo>,</mo><msubsup><mi>x</mi><mn>3</mn><mi>R</mi></msubsup><mo>⩾</mo><mn>0</mn><mo>}</mo></math> .

<h31 id="AN0170719011-17">First: Constraint Method</h31>

Using Theorems (1), (2), and (3), the three-level programming problem with RI parameters and RI decision variables in the constraints and objective functions and is converted into a crisp model as follows:

SP_st_sp_Level<bold>DM</bold>:

Subject to

6x<subs>1</subs> + 5x<subs>2</subs> ⩽ 12, 5x<subs>1</subs> - 7x<subs>2</subs> + 7x<subs>3</subs> ⩽ 9, 7x<subs>1</subs> + 5x<subs>2</subs> ⩾ 4,

<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mi>i</mi><mi>R</mi></msubsup><mo>⩾</mo><mn>0</mn><mo>,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></math> .

whose solution is <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>f</mi><mo>-</mo></munder><mn>1</mn><mi mathvariant="italic">LF</mi><mo>=</mo><mn>10.5714</mn><mo>,</mo></math>

<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mn>1</mn><mi mathvariant="italic">FL</mi><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mn>2</mn><mi mathvariant="italic">FL</mi><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mn>3</mn><mi mathvariant="italic">FL</mi></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3.285714</mn></mrow><mo>)</mo></mrow></math> .

Subject to

3x<subs>1</subs> + 3x<subs>2</subs> ⩽ 14, 3x<subs>1</subs> - 5x<subs>2</subs> + 6x<subs>3</subs> ⩽ 11, 5x<subs>1</subs> + 3x<subs>2</subs> ⩾ 8,

whose solution is <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>f</mi><mo>-</mo></munder><mn>1</mn><mi mathvariant="italic">UF</mi><mo>=</mo><mn>30.4444</mn></math> ,

<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mn>1</mn><mi mathvariant="italic">FU</mi><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mn>2</mn><mi mathvariant="italic">FU</mi><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mn>3</mn><mi mathvariant="italic">FU</mi></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>4.6667</mn><mo>,</mo><mn>5.7222</mn></mrow><mo>)</mo></mrow></math> .

P <subs>13</subs>: <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mn>1</mn><mi>L</mi></msubsup><mo>:</mo><mo>=</mo><mo>max</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mn>2</mn><msub><mi>x</mi><mn>3</mn></msub><mo>+</mo><mn>1</mn><mo>,</mo></math>

Subject to

7x<subs>1</subs> + 6x<subs>2</subs> ⩽ 10, 8x<subs>1</subs> - 9x<subs>2</subs> + 9x<subs>3</subs> ⩽ 8, 9x<subs>1</subs> + 8x<subs>2</subs> ⩾ 3,

whose solution is <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mn>1</mn><mi mathvariant="italic">LF</mi></msubsup><mo>=</mo><mn>7.7778</mn><mo>,</mo></math>

<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mn>1</mn><mi mathvariant="italic">FL</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mn>2</mn><mi mathvariant="italic">FL</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mn>3</mn><mi mathvariant="italic">FL</mi></msubsup></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>1.6667</mn><mo>,</mo><mn>2.5556</mn></mrow><mo>)</mo></mrow></math> .

P <subs>14</subs>: <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mn>1</mn><mi>U</mi></msubsup><mo>:</mo><mo>=</mo><mo>max</mo><mn>4</mn><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mn>6</mn><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mn>5</mn><msub><mi>x</mi><mn>3</mn></msub><mo>+</mo><mn>7</mn><mo>,</mo></math>

Subject to

x <subs>1</subs> + 2x<subs>2</subs> ⩽ 16,

x <subs>1</subs> - 2x<subs>2</subs> + 3x<subs>3</subs> ⩽ 14,

5x<subs>1</subs> + 3x<subs>2</subs> ⩾ 16,

<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mn>1</mn><mi mathvariant="italic">FU</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mn>2</mn><mi mathvariant="italic">FU</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mn>3</mn><mi mathvariant="italic">FU</mi></msubsup></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>10</mn></mrow><mo>)</mo></mrow></math> .

Then, set <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mn>1</mn><mi>F</mi></msubsup><mo>=</mo><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow></math> to the 2<sups>nd</sups>Level DM constraints. The problem of the 2<sups>nd</sups>Level DM by using interval method [[19]] and slice sum method [[25]] can be formulated as follows:

2<sups>nd</sups>Level<bold>DM</bold>:

P <subs>21</subs>: <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>f</mi><mo>-</mo></munder><mn>2</mn><mi>L</mi><mo>:</mo><mo>=</mo><mo>max</mo><mn>2</mn><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mn>5</mn><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mn>3</mn><msub><mi>x</mi><mn>3</mn></msub><mo>+</mo><mn>5</mn><mo>,</mo></math>

subject to

6x<subs>1</subs> + 5x<subs>2</subs> ⩽ 12, 5x<subs>1</subs> - 7x<subs>2</subs> + 7x<subs>3</subs> ⩽ 9, 7x<subs>1</subs> + 5x<subs>2</subs> ⩾ 4,

x <subs>1</subs> = 0, <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mi>i</mi><mi>R</mi></msubsup><mo>⩾</mo><mn>0</mn><mo>,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></math> .

whose solution is <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>f</mi><mo>-</mo></munder><mn>2</mn><mi mathvariant="italic">LS</mi><mo>=</mo><mn>24.8571</mn></math> ,

<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mn>1</mn><mi mathvariant="italic">FL</mi><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mn>2</mn><mi mathvariant="italic">SL</mi><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mn>3</mn><mi mathvariant="italic">SL</mi></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3.285714</mn></mrow><mo>)</mo></mrow></math> .

P <subs>22</subs>: <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>f</mi><mo>-</mo></munder><mn>2</mn><mi>U</mi><mo>:</mo><mo>=</mo><mo>max</mo><mn>3</mn><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mn>7</mn><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mn>5</mn><msub><mi>x</mi><mn>3</mn></msub><mo>+</mo><mn>7</mn><mo>,</mo></math>

Subject to

3x<subs>1</subs> + 3x<subs>2</subs> ⩽ 14, 3x<subs>1</subs> - 5x<subs>2</subs> + 6x<subs>3</subs> ⩽ 11, 5x<subs>1</subs> + 3x<subs>2</subs> ⩾ 8,

whose solution is <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>f</mi><mo>-</mo></munder><mn>2</mn><mi mathvariant="italic">US</mi><mo>=</mo><mn>68.2778</mn></math> ,

<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mn>1</mn><mi mathvariant="italic">FU</mi><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mn>2</mn><mi mathvariant="italic">SU</mi><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mn>3</mn><mi mathvariant="italic">SU</mi></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>4.6667</mn><mo>,</mo><mn>5.7222</mn></mrow><mo>)</mo></mrow></math> .

P <subs>23</subs>: <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mn>2</mn><mi>L</mi></msubsup><mo>:</mo><mo>=</mo><mo>max</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mn>3</mn><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mn>2</mn><msub><mi>x</mi><mn>3</mn></msub><mo>+</mo><mn>2</mn><mo>,</mo></math>

Subject to

7x<subs>1</subs> + 6x<subs>2</subs> ⩽ 10, 8x<subs>1</subs> - 9x<subs>2</subs> + 9x<subs>3</subs> ⩽ 8, 9x<subs>1</subs> + 8x<subs>2</subs> ⩾ 3,

whose solution is <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mn>2</mn><mi mathvariant="italic">LS</mi></msubsup><mo>=</mo><mn>12.1111</mn></math> ,

P <subs>24</subs>: <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mn>2</mn><mi>U</mi></msubsup><mo>:</mo><mo>=</mo><mo>max</mo><mn>4</mn><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mn>8</mn><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mn>8</mn><msub><mi>x</mi><mn>3</mn></msub><mo>+</mo><mn>10</mn><mo>,</mo></math>

Subject to

x <subs>1</subs> + 2x<subs>2</subs> ⩽ 16,

x <subs>1</subs> - 2x<subs>2</subs> + 3x<subs>3</subs> ⩽ 14,

5x<subs>1</subs> + 3x<subs>2</subs> ⩾ 16,

Then, set <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mn>1</mn><mi>F</mi></msubsup><mo>=</mo><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mn>2</mn><mi>S</mi></msubsup><mo>=</mo><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><mn>2</mn><mo>,</mo><mn>4.6667</mn></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><mn>1.6667</mn><mo>,</mo><mn>8</mn></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow></math> to the 3<sups>rd</sups>L DM constraints.

The 3<sups>rd</sups>L DM problem by using interval method [[19]] and slice sum method [[25]] can be formulated as follows:

3<sups>rd</sups>Level<bold>DM:</bold>

P <subs>31</subs>: <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>f</mi><mo>-</mo></munder><mn>3</mn><mi>L</mi><mo>:</mo><mo>=</mo><mo>max</mo><mn>3</mn><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mn>4</mn><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mn>5</mn><msub><mi>x</mi><mn>3</mn></msub><mo>+</mo><mn>3</mn><mo>,</mo></math>

Subject to

6x<subs>1</subs> + 5x<subs>2</subs> ⩽ 12, 5x<subs>1</subs> - 7x<subs>2</subs> + 7x<subs>3</subs> ⩽ 9, 7x<subs>1</subs> + 5x<subs>2</subs> ⩾ 4,

x <subs>1</subs> = 0, x<subs>2</subs> = 2, <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mi>i</mi><mi>R</mi></msubsup><mo>⩾</mo><mn>0</mn><mo>,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></math> .

whose solution is <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>f</mi><mo>-</mo></munder><mn>3</mn><mi mathvariant="italic">LT</mi><mo>=</mo><mn>27.4286</mn></math> ,

P <subs>32</subs>: <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>f</mi><mo>-</mo></munder><mn>3</mn><mi>U</mi><mo>:</mo><mo>=</mo><mo>max</mo><mn>5</mn><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mn>5</mn><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mn>8</mn><msub><mi>x</mi><mn>3</mn></msub><mo>+</mo><mn>6</mn><mo>,</mo></math>

Subject to

3x<subs>1</subs> + 3x<subs>2</subs> ⩽ 14, 3x<subs>1</subs> - 5x<subs>2</subs> + 6x<subs>3</subs> ⩽ 11, 5x<subs>1</subs> + 3x<subs>2</subs> ⩾ 8,

x <subs>1</subs> = 0, x<subs>2</subs> = 4.6667, <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mi>i</mi><mi>R</mi></msubsup><mo>⩾</mo><mn>0</mn><mo>,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></math> .

whose solution is <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>f</mi><mo>-</mo></munder><mn>3</mn><mi mathvariant="italic">UT</mi><mo>=</mo><mn>75.1111</mn></math> ,

<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mn>1</mn><mi mathvariant="italic">FU</mi><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mn>2</mn><mi mathvariant="italic">SU</mi><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mn>3</mn><mi mathvariant="italic">TU</mi></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>4.6667</mn><mo>,</mo><mn>5.7222</mn></mrow><mo>)</mo></mrow></math> .

P <subs>33</subs>: <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mn>3</mn><mi>L</mi></msubsup><mo>:</mo><mo>=</mo><mo>max</mo><mn>2</mn><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mn>3</mn><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mn>3</mn><msub><mi>x</mi><mn>3</mn></msub><mo>+</mo><mn>1</mn><mo>,</mo></math>

Subject to

7x<subs>1</subs> + 6x<subs>2</subs> ⩽ 10, 8x<subs>1</subs> - 9x<subs>2</subs> + 9x<subs>3</subs> ⩽ 8, 9x<subs>1</subs> + 8x<subs>2</subs> ⩾ 3,

x <subs>1</subs> = 0, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>2</mn></msub><mo>=</mo><mn>1.6667</mn><mo>,</mo><msubsup><mi>x</mi><mi>i</mi><mi>R</mi></msubsup><mo>⩾</mo><mn>0</mn><mo>,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></math> .

whose solution is <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mn>3</mn><mi mathvariant="italic">LT</mi></msubsup><mo>=</mo><mn>13.6667</mn></math> ,

<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mn>1</mn><mi mathvariant="italic">FL</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mn>2</mn><mi mathvariant="italic">SL</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mn>3</mn><mi mathvariant="italic">TL</mi></msubsup></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>1.6667</mn><mo>,</mo><mn>2.5556</mn></mrow><mo>)</mo></mrow></math> .

P <subs>34</subs>: <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mn>3</mn><mi>U</mi></msubsup><mo>:</mo><mo>=</mo><mo>max</mo><mn>7</mn><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mn>6</mn><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mn>9</mn><msub><mi>x</mi><mn>3</mn></msub><mo>+</mo><mn>8</mn><mo>,</mo></math>

subject to

x <subs>1</subs> + 2x<subs>2</subs> ⩽ 16,

x <subs>1</subs> - 2x<subs>2</subs> + 3x<subs>3</subs> ⩽ 14,

5x<subs>1</subs> + 3x<subs>2</subs> ⩾ 16,

x <subs>1</subs> = 0, x<subs>2</subs> = 8, <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mi>i</mi><mi>R</mi></msubsup><mo>⩾</mo><mn>0</mn><mo>,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></math> .

Finally, the optimal interval values of objective functions are given in Table 2.

Table 2 The constraint method-based optimal interval values of objective functions

<table><colgroup><col align="left" /><col align="center" /><col align="center" /></colgroup><thead valign="top"><tr><td /><td>The <bold>SO-</bold>range</td><td>The <bold>PO-</bold>range</td></tr></thead><tbody><tr><td>1<sup>st</sup><bold>L</bold><bold>DM</bold></td><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>[</mo><mrow><munder><mi>f</mi><mo>-</mo></munder><mn>1</mn><mi mathvariant="italic">LF</mi><mo>,</mo><munder><mi>f</mi><mo>-</mo></munder><mn>1</mn><mi mathvariant="italic">UF</mi></mrow><mo>]</mo></mrow><mo xmlns="">=</mo><mrow xmlns=""><mo>[</mo><mrow><mn>10.5714</mn><mo>,</mo><mn>30.444</mn></mrow><mo>]</mo></mrow></math></p></td><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>[</mo><mrow><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mn>1</mn><mi mathvariant="italic">LF</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mn>1</mn><mi mathvariant="italic">UF</mi></msubsup></mrow><mo>]</mo></mrow><mo xmlns="">=</mo><mrow xmlns=""><mo>[</mo><mrow><mn>7.7778</mn><mo>,</mo><mn>105</mn></mrow><mo>]</mo></mrow></math></p></td></tr><tr><td>2<sup>nd</sup><bold>L</bold><bold>DM</bold></td><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>[</mo><mrow><munder><mi>f</mi><mo>-</mo></munder><mn>2</mn><mi mathvariant="italic">LS</mi><mo>,</mo><munder><mi>f</mi><mo>-</mo></munder><mn>2</mn><mi mathvariant="italic">US</mi></mrow><mo>]</mo></mrow><mo xmlns="">=</mo><mrow xmlns=""><mo>[</mo><mrow><mn>24.8571</mn><mo>,</mo><mn>68.2778</mn></mrow><mo>]</mo></mrow></math></p></td><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>[</mo><mrow><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mn>2</mn><mi mathvariant="italic">LS</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mn>2</mn><mi mathvariant="italic">US</mi></msubsup></mrow><mo>]</mo></mrow><mo xmlns="">=</mo><mrow xmlns=""><mo>[</mo><mrow><mn>12.1111</mn><mo>,</mo><mn>154</mn></mrow><mo>]</mo></mrow></math></p></td></tr><tr><td>3<sup>rd</sup><bold>L</bold><bold>DM</bold></td><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>[</mo><mrow><munder><mi>f</mi><mo>-</mo></munder><mn>3</mn><mi mathvariant="italic">LT</mi><mo>,</mo><munder><mi>f</mi><mo>-</mo></munder><mn>3</mn><mi mathvariant="italic">UT</mi></mrow><mo>]</mo></mrow><mo xmlns="">=</mo><mrow xmlns=""><mo>[</mo><mrow><mn>27.4286</mn><mo>,</mo><mn>75.111</mn></mrow><mo>]</mo></mrow></math></p></td><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>[</mo><mrow><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mn>3</mn><mi mathvariant="italic">LT</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mn>3</mn><mi mathvariant="italic">UT</mi></msubsup></mrow><mo>]</mo></mrow><mo xmlns="">=</mo><mrow xmlns=""><mo>[</mo><mrow><mn>13.6667</mn><mo>,</mo><mn>146</mn></mrow><mo>]</mo></mrow></math></p></td></tr></tbody></table>

<h31 id="AN0170719011-18">Second: interactive model</h31>

Firstly, based on the optimal RI solutions of the decision variables obtained in P<subs>rs</subs>, (r = 1, 2 ; s = 1, 2, 3, 4), the 1<sups>st</sups>L DM tests whether the obtained solution <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mn>1</mn><mi mathvariant="italic">FL</mi><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mn>2</mn><mi mathvariant="italic">SL</mi><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mn>3</mn><mi mathvariant="italic">SL</mi></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3.2857</mn></mrow><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mn>1</mn><mi mathvariant="italic">FU</mi><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mn>2</mn><mi mathvariant="italic">SU</mi><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mn>3</mn><mi mathvariant="italic">SU</mi></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>4.667</mn><mo>,</mo><mn>5.7222</mn></mrow><mo>)</mo></mrow><mo>,</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mn>1</mn><mi mathvariant="italic">FL</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mn>2</mn><mi mathvariant="italic">SL</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mn>3</mn><mi mathvariant="italic">SL</mi></msubsup></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>1.667</mn><mo>,</mo><mn>2.556</mn></mrow><mo>)</mo></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mn>1</mn><mi mathvariant="italic">FU</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mn>2</mn><mi mathvariant="italic">SU</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mn>3</mn><mi mathvariant="italic">SU</mi></msubsup></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>10</mn></mrow><mo>)</mo></mrow></math> is a preferred solution that is agreeable to them, or it may be changed, by the function of the 1<sups>st</sups>Level DM that measures satisfactoriness:

So <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mn>1</mn><mi>F</mi></msubsup><mo>=</mo><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mn>2</mn><mi>S</mi></msubsup><mo>=</mo><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><mn>2</mn><mo>,</mo><mn>4.667</mn></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><mn>1.667</mn><mo>,</mo><mn>8</mn></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mn>3</mn><mi>S</mi></msubsup><mo>=</mo><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><mn>3.285714</mn><mo>,</mo><mn>5.7222</mn></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><mn>2.5556</mn><mo>,</mo><mn>10</mn></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow></math> are a satisfactory solution to the 1<sups>st</sups>Level DM, where δ<sups>F</sups> = ([0.5, 0.5] , [0.5, 0.5]).

Secondly, the 2<sups>nd</sups>Level DM tests whether the obtained solution <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mn>1</mn><mi mathvariant="italic">FL</mi><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mn>2</mn><mi mathvariant="italic">SL</mi><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mn>3</mn><mi mathvariant="italic">TL</mi></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3.2857</mn></mrow><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mn>1</mn><mi mathvariant="italic">FU</mi><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mn>2</mn><mi mathvariant="italic">SU</mi><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mn>3</mn><mi mathvariant="italic">TU</mi></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>4.667</mn><mo>,</mo><mn>5.722</mn></mrow><mo>)</mo></mrow><mo>,</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mn>1</mn><mi mathvariant="italic">FL</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mn>2</mn><mi mathvariant="italic">SL</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mn>3</mn><mi mathvariant="italic">TL</mi></msubsup></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>1.667</mn><mo>,</mo><mn>2.556</mn></mrow><mo>)</mo></mrow><mo>,</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mn>1</mn><mi mathvariant="italic">FU</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mn>2</mn><mi mathvariant="italic">SU</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mn>3</mn><mi mathvariant="italic">SU</mi></msubsup></mrow><mo>)</mo></math> = (0, 8, 10) are a preferred solution that is agreeable to them, or it may be changed, by the function of the 2<sups>nd</sups>Level DM that measures satisfactoriness:

So <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mn>1</mn><mi>F</mi></msubsup><mo>=</mo><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mn>2</mn><mi>S</mi></msubsup><mo>=</mo><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><mn>2</mn><mo>,</mo><mn>4.667</mn></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><mn>1.667</mn><mo>,</mo><mn>8</mn></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mn>3</mn><mi>T</mi></msubsup><mo>=</mo><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mrow><mn>3.2857</mn><mo>,</mo><mn>5.722</mn></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><mn>2.5556</mn><mo>,</mo><mn>10</mn></mrow><mo>]</mo></mrow></mrow><mo>)</mo></mrow></math> are a satisfactory solution to the 2<sups>nd</sups>Level DM, where δ<sups>S</sups> = ([0.5, 0.5] , [0.5, 0.5]).

Finally, the optimal interval values of objective functions are given in Table 3.

Table 3 Based on an interactive model, the optimum interval values for objective functions

<h31 id="AN0170719011-19">Third: Fuzzy Approach</h31>

Based on the optimal RI solutions of the linear programming problems P<subs>rs</subs>, (r = 1, 2 ; s = 1, 2, 3, 4) obtained in the constraint method and interactive model, formulate the membership function for the rth-Level DM (r = 1, 2, 3) as Problems (18)–(21).

Assume that the 1<sups>st</sups>Level DM'S control decision <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mn>1</mn><mi>F</mi></msubsup></math> with tolerance <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>[</mo><mrow><munder><mi>t</mi><mo>-</mo></munder><mn>1</mn><mi>L</mi><mo>,</mo><munder><mi>t</mi><mo>-</mo></munder><mn>1</mn><mi>U</mi></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mn>1</mn><mi>L</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mn>1</mn><mi>U</mi></msubsup></mrow><mo>]</mo></mrow><mo>=</mo></math> ([2, 3] , [1, 4]), and that the 2<sups>nd</sups>Level DM'S control decision <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mn>2</mn><mi>S</mi></msubsup></math> with tolerance <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>[</mo><mrow><munder><mi>t</mi><mo>-</mo></munder><mn>2</mn><mi>L</mi><mo>,</mo><munder><mi>t</mi><mo>-</mo></munder><mn>2</mn><mi>U</mi></mrow><mo>]</mo></mrow><mo>,</mo><mrow><mo>[</mo><mrow><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mn>2</mn><mi>L</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>t</mi><mo>¯</mo></mover><mn>2</mn><mi>U</mi></msubsup></mrow><mo>]</mo></mrow><mo>=</mo></math> ([2, 2] , [2, 2]), and then formulate the membership function for the RI decision variables <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>x</mi><mi>j</mi><mi mathvariant="italic">rth</mi></msubsup><mo>,</mo><mrow><mo>(</mo><mrow><mi>j</mi><mo>,</mo><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow><mo>)</mo></mrow></math> as Problems (14)–(17).

Solve the following Tchebycheff problems to produce a solution that will satisfy all DMs:

subject to

6x<subs>1</subs> + 5x<subs>2</subs> ⩽ 12, 5x<subs>1</subs> - 7x<subs>2</subs> + 7x<subs>3</subs> ⩽ 9, 7x<subs>1</subs> + 5x<subs>2</subs> ⩾ 4,

x <subs>i</subs> ⩾ 0, (i = 1, 2, 3),

t <subs>i</subs> > 0, (i = 1, 2) ,

whose solution is <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msubsup><munder accentunder="true"><mi>x</mi><mo>_</mo></munder><mn>1</mn><mi>L</mi></msubsup><mo>,</mo><msubsup><munder accentunder="true"><mi>x</mi><mo>_</mo></munder><mn>2</mn><mi>L</mi></msubsup><mo>,</mo><msubsup><munder accentunder="true"><mi>x</mi><mo>_</mo></munder><mn>3</mn><mi>L</mi></msubsup><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mtext> </mtext><msup><munder accentunder="true"><mi>ω</mi><mo>_</mo></munder><mi>L</mi></msup><mo>=</mo><mn>1.</mn></mrow></math>

subject to

7x<subs>1</subs> + 6x<subs>2</subs> ⩽ 10, 8x<subs>1</subs> - 9x<subs>2</subs> + 9x<subs>3</subs> ⩽ 8, 9x<subs>1</subs> + 8x<subs>2</subs> ⩾ 3,

x <subs>i</subs> ⩾ 0, (i = 1, 2, 3),

t <subs>i</subs> > 0, (i = 1, 2) ,

whose solution is

<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mn>1</mn><mi>L</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mn>2</mn><mi>L</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mn>3</mn><mi>L</mi></msubsup></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>1.6667</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow><mo>,</mo><msup><mover accent="true"><mi>ω</mi><mo>¯</mo></mover><mi>L</mi></msup><mo>=</mo><mn>1</mn></math> .

subject to

<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msub><mi>x</mi><mn>2</mn></msub><mo>-</mo><mn>2</mn><msubsup><mi>ω</mi><mo>-</mo><mi>U</mi></msubsup><mo>⩾</mo><mo>-</mo><mn>6.6667</mn></math> ,

3x<subs>1</subs> + 3x<subs>2</subs> ⩽ 14, 3x<subs>1</subs> - 5x<subs>2</subs> + 6x<subs>3</subs> ⩽ 11, 5x<subs>1</subs> + 3x<subs>2</subs> ⩾ 8,

x <subs>i</subs> ⩾ 0, (i = 1, 2, 3),

t <subs>i</subs> > 0, (i = 1, 2) ,

whose solution is

<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><mrow><munder><mi>x</mi><mo>-</mo></munder><mn>1</mn><mi>U</mi><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mn>2</mn><mi>U</mi><mo>,</mo><munder><mi>x</mi><mo>-</mo></munder><mn>3</mn><mi>U</mi></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>4.6667</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow><mo>,</mo><msubsup><mi>ω</mi><mrow><mo>-</mo></mrow><mi>U</mi></msubsup><mo>=</mo><mn>1</mn></math> .

subject to

x <subs>1</subs> + 2x<subs>2</subs> ⩽ 16,

x <subs>1</subs> - 2x<subs>2</subs> + 3x<subs>3</subs> ⩽ 14,

5x<subs>1</subs> + 3x<subs>2</subs> ⩾ 16,

x <subs>i</subs> ⩾ 0, (i = 1, 2, 3),

t <subs>i</subs> > 0, (i = 1, 2) ,

whose solution is

<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><mrow><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mn>1</mn><mi>U</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mn>2</mn><mi>U</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>x</mi><mo>¯</mo></mover><mn>3</mn><mi>U</mi></msubsup></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>8</mn><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow><mo>,</mo><msup><mover accent="true"><mi>ω</mi><mo>¯</mo></mover><mi>U</mi></msup><mo>=</mo><mn>1</mn></math> .

Finally, the optimal interval values of objective functions are given in Table 4.

Table 4 The optimal interval values of objective functions based on fuzzy approach

<table><colgroup><col align="left" /><col align="center" /><col align="center" /></colgroup><thead valign="top"><tr><td /><td>The <bold>SO-</bold>range</td><td>The <bold>PO-</bold>range</td></tr></thead><tbody><tr><td>1<sup>st</sup><bold>L</bold><bold>DM</bold></td><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>[</mo><mrow><munder><mi>f</mi><mo>-</mo></munder><mn>1</mn><mi mathvariant="italic">LF</mi><mo>,</mo><munder><mi>f</mi><mo>-</mo></munder><mn>1</mn><mi mathvariant="italic">UF</mi></mrow><mo>]</mo></mrow><mo xmlns="">=</mo><mrow xmlns=""><mo>[</mo><mrow><mn>4</mn><mo>,</mo><mn>19.0001</mn></mrow><mo>]</mo></mrow></math></p></td><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>[</mo><mrow><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mn>1</mn><mi mathvariant="italic">LF</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mn>1</mn><mi mathvariant="italic">UF</mi></msubsup></mrow><mo>]</mo></mrow><mo xmlns="">=</mo><mrow xmlns=""><mo>[</mo><mrow><mn>2.6667</mn><mo>,</mo><mn>55</mn></mrow><mo>]</mo></mrow></math></p></td></tr><tr><td>2<sup>nd</sup><bold>L</bold><bold>DM</bold></td><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>[</mo><mrow><munder><mi>f</mi><mo>-</mo></munder><mn>2</mn><mi mathvariant="italic">LS</mi><mo>,</mo><munder><mi>f</mi><mo>-</mo></munder><mn>2</mn><mi mathvariant="italic">US</mi></mrow><mo>]</mo></mrow><mo xmlns="">=</mo><mrow xmlns=""><mo>[</mo><mrow><mn>15</mn><mo>,</mo><mn>39.6667</mn></mrow><mo>]</mo></mrow></math></p></td><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>[</mo><mrow><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mn>2</mn><mi mathvariant="italic">LS</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mn>2</mn><mi mathvariant="italic">US</mi></msubsup></mrow><mo>]</mo></mrow><mo xmlns="">=</mo><mrow xmlns=""><mo>[</mo><mrow><mn>7.0001</mn><mo>,</mo><mn>74</mn></mrow><mo>]</mo></mrow></math></p></td></tr><tr><td>3<sup>rd</sup><bold>L</bold><bold>DM</bold></td><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>[</mo><mrow><munder><mi>f</mi><mo>-</mo></munder><mn>3</mn><mi mathvariant="italic">LT</mi><mo>,</mo><munder><mi>f</mi><mo>-</mo></munder><mn>3</mn><mi mathvariant="italic">UT</mi></mrow><mo>]</mo></mrow><mo xmlns="">=</mo><mrow xmlns=""><mo>[</mo><mrow><mn>11</mn><mo>,</mo><mn>29.3335</mn></mrow><mo>]</mo></mrow></math></p></td><td><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>[</mo><mrow><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mn>3</mn><mi mathvariant="italic">LT</mi></msubsup><mo>,</mo><msubsup><mover accent="true"><mi>f</mi><mo>¯</mo></mover><mn>3</mn><mi mathvariant="italic">UT</mi></msubsup></mrow><mo>]</mo></mrow><mo xmlns="">=</mo><mrow xmlns=""><mo>[</mo><mrow><mn>6.0001</mn><mo>,</mo><mn>56</mn></mrow><mo>]</mo></mrow></math></p></td></tr></tbody></table>

8 Comparison

The results in Tables 2 and 3 show that the constraint method and the interactive model produced identical solutions to the MLFRILP problem and were superior to the outcomes in Table 4 provided by the fuzzy approach.

Conclusion

This paper has provided an effective overview of current rough multi-level decision-making processes and combinations of improvements associated with procedures in seven major categories, Moreover, the multi-level linear full-interval programming (MLFRILP) problem has been solved. Using the interval method and the slice sum method, the problem was first transformed into its crisp equivalent. Furthermore, were used three methods to solve the MLFRILP problem. Firstly, it was solved by using the constraint method, secondly by using an interactive model, and thirdly by using the fuzzy approach. To check the validity of the methods, one numerical example was provided. The obtained results clarify that the proposed techniques create a powerful tool that is required for solving the multi-level fully rough interval linear programming (MLFRILP) problem. By taking roughness into account in the model's formulation, the work will be able to help solve real-life and industrial problems that are typically complicated, uncertain, and constantly subject to change. For future research, one possible direction is to apply the proposed approaches to deal with real-world decision-making situations, such as a multi-level logistics planning problem in fully rough environments. Finally, a comparison was made between the approaches used in the three methods to find out the best way to solve MLFRILP, and the results showed that the proposed approaches for the constraint method and the interactive model were better than the results of the proposed method from the fuzzy approach.

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By E. Fathy; E. Ammar and M.A. Helmy

Reported by Author; Author; Author

Result: Three different optimization techniques for solving the fully rough interval multi-level linear programming problem

Further Information

Three different optimization techniques for solving the fully rough interval multi-level linear programming problem

1 Introduction

2 A survey of research on problems associated with rough multi-level programming problems in...

3 Basic preliminaries

4 Problem formulation and solution concept

5 The transformation of MLFRILP problem into a crisp model

5.1 The transformation of RI coefficient

5.2 The transformation of RI variables

6 The optimization approaches on MLFRILP Problem

6.1 The constraint method for MLFRILP problem

6.1.1 A flowchart for solving MLFRIL

6.2 An interactive model for MLFRILP problem

6.2.1 A flowchart for solving MLFRILP

6.3 Optimization method of the fuzzy approach

6.3.1 A flowchart for solving MLFRILP

7 Numerical example

8 Comparison

Conclusion

References

Links

Additional functions