A Study on Fully Interval Valued Intuitionistic Fuzzy Multilevel Quadratic Programming Problem

Abstract

Uncertainty is a parameter in optimization problems encountered in real-world scenarios. Identifying a particular category of models for multilevel linear programming under uncertainty is challenging. The emphasis of this research is the examination of uncertainty in multilevel quadratic programming problems. The primary aim of this initiative is to address the FIVIFMQP problem, which denotes completely interval-valued intuitionistic fuzzy multilevel quadratic programming. To convert the specified quadratic objective function into an equivalent linear objective, initially employ a linearization technique on the objective function. The anticipated value function is utilized for all elements of the objective function and constraints to derive an equivalent crisp model. The crisp model is subsequently resolved utilizing an augmented fuzzy methodology grounded in the anticipated value function. Resolving the multilevel programming problem can yield a compromise solution to the FIVIFMQP challenge. A numerical example is included to enhance comprehension of the solution approach of the proposed method.

Keywords

interval valued intuitionistic fuzzy numbers quadratic programming multilevel programming expected value function

1. Introduction

Decision-making problems in real-world applications such as engineering, economics, and management are often characterized by uncertainty, hierarchical structures, and nonlinear relationships. Traditional optimization models, particularly classical linear and quadratic programming approaches, typically assume precise data and deterministic environments. However, such assumptions are often unrealistic, as real-world data frequently involve vagueness, imprecision, and incomplete information.

To address uncertainty, various fuzzy-based approaches have been developed. In particular, intuitionistic fuzzy sets extend classical fuzzy sets by incorporating both membership and non-membership degrees, allowing for a more flexible representation of uncertainty. Furthermore, interval-valued intuitionistic fuzzy (IVIF) sets provide an additional level of generality by representing these degrees as intervals, thereby capturing hesitation and ambiguity more effectively.

In parallel, multilevel programming has emerged as an important tool for modeling hierarchical decision-making problems involving multiple decision makers with potentially conflicting objectives. These models are widely used in areas such as supply chain management, transportation systems, and economic planning. On the other hand, quadratic programming plays a crucial role in modeling nonlinear relationships, particularly in problems involving cost, risk, and resource allocation.

Despite these developments, existing studies often address only limited aspects of the problem. Many works consider fuzzy or interval uncertainty without incorporating intuitionistic characteristics, while others focus on multilevel programming structures but assume linear objective functions. Additionally, only a limited number of studies investigate quadratic programming under uncertainty, and these approaches typically do not fully utilize interval-valued intuitionistic fuzzy representations.

To the best of our knowledge, existing approaches do not provide a unified framework that simultaneously integrates interval-valued intuitionistic fuzzy representation, multilevel hierarchical decision structures, and quadratic objective functions. This limitation restricts their ability to model complex real-world problems involving uncertainty, hierarchy, and nonlinearity within a single framework.

Although real-world problems often involve hybrid uncertainty combining probabilistic and fuzzy characteristics, the present study focuses specifically on interval-valued intuitionistic fuzzy uncertainty. This choice is motivated by the ability of IVIF sets to capture multiple dimensions of uncertainty, including membership, non-membership, and hesitation, within a unified and flexible framework. Moreover, IVIF models do not require probabilistic assumptions, which may not always be available in practice. While hybrid uncertainty models may provide a more comprehensive representation, their integration within a multilevel quadratic programming framework would significantly increase model complexity and computational burden. Therefore, this study aims to strike a balance between expressive uncertainty modeling and computational tractability. Extending the proposed approach to hybrid uncertainty environments remains an important direction for future research.

To address the above research gap, this paper proposes a Fully Interval-Valued Intuitionistic Fuzzy Multilevel Quadratic Programming (FIVIFMQP) model. The main contributions of this work can be summarized as follows:

A novel formulation of a multilevel quadratic programming problem under an interval-valued intuitionistic fuzzy environment.

A unified solution methodology combining linearization and expected value transformation to obtain a tractable crisp equivalent model.

An integrated framework that simultaneously handles uncertainty, hierarchical decision-making, and nonlinear objective functions.

A numerical example demonstrating the applicability and effectiveness of the proposed approach.

The remainder of the paper is organized as follows. Section 2 presents the basic concepts and definitions related to interval-valued intuitionistic fuzzy sets. Section 3 formulates the proposed model. Section 4 describes the solution methodology. Section 5 provides a numerical example to illustrate the proposed approach. Section 6 discusses the results and implications. Finally, Section 7 concludes the paper and outlines directions for future research.

2. Literature Review

In intuitionistic fuzzy set theory, the membership degrees of acceptance and rejection are assessed, representing an extension of classical fuzzy set theory (Atanassov, 2016). An assignment from X to closed sub-intervals of the real interval [0, 1] constitutes an interval-valued intuitionistic fuzzy (IVIF) set on the universe X (Wan & Dong, 2020). This variant of fuzzy set has been thoroughly examined, both regarding its theoretical attributes and its numerous applications (Jebadass & Balasubramaniam, 2024; Alolaiyan et al., 2024; Zhou et al., 2024; Mandal et al., 2024; Bilişik et al., 2024; Alghazzawi et al., 2024). Dong and Wan (2024) proposed a novel IVIF Best-worst technique characterized by additive consistency. In an IVIF context, Shivani and Rani (2024) examined the multi-objective, multi-item, four-dimensional green transportation problem. Malik and Gupta (2024) employed the Hamming distance concept to analyze division and subtraction operations within any set or value of IVIF. Acar et al. (2024) analyzed and evaluated four hydrogen storage solutions via the IVIF analytic hierarchy process. Singer and Özşahin (2024) introduced a framework for decision-making utilizing the IVIF analytic hierarchy approach to discover and assess the significant elements influencing customers’ decisions to purchase non-wood forest products.

When a decision-maker (DM) is assigned the responsibility of optimizing one or more objective functions at each tier of the decision-making process within a hierarchical organization, it results in a challenge referred to as multi-level (ML) programming challenges. Moreover, each decision maker separately regulates a set of decision criteria (Anandalingam, 1988; Zhang et al., 2015). In the stochastic machine learning lot sizing problem with a service level, and in a generic context where independent demand for components is conceivable, Goshu and Kassa (2024) proposed a systematic approach to evaluate the value of enhancing flexibility. Qin et al. (2024) proposed a matheuristic approach to tackle the ML capacitated lot-sizing problem including backorder and substitution. Tong et al. (2024) introduced a comprehensive machine learning complementary scheduling methodology that facilitates dynamic scheduling in shared manufacturing by employing a digital twin model grounded in information freshness to detect disturbances. Due to the detrimental environmental impact of fossil fuel-powered generation units, Fatemi et al. (2023) developed a stochastic machine learning multi-objective approach to facilitate energy market clearing across micro-grids. Yao et al. (2023) created a machine learning optimization model for the real-time operation of integrated energy systems encompassing carbon capture, energy storage, and renewable energy sources.

An effective method for utilizing limited resources is through a mathematical model known as quadratic programming (QP). This has led to the emergence of numerous practical outcomes and intriguing applications (Mahajan et al., 2024; Du et al., 2024; Shen et al., 2024; Nguyen et al., 2024). Kushwah and Sharma (2024) discussed an approach that identifies all viable methods for solving a multi-objective integer quadratic programming issue. Cuong et al. (2024) defined certain characteristics of the Proximal Difference-of-convex functions decomposition method within the context of linear constraints in indefinite quadratic programming. Mangalore et al. (2024) demonstrated a solution for addressing convex continuous optimization problems with linear constraints and quadratic cost functions using the scalable neuromorphic research device, Intel Loihi 2. Cifuentes et al. (2024) examine sensitivity analysis for Mixed Binary Quadratic Programming for diverse right-hand sides. Boudjellal and Benterki (2024) proposed and analyzed an innovative full-Newton step feasible interior point method for convex quadratic programming. Many studies utilized fuzzy approaches, neutrosophic approaches and rough approaches to programming problems, whereas some researchers adopted an Interval Valued intuitionistic fuzzy approaches that merely used a linear membership function.

However, as Table 1 demonstrates, this work provides a highly detailed and targeted literature review on the uncertainty, and certainty in multilevel quadratic programming (MQP) problems in different uncertainty environments. Able classifies previous research precisely based on the nature of uncertainty, the method of handling uncertainty, and whether the uncertainty is in parameters or variables whether it is, multilevel and Quadratic.

Table 1.
Research Studies Performed by Multiple Authors on MQP in various Uncertainty Environments.

Nature of Uncertainty

Author Year Type Coefficients Variables Quadratic Multi-Level Solution Method

Sakawa (1983) 1983 Fuzzy Environment ✓ × × × Interactive fuzzy programming

Li and Lee (1991) 1991 Fuzzy Environment ✓ × × × Exponential membership function

Hulsurkar et al. (1997) 1997 Fuzzy Environment ✓ × × × Fuzzy stochastic programming

Jiménez and Verdegay (1999) 1999 Interval Environment ✓ × × × Interval programming

Bit (2004) 2004 Fuzzy Environment ✓ × × × Hyperbolic membership functions

Iskander (2006) 2006 Stochastic fuzzy Environment ✓ × × × Stochastic fuzzy goal programming

Yang and Feng (2007) 2007 Stochastic Environment ✓ × × × Stochastic programming

Ojha et al. (2010) 2010 Stochastic Environment ✓ × × × Stochastic programming with AHP

Pramanik et al. (2013) 2013 Random Fuzzy Environment ✓ × × × Generalized reduced gradient method

Giri et al. (2014) 2014 Fuzzy Stochastic Environment ✓ × × × Fuzzy goal programming

Pramanik et al. (2016) 2016 Fuzzy Rough Environment ✓ × × × An interactive fuzzy method

Sinha et al. (2016) 2016 Trapezoidal Interval Type-2 Fuzzy Environment ✓ × × × the expected value

Dalman and Sivri (2017) 2017 Interval Environment ✓ × × × Fuzzy programming,

Kumar (2018) 2018 Intuitionistic Fuzzy Environment ✓ ✓ × × PSK method

Das et al. (2019) 2019 Type-2 Fuzzy Environment ✓ × × × Weighted mean programming technique

Singh et al. (2019) 2019 Stochastic Environment ✓ × × × Chance-constraint programming

Nouriani and Ezzati (2020) 2020 Fuzzy Environment ✓ × × × Simpson quadrature rule

Anithakumari et al. (2021) 2021 Rough Interval Integers Environment ✓ ✓ × × Modified slice-sum method

Fathy et al. (2022) 2022 Rough Interval Environment ✓ × × ✓ Interval method

Choudhary and Yadav (2022) 2022 Interval Valued Intuitionistic Fuzzy Environment ✓ × × × An algorithm is developed by author

Tamilarasi and Paulraj (2022) 2022 Neutrosophic Environmen ✓ × × × Mellin transform based de-neutrosophication

Fathy and Ammar (2023) 2023 Neutrosophic environments ✓ × × ✓ Harmonic mean technique

Fathy et al. (2023a) 2023 Intuitionistic Fuzzy Environment ✓ ✓ × ✓ A simple approach based on fuzzy harmonic mean operators

Kousar et al. (2023) 2023 Neutrosophic Fuzzy Environment ✓ × × × neutrosophic programming

Fathy et al. (2023b) 2023 Rough Interval Environment ✓ ✓ × ✓ Modified slice-sum method

This Study IVIF Environment ✓ ✓ ✓ ✓ The expected value

		Nature of Uncertainty
Sakawa (1983)	1983	Fuzzy Environment	✓	×	×	×	Interactive fuzzy programming
Li and Lee (1991)	1991	Fuzzy Environment	✓	×	×	×	Exponential membership function
Hulsurkar et al. (1997)	1997	Fuzzy Environment	✓	×	×	×	Fuzzy stochastic programming
Jiménez and Verdegay (1999)	1999	Interval Environment	✓	×	×	×	Interval programming
Bit (2004)	2004	Fuzzy Environment	✓	×	×	×	Hyperbolic membership functions
Iskander (2006)	2006	Stochastic fuzzy Environment	✓	×	×	×	Stochastic fuzzy goal programming
Yang and Feng (2007)	2007	Stochastic Environment	✓	×	×	×	Stochastic programming
Ojha et al. (2010)	2010	Stochastic Environment	✓	×	×	×	Stochastic programming with AHP
Pramanik et al. (2013)	2013	Random Fuzzy Environment	✓	×	×	×	Generalized reduced gradient method
Giri et al. (2014)	2014	Fuzzy Stochastic Environment	✓	×	×	×	Fuzzy goal programming
Pramanik et al. (2016)	2016	Fuzzy Rough Environment	✓	×	×	×	An interactive fuzzy method
Sinha et al. (2016)	2016	Trapezoidal Interval Type-2 Fuzzy Environment	✓	×	×	×	the expected value
Dalman and Sivri (2017)	2017	Interval Environment	✓	×	×	×	Fuzzy programming,
Kumar (2018)	2018	Intuitionistic Fuzzy Environment	✓	✓	×	×	PSK method
Das et al. (2019)	2019	Type-2 Fuzzy Environment	✓	×	×	×	Weighted mean programming technique
Singh et al. (2019)	2019	Stochastic Environment	✓	×	×	×	Chance-constraint programming
Nouriani and Ezzati (2020)	2020	Fuzzy Environment	✓	×	×	×	Simpson quadrature rule
Anithakumari et al. (2021)	2021	Rough Interval Integers Environment	✓	✓	×	×	Modified slice-sum method
Fathy et al. (2022)	2022	Rough Interval Environment	✓	×	×	✓	Interval method
Choudhary and Yadav (2022)	2022	Interval Valued Intuitionistic Fuzzy Environment	✓	×	×	×	An algorithm is developed by author
Tamilarasi and Paulraj (2022)	2022	Neutrosophic Environmen	✓	×	×	×	Mellin transform based de-neutrosophication
Fathy and Ammar (2023)	2023	Neutrosophic environments	✓	×	×	✓	Harmonic mean technique
Fathy et al. (2023a)	2023	Intuitionistic Fuzzy Environment	✓	✓	×	✓	A simple approach based on fuzzy harmonic mean operators
Kousar et al. (2023)	2023	Neutrosophic Fuzzy Environment	✓	×	×	×	neutrosophic programming
Fathy et al. (2023b)	2023	Rough Interval Environment	✓	✓	×	✓	Modified slice-sum method
This Study		IVIF Environment	✓	✓	✓	✓	The expected value

Table 1 presents a structured comparison of existing studies based on key characteristics, including the type of uncertainty representation, problem structure, and objective function. It can be observed that most existing works focus on limited aspects of the problem. In particular, a significant portion of the literature addresses fuzzy or interval uncertainty without incorporating intuitionistic features, thereby neglecting the representation of hesitation. Moreover, several studies consider multilevel programming structures but are restricted to linear objective functions. Although a few contributions have explored intuitionistic or interval-valued fuzzy models, they do not simultaneously account for multilevel decision-making and quadratic objectives. As a result, these approaches are unable to fully capture the complexity of real-world problems involving hierarchical interactions, nonlinear relationships, and rich uncertainty structures.

Based on the comparison in Table 1, it is evident that there is a lack of models that simultaneously integrate:

Interval-valued intuitionistic fuzzy representation,

Multilevel hierarchical structure, and

Quadratic objective functions

within a unified optimization framework.

This limitation restricts the ability of existing approaches to effectively model complex real-world problems that involve uncertainty, hierarchy, and nonlinear relationships. To address the identified research gap, the present study proposes a Fully Interval-Valued Intuitionistic Fuzzy Multilevel Quadratic Programming (FIVIFMQP) model. Unlike existing approaches that treat these aspects separately, the proposed model integrates them within a unified framework. This integration enables a more comprehensive representation of decision-making problems by simultaneously capturing uncertainty through IVIF sets, hierarchical interactions through multilevel programming, and nonlinear relationships through quadratic objective functions.

Therefore, the contribution of this work lies not in introducing entirely new individual components, but in effectively combining multiple advanced features into a coherent and practically applicable optimization framework.

Section 2 of this study delineates the definitions of IVIF sets, their operations, and associated relations. Section 3 introduces the FIVIMQP problem. Section 4 presents the technical solution for the topic under consideration. Section 5 presents a practical example to illustrate the applicability of the proposed solution methodology. Section 6 provides a discourse. The final segment closes the work.

3. Fundamental Concepts

This section defines IVIF sets, IVIF numbers, their operations, and related relationships.

3.1 IVIF Sets, IVIF Numbers, Their Operations

Definition 1
(Atanassov, 1999) An IVIF set ${\tilde{A}}^{I}$ in X is defined as a collection of ordered triples ${\tilde{A}}^{I} = {(x, μ_{{\tilde{A}}^{I}} (x), v_{{\tilde{A}}^{I}} (x)) : x \in X}$ , where $μ_{{\tilde{A}}^{I}} : X \to [0, 1]$ and $v_{{\tilde{A}}^{I}} : X \to [0, 1]$ denote the degrees of membership and non-membership of $x \in X$ , respectively, with the constraints that $\forall x \in X$ , $0 \leq μ_{{\tilde{A}}^{I}} (x), + v_{{\tilde{A}}^{I}} (x) \leq 1$ .
Definition 2
(Atanassov, 1999) An interval-valued triangular intuitionistic fuzzy number (IVTIFN) is represented by: ${\tilde{a}}^{I} = {\begin{matrix} (a_{1}^{U}, a_{1}^{L}, a_{2}, a_{3}^{L}, a_{3}^{U}), \\ (a_{1}^{' L}, a_{1}^{' U}, a_{2}, a_{3}^{' U}, a_{3}^{' L}) \end{matrix}}$ , $a_{1}^{' L} \leq a_{1}^{' U} \leq a_{1}^{U} \leq a_{1}^{L} \leq a_{2} \leq a_{3}^{L} \leq a_{3}^{U} \leq a_{3}^{' U} \leq a_{3}^{' L}$ . The MS and NMS degrees are delineated as follows:
$\begin{matrix} Membership Degrees \\ Lower MS functions & Upper MS functions \\ μ_{{\tilde{a}}^{I}}^{L} (x) = {\begin{matrix} 1, i f x = a_{2} \\ \frac{x - a_{1}^{L}}{a_{2} - a_{1}^{L}}, i f a_{1}^{L} < x < a_{2} \\ \frac{x - a_{1}^{L}}{a_{2} - a_{1}^{L}}, i f a_{2} < x < a_{3}^{L} \\ 0, o t h e r w i s e \end{matrix} & μ_{{\tilde{a}}^{I}}^{U} (x) = {\begin{matrix} 1, i f x = a_{2} \\ \frac{x - a_{1}^{U}}{a_{2} - a_{1}^{U}}, i f a_{1}^{U} < x < a_{2} \\ \frac{a_{3}^{U} - x}{a_{3}^{U} - a_{2}}, i f a_{2} < x < a_{3}^{U} \\ 0, o t h e r w i s e \end{matrix} \\ Non - membership Degrees \\ Lower NMS functions & Upper NMS functions \\ v_{{\tilde{a}}^{I}}^{L} (x) = {\begin{matrix} 1, i f x = a_{2} \\ \frac{a_{2} - x}{a_{2} - a_{1}^{' L}}, i f a_{1}^{' L} < x < a_{2} \\ \frac{a_{2} - x}{a_{2} - a_{3}^{' L}}, i f a_{2} < x < a_{3}^{' L} \\ 0, o t h e r w i s e \end{matrix} & v_{{\tilde{a}}^{I}}^{U} (x) = {\begin{matrix} 1, i f x = a_{2} \\ \frac{x - a_{2}}{a_{1}^{' U} - a_{2}}, i f a_{1}^{' U} < x < a_{2} \\ \frac{x - a_{2}}{a_{3}^{' U} - a_{2}}, i f a_{2} < x < a_{3}^{' U} \\ 0, o t h e r w i s e \end{matrix} \end{matrix}$

Definition 3
(Atanassov, 1999) Arithmetic operations can be performed between two IVTIFNs ${\tilde{a}}^{I} = {\begin{matrix} (a_{1}^{U}, a_{1}^{L}, a_{2}, a_{3}^{L}, a_{3}^{U}), \\ (a_{1}^{' L}, a_{1}^{' U}, a_{2}, a_{3}^{' U}, a_{3}^{' L}) \end{matrix}}$ and ${\tilde{c}}^{I} = {\begin{matrix} (c_{1}^{U}, c_{1}^{L}, c_{2}, c_{3}^{L}, c_{3}^{U}), \\ (c_{1}^{' L}, c_{1}^{' U}, c_{2}, c_{3}^{' U}, c_{3}^{' L}) \end{matrix}}$ as follows: (a)
${\tilde{a}}^{I} \oplus {\tilde{c}}^{I} = {\begin{matrix} (a_{1}^{U} + c_{1}^{U}, a_{1}^{L} + c_{1}^{L}, a_{2} + c_{2}, a_{3}^{L} + c_{3}^{L}, a_{3}^{U} + c_{3}^{U}), \\ (a_{1}^{' L} + c_{1}^{' L}, a_{1}^{' U} + c_{1}^{' U}, a_{2} + c_{2}, a_{3}^{' U} + c_{3}^{' U}, a_{3}^{' L} + c_{3}^{' L}) \end{matrix}}$ .
(b)
$k {\tilde{a}}^{I} = {\begin{matrix} {\begin{matrix} (k a_{1}^{U}, k a_{1}^{L}, k a_{2}, k a_{3}^{L}, k a_{3}^{U}), \\ (k a_{1}^{' L}, k a_{1}^{' U}, k a_{2}, k a_{3}^{' U}, k a_{3}^{' L}) \end{matrix}}, & i f k \geq 0 \\ {\begin{matrix} (k a_{3}^{U}, k a_{3}^{L}, k a_{2}, k a_{1}^{L}, k a_{1}^{U}), \\ (k a_{3}^{' L}, k a_{3}^{' U}, k a_{2}, k a_{1}^{' U}, k a_{1}^{' L}) \end{matrix}}, & i f k < 0 \end{matrix}$
(c)
${\tilde{a}}^{I} ⊖ {\tilde{c}}^{I} = {\begin{matrix} (a_{1}^{U} - c_{3}^{U}, a_{1}^{L} - c_{3}^{L}, a_{2} - c_{2}, a_{3}^{L} - c_{1}^{L}, a_{3}^{U} - c_{1}^{U}), \\ (a_{1}^{' L} - c_{3}^{' L}, a_{1}^{' U} - c_{3}^{' U}, a_{2} - c_{2}, a_{3}^{' U} - c_{1}^{' U}, a_{3}^{' L} - c_{1}^{' L}) \end{matrix}}$
(d)
${\tilde{a}}^{I} \otimes {\tilde{c}}^{I} = {\begin{matrix} (a_{1}^{U} c_{1}^{U}, a_{1}^{L} c_{1}^{L}, a_{2} c_{2}, a_{3}^{L} c_{3}^{L}, a_{3}^{U} c_{3}^{U}), \\ (a_{1}^{' L} c_{1}^{' L}, a_{1}^{' U} c_{1}^{' U}, a_{2} c_{2}, a_{3}^{' U} c_{3}^{' U}, a_{3}^{' L} c_{3}^{' L}) \end{matrix}}, i f a_{1}^{' L}, c_{1}^{' L} \geq 0$

Definition 4
(Atanassov, 1999) Two IVTIFNs ${\tilde{a}}^{I} = {\begin{matrix} (a_{1}^{U}, a_{1}^{L}, a_{2}, a_{3}^{L}, a_{3}^{U}), \\ (a_{1}^{' L}, a_{1}^{' U}, a_{2}, a_{3}^{' U}, a_{3}^{' L}) \end{matrix}}$ and ${\tilde{c}}^{I} = {\begin{matrix} (c_{1}^{U}, c_{1}^{L}, c_{2}, c_{3}^{L}, c_{3}^{U}), \\ (c_{1}^{' L}, c_{1}^{' U}, c_{2}, c_{3}^{' U}, c_{3}^{' L}) \end{matrix}}$ are deemed equal, ${\tilde{a}}^{I} ≃ {\tilde{c}}^{I}$ , if and only if $a_{1}^{U} = c_{1}^{U}$ , $a_{1}^{L} = c_{1}^{L}$ , $a_{3}^{L} = c_{3}^{L}$ , $a_{3}^{U} = c_{3}^{U}$ , $a_{1}^{' L} = c_{1}^{' L}$ , $a_{1}^{' U} = c_{1}^{' U}$ , $a_{3}^{' U} = c_{3}^{' U}$ , and $a_{3}^{' L} = c_{3}^{' L}$ .
3.2 Expected Value of IVIF Numbers

To facilitate computation, an expected value function is used to convert IVIF numbers into crisp values.

Definition 5
(Atanassov, 1999) Let ${\tilde{a}}^{I} = {\begin{matrix} (a_{1}^{U}, a_{1}^{L}, a_{2}, a_{3}^{L}, a_{3}^{U}), \\ (a_{1}^{' L}, a_{1}^{' U}, a_{2}, a_{3}^{' U}, a_{3}^{' L}) \end{matrix}}$ be an IVTIFN, the subsequent formula delineates its expected value:
$E V ({\tilde{a}}^{I}) = \frac{a_{1}^{U} + a_{1}^{L} + a_{1}^{' L} + a_{1}^{' U} + 8 a_{2} + a_{3}^{L} + a_{3}^{U} + a_{3}^{' U} + a_{3}^{' L}}{16}$

Definition 6
(Atanassov, 1999) Let $\tilde{A}$ and $\tilde{B}$ be two IVTIFNs. Then,
$\begin{aligned} {\tilde{a}}^{I} & \leq {\tilde{c}}^{I} \Leftrightarrow E V ({\tilde{a}}^{I}) \leq E V ({\tilde{c}}^{I}) \end{aligned}$

$\begin{aligned} {\tilde{a}}^{I} & \geq {\tilde{c}}^{I} \Leftrightarrow E V ({\tilde{a}}^{I}) \geq E V ({\tilde{c}}^{I}) \end{aligned}$

This function provides a representative crisp value by balancing the effects of membership and non-membership degrees.

It is important to note that the expected value does not eliminate uncertainty arbitrarily; rather, it aggregates the interval information into a single value while implicitly preserving the influence of hesitation and variability.
4. Formulation of the FIVIFMQP Problem

In this section, we present a well-structured formulation of the fully interval-valued intuitionistic fuzzy multilevel quadratic programming (FIVIFMQP) problem.

4.1 Notation and Preliminaries

Consider a multilevel decision-making system consisting of p hierarchical decision makers (DMs). Each decision maker $D M_{i}$ , $i = 1, 2, \dots, p$ , controls a decision vector ${\tilde{x}}_{i}^{I} \in R^{n}$ where all variables are expressed as interval-valued intuitionistic fuzzy (IVIF) numbers.

Let:

${\tilde{x}}^{I} = (x_{1}^{I}, x_{2}^{I}, \dots, x_{p}^{I})$ : Vector of IVIF decision variable s

${\tilde{c}}_{i}^{I}$ : IVIF coefficient vector of linear terms

${\tilde{D}}_{i}^{I}$ : IVIF matrix of quadratic coefficient s

${\tilde{a}}^{I}$ : IVIF constraint matrix

${\tilde{b}}^{I}$ : IVIF right-hand-side vecto r

All parameters and variables are represented using IVIF numbers as defined in Section 2.

4.2 Mathematical Model

At each level i, the decision maker $D M_{i}$ solves the following optimization problem:

[D M_{i}] : max_{{\tilde{x}}_{i}^{I}} {\tilde{F}}_{i}^{I} ({\tilde{x}}^{I}) = \underset{{\tilde{x}}_{1}^{I}}{m a x} (\sum_{j = 1}^{n} {\tilde{c}}_{i j}^{I} \otimes {\tilde{x}}_{j}^{I} \oplus \frac{1}{2} \sum_{k = 1}^{n} \sum_{j = 1}^{n} {\tilde{x}}_{k}^{I} \otimes {\tilde{D}}_{i k j}^{I} \otimes {\tilde{x}}_{j}^{I})

where ${\tilde{x}}_{i + 1}^{I}, {\tilde{x}}_{i + 2}^{I}, \dots, {\tilde{x}}_{p}^{I}$ solve.

Subject to:

{\tilde{x}}^{I} \in {\tilde{G}}^{I} = {{\tilde{x}}^{I} \in R^{n} | \begin{matrix} \sum_{j = 1}^{n} {\tilde{a}}_{s j}^{I} \otimes {\tilde{x}}_{j}^{I} \leq {\tilde{b}}_{s}^{I}, s = 1, 2, \dots, m \\ {\tilde{x}}_{j}^{I} \geq 0 \end{matrix}}

(1)

$\otimes$ denotes IVIF multiplication

$\oplus$ denotes IVIF addition

The hierarchical nature of the problem implies that:

The decision of $D M_{i}$ depends on the decisions of lower-level decision makers $D M_{i + 1}, \dots, D M_{p}$

Each level solves its own objective function while anticipating the reactions of subsequent levels

Thus, the problem follows a leader–follower structure across multiple levels.

4.3 Remarks on IVIF Representation

All parameters and variables in the model are expressed as interval-valued intuitionistic fuzzy numbers (IVIFNs), which allow:

Simultaneous modeling of membership and non-membership degrees

Representation of uncertainty using interval data

Inclusion of hesitation margins

This formulation enhances the ability of the model to represent real-world uncertainty more effectively than classical crisp or fuzzy approaches.

5. Methodology

The FIVIFMQP issue is addressed in three phases. Phase I involves converting the basic issue into a corresponding linear programming (FIVIFMLP) problem. Phase II tackles and transforms the FIVIFMLP issue into a definitive model. Phase III entails employing a linear MS function within the IVIF framework to adjust the fuzzy programming methodology for the optimization of the precise linear programming issue.

Let IVTIFNs, each possessing the subsequent values, represent all IVIF parameters and variables for the objective functions and constraints in equation (1). Then an FIVIFMQP problem can be organized as follows at the $D M_{i}$ :

[D M_{i}] : max_{{\tilde{x}}_{i}^{I}} {\tilde{F}}_{i}^{I} ({\tilde{x}}^{I}) = max_{{\tilde{x}}_{i}^{I}} ({(F_{1 i}^{U}, F_{1 i}^{L}, F_{2 i}, F_{3 i}^{L}, F_{3 i}^{U}), (F_{1 i}^{' L}, F_{1 i}^{' U}, F_{2 i}, F_{3 i}^{' U}, F_{3 i}^{' L})})

where ${\tilde{x}}_{i + 1}^{I}, {\tilde{x}}_{i + 2}^{I}, \dots, {\tilde{x}}_{p}^{I}$ solve.

subject to

{\tilde{G}}^{I} = {\begin{matrix} {\tilde{x}}^{I} \in R^{n} | \sum_{j = 1}^{n} {\begin{matrix} (a_{1 s j}^{U} x_{1 j}^{U}, a_{1 s j}^{L} x_{1 j}^{L}, a_{2 s j} x_{2 j}, a_{3 s j}^{L} x_{3 j}^{L}, a_{3 s j}^{U} x_{3 j}^{U}), \\ (a_{1 s j}^{' L} x_{1 j}^{' L}, a_{1 s j}^{' U} x_{1 j}^{' U}, a_{2 s j} x_{2 j}, a_{3 s j}^{' U} x_{3 j}^{' U}, a_{3 s j}^{' L} x_{3 j}^{' L}) \end{matrix}} \leq \\ {(b_{1 s}^{U}, b_{1 s}^{L}, b_{2 s}, b_{3 s}^{L}, b_{3 s}^{U}), (b_{1 s}^{' L}, b_{1 s}^{' U}, b_{2 s}, b_{3 s}^{' U}, b_{3 s}^{' L})}, \\ x_{1 k}^{' L} \leq x_{1 k}^{' U} \leq x_{1 k}^{U} \leq x_{1 k}^{L} \leq x_{2 k} \leq x_{3 k}^{L} \leq x_{3 k}^{U} \leq x_{3 k}^{' U} \leq x_{3 k}^{' L}, \\ x_{1 j}^{' L} \geq 0. \end{matrix}}

(2)

Where

\begin{aligned} F_{1 i}^{U} & = \sum_{j = 1}^{n} c_{1 i j}^{U} x_{1 j}^{U} + \frac{1}{2} \sum_{k = 1}^{n} \sum_{j = 1}^{n} x_{1 k}^{U} d_{1 i k j}^{U} x_{1 j}^{U} \end{aligned}

\begin{aligned} F_{1 i}^{L} & = \sum_{j = 1}^{n} c_{1 i j}^{L} x_{1 j}^{L} + \frac{1}{2} \sum_{k = 1}^{n} \sum_{j = 1}^{n} x_{1 k}^{L} d_{1 i k j}^{L} x_{1 j}^{L} \end{aligned}

\begin{aligned} F_{2 i} & = \sum_{j = 1}^{n} c_{2 i j} x_{2 j} + \frac{1}{2} \sum_{k = 1}^{n} \sum_{j = 1}^{n} x_{2 j} d_{2 i k j} x_{2 j} \end{aligned}

\begin{aligned} F_{3 i}^{L} & = \sum_{j = 1}^{n} c_{3 i j}^{L} x_{3 j}^{L} + \frac{1}{2} \sum_{k = 1}^{n} \sum_{j = 1}^{n} x_{3 k}^{L} d_{3 i k j}^{L} x_{3 j}^{L} \end{aligned}

\begin{aligned} F_{3 i}^{U} & = \sum_{j = 1}^{n} c_{3 i j}^{U} x_{3 j}^{U} + \frac{1}{2} \sum_{k = 1}^{n} \sum_{j = 1}^{n} x_{3 k}^{U} d_{3 i k j}^{U} x_{3 j}^{U} \end{aligned}

\begin{aligned} F_{1 i}^{' L} & = \sum_{j = 1}^{n} c_{1 i j}^{' L} x_{1 j}^{' L} + \frac{1}{2} \sum_{k = 1}^{n} \sum_{j = 1}^{n} x_{1 k}^{' L} d_{1 i k j}^{' L} x_{1 j}^{' L} \end{aligned}

\begin{aligned} F_{1 i}^{' U} & = \sum_{j = 1}^{n} c_{1 i j}^{' U} x_{1 j}^{' U} + \frac{1}{2} \sum_{k = 1}^{n} \sum_{j = 1}^{n} x_{1 k}^{' U} d_{1 i k j}^{' U} x_{1 j}^{' U} \end{aligned}

\begin{aligned} F_{3 i}^{' U} & = \sum_{j = 1}^{n} c_{3 i j}^{' U} x_{3 j}^{' U} + \frac{1}{2} \sum_{k = 1}^{n} \sum_{j = 1}^{n} x_{3 k}^{' U} d_{3 i k j}^{' U} x_{3 j}^{' U} \end{aligned}

\begin{aligned} F_{3 i}^{' L} & = \sum_{j = 1}^{n} c_{3 i j}^{' L} x_{3 j}^{' L} + \frac{1}{2} \sum_{k = 1}^{n} \sum_{j = 1}^{n} x_{3 k}^{' L} d_{3 i k j}^{' L} x_{3 j}^{' L} \end{aligned}

5.1 Linearization Technique

A linearization technique for objective functions has been devised to address the challenges posed by the current quadratic objectives.

We utilize novel $n^{2} \times 1$ vectors of decision variables $y_{1 k j}^{U}$ , $y_{1 k j}^{L}$ , $y_{2 k j}$ , $y_{3 k j}^{L}$ , $y_{3 k j}^{U}$ , $y_{1 k j}^{' L}$ , $y_{1 k j}^{' U}$ , $y_{3 k j}^{' U}$ , and $y_{3 k j}^{' L}$ to linearize the quadratic term in the FIVIFMQP. Assume $y_{1 k j}^{U} = x_{1 k}^{U} x_{1 j}^{U}$ , $y_{1 k j}^{L} = x_{1 k}^{L} x_{1 j}^{L}$ , $y_{2 k j} = x_{2 k} x_{2 j}$ , $y_{3 k j}^{L} = x_{3 k}^{L} x_{3 j}^{L}$ , $y_{3 k j}^{U} = x_{3 k}^{U} x_{3 j}^{U}$ , $y_{1 k j}^{' L} = x_{1 k}^{' L} x_{1 j}^{' L}$ , $y_{1 k j}^{' U} = x_{1 k}^{' U} x_{1 j}^{' U}$ , $y_{3 k j}^{' U} = x_{3 k}^{' U} x_{3 j}^{' U}$ , and $y_{3 k j}^{' L} = x_{3 k}^{' L} x_{3 j}^{' L}$ .

Let $h_{1 i k j}^{U}$ , $h_{1 i k j}^{L}$ , $h_{2 i k j}$ , $h_{3 i k j}^{L}$ , $h_{3 i k j}^{U}$ , $h_{1 i k j}^{' L}$ , $h_{1 i k j}^{' U}$ , $h_{3 i k j}^{' U}$ , and $h_{3 i k j}^{' L}$ be $1 \times n^{2}$ constant vectors. In this instance: $h_{1 i k j}^{U} = \frac{1}{2} d_{1 i j k}^{U}$ , $h_{1 i k j}^{L} = \frac{1}{2} d_{1 i j k}^{L}$ , $h_{2 i k j} = \frac{1}{2} d_{2 i j k}$ , $h_{3 i k j}^{L} = \frac{1}{2} d_{3 i j k}^{L}$ , $h_{3 i k j}^{U} = \frac{1}{2} d_{3 i j k}^{L}$ , $h_{1 i k j}^{' L} = \frac{1}{2} d_{1 i j k}^{' L}$ , $h_{1 i k j}^{' U} = \frac{1}{2} d_{1 i j k}^{' U}$ , $h_{3 i k j}^{' U} = \frac{1}{2} d_{3 i j k}^{' U}$ , and $h_{3 i k j}^{' L} = \frac{1}{2} d_{3 i j k}^{' L}$ .

The FIVIFMLP issue with linear objective functions can thus be reformulated as follows at the $D M_{i}$ :

[D M_{i}] : max_{{\tilde{x}}_{i}^{I}} {\tilde{F}}_{i}^{I} ({\tilde{x}}^{I}, {\tilde{y}}^{I}) = max_{{\tilde{x}}_{i}^{I}} ({(z_{1 i}^{U}, z_{1 i}^{L}, z_{2 i}, z_{3 i}^{L}, z_{3 i}^{U}), (z_{1 i}^{' L}, z_{1 i}^{' U}, z_{2 i}, z_{3 i}^{' U}, z_{3 i}^{' L})})

where ${\tilde{x}}_{i + 1}^{I}, {\tilde{x}}_{i + 2}^{I}, \dots, {\tilde{x}}_{p}^{I}$ solve.

subject to

{\tilde{G}}^{I} = {\begin{matrix} {\tilde{x}}^{I} \in R^{n} | \sum_{j = 1}^{n} {\begin{matrix} (a_{1 s j}^{U} x_{1 j}^{U}, a_{1 s j}^{L} x_{1 j}^{L}, a_{2 s j} x_{2 j}, a_{3 s j}^{L} x_{3 j}^{L}, a_{3 s j}^{U} x_{3 j}^{U}), \\ (a_{1 s j}^{' L} x_{1 j}^{' L}, a_{1 s j}^{' U} x_{1 j}^{' U}, a_{2 s j} x_{2 j}, a_{3 s j}^{' U} x_{3 j}^{' U}, a_{3 s j}^{' L} x_{3 j}^{' L}) \end{matrix}} \leq \\ {(b_{1 s}^{U}, b_{1 s}^{L}, b_{2 s}, b_{3 s}^{L}, b_{3 s}^{U}), (b_{1 s}^{' L}, b_{1 s}^{' U}, b_{2 s}, b_{3 s}^{' U}, b_{3 s}^{' L})}, \\ x_{1 j}^{' L} \leq x_{1 j}^{' U} \leq x_{1 j}^{U} \leq x_{1 j}^{L} \leq x_{2 j} \leq x_{3 j}^{L} \leq x_{3 j}^{U} \leq x_{3 j}^{' U} \leq x_{3 j}^{' L}, \\ y_{1 k j}^{U} = x_{1 k}^{U} x_{1 j}^{U}, y_{1 k j}^{L} = x_{1 k}^{L} x_{1 j}^{L}, y_{2 k j} = x_{2 k} x_{2 j}, y_{3 k j}^{L} = x_{3 k}^{L} x_{3 j}^{L}, \\ y_{3 k j}^{U} = x_{3 k}^{U} x_{3 j}^{U}, y_{1 k j}^{' L} = x_{1 k}^{' L} x_{1 j}^{' L}, y_{1 k j}^{' U} = x_{1 k}^{' U} x_{1 j}^{' U}, y_{3 k j}^{' U} = x_{3 k}^{' U} x_{3 j}^{' U}, y_{3 k j}^{' L} = x_{3 k}^{' L} x_{3 j}^{' L}, \\ x_{1 j}^{' L} \geq 0. \end{matrix}}

(3)

where

\begin{aligned} z_{1 i}^{U} & = \sum_{j = 1}^{n} c_{1 i j}^{U} x_{1 j}^{U} + \sum_{k = 1}^{n} \sum_{j = 1}^{n} h_{1 i k j}^{U} y_{1 k j}^{U} \end{aligned}

\begin{aligned} z_{1 i}^{L} & = \sum_{j = 1}^{n} c_{1 i j}^{L} x_{1 j}^{L} + \sum_{k = 1}^{n} \sum_{j = 1}^{n} h_{1 i k j}^{L} y_{1 k j}^{L} \end{aligned}

\begin{aligned} z_{2 i} & = \sum_{j = 1}^{n} c_{2 i j} x_{2 j} + \sum_{k = 1}^{n} \sum_{j = 1}^{n} h_{2 i k j} y_{2 k j} \end{aligned}

\begin{aligned} z_{3 i}^{L} & = \sum_{j = 1}^{n} c_{3 i j}^{L} x_{3 j}^{L} + \sum_{k = 1}^{n} \sum_{j = 1}^{n} h_{3 i k j}^{L} y_{3 k j}^{L} \end{aligned}

\begin{aligned} z_{3 i}^{U} & = \sum_{j = 1}^{n} c_{3 i j}^{U} x_{3 j}^{U} + \sum_{k = 1}^{n} \sum_{j = 1}^{n} h_{3 i k j}^{U} y_{3 k j}^{U} \end{aligned}

\begin{aligned} z_{1 i}^{' L} & = \sum_{j = 1}^{n} c_{1 i j}^{' L} x_{1 j}^{' L} + \sum_{k = 1}^{n} \sum_{j = 1}^{n} h_{1 i k j}^{' L} y_{1 k j}^{' L} \end{aligned}

\begin{aligned} z_{1 i}^{' U} & = \sum_{j = 1}^{n} c_{1 i j}^{' U} x_{1 j}^{' U} + \sum_{k = 1}^{n} \sum_{j = 1}^{n} h_{1 i k j}^{' U} y_{1 k j}^{' U} \end{aligned}

\begin{aligned} z_{3 i}^{' U} & = \sum_{j = 1}^{n} c_{3 i j}^{' U} x_{3 j}^{' U} + \sum_{k = 1}^{n} \sum_{j = 1}^{n} h_{3 i k j}^{' U} y_{3 k j}^{' U} \end{aligned}

\begin{aligned} z_{3 i}^{' L} & = \sum_{j = 1}^{n} c_{3 i j}^{' L} x_{3 j}^{' L} + \sum_{k = 1}^{n} \sum_{j = 1}^{n} h_{3 i k j}^{' L} y_{3 k j}^{' L} \end{aligned}

Proposition

The linearization transformation defined by introducing auxiliary variables ${\tilde{y}}_{k j}^{I} = {\tilde{x}}_{k}^{I} {\tilde{x}}_{j}^{I}$

Proof

Step 1: Original Quadratic Form

Consider the quadratic term in the objective function: $\frac{1}{2} \sum_{k = 1}^{n} \sum_{j = 1}^{n} {\tilde{x}}_{k}^{I} \otimes {\tilde{D}}_{k j}^{I} \otimes {\tilde{x}}_{j}^{I}$

This term is nonlinear due to the product ${\tilde{x}}_{k}^{I} {\tilde{x}}_{j}^{I}$ .

Step 2: Introduction of auxiliary variables

Define new variables: ${\tilde{y}}_{k j}^{I} = {\tilde{x}}_{k}^{I} {\tilde{x}}_{j}^{I}$ , $\forall k, j = 1, 2, \dots n$

Substituting into the quadratic term gives: $\frac{1}{2} \sum_{k = 1}^{n} \sum_{j = 1}^{n} {\tilde{D}}_{k j}^{I} \otimes {\tilde{y}}_{k j}$ which is linear in ${\tilde{y}}_{k j}^{I}$ .

Step 3: Equivalence via linking constraints

To ensure equivalence, the following constraints must be imposed: ${\tilde{y}}_{k j}^{I} = {\tilde{x}}_{k}^{I} {\tilde{x}}_{j}^{I}$ , $\forall k, j$

These constraints guarantee that:

Every feasible solution in the original problem corresponds to a feasible solution in the transformed problem.

No spurious solutions are introduced.

Step 4: Feasibility Preservation

Let ${\tilde{x}}^{I}$ be any feasible solution of the original problem.

Define ${\tilde{y}}_{k j}^{I} = {\tilde{x}}_{k}^{I} {\tilde{x}}_{j}^{I}$ . Then:

All constraints of the transformed problem are satisfied.

Thus, $({\tilde{x}}^{I}, {\tilde{y}}^{I})$ is feasible in the linearized model.

Conversely, if $({\tilde{x}}^{I}, {\tilde{y}}^{I})$ satisfies ${\tilde{y}}_{k j}^{I} = {\tilde{x}}_{k}^{I} {\tilde{x}}_{j}^{I}$ , then:

Substituting back yields a feasible solution of the original quadratic model.

Step 5: Objective Value Preservation

For any feasible ${\tilde{x}}^{I}$ , we have: $\frac{1}{2} \sum_{k = 1}^{n} \sum_{j = 1}^{n} {\tilde{x}}_{k}^{I} \otimes {\tilde{D}}_{k j}^{I} \otimes {\tilde{x}}_{j}^{I} = \frac{1}{2} \sum_{j = 1}^{n} {\tilde{D}}_{k j}^{I} {\tilde{y}}_{k j}^{I}$

Thus, the objective value remains unchanged under the transformation.

Step 6: Convexity Consideration

If the matrix ${\tilde{D}}_{k j}^{I}$ is positive semi-definite, then:

The original quadratic problem is convex.

The linearized formulation preserves convexity through exact representation of ${\tilde{y}}_{k j}^{I} = {\tilde{x}}_{k}^{I} {\tilde{x}}_{j}^{I}$ .

Conclusion

Since:

Feasibility is preserved,

Objective values are unchanged, and

A one-to-one correspondence exists between solutions,

The linearized model is equivalent to the original quadratic model.

Hence, the linearization is valid

Remark

The proposed linearization is exact but increases the problem size from n to $n^{2}$ variables.

5.2 Equivalent Crisp Model

The proposed model is formulated using interval-valued intuitionistic fuzzy (IVIF) data, which incorporates membership, non-membership, and hesitation in interval form. While this representation provides a rich description of uncertainty, it is computationally challenging to solve directly. To overcome this difficulty, an expected value function (Bharati & Singh, 2018) is employed to transform the IVIF model into an equivalent crisp formulation.

The expected value serves as an aggregation measure that combines the lower and upper bounds of membership and non-membership degrees into a single representative value. It is important to note that this transformation does not arbitrarily eliminate uncertainty. Instead, it preserves the essential characteristics of IVIF data by providing a balanced representation of the underlying uncertainty structure. In particular, the effects of hesitation and interval variability are implicitly reflected in the resulting crisp model. Therefore, the use of the expected value function represents a trade-off between preserving the richness of IVIF information and ensuring computational tractability.

Employing the technique outlined in (Bharati & Singh, 2018), a FIVIFMLP problem is transformed into its crisp version, specifically the (MLP) problem, as detailed below:

[D M_{i}] : max_{x_{i}} Z_{i} (x, y) = max_{x_{i}} (E V {(z_{1 i}^{U}, z_{1 i}^{L}, z_{2 i}, z_{3 i}^{L}, z_{3 i}^{U}), (z_{1 i}^{' L}, z_{1 i}^{' U}, z_{2 i}, z_{3 i}^{' U}, z_{3 i}^{' L})})

where $x_{i + 1}, x_{i + 2}, \dots, x_{p}$ solve.

subject to

G = {\begin{matrix} x \in R^{n} | \sum_{j = 1}^{n} E V {\begin{matrix} (a_{1 s j}^{U} x_{1 j}^{U}, a_{1 s j}^{L} x_{1 j}^{L}, a_{2 s j} x_{2 j}, a_{3 s j}^{L} x_{3 j}^{L}, a_{3 s j}^{U} x_{3 j}^{U}), \\ (a_{1 s j}^{' L} x_{1 j}^{' L}, a_{1 s j}^{' U} x_{1 j}^{' U}, a_{2 s j} x_{2 j}, a_{3 s j}^{' U} x_{3 j}^{' U}, a_{3 s j}^{' L} x_{3 j}^{' L}) \end{matrix}} \leq \\ E V {(b_{1 s}^{U}, b_{1 s}^{L}, b_{2 s}, b_{3 s}^{L}, b_{3 s}^{U}), (b_{1 s}^{' L}, b_{1 s}^{' U}, b_{2 s}, b_{3 s}^{' U}, b_{3 s}^{' L})}, \\ x_{1 j}^{' L} \leq x_{1 j}^{' U} \leq x_{1 j}^{U} \leq x_{1 j}^{L} \leq x_{2 j} \leq x_{3 j}^{L} \leq x_{3 j}^{U} \leq x_{3 j}^{' U} \leq x_{3 j}^{' L}, \\ y_{1 k j}^{U} = x_{1 k}^{U} x_{1 j}^{U}, y_{1 k j}^{L} = x_{1 k}^{L} x_{1 j}^{L}, y_{2 k j} = x_{2 k} x_{2 j}, y_{3 k j}^{L} = x_{3 k}^{L} x_{3 j}^{L}, \\ y_{3 k j}^{U} = x_{3 k}^{U} x_{3 j}^{U}, y_{1 k j}^{' L} = x_{1 k}^{' L} x_{1 j}^{' L}, y_{1 k j}^{' U} = x_{1 k}^{' U} x_{1 j}^{' U}, y_{3 k j}^{' U} = x_{3 k}^{' U} x_{3 j}^{' U}, y_{3 k j}^{' L} = x_{3 k}^{' L} x_{3 j}^{' L}, \\ x_{1 j}^{' L} \geq 0. \end{matrix}}

(4)

Theorem 1

Problems (3) and (4) are equivalent.

Proof

To prove the equivalence between Problems (3) and (4), we show that both formulations yield identical feasible regions and preserve the ordering of the objective function values.

Step 1: Feasibility Preservation

Assume that ${\tilde{x}}^{I} \in {\tilde{G}}^{I}$ is a feasible solution of Problem (3).

Then, by definition of the constraints, we have:

$\sum_{j = 1}^{n} {\tilde{a}}_{sj}^{I} \otimes {\tilde{x}}_{j}^{I} \leq {\tilde{b}}_{s}^{I}, s = 1, 2, \dots, m$

${\tilde{x}}_{j}^{I} \geq 0$

Applying the expected value operator $EV (.)$ to both sides of the inequality, and using Definition 6, we obtain:

EV (\sum_{j = 1}^{n} {\tilde{a}}_{sj}^{I} \otimes {\tilde{x}}_{j}^{I}) \leq EV ({\tilde{b}}_{s}^{I}), s = 1, 2, \dots, m

Since the expected value function is monotonic (order-preserving), it follows that:

\sum_{j = 1}^{n} EV ({\tilde{a}}_{sj}^{I} \otimes {\tilde{x}}_{j}^{I}) \leq EV ({\tilde{b}}_{s}^{I}), s = 1, 2, \dots, m

Thus, the transformed solution ${\tilde{x}}^{I}$ satisfies the constraints of Problem (4), implying that:

Every feasible solution of Problem (3) is feasible in Problem (4).

Step 2: Objective Function Consistency

The objective function in Problem (3) is expressed in IVIF form: ${\tilde{F}}_{i}^{I} ({\tilde{x}}^{I}, {\tilde{y}}^{I})$

By applying the expected value operator, we obtain the crisp objective function: $Z_{i} (x, y) = EV ({\tilde{F}}_{i}^{I} ({\tilde{x}}^{I}, {\tilde{y}}^{I}))$

Since the expected value aggregates all IVIF components into a scalar while preserving their relative magnitudes, the ordering of solutions is maintained.

That is, for any two feasible solutions ${\tilde{x}}_{1}^{I}$ and ${\tilde{x}}_{2}^{I}$ :

{\tilde{F}}_{i}^{I} ({\tilde{x}}_{1}^{I}) \leq {\tilde{F}}_{i}^{I} ({\tilde{x}}_{2}^{I}) \Rightarrow EV ({\tilde{F}}_{i}^{I} ({\tilde{x}}_{1}^{I})) \leq EV ({\tilde{F}}_{i}^{I} ({\tilde{x}}_{2}^{I}))

Thus, the optimization direction is preserved.

Step 3: Reverse Direction

Conversely, assume that $x$ is a feasible solution of Problem (4).

Then there exists a corresponding IVIF representation ${\tilde{x}}^{I}$ such that:

EV ({\tilde{x}}^{I}) = x

Since the expected value is derived directly from IVIF parameters, we can reconstruct a consistent IVIF solution satisfying Problem (3).

Thus every feasible solution of Problem (4) corresponds to a feasible IVIF solution in Problem (3).

Conclusion

Since:

Feasible regions are preserved, and

Objective function ordering is maintained,

It follows that Problems (3) and (4) are equivalent.

Hence, the theorem is proved

Interpretation and Practical Significance of Theorem

Theorem 1 establishes the equivalence between the original IVIF multilevel quadratic programming model and its transformed crisp counterpart. This guarantees that no feasible solutions are lost and that the optimization direction is preserved.

The theorem provides the theoretical foundation that justifies solving the transformed model instead of the original IVIF model, which is difficult to handle directly.

Theorem 1 serves as a key step in the proposed solution framework, ensuring that the transformation process (expected value and linearization) leads to a valid and consistent optimization problem.

We clarified that all numerical results presented in the paper rely on this equivalence, which ensures their correctness and reliability.

Theorem 1 provides the theoretical justification for transforming the original IVIF multilevel quadratic programming problem into an equivalent crisp model. This equivalence ensures that solving the transformed problem yields solutions that are consistent with the original IVIF formulation. Therefore, the theorem plays a crucial role in enabling the practical implementation of the proposed approach.

Remark: The equivalence relies on the monotonicity and linearity of the expected value function, which ensures that no feasible solution is lost during transformation.

5.3 Expanded Fuzzy Programming Approach

This phase presents the solution concept for the MLP problem by adapting the fuzzy programming method (Osman et al., 2004) within an IVIF context utilizing a linear MS function. Initially, $D M_{i}$ acquires the optimal IVIF solution represented as ${\tilde{Z}}_{i}^{I i t h} = {(z_{1 i}^{U i t h}, z_{1 i}^{L i t h}, z_{s i}^{i t h}, z_{3 i}^{L i t h}, z_{3 i}^{U i t h}), (z_{1 i}^{' L i t h}, z_{1 i}^{' U i t h}, z_{s i}^{i t h}, z_{3 i}^{' U i t h}, z_{3 i}^{' L i t h})}$ and the optimal IVIF decision variables denoted as ${\tilde{x}}_{j}^{I i t h} = {(x_{1 j}^{U i t h}, x_{1 j}^{L i t h}, x_{2 j}^{i t h}, x_{3 j}^{L i t h}, x_{3 j}^{U i t h}), (x_{1 j}^{' L i t h}, x_{1 j}^{' U i t h}, x_{2 j}^{i t h}, x_{3 j}^{' U i t h}, x_{3 j}^{' L i t h})}$ and ${\tilde{y}}_{k j}^{I i t h} = {(y_{1 k j}^{U i t h}, y_{1 k j}^{L i t h}, y_{2 k j}^{i t h}, y_{3 k j}^{L i t h}, y_{3 k j}^{U i t h}), (y_{1 k j}^{' L i t h}, y_{1 k j}^{' U i t h}, y_{2 k j}^{i t h}, y_{3 k j}^{' U i t h}, y_{3 k j}^{' L i t h})}$ , $(j, k = 1, 2, \dots, n; i = 1, 2, \dots, p)$ .

Let ${\tilde{t}}_{j}^{I} = {(t_{1 j}^{U}, t_{1 j}^{L}, t_{2 j}, t_{3 j}^{L}, t_{3 j}^{U}), (t_{1 j}^{' L}, t_{1 j}^{' U}, t_{2 j}, t_{3 j}^{' U}, t_{3 j}^{' L})}$ represent the IVIF tolerance values of decision variables ${\tilde{x}}_{j}^{I} = {(x_{1 j}^{U}, x_{1 j}^{L}, x_{2 j}, x_{3 j}^{L}, x_{3 j}^{U}), (x_{1 j}^{' L}, x_{1 j}^{' U}, x_{2 j}, x_{3 j}^{' U}, x_{3 j}^{' L})}$ , where $j = 1, \dots, n$ and $r = 1, \dots, k - 1$ , as recognized by the $D M_{i}$ . Generally, ${\tilde{x}}_{j}^{I}$ resides within the range of ${\tilde{x}}_{j}^{I i t h} - {\tilde{t}}_{j}^{I}$ and ${\tilde{x}}_{j}^{I i t h} + {\tilde{t}}_{j}^{I}$ . DMs possess the authority to adjust the position of ${\tilde{x}}_{j}^{I i t h}$ , either to the left or right, solely in accordance with the orders and preferences of the higher-level DMs. Provided that ${\tilde{x}}_{j}^{I i t h} \geq {\tilde{t}}_{j}^{I}$ , we can utilize the tolerances as variables (ensuring the variables’ values remain positive). The ideal value of the choice variables signifies the top limit of tolerance or the intended level of attainment about the control variables. Subsequently, given the IVIF decision variables ${\tilde{x}}_{j}^{I i t h}$ the MS functions can be articulated as follows:

μ_{{\tilde{x}}_{j}^{I}} ({\tilde{x}}_{j}^{I}) = {\begin{array}{cc} \frac{E V ({\tilde{x}}_{j}^{I}) - (E V ({\tilde{x}}_{j}^{I i t h}) - E V ({\tilde{t}}_{j}^{I}))}{E V (t_{j}^{I})}, & i f E V ({\tilde{x}}_{j}^{I i t h}) - E V ({\tilde{t}}_{j}^{I}) \leq E V ({\tilde{x}}_{j}^{I}) \leq E V ({\tilde{x}}_{j}^{I i t h}), \\ \frac{(E V ({\tilde{x}}_{j}^{I i t h}) + E V ({\tilde{t}}_{j}^{I})) - E V ({\tilde{x}}_{j}^{I})}{E V ({\tilde{t}}_{j}^{I})}, & i f E V ({\tilde{x}}_{j}^{I i t h}) \leq E V ({\tilde{x}}_{j}^{I}) \leq E V ({\tilde{x}}_{j}^{I i t h}) - E V ({\tilde{t}}_{j}^{I}) . \end{array}

(5)

Formulate the MS functions of the ith-LDM $(i = 1, 2, \dots, p)$ as follows:

μ_{Z_{i}} (Z_{i} (x, y)) = {\begin{array}{cc} 1, & Z_{i} (x, y) \geq Z_{i} ({\tilde{x}}^{I i t h}, {\tilde{y}}^{I i t h}), \\ \frac{Z_{i} (x, y) - Z_{i} ({\tilde{x}}^{I (i + 1) t h}, {\tilde{y}}^{I (i + 1) t h})}{Z_{i} ({\tilde{x}}^{I i t h}, {\tilde{y}}^{I i t h}) - Z_{i} ({\tilde{x}}^{I (i + 1) t h}, {\tilde{y}}^{I (i + 1) t h})}, & Z_{i} ({\tilde{x}}^{I (i + 1) t h}, {\tilde{y}}^{I (i + 1) t h}) < Z_{i} (x, y) < Z_{i} ({\tilde{x}}^{I i t h}, {\tilde{y}}^{I i t h}) \\ 0, & Z_{i} (x, y) \leq Z_{i} ({\tilde{x}}^{I (i + 1) t h}, {\tilde{y}}^{I (i + 1) t h}) \end{array}

(6)

When $i = p$ , the MS functions of the pth-LDM can be articulated as follows:

μ_{Z_{p}} (Z_{p} (x, y)) = {\begin{array}{cc} 1, & Z_{p} (x, y) \geq Z_{p} ({\tilde{x}}^{I p t h}, {\tilde{y}}^{I p t h}), \\ \frac{Z_{p} (x, y) - Z_{p} ({\tilde{x}}^{I (p - 1) t h}, {\tilde{y}}^{I (p - 1) t h})}{Z_{p} ({\tilde{x}}^{I p t h}, {\tilde{y}}^{I p t h}) - Z_{p} ({\tilde{x}}^{I (p - 1) t h}, {\tilde{y}}^{I (p - 1) t h})}, & Z_{p} ({\tilde{x}}^{I (p - 1) t h}, {\tilde{y}}^{I (p - 1) t h}) < Z_{p} (x, y) < Z_{p} ({\tilde{x}}^{I p t h}, {\tilde{y}}^{I p t h}) \\ 0, & Z_{p} (x, y) \leq Z_{p} ({\tilde{x}}^{I (p - 1) t h}, {\tilde{y}}^{I (p - 1) t h}) \end{array}

(7)

Ultimately, to provide an optimal solution that meets the requirements of all decision-makers (DMs), the subsequent Tchebycheff (Tch) problems (Osman et al., 2004) will be examined:

max λ,

subject to

x \in G,

\begin{aligned} μ_{Z_{i}} (Z_{i} (x, y)) & \geq λ, (i = 1, 2, \dots, p) \end{aligned}

\begin{aligned} \frac{E V ({\tilde{x}}_{j}^{I}) - (E V ({\tilde{x}}_{j}^{I i t h}) - E V ({\tilde{t}}_{j}^{I}))}{E V ({\tilde{t}}_{j}^{I})} \geq λ, (j = 1, 2, \dots, n - 1) \end{aligned}

\begin{aligned} \frac{(E V ({\tilde{x}}_{j}^{I i t h}) + E V ({\tilde{t}}_{j}^{I})) - E V ({\tilde{x}}_{j}^{I})}{E V ({\tilde{t}}_{j}^{I})} \geq λ, (j = 1, 2, \dots, n - 1) \end{aligned}

\begin{aligned} {\tilde{t}}_{j}^{I} > 0, (j = 1, 2, \dots, n - 1) \end{aligned}

\begin{aligned} λ \in [0, 1] . \end{aligned}

(8)

If the decision-makers are content with the resolution derived from addressing problem (8), then a satisficing solution has been achieved. If a satisfactory solution cannot be achieved, senior DMs should provide new tolerance limits for the control variables. The chosen tolerance levels compel the best solution to supplant the satisfactory answer. The resolution to problem (8) becomes satisfactory for all level DMs when considering a set of potential relaxations offered by the higher-level DMs. This method may be reiterated until an acceptable outcome is achieved.

5.4 Interpretation and Evaluation of the Compromise Solution

The compromise solution represents a balanced decision that simultaneously considers the objectives of all decision makers in the multilevel structure. It is obtained by maximizing an overall satisfaction measure derived from the fuzzy programming formulation. Unlike classical optimal solutions, the compromise solution does not necessarily optimize any single objective individually. Instead, it reflects a trade-off among competing objectives, ensuring that no decision level is significantly disadvantaged. The quality of the compromise solution can be interpreted in terms of the achieved satisfaction levels, which indicate how well each objective is fulfilled. Moreover, the obtained solution depends on model parameters and may vary under different preference structures, highlighting its sensitivity to decision-making conditions. Thus, the compromise solution provides a realistic and flexible outcome for complex optimization problems involving multiple objectives and uncertainty.

5.5 Complexity Analysis

1.
Problem size expansion:

The linearization process increases the number of variables from n to $n^{2}$ , and the number of constraints increases accordingly due to the linking relations.
2.
Computational complexity:

The resulting problem becomes a large-scale linear (or fuzzy linear) programming problem.

Using standard solvers, the worst-case computational complexity grows polynomially with the number of variables and constraints.
3.
Trade-off discussion:

We clarify that the proposed method involves a trade-off between:

Modeling accuracy (exact handling of quadratic terms), and computational cost.
4.
Practical applicability:

The method is suitable for small- to medium-scale problems, while large-scale problems may require:

Decomposition techniques, or approximation methods.
5.
Future work direction:

We have highlighted the development of more efficient or reduced-complexity formulations as an important direction for future research.
5.6 Algorithm for Solution

The subsequent stages propose a novel method to deliver an accurate fuzzy Pareto optimal solution to the FIVIFMQP problem.

Step 1. Formulate the FIVIFMQP Problem As Expressed in Equation (1)

Step 2. Let IVTIFNs represent all IVIF parameters and variables pertaining to the objective functions and constraints in equation (1).

Step 3. Initialize i to 1.

Step 4. Generate the $D M_{i}$ issue, then advance to Step 6.

Step 5. Advance to Step 10 if the $D M_{i}$ identifies the optimal solution. If not, advance to Step 5.

Step 6. $D M_{i}$ employs a linearization method to linearize the quadratic component of objective functions.

Step 7. $D M_{i}$ developed an expected value (Bharati & Singh, 2018) to convert IVIF data into a precise model.

Step 8. Advance to Step 9 upon addressing the resultant crisp-linear programming challenges to ascertain the optimal answer to the $D M_{i}$ problem.

Step 9. The $D M_{i}$ attains the optimal solution ${\tilde{x}}_{j}^{I i t h} = {(x_{1 j}^{U i t h}, x_{1 j}^{L i t h}, x_{2 j}^{i t h}, x_{3 j}^{L i t h}, x_{3 j}^{U i t h}), (x_{1 j}^{' L i t h}, x_{1 j}^{' U i t h}, x_{2 j}^{i t h}, x_{3 j}^{' U i t h}, x_{3 j}^{' L i t h})}$ and ${\tilde{y}}_{k j}^{I i t h} = {(y_{1 k j}^{U i t h}, y_{1 k j}^{L i t h}, y_{2 k j}^{i t h}, y_{3 k j}^{L i t h}, y_{3 k j}^{U i t h}), (y_{1 k j}^{' L i t h}, y_{1 k j}^{' U i t h}, y_{2 k j}^{i t h}, y_{3 k j}^{' U i t h}, y_{3 k j}^{' L i t h})}$ , $(j, k = 1, 2, \dots, n; i = 1, 2, \dots, p)$

Step 10. Advance to Step 12 if i equals p; else, advance to Step 11.

Step 11. Advance to Step 12 subsequent to incrementing i by 1.

Step 12. Determine the maximum tolerance for the $D M_{i}$ and the values of the control decision variables.

Step 13. Formulate the MS function of the $D M_{i}$ as specified in Equations (5) to (7)

Step 14. Formulate Tch problems utilizing Equations (8) for each DMs problem.

Step 15. Augment the tolerance and advance to Step 12 if $λ < 0$ ; otherwise, move to Step 16.

Step 16. The ${\tilde{x}}_{j}^{I i t h} = {(x_{1 j}^{U i t h}, x_{1 j}^{L i t h}, x_{2 j}^{i t h}, x_{3 j}^{L i t h}, x_{3 j}^{U i t h}), (x_{1 j}^{' L i t h}, x_{1 j}^{' U i t h}, x_{2 j}^{i t h}, x_{3 j}^{' U i t h}, x_{3 j}^{' L i t h})}$ constitutes a preferred solution to the $D M_{i}$ .

Step 17. Advance to Step 18 upon acquiring the optimal solution for the FIVIFMQP issue.

Step 18. Stop.

5.7 A Flowchart of the Proposed Methodology

The proposed technique is illustrated in the flowchart below (Fig. 1).

Figure 1.

A flowchart delineates the procedure of the suggested approach.

6. Exemplification by Numerical Analysis

The obtained results demonstrate that the proposed model is capable of capturing the interaction between multiple decision levels under uncertainty. In contrast, classical crisp or standard fuzzy approaches would fail to represent hesitation and interval uncertainty, leading to less informative solutions.

\begin{aligned} D M_{1} : max_{{\tilde{x}}_{1}^{I}} {\tilde{f}}_{1}^{I} ({\tilde{x}}^{I}) & = {\begin{matrix} (168, 176, 185, 193, 202), \\ (150, 160, 185, 210, 220) \end{matrix}} Θ {\begin{matrix} (5, 7, 8, 10, 13), \\ (1, 3, 8, 17, 20) \end{matrix}} \otimes {\tilde{x}}_{1}^{I} Θ {\begin{matrix} (15, 20, 26, 30, 37), \\ (4, 9, 26, 42, 49) \end{matrix}} \\ \otimes {\tilde{x}}_{3}^{I} \oplus {\tilde{x}}_{1}^{I} \otimes {\tilde{x}}_{1}^{I} \oplus {\tilde{x}}_{3}^{I} \otimes {\tilde{x}}_{3}^{I} \end{aligned}

\begin{aligned} D M_{2} : max_{{\tilde{x}}_{2}^{I}} {\tilde{f}}_{2}^{I} ({\tilde{x}}^{I}) & = {\begin{matrix} (1200, 1220, 1240, 1260, 1280), \\ (1160, 1180, 1240, 1300, 1320) \end{matrix}} Θ {\begin{matrix} (55, 63, 70, 78, 85), \\ (40, 48, 70, 92, 100) \end{matrix}} \otimes {\tilde{x}}_{2}^{I} Θ {\begin{matrix} (2, 3, 4, 5, 7), \\ (1, 2, 4, 8, 10) \end{matrix}} \\ \otimes {\tilde{x}}_{4}^{I} \oplus {\tilde{x}}_{2}^{I} \otimes {\tilde{x}}_{2}^{I} \oplus {\tilde{x}}_{4}^{I} \otimes {\tilde{x}}_{4}^{I} \end{aligned}

\begin{aligned} D M_{3} : max_{{\tilde{x}}_{3}^{I}, {\tilde{x}}_{4}^{I}} {\tilde{f}}_{3}^{I} ({\tilde{x}}^{I}) & = {\begin{matrix} (184, 193, 200, 203, 208), \\ (172, 178, 200, 211, 215) \end{matrix}} \otimes {\tilde{x}}_{1}^{I} \oplus {\begin{matrix} (184, 193, 200, 203, 208), \\ (172, 178, 200, 211, 215) \end{matrix}} \\ \otimes {\tilde{x}}_{2}^{I} \oplus {\begin{matrix} (150, 155, 160, 165, 170), \\ (140, 145, 160, 175, 180) \end{matrix}} \otimes {\tilde{x}}_{3}^{I} \oplus {\begin{matrix} (150, 155, 160, 165, 170), \\ (140, 145, 160, 175, 180) \end{matrix}} \\ \otimes {\tilde{x}}_{4}^{I} ⊖ {\begin{matrix} (1, 2, 2, 2, 3), \\ (1, 1, 2, 5, 6) \end{matrix}} \otimes {\tilde{x}}_{1}^{I} \otimes {\tilde{x}}_{2}^{I} ⊖ {\begin{matrix} (1, 2, 2, 2, 3), \\ (0, 1, 2, 5, 6) \end{matrix}} \\ \otimes {\tilde{x}}_{3}^{I} \otimes {\tilde{x}}_{4}^{I} Θ {\tilde{x}}_{1}^{I} \otimes {\tilde{x}}_{1}^{I} Θ {\tilde{x}}_{2}^{I} \otimes {\tilde{x}}_{2}^{I} Θ {\tilde{x}}_{3}^{I} \otimes {\tilde{x}}_{3}^{I} Θ {\tilde{x}}_{4}^{I} \otimes {\tilde{x}}_{4}^{I} \end{aligned}

Subject to

\begin{aligned} {\tilde{x}}_{1}^{I} \oplus {\tilde{x}}_{2}^{I} \oplus {\tilde{x}}_{3}^{I} \oplus {\tilde{x}}_{4}^{I} \leq {\begin{matrix} (30, 36, 40, 43, 47), \\ (20, 24, 40, 50, 55) \end{matrix}} \end{aligned}

\begin{aligned} {\begin{matrix} (2, 3, 4, 5, 7), \\ (1, 2, 4, 8, 10) \end{matrix}} \otimes {\tilde{x}}_{1}^{I} \oplus {\begin{matrix} (4, 5, 7, 9, 12), \\ (1, 3, 7, 13, 15) \end{matrix}} \otimes {\tilde{x}}_{3}^{I} \leq {(5, 8, 10, 11, 13), (1, 3, 10, 16, 20)} \end{aligned}

\begin{aligned} {\begin{matrix} (4, 5, 6, 7, 8), \\ (2, 3, 6, 9, 10) \end{matrix}} \otimes {\tilde{x}}_{2}^{I} \oplus {\begin{matrix} (2, 3, 3, 4, 5), \\ (1, 1, 3, 6, 8) \end{matrix}} \otimes {\tilde{x}}_{4}^{I} \leq {(5, 8, 10, 11, 13), (1, 3, 10, 16, 20) \end{aligned}

{\tilde{x}}_{1}^{I} \leq {(5, 8, 10, 11, 13), (1, 3, 10, 16, 20)}, {\tilde{x}}_{3}^{I} \leq {(3, 4, 5, 6, 7), (1, 2, 5, 8, 9)}

{\tilde{x}}_{2}^{I} \leq {(12, 14, 15, 16, 19), (8, 9, 15, 21, 25)}, {\tilde{x}}_{4}^{I} \leq {(16, 18, 20, 22, 24), (12, 14, 20, 26, 28)}

{\tilde{x}}_{1}^{I}, {\tilde{x}}_{2}^{I}, {\tilde{x}}_{3}^{I}, {\tilde{x}}_{4}^{I} \geq {\tilde{0}}^{I}

Assume the subsequent representation for the IVIF decision variables:

\begin{aligned} {\tilde{x}}_{1}^{I} & = {(x_{11}^{U}, x_{11}^{L}, x_{21}, x_{31}^{L}, x_{31}^{U}), (x_{11}^{^{'} L}, x_{11}^{^{'} U}, x_{21}, x_{31}^{^{'} U}, x_{31}^{^{'} L})}, \\ {\tilde{x}}_{2}^{I} & = {(x_{12}^{U}, x_{12}^{L}, x_{22}, x_{32}^{L}, x_{32}^{U}), (x_{12}^{^{'} L}, x_{12}^{^{'} U}, x_{22}, x_{32}^{^{'} U}, x_{32}^{^{'} L})}, \\ {\tilde{x}}_{3}^{I} & = {(x_{13}^{U}, x_{13}^{L}, x_{23}, x_{33}^{L}, x_{33}^{U}), (x_{13}^{^{'} L}, x_{13}^{^{'} U}, x_{23}, x_{33}^{^{'} U}, x_{33}^{^{'} L})}, \\ {\tilde{x}}_{4}^{I} & = {(x_{14}^{U}, x_{14}^{L}, x_{24}, x_{34}^{L}, x_{34}^{U}), (x_{14}^{^{'} L}, x_{14}^{^{'} U}, x_{24}, x_{34}^{^{'} U}, x_{34}^{^{'} L})} . \end{aligned}

Subsequently, succinctly articulate the multilevel optimization utilizing arithmetic procedures, the linearization method, and the expected value.

\begin{aligned} D M_{1} : max_{{\tilde{x}}_{1}^{I}} Z_{1} (x, y) & = 190.9375 - 0.8125 x_{31}^{U} - 2.3125 x_{33}^{U} + 0.0625 y_{111}^{U} + 0.0625 y_{133}^{U} - 0.625 x_{31}^{L} - 1.875 x_{33}^{L} \\ + 0.0625 y_{111}^{L} + 0.0625 y_{133}^{L} - 4 x_{21} - 13 x_{23} + 0.5 y_{211} + 0.5 y_{233} - 0.4375 x_{11}^{L} - 1.25 x_{13}^{L} \\ + 0.0625 y_{311}^{L} + 0.0625 y_{333}^{L} - 0.3125 x_{11}^{U} - 0.9375 x_{13}^{U} + 0.0625 y_{311}^{U} + 0.0625 y_{333}^{U} - 1.25 x_{31}^{^{'} L} \\ - 3.0625 x_{33}^{^{'} L} + 0.0625 y_{111}^{^{'} L} + 0.0625 y_{133}^{^{'} L} - 1.0625 x_{13}^{^{'} U} - 2.625 x_{33}^{^{'} U} + 0.0625 y_{111}^{^{'} U} \\ + 0.0625 y_{133}^{^{'} U} - 0.1875 x_{11}^{^{'} U} - 0.5625 x_{13}^{^{'} U} + 0.0625 y_{311}^{^{'} U} + 0.0625 y_{333}^{^{'} U} - 0.0625 x_{11}^{^{'} L} - 0.25 x_{13}^{^{'} L} \\ + 0.0625 y_{311}^{^{'} L} + 0.0625 y_{333}^{^{'} L} \end{aligned}

\begin{aligned} D M_{2} : max_{{\tilde{x}}_{2}^{I}} Z_{2} (x, y) & = 680 - 5.3125 x_{32}^{U} - 0.4375 x_{34}^{U} + 0.0625 y_{122}^{U} + 0.0625 y_{144}^{U} - 4.875 x_{32}^{L} - 0.3125 x_{34}^{L} \\ + 0.0625 y_{122}^{L} + 0.0625 y_{144}^{L} - 35 x_{22} - 2 x_{24} + 0.5 y_{222} + 0.5 y_{244} - 3.9375 x_{21}^{L} - 0.1875 x_{14}^{L} \\ + 0.0625 y_{322}^{L} + 0.0625 y_{344}^{L} - 3.4375 x_{12}^{U} - 0.125 x_{14}^{U} + 0.0625 y_{322}^{U} + 0.0625 y_{344}^{U} - 6.25 x_{32}^{^{'} L} \\ - 0.625 x_{34}^{^{'} L} + 0.0625 y_{122}^{^{'} L} + 0.0625 y_{144}^{^{'} L} - 5.75 x_{32}^{^{'} U} - 0.5 x_{34}^{^{'} U} + 0.0625 y_{122}^{^{'} U} + 0.0625 y_{144}^{^{'} U} \\ - 3 x_{12}^{^{'} U} - 0.125 x_{14}^{^{'} U} + 0.0625 y_{322}^{^{'} U} + 0.0625 y_{344}^{^{'} U} - 2.5 x_{12}^{^{'} L} - 0.0625 x_{14}^{^{'} L} + 0.0625 y_{322}^{^{'} L} \\ + 0.0625 y_{344}^{^{'} L} \end{aligned}

\begin{aligned} D M_{3} : max_{{\tilde{x}}_{3}^{I}, {\tilde{x}}_{4}^{I}} Z_{3} (x) & = 11.5 x_{11}^{U} + 11.5 x_{12}^{U} + 9.375 x_{13}^{U} + 9.375 x_{14}^{U} - 0.1875 y_{312}^{U} - 0.1875 y_{334}^{U} - 0.0625 y_{311}^{U} \\ - 0.0625 y_{322}^{U} - 0.0625 y_{333}^{U} - 0.0625 y_{344}^{U} + 12.0625 x_{11}^{L} + 12.0625 x_{12}^{L} + 9.6875 x_{13}^{L} \\ + 9.6875 x_{14}^{L} - 0.125 y_{312}^{L} - 0.125 y_{334}^{L} - 0.0625 y_{311}^{L} - 0.0625 y_{322}^{L} - 0.0625 y_{333}^{L} \\ - 0.0625 y_{344}^{L} + 100 x_{21} + 100 x_{22} + 80 x_{23} + 80 x_{24} - y_{212} - y_{234} - 0.5 y_{211} - 0.5 y_{222} \\ - 0.5 y_{233} - 0.5 y_{244} + 12.6875 x_{31}^{L} + 12.6875 x_{32}^{L} + 10.3125 x_{33}^{L} + 10.3125 x_{34}^{L} - 0.125 y_{112}^{L} \\ - 0.125 y_{134}^{L} - 0.0625 y_{111}^{L} - 0.0625 y_{122}^{L} - 0.0625 y_{133}^{L} - 0.0625 y_{144}^{L} + 13 x_{31}^{U} + 13 x_{32}^{U} \\ + 10.625 x_{33}^{U} + 10.625 x_{34}^{U} - 0.0625 y_{112}^{U} - 0.0625 y_{134}^{U} - 0.0625 y_{111}^{U} - 0.0625 y_{122}^{U} \\ - 0.0625 y_{133}^{U} - 0.0625 y_{144}^{U} + 10.75 x_{11}^{^{'} L} + 10.75 x_{12}^{^{'} L} + 8.75 x_{13}^{^{'} L} + 8.75 x_{14}^{^{'} L} - 0.375 y_{312}^{^{'} L} \\ - 0.375 y_{334}^{^{'} L} - 0.0625 y_{311}^{^{'} L} - 0.0625 y_{322}^{^{'} L} - 0.0625 y_{333}^{^{'} L} - 0.0625 y_{344}^{^{'} L} + 11.125 x_{11}^{^{'} U} \\ + 11.125 x_{12}^{^{'} U} + 9.0625 x_{13}^{^{'} U} + 9.0625 x_{14}^{^{'} U} - 0.3125 y_{312}^{^{'} U} - 0.3125 y_{334}^{^{'} U} - 0.0625 y_{311}^{^{'} U} \\ - 0.0625 y_{322}^{^{'} U} - 0.0625 y_{333}^{^{'} U} - 0.0625 y_{344}^{^{'} U} + 13.1875 x_{31}^{^{'} U} + 13.1875 x_{32}^{^{'} U} + 10.9375 x_{33}^{^{'} U} \\ + 10.9375 x_{34}^{^{'} U} - 0.0625 y_{112}^{^{'} U} - 0.0625 y_{134}^{^{'} U} - 0.0625 y_{111}^{^{'} U} - 0.0625 y_{122}^{^{'} U} - 0.0625 y_{133}^{^{'} U} \\ - 0.0625 y_{144}^{^{'} U} + 13.4375 x_{31}^{^{'} L} + 13.4375 x_{32}^{^{'} L} + 11.25 x_{33}^{^{'} L} + 11.25 x_{34}^{^{'} L} - 0.0625 y_{112}^{^{'} L} \\ - 0.0625 y_{133}^{^{'} L} - 0.0625 y_{111}^{^{'} L} - 0.0625 y_{122}^{^{'} L} - 0.0625 y_{133}^{^{'} L} - 0.0625 y_{144}^{^{'} L} \end{aligned}

Subject to

\begin{aligned} G & = {x_{11}^{U} + x_{12}^{U} + x_{13}^{U} + x_{14}^{U} + x_{11}^{L} + x_{12}^{L} + x_{13}^{L} + x_{14}^{L} + 8 x_{21} + 8 x_{22} + 8 x_{23} + 8 x_{24} + x_{31}^{L} + x_{32}^{L} + x_{33}^{L} + x_{34}^{L} \\ + x_{31}^{U} + x_{32}^{U} + x_{33}^{U} + x_{34}^{U} + x_{11}^{^{'} L} + x_{12}^{^{'} L} + x_{13}^{^{'} L} + x_{14}^{^{'} L} + x_{11}^{^{'} U} + x_{12}^{^{'} U} + x_{13}^{^{'} U} + x_{14}^{^{'} U} + x_{31}^{^{'} U} + x_{32}^{^{'} U} + x_{33}^{^{'} U} \\ + x_{34}^{^{'} U} + x_{31}^{^{'} L} + x_{32}^{^{'} L} + x_{33}^{^{'} L} + x_{34}^{^{'} L} \leq 625 \end{aligned}

\begin{aligned} 2 x_{11}^{U} + 4 x_{13}^{U} + 3 x_{11}^{L} + 5 x_{13}^{L} + 32 x_{21} + 56 x_{23} + 5 x_{31}^{L} + 9 x_{33}^{L} + 7 x_{31}^{U} + 12 x_{33}^{U} + x_{11}^{^{'} L} + x_{31}^{^{'} L} + 2 x_{11}^{^{'} U} + 3 x_{13}^{^{'} U} + 8 x_{31}^{^{'} U} \\ + 13 x_{33}^{^{'} U} + 10 x_{31}^{^{'} L} + 15 x_{33}^{^{'} L} \leq 157 \end{aligned}

\begin{aligned} 4 x_{12}^{U} + 2 x_{14}^{U} + 5 x_{12}^{L} + 3 x_{14}^{L} + 48 x_{22} + 18 x_{24} + 7 x_{32}^{L} + 4 x_{34}^{L} + 8 x_{32}^{U} + 5 x_{34}^{U} + 2 x_{12}^{^{'} L} + x_{14}^{^{'} L} + 3 x_{12}^{^{'} U} + x_{14}^{^{'} U} \\ + 9 x_{32}^{^{'} U} + 6 x_{34}^{^{'} U} + 10 x_{32}^{^{'} L} + 8 x_{34}^{^{'} L} \leq 157 \end{aligned}

\begin{aligned} x_{11}^{U} + x_{11}^{L} + 8 x_{21} + x_{31}^{L} + x_{31}^{U} + x_{11}^{^{'} L} + x_{11}^{^{'} U} + x_{31}^{^{'} U} + x_{31}^{^{'} L} \leq 157 \end{aligned}

\begin{aligned} x_{13}^{U} + x_{13}^{L} + 8 x_{23} + x_{33}^{L} + x_{33}^{U} + x_{13}^{^{'} L} + x_{13}^{^{'} U} + x_{33}^{^{'} U} + x_{33}^{^{'} L} \leq 80 \end{aligned}

\begin{aligned} x_{12}^{U} + x_{12}^{L} + 8 x_{22} + x_{32}^{L} + x_{32}^{U} + x_{12}^{^{'} L} + x_{12}^{^{'} U} + x_{32}^{^{'} U} + x_{32}^{^{'} L} \leq 244 \end{aligned}

\begin{aligned} x_{14}^{U} + x_{14}^{L} + 8 x_{24} + x_{34}^{L} + x_{34}^{U} + x_{14}^{^{'} L} + x_{14}^{^{'} U} + x_{34}^{^{'} U} + x_{34}^{^{'} L} \leq 320 \end{aligned}

\begin{aligned} x_{11}^{^{'} L} \leq x_{11}^{^{'} U} \leq x_{11}^{U} \leq x_{11}^{L} \leq x_{21} \leq x_{31}^{L} \leq x_{31}^{U} \leq x_{31}^{^{'} U} \leq x_{31}^{^{'} L} \end{aligned}

\begin{aligned} x_{12}^{^{'} L} \leq x_{12}^{^{'} U} \leq x_{12}^{U} \leq x_{12}^{L} \leq x_{22} \leq x_{32}^{L} \leq x_{32}^{U} \leq x_{32}^{^{'} U} \leq x_{32}^{^{'} L} \end{aligned}

\begin{aligned} x_{13}^{^{'} L} \leq x_{13}^{^{'} U} \leq x_{13}^{U} \leq x_{13}^{L} \leq x_{23} \leq x_{33}^{L} \leq x_{33}^{U} \leq x_{33}^{^{'} U} \leq x_{33}^{^{'} L} \end{aligned}

\begin{aligned} x_{14}^{^{'} L} \leq x_{14}^{^{'} U} \leq x_{14}^{U} \leq x_{14}^{L} \leq x_{24} \leq x_{34}^{L} \leq x_{34}^{U} \leq x_{34}^{^{'} U} \leq x_{34}^{^{'} L} \end{aligned}

\begin{aligned} x_{11}^{^{'} L}, x_{12}^{^{'} L}, x_{13}^{^{'} L}, x_{14}^{^{'} L} \geq 0} \end{aligned}

Table 2 presents the precise optimal solutions for the choice variables and objective functions at each level.

Table 2
The Precise Optimal Solutions for the Objective Functions and Decision Factors.

$D M_{1}$ $D M_{2}$ $D M_{3}$

${\tilde{x}}_{1}^{I}$ ${\begin{matrix} (0, 0, 0, 0, 0), \\ (0, 0, 0, 0, 0) \end{matrix}}$ ${\begin{matrix} (0, 0, 0, 0, 0), \\ (0, 0, 0, 0, 0) \end{matrix}}$ ${\begin{matrix} (0, 0, 0, 0, 0), \\ (0, 0, 0, 0, 0) \end{matrix}}$

${\tilde{x}}_{2}^{I}$ ${\begin{matrix} (0, 0, 0, 0, 0), \\ (0, 0, 0, 0, 0) \end{matrix}}$ ${\begin{matrix} (0, 0, 0, 0, 0), \\ (0, 0, 0, 0, 0) \end{matrix}}$ ${\begin{matrix} (2.2429, 2.2429, 2.2429, 2.2429, 2.2429), \\ (2.2429, 2.2429, 2.2429, 2.2429, 2.2429) \end{matrix}}$

${\tilde{x}}_{3}^{I}$ ${\begin{matrix} (0, 0, 0, 0, 0), \\ (0, 0, 0, 0, 0) \end{matrix}}$ ${\begin{matrix} (0, 0, 0, 0, 0), \\ (0, 0, 0, 0, 0) \end{matrix}}$ ${\begin{matrix} (2.9074, 2.9074, 2.9074, 2.9074, 2.9074), \\ (2.9074, 2.9074, 2.9074, 2.9074, 2.9074) \end{matrix}}$

${\tilde{x}}_{4}^{I}$ ${\begin{matrix} (0, 0, 0, 0, 0), \\ (0, 0, 0, 0, 0) \end{matrix}}$ ${\begin{matrix} (0, 0, 0, 0, 0), \\ (0, 0, 0, 0, 0) \end{matrix}}$ ${\begin{matrix} (0, 0, 0, 0, 0), \\ (0, 0, 0, 0, 0) \end{matrix}}$

$f *$ $190.9375$ $680$ $895.2268$

$f^{-}$ $190.9375$ $527.89136$ 0

	$D M_{1}$	$D M_{2}$	$D M_{3}$
${\tilde{x}}_{1}^{I}$	${\begin{matrix} (0, 0, 0, 0, 0), \\ (0, 0, 0, 0, 0) \end{matrix}}$	${\begin{matrix} (0, 0, 0, 0, 0), \\ (0, 0, 0, 0, 0) \end{matrix}}$	${\begin{matrix} (0, 0, 0, 0, 0), \\ (0, 0, 0, 0, 0) \end{matrix}}$
${\tilde{x}}_{2}^{I}$	${\begin{matrix} (0, 0, 0, 0, 0), \\ (0, 0, 0, 0, 0) \end{matrix}}$	${\begin{matrix} (0, 0, 0, 0, 0), \\ (0, 0, 0, 0, 0) \end{matrix}}$	${\begin{matrix} (2.2429, 2.2429, 2.2429, 2.2429, 2.2429), \\ (2.2429, 2.2429, 2.2429, 2.2429, 2.2429) \end{matrix}}$
${\tilde{x}}_{3}^{I}$	${\begin{matrix} (0, 0, 0, 0, 0), \\ (0, 0, 0, 0, 0) \end{matrix}}$	${\begin{matrix} (0, 0, 0, 0, 0), \\ (0, 0, 0, 0, 0) \end{matrix}}$	${\begin{matrix} (2.9074, 2.9074, 2.9074, 2.9074, 2.9074), \\ (2.9074, 2.9074, 2.9074, 2.9074, 2.9074) \end{matrix}}$
${\tilde{x}}_{4}^{I}$	${\begin{matrix} (0, 0, 0, 0, 0), \\ (0, 0, 0, 0, 0) \end{matrix}}$	${\begin{matrix} (0, 0, 0, 0, 0), \\ (0, 0, 0, 0, 0) \end{matrix}}$	${\begin{matrix} (0, 0, 0, 0, 0), \\ (0, 0, 0, 0, 0) \end{matrix}}$
$f *$	$190.9375$	$680$	$895.2268$
$f^{-}$	$190.9375$	$527.89136$	0

Develop the MS function for the ith -LDM (i = 1,2,3) as delineated in problems (4)-(5) and address the Tch problem as outlined in problem (6) to achieve an acceptable solution that meets the requirements of all DMs. The preferred solution for the FIVMQP problem is ${\tilde{x}}_{1}^{I *} = {(0, 0, 0, 0, 0), (0, 0, 0, 0, 0)}$ , ${\tilde{x}}_{2}^{I *} = {\begin{matrix} (0.0164, 0.0164, 0.0164, 0.0164, 0.0164), \\ (0.0164, 0.0164, 0.0164, 0.0164, 0.0164) \end{matrix}}$ , ${\tilde{x}}_{3}^{I *} = {\begin{matrix} (2.9074, 2.9074, 2.9074, 2.9074, 2.9074), \\ (2.9074, 2.9074, 2.9074, 2.9074, 2.9074) \end{matrix}}$ , and ${\tilde{x}}_{4}^{I *} {\begin{matrix} (1.3208, 1.3208, 1.3208, 1.3208, 1.3208), \\ (1.3208, 1.3208, 1.3208, 1.3208, 1.3208) \end{matrix}}$ with optimal objective functions values are $f_{1} * = 124.1614$ , $f_{2} * = 674.819$ , and $f_{3} * = 660.6723$ .

7. Discussion

The FIVIMQP problem is now a widely utilized method across various fields, including engineering, finance, and economics. The resolution of the FIVIMQP problem is exceedingly challenging by conventional methods due to:

The utilization of IVIF numbers for decision variables and coefficients in the objective functions and constraints.

The quadratic nature of the objective functions.

The decision-making framework is hierarchical and comprises distinct, often opposing tiers.

The proposed approach introduces a structured framework for solving fully interval-valued intuitionistic fuzzy multilevel quadratic programming (FIVIFMQP) problems. The method integrates three key components: linearization of quadratic terms, transformation using the expected value function, and application of fuzzy programming techniques.

One of the main strengths of the proposed method lies in its ability to simultaneously handle:

Interval-valued intuitionistic fuzzy data,

Multilevel hierarchical decision structures, and

Quadratic objective functions.

This integration distinguishes the model from many existing approaches that typically address these features separately.

The linearization technique enables the transformation of nonlinear quadratic terms into an equivalent linear form, which facilitates the use of standard optimization solvers. However, this transformation increases the dimensionality of the problem, leading to higher computational cost. Therefore, a trade-off exists between modeling accuracy and computational efficiency.

The use of the expected value function provides a practical way to convert IVIF data into a solvable crisp model. While this step simplifies the computational process, it also represents a balance between preserving uncertainty information and ensuring tractability.

Despite these advantages, some limitations remain. The increase in problem size due to the introduction of auxiliary variables may restrict applicability to large-scale problems. Additionally, the reliance on expected value transformation may reduce the richness of uncertainty representation.

Future research may focus on:

Developing more efficient linearization or approximation techniques,

Extending the model to hybrid uncertainty environments.

Applying the method to real-world large-scale optimization problems.

8. Conclusion

Although hybrid uncertainty models combining probabilistic and fuzzy information are important, this study focuses on IVIF uncertainty as a balance between expressive modeling and computational tractability. This work delineates a method for addressing the FIVIFMQP problem utilizing IVIF numbers for all parameters and decision factors. The author initially employed a linearization technique to convert the quadratic objective function into a linear format. Subsequently, we applied the anticipated value function to all coefficients of the objective function and constraints to convert the issue into its precise counterpart. The extended fuzzy technique simplifies the crisp issue to a singular quadratic programming problem. A compromise solution to the FIVIFMQP problem was ultimately achieved by addressing the multilevel programming issue. The author provided an example to illustrate the application of the proposed method in addressing the FIVIFMQP problem. The presented example is designed to illustrate the applicability of the proposed model. However, further validation on large-scale and real-world problems is required to fully assess the computational performance and scalability of the approach. This remains an important direction for future research. In additional this study provided a focused review of uncertainty models in multilevel quadratic programming, systematically classifying previous contributions according to the type of uncertainty, the approaches used to handle it, and whether uncertainty affects parameters or decision variables. This targeted review reinforces the manuscript's claim of novelty: to the best of our knowledge, there is no existing algorithm specifically designed for the FIVIFMQP problem. By clearly locating this gap in the literature and proposing a practical solution, the paper not only fills an important methodological void but also outlines concrete directions for future research and application. Future research may focus on extending the model to hybrid uncertainty environments, improving computational efficiency, and applying the approach to large-scale real-world problems, and applying the approach to large-scale real-world problems, and an empirical benchmarking study comparing the proposed uncertainty-handling framework against recently developed techniques in related optimization settings to further substantiate its computational and solution-quality performance.

Footnotes

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

No data was used, generated, or analysed in the course of this study. Consequently, the Data Availability section is not applicable.

ORCID iDs

Eman Fathy

Mohamed Abdelhamid Helmy

References

Acar

Haktanır

Temur

G. T.

Beskese

(2024). Sustainable stationary hydrogen storage application selection with interval-valued intuitionistic fuzzy AHP. Int. J. Hydrogen Energy, 49, 619–634. https://doi.org/10.1016/j.ijhydene.2023.10.081

Alghazzawi

Alolaiyan

Ashfaq

Shuaib

Khalifa

H. A. E. W.

Gomaa

H. G.

Xin

(2024). Selecting an optimal approach to reduce energy crises under interval-valued intuitionistic fuzzy environment. Sci. Rep., 14(1), 8713. https://doi.org/10.1038/s41598-024-57164-1

Alolaiyan

Razaq

Ashfaq

Alghazzawi

Shuaib

Liu

J. B.

(2024). Improving similarity measures for modeling real-world issues with interval-valued intuitionistic fuzzy sets. IEEE Access, 12, 10482–10496. https://doi.org/10.1109/ACCESS.2024.3351205

Anandalingam

(1988). A mathematical programming model of decentralized multi-level systems. J. Oper. Res. Soc, 39(11), 1021–1033. https://doi.org/10.1057/jors.1988.172

Anithakumari

Venkateswarlu

Akilbasha

(2021). Optimizing a fully rough interval integer solid transportation problems. J. Intell. Fuzzy Syst, 41(1), 2429–2439. https://doi.org/10.3233/JIFS-202373

Atanassov

(2016). Intuitionistic fuzzy sets. Int. J. Bioautomation, 20(1), 1–6. https://doi.org/10.1016/S0165-0114(86)80034-3.

Atanassov

K. T.

(1999). Interval valued intuitionistic fuzzy sets. Physica-Verlag HD, 35, 139–177. https://doi.org/10.1007/978-3-7908-1870-3_2.

Bharati

S. K.

Singh

S. R.

(2018). Transportation problem under interval-valued intuitionistic fuzzy environment. Int. J. Fuzzy Syst, 20(5), 1511–1522. https://doi.org/10.1007/s40815-018-0470-y

Bilişik

ÖN

Duman

N. H.

Taş

(2024). A novel interval-valued intuitionistic fuzzy CRITIC–TOPSIS methodology: Application for transportation mode selection problem for a glass production company. Expert Syst. Appl., 235, 121134. https://doi.org/10.1016/j.eswa.2023.121134

10.

Bit

A. K.

(2004). Fuzzy programming with hyperbolic membership functions for multiobjective capacitated transportation problem. Opsearch, 41(2), 106–120. https://doi.org/10.1007/BF03398837

11.

Boudjellal

Benterki

(2024). A new full-newton step feasible interior point method for convex quadratic programming. Optimization, 73(5), 1571–1588. https://doi.org/10.1080/02331934.2023.2169047

12.

Choudhary

Yadav

S. P.

(2022). An approach to solve interval valued intuitionistic fuzzy transportation problem of type-2. Int. J. Syst. Assur. Eng. Manag, 13(6), 2992–3001. https://doi.org/10.1007/s13198-022-01771-6

13.

Cifuentes

Dey

S. S.

( 2024). Sensitivity analysis for mixed binary quadratic programming. Int. Conf. Integer Program. Combin. Optim., Cham: Springer Nature, pp. 446–459.

14.

Cuong

T. H.

Lim

Yen

N. D.

(2024). On a solution method in indefinite quadratic programming under linear constraints. Optimization, 73(4), 1087–1112. https://doi.org/10.1080/02331934.2022.2141056

15.

Dalman

Sivri

(2017). Multi-objective solid transportation problem in uncertain environment. Ir. J. Sci. Technol., Trans. A: Sci, 41(2), 505–514. https://doi.org/10.1007/s40995-017-0254-5

16.

Das

Bera

U. K.

Maiti

(2019). A solid transportation problem in uncertain environment involving type-2 fuzzy variable. Neural Comput. Appl., 31(9), 4903–4927. https://doi.org/10.1007/s00521-018-03988-8

17.

Dong

J. Y.

Wan

S. P.

(2024). Interval-valued intuitionistic fuzzy best–worst method with additive consistency. Expert Syst. Appl., 236, 121213. https://doi.org/10.1016/j.eswa.2023.121213

18.

Jin

(2024). A zeroing-feedback gradient-based neural dynamics model for solving dynamic quadratic programming problems with linear equation constraints in finite time. Neural Comput. Appl, 36(26), 16395–16409. https://doi.org/10.1007/s00521-024-09762-3

19.

Fatemi

Ketabi

Mansouri

S. A.

(2023). A multi-level multi-objective strategy for eco-environmental management of electricity market among micro-grids under high penetration of smart homes, PEVs and energy storage devices. J. Energy Storage, 67, 107632. https://doi.org/10.1016/j.est.2023.107632

20.

Fathy

Ammar

(2023). On neutrosophic multi-level multi-objective linear programming problem with application in transportation problem. J. Intell. Fuzzy Syst, 44(2), 2251–2267. https://doi.org/10.3233/JIFS-211374

21.

Fathy

Ammar

Helmy

M. A.

(2022). On solving the multilevel rough interval linear programming problem. J. Intell. Fuzzy Syst, 42(4), 3011–3028. https://doi.org/10.3233/JIFS-210694

22.

Fathy

Ammar

Helmy

M. A.

(2023a). Fully intuitionistic fuzzy multi-level linear fractional programming problem. Alex. Eng. J, 77, 684–694. https://doi.org/10.1016/j.aej.2023.07.018

23.

Fathy

Ammar

Helmy

M. A.

(2023b). Three different optimization techniques for solving the fully rough interval multi-level linear programming problem. J. Intell. Fuzzy Syst, 45(2), 2593–2610. https://doi.org/10.3233/JIFS-230057

24.

Giri

P. K.

Maiti

M. K.

Maiti

(2014). Fuzzy stochastic solid transportation problem using fuzzy goal programming approach. Comput. Ind. Eng, 72, 160–168. https://doi.org/10.1016/j.cie.2014.03.001

25.

Goshu

N. N.

Kassa

S. M.

(2024). A solution method for stochastic multilevel programming problems: A systematic sampling evolutionary approach. Oper. Res. Decis, 34(1), 149–174. https://doi.org/10.37190/ord240108

26.

Hulsurkar

Biswal

M. P.

Sinha

S. B.

(1997). Fuzzy programming approach to multi-objective stochastic linear programming problems. Fuzzy Sets Syst, 88(2), 173–181. https://doi.org/10.1016/S0165-0114(96)00056-5

27.

Iskander

M. G.

(2006). Exponential membership function in stochastic fuzzy goal programming. Appl. Math. Comput, 173(2), 782–791. https://doi.org/10.1016/j.amc.2005.04.014

28.

Jebadass

J. R.

Balasubramaniam

(2024). Color image enhancement technique based on interval-valued intuitionistic fuzzy set. Inf. Sci., 653, 119811. https://doi.org/10.1016/j.ins.2023.119811

29.

Jiménez

Verdegay

J. L.

(1999). An evolutionary algorithm for interval solid transportation problems. Evol. Comput, 7(1), 103–107. https://doi.org/10.1162/evco.1999.7.1.103

30.

Kousar

Sangi

M. N.

Kausar

Pamučar

Ozbilge

Cagin

(2023). Multi-objective optimization model for uncertain crop production under neutrosophic fuzzy environment: A case study. AIMS Math, 8(3), 7584–7605. https://doi.org/10.3934/math.2023380

31.

Kumar

P. S.

(2018). PSK Method for solving intuitionistic fuzzy solid transportation problems. Int. J. Fuzzy Syst. Appl, 7(4), 62–99. https://doi.org/10.4018/IJFSA.2018100104

32.

Kushwah

Sharma

(2024). An algorithm to solve multi-objective integer quadratic programming problem. Ann. Oper. Res, 332(1), 433–459. https://doi.org/10.1007/s10479-022-05123-2

33.

R. J.

Lee

E. S.

(1991). An exponential membership function for fuzzy multiple objective linear programming. Comput. Math. Appl, 22(12), 55–60. https://doi.org/10.1016/0898-1221(91)90147-V

34.

Mahajan

Gupta

S. K.

Ahmad

(2024). On fuzzy fractional quadratic programming problems with an application in the tourism sector. Appl. Soft Comput, 150, 111069. https://doi.org/10.1016/j.asoc.2023.111069

35.

Malik

Gupta

S. K.

(2024). On basic arithmetic operations for interval-valued intuitionistic fuzzy sets using the hamming distance with their application in decision making. Expert Syst. Appl., 239, 122429. https://doi.org/10.1016/j.eswa.2023.122429

36.

Mandal

Gazi

K. H.

Salahshour

Mondal

S. P.

Bhattacharya

Saha

A. K.

(2024). Application of interval-valued intuitionistic fuzzy uncertain MCDM methodology for ph.D. Supervisor selection problem. Results Control Optim, 15, 100411. https://doi.org/10.1016/j.rico.2024.100411

37.

Mangalore

A. R.

Fonseca

G. A.

Risbud

S. R.

Stratmann

Wild

(2024). Neuromorphic quadratic programming for efficient and scalable model predictive control: towards advancing speed and energy efficiency in robotic control. IEEE Robot. Autom. Mag, 32(2), 69–79. https://doi.org/10.1109/MRA.2024.3415005

38.

Nguyen

Q. M.

Onur

Alpak

F. O.

(2024). Multi-objective optimization of subsurface CO₂ capture, utilization, and storage using sequential quadratic programming with stochastic gradients. Comput. Geosci., 28(2), 195–210. https://doi.org/10.1007/s10596-023-10213-6

39.

Nouriani

Ezzati

(2020). Application of simpson quadrature rule and iterative method for solving nonlinear fuzzy delay integral equations. Fuzzy Sets Syst, 400, 147–161. https://doi.org/10.1016/j.fss.2020.01.007

40.

Ojha

Das

Mondal

Maiti

(2010). A stochastic discounted multi-objective solid transportation problem for breakable items using AHP. Appl. Math. Modell, 34(8), 2256–2271. https://doi.org/10.1016/j.apm.2009.10.034

41.

Osman

M. S.

Abo-Sinna

M. A.

Amer

A. H.

Emam

(2004). A multi-level non-linear multi-objective decision-making under fuzziness. Appl. Math. Comput, 153(1), 239–252. https://doi.org/10.1016/S0096-3003(03)00628-3

42.

Pramanik

Jana

D. K.

Maiti

(2013). Multi-objective solid transportation problem in imprecise environments. J. Transp. Secur, 6(2), 131–150. https://doi.org/10.1007/s12198-013-0108-0

43.

Pramanik

Jana

D. K.

Maiti

(2016). Bi-criteria solid transportation problem with substitutable and damageable items in disaster response operations on fuzzy rough environment. Socio-Econ. Plan. Sci, 55, 1–13. https://doi.org/10.1016/j.seps.2016.04.002

44.

Qin

Huang

(2024). A matheuristic approach for the multi-level capacitated lot-sizing problem with substitution and backorder. Int. J. Prod. Res, 62(13), 4645–4673. https://doi.org/10.1080/00207543.2023.2270076

45.

Sakawa

(1983). Interactive computer programs for fuzzy linear programming with multiple objectives. Int. J. Man-Mach. Stud, 18(5), 489–503. https://doi.org/10.1016/S0020-7373(83)80022-4

46.

Shen

Huang

(2024). An efficient and provable sequential quadratic programming method for American and swing option pricing. Eur. J. Oper. Res, 316(1), 19–35. https://doi.org/10.1016/j.ejor.2023.11.012

47.

Shivani

D. R.

(2024). Multi-objective multi-item four dimensional green transportation problem in interval-valued intuitionistic fuzzy environment. Int. J. Syst. Assur. Eng. Manag, 15(2), 727–744. https://doi.org/10.1007/s13198-022-01794-z

48.

Singer

Özşahin

(2024). An interval-valued intuitionistic fuzzy AHP model for understanding consumer decision-making in non-wood forest product purchases. J. Retail. Consum. Serv, 78, 103734. https://doi.org/10.1016/j.jretconser.2024.103734

49.

Singh

Pradhan

Biswal

M. P.

(2019). Multi-objective solid transportation problem under stochastic environment. Sādhanā, 44(5), 105. https://doi.org/10.1007/s12046-019-1094-0

50.

Sinha

Das

Bera

U. K.

(2016). Profit maximization solid transportation problem with trapezoidal interval type-2 fuzzy numbers. Int. J. Appl. Comput. Math, 2(1), 41–56. https://doi.org/10.1007/s40819-015-0044-8

51.

Tamilarasi

Paulraj

(2022). An improved solution for the neutrosophic linear programming problems based on Mellin's transform. Soft Comput, 26(17), 8497–8507. https://doi.org/10.1007/s00500-022-07252-z

52.

Tong

Zhu

(2024). An information-freshness-based digital twin model to support multi-level complementary dynamic scheduling in shared manufacturing. Adv. Eng. Inform., 61, 102525. https://doi.org/10.1016/j.aei.2024.102525

53.

Wan

Dong

(2020). Decision making theories and methods based on interval-valued intuitionistic fuzzy sets. Springer Nature.

54.

Yang

Feng

(2007). A bicriteria solid transportation problem with fixed charge under stochastic environment. Appl. Math. Modell, 31(12), 2668–2683. https://doi.org/10.1016/j.apm.2006.10.011

55.

Yao

Liu

Chang

Yang

(2023). Multi-level model predictive control based multi-objective optimal energy management of integrated energy systems considering uncertainty. Renew. Energy, 212, 523–537. https://doi.org/10.1016/j.renene.2023.05.082

56.

Zhang

Gao

(2015). Multi-level decision making. Models, Methods Appl, xvi–377. https://doi.org/10.1007/978-3-662-46059-7

57.

Zhou

Zhang

Chen

(2024). A water–land–energy–carbon nexus evaluation of agricultural sustainability under multiple uncertainties: Application of a multi-attribute group decision method determined by an interval-valued intuitionistic fuzzy set. Expert Syst. Appl., 242, 122833. https://doi.org/10.1016/j.eswa.2023.122833

		Nature of Uncertainty
Author	Year	Type	Coefficients	Variables	Quadratic	Multi-Level	Solution Method
Sakawa (1983)	1983	Fuzzy Environment	✓	×	×	×	Interactive fuzzy programming
Li and Lee (1991)	1991	Fuzzy Environment	✓	×	×	×	Exponential membership function
Hulsurkar et al. (1997)	1997	Fuzzy Environment	✓	×	×	×	Fuzzy stochastic programming
Jiménez and Verdegay (1999)	1999	Interval Environment	✓	×	×	×	Interval programming
Bit (2004)	2004	Fuzzy Environment	✓	×	×	×	Hyperbolic membership functions
Iskander (2006)	2006	Stochastic fuzzy Environment	✓	×	×	×	Stochastic fuzzy goal programming
Yang and Feng (2007)	2007	Stochastic Environment	✓	×	×	×	Stochastic programming
Ojha et al. (2010)	2010	Stochastic Environment	✓	×	×	×	Stochastic programming with AHP
Pramanik et al. (2013)	2013	Random Fuzzy Environment	✓	×	×	×	Generalized reduced gradient method
Giri et al. (2014)	2014	Fuzzy Stochastic Environment	✓	×	×	×	Fuzzy goal programming
Pramanik et al. (2016)	2016	Fuzzy Rough Environment	✓	×	×	×	An interactive fuzzy method
Sinha et al. (2016)	2016	Trapezoidal Interval Type-2 Fuzzy Environment	✓	×	×	×	the expected value
Dalman and Sivri (2017)	2017	Interval Environment	✓	×	×	×	Fuzzy programming,
Kumar (2018)	2018	Intuitionistic Fuzzy Environment	✓	✓	×	×	PSK method
Das et al. (2019)	2019	Type-2 Fuzzy Environment	✓	×	×	×	Weighted mean programming technique
Singh et al. (2019)	2019	Stochastic Environment	✓	×	×	×	Chance-constraint programming
Nouriani and Ezzati (2020)	2020	Fuzzy Environment	✓	×	×	×	Simpson quadrature rule
Anithakumari et al. (2021)	2021	Rough Interval Integers Environment	✓	✓	×	×	Modified slice-sum method
Fathy et al. (2022)	2022	Rough Interval Environment	✓	×	×	✓	Interval method
Choudhary and Yadav (2022)	2022	Interval Valued Intuitionistic Fuzzy Environment	✓	×	×	×	An algorithm is developed by author
Tamilarasi and Paulraj (2022)	2022	Neutrosophic Environmen	✓	×	×	×	Mellin transform based de-neutrosophication
Fathy and Ammar (2023)	2023	Neutrosophic environments	✓	×	×	✓	Harmonic mean technique
Fathy et al. (2023a)	2023	Intuitionistic Fuzzy Environment	✓	✓	×	✓	A simple approach based on fuzzy harmonic mean operators
Kousar et al. (2023)	2023	Neutrosophic Fuzzy Environment	✓	×	×	×	neutrosophic programming
Fathy et al. (2023b)	2023	Rough Interval Environment	✓	✓	×	✓	Modified slice-sum method
This Study		IVIF Environment	✓	✓	✓	✓	The expected value