A modified fuzzy approach for the fully fuzzy multi-objective and multi-level integer quadratic programming problems based on a decomposition technique

Abstract

This paper focuses on extending and applying a fuzzy approach for utilization with the fully fuzzy multi-objective and multi-level integer quadratic programming (FFMMQP) problems. First, the decomposition technique is used to convert the fuzzy problem for each level into three crisp multi-objective integer quadratic (MQP) problems namely, Middle-MQP, Upper-MQP and Lower-MQP problem. Each crisp problem has its own variables. Furthermore, the functions of each problem have several quadratic functions. Then by considering the individual solution of each objective function, the middle, upper and lower membership functions are constructed. Second, the concept of the tolerance membership function and multi-objective optimization in the decomposition form is used to establish decomposed Tchebycheff problems to achieve the Pareto optimal fuzzy solution for the FFMMQP problems. An example is provided to prove the theoretical results.

Keywords

Fully fuzzy programming multi-level programming multi-objective programming quadratic programming fuzzy approach

1 Introduction

The research on fuzzy linear programming (FLP) has risen highly since Bellman and Zadeh [4] proposed the concept of decision making in fuzzy environment. The most common approach to treat the challenge of solving the (FLP) problems is to change the fuzzy linear programming into the corresponding crisp linear programming [21 , 26]. By allowing all of the parameters and variables in the FLP problem to be fuzzy, we obtain a fully fuzzy linear programming (FFLP) problem [1 , 9]. Hepzibah and Umamaheswari [14] presented a new method for solving the fuzzy quadratic programming problem by converting it into a fuzzy linear complimentarity problem. A computational algorithm based on the degree of deviation is proposed to obtain a Pareto-optimal solution of the fully fuzzy multi-objective linear programming (FFMOLP) problem [5]. Sharma and Aggarwal [25] proposed a method to solve the FFMOLP problem based on the nearest interval approximation of the fuzzy number and interval programming. A fuzzy compromise solution for the FFMOLP problems was obtained by Hamadameen and Hassan [10]. Arana-Jimenez [3] presented a new method based on a multiple objective linear programming (MOLP) problem to determine a non-dominated solution for the FFLP problem.

The multi-level programming problems are attractive for many researchers because of their application in several areas such as competitive economic organization, biofuel production, supply chains, government systems, and so on. Several mathematical models for such problems have been exhibited [2 , 28]. Han et al. [12], in their work on the multi-level decision making, used the particle swarm optimization as an approach for solving the bi-level and tri-level programming problems. The authors also presented a survey of the theoretical research results and the related technique developments to multilevel decision-making [20]. Kumar and Rakshit [19] introduced the fuzzy goal programming (FGP) procedure for a fuzzy three level quadratic fractional programming problem and also for a fuzzy multi-level quadratic fractional programming problem [18]. They also modified the FGP in [17] to solve the bi-level quadratic fractional programming problem. Hosseini [13] developed an approach based on the Taylor series to solve the non-linear quadratic multi-level programming problem.

Due to the lack of efficient algorithms for tackling fully fuzzy multi-level programming problems, we have been motivated to work on developing a new algorithm or improving current methods to solve these types of problems more efficiently. The work on this paper is based on the concepts which were discussed in [22]. In [22], the authors discussed the fuzzy approach to deal with the three-level nonlinear multi-objective decision making. However, the existing method [22] cannot be used to find the Pareto optimal fuzzy solution of the FFMMQP problems. In this paper, we extend the same idea by explaining new methodological developments in fully fuzzy data.

The modified fuzzy approach will be expected to be applicable to real-life decision problems in many areas, such as engineering management, supply chain management, risk investment, transportation problems, and others.

This paper is organized into five sections. In the next section, some necessary definitions and arithmetic operations of triangular fuzzy numbers are reviewed. In Section 3, the fully fuzzy multi-objective and multi-level integer quadratic programming problems are formulated and a detailed methodology to solve this kind of problems is explained. An illustrative numerical example about the proposed method is presented in Section 4. Finally, a conclusion is provided in Section 5.

2 Preliminary concept

In this section, some necessary concepts of fuzzy sets and triangular fuzzy number are presented.

Definition 2.1 [8] A fuzzy number $\tilde{b} = (b_{1}, b_{2}, b_{3})$ is said to be a triangular fuzzy number if its membership function $μ_{\tilde{b}} (x)$ is defined as: $μ_{\tilde{b}} (x) = {\begin{matrix} \frac{x - b_{1}}{b_{2} - b_{1}}, b_{1} \leq x \leq b_{2}, \\ 1, x = b_{2}, \\ \frac{x - b_{3}}{b_{2} - b_{3}}, b_{2} \leq x \leq b_{3}, \\ 0, otherwise . \end{matrix}$

Definition 2.2 [16] Let $\tilde{b} = (b_{1}, b_{2}, b_{3})$ and $\tilde{c} = (c_{1}, c_{2}, c_{3})$ be two triangular fuzzy numbers. Then, the basic arithmetic operations will be defined as follows:

Addition: $\tilde{b} + \tilde{c} = (b_{1} + c_{1}, b_{2} + c_{2}, b_{3} + c_{3})$ ,

Subtraction: $\tilde{b} - \tilde{c} = (b_{1} - c_{3}, b_{2} - c_{2}, b_{3} - c_{1})$ ,

Scalar multiplication: $k \tilde{b} = ({kb}_{1}, {kb}_{2}, {kb}_{3})$ if k ≥ 0,

$k \tilde{b} = ({kb}_{3}, {kb}_{2}, {kb}_{1})$ if k < 0,

Multiplication: $\tilde{b} \times \tilde{c} = (b_{1} c_{1}, b_{2} c_{2}, b_{3} c_{3}), b_{1} \geq 0$ ,

$\tilde{b} \times \tilde{c} = (b_{1} c_{3}, b_{2} c_{2}, b_{3} c_{3}), b_{1} < 0$ ,

b₃ ≥ 0,

$\tilde{b} \times \tilde{c} = (b_{1} c_{3}, b_{2} c_{2}, b_{3} c_{1}), b_{3} < 0$

3 Methodology

3.1 Formulation of the fully fuzzy multi-objective and multi-level integer quadratic programming problems

Let’s consider the following fully fuzzy multi-objective and multi-level integer quadratic programming (FFMMQP) problems:

1st level decision maker [DM₁]: $\begin{matrix} max_{{\tilde{x}}_{1}} {\tilde{F}}_{1} (\tilde{x}) = max_{{\tilde{x}}_{1}} ({\tilde{f}}_{11} (\tilde{x}), {\tilde{f}}_{12} (\tilde{x}), \dots, {\tilde{f}}_{1 m_{1}} (\tilde{x})) \\ where {\tilde{x}}_{2}, {\tilde{x}}_{3}, \dots, {\tilde{x}}_{p} solves, \end{matrix}$

2nd level decision maker [DM₂]: $max_{{\tilde{x}}_{2}} {\tilde{F}}_{2} (\tilde{x}) = max_{{\tilde{x}}_{2}} ({\tilde{f}}_{21} (\tilde{x}), {\tilde{f}}_{22} (\tilde{x}), \dots, {\tilde{f}}_{2 m_{2}} (\tilde{x})),$

$where {\tilde{x}}_{3}, {\tilde{x}}_{4}, \dots, {\tilde{x}}_{p} solves$ ,

⋮

$where {\tilde{x}}_{p} solves$ ,

Pth level decision maker [DM_p]: $max_{{\tilde{x}}_{p}} {\tilde{F}}_{p} (\tilde{x}) = max_{{\tilde{x}}_{p}} ({\tilde{f}}_{p 1} (\tilde{x}), {\tilde{f}}_{p 2} (\tilde{x}), \dots, {\tilde{f}}_{{pm}_{p}} (\tilde{x})),$ subject to $\begin{matrix} \tilde{x} \in \tilde{G} = \\ {\tilde{x} \in R^{n} | \begin{matrix} \sum_{j = 1}^{n} {\tilde{A}}_{sj} \otimes {\tilde{x}}_{j} \leq {\tilde{b}}_{s}, (s = 1, 2, \dots, m) \\ {\tilde{x}}_{j} \geq 0 and integer for all j . \end{matrix}} \end{matrix}$ (1)

Where ${\tilde{x}}_{i} \in R^{n_{i}}, n_{i} \geq 1, (i = 1, 2, \dots, p)$ are the vector fuzzy decision variables demonstrating the control of each of the level decision makers, $\tilde{x} = (x_{1}, x_{2}, \dots, x_{p})$ , $\sum_{i = 1}^{p} n_{i} = n$ . $\tilde{G}$ is the fuzzy feasible choice, ${\tilde{A}}_{sj}$ is m × n fuzzy coefficients matrix, ${\tilde{b}}_{s}, (s = 1, 2, \dots, m)$ is a fuzzy vector and ${\tilde{F}}_{i} (\tilde{x}) = ({\tilde{f}}_{i 1} (\tilde{x}), {\tilde{f}}_{i 2} (\tilde{x}), \dots, {\tilde{f}}_{{im}_{i}} (\tilde{x}))$ , (i = 1, 2, …, p) is the vector of m_ispecific fuzzy quadratic objective functions of the ith level decision maker,

$\begin{matrix} {\tilde{f}}_{ir} (\tilde{x}) = \sum_{j = 1}^{n} {\tilde{c}}_{j}^{ir} \otimes {\tilde{x}}_{j} \oplus \frac{1}{2} \sum_{j = 1}^{n} \\ \sum_{k = 1}^{n} {\tilde{x}}_{k} \otimes {\tilde{L}}_{kj}^{ir} \otimes {\tilde{x}}_{j}, (i = 1, 2, \dots, p) \end{matrix}$ (2)

r = 1, 2, …, m₁ for the 1st level objective functions,

r = 1, 2, …, m₂ for the 2nd level objective functions,

⋮

r = 1, 2, …, m_p for the pth level objective functions.

Where ${\tilde{c}}_{j}^{ir}$ is the 1 × n fuzzy matrix representing the coefficients of the linear terms, and ${\tilde{L}}_{kj}^{ir}$ is the n × n fuzzy matrix expressing the coefficients of the quadratic terms.

3.2 Defuzzification process

Jayalakshmi and Pandian [15] presented the bound and decomposition method to determine an optimal fuzzy solution for the fully fuzzy linear programming (FFLP) problems. Conforming to this method, the FFMMQP problems are decomposed into three crisp multi-objective integer quadratic programming MQP problems with bounded variable constraints for each level, and the three crisp MQP problems are solved one at a time.

Let all of the fuzzy parameters and the fuzzy variables of both objective functions and constraints for each level in Equations (1) and (2) be represented by triangular fuzzy numbers. Each of these has the following values: $\begin{matrix} {\tilde{F}}_{i} = (u_{i}^{1}, u_{i}^{2}, u_{i}^{3}), {\tilde{f}}_{ir} = (z_{ir}^{1}, z_{ir}^{2}, z_{ir}^{3}), {\tilde{c}}_{j}^{ir} = (α_{j}^{ir}, β_{j}^{ir}, γ_{j}^{ir}), \\ {\tilde{L}}_{kj}^{ir} = (h_{kj}^{ir}, v_{kj}^{ir}, w_{kj}^{ir}), {\tilde{x}}_{j} = (x_{j}, y_{j}, t_{j}), {\tilde{x}}_{k} = (x_{k}, y_{k}, t_{k}), \\ {\tilde{A}}_{sj} = (a_{sj}^{1}, a_{sj}^{2}, a_{sj}^{3}) and {\tilde{b}}_{s} = (b_{s}^{1}, b_{s}^{2}, b_{s}^{3}) \end{matrix}$

Hence, Equation (1) can be reformulated as follows:

1st level decision maker[DM₁]: $\begin{matrix} max_{(x_{1}, y_{1}, t_{1})} (u_{1}^{1}, u_{1}^{2}, u_{1}^{3}) = max_{(x_{1}, y_{1}, t_{1})} (z_{1 r}^{1}, z_{1 r}^{2}, z_{1 r}^{3}), \\ r = 1, 2, \dots, m_{1} \end{matrix}$

where (x₂, y₂, t₂) , (x₃, y₃, t₃) , …, (x_p, y_p, t_p) solves

2nd level decision maker[DM₂]: $\begin{matrix} max_{(x_{2}, y_{2}, t_{2})} (u_{2}^{1}, u_{2}^{2}, u_{2}^{3}) = max_{(x_{2}, y_{2}, t_{2})} (z_{2 r}^{1}, z_{2 r}^{2}, z_{2 r}^{3}), \\ r = 1, 2, \dots, m_{2} \end{matrix}$

where (x₃, y₃, t₃) , (x₄, y₄, t₄) , …, (x_p, y_p, t_p) solves

⋮

where (x_p, y_p, t_p) solves,

Pth level decision maker[DM_p]: $\begin{matrix} max_{(x_{p}, y_{p}, t_{p})} (u_{p}^{1}, u_{p}^{2}, u_{p}^{3}) = max_{(x_{p}, y_{p}, t_{p})} (z_{pr}^{1}, z_{pr}^{2}, z_{pr}^{3}), \\ r = 1, 2, \dots, m_{p} \end{matrix}$

subject to $x \in G = {x \in R^{n} |$ $\begin{matrix} \sum_{j = 1}^{n} (a_{sj}^{1}, a_{sj}^{2}, a_{sj}^{3}) \otimes (x_{j}, y_{j}, t_{j}) \leq (b_{s}^{1}, b_{s}^{2}, b_{s}^{3}), \\ (s = 1, 2, \dots, m) \end{matrix}$ $x_{j}, y_{j}, t_{j} \geq 0 and integer for all j}$ (3)

And the crisp form of the quadratic objective functions of the ith-LDM is obtained as follows: $\begin{matrix} {\tilde{f}}_{ir} = (z_{ir}^{1}, z_{ir}^{2}, z_{ir}^{3}) = \\ \sum_{j = 1}^{n} (α_{j}^{ir}, β_{j}^{ir}, γ_{j}^{ir}) \otimes (x_{j}, y_{j}, t_{j}) \oplus \\ \frac{1}{2} \sum_{j = 1}^{n} \sum_{k = 1}^{n} (x_{k}, y_{k}, t_{k}) \otimes (h_{kj}^{ir}, v_{kj}^{ir}, w_{kj}^{ir}) \otimes (x_{j}, y_{j}, t_{j}) \end{matrix}$ (4)

Now, using the arithmetic operations as stated in Definition 2.2, Equation (3) can be changed into the following deterministic MQP problem for each level:

ith level decision maker [DM_i] , (i = 1, 2, …, p): $\begin{matrix} max_{x_{i}} u_{i}^{1} = max_{x_{i}} z_{ir}^{1} = \\ \sum_{j = 1}^{n} lower value of ((α_{j}^{ir}, β_{j}^{ir}, γ_{j}^{ir}) \otimes (x_{j}, y_{j}, t_{j})) \\ \oplus \frac{1}{2} \sum_{j = 1}^{n} \sum_{k = 1}^{n} lower value of \\ ((x_{k}, y_{k}, t_{k}) \otimes (h_{kj}^{ir}, v_{kj}^{ir}, w_{kj}^{ir}) \otimes (x_{j}, y_{j}, t_{j})) \end{matrix}$ $\begin{matrix} max_{y_{i}} u_{i}^{2} = max_{y_{i}} z_{ir}^{2} = \\ \sum_{j = 1}^{n} middle value of \\ ((α_{j}^{ir}, β_{j}^{ir}, γ_{j}^{ir}) \otimes (x_{j}, y_{j}, t_{j})) \\ \oplus \frac{1}{2} \sum_{j = 1}^{n} \sum_{k = 1}^{n} middle value of \\ ((x_{k}, y_{k}, t_{k}) \otimes (h_{kj}^{ir}, v_{kj}^{ir}, w_{kj}^{ir}) \otimes (x_{j}, y_{j}, t_{j})) \end{matrix}$ $\begin{matrix} max_{t_{i}} u_{i}^{3} = max_{t_{i}} z_{ir}^{3} = \\ \sum_{j = 1}^{n} upper value of ((α_{j}^{ir}, β_{j}^{ir}, γ_{j}^{ir}) \otimes (x_{j}, y_{j}, t_{j})) \\ \oplus \frac{1}{2} \sum_{j = 1}^{n} \sum_{k = 1}^{n} upper value of \\ ((x_{k}, y_{k}, t_{k}) \otimes (h_{kj}^{ir}, v_{kj}^{ir}, w_{kj}^{ir}) \otimes (x_{j}, y_{j}, t_{j})) \end{matrix}$

where (x_j, y_j, t_j) solves, (j = i + 1, 2, …, p), (r = 1, 2, …, m_i)

subject to $\begin{matrix} x \in G = {\sum_{j = 1}^{n} lower value of \\ (a_{sj}^{1}, a_{sj}^{2}, a_{sj}^{3}) \otimes (x_{j}, y_{j}, t_{j}) \leq (b_{s}^{1}, b_{s}^{2}, b_{s}^{3}), \\ \sum_{j = 1}^{n} middle value of \\ (a_{sj}^{1}, a_{sj}^{2}, a_{sj}^{3}) \otimes (x_{j}, y_{j}, t_{j}) \leq (b_{s}^{1}, b_{s}^{2}, b_{s}^{3}), \\ \sum_{j = 1}^{n} upper value of \\ (a_{sj}^{1}, a_{sj}^{2}, a_{sj}^{3}) \otimes (x_{j}, y_{j}, t_{j}) \leq (b_{s}^{1}, b_{s}^{2}, b_{s}^{3}), \\ x_{j} \leq y_{j} \leq t_{j}, \\ x_{j}, y_{j}, t_{j} \geq 0 and integer for all j} \end{matrix}$ (5)

The relationship x_j ≤ y_j ≤ t_j, (j = 1, 2, …, n), and is called a bounded constraint.

From the decomposition of Equation (5), three crisp MQP problems, namely, the middle MQP problem (M-MQP), the upper MQP problem (U-MQP) and the lower MQP problem (L-MQP) are constructed as follows:

ith level decision maker [DM_i] , (i = 1, 2, …, p): $\begin{matrix} (M - MQP) : \\ max_{y_{i}} u_{i}^{2} = max_{y_{i}} z_{ir}^{2} = \\ \sum_{j = 1}^{n} middle value of ((α_{j}^{ir}, β_{j}^{ir}, γ_{j}^{ir}) \otimes (x_{j}, y_{j}, t_{j})) \\ \oplus \frac{1}{2} \sum_{j = 1}^{n} \sum_{k = 1}^{n} middle value of \\ ((x_{k}, y_{k}, t_{k}) \otimes (h_{kj}^{ir}, v_{kj}^{ir}, w_{kj}^{ir}) \otimes (x_{j}, y_{j}, t_{j})), \end{matrix}$

where y_j solves, (j = i + 1, 2, …, p)

(r = 1, 2, …, m_i),

subject to $\begin{matrix} x \in G^{M} = {\sum_{j = 1}^{n} middle value of \\ ((a_{sj}^{1}, a_{sj}^{2}, a_{sj}^{3}) \otimes (x_{j}, y_{j}, t_{j}) \leq (b_{s}^{1}, b_{s}^{2}, b_{s}^{3})), \\ x_{j} \leq y_{j} \leq t_{j}, \\ x_{j}, y_{j}, t_{j} \geq 0 and integer for all j} \end{matrix}$ (6) $\begin{matrix} (U - MQP) : \\ max_{t_{i}} u_{i}^{3} = max_{t_{i}} z_{ir}^{3} = \\ \sum_{j = 1}^{n} upper value of ((α_{j}^{ir}, β_{j}^{ir}, γ_{j}^{ir}) \otimes (x_{j}, y_{j}, t_{j})) \\ \oplus \frac{1}{2} \sum_{j = 1}^{n} \sum_{k = 1}^{n} upper value of \\ ((x_{k}, y_{k}, t_{k}) \otimes (h_{kj}^{ir}, v_{kj}^{ir}, w_{kj}^{ir}) \otimes (x_{j}, y_{j}, t_{j})), \end{matrix}$

where (t_j) solves, (j = i + 1, 2, …, p),

(r = 1, 2, …, m_i)

subject to

$\begin{matrix} x \in G^{U} = {\sum_{j = 1}^{n} upper value of \\ ((a_{sj}^{1}, a_{sj}^{2}, a_{sj}^{3}) \otimes (x_{j}, y_{j}, t_{j}) \leq (b_{s}^{1}, b_{s}^{2}, b_{s}^{3})), \\ (\sum_{j = 1}^{n} upper value of ((α_{j}^{ir}, β_{j}^{ir}, γ_{j}^{ir}) \otimes (x_{j}, y_{j}, t_{j})) \\ \oplus \frac{1}{2} \sum_{j = 1}^{n} \sum_{k = 1}^{n} upper value of \\ ((x_{k}, y_{k}, t_{k}) \otimes (h_{kj}^{ir}, v_{kj}^{ir}, w_{kj}^{ir}) \otimes (x_{j}, y_{j}, t_{j}))) \\ \geq z_{i r}^{\circ 2}, x_{j} \leq y_{j} \leq t_{j}, \\ x_{j}, y_{j}, t_{j} \geq 0 and integer for all j} \end{matrix}$ (7)

Furthermore, $\begin{matrix} (L - MQP) : \\ max_{x_{i}} u_{i}^{1} = max_{x_{i}} z_{ir}^{1} = \\ \sum_{j = 1}^{n} lower value of ((α_{j}^{ir}, β_{j}^{ir}, γ_{j}^{ir}) \otimes (x_{j}, y_{j}, t_{j})) \\ \oplus \frac{1}{2} \sum_{j = 1}^{n} \sum_{k = 1}^{n} lower value of \\ ((x_{k}, y_{k}, t_{k}) \otimes (h_{kj}^{ir}, v_{kj}^{ir}, w_{kj}^{ir}) \otimes (x_{j}, y_{j}, t_{j})) \end{matrix}$

where x_j solves, (j = i + 1, 2, …, p)

(r = 1, 2, …, m_i)

subject to $\begin{matrix} x \in G^{L} = {\sum_{j = 1}^{n} lower value of \\ ((a_{sj}^{1}, a_{sj}^{2}, a_{sj}^{3}) \otimes (x_{j}, y_{j}, t_{j}) \leq (b_{s}^{1}, b_{s}^{2}, b_{s}^{3})), \\ (\sum_{j = 1}^{n} lower value of \\ ((α_{j}^{ir}, β_{j}^{ir}, γ_{j}^{ir}) \otimes (x_{j}, y_{j}, t_{j})) \end{matrix}$

$\begin{matrix} \oplus \frac{1}{2} \sum_{j = 1}^{n} \sum_{k = 1}^{n} lower value of \\ ((x_{k}, y_{k}, t_{k}) \otimes (h_{kj}^{ir}, v_{kj}^{ir}, w_{kj}^{ir}) \otimes (x_{j}, y_{j}, t_{j}))) \\ \geq z_{i r}^{\circ 2}, x_{j} \leq y_{j} \leq t_{j}, \\ x_{j}, y_{j}, t_{j} \geq 0 and integer for all j} \end{matrix}$ (8) Where $z_{i r}^{\circ 2}$ , (r = 1, 2, …, m_i ; i = 1, 2, …, p) is the optimal objective value of the M-MQP problem.

3.3 Fuzzy programming formulation

To prepare the fuzzy programming model [22] to achieve the Pareto optimal fuzzy solution of Equation (5), all of the objective functions at each level and decision vectors are reconstructed into fuzzy goals by selecting imprecise aspiration levels for them. Then, they are characterized by the associated membership functions by describing tolerance limits for the achievement of the aspired levels of the corresponding fuzzy goals.

3.3.1 Construction of membership functions

Let $f_{ir}^{b} = (z_{ir}^{b 1}, z_{ir}^{b 2}, z_{ir}^{b 3})$ and $f_{ir}^{w} = (z_{ir}^{w 1}, z_{ir}^{w 2}, z_{ir}^{w 3})$ , (r = 1, 2, …, m_i ; i = 1, 2, …, p) be the individual best solutions and the individual worst solutions for each objective function, respectively. Where $z_{ir}^{b 2} = max_{x \in G^{M}} z_{ir}^{2}, z_{ir}^{w 2} = min_{x \in G^{M}} z_{ir}^{2}$ (9) $z_{ir}^{b 1} = max_{x \in G^{L}} z_{ir}^{1}, z_{ir}^{w 1} = min_{x \in G^{L}} z_{ir}^{1}$ (10) $z_{ir}^{b 3} = max_{x \in G^{U}} z_{ir}^{3}, z_{ir}^{w 3} = min_{x \in G^{U}} z_{ir}^{3}$ (11)

As obtained above, the optimal solution of each objective function is described by the triangular fuzzy number. The membership function for the defined fuzzy decision of the rth objective function at the ith level is decomposed into three sub-membership functions, namely, the middle-membership functions (M-MF), the upper-membership functions (U-MF) and the lower-membership functions (L-MF).

The sub-membership functions for the rth objective function at the ith level are formulated in the following manner: $\begin{matrix} (M - MF) : μ_{z_{ir}^{2}}^{M} (z_{ir}^{2} (x)) = \\ {\begin{matrix} 1, z_{ir}^{2} (x) \geq z_{ir}^{b 2}, \\ \frac{z_{ir}^{2} (x) - z_{ir}^{w 2}}{z_{ir}^{b 2} - z_{ir}^{w 2}}, z_{ir}^{w 2} \leq z_{ir}^{2} (x) \leq z_{ir}^{b 2}, \\ 0, z_{ir}^{w 2} > z_{ir}^{2} (x) \end{matrix} \end{matrix}$ (12)

$\begin{matrix} (U - MF) : μ_{z_{ir}^{3}}^{U} (z_{ir}^{3} (x)) = \\ {\begin{matrix} 1, z_{ir}^{3} (x) \geq z_{ir}^{b 3}, \\ \frac{z_{ir}^{3} (x) - z_{ir}^{w 3}}{z_{ir}^{b 3} - z_{ir}^{w 3}}, z_{ir}^{w 3} \leq z_{ir}^{3} (x) \leq z_{ir}^{b 3}, \\ 0, z_{ir}^{w 3} > z_{ir}^{3} (x) \end{matrix} \end{matrix}$ (13)

and

$\begin{matrix} (L - MF) : μ_{z_{ir}^{1}}^{L} (z_{ir}^{1} (x)) = \\ {\begin{matrix} 1, z_{ir}^{1} (x) \geq z_{ir}^{b 1}, \\ \frac{z_{ir}^{1} (x) - z_{ir}^{w 1}}{z_{ir}^{b 1} - z_{ir}^{w 1}}, z_{ir}^{w 1} \leq z_{ir}^{1} (x) \leq z_{ir}^{b 1}, \\ 0, z_{ir}^{w 1} > z_{ir}^{1} (x) \end{matrix} \end{matrix}$ (14)

Now, we can obtain the solution of the ith level decision maker ith-LDM problem by solving the following sub-Tchebycheff problems, namely, the middle-Tchebycheff (M-Tch) problems, the upper-Tchebycheff (U-Tch) problems and the lower-Tchebycheff (L-Tch) problems.

$\begin{matrix} (M - Tch) : max λ_{ir}^{2}, \\ subject to \\ x \in G^{M}, \\ μ_{z_{ir}^{2}}^{M} (z_{ir}^{2} (x)) \geq λ_{ir}^{2}, \\ λ_{ir}^{2} \in [0, 1] \end{matrix}$ (15)

$\begin{matrix} (U - Tch) : max λ_{ir}^{3} \\ subject to \\ x \in G^{U}, \\ μ_{z_{ir}^{3}}^{U} (z_{ir}^{3} (x)) \geq λ_{ir}^{3}, \\ λ_{ir}^{3} \geq λ_{ir}^{* 2}, \\ λ_{ir}^{3} \in [0, 1] \end{matrix}$ (16) and

$\begin{matrix} (L - Tch) : max λ_{ir}^{1} \\ subject to \\ x \in G^{L}, \\ μ_{z_{ir}^{1}}^{L} (z_{ir}^{1} (x)) \geq λ_{ir}^{1}, \\ λ_{ir}^{1} \leq λ_{ir}^{* 2}, \\ λ_{ir}^{1} \in [0, 1], \end{matrix}$ (17)

Thus, the solutions of the i^th-LDM are $(x_{j}^{i}, y_{j}^{i}, t_{j}^{i}), (j = 1, 2, \dots, n; i = 1, 2, \dots, p)$ with objective values $f_{ir}^{i} = (z_{ir}^{iL}, z_{ir}^{iM}, z_{ir}^{iU})$ and with membership function values $λ_{ir}^{*} = (λ_{ir}^{* 1}, λ_{ir}^{* 2}, λ_{ir}^{* 3})$ , (r = 1, 2, …, m_i ; i = 1, 2, …, p)

3.3.2 Decomposed fuzzy decision approach

Let $(x_{j}^{i}, z_{ir}^{iL})$ , $(y_{j}^{i}, z_{ir}^{iM})$ and $(t_{j}^{i}, z_{ir}^{iU})$ be the lower optimal solutions, the middle optimal solutions and the upper optimal solutions of the ith-LDM, respectively, and where $z_{ir}^{iL} \leq z_{ir}^{iM} \leq z_{ir}^{iU}$ ,(r = 1, 2, …, m_i ; i = 1, 2, …, p) and $(x_{j}^{i} \leq y_{j}^{i} \leq t_{j}^{i}), (j = 1, 2, \dots, n)$ . Since execution of the decision is successive from an upper to lower level, it can be assumed that the values $z_{ir}^{(i + 1) L} [= z_{(i + 1) r}^{1} (x_{j}^{i + 1})] < z_{ir}^{iM},$ $z_{ir}^{(i + 1) L} [= z_{(i + 1) r}^{1} (x_{j}^{i + 1})] < z_{ir}^{iM}$ and $z_{ir}^{(i + 1) M} [= z_{(i + 1) r}^{2} (y_{j}^{i + 1})] < z_{ir}^{iU}$ are the lower tolerance limits of the lower branch, the middle branch and the upper branch of the sequential upper-level DMs, respectively.

Then, the membership functions of the ith-LDM can be assumed as follows: $\begin{matrix} (M - MF)^{'} : μ_{M}^{'} (z_{ir}^{2} (x)) = \\ {\begin{matrix} 1, z_{ir}^{2} (x) \geq z_{ir}^{iM}, \\ \frac{z_{ir}^{2} (x) - z_{ir}^{(i + 1) L}}{z_{ir}^{iM} - z_{ir}^{(i + 1) L}}, z_{ir}^{(i + 1) L} \leq z_{ir}^{2} (x) \leq z_{ir}^{iM}, \\ 0, z_{ir}^{2} > z_{ir}^{(i + 1) L} \end{matrix} \end{matrix}$ (18) $\begin{matrix} (U - MF)^{'} : μ_{U}^{'} (z_{ir}^{3} (x)) = \\ {\begin{matrix} 1, z_{ir}^{3} (x) \geq z_{ir}^{iU}, \\ \frac{z_{ir}^{3} (x) - z_{ir}^{(i + 1) M}}{z_{ir}^{iU} - z_{ir}^{(i + 1) M}}, z_{ir}^{(i + 1) M} \leq z_{ir}^{3} (x) \leq z_{ir}^{iU}, \\ 0, z_{ir}^{3} > z_{ir}^{(i + 1) M} \end{matrix} \end{matrix}$ (19)

and $\begin{matrix} (L - MF)^{'} : μ_{L}^{'} (z_{ir}^{1} (x)) = \\ {\begin{matrix} 1, z_{ir}^{1} (x) \geq z_{ir}^{iM}, \\ \frac{z_{ir}^{1} (x) - z_{ir}^{(i + 1) L}}{z_{ir}^{iM} - z_{ir}^{(i + 1) L}}, z_{ir}^{(i + 1) L} \leq z_{ir}^{1} (x) \leq z_{ir}^{iM}, \\ 0, z_{ir}^{1} > z_{ir}^{(i + 1) L} \end{matrix} \end{matrix}$ (20)

Let $(R_{iL}^{-}, R_{iL}^{+})$ , $(R_{iM}^{-}, R_{iM}^{+})$ and $(R_{iU}^{-}, R_{iU}^{+})$ be the negative and positive tolerance values on the lower decision vectors x_i, the middle decision vectors y_i and the upper decision vectors t_i, (i = 1, 2, …, p - 1) recognized by the ith-LDM, respectively. Then, the membership functions for the decision vectors x_i, y_i and t_i can be formulated as follows: $μ_{x_{i}} (x_{i}) = {\begin{matrix} \frac{x_{i} - (x_{i}^{i} - R_{iL}^{-})}{R_{iL}^{-}}, x_{i}^{i} - R_{iL}^{-} \leq x_{i} \leq x_{i}^{i}, \\ \frac{x_{i}^{i} + R_{iL}^{+}) - x_{i}}{R_{iL}^{+}}, x_{i}^{i} \leq x_{i} \leq x_{i}^{i} + R_{iL}^{+} \end{matrix}$ (21) $μ_{y_{i}} (y_{i}) = {\begin{matrix} \frac{y_{i} - (y_{i}^{i} - R_{iM}^{-})}{R_{iM}^{-}}, y_{i}^{i} - R_{iM}^{-} \leq y_{i} \leq y_{i}^{i}, \\ \frac{y_{i}^{i} + R_{iM}^{+}) - y_{i}}{R_{iM}^{+}}, y_{i}^{i} \leq y_{i} \leq y_{i}^{i} + R_{iM}^{+} \end{matrix}$ (22)

and $μ_{t_{i}} (t_{i}) = {\begin{matrix} \frac{t_{i} - (t_{i}^{i} - R_{iU}^{-})}{R_{iU}^{-}}, t_{i}^{i} - R_{iU}^{-} \leq t_{i} \leq t_{i}^{i}, \\ \frac{t_{i}^{i} + R_{iU}^{+}) - t_{i}}{R_{iU}^{+}}, t_{i}^{i} \leq t_{i} \leq t_{i}^{i} + R_{iU}^{+} \end{matrix}$ (23)

Finally, to obtain a satisfactory solution, which is also a Pareto optimal solution (in the fuzzy form) with overall satisfaction, $\tilde{δ} = (δ^{1}, δ^{2}, δ^{3})$ , for all DMs, the following sub-Tchebycheff problems are solved: $\begin{matrix} {(M - Tch)}_{T} : max δ^{2}, \\ subject to \end{matrix}$

$\begin{matrix} x \in G^{M} \\ \frac{y_{i} - (y_{i}^{i} - R_{iM}^{-})}{R_{iM}^{-}} \geq δ^{2}, \\ \frac{(y_{i}^{i} + R_{iM}^{+}) - y_{i}}{R_{iM}^{+}} \geq δ^{2}, \end{matrix}$

$\begin{matrix} μ_{M}^{'} (z_{ir}^{2} (x)) \geq δ^{2}, \\ R_{iM}^{-}, R_{iM}^{+} > 0, \\ δ^{2} \in [0, 1] \end{matrix}$ (24)

$\begin{matrix} {(U - Tch)}_{T} : max δ^{3}, \\ subject to \\ x \in G^{U}, \\ \frac{t_{i} - (t_{i}^{i} - R_{iU}^{-})}{R_{iU}^{-}} \geq δ^{3}, \\ \frac{(t_{i}^{i} + R_{iU}^{+}) - t_{i}}{R_{iU}^{+}} \geq δ^{3}, \\ μ_{U}^{'} (z_{ir}^{3} (x)) \geq δ^{3}, \\ R_{iU}^{-}, R_{iU}^{+} > 0, \\ δ^{3} \geq δ^{* 2}, \\ δ^{3} \in [0, 1] \end{matrix}$ (25)

and

$\begin{matrix} {(L - Tch)}_{T} : max δ^{1} \\ subject to \\ x \in G^{L}, \\ \frac{x_{i} - (x_{i}^{i} - R_{iL}^{-})}{R_{iL}^{-}} \geq δ^{1}, \\ \frac{(x_{i}^{i} + R_{iL}^{+}) - x_{i}}{R_{iL}^{+}} \geq δ^{1}, \\ μ_{L}^{'} (z_{ir}^{1} (x)) \geq δ^{1}, \\ R_{iL}^{-}, R_{iL}^{+} > 0, \\ δ^{1} \leq δ^{* 2}, \\ δ^{1} \in [0, 1] \end{matrix}$ (26)

If the solution is not satisfactory for any one of the DMs, the membership functions or the tolerances of the DMs are re-adjusted. Again, this process can be continued until a satisfactory solution is achieved.

3.4 A flowchart for the proposed method

4 Numerical example

The following example demonstrates the computational procedure of the fully fuzzy multi-objective and multi-level integer quadratic programming FFMMQP problems:

[FLDM]: $max_{{\tilde{x}}_{1}} {\tilde{F}}_{1} (\tilde{x}) = max_{{\tilde{x}}_{1}} (({\tilde{f}}_{1} (\tilde{x}), {\tilde{f}}_{2} (\tilde{x}), {\tilde{f}}_{3} (\tilde{x})),$

where ${\tilde{x}}_{2}, {\tilde{x}}_{3}$ solves,

[SLDM]: $max_{{\tilde{x}}_{2}} {\tilde{F}}_{2} (\tilde{x}) = max_{{\tilde{x}}_{2}} (({\tilde{f}}_{4} (\tilde{x}), {\tilde{f}}_{5} (\tilde{x})),$

where ${\tilde{x}}_{3}$ solves,

[TLDM]: $max_{{\tilde{x}}_{3}} {\tilde{F}}_{3} (\tilde{x}) = max_{{\tilde{x}}_{3}} (({\tilde{f}}_{6} (\tilde{x}), {\tilde{f}}_{7} (\tilde{x}), {\tilde{f}}_{8} (\tilde{x})),$ subject to $\begin{matrix} \tilde{G} = {(2, 4, 5) \otimes {\tilde{x}}_{1} \oplus (1, 5, 7) \otimes {\tilde{x}}_{2} \\ \oplus (3, 6, 8) \otimes {\tilde{x}}_{3} \leq (30, 40, 50), \\ (3, 7, 10) \otimes {\tilde{x}}_{1} \oplus (2, 5, 6) \otimes {\tilde{x}}_{2} \\ \oplus (4, 8, 9) \otimes {\tilde{x}}_{3} \leq (21, 27, 50), \\ (5, 9, 12) \otimes {\tilde{x}}_{1} \oplus (1, 3, 4) \otimes {\tilde{x}}_{2} \\ \oplus (6, 8, 11) \otimes {\tilde{x}}_{3} \leq (29, 36, 58) \\ {\tilde{x}}_{1}, {\tilde{x}}_{2}, {\tilde{x}}_{3} \geq 0 and integer} \end{matrix}$ where $\begin{matrix} {\tilde{f}}_{1} = (6, 9, 13) \otimes {\tilde{x}}_{1} \oplus & (3, 4, 5) \otimes {\tilde{x}}_{2} \oplus (4, 6, 7) \\ \otimes {\tilde{x}}_{3} \oplus ({\tilde{x}}_{1} {\tilde{x}}_{2}) \\ \otimes (\begin{matrix} \begin{matrix} (8, \\ (1, \end{matrix} & \begin{matrix} 12, \\ 1, \end{matrix} & \begin{matrix} 17) \\ 3) \end{matrix} \end{matrix} \begin{matrix} \begin{matrix} (6, \\ (1, \end{matrix} & \begin{matrix} 9, \\ 2, \end{matrix} & \begin{matrix} 13) \\ 4) \end{matrix} \end{matrix}) \\ \otimes (\begin{matrix} {\tilde{x}}_{1} \\ {\tilde{x}}_{2} \end{matrix}), \\ {\tilde{f}}_{2} = (8, 11, 14) \otimes {\tilde{x}}_{1} \oplus & (5, 7, 8) \otimes {\tilde{x}}_{2} \oplus (2, 4, 7) \\ \otimes {\tilde{x}}_{3} \otimes {\tilde{x}}_{3}, \\ {\tilde{f}}_{3} = (10, 12, 15) \otimes {\tilde{x}}_{1} & \oplus (1, 3, 6) \otimes {\tilde{x}}_{2} \oplus (3, 5, 7) \\ \otimes {\tilde{x}}_{3} \oplus ({\tilde{x}}_{2} {\tilde{x}}_{3}) \\ \otimes (\begin{matrix} \begin{matrix} (3, \\ (3, \end{matrix} & \begin{matrix} 5, \\ 4, \end{matrix} & \begin{matrix} 7) \\ 4) \end{matrix} \end{matrix} \begin{matrix} \begin{matrix} (2, \\ (0, \end{matrix} & \begin{matrix} 6, \\ 2, \end{matrix} & \begin{matrix} 7) \\ 3) \end{matrix} \end{matrix}) \\ \otimes (\begin{matrix} {\tilde{x}}_{2} \\ {\tilde{x}}_{3} \end{matrix}), \end{matrix}$ $\begin{matrix} {\tilde{f}}_{4} = (4, 5, 8) \otimes {\tilde{x}}_{3} \oplus & (6, 7, 10) \otimes {\tilde{x}}_{1} \otimes {\tilde{x}}_{1} \\ \oplus (12, 14, 17) \otimes {\tilde{x}}_{2} \otimes {\tilde{x}}_{2}, \\ {\tilde{f}}_{5} = (1, 2, 4) \otimes {\tilde{x}}_{1} \oplus & (9, 12, 16) \otimes {\tilde{x}}_{2} \otimes {\tilde{x}}_{2} \\ \oplus ({\tilde{x}}_{1} {\tilde{x}}_{3}) \\ \otimes (\begin{matrix} \begin{matrix} (9, \\ (1, \end{matrix} & \begin{matrix} 12, \\ 2, \end{matrix} & \begin{matrix} 16) \\ 6) \end{matrix} \end{matrix} \begin{matrix} \begin{matrix} (0, \\ (9, \end{matrix} & \begin{matrix} 1, \\ 10, \end{matrix} & \begin{matrix} 3) \\ 14) \end{matrix} \end{matrix}) \\ \otimes (\begin{matrix} {\tilde{x}}_{1} \\ {\tilde{x}}_{3} \end{matrix}) \end{matrix}$ $\begin{matrix} {\tilde{f}}_{6} = (2, 5, 6) \otimes {\tilde{x}}_{1} \oplus (5, 7, 12) \otimes {\tilde{x}}_{2} \oplus ({\tilde{x}}_{1} {\tilde{x}}_{3}) \\ \otimes (\begin{matrix} \begin{matrix} (5, \\ (0, \end{matrix} & \begin{matrix} 6, \\ 1, \end{matrix} & \begin{matrix} 8) \\ 2) \end{matrix} \end{matrix} \begin{matrix} \begin{matrix} (1, \\ (3, \end{matrix} & \begin{matrix} 2, \\ 5, \end{matrix} & \begin{matrix} 4) \\ 5) \end{matrix} \end{matrix}) \otimes (\begin{matrix} {\tilde{x}}_{2} \\ {\tilde{x}}_{3} \end{matrix}) \\ \oplus ({\tilde{x}}_{1} {\tilde{x}}_{3}) \\ \otimes (\begin{matrix} \begin{matrix} (2, \\ (1, \end{matrix} & \begin{matrix} 4, \\ 2, \end{matrix} & \begin{matrix} 7) \\ 4) \end{matrix} \end{matrix} \begin{matrix} \begin{matrix} (8, \\ (0, \end{matrix} & \begin{matrix} 9, \\ 3, \end{matrix} & \begin{matrix} 11) \\ 3) \end{matrix} \end{matrix}) \otimes (\begin{matrix} {\tilde{x}}_{1} \\ {\tilde{x}}_{2} \end{matrix}), \end{matrix}$

$\begin{matrix} {\tilde{f}}_{7} = (7, 8, 10) \otimes {\tilde{x}}_{1} \otimes & {\tilde{x}}_{1} \oplus (7, 9, 12) \otimes {\tilde{x}}_{2} \otimes {\tilde{x}}_{2} \\ \oplus (15, 18, 20) \otimes {\tilde{x}}_{3} \otimes {\tilde{x}}_{3}, \\ {\tilde{f}}_{8} = (8, 9, 11) \otimes {\tilde{x}}_{1} \oplus & (9, 11, 13) \otimes {\tilde{x}}_{2} \\ \oplus (19, 22, 25) \otimes {\tilde{x}}_{3} \\ \oplus (3, 6, 8) \otimes {\tilde{x}}_{1} \otimes {\tilde{x}}_{1} \oplus ({\tilde{x}}_{2} {\tilde{x}}_{3}) \\ \otimes (\begin{matrix} \begin{matrix} (11, \\ (3, \end{matrix} & \begin{matrix} 14, \\ 5, \end{matrix} & \begin{matrix} 16) \\ 7) \end{matrix} \end{matrix} \begin{matrix} \begin{matrix} (4, \\ (1, \end{matrix} & \begin{matrix} 5, \\ 3, \end{matrix} & \begin{matrix} 8) \\ 4) \end{matrix} \end{matrix}) \\ \otimes (\begin{matrix} {\tilde{x}}_{2} \\ {\tilde{x}}_{3} \end{matrix}) \end{matrix}$

The first step in the solution procedure is to use the bound and decomposition method [15] to transform the original FFMMQP problem to the MMQP problem with crisp coefficients and crisp decision variables. Let ${\tilde{x}}_{j} = (x_{j}, y_{j}, t_{j})$ , (j = 1, 2, 3) and ${\tilde{f}}_{i} = (z_{i 1}, z_{i 2}, z_{i 3})$ , (i = 1, 2, …, 8).

First, the FLDM can be re-formulated as follows:

[FLDM]: $\begin{matrix} (L - MQP) : max_{x_{1}} (z_{11}, z_{21}, z_{31}) \\ = (6 x_{1} + 3 x_{2} + 4 x_{3} + 8 x_{1}^{2} + 7 x_{1} x_{2} \\ + x_{2}^{2}, 8 x_{1} + 5 x_{2} + 2 x_{3}^{2}, 10 x_{1} + x_{2} \\ + 3 x_{3} + 3 x_{2}^{2} + 5 x_{2} x_{3}), \end{matrix}$ $\begin{matrix} (M - MQP) : max_{y_{1}} (z_{12}, z_{22}, z_{32}) \\ = (9 y_{1} + 4 y_{2} + 6 y_{3} + 12 y_{1}^{2} \\ + 10 y_{1} y_{2} + 2 y_{2}^{2}, 11 y_{1} + 7 y_{2} \\ + 4 y_{3}^{2}, 12 y_{1} + 3 y_{2} + 5 y_{3} + 5 y_{2}^{2} \\ + 10 y_{2} y_{3} + 2 y_{3}^{2}) \end{matrix}$ $\begin{matrix} (U - MQP) : max_{t_{1}} (z_{13}, z_{23}, z_{33}) \\ = (13 t_{1} + 5 t_{2} + 7 t_{3} + 17 t_{1}^{2} \\ + 16 t_{1} t_{2} + 4 t_{2}^{2}, 14 t_{1} + 8 t_{2} \\ + 7 t_{3}^{2}, 15 t_{1} + 6 t_{2} + 7 t_{3} + 7 t_{2}^{2} \\ + 11 t_{2} t_{3} + 3 t_{3}^{2}) \end{matrix}$

where (x₂, y₂, t₂) , (x₃, y₃, t₃) solves,

subject to $\begin{matrix} G = {G^{L} = {2 x_{1} + x_{2} + 3 x_{3} \leq 30, \\ 3 x_{1} + 2 x_{2} + 4 x_{3} \leq 21, \\ 5 x_{1} + x_{2} + 6 x_{3} \leq 29, \\ x_{j \leq} y_{j} \leq t_{j}, j = 1, 2, 3, \\ x_{j} \geq 0 and integer}, \\ G^{M} = {4 y_{1} + 5 y_{2} + 6 y_{3} \leq 40, \\ 7 y_{1} + 5 y_{2} + 8 y_{3} \leq 27, \\ 9 y_{1} + 3 y_{2} + 8 y_{3} \leq 36, \\ x_{j} \leq y_{j} \leq t_{j}, j = 1, 2, 3, \\ y_{j} \geq 0 and integer}, \\ G^{U} = {5 t_{1} + 7 t_{2} + 8 t_{3} \leq 50, \\ 10 t_{1} + 6 t_{2} + 9 t_{3} \leq 50, \\ 12 t_{1} + 4 t_{2} + 11 t_{3} \leq 58, \\ x_{j} \leq y_{j} \leq t_{j}, j = 1, 2, 3, \\ t_{j} \geq 0 and integer}} . \end{matrix}$

The individual optimal solutions of the decision variables and the objective functions of the FLDM are provided in Table 1.

Table 1
The individual optimal solutions of the decision variables and the objective functions of the FLDM

L-MQP

$z_{11}^{b}$ $z_{11}^{w}$ $z_{21}^{b}$ $z_{21}^{w}$ $z_{31}^{b}$ $z_{31}^{w}$

x ₁ 3 0 3 0 0 0

x ₂ 1 0 1 0 5 0

x ₃ 0 0 0 0 0 0

Optimal solution 115 0 29 0 80 0

M-MQP

$z_{12}^{b}$ $z_{12}^{w}$ $z_{22}^{b}$ $z_{22}^{w}$ $z_{32}^{b}$ $z_{32}^{w}$

y ₁ 3 0 3 0 0 0

y ₂ 1 0 1 0 5 0

y ₃ 0 0 0 0 0 0

Optimal solution 171 0 40 0 140 0

U-MQP

$z_{13}^{b}$ $z_{13}^{w}$ $z_{23}^{b}$ $z_{23}^{w}$ $z_{33}^{b}$ $z_{33}^{w}$

t ₁ 4 0 3 0 0 0

t ₂ 1 0 3 0 7 0

t ₃ 0 0 0 0 0 0

Optimal solution 397 0 66 0 385 0

L-MQP
$z_{11}^{b}$	$z_{11}^{w}$	$z_{21}^{b}$	$z_{21}^{w}$	$z_{31}^{b}$	$z_{31}^{w}$
x ₁	3	0	3	0	0	0
x ₂	1	0	1	0	5	0
x ₃	0	0	0	0	0	0
Optimal solution	115	0	29	0	80	0
M-MQP
$z_{12}^{b}$	$z_{12}^{w}$	$z_{22}^{b}$	$z_{22}^{w}$	$z_{32}^{b}$	$z_{32}^{w}$
y ₁	3	0	3	0	0	0
y ₂	1	0	1	0	5	0
y ₃	0	0	0	0	0	0
Optimal solution	171	0	40	0	140	0
U-MQP
$z_{13}^{b}$	$z_{13}^{w}$	$z_{23}^{b}$	$z_{23}^{w}$	$z_{33}^{b}$	$z_{33}^{w}$
t ₁	4	0	3	0	0	0
t ₂	1	0	3	0	7	0
t ₃	0	0	0	0	0	0
Optimal solution	397	0	66	0	385	0

The individual optimal fuzzy solutions of the FLDM are ${\tilde{f}}_{1}^{b} = (115, 171, 397),$ ${\tilde{f}}_{2}^{b} = (29, 40, 66),$ ${\tilde{f}}_{3}^{b} = (80, 140, 385),$ ${\tilde{f}}_{1}^{w} = (0, 0, 0),$ ${\tilde{f}}_{2}^{w} = (0, 0, 0),$ and ${\tilde{f}}_{3}^{w} = (0, 0, 0) .$

By using Equations (12) and (14), we build the sub-membership functions $μ_{z_{11}}^{L} (Z_{11}),$ $μ_{z_{21}}^{L} (Z_{21}),$ $μ_{z_{31}}^{L} (Z_{31}),$ $μ_{z_{12}}^{M} (Z_{12}),$ $μ_{z_{22}}^{M} (Z_{22}),$ $μ_{z_{32}}^{M} (Z_{32}),$ $μ_{z_{13}}^{U} (Z_{13}),$ $μ_{z_{23}}^{U} (Z_{23}),$ and $μ_{z_{33}}^{U} (Z_{33}),$ and then solve Equations (15)–(17) as follows: $\begin{matrix} (M - Tch) : max λ^{2}, \\ subject to \\ y_{1}, y_{2}, y_{3} \in G^{M}, \\ 9 y_{1} + 4 y_{2} + 6 y_{3} + 12 y_{1}^{2} + 10 y_{1} y_{2} + 2 y_{2}^{2} \geq 171 λ^{2}, \\ 11 y_{1} + 7 y_{2} + 4 y_{3}^{2} \geq 40 λ^{2}, \\ 12 y_{1} + 3 y_{2} + 5 y_{3} + 5 y_{2}^{2} + 10 y_{2} y_{3} + 2 y_{3}^{2} \geq 140 λ^{2}, \\ λ^{2} \in [0, 1] . \end{matrix}$ $\begin{matrix} (U - Tch) : max λ^{3}, \\ subject to \\ t_{1}, t_{2}, t_{3} \in G^{U}, \\ 13 t_{1} + 5 t_{2} + 7 t_{3} + 17 t_{1}^{2} + 16 t_{1} t_{2} + 4 t_{2}^{2} \geq 397 λ^{3}, \\ 14 t_{1} + 8 t_{2} + 7 t_{3}^{2} \geq 66 λ^{3}, \\ 15 t_{1} + 6 t_{2} + 7 t_{3} + 7 t_{2}^{2} + 11 t_{2} t_{3} + 3 t_{3}^{2} \geq 385 λ^{3}, \\ λ^{3} \geq λ^{* 2}, \\ λ^{3} \in [0, 1] . \end{matrix}$ and $\begin{matrix} (L - Tch) : max λ^{1}, \\ subject to \\ x_{1}, x_{2}, x_{3} \in G^{L}, \\ 6 x_{1} + 3 x_{2} + 4 x_{3} + 8 x_{1}^{2} + 7 x_{1} x_{2} + x_{2}^{2} \geq 115 λ^{1}, \\ 8 x_{1} + 5 x_{2} + 2 x_{3}^{2} \geq 29 λ^{1}, \\ 10 x_{1} + x_{2} + 3 x_{3} + 3 x_{2}^{2} + 5 x_{2} x_{3} \geq 80 λ^{1}, \\ λ^{1} \leq λ^{* 2}, \\ λ^{1} \in [0, 1] \end{matrix}$

whose fuzzy solutions are ${\tilde{x}}_{1}^{F} = (1, 1, 1),$ ${\tilde{x}}_{2}^{F} = (4, 4, 6),$ ${\tilde{x}}_{3}^{F} = (0, 0, 0),$ ${\tilde{f}}_{1}^{F} = (70, 109, 300),$ ${\tilde{f}}_{2}^{F} = (28, 39, 62),$ ${\tilde{f}}_{3}^{F} = (62, 104, 303),$ and ${\tilde{λ}}^{*} = (λ^{* 1}, λ^{* 2}, λ^{* 3}) = (0.6087, 0.6374, 0.7557)$ .

Second, the SLDM can be re-written in the following form:

[SLDM]: $\begin{matrix} (L - MQP) : max_{x_{2}} (z_{41}, z_{51}) = \\ (4 x_{3} + 6 x_{1}^{2} + 12 x_{2}^{2}, x_{1} + 9 x_{2}^{2} \\ + 9 x_{1}^{2} + x_{1} x_{3} + 9 x_{3}^{2}), \\ (M - MQP) : max_{y_{2}} (z_{42}, z_{52}) = \\ (5 y_{3} + 7 y_{1}^{2} + 14 y_{2}^{2}, 2 y_{1} + 12 y_{2}^{2} \\ + 12 y_{1}^{2} + 3 y_{1} y_{3} + 10 y_{3}^{2}), \\ (U - MQP) : max_{t_{2}} (z_{43}, z_{53}) = \\ (8 t_{3} + 10 t_{1}^{2} + 17 t_{2}^{2}, 4 t_{1} + 16 t_{2}^{2} \\ + 16 t_{1}^{2} + 9 t_{1} t_{3} + 14 t_{3}^{2}), \\ where (x_{3}, y_{3}, t_{3}) solves, \\ subject to \\ x \in G \end{matrix}$

The individual optimal solutions of the decision variables and the objective functions of the SLDM are summarized in Table 2.

Table 2

The individual optimal solutions of the decision variables and the objective functions of the SLDM

L-MQP
$z_{41}^{b}$	$z_{41}^{w}$	$z_{51}^{b}$	$z_{51}^{w}$
x ₁	0	0	0	0
x ₂	5	0	5	0
x ₃	0	0	0	0
Optimal solution	300	0	225	0
M-MQP
$z_{42}^{b}$	$z_{42}^{w}$	$z_{52}^{b}$	$z_{52}^{w}$
y ₁	0	0	0	0
y ₂	5	0	5	0
y ₃	0	0	0	0
Optimal solution	350	0	300	0
U-MQP
$z_{43}^{b}$	$z_{43}^{w}$	$z_{53}^{b}$	$z_{53}^{w}$
t ₁	0	0	0	0
t ₂	7	0	7	0
t ₃	0	0	0	0
Optimal solution	833	0	784	0

However, as obtained in FLDM, the SLDM fuzzy solutions are ${\tilde{x}}_{1}^{S} = (0, 0, 0),$ ${\tilde{x}}_{2}^{S} = (5, 5, 7),$ ${\tilde{x}}_{3}^{S} = (0, 0, 0),$ ${\tilde{f}}_{4}^{S} = (0, 350, 833),$ ${\tilde{f}}_{5}^{S} = (0, 300, 784),$ and ${\tilde{β}}^{*} = (β^{* 1}, β^{* 2}, β^{* 3}) = (1, 1, 1) .$

Third, the TLDM can be formulated in a similar way as follows:

[TLDM]: $\begin{matrix} (L - MQP) : max_{x_{3}} (z_{61}, z_{71}, z_{81}) \\ = (2 x_{1} + 5 x_{2} + 13 x_{1} x_{2} + 2 x_{1} x_{3} \\ + 2 x_{1}^{2} + 3 x_{3}^{2}, 7 x_{1}^{2} + 7 x_{2}^{2} \\ + 15 x_{3}^{2}, 8 x_{1} + 9 x_{2} + 19 x_{3} \\ + 3 x_{1} x_{2} + 11 x_{2}^{2} + 7 x_{2} x_{3} + x_{3}^{2}), \\ (M - MQP) : max_{y_{3}} (z_{62}, z_{72}, z_{82}) \\ = (5 y_{1} + 7 y_{2} + 15 y_{1} y_{2} + 4 y_{1} y_{3} \\ + 4 y_{2} y_{3} + 4 y_{1}^{2} + 5 y_{3}^{2}, 8 y_{1}^{2} + 9 y_{2}^{2} \\ + 18 y_{3}^{2}, 9 y_{1} + 11 y_{2} + 22 y_{3} \\ + 6 y_{1} y_{2} + 14 y_{2}^{2} + 10 y_{2} y_{3} \\ + 3 y_{3}^{2}), \\ (U - MQP) : max_{t_{3}} (z_{63}, z_{73}, z_{83}) \\ = (6 t_{1} + 12 t_{2} + 19 t_{1} t_{2} + 8 t_{1} t_{3} \\ + 5 t_{2} t_{3} + 7 t_{1}^{2} + 5 t_{3}^{2}, 10 t_{1}^{2} + 12 t_{2}^{2} \\ + 20 t_{3}^{2}, 11 t_{1} + 13 t_{2} + 25 t_{3} \\ + 8 t_{1} t_{2} + 16 t_{2}^{2} + 15 t_{2} t_{3} + 4 t_{3}^{2}), \\ subject to \\ x \in G \end{matrix}$

The individual optimal solutions of the decision variables and the objective functions of the TLDM are summarized in Table 3.

Table 3

The individual optimal solutions of the decision variables and the objective functions of the TLDM

L-MQP
$z_{61}^{b}$	$z_{61}^{w}$	$z_{71}^{b}$	$z_{71}^{w}$	$z_{81}^{b}$	$z_{81}^{w}$
x ₁	3	0	0	0	0	0
x ₂	1	0	5	0	5	0
x ₃	0	0	0	0	0	0
Optimal solution	68	0	175	0	320	0
M-MQP
$z_{62}^{b}$	$z_{62}^{w}$	$z_{72}^{b}$	$z_{72}^{w}$	$z_{82}^{b}$	$z_{82}^{w}$
y ₁	3	0	0	0	0	0
y ₂	1	0	5	0	5	0
y ₃	0	0	0	0	0	0
Optimal solution	103	0	225	0	405	0
U-MQP
$z_{63}^{b}$	$z_{63}^{w}$	$z_{73}^{b}$	$z_{73}^{w}$	$z_{83}^{b}$	$z_{83}^{w}$
t ₁	3	0	0	0	0	0
t ₂	3	0	7	0	7	0
t ₃	0	0	0	0	0	0
Optimal solution	288	0	588	0	875	0

The TLDM can be treated in a similar way as discussed in the FLDM. Then, the TLDM fuzzy solutions are ${\tilde{x}}_{1}^{T} = (1, 1, 1),$ ${\tilde{x}}_{2}^{T} = (4, 4, 6),$ ${\tilde{x}}_{3}^{T} = (0, 0, 0),$ ${\tilde{f}}_{6}^{T} = (0, 97, 199),$ ${\tilde{f}}_{7}^{T} = (0, 152, 442),$ ${\tilde{f}}_{8}^{T} = (0, 301, 713)$ and ${\tilde{γ}}^{*} = (γ^{* 1}, γ^{* 2}, γ^{* 3}) = (0.6756, 0.6756, 0.7153) .$

Let (1, 4) , (1, 3) and (1, 3) be the tolerance values on the lower decision vectors x₁, x₂, the middle decision vectors y₁, y₂, and the upper decision vectors t₁, t₂, respectively, as considered by the FLDM and SLDM. By using Equations (24) and (26), the TLDM solves the following sub-Tchebycheff problems: $\begin{matrix} {(M - Tch)}_{T} : max δ^{2}, \\ subject to \\ y_{1}, y_{2}, y_{3} \in G^{M}, \\ 9 y_{1} + 4 y_{2} + 6 y_{3} + 12 y_{1}^{2} + 10 y_{1} y_{2} + 2 y_{2}^{2} - \\ 109 δ^{2} \geq 0, \\ 11 y_{1} + 7 y_{2} + 4 y_{3}^{2} - 39 δ^{2} \geq 0, \\ 12 y_{1} + 3 y_{2} + 5 y_{3} + 5 y_{2}^{2} + 10 y_{2} y_{3} + 2 y_{3}^{2} - \\ 104 δ^{2} \geq 0, \\ 5 y_{3} + 7 y_{1}^{2} + 14 y_{2}^{2} - 350 δ^{2} \geq 0, \\ 2 y_{1} + 12 y_{2}^{2} + 12 y_{1}^{2} + 3 y_{1} y_{3} + 10 y_{3}^{2} - 300 δ^{2} \geq 0, \\ 5 y_{1} + 7 y_{2} + 15 y_{1} y_{2} + 4 y_{1} y_{3} + 4 y_{2} y_{3} + 4 y_{1}^{2} + \\ 5 y_{3}^{2} - 97 δ^{2} \geq 0, \\ 8 y_{1}^{2} + 9 y_{2}^{2} + 18 y_{3}^{2} - 152 δ^{2} \geq 0, \\ 9 y_{1} + 11 y_{2} + 22 y_{3} + 6 y_{1} y_{2} + 14 y_{2}^{2} + 10 y_{2} y_{3} + \\ 3 y_{3}^{2} - 301 δ^{2} \geq 0, \\ y_{1} + δ^{2} \leq 2, \\ y_{1} - δ^{2} \geq 0, \\ y_{2} + 3 δ^{2} \leq 8, \\ y_{2} - 3 δ^{2} \geq 2, \\ δ^{2} \in [0, 1] . \end{matrix}$

$\begin{matrix} {(U - Tch)}_{T} : max δ^{3}, \\ subject to \\ t_{1}, t_{2}, t_{3} \in G^{U}, \\ 13 t_{1} + 5 t_{2} + 7 t_{3} + 17 t_{1}^{2} + 16 t_{1} t_{2} + 4 t_{2}^{2} - \\ 230 δ^{3} \geq 70, \\ 14 t_{1} + 8 t_{2} + 7 t_{3}^{2} - 27 δ^{3} \geq 35, \\ 15 t_{1} + 6 t_{2} + 7 t_{3} + 7 t_{2}^{2} + 11 t_{2} t_{3} + 3 t_{3}^{2} - 163 δ^{3} \geq \\ 140, \\ 8 t_{3} + 10 t_{1}^{2} + 17 t_{2}^{2} - 602 δ^{3} \geq 231, \\ 4 t_{1} + 16 t_{2}^{2} + 16 t_{1}^{2} + 9 t_{1} t_{3} + 14 t_{3}^{2} - 578 δ^{3} \geq 206, \end{matrix}$ $\begin{matrix} 6 t_{1} + 12 t_{2} + 19 t_{1} t_{2} + 8 t_{1} t_{3} + 5 t_{2} t_{3} + 7 t_{1}^{2} + 5 t_{3}^{2} - \\ 164 δ^{3} \geq 35, \\ 10 t_{1}^{2} + 12 t_{2}^{2} + 20 t_{3}^{2} - 217 δ^{3} \geq 225, \\ 11 t_{1} + 13 t_{2} + 25 t_{3} + 8 t_{1} t_{2} + 16 t_{2}^{2} + 15 t_{2} t_{3} + \\ 4 t_{3}^{2} - 308 δ^{3} \geq 405, \\ t_{1} + δ^{3} \leq 2, \\ t_{1} - δ^{3} \geq 0, \\ t_{2} + 3 δ^{3} \leq 10, \\ t_{2} - 3 δ^{3} \geq 4, \\ δ^{3} \geq δ^{* 2}, \\ δ^{3} \in [0, 1] . \end{matrix}$ and $\begin{matrix} {(L - Tch)}_{T} : max δ^{1}, \\ subject to \\ x_{1}, x_{2}, x_{3} \in G^{L}, \\ 6 x_{1} + 3 x_{2} + 4 x_{3} + 8 x_{1}^{2} + 7 x_{1} x_{2} + x_{2}^{2} - 109 δ^{1} \geq 0, \\ 8 x_{1} + 5 x_{2} + 2 x_{3}^{2} - 39 δ^{1} \geq 0, \\ 10 x_{1} + x_{2} + 3 x_{3} + 3 x_{2}^{2} + 5 x_{2} x_{3} - 104 δ^{1} \geq 0, \\ 4 x_{3} + 6 x_{1}^{2} + 12 x_{2}^{2} - 350 δ^{1} \geq 0, \\ x_{1} + 9 x_{2}^{2} + 9 x_{1}^{2} + x_{1} x_{3} + 9 x_{3}^{2} - 300 δ^{1} \geq 0, \\ 2 x_{1} + 5 x_{2} + 13 x_{1} x_{2} + 2 x_{1} x_{3} + 2 x_{1}^{2} + 3 x_{3}^{2} - \\ 97 δ^{1} \geq 0, \\ 7 x_{1}^{2} + 7 x_{2}^{2} + 15 x_{3}^{2} - 152 δ^{1} \geq 0, \\ 8 x_{1} + 9 x_{2} + 19 x_{3} + 3 x_{1} x_{2} + 11 x_{2}^{2} + 7 x_{2} x_{3} + x_{3}^{2} - \\ 301 δ^{1} \geq 0, \\ x_{1} + δ^{1} \leq 2, \\ x_{1} - δ^{1} \geq 0, \\ x_{2} + 4 δ^{1} \leq 9, \\ x_{2} - 4 δ^{1} \geq 1, \\ δ^{1} \leq δ^{* 2}, \\ δ^{1} \in [0, 1] . \end{matrix}$

Then, the Pareto optimal fuzzy solution of the FFMMQP problem using a decomposed fuzzy approach is obtained as follows:

${\tilde{x}}_{1}^{*} = (1, 1, 1)$ , ${\tilde{x}}_{2}^{*} = (4, 4, 6)$ , ${\tilde{x}}_{3}^{*} = (0, 0, 0)$ , ${\tilde{f}}_{1}^{*} = (70, 109, 300)$ , ${\tilde{f}}_{2}^{*} = (28, 39, 62)$ , ${\tilde{f}}_{3}^{*} = (62, 104, 303)$ , ${\tilde{f}}_{4}^{*} = (198, 231, 622)$ , ${\tilde{f}}_{5}^{*} = (154, 206, 596)$ , ${\tilde{f}}_{6}^{*} = (76, 97, 199)$ , ${\tilde{f}}_{7}^{*} = (119, 152, 442)$ , ${\tilde{f}}_{8}^{*} = (232, 301, 713)$ , and ${\tilde{δ}}^{*} = (δ^{* 1}, δ^{* 2}, δ^{* 3}) = (0.5133, 0.6400, 0.6495)$ .

5 Conclusion

In this paper, a new modified fuzzy approach was modeled to deal with the fully fuzzy multi-objective and multi-level integer quadratic programming (FFMMQP) problems. In the proposed approach, each fuzzy problem was converted to three crisp multi-objective quadratic problems based on the decomposition technique. Then the concept of a tolerance membership functions and multi-objective optimization in the decomposition form was used to generate the Pareto optimal fuzzy solution for the given FFMMQP problems. To check the validity of the method, one numerical example was provided. Obtained results clarify that the proposed technique creates a powerful tool which demanded for solving the fully fuzzy multi-level problems. The work will be able to help solve real life and industrial problems which are usually complicated, uncertain and continuously subject to changes, by considering the fuzziness in the formulation of the model. For future research, one possible direction is to apply the proposed approach to deal with the real world decision making situation such as a multi-level logistics planning problem under fully fuzzy environments.

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