Solving bi-level programming problem with fuzzy random variable coefficients

Abstract

This paper represents the bi-level linear programming problem in an imprecise and uncertain mixed environment. The aim of this paper is to introduce coefficients of leader and the follower’s objective function and the constraints as the fuzzy random variable. To determine the optimal value of the leader and follower’s objectives a new methodology is developed for bi-level linear programming model in the presence of fuzzy random variable. A numerical example is solved to demonstrate the methodology.

Keywords

Bi-level programming fuzzy random variable two stage stochastic programming fuzzy goal programming

1 Introduction

In a hierarchical administrative structure, Decision-makers (DMs) often deal with conflicting objectives. At one level of the hierarchy, a decision maker may have his own objective and decision space and due to the different level of the hierarchy, it may be influenced by the other levels. Fundamentally, Hierarchical organization deals with decentralized programming problem with multiple decision makers which can be referred to multi-level programming (MLP). Bi-level programming problem (BLPP) is the special case of MLP. There are two different hierarchical levels with two decision makers in BLPP. Both the decision maker controls their variables independently. The upper-level DM is called the leader who can affect the objective function of lower level (DM of the lower level is called follower). Both the decision maker wants to optimize their objective function and is affected by the decisions of one another. The nested structure of the overall problem requires that a solution to the upper-level problem may be feasible only if it is an optimal solution to the lower level problem. The bi-level programming is quite difficult to deal due to its inherent non-convexity. The major part of research on the bi-level programming problem is still concerned in the deterministic case. In the literature, BLPP has been first formulated by Candler and Townsley [2] and Fortuny-Amat and Mc Carl [4].

BLPP [2 , 9] can be formulated as: $max_{X} F_{1} (X, Y) = aX + bY$ where Y solves $\begin{matrix} max_{Y} F_{2} (X, Y) = cX + dY \\ s . t AX + BY \leq r, \end{matrix}$

Here $a, c \in ℝ^{n_{1}}$ , $b, d \in ℝ^{n_{2}}$ , $r \in ℝ^{m}$ , A is an m × n₁, matrix, B is an m × n₂ matrix. $(X, Y) \in ℝ^{n}$ is a vector of decision variables which can be controlled by the decision makers. $X \in ℝ^{n_{1}}$ is a vector control by leader and $Y \in ℝ^{n_{2}}$ by follower, where n₁ + n₂ = n. F₁ & F₂ are the objectives functions of leader and follower respectively. In the past several approaches have been studied by Bard [5, 7], Bialas and Wen and Hsu [6, 8], Karwan [9] and others.

Bi-level optimization problems are commonly found in the domain of transportation, economics,environmental economics, decision science, engineering, business etc. However, in real world situations, uncertainty and impreciseness are involved in defining the parameters. It is difficult to fix parameters in the objective functions and constraints in this case. Both the decision-makers have to take a decision even if they do not know the parameter of the problem with full certainty hence bi-level programming problem with fuzzy parameter and Stochastic bi-level programming problem bi-level programming with fuzzy random variable coefficients has been developed separately. Sakawa et al. [10] described the bi-level programming problem with fuzzy parameter and introduced a fuzzy programming method to solve it. Zang and Lu [11] designed a fuzzy number based Kuhn-Tucker condition to solve bi-level programming problem with the fuzzy parameter. Some multi-objective bi-level programming also has been studied with fuzzy parameter [12, 13]. For the randomness, Nishizaki et al. [14] solved the bi-level programming problem with random variable coefficients. They considered the variance of the objective function of the leader and means of the objective function of the follower to find the deterministic programming problem. Stochastic bi-level programming problem has been solved by Kosuch et al. [15] with probabilistic knapsack constraints, which can be used to jointly optimize network resources and service pricing.

But in the real-world decision-making situations, simultaneous existence of randomness and fuzziness is a common requirement. An interactive fuzzy programming approach by using fuzzy goals with fractile criterion optimization has been developed by Sakawa et al. [16] to deal with the fuzzy random bi-level programming problem. In 2012 Sakawa and Katagiri [17] used level sets and fractile criterion optimization to solve a fuzzy random bi-level programming problem. Sakawa et al. [18] used level set and probability maximization to solve a fuzzy random bi-level programming problem. Recently, Sakawa and Matsui [19] developed an interactive fuzzy programming approach to solve fuzzy random bi-level linear programming problem through probability maximization with possibility. Ren and wang [30, 31] considered a kind of bi-level linear programming problem where the coefficients of objective functions are fuzzy random variables. The approach first transforms the fuzzy random bi-level programming problem into an α-stochastic interval bi-level linear programming problem. The α-stochastic interval bi-level linear programming problem can be converted into a deterministic multi-objective bi-level linear programming problem. Assuming cooperative behavior of the decision makers, two-level linear programming problems involving random variables in constraints are considered by Sakawa et al. [32]. Using the concept of simple recourse, the formulated stochastic two-level simple recourse problems are transformed into deterministic two-level programming ones.

In the literature, Fuzzy random variable has been considered only in the objective functions of the fuzzy random bi-level linear programming problem. Inspired by Chakraborty’s [20] paper on solving geometric programming problem in the mixed environment, In this paper, Fuzzy Stochastic bi-level programming has been considered to deal with fuzziness and randomness simultaneously. Here the coefficients of both the objective functions as well as constraints are fuzzy random variable. H. kwakernaak [21, 22] and Puri [23] presented the concept of fuzzy random variable, fuzzy expectation and its fuzzy variance.

The paper is organized as follows. The fuzzy number and Fuzzy random variable are given in Section 2. Next, in Section 3, Fuzzy Stochastic bi-level programming problem, and methodology of solving this problem is considered. Section 4 deals with a numerical example and the conclusion has been made in Section 5.

2 Preliminaries

2.1 Basic definitions and notations [24, 25]

Definition 2.1. (Fuzzy number) A Fuzzy number $\tilde{M} = (m, α, β)$ is a fuzzy set of real line $ℝ$ such that

$μ_{\tilde{M}} (t)$ is upper semi continuous.

$\exists α, β \in ℝ$ such that $μ_{\tilde{M}} (t) = 0$ for t ∉ (m - α, m + β).

$μ_{\tilde{M}} (t)$ is increasing on (m - α, m) and is decreasing on (m, m + β) and $μ_{\tilde{M}} (m) = 1$ .

Definition 2.2. (α - cut of a fuzzy number) The α - cut of a fuzzy number $\tilde{M}$ is denoted by ${\tilde{M}}_{α}$ and is defined by $M_{α} = {x \in ℝ, μ_{\tilde{M}} (x) \geq α}$ . One also defines the strong α - cut $M_{α} = {x \in ℝ, μ_{\tilde{M}} (x) > α}$ .

Definition 2.3. (Reference Function) A function, usually denoted by “L” or “R”, is a reference function of a fuzzy number iff

L (x) = L (- x),

L (0) =1,

L is non-increasing on [0, ∞).

Definition 2.4. (LR-type fuzzy number) A fuzzy number $\tilde{M}$ is of LR - type if there exist reference functions L(for left), R(for right), and scalars α > 0, β > 0 with

$μ_{\tilde{M}} (x) = {\begin{matrix} L (\frac{m - x}{α}), & for x \leq m \\ R (\frac{x - m}{β}), & for x \geq m \end{matrix}$ (1)m, called the mean value of $\tilde{M}$ , is a real number and α and β are called the left and right spreads, respectively. Symbolically $\tilde{M}$ is denoted by (m, α, β) _LR.

For reference function L, different function can be chosen. Dubois and Prade in 1988 mention, for instance, L (x) = max(0, 1 - x) ^p, L (x) = max(0, 1 - x^p), with p > 0, L (x) = e^-x or L (x) = e^-x². If m is not a real number but an interval $[\underline{m}, \bar{m}]$ then the fuzzy set $\tilde{M}$ is not a fuzzy number but a fuzzy interval.

For LR fuzzy number the computations necessary for the arithmetic operations are considerably simplified: the exact formulas can be given for ⊕ and ⊖ . Let $\tilde{A} = (a, α, β)_{LR}, \tilde{B} = (b, γ, δ)_{LR}$ be two fuzzy number of LR-type. Then,

(a, α, β) _LR ⊕ (b, γ, δ) _LR = (a + b, α + γ, β + δ) _LR

- (a, α, β) _LR = (- a, β, α) _LR

(a, α, β) _LR ⊖ (b, γ, δ) _LR = (a - b, α + δ, β + γ) _LR

2.2 Fuzzy random variable, its expectation and variance

Definition 2.5. H. Kwakernaak [21, 22] and M.L. Puri and D.A. Ralescu [23] introduced the notion of a fuzzy random variable. Let (Ω, A, P) be probability space where Ω is the sample space, A is σ - algebra on Ω and P is the probability measure. Let $F (ℝ)$ be the set of all fuzzy numbers on $ℝ$ . A function $\tilde{ξ} : Ω \to ℝ$ is said to be fuzzy random variable if ${\underline{ξ}}_{α}$ and ${\bar{ξ}}_{α}$ are two real valued random variables for all α ∈ [o, 1] which are defined as follows: $\begin{matrix} {\underline{ξ}}_{α} (x) = inf (x \in ℝ : μ_{\tilde{ξ}} (x) \geq α) \end{matrix}$ and $\begin{matrix} {\bar{ξ}}_{α} (x) = sup (x \in ℝ : μ_{\tilde{ξ}} (x) \geq α) . \end{matrix}$

So if $\tilde{ξ}$ is a fuzzy random variable, then $[\tilde{ξ}]_{α} = [{\underline{ξ}}_{α}, {\bar{ξ}}_{α}]$ , α ∈ [0, 1] is a random closed interval set.

Bur according to Puri and Ralescu [23], the fuzzy random variable is directly generated as a fuzzy valued variable. For the probability space (Ω, A, P), a function $\tilde{ξ} : Ω \to F (ℝ^{n})$ is said to be fuzzy random variable if for all α ∈ [0, 1] the set value mappings ${\tilde{ξ}}_{α} : Ω \to C (ℝ^{n})$ are random sets where $C (ℝ^{n})$ denotes the set of all compact subset of $ℝ^{n}$ .

H. Kwakernaak defined fuzzy random variable such that for each α ∈ [0, 1] ${\tilde{ξ}}_{α}$ are real valued random variables. On the other hand by definition given by Puri and Ralescu α - cuts are set value mappings. Though their approaches are different, Gil et al. [26] proved that both the definitions are equivalent.

Definition 2.6. (Discrete Fuzzy Random Variable) Suppose (Ω, A, P) be probability space and $\tilde{ξ}$ be is a fuzzy random variable, $\tilde{ξ}$ is said to be discrete fuzzy random variable if Ω is a countable set.

Definition 2.7. (Expectation and Variance of fuzzy random variable) Expectation of fuzzy random variable $\tilde{ξ}$ is denoted by $E (\tilde{ξ})$ and defined by $E (\tilde{ξ}) = \int \tilde{ξ} dP = {(\int {\underline{ξ}}_{α} dP, \int {\bar{ξ}}_{α} dP) : 0 \leq α \leq 1}$

If $\tilde{ξ}$ is a discrete fuzzy random variable such that $P (\tilde{ξ} = \tilde{ξ} (s)) = {\tilde{p}}_{s}$ , s = 1, 2 … S, then its expectation and variance are defined by

$\begin{matrix} E [\tilde{ξ}] = \sum_{s = 1}^{S} {\tilde{p}}_{s} ⊙ \tilde{ξ} (s), \\ Var [\tilde{ξ}] = E [(\tilde{ξ} - E [\tilde{ξ}])^{2}] \end{matrix}$ (2)

3 Fuzzy stochastic bi-level programming problem

We consider the bi-level programming problem where the coefficients of both the objective functions as well as constraints are fuzzy random variable and it is defined as: $\tilde{max_{X}} {\tilde{F}}_{1} (X, Y) \tilde{=} \sum_{i = 1}^{n} {\tilde{A}}_{i} x_{i} + \sum_{j = 1}^{m} {\tilde{B}}_{j} y_{j}$ (3) where Y solves

$\begin{matrix} \tilde{max_{Y}} {\tilde{F}}_{2} (X, Y) \tilde{=} \sum_{i = 1}^{n} {\tilde{C}}_{i} x_{i} + \sum_{j = 1}^{m} {\tilde{D}}_{j} y_{j} \\ {\tilde{G}}_{k} (X, Y) \tilde{=} \sum_{i = 1}^{n} {\tilde{M}}_{ik} x_{i} + \sum_{j = 1}^{m} {\tilde{N}}_{jk} y_{j} \tilde{\leq} {\tilde{r}}_{k}, \\ x_{1}, x_{2} \dots x_{n} \geq 0, y_{1}, y_{2} \dots y_{m} \geq 0 . \\ i = 1, 2, \dots n, j = 1, 2, \dots m, k = 1, 2 \dots p . \end{matrix}$ (4)

Here $X \in ℝ^{n}$ and ${\tilde{F}}_{1}$ , $Y \in ℝ^{m}$ and ${\tilde{F}}_{2}$ are the decision variables and objective functions for leader and follower respectively. ${\tilde{G}}_{k}, k = 1, 2 \dots p .$ represents the number of constraints for Fuzzy Stochastic bi-level programming problem. ${\tilde{A}}_{i}$ , ${\tilde{C}}_{i}$ , ${\tilde{M}}_{ik}$ for i = 1, 2 … n, k = 1, 2 … p, ${\tilde{B}}_{j}$ , ${\tilde{D}}_{j}$ , ${\tilde{N}}_{jk}$ ,for j = 1, 2 … m, k = 1, 2 … p are positive fuzzy random variable with finite support. For the sake of simplicity all the fuzzy random variable and ${\tilde{r}}_{k}, k = 1, 2 \dots p$ can be considered as triangular type fuzzy number. We have considered that all the fuzzy random variables have finite support, i.e all the fuzzy random variables have finite number of realization or scenarios with respective fuzzy probabilities. If ${\tilde{A}}_{i}$ is a fuzzy random variable, has finite number of realizations and scenarios ${\tilde{a}}_{is}$ (say) with respective fuzzy probabilities say ${\tilde{p}}_{as}$ where s ∈ {1, 2 … S} such that $\sum_{s = 1}^{S} {\tilde{p}}_{as} = \tilde{1}$ [3]. Similarly we can define the notation of scenarios and fuzzy probabilities for all fuzzy random variables. Such type of problem can be considered as fuzzy stochastic bi-level programming problem.

Further assumptions may be considered to define the above model as a fuzzy two-stage stochastic bi-level programming problem. At first Stage, we convert the above model in fuzzy programming problem. Later on the second stage, the fuzzy programming problem is converted into deterministic programming problem using Bellmen and Zadeh [27] min operator and the upper bound of the α - cut of the leader’s objective function. We use the fuzzy goal programming approach [1] to solve deterministic programming problem at the second stage.

3.1 The optimization procedure

To take on the above problem (3) first we need to define $max^{˜} {\tilde{F}}_{1} (X, Y)$ and $max^{˜} {\tilde{F}}_{2} (X, Y)$ . It is possible to model the above fuzzy stochastic bi-level programming problem into fuzzy equivalent form i.e. Fuzzy BLPP. A fuzzy equivalent objective functions for leader and follower may be built as: $\tilde{max_{X}} k_{1} . E ({\tilde{F}}_{1} (X, Y)) + k_{2} . Var ({\tilde{F}}_{1} (X, Y))$ (5) $\tilde{max_{Y}} k_{1} . E ({\tilde{F}}_{2} (X, Y)) + k_{2} . Var ({\tilde{F}}_{2} (X, Y))$ (6) where the numerical values of k₁ ≥ 0 and k₂ ≥ 0 indicate the relative importance of expected value and variance of fuzzy random objective functions. Mean and variance of objective functions are fuzzy numbers $E ({\tilde{F}}_{1} (X, Y))$ and $E ({\tilde{F}}_{2} (X, Y))$ can be defined as: $\begin{matrix} E ({\tilde{F}}_{1} (X, Y)) & = & \sum_{s = 1}^{S} {\tilde{p}}_{s} ⊙ {\tilde{F}}_{1} (X, Y, {\tilde{a}}_{is}, {\tilde{b}}_{js}) \\ E ({\tilde{F}}_{2} (X, Y)) & = & \sum_{s = 1}^{S} {\tilde{p}}_{s} ⊙ {\tilde{F}}_{2} (X, Y, {\tilde{c}}_{is}, {\tilde{d}}_{js}) \end{matrix}$

After calculating expectations of each objective functions, Variance can be calculated using (2).

The constraints set may also be newly defined as

$\begin{matrix} k_{1} . E ({\tilde{G}}_{ks} (X, Y)) + k_{2} . Var ({\tilde{G}}_{ks} (X, Y)) \tilde{\leq} {\tilde{r}}_{ks} \\ for all k = 1, 2 \dots p, s = 1, 2 \dots S . \end{matrix}$ (7)

Thus the final form of fuzzy equivalent fuzzy stochastic BLPP in fuzzy BLPP is given as follows:

$\begin{matrix} \tilde{max_{X}} k_{1} . E ({\tilde{F}}_{1} (X, Y)) + k_{2} . Var ({\tilde{F}}_{1} (X, Y)) \\ where Y solves \\ \tilde{max_{Y}} k_{1} . E ({\tilde{F}}_{2} (X, Y)) + k_{2} . Var ({\tilde{F}}_{2} (X, Y)) \\ s . t k_{1} . E ({\tilde{G}}_{ks} (X, Y)) + k_{2} . Var ({\tilde{G}}_{ks} (X, Y)) \tilde{\leq} {\tilde{r}}_{ks}, \\ i = 1, 2, \dots n, j = 1, 2, \dots m \\ k = 1, 2 \dots p, s = 1, 2 \dots S . \end{matrix}$ (8)

Now we have a fuzzy BLPP, to handling with the fuzzy constraints Rommelfanger [28] method and for finding the aspiration level of leader or follower Chakraborty [29] method can be used.

3.2 Fuzzy inequality constraint to crisp inequality constraints

Rommelfanger [28] has suggested that how a fuzzy inequality can be replaced by a combination of crisp inequality and objective function. If $\tilde{A} (x) \tilde{\leq} \tilde{B}$ be a fuzzy inequality where $\tilde{A} (x) = (a (x), α (x), β (x))_{LR}$ and $\tilde{B} = (b, η, γ)_{LR}$ then the corresponding crisp inequality can be given as, $\tilde{A} (x) \tilde{\leq} \tilde{B} \Rightarrow {\begin{matrix} a (x) + β (x) R^{- 1} (ɛ) \leq b + ν R^{- 1} (ɛ) \\ μ_{\tilde{D} (x)} ⟶ maximize \end{matrix}$

Where, ɛ ∈ [0, 1] is chosen by decision maker and $μ_{\tilde{(} D) (x)}$ is defined as $μ_{\tilde{D}} (x) = {\begin{matrix} 1 & a (x) \leq b \\ \frac{b + ν R^{- 1} (ɛ) - a (x)}{ν R^{- 1} (ɛ)} & b \leq a (x) \leq b + ν R^{- 1} (ɛ) \\ 0 & a (x) \geq b + ν R^{- 1} (ɛ) \end{matrix}$

In our model, we have fuzzy constraints in 8. Let us consider the form $\begin{matrix} {\tilde{G}}_{ks} (X, Y) = (G_{ks} (X, Y), α_{ks}^{G} (X, Y), β_{ks}^{G} (X, Y))_{LR} \\ = {\tilde{k}}_{1} . E ({\tilde{G}}_{ks} (X, Y)) + k_{2} . Var ({\tilde{G}}_{ks} (X, Y)) \tilde{\leq} {\tilde{r}}_{ks} \end{matrix}$ and ${\tilde{r}}_{ks} = (r_{ks}, 0, δ_{ks})_{LR}$

For ɛ = 0, R^-1 (ɛ) =1 So the fuzzy constraints can be written as $\begin{matrix} G_{ks} (X, Y) + β_{ks}^{G} (X, Y) \leq r_{ks} + δ_{ks} \\ k = 1, 2 \dots p, s = 1, 2 \dots S . \end{matrix}$ (9) $μ_{\tilde{D}} (X, Y) ⟶ maximize$ where $\begin{matrix} μ_{\tilde{D}} (X, Y) \\ = {\begin{matrix} 1 & G_{ks} (X, Y) \leq r_{ks} \\ \frac{r_{ks} + δ_{ks} - G_{ks} (X, Y)}{δ_{ks}} & r_{ks} \leq G_{ks} (X, Y) \leq r_{ks} + δ_{ks} \\ 0 & G_{ks} (X, Y) \geq r_{ks} + δ_{ks} \end{matrix} \end{matrix}$

So if there are p number of fuzzy constraints in 8. These fuzzy constraints are converted to 2 . p number of crisp inequalities.

3.3 Maximization of objective functions

From the Equation 6 let us consider

$\begin{matrix} (F_{2} (X, Y), α_{2} (X, Y), β_{2} (X, Y))_{LR} \\ \equiv \tilde{max_{X}} k_{1} . E ({\tilde{F}}_{2} (X, Y)) + k_{2} . Var ({\tilde{F}}_{2} (X, Y)) \end{matrix}$ (10)

Which is a fuzzy objective function. To convert this objective function into crisp, since our goal is to maximize the objective one can select the upper bound of α - cut of this LR - type fuzzy number. Which can be written as:

$\begin{matrix} [F_{2} (X, Y)]_{α}^{U} \\ = k_{1} . [E ({\tilde{F}}_{2} (X, Y))]_{α}^{U} + k_{2} . [Var ({\tilde{F}}_{2} (X, Y))]_{α}^{U} \end{matrix}$ (11)

While in case of leader’s objective function, from the Equation 5, let us consider

$\begin{matrix} (F_{1} (X, Y), α_{1} (X, Y), β_{1} (X, Y))_{LR} \\ \equiv \tilde{max_{X}} k_{1} . E ({\tilde{F}}_{1} (X, Y)) + k_{2} . Var ({\tilde{F}}_{1} (X, Y)) \end{matrix}$ (12) is the objective function of the follower. Suppose leader wants to find an aspiration level [29] $\tilde{N} = (n, 0, δ)_{LR}$ such that (F₁ (X, Y) , α₁ (X, Y) , β₁ (X, Y)) _LR $\tilde{\geq} \tilde{N}$ Since this is a fuzzy inequality, the crisp equivalent form of this can be written as ${\begin{matrix} F_{1} (X, Y) + β_{1} (X, Y) \leq n + δ \\ μ_{{\tilde{D}}_{1} (X, Y)} ⟶ maximize \end{matrix}$ where $μ_{{\tilde{D}}_{1}} (X, Y) = {\begin{matrix} 1 & F_{1} (X, Y) \leq n \\ \frac{n + δ - F_{1} (X, Y)}{δ} & n \leq F_{1} (X, Y) \leq n + δ \\ 0 & F_{1} (X, Y) \geq n + δ \end{matrix}$ for n solve

$\begin{matrix} max F_{1} (X, Y) Subject to 9 \\ x_{1}, x_{2} \dots x_{n} \geq 0, y_{1}, y_{2} \dots y_{m} \geq 0 . \\ i = 1, 2, \dots n, j = 1, 2, \dots m, k = 1, 2 \dots p . \end{matrix}$ (13)

For n + δ

$\begin{matrix} max F_{1} (X, Y) + β_{1} (X, Y) Subject to 9 \\ x_{1}, x_{2} \dots x_{n} \geq 0, y_{1}, y_{2} \dots y_{m} \geq 0 . \\ i = 1, 2, \dots n, j = 1, 2, \dots m, k = 1, 2 \dots p . \end{matrix}$ (14)

3.4 Equivalent deterministic model

The final form of the equivalent crisp model can be written as, using 11 and Bellmen-Zadeh’s [27] min operator.

$\begin{matrix} max λ \\ max_{Y} k_{1} . [E ({\tilde{F}}_{2} (X, Y))]_{α}^{U} + k_{2} . [Var ({\tilde{F}}_{2} (X, Y))]_{α}^{U} \\ Subject to \\ F_{1} (X, Y) + λ . δ \leq n + δ \\ F_{1} (X, Y) + β_{1} (X, Y) \leq n + δ \\ G_{k} (X, Y) + λ . δ_{k} \leq r_{k} + δ_{k} \\ G_{ks} (X, Y) + β_{ks} (X, Y) . r_{k} \leq r_{k} + δ_{ks} . r_{k} \\ i = 1, 2, \dots n, j = 1, 2, \dots m, k = 1, 2 \dots p . \end{matrix}$ (15)

Now this is a BLPP which can be solved using fuzzy goal programming approach. Let X^* and Y^* be the optimal solution of this model, then $k_{1} . E ({\tilde{F}}_{1} (X^{*}, Y^{*}))$ + $k_{2} . Var ({\tilde{F}}_{1} (X^{*}, Y^{*}))$ and $k_{1} . E ({\tilde{F}}_{2} (X^{*}, Y^{*}))$ + $k_{2} . Var ({\tilde{F}}_{2} (X^{*}, Y^{*}))$ is the equivalent optimal objective value of leader and follower respectively.

4 Numerical example

Let us consider the following fuzzy stochastic bi-level programming problem:

$\begin{matrix} \tilde{max_{X}} {\tilde{F}}_{1} (X, Y) \tilde{=} {\tilde{c}}_{1} x_{1} + {\tilde{c}}_{2} x_{2} + {\tilde{c}}_{3} y_{1} \\ where Y solves \\ \tilde{max_{Y}} {\tilde{F}}_{2} (X, Y) \tilde{=} {\tilde{c}}_{4} x_{2} + {\tilde{c}}_{1} y_{1} - {\tilde{c}}_{2} y_{2} \\ s . t {\tilde{G}}_{1} (X, Y) \tilde{=} {\tilde{c}}_{2} x_{1} + {\tilde{c}}_{2} x_{2} + {\tilde{c}}_{4} y_{1} + {\tilde{c}}_{3} y_{2} \tilde{\leq} \tilde{8}, \\ {\tilde{G}}_{2} (X, Y) \tilde{=} \tilde{1} x_{1} + \tilde{1} x_{2} + \tilde{1} y_{1} + \tilde{\leq} \tilde{2}, \\ {\tilde{G}}_{3} (X, Y) \tilde{=} \tilde{1} x_{2} + \tilde{1} y_{1} + \tilde{1} y_{2} + \tilde{\leq} \tilde{3}, \\ G_{4} (X, Y) = x_{1} \leq 4, \\ G_{5} (X, Y) = x_{2} \leq 4, \\ G_{6} (X, Y) = y_{1} \leq 2, \\ G_{7} (X, Y) = y_{2} \leq 2, \\ x_{1}, x_{2} \geq 0, y_{1}, y_{2} \geq 0 . \end{matrix}$ (16) where ${\tilde{c}}_{1}$ , ${\tilde{c}}_{2}$ , ${\tilde{c}}_{3}$ , ${\tilde{c}}_{4}$ are positive discrete random variable and $\tilde{1} = (1, 0, 0.2)_{LR}$ . Now let us assume there are two points in the sample space Ω = {ω₁, ω₂} where ω₁, ω₂ are sample points and ${\tilde{c}}_{1}$ , ${\tilde{c}}_{2}$ , ${\tilde{c}}_{3}$ , ${\tilde{c}}_{4}$ are independently distributed with fuzzy probability distribution given in the table.

We use the optimization procedure given in the Section 3.1 to convert the leader’s and follower’s objective function in fuzzy equivalent form.

Considering k₁ = 1 and k₂ = 0, the leader’s objective function can be calculated in fuzzy equivalent form as follows: $\begin{matrix} \tilde{max_{X}} E [{\tilde{F}}_{1} (X, Y)] = \sum_{s = 1}^{2} {\tilde{p}}_{s} ⊙ {\tilde{F}}_{1 s} (X, Y) \\ = (F_{1} (X, Y), α_{1} (X, Y), β_{1} (X, Y))_{LR} \end{matrix}$ where $\begin{matrix} {\tilde{F}}_{1} (X, Y) = 8 x_{1} + 5.6 x_{2} + 8.8 y_{1} \\ {\tilde{α}}_{1} (X, Y) = 3.9 x_{1} + 1.7 x_{2} + 2.8 y_{1} \\ {\tilde{β}}_{1} (X, Y) = 4.2 x_{1} + 1.76 x_{2} + 2.84 y_{1} . \end{matrix}$

Similarly, the follower’s objective function and constraints can also be calculated in fuzzy equivalent form and the final model in fuzzy form can be written as:

$\begin{matrix} \tilde{max_{X}} (8 x_{1} + 5.6 x_{2} + 8.8 y_{1}, 3.9 x_{1} + 1.7 x_{2} + 2.8 y_{1}, \\ 4.2 x_{1} + 1.76 x_{2} + 2.84 y_{1})_{LR} \\ where Y solves \\ \tilde{max_{Y}} (1.8 x_{2} + 8 y_{1} - 5.6 y_{2}, 1.75 x_{2} + 3.95 y_{1} \\ + 1.5 y_{2}, 1.9 x_{2} + 4.4 y_{1} + 1.44 y_{2})_{LR} \\ Subject to \\ {\tilde{G}}_{1} = (5.6 x_{1} + 5.6 x_{2} + 7.2 y_{1} + 8.8 y_{2}, 1.7 x_{1} \\ + 1.7 x_{2} + 1.75 y_{1} + 2.8 y_{2}, 1.7 x_{1} + 1.7 x_{2} + 1.9 y_{1} \\ + 2.8 y_{2})_{LR} \leq \tilde{8} \\ 2 x_{1} + 2 x_{2} + 1.5 y_{1} + 3 y_{2})_{LR} \leq (8, 0, 0.2)_{LR} \\ (x_{1} + x_{2} + y_{1}, 0, 0.2 (x_{1} + x_{2} + y_{1}))_{LR} \\ \leq (2, 0, 0.2)_{LR} \\ (x_{2} + y_{1} + y_{2}, 0, 0.2 (x_{2} + y_{1} + y_{2}))_{LR} \\ \leq (3, 0, 0.2)_{LR} \\ x_{1} \leq 4 \\ x_{2} \leq 4 \\ y_{1} \leq 2 \\ y_{2} \leq 2 \\ x_{1}, x_{2} \geq 0, y_{1}, y_{2} \geq 0 . \end{matrix}$ (17)

Now the procedure given in 3.2 has been used to convert the fuzzy inequalities to crisp inequalities.

The condition ${\tilde{G}}_{1} (X, Y) = (5.6 x_{1} + 5.6 x_{2} + 7.2 y_{1} + 8.8 y_{2}, 1.7 x_{1} + 1.7 x_{2} + 1.75 y_{1} + 2.8 y_{2}, 1.7 x_{1} + 1.7 x_{2} + 1.9 y_{1} + 2.8 y_{2})_{LR} \leq (8, 0, 0.2)_{LR}$ is equivalent to $\begin{matrix} {\tilde{G}}_{1} (X, Y) \\ = {\begin{matrix} 7.3 x_{1} + 7.3 x_{2} + 9.1 y_{1} + 11.64 y_{2} \leq 8.2 \\ μ_{{\tilde{G}}_{1}} ⟶ maximize \end{matrix} \end{matrix}$ where $μ_{{\tilde{G}}_{1}} = = {\begin{matrix} 1, & G_{1} \leq 8 \\ \frac{8.2 - G_{1}}{0.2}, & 8 \leq G_{1} \leq 8.2 \\ 0, & G_{1} > 8.2 \end{matrix}$

Here, G₁ (X, Y) =5.6x₁ + 5.6x₂ + 7.2y₁ + 8.8y₂ ${\tilde{G}}_{2} (X, Y) = {\begin{matrix} 1.2 (x_{1} + x_{2} + y_{1}) \leq 2.2 \\ μ_{{\tilde{G}}_{2}} ⟶ maximize \end{matrix}$ where $μ_{{\tilde{G}}_{2}} = {\begin{matrix} 1, & G_{2} \leq 2 \\ \frac{2.2 - G_{2}}{0.2}, & 2 \leq G_{2} \leq 2.2 \\ 0, & G_{2} > 2.2 \end{matrix}$

Here, G₂ (X, Y) = x₁ + x₂ + y₁ ${\tilde{G}}_{3} (X, Y) = {\begin{matrix} 1.2 (x_{2} + y_{1} + y_{2}) \leq 3.2 \\ μ_{{\tilde{G}}_{3}} ⟶ maximize \end{matrix}$ where $μ_{{\tilde{G}}_{3}} = {\begin{matrix} 1, & G_{3} \leq 3 \\ \frac{3.2 - G_{3}}{0.2}, & 3 \leq G_{3} \leq 3.2 \\ 0, & G_{3} > 3.2 \end{matrix}$

Here, G₂ (X, Y) = x₂ + y₁ + y₂.

Using 3.3 and 3.4, we convert the above fuzzy model into the deterministic model and given as:

$\begin{matrix} max λ \\ where Y solves \\ max_{Y} 2.75 x_{2} + 10.2 y_{1} - 4.88 y_{2} \\ Subject to \\ 8 x_{1} + 5.6 x_{2} + 8.8 y_{1} + 1.9 λ \leq 11.71 \\ 12.2 x_{1} + 7.36 x_{2} + 11.6 y_{1} \leq 13.70 \\ 7.3 x_{1} + 7.3 y_{2} + 9.1 y_{1} + 11.64 y_{2} \leq 8.2 \\ 5.6 x_{1} + 5.6 x_{2} + 7.2 y_{1} + 8.8 y_{2} + 0.2 λ \leq 8.2 \\ x_{1} + x_{2} + y_{1} + . 2 λ \leq 2.2 \\ x_{2} + y_{1} + y_{2} + . 2 λ \leq 2.2 \\ x_{1} + x_{2} + y_{1} \leq 1.8 \\ x_{2} + y_{1} + y_{2} \leq 2.7 \\ x_{1} \leq 4 \\ x_{2} \leq 4 \\ y_{1} \leq 2 \\ y_{2} \leq 2 \\ x_{1}, x_{2} \geq 0, y_{1}, y_{2} \geq 0 . \end{matrix}$ (18)

Using Goal programming approach $\begin{matrix} min \frac{1}{6.16} d_{1}^{+} + \frac{1}{0.62} d_{2}^{-} + \frac{1}{9.19} d_{3}^{+} + \frac{1}{0.4} d_{4}^{-} \\ Subject to \\ \frac{λ}{6.16} + d_{1}^{-} - d_{1}^{+} = 1 \\ \frac{x_{1} - 0.5}{0.62} + d_{2}^{-} - d_{2}^{+} = 1 \\ \frac{2.75 x_{2} + 10.2 y_{1} - 4.88 y_{2}}{8.28} + d_{3}^{-} - d_{3}^{+} = 1 \\ \frac{y 1 - 0.5}{0.4} + d_{4}^{-} - d_{4}^{+} = 1 \\ S \end{matrix}$ where S represents the constraints of equations (18) and $d_{i}^{-}$ , $d_{i}^{+}$ are the deviational variables for i = 1, 2 … 4 .

On solving we get x₁ = 0.0013, x₂ = 0, y₁ = 0.9, y₂ = 0 with leader’s objective value (7.93, 2.525, 2.555) _LR and follower’s objective value (7.2, 3.5, 3.96) _LR.

Thus the fuzzy objective function of leader is replaced with one crisp inequality and one objective function as maximization of membership function.

5 Conclusion

In this paper, we have dealt with bi-level programming problem involving fuzzy random variable coefficients and named it as the fuzzy stochastic bi-level programming problem. The fuzzy programming problem has been formulated by handling the expectation and variance of the objective functions of the leader as well as follower’s. Once we get the fuzzy programming model, using the aspiration level and α - cut of leader’s objective function, the fuzzy programming problem is converted to the deterministic bi-level programming problem. The proposed methodology provides an optimal solution of fuzzy stochastic bi-level programming problem.

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