Decentralized multi-objective bilevel decision making with fuzzy demands
Introduction
Organizational decision making often involves two levels of decision makers, uncertain information, and multiple conflicting objectives. With the complex decision making environment, knowledge-based intelligent systems, including fuzzy sets and fuzzy logic, neural networks, optimization algorithms, etc., provide effective assistant for decision problem recognition, modeling, and solving.
Bilevel programming (BP) arises where decisions are made in a two-level hierarchical order and each decision maker has no direct control upon the decision of the other, but actions taken by one decision maker effect returns from the other [1], [2], [3], [4], [5], [6], [7], [8], [10], [26], [27]. Decision maker at the upper level is termed as the leader, and at the lower level, the follower. The leader and the follower each tries to optimize his/her own objective function, but the decision affects the objective values of the other level [12]. As decision environments become more complex, two issues below need to be considered when model a real-world bilevel decision problem and find a optimal solution for the problem.
First, the upper level or the lower level or both of a bilevel decision have multiple conflicting objectives which should be considered simultaneously by the decision makers. For example, a coordinator of a multi-division firm considers three objectives in making an aggregate production plan: maximize net profits, maximize quality of products, and maximize worker satisfaction. The three objectives are in conflict with each other, but must be considered simultaneously by the coordinators. Any improvement in one objective may be achieved only at the expense of others. One level multi-objective decision-making problem has been well-studied by researchers such as Hwang and Masud [9]. But in a bilevel decision model, the selection of a solution by the leader is also affected by his/her followers’ optimal reactions. Therefore, to find a solution for the leader who has multiple objectives needs to consider both the solution of the leader’s multiple objectives and his/her follower’s decision.
Second, decision makers have fuzzy demands. When formulate some bilevel decision problems, the parameters of the objective functions and the constraints of the leader and the followers are required to be obtained through some experiments and/or some experts’ understanding of the nature of the parameters. It has been observed that, in most real-world situations, for example, power market and business management, the possible values of these parameters are often only imprecisely or ambiguously known to the experts and cannot be described by precise values. With this observation, it would be certainly more appropriate to interpret the experts’ understanding of the parameters as fuzzy numerical data which can be represented by means of fuzzy sets [28].
Research on BP with fuzzy parameters, call fuzzy BP, have been reported in literatures. For example, Sakawa et al. [14], [15], [16], [17], [18], [19], [20], [21] formulated cooperative fuzzy BP problems and proposed an interactive fuzzy programming approach for solving the problem. In their approach, the concepts of α-bilevel programming was introduced based on the basis of fuzzy number α-level sets. At the same time, some researches applied fuzzy set technique to deal with BP problems. For example, Shih and Lee [24] applied fuzzy set theory to overcome the computational difficulties in solving bilevel problems, and Shina [25] applied fuzzy mathematical programming approach to obtain the solution of multi-level linear programming problems.
Our research aims to apply fuzzy set technique to deal with fuzzy linear BP (FLBP) problems when the leader or the follower or both have multiple objectives. Based on the extended solution concept and related theorems of BP [11], [22], [23], we have first solved FLBP problems with a specialized forms of membership functions, triangular form, in the fuzzy parameters [29], [30], [31], and then in the general form of fuzzy numbers [32]. This paper extends our previous research by allowing the leader and the follower to have multiple objectives with fuzzy parameters, called a fuzzy multi-objective linear bilevel programming (FMOLBP) problem. This paper in particular develops an approximation branch-and-bound algorithm to solve the FMOLBP problem.
Following Section 1, Section 2 gives some basic concepts and theorems regarding to FMOLBP problems. Section 3 presents a FMOLBP model, related definitions, transformation theorems and properties. A general fuzzy number based approximation branch-and-bound algorithm for solving FMOLBP problems are proposed in Section 4. Two case-based examples are shown in Section 5 for illustrating the proposed model and algorithm. Conclusions and further study are discussed in Section 6.
Section snippets
Preliminaries
In this section, we present some basic concepts, definitions, and theorems that are to be used in the subsequent sections. The work presented in this section can also be found from our recent papers in [31], [32].
Let be the set of all real numbers, be n-dimensional Euclidean space, and x = (x1,x2, … ,xn)T, be any two vectors, where xi, , i = 1, 2, … , n and T denotes the transpose of the vector. Then we denote the inner product of x and y by 〈x, y〉. For any two vectors x,
A model and solution concepts for fuzzy multi-objective linear bilevel programming model
Based on our discussion in previous sections, a FMOLBP problem can be modeled as following:
For , , , and ,where , , , , i = 1, 2, … , s, j = 1, 2, … , t, , , Ã1 = (ãij)p×n, , , , , , , .
For the
An approximation branch-and-bound algorithm
This section proposes an approximation branch-and-bound algorithm for solving the FMOLBP problems.
We first write all the inequalities (except of the leader’s variables) of (4′a), (4′b), (4′c), (4′d) as gi(x,y) ⩾ 0, i = 1, … ,p + q + m, and note that complementary slackness simply means uigi(x,y) = 0, (i = 1, … ,p + q + m). We suppress the complementary term and solve the resulted linear sub-problem. At each time of iteration the condition (5e) is checked. If it is satisfied, the corresponding point is in the
Case-based examples
We first apply the proposed approximation branch-and-bound algorithm to solve a simple FMOLBP problem to illustrate how the algorithm is used. Example 22 Consider the following FMOLBP problem with , , and X = {x ⩾ 0}, Y = {y ⩾ 0},where
Conclusions and further study
A bilevel decision problem may have multiple objective functions and fuzzy parameters can appear in both the objectives and the constraints of the leader and the follower. The main research issue is how to derive an optimal solution for such a FMOLBP problem. This paper proposes a fuzzy number based approximation branch-and-bound algorithm to this issue. A cased based example and a numeral example are then given to illustrate the proposed FMOLBP model and the approximation branch-and-bound
Acknowledgement
The work presented in this paper was supported by Australian Research Council (ARC) under discovery Grants DP0557154.
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