Discrete Dynamics in Nature and Society

Volume 2017 (2017), Article ID 2870420, 10 pages

https://doi.org/10.1155/2017/2870420

## A New Method for Solving Multiobjective Bilevel Programs

^{1}Business School, University of Shanghai for Science and Technology, 516 Jungong Road, Shanghai 200093, China^{2}College of Science, University of Shanghai for Science and Technology, Shanghai, China

Correspondence should be addressed to Ying Ji; moc.621@1891_gniyij

Received 14 May 2016; Revised 15 August 2016; Accepted 28 December 2016; Published 23 March 2017

Academic Editor: Kamel Barkaoui

Copyright © 2017 Ying Ji et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We study a class of multiobjective bilevel programs with the weights of objectives being uncertain and assumed to belong to convex and compact set. To the best of our knowledge, there is no study about this class of problems. We use a worst-case weighted approach to solve this class of problems. Our “worst-case weighted multiobjective bilevel programs” model supposes that each player (leader or follower) has a set of weights to their objectives and wishes to minimize their maximum weighted sum objective where the maximization is with respect to the set of weights. This new model gives rise to a new Pareto optimum concept, which we call “robust-weighted Pareto optimum”; for the worst-case weighted multiobjective optimization with the weight set of each player given as a polytope, we show that a robust-weighted Pareto optimum can be obtained by solving mathematical programing with equilibrium constraints (MPEC). For an application, we illustrate the usefulness of the worst-case weighted multiobjective optimization to a supply chain risk management under demand uncertainty. By the comparison with the existing weighted approach, we show that our method is more robust and can be more efficiently applied to real-world problems.

#### 1. Introduction

Multiobjective bilevel programs are a generalization of the scalar criterion bilevel programs [1] and used to model situations where decision-makers at both upper and lower levels, called players, take actions by considering their individual multiple objectives [2–4]. Many papers have been published in the last two decades about (multiobjective) bilevel programs because of their possible applications [1, 4, 5].

In this paper, we consider a special class of multiobjective bilevel programs with the weights of objectives being uncertain and assumed to belong to convex and compact set. To the best our knowledge, there is no study about this class of problems and the existing approaches in multiobjective bilevel programs can not solve this class of problems. Therefore, new methods are necessary to be proposed. In this paper, we present a robust-weighted approach to solve this class of multiobjective bilevel programs. In our model, we suppose that players at both upper and lower levels are risk-averse and a player uses the robust optimization approach to manage the weight uncertainty, assuming that the other player is a robust optimizer as well. Note that the robust optimization approach is not concerning problem data or the other players’ strategies but concerning the weights to the objectives. We note that this model is closely related to robust optimization [6, 7]. It can be seen as a natural extension of the robust modeling technique by replacing a decision variable with a general deterministic function [6, 7]. However, our model is also different from the studies in [6, 7] that we study the bilevel problem, while the model in [6, 7] is single level.

Multiobjective optimization has been extensively studied and a number of different approaches have also been proposed [8]. In multiobjective problems, since there are several competing objectives to be considered and it is not possible to simultaneously optimize all objectives, a commonly accepted approach for coping with this setting is the weighted approach by assigning a nonnegative weight by considering the importance of the corresponding objective function. And then the decision-maker can make a decision by optimizing a weighted sum objective [9–12]. A weighted Pareto optimal point can be obtained. In the setting of multiobjective models including input from multiple experts, a decision can often be obtained only after considering different opinions from different experts [11].

However current approaches to model-based multiexpert multiobjective decision-making have several shortcomings. How to choose a solution among (possibly infinite) generated Pareto optimal solutions? There is no guidance. In multiexpert optimization, the relative weights given by experts can differ significantly as experts with differing opinions often assign different relative weights to objectives. As shown in applications, the weights are not known in advance and the player has to choose them. Ambiguity often exists in the choice of the weights to objectives, as it is not easy to decide relative weights for each objective. In addition, as shown in multiobjective optimization in the literature, relative weights given by the same decision-maker may rely on the elicitation methods [12, 13]. In the bilevel setting considered in this paper, it is more difficult for the decision-maker at the upper level to exactly assume the weights of the decision-maker’s objectives at the lower level. Therefore, in this paper, we focus on a special class of multiobjective bilevel programs with the weights of objectives being uncertain and assumed to belong to convex and compact set. There is no study about this class of problems and there no methods in multiobjective bilevel programs can solve this class of problems. It is necessary to provide a new approach to cope with these issues.

Hence, our motivation to utilize a worst-case weighted approach is that it provides an alternative way to deal with the weights ambiguity. Furthermore, if each player in multiobjective bilevel problems chooses the worst-case weighted approach, then we show that the computation for robust-weighted Pareto optimum, with the choice of polytope weight set for every player, is reformulated as a solution to mathematical programing with equilibrium constraints (MPEC) which can be solved by the existing methods (e.g., the sequential quadratic programing (SQP) methods).

Though multiobjective bilevel programs have not attracted much attention in the literature, there are some interesting potential applications. One example is the multicriteria Stackelberg competition of a supply chain containing a manufacturer who supplies a set of products to a risk-averse retailer satisfying uncertain consumer demand. In our model, the manufacturer decides on the quantity for each product so as to maximize the profit and minimize the cost simultaneously by forecasting the order quantity from the retailer and the wholesale prices resulting from market clearing conditions. The retailer also decides their wholesale market order quantity for each product in order to simultaneously maximize the mean profit and minimize the standard deviation of the profit.

Since the possible applications, multiobjective bilevel programs have attracted some attention. For example, Yin [14] considers a multiobjective bilevel model for transportation planning and management problems, where genetic algorithms are proposed to solve the resulting model. Deb and Sinha [15] present evolutionary algorithms for solving multiobjective optimization problems. Pieume et al. [16] develop two methods for solving bilevel linear multiobjective optimization problems. Eichfelder [17] gives a solution method for solving nonlinear multiobjective bilevel problems based on a scalarization approach and the sensitivity analysis of adaptive parameters. To reduce traffic congestion as well as to improve workforce productivity, a bilevel multiobjective model is proposed for an urban logistics metropolis [18]. A discrete approach is proposed to solve the resulting model. The convergence result and numerical tests are also provided. Some special multiobjective bilevel programs with linear objectives are also studied extensively. For example, a linear bilevel optimization problem with multiple objectives at the upper level is studied in [19], where the original problem reduces to solving a series of linear bilevel problems with a single objective function at each level; a linear bilevel optimization problem with multiple objectives at the lower level is studied in [20], where the original problem is reformulated as an optimization problem over a nonconvex region given by a union of faces of the polyhedron defined by all constraints.

Our study is different from the above papers in that we focus on a special class of multiobjective bilevel nonlinear programs—the weights of objectives are uncertain and assumed to belong to convex and compact set. To deal with this uncertainty, a robust optimization approach is used. Then we reformulated the original problem as a robust bilevel nonlinear optimization problem. To the best of our knowledge, this is the first paper to consider these special multiobjective bilevel problems.

We feel that the primary contributions of this paper are as follows. We deal with a special class of multiobjective bilevel programs, where the weights of objectives are uncertain and assumed to belong to convex and compact set. To the best of our knowledge, this is the first paper to consider such multiobjective bilevel programs. In our model, we suppose that players at both upper and lower levels are risk-averse and a player uses the robust optimization approach to manage the weight uncertainty, assuming that the other player is a robust optimizer as well. As we know that, there is no method to solve such problem. We propose a worst-case weighted approach for solving such multiobjective bilevel programs, extending the notion of robust-weighted multiobjective optimization models [21] to multiobjective bilevel problems. We show that a robust-weighted Pareto optimal point can be calculated by solving MPEC when the weight sets chosen by the players are polytopes. We note that the MPEC has been extensively studied and can be solved by the sequential quadratic programing (SQP) method [22]. We demonstrate the usefulness of the worst-case weighted method in a bilevel multiobjective competition problem within supply chain. We note that, compared with the existing weighted approach [9], our method is more “robust" in that there are different solutions for choosing different weights by using weighted approach [9], but there is a unique solution by using our approach.

Throughout this paper we use the following notations. is the set of real numbers; denotes the set of nonnegative real numbers; and is the set of strictly positive real numbers. Let For any , denotewhere is the index set. Given any vector function , by , we indicate that is a continuously differentiable function from to and we use to denote the gradient of the function at ; for simplicity, in this paper we use superscript (resp., ) to denote decision-makers in the upper level (resp., decision-makers in the lower level) parameters, decision variables, cost functions, and weights of payoff functions. We note that the different mechanisms (e.g., bilevel, equilibrium, multiobjective optimization, weighting, and Pareto) are used in this paper. To specially understand the links between all these components, we provide a figure (see Figure 1) to illustrate this.