Bilevel Multiobjective Programming Problem Research Articles

Sustainable road network design considering hydrogen fuel cell vehicles

Environmental pollution and energy shortages have brought about an increased focus on new energy vehicles. Hydrogen fuel cell vehicles (HFCVs) have experienced rapid development due to the potential to alleviate energy pressures and reduce pollution emissions. Near zero-carbon emissions offer a promising avenue for promoting sustainable transportation development. To evaluate the impact of HFCVs on the transportation environment, this paper investigates the problem of sustainable transportation network design including HFCVs. Specifically, the problem is formulated as a bi-level multi-objective programming problem, with the upper level aimed at determining the optimal network design scheme considering multiple objectives, while the lower level addresses the mixed traffic flow that comprises both HFCVs and fuel vehicles. To solve the multi-objective sustainable network design problem, an integrated solution framework that combines the particle swarm optimization (PSO) algorithm with the Frank-Wolfe algorithm (FW) is developed. Specifically, the PSO algorithm is utilized to solve the upper-level model and identify the optimal network design schemes, while FW algorithm is adopted to handle mixed traffic flow assignments. Finally, the proposed model and algorithm are implemented in two numerical experiment to demonstrate their effectiveness and efficiency.

An acceptability-based multi-objective traffic flow adjustment method for environmental sustainability and equity

As a product of digital development, connected autonomous vehicles (CAVs) offer a unique prospective solution to alleviate the possible performance deterioration of road networks under the mixed environment of human-driven vehicles (HVs) and CAVs. In this paper, we propose a traffic flow adjustment method (TFAM) that treats CAVs as mobile regulators with the purpose to reshape traffic flow distribution on road networks by guiding rather than controlling CAVs. More specifically, we deploy subsidy nodes to briefly outline travel routes and achieve higher acceptability than traditional route-based control schemes. The TFAM is a multi-objective bi-level programming problem where the upper-level problem optimizes the network performance through regulating location and subsidy on the subsidy nodes. The lower-level problem is a dual dynamic traffic assignment (DDTA) model. Apart from the total travel time cost (TTTC), total emission cost (TEC) and network equity (NE) are also introduced as optimization objectives to highlight environmental sustainability and acceptability. To obtain the Pareto solution frontier, a meta-heuristic algorithm with an improved encoding process is proposed. Results of two numerical case studies demonstrate the effects of TFAM on traffic flow distribution and network performance, which yields valuable insights on the optimization of urban traffic systems.

Solving a multi-objective chance constrained hierarchical optimization problem under intuitionistic fuzzy environment with its application

Various optimization approaches have been developed and used for generating optimal solutions for different industry related optimization problems. The semantic representation of imprecise coefficients and various types of uncertainties arising in real life optimization problems are still a challenging task and require attention of academicians as well as professionals. Decision makers (DMs) often face problems in presenting the linguistically expressed constraints. The impreciseness of information also plays a key role in modelling and thus in decision making. The impreciseness of coefficients is usually handled by using various elements of fuzzy set theory. The uncertainty associated with constraints in optimization problem is handled by using stochastic programming approaches. These coefficients are usually revealed by a logical DM, who takes into account the scope of hesitation along with the imprecise nature of data. Such coefficients can be very well represented by using Intuitionistic fuzzy numbers. In this work, an approach for solving multi-objective bi-level chance constrained optimization problem in an intuitionistic fuzzy environment has been presented. The technological coefficients and the coefficients of objective functions are represented by Triangular Intuitionistic fuzzy numbers whereas normal random variables are used to represent the coefficient of resources. The concept of component wise optimization is used for the reduction of intuitionistic fuzzy objective into their equivalent deterministic form. The concept of alpha-beta cut, chance constrained programming and interval programming are then used to convert the problem into a deterministic multi-objective bi-level linear programming problem. The resultant deterministic optimization problem is then solved by using TOPSIS method. The production planning problem of a firm has been modelled and solved by using the proposed approach.

Solving Bi-Level Multi-Objective Quadratic Programming Problem with Neutrosophic Parameters in Objective Functions and Constraints Using Fuzzy Approach

Solving Bi-Level Multi-Objective Quadratic Programming Problem with Neutrosophic Parameters in Objective Functions and Constraints Using Fuzzy Approach

A novel multi-objective bi-level programming problem under intuitionistic fuzzy environment and its application in production planning problem

This paper presents an optimization method to solve a multi-objective model of a bi-level linear programming problem with intuitionistic fuzzy coefficients. The idea is based on TOPSIS (technique for order preference by similarity to ideal solution) method. TOPSIS method is a multiple criteria method that identifies a satisfactory solution from a given set of alternatives based on the minimization of distance from an ideal point and maximization of distance from the nadir point simultaneously. A new model of multi-objective bi-level programming problem in an intuitionistic fuzzy environment has been considered. The problem is first reduced to a conventional multi-objective bi-level linear programming problem using accuracy function. Then the modified TOPSIS method is proposed to solve the problem at both the leader and the follower level where various linear/non-linear membership functions are used to represent the flexibility in the approach of decision-makers (DMs). The problem is solved hierarchically, i.e., first the problem at the leader level is solved and then the feasible region is extended by relaxing the decision variables controlled by the leader. The feasible region is extended to obtain a satisfactory solution for the DMs at both levels. Finally, the application of the proposed approach in the production planning of a company has been presented. An illustrative numerical example is also given to explain the methodology defined in this paper.

Optimality conditions for nonsmooth multiobjective bilevel optimization using tangential subdifferentials

In combining the value function approach and tangential subdifferentials, we establish necessary optimality conditions of a nonsmooth multiobjective bilevel programming problem under a suitable constraint qualification. The upper level objectives and constraint functions are neither assumed to be necessarily locally Lipschitz nor convex.

K-means cluster interactive algorithm-based evolutionary approach for solving bilevel multi-objective programming problems

Solving bilevel multi-objective programming problems is one of the hardest tasks facing researchers in the optimization community. Bilevel multi-objective programming problems is an optimization problem consists of two interconnected hierarchical multi-objective programming problems: upper-level problem and lower-level problem. Difficulty in solving bilevel multi-objective programming problems is the need to solve lower-level multi-objective programming problem to know the feasible space of the upper-level problem. The proposed algorithm consists of two nested artificial multi-objective algorithms. One algorithm is for the upper-level problem and the other is for the lower-level problem. Also, the proposed algorithm is enriched with a k-means cluster scheme in two phases. The first phase is before starting two nested algorithms to help the algorithm to start with more appropriates solutions to the bi-level problem. The second phase is within the two nested algorithms to guide the algorithm to the most preferred solutions to the upper-level decision-maker. The performance of the proposed algorithm has been evaluated on different test problems including low dimension and high dimension test problems. The experimental results show that the proposed algorithm is a feasible and efficient method for solving the bilevel multi-objective programming problem.

Optimality conditions and duality in terms of convexificators for multiobjective bilevel programming problem with equilibrium constraints

This paper is devoted to the investigation of a nonsmooth multiobjective bilevel programming problem with equilibrium constraints ((MBPP) for short) in terms of convexificators in finite-dimensional spaces. We present necessary optimality conditions for the local weak efficient solution to such problem. Under the Mangasarian–Fromovitz and generalized standard Abadie type constraint qualification in the sense of convexificators, we establish as an application the Wolfe and Mond-Weir type dual problem for the problem (MBPP). Besides, we provide strong and weak duality theorems for the original problem and its Wolfe and Mond–Weir type dual problem under suitable assumptions on the $$\partial ^*$$ -convexity and the upper semi-regularity of objective and constraint functions. Illustrative examples are also proposed to demonstrate the main results of the paper.

Evolutionary algorithm using surrogate models for solving bilevel multiobjective programming problems.

A bilevel programming problem with multiple objectives at the leader's and/or follower's levels, known as a bilevel multiobjective programming problem (BMPP), is extraordinarily hard as this problem accumulates the computational complexity of both hierarchical structures and multiobjective optimisation. As a strongly NP-hard problem, the BMPP incurs a significant computational cost in obtaining non-dominated solutions at both levels, and few studies have addressed this issue. In this study, an evolutionary algorithm is developed using surrogate optimisation models to solve such problems. First, a dynamic weighted sum method is adopted to address the follower's multiple objective cases, in which the follower's problem is categorised into several single-objective ones. Next, for each the leader's variable values, the optimal solutions to the transformed follower's programs can be approximated by adaptively improved surrogate models instead of solving the follower's problems. Finally, these techniques are embedded in MOEA/D, by which the leader's non-dominated solutions can be obtained. In addition, a heuristic crossover operator is designed using gradient information in the evolutionary procedure. The proposed algorithm is executed on some computational examples including linear and nonlinear cases, and the simulation results demonstrate the efficiency of the approach.

A novel algorithm for generating Pareto frontier of bi-level multi-objective rough nonlinear programming problem

This paper discusses a new algorithm for generating the Pareto frontier for bi-level multi-objective rough nonlinear programming problem (BL-MRNPP). In this algorithm, the uncertainty exists in constraints which are modeled as a rough set. Initially, BL-MRNPP is transformed into four deterministic models. The weighted method and the Karush-Kuhn-Tucker optimality condition are combined to obtain the Pareto front of each model. The nature of the problem solutions is characterized according to newly proposed definitions. The location of efficient solutions depending on the lower/upper approximation set is discussed. The aim of the proposed solution procedure for the BL-MRNPP is to avoid solving four problems. A numerical example is solved to indicate the applicability of the proposed algorithm.

A comparative study of two optimization approaches for solving bi-level multi-objective linear fractional programming problem

Despite the important role of bi-level multi-objective linear fractional programming (BL-MOLFP) problem for many hierarchical organizations, a very little success has been achieved to deal with this problem. This paper presents a comparative study between two computational approaches, namely fuzzy TOPSIS (technique for order preference by similarity to ideal solution) approach and Jaya (a Sanskrit word meaning victory) approach, for solving BL-MOLFP problem. The fuzzy TOPSIS (FTOPSIS) approach aims to obtain the satisfactory solution of BL-MOLFP problem by using linearization process as well as formulating the membership functions for the distances of positive ideal solution (PIS) and negative ideal solution (NIS) for each level, respectively. In this sense, the deadlock situations among levels are avoided by establishing the membership functions for the upper level decision variables vector with possible tolerances. On the other hand, Jaya algorithm is proposed for solving BL-MOLFP problem based on nested structure scheme to optimize both levels hierarchically. An illustrative example is presented to describe the proposed approaches. In addition, the performances among the proposed approaches are assessed based on ranking strategy of the alternatives to affirm the superior approach. Based on the examined simulation, Jaya algorithm is preferable than the FTOPSIS approach.

Toward sustainable optimization with stackelberg game between green product family and downstream supply chain

Green products are increasingly considered by companies, owing to the significant attention of government regulations, customers’ requests and competitors. Here, we deal with Green Product Families (GPFs) by selecting green components, modules and products which are produced based on the assembling to order (ATO) approach to cover diverse customer needs. Designing a GPFs is important but not sufficient for sustainable optimization. In fact, we need to simultaneously consider the supply chain of a GPF to control the pollution generated in the downstream supply chain. For a sustainable optimization approach, this joint configuration is usually structured based on economic, environmental and social criteria, which are often conflicting. Maximizing the total profit, maximizing the product greenness and minimizing the total supply chain cost with environmental considerations constitute our main objectives. We apply the Leader-Follower Stackelberg game in order to present a joint configuration of the GPF and its green supply chain (GSC). Maximizing the total profit (an economic criterion) and maximizing product greenness (a social criterion) are the two objectives of the leader problem to attain the GPF. This is done by determining the optimal selection of components, modules, and product variants. The follower of the joint problem seeks to configure the supply chain by minimizing the GSC costs considering carbon emissions (an environmental criterion). This includes the optimal selection of suppliers, manufacturers, the assembly plant, distribution centers, and retailers. A bi-level multi-objective linear programming problem (B-MOLP) is used to model the Stackelberg game of the GPF-GSC. A new particle swarm optimization algorithm, named as bi-level multi-objective PSO (B-MOPSO), is developed to solve the proposed bi-level multi-objective model. To show the validity of the proposed model and the efficiency of our algorithm, a case study at a mobile phone company is worked through. Finally, certain results with some managerial implications are obtained through a sensitivity analysis.

A bi-level programming approach for improving relief logistics operations: A real case in Kermanshah earthquake

Due to its location on the world dry belt, Iran is known as a disaster-prone country. Regarding such a problem, planning for relief operations in disaster situations seems significant and undeniable. The literature on disaster management has mostly defined their problems in a single-level framework, ignoring the bi-level aspect of such problems in both preparedness and response phases. However, in the context of disaster in the real world, there is normally more than one decision-maker that each of them may have their own objective. In this study, a bi-level multi-objective programming approach is extended to design a pre-positioning and distribution network under disastrous conditions considering both preparedness and response phases. The extended framework takes into account simultaneous planning for the distribution of drug, blood and relief items among disaster victims, and transferring the injured to hospital in order to diminish the adverse effects of potential disasters in the response phase. Additionally, to determine suitable candidate places for locating relief centers in the preparedness phase, a modified version of data envelopment analysis model is utilized, something that is not frequently done for facility location planning in the disaster management field. On the other hand, to solve the bi-level multi-objective programming problem with decision-makers at the upper level (the affected country) and the lower level (the international organization), a fuzzy goal programming algorithm is proposed. Furthermore, a consolidated method with respect to a novel fuzzy measure is developed to keep the considered network safe against the inherent uncertainty. Finally, the extended framework is utilized to solve a case study in Kermanshah, the province of Iran.

Co-reconfiguration of product family and supply chain using leader–follower Stackelberg game theory: Bi-level multi-objective optimization

This article deals with product Family: a group of products with similar modules produced based on assembling to order (ATO) approach to cover diverse customer needs. The demand level and the customer requirements of these products are dynamically changing, which necessitate a novel model for co-reconfiguration of the product family (PF) and the supply chain (SC). Therefore, this article aims to apply Leader–follower Stackelberg Game Theory in order to present a co-reconfiguration of PF and SC based on three objectives: maximizing the total profit, maximizing customer utility and minimizing the supply chain cost in a bi-level structure. Maximizing the total profit and maximizing customer utility are the two objectives of the Leader problem for product family reconfiguration, which results in the optimal selection of components, modules, and product variants. The upper-level problem is considered as a multi-objective problem with the aforementioned two objectives. The lower level of this problem intends to reconfigure the supply chain with the objective of minimizing the supply chain costs, and therefore, to reach the optimal selection of suppliers, manufacturers, assembly plant, distribution centers, and retailers. A bi-level multi-objective linear programming problem (B-MOLP) is used to model the game of the leader–follower. A new particle swarm optimization algorithm, called bi-level multi-objective PSO (B-MOPSO), is developed to solve the proposed bi-level multi-objective model. To show the validity of the proposed model and the efficiency of our algorithm, a case study at a mountain bike industry is investigated. Finally, results in some managerial implications are obtained through sensitivity analysis.

Solving a class of multiobjective bilevel problems by DC programming

In this paper, we consider a class of multiobjective bilevel programming problems in which the first level objective function is assumed to be a vector valued DC function and the second level problem is a linear multiobjective program. The problem is transformed into a standard single optimisation problem by using a preference function. We give a characterisation to the induced region and reformulate the problem as a problem of optimising a function over the efficient set. Next, a well-known representation of the efficient set is used which will allow to transform the problem, using an exact penalisation, into a DC program. Finally, we apply the DC algorithm to solve the resulting DC program.

Efficiency conditions for multiobjective bilevel programming problems via convexificators

Efficiency conditions for multiobjective bilevel programming problems via convexificators

Fuzzy rough bi-level multi-objective nonlinear programming problems

Fuzzy rough bi-level multi-objective nonlinear programming problem (FRBMNPP) moved toward becoming rise normally in various real applications. In this article we develop bi-level multi-objective nonlinear programming problem (BMNPP), in which the objective functions have fuzzy nature and the constraints represented as a rough set. The fuzzy objective functions converted into deterministic ones by utilizing the α-cut methodology. Thus the FRBMNPP become a rough BMNPP which is transformed into two problems corresponding to the upper and lower approximation models. The Karush-Kuhn-Tucker (KKT) method and two models of technique of order preferences by similarity to ideal solution (TOPSIS) approach are developed to solve such problem. At last, applicability and efficiency of the two TOPSIS models and KKT method, suggested in this study, are presented through an algorithm and a numerical illustration.

Penalty method-based equilibrium point approach for solving the linear bilevel multiobjective programming problem

In this paper, a linear bilevel multiobjective programming problem is concerned. Based on the method of replacing the lower level problem with its optimality conditions, and taking the complementary constraints as the penalty term of the upper level objectives, we obtain the exact penalized multiobjective programming problem \begin{document}$ (P_{K}) $\end{document} . The concept of equilibrium point of problem \begin{document}$ (P_{K}) $\end{document} is introduced and its properties are analyzed. Thereafter, we propose a penalty method-based equilibrium point algorithm, which only needs to solve a series of linear programming problems, for the linear bilevel multiobjective programming problem. Numerical results showing viability of the equilibrium point approach are presented.

A fuzzy goal programming approach to solving decentralized bi-level multi-objective linear fractional programming problems

This paper presents a new approach for solving decentralized bi-level multi-objective linear fractional programming problems. The main goal was to find a simple algorithm with high confidence of decision-makers in the results. First, all the linear fractional programming models on the given set of constraints were solved separately. Next, all the linear fractional objective functions were linearized, membership functions of objective functions and decision variables controlled by decision-makers at the highest level calculated, and a fuzzy multi-objective linear programming model formed and solved as linear goal programming problem by using simplex algorithm. The efficiency of the proposed algorithm was investigated using an economic example, and the obtained results compared with those obtained using an existing method.

A Fuzzy Goal Programming Approach for Solving Chance Constrained Bi-Level Multi-Objective Quadratic Fractional Programming Problem

A Fuzzy Goal Programming Approach for Solving Chance Constrained Bi-Level Multi-Objective Quadratic Fractional Programming Problem