Bi-level Multi-objective Programming Research Articles

Interactive Fuzzy Goal Programming Based on Jacobian Matrix to Solve Decentralized Bi-level Multi-objective Fractional Programming Problems

This paper attempts to shed light on solving decentralized bi-level multi-objective fractional programming problems (DBL-MOFPP) with single decision maker at the first level and multiple decision makers at the second level. In this paper, we proposed a fuzzy goal programming (FGP) based on Jacobian matrix for DBL-MOFPP. In the proposed approach, membership functions are associated for the fuzzy goals of all objectives at two levels and they are linearized using a Jacobian matrix. Then FGP approach is used to achieve highest degree of each of the membership goals by obtaining the most satisfactory solution for all decision makers. We used known numerical example and practical application in order to show the efficiency and superiority of the proposed approach. Sensitivity analysis with variation of tolerance values on decision functions is performed to present how the solution is sensitive to the change of tolerance values.

Supply Chain Coordination Under Ramp-Type Price and Effort Induced Demand Considering Revenue Sharing Contract

In this paper, a revenue sharing contract is designed to coordinate a distribution channel where the demand of the product is ramp-type price and effort sensitive. It is shown that traditional revenue sharing contract does not coordinate the system. As an alternative, two new mechanisms are proposed (i) revenue sharing with coordinated effort of the retailer alone and (ii) both revenue and effort sharing contract. In addition, a crucial modification of revenue sharing fraction is also proposed. To enhance the applicability of revenue sharing contract, the contract parameters are determined by using bi-level multi-objective fuzzy goal programming technique where manufacturer sets the wholesale price greater than the marginal cost. Numerical examples are presented to illustrate all the models.

Administrative and market-based allocation mechanism for regional water resources planning

The aim of this paper is to present an administrative and market-based optimization method for solving a problem of regional water resources allocation by considering a hierarchical structure under multiple uncertainties. To accomplish this, a multi-objective bi-level programming model is developed based on the water right distribution in a river basin. In this model, the stream flow (i.e., water supply) and water demand are considered as a fuzzy random variable and a random fuzzy variable, respectively. The regional authority, the leader in the hierarchy, seeks to maximize the total benefit to society while simultaneously minimizing pollution emissions. The sub-areas, the followers in the hierarchy, seek to maximize their own economic benefits. To deal with the inherent uncertainty, a transformation of variables into fuzzy variables is done, and through the expected value operation, the fuzzy variables are subsequently transformed into determined ones. For solving the complex non-linear bi-level programming model, a bi-level interactive method based on satisfactory solution with global–local–neighbor adaptive particle swarm optimization (GLN-aPSO) is designed as a combined solution method. A case study is presented to demonstrate the applicability and efficiency of this method. The interactive solutions associated with different minimal satisfactory degrees of the two objectives in the upper level have been generated. They can help the regional authority and the sub-areas to identify desired water allocation schemes according to their preferences and practical conditions, as well as facilitate in-depth analyses of tradeoffs between the objectives in the two levels. Finally, to verify that it is reasonable to use bi-level programming the results are compared with those of using single level programming.

An exact penalty function approach for solving the linear bilevel multiobjective programming problem

In this paper, a new penalty function approach is proposed for the linear bilevel multiobjective programming problem. Using the optimality conditions of the lower level problem, we transform the linear bilevel multiobjective programming problem into the corresponding linear multiobjective programming problem with complementary constraint. The complementary constraint is appended to the upper level objectives with a penalty. Then, we give via an exact penalty method an existence theorem of Pareto optimal solutions and propose an algorithm for the linear bilevel multiobjective programming problem. Numerical results showing viability of the penalty function approach are presented.

Multidemand Multisource Order Quantity Allocation with Multiple Transportation Alternatives

This paper focuses on a multidemand multisource order quantity allocation problem with multiple transportation alternatives. To solve this problem, a bilevel multiobjective programming model under a mixed uncertain environment is proposed. Two levels of decision makers are considered in the model. On the upper level, the purchaser aims to allocate order quantity to multiple suppliers for each demand node with the consideration of three objectives: total purchase cost minimization, total delay risk minimization, and total defect risk minimization. On the lower level, each supplier attempts to optimize the transportation alternatives with total transportation and penalty costs minimization as the objective. In contrast to prior studies, considering the information asymmetry in the bilevel decision, random and fuzzy random variables are used to model uncertain parameters of the construction company and the suppliers. To solve the bilevel model, a solution method based on Kuhn-Tucker conditions, sectional genetic algorithm, and fuzzy random simulation is proposed. Finally, the applicability of the proposed model and algorithm is evaluated through a practical case from a large scale construction project. The results show that the proposed model and algorithm are efficient in dealing with practical order quantity allocation problems.

A Smoothing Method for Solving Bilevel Multiobjective Programming Problems

In this paper, a bilevel multiobjective programming problem, where the lower level is a convex parameter multiobjective program, is concerned. Using the KKT optimality conditions of the lower level problem, this kind of problem is transformed into an equivalent one-level nonsmooth multiobjective optimization problem. Then, a sequence of smooth multiobjective problems that progressively approximate the nonsmooth multiobjective problem is introduced. It is shown that the Pareto optimal solutions (stationary points) of the approximate problems converge to a Pareto optimal solution (stationary point) of the original bilevel multiobjective programming problem. Numerical results showing the viability of the smoothing approach are reported.

Fuzzy Bi-Level Multi-Objective Fractional Integer Programming

We present an algorithm to solve a bi-level multi-objective fractional integer programming problem involving fuzzy numbers in the right-hand side of the constraints. The suggested algorithm combine the method of Taylor series together with the Kuhn Tucker conditions to solve fuzzy bi-level multi-objective fractional integer programming problem (FBLMOFIPP) then Gomory cuts are added till the integer solution is obtained. An illustrative example is discussed to demonstrate the correctness of the proposed solution method.

Optimality Conditions for Multiobjective Optimization Problem Constrained by Parameterized Variational Inequalities

We study a multiobjective optimality problem constrained by parameterized variational inequalities. By separation theorem for convex sets, we translate the multiobjective optimality problem into single objective optimality problem, and obtain the first-order optimality conditions of this problem. Under the calmness conditions, an efficient upper estimate of coderivative for a composite set-valued mapping is derived. At last, we apply that result to the multiobjective bilevel programming problem and MPEC with Nash equilibrium constraints.

Retrofitting Transportation Network Using a Fuzzy Random Multiobjective Bilevel Model to Hedge against Seismic Risk

This paper focuses on the problem of hedging against seismic risk through the retrofit of transportation systems in large-scale construction projects (LSCP). A fuzzy random multiobjective bilevel programming model is formulated with the objectives of the retrofit costs and the benefits on two separate levels. After establishing the model, a fuzzy random variable transformation approach and fuzzy variable approximation decomposition are used to deal with the uncertainty. An approximation decomposition-based multi-objective AGLNPSO is developed to solve the model. The results of a case study validate the efficiency of the proposed approach.

A new multi-objective bi-level programming model for energy and locality aware multi-job scheduling in cloud computing

How to reduce power consumption of data centers has received worldwide attention. By combining the energy-aware data placement policy and locality-aware multi-job scheduling scheme, we propose a new multi-objective bi-level programming model based on MapReduce to improve the energy efficiency of servers. First, the variation of energy consumption with the performance of servers is taken into account; second, data locality can be adjusted dynamically according to current network state; last but not least, considering that task-scheduling strategies depend directly on data placement policies, we formulate the problem as an integer bi-level programming model. In order to solve the model efficiently, specific-design encoding and decoding methods are introduced. Based on these, a new effective multi-objective genetic algorithm based on MOEA/D is proposed. As there are usually tens of thousands of tasks to be scheduled in the cloud, this is a large-scale optimization problem and a local search operator is designed to accelerate convergent speed of the proposed algorithm. Finally, numerical experiments indicate the effectiveness of the proposed model and algorithm.

Control Risk for Multimode Resource-Constrained Project Scheduling Problems under Hybrid Uncertainty

AbstractRisk control in project management has been receiving increasing attention in recent years. This paper proposes an approach based on a risk-assessment program and demonstrates how this can be extended to enhance the multimode resource-constrained project-scheduling problem (MRCPSP) to realize risk control and effective scheduling. The main contribution of this paper is to provide a means to control scheduling and material-purchasing risks by using a bilevel decision framework to identify risk and conduct a focused evaluation. Specifically, this paper establishes a hybrid uncertain multiobjective bilevel programming model and introduces an expected-value operator to deal with the hybrid uncertainty. A multiswarm differential-updating particle swarm optimization is developed to support the model calculations. Worthiness and efficiency of the proposed approach is validated using a case study. This approach can help improve scheduling strategies and ensure realistic project-planning.

Solving high dimensional bilevel multiobjective programming problem using a hybrid particle swarm optimization algorithm with crossover operator

In this paper, a hybrid particle swarm optimization with crossover operator (denoted as C-PSO) is proposed, in which a crossover operator is adopted for enhancing the information exchange between particles to prevent premature convergence of the swarm. The C-PSO algorithm is employed for solving high dimensional bilevel multiobjective programming problem (HDBLMPP) in this study, which performs better than the existing method with respect to the generational distance and has almost the same performance with respect to the spacing. Finally, we use four test problems and a practical application to measure and evaluate the proposed algorithm. Our results indicate that the proposed algorithm is highly competitive with respect to the algorithm representative of the state-of-the-art in high dimensional bilevel multiobjective optimization.

Bi-level multi-objective programming problem with fuzzy demands: a fuzzy goal programming algorithm

In this paper, a fuzzy goal programming (FGP) algorithm for solving bi-level multi-objective programming problems with fuzzy demands is presented. These fuzzy demands reflect the experts' imprecise or fuzzy understandings of the nature of parameters in the problem formulation process are assumed to be characterized as fuzzy numbers. Using the level sets of fuzzy parameters, the corresponding non fuzzy bi-level programming problem is introduced. In the proposed algorithm, the membership functions for the defined fuzzy goals of all objective functions at the two levels, as well as the membership functions for the vector of decision variables controlled by FLDM are developed in the model formulation of the problem. Then FGP algorithm is used to achieve the highest degree of each of the membership goals by minimizing their deviational variables and thereby obtaining the most satisfactory solution for all decision makers. Illustrative numerical example is given to demonstrate the proposed algorithm.

Considering decision decentralizations to solve bi-level multi-objective decision-making problems: A fuzzy approach

This paper investigates decentralized bi-level multi-objective linear programming (DBL-MOLP) problems with a single decision-maker (DM) at the higher level and more than one DM at the lower level. Each DM can have more than one objective function, which is formulated as a fuzzy goal. To characterize the decision decentralization in a DBL-MOLP problem, this paper proposes an assignment scheme of relative satisfaction for the higher-level DM to ensure his leadership and therefore prevent the paradox problem reported in the literature, where lower-level DMs have higher satisfaction degrees than that of the higher-level DM. Through the assignment scheme, if the higher-level DM is not satisfied with the resulting solutions of objective functions, the re-solving process is easily conducted by adjusting the level of relative satisfaction for the associated lower-level DMs. A linearization transformation approach is also presented to facilitate the solution process. To emphasize some important fuzzy goals, a weighting scheme is considered in this paper. A numerical example is used for illustration, and comparisons with existing approaches are conducted to demonstrate the feasibility of the proposed method.

Using Bi-level Multi-Objective Programming in Passenger Structure Optimization for Comprehensive Transportation Channel

The economic structure is the lifeblood of national economic development. Passenger structure within comprehensive transportation channel (CTC) is the key to play a higher efficiency and to operate normally of an integrated transport system. This paper analyzes the ideas of PTS optimization within CTC using bi-level planning strategies. And then a bi-level multi-objective programming model is established in this paper, in which the upper level takes the builders and government’s concern into account, such as system efficiency, transport capacity, capital investment. The lower level is a network equilibrium model based on a generalized cost function, while Channel saturation entropy index is introduced in the response function. A particle swarm optimization algorithm is designed to efficiently solve this NP-hard model. Finally, numerical implementations based on Beijing-Shanghai Channel are carried out to account for the performance of our model and algorithm.

Stability of multi-objective bi-level linear programming problems under fuzziness

This paper deals with multi-objective bi-level linear programming problems under fuzzy environment. In the proposed method, tentative solutions are obtained and evaluated by using the partial information on preference of the decision-makers at each level. The existing results concerning the qualitative analysis of some basic notions in parametric linear programming problems are reformulated to study the stability of multi-objective bi-level linear programming problems. An algorithm for obtaining any subset of the parametric space, which has the same corresponding Pareto optimal solution, is presented. Also, this paper established the model for the supply-demand interaction in the age of electronic commerce (EC). First of all, the study uses the individual objectives of both parties as the foundation of the supply-demand interaction. Subsequently, it divides the interaction, in the age of electronic commerce, into the following two classifications: (i) Market transactions, with the primary focus on the supply demand relationship in the marketplace; and (ii) Information service, with the primary focus on the provider and the user of information service. By applying the bi-level programming technique of interaction process, the study will develop an analytical process to explain how supply-demand interaction achieves a compromise or why the process fails. Finally, a numerical example of information service is provided for the sake of illustration.

Bilevel Multiobjective Programming Applied to Water Resources Allocation

Water allocation is an essential programming to support the sustainable development of Wuwei Basin, Gansu Province, China. To satisfy the demands of the decision makers (DMs) of each subarea and the total area, a bilevel multiobjective linear programming (BLMOLP) model is proposed. In the BLMOLP, DMs have a hierarchy of two levels—the upper level and the lower level DMs. In this paper, a fuzzy goal programming (FGP) approach is applied to solve the BLMOLP. Firstly, the upper level is solved and used as the tolerance for the lower level. Then the weights of each objective function in the lower level are evaluated. Finally, a satisfied optimization solution of the problem was calculated. The result suggests that the FGP is a simple and feasible approach to BLMOLP problems. The proposed method was applied to a case study for water resources allocation in Wuwei Basin. For four scenarios under consideration, the model can effectively balance the benefits among all regions and sections according to the priority of the upper level decision makers. The results indicate that comprehensive solutions have been obtained.

Generating Efficient Solutions in Bilevel Multi-Objective Programming Problems

In this paper, we address bilevel multi-objective programming problems (BMPP) in which the decision maker at each level has multiple objective functions conflicting with each other. Given a BMPP, we show how to construct two artificial multiobjective programming problems such that any point that is efficient for both the two problems is an efficient solution of the BMPP. Some necessary and sufficient conditions for which the obtained result is applicable are provided. A complete procedure of the implementation of an algorithm for generating efficient solutions for the linear case of BMPP is presented. A numerical example is provided to illustrate how the algorithm operates.

Fuzzy linear fractional bi-level multi-objective programming problems

The Kuhn-Tuker condition has become nowadays an important tool in the hands of investigation for checking the optimality in optimization literature. In the present paper with use of a Taylor series and Kuhn-Tucker conditions approach, we solve a fuzzy linear fractional bilevel multi-objective programming (FLFBL-MOP) problem. The Taylor series is an expansion of a series that represents a function. In the proposed approach, membership functions associated with each level(s) of the objective(s) of FLFBL-MOP problems are transformed and unied by using a Taylor series approach. By using the Kuhn-Tucker conditions, the problem is reduced to a single objective and nally, numericalexample is given to illustrates the efficiency and superiority of the proposed approach.

Modeling and Simulation of Bus Priority Oriented Urban Road Network Elastic Equilibrium

As the poor applicability of rail transit and elevated road limited by economy, population and many other factors, the waste of road resources caused by bus lane, maximizing application of available road resources and bus facilities, reducing the environmental pollution is the critical issue faced by current situation of urban transportation development in China. Guided by Bus Priority Oriented, utilized comprehensive techniques, such as ABC classification, bi-level programming model, sensitivity analysis, this paper constructed a theoretical frame of elastic equilibrium based on system optimization. Taking time and environment as optimized objectives, This paper established the Bi-Level multiobjective programming model based on travel time, and programmed with MATLAB for the calculation. Finally, the case study is illustrated. the software VISSIM was adopted to realize the simulation of the volume before and after the implementation of elastic plan in urban areas of PingDingshan. The simulation results showed the validity and feasibility of the modeling and simulation of Bus Priority Oriented urban road network elastic equilibrium.