Bilevel Multiobjective Optimization Problem Research Articles

Valuing water quality tradeoffs at different spatial scales: An integrated approach using bilevel optimization

Abstract This study evaluates the tradeoff between agricultural production and water quality at both the watershed scale and the farm scale, using an integrated economic-biophysical hybrid genetic algorithm. We apply a multi-input, multi-output profit maximization model to detailed farm-level production data from the Oregon Willamette Valley to predict each producer's response to a targeted fertilizer tax policy. Their resulting production decisions are included in a biophysical model of basin-level soil and water quality. Building on a general regulation problem for nonpoint pollution, we use a hybrid genetic algorithm to integrate the economic and biophysical models into one bilevel multiobjective optimization problem, the joint maximization of farm profits and minimization of Nitrate runoff resulting from fertilizer usage. This approach allows us to more fully endogenize fertilizer reduction cost, rather than assume an average cost relationship. The solution set of tax rates generates the Pareto optimal frontier at the watershed level. We then measure the tradeoffs between maximum profit and Nitrogen loading for individual farms, subject to the solution fertilizer tax policy. We find considerable variation in tradeoff values across the basin, which could be used to target incentives for reducing Nitrogen loading to agricultural producers under non-uniform control strategies.

A multi-objective bi-level location planning problem for stone industrial parks

This paper focuses on a stone industrial park location problem with a hierarchical structure consisting of a local government and several stone enterprises under a random environment. In contrast to previous studies, conflicts between the local authority and the stone enterprises are considered. The local government, being the leader in the hierarchy, aims to minimize both total pollution emissions and total development and operating costs. The stone enterprises, as the followers in the hierarchy, only aim to minimize total costs. In addition, unit production cost and unit transportation cost are considered random variables. This complicated multi-objective bi-level optimization problem poses several challenges, including randomness, two-level decision making, conflicting objectives, and difficulty in searching for the optimal solutions. Various approaches are employed to tackle these challenges. In order to make the model trackable, expected value operator is used to deal with the random variables in the objective functions and a chance constraint-checking method is employed to deal with such variables in the constraints. The problem is solved using a bi-level interactive method based on a satisfactory solution and Adaptive Chaotic Particle Swarm Optimization (ACPSO). Finally, a case study is conducted to demonstrate the practicality and efficiency of the proposed model and solution algorithm. The performance of the proposed bi-level model and ACPSO algorithm was highlighted by comparing to a single-level model and basic PSO and GA algorithms.

Genetic Algorithm-Based Approach for Estimating Commodity OD Matrix

This paper proposes an estimation approach for a commodity origin-destination matrix by using a sample commodity origin-destination (OD) matrix from a Commodity Flow Survey and also mode-specific OD matrices obtained from a transportation record of freight carriers. The proposed approach is formulated as a multi-objective bi-level optimization problem in which the upper-level seeks to minimize the sum of square deviation from the target matrices, while in the lower-level, user-equilibrium assignments of commodity OD matrix from a CFS and mode-specific OD matrices are performed. The developed model and the Genetic Algorithms-based solution algorithm were validated and tested on an intermodal transportation network of Korea. The results show that the model is able to produce an acceptable commodity OD matrix, implying that the proposed approach is applicable for a real-world problem.

Strategic Bidding in Transmission Constrained Electricity Markets Using Artificial Bee Colony Algorithm

This article presents a methodology for a supplier's optimal bidding strategy considering network constraints and rivals' bidding behaviors. This has been formulated as a bi-level multi objective optimization problem, in which the lower-level problem represents the market clearing process by the system operator for minimization of customers' payments. The upper-level problem is a multi-objective formulation that simultaneously considers profit maximization and risk minimization. Pareto-optimal solutions of the bi-level multi-objective optimization problem have been obtained using the ε-constraint method. A fuzzy decision-making process has been used to derive the most efficient solution among the Pareto-optimal solutions. Upper-level objective functions are highly non-linear; therefore, the artificial bee colony algorithm, a relatively new heuristic approach, has been applied to obtain the best solution to the bi-level multi objective optimization problem for a day-ahead hourly market. The effectiveness of the proposed method has been established on a 2-bus system and the IEEE-30 bus test system. The results, obtained using the artificial bee colony algorithm, have been compared with those obtained using a genetic algorithm based approach.

A PARTIAL LINEARIZATION METHOD FOR MULTI-OBJECTIVE CONTINUOUS NETWORK DESIGN PROBLEM WITH ENVIRONMENTAL CONSIDERATIONS

Nowadays, the environmental impact of transportation project and, especially air pollution impacts, are major concerns in evaluating transportation projects. Based on this concern, beside definition of traditional objective functions like total travel time and total investment cost, different type of environmental related function is considered as objective function in urban network design. In this paper, the continuous network design problem (CNDP) is formulated as a multi-objective bi-level optimization problem. The objective function of the upper level problem is a weighted combination of total travel time, total investment cost and total vehicular emission in the network. The lower level problem is the traffic assignment problem which would predict the vehicular flow on each link in the network. A new solution algorithm is proposed for solving the bi-level optimization problem by the partial linearization of the lower level problem. The solution algorithm was applied to the city of Sioux Falls, a well known transportation network.

Necessary Optimality Conditions and a New Approach to Multiobjective Bilevel Optimization Problems

Multiobjective optimization problems typically have conflicting objectives, and a gain in one objective very often is an expense in another. Using the concept of Pareto optimality, we investigate a multiobjective bilevel optimization problem (say, P). Our approach consists of proving that P is locally equivalent to a single level optimization problem, where the nonsmooth Mangasarian–Fromovitz constraint qualification may hold at any feasible solution. With the help of a special scalarization function introduced in optimization by Hiriart–Urruty, we convert our single level optimization problem into another problem and give necessary optimality conditions for the initial multiobjective bilevel optimization problem P.

Solution Concepts and New Optimality Conditions in Bilevel Multiobjective Programming

In this paper, new sufficient optimality theorems for a solution of a differentiable bilevel multiobjective optimization problem (BMOP) are established. We start with a discussion on solution concepts in bilevel multiobjective programming; a theorem giving necessary and sufficient conditions for a decision vector to be called a solution of the BMOP and a proposition giving the relations between four types of solutions of a BMOP are presented and proved. Then, under the pseudoconvexity assumptions on the upper and lower level objective functions and the quasiconvexity assumptions on the constraints functions, we establish and prove two new sufficient optimality theorems for a solution of a general BMOP with coupled upper level constraints. Two corollary of these theorems, in the case where the upper and lower level objectives and constraints functions are convex are presented.

Connections Between Single-Level and Bilevel Multiobjective Optimization

The relationship between bilevel optimization and multiobjective optimization has been studied by several authors, and there have been repeated attempts to establish a link between the two. We unify the results from the literature and generalize them for bilevel multiobjective optimization. We formulate sufficient conditions for an arbitrary binary relation to guarantee equality between the efficient set produced by the relation and the set of optimal solutions to a bilevel problem. In addition, we present specially structured bilevel multiobjective optimization problems motivated by real-life applications and an accompanying binary relation permitting their reduction to single-level multiobjective optimization problems.

Optimal bidding strategy for all market players in a wholesale power market considering demand response programs

In a wholesale power market, generation companies (GENCOs) and distribution companies (DISCOs), as market players, try to maximize their payoffs by implementation of optimal strategies. This paper presents a method for determining GENCOs and DISCOs' optimal bidding strategies considering other participants' bidding and power systems operating conditions. It is assumed that DISCOs bid their distributed generations (DG) and GENCOs bid their generators in a day-ahead wholesale energy market. Meanwhile we assume that DISCOs bid interruptible loads (ILs) in market on behalf of their end customers. The proposed methodology is modeled as a bi-level multi-objective optimization problem with the upper subproblem representing individual GENCOs and DISCOs and the lower subproblem representing the Independent System Operator (ISO). The upper level maximizes the individual market participant's payoffs and the lower one solves the ISO's market clearing problem for minimizing operation costs. Genetic algorithm (GA) and fuzzy satisfying method (FSM) are employed for solving the proposed model and determining optimal bidding strategies. An eight-bus system is employed to illustrate the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.

Multiobjective bilevel optimization

In this work nonlinear non-convex multiobjective bilevel optimization problems are discussed using an optimistic approach. It is shown that the set of feasible points of the upper level function, the so-called induced set, can be expressed as the set of minimal solutions of a multiobjective optimization problem. This artificial problem is solved by using a scalarization approach by Pascoletti and Serafini combined with an adaptive parameter control based on sensitivity results for this problem. The bilevel optimization problem is then solved by an iterative process using again sensitivity theorems for exploring the induced set and the whole efficient set is approximated. For the case of bicriteria optimization problems on both levels and for a one dimensional upper level variable, an algorithm is presented for the first time and applied to two problems: a theoretical example and a problem arising in applications.