A Hybrid Multiobjective Evolutionary Algorithm for Solving Vehicle Routing Problem with Time Windows

Vehicle routing problem with time windows (VRPTW) involves the routing of a set of vehicles with limited capacity from a central depot to a set of geographically dispersed customers with known demands and predefined time windows. The problem is solved by optimizing routes for the vehicles so as to meet all given constraints as well as to minimize the objectives of traveling distance and number of vehicles. This paper proposes a hybrid multiobjective evolutionary algorithm (HMOEA) that incorporates various heuristics for local exploitation in the evolutionary search and the concept of Pareto's optimality for solving multiobjective optimization in VRPTW. The proposed HMOEA is featured with specialized genetic operators and variable-length chromosome representation to accommodate the sequence-oriented optimization in VRPTW. Unlike existing VRPTW approaches that often aggregate multiple criteria and constraints into a compromise function, the proposed HMOEA optimizes all routing constraints and objectives simultaneously, which improves the routing solutions in many aspects, such as lower routing cost, wider scattering area and better convergence trace. The HMOEA is applied to solve the benchmark Solomon's 56 VRPTW 100-customer instances, which yields 20 routing solutions better than or competitive as compared to the best solutions published in literature.     

Solving multiobjective vehicle routing problem with stochastic demand via evolutionary computation

This paper considers the routing of vehicles with limited capacity from a central depot to a set of geographically dispersed customers where actual demand is revealed only when the vehicle arrives at the customer. The solution to this vehicle routing problem with stochastic demand (VRPSD) involves the optimization of complete routing schedules with minimum travel distance, driver remuneration, and number of vehicles, subject to a number of constraints such as time windows and vehicle capacity. To solve such a multiobjective and multi-modal combinatorial optimization problem, this paper presents a multiobjective evolutionary algorithm that incorporates two VRPSD-specific heuristics for local exploitation and a route simulation method to evaluate the fitness of solutions. A new way of assessing the quality of solutions to the VRPSD on top of comparing their expected costs is also proposed. It is shown that the algorithm is capable of finding useful tradeoff solutions for the VRPSD and the solutions are robust to the stochastic nature of the problem. The developed algorithm is further validated on a few VRPSD instances adapted from Solomon’s vehicle routing problem with time windows (VRPTW) benchmark problems.     

A two-phase algorithm for the partial accessibility constrained vehicle routing problem

In the partial accessibility constrained vehicle routing problem, a route can be covered by two types of vehicles, i.e. truck or truck + trailer. Some customers are accessible by both vehicle types, whereas others solely by trucks. After introducing an integer programming formulation for the problem, we describe a two-phase heuristic method which extends a classical vehicle routing algorithm. Since it is necessary to solve a combinatorial problem that has some similarities with the generalized assignment problem, we propose an enumerative procedure in which bounds are obtained from a Lagrangian relaxation. The routine provides very encouraging results on a set of test problems.     

A matheuristic for the truck and trailer routing problem

In the truck and trailer routing problems (TTRPs) a fleet of trucks and trailers serves a set of customers. Some customers with accessibility constraints must be served just by truck, while others can be served either by truck or by a complete vehicle (a truck pulling a trailer). We propose a simple, yet effective, two-phase matheuristic that uses the routes of the local optima of a hybrid GRASP × ILS as columns in a set-partitioning formulation of the TTRP. Using this matheuristic we solved both the classical TTRP with fixed fleet and the new variant with unlimited fleet. This matheuristic outperforms state-of-the-art methods both in terms of solution quality and computing time. While the best variant of the matheuristic found new best-known solutions for several test instances from the literature, the fastest variant of the matheuristic achieved results of comparable quality to those of all previous method from the literature with an average speed-up of at least 2.5.     

随机混流U型拆卸线平衡排序问题多目标进化算法优化

针对混流U型拆卸线平衡排序问题,考虑拆卸时间不确定,建立了该问题最小拆卸线平均闲置率、尽早拆卸危害和高需求零部件、最小化平均方向改变次数的多目标优化模型,并提出一种基于分解和动态邻域搜索的混合多目标进化算法(Hybrid Multi-objective Evolutionary Algorithm Based on Decomposition, HMOEA/D)。该算法通过采用弹性任务分配策略、动态邻域结构和动态调整权重以保证解的可行性并搜索得到分布较好的非劣解集。最后,仿真求解实验设计技术(DOE)生成的测试算例,结果表明HMOEA/D较其它算法能得到更接近Pareto最优、分布更好的近似解集。     

Implementation of scatter search for multi-objective optimization: a comparative study

Interest in the design of efficient meta-heuristics for the application to combinatorial optimization problems is growing rapidly. The optimal design of water distribution networks is an important optimization problem which consists of finding the best way of conveying water from the sources to the users, thus satisfying their requirements. The efficient design of looped networks is a much more complex problem than the design of branched ones, but their greater reliability can compensate for the increase in cost when closing some loops. Mathematically, this is a non-linear optimization problem, constrained to a combinatorial space, since the diameters are discrete and it has a very large number of local solutions. Many works have dealt with the minimization of the cost of the network but few have considered their cost and reliability simultaneously. The aim of this paper is to evaluate the performance of an implementation of Scatter Search in a multi-objective formulation of this problem. Results obtained in three benchmark networks show that the method here proposed performs accurately well in comparison with other multi-objective approaches also implemented.     

Multi-objective vehicle routing problems

Routing problems, such as the traveling salesman problem and the vehicle routing problem, are widely studied both because of their classic academic appeal and their numerous real-life applications. Similarly, the field of multi-objective optimization is attracting more and more attention, notably because it offers new opportunities for defining problems. This article surveys the existing research related to multi-objective optimization in routing problems. It examines routing problems in terms of their definitions, their objectives, and the multi-objective algorithms proposed for solving them.     

求解最小Steiner树的蚁群优化算法及其收敛性

最小Steiner树问题是NP难问题,它在通信网络等许多实际问题中有着广泛的应用．蚁群优化算法是最近提出的求解复杂组合优化问题的启发式算法．本文以无线传感器网络中的核心问题之一,路由问题为例,给出了求解最小Steiner树的蚁群优化算法的框架．把算法的迭代过程看作是离散时间的马尔科夫过程,证明了在一定的条件下,该算法所产生的解能以任意接近于1的概率收敛到路由问题的最优解．     

The small world of efficient solutions: empirical evidence from the bi-objective {0,1}-knapsack problem

The small world phenomenon, Milgram (1967) has inspired the study of real networks such as cellular networks, telephone call networks, citation networks, power and neural networks, etc. The present work is about the study of the graphs produced by efficient solutions of the bi-objective {0,1}-knapsack problem. The experiments show that these graphs exhibit properties of small world networks. The importance of the supported and non-supported solutions in the entire efficient graph is investigated. The present research could be useful for developing more effective search strategies in both exact and approximate solution methods of {0,1} multi-objective combinatorial optimization problems.     

Pareto Simulated Annealing for Fuzzy Multi-Objective Combinatorial Optimization

The paper presents a metaheuristic method for solving fuzzy multi-objective combinatorial optimization problems. It extends the Pareto simulated annealing (PSA) method proposed originally for the crisp multi-objective combinatorial (MOCO) problems and is called fuzzy Pareto simulated annealing (FPSA). The method does not transform the original fuzzy MOCO problem to an auxiliary deterministic problem but works in the original fuzzy objective space. Its goal is to find a set of approximately efficient solutions being a good approximation of the whole set of efficient solutions defined in the fuzzy objective space. The extension of PSA to FPSA requires the definition of the dominance in the fuzzy objective space, modification of rules for calculating probability of accepting a new solution and application of a defuzzification operator for updating the average position of a solution in the objective space. The use of the FPSA method is illustrated by its application to an agricultural multi-objective project scheduling problem.     

Genetic local search for multi-objective combinatorial optimization

The paper presents a new genetic local search (GLS) algorithm for multi-objective combinatorial optimization (MOCO). The goal of the algorithm is to generate in a short time a set of approximately efficient solutions that will allow the decision maker to choose a good compromise solution. In each iteration, the algorithm draws at random a utility function and constructs a temporary population composed of a number of best solutions among the prior generated solutions. Then, a pair of solutions selected at random from the temporary population is recombined. Local search procedure is applied to each offspring. Results of the presented experiment indicate that the algorithm outperforms other multi-objective methods based on GLS and a Pareto ranking-based multi-objective genetic algorithm (GA) on travelling salesperson problem (TSP).     

ParEGO extensions for multi-objective optimization of expensive evaluation functions

This paper deals with multi-objective optimization in the case of expensive objective functions. Such a problem arises frequently in engineering applications where the main purpose is to find a set of optimal solutions in a limited global processing time. Several algorithms use linearly combined criteria to use directly mono-objective algorithms. Nevertheless, other algorithms, such as multi-objective evolutionary algorithm (MOEA) and model-based algorithms, propose a strategy based on Pareto dominance to optimize efficiently all criteria. A widely used model-based algorithm for multi-objective optimization is Pareto efficient global optimization (ParEGO). It combines linearly the objective functions with several random weights and maximizes the expected improvement (EI) criterion. However, this algorithm tends to favor parameter values suitable for the reduction of the surrogate model error, rather than finding non-dominated solutions. The contribution of this article is to propose an extension of the ParEGO algorithm for finding the Pareto Front by introducing a double Kriging strategy. Such an innovation allows to calculate a modified EI criterion that jointly accounts for the objective function approximation error and the probability to find Pareto Set solutions. The main feature of the resulting algorithm is to enhance the convergence speed and thus to reduce the total number of function evaluations. This new algorithm is compared against ParEGO and several MOEA algorithms on a standard benchmark problems. Finally, an automotive engineering problem allowing to illustrate the applicability of the proposed approach is given as an example of a real application: the parameter setting of an indirect tire pressure monitoring system.     

Utilizing expected improvement and generalized data envelopment analysis in multi-objective genetic algorithms

Meta-heuristic methods such as genetic algorithms (GA) and particle swarm optimization (PSO) have been extended to multi-objective optimization problems, and have been observed to be useful for finding good approximate Pareto optimal solutions. In order to improve the convergence and the diversity in the search of solutions using meta-heuristic methods, this paper suggests a new method to make offspring by utilizing the expected improvement (EI) and generalized data envelopment analysis (GDEA). In addition, the effectiveness of the proposed method will be investigated through several numerical examples in comparison with the conventional multi-objective GA and PSO methods.     

K-PPM: A new exact method to solve multi-objective combinatorial optimization problems

In this paper we propose an exact method able to solve multi-objective combinatorial optimization problems. This method is an extension, for any number of objectives, of the 2-Parallel Partitioning Method (2-PPM) we previously proposed. Like 2-PPM, this method is based on splitting of the search space into several areas, leading to elementary searches. The efficiency of the proposed method is evaluated using a multi-objective flow-shop problem.     

SOME COMBINATORIAL OPTIMIZATION PROBLEMS ARISING FROM VLSI CIRCUIT DESIGN

This paper is basically a survey to show a number of combinatorlal optimization problems arising from VLSI clrcult design.Some of them including the existence problem,minimax problem,net representation,bend minimization,area minimization,placement problem,routing problem,etc,are especially discussed with new results and theoretical ideas for treating them.Finally,a number of problems for further research are mentioned.     

Inverse multi-objective combinatorial optimization

Inverse multi-objective combinatorial optimization consists of finding a minimal adjustment of the objective functions coefficients such that a given set of feasible solutions becomes efficient. An algorithm is proposed for rendering a given feasible solution into an efficient one. This is a simplified version of the inverse problem when the cardinality of the set is equal to one. The adjustment is measured by the Chebyshev distance. It is shown how to build an optimal adjustment in linear time based on this distance, and why it is right to perform a binary search for determining the optimal distance. These results led us to develop an approach based on the resolution of mixed-integer linear programs. A second approach based on a branch-and-bound is proposed to handle any distance function that can be linearized. Finally, the initial inverse problem is solved by a cutting plane algorithm.     

On solving multiobjective bin packing problems using evolutionary particle swarm optimization

The bin packing problem is widely found in applications such as loading of tractor trailer trucks, cargo airplanes and ships, where a balanced load provides better fuel efficiency and safer ride. In these applications, there are often conflicting criteria to be satisfied, i.e., to minimize the bins used and to balance the load of each bin, subject to a number of practical constraints. Unlike existing studies that only consider the issue of minimum bins, a multiobjective two-dimensional mathematical model for bin packing problems with multiple constraints (MOBPP-2D) is formulated in this paper. To solve MOBPP-2D problems, a multiobjective evolutionary particle swarm optimization algorithm (MOEPSO) is proposed. Without the need of combining both objectives into a composite scalar weighting function, MOEPSO incorporates the concept of Pareto’s optimality to evolve a family of solutions along the trade-off surface. Extensive numerical investigations are performed on various test instances, and their performances are compared both quantitatively and statistically with other optimization methods to illustrate the effectiveness and efficiency of MOEPSO in solving multiobjective bin packing problems.     

A heuristic algorithm for the free newspaper delivery problem

This paper addresses the problem of finding an effective distribution plan to deliver free newspapers from a production plant to subway, bus, or tram stations. The overall goal is to combine two factors: first, the free newspaper producing company wants to minimize the number of vehicle trips needed to distribute all newspapers produced at the production plant. Second, the company is interested in minimizing the time needed to consume all newspapers, i.e., the time needed to get all the newspapers taken by the final readers. The resulting routing problem combines aspects of the vehicle routing problem with time windows, the inventory routing problem, and additional constraints related to the production schedule. We propose a formulation and different heuristic approaches, as well as a hybrid method. Computational tests with real world data show that the hybrid method is the best in various problem settings.     

Finding representations for an unconstrained bi-objective combinatorial optimization problem

Typically, multi-objective optimization problems give rise to a large number of optimal solutions. However, this information can be overwhelming to a decision maker. This article introduces a technique to find a representative subset of optimal solutions, of a given bounded cardinality for an unconstrained bi-objective combinatorial optimization problem in terms of \(\epsilon \)-indicator. This technique extends the Nemhauser–Ullman algorithm for the knapsack problem and allows to find a representative subset in a single run. We present a discussion on the representation quality achieved by this technique, both from a theoretical and numerical perspective, with respect to an optimal representation.     

Approximating the Pareto optimal set using a reduced set of objective functions

Real-world applications of multi-objective optimization often involve numerous objective functions. But while such problems are in general computationally intractable, it is seldom necessary to determine the Pareto optimal set exactly. A significantly smaller computational burden thus motivates the loss of precision if the size of the loss can be estimated. We describe a method for finding an optimal reduction of the set of objectives yielding a smaller problem whose Pareto optimal set w.r.t. a discrete subset of the decision space is as close as possible to that of the original set of objectives. Utilizing a new characterization of Pareto optimality and presuming a finite decision space, we derive a program whose solution represents an optimal reduction. We also propose an approximate, computationally less demanding formulation which utilizes correlations between the objectives and separates into two parts. Numerical results from an industrial instance concerning the configuration of heavy-duty trucks are also reported, demonstrating the usefulness of the method developed. The results show that multi-objective optimization problems can be significantly simplified with an induced error which can be measured.