Solving Location-allocation Problems with Rectilinear Distances by Simulated Annealing

The objective of this study is to use the simulated annealing method to solve minisum location-allocation problems with rectilinear distances. The major advantage of the simulated annealing method is that it is a very general and efficient algorithm for solving combinatorial optimization problems with know objective functions. In this study, a simulated annealing algorithm was developed to solve the location-allocation problems, and its performance was compared with two other popular methods for solving location-allocation problems. The results show that simulated annealing is a good alternative to the two methods, as measured by both the solution quality and the computational time.     

A repeated matching heuristic for the single-source capacitated facility location problem

Facility location models form an important class of integer programming problems, with application in many areas such as the distribution and transportation industries. An important class of solution methods for these problems are so-called Lagrangean heuristics which have been shown to produce high quality solutions and which are at the same time robust. The general facility location problem can be divided into a number of special problems depending on the properties assumed. In the capacitated location problem each facility has a specific capacity on the service it provides. We describe a new solution approach for the capacitated facility location problem when each customer is served by a single facility. The approach is based on a repeated matching algorithm which essentially solves a series of matching problems until certain convergence criteria are satisfied. The method generates feasible solutions in each iteration in contrast to Lagrangean heuristics where problem dependent heuristics must be used to construct a feasible solution. Numerical results show that the approach produces solutions which are of similar and often better than those produced using the best Lagrangean heuristics.     

Lagrangian heuristics for the two-echelon,single-source,capacitated facility location problem

Facility location problems form an important class of integer programming problems, with application in the distribution and transportation industries. In this paper we are concerned with a particular type of facility location problem in which there exist two echelons of facilities. Each facility in the second echelon has limited capacity and can be supplied by only one facility (or depot) in the first echelon. Each customer is serviced by only one facility in the second echelon. The number and location of facilities in both echelons together with the allocation of customers to the second-echelon facilities are to be determined simultaneously. We propose a mathematical model for this problem and consider six heuristics based on Lagrangian relaxation for its solution. To solve the dual problem we make use of a subgradient optimization procedure. We present numerical results for a large suite of test problems. These indicate that the lower-bounds obtained from some relaxations have a duality gap which frequently is one third of the one obtained from traditional linear programming relaxation. Furthermore, the overall solution time for the heuristics are less than the time to solve the LP relaxation.     

A GRASP for Coloring Sparse Graphs

We first present a literature review of heuristics and metaheuristics developed for the problem of coloring graphs. We then present a Greedy Randomized Adaptive Search Procedure (GRASP) for coloring sparse graphs. The procedure is tested on graphs of known chromatic number, as well as random graphs with edge probability 0.1 having from 50 to 500 vertices. Empirical results indicate that the proposed GRASP implementation compares favorably to classical heuristics and implementations of simulated annealing and tabu search. GRASP is also found to be competitive with a genetic algorithm that is considered one of the best currently available for graph coloring.     

A continuous approximation approach for the integrated facility-inventory allocation problem

In today’s retail business many companies have a complex distribution network with several national and regional distribution centers. This article studies an integrated facility location and inventory allocation problem for designing a distribution network with multiple distribution centers and retailers. The key decisions are where to locate the regional distribution centers (RDCs), how to assign retail stores to RDCs and what should be the inventory policy at the different locations such that the total network cost is minimized. Due to the complexity of the problem, a continuous approximation (CA) model is used to represent the network. Nonlinear programming techniques are developed to solve the optimization problems. The main contribution of this work lies in developing a new CA modeling technique when the discrete data cannot be modeled by a continuous function and applying this technique to solve an integrated facility location-allocation and inventory-management problem. Our methodology is illustrated with the network from a leading US retailer. Numerical analysis suggests that the total cost is significantly lower in the case of the integrated model as compared with the non-integrated model, where the location-allocation and inventory-management problems are considered separately. This paper also studies the effects of changing parameter values on the optimal solutions and to point out some management implications.     

Combining simulated annealing with local search heuristics

We introduce a meta-heuristic to combine simulated annealing with local search methods for CO problems. This new class of Markov chains leads to significantly more powerful optimization methods than either simulated annealing or local search. The main idea is to embed deterministic local search techniques into simulated annealing so that the chain explores only local optima. It makes large, global changes, even at low temperatures, thus overcoming large barriers in configuration space. We have tested this meta-heuristic for the traveling salesman and graph partitioning problems. Tests on instances from public libraries and random ensembles quantify the power of the method. Our algorithm is able to solve large instances to optimality, improving upon local search methods very significantly. For the traveling salesman problem with randomly distributed cities, in a square, the procedure improves on 3-opt by 1.6%, and on Lin-Kernighan local search by 1.3%. For the partitioning of sparse random graphs of average degree equal to 5, the improvement over Kernighan-Lin local search is 8.9%. For both CO problems, we obtain new best heuristics.     

On solving complex multi-period location models using simulated annealing

This paper describes a study aimed at evaluating the capabilities of simulated annealing in dealing with complex, real-world multi-period location problems raised by school network planning in Portugal. The problems were formulated as mixed-integer linear optimization models. The models allow for facility closure or size reduction besides facility opening and size expansion, with sizes possibly limited to a set of pre-defined standards. They assume facility costs to be divided into a fix component and two variable components, respectively dependent on facility size and facility attendance. Results obtained through the study indicate that simulated annealing can be a useful tool for solving these kinds of models.     

考虑共享不确定因素的应急设施最大覆盖选址模型

为提升应急设施的服务质量和抵御中断风险的能力,研究应急设施最大覆盖选址-分配决策问题。扩展无容量限制的固定费用的可靠性选址决策模型,建立考虑共享不确定因素的应急设施最大覆盖选址优化模型,通过在目标和约束中引入budget不确定集刻画共享不确定因素,基于Bertsimas和Sim鲁棒优化方法建立混合整数规划模型,并将非线性问题转化为易于求解的鲁棒等价模型,利用带混沌搜索策略的改进灰狼优化算法求解模型,并对不确定鲁棒水平和中断概率进行敏感性分析。最后通过案例及数据仿真结果的对比分析,验证了模型的合理性和有效性,并给出最优的选址分配布局。     

Simulated annealing in the presence of noise

In many practical optimization problems, evaluation of a solution is subject to noise, e.g., due to stochastic simulations or measuring errors. Therefore, heuristics are needed that are capable of handling such noise. This paper first reviews the state-of-the-art in applying simulated annealing to noisy optimization problems. Then, two new algorithmic variants are proposed: an improved version of stochastic annealing that allows for arbitrary annealing schedules, and a new approach called simulated annealing in noisy environments (SANE). The latter integrates ideas from statistical sequential selection in order to reduce the number of samples required for making an acceptance decision with sufficient statistical confidence. Finally, SANE is shown to significantly outperform other state-of-the-art simulated annealing techniques on a stochastic travelling salesperson problem.     

Applying simulated annealing to location-planning models

Simulated annealing is a computational approach that simulates an annealing schedule used in producing glass and metals. Originally developed by Metropolis et al. in 1953, it has since been applied to a number of integer programming problems, including the p-median location-allocation problem. However, previously reported results by Golden and Skiscim in 1986 were less than encouraging. This article addresses the design of a simulated-annealing approach for the p-median and maximal covering location problems. This design has produced very good solutions in modest amounts of computer time. Comparisons with an interchange heuristic demonstrate that simulated annealing has potential as a solution technique for solving location-planning problems and further research should be encouraged.     

A Two Stage Search Heuristic for Scheduling Payments in Projects

When the Net Present Value (NPV) of a project is used as a measure of its financial performance, effective management of cash flows over the duration of the project is critical for improved profitability. Progress payments are a major component of project cash flows. In many project environments, the contractor can negotiate payment terms. Payments are typically tied to completion of project activities and therefore have significant impact on the schedule of activities and the timing of the payments. In this paper, we consider the problem of simultaneously determining the amount, timing and location of progress payments in projects to maximize NPV. Due to the combinatorial nature of the problem, heuristics are a practical approach to solving the problem. We propose a two-stage heuristic where simulated annealing is used in the first stage to determine a set of payments. In the second stage, activities are rescheduled to improve project NPV. We compare the performance of this general purpose heuristic with other problem-dependent heuristics from the literature. Our results indicate that the simulated annealing heuristic significantly outperforms the parameter-based heuristics. Although rescheduling in the second stage improves NPV, increases are relatively small in magnitude. While the specific parameters settings suggested by the simulated annealing heuristic in this study may have limited generalizability at this time due to the narrow range of problems tested, our analysis suggests that a pure simulated annealing approach is a very attractive alternative for obtaining good heuristic solutions to the complex problem of scheduling payments in projects.     

Integrating Interval Estimates of Global Optima and Local Search Methods for Combinatorial Optimization Problems

The problem of estimating the global optimal values of intractable combinatorial optimization problems is of interest to researchers developing and evaluating heuristics for these problems. In this paper we present a method for combining statistical optimum prediction techniques with local search methods such as simulated annealing and tabu search and illustrate the approach on a single machine scheduling problem. Computational experiments show that the approach yields useful estimates of optimal values with very reasonable computational effort.     

Search Heuristics for Resource Constrained Project Scheduling

We develop a search procedure for project scheduling problems with multiple resource constraints as well as precedence constraints. The procedure is applied to three popular search heuristics, simulated annealing, tabu search and genetic algorithms. In the heuristics, a solution is represented with a string of numbers each of which denotes priority of each activity. The priorities are used to select an activity for scheduling among competing ones. The search heuristics with this encoding method can always generate feasible neighbourhood solutions for a given solution. Moreover, this encoding method is very flexible in that problems with objective functions of a general functional form (such as a nonlinear function) and complex constraints can be considered without much difficulty. Results of computational tests on the performance of the search heuristics showed that the search heuristics, especially the simulated annealing and tabu search algorithms worked better than existing heuristics.     

New Reflection Generator for Simulated Annealing in Mixed-Integer/Continuous Global Optimization

To reduce the well-known jamming problem in global optimization algorithms, we propose a new generator for the simulated annealing algorithm based on the idea of reflection. Furthermore, we give conditions under which the sequence of points generated by this simulated annealing algorithm converges in probability to the global optimum for mixed-integer/continuous global optimization problems. Finally, we present numerical results on some artificial test problems as well as on a composite structural design problem.     

On the Hardness of the Quadratic Assignment Problem with Metaheuristics

Meta-heuristics are a powerful way to approximately solve hard combinatorial optimization problems. However, for a problem, the quality of results can vary considerably from one instance to another. Understanding such a behaviour is important from a theoretical point of view, but also has practical applications such as for the generation of instances during the evaluation stage of a heuristic.In this paper we propose a new complexity measure for the Quadratic Assignment Problem in the context of metaheuristics based on local search, e.g. simulated annealing. We show how the ruggedness coefficient previously introduced by the authors, in conjunction with the well known concept of dominance, provides important features of the search space explored during a local search algorithm, and gives a rather precise idea of the complexity of an instance for these heuristics. We comment previous experimental studies concerning tabu search methods and genetic algorithms with local search in the light of our complexity measure. New computational results with simulated annealing and taboo search are presented.     

Capacitated,balanced, sequential location-allocation problems on chains and trees

This paper considers finite horizon, multiperiod, sequential, minisum location-allocation problems on chain graphs and tree networks. The demand has both deterministic and probabilistic components, and increases dynamically from period to period. The problem is to locate one additionalcapacitated facility in each of thep specified periods, and to determine the service allocations of the facilities, in order to optimally satisfy the demand on the network. In this context, two types of objective criteria or location strategies are addressed. The first is a myopic strategy in which the present period cost is minimized sequentially for each period, and the second is a discounted present worth strategy. For the chain graph, we analyze ap-facility problem under both these criteria, while for the tree graph, we analyze a 3-facility myopic problem, and a 2-facility discounted present worth problem. All these problems are nonconvex, and we specify a finite set of candidate solutions which may be compared in order to determine a global optimal solution.     

An exact algorithm for the capacitated facility location problems with single sourcing

Facility location problems are often encountered in many areas such as distribution, transportation and telecommunication. We describe a new solution approach for the capacitated facility location problem in which each customer is served by a single facility. An important class of heuristic solution methods for these problems are Lagrangian heuristics which have been shown to produce high quality solutions and at the same time be quite robust. A primal heuristic, based on a repeated matching algorithm which essentially solves a series of matching problems until certain convergence criteria are satisfied, is incorporated into the Lagrangian heuristic. Finally, a branch-and-bound method, based on the Lagrangian heuristic is developed, and compared computationally to the commercial code CPLEX. The computational results indicate that the proposed method is very efficient.     

Optimizing simulated annealing schedules with genetic programming

Combinatorial optimization problems are encountered in many areas of science and engineering. Most of these problems are too difficult to be solved optimally, and hence heuristics are used to obtain “good” solutions in reasonable time. One heuristic that has been successfully applied to a variety of problems is Simulated Annealing. The performance of Simulated Annealing depends on the appropriate choice of a key parameter, the annealing schedule. Researchers usually experiment with some manually created annealing schedules and then use the one that performs best in their algorithms.This work replaces this manual search by Genetic Programming, a method based on natural evolution. We demonstrate the potential of this new approach by optimizing the annealing schedule for a well-known combinatorial optimization problem, the Quadratic Assignment Problem. We introduce two new algorithms for solving the Quadratic Assignment Problem that perform extremely well and outperform existing Simulated Annealing algorithms.     

Determination of lockbox collection points via simulated annealing

This research presents a heuristic to solve the lockbox location problem via a search-based technique known as simulated annealing. In the past, more traditional mathematical programming techniques have been used to address this problem, but with limited success due to its combinatorial nature. Because simulated annealing is a search-based technique, an optimal solution is not guaranteed, but past research has demonstrated that search-based heuristics can provide reasonable solutions without the difficulties associated with the more traditional formulations. In this paper, the simulated annealing methodology is used to solve a large lockbox location problem at several differing levels of cost. The results compare favourably to solutions obtained from a K-means clustering heuristic.     

Rounding algorithms for covering problems

In the last 25 years approximation algorithms for discrete optimization problems have been in the center of research in the fields of mathematical programming and computer science. Recent results from computer science have identified barriers to the degree of approximability of discrete optimization problems unless P = NP. As a result, as far as negative results are concerned a unifying picture is emerging. On the other hand, as far as particular approximation algorithms for different problems are concerned, the picture is not very clear. Different algorithms work for different problems and the insights gained from a successful analysis of a particular problem rarely transfer to another.Our goal in this paper is to present a framework for the approximation of a class of integer programming problems (covering problems) through generic heuristics all based on rounding (deterministic using primal and dual information or randomized but with nonlinear rounding functions) of the optimal solution of a linear programming (LP) relaxation. We apply these generic heuristics to obtain in a systematic way many known as well as new results for the set covering, facility location, general covering, network design and cut covering problems. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Research partially supported by a Presidential Young Investigator Award DDM-9158118 with matching funds from Draper Laboratory.Research partially supported by a Deans Summer Fellowship of the College of Business of the Ohio State University.