A Lagrangian Relaxation Heuristic for Capacitated Facility Location with Single-Source Constraints

Facility location models are applicable to problems in many diverse areas, such as distribution systems and communication networks. In capacitated facility location problems, a number of facilities with given capacities must be chosen from among a set of possible facility locations and then customers assigned to them. We describe a Lagrangian relaxation heuristic algorithm for capacitated problems in which each customer is served by a single facility. By relaxing the capacity constraints, the uncapacitated facility location problem is obtained as a subproblem and solved by the well-known dual ascent algorithm. The Lagrangian relaxations are complemented by an add heuristic, which is used to obtain an initial feasible solution. Further, a final adjustment heuristic is used to attempt to improve the best solution generated by the relaxations. Computational results are reported on examples generated from the Kuehn and Hamburger test problems.     

A Lagrangian heuristic for concave cost facility location problems: the plant location and technology acquisition problem

We propose a Lagrangian heuristic for facility location problems with concave cost functions and apply it to solve the plant location and technology acquisition problem. The problem is decomposed into a mixed integer subproblem and a set of trivial single-variable concave minimization subproblems. We are able to give a closed-form expression for the optimal Lagrangian multipliers such that the Lagrangian bound is obtained in a single iteration. Since the solution of the first subproblem is feasible to the original problem, a feasible solution and an upper bound are readily available. The Lagrangian heuristic can be embedded in a branch-and-bound scheme to close the optimality gap. Computational results show that the approach is capable of reaching high quality solutions efficiently. The proposed approach can be tailored to solve many concave-cost facility location problems.     

Lagrangian-relaxation-based solution procedures for a multiproduct capacitated facility location problem with choice of facility type

This paper presents exact and heuristic solution procedures for a multiproduct capacitated facility location (MPCFL) problem in which the demand for a number of different product families must be supplied from a set of facility sites, and each site offers a choice of facility types exhibiting different capacities. MPCFL generalizes both the uncapacitated (or simple) facility location (UFL) problem and the pure-integer capacitated facility location problem. We define a branch-and-bound algorithm for MPCFL that utilizes bounds formed by a Lagrangian relaxation of MPCFL which decomposes the problem into UFL subproblems and easily solvable 0-1 knapsack subproblems. The UFL subproblems are solved by the dual-based procedure of Erlenkotter. We also present a subgradient optimization-Lagrangian relaxation-based heuristic for MPCFL. Computational experience with the algorithm and heuristic are reported. The MPCFL heuristic is seen to be extremely effective, generating solutions to the test problems that are on average within 2% of optimality, and the branch-and-bound algorithm is successful in solving all of the test problems to optimality.     

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.     

A column generation heuristic for congested facility location problem with clearing functions

Nonlinear clearing functions, an idea initially suggested to reflect congestion effects in production planning, are used to express throughput of facilities prone to congestion in a facility location problem where each demand site is served by exactly one facility. The traditional constant capacity constraint for a facility is replaced with the nonlinear clearing function. The resulting nonlinear integer problem is solved by a column generation heuristic in which initial columns for the restricted master problem are generated by known existing algorithms and additional columns by a previously developed dynamic programming algorithm. Computational experimentation in terms of dual gap and CPU time based on both randomly generated and published data sets show not only clear dominance of the column generation over a Lagrangian heuristic previously developed, but also the high quality of results from the suggested heuristic for large problems.     

A genetic algorithm for a single product network design model with lead time and safety stock considerations

We consider a two-stage supply chain with a production facility that replenishes a single product at retailers. The objective is to locate distribution centers in the network such that the sum of facility location, pipeline inventory, and safety stock costs is minimized. We explicitly model the relationship between the flows in the network, lead times, and safety stock levels. We use genetic algorithms to solve the model and compare their performance to that of a Lagrangian heuristic developed in earlier work. A novel chromosome representation that combines binary vectors with random keys provides solutions of similar quality to those from the Lagrangian heuristic. The model is then extended to incorporate arbitrary demand variance at the retailers. This modification destroys the structure upon which the Lagrangian heuristic is based, but is easily incorporated into the genetic algorithm. The genetic algorithm yields significantly better solutions than a greedy heuristic for this modification and has reasonable computational requirements.     

An integrated approach to dynamic plant location and capacity planning

We consider a dynamic capacitated plant location problem in which capacities of opened plants are determined by acquisition and/or disposal of multiple types of facilities. We determine the opening schedule of plants, allocations of customers' demands and plans for acquisition and/or disposal of plant capacities that minimise the sum of discounted fixed costs for opening plants, delivery costs of products, and acquisition and operation costs of facilities. The dynamic capacitated plant location problem is formulated as a mixed integer linear program and solved by a heuristic algorithm based on Lagrangian relaxation and a cut and branch algorithm which uses Gomory cuts. Several solution properties of the relaxed problem are found and used to develop efficient solution procedures for the relaxed problem. A subgradient optimisation method is employed to obtain better lower bounds. The heuristic algorithm is tested on randomly generated test problems and results show that the algorithm finds good solutions in a reasonable amount of computation time.     

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.     

一类多商品设施选址问题的基于线性松弛解的启发式方法

多商品设施选址问题是众多设施选址问题中一类重要而困难的问题.在这一问题中,顾客的需求可能包含不止一种商品.对于大规模问题,成熟的商业求解器往往不能在满意的时间内找到高质量的可行解.研究了无容量限制的单货源多商品设施选址问题的一般形式,并给出了应用于此类问题的两个启发式方法.这两个方法基于原选址问题的线性规划松弛问题的最优解,分别通过求解紧问题和邻域搜索的方式给出了原问题的一个可行上界.理论分析指出所提方法可以实施于任意可行问题的实例.数值结果表明所提方法可以显著地提高求解器求解此类设施选址问题的求解效率.     

Optimal and heuristic algorithms for multiproduct uncapacitated facility location

We consider a generalization of the classic uncapacitated facility location problem (UFLP) in which customers require multiple products. We call this the multiproduct uncapacitated facility location problem (MUFLP). In MUFLP, in addition to a fixed cost for opening a facility, a fixed cost is incurred for each product that a facility is equipped to handle. Also, an assignment cost is incurred for satisfying a customer's requirement for a particular product at a chosen facility. We describe a branch-and-bound algorithm for MUFLP. Lower bounds are obtained by solving a UFLP subproblem for each product using a dual ascent routine. We also describe a heuristic branch-and-bound procedure in which the solutions to the subproblems at a given node might not generate a true lower bound. To generate a feasible solution, a ‘superposition’ heuristic based on solving UFLP subproblems for each product, as well as a ‘drop’ heuristic that eliminates facilities and equipment from the solution in a step-by-step manner, are given. Computational results are reported.     

A facility neighborhood search heuristic for capacitated facility location with single-source constraints and flexible demand

We consider a generalization of the well-known capacitated facility location problem with single source constraints in which customer demand contains a flexible dimension. This work focuses on providing fast and practically implementable optimization-based heuristic solution methods for very large scale problem instances. We offer a unique approach that utilizes a high-quality efficient heuristic within a neighborhood search to address the combined assignment and fixed-charge structure of the underlying optimization problem. We also study the potential benefits of combining our approach with a so-called very large-scale neighborhood search (VLSN) method. As our computational test results indicate, our work offers an attractive solution approach that can be tailored to successfully solve a broad class of problem instances for facility location and similar fixed-charge problems.     

A tabu search heuristic procedure for the capacitated facility location problem

A tabu search heuristic procedure is developed, implemented and computationally tested for the capacitated facility location problem. The procedure uses different memory structures. Visited solutions are stored in a primogenitary linked quad tree. For each facility, the recent move at which the facility changed its status and the frequency it has been open are also stored. These memory structures are used to guide the main search process as well as the diversification and intensification processes. Lower bounds on the decreases of total cost are used to measure the attractiveness of the moves and to select moves in the search process. A specialized network algorithm is developed to exploit the problem structure in solving transportation problems. Criterion altering, solution reconciling and path relinking are used to perform intensification functions. The performance of the procedure is tested through computational experiments using test problems from the literature and new test problems randomly generated. It found optimal solutions for almost all test problems from the literature. As compared to the heuristic method of Lagrangean relaxation with improved subgradient scheme, the tabu search heuristic procedure found much better solutions using much less CPU time.     

The return plant location problem: Modelling and resolution

The formulation and analysis of a new plant location problem is presented. The problem studied, herein referred to as the Return Plant Location Problem (RPLP), is that of cost minimization in a system of suppliers and customers in which there exists a return product from each customer. Lagrangian decomposition based heuristic and exact solution methods are given. The methods are applied to test problems with different structures and compared with the classical subgradient optimization approach.     

Improving solution of discrete competitive facility location problems

We consider discrete competitive facility location problems in this paper. Such problems could be viewed as a search of nodes in a network, composed of candidate and customer demand nodes, which connections correspond to attractiveness between customers and facilities located at the candidate nodes. The number of customers is usually very large. For some models of customer behavior exact solution approaches could be used. However, for other models and/or when the size of problem is too high to solve exactly, heuristic algorithms may be used. The solution of discrete competitive facility location problems using genetic algorithms is considered in this paper. The new strategies for dynamic adjustment of some parameters of genetic algorithm, such as probabilities for the crossover and mutation operations are proposed and applied to improve the canonical genetic algorithm. The algorithm is also specially adopted to solve discrete competitive facility location problems by proposing a strategy for selection of the most promising values of the variables in the mutation procedure. The developed genetic algorithm is demonstrated by solving instances of competitive facility location problems for an entering firm.     

A Lagrangian relaxation approach to large-scale flow interception problems

The paper presents a tight Lagrangian bound and an efficient dual heuristic for the flow interception problem. The proposed Lagrangian relaxation decomposes the problem into two subproblems that are easy to solve. Information from one of the subproblems is used within a dual heuristic to construct feasible solutions and is used to generate valid cuts that strengthen the relaxation. Both the heuristic and the relaxation are integrated into a cutting plane method where the Lagrangian bound is calculated using a subgradient algorithm. In the course of the algorithm, a valid cut is added and integrated efficiently in the second subproblem and is updated whenever the heuristic solution improves. The algorithm is tested on randomly generated test problems with up to 500 vertices, 12,483 paths, and 43 facilities. The algorithm finds a proven optimal solution in more than 75% of the cases, while the feasible solution is on average within 0.06% from the upper bound.     

Outbound supply chain network design with mode selection,lead times and capacitated vehicle distribution centers

Most distribution network design models considered to date have focused on minimizing fixed costs of facility location and transportation costs. Measures of customer satisfaction driven by the operational dynamics such as lead times have seldom been considered. We consider the design of an outbound supply chain network considering lead times, location of distribution facilities and choice of transportation mode. We present a Lagrangian heuristic that gives excellent solution quality in reasonable computational time. Scenario analyses are conducted on industrial data using this algorithm to observe how the supply chain behaves under different parameter values.     

范数下无容量限制设施选址逆问题的求解方法

一个优化问题的逆问题是这样一类问题,在给定该优化问题的一个可行解时,通过最小化目标函数中参数的改变量(在某个范数下)使得该可行解成为改变参数后的该优化问题的最优解。对于本是NP-难问题的无容量限制设施选址问题,证明了其逆问题仍是NP-难的。研究了使用经典的行生成算法对无容量限制设施选址的逆问题进行计算,并给出了求得逆问题上下界的启发式方法。两种方法分别基于对子问题的线性松弛求解给出上界和利用邻域搜索以及设置迭代循环次数的方式给出下界。数值结果表明线性松弛法得到的上界与最优值差距较小,但求解效率提升不大;而启发式方法得到的下界与最优值差距极小,极大地提高了求解该逆问题的效率。     

On solving the discrete location problems when the facilities are prone to failure

The classical discrete location problem is extended here, where the candidate facilities are subject to failure. The unreliable location problem is defined by introducing the probability that a facility may become inactive. The formulation and the solution procedure have been motivated by an application to model and solve a large size problem for locating base stations in a cellular communication network. We formulate the unreliable discrete location problems as 0–1 integer programming models, and implement an enhanced dual-based solution method to determine locations of these facilities to minimize the sum of fixed cost and expected operating (transportation) cost. Computational tests of some well-known problems have shown that the heuristic is efficient and effective for solving these unreliable location problems.     

Simulated N-Body: New Particle Physics-Based Heuristics for a Euclidean Location-Allocation Problem

The general facility location problem and its variants, including most location-allocation and P-median problems, are known to be NP-hard combinatorial optimization problems. Consequently, there is now a substantial body of literature on heuristic algorithms for a variety of location problems, among which can be found several versions of the well-known simulated annealing algorithm. This paper presents an optimization paradigm that, like simulated annealing, is based on a particle physics analogy but is markedly different from simulated annealing. Two heuristics based on this paradigm are presented and compared to simulated annealing for a capacitated facility location problem on Euclidean graphs. Experimental results based on randomly generated graphs suggest that one of the heuristics outperforms simulated annealing both in cost minimization as well as execution time. The particular version of location problem considered here, a location-allocation problem, involves determining locations and associated regions for a fixed number of facilities when the region sizes are given. Intended applications of this work include location problems with congestion costs as well as graph and network partitioning problems.     

The multi-source Weber problem with constant opening cost

A constant fixed cost of establishing a facility is introduced within the framework of minisum facility location in the continuous space. The solution method developed uses a multi-phase heuristic that first solves a discrete version of the problem by existing methods to obtain an estimate of the optimal number of facilities. Some results are presented for test problems taken from the literature and compared with best-known solutions of the multi-source Weber problem with the addition of the appropriate fixed costs.