An algorithm for the capacitated, multi-commodity multi-period facility location problem

We consider the dynamic facility location problem with generalized modular capacities, a multiperiod facility location problem in which the costs for capacity changes may differ for all pairs of capacity levels. The problem embeds a complex cost structure and generalizes several existing facility location problems, such as those that allow temporary facility closing or capacity expansion and reduction. As the model may become very large, general-purpose mixed-integer programming (MIP) solvers are limited to solving instances of small to medium size. In this paper, we extend the generalized model to the case of multiple commodities. We propose Lagrangian heuristics, based on subgradient and bundle methods, to find good quality solutions for large-scale instances with up to 250 facility locations and 1,000 customers. To improve the final solution quality, a restricted MIP model is solved based on the information collected through the solution of the Lagrangian dual. Computational results show that the Lagrangian-based heuristics provide highly reliable results for all problem variants considered. They produce good quality solutions in short computing times even for instances where state-of-the-art MIP solvers do not find feasible solutions. The strength of the formulation also allows the method to provide tight bounds on the optimal value. Data and the online appendix are available at https://doi.org/10.1287/ijoc.2016.0738 .

The multiple facility location problem (MFLP) is to locate certain facilities so as to serve optimally a given set of customers, whose locations and requirements are known. When the facility locations have to be selected from a given set of candidate locations, the corresponding location problem becomes a discrete multiple facility location problem (DMFLP). In this study we consider a special case of DMFLP where multiple facilities that are of different type are to be placed (location decision) and assign customers to these facilities (allocation or assignment). The case will be discussed based on with and without interactions among new facilities. We then propose new heuristic solution methods and branch-and-bound algorithms. Computational results on randomly generated data in comparison with optimal solutions indicate that the new methods are both accurate and efficient.

Dynamic reserve site selection under contagion risk of deforestation

Examining the Performance of Heuristics for the Disassemble-to-Order Problem under Rolling Planning using Actual Product Structures

In this study, we use a systematic approach named the generalized unified method (GUM) to construct the general exact solutions of the derivative nonlinear Schrödinger (DNLS) family that also includes perturbed terms, which are the Kaup–Newell equation, the Chen–Lee–Liu equation, and the Gerdjikov–Ivanov equation. The GUM provides more general exact solutions with free parameters for nonlinear partial differential equations such that some solutions obtained by different exact solution methods, including the hyperbolic function solutions, the trigonometric function solutions, and the exponential solutions, are derived from these solutions by giving special values to these free parameters. Additionally, the used method reduces a large number of calculations compared to other exact solution methods, enabling computations to be made in a short, effortless, and elegant way. We investigate the DNLS family in this work because of its extensive applications in nonlinear optics. Particularly, the obtained optical soliton solutions of the DNLS family are useful for describing waves in optics and facilitating the interpretation of the propagation of solitons through optical fibers. Furthermore, this work not only contributes significantly to the advancement of soliton dynamics and their applications in photonic systems but also be productively used for more equations that occur in mathematical physics and engineering problems. Finally, 2D and 3D graphs of some derived solutions are plotted to illustrate behaviors of optical soliton.

Researchers have studied discrete location problems for a long time because of their importance in practice. The Discrete Ordered Median Problem (DOMP) generalizes discrete facility location problems. The DOMP generalizes the main facility location problems' objective functions such as the p-median, p-center and p-centdian location problems. As these problems, also known as the problems of location-allocation, have NP-hard structure, it is inevitable to use heuristic methods for solution. In this study, a metaheuristic algorithmic suggestion will be put forward by examining the DOMP to find optimal solutions. For that purpose, we proposed a Simulated Annealing (SA) metaheuristic with K-means Clustering Algorithm in initialization for the DOMP. Novel approaches for initial solution and K-exchange algorithm-based neighborhoods for local search were analysed. In addition, best level of selected parameters were determined by Taguchi method. Forty common p-median instances derived from OR-LIB were used to test the SA performance, and the results were compared with three state-of-art algorithms in the literature. According to the computational results, 21 best solutions were obtained on instances despite gap values and CPU times increasing proportionally to the scale of the instances. In a conclusion, the proposed clustering-based SA algorithm is competitive and can be a robust alternative for the DOMP.

Postboxes generally represent a cost for a postal service provider since they require to be maintained and daily visited by an operator to collect the letters. On the other hand, for the users, they are the main access points to the postal network and so the reduction of their number could result in a deterioration of the quality of the service for them. This work addresses a real problem involving the reduction of the number of postboxes located in rural areas to reorganize the collection operations of a postal service provider. We formulate this problem as a location-routing problem where the number of postmen employed is minimized guaranteeing the user accessibility. Exact and heuristic solution methods have been developed, tested and compared to existing solution methods on instances based on real data of rural instances showing the effectiveness of the proposed methodologies.

In the facility location problem usually reducing total transferring cost and time are common objectives. Designing of a network with hub facilities can improve network efficiency. In this study a new model is presented for P-hub covering location problem. In the p-hub covering problem it is attempted to locate hubs and allocate customers to established hubs while allocated nodes to hubs are inside of related hubs covering radius. In this paper we attempt to consider capability of established hubs according to their corresponding capabilities on sorting operation. Moreover the reliability of network is considered in the proposed model as an objective. The proposed multi objective model is solved by e-constraint algorithm for small size instances. For large scale instances a non-dominated sorting genetic algorithm (NSGA-II) is presented to obtain Pareto solutions and its performance is compared with results of e-constraint algorithm. The model and solution algorithm were analyzed by more numerical examples and some sensitivity analysis have been done. The analysis confirms that the network extracted by the proposed model is more efficient than classic networks.

Chapter 7 - Network Design

We propose a bi-level multi-objective model to solve the multi-facility location problem with traffic equilibrium constraints. The main facility location problem within our proposed model consists of locating a set of buildings with varying sensitivity thresholds due to the negative impacts propagating from an existing semi-obnoxious facility. The traffic routing problem is modelled as a user equilibrium which is embedded using its Karush-Kuhn-Tucker optimality conditions. We use the convex scalarisation approach to deal with multiple objectives. Two solution methods are then contrasted: in the first method we solve our linearised model using an off-the shelf Mixed Integer Programming solver. In the second solution approach we use Benders Decomposition algorithm to improve computational tractability. Numerical results highlight the superiority of the decomposition approach when solving a realistic-sized instance.

In this paper we present different exact and heuristic optimization methods for scheduling planes which want to land (and start) at an airport - the Aircraft Sequencing Problem (ASP). We compare two known integer programming formulations with four new exact and heuristic solution methods regarding quality, speed and flexibility.KeywordsSimulated Annealing AlgorithmConstraint ProgrammingValid SolutionMixed Integer Programming FormulationPlane LandThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper introduces a variant of the dynamic Facility Location Problem with modular capacities. Inspired by the challenges faced by cellular telecommunication networks, the problem seeks to optimize the placement and equipment of facilities to meet the fluctuating demand of clients while minimizing installation and operational costs. Selected facilities must provide coverage to the whole considered area. In addition, clients must be served by the active facility that provides the best quality of service. We propose two integer linear programming formulations and analyze their theoretical properties in terms of lower bound. Furthermore, we propose a reinforcement for one formulation and valid inequalities. To evaluate the performance of the proposed formulations, we performed computational experiments with a commercial solver on instances with up to 20 candidate sites and 60 clients. The formulations proved to be very sensitive to the size of the instances. The analysis, both theoretical and numerical, provides meaningful insights on the formulations performance, and it constitutes a key starting point for developing further approaches, either exact methods or heuristics.

Verification & Validation (V&V) attract more and more interests now and are extensively used in computational electromagnetics. In this paper, the method of exact solutions (MES) which is important in code verification is introduced by explaining the flow chartand the important glossaries and concepts of this method are explicated too. The process of global error evaluating by the approach of L 2 norm of the numerical simulation is commentated. Finite-difference time-domain (FDTD) arithmetic is chosen for the application of code verification by MES and the convergence rate and order-of-accuracy of FDTD are analyzed. The exact solution is selected and code verification using MES is given by comparing the observed order-of-accuracy of the numerical simulation result with the theoretical analysis result. This verification method would be useful in FDTD software development.

In the last two decades, water consumption in Germany has been decreasing which causes water tanks and pipes in water distribution systems to work inefficiently. This paper presents a mathematical optimization model to optimize water tanks in a water distribution system. Due to the hydraulic properties in water distribution systems the model is a non-convex Mixed Integer Quadratically Constrained Program (MIQCP). For problem instances of realistic size, the model cannot be solved within reasonable time with exact solution methods. We use different heuristic solution methods to solve the problem, such as a Simulated Annealing (SA) algorithm, a Shuffled Complex Evolution (SCE) algorithm as well as a Shuffled Frog-Leaping Algorithm (SFLA). These methods are combined with a hydraulic simulation to evaluate the solutions. The results of each method are compared to an exact solution method and discussed in this paper.

We consider a facility-location problem that abstracts settings where the cost of serving the clients assigned to a facility is incurred by the facility. Formally, we consider the minimum-load k-facility location (MLkFL) problem, which is defined as follows. We have a set F of facilities, a set C of clients, and an integer k > 0. Assigning client j to a facility f incurs a connection cost d(f, j). The goal is to open a set F' of k facilities, and assign each client j to a facility f(j) in F' so as to minimize maximum, over all facilities in F', of the sum of distances of clients j assigned to F' to F'. We call this sum the load of facility f. This problem was studied under the name of min-max star cover in [6, 2], who (among other results) gave bicriteria approximation algorithms for MLkFL for when F = C. MLkFL is rather poorly understood, and only an O(k)-approximation is currently known for MLkFL, even for line metrics. Our main result is the first polynomial time approximation scheme (PTAS) for MLkFL on line metrics (note that no non-trivial true approximation of any kind was known for this metric). Complementing this, we prove that MLkFL is strongly NP-hard on line metrics. We also devise a quasi-PTAS for MLkFL on tree metrics. MLkFL turns out to be surprisingly challenging even on line metrics, and resilient to attack by the variety of techniques that have been successfully applied to facility-location problems. For instance, we show that: (a) even a configuration-style LP-relaxation has a bad integrality gap; and (b) a multi-swap k-median style local-search heuristic has a bad locality gap. Thus, we need to devise various novel techniques to attack MLkFL. Our PTAS for line metrics consists of two main ingredients. First, we prove that there always exists a near-optimal solution possessing some nice structural properties. A novel aspect of this proof is that we first move to a mixed-integer LP (MILP) encoding the problem, and argue that a MILP-solution minimizing a certain potential function possesses the desired structure, and then use a rounding algorithm for the generalized-assignment problem to transfer this structure to the rounded integer solution. Complementing this, we show that these structural properties enable one to find such a structured solution via dynamic programming.