The capacitated multi-facility location–allocation problem with probabilistic customer location and demand: two hybrid meta-heuristic algorithms

A new mathematical model for the capacitated multi-facility location–allocation problem with probabilistic customers' locations and demands is developed in this article. The model is formulated into the frameworks of the expected value model (EVM) and the chance-constrained programming (CCP) based on two different distance measures. In order to solve the model, two hybrid intelligent algorithms are proposed, where the simplex algorithm and stochastic simulation are the bases for both algorithms. However, in the first algorithm, named SSGA, a special type of genetic algorithm is combined and in the second, SSVDO, a vibration-damping optimisation (VDO) algorithm is united. The Taguchi method is employed to tune the parameters of the two proposed algorithms. Finally, some numerical examples are given to illustrate the applications of the proposed methodologies and to compare their performances.     

Fix-and-Relax-Coordination for a multi-period location–allocation problem under uncertainty

A multi-period discrete facility location problem is introduced for a risk neutral strategy with uncertainty in the costs and some of the requirements along the planning horizon. A compact 0–1 formulation for the Deterministic Equivalent Model of the problem under two alternative strategies for the location decisions is presented. Furthermore, a new algorithmic matheuristic, Fix-and-Relax-Coordination, is introduced. This solution scheme is based on a specialization of the Branch-and-Fix Coordination methodology, which exploits the Nonanticipativity Constraints and uses the Twin Node Family concept. The results of an extensive computational experience allow to compare the alternative modeling strategies and assess the effectiveness of the proposed approach versus the plain use of a state-of-the-art MIP solver.     

A multi-dimensional shooting algorithm for the two-facility location–allocation problem with dense demand

We develop an efficient allocation-based solution framework for a class of two-facility location–allocation problems with dense demand data. By formulating the problem as a multi-dimensional boundary value problem, we show that previous results for the discrete demand case can be extended to problems with highly dense demand data. Further, this approach can be generalized to non-convex allocation decisions. This formulation is illustrated for the Euclidean metric case by representing the affine bisector with two points. A specialized multi-dimensional shooting algorithm is presented and illustrated on an example. Comparisons with two alternative methods through a computational study confirm the efficiency of the proposed methodology.     

Discrete approximation heuristics for the capacitated continuous location–allocation problem with probabilistic customer locations

The capacitated continuous location–allocation problem, also called capacitated multisource Weber problem (CMWP), is concerned with locating m facilities in the Euclidean plane, and allocating their capacity to n customers at minimum total cost. The deterministic version of the problem, which assumes that customer locations and demands are known with certainty, is a nonconvex optimization problem. In this work, we focus on a probabilistic extension referred to as the probabilistic CMWP (PCMWP), and consider the situation in which customer locations are randomly distributed according to a bivariate probability distribution. We first formulate the discrete approximation of the problem as a mixed-integer linear programming model in which facilities can be located on a set of candidate points. Then we present three heuristics to solve the problem. Since optimal solutions cannot be found, we assess the performance of the heuristics using the results obtained by an alternate location–allocation heuristic that is originally developed for the deterministic version of the problem and tailored by us for the PCMWP. The new heuristics depend on the evaluation of the expected distances between facilities and customers, which is possible only for a few number of distance function and probability distribution combinations. We therefore propose approximation methods which make the heuristics applicable for any distance function and probability distribution of customer coordinates.     

A multi-objective model for facility location–allocation problem with immobile servers within queuing framework

This research investigates a practical bi-objective model for the facility location–allocation (BOFLA) problem with immobile servers and stochastic demand within the M/M/1/K queue system. The first goal of the research is to develop a mathematical model in which customers and service providers are considered as perspectives. The objectives of the developed model are minimization of the total cost of server provider and minimization of the total time of customers. This model has different real world applications, including locating bank automated teller machines (ATMs), different types of vendor machines, etc. For solving the model, two popular multi-objective evolutionary algorithms (MOEA) of the literature are implemented. The first algorithm is non-dominated sorted genetic algorithm (NSGA-II) and the second one is non-dominated ranked genetic algorithm (NRGA). Moreover, to illustrate the effectiveness of the proposed algorithms, some numerical examples are presented and analyzed statistically. The results indicate that the proposed algorithms provide an effective means to solve the problems.     

Uncertain location–allocation decisions for a bi-objective two-stage supply chain network design problem with environmental impacts

In the cases that the historical data of an uncertain event is not available, belief degree-based uncertainty theory is a useful tool to reflect such uncertainty. This study focuses on uncertain bi-objective supply chain network design problem with cost and environmental impacts under uncertainty. As such network may be designed for the first time in a geographical region, this problem is modelled by the concepts of belief degree-based uncertainty theory. This article is almost the first study on belief degree-based uncertain supply chain network design problem with environmental impacts. Two approaches such as expected value model and chance-constrained model are applied to convert the proposed uncertain problem to its crisp form. The obtained crisp forms are solved by some multi-objective optimization approaches of the literature such as TH, Niroomand, MMNV. A deep computational study with several test problems are performed to study the performance of the crisp models and the solution approaches. According to the results, the obtained crisp formulations are highly sensitive to the changes in the value of the cost parameters. On the other hand, Niroomand and MMNV solution approaches perform better than other solution approaches from the solution quality point of view.     

A two-level location–allocation problem in designing local access fiber optic networks

This paper deals with a two-level facility location–allocation problem on tree topology arising from the design of access networks. The access network design problem seeks to find an optimal location of switch and allocation of demands such that the total cost of switch and fiber cable is minimized, while satisfying switch port constraint, switch capacity constraint, and no-split routing constraint. The problem is formulated as a mixed-integer programming model and alternative formulations are developed by the reformulation-linearization technique (RLT) for improving computational effectiveness. By exploiting the tree structure of the problem, we develop some valid inequalities that have complementary strength and devise separation problems for finding effective valid inequalities that cut off fractional LP solutions. Also, we develop an effective MIP-based tree partitioning heuristic for finding good quality solutions for large size problems. Computational results demonstrate the efficacy of the valid inequalities and the proposed heuristic.     

A revised ant algorithm for solving location–allocation problem with risky demand in a multi-echelon supply chain network

This study addresses a capacitated facility location and task allocation problem of a multi-echelon supply chain against risky demands. Two and three-echelon networks are considered to maximize profit. The study represents the problem by a bi-level stochastic programming model. The revised ant algorithm proposed in the study improves the existing ant algorithm by using new design of heuristic desirability and efficient greedy heuristics to solve the problem. A set of computational experiments is reported to not only allow to fine-tune the parameters of the algorithm but also to evaluate its performance for solving the problem proposed. Experiments reveal that the proposed solution algorithm can reach 95–99% of the optimal solution against risky demands while consuming only 1000th of the computational time for large-sized problems as compared to an optimization-based tool.     

A study on facility location–allocation problem in mixed environment of randomness and fuzziness

In logistics system, facility location–allocation problem, which can be used to determine the mode, the structure and the form of the whole logistics system, is a very important decision problem in the logistics network. It involves locating plants and distribution centers, and determining the best strategy for allocation the product from the plants to the distribution centers and from the distribution centers to the customers. Often uncertainty may be associated with demand, supply or various relevant costs. In many cases, randomness and fuzziness simultaneously appear in a system, in order to describe this phenomenon; we introduce the concept of hybrid variable and propose a mixed-integer programming model for random fuzzy facility location–allocation problem. By expected value and chance constraint programming technique, this model is reduced to a deterministic model. Furthermore, a priority-based genetic algorithm is designed for solving the proposed programming model and the efficacy and the efficiency of this method and algorithm are demonstrated by a numerical example. Till now, few has formulated or attacked the FLA problems in the above manner. Furthermore, the techniques illustrated in this paper can easily be applied to other SCN problems. Therefore, these techniques are the appropriate tools to tackle other supply chain network problems in realistic environments.     

Optimal allocation–consumption problem for a portfolio with an illiquid asset

During financial crises investors manage portfolios with low liquidity, where the paper-value of an asset differs from the price proposed by the buyer. We consider an optimization problem for a portfolio with an illiquid, a risky and a risk-free asset. We work in the Merton's optimal consumption framework with continuous time. The liquid part of the investment is described by a standard Black–Scholes market. The illiquid asset is sold at a random moment with prescribed distribution and generates additional liquid wealth dependent on its paper-value. The investor has a hyperbolic absolute risk aversion also denoted as HARA-type utility function, in particular, the logarithmic utility function as a limit case. We study two different distributions of the liquidation time of the illiquid asset – a classical exponential distribution and a more practically relevant Weibull distribution. Under certain conditions we show the smoothness of the viscosity solution and obtain closed formulae relevant for numerics.     

Hierarchical maximal-coverage location–allocation: Case of generalized search-and-rescue

We offer a variant of the maximal covering location problem to locate up to p$p$ signal-receiving stations. The “demands,” called geolocations, to be covered by these stations are distress signals and/or transmissions from any targets. The problem is complicated by several factors. First, to find a signal location, the signal must be received by at least three stations—two lines of bearing for triangulation and a third for accuracy. Second, signal frequencies vary by source and the included stations do not necessarily receive all frequencies. One must decide which listening frequencies are allocated to which stations. Finally, the range or coverage area of a station varies stochastically because of meteorological conditions. This problem is modeled using a multiobjective (or multicriteria) linear integer program (MOLIP), which is an approximation of a highly nonlinear integer program. As a solution algorithm, the MOLIP is converted to a two-stage network-flow formulation that reduces the number of explicitly enumerated integer variables. Non-inferior solutions of the MOLIP are evaluated by a value function, which identifies solutions that are similar to the more accurate nonlinear model. In all case studies, the “best” non-inferior solutions were about one to four standard deviations better than the sample mean of thousands of randomly located receivers with heuristic frequency assignments. We also show that a two-stage network-flow algorithm is a practical solution to an intractable nonlinear integer model. Most importantly, the procedure has been implemented in the field.     

Efficient solution of a class of location–allocation problems with stochastic demand and congestion

We consider a class of location–allocation problems with immobile servers, stochastic demand and congestion that arises in several planning contexts: location of emergency medical clinics; preventive healthcare centers; refuse collection and disposal centers; stores and service centers; bank branches and automated banking machines; internet mirror sites; web service providers (servers); and distribution centers in supply chains. The problem seeks to simultaneously locate service facilities, equip them with appropriate capacities, and allocate user demand to these facilities such that the total cost, which consists of the fixed cost of opening facilities with sufficient capacities, the access cost of users׳ travel to facilities, and the queuing delay cost, is minimized. Under Poisson user demand arrivals and general service time distributions, the problem is set up as a network of independent M/G/1 queues, whose locations, capacities and service zones need to be determined. The resulting mathematical model is a non-linear integer program. Using simple transformation and piecewise linear approximation, the model is linearized and solved to ϵ-optimality using a constraint generation method. Computational results are presented for instances up to 400 users, 25 potential service facilities, and 5 capacity levels with different coefficients of variation of service times and average queueing delay costs per customer. The results indicate that the proposed solution method is efficient in solving a wide range of problem instances.     

The file allocation problem—A queueing network optimization approach

A set of customers use a connected network of computer installations, each accessing the network from a particular node. These customers share information contained in a set of data files. A typical customer's need is characterized by a request requiring a subset of these files being accessed in a Markovian sequence. The cycle time for this customer is the total time taken on the average to complete his request sequence. The objective is to locate a single copy of each of the files in such a way that a weighted sum of these response times is minimized. The problem is modelled as a Closed Queueing Network optimization problem. Models are developed for both single and multiple chain cases. An incremental analysis approach is used to solve the single chain case. For the multiple chain case, it is shown how this model approximates to a set partitioning problem under certain conditions. Efficient heuristics are developed to solve this partitioning problem. With certain simplifying assumptions, the associated communication problem is then included in the model.     

Solving the multi-compartment capacitated location routing problem with pickup–delivery routes and stochastic demands

This paper considers an advanced capacitated location routing problem in a distribution network with multiple pickup and delivery routes, and each customer placing random multi-item demands on it. The pickup and delivery services need two fleets of vehicles and will form two different sets of routes. However, the unpredictability of variation in the multi-item demands makes the routing of multi-compartment vehicles to accommodate such demands complex. To solve this multifaceted problem, a new process employing the TABU search is proposed in this research study. This proposed approach includes three stages: location selection, customer assignment, and vehicle routing. The innovative concept is to divide all customers into assignment-determined and assignment-undetermined groups in order to narrow down the search area of a solution domain so the TABU search can be more efficient and effective. Two sets of benchmarks are then generated to verify the quality of the proposed method. According to the experiment results, the proposed solution process can both resolve the problems and yield good results in a reasonable amount of computing time. The analysis of the solution process parameters is also provided. In addition, the comparisons between stochastic demand and deterministic demand cases are calculated and discussed as well.     

Bi-objective vibration damping optimization for congested location–pricing problem

This paper presents a bi-objective mathematical programming model for the restricted facility location problem, under a congestion and pricing policy. Motivated by various applications such as locating server on internet mirror sites and communication networks, this research investigates congested systems with immobile servers and stochastic demand as M/M/m/k queues. For this problem, we consider two simultaneous perspectives; (1) customers who desire to limit waiting time for service and (2) service providers who intend to increase profits. We formulate a bi-objective facility location problem with two objective functions: (i) maximizing total profit of the whole system and (ii) minimizing the sum of waiting time in queues; the model type is mixed-integer nonlinear. Then, a multi-objective optimization algorithm based on vibration theory (so-called multi-objective vibration damping optimization (MOVDO)), is developed to solve the model. Moreover, the Taguchi method is also implemented, using a response metric to tune the parameters. The results are analyzed and compared with a non-dominated sorting genetic algorithm (NSGA-II) as a well-developed multi-objective evolutionary optimization algorithm. Computational results demonstrate the efficiency of the proposed MOVDO to solve large-scale problems.     

Formulations and relaxations for a multi-echelon capacitated location–distribution problem

We consider a multi-echelon location–distribution problem arising from an actual application in fast delivery service. We present and compare two formulations for this problem: an arc-based model and a path-based model. We show that the linear programming (LP) relaxation of the path-based model provides a better bound than the LP relaxation of the arc-based model. We also compare the so-called binary relaxations of the models, which are obtained by relaxing the integrality constraints for the general integer variables, but not for the 0–1 variables. We show that the binary relaxations of the two models always provide the same bound, but that the path-based binary relaxation appears preferable from a computational point of view, since it can be reformulated as an equivalent simple plant location problem (SPLP), for which several efficient algorithms exist. We also show that the LP relaxation of this SPLP reformulation provides a better bound than the LP relaxation of the path-based model.     

The integrated production–inventory–distribution–routing problem

The integration of production and distribution decisions presents a challenging problem for manufacturers trying to optimize their supply chain. At the planning level, the immediate goal is to coordinate production, inventory, and delivery to meet customer demand so that the corresponding costs are minimized. Achieving this goal provides the foundations for streamlining the logistics network and for integrating other operational and financial components of the system. In this paper, a model is presented that includes a single production facility, a set of customers with time varying demand, a finite planning horizon, and a fleet of vehicles for making the deliveries. Demand can be satisfied from either inventory held at the customer sites or from daily product distribution. In the most restrictive case, a vehicle routing problem must be solved for each time period. The decision to visit a customer on a particular day could be to restock inventory, meet that day’s demand or both. In a less restrictive case, the routing component of the model is replaced with an allocation component only. A procedure centering on reactive tabu search is developed for solving the full problem. After a solution is found, path relinking is applied to improve the results. A novel feature of the methodology is the use of an allocation model in the form of a mixed integer program to find good feasible solutions that serve as starting points for the tabu search. Lower bounds on the optimum are obtained by solving a modified version of the allocation model. Computational testing on a set of 90 benchmark instances with up to 200 customers and 20 time periods demonstrates the effectiveness of the approach. In all cases, improvements ranging from 10–20% were realized when compared to those obtained from an existing greedy randomized adaptive search procedure (GRASP). This often came at a three- to five-fold increase in runtime, however.     

Burglars blocked by barriers? The impact of physical and social barriers on residential burglars' target location choices in China

Based on an offender spatial decision-making perspective, this burglary target location choice study aims to understand how physical and social barriers affect why residential burglars commit their crimes at particular locations in a major Chinese city. Using data on 3860 residential burglaries committed by 3772 burglars between January 2012 and June 2016 in ZG city, China, conditional logit (discrete choice) models were estimated to assess residential burglars' target location choice preferences. Three types of physical barriers were distinguished: major roads with access control, major roads without access control, and major rivers. Social barriers were constructed based on the Hukou system to reflect how local and nonlocal residents live segregated lives. Results show that residential burglars are less likely to target areas for which they have to cross a physical barrier and even less likely to do so if they have to cross multiple rivers. Local burglars are more likely to target communities with a majority of local residents than communities with a majority nonlocal population or a mixed community. Such a social barrier was less pronounced for nonlocal burglars. These findings add new insight that physical and social barriers affect, to various degrees, where residential burglars in China commit their crimes.     

FA–QABC–MRTA: a solution for solving the multi-robot task allocation problem

Intelligent Service Robotics - The problem of task allocation in a multi-robot system is the situation where we have a set of tasks and a number of robots; then each task is assigned to the...     

A location–inventory supply chain problem: Reformulation and piecewise linearization

In this paper, we study a two-echelon inventory management problem with multiple warehouses and retailers. The problem is a natural extension to the well-known one-warehouse multi-retailer inventory problem. The problem is formulated as a mixed integer non-linear program such that its continuous relaxation is non-convex. We propose an equivalent formulation with fewer non-linear terms in the objective function so that the continuous relaxation of the new model is a convex optimization problem. We use piecewise linearization to transform the resulting MINLP to a mixed integer program and we solve it using CPLEX. Through numerical experiments, we compare the solutions obtained by solving the new formulation using CPLEX with two previously published Lagrangian relaxation based heuristics to solve the original mixed integer non-linear program. We demonstrate that the new approach is capable of providing almost the same solutions without the need of using specialized algorithms. This important contribution further implies that additional variants of the problem, such as multiple products, capacitated warehouses and routing, can be added to result in a problem that will again be solvable by commercial optimization software, while the respective Lagrangian heuristics will fail to solve such variants or extended problems.