Elsevier

Applied Soft Computing

Volume 34, September 2015, Pages 551-571
Applied Soft Computing

A capacitated location-allocation problem with stochastic demands using sub-sources: An empirical study

https://doi.org/10.1016/j.asoc.2015.05.020Get rights and content

Abstract

In a recent work, Alizadeh et al. (2013) studied a capacitated multi-facility location-allocation problem in which customers had stochastic demands based on the Bernoulli distribution function. Authors considered capacitated sub-sources of facilities to satisfy customer demands. In this discrete stochastic problem, the goal was to find optimal locations of facilities among candidate locations and optimal allocations of existing customers to operating facilities so that the total sum of fixed costs of operating facilities, allocation costs of customers and expected values of servicing and outsourcing costs was minimized. The model was formulated as a mixed-integer nonlinear programming problem. Since finding an optimal solution may require an excessive amount of time depending on the nonlinear constraints, here we transform the nonlinear constraints of the problem to linear ones to obtain a simple formulation of the model. An empirical study of an automobile manufacturer, namely Geelran Motor and three sets of test problems of small, medium and large sizes were considered to show the applicability of the presented model and efficiency of the proposed meta-heuristic algorithms. Numerical results show that the LINGO 9.0 software package is capable of solving the empirical study and small problems. For medium and large problems, we propose two meta-heuristic algorithms, a genetic algorithm (GA) and a discrete version of the colonial competitive algorithm (CCA). Computational investigations illustrate the efficiency of the proposed algorithms in obtaining effective solutions.

Introduction

In facility location problems, finding optimal facility locations of a number of candidate locations is considered so that the cost of satisfying customer demands is minimized. This problem was originally proposed by Weber [2] who presented location theory and solved a single warehouse location problem by minimization of total distances between the warehouse and customers. Francis et al. [3] proposed some general models, including single/multi facility location problems, quadratic assignment location problems and covering problems. Wagner et al. [4] proposed an uncapacitated facility location model in a stochastic environment with several risk factors. Using the mean-variance approach, authors optimized “Value-at-Risk” (VaR) measure in a location problem. Also, they presented a branch-and-bound algorithm for solving small and medium problems. Considering a new concept for the maximal covering model, Drezner et al. [5] solved a single facility location covering problem. For the more realistic case, they assumed that the short and long distances were random variables. Also, they proposed an algorithm to find optimal location of a facility in the plane that covers maximum demands. In order to satisfy customer demands, Ho [6] studied a single source facility location problem for locating capacitated facilities. To avoid getting stuck in local optima, authors proposed an integrated tabu search with perturbation operators. Beraldi and Bruni [7] contributed the model of Kouvelis and Yu [8]. They introduced a stochastic programming model for designing and planning of emergency medical services. To solve the problem, the authors presented an exact and three heuristics methods. Pilotta and Torres [9] generalized the Weber location problem by considering box constraints. The authors constructed a fixed-point iteration method with projections on the constraints. Also, they proposed a projected Weiszfeld algorithm to solve the box-constrainted Weber problem. Considering the variable neighborhood search (VNS) meta-heuristic, LP-rounding heuristic, k-opt neighborhood concept, and an exact method, Rahmaniani and Ghaderi [10] introduced a comprehensive framework for solving a facility location and the network design problem. To strengthen the linear relaxation of the problem, authors considered some valid inequalities and presented an efficient procedure to generate high quality initial solutions.

In a location-allocation (LA) problem, locating a number of facilities among candidate locations and allocating customers to facilities are considered such that the cost of satisfying customer demands is minimized. This problem was originally proposed by Cooper [11] in which a prevalent model of location-allocation with two new facilities and seven demand points was presented. A heuristic method was proposed by Cooper [12] for the classic LA problem to obtain good initial solutions for exact algorithms. In deterministic versions of multi-facility location problems, requests for demands are assumed to be well-known, which are not satisfied in most applications. When we are concerned with determination of customer demands, we can never be certain which customer has request for demand. Therefore, a reasonable assumption is that requests for demands occur randomly. This gives rise to probabilistic versions of multi-facility location problem. Considering a capacitated LA problem with stochastic demands, Zhou and Liu [13] formulated three stochastic models for the problem. Authors proposed a hybrid intelligent algorithm consisting of a network simplex algorithm, stochastic simulation and genetic algorithm for solving the problem. Considering a capacitated LA problem with fuzzy demands, Zhou and Liu [14] proposed three types of fuzzy programming models including fuzzy expected cost minimization, fuzzy α-cost minimization and credibility maximization according to different decision criteria. For solving these models, some hybrid intelligent algorithms were considered. A stochastic uncapacitated transportation plant LA model with the goal of maximizing the expected profit was considered by Logendran and Terrell [15]. Authors proposed some heuristics to solve the problem. Vidyarthi and Jayaswal [16] proposed a non-linear integer programming model to solve an LA problem with immobile servers, stochastic demands and congestions. Authors considered that customer demands occur continuously over time according to an independent Poisson process and service times at facilities follow an exponential distribution. They set up the problem as a network of independent M/G/1 queues in which locations, capacities and service zones needed to be determined. Using simple transformations and piecewise linear approximations, they linearized the model and solved the problem with an ϵ-constraint generation method. A robust optimization model for the logistics center LA problem was proposed by Wang et al. [17] in an uncertain environment. Then, relations among the robust optimization model, stochastic optimization model and deterministic optimization model were exposed. Also, they considered two algorithms, namely an enumeration method and a genetic algorithm for solving the problem. Bojić et al. [18] proposed an LA problem to find the capacity, type and location of solid biomass power plants for the Vojvodina region with defined biomass potentials. Authors considered the objective function based on minimal electricity generation costs and investigated financial effects of increasing upper limits of the capacities of the power plants. A joint facility LA and inventory problem with stochastic demands was proposed by Yao et al. [19]. The problem was to find locations of warehouses, allocation of customers and inventory levels of warehouses. To develop a heuristic method, the authors integrated approximation and transformation techniques. A capacitated facility location problem with constrained backlogging probabilities was presented by Silva and De la Figuera [20]. Considering the stochastic demands and formulating each facility as an independent queue, the authors proposed a heuristic method to solve the problem. This heuristic approach was based on a reactive greedy adaptive search method. Using a quad tree-based method, Salhi and Irawan [21] introduced a data compression approach for allocating very large demand points to nearest facilities while eliminating the aggregated error. Authors first used the proposed method to solve the allocation problems and then extended it to solve a class of discrete facility location problems, namely the p-median and the vertex p-center problems. A facility LA problem with random fuzzy demands was proposed by Wen and Kang [22]. To solve the random fuzzy models, authors integrated the simplex algorithm, a random fuzzy simulation and a genetic algorithm. Zahiri et al. [23] proposed a mixed-integer linear programming model for a multi-period LA problem in an organ transplant supply chain under inherent uncertain inputs. Because of the imprecise nature of the studied network, they considered the model under uncertainty and elevated efficiency of the studied supply chain network by considering an objective function minimization. Considering the sufficient and safe blood supply as a challenge, Sha and Huang [24] proposed a multi-period LA model for an emergency blood supply scheduling problem and a heuristic algorithm based on Lagrangian relaxation. Mousavi and Niaki [25] presented an LA problem with fuzzy demands in which locations of customers were uncertain and followed a normal probability distribution. Authors used two closed-form Euclidean and squared Euclidean expressions to estimate the expected distance among customers and facilities. Also, they applied a hybrid intelligent algorithm based on the simplex algorithm, fuzzy simulation and a modified genetic algorithm.

Several stochastic optimization models have considered Bernoulli demands in LA problems. In many logistic applications such as delivery services, door-to-door mail services and home assistance services, service request of a customer is stated in terms of a ‘quantity’ that shows the amount of resources of a service center that is consumed if the customer is served. In this situation, request for service of a customer is unitary in which each service request utilizes one resource unit from the service center. Customer demand requests in possible scenarios can be defined as binary vectors. In our work, demands are considered to be stochastic and the request vectors to be Bernoulli random variables. Considering the stochastic generalized assignment problem, Albareda-Sambola et al. [26] proposed an exact algorithm to optimize the expected cost of a recourse function. Later, Albareda-Sambola et al. [27] presented a location-routing problem with Bernoulli demands where several heuristics and lower bounds for minimizing the expected value of the recourse function were introduced. Considering a discrete capacitated facility location problem with Bernoulli demands, Albareda-Sambola et al. [28] proposed two different recourse actions joined with a two-stage stochastic programming model. For each recourse action, authors provided an approximation method to compute the expected value estimating the impact of randomness on the problem. Berman and Simichi-Levi [41] studied a single-vehicle location-routing problem with Bernoulli demand and proposed a lower bound on the value of the optimal a priori tour. Considering an integrated machine allocation and layout problem with fuzzy demands, Hosseini Nasab [29] proposed a hybrid intelligent algorithm, including a genetic algorithm and a fuzzy simulation approach. Authors solved a set of numerical examples to compare efficiencies of the hybrid genetic algorithm, Lingo 8.0, SA, PSO and ACO. Imperialist competitive algorithm (ICA) was first proposed by Atashpaz-Gargari and Lucas [30] for solving engineering and optimization problems. ICA is a new meta-heuristic optimization algorithm based on human's social evolution while genetic algorithm (GA) was developed based on biological evolution. Colonial competitive algorithm (CCA) was later used by Atashpaz-Gargari et al. [31] to design a proportional integral derivative (PID) controller. Several researchers extensively used CCA to solve different kinds of optimization problems: design of optimal antenna array [32], design of decentralized PID controller [33], characterization of material property from sharp indentation test [42], design of a fuzzy controller for the vehicle dynamics [34] and game theory and multi-objective optimization problems [45]. Solving an integrated product mix-outsourcing optimization problem, Nazari-Shirkouhi et al. [44] compared the obtained results by ICA with the results of TOC and standard accounting approaches. Pereira et al. [35] presented a probabilistic maximal covering LA problem in which a hybrid algorithm combining a meta-heuristic and an integer exact method was proposed to solve the problem. To efficiently solve small and medium problems, authors formulated a linear programming model. Also, they proposed a flexible adaptive large neighborhood search heuristic to solve large problems. Shiripour et al. [36] proposed a multi-facility location problem with uniformly distributed starting point of the line barriers. They applied two meta-heuristics, namely GA and ICA for solving the problem. Comparison of the results signified preference of ICA over GA in terms of computing time and solution accuracy. Mozafari et al. [37] considered ICA to optimize the intermediate epoxy adhesive layer bonded between two dissimilar strips of material. Also, authors compared the results of ICA with a finite element method (FEM) and GA. Comparisons showed ICA to be better than others in designing adhesive joints in composite materials. Amiri-Aref et al. [38] presented a nonlinear programming model for a multi-period rectilinear distance center location-dependent relocation problem in the presence of a probabilistic barrier. Using an optimization software, they obtained optimal solutions of small problems. Also, two meta-heuristic algorithms, namely genetic algorithm (GA) and imperialist competitive algorithm (ICA) were applied to solve large problems. Alizadeh et al. [1] proposed a capacitated LA problem with stochastic demands. Capacitated sub-sources of facilities were considered to satisfy the Bernoulli customer demands. In order to simplify the complexity, authors transformed the non-homogeneous case having the customer request services with the probabilities pj to a homogenous case with all customers having the same probability of request for demand.

Because of the nonlinear constraints finding a solution of the model in Alizadeh et al. [1] was quite time-consuming. In order to reduce the computing time, here we simplify the proposed model by transforming the nonlinear constraints to linear ones. Also, two meta-heuristic algorithms, namely GA and CCA are used to find optimal locations of facilities among a set of candidate locations and optimal allocations of customers in a simplified capacitated LA problem with Bernoulli demands using sub-sources (SCLAPBDS). Our goal in the proposed simplified model is to minimize the total sum of fixed costs of operating facilities, allocation costs of customers and expected values of servicing and outsourcing costs. The rest of our work is structured as follows. The problem definitions are given in Section 2. In Section 3, the mathematical programming model for SCLAPBDS is considered. Then, the model is transformed into a simplified version. Section 4 introduces two meta-heuristic algorithms for solving large problems. In Section 5, we work through an empirical study of a vehicles manufacturer, namely Geelran Motor, in the city of Amol in Iran. Also, various sets of problems of small, medium and large sizes are tested to show the efficiency and accuracy of the meta-heuristic algorithms. Section 6 concludes with some directions for future research. Finally, the computed values for some parameters are given in the Appendix.

Section snippets

Notations

The following notations are defined to formulate the proposed model:

iindex of candidate location for establishment of facilities, i1,2,,I,
jindex of existing customer, j1,2,,J,
θj1 if customer j has request for demand, and 0 otherwise,
pjprobability of request for demand of jth customer,
fifixed cost for operating facility i,
liminimum number of customers to be allocated to facility i if it is to be operating,
kimaximum number of customers to be served from facility i if it is to be operating,
ui

The mathematical model

The purpose of the nonlinear model in Alizadeh et al. [1] was to locate a set of facilities among candidate locations and to allocate existing customers with probabilistic demands to capacitated facilities and sub-sources. The objective of the model was to minimize the total sum of costs of operating facilities, costs of allocation of customers and costs of service and the outsourcing process. Some necessary binary variables are defined as followings:τi=1,kisi0,0,oherwise,iI\i,λi=1,si

A genetic algorithm

A genetic algorithm (GA) is an evolutionary search for finding an approximate optimizer starting with an initial set of solutions for the considered problem. The initial set of solutions is called initial population. Each individual in a population is known as a chromosome, for which an array of problem variables is constructed. The value for each chromosome is computed by a pre-specified fitness function. The chromosomes are developed in successive iterations, named as generations. A new

Empirical study

For a better illustration of SCLAPBDS and the proposed GA and CCA, here an empirical study of a vehicle distributor, namely Geelran Motor (GM) in city of Amol in Iran is considered. GM is the exclusive representative of Geely company in Iran. Geely is a Chinese auto manufacturer company with products: Emergrand 7, Emergrand X7 and Emergrand 7RV. GM aims to distribute the bestselling car of Geely, namely Emergrand 7 in Amol. To satisfy the customer requests, Geelran Motor wants to find optimal

Conclusion

Here, by linearizing the nonlinear constraints, we simplified and solved a capacitated multi-facility location-allocation problem, proposed by Alizadeh et al., where the customer demand requests were considered to be stochastic. The goal was to minimize the total sum of fixed costs of locating facilities plus costs of allocating customers plus the expected values of servicing and outsourcing costs. Solving medium and large problems by the LINGO software package being too time consuming, two

Acknowledgements

The first, second and fourth authors acknowledge Mazandaran University of Science and Technology and the third author thanks Sharif University of Technology for supporting this work.

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