Facility Location Costs Research Articles

Joint location–inventory problem with economies and diseconomies of scale: A general model and an efficient column generation algorithm

This paper studies the location–inventory problem, which jointly optimizes the location, inventory, and allocation of distribution centers (DCs) to serve uncertain demand of multiple retailers. Problems of this kind have been extensively studied to carefully tradeoff various costs, including the fixed facility location cost and the variable facility operating cost Most existing works assumed the facility operating cost to be either a linear or a concave function of the volume of demand allocated. However, practical evidence suggests that when the volume of allocated demand exceeds a certain level, the marginal cost may increase due to overtime charge or facility congestion, leading to a complicated cost function. In this paper, we capture the facility operating cost as a general inverse S-shaped function of the volume of allocated demand, which is first concave due to economies of scale and then convex due to diseconomies of scale. We formulate this problem as a set-covering model and propose a column generation (CG) algorithm to solve its linear relaxation. The corresponding pricing problem has nice structural properties and can be solved efficiently by a branch-and-bound (B&B) method. The efficiency of the proposed algorithm is validated through extensive numerical studies, where two types of inverse S-shaped functions are considered to capture facility operating cost under various situations. Sensitivity analysis is also carried out to reveal managerial insights and analyze the delicate tradeoffs between several cost components.

A General Model and Efficient Algorithms for Reliable Facility Location Problem Under Uncertain Disruptions

This paper studies the reliable uncapacitated facility location problem in which facilities are subject to uncertain disruptions. A two-stage distributionally robust model is formulated, which optimizes the facility location decisions so as to minimize the fixed facility location cost and the expected transportation cost of serving customers under the worst-case disruption distribution. The model is formulated in a general form, where the uncertain joint distribution of disruptions is partially characterized and is allowed to have any prespecified dependency structure. This model extends several related models in the literature, including the stochastic one with explicitly given disruption distribution and the robust one with moment information on disruptions. An efficient cutting plane algorithm is proposed to solve this model, where the separation problem is solved respectively by a polynomial-time algorithm in the stochastic case and by a column generation approach in the robust case. Extensive numerical study shows that the proposed cutting plane algorithm not only outperforms the best-known algorithm in the literature for the stochastic problem under independent disruptions but also efficiently solves the robust problem under correlated disruptions. The practical performance of the robust models is verified in a simulation based on historical typhoon data in China. The numerical results further indicate that the robust model with even a small amount of information on disruption correlation can mitigate the conservativeness and improve the location decision significantly. Summary of Contribution: In this paper, we study the reliable uncapacitated facility location problem under uncertain facility disruptions. The problem is formulated as a two-stage distributionally robust model, which generalizes several related models in the literature, including the stochastic one with explicitly given disruption distribution and the robust one with moment information on disruptions. To solve this generalized model, we propose a cutting plane algorithm, where the separation problem is solved respectively by a polynomial-time algorithm in the stochastic case and by a column generation approach in the robust case. The efficiency and effectiveness of the proposed algorithm are validated through extensive numerical experiments. We also conduct a data-driven simulation based on historical typhoon data in China to verify the practical performance of the proposed robust model. The numerical results further reveal insights into the value of information on disruption correlation in improving the robust location decisions.

A Lagrangian relaxation approach to fuzzy robust multi-objective facility location network design problem

This study considers a multi-objective combined budget constrained facility location/network design problem (FL/NDP) in which the system uncertainty is considered. The most obvious practical examples of the problem are territorial designing and locating of academies, airline networks, and medical service centers. In order to assure the network reliability versus uncertainty, an efficient robust optimization approach is applied to model the proposed problem. The formulation is minimizing the total expected costs, including, transshipment costs, facility location (FL) costs, fixed cost of road/link utilization as well as minimizing the total penalties of uncovered demand nodes. Then, in order to consider of several system uncertainty, the proposed model is changed to a fuzzy robust model by suitable approaches. An efficient Sub-gradient based Lagrangian relaxation algorithm is applied. In addition, a practical example is studied. At the following, a series of experiments, including several test problems, is designed and solved to evaluate of the performance of the algorithm. The obtained results emphasize that considering of practical factors (e.g., several uncertainties, system disruptions, and customer satisfaction) in modelling of the problem can lead to significant improvement of the system yield and subsequently more efficient utilization of the established network.

A simulation based research on chance constrained programming in robust facility location problem

Abstract Since facility location decisions problem include long-term character and potential parameter variations, it is important to consider uncertainty in its modeling. This paper examines robust facility location problem considering supply uncertainty, in which we assume the supply of the facility in the actual operation is not equal to the supply initially established, the supply is subject to random fluctuation. The chance constraints are introduced when formulating the robust facility location model to make sure the system operate properly with a certain probability while the supply fluctuates. The chance constraints are approximated safely by using Hoeffding’s inequality and the problem is transformed to a general deterministic linear programming. Furthermore, how the facility location cost change with confidence level is investigated through a numerical example. The sensitivity analysis is conducted for important parameters of the model and we get the main factors that affect the facility location cost.

Integrated location-inventory modelling under forward and reverse product flows in the used merchandise retail sector: A multi-echelon formulation

This study presents a joint three-echelon location inventory model for a donation-demand driven industry in which the main warehouse (MW), distribution centers (DC), retail stores (RS) and donation-only centers (ADCs) exist. This unique inventory-location problem involves demand and supply uncertainties, coverage radius limitations, service level requirements, and multiple products consideration. Each retailer has two classes of products flowing from the assigned DC due to demands minus donations occurring in that retailer. The proposed model simultaneously determines the number of DCs to open, DC locations, and assignments of retailers to the open DCs for particular product types. The objective is to minimize the total annual cost including: facility location costs, transportation costs, inventory costs, and the lost sale costs. Due to the complexity of the problem, the proposed model structure allows for relaxing complicating constraints through recourse to Lagrangian relaxation. The use of robust branch-cut and price heuristics solves the mixed integer nonlinear problem to obtain a lower bound and a distance-based heuristic to get an upper bound. We formulate essential features of this novel problem, solve several numerical example problems and evaluate solution performance. We believe this is a novel problem environment, and that this initial study extends integrated location-inventory modeling to a new context.

The Ring Tree Facility Location Problem

In this work we discuss a facility location variant of the capacitated ring tree problem. The new model generalizes vehicle routing problems and Steiner tree problems, yielding applicability in telecommunication and transportation networks. In this ring tree facility location problem (RTFLP) two layers of networks have to be designed to connect two different types of customers to a central depot. The first, or inner layer, consists of cycles that intersect in the depot and collect all type 2 customers, and some of the type 1 customers. An outer layer is represented by a forest that contains the remaining type 1 customers such that each tree shares exactly one vertex with the inner layer. Capacity bounds apply to the number of connected substructures emanating from the depot, the number of customers in each of these so-called ring trees, and in each tree of the forest. Additional optional Steiner vertices can be used to reduce the overall costs, which are layer-dependent edge costs and facility location costs at the vertices in which the two layers coincide. Our contribution is the introduction of the RTFLP, the development of two mathematical formulations, and preliminary computational results for the first RTFLP test set derived from instances from the literature.

A Reliable Budget-Constrained FL/ND Problem with Unreliable Facilities

The combined facility location and network design problem is an important practical problem for locating public and private facilities. Moreover, incorporating aspects of reliability into the modeling of facility location problems is an effective way to hedge against disruptions in the system. In this paper, we consider a combined facility location/network design problem that considers system reliability. This problem has a number of applications, many of which fall into the category of service systems, such as regional planning and locating schools, health care service centers and airline networks. Our model also includes an investment budget constraint. We propose a mixed integer programming formulation to model this problem, as well as an efficient heuristic based on the problem’s LP relaxation. Numerical results demonstrate that the proposed heuristic significantly outperforms CPLEX in terms of solution speed, while still maintaining excellent solution quality. Our results also suggest a favorable tradeoff between the “nominal cost” (including fixed facility location costs and link construction costs, as well as transportation costs) and system reliability; that is, substantial improvements in reliability are often possible with only slight increases in the total cost of investment and transportation.

Characterization of facility assignment costs for a location-inventory model under truckload distribution

We consider a two-stage distribution system, where the first stage consists of potential distribution centres (DCs) and the second stage consists of geographically dispersed existing retailers. Our goal is to determine the set of open DCs and assignment of open DCs to retailers simultaneously with inventory decisions of retailers. In addition to the DC-specific fixed facility location costs, we explicitly model the inventory replenishment and holding costs at the retailers and truckload transportation costs between the DCs and the retailers. The transportation costs are subject to truck/cargo capacity, leading to an integrated location-inventory problem with explicit cargo costs. We develop a mixed-integer nonlinear model and analyse its structural properties leading to exact expressions for the so-called implied facility assignment costs and imputed per-unit per-mile transportation costs. These expressions analytically demonstrate the interplay between strategic location and tactical inventory/transportation decisions in terms of resulting operational costs. Although both the theory and practice of integrated logistics have recognized the fact that strategic and tactical decisions are interrelated, to the best of our knowledge, our paper is the first to offer closed-form results demonstrating the relationship explicitly. We propose an efficient solution approach utilizing the implied facility assignment costs and we demonstrate that significant savings are realizable when the inventory decisions and cargo costs are modelled explicitly for facility location purposes.

A Node-based Mathematical Model towards the Location Routing Problem with Intermediate Replenishment Facilities under Capacity Constraint

In this paper, we study the location routing problem with intermediate replenishment facilities (LRPIRF), an extension of the location routing problem (LRP), where the vehicles can replenish at some intermediate facilities. Vehicles leave the depot with load on-board, serve customers until out of load, may return to an intermediate facility to replenish, and finally return to the depot, completing their route. In this paper, we initiate a mathematical mixed integer programming model with new kind of subtour elimination constraints for this problem. Moreover, the facility location phase is considered besides vehicle routing phase in our problem. The objective of the problem is to find routes for vehicles to serve all customers at a minimal cost in terms of total travel cost and total facility location cost, without violating the capacity constraint of the vehicles. The solution to the LRPIRF is obtained through CPLEX solver in commercial software GAMS 23.5.1 ,Genetic Algorithm and Tabu Search algorithm. Computational results are obtained on a set of randomly generated instances and indicate the effectiveness of the proposed algorithms in terms of solution time and quality.

Constrained optimisation of distribution network using new hybrid PSO optimiser

The efficient and effective movement of goods from raw material sites to processing facilities, component fabrication plants, finished goods assembly plants, distribution centres, retailers and customers are critical in today’s competitive environment. These in turn depend on facility location decisions taken during strategic planning. These facilities are expensive to construct and difficult to modify. Inefficient locations for production and assembly plants as well as distribution centres will result in excess costs being incurred throughout the lifetime of the facilities, no matter how well the production plans, transportation options, inventory management, and information sharing decisions are optimised in response to changing conditions. This paper addresses design of network model for facility location and capacity allocation where in a set of customer locations with demands and a set of candidate facility locations will be known in advance. If a facility is located at a candidate site, a known fixed location cost is incurred. There is a known unit shipment cost between each candidate site and each customer location. The problem is to find the locations of the facilities and the shipment pattern between the facilities and the customers to minimise the combined facility location and shipment costs subject to a requirement that all customer demands be met. To optimise this, the distribution network model is mathematically represented and solved using hybrid PSO algorithm.

Bi‐objective optimization of distribution scheduling using MOPSO optimizer

PurposeThe paper aims at the bi‐objective optimization of a two‐echelon distribution network model for facility location and capacity allocation where in a set of customer locations with demands and a set of candidate facility locations will be known in advance. The problem is to find the locations of the facilities and the shipment pattern between the facilities and the distribution centers (DCs) to minimize the combined facility location and shipment costs subject to a requirement that maximum customer demands be met.Design/methodology/approachTo optimize the two objectives simultaneously, the location and distribution two‐echelon network model is mathematically represented in this paper considering the associated constraints, capacity, production and shipment costs and solved using hybrid multi‐objective particle swarm optimization (MOPSO) algorithm.FindingsThis paper shows that the heuristic based hybrid MOPSO algorithm can be used as an optimizer for characterizing the Pareto optimal front by computing well‐distributed non‐dominated solutions. These aolutions represent trade‐off solutions out of which an appropriate solution can be chosen according to industrial requirement.Originality/valueVery few applications of hybrid MOPSO are mentioned in literature in the area of supply chain management. This paper addresses one of such applications.

Efficient algorithms for the conditional covering problem

We consider the conditional covering problem in an undirected network, in which each vertex represents a demand point that must be covered by a facility as well as a potential facility site. Each facility can cover all vertices within a given coverage radius, except the vertex at which the facility is located. The objective is to locate facilities to cover all vertices such that the total facility location cost is minimized. In this paper, new upper bounds are proposed for the conditional covering problem on paths, cycles, extended stars, and trees. In particular, we provide an O(nlogn)-time algorithm for paths, an O(n2logn)-time algorithm for cycles, an O(n1.5logn)-time algorithm for extended stars, and an O(n3)-time algorithm for trees. Our algorithms for paths, extended stars, and trees improve the previous upper bounds from O(n2), O(n2), and O(n4), respectively.

Integrated Production-Inventory-Distribution System Design with Risk Pooling: Model Formulation and Heuristic Solution

In this paper, we consider a multiproduct two-echelon production-inventory-distribution system design model that captures risk-pooling effects by consolidating the safety-stock inventory of the retailers at distribution centers (DCs). We propose a model that determines plant and DC locations, shipment levels from plants to the DCs, safety-stock levels at DCs, and the assignment of retailers to DCs by minimizing the sum of fixed facility location costs, transportation costs, and safety-stock costs. The model is formulated as a nonlinear mixed-integer programming problem and linearized using piecewise-linear functions. The formulation is strengthened using redundant constraints. Lagrangean relaxation is applied to decompose the problem by echelon. A lower bound is provided by the Lagrangean relaxation, while a heuristic is proposed that uses the solution of the subproblems to construct an overall feasible solution. Computational results reveal that the Lagrangean relaxation provides a sharp lower bound and a heuristic solution that is within 5% of the optimal solution.

Algorithms for solving the conditional covering problem on paths

AbstractConsider the conditional covering problem on an undirected graph, where each node represents a site that must be covered by a facility, and facilities may only be established at these nodes. Each facility can cover all sites that lie within some common covering radius, except the site at which it is located. Although this problem is difficult to solve on general graphs, there exist special structures on which the problem is easily solvable. In this paper, we consider the special case in which the graph is a simple path. For the case in which facility location costs do not vary based on the site, we derive characteristics of the problem that lead to a linear‐time shortest path algorithm for solving the problem. When the facility location costs vary according to the site, we provide a more complex, but still polynomial‐time, dynamic programming algorithm to find the optimal solution. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.

Dynamic programming algorithms for the conditional covering problem on path and extended star graphs

The Conditional Covering Problem (CCP) is a facility location problem on a graph, where the set of nodes represents demand points and potential facility locations. The key aspect of the CCP is that each facility covers all nodes within a given facility-specific coverage radius, except for the node at which it is located. The objective of this problem is to minimize the sum of the facility location costs required to cover all demand points. We first discuss the worst-case complexity of the CCP by examining literature related to the total domination problem, which is a special case of the CCP. Next, we examine the special case of path graphs and provide an O(n2) algorithm for its solution. Finally, we leverage information obtained from this procedure to derive an optimal algorithm for “extended star” graphs (multiple paths having one node in common), without increasing the worst-case complexity of the algorithm. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 46(4), 177–185 2005

Warehouse-Retailer Network Design Problem

In this paper, we study the distribution network design problem integrating transportation and infinite horizon multiechelon inventory cost function. We consider the trade-off between inventory cost, direct shipment cost, and facility location cost in such a system. The problem is to determine how many warehouses to set up, where to locate them, how to serve the retailers using these warehouses, and to determine the optimal inventory policies for the warehouses and retailers. The objective is to minimize the total multiechelon inventory, transportation, and facility location costs. To the best of our knowledge, none of the papers in the area of distribution network design has explicitly addressed the issues of the 2-echelon inventory cost function arising from coordination of replenishment activities between the warehouses and the retailers. We structure this problem as a set-partitioning integer-programming model and solve it using column generation. The pricing subproblem that arises from the column generation algorithm gives rise to a new class of the submodular function minimization problem. We show that this pricing subproblem can be solved in O(nlog n) time, where n is the number of retailers. Computational results show that the moderate size distribution network design problem can be solved efficiently via this approach.