Mark S. Daskin

Strategic facility location: A review

Abstract Facility location decisions are a critical element in strategic planning for a wide range of private and public firms. The ramifications of siting facilities are broadly based and long-lasting, impacting numerous operational and logistical decisions. High costs associated with property acquisition and facility construction make facility location or relocation projects long-term investments. To make such undertakings profitable, firms plan for new facilities to remain in place and in operation for an extended time period. Thus, decision makers must select sites that will not simply perform well according to the current system state, but that will continue to be profitable for the facilitys lifetime, even as environmental factors change, populations shift, and market trends evolve. Finding robust facility locations is thus a difficult task, demanding that decision makers account for uncertain future events. The complexity of this problem has limited much of the facility location literature to simplified static and deterministic models. Although a few researchers initiated the study of stochastic and dynamic aspects of facility location many years ago, most of the research dedicated to these issues has been published in recent years. In this review, we report on literature which explicitly addresses the strategic nature of facility location problems by considering either stochastic or dynamic problem characteristics. Dynamic formulations focus on the difficult timing issues involved in locating a facility (or facilities) over an extended horizon. Stochastic formulations attempt to capture the uncertainty in problem input parameters such as forecast demand or distance values. The stochastic literature is divided into two classes: that which explicitly considers the probability distribution of uncertain parameters, and that which captures uncertainty through scenario planning. A wide range of model formulations and solution approaches are discussed, with applications ranging across numerous industries.

A Joint Location-Inventory Model

We consider a joint location-inventory problem involving a single supplier and multiple retailers. Associated with each retailer is some variable demand. Due to this variability, some amount of safety stock must be maintained to achieve suitable service levels. However, risk-pooling benefits may be achieved by allowing some retailers to serve as distribution centers (and therefore inventory storage locations) for other retailers. The problem is to determine which retailers should serve as distribution centers and how to allocate the other retailers to the distribution centers. We formulate this problem as a nonlinear integer-programming model. We then restructure this model into a set-covering integer-programming model. The pricing problem that must be solved as part of the column generation algorithm for the set-covering model involves a nonlinear term in the retailerdistribution-center allocation terms. We show that this pricing problem can (theoretically) be solved efficiently, in general, and we show how to solve it practically in two important cases. We present computational results on several instances of sizes ranging from 33 to 150 retailers. In all cases, the lower bound from the linear-programming relaxation to the set-covering model gives the optimal solution.

Carbon Footprint and the Management of Supply Chains: Insights From Simple Models

Using relatively simple and widely used models, we illustrate how carbon emission concerns could be integrated into operational decision-making with regard to procurement, production, and inventory management. We show how, by associating carbon emission parameters with various decision variables, traditional models can be modified to support decision-making that accounts for both cost and carbon footprint. We examine how the values of these parameters as well as the parameters of regulatory emission control policies affect cost and emissions. We use the models to study the extent to which carbon reduction requirements can be addressed by operational adjustments, as an alternative (or a supplement) to costly investments in carbon-reducing technologies. We also use the models to investigate the impact of collaboration among firms within the same supply chain on their costs and carbon emissions and study the incentives firms might have in seeking such cooperation. We provide a series of insights that highlight the impact of operational decisions on carbon emissions and the importance of operational models in evaluating the impact of different regulatory policies and in assessing the benefits of investments in more carbon efficient technologies. Note to Practitioners-Firms worldwide, responding to the threat of government legislation or to concerns raised by their own consumers or shareholders, are undertaking initiatives to reduce their carbon footprint. It is the conventional thinking that such initiatives will require either capital investments or a switch to more expensive sources of energy or input material. In this paper, we show that firms could effectively reduce their carbon emissions without significantly increasing their costs by making only operational adjustments and by collaborating with other members of their supply chain. We describe optimization models that can be used by firms to support operational decision making and supply chain collaboration, while taking into account carbon emissions. We analyze the effect of different emission regulations, including strict emission caps, taxes on emissions, cap-and-offset, and cap-and-trade, on supply chain management decisions. In particular, we show that the presence of emission regulation can significantly increase the value of supply chain collaboration.

AN INVENTORY-LOCATION MODEL: FORMULATION, SOLUTION ALGORITHM AND COMPUTATIONAL RESULTS

We introduce a distribution center (DC) location model that incorporates working inventory and safety stock inventory costs at the distribution centers. In addition, the model incorporates transport costs from the suppliers to the DCs that explicitly reflect economies of scale through the use of a fixed cost term. The model is formulated as a non-linear integer-programming problem. Model properties are outlined. A Lagrangian relaxation solution algorithm is proposed. By exploiting the structure of the problem we can find a low-order polynomial algorithm for the non-linear integer programming problem that must be solved in solving the Lagrangian relaxation subproblems. A number of heuristics are outlined for finding good feasible solutions. In addition, we describe two variable forcing rules that prove to be very effective at forcing candidate sites into and out of the solution. The algorithms are tested on problems with 88 and 150 retailers. Computation times are consistently below one minute and compare favorably with those of an earlier proposed set partitioning approach for this model (Shen, 2000; Shen, Coullard and Daskin, 2000). Finally, we discuss the sensitivity of the results to changes in key parameters including the fixed cost of placing orders. Significant reductions in these costs might be expected from e-commerce technologies. The model suggests that as these costs decrease it is optimal to locate additional facilities.

Time Dependent Vehicle Routing Problems: Formulations, Properties and Heuristic Algorithms

The time dependent vehicle routing problem (TDVRP) is defined as follows. A vehicle fleet of fixed capacities serves customers of fixed demands from a central depot. Customers are assigned to vehicles and the vehicles routed so that the total time of the routes is minimized. The travel time between two customers or between a customer and the depot depends on the distance between the points and time of day. Time windows for serving the customers may also be present. The time dependent traveling salesman problem (TDTSP) is a special case of the TDVRP in which only one vehicle of infinite capacity is available. Mixed integer linear programming formulations of the TDVRP and the TDTSP are presented that treat the travel time functions as step functions. The characteristics and properties of the TDVRP preclude modification of most of the algorithms that have been developed for the vehicle routing problem. Several simple heuristic algorithms are given for the TDTSP and TDVRP without time windows based on the nearest-neighbor heuristic. A mathematical-programming-based heuristic for the TDTSP without time windows using cutting planes is also briefly discussed. Test results on small, randomly generated problems are reported.

A bibliography for some fundamental problem categories in discrete location science

Following a brief taxonomy of the broad field of facility location modeling, this paper provides an annotated bibliography of recent papers in two branches of discrete location theory and modeling. In particular, we review papers related to (1) the median and plant location models and (2) to center and covering models. We show how the contributions of the papers we review are embedded in the field. A summary and outlook conclude the paper.

The stochastic location model with risk pooling

Abstract In this paper, we present a stochastic version of the location model with risk pooling (LMRP) that optimizes location, inventory, and allocation decisions under random parameters described by discrete scenarios. The goal of our model (called the stochastic LMRP, or SLMRP) is to find solutions that minimize the expected total cost (including location, transportation, and inventory costs) of the system across all scenarios. The location model explicitly handles the economies of scale and risk-pooling effects that result from consolidating inventory sites. The SLMRP framework can also be used to solve multi-commodity and multi-period problems. We present a Lagrangian-relaxation–based exact algorithm for the SLMRP. The Lagrangian subproblem is a non-linear integer program, but it can be solved by a low-order polynomial algorithm. We discuss simple variable-fixing routines that can drastically reduce the size of the problem. We present quantitative and qualitative computational results on problems with up to 150 nodes and 9 scenarios, describing both algorithm performance and solution behavior as key parameters change.

A WAREHOUSE LOCATION-ROUTING PROBLEM

The interdependence between distribution center location and vehicle routing has been recognized by both academics and practitioners. However, only few attempts have been made to incorporate routing in location analysis. This paper defines the Warehouse Location-Routing Problem (WLRP) as one of simultaneously solving the DC location and vehicle routing problems. We present a mixed integer programming formulation of the WLRP. Based on this formulation, it can be seen that the WLRP is a generalization of well-known and difficult location and routing problems, such as the Location-Allocation Problem and the Multi-depot Vehicle Dispatch Problem. It is therefore a large and complex problem which cannot be solved using existing mixed-integer programming techniques. We present a heuristic solution method for the WLRP, based on decomposing the problem into three subproblems. The proposed method solves the subproblems in a sequential manner while accounting for the dependence between them. We discuss a large-scale application of the proposed method to a national distribution company at a regional level.

A random-key genetic algorithm for the generalized traveling salesman problem

The generalized traveling salesman problem is a variation of the well-known traveling salesman problem in which the set of nodes is divided into clusters; the objective is to find a minimum-cost tour passing through one node from each cluster. We present an effective heuristic for this problem. The method combines a genetic algorithm (GA) with a local tour improvement heuristic. Solutions are encoded using random keys, which circumvent the feasibility problems encountered when using traditional GA encodings. On a set of 41 standard test problems with symmetric distances and up to 442 nodes, the heuristic found solutions that were optimal in most cases and were within 1% of optimality in all but the largest problems, with computation times generally within 10 seconds. The heuristic is competitive with other heuristics published to date in both solution quality and computation time.

Capacitated facility location/network design problems

Abstract We introduce a combined facility location/network design problem in which facilities have constraining capacities on the amount of demand they can serve. This model has a number of applications in regional planning, distribution, telecommunications, energy management, and other areas. Our model includes the classical capacitated facility location problem (CFLP) on a network as a special case. We present a mixed integer programming formulation of the problem, and several classes of valid inequalities are derived to strengthen its LP relaxation. Computational experience with problems with up to 40 nodes and 160 candidate links is reported, and a sensitivity analysis provides insight into the behavior of the model in response to changes in key problem parameters.