Production, Manufacturing, Transportation and Logistics
The two-echelon stochastic multi-period capacitated location-routing problem

https://doi.org/10.1016/j.ejor.2022.07.022Get rights and content

Highlights

  • Introduce the two echelon stochastic multi period capacitated location routing problem.

  • Present a two stage stochastic programming formulation.

  • Propose a logic based B enders decomposition approach to solve the model.

  • Impact of the stochastic and multi period settings are investigated.

  • Evaluate the quality of the generated designs and stochastic solutions.

Abstract

Given the emergence of two-echelon distribution systems in several practical contexts, this paper tackles, at the strategic level, a distribution network design problem under uncertainty. This problem is defined as the two-echelon stochastic multi-period capacitated location-routing problem (2E-SM-CLRP). It considers a network partitioned into two capacitated distribution echelons: each echelon involves a specific location-assignment-transportation schema that must cope with the future demand. It aims to decide the number and location of warehousing/storage platforms (WPs) and distribution/fulfillment platforms (DPs), and on the capacity allocated from first echelon to second echelon platforms. In the second echelon, the goal is to construct vehicle routes that visit ship-to locations (SLs) from operating distribution platforms under a stochastic and time-varying demand and varying costs. This problem is modeled as a two-stage stochastic program with integer recourse, where the first-stage includes location and capacity decisions to be fixed at each period over the planning horizon, while routing decisions of the second echelon are determined in the recourse problem. We propose a logic-based Benders decomposition approach to solve this model. In the proposed approach, the location and capacity decisions are taken by solving the Benders master problem. After these first-stage decisions are fixed, the resulting sub-problem is a capacitated vehicle-routing problem with capacitated multiple depots (CVRP-CMD) that is solved by a branch-cut-and-price algorithm. Computational experiments show that instances of realistic size can be solved optimally within a reasonable time and provide relevant managerial insights on the impact of the stochastic and multi-period settings on the 2E-CLRP.

Introduction

At the strategic planning level of distribution networks, the design decisions involve the determination of the number, location and capacity of the distribution platforms required to satisfy the evolving demand of a customers base. It also determines the mission of these platforms in terms of the subset of customers they must supply and of the interconnection between the facilities. This design problem takes basis on the well-known Facility location problem (FLP) (Geoffrion, Graves, 1974, Klose, Drexl, 2005), and is also extensively tackled in the Location-Routing problem (LRP) that integrates transportation decisions (Nagy, Salhi, 2007, Prodhon, Prins, 2014). These distribution networks are dominated by a single-echelon structure, in which the network topology includes a layer of warehouse platforms (WPs) where the inventory is held, and a layer of customers’ ship-to locations (SLs). Products are shipped from WPs directly to SLs via large trucks that possibly visit multiple SLs (Hemmelmayr, Cordeau, Crainic, 2012, Nagy, Salhi, 2007). An example of the one-echelon capacitated LRP (1E-CLRP) is illustrated in Fig. 1a. In recent years, the rise of online sales coupled with the consumers desire of speed, the increased attention to sustainability in urban logistics and the evolution of logistics assets capability, changed drastically the distribution landscape. Most of practitioners admit the limits of single echelon distribution networks and have nowadays turned their attention to two-echelon distribution structures with additional fulfillment platforms to meet today’s challenges. Several retailers realize that the location of their centralized warehouses, calibrated for logistics efficiency, is not necessarily optimized to provide fast deliveries, or to efficiently operate fulfillment and urban shipment services.

The growth of e-commerce and the emergence of omnichannel retailing have drastically favored a high proximity to customers’ ship-to locations (SLs), which impact the last-mile delivery cost (Chopra, 2018). As mentioned in Savelsbergh & Van Woensel (2016), the increase of direct-to-consumer deliveries and the emergence of new freight movement lead to additional fulfillment complexity and transportation complexity. For instance, Parcel delivery companies are experiencing important stretch of their distribution systems with the inclusion of a new echelon of urban fulfilment hubs (Winkenbach, Kleindorfer, & Spinler, 2016). In the same way, several retailers such as Walmart, JD.com or Amazon have adapted their distribution networks by adding an advanced echelon of distribution/fulfillment platforms, mostly in urban areas, and often relying on retail stores as ship-from or pick-up locations. According to Weinswig (2018), Walmart turned several Sam’s Club stores into e-commerce fulfillment centers to support the rapid e-commerce growth. In link with that, urban last-mile delivery faces great challenges to satisfy growing customer needs for a punctual and a precise delivery location, either their home, office, car, smart locker or a store (Savelsbergh & Van Woensel, 2016). In several business contexts, this issue tends to increase the number of inventory locations and echelons in the distribution system. Crainic & Montreuil (2016) underline the current expansion to multi-tier urban distribution systems and proposed an interconnected logistics framework. Urban parcels delivery is probably the most significant example of the shift from a one-echelon to a two-echelon distribution network setting (Crainic, Errico, Rei, Ricciardi, 2016, Janjevic, Winkenbach, Merchán, 2019). This is achieved by creating peripheral distribution/consolidation centers dedicated to transferring and consolidating freight from back-level platforms/hubs. Recent papers in omnichannel retailing also support the growing tendency for a two-echelon distribution network (Arslan, Klibi, Montreuil, 2021, Janjevic, Merchán, Winkenbach, 2021). Given these challenges, it is legitimate for the distribution network designer to question to which extent current models are sufficient to provide good quality designs.

Current distribution network design models found in the literature have three main shortcomings when compared to the strategic needs of distribution businesses. First, most of related works rely mainly on a single echelon distribution structure (Melo, Nickel, & Saldanha-da Gama, 2009), whereas several distribution systems are more suitable for a two-echelon network structure as highlighted with a number of examples above and in the reviews of Savelsbergh & Van Woensel (2016), Cuda, Guastaroba, & Speranza (2015). We refer hereafter in general to the two-echelon Location-Routing problem by 2E-LRP, and to its specific capacitated version by 2E-CLRP. When it comes to the 2E-LRP, the literature is still scarce. Deterministic versions of the 2E-LRP are introduced in Boccia, Crainic, Sforza, & Sterle (2010) and Contardo, Hemmelmayr, & Crainic (2012).

Second, the dynamics of business operations nowadays calls for a decision process that integrates a more refined granularity of the operations than traditional high level of aggregation (Geoffrion & Graves, 1974). Once designed, the distribution network is in operation on a daily basis and has to deal with detailed customer locations and varying demand/orders. These operational details are found in routing models (VRPs) but unfortunately not often integrated when location decisions have to be taken (Klibi, Martel, & Guitouni, 2016). Giving the complexity of VRPs, most 2E-LRPs rely on continuous approximations based on route length estimation (RLE), but this work shows that, with recent advances, realistic size 2E-CLRPs integrating routing decisions can be solved optimally within a reasonable time. Further, we note that most existent 2E-LRP and most LRP modeling approaches implicitly assume that location and routing decisions are made simultaneously for the planning horizon, without considering the hierarchical structure of the strategic problem. Such framework, introduced by Schneeweiss (2003), relies on the time lag and the top-down relation between location and operational decisions, as applied in Klibi, Lasalle, Martel, & Ichoua (2010); Klibi et al. (2016) and Ben Mohamed, Klibi, & Vanderbeck (2020).

Third, the business environment is clearly dominated by a time-varying-stochastic setting. Accordingly, the traditional deterministic-static representation of the planning horizon is due to be replaced by a more realistic stochastic and multi-period characterization of the planning horizon. Since the 2E-CLRP is a strategic decision process that must be designed to last for several years, the horizon must be partitioned into a set of periods shaping the uncertainty and variability in time. The location and capacity decisions should be planned as a set of sequential decisions to be implemented at different design periods of the horizon (a year, for example) and promoting the structural adaptability of the network. Existing works focus on static and deterministic settings for the 2E-LRPs. A stochastic 2E-location model with RLE is proposed by Snoeck, Winkenbach, & Mascarino (2018) and a multi-stage 2E-location-allocation model is introduced in Ben Mohamed et al. (2020). Including the features stressed above, gives rise to the two-echelon stochastic multiperiod capacitated location-routing problem, denoted by (2E-SM-CLRP). As far as we know, the 2E-SM-CLRP has not been addressed yet.

Figure 1 b illustrates a typical 2E-CLRP partitioned into two capacitated distribution echelons. Each echelon in the 2E-CLRP involves a specific location-assignment-transportation schema that must cope with the future demand. It aims to decide the number and location of warehousing/storage platforms (WPs) and distribution/fulfillment platforms (DPs), and on the capacity allocated from first echelon to second echelon platforms. It also decides the transportation activity between platforms. Direct routes with full truckloads transportation option, relying on consolidation policy, are considered in the first echelon between opened WPs and selected DPs. This is common in practice, where the distributor, using its own vehicle fleet or a contract carrier, considers a single-destination full trailer to deliver a regional/urban DP on a scheduled basis. In the second echelon, the transportation activity is shaped by multi-drop routes, as DPs are generally devoted to more fragmented urban services (Fig. 1b). These are characterized by a high density and a limited accessibility which favor the use of small-sized trucks than large trucks, while WPs are located in peri-urban areas to allow efficient inbound logistics operations. Accordingly, the division into the first and second echelon is done in the usual geographical sense (Behrends, 2016, Crainic, Ricciardi, Storchi, 2004). Our problem is purely strategic: the emphasis is on the location of platforms and their mission (allocation, capacity), whereas the operational routing decisions are considered to better estimate the quality of the strategic design decisions (Klibi et al., 2016). These latter are taken prior to their deployment period and after an implementation period, they will be available for use at the operational level. This stresses the need for a solution procedure that provides good-quality design solutions with a precise evaluation, that may be at the expense of the running time. Our approach builds on recent progress on stochastic modeling for location-allocation problems and on exact algorithms for vehicle routing problems to make it feasible.

The contribution of this study is threefold. Firstly, this work is the first to introduce the 2E-SM-CLRP under uncertain and time-varying demand and cost, a hierarchical decision problem that is characterized by a temporal hierarchy between design and transportation decisions. Secondly, we present a two-stage stochastic program with integer recourse for the 2E-SM-CLRP that captures the temporal hierarchy and the multi-stage setting of the decision process. A scenario-based approach is used to describe the uncertain SLs demands: it relies on a set of multi-period scenarios generated with a Monte-Carlo approach. In addition, 2E-SM-CLRP inherits complexity of the LRP (Laporte, 1988) and adds a combinatorial-stochastic structure (Shapiro & Nemirovski, 2005), which makes it a very hard optimization problem. As a third contribution, we propose a first exact approach to solve this problem by means of a logic-based Benders decomposition approach (Benders, 1962, Hooker, 2019) and a sample average approximation (SAA) (Shapiro, Dentcheva, & Ruszczynski, 2009). It calls the branch-cut-and-price algorithm of Sadykov, Uchoa, & Pessoa (2021) to solve Benders sub-problems in a parallel computing scheme. Two families of Benders cuts are proposed to cut off infeasible solutions and help converging to an optimal solution of the 2E-SM-CLRP. The extensive computational experiments on a large set of instances, emphasize the performance of the proposed algorithm on solving medium-scale instances optimally, and on getting good feasible solutions, i.e., with less than 0.5% gap from a lower bound, for larger instances with up to 50 SLs and 25 demand scenarios over a 5-year planning horizon. Finally, we share the insights we derived from the computational experiments about the impact of the stochastic and multi-period settings on the 2E-CLRP.

The reminder of this paper is organized as follows. Section 2 surveys the related work on the 2E-CLRP. Section 3 introduces the mathematical formulation of the 2E-SM-CLRP. Section 4 presents the proposed exact solution approach based on Benders decomposition. The computational results are presented and analyzed in Section 5. Section 6 provides conclusion and future research avenues. The appendix presents complementary information on the algorithm and detailed computational results from the test-bed.

Section snippets

Literature review

The literature on distribution network design problems is evolving towards the study of more complex and integrated models. Therein, strategic decisions and operational decisions are strongly interrelated (Ambrosino, Scutellà, 2005, Crainic, Laporte, 1997). More specifically, the strategic level involves location and capacity planning decisions and the operational level implies transportation decisions. The integration of the two decision levels into a location-routing problem (LRP), often

Mathematical formulation

The two-echelon stochastic multi-period capacitated location-routing problem (2E-SM-CLRP) is a hierarchical decision problem. This stems from the temporal hierarchy between the design decisions and the operational transportation decisions. The design decisions, i.e, location and capacity, are made on yearly basis before the realization of the uncertain parameters, whereas the operational routing decisions are determined when uncertainty is revealed. To catch the temporal hierarchy, we formulate

Logic-based Benders decomposition approach

We solve the 2E-SM-CLRP using a logic-based Benders decomposition (LBBD) approach (Hooker, 2019). LBBD is a generalization of classical Benders decomposition (Benders, 1962) that allows the subproblem to be any optimization problem rather than specifically a linear or non-linear programming problem.

In order to use the LBBD approach, we slightly modify formulation (1)–(19). We introduce additional continuous variablesθtω0tT,ωΩt.We also replace the component tTEΩt[ϕtω(x)] in objective

Computational results

In this section, we present our experimental results. First, we present the instances used in the experiments. Then, we report and discuss the obtained results.

Our approach is implemented in the C++ and compiled with GCC 5.3.0. BaPCod package (Sadykov & Vanderbeck, 2021) is used to handle the branch-cut-and-price framework. The code from (Sadykov et al., 2021) is used to solve the resource constrained shortest path pricing problems. We use Cplex 12.8.0 as the linear programming solver in column

Conclusion

In this paper, we introduced the two-echelon stochastic multi-period capacitated location-routing problem (2E-SM-CLRP). The problem is characterized as a hierarchical decision process involving a design level in which network location and capacity allocation decision are taken, and an operational level dealing with transportation decisions of the second echelon. A stochastic multi-period characterization of the planning horizon is considered, shaping the evolution of the uncertain SL demand and

Acknowledgements

Experiments presented in this paper were carried out using the PlaFRIM experimental testbed, supported by Inria, CNRS (LABRI and IMB), UniversitȨ de Bordeaux, Bordeaux INP and Conseil RȨgional d’Aquitaine (see https://www.plafrim.fr).

HS gratefully acknowledges the financial support of the FMJH “Program Gaspard Monge for optimization and operations research and their interactions with data science”, and EDF.

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