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A survey on the inventory-routing problem with stochastic lead times and demands
Accepted Manuscript A survey on the inventory-routing problem with stochastic lead times and demands
Raúl F. Roldán, Rosa Basagoiti, Leandro C. Coelho
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S1570-8683(16)30064-7 http://dx.doi.org/10.1016/j.jal.2016.11.010 JAL 441
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Journal of Applied Logic
Please cite this article in press as: R.F. Roldán et al., A survey on the inventory-routing problem with stochastic lead times and demands, J. Appl. Log. (2016), http://dx.doi.org/10.1016/j.jal.2016.11.010
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Highlights • We review of papers working with stochastic demand and stochastic lead times focusing on its stochastic and multi-depot aspects • We identify critical factors for the performance of many logistics activities and industries. • We have shown that studying the behavior of the demand and the lead time are essential in order to achieve a useful representation of the system to take proper decisions. • We highlight some characteristics and solution methodologies for multi-depot problems, most of which borrow ideas from the vehicle routing literature and adapt them to consider inventory management.
A survey on the inventory-routing problem with stochastic lead times and demands Ra´ ul F. Rold´ana,b,∗, Rosa Basagoitia , Leandro C. Coelhoc a
Electronics and Computing Department, Mondragon University, Goiru Kalea, 2, Arrasate 20500, Spain b Engineering Faculty,Compensar Unipanamericana Fundaci´ on Universitaria, Avenida (Calle) 32 No. 17 - 30, Bogot´ a D.C., Colombia c Interuniveristy Research Centre on Enterprise Networks, Logistics and Transportation (CIRRELT) and Facult´e des sciences de l’administration, Universit´e Laval, Qu´ebec, Canada
Abstract The integration of the different processes and players that compose the supply chain (SC) is essential to obtain a better coordination level. Inventory control and distribution management are the two processes that researchers have identified as the key to gain or lose in efficiency and effectiveness in the field of logistics, with a direct effect on the synchronization and overall performance of SCs. In practical situations demand is often not deterministic, and lead times are also variable, yielding a complex stochastic problem. In order to analyze the recent developments in the integration of these processes, this paper analyzes the state of the art of the information management in the SC, the relationship between inventory policies and available demand information, and the use of optimization methods to provide good solutions for the problem in single and multi depot versions. Keywords: stochastic demand, inventory-routing problem, stochastic lead time, inventory policy, survey. 1. Introduction A supply chain (SC) is defined as the system of organizations and flows of products, information and money spanning over the product’s entire life cycle, from conception to consumption and final disposal. It involves different actors such as suppliers, producers, distributors, transporters and customers, among others, all of them involved directly or indirectly in satisfying the request of a final client. Coordination and integration of SC activities are now recognized as critical to obtain competitive advantage [42]. Guasch and Kogan [27] observe that the logistics performance directly affect the cost ∗
Corresponding author. Email addresses: [email protected] (Ra´ ul F. Rold´an), [email protected] (Rosa Basagoiti), [email protected] (Leandro C. Coelho)
Preprint submitted to Journal of Applied Logic
November 9, 2016
of the products and hence, the overall performance of an industry. These authors compare two economic organizations in order to find the impact of logistics cost on the price of a product, and conclude that the costs associated with inventory management represent about 19% of the price of a product in countries with poor logistics systems, compared to 8% in countries with efficient logistics networks. Guasch [26] conclude that the reduction in logistics costs can be evidenced by two indicators, namely 1) the increase in the proportion of demand of a product, and 2) the increase in the proportion of employment that can be generated for some sectors of the economy. As a consequence, the increase in product price obstructs the competitiveness and complicates the maintenance of inventory. Logistics costs can be categorized into administrative, warehousing, inventory, transportation, and licenses costs. Guasch [26] identifies that more than 69% of these are directly related to transportation and inventory. Inventory management alone represents a proportion of the net operating assets of approximately 37% in production, 62% in distribution, and 56% in retail [59]. In addition, it is important to note that inventory control has to balance conflicting objectives due to two main reasons: 1) economies of scale and purchasing large batch sizes, and 2) uncertainty in supply and demand, which inevitably create safety stocks. In this paper, our goal is to provide an overview of the literature on problems related to the coordination of the inventory and its distribution. We provide a thorough review of papers dealing with stochastic demand and stochastic lead times, as these are the factors identified as critical for the performance of many logistics activities and industries. The remainder of this paper is organized as follows. Section 2 reviews the information management between different actors in the SC, since this will determine the evolution and quality of the information. It also analyzes inventory policies and their relation to the demand, in order to properly manage inventory levels. Stochastic demand and lead times are studied in Section 3. Several optimization methods are revised in Section 4. In Section 5, we present some versionof the inventory-routing problem (IRP) with special attention to multi-depot IRPs (MDIRP). In Section 6, we discuss the key elements presented with some methodological aspects. Our conclusions and final remarks are presented in Section 7. 2. Information management and the inventory policies The management and coordination of the information between the processes in the SC is crucial in order to make good decisions. In Gavirneni et al. [24], the flow of information between a supplier and a customer was analyzed and three situations compared: 1) no information for the supplier before the request arrives, 2) the supplier knows the policies that the customer uses as well as the final distribution processes, and 3) the supplier has all the information about the state of the customer’s inventory levels and decision process. The cost analysis indicates that the second configuration reduces 50% of the cost compared to the first one. When the second and the third 2
cases are compared, the cost reductions vary between 1% and 35%. According to Psaraftis [45], the four dimensions of the information are evolution, quality, availability and processing. Just the first and second components add randomness and stochasticity. The evolution that the information experiences over time highlights that it can change during the execution of the preliminary planning, and its quality reflects the possibility of the existence of some amount of uncertainty and asymmetric information between actors or entities. Demand information experiences variability and amplification along the SC. This is known as the bullwhip effect, which has been widely studied. For instance, Giard and Sali [25] argued that the bullwhip effect is the main reason for the loss of efficiency and efficacy in SCs. Chopra and Meindl [14] state that it can be damped by an improvement on the operative performance and the design of rationing schemes in the products that present shortage. In this way, some of the proposals include reducing the replenishment lead-time, reducing the lot size, taking into account historical data, and the interchange of information to limit the variability [18]. Accurate and timely information management can help improving the performance of the SC. According to Wagner [60], although the SC involves several activities such as purchase, production, location, marketing, inventory control and distribution, the foundations of integration in the SC are in the last two, which are focused in the efficiency of the channel and the coordination of the performance of the individual entities for the satisfaction of the final client. However, uncertainty plays a crucial role, making the environment less predictable. One way to reduce the effect of variability in the information is to assign the responsibility for management between activities to a single actor. This is achieved with strategies such as the vendor-managed inventory (VMI), which requires information to be shared. According to Beyer et al. [11], the characteristics of inventory control systems are determined by the efficiency measures with which to evaluate the performance over time, as well as by factors taken into account when modeling the system, such as cost structure, demand, lead time, review time, excess demand, deteriorating inventory, and ad hoc constraints. With respect to the demand, its analysis and estimation can reduce the risk of surplus or shortages. In this context, two issues become relevant: when and how much to replenish. Demand characteristics can be broadly classified into certain and uncertain. When demand is deterministic and all its parameters are known, the economic order quantity (EOQ) [28] and its variations are used to solve the problem of replenishment timing and quantities. When uncertainty is significant, it must be explicitly taken into account, and modeling alternatives include one-time decision models, continuous decision models, and periodical decision models. When decisions are taken either continuously or periodically it is necessary to establish a policy for ordering. The choice of which inventory policy to apply largely influences the cost of the optimization process. Typically, three key parameters are used to control the inventory policy: when replenish, how much to replenish, and how often the inventory level is reviewed. For the periodic review inventory system, Wensing [62] describes three 3
policies. The first is the order-up-to level (OU) which refers to a (t, S) system. Here, in each period t, the quantity delivered is that to fill the inventory capacity up to S. Other policies include the (t, s, S) and the (t, s, q). In the former, the customer is served if the inventory level is less than s. In the latter, the replenishment level q is flexible but bounded by the storage capacity. These policies should be articulated with strategies for customer selection, because sometimes it is not possible to serve all customers due to vehicle capacities, and in such cases, it is necessary to prioritize them. 3. Stochastic demand and lead time modeling Axs¨ater [6] describe that stochastic demands can be modeled as continuous variables if their average is big enough; however, for small stochastic demands, it is easier to model them using discrete models. Under the assumption that individual demand events occur independently, two general modeling techniques arise. The first option is to use the Poisson probability distribution, which is appropriate when the expected demand in a time interval is small; the second option is to use normal probability distribution when the expected demand is large enough [31]. Some studies considering the demand as a random variable are summarized below. In Chao and Zhou [12], an inventory system with continuous review in infinite horizon is considered. The sales price and inventory replenishment are determined simultaneously. The demand process is modeled by a Poisson probability distribution, with an arrival rate dependent on the price. Another model for an inventory system with two suppliers is proposed by Song and Zipkin [56], where one supplier responds best to demand than the other. One of the nodes has a capacity limit, meaning that it can only satisfy the demand up to that point. Both suppliers work with constant lead times, and the demand is modeled as a continuous time variable following a Poisson probability distribution with linear ordering costs. Bertazzi et al. [10] study a case in which the supplier has a limited production capacity and the demand from the customers is stochastic. These demands are modeled as discrete random variables and deliveries are performed using transportation procurement. In Berling and Marklund [7], the authors assume a normally distributed customer demand, and lead times are estimated based on the magnitude of the demand. A model that approximates the true distribution of the lead-times by means three standard distributions is used. The employed distributions are the negative binomial distribution, a discrete approximation of the normal distribution, and a discrete approximation of the gamma distribution. Queueing models and Markov process have been used for representing inventory systems, in special in those considering a single supplier. The queueing model found most often in the IRP literature is the M/M/1, in which the demands arrive according to a Poisson distribution and service times are modeled by an exponential probability distribution. For instance, Saffari et al. [50, 49] and Saffari and Haji [48] used this model to obtain reorder points and optimal replenishment quantities for several cases of an (s, q) inventory policy. In Schwarz et al. [53], the previous setting is extended 4
to consider several inventory policies. Finally, Kleywegt et al. [33, 32] and Larsen and Turkensteen [35] formulated the IRP as a Markov decision process. The objective is to determine a distribution policy that maximizes the expected discounted value (revenues minus costs) over an infinite time horizon. Other studies focus on the revision of inventory policies, i.e., determining an accurate methodology for analyzing and evaluating inventory costs by means different demand models. Axs¨ater [5] compares the performance of such systems with Poisson and compound Poisson distributions. The objective was to evaluate the total system costs for different inventory policies. Farvid and Rosling [23] observed that when the demand follows an idealized stochastic process, lead-times may be stochastic and still an optimality condition for the discounted (s, q) policy can be achieved. Regarding the lead time, its impact on the inventory cost was analyzed in a model for single product in continuous time by Song [55]. The variables of interest are the inventory level and the behavior of average long-term cost. Sapna Isotupa and Samanta [52] analyzed a lost sales (s, q) inventory policy with two types of customers and stochastic lead times. Demands from each type of customer arrives according to two independent Poisson processes and the lead times are modeled using Erlang distribution and constant lead times. Their numerical results for the total costs, illustrated the fact that there are situations where inventory rationing policies are better than treating both customers alike. 4. Optimization methods for the inventory and distribution Optimization methods for stochastic IRPs require information about the current and historical inventory levels, the behavior of the demand, the location of the nodes, the transportation costs, as well as the capacity and availability of vehicles and drivers. All this information is required to provide a solution and to properly compute its cost. Coordination between inventory and transport in this stochastic problem have been modeled in different ways in the literature. For example, one consists in adding inventory constraints to the transportation problem [34], while the other approach considers the vehicle as the “producer” of the inventory, modeling it as a variant of a production problem [47]. In both of these methods, the goal is to compute the marginal profit (revenue minus delivery costs) for each customer added to a route, and the delivery cost (routes, customers selection and the quantities allocated for each customer). The IRP works as an integration component, and some authors claim that it is the fundamental problem needed to operate a vendor-management inventory strategy [32, 21]. Overviews of the IRP can be found in Moin and Salhi [42], Bertazzi et al. [8] and Coelho et al. [20]. Heuristics and metaheuristics have been used to provide solutions to this complex problem, and evolutionary algorithms in particular are widely used. Simic and Simic [54] state that for complex optimization problems such as the IRP, hybrid methods with techniques such as artificial neural networks, genetic algorithms, tabu search, simulated annealing and evolutionary algorithms can be successfully applied. Some of 5
the techniques used to solve IRP are summarized next. Genetic algorithms have been employed by Christiansen et al. [15] and Liu and Lee [38], who clustered customers in geographical areas to serve them together. Local search operators were explored by Javid and Azad [30] and Qin et al. [46], who changed the delivery schedule for customers and adjusted the quantities deliveries accordingly. Li et al. [37], Liu and Lin [39] and Sajjadi and Cheraghi [51] used simulated annealing to integrate location decisions into the IRP. Adaptive large neighborhood search [19] and a hybrid of mathematical programming and local search [3] have also been used. Finally, exact methods relying on branch-and-cut [2, 16, 17] and branch-cut-and-price [22] have also been developed. An alternative to face very large complex problems is to decompose them. Using this idea, Archetti et al. [4] proposes a model that includes inventory control, routing and delivery scheduling. The model is solved by decomposing the problem into two phases, the first one creating a scheduling and the second one the designing of the routes. The second phase uses a VRP model with time windows. Variable neighborhood search is used to improve the solutions. In Christiansen et al. [15], the method is composed of two main components: a heuristic construction algorithm and a genetic algorithm. The construction algorithm builds a plan from scratch. It is deterministic, but has parameters that can be varied to produce different plans. The genetic algorithm is used to search for solutions that produce good plans by the construction heuristic. Another application of a genetic algorithms is the clustering of customers into m groups in accordance with the number of vehicles available. Cheng and Wang [13] exploits this clustering information, which is then passed to sub-problems optimizing its own routing sequence for replenishment. Metaheuristics such as simulated annealing and local search are used to evaluate and improve initial solutions. In Li et al. [36], Liu and Lin [39] and Sajjadi and Cheraghi [51] simulated annealing is used to improve initial solutions obtained from other heuristics and metaheuristics. In Qin et al. [46], local search methods are used to insert and remove new replenishment points into an existing schedule. Li et al. [36] focused the study on minimizing travel times in a context related to law regulations about hours of service. The problem addressed considers a single supplier, several customers, an homogeneous fleet and an estimation of a deterministic demand. To evaluate the performance of the system, a Lagrangian relaxation algorithm was used in order to obtain a lower bound for the solution of the problem. Compared to this, the tabu search algorithm proposed proves to be close to the lower limits of the problems for small to medium size instances. The model presented in Agra et al. [1] includes multiple products, multiple suppliers, multiple customers and a heterogeneous fleet. Three methods were developed, which include a rolling horizon (RH) algorithm, local branching (LB), and feasibility pump (FP). In the case of RH the planning horizon was decomposed into smaller time horizons. On the other hand, LB seeks for local optimal solutions by restricting the number of variables that can change their values, and FP seeks for initial feasible solution. The results show better solutions than those obtained with a single heuristic. When the IRP is defined with a stochastic demand, Bertazzi et al. [9] propose a 6
model consisting of one supplier and a set of customers. A hybrid algorithm is used to solve the problem. The estimated costs are obtained joining the exact solution of a mixed integer linear programming with the branch-and-cut heuristic. Another important variability term faced by an IRP model is the variability in the travel times, which requires extra work on non-deterministic and probabilistic approaches for some instances. In Reiman et al. [47], the travel time between customers is represented as a random variable. When dealing with complex problems, such as the IRP, it is common to place a set of instances or testing problems available for other researchers to compare their methods. In what follows we describe the existing benchmarks as well as their origins and sources. Papageorgiou et al. [44] creates a library composed of test instances for the maritime IRP, providing a framework with common characteristics for this type of problems. Resources are available online1 . In the specific case of road-based IRPs, there exist a few sets proposed by the researchers belonging to CIRRELT2 , SCL3 , OR@Brescia4 , and Logistics Management Department of Helmut-Schmidt-Universitat5 . Other resources can be found in the webpage of Y. Adulyasak6 . 5. Multiple Depot IRP In this section we briefly review the existing research on multi-depot IRPs (MDIRP). Our focus is on the most relevant elements for dynamic and stochastic IRPs. Considering that the use of clustering methods is important to solve the IRP problems as can be observed in the literature review presented in Section 4, we now mention some of the techniques applied for MDIRPs. Generally this problem is decomposed into its two natural components: (1) clustering of edges into feasible routes and (2) actual route construction, with possible feedback loops between the two stages. To bring an overview of the using of these techniques, we trough of the some implementations will describe by means recent articles how the clustering is applied. For instance, Luo and Chen [40] and Luo and Chen [41] implemented an algorithm that generates clusters randomly to perform the clustering analyses considering the depots as the centroids of the clusters for all the customers. Afterwards, they implemented the local depth search for every cluster, and then, the readjustment of the solutions was performed. In a next step, a new clustering analysis was performed to generate new clusters according to the best solution achieved by the preceding process. The improved path information was inherited to the new clusters, and local search for every cluster is used again iteratively. The process continues until the convergence criteria are satisfied. 1
http://mirplib.scl.gatech.edu/ http://www.leandro-coelho.com/instances/ 3 http://www.tli.gatech.edu/research/casestudies/irp2/ 4 https://sites.google.com/site/orbrescia/home 5 http://www.hsu-hh.de/logistik/ 6 https://sites.google.com /site/yossiriadulyasak/ 2
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Similar process was followed by Zeng et al. [66] and He et al. [29], who classified the customers in certain and uncertain assignment to a depot, according to the distances of that customers to depots. Their method creates an iterative modification of those assignments. When each customer corresponds to only one depot, the multi-depot vehicle routing problem is solved as a single depot vehicle routing problem for each depot in the system. Xu and Xiao [63] and Y¨ ucenur and Demirel [65] used a technique that allowed implementing a new type of geometric shape based genetic clustering algorithm which could be used effectively to route vehicles if the new shapes have the capability to adapt to the route shapes, resulting in the minimization of the routing cost. The GA is used to adaptively search for the attributes of a set of shapes (example circles) that clusters customers using the routing cost as the fitness value for the individual chromosomes. Wang [61] used a typical procedure which consist on the decomposition of a multi suppliers to a single supplier IRP problem. A heuristic was used to simplify the multidepot problem into a single depot problem. The maximal route is improved with the highest number of customers and similarly also, by exchanging customers between routes. Other techniques besides grouping have been used to deal with the MDIRP problem. In Stodola and Mazal [58] and Stodola and Mazal [57], a tactical model comprises the models of optimal supply distribution on the battlefield and of optimal reconnaissance by unmanned aerial vehicles used by the military. They used ant colony optimization algorithm with five special forms to select the supplier that should cluster the customers. These forms include selection of suppliers through a random manner, shortest distance, probabilities according the distance traveled so far and pheromone trails. Finally, Nananukul [43] illustrated how customers’ demands pattern and holding costs could affect their clustering decision. A basic model for clustering customers called multi-period clustering problem was introduced, taking into consideration the demand pattern and holding costs. In this method, an enhanced K-means algorithm is used to construct an initial solution. A novel feature of the algorithm is to create adaptive core clusters which are used in the clustering process instead of the original data points. The neighborhoods of the solution space consist of two types of moves: reassigning customers to clusters and rescheduling the delivery quantity from one period to another. You et al. [64] used clustering and location-based heuristics to group the customers into a number of small clusters and solved the routing problem within each cluster independently. By iteratively changing the customers in the clusters, they obtained a near-optimal solution within the required computational time. The clustering method was integrated into a multi-period two-stage stochastic mixed-integer nonlinear programming model that considered the uncertain demand as random variable. 6. Discussion In Table 1, some key elements have been identified and some methodological aspects are compared. The emphasis on the optimization topic is clear by the number of papers 8
we have identified in this research, and is justified by the need to obtain very good and near-optimal solutions for this practical problem. However, in order to be successful one needs to use the right tools in terms of mathematical modeling, heuristics, and solution algorithms used. Probabilistic methods used before optimization, such as queueing and forecasts, have been successfully used to reduce the uncertainty (e.g., on the demand) and can enhance the quality of the solution obtained later. The moment at which information is revealed determines the changes in the planning process. This characteristic is transversal to the whole planning process. The information that has been revealed period by period is used to replenish the customers according to the resources available at that moment. To the extent that demand information is obtained, it must be incorporated in the replenishment plan. However, it is important to take into account several methods and eventually to prioritize some particular customers due to capacity issues not allowing the decision maker to satisfy all the newly revealed demand. Finally, another important issue is that in the DSRIP, time and demand generally are considered dynamic variables, for this reason, one must consider them accordingly in the modeling phase of the problem. 7. Conclusions In this paper we have presented an analysis of the scientific literature on the IRP focusing on its stochastic and multi-depot aspects. We have first presented a review of information management and inventory policies techniques as key to IRP problems, since they determine not only the modeling of the problem, but also the availability of information, the objective function and the constraints used. We have shown that studying the behavior of the demand and the lead time are essential in order to achieve a useful representation of the system to take proper decisions. Thus, we have analyzed a special case of demand as random variable, and we provided a summary of probability distributions, queuing models and Markov decision process used by researchers. We have also reviewed and described the most used algorithms proposed for the IRP. We finalize this survey by highlighting some characteristics and solution methodologies for multi-depot problems, most of which borrow ideas from the vehicle routing literature and adapt them to consider inventory management. Acknowledgments This research was partly supported by grant 2014-05764 from the Canadian Natural Sciences and Engineering Research Council. This support is gratefully acknowledged. We thank an associated editor and two anonymous referees who provided valuable comments.
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Modeling stochastic problem
Leadtime
Table 1: An identification of key elements and their methodological aspects
Multi-depot multi-vehicle IRP
Inventory policy
Optimization methods Exact methods
VMI Strategy
Methodological Aspects
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Time dimension Problem may be open ended Future information Queueing considerations Focus on near-term events Information update mechanisms Resequencing and reassignment decisions Indefinite deferment mechanisms Faster computation times Dynamic inputs in the objective Dynamic inputs in timing constraints Change previously established plan of replenishment
Optimization methods Heuristic methods
Variability demand information
Flow of information between actors in SC
Evolution and quality of the information
Key Elements Analyzed
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[40] J. Luo and M.-R. Chen. Improved shuffled frog leaping algorithm and its multiphase model for multi-depot vehicle routing problem. Expert Systems with Applications, 41(5):2535–2545, 2014. [41] J. Luo and M.-R. Chen. Multi-phase modified shuffled frog leaping algorithm with extremal optimization for the MDVRP and the MDVRPTW. Computers and Industrial Engineering, 72(1):84–97, 2014. [42] N. H. Moin and S. Salhi. Inventory routing problems: A logistical overview. Journal of the Operational Research Society, 58(9):1185–1194, 2007. [43] N. Nananukul. Clustering model and algorithm for production inventory and distribution problem. Applied Mathematical Modelling, 37(24):9846–9857, 2013. [44] D. Papageorgiou, G. Nemhauser, J. Sokol, M.-S. Cheon, and A. Keha. MIRPLib – A library of maritime inventory routing problem instances: Survey, core model, and benchmark results. European Journal of Operational Research, 235(2):350– 366, 2013. [45] H.N. Psaraftis. Dynamic vehicle routing: Status and prospects. Annals of Operations Research, 61(1):143–164, 1995. [46] L. Qin, L. Miao, Q. Ruan, and Y. Zhang. A local search method for periodic inventory routing problem. Expert Systems with Applications, 41(2):765–778, 2014. [47] M. I. Reiman, R. Rubio, and L. M. Wein. Heavy traffic analysis of the dynamic stochastic inventory-routing problem. Transportation Science, 4(33):361– 380, 1999. [48] M. Saffari and R. Haji. Queueing system with inventory for two-echelon supply chain. In International Conference on Computers and Industrial Engineering, Troyes, pages 835–838. Troyes, 2009. [49] M. Saffari, R. Haji, and F. Hassanzadeh. A queueing system with inventory and mixed exponentially distributed lead times. International Journal of Advanced Manufacturing Technology, 53(9-12):1231–1237, 2011. [50] M. Saffari, S. Asmussen, and R. Haji. The M/M/1 queue with inventory, lost sale, and general lead times. Queueing Systems, 75(1):65–77, 2013. [51] S. R. Sajjadi and S. H. Cheraghi. Multi-products location-routing problem integrated with inventory under stochastic demand. International Journal of Industrial and Systems Engineering, 7(4):454–476, 2011. [52] K. P. Sapna Isotupa and S. K. Samanta. A continuous review (s, Q) inventory system with priority customers and arbitrarily distributed lead times. Mathematical and Computer Modelling, 57(5-6):1259–1269, 2013. 14
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Raúl F. Roldán, Rosa Basagoiti, Leandro C. Coelho
PII: DOI: Reference:
S1570-8683(16)30064-7 http://dx.doi.org/10.1016/j.jal.2016.11.010 JAL 441
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Journal of Applied Logic
Please cite this article in press as: R.F. Roldán et al., A survey on the inventory-routing problem with stochastic lead times and demands, J. Appl. Log. (2016), http://dx.doi.org/10.1016/j.jal.2016.11.010
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Highlights • We review of papers working with stochastic demand and stochastic lead times focusing on its stochastic and multi-depot aspects • We identify critical factors for the performance of many logistics activities and industries. • We have shown that studying the behavior of the demand and the lead time are essential in order to achieve a useful representation of the system to take proper decisions. • We highlight some characteristics and solution methodologies for multi-depot problems, most of which borrow ideas from the vehicle routing literature and adapt them to consider inventory management.
A survey on the inventory-routing problem with stochastic lead times and demands Ra´ ul F. Rold´ana,b,∗, Rosa Basagoitia , Leandro C. Coelhoc a
Electronics and Computing Department, Mondragon University, Goiru Kalea, 2, Arrasate 20500, Spain b Engineering Faculty,Compensar Unipanamericana Fundaci´ on Universitaria, Avenida (Calle) 32 No. 17 - 30, Bogot´ a D.C., Colombia c Interuniveristy Research Centre on Enterprise Networks, Logistics and Transportation (CIRRELT) and Facult´e des sciences de l’administration, Universit´e Laval, Qu´ebec, Canada
Abstract The integration of the different processes and players that compose the supply chain (SC) is essential to obtain a better coordination level. Inventory control and distribution management are the two processes that researchers have identified as the key to gain or lose in efficiency and effectiveness in the field of logistics, with a direct effect on the synchronization and overall performance of SCs. In practical situations demand is often not deterministic, and lead times are also variable, yielding a complex stochastic problem. In order to analyze the recent developments in the integration of these processes, this paper analyzes the state of the art of the information management in the SC, the relationship between inventory policies and available demand information, and the use of optimization methods to provide good solutions for the problem in single and multi depot versions. Keywords: stochastic demand, inventory-routing problem, stochastic lead time, inventory policy, survey. 1. Introduction A supply chain (SC) is defined as the system of organizations and flows of products, information and money spanning over the product’s entire life cycle, from conception to consumption and final disposal. It involves different actors such as suppliers, producers, distributors, transporters and customers, among others, all of them involved directly or indirectly in satisfying the request of a final client. Coordination and integration of SC activities are now recognized as critical to obtain competitive advantage [42]. Guasch and Kogan [27] observe that the logistics performance directly affect the cost ∗
Corresponding author. Email addresses: [email protected] (Ra´ ul F. Rold´an), [email protected] (Rosa Basagoiti), [email protected] (Leandro C. Coelho)
Preprint submitted to Journal of Applied Logic
November 9, 2016
of the products and hence, the overall performance of an industry. These authors compare two economic organizations in order to find the impact of logistics cost on the price of a product, and conclude that the costs associated with inventory management represent about 19% of the price of a product in countries with poor logistics systems, compared to 8% in countries with efficient logistics networks. Guasch [26] conclude that the reduction in logistics costs can be evidenced by two indicators, namely 1) the increase in the proportion of demand of a product, and 2) the increase in the proportion of employment that can be generated for some sectors of the economy. As a consequence, the increase in product price obstructs the competitiveness and complicates the maintenance of inventory. Logistics costs can be categorized into administrative, warehousing, inventory, transportation, and licenses costs. Guasch [26] identifies that more than 69% of these are directly related to transportation and inventory. Inventory management alone represents a proportion of the net operating assets of approximately 37% in production, 62% in distribution, and 56% in retail [59]. In addition, it is important to note that inventory control has to balance conflicting objectives due to two main reasons: 1) economies of scale and purchasing large batch sizes, and 2) uncertainty in supply and demand, which inevitably create safety stocks. In this paper, our goal is to provide an overview of the literature on problems related to the coordination of the inventory and its distribution. We provide a thorough review of papers dealing with stochastic demand and stochastic lead times, as these are the factors identified as critical for the performance of many logistics activities and industries. The remainder of this paper is organized as follows. Section 2 reviews the information management between different actors in the SC, since this will determine the evolution and quality of the information. It also analyzes inventory policies and their relation to the demand, in order to properly manage inventory levels. Stochastic demand and lead times are studied in Section 3. Several optimization methods are revised in Section 4. In Section 5, we present some versionof the inventory-routing problem (IRP) with special attention to multi-depot IRPs (MDIRP). In Section 6, we discuss the key elements presented with some methodological aspects. Our conclusions and final remarks are presented in Section 7. 2. Information management and the inventory policies The management and coordination of the information between the processes in the SC is crucial in order to make good decisions. In Gavirneni et al. [24], the flow of information between a supplier and a customer was analyzed and three situations compared: 1) no information for the supplier before the request arrives, 2) the supplier knows the policies that the customer uses as well as the final distribution processes, and 3) the supplier has all the information about the state of the customer’s inventory levels and decision process. The cost analysis indicates that the second configuration reduces 50% of the cost compared to the first one. When the second and the third 2
cases are compared, the cost reductions vary between 1% and 35%. According to Psaraftis [45], the four dimensions of the information are evolution, quality, availability and processing. Just the first and second components add randomness and stochasticity. The evolution that the information experiences over time highlights that it can change during the execution of the preliminary planning, and its quality reflects the possibility of the existence of some amount of uncertainty and asymmetric information between actors or entities. Demand information experiences variability and amplification along the SC. This is known as the bullwhip effect, which has been widely studied. For instance, Giard and Sali [25] argued that the bullwhip effect is the main reason for the loss of efficiency and efficacy in SCs. Chopra and Meindl [14] state that it can be damped by an improvement on the operative performance and the design of rationing schemes in the products that present shortage. In this way, some of the proposals include reducing the replenishment lead-time, reducing the lot size, taking into account historical data, and the interchange of information to limit the variability [18]. Accurate and timely information management can help improving the performance of the SC. According to Wagner [60], although the SC involves several activities such as purchase, production, location, marketing, inventory control and distribution, the foundations of integration in the SC are in the last two, which are focused in the efficiency of the channel and the coordination of the performance of the individual entities for the satisfaction of the final client. However, uncertainty plays a crucial role, making the environment less predictable. One way to reduce the effect of variability in the information is to assign the responsibility for management between activities to a single actor. This is achieved with strategies such as the vendor-managed inventory (VMI), which requires information to be shared. According to Beyer et al. [11], the characteristics of inventory control systems are determined by the efficiency measures with which to evaluate the performance over time, as well as by factors taken into account when modeling the system, such as cost structure, demand, lead time, review time, excess demand, deteriorating inventory, and ad hoc constraints. With respect to the demand, its analysis and estimation can reduce the risk of surplus or shortages. In this context, two issues become relevant: when and how much to replenish. Demand characteristics can be broadly classified into certain and uncertain. When demand is deterministic and all its parameters are known, the economic order quantity (EOQ) [28] and its variations are used to solve the problem of replenishment timing and quantities. When uncertainty is significant, it must be explicitly taken into account, and modeling alternatives include one-time decision models, continuous decision models, and periodical decision models. When decisions are taken either continuously or periodically it is necessary to establish a policy for ordering. The choice of which inventory policy to apply largely influences the cost of the optimization process. Typically, three key parameters are used to control the inventory policy: when replenish, how much to replenish, and how often the inventory level is reviewed. For the periodic review inventory system, Wensing [62] describes three 3
policies. The first is the order-up-to level (OU) which refers to a (t, S) system. Here, in each period t, the quantity delivered is that to fill the inventory capacity up to S. Other policies include the (t, s, S) and the (t, s, q). In the former, the customer is served if the inventory level is less than s. In the latter, the replenishment level q is flexible but bounded by the storage capacity. These policies should be articulated with strategies for customer selection, because sometimes it is not possible to serve all customers due to vehicle capacities, and in such cases, it is necessary to prioritize them. 3. Stochastic demand and lead time modeling Axs¨ater [6] describe that stochastic demands can be modeled as continuous variables if their average is big enough; however, for small stochastic demands, it is easier to model them using discrete models. Under the assumption that individual demand events occur independently, two general modeling techniques arise. The first option is to use the Poisson probability distribution, which is appropriate when the expected demand in a time interval is small; the second option is to use normal probability distribution when the expected demand is large enough [31]. Some studies considering the demand as a random variable are summarized below. In Chao and Zhou [12], an inventory system with continuous review in infinite horizon is considered. The sales price and inventory replenishment are determined simultaneously. The demand process is modeled by a Poisson probability distribution, with an arrival rate dependent on the price. Another model for an inventory system with two suppliers is proposed by Song and Zipkin [56], where one supplier responds best to demand than the other. One of the nodes has a capacity limit, meaning that it can only satisfy the demand up to that point. Both suppliers work with constant lead times, and the demand is modeled as a continuous time variable following a Poisson probability distribution with linear ordering costs. Bertazzi et al. [10] study a case in which the supplier has a limited production capacity and the demand from the customers is stochastic. These demands are modeled as discrete random variables and deliveries are performed using transportation procurement. In Berling and Marklund [7], the authors assume a normally distributed customer demand, and lead times are estimated based on the magnitude of the demand. A model that approximates the true distribution of the lead-times by means three standard distributions is used. The employed distributions are the negative binomial distribution, a discrete approximation of the normal distribution, and a discrete approximation of the gamma distribution. Queueing models and Markov process have been used for representing inventory systems, in special in those considering a single supplier. The queueing model found most often in the IRP literature is the M/M/1, in which the demands arrive according to a Poisson distribution and service times are modeled by an exponential probability distribution. For instance, Saffari et al. [50, 49] and Saffari and Haji [48] used this model to obtain reorder points and optimal replenishment quantities for several cases of an (s, q) inventory policy. In Schwarz et al. [53], the previous setting is extended 4
to consider several inventory policies. Finally, Kleywegt et al. [33, 32] and Larsen and Turkensteen [35] formulated the IRP as a Markov decision process. The objective is to determine a distribution policy that maximizes the expected discounted value (revenues minus costs) over an infinite time horizon. Other studies focus on the revision of inventory policies, i.e., determining an accurate methodology for analyzing and evaluating inventory costs by means different demand models. Axs¨ater [5] compares the performance of such systems with Poisson and compound Poisson distributions. The objective was to evaluate the total system costs for different inventory policies. Farvid and Rosling [23] observed that when the demand follows an idealized stochastic process, lead-times may be stochastic and still an optimality condition for the discounted (s, q) policy can be achieved. Regarding the lead time, its impact on the inventory cost was analyzed in a model for single product in continuous time by Song [55]. The variables of interest are the inventory level and the behavior of average long-term cost. Sapna Isotupa and Samanta [52] analyzed a lost sales (s, q) inventory policy with two types of customers and stochastic lead times. Demands from each type of customer arrives according to two independent Poisson processes and the lead times are modeled using Erlang distribution and constant lead times. Their numerical results for the total costs, illustrated the fact that there are situations where inventory rationing policies are better than treating both customers alike. 4. Optimization methods for the inventory and distribution Optimization methods for stochastic IRPs require information about the current and historical inventory levels, the behavior of the demand, the location of the nodes, the transportation costs, as well as the capacity and availability of vehicles and drivers. All this information is required to provide a solution and to properly compute its cost. Coordination between inventory and transport in this stochastic problem have been modeled in different ways in the literature. For example, one consists in adding inventory constraints to the transportation problem [34], while the other approach considers the vehicle as the “producer” of the inventory, modeling it as a variant of a production problem [47]. In both of these methods, the goal is to compute the marginal profit (revenue minus delivery costs) for each customer added to a route, and the delivery cost (routes, customers selection and the quantities allocated for each customer). The IRP works as an integration component, and some authors claim that it is the fundamental problem needed to operate a vendor-management inventory strategy [32, 21]. Overviews of the IRP can be found in Moin and Salhi [42], Bertazzi et al. [8] and Coelho et al. [20]. Heuristics and metaheuristics have been used to provide solutions to this complex problem, and evolutionary algorithms in particular are widely used. Simic and Simic [54] state that for complex optimization problems such as the IRP, hybrid methods with techniques such as artificial neural networks, genetic algorithms, tabu search, simulated annealing and evolutionary algorithms can be successfully applied. Some of 5
the techniques used to solve IRP are summarized next. Genetic algorithms have been employed by Christiansen et al. [15] and Liu and Lee [38], who clustered customers in geographical areas to serve them together. Local search operators were explored by Javid and Azad [30] and Qin et al. [46], who changed the delivery schedule for customers and adjusted the quantities deliveries accordingly. Li et al. [37], Liu and Lin [39] and Sajjadi and Cheraghi [51] used simulated annealing to integrate location decisions into the IRP. Adaptive large neighborhood search [19] and a hybrid of mathematical programming and local search [3] have also been used. Finally, exact methods relying on branch-and-cut [2, 16, 17] and branch-cut-and-price [22] have also been developed. An alternative to face very large complex problems is to decompose them. Using this idea, Archetti et al. [4] proposes a model that includes inventory control, routing and delivery scheduling. The model is solved by decomposing the problem into two phases, the first one creating a scheduling and the second one the designing of the routes. The second phase uses a VRP model with time windows. Variable neighborhood search is used to improve the solutions. In Christiansen et al. [15], the method is composed of two main components: a heuristic construction algorithm and a genetic algorithm. The construction algorithm builds a plan from scratch. It is deterministic, but has parameters that can be varied to produce different plans. The genetic algorithm is used to search for solutions that produce good plans by the construction heuristic. Another application of a genetic algorithms is the clustering of customers into m groups in accordance with the number of vehicles available. Cheng and Wang [13] exploits this clustering information, which is then passed to sub-problems optimizing its own routing sequence for replenishment. Metaheuristics such as simulated annealing and local search are used to evaluate and improve initial solutions. In Li et al. [36], Liu and Lin [39] and Sajjadi and Cheraghi [51] simulated annealing is used to improve initial solutions obtained from other heuristics and metaheuristics. In Qin et al. [46], local search methods are used to insert and remove new replenishment points into an existing schedule. Li et al. [36] focused the study on minimizing travel times in a context related to law regulations about hours of service. The problem addressed considers a single supplier, several customers, an homogeneous fleet and an estimation of a deterministic demand. To evaluate the performance of the system, a Lagrangian relaxation algorithm was used in order to obtain a lower bound for the solution of the problem. Compared to this, the tabu search algorithm proposed proves to be close to the lower limits of the problems for small to medium size instances. The model presented in Agra et al. [1] includes multiple products, multiple suppliers, multiple customers and a heterogeneous fleet. Three methods were developed, which include a rolling horizon (RH) algorithm, local branching (LB), and feasibility pump (FP). In the case of RH the planning horizon was decomposed into smaller time horizons. On the other hand, LB seeks for local optimal solutions by restricting the number of variables that can change their values, and FP seeks for initial feasible solution. The results show better solutions than those obtained with a single heuristic. When the IRP is defined with a stochastic demand, Bertazzi et al. [9] propose a 6
model consisting of one supplier and a set of customers. A hybrid algorithm is used to solve the problem. The estimated costs are obtained joining the exact solution of a mixed integer linear programming with the branch-and-cut heuristic. Another important variability term faced by an IRP model is the variability in the travel times, which requires extra work on non-deterministic and probabilistic approaches for some instances. In Reiman et al. [47], the travel time between customers is represented as a random variable. When dealing with complex problems, such as the IRP, it is common to place a set of instances or testing problems available for other researchers to compare their methods. In what follows we describe the existing benchmarks as well as their origins and sources. Papageorgiou et al. [44] creates a library composed of test instances for the maritime IRP, providing a framework with common characteristics for this type of problems. Resources are available online1 . In the specific case of road-based IRPs, there exist a few sets proposed by the researchers belonging to CIRRELT2 , SCL3 , OR@Brescia4 , and Logistics Management Department of Helmut-Schmidt-Universitat5 . Other resources can be found in the webpage of Y. Adulyasak6 . 5. Multiple Depot IRP In this section we briefly review the existing research on multi-depot IRPs (MDIRP). Our focus is on the most relevant elements for dynamic and stochastic IRPs. Considering that the use of clustering methods is important to solve the IRP problems as can be observed in the literature review presented in Section 4, we now mention some of the techniques applied for MDIRPs. Generally this problem is decomposed into its two natural components: (1) clustering of edges into feasible routes and (2) actual route construction, with possible feedback loops between the two stages. To bring an overview of the using of these techniques, we trough of the some implementations will describe by means recent articles how the clustering is applied. For instance, Luo and Chen [40] and Luo and Chen [41] implemented an algorithm that generates clusters randomly to perform the clustering analyses considering the depots as the centroids of the clusters for all the customers. Afterwards, they implemented the local depth search for every cluster, and then, the readjustment of the solutions was performed. In a next step, a new clustering analysis was performed to generate new clusters according to the best solution achieved by the preceding process. The improved path information was inherited to the new clusters, and local search for every cluster is used again iteratively. The process continues until the convergence criteria are satisfied. 1
http://mirplib.scl.gatech.edu/ http://www.leandro-coelho.com/instances/ 3 http://www.tli.gatech.edu/research/casestudies/irp2/ 4 https://sites.google.com/site/orbrescia/home 5 http://www.hsu-hh.de/logistik/ 6 https://sites.google.com /site/yossiriadulyasak/ 2
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Similar process was followed by Zeng et al. [66] and He et al. [29], who classified the customers in certain and uncertain assignment to a depot, according to the distances of that customers to depots. Their method creates an iterative modification of those assignments. When each customer corresponds to only one depot, the multi-depot vehicle routing problem is solved as a single depot vehicle routing problem for each depot in the system. Xu and Xiao [63] and Y¨ ucenur and Demirel [65] used a technique that allowed implementing a new type of geometric shape based genetic clustering algorithm which could be used effectively to route vehicles if the new shapes have the capability to adapt to the route shapes, resulting in the minimization of the routing cost. The GA is used to adaptively search for the attributes of a set of shapes (example circles) that clusters customers using the routing cost as the fitness value for the individual chromosomes. Wang [61] used a typical procedure which consist on the decomposition of a multi suppliers to a single supplier IRP problem. A heuristic was used to simplify the multidepot problem into a single depot problem. The maximal route is improved with the highest number of customers and similarly also, by exchanging customers between routes. Other techniques besides grouping have been used to deal with the MDIRP problem. In Stodola and Mazal [58] and Stodola and Mazal [57], a tactical model comprises the models of optimal supply distribution on the battlefield and of optimal reconnaissance by unmanned aerial vehicles used by the military. They used ant colony optimization algorithm with five special forms to select the supplier that should cluster the customers. These forms include selection of suppliers through a random manner, shortest distance, probabilities according the distance traveled so far and pheromone trails. Finally, Nananukul [43] illustrated how customers’ demands pattern and holding costs could affect their clustering decision. A basic model for clustering customers called multi-period clustering problem was introduced, taking into consideration the demand pattern and holding costs. In this method, an enhanced K-means algorithm is used to construct an initial solution. A novel feature of the algorithm is to create adaptive core clusters which are used in the clustering process instead of the original data points. The neighborhoods of the solution space consist of two types of moves: reassigning customers to clusters and rescheduling the delivery quantity from one period to another. You et al. [64] used clustering and location-based heuristics to group the customers into a number of small clusters and solved the routing problem within each cluster independently. By iteratively changing the customers in the clusters, they obtained a near-optimal solution within the required computational time. The clustering method was integrated into a multi-period two-stage stochastic mixed-integer nonlinear programming model that considered the uncertain demand as random variable. 6. Discussion In Table 1, some key elements have been identified and some methodological aspects are compared. The emphasis on the optimization topic is clear by the number of papers 8
we have identified in this research, and is justified by the need to obtain very good and near-optimal solutions for this practical problem. However, in order to be successful one needs to use the right tools in terms of mathematical modeling, heuristics, and solution algorithms used. Probabilistic methods used before optimization, such as queueing and forecasts, have been successfully used to reduce the uncertainty (e.g., on the demand) and can enhance the quality of the solution obtained later. The moment at which information is revealed determines the changes in the planning process. This characteristic is transversal to the whole planning process. The information that has been revealed period by period is used to replenish the customers according to the resources available at that moment. To the extent that demand information is obtained, it must be incorporated in the replenishment plan. However, it is important to take into account several methods and eventually to prioritize some particular customers due to capacity issues not allowing the decision maker to satisfy all the newly revealed demand. Finally, another important issue is that in the DSRIP, time and demand generally are considered dynamic variables, for this reason, one must consider them accordingly in the modeling phase of the problem. 7. Conclusions In this paper we have presented an analysis of the scientific literature on the IRP focusing on its stochastic and multi-depot aspects. We have first presented a review of information management and inventory policies techniques as key to IRP problems, since they determine not only the modeling of the problem, but also the availability of information, the objective function and the constraints used. We have shown that studying the behavior of the demand and the lead time are essential in order to achieve a useful representation of the system to take proper decisions. Thus, we have analyzed a special case of demand as random variable, and we provided a summary of probability distributions, queuing models and Markov decision process used by researchers. We have also reviewed and described the most used algorithms proposed for the IRP. We finalize this survey by highlighting some characteristics and solution methodologies for multi-depot problems, most of which borrow ideas from the vehicle routing literature and adapt them to consider inventory management. Acknowledgments This research was partly supported by grant 2014-05764 from the Canadian Natural Sciences and Engineering Research Council. This support is gratefully acknowledged. We thank an associated editor and two anonymous referees who provided valuable comments.
9
Modeling stochastic problem
Leadtime
Table 1: An identification of key elements and their methodological aspects
Multi-depot multi-vehicle IRP
Inventory policy
Optimization methods Exact methods
VMI Strategy
Methodological Aspects
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Time dimension Problem may be open ended Future information Queueing considerations Focus on near-term events Information update mechanisms Resequencing and reassignment decisions Indefinite deferment mechanisms Faster computation times Dynamic inputs in the objective Dynamic inputs in timing constraints Change previously established plan of replenishment
Optimization methods Heuristic methods
Variability demand information
Flow of information between actors in SC
Evolution and quality of the information
Key Elements Analyzed
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