Simheuristics

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A simheuristic algorithm for the Single-Period Stochastic Inventory-Routing Problem with stock-outs

Angel A. Juan a,⇑, Scott E. Grasman b, Jose Caceres-Cruz a,1, Tolga Bektas� c a Department of Computer Science, Multimedia, and Telecommunication, IN3-Open University of Catalonia, 08018 Barcelona, Spain b Department of Industrial and Systems Engineering, Rochester Institute of Technology, USA c Southampton Management School and Centre for Operational Research, Management Science and Information Systems (CORMSIS), University of Southampton, UK

a r t i c l e i n f o

Article history: Available online 7 December 2013

Keywords: Inventory-Routing Problem Stochastic demands Stock-outs Simulation–optimization Simheuristics Randomized heuristics

a b s t r a c t

This paper describes a ‘simheuristic’ algorithm – one which combines simulation with heuristics – for solving a stochastic variant of the well-known Inventory-Routing Problem. The variant discussed here is integrated by a vehicle routing problem and several inventory problems characterized by stochastic demands. Initial stock levels and potential stock-outs are also considered, as well as a set of alternative refill policies for each retail center. The goal is to find the personalized refill policies and associated routing plan that minimize, at each single period, the expected total costs of the system, i.e., the sum of inventory and routing costs. After motivating it, a detailed description of the problem is provided. Then, a review of the related literature is performed and our simulation–optimization approach is introduced. The paper presents a set of numerical experiments comparing the proposed method against different refill strategies and discusses how total costs evolve as the level of system uncertainty and the inventory-holding costs per unit are varied.

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1. Introduction

One of the most important paradigms in supply chain management is to move from sequential decision making toward integrated decision making, where all parties in the supply chain determine the best policy for the entire system. This is in contrast to sequentially optimized decisions in supply chains, where each party determines its own course of action inde- pendently of other actors in the chain. Inventory and transportation systems are good examples of sequential decision mak- ing. Chien et al. [22] discuss the importance of considering both the inventory allocation and the vehicle routing when making logistical decisions. Driven by business practices, such as vendor managed inventory (VMI), integrated inventory and transportation systems have received much attention [39]. VMI is a supply chain centralized control initiative where the supplier is authorized to manage inventories of the retailers and to make decisions such as when and how much inven- tory to ship to the retailer. VMI is seen as an effective means of managing inventory through leveraging advanced technology and trading-partner relationships to enable the flow of information and inventory throughout the entire supply chain.

Despite the potential benefits, only a relatively small number of works have analytically approached the issue of integrat- ing inventory and transportation (vehicle routing) decisions, probably due to the inherent complexity resulting from the integration. This issue is known in the literature as the Inventory-Routing Problem or IRP [17]. Formulations with exact or approximate solution procedures are still needed to assist with the widespread adoption of VMI and use of synchronized

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⇑ Corresponding author. Tel.: +34 933263627. E-mail addresses: [email protected] (A.A. Juan), [email protected] (S.E. Grasman), [email protected] (J. Caceres-Cruz), [email protected] (T. Bektas�).

1 Tel.: +34 933263627.

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inventory and transportation systems. In this paper, we consider a single-period IRP consisting of multiple retail centers with stochastic demands and a single distribution depot. In our modeling, potential stock-outs are taken into account as final demands at the retail centers are assumed to be random variables. In a decentralized version of this problem each retailer would utilize the inventory (refill) policy that either minimizes its own expected costs or achieves a prescribed service level. Supply requests would then be transferred to the distribution depot, so that it can design the corresponding delivery routes. On the contrary, in the centralized version that we are addressing, no assumption is made about the replenishment policy at an individual retailer. The distribution depot will analyze the inventory position of the retailers and make joint refill and routing decisions aimed at minimizing the total cost to the system.

Another aspect to note is that most of the existing literature has considered the IRP as a long-term, multi-period problem [16]. This is especially the case when the final demands at the retail centers are assumed to be deterministic. However, research is still needed into the study of the single-period problem, particularly in scenarios characterized by the existence of: (a) information and communication tools, which are able to efficiently track and report retailers’ stock levels at the end of each period and (b) random demands with a high variability, which make it difficult to forecast future inventory levels. Under these conditions – which seem common in real-life IRP applications – long-run planning could be a much more inefficient policy than just solving the problem with updated data at the end of each period. Fig. 1 illustrates the different agents associated with the Stochastic IRP.

Each retail center (RC) holds inventory for a single product, which is managed by the central depot. For each RC, the inventory level at the end of a period is given by the difference between: (a) the initial stock level plus the quantity received from the depot at the beginning of the period and (b) the end-client demands during that period. End-client demands are stochastic in nature, but we assume that, for each RC, it is possible to use historical data to model demands through a the- oretical or empirical probability distribution. We make no particular assumptions made on the type of distribution used to model these demands as long as it has an associated mean value. At the end of each period there may be costs associated with either inventory holding or inventory stock-outs. These costs should be incorporated into the decision-making process and are therefore added to the distribution or routing costs, which are usually based on traveling distances or times. At the end of each period, inventory levels are registered by the RC and updated by the central depot, so that a new routing strategy is defined for the new period taking into account the current inventory levels and the probability distributions representing end-client demands for that period. Our goal is then to find the personalized refill policies and an associated routing plan that minimize, in a given period, the expected total costs of the system.

In order to solve the aforementioned IRP variant, we propose what we refer to as a ‘simheuristic’ algorithm, one which combines simulation with a randomized heuristic for optimizing system performance. This can be seen as a special case of a simulation–optimization approach [3]. The algorithm initially makes use of Monte Carlo simulation to estimate the expected inventory costs associated with each RC-policy combination. Next, it employs a fast routing heuristic to compute the total costs – inventory plus routing – associated with several refill strategies. The best of these strategies is then utilized as an initial base solution in a multi-start randomized heuristic. After randomly sorting the RCs, this heuristic iteratively improves the base solution by trying different refill policies for each RC and selecting the one with the lowest total costs. Our methodology has similarities with some previous work, especially with those considering stochastic demands, stock-outs, and rollout periods. These works will be discussed in detail in the next section.

Fig. 1. Different agents of the Stochastic Inventory-Routing Problem.

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The remainder of the article is structured as follows: Section 2 presents a review of the relevant literature; Section 3 describes in more detail the Stochastic IRP variant we analyze; Section 4 gives an explanation of our approach; Sections 5–7 present and discuss some numerical experiments that serve to both illustrate and validate our approach; finally, Section 8 summarizes the main contributions and results of this work.

2. Literature review

Several studies have combined simulation and optimization approaches to find solutions to complex optimization prob- lems [47,42,31]. The supply chain process has been a popular target for this type of techniques. The work of Eskandari et al. [27] focuses on channel coordination of the supply chain process and highlights the sensitive influence of stochastic demands on supplier and retailer perspectives. Then, a decision support tool based on simulation–optimization is proposed. The authors state that, unlike traditional mathematical techniques, the use of simulation–optimization modelling helps to deal with more realistic–complex scenarios. For instance, the scheduling problem in complex assembly lines is studied in Angelidis et al. [4], where the authors propose a decentralized heuristic based on simulation. Also, the particular inventory problem has been addressed using simulation–optimization techniques by Alizadeh et al. [2], which use models with sto- chastic lead times, deteriorating items, and Poisson demands, with a focus on minimizing long-run total expected costs allowing shortages. For this, three stochastic parameters are included in their simulation model: items life time, demands, and lead time.

In the past years, several approaches have been proposed for different variants of the Inventory-Routing Problem. The main factors to take into account when classifying the different works are: (a) whether they consider deterministic or stochastic demands; (b) whether they consider single- or multiple-periods (including an infinite horizon); (c) whether they allow inventory shortages or not; (d) whether they consider single or multiple products; (e) whether they use the same refill policy for all nodes or personalized replenishment policies for each node; and (f) whether they use exact or approximate methods to solve the problem. We have divided our literature review according to the first – and probably most relevant – criteria, i.e., whether the demands are considered to have a deterministic or a stochastic nature.

2.1. The IRP with deterministic demands

Regarding the IRP with deterministic demands, various modeling and solution approaches have been proposed, the majority of which are summarized in Bertazzi et al. [12]. Some use integer or dynamic programming approaches, e.g., Chien et al. [22], Campbell et al. [17], and Campbell et al. [16], while others utilize a heuristic or metaheuristic approach, e.g., Anily and Federgruen [5], Anily and Federgruen [6], Bramel and Simchi-Levi [15], Chan et al. [21], Bertazzi et al. [13], and Cousineau-Ouimet [25]. Other sequential approaches include Campbell and Savelsbergh [18], which presents a two-phase approach including support of GRASP-like randomization ‘‘as a powerful tool to improve the performance of insertion heuristics’’. Campbell and Savelsbergh [19] and Campbell and Savelsbergh [20] consider extensions.

2.2. The IRP with stochastic demands

One of the first works on the IRP with stochastic demands is Federgruen and Zipkin [28], which addresses the single- period combined problem of ‘‘allocating a scarce resource available at some central depot among several locations, each experiencing a random demand pattern, and planning deliveries using a fleet of vehicles’’. Transportation, holding and shortage costs are considered by the authors as ‘‘an extension of the standard vehicle routing problem’’. They provide several examples of potential applications, including: (a) deliveries of fuel oil to automotive service stations; (b) periodic replenishment of gas tanks at customer locations; or (c) coordinated allocation and supply to various locations of a perishable product such as blood. They propose a mathematical model and design a modified interchange heuristic, as well as an exact algorithm, to solve some randomly generated instances with up to 75 nodes. In Federgruen and Zipkin [29] the authors prove the fast convergence to ‘good’ solutions, under standard assumptions of (s, S) policies, using a pol- icy-iteration algorithm and tested their approach with 4 sets of 192 instances varying the mean and variance of demand distribution – one for each set. Good results are obtained in a reasonable computing time. Trudeau and Dror [46] address a multi-period version of the IRP with stochastic demands in which stock-outs are also allowed. In their computational experiments they consider 12 weekly periods and 2077 customers, making use of simulation to randomly generate demands. It is assumed that each customer’s random demand is not reported until the customer is visited by a vehicle, which leads to the possibility of route failures. Another interesting application is found in Bard et al. [9], where the authors study the IRP with satellite facilities – depots geographically scattered throughout the service area, which permit drivers to refill their vehicles during a shift. The authors use a randomized version of the classical Clarke & Wright Savings (CWS) heuristic [23] to solve routing instances with up to 500 nodes in about two hours. They show that this randomized heuristic outperforms other algorithms, including GRASP.

Barnes-Schuster and Bassok [10] address an infinite-horizon scenario in which demands are stochastic. From the compu- tational results they conclude that direct shipping (one truck delivering one retailer and then returning to the depot) can be a simple yet effective strategy whenever the capacity of the truck is close to the customer’s average demand. Reiman et al. [44]

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consider an IRP with stochastic demands and one vehicle covering a region composed of several customers, i.e., they assume that a previous process has been carried out in which customers have been assigned to different regions, each region covered by a single vehicle. Then they compare three strategies: the first based on direct shipping, the second based on a pre- specified tour (i.e., a Traveling Salesman Problem), and the third based on the dynamic choice between the TSP and direct shipping options. They conclude that the direct shipping strategy is the preferred strategy in most situations. Gaur and Fisher [30] describe a similar application with stochastic demands and a heterogeneous fleet. In order to simplify their problem they split the set of customers into several disjoint regions. Originally, only partitions with at most two customers are allowed, but once the partitions are created they try to improve them using a heuristic. However, they do not really consider inventory costs since in their case the relevant costs are routing costs. Another real-life IRP application is presented by Custodio and Oliveira [26]. After discussing the different strategies commonly employed in the literature (fixed partition, direct shipping, and ratios of integers), they propose a classical heuristic for solving the case study. In Yu et al. [48], the authors analyze the multi-period Stochastic IRP with split delivery. Their approach aims at transforming the stochastic model into a deterministic one and then using Lagrangian relaxation to decompose this latter model into inventory and routing subproblems. Using this approach they solve randomly generated instances with up to 100 nodes in reasonable computing times. Jarugumilli et al. [36] make use of a modified version of the A⁄ algorithm to solve the Stochastic IRP with a single vehicle, while Jarugumilli and Grasman [35], extends the problem to incorporate real-time information.

Berman and Larson [11] focus on the problem associated with the distribution of industrial gases to replenish customer tanks. They model customers’ demands as random processes and propose four dynamic-programming algorithms for solving the associated problem. In Kleywegt et al. [40] the authors address the IRP with stochastic demands over an infinite horizon. They formulate the aforementioned problem as a discrete time Markov decision process and propose several approximation methods to solve it. Kleywegt et al. [41] extend their previous work with a Markov process considering multiple resources for improving safety and reduce contamination between products on transportation and storage phases. Then, approximate methods are proposed to solve instances with up to 20 nodes in reasonable computing times. This is one of the few articles that provides complete information (e.g., nodes coordinates) on the tested instances. Adelman [1] uses linear programming and Markov processes for solving the IRP with stochastic demands and infinite horizon. Unfortunately, the largest instances included in the experimental section – which were randomly generated – contain no more than 40 customers.

The work of Jaillet et al. [34] has noticeable similarities with our IRP model: these authors support the convenience of considering a short-time rolling horizon framework when dealing with the Stochastic IRP. In their words: ‘‘For typical IRPs, a customer’s consumption rate is difficult to predict with certainty and can only be represented at best by a random variable with known probability distribution. Planning the entire annual distribution scheme in advance would, however, be unreliable and prone to many needed adjustments’’. Also, they assume the possibility of stock-outs; in particular ‘‘. . .as soon as the [customer’s] tank becomes empty, an immediate, and thus costly, special delivery is made’’. However, they also assume that the central depot cannot trace the inventory levels and that these are only revealed after each customer is visited by a truck. Thus, delivery strategies based upon end-of-period inventory levels are not considered in their work, while they are a fundamental part of our approach. They also propose several replenishment strategies for a finite horizon.

Probably the works most closely related to our work are those of Hvattum et al. [33] and Bertazzi et al. [14]. Hvattum et al. [33] address the Stochastic IRP with an infinite horizon as a Markov process. They formulate a scenario tree in order to exam- ine a finite horizon as a good approximation to the infinite horizon model. Again, since solving the Markov process is imprac- tical for all but the smallest instances, they employ a GRASP heuristic. Finally, Bertazzi et al. [14] undertake a Stochastic IRP with stock-outs and a finite horizon. They assume an order-up-to level policy, i.e., ‘‘the quantity sent to each retailer is such that its inventory level reaches a pre-determined level whenever the retailer is served’’. They present a dynamic program- ming model and propose a hybrid rollout algorithm. In order to validate their approach they use a randomly generated set of instances with up to 50 nodes and 6 periods.

Our approach shows some significant differences and original contributions: (a) we consider several replenishment pol- icies – personalized for each retail center – instead of just an order-to-level policy; (b) we use a simulation–optimization approach, which allows us to employ any probability distribution to model stochastic demands at each retail center and also to consider possible stock-outs; (c) we promote the use of multi-start randomized heuristics, which can be run in a parallel- computing environment if ‘real-time’ solutions are needed for large-scale problems; (d) we analyze how total costs evolve as different levels of system uncertainty (i.e., the variance of the demands) and inventory-holding costs per unit are considered; and (e) we propose a completely described set of instances, which can be employed by other researchers as well-defined benchmarks.

3. A formal description of the Stochastic IRP

The single-period Stochastic IRP that is considered in this paper can formally be described as follows. The problem is defined on a complete and undirected graph G = (V, A), where V = V� U {0}, node 0 is the depot, V� = {1, 2, . . . , n} is the set of RC nodes, and A = {(i, j) | i e V, j e V} is the set of arcs connecting those nodes. The parameters and assumptions of the problem are given as follows:

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� For each i e V⁄, both the current inventory level, Li P 0, as well as the maximum allowable inventory level, bLi > 0, are known. � Each i e V⁄ has to serve several customers for whom the aggregated demand is a random variable, Di P 0, following a

known probability distribution with E[Di] = di > 0. � A fleet of k homogeneous vehicles is used to perform the routing, each vehicle with a given maximum capacity Q > 0. It

will be assumed that Q P Di for each i e V ⁄.

� For each (i, j) e A, the cost of traveling from node i to node j is known and is shown by cij > 0. A symmetric cost matrix is assumed, i.e., cij = cji.

A decentralized-policy would dictate that inventory (refill) decisions would be made first by each RC in order to minimize their own expected inventory costs. Once desired inventory levels are fixed, routing decisions would be made by the depot in order to serve the individual orders. In other words, each RC would choose to have an inventory level, L�i (decision variable), which minimizes its expected inventory costs, i.e., without considering routing costs. Therefore, the quantity of product it would request to the depot is given by qi ¼ L

� i � Li if L

� i > Li and 0 otherwise. Once these qi values are determined for each

RC they would be considered as parameter inputs for the associated vehicle routing problem. The centralized approach assumed in this paper aims at jointly determining the amount of inventory each RC i e V� is sup-

plied with, denoted qi, and the routing plan in order to minimize the total expected costs the routing costs and the expected inventory costs. The latter is calculated by summing f(qi) over all i e V

� which is the inventory cost for the ith RC written as a function of the decision variables qi. Without loss of generality, this paper assumes the following structure for the inventory cost function:

fðqiÞ¼ ksi if si P 0 2c0i if si < 0

� ð1Þ

where k P 0 represents the cost of holding a unit of product in stock at the end of the period that is assumed to be known, and si represents the surplus at the end of the period, i.e., si = Li + qi � Di, for all i e V�. This calculation assumes that the cost of a stock-out at a RC is the cost of sending a new vehicle from the depot to that RC, i.e., a roundtrip.

It is possible to formulate this problem as a mixed-integer stochastic program, a class of formulations known to be difficult to solve. Even when delivery quantities qi are known, the model becomes that of the Capacitated Vehicle Routing Problem (CVRP) [7,45], which is NP-Hard. The fact that delivery quantities are decision variables in our model adds another layer of complexity. It is for this reason that we develop a hybrid solution approach combining simulation with heuristics. Our methodology is described in the following sections.

4. The simheuristic algorithm

One of the main ideas behind the algorithm is to consider a single period and assume that updated information on current inventory levels is obtained at the end of each period. The end-of-period inventory levels might be very difficult to forecast in reality, especially when the probability distributions modeling the random demands are characterized by high variances. One option is to use a plan-one-step-ahead policy, i.e., plan for the period ahead and then update the current inventory levels before planning again. We assume that the depot has enough stock to satisfy the RC demands in each period.

In this context, we propose a hybrid approach that combines Monte Carlo Simulation (MCS) with a multi-start random- ized heuristic. We refer to this approach as a simheuristic algorithm for which a flowchart is shown in Fig. 2. MCS can be defined as a set of techniques that make use of random numbers and probability distributions to solve certain stochastic and deterministic problems [43]. When properly combined with heuristic techniques, MCS has proven to be useful for solv- ing stochastic vehicle routing problems [38]. The approach we use in this paper is also based on the CWS heuristic [23] pro- posed for the CVRP. Since this heuristic is extremely fast and reasonably efficient, we employ the CWS heuristic for solving the multiple vehicle-routing problems that arise during the resolution process as different refill policies are considered for each RC. As the experimental section will show, this allows us to obtain ‘good’ solutions for the Stochastic IRP in just a few seconds. However, if time is not an important issue, more efficient routing algorithms can be used instead. In particular, any routing algorithm could be applied to a reduced list of top solutions generated by our approach in order to further improve the final routing plan (see, e.g., Juan et al. [37]).

As Fig. 2 shows, our approach starts by considering a discrete number of p centralized refill policies for each retail center. In this paper we will consider the following ‘natural’ policies for each RC: (a) no refill; (b) refill up to one quarter of its capacity (1/4-refill policy); (c) refill up to half of its capacity (1/2-refill policy); (d) refill up to three quarters of its capacity (3/4-refill policy); (e) full refill; and (f) refill up to the optimal inventory level – this policy is related to a decentralized strategy in which each retail center decides its refill level without looking at routing costs. Our methodology is able and flexible enough to accommodate more intermediate policies if necessary. Of course, examining more intermediate policies i.e., by using higher granularity, could lead to slightly better solutions, but it will also increase somewhat the computational effort. For each RC-policy combination, Monte Carlo simulation is used to obtain estimates of the expected inventory costs associated with it – including both surplus and shortage costs.

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For illustrative purposes, Table 1 shows a simplified numerical example of how these estimates are computed for one RC. The following numerical assumptions are made: (a) the cost for each unit in stock at the end of the period, k, is 0.05; (b) the cost of a roundtrip from the RC to the depot, 2c0i, is 1; (c) the expected demand, di, is 20; (d) the maximum capacity of the RC,

Fig. 2. Flowchart of the simheuristic algorithm.

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L _

i , is 40; and (e) the current inventory level, Li, is 10. Notice that qi (units to serve) is always a non-negative value given by the difference between L

_

i and Li – or zero if the current level is greater than L _

i. These qi values are the ones considered during the routing stage. The surplus, si, is obtained as the difference between the total units in the RC (served plus current) and the random demand. Notice also that only three random demands, Di, have been generated in this example. In the actual code, however, the probability distribution that models customers’ demands at each RC is used to randomly generate 1e6 obser- vations – this only takes about 1.5 s in a standard computer and generates much more accurate estimates of the stock costs associated with each RC-policy combination.

By applying the same refill policy to all RCs, we consider the same global refill strategies as described earlier. The routing costs belonging to each of these strategies are computed using the CWS heuristic. It is now possible to estimate the expected total costs (inventory plus routing costs) related to each of the aforementioned refill strategies. The strategy with the lowest total costs is set as the best solution found so far, and a randomized multi-start process is started. This multi-start process will be running as long as the maximum computation time has not been reached. At each iteration of this process, the base solution is reset to the previously selected ‘basic’ strategy, and the list of RCs is shuffled. This random sorting of the list of RCs contributes to randomize the local search process, which can be then encapsulated inside a multi-start framework to reduce the risk of getting trapped in a local minima – another advantage of this randomization feature is the possibility of running multiple threads of the algorithm in parallel. For each RC in the sorted list, all of its refill policies are considered and the resulting global refill strategy – new ‘map’ of RC-policy combinations – is evaluated. The routing costs of the new strategy are recomputed employing the fast CWS routing heuristic. Thus, for each RC the policy with the lowest total costs is selected, and both the base and the best solutions are updated if appropriate. Eventually, the best solution found so far is returned as output. This solution includes both the personalized refill policies for each RC as well as the associated routing plan.

As with any other approximate approach, this method is not guaranteed to obtain an optimal solution but it can produce feasible and ‘good’ (or even pseudo-optimal) solutions to the Stochastic IRP in a reasonable amount of time, which is not a trivial task considering the complexity of the problem.

5. Designing the numerical experiments

In the CVRP literature there exists a classical set of well-known benchmarks commonly used to test new CVRP algorithms. However, this is not exactly the case for the single-period IRP with stochastic demands and stock-outs discussed in this paper. In fact, for this and other IRP versions it is a usual practice that each paper presents a different set of randomly generated benchmarks, i.e., without providing the exact values obtained during the randomization process, which makes it impossible to reproduce the exact results and which makes it difficult to perform direct and fair comparisons among different approaches. Moreover, in some other cases the proposed set of instances is no longer available, as with the expired link presented in Campbell et al. [17]. Finally, there exist other benchmark instances using a truncated Normal probability distribution, with uniformly generated mean and variance, to model customer demands [24]. Our approach extends to instances using any probability distribution.

For these reasons, and with the goal of providing complete information about the set of benchmarks employed so that other researchers can use them, we have developed our own set of data by generalizing the well-known datasets A and B from the CVRP literature [8], available at http://www.branchandcut.org/VRP. These datasets consist of 27 small and med- ium-size test instances with the number of nodes ranging from 32 to 80, where each node represents an RC. We have con- verted the fixed demands di for each node i = 1, 2, . . . , n given by these instances into probabilistic demand Di with E[Di] = di. For the numerical experiments of this paper we will assume that Di will follow a LogNormal distribution with E[Di] = di, although, they can follow any probability distribution so long as it has a mean, such as Weibull or Gamma distributions. The LogNormal distribution has been chosen because it should be preferred over the Normal distribution when modeling positive demands. The use of continuous distributions to model stochastic demands is a reasonable practice in the vehicle routing arena [32,38], particularly when some of the practical routing applications concern the delivery of products in a gas or liquid state. We also consider three different levels of variance, i.e., Var[Di] = k di, with k = 0.25 defining a ‘low’ variance scenario, k = 0.5 defining an ‘intermediate’ variance scenario, and k = 0.75 defining a ‘high’ variance scenario.

Table 1 Estimating expected inventory costs for the ith retail center.

Refill policy L�i qi Di1 si1 Stock costs 1 Di2 si2 Stock costs 2 Di3 Si3 Stock costs 3 Avg. stock costs

Full- 40.0 30.0 15.5 24.5 1.22 21.0 19.0 0.95 19.4 20.6 1.03 1.07 3/4- 30.0 20.0 14.5 0.72 9.0 0.45 10.6 0.53 0.57 1/2- 20.0 10.0 4.5 0.22 �1.0 1.00 0.6 0.03 0.42 1=4 - 10.0 0.0 �5.5 1.00 �11.0 1.00 �9.4 1.00 1.00 No- 0.0 0.0 �5.5 1.00 �11.0 1.00 �9.4 1.00 1.00

Initial assumptions: k = 0.05; 2c0i = 1; di = 20; L _

i ¼ 40; Li = 10.

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As far as the inventory is concerned, the following assumptions are made in order to define a numerical example to experiment with. These are not assumptions on the model but to define a numerical example so that we can illustrate the potential of our approach with some benchmarks:

(a) For each RC i, its maximum inventory capacity is defined as bLi ¼ 2di. As it usually happens in practice, retailers with higher expected demands will also have higher inventory capacities.

(b) Trying to imitate a realistic scenario in which it is likely that different retailers will present different starting stock levels, initial inventory level at retailer i, Li, is assigned according to the following expression:

Li ¼

0 if i is odd and multiple of 3 ðe:g: L3; L9; L15; . . .Þ di 2 if i is odd and not multiple of 3 ðe:g: L1; L5; L7; . . .Þ di if i is even and multiple of 4 ðe:g: L4; L8; L12; . . .Þ 3di 2 if i is even and not multiple of 4 ðe:g: L2; L6; L10; . . .Þ

8>>>>< >>>>:

ð2Þ

Notice that instead of using these pre-defined Li values as initial inventory levels, we could simply have selected these initial values at random. But this would have prevented reproducing the computational experiments with the same initial data. Although the inventory costs are obtained throughout simulation and they can vary a little bit due to random error, the routing costs can be replicated by using the same initial data and heuristic.

Finally, regarding the inventory costs, we have adopted the convention that they should be of a similar order of magnitude as the routing costs. Otherwise, the problem would basically reduce to solving a CVRP – if the routing costs are the predominant component of total costs – or to applying the decentralized strategy – if the inventory costs are much higher than the routing costs. Thus, it might make sense not to serve some retailers under certain conditions. In particular, we might decide not to serve retailers with a low probability of suffering a stock-out, e.g., those in which the expected demand is much lower than the current inventory levels, and also those with low penalty costs in case they suffer from a stock-out, e.g., those closer to the depot. In order to attain this goal we have used in our experiments the previously explained expression in Section 3 for defining the inventory costs associated with each retailer, i.e., f(qi) = k si if si P 0, and f(qi) = 2c0i otherwise. As it is usual in the CVRP literature, our routing costs will be based on Euclidean distances. Notice that the parameter k represents the cost per unit of stock at the end of the period. Also notice that whenever a stock-out occurs, a ‘penalty’ cost is incurred since a new vehicle must be sent from the depot to the retailer to solve the shortage issue. In our numerical experiments we have used the following values of k: 0.01, 0.25, 0.5, 0.75, and 1. The two uttermost values represent extreme scenarios with ‘low’ and ‘high’ inventory costs per unit, respectively. In effect, as it will be show in the next section, for k = 0.01 the inventory costs are much smaller than the routing costs, while for k = 1 both costs are of approx- imately the same size. Of course, the election of the aforementioned values for k depends upon the set of benchmarks selected, since they determine the RCs coordinates and, therefore, the size of the routing costs as well as the ‘penalty’ costs in case of stock-outs.

6. Computational results

The algorithm was implemented as a Java application and used to run the 27 CVRP instances described above for each combination of the parameters k and k. This makes a total of 27 � 3 � 5 = 405 test instances in total. Each of these tests was run for a maximum computing time of 30 s on a standard personal computer (Intel i5-2400 CPU at 3.10 GHz with 3.16 GB RAM). This time period does not include the 1e6 simulation iterations needed to estimate the inventory costs for each RC-policy combination, which is about 1.5 s of additional computation time for each RC. The limit on the computing time is placed to obtain ‘good’ results in ‘short’ computing times, e.g., a time small enough for the algorithm to be executed fairly regularly or whenever forecasts on demands are updated. Table 2 shows a summary of the results obtained in our experiments for the six strategies defined before (no-refill, 1/4-refill, 1/2-refill, 3/4-refill, full-refill, and decentralized), as well as our best solution (OBS). For each of the replenishment strategies, both the average values and standard deviations (in cursive) of inventory, routing and total expected costs are provided. On average, our best solution has been obtained in about 5 s.

7. Discussion of the results

Fig. 3 shows a multiple boxplot representing the percentage gaps between the total costs obtained with each ‘basic’ strat- egy and the total costs provided by our best solution (OBS). Notice that all gaps are negative, which means that OBS outper- forms all other strategies in every test. The average gaps range from �68% (1/4-refill strategy) to �7% (3/4-refill and decentralized strategies). As expected, the performances of the no-refill, 1/4-refill, and 1/2-refill strategies are quite poor. This is due to the accumulated ‘penalty’ costs generated by frequent stock-outs. Less extreme strategies, such as the 3/4-refill and the decentralized ones, show a much better performance. However, the average gap with respect to our best solution is still noticeable – and also significant, as it will be discussed later.

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Table 2 Averages and standard deviations (in italics) of 27 instances run for different values of k, k, and refill strategies (max. time = 30 s., avg. time for OBS = 5 s.).

k/k No-refill strategy 1/4-refill strategy 1/2-refill strategy 3/4-refill strategy Full-refill strategy Decentralized strategy Our Best Solution

Stock Routing Total Stock Routing Total Stock Routing Total Stock Routing Total Stock Routing Total Stock Routing Total Stock Routing Total

0.01 3061.12 0.00 3061.12 3058.00 284.24 3342.24 1752.69 513.77 2266.46 78.41 798.95 877.35 16.68 1170.95 1187.63 14.84 931.13 945.97 49.54 773.74 823.29 0.25 1279.81 0.00 1279.81 1277.21 42.28 1304.33 732.75 137.28 855.90 50.13 255.86 289.31 8.23 406.51 411.79 8.02 308.01 312.93 47.39 259.84 279.79 0.01 3052.70 0.00 3052.70 3041.34 284.24 3325.58 1732.08 513.77 2245.85 176.88 798.95 975.83 39.44 1170.95 1210.39 39.24 1146.26 1185.50 114.57 776.13 890.70 0.50 1269.70 0.00 1269.70 1261.87 42.28 1289.02 723.97 137.28 846.94 98.07 255.86 329.64 23.93 406.51 422.68 23.98 405.15 421.33 85.97 261.29 319.00 0.01 3043.41 0.00 3043.41 3022.29 284.24 3306.53 1720.65 513.77 2234.42 267.47 798.95 1066.42 63.80 1170.95 1234.75 63.71 1161.00 1224.71 175.00 786.60 961.60 0.75 1260.79 0.00 1260.79 1247.96 42.28 1275.16 718.55 137.28 841.40 135.40 255.86 365.06 38.44 406.51 433.60 38.48 404.69 431.88 124.75 255.75 355.19 0.25 3082.40 0.00 3082.40 3079.28 284.24 3363.52 1777.96 513.77 2291.73 157.66 798.95 956.61 175.09 1170.95 1346.04 104.57 834.14 938.71 125.80 781.57 907.37 0.25 1284.28 0.00 1284.28 1281.69 42.28 1308.81 737.90 137.28 861.11 64.67 255.86 306.61 51.07 406.51 448.01 30.76 274.25 299.07 38.85 274.32 294.89 0.25 3074.86 0.00 3074.86 3063.51 284.24 3347.75 1760.15 513.77 2273.92 256.51 798.95 155.46 197.89 1170.95 1368.84 149.25 896.46 1045.71 175.64 801.07 976.71 0.50 1274.40 0.00 1274.40 1266.60 42.28 1293.76 729.95 137.28 852.99 112.74 255.86 347.05 61.93 406.51 458.87 49.00 297.70 338.63 59.69 271.58 317.46 0.25 3066.33 0.00 3066.33 3045.22 284.24 3329.46 1750.67 513.77 2264.44 347.67 798.95 1146.62 222.40 1170.95 1393.35 193.71 970.07 1163.78 243.38 805.31 1048.69 0.75 1265.68 0.00 1265.68 1252.85 42.28 1280.05 725.19 137.28 848.11 150.66 255.86 382.64 74.12 406.51 469.77 68.57 328.11 386.79 83.16 280.39 349.67 0.50 3104.53 0.00 3104.53 3101.40 284.24 3385.64 1804.46 513.77 2318.23 240.29 798.95 1039.23 340.08 1170.95 1511.02 192.32 827.79 1020.10 212.93 778.60 991.53 0.25 1288.97 0.00 1288.97 1286.38 42.28 1313.51 743.76 137.28 867.02 84.14 255.86 325.54 99.07 406.51 487.56 56.63 270.55 316.35 57.51 274.27 315.91 0.50 3097.98 0.00 3097.98 3086.64 284.24 3370.88 1789.24 513.77 2303.01 339.52 798.95 1138.47 362.97 1170.95 1533.92 246.46 860.10 1106.56 270.47 791.54 1062.02 0.50 1279.31 0.00 1279.31 1271.51 42.28 1298.68 736.34 137.28 859.43 130.69 255.86 365.91 108.59 406.51 498.33 75.06 288.13 352.47 75.16 276.67 337.30 0.50 3090.26 0.00 3090.26 3069.18 284.24 3353.42 1781.92 513.77 2295.69 431.17 798.95 1230.12 387.54 1170.95 1558.49 302.28 912.52 1214.80 341.53 796.03 1137.56 0.75 1270.89 0.00 1270.89 1258.08 42.28 1285.30 732.02 137.28 855.00 168.43 255.86 401.49 119.30 406.51 509.14 96.79 305.33 388.50 109.64 273.95 370.79 0.75 3126.66 0.00 3126.66 3123.54 284.24 3407.78 1830.87 513.77 2344.64 322.86 798.95 1121.81 505.07 1170.95 1676.03 278.40 823.01 1101.42 301.52 773.39 1074.91 0.25 1293.74 0.00 1293.74 1291.16 42.28 1318.30 749.52 137.28 872.83 105.49 255.86 345.02 147.22 406.51 528.57 82.54 270.26 337.30 79.74 278.03 338.76 0.75 3121.02 0.00 3121.02 3109.68 284.24 3393.92 1818.44 513.77 2332.21 422.43 798.95 1221.38 528.01 1170.95 1698.96 339.02 840.66 1179.68 359.45 789.79 1149.25 0.50 1284.26 0.00 1284.26 1276.46 42.28 1303.64 742.71 137.28 865.85 150.38 255.86 385.35 156.24 406.51 539.24 101.74 275.99 363.44 100.36 278.19 361.35 0.75 3114.08 0.00 3114.08 3093.00 284.24 3377.24 1813.17 513.77 2326.94 514.73 798.95 1313.67 552.75 1170.95 1723.70 402.04 870.53 1272.56 426.34 798.74 1225.08 0.75 1275.86 0.00 1275.86 1263.05 42.28 1290.27 738.85 137.28 861.88 187.83 255.86 420.99 166.31 406.51 550.02 124.19 290.15 398.72 131.16 271.48 386.32 1.00 3148.79 0.00 3148.79 3145.66 284.24 3429.90 1857.37 513.77 2371.14 405.49 798.95 1204.43 670.07 1170.95 1841.02 363.31 821.95 1185.27 389.47 765.41 1154.88 0.25 1298.48 0.00 1298.48 1295.89 42.28 1323.05 755.48 137.28 878.81 127.92 255.86 365.16 195.42 406.51 570.70 108.23 269.14 358.10 108.63 273.39 359.88 1.00 3144.13 0.00 3144.13 3132.81 284.24 3417.05 1847.53 513.77 2361.30 505.45 798.95 1304.40 693.08 1170.95 1864.03 429.05 836.13 1265.18 451.36 783.17 1234.52 0.50 1289.21 0.00 1289.21 1281.42 42.28 1308.61 749.21 137.28 872.39 171.29 255.86 405.33 204.16 406.51 581.25 128.33 279.20 389.52 126.51 281.08 387.59 1.00 3138.01 0.00 3138.01 3116.97 284.24 3401.21 1844.42 513.77 2358.19 598.23 798.95 1397.18 717.88 1170.95 1888.83 497.23 857.91 1355.14 522.37 788.88 1311.25 0.75 1281.12 0.00 1281.12 1268.34 42.28 1295.57 745.81 137.28 868.88 208.15 255.86 440.90 213.81 406.51 591.92 151.22 293.04 427.12 157.13 276.11 411.39

All 3097.75 0.00 3097.75 3085.90 284.24 3370.14 1792.11 513.77 2305.88 337.65 798.95 1136.60 364.85 1170.95 1535.80 241.03 905.98 1147.00 277.29 786.00 1063.29

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Fig. 4 exhibits the average distribution of routing and inventory costs for each explored strategy. Note that direct routing costs are non-existent or relatively small in the case of the first three strategies. Nonetheless, since these strategies suffer frequent stock-outs their inventory costs include numerous roundtrips to the depot as specified in the inventory costs function. The remaining strategies show average inventory costs of similar magnitudes. However, the full-refill strategy has to face higher routing costs since all RCs must be served up to their maximum capacity (worst-case routing scenario). The average stock costs of the full-refill strategy are slightly higher than those of the 3/4-refill strategy, which implies that

Fig. 3. Performance comparison of different strategies.

Fig. 4. Average routing and inventory costs of different strategies.

Fig. 5. Boxplots showing the evolution of OBS total costs as k and k vary.

A.A. Juan et al. / Simulation Modelling Practice and Theory 46 (2014) 40–52 49

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the end-of-period surplus is excessive in the former strategy. Using a significance level a = 0.05, an ANOVA test was performed in order to determine that the OBS average total costs are significantly different from those of the 3/4-refill and the decentralized strategies (p-value = 0.003, with non-overlapping 95% confidence intervals).

Fig. 6. Surface plot showing the evolution of OBS average stock costs as k and k vary.

Fig. 7. Comparison between the 3/4-refill strategy (up) and the OBS (down) for the A-n63-k9.

50 A.A. Juan et al. / Simulation Modelling Practice and Theory 46 (2014) 40–52

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We were also interested in analyzing the effects, over OBS total and partial costs, of joint variations in the system uncer- tainty level (i.e., the value of the parameter k) and the inventory-holding costs per unit (i.e., the value of the parameter k). Fig. 5 shows multiple boxplots, each of them representing results for a given level of k and k. Note that, for each level of k, as the level of k increases the OBS total costs shift upwards,i.e., higher inventory costs per unit implies higher total costs, rang- ing from 823.29 to 1311.25. Similarly, as the level of k increases, these costs also shift upwards, i.e., higher variability in demands implies higher total costs. According to the results obtained in a two-way ANOVA tests, average total costs are significantly affected by variations in any of these two parameters (p-value = 0.003 for k and p-value = 0.000 for k, with no interaction between both factors).

Fig. 6 depicts the evolution of OBS average stock costs as k and k vary. Note that for low values of both parameters (k = 0.01 and k = 0.25), inventory costs are almost inexistent: 49.54, which represent just 6% of total costs.

Notice also that average inventory costs increase almost linearly as either parameter increases. Thus, for the highest levels of both parameters (k = 1.00 and k = 0.75), the average inventory costs rise up to 522.37, which represent 40% of total costs.

Finally, Fig. 7 depicts two different solutions for the A-n63-k9 instance with k = 0.01 and k = 0.75. Squares (j) represent customers under a full-refill policy; diamonds (�) show customers under a 3/4-refill policy; and stars (�) represent non- served customers.

The upper part of Fig. 7 shows the solution provided by the 3/4-refill strategy, which expected total costs rise up to 1740.01 (expected stock costs = 446.61 and routing costs = 1293.40). The bottom part of Fig. 7 depicts the best solution found by our method within the allowed computing time of 30 s. Here, the expected total costs rise up to 1527.31 (expected stock costs = 225.01 and routing costs = 1302.30). The decentralized solution for this instance has higher expected total costs (2048.28, with expected stock costs = 107.20 and routing costs = 1941.07). Notice that, by serving different quantities to dif- ferent customers, the solution obtained with our algorithm is able to reduce total costs by 12% and 25% when compared with the solutions provided by the 3/4-refill and the decentralized strategies, respectively.

8. Conclusions

We have presented a ‘simheuristic’ algorithm for solving the Single-Period Stochastic Inventory-Routing Problem with Stock-outs. This is a challenging research area because it introduces random behavior into a problem combining two steps of supply chain management, inventory control and distribution planning. The proposed approach combines Monte Carlo simulation with a routing heuristic inside a multi-start randomized procedure. By doing so, it allows solving both the routing and the inventory problems in an integrated way. One of the main contributions of our methodology is that it can consider personalized refill policies for each retail center, which contributes to significantly reduce total costs over other approaches using standard refill policies. Another important contribution is that our approach can be used with any probability distri- bution, which means that positive demands in retail centers are not assumed to follow a Normal distribution – which is an unrealistic assumption usually employed in the existing literature. A set of benchmarks for the considered problem were developed and a realistic expression to model inventory costs was also proposed. A complete set of tests has been performed to illustrate the methodology and analyze how costs vary as different uncertainty and costs scenarios are considered.

Acknowledgments

This work has been partially supported by the Spanish Ministry of Science and Innovation (TRA2010-21644-C03). It has been developed in the context of the IN3-ICSO program and the CYTED-HAROSA network (http://dpcs.uoc.edu).

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