The inventory routing problem: the value of integration

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1 Intl. Trans. in Op. Res. 23 (2016) DOI: /itor INTERNATIONAL TRANSACTIONS IN OPERATIONAL RESEARCH The inventory routing problem: the value of integration Claudia Archetti and M. Grazia Speranza Department of Economics and Management, University of Brescia, I Brescia, Italy [Archetti]; [Speranza] Received 5 October 2015; received in revised form 15 October 2015; accepted 24 October 2015 Abstract We consider an inventory routing problem in which a supplier delivers goods to customers over a given planning period. Before the advent of the supply chain management concept, customers usually applied a (s, S) policy for the inventory management. The supplier then, on the basis of the distribution schedule determined by the customers, organized the distribution routes. In an integrated approach, the supplier has access to the inventory levels at the customers and knowledge of their demand process. On the basis of this information the supplier organizes a complete distribution plan, determining the days of visit, quantities to deliver, and distribution routes. This integrated policy is called vendor-managed inventory policy. We solve the optimization problems arising in the traditional and integrated management, and analyze the two approaches, comparing the costs and characteristics of the different solutions. Keywords: inventory routing; integration; retailer-managed inventory; vendor-managed inventory 1. Introduction Logistics management is the part of supply chain management that plans, implements, and controls the forward and reverse flow and storage of goods, services, and related information between the point of origin and point of consumption in order to meet customers requirements. Logistics comprises all the activities related to the functioning of a production distribution system. Such activities are linked and need to be coordinated to guarantee a good system performance. This concept is widely recognized and has advanced significantly with the advent of the supply chain management concept. However, the most common practices adopted have been based on decomposing the system in parts and handling the single parts independently or sequentially. The main reason for this is related to the fact that integration is very difficult to pursue, often leading to extremely complex management problems. On the other hand, decomposition leads to worsening of the system performance. Also scientific research has traditionally adopted a decomposition approach of the system in parts and then invested efforts to optimally solve the problems arising in each single part. Published by John Wiley & Sons Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main St, Malden, MA02148, USA.

2 394 C. Archetti and M.G. Speranza / Intl. Trans. in Op. Res. 23 (2016) In the last years, the advances in technology, information systems, decision support tools, and scientific research have favored a trend that leads to the management and optimization of more and more integrated parts of the logistic system. The incentive for going in this direction comes from the economic advantages that may be achieved through integrated policies. On the other hand, integration implies additional costs and possibly organizational challenges. Thus, studies aimed at evaluating the benefits of integrated policies are extremely valuable. In the literature, a number of contributions appeared on the analysis of the benefits achieved by introducing collaborative initiatives and, in particular, vendor-managed inventory (VMI) in supply chain management (see Kanda and Deshmukh, 2008; Sari, 2008; Lyu et al., 2010; Marquès et al., 2010; Wadhwa et al., 2010; Danese, 2011; and the references therein). These papers present empirical studies or the results of simulations and substantially differ from this paper where an analytical evaluation of the savings achieved with an integrated policy is proposed. Moreover, the distribution cost, when considered, is estimated and not calculated as the routing cost, that is, the cost of the distance traveled by the vehicles. Practical applications of VMI policies and analysis of the corresponding performance are also provided in the literature. For example, in Yu et al. (2001) the authors show a case study of a Hong Kong multinational company that manufactures and distributes electronics components and whose customers are mainly located in Europe. In Tyan and Wee (2003), the authors provide a survey of practical implementations of VMI strategies in the Taiwanese grocery industry, while the electronic industry is analyzed in Kuk (2004). In this paper, we aim at quantifying the benefits coming from the integration by closely analyzing a specific problem. We focus on inventory and distribution management and observe the two problems separately and in an integrated manner. We consider a deterministic setting that allows us to model and find the optimal solution of the resulting problems. A supplier delivers goods to customers over a given planning period. Customers must be served in such a way that their inventory capacity is not exceeded and no stockout occurs. Before the advent of the supply chain management concept, customers usually applied and still apply in many cases a (s, S) policy for the inventory management. According to this policy, which is called retailer-managed inventory (RMI), an order is issued at a time such that when the delivery arrives no inventory is held in stock. The supplier then, on the basis of the distribution schedule determined by the customers in terms of times and quantities, organizes the distribution routes. In an integrated approach, the supplier has access to the inventory levels with the customers and knowledge of their demand process. On the basis of this information the supplier organizes a complete distribution plan, determining the days of visit, quantities to deliver, and distribution routes. This integrated management policy is called VMI. Thus, the two policies compared are as follows: 1. The RMI policy: Each customer (retailer) decides delivery times and quantities. The supplier organizes the distribution routes, taking delivery times and quantities as constraints. 2. The VMI policy: The supplier monitors the inventory levels of the customers and determines delivery times and quantities, in a way to avoid the occurrence of any stockout, and the routes. The RMI policy corresponds to a sequential approach to inventory and distribution management. The inventory decisions at the customers level are taken by the customers. This implies that the supplier decision space is reduced to organizing the customers to be served in a day in routes. This policy may also be seen as a decentralized policy, where each customer is in charge of the inventory

3 C. Archetti and M.G. Speranza / Intl. Trans. in Op. Res. 23 (2016) decisions (when and how much to order). The supplier represents the centralized stage where the routes are organized. The VMI policy corresponds to an integrated or fully centralized approach where all decisions (when to serve customers, how much to deliver, and the routes) are taken jointly by a single decision maker the supplier. In the RMI policy a customer apparently takes the best possible decision, by reducing the inventory cost to minimum. On the other hand, the cost of a poor distribution plan for the supplier will push the supplier to impose a price of the goods higher than it might be in a VMI policy. To compare the two policies, we solve the optimization problems arising under the RMI and VMI policies. Under the RMI policy, after customers have decided times and quantities to be delivered, the supplier will minimize the distribution costs by solving a routing problem each day (see Toth and Vigo, 2014). An inventory routing problem (IRP) has instead to be solved for the VMI policy (for a general introduction to inventory routing problems, we refer to Bertazzi and Speranza, 2012, 2013; and for surveys we refer to Bertazzi et al., 2008; Coelho et al., 2013). We adopt the commonly used setting for an iirp (see Adulyasak et al., 2012, 2013; Coelho et al., 2012; Coelho and Laporte, 2013; Archetti et al., 2014a), find the best possible solutions under the two policies, and compare them. The results show that on the tested instances the VMI policy generates average and maximum savings, in terms of global system cost, of 23.97% and of 33.63%, respectively. This is mainly due to a decrease in the transportation cost, which is achieved by a better organization of the delivery routes due to the centralized policy. In fact, VMI solutions are characterized by a substantially smaller number of delivery routes and a higher average vehicle load. At least part of the savings achieved by the VMI policy is due to the fact that in the classical inventory routing setting, there is no inventory at the end of the horizon. This implies, in particular, that the total quantity distributed under the RMI policy is greater than the total quantity distributed under the VMI policy. In order to have a more even comparison of the two policies, we introduce a new iirp where the final inventory level at each customer is given. We then impose the constraint that the final inventory levels must be equal to the final levels of the RMI policy. This condition somehow penalizes the VMI policy but allows us to claim that while the savings achieved by the VMI policy overestimate the real savings, the savings achieved by the modified VMI policy underestimate them. The average and maximum savings of the modified VMI policy are 9.49% and 20.23%, respectively. We complement the computational comparison with worst-case results that show, in particular, that unlimited savings are possible when a VMI policy is applied with respect to an RMI policy. The paper is organized as follows. In Section 2, we present the problem setting, and analysis and comparison of the policies. Section 3 is devoted to the worst-case analysis. In Section 4, we briefly describe the approaches used for solving the optimization problems arising under each of the two policies. In Section 5, we report the results of the computational study. Section 6 concludes the paper. 2. Problem description and representation A supplier has to serve his/her customers over a given discrete time horizon, which is assumed for the sake of simplicity to be structured in days. Each customer faces a daily demand, possibly different from day to day, and no stockout is allowed. Moreover, each customer has a maximum inventory capacity. We assume that each customer can be served at most once, that is, by one vehicle

4 396 C. Archetti and M.G. Speranza / Intl. Trans. in Op. Res. 23 (2016) at most per day. We assume that the supplier has an infinite availability of goods for each day of the horizon. This assumption is related to the fact that we focus on the distribution phase and do not include the analyzed production or replenishment decisions of the supplier in the system. This in turn implies that we do not consider any inventory-holding cost at the supplier. We also assume that an unlimited number of homogeneous capacitated vehicles are available to deliver goods to the customers. This assumption is justified by a setting in which the supplier engages an external carrier to distribute the goods from its depot to the customers. The fare paid by the supplier for the distribution is proportional to the distance traveled by the carrier to serve all customers over the entire horizon. The total system cost comprises two terms: the transportation cost, that is, the total distance traveled, and the inventory-holding cost at the customers. Formally, we are given a time horizon of H days and a complete undirected graph G = (N, E ), where node 0 is the depot from where the vehicles start and end their routes. Nodes in N ={1,...,n} are the customers. A daily unitary inventory holding cost h i, an initial inventory level I i0,anda maximum inventory level U i are associated with each customer i N.LetT denote the set of days {1,...,H}. Each customer i N faces a daily demand r it, t T.Eachedge(i, j) E represents the possibility to travel between locations i and j at nonnegative cost c ij. An unlimited fleet of identical vehicles with capacity Q is available to provide the service. Inventory costs are incurred at the customers and transportation costs have to be paid each day where some customers are served. No stockout is allowed at the customers and the quantities delivered by each vehicle in each route cannot exceed the vehicle capacity RMI policy In the RMI policy, each customer decides the delivery times and quantities. The supplier is then in charge of organizing the delivery routes each day. The goal of the supplier is to minimize the transportation cost of the delivery routes. The problem is solved in two stages. In the first stage, the customers take their decisions, while in the second stage the supplier optimizes the routes: 1. Delivery schedule stage: Each customer determines the delivery schedule, that is, the delivery days and quantities to be delivered each day of service. 2. Route optimization stage: On the basis of the delivery schedule, the routes that have to be performed each day are determined in order to minimize the transportation cost. In the first stage, each customer will apply his/her (s, S) policy. When the inventory level reaches or falls below level s, the quantity that brings the inventory level up to S is ordered. This is the classical order-up-to-level policy. The values s and S are customer dependent and determined by the customer. According to this policy, a quantity equal to S s has to be delivered each day, where the inventory level is equal to s. The delivered quantity is 0 on all the remaining days. Next, we set s = 0. The problem to be solved by the supplier each day is a classical capacitated vehicle routing problem (VRP), where the customers who have to be served are those who need to be served, that is, those for which the inventory level has reached level 0.

5 2.2. VMI policies C. Archetti and M.G. Speranza / Intl. Trans. in Op. Res. 23 (2016) In the VMI policy, the objective of the supplier, with the role of centralized decision maker, is to minimize the total cost that includes the inventory cost at the customers and transportation cost. The problem to be solved is an IRP. The class of IRP problems covers the problems in which decisions on when and how much to deliver have to be taken simultaneously compared to the decisions on how to organize the routes. The single-vehicle version of the IRP problem, we are interested here, was studied in Bertazzi et al. (2002), Archetti et al. (2007, 2012), and Solyalı and Süral (2011). More recently, a number of contributions have been devoted to the multiple-vehicle version (see Adulyasak et al., 2012, 2013; Coelho et al., 2012; Coelho and Laporte, 2013; Archetti et al., 2014a). In all the studies mentioned above, it is assumed that the supplier has a fixed daily production rate and faces an inventoryholding cost. As already mentioned, we consider an unlimited product availability with the supplier each day, that is, no fixed production rate, and as a consequence no inventory-holding cost. Thus, the supplier has no limit on the quantities to deliver. Obviously, the delivered quantities impact the transportation costs and inventory costs at the customers. Also, we do not consider any bound on the size of the fleet of vehicles. For the sake of completeness, we report the mathematical formulation of the problem. In order to model the problem, we have to identify an upper bound on the number of vehicles needed to distribute the goods. We denote the set of vehicles as K. A trivial upper bound on the maximum fleet size needed is K =n. The formulation uses the following decision variables: Iit is a nonnegative variable indicating the inventory level at customer i N at the end of day t T. z k it is a binary variable equal to 1 if customer i N is visited at day t T by vehicle k K. q k it is a nonnegative variable representing the quantity delivered to customer i N at day t T by vehicle k K. y kt ij represents the number of times the edge (i, j) E is traversed by vehicle k K in day t T. Itisabinaryvariablefori 1 while it may take values 0, 1, or 2 for i = 0 (value 2 is chosen if vehicle k visits at day t customer i only). The formulation of the classical IRP is as follows: min h i I it + c ij y kt ij i N t T k K (i, j) E t T (1a) I it = I i,t 1 r it + k K q k it i N, t T (1b) I it 0 i N, t T (1c) q k it U i I it 1 i N, t T (1d) k K q k it U i z k it i N, k K, t T (1e)

6 398 C. Archetti and M.G. Speranza / Intl. Trans. in Op. Res. 23 (2016) q k it Qz k 0t k K, t T (1f) i N z k it 1 i N, t T (1g) k K y kt ij = 2z k it i N, k K, t T (1h) j:(i, j) E y kt ij z k it z k st (i, j) E(S) i S S N,s S, k K,t T (1i) z k it {0, 1} i N, k K, t T (1j) q k it 0 i N, k K, t T (1k) y kt ij {0, 1} {i, j} E, k K, t T (1l) y kt 0 j {0, 1, 2} j N, k K, t T, (1m) where E(S) ={(i, j) E i S, j S}. The objective function (1a) calls for the minimization of the total cost. Constraints (1b) and (1c) determine the evolution of the inventory level over time and avoid stockout. Constraints (1d) and (1e) ensure that if a customer is visited, the quantity delivered is such that the maximum inventory level does not exceed. Constraints (1f) are the vehicle capacity constraints. Inequalities (1g) impose that if a customer is visited on a given day, then the entire quantity has to be delivered by a single vehicle. Constraints (1h) and (1i) are routing constraints while (1j) (1m) are variable definitions. The optimal solution under the VMI policy is found by solving formulation (1). Formulation (1) allows any quantity to be delivered to a customer at day t, t T, and, thus, as the inventory cost is minimized in the objective function, the optimal final inventory level at any customer is null. We consider here a new variant of this classical IRP, where the final inventory levels at customers are given. This variant allows us to make a more even comparison with the RMI policy. We refer to this new IRP as the IRP with given final levels. The formulation of the latter problem is obtained by adding the constraints to formulation (1) I it = I F it i N, (2) where IiT F is the final inventory level at customer i. The adjusted-vmi (adj-vmi) is called the VMI policy with the additional constraint that the final inventory level at each customer i equals the final inventory level of the RMI policy IiT RMI.To find the optimal solution of the adj-vmi, we solve the IRP with given final levels with I F it = I RMI it i N. (3) Thus, the same quantity is delivered to each customer under both policies.

7 3. Worst-case analysis C. Archetti and M.G. Speranza / Intl. Trans. in Op. Res. 23 (2016) In this section, we present a worst-case analysis that compares the different policies described in the previous section. We will show the maximum savings that can be achieved by applying the VMI policy with respect to RMI and adj-vmi, respectively, and by applying adj-vmi with respect to RMI. Let z(p) be the value of the optimal solution obtained when applying policy P. Theorem 1. The ratio z(rmi ) is unbounded. z(vmi) Proof. Let us consider the following instance. H = Q = n, h i = 0, U i = 1, and I i0 = 0, i N. All customers are co-located, that is, the distance between any pair of customers is 0. Let d denote the distance between any customer and the depot. Customer demands are as follows: r it = 1fort = i and 0 otherwise, i N. In the RMI policy, each customer i requires an order of 1 on day t = i. Thus, n routes are needed to serve a single customer each with a total cost equal to 2dn. In the VMI policy, a single route at day 1 serves all customers by delivering one unit each, for a total cost of 2d. = n may be greater than any given value and tends to infinity when n tends to infinity. Thus, z(rmi ) z(vmi) z(rmi ) Theorem 2. The ratio is unbounded. z(ad j VMI) Proof. The same instance used in the proof of Theorem 1 proves the result as the solution of the adj-vmi policy is identical to the solution of the VMI policy. Theorem 3. The ratio z(ad j VMI) z(vmi) is unbounded. Proof. Let us consider an instance where T = 2, U i = Q = n, h i = 0, r i1 = 1, r i2 = 0, and I i0 = 0, i N. All customers are co-located. Let d be the distance between each customer and the depot. In the RMI policy, a quantity equal to Q is delivered to each customer at day 1 and the final inventory levelisequaltoq 1 for each customer. The same quantity is delivered when applying the adj-vmi policy. Thus, n routes are performed at day 1 visiting a single customer and delivering Q units each for a total cost of 2dn. When applying the VMI policy, a single route is performed at day 1 visiting all customers and delivering one unit each for a total cost of 2d. These results show extreme cases but indicate that large savings can be achieved. The most interesting results concern the savings that can be achieved by applying the VMI and adj-vmi policies with respect to the RMI. Next, we will address the issue of the savings that can be achieved on specific and randomly generated instances. 4. Solution algorithms Formulation (1) and formulation (1) with the additional constraints (3) have to be solved in order to find the optimal solution under the VMI policy and adj-vmi policy, respectively. Formulation (1) is known to be computationally very hard and no state-of-the-art exact method is able to solve instances of interesting size. Thus, we will use the state-of-the-art heuristic method presented in Archetti et al. (2014b), which was shown to generate errors with respect to the optimal solutions of less than 1% on average. This heuristic has been adapted to solve formulation (1) with the additional constraints (3).

8 400 C. Archetti and M.G. Speranza / Intl. Trans. in Op. Res. 23 (2016) The delivery schedule for the RMI policy is determined by setting the delivery quantities to each customer equal to the maximum inventory level whenever the inventory level equals 0. In the remaining days, the delivery quantity is set to 0. In the second stage, a VRP is solved for each day. The VRP is solved to optimality through a branch-and-cut algorithm on the classical aggregated formulation (for more details, see Toth and Vigo, 2014). Clearly, although the heuristic of Archetti et al. (2014b) was shown to give very good solutions, the solutions found for the VMI policies are not optimal in general, while the solutions for the RMI are optimal. This means that when we will compare the solutions found for the VMI policies with the solutions found for the RMI policy and calculate the savings due to the integration we underestimate the maximum possible savings. 5. Computational study In this section, we present a computational study focused on the analysis ofthe VMI policy benefits with respect to the classical RMI policy Test instances The tests are performed on a subset of the benchmark instances created and tested for the multivehicle iirp (see Adulyasak et al., 2012, 2013; Coelho et al., 2012; Coelho and Laporte, 2013; Archetti et al., 2014a). The instances have the following characteristics: Planning horizon: 6 days. Number of customers: 10, 30, and 50. Inventory costs at the customers: randomly generated at the intervals [0.01; 0.05] and [0.1; 0.5]. Transportation costs: equal to the Euclidean distances between the location of the customers and supplier, which are defined through Euclidean coordinates. Customer demands rit : constant overtime, that is, r it = r i, t T, and randomly generated as an integer number at the interval [10, 100]. Maximum inventory level Ui : equal to r i g i,whereg i is randomly selected from the set {2, 3} and represents the number of days needed in order to consume the quantity U i. These instances are derived from the instances proposed in Archetti et al. (2007) for the singlevehicle case, where the vehicle capacity was set to 3 2 i N r i. For the multiple-vehicle case, the vehicle capacity was divided by the fleet size, while the remaining characteristics were not changed. In the above-mentioned papers, a fleet size of K =2,...,5 was considered. Even though we consider an unlimited fleet size, we tested all instances in order to assess the impact of different vehicle capacities. Vehicle capacities equal to 3 4 D, 1 2 D, 3 8 D, and 3 10 D are, thus, considered, where D = i N r i.asan upper bound on the fleet size, we consider a fleet size equal to the double of the fleet size of the original instances. For each of the previous characteristics, five randomly generated instances were tested for a total of 120 instances.

9 C. Archetti and M.G. Speranza / Intl. Trans. in Op. Res. 23 (2016) Table 1 Statistics on the first and second instances First instance Transportation Maximum No. of Total cost Inventory cost cost of vehicles load routes VMI RMI 10, , adj-vmi Second instance VMI RMI adj-vmi Comparison of VMI and RMI policies: detailed analysis of two instances First, we analyze in detail the solutions obtained through the VMI, RMI, and adj-vmi policy for two instances with 30 customers and low inventory cost. For both instances,. In the first 4 instance, the RMI and adj-vmi policies use the same number of routes but still the adj-vmi offers a saving of 9.14% in terms of total cost. This shows that by simply exploiting the flexibility of the integrated policy, remarkable savings can be achieved. In the second instance instead, the adj-vmi policy uses a lower number of routes with respect to the RMI. Table 1 reports the total cost, inventory cost, and transportation cost of the solutions found under the three policies for both instances, as well as the maximum number of vehicles used in a single day, the average vehicle load, and the number of routes. The routes performed by each policy each day for the first instance are shown in Fig. 1. The VMI policy performs in total of six routes, while under the RMI policy we obtain a total of nine routes. The routes performed on days 3 and 6, as well as the single route performed on days 2, 4, and 5, are identical. This is due to the same subset of customers reaching an inventory level equal to 0. The adj-vmi policy performs a total of nine routes. The routes performed on days 2, 3, and 4 are the same as the routes performed by the RMI policy. Note that all policies do not deliver any quantity at day 1, as the inventory level of all customers is sufficient to cover the demand of the first day. The number of routes used in the RMI and adj-vmi policies is 50% higher than the route used under the VMI policy for two reasons: The VMI policy better exploits the vehicle capacity, and the RMI and adj-vmi policies deliver a higher quantity, namely 2412 units more (the vehicle capacity is 1381 and total daily demand over all customers is 1842). Note that the adj-vmi policy generates a saving of 9.14% in terms of total cost with respect to the RMI policy even though the routes of the first 4 days (including day 1) are identical. The routes performed by each policy in each day for the second instance are shown in Fig. 2. The VMI policy makes a total of six routes, the RMI policy 11 routes, while the adj-vmi policy makes a total of nine routes. As in the previous instance, no policy delivers any quantity at day 1 as the inventory level of all customers is sufficient to cover the demand of the first day. The VMI performs six routes and adj-vmi policy performs nine routes instead of the 11 routes performed by the RMI policy with a saving of 25% and 12%, respectively.

10 402 C. Archetti and M.G. Speranza / Intl. Trans. in Op. Res. 23 (2016) Day 6 Day 5 Day 4 Day 3 Day 2 Fig. 1. Routes performed in the VMI, RMI, and adj-vmi solutions on the first instance Comparison of VMI and RMI policies The results of the computational study are summarized in Tables 2 and 3. Table 2 reports the solution costs while Table 3 reports the solution characteristics. In both tables, we show the results classified by inventory costs (high and low), number of customers, and vehicle capacity. In the last row, the results over all instances are reported in bold. Also, in Tables 2 and 3, columns 1 4 refer to average values while columns 5 7 refer to maximum values. All percentages reported in these tables are calculated as follows. Let x(p) be the value of the evaluated measure x (cost, maximum number of vehicles per day, etc.) in the solution under policy P. Then, the percentage is calculated as 100 x(vmi) x(rmi ) x(rmi ).

11 C. Archetti and M.G. Speranza / Intl. Trans. in Op. Res. 23 (2016) Day 6 Day 5 Day 4 Day 3 Day 2 Fig. 2. Routes performed in the VMI, RMI, and adj-vmi solutions on the second instance. Table 2 reports the average and maximum savings achieved by the VMI policy with respect to the RMI policy, in terms of total cost, inventory cost, and transportation cost, respectively. Table 3 compares the two policies in terms of maximum number of vehicles used in a single day, average vehicle load, and total number of routes in the planning horizon. As we aim at assessing the benefits of the VMI policy with respect to the RMI policy, columns 5 and 7, which give the maximum number of vehicles per day and the number of routes, report the smallest value over the instances of the row, while column 6, which gives the vehicle load, reports the largest value. Table 2 shows that the VMI policy can achieve remarkable savings with respect to the RMI policy. The total saving is on average more than 20% on each class of instances and the maximum saving may be as large as 33.63%. This saving is mainly due to transportation costs. The transportation

12 404 C. Archetti and M.G. Speranza / Intl. Trans. in Op. Res. 23 (2016) Table 2 Solution cost: percentage cost reduction of VMI with respect to RMI Maximum Total cost Inventory cost Transportation cost Total cost Inventory cost Transportation cost High inventory cost Low inventory cost n = n = n = Q = 1 D All Table 3 Solution characteristics: percentage difference of VMI with respect to RMI Maximum Maximum no. of vehicles load No. of routes Maximum no. of vehicles load No. of routes High inv. cost Low inv. cost n = n = n = Q = 1 D All cost represents on average 86% and 98% of the total cost of the RMI and VMI solutions for the cases with high and low inventory cost, respectively. This is reflected in the fact that the average and maximum savings on transportation costs are similar to the savings on total costs. The savings in inventory costs are higher, with an average of 30.05% and a maximum of 51.92%. In general, the savings do not seem to depend on the number of customers, while the total cost and transportation cost savings decrease when the vehicle capacity decreases. The inventory cost savings are higher for the class of instances with high inventory cost. Table 3 provides information on the characteristics of the solutions found by the two policies. The VMI policy requires a much smaller number of routes and, moreover, these routes are used more efficiently, as the average load is much higher. Also, the maximum number of vehicles used in a single day is less. At least part of benefits is due to the fact that although the RMI policy delivers what is requested by the customers, the VMI policy delivers what is necessary to satisfy the demand of the customers

13 C. Archetti and M.G. Speranza / Intl. Trans. in Op. Res. 23 (2016) Table 4 Solution cost: percentage saving of adj-vmi with respect to RMI Maximum Total cost Inventory cost Transportation cost Total cost Inventory cost Transportation cost High inventory cost Low inventory cost n = n = n = Q = 1 D All Table 5 Solution characteristics: percentage difference of adj-vmi with respect to RMI Maximum Maximum no. of vehicles load No. of routes Maximum no. of vehicles load No. of routes High inventory cost Low inventory cost n = n = n = Q = 1 D All at minimum cost. This implies that at the end of the planning horizon, under the VMI policy the customers have no inventory, while this is not the case under the RMI policy Comparison of VMI and RMI policies with same final inventory levels In this section, we compare the RMI with adj-vmi policy, that is, under the condition that the final inventory level at each customer equals the inventory level of the RMI policy. The results of the comparison between the adj-vmi and RMI are shown in Tables 4 and 5, which have the same structure of Tables 2 and 3, respectively. Comparing Tables 2 and 4, we can see that the average percentage saving in terms of total cost decreases from 23.97% to 9.49%. However, even in presence of the artificial constraint imposed

14 406 C. Archetti and M.G. Speranza / Intl. Trans. in Op. Res. 23 (2016) on the VMI, still a saving of almost 10% is achieved. In Table 5, we see that the solutions found under the adj-vmi policy use a lower number of routes and exploit more efficiently the vehicle capacity, as the average load is higher. The maximum number of vehicles used in a single day is on average higher than under the RMI. This is consistent with the assumption that the fleet of vehicles is unlimited, which means that in order to exploit the advantages of the integration, the distribution is concentrated on specific days. 6. Conclusions In this paper, we analyzed the benefits of the integration of distribution and inventory management comparing the solutions obtained by integrated VMI policies with the solutions obtained by a sequential RMI policy, where the customers first decide their own replenishment schedule and the transportation plan is then built on the basis of the decisions taken by the customers. As in the classical IRP, solved to find the optimal solution under the VMI policy, the final inventory level at each customer is null and thus the total quantity distributed is smaller than in the RMI policy; we also introduced a variant of such problem in which the final inventory level at each customer is given. The results show that the VMI policies can achieve remarkable savings both in terms of solution cost and number of vehicles used. The savings that can be achieved with an integrated policy are relevant, on average 9.49% in terms of total cost and 9.06% in terms of number of vehicles, even under the condition that the final inventory levels for a VMI policy equal the final inventory levels of the RMI policy. As these savings are achieved by using a heuristic algorithm for the integrated policies, while an exact algorithm is used for the sequential policy, the savings underestimate the real savings that can be achieved. This also shows that a heuristic solution of the integrated problem is better than the solution obtained by solving to optimality a sequence of simpler problems. In other words, the benefits of considering the integrated problem overcome the limitations due to the problem complexity, provided the quality of the heuristic used for the integrated problem is reasonable. The heuristic used in this paper was shown to generate an error of less than 1% on average. In general, most heuristics generate on average errors mostly below 10%. Future research should be devoted to finding the exact savings due to the VMI and adj-vmi policies, by solving the integrated problem optimally rather than heuristically. It would be interesting also to study more complex supply chains that also incorporate supplier production as well as further actors involved in the system. References Adulyasak, Y., Cordeau, J.-F., Jans, R., Optimization-based adaptive large neighborhood search for the production routing problem. Transportation Science 48, 1, Adulyasak, Y., Cordeau, J.-F., Jans, R., Formulations and branch-and-cut algorithms for multivehicle production and inventory routing problems. INFORMS Journal on Computing 26, 1, Archetti, C., Bertazzi, L., Hertz, A., Speranza, M.G., A hybrid heuristic for an inventory-routing problem. IN- FORMS Journal on Computing 24, Archetti, C., Bertazzi, L., Laporte, G., Speranza, M.G., A branch-and-cut algorithm for a vendor managed inventory routing problem. Transportation Science 41,

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