We consider a distribution problem in which a set of products has to be shipped from

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1 in an Inventory Routing Problem Luca Bertazzi Giuseppe Paletta M. Grazia Speranza Dip. di Metodi Quantitativi, Università di Brescia, Italy Dip. di Economia Politica, Università della Calabria, Italy Dip. di Metodi Quantitativi, Università di Brescia, Italy We consider a distribution problem in which a set of products has to be shipped from a supplier to several retailers in a given time horizon. Shipments from the supplier to the retailers are performed by a vehicle of given capacity and cost. Each retailer determines a minimum and a maximum level of the inventory of each product, and each must be visited before its inventory reaches the minimum level. Every time a retailer is visited, the quantity of each product delivered by the supplier is such that the maximum level of the inventory is reached at the retailer. The problem is to determine for each discrete time instant the retailers to be visited and the route of the vehicle. Various objective functions corresponding to different decision policies, and possibly to different decision makers, are considered. We present a heuristic algorithm and compare the solutions obtained with the different objective functions on a set of randomly generated problem instances. Introduction A company is a system with complex internalmanagement problems and complex relations with the suppliers and the customers. The internal management and the relations with the external world are strongly interrelated, and correct management of the company requires an understanding of these relations. In the last decade the importance of these relations has been widely recognized and the expression supply-chain management, which emphasizes the view of the company as part of the supply chain, has become common. Sometimes the expression coordinated supply-chain management is used with an additional emphasis on the coordination among the different components of the supply chain. The availability of data and information systems that derives from the advances in technology and communication systems has created the conditions for the coordination inside the supply chain. On the modeling side, there is a strong need for models that explicitly consider the coordination issues. A review of models can be found in Thomas and Griffin (1996). Three main management areas of the physical flow require coordination: production, inventory, and transportation. An interesting introduction to the problem of coordinating production and transportation issues can be found in Hall and Potts (2000). Sometimes, when the company is not a factory, the production area may not exist. One of the critical coordination problems that all companies face is the coordination of transportation issues with inventory control issues. Typical examples are given by the internal distribution systems, in which the supplier and the retailers are different echelons of a single company, and by external distribution systems, in which the supplier replenishes the retailers with respect to a given service level. Models that integrate vehicle routing with inventory control problems are called inventory-routing models (see Federgruen and Zipkin 1984 for the first optimization model that combines inventory and routing costs and Campbell et al for a recent overview) and can be classified as follows (see Bramel and Simchi-Levi 1997): single-period models with stochastic demand, /02/3601/0119$ electronic ISSN Transportation Science 2002 INFORMS Vol. 36, No. 1, February 2002 pp

2 multiperiod models with deterministic demand, and infinite-horizon models with deterministic demand. An example of single-period models with stochastic demand can be found in Federgruen and Zipkin (1984). In this model, the quantity of product to ship to each retailer is determined on the basis of the level of the inventory at the retailer. Then, the retailers are assigned to the vehicles and the routes are determined. The multiperiod models with deterministic demand are deterministic models in which several shipments can be performed over a time horizon. In these models, simpler models are often used as subproblems. For instance, in Dror and Ball (1987) singleperiod models are used as subproblems, while in Bertazzi et al. (1997) direct shipping models are used. Finally, in the infinite-horizon models with deterministic demand the product is absorbed by each retailer at a given constant rate and the problem is to determine an infinite-horizon shipping policy that minimizes the sum of inventory and vehicle-routing costs. Examples of these models can be found in Anily and Federgruen (1990) and in Chan and Simchi-Levi (1998). We study a multiperiod model with deterministic demand in which a set of products is shipped from a common supplier to several retailers. Each product is made available at the supplier and absorbed by the retailers in a deterministic and time-varying way. By deterministic way we mean that the quantity of each product that will be made available at the supplier and the quantity that will be absorbed by the retailers are known in each discrete time instant. By time-varying way we mean that the quantity of each product made available and absorbed in each time instant can be different from the one made available and absorbed in a different time instant. The products are shipped from the supplier to the retailers by a vehicle of given transportation capacity and cost. Shipments can be performed only in the discrete time instants (delivery time instants) that belong to a given time horizon. For each product, a starting level of the inventory is given both for the supplier and for each retailer, and the level of the inventory at the end of the time horizon can be different from the starting one. Therefore, the problem is not periodic. Each retailer determines a lower and an upper level of the inventory of each product and can be visited several times during the time horizon. Every time a retailer is visited the quantity delivered is such that the maximum level of inventory is reached. This inventory policy is inspired, in a deterministic setting, by the classical order-up-to level policy, widely studied in inventory theory (see Axsäter and Rosling 1994 for an overview of inventory policies in multilevel systems). A unique decision maker, typically, a logistics manager, has to determine a shipping policy that minimizes the sum of transportation costs and inventory costs both at the supplier and at the retailers. A shipping policy consists of determining for each delivery time instant the set of retailers to visit, the quantity of each product to ship to each retailer, and the route of the vehicle. The constraints of the problem guarantee that in each route the capacity of each vehicle is not exceeded and that the level of the inventory both at the supplier and at each retailer is never lower than the minimum level. This problem, referred to as deterministic orderup-to level inventory-routing problem, is obviously NP-hard, because it reduces to the TSP in the class of instances in which the time horizon is one, the inventory costs are zero, the capacity of the vehicle is infinite, and all the retailers need to be served. The scope of this paper is twofold. On one hand, we aim at solving the above problem. On the other hand, we intend to study the impact of the objective function on the problem solution. Because the problem is very complex and the exact solution would be impractical, we propose a constructive heuristic algorithm in which at each iteration a retailer is inserted in the solution. For each retailer, the heuristic builds a network to represent the incremental costs due to the insertion of the retailer in the solution. The shortest path of the network identifies a min cost policy for adding the retailer to the solution. To evaluate the impact of the objective function on the problem solution, we consider different objective functions that include some of the cost components only: the transportation cost only, the inventory costs at the retailers only, and the sum of the transportation cost and the inventory cost at the supplier. In this way we obtain three variants of the problem, never studied in the literature to the best of our knowledge. We try here to 120 Transportation Science/Vol. 36, No. 1, February 2002

3 give a partial answer to the question of how important the coordination issue is, evaluating which of the solutions is closer to the solution in which the total cost is minimized, the impact of each cost on the total cost, and how the objective function affects each type of cost. To have a better understanding of the model, in this paper we investigate the case of a single product and a single vehicle. The model and the proposed heuristic can be extended easily to the case of multiple products and vehicles. The paper is organized as follows. In 1 the problem is described and in 2 the heuristic algorithm is presented. The variants of the problem, with different objective functions, are introduced in 3. Finally, in 4, the computational results obtained on randomly generated problem instances are shown and discussed. 1. Problem Description We consider a logistic network in which a product is shipped from a common supplier 0 to a set M = 1 2 n of retailers over a given time horizon H. In each discrete time instant t = 1 2 H a quantity r it of the product is absorbed at the retailer i M and a quantity r 0t is made available at the supplier. Each retailer i M defines a minimum level L i and a maximum level U i of the inventory of the product. If retailer i is visited at time t, then the quantity x it of product shipped to retailer i is such that the level of the inventory in i reaches its maximum level U i (order-up-to level policy). More precisely, if we denote by I it the level of the inventory at retailer i at time t, then x it is either equal to U i I it if a shipment to i is performed at time t, or equal to 0 otherwise. Shipments from the supplier to the retailers can be performed in each time instant t by a vehicle with given capacity C, and routing is allowed. In each route the vehicle visits the retailers that must be served at the same time instant. We assume that any combination of customers can be visited in a single delivery time instant. The transportation cost c ij from i to j, with i j M = M 0, is known. Therefore, given a route, the corresponding transportation cost is simply obtained by summing up the cost of the arcs that belong to the route. An inventory cost is charged both at the supplier and at the retailers. If we denote by h i the unit inventory cost at node i M, then the total inventory cost over the time horizon can be computed as follows. At the supplier 0 the level of the inventory at time t + 1 is given by the level at time t, plus the quantity of product made available at time t, minus the total quantity shipped to the retailers at time t, that is B t+1 = B t + r 0t i M x it where B 0 (the starting level of the inventory) is given, r 00 = 0 and x i0 = 0, i M. Therefore, the total inventory cost at the supplier is t h 0B t, where = H + 1. At each retailer i M, the level of the inventory at time t + 1 is given by the level at time t plus the quantity of product shipped from the supplier to the retailer i at time t minus the quantity of product absorbed at time t; that is, I it+1 = I it + x it r it where I i0 (the starting level of the inventory) is given, and r i0 = x i0 = 0. Therefore, the total inventory cost at the retailer i is t h ii it. The time instant H + 1is included in the computation of the inventory cost to take into account the consequences of the operations performed at time H. The problem is to determine for each retailer i M a set i of delivery time instants, and for each time instant t a route R t that visits all the retailers served at time t such that the sum of the transportation cost and of the inventory cost both at the supplier and at the retailers is minimized and the following constraints are satisfied: (1) Capacity Constraints. They guarantee that the total quantity of the product loaded on the vehicle is not greater than the transportation capacity: x it C t (1) i M (2) Stockout Constraints at the Supplier. They guarantee that the level of the inventory B t is nonnegative in each time instant t : x it B t t (2) i M Transportation Science/Vol. 36, No. 1, February

4 (3) Stockout Constraints at the Retailers. They guarantee that for each retailer i M the level of the inventory I it in each time instant t is not lower than the minimum level L i : I it L i t i M (3) Note that the problem in which several products have to be shipped from the supplier to the retailers can be handled easily by this model. It is sufficient to replace each retailer i with a set of retailers, one for each product absorbed by the retailer i, setting to zero the transportation cost between nodes of the same set and setting to the transportation cost between the retailers the transportation cost between nodes of different sets. 2. A Heuristic Algorithm We propose a two-step heuristic algorithm to solve the order-up-to level inventory-routing problem. In the algorithm the retailers are ranked in the nondecreasing order of the average number of time units needed to consume the quantity U i L i and the retailers with the same number of time units are ranked in the nonincreasing order of U i L i. In the initialization phase of the algorithm, referred to as Start, a feasible solution of the problem is built by an iterative procedure that inserts a retailer at each iteration. When retailer i is considered, a set of delivery time instants is determined by solving a shortest-path problem on an acyclic network in which every node is a possible delivery time instant (procedure Assign). Then, for each of the selected delivery time instants, the retailer is inserted in the route traveled by the vehicle by applying the well-known rule of insertion at cheapest cost (procedure Insert). In the second phase of the algorithm, referred to as Improve, the current solution is improved iteratively. At each iteration, a pair of retailers is temporarily removed from the current solution by using the procedure Remove. Then, each of the removed retailers is inserted in the current solution as in the procedure Start. Finally, if this reduces the total cost, then the solution is modified accordingly. The second step of the algorithm is repeated as long as an improvement in the total cost is reached. The algorithm can be formally described as follows. Heuristic Algorithm 0. Sort the set of retailers M in the nondecreasing order of the ratio between U i L i and t r it. If there exist retailers with the same 1 H ratio, sort them in the nonincreasing order of U i L i. Rename the retailers accordingly. 1. Start. For s = 1 2 n Determine for the retailer s M a set s of delivery time instants by using the procedure Assign. For each time instant t s insert the retailer s in the route R t traveled by the vehicle at time t by using the procedure Insert. 2. Improve. Let TC be the total cost of the solution. (a) For s = 1 2 n For i = n n 1 1 and i s R t = R t, t. Remove the retailer s from the routes R t by using the procedure Remove. Do the same for the retailer i. Determine for i a new set of delivery time instants by using the procedure Assign and insert the retailer in the routes R t by using the procedure Insert. Do the same for the retailer s. Let TC be the cost of the obtained solution. If TC < TC, then adopt the new solution. (b) If a new solution has been adopted for at least a pair of retailers, then go to (a) Determiningthe Delivery Time Instants In this section we describe the procedure Assign which is used to determine a feasible set of delivery time instants for each retailer s. This procedure works on an acyclic network G s s s s s, in which each element of the set s is a node that corresponds to a discrete time instant between 0 and H + 1 and each element a s kt of the set s is an arc that exists if no stockout occurs in s whenever s is not visited between k and t. Therefore, each path on the network between 0 and H + 1 represents a set of delivery time instants 122 Transportation Science/Vol. 36, No. 1, February 2002

5 for s that satisfy the stockout constraints (3). Each element q s kt of the set s is a weight on the arc a s kt that represents the quantity of product to deliver at time t, and each element p s kt of the set s is a weight on the arc a s kt used in order to determine the shortest path between 0 and H +1 on the network, i.e. a good set of delivery time instants for s. Let us describe in more detail the sets s, s, and s. The set s has for elements the arcs that satisfy the stockout constraints (3) at the retailer s; in particular, the arc a s 0t,1 t H +1, exists if t j=1 r sj 1 I s0 L s and the arc a s kt,1 k<t H + 1, exists if t j=k+1 r sj 1 U s L s. Note that if the arc a s 0H+1 exists, then a feasible policy is to not visit the retailer during the time horizon. The set s is a set of weights in which each element q s kt, associated with the arc a s kt, represents the quantity of product to ship to s at time t. Given that an order-up-to level policy is adopted, then the quantity q s kt is such that the maximum level of the inventory U s is reached in s, that is, q s kt = t j=k+1 r sj 1 for each arc a s kt with 1 k<t H and q0t s = U s I s0 + t j=1 r sj 1 for each arc a s 0t with 1 t H. Note that q s kh+1,0 k H is obviously equal to 0, given that a shipment cannot be performed in H + 1. Finally, the set s is a set of weights in which each element p s kt associated with the arc a s kt represents the estimate of the variation in the total cost obtained by including in the current solution a visit of the retailer s at time t, given that the previous visit has been at time k. For each arc a s kt the weight p s kt is computed on the basis of the partial solution generated by the algorithm before applying this procedure, that is, on the basis of the route R t traveled by the vehicle at time t, of the level of the inventory B t at the supplier, and of the level of inventory I st at the retailer s at time t. If this partial solution does not include any retailer, then each route R t, t, is empty and the level of the inventory B t at the supplier is equal to the level obtained if no shipments occur up to time t, that is, B t = B 0 + t j=1 r 0j 1, for each time instant t. Given the partial solution, the weight p s kt is computed as the sum of three components. The first one, c t s,is the estimate of the variation in the transportation cost obtained if the retailer s is served at time t. This estimate is 2c 0s if no retailers are visited at time t in the partial solution, i.e., if R t =. Otherwise, c t s is computed by taking into account that the retailer s has to be inserted between two of the nodes of the route R t ; given that in the algorithm the rule of insertion at cheapest cost is used (see, for instance, Rosenkrantz et al. 1977), then s would be inserted between the node i R t and its successor su i R t such that i = arg min c i s + c s su i c i su i i R t therefore, the estimate c t s of the variation in the transportation cost is c i s + c s su i c i su i. Obviously, if the capacity constraint (1) at time t is violated, then c t s =+. The second component of the weight ps kt is the estimate B s kt of the variation in the inventory cost at the supplier. This estimate is computed by considering that if a quantity q s kt of product is shipped to the retailer s at time t, then the level of the inventory B t of the supplier decreases by a quantity q s kt for all the time instants between t + 1 and H + 1. Therefore, B s kt = h 0 H +1 t q s kt. Obviously, if the stockout constraints (2) at the supplier are violated, then B s kt =+. Finally, the third component of the weight p s kt is the estimate Ĩ s kt of the variation in the inventory cost at the retailer s. This estimate is computed by considering that every time the retailer s is visited the level of the inventory in s reaches its maximum value U s and that it then decreases during the time on the basis of the quantities absorbed in s. Therefore, if a shipment to s is performed in t and the previous shipment has been in k, then the estimate Ĩ s kt of the variation in the inventory cost in s is h t s j=k+1 U s j l=k+1 r sl 1, while, if the shipment performed at time t is the first shipment to s during the time horizon, then Ĩ s kt = h t s j=1 I s0 j l=1 r sl 1. In conclusion, the weight p s kt associated to the arc a s kt is p s kt = cs t + B s kt + Ĩ s kt Once the weight p s kt is computed for each arc a s kt s, the procedure determines the shortest path between 0 and H +1 by using an algorithm for acyclic networks (see, for instance, Hu 1982) to obtain a set of delivery time instants for s that allows us to minimize the total cost. Finally, the procedure includes in the set s of the selected delivery time instants for s the intermediate nodes that belong to the shortest path. The procedure Assign can be formally described as follows. Transportation Science/Vol. 36, No. 1, February

6 Procedure Assign Build the acyclic network G s s s s s. Determine the shortest path between 0 and H +1 on the basis of the weights in s. If the length of the shortest path is finite, then include in the set s the intermediate nodes that belong to the shortest path; otherwise, the algorithm is not able to find a feasible solution. Note that the total quantity Qt s of product shipped to s up to each delivery time instant t, 1 t H, is independent of the path between 0 and t selected on the network G s s s s s. In fact, if t 1 t 2 t n are the delivery time instants selected up to time t, with t = t n, then the total quantity Qt s shipped to s up to time t is q0t s 1 +qt s 1 t 2 + +qt s n 1 t n, that is equal to U s I s0 + t j=1 r sj 1, asq0t s 1 = U s I s0 + t 1 j=1 r sj 1 and qt s m t m+1 = tm+1 j=1+t m r sj 1,1 m<n. Therefore, Qt s is independent of the selected delivery time instants t 1 t 2 t n 1. Moreover, note that the total quantity K s of product shipped to s during the time horizon depends on the last delivery time instant ˆt selected for s; in fact, K s = Q sˆt. Therefore, Ks can be different from the total quantity t r st of product absorbed from s during the time horizon Insertingand Removinga Retailer In this section we describe the procedures Insert and Remove which are used during the algorithm to insert and to remove, respectively, a retailer s from the route R t traveled by the vehicle at time t. Let us first consider the procedure Insert. As described in the previous section, two different situations can happen when the retailer s has to be inserted in the route R t. The first one happens when the route R t is empty; in this case, the insertion of the retailer gives a route composed of only the arcs 0 s and s 0. The second one happens when the route R t already contains some retailers; in this case, the rule of insertion at cheapest cost described in the previous section is used. The procedure can be formally described as follows. Procedure Insert If R t =, then R t = 0 s 0. Else Select the retailer i such that i = arg min i R t c i s + c s su i c i su i Remove from R t the arc i su i. Introduce in R t the arcs i s and s su i. The insertion of the retailer s in the route R t implies an increase in the total quantity of the product loaded on the vehicle equal to q s kt, a variation in the transportation cost equal to either 2c 0s if the route R t was empty before inserting s or c i s+c s su i c i su i otherwise, and a reduction of the level of the inventory at the supplier equal to B j+1 = B j+1 q s kt j = t H Let us now describe the procedure Remove used during the algorithm to remove the retailer s from the route R t. Two different situations can happen, depending on the fact that s is the only retailer visited in the route or not before removing it. Let pr s be the predecessor to the retailer s in the route R t. Procedure Remove If R t = 0 s 0, then remove the arcs 0 s and s 0. Else Remove the arcs pr s s and s su s. Introduce the arc pr s su s. The decrease in the total quantity of the product loaded on the vehicle is q s kt, while the variation in the transportation cost is 2c 0s if only the retailer s was in the route before to remove it, and is c pr s su s c pr s s c s su s otherwise. Finally, the level of the inventory at the supplier becomes B j+1 = B j+1 + q s kt j = t H 3. Variants of the Problem In this section we consider variants of the problem described in 1, in which the aim of the decision maker is not to minimize the overall cost but to minimize some of the cost components only. All the variants share the same set of data and the same set of 124 Transportation Science/Vol. 36, No. 1, February 2002

7 constraints of the problem described above. The only difference is in the objective function. In the first variant the aim of the decision maker is to minimize the sum of the inventory cost at the supplier and of the transportation cost only, without any regard to the inventory costs at the retailers. This happens when the shipping policy is defined by the supplier who organizes the routing and determines at its discretion when to visit each retailer, under the condition that the level of the inventory of each retailer is never lower than the given minimum. In this case, the supplier has a global view of his own internal costs but does not care about the retailers costs. This may be due to a lack of interest in the retailers costs when supplier and retailers are different companies; instead, it may be simply due to a lack of coordination in the case in which the supplier and the retailers are part of the same company. This first variant is NP-hard because it has the TSP as a particular problem and can be heuristically solved by a modified version of the algorithm described in the previous section. The only required change is in the procedure Assign where the weights p s kt have to be computed here as the sum of the estimate c t s of the variation in the transportation cost and of the estimate B s kt of the variation in the inventory cost at the supplier only. The total cost TC is computed accordingly. A second interesting variant is obtained when the objective function includes only the transportation cost. This may happen when the transportation is outsourced and the agreement between the supplier and the transportation company defines the basic principles of the service, namely that the stockout situations have to be avoided. However, the transportation company can organize the deliveries on the basis of its own costs. Then, the inventory costs at the supplier and at the retailers follow from the shipping policy of the transportation company. In the case in which the whole system is owned by the same company, the focus on the transportation cost only may again be simply due to a lack of global view and to a lack of coordination. This variant remains NPhard since the routing problem remains to be solved here and can be heuristically solved by modifying the weights p s kt of the procedure Assign. Here, the weights include the estimate c t s of the variation in the transportation cost only, and the total cost TC is computed accordingly. The third variant puts the focus on the retailers costs and considers in the objective function the inventory costs of the retailers only. The optimal solution here tends to serve each retailer when the inventory reaches the minimum level, as stated by the classical order-up-to level policy. The only reason why the optimal solution may happen to serve a retailer before it is necessary is the capacity of the vehicle or the stockout constraints at the supplier. Because an optimal solution only determines the delivery time instants and leaves the routing undetermined, we consider the transportation cost as a second objective. This variant corresponds to the situation where the retailers can determine the delivery times and the routing is optimized afterwards. This hierarchical problem is again NP-hard and can be heuristically solved by computing in the procedure Assign the weights p s kt as the estimate Ĩ s kt of the variation in the inventory cost at the retailer s only and the total cost TC accordingly. The original problem and the above variants model different organizational situations of a company, with a different emphasis on what is important in the objective of the decision. In the following section, we compare the solutions obtained in the various cases, trying to derive some conclusions about the impact of the objective on the solution, both in terms of total cost and in terms of single components of the total cost. 4. Computational Results The heuristic algorithm described in 2 has been implemented in Fortran and used to solve the problems described in 1 and 3 in a set of computational experiments. We first describe how the random instances have been generated. Then we show a series of computational results to analyze the solution obtained for the problem in which the total cost is minimized and to compare the solutions obtained by the different variants of the problem. Finally, we computationally evaluate the quality of the solution generated by the heuristic algorithm. Transportation Science/Vol. 36, No. 1, February

8 4.1. Generation of the Instances The instances have been generated on the basis of the following data: Number of retailers n: 50; Time horizon H: 30; Quantity of product r it absorbed by retailer i at time t: Constant over time, i.e., r it = r i, t, and randomly generated as an integer number in the interval [10, 100]; Quantity of product r 0t made available at the supplier at time t: i M r i ; Minimum level L i of the inventory at retailer i: Randomly generated as an integer number in the interval [50, 150]; Maximum level U i of the inventory at retailer i: L i + r i g i, where g i is randomly selected from the set and represents the number of time units needed in order to consume the quantity U i L i ; Starting level I i0 of the inventory at the retailer i: U i r i ; Starting level B 0 of the inventory at the supplier: i M U i L i ; Inventory cost at retailer i M, h i : Randomly generated in the intervals [0.1, 0.5] and [0.6, 1]; Inventory cost at the supplier h 0 : 0.3 and 0.8; Transportation capacity C: i M r i, 2 i M r i, and 3 i M r i ; Transportation cost c ij : x i x j 2 + y i y j 2, where the points x i y i and x j y j are obtained by randomly generating each coordinate as an integer number in the interval [0, 500] and in the interval [0, 1000]. In all cases, random selections have been performed in accordance with a uniform distribution. The computations have been carried out on an Intel Pentium II personal computer. Other experiments have been carried out on other sets of instances, such as instances with L i = 0 and r i randomly generated as an integer number in the interval [5, 25]. Because the results obtained were similar, we do not report them Computational Analysis of the Solution We now focus on the original problem, referred to as IS + IR+ T, in which the total cost is minimized. Two hundred and forty instances have been generated on the basis of the data presented in 4.1; in particular, 10 instances have been randomly generated for each of the 24 combinations obtained on the basis of the different values or intervals of the last four parameters. The computational results allow us to evaluate the distribution of the total cost on the cost components (transportation cost, inventory cost at the retailers, and inventory cost at the supplier) and to carry out a sensitivity analysis on the total cost. The obtained results are shown in Tables 1 and 2 and in Figure 1. Table 1 allows us to evaluate how the total cost is distributed on the different cost components in three pairs of classes of instances, where a pair of classes covers all the tested instances. Each class is composed of all the instances for which a given parameter has been generated in the same interval or has the same value. Three different parameters have been considered: the transportation cost c ij, the inventory cost at the retailers h i, and the inventory cost at the supplier h 0. For each of the parameters, two intervals or values are considered: [0, 500] and [0, 1000] for the transportation cost, [0.1, 0.5] and [0.6, 1] for the inventory cost at the retailers, and, finally, 0.3 and 0.8 for the inventory cost at the supplier. In the following, the first interval or value of each of the parameters will be called low and the second high. The table is organized as follows. Each row corresponds to a class of instances; the first column gives the parameter that defines the class, the second one the interval in which the parameter has been generated or its value, and the remaining three columns show the corresponding average transportation cost, inventory cost at the Table 1 Average Costs Average Costs Parameters Transportation Inv. Retailers Inv. Supplier 4 c ij [0, 500] c ij [0, 1000] h i [0.1, 0.5] h i [0.6, 1] h h Transportation Science/Vol. 36, No. 1, February 2002

9 Figure 1 retailers, and inventory cost at the supplier, respectively. In each class of instances the inventory cost at the retailers is the main part of the total cost, followed by the inventory cost at the supplier and then by the transportation cost. In the first two rows of the table the average costs obtained in the classes of instances with low and high transportation cost are compared. The instances with high transportation cost have slightly lower inventory cost at the retailers but higher inventory cost at the supplier. This is an effect that can be explained by observing that an increase of the transportation cost tends to reduce the number of times a retailer is served and, therefore, to reduce the transportation cost and to increase the inventory cost at the supplier. Rows 3 and 4 allow us to compare the average costs obtained in the classes of instances with low and high inventory cost at the retailers. The results show that the instances with high inventory cost at the retailers have lower transportation cost, but higher inventory cost at the supplier. This is due to the order-up-to level policy; in fact, contrary to the classical inventory models, here the inventory cost at the retailers is minimized by serving the retailers rarely over time. Finally, in rows 5 and 6 the average costs obtained in the classes of instances with low and high inventory cost at the supplier are compared. The results show that the instances with high inventory cost at the supplier have higher transportation cost and higher inventory cost at the retailers. This is due to the fact that the supplier tends to ship more frequently. Figure 1 allows us to evaluate the impact of each cost component on the total cost in the classes of instances described above. The figure is composed of three figures: The first one (1a) compares the classes of instances with low and high transportation cost, the second one (1b) the classes of instances with low and high inventory cost at the retailers, and the third one (1c) the classes of instances with low and high inventory cost at the supplier. In each figure the comparison between the two classes of instances is performed by computing, for each class, the percentage of each cost component in the total cost. The first figure shows that in the class of instances with high transportation cost the impact of the inventory cost at the retailers on the total cost is significantly lower with respect to the class with low transportation cost; the second figure shows that in the class of instances with high inventory cost at the retailers the impact of the transportation cost on the total cost is significantly lower with respect to the class with low inventory cost at the retailers; finally, the third figure shows that the impact of the inventory cost at the retailers and of the transportation cost does not significantly change in the classes of instances with low and high inventory cost at the supplier. In Table 2 a sensitivity analysis on the total cost in a given instance is presented. Each row of the table corresponds to a parameter of the instance and shows the Transportation Science/Vol. 36, No. 1, February

10 Table 2 Sensitivity Analysis of the Total Cost c ij h i h L i C percent variation of the total cost obtained by reducing and increasing, respectively, the parameter of a given percentage of its original value. The parameters of the instance have been generated as in 4.1, with the exception of the ones that are involved in the sensitivity analysis, namely the transportation cost, the inventory cost at the retailers, the inventory cost at the supplier, the minimum level of the inventory at the retailers and, finally, the transportation capacity. These parameters have been obtained as follows: First, the transportation costs have been generated in the interval [0, 500], the inventory cost at the retailers and at the supplier are set equal to 1, the minimum level of the inventory at the retailers is generated in the interval [50, 150] and, finally, the transportation capacity is set equal to i r i. Then, each of these values has been multiplied by a factor, which takes value 0.5 for all the parameters, with the exception of the transportation capacity for which takes value 1.5. The total cost of this instance is 634,014. The sensitivity analysis has been carried out by computing the percent increase of the cost obtained by changing, for one parameter at a time, the value of with respect to the cost of the basic instance. In particular, has been set equal to ± 0 4, corresponding to an increase and a decrease, respectively, of 80% for the transportation costs, the inventory costs, and the minimum level of the inventory at the retailers, and of about 26.7% for the transportation capacity. The first three rows of the table show the variations obtained in the total cost by varying the transportation cost, the inventory cost at the retailers, and, finally, the inventory cost at the supplier by 80% of their original value, respectively. The variations in the transportation cost have a very low impact on the total cost, while the contrary happens for the same variations in the inventory cost at the supplier and, in particular, in the inventory cost at the retailers. This behavior reflects the fact that the different cost components have a significantly different importance in the total cost. The last two rows of the table show the variations obtained in the total cost by varying by about 80% and 26.7% the minimum level of the inventory at the retailers and the transportation capacity, respectively. The results show that, as expected, these parameters do not significantly affect the total cost and that the total cost increases when the former parameter increases and when the latter one decreases Computational Comparison of the Solutions Obtained with Different Objective Functions We now compare the solutions obtained for the problem IS + IR+ T with the solutions obtained for its variants, namely, problem IS + T, in which the sum of the inventory cost at the supplier and of the transportation cost only is minimized; problem T, in which the transportation cost is minimized only; and problem IR, in which first the inventory cost at the retailers and then the transportation cost are minimized. The analysis has been carried out on the instances generated in 4.2. The obtained results are shown in Tables 3 5 and in Figure 2. The first row of Table 3 presents the average total cost of the different problems on all the instances. The results show that the average total cost of problem T, in which only the transportation cost is minimized, is closer to the average total cost of problem IS+IR+T than the total cost generated by problem IS + T and by problem IR. The last three rows of Table 3 show the importance of each type of cost in the total cost. While Table 3 is especially interesting when read column by column, Table 4 has to be read row by row. It gives, for each type of cost (total cost, transportation cost, inventory cost at the retailers, and, finally, inventory cost at the supplier), the average percent increase Table 3 Average Costs IS+ IR+ T IR T IS+ T 4 Total cost Transportation cost Inv. cost retailers Inv. cost supplier Transportation Science/Vol. 36, No. 1, February 2002

11 Table 4 Percent Increase of the Costs IS+ IR+ T IR T IS+ T 4 Total cost Transportation cost Inv. cost retailers Inv. cost supplier of the cost of each problem with respect to the minimum corresponding cost obtained by solving the four problems. The different problems, with substantially different objective functions, have obtained total costs which are not very different from each other, with a maximum increase of 14% corresponding to problem IR. The percentage of 0.1% corresponding to problem IS + IR+ T is due to the fact that in a small subset of the tested instances the best solution has been obtained by a variant of the problem instead of by the problem IS+IR+T. This is because the problems are heuristically solved. The last three rows of Table 4 allow us to evaluate how the goal of the decision maker affects each type of cost. The problems IR and T have obtained, as expected, the minimum inventory cost at the retailers and the minimum transportation cost, respectively, on each instance. Moreover, the problem IS+ T has obtained the minimum inventory cost at the supplier on almost all the instances. Contrary to the situation of the total cost, which is similar in the different problems, the different cost components give substantially different contributions to the total cost in the different problems. Table 5 gives the average number of visits and the average quantity of product delivered during the time horizon obtained in the different problems. The results show that problem IR gives the minimum number of visits, followed by problem T, then by problem IS + IR+ T and, finally, by problem IS + T. The reason can be found in the trade-off existing between the inventory cost at the retailers and Table 5 Average Number of Visits and Delivery Quantity IS+ IR+ T IR T IS+ T Number of visits Delivery quantity the transportation cost on one hand and the inventory cost at the supplier on the other hand. When the inventory cost at the supplier is not included in the objective function (see problems IR and T ) the retailers are more rarely visited than in the other cases. The problems are ranked in the same order when the total quantity of product delivered during the time horizon is considered. Because in the problem formulation no condition is imposed on the inventory at the end of the horizon, the different problems create different ending situations. Since a visit brings the level of the inventory of the visited retailers to the maximum level, when the visits are frequent the level of the inventory at the retailers tends to be high at the end of the time horizon. This implies that, as the initial inventory situation and the quantities absorbed by the retailers on the horizon are identical for all problems, in the problems with more frequent visits the total delivered quantity is higher. In Figure 2 we consider four classes of instances obtained on the basis of the combinations of low and high values of inventory cost at the supplier and of the inventory cost at the retailers, as defined in the previous section. Figures 2a and 2d show that in the instances in which the inventory cost at the retailers and the inventory cost at the supplier are both low or high, the results are very similar to the ones shown in Table 3. Figure 2b shows that when the inventory cost at the retailers is low and the inventory cost at the supplier is high the problem IS + T and the problem IS + IR+ T generate, as expected, very similar solutions. Moreover, as expected, the inventory cost at the supplier is a major part of the total cost. A different situation can be observed in Figure 2c. Figure 3 shows the percent distribution of the total cost on the various cost components obtained by the problems IR T, and IS + T on the class of instances with low transportation cost (in the interval [0, 500]) with respect to the distribution obtained in the class of instances with high transportation cost (in the interval [0, 1000]). In each of these problems the increase in the transportation cost implies a reduction of the impact of the inventory cost at the retailers, as shown in Figure 1a for the problem IS + IR+ T. Finally, in Table 6 we present a sensitivity analysis on the number of visits obtained in the problems IS + IR+ T and IS + T in a given instance. Each Transportation Science/Vol. 36, No. 1, February

12 Figure 2 Figure Transportation Science/Vol. 36, No. 1, February 2002

13 Table 6 Sensitivity Analysis of the Number of Visits IS+ IR+ T IS+ T c ij h i h L i C row corresponds to a parameter of the instance and gives the number of visits obtained in the different problems by decreasing and then increasing, respectively, the parameter of a given percentage of its original value. The sensitivity analysis has been carried out as described in 4.2 on the same instance. The number of visits obtained in the original instance is 325 for the problem IS+IR+T and 1,185 for the problem IS+T. The table clearly says that the number of visits strongly depends on the values of the parameters and on the objective function. In particular, the number of visits in the problem IS + T is substantially greater than in the problem IS+IR+T. The reason is that, as shown in particular in the second row of the table, the number of visits decreases when the inventory cost at the retailers increases. This is due to the fact that the application of the order-up-to level policy implies that the inventory cost at the retailers is minimized when the retailers are served rarely over time. Moreover, as expected, the number of visits increases when the transportation cost decreases and when the inventory cost at the supplier increases, has no variations with respect to the minimum level of the inventory at the retailers, and has relatively small variations with respect to the transportation capacity Performance Evaluation Finally, we show the results obtained in a computational experiment carried out to evaluate the quality of the solution generated by the heuristic algorithm in the problem IS + IR+ T. Since the complexity of the problem precludes the possibility of solving it optimally and no lower bounds are known, we compare the cost of the solution generated by the heuristic algorithm with the optimal cost of two intuitive policies. The first policy, referred to as Every, is to visit in each discrete time instant all the retailers on the basis of an optimal route. The second policy, referred to as Latest, is to visit in each delivery time instant t the set of retailers that will have stockout at time t + 1if not served at time t. The retailers served at time t are visited on the basis of an optimal route. Eight instances have been generated according to the different intervals and values of the parameters described in 4.1, with the only exception of the transportation capacity C, which we assumed to be large enough to guarantee a feasible solution for the policies Every and Latest, and the transportation costs c ij, which have been generated as x i x j 2 + y i y j , since the optimal solution of the TSP problems has been obtained by applying an existing exact algorithm in which the transportation costs are computed as indicated above. The computational results are shown in Table 7. The first three columns give the intervals in which the transportation costs c ij and the inventory costs at the retailers h i have been generated and the value of the inventory cost at the supplier h 0, respectively; the last two columns give the percent increase error of the optimal cost of the policies Every and Latest with respect to the cost of the solution generated by the heuristic algorithm. The results show that the solution obtained by the heuristic algorithm always outperforms the optimal solution of the two intuitive policies. In fact, the policy Every, which requires the optimal solution of one TSP per instance, gives an average error of about 14% with respect to the cost of the heuristic solution, Table 7 Percentage Increase Error of Two Intuitive Policies c ij h i h 0 Every Latest Transportation Science/Vol. 36, No. 1, February

14 while the policy Latest, which gives an average error of about 5%, requires the solution of 30 exact TSPs per instance. Note that the maximum error generated by the policy Every is obtained when the inventory cost at the retailers and the transportation cost are large and the inventory cost at the supplier is small; instead, the maximum error generated by the policy Latest is obtained when the inventory cost at the retailers and the transportation cost are small and the inventory cost at the supplier is large. Conclusions We studied a problem in which a deterministic orderup-to level policy is adopted for the minimization of the costs in a distribution system. We considered different problems obtained by different goals of the decision maker and we used a heuristic algorithm to solve each problem and to compare the solutions obtained by the different problems on randomly generated problem instances. The results show how relevant the goal is on the obtained solution. A future research issue consists of studying the production policy of the supplier and its influence on the system inventory. Acknowledgments The authors wish to thank two anonymous referees for their useful comments and suggestions. References Anily, S., A. Federgruen One warehouse multiple retailer systems with vehicle routing costs. Management Sci Axsäter, S., K. Rosling Multi-level production-inventory control: Material requirements planning or reorder point policies? Euro. J. Oper. Res Bertazzi, L., M. G. Speranza, W. Ukovich Minimization of logistic costs with given frequencies. Transportation Res. B Bramel, J., D. Simchi-Levi The Logic of Logistics. Springer, New York. Campbell, A., L. Clarke, A. Kleywegt, M. Savelsbergh The inventory routing problem. T. G. Crainic, G. Laporte, eds. Fleet Management and Logistics. Kluwer Academic Publishers, London, UK Chan, L. M. A., D. Simchi-Levi Probabilistic analyses and algorithms for three-level distribution systems. Management Sci Dror, M., M. Ball Inventory/routing: Reduction from an annual to a short-period problem. Naval Res. Logist. Quart Federgruen, A., P. Zipkin A combined vehicle routing and inventory allocation problem. Oper. Res Hall, N. G., C. N. Potts Supply chain scheduling: Batching and delivery. Working paper, University of Southampton. Hu, T. C Combinatorial Algorithms. Addison-Wesley, Reading, MA. Rosenkrantz, D. J., R. E. Stearns, P. M. Lewis An analysis of several heuristics for the traveling salesman problem. SIAM J. Comput Thomas, D. J., P. M. Griffin Coordinated supply chain management. Euro. J. Oper. Res Received: December 2000; revision received: September 2001; accepted: September Transportation Science/Vol. 36, No. 1, February 2002