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A single supplier–single retailer system with an order-up-to level inventory policy
Operations Research Letters 36 (2008) 543–546
Contents lists available at ScienceDirect
Operations Research Letters journal homepage: www.elsevier.com/locate/orl
A single supplier–single retailer system with an order-up-to level inventory policy Oğuz Solyalı, Haldun Süral ∗ Department of Industrial Engineering, Middle East Technical University, Ankara, Turkey
article
info
Article history: Received 6 February 2006 Accepted 9 May 2008 Available online 11 June 2008 Keywords: Lot-sizing Order-up-to level inventory policy Dynamic programming
a b s t r a c t We consider a two-level vendor-managed system in which external demand occurs only at a retailer and a supplier replenishes the retailer employing an order-up-to S policy over T periods. We present an O(T 3 ) algorithm to coordinate the system when S is known. We also show that S can be optimized in O(aT 3 ) time for an input parameter a. © 2008 Elsevier B.V. All rights reserved.
1. Introduction We consider a two-level vendor-managed system in which a supplier (vendor) replenishes a retailer facing deterministic timevarying demand of a single product over a finite time horizon. The retailer has a prespecified maximum stocking level S and its inventory is brought back up to S whenever replenished. The problem is to determine when to visit the retailer, and when to place an order, and in what quantity, for the supplier over T periods, so that the sum of fixed order and inventory holding costs at both locations, as well as the transportation costs are minimized. No limit is assumed on order sizes and there are no replenishment lead times. The retailer’s inventory control policy is called the deterministic order-up-to S policy, which was first introduced by Bertazzi et al. [1] in an inventory routing application (see [3] for an overview on inventory routing). Later, Bertazzi et al. [2], Pınar and Süral [6], and Solyalı[7] addressed several variants of [1]. The special case of the problems studied in [1,2,6,7] that we consider is related to multi-level lot-sizing problems. Zangwill [10] proposes a dynamic programming (DP) algorithm for an uncapacitated multi-level lot-sizing problem with concave costs at all L levels in series. The algorithm runs in O(LT 4 ) time and reduces to O(T 3 ) time if the problem has two levels (see [8]). The DP algorithm solves a minimum-cost network flow problem that characterizes the optimal extreme point solutions over a convex region for concave objective functions. Van Hoesel et al. [8] generalize these results for problems with a production capacity in the first
∗
Corresponding author. E-mail address: [email protected] (H. Süral).
0167-6377/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2008.05.005
level. Lee et al. [5] consider a lot-sizing problem in a two-level serial supply chain. However, none of these approaches can be easily extended to our problem as we impose an inventory control policy at the retailer, namely that its inventory level must be raised up to S at replenishment. Moreover, the zero inventory ordering property discussed in [9,10] does not hold and the zero inventory quantity at the end of T cannot be guaranteed due to feasibility. In addition, the problems in [5,8] are quite different from our problem since [5] models a stepwise shipment cost function while [8] deals with capacity and a general concave cost function. The order-up-to S policy is one of the most basic inventory control policies and is often applied in practice, for example the distribution of industrial gases (see [4]) and the replenishment of vending machine products, where tanks or vending machines are filled up to their capacity whenever replenished. Another example is a situation where a retailer has a limited shelf-space for a product. The manufacturer of the product replenishes the retailer store’s shelves. As in [1,2,5–10], we assume that the demand is dynamic but known with certainty, since production and distribution systems can usually be approximated by deterministic demands. Our problem is important both in its own right and as a subproblem in decomposition solution approaches to the multiple retailer versions of the problem, which are growing in importance as vendor-managed resupply becomes more prevalent in industry. In Section 2, we first define the two-level vendor-managed system problem with an order-up-to level inventory policy. We next develop an alternative solution method with the same efficiency as in [1] for the (one-level) retailer problem, and then extend this method to solve the integrated (two-level) problem in polynomial time. Section 3 discusses how to optimize S and concludes the paper.
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O. Solyalı, H. Süral / Operations Research Letters 36 (2008) 543–546
2. Problem description and solution approach
Let Nt be the set of nodes and At be the set of arcs for period t. Pk Define Drk = m=r dm (assuming Drk = 0 if k < r). Then
2.1. Problem description
N0 = I10
A single product is delivered from a supplier to a retailer over a set τ = {1, 2, . . . , T } of discrete time periods. The retailer faces external deterministic dynamic demand dt in each time period t ∈ τ and has a prespecified maximum inventory level S of the product. The supplier controls the inventories at the retailer and must replenish these by raising its inventory level up to S before the retailer’s stocks run out. Let I1t (I0t ) be the inventory levels of the retailer (supplier) at the end of time t. At any period t, the retailer receives either nothing or a delivery Qt = S − I1,t −1 from the supplier. At each t, the supplier may place an order of size Pt of the product to replenish its own stock. Since there are no replenishment lead times, any portion of Pt can be shipped to the retailer to serve demand at t. A fixed cost f0t (f1t ) is incurred at every time t that the supplier (retailer) places an order. There are transportation costs from the supplier (retailer) to the retailer (supplier), which are part of the retailer’s fixed cost. Inventory holding costs h0t and h1t are charged for each unit of product at the end of period t at the supplier and retailer locations, respectively. Let y0t = 1(y1t = 1) if a replenishment for the supplier (retailer) is made in period t ∈ τ and 0 otherwise. The problem is to find the optimal resupply policy that minimizes the total of fixed replenishment and inventory holding costs, i.e., Minimize PT t =1 (f0t y0t + f1t y1t + h0t I0t + h1t I1t ). This problem is called the integrated lot-sizing and order-up-to level problem (ILOP), and its mathematical model can be easily adapted from Pınar and Süral [6]. Below, we transform ILOP into an equivalent network problem. First, we construct only the retailer’s network. A heuristic in [1] for the multiple-retailer variant of the problem repeatedly solves the single-retailer network problem, where the nodes denote periods while the arcs represent replenishment quantities. Alternatively, we change the node definition and let the nodes denote ending inventory levels because this explicit knowledge helps us incorporate the supplier into the network. Second, we extend the retailer’s network to account for the integrated replenishments at both levels; then finding the shortest path over the extended network yields the optimal solution to the ILOP. 2.2. Replenishment of the retailer In the retailer’s network, the nodes denote the inventory level of the retailer at the end of a period and the arcs denote the amount of product supplied to the retailer at the beginning of a period. The network has two arcs emanating from each node since, in every period, the retailer either receives nothing or a delivery that raises its inventory level up to S. We let the very first node (origin) represent the initial inventory level I10 . The first arc from the origin indicates a delivery of S − I10 , and is connected to the node representing the inventory level at the end of period t = 1, which is equal to S − d1 . The second arc indicates that no delivery is made and enters a second node where the inventory level equals I10 − d1 . We thus have two nodes at t = 1 and connect them to the related nodes at t = 2. This procedure continues until all time periods are examined, so that all possible nodes (states) are created. Note that the states with negative inventory levels are disregarded since backlogging is not allowed, and all arcs with a delivery at time period t are connected to the same (destination) node at t + 1. These remarks lead to the following network in which the number of nodes at t is at most one more than that of t − 1.
and
Nt = {S − Drt |1 ≤ r ≤ t } ∪ {I10 − D1t }
∀t = 1, 2, . . . , T .
(1)
At = {(I1,t −1 , I1,t −1 − dt )}
∪{(I1,t −1 , S − dt )} I1,t −1 ∈ Nt −1
∀t = 1, 2, . . . , T .
(2)
The set At defines the arcs from nodes at time t − 1 to nodes at t. The quantity Qt on arcs can be computed using Qt = I1t − I1,t −1 + dt where I1,t −1 ∈ Nt −1 and I1t ∈ Nt . An example network for a fourperiod problem using (1) and (2) is presented in Fig. 1. Since all feasible nodes and arcs are in play, it suffices to consider only the inventory levels Nt (t = 1, . . . , T ) for an optimal solution to the problem. The number of nodes equals (1 + 2 +· · ·+ (T +1))−1, the number of arcs equals (2+4+· · ·+2T )−1, and thus the network has O(T 2 ) nodes and arcs. Finding the minimum-cost path on the network solves the single-retailer problem. Let Ft (I ) be the minimum cost of having inventory I at the end of period t, where I ∈ Nt for t = 1, 2, . . . , T , and let gt (Qt ) be the cost of Qt , where gt (Qt ) = f1t if Qt > 0 and 0 otherwise. Let F0 (I10 ) = 0. For each t = 1, 2, . . . , T , we have I ∈ Nt and Ft (I ) = h1t I
+ min Ft −1 (¯I ) + gt (Qt ) (¯I , I ) ∈ At and ¯I ∈ Nt −1 .
(3)
The optimal solution is computed from min {FT (I ) |I ∈ NT } in O(T 2 ) time, which is the same efficiency as in [1]. 2.3. Integrated supplier and retailer replenishments Now, we incorporate decisions made by the supplier into the retailer’s network. In the extended network, a node is defined by two attributes. The first (second) attribute denotes the inventory level of the retailer (supplier) at the end of a period. It is easy to see that the characteristic of an optimal replenishment policy at the supplier is the same as the well-known Wagner–Whitin property [9]. Regardless of how the retailer is resupplied, the supplier replenishment problem is equivalent to the single-level uncapacitated lot-sizing problem in [9] in that the amount ordered by the supplier in a period is the sum of the demands of some future periods (i.e., the supplier orders at time period t only when I0,t −1 = 0). However, in our case, the supplier faces different order quantities depending on the replenishments made to the retailer. We define the second attribute of the extended network for all possible ‘‘realizable’’ demands by following the paths of the retailer’s network. We assume I00 = 0 for simplicity, and I0T = 0 without loss of generality. The extended network has T nodes for each node of the retailer’s network at time period t = 1 and it thus has 2T nodes in total at t = 1. The first T nodes require that a delivery be made to the retailer at t = 1. Then the supplier may place an order of size S − I10 (demand of the retailer at t = 1), or S − (I10 − D11 ) (demand of the retailer from t = 1 to t = 2), . . . , or S − (I10 − D1,T −1 ) (sum of demands of the retailer over T ). The next T nodes denote the case of no delivery. Then the supplier orders either 0, or S − (I10 − D11 ), . . . , or S − (I10 − D1,T −1 ). At t = 2, we have T − 1 nodes for each node of the retailer’s network, since the supplier may place an order considering the demands of the retailer from t = 2. This leads to 3(T − 1) nodes at t = 2. The procedure is repeated for the remaining t to generate all nodes accordingly. The nodes at time period t − 1 with their second attributes equal to 0 are connected to 2(T − t + 1) nodes at t, since it is possible for the supplier to order the sum of the demands for a set of future periods (the multiplier 2 is due to receiving delivery or not). The nodes with their second attributes corresponding to
O. Solyalı, H. Süral / Operations Research Letters 36 (2008) 543–546
545
Fig. 1. Initial network for T = 4.1
Fig. 2. Extended network for T = 3.2
the inventory carried for (more than) the next period’s demand are connected to only one (two) node(s). Let Nte = (I1t , I0t ) be the set of nodes and Aet be the set of arcs of the extended network in t. Similar to Section 2.2, the relevant sets can be defined as follows. N0e = (I10 , 0) and Nte
= {(S − Drt , {0} ∪ {Drk |t ≤ k ≤ T − 1 }) |1 ≤ r ≤ t } ∪ {(I10 − D1t , {0} ∪ {S − (I10 − D1r ) | t ≤ r ≤ T − 1})}
1 The node in dashed lines refers to the ignored (trivial) case.
∀t = 1, 2, . . . , T . For each t = 1, 2, . . . , T and (I1,t −1 , I0,t −1 ) ∈ Nte−1 , If I0,t −1 = 0, then Aet = (I1,t −1 , 0), (S − dt , {Dtr |t − 1 ≤ r ≤ T − 1 }) ∪ (I1,t −1 , 0), (I1,t −1 − dt , {0} ∪{S − (I1,t −1 − Dtr ) |t ≤ r ≤ T − 1 })
(4)
2 In this figure, I and I denote S − (I − D ) and S − (I − D ), respectively. a b 10 11 10 12 The node in dashed lines refers to the ignored (trivial) case.
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O. Solyalı, H. Süral / Operations Research Letters 36 (2008) 543–546
Else if I1,t −1 + I0,t −1 6= S , then Aet
= (I1,t −1 , I0,t −1 ), (I1,t −1 − dt , I0,t −1 ) ∪ (I1,t −1 , I0,t −1 ), (S − dt , I0,t −1 − (S − I1,t −1 )) Else Aet = (I1,t −1 , I0,t −1 ), (S − dt , 0) .
Ft (I1 , I0 ) = h1t I1 + h0t I0 + min Ft −1 (¯I1 , ¯I0 ) + gt (Qt , Pt ) |
{(¯I1 , ¯I0 ), (I1 , I0 )} ∈ Aet and (¯I1 , ¯I0 ) ∈ Nte−1 , (5)
The quantities (Qt , Pt ) on arcs can be easily computed using Qt = I1t − I1,t −1 + dt and Pt = I0t − I0,t −1 + Qt , where (I1,t −1 , I0,t −1 ) ∈ Nte−1 and (I1t , I0t ) ∈ Nte . The entire network has 1 +
PT
t =1
[(t + 1)(T − t + 1)] − 1 nodes and 2T +
t =2 t [2(T − t + 1) + 2(T − t ) + 1] − 1 arcs. Hence, it is of size O(T 3 ). Using (4) and (5), an example network for a three-period problem is given in Fig. 2.
PT
Theorem. For an optimum solution to ILOP, it suffices to consider only the inventory levels of both the supplier and the retailer specified by Nte . Proof. It will be shown that a state not belonging to Nte cannot be in the optimal solution. Since Nt constitutes all possible states at the retailer, it is adequate to consider the inventory levels at the supplier given that of the retailer only. Let X = (I1t , I0t ) be a state in a period t (1 ≤ t < T ) constructed by ordering an amount different from the sum of the demands of the retailer in some future periods (i.e., an amount violating the Wagner–Whitin property). Then there are three cases to consider: Case 1. I0t < S − I1t . Suppose the retailer needs replenishment at t + 1. Then the supplier must place an order. It is obvious that whatever the amount ordered is, a path involving X is more costly than a path with state (I1t , 0) ∈ Nte in place of X due to the excess inventory cost incurred. The same argument also holds if the retailer does not need replenishment at t + 1. Case 2. S − I1t < I0t < S − I1t + Dt +1,T −1 . (i) Suppose the retailer does not need replenishment at t + 1 but needs one in a period j > t + 1, and I0j < S − I1j . Then a path with X is more expensive than a path through (I1t , 0) ∈ Nte as shown in Case 1. (ii) Suppose the retailer needs replenishment(s) in future period(s), and that after the replenishment(s) the inventory level of the supplier falls below S − I1k in a period, say k. Then a path through a state having just the sum of the retailer’s replenishment amount at time period t and (I1k , 0) ∈ Nke at k is cheaper than a path through X and (I1k , I0k ), where I0k < S − I1k . The reason is that excess inventory is carried between t and k, and between k and T in the path with X and (I1k , I0k ) as shown in Case 1. Case 3. I0t > S − I1t + Dt +1,T −1 . Since the supplier holds inventory that exceeds the need of the retailer, a path involving X is obviously non-optimal. Hence, a state not conforming to the Wagner–Whitin property, thus not belonging to Nte , cannot be in the optimal solution. The minimum cost path on the extended network can be found in O(T 3 ) time as follows. Let F0 (I10 , 0) = 0. For each t = 1, 2, . . . , T , we have (I1 , I0 ) ∈ Nte and
(6)
where Ft (I1 , I0 ) is the minimum cost of having inventories I1 at the retailer and I0 at the supplier at the end of period t, and gt (Qt , Pt ) is the cost of having a delivery of size Qt to the retailer and Pt to the supplier. If Qt > 0, gt includes f1t , and is 0 otherwise. If Pt > 0, gt also includes f0t , and is 0 otherwise. Then the optimal solution value to ILOP is min FT (I1 , I0 ) |(I1 , I0 ) ∈ NTe . 3. Conclusion There is no known fast method for setting the order-up-to level S in the single supplier-single retailer problem ILOP. If the demand pattern or the costs are changing with time, however, S needs to be adjusted. In general, S is a finite number and can be bounded as dmax = max{d1 , d2 , . . . , dT } ≤ S ≤ D1T . The lower limit arises because we do not allow backorders and the upper limit eliminates the excess inventory. The bounding leads to a pseudo-polynomial algorithm with O(aT 3 ) to find the optimal value of S in solving ILOP, where a = D1T − dmax + 1. Acknowledgments The authors thank Meltem Denizel (Sabancı University), David Goldsman (Georgia Institute of Technology), and an anonymous referee for helpful comments and suggestions. References [1] L. Bertazzi, G. Paletta, M.G. Speranza, Deterministic order-up-to level policies in an inventory routing problem, Transportation Science 36 (2002) 119–132. [2] L. Bertazzi, G. Paletta, M.G. Speranza, Minimizing the total cost in an integrated vendor managed inventory system, Journal of Heuristics 11 (2005) 393–419. [3] A. Campbell, L. Clarke, A. Kleywegt, M.W.P. Savelsbergh, The inventory routing problem, in: T.G. Crainic, G. Laporte (Eds.), Fleet Management and Logistics, Kluwer Academic Publishers, London, UK, 1998, pp. 95–113. [4] M. Dror, M. Ball, Inventory/Routing: Reduction from an annual to a shortperiod problem, Naval Research Logistics 34 (1987) 891–905. [5] C.Y. Lee, S. Çetinkaya, W. Jaruphongsa, A dynamic model for inventory lot sizing and outbound shipment scheduling at a third-party warehouse, Operations Research 51 (2003) 735–747. [6] Ö. Pınar, H. Süral, Coordinating inventory and transportation in vendor managed systems, in: R. Meller, et al., (Eds). Proceedings of the Material Handling Research Colloquium 2006, 2006, pp. 459–474. [7] O. Solyalı, An integrated inventory control and vehicle routing problem. M.Sc. Thesis, Department of Industrial Engineering, Middle East Technical University, Ankara, Turkey, 2005. [8] S. van Hoesel, H.E. Romeijn, D.R. Morales, A.P.M. Wagelmans, Integrated lot sizing in serial supply chains with production capacities, Management Science 51 (2005) 1706–1719. [9] H.M. Wagner, T.M. Whitin, Dynamic version of the economic lot size model, Management Science 5 (1958) 89–96. [10] W.I. Zangwill, A backlogging model and a multi-echelon model of a dynamic economic lot size production system — a network approach, Management Science 15 (1969) 506–527.
Contents lists available at ScienceDirect
Operations Research Letters journal homepage: www.elsevier.com/locate/orl
A single supplier–single retailer system with an order-up-to level inventory policy Oğuz Solyalı, Haldun Süral ∗ Department of Industrial Engineering, Middle East Technical University, Ankara, Turkey
article
info
Article history: Received 6 February 2006 Accepted 9 May 2008 Available online 11 June 2008 Keywords: Lot-sizing Order-up-to level inventory policy Dynamic programming
a b s t r a c t We consider a two-level vendor-managed system in which external demand occurs only at a retailer and a supplier replenishes the retailer employing an order-up-to S policy over T periods. We present an O(T 3 ) algorithm to coordinate the system when S is known. We also show that S can be optimized in O(aT 3 ) time for an input parameter a. © 2008 Elsevier B.V. All rights reserved.
1. Introduction We consider a two-level vendor-managed system in which a supplier (vendor) replenishes a retailer facing deterministic timevarying demand of a single product over a finite time horizon. The retailer has a prespecified maximum stocking level S and its inventory is brought back up to S whenever replenished. The problem is to determine when to visit the retailer, and when to place an order, and in what quantity, for the supplier over T periods, so that the sum of fixed order and inventory holding costs at both locations, as well as the transportation costs are minimized. No limit is assumed on order sizes and there are no replenishment lead times. The retailer’s inventory control policy is called the deterministic order-up-to S policy, which was first introduced by Bertazzi et al. [1] in an inventory routing application (see [3] for an overview on inventory routing). Later, Bertazzi et al. [2], Pınar and Süral [6], and Solyalı[7] addressed several variants of [1]. The special case of the problems studied in [1,2,6,7] that we consider is related to multi-level lot-sizing problems. Zangwill [10] proposes a dynamic programming (DP) algorithm for an uncapacitated multi-level lot-sizing problem with concave costs at all L levels in series. The algorithm runs in O(LT 4 ) time and reduces to O(T 3 ) time if the problem has two levels (see [8]). The DP algorithm solves a minimum-cost network flow problem that characterizes the optimal extreme point solutions over a convex region for concave objective functions. Van Hoesel et al. [8] generalize these results for problems with a production capacity in the first
∗
Corresponding author. E-mail address: [email protected] (H. Süral).
0167-6377/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2008.05.005
level. Lee et al. [5] consider a lot-sizing problem in a two-level serial supply chain. However, none of these approaches can be easily extended to our problem as we impose an inventory control policy at the retailer, namely that its inventory level must be raised up to S at replenishment. Moreover, the zero inventory ordering property discussed in [9,10] does not hold and the zero inventory quantity at the end of T cannot be guaranteed due to feasibility. In addition, the problems in [5,8] are quite different from our problem since [5] models a stepwise shipment cost function while [8] deals with capacity and a general concave cost function. The order-up-to S policy is one of the most basic inventory control policies and is often applied in practice, for example the distribution of industrial gases (see [4]) and the replenishment of vending machine products, where tanks or vending machines are filled up to their capacity whenever replenished. Another example is a situation where a retailer has a limited shelf-space for a product. The manufacturer of the product replenishes the retailer store’s shelves. As in [1,2,5–10], we assume that the demand is dynamic but known with certainty, since production and distribution systems can usually be approximated by deterministic demands. Our problem is important both in its own right and as a subproblem in decomposition solution approaches to the multiple retailer versions of the problem, which are growing in importance as vendor-managed resupply becomes more prevalent in industry. In Section 2, we first define the two-level vendor-managed system problem with an order-up-to level inventory policy. We next develop an alternative solution method with the same efficiency as in [1] for the (one-level) retailer problem, and then extend this method to solve the integrated (two-level) problem in polynomial time. Section 3 discusses how to optimize S and concludes the paper.
544
O. Solyalı, H. Süral / Operations Research Letters 36 (2008) 543–546
2. Problem description and solution approach
Let Nt be the set of nodes and At be the set of arcs for period t. Pk Define Drk = m=r dm (assuming Drk = 0 if k < r). Then
2.1. Problem description
N0 = I10
A single product is delivered from a supplier to a retailer over a set τ = {1, 2, . . . , T } of discrete time periods. The retailer faces external deterministic dynamic demand dt in each time period t ∈ τ and has a prespecified maximum inventory level S of the product. The supplier controls the inventories at the retailer and must replenish these by raising its inventory level up to S before the retailer’s stocks run out. Let I1t (I0t ) be the inventory levels of the retailer (supplier) at the end of time t. At any period t, the retailer receives either nothing or a delivery Qt = S − I1,t −1 from the supplier. At each t, the supplier may place an order of size Pt of the product to replenish its own stock. Since there are no replenishment lead times, any portion of Pt can be shipped to the retailer to serve demand at t. A fixed cost f0t (f1t ) is incurred at every time t that the supplier (retailer) places an order. There are transportation costs from the supplier (retailer) to the retailer (supplier), which are part of the retailer’s fixed cost. Inventory holding costs h0t and h1t are charged for each unit of product at the end of period t at the supplier and retailer locations, respectively. Let y0t = 1(y1t = 1) if a replenishment for the supplier (retailer) is made in period t ∈ τ and 0 otherwise. The problem is to find the optimal resupply policy that minimizes the total of fixed replenishment and inventory holding costs, i.e., Minimize PT t =1 (f0t y0t + f1t y1t + h0t I0t + h1t I1t ). This problem is called the integrated lot-sizing and order-up-to level problem (ILOP), and its mathematical model can be easily adapted from Pınar and Süral [6]. Below, we transform ILOP into an equivalent network problem. First, we construct only the retailer’s network. A heuristic in [1] for the multiple-retailer variant of the problem repeatedly solves the single-retailer network problem, where the nodes denote periods while the arcs represent replenishment quantities. Alternatively, we change the node definition and let the nodes denote ending inventory levels because this explicit knowledge helps us incorporate the supplier into the network. Second, we extend the retailer’s network to account for the integrated replenishments at both levels; then finding the shortest path over the extended network yields the optimal solution to the ILOP. 2.2. Replenishment of the retailer In the retailer’s network, the nodes denote the inventory level of the retailer at the end of a period and the arcs denote the amount of product supplied to the retailer at the beginning of a period. The network has two arcs emanating from each node since, in every period, the retailer either receives nothing or a delivery that raises its inventory level up to S. We let the very first node (origin) represent the initial inventory level I10 . The first arc from the origin indicates a delivery of S − I10 , and is connected to the node representing the inventory level at the end of period t = 1, which is equal to S − d1 . The second arc indicates that no delivery is made and enters a second node where the inventory level equals I10 − d1 . We thus have two nodes at t = 1 and connect them to the related nodes at t = 2. This procedure continues until all time periods are examined, so that all possible nodes (states) are created. Note that the states with negative inventory levels are disregarded since backlogging is not allowed, and all arcs with a delivery at time period t are connected to the same (destination) node at t + 1. These remarks lead to the following network in which the number of nodes at t is at most one more than that of t − 1.
and
Nt = {S − Drt |1 ≤ r ≤ t } ∪ {I10 − D1t }
∀t = 1, 2, . . . , T .
(1)
At = {(I1,t −1 , I1,t −1 − dt )}
∪{(I1,t −1 , S − dt )} I1,t −1 ∈ Nt −1
∀t = 1, 2, . . . , T .
(2)
The set At defines the arcs from nodes at time t − 1 to nodes at t. The quantity Qt on arcs can be computed using Qt = I1t − I1,t −1 + dt where I1,t −1 ∈ Nt −1 and I1t ∈ Nt . An example network for a fourperiod problem using (1) and (2) is presented in Fig. 1. Since all feasible nodes and arcs are in play, it suffices to consider only the inventory levels Nt (t = 1, . . . , T ) for an optimal solution to the problem. The number of nodes equals (1 + 2 +· · ·+ (T +1))−1, the number of arcs equals (2+4+· · ·+2T )−1, and thus the network has O(T 2 ) nodes and arcs. Finding the minimum-cost path on the network solves the single-retailer problem. Let Ft (I ) be the minimum cost of having inventory I at the end of period t, where I ∈ Nt for t = 1, 2, . . . , T , and let gt (Qt ) be the cost of Qt , where gt (Qt ) = f1t if Qt > 0 and 0 otherwise. Let F0 (I10 ) = 0. For each t = 1, 2, . . . , T , we have I ∈ Nt and Ft (I ) = h1t I
+ min Ft −1 (¯I ) + gt (Qt ) (¯I , I ) ∈ At and ¯I ∈ Nt −1 .
(3)
The optimal solution is computed from min {FT (I ) |I ∈ NT } in O(T 2 ) time, which is the same efficiency as in [1]. 2.3. Integrated supplier and retailer replenishments Now, we incorporate decisions made by the supplier into the retailer’s network. In the extended network, a node is defined by two attributes. The first (second) attribute denotes the inventory level of the retailer (supplier) at the end of a period. It is easy to see that the characteristic of an optimal replenishment policy at the supplier is the same as the well-known Wagner–Whitin property [9]. Regardless of how the retailer is resupplied, the supplier replenishment problem is equivalent to the single-level uncapacitated lot-sizing problem in [9] in that the amount ordered by the supplier in a period is the sum of the demands of some future periods (i.e., the supplier orders at time period t only when I0,t −1 = 0). However, in our case, the supplier faces different order quantities depending on the replenishments made to the retailer. We define the second attribute of the extended network for all possible ‘‘realizable’’ demands by following the paths of the retailer’s network. We assume I00 = 0 for simplicity, and I0T = 0 without loss of generality. The extended network has T nodes for each node of the retailer’s network at time period t = 1 and it thus has 2T nodes in total at t = 1. The first T nodes require that a delivery be made to the retailer at t = 1. Then the supplier may place an order of size S − I10 (demand of the retailer at t = 1), or S − (I10 − D11 ) (demand of the retailer from t = 1 to t = 2), . . . , or S − (I10 − D1,T −1 ) (sum of demands of the retailer over T ). The next T nodes denote the case of no delivery. Then the supplier orders either 0, or S − (I10 − D11 ), . . . , or S − (I10 − D1,T −1 ). At t = 2, we have T − 1 nodes for each node of the retailer’s network, since the supplier may place an order considering the demands of the retailer from t = 2. This leads to 3(T − 1) nodes at t = 2. The procedure is repeated for the remaining t to generate all nodes accordingly. The nodes at time period t − 1 with their second attributes equal to 0 are connected to 2(T − t + 1) nodes at t, since it is possible for the supplier to order the sum of the demands for a set of future periods (the multiplier 2 is due to receiving delivery or not). The nodes with their second attributes corresponding to
O. Solyalı, H. Süral / Operations Research Letters 36 (2008) 543–546
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Fig. 1. Initial network for T = 4.1
Fig. 2. Extended network for T = 3.2
the inventory carried for (more than) the next period’s demand are connected to only one (two) node(s). Let Nte = (I1t , I0t ) be the set of nodes and Aet be the set of arcs of the extended network in t. Similar to Section 2.2, the relevant sets can be defined as follows. N0e = (I10 , 0) and Nte
= {(S − Drt , {0} ∪ {Drk |t ≤ k ≤ T − 1 }) |1 ≤ r ≤ t } ∪ {(I10 − D1t , {0} ∪ {S − (I10 − D1r ) | t ≤ r ≤ T − 1})}
1 The node in dashed lines refers to the ignored (trivial) case.
∀t = 1, 2, . . . , T . For each t = 1, 2, . . . , T and (I1,t −1 , I0,t −1 ) ∈ Nte−1 , If I0,t −1 = 0, then Aet = (I1,t −1 , 0), (S − dt , {Dtr |t − 1 ≤ r ≤ T − 1 }) ∪ (I1,t −1 , 0), (I1,t −1 − dt , {0} ∪{S − (I1,t −1 − Dtr ) |t ≤ r ≤ T − 1 })
(4)
2 In this figure, I and I denote S − (I − D ) and S − (I − D ), respectively. a b 10 11 10 12 The node in dashed lines refers to the ignored (trivial) case.
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O. Solyalı, H. Süral / Operations Research Letters 36 (2008) 543–546
Else if I1,t −1 + I0,t −1 6= S , then Aet
= (I1,t −1 , I0,t −1 ), (I1,t −1 − dt , I0,t −1 ) ∪ (I1,t −1 , I0,t −1 ), (S − dt , I0,t −1 − (S − I1,t −1 )) Else Aet = (I1,t −1 , I0,t −1 ), (S − dt , 0) .
Ft (I1 , I0 ) = h1t I1 + h0t I0 + min Ft −1 (¯I1 , ¯I0 ) + gt (Qt , Pt ) |
{(¯I1 , ¯I0 ), (I1 , I0 )} ∈ Aet and (¯I1 , ¯I0 ) ∈ Nte−1 , (5)
The quantities (Qt , Pt ) on arcs can be easily computed using Qt = I1t − I1,t −1 + dt and Pt = I0t − I0,t −1 + Qt , where (I1,t −1 , I0,t −1 ) ∈ Nte−1 and (I1t , I0t ) ∈ Nte . The entire network has 1 +
PT
t =1
[(t + 1)(T − t + 1)] − 1 nodes and 2T +
t =2 t [2(T − t + 1) + 2(T − t ) + 1] − 1 arcs. Hence, it is of size O(T 3 ). Using (4) and (5), an example network for a three-period problem is given in Fig. 2.
PT
Theorem. For an optimum solution to ILOP, it suffices to consider only the inventory levels of both the supplier and the retailer specified by Nte . Proof. It will be shown that a state not belonging to Nte cannot be in the optimal solution. Since Nt constitutes all possible states at the retailer, it is adequate to consider the inventory levels at the supplier given that of the retailer only. Let X = (I1t , I0t ) be a state in a period t (1 ≤ t < T ) constructed by ordering an amount different from the sum of the demands of the retailer in some future periods (i.e., an amount violating the Wagner–Whitin property). Then there are three cases to consider: Case 1. I0t < S − I1t . Suppose the retailer needs replenishment at t + 1. Then the supplier must place an order. It is obvious that whatever the amount ordered is, a path involving X is more costly than a path with state (I1t , 0) ∈ Nte in place of X due to the excess inventory cost incurred. The same argument also holds if the retailer does not need replenishment at t + 1. Case 2. S − I1t < I0t < S − I1t + Dt +1,T −1 . (i) Suppose the retailer does not need replenishment at t + 1 but needs one in a period j > t + 1, and I0j < S − I1j . Then a path with X is more expensive than a path through (I1t , 0) ∈ Nte as shown in Case 1. (ii) Suppose the retailer needs replenishment(s) in future period(s), and that after the replenishment(s) the inventory level of the supplier falls below S − I1k in a period, say k. Then a path through a state having just the sum of the retailer’s replenishment amount at time period t and (I1k , 0) ∈ Nke at k is cheaper than a path through X and (I1k , I0k ), where I0k < S − I1k . The reason is that excess inventory is carried between t and k, and between k and T in the path with X and (I1k , I0k ) as shown in Case 1. Case 3. I0t > S − I1t + Dt +1,T −1 . Since the supplier holds inventory that exceeds the need of the retailer, a path involving X is obviously non-optimal. Hence, a state not conforming to the Wagner–Whitin property, thus not belonging to Nte , cannot be in the optimal solution. The minimum cost path on the extended network can be found in O(T 3 ) time as follows. Let F0 (I10 , 0) = 0. For each t = 1, 2, . . . , T , we have (I1 , I0 ) ∈ Nte and
(6)
where Ft (I1 , I0 ) is the minimum cost of having inventories I1 at the retailer and I0 at the supplier at the end of period t, and gt (Qt , Pt ) is the cost of having a delivery of size Qt to the retailer and Pt to the supplier. If Qt > 0, gt includes f1t , and is 0 otherwise. If Pt > 0, gt also includes f0t , and is 0 otherwise. Then the optimal solution value to ILOP is min FT (I1 , I0 ) |(I1 , I0 ) ∈ NTe . 3. Conclusion There is no known fast method for setting the order-up-to level S in the single supplier-single retailer problem ILOP. If the demand pattern or the costs are changing with time, however, S needs to be adjusted. In general, S is a finite number and can be bounded as dmax = max{d1 , d2 , . . . , dT } ≤ S ≤ D1T . The lower limit arises because we do not allow backorders and the upper limit eliminates the excess inventory. The bounding leads to a pseudo-polynomial algorithm with O(aT 3 ) to find the optimal value of S in solving ILOP, where a = D1T − dmax + 1. Acknowledgments The authors thank Meltem Denizel (Sabancı University), David Goldsman (Georgia Institute of Technology), and an anonymous referee for helpful comments and suggestions. References [1] L. Bertazzi, G. Paletta, M.G. Speranza, Deterministic order-up-to level policies in an inventory routing problem, Transportation Science 36 (2002) 119–132. [2] L. Bertazzi, G. Paletta, M.G. Speranza, Minimizing the total cost in an integrated vendor managed inventory system, Journal of Heuristics 11 (2005) 393–419. [3] A. Campbell, L. Clarke, A. Kleywegt, M.W.P. Savelsbergh, The inventory routing problem, in: T.G. Crainic, G. Laporte (Eds.), Fleet Management and Logistics, Kluwer Academic Publishers, London, UK, 1998, pp. 95–113. [4] M. Dror, M. Ball, Inventory/Routing: Reduction from an annual to a shortperiod problem, Naval Research Logistics 34 (1987) 891–905. [5] C.Y. Lee, S. Çetinkaya, W. Jaruphongsa, A dynamic model for inventory lot sizing and outbound shipment scheduling at a third-party warehouse, Operations Research 51 (2003) 735–747. [6] Ö. Pınar, H. Süral, Coordinating inventory and transportation in vendor managed systems, in: R. Meller, et al., (Eds). Proceedings of the Material Handling Research Colloquium 2006, 2006, pp. 459–474. [7] O. Solyalı, An integrated inventory control and vehicle routing problem. M.Sc. Thesis, Department of Industrial Engineering, Middle East Technical University, Ankara, Turkey, 2005. [8] S. van Hoesel, H.E. Romeijn, D.R. Morales, A.P.M. Wagelmans, Integrated lot sizing in serial supply chains with production capacities, Management Science 51 (2005) 1706–1719. [9] H.M. Wagner, T.M. Whitin, Dynamic version of the economic lot size model, Management Science 5 (1958) 89–96. [10] W.I. Zangwill, A backlogging model and a multi-echelon model of a dynamic economic lot size production system — a network approach, Management Science 15 (1969) 506–527.