Regular and urgent ordering with lateral transshipment in a distribution network with stochastic pre-ordered and peddling demands

Chalida Donjuk; Jirajat Settasuk; Po-ngarm Somkun Article Regular and urgent ordering with lateral transshipment in a distribution network with stochastic pre-ordered and peddling demands Asian Journal of Shipping and Logistics (AJSL) Provided in Cooperation with: Korean Association of Shipping and Logistics, Seoul Suggested Citation: Chalida Donjuk; Jirajat Settasuk; Po-ngarm Somkun (2025) : Regular and urgent ordering with lateral transshipment in a distribution network with stochastic pre-ordered and peddling demands, Asian Journal of Shipping and Logistics (AJSL), ISSN 2352-4871, Elsevier, Amsterdam, Vol. 41, Iss. 1, pp. 61-74, https://doi.org/10.1016/j.ajsl.2025.01.004 This Version is available at: https://hdl.handle.net/10419/329757 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/ Regular and urgent ordering with lateral transshipment in a distribution network with stochastic pre-ordered and peddling demands Chalida Donjuk a , Jirajat Settasuk b , Po-ngarm Somkun a,* a Department of Industrial Engineering, Faculty of Engineering, Naresuan University, Phitsanulok, Thailand b Thirapat Drink Company Limited, Thailand ARTICLE INFO Keywords: Inventory replenishment policy (R, s, S) policy Demand classes Mathematical model Spreadsheet model Logistics ABSTRACT We propose inventory replenishment and lateral transshipment policies for a distribution network. The distribution centers place weekly regular orders and daily urgent orders to the factory. Two classes of stochastic demands, pre-ordered and peddling, cause either backlogging or lost sales when they are not satisfied. The policy was simulated using data from a case study. The results showed that the proposed policies reduced the total cost, which includes transportation costs for satisfied demands and lateral transshipment, back order and lost sale costs for unsatisfied demands, and inventory holding costs. The best policy reduced long-run total cost and was proved robust from sensitivity analysis. Finally, implementing the policy with the case study confirmed 9.97 % total cost reduction and its practical application. 1. Introduction Inventory replenishment policies are critical to maintaining safe inventory levels to meet customer demand and prevent overstocking or stock shortages. A trading business’s distribution network managing various products requires efficient inventory replenishment policies that ensure high customer satisfaction while keeping costs low. In a conventional supply chain, distribution centers (DCs) place orders with factories to replenish their stocks. However, when distribution centers can perform lateral transshipment, a process where inventory is moved between neighboring locations to meet demand, the practice becomes more complex, requiring careful coordination of inventory replenishment policies and inbound/outbound transshipment decisions (Paterson et al., 2011). The ordering decision of the distribution network can be even more sophisticated when the DCs can place urgent orders during the regular ordering cycle. Urgent orders are placed to the upper echelon of the supply chain when the inventory level is critically low, and the regular orders that have already been placed are not arriving soon enough. An urgent order has a shorter lead time and typically costs more than a regular one (Chiang, 2010; Johansen, 2019; Tagaras and Vlachos, 2001). Therefore, these expedited shipments are expected to satisfy incoming demand and avoid stockout. Previous studies have shown that the combination of regular and urgent ordering for inventory replenishment can reduce the cost and improve the service level in periodic (Johansen, 2019; Tagaras and Vlachos, 2001) and continuous (Chiang, 2002; Chiang, 2010) systems. Demand classes in the inventory system generally represent the level of importance of demand sources. However, in some studies, the classes only define the sources of demands, such as online or offline demands (Qu et al., 2022) and local or switched demands (Liao et al., 2020). Thus, the unmet demands from different sources are considered the same. For example, they could all be considered as lost sales. However, in some studies, the unsatisfied demands of different classes result in different treatments and costs. Some unmet demands are regarded as lost sales, while others are backlogged based on their priorities (Malligarjunan, 2016; Zhou and Zhao, 2010). This arrangement, necessary for real system implementation, elevates the difficulty of inventory replenishment decisions, underscoring the importance of understanding these complexities. Our study framework is based on all the above decisions that the distribution network found in practice must make, including regular, urgent, and lateral ordering. Additionally, the demands faced by the DCs in this study can be separated into pre-ordered and peddling demands. The pre-ordered demand is known at the beginning of each period before the lateral transshipment occurs. These demands are from steady customers. If the stock is out, these customers will wait, and the new stock will be sent to them first. Therefore, the unmet pre-ordered * Corresponding author. E-mail address: [email protected] (P.-n. Somkun). Contents lists available at ScienceDirect The Asian Journal of Shipping and Logistics journal homepage: www.elsevier.com/locate/ajsl https://doi.org/10.1016/j.ajsl.2025.01.004 Received 21 October 2024; Accepted 27 January 2025 The Asian Journal of Shipping and Logistics 41 (2025) 61–74 Available online 12 February 2025 2092-5212/© 2025 The Author(s). Published by Elsevier B.V. on behalf of The Korean Association of Shipping and Logistics, Inc. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ). demand is backlogged and applied to the back order cost. The DC also has another selling approach, herein called peddling. The act of peddling is defined as the movement of a company’s vehicle that carries the DC’s products on a tour to offer a sale to its customers that are local retail and wholesale stores (adapted from Ruan et al. 2012 and Merriam-Webster 2024). The demand from these customers is called peddling demand, which has yet to be known in advance. The peddling demand is realized after the lateral transshipment is operated. These demands are of the retail and wholesale stores the company’s vehicles visit along its tour in the locality. Because these peddling customers are considered less important and the vehicle’s route might sometimes differ each day, unsatisfied peddling demand is considered a lost sale and associated with the lost sale cost. In this paper, we simulate a case study of a distribution network of distribution centers (DCs) that deliver multiple products to their customers: local wholesale and retail stores. The DCs are non-identical based on unit costs, shipping lead times, and capacities. The cost components considered in this study included inventory holding costs, shipping costs from the DCs to customers and between DCs, and opportunity costs from unsatisfied demands. We describe the relationship between entities of the system using mathematical modeling. Our main objective is determining the inventory replenishment policy the DC placed on the factory that works with the lateral transshipment policy. The spreadsheet model compares sets of proposed policies with the current rule applied by the case study. Our proposed policies consider the system’s complexity unique from previous studies, as depicted in Fig. 1. First, two types of periodic ordering that the DCs send to the factory are considered. Each DC places a regular order every week, while an urgent order with a shorter lead time can be placed daily. Our inventory policy adapts a well-known (R, s, S) control for both regular and urgent ordering. In this (R, s, S) inventory policy, every R interval, if the inventory level falls below the re-order level, s, an order is placed to fill up the order-up-to level, S (Visentin et al., 2023). This control has proven to be efficient in terms of costs and practical in actual situations (Cabrera et al., 2013; Esmaili et al., 2019; Johansson et al., 2020; Visentin et al., 2021). The simplicity of the (R, s, S) policy allows practitioners to apply it to any available software. Our proposed policies include the Basic (R, s, S) policy and PIP (R, s, S) policy, where the latter considers the pipeline delivery into a reorder decision. Weekly and daily demand data are utilized with these policies for regular and urgent ordering. Second, the DCs confront two classes of stochastic customer demands: pre-ordered and peddling demands. When the demands are unsatisfied, there will be backlog and lost sale cases in the same system. Most studies consider each case separately, while some similarly handle the unsatisfied demand from different classes. Also, literature that considers peddling demands is scarce, while this type of demand exists in the business environment. Therefore, our work can fulfill these research gaps. Third, we consider bidirectional lateral transshipment between the DCs. Each pair of distribution centers could request or be requested to transfer products based on a decentralized decision scheme. This transfer is a complete pooling (Paterson et al., 2011), where the DC first takes care of their pre-ordered demands of the current period, and then the rest of the inventory can be transferred. Our three proposed lateral transshipment policies are straightforward and designed based on the presence of the two stochastic demand classes: pre-ordered and peddling. In conclusion, this study contributes to inventory policy research by addressing a unique problem scope that occurs in practice. Additionally, the proposed inventory policy is simple and can be implemented using spreadsheet software, which is typically available in most organizations. Moreover, the best set of proposed regular, urgent, and lateral transshipment policies was applied to the case study to validate the model results. 2. Literature review The distribution network with lateral transshipment, a unique area of study, is characterized by the number of items, echelons, and stock points being considered, the identicalness of the distribution centers, and the inventory replenishment timing (Paterson et al., 2011). Apart from that, existing works that relate to the scope of our study can be identified by the three subjects, which include utilization of lateral transshipment, practice of regular and urgent ordering, and consideration of demand classes. Lateral transshipment within a distribution network involves Demand classes Pre-ordered demand Lateral transshipment request Peddling demand Costs associated to the satisfied demands Transportation cost Lateral transshipment cost Peddling transportation cost Costs associated to the unsatisfied demands Back order cost Lost sale cost Propose policies for Regular ordering Urgent ordering Lateral transshipment Ordering types Placed to Review period Lead time Regular Factory Weekly L1 Urgent Factory Daily L 2 Lateral Neighboring Daily L3 transshipment DCs L1> L2> L3 Inventory holding cost Fig. 1. Problem framework. C. Donjuk et al. The Asian Journal of Shipping and Logistics 41 (2025) 61–74 62 transferring inventory between stock points at the same level to reduce inventory costs and improve flexibility and customer service (Paterson et al., 2011). This transfer is made to increase inventory levels in order to prevent shortages. If the transfer occurs before demand arrives, it is considered proactive lateral transshipment. If the transfer is done after the demand is known, it is classified as reactive lateral transshipment. These options depend on the transshipment cost (Paterson et al., 2011). Stock points involved in the transshipment typically operate under two types of inventory pooling: complete pooling, where they fulfill the request based on availability, and partial pooling, where they allocate inventory based on future usage. Several research studies have proposed different lateral transshipment models. For example, Chen and Lu (2010) compared three lateral transshipment scenarios: non-lateral, unidirectional, and bidirectional transshipment. Partial pooling lateral transshipment occurs when one retailer lacks sufficient inventory and the other only transfers the excess amount. A periodic order-up-to policy was applied in regular replenishment with the upstream player. The results indicated that both unidirectional and bidirectional transshipment could reduce total inventory and increase customer satisfaction rates compared to the non-lateral transshipment scenario, but only when the demand at both ends is the same. Dhahri et al. (2022) proposed a joint production-transshipment control policy for a two-location system. This system allows transshipment between two failure-prone production facilities with different capacities. Kundu and Rossini (2023) also studied inventory replenishment policy in a distribution network of multiple products. They found that lateral transshipment combined with a periodic delivery-continuous reorder policy could reduce logistics costs, increase delivery truck saturation, and maintain service levels. Most research works omitted urgent ordering in their studies, although inventory systems with combination of regular and urgent ordering are often found in practice (Chiang, 2010). Tagaras and Vlachos (2001) introduced a simple rule that combined an urgent ordering within the familiar base stock policy. The base stock’s target level was used for the regular order to raise the inventory level when it is at the time to order, and the inventory level is at or below the reorder point. At a pre-specified point during the regular order’s lead time, if the inventory level falls below the base stock’s reorder point, an urgent order is placed at the quantity that raises the inventory level to the reorder point. This rule, which operates within the framework of the base stock policy, does not need any new parameters for the control. The advantages from the base stock control know-how can efficiently operate this additional urgent ordering in practice. Johansen (2019) also examined a combination of well-known policies. The reorder point and fixed order quantity were applied for the regular orders, while the reorder point and target stock level were used for the urgent orders. Markov decision model and neighborhood search were used to compute the combined policies. Chiang (2010) proposed a practical method by incorporating an urgent order feature into a standard continuous reorder point and fixed-order quantity policy. This was achieved through the use of a new parameter, the expedite-up-to level. A regular order is placed once the inventory level reaches the reorder point. Subsequently, the inventory level is checked at a fixed time. If the inventory level is lower than the expedite-up-to level, an urgent order is placed to raise the stock to the expedite-up-to level. Importantly, a portion of the regular order is treated as an urgent order. The remaining, which is not expedited, arrives at a later time. This method ensures that the total product from regular and urgent shipments remains equal to the fixed order quantity, thereby maintaining a steady supply of inventory. In our study, we modify a familiar (R, s, S) policy to propose a combined regular and urgent ordering policy that is unique from previous articles. Avci (2019) investigates the effect of lateral transshipment and expedited shipping on disruption management. A simulation model and a differential evolution algorithm are applied to achieve the optimal parameters for retailers’inventory policies with centralized decision-making. The importance of different sources of demand in an inventory system can vary. For example, requests for spare parts due to machine breakdowns or preventive maintenance, as well as orders from customers with different levels of importance, all contribute to this classification (Fadılo˘ glu &Bulut, 2010). Therefore, the treatment or costs of unsatisfied demands may differ based on their importance or demand class. For instance, Malligarjunan, (2016) study examines a continuous inventory system with two demand classes of the Poisson process. In a stockout, the unmet demand may be considered backlogs or lost sales, depending on its class. Zhou and Zhao (2010) also address multiple demand classes, categorizing them as either backlogs or losses of sales during shortages. They explain that offline demand tends not to wait, resulting in lost sales, while online demand tends to wait, resulting in back order. The optimal periodic replenishment for these demands was a state-dependent (R, s, S) policy. Qu et al. (2022) investigate these types of demands in omnichannel multi-echelon distribution networks. Their model also allows emergency lateral transshipment at the store level. They developed a joint decision on inventory replenishment and order allocation to meet online and offline demands by adopting a genetic algorithm. However, the unmet demands of the two classes are all treated indifferently as lost sales. Liao et al. (2020) focus on the benefit of reactive lateral transshipment of a single-period decentralized decision of two retailers with random demand switching. Customers facing stockout at a store have three options: wait for lateral transshipment from another store, switch to buy at another store, or give up buying. The demands could be classified as local and switched, whereas the unmet demands are treated similarly. The lateral transshipment request from the shortage store is as much as a percentage of the lack amount. van Wijk et al. (2019) proposed optimal spare parts management with two stock points in technical systems based on multiple demand classes. The classes are defined based on downtime costs; thus, stock rationing is required when the demand for a spare part arrives. The fulfillment options include using one’s stocks, requesting lateral transshipment, or repairing or ordering expedited shipment from other sources. Previous studies have yet to explore a dimension similar to ours, which includes a joint consideration of the regular and urgent inventory replenishment and lateral transshipment policies for a distribution network with multiple products and demand classes as shown in Table 1. 3. Methods This research methodology consists of four parts. In Section 3.1, we first describe the mathematical relationship of the inventory system used in our spreadsheet model. Then, in Section 3.2, we propose our regular and urgent inventory replenishment and lateral transshipment policies. The case study and the model input are introduced in Section 3.3 and the numerical experiment setting is explained in Section 3.4. 3.1. Mathematical and spreadsheet modeling 3.1.1. Indices i, j Distribution Center (DC) tTime pProduct 3.1.2. Sets i* Primary distribution center t* The effective time for regular ordering 3.1.3. Parameters ω j Transportation cost from DC jto the retail stores (Currency/Unit) δjTransportation cost for inbound lateral transshipment for DC J(Currency/ Trip) (continued on next page) C. Donjuk et al. The Asian Journal of Shipping and Logistics 41 (2025) 61–74 63 Table 1 Literature related to the study scope. Articles Distribution network (Number of echelons) (Number of stock points) Multiple products / Items Demand classes Inventory replenishment policy Lateral transshipment Cost components Study Details Effects from unmet demands Regular Urgent Proactive Reactive Partial pooling Complete pooling Vehicle capacity Avci (2019) Yes (2) (N) No No - - Periodic Periodic ✓✓No Holding; Back order; Purchase; Lateral; Expedited Burton and Banerjee (2005) Yes (2) (N) No No - - Periodic No ✓ ✓ ✓ No Lateral Chen and Lu (2010) Yes (2) (2) No No - - Periodic No ✓✓No - Chiang (2010) Yes (1) (1) No No - - Continuous Continuous - - - - - Fixed ordering; Holding Dhahri et al. (2022) Yes (2) (2): Manufacturer and retailers No No - - Periodic / Continuous No ✓✓Yes Holding; Back order; Fixed and variable lateral Johansen (2019) Yes (1) (1) No No - - Periodic Periodic - - - - - Fixed and variable ordering; Holding; Back order Kundu and Rossini (2023) Yes (2) (N) Yes No - - Periodic / Continuous No ✓✓Yes Holding; Lateral Liao et al. (2020) Yes (1) (2) No Yes: 2 Local demand; Switched demand Same Periodic (Single period) No ✓ ✓ No Procurement; Lateral Olsson (2010) Yes (1) (N) No No - - Continuous No ✓ ✓ No Holding; Lateral; Back order; Lost sale Qu et al. (2022) Yes (3) (N) No Yes: 2 Online; Offline Same: Lost sale Periodic No ✓✓No Replenishment; Lateral; Holding; Stock-out van Wijk et al. (2019) No (1) (2): Advanced technical system No Yes: Multiple High to low priority Different: High to low costs for lateral transshipment and emergency procedure Continuous Continuous ✓ ✓ ✓ No Lateral; Emergency procedures; Downtimes of the systems Wei et al. (2022) Yes (1) (N) No No - - Periodic No ✓✓ ✓ No Holding; Replenishment; Lateral; Penalty for unsold items This study Yes (2) (7) Yes Yes: 3 Preordering; Peddling; Lateral request Different: Back order; Lost sale; No penalty Periodic: Weekly Periodic: Daily ✓ ✓ ✓Yes Transportation for pre-ordered and peddling; Holding; Back order; Lost sale; Lateral C. Donjuk et al. The Asian Journal of Shipping and Logistics 41 (2025) 61–74 64 (continued) βpBack order cost of product p(Currency/Unit) γpLost sale cost of product p(Currency/Unit) α pInventory holding cost of product p(Currency /Unit/Unit of time) Lj RRegular delivery lead time for DC j(Unit of time) Lj UUrgent delivery lead time for DC j(Unit of time) VCap Maximum capacity of a vehicle used for delivery lateral transshipment (Unit/Trip) 3.1.4. Intermediary variables for each product p at DC j at time t All variables are in units. Ej (p,t)Incoming regular delivery from the factory Fj (p,t)Incoming urgent delivery from the factory Gj (p,t)Availability after receiving the delivery from the factory Dj (p,t)Customers’pre-orders Tij (p,t)Inbound delivery of lateral transshipment of product pfrom DC jto the primary DC iat time t Wj (p,t)Availability after receiving incoming lateral transshipment Jj (p,t)Outgoing delivery for customers’pre-orders Bj (p,t)Back order of pre-ordered demand Aj (p,t)Availability after outgoing delivery for customers’pre-orders Mj (p,t)Customer peddling demands Hj (p,t)Availability after outgoing delivery for peddling demands Vij (p,t)Outbound delivery of lateral transshipment of product pfrom primary DC ito DC jat time t Nj (p,t)Satisfied peddling demand Lj (p,t)Lost sales of the peddling demand Ij (p,t)Inventory level at the end of period Rj tNumber of lateral transshipment trips (Trip) 3.1.5. Decision variables for the inventory replenishment policy for each product p at DC j at time t All variables are in units. Sj (p,t)Regular order placed to the factory Uj (p,t)Urgent order placed to the factory Kij (p,t)Order placed by a primary DC ifor lateral transshipment of product pto DC j at time t Oij (p,t)Order placed by a minor DC jfor lateral transshipment of product pto DC iat time t 3.1.6. The total cost of the distribution network TC Annual total cost of the distribution network in monetary units includes six cost terms: TRC Transportation costs for delivery of pre-ordered products from the DCs to the retailers; PC Peddling costs for delivery of products from the DCs to the retailers to serve peddling demands; LTC Lateral transshipment costs for delivery between DCs; BC Back order costs of the unmet pre-ordered demands; LC Lost sale costs of the unmet peddling demands; IC Inventory costs for holding products in stock, as described in Eq. (1) and Eq. (2) . TC =TRC+PC+LTC +BC +LC +IC (1) TC =∑p∑j ω j(∑ t Jj (p,t))+∑p∑j ω j(∑ t Nj (p,t))+∑jδj(∑ t Rj t) +∑p∑jβp(∑ t Bj (p,t))+∑p∑jγp(∑ t Lj (p,t)) +∑p∑j α p(∑ t Ij (p,t)) (2) We remark that the DCs have decentralized decisions and the lateral transshipment costs are for receiving party to pay. 3.1.7. Mathematical and Spreadsheet model The sequence of activities, decisions, and the flow of orders and products are set in our model for each DC jfor a particular product pat period t. These are depicted in Fig. 2. At the beginning of each period, the DC received incoming deliveries (Ej (p,t)and/or Fj (p,t)) that correspond to orders that had been placed as either regular orders (Sj (p,t)) in Eq. (3) or as urgent orders (Uj (p,t)) in Eq. (4). Then, the inventory levels are updated by Eq. (5). The primary DC has the priority in placing a lateral transshipment request with a neighboring DC. The request will only be made when its stock cannot satisfy pre-order demand. The receiving lateral delivery (Tij (p,t)), however, depends on the availability of the partner DC, as shown in Eq. (6). The primary DC updates its inventory again (Eq. (7)) before dispatching its delivery to the retailers. The pre-order demand is served first (Eq. (8)). Then, the primary DC updates its stock (Eq. (11)) and serves the peddling demand (Eq. (13)). For the minor DC, its pre-order demand is served before deciding the transshipment for the primary DC’s request (Eq. (9)). Subsequently, the minor DC places a lateral request to the primary DC after all of the primary DC’s demands have already been handled. This request could be responded to if the primary DC did not place any lateral request in this period. The incoming delivery of the lateral transshipment to the minor DC is calculated by Eq. (17). After updating its stock (Eq. (12)), the minor DC serves the peddling demands (Eq. (14)). We assume that all available stock from the DC serves the peddling demand. The backlog of the pre-order demand is calculated by Eq. (10), while the lost sale of the peddling demand follows Eq. (15). Finally, the inventory level at the end of each period, right before placing any order with the factory, is presented by Eq. (16),Eq. (18), and Eq. (19). It is assumed that the factory will fully satisfy all orders that the DC places. All products share the vehicle. Each lateral transshipment trip cannot be larger than the vehicle’s capacity, and the number of trips required is calculated from Eq. (20). Ej (p,t)=Sj (p,t−Lj R),∀j,p,t(3) Fj (p,t)=Uj (p,t−Lj U),∀j,p,t(4) Gj (p,t)=Ej (p,t)+Fj (p,t)+Ij (p,t−1),∀j,p,t(5) Tij (p,t)=⎧ ⎨ ⎩ Kij (p,t),Kij (p,t)≤Gj (p,t)−Jj (p,t) Gj (p,t)−Jj (p,t),else ,∀i∈i∗,j,p,t(6) Wj (p,t)=Gj (p,t)+Tij (p,t),∀j∈i∗,p,t(7) Jj (p,t)=⎧ ⎨ ⎩ Dj (p,t),Dj (p,t)≤Wj (p,t) Wj (p,t),else ,∀j∈i∗,p,t(8) Jj (p,t)=⎧ ⎨ ⎩ Dj (p,t),Dj (p,t)≤Gj (p,t) Gj (p,t),else ,∀j∕∈ i∗,p,t(9) Bj (p,t)=Dj (p,t)−Jj (p,t),∀j,p,t(10) Aj (p,t)=Wj (p,t)−Jj (p,t),∀j∈i∗,p,t(11) C. Donjuk et al. The Asian Journal of Shipping and Logistics 41 (2025) 61–74 65 Aj (p,t)=Gj (p,t)−Jj (p,t),∀j∕∈ i∗,p,t(12) Nj (p,t)=⎧ ⎨ ⎩ Mj (p,t),Mj (p,t)≤Aj (p,t) Aj (p,t),else ,∀j∈i∗,p,t(13) Nj (p,t)=⎧ ⎨ ⎩ Mj (p,t),Mj (p,t)≤Aj (p,t)−Tij (p,t)+Vij (p,t) Aj (p,t)−Tij (p,t)+Vij (p,t),else ,∀j∕∈ i∗,p,t (14) Lj (p,t)=Mj (p,t)−Nj (p,t),∀j,p,t(15) Hj (p,t)=Aj (p,t)−Nj (p,t),∀j,p,t(16) Vij (p,t)=⎧ ⎨ ⎩ Oij (p,t),Oij (p,t)≤Hi (p,t) Hi (p,t),else ,∀j∕∈ i∗,p,t(17) Ij (p,t)=Hj (p,t)−Vij (p,t),∀j∈i∗,p,t(18) For each DC j, product p, and time t Receive incoming regular (E j (p, t) ) and/or urgent (F j (p, t) ) delivery from the factory Require lateral transshipment? Calculate lost sale (L j (p, t) ) Yes No Calculate back order (B j (p, t) ) Update stock availability (G j (p, t) ) Order for lateral transshipment (K ij (p, t) ) Inbound delivery of lateral transshipment (T ij (p, t) ) Update stock availability (W j (p, t) ) Lateral request from another DC (O ij (p, t) ) Incoming lateral transshipment request? Outbound lateral transshipment (V ij (p, t) ) Place regular order (S j (p, t) ) or Urgent order (U j (p, t) ) Yes No Delivery pre-order (J j (p, t) ) to customer Update stock availability (A j (p, t) ) Delivery peddle (N j (p, t) ) delivery to customer Update stock availability (H j (p, t) ) Update stock availability (I j (p, t) ) Factory Start End Fig. 2. Flow diagram of the spreadsheet model. C. Donjuk et al. The Asian Journal of Shipping and Logistics 41 (2025) 61–74 66 Ij (p,t)=Hj (p,t),∀j∕∈ i∗,p,t(19) Rj t=⌈∑ p Tij (p,t) VCap ⌉,∀j,t(20) 3.2. Proposed inventory replenishment and lateral transshipment policies 3.2.1. Nomenclature for the inventory replenishment policy for each DC j and product p Sj (p,week)Target inventory level for regular ordering (unit) sj (p,week)Re-order point for regular ordering (unit) Dj (p,week)Average demand calculated from past weekly data (unit/week) Sj D(p,week)The standard deviation of the demand calculated from past weekly data (unit/week) R(p,week)Review period for regular ordering (week) CSL Desired cycle service level [0,1] (Chopra &Meindl, 2007) F−1 s(.)The inverse of the standard normal distribution Sj (p,day)Target inventory level for urgent ordering (unit) sj (p,day)Re-order point for urgent ordering (unit) Dj (p,day)Average demand calculated from past daily data (unit/day) Sj D(p,day)The standard deviation of the demand calculated from a set of past daily data (unit/day) R(p,day)Review period for urgent ordering (day) π Number of working days in a week (day) 3.2.2. Proposed policies 3.2.2.1. Regular ordering (Sj (p,t)) Basic (R, s, S) Policy. The Basic (R, s, S) Policy placed an order to raise the inventory level to the target level (Sj (p,week)). The regular order can be placed only at the specific time (t∗), and when the inventory level is at or below the re-order point (sj (p,week)), as in Eq. (21). The target level and re-order point are calculated by Eq. (22) and Eq. (23), respectively. Sj (p,t)={Sj (p,week)−Ij (p,t),Ij (p,t)≤sj (p,week) 0,else ,∀j,p,t∈t∗(21) Sj (p,week)=Dj (p,week)•(Lj R+R(p,week))+F−1 s(CSL) •  Lj R √•Sj D(p,week),∀j,p (22) sj (p,week)=Dj (p,week)•(Lj R)+F−1 s(CSL) •  Lj R √•Sj D(p,week),∀j,p(23) PIP (R, s, S) Policy. The PIP (R, s, S) Policy operates the same way as the Basic (R, s, S); except the condition for re-ordering is adjusted by considering pipeline inventory. The pipeline inventory is the order that has already been placed but not yet received. In our policy, we consider the pipeline inventory that is expected to be received by the DC in the next period and refers to them as PIP. Thus, if the current period is t, the PIP will be arrived at period t+1. In other words, this PIP is a regular order to the factory that has been placed at time t−Lj R+1 and/or an urgent order placed at time t−Lj U+1. For Lj Rand Lj Uthat are less than or equal to one, the PIP for the calculation is zero. The PIP (R, s, S) Policy is described by Eq. (24). Sj (p,t)=⎧ ⎨ ⎩ Sj (p,week)−Ij (p,t),Ij (p,t)+Sj (p,t−Lj R+1)+Uj (p,t−Lj U+1)≤sj (p,week) 0,else ,∀j,p,t∈t∗ (24) 3.2.2.2. Urgent ordering (Uj (p,t)).Urgent orders can be placed daily. Again, we applied the Basic (R, s, S) Policy and PIP (R, s, S) Policy that considered the daily data of demands as shown in Eq. (25) and Eq. (28), respectively. We remark that this urgent order can only be placed if the regular order has not been placed within that period. Basic (R, s, S) Policy Uj (p,t)={Sj (p,day)−Ij (p,t),Ij (p,t)≤sj (p,day)and Sj (p,t)=0 0,else ,∀j,p,t(25) Sj (p,day)=Dj (p,day)•(Lj U+R(p,day))+F−1 s(CSL) •  Lj U √•Sj D(p,day),∀j,p(26) sj (p,day)=Dj (p,day)•(Lj U)+F−1 s(CSL) •  Lj U √•Sj D(p,day),∀j,p(27) PIP (R, s, S) Policy. Again for Lj Rand Lj Uthat are less than or equal to one, the PIP for the calculation is zero. Fig. 3 depicts how the Basic (R, s, S) policy for regular and urgent ordering work. The urgent ordering is reviewed daily; the raised amounts aand care of urgent orders, Uj (p,t). The regular ordering is reviewed weekly; bis the amount of a regular order, Sj (p,t). Fig. 4 presents the application of the PIP (R, s, S) policy for regular and urgent ordering. The urgent orders, Uj (p,t), are aand c. The regular order, Sj (p,t), is b. If the Basic (R, s, S) policy is applied herein, the number of orders will be 5 times instead of 3. Thus, the consideration of pipeline inventory in the PIP (R, s, S) policy reduces the number of orders and the average inventory level. 3.2.2.3. Lateral transshipment ordering (Kij (p,t)and Oij (p,t)). The DC can place a lateral transshipment order to its neighboring DCs daily. The lateral transshipment order will be responded to according to the availability of the other DCs. There are primary DCs that have more priority than the other DCs and have a chance to place the lateral transshipment order first. The ordering rules have been intentionally designed for simplicity, making them easy to incorporate into a spreadsheet. This straightforward approach empowers small and medium enterprises, giving them the confidence to apply these rules to their spreadsheets easily. We focused on satisfying pre-ordered demands generally from the company’s regular customers. Therefore, the lateral order would be placed only when the available amount at the beginning of the period is Uj (p,t)=⎧ ⎨ ⎩ Sj (p,day)−Ij (p,t),Ij (p,t)+Sj (p,t−Lj R+1)+Uj (p,t−Lj U+1)≤sj (p,day)and Sj (p,t)=0 0,else ,∀j,p,t(28) C. Donjuk et al. The Asian Journal of Shipping and Logistics 41 (2025) 61–74 67 insufficient for the pre-ordered demands, Di (p,t)>Gi (p,t). “As much”: The 1st lateral transshipment rule orders the amount to fulfill the pre-ordered demand, as in Eq. (29). Kij (p,t)=Di (p,t)−Gi (p,t),∀i∈i∗,p,t(29) “Average pre-ordered”: The 2nd lateral rule orders the amount equal to the moving average of the daily pre-ordered demand of the past week, as described in Eq. (30). Kij (p,t)=∑ k= π k=1 Di (p,t−k) π ,∀i∈i∗,p,t(30) “Average all”: The 3rd lateral rule orders an amount equal to the moving average of the daily total demand of the past week. This total demand consists of pre-ordered and peddling demands, as described in Eq. (31). Kij (p,t)=∑ k= π k=1(Di (p,t−k)+Mi (p,t−k)) π ,∀i∈i∗,p,t(31) The minor DCs apply the same rule for lateral ordering, (Oij (p,t)), but this order can be considered after the primary DC lateral transshipment decision. 3.3. Our case study and parameter setting Our simulation setting and inputs are based on a case study. This case study is a distribution network of a bottled soft drink business comprising seven DCs serving approximately 8,000 local wholesale and retail stores in several cities in the north of Thailand. All DCs place their regular orders to a factory every Wednesday. The delivery will be received at the DC in four working days. The DCs currently use past sale figures to decide the order amount. For each product, the average sales in the last month of this year and the previous year are used for this month’s order. The monthly figure is divided by four or five to get the amount for weekly regular orders. An urgent order can be placed any day during the week. The company does not have a fixed rule for this urgent order. Based on our interview with the manager and investigation of the record, our model assumes that an urgent order is placed according to a reorder point and fixed order quantity policy. The parameters were set based on the company’s past two-year sales data. Emergency lateral transshipment is allowed between DCs in a neighboring area, as shown in Fig. 5. The requested amount is generally just enough to fulfill the pre-order required for that day. There are three groups of DCs. In each group, we list the DC by its priority for lateral ordering, which is based on their maximum capacity, i.e.: 1. Chiang Rai and Phayao. 2. Uttaradit, Phrae, and Nan. 3. Phetchabun (City branch) and Phetchabun (Bueng Sampan branch). Two-year sales data of 64 products were collected for pre-ordered and peddling demands. We ranked the products by the total value, resulting in the top products with 80 % cumulative value while the cumulative volume reached 50 %. Thus, our model focused only on these products, p={1, 2, 3, 4, 5}. The truck capacity (VCap) was calculated for these five products by a percentage of the total volume. Parameters setting: Lj R=4 days or 4/6 week, for all DC j Lj U=1 day for Phetchabun (City branch) and Phetchabun (Bueng Fig. 3. Proposed Basic (R, s, S) policy for regular and urgent ordering. C. Donjuk et al. The Asian Journal of Shipping and Logistics 41 (2025) 61–74 68