Multi-warehouse package consolidation for split orders in online retailing

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Multi-warehouse package consolidation for split orders in online retailing Yuankai Zhang a, Wei-Hua Lin b, Minfang Huang c, Xiangpei Hu d,∗ a

Institute of Systems Engineering, Dalian University of Technology, Dalian, Liaoning 116024, China Department of Systems and Industrial Engineering, The University of Arizona, Tucson, AZ 85721, USA c School of Economics and Management, North China Electric Power University, Beijing 102206, China d School of Management, Zhejiang University, Hangzhou, Zhejiang 310058, China b

a r t i c l e

i n f o

Article history: Received 31 August 2018 Accepted 4 July 2019 Available online xxx Keywords: Logistics Package consolidation Split order Multi-commodity network flow Logic-based Benders’ decomposition

a b s t r a c t With the rapid growth of the online market in recent years, order splitting has become a great challenge to online retailers for fulfilling multi-item orders in a multi-warehouse storage network. Order splitting can lead to higher shipping costs, the use of more packages, and possible dissatisfaction from customers. This paper presents a multi-warehouse package consolidation approach aimed at consolidating multiple suborders’ stock-keeping units (SKUs) through transshipments among warehouses. A combined multicommodity network flow model is proposed to determine the consolidation warehouses for each order and make transshipment decisions for individual SKUs. An enhanced logic-based Benders’ decomposition algorithm is proposed to decompose the model into a general multi-commodity network flow master problem and a set of bin packing with conflicts sub-problems. Two proposed Benders’ cuts guarantee the algorithm to converge to optimality. The proposed algorithm can generate the near-optimal result with only about 25% of the CPU time required by CPLEX to solve the proposed model. Numerical experiments reveal that the proposed package consolidation approach outperforms the order splitting fulfillment approach in reducing the total costs, the number of packages, and the delivery times, especially for cases with a small number of SKUs in each suborder which are typical for online retailers. Sensitivity analyses are performed to provide managerial insights of adopting the proposed approach in the real world where order splitting is a common phenomenon. © 2019 Elsevier B.V. All rights reserved.

1. Introduction The booming development of e-commerce has provided more options for shopping but posed a great challenge to logistics systems (Boysen, de Koster & Weidinger, 2019; Ishfaq & Bajwa, 2019; Steinker, Hoberg & Thonemann, 2017), see for instance, China. It has become the world’s largest market for e-commerce. Online sales in China hit $1.1 trillion in 2017, and are predicted to grow to a 24% share of total retail sales by 2020 (The Economist, 2017). However, the split-order fulfillment problem has become more pronounced. Order splitting greatly increases operational costs for online retailers and decreases customer satisfaction (Co, Miller & Xu, 2007). For example, a large online retailer – Yihaodian.com (was acquired by JD.com), had to split about 13–18% of over several million daily orders. It was estimated that each split suborder ∗ Corresponding author at: School of Management, Zhejiang University, Hangzhou, Zhejiang 310058, China. E-mail addresses: [email protected] (Y. Zhang), [email protected] (W.-H. Lin), [email protected] (M. Huang), [email protected] (X. Hu).

adds an additional shipping cost of approximately 1.9 US dollars (Catalán & Fisher, 2012). Moreover, the environmental impact of discarded packages has become broader in recent years. The negative impact of order splitting has become an issue for online retailers not only in China but in many other countries as well (Acimovic & Graves, 2015; Xu, Allgor & Graves, 2009; Zhang, Sun, Hu & Zhao, 2019). It’s also a challenge for omnichannel commerce. A Radial & EKN research study revealed that the profitability of some 31 percent of omnichannel retailers has been affected by split orders (Radial, 2017). It has become important to deal with the issue of order splitting for the further development of online retailers. The order splitting problem (also known as split delivery) means that multi-item orders are split into several suborders fulfilled by multiple warehouses separately. The suborders are packed in different packages and delivered with multiple shipments. Order splitting is becoming particularly prominent for large-scale online retailers with the multi-item order feature and the “multiple category warehouses” configuration. The average order size is relatively large since online retailers, especially online supermarkets, usually

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Please cite this article as: Y. Zhang, W.-H. Lin and M. Huang et al., Multi-warehouse package consolidation for split orders in online retailing, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2019.07.004

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set a threshold of purchasing for free delivery (e.g., the threshold is set to $35 on Walmart.com for customers to enjoy free delivery) and the price for a single item is mostly lower than the threshold. For example, Yihaodian.com (part of JD.com) averages 16.7 items per order, which is 8 to 10 times greater than that of general online retailers, while over half of the orders for Amazon.com are single-item orders (Xu et al., 2009). The online retailers sell a wide variety of stock-keeping units (SKUs), e.g., over five million SKUs in JD.com. Due to the limited capacity of any single warehouse, SKUs have to be stocked in multiple category warehouses where there is no storage overlap among individual warehouses. Generally, there are two modes of warehouse fulfillment in a region, the Amazon.com mode and the JD.com mode (Zhang, Huang, Hu & Sun, 2018). In the Amazon.com mode, warehouses are spread in multiple cities to serve customers around the country, making it possible to keep a wide scope of inventory but potentially at a cost of a longer order fulfillment time. In the JD.com mode, however, the whole market in a country is divided into regions served by several regional fulfillment centers. Each fulfillment center, formed by a network of multiple warehouses, serves only customers in that region to ensure a speedy delivery. In this paper, we mainly consider the latter mode since it is more likely to create order splitting. Unfortunately, only a small portion of split orders can be reduced through existing approaches (see literature review in detail), because orders can be of any combination of millions of SKUs stored in multiple warehouses. This paper proposes a multi-warehouse package consolidation approach, which allows SKUs from different warehouses to be combined through transshipments before being shipped to end customers. This will reduce the shipping cost by reducing the number of packages used for each order. However, the added transshipment process by package consolidation will induce additional transshipment costs. To minimize the total order fulfillment cost, a tradeoff between the transshipment cost and the shipping cost needs to be considered. We will address the tradeoff problem by considering a series of research questions such as how to decide splitting or consolidation for each order, how to choose the specific consolidation warehouses for individual SKUs in the suborders and decide the corresponding transshipments, and how to determine the packing scheme for each shipment. The decisions for multi-warehouse package consolidation mainly present two challenges to model formulation and solving, which are not easily amenable to standard techniques. First, although the problem we investigate can be treated as a special kind of multi-commodity network flow problem, the problem fails to be represented by existing models since it contains two kinds of flows, the SKU flow and the package flow. The SKU flow can be merged into the package flow subject to a set of constraints, including the weight and volume constraints, and the conflicts of packing SKUs in each shipment. Second, the solution space of the problem we deal with is extremely large, making it difficult to apply the standard algorithms. For example, only for the transshipment decision of one order with ns SKUs, the solution space will increase from 2ns −1 (if there is only one consolidation warehouse) to nk ns (if there are nk consolidation warehouses). Applying a commercial solver (e.g., CPLEX) directly to this model is computationally infeasible for large-scale problems. In this paper, we propose a combined multi-commodity network flow model and an enhanced logic-based Benders’ decomposition algorithm to effectively generate schemes for the multi-warehouse package consolidation problem. The main contributions of our work can be summarized as follows: First, a new variant of the package consolidation problem for online retailing is formulated, in which the consolidation of SKUs for a particular order can be performed in any warehouse instead of a designated

one. Second, a combined multi-commodity network flow model is developed. The proposed model combines a general linear multi-commodity network with a fixed-charge multi-commodity network through a packing process, which can formulate two kinds of flows (SKUs and packages) together in a single network. Third, our enhanced logic-based Benders’ decomposition algorithm can solve the combined model in a more efficient way. The proposed algorithm can generate the near-optimal result with only about 25% of the CPU time required by CPLEX to solve the proposed multi-commodity network flow model. Fourth, various management insights to apply package consolidation are revealed, in which the proposed approach outperforms the existing order splitting fulfillment approach in reducing the total costs, the number of packages, and the delivery times. The analysis further reveals the conditions under which package consolidation would perform particularly well. The remainder of the paper is organized as follows. A literature review for the related work is given in Section 2. In Section 3, we analyze the network structure of package consolidation and costs obtained from the order fulfillment process. Section 4 presents the formulation of a multi-commodity network flow model. A logic-based Benders’ decomposition algorithm to solve the model is presented in Section 5. In Section 6, numerical analyses are conducted to demonstrate the performance of the proposed model and the algorithm. The insights obtained are discussed. These are followed by a conclusion and a discussion of possible future research directions shown in Section 7. 2. Literature review The split-order fulfillment problem has become one of the main challenges for fulfilling multi-item orders in online retailing. There are mainly three processes in order fulfillment relating to split orders: assortment allocation, order allocation, and consolidation. Assortment allocation is to reduce the number of split orders from the strategic perspective, which allocates closely related SKUs to the same warehouses according to the past record of sales. Kök, Fisher and Vaidyanathan (2015) provide an excellent review of the assortment allocation problem. In online retailing, a large number of SKUs have to be stored in warehouses considering the capacity constraint. By clustering frequently ordered SKUs, Co et al. (2007) develop a two-step heuristics method to optimally allocate SKUs in each warehouse to minimize order splitting and maximize flexibility. Catalán and Fisher (2012) subsequently prove that the problem of SKUs assignment in warehouses to minimize the number of split orders is an NP-hard problem. Recently, Acimovic and Graves (2017) propose a heuristic to better allocate inventory in different warehouses accounting for possible spillover. However, the strategy of assortment allocation can only play a limited role, since it is restricted by the capacity of each warehouse and cannot be used frequently for daily dynamic orders. In online retailing, order allocation is to reduce the number of split orders in operation, assuming that one SKU can be stored in several warehouses. Xu et al. (2009) have realized the importance of shipping multi-item orders in fewer boxes to lower the shipping costs. They reevaluate the real-time order assignments periodically mainly through avoiding order splitting for multi-item orders. Jasin and Sinha (2015) prove that allocation of items to fulfillment centers for a single multi-item order is NP-hard, and propose an LP-based correlated rounding scheme for multi-item order fulfillment. Acimovic and Graves (2015) examine the impact of the forward-looking fulfillment decisions on outbound shipping costs, and make real-time order fulfillment decisions considering order splitting in future. Torabi, Hassini and Jeihoonian (2015) propose a Benders’ decomposition-based approach to find optimal order fulfillment solutions including source allocation, inventory

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transshipment, and customer order transfer. Lei, Jasin and Sinha (2018) consider a joint pricing and fulfillment problem for ecommerce retailers to maximize the total expected profits. Order allocation strategy is used to reduce order splitting for online retailers with a network of multiple warehouses with overlapping storage of SKUs. However, the SKUs storage in different category warehouses typically does not overlap in online retailing. Order splitting has become an inevitable phenomenon even with a sound allocation strategy. Therefore, online retailers still have to fulfill a huge amount of split orders from the daily operational perspective. Consolidation strategy, such as shipment consolidation, has been widely used in the traditional areas because it can reduce the number of shipments (e.g., trucks), lower transportation costs and lower carbon emissions (Hall, 1987; Hanbazazah, Abril, Erkoc & Shaikh, 2019; Stenius, Karaarslan, Marklund & De Kok, 2016). According to Ramakrishna, Sharafali and Lim (2015), one way to avoid split deliveries is to consolidate a customer’s order in a warehouse near the customer by making transshipments of items from other warehouses. Facing the split-order fulfillment problem, Zhang et al. (2018) propose the single-warehouse package consolidation approach which has the potential to lower the order fulfillment cost. However, with the rapid development of online retailing, multiple warehouses are geographically dispersed in large cities (For example, JD.com has over 10 warehouses spread out in several logistics parks in Shanghai) and the well-equipped warehouses have the full order fulfillment process capability, which calls for multiple warehouses to conduct package consolidation. This work is an extension of Zhang et al. (2018). Compared with our previous work (Zhang et al., 2018), package consolidation in this work can be accomplished in any warehouse instead of a prespecified warehouse in a multi-warehouse environment. The delivery distance is ignored in Zhang et al. (2018) for the warehouses located in a single logistic park. This work considers the delivery distance in the objective of minimizing the total shipping costs, which is more typical in online retailing practice with multiple dispersed warehouses. Instead of formulating the problem with two models as in Zhang et al. (2018), a consolidation model and a packing model, this work formulates a combined multi-commodity network flow model, simultaneously capturing the SKU flow and the package flow in the network. Moreover, we develop a new exact solution method which can generate the optimal package consolidation solutions instead of the heuristic one in Zhang et al. (2018). The proposed logic-based Benders’ decomposition algorithm can also be extended easily to yield near-optimal results for large-scale problems. Furthermore, this research verifies the package consolidation from additional perspectives, e.g., the order fulfillment time. 3. Problem description and analysis 3.1. Problem description Consider an online retailer with several category warehouses in a region. Each warehouse has order fulfillment ability and can act as a consolidation warehouse. Each order that contains multiple SKUs can be split into suborders according to the storage of SKUs in each warehouse. For each SKU in each order, it can be shipped directly from its warehouse to the customer or transshipped to another consolidation warehouse for consolidation first, packed with other SKUs and shipped to the customer. For each multi-item order, the demand for each type of SKU is no more than one unit. In reality, the request for multiple units of one SKU in a single order could happen. This can easily be handled with duplicate SKUs in the proposed model. With the objective of minimizing the total order fulfillment cost, we address the following three questions: (1) Split fulfillment or package consolidation for split orders in SKU level? (2) For package consolidation for a multi-item

3

Fig. 1. Illustration of the package consolidation process for split orders.

order, which warehouse(s) should be chosen as the consolidation warehouse(s) for consolidation and which SKU(s) in the split orders should be transshipped to which warehouse? (3) Which SKU(s) should be packed in one package? Here is an example to illustrate the package consolidation process of split orders, as shown in Fig. 1. There are three consolidation warehouses, labeled as k1 , k2 , and k3 . SKU1 is only stored in warehouse k1 while SKU2 is only stored in warehouse k2 . Customer m places an order labeled as Order1, which contains two SKUs: SKU1 and SKU2. Order1 will be split into two suborders, Suborder1 for SKU1 and Suborder2 for SKU2, which will be allocated to warehouse k1 and k2 , separately. The total costs are the sum of shipping costs and transshipment costs. We assume that the transshipment costs between three warehouses are: ctran = ctran = 3, ctran = ctran = 2, ctran = ctran = 1.5. k1 k2 k2 k1 k1 k3 k3 k1 k2 k3 k3 k2 The shipping costs here contain two parts: the fixed cost, fk1m = 9, fk2m = 11, fk3m = 10, and the variable cost for delivering ship ship ship a SKU (unit variable shipping cost), ck1m = 1, ck2m = 1, ck3m = 1. With the order splitting fulfillment approach, SKU1 and SKU2 will be separately packed and fulfilled with two packages, which results in the total order fulfillment ship ship cost= fk1m + ck1m + fk2m + ck2m =22. Also, SKUs can be transshipped and consolidated in one package with package consolidation approach. Based on this approach, three consolidation schemes might be generated: CW1 (Consolidation at Warehouse 1): SKU2 is transshipped to warehouse k1 and consolidated with SKU1 into one package, which is shipped from warehouse k1 to the customer. The ship total cost of CW1= fk1m + 2ck1m + ctran =14. k2 k1 CW2 (Consolidation at Warehouse 2): The total cost of ship CW2= fk2m + 2ck2m + ctran =16. k1 k2 CW3 (Consolidation at Warehouse 3): The total cost of ship CW3=ctran +ctran + fk3m + 2ck3m =15.5. k1 k3 k2 k3 The best result is CW1. Package consolidation has the potential to reduce total shipping costs by reducing the number of shipments. Meanwhile, it can induce transshipment costs. Models are needed to balance the two costs to find an order fulfillment scheme with the lowest total cost. Another important issue is that the decision of packing SKUs into packages has to take the conflict graph of SKUs into consideration. The conflicts between SKUs prevent some SKUs from being packed in the same package. An example in Fig. 2 shows that only one package is needed to pack the SKUs in two split orders together if there is no conflict between SKUs, while these SKUs should be packed into at least two packages if one conflict exists in these SKUs. For online retailers, some SKUs cannot be packed together in a package because of possible contamination, such

Please cite this article as: Y. Zhang, W.-H. Lin and M. Huang et al., Multi-warehouse package consolidation for split orders in online retailing, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2019.07.004

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Fig. 2. An example showing the influence of SKU conflicts.

as liquid laundry detergent and foods. The package consolidation decision will be tougher if the conflicts of SKUs are considered. 3.2. Problem analysis 3.2.1. Network structure analysis The multi-commodity network flow problem is for sending distinct commodities from their sources to their respective destinations along the arcs of an underlying network (Assad, 1978), which is similar to sending SKUs from warehouses to end customers in our problem. We formulate the multi-warehouse package consolidation problem as a combined fixed-charge multicommodity network flow problem, which is a combination of a general linear multi-commodity network capturing the transshipment of SKUs among the warehouses, and a fixed-charge multi-commodity network capturing the packages shipped from the warehouses to the customers. The cost structure of the combined network is in some way related to the cost structures of two kinds of classical multi-commodity network flow problems. The cost of the general linear multi-commodity network flow problem is linear to the number of the flows in the arcs, while the cost of the fixed-charge multi-commodity network flow problem associated with an arc is the sum of a fixed cost derived from its use and a variable cost proportional to the flow going through it (Errico, Crainic, Malucelli & Nonato, 2017; Paraskevopoulos, Bektas¸ , Crainic & Potts, 2016; Yaghini, Karimi, Rahbar & Sharifitabar, 2015). Compared with classical multi-commodity networks, there are three special features for the combined multi-commodity network. ➢ First, it has no bundling capacity constraint on all SKUs in the flow arc since there is no quantity decision to be made for SKUs. However, package capacity exists for the SKUs to be packed together. There is a packing process that transfers the SKUs in the transshipment network to the packages in the shipping network. SKUs in the same package should satisfy the weight and volume capacity constraints of packages. ➢ Second, different from the classical network flow problem, there may exist conflicts between SKUs in the combined network flow, which prevents these SKUs from being packed in the same package. ➢ Third, compared with the traditional fixed-charge network flow problem, the fixed cost of one arc not only depends on whether we use the arc or not, but also varies with the

number of packages using the arc as demonstrated by Ahire, Malhotra and Jensen (2015) and Xu et al. (2009). Considering that multiple packages belonging to one customer usually are shipped together, there also exists a scale economy effect on the arc cost, which will be incorporated in the formulation. 3.2.2. Cost analysis based on order fulfillment process Cost measurement is based on the order fulfillment process. In this paper, the measurement of each kind of cost is taken based on our investigation of the practice of several online retailers. We compare the costs between the existing practice of the order splitting fulfillment and the proposed package consolidation fulfillment. Order splitting fulfillment (OSF): SKUs are packed in their warehouses without transshipment and consolidation. Therefore, SKUs in a single order are following the process of picking, packing, and shipping. The total order fulfillment cost of OSF is equal to the summation of picking, packing, and shipping costs. Package consolidation fulfillment (PCF): Some SKUs are transshipped to other warehouses to be consolidated, which creates a new process, the transshipment process, between the picking and packing processes. Note that in the proposed package consolidation approach, there are still some SKUs following the order splitting fulfillment to minimize the total cost. The total order fulfillment cost of PCF is equal to the summation of picking, transshipment, packing, and shipping costs. The picking and packing costs are determined by the number of SKUs in an order. In practice, these two costs mainly contain workers’ piecework wages that are calculated by the number of fulfilling SKUs. While comparing both total costs in OSF and PCF, we ignore the picking and packing costs since they are equal in these two order fulfillment methods. Therefore, we calculate the total order fulfillment cost with the following equation: The total order fulfillment cost Ctotal = transshipment cost tran C + shipping cost C ship . The transshipment cost is the result of the number of transshipped SKUs multiplying arc transshipment cost for per unit SKU. The shipping cost is divided into two parts, the fixed cost and the variable cost, similar to the one used in Jasin and Sinha (2015). It is also a typical cost structure found in practice. The fixed cost represents the costs related to one time delivery between two points (e.g., being evaluated by distances) and the variable cost represents the complexity of the delivery, e.g.,

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the number of SKUs and the number of packages. As introduced before, we improve the fixed shipping cost formula by adding its relationship with the number of packages if these packages sharing the same arc. Therefore, shipping cost = fixed shipping cost + variable shipping cost = arc fixed shipping cost ∗ ( (1 − α ) + α ∗ the number of packages on the arc) + unit variable shipping cost ∗ the number of shipped SKUs, where the cost parameter α (0 < α < 1 ) represents the scale economy effect. Lower limit α = 0 implies that there is no relationship between the fixed shipping cost and the number of packages which is the general setting for the fixed-charge multi-commodity network, while upper limit α = 1 implies that the fixed shipping cost has a linear relation with the number of packages, which is the same as in Costa (2005). 4. Problem formulation 4.1. Notations G = (Z, A )

Let be a directed network, where Z is the set of vertices and A is the set of arcs. A is the transshipment arc set while B is the shipping arc set. We have A =A ∪ B. Let K = {k1 , k2 , ..., k|K | } be a set of warehouses and S= {s1 , s2 , ..., s|S| } be a set of SKUs ordered by customer m. Then we have A = {(k1 , k2 ), (k2 , k1 )...(k1 , k|K | ), (k|K | , k1 )} , B = {(k1 , m ), (k2 , m )...(k|K | , m )} and Z = K ∪ {m}. Each SKU s is specified by a source vertex as ∈ K and a destination vertex es = m. The flow differential for a vertex j and an SKU s is expressed by ds j , which is defined as:



ds j =

1, j = as −1, j = es . 0, otherwise

ws and vs are the weight and volume of SKU s. W and V are the weight and volume constraint of packages. ctran is the jk transshipment cost of unit SKU from warehouse j to wareship house k. f jk and c jk are the fixed and variable shipping cost

of packages from warehouse j to warehouse k. α (0 < α < 1 ) is the cost parameter of multiple packages in one shipment. S jk = {s|s ∈ S, xs jk = 1}, ∀( j, k ) ∈ B is the set of SKUs to be packed in warehouse j and shipped to customer k, and n = |S jk | is the number of SKUs in S jk . E = {(s1 , s2 )|s1 , s2 ∈ S, hs1 s2 = 1} represents the conflict graph of SKUs, where SKU s1 and SKU s2 cannot be assigned in the same shipment with the conflicting constraint hs1 s2 = 1, s1 , s2 ∈ S, s1 = s2 . Decision variables are as following: xs jk ∈ {0, 1}, SKU s flows from node j to node k if xs jk = 1, ( j, k ) ∈ A ; y jk ∈ {0, 1}, at least one package flowing from warehouse j to customer k if y jk =1; br jk ∈ {0, 1}, the package r is used in the flow from warehouse j to customer k if br jk = 1; psr jk ∈ {0, 1}, the SKU s is packed in the package r, which is in the flow from warehouse j to customer k if psr jk = 1. 4.2. Multi-commodity network flow model We propose a multi-commodity network flow model for the problem in which arcs represent transitions of SKUs in a particular network graph. In the model, each SKU is a commodity and the vertex nodes stand for the warehouses and the customer nodes. Determining the fulfillment scheme for SKUs in an order is to find flowing paths for these SKUs in an appropriate network graph. Different from the transshipment phase, in the shipping phase, SKUs are delivered as packages in which one or several SKUs are combined together. The SKU flows are transferred into packages

5

flows in warehouse nodes, which should be formulated with the package constraints of weight, volume and SKU conflicts. The resulting model is as follows:

Min

 

ctran jk xs jk +

s∈S ( j,k )∈A

+

 

s∈S ( j,k )∈B





br jk

r=1

(1)

∀s ∈ S, ( j, k ) ∈ A

∀s ∈ S, ( j, k ) ∈ B 

xs jk −

( j,k )∈A n 

f jk (1 − α )y jk + α

( j,k )∈B

n 

cship xs jk jk

s.t. xs jk ≤ ds j , xs jk ≤ y jk ,





xsk j = ds j ,

(2) (3)

∀s ∈ S, j ∈ Z

(4)

(k, j )∈A

psr jk ≤ xs jk ,

∀s ∈ S, ( j, k ) ∈ B

(5)

r =1 n  

psr jk = 1,

∀s ∈ S

(6)

r =1 ( j,k )∈B



psr jk ws ≤ br jkW,

∀1 ≤ r ≤ n, ( j, k ) ∈ B

(7)

s∈S



psr jk vs ≤ br jkV,

∀1 ≤ r ≤ n, ( j, k ) ∈ B

(8)

s∈S

ps1 r jk + ps2 r jk ≤ br jk ,

∀(s1 , s2 ) ∈ E, 1 ≤ r ≤ n, ( j, k ) ∈ B

psr jk ,br jk ∈ {0, 1}, ∀s ∈ S, 1 ≤ r ≤ n, ( j, k ) ∈ B

(9) (10)

xs jk ∈ {0, 1},

∀s ∈ S, ( j, k ) ∈ A

(11)

y jk ∈ {0, 1},

∀( j, k ) ∈ B

(12)

The objective function contains two parts: the transship  tran ment cost s∈S ( j,k )∈A c jk xs jk in the transshipment network,  n and the shipping cost ( j,k )∈B f jk ((1 − α )y jk + α r=1 br jk ) +   ship s∈S ( j,k )∈B c jk xs jk in the shipping network. Constraint (2) ensures that the SKU can be transshipped from the warehouse where it is stored only. Constraint (3) represents the relationship of the use of shipping arcs and SKUs flowing through these arcs. Constraint (4) ensures that the inflow and outflow satisfy the supply or demand requirements at all the nodes in the combined network. Constraint (5) states that in the shipping network, the package flow exists in the arc only when its corresponding packed SKUs flow in the same arc. Constraint (6) ensures that each SKU can only be packed in one package. Constraints (7) and (8) represent the weight and volume constraints of the packages. Constraint (9) states the conflicting SKUs cannot be assigned in the same packages. Constraints (10–12) define the domain of variables. The problem of multi-warehouse package consolidation for split orders is NP-hard since a special case of this problem, the package consolidation for split orders with one consolidation warehouse, has been proved to be an NP-hard problem by Zhang et al. (2018). The number of possible fulfillment schemes for split orders will increase exponentially with the increase in the number of SKUs in the order. Applying a commercial solver (e.g., CPLEX) directly to this model usually takes lots of running time and is not feasible for large-scale problems. However, we found that this formulation has the block diagonal structure, which allows us to solve the original problem by solving a sequence of smaller decoupled problems. We explore in the next section a decomposition solution approach to efficiently solve the model.

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Start

Solve multi-commodity network flow master problem

MP feasible?

Yes

Improved FFD heuristic

MP Solution=FFD Solutions? No

No

Yes

Finish

Yes

Solve bin packing with conflicts sub problems

Cut propagation

No

MP Solution=SP Solutions?

Add Benders cuts to MP

Fig. 3. The flowchart of the enhanced LBBD for package consolidation.

5. An enhanced logic-based Benders’ decomposition algorithm We propose an enhanced logic-based Benders’ decomposition (LBBD) algorithm to solve the combined fixed-charge multicommodity network flow model. The structure of the fixed-charge network design problem presents a natural decomposition scheme for the Benders approach (Costa, 2005). However, the classical Benders approach cannot be directed to solve the 0–1 integer programming (IP) model (Mariel & Minner, 2017; Rahmaniani, Crainic, Gendreau & Rei, 2017). Fortunately, it is substantially generalized by LBBD (Chu & Xia, 2004; Hooker & Ottosson, 2003), in which the sub-problem (SP) can in principle be any optimization or constraint satisfaction problem instead of Linear Programming. As a versatile decomposition technique, LBBD has been successfully applied to a wide variety of applications (Delorme, Iori & Martello, 2017; Fazel-Zarandi & Beck, 2012; Riedler & Raidl, 2018; Roshanaei, Luong, Aleman & Urbach, 2017). But there are still several challenges in developing effective LBBD approaches (Roshanaei et al., 2017): (1) The incorporation of a tight SP relaxation into the master problem (MP) to direct the master search towards solutions that are likely to satisfy the SPs; (2) Generation of strong Benders’ cuts; (3) Frequency of solving SPs. The enhanced LBBD proposed can overcome some of these challenges, such as how to decompose the model, how to generate the Benders’ cut, and how to speed up the algorithm. The flowchart of the proposed LBBD is illustrated in Fig. 3. The LBBD decomposes the original model into a multi-commodity network flow MP and multiple SPs of bin packing problems with conflicts. This decomposition can utilize the block diagonal structure of the original model, which makes the decomposed models much easier to solve.

Benders’ cut is then generated after solving the SPs and added to the MP in each iteration to obtain the optimal value. Two new Benders’ cuts guarantee the enhanced LBBD to converge to optimality. Additionally, two strategies (two cut propagations and a first fit decreasing heuristic) are applied into LBBD to reduce the number of iterations and improve the efficiency in solving the model. 5.1. Multi-commodity network flow master problem The MP determines which warehouses to consolidate and allocates SKUs in each suborder to the consolidation warehouses, leaving the number of packages delivered by each warehouse to be determined by the SPs. The MP can also compute a lower bound on the number of packages delivered by each warehouse. The MP is obtained through a relaxation of the 0–1 IP model in which the three-indexed flow decision binary variable (br jk ) is transformed  into a two-indexed flow decision variable (b jk ,b jk = nr=1 br jk ), and the binary variable for deciding which SKU to be packed in which package ( psr jk ) is removed. b jk is a variable reflecting the lower bound on the number of packages flowing from warehouse j to customer k without considering specific SKUs-to-packages packing decisions. MP is much easier to solve without variable psr jk . We can make the SKUs-to-packages packing decisions separately for each SKU sets in each warehouse with several SPs. The MP is formulated as follows:

Min

 

ctran jk xs jk +

s∈S ( j,k )∈A

+

 

s∈S ( j,k )∈B

cship xs jk jk



f jk ((1 − α )y jk + α b jk )

( j,k )∈B

(13)

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7

is over-capacitated in weight and volume. Constraint (21) enforces that the SKUs with conflicts cannot be packed in the same package. The upper and lower bounds on the number of packages are represented by constraint (23). 5.3. Upper bounds for sub-problems

Fig. 4. The relationship between MP and multiple SPs.

 s.t. b jk ≥

ws xs jk , W

s∈S

 s∈S

vs xs jk

V

 ,

∀( j, k ) ∈ B

∀( j, k ) ∈ B

b jk ≥ 0, b jk ∈ N,

(14) (15)

Constraints (2)–(4), (11)–(12)

Cuts

(16)

Constraint (14) is the relaxation of the SP that defines the minimum number of packages shipped from each warehouse. Constraint (16) is the Benders cuts added to the MP each time the MP is not able to find a feasible solution to one of the SPs.

Through decomposition, we may have to solve lots of SPs since the packing problem for each order can be decomposed into several packing problems in different warehouses. Prior to solving SPs, we can get an upper bound on the minimum number of packages (bLF ) required for each SP, which can reduce the number of solving jk SPs directly. The upper bounds are obtained by using the modified first fit decreasing (FFD) heuristic algorithm. To deal with the absence of SKU conflicts, the classical FFD heuristic is improved with a local search improvement based on the bin shuffling strategy, and the detailed steps of the algorithm can be found in Zhang et al. (2018). The relationship b jk ≤ bGC ≤ bLF holds among the solutions jk jk of the MP, SP, and FFD. The advantage of using the upper bounds is that we detect optimal SPs without solving them if b jk =bLF . jk The benefits of the upper bound can be more substantial if many decomposed SPs to be solved. The upper bounds can also tighten the BPPC model when FFD does not find the same value as the MP. 5.4. Benders’ cuts

5.2. Bin packing with conflicts sub-problems Given the set of SKUs S jk to be packed in warehouse j and shipped to customer k, the goal of the SPs is to assign SKUs into packages with the objective to minimize the number of packages used while satisfying constraints of the weight, volume capacity of packages and the conflicts of SKUs. The conflict graph for sets S jk can be extracted from the original graph E as E jk = {(s1 , s2 )|s1 , s2 ∈ S jk , (s1 , s2 ) ∈ E }. The relationship between MP and multiple SPs can be illustrated as in Fig. 4. SP in each warehouse is a Bin Packing Problem with Conflicts (BPPC). SP in warehouse j ∈ K can be modeled as BP PC j . We denote br ∈ {0, 1}, 1 ≤ r ≤ n as package r used if br = 1 and psr ∈ {0, 1} as the SKU s packed in package r if psr = 1. b jk is the number of packages generated in the MP. bGC is the minimal jk number of packages needed in arc ( j, k ) ∈ B, and bLF is the correjk sponding upper bound, which will be introduced later. BP PC j is formulated as follows:

Min bGC jk =

n 

br

(17)

∀s ∈ S jk

(18)

r=1

s.t.

n 

psr = 1,

r=1



psr ws ≤ br W,

∀1 ≤ r ≤ n

(19)

s∈S jk



psr vs ≤ br V,

∀1 ≤ r ≤ n

(20)

s∈S jk

ps1 r + ps2 r ≤ br ,

∀(s1 , s2 ) ∈ E jk , 1 ≤ r ≤ n

(21)

psr ,br ∈ {0, 1}, ∀s ∈ S jk , 1 ≤ r ≤ n

(22)

LF b jk ≤ bGC jk ≤ b jk

(23)

Constraint (18) ensures that each SKU is assigned to exactly one package. Constraints (19) and (20) represent that no package

The generation of Benders cuts is an essential part of LBBD. Unlike classical Benders’ decomposition, which relies on the solution of its SP dual to derive a Benders’ cut, LBBD provides no standard scheme for generating Benders’ cuts and cuts must be devised for each problem class uniquely (Roshanaei et al., 2017). In each iteration of LBBD, Benders cuts are added to the MP for any SP that has a mismatch between b jk and bGC . Inspired by jk the Benders’ cut proposed by Fazel-Zarandi and Beck (2012), we propose two types of cuts, which considers incorporating different information of the mismatch to the MP to guide the master search towards global optimality. Assume that in one particular iteration h, the solution to the MP packs a set S jkh = {s|xhs jk = 1} of SKUs into b jk number of packages and ships packages from warehouse j to customer k. The solution of BPPC indicates that at least b∗jkh packages are needed to pack S jkh SKUs considering the constraints of weight, volume and conflicts. We have b∗jkh > b jk (S jkh ). The cuts then should specify that if S jkh or a superset of S jkh is again packed at warehouse j and shipped to customer k, then the number of packages must be greater than or equal to b∗jkh . The two kinds of cuts are generated to guarantee it in the transshipment network flow and the shipping network flow of SKUs, respectively. (1) Cut 1 The first cut (Cut1) is generated from the shipping network flow. The cuts after iteration h are

b jk ≥ b∗jkh −



(1 − xs jk ), ∀( j, k ) ∈ Bh .

s∈S jkh

Bh is the set of arcs for which the gap exists between BPPC and MP in iteration h, and b∗jkh is the minimum number of packages needed for the SKU set S jkh . Cut 1 is a valid cut (see Theorem 1), defined as a logical expression that has two properties (Chu & Xia, 2004): (1) the cut must eliminate the current MP solution if it is not globally feasible, and (2) the cut must not remove any globally feasible solution. Since Properties 1 and 2 are satisfied and b jk has a finite domain, the logic-based Benders’ decomposition approach with Cut 1 will converge to optimality in a finite number of steps.

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Table 1 Problem sets.

Theorem 1. The proposed Benders Cut 1 is valid. Proof. See Appendix A. (2) Cut 2 The second cut (Cut 2) is generated from the transshipment network flow. The cuts after iteration h are

b jk ≥ b∗jkh −

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( 1 − xs j  j ) −

s∈S jkh , ( j , j )∈A ds j =1



xs j j  ,

∀( j, k ) ∈ Bh .

s∈S jkh , ( j, j )∈A ds j =1

The summation term in this constraint is the maximal decrease in the number of packages needed, given that some of the SKUs may be transshipped to other warehouses. The largest possible reduction in the number of packages in changing the flow of one SKU is 1. There are some differences for the SKUs because they are stored at different warehouses. Some SKUs that are transshipped to warehouse j and stored at different warehouses can be represented in s ∈ S jkh , ( j , j ) ∈ A, ds j = 1, and other SKUs that stored at warehouse j can be represented in s ∈ S jkh , ( j, j ) ∈ A, ds j = 1. Theorem 2. The proposed Benders Cut 2 is valid. Proof. See Appendix B. 5.5. Cut propagation Cut propagation (Roshanaei et al., 2017) is proposed to remove more infeasible solutions and reduce the number of iterations to get the feasible solution by adding more similar Benders’ cuts. It would work because if a set of SKUs S jkh cannot be packed in warehouse j with b jk packages, the same set of SKUs also cannot be packed in other warehouses within b jk packages. With this, up to |K | Benders’ cuts can possibly be generated from a single infeasible SP. However, more cuts, which are added to the MP as constraints, may remove more infeasible solutions and can also make it more difficult to solve the MP. Therefore, based on the traditional Cut propagation 1, we also propose Cut propagation 2, which only copies Benders’ cuts to the warehouses where the SKUs in S jkh are initially stored. Cut propagation 1: Generate the Benders’ cut for all other warehouses ( j ∈ K). Cut propagation 2: Generate the Benders’ cut for the warehouses ( j ∈ K, ds j = 1, s ∈ S jkh ). 6. Numerical experiments In this section, we demonstrate the efficiency of the proposed model and algorithm via some numerical examples, as well as provide insights into the circumstances under which package consolidation performs particularly well. We simulate the instance data according to the realistic data and real practice of our industrial partner, one of the largest online retailers in China. The proposed LBBD algorithm was implemented with C# using CPLEX 12.6.2 solver. All experiments were performed on an Intel Core i5-5200 U 2.2 gigahertz processor with 4 gigabyte RAM and the Windows 10 operating system. The details of the results are explained below. 6.1. Generation of problem sets The problem sets were initially generated with K = 10 and then extend to K = 20 for the large-scale problems. Each warehouse can perform package consolidation and SKUs are randomly generated in different warehouses. The total number of SKUs is set as 10 0 0, which can be easily extended because the number of ordered SKUs in each order would affect the performance of the algorithm instead of the total number of SKUs. The number of multi-item orders is set to 100, which represents a much larger order scale

Problem set

# of SKUs

# of split orders

Problem sets

# of SKUs

1 2 3 4 5

2 5 5 10 10

2 2 5 2 5

6 7 8 9 10

10 15 15 20 20

# of split orders 10 5 10 5 10

for online retailers since we exclude the single-item orders and unsplit multi-item orders and assume all the orders are split multi-item orders. Also, scaling the order scale to the practical size (e.g., 10,0 0 0 in a period) seems to be easy because the proposed algorithm can generate schemes for 100 orders within a relatively short time, and different orders can be solved separately under a distributed computing environment. The weight, volume, and conflicts of SKUs were generated from U (0.1, 2), U (0.1, 0.5) and a probability of 0.2 in the same warehouse and 0.4 among warehouses. The range of weight and volume of SKUs is estimated by most small and medium goods sold by the online retailers. The coefficients of the conflicts of SKUs are generated on the similarities of items in the same category and different categories. The fixed and the variable shipping costs, and the ratio of the unit transshipment cost to the maximum fixed shipping cost follow U (6, 8 ), U (1, 2 ) and U (0.25, 0.38), respectively. The range of shipping costs tested is chosen based on the regular prices charged by shipping companies. As shown in Table 1, we group the problem instances into ten different sizes based on the numbers of SKUs and suborders per order considering the complexity of this problem is mainly determined by these two factors. The number of suborders in each order is {2, 5, 10}, corresponding to low, medium, and high order splitting scenarios, respectively. The number of SKUs per order ranges from 2 to 20, which will be extended to 100 for the large-scale problem. A split order generation algorithm is provided in Appendix C. There are 10 instances for each problem set, and 5 instances for each large-scale problem set. These generated instances are available on the website: https://sites.google.com/view/onlineretailing/. When testing the LBBD algorithm, we set the maximum run time limit for each order to 108 seconds. For each instance with 100 orders, the maximum run time will not exceed 3 hours since orders in each instance can be decomposed and solved one by one. 6.2. Computational results 6.2.1. Computational efficiency of LBBD We compare IP+CPLEX (the original IP model solved directly by CPLEX) optimal solutions to the LBBD approaches (LBBDs). No time limits were set for IP+CPLEX to get the optimal solutions so that they can be used as a benchmark for evaluating the performance of the other algorithms in terms of solution quality. LBBDs vary with six kinds of combinations of two cuts (Cut 1 and Cut 2) and two propagations (Propagation 1 and Propagation 2). The averages of the instances for each problem set are summarized in Appendix D. Since all six LBBDs have similar performance, we use the results of Cut1+Propagation 1 to demonstrate the computational efficiency of LBBD. Cut 1 + Propagation 1 will also be used for the experiments shown from Sections 6.2.2–6.3.4. As shown in Table 2, the proposed LBBD can generate nearoptimal solutions with only 25.56% of the CPU time required by CPLEX to solve the proposed multi-commodity network flow model, showing that LBBD runs much faster than IP + CPLEX. The average gap between the results from LBBD and the optimal results is lower than 0.005%. The gap is very low since only a

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Table 2 The comparison of LBBD and IP model. Problem set

1 2 3 4 5 6 7 8 9 10 Average

LBBD (Cut1+Propogation1)

IP

Avg. CPU time(second)

Avg. SP time%

Avg. iteration count (order level)

Avg. optimality gap

Avg. # of optimal solutions

0.80 3.85 10.26 19.21 17.04 179.14 45.97 158.83 163.83 195.62 25.56%

0.20% 0.58% 0.46% 12.58% 0.77% 0.31% 3.35% 0.36% 6.63% 0.67% 2.59%

1.01 1.02 1.04 1.39 1.11 1.18 1.38 1.22 2.67 1.51 1.35

0.000% 0.000% 0.000% 0.000% 0.000% 0.000% 0.002% 0.000% 0.029% 0.014% 0.004%

100/100 100/100 100/100 100/100 100/100 100/100 99.9/100 100/100 99.4/100 99.7/100 99.9/100

Avg. CPU time(second) 1.28 15.62 38.47 67.25 114.02 631.53 263.05 1206.36 562.07 1969.44 –

Table 3 Comparison between OSF and PCF. Problem set

1 2 3 4 5 6 7 8 9 10

OSF (average)

PCF (average)

Total cost

Package number

Delivery times

Total cost

Package number

Delivery times

1689.33 2182.27 4220.84 3432.51 5011.69 8449.80 6061.46 9203.78 7274.84 10,060.25

200 208.2 500 299.9 509.4 1000 562.6 1003.1 651.4 1024.2

200 200 500 200 500 1000 500 1000 500 1000

1166.82 2007.87 2396.50 3333.23 3908.40 4423.76 5359.96 5990.13 6734.70 7523.93

100.5 154.3 153.7 264.2 253.5 249.6 361.1 349.2 471.6 453.9

100.3 147.1 149.1 193.9 246.2 247.5 333.1 342.2 399.5 440

few orders cannot be solved to optimality due to the run time limits set for the experiments, as indicated in the column “Avg # of optimal solutions”. The column “Avg iteration count” indicates the average of the iteration count for each order. It shows that the number of iterations can be higher if optimal solutions are not found. For the comparison of MP and SP, the MP takes up most of the running time of LBBD since the average SP running time is 2.59%. Additional improvements of LBBD may include how to better solve the master problem. 6.2.2. Order splitting fulfillment vs. package consolidation fulfillment We compare OSF and PCF in terms of order fulfillment costs, the number of packages, and the delivery times. Table 3 shows that compared with OSF, PCF has the potential to reduce costs by 2.89% to 47.65% varying by the problem sets. As we mentioned before, the problem sets are determined by the total number of SKUs and the number of suborders per order. The ratio of these two factors is the average SKUs per suborder. As shown in Fig. 5, we find that the cost saving by PCF decreases with the increase of the average SKUs per suborder. This is because each suborder with a small number of SKUs cannot achieve its scale effect to deliver as a single shipment, which needs package consolidation. It implies that package consolidation is more economical and can reduce more costs if there is a small number of SKUs in each suborder. In fact, a small number of SKUs in each suborder has two scenarios, one is that the number of SKUs in an order is static, but the number of split orders is higher; the other one is that the number of split orders for an order is static, but the number of SKUs is lower. With PCF, the number of packages per order saved is between 0.36 and 7.50 based on different problem sets. Taking problem set 3 as an example. This implies a saving of over 10 million (10,0 0 0 ∗ 365 ∗ 3.46 = 12,629,0 0 0) package materials in one year if there are 10,0 0 0 orders per day in an online retailer. From the customer perspective, we compare the delivery times for each order. As we can see, the customer would receive fewer shipments

Cost savings

Saving packages per order

Saving delivery times per order

30.93% 7.99% 43.22% 2.89% 22.01% 47.65% 11.57% 34.92% 7.42% 25.21%

1.00 0.54 3.46 0.36 2.56 7.50 2.02 6.54 1.80 5.70

1.00 0.53 3.51 0.06 2.54 7.53 1.67 6.58 1.01 5.60

Table 4 Comparison of order fulfillment time per order between OSF and PCF. Problem set

1 2 3 4 5 6 7 8 9 10

Average order fulfillment time per order POF (minute)

PCM (minute)

127.55 129.23 129.11 131.88 130.33 129.66 131.55 130.44 132.72 131.14

179.30 157.04 179.95 146.98 181.11 180.04 181.26 181.08 180.64 181.76

Gap

40.57% 21.52% 39.38% 11.45% 38.97% 38.86% 37.78% 38.82% 36.11% 38.60%

Gap=(PCF - OSF)/ OSF ∗ 100%.

(saving 0.06∼7.53) if we conduct package consolidation, which is beneficial to improve customer satisfaction facing splitting order. The saving delivery times would be significantly increasing if the number of suborders in an order increases. 6.2.3. Insights on order fulfillment time Another important indicator for comparison is the order fulfillment time because package consolidation may increase the order fulfillment time since additional transshipment may be required. Here, we try to answer two questions: (1) How much additional order fulfillment time will be added by PCF compared with OSF? (2) Is the additional time introduced reasonable for online retailing practice? The detailed comparison can be found in Appendix E. We show only a summary of the results in Table 4. We can see that PCF will increase averagely 34.20% of the order fulfillment time compared with OSF. It indicates that PCF is not suitable for urgent orders. However, for regular orders, the increased order fulfillment time is still acceptable for online retailers. According to our investigation, the shipping vehicles leaving from warehouses usually

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Fig. 5. The relationship between cost savings and the average SKUs per suborder.

Table 5 The results for the large LBBD. Problem set

1 2 3 4 5 6 7 8 9 10

Large LBBD (average)

LBBD (average)

Gaps in avg costs

Cost

CPU time (second)

Cost

CPU time (second)

1175.86 2012.50 2403.30 3361.18 3929.73 4450.01 5395.91 6017.98 6811.16 7577.69

0.62 1.50 4.75 2.70 6.48 138.82 10.89 117.28 16.59 105.32

1166.82 2007.87 2396.50 3333.23 3908.40 4423.76 5359.96 5990.13 6734.70 7523.93

0.80 3.85 10.26 19.21 17.04 179.14 45.97 158.83 163.83 195.62

0.77% 0.23% 0.28% 0.83% 0.54% 0.59% 0.67% 0.46% 1.12% 0.71%

Gaps in avg. costs = (Large LBBD- LBBD)/LBBD∗ 100%.

follow a regular dispatching schedule of a day, e.g., 8 am, 1 pm, 6 pm. That means the transshipment process and packing process for the split orders will have little influence on the delivery time to the customers if they can be completed by these deadlines. But this would affect orders that are picked at a time near to the deadlines. In that case, these orders would be treated as urgent orders. Therefore, the added order fulfillment time in PCF is reasonable and acceptable for most of the regular orders in online retailers, while urgent orders may require OSF for speedy delivery. 6.2.4. Large LBBD for large-scale problems For large-scale problems in practice, sometimes we only need to find a near-optimal instead of the optimal solution within a reasonable amount of time. Based on the introduction of the LBBD algorithm before, we notice that the proposed algorithm can derive a globally feasible and suboptimal solution at each iteration. In generating a cut, we find the minimum number of packages needed at each warehouse. The number generated by SPs constitutes a feasible solution even though fewer packages were generated in the MP. Thus, we have a globally feasible solution at the end of each iteration. Here, we first compare the results generated by LBBD with only one iteration (denoted as Large LBBD) with previous results. As can be seen in Table 5, on average, the Large LBBD approach finds solutions with a gap of 0.61%. However, it only requires less than half of the average CPU time (44.37%) of the LBBD algorithm. We next generate some larger problem sets to test the Large LBBD algorithm. The number of warehouses is extended to 20, and four types of problem sets are generated based on the combination of 50/100 SKUs and 10/20 suborders per order. The problem sets

Table 6 The results of Large LBBD for large-scale problems. Large problem set

# of Warehouses

# of SKUs

# of split orders

Gaps in avg. costs

Large1 Large2 Large3 Large4

20 20 20 20

50 50 100 100

10 20 10 20

1.67% 0.59% 0.58% 0.53%

Avg CPU time/second 237.44 1698.11 751.02 2716.85

Gaps in avg. costs are based on the results between LBBD with 3 hours limit and Large LBBD.

are sufficient for large-scale because the order size is greater than the typical order size (between 20 and 55 items) in the grocery e-commerce recently reported by Sinha and Weitzel (2015). We use the LBBD algorithm (Cut1+Pro1) running with a 3 hours limit to assess the effectiveness of Large LBBD. Table 6 exhibits that the average gap ranges from 0.53% to 1.67% with an average CPU time ranging from 237.44 second to 2716.85 second (2.20%∼25.16% of the time limit by LBBD). The Large LBBD is effective since results by LBBD exhibit a substantial decrease at the beginning (Large LBBD) and then the decrease trend continues but at a slower rate. This pattern is consistent for all the results obtained by LBBD with different running time limits as shown in Fig. 6. 6.3. Managerial insights 6.3.1. Effect of the transshipment cost We increase the ratio between the unit transshipment cost and the maximum fixed shipping cost from U (0, 0.13) to U (1.25, 1.38) in problem set 5 to examine the impact of the unit transshipment

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Fig. 6. LBBD with different running time limits.

Fig. 7. Cost savings and total number of packages vs. the ratio.

cost on the total number of packages and the percentage of cost reduction. Fig. 7 exhibits that compared with OSF, the average cost savings by PCF are reduced as the ratio increases. However, the total number of packages remains almost the same when the ratio is below a certain level but increases steadily once the ratio passes that level. This implies that there exists a range of ratios within which the benefit of package consolidation is almost insensitive to the unit transshipment cost. When the unit transshipment cost is higher than the fixed shipping cost, the cost savings are reduced to 0 and the total number of packages remains the same.

6.3.2. Effect of SKU conflicts We set the SKU conflict rates ranging from 0.1 to 0.4 (INCONFLICT, the conflicts in the same warehouse) and 0.4 to 0.9 (OUTCONFLICT, the conflicts among warehouses) based on problem set 5. Fig. 8 indicates that the average order fulfillment cost almost remains the same with the increase of INCONFLICT but exhibits a large increase as OUTCONFLICT increases. Zhang et al. (2018) confirm that it is difficult to apply the package consolidation approach for high OUTCONFLICT for single consolidation warehouse situation. The results here show that this also applies

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Fig. 8. The total costs vs. different conflict types.

Fig. 9. The cost savings by PCF vs. the number of SKUs per order.

Fig. 10. The total costs and cost savings vs. the number of split orders per order.

to the situation in which multiple warehouses are used as the consolidation warehouses. 6.3.3. Impact of the number of split orders and SKUs For three kinds of order splitting scenarios (Low Splitting (Split order count = 2), Medium Splitting (Split order count = 5), High Splitting (Split order count = 10)), we calculated average cost savings by PCF compared with OSF with respect to the number of SKUs per order. As shown in Fig. 9, cost savings decrease as the

increase of the number of SKUs per order for three scenarios; the higher splitting scenario always has higher cost savings potential. Moreover, we test the sensitivity by varying the number of suborders per order based on problem sets 9 and 10. Fig. 10 shows that both the average total costs and average cost savings by PCF compared with OSF increase as the increase of the number of suborders per order. It again implies that package consolidation will play a more important role with a higher number of suborders per order.

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13

Fig. 11. The cost savings by PCF compared with OSF vs. the change of α value.

6.3.4. The role of α In our model, parameter α is closely related to the shipping cost. Here we test the level of sensitivity for α based on problem set 5. Fig. 11 shows that with the increase in α , the average cost saving by PCF compared with OSF decreases. Despite the decrease, the amount of saving is still very significant. 7. Conclusions and future work Despite the split-order fulfillment problem has become one of the most intractable order fulfillment problems emerging from the practice of online retailing in recent years, the area has received relatively low attention in the literature. Our research addresses a promising order fulfillment approach: multi-warehouse package consolidation, which has the potential to reduce the number of shipments caused by split orders. We develop a combined multicommodity network flow model to handle multiple SKUs of an order flow from warehouses to the customer through consolidation warehouses. We present in this work an enhanced logic-based Benders’ decomposition algorithm, including two Benders’ cuts, two cut propagations, and a FFD heuristic, to effectively solve the model. We demonstrate that the proposed LBBD algorithm can generate the near-optimal result with only about 25% of the CPU time required by CPLEX to solve the proposed multi-commodity network flow model, and it can be extended to effectively generate near-optimal results for the large-scale problem with less than half of the computation time required to solve the problem to optimality. Numerical experiments indicate that compared with the order splitting fulfillment approach, the multi-warehouse package consolidation approach can greatly reduce the fulfillment cost and the environmental impact of packages by decreasing the number of packages, and potentially improve customer satisfaction by reducing the delivery times. The level of cost reduction is more pronounced when the number of SKUs in each suborder is moderately small, a case quite typical for daily orders handled by online retailers. Meanwhile, practically speaking, the increased order fulfillment time resulting from package consolidation is within the acceptable range for most of the multi-item orders. Moreover, sensitivity analyses are performed to generate managerial insights for applying package consolidation in practice. Several promising directions for future research remain. First, the LBBD algorithm can still be improved via the addition of new Benders’ cuts, SP relaxations and other heuristic strategies. The multi-commodity network flow model is solved based on CPLEX with default preprocessing techniques (IBM, 2015). Prior

to using the LBBD algorithm, the model may be further improved by some preprocessing techniques, which needs to address the running time for preprocessing and the savings in the solution time for the whole problem (Martin, 2001). The common packing problems in online retailing have a small number of items. The decomposed SPs can be solved easily by any state-of-the-art mixed integer programming solver. In some situations, a large number of items may emerge in the packing problem. One possible solution is to incorporate the arc-flow formulation and the graph compression algorithm by Brandão and Pedroso (2016) into the LBBD framework to solve the SPs more efficiently. It would be interesting to explore if it is effective to incorporate their algorithm through updating the graph for each SP after each iteration in LBBD. Moreover, our results are based on the network of multiple category warehouses with non-overlapping categories. A network of multiple warehouses with overlapping storage of SKUs among these warehouses may be considered, which may need a joint optimization of order allocation and package consolidation. Moreover, timely delivery is becoming more and more important these days. The model discussed can be extended to involve the specific delivery time window requested by customers, which may greatly increase the difficulty level of the model and more sophisticated algorithms would be needed. Furthermore, the consolidation for multiple customers with limited warehouses’ capacities can be further explored by addressing the benefit of package consolidation schemes among multiple customers. Since different customers may have different degrees of satisfaction for multiple shipments by split orders, another research direction to take is to come up with a better order fulfillment policy which explicitly addresses the satisfaction level of customers. Acknowledgment We are grateful for the careful and detailed feedback provided by two anonymous referees. This research was supported by program of the National Natural Science Foundation of China (71571067), the innovative research group program of the National Natural Science Foundation of China (71421001), and China Scholarship Council (201706060098). Appendices Appendix A. Proof of Theorem 1 Theorem 1. The proposed Benders Cut 1 is valid.

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Proof. To prove the validity of Cut 1, we need to show that both properties 1 and 2 are satisfied. We first prove Property 1. Recall that S jkh is the set of SKUs that are packed at warehouse j and shipped to customer k in iteration h, and b∗jkh is the minimum number of packages needed for the set S jkh as found by the BPPC model in iteration h. We denote b jkh to be the number of packages generated by the MP in iteration h. If the solution of MP is infeasible in the BPPC model, then b jkh < b∗jkh , which will result  in Cut 1: b jk ≥ b∗jkh − s∈S (1 − xs jk ). If the same set of SKUs is jkh

again packed at warehouse j and shipped to the customer k, then  (1 − xs jk ) = 0. It follows that b jk ≥ b∗jkh > b jkh , which shows s∈S jkh

that the same set of SKUs should result in a larger number of packages generated in the MP, and conversely, the same number of packages generated in the MP should result in a different set of SKUs. It also shows that Cut 1 excludes the current MP solution from all subsequent MP solutions. Therefore, Cut 1 satisfies property 1.



d

(S

∪S

)

ages for the set. Consider S jkt ∪ S jkh , and assume that b jk jkt jkh is the minimum number of packages for the set S jkt ∪ S jkh . We can (S

∪S

)

easily get b jk jkt jkh ≥ b∗jkh , because b∗jkh is the minimum number of packages for the subset S jkh . Consider also that S jkt ∪ S jkh is the union set of the sets S jkt and S jkh \S jkt . We then can get  ( S ∪S ) b jk jkt jkh ≤ b jkt + s∈{S \S } (1 − xs jk ). Combining the above jkh jkt  two formulas, we have b∗jkh ≤ b jkt + s∈{S \S } (1 − xs jk ). Since jkh

jkt

there is no difference for SKU s ∈ {S jkh ∩ S jkt } between iteration  h and t, then s∈{S jkh ∩S jkt } (1 − xs jk )=0. Therefore, we can get  b jkt ≥ b∗jkh − s∈S (1 − xs jk ), and Cut 1 does not remove any jkh

globally feasible solutions (Property 2).

Appendix B. Proof of Theorem 2



=1 s j

s∈S jkh , ( j, j  )∈A

xs j j = 0, ∀( j, k ) ∈ Bh .

The

ds j =1

current solution will be eliminated because Cut 2 needs at least one more package (b jk ) or subtract at least one SKU from set S jkh . We next show that the Benders’ Cut 2 does not remove any globally feasible solutions in future iterations (Property 2). Assume a globally feasible solution is found in iteration t > h. Let S jkt be the set of SKUs packed at warehouse j and shipped to customer k in iteration t, and b jkt is the needed packages for the set. Assume that b jk

(S jkt ∪S jkh )

is the minimum number of packages for set

S jkt ∪ S jkh . Then we have b jk

(S jkt ∪S jkh )

≥ b∗jkh . Since S jkt ∪ S jkh is the (S

union of sets S jkt and set S jkh \S jkt , we can get b jk jkt   b jkt + s∈{S \S }, ( j , j )∈A (1 − xs j j ) + s∈{S \S }, ( j, j )∈A xs j j . d

s j

jkh =1

jkt

jkh ds j =1

∪S jkh )

≤

jkt

Combining the above two formulas, we have b∗jkh ≤ b jkt +   s∈{S \S }, ( j  , j )∈A ( 1 − xs j  j ) + s∈{S \S }, ( j, j )∈A xs j j  . Since there d

We next show that Cut 1 does not remove any globally feasible solution. Assume that a globally feasible solution is found in iteration t > h. Let S jkt be the set of SKUs packed at warehouse j and shipped to customer k in iteration t, and b jkt is the needed pack-

( 1 − xs j  j ) +

s∈S jkh , ( j  , j )∈A

s j

jkh =1

jkt

jkh ds j =1

jkt

is no change for the SKU s ∈ {S jkh ∩ S jkt } from iteration h to t, we   still have s∈{S ∩S }, ( j  , j )∈A ( 1 − xs j  j ) + s∈{S ∩S }, ( j, j  )∈A xs j j  = 0. d

Therefore, 

s j

jkh =1

we

jkt

can

jkh d =1

get

b jkt ≥ b∗jkh −

s∈S jkh , ( j  , j )∈A d

s∈S jkh , ( j, j  )∈A

jkt

s j

( 1 − xs j  j ) −

=1 s j

xs j j , and property 2 can be satisfied by Cut 2.

ds j =1

Appendix C. Split order generation algorithm We propose a split order generation algorithm to generate the suborders for each order which contains [SplitOrderNum] suborders and [SKUNum] SKUs: Step 1. From the warehouse set K, we generate the sub warehouse set containing [SplitOrderNum] warehouses; Step 2. For each warehouse in the sub warehouse set, we randomly generate one SKU for the order; Step 3. The remained number of SKUs ([SKUNum] - [SplitOrderNum]) are randomly generated from the sub warehouse set. Appendix D. Comparison of two cuts and two propagations in LBBD

Theorem 2. The proposed Benders Cut 2 is valid. Proof. As same as Cut 1, we still need to verify properties 1 and 2 are satisfied for Cut 2. We first show that Cut 2 eliminates the current MP sub-optimal solution (Property 1). Recall that S jkh is the set of SKUs are packed at warehouse j and shipped to customer k in iteration h, which results in an infeasible MP solution. For SKUs which are stored and packed at warehouse j, we have xs j j =0, ∀( j, j ) ∈ A, s ∈ S jkh , ds j = 1 . For SKUs which are transshipped and packed at warehouse j, we have 1 − xs j j =0, ∀( j , j ) ∈ A, s ∈ S jkh , ds j = 1 . Therefore, we can get

Table 7 shows the results of different versions of LBBD algorithm. For the two kinds of cuts, the average CPU time of Cut2 is slightly lower than that of Cut1 (the average gap is about 0.92%). The results of the two cuts are similar since the shipping network flow (Cut1) is closely connected with the transshipment network flow (Cut2) by the warehouse nodes in the multi-commodity network flow model. As for propagations, the average CPU times for propagations 1 and 2 decrease 1.71% based on Cut1 and Cut2. The

Table 7 Results of two cuts and two propagations in LBBDs.

Avg. CPU time(s)

Problem set

Cut1

Cut1+Pro1

Cut1+Pro2

Cut2

Cut2+Pro1

Cut2+Pro2

IP

1 2 3 4 5 6 7 8 9 10

0.92 3.95 11.53 18.15 17.20 199.20 44.02 157.71 155.25 198.17

0.80 3.85 10.26 19.21 17.04 179.14 45.97 158.83 163.83 195.62

0.83 3.78 10.64 18.29 17.05 188.28 43.36 158.85 163.04 198.69

0.80 3.92 11.43 18.88 17.17 198.90 43.93 159.43 161.08 192.39

0.75 3.88 10.18 19.99 16.80 178.57 44.06 160.11 170.17 196.28

0.77 3.83 10.91 19.07 17.10 188.86 43.86 161.68 168.31 195.66

1.28 15.62 38.47 67.25 114.02 631.53 263.05 1206.36 562.07 1969.44

Average

26.81%

25.56%

25.73%

26.01%

25.33%

25.61% (continued on next page)

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15

Table 7 (continued)

Avg. SP time%

Avg. iteration count

Avg. optimality gap

Avg. # of optimal solutions

Problem set

Cut1

Cut1+Pro1

Cut1+Pro2

Cut2

Cut2+Pro1

Cut2+Pro2

1 2 3 4 5 6 7 8 9 10

1.20% 0.56% 0.79% 12.85% 0.80% 0.53% 3.83% 0.34% 7.51% 0.66%

0.20% 0.58% 0.46% 12.58% 0.77% 0.31% 3.35% 0.36% 6.63% 0.67%

0.76% 0.51% 0.46% 12.35% 0.77% 0.31% 3.64% 0.39% 6.73% 0.66%

1.18% 0.69% 0.88% 12.30% 0.80% 0.53% 4.05% 0.35% 6.94% 0.70%

0.42% 0.37% 0.40% 12.47% 0.85% 0.33% 3.72% 0.38% 6.28% 0.65%

1.24% 0.54% 0.56% 12.46% 0.85% 0.31% 3.92% 0.36% 6.34% 0.67%

Average 1 2 3 4 5 6 7 8 9 10

2.91% 1.04 1.03 1.12 1.40 1.11 1.25 1.39 1.22 2.77 1.51

2.59% 1.01 1.02 1.04 1.39 1.11 1.18 1.38 1.22 2.67 1.51

2.66% 1.04 1.03 1.07 1.39 1.11 1.20 1.36 1.22 2.68 1.51

2.84% 1.04 1.03 1.12 1.39 1.11 1.25 1.39 1.22 2.74 1.50

2.59% 1.01 1.02 1.04 1.39 1.10 1.18 1.37 1.23 2.63 1.50

2.72% 1.04 1.03 1.07 1.39 1.11 1.20 1.37 1.22 2.66 1.50

Average 1 2 3 4 5 6 7 8 9 10

1.38 0.000% 0.000% 0.000% 0.000% 0.000% 0.003% 0.002% 0.000% 0.019% 0.014%

1.35 0.000% 0.000% 0.000% 0.000% 0.000% 0.000% 0.002% 0.000% 0.029% 0.014%

1.36 0.000% 0.000% 0.000% 0.000% 0.000% 0.000% 0.002% 0.000% 0.029% 0.014%

1.38 0.000% 0.000% 0.000% 0.000% 0.000% 0.003% 0.002% 0.000% 0.019% 0.008%

1.35 0.000% 0.000% 0.000% 0.000% 0.000% 0.000% 0.002% 0.000% 0.029% 0.014%

1.36 0.000% 0.000% 0.000% 0.000% 0.000% 0.003% 0.002% 0.000% 0.029% 0.014%

Average 1 2 3 4 5 6 7 8 9 10

0.004% 100/100 100/100 100/100 100/100 100/100 99.9/100 99.9/100 100/100 99.6/100 99.7/100

0.004% 100/100 100/100 100/100 100/100 100/100 100/100 99.9/100 100/100 99.4/100 99.7/100

0.004% 100/100 100/100 100/100 100/100 100/100 100/100 99.9/100 100/100 99.4/100 99.7/100

0.003% 100/100 100/100 100/100 100/100 100/100 99.9/100 99.9/100 100/100 99.6/100 99.8/100

0.004% 100/100 100/100 100/100 100/100 100/100 100/100 99.9/100 100/100 99.4/100 99.7/100

0.005% 100/100 100/100 100/100 100/100 100/100 99.9/100 99.9/100 100/100 99.4/100 99.7/100

Average

99.91/100

99.9/100

99.9/100

99.92/100

99.9/100

99.89/100

IP

Pro1: Propagation 1; Pro2: Propagation 2.

improvement is limited probably due to the small iteration count that is shown in Table 7. Therefore, when using propagation rules, one perhaps should focus on balancing the increase in complexity of the MP due to the more cuts generated by cut propagation and the decrease in iteration times since cut propagation eliminates infeasible solutions. Appendix E. Order fulfillment time comparison Similar to the order fulfillment cost, order fulfillment time is also analyzed based on the fulfillment process. We ignore the order picking time considering its consistency in the OSF and PCF. The start time is set as the finish time of order picking, whereas the end time is the finish time of shipments for all packages. In the OSF, the fulfillment time of an order is the maximum fulfillment time for each suborder split from the order. It can be represented by the following equation: The order fulfillment time of OSF = Max{packing time + shipping time, for each shipment}. In the PCF, the fulfillment time for an order is the maximum fulfillment time for each shipment. There may be a waiting time if the SKUs from different warehouses are consolidated. That will result in that the transshipment time should be the maximum

transshipment time of each SKU in the shipment. The fulfillment time can be represented by the following equation: The order fulfillment time of PCF = Max{Max{transshipment time, for each SKU in the package} + packing time + shipping time, for each shipment}. The packing time for each shipment is the sum of the package setup time and the package processing time. Assume that t f pack is the unit package setup time and t v pack is the unit package processing time. The packing time for each shipment can be represented by the following equation: Packing time=t f pack ∗ the number of packages + t v pack ∗ the number of SKUs, for each shipment. We also propose an algorithm for calculating the order fulfillment time. The main idea is to follow the number of SKUs and packages in each shipping arc and then retrospect the possible transshipment arcs before each shipping arc. Let T jk represents the packing and shipping time for one shipment containing arc ( j, k ) ∈ B, T jktran is the maximum transshipment time for each arc

( j, k ) ∈ B, and MaxT represents the order fulfillment time for one order. The detailed steps of the algorithm are as follows:

Step 1. For each arc ( j, k ) ∈ B with y jk = 1, get the SKU set Sjk = {s|s ∈ S, xs jk = 1};

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Step 2. If Sjk is not null, T jk ← t jk + t f pack b jk + t v pack |Sjk |; else return to Step 1; tran =0. For each SKU in S , if d   Step 3. Set T jk s j  = 1 ( j ∈ K ) and j = jk ship

tran ← max{T tran , t tran }; j, T jk jk j j

tran | ( j, k ) ∈ B}. Step 4. MaxT = max{T jk +T jk

For the numerical experiments, the relevant parameters are generated as follows. We set t f pack = 0.5, t v pack = 0.1. The transshipment time t tran , ( j , j ) ∈ A among warehouses, the shipping time j j ship

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