Order fulfillment policies for ship-from-store implementation in omni-channel retailing

Order Fulfillment Policies for Ship-from-Store Implementation in Omni-Channel Retailing

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Order Fulfillment Policies for Ship-from-Store Implementation in Omni-Channel Retailing Armagan Bayram, Bahriye Cesaret PII: DOI: Reference:

S0377-2217(20)30032-1 https://doi.org/10.1016/j.ejor.2020.01.011 EOR 16265

To appear in:

European Journal of Operational Research

Received date: Accepted date:

9 October 2018 3 January 2020

Please cite this article as: Armagan Bayram, Bahriye Cesaret, Order Fulfillment Policies for Ship-fromStore Implementation in Omni-Channel Retailing, European Journal of Operational Research (2020), doi: https://doi.org/10.1016/j.ejor.2020.01.011

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Highlights • • • • •

We investigate dynamic order fulfillment decisions in ship-from-store. We develop an optimal cost threshold policy to guide retailers in their decisions. We provide insights on the value of ship-from-store implementation. We propose an efficient heuristic as a smart way of implementing ship-from-store. Optimal ship-from-store implementation does not hurt the profits of the retailer.

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Order Fulfillment Policies for Ship-from-Store Implementation in Omni-Channel Retailing Armagan Bayram College of Engineering & Computer Science, University of Michigan at Dearborn, Dearborn, MI 48128, USA, [email protected],

Bahriye Cesaret * Faculty of Business, Ozyegin University, Cekmekoy, Istanbul, 34794, Turkey, [email protected],

Abstract One of the recent trends in omni-channel retailing is ship-from-store which allows a retailer to fulfill online orders by using inventory from a nearby store. The benefits of this fulfillment model include faster delivery, lower shipment costs, higher in-stock probability, increased sales and customer satisfaction, etc. Despite its many benefits, this fulfillment model introduces many new operational challenges to the retailer, including the need to identify from which location to fulfill an online order when it arrives. In this study, we consider a retailer having both online and store operations, with each channel carrying its own inventory. Store orders are fulfilled from store inventories, whereas an online order can be shipped either from an online fulfillment center or from any other store that maximizes the retailer’s overall profit. Our study investigates dynamic fulfillment decisions: from which location to fulfill an online order when it arrives. We incorporate the uncertainty both in demand and in the cost of shipment to individual customers to characterize the optimal cross-channel fulfillment policy. Due to the optimal policy being computationally intractable for large-sized problems, we construct an intuitive heuristic policy to guide the retailers in their fulfillment decisions. We find that the proposed heuristic method is effective and obtains solutions within a reasonable amount of time for the cross-channel fulfillment problem. Key words : Retailing, ship-from-store, omni-channel retailing, order fulfillment, dynamic programming.

“We’re no longer going to need fulfillment centers anymore. We’ve got 800 of them, and they’re called Macys stores. ” – Former Macy’s CEO Terry Lundgren (Wall Street Journal 2012).

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Introduction

The popularity of online shopping continues to rise among consumers, due to its convenience, the greater selection of products being available online, the ability to obtain more information about products through customer reviews, etc. According to a recent report from Forrester Research (2016), US online retail sales will exceed $530 billion by 2020, up from $335 billion in 2015, with an average annual growth rate of 9.3%. The number of online customers in the US is predicted to reach 270 million in 2020, up from 244 million in 2015. Not only will the number of customers increase, but also the amount spent by each online customer is expected to reach $2500 by 2020, a 7% annual increase from 2015. On the other hand, physical store sales – still accounting for the majority of total retail sales – have an estimated growth rate of just 2.8%, a far slower rate than the average rate of growth for the overall industry (Business Insider 2017). This implies that while in-store sales are still important, finding ways to exploit rising online sales, such as by the addition of new channels or investing in cross-channel strategies by blending online and offline channels, will be increasingly important. Indeed, this is the recent evolutionary state of the retail industry. Lately, many retailers (e.g., Bloomingdale’s, Barnes & Noble, Walmart) have transformed quickly and embraced new strategies for integrating their online and offline channels to be able to meet customers’ demand across different channels more efficiently. The omni-channel (meaning “all channels”) approach is such a strategy that encourages shopping across channels. It is a fully integrated approach that allows customers to purchase products from anywhere and return them anywhere and allows retailers to fulfill orders from anywhere, thus offering customers more flexibility and a unified shopping experience. Currently, 91% * Corresponding author

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of retailers either have or are planning to invest in omni-channel technologies, while only four years ago, less than half of retailers were pursuing an omni-channel approach (Carroll 2018). Omni-channel retailing unlocks tremendous opportunities for a retailer to leverage its store network to support its online sales. Ship-from-store, which allows retailers to fulfill an online order from a nearby store rather than from a fulfillment center (FC), is one of the omni-channel implementations. Among several omni-channel initiatives, ship-from-store is rated with the highest priority (18%) according to retailers (Forrester Research 2014), and about one of every three omni-channel retailers adopted a ship-from-store program in 2016, according to Forrester Research (Roggio 2017). Stimulated by these observations, we study the order fulfillment problem that an omni-channel retailer faces in ship-from-store implementation. Retailers can benefit from ship-from-store in several ways. First, orders can be shipped faster and often cheaper than shipping from FCs, as local stores are expected to be closer to the customers than FCs. Second, it can positively impact gross margins by preventing stores from marking down the price of a product, since store inventory can be used to fulfill online full-priced orders. Moreover, retailers can avoid online stock-outs by fulfilling orders from a store, which results in an increase in sales and customers’ satisfaction (Pappas 2014). Motivated by its several benefits, we consider implementation of ship-from-store by a retailer that sells seasonal products, such as a single-season fashion good, with long manufacturing lead times relative to the selling season. Customers can buy this fashion good (e.g., a shirt) through the online and store channels of the retailer, and the customers are expected to buy one unit of the product at their purchase. Due to the differences in fulfillment processes for online and store customers, the cost of handling these operations varies across different channels. For example, a store customer chooses the shirt in the store and then takes it home immediately, while an online order needs to be picked, packed, and shipped to the individual customer. In the fashion industry, the orders from the manufacturer are placed several months ahead of the season, so the initial inventory of the stores and the FC needs to be determined in advance. With such seasonal fashion products, there is high uncertainty in demand and only a single ordering opportunity. Having a mismatch in supply and demand is a frequent problem in the fashion industry, which can potentially lead to stock-outs and leftover inventory. Furthermore, shipping costs caused due to uncertainties in locations of customers can have a huge impact on the retailer’s profit. Thus, it is important for retailers to have an efficient fulfillment policy to minimize the negative impacts of the uncertainties on profits, leftover inventory, and stock-outs. The retailer can utilize its established store network to support its online sales via ship-from-store implementation. However, this flexibility complicates the fulfillment decisions of the retailer. More specifically, the retailer now needs to decide from which location to fulfill online orders as they arrive, under the uncertainty of demand and location of the customers. In this paper, we study this fulfillment problem that an omni-channel retailer faces. This study aims to address the following research questions: 1. Given a fixed amount of initial store and online inventory, what should be the optimal fulfillment strategy of a retailer that utilizes a ship-from-store strategy? 2. What is the value of ship-from-store implementation in terms of total profits? How does this implementation affect the retailer’s total sales, per unit shipment costs, total number of stock-outs that the retailer observes, and leftover inventory at the end of the selling season? The contribution of our study can be summarized as follows. First, we capture a stochastic dynamic decision structure faced by an omni-channel retailer by building a finite horizon dynamic programming model. Second, we present the structural properties of the optimal profit function and characterize the optimal order fulfillment policy for the retailer that maximizes the retailer’s total profit gained from

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total sales across all channels. We define a cost threshold policy to decide from which location to fulfill online orders. Third, we present an optimal independent channel fulfillment policy in which ship-fromstore operations are not allowed. We use this policy to provide insights for retailers on the value of ship-from-store implementation and also to provide guidance for which circumstances ship-from-store implementation creates more value to the retailers. Fourth, since the implementation of the optimal cross-channel fulfillment policy for large-sized problems is difficult, we present a myopic policy for ship-from-store, which is currently in use in practice, and also propose an integrative fulfillment policy that combines the strengths of myopic and independent fulfillment policies. We perform numerical analyses to compare the performance of the two heuristics and the optimal fulfillment policy relative to the independent channel policy. Our computational results indicate that the proposed heuristic outperforms the myopic and the independent channel fulfillment policies in terms of total profits and obtains a solution in a reasonable amount of time (i.e., less than 1 minute per scenario) for the problem instances that we consider. Our study provides the following insights for the retailers: An optimal implementation of ship-fromstore does not hurt the profits of the retailer and therefore is desirable, but it is not practical when many stores are included in the implementation. On the other hand, a myopic implementation of the strategy may hurt the retailer’s profit. Therefore, a smart and practical implementation of ship-fromstore is required to enjoy the benefits of the strategy. The value of ship-from-store implementation increases as (i) the inventory level of a store deviates from the expected demand of that store and (ii) the arrival rate of online customers increases. Moreover, an optimal implementation of ship-fromstore becomes more desirable when the (expected) net profit margin difference between online and ship-from-store fulfillment options decreases and when the distance of the FC to customers increases. Operating the channels independently is optimal when inventory in every store is very low compared to their respective expected demand. In these cases, it is best to fulfill all online demand from the FC. Myopic policy is optimal when inventory in every store is plenty. In these cases, it is best to fulfill an online demand from the closest fulfillment location having positive inventory. As a result of putting the correct policy into implementation, our proposed heuristic also achieves optimality in these cases. The remainder of this paper is structured as follows. In Section 2, we review the related literature. In Section 3, we describe the omni-channel fulfillment model and characterize the optimal cross-channel fulfillment policy. Section 4 presents the independent channel fulfillment policy and discusses the value of ship-from-store. Section 5 presents myopic policy and our proposed heuristic method, while Section 6 discusses the computational results. Finally, our conclusions are outlined in Section 7.

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Literature Review

We first discuss previous studies on online and omni-channel retailing that are relevant to our study. There exist several papers that study cross-channel interactions for fulfillment of online sales. Among these papers, Bendoly et al. (2007) investigate settings where either all or none of the retail stores handle online fulfillment, and they build models that integrate cross-channel inventories for fulfillment of online sales. The authors mainly focus on a comparison of centralized (i.e., online fulfillment is handled by a central facility) and decentralized (i.e., online fulfillment is handled only by stores) distribution network structures instead of when and from where to fulfill an online order. Chen et al. (2011) examine a setting with an online retailer and two physical retailers. In addition to serving their in-store customers, the physical retailers act as drop-shippers for the online retailer, which carries no inventory of its own. Our setting differs from theirs in the sense that they consider cross-channel

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operations across independent retailers having separate profit functions, whereas we examine crosschannel interactions of a single retailer. Acimovic and Graves (2014) investigate decision rules on where and how the orders should be fulfilled to minimize the outbound shipping costs of an online retailer. Related to another trending omni-channel strategy, Gao and Su (2016) investigate policy rules for buy-online-and-pick-up-in-store operations, in which a product bought online is shipped to a store for the customer to pick up, thereby reducing the waiting time for the customer to receive the product and avoiding the potential shipping costs to the customer. In another relevant study, DeValve et al. (2018) build a stochastic programming model to decide on which distribution center to use to fulfill online orders. They do not consider any physical stores, while our model includes both an FC and physical stores with various cost structures and profit margins. Our problem setting, model structure, and research questions are different from their study, as we investigate order fulfillment decisions for a ship-from-store strategy. The following papers consider similar settings to ship-from-store. Harsha et al. (2016) study price optimization on an omni-channel network when there exist cross-channel interactions in both demand and supply. Similarly, Koc et al. (2017) propose a multi-objective mixed integer programming model to fulfill online orders from stores or FCs. Our study differs from these two papers in its modeling approach, as we develop a dynamic programming model to help an omni-channel retailer with its fulfillment decisions. Govindarajan et al. (2018) build a stochastic programming model to find optimal cross-channel fulfillment decisions with the goal of minimizing their cost functions. They assume that online orders are fulfilled in batches, whereas our model makes fulfillment decisions immediately as each order arrives. Karp (2017) builds a two-stage stochastic programming model, with the first stage of the model handling the acceptance decision for each online order and the second stage investigating how to fulfill all accepted online orders. In contrast, we make simultaneous acceptance and fulfillment decisions when an online order arrives, without knowing future orders with certainty. To the best of our knowledge, the study provided by Jalilipour Alishah et al. (2015) has the closest model setting to ours. In their setting, the fulfillment network consists of only one store and one FC, and online demand is routed to the offline store only when the FC runs out of stock, which further simplifies the problem. Our study considers multiple stores; therefore, an additional decision on which store to use in fulfillment of an incoming online order has to be made. Our cost structure is also different from theirs, where we split their additional cost and define shipment cost, handling cost, and overhead cost separately. This allows us to incorporate the uncertainty in shipment costs as well as the asymmetries in shipment costs associated with the FC and the stores. Similar to the results of Jalilipour Alishah et al. (2015), we establish the structural properties of the profit function and the threshold policy. Our study contributes to theirs by theoretically showing the value of ship-from-store implementation under different settings. The authors have updated their working paper by considering multiple stores (Jalilipour Alishah et al. 2018). However, their theoretical findings are still for the stylized model where there is one offline store. In their numerical analyses, they investigate up to four stores. Our study contributes to their results by considering N stores in our theoretical findings. Another stream of literature that is relevant to our study is rationing inventory to multiple customer classes in a stochastic demand environment. Most of the studies in this stream focus on multi-period, single-product, single-location inventory settings (see, e.g., Nahmias and Demmy 1981, Moon and Kang 1998). Kleijn and Dekker (1998) provide a survey of early papers in this stream. In our setting, a retailer operating N stores faces N + 1 (i.e., N store and one online) independent demand classes, and the net profit earned from an online demand may differ depending on from which location the online demand is fulfilled. Thus, our model focuses on rationing of multi-location inventory to multiple

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customer classes with a single inventory replenishment opportunity. In our setting, inventory rationing decisions depend on net profits (including the net profit of an online order, which depends on a customer’s distance to each possible fulfillment location) and inventory level at each location (which needs to be updated depending on customer arrivals) and the decision made in the previous period. All of these issues complicate the problem that we study and inhibit the application of the solution approaches available in the literature. Finally, the research on lateral transshipment, in which inventory is allowed to be transshipped among several locations to better match demand, is also relevant to our study. Paterson et al. (2011) provide a review of this literature. Lateral transshipment can be categorized as real or virtual. In the former practice, inventory moves among the locations, whereas in the latter, demand is redirected to another location and is satisfied from this location, without actual inventory being transferred between the locations. Yang and Qin (2007) introduce the concept of virtual transshipment and focus on two locations. Our model can be viewed as a virtual transshipment model with N + 1 independent demand classes and only online demand possibly being redirected to one of the N store locations.

3.

An Omni-Channel Fulfillment Model

We consider a retailer that sells a seasonal product through both its online and offline (store) channels. There exist no stock replenishment opportunities during the selling season, due to the presence of a long lead time. The retailer’s fulfillment network may consist of multiple stores that support offline sales and multiple FCs that support online sales. We focus on a special case of the problem with multiple stores, each serving a different region, and one FC. We build a discrete-time, finite-horizon Markov decision process (MDP) model to determine the optimal order fulfillment decisions. Before the selling season starts, the retailer observes inventory positions at the FC and at each of the stores and places replenishment orders with its supplier(s) accordingly. Let I = {1, 2, . . .} represent the set of stores and J = I ∪ {0} represent the set of all channel locations, where {0} refers to the FC. We consider a finite-horizon MDP over T periods in a discrete-time setting. The time horizon is divided into T short intervals in such a way that, during each interval t, the probability of more than one customer of any type arriving is negligibly small, where t = 1, 2, . . . , T . Demand at each store i ∈ I (resp., at the FC) follows a Poisson process with arrival rate λi (resp., λ0 ). Thus, the rates λj also represent the probability of a unit demand in period t1 for each location j ∈ J , since we consider independent Poisson process arrivals. More specifically, for a given period, a store i customer arrives with probability λi , an online customer arrives with probability λ0 , or no customer arrives P with probability (1 − λ0 − i∈I λi ). We assume that the demands across all locations are independent; thus, the retailer faces |J | independent demands. We also assume that no customer switches across the channels. In other words, a customer placing an order through the online channel and observing that the product is out-of-stock does not intend to visit any of the stores and vice versa. Note that in our setting, stores serve different geographical regions, and therefore switching within the offline channel is also not possible. Suppose that at the beginning of period t, there exist xj (t) units available at location j. Then, we represent the state of the system by (t, x0 , x1 , . . . , x|I| ).2 In each period t, the retailer observes the current state of the system and makes a fulfillment decision to maximize its profit-to-go (i.e., the profit from selling its remaining units across all locations over the remaining selling season). If no 1

We use fixed arrival probabilities for each t; thus, instead of λj (t), we use the notation λj .

2

For ease of exposition, we remove the index t from the state and decision variables whenever it is clear from the context.

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customer arrives in period t, then there is no fulfillment decision to make. If a store customer arrives, the retailer fulfills the demand from the corresponding store inventory as long as the inventory level is positive. If an online customer arrives, then the retailer has the following three fulfillment options: (1) fulfill the order from the FC, (2) fulfill the order from one of the stores, or (3) do not fulfill and thus reject the order3 . The last option may occur when serving a customer is not profitable, or it is profitable but the profit that this customer brings is not high enough to compensate for reserving this unit for future customers. The selling prices are pj , and we assume that prices are fixed over the selling season. The unit operating costs are cpj and include the cost of goods sold, rent, labor, overhead, etc. Fixed ordering costs are assumed to be zero, and the salvage value of the product at the end of the selling season is normalized to zero. The retailer incurs a handling cost and a shipment cost in fulfilling an online demand. Handling cost is the cost associated with, e.g., picking, handling, and packaging of an online order, either at the FC or at a store. We denote the handling cost incurred at location j by chj . We assume that handling cost incurred at any store is typically higher than that at the FC (i.e., chi ≥ ch0 ), since stores are not designed to handle online orders. The shipment cost for an online order depends on the distance between the customer’s location and the fulfillment location from which this customer is to be fulfilled. We denote the shipment cost of a customer to location j by csj , which has a cumulative distribution function Gj (· ). All shipment costs are uncertain, and the retailer learns csj for each location j at the time of an online arrival. To ensure that fulfilling an online order from both channels is at least an option, we assume that all handling costs are smaller than the minimum of the gross profit margins p0 − cpj . Given shipping costs csj from each fulfillment location j to the customer’s location, the retailer calculates the expected profit of fulfilling the online order from each location j, compares them, and chooses accordingly. We now proceed to define the optimal cross-channel fulfillment policy. 3.1. Optimal Cross-Channel Fulfillment Policy (OCCFP) We develop a dynamic solution methodology to determine the optimal cross-channel fulfillment decisions of a retailer that is faced with |J | independent demand classes. The net profit of a sale occurring at store i is given by pi − cpi , whereas the net profit of a sale occurring online differs depending on from where this order is fulfilled, and it is denoted by p0 − cpj − chj − csj , given that the online order is fulfilled from location j ∈ J . We assume that selling the product in-store is more profitable than selling the product online via any fulfillment option. Thus, a store customer will always be fulfilled as long as the corresponding store inventory is positive. The retailer’s problem is whether to accept or reject an online customer and, if accepted, from which location to fulfill this customer. Let V ∗ (t, x0 , x1 , . . . , x|I| ) denote the retailers maximum expected profit from t to the final period T , when the system status is (t, x0 , x1 , . . . , x|I| ). In the final period (i.e., t = T ), at most one customer may arrive. If xj = 0, ∀j ∈ J , all locations are stocked-out; thus, if a demand (online or store) is realized it cannot be fulfilled. If x0 > 0, and xi = 0, ∀i ∈ I , all stores are stocked-out. Thus, if a store demand is realized, it cannot be fulfilled. If an online demand is realized, it will be fulfilled from the FC as long as fulfilling this demand from the FC is profitable, i.e., p0 − cp0 − ch0 − cs0 ≥ 0. Then, the retailer earns p0 − cp0 − ch0 − cs0 . If x0 = 0, xi > 0 for some i ∈ I , then the FC is stocked-out. If (say) a store i demand is realized with probability λi and xi > 0, the demand will be fulfilled from store 3

One may consider that rejecting an online customer for a potential future customer is not very common in practice. However, the inclusion of this option handles a more general and complicated setting that is also applicable to the setting that ignores this option. Thus, consideration of this option also allows us to analyze the difference between “reject” and “no reject” policies. A discussion on comparison of “reject” and “no reject” policies is provided in Appendix A.

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i, and the retailer obtains pi − cpi in this case. If an online demand is realized with probability λ0 , it will be fulfilled from the most profitable store that has positive inventory, say store i, as long as the associated profit is positive, i.e., p0 − cpi − chi − csi ≥ 0, and the retailer earns the corresponding profit in this case. Otherwise, this demand will be rejected, and the retailer’s associated profit is zero. If xj > 0, ∀j ∈ J , a potential store demand will be fulfilled with a profit of pi − cpi ; a potential online demand will be fulfilled from a location that provides the highest (positive) net profit across all locations and will be rejected otherwise. Let yj (t) equal one if xj (t) (i.e., the inventory of location j in period t) is positive and zero otherwise. Then, the optimal value function in the last period V ∗ (T, x0 , x1 , . . . , x|I| ) can be stated as follows:    X
V ∗ (T, x0 , x1 , . . . , x|I| ) = λ0 Ecsj max yj (p0 − cpj − chj − csj ), 0 + yi λi pi − cpi j∈J

(1)

i∈I

P For ease of exposition, we define λ0 = 1 − j∈J λj and ρj = p0 − cpj − chj − csj , ∀j ∈ J . Then, for any period t = 1, 2, . . . , T − 1, the optimal value function in t V (t, x0 , x1 , . . . , x|I| ) is defined as follows: V ∗ (t, x0 , x1 , . . . , xi , . . . , x|I| ) = λ0 V ∗ (t + 1, x0 , . . . , xi , . . . , x|I| )    ∗ ∗ + λ0 Eρj max yj ρj + V (t + 1, x0 , . . . , xj − yj , . . . , x|I| ), V (t + 1, x0 , . . . , xj . . . , x|I| ) j∈J  X  + λi yi (pi − cpi ) + V ∗ (t + 1, x0 , . . . , xi − yi , . . . , x|I| )

(2)

i∈I

According to Equation 2, if a store i customer arrives in period t with probability λi , then the retailer’s expected profit (present and future) is yi (pi − cpi ) + V ∗ (t + 1, x0 , . . . , xi − yi , . . . , x|I| ). If an online demand occurs with probability λ0 , there are three possibilities: (i) demand is rejected and the corresponding expected profit is V ∗ (t + 1, x0 , . . . , xj , . . . , x|I| ); (ii) demand is fulfilled from the FC and the corresponding expected profit is y0 ρ0 + V (t + 1, x0 − y0 , . . . , xi , . . . , x|I| ); (iii) demand is fulfilled from store i and the corresponding expected profit is yi ρi + V ∗ (t + 1, x0 , . . . , xi − yi , . . . , x|I| ). Finally, if no demand occurs with probability λ0 , then the expected profit is V ∗ (t + 1, x0 , . . . , xi , . . . , x|I| ). Theorem 1 below states the structural properties of the optimal profit function. The proofs of all technical results are provided in Appendix F. Theorem 1. The optimal profit-to-go function V ∗ (t, x0 , . . . , xi , . . . , x|I| ) presents the following structural properties: a) For a given period t, online inventory level x0 , and store inventory levels xk , ∀k ∈ I\{i}, V ∗ (t, x0 , . . . , xi , . . . , x|I| ) is non-decreasing concave in xi . In other words, the expected marginal profit of xi (i.e., V ∗ (t, x0 , . . . , xi , . . . , x|I| ) − V ∗ (t, x0 , . . . , xi − 1, . . . , x|I| )) is non-negative and non-increasing in store inventory level xi , ∀i ∈ I . b) For a given period t, and store inventory levels xi , ∀i ∈ I , V ∗ (t, x0 , . . . , xi , . . . , x|I| ) is non-decreasing concave in x0 . In other words, the expected marginal profit of x0 (i.e., V ∗ (t, x0 , . . . , xi , . . . , x|I| ) − V ∗ (t, x0 − 1, . . . , xi , . . . , x|I| )) is non-negative and non-increasing in online inventory level x0 . c) For given inventory levels xj , ∀j ∈ J , the expected marginal profit with respect to xj (i.e., ∗ V (t, . . . , xj , . . . , x|I| ) − V ∗ (t, . . . , xj − 1, . . . , x|I| )) is non-increasing in period t. d) For given inventory levels xj , ∀j ∈ J , V ∗ (t, x0 , . . . , xj , . . . , x|I| ) is non-increasing concave in t. In other words, the difference V ∗ (t, x0 , . . . , xj , . . . , x|I| ) − V ∗ (t + 1, x0 , . . . , xj , . . . , x|I| )) is non-negative and non-decreasing in period t.

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Theorems 1a and 1b establish that the expected profit is greater for higher inventory levels. However, the increase in the expected profit per unit inventory decreases as inventory level increases. Theorem 1c indicates that the expected marginal profit with respect to inventory level xj , ∀j ∈ J (i.e., online or store inventory) decreases toward the end of the selling season. More specifically, the value of having one more unit in the inventory toward the end of the selling season is lower compared to having that unit at the beginning of the selling season, which implies that the retailer is more likely to sell units as the selling deadline gets closer. Theorem 1d states that for given inventory levels, the expected profit decreases over time, since the retailer would have fewer periods and thus fewer expected customers to whom to sell these units. To characterize the optimal fulfillment policy, we define ψj (t, x0 , . . . , xj , . . . , xI ) as an individual cost threshold of location j in period t for given inventory levels xj , ∀j ∈ J . Any shipment cost csj that is below ψj (t, x0 , . . . , xj , . . . , xI ) makes location j a more profitable fulfillment option than reserving the current unit for future customers in period t. Note that in our setting, it is possible to reject a profitable customer in anticipation of more profitable customers arriving in the future. Referring to the term involving the maximum operator in Equation 2, Equation 3 below defines the cost thresholds for positive inventory levels (i.e., xj (t) > 0, ∀j ∈ J ). Equation 4 defines ψjk (t, x0 , . . . , xj , . . . , xI ) as a pairwise cost threshold of locations j and k in period t, where j, k ∈ J and t ∈ {1, 2, . . . , T − 1}. The pairwise cost thresholds further help the retailer to choose between locations j and k. If the cost difference csj − csk ≤ ψjk (t, x0 , . . . , xj , . . . , xI ), fulfilling the order from location j is more profitable. Cost Thresholds: ψj (t, x0 , . . . , xj , . . . , xI ) = (p0 − cpj − chj ) + V ∗ (t + 1, x0 , . . . , xj − 1, . . . , x|I| ) − V ∗ (t + 1, x0 , . . . , xj , . . . , x|I| )

ψjk (t, x0 , . . . , xj , . . . , xI ) = ψj (t, x0 , . . . , xj , . . . , xI ) − ψk (t, x0 , . . . , xj , . . . , xI )

= cpk − cpj + chk − chj + V ∗ (t + 1, x0 , . . . , xj − 1, . . . , x|I| )

− V ∗ (t + 1, x0 , . . . , xk − 1, . . . , x|I| )

∀j ∈ J

(3)

∀j, k ∈ J

(4)

Theorem 2 states some structural properties of the cost threshold ψj (t, x0 , . . . , xj , . . . , xI ). Theorem 2. The optimal cost threshold ψj (t, x0 , . . . , xj , . . . , xI ) has the following structural properties: a) For a given period t, and inventory levels xk , ∀k ∈ J \{j }, ψj (t, x0 , . . . , xj , . . . , xI ) is nondecreasing in xj , ∀j ∈ J . b) For given inventory levels xj , ∀j ∈ J , ψj (t, x0 , . . . , xj , . . . , xI ) is non-decreasing in t, ∀t ∈ {1, 2, . . . , T − 1}. Theorem 2a states that fulfilling an order from location j is more likely to be profitable for higher inventory levels, as the cost threshold value increases. Similarly, Theorem 2b establishes that the individual cost threshold increases over time, indicating that the retailer is more likely to fulfill orders toward the end of the selling season compared to the beginning of the season. By using the individual and the pairwise cost thresholds, we specify the following decision rule for the retailer: Decision Rule 1. When the inventory level of each location at the beginning of period t ∈ {1, 2, . . . , T − 1} is positive (i.e., xj (t) > 0, ∀j ∈ J ), if an online customer arrives in this period, then the retailer carrying out a cross-channel fulfillment policy should make its decision as follows: 1. Do not fulfill the order if csj > ψj (t, x0 , . . . , xj , . . . , xI ); ∀j ∈ J .

10

2. Fulfill the order from store i if csi ≤ ψi (t, x0 , . . . , xi , . . . , xI ), and csj − csi > ψji (t, x0 , . . . , xi , . . . , xI ); ∀j ∈ J . 3. Fulfill the order from the FC if cs0 ≤ ψ0 (t, x0 , . . . , xi , . . . , xI ), and cs0 − csi ≤ ψ0i (t, x0 , . . . , xi , . . . , xI ); ∀i ∈ I . Decision rule 1.1 characterizes a case where it is not profitable to fulfill an online order from the FC or from any of the stores in current period t, since the expected marginal profit of reserving the current unit at each location for future customers is greater than selling this unit with the corresponding fulfillment option. If the condition described in decision rule 1.2 is met, then fulfilling an online order from store i in current period t is not only more profitable than the expected marginal profit of reserving the current unit at store i for future customers but is also the most profitable fulfillment option among all locations. Finally, decision rule 1.3 corresponds to a case where the expected marginal profit of fulfilling the order from the FC in current period t is the most profitable fulfillment option, and it also brings a higher expected profit than reserving the current unit at FC for future customers. For the general case with a fulfillment network consisting of multiple stores and multiple FCs, our value function representation would change as follows: V ∗ (t, x01 , . . . , x0W , x1 , . . . , x|I| ), where x01 , . . . , x0W , indicates current inventory levels at the wth FC, for w = 1, 2, . . . , W . We now need to define new values for cost thresholds; however, the threshold policy would still be valid. Based on decision rule 1, which characterizes the optimal cross-channel fulfillment policy for a given set of inventory levels, we define a critical period tˆ after which it is always profitable to fulfill an online order from location j. Similarly, for a fixed period t and a given set of inventory levels except location j, we define a critical inventory level x ˆj , ∀j ∈ J , such that for inventory levels higher than the critical level, it is always profitable to fulfill an online order from location j. Through Proposition 1, we describe the properties of critical period and critical inventory levels. Proposition 1. The structural properties of the critical time tˆ and critical inventory levels x ˆj are as follows: a) For a given period t, and inventory levels xk , ∀k ∈ J \{j }, there exists a critical inventory level x ˆj (t, x0 , . . . , xk , . . . , xj , . . . , xI ) for each value of j, which is non-increasing in t, such that fulfilling the online order from location j is profitable for xj ≥ x ˆj (t, x0 , . . . , xk , . . . , xj , . . . , xI ) and not profitable for xj < x ˆj (t, x0 , . . . , xk , . . . , xj , . . . , xI ). b) For given inventory levels xk , ∀k ∈ J , there exists a critical period tˆ, which is non-increasing in any xj , such that fulfilling the online order from location j is profitable for t ≥ tˆ and not profitable for t < tˆ. Proposition 1 is a conclusion of the threshold policy stated in Theorem 2 and states that a retailer can make an accept or reject decision for an online order by considering the current inventory levels and period. For example, for a given set of inventory levels xk , ∀k ∈ J , if it is profitable to fulfill an online order from location j in period t, according to Proposition 1 it will also be profitable to fulfill an online order from location j after period t (i.e., from period t + 1 to period T ).

4.

A Benchmark Policy: Optimal Independent Channel Fulfillment Policy (OICFP)

One of the most common current practices in the retail industry is to operate multiple (e.g., online and offline) channels separately. In this policy, the retailer dedicates the FC to online demand and store inventories to in-store demand. Thus, these two channels are managed independently such that there exists no inventory sharing and no interaction among the channels. In this section, we consider

11

the independent channel fulfillment policy as a benchmark to evaluate the value of OCCFP in terms of retailer’s profit. Similar to OCCFP, we seek an optimal fulfillment policy for retailers operating their multiple channels independently. To this end, we state separate profit functions for store and online channels. For the store channel, we define Di as the demand of store i, where it follows a Poisson process with arrival rate λi . Any store demand is fulfilled from the corresponding store inventory as long as the corresponding store inventory level is positive. Let Π∗s (t, x1 , . . . , xi , . . . , x|I| ) denote the profit function for the store channel; then, given the inventory levels xi at store i in period t, the expected profit-to-go function of the store channel in period t can be described as follows: Π∗s (t, x1 , . . . , xi , . . . , x|I| ) =

h X
i (pi − cpi )E min Di , xi

(5)

i∈I

Equation 5 states that the expected profit of the store channel is the sum over all stores of the expected store profits obtained from sales. We calculate the number of sales by considering the minimum of the demand Di and available store inventory xi in period t. An online order realized in period t, on the other hand, can be rejected or fulfilled from the FC depending on the realized shipment cost cs0 . Similar to OCCFP, we consider a finite-horizon MDP model over T periods to express the profit function of the online channel. Let Π∗o (t, x0 ) denote the retailer’s maximum expected profit from period t to the final period T when the system status is (t, x0 ). Then, the expected profit-to-go function for the online channel is described as follows:    ∗ Πo (T, x0 ) = λ0 Eρ0 max y0 ρ0 , 0 (6)     Π∗o (t, x0 ) = λ0 Π∗o (t + 1, x0 ) + λ0 Eρ0 max y0 ρ0 + Π∗o (t + 1, x0 − y0 ), Π∗o (t + 1, x0 ) (7) According to Equation 6, when an online customer arrives in the final period T , the retailer fulfills the order if it is profitable (i.e., ρ0 ≥ 0) and the FC has positive inventory. Then, the profit of the online channel is ρ0 , and it is zero otherwise. According to Equation 7, if an online customer arrives in period t with probability λ0 , there are two possibilities: (i) the demand is rejected and the corresponding expected profit is Π∗o (t + 1, x0 ); (ii) the demand is fulfilled from the FC and the corresponding expected profit is ρ0 + Π∗o (t + 1, x0 − 1). Finally, if no demand occurs in period t with probability λ0 , then the expected profit is Π∗o (t + 1, x0 ). The total profit-to-go function Π∗ (t, x0 , . . . , xi , . . . , x|I| ) of the retailer operating the two channels independently can be obtained by summing up the store and the online channel profits, that is: Π∗ (t, x0 , . . . , xi , . . . , x|I| ) = Π∗s (t, x1 , . . . , xi , . . . , x|I| ) + Π∗o (t, x0 )

(8)

Similar to OCCFP, by considering the term involving the maximum operator in Equation 7, we define the following decision rule for the retailer that carries out the independent channel policy. Decision Rule 2. When the inventory level of the FC at the beginning of period t ∈ {1, 2, ..., T − 1} is positive (i.e., x0 (t) > 0), and an online customer arrives, then a retailer following an independent channel policy should fulfill the online order if the following inequality holds: cs0 ≤ p0 − cp0 − ch0 + Π∗o (t + 1, x0 − 1) − Π∗o (t + 1, x0 ). Decision rule 2, together with the fact that any store customer will be fulfilled from the corresponding store inventory unless the associated store inventory is zero, describes an optimal fulfillment strategy for a retailer that operates multiple channels independently.

12

4.1. Value of Ship-from-Store Implementation: Comparison of OCCFP and OICFP It is important for a retailer to evaluate the impact of the integration of its multiple channels on profits and also on resulting sales, leftover inventory, etc. To evaluate the value of ship-from-store implementation in terms of profit, we calculate the profit difference between the two optimal policies
that are presented above. To this end, let ∆ Π∗ (t, x0 , . . . , xi , . . . , x|I| ) denote the expected profit difference between OCCFP and OICFP for given period t and a set of inventory levels xj , ∀j ∈ J . This difference can be represented as follows:
∆ Π∗ (t, x0 , . . . , xi , . . . , x|I| ) = V ∗ (t, x0 , . . . , xi , . . . , x|I| ) − Π∗ (t, x0 , . . . , xi , . . . , x|I| ) (9)

Examining the behavior of the expected profit difference in Equation 9 can provide insights for retailers about the conditions in which ship-from-store implementation yields the most benefits. Theorem 3 below characterizes some structural properties of this expected profit difference.
Theorem 3. ∆ Π∗ (t, x0 , . . . , xi , . . . , x|I| ) exhibits the following structural properties: a) For a given period t, and inventory levels xj , ∀j ∈ J \{i}, when xi ∈ [0, Di ),
• ∆ Π∗ (t, x0 , . . . , xi , . . . , x|I| ) is non-increasing concave in xi , if the expected marginal profit of xi is less than or equal to (pi − cpi ), ∀i ∈ I .
• ∆ Π∗ (t, x0 , . . . , xi , . . . , x|I| ) is non-decreasing concave in xi , if the expected marginal profit of xi is greater than (pi − cpi ), ∀i ∈ I . b) For a given period t, and inventory levels xj , ∀j ∈ J \{i}, when xi = Di ,
∆ Π∗ (t, x0 , . . . , xi , . . . , x|I| ) takes its lowest value if the expected marginal profit of xi is less than or equal to (pi − cpi ), ∀i ∈ I . c) For a given period t, and inventory levels xj , ∀j ∈ J \{i}, when xi ∈ [Di , ∞),
∆ Π∗ (t, x0 , . . . , xi , . . . , x|I| ) is non-decreasing concave in xi , ∀i ∈ I . d) For a given period t, and inventory levels xj , ∀j ∈ J \{i}, when xi ∈ [0, ∞),
∆ Π∗ (t, x0 , . . . , xi , . . . , x|I| ) ≥ 0.
e) For a given period t, and inventory levels xj , ∀j ∈ J , ∆ Π∗ (t, x0 , . . . , xi , . . . , x|I| ) is nondecreasing in λ0 . f ) Given a set of stores ready to use for ship-from-store, the retailer’s profit is non-decreasing in the number of stores used in ship-from-store.

Theorem 3a first shows the impact of store inventory on the value of ship-from-store implementation. In our setting, the maximum profit that can be earned from an inventory of store i in any period t is pi − cpi ; hence, the expected marginal profit of such a unit is at most pi − cpi . Thus, when inventory
level xi increases up to Di , the value of ship-from-store implementation ∆ Π∗ (t, x0 , . . . , xi , . . . , x|I| ) decreases. Theorem 3b states that for any given period and inventory levels of all locations except location j, the benefits realized via ship-from-store reaches its lowest value when store j inventory equals the expected demand of that store. Theorem 3c describes the relationship between the value of ship-from-store implementation and a store’s inventory when inventory of the corresponding store is higher than the expected demand of that store. In this case, there is more likely to be leftover inventory at the considered store under the independent channel policy, whereas some of these units can be used to fulfill online demand under a cross-channel fulfillment policy. Hence, the value of ship-fromstore implementation increases as store inventory level xi increases, given that xi ∈ [Di , ∞). Theorem 3d shows that implementing ship-from-store optimally does not hurt the profits of the retailer in comparison to operating the channels independently. Overall, our analytical findings show that the value of ship-from-store implementation (the gap between OCCFP and OICFP) is at least zero and

13

increases when the inventory level of a store deviates from the expected demand of that store. We then analyze the impact of online customer arrivals on the value of ship-from-store implementation in Theorem 3e. If there are no online customer arrivals, our results show that OCCFP would be exactly the same as OICFP, and thus the expected profit difference between these policies would be equal to zero. The value of ship-from-store implementation increases as the arrival rate of online customers increases. Theorem 3f shows that given a set of stores ready to be used in ship-from-store, it is always profitable to include a store as a candidate fulfillment location for ship-from-store. However, we note that our model does not include the investment cost that is required to prepare a store for ship-fromstore operations. Nonetheless, to capture the investment cost, we define cinv as the investment cost i of store i and then consider the gain of making a store available for ship-from-store operations. In any period, the maximum gain that can be obtained from the use of store i in ship-from-store can be expressed as: ρsi = p0 − cpi − chi −csi which represents the maximum possible value of the net profit when the uncertain shipment cost takes its lowest value (i.e., csi ). Let T 0 be the number of periods considered in the selling season. Then, preparing N stores for ship-from-store operations will not be beneficial if the investment cost is more than the maximum expected benefits over T 0 periods, i.e., PN inv > T 0 ρsi . i=1 ci

5.

Heuristic Policies

Dynamic programming models are usually difficult to solve for large instances due to their curse of dimensionality. In our model, as the number of stores, number of periods, and inventory levels increase, obtaining the optimal value function and therefore determining the optimal cross-channel fulfillment policy become computationally intractable. Based on our analyses in the following section (Section 6), we find that OICFP obtains a solution in a reasonable amount of time for large-sized problems; however, it does not provide ship-from-store flexibility. In this section, to address computational and practical challenges, we describe a myopic fulfillment policy that is currently in use in practice and also propose an integrative fulfillment policy that combines the strengths of OICFP and myopic policy. 5.1. Myopic Cross-Channel Fulfillment Policy (MCCFP) In this policy, retailers use a cross-channel fulfillment option, and with this functionality they may fulfill an online demand from a store rather than from the FC. Many retail organizations use a myopic policy and fulfill each online demand the cheapest way possible based on their current inventory levels, without considering any future cost implications (Acimovic and Graves 2014). We use this policy to display the value of a not optimal but simple implementation of ship-from-store. In MCCFP, a customer arriving at store i will always be served as long as there is positive inventory in store i. The retailer’s fulfillment decision for an online demand is determined as follows: an online customer will be fulfilled from the location that achieves the highest current net profit among all the locations that have positive inventory, given that the highest current net profit is positive. If the highest current net profit is negative, then the online customer will be rejected. We note that the myopic fulfillment policy described here does not account for remaining inventory positions at fulfillment locations or the remaining periods until the end of the selling season. We define the following decision rule for the retailer that uses a myopic fulfillment policy: Decision Rule 3. When inventory level at location j ∈ J at the beginning of period t ∈ {1, 2, ..., T } is positive (i.e., xj (t) > 0), and an online customer arrives, the retailer following a myopic policy should fulfill the online order from location j if the following inequality holds: ρj ≥ ρk , ∀k ∈ J \{j }, which simplifies into csj + chj + cpj ≤ csk + chk + cpk ; ∀k ∈ J \{j }.

14

5.2. An Integrative Cross-Channel Fulfillment Policy (ICCFP) In our numerical analyses, we observe that the strengths of MCCFP and OICFP in yielding higher profits are complementary to each other. Thus, in this section, we propose an alternative heuristic approach, which we refer to as ICCFP. This policy aims to integrate the strengths of MCCFP and OICFP without sacrificing their time-efficiency. Consistent with all fulfillment policies described above, in our proposed heuristic if no order arrives there is no fulfillment decision to make, and if a store customer arrives the order will be fulfilled from the corresponding store inventory as long as the inventory level is positive. From the retailer’s perspective, it makes sense to direct inventory of a store to online demand when there is a high chance that this store will experience leftover units at the end of the selling season. The decision process of our proposed heuristic for an online order utilizes this intuition and is described in Algorithm 1.

Note that a comparison of remaining inventory at a location with this location’s expected number of future customers informs the retailer about the stock-out probability of this location and also the scarcity level of this location’s inventory. Now suppose that an online customer has arrived in period t ∈ {1, 2, . . . , T − 1}. Our proposed heuristic computes a scarcity indicator ratio sj , which is the ratio of remaining inventory of location j to the expected number of customers of this location in the
x (t) remaining periods i.e., sj = λj (Tj−t+1) , for each location j ∈ J . If the current scarcity indicator ratio
x0 (t) of the FC i.e., s0 = λ0 (T is greater than a certain threshold value Θ0 , then FC is considered to −t+1) have sufficient inventory to fulfill future online orders. However, it might still be appealing to direct inventory of some stores to online demand if these stores are also considered to have enough inventory to satisfy their in-store demand. On the other hand, if the scarcity indicator ratio of the FC is less than or equal to Θ0 , then FC is expected to be stocked-out in future periods. In this case, it will be more appealing to direct online demand to the inventory of stores that are considered to have enough inventory to satisfy their in-store demand, compared to the previous case where the scarcity indicator ratio of FC is greater than Θ0 . Thus, there might be an asymmetry in the need to direct store inventory to online demand, depending on whether or not the FC has enough inventory to satisfy online demand. To capture this asymmetry, we define different threshold values of Θl ; l ∈ {1, 2} for stores’ scarcity indicator ratios, where Θ1 is used when the FC has sufficient inventory (i.e., s0 > Θ0 ) while Θ2 is used when the FC has insufficient inventory (i.e., s0 ≤ Θ0 ). Logically, the store threshold values (i.e., Θ1 and Θ2 ) should be at least one, since it is not profitable to replace a store sale with any fulfillment option of an online sale. The value of Θ0 should be at most one, because allowing

15

fulfillment of an online demand from a store cannot hurt the profit of a certain online sale, as we compare the profits of possible fulfillment options (including the FC) for this online demand. If the scarcity indicator ratio of store i is at most Θ1 (resp., Θ2 ), ∀i ∈ I when the scarcity indicator ratio of FC is greater than Θ0 (resp., at most Θ0 ), then this indicates that all stores are likely to be stocked-out at the end of the selling season. In this case, directing store inventories to online demand does not make sense, and thus OICFP should be followed. More specifically, if ρ0 ≥ Π∗o (t + 1, x0 ) − Π∗o (t + 1, x0 − 1) and the online inventory level is positive (i.e., x0 > 0), an online order should be fulfilled from the FC. Otherwise, the online order should be rejected. On the other hand, if the scarcity indicator ratio of store i is greater than a certain threshold value Θl for some i ∈ I , then directing some inventory of these stores to online demand makes sense. Thus, MCCFP should then be followed. More specifically, MCCFP fulfills the order from location j that provides the highest positive current net profit ρj among the candidate locations (i.e., FC and stores for which the scarcity indicator ratio sj is greater than Θl ); otherwise, it rejects the order. In the final period T , the algorithm only compares the current net profit ρj of all locations that have positive inventory level. The order is fulfilled from the location having the maximum current net profit, given that it is positive. If the maximum of the current net profits is negative or all locations are stocked-out, then the online order will be rejected.

6.

Computational Study

In this section, we present the results of our computational study. More specifically, we discuss the performance of the four policies described above and evaluate the value of ship-from-store implementation in terms of profit, total sales, number of stock-outs, and leftover inventory levels. 6.1. Parameter Setting and Scenario Description While setting the parameters for our computational study, we aim to make sure that selling a product in store is the most profitable and ship-from-store is the least profitable fulfillment option.4 We consider a retailer that adopts the same pricing strategy for online and offline channels and charges $100 for the product, i.e., pj = $100, ∀j ∈ J . To capture the difference in profit margins across online and offline channels with various fulfillment options, we set the unit operating cost for the online channel at cp0 = $65 and for the offline channel at cpi = $70, ∀i ∈ I (i.e., we assume that the unit operating cost is approximately the same across all stores). Handling costs are set for the online and offline channels at ch0 = $5 and chi = $15, respectively, where ch0 ≤ chi , since stores are not designed to support efficient shipment of products. We use triangular distribution to model the distance between customers and a fulfillment location j, ∀j ∈ J . Triangular distribution is well suited for modeling distances, and thus for shipment costs, because it is defined for non-negative values and is consistent with the distance and shipment time/cost distributions used in prior related studies (Herbon 2018). Thus, when the retailer operates N stores in total, N + 1 distributions need to be specified. To simplify the specification of the distributions, especially for larger sizes of N , we assume that there are clusters of stores, each cluster sharing the same distribution. We also assume that cost distributions specified for all the locations are independent.5 For our computational study, we set the mean of the shipment cost distribution 4

This ranking of the profit margins across the channels is in line with a recent illustrative model presented by the retail consultancy AlixPartners to show the profit differential across various channels (Reagan 2017). 5

In reality, the independence assumption on cost distributions of different locations may not hold for a company, and thus the results may differ from company to company depending on their actual network structure. We use the independence assumption in our analyses mainly for tractability. However, a discussion on how the result would differ under negatively and positively correlated cost distribution functions is provided for N = 1 in Appendix B.

16

Gj (· ) for the FC at 10 and for all stores at 5, considering the fact that stores are located closer to the customers than the FC is. Note that although the means are the same for the distributions, their variability may differ across the stores. These numbers are selected to ensure that the profit margins across the channels are, on average, positive and also that the resulting profit margins comply with the logic provided above. Table 1 summarizes the main parameters for our computational study and indicates that if a store sale occurs, the retailer earns $30; if an online order is fulfilled from the FC (resp., from a store), then the retailer earns $20 (resp., $10). Table 1

cpj

pj FC 100 65 Store i 100 70

pj − cpj 35 30

Summary of parameters in $

Profit margin for Profit margin for Mean Gj (· ) independent channel cross-channel fulfillment 5 10 20 15 5 30 10 chj

We consider eight different store sizes, three being small-sized (i.e., N = 1, 2, 3), another three being medium-sized (i.e., N = 10, 15, 20), and the remaining two being large-sized problems (i.e., N = 100, 200). The selling season is assumed to be divided into T = 100, T = 400, and T = 2000 periods for the small-, medium-, and large-sized instances, respectively. Let X T denote the total available
P initial inventory across the two channels i.e., j xj (0) and CusT denote the expected total number P of customers that the retailer will observe across the two channels over T periods (i.e., j λj T ). Then, XT the ratio Cus T can be considered as the scarcity indicator ratio of the total inventory. To observe the performance of the policies under various scarcity levels, we vary X T at three and CusT at two different levels. We generated several scenarios by varying arrival probabilities λj and initial inventory levels xj (0) at each location j to capture the impact of asymmetry in arrival probabilities and initial inventory levels. For each combination of X T and CusT , nine scenarios are generated for N = 1, thirty scenarios are generated for N = 2, 3, and forty-five scenarios are generated for all remaining values of N .6 Table 2 summarizes the values of X T and CusT and the resulting total number of scenarios generated for each value of N .7 Table 2

N

Summary of X T and CusT values for each N

XT

1 40, 60, 100 2 45, 60, 100 3 40, 60 10 160, 240, 360 15 160, 240, 360 20 160, 240, 360 100 840, 1240, 1840 200 840, 1240, 1840

P

j

0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,

λj

T

CusT

0.8 100 50, 0.8 100 50, 0.8 100 50, 0.8 400 200, 0.8 400 200, 0.8 400 200, 0.8 2000 1000, 0.8 2000 1000,

80 80 80 320 320 320 1600 1600

Total # of scenarios 54 180 120 270 270 270 270 270

For each scenario, we generate 1000 instances,8 where each instance represents a different realization of customer arrivals over time according to the corresponding arrival rates λj , ∀j ∈ J . We calculate the 6

The number of scenarios to examine the asymmetry in arrival probabilities and initial inventory levels differ depending on the store size. 7

A table describing the detailed scenarios that we consider for N = 2 is provided in Appendix C.

8

The number of instances is set to ensure that the OCCFP is solved in a reasonable amount of time.

17

averages for the total sales, shipment cost per sale, total number of stock-outs, leftover inventory, and resulting total profits over these 1000 instances. We use these averages to discuss the performance of the policies. We consider OICFP as our base policy and present our results by showing the percentage deviation from this policy using the following formula: Percentage Deviation =

Performance of the Corresponding Policy − Performance of OICFP Performance of OICFP

(10)

For small values of N , the performance of the retailer under OICFP is compared with those under MCCFP, OCCFP, and ICCFP (see Table 3). For medium and large values of N , solving OCCFP in a reasonable amount of time is not feasible; therefore, we compare the performance of the retailer under OICFP with those under MCCFP and ICCFP (see Tables 4-5). The results are presented in terms of the minimum, the maximum, and the average percentage deviations from OICFP over all scenarios generated for the corresponding combination of X T and CusT . In other words, the minimums, the maximums, and the averages reported in Tables 3-5 are over nine, thirty, and forty-five scenarios, respectively, for N = 1, N = 2, 3, and N = 10, 15, 20, 100, 200. For instance, a negative percentage deviation in profits indicates that OICFP performs better, whereas a positive percentage deviation indicates that the corresponding policy performs better.9 We first discuss the performance of the optimal implementation of ship-from-store strategy over the independent channel strategy and then discuss the performance of the proposed heuristics separately. • Value of Ship-from-Store Implementation: We see from Table 3 that (for small values of N ) over all combinations of X T and CusT , OCCFP achieves a 6.23% increase in total profits. When we examine the results in Table 3 in detail, we see that OCCFP results in higher expected total profit than OICFP in all scenarios, with minimum deviations always being non-negative. This numerically confirms the result we proved earlier (in Theorem 3d) that implementing ship-from-store optimally does not hurt the profits of the retailer. OCCFP results in a 15.16% increase in total sales, which results in a 38.21% decrease in total leftover inventory compared to OICFP. We also observe that the number of customers who are rejected due to inventory being depleted (i.e., the number of stockouts) is 18.80% lower in OCCFP compared to OICFP. This indicates that OCCFP mitigates the FC stock-outs by allowing fulfillment of online orders from stores. • Performance of heuristic policies: In our computational study, for ICCFP, we set all threshold values on scarcity indicator ratios to one, i.e., Θ0 = Θ1 = Θ2 = 1. A sensitivity analysis for threshold values Θ0 , Θ1 , and Θ2 is provided in Appendix B. We see from Table 3 that for small values of N , over all combinations of X T and CusT , MCCFP and ICCFP respectively achieve a 0.75% and 3.46% increase in total profits. When we examine the percentage deviation in profits in detail, we observe that ICCFP outperforms MCCFP in most scenarios, with both the minimum and the average deviations of the former policy mostly (except for 4 out of the 16 scenarios) being higher than those of the latter policy. Both MCCFP and ICCFP result in an approximately 15.40% increase in total sales, resulting in a 45.79% and 40.20% decrease in total leftover inventory, respectively. Although MCCFP achieves a lower shipment cost per sale compared to ICCFP, the latter policy turns its sales into higher total profits. This indicates that ICCFP is more selective in its customer acceptance decisions and accepts more profitable customers than MCCFP. We also observe that the number of customers who are rejected due to inventory being depleted (i.e., the number of stock-outs) is 16.55% higher in MCCFP compared to OICFP, whereas this percentage is 107.91% higher in ICCFP compared to OICFP. 9

An analysis of the negative and the positive percentage deviations separately is provided in Appendix D.

18

A detailed examination of results in Table 3 reveals that although MCCFP and ICCFP, on average, perform better than OICFP, they perform worse than OICFP in several scenarios, with minimum deviations for both policies always being non-positive. This emphasizes the fact that not necessarily

3

2

1

N

60

40

100

60

45

100

60

40

XT

80

50

80

50

80

50

80

50

80

50

80

50

80

50

80

50

CusT

Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Avg.

ICCFP -1.60% 27.79% 3.36% -5.42% 0.76% -0.52% 0.00% 34.23% 5.64% -2.30% 17.35% 1.60% 0.00% 14.05% 1.71% 0.00% 25.44% 4.06% -0.45% 23.85% 5.83% -6.81% 37.69% 2.87% 0.00% 14.08% 3.63% -3.24% 19.95% 3.71% 0.00% 8.85% 1.75% 0.00% 36.32% 7.61% -1.86% 29.58% 3.20% -6.73% 13.50% -0.47% 0.00% 38.28% 6.93% -3.97% 30.60% 4.40% 3.46%

Table 3

Total profit MCCFP OCCFP -6.78% 0.00% 24.88% 40.73% 1.44% 6.78% -20.77% 0.00% 0.00% 13.74% -8.76% 1.69% -0.01% 0.00% 34.87% 42.12% 5.79% 7.31% -11.50% 0.00% 11.23% 29.58% -1.79% 4.66% 0.00% 0.00% 14.54% 19.94% 1.78% 2.43% -0.01% 0.00% 25.97% 32.46% 4.19% 5.51% -2.24% 0.00% 19.15% 33.65% 3.96% 8.74% -22.43% 0.00% 17.18% 50.24% -5.49% 8.64% 0.00% 0.00% 14.37% 18.78% 3.52% 5.22% -10.47% 0.00% 10.54% 31.91% 0.37% 8.90% 0.00% 0.00% 9.18% 11.95% 1.83% 2.24% 0.00% 0.00% 36.96% 43.29% 7.07% 9.32% -5.20% 0.00% 22.21% 38.33% 1.14% 6.66% -22.48% 0.00% 5.18% 17.51% -10.20% 3.63% 0.00% 0.00% 32.23% 45.40% 6.39% 8.87% -9.42% 0.00% 23.53% 40.56% 0.76% 9.06% 0.75% 6.23%

Shipment cost per sale MCCFP OCCFP ICCFP -1.43% -26.82% -0.77% 7.72% 0.00% 4.92% 1.32% -7.08% 1.37% 0.01% -26.55% -0.01% 29.15% 0.00% 25.47% 14.90% -3.34% 5.31% -6.44% -24.08% -6.18% -0.01% 0.00% 0.00% -1.95% -7.14% -1.82% 0.00% -27.49% 0.00% 12.17% 0.00% 13.89% 4.51% -6.23% 3.47% -3.99% -19.66% -3.59% -0.01% 0.00% 0.00% -0.76% -3.04% -0.68% -4.14% -22.93% -3.84% 0.00% 0.00% 0.00% -1.12% -6.17% -0.98% -16.66% -30.97% -14.44% 0.49% 1.63% 0.87% -6.88% -13.11% -5.31% -6.32% -34.97% -0.37% 17.91% 0.00% 23.67% 4.09% -11.54% 8.10% -11.59% -27.65% -11.23% 0.00% 0.00% 0.00% -4.23% -10.12% -3.93% -5.08% -30.42% -2.19% 7.67% 0.00% 11.30% 0.07% -12.16% 2.46% -8.71% -22.28% -8.25% 0.00% 0.00% 0.00% -2.16% -5.38% -2.02% -12.96% -30.21% -12.66% 0.00% 0.00% 0.00% -4.33% -11.80% -3.91% -21.72% -38.81% -18.58% 2.53% 0.01% 2.57% -8.45% -13.89% -5.48% -15.54% -37.28% -8.19% 22.95% 0.00% 33.48% 1.27% -12.10% 7.32% -24.39% -37.78% -23.31% 0.00% 0.00% 0.00% -8.60% -14.63% -8.14% -18.87% -38.09% -12.25% 6.46% 0.00% 11.12% -5.91% -16.82% -1.70% -1.14% -9.66% -0.37% MCCFP 0.00% 99.59% 16.89% 0.00% 32.84% 4.17% -0.01% 101.59% 17.28% 0.00% 70.83% 11.23% 0.00% 44.38% 5.39% -0.01% 78.75% 13.14% 0.00% 81.76% 23.15% 0.01% 128.02% 21.86% 0.00% 43.40% 11.65% 0.00% 79.11% 21.75% 0.00% 26.55% 5.30% 0.00% 101.21% 23.16% 0.00% 97.28% 18.65% 0.04% 43.13% 9.74% 0.00% 102.15% 19.83% -0.01% 100.18% 23.01% 15.39%

Total sales OCCFP 0.00% 98.13% 16.29% 0.00% 32.41% 4.01% 0.00% 101.56% 17.28% 0.00% 70.34% 11.01% 0.00% 44.38% 5.39% 0.00% 78.75% 13.14% 0.00% 77.39% 21.97% 0.00% 127.68% 21.51% 0.00% 44.16% 12.20% 0.00% 78.74% 21.36% 0.00% 26.55% 5.22% 0.00% 101.35% 21.79% 0.00% 97.68% 17.82% 0.00% 46.52% 9.48% 0.00% 111.78% 21.19% 0.00% 99.12% 22.90% 15.16% ICCFP 0.00% 99.51% 16.78% 0.00% 32.84% 4.08% 0.00% 100.01% 16.91% 0.00% 70.83% 11.17% 0.00% 43.23% 5.23% 0.00% 77.41% 12.82% 0.00% 85.67% 23.69% 0.00% 128.02% 21.82% 0.00% 42.61% 11.54% 0.00% 79.12% 21.70% 0.00% 25.79% 5.12% 0.00% 99.70% 23.34% 0.00% 99.18% 18.86% 0.00% 47.65% 9.86% 0.00% 107.52% 20.16% 0.00% 100.19% 23.40% 15.40%

# of stock-outs MCCFP OCCFP ICCFP 0.00% -97.87% 0.00% 174.28% 0.00% 393.36% 53.18% -42.84% 114.70% 0.00% -97.60% 0.00% 390.63% 0.00% 312.04% 112.78% -31.52% 65.75% -99.11% -99.89% 0.00% 0.03% 0.00% 336.61% -46.97% -49.18% 99.60% 0.00% -97.98% 0.00% 381.55% 0.00% 660.88% 110.16% -38.52% 162.26% -100.00% -99.95% 0.00% 0.01% 0.00% 351.09% -33.33% -33.32% 48.26% -100.00% -99.96% 0.00% 1.66% 0.00% 568.25% -49.20% -49.36% 153.57% -40.09% -6.58% 0.00% 39.87% 53.48% 297.59% -5.34% 4.85% 83.25% 0.16% -9.15% 0.00% 372.41% 20.34% 472.82% 83.81% 1.91% 113.54% -94.08% -42.59% 0.00% 0.01% 0.11% 278.30% -40.36% -18.30% 94.37% -1.52% -2.57% 0.00% 373.59% 31.13% 679.76% 90.47% 5.19% 208.25% -100.00% -100.00% 0.00% 0.08% 0.00% 231.63% -45.88% -25.37% 54.71% -99.61% -62.03% 0.00% 0.13% 0.00% 651.61% -41.71% -24.78% 217.45% -28.36% -1.91% 0.00% 61.38% 30.87% 270.04% 7.94% 6.53% 68.01% -5.90% -0.98% 0.00% 186.57% 9.35% 230.70% 56.63% 2.05% 53.09% -92.08% -34.03% 0.00% 0.02% 0.00% 272.40% -32.64% -12.89% 70.85% -10.70% -6.80% 0.00% 195.95% 26.57% 422.32% 45.21% 4.67% 118.99% 16.55% -18.80% 107.91%

Performance of MCCFP, OCCFP, and ICCFP against OICFP, for N = 1, 2, 3. Leftover inventory MCCFP OCCFP ICCFP -99.88% -98.42% -99.81% 0.01% 0.00% 0.00% -51.50% -43.36% -47.33% -100.00% -98.70% -100.00% 0.00% 0.00% 0.00% -76.97% -29.83% -31.38% -72.68% -72.66% -71.55% 0.02% 0.00% 0.00% -20.64% -20.61% -20.04% -100.00% -99.30% -100.00% 0.02% 0.00% 0.00% -59.57% -40.21% -42.06% -23.87% -23.87% -23.25% 0.00% 0.00% 0.00% -3.09% -3.09% -2.98% -64.73% -64.73% -63.63% 0.01% 0.00% 0.01% -16.27% -16.27% -15.78% -97.46% -95.84% -97.58% 0.00% 0.00% 0.00% -56.62% -53.59% -57.33% -100.00% -99.73% -100.00% -0.10% 0.00% 0.00% -83.31% -70.40% -75.59% -60.86% -61.83% -59.75% 0.00% 0.00% 0.00% -22.28% -23.32% -21.97% -100.00% -99.51% -100.00% 0.00% 0.00% 0.00% -74.46% -67.81% -71.35% -17.66% -17.66% -17.16% 0.00% 0.00% 0.00% -3.80% -3.75% -3.67% -67.51% -67.61% -66.50% 0.01% 0.00% 0.00% -25.43% -24.75% -25.42% -99.28% -96.89% -99.38% 0.00% 0.00% 0.00% -54.39% -49.76% -54.65% -100.00% -98.06% -100.00% -0.52% 0.00% 0.00% -85.91% -69.42% -75.91% -66.58% -71.54% -65.80% 0.01% 0.00% 0.00% -23.00% -24.46% -23.15% -100.00% -98.92% -100.00% 0.06% 0.00% 0.00% -75.46% -70.68% -74.61% -45.79% -38.21% -40.20%

19

20

an optimal but a smart implementation of the ship-from-store strategy is required to guarantee an increase in profits. Moreover, the performance of both MCCFP and ICCFP deteriorate for low values XT of the scarcity ratio Cus T . The box plots that show the distributions of percentage profit deviations of each policy from OICFP are provided in Appendix D, and they also confirm this result. For medium values of N (see Table 4), ICCFP achieves a 1.25% increase in total profits compared to OICFP, whereas the corresponding percentage is 0.50% for MCCFP. Although these percentages may seem small in magnitude, they are actually very important for the retail industry, for which the average profit margins are very thin. When we examine the results in detail, for each combination of X T and CusT , ICCFP always outperforms MCCFP in terms of total profits, with both the average and the minimum deviations of ICCFP always being higher than those of MCCFP. Similar to their performance for small values of N , both MCCFP and ICCFP achieve higher total sales, lower leftover inventory at the end of the selling season, and lower shipment cost per sale compared to OICFP for medium values of N . Although total sales are lower for ICCFP than for MCCFP, it seems that by accepting customers in a more selective manner, ICCFP again turns its resulting lower sales into higher profits than MCCFP.

Table 4 N

XT

CusT 200

160 320

200 10

240 320

200 360 320

200 160 320

200 15

240 320

200 360 320

200 160 320 200 20

240 320 200 360 320

Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Avg.

Performance of MCCFP and ICCFP against OICFP, for N = 10, 15, 20. Total profit MCCFP ICCFP -5.60% -0.56% 7.38% 10.25% 0.56% 1.58% -10.77% -0.01% 4.29% 7.34% -0.83% 1.22% 0.00% 0.00% 7.57% 9.57% 1.30% 1.65% -8.54% -0.39% 6.28% 10.42% 0.09% 1.63% 0.00% 0.00% 4.75% 4.76% 0.74% 0.78% 0.00% 0.00% 4.02% 6.66% 0.65% 1.04% -4.53% -0.50% 6.78% 8.85% 0.68% 1.56% -11.24% -0.03% 3.14% 5.43% -1.09% 1.40% 0.00% 0.00% 6.06% 6.97% 1.42% 1.61% -7.09% -0.41% 5.98% 7.80% 0.09% 1.48% 0.00% 0.00% 4.58% 4.71% 0.86% 0.89% -0.01% 0.00% 5.16% 6.15% 0.94% 1.15% -4.98% -0.40% 4.81% 8.04% 0.48% 0.83% -11.12% -0.92% 4.46% 5.84% -0.56% 0.84% -0.01% 0.00% 5.85% 7.00% 0.95% 1.10% -6.42% -0.41% 5.34% 7.93% 0.65% 1.38% -0.01% 0.00% 5.27% 5.35% 1.00% 1.03% -0.01% 0.00% 5.28% 7.29% 1.10% 1.42% 0.50% 1.25%

Shipment cost per sale MCCFP ICCFP -30.35% -28.50% -0.09% 0.00% -9.87% -6.75% -23.93% -18.68% 15.43% 16.28% -7.07% -2.39% -32.49% -31.99% -0.11% -0.02% -9.39% -8.96% -29.45% -26.16% -0.09% 7.70% -10.92% -6.65% -20.67% -20.55% -0.15% -0.13% -4.27% -4.20% -26.06% -24.69% -0.09% 0.00% -6.91% -5.74% -33.19% -31.18% -0.12% 0.00% -12.83% -9.21% -27.81% -24.07% 5.97% 5.67% -11.68% -5.12% -28.70% -28.29% -0.14% -0.09% -10.22% -9.73% -32.42% -30.25% -0.18% 4.16% -12.77% -7.71% -22.71% -22.58% -0.14% -0.09% -5.63% -5.47% -29.32% -28.73% -0.16% 0.00% -9.49% -7.99% -38.12% -33.89% -0.17% 0.00% -8.50% -5.72% -37.45% -35.94% 11.77% 22.60% -8.12% -3.87% -35.97% -35.23% -0.19% -0.03% -8.24% -7.54% -36.05% -32.48% -0.18% 3.12% -11.25% -7.31% -27.87% -27.74% -0.19% -0.02% -6.68% -6.56% -33.89% -32.35% -0.17% 0.00% -10.82% -9.57% -9.15% -6.69%

Total sales MCCFP ICCFP 0.00% 0.00% 26.44% 29.32% 5.50% 5.32% 0.00% 0.00% 29.68% 25.89% 6.16% 5.02% 0.00% 0.00% 24.27% 26.26% 4.54% 4.85% 0.00% 0.00% 26.40% 30.49% 5.35% 5.50% 0.00% 0.00% 13.22% 13.35% 2.15% 2.18% 0.00% 0.00% 15.33% 18.98% 3.11% 3.31% 0.00% 0.00% 24.90% 26.90% 5.99% 5.65% -0.01% 0.00% 22.64% 20.69% 6.28% 5.32% 0.00% 0.00% 17.92% 18.81% 4.60% 4.71% -0.01% 0.00% 21.44% 23.02% 5.37% 5.21% 0.00% 0.00% 12.81% 12.90% 2.45% 2.46% -0.01% 0.00% 15.76% 16.74% 3.84% 3.66% -0.02% 0.00% 22.66% 25.43% 3.72% 3.07% -0.01% 0.00% 24.98% 18.75% 4.98% 3.83% -0.01% 0.00% 17.86% 19.45% 3.40% 3.40% -0.01% 0.00% 22.56% 24.77% 5.44% 4.76% -0.01% 0.00% 13.97% 14.03% 2.76% 2.78% -0.01% 0.00% 17.29% 19.32% 4.25% 4.33% 4.44% 4.19%

# of stock-outs MCCFP ICCFP -5.96% 0.00% 39.40% 66.27% 1.52% 9.62% -0.74% 0.00% 29.00% 27.55% 3.92% 6.09% -50.19% 0.00% 0.05% 248.22% -5.87% 25.10% -3.61% 0.00% 54.51% 46.17% 3.56% 9.04% -99.84% 0.00% 0.02% 297.42% -11.38% 22.87% -20.53% 0.00% 15.30% 396.70% -1.34% 30.48% -5.03% 0.00% 29.29% 62.25% 1.27% 11.78% -1.36% 0.00% 28.94% 27.46% 4.57% 6.69% -23.29% 0.00% 0.03% 252.73% -5.44% 28.02% -2.60% 0.00% 47.46% 42.09% 4.25% 11.09% -97.65% 0.00% 0.05% 310.53% -15.56% 32.92% -8.74% 0.00% 21.97% 337.49% -0.60% 31.32% -5.06% 0.00% 19.00% 52.42% 0.16% 7.05% -2.39% 0.00% 24.74% 26.66% 3.11% 5.20% -17.74% 0.00% 2.18% 204.25% -2.94% 19.22% -2.93% 0.00% 35.48% 43.02% 1.89% 8.87% -94.86% 0.00% 0.04% 316.49% -13.81% 33.65% -6.91% 0.00% 16.07% 293.73% -0.79% 28.62% -1.86% 18.20%

Leftover inventory MCCFP ICCFP -92.96% -69.48% 0.05% 0.00% -22.22% -16.77% -100.00% -96.38% 0.04% 0.00% -48.52% -29.32% -38.30% -39.47% 0.01% 0.01% -7.30% -7.76% -99.48% -86.20% 0.07% 0.00% -28.51% -19.86% -11.78% -11.73% 0.00% 0.00% -1.91% -1.93% -41.95% -44.16% 0.02% 0.01% -7.32% -7.70% -86.49% -55.44% 0.04% 0.00% -23.71% -18.08% -100.00% -90.48% 0.05% 0.00% -52.34% -29.58% -31.88% -32.93% 0.01% 0.00% -8.30% -8.47% -98.36% -69.92% 0.06% 0.00% -30.47% -21.59% -11.47% -11.43% 0.00% 0.00% -2.25% -2.26% -37.05% -38.40% 0.02% 0.01% -9.30% -8.88% -77.98% -63.41% 0.10% 0.01% -14.94% -10.51% -100.00% -82.32% 0.18% 0.01% -44.56% -22.38% -30.87% -31.41% 0.03% 0.02% -5.98% -6.00% -97.21% -72.72% 0.11% 0.01% -24.36% -16.80% -12.78% -12.77% 0.01% 0.00% -2.49% -2.52% -37.59% -38.43% 0.04% 0.02% -9.03% -9.16% -19.08% -13.31%

21

We see from Table 5 that the performance of MCCFP and ICCFP for large values of N is similar to their performance for small and medium values of N (Tables 3-4). That is, ICCFP (resp., MCCFP) achieves an 8.52% (resp., 3.69%) increase in total profits compared to OICFP. A detailed examination of results in Table 5 reveals that for each combination of X T and CusT , ICCFP outperforms MCCFP in terms of total profits, with both the average and the minimum deviations of ICCFP being higher than those of MCCFP in all scenarios but one. MCCFP achieves a 21.34% decrease in shipment cost per sale and a 41.10% increase in sales, which results in a 3.35% decrease in leftover inventory compared to OICFP. Similarly, ICCFP results in a 15.29% decrease in shipment cost, a 30.62% increase in total sales, and a 2.51% decrease in leftover inventory at the end of the selling season. Although total sales are lower and shipment cost per sale is higher for ICCFP than for MCCFP, it seems that by being more selective in customer acceptance, ICCFP again turns its resulting lower sales into higher profits than MCCFP. We also observe that the number of customers who are rejected due to inventory being depleted is 3.45% and 6.80% higher for MCCFP and ICCFP, respectively, compared to OICFP. We note that for all small-, medium-, and large-sized instances, computational run time of a scenario for MCCFP, OICFP, and ICCFP is less than 1 minute. We summarize the computational run time results of each policy in Appendix E.

Table 5 N

XT

CusT 1000

840 1600

1000 100

1240 1600

1000 1840 1600

1000 840 1600

1000 200

1240 1600

1000 1840 1600

Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg. Avg.

Performance of MCCFP and ICCFP against OICFP, for N = 100, 200.

Total profit MCCFP ICCFP -28.46% -2.31% 13.97% 19.20% -3.03% 3.68% -33.38% -2.51% 6.59% 10.74% -7.43% 2.44% -19.82% -2.11% 42.60% 38.40% 7.30% 9.59% -29.23% -1.65% 16.49% 19.33% -1.81% 5.62% -4.49% 0.00% 66.06% 69.22% 16.67% 17.66% -26.66% -2.63% 48.99% 41.53% 8.65% 11.31% -33.53% -7.58% 18.56% 25.74% -2.77% 2.21% -40.96% -5.13% 11.38% 20.55% -10.03% 2.53% -29.74% -5.93% 54.78% 44.90% 7.04% 7.87% -36.15% -3.33% 35.53% 33.65% -2.82% 6.18% -16.42% -1.85% 118.07% 79.91% 22.08% 18.31% -29.68% -5.31% 72.41% 65.86% 10.49% 14.85% 3.69% 8.52%

Shipment cost per sale MCCFP ICCFP -42.30% -41.18% 8.87% 14.48% -19.03% -10.84% -44.55% -37.83% 20.18% 29.69% -15.78% -4.88% -51.91% -50.39% 0.00% 16.07% -25.32% -21.38% -45.55% -43.36% 0.03% 16.11% -24.71% -12.92% -51.99% -51.75% 0.00% 0.00% -28.28% -27.36% -51.59% -49.86% -0.53% 26.27% -30.21% -24.20% -52.92% -48.70% 17.71% 29.63% -14.04% -7.53% -50.30% -38.23% 30.73% 36.38% -9.29% -3.55% -54.94% -53.91% 3.29% 35.37% -19.56% -15.09% -52.24% -44.59% 12.91% 28.20% -17.97% -10.54% -61.86% -60.49% 0.00% 2.89% -24.94% -23.25% -56.20% -51.96% -3.13% 28.20% -27.01% -21.95% -21.34% -15.29%

Total MCCFP 0.00% 85.01% 22.45% 0.03% 65.91% 21.02% 0.00% 153.29% 37.50% 0.07% 93.38% 30.53% 0.00% 169.08% 48.58% 0.16% 162.28% 45.62% 0.00% 135.92% 33.09% 0.00% 112.05% 25.51% 0.00% 213.54% 50.80% 0.74% 156.81% 39.68% 0.00% 332.60% 75.92% 3.22% 211.69% 62.52% 41.10%

sales ICCFP 0.00% 67.15% 17.70% 0.00% 40.85% 11.50% 0.00% 116.37% 31.78% 0.00% 65.27% 20.47% 0.00% 172.00% 46.73% 0.16% 113.68% 35.06% -0.27% 95.91% 19.57% 0.00% 82.65% 16.22% 0.00% 142.15% 34.49% 0.00% 110.70% 27.38% 0.00% 219.79% 55.59% 0.90% 183.73% 50.93% 30.62%

# of stock-outs MCCFP ICCFP -1.31% 0.00% 17.13% 19.38% 4.34% 6.36% -0.38% 0.00% 18.04% 18.18% 5.49% 3.99% -3.63% 0.00% 14.91% 22.94% 1.83% 8.86% -0.79% 0.00% 18.14% 19.15% 4.48% 5.92% -5.81% 0.00% 9.38% 22.06% -0.56% 8.99% -4.10% 0.01% 16.92% 23.04% 2.29% 9.01% -1.61% 0.00% 17.11% 19.83% 4.66% 5.56% -0.70% 0.00% 17.06% 17.31% 5.75% 3.39% -3.33% 0.00% 16.24% 21.16% 3.31% 7.41% -1.58% 0.00% 16.88% 18.10% 5.03% 5.31% -7.25% 0.00% 13.22% 21.24% 1.15% 8.59% -2.40% 0.06% 15.76% 20.23% 3.64% 8.19% 3.45% 6.80%

Leftover inventory MCCFP ICCFP -10.66% -8.27% 0.00% 0.00% -3.53% -2.77% -9.23% -6.17% -0.01% 0.00% -3.44% -1.87% -13.21% -10.41% 0.00% 0.00% -4.46% -3.77% -11.39% -8.48% -0.01% 0.00% -4.08% -2.70% -10.53% -10.58% 0.00% 0.00% -4.01% -3.86% -13.84% -10.87% -0.03% -0.03% -4.72% -3.67% -6.61% -5.00% 0.00% 0.01% -2.36% -1.45% -6.10% -4.30% 0.00% 0.00% -2.04% -1.33% -7.72% -5.35% 0.00% 0.00% -2.81% -1.97% -6.98% -5.16% -0.04% 0.00% -2.58% -1.85% -8.47% -6.13% 0.00% 0.00% -3.03% -2.30% -7.46% -6.09% -0.29% -0.05% -3.13% -2.57% -3.35% -2.51%

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6.2. Sensitivity Analysis for the Parameters of Our Computational Study In this section, we perform sensitivity analyses on the following model parameters: (i) profit margins, (ii) shipping costs, (iii) online customer arrivals, and (iv) initial inventory levels. We provide sensitivity results of (i)-(ii) for small- and medium-sized problems and sensitivity results of (iii)-(iv) for smallsized problems only. 6.2.1. Profit Margins of Different Fulfillment Options An important input to make fulfillment decisions is the net revenue obtained from each customer, i.e., pi − cpi or p0 − cpj − chj − csj . Since csj values are uncertain and we assume that the unit operating (cpj ) and handling (chj ) costs are approximately the same across all stores, it makes sense to focus on the sensitivity of the following profit margins: pi − cpi for store fulfillment, p0 − cp0 − ch0 for online fulfillment (from the FC) and p0 − cpi − chi for ship-from-store fulfillment. Recall from Section 6.1 that while setting the parameters for our computational study, we aim to make sure that pi − cpi ≥ p0 − cp0 − ch0 − E[cs0 ] ≥ p0 − cpi − chi − E[csi ] holds. For the original parameters in our study, we have $30 ≥ $20 ≥ $10, i.e., the associated profit of online fulfillment lies at the midpoint of the (expected) profit distance between store and ship-from-store fulfillment. To observe the impact of the asymmetry in profit margins associated with each fulfillment option, we shift the profit of online fulfillment (i) closer to store fulfillment and (ii) closer to ship-from-store fulfillment. This will allow us to change the attractiveness of each fulfillment option with respect to each other on the same expected profit distance. Changing only cp0 at two levels allows us to achieve this result; thus, we consider the following new values of the unit operating cost: cp0 = 60 and cp0 = 70. Table 6 summarizes the new values of cp0 and their resulting profit margins for each fulfillment option. Table 6

Summary of parameters in $ for profit margin sensitivity

cp0

Store fulfillment Online (FC) fulfillment Ship-from-store 65 30 20 10 60 30 25 10 70 30 15 10 * Note: The original parameters are in bold.

Figure 1 displays the results of our sensitivity analysis on profit margins. It shows that for all three possible profit margin scenarios, the average percentage deviations from OICFP (over all scenarios) are positive for all three policies (i.e., MCCFP, ICCFP, and OCCFP). This indicates that regardless of the profit asymmetries of the fulfillment options, all three policies, on average, perform better than OICFP. This result holds for both small and medium values of N ; see Figures 1a and 1b. For small values of N , the performance of ICCFP and OCCFP are similar. That is, as the difference between profit margins of FC (online) and ship-from-store fulfillment decreases (i.e., cp0 increases from 60 to 70), and the performance of both policies compared to OICFP increases, with higher percentage deviations from OICFP. This result is intuitive, because when the profit margin of shipfrom-store fulfillment increases with respect to online fulfillment (without a change with respect to store fulfillment), then ship-from-store becomes more attractive, thus the results would deviate more from OICFP. This performance of ICCFP is also valid for medium values of N . Although the performance of MCCFP for N = 2 is also similar to the performance of ICCFP and OICFP that is discussed above, its performance for medium values of N and also for N = 1 is different. For these cases, when the profit margin of online fulfillment increases or decreases with respect to ship-from-store fulfillment compared to the equal profit margin case, the performance of MCCFP gets closer to OICFP, with lower percentage deviations from OICFP.

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Figure 1

The percentage profit deviations of MCCFP (M), ICCFP (H), and OCCFP (O) with respect to OICFP when cp0 = 60, cp0 = 65 (base case), and cp0 = 70.

Figure 2

The percentage profit deviations of MCCFP (M), ICCFP (H), and OCCFP (O) with respect to OICFP when G0 (· ) = 20 and Gi (· ) = 5 with symmetric stores, G0 (· ) = 10 and Gi (· ) = 5 with symmetric stores (base case), and G0 (· ) = 10 and Gi (· ) = 5 with asymmetric stores.

6.2.2. Shipping Costs Recall that we denote shipment cost of a customer to location j by csj , which has a cumulative distribution function Gj (· ). For our computational study (Section 6.1), we set the mean of the shipment cost distribution Gj (· ) for the FC at 10 and for all stores at 5. Since this base case (i.e., Case 1) considers the same mean for all stores, we refer to this scenario as a “symmetric stores” case. In our cost sensitivity analysis, we consider the following two modifications to the base case: • Case 2: We set the mean of the shipment cost distribution for the FC at 20 (i.e., G0 (· ) = 20), thus examining the impact of having an FC that is located farther away from the customers. • Case 3: We consider different means of shipment cost distributions for stores by keeping the overall mean of the shipment cost distribution for stores at 5, and refer to this case as an “asymmetric stores” case. For instance, for N = 2, we suppose that G1 (· ) = 3 and G2 (· ) = 7, and therefore examine the impact of having some stores that are closer to the customers than others. Figure 2 displays the percentage profit deviations from OICFP for all three policies in all three shipment cost cases that we consider. Note that no data are available for the asymmetric store case

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when N = 1, since there is only one store. In the base case, although MCCFP outperforms OICFP for all values of N , its performance deteriorates in Case 2. Thus, the performance of MCCFP deteriorates when the distance of the FC to the customers increases (i.e., moves from Case 1 to 2). On the other hand, the average performance of both ICCFP and OCCFP improves in Case 2 compared to the base case. For the asymmetric stores case, for N = 2, the performance of both MCCFP and ICCFP deteriorates, while the performance of OCCFP remains almost the same. For medium-sized instances, MCCFP performs better, whereas the performance of ICCFP varies. 6.2.3. Online Customer Arrivals We next examine the impact of λ0 on the performance of the policies. For varying λ0 values, Table 7 presents the minimum, the maximum, and the average deviations of each policy from OICFP for N = 2. According to Table 7, the performance of MCCFP, OCCFP, and ICCFP with respect to OICFP improves as online customer arrivals (i.e., λ0 T ) increase. Since OICFP fulfills an online order from only the FC, the value of ship-from-store implementation can be observed better as the number of online customers increases. Thus, the performance of policies that allow shipping from a store increases when there are more online customers. We note that our results hold for other values of N as well. Table 7 XT

CusT

The impact of λ0 on performance of the policies with respect to OICFP, when N = 2. λ0 T 10

50 30 45

20 80 60

Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg.

MCCFP -2.24% 4.07% 0.52% -1.33% 19.15% 7.76% -11.96% -0.01% -4.87% -15.43% 17.18% -3.48%

Total profit OCCFP ICCFP 0.00% -0.09% 6.82% 5.68% 1.66% 1.20% 1.35% -0.45% 33.65% 23.85% 15.33% 10.17% 0.00% -1.38% 1.06% 0.00% 0.21% -0.28% 2.05% -6.81% 50.24% 37.69% 20.16% 7.70%

XT

CusT

λ0 T 10

50 30 60

20 80 60

Min Max Avg. Min Max Avg. Min Max Avg. Min Max Avg.

Total profit MCCFP OCCFP 0.00% 0.00% 0.06% 0.09% 0.02% 0.03% 1.37% 2.09% 14.37% 18.78% 7.23% 10.31% -2.53% 0.00% 0.00% 1.02% -0.74% 0.21% -4.56% 8.56% 10.54% 31.91% 3.81% 20.32%

ICCFP 0.00% 0.07% 0.02% 1.28% 14.08% 7.29% -0.06% 0.23% 0.03% -1.22% 19.95% 9.09%

6.2.4. Initial Inventory Levels To address the impact of inventory allocation decisions on the relative performance of different algorithms, we perform additional numerical experiments. In our analyses, we consider N = 1 and N = 2 cases in which the comparison with the optimal policy is tractable. We vary the inventory level of one location while keeping others the same. We present our results for N = 1 in the following figure and include the results for 3(b), N = 2weinconsider Appendix B. the expected number of both store and online customers In Figures 3(a) and that is 25. In Figure 3(a), FC inventory is fixed at 25 (i.e., at the expected number of online customers) and store inventory level is varied, while in Figure 3(b) store inventory is fixed at 25 (i.e., at the expected number of store customers) and the FC inventory level is varied. Figure 3(a) shows that OICFP does not perform well when the store inventory is greater than the expected number of store customers (similar to our theoretical results). MCCFP has varying performances, but it performs worse when the store inventory level is around and equal to the expected number of store customers. ICCFP is the best among these policies with respect to a change in the store inventory level and combines the strengths of both MCCFP and OICFP (with a gap of less than 0.5%). When the store inventory level is more than the expected number of store customers, ICCFP starts to deviate more from the optimal policy. Figure 3(b) shows that for all policies, the percentage deviation from OCCFP decreases as the

25

(a) Change in store inventory x1 Figure 3

(b) Change in FC inventory x0

Change in the percentage profit deviation from OCCFP for varying initial inventory levels when N = 1

FC inventory level increases and reaches its minimum value approximately when the FC inventory level is slightly higher than the expected number of online customers. We note that a retailer ordering even all quantities before the selling season may have multiple (periodic) replenishment opportunities to allocate the initially ordered quantity to its fulfillment locations. A retailer having this opportunity can observe its sales better over time and thus can do a better job of re-allocating its remaining inventory across the fulfillment locations as the selling season progresses. Under such an availability, the performance of OICFP increases over time, and one would expect the benefits realized via ship-from-store implementation to decrease. The performance of MCCFP and ICCFP compared to OICFP decreases as well, with higher chances of both policies resulting in worse performance than OICFP. Thus, the implementation of ship-from-store becomes riskier for the retailer, resulting in a higher need for optimal implementation of the strategy.

7.

Conclusion

Omni-channel initiatives provide retailers numerous opportunities and allow them to outpace their competition. One of the most commonly implemented omni-channel approaches is ship-from-store, which connects the retailer’s online channel with its stores by exposing store inventory to online customers and thus turning the stores of a retailer into mini FCs. Today, several retailers have successfully implemented ship-from-store strategies in their operations (e.g., Best Buy, Macy’s, Walmart) to meet customer demand effectively. However, from an operations perspective, it is important to assess the benefits of a ship-from-store strategy on several performance measures, including profits, sales, leftover inventory that would be marked down at the end of the selling season, and derive efficient and easy-to-implement order fulfillment policy rules for the retailers. In this study, we investigate dynamic order fulfillment policies for retailers putting ship-from-store implementation into practice. We show the structural properties of the optimal profit function and develop an optimal threshold policy that can guide retailers in making their fulfillment decisions. We also outline the cases where implementation of the optimal cross-channel fulfillment policy is more desirable than the optimal independent channel policy through analytical analyses. We find that the value of ship-from-store implementation increases as the inventory level of a store deviates from the expected demand of that store and as the expected online demand increases. Furthermore, to provide numerical insights on the potential benefits of ship-from-store, we compare the performance of the optimal implementation of the policy with the performances of two fulfillment policies that are used in practice. Our results indicate that ship-from-store strategy is not a free lunch. That is, although

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the optimal implementation of the policy guarantees an increase in retailers’ profit and therefore is desirable, the optimal implementation is not practical for large-sized problems. On the other hand, when ship-from-store implementation is done recklessly, it may hurt the retailer’s profit. One key takeaway from our results is that a smart and a practical implementation of ship-from-store is required to enjoy the benefits of the strategy. Then, we propose an intuitive heuristic policy as a smart way of implementing the strategy. Through computational studies, we compare the proposed heuristic with myopic and independent channel policies as well as with the optimal policy under several scenarios. Our results indicate that the proposed heuristic outperforms both myopic and independent channel policies in terms of total profits. We conclude with a discussion of the limitations of our paper and directions for future research. First, our results show that the relationship between the initial inventory level of a location and its expected demand affects the value of ship-from-store implementation. Thus, it seems worthwhile to relax the assumption of fixed initial inventory levels and to study the optimal initial inventory levels of fulfillment locations and the gradual allocation of the inventory to the locations. Second, some of the customers may consider switching across the sales channels once they observe that their intended purchase channel is out-of-stock. Examining how the results would change once the customers’ switching behavior is incorporated might provide further insights for retailers. Third, the results of this study can be extended to multi-product settings. In such a setting, an online order may consist of more than one item, a so-called multi-item order. Due to at least one of the items either not being carried or having a low level of inventory at a certain location, the retailer may not be able to fulfill all the items included in a multi-item order from the same location when it arrives. This results in split shipments and therefore additional decisions for the retailer on how to split such orders and from which locations to fulfill each of the resulting split orders. Fourth, since the fulfillment network may consist of multiple stores and multiple FCs, multiple FCs can also be incorporated into the model.

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