## Abstract

The Internet has revolutionized the retailing landscape and how goods and services are sold and distributed to consumers. One avenue of significant growth in online selling comes from multichannel retailers who offer products in stores as well as over the Web. These hybrids may leverage their “brick” locations by allowing customers to pick up or return orders purchased online at retail stores. This option lets Web-based buyers avoid added shipping costs and long package carrier lead times, albeit at a cost to retailers. To examine the viability of this strategy, we develop a mathematical model that examines the cost and value of providing in-store pickup and return. The model is used to determine the best subset of brick-and-mortar stores to handle in-store pickup and return demand. One of the principal takeaways is that not all retail stores should be offering in-store pickups and/or returns. Our computational results show optimizing the set of pickup and return locations may reduce system cost over baseline marketing policies where these services are set up at all or none of a retailer’s stores. In addition, we show that retailers can significantly improve some consumer benefits at little extra cost.

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## Appendices

### Appendix 1

### Derivation of F (Total cost under in-store pickups and returns of online sales)

This section develops the total cost (holding, backorder, shipping, and handling cost) for the system depicted in Fig. 1 where a portion of Internet orders may be picked up at or returned to the local stores. First, to formulate the expected holding plus backorder cost we make use of the allocation assumption in Eppen and Schrage (1981). Specifically, we assume that in each allocation period the central warehouse receives sufficient inventory so that the system can be brought into balance after the allocation.

Let (*l*
_{
i
} + 1) ⋅ *μ*
_{
i
} and \( a\cdot {\sigma}_i\cdot \sqrt{l_i+1} \) denote the expected demand and safety stock at site *i* over *l*
_{
i
} + 1 periods (for some *a*). Then, the base stock level (identified at the start of each period) that equalizes the probability of a stock-out at retail or e-fulfillment site *i* in period *l*
_{
i
} + 1 is: \( {r}_i=\left({l}_i+1\right)\cdot {\mu}_i+a\cdot {\sigma}_i\cdot \sqrt{l_i+1} \) for some *a*, where

*l*
_{
i
} = *l*
_{1} for *i* = 1…*J*; *L*
_{1} for *i* = *J* + 1.

Assume that unit holding and backorder costs, *h* and *p*, are constant across all locations such that the same fractile point is optimal at each source of demand. Let *z* be the *p*/(*p*+*h*)th fractile of the standard normal distribution. Using the approach in Eppen (1979) and Bollapragada et al. (1998) and accounting for the lead times involved, the expected total holding plus backorder cost per period can now be calculated as:

where *K* = *h* ⋅ *z* + (*h* + *p*) ⋅ *R*(*z*), \( Y=\left[{\displaystyle \sum_{i=1}^J\sqrt{\left({l}_i+1\right)\cdot \left({\sigma}_i^2\right)}}+\sqrt{\left({l}_{J+1}+1\right)\cdot \left({\sigma}_{J+1}^2\right)}\right] \), and *R(z)* is the unit Normal right-tail linear loss function. In the above equation the ratio of system ending stock to the standard deviation of ending stock is:

and *y*
_{
cw
} is the order-up-to quantity at the central warehouse. Solving for *y*
_{
cw
} in the above equation for *z* yields

Next, the system-wide expected fixed cost of operating pick-up and return centers at the sites is

Additionally, the cost associated with lost pickup and return demand is obtained by multiplying the expected (1) pickup demand that materializes as online sales, (2) pickup demand that materializes as regular in-store sales at the desired pickup location, (3) pickup demand that materializes as pickup sales at an alternate store, (4) pickup demand that is lost, and (5) in-store return demand that is handled at the central warehouse; by corresponding unit lost sale and goodwill costs; *g*
_{
pe
}
*, g*
_{
ps
}
*, g*
_{
po
}
*, g*
_{
pc
} and *g*
_{
r
}.

Finally, the shipping and handling cost is obtained by adding the expected cost of transporting and handling product between the online fulfillment center and the online customer regions to the expected cost of transporting and handling product between the central warehouse and the stores:

Adding these four cost components yields the system-wide fixed operating, shipping, lost sale, and inventory cost per period objective function under the in-store pickup and return of online sales:

### Appendix 2

### Estimates of customer alternate purchasing behavior *γ*
_{
pe,i
}, *γ*
_{
ps,i
}, *γ*
_{
po,i
}, and *γ*
_{
pc,i
}

When in-store pickup is not offered at store *i* some fraction of that store’s pickup demand will materialize as regular online sales (*γ*
_{
pe,i
}), some fraction will materialize as in-store sales (*γ*
_{
ps,i
}), some fraction will materialize as pickup sales at an alternate store (*γ*
_{
po,i
}), and some fraction will be lost (*γ*
_{
pc,i
}). Equations (23), (24), (25), and (26) below establish customer buying behavior when the pickup option is not available at their local store (i.e., probabilities the customer will attempt to purchase the product as a regular in-store sale, pick the product up at the retailer’s next closest pickup store, convert the pickup demand to an online sale at the retailer, and choose not to purchase the product from the retailer).

Equation (23) sets the probability that pickup demand materializes as regular sales when in-store pickup is not offered to τ. Equation (24) ensures that the probability any pickup customer purchases from a nearby store when their preferred location is not available decreases with distance to the nearest pickup site. Equations (25) and (26) then set the relative likelihood for unmet pickup demand to materialize online or at a competitor respectively. Note that the model is general to allow alternate expressions for these switching distributions. For clarity the switching distributions under the four levels of λ/τ considered in the experimental design are illustrated in Fig. 4. It seems natural that a common reaction for a customer whose preferred store is not offered for pickup would be to go to the next closest store. For that reason, each of the four scenarios include a certain probability (γ_{po,i}) to accommodate that behavior. In Fig. 4a, we include a probability that the customer switches to a competitor (γ_{pc,i}). Since proximity is a factor in the customer’s behavior, the probabilities for whether the customer will go to the marketer’s next closest location or to the competitor change as travel distance changes. For instance, in Fig. 4a, if the customer is zero miles from the next nearest pickup store, there is a 100% probability that they will go to that store. If the customer is 10 miles away, the customer is equally likely to go to the next closest location for pickup or to buy from a competitor. At 20 miles from the marketer’s next pickup location, the customer will buy from the competitor 100% of the time. Note that we are not suggesting these probabilities govern all customer behavior. Rather we present a range of cases for how customers might react. Any probabilities could be inserted into our model depending on specific marketer/customer characteristics. Figure 4b shows a scenario where customers switch between in-store pickup at another of the marketer’s locations and choosing to have the product delivered by direct-to-home. Again, the probabilities of these two behaviors (γ_{po,i,} and γ_{pe,i}) are based on the customer’s distance from the marketer’s next closest location that offers in-store pickup. The further the next closest store is away from the customer, the more likely they will choose to have it shipped to their home. Figure 4c allows for both competition and direct-to-home choices in addition to the next closest pickup location choice. For instance, at 10 miles, we suggest that there is a 50% probability that the customer will go to that location, a 25% probability that the customer goes to the competitor, and a 25% probability that the customer chooses direct-to-home. Finally, Fig. 4d allows for four behaviors. In addition to those behaviors above, we allow for the customer to go to the local store (the same one that they requested in-store pickup) and purchase their item(s) as an in-store sale. This channel switching, we assume, does not depend on distance and thus is shown in the top portion of Fig. 4d as a 20% probability (γ_{ps,i}) regardless of distance.

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Mahar, S., Wright, P.D., Bretthauer, K.M. *et al.* Optimizing marketer costs and consumer benefits across “clicks” and “bricks”.
*J. of the Acad. Mark. Sci.* **42**, 619–641 (2014). https://doi.org/10.1007/s11747-014-0367-8

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DOI: https://doi.org/10.1007/s11747-014-0367-8