New Shipping Headaches; Classic Solution with a Twist

Online retailers have a challenge. Say a customer in Northwest Indiana orders a package of paper towels from Amazon, which operates dozens of logistics centers in the region. Should Amazon send the towels from a center outside of Chicago that is closer but flooded with orders, or ship from a center that is farther away but less strained by demand? This is a tough question on its own and is further complicated by interactions with the upstream decision of where to stock paper towels in the first place.

There are an astronomical number of policies that an online retailer could use to attempt to answer the question, each offering a different recipe for how many towels to order, where to stock them, and from where they should be shipped. But research by Chicago Booth’s Levi DeValve and North Carolina State University PhD student Jabari Myles finds that the best strategy for these companies, which have in many ways rewritten the rules of retail, is to put a twist on a classic solution, the base stock policy.

Operations research has a long tradition of examining how companies should manage their inventory, beginning with a world in which a single retailer with a single location served customers who shopped in person. One of the most celebrated results in early operations work, says DeValve, explained the base stock policy—the optimal inventory-stocking policy under these conditions.

Its central idea is that a retailer seeking to maximize profit should focus on maintaining a target (or “base”) stock level for inventory. Say a stand-alone store calculates the base-stock level for wickets to be 10. If a customer orders five wickets, the store orders five more from its supplier to bring the inventory level back to 10. Research has soundly established this policy for the single-store system. But there hasn’t been a theoretical justification for using the base stock policy in the complex modern setting of online retailers with warehouses, inventory, and customers spread across the globe.

To model this, the researchers considered a hypothetical retailer that needs to figure out how much inventory to stock and replenish for every item it carries, in every location. It can choose a general replenishment and fulfillment policy, but it’s impossible to calculate the optimal solution because of all the variables at play. Those include fluctuating demand and transportation costs, as well as correlations of demand for different items and different locations. (As in, will someone ordering paper towels be likely to order toilet paper too? And does increased demand in Chicago suggest higher demand is likely in NW Indiana as well?)

Another important variable is time dependence: At any point in time, inventory availability and feasible fulfillment configurations depend on past, related decisions. The combination of variables along these demand, network, and time dimensions create a massive number of decisions that need to be considered.

The researchers addressed this challenge by developing a novel approximation of the retailer’s operating costs that is both relatively easy to compute and guaranteed to be close to optimal. The basic idea of the approximation is to simplify one of the three variable dimensions, though the researchers determined that it’s far better to simplify the variable of time rather than demand or network. Why? A retailer can’t credibly ignore demand fluctuations, because a company that ostensibly knows exactly what demand is and will be would never incur holding costs or lose a sale. It’s equally unrealistic to pretend that the network is a collection of classic single-store systems.

So the researchers instead simplified the time variable by ignoring the dependency of current decisions on past decisions. “Clearly, this leads to some loss of accuracy in representing the retailer’s true operating costs, since in reality these time dependencies determine feasible decisions,” says DeValve. However, the simplified model reduces to what’s called a stochastic program, which can be solved efficiently by modern algorithms. Further, the stochastic program accurately captures all the intricate demand fluctuations and network pooling effects, and identifies ideal inventory targets across the network.

This discovery led to the key question: Could the inventory targets of the stochastic program be used as base stock levels? Classic operations theory couldn’t answer that, but DeValve and Myles developed a theory to explain why the answer is yes. It employs a novel cost-accounting technique to compare the base stock policy with an optimal policy—and proves that the cost of the base stock policy is guaranteed to be close to that of the optimal one.

With this theory established, DeValve and Myles designed and simulated a variety of scenarios in which hypothetical e-commerce retailers used the relaxed model to manage their inventory. In the vast majority of these realistic, simulated cases, the cost difference relative to an optimal policy was about 1 percent, validating the practical performance of their design.

Companies can more or less plug this work into current operations, which DeValve demonstrates in another paper—with Yanyang Zhao of the supply-chain management company Deposco (and a graduate of Booth’s PhD program), Booth’s John R. Birge, and Robert Inman of General Motors. They applied their model to the inventory of an automobile manufacturer and find that it generated significant cost savings during both normal operation and periods affected by supply-chain disruptions.