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by Austin Bren

In high-fashion sales markets, management naturally seeks to order inventory so as to maximize profits. By their nature, however, fashion goods can depreciate quickly, so it’s important to order sufficient inventory to cover the demand for a given sales period, but still avoid overly high levels of stock so as to avoid loss of value due to depreciation. Moreover, since these products are often highly novel, there are few parallels that can be drawn to other products, and consequently accurate demand forecasts for a new item or product line are often not readily available.

A manager purchases fashion-forward shoes for a demand period. Any demand above the quantity ordered missed sales opportunity whereas any remaining inventory is sold back at a loss.

The framework for studying this kind of scenario is commonly known as the “newsvendor problem.” It describes the decision making surrounding the sale of a perishable good (such as a daily newspaper), and acts as a foundation for many inventory control problems with stochastic elements.

In the traditional newsvendor problem (and in most decision-making models), a manager’s goal is to maximize his/her average profits, or, equivalently, minimize the mean of the underage cost (cost per unit of inventory under demand, due to lost sales) and overage cost (cost per unit of inventory over demand, due to depreciation). This average (or expected value) is calculated with respect to a known density of demand, characterized as a function ƒ. Therefore, the manager’s goal is to identify the order quantity 𝑥 that minimizes the expected cost, which is a combination of underage (𝑢) and overage (ℎ) costs due to inventory mismatches with random demand 𝐷, expressed as

where the inner operator represents the expected value with respect to demand density ƒ.

The newsvendor problem has a well-known closed-form solution: the optimal, cost-minimizing order quantity is the 𝑢/(𝑢+ℎ) percentile of the demand distribution determined by ƒ.

Photo by G. Crescoli on Unsplash

Despite the undeniable success of this result, in both its analytical ease and its effectiveness in stable, recurring decision-making environments, it relies on the accurate estimation of underlying random elements (demand), and hence is incomplete in its ability to express risks due to misspecification in the underlying distribution guiding the model.

In general, these kinds of uncertainties can arise for a couple main reasons:

- a dearth of historical data, whether due to initialization or a lack of recording

- a highly dynamic, quickly-evolving environment

- a highly variable process

This is the case in our high-fashion retail environment, where there is little data with which to forecast incoming demands due to product novelty.

To mitigate the effect of inaccuracies in their estimates, some managers choose to utilize alternative strategies via objectives based on percentiles or regret-based approaches, both of which are aimed at providing a level of protection against poor outcomes. However, when the underlying uncertainty lies in the random elements of the model itself, these strategies are inadequate since they rely on the belief that the model is correctly specified (i.e., they trust their model). Reducing the risks due to misspecifications in the model itself requires an alternative approach that can express a distrust in any individual model.

For example, consider a product that has a purchase cost of $50 and a sale cost of $60, which can be sold at the end of the sales period at $30 (therefore inducing an underage cost of 𝑢 = $10 and an overage cost of ℎ = $20). From prior experience with other similar goods, suppose a manager estimates his/her demand should behave like a normal random variable with a mean of 100 units and a standard deviation of 10 units. In accordance with the optimal newsvendor order quantity described above, they decide to order approximately 96 units to maximize their expected profit. However, since the true demand distribution is subject to uncertainty, it is possible that the true distribution for demand instead takes on a bimodal form shown below:

Potential demand distributions of a bimodal form. Classified according to their peaks’ distances from D=$100.

When the demand distribution deviates from the original estimate, the manager no longer enjoys a policy that minimizes average costs. Ideally, we would like to protect against these kinds of errors in demand misspecification, which entails the consideration of uncertainties in the estimated model. One such approach utilizes distributionally-robust optimization, which sets up an optimization model consisting of a two-player zero-sum game in which the manager pits himself/herself against a robust antagonistic agent who is responsible for choosing the underlying stochastic framework for the game. Letting 𝑆 denote the set of all densities available to the antagonist, our objective in this game can be described mathematically as:

Thus, the manager attempts to order inventory with the knowledge that the robust agent is working against them by choosing the worst-case distribution associated with whatever order quantity is chosen. The key component in this expression is 𝑆, which controls the robust agent’s power and hence the level of pessimism with respect to the ambiguity in demand distribution. A large, diverse 𝑆 can protect against many poor scenarios; however, it may also hold highly unusual demand distributions, leading to overly pessimistic, unrealistic solutions for real applications. In contrast, a highly constricted or poorly constructed 𝑆 that does not accurately represent the level of uncertainty in demand can also result in poor ordering decisions.

For example, one might consider a moment-based 𝑆 by allowing it to consist of the all demand distributions that have mean μ and standard deviation 𝜎. In this way, a manager could estimate the mean and standard deviation of demand, let 𝑆 consist of a hugely diverse pool of distributions that match these estimates, and develop a policy that protects against the worst-case scenario from these distributions. As shown in Scarf (1957) and Moon (1993), an optimal solution for the manager yields an order quantity of

if 𝜎²/μ² < 𝑢/(𝑢+ℎ) and zero otherwise. Not only does this moment-based strategy allow the manager to protect against every possible distribution with a given mean/variance structure, but like the traditional newsvendor solution, it offers a surprisingly effortless, closed-form representation.

However, we must ask also ourselves if 𝑆 is a fair representation of the underlying ambiguity of our demand distribution. It turns out that the worst-case demand distribution with such a diverse 𝑆 is one that features only two outcomes with positive probability! Hence, in this case, the robust agent chooses a demand distribution that can only result in two demand outcomes: one with low demand, and one whose demand is high.

Since this is unrealistic when juxtaposed with our real setting (where we expect a wide variety of outcomes, like our normal demand estimate), it is no surprise that the traditional policy outperforms the moment-constrained policy when the true demand is of the bimodal form shown above.

Costs under the moment-constrained and traditional newsvendor solutions when the underlying density is bimodal with varying peak distance from D=$100.

This motivates our desire for a more reasonable S that could somehow express different levels of uncertainty surrounding an initial estimate of the underlying distribution.

With such an S, a manager could protect himself/herself against more realistic potential demand distributions instead of focusing on the unrealistic (for our application) moment-based strategy discussed above.

One approach that follows this line of reasoning frames S as the set of all densities that lie some “distance’’ away from an estimate density ƒ, where distance is calculated via the Kullback Leibler (KL) divergence criteria.

KL-divergence is defined between two densities ƒ and 𝑔 as

Despite the fact that KL-divergence is not a true distance measure, it allows us to construct S with respect to a level of uncertainty defined by the distance from ƒ by letting S be the set of all densities 𝑔 such that the KL-divergence between 𝑔 and ƒ is smaller than some distance η. Here, η controls the maximum “distance’’ from estimate density ƒ. Surprisingly, as shown in Hu & Hong, these problems can result in easily-solvable convex optimization problems, and can have closed forms in many special cases. With some work (suppressed for the sake of brevity), one can easily determine order quantities as well as the worst-case densities chosen by the robust agent when the estimate density ƒ is normal. For example, when ƒ is normal with mean 100 and standard deviation 10 (as per our running example), we can show the underlying worst-case densities for a variety of distances below.

The worst-case densities chosen by the robust agent with respect to various KL-divergence sets $S$.

These densities appear to form a much more reasonable outcome for our real-world scenario since they do not sharply diverge from the initial estimate density ƒ. Furthermore, the level of uncertainty can be easily modified by simply adjusting the size of S via the distance η, resulting in either a more diverse set of demand outcomes or a tighter fit to ƒ, which allows for the manager to observe how ordering behavior should change as the level of uncertainty changes. As expected, with more reasonable ambiguity sets, robust policies provide much better solutions in the presence of ambiguities as compared to the overly pessimistic moment-based S.

The robust policies induced by KL-divergence can outperform the traditional newsvendor policy and moment-based robust approach when the true demand is bimodal.

In settings with little supporting data, it is important to consider the effect of decisions under a variety of potential scenarios since the estimated model may not be fully reliable. However, despite the value in designing policies that can protect against poor outcomes, this requires an understanding of the level of uncertainty in the real scenario for any reasonable analysis. Hence, when robust policies can be generated via easily configurable S, we can improve our ability to describe how decisions and outcomes change as the level of uncertainty changes. This in turn can help to provide managers with tools to analyze risks and determine policies best suited for the application at hand.

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