Forecasting time-phased demand

Forecasting methods generally fall into the class of methods known as time series methods, primarily exponential smoothing, or causal methods, where price is taken to be (one of) the causal factors. In pricing science applications, it is necessary to produce forecasts of demand at the level of granularity at which pricing decisions are made. This introduces both modeling and computation complexity not addressed in standard treatments of forecasting methods. Also, in cases where capacity constraints are present, methods are required to account for the censoring of demand that occurs when demand exceeds capacity. In cases where bookings are closed because they have reached the maximum authorization, one must estimate what the "true" demand would have been had bookings been accepted during those closed periods.

Forecasting granular demand

Often, there may be insufficient historical instances of the series of interest to produce a reliable demand forecast. For an airline, this might happen for flights to new markets, where no history is available to reference. For a retailer, it may simply be sparse data on sales of a particular SKU. A widely used method used to produce the necessary forecasts in such cases is sometimes referred to as "aggregate and distribute." This method decomposes the forecast into two components, a forecast of a more aggregated series and a forecast of how that more aggregated demand is distributed across its components, viz:

${\displaystyle Q_{t+n}^{i}=Q_{t+n}^{I}*sI_{t+n}^{i},i\in I;}$

where ${\displaystyle i}$ is the particular low-level series of interest, ${\displaystyle I}$ is the aggregate of related series (e.g. all itineraries serving a particular origin-destination, or all sizes and colors of the particular style of shirt), ${\displaystyle Q_{t+n}^{I}}$ is the forecast of the aggregate, and ${\displaystyle sI_{t+n}^{i}}$ is the forecast of ${\displaystyle i}$'s share of ${\displaystyle I}$. Both ${\displaystyle Q_{t+n}^{I}}$ and ${\displaystyle sI_{t+n}^{i}}$ may be produced using standard exponential smoothing methods.^[3]

Accounting for censoring

When the application balances demand against supply through direct control of product availability, as is common in many yield management applications, producing good time-phased forecasts requires either capturing the demand which doesn't result in a sale or booking directly (often referred to as "turndowns" or "loss data"); or using some scientific method to estimate the unobserved demand. Conventionally, these methods are referred to as "unconstraining methods", include manual adjustment, averaging methods, Expectation Maximization (EM) methods, exponential smoothing methods.^[4]

Causal methods

When the application uses prices as the control instrument, setting prices to modulate sales, producing good time-phased forecasts may require using causal methods (sometimes referred to as econometric methods) to account for the relationship between the prices in effect at a point in time and the observed sales at that point in time. In this way, the relationship between price and sales volume, often referred to as the "price response effect," can be used to separate the underlying time-phased demand from the sales effects of price changes. Since the objective of these applications of pricing science is precisely to take best advantage of the sale volume effects of price changes, accounting for these effects can be a significant focus of the scientific work in support of these applications. The problem of identifying and estimating these effects is not trivial since, in addition to the price of a specific product, sales volume is affected by numerous other effects, some of which are under the control of the firm (e.g. advertising, prices of related goods) and other which are outside the control of the firm (e.g. competitors' prices, seasonality). In the domain of pricing science, these methods are typically referred to as Market Response Models.^[5]

Heuristic methods

The most well-known (and likely, most broadly applied) heuristic method for a large class of yield management problems is known as the Expected Marginal Seat Revenue (EMSR) algorithm.^[6] This heuristic provides a decision rule for allocating inventory for sale at lower prices, as a function of the demand at higher prices and the differences in prices. Phillips ^[1] discusses extensions of the EMSR heuristic.

Numerical optimization methods

Many optimization problems are formulated as constrained or unconstrained mathematical programs, either linear programs (LP) or mixed integer programs (MIP), for which many solution techniques and commercial solvers are available.

Strict mathematical methods

If the market response model is formulated within a certain class, and point estimates of the model parameters obtained, the optimal solution can be obtained analytically exploiting the special structure of the problem.