Efficient computer experiment-based optimization through variable selection

Abstract

A computer experiment-based optimization approach employs design of experiments and statistical modeling to represent a complex objective function that can only be evaluated pointwise by running a computer model. In large-scale applications, the number of variables is huge, and direct use of computer experiments would require an exceedingly large experimental design and, consequently, significant computational effort. If a large portion of the variables have little impact on the objective, then there is a need to eliminate these before performing the complete set of computer experiments. This is a variable selection task. The ideal variable selection method for this task should handle unknown nonlinear structure, should be computationally fast, and would be conducted after a small number of computer experiment runs, likely fewer runs (N) than the number of variables (P). Conventional variable selection techniques are based on assumed linear model forms and cannot be applied in this “large P and small N” problem. In this paper, we present a framework that adds a variable selection step prior to computer experiment-based optimization, and we consider data mining methods, using principal components analysis and multiple testing based on false discovery rate, that are appropriate for our variable selection task. An airline fleet assignment case study is used to illustrate our approach.

Appendix: Airline fleet assignment model formulation

The optimization formulation from Pilla et al. (2008) is reproduced here for the readers’ reference.

Let L be the set of flight legs (indexed by l). Let F denote the set of fleet types (indexed by f), and G be the set of crew-compatible families (indexed by g), which can be used for each of the legs l∈ L. Since we assign crew-compatible families in the first stage, for each leg l∈L and for each crew-compatible family type g∈G, let a binary variable x _gl be defined such that

In the second stage, we assign specific aircraft within the crew-compatible family. As such, for each leg l∈L, for each aircraft type f∈F, and for each scenario ξ∈Ξ, let a binary variable $x^{\xi}_{fl}$ be defined such that

Since a combined FAM and PMM model is used, let the decision variable $z^{\xi}_{i}$ represent the number of booked passengers for itinerary-fare class i in scenario ξ.

For combined FAM and PMM, consider the following additional parameters:

• S = set of stations, indexed by s,

• I = set of itinerary-fare classes, indexed by i,

• V = set of nodes in the entire network, indexed by v,

• f(v) = fleet type associated with node v,

• A _v = set of flights arriving at node v,

• D _v = set of flights departing at node v,

• M _f = number of aircraft of type f,

• f _i = fare for itinerary-fare class i,

• C _fl = cost if aircraft type f is assigned to flight leg l,

• $a^{\xi}_{v^{+}}$ = value of ground arc leaving node v for scenario ξ,

• $a^{\xi}_{v^{-}}$ = value of ground arc entering node v for scenario ξ,

• O _f = set of arcs that include the plane count hour for fleet type f, indexed by o,

• L ₀ = set of flight legs in air at the plane count hour,

• Cap _f = capacity of aircraft type f,

• $D^{\xi}_{i}$ = demand for itinerary-fare class i in scenario ξ.

The two-stage formulation can be represented as:

(4)

(5)

(6)

(7)

(8)

(9)

The objective is to maximize profit (revenue − cost) in the second stage by assigning aircraft within the crew-compatible allocation made in the first stage. The block time of a flight leg l is defined as the length of time from the moment the plane leaves the origin station until it arrives at the destination station. Let b _l be the scheduled block time for flight leg l. The cost for each flight leg is calculated as a function of block time and operating cost of a particular fleet type per block hour, and is given by:

$$C_{fl} = b_{l} * ({\mbox{Operating cost per block hour}})_{f}.$$

Constraints in set (4) represent the balance constraints needed to maintain the circulation of aircraft throughout the network. Cover constraints (5) guarantee that aircraft within the crew-compatible family (assigned in the first stage) are allocated. For formulating the plane count constraints (6), we need to count the number of aircraft of each fleet being used at a particular point of the day (generally when there are fewer planes in the air). As such the ground arcs that cross the time line at the plane count hour and the flights in air during that time are summed to assure that the total number of aircraft of a particular fleet type do not exceed the number available. Constraints (7) impose the seat capacity limits, i.e., the sum of all the booked passengers on different itineraries for a flight l should not exceed the capacity of the aircraft assigned and constraint (8) to meet the forecasted demand.