Pergamon Computers ind. Engng Vol. 33, Nos 1-2, pp. 51-54, 1997

© 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved

0360-8352/97 $17.00 + 0.00 PII: S0360-8352(97)0N39-9

A CUTTING-PLANE PROCEDURE FOR MAXIMIZING REVENUES IN YIELD MANAGEMENT

Alberto Garcia-Diaz, Texas A&M Umversity Ahmet Kuyumcu, Aeronomics Incorporated

ABSTRACT

A graph-theory approach for allocating seats and setting optimal prices in an origin-destination (O-D) network is developed. Input includes demand forecasts, computer reservation system restrictions, and aircraft capacities. The concept of a split graph is used to derive cutting planes. © 1997 Elsevier Science Ltd

KEYWORDS

Revenue management; split-graphs; perfect graphs; seat allocation; pricing; integer programming.

INTRODUCTION

Yield or revenue ~ is a marketing tool used by airlines to establish pricing and seat allocation strategies that maximize revenues. Pricing and seat allocation policies must be formulated in consistency with restrictions imposed by the airline's ~nputer reservation system (CRS). Current revenue nmnagement procedures either allocate seats based on predetermined price levels or set prices based on predetermined seat allocation levels. Furthermore, most seat allocation models maximize revenue considering a single flight leg as being independent of others.

This article assumes that the CRS uses a non-nested seat allocation system; that is, reservation requests for any fare class will be denied if the booking limit of the fare class is exceeded. In addition, it is assumed that the total passenger demand for each O-D itinerary and fare class combination is random, mutually independent, and normally distributed. This implies that the number of fare classes m a price structure does not alter the underlying demand distribution of a certain fare class.

Kimes (1989) and Weatherford and Bodily (1992) have written literature reviews in the area of revenue management. Price structures are predetermined in most of the reviewed procedures. Several other articles in the relevant literature address multi-leg seat inventory control problems for single-fare scenarios, as well as pricing studies considering behavioral aspects. Cross (1997) discusses core concepts of revenue management in a non-technical fashion. Kuyumcu (1996), and Kuyumcu and Crarcia-Diaz (1996) have developed relevant formulations used in this article.

FORMULATION CONCEPTS

Let set K(i) contain all fare classes in O-D itinerary i. The problem is to select a price structure (a subset of K(i)) and allocate sea~s to each fare class in the price structure such that the total expected revenue is maximized for the entire O-D network. It is assumed that even if a large number of fare

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classes may exist in K(i), a limited number of fare classes can be offered through the CRS. Since a price structure with more fare classes eaptures more revenue by satisfying more customers, the number of fare classes in a price structure will be equal to its maximum size; that is, the smaller subsets need not be further considered.

The expected marginal seat revenue derived from the allocation of seat j for a partic-lar fare class is computed as the ticket price multiplied by the probability that the demand will exceed j-I seats. The model has four groups of constraints. The first group of constraints will insure that exactly one price structure is selected for each O-D itinerary. A second group of constraints will not allow the seat allocation to each fare class to be possible if it is not in a specified price structure. A third group of constraints will establish that for every flight leg the total number of passenger assignments cannot exceed the residual capacity of the aircraft. A final group of constraints will require all decision variables to be binary.

Consider the first and second groups of constraints. For a given O-D itinerary i and seat j, it is always possible to define a graph G=-(V,E) and a node-packing problem as follows. Let the binary decision variable Xijk (seat variable) indicate wiietherj-th seat is aecepted (Xijk=l) or rejected (X~jk=0) for the allocation of passengers to fare class k and O-D itinerary i. Additionally, the binary decision variable W~ (pricing variable) specifies if price structure m of O-D itinerary i is selected 0N~=I) or rejected 0V~=0). Let each node represent a 0-1 variable Xijk or W~ with an edge in E. There is one edge between every pair of decision variables if they appear in the same constraint. The set of nodes V can be partitioned into nodes that match Xijk and W~m. The first group of constraints indicates that there exists an edge between every pair of W~'s for a given O-D itinerary i, which generates a complete subgraph or clique in graph G. In addition, the second group of constraints indicates that there is an edge from W~ to Xijk if price structure m does not contain fair class k. The set of feasible 0-1 solutions coincides with the set of node packings in graph G.

Figure 1 illustrates a graph G with two fare classes in the CRS and four fare classes (IK(i)l---4) for a given O-D itinerary i and seatj. In Figure 1 fare classes in each price structure m are represented by a set next to each node W~. Since by definition graph G is a spit graph, and split graphs are perfect (Golumbic, 1980), maximal clique and non-negativity constraints suffice to describe the convex hull of 0-1 solutions to the problem containing the first and second groups of constraints (Padberg, 1973). Note that there is an edge from W~ to Xijk if price structure m does not contain fare class k. It is also noted that the capacity constraints are required for the overall formulation of the problem.

{1,2}

{1,3}

{1,4}

{2,3}

{2,4}

{3,4}

Figure 1. Sample Graph for O-D Itinerary i and Seatj.

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OVERVIEW OF SOLUTION PROCEDURE

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The proposed algorithm uses cut6"ng planes representing the first and second sets of constraints, which are derived from split graphs. The procedure consists of three steps (Kuyumcu, 1996; Kuyumcu and A. Garcia-Diaz, 1996). The first step eliminates seat variables with low booking probabilities and expected marginal revenue values. The second step eliminates a single price structure variable from the first group of constraints to attain a full dimensional polybedra (Hoffinan and Padberg, 1993). The last step initiates a brunch-and-bound procedure, which takes advantage of CPLEX's rounding heuristic procedure to get a lower bound, special ordered-set constraints for branchin~ and a best- bound search method for backtracking.

COMPUTATIONAL RESULTS

A straightforward integer programming model of the problem can be formulated, but it may require exceptionally high computer execution times in an O-D network involving as few as two flights and three O-D itineraries. If the derivations of the first and second constraints are based on the complete subgraphs of the split graph as described above, the solution time improves significantly.

All procedures were ceded in FORTRAN 77 and linked thorough the CPLEX Linear Optimizer 3.0 running on IBM AIX Version 3.2 for RISC System/6000. Price structure capacities corresponding to O-D itineraries varied from 4 to 8. The relationship between CPU times and number of fare classes in each price structure is depicted in Figure 2. Figure 3 shows the relationship between optimal objective function values and number of fare classes in each price structure. As can be seen in this figure, an increase in number of fare classes leads to an increase in total expected revenue. However, it should be noted that the revenue increase can only be realized under the assumptions of the proposed models. Specifically, it is assumed that the number and type of fare classes in a price structure do not change the underlying demand distribution of a certain fare class.

Six additional problems representing an increasing number of flight legs and O-D itineraries were cousidered, setting the number of fare classes in each price structure equal to 8, with CRS capacity is equal to 4. Figure 4 shows the relationship between problems representing increasing numbers of flight legs and O-D itineraries and CPU times. As can be concluded from this figure, the propo-~i procedure solves pricing and seat allocation problem under consideration quite efficiently for problems involving as many as 22 flight legs and 128 O-D itineraries. It is noted that memory requirements exceeded the capacity of the workstation for larger problems.

300-

~ 250

2OO

150

100

50 0

J I I I

5 6 7 8

N u m b e r o f Fare Classes in Each Price Structure

Figure 2. CPU Time as a Function of Number of Fare Classes in Each Price Structure

1

J !

4 5 6 7 8

$375,000 $370 ,000 $365 ,000 $360 ,000 $355 ,000 $350 ,000 $345 ,000 $340 ,000 $335 ,000 $330 ,000 $325 ,000

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N u m b e r o f Fare C l a s s e s in Each Pr ice Structure

Figure 3. Total Expected Revenue as a Function of Number of Fare Classes

300

250 ~ 200

150

100

v I 5000 10000 15000 20000 25000 30000 35000

Total Number of Variablm

Figure 4. CPU Times as a Function of Number of Variables

REFERENCES

CROSS, R. G., 1997. Revenue Management: Hard-Core Tactics for Market Domination. Broadway Books, New York, N-Y.

GOLUMBIC, M. C. 1980. Algorithmic Graph Theory and Perfect Graphs.Acadenfi¢ Press, New York, N'Y.

HOFFMAN, K. L., AND M. PADBERG. 1993. Solving Airline Crew Scheduling Problems by Brunch-and-Cut, Management Science 39, 657-682.

KIMES, S.E. 1989. Yield Management: A Tool for Capacity-Constrained Service Firms. Journal of Operations Management 8, 348-363.

KUYUMCU, A. 1996. A Split-Graph and Cutting-Plane Approach to Optimal Pricing and Scat Allocation in Origin-Destination Networks, Ph.D. Dissertation, Texas A&M University, Colloge Station, TX.

KUYUMCU, A. AND A. GARCIA-DIAZ. 1996. A Polyhedral Graph-Theory Approach to Airline Yield Management, INEN/OR/WP/01/02/96, Texas A&M University, Department of Industrial r gin ing.

PADBERG, M. 1973. On the Facial Structure of Set Packing Polyhedra. Mathematical Programming 5, 199-215.

WEATHERFORD, L. 1~, AND S. E. BODILY. 1992. A Taxonomy and Research Ovorviow of Perishable-Asset Revenue Management: Yield Management, Overbooking, and Pricing. Operations Research 40, 831-844.6