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An Effective Deflected Subgradient Optimization Scheme for Implementing Column Generation for Large-Scale Airline Crew Scheduling Problems

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An Effective Deflected Subgradient Optimization Scheme for Implementing Column Generation for Large-Scale Airline Crew Scheduling Problems This article was downloaded by: [128.173.125.76] On: 21 February 2014, At: 12:09 Publisher: Institute for Operations Research and the Management Sciences (INFORMS) INFORMS is located in Maryland, USA INFORMS Journal on Computing Publication details, including instructions for authors and subscription information: http://pubsonline.informs.org An Effective Deflected Subgradient Optimization Scheme for Implementing Column Generation for Large-Scale Airline Crew Scheduling Problems Shivaram Subramanian, Hanif D. Sherali, To cite this article: Shivaram Subramanian, Hanif D. Sherali, (2008) An Effective Deflected Subgradient Optimization Scheme for Implementing Column Generation for Large-Scale Airline Crew Scheduling Problems. INFORMS Journal on Computing 20(4):565-578. http:// dx.doi.org/10.1287/ijoc.1080.0267 Full terms and conditions of use: http://pubsonline.informs.org/page/terms-and-conditions This article may be used only for the purposes of research, teaching, and/or private study. Commercial use or systematic downloading (by robots or other automatic processes) is prohibited without explicit Publisher approval. For more information, contact [email protected]. The Publisher does not warrant or guarantee the article’s accuracy, completeness, merchantability, fitness for a particular purpose, or non-infringement. Descriptions of, or references to, products or publications, or inclusion of an advertisement in this article, neither constitutes nor implies a guarantee, endorsement, or support of claims made of that product, publication, or service. Copyright © 2008, INFORMS Please scroll down for article—it is on subsequent pages INFORMS is the largest professional society in the world for professionals in the fields of operations research, management science, and analytics. For more information on INFORMS, its publications, membership, or meetings visit http://www.informs.org INFORMS Journal on Computing informs Vol. 20, No. 4, Fall 2008, pp. 565–578 ® issn 1091-9856 eissn 1526-5528 08 2004 0565 doi 10.1287/ijoc.1080.0267 © 2008 INFORMS AnEffective Deflected SubgradientOptimization Scheme for Implementing Column Generation for Large-Scale Airline Crew Scheduling Problems Shivaram Subramanian Enterprise Optimization, United Airlines (WHQKB), Elk Grove Village, Illinois 60007, [email protected] Hanif D. Sherali Grado Department of Industrial and Systems Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, [email protected] e present a new deflected subgradient scheme for generating good quality dual solutions for linear pro- Wgramming (LP) problems and utilize this within the context of large-scale airline crew planning problems that arise inpractice. The motivationfor the developmentof this method came from the failure of a black-box- type approach implemented at United Airlines for solving such problems using column generation in concert with a commercial LP solver, where the software was observed to stall while yet remote from optimality. We identify a phenomenon called dual noise to explain this stalling behavior and present an analysis of the desirable properties of dual solutions in this context. The proposed deflected subgradient approach has been embed- ded within the crew pairing solver at United Airlines and tested using historical data sets. Our computational experience suggests a strong correlation between the dual noise phenomenon and the quality of the final solu- tion produced, as well as with the accompanying algorithmic performance. Although we observed that our deflected subgradientscheme yielded anaverage speed-up factor of 10 for the columngenerationschemeover the commercial solver, the average reductioninthe optimality gap over the same numberof iterationswas better by a factor of 26, alongwith anaverage reductioninthe dual noiseby a factor of 30. The results from the column generation implementation suggest that significant benefits can be obtained by using the deflected subgradient-based scheme instead of a black-box-type or standard solver approach to solve the intermediate linearprograms that arise withinthe columngenerationscheme. Key words: airline crew planning; set-partitioning problems; deflected subgradient approach; dual noise; variable target value method History: Accepted by William Cook, former Area Editor for Design and Analysis of Algorithms; received February 2006; revised April 2007; accepted November 2007. Published online in Articles in Advance May 15, 2008. 1. Introduction of some r inequality restrictions are side constraints Consider the following discrete optimization prob- that arise out of certain designated goals pertaining lem, which represents a set-partitioning problem with to alternative multiobjective requirements, beyond the side constraints (SPP-SC) (see Nemhauser and Wolsey cost-minimization objective of SPP-SC. 1999): A typical airline planning instance of SPP-SC is the multiobjective airline crew pairing optimization prob- SPP-SC: Minimize cT x lem, where the columns Aj of A are generated to sat- subject to Ax = e isfy delineated crew pairing legality restrictions. More (1) Dx ≤ d specifically, this problem seeks anoptimal assign- ment of crew pairings to different flight segments, x ∈ Bn Downloaded from informs.org by [128.173.125.76] on 21 February 2014, at 12:09 . For personal use only, all rights reserved. subject to various contractual and FAA-mandated Here, the variables x ∈ Bn represent binary decision crew scheduling regulations. In practice, airlines are variables, with anassociated cost vector c. The first set interested in finding a (near-optimal) solution that of equality constraints in problem SPP-SC represent would satisfy certain utilization, manpower availabil- ity, and quality-of-work-life (QWL) requirements such set-partitioning constraints with the coefficients aij of the m × n matrix A being either zero or one, depend- as schedule regularity, inadditionto minimizingcrew ing on whether variable xj participates in constraint i costs that include hotel, per diem, and traditional met- or not, and where e is a vector of ones. The second set rics such as flight-time-credit (FTC), and a variety of 565 Subramanian and Sherali: Effective Deflected Subgradient Optimization Scheme 566 INFORMS Journal on Computing 20(4), pp. 565–578, © 2008 INFORMS other segment and duty-based penalties and incen- group of Boeing, Inc. reports results that demonstrate tives. Vance et al. (1997) provide more details on these their considerable success in solving massively sized crew pairing costs, and Klabjan et al. (2001, 2002) railroad crew scheduling problems using column gen- consider such crew pairing problems with additional erationtechniques(Kohl 2003). regularity and time-window restrictions, as well as Given the nature of the problem and various other include some aspects related to the aircraft-routing business requirements, a sequential planning process problem. has been traditionally followed, starting with a gross- We will restrict the scope of our study inthis paper day approximationwhere every flight is assumed to to the pure set-partitioning problem SPP, whose struc- operate every day, and ending with a final solution ture poses the mainchallengeinsolvinglarge-scale that represents crew pairings that cover each of the instances to near optimality. The current analysis can flight segments by date. However, recent advances be readily extended to deal with the more restricted in computer hardware and new column generation and realistic model SPP-SC. In practice, to main- algorithms and techniques have facilitated obviat- tain feasibility, we add nonnegative slack variables s, ing traditional sequential and suboptimal crew plan- along with a corresponding large positive undercov- ning approaches, and to directly solve the multiob- erage penalty P to the objective function, where P is jective crew scheduling problem in its entirety within typically chosento be considerablylarger thanthe a reasonable amount of computational time. Further- cost of any legal pairing that can be generated for more, this success has enabled the solution of large the givenproblem. This results inthe followingaug- optimizationproblems that arise from integratedair- mented formulation of SPP: line planning models that cross traditional boundaries SPP: Minimize cT x + Ps between different airline business areas. The problem instances we consider in this paper subject to Ax + s = e (2) are from weekly and monthly crew pairing prob- s ≥ 0x∈ Bn lems that pertain to United Airlines’s flight sched- ules. These are large-scale SPP instances that directly We now present a brief summary of relevant crew address a typical week’s abstracted problem or the pairing concepts used in crew scheduling applica- scheduling problem for an entire month. The lat- tions. A crew duty represents an approximate day’s ter choice is made whena large numberof dated worth of work and consists of a legal sequence of exceptions are present in the schedule, which usu- flight segments and flight connections. Layovers repre- ally occurs during holiday seasons or during months sent rest periods between successive duty schedules that contain several scheduling changes. In fact, the and these accrue both crew and hotel costs. A crew intensely competitive nature of the airline industry pairing is a sequence of crew duties, along with any has typically resulted inanincrease inthe frequency intermediate layovers, which begins and ends at the of such demand-driven dynamic scheduling changes same
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