We discuss formulations of integer programs with a huge number of variables and their solution by column generation methods, i.e., implicit pricing of nonbasic variables to generate new columns or to prove LP optimality at a node of the branch-and-bound tree. We present classes of models for which this approach decomposes the problem, provides tighter LP relaxations, and eliminates symmetry. We then discuss computational issues and implementation of column generation, branch-and-bound algorithms, including special branching rules and efficient ways to solve the LP relaxation. We also discuss the relationship with Lagrangian duality.

Consumers' attempts to control their unwanted consumption impulses influence many everyday purchases with broad implications for marketers' pricing policies. Addressing theoreticians and practitioners alike, this paper uses multiple empirical methods to show that consumers voluntarily and strategically ration their purchase quantities of goods that are likely to be consumed on impulse and that therefore may pose self-control problems. For example, many regular smokers buy their cigarettes by the pack, although they could easily afford to buy 10-pack cartons. These smokers knowingly forgo sizable per-unit savings from quantity discounts, which they could realize if they bought cartons; by rationing their purchase quantities, they also self-impose additional transactions costs on marginal consumption, which makes excessive smoking overly difficult and costly. Such strategic self-imposition of constraints is intuitively appealing yet theoretically problematic. The marketing literature lacks operationalizations and empirical tests of such consumption self-control strategies and of their managerial implications. This paper provides experimental evidence of the operation of consumer self-control and empirically illustrates its direct implications for the pricing of consumer goods. Moreover, the paper develops a conceptual framework for the design of empirical tests of such self-imposed constraints on consumption in consumer goods markets. Within matched pairs of products, we distinguish relative “virtue” and “vice” goods whose preference ordering changes with whether consumers evaluate immediate or delayed consumption consequences. For example, ignoring long-term health effects, many smokers prefer regular (relative vice) to light (relative virtue) cigarettes, because they prefer the taste of the former. However, ignoring these short-term taste differences, the same smokers prefer light to regular cigarettes when they consider the long-term health effects of smoking. These preference orders can lead to dynamically inconsistent consumption choices by consumers whose tradeoffs between the immediate and delayed consequences of consumption depend on the time lag between purchase and consumption. This creates a potential self-control problem, because these consumers will be tempted to overconsume the vices they have in stock at home. Purchase quantity rationing helps them solve the self-control problem by limiting their stock and hence their consumption opportunities. Such rationing implies that, per purchase occasion, vice consumers will be less likely than virtue consumers to buy larger quantities in response to unit price reductions such as quantity discounts. We first test this prediction in two laboratory experiments. We then examine the external validity of the results at the retail level with a field survey of quantity discounts and with a scanner data analysis of chain-wide store-level demand across a variety of different pairs of matched vice (regular) and virtue (reduced fat, calorie, or caffeine, etc.) product ca...

We present a column-generation model and branch-and-price-and-cut algorithm for origin-destination integer multicommodity flow problems. The origin-destination integer multicommodity flow problem is a constrained version of the linear multicommodity flow problem in which flow of a commodity (defined in this case by an origin-destination pair) may use only one path from origin to destination. Branch-and-price-and-cut is a variant of branch-and-bound, with bounds provided by solving linear programs using column-and-cut generation at nodes of the branch-and-bound tree. Because our model contains one variable for each origindestination path, for every commodity, the linear programming relaxations at nodes of the branch-and-bound tree are solved using column generation, i.e., implicit pricing of nonbasic variables to generate new columns or to prove LP optimality. We devise a new branching rule that allows columns to be generated efficiently at each node of the branch-and-bound tree. Then, we describe cuts (cover inequalities) that can be generated at each node of the branch-and-bound tree. These cuts help to strengthen the linear programming relaxation and to mitigate the effects of problem symmetry. We detail the implementation of our combined columnand-cut generation method and present computational results for a set of test problems arising from telecommunications applications. We illustrate the value of our branching rule when used to find a heuristic solution and compare branch-and-price and branch-and-price-and-cut methods to find optimal solutions for highly capacitated problems.

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Airlines typically construct their schedules assuming that every flight leg will depart and arrive as planned. Because this optimistic scenario rarely occurs, these plans are frequently disrupted and airlines often incur significant costs in addition to those originally planned. Flight delays and schedule disruptions also cause passenger delays and disruptions. A more robust plan can reduce the occurrence and impact of these delays, thereby reducing costs. In this paper, we present two new approaches to minimize passenger disruptions and achieve robust airline schedule plans. The first approach involves routing aircraft, and the second involves retiming flight departure times. Because each airplane usually flies a sequence of flight legs, delay of one flight leg might propagate along the aircraft route to downstream flight legs and cause further delays and disruptions. We propose a new approach to reduce delay propagation by intelligently routing aircraft. We formulate this problem as a mixed-integer programming problem with stochastically generated inputs. An algorithmic solution approach is presented. Computational results obtained using data from a major U.S. airline show that our approach can reduce delay propagation significantly, thus improving on-time performance and reducing the numbers of passengers disrupted. Our second area of research considers passengers who miss their flight legs due to insufficient connection time. We develop a new approach to minimize the number of passenger misconnections by retiming the departure times of flight legs within a small time window. We formulate the problem and an algorithmic solution approach is presented. Computational results obtained using data from a major U.S. airline show that this approach can substantially reduce the number of passenger misconnections without significantly increasing operational costs.

Given a schedule of flight legs to be flown by an airline, the fleet assignment problem is to determine the minimum cost assignment of flights to aircraft types, called fleets, such that each scheduled flight is assigned to exactly one fleet, and the resulting assignment is feasible to fly given a limited number of aircraft in each fleet. Then the airline must determine a sequence of flights, or routes, to be flown by individual aircraft such that assigned flights are included in exactly one route, and all aircraft can be maintained as necessary. This is referred to as the aircraft routing problem. In this paper, we present a single model and solution approach to solve simultaneously the fleet assignment and aircraft routing problems. Our approach is robust in that it can capture costs associated with aircraft connections and complicating constraints such as maintenance requirements. By setting the number of fleets to one, our approach can be used to solve the aircraft routing problem alone. We show how to extend our model and solution approach to solve aircraft routing problems with additional constraints requiring equal aircraft utilization. With data provided by airlines, we provide computational results for the combined fleet assignment and aircraft routing problems without equal utilization requirements and for aircraft routing problems requiring equal aircraft utilization.

T his paper presents an overview of several important areas of operations research applications in the air transport industry. Specific areas covered are: the various stages of aircraft and crew schedule planning; revenue management, including overbooking and legbased and network-based seat inventory management; and the planning and operations of aviation infrastructure (airports and air traffic management). For each of these areas, the paper provides a historical perspective on OR contributions, as well as a brief summary of the state of the art. It also identifies some of the main challenges for future research.

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