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.

Abstract. We present an algorithm for the binary cutting stock problem that employs both column generation and branch-and-bound to obtain optimal integer solutions. We formulate a branching rule that can be incorporated into the subproblem to allow column generation at any node in the branch-and-bound tree. Implementation details and computational experience are discussed.

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On major domestic railroads, a typical general merchandise shipment may pass through many classification yards on its route from origin to destination. At these yards, the incoming traffic, which may consist of a number of individual shipments, is reclassified (sorted and grouped together) to be placed on outgoing trains. Each reclassification incurs costs due to handling and delay. To prevent shipments from being reclassified at every yard they pass through, several shipments may be grouped together to form a block. A block has associated with an origin–destination pair that may or may not be the origin or destination of any of the individual cars contained in the block. The objective of the railroad blocking problem is to choose which blocks to build at each yard and to assign sequences of blocks to deliver each shipment to minimize total mileage, handling, and delay costs. We model the railroad blocking problem as a network design problem in which yards are represented by nodes and blocks by arcs. Our model is intended as a strategic decision-making tool. We develop a column generation, branch-and-bound algorithm in which attractive paths for each shipment are generated by solving a shortest path problem. Our solution approach is unique in constraining the classification resources of each yard and simultaneously solving for different priority classes of shipments. We implement our algorithm and find near-optimal solutions in about one hour for the blocking problem of a large domestic railroad, in which the paths that shipments may take in the physical network are restricted. The resulting network design problem has 150 nodes, 1300 commodities, and 6800 possible arcs (blocks). We test the robustness of our solution on 19 test instances that are variations of the data for the real-world problems. If shipments are restricted to following one of a limited number of paths in the rail network, then, in four hours or less, our algorithm finds solutions within 0.4% of optimal for all test cases. Furthermore, the solutions obtained are no more than 3.9% from optimal even if all possible paths are allowed.

In this study, we formulate the railroad blocking problems as a network design problem with maximum degree and flow constraints on the nodes and propose a heuristic Lagrangian relaxation approach to solve the problem. The newapproach decomposes the complicated mixed integer programming problem into two simple subproblems so that the storage requirement and computational effort are greatly reduced. A set of inequalities are added to one subproblem to tighten the lower bounds and facilitate generating feasible solutions. Subgradient optimization is used to solve the Lagrangian dual. An advanced dual feasible solution is generated to speed up the convergence of the subgradient method. The model is tested on blocking problems from a major railroad, and the results show that the blocking plans generated have the potential to reduce the railroad's operating costs by millions of dollars annually.