Method and apparatus for generating profile of solutions trading off number of activities utilized and objective value for bilinear integer optimization models

Abstract

A method (and system) of generating at least one of a solution and a profile of solutions for a problem, includes trading off a reduction of an objective of the problem against a number of activities utilized in a solution.

Description

BACKGROUND OF THE INVENTION [0001] 1. Field of the Invention [0002] The present invention generally relates to a method and apparatus for integer linear programming, and more particularly to a method and apparatus for generating a profile of solutions that trades off a number of activities utilized against an objective value for a bilinear integer optimization model. The bilinear integer optimization model seeks to determine the activities (encoded by integers) and the integer utilizations of these activities, so as to minimize the objective value. For clarity, the exemplary method (and system) is developed and discussed in an exemplary case of a cutting stock problem. In this case, a profile of solutions is generated to trade off the number of patterns used to cut raw units of stock material while minimizing the amount of material wasted. [0003] 2. Description of the Related Art [0004] When attempting to solve a specific problem, it is desirable to provide a profile of possible sol...

AI Technical Summary

Benefits of technology

[0012] It is another exemplary feature to provide a method and system that limits the number of distinct activities utilized in the solution, which will reduce the overall cost of the solution.

[0013] It is another exemplary feature to provide a method and system wherein it is possible to generate new activities while the profile of the solution is being generated.

[0033] The method (and system) of the present invention also includes combinatorially-motivated constraints to tighten the formulation. The present method relies on the solution of a sequence of integer linear programs. Standard methods for solving such problems rely on solving further sequences of linear programming relaxations (where we ignore the integer restrictions). The efficiency of the entire process depends on how well the linear programming relaxation approximates the integer program (in terms of optimal objective value and nearness of the feasible region of the linear programming relaxation to the smallest convex set that contains the integer feasible points). Tightening the formulation refers to adjoining further linear inequalities that make the linear programming relaxation a sharper approximation of the integer program. Additional linear constraints are provided to eliminate symmetry between the new activities, and to preclude generating new activities that are already included in the set of known activities.

[0034] Additionally, the method (and system) of the present invention includes the facility to dynamically change the number of bits for representing utilizations of new activities. As new activities are generated, their optimal utilizations tend to decrease as we allow the total number of activities to increase. Moreover, as the number of activities increases, the difficult of the integer linear programs increase. Computational efficiencies can be realized be allowing the number of bits to vary (usually decrease) dynamically.

[0035] The present invention addresses these and other shortcomings of the conventional methods by formulating a holistic model that simultaneously seeks a small number of new activities, their utilizations and the utilizations of the known set of activities. That is, the unified model of the present invention generates new patterns in situ. Another benefit of the present invention is that it solves a sequence of models in a local-search framework which is ideal for working with constraints that are not modeled effectively in an integer linear programming (ILP) setting, but are relatively simple (e.g., a limit on the number of activities utilized), and (2) is well suited to efficiently generating a profile of solutions trading off a secondary objective (e.g., the number of distinct activities utilized) versus a primary objective.

[0036] With the above and other unique and unobvious exemplary aspects of the present invention, it is possible to balance the quality of a solution and the speed in which the solution is generated.

Problems solved by technology

In the cutting stock problem, a user has a supply of stock rolls.

Additionally, the cutting machines used to cut the stock material have a limited number of knives, which limits the type of patterns that may be utilized.

It is disadvantageous to provide the user with a single solution to the cutting stock problem, or any other problem.

In the cutting stock problem, it is important to limit the number of distinct cutting patterns, because there is a changeover cost associated with setting up the stock cutting machine to cut a different pattern.

Since the user desires an integer solution to the master problem, prices are not sufficient to describe the optimal coverage of demand based on known activities to the subproblem.

Moreover, this method does not lend itself to efficiently providing a profile of solutions with the desired tradeoff.

The Branch-and-Price method tends to provide good solutions, but because of the substantial computational requirements, can be a very slow process.

On the other hand, the Gilmore and Gomory process is a fast process, but may not provide good overall solutions.

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