The Role of Software in Optimization and Operations Research

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OPTIMIZATION AND OPERATIONS RESEARCH – The Role of Software in Optimization and Operations Research - Harvey J. Greenberg Figure 3: Anatomy of a Model Finally, learning is another level of relations that can be part of an artificially intelligent environment for model formulation and management. It includes learning relations among objects from experience with some scenarios, even without human feedback. One example is when it detects an infeasibility and diagnoses its cause. This could lead to an automatic check for those conditions in future scenarios. The diagram in Figure 3 is simplified. It does not show, for example, whether an object or relation is primitive or implied; it does not show attributes or other elements that require discussion if we were to consider implementation. Its purpose is simply to highlight an anatomy in tune with the role of software in optimization, starting with model formulation. Also, the modeling process contains more than models; it also contains guidance for solvers, model management aids, and rules to support analysis of instances. 4.1. Expressive Power The expressive power of a language is its ability to represent different kinds of objects and relations. Theoretically, the languages in use today have the same expressive power, ©Encyclopedia of Life Support Systems (EOLSS)
OPTIMIZATION AND OPERATIONS RESEARCH – The Role of Software in Optimization and Operations Research - Harvey J. Greenberg but practically, they do differ. Some languages emphasize that they are algebraic, but what does this mean? What languages are not algebraic? Essentially, an algebraic language is one that allows you to express the optimization model the way you would on a blackboard or in a textbook. It is some variant of the following: Let x(i, j, k) denote the level of transportation of the k th material from i to j … There is limited supply: x(, i j, k) S() i ≤ ∑ … jk , Languages like AMPL and GAMS are like this. Older languages, like MAGEN/OMNI and RPMS/GAMMA, are process oriented. They would express activities, like the following: Let x(i, k) denote the level of operation of a distillation unit in region i at mode k … It uses materials M(i, k) and has product yields Y (i, k); capacity is limited by C(i, k). These have been unnecessarily contentious, and now there is another paradigm, based on logical expressions, called logic programming. We commonly refer to constrained logic programming to mean logical expressions of what to do, subject to constraints like: Choose 1 from the set... 4.2. Logic Programming and Optimization There has been a gradual merger of operations research methodology with logic programming. The role of the software is to provide a different paradigm for expressing combinatorial optimization problems, like scheduling. Rather than require the modeler to specify some system of inequalities; constraints, such as job i must precede job j, can be expressed more naturally. To illustrate, consider the traveling salesman problem. There are many integer programming formulations, but constraint logic programming can represent it differently. Let yi be from the set of cities, {1, …, n}. To express the constraint that every city be visited exactly once, the logical constraint is all different {yi}, and that is precisely all one must specify! This is beyond the expressive power of integer programming, adding new capabilities to optimization modeling, as well as new opportunities to express bad formulations. 5. Computer-Assisted Analysis Obtaining a solution is not enough, and we have just described the role of software in formulating and managing models. In addition, the role of optimization software must support analysis. At least in linear programming, we often solve larger problems than we can understand. It is not only post-optimal analysis that is important, but also debugging tools to deal with anomalies and infeasibility diagnosis. This is a underdeveloped area, which began to emerge in the late 1970s. The only general-purpose software currently available are ANALYZE, GMSCHK and MProbe. Some modeling systems, like ASCEND, MIMI and XPRESS-LP, have some analysis aids. ©Encyclopedia of Life Support Systems (EOLSS)
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