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Algebraic languages are at the heart of many successful optimization modeling systems, yet they have been used with only limited success for combinatorial (or discrete) optimization. We show in this paper, through a series of examples, how an algebraic modeling language might be extended to help with a greater variety of combinatorial optimization problems. We consider specifically those problems that are readily expressed as the choice of a subset from a certain set of objects, rather than as the assignment of numerical values to variables. Since there is no practicable universal algorithm for problems of this kind, we explore a hybrid approach that employs a general-purpose subset enumeration scheme together with problem-specific directives to guide an efficient search.

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T his paper presents a series of simple but realistic examples in which common algebraic indexing conventions are not so convenient. In particular, it analyzes the difficulties caused by the conventions that each model component must have a fixed number of indices and that the order of the indices is significant to their meaning. To deal with these difficulties compensating extensions to algebraic notation are proposed. The proposed notation is compared to existing notation in terms of the human abilities to understand, maintain and verify model descriptions.{Mathematical Programming; Modeling Language; Index Sets)