Computer assisted modelling of linear, integer and separable programming problems

For mathematical programming (MP) to have greater impact upon the decision making process, MP software systems must offer suitable support in terms of model communication and modelling techniques . In this paper modelling techniques that allow logical restrictions to be modelled in integer programming terms are described and their implications discussed. In addition it is demonstrated that many classes of non-linearities which are not variable separable may be reformulated in piecewise linear form. It is shown that analysis of bounds is necessary in the following three important contexts: model reduction, formulation of logical restrictions as 0-1 mixed integer programs and reformulation of nonlinear programs as variable separable programs, It is observed that as well as incorporating an interface between the modeller and the optimiser there is a need to make available to the modeller software facilities which support the modelling techniques described here

Linear, integer separable and fuzzy programming problems: a united approach towards automatic reformulation

For mathematical programming (MP) to have greater impact as a decision tool, MP software systems must offer suitable support in terms of model communication and modelling techniques. In this paper modelling techniques that allow logical restrictions to be modelled in integer programming terms are described and their implications discussed. In addition it is demonstrated that many classes of non-linearities which are not variable separable may be after suitable algebraic manipulation put in a variable separable form. The methods of reformulating the fuzzy linear programming problem as a Max-Min problem is also introduced. It is shown that analysis of bounds plays a key role in the following four important contexts: model reduction, reformulation of logical restrictions as 0-1 mixed integer programs, reformulation of nonlinear programs as variable separable programs and reformulation of fuzzy linear programs. It is observed that as well as incorporating an interface between the modeller and the optimiser there is a need to make available to the modeller software facilities which support the model reformulation techniques described here

Modelling and Solving the Stable Marriage Problem Using Constraint Programming

We study the Stable Marriage problem (SM), which is a combinatorial problem that arises in many practical applications. We present two new models of an instance I of SM with n men and n women as an instance J of a Constraint Satisfaction Problem. We prove that establishing arc consistency in J yields the same structure as given by the established Extended Gale/Shapley algorithm for SM as applied to I. Consequently, a solution (stable matching) of I can be derived without search. Furthermore we show that, in both encodings, all stable matchings in I may be enumerated in a failure-free manner. Our first encoding is of O(n^3) complexity and is very natural, whilst our second model, of O(n^2) complexity (which is optimal), is a development of the Boolean encoding in [6], establishing a greater level of structure

A Constraint Programming Approach to the Hospitals / Residents Problem

An instance I of the Hospitals / Residents problem (HR) involves a set of residents (graduating medical students) and a set of hospitals, where each hospital has a given capacity. The residents have preferences for the hospitals, as do hospitals for residents. A solution of I is a stable matching, which is an assignment of residents to hospitals that respects the capacity conditions and preference lists in a precise way. In this paper we present constraint encodings for HR that give rise to important structural properties. We also present a computational study using both randomly-generated and real-world instances. Our study suggests that Constraint Programming is indeed an applicable technology for solving this problem, in terms of both theory and practice

A constraint programming approach to the hospitals/residents problem

An instance I of the Hospitals/Residents problem (HR) involves a set of residents (graduating medical students) and a set of hospitals, where each hospital has a given capacity. The residents have preferences for the hospitals, as do hospitals for residents. A solution of I is a <i>stable matching</i>, which is an assignment of residents to hospitals that respects the capacity conditions and preference lists in a precise way. In this paper we present constraint encodings for HR that give rise to important structural properties. We also present a computational study using both randomly-generated and real-world instances. We provide additional motivation for our models by indicating how side constraints can be added easily in order to solve hard variants of HR