A Hybrid Method for the Modelling and Optimisation of Constrained Search Problems

Abstract

The paper presents a concept and the outline of the implementation of a hybrid approach to modelling and solving constrained problems. Two environments of mathematical programming (in particular, integer programming) and declarative programming (in particular, constraint logic programming) were integrated. The strengths of integer programming and constraint logic programming, in which constraints are treated in a different way and different methods are implemented, were combined to use the strengths of both. The hybrid method is not worse than either of its components used independently. The proposed approach is particularly important for the decision models with an objective function and many discrete decision variables added up in multiple constraints. To validate the proposed approach, two illustrative examples are presented and solved. The first example is the authors’ original model of cost optimisation in the supply chain with multimodal transportation. The second one is the two-echelon variant of the well-known capacitated vehicle routing problem.

A declarative decision support framework for supply chain problems

“…Unfortunately, high complexity and the multiple types of constraints of decision-making models as well as combinatorial nature contribute to poor efficiency of modeling in OR methods and inefficient optimization in CLP. Therefore, a new approach to modeling and solving such problems was developed [7,8,9,10]. A declarative environment was chosen as the best structure for this approach especially in modeling [5].…”

“…Instead of analyzing all possible trnasport connections from the factory to the distribution center and then from the center to the customer, only the feasible connections (factory-center-customer) were generated and named routes. This resulted in the removal of certain indices and in the aggregation of other indices for decision variables, parameters, etc., which eventually led to the reduction in the number of decision variables and constraints [7,8]. The new set of decision variable, constraints and facts was the basis for creating the CLP T model.…”

A declarative decision support framework for scheduling groups of orders

“…Unfortunately, high complexity and the multiple types of constraints of decision-making models as well as combinatorial nature contribute to poor efficiency of modeling in OR methods and inefficient optimization in CLP. Therefore, a new approach to modeling and solving such problems was developed [6,7,8]. A declarative environment was chosen as the best structure for this approach especially in modeling [3,5,9,10].…”

“…The term of delivery to the point a defines the constraint (5). Constraint (6) ensures that the number of available resources k in period g is not exceeded. Constraint (7) provides resource occupancy for the time of the order execution.…”

“…Based on them, the primary model of the problem is generated as a CLP model, which is then presolved. The built-in CLP method (constraint propagation [13]) and the method of problem transformation designed by the authors [6,9] (Section 3A) are used for this purpose. Presolving procedure results on the transformed model CLP T .…”