Open constraint programming, including open constraint satisfaction (Open COP) and open constraint optimization (Open COP), is an extended constraint programming framework designed to model and solve practical problems with openworld settings. We extend open constraint programming to the Open Constraint-Based Inference (Open CBI) framework based on the unified semiring-based CBI framework. The ...

Convexity theory and duality theory are important issues in math- ematical programming. Within the framework of credibility theory, this paper rst introduces the concept of convex fuzzy variables and some basic criteria. Furthermore, a convexity theorem for fuzzy chance constrained programming is proved by adding some convexity conditions on the objective and constraint functions. Finally,...

Sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers. Here, nonsmooth approximate gradient projection and complementary approximate Karush-Kuhn-Tucker conditions are presented. These sequential optimality conditions are satisfied by local minimizers of optimization problems independently of the fulfillment of constrai...

Constraint programming is rapidly becoming the technology of choice for modeling and solving complex combinatorial problems. However, users of constraint programming technology need significant expertise in order to model their problem appropriately. The lack of availability of such expertise can be a significant bottleneck to the broader uptake of constraint technology in the real world. In th...

One of the most appealing features of constraint programming is its rich constraint language for expressing combinatorial optimization problems. This paper demonstrates that traditional combinators from constraint programming have natural counterparts for local search, although their underlying computational model is radically different. In particular, the paper shows that constraint combinator...

Constraint programming is another form of programming and, as such, should be supported by appropriate environments that support debugging, explanation and optimization. We are building tools needed in such an environment and using them in the context of geometric constraint programming for graphics and mechanical design. In this paper, we present the components of such an environment and their...

The purpose of this article is to model and solve an integrated location, routing and inventory problem (LRIP) in cash-in-transit (CIT) sector. In real operation of cash transportation, to decrease total cost and to reduce risk of robbery of such high-value commodity. There must be substantial variation, making problem difficult to formulate. In this paper, to better fit real life applications ...

Texts in constraint programming are a little like buses. You wait forever for one to come along, and then two come along at once. In this case, there has been a large gap in the market for a theoretical introduction to constraint programming ever since Edward Tsang’s Foundations of Constraint Satisfaction (1993) went out of print.1 Therefore, we are very pleased to see two books written by two ...

The use of predicates to state constraints in Constraint Satisfaction is explained. If, as an alternative, the traditional approach of Integer Programming (IP) is used, it is desirable to model these constraints so as to give as tight a linear programming relaxation as possible. One of the most commonly used predicates is the “all_different” predicate. Different applications of this are describ...