A Stochastic Approach to Solving Fuzzy Constraint Satisfaction Problems

Traditionally, constraint satisfaction problems (CSP's) 1] are so deened that \all the constraints are satissed simultaneously." However, this is not always true. Many CSP's in real-life are \soft CSP's," i.e., an assignment of values to the variables is considered to be a solution even if some constraints are violated. Some of the practical CSP's are fuzzy: they are fully satissed by some value assignments to the variables in the constraint, and they are considered to be \partially" or \less" satissed, instead of \violated," by some other assignments. Sometimes a real-life CSP may consist of a mixture of hard constraints and soft constraints. In these cases we are required to nd assignments that fully satisfy the hard constraints and fully or partially satisfy the soft constraints. A constraint satisfaction problem is deened as a tuple (Z; D; C c). Z is a nite set of variables and D is a nite set of domains one associated with each variable in Z. C c is a set of constraints. Each constraint is a crisp relation among the domains of a subset of the variables in Z. Each constraint restricts the combination of values that these variables can take. The goal of a CSP is to nd a consistent assignment of values to the variables in Z that satisses all the constraints in C c. A fuzzy constraint satisfaction problem (FCSP) is deened as a tuple (Z; D; C f). C f is a set of fuzzy constraints. Each fuzzy constraint is a fuzzy relation among the domains of a subset of the variables in Z. Satisfaction index of a fuzzy constraint tells us to what extent a constraint is satissed. Solution index of an FCSP (Z; D; C f) shows its overall satisfaction. It is based on the satisfaction indexes of all the constraints in C f and obtained by a user-deened function called satisfaction function. Threshold is a user-deened lower bound of the acceptable solution index of an FCSP. The goal of an FCSP (Z; D; C f) is to nd an assignment of values to all variables in Z so that the solution index is not less than the threshold. The diierence between FCSP and CSP lies on the set of constraints they involve. For a CSP (Z; D; C c), the constraints in C c are Boolean. An assignment of a tuple to the variables in C c return …