Stable Solutions Dealing with Dynamics in Scheduling Based on Dynamic Constraint Satisfaction Problems

The real-life scheduling problems are often over-constrained, and there is often one intractable case where unexpected events are added and a sudden change is required. In this paper, we describe such problems as the Dynamic Valued Constraint Satisfaction Problem (DyVCSP). In DyVCSP, although the previous schedule would be modified when there is some change, the new schedule should be similar to the current one. We propose a method for achieving solution stability, which maintains portions of the current schedule using the provisional soft constraint. The provisional constraint ensures each variable keeps its value as much as possible. In this paper, we formalize the nurse scheduling problem as DyVCSP and use our method to achieve solution stability. 1 Nurse Scheduling as a Dynamic VCSP In a DyVCSP, the nurse scheduling problem could be defined as a sequence of VCSPs, each of which represents the problem at each time step. Each VCSP is changed to the next VCSP by loss or gain of values, variables, or constraints. A VCSP at a time step i is defined by VP i = (Xi, Di, Ci, S, φ), where Xi = {x(1,1), x(1,2), ..., x(s,t), ...}. x(s,t) is the working shift of nurse s on day t. D i is the set of domain. d(s,t) is the domain of x(s,t). On the nurse roster, d(s,t) is {free,morning, evening, night}. S is the valuation structure defined by E = [0, 9], =>, ⊥ = 0, = 9, ⊗ = +, and the valuation function φ is simply summation of values. C i is the set of constraints. The constraints are described in the following form: constraint(lower lim, upper lim, assignment list, weight). This constraint is satisfied if the number of elements in the current assignment corresponding to those in the assignment list are more than the lower lim and less than the upper lim. This is an example of a personal constraint: constraint(1, 3, {x(s,1) = morning, ..., x(s,t) = morning, ...}, 4) This is satisfied if the number of corresponding elements between the current assignment and the assignment list is 1 to 3. Then, a DyVCSP could be described as follows: DP = {VP0,VP1, ...,VPi, ...} The problem with solution stability [1, 2] in DyVCSP can be defined as the problem of sequentially computing a solution for each of the VCSPs {VP 0,VP1, ...,VP i, ...} given some existing static constraint satisfaction algorithms. 2 Re-scheduling with Dynamic VCSP We focus on a case where a nurse suddenly needs to change his/her schedule and rescheduling is needed. As mentioned above, we deal with the solution stability. In this paper, the solution stability is defined as follows: solution stability : The results after re-scheduling should be similar to those of previously completed scheduling because users are confused if there are extensive changes. In our method for solution stability, we introduce a provisional constraint. The provisional constraint is used to maintain the previous value of each variables as much as possible. For example, when v(i,j) is assigned to the variable x(i,j) in the previous problem, the provisional constraint which is used to keep the value v (i,j) is: constraint(1, 1, {x(i,j)=v(i,j)}, w) where, w is the weight of the provisional constraint. The value of w is predefined. The process of re-scheduling for achieving solution stability is as follows: Step 1: New constraints for sudden requests are generated and the constraints Cnew are added to current problem VP i. The problem then changes VP i to VPi+1. The new constraints Cnew is the cause of re-scheduling. Step 2: The provisional constraints C i prov to maintain the current solution are generated and added to the set of provisional constraints Cprov, which consists of all provisional constraints. Cprov is as follows: