Integer Programming: Using Branch and Bound to Solve the Nurse Scheduling Problem

The nurse scheduling problem (NSP) is a complex optimisation problem regarding the allocation of nurses to duty rosters in hospitals. The objective is to ensure that there are sufficient nurses on duty while considering individual preferences with respect to work patterns, requests for leave and financial restrictions, in such a way that all employees are treated equally. In this paper, we extend our novel approach to solving the NSP by transforming it through information granulation. The approach is general enough to be applied within a wide range of benchmark instances and the majority of these instances have real world applications. They have been collected from a variety of sources including industrial collaborators, other researchers and previous publications. Domain transformation is an approach to solving complex problems that relies on a simplification of the original problem. First, the solution to a problem and the refined solution favouring simplification are introduced. The approach we use involves information granulation of shift types to transforms the problem into a smaller solution domain. Next, schedules derived from smaller problem domains are converted into the original problem domain. The conversion takes care of the constraints which were not represented in the smaller domain. The problem is then solved via integer programming (IP). IP is formulated to solve the transformed scheduling problem using the branch and bound (IP-BB) algorithm. We have used the GNU Octave, open source mathematical modelling and simulation software for Windows to solve this problem. We tested the incorporation of (IP-BB) within our proposed methodology to solve the NSP. The (IP-BB) outperforms in most cases other approaches when tested with the different demands and number of nurses. The results facilitated the development of a cost–benefit analysis across different levels of staffing.