─ 2 ─vicious circle. To prevent this outcome, many companies now shorten working hours by subdividing each the ratio of part-time workers to the total labor force has been increasing.The problem of finding “the most effective way of allocating jobs to workers” when creating a work schedule is called an SSP in mathematical programming. In particular, the nurse scheduling problem (NSP)—creating a work schedule of nurses at a medical facility— has been studied as a typical SSP and the most attractive problem in combinational optimization problems. The model for a real problem has been created such that a method of obtaining the excellent solution under complicated conditions to some extent has been proposed1). As mentioned above, it is also difcult to give concrete gures because the calculation scale depends on a basis.Conventional SSPs represented by NSP have allocated shift patterns such as “day shifts” and “night shifts” to workers on the assumption of full-time duties. In a workplace where part-time workers account for most of the total workforce, different starting times for work and working hours are necessary, so the number of shift patterns is quite large. Moreover, each worker’s limitation for working hours and work pace, including service time zones and total working hours due to their unique circumstances, which were not investigated in previous studies, must be considered. Therefore, working conditions are quite complicated. An SSP can be solved as an optimization problem containing integer variables: the majority of SSPs are NP-hard. Estimating the relationship between the scale of a problem and the time when the computational complexity begins to increase is difficult, however. In the case of handling the same problem, the scale of its solution differs entirely according to its structure, in which numeric values are actually entered. Recently, due to the increase in computing power and the advancement of optimization algorithms, the performances of general-purpose solvers have been improved. As a result, large-scale problems can now be handled. While exact solution methods are seldom applied to large-scale problems with complicated conditions, such as integer programming problems (SSPs), using approximate methods with heuristics for these problems is realistic. Genetic algorithms (GAs) have been used as meta-heuristics. To solve an SSP consisting mainly of part-time workers, In which the size of the method is necessary.3. Hybrid solution methods using swarm intelligenceMany researchers have recently examined swarm intelligence, which is based on the distinctive behaviors of animals and insects. Swarm intelligence is effective in solving a complicated and multivariate optimization problem. An articial bee colony (ABC)2) and rey algorithm (FA)3) are typical examples of swarm intelligence.ABC comprises three phases, based on the behaviors of three articial bee groups of employed, onlooker, and scout. In the phases of employed and onlooker bees, a local search is performed near the solution candidates. In the phase of scout bees, a solution candidate, which is not useful in the progress of the search, is abandoned and a new solution candidate is inserted into the search space to imitate the behavior of scout bees, which abandon an empty food source in foraging behavior.FA was based on rey ashing patterns to perform the search in accordance with the following three regulations: (1) a rey is attracted by another rey; (2) the force of a rey to attract another rey is proportional to the light intensity—a dark rey is attracted by a bright rey and the light intensity decreases as the distance between reies increases; and (3) the light intensity of a rey is determined based on the objective function.These solution methods have attracted the attention of overseas researchers. However, in Japan, there are no cases of applying optimization problems containing integer variables, such as scheduling problems. However, a study was conducted in which ABC and FA were used, and benchmark functions were solved as integer programming problems, although ABC could indicate the possibility of swarm intelligence, FA could not obtain the solution of a certain problem, which satised every constraint. Therefore, challenges remain4).There are solution methods to which swarm intelligence is applicable and inapplicable, to according to the problem. To develop a solution method applicable to a general-purpose model, the characteristics of each solution method should be utilized. Studies have sought to improve the search performance of a basic method by compensating for its defects in the form of combining algorithms in other methods to create a hybrid method, in which a) a swarm intelligence algorithm is combined with an evolutionary algorithm or b) a swarm intelligence algorithm is combined with another swarm intelligence algorithm.An arithmetic crossover-based ABC algorithm (AC-ABC) is a hybrid method in which ABC is combined with GA. In the phases of employed and onlooker bees in ABC, a search is performed by selecting one dimension of each individual at random. As a result, the diversity of individuals in a group can be maintained without falling into a local optima. However, it takes time to search
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