─ 3 ─for a decent solution. In AC-ABC, by using arithmetic crossover instead of search by onlooker bees, a search is performed across multiple dimensions and the searching speed is accelerated5).In a global search type articial bee colony (GS-ABC), in which ABC is combined with GA, arithmetic crossover is not used, unlike AC-ABC; but the mutation probability α, the movement of which is similar to that of the mutation rate of GA, is introduced into the search of employed and onlooker bees. In the phases of employed and onlooker bees in the original ABC, local search is performed near solution candidates using Formula (1).vij＝xij＋φij・（xij－xkj）……………………………………(1)（i＝1, 2, ..., NF）In this formula, NF represents the number of food sources, φij represents the uniform random number of [-1,1], k is a food source for bees other than randomly selected i, which is selected at random, and j represents a dimension, which is selected at random. In the comparison of vij with xij, when the evaluation by vij is better than that by xij, xij is renewed by vij.In GS-ABC, all the dimensions of individuals are searched using Formula (2) instead of Formula (1).vij＝｛xij＋φij・（xij－xkj） if γij＜α xij others……………………(2)（i＝1, 2, ..., NF, j＝1, 2, ..., D）In this formula, D represents the number of dimensions, γij represents the uniform random number of [0,1], and α represents the search parameter of 0.0‒1.0.By using the mutation probability α, a movement similar to the mutation rate of GA, the diversity of a population can be highly maintained. By selecting a large number of dimensions to be searched at one time using the search parameter, the convergence performance is improved6). In overseas countries, a method in which the arithmetic crossover of GA is incorporated into ABC has been proposed. In a crossover based ABC algorithm (CbABC), the phase of arithmetic crossover is added after search by employed bees (before search by onlooker bees) in order to accelerate the searching speed7). A hybrid method has been proposed, in which ABC is combined with differential evolution (DE)8).DE sometimes falls into a local solution in the early stage. Although ABC is difcult to fall into a local solution, its search takes a long time. In particular, the searching speed decreases in the late stage of calculation. In the hybrid differential articial bee colony algorithm (HDABCA), which is a hybrid of ABC and DE, (1) initialization, (2) search by employed bees, (3) search by onlooker bees, and (4) search by scout bees are completed using ABC, and (5) the best food source is renewed. Before returning back to (2), the best tness is delivered to DE as the initial value, and search is performed multiple times which are equivalent to the number of individuals.Specically, a mutant vector is created at each search point in the mutation section, which corresponds to mutation in GA, the candidate vector of each search point is created in the crossover section, which corresponds to the arithmetic crossover in GA, the created candidate vectors are selected, the termination condition is judged similar to ABC and returns back to (2)9).A hybrid method in which a swarm intelligence algorithm is combined with another swarm intelligence algorithm has been studied. In a modied articial bee colony (MABC), in which ABC is combined with FA, search by employed bees, search by onlooker bees, and search by scout bees are improved based on ABC in order to accelerate the searching speed. In MABC, a formula used for search by employed bees and a formula used for search by onlooker bees are modied to use the value of the best food source. Moreover, a formula, which is similar to the movement formula in FA and considers the distance between a scout bee and the position of a food source with a better value, is used for search by scout bees10).4. Method to be proposedIn almost all previous studies, hybrid methods were veried using benchmark functions of linear programming problems. Some studies conducted in overseas countries used hybrid methods to solve a traveling salesman problem (TSP). However, only a few studies applied hybrid methods to integer programming problems (SSPs, or Integer programming problem represented by SSPs). In ABC, one variable (dimension) is randomly selected in one search and recalculation (search) is performed only for the randomly selected variable. Therefore, ABC takes time. In previous studies, to compensate for the defect of ABC, multiple dimensions were selected using GA or an algorithm similar to GA. By selecting multiple dimensions as search targets at once using search parameters, many solution methods improved their performances to reach optimum solutions.The solution of an SSP (integer programming problem) is more difcult to obtain than that of a linear programming problem. Therefore, the validity of a method to solve an SSP must be veried.Determining the suitable solution method for a general-purpose model is difcult. Whether hybrid methods, which were previously studied in Japan and elsewhere, are suitable
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