The Multiple Traveling Salesmen Problem (mTSP) is of the famous and classical problems of research in operations and is accounted as one of the most famous and widely used problems of combinational optimization. Most of the complex problems can be modeled as the mTSP and then be solved. The mTSP is a NP-Complete one; therefore, it is not possible to use the exact algorithms for solving it instead the heuristics methods are often applied for solving such problems.In this paper, a new hybrid algorithm, called GELS-GA, has been presented for solving the mTSP. The utility of GELS-GA is compared with some related works such as GA and ACO and achieves optimality even in highly complex scenarios.Although, the proposed algorithm is simple, it includes an appropriate time of completion and the least traversed distance among existing algorithms.

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The problem of open-shop scheduling includes a set of activities which must be performed on a limited set of machines. The goal of scheduling in open-shop is the presentation of a scheduled program for performance of the whole operation, so that the ending performance time of all job operations will be minimised. The open-shop scheduling problem can be solved in polynomial time when all nonzero processing times are equal, becoming equivalent to edge coloring that has the jobs and workstations as its vertices and that has an edge for every job-workstation pair with a nonzero processing time. For three or more workstations, or three or more jobs, with varying processing times, open-shop scheduling is NP-hard. Different algorithms have been presented for open-shop scheduling so far. However, most of these algorithms have not considered the machine maintenance problem. Whilst in production level, each machine needs maintenance, and this directly influences the assurance reliability of the system. In this paper, a new genetic-based algorithm to solve the open-shop scheduling problem, namely OSGA, is developed. OSGA considers machine maintenance. To confirm the performance of OSGA, it is compared with DGA, SAGA and TSGA algorithms. It is observed that OSGA performs quite well in terms of solution quality and efficiency in small and medium enterprises (SMEs). The results support the efficiency of the proposed method for solving the open-shop scheduling problem, particularly considering machine maintenance especially in SMEs'.

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