An assignment problem (AP) usually deals with how a set of persons/tasks can be assigned to a set of tasks/persons on a one-to-one basis in an optimal manner. It has been observed that balancing among the persons and jobs in several real-world situations is very hard, thus such scenarios can be seen as unbalanced assignment models (UAP) being a lack of workforce. The solution techniques presented in the literature for solving UAP’s depend on the assumption to allocate some of the tasks to fictitious persons; those tasks assigned to dummy persons are ignored at the end. However, some situations in which it is inevitable to assign more tasks to a single person. This paper addresses a practical variant of UAP called k-cardinality unbalanced assignment problem (k-UAP), in which only of persons are asked to perform jobs and all the persons should perform at least one and at most jobs. The k-UAP aims to determine the optimal assignment between persons and jobs. To tackle this problem optimally, an enumerative Lexi-search algorithm (LSA) is proposed. A comparative study is carried out to measure the efficiency of the proposed algorithm. The computational results indicate that the suggested LSA is having the great capability of solving the smaller and moderate instances optimally.

A practical distribution system that arises in the context of delivering liquefied petroleum gas (LPG) through cylinders is considered in this study. To meet all the challenging constraints, the model is explicitly considered as a simultaneous pickup and delivery single commodity truncated vehicle routing problem with the homogeneous fleet of vehicles. The aim of this problem is to find the optimal routes for the set of vehicles locating at the distributing agency (DA), which offers simultaneous pickup and delivery operations over single commodity (i.e. LPG cylinders) to a fixed subset (need not serve all delivery centers) of delivery centers at rural level. The model is designed using zero-one integer linear programming. For proper treatment of the present model, an exact Lexi-search algorithm (LSA) has been developed. A comparative study is performed between the LSA and existing results for the relaxed version of the present model. Further, the efficiency of the LSA is tested through numerical experiments over small and medium CVRP benchmark test instances. The extensive computational results have shown that the LSA is productive and revealed that the real solutions have more consistent than the integral solutions in the presence of truncation constraint.

The multiple travelling salesman problem (MTSP) is one of the widely studied combinatorial optimization problems with various theoretical and practical applications. However, most of the studies intended to deal with classical MTSP, very limited attention has been given to an open multiple travelling salesman problem and its variants. In this paper, an open multiple travelling salesman problem with load balancing constraint (OMTSPLB) is addressed. The OMTSPLB differs from the conventional MTSP, in which all the salesmen start from the central depot and need not come back to it after visiting the given number of cities by accomplishing the load balance constraint, which helps in fairly distributing the task among all salesmen. The problem aims to minimize the overall traversal distance/cost for operating open tours subject to the load balancing constraint. A zero-one integer linear programming (0-1 ILP) model and an efficient metaheuristic genetic algorithm (GA), is established for the OMTSPLB. Since no existing study on OMTSPLB, the proposed GA is tested on the relaxed version of the present model, comparative results are reported. The comparative results show that the proposed GA is competent over the existing algorithms. Furthermore, extensive experiments are carried out on OMTSPLB and the results show that proposed GA can find the global solution effectively.

This paper introduces a novel practical variant, namely an open close multiple travelling salesmen problem with single depot (OCMTSP) that concerns the generalization of classical travelling salesman problem (TSP). In OCMTSP, the overall salesmen can be categorized into internal/permanent and external/outsourcing ones, where all the salesmen are positioned at the depot city. The primary objective of this problem is to design the optimal route such that all salesmen start from the depot/base city, and then visit a given set of cities. Each city is to be visited precisely once by exactly one salesman, and only the internal salesmen have to return to the depot city whereas the external ones need not return. To find optimal solutions, an exact pattern recognition technique based Lexi-search algorithm (LSA) is developed which has been subjected in Matlab. Comparative computational results of the LSA have been made with the existing methods for general multiple travelling salesman problem (MTSP). Further, to test the performance of LSA, computational experiments have been carried out on some benchmark as well as randomly generated test instances for OCMTSP, and results are reported. The overall computational results demonstrate that the proposed LSA is efficient in providing optimal and sub-optimal solutions within the considerable CPU times.

In open travelling salesman subset-tour problem (OTSSP), the salesman needs to traverse a set of k (≤n) out of n cities and after visiting the last city, the salesman does not necessarily return to the central depot. The goal is to minimize the overall traversal distance of covering k cities. The OTSSP model comprises two types of problems such as subset selection and permutation of the cities. Firstly, the problem of selection takes place as the salesman’s tours do not contain all the cities. On the other hand, the next problem is about to determine the optimal sequence of the cities from the selected subset of cities. To deal with this problem efficiently, a hybrid nearest neighbor technique based crossover-free Genetic algorithm (GA) with complex mutation strategies is proposed. To the best of the author’s knowledge, this is the first hybrid GA for the OTSSP. As there are no existing studies on OTSSP yet, benchmark instances are not available for OTSSP. For computational experiments, a set of test instances is created by using TSPLIB. The extensive computational results show that the proposed algorithm is having great potential in achieving better results for the OTSSP. Our proposed GA being the first evolutionary-based algorithm that will help as the baseline for future research on OTSSP.