# A priority based unbalanced time minimization assignment problem

## Abstract

This paper discusses a priority based unbalanced time minimization assignment problem which deals with the allocation of *n* jobs to \(m~(<n)\) persons such that the project is executed in two stages, viz. Stage-I and Stage-II. Stage-I is composed of \(n_1(<m)\) primary jobs and Stage-II is composed of the remaining \((n-n_1)\) secondary jobs which are commenced only after Stage-I jobs are completed. Each person has to do at least one job whereas each job is to be assigned to exactly one person. It is assumed that the nature of primary jobs is such that one person can perform exactly one job whereas a person is free to perform more than one job in Stage-II. Also, persons assigned to primary jobs cannot perform secondary jobs. In a particular stage, all persons start performing the jobs simultaneously. However, if a person is performing more than one job, he does them one after the other. The objective of the proposed study is to find the feasible assignment that minimizes the overall completion time (i.e., the sum of Stage-I and Stage-II time) for the two stage implementation of the project. In this paper, an iterative algorithm is proposed that solves a constrained unbalanced time minimization assignment problem at each iteration and generates a pair of Stage-I and Stage-II times. In order to solve this constrained problem, a solution strategy is developed in the current paper. An alternative combinations based method to solve the priority based unbalanced problem is also analysed and a comparative study is carried out. Numerical demonstrations are provided in support of the theory.

## Keywords

Assignment Time minimization Unbalanced Priority Constrained## Notes

### Acknowledgements

We would like to thank the anonymous reviewers for their valuable suggestions and comments which helped in improving the manuscript considerably. The first author is thankful to Council of Scientific and Industrial Research, India (Sanction No. 09/135/(0724)/2015-EMR-I) for providing financial support for carrying out this research.

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