This paper studies a time minimizing assignment problem (TMAP) also known as the bottleneck assignment problem in which out of the given n jobs only n′ (< n) are to be assigned to m (< n′) facilities each of which is constrained to do a minimum number of jobs along with a maximum number that each may do being specified. All facilities start working on the jobs simultaneously but a facility can not work on more than one job at a time and each chosen job is to be assigned to a unique facility. A facility performs the jobs one after the other if it is to do more than one job. A lexi-search approach is proposed to find an optimal feasible assignment of the facilities so as to minimize the total time of completion of n′ out of n jobs.
