Cross trained workers with switching points, switching cost and optional feedback

In this paper a service facility with the front room and the back room has limited number of servers. Two models are analysed. In the first model, given the number of available workers, we determine an optimal policy so that the expected number of customers in queue is minimized subject to a constraint ensuring that there is a sufficient work force to fulfil the functions in the back room. In the second model, the number of workers needed is minimized subject to a additional constraint requiring that the expected number of customers waiting in queue is bounded above by a given threshold value. After the completion of front room service if the customer is unsatisfied with its service for certain reason or if it received unsuccessful service, the customer may immediately join the second optional service and may leave the queue. Exact solution procedures are developed for both models.