Determination of optimal reserve inventory between machines in series with stochastic breakdown durations having scbz property

The determination of optimal inventory theory several variations of inventory problem are discussed. The objective is to derive the optimal re-order level or optimal stock level. In this paper the optimal reserve inventory between two machines in series attempted. The output of machine M1 is the input for machine M_2^a and〖 M〗_2^b. Whenever the breakdown of the machine M1 is at fault i.e not working, then the machine M_2^a and M_2^b may go to an idle stage. The ideal time of M_2^a and M_2^b is very expensive and hence to prevent the ideal time of M_2^a and M_2^b a reserve inventory is maintained in between M1 and M_2^a, M_2^b. The breakdown durations τ of machine M1 is a random variable. This random variable is assumed to possess the called Setting the Clock Back to Zero (SCBZ) property for two random variables namely the breakdowns duration and interarrival times between repairs of M1 the optimal reserve inventory size is obtained. Numerical illustration also substantiated.