Optimal delivery due date for a supplier with an unreliable machine under outsourced maintenance

In this paper, we analyse the due-date-to-promise problem for a supplier who supplies parts to his customer. The parts are produced on an unreliable machine with constant production rate. The machine is leased from a contractor who is also entrusted with its maintenance. The supplier's decision is the optimal due date to promise to his customer, taking into account the holding and tardiness costs on the customer side, as well as the transfer payment to the contractor for the lease and maintenance of the equipment. The contractor in turn has to decide on the frequency of pre-ventive maintenance. Both maximize their own profits. For this finite horizon optimization problem, we employ a Non-Renewal-Theory based approach from the literature to derive the performance characteristics in the transient regime - first for a general setting involving general distributions and then, for a special case. We then use Game Theory to analyse the underlying two-member game for both the non-cooperative and cooperative cases. A numerical example drawn from the literature is used to illustrate the special case. We show that for the special case, the game has a unique Nash equilibrium. Another significant result is that the Nash solution dominates the Stackelberg solution when the supplier is the leader. The supplier is worse off when the contractor is the leader. The example further shows that the supplier and the contractor stand to gain under cooperation which is well documented in the literature. We point to relevant literature for strategies to enforce this cooperation.