Timely maintenance of degrading components is critical to the continuing operation of a system. By implementing prognostics, it is possible for the operator to maintain the system in the right place at the right time. However, the complexity of real-world operating environment makes near-zero downtime difficult to achieve, partly because of a possible shortage of required service parts. To coordinate with a prognostics-based maintenance schedule, it is necessary for the operator to decide when to order the service parts and how to compete with other operators in service part procurement. In this paper, we investigate a situation where two operators are to make prognostics-based replacement decisions and strategically compete for a service part that both operators need at around the same time. A Stackelberg game is formulated in this context. A sequential, constrained maximin space-filling experimental-design approach is developed to facilitate the implementation of backward induction. This approach is efficient in searching the Nash equilibrium when the follower's best response to the leader's strategies has no closed-form expression. A numerical study on wind turbine operation is provided to demonstrate the use of the joint decision-making tool in solving such complex, yet realistic maintenance and service part logistic problems.