Proof of Property 1

1

APPENDICES.

Appendix-A

Proof of Property 1:

When and we have: . Note that is increasing on its whole domain, and thus is convex in .

For , we have:and.

Similarly, for , we have: and .

Therefore is convex in .

Proof of Property 2:

In order to prove the joint convexity, we need to show that the Hessian matrix of is positive semidefinite.

since , therefore is jointly convex in and .

The following solution procedure uses the Properties 1 and 2 of in order to find the single best capacity level for a given base stock level and find the best for a given capacity level iteratively. The results of the numerical study that is conducted are presented in the main body of the article. The optimal costs for problem (1) are obtained from the Search Procedure-I.

Search Procedure-I

1. Let = denote the that satisfies: . Then we have = and .

2. For n=1,2,…. do while , for small enough

a. Compute which is the that satisfies: .

b. Compute which is the that satisfies: . Then we have =

3. Take the

Appendix-B

Theorem 1: The underlying MDP in (6) is communicating for .

Proof:Given , , is the probability that the state will be in the next period given the current state is and action is . Suppose that in the policy, is chosen .

For , we know that involves all possible sample paths. Therefore, is greater than the probability of a specific path, in which no new defective unit has arrived and an exact number of () defective units are repaired. Hence we have:

and

Similarly, for , we know that is greater than the probability of a specific unique path, in which no new defective unit has been repaired and () defective units have arrived. Hence we have:

and

Hence, it is shown that s.t. , therefore the MDP in (9) is communicating which implies that it is also weakly communicating (Puterman 1994).

Theorem 2:

Under average cost criteria, an optimal deterministic policy exists for problem (9)

Proof: The reader is referred to Theorem 8.3.2 and 8.5.3 from (Puterman 1994), which show the existence of a deterministic optimal policy in (weakly) communicating MDPs.

Appendix-CAccuracy of Finite Waiting Room Approximation

The accuracy of our finite waiting room approximation is examined by comparing the total cost rate of the proposed analytical model (having a finite waiting room of 40) with the cost rate obtained by simulating the real environment having a repair shop that has an infinite waiting room. In our simulations, we used a run length of defective unit arrivals (when ) in a single replication, where the average total cost rate is calculated under a policy: .

We investigated a total of different scenarios with different and and resulting optimal policy parameters from Search Procedure-II. The percentage error, , of using the approximation for in a scenario can be found as:

Here, is the total relevant costs obtained from the simulation and is the total relevant costs obtained from the analytical model.

Table C1 Accuracy of the approximation for the values

Table C1 summarizes the accuracy of the approximations. This table shows the absolute value, minimum, median and the maximum for the percentage errors are listed, respectively. We can see that the approximation can mimic the performance of the original, infinite waiting room environment almost perfectly, which demonstrates the accuracy of our method.

The accuracy holds even for very high traffic ratios (when ), because, under two-level policy, the capacity is adapted according to the workload and even though the average deployed capacity is very close to , the number of defective parts waiting never accumulates close to , because the it is already switched to a higher level before the workload reaches to jeopardizing levels. The results of the simulation study are explicated further in the end of this section.