Multilevel rationing policy for spare parts when demand is state dependent

Abstract

The multilevel rationing (MR) policy is the optimal inventory control policy for single-item M / M / 1 make-to-stock queues serving different priority classes when demand rate is constant and backlogging is allowed. Make-to-repair queues serving different fleets differ from make-to-stock queues because in the setting of the former, each fleet comprises finitely many machines. This renders the characterization of the optimal control policy of the spare part inventory system difficult. In this paper, we implement the MR policy for such a repair shop/spare part inventory system. The state-dependent arrival rates of broken components at the repair shop necessitate a different queueing-based solution for applying the MR policy from that used for make-to-stock queues. We find the optimal control parameters and the cost of the MR policy; we, then compare its performance to that of the hybrid FCFS and hybrid priority policies described in the literature. We find that the MR policy performs close to the optimal policy and outperforms the hybrid policies.

Appendices

Appendix A: Proofs

Proof of Theorem 1

Equation (5) is a direct result of the two possible trajectories the inventory level can follow starting from state $L_k-1$ until reaching state $L_k$ for the first time.

Fig. 4

A sample path of the interruption time for class k

As seen in Fig. 4, each time the inventory moves from $L_{k-1}$ to $L_{k-1}+1$, with probability $Q_{L_\mathbf {{k-1}}+1,L_\mathbf {{k-1}}}$ ($Q_{L_\mathbf {{k-1}}+1,L_k}$), the inventory level, before reaching $L_{k}$, returns to state $L_{k-1}$ in $T_{L_\mathbf {{k-1}}+1,L_\mathbf {{k-1}}}$ units, and another subcycle of length $T_u$ starts (the inventory level reaches $L_k$ ending $T_u$ in $T_{L_\mathbf {{k-1}}+1,L_k}$ time units in a last cycle). This gives us Eq. (6). Since all states of the underlying birth-and-death process are recurrent, the system goes through a random but a finite number of subcycles, each one of length $T_u$.

Finally, in Eq. (7), $Q_{L_\mathbf {{k-1}}+1,L_\mathbf {{k-1}}}$ is the probability of reaching (the absorbing) state $L_{k-1}$ from state $L_{k-1}+1$ before reaching (the absorbing) state $L_{k}$ in a Gambler’s ruin problem. $\square$

Proof of Corollary 1

We make the following analogy between the original system and the $M/M/1//N_{k-1}+1$ queue: When the inventory level hits $L_{k-1}$ for the first time, there are $N_{k-1}$ operational machines in the original system and the server is busy (one customer out of $N_{k-1}$+1 customers initiates a busy period in the $M/M/1//N_{k-1}+1$ queue). An arrival of classes 1 to $k-2$ drops the inventory level at a rate of $\Lambda _{k-2}$ in the original system (the server fails in the $M/M/1//N_{k-1}+1$ queue at rate $\Lambda _{k-2}$), and it takes $D_{k-1}$ time units before the inventory reaches $L_{k-1}$ again (before the server interruption ends in the $M/M/1//N_{k-1}+1$ queue). During this time, each type $k-1$ machine may fail at a rate of $\lambda _{k-1}$ (additional customers, each with a rate of $\lambda _{k-1}$, may arrive at the $M/M/1//N_{k-1}+1$ queue). When any down machines in the original system (if there are down machines) are supplied with a fixed component while the inventory level is at $L_{k-1}$ and one more component is fixed (corresponding to having all $N_{k-1}$+1 customers out of the $M/M/1//N_{k-1}+1$ queue), $T_{L_\mathbf {{k-1}},L_\mathbf {{k-1}}+1}$ (the busy period in the $M/M/1//N_{k-1}+1$ queue) ends. The moments of the busy period in the $M/M/1//N_{k-1}+1$ queue, hence those of $T_{L_\mathbf {{k-1}},L_\mathbf {{k-1}}+1}$, can be found in Sahba, Balcıog̃lu, and Banjevic (2013). $\square$

Proof of Theorem 2

We introduce the following events and r.v.s to present the proof:

$A_{i,j}$ :

The event of reaching state j from state i in a single step of transition,

$A_{i,\circ ,k}$ :

The event of eventually reaching state $k=0,m$ after exiting state i,

$X_i$ :

The time to reach state 0 or m from state i ($X_k=0$ for $k=0,m$).

Let I(E) denote the indicator function which equals 1 if event E is true and 0 otherwise. Then,

$$\begin{aligned} X_i=X_i I(A_{i,\circ ,0})+X_iI(A_{i,\circ ,m}), i \ne 0,m. \end{aligned}$$

Exiting state i, the system can be in any state after the first transition, thus implying that $\sum _k I(A_{i,k})=1$. Let the random variables $X'_k$ and $X_k$ be independent and identically distributed ($k \ne 0,m$ and $X'_0=X'_m=0$). Then,

$$\begin{aligned} X_i=\sum _k{X_i I(A_{i,k})} = \sum _k{(Y_i+{X'}_k)I(A_{i,k})}=Y_i \sum _{k}{I(A_{i,k})} + \sum _{k \ne 0,m}{I(A_{i,k}){X'}_k}. \end{aligned}$$

If the first state entered after leaving state i is either 0 or m, the remaining time to reach state 0 is zero. Otherwise, it is

$$\begin{aligned} X_iI (A_{i,\circ ,0})=Y_iI(A_{i,\circ ,0})+\sum _{k\ne 0,k\ne m}I(A_{i,k})X'_k I(A'_{k,\circ ,0}). \end{aligned}$$

By definition, $\overline{L}^{(n)}_i=E[X_i^n|A_{i,\circ ,0}]=E[X_i^nI (A_{i,\circ ,0})]/Q_i$ (recall that $Q_i$ is the probability of $A_{i,\circ ,0}$ being true). Using the fact that for any random variable $X_i$ and disjoint events $B_i$, $[I(B_i)]^n=I(B_i)$ and $[\sum _i X_i I(B_i)]^n=\sum _i X^n_i I(B_i)$, and that in our case, $I(A_{i,\circ ,0}) I(A_{i,k}) I(A'_{k,\circ ,0})=I(A_{i,k}) I(A'_{k,\circ ,0})$ for $k \ne 0,m$, we have

$$\begin{aligned} E[\left( X^n_i I (A_{i,\circ ,0})\right) ]= & {} E[\left( X_iI (A_{i,\circ ,0})\right) ^n]\\= & {} E\left[ \left( Y_iI(A_{i,\circ ,0})+ \sum _{k\ne 0,k\ne m}X'_kI(A_{i,k})I(A'_{k,\circ ,0})\right) ^n\right] \\= & {} E[Y_i^n]E[I(A_{i,\circ ,0})] + \sum _{k\ne 0,k\ne m}E[I(A_{i,k})]E[\left( (X'_k)^n I(A'_{k,\circ ,0})\right) ] \\&+\sum ^{n-1}_{l=1}\left( \begin{array}{c} n \\ l \end{array} \right) \left( E[Y_i^l] \sum _{k\ne 0,k\ne m} E[{X'_k}^{n-l}I(A_{i,k})I(A'_{k,\circ ,0})]\right) . \end{aligned}$$

Note that $E[I(A_{i,\circ ,0})]=Q_i$ and $E[I(A_{i,k})]=p_{i,k}$. Also,

$$\begin{aligned} E[{X'_k}^{n-l}I(A_{i,k})I(A'_{k,\circ ,0})]= & {} E[{X'_k}^{n-l}I(A'_{k,\circ ,0})|A_{i,k}]P(A_{i,k})\\= & {} E[{X_k}^{n-l}I(A_{k,\circ ,0})]p_{i,k}. \end{aligned}$$

Then,

$$\begin{aligned} E[\left( X_i^n I (A_{i,\circ ,0})\right) ]= & {} E[Y_i^n]Q_i + \sum _{k\ne 0,k\ne m}p_{i,k}E[X_k^n|A_{k,\circ ,0}]Q_k \\&+\sum ^{n-1}_{l=1}\left( \begin{array}{c} n \\ l \end{array} \right) \left( E[Y_i^l]\sum _{k\ne 0,k\ne m}p_{i,k}E[X_k^{n-l}|A_{k,\circ ,0}]Q_k\right) . \end{aligned}$$

Dividing both sides by $Q_i$ yields Eq. (8). $\square$

Proof of Corollary 2

Consider the birth-and-death process capturing the changes of the inventory level between levels $L_{k}$ and $L_{k-1}$. This process has $m(=L_k-L_{k-1})+1$ states. If we consider the time it takes until the inventory level reaches $L_k$ (to be interpreted as state 0) before hitting $L_{k-1}$ (to be interpreted as state m) starting from the inventory level $L_k-1, L_k-2,\ldots ,L_{k-1}+1$ (to be interpreted as states $1,\ldots ,m-1$, respectively), from Eq. (8), we get Eqs. (9) and (10). The probabilities $Q_i$ and $p_{i,i-1}$ follow similarly. The duration in each state follows an exponential distribution with rate $\mu +\Lambda _{k-1}$; hence, we have Eq. (11). $\square$

Proof of Corollary 3

The proof is similar to that of Corollary 2. We consider the time it takes until the inventory level hits $L_{k-1}$ (to be interpreted as state 0) before reaching $L_k$ (to be interpreted as state m), starting from the inventory level $L_{k-1}+1, L_{k-1}+2,\ldots , L_k-1$ to be interpreted as states $1,\ldots ,m-1$, respectively. $\square$

Proof of Corollary 4

From Eq. (8), the system of equations for the first moment of the absorption time r.v. from state i is

$$\begin{aligned} \overline{L}^{(1)}_i=E(Y_i)+\sum _{k\ne 0,k\ne m}\frac{Q_k}{Q_i}p_{i,k}\overline{L}^{(1)}_k, \,1 \le i \le m-1, \end{aligned}$$

which is used alongside Eq. (8) to obtain the system of equations for the second moment as

$$\begin{aligned} \overline{L}^{(2)}_i=E(Y^2_i)+2E(Y_i)\left( \overline{L}^{(1)}_i-E(Y_i)\right) +\sum _{k\ne 0,k\ne m}\frac{Q_k}{Q_i}p_{i,k}\overline{L}^{(2)}_k, \,1 \le i \le m-1. \end{aligned}$$

Using the previous two equations together with Eq. (8), the system of equations for the third moment is

$$\begin{aligned} \overline{L}^{(3)}_i= & {} E(Y^3_i)+\left( 3E(Y^2_i)-6E^2(Y_i)\right) \left( \overline{L}^{(1)}_i-E(Y_i)\right) \\&+\,3E(Y_i)\left( \overline{L}^{(2)}_i-E(Y^2_i)\right) + \sum _{k\ne 0,k\ne m}\frac{Q_k}{Q_i}p_{i,k}\overline{L}^{(3)}_k,\\&1\le i \le m-1. \end{aligned}$$

Let m be $L_k-L_{k-1}$ and $P_u=1-P_d=\mu /(\mu +\Lambda _\mathrm{k-1})$. Then, any of the above equations can be rewritten for $n=1,2,3$ as

$$\begin{aligned} \overline{L}^{(n)}_1= & {} C^{(n)}_1+P_uH^{-1}_2\overline{L}^{(n)}_2,\\ \overline{L}^{(n)}_i= & {} C^{(n)}_i+P_dH_i\overline{L}^{(n)}_{i-1}+P_uH^{-1}_{i+1}\overline{L}^{(n)}_{i+1},\,\,\,\,1<i<m-1,\\ \overline{L}^{(n)}_{m-1}= & {} C^{(n)}_{m-1}+P_dH_{m-1}\overline{L}^{(n)}_{m-2}. \end{aligned}$$

Defining $b^{(n)}_{m-1}=C^{(n)}_{m-1}$ and $d_{m-1}=P_dH_{m-1}$, the equations given above become

$$\begin{aligned} \overline{L}^{(n)}_{m-1}=b^{(n)}_{m-1}+d_{m-1}\overline{L}^{(n)}_{m-2}. \end{aligned}$$

Hence, for $i=m-1$ down to 2,

$$\begin{aligned} \overline{L}^{(n)}_i=C^{(n)}_i+P_dH_i\overline{L}^{(n)}_{i-1}+P_uH^{-1}_{i+1}\left( {b^{(n)}_{i+1}}+d_{i+1}\overline{L}^{(n)}_i\right) , \end{aligned}$$

or

$$\begin{aligned} \overline{L}^{(n)}_i=\frac{C^{(n)}_i+P_uH^{-1}_{i+1}b^{(n)}_{i+1}}{1-P_uH^{-1}_{i+1}d_{i+1}}+\frac{P_dH_i}{1-P_uH^{-1}_{i+1}d_{i+1}}\overline{L}^{(n)}_{i-1}, \end{aligned}$$

(A.15)

Defining

$$\begin{aligned} b_i^{(n)}=\frac{C^{(n)}_i+P_uH^{-1}_{i+1}b^{(n)}_{i+1}}{1-P_uH^{-1}_{i+1}d_{i+1}},\,\,\,\,d_i=\frac{P_dH_i}{1-P_uH^{-1}_{i+1}d_{i+1}}, \end{aligned}$$

we next show that $d_i=1$ for $1\le i\le m-1$

$$\begin{aligned} d_{m-1}=P_dH_{m-1}=\frac{\Lambda _{k-1}}{\Lambda _{k-1}+\mu }\frac{1-\left( \frac{\mu }{\Lambda _{k-1}}\right) ^2}{1-\left( \frac{\mu }{\Lambda _{k-1}}\right) }=1, \end{aligned}$$

and similarly, for $i=m-1$ to 2, we can show that

$$\begin{aligned} d_{i-1}=\frac{P_dH_{i-1}}{1-P_uH^{-1}_id_i}=1. \end{aligned}$$

With these, we have Eq. (13). Using it in Eq. (A.15) and noting $b^{(n)}_{m-1}=C^{(n)}_{m-1}$, we obtain Eq. (12). Moreover,

$$\begin{aligned} \overline{L}^{(n)}_i=\sum ^i_{j=1}b^{(n)}_j, \,\,\,\,i=1,\ldots ,m. \end{aligned}$$

$\square$

Appendix B: Tables

See Tables 2, 3, 4, 5, 6, 7, 8, 9.

Table 2 Parameters of the examples cases 1–18

Table 3 Parameters of the examples cases 19–36

Table 4 Optimal inventory rationing levels and $C_\mathrm{MR}^*$ of the MR policy cases 1–18

Table 5 Optimal inventory rationing levels and $C_\mathrm{MR}^*$ of the MR policy cases 19–36

Table 6 Comparison of $\varepsilon$-optimal policy and the MR policy cases 1–18

Table 7 Comparison of $\varepsilon$-optimal policy and the MR policy cases 19–36

Table 8 Optimal HF and HP policies cases 1–18

Table 9 Optimal HF and HP policies cases 19–36