Joint maintenance and just-in-time spare parts provisioning policy for a multi-unit production system

Abstract

In this paper, two new joint maintenance and spare parts provisioning policies for a multi-unit production system are proposed. The production system consists of N identical, independent units, each subject to gradual deterioration. Production rates of the units depend on their operating states. Every maintenance action incurs a high set-up cost which includes the cost of sending a crew to the field, and it is therefore cost effective to maintain several units at the same time. The paper’s main contribution is an analytical modeling of a multi-unit production system and the development of effective joint maintenance and spare parts ordering policies for such system. The states of the units are observable through regular inspections and the maintenance and spare part ordering decisions depend on the number of the failed units and the number of available spare parts. The process is formulated as a semi-Markov decision process with the optimality criterion being the minimization of the total long-run expected average cost per unit time. The objective is to determine the optimal levels of the number of failed units to place an order for the spare parts or to initiate group maintenance, and to find the optimal inspection interval. Numerical examples are provided to illustrate the proposed optimization model and to compare the two maintenance policies. Sensitivity analysis is conducted to analyze the effect of several cost components on the optimal levels and on the long run expected average cost rate.

Appendices

Appendix A: Calculation of transition probabilities for each unit

The instantaneous deterioration transition rates $q_{ij}$, $i,j\in {\varOmega }$ for each unit are defined by:

$$\begin{aligned} q_{ij}= & {} lim_{u\rightarrow 0}\frac{P(X_{t+u}=j|X_t=i)}{u}<+\infty , \ \ j\ne i \ \nonumber \\&and \ \ \ q_{ii}=-\sum _{j\ne i}q_{ij} \end{aligned}$$

(12)

where $i,\ j\ \in \{0,1,F\}$, and the state transition matrix P(t) can be written as follows:

$$\begin{aligned} P(t)= \begin{bmatrix} e^{-\nu _0 t}&\quad \dfrac{q_{01}(e^{-\nu _1 t}-e^{-\nu _0 t})}{\nu _0-\nu _1}&\quad 1-e^{-\nu _0 t} - \dfrac{q_{01}(e^{-\nu _1 t}-e^{-\nu _0 t})}{\nu _0-\nu _1} \\ 0&\quad e^{-\nu _1 t}&\quad 1-e^{-\nu _1 t}\\ 0&\quad 0&\quad 1 \end{bmatrix} , \end{aligned}$$

(13)

where the exponential distribution parameters of the sojourn times are denoted by $\nu _0=q_{01}+q_{0F}$ and $\nu _1=q_{1F}$.

We assume that the state process is non-decreasing with probability 1, i.e., $q_{ij}=0$ for all $j<i$ and failure state is absorbing. The Kolmogorov’s backward differential equations for given transition rates can be written as follows:

$$\begin{aligned} P'_{00}(t)= & {} -\nu _0 P_{00}(t)\\ P'_{01}(t)= & {} q_{01}P_{11}(t)-\nu _{0} P_{01}(t)\\ P'_{02}(t)= & {} q_{01}P_{12}(t)+q_{02}(t)P_{22}(t)-\nu _0 P_{02}(t)\\ P'_{11}(t)= & {} -\nu _1 P_{11}(t)\\ P'_{12}(t)= & {} \nu _1P_{22}(t)-\nu _1 P_{12}(t) \end{aligned}$$

Extensive details on the Laplace transform can be found in Debnath and Bhatta (2014). Taking Laplace transform of the above equations, the following equations are obtained:

$$\begin{aligned}&s{\overline{P}}_{00}(s)-1=-\nu _0 {\overline{P}}_{00}(s)\\&s{\overline{P}}_{01}(s)=q_{01}{\overline{P}}_{11}(s)-\nu _0 {\overline{P}}_{01}(s)\\&s {\overline{P}}_{02}(s)=q_{01} {\overline{P}}_{12}(s)+\frac{q_{02}}{s} -\nu _{0} {\overline{P}}_{02}(s)\\&s{\overline{P}}_{11}(s)-1=-\nu _1 {\overline{P}}_{11}(s)\\&s{\overline{P}}_{12}(s)=\frac{\nu _1}{s}-\nu _1 {\overline{P}}_{12}(s), \end{aligned}$$

where ${\overline{P}}_{ij}(s)$ is the Laplace transform of the transition probability $P_{ij}(t)$, which is written as: ${\overline{P}}_{ij}(s)=\int _{0}^{\infty } e^{-st} P_{ij}(t)dt$. By solving the above equations we get:

$$\begin{aligned}&{\overline{P}}_{00}(s)=\frac{1}{s+\nu _0}, \ \ \ {\overline{P}}_{01}(s)=\frac{q_{01}}{(s+\nu _0)(s+\nu _1)}, \ \ \ {\overline{P}}_{02}(s)=\frac{\nu _0\nu _1+q_{02}s}{s(s+\nu _0)(s+\nu _1)}\\&{\overline{P}}_{11}(s)=\frac{1}{s+\nu _1}, \ \ \ {\overline{P}}_{12}(s)=\frac{\nu _1}{s(s+\nu _1)}. \end{aligned}$$

By taking the inverse Laplace transform of the above equations, we derive the transition probabilities as follows:

$$\begin{aligned}&P_{00}(t)=e^{-\nu _0t},\ \ \ P_{01}(t)=\frac{q_{01}(e^{-\nu _1t}-e^{-\nu _0t})}{\nu _0-\nu _1},\\&P_{02}(t)=1-P_{00}(t)-P_{01}(t)=1-e^{-\nu _0t}-\frac{q_{01}(e^{-\nu _1t}-e^{-\nu _0t})}{\nu _0-\nu _1}\\&P_{11}(t)=e^{-\nu _1t}, \ \ P_{12}(t)=1-e^{-\nu _1t}. \end{aligned}$$

Appendix B: Proof of Theorem 5

Assume that the state of the system is $(n_F,n_1,S)$ and the decision is to perform maintenance. If the state of the system changes to $(n_F+i,n_1',S), \ i\ge 0$ during $T_R$ (or $T_P$) an emergency order of size $n_F+i-S$ is placed. The expected cost of emergency ordering in state $(n_F,n_1,S)$ for different values of i is as follows:

$$\begin{aligned} E(C_{EM}|T_R, (n_F,n_1,S))&=\sum _{(n_F+i,n_1',S)} E(C_{EM}|(n_F+i,n_1',S))\nonumber \\&\quad \times \int _{0}^{\infty } P_{(n_F,n_1,S)(n_F+i,n_1',S)}(u) f_{T_{R}}(u)\ du\nonumber \\&=\sum _{i=x}^{N-n_F-n_1'}\sum _{n_1'=max\{n_1-i,0\}}^{n_1+n_0-i} C_{es}(n_F+i-S)\nonumber \\&\quad \times \int _{0}^{\infty } P_{(n_F,n_1,S)(n_F+i,n_1',S)}(u) f_{T_{R}}(u)\ du\nonumber \\&=\!\!\sum _{i=x}^{N-n_F-n_1'}\!\!\sum _{n_1'=max\{n_1-i,0\}}^{n_1+n_0-i}\!\!\sum _{j=x}^{min\{i,n_1\}}\!C_{es}(n_F+i-S) \nonumber \\&\quad \times \!\left( {\begin{array}{c}n_0\\ i-\!j\end{array}}\right) \!\left( {\begin{array}{c}n_1\\ j\end{array}}\right) \!\left( {\begin{array}{c}n_0-i+j\\ n_1'\!-n_1\!+j\end{array}}\right) \!\int _{0}^{\infty }\!\! P_{00}(u)^{n_0'} P_{01}(u)^{n_1'\!-n_1+j}\nonumber \\&\quad \times P_{0F}(u)^{i-j}P_{11}(u)^{n_1-j}P_{1F}(u)^j \cdot f_{T_{R}}(u)\ du \end{aligned}$$

(14)

where $x=0$, for $n_1' \ge n_1$, and $x=n_1-n_1'$, for $n_1'<n_1$.

For the first maintenance policy, the expected cost of corrective replacement in state $(n_F,n_1,S)$ is equal to:

$$\begin{aligned}&E(C|T_R, (n_F,n_1,S))=C_{KM} +C_I (N-n_F)+ E(C_H|T_R)\nonumber \\&\qquad +E(CR|T_R) + E(C_{EM}|T_R)+ E(C_{CM}|T_R) \nonumber \\&\quad =\! C_{KM}+ C_I (N-n_F)+\int _{0}^{\infty } C_H \cdot S \cdot f_{T_R}(u)du \ \ \ \ \ \ \ \ \nonumber \\ E(\text{ Cost } \text{(profit) } \text{ from } \text{ production })&{\left\{ \begin{array}{ll} + \sum _{i=x}^{N-n_F-n_1} \sum _{n_1'=max\{n_1-i,0\}}^{n_1+n_0-i}\sum _{j=x}^{min\{i,n_1\}} C_{rt}(n_F+i,n_1',S') \nonumber \\ \times \left( {\begin{array}{c}n_0\\ i-j\end{array}}\right) \left( {\begin{array}{c}n_1\\ j\end{array}}\right) \left( {\begin{array}{c}n_0-i+j\\ n_1'-n_1+j\end{array}}\right) \int _{0}^{\infty }\int _{0}^{u}P_{00}(t)^{n_0'} \nonumber \\ \times P_{01}(t)^{n_1'-n_1+j} P_{0F}(t)^{i-j}P_{11}(t)^{n_1-j}P_{1F}(t)^{j} f_{T_R}(u) dtdu \nonumber \\ \end{array}\right. }\\ E(\text {Cost of emergency ordering})&{\left\{ \begin{array}{ll} +\sum _{i=x}^{N-n_F-n_1'}\sum _{n_1'=max\{n_1-i,0\}}^{n_1+n_0-i}\sum _{j=x}^{min\{i,n_1\}}C_{es}(n_F+i-S)\nonumber \\ \times \left( {\begin{array}{c}n_0\\ i-j\end{array}}\right) \left( {\begin{array}{c}n_1\\ j\end{array}}\right) \left( {\begin{array}{c}n_0-i+j\\ n_1'-n_1+j\end{array}}\right) \int _{0}^{\infty } P_{00}(u)^{n_0'}\nonumber \\ \times P_{01}(u)^{n_1'-n_1+j} P_{0F}(u)^{i-j}P_{11}(u)^{n_1-j}P_{1F}(u)^j f_{T_{R}}(u)\ du\nonumber \\ \end{array}\right. }\\ E(\text {Cost of failure replacement})&{\left\{ \begin{array}{ll} +\sum _{i=x}^{N-n_F-n_1'}\sum _{n_1'=max\{n_1-i,0\}}^{n_1+n_0-i} \sum _{j=x}^{min\{i,n_1\}} C_F(n_F+i)\nonumber \\ \times \left( {\begin{array}{c}n_0\\ i-j\end{array}}\right) \left( {\begin{array}{c}n_1\\ j\end{array}}\right) \left( {\begin{array}{c}n_0-i+j\\ n_1'-n_1+j\end{array}}\right) \int _{0}^{\infty } P_{00}(u)^{n_0'}\nonumber \\ \times P_{01}(u)^{n_1'-n_1+j} P_{0F}(u)^{i-j}P_{11}(u)^{n_1-j}P_{1F}(u)^j f_{T_{R}}(u)\ du\nonumber \\ \end{array}\right. }\\&\quad = C_{KM}+ C_I (N-n_F)+\int _{0}^{\infty } C_H \cdot S\cdot f_{T_R}(u)du\nonumber \\&\qquad +\sum _{i=x}^{N-n_F-n_1'}\sum _{n_1'=max\{n_1-i,0\}}^{n_1+n_0-i}\sum _{j=x}^{min\{i,n_1\}}\left( {\begin{array}{c}n_0\\ i-j\end{array}}\right) \left( {\begin{array}{c}n_1\\ j\end{array}}\right) \left( {\begin{array}{c}n_0-i+j\\ n_1'-n_1+j\end{array}}\right) \nonumber \\&\qquad \times \Bigg [\bigg (C_{es}(n_F+i-S)+C_F(n_F+i)\bigg ) \int _{0}^{\infty } P_{00}(u)^{n_0'}P_{01}(u)^{n_1'-n_1+j} \nonumber \\&\qquad \times P_{0F}(u)^{i-j}P_{11}(u)^{n_1-j}P_{1F}(u)^j f_{T_{R}}(u)\ du\nonumber \\&\qquad +C_{rt}(n_F+i,n_1',S') \int _{0}^{\infty }\int _{0}^{u}P_{00}(t)^{n_0'} P_{01}(t)^{n_1'-n_1+j}\nonumber \\&\qquad \times P_{0F}(t)^{i-j}P_{11}(t)^{n_1-j}P_{1F}(t)^{j} f_{T_R}(u) dtdu\Bigg ] \end{aligned}$$

(15)

where $x=0$ for $n_1' \ge n_1$, and $x=n_1-n_1'$ for $n_1'<n_1$.

The expected cost of corrective replacement and preventive maintenance for the second policy in state $(n_F,n_1,S)$ is equal to:

$$\begin{aligned} E(C_{CM}+C_{PM}|T_P,(n_F,n_1,S))&=\sum _{(n_F+i,n_1',S)} E(C_{CM}+C_{PM}|(n_F+i,n_1',S)) \nonumber \\&\quad \times \int _{0}^{\infty }\!\! P_{(n_F,n_1,S)(n_F+i,n_1',S)}(u) f_{T_{P}}(u)\ du\nonumber \\&=\sum _{i=x}^{N-n_F-n_1'}\sum _{n_1'=max\{n_1-i,0\}}^{n_1+n_0-i}(C_F(n_F+i)+C_P (n_1'))\nonumber \\&\quad \times \int _{0}^{\infty } P_{(n_F,n_1,S)(n_F+i,n_1',S)}(u) f_{T_{P}}(u)\ du\nonumber \\&=\!\sum _{i=x}^{N-n_F-n_1'}\!\sum _{n_1'=max\{n_1-i,0\}}^{n_1+n_0-i}\! \sum _{j=x}^{min\{i,n_1\}}\! (C_F(n_F+i)+C_P (n_1'))\nonumber \\&\quad \times \left( {\begin{array}{c}n_0\\ i-j\end{array}}\right) \left( {\begin{array}{c}n_1\\ j\end{array}}\right) \left( {\begin{array}{c}n_0-i+j\\ n_1'-n_1+j\end{array}}\right) \int _{0}^{\infty } P_{00}(u)^{n_0'}P_{01}(u)^{n_1'-n_1+j}\nonumber \\&\quad \times P_{0F}(u)^{i-j}P_{11}(u)^{n_1-j}P_{1F}(u)^j \times f_{T_{P}}(u)\ du \end{aligned}$$

(16)

where $x=0$ for $n_1' \ge n_1$, and $x=n_1-n_1'$ for $n_1'<n_1$.

Appendix C: Extension to more than 3 state deterioration process

First, we will show how to extend the model to a more complex case of a 4 state deterioration process (we consider 4 states for each unit of the system). This change affects the number of system states, transition probabilities, production rate and the expected cost. In this section we provide the formula to calculate the transition probabilities. The calculation of the expected cost is similar to to the previous case (3 state deterioration process).

Consider a system consisting of N units where deterioration process of each unit is modeled as a 4-state continuous time homogeneous Markov chain with three working states and a failure state F which is absorbing, $\varOmega =\{0,1,2,F\}$. The state space for the whole system can be defined as $W =\{(n_F,n_2,n_1, S)\mid n_1+n_2+n_F\le N, n_1, n_2, n_F, S\ge 0\}$, where $n_F$ , $n_2$, and $n_1$ represent the number of units in the failure state, state 2, and state 1, respectively, and S represents the on-hand inventory. Based on Remark 2, the number of available spare parts in the system at each inspection time can be 0 or R, so the number of states for this system is equal to $\frac{(N+1)(N+2)(N+3)\times 2}{3!}$.

The transition probability function when the decision is to continue operation is as follows:

$$\begin{aligned}&P_{(n_F,n_2,n_1,S)(n_F+i,n_2',n_1', S)}(\varDelta )=\nonumber \\&\sum _{k=max\{0,n_2-n_2'\}}^{min\{i,n_2\}} \sum _{j=max\{0,i-k-n_0\}}^{min\{i-k,n_1\}}\sum _{m=max\{0,n_1-n_1'-j\}}^{min\{n_1-j,n_2'-n_2+k\}} \left( {\begin{array}{c}n_0\\ i-j-k\end{array}}\right) \left( {\begin{array}{c}n_0-i+k+j\\ n_1'-n_1+m+j\end{array}}\right) \nonumber \\&\left( {\begin{array}{c}n_0-i+k-n_1'+n_1-m\\ n_2'-n_2+k-m\end{array}}\right) \left( {\begin{array}{c}n_1\\ j\end{array}}\right) \left( {\begin{array}{c}n_1-j\\ m\end{array}}\right) \left( {\begin{array}{c}n_2\\ k\end{array}}\right) P_{00}(\varDelta )^{n_0'} P_{01}(\varDelta )^{n_1'-n_1+j+m}\nonumber \\&P_{02}(\varDelta )^{n_2'-n_2+k-m} P_{0F}(\varDelta )^{i-j-k} P_{11}(\varDelta )^{n_1-m-j}P_{12}(\varDelta )^{m} P_{1F}(\varDelta )^{j} P_{22}(\varDelta )^{n_2-k}P_{2F}(\varDelta )^{k}\nonumber \\ \end{aligned}$$

(17)

Figure 3 shows the number of units which change their state in the next time interval of length $\varDelta$ when deterioration process of units has 4 states.

Fig. 3

Number of units changing state in a time interval of length $\varDelta$

The transition probability when the decision is to perform maintenance is given by the following formulas:

• For the first maintenance policy,

  $$\begin{aligned}&P_{(n_F,n_2,n_1,S)(0,n_2',n_1', 0)}(T_R)= \nonumber \\&\sum _{i=max\{0, n_1+n_2-n_1'-n_2'\}}^{N-n_F-n_1'-n_2'}\sum _{k=max\{0,n_2-n_2'\}}^{min\{i,n_2\}} \sum _{j=max\{0,i-k-n_0\}}^{min\{i-k,n_1\}}\sum _{m=max\{0,n_1-n_1'-j\}}^{min\{n_1-j,n_2'-n_2+k\}} \left( {\begin{array}{c}n_0\\ i-j-k\end{array}}\right) \nonumber \\&\left( {\begin{array}{c}n_0-i+k+j\\ n_1'-n_1+m+j\end{array}}\right) \left( {\begin{array}{c}n_0-i+k-n_1'+n_1-m\\ n_2'-n_2+k-m\end{array}}\right) \left( {\begin{array}{c}n_1\\ j\end{array}}\right) \left( {\begin{array}{c}n_1-j\\ m\end{array}}\right) \left( {\begin{array}{c}n_2\\ k\end{array}}\right) \nonumber \\&\left( \int _0^\infty P_{00}(u)^{n_0'}P_{01}(u)^{n_1'-n_1+j+m} P_{02}(u)^{n_2'-n_2+k-m} P_{0F}(u)^{i-j-k} P_{11}(u)^{n_1-m-j}\right. \nonumber \\&\left. P_{12}(u)^{m} P_{1F}(u)^{j} P_{22}(u)^{n_2-k}P_{2F}(u)^{k} \times f_{T_R} (u) du\right) \end{aligned}$$

  (18)

• For the second maintenance policy,

  $$\begin{aligned} P_{(n_F,n_2,n_1,S)(0,0,0, 0)}(T_P)=1 \end{aligned}$$

As can be seen, the number of system states increases dramatically and the calculation of transition probability becomes quite complicated by adding new states. We assume that a transition to a different state may or may not occur during the next $\varDelta$ time units, which makes it difficult to calculate transition probabilities for the system. Some simplification is possible for reliable systems, for which transitions to some states can be eliminated. So, we can consider a system consisting of N units where deterioration process of each unit is modeled as an m-state continuous time homogeneous Markov chain with $m-1$ working states and a failure state F which is absorbing, $\varOmega =\{0,1,2,\ldots ,m-1,F\}$. The multi-state structure is considered where transitions are allowed either to the neighbor state or to the failure state. For highly reliable systems, we can omit any intermediate transition between states during each inspection interval (see Fig. 4).

Fig. 4

Number of units changing state in a time interval of length $\varDelta$

The state space for the whole system can be defined as $W =\{(n_F,n_{m-1},\ldots ,n_1, S)\mid n_1+\cdots +n_F\le N, n_1,\ldots , n_F, S\ge 0\}$, where $n_F$ , $n_i\ i\in \{0,1,\ldots ,m-1\}$ represent the number of units in the failure state, and state i, respectively, and S represents the on-hand inventory.

The transition probability function when the decision is to continue operation is given by:

$$\begin{aligned}&P_{(n_F,n_{m-1},\ldots ,n_1, S)(n_F+i,n'_{m-1},\ldots ,n'_1, S)}(\varDelta )=\\&\sum _{Y_{n-1}=max\{0, n_{m-1}-n'_{m-1}\}}^{min\{i,n_{m-1}, n_{m-1}-n'_{m-1}+n_{m-2}\}} \sum _{Y_{m-2}=max\{0, n_{m-2}-n'_{m-2}\}}^{min\{i-Y_{m-1},n_{m-2}, n_{m-2}-n'_{m-2}+n_{m-3}\}}\ldots \sum _{Y_1=max\{0, n_{1}-n'_{1}\}}^{min\{i-Y_{m-1}-\cdots -Y_2,n_{1}, n_{1}-n'_{1}+n_{0}\}}\\&\left( {\begin{array}{c}n_0\\ Z_0\end{array}}\right) \left( {\begin{array}{c}n_0-Z_0\\ i-Y_1-Y_2-\cdots -Y_{m-1}\end{array}}\right) \left( {\begin{array}{c}n_1\\ Z_1\end{array}}\right) \left( {\begin{array}{c}n_1-Z_1\\ Y_1\end{array}}\right) \ldots \left( {\begin{array}{c}n_{m-1}\\ Y_{m-1}\end{array}}\right) \\&P_{0,0}^{n_0'}\ P_{(0,1)}^{Z_0}\ P_{(0,F)}^{Y_0}\ P_{1,1}^{n_1-Y_1-Z_1}\ P_{(1,2)}^{Z_1}\ P_{(1,F)}^{Y_1}\ldots \ P_{m-1,m-1}^{m-1-Y_{m-1}}\ P_{(m-1,F)}^{Y_{m-1}} \end{aligned}$$

where

$$\begin{aligned} \begin{array}{c} Z_{m-2}=n'_{m-1}-n_{m-1}+Y_{m-1}\\ Z_{m-3}=n'_{m-2}-n_{m-2}+Y_{m-2}+Z_{n-2}\\ \ldots \\ Z_1=n_2'-n_2+Y_2+Z_2\\ Z_0=n_1'-n_1+Y_1+Z_1. \end{array} \end{aligned}$$