2.1. Cost function at system failure

Suppose that the system starts working at $t=0$ and fails at a random time after $t=0$ . Assume that when the system fails we have a cost $c_i$ for each failed component of type i to replace it with a new one and a cost $c_i^*$ for each unfailed component to refresh it so that it becomes as good as new, where we assume that $c_i\geq c^*_i$ , $i=1, \ldots, L$ . Furthermore, we assume that $c^{**}$ denotes the fixed overall cost for system failure. With $T_R$ as the lifetime of the system after redundancy, let the random variable $X_i(T_R)$ denote the number of failed components of type i at the time of system failure, $i=1,\dots,L$ . Then the mean cost rate function for a failed system is defined as

(2.1) \begin{align}\mathrm{Cost}_1(\mathbf{v})=\dfrac{\sum_{i=1}^Lc_i\mathbb{E}(X_i(T_R))+\sum_{i=1}^L c_i^*\mathbb{E}(n_i(v_i+1)-X_i(T_R))+c^{**}}{\mathbb{E}(T_R)},\end{align}

where $\mathbf{v}=(v_1,\ldots, v_L)$ . The numerator is the expected cost of the system failure, and the denominator is the mean time to failure (MTTF) of the system, so $\mathrm{Cost}_1$ becomes the mean cost per unit of time. Note that in the system after redundancy, there are altogether $n_i(v_i+1)$ components of type i, $i=1,2, \ldots, L$ . The relation (2.1) can be rewritten in terms of the lifetime of the original system without any redundancy, T, as

(2.2) \begin{align}\mathrm{Cost}_1(\mathbf{v})&=\dfrac{\sum_{i=1}^Lc_i(v_i+1)\mathbb{E}(X_i(T))+\sum_{i=1}^L c_i^*(v_i+1)\mathbb{E}(n_i-X_i(T))+c^{**}}{\mathbb{E}(T_R)}\nonumber\\[5pt]&=\dfrac{\sum_{i=1}^L(c_i-c_i^*)(v_i+1)\mathbb{E}(X_i(T))+\sum_{i=1}^L c_i^*(v_i+1)n_i+c^{**}}{\mathbb{E}( T_R)}.\end{align}

Lemma 2.1. The quantity $\mathbb{E}( X_i(T))$ in (2.2) can be expressed as follows:

\begin{align*}&\mathbb{E}( X_i(T))\\[5pt] & =n_i\int_{0}^{\infty} \lim_{\delta\rightarrow 0}\dfrac{1}{\delta}\sum_{m_1=0}^{n_1}\!\cdots\! \sum_{m_i=0}^{n_i-1}\!\cdots\! \sum_{m_L=0}^{n_L}\Phi(m_1,\ldots,m_{i-1},m_i+1,m_{i+1},\ldots, m_L)\\[5pt] & \quad \times \binom{n_1}{m_1}\!\cdots\! \binom{n_i-1}{m_i}\!\cdots\! \binom{n_L}{m_L} A_{\mathbf{m}}^{(i)}(t,\delta) \,{\mathrm{d}} t,\end{align*}

where

(2.3) \begin{align}A_{\mathbf{m}}^{(i)}(t,\delta)&=\mathbb{P}\bigl({T_1^{(1)}>t},\ldots,{T_{m_1}^{(1)}>t},{T_{m_1+1}^{(1)}\leq t},\ldots, {T_{n_1}^{(1)}\leq t}, \nonumber\\[5pt]&\quad \ldots, t<T_1^{(i)}\leq t+\delta, {T_2^{(i)}>t},\ldots,{T_{m_i+1}^{(i)}>t},{T_{m_i+2}^{(i)}\leq t},\ldots, {T_{n_i}^{(i)}\leq t}, \nonumber\\[5pt]&\quad \ldots, {T_1^{(L)}>t},\ldots,{T_{m_L}^{(L)}>t},{T_{m_L+1}^{(L)}\leq t},\ldots, {T_{n_L}^{(L)}\leq t}\bigr).\end{align}

Proof.

\begin{align*} \mathbb{E}( X_i(T))&=\mathbb{E}\Biggl( \sum_{j=1}^{n_i}I\bigl(T_j^{(i)}\leq T\bigr)\Biggr)\\[5pt] &=\sum_{j=1}^{n_i}\mathbb{P}\bigl(T_j^{(i)}\leq T\bigr)=n_i\mathbb{P}\bigl(T_1^{(i)}\leq T\bigr)\\[5pt]&=n_i\int_{0}^{\infty} \lim_{\delta\rightarrow 0}\dfrac{\mathbb{P}\bigl(T>t,t<T_1^{(i)}\leq t+\delta\bigr)}{\delta} \,{\mathrm{d}} t,\end{align*}

where the third equality follows from the exchangeability of the components of type i, $i=1, \dots, L$ . By conditioning on the number of live components of each type, we obtain

(2.4) \begin{align}&\mathbb{P}\bigl(T>t,t<T_1^{(i)}\leq t+\delta\bigr)\nonumber\\[5pt] & =\sum_{m_1=0}^{n_1}\!\cdots\!\sum_{m_i=0}^{n_i-1}\!\cdots\!\sum_{m_L=0}^{n_L}\mathbb{P}\bigl(T>t,t<T_1^{(i)}\leq t+\delta, C_j(t)=m_j, j=1,\ldots, L \bigr) \nonumber\\& =\sum_{m_1=0}^{n_1}\!\cdots\!\sum_{m_i=0}^{n_i-1}\!\cdots\!\sum_{m_L=0}^{n_L}\Phi(m_1,\ldots,m_{i-1},m_i+1,m_{i+1},\ldots, m_L) \binom{n_1}{m_1}\!\cdots\! \binom{n_i-1}{m_i}\!\cdots\!\binom{n_L}{m_L}\nonumber\\[5pt]& \quad \times A_{\mathbf{m}}^{(i)}(t,\delta).\end{align}

The last equality in (2.4) holds because the components of the same type have a common failure time distribution.

In the following theorem, (2.3) is represented based on the survival copula of the component lifetimes.

Theorem 2.1. Using the inclusion–exclusion rule, $A_{\mathbf{m}}^{(i)}(t,\delta)$ can be represented as follows:

\begin{align*}&A_{\mathbf{m}}^{(i)}(t, \delta)\\[5pt]& ={\sum_{j_1=0}^{n_1-m_1}\!\cdots\!\sum_{j_i=0}^{n_i-m_i-1}\!\cdots\!\sum_{j_L=0}^{n_L-m_L}}(\!-\!1)^{j_1+\cdots + j_L}\binom{n_1-m_1}{j_1}\!\cdots\! \binom{n_i-m_i-1}{j_i}\!\cdots\!\binom{n_L-m_L}{j_L}\\& \quad \times \biggl[\hat{C}\Bigl(\underbrace{\bar{F}_1(t)}_{m_1+j_1},\underbrace{1}_{n_1-(m_1+j_1)},\ldots,\underbrace{\bar{F}_i(t)}_{m_i+j_i+1},\underbrace{1}_{n_i-(m_i+j_i+1)},\ldots,\underbrace{\bar{F}_L(t)}_{m_L+j_L}, \underbrace{1}_{n_L-(m_L+j_L)}\Bigr)\\[5pt]& \quad -\hat{C}\Bigl(\underbrace{\bar{F}_1(t)}_{m_1+j_1},\underbrace{1}_{n_1-(m_1+j_1)},\ldots,\underbrace{\bar{F}_i(t)}_{m_i+j_i},\bar{F}_i(t+\delta),\underbrace{1}_{n_i-(m_i+j_i+1)}, \ldots,\underbrace{\bar{F}_L(t)}_{m_L+j_L},\underbrace{1}_{n_L-(m_L+j_L)}\Bigr)\biggr].\end{align*}

Note that in the particular case of independence of all components, we get

(2.5) \begin{align}&\mathbb{E}( X_i(T))\nonumber\\[5pt] & =n_i\sum_{m_1=0}^{n_1}\!\cdots\!\sum_{m_i=0}^{n_i-1}\!\cdots\!\sum_{m_L=0}^{n_L}\Phi(m_1,\ldots,m_{i-1},m_i+1,m_{i+1},\ldots, m_L) \notag \\[5pt] & \quad \times \binom{n_1}{m_1}\!\cdots\! \binom{n_i-1}{m_i}\!\cdots\!\binom{n_L}{m_L}\nonumber\\[5pt]& \quad \times\int_{0}^{\infty}\bar{F}_1^{m_1}(t)F_1^{n_1-m_1}(t)\!\cdots\!\bar{F}_i^{m_i}(t)F_i^{n_i-m_i-1}(t)\!\cdots\! \bar{F}_L^{m_L}(t)F_L^{n_L-m_L}(t) \,{\mathrm{d}} F_i(t).\end{align}

In order to minimize the mean cost rate function $\mathrm{Cost}_1(\mathbf{v})$ , we impose the constraint that there are at most $M_i$ components of type i as spares, $i=1,\dots,L$ . This means that the number of the components that can be connected in parallel at the ith group satisfies the inequality $n_iv_i\leq M_i$ , $i=1,\ldots, L$ . To determine the optimal values of $v_i$ , we do the following: for given values of $n_i, c_i, c_i^*$ , and $M_i$ , $i=1, \ldots, L$ , and $c^{**}$ , we evaluate $\mathrm{Cost}_1(\mathbf{v})$ for all possible choices of $v_1, \ldots, v_L$ such that for all i, $n_iv_i\leq M_i$ . Then the optimal values of $v_1, \ldots, v_L$ can be determined as the values for which the corresponding mean cost rate function $\mathrm{Cost}_1(\mathbf{v})$ is minimum.

Remark 2.2. If the system has a k-out-of-n structure with independent components from multiple type components, then (2.5) reduces to the result of Eryilmaz [Reference Eryilmaz7]. This is so because for such systems the survival signature is obviously given by

\begin{align*}\Phi(l_1,\ldots, l_L)=\begin{cases}1, & \sum_{j=1}^L l_j\geq k,\\0, & \text{otherwise,}\end{cases}\end{align*}

i.e. the system works if at least k components are alive.

2.2. Cost function based on preventive replacement

In this section we propose a kind of age replacement preventive maintenance policy for the system with multiple types of components described in Section 1.2. The policy of renewing the system performed by the operator is such that it is replaced at failure time or at a predetermined time $\tau$ , whichever occurs first. There are many papers on age replacement strategy; the interested reader can refer to Zhao et al. [Reference Zhao, Al-Khalifa, Hamouda and Nakagawa36], Ashrafi and Asadi [Reference Ashrafi and Asadi1], and Mizutani et al. [Reference Mizutani, Zhao and Nakagawa23], for example. Mannai and Gasmi [Reference Mannai and Gasmi22] found the optimal configuration of a k-out-of-n system so that the expected total costs of the system under some generalized age replacement policies are minimized.

Here, suppose that the operator has $M_i$ components of type i available as spares, and he/she decides to add $v_i$ components to each of the components of type i, where $n_iv_i\leq M_i$ . Under the implemented policy here, the aim is to find the optimal number of v such that the mean cost rate we impose below is minimized.

If the replacement occurs after the system failure, i.e. $T_R\leq \tau$ , then, considering the costs $c_i$ , $c_i^*$ , and $c^{**}$ as defined in the previous subsection, the average cost of renewing the system is obtained as

\begin{align*}M_1(\mathbf{v})&=\sum_{i=1}^L c_i \mathbb{E}( X_i(T_R)\mid T_R\leq \tau)+\sum_{i=1}^L c^*_i \mathbb{E}( n_i(v_i+1)-X_i(T_R)\mid T_R\leq \tau)+c^{**}\nonumber\\[5pt]&=\sum_{i=1}^L (v_i+1)c_i \mathbb{E}( X_i(T)\mid T\leq\tau)+\sum_{i=1}^L (v_i+1)c^*_i \mathbb{E}( n_i-X_i(T)\mid T\leq\tau)+c^{**}\nonumber\\[5pt]&=\sum_{i=1}^L (c_i-c^*_i)(v_i+1) \mathbb{E}( X_i(T)\mid T\leq\tau)+\sum_{i=1}^L (v_i+1)c^*_i n_i+c^{**},\end{align*}

where T is the lifetime of the system before redundancy allocation.

If the system is replaced before failure, i.e. $T_R>\tau$ , then, by the costs $c_i$ and $c_i^*$ , $i=1, \ldots, L$ for renewing the failed components and refreshing the alive components of type i, respectively, the system will be as good as the new condition. Let $N_i(\tau)$ be the number of failed components of type i on $[0, \tau]$ . Then the average cost of renewing the system is defined as

\begin{align*}M_2(\mathbf{v})&=\sum_{i=1}^L c_i \mathbb{E}( N_i(\tau)\mid T_R>\tau)+\sum_{i=1}^L c^*_i \mathbb{E}( n_i(v_i+1))-N_i(\tau)\mid T_R>\tau)\nonumber\\[5pt]&=\sum_{i=1}^L (c_i-c^*_i)(v_i+1) \mathbb{E}( N_i(\tau)\mid T>\tau)+\sum_{i=1}^L (v_i+1)c^*_i n_i.\end{align*}

Consequently, the mean cost rate function of the system renewing at time $\min\!(\tau, T_R)$ is achieved as

(2.6) \begin{align}\mathrm{Cost}_2(\mathbf{v})=\dfrac{M_1(\mathbf{v})\,\mathbb{P}(T_R\leq \tau)+M_2(\mathbf{v})\,\mathbb{P}(T_R>\tau)}{\mathbb{E}(\!\min\!(\tau,T_R))},\end{align}

where it is attained that

\begin{align*}\mathbb{E}(\!\min\!(\tau,T_R))=\int_{0}^{\tau}\bar{F}_{T_R}(y) \,{\mathrm{d}} y.\end{align*}

To compute (2.6), we need to calculate $\mathbb{E}( N_i(\tau)\mid T>\tau)$ and $\mathbb{E}( X_i(T)\mid T\leq \tau)$ . For the first one, we have

(2.7) \begin{align}&\mathbb{E}( N_i(\tau)\mid T>\tau)\notag \\[5pt]& =\dfrac{1}{\bar{F}_T(\tau)}\sum_{j_i=0}^{n_i} j_i \mathbb{P}(N_i(\tau)=j_i, T>\tau)\nonumber\\[5pt]& =\dfrac{1}{\bar{F}_T(\tau)}\sum_{j_1=0}^{n_1}\!\cdots\!\sum_{j_L=0}^{n_L} j_i \mathbb{P}(T>\tau \mid N_1(\tau)=j_1,\ldots, N_L(\tau)=j_L)\,\mathbb{P}(N_1(\tau)=j_1,\ldots, N_L(\tau)=j_L)\nonumber\\[5pt]& =\dfrac{1}{\bar{F}_T(\tau)}\sum_{j_1=0}^{n_1}\!\cdots\!\sum_{j_L=0}^{n_L} j_i \Phi(n_1-j_1,\ldots, n_L-j_L)\binom{n_1}{j_1}\!\cdots\! \binom{n_L}{j_L}B(\tau,j_1,\dots,j_L),\end{align}

where

(2.8) \begin{align}B(\tau,j_1,\dots,j_L)&=\mathbb{P}\bigl(T_1^{(1)}\leq\tau, \ldots, T_{j_1}^{(1)}\leq\tau, T_{j_1+1}^{(1)}>\tau, \ldots, T_{n_1}^{(1)}>\tau, \nonumber\\[5pt]&\quad \ldots, T_1^{(L)}\leq\tau, \ldots, T_{j_L}^{(L)}\leq\tau, T_{j_L+1}^{(L)}>\tau, \ldots, T_{n_L}^{(L)}>\tau \bigr).\end{align}

Using a method similar to that used in Lemma 2.1, we can calculate $\mathbb{E}( X_i(T)\mid T\leq \tau)$ , $i=1,\dots,L$ as follows:

\begin{align*}\mathbb{E}( X_i(T)\mid T\leq \tau)=n_i\mathbb{P}\bigl(T_1^{(i)}\leq T\mid T\leq \tau\bigr)=n_i\dfrac{\mathbb{P}\bigl(T_1^{(i)}\leq T,T\leq \tau\bigr)}{1-\mathbb{P}(T>\tau)}, \quad j=1,\dots,L.\end{align*}

Now we can write

\begin{align*}\mathbb{P}\bigl(T_1^{(i)}\leq T,T\leq \tau\bigr)=\int_{0}^{\tau}\lim_{\delta\rightarrow 0}\dfrac{\mathbb{P}\bigl(s<T\leq \tau, s<T_1^{(i)}\leq s+\delta\bigr)}{\delta} \,{\mathrm{d}} s,\quad i=1,\dots,L,\end{align*}

for which we have

\begin{align*}&\mathbb{P}\bigl(s<T\leq \tau, s<T_1^{(i)}\leq s+\delta\bigr)\nonumber\\[5pt] & =\sum_{m_1=0}^{n_1}\!\cdots\!\sum_{m_i=0}^{n_i-1}\!\cdots\!\sum_{m_L=0}^{n_L}\;\sum_{l_1=0}^{m_1}\!\cdots\!\sum_{l_L=0}^{m_L}\mathbb{P}\bigl(s<T\leq \tau\mid s<T_1^{(i)}\leq s+\delta, C_1(\tau)=l_1, \\[5pt] & \quad \ldots, C_i(\tau)=l_i, C_L(\tau)=l_L, C_1(s)=m_1,\ldots, C_i(s)=m_i,\ldots, C_L(s)=m_L\bigr) \, \\[5pt] & \quad \times \mathbb{P}\bigl(s<T_1^{(i)}\leq s+\delta,C_1(\tau)=l_1, \ldots, C_i(\tau)=l_i,C_L(\tau)=l_L, C_1(s)=m_1,\ldots, C_i(s)=m_i,\\[5pt] & \quad \ldots, C_L(s)=m_L)\bigr)\nonumber\\[5pt] & =\sum_{m_1=0}^{n_1}\!\cdots\!\sum_{m_i=0}^{n_i-1}\!\cdots\!\sum_{m_L=0}^{n_L}\;\sum_{l_1=0}^{m_1}\!\cdots\!\sum_{l_L=0}^{m_L}[\Phi(m_1,\ldots, m_L)-\Phi(l_1,\ldots, l_L)]\Biggl[\prod_{j=1,j\neq i}^L\binom{n_j}{m_j}\binom{m_j}{l_j}\Biggr]\nonumber\\[5pt]& \quad \times \binom{n_i-1}{m_i}\binom{m_i}{l_i}A_{\mathbf{m}, \mathbf{l}}^{(i)}(s,s+\delta, \tau),\end{align*}

where

(2.9) \begin{align}&A_{\mathbf{m}, \mathbf{l}}^{(i)}(s,s+\delta, \tau)\nonumber\\[5pt]& =\mathbb{P}\bigl(T_1^{(1)}>\tau, \ldots, T_{l_1}^{(1)}>\tau, s<T_{l_1+1}^{(1)}\leq \tau, \ldots, s<T_{m_1}^{(1)}\leq \tau, T_{m_1+1}^{(1)}\leq s,\ldots, T_{n_1}^{(1)}\leq s, \ldots,\nonumber \\[5pt] & \quad T_1^{(i)}>\tau, \ldots, T_{l_i}^{(i)}>\tau, s<T_{l_i+1}^{(i)}\leq\tau, \ldots, s<T_{m_i}^{(i)}\leq\tau, s<T_{m_i+1}^{(i)}\leq s+\delta,\notag \\[5pt] & \quad T_{m_i+2}^{(i)}\leq s,\ldots, T_{n_i}^{(i)}\leq s, \ldots, T_1^{(L)}>\tau, \ldots, T_{l_L}^{(L)}>\tau, s<T_{l_L+1}^{(L)}\leq \tau, \ldots, s<T_{m_L}^{(L)}\leq \tau, \notag \\[5pt] & \quad T_{m_L+1}^{(L)}\leq s,\ldots, T_{n_L}^{(L)}\leq s \bigr).\end{align}

In the following theorem, the probabilities in (2.8) and (2.9) are represented based on the survival copula of component lifetimes.

Theorem 2.2. Using the inclusion–exclusion rule, we obtain the following expressions for $B(\tau,j_1,\dots,j_L)$ and $A_{\mathbf{m}, \mathbf{l}}^{(i)}(s,s+\delta, \tau)$ , respectively:

\begin{align*}&B(\tau,j_1,\dots,j_L)\\[5pt]& ={\sum_{b_1=0}^{j_1}\!\cdots\!\sum_{b_L=0}^{j_L}}(\!-\!1)^{b_1+\cdots+ b_L}\binom{j_1}{b_1}\!\cdots\! \binom{j_L}{b_L} \hat{C}\Bigl(\underbrace{\bar{F}_1(\tau)}_{n_1-j_1+b_1},\underbrace{1}_{j_1-b_1},\ldots,\underbrace{\bar{F}_L(\tau)}_{n_L-j_L+b_L},\underbrace{1}_{j_L-b_L)}\Bigr)\end{align*}

and

\begin{align*}&A_{\mathbf{m}, \mathbf{l}}^{(i)}(s,s+\delta, \tau)\\[5pt]& ={\sum_{j_1=0}^{n_1-m_1}\!\cdots\!\sum_{j_i=0}^{n_i-m_i-1}\!\cdots\!\sum_{j_L=0}^{n_L-m_L}}(\!-\!1)^{\,j_1+\cdots + j_L}\binom{n_1-m_1}{j_1}\!\cdots\! \binom{n_i-m_i-1}{j_i}\!\cdots\!\binom{n_L-m_L}{j_L}\\& \quad \times {\sum_{d_1=0}^{m_1-l_1}\!\cdots\!\sum_{d_i=0}^{m_i-l_i}\!\cdots\!\sum_{d_L=0}^{m_L-l_L}}(\!-\!1)^{d_1+\cdots + d_L}\binom{m_1-l_1}{d_1}\!\cdots\! \binom{m_L-l_L}{d_L}\\ &\quad \times \biggl[\hat{C}\Bigl(\underbrace{\bar{F}_k(\tau)}_{l_k+d_k}, \underbrace{\bar{F}_k(s)}_{m_k-l_k+j_k-d_k}\!,\underbrace{1}_{n_k-m_k-j_k}\!\!,1 \leq k \leq L, k\neq i, \underbrace{\bar{F}_i(\tau)}_{l_i+d_i}, \!\underbrace{\bar{F}_i(s)}_{m_i-l_i+j_i-d_i+1}\!, \underbrace{1}_{n_i-m_i-j_i-1}\Bigr)\\[5pt] &\quad -\hat{C}\Bigl(\underbrace{\bar{F}_k(\tau)}_{l_k+d_k}, \underbrace{\bar{F}_k(s)}_{m_k-l_k+j_k-d_k}\!,\underbrace{1}_{n_k-m_k-j_k}\!\!,1 \leq k \leq L, k\neq i,\underbrace{\bar{F}_i(\tau)}_{l_i+d_i}, \!\underbrace{\bar{F}_i(s)}_{m_i-l_i+j_i-d_i}\!\!, \bar{F}_i(s+\delta), \!\underbrace{1}_{n_i-m_i-j_i-1} \Bigr)\biggr].\end{align*}

Corollary 2.1. For the particular case of independent components, it can be deduced that

(2.10) \begin{align}\mathbb{E}( N_i(\tau)\mid T>\tau)=\dfrac{1}{\bar{F}(\tau)}\sum_{j_1=0}^{n_1}\!\cdots\!\sum_{j_L=0}^{n_L} j_i\Phi(n_1-j_1,\ldots, n_L-j_L)\prod_{l=1}^L \binom{n_l}{j_l}F_l^{j_l}(\tau)\bar{F}_l^{n_l-j_l}(\tau).\end{align}

Also, in this case we have

\begin{align*}A_{\mathbf{m}, \mathbf{l}}^{(i)}(s,s+\delta, \tau)&=\Biggl\{\prod_{j=1,j\neq i}^L F_j^{n_j-m_j}(s)[\bar{F}_j(\tau)-\bar{F}_j(s)]^{m_j-l_j}\bar{F}_j^{l_j}(\tau)\Biggr\}\\[5pt]&\quad \times[\bar{F}_i(s)-\bar{F}_i(s+\delta)]F_i^{n_i-m_i-1}(s)[\bar{F}_i(\tau)-\bar{F}_i(s)]^{m_i-l_i}\bar{F}_i^{l_i}(\tau),\end{align*}

which in turn implies that

(2.11) \begin{align} &\mathbb{E}( X_i(T)\mid T\leq \tau) \notag \\[5pt] & =\dfrac{n_i}{1-\bar{F}_{T}(\tau)}\sum_{m_1=0}^{n_1}\!\cdots\!\sum_{m_i=0}^{n_i-1}\!\cdots\!\sum_{m_L=0}^{n_L}\sum_{l_1=0}^{m_1}\!\cdots\!\sum_{l_L=0}^{m_L}[\Phi(m_1,\ldots, m_L)-\Phi(l_1,\ldots, l_L)]\nonumber\\[5pt]& \quad \times\Biggl[\prod_{j=1,j\neq i}^L\binom{n_j}{m_j}\binom{m_j}{l_j}\Biggr] \binom{n_i-1}{m_i}\binom{m_i}{l_i}\notag \\[5pt]& \quad \times \int_{0}^{\tau}\Biggl\{\prod_{j=1,j\neq i}^L F_j^{n_j-m_j}(s)[\bar{F}_j(\tau)-\bar{F}_j(s)]^{m_j-l_j}\bar{F}_j^{l_j}(\tau)\Biggr\}\nonumber\\[5pt]& \quad \times F_i^{n_i-m_i-1}(s)[\bar{F}_i(\tau)-\bar{F}_i(s)]^{m_i-l_i}\bar{F}_i^{l_i}(\tau) \,{\mathrm{d}} F_i(s).\end{align}

It is worth noting that if the structure of the system is k-out-of-n whose components are independent, then (2.10) and (2.11) are reduced to the results that appeared in Eryilmaz [Reference Eryilmaz7].

For given values $n_i, c_i, c_i^*, M_i$ , $i=1,\ldots, L$ , $c^{**}$ and $\tau$ , we aim to determine the optimal values of $v_i$ under the constraints $n_iv_i\leq M_i$ , $ i=1, \ldots, L$ , such that the mean cost rate function $\mathrm{Cost}_2$ is minimized.

In the following, we present two examples to examine the aforementioned theoretical results.

Example 2.1. Consider the system depicted in Figure 1, given in Feng et al. [Reference Feng, Patelli, Beer and Coolen16] and Eryilmaz et al. [Reference Eryilmaz, Coolen and Coolen-Maturi10]. The system consists of six components in which components 1, 2, and 5 are of type 1 and components 3, 4, and 6 are of type 2. The survival signature of the system is presented in Table 1. Assume that the dependency structure of the component lifetimes is modeled by a parametric family of copulas known as the Gumbel–Hougaard family, defined as

\begin{align*}&\hat{C}(u_1,\ldots, u_n)=\exp\bigl(-[(\!-\!\ln u_1)^{\alpha})+\cdots+(\!-\!\ln u_n)^{\alpha}) ] ^{1/\alpha} \bigr), \end{align*}

where $\alpha\geq 1$ is the dependency parameter in the family. The value $\alpha=1$ corresponds to the independent condition. Let the component lifetimes of the two types follow exponential distributions with reliability functions $\bar{F}_i(t)={\mathrm{e}}^{-t\theta_i}$ , where we assume that $\theta_1=0.2$ and $\theta_2=0.3$ . If there are $M_1=9$ and $M_2=6$ components from type 1 and type 2, respectively, as spares, then $v_1\in\{0,1,2,3\}$ and $ v_2\in\{0,1,2\}$ . To find the optimal number of redundant components for each type, we use the following values for the replacement costs: $c_1=3$ , $c_2=2$ , $c_1^*=1.5$ , $c_2^*=1$ , and $ c^{**}=10$ . To compute the numerator of (2.2), we need to compute $\mathbb{E}( X_i(T)), i=1, 2$ . From Lemma 2.1 and Theorem 2.1, we have

\begin{align*}&\mathbb{E}( X_1(T))=n_1\int_{0}^{\infty} \lim_{\delta\rightarrow 0}\dfrac{1}{\delta}\sum_{m_1=0}^{n_1-1}\sum_{m_2=0}^{n_2}\Phi(m_1+1, m_2) \binom{n_1-1}{m_1}\binom{n_2}{m_2}A_{\mathbf{m}}^{(1)}(t,\delta) \,{\mathrm{d}} t,\end{align*}

where

\begin{align*}A_{\mathbf{m}}^{(1)}(t, \delta)&={\sum_{j_1=0}^{n_1-m_1-1} \sum_{j_2=0}^{n_2-m_2}(\!-\!1)^{j_1+ j_2}\binom{n_1-m_1-1}{j_1}\binom{n_2-m_2}{j_2}} \\[5pt]&\quad \times \bigl[ {\mathrm{e}}^{-[(m_1+j_1+1)(t \theta_1)^{\alpha}+(m_2+j_2)(t \theta_2)^{\alpha}]^{1/\alpha}}-{\mathrm{e}}^{-[(m_1+j_1)(t \theta_1)^{\alpha}+((t+\delta)\theta_1)^{\alpha}+(m_2+j_2)(t \theta_2)^{\alpha}]^{1/\alpha}}\bigr].\end{align*}

Figure 1.

The system in Example 2.1 with two types of components.

Table 1.

Survival signature of the system in Figure 1.

Thus we get

\begin{align*}&\mathbb{E}( X_1(T))\\[5pt] & =n_1 \sum_{m_1=0}^{n_1-1}\sum_{m_2=0}^{n_2}\binom{n_1-1}{m_1}\binom{n_2}{m_2}\Phi(m_1+1, m_2)\\[5pt] & \quad \times \sum_{j_1=0}^{n_1-m_1-1} \sum_{j_2=0}^{n_2-m_2} (\!-\!1)^{j_1+ j_2}\binom{n_1-m_1-1}{j_1}\binom{n_2-m_2}{j_2}\\[5pt]& \quad \times \theta_1^{\alpha}\bigl[(m_1+j_1+1)\theta_1^{\alpha}+(m_2+j_2)\theta_2^{\alpha}\bigr]^{-1}.\end{align*}

Similarly, we have

\begin{align*}&\mathbb{E}( X_2(T))\\[5pt] & =n_2 \sum_{m_1=0}^{n_1}\sum_{m_2=0}^{n_2-1}\binom{n_1}{m_1}\binom{n_2-1}{m_2}\Phi(m_1, m_2+1)\\[5pt] & \quad\times \sum_{j_1=0}^{n_1-m_1} \sum_{j_2=0}^{n_2-m_2-1} (\!-\!1)^{j_1+ j_2}\binom{n_1-m_1}{j_1}\binom{n_2-m_2-1}{j_2}\\[5pt]& \quad \times\theta_2^{\alpha}\bigl[(m_1+j_1)\theta_1^{\alpha}+(m_2+j_2+1)\theta_2^{\alpha}\bigr]^{-1}.\end{align*}

Also, the denominator of (2.2) can be written as follows:

\begin{align*} \mathbb{E}( T_R)&=\int_0^{\infty} \bar{F}_{T_R}(t) \,{\mathrm{d}} t\\[5pt] &={\sum_{l_1=0}^{n_1}\sum_{l_2=0}^{n_2}}\binom{n_1}{l_1} \binom{n_2}{l_2}\Phi(l_1, l_2){\sum_{i_1=0}^{n_1-l_1}\sum_{i_2=0}^{n_2-l_2}}(\!-\!1)^{i_1+ i_2}\binom{n_1-l_1}{i_1} \binom{n_2-l_2}{i_2} \nonumber\\[5pt]&\quad \times\int_0^{\infty} {\mathrm{e}}^{- [(i_1+l_1)(\!-\!\ln(1-(1-\exp[-t \theta_1])^{v_1+1}))^{\alpha}+(i_2+l_2)(\!-\!\ln(1-(1-\exp[-t \theta_2])^{v_2+1}))^{\alpha} ]^{1/\alpha}} \,{\mathrm{d}} t,\end{align*}

which should be evaluated numerically by suitable softwares such as Mathematica. The values of $\mathrm{Cost}_1(\mathbf{v})$ for different combinations of $v_1$ and $ v_2$ are presented in Table 2 for two values $\alpha=2$ (dependent components) and $\alpha=1$ (independent components). It is seen that $v_1=2$ and $v_2=0$ are the optimal choices for the number of redundant components of types 1 and 2, respectively, under the criterion $\mathrm{Cost}_1$ in the case $\alpha=2$ , and $v_1=3$ and $v_2=0$ are the optimal numbers in the case $\alpha=1$ .

Table 2.

The values of $\mathrm{Cost}_1(\mathbf{v})$ and $\mathrm{Cost}_2(\mathbf{v})$ in Example 2.1 in the case of dependent components $(\alpha=2)$ and independent components $(\alpha=1)$ .

Suppose that the described system is maintained under the aforementioned age replacement policy, where we assume that $\tau=2$ , i.e. the replacement time of the system is $\min\!(T_R, 2)$ . From equations (2.7) and (2.8) the mean number of failed components of ith type at time $\tau$ , before system failure, is evaluated by the following expression:

\begin{align*}&\mathbb{E}( N_i(\tau)\mid T>\tau)\\[5pt]& =\dfrac{1}{\bar{F}_T(\tau)}\sum_{j_1=0}^{n_1}\sum_{j_2=0}^{n_2} j_i \Phi(n_1-j_1, n_2-j_2)\binom{n_1}{j_1} \binom{n_2}{j_2}{\sum_{b_1=0}^{j_1}\sum_{b_2=0}^{j_2}}(\!-\!1)^{b_1+ b_2}\binom{j_1}{b_1} \binom{j_2}{b_2} \\[5pt]& \quad \times \exp\bigl[-\tau\bigl( \theta_1^{\alpha}(n_1-j_1+b_1)+\theta_2^{\alpha}(n_2-j_2+b_2) \bigr)^{1/\alpha}\bigr], \quad i=1, 2,\end{align*}

where from (1.3) we get

\begin{align*}\bar{F}_T(\tau)&={\sum_{l_1=0}^{n_1}\sum_{l_2=0}^{n_2}}{\sum_{i_1=0}^{n_1-l_1}\sum_{i_2=0}^{n_2-l_2}}(\!-\!1)^{i_1+ i_1}\binom{n_1}{l_1} \binom{n_2}{l_2}\binom{n_1-l_1}{i_1} \binom{n_2-l_2}{i_2}\Phi(l_1, l_2)\nonumber\\[5pt]&\quad \times \exp\bigl[-\tau\bigl((i_1+l_1)\theta_1^{\alpha}+(i_2+l_2)\theta_2^{\alpha} \bigr)^{1/\alpha}\bigr].\end{align*}

Next, for the mean number of failed components at the time of the system failure given that the system has failed before $\tau$ , we have

\begin{align*}&\mathbb{E}( X_1(T)\mid T\leq \tau)\\[5pt]& =\dfrac{n_1}{1-\bar{F}_T(\tau)}\sum_{m_1=0}^{n_1-1}\sum_{m_2=0}^{n_2}\sum_{l_1=0}^{m_1}\sum_{l_2=0}^{m_2}[\Phi(m_1, m_2)-\Phi(l_1, l_2)]\binom{n_1-1}{m_1}\binom{m_1}{l_1}\binom{n_2}{m_2}\binom{m_2}{l_2}\\& \quad \times {\sum_{j_1=0}^{n_1-m_1-1}\sum_{j_2=0}^{n_2-m_2}}(\!-\!1)^{j_1+ j_2}\binom{n_1-m_1-1}{j_1} \binom{n_2-m_2}{j_2}\\[5pt] & \quad\times {\sum_{d_1=0}^{m_1-l_1}\sum_{d_2=0}^{m_2-l_2}}(\!-\!1)^{d_1+d_2}\binom{m_1-l_1}{d_1} \binom{m_2-l_2}{d_2}\\[5pt]& \quad \times \dfrac{\theta_1^{\alpha}\bigl[ {\mathrm{e}}^{-\tau((l_1+d_1)\theta_1^{\alpha}+(l_2+d_2)\theta_2^{\alpha} )^{1/\alpha}}- {\mathrm{e}}^{-\tau((m_1+j_1+1)\theta_1^{\alpha}+(m_2+j_2)\theta_2^{\alpha} )^{1/\alpha}}\bigr]}{(m_1-l_1+j_1-d_1+1)\theta_1^{\alpha}+(m_2-l_2+j_2-d_2)\theta_2^{\alpha}}\end{align*}

and

\begin{align*}&\mathbb{E}( X_2(T)\mid T\leq \tau)\\[5pt]& =\dfrac{n_2}{1-\bar{F}_T(\tau)}\sum_{m_1=0}^{n_1}\sum_{m_2=0}^{n_2-1}\sum_{l_1=0}^{m_1}\sum_{l_2=0}^{m_2}[\Phi(m_1, m_2)-\Phi(l_1, l_2)]\binom{n_1}{m_1}\binom{m_1}{l_1}\binom{n_2-1}{m_2}\binom{m_2}{l_2}\\& \quad \times {\sum_{j_1=0}^{n_1-m_1}\sum_{j_2=0}^{n_2-m_2-1}}(\!-\!1)^{j_1+ j_2}\binom{n_1-m_1}{j_1} \binom{n_2-m_2-1}{j_2} \\ & \quad\times {\sum_{d_1=0}^{m_1-l_1}\sum_{d_2=0}^{m_2-l_2}}(\!-\!1)^{d_1+d_2}\binom{m_1-l_1}{d_1} \binom{m_2-l_2}{d_2}\\[5pt]& \quad \times \dfrac{\theta_2^{\alpha}\bigl[ {\mathrm{e}}^{-\tau((l_1+d_1)\theta_1^{\alpha}+(l_2+d_2)\theta_2^{\alpha} )^{1/\alpha}}- {\mathrm{e}}^{-\tau((m_1+j_1)\theta_1^{\alpha}+(m_2+j_2+1)\theta_2^{\alpha} )^{1/\alpha}}\bigr]}{(m_1-l_1+j_1-d_1)\theta_1^{\alpha}+(m_2-l_2+j_2-d_2+1)\theta_2^{\alpha}}.\end{align*}

By substituting these results in (2.6), the mean cost rate of replacement strategy can be evaluated. In Table 2 the values of $\mathrm{Cost}_2(\mathbf{v})$ are calculated for different combinations of $v_1$ and $v_2$ , for both dependent and independent situations. It follows from the results of the table that for $v_1=1$ and $v_2=0$ , the mean cost rate $\mathrm{Cost}_2(\mathbf{v})$ is minimized, in both cases $\alpha=1,2$ .

Table 3.

The optimum values of $\mathbf{v}$ by minimizing $\mathrm{Cost}_i(v_1,v_2)$ , $i=1,2$ , for different $\alpha$ in Example 2.1.

Table 4.

The optimum values of $\mathbf{v}$ by minimizing $\mathrm{Cost}_1$ for different costs in Example 2.1.

In order to investigate the robustness of our strategies concerning the model parameters, we calculate some numerical results based on these parameters. The results in Table 3 shows the effect of the dependency parameter $\alpha$ on the optimal values of $v_1$ and $v_2$ , for different values of $\alpha$ . As can be seen, when $\alpha$ increases (i.e. we get far from independence), the number of redundant components decreases based on the objective function $\mathrm{Cost}_1(v_1,v_2)$ but remain unchanged under the $\mathrm{Cost}_2(v_1,v_2)$ . This makes sense since under more dependency the MTTF is increased and the need to spare components reduces. Also, it should be noted that the higher the $\alpha$ , the lower the mean cost rates. To explore the sensitivity of the proposed models with respect to component costs, $\mathbf{c}=(c_1,c_2)$ and $\mathbf{c}^*=(c_1^*,c_2^*)$ , we have provided some numerical results in Table 4 for $\alpha=2$ . In the top left panel of the table, we observe that for fixed values of $c_1^*=1.5$ and $c_2^*=1$ , the increase in costs $c_1$ and $c_2$ results in a reduction in the number of optimal values of $v_1$ and $v_2$ . In the bottom left panel of the table, the costs $c_i$ and $c^*_i$ , $i=1,2$ , of the two types are swapped. In this case, when the costs $c_1^*$ and $c_2^*$ are fixed as $c_1^*=1$ and $c_2^*=1.5$ , we again observe that the increase in the costs $c_1$ and $c_2$ results in a decline in the number of optimal values of $v_1$ and $v_2$ . In the top right panel of the table it can be seen that for fixed values of renewing failed components as $c_1=6, c_2=5.5$ , the decrease in the costs $c_1^*$ and $c_2^*$ results in an increase to the number of optimal values of $v_1$ and $v_2$ . As shown in the bottom right panel, the same result holds by swapping the costs of the components of type 1 and type 2. It is seen that in all four parts of Table 4 the value of costs and the numbers of spare components are inversely related to each other. Table 5 shows the behavior of the number of redundant components from another point of view. In the left panel of the table we have kept $c_2 (c^*_2)$ constant and have increased the values of $c_1 (c^*_1)$ . In fact, we have assumed that $c_1=\omega c_2$ and $c_1^*=\omega c_2^*$ for $\omega=1.5,2,3, \ldots, 10$ . As seen, when $\omega$ increases the optimal value of the redundant component $v_1$ decreases and the optimal value of $v_2$ increases. In the right panel of the table we exchange the costs of type 1 and type 2, i.e. we assume $c_2=\omega c_1$ and $c_2^*=\omega c_1^*$ . In this case we observe no changes in the number of redundant components $v_1, v_2$ when $\omega$ increases.

Table 5.

The optimum values of $\mathbf{v}$ by minimizing $\mathrm{Cost}_1$ for different costs in Example 2.1.

In this example the distributions of the component lifetimes are ordered such that $\bar{F}_1(t)\geq \bar{F}_2(t)$ , for all $t>0$ , i.e. the reliability (and subsequently the MTTF) of the components of type 1 is more than type 2. Note that according to the system structure, it is revealed that the components of type 1 are generally in more critical positions than those of type 2. Hence one should intuitively expect that the optimal solution, according to the cost criterion, would be the case in which one allocates more components of type 1 than type 2.

To see whether this fact affects the number of $v_i$ , we let $\bar{F}_2^*(t)={\mathrm{e}}^{-0.07 t^{2}}$ , $t>0$ , be the reliability function of the components of type 2. In this case the two reliability functions cross each other such that $\bar{F}_1(t)<\bar{F}_2^*(t)$ for $t<2.86$ and $\bar{F}_1(t)>\bar{F}_2^*(t)$ for $t>2.86$ . Note that the MTTF for components of type 2 in this new case is the same as the previous one. By fixing the other parameters as before, we get the results given in Table 6. We see that although the distributions cross each other, the optimal numbers of components in Table 6 are mostly the same as those in Table 3, perhaps since the MTTFs have not been changed in either case.

Table 6.

The optimum values of $\mathbf{v}$ by minimizing $\mathrm{Cost}_1(v_1,v_2)$ , for different values of $\alpha$ in Example 2.1.

As a final point, to see the effect of the survival copula on the optimal numbers of $v_1$ and $v_2$ , we suppose that the dependency structure is followed by the Clayton copula with the following form:

\begin{align*}\hat{C}(u_1,\ldots, u_n)=\bigl(u_1^{-1/\alpha}+\cdots + u_n^{-1/\alpha}-n+1 \bigr)^{-\alpha}, \quad \alpha>0.\end{align*}

The parameter $\alpha$ manages the dependency degree of the copula, and the limiting case $\alpha=0$ gives the independence. We obtain the optimal values of redundant components for some values of $\alpha$ in Table 7. As can be seen, by increasing $\alpha$ , the values of $v_1$ and $v_2$ and also the mean cost rate show increase, which is in contradiction to the results in Table 3. Hence the output of the optimization problem strongly pertains to the functional structure of dependence, not only to the dependency parameter.

Table 7.

The optimum values of $\mathbf{v}$ by minimizing $\mathrm{Cost}_1(v_1,v_2)$ for different $\alpha$ under the Clayton copula in Example 2.1.

In the following example we consider an 8-component system consisting of three types of components. For fixed values of $v_i$ , we minimize the function $\mathrm{Cost}_1$ and also the function $\mathrm{Cost}_2$ in the case that the replacement time of the unfailed system, $\tau$ , is considered as the variable of interest.

Example 2.2. Consider the system depicted in Figure 2, given in Huang et al. [Reference Huang, Coolen and Coolen-Maturi19]. The system has eight components from which three components (1, 2, and 3) are of type 1, three components (4, 5, and 7) are of type 2, and two components (6 and 8) are of type 3. The values of the system survival signature are computed in [Reference Huang, Coolen and Coolen-Maturi19], to which we refer the reader for the details.

Figure 2.

The system in Example 2.2.

Suppose here that all the components are independent, where the components of type i have common Weibull reliability functions $\bar{F}_i(t)={\mathrm{e}}^{-\beta_i t^{\alpha_i}}$ , $\alpha_i, \beta_i>0$ for $i=1,2,3$ . In Table 8 we present the mean cost rate $\mathrm{Cost}_1(v_1, v_2,v_3)$ for given values of $\beta_1=3$ , $\beta_2=4$ , $ \beta_3=2$ , $\alpha_1=2$ , $\alpha_2=3$ , $ \alpha_3=1$ when the cost parameters are $c_1=1.5$ , $ c_2=1$ , $c_3=2, c_1^*=0.75$ , $c_2^*=0.4$ , $c_3^*=1$ , and $c^{**}=10$ . Assume that we have $M_1=7$ , $M_2=4$ , and $M_3=5$ components from types 1, 2, and 3, respectively, as spares. Hence we can choose $v_1\in \{0, 1, 2 \}$ , $v_2\in \{0, 1\}$ , and $v_3\in \{0, 1, 2\}$ as the redundancy for each type, respectively.

Table 8.

The values of $\mathrm{Cost}_1(\mathbf{v})$ and $\mathrm{Cost}_2(\mathbf{v})$ in Example 2.2.

In the left panel of Table 8, the values of $\mathrm{Cost}_1$ are computed for different combinations of $v_i$ . As can be seen, by adding $v_1=0$ , $v_2=1$ , and $ v_3=0$ as the redundant components to groups 1, 2, and 3, respectively, we get the minimum value for the mean cost rate $\mathrm{Cost}_1(v_1, v_2,v_3)$ .

Under the assumption that $\tau$ is the variable of interest, in the right panel of the table we have minimized $\mathrm{Cost}_2(v_1, v_2,v_3)$ in terms of $\tau$ and have reported the optimum value of $\tau$ , in the case that the values of $v_i$ are kept fixed and known. It is observed that among all minimized values of $\mathrm{Cost}_2$ , the least value is obtained for the case that the number of redundant components are $v_1=0$ , $v_2=1$ , and $ v_3=0$ , for which we have $\tau=0.375$ .

3.1. Cost function at system failure

In a similar manner to Section 2.1, the mean cost rate function for system failure is defined as

(3.1) \begin{align}\mathrm{Cost}_3(\mathbf{n})=\dfrac{\sum_{i=1}^Lc_i\mathbb{E}( X_i(T))+\sum_{i=1}^L c_i^*\mathbb{E}( n_i-X_i(T))+c^{**}}{\mathbb{E}( T)},\end{align}

where $\mathbf{n}=(n_1,\ldots, n_L)$ , and

(3.2) \begin{align}\mathbb{E}( X_i(T))=n_i \sum_{m_1=1}^{n_1}\!\cdots\!\sum_{m_i=0}^{n_i-1}\!\cdots\!\sum_{m_L=1}^{n_L}\binom{n_1}{m_1}\!\cdots\! \binom{n_i-1}{m_i}\!\cdots\!\binom{n_L}{m_L}\int_{0}^{\infty}\lim_{\delta\rightarrow 0}\dfrac{A_{\mathbf{m}}^{(i)}(t,\delta)}{\delta} \,{\mathrm{d}} t,\end{align}

in which $A_{\mathbf{m}}^{(i)}(t,\delta)$ is introduced in (2.3). For the independent components, (3.2) reduces to the following expression:

(3.3) \begin{align}\mathbb{E}( X_i(T))=n_i \int_0^{\infty} \prod_{l=1,l\neq i}^L (1-(1-\bar{F}_l(t))^{n_l}) \,{\mathrm{d}} F_{i}(t).\end{align}

If $L=1$ then the system becomes a parallel system with $n_1$ components, and hence in this case $\mathbb{E}( X_1(T))=n_1$ .

For the considered series-parallel system in which the components of subsystems are independent, Eryilmaz et al. [Reference Eryilmaz, Özkurt and Rekan11] gained a similar result for $\mathbb{E}( X_i(T))$ in (3.3). Subsequently, they found the optimal numbers of components in each subsystem based on the minimization of cost function (3.1) under the constraints on the total allotted cost for replacing failed components and the total allotted cost for rejuvenation of unfailed ones. Hence our results in this subsection may be considered as an extension of their work to the case of dependent components. Also, Dembinska and Eryilmaz [Reference Dembinska and Eryilmaz5] discussed a similar problem for the case that the lifetime distributions of components are discrete; in particular, they obtained some results for discrete phase-type distribution.

3.2. Cost function based on preventive replacement

The mean cost rate function of the system for age replacement at time $\min\!(\tau, T)$ is defined as

\begin{align*}\mathrm{Cost}_4(\mathbf{n})=\dfrac{M_1(\mathbf{n})\,\mathbb{P}(T\leq \tau)+M_2(\mathbf{n})\,\mathbb{P}(T> \tau)}{\mathbb{E}(\!\min\!(\tau,T))},\end{align*}

where

\begin{align*}M_1(\mathbf{n})=\sum_{i=1}^L c_i \mathbb{E}( X_i(T)\mid T\leq \tau)+\sum_{i=1}^L c^*_i \mathbb{E}( n_i-X_i{(T)}\mid T\leq \tau)+c^{**}\end{align*}

and

\begin{align*}M_2(\mathbf{n})=\sum_{i=1}^L c_i \mathbb{E}( N_i{(\tau)}\mid T>\tau)+\sum_{i=1}^L c^*_i \mathbb{E}( n_i-N_i{(\tau)}\mid T>\tau).\end{align*}

Using the formula for the survival signature of the series-parallel system, from the results given in Section 2.2, we get the following expressions:

\begin{align*}\mathbb{E}( N_i(\tau)\mid T>\tau)=\dfrac{1}{\bar{F}(\tau)}\sum_{j_1=0}^{n_1-1}\!\cdots\!\sum_{j_L=0}^{n_L-1} j_i \binom{n_1}{j_1}\!\cdots\! \binom{n_L}{j_L}B(\tau, j_1,\dots,j_L)\end{align*}

and for $n_i\geq 2$

(3.4) \begin{align}&\mathbb{E}( X_i(T)\mid T\leq \tau)=\dfrac{n_i}{1-\bar{F}_T(\tau)} \Biggl[\sum_{m_1=1}^{n_1}\!\cdots\!\sum_{m_i=1}^{n_i-1}\!\cdots\!\sum_{m_L=1}^{n_L}{\sum_{l_1=0}^{m_1}\!\cdots\!\sum_{l_L=0}^{m_L}}\Biggl[\prod_{j=1,j\neq i}^L\binom{n_j}{m_j}\binom{m_j}{l_j}\Biggr] \\[5pt]& \times\binom{n_i-1}{m_i}\binom{m_i}{l_i}\int_{0}^{\tau}\lim_{\delta\rightarrow 0}\dfrac{1}{\delta} A_{\mathbf{m}, \mathbf{l}}^{(i)}(s,s+\delta, \tau) \,{\mathrm{d}} s - \sum_{m_1=1}^{n_1}\!\cdots\!\sum_{m_i=1}^{n_i-1}\!\cdots\!\sum_{m_L=1}^{n_L}{\sum_{l_1=1}^{m_1}\!\cdots\!\sum_{l_L=1}^{m_L}}\nonumber\\[5pt]& \quad\Biggl[\prod_{j=1,j\neq i}^L\binom{n_j}{m_j}\binom{m_j}{l_j}\Biggr]\binom{n_i-1}{m_i}\binom{m_i}{l_i}\int_{0}^{\tau}\lim_{\delta\rightarrow 0}\dfrac{1}{\delta} A_{\mathbf{m}, \mathbf{l}}^{(i)}(s,s+\delta, \tau) \,{\mathrm{d}} s \Biggr].\end{align}

If $n_i=1$ then it is easily deduced that

\begin{equation*}\mathbb{E}( X_i(T)\mid T\leq \tau)=\dfrac{{F}_i(\tau)}{1-\bar{F}_T(\tau)}.\end{equation*}

Example 3.1. Consider a series-parallel system with $L=3$ subsystems and assume that there are $M_1=2, M_2=3$ , and $M_3=3$ components from types 1, 2, and 3, respectively, to construct the system. Suppose that the joint reliability function of the component lifetimes follow the multivariate Pareto model given by

\begin{align*}&\mathbb{P}\bigl( T_1^{(1)}>t_1^{(1)}, \ldots, T_{n_1}^{(1)}>t_{n_1}^{(1)}, \ldots, T_1^{(L)}>t_L^{(L)}, \ldots, T_{n_L}^{(L)}>t_{n_L}^{(L)}\bigr)\\[5pt] &\quad =\Biggl[ 1+\theta_1\sum_{i=1}^{n_1}t_i^{(1)}+\cdots+\theta_L\sum_{i=1}^{n_L}t_i^{(L)} \Biggr]^{-\alpha} \end{align*}

for $\theta_i>0, i=1, \ldots, L$ , and $\alpha>0$ . In fact the corresponding survival copula is

\begin{align*}\hat{C}(u_1,\ldots, u_n)=\bigl(u_1^{-1/\alpha}+\cdots + u_n^{-1/\alpha}-n+1 \bigr)^{-\alpha},\end{align*}

and the marginal reliability functions of the components in the subsystems are $\bar{F}_i(t)=(1+\theta_i t)^{-\alpha}$ , $i=1,2, \ldots, L$ .

First note that for the described system we have

\begin{align*}&\int_{0}^{\infty}\lim_{\delta\rightarrow 0}\dfrac{A_{\mathbf{m}}^{(i)}(t,\delta)}{\delta} \,{\mathrm{d}} t\\[5pt]& ={\sum_{j_1=0}^{n_1-m_1}\!\cdots\!\sum_{j_i=0}^{n_i-m_i-1}\!\cdots\!\sum_{j_L=0}^{n_L-m_L}}(\!-\!1)^{j_1+\cdots + j_L}\binom{n_1-m_1}{j_1}\!\cdots\! \binom{n_i-m_i-1}{j_i}\!\cdots\!\binom{n_L-m_L}{j_L}\\[5pt]& \quad \times \dfrac{\theta_i}{\theta_i(m_i+j_i+1)+\sum_{l=1, l\neq i}^{L}\theta_l(m_l+j_l)}.\end{align*}

By replacing these expressions in (3.2), $\mathbb{E}( X_i(T))$ , $i=1, 2, 3$ are obtained.

Let ${\mathbf{\theta}}=(0.4, 0.2, 0.3)$ , $\mathbf{c}=(1.5, 2, 3)$ , $\mathbf{c}^*=(0.3, 0.75, 1)$ , $c^{**}=8$ , and $\alpha=2$ . The values of the mean cost rate function $\mathrm{Cost}_3$ are obtained for all combinations of $n_1, n_2$ , and $n_3$ , such that $n_1\in\{1, 2\}$ , $n_2\in\{1, 2, 3\}$ , and $n_3\in\{1,2,3\}$ . Also, under the age replacement strategy in $\tau=1$ , the values of mean cost rate function $\mathrm{Cost}_4$ are calculated for different values $n_1$ , $n_2$ , and $n_3$ .

Table 9.

The values of $\mathrm{Cost}_3(\mathbf{n})$ and $\mathrm{Cost}_4(\mathbf{n})$ in Example 3.1.

To compute $\mathrm{Cost}_4(\tau)$ , we use the following simplified expressions:

\begin{align*} &\mathbb{E}( N_i(\tau)\mid T>\tau)\\[5pt] & =\dfrac{1}{\bar{F}_T(\tau)}\sum_{j_1=0}^{n_1-1}\sum_{j_2=0}^{n_2-1}\sum_{j_3=0}^{n_3-1} j_i \binom{n_1}{j_1} \binom{n_2}{j_2} \binom{n_3}{j_3}{\sum_{b_1=0}^{j_1}\sum_{b_2=0}^{j_2}\sum_{b_3=0}^{j_3}}(\!-\!1)^{b_1+ b_2+ b_L}\\[5pt]& \quad \binom{j_1}{b_1}\binom{j_2}{b_2}\binom{j_3}{b_3}[1+\theta_1\tau(n_1-j_1+b_1)+\theta_2\tau(n_2-j_2+b_2)+\theta_3\tau(n_3-j_3+b_3)]^{-\alpha},\end{align*}

for $i=1, 2, 3$ , where

\begin{align*}&\bar{F}_T(\tau)\\[5pt]& ={\sum_{l_1=1}^{n_1}\sum_{l_2=1}^{n_2}\sum_{l_3=1}^{n_3}}{\sum_{i_1=0}^{n_1-l_1}\sum_{i_2=0}^{n_2-l_2}\sum_{i_3=0}^{n_3-l_3}}(\!-\!1)^{i_1+ i_2+ i_3}\binom{n_1}{l_1}\binom{n_2}{l_2}\binom{n_3}{l_3}\binom{n_1-l_1}{i_1}\binom{n_2-l_2}{i_2} \binom{n_3-l_3}{i_3}\nonumber\\[5pt]& \quad (1+\theta_1 \tau(i_1+l_1)+\theta_2 \tau(i_2+l_2)+\theta_3 \tau(i_3+l_3))^{-\alpha}.\end{align*}

Also, we obtain $\mathbb{E}( X_i(T)\mid T\leq \tau), i=1, 2, 3$ by placing the following quantity in (3.4):

\begin{align*}&\int_{0}^{\tau}\lim_{\delta\rightarrow 0}\dfrac{1}{\delta} A_{\mathbf{m}, \mathbf{l}}^{(i)}(s,s+\delta, \tau) \,{\mathrm{d}}s \\[5pt]& ={\sum_{j_1=0}^{n_1-m_1}\!\cdots\!\sum_{j_i=0}^{n_i-m_i-1}\!\cdots\!\sum_{j_L=0}^{n_L-m_L}}(\!-\!1)^{\,j_1+\cdots + j_L}\binom{n_1-m_1}{j_1}\!\cdots\! \binom{n_i-m_i-1}{j_i}\!\cdots\!\binom{n_L-m_L}{j_L}\\& \quad \times {\sum_{d_1=0}^{m_1-l_1}\!\cdots\!\sum_{d_i=0}^{m_i-l_i}\!\cdots\!\sum_{d_L=0}^{m_L-l_L}}(\!-\!1)^{d_1+\cdots + d_L}\binom{m_1-l_1}{d_1}\!\cdots\! \binom{m_L-l_L}{d_L}\\[5pt]& \quad \times \dfrac{\theta_i\bigl[\bigl(1+\tau(\!\sum_{k=1}^L \theta_k(l_k+d_k)) \bigr)^{-\alpha}-\bigl(1+\tau(\theta_i(m_i+j_i+1)+\sum_{k=1, \neq i}^L \theta_k(m_k+j_k))\bigr)^{-\alpha}\bigr]}{\theta_i(m_i-l_i+j_i-d_i+1)+\sum_{k=1, \neq i}^L \theta_k(m_k-l_k+j_k-d_k)}.\end{align*}

The results are given in Table 9. It is seen from the results that based on the objective function $\mathrm{Cost}_3(.)$ the optimal series-parallel system has $n_1=2$ , $n_2=3$ , $n_3=2$ components. Since the reliability of type 2 is greater than the other two types, it is expected that more components for the second subsystem will lead to a reduction in the mean cost rate of system failure. Also, $ n_1 = 2$ , $n_2 = 2$ , $n_3 = 2 $ are the optimal number of components in the subsystems so as to minimize the average cost rate of the age replacement policy.