Optimal two-threshold replacement policy in a cumulative damage model

Abstract

In this paper, a two-unit system with failure interactions is studied. The system is subject to two types of shocks (I and II) and the probabilities of these two shock types are age-dependent. A type I shock, rectified by a minimal repair, causes a minor failure of unit A and type II shock causes a complete system failure that calls for a replacement. Each unit A minor failure also results in an amount of damage to unit B. The damages to unit B caused by type I shocks can be accumulated to trigger a preventive replacement or a corrective replacement action. Besides, unit B with cumulative damage of level z may become minor failed with probability $\pi (z)$ at each unit A minor failure and rectified by a minimal repair. We consider a more general replacement policy. Under this policy, the system is preventively replaced at the Nth type I shock, or at the time when the total damage to unit B exceeds a pre-specified level Z (but less than a failure level K where $K>Z$) or is replaced correctively at first type II shock or when the total damage to unit B exceeding a failure level K, whichever occurs first. To minimize the expected cost per unit time, the optimal replacement policy $(N^{*}$, $Z^{*})$ is derived analytically and determined numerically. We also show that several previous maintenance models in the literature are special cases of our model.

Appendix

1.1 Proof of (6) + (7) + (8) + (9) = 1

$$\begin{aligned}&P_N +P_Z +P_K +P_{II}\\&\quad = G^{\left( N \right) }\left( Z \right) \int _0^\infty {{\bar{F}}_P (t)q(t)r(t)P_{N-1} (t)dt} +\sum _{j=0}^{N-1} {G^{(j)}\left( Z \right) \int _0^\infty {P_j (t)dF_P (t)} }\\&\qquad + \sum _{j=0}^{N-1} {\int _0^Z {\left[ {G_{j+1} \left( {K-y} \right) -G_{j+1} \left( {Z-y} \right) } \right] dG^{\left( j \right) }\left( y \right) } \times \int _0^\infty {{\bar{F}}_P (t)q(t)r(t)P_j (t)dt} }\\&\qquad +\sum _{j=0}^{N-1} {\int _0^Z {\bar{{G}}_{j+1} \left( {K-y} \right) dG^{(j)}\left( y \right) } \times \int _0^\infty {{\bar{F}}_P (t)q(t)r(t)P_j (t)dt}}\\&\quad = G^{\left( N \right) }\left( Z \right) \int _0^\infty {{\bar{F}}_P (t)q(t)r(t)P_{N-1} (t)dt} +\sum _{j=0}^{N-1} {G^{(j)}\left( Z \right) \int _0^\infty {P_j (t)dF_P (t)} }\\&\qquad +\sum _{j=0}^{N-1} {\left[ {G^{(j)}\left( Z \right) -G^{(j+1)}\left( Z \right) } \right] \int _0^\infty {{\bar{F}}_P (t)q(t)r(t)P_j (t)dt} }\\&\quad = G^{\left( N \right) }\left( Z \right) \int _0^\infty {{\bar{F}}_P (t)q(t)r(t)P_{N-1} (t)dt} +\sum _{j=0}^{N-1} {G^{(j)}\left( Z \right) \int _0^\infty {{\bar{F}}_P (t)q(t)r(t)P_j (t)dt}}\\&\qquad -\sum _{j=1}^N {G^{(j)}\left( Z \right) \int _0^\infty {{\bar{F}}_P (t)q(t)r(t)P_{j-1} (t)dt} } +\sum _{j=0}^{N-1} {G^{(j)}\left( Z \right) \int _0^\infty {P_j (t)dF_P (t)}}\\ \end{aligned}$$

$$\begin{aligned}&\quad \!= \!\sum _{j=0}^{N-1} {G^{(j)}\left( Z \right) \int _0^\infty {{\bar{F}}_P (t)q(t)r(t)P_j (t)dt} } \!-\!\sum _{j=1}^{N-1} {G^{(j)}\left( Z \right) \int _0^\infty {{\bar{F}}_P (t)q(t)r(t)P_{j-1} (t)dt}}\\&\qquad + \sum _{j=0}^{N-1} {G^{(j)}\left( Z \right) \int _0^\infty {P_j (t)dF_P (t)}}\\&\quad = \sum _{j=0}^{N-1} {G^{(j)}\left( Z \right) \int _0^\infty {P_j (t)dF_P (t)} } -\sum _{j=0}^{N-1} {G^{(j)}\left( Z \right) \int _0^\infty {{\bar{F}}_P (t)dP_j (t)}}\\&\quad = \sum _{j=0}^{N-1} {G^{(j)}\left( Z \right) \int _0^\infty {P_j (t)dF_P (t)} } -\sum _{j=0}^{N-1} {G^{(j)}\left( Z \right) \left[ {\int _0^\infty {P_j (t)dF_P (t)} -{\bar{F}}_P \left( 0 \right) P_j \left( 0 \right) } \right] }\\&\quad = 1 \end{aligned}$$

1.2 Derivation of (12)

$$\begin{aligned} E\left( {U_1 } \right)= & {} G^{(N)}\left( Z \right) \int _0^\infty {t{\bar{F}}_P (t)q(t)r(t)P_{N-1} (t)dt} +\sum _{j=0}^{N-1} {G^{(j)}\left( Z \right) \int _0^\infty {tP_j (t)dF_P (t)} }\\&+ \sum _{j=0}^{N-1} \int _0^Z {\left[ {G_{j+1} \left( {K-y} \right) -G_{j+1} \left( {Z-y} \right) } \right] dG^{\left( j \right) }\left( y \right) }\\&\times \int _0^\infty {t{\bar{F}}_P (t)q(t)r(t)P_j (t)dt}\\&+ \sum _{j=0}^{N-1} {\int _0^Z {\bar{{G}}_{j+1} \left( {K-y} \right) dG^{(j)}\left( y \right) } \times \int _0^\infty {t{\bar{F}}_P (t)q(t)r(t)P_j (t)dt} }\\ \!= & {} \!G^{(N)}\left( Z \right) \int _0^\infty {t{\bar{F}}_P (t)q(t)r(t)P_{N-1} (t)dt} +\sum _{j=0}^{N-1} {G^{(j)}\left( Z \right) \int _0^\infty {tP_j (t)dF_P (t)} }\\&+ \sum _{j=0}^{N-1} {\left[ {G^{(j)}\left( Z \right) -G^{(j+1)}\left( Z \right) } \right] \int _0^\infty {t{\bar{F}}_P (t)q(t)r(t)P_j (t)dt} }\\= & {} G^{(N)}\left( Z \right) \int _0^\infty {t{\bar{F}}_P (t)q(t)r(t)P_{N-1} (t)dt}\\&+ \sum _{j=0}^{N-1} {G^{(j)}\left( Z \right) \int _0^\infty {t{\bar{F}}_P (t)q(t)r(t)P_j (t)dt}}\\&-\sum _{j=1}^N {G^{(j)}\left( Z \right) \int _0^\infty {t{\bar{F}}_P (t)q(t)r(t)P_{j-1} (t)dt} } \!+\!\sum _{j=0}^{N-1} {G^{(j)}\left( Z \right) \int _0^\infty {tP_j (t)dF_P (t)}}\\ \end{aligned}$$

$$\begin{aligned}= & {} \sum _{j=0}^{N-1} {G^{(j)}\left( Z \right) \int _0^\infty {t{\bar{F}}_P (t)q(t)r(t)P_j (t)dt}}\\&-\sum _{j=1}^{N-1} {G^{(j)}\left( Z \right) \int _0^\infty {t{\bar{F}}_P (t)q(t)r(t)P_{j-1} (t)dt} } +\sum _{j=0}^{N-1} {G^{(j)}\left( Z \right) \int _0^\infty {tP_j (t)dF_P (t)}}\\= & {} -\sum _{j=0}^{N-1} {G^{(j)}\left( Z \right) \int _0^\infty {t{\bar{F}}_P (t)dP_j (t)} } +\sum _{j=0}^{N-1} {G^{(j)}\left( Z \right) \int _0^\infty {tP_j (t)dF_P (t)}}\\= & {} \sum _{j=0}^{N-1} {G^{(j)}\left( Z \right) \int _0^\infty {{\bar{F}}_P (t)P_j (t)dt} } -\sum _{j=0}^{N-1} {G^{(j)}\left( Z \right) \int _0^\infty {tP_j (t)dF_P (t)}}\\&+\sum _{j=0}^{N-1} {G^{(j)}\left( Z \right) \int _0^\infty {tP_j (t)dF_P (t)} }\\= & {} \sum _{j=0}^{N-1} {G^{(j)}\left( Z \right) \int _0^\infty {{\bar{F}}_P (t)P_j (t)dt} } \end{aligned}$$

1.3 Derivation of (14)

$$\begin{aligned} E\left( {R_1 } \right)= & {} c_1 P_N +c_2 P_Z +c_3 P_{II} +c_4 P_K +\sum _{j=1}^{N-1} {G^{\left( j \right) }\left( Z \right) \int _0^\infty {\alpha _j (t){\bar{F}}_P (t)q(t)r(t)P_{j-1} (t)dt}}\\&+\sum _{j=1}^{N-1} {\int _0^Z {\pi \left( y \right) \eta \left( y \right) dG^{\left( j \right) }\left( y \right) } \times \int _0^\infty {{\bar{F}}_P (t)q(t)r(t)P_{j-1} (t)dt}}\\= & {} c_1 G^{\left( N \right) }\left( Z \right) \int _0^\infty {{\bar{F}}_P (t)q(t)r(t)P_{N-1} (t)dt} +c_2 \left( {1-P_N -P_{II} -P_K } \right) \\&+\, c_3 \sum _{j=0}^{N-1} {G^{(j)}\left( Z \right) \int _0^\infty {P_j (t)dF_P (t)}}\\&+\, c_4 \sum _{j=0}^{N-1} {\int _0^Z {\bar{{G}}_{j+1} \left( {K-y} \right) dG^{(j)}\left( y \right) } \times \int _0^\infty {{\bar{F}}_P (t)q(t)r(t)P_j (t)dt}}\\&+ \sum _{j=1}^{N-1} {G^{\left( j \right) }\left( Z \right) \int _0^\infty {\alpha _j (t){\bar{F}}_P (t)q(t)r(t)P_{j-1} (t)dt}}\\&+ \sum _{j=1}^{N-1} {\int _0^Z {\pi \left( y \right) \eta \left( y \right) dG^{\left( j \right) }\left( y \right) } \times \int _0^\infty {{\bar{F}}_P (t)q(t)r(t)P_{j-1} (t)dt}}\\ \end{aligned}$$

$$\begin{aligned}= & {} c_2 +\left( {c_1 -c_2 } \right) G^{\left( N \right) }\left( Z \right) \int _0^\infty {{\bar{F}}_P (t)q(t)r(t)P_{N-1} (t)dt}\\&+ \left( {c_3 -c_2 } \right) \sum _{j=0}^{N-1} {G^{(j)}\left( Z \right) \int _0^\infty {P_j (t)dF_P (t)}}\\&+ \left( {c_4 -c_2 } \right) \sum _{j=0}^{N-1} {\int _0^Z {\bar{{G}}_{j+1} \left( {K-y} \right) dG^{(j)}\left( y \right) } \times \int _0^\infty {{\bar{F}}_P (t)q(t)r(t)P_j (t)dt}}\\&+ \sum _{j=1}^{N-1} {G^{\left( j \right) }\left( Z \right) \int _0^\infty {\alpha _j (t){\bar{F}}_P (t)q(t)r(t)P_{j-1} (t)dt}}\\&+ \sum _{j=1}^{N-1} {\int _0^Z {\pi \left( y \right) \eta \left( y \right) dG^{\left( j \right) }\left( y \right) } \times \int _0^\infty {{\bar{F}}_P (t)q(t)r(t)P_{j-1} (t)dt} } \end{aligned}$$

1.4 Proof of Lemma 1-(i)

First, we rewrite $A_N$ as follows.

$$\begin{aligned} A_N= & {} \frac{G^{\left( N \right) }\left( Z \right) \int _0^\infty {{\bar{F}}_P (t)q(t)r(t)P_{N-1} (t)dt} -G^{\left( {N+1} \right) }\left( Z \right) \int _0^\infty {{\bar{F}}_P (t)q(t)r(t)P_N (t)dt} }{G^{\left( N \right) }\left( Z \right) \int _0^\infty {{\bar{F}}_P (t)P_N (t)dt}}\\= & {} \left[ {1-\frac{G^{\left( {N+1} \right) }\left( Z \right) }{G^{\left( N \right) }\left( Z \right) }+\frac{G^{\left( {N+1} \right) }\left( Z \right) }{G^{\left( N \right) }\left( Z \right) }} \right] \frac{\int _0^\infty {{\bar{F}}_P (t)r(t)P_N (t)dt} }{\int _0^\infty {{\bar{F}}_P (t)P_N (t)dt}}\\&-\frac{G^{\left( {N+1} \right) }\left( Z \right) \int _0^\infty {{\bar{F}}_P (t)q(t)r(t)P_N (t)dt} }{G^{\left( N \right) }\left( Z \right) \int _0^\infty {{\bar{F}}_P (t)P_N (t)dt}}\\= & {} \left[ {1-\frac{G^{\left( {N+1} \right) }\left( Z \right) }{G^{\left( N \right) }\left( Z \right) }} \right] \left[ {\frac{\int _0^\infty {{\bar{F}}_P (t)r(t)P_N (t)dt} }{\int _0^\infty {{\bar{F}}_P (t)P_N (t)dt} }} \right] \\&-\left\{ {\frac{G^{\left( {N+1} \right) }\left( Z \right) }{G^{\left( N \right) }\left( Z \right) }\left[ {-\frac{\int _0^\infty {{\bar{F}}_P (t)p(t)r(t)P_N (t)dt} }{\int _0^\infty {{\bar{F}}_P (t)P_N (t)dt} }} \right] } \right\} \end{aligned}$$

We first proof ${\int _0^\infty {{\bar{F}}_P (t)r(t)P_N (t)dt} }/{\int _0^\infty {{\bar{F}}_P (t)P_N (t)dt} }$ is increasing in N.

Let

$$\begin{aligned} J_1 (T)= & {} \int _0^T {{\bar{F}}_P (t)r(t)P_{N+1} (t)dt} \int _0^T {{\bar{F}}_P (t)P_N (t)dt}\\&-\int _0^T {{\bar{F}}_P (t)r(t)P_N (t)dt} \int _0^T {{\bar{F}}_P (t)P_{N+1} (t)dt} \end{aligned}$$

Differentiating $J_1 (T)$ with respect to T, then we obtain

$$\begin{aligned} J_1^{\prime } (T)= & {} \int _0^T {{\bar{F}}_P (T)r(T)P_{N+1} (T){\bar{F}}_P (t)P_N (t)dt} +\int _0^T {{\bar{F}}_P (t)r(t)P_{N+1} (t){\bar{F}}_P (T)P_N (T)dt}\\&-\int _0^T {{\bar{F}}_P (T)r(T)P_N (T){\bar{F}}_P (t)P_{N+1} (t)dt} -\int _0^T {{\bar{F}}_P (t)r(t)P_N (t){\bar{F}}_P (T)P_{N+1} (T)dt}\\= & {} {\bar{F}}_P (T)\int _0^T {{\bar{F}}_P (t)\left[ {r(T)-r(t)} \right] \left[ {P_{N+1} (T)P_N (t)-P_{N+1} (t)P_N (T)} \right] dt} >0 \end{aligned}$$

since r(t) is increasing with t, we obtain $J_1 (0)=0$ and $J_1^{\prime } (T)>0$. Hence, $J_1 (T)>0$ for all $T>0$ and ${\int _0^\infty {{\bar{F}}_P (t)r(t)P_N (t)dt} }/{\int _0^\infty {{\bar{F}}_P (t)P_N (t)dt}}$ is increasing in N.

Evidently, for and N,

$$\begin{aligned} \frac{\int _0^\infty {{\bar{F}}_P (t)r(t)P_N (t)dt} }{\int _0^\infty {{\bar{F}}_P (t)P_N (t)dt} }\le r\left( \infty \right) \end{aligned}$$

For any $T\in \left( {0,\infty } \right)$, we have

$$\begin{aligned}&\frac{\int _0^\infty {{\bar{F}}_P (t)r(t)P_N (t)dt} }{\int _0^\infty {{\bar{F}}_P (t)P_N (t)dt} }\ge \frac{\int _T^\infty {{\bar{F}}_P (t)r(t)P_N (t)dt} }{\int _0^\infty {{\bar{F}}_P (t)P_N (t)dt} }\\&\quad \ge \frac{r(T)\int _T^\infty {{\bar{F}}_P (t)P_N (t)dt} }{\int _0^T {{\bar{F}}_P (t)P_N (t)dt} +\int _T^\infty {{\bar{F}}_P (t)P_N (t)dt} }\ge \frac{r(T)}{1+\left\{ {\frac{\int _0^T {{\bar{F}}_P (t)P_N (t)dt} }{\int _T^\infty {{\bar{F}}_P (t)P_N (t)dt} }} \right\} } \end{aligned}$$

Further, the bracket of the denominator is, for $T_1 \in \left( {T,\infty } \right)$,

$$\begin{aligned} \frac{\int _0^T {{\bar{F}}_P (t)P_N (t)dt} }{\int _T^\infty {{\bar{F}}_P (t)P_N (t)dt} }\le & {} \frac{\int _0^T {\exp \left\{ {-\int _0^t {r(x)dx} } \right\} \left( {\int _0^t {q(x)r(x)dx} } \right) ^{N}dt} }{\int _{T_1 }^\infty {\exp \left\{ {-\int _0^t {r(x)dx} } \right\} \left( {\int _0^t {q(x)r(x)dx} } \right) ^{N}dt} }\\\le & {} \frac{\int _0^T {\exp \left\{ {-\int _0^t {r(x)dx} } \right\} dt} }{\left( {\frac{\int _0^{T_1 } {q(x)r(x)dx} }{\int _0^T {q(x)r(x)dx} }} \right) ^{N}\int _{T_1 }^\infty {\exp \left\{ {-\int _0^t {r(x)dx} } \right\} dt} }\rightarrow 0\hbox { as }N\rightarrow \infty \end{aligned}$$

Thus,

$$\begin{aligned} r\left( \infty \right) =\mathop {\lim }\limits _{N\rightarrow \infty } r\left( \infty \right) \ge \mathop {\lim }\limits _{N\rightarrow \infty } \frac{\int _0^\infty {{\bar{F}}_P (t)r(t)P_N (t)dt} }{\int _0^\infty {{\bar{F}}_P (t)P_N (t)dt} }\ge r(T) \end{aligned}$$

Since T is arbitrary in $\left( {0,\infty } \right)$, we have

$$\begin{aligned} \mathop {\lim }\limits _{N\rightarrow \infty } \frac{\int _0^\infty {{\bar{F}}_P (t)r(t)P_N (t)dt} }{\int _0^\infty {{\bar{F}}_P (t)P_N (t)dt} }=r\left( \infty \right) \end{aligned}$$

From the above result and Lemma 1-(ii), we can show that $A_N$ is increasing in N, and diverges to $\infty$ as $N \rightarrow \infty$ if $G^{(N)}(\cdot )$ is non-increasing in N and ${G^{({N+1})}(\cdot )}/{G^{(N)}(\cdot )}$ is non-increasing in N.

1.5 Proof of Lemma 1-(ii)

Let

$$\begin{aligned} J_2 (T)= & {} \int _0^T {{\bar{F}}_P (t)p(t)r(t)P_{N+1} (t)dt} \int _0^T {{\bar{F}}_P (t)P_N (t)dt}\\&-\int _0^T {{\bar{F}}_P (t)p(t)r(t)P_N (t)dt} \int _0^T {{\bar{F}}_P (t)P_{N+1} (t)dt} \end{aligned}$$

Differentiating $J_2 (T)$ with respect to T, then we obtain

$$\begin{aligned} J_2^{\prime } (T)= & {} \int _0^T {{\bar{F}}_P (T)p(T)r(T)P_{N+1} (T){\bar{F}}_P (t)P_N (t)dt}\\&+\int _0^T {{\bar{F}}_P (t)p(t)r(t)P_{N+1} (t){\bar{F}}_P (T)P_N (T)dt}\\&-\int _0^T {{\bar{F}}_P (T)p(T)r(T)P_N (T){\bar{F}}_P (t)P_{N+1} (t)dt}\\&-\int _0^T {{\bar{F}}_P (t)p(t)r(t)P_N (t){\bar{F}}_P (T)P_{N+1} (T)dt}\\= & {} {\bar{F}}_P (T)\int _0^T {\bar{F}}_P (t)\left[ {p(T)r(T)-p(t)r(t)} \right] \left[ P_{N+1} (T)P_N (t)\right. \\&\left. -P_{N+1} (t)P_N (T) \right] dt >0 \end{aligned}$$

since r(t) is increasing with t and p(t) is non-decreasing with t, we obtain $J_2 \left( 0 \right) =0$ and $J_2^{\prime } (T)>0$. Hence, $J_2 (T)>0$ for all $T>0$ and $B_N$ is increasing in N.

Evidently, for and N,

$$\begin{aligned} \frac{\int _0^\infty {{\bar{F}}_P (t)p(t)r(t)P_N (t)dt} }{\int _0^\infty {{\bar{F}}_P (t)P_N (t)dt} }\le p\left( \infty \right) r\left( \infty \right) \end{aligned}$$

For any $T\in \left( {0,\infty } \right)$, we have

$$\begin{aligned}&\frac{\int _0^\infty {{\bar{F}}_P (t)p(t)r(t)P_N (t)dt} }{\int _0^\infty {{\bar{F}}_P (t)P_N (t)dt} }\ge \frac{\int _T^\infty {{\bar{F}}_P (t)p(t)r(t)P_N (t)dt} }{\int _0^\infty {{\bar{F}}_P (t)P_N (t)dt} }\\&\ge \frac{p(T)r(T)\int _T^\infty {{\bar{F}}_P (t)P_N (t)dt} }{\int _0^T {{\bar{F}}_P (t)P_N (t)dt} +\int _T^\infty {{\bar{F}}_P (t)P_N (t)dt} }\ge \frac{p(T)r(T)}{1+\left\{ {\frac{\int _0^T {{\bar{F}}_P (t)P_N (t)dt} }{\int _T^\infty {{\bar{F}}_P (t)P_N (t)dt} }} \right\} } \end{aligned}$$

where ${\int _0^T {{\bar{F}}_P (t)P_N (t)dt} }/{\int _T^\infty {{\bar{F}}_P (t)P_N (t)dt} }$ is as given in the above and has the same property. Thus,

$$\begin{aligned} p\left( \infty \right) r\left( \infty \right) =\mathop {\lim }\limits _{N\rightarrow \infty } p\left( \infty \right) r\left( \infty \right) \ge \mathop {\lim }\limits _{N\rightarrow \infty } \frac{\int _0^\infty {{\bar{F}}_P (t)p(t)r(t)P_N (t)dt} }{\int _0^\infty {{\bar{F}}_P (t)P_N (t)dt} }\ge p(T)r(T) \end{aligned}$$

Because T is arbitrary in $\left( {0,\infty } \right)$, we have $\lim \limits _{N\rightarrow \infty } B_N =p\left( \infty \right) r\left( \infty \right)$.

1.6 Proof of Lemma 1-(iii)

Using the skills of the integration by parts, we have

$$\begin{aligned} C_N= & {} \frac{\int _0^Z {\bar{{G}}_{N+1} \left( {K-y} \right) dG^{\left( N \right) }\left( y \right) } }{G^{\left( N \right) }\left( Z \right) }\frac{\int _0^\infty {{\bar{F}}_P (t)q(t)r(t)P_N (t)dt} }{\int _0^\infty {{\bar{F}}_P (t)P_N (t)dt}}\\= & {} \frac{\int _0^Z {\bar{{G}}_{N+1} \left( {K-y} \right) dG^{\left( N \right) }\left( y \right) } }{G^{\left( N \right) }\left( Z \right) }\frac{\int _0^\infty {{\bar{F}}_P (t)r(t)P_{N+1} (t)dt} }{\int _0^\infty {{\bar{F}}_P (t)P_N (t)dt}} \end{aligned}$$

Let

$$\begin{aligned} J_3 (T)= & {} \int _0^T {{\bar{F}}_P (t)r(t)P_{N+2} (t)dt} \hbox { }\times \int _0^T {{\bar{F}}_P (t)P_N (t)dt}\\&-\int _0^T {{\bar{F}}_P (t)r(t)P_{N+1} (t)dt} \quad \times \int _0^T {{\bar{F}}_P (t)P_{N+1} (t)dt} \end{aligned}$$

Differentiating $J_3 (T)$ with respect to T, then we obtain

$$\begin{aligned} J_3 (T)= & {} \int _0^T {{\bar{F}}_P (T)r(T)P_{N+2} (T){\bar{F}}_P (t)P_N (t)dt} +\int _0^T {{\bar{F}}_P (t)r(t)P_{N+2} (t){\bar{F}}_P (T)P_N (T)dt}\\&-\int _0^T {{\bar{F}}_P (T)r(T)P_{N+1} (T){\bar{F}}_P (t)P_{N+1} (t)dt}\\&-\int _0^T {{\bar{F}}_P (t)r(t)P_{N+1} (t){\bar{F}}_P (T)P_{N+1} (T)dt}\\= & {} \int _0^T {{\bar{F}}_P (T)r(T){\bar{F}}_P (t)\left[ {P_{N+2} (T)P_N (t)-P_{N+1} (T)P_{N+1} (t)} \right] dt}\\&+\int _0^T {{\bar{F}}_P (T)r(t){\bar{F}}_P (t)\left[ {P_{N+2} (t)P_N (T)-P_{N+1} (T)P_{N+1} (t)} \right] dt}\\= & {} \int _0^T {{\bar{F}}_P (T){\bar{F}}_P (t)\left[ {r(T)P_{N+1} (T)P_N (t)-r(t)P_{N+1} (t)P_{N+1} (T)} \right] }\\&\times \left[ {\frac{P_{N+2} (T)}{P_{N+1} (T)}-\frac{P_{N+2} (t)}{P_{N+1} (t)}} \right] dt\\&+\int _0^T {{\bar{F}}_P (T){\bar{F}}_P (t)r(T)P_{N+1} (T)P_N (T)\left[ {\frac{P_{N+2} (t)}{P_{N+1} (t)}-\frac{P_{N+1} (t)}{P_N (t)}} \right] dt}\\&+\int _0^T {{\bar{F}}_P (T){\bar{F}}_P (t)r(T)P_{N+1} (t)P_N (T)\left[ {\frac{P_{N+2} (T)}{P_{N+1} (T)}-\frac{P_{N+1} (T)}{P_N (T)}} \right] dt} \end{aligned}$$

since r(t) is increasing with t and p(t) is non-decreasing with t, we obtain $J_3 (0)=0$ and $J_3^{\prime } (T)>0$. Hence, $J_3 (T)>0$ for all $T>0$ and ${\int _0^\infty {{\bar{F}}_P (t)r(t)P_{N+1} (t)dt} }/{\int _0^\infty {{\bar{F}}_P (t)P_N (t)dt} }$ is increasing in N. Hence, $C_N$ is increasing in N if ${\int _0^Z {\bar{{G}}_{N+1} ({K-y})dG^{(N)}(y)} }/{G^{(N)}(Z)}$ is non-decreasing in N.

From the above result and Lemma 1-(ii), we can show that $\lim \limits _{N \rightarrow \infty }$ $C_N =\lim \limits _{N \rightarrow \infty }$ $\frac{\int _0^Z {\bar{{G}}_{N+1} ({K-y})dG^{(N)}(y)} }{G^{(N)}(Z)}$ $\frac{\int _0^\infty {{\bar{F}}_P (t)q(t)r(t)P_N (t)dt} }{\int _0^\infty {{\bar{F}}_P (t)P_N (t)dt} }$

$$\begin{aligned}= & {} \mathop {\lim }\limits _{N\rightarrow \infty } \frac{\int _0^Z {\bar{{G}}_{N+1} \left( {K-y} \right) dG^{\left( N \right) }\left( y \right) } }{G^{\left( N \right) }\left( Z \right) }\\&\times \mathop {\lim }\limits _{N\rightarrow \infty } \frac{\int _0^\infty {{\bar{F}}_P (t)r(t)P_N (t)dt} -\int _0^\infty {{\bar{F}}_P (t)p(t)r(t)P_N (t)dt} }{\int _0^\infty {{\bar{F}}_P (t)P_N (t)dt}}\\= & {} \mathop {\lim }\limits _{N\rightarrow \infty } \frac{\int _0^Z {\bar{{G}}_{N+1} \left( {K-y} \right) dG^{\left( N \right) }\left( y \right) } }{G^{\left( N \right) }\left( Z \right) }\\&\times \left[ {\mathop {\lim }\limits _{N\rightarrow \infty } \frac{\int _0^\infty {{\bar{F}}_P (t)r(t)P_N (t)dt} }{\int _0^\infty {{\bar{F}}_P (t)P_N (t)dt} }-\mathop {\lim }\limits _{N\rightarrow \infty } \frac{\int _0^\infty {{\bar{F}}_P (t)p(t)r(t)P_N (t)dt} }{\int _0^\infty {{\bar{F}}_P (t)P_N (t)dt} }} \right] \\= & {} \mathop {\lim }\limits _{N\rightarrow \infty } \frac{\int _0^Z {\bar{{G}}_{N+1} \left( {K-y} \right) dG^{\left( N \right) }\left( y \right) } }{G^{\left( N \right) }\left( Z \right) } \times \left[ {r\left( \infty \right) -p\left( \infty \right) r\left( \infty \right) } \right] \\= & {} \mathop {\lim }\limits _{N\rightarrow \infty } \frac{\int _0^Z {\bar{{G}}_{N+1} \left( {K-y} \right) dG^{\left( N \right) }\left( y \right) } }{G^{\left( N \right) }\left( Z \right) } \times q\left( \infty \right) r\left( \infty \right) \end{aligned}$$

1.7 Proof of Lemma 1-(iv)

Let

$$\begin{aligned} J_4 (T)= & {} \int _0^T {\left[ {\alpha _{N+1} (t)r(t)-\alpha _{N+1}^{\prime } (t)+\phi _{N+1} \left( Z \right) r(t)} \right] {\bar{F}}_P (t)P_{N+1} (t)dt}\\&\times \int _0^T {{\bar{F}}_P (t)P_N (t)dt}\\&-\int _0^T {\left[ {\alpha _N (t)r(t)-\alpha _N^{\prime } (t)+\phi _N \left( Z \right) r(t)} \right] {\bar{F}}_P (t)P_N (t)dt}\\&\times \int _0^T {{\bar{F}}_P (t)P_{N+1} (t)dt} \end{aligned}$$

Differentiating $J_4 (T)$ with respect to T, then we obtain

$$\begin{aligned} J_4^{\prime } (T)= & {} \int _0^T {\left[ {\alpha _{N+1} \left( \hbox {T} \right) r(T)-\alpha _{N+1}^{\prime } (T)+\phi _{N+1} \left( Z \right) r(T)} \right] {\bar{F}}_P (T)P_{N+1} (T){\bar{F}}_P (t)P_N (t)dt}\\&+\int _0^T {\left[ {\alpha _{N+1} (t)r(t)-\alpha _{N+1}^{\prime } (t)+\phi _{N+1} \left( Z \right) r(t)} \right] {\bar{F}}_P (t)P_{N+1} (t){\bar{F}}_P (T)P_N (T)dt}\\&-\int _0^T {\left[ {\alpha _N (T)r(T)-\alpha _N^{\prime } (T)+\phi _N \left( Z \right) r(T)} \right] {\bar{F}}_P (T)P_N (T){\bar{F}}_P (t)P_{N+1} (t)dt}\\&-\int _0^T {\left[ {\alpha _N (t)r(t)-\alpha _N^{\prime } (t)+\phi _N \left( Z \right) r(t)} \right] {\bar{F}}_P (t)P_N (t){\bar{F}}_P (T)P_{N+1} (T)dt}\\= & {} {\bar{F}}_P (T)\int _0^T {{\bar{F}}_P (t)\left\{ {\left[ {\alpha _{N+1} \left( \hbox {T} \right) r(T)-\alpha _{N+1}^{\prime } (T)+\phi _{N+1} \left( Z \right) r(T)} \right] } \right. }\\&\left. {-\left[ {\alpha _N (t)r(t)-\alpha _N^{\prime } (t)+\phi _N \left( Z \right) r(t)} \right] } \right\} P_{N+1} (T)P_N (t)dt\\&-\left\{ {{\bar{F}}_P (T)\int _0^T {{\bar{F}}_P (t)\left\{ {\left[ {\alpha _N \left( \hbox {T} \right) r(T)-\alpha _N^{\prime } (T)+\phi _N \left( Z \right) r(T)} \right] } \right. } } \right. \\&\left. {\left. {-\left[ {\alpha _{N+1} (t)r(t)-\alpha _{N+1}^{\prime } (t)+\phi _{N+1} \left( Z \right) r(t)} \right] } \right\} P_{N+1} (t)P_N (T)dt} \right\} \\\ge & {} {\bar{F}}_P (T)\int _0^T {{\bar{F}}_P (t)\left\{ {\left[ {\alpha _{N+1} \left( \hbox {T} \right) r(T)-\alpha _{N+1}^{\prime } (T)+\phi _{N+1} \left( Z \right) r(T)} \right] } \right. }\\&\left. {-\left[ {\alpha _{N+1} (t)r(t)-\alpha _{N+1}^{\prime } (t)+\phi _{N+1} \left( Z \right) r(t)} \right] } \right\} P_{N+1} (T)P_N (t)dt\\&-\left\{ {{\bar{F}}_P (T)\int _0^T {{\bar{F}}_P (t)\left\{ {\left[ {\alpha _{N+1} \left( \hbox {T} \right) r(T)-\alpha _{N+1}^{\prime } (T)+\phi _{N+1} \left( Z \right) r(T)} \right] } \right. } } \right. \\&\left. {\left. {-\left[ {\alpha _{N+1} (t)r(t)-\alpha _{N+1}^{\prime } (t)+\phi _{N+1} \left( Z \right) r(t)} \right] } \right\} P_{N+1} (t)P_N (T)dt} \right\} \\= & {} {\bar{F}}_P (T)\int _0^T {{\bar{F}}_P (t)\left\{ {\left[ {\alpha _{N+1} \left( \hbox {T} \right) r(T)-\alpha _{N+1}^{\prime } (T)+\phi _{N+1} \left( Z \right) r(T)} \right] } \right. }\\&\left. {-\left[ {\alpha _{N+1} (t)r(t)-\alpha _{N+1}^{\prime } (t)+\phi _{N+1} \left( Z \right) r(t)} \right] } \right\} \left[ P_{N+1} (T)P_N (t)\right. \\&\left. -P_{N+1} (t)P_N (T) \right] dt\\&>0 \end{aligned}$$

since r(t) is increasing with t and $\alpha _j (t)r(t)-\alpha _j^{\prime } (t)+\phi _j \left( Z \right) r(t)$ is non-decreasing with t, we obtain $J_4 \left( 0 \right) =0$ and $J_4^{\prime } (T)>0$. Hence, $J_4 (T)>0$ for all $T>0$ and $D_N$ is increasing in N.

Evidently, for and N,

$$\begin{aligned} D_N= & {} \frac{\int _0^\infty {\left[ {\alpha _N (t)r(t)-\alpha _N^{\prime } (t)+\phi _N \left( Z \right) r(t)} \right] {\bar{F}}_P (t)P_N (t)dt}}{\int _0^\infty {{\bar{F}}_P (t)P_N (t)dt}}\\\le & {} \alpha _\infty \left( \infty \right) r\left( \infty \right) -\alpha _\infty ^{\prime } \left( \infty \right) +\phi _\infty \left( Z \right) r\left( \infty \right) \end{aligned}$$

For any $T\in \left( {0,\infty } \right)$, we have

$$\begin{aligned}&\frac{\int _0^\infty {\left[ {\alpha _N (t)r(t)-\alpha _N^{\prime } (t)+\phi _N \left( Z \right) r(t)} \right] {\bar{F}}_P (t)P_N (t)dt}}{\int _0^T {{\bar{F}}_P (t)P_N (t)dt}}\\&\quad \ge \frac{\int _T^\infty {\left[ {\alpha _N (t)r(t)-\alpha _N^{\prime } (t)+\phi _N \left( Z \right) r(t)} \right] {\bar{F}}_P (t)P_N (t)dt}}{\int _0^T {{\bar{F}}_P (t)P_N (t)dt}}\\&\quad \ge \frac{\left[ {\alpha _N (T)r(T)-\alpha _N^{\prime } (T)+\phi _N \left( Z \right) r(T)} \right] \int _T^\infty {{\bar{F}}_P (t)P_N (t)dt} }{\int _0^T {{\bar{F}}_P (t)P_N (t)dt} +\int _T^\infty {{\bar{F}}_P (t)P_N (t)dt}}\\&\quad \ge \frac{\alpha _N (T)r(T)-\alpha _N^{\prime } (T)+\phi _N \left( Z \right) r(T)}{1+\left\{ {\frac{\int _0^T {{\bar{F}}_P (t)P_N (t)dt} }{\int _T^\infty {{\bar{F}}_P (t)P_N (t)dt} }} \right\} } \end{aligned}$$

where ${\int _0^T {{\bar{F}}_P (t)P_N (t)dt} }/{\int _T^\infty {{\bar{F}}_P (t)P_N (t)dt} }$ is as given in the above and has the same property. Thus,

$$\begin{aligned}&\alpha _\infty (\infty )r(\infty )-\alpha _\infty ^{\prime } (\infty )+\phi _\infty \left( Z \right) r(\infty ) \\&\quad =\mathop {\lim }\limits _{N\rightarrow \infty } [\alpha _N (\infty )r(\infty )-\alpha _N^{\prime } (\infty )+\phi _N \left( Z \right) r(\infty )]\\&\quad \ge \mathop {\lim }\limits _{N\rightarrow \infty } \frac{\int _{\hbox { }0}^{\hbox { }T} {\left[ {\alpha _N (t)r(t)-\alpha _N^{\prime } (t)+\phi _N \left( Z \right) r(t)} \right] {\bar{F}}_p (t)P_N (t)\hbox {d}t}}{\int _{\hbox { }0}^{\hbox { }T} {{\bar{F}}_p (t)P_N (t)\hbox {d}t}}\\&\quad \ge \alpha _\infty (T)r(T)-\alpha _\infty ^{\prime } (T_1)+\phi _\infty \left( Z \right) r(T). \end{aligned}$$

Because T is arbitrary in $({0,\infty })$, we have

$$\begin{aligned} \mathop {\lim }\limits _{N\rightarrow \infty } D_N =\alpha _\infty (\infty )r(\infty )-\alpha _\infty ^{\prime } (\infty )+\phi _\infty \left( Z \right) r(\infty ). \end{aligned}$$