2.1. Stochastic orders

We first give the definitions of some stochastic orders between two random variables. For a random variable $X$, let us denote $F_X, {{\bar F}}_X, h_X$ and $\tilde r_X$ the distribution function, the survival function, the hazard rate function and the reversed hazard rate function, respectively.

It is well-known that the (reversed) hazard rate order implies the usual stochastic order, while the reversed statement is not true in general. For more comprehensive discussions on various stochastic orders and their applications, one may refer to the monographs by Shaked and Shanthikumar [Reference Shaked and Shanthikumar40] and Belzunce et al. [Reference Belzunce, Riquelme and Mulero3].

It is well-known that the Exponential, Weibull, Lomax and Pareto distributions are special cases of the PH model, while Fréchet distribution is an example of the PRH model. For more detailed applications of the PH model, we refer the readers to Cox [Reference Cox9], Collett [Reference Collett8], Li et al. [Reference Li, Yan and Zuo33], Yan et al. [Reference Yan, Lu and Li48] and Jarrahiferiz et al. [Reference Jarrahiferiz, Kayid and Izadkhah26], and for more comprehensive discussions on the PRH model, one may refer to Kalbfleisch and Lawless [Reference Kalbfleisch and Lawless27], Gupta and Gupta [Reference Gupta and Gupta24] and Belzunce and Martínez-Riquelme [Reference Belzunce and Martínez-Riquelme2].

2.2. Majorization order

The notion of majorization order is a key tool in establishing various inequalities arising from many research areas. For any real two vectors $\boldsymbol {x}=(x_{1}, x_{2},\ldots , x_{n})$ and $\boldsymbol {y}=(y_{1}, y_{2},\ldots , y_{n})$, denote $x_{1:n}\leq x_{2:n}\leq \cdots \leq x_{n:n}$ and $y_{1:n}\leq y_{2:n}\leq \cdots \leq y_{n:n}$ be the increasing arrangements of the components of $\boldsymbol {x}$ and $\boldsymbol {y}$, respectively.

For an elaborate discussion on the theory of majorization orders and their applications, one may refer to Marshall et al. [Reference Marshall, Olkin and Arnold37] and Kundu et al. [Reference Kundu, Chowdhury, Nanda and Hazra28].

Lemma 1 ([Reference Marshall, Olkin and Arnold37])

Let $\varphi$ be a continuously differentiable function on $\mathbb {A} \subseteq \mathbb {R}^{n}$. Then, $\varphi$ is said to be Schur-convex [Schur-concave] on $\mathbb {A}$ if and only if $\varphi$ is symmetric on $\mathbb {A}$, and for all $\boldsymbol {z} \in \mathbb {A}$ and $1 \le i \neq j \le n$,

$$(z_i-z_j) \left(\frac{\partial \varphi(\boldsymbol{z}) }{\partial z_i}-\frac{\partial\varphi(\boldsymbol{z}) }{\partial z_j}\right) \ge[\le] 0.$$

2.3. Copula

For a random vector $\boldsymbol {X}=(X_1,X_2,\ldots ,X_n)$ with joint cumulative distribution function $H$ and univariate marginal distribution functions $F_1,F_2,\ldots ,F_n$, its copula is a distribution function $C:[0,1]^{n}\to [0,1]$, such that

$$H(\boldsymbol{x})=C(F_1(x_1),F_2(x_2),\ldots,F_{n}(x_n)).$$

Similarly, a survival copula is a joint cumulative survival function $\hat {C} :[0,1]^{n}\to [0,1]$, such that

$${{\bar H}}(\boldsymbol{x})=\mathbb{P}(X_1>x_1,X_2>x_2,\ldots,X_n>x_n)=\hat{C}({{\bar F}}_1(x_1),{{\bar F}}_2(x_2),\ldots,{{\bar F}}_n(x_n)),$$

where ${{\bar F}}_i=1-F_i$ $(i=1,2,\ldots , n)$ is the marginal survival function and ${{\bar H}}$ is the joint survival function.

When $C$ is Schur-concave, $C$ is said to be the Schur-concave copula. The family of the Schur-concave copula includes some important copulas, such as Archimedean copula and Farlie–Gumble–Morgenstern (FGM). For a decreasing and continuous function $\psi :\mathbb {R}^{+}\to [0,1]$ such that $\psi (0)=1$, $\psi (+\infty )=0$, $(-1)^{k}\psi ^{(k)}(x)\geq 0$, $k=0,1,\ldots ,n-2$, and $(-1)^{n-2}\psi ^{(n-2)}(x)$ is decreasing and convex. Then,

$$C_{\psi}(u_{1},u_{2},\cdots,u_{n})=\psi\left(\sum_{i=1}^{n}\psi^{{-}1}(u_{i})\right),\quad u_{i} \in [0,1]$$

is called Archimedean copula. If we take $\psi _\theta ^{-1}(t) = t^{\theta }-1$ as a generator, then we get the Clayton copula (cf. [Reference Cuvelier and Noirhomme-Fraiture10])

(1)\begin{equation} C_{\theta}(u_{1},u_{2},\cdots,u_{n})=\left(\sum_{i=1}^{n} u_{i}^{-\theta} -n+1 \right)^{{-}1/\theta}, \quad u_{i} \in [0,1 ] \text{ and } \theta \in (0, \infty). \end{equation}

In fact, Archimedean copula includes Clayton family, Ali-Mikhail-Had (AMH) family and Gumbel family, etc.

As pointed in Durante and Sempi [Reference Durante and Sempi18], FGM family defined as

(2)\begin{equation} C_{\theta}(u_{1},u_{2},\ldots,u_{n})=\prod_{i=1}^{n}u_{i}+\theta\prod_{i=1}^{n}u_{i}(1-u_{i}), \quad u_{i} \in [0,1 ] \text{ and }\theta\in[{-}1,1], \end{equation}

which must be Schur-concave but not be Archimedean copula, and every Archimedean copula must be Schur-concave, but the Schur-concave copula is not necessarily an Archimedean copula. Hence, the Schur-concave copula characterize more extensive dependency to some extent. For detailed discussions on copulas and its applications, one may refer to Durante and Sempi [Reference Durante and Sempi18], Nelsen [Reference Nelsen39] and Durante and Papini [Reference Durante and Papini17].

Next, we recall the definition of the concordance order.

The concordance order is referred to as the positive quadrant dependent order. For an elaborate discussion of the positive quadrant dependent and concordance order, one may refer to Lehmann [Reference Lehmann31], Dhaene and Goovaerts [Reference Dhaene and Goovaerts13,Reference Dhaene and Goovaerts14] , and Nelsen [Reference Nelsen39].

The following lemmas play a vital role in establishing the main results.

Proof. Suppose $u=1-\alpha ^{x},\ v=1-\alpha ^{y}$, without loss of generality, assume $x\ge y$, which implies $u \ge v$. From the Schur-concavity of $C(u, v)$, we have

$$\frac{\partial C(u,v) }{\partial u }-\frac{\partial C(u,v)}{\partial v}\le 0.$$

Note that ${\partial C(u,v)}/{\partial u}$ or ${\partial C(u,v)}/{\partial v}$ is non-negative, it holds that,

\begin{align*} \frac{\partial C(u,v) }{\partial x }-\frac{\partial C(u,v) }{\partial y} & = \frac{\partial C(u,v) }{\partial u }(-\alpha^{x} \ln \alpha )-\frac{\partial C(u,v) }{\partial v}(-\alpha^{y} \ln \alpha)\\ & =\left[\frac{\partial C(u,v) }{\partial v} \big(\alpha^{y}-\alpha^{x}\big) +\left(\frac{\partial C(u,v) }{\partial v}-\frac{\partial C(u,v) }{\partial u }\right) \alpha^{x} \right] \ln \alpha \le 0. \end{align*}

And thus

$$(x-y) \left(\frac{\partial C(u,v)}{\partial x }-\frac{\partial C(u,v)}{\partial y} \right) \le 0.$$

It follows from Lemma 1 that $C(1-\alpha ^{x}, 1-\alpha ^{y})$ is also Schur-concave. The proof is completed.

Combining Proposition C.1 in Part I in [Reference Marshall, Olkin and Arnold37] p. 92 with Lemma 1 in Laniado and Lillo [Reference Laniado and Lillo30], we obtain immediately the following Lemma 4.

Throughout this article, all random variables are assumed to be positive and absolutely continuous, the terms increasing and decreasing are used instead of monotone nondecreasing and monotone nonincreasing, respectively.

3.1. Dependence case

In this subsection, we consider the case of two subsystems are dependent. Denote the reliability function of the resulting two-parallel–series system under allocation policy $a_k$ by ${{\bar H}}_S(k;x)$, $k=1,2,\ldots , n$. Then,

$${{\bar H}}_S(k;x)=1-C_k(1- {{\bar F}} ^{\sum_{i=1}^{k}\alpha_i+\sum_{i=k+1}^{n}\beta_i}, 1- {{\bar F}}^{\sum_{i=1}^{k}\beta_i+\sum_{i=k+1}^{n}\alpha_i}).$$

Theorem 1 presents some sufficient conditions for the lifetime of the two-parallel–series system under different allocation policies in terms of the usual stochastic order.

Proof. The survival function of the resulting two-parallel–series system under allocation policy $a_{k_j}$ can be written as

\begin{align*} {{\bar H}}_{S} (k_j;x)& =1-C_{k_j}(1- {{\bar F}} ^{\sum_{i=1}^{k_j}\alpha_{i:n}+\sum_{i=k_j+1}^{n}\beta_{i:n}}, 1- {{\bar F}}^{\sum_{i=1}^{k_j}\beta_{i:n}+\sum_{i=k_j+1}^{n}\alpha_{i:n}} )\\ & =1-C_{k_j}(1-{{\bar F}}^{ s_{j1}}, 1-{{\bar F}}^{s_{j2}}), \end{align*}

where $s_{j1}={\sum _{i=1}^{k_j}\alpha _{i:n}+\sum _{i=k_j+1}^{n}\beta _{i:n}}$ and $s_{j2}={\sum _{i=1}^{k_j}\beta _{i:n}+\sum _{i=k_j+1}^{n}\alpha _{i:n}},\ j=1,2$. From Lemma 2, the Schur-concavity of $C_{k_1}$ implies that $C_{k_j}(1-{{\bar F}}^{ s_{j1}}, 1-{{\bar F}}^{s_{j2}})$ is also Schur-concave. According to $\boldsymbol {\alpha }\overset {\textrm {{m}}}{\succeq }\boldsymbol {\beta }$, it holds that

$$\sum_{i=1}^{k_j}\alpha_{i:n} \le \sum_{i=1}^{k_j}\beta_{i:n} ,\quad \sum_{i=k_j+1}^{n}\beta_{i:n} \le \sum_{i=k_j+1}^{n}\alpha_{i:n}, \quad j=1,2,$$

thus,

(3)\begin{equation} {\sum_{i=1}^{k_j}\alpha_{i:n}+\sum_{i=k_j+1}^{n}\beta_{i:n}} \le {\sum_{i=1}^{k_j}\beta_{i:n}+\sum_{i=k_j+1}^{n}\alpha_{i:n}}, \quad j=1,2. \end{equation}

Based on conditions $k_1 < k_2$ and $\sum _{i=k_1+1}^{k_2}\alpha _{i:n} \ge \sum _{i=k_1+1}^{k_2}\beta _{i:n}$, we have

$$\sum_{i=1}^{k_1}\alpha_{i:n}+ \sum_{i=k_1+1}^{k_2}\beta_{i:n} +\sum_{i=k_2+1}^{n}\beta_{i:n} \le \sum_{i=1}^{k_1}\alpha_{i:n}+ \sum_{i=k_1+1}^{k_2}\alpha_{i:n} +\sum_{i=k_2+1}^{n}\beta_{i:n},$$

which is equivalent to

(4)\begin{equation} \sum_{i=1}^{k_1}\alpha_{i:n} +\sum_{i=k_1+1}^{n}\beta_{i:n} \le \sum_{i=1}^{k_2}\alpha_{i:n}+ \sum_{i=k_2+1}^{n}\beta_{i:n} . \end{equation}

According to (3) and (4), we obtain the following majorization order

(5)\begin{align} & \left( \sum_{i=1}^{k_1}\alpha_{i:n}+\sum_{i=k_1+1}^{n}\beta_{i:n}, \sum_{i=1}^{k_1}\beta_{i:n}+ \sum_{i=k_1+1}^{n}\alpha_{i:n}\right) \nonumber\\ & \quad \stackrel{\textrm{{m}}}{\succeq} \left( \sum_{i=1}^{k_2}\alpha_{i:n}+\sum_{i=k_2+1}^{n}\beta_{i:n}, \sum_{i=1}^{k_2}\beta_{i:n}+ \sum_{i=k_2+1}^{n}\alpha_{i:n} \right), \end{align}

hence, we have

$$( s_{11}, s_{12} ) \stackrel{\textrm{{m}}}{\succeq} ( s_{21}, s_{22} ),$$

combining Schur-concavity of $C_{k_j}(1-{{\bar F}}^{ s_{j1}}, 1-{{\bar F}}^{s_{j2}})$ with the majorization order, it holds that

\begin{align*} {{\bar H}}_{S} ({k_1 };x)& =1-C_{k_1}(1-{{\bar F}}^{ s_{11}}, 1-{{\bar F}}^{s_{12}})\\ & \ge 1-C_{k_1}(1-{{\bar F}}^{ s_{21}}, 1-{{\bar F}}^{s_{22}}) \\ & \ge 1-C_{k_2}(1-{{\bar F}}^{ s_{21}}, 1-{{\bar F}}^{s_{22}}) ={{\bar H}}_{S} ({k_2 };x), \end{align*}

the last inequality follows from the assumption $C_{k_1} \prec C_{k_2}$. The proof is completed.

The following Example 1(i) illustrates the result of Theorem 1, and Example 1(ii) shows that the condition $\sum _{i=k_1+1}^{k_2}\alpha _{i:n} \ge \sum _{i=k_1+1}^{k_2}\beta _{i:n}$ can not be dropped.

Example 1 Under the assumptions of Theorem 1, let ${{\bar F}}(x)=e^{-\lambda x}$, suppose subsystems $S_1$ and $S_2$ have FGM copula as (2)

$$C_{\theta_j}(u_{1},u_{2})=u_{1}u_{2}(1+\theta_j(1-u_{1})(1-u_{2})), \quad u_{1}, u_{2} \in [0,1 ] \text{ and }\theta_j\in[{-}1,1],$$

$j=1,2,3,4$. Assume $\lambda =0.008, \theta _1=-0.9, \theta _2=-0.3, \theta _3=0.2$, and $\theta _4=0.9$. For convenience, let $k_1=1, k_2=2,k_3=3,k_4=4$ in all examples of this paper. To plot the entire survival curves of $S_{k_1}$ and $S_{k_2}$ on $[0, \infty )$, we perform the transformation $e^{-x}: [0, \infty ) \to (0, 1]$. Then, it is obvious that $S_{k_1} \le _{\textrm {st}} S_{k_2}$ is equivalent to $e^{-S_{k_1}} \ge _{\textrm {st}} e^{-S_{k_2}}$.

1. (i) Seting $\boldsymbol {\alpha }=(1,8, 11, 23)\stackrel {\textrm {{m}}}{\succeq }(6,7,10,20)=\boldsymbol {\beta },$ from Figure 2(a) we can see that the difference function ${{\bar H}}_{S}(k_{i};x)- {{\bar H}}_{S}(k_{i+1};x)$ is always non-negative, $i=1,2,3$, for all $x=-\ln u$ and $u\in (0,1]$. Hence, ${{\bar H}}_{S}({k_1};x) \ge {{\bar H}}_{S}({k_2};x) \ge {{\bar H}}_{S}({k_3};x) \ge {{\bar H}}_{S}({k_4};x)$, which is in accordance with Theorem 1.

2. (ii) Taking $\boldsymbol {\alpha }=(1,8, 11, 23)\stackrel {\textrm {{m}}} {\succeq }(2,9,12,20)=\boldsymbol {\beta }$ with remaining conditions unchanged, notice that $\sum _{i=2}^{3}\alpha _{i:n} < \sum _{i=2}^{3}\beta _{i:n}$. Figure 2(b) displays the curves of ${{\bar H}}_{U}(k_{1};x)- {{\bar H}}_{U}(k_{3};x)$, which is crossing with the line $y=0$, this shows that the usual stochastic order does not hold between $S_{k_1}$ and $S_{k_2}$. Therefore, the condition $\sum _{i=k_1+1}^{k_2}\alpha _{i:n} \ge \sum _{i=k_1+1}^{k_2}\beta _{i:n}$ can not be dropped.

FIGURE 2. Plots of the difference functions ${{\bar H}}_{S}(k_{i};-\ln u)- {{\bar H}}_{S}(k_{i+1};-\ln u)$, $i=1,2,3$, for all $u\in (0,1]$.

Combining Theorem 1 with the transitivity of the usual stochastic order, we have the following corollary.

Corollary 1 For $\boldsymbol {X} \sim \textrm {PH} (\boldsymbol {\alpha }, {{\bar F}})$ and $\boldsymbol {Y} \sim \textrm {PH} (\boldsymbol {\beta }, {{\bar F}})$, suppose that $n-1$ copulas of $C_{j}\ (j=1,2,\ldots , n)$ are Schur-concave, and $C_{k}\leq C_{k+1}(k=1,2,\ldots ,n-1)$. If $\boldsymbol {\alpha }\stackrel {\textrm {{m}}}{\succeq }\boldsymbol {\beta }$ and $\alpha _{i:n} \ge \beta _{i:n}$, $i=2,3,\ldots ,n$, then

$$S_1 \ge_{\textrm{st}} S_2 \ge_{\textrm{st}} \cdots \ge_{\textrm{st}} S_n.$$

Corollary 1 shows that $a_1$ is the optimal allocation policy and $a_n$ is the worst allocation policy within all possible allocations.

3.2. Independence case

In this part, we consider that the case of two subsystems are independent. The reliability function of the resulting two-parallel–series system can be written as

$${{\bar H}}_S(k;x) =1-(1- {{\bar F}} ^{\sum_{i=1}^{k}\alpha_i+\sum_{i=k+1}^{n}\beta_i}) (1-{{\bar F}}^{\sum_{i=1}^{k}\beta_i+\sum_{i=k+1}^{n}\alpha_i}).$$

Theorem 2 provides some sufficient conditions to compare the lifetime of two-parallel–series system with respect to the hazard rate and the reversed hazard rate orders.

Proof. (i) Note that the survival function of ${S_{k_j}}\ (j=1,2)$ can be expressed by

$${{\bar H}}_{S}(k_j;x)={{\bar F}}^{\sum_{i=1}^{k_j}\alpha_{i:n}+\sum_{i=k_j+1}^{n}\beta_{i:n}}+ {{\bar F}}^{\sum_{i=1}^{k_j}\beta_{i:n}+\sum_{i=k_j+1}^{n}\alpha_{i:n}}- {{\bar F}}^{\sum_{i=1}^{n}\alpha_{i:n}+\sum_{i=1}^{n}\beta_{i:n}}.$$

To reach the desired result, it suffices to show that

\begin{align*} R_S(k_1,k_2; x) & =\frac {{{\bar F}} ^{\sum_{i=1}^{k_1}\alpha_{i:n}+\sum_{i=k_1+1}^{n}\beta_{i:n}}+ {{\bar F}}^{\sum_{i=1}^{k_1}\beta_{i:n}+\sum_{i=k_1+1}^{n}\alpha_{i:n}}- {{\bar F}}^{\sum_{i=1}^{n}\alpha_{i:n}+\sum_{i=1}^{n}\beta_{i:n}}} {{{\bar F}} ^{\sum_{i=1}^{k_2}\alpha_{i:n}+\sum_{i=k_2+1}^{n}\beta_{i:n}}+ {{\bar F}}^{\sum_{i=1}^{k_2}\beta_{i:n}+\sum_{i=k_2+1}^{n}\alpha_{i:n}}- {{\bar F}}^{\sum_{i=1}^{n}\alpha_{i:n}+\sum_{i=1}^{n}\beta_{i:n}}}\\ & =\frac {p(x)^{t_\delta }+p(x)^{1-t_\delta }-p(x)} {p(x)^{t_\lambda}+p(x)^{1-t_\lambda}-p(x)} \end{align*}

is increasing in $x \in \mathbb {R}^{+}$, where

$$p(x)={{\bar F}}^{\sum_{i=1}^{n}\alpha_{i:n}+\sum_{i=1}^{n}\beta_{i:n}}, \quad t_\delta=\frac{\sum_{i=1}^{k_1}\alpha_{i:n}+\sum_{i=k_1+1}^{n}\beta_{i:n}}{\sum_{i=1}^{n}\alpha_{i:n}+\sum_{i=1}^{n}\beta_{i:n}}$$

and

$$t_\lambda=\frac{\sum_{i=1}^{k_2}\alpha_{i:n}+\sum_{i=k_2+1}^{n}\beta_{i:n}}{\sum_{i=1}^{n}\alpha_{i:n}+\sum_{i=1}^{n}\beta_{i:n}}.$$

Note that (3) and (4) imply $t_\delta \le t_\lambda \le 1/2,$ it holds from Lemma 3 that $R_S(k_1,k_2; x)$ is decreasing in $p(x)$, and consider that $p(x)$ is decreasing in $x\in \mathbb {R}^{+}$, hence, $R_S(k_1,k_2; x)$ is increasing in $x \in \mathbb {R}^{+}$, which completes the proof of (i) .

(ii) Denote the distribution function and the reversed hazard rate function of $S$ under allocation policy $a_{k_j}$ by $H_{S}(k_j; x)$ and $\tilde {r}_{S}(k_j;x)$, respectively. Then,

$$H_{S}(k_j; x)=(1- {{\bar F}} ^{\sum_{i=1}^{k_j}\alpha_{i:n}+\sum_{i=k_j+1}^{n}\beta_{i:n}}) (1-{{\bar F}}^{\sum_{i=1}^{k_j}\beta_{i:n}+\sum_{i=k_j+1}^{n}\alpha_{i:n}})$$

and

\begin{align*} \tilde{r}_{S}(k_j;x)& =\frac{\mathrm{d}}{\mathrm{d} x} [\ln (1- {{\bar F}} ^{\sum_{i=1}^{k_j}\alpha_{i:n}+\sum_{i=k_j+1}^{n}\beta_{i:n}})+ \ln (1-{{\bar F}}^{\sum_{i=1}^{k_j}\beta_{i:n}+\sum_{i=k_j+1}^{n}\alpha_{i:n}})]\\ & =\frac{f(x) }{{{\bar F}}(x)} \left(\frac{(\sum_{i=1}^{k_j}\alpha_{i:n}+\sum_{i=k_j+1}^{n}\beta_{i:n}){{\bar F}}^{\sum_{i=1}^{k_j}\alpha_{i:n}+\sum_{i=k_j+1}^{n}\beta_{i:n}}}{1- {{\bar F}}^{\sum_{i=1}^{k_j}\alpha_{i:n}+\sum_{i=k_j+1}^{n}\beta_{i:n}}}\right)\\ & \quad + \frac{f(x) }{{{\bar F}}(x)} \left(\frac{( \sum_{i=1}^{k_j}\beta_{i:n}+\sum_{i=k_j+1}^{n}\alpha_{i:n}) {{\bar F}}^{\sum_{i=1}^{k_j}\beta_{i:n}+\sum_{i=k_j+1}^{n}\alpha_{i:n}}}{1- {{\bar F}}^{\sum_{i=1}^{k_j}\beta_{i:n}+\sum_{i=k_j+1}^{n}\alpha_{i:n}}}\right)\\ & =r(x)\sum_{m=1}^{2} \frac{t_{jm}{{\bar F}}^{t_{jm}}}{1-{{\bar F}}^{t_{jm}}}, \end{align*}

where $t_{j1}=\sum _{i=1}^{k_j}\alpha _{i:n}+\sum _{i=k_j+1}^{n}\beta _{i:n}$ and $t_{j2}=\sum _{i=1}^{k_j}\beta _{i:n}+\sum _{i=k_j+1}^{n}\alpha _{i:n},\ j=1,2$.

For any fixed $x\in \mathbb {R}^{+}$, it follows from Lemma 4 that $\sum _{m=1}^{2} {t_{jm}{{\bar F}}^{t_{jm}}}/{(1-{{\bar F}}^{t_{jm}})}$ is Schur-convex in $(t_{j1}, t_{j2})$, and thus $\tilde {r}_{S}(k_j;x)$ is also Schur-convex. Note that (5) implies $( t_{11}, t_{12} ) \stackrel {\textrm {{m}}}{\succeq } ( t_{21}, t_{22} )$, we have

$$\tilde{r}_{S}(k_1;x)=r(x)\sum_{m=1}^{2} \frac{t_{1m}{{\bar F}}^{t_{1m}}}{1-{{\bar F}}^{t_{1m}}} \ge r(x)\sum_{m=1}^{2} \frac{t_{2m}{{\bar F}}^{t_{2m}}}{1-{{\bar F}}^{t_{2m}}}=\tilde{r}_{S}(k_2;x),$$

for all $x \in \mathbb {R}^{+}$. Then, the theorem is proved.

The next example illustrates the theoretical result of Theorem 2.

Example 2 Under the setups of Theorem 2, assume ${{\bar F}}(x)=e^{-\lambda x}$, set $\boldsymbol {\alpha }=(0.1,0.8, 1.1, 2.3) \stackrel{\textrm {{m}}}{\succeq }(0.6,0.7,1,2)=\boldsymbol {\beta }, \lambda =0.8$. Figure 3(a) and 3(b) plots the difference functions $h_{S}(k_{i+1};x)-h_{S}({k_i};x)$ and $\tilde {r}_{S}(k_{i};x)-\tilde {r}_{S}({k_{i+1}};x)$, respectively, $i=1,2,3,$ for all $x=-\ln u$ and $u\in (0,1]$. They are always non-negative, which demonstrates the theoretical results of Theorem 2.

FIGURE 3. Plots of the difference functions $h_{S}(k_{i+1};x)-h_{S}({k_i};x)$ and $\tilde {r}_{S}(k_{i};x)-\tilde {r}_{S}({k_{i+1}};x),\ i=1,2,3$, for all $x=-\ln u$ and $u\in (0,1]$.

The following corollary can be obtained from Theorem 2 and the transitivity of the hazard rate order and the reversed hazard rate order.

In accordance with Corollary 2, $a_1$ is the optimal allocation policy and $a_n$ is the worst allocation policy within all possible allocations.

4.1. Dependent case

In this subsection, we focus that the case of two subsystems are dependent. Denote the reliability function of the resulting two-series–parallel system under the allocation policy $b_k$ by ${{\bar H}}_U(k;x)$, $k=1,2,\ldots , n$. Then,

$${{\bar H}}_U(k;x)=\hat C_k(1- F^{\sum_{i=1}^{k}\alpha_i+\sum_{i=k+1}^{n}\beta_i}, 1- F^{\sum_{i=1}^{k}\beta_i+\sum_{i=k+1}^{n}\alpha_i} ).$$

The following result establishes the usual stochastic order for the two-series–parallel system. The proof can be obtained along the same way with that of Theorem 1, and thus omitted here.

By the transitivity of the usual stochastic orders, the following corollary follows immediately from Theorem 3.

Corollary 3 For $\boldsymbol {X} \sim \textrm {PRH} (\boldsymbol {\alpha }, F)$ and $\boldsymbol {Y} \sim \textrm {PRH}(\boldsymbol {\beta }, F)$, suppose that $n-1$ copulas of $\hat C_{j}\ (j=1,\ldots , n)$ are Schur-concave, and $C_{k}\leq C_{k+1}\ (k=1,2,\ldots ,n-1)$. If $\boldsymbol {\alpha }\stackrel {\textrm {{m}}}{\succeq }\boldsymbol {\beta }$ and $\alpha _{i:n} \ge \beta _{i:n}$, i = 2,3,…,n, then

$$U_1 \le_{\textrm{st}} U_2 \le_{\textrm{st}} \cdots \le_{\textrm{st}} U_n.$$

In accordance with Corollary 3, $b_1$ is the worst allocation policy and $b_n$ is the optimal allocation policy within all possible allocations.

4.2. Independence case

When the lifetimes of two subsystems are independent, the reliability function of the two-series–parallel system can be expressed as

$${{\bar H}}_U(k;x)= (1- F^{\sum_{i=1}^{k}\alpha_i+\sum_{i=k+1}^{n}\beta_i} ) (1- F^{\sum_{i=1}^{k}\beta_i+\sum_{i=k+1}^{n}\alpha_i} ).$$

Now, we develop some sufficient conditions for the reversed hazard rate and the hazard rate orders of the two-series–parallel system.

Proof. (i) Note that the distribution functions of $U_{lk}$ and $U_{rk}$ are given by

$$F_{U_l}(k;x)=[F(x)]^{\sum_{i=1}^{k}\alpha_{i:n}+\sum_{i=k+1}^{n}\beta_{i:n}} \quad \text{and}\quad F_{U_r}(k;x)=[F(x)]^{\sum_{i=1}^{k}\beta_{i:n}+\sum_{i=k+1}^{n}\alpha_{i:n}},$$

respectively. In order to obtain the desired result, let $\tilde {U}_{lk}=1/U_{lk}$ and $\tilde {U}_{rk}=1/U_{rk}$, then

\begin{align*} {{\bar F}}_{\tilde{U}_l}(k;x)&= \mathbb{P}(1/U_{lk}>x) =\mathbb{P}\left(U_{lk}<\frac{1}{x}\right)\\ & =F_{{ U}_l}\left(k;\frac{1}{x}\right)=\left[F\left(\frac{1}{x}\right)\right]^{\sum_{i=1}^{k}\alpha_{i:n}+\sum_{i=k+1}^{n}\beta_{i:n}} \end{align*}

and

$$F\left(\frac{1}{x}\right)=\mathbb{P}\left(X \le \frac{1}{x}\right)=\mathbb{P}(1/X\ge x)={{\bar F}}_0(x).$$

Hence, the survival functions of $\tilde {U}_{lk}$ and $\tilde {U}_{rk}$ can be expressed by

$${{\bar F}}_{\tilde{U}_l}(k;x)=[{{\bar F}}_0(x)]^{\sum_{i=1}^{k}\alpha_{i:n}+\sum_{i=k+1}^{n}\beta_{i:n}}$$

and

$${{\bar F}}_{\tilde{U}_r}(k;x)=[{{\bar F}}_0(x)]^{\sum_{i=1}^{k}\beta_{i:n}+\sum_{i=k+1}^{n}\alpha_{i:n}},$$

respectively. Obviously, $\tilde {U}_{lk}$ and $\tilde {U}_{rk}$ follow the PH model. Let $\tilde U=\max \{\tilde U_{lk}, \tilde U_{rk}\}$, according to (i) of Theorem 2, we have $\tilde U_{k_1} \ge _{\textrm {hr}} \tilde U_{k_2}$, which is equivalent to

$$R_{\tilde U}(k_1,k_2; x) =\frac{{{\bar H}}_{\tilde{U}}({k_1};x)}{{{\bar H}}_{\tilde{U}}({k_2};x)} =\frac{H_{{U}}\left({k_1};\frac{1}{x}\right)}{H_{{U}}\left({k_2};\frac{1}{x}\right)} =R_{U}\left(k_1,k_2; \frac{1}{x}\right)$$

is increasing in $x \in \mathbb {R}^{+}$, and thus

$$R_{U}(k_1,k_2; x)=\frac{H_{{U}}({k_1};x)}{H_{{U}}({k_2};x)}$$

is decreasing in $x \in \mathbb {R}^{+}$, that is, $U_{k_1 } \le _{\textrm {rh}} U_{k_2}$. The desired result (i) is proved.

(ii) The proof of (ii) is similar to that of (i), and thus is omitted here.

Example 3 Under the assumptions of Theorems 3 and 4, let $F(x)$ be a Fréchet distribution, that is, $F(x)=e^{-({x}/{\lambda })^{-\gamma }}$. Set $\boldsymbol {\alpha }=(0.1, 0.9, 1.3, 1.7)\stackrel {\textrm {{m}}}{\succeq }(0.4,0.8,1.2,1.6)=\boldsymbol {\beta }$.

1. (i) Assume that subsystems $U_1$ and $U_2$ have the Clayton survival copula as (1)

   $$\hat C_{\theta_j}(u_{1},u_{2}) =(u_1^{-\theta_j}+u_2^{-\theta_j}-1)^{{-}1/\theta_j}, \quad u_{1} ,u_{2}\in [0,1] \text{ and } \theta_j\in(0,+\infty).$$
   Setting $\gamma =0.3, \lambda =2, \theta _1=0.1, \theta _2=0.2, \theta _3=0 .7$, and $\theta _4=0.8$. From Figure 5(a), we can see that all the difference functions ${{\bar H}}_{U}(k_{i+1};x)- {{\bar H}}_{U}(k_{i};x)$ are always non-negative, $i=1,2,3$, which justifies the validity of the result in Theorem 3.

2. (ii) Taking $\gamma _1=0.2, \lambda _1=0.1$ with remaining conditions unchanged. Figure 5(b) and 5(c) presents the plots of the difference functions $h_{U}(k_{i};x)-h_{U}(k_{i+1};x)$ and $\tilde {r}_{U}(k_{i+1};x)-\tilde {r}_{U}({k_{i}};x)$, respectively, which are always non-negative, $i=1,2,3$. Thus, the result of Theorem 4 is verified.

FIGURE 5. Plots of the difference functions ${{\bar H}}_{U}(k_{i+1};x)- {{\bar H}}_{U}(k_{i};x)$, $h_{U}(k_{i};x)-h_{U}(k_{i+1};x)$ and $\tilde {r}_{U}(k_{i+1};x)-\tilde {r}_{U}({k_{i}};x),\ i=1,2,3$, for all $x=-\ln u$ and $u\in (0,1]$.

Similar to Corollary 2, from Theorem 4, we have the following Corollary.

According to Corollary 4, $b_1$ is the worst allocation policy and $b_n$ is the optimal allocation policy within all possible allocations.

5.1. Viewpoint of the minimal path sets

In this subsection, we investigate the allocation policies of a parallel–series system with multiple subsystems in the viewpoint of the minimal path sets. Consider a parallel–series system having structure function $\tau$, and minimal path sets $P_1, P_2,\ldots , P_l$. For convenience, denote the lifetime of minimal set $P_j$ by $W_j$, where $W_j=\min _{i\in P_{j}}X_{i}$, $1\le j\le l$. Suppose that there are two classes of independent components $A_1,A_2,\ldots ,A_n$ and $B_1,B_2,\ldots ,B_n$ with lifetimes vectors $\boldsymbol {X}=(X_1, X_2,\ldots ,X_n)$ and $\boldsymbol {Y}=(Y_1, Y_2,\ldots ,Y_n)$ to be allocated, respectively. For two disjoint minimal path sets $P_{p}$ and $P_{q}$ ($1 \le p < q \le l$), denote $a_{\tau k}$ the policy that allocate $k$ components $A_i(i=1,2,\ldots ,k)$ and $n-k$ components $B_i(i=k+1,k+2,\ldots ,n)$ to minimal path sets $P_{p}$, and denote the corresponding copula of allocation policy $a_{\tau k}$ by $C_{ k}$, $k=1,2,\ldots ,n,$ then the lifetime of the resulting system under allocation policy $a_{\tau k}$ can be expressed by

(6)\begin{align} W_{\tau k}& = \max \left\{\max_{1\le m\le l,m\neq p, q} \{W_{m}\}, \max\{W_p, W_q \} \right\} \nonumber\\ & = \max \left\{\max_{1\le m\le l,m\neq p, q} \{W_{m}\}, S_k \right\}, \end{align}

where $S_k=\max \{W_p, W_q \}$.

In fact, there are situations where two disjoint minimal paths, two crossing minimal paths and a minimal path contains or is contained by another minimal path. Hence, owing to the complexity of modeling, we are ready to develop the main results in the following context:

1. A1: ${\rm A}1{\rm :}\; P_{1},P_{2},\ldots ,P_{l}$ are disjoint minimal path sets, that is, for $1\leq i\neq j\leq l, P_i\bigcap P_j=\emptyset$;

2. A2: A2: There exist two minimal path sets $P_{p}$ and $P_{q}$ having the same numbers of components;

3. A3: A3: The component lifetime vectors in the minimal path sets $P_{p}$ and $P_{q}$ both follow the PH model, that is, $\boldsymbol {X} \sim \textrm {PH} (\boldsymbol {\alpha }_{P_p}, {{\bar F}})$ and $\boldsymbol {Y} \sim \textrm {PH}(\boldsymbol {\beta }_{P_q}, {{\bar F}})$.

Next, we present some sufficient conditions in the sense of the usual stochastic order for two disjoint minimal path sets.

Proof. The idea of the proof is borrowed from Theorem 1 of Belzunce et al. [Reference Belzunce, Martínez-Riquelme and Ruiz4]. According to (6), the lifetime of the parallel–series system under two different allocation policies $a_{\tau k_1}$ and $a_{\tau k_2}$ can be written as

$$W_{\tau k_j} =\max \left\{\max_{1\le m\le l,m\neq p, q} \{W_{m}\}, S_{k_j} \right\}, \quad j=1,2.$$

Since the structure function $\tau$ is an increasing function, thus, according to Theorem 1 and the usual stochastic order with respect to the preservation property of increasing function, it follows that

$$\max \left\{\max_{1\le m\le l,m\neq p, q} \{W_{m}\}, S_{k_1} \right\} \ge_{\textrm{st}} \max \left\{\max_{1\le m\le l,m\neq p, q} \{W_{m}\}, S_{k_2} \right\},$$

which implies $W_{\tau k_1} \ge _{\textrm {st}} W_{\tau k_2}$, yielding the desired result.

In combination with the result of Theorem 2, it is easy to obtain the following Theorem 6 in terms of the reversed hazard rate order, the proof is similar to that of Theorem 5, thus is omitted here.

Theorem 6 Under the context of A1, A2 and A3. If $\boldsymbol {\alpha }_{P_p}\stackrel {\textrm {{m}}}{\succeq }\boldsymbol {\beta }_{P_q}$ and $\sum _{i=k_1+1}^{k_2} \alpha _{i:n} \ge \sum _{i=k_1+1}^{k_2}\beta _{i:n}$, then $k_1 < k_2$ implies

$$W_{\tau k_1 } \ge_{\textrm{rh}} W_{\tau k_2}.$$

The following example is presented to illustrate Theorems 5 and 6.

Example 4 Consider a parallel–series system as shown in Figure 6, its lifetime is

$$W= \max \{\min\{X_1, X_4 \}, X_3, \min\{X_2, X_5\} \}.$$

FIGURE 6. Parallel–series system.

The minimal path sets of the system are $P_1 = \{1, 4\}$, $P_2 = \{3\}$ and $P_3 = \{2,5\}$. Let $\bar F_i(x)=[\bar F(x)]^{a_i}$ be the survival function of $X_i$, where $\bar F(x)=e^{-\lambda x}$, $i=1,2,3,4,5$. Set $\lambda =0.02$, $a_1=3, a_2=4, a_3=2, a_4=6$, $a_5=5$. For two minimal path sets $P_1$ and $P_3$, we can check that $\boldsymbol {\alpha }_{P_1}=(3, 6)\stackrel {\textrm {{m}}}{\succeq }(4,5)=\boldsymbol {\beta }_{P_3}$.

(i) Suppose that $P_{1}$, $P_{2}$ and $P_3$ has FGM copula

$$C_{\theta_j}(u_{1},u_{2},u_{3})=\prod_{i=1}^{3}u_{i}+\theta_j\prod_{i=1}^{3}u_{i}(1-u_{i}), \quad u_{i} \in [0,1 ] \text{ and }\theta_j\in[{-}1,1].$$

Set $\theta _1=-0.8$, $\theta _2=0.9$. As Figure 7(a) displays, $W_{\tau k_1 } \ge _{\textrm {st}} W_{\tau k_2}$.

FIGURE 7. Plots of the difference functions (a) $\bar H_{W}(1;-\ln u)- \bar H_{W}(2;-\ln u)$ and (b) $\tilde {r}_{W}(1;-\ln u)-\tilde {r}_{W}(2;-\ln u)$, for all $x=-\ln u$ and $u\in (0,1]$.

(ii) As shown in Figure 7(b), $\tilde {r}_{W}(1;-\ln u)-\tilde {r}_{W}(2;-\ln u)$ is always non-negative.

5.2. Viewpoint of the minimal cut sets

In parallel to the previous subsection, here, we study the allocation policies of a series–parallel system in the viewpoint of the minimal cut sets. Consider a series–parallel system having structure function $\tau$, and minimal cut sets $C_1, C_2,\ldots , C_s$. For converience, we denote the lifetime of minimal set $C_j$ by $Z_j$, where $Z_j=\max _{i\in C_{j}}X_{i}$, $1\le j\le s$. For two disjoint minimal cut sets $C_{p}$ and $C_{q}$ ($1 \le p < q \le s$), denote the allocation policy with $k$ components $A_i\ (i=1,2,\ldots ,k)$ and $n-k$ components $B_i\ (i=k+1,k+2,\ldots ,n)$ being allocated to minimal cut sets $C_{p}$ by $b_{\tau k}$, and denote the corresponding survival copula of allocation policy $b_{\tau k}$ by $\hat C_{ k}$, $k=1,2,\ldots ,n,$ then lifetime of the resulting system under allocation policy $b_{\tau k}$ can be written as

(7)\begin{align} Z_{\tau k} & = \min \left\{\min_{1\le m\le s,m\neq p, q} \{Z_{m}\}, \min\{Z_p, Z_q \} \right\} \nonumber\\ & = \min \left\{\min_{1\le m\le s,m\neq p, q} \{Z_{m}\}, U_k \right\}, \end{align}

where $U_k=\min \{Z_p, Z_q \}$.

In general, for two minimal cuts, they may be disjoint, crossing or a minimal cut contains or is contained another minimal cut. Hence, owing to the complexity of modeling, we investigate the allocation policies of the coherent system in the following context:

1. B1: ${\rm B}1{\rm :} \;C_1, C_2,\ldots , C_s$ are disjoint minimal cut sets, that is, for $1\leq i\neq j\leq l, C_i\cap C_j=\emptyset$;

2. B2: B2: There exist two minimal cut sets $C_{p}$ and $C_{q}$ having the same numbers of components;

3. B3: B3: The component lifetime vectors in the minimal cut sets $C_{p}$ and $C_{q}$ both follow the PRH model, that is, $\boldsymbol {X} \sim \textrm {PRH} (\boldsymbol {\alpha }_{C_p}, F)$ and $\boldsymbol {Y} \sim \textrm {PRH}(\boldsymbol {\beta }_{C_q}, F)$.

Proof. By means of (7), the lifetimes of the series–parallel system under two different allocation policies $b_{\tau {k_1}}$ and $b_{\tau {k_2}}$ can be expressed by

$$Z_{k_j} =\min \left\{\min_{1\le m\le s,m\neq p, q} \{Z_{m}\}, U_{k_j} \right\}, \quad j=1,2.$$

Owing to the structure function, $\tau$ is an increasing function, hence, combining Theorem 4 with the usual stochastic order with respect to the preservation property of increasing function, it holds that

$$\min \left\{\min_{1\le m\le s,m\neq p, q} \{Z_{m}\}, U_{k_1} \right\} \le_{\textrm{st}} \min \left\{\min_{1\le m\le s,m\neq p, q} \{Z_{m}\}, U_{k_2} \right\},$$

which means that $Z_{\tau k_1} \le _{\textrm {st}} Z_{\tau k_2}$. Hence, the result is proved.

The following theorem provides the results with respect to the hazard rate order. The proof can be obtained along the same line as in Theorem 7 and is hence omitted.

Theorem 8 Under the context of B1, B2 and B3. If $\boldsymbol {\alpha }_{C_p}\stackrel {\textrm {{m}}}{\succeq }\boldsymbol {\beta }_{C_q}$ and $\sum _{i=k_1+1}^{k_2} \alpha _{i:n} \ge \sum _{i=k_1+1}^{k_2}\beta _{i:n}$, then $k_1 < k_2$ implies

$$Z_{\tau k_1 } \le_{\textrm{hr}} Z_{\tau k_2}.$$

The next example gives some illustrations of Theorems 7 and 8.

Example 5 Consider a series–parallel system as shown in Figure 8, its lifetime is

$$Z= \min \{\max\{X_1, X_2 \}, X_3, \max\{ X_4, X_5 \} \}.$$

The minimal cut sets are $C_1 = \{1, 2\}$, $C_2 = \{3\}$ and $C_3 = \{4, 5\}$. Let $F_i(x)=[F(x)]^{b_i}$ be the distribution function of $X_i$, where $F(x)=e^{-({x}/{\lambda })^{-\gamma }}$, $i=1,2,3,4,5$. Set $\lambda =1$, $\gamma =0.1$, $b_1=3$, $b_2=6$, $b_3=2$, $b_4=4$, $b_5=5$. For two minimal cut sets $C_1$ and $C_3$, it is easily checked that $\boldsymbol {\alpha }_{C_1}=(3, 6)\stackrel{ \textrm {{m}}}{\succeq }(4,5)=\boldsymbol {\beta }_{C_3}$.

FIGURE 8. Series–parallel system.

(i) Suppose that $C_1$, $C_2$ and $C_3$ have Clayton copula

$$\hat C_{\theta_j}(u_{1},u_{2},u_{3})=\left(\sum_{i=1}^{3} u_{i}^{-\theta_j}-2 \right)^{{-}1/\theta_j}, \quad u_{i} \in [0,1 ],\ \theta_j \in (0, \infty),\ j=1, 2.$$

Set $\theta _1=6$, $\theta _2=8$. As displayed in Figure 9(a), $\bar H_{Z}(2;x)- \bar H_{Z}(1;x)$ is always non-negative.

FIGURE 9. Plots of the difference functions (a) $\bar H_{Z}(2;x)- \bar H_{Z}(1;x)$ and (b) $h_{Z}(1;x)-h_{Z}(2;x)$, for all $x=-\ln u$ and $u\in (0,1]$.

(ii) As Figure 9(b) shows, $h_{Z}(1;x)-h_{Z}(2;x)$ is always non-negative.