Appendix A: Proofs of the main results
First, we express our heartfelt thanks to Dr. Weiyong Ding for the careful reading on our previous work (Yan et al. 2021), Optimal allocation of relevations in coherent systems, Journal of Applied Probability, 58, 1152–1169), and pointing out that the proof of Theorem 1 is not correct under the assumption \(T_1 \le _{hr} T_2\). The main problem is that the application of Lemma 1 is not correct in the end of that proof. As a remedy, the following Lemma 4 is a corrected version of the result with the assumption \(T_1 \le _{hr} T_2\) replaced by \(T_1 \le _{lr} T_2\). For this case, the proof can be proven separately without the help of Lemma 1.
Lemma 4
Let \(T_1,T_2, \dots , T_n\) be the n components lifetimes with survival functions \(\overline{F}_1, \overline{F}_2, \dots ,\) \( \overline{F}_n\) in a series system, respectively. Let \(S_1\) and \(S_2\) be the lifetimes of two relevations with survival functions \( \overline{G}_1\) and \(\overline{G}_2\), respectively. Denote
$$\begin{aligned} V_1=\min \Big \{ T_1\#S_1, T_2\#S_2, T_3, \dots , T_n \Big \} \end{aligned}$$
and
$$\begin{aligned} V_2=\min \Big \{T_1\#S_2, T_2\#S_1, T_3, \dots , T_n \Big \}. \end{aligned}$$
If \(T_1 \le _{lr} T_2\) and \(S_1 \ge _{hr} S_2\), then \(V_1 \ge _{st} V_2.\)
Proof
The survival functions of \(V_1\) and \(V_2\) can be expressed as
$$\begin{aligned} \overline{H}_{V_1}(t) =\prod _{ l=3 }^n \overline{F}_l(t) \Bigg \{ \Big ( \overline{F}_1(t) + \overline{G}_1(t)\int _0^t \frac{f_1( u )}{\overline{G}_1( u )} \textrm{d}u \Big ) \Big ( \overline{F}_2(t) + \overline{G}_2(t)\int _0^t \frac{f_2( u )}{\overline{G}_2( u )} \textrm{d}u \Big ) \Bigg \} \end{aligned}$$
and
$$\begin{aligned} \overline{H}_{V_2}(t) =\prod _{ l=3 }^n \overline{F}_l(t) \Bigg \{ \Big ( \overline{F}_1(t) + \overline{G}_2(t)\int _0^t \frac{f_1( u )}{\overline{G}_2( u )} \textrm{d}u \Big ) \Big ( \overline{F}_2(t) + \overline{G}_1(t)\int _0^t \frac{f_2( u )}{\overline{G}_1( u )} \textrm{d}u \Big ) \Bigg \}, \end{aligned}$$
respectively. Observe that
$$\begin{aligned}{} & {} \overline{H}_{V_1}(t)-\overline{H}_{V_2}(t)\\{} & {} \quad = \overline{F}_1(t)\overline{F}_2(t) + \overline{F}_1(t){{\overline{G}}_2}(t)\int _0^t {\frac{{f_2(u)}}{{\overline{G}_2(u)}}} \textrm{d}u + \overline{F}_2(t)\overline{G}_1(t)\int _0^t {\frac{{ f_1(u)}}{{\overline{G}_1(u)}}} \textrm{d}u \\{} & {} \qquad + \overline{G}_1(t)\overline{G}_2(t)\int _0^t {\frac{{ f_1(u)}}{{\overline{G}_1(u)}}} \textrm{d}u\int _0^t {\frac{{f_2(u)}}{{\overline{G}_2(u)}}} \textrm{d}u \\{} & {} \qquad -\Big ( \overline{F}_1(t)\overline{F}_2(t) + \overline{F}_1(t)\overline{G}_1(t)\int _0^t {\frac{{ f_2(u)}}{{\overline{G}_1(u)}}} \textrm{d}u + \overline{F}_2(t)\overline{G}_2(t)\int _0^t {\frac{{ f_1(u)}}{{\overline{G}_2(u)}}} \textrm{d}u \\{} & {} \qquad + \overline{G}_1(t)\overline{G}_2(t)\int _0^t {\frac{{ f_1(u)}}{{\overline{G}_2(u)}}} \textrm{d}u\int _0^t {\frac{{ f_2(u)}}{{\overline{G}_1(u)}}} \textrm{d}u \Big ) \\{} & {} \quad = \int _0^t {\overline{F}_1(t) f_2(u) \left[ {\frac{{\overline{G}_2(t)}}{{\overline{G}_2(u)}} - \frac{{\overline{G}_1(t)}}{{\overline{G}_1(u)}}} \right] } \textrm{d}u + \int _0^t {\overline{F}_2(t) f_1(u) \left[ {\frac{{\overline{G}_1(t)}}{{\overline{G}_1(u)}} - \frac{{\overline{G}_2(t)}}{{\overline{G}_2(u)}}} \right] } \textrm{d}u \\{} & {} \qquad + \overline{G}_1(t)\overline{G}_2(t) \left[ {\int _0^t {\frac{{ f_1(u)}}{{\overline{G}_1(u)}}} \textrm{d}u\int _0^t {\frac{{ f_2(u)}}{{\overline{G}_2(u)}}} \textrm{d}u - \int _0^t {\frac{{ f_1(u)}}{{\overline{G}_2(u)}}} \textrm{d}u\int _0^t {\frac{{ f_2(u)}}{{\overline{G}_1(u)}}} \textrm{d}u} \right] \\{} & {} \quad =: \phi _1(t)+\phi _2(t), \end{aligned}$$
where
$$\begin{aligned} \phi _1(t) = \int _0^t { \Big [{\overline{F}_2(t)f_1(u) - \overline{F}_1(t)f_2(u)} \Big ] \left[ {\frac{\overline{G}_1(t)}{\overline{G}_1(u)} - \frac{\overline{G}_2(t)}{\overline{G}_2(u)}} \right] } \textrm{d}u \end{aligned}$$
and
$$\begin{aligned} \phi _2(t)=\overline{G}_1(t)\overline{G}_2(t) \left[ {\int _0^t {\frac{f_1(u)}{\overline{G}_1(u)}} \textrm{d}u\int _0^t {\frac{f_2(u)}{\overline{G}_2(u)}} \textrm{d}u - \int _0^t {\frac{f_1(u)}{\overline{G}_2(u)}} \textrm{d}u\int _0^t {\frac{f_2(u)}{\overline{G}_1(u)}} \textrm{d}u} \right] . \end{aligned}$$
Note that \(T_1 \le _{lr} T_2\) implies \( {\overline{F}_2(t)f_1(u) - \overline{F}_1(t)f_2(u)} \ge 0\), for all \( 0 \le u \le t\). On the other hand, \(S_1 \ge _{hr} S_2\) is equivalent to
$$\begin{aligned} \frac{\overline{G}_1(t)}{\overline{G}_2(t)} \ge \frac{\overline{G}_1(u)}{\overline{G}_2(u)},\quad \text { for all } 0 \le u \le t, \end{aligned}$$
that is, \({\frac{\overline{G}_1(t)}{\overline{G}_1(u)} - \frac{\overline{G}_2(t)}{\overline{G}_2(u)}}\) is also non-negative, for all \( 0 \le u \le t\). Hence, \(\phi _1(t)\) is non-negative for all \(t\in {\mathbb {R}^{}}_+\).
Observe that
$$\begin{aligned}{} & {} \phi _2(t) \overset{\textrm{sign}}{=}\ {\int _0^t {\frac{f_1(u)}{\overline{G}_1(u)}} \textrm{d}u\int _0^t {\frac{f_2(u)}{\overline{G}_2(u)}} \textrm{d}u - \int _0^t {\frac{f_1(u)}{\overline{G}_2(u)}} \textrm{d}u\int _0^t {\frac{f_2(u)}{\overline{G}_1(u)}} \textrm{d}u}\\{} & {} \quad =\int _0^t \int _0^t \frac{f_1(u_1) f_2(u_2)}{\overline{G}_1(u_1) \overline{G}_2(u_2)}\textrm{d}u_1 \textrm{d}u_2 -\int _0^t \int _0^t \frac{f_1(u_1) f_2(u_2)}{\overline{G}_2(u_1) \overline{G}_1(u_2)}\textrm{d}u_1 \textrm{d}u_2\\{} & {} \quad =\int _0^t \int _0^t f_1(u_1) f_2(u_2) \bigg [\frac{1}{\overline{G}_1(u_1) \overline{G}_2(u_2)}-\frac{1}{\overline{G}_1(u_2) \overline{G}_2(u_1)} \bigg ]\textrm{d}u_1 \textrm{d}u_2 \\{} & {} \quad =\int _0^{u_1} \int _{u_2}^t f_1(u_1) f_2(u_2) \bigg [\frac{1}{\overline{G}_1(u_1) \overline{G}_2(u_2)}-\frac{1}{\overline{G}_1(u_2) \overline{G}_2(u_1)} \bigg ]\textrm{d}u_1 \textrm{d}u_2 \\{} & {} \qquad +\int _0^{u_1} \int _{u_2}^t {f_1(u_2) f_2(u_1) } \bigg [\frac{1}{\overline{G}_1(u_2) \overline{G}_2(u_1)}-\frac{1}{\overline{G}_1(u_1) \overline{G}_2(u_2)} \bigg ]\textrm{d}u_1 \textrm{d}u_2\\{} & {} \quad =\int _0^{u_1} \int _{u_2}^t \bigg [f_1(u_1) f_2(u_2)-f_1(u_2) f_2(u_1)\bigg ] \bigg [\frac{1}{\overline{G}_1(u_1) \overline{G}_2(u_2)} -\frac{1}{\overline{G}_1(u_2) \overline{G}_2(u_1)} \bigg ]\textrm{d}u_1 \textrm{d}u_2. \end{aligned}$$
For any \(u_1 \le u_2\), note that \(T_1 \le _{lr} T_2\) and \(S_1 \ge _{hr} S_2\) imply
$$\begin{aligned} f_1(u_1) f_2(u_2)-f_1(u_2) f_2(u_1)\ge 0 \end{aligned}$$
and
$$\begin{aligned} \frac{1}{\overline{G}_1(u_1) \overline{G}_2(u_2)}-\frac{1}{\overline{G}_1(u_2) \overline{G}_2(u_1)} \ge 0, \end{aligned}$$
respectively. Hence, \(\phi _2(t)\) is non-negative for all \(t\in {\mathbb {R}}_+\). To sum up, for all \(t\in {\mathbb {R}}_+\),
$$\begin{aligned} \overline{H}_{V_1}(t) - \overline{H}_{V_2}(t) =\phi _1(t)+\phi _2(t) \ge 0, \end{aligned}$$
and thus the proof is completed. \(\square \)
Now, we are ready to present the proofs of the main results in this paper. Some of the proofs will depend on the corrected result in Lemma 4.
1.1 Appendix A.1: Proof of Theorem 1
Proof
Denote the distribution functions of \(\tau (X_k\#Y_1, X_l\#Y_2, \varvec{X}^{(k,l)})\) and \(\tau (X_k\#Y_2, X_l\#Y_1, \varvec{X}^{(k,l)})\) by \( F_{\tau _1}\) and \(F_{\tau _2}\), respectively. It suffices to prove that \( F_{\tau _1}(t) \le F_{\tau _2}(t)\), for all \(t \in {\mathbb {R}}_+\). According to Equation (3), we have
$$\begin{aligned} \tau (X_k\#Y_1, X_l\#Y_2, \varvec{X}^{(k,l)})= & {} \max \Big \{\min \{X_k\#Y_1, X_l\#Y_2,\underset{{\mathcal {P}} \in {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne k,l}{\min }X_i\},\\{} & {} \min \{X_l\#Y_2,\underset{{\mathcal {P}} \in {\mathcal {A}}_l\backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i\}, \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}}}{\min }X_i \Big \}. \end{aligned}$$
Then, it can be obtained that
$$\begin{aligned} F_{\tau _1}(t)= & {} {\mathbb {P}}(\tau (X_k\#Y_1, X_l\#Y_2, \varvec{X}^{(k,l)}) \le t)\\= & {} {\mathbb {P}}(\min \{X_k\#Y_1, X_l\#Y_2,\underset{{\mathcal {P}} \in {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne k,l}{\min }X_i\}\\{} & {} \le t, \min \{X_l\#Y_2,\underset{{\mathcal {P}} \in {\mathcal {A}}_l\backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i\}\le t, \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}}}{\min }X_i \le t)\\=: & {} {\mathbb {P}}(E_{kl}\cap E_{{\bar{k}}l} \cap E_{{\bar{k}} {\bar{l}}}), \end{aligned}$$
where \(E_{{\bar{k}} {\bar{l}}}=\Big \{ \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}}}{\min }X_i \le t \Big \}\), \(E_{kl}=\Big \{\min \{X_k\#Y_1, X_l\#Y_2,\underset{{\mathcal {P}} \in {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne k,l}{\min }X_i\} \le t \Big \}\), and \(E_{{\bar{k}}l}=\Big \{ \min \{X_l\#Y_2,\underset{{\mathcal {P}} \in {\mathcal {A}}_l\backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i\}\le t \Big \}\). Based on the decomposition method in Wu et al. (2022), we have
$$\begin{aligned} {\mathbb {P}}(E_{kl}\cap E_{{\bar{k}}l} \cap E_{{\bar{k}} {\bar{l}}})= & {} {\mathbb {P}}(E_{{\bar{k}} {\bar{l}}}) -{\mathbb {P}}({{\overline{E}}_{kl}}\cap {E_{\bar{k} {\bar{l}}} }) \nonumber \\{} & {} -{\mathbb {P}}({{\overline{E}}_{{\bar{k}}l}}\cap {E_{\bar{k} {\bar{l}}} } ) +{\mathbb {P}}({{\overline{E}}_{kl}}\cap {E_{\bar{k} {\bar{l}}} } \cap {{\overline{E}}_{{\bar{k}}l}} ). \end{aligned}$$
(A.1)
First, one can calculate that
$$\begin{aligned} {\mathbb {P}}({{\overline{E}}_{kl}}\cap {E_{\bar{k} {\bar{l}}} })= & {} {\mathbb {P}}(\min \{X_k\#Y_1, X_l\#Y_2,\underset{{\mathcal {P}} \in {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne k,l}{\min }X_i\}> t, \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}} }{\min }X_i \le t)\\= & {} {\mathbb {P}}( X_k\#Y_1> t){\mathbb {P}}( X_l\#Y_2> t) {\mathbb {P}}(\underset{{\mathcal {P}} \in {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne k,l}{\min }X_i> t, \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}} }{\min }X_i \le t),\\ {\mathbb {P}}({{\overline{E}}_{{\bar{k}}l}}\cap {E_{\bar{k} {\bar{l}}} })= & {} {\mathbb {P}}(\min \{ X_l\#Y_2,\underset{{\mathcal {P}} \in {\mathcal {A}}_l \backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i\}> t, \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}} }{\min }X_i \le t)\\= & {} {\mathbb {P}}( X_l\#Y_2> t) {\mathbb {P}}( \underset{{\mathcal {P}} \in {\mathcal {A}}_l \backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i> t, \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}} }{\min }X_i \le t),\\ {\mathbb {P}}({{\overline{E}}_{kl}}\cap {E_{\bar{k} {\bar{l}}} } \cap {{\overline{E}}_{{\bar{k}}l}} )= & {} {\mathbb {P}}(\min \{X_k\#Y_1, X_l\#Y_2,\underset{{\mathcal {P}} \in {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne k,l}{\min }X_i\}> t,\\{} & {} \min \{X_l\#Y_2,\underset{{\mathcal {P}} \in {\mathcal {A}}_l\backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i\}> t, \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}}}{\min }X_i \le t)\\= & {} {\mathbb {P}}( X_k\#Y_1> t){\mathbb {P}}( X_l\#Y_2> t) {\mathbb {P}}(\underset{{\mathcal {P}} \in {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne k,l}{\min }X_i> t,\\{} & {} \underset{{\mathcal {P}} \in {\mathcal {A}}_l\backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i > t, \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}}}{\min }X_i \le t). \end{aligned}$$
Then the distribution function of \(\tau (X_k\#Y_1, X_l\#Y_2, \varvec{X}^{(k,l)})\) can be rewritten as
$$\begin{aligned} F_{\tau _1}(t)= & {} {\mathbb {P}}\left( \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}}}{\min }X_i \le t\right) -{\mathbb {P}}( X_k\#Y_1> t){\mathbb {P}}\\ {}{} & {} \times ( X_l\#Y_2> t) {\mathbb {P}}\left( \underset{{\mathcal {P}} \in {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne k,l}{\min }X_i> t, \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}} }{\min }X_i \le t\right) \\{} & {} -{\mathbb {P}}( X_l\#Y_2> t) {\mathbb {P}}\left( \underset{{\mathcal {P}} \in {\mathcal {A}}_l \backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i> t, \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}} }{\min }X_i \le t\right) \\{} & {} +{\mathbb {P}}( X_k\#Y_1> t){\mathbb {P}}( X_l\#Y_2> t) {\mathbb {P}}\\ {}{} & {} \times \left( \underset{{\mathcal {P}} \in {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne k,l}{\min }X_i> t, \underset{{\mathcal {P}} \in {\mathcal {A}}_l\backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i> t, \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}}}{\min }X_i \le t\right) \\= & {} {\mathbb {P}}\left( \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}}}{\min }X_i \le t\right) -{\mathbb {P}}( X_k\#Y_1> t){\mathbb {P}}( X_l\#Y_2> t) {\mathbb {P}}\\ {}{} & {} \times (\underset{{\mathcal {P}} \in {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne k,l}{\min }X_i> t, \underset{{\mathcal {P}} \in {\mathcal {A}}_l\backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i \le t, \\{} & {} \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}}}{\min }X_i \le t) -{\mathbb {P}}( X_l\#Y_2> t) {\mathbb {P}}( \underset{{\mathcal {P}} \in {\mathcal {A}}_l \backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i > t, \\ {}{} & {} \times \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}} }{\min }X_i \le t). \end{aligned}$$
Similarly, the distribution function of \(\tau (X_k\#Y_2, X_l\#Y_1, \varvec{X}^{(k,l)})\) is given by
$$\begin{aligned} F_{\tau _2}(t)= & {} {\mathbb {P}}(\tau (X_k\#Y_2, X_l\#Y_1, \varvec{X}^{(k,l)}) \le t)\\= & {} {\mathbb {P}}(\underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}}}{\min }X_i \le t)\\{} & {} -{\mathbb {P}}( X_k\#Y_2> t){\mathbb {P}}( X_l\#Y_1> t) {\mathbb {P}}(\underset{{\mathcal {P}} \in {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne k,l}{\min }X_i> t, \underset{{\mathcal {P}} \in {\mathcal {A}}_l\backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i \le t, \\{} & {} \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}}}{\min }X_i \le t)\\{} & {} -{\mathbb {P}}( X_l\#Y_1> t) {\mathbb {P}}( \underset{{\mathcal {P}} \in {\mathcal {A}}_l \backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i > t, \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}} }{\min }X_i \le t). \end{aligned}$$
Therefore, we have
$$\begin{aligned}{} & {} F_{\tau _2}(t)-F_{\tau _1}(t)\\{} & {} \quad ={\mathbb {P}}(\underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}}}{\min }X_i \le t) \\{} & {} \qquad -{\mathbb {P}}( X_k\#Y_2> t){\mathbb {P}}( X_l\#Y_1> t) {\mathbb {P}}(\underset{{\mathcal {P}} \in {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne k,l}{\min }X_i> t, \underset{{\mathcal {P}} \in {\mathcal {A}}_l\backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i \le t, \\{} & {} \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}}}{\min }X_i \!\le \! t) -{\mathbb {P}} ( X_l\#Y_1\!>\! t) {\mathbb {P}}( \underset{{\mathcal {P}} \in {\mathcal {A}}_l \backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i \!>\! t, \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}} }{\min }X_i \!\le \! t)\\{} & {} \qquad -\left[ {\mathbb {P}}(\underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}}}{\min }X_i \le t) -{\mathbb {P}}( X_k\#Y_1> t){\mathbb {P}}( X_l\#Y_2> t) {\mathbb {P}} (\underset{{\mathcal {P}} \in {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne k,l}{\min }X_i> t, \underset{{\mathcal {P}} \in {\mathcal {A}}_l\backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i \le t, \right. \\{} & {} \qquad \left. \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}}}{\min }X_i \le t )-{\mathbb {P}}( X_l\#Y_2> t) {\mathbb {P}} ( \underset{{\mathcal {P}} \in {\mathcal {A}}_l \backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i> t, \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}} }{\min }X_i \le t) \right] \\{} & {} \quad =\Big [{\mathbb {P}}( X_k\#Y_1> t){\mathbb {P}}( X_l\#Y_2> t)-{\mathbb {P}}( X_k\#Y_2> t){\mathbb {P}}( X_l\#Y_1> t) \Big ]\\{} & {} \quad \times {\mathbb {P}}(\underset{{\mathcal {P}} \in {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne k,l}{\min }X_i> t, \underset{{\mathcal {P}} \in {\mathcal {A}}_l\backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i \le t, \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}}}{\min }X_i \le t)\\{} & {} \qquad +\Big [{\mathbb {P}}(X_l\#Y_2>t)-{\mathbb {P}}(X_l\#Y_1>t) \Big ]{\mathbb {P}}( \underset{{\mathcal {P}} \in {\mathcal {A}}_l \backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i> t, \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}} }{\min }X_i \le t)\\{} & {} \quad =: D_{1}(t) {\mathbb {P}}(\underset{{\mathcal {P}} \in {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne k,l}{\min }X_i> t, \underset{{\mathcal {P}} \in {\mathcal {A}}_l\backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i \le t, \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}}}{\min }X_i \le t)\\{} & {} \qquad +D_{2}(t){\mathbb {P}}( \underset{{\mathcal {P}} \in {\mathcal {A}}_l \backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i > t, \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}} }{\min }X_i \le t), \end{aligned}$$
where \(D_{1}(t)={\mathbb {P}}( X_k\#Y_1> t){\mathbb {P}}( X_l\#Y_2> t)-{\mathbb {P}}( X_k\#Y_2> t){\mathbb {P}}( X_l\#Y_1> t)\) and \(D_{2}(t)= {\mathbb {P}}(X_l\#Y_2>t)-{\mathbb {P}}(X_l\#Y_1>t)\).
On one hand, according to the assumptions \(X_k \ge _{lr} X_l\) and \(Y_1 \le _{hr} Y_2\), and the result of Lemma 4, we have \(D_{1}(t)\ge 0\). On the other hand, \(Y_1 \le _{hr} Y_2\) implies
$$\begin{aligned} D_{2}(t)= & {} {\mathbb {P}}(X_l\#Y_2>t)-{\mathbb {P}}(X_l\#Y_1>t) =\overline{F}_l(t)\\{} & {} +\overline{G}_2(t)\int _0^t\frac{f_l(u)}{\overline{G}_2(u)}\textrm{d}u-\Big (\overline{F}_l(t)+\overline{G}_1(t)\int _0^t\frac{f_l(u)}{\overline{G}_1(u)}\textrm{d}u \Big ) \\= & {} \int _0^t \bigg (\frac{\overline{G}_2(t)f_l(u)}{\overline{G}_2(u)} -\frac{\overline{G}_1(t)f_l(u)}{\overline{G}_1(u)}\bigg )\textrm{d}u \ge 0. \end{aligned}$$
Then, it follows that both \(D_{1}(t)\) and \(D_{2}(t)\) are non-negative, meaning that \(F_{\tau _2}(t)\ge F_{\tau _1}(t)\), for all \(t \in {\mathbb {R}}_+\). Thus, the desired result is proved. \(\square \)
1.2 Appendix A.2: Proof of Theorem 2
Proof
The survival functions of \(\tau (X_k\#Y_1, X_l\#Y_2, \varvec{X}^{(k,l)})\) and \(\tau (X_k\#Y_2, X_l\#Y_1, \varvec{X}^{(k,l)})\) denote as \(\overline{F}_{\tau _1}\) and \({\overline{F}}_{\tau _2}\), respectively. Based on the minimal cut set decomposition in Equation (3) and \({\mathcal {S}}_k \subseteq {\mathcal {S}}_l\), we have
$$\begin{aligned} \tau (X_k\#Y_1, X_l\#Y_2, \varvec{X}^{(k,l)})= & {} \min \Big \{\max \{X_k\#Y_1, X_l\#Y_2,\underset{{\mathcal {C}} \in {\mathcal {S}}_k }{\min }\underset{i \in {\mathcal {C}}, i \ne k,l}{\max }X_i\}, \\{} & {} \max \{X_l\#Y_2,\underset{{\mathcal {C}} \in {\mathcal {S}}_l\backslash {\mathcal {S}}_k }{\min }\underset{i \in {\mathcal {C}}, i \ne l}{\max }X_i\}, \underset{{\mathcal {C}} \in {\mathcal {S}}\backslash {\mathcal {S}}_l }{\min }\underset{i \in {\mathcal {C}}}{\max }X_i \Big \}. \end{aligned}$$
using similar arguments with Equation (A.1), for any \(t\in {\mathbb {R}}_+\), we have
$$\begin{aligned} {\overline{F}}_{\tau _1}(t)= & {} {\mathbb {P}}(\max \{X_k\#Y_1, X_l\#Y_2,\underset{{\mathcal {C}} \in {\mathcal {S}}_k }{\min }\underset{i \in {\mathcal {C}}, i \ne k,l}{\max }X_i\}>t, \\{} & {} \max \{X_l\#Y_2,\underset{{\mathcal {C}} \in {\mathcal {S}}_l\backslash {\mathcal {S}}_k }{\min }\underset{i \in {\mathcal {C}}, i \ne l}{\max }X_i\}>t, \underset{{\mathcal {C}} \in {\mathcal {S}}\backslash {\mathcal {S}}_l }{\min }\underset{i \in {\mathcal {C}}}{\max }X_i>t)\\= & {} {\mathbb {P}}(\underset{{\mathcal {C}} \in {\mathcal {S}}\backslash {\mathcal {S}}_l }{\min }\underset{i \in {\mathcal {C}}}{\max }X_i>t) -{\mathbb {P}}( X_k\#Y_1\le t){\mathbb {P}}( X_l\#Y_2\le t) {\mathbb {P}}(\underset{{\mathcal {C}} \in {\mathcal {S}}_k }{\min }\underset{i \in {\mathcal {C}}, i \ne k,l}{\max }X_i \le t,\\{} & {} \quad \underset{{\mathcal {C}} \in {\mathcal {S}}\backslash {\mathcal {S}}_l }{\min }\underset{i \in {\mathcal {C}} }{\max }X_i> t)\\{} & {} -{\mathbb {P}}( X_l\#Y_2\le t) {\mathbb {P}}( \underset{{\mathcal {C}} \in {\mathcal {S}}_l \backslash {\mathcal {S}}_k }{\min }\underset{i \in {\mathcal {C}}, i \ne l}{\max }X_i \le t, \underset{{\mathcal {C}} \in {\mathcal {S}}\backslash {\mathcal {S}}_l }{\min }\underset{i \in {\mathcal {C}} }{\max }X_i> t)\\{} & {} +{\mathbb {P}}( X_k\#Y_1\le t){\mathbb {P}}( X_l\#Y_2\le t) {\mathbb {P}}(\underset{{\mathcal {C}} \in {\mathcal {S}}_k }{\min }\underset{i \in {\mathcal {C}}, i \ne k,l}{\max }X_i \le t,\\{} & {} \quad \underset{{\mathcal {C}} \in {\mathcal {S}}_l\backslash {\mathcal {S}}_k }{\min }\underset{i \in {\mathcal {C}}, i \ne l}{\max }X_i \le t, \underset{{\mathcal {C}} \in {\mathcal {S}}\backslash {\mathcal {S}}_l }{\min }\underset{i \in {\mathcal {C}}}{\max }X_i> t)\\= & {} {\mathbb {P}}(\underset{{\mathcal {C}} \in {\mathcal {S}}\backslash {\mathcal {S}}_l }{\min }\underset{i \in {\mathcal {C}}}{\max }X_i>t) -{\mathbb {P}}( X_k\#Y_1\le t){\mathbb {P}}( X_l\#Y_2\le t) {\mathbb {P}}(\underset{{\mathcal {C}} \in {\mathcal {S}}_k }{\min }\underset{i \in {\mathcal {C}}, i \ne k,l}{\max }X_i \le t,\\{} & {} \quad \underset{{\mathcal {C}} \in {\mathcal {S}}_l\backslash {\mathcal {S}}_k }{\min }\underset{i \in {\mathcal {C}}, i \ne l}{\max }X_i> t, \underset{{\mathcal {C}} \in {\mathcal {S}}\backslash {\mathcal {S}}_l }{\min }\underset{i \in {\mathcal {C}}}{\max }X_i> t) \\{} & {} -{\mathbb {P}}( X_l\#Y_2\le t) {\mathbb {P}}( \underset{{\mathcal {C}} \in {\mathcal {S}}_l \backslash {\mathcal {S}}_k }{\min }\underset{i \in {\mathcal {C}}, i \ne l}{\max }X_i \le t, \underset{{\mathcal {C}} \in {\mathcal {S}}\backslash {\mathcal {S}}_l }{\min }\underset{i \in {\mathcal {C}} }{\max }X_i > t). \end{aligned}$$
Similarly,
$$\begin{aligned} {\overline{F}}_{\tau _2}(t)= & {} {\mathbb {P}}(\underset{{\mathcal {C}} \in {\mathcal {S}}\backslash {\mathcal {S}}_l }{\min }\underset{i \in {\mathcal {C}}}{\max }X_i>t) \\{} & {} -{\mathbb {P}}( X_k\#Y_2\le t){\mathbb {P}}( X_l\#Y_1\le t) {\mathbb {P}}(\underset{{\mathcal {C}} \in {\mathcal {S}}_k }{\min }\underset{i \in {\mathcal {C}}, i \ne k,l}{\max }X_i \le t, \underset{{\mathcal {C}} \in {\mathcal {S}}_l\backslash {\mathcal {S}}_k }{\min }\underset{i \in {\mathcal {C}}, i \ne l}{\max }X_i> t, \\{} & {} \underset{{\mathcal {C}} \in {\mathcal {S}}\backslash {\mathcal {S}}_l }{\min }\underset{i \in {\mathcal {C}}}{\max }X_i> t) -{\mathbb {P}}( X_l\#Y_1\le t) {\mathbb {P}}( \underset{{\mathcal {C}} \in {\mathcal {S}}_l \backslash {\mathcal {S}}_k }{\min }\underset{i \in {\mathcal {C}}, i \ne l}{\max }X_i \le t, \underset{{\mathcal {C}} \in {\mathcal {S}}\backslash {\mathcal {S}}_l }{\min }\underset{i \in {\mathcal {C}} }{\max }X_i > t). \end{aligned}$$
Hence,
$$\begin{aligned}{} & {} {\overline{F}}_{\tau _2}(t)-{\overline{F}}_{\tau _1}(t)\\{} & {} \quad =\Big [{\mathbb {P}}( X_k\#Y_1\le t){\mathbb {P}}( X_l\#Y_2\le t)-{\mathbb {P}}( X_k\#Y_2\le t) {\mathbb {P}}( X_l\#Y_1\le t) \Big ]\\{} & {} \qquad \times {\mathbb {P}}(\underset{{\mathcal {C}} \in {\mathcal {S}}_k }{\min }\underset{i \in {\mathcal {C}}, i \ne k,l}{\max }X_i \le t, \underset{{\mathcal {C}} \in {\mathcal {S}}_l\backslash {\mathcal {S}}_k }{\min }\underset{i \in {\mathcal {C}}, i \ne l}{\max }X_i> t, \underset{{\mathcal {C}} \in {\mathcal {S}}\backslash {\mathcal {S}}_l }{\min }\underset{i \in {\mathcal {C}}}{\max }X_i> t)\\{} & {} \qquad +\Big [{\mathbb {P}}(X_l\#Y_2\le t)-{\mathbb {P}}(X_l\#Y_1\le t) \Big ] {\mathbb {P}}( \underset{{\mathcal {C}} \in {\mathcal {S}}_l \backslash {\mathcal {S}}_k }{\min }\underset{i \in {\mathcal {C}}, i \ne l}{\max }X_i \le t, \underset{{\mathcal {C}} \in {\mathcal {S}}\backslash {\mathcal {S}}_l }{\min }\underset{i \in {\mathcal {C}} }{\max }X_i> t)\\{} & {} \quad =:H_{1}(t) {\mathbb {P}}(\underset{{\mathcal {C}} \in {\mathcal {S}}_k }{\min }\underset{i \in {\mathcal {C}}, i \ne k,l}{\max }X_i \le t, \underset{{\mathcal {C}} \in {\mathcal {S}}_l\backslash {\mathcal {S}}_k }{\min }\underset{i \in {\mathcal {C}}, i \ne l}{\max }X_i> t, \underset{{\mathcal {C}} \in {\mathcal {S}}\backslash {\mathcal {S}}_l }{\min }\underset{i \in {\mathcal {C}}}{\max }X_i> t)\\{} & {} \qquad +H_{2}(t){\mathbb {P}}( \underset{{\mathcal {C}} \in {\mathcal {S}}_l \backslash {\mathcal {S}}_k }{\min }\underset{i \in {\mathcal {C}}, i \ne l}{\max }X_i \le t, \underset{{\mathcal {C}} \in {\mathcal {S}}\backslash {\mathcal {S}}_l }{\min }\underset{i \in {\mathcal {C}} }{\max }X_i > t), \end{aligned}$$
where
$$\begin{aligned} H_{1}(t)= & {} {\mathbb {P}}( X_k\#Y_1\le t){\mathbb {P}}( X_l\#Y_2\le t)\\{} & {} -{\mathbb {P}}( X_k\#Y_2\le t){\mathbb {P}}( X_l\#Y_1\le t),\\ H_{2}(t)= & {} {\mathbb {P}}(X_l\#Y_2\le t)-{\mathbb {P}}(X_l\#Y_1\le t). \end{aligned}$$
On the one hand,
$$\begin{aligned} F_{X_k\#Y_1}(t) F_{X_l\#Y_2}(t)\ge F_{X_k\#Y_2}(t) F_{X_l\#Y_1}(t), \end{aligned}$$
implies \(H_{1}(t)\ge 0\). On the other hand, \(Y_1 \ge _{hr} Y_2\) implies
$$\begin{aligned} \frac{\overline{G}_1(t) }{\overline{G}_1(u)} -\frac{\overline{G}_2(t)}{\overline{G}_2(u)}\ge 0, \text { for all } u \le t, \end{aligned}$$
hence, we have
$$\begin{aligned} H_{2}(t)= & {} {\mathbb {P}}(X_l\#Y_2\le t)-{\mathbb {P}}(X_l\#Y_1\le t) = F_l(t)\\{} & {} -\overline{G}_2(t)\int _0^t\frac{f_l(u)}{\overline{G}_2(u)}\textrm{d}u -\Big ( F_l(t)-\overline{G}_1(t)\int _0^t\frac{f_l(u)}{\overline{G}_1(u)}\textrm{d}u \Big ) \\= & {} \int _0^t \bigg ( \frac{\overline{G}_1(t)f_l(u)}{\overline{G}_1(u)} -\frac{\overline{G}_2(t)f_l(u)}{\overline{G}_2(u)} \bigg )\textrm{d}u \ge 0. \end{aligned}$$
Therefore, it follows that \(H_{1}(t)\) and \(H_{2}(t)\) are non-negative, which means that \({\bar{F}}_{\tau _1}(t)\le \bar{F}_{\tau _2}(t)\), for all \(t \in {\mathbb {R}}_+\), the desired result is proved. \(\square \)
1.3 Appendix A.3: Proof of Theorem 3
Proof
We denote the survival functions of \(\tau (X_k\#Y_1, X_l\#Y_2, \varvec{X}^{(k,l)})\) and \(\tau (X_k\#Y_2, X_l\#Y_1, \varvec{X}^{(k,l)})\) by \({\overline{F}}_{\tau _1}\) and \({\overline{F}}_{\tau _2}\), respectively. Note that
$$\begin{aligned} {\overline{F}}_{\tau _1}(t)= & {} {\overline{F}}_{X_k\#Y_1}(t) \overline{F}_{X_l\#Y_2}(t) [{\mathbb {P}} (B_{kl})-{\mathbb {P}} (B_{kl} B_{k\bar{l}})-{\mathbb {P}} (B_{kl} B_{{\bar{k}}l})\\{} & {} -{\mathbb {P}} (B_{kl} A_{{\bar{k}}{\bar{l}}}) -{\mathbb {P}} (B_{k {\bar{l}}} A_{{\bar{k}} l})\\{} & {} +{\mathbb {P}} (B_{kl} B_{k{\bar{l}}} B_{{\bar{k}}l})+ {\mathbb {P}} (B_{kl} B_{{\bar{k}} l} B_{{\bar{k}} {\bar{l}}}) + {\mathbb {P}} (B_{kl} B_{ k {\bar{l}}} A_{{\bar{k}}{\bar{l}}})\\{} & {} + {\mathbb {P}} (B_{k {\bar{l}}} B_{{\bar{k}} l} A_{{\bar{k}} {\bar{l}}}) - {\mathbb {P}} (B_{kl} B_{ k {\bar{l}}} B_{{\bar{k}} l} A_{{\bar{k}}{\bar{l}}})]\\{} & {} + {\overline{F}}_{X_k\#Y_1}(t) [{\mathbb {P}} (B_{k\bar{l}})-{\mathbb {P}} (B_{k {\bar{l}}} B_{{\bar{k}} {\bar{l}}})] + \overline{F}_{X_l\#Y_2}(t)[{\mathbb {P}} (B_{{\bar{k}} l})-{\mathbb {P}} (B_{{\bar{k}} l} B_{{\bar{k}} {\bar{l}}}) ] + {\mathbb {P}} (A_{{\bar{k}}{\bar{l}}}). \end{aligned}$$
Similarly,
$$\begin{aligned} {\overline{F}}_{\tau _2}(t)= & {} {\overline{F}}_{X_k\#Y_2}(t) \overline{F}_{X_l\#Y_1}(t) [{\mathbb {P}} (B_{kl})-{\mathbb {P}} (B_{kl} B_{k\bar{l}})-{\mathbb {P}} (B_{kl} B_{{\bar{k}}l})-{\mathbb {P}} (B_{kl} A_{\bar{k}{\bar{l}}}) -{\mathbb {P}} (B_{k {\bar{l}}} A_{{\bar{k}} l})\\{} & {} +{\mathbb {P}} (B_{kl} B_{k{\bar{l}}} B_{{\bar{k}}l})+ {\mathbb {P}} (B_{kl} B_{{\bar{k}} l} B_{{\bar{k}} {\bar{l}}}) + {\mathbb {P}} (B_{kl} B_{ k {\bar{l}}} A_{{\bar{k}}{\bar{l}}}) \\{} & {} + {\mathbb {P}} (B_{k {\bar{l}}} B_{{\bar{k}} l} A_{{\bar{k}} {\bar{l}}}) - {\mathbb {P}} (B_{kl} B_{ k {\bar{l}}} B_{{\bar{k}} l} A_{{\bar{k}}\bar{l}})]\\{} & {} + {\overline{F}}_{X_k\#Y_2}(t) [{\mathbb {P}} (B_{k\bar{l}})-{\mathbb {P}} (B_{k {\bar{l}}} B_{{\bar{k}} {\bar{l}}})] + {\overline{F}}_{X_l\#Y_1}(t)[{\mathbb {P}} (B_{{\bar{k}} l})-{\mathbb {P}} (B_{{\bar{k}} l} B_{{\bar{k}} {\bar{l}}}) ] + {\mathbb {P}} (A_{{\bar{k}}\bar{l}}) \end{aligned}$$
We need to show that \({\overline{F}}_{\tau _1}(t)\le \overline{F}_{\tau _2}(t)\), for all \(t\in {\mathbb {R}}_+\). Note that
$$\begin{aligned}{} & {} {\overline{F}}_{\tau _1}(t)-{\overline{F}}_{\tau _2}(t)\\{} & {} \quad = [{\overline{F}}_{X_k\#Y_1}(t) {\overline{F}}_{X_l\#Y_2}(t) -{\overline{F}}_{X_k\#Y_2}(t) {\overline{F}}_{X_l\#Y_1}(t)] [{\mathbb {P}} (B_{kl})-{\mathbb {P}} (B_{kl} B_{k{\bar{l}}})\\{} & {} \qquad -{\mathbb {P}} (B_{kl} B_{{\bar{k}}l})-{\mathbb {P}} (B_{kl} A_{{\bar{k}}{\bar{l}}}) -{\mathbb {P}} (B_{k {\bar{l}}} A_{{\bar{k}} l}) +{\mathbb {P}} (B_{kl} B_{k{\bar{l}}} B_{{\bar{k}}l})+ {\mathbb {P}} (B_{kl} B_{{\bar{k}} l} B_{{\bar{k}} {\bar{l}}})\\{} & {} \qquad + {\mathbb {P}} (B_{kl} B_{ k {\bar{l}}} A_{{\bar{k}}{\bar{l}}}) + {\mathbb {P}} (B_{k {\bar{l}}} B_{{\bar{k}} l} A_{{\bar{k}} {\bar{l}}}) - {\mathbb {P}} (B_{kl} B_{ k {\bar{l}}} B_{{\bar{k}} l} A_{{\bar{k}}{\bar{l}}})]\\{} & {} \qquad + [{\overline{F}}_{X_k\#Y_1}(t)-{\overline{F}}_{X_k\#Y_2}(t)] [{\mathbb {P}} (B_{k{\bar{l}}}) -{\mathbb {P}} (B_{k {\bar{l}}} B_{{\bar{k}} {\bar{l}}})]\\{} & {} \qquad + [{\overline{F}}_{X_l\#Y_2}(t)-{\overline{F}}_{X_l\#Y_1}(t)][{\mathbb {P}} (B_{{\bar{k}} l}) -{\mathbb {P}} (B_{{\bar{k}} l} A_{{\bar{k}} {\bar{l}}}) ] \\{} & {} \quad =:J_{1}(t) +J_{2}(t), \end{aligned}$$
where
$$\begin{aligned} J_{1}(t)= & {} [{\overline{F}}_{X_k\#Y_1}(t) {\overline{F}}_{X_l\#Y_2}(t)-{\overline{F}}_{X_k\#Y_2}(t) {\overline{F}}_{X_l\#Y_1}(t)] [{\mathbb {P}} (B_{kl})-{\mathbb {P}} (B_{kl} B_{k{\bar{l}}})\\{} & {} -{\mathbb {P}} (B_{kl} B_{{\bar{k}}l})-{\mathbb {P}} (B_{kl} A_{{\bar{k}}{\bar{l}}}) -{\mathbb {P}} (B_{k {\bar{l}}} A_{{\bar{k}} l}) +{\mathbb {P}} (B_{kl} B_{k{\bar{l}}} B_{{\bar{k}}l}) + {\mathbb {P}} (B_{kl} B_{{\bar{k}} l} B_{{\bar{k}} {\bar{l}}})\\{} & {} + {\mathbb {P}} (B_{kl} B_{ k {\bar{l}}} A_{{\bar{k}}{\bar{l}}}) + {\mathbb {P}} (B_{k {\bar{l}}} B_{{\bar{k}} l} A_{{\bar{k}} {\bar{l}}}) - {\mathbb {P}} (B_{kl} B_{ k {\bar{l}}} B_{{\bar{k}} l} A_{{\bar{k}}{\bar{l}}})],\\ J_{2}(t)= & {} [{\overline{F}}_{X_k\#Y_1}(t)-{\overline{F}}_{X_k\#Y_2}(t)] [{\mathbb {P}} (B_{k{\bar{l}}}) -{\mathbb {P}} (B_{k {\bar{l}}} B_{{\bar{k}} {\bar{l}}})] \\{} & {} \quad + [{\overline{F}}_{X_l\#Y_2}(t) -{\overline{F}}_{X_l\#Y_1}(t)][{\mathbb {P}} (B_{{\bar{k}} l})-{\mathbb {P}} (B_{{\bar{k}} l} A_{{\bar{k}} {\bar{l}}}) ]. \end{aligned}$$
On the one hand, according to Lemma 4, \(X_k \le _{lr} X_l\) and \(Y_1 \ge _{hr} Y_2\) imply
$$\begin{aligned} {\overline{F}}_{X_k\#Y_1}(t) {\overline{F}}_{X_l\#Y_2}(t)\ge {\overline{F}}_{X_k\#Y_2}(t) {\overline{F}}_{X_l\#Y_1}(t), \end{aligned}$$
and
$$\begin{aligned}{} & {} \partial _{k,l}h({\overline{F}}_1(t), \dots , {\overline{F}}_n(t))\\{} & {} \quad ={\mathbb {P}} (B_{kl})-{\mathbb {P}} (B_{kl} B_{k{\bar{l}}}) -{\mathbb {P}} (B_{kl} B_{{\bar{k}}l})-{\mathbb {P}} (B_{kl} A_{{\bar{k}}{\bar{l}}}) -{\mathbb {P}} (B_{k {\bar{l}}} A_{{\bar{k}} l}) +{\mathbb {P}} (B_{kl} B_{k{\bar{l}}} B_{{\bar{k}}l})\\{} & {} \qquad + {\mathbb {P}} (B_{kl} B_{{\bar{k}} l} B_{{\bar{k}} {\bar{l}}})+ {\mathbb {P}} (B_{kl} B_{ k {\bar{l}}} A_{{\bar{k}}{\bar{l}}}) + {\mathbb {P}} (B_{k {\bar{l}}} B_{{\bar{k}} l} A_{{\bar{k}} {\bar{l}}}) - {\mathbb {P}} (B_{kl} B_{ k {\bar{l}}} B_{{\bar{k}} l} A_{{\bar{k}}{\bar{l}}}) \ge 0, \end{aligned}$$
which implies that \(J_{1}(t)\ge 0\). On the other hand,
$$\begin{aligned} J_{2}(t)= & {} [{\overline{F}}_{X_k\#Y_1}(t)-{\overline{F}}_{X_k\#Y_2}(t)] [{\mathbb {P}} (B_{k{\bar{l}}})-{\mathbb {P}} (B_{k {\bar{l}}} B_{{\bar{k}} {\bar{l}}})] \\{} & {} + [{\overline{F}}_{X_l\#Y_2}(t)-{\overline{F}}_{X_l\#Y_1}(t)][{\mathbb {P}} (B_{{\bar{k}} l}) -{\mathbb {P}} (B_{{\bar{k}} l} A_{{\bar{k}} {\bar{l}}}) ] \\= & {} [{\overline{F}}_{X_k\#Y_1}(t)-{\overline{F}}_{X_k\#Y_2}(t)+{\overline{F}}_{X_l\#Y_2}(t)-{\overline{F}}_{X_l\#Y_1}(t)] [{\mathbb {P}} (B_{k{\bar{l}}})-{\mathbb {P}} (B_{k {\bar{l}}} B_{{\bar{k}} {\bar{l}}})]\\{} & {} +[{\overline{F}}_{X_l\#Y_2}(t)-{\overline{F}}_{X_l\#Y_1}(t)] [-{\mathbb {P}} (B_{k{\bar{l}}}) +{\mathbb {P}} (B_{k {\bar{l}}} B_{{\bar{k}} {\bar{l}}})+{\mathbb {P}} (B_{{\bar{k}} l})-{\mathbb {P}} (B_{{\bar{k}} l} A_{{\bar{k}} {\bar{l}}}) ]\\=: & {} J_{3}(t) +J_{4}(t), \end{aligned}$$
where
$$\begin{aligned} J_{3}(t)= & {} {\overline{F}}_{X_k\#Y_1}(t)-{\overline{F}}_{X_k\#Y_2}(t)+{\overline{F}}_{X_l\#Y_2}(t) -{\overline{F}}_{X_l\#Y_1}(t),\\ J_{4}(t)= & {} [{\overline{F}}_{X_l\#Y_2}(t)-{\overline{F}}_{X_l\#Y_1}(t)] [-{\mathbb {P}} (B_{k{\bar{l}}})+{\mathbb {P}} (B_{k {\bar{l}}} B_{{\bar{k}} {\bar{l}}})+{\mathbb {P}} (B_{{\bar{k}} l}) -{\mathbb {P}} (B_{{\bar{k}} l} A_{{\bar{k}} {\bar{l}}}) ]. \end{aligned}$$
Condition (iii) implies that \(J_{3}(t) \ge 0\), for all \(t \in {\mathbb {R}}_+\). Meanwhile, \(Y_1 \ge _{hr} Y_2\) implies
$$\begin{aligned}{} & {} {\mathbb {P}}(X_l\#Y_2>t)-{\mathbb {P}}(X_l\#Y_1>t) = \overline{F}_l(t)\\{} & {} \qquad +\overline{G}_2(t)\int _0^t\frac{f_l(u)}{\overline{G}_2(u)}\textrm{d}u-\Big (\overline{F}_l(t) +\overline{G}_1(t)\int _0^t\frac{f_l(u)}{\overline{G}_1(u)}\textrm{d}u \Big ) \\{} & {} \quad =\int _0^t \bigg (\frac{\overline{G}_2(t)f_l(u)}{\overline{G}_2(u)}-\frac{{\overline{G}}_1(t)f_l(u)}{{\overline{G}}_1(u)}\bigg ) \textrm{d}u \le 0. \end{aligned}$$
Note that \(-{\mathbb {P}} (B_{k{\bar{l}}})+{\mathbb {P}} (B_{k {\bar{l}}} B_{{\bar{k}} {\bar{l}}}) +{\mathbb {P}} (B_{{\bar{k}} l})-{\mathbb {P}} (B_{{\bar{k}} l} A_{{\bar{k}} {\bar{l}}})\le 0\), and thus \(J_{4}(t) \ge 0\). Therefore, it follows that \(J_{1}(t)\), \(J_{3}(t)\) and \(J_{4}(t)\) are non-negative, which means that \({\overline{F}}_{\tau _1}(t)\ \ge {\overline{F}}_{\tau _2}(t)\), for all \(t \in {\mathbb {R}}_+\). Hence, the desired result is proved. \(\square \)
1.4 Appendix A.4: Proof of Theorem 4
Proof
The survival functions of \(\tau (X_k\#Y_1, X_l\#Y_2, \varvec{X}^{(k,l)})\) and \(\tau (X_k\#Y_2, X_l\#Y_1, \varvec{X}^{(k,l)})\) denote as \( F_{\tau _1}\) and \( F_{\tau _2}\), respectively. Note that
$$\begin{aligned} F_{\tau _1}(t)= & {} F_{X_k\#Y_1}(t) F_{X_l\#Y_2}(t) [{\mathbb {P}} ({\tilde{B}}_{kl}) -{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{k{\bar{l}}})\\{} & {} \quad -{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{{\bar{k}}l}) -{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{A}}_{{\bar{k}}{\bar{l}}}) -{\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{A}}_{{\bar{k}} l})\\{} & {} survivalfun +{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{k{\bar{l}}} {\tilde{B}}_{{\bar{k}}l}) + {\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{{\bar{k}} l} {\tilde{B}}_{{\bar{k}} {\bar{l}}}) + {\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{ k {\bar{l}}} {\tilde{A}}_{{\bar{k}}{\bar{l}}})\\{} & {} \quad + {\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{B}}_{{\bar{k}} l} {\tilde{A}}_{{\bar{k}} {\bar{l}}}) - {\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{ k {\bar{l}}} {\tilde{B}}_{{\bar{k}} l} {\tilde{A}}_{{\bar{k}}{\bar{l}}})]\\{} & {} survivalfun + F_{X_k\#Y_1}(t) [{\mathbb {P}} ({\tilde{B}}_{k{\bar{l}}})-{\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{B}}_{{\bar{k}} {\bar{l}}})] \\{} & {} \quad + F_{X_l\#Y_2}(t)[{\mathbb {P}} ({\tilde{B}}_{{\bar{k}} l}) -{\mathbb {P}} ({\tilde{B}}_{{\bar{k}} l} {\tilde{B}}_{{\bar{k}} {\bar{l}}}) ] + {\mathbb {P}} ({\tilde{A}}_{{\bar{k}}{\bar{l}}}) \end{aligned}$$
Similarly,
$$\begin{aligned} F_{\tau _2}(t)= & {} F_{X_k\#Y_2}(t) F_{X_l\#Y_1}(t) [{\mathbb {P}} ({\tilde{B}}_{kl}) -{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{k{\bar{l}}})-{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{{\bar{k}}l})\\{} & {} -{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{A}}_{{\bar{k}}{\bar{l}}}) -{\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{A}}_{{\bar{k}} l})\\{} & {} +{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{k{\bar{l}}} {\tilde{B}}_{{\bar{k}}l}) + {\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{{\bar{k}} l} {\tilde{B}}_{{\bar{k}} {\bar{l}}}) + {\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{ k {\bar{l}}} {\tilde{A}}_{{\bar{k}}{\bar{l}}})\\{} & {} + {\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{B}}_{{\bar{k}} l} {\tilde{A}}_{{\bar{k}} {\bar{l}}}) - {\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{ k {\bar{l}}} {\tilde{B}}_{{\bar{k}} l} {\tilde{A}}_{{\bar{k}}{\bar{l}}})]\\{} & {} + F_{X_k\#Y_2}(t) [{\mathbb {P}} ({\tilde{B}}_{k{\bar{l}}})-{\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{B}}_{{\bar{k}} {\bar{l}}})]\\{} & {} + F_{X_l\#Y_1}(t)[{\mathbb {P}} ({\tilde{B}}_{{\bar{k}} l})-{\mathbb {P}} ({\tilde{B}}_{{\bar{k}} l} {\tilde{B}}_{{\bar{k}} {\bar{l}}}) ] + {\mathbb {P}} ({\tilde{A}}_{{\bar{k}}{\bar{l}}}) \end{aligned}$$
In order to obtain the desired result, it is sufficient to show that \( F_{\tau _1}(t)\le F_{\tau _2}(t)\), hence, we have
$$\begin{aligned} F_{\tau _1}(t)- F_{\tau _2}(t)= & {} [ F_{X_k\#Y_1}(t) F_{X_l\#Y_2}(t)- F_{X_k\#Y_2}(t) F_{X_l\#Y_1}(t)] [{\mathbb {P}} ({\tilde{B}}_{kl}) -{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{k{\bar{l}}})\\{} & {} -{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{{\bar{k}}l})-{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{A}}_{{\bar{k}}{\bar{l}}}) -{\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{A}}_{{\bar{k}} l}) \\{} & {} +{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{k{\bar{l}}} {\tilde{B}}_{{\bar{k}}l})+ {\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{{\bar{k}} l} {\tilde{B}}_{{\bar{k}} {\bar{l}}})\\{} & {} + {\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{ k {\bar{l}}} {\tilde{A}}_{{\bar{k}}{\bar{l}}}) + {\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{B}}_{{\bar{k}} l} {\tilde{A}}_{{\bar{k}} {\bar{l}}}) - {\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{ k {\bar{l}}} {\tilde{B}}_{{\bar{k}} l} {\tilde{A}}_{{\bar{k}}{\bar{l}}})]\\{} & {} + [ F_{X_k\#Y_1}(t)- F_{X_k\#Y_2}(t)] [{\mathbb {P}} ({\tilde{B}}_{k{\bar{l}}}) -{\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{B}}_{{\bar{k}} {\bar{l}}})] + [ F_{X_l\#Y_2}(t)\\{} & {} - F_{X_l\#Y_1}(t)][{\mathbb {P}} ({\tilde{B}}_{{\bar{k}} l})-{\mathbb {P}} ({\tilde{B}}_{{\bar{k}} l} {\tilde{A}}_{{\bar{k}} {\bar{l}}}) ] \\=: & {} L_{1}(t) +L_{2}(t), \end{aligned}$$
where
$$\begin{aligned} L_{1}(t)= & {} [ F_{X_k\#Y_1}(t) F_{X_l\#Y_2}(t)- F_{X_k\#Y_2}(t) F_{X_l\#Y_1}(t)] [{\mathbb {P}} ({\tilde{B}}_{kl}) -{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{k{\bar{l}}})\\{} & {} -{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{{\bar{k}}l})-{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{A}}_{{\bar{k}}{\bar{l}}}) -{\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{A}}_{{\bar{k}} l}) \\{} & {} \quad +{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{k{\bar{l}}} {\tilde{B}}_{{\bar{k}}l})+ {\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{{\bar{k}} l} {\tilde{B}}_{{\bar{k}} {\bar{l}}})\\{} & {} + {\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{ k {\bar{l}}} {\tilde{A}}_{{\bar{k}}{\bar{l}}}) + {\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{B}}_{{\bar{k}} l} {\tilde{A}}_{{\bar{k}} {\bar{l}}}) - {\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{ k {\bar{l}}} {\tilde{B}}_{{\bar{k}} l} {\tilde{A}}_{{\bar{k}}{\bar{l}}})],\\ L_{2}(t)= & {} [ F_{X_k\#Y_1}(t)- F_{X_k\#Y_2}(t)] [{\mathbb {P}} ({\tilde{B}}_{k{\bar{l}}}) -{\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{B}}_{{\bar{k}} {\bar{l}}})] \\{} & {} \quad + [ F_{X_l\#Y_2}(t)- F_{X_l\#Y_1}(t)] [{\mathbb {P}} ({\tilde{B}}_{{\bar{k}} l})-{\mathbb {P}} ({\tilde{B}}_{{\bar{k}} l} {\tilde{A}}_{{\bar{k}} {\bar{l}}}) ]. \end{aligned}$$
On the one hand, according to conditions (i) and (ii), we have
$$\begin{aligned}{} & {} -\partial _{k,l}{\tilde{h}}(F_1(t), \dots , F_n(t))\\{} & {} \quad ={\mathbb {P}} ({\tilde{B}}_{kl})-{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{k{\bar{l}}}) -{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{{\bar{k}}l})-{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{A}}_{{\bar{k}}{\bar{l}}}) -{\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{A}}_{{\bar{k}} l}) +{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{k{\bar{l}}} {\tilde{B}}_{{\bar{k}}l})\\{} & {} \qquad + {\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{{\bar{k}} l} {\tilde{B}}_{{\bar{k}} {\bar{l}}})+ {\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{ k {\bar{l}}} {\tilde{A}}_{{\bar{k}}{\bar{l}}}) + {\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{B}}_{{\bar{k}} l} {\tilde{A}}_{{\bar{k}} {\bar{l}}}) - {\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{ k {\bar{l}}} {\tilde{B}}_{{\bar{k}} l} {\tilde{A}}_{{\bar{k}}{\bar{l}}}) \le 0, \end{aligned}$$
Combining this inequality with condition (iii), we have \(L_{1}(t)\le 0\). On the other hand,
$$\begin{aligned} L_{2}(t)= & {} [ F_{X_k\#Y_1}(t)- F_{X_k\#Y_2}(t)+ F_{X_l\#Y_2}(t)- F_{X_l\#Y_1}(t)] [{\mathbb {P}} ({\tilde{B}}_{k{\bar{l}}})-{\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{B}}_{{\bar{k}} {\bar{l}}})]\\{} & {} +[ F_{X_l\#Y_2}(t)- F_{X_l\#Y_1}(t)] [-{\mathbb {P}} ({\tilde{B}}_{k{\bar{l}}})+{\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{B}}_{{\bar{k}} {\bar{l}}})+{\mathbb {P}} ({\tilde{B}}_{{\bar{k}} l})-{\mathbb {P}} ({\tilde{B}}_{{\bar{k}} l} {\tilde{A}}_{{\bar{k}} \bar{l}}) ]\\=: & {} L_{3}(t) +L_{4}(t), \end{aligned}$$
where
$$\begin{aligned} L_{3}(t)= & {} [ F_{X_k\#Y_1}(t)- F_{X_k\#Y_2}(t)+ F_{X_l\#Y_2}(t)- F_{X_l\#Y_1}(t)] [{\mathbb {P}} ({\tilde{B}}_{k{\bar{l}}})-{\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{B}}_{{\bar{k}} {\bar{l}}})] \\ L_{4}(t)= & {} [{\overline{F}}_{X_l\#Y_1}(t)- {\overline{F}}_{X_l\#Y_2}(t) ] [-{\mathbb {P}} ({\tilde{B}}_{k{\bar{l}}})+{\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{B}}_{{\bar{k}} {\bar{l}}})+{\mathbb {P}} ({\tilde{B}}_{\bar{k} l})-{\mathbb {P}} ({\tilde{B}}_{{\bar{k}} l} {\tilde{A}}_{{\bar{k}} \bar{l}}) ]. \end{aligned}$$
Observe that
$$\begin{aligned} {\overline{F}}_{X_k\#Y_1}(t)-{\overline{F}}_{X_k\#Y_2}(t)\ge \overline{F}_{X_l\#Y_1}(t) -{\overline{F}}_{X_l\#Y_2}(t) \end{aligned}$$
means
$$\begin{aligned} F_{X_k\#Y_1}(t)- F_{X_k\#Y_2}(t)+ F_{X_l\#Y_2}(t)- F_{X_l\#Y_1}(t) \le 0, \end{aligned}$$
hence, \(L_3(t)\le 0\). Meanwhile, \(Y_1 \ge _{hr} Y_2\) implies \({\overline{F}}_{X_l\#Y_1}(t)- {\overline{F}}_{X_l\#Y_2}(t) \ge 0.\) Combining the above inequality with \(-{\mathbb {P}} ({\tilde{B}}_{k{\bar{l}}})+{\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{B}}_{{\bar{k}} {\bar{l}}})+{\mathbb {P}} ({\tilde{B}}_{{\bar{k}} l})-{\mathbb {P}} ({\tilde{B}}_{{\bar{k}} l} {\tilde{A}}_{{\bar{k}} \bar{l}})\le 0\), hence we have \(L_{4}(t) \le 0\).
Therefore, it follows that \(L_{1}(t)\), \(L_{3}(t)\) and \(L_{4}(t)\) are non-positive, which means that \(F_{\tau _1}(t)\le F_{\tau _2}(t)\), for all \(t \in {\mathbb {R}}_+\), the desired result is proved. \(\square \)
1.5 Appendix A.5: Proof of Theorem 5
Proof
Let \(F_{\tau _1}\) and \(F_{\tau _2}\) be the distributions of \(\tau (X_k\#^{r_k}X_k, X_l\#^{r_l}X_l, \varvec{X}^{(k,l)})\) and \(\tau (X_k\#^{r_k^*}X_k, X_l\#^{r_l^*}X_l,\) \( \varvec{X}^{(k,l)})\), respectively. It suffices to prove that \(F_{\tau _1}(t) \ge F_{\tau _2}(t)\), for all \(t \in {\mathbb {R}}_+\). For the allocation policy \((r_k,r_l)\), the system’s lifetime can be written by
$$\begin{aligned}{} & {} \tau (X_k\#^{r_k}X_k, X_l\#^{r_l}X_l, \varvec{X}^{(k,l)})\\{} & {} \quad =\max \Big \{\min \{X_k\#^{r_k}X_k, X_l\#^{r_l}X_l, \underset{{\mathcal {P}}\in {\mathcal {A}}_k}{\max }\underset{i \in {\mathcal {P}}, i \ne k,l}{\min }X_i\}, \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}}{\min }X_i \Big \}. \end{aligned}$$
Then, it can be obtained that
$$\begin{aligned} F_{\tau _1}(t) =[1- {\mathbb {P}}(X_k\#^{r_k}X_k>t) {\mathbb {P}}(X_l\#^{r_l}X_l>t){\mathbb {P}}(\underset{{\mathcal {P}}\in {\mathcal {A}}_k}{\max }\underset{i \in {\mathcal {P}}, i \ne k,l}{\min }X_i > t)] {\mathbb {P}}(\underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}}{\min }X_i \le t). \end{aligned}$$
Similar, we have
$$\begin{aligned} F_{\tau _2}(t) =[1- {\mathbb {P}}(X_k\#^{r_k^*}X_k>t) {\mathbb {P}}(X_l\#^{r_l^*}X_l>t) {\mathbb {P}}(\underset{{\mathcal {P}}\in {\mathcal {A}}_k}{\max }\underset{i \in {\mathcal {P}}, i \ne k,l}{\min }X_i > t)] {\mathbb {P}}(\underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}}{\min }X_i \le t). \end{aligned}$$
According to Corollary 4.3 of Zhang and Zhao (2019), the condition \((r_k, r_l) \overset{\textrm{m}}{\succeq }\ (r_k^*, r_l^*)\) implies that
$$\begin{aligned} F_{\tau _1}(t)-F_{\tau _2}(t) \overset{\textrm{sign}}{=} {\mathbb {P}}(X_k\#^{r_k^*}X_k>t) {\mathbb {P}}(X_l\#^{r_l^*}X_l>t) -{\mathbb {P}}(X_k\#^{r_k}X_k>t) {\mathbb {P}}(X_l\#^{r_l}X_l>t) \ge 0, \end{aligned}$$
which completes the proof. \(\square \)
1.6 Appendix A.6: Proof of Theorem 7
Proof
The distribution function of \(\tau (X_k\#^{r_k}X_k, X_l\#^{r_l}X_l, \varvec{X}^{(k,l)})\) can be written as
$$\begin{aligned} F_{\tau _1}(t)= & {} {\mathbb {P}}(\underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}}}{\min }X_i \le t)-{\mathbb {P}}(X_k\#^{r_k}X_k> t){\mathbb {P}}( X_l\#^{r_l}X_l> t)\\{} & {} \times {\mathbb {P}}(\underset{{\mathcal {P}} \in {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne k,l}{\min }X_i> t, \underset{{\mathcal {P}} \in {\mathcal {A}}_l\backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i \le t, \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}}}{\min }X_i \le t)\\{} & {} -{\mathbb {P}}( X_l\#^{r_l}X_l> t) {\mathbb {P}}( \underset{{\mathcal {P}} \in {\mathcal {A}}_l \backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i > t, \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}} }{\min }X_i \le t). \end{aligned}$$
Similarly, the distribution function of \(\tau (X_k\#^{r_k^*}X_k, X_l\#^{r_l^*}X_l, \varvec{X}^{(k,l)})\) is given by
$$\begin{aligned} F_{\tau _2}(t)= & {} {\mathbb {P}}\left( \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}}}{\min }X_i \le t\right) -{\mathbb {P}}(X_k\#^{r_k^*}X_k> t){\mathbb {P}}( X_l\#^{r_l^*}X_l> t)\\{} & {} \times {\mathbb {P}}\left( \underset{{\mathcal {P}} \in {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne k,l}{\min }X_i> t, \underset{{\mathcal {P}} \in {\mathcal {A}}_l\backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i \le t, \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}}}{\min }X_i \le t\right) \\{} & {} -{\mathbb {P}}( X_l\#^{r_l^*}X_l> t) {\mathbb {P}} \left( \underset{{\mathcal {P}} \in {\mathcal {A}}_l \backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i > t, \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}} }{\min }X_i \le t\right) . \end{aligned}$$
Hence, we have
$$\begin{aligned}{} & {} F_{\tau _1}(t)-F_{\tau _2}(t)\\ {}{} & {} \quad =\Big [{\mathbb {P}}(X_k\#^{r_k^*}X_k> t){\mathbb {P}}( X_l\#^{r_l^*}X_l> t) -{\mathbb {P}}(X_k\#^{r_k}X_k> t){\mathbb {P}}( X_l\#^{r_l}X_l> t) \Big ]\\{} & {} \qquad \times {\mathbb {P}}(\underset{{\mathcal {P}} \in {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne k,l}{\min }X_i> t, \underset{{\mathcal {P}} \in {\mathcal {A}}_l\backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i \le t, \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}}}{\min }X_i \le t)\\{} & {} \qquad +\Big [{\mathbb {P}}( X_l\#^{r_l^*}X_l> t)-{\mathbb {P}}( X_l\#^{r_l}X_l> t) \Big ] {\mathbb {P}}( \underset{{\mathcal {P}} \in {\mathcal {A}}_l \backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i> t, \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}} }{\min }X_i \le t)\\{} & {} \quad =: M_{1}(t) {\mathbb {P}}(\underset{{\mathcal {P}} \in {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne k,l}{\min }X_i> t, \underset{{\mathcal {P}} \in {\mathcal {A}}_l\backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i \le t, \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}}}{\min }X_i \le t)\\{} & {} \qquad +M_{2}(t){\mathbb {P}}( \underset{{\mathcal {P}} \in {\mathcal {A}}_l \backslash {\mathcal {A}}_k }{\max }\underset{i \in {\mathcal {P}}, i \ne l}{\min }X_i > t, \underset{{\mathcal {P}} \in {\mathcal {A}}\backslash {\mathcal {A}}_l }{\max }\underset{i \in {\mathcal {P}} }{\min }X_i \le t). \end{aligned}$$
where
$$\begin{aligned} M_{1}(t)={\mathbb {P}}(X_k\#^{r_k^*}X_k> t){\mathbb {P}}( X_l\#^{r_l^*}X_l> t)-{\mathbb {P}}(X_k\#^{r_k}X_k> t){\mathbb {P}}( X_l\#^{r_l}X_l > t), \end{aligned}$$
and \(M_{2}(t)={\mathbb {P}}( X_l\#^{r_l^*}X_l> t)-{\mathbb {P}}( X_l\#^{r_l}X_l > t)\). On one hand, according to Corollary 4.3 of Zhang and Zhao (2019), \((r_k, r_l) \overset{\textrm{m}}{\succeq }\ (r_k^*, r_l^*)\) implies that \(M_{1}(t)\ge 0\) for all \(t\in {\mathbb {R}}_+\). On the other hand, \(r_l^* \ge r_l\) suggests that \(M_{2}(t)\ge 0\) for all \(t\in {\mathbb {R}}_+\). Hence, the proof is finished. \(\square \)
1.7 Appendix A.7: Proof of Theorem 9
Proof
Let \({\overline{F}}_{\tau _1}\) and \({\overline{F}}_{\tau _2}\) be the survival functions of \(\tau (X_k\#^{r_k}X_k, X_l\#^{r_l}X_l, \varvec{X}^{(k,l)})\) and \(\tau (X_k\#^{r_k^*}X_k, X_l\#^{r_l^*}X_l, \varvec{X}^{(k,l)}),\) respectively. Note that
$$\begin{aligned} {\overline{F}}_{\tau _1}(t)-{\overline{F}}_{\tau _2}(t)= & {} [{\mathbb {P}}(X_k\#^{r_k}X_k>t){\mathbb {P}}(X_l\#^{r_l}X_l>t)\\{} & {} -{\mathbb {P}}(X_k\#^{r_k^*}X_k>t) {\mathbb {P}}(X_l\#^{r_l^*}X_l>t)] [{\mathbb {P}} (B_{kl})\\{} & {} -{\mathbb {P}} (B_{kl} B_{k{\bar{l}}})-{\mathbb {P}} (B_{kl} B_{\bar{k}l})-{\mathbb {P}} (B_{kl} A_{{\bar{k}}{\bar{l}}}) -{\mathbb {P}} (B_{k \bar{l}} A_{{\bar{k}} l})\\{} & {} +{\mathbb {P}} (B_{kl} B_{k{\bar{l}}} B_{{\bar{k}}l})+ {\mathbb {P}} (B_{kl} B_{{\bar{k}} l} B_{{\bar{k}} {\bar{l}}})\\{} & {} + {\mathbb {P}} (B_{kl} B_{ k {\bar{l}}} A_{{\bar{k}}{\bar{l}}}) + {\mathbb {P}} (B_{k {\bar{l}}} B_{{\bar{k}} l} A_{{\bar{k}} {\bar{l}}}) - {\mathbb {P}} (B_{kl} B_{ k {\bar{l}}} B_{{\bar{k}} l} A_{{\bar{k}}{\bar{l}}})]\\{} & {} + [{\mathbb {P}}(X_k\#^{r_k}X_k>t)-{\mathbb {P}}(X_k\#^{r_k^*}X_k>t)] [{\mathbb {P}} (B_{k{\bar{l}}})-{\mathbb {P}} (B_{k {\bar{l}}} B_{{\bar{k}} {\bar{l}}})] \\{} & {} + [ {\mathbb {P}}(X_l\#^{r_l}X_l>t)-{\mathbb {P}}(X_l\#^{r_l^*}X_l >t)][{\mathbb {P}} (B_{{\bar{k}} l})-{\mathbb {P}} (B_{{\bar{k}} l} A_{\bar{k} {\bar{l}}}) ] \\=: & {} M_{1}(t) +M_{2}(t), \end{aligned}$$
where
$$\begin{aligned} M_{1}(t)= & {} [{\mathbb {P}}(X_k\#^{r_k}X_k>t){\mathbb {P}}(X_l\#^{r_l}X_l>t)-{\mathbb {P}}(X_k\#^{r_k^*}X_k>t) {\mathbb {P}}(X_l\#^{r_l^*}X_l>t)] [{\mathbb {P}} (B_{kl})\\{} & {} -{\mathbb {P}} (B_{kl} B_{k{\bar{l}}})-{\mathbb {P}} (B_{kl} B_{\bar{k}l})-{\mathbb {P}} (B_{kl} A_{{\bar{k}}{\bar{l}}}) -{\mathbb {P}} (B_{k \bar{l}} A_{{\bar{k}} l}) +{\mathbb {P}} (B_{kl} B_{k{\bar{l}}} B_{{\bar{k}}l})\\{} & {} + {\mathbb {P}} (B_{kl} B_{{\bar{k}} l} B_{{\bar{k}} {\bar{l}}}) + {\mathbb {P}} (B_{kl} B_{ k {\bar{l}}} A_{{\bar{k}}{\bar{l}}}) + {\mathbb {P}} (B_{k {\bar{l}}} B_{{\bar{k}} l} A_{{\bar{k}} {\bar{l}}}) - {\mathbb {P}} (B_{kl} B_{ k {\bar{l}}} B_{{\bar{k}} l} A_{{\bar{k}}{\bar{l}}})],\\ M_{2}(t)= & {} [{\mathbb {P}}(X_k\#^{r_k}X_k>t)-{\mathbb {P}}(X_k\#^{r_k^*}X_k>t)] [{\mathbb {P}} (B_{k{\bar{l}}})-{\mathbb {P}} (B_{k {\bar{l}}} B_{{\bar{k}} {\bar{l}}})] \\{} & {} + [ {\mathbb {P}}(X_l\#^{r_l}X_l>t)-{\mathbb {P}}(X_l\#^{r_l^*}X_l >t)][{\mathbb {P}} (B_{{\bar{k}} l})-{\mathbb {P}} (B_{{\bar{k}} l} A_{\bar{k} {\bar{l}}}) ]. \end{aligned}$$
On one hand, according to Corollary 4.3 of Zhang and Zhao (2019), \((r_k^*, r_l^*)\overset{\textrm{m}}{\preceq }\ (r_k, r_l) \) implies
$$\begin{aligned} {\mathbb {P}}(X_k\#^{r_k}X_k>t){\mathbb {P}}(X_l\#^{r_l}X_l>t)-{\mathbb {P}}(X_k\#^{r_k^*}X_k>t) {\mathbb {P}}(X_l\#^{r_l^*}X_l>t) \le 0. \end{aligned}$$
Then by applying condition (ii), we have \(M_{1}(t)\le 0\). On the other hand,
$$\begin{aligned} M_{2}(t)= & {} [{\mathbb {P}}(X_k\#^{r_k}X_k>t)-{\mathbb {P}}(X_k\#^{r_k^*}X_k>t)] [{\mathbb {P}} (B_{k{\bar{l}}})-{\mathbb {P}} (B_{k {\bar{l}}} B_{{\bar{k}} {\bar{l}}})\\{} & {} -{\mathbb {P}} (B_{{\bar{k}} l})+{\mathbb {P}} (B_{{\bar{k}} l} A_{{\bar{k}} {\bar{l}}}) ] \\{} & {} +[{\mathbb {P}}(X_k\#^{r_k}X_k>t)-{\mathbb {P}}(X_k\#^{r_k^*}X_k>t)\\{} & {} +[{\mathbb {P}}(X_l\#^{r_l}X_l>t)-{\mathbb {P}}(X_l\#^{r_l^*}X_l >t)] [{\mathbb {P}} (B_{k{\bar{l}}})-{\mathbb {P}} (B_{k {\bar{l}}} B_{{\bar{k}} {\bar{l}}})]\\=: & {} M_{3}(t) +M_{4}(t), \end{aligned}$$
where
$$\begin{aligned} M_{3}(t)= & {} [{\mathbb {P}}(X_k\#^{r_k}X_k>t)-{\mathbb {P}}(X_k\#^{r_k^*}X_k>t)] [{\mathbb {P}} (B_{k{\bar{l}}})-{\mathbb {P}} (B_{k {\bar{l}}} B_{{\bar{k}} {\bar{l}}})\\{} & {} -{\mathbb {P}} (B_{{\bar{k}} l})+{\mathbb {P}} (B_{{\bar{k}} l} A_{{\bar{k}} {\bar{l}}}) ],\\ M_{4}(t)= & {} [{\mathbb {P}}(X_k\#^{r_k}X_k>t)-{\mathbb {P}}(X_k\#^{r_k^*}X_k>t)+ {\mathbb {P}}(X_l\#^{r_l}X_l>t)\\{} & {} -{\mathbb {P}}(X_l\#^{r_l^*}X_l >t)] [{\mathbb {P}} (B_{k{\bar{l}}})-{\mathbb {P}} (B_{k {\bar{l}}} B_{{\bar{k}} {\bar{l}}})]. \end{aligned}$$
Due to the majorization order, it can be obtained that \(r_k\le r_k^*\), which further implies
$$\begin{aligned} {\mathbb {P}}(X_k\#^{r_k}X_k>t)-{\mathbb {P}}(X_k\#^{r_k^*}X_k>t) \le 0. \end{aligned}$$
Combining this inequality and condition (ii), we have \(M_{3}(t)\le 0\). Besides, condition (iii) implies
$$\begin{aligned} {\mathbb {P}}(X_k\#^{r_k}X_k>t)-{\mathbb {P}}(X_k\#^{r_k^*}X_k>t)+ {\mathbb {P}}(X_l\#^{r_l}X_l>t)-{\mathbb {P}}(X_l\#^{r_l^*}X_l >t) \le 0, \end{aligned}$$
which further concludes that \(M_{4}(t)\le 0\). Therefore, it follows that \(M_{1}(t)\), \(M_{3}(t)\) and \(M_{4}(t)\) are non-positive, which means that \({\overline{F}}_{\tau _1}(t)\le {\overline{F}}_{\tau _2}(t)\), for all \(t \in {\mathbb {R}}_+\). Thus, the desired result is proved. \(\square \)
1.8 Appendix A.8: Proof of Theorem 10
Proof
Let \(F_{\tau _1}\) and \(F_{\tau _2}\) be distributions of \(\tau (X_k\#^{r_k}X_k, X_l\#^{r_l}X_l, \varvec{X}^{(k,l)})\) and \(\tau (X_k\#^{r_k^*}X_k, X_l\#^{r_l^*}X_l, \varvec{X}^{(k,l)}),\) respectively. Note that
$$\begin{aligned}{} & {} F_{\tau _1}(t)- F_{\tau _2}(t) \\{} & {} \quad = [{\mathbb {P}}(X_k\#^{r_k}X_k\le t){\mathbb {P}}(X_l\#^{r_l}X_l \le t) -{\mathbb {P}}(X_k\#^{r_k^*}X_k\le t) {\mathbb {P}}(X_l\#^{r_l^*}X_l\le t)] [{\mathbb {P}} ({\tilde{B}}_{kl})\\{} & {} \qquad -{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{k{\bar{l}}})-{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{{\bar{k}}l}) -{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{A}}_{{\bar{k}}{\bar{l}}}) -{\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{A}}_{{\bar{k}} l}) +{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{k{\bar{l}}} {\tilde{B}}_{{\bar{k}}l}) + {\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{{\bar{k}} l} {\tilde{B}}_{{\bar{k}} {\bar{l}}})\\{} & {} \qquad + {\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{ k {\bar{l}}} {\tilde{A}}_{{\bar{k}}{\bar{l}}}) + {\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{B}}_{{\bar{k}} l} {\tilde{A}}_{{\bar{k}} {\bar{l}}}) - {\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{ k {\bar{l}}} {\tilde{B}}_{{\bar{k}} l} {\tilde{A}}_{{\bar{k}}{\bar{l}}})]\\{} & {} \qquad + [{\mathbb {P}}(X_k\#^{r_k}X_k\le t)-{\mathbb {P}}(X_k\#^{r_k^*}X_k\le t)] [{\mathbb {P}} ({\tilde{B}}_{k{\bar{l}}})-{\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{B}}_{{\bar{k}} {\bar{l}}})] \\{} & {} \qquad + [ {\mathbb {P}}(X_l\#^{r_l}X_l\le t)-{\mathbb {P}}(X_l\#^{r_l^*}X_l \le t)] [{\mathbb {P}} ({\tilde{B}}_{{\bar{k}} l})-{\mathbb {P}} ({\tilde{B}}_{{\bar{k}} l} {\tilde{A}}_{{\bar{k}} {\bar{l}}}) ] \\{} & {} \quad =:N_{1}(t)+N_{2}(t), \end{aligned}$$
where
$$\begin{aligned} N_{1}(t)= & {} [{\mathbb {P}}(X_k\#^{r_k}X_k\!\le \! t){\mathbb {P}}(X_l \#^{r_l}X_l\!\le \! t) -{\mathbb {P}}(X_k\#^{r_k^*}X_k\!\le \! t) {\mathbb {P}}(X_l\#^{r_l^*}X_l\!\le \! t)] [{\mathbb {P}} ({\tilde{B}}_{kl})\\{} & {} -{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{k{\bar{l}}})-{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{{\bar{k}}l}) -{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{A}}_{{\bar{k}}{\bar{l}}}) -{\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{A}}_{{\bar{k}} l}) +{\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{k{\bar{l}}} {\tilde{B}}_{{\bar{k}}l})\\{} & {} + {\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{{\bar{k}} l} {\tilde{B}}_{{\bar{k}} {\bar{l}}}) + {\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{ k {\bar{l}}} {\tilde{A}}_{{\bar{k}}{\bar{l}}}) + {\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{B}}_{{\bar{k}} l} {\tilde{A}}_{{\bar{k}} {\bar{l}}}) - {\mathbb {P}} ({\tilde{B}}_{kl} {\tilde{B}}_{ k {\bar{l}}} {\tilde{B}}_{{\bar{k}} l} {\tilde{A}}_{{\bar{k}}{\bar{l}}})],\\ N_{2}(t)= & {} [{\mathbb {P}}(X_k\#^{r_k}X_k\le t)-{\mathbb {P}}(X_k\#^{r_k^*}X_k\le t)] [{\mathbb {P}} ({\tilde{B}}_{k{\bar{l}}}) -{\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{B}}_{{\bar{k}} {\bar{l}}})] \\{} & {} + [ {\mathbb {P}}(X_l\#^{r_l}X_l\le t)-{\mathbb {P}}(X_l\#^{r_l^*}X_l \le t)][{\mathbb {P}} ({\tilde{B}}_{{\bar{k}} l}) -{\mathbb {P}} ({\tilde{B}}_{{\bar{k}} l} {\tilde{A}}_{{\bar{k}} {\bar{l}}}) ]. \end{aligned}$$
First, according to Corollary 4.3 of Zhang and Zhao (2019), \((r_k, r_l)\overset{\textrm{m}}{\succeq }\ (r_k^*, r_l^*)\) implies
$$\begin{aligned} {\mathbb {P}}(X_k\#^{r_k}X_k\le t){\mathbb {P}}(X_l\#^{r_l}X_l\le t)-{\mathbb {P}}(X_k\#^{r_k^*}X_k\le t) {\mathbb {P}}(X_l\#^{r_l^*}X_l\le t) \le 0. \end{aligned}$$
Then by using condition (ii), we have \(N_{1}(t)\le 0\). On the other hand,
$$\begin{aligned} N_{2}(t)= & {} [{\mathbb {P}}(X_k\#^{r_k}X_k \le t)-{\mathbb {P}}(X_k\#^{r_k^*}X_k \le t)] [{\mathbb {P}} ({\tilde{B}}_{k{\bar{l}}})-{\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{B}}_{{\bar{k}} {\bar{l}}})\\{} & {} -{\mathbb {P}} ({\tilde{B}}_{{\bar{k}} l})+{\mathbb {P}} ({\tilde{B}}_{{\bar{k}} l} {\tilde{A}}_{{\bar{k}} \bar{l}}) ] \\{} & {} +[{\mathbb {P}}(X_k\#^{r_k}X_k \le t)-{\mathbb {P}}(X_k\#^{r_k^*}X_k \le t)+[ {\mathbb {P}}(X_l\#^{r_l}X_l \le t)\\{} & {} -{\mathbb {P}}(X_l\#^{r_l^*}X_l \le t)] [{\mathbb {P}} ({\tilde{B}}_{k{\bar{l}}})-{\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{B}}_{{\bar{k}} {\bar{l}}})]\\=: & {} N_{3}(t) +N_{4}(t), \end{aligned}$$
where
$$\begin{aligned} N_{3}(t)= & {} [{\mathbb {P}}(X_k\#^{r_k^*}X_k> t)-{\mathbb {P}}(X_k\#^{r_k}X_k > t)] [{\mathbb {P}} ({\tilde{B}}_{k{\bar{l}}})\\{} & {} \quad -{\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{B}}_{{\bar{k}} {\bar{l}}})-{\mathbb {P}} ({\tilde{B}}_{{\bar{k}} l})+{\mathbb {P}} ({\tilde{B}}_{{\bar{k}} l} {\tilde{A}}_{{\bar{k}} \bar{l}}) ],\\ N_{4}(t)= & {} [{\mathbb {P}}(X_k\#^{r_k}X_k \le t)-{\mathbb {P}}(X_k\#^{r_k^*}X_k \le t)+ {\mathbb {P}}(X_l\#^{r_l}X_l \le t)\\{} & {} \quad -{\mathbb {P}}(X_l\#^{r_l^*}X_l \le t)] [{\mathbb {P}} ({\tilde{B}}_{k{\bar{l}}})-{\mathbb {P}} ({\tilde{B}}_{k {\bar{l}}} {\tilde{B}}_{{\bar{k}} {\bar{l}}})]. \end{aligned}$$
Note that \(r_k\ge r_k^*\), and this implies
$$\begin{aligned} {\mathbb {P}}(X_k\#^{r_k^*}X_k>t)-{\mathbb {P}}(X_k\#^{r_k}X_k>t) \le 0. \end{aligned}$$
With this inequality and condition (ii), we have \(N_{3}(t)\le 0\). Besides, condition (iii) implies
$$\begin{aligned} {\mathbb {P}}(X_k\#^{r_k}X_k \le t)-{\mathbb {P}}(X_k\#^{r_k^*}X_k \le t)+ {\mathbb {P}}(X_l\#^{r_l}X_l \le t)-{\mathbb {P}}(X_l\#^{r_l^*}X_l \le t)\le 0, \end{aligned}$$
which concludes that \(N_{4}(t)\le 0\). Therefore, it follows that \(N_{1}(t)\), \(N_{3}(t)\) and \(N_{4}(t)\) are non-positive, which means that \(F_{\tau _1}(t)\le F_{\tau _2}(t)\), for all \(t \in {\mathbb {R}}_+\). Thus, the proof is finished. \(\square \)
