Availability contracts under hierarchical maintenance

Abstract

Availability contracting is a prominent procurement strategy among many defense and other capital equipment suppliers and their customers. Under availability contracting, the revenue generated for a service provider is a function of equipment/system availability over the predetermined contract period. A major challenge in implementing availability contracting is predicting the availability of equipment, defining the target availability, and determining the best contract duration based on the target availability. In this paper, we extend the classical operational availability model for estimating equipment availability under hierarchical maintenance, which is frequently used for complex equipment such as aircraft. Under an availability contract, bounds have been proposed on system-level operational availability. The current study also develops optimization models for finding the optimal availability contract duration under different scenarios: stochastic and non-stochastic contract durations; contract written at parts level, system level; minimizing total maintenance costs with minimum level of target operational availability as a constraint etc. The study also shows that the optimal contract duration is a non-decreasing function of spares on hand, and the inherent availability of system part. The results from the study have been validated using numerical study with both simulated and real data.

Appendices

Appendix A

Proof of Theorem 1

For proving the lower bound in Theorem 1, we use the Cauchy–Schwartz inequality. The inequality is given as follows:

$$ \sum\limits_{i = 1}^{n} {a_{i} b_{i} } \le \left( {\sum\limits_{i = 1}^{n} {a_{i}^{2} } } \right)^{\frac{1}{2}} \left( {\sum\limits_{i = 1}^{n} {b_{i}^{2} } } \right)^{\frac{1}{2}} $$

(A1)

The $a{\prime} s$ and $b{\prime} s$ are arbitrary sequences of real numbers.

For this proof, we define $P_{1}$ as follows:

$$ \log_{e} \left( {\frac{1}{{P_{1} }}} \right) = \sum\limits_{i = 1}^{M} {\log_{e} \left( {1 + \frac{{MTTRS_{i} }}{T} \times \left( {\frac{T}{{\theta_{i} }}} \right)^{{\beta_{i} }} } \right)} $$

(A2)

From Eq. (17), following equation (A3) is obtained:

$$ \frac{1}{{P_{0} }} = \frac{1}{{P_{1} }} \times K $$

(A3)

Considering $a_{i} =$$\log_{e} \left( {1 + \frac{{MTTRS_{i} }}{T} \times \left( {\frac{T}{{\theta_{i} }}} \right)^{{\beta_{i} }} } \right)$ and $b_{i} = 1$, (A4) is obtained:

$$ \log_{e} \left( {\frac{1}{{P_{1} }}} \right) \le \left( {\left( {M \times \sum\limits_{i = 1}^{M} {\left( {\log_{e} \left( {1 + \frac{{MTTRS_{i} }}{T} \times \left( {\frac{T}{{\theta_{i} }}} \right)^{{\beta_{i} }} } \right)} \right)}^{2} } \right)^{\frac{1}{2}} } \right) $$

(A4)

From (A4), Eq. (A5) is obtained:

$$ \left( {\frac{1}{{P_{1} }}} \right) \le \exp \left( {\left( {M \times \sum\limits_{i = 1}^{M} {\left( {\log_{e} \left( {1 + \frac{{MTTRS_{i} }}{T} \times \left( {\frac{T}{{\theta_{i} }}} \right)^{{\beta_{i} }} } \right)} \right)}^{2} } \right)^{\frac{1}{2}} } \right) = P_{2} $$

(A5)

Since $K > 0$, on multiplying both sides of (A5) with $K$, the sign of the inequality is preserved and Eq. (A6) is obtained:

$$ \frac{1}{{P_{0} }} = \frac{1}{{P_{1} }} \times K \le K \times P_{2} $$

(A6)

Equation (A6) leads to lower bound of Theorem 1. For the upper bound of Theorem 1, Mahler’s inequality is used:

We have $\prod\limits_{i = 1}^{N} {\left( {x_{i} + y_{i} } \right)^{\frac{1}{N}} } \ge \prod\limits_{i = 1}^{N} {\left( {x_{i} } \right)^{\frac{1}{N}} } + \prod\limits_{i = 1}^{N} {\left( {y_{i} } \right)^{\frac{1}{N}} }$ where $x_{i} \ge 0,y_{i} \ge 0$ for all i.

This inequality can be rewritten as:

$$ \prod\limits_{i = 1}^{N} {\left( {x_{i} + y_{i} } \right)} \ge \left( {\prod\limits_{i = 1}^{N} {\left( {x_{i} } \right)^{\frac{1}{N}} } + \prod\limits_{i = 1}^{N} {\left( {y_{i} } \right)^{\frac{1}{N}} } } \right)^{N} $$

(A7)

Taking $x_{i} = 1,\,\,y_{i} = \frac{{MTTRS_{i} }}{T} \times \left( {\frac{T}{{\theta_{i} }}} \right)^{{\beta_{i} }}$, Eq. (A8) is obtained:

$$ \frac{1}{{P_{1} }} \ge \left( {1 + \prod\limits_{i = 1}^{M} {\left( {\frac{{MTTRS_{i} }}{T} \times \left( {\frac{T}{{\theta_{i} }}} \right)^{{\beta_{i} }} } \right)^{\frac{1}{M}} } } \right)^{M} = P_{3} $$

(A8)

Again, since $K > 0$, it can be seen that, multiplying both sides of (A8) with $K$ leads to the upper bound of Theorem 1.

Validity of bounds:

It can be seen that $0 < \frac{1}{K} < 1$.

Also $\left( {1 + \prod\limits_{i = 1}^{M} {\left( {\frac{{MTTRS_{i} }}{T} \times \left( {\frac{T}{{\theta_{i} }}} \right)^{{\beta_{i} }} } \right)^{\frac{1}{M}} } } \right)^{M} = P_{3} > 1$. Hence, it is clear that $\frac{1}{{P_{3} }} < 1$. Again, it can be seen that $\frac{1}{{P_{3} }} > 0$. Using these, it can be trivially shown that the upper bound is valid i.e. takes values between 0 and 1.

For the lower bound, the following is obtained:

$$ \begin{aligned} \log_{e} \left( {1 + \frac{{MTTRS_{i} }}{T} \times \left( {\frac{T}{{\theta_{i} }}} \right)^{{\beta_{i} }} } \right) > 0 \Rightarrow \left( {\log_{e} \left( {1 + \frac{{MTTRS_{i} }}{T} \times \left( {\frac{T}{{\theta_{i} }}} \right)^{{\beta_{i} }} } \right)} \right)^{2} > 0 \hfill \\ \Rightarrow M \times \sum\limits_{i = 1}^{M} {\left( {\log_{e} \left( {1 + \frac{{MTTRS_{i} }}{T} \times \left( {\frac{T}{{\theta_{i} }}} \right)^{{\beta_{i} }} } \right)} \right)^{2} > 0} \hfill \\ \Rightarrow \left( {M \times \sum\limits_{i = 1}^{M} {\left( {\log_{e} \left( {1 + \frac{{MTTRS_{i} }}{T} \times \left( {\frac{T}{{\theta_{i} }}} \right)^{{\beta_{i} }} } \right)} \right)^{2} } } \right)^{1/2} > 0 \hfill \\ \Rightarrow \exp \left( {M \times \sum\limits_{i = 1}^{M} {\left( {\log_{e} \left( {1 + \frac{{MTTRS_{i} }}{T} \times \left( {\frac{T}{{\theta_{i} }}} \right)^{{\beta_{i} }} } \right)} \right)^{2} } } \right)^{1/2} > 1 \Rightarrow P_{2} > 1 \hfill \\ \end{aligned} $$

Using this, it can be shown that the lower bound is also valid.

Proof of Theorem 2

Differentiating Eq. (16) w.r.t.$T$, following equation (A9) is obtained:

$$ \frac{{\partial A_{0}^{ - 1} }}{\partial T} = - \frac{Z}{{T^{2} }} + \frac{1}{T} \times \frac{\partial Z}{{\partial T}} + \left( {\beta - 1} \right) \times \frac{MTTRS}{{\theta^{\beta } }} \times T^{\beta - 2} $$

(A9)

$$ \frac{{\partial^{2} A_{0}^{ - 1} }}{{\partial T^{2} }} = \left( {\beta - 1} \right) \times \left( {\beta - 2} \right) \times \frac{MTTRS}{{\theta^{\beta } }} \times T^{\beta - 3} + \frac{1}{T} \times \frac{{\partial^{2} Z}}{{\partial T^{2} }} + \frac{2Z}{{T^{3} }} - \frac{2}{{T^{2} }} \times \frac{\partial Z}{{\partial T}} $$

(A10)

Equating (A9) to zero, it can be seen that:

$$ \frac{MTTRS}{{\theta^{\beta } }} \times \left( {\beta - 1} \right) \times T^{\beta } + T \times \left( {\frac{\partial Z}{{\partial T}}} \right) = Z $$

(A11)

Substituting $\frac{\partial Z}{{\partial T}}$ from Eq. (A11) in Eq. (A10), it can be seen that:

We have

$$ \frac{{\partial^{2} A_{0}^{ - 1} }}{{\partial T^{2} }} = \left\{ {\left( {\beta - 1} \right) \times \beta \times \frac{MTTRS}{{\theta^{\beta } }} \times T^{\beta - 3} + \frac{1}{T} \times \frac{{\partial^{2} Z}}{{\partial T^{2} }}} \right\} > 0\quad {\text{when}}\quad \beta > 1,\quad T > 0. $$

(A12)

At all values of $T$ that satisfy Eq. (A11), $A_{0}^{ - 1}$ is convex. Thus, unique minimum is obtained. Hence, the theorem is proved.

The Corollary is a special case of Theorem 2. When $Z = a + bT$, it is seen that $\frac{\partial Z}{{\partial T}} = b$ and $\frac{{\partial^{2} Z}}{{\partial T^{2} }} = 0$. Using these, the Corollary can be proved easily.

Proof of Theorem 3

From Eq. (17), the following Eq. (A13) is obtained:

$$ \log_{e} \left( {\frac{1}{{P_{0} }}} \right) = \sum\limits_{i = 1}^{M} {\log_{e} \left\{ {1 + \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times T^{{\beta_{i} - 1}} } \right\}} + \log_{e} \left( {1 + \frac{Z}{T}} \right) $$

(A13)

Differentiating (A13) w.r.t.$T$, Eq. (A14) is obtained as follows:

$$\begin{aligned} \frac{{\partial \log_{e} \left( {\frac{1}{{P_{0} }}} \right)}}{\partial T} &= \left\{ {\sum\limits_{i = 1}^{M} {\frac{1}{{\left( {1 + \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times T^{{\beta_{i} - 1}} } \right)}}} \times \left( {\beta_{i} - 1} \right) \times T^{{\beta_{i} - 2}} \times \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }}} \right\} \\& \quad + \frac{1}{{\left( {1 + \frac{Z}{T}} \right)}} \times \left( { - \frac{Z}{{T^{2} }} + \frac{1}{T} \times \frac{\partial Z}{{\partial T}}} \right) \end{aligned}$$

(A14)

Differentiating (A14) once again w.r.t $T$ and simplifying leads to Eq. (A15) as follows:

$$ \begin{aligned} \frac{{\partial^{2} \log_{e} \left( {\frac{1}{{P_{0} }}} \right)}}{{\partial T^{2} }}& = \left\{ {\sum\limits_{i = 1}^{M} {\frac{1}{{\left( {1 + \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times T^{{\beta_{i} - 1}} } \right)^{2} }}} \times \left( {\beta_{i} - 1} \right) \times T^{{\beta_{i} - 3}} }\right. \\ & \quad \left.{\times \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left[ {\left( {\beta_{i} - 2} \right) - T^{{\beta_{i} - 1}} \times \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }}} \right]} \right\} \hfill \\ & \quad + \frac{1}{{\left( {1 + \frac{Z}{T}} \right)^{2} }} \times \left\{ {\left( {\frac{2Z}{{T^{3} }} + \frac{{Z^{2} }}{{T^{4} }}} \right) - \frac{1}{T} \times \left( {2 + \frac{\partial Z}{{\partial T}}} \right) \times \frac{1}{T} }\right. \\ & \quad \left.{ \times \frac{\partial Z}{{\partial T}} + \left( {1 + \frac{Z}{T}} \right) \times \frac{1}{T} \times \frac{{\partial^{2} Z}}{{\partial T^{2} }}} \right\} \hfill \\ \end{aligned} $$

(A15)

Equating (A14) to zero leads to Eq. (A16):

$$\begin{aligned} \left( {1 + \frac{Z}{T}} \right)& \times \left\{ {\sum\limits_{i = 1}^{M} {\frac{1}{{\left( {1 + \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times T^{{\beta_{i} - 1}} } \right)}}} \times \left( {\beta_{i} - 1} \right) \times T^{{\beta_{i} - 2}} \times \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }}} \right\} \\ & \quad - \frac{Z}{{T^{2} }} = - \frac{1}{T} \times \frac{\partial Z}{{\partial T}} \end{aligned}$$

(A16)

Using (A16) in (A15), it can be seen that:

$$ \begin{aligned} \frac{{\partial^{2} \log_{e} \left( {\frac{1}{{P_{0} }}} \right)}}{{\partial T^{2} }} = \left\{ \begin{aligned} \sum\limits_{i = 1}^{M} {\frac{1}{{\left( {1 + \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times T^{{\beta_{i} - 1}} } \right)^{2} }}} \times \left( {\beta_{i} - 1} \right) \times T^{{\beta_{i} - 3}} \times \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \hfill \\ \quad\times \left( {1 + \frac{Z}{T}} \right)\left[ {\left( {\beta_{i} - 2} \right)\left( {1 + \frac{Z}{T}} \right) + \left( {2 + \frac{\partial Z}{{\partial T}}} \right) + T^{{\beta_{i} - 1}} }\right. \hfill \\ \quad\times \left.{\frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left\{ {1 - \left( {\frac{Z}{T} - \frac{\partial Z}{{\partial T}}} \right)} \right\}} \right] \hfill \\ \end{aligned} \right\} \hfill \quad + \frac{1}{{\left( {1 + \frac{Z}{T}} \right)^{2} }} \times \left\{ {\frac{Z}{{T^{3} }} \times \left( {\frac{Z}{T} - \frac{\partial Z}{{\partial T}}} \right) + \left( {1 + \frac{Z}{T}} \right) \times \frac{1}{T} \times \frac{{\partial^{2} Z}}{{\partial T^{2} }}} \right\} \hfill \\ \end{aligned} $$

(A17)

The R.H.S. expression in Eq. (A17) is positive because $1 - \frac{Z}{T} + \frac{\partial Z}{{\partial T}} > 0$ by properties of $Z\left( T \right)$. Further, $\frac{Z}{T} - \frac{\partial Z}{{\partial T}} > 0$ for $Z = a + bT$. Hence, at all the values of $T$ satisfying Eq. (A16), it can be seen that $\log_{e} \left( {\frac{1}{{P_{0} }}} \right)$ is convex. Since $\log_{e} \left( {\frac{1}{{P_{0} }}} \right)$ is convex at all stationary points, it implies there is only one such point and unique minimum is attained at that point. Hence, Theorem 3 is proved.

Proof of Theorem 4

Equation (16) is rewritten as follows:

$$ A_{0}^{ - 1} = 1 + \frac{Z}{T} + \left\{ {\frac{{T^{\beta - 1} }}{{\theta^{\beta } }} \times \left( {\left( {\frac{{1 - A_{inh} }}{{A_{inh} }}} \right)\tau + MLDT(s)} \right)} \right\} $$

(A18)

From the derivation of Theorem 2, using $Z = a + bT,a > 0,b > 0$, it can be shown that:

$$ T^{*} = \left( {\frac{a}{\beta - 1}} \right)^{1/\beta } \times \theta \times \left( {\left( {\frac{{1 - A_{inh} }}{{A_{inh} }}} \right)\tau + MLDT(s)} \right)^{ - 1/\beta } $$

$$\begin{aligned} \frac{{\partial T^{*} }}{\partial s}& = \left\{ {\left( {\frac{a}{\beta - 1}} \right)^{1/\beta } \times \theta \times \left( { - 1/\beta } \right) \times \left( {\left( {\frac{{1 - A_{inh} }}{{A_{inh} }}} \right)\tau + MLDT(s)} \right)^{ - ((1 + \beta )/\beta )} }\right. \\ & \quad \left.{\times \frac{\partial MLDT(s)}{{\partial s}}} \right\} \ge 0 \end{aligned}$$

Also, $\frac{{\partial T^{*} }}{{\partial A_{inh} }} = \Big\{ \left( {\frac{a}{\beta - 1}} \right)^{1/\beta } \times \theta \times \left( { - 1/\beta } \right) \times \left( {\left( {\frac{{1 - A_{inh} }}{{A_{inh} }}} \right)\tau + MLDT(s)} \right)^{ - ((1 + \beta )/\beta )} \times \tau \times - \left( {\frac{1}{{A_{inh} }}} \right)^{2} \Big\} > 0$.

Proof of Theorem 5

Equation (16) is rewritten by replacing $T$ with $E\left( T \right)$ as follows:

$$ A_{0}^{ - 1} = 1 + \frac{Z}{E\left( T \right)} + \left\{ {\left( {E\left( T \right)} \right)^{\beta - 1} \times \frac{MTTRS}{{\theta^{\beta } }}} \right\} $$

(A19)

$$ \frac{{\partial A_{0}^{ - 1} }}{\partial C} = \left( {\beta - 1} \right) \times \left( {E\left( T \right)} \right)^{\beta - 2} \times \frac{\partial E\left( T \right)}{{\partial C}} \times \frac{MTTRS}{{\theta^{\beta } }} + \frac{1}{E\left( T \right)} \times \frac{\partial Z}{{\partial C}} - \frac{\partial E\left( T \right)}{{\partial C}} \times \frac{Z}{{\left( {E\left( T \right)} \right)^{2} }} $$

(A20)

$$ \begin{aligned} & \frac{{\partial^{2} A_{0}^{ - 1} }}{{\partial C^{2} }} = \left( {\beta - 1} \right) \times \left( {\beta - 2} \right) \times \left( {E\left( T \right)} \right)^{\beta - 3} \times \left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right)^{2} \times \frac{MTTRS}{{\theta^{\beta } }} \hfill \\ &\quad + \,\left( {\beta - 1} \right) \times \left( {E\left( T \right)} \right)^{\beta - 2} \times \left( {\frac{{\partial^{2} E\left( T \right)}}{{\partial C^{2} }}} \right) \times \frac{MTTRS}{{\theta^{\beta } }} \hfill \\ & \quad - \frac{\partial Z}{{\partial C}} \times \frac{1}{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{\partial E\left( T \right)}{{\partial C}} + \frac{1}{E\left( T \right)} \times \frac{{\partial^{2} Z}}{{\partial C^{2} }} \hfill \\ &\quad - \frac{Z}{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{{\partial^{2} E\left( T \right)}}{{\partial C^{2} }} - \frac{1}{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{\partial E\left( T \right)}{{\partial C}} \times \frac{\partial Z}{{\partial C}} + \frac{2Z}{{\left( {E\left( T \right)} \right)^{3} }} \times \left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right)^{2} \hfill \\ \end{aligned} $$

(A21)

By equating the first derivative in (A20) to zero, Eq. (A22) is obtained:

$$ \begin{aligned} \left( {\beta - 1} \right) \times \left( {\beta - 2} \right) \times \left( {E\left( T \right)} \right)^{\beta - 3} \times \left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right)^{2} \times \frac{MTTRS}{{\theta^{\beta } }} = \hfill \\ \left( {\beta - 2} \right) \times \frac{Z}{{\left( {E\left( T \right)} \right)^{3} }} \times \left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right)^{2} - \left( {\beta - 2} \right) \times \frac{1}{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{\partial Z}{{\partial C}} \times \frac{\partial E\left( T \right)}{{\partial C}} \hfill \\ \end{aligned} $$

(A22)

Substituting the expression from Eqs. (A22) in (A21), it can be seen that:

$$ \begin{aligned} \frac{{\partial^{2} A_{0}^{ - 1} }}{{\partial C^{2} }}& = \,\left( {\beta - 1} \right) \times \left( {E\left( T \right)} \right)^{\beta - 2} \times \left( {\frac{{\partial^{2} E\left( T \right)}}{{\partial C^{2} }}} \right) \times \frac{MTTRS}{{\theta^{\beta } }} + \frac{1}{E\left( T \right)} \times \frac{{\partial^{2} Z}}{{\partial C^{2} }} \hfill \\ &\quad - \frac{Z}{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{{\partial^{2} E\left( T \right)}}{{\partial C^{2} }} - \frac{\beta }{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{\partial E\left( T \right)}{{\partial C}} \times \frac{\partial Z}{{\partial C}} + \beta \times \frac{Z}{{\left( {E\left( T \right)} \right)^{3} }} \times \left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right)^{2} \hfill \\ \end{aligned} $$

(A23)

The first term on RHS of Eq. (A23) is non-negative for $0 < \beta < 1$. Also $\frac{\partial Z}{{\partial C}} = b\frac{\partial E\left( T \right)}{{\partial C}} > 0;\frac{{\partial^{2} Z}}{{\partial C^{2} }} = b\frac{{\partial^{2} E\left( T \right)}}{{\partial C^{2} }} \le 0$. Grouping the 4th and 5th terms of (A23) and simplifying leads to $\frac{\beta }{{\left( {E\left( T \right)} \right)^{2} }} \times \left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right)^{2} \times \frac{a}{E\left( T \right)} > 0$. Again grouping 2^nd and 3^rd terms leads to $- \frac{{\partial^{2} E\left( T \right)}}{{\partial C^{2} }} \times \frac{1}{E\left( T \right)} \times \frac{a}{E\left( T \right)} \ge 0$. Hence, $\frac{{\partial^{2} A_{0}^{ - 1} }}{{\partial C^{2} }} > 0$ and $A_{0}^{ - 1}$ has a unique minimum at $C$ satisfying Eq. (A22).

Proof of Theorem 6

Equation (17) is rewritten by replacing $T$ with $E\left( T \right)$ and taking the natural logarithm as follows:

$$ \log_{e} \left( {\frac{1}{{P_{0} }}} \right) = \left\{ {\sum\limits_{i = 1}^{M} {\log_{e} \left\{ {1 + \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left( {E\left( T \right)} \right)^{{\beta_{i} - 1}} } \right\}} } \right\} + \log_{e} \left( {1 + \frac{Z}{E\left( T \right)}} \right) $$

(A24)

$$ \begin{aligned} \frac{{\partial \log_{e} \left( {\frac{1}{{P_{0} }}} \right)}}{\partial C} &= \left\{ {\sum\limits_{i = 1}^{M} {\frac{1}{{\left\{ {1 + \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left( {E\left( T \right)} \right)^{{\beta_{i} - 1}} } \right\}}}} }\right. \\ &\quad \left.{\times \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left( {\beta_{i} - 1} \right) \times \left( {E\left( T \right)} \right)^{{\beta_{i} - 2}} \times \frac{\partial E\left( T \right)}{{\partial C}}} \right\} \hfill \\ &\quad +\, \frac{1}{{\left( {1 + \frac{Z}{E\left( T \right)}} \right)}} \times \left\{ {\frac{1}{E\left( T \right)} \times \frac{\partial Z}{{\partial C}} - \frac{Z}{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{\partial E\left( T \right)}{{\partial C}}} \right\} \hfill \\ \end{aligned} $$

(A25)

$$ \frac{{\partial^{2} \log_{e} \left( {\frac{1}{{P_{0} }}} \right)}}{{\partial C^{2} }} = A + B $$

(A26)

$$ \begin{aligned} A &= \sum\limits_{i = 1}^{M} {\frac{1}{{\left\{ {1 + \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left( {E\left( T \right)} \right)^{{\beta_{i} - 1}} } \right\}}}} \times \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \hfill \\ &\times \left\{ \begin{aligned} \left( {\beta_{i} - 1} \right) \times \left( {\beta_{i} - 2} \right) \times \left( {E\left( T \right)} \right)^{{\beta_{i} - 3}} \times \left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right)^{2} + \hfill \\ \left( {\beta_{i} - 1} \right) \times \left( {E\left( T \right)} \right)^{{\beta_{i} - 2}} \times \frac{{\partial^{2} E\left( T \right)}}{{\partial C^{2} }} \hfill \\ \end{aligned} \right\} \hfill \\ &\quad - \frac{1}{{\left\{ {1 + \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left( {E\left( T \right)} \right)^{{\beta_{i} - 1}} } \right\}^{2} }} \times \left\{ {\frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left( {\beta_{i} - 1} \right) \times \left( {E\left( T \right)} \right)^{{\beta_{i} - 2}} \times \frac{\partial E\left( T \right)}{{\partial C}}} \right\}^{2} \hfill \\ \end{aligned} $$

(A27)

$$ \begin{aligned} B& = - \frac{1}{{\left( {1 + \frac{Z}{E\left( T \right)}} \right)^{2} }} \times \left\{ { - \frac{Z}{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{\partial E\left( T \right)}{{\partial C}} + \frac{1}{E\left( T \right)} \times \frac{\partial Z}{{\partial C}}} \right\}^{2} + \frac{1}{{\left( {1 + \frac{Z}{E\left( T \right)}} \right)}} \hfill \\ &\quad \times\left\{ {\frac{1}{E\left( T \right)} \times \frac{{\partial^{2} Z}}{{\partial C^{2} }} - \frac{2}{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{\partial E\left( T \right)}{{\partial C}} \times \frac{\partial Z}{{\partial C}} - \frac{Z}{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{{\partial^{2} E\left( T \right)}}{{\partial C^{2} }} + \frac{2Z}{{\left( {E\left( T \right)} \right)^{3} }} \times \left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right)^{2} } \right\} \hfill \\ \end{aligned} $$

(A28)

Simplifying, Eq. (A29) is obtained:

$$ \begin{aligned} A& = \sum\limits_{i = 1}^{M} {\frac{1}{{\left\{ {1 + \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left( {E\left( T \right)} \right)^{{\beta_{i} - 1}} } \right\}^{2} }}} \times \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left( {\beta_{i} - 1} \right) \times \left( {E\left( T \right)} \right)^{{\beta_{i} - 3}} \times \left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right)^{2} \times \hfill \\ &\quad \left\{ {\left( {\beta_{i} - 2} \right) - \left( {E\left( T \right)} \right)^{{\beta_{i} - 1}} \times \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }}} \right\} + \hfill \\ &\quad \sum\limits_{i = 1}^{M} {\frac{1}{{\left\{ {1 + \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left( {E\left( T \right)} \right)^{{\beta_{i} - 1}} } \right\}}} \times \left\{ {\frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left( {\beta_{i} - 1} \right) \times \left( {E\left( T \right)} \right)^{{\beta_{i} - 2}} } \right\} \times \frac{{\partial^{2} E\left( T \right)}}{{\partial C^{2} }}} \hfill \\ \end{aligned} $$

(A29)

$$\begin{aligned} B &= \frac{1}{{\left( {1 + \frac{Z}{E\left( T \right)}} \right)^{2} }} \\ & \quad\times \left\{ \begin{aligned} &\frac{1}{E\left( T \right)} \times \frac{{\partial^{2} Z}}{{\partial C^{2} }} - \frac{2}{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{\partial E\left( T \right)}{{\partial C}} \times \frac{\partial Z}{{\partial C}} - \frac{Z}{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{{\partial^{2} E\left( T \right)}}{{\partial C^{2} }} + \frac{2Z}{{\left( {E\left( T \right)} \right)^{3} }} \times \left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right)^{2} \hfill \\ &\quad+ \frac{Z}{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{{\partial^{2} Z}}{{\partial C^{2} }} + \frac{{Z^{2} }}{{\left( {E\left( T \right)} \right)^{4} }} \times \left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right)^{2} - \frac{{Z^{2} }}{{\left( {E\left( T \right)} \right)^{3} }} \times \frac{{\partial^{2} E\left( T \right)}}{{\partial C^{2} }} - \frac{1}{{\left( {E\left( T \right)} \right)^{2} }} \times \left( {\frac{\partial Z}{{\partial C}}} \right)^{2} \hfill \\ \end{aligned} \right\} \end{aligned}$$

(A30)

Equating (A25) to zero, the following is obtained:

$$ \begin{aligned}& \left\{ {\sum\limits_{i = 1}^{M} {\frac{1}{{\left\{ {1 + \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left( {E\left( T \right)} \right)^{{\beta_{i} - 1}} } \right\}}}} \times \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left( {\beta_{i} - 1} \right) \times \left( {E\left( T \right)} \right)^{{\beta_{i} - 2}} } \right\} \hfill \\ &\quad = \frac{1}{{\frac{\partial E\left( T \right)}{{\partial C}} \times \left( {1 + \frac{Z}{E\left( T \right)}} \right)}} \times \left\{ { - \frac{1}{E\left( T \right)} \times \frac{\partial Z}{{\partial C}} + \frac{Z}{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{\partial E\left( T \right)}{{\partial C}}} \right\} \hfill \\ \end{aligned} $$

(A31)

Substituting (A31) in the second part of the expression in (A29) and simplifying leads to:

$$ \frac{{\partial^{2} \log_{e} \left( {\frac{1}{{P_{0} }}} \right)}}{{\partial C^{2} }} = A_{1} + A_{2} - \frac{1}{{\left( {1 + \frac{Z}{E\left( T \right)}} \right)^{2} }} \times \frac{1}{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{\partial Z}{{\partial C}} \times \left( {2 \times \frac{\partial E\left( T \right)}{{\partial C}} + \frac{\partial Z}{{\partial C}}} \right) $$

(A32)

$$ \begin{aligned} A_{1} &= \sum\limits_{i = 1}^{M} {\frac{1}{{\left\{ {1 + \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left( {E\left( T \right)} \right)^{{\beta_{i} - 1}} } \right\}^{2} }}} \times \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left( {\beta_{i} - 1} \right) \times \left( {E\left( T \right)} \right)^{{\beta_{i} - 3}} \times \left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right)^{2}\hfill \\ &\quad \times \left\{ {\left( {\beta_{i} - 2} \right) - \left( {E\left( T \right)} \right)^{{\beta_{i} - 1}} \times \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }}} \right\} \hfill \\ \end{aligned} $$

(A33)

$$ \begin{aligned}A_{2} &= \frac{1}{{\left( {1 + \frac{Z}{E\left( T \right)}} \right)^{2} }} \\ & \quad \times \left\{ \begin{aligned} &- \frac{1}{E\left( T \right)} \times \frac{{\partial^{2} E\left( T \right)}}{{\partial C^{2} }} \times \frac{\partial Z}{{\partial E\left( T \right)}} - \frac{Z}{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{{\partial^{2} E\left( T \right)}}{{\partial C^{2} }} \times \frac{\partial Z}{{\partial E\left( T \right)}} + \frac{1}{E\left( T \right)} \times \frac{{\partial^{2} Z}}{{\partial C^{2} }} \hfill \\ &\quad + \frac{2Z}{{\left( {E\left( T \right)} \right)^{3} }} \times \left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right)^{2} + \frac{Z}{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{{\partial^{2} Z}}{{\partial C^{2} }} + \frac{{Z^{2} }}{{\left( {E\left( T \right)} \right)^{4} }} \times \left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right)^{2} \hfill \\ \end{aligned} \right\} \end{aligned}$$

(A34)

Equation (A31) can be written differently as follows:

$$ \begin{aligned} &\frac{1}{{\left( {1 + \frac{Z}{E\left( T \right)}} \right)}} \times \frac{Z}{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{\partial E\left( T \right)}{{\partial C}} \hfill \\ &\qquad - \left\{ {\sum\limits_{i = 1}^{M} {\frac{1}{{\left\{ {1 + \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left( {E\left( T \right)} \right)^{{\beta_{i} - 1}} } \right\}}}} \times \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left( {\beta_{i} - 1} \right) \times \left( {E\left( T \right)} \right)^{{\beta_{i} - 2}} \times \frac{\partial E\left( T \right)}{{\partial C}}} \right\} \hfill \\ &\quad = \frac{1}{{\left( {1 + \frac{Z}{E\left( T \right)}} \right)}} \times \frac{1}{E\left( T \right)} \times \frac{\partial Z}{{\partial C}} \hfill \\ \end{aligned} $$

(A35)

Substituting (A35) in (A32), following is obtained:

$$ \frac{{\partial^{2} \log_{e} \left( {\frac{1}{{P_{0} }}} \right)}}{{\partial C^{2} }} = A_{1} + A_{2} + A_{3} - A_{4} $$

(A36)

$$\begin{aligned} A_{3} &= \left\{ {\sum\limits_{i = 1}^{M} {\frac{1}{{\left\{ {1 + \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left( {E\left( T \right)} \right)^{{\beta_{i} - 1}} } \right\}}}} \times \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left( {\beta_{i} - 1} \right) \times \left( {E\left( T \right)} \right)^{{\beta_{i} - 2}} \times \frac{\partial E\left( T \right)}{{\partial C}}} \right\} \\ & \quad \times \frac{{\left\{ {\frac{\partial Z}{{\partial C}} + 2 \times \frac{\partial E\left( T \right)}{{\partial C}}} \right\}}}{{\left( {E\left( T \right)} \right) \times \left( {1 + \frac{Z}{E\left( T \right)}} \right)}}\end{aligned} $$

(A37)

$$ A_{4} = \left\{ {\frac{1}{{\left( {1 + \frac{Z}{E\left( T \right)}} \right)}} \times \frac{Z}{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{\partial E\left( T \right)}{{\partial C}}} \right\} \times \frac{{\left\{ {\frac{\partial Z}{{\partial C}} + 2 \times \frac{\partial E\left( T \right)}{{\partial C}}} \right\}}}{{\left( {E\left( T \right)} \right) \times \left( {1 + \frac{Z}{E\left( T \right)}} \right)}} $$

(A38)

$$\begin{aligned} A_{2} - A_{4}& = \frac{1}{{\left( {1 + \frac{Z}{E\left( T \right)}} \right)^{2} }} \\ & \quad \times \left\{ \begin{aligned} &- \frac{1}{E\left( T \right)} \times \frac{{\partial^{2} E\left( T \right)}}{{\partial C^{2} }} \times \frac{\partial Z}{{\partial E\left( T \right)}} - \frac{Z}{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{{\partial^{2} E\left( T \right)}}{{\partial C^{2} }} \times \frac{\partial Z}{{\partial E\left( T \right)}} + \frac{1}{E\left( T \right)} \times \frac{{\partial^{2} Z}}{{\partial C^{2} }} \hfill \\ &\quad + \frac{Z}{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{{\partial^{2} Z}}{{\partial C^{2} }} + \frac{{Z^{2} }}{{\left( {E\left( T \right)} \right)^{4} }} \times \left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right)^{2} - \frac{Z}{{\left( {E\left( T \right)} \right)^{3} }} \times \left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right) \times \left( {\frac{\partial Z}{{\partial C}}} \right) \hfill \\ \end{aligned} \right\} \end{aligned}$$

(A39)

Using $\frac{\partial E[T]}{{\partial C}} > 0$ and $\frac{{\partial^{2} E[T]}}{{\partial C^{2} }} \le 0$ along with the assumptions in Theorem 6, it can be seen that the 1^st and 3^rd terms on RHS of (A39) cancels. Similarly the 2^nd and 4th terms cancel. Grouping last two terms leads to

$$ \,\frac{Z}{{\left( {E\left( T \right)} \right)^{3} }} \times \left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right)^{2} \left\{ {\frac{Z}{E\left( T \right)} - \frac{\partial Z}{{\partial E\left( T \right)}}} \right\} > 0.\;{\text{Hence}}.\;A_{2} - A_{4} > 0 $$

$$ \begin{gathered} A_{1} + A_{3} = \left\{ {\sum\limits_{{i = 1}}^{M} {\frac{1}{{\left\{ {1 + \frac{{MTTRS_{i} }}{{\theta _{i}^{{\beta _{i} }} }} \times \left( {E\left( T \right)} \right)^{{\beta _{i} - 1}} } \right\}}}} \times \frac{{MTTRS_{i} }}{{\theta _{i}^{{\beta _{i} }} }} \times \left( {\beta _{i} - 1} \right) \times \left( {E\left( T \right)} \right)^{{\beta _{i} - 3}} \times \frac{{\partial E\left( T \right)}}{{\partial C}}} \right\} \times \hfill \\ \frac{1}{{\left( {1 + \frac{Z}{{E\left( T \right)}}} \right)}} \times \left\{ \begin{gathered} \left( {1 + \frac{Z}{{E\left( T \right)}}} \right) \times \left( {\beta _{i} - 2} \right) \times \frac{{\partial E\left( T \right)}}{{\partial C}} + \left\{ {\frac{{\partial Z}}{{\partial C}} + 2 \times \frac{{\partial E\left( T \right)}}{{\partial C}}} \right\} + \hfill \\ \frac{{MTTRS_{i} }}{{\theta _{i}^{{\beta _{i} }} }} \times \left( {E\left( T \right)} \right)^{{\beta _{i} - 1}} \times \left\{ {\frac{{\partial Z}}{{\partial C}} + \frac{{\partial E\left( T \right)}}{{\partial C}} \times \left( {1 - \frac{Z}{{E\left( T \right)}}} \right)} \right\} \hfill \\ \end{gathered} \right\} \hfill \\ \end{gathered} $$

(A40)

Using $\frac{\partial E[T]}{{\partial C}} > 0$ and $\frac{{\partial^{2} E[T]}}{{\partial C^{2} }} \le 0$ along with the assumptions in Theorem 6, it can be seen that $A_{1} + A_{3} > 0$.

Thus, it is seen that $\frac{{\partial^{2} \log_{e} \left( {\frac{1}{{P_{0} }}} \right)}}{{\partial C^{2} }} > 0$ at the points where Eq. (A25) is zero i.e. stationary points. Hence, $\log_{e} \left( {\frac{1}{{P_{0} }}} \right)$ has a unique minimum for some $C$.

Appendix B

We extend the theorems in our paper for the quadratic form of $Z\left( T \right) = a + bT^{2}$ or $Z(T) = a + b\left( {E\left( T \right)} \right)^{2}$ where $a > 0;b > 0$.

Theorem 1 is not influenced by the functional form of $Z\left( T \right)$.

Theorem 2B. Since the Theorem 2has been derived for a general increasing, convex function $Z\left( T \right)$ the derivation steps upto Eq. (A12) remains the same.

$$ \frac{MTTRS}{{\theta^{\beta } }} \times \left( {\beta - 1} \right) \times T^{\beta } + T \times \left( {\frac{\partial Z}{{\partial T}}} \right) = Z $$

(A11)

When we consider the special case of $Z\left( T \right) = a + bT^{2}$ we have:

$$ \frac{\partial Z}{{\partial T}} = 2bT;\frac{{\partial^{2} Z}}{{\partial T^{2} }} = 2b $$

(B1)

Substituting these in Eq. (A11), we see that the first order condition is given by the following equation:

$$ \frac{MTTRS}{{\theta^{\beta } }}\left( {\beta - 1} \right)T^{\beta } + bT^{2} = a $$

(B2)

Thus we do not obtain a closed form expression for the optimal contract duration unless $\beta = 2$. The optimal contract duration for other $\beta$ values may be obtained by using numerical methods.

When we substitute the values from (B1) in the Eq. (A12) we get the following:

$$ \frac{{\partial^{2} A_{0}^{ - 1} }}{{\partial T^{2} }} = \frac{MTTRS}{{\theta^{\beta } }}\beta \left( {\beta - 1} \right)T^{\beta - 3} + \frac{2b}{T} $$

(B3)

The right hand side of the above equation is positive for $\beta > 1$.

Thus theorem 2 holds for the new quadratic form of $Z\left( T \right)$.

Theorem 3B. Similar to Theorem 2, since all the Eqs. [(A13)–(A17)] in Theorem 3have been derived for a general $Z\left( T \right)$, they remain same for the new quadratic form of $Z\left( T \right)$.

$$ \begin{aligned} \frac{{\partial^{2} \log_{e} \left( {\frac{1}{{P_{0} }}} \right)}}{{\partial T^{2} }} \hfill \\ \quad = \left\{ \begin{aligned} \sum\limits_{i = 1}^{M} {\frac{1}{{\left( {1 + \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times T^{{\beta_{i} - 1}} } \right)^{2} }}} \times \left( {\beta_{i} - 1} \right) \times T^{{\beta_{i} - 3}} \times \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \hfill \\ \left( {1 + \frac{Z}{T}} \right)\left[ {\left( {\beta_{i} - 2} \right)\left( {1 + \frac{Z}{T}} \right) + \left( {2 + \frac{\partial Z}{{\partial T}}} \right) + T^{{\beta_{i} - 1}} \times \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left\{ {1 - \left( {\frac{Z}{T} - \frac{\partial Z}{{\partial T}}} \right)} \right\}} \right] \hfill \\ \end{aligned} \right\} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \frac{1}{{\left( {1 + \frac{Z}{T}} \right)^{2} }} \times \left\{ {\frac{Z}{{T^{3} }} \times \left( {\frac{Z}{T} - \frac{\partial Z}{{\partial T}}} \right) + \left( {1 + \frac{Z}{T}} \right) \times \frac{1}{T} \times \frac{{\partial^{2} Z}}{{\partial T^{2} }}} \right\} \hfill \\ \end{aligned} $$

(A17)

In the Eq. (A17), the last term $\frac{1}{{\left( {1 + \frac{Z}{T}} \right)^{2} }} \times \left\{ {\frac{Z}{{T^{3} }} \times \left( {\frac{Z}{T} - \frac{\partial Z}{{\partial T}}} \right) + \left( {1 + \frac{Z}{T}} \right) \times \frac{1}{T} \times \frac{{\partial^{2} Z}}{{\partial T^{2} }}} \right\}$ with values substituted from (B1) above converts to the following:

$\frac{1}{{\left( {1 + \frac{a}{T} + bT} \right)^{2} }} \times \left( {\frac{{a^{2} }}{{T^{4} }} + \frac{2b}{T} + \frac{2ab}{{T^{2} }} + b^{2} } \right)$. This is greater than 0 since $a > 0;b > 0;T > 0$.

Since $\beta_{i} > 2$ the only other term that we need to consider in Eq. (A17) is $\left\{ {1 - \left( {\frac{Z}{T} - \frac{\partial Z}{{\partial T}}} \right)} \right\}$.

Since $\frac{Z\left( T \right)}{T} < 1$, as per properties of the $Z\left( T \right)$ function, the term $\left\{ {1 - \left( {\frac{Z}{T} - \frac{\partial Z}{{\partial T}}} \right)} \right\}$ is also positive. However, let us check the feasibility of this under the new values in (B1).

On substituting the values from (B1) in $\left\{ {1 - \left( {\frac{Z}{T} - \frac{\partial Z}{{\partial T}}} \right)} \right\}$ we get the following: $\left\{ {1 - \left( {\frac{Z}{T} - \frac{\partial Z}{{\partial T}}} \right)} \right\} > 0 \Leftrightarrow T > \frac{{\sqrt {1 + 4ab} - 1}}{2b} > 0$. Hence $\left\{ {1 - \left( {\frac{Z}{T} - \frac{\partial Z}{{\partial T}}} \right)} \right\} > 0.$

Hence, $\frac{{\partial^{2} \log_{e} \left( {\frac{1}{{P_{0} }}} \right)}}{{\partial T^{2} }} > 0$ in Eq. (A17). Thus Theorem 3 is also satisfied under the new $Z\left( T \right) = a + bT^{2}$.

Theorem 4B. The optimal contract duration $T^{*}$ is obtained from equation the Eq. (B2) as follows:

$$ \frac{MTTRS}{{\theta^{\beta } }}\left( {\beta - 1} \right)\left( {T^{*} } \right)^{\beta } + b\left( {T^{*} } \right)^{2} = a $$

(B4)

This may be further re-written as follows:

$$ \frac{{\left\{ {\left( {\frac{{1 - A_{inh} }}{{A_{inh} }}} \right)\tau + MLDT(s)} \right\}}}{{\theta^{\beta } }}\left( {\beta - 1} \right)\left( {T^{*} } \right)^{\beta } + b\left( {T^{*} } \right)^{2} = a $$

(B5)

By performing $\frac{{\partial T^{*} }}{\partial s}$ on Eq. (B5) and simplifying we obtain the following relation:

$$ \begin{aligned}\frac{{\partial T^{*} }}{\partial s} &= \frac{{ - \left\{ {\frac{{\left( {\beta - 1} \right)}}{{\theta^{\beta } }} \times \left( {T^{*} } \right)^{\beta } \times \frac{\partial MLDT(s)}{{\partial s}}} \right\}}}{{\left\{ {2bT^{*} + \left\{ {\left( {\frac{{1 - A_{inh} }}{{A_{inh} }}} \right)\tau + MLDT(s)} \right\} \times \frac{{\left( {\beta - 1} \right)}}{{\theta^{\beta } }} \times \beta \times \left( {T^{*} } \right)^{\beta - 1} } \right\}}} \ge 0\\ & \quad {\text{When}}\quad \beta > 1;a > 0;b > 0 \end{aligned}$$

By performing $\frac{{\partial T^{*} }}{{\partial A_{inh} }}$ on Eq. (B5) and simplifying we obtain the following relation:

$$\begin{aligned} \frac{{\partial T^{*} }}{{\partial A_{inh} }} &= \frac{{\left\{ {\frac{{\left( {\beta - 1} \right)}}{{\theta^{\beta } }} \times \left( {T^{*} } \right)^{\beta } \times \tau \times \left( {\frac{1}{{A_{inh} }}} \right)^{2} } \right\}}}{{\left\{ {2bT^{*} + \left\{ {\left( {\frac{{1 - A_{inh} }}{{A_{inh} }}} \right)\tau + MLDT(s)} \right\} \times \frac{{\left( {\beta - 1} \right)}}{{\theta^{\beta } }} \times \beta \times \left( {T^{*} } \right)^{\beta - 1} } \right\}}} > 0\\ & \quad {\text{when}}\quad \beta > 1;a > 0;b > 0 \end{aligned}$$

Thus Theorem 4 is also satisfied under the new $Z\left( T \right) = a + bT^{2}$.

Theorem 5B. For Theorem 5we consider $Z(T) = a + b\left( {E\left( T \right)} \right)^{2}$ where $a > 0;b > 0$. Similar to the previous theorems, Eqs. (A19)–(A23) have been derived for a general $Z\left( T \right)$. So we will substitute the new functional form of $Z\left( T \right)$ and its derivatives in Eq. (A23) and draw conclusions from there.

$$ \begin{aligned} \frac{{\partial^{2} A_{0}^{ - 1} }}{{\partial C^{2} }} = \,\left( {\beta - 1} \right) \times \left( {E\left( T \right)} \right)^{\beta - 2} \times \left( {\frac{{\partial^{2} E\left( T \right)}}{{\partial C^{2} }}} \right) \times \frac{MTTRS}{{\theta^{\beta } }} + \frac{1}{E\left( T \right)} \times \frac{{\partial^{2} Z}}{{\partial C^{2} }} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \frac{Z}{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{{\partial^{2} E\left( T \right)}}{{\partial C^{2} }} - \frac{\beta }{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{\partial E\left( T \right)}{{\partial C}} \times \frac{\partial Z}{{\partial C}} + \beta \times \frac{Z}{{\left( {E\left( T \right)} \right)^{3} }} \times \left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right)^{2} \hfill \\ \end{aligned} $$

(A23)

The first term in Eq. (A23) is positive for $0 < \beta < 1$.

Using $\frac{\partial Z}{{\partial E\left( T \right)}} = 2bE\left( T \right);\frac{\partial Z}{{\partial C}} = 2bE\left( T \right)\frac{\partial E\left( T \right)}{{\partial C}};\frac{{\partial^{2} Z}}{{\partial C^{2} }} = 2b\left\{ {E\left( T \right)\frac{{\partial^{2} E\left( T \right)}}{{\partial C^{2} }} + \left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right)^{2} } \right\}$ in the remaining terms of Eq. (A23) and simplifying we get the following:

$$ \frac{2b}{{E\left( T \right)}} \times \left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right)^{2} + \left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right)^{2} \times \frac{\beta }{E\left( T \right)} \times \left( {\frac{a}{{\left( {E\left( T \right)} \right)^{2} }} - b} \right) + \frac{{\partial^{2} E\left( T \right)}}{{\partial C^{2} }} \times \left\{ { - \frac{a}{{\left( {E\left( T \right)} \right)^{2} }} + b} \right\} $$

(B6)

In the Eq. (B6), the first term is positive. The 2^nd and 3^rd terms are positive provided $E\left( T \right) < \sqrt{\frac{a}{b}}$. Consequently, Eq. (A23) is positive under $Z(T) = a + b\left( {E\left( T \right)} \right)^{2}$ provided $E\left( T \right) < \sqrt{\frac{a}{b}}$. This is not a very restrictive condition. Hence, Theorem 5 is also satisfied under the new $Z\left( T \right) = a + b\left( {E\left( T \right)} \right)^{2}$.

Theorem 6B. For Theorem 6we consider $Z(T) = a + b\left( {E\left( T \right)} \right)^{2}$ where $a > 0;b > 0$. Similar to the previous theorems, Eqs. (A24)–(A40) have been derived for a general $Z\left( T \right)$. So we will substitute the new functional form of $Z\left( T \right)$ and its derivatives in Eqs. (A39)–(A40) and draw conclusions from there.

$$ \frac{{\partial^{2} \log_{e} \left( {\frac{1}{{P_{0} }}} \right)}}{{\partial C^{2} }} = A_{1} + A_{2} + A_{3} - A_{4} $$

(A36)

$$ \begin{aligned} A_{1} = \sum\limits_{i = 1}^{M} {\frac{1}{{\left\{ {1 + \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left( {E\left( T \right)} \right)^{{\beta_{i} - 1}} } \right\}^{2} }}} \times \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left( {\beta_{i} - 1} \right) \times \left( {E\left( T \right)} \right)^{{\beta_{i} - 3}} \times \left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right)^{2} \times \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\left\{ {\left( {\beta_{i} - 2} \right) - \left( {E\left( T \right)} \right)^{{\beta_{i} - 1}} \times \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }}} \right\} \hfill \\ \end{aligned} $$

(A33)

$$\begin{aligned} A_{2} &= \frac{1}{{\left( {1 + \frac{Z}{E\left( T \right)}} \right)^{2} }} \\ & \quad \times \left\{ \begin{aligned} - \frac{1}{E\left( T \right)} \times \frac{{\partial^{2} E\left( T \right)}}{{\partial C^{2} }} \times \frac{\partial Z}{{\partial E\left( T \right)}} - \frac{Z}{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{{\partial^{2} E\left( T \right)}}{{\partial C^{2} }} \times \frac{\partial Z}{{\partial E\left( T \right)}} + \frac{1}{E\left( T \right)} \times \frac{{\partial^{2} Z}}{{\partial C^{2} }} \hfill \\ + \frac{2Z}{{\left( {E\left( T \right)} \right)^{3} }} \times \left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right)^{2} + \frac{Z}{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{{\partial^{2} Z}}{{\partial C^{2} }} + \frac{{Z^{2} }}{{\left( {E\left( T \right)} \right)^{4} }} \times \left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right)^{2} \hfill \\ \end{aligned} \right\} \end{aligned}$$

(A34)

$$\begin{aligned} A_{3} &= \left\{ {\sum\limits_{i = 1}^{M} {\frac{1}{{\left\{ {1 + \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left( {E\left( T \right)} \right)^{{\beta_{i} - 1}} } \right\}}}} \times \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left( {\beta_{i} - 1} \right) \times \left( {E\left( T \right)} \right)^{{\beta_{i} - 2}} \times \frac{\partial E\left( T \right)}{{\partial C}}} \right\} \\ &\quad \times \frac{{\left\{ {\frac{\partial Z}{{\partial C}} + 2 \times \frac{\partial E\left( T \right)}{{\partial C}}} \right\}}}{{\left( {E\left( T \right)} \right) \times \left( {1 + \frac{Z}{E\left( T \right)}} \right)}} \end{aligned}$$

(A37)

$$ A_{4} = \left\{ {\frac{1}{{\left( {1 + \frac{Z}{E\left( T \right)}} \right)}} \times \frac{Z}{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{\partial E\left( T \right)}{{\partial C}}} \right\} \times \frac{{\left\{ {\frac{\partial Z}{{\partial C}} + 2 \times \frac{\partial E\left( T \right)}{{\partial C}}} \right\}}}{{\left( {E\left( T \right)} \right) \times \left( {1 + \frac{Z}{E\left( T \right)}} \right)}} $$

(A38)

$$ \begin{aligned}A_{2} - A_{4} &= \frac{1}{{\left( {1 + \frac{Z}{E\left( T \right)}} \right)^{2} }} \\&\quad \times \left\{ \begin{aligned} - \frac{1}{E\left( T \right)} \times \frac{{\partial^{2} E\left( T \right)}}{{\partial C^{2} }} \times \frac{\partial Z}{{\partial E\left( T \right)}} - \frac{Z}{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{{\partial^{2} E\left( T \right)}}{{\partial C^{2} }} \times \frac{\partial Z}{{\partial E\left( T \right)}} + \frac{1}{E\left( T \right)} \times \frac{{\partial^{2} Z}}{{\partial C^{2} }} \hfill \\ + \frac{Z}{{\left( {E\left( T \right)} \right)^{2} }} \times \frac{{\partial^{2} Z}}{{\partial C^{2} }} + \frac{{Z^{2} }}{{\left( {E\left( T \right)} \right)^{4} }} \times \left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right)^{2} - \frac{Z}{{\left( {E\left( T \right)} \right)^{3} }} \times \left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right) \times \left( {\frac{\partial Z}{{\partial C}}} \right) \hfill \\ \end{aligned} \right\} \end{aligned}$$

(A39)

Using $\frac{\partial Z}{{\partial E\left( T \right)}} = 2bE\left( T \right);\frac{\partial Z}{{\partial C}} = 2bE\left( T \right)\frac{\partial E\left( T \right)}{{\partial C}};\frac{{\partial^{2} Z}}{{\partial C^{2} }} = 2b\left\{ {E\left( T \right)\frac{{\partial^{2} E\left( T \right)}}{{\partial C^{2} }} + \left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right)^{2} } \right\}$ in $A_{2} - A_{4}$ and simplifying the following results are seen:

$$ A_{2} - A_{4} = \frac{1}{{\left( {1 + \frac{Z}{E\left( T \right)}} \right)^{2} }} \times \left\{ {\left( {\frac{\partial E\left( T \right)}{{\partial C}}} \right)^{2} \times \left[ {\frac{aZ}{{\left( {E\left( T \right)} \right)^{4} }} + \frac{2b}{{E\left( T \right)}} + \frac{bZ}{{\left( {E\left( T \right)} \right)^{2} }}} \right]} \right\} > 0 $$

Hence $A_{2} - A_{4} > 0$.

$$ \begin{aligned} A_{1} + A_{3} = \left\{ {\sum\limits_{i = 1}^{M} {\frac{1}{{\left\{ {1 + \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left( {E\left( T \right)} \right)^{{\beta_{i} - 1}} } \right\}}}} \times \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left( {\beta_{i} - 1} \right) \times \left( {E\left( T \right)} \right)^{{\beta_{i} - 3}} \times \frac{\partial E\left( T \right)}{{\partial C}}} \right\} \times \hfill \\ \frac{1}{{\left( {1 + \frac{Z}{E\left( T \right)}} \right)}} \times \left\{ \begin{aligned} \left( {1 + \frac{Z}{E\left( T \right)}} \right) \times \left( {\beta_{i} - 2} \right) \times \frac{\partial E\left( T \right)}{{\partial C}} + \left\{ {\frac{\partial Z}{{\partial C}} + 2 \times \frac{\partial E\left( T \right)}{{\partial C}}} \right\} + \hfill \\ \frac{{MTTRS_{i} }}{{\theta_{i}^{{\beta_{i} }} }} \times \left( {E\left( T \right)} \right)^{{\beta_{i} - 1}} \times \left\{ {\frac{\partial Z}{{\partial C}} + \frac{\partial E\left( T \right)}{{\partial C}} \times \left( {1 - \frac{Z}{E\left( T \right)}} \right)} \right\} \hfill \\ \end{aligned} \right\} \hfill \\ \end{aligned} $$

(A40)

We know that $\frac{\partial Z}{{\partial C}} = 2bE\left( T \right)\frac{\partial E\left( T \right)}{{\partial C}};\frac{\partial E\left( T \right)}{{\partial C}} > 0$. We also consider that $\beta_{i} > 2$. Looking at the feasibility of $\frac{Z}{E\left( T \right)} < 1$ we find that this is true provided $E\left( T \right) > \frac{{\sqrt {1 + 4ab} - 1}}{2b} > 0$ which is reasonable.

Since all the terms in Eq. (A40) are positive, it can be seen that $A_{1} + A_{3} > 0$.

Thus $\frac{{\partial^{2} \log_{e} \left( {\frac{1}{{P_{0} }}} \right)}}{{\partial C^{2} }} > 0$ and Theorem 6 is true for $Z(T) = a + b\left( {E\left( T \right)} \right)^{2}$.