A comparison of fixed and variable capacity-addition policies for outpatient capacity allocation

Abstract

Residents of mainland China always come to the first-class hospitals. Facing limitations to medical resources, hospital managers consider adding some potential capacity beyond the regular daily capacity to meet demands and improve the profit. This paper systematically studies the capacity-addition policy, and compares two kinds of them: fixed capacity-addition policy (F-CAP) and variable capacity-addition policy (V-CAP). The former sets the additional capacity by simple calculations according to experiences or history statistics. Under the V-CAP policy, the additional capacity is determined by solving an optimization model rather than a constant. The V-CAP definitely achieves the optimal expected profit, which can be regarded as the benchmark that leads to the best improvement. Since the F-CAP is widely used in many first-class hospitals in practice, this paper attempts to explore in which situations the F-CAP can achieve a similar improvement of the V-CAP, and in what environments the additional capacity should be a variable. In addition to focusing on additional capacity, both policies allocate the regular capacity to two types of patient: routine patients and same-day patients, who have different no-show probabilities. We formulate linear integer programming models of F-CAP and V-CAP to maximize the expected profit. Several propositions and corollaries are proved to cut off the solution space and accelerate the search process. The optimal additional capacity can be directly determined by these properties, and the optimal value is non-decreasing with the regular capacity allocated to routine patients. Numerical experiments indicate that the optimal total supply of regular capacity and additional capacity is always less than the expected total demand. The F-CAP is recommended to the environments with low no-show probabilities of routine patients and same-day patients, moderate expected total demand, correlation coefficient of demands and regular capacity, where the reduction of the expected profit is less than 5% of the F-CAP compared to the V-CAP.

Appendix

Proofs of the theorems and corollaries in Sects. 4 and 5

1. Proof of Proposition 1

If there is an integer ${C_R} \in \left\{ {1,2, \ldots ,N - 1} \right\}$ that satisfies $E_\pi ^F\left( {{C_R}} \right) - E_\pi ^F\left( {{C_R}\! -\! 1} \right) \! > 0$ and $E_\pi ^F\left( {{C_R} + 1} \right) - E_\pi ^F\left( {{C_R}} \right) < 0$, ${C_R}$ is a local optimal solution to the programming P-F. There are equations (A.1) to (A.3),

$$\begin{aligned}&E_R^F\left( {{C_R} + 1} \right) - E_R^F\left( {{C_R}} \right) = {P_R}\left( {{D_R} > {C_R} - 1} \right) \end{aligned}$$

(A.1)

$$\begin{aligned}&E_O^F\left( {{C_R} + 1} \right) - E_O^F\left( {{C_R}} \right) = - P\left( {{D_R}> {C_R},{D_O} > N + {N_A} - {C_R} - 1} \right) \nonumber \\ \end{aligned}$$

(A.2)

$$\begin{aligned}&E_c^F\left( {{C_R} + 1} \right) - E_c^F\left( {{C_R}} \right) = P\left( {{D_R}> {C_R},{D_O}> N - {C_R} - 1} \right) \nonumber \\&\quad -\,P\left( {{D_R}> {C_R},{D_O} > N + {N_A} - {C_R} - 1} \right) \end{aligned}$$

(A.3)

Therefore, $E_\pi ^F\left( {{C_R}} \right) - E_\pi ^F\left( {{C_R} - 1} \right) > 0$ and $E_\pi ^F\left( {{C_R} + 1} \right) - E_\pi ^F\left( {{C_R}} \right) < 0$ are equivalent to inequalities (A.4) and (A.5).

$$\begin{aligned}&\left( {1 - {\theta _R} + {\alpha _R}} \right) \, {P_R}\left( {{D_R}> {C_R} - 1} \right) \nonumber \\&\quad>\beta \, P\left( {{D_R}> {C_R} - 1,{D_O}> N - {C_R}} \right) \nonumber \\&\qquad + \left( {1 - {\theta _O} + {\alpha _O} - \beta } \right) \, P\left( {{D_R}> {C_R} - 1,{D_O} > N + {N_A} - {C_R}} \right) \end{aligned}$$

(A.4)

$$\begin{aligned}&\left( {1 - {\theta _R} + {\alpha _R}} \right) \, {P_R}\left( {{D_R}> {C_R}} \right) \nonumber \\&\quad < \beta \, P\left( {{D_R}> {C_R},{D_O}> N - {C_R} - 1} \right) \nonumber \\&\qquad \mathrm{{ + }} \left( {1 - {\theta _O} + {\alpha _O} - \beta } \right) \, P\left( {{D_R}> {C_R},{D_O} > N + {N_A} - {C_R} - 1} \right) \end{aligned}$$

(A.5)

Since $\left( {1 - {\theta _R} + {\alpha _R}} \right) \, {P_R}\left( {{D_R}> {C_R} - 1} \right) > 0$ and $\left( {1 - {\theta _R} + {\alpha _R}} \right) \, {P_R}\left( {{D_R}> {C_R}} \right) > 0$, divide the two, respectively, in (A.4) and (A.5). Then, the left-hand sides are equal to 1, and the right-hand sides are conditional probabilities multiplied by ratios of revenue and cost weights. Hence inequalities (A.4) and (A.5) are equivalent to inequalities (15) and (16) in Proposition 1. If an integer ${C_R} \in \left\{ {1,2, \ldots ,N - 1} \right\}$ satisfies (15) and (16), it is a local optimal solution.

If ${C_R}=0$ is a local optimal solution, then $E_\pi ^F\left( 1 \right) - E_\pi ^F\left( 0 \right) < 0$, which is equivalent to $E_\pi ^F\left( {{C_R} + 1} \right) - E_\pi ^F\left( {{C_R}} \right) < 0$ when ${C_R}=0$, is required. Thus, if inequality (A.5) holds when ${C_R}=0$, then ${C_R}=0$ is a local optimal solution.

If ${C_R}=N$ is a local optimal solution, then $E_\pi ^F\left( N \right) - E_\pi ^F\left( {N - 1} \right) > 0$, which is equivalent to $E_\pi ^F\left( {{C_R}} \right) - E_\pi ^F\left( {{C_R} - 1} \right) > 0$ when ${C_R}=N$, is required. Thus, if inequality (A.4) holds when ${C_R}=N$, then ${C_R}=N$ is a local optimal solution.

2. Proof of Corollary 1

Let ${i^ * }$ denote the local optimal solution satisfying Proposition 1. When $P\left( {{D_O}> H}\right. \left. {- {C_R}|{D_R}> {C_R}} \right) < P\left( {{D_O}> H - {C_R} - 1|{D_R} > {C_R} + 1} \right)$

(1). If ${i^ * } \in \left\{ {1,2, \ldots ,N - 1} \right\}$, then we have

$$\begin{aligned}&\displaystyle \frac{{1 - {\theta _O} + {\alpha _O} - \beta }}{{1 - {\theta _R} + {\alpha _R}}} \, P\left( {{D_O}> N + {N_A} - {i^ * }|{D_R}> {i^ * } - 1} \right) \nonumber \\&\quad \displaystyle +\,\frac{\beta }{{1 - {\theta _R} + {\alpha _R}}} \, P\left( {{D_O}> N - {i^ * }|{D_R} > {i^ * } - 1} \right) < 1 \end{aligned}$$

(A.6)

$$\begin{aligned}&\displaystyle \frac{{1 - {\theta _O} + {\alpha _O} - \beta }}{{1 - {\theta _R} + {\alpha _R}}} \, P\left( {{D_O}> N + {N_A} - {i^ * } - 1|{D_R}> {i^ * }} \right) \nonumber \\&\quad \displaystyle +\,\frac{\beta }{{1 - {\theta _R} + {\alpha _R}}} \, P\left( {{D_O}> N - {i^ * } - 1|{D_R}> {i^ * }} \right) > 1 \end{aligned}$$

(A.7)

1) For all $i \in \left\{ {0,1, \ldots ,{i^ * } - 1} \right\}$, the sum of weighted conditional probabilities holds that

$$\begin{aligned}&\displaystyle \frac{{1 - {\theta _O} + {\alpha _O} - \beta }}{{1 - {\theta _R} + {\alpha _R}}} \, P\left( {{D_O}> N + {N_A} - i|{D_R}> i - 1} \right) \nonumber \\&\quad \displaystyle + \frac{\beta }{{1 - {\theta _R} + {\alpha _R}}} \, P\left( {{D_O}> N - i|{D_R}> i - 1} \right) \nonumber \\&\displaystyle< \frac{{1 - {\theta _O} + {\alpha _O} - \beta }}{{1 - {\theta _R} + {\alpha _R}}} \, P\left( {{D_O}> N + {N_A} - {i^ * }|{D_R}> {i^ * } - 1} \right) \nonumber \\&\quad \displaystyle +\,\frac{\beta }{{1 - {\theta _R} + {\alpha _R}}} \, P\left( {{D_O}> N - {i^ * }|{D_R} > {i^ * } - 1} \right) < 1 \end{aligned}$$

(A.8)

Hence $i \in \left\{ {1,2, \ldots ,{i^ * } - 1} \right\}$ and $i=0$ cannot satisfy conditions (i) and (ii), and $i \in \left\{ {0,1, \ldots ,{i^ * } - 1} \right\}$ are not local optimal solutions;

2) For all $i \in \left\{ {{i^ * } + 1, \ldots ,N - 1,N} \right\}$, the sum of weighted conditional probabilities holds that

$$\begin{aligned}&\displaystyle \frac{{1 - {\theta _O} + {\alpha _O} - \beta }}{{1 - {\theta _R} + {\alpha _R}}} \, P\left( {{D_O}> N + {N_A} - i - 1|{D_R}> i} \right) \nonumber \\&\displaystyle \quad +\,\frac{\beta }{{1 - {\theta _R} + {\alpha _R}}} \, P\left( {{D_O}> N - i - 1|{D_R}> i} \right) \nonumber \\&\displaystyle> \frac{{1 - {\theta _O} + {\alpha _O} - \beta }}{{1 - {\theta _R} + {\alpha _R}}} \, P\left( {{D_O}> N + {N_A} - {i^ * } - 1|{D_R}> {i^ * }} \right) \nonumber \\&\displaystyle \quad +\,\frac{\beta }{{1 - {\theta _R} + {\alpha _R}}} \, P\left( {{D_O}> N - {i^ * } - 1|{D_R}> {i^ * }} \right) > 1 \end{aligned}$$

(A.9)

Therefore, $i \in \left\{ {{i^ * } + 1,{i^ * } + 2, \ldots ,N - 1} \right\}$ and $i=N$ cannot satisfy conditions (i) and (iii), and $i \in \left\{ {{i^ * } + 1, \ldots ,N - 1,N} \right\}$ are not local optimal solutions.

So $i^*$is the only optimal solution for all $i \in \left\{ {1,2, \ldots ,N} \right\}$.

(2). If $i^*=0$, we have inequality (A.7). All $i \in \left\{ {1,2, \ldots ,N} \right\}$ satisfy inequality (A.9). Hence $i \in \left\{ {1,2, \ldots ,N - 1} \right\}$ and $i=N$ cannot satisfy conditions (i) and (iii), and $i \in \left\{ {1,2, \ldots ,N} \right\}$ are not local optimal solutions.

(3). If $i^*=N$, we have inequality (A.6). All $i \in \left\{ {0,1, \ldots ,N-1} \right\}$ satisfy inequality (A.8). Hence $i \in \left\{ {1,2, \ldots ,N - 1} \right\}$ and $i=0$ cannot satisfy conditions (i) and (ii), and $i \in \left\{ {0,1, \ldots ,N-1} \right\}$ are not local optimal solutions.

In summary, if $P\left( {{D_O}> H - {C_R}|{D_R}> {C_R}} \right) < P\left( {{D_O}> H - {C_R} - 1|{D_R} >}\right. \left. {{C_R} + 1} \right)$, the solution satisfying the conditions in Proposition 1 is the only optimal solution in the feasible space.

3. Proof of Proposition 2

If the demands of two types of patients are independent, inequalities (15) and (16) are equivalent to (A.10) and (A.11).

$$\begin{aligned}&\displaystyle \frac{{1 - {\theta _O} + {\alpha _O} - \beta }}{{1 - {\theta _R} + {\alpha _R}}} \, {P_O}\left( {{D_O}> N + {N_A} - {C_R}} \right) \nonumber \\&\displaystyle \quad + \frac{\beta }{{1 - {\theta _R} + {\alpha _R}}} \, {P_O}\left( {{D_O} > N - {C_R}} \right) < 1 \end{aligned}$$

(A.10)

$$\begin{aligned}&\displaystyle \frac{{1 - {\theta _O} + {\alpha _O} - \beta }}{{1 - {\theta _R} + {\alpha _R}}} \, {P_O}\left( {{D_O}> N + {N_A} - {C_R} - 1} \right) \nonumber \\&\displaystyle \quad + \frac{\beta }{{1 - {\theta _R} + {\alpha _R}}} \, {P_O}\left( {{D_O}> N - {C_R} - 1} \right) > 1 \end{aligned}$$

(A.11)

It is obvious that ${P_O}\left( {{D_O} > H - {C_R}} \right)$ increases monotonically with ${C_R}$. As in the proof of Corollary 1, there is only one solution that satisfies the optimal conditions, such as in (A.10) and (A.11). Therefore, when $E_\pi ^F\left( {C_R^ * } \right) - E_\pi ^F\left( {C_R^ * - 1} \right) > 0$ and $E_\pi ^F\left( {C_R^ * + 1} \right) - E_\pi ^F\left( {C_R^ * } \right) < 0$, the expected profit decreases and does not increase again for all ${C_R} \in \left\{ {C_R^ * + 1,C_R^ * + 2, \ldots ,N} \right\}$. Hence $E_\pi ^F\left( {{C_R}} \right)$ is a unimodal function of ${C_R}$.

4. Proof of Corollary 2

Inequalities (A.10) and (A.11) are equivalent to inequalities (A.13) and (A.12), respectively, by multiplying $1 - {\theta _R} + {\alpha _R}$ to the right-hand side, and conversely presenting the probabilities that same-day patient demand is not greater than the corresponding supply.

$$\begin{aligned}&\left( {1 - {\theta _O} + {\alpha _O} - \beta } \right) \, {P_O}\left( {{D_O} \le N + {N_A} - {C_R} - 1} \right) \nonumber \\&\quad +\,\beta \, {P_O}\left( {{D_O} \le N - {C_R} - 1} \right) < {\theta _R} - {\theta _O} + {\alpha _O} - {\alpha _R} \end{aligned}$$

(A.12)

$$\begin{aligned}&\left( {1 - {\theta _O} + {\alpha _O} - \beta } \right) \, {P_O}\left( {{D_O} \le N + {N_A} - {C_R}} \right) \nonumber \\&\quad +\,\beta \, {P_O}\left( {{D_O} \le N - {C_R}} \right) > {\theta _R} - {\theta _O} + {\alpha _O} - {\alpha _R} \end{aligned}$$

(A.13)

Therefore, similar to the conditions in Proposition 1, when the demands of two types of patients are independent, the conditions determining whether a solution is global optimal are:

• i. when ${C_R} \in \left\{ {1,2, \ldots ,N - 1} \right\}$, both inequalities (A.12) and (A.13) hold;

• ii. when ${C_R}=0$, inequality (A.12) holds;

• iii. when ${C_R}=N$, inequality (A.13) holds.

5. Proof of Proposition3

For a certain value ${N_R}$, when the additional capacity increase from ${C_A}$ to ${C_A}+1$, the increment in the expected number of same-day appointments and expected doctors overwork are

$$\begin{aligned} \varDelta E_O^D\left( {{C_A},{N_R}} \right)= & {} E_O^D\left( {{C_A} + 1,{N_R}} \right) - E_O^D\left( {{C_A},{N_R}} \right) \nonumber \\= & {} P\left( {{D_R} \le {N_R},{D_O}> N + {C_A} - {d_R}} \right) \nonumber \\&+\,P\left( {{D_R}> {N_R},{D_O} > N + {C_A} - {N_R}} \right) \end{aligned}$$

(A.14)

$$\begin{aligned} \varDelta E_c^D\left( {{C_A},{N_R}} \right)= & {} E_c^D\left( {{C_A} + 1,{N_R}} \right) - E_c^D\left( {{C_A},{N_R}} \right) \nonumber \\= & {} P\left( {{D_R} \le {N_R},{D_O}> N + {C_A} - {d_R}} \right) \nonumber \\&+\,P\left( {{D_R}> {N_R},{D_O} > N + {C_A} - {N_R}} \right) \end{aligned}$$

(A.15)

Since $\varDelta E_O^D\left( {{C_A},{N_R}} \right) > 0$ and $\varDelta E_c^D\left( {{C_A},{N_R}} \right) > 0$, $\varDelta E_O^D\left( {{C_A},{N_R}} \right)$ and $\varDelta E_c^D\left( {{C_A},{N_R}} \right)$ monotonically increase with ${C_A}$. Additionally, $\varDelta E_O^D\left( {{C_A},{N_R}} \right) = \varDelta E_c^D\left( {{C_A},{N_R}} \right)$, and intuitively, the increment monotonically decreases with ${C_A}$. Therefore, $\varDelta E_O^D\left( {{C_A},{N_R}} \right)$ and $\varDelta E_c^D\left( {{C_A},{N_R}} \right)$ have the same increment with monotonic decrease.

6. Proof of Proposition 4

For a certain value of ${N_R}$, the increment in expected profit from ${C_A}$ to ${C_A}+1$ is shown as expression (A.16).

$$\begin{aligned} \varDelta E_\pi ^D\left( {{C_A},{N_R}} \right)= & {} E_\pi ^D\left( {{C_A} + 1,{N_R}} \right) - E_\pi ^D\left( {{C_A},{N_R}} \right) \nonumber \\= & {} \left( {1 - {\theta _O} + {\alpha _O}} \right) \, \varDelta E_O^D\left( {{C_A},{N_R}} \right) - \beta \, \varDelta E_c^D\left( {{C_A},{N_R}} \right) - \gamma \nonumber \\= & {} - \gamma + \left( {1 - {\theta _O} + {\alpha _O} - \beta } \right) \nonumber \\&\times \left\{ {P\left( {{D_R} \le {N_R},{D_O}> N + {C_A} - {d_R}} \right) } \right. \nonumber \\&\left. { + P\left( {{D_R}> {N_R}, {D_O} > N + {C_A} - {N_R}} \right) } \right\} \end{aligned}$$

(A.16)

$\left\{ {P\left( {{D_R} \le {N_R},{D_O}> N + {C_A} - {d_R}} \right) + P\left( {{D_R}> {N_R},{D_O} > N + {C_A} - {N_R}} \right) } \right\}$ decreases with ${C_A}$, $\mathrm{{ }}\left( {1 - {\theta _O} + {\alpha _O} - \beta } \right) > 0$, and therefore, if $\varDelta E_\pi ^D\left( {{C_A},{N_R}} \right) < 0$ for all ${C_A} \ge 0$, then

$$\begin{aligned}&\varDelta E_\pi ^D\left( {{C_A} + 1,{N_R}} \right) \nonumber \\&\quad = - \gamma + \left( {1 - {\theta _O} + {\alpha _O} - \beta } \right) \times \left\{ {P\left( {{D_R} \le {N_R},{D_O}> N + {C_A} + 1 - {d_R}} \right) } \right. \nonumber \\&\qquad \left. { + P\left( {{D_R}> {N_R},{D_O}> N + {C_A} + 1 - {N_R}} \right) } \right\} \nonumber \\&\quad< - \gamma + \left( {1 - {\theta _O} + {\alpha _O} - \beta } \right) \times \left\{ {P\left( {{D_R} \le {N_R},{D_O}> N + {C_A} - {d_R}} \right) } \right. \nonumber \\&\qquad \left. { + P\left( {{D_R}> {N_R},{D_O} > N + {C_A} - {N_R}} \right) } \right\} \nonumber \\&\quad = \varDelta E_\pi ^D\left( {{C_A},{N_R}} \right) < 0 \end{aligned}$$

(A.17)

Once the increment in $E_\pi ^D\left( {{C_A},{N_R}} \right)$ becomes smaller than 0, it is always negative.

7. Proof of Corollary 3

For a certain value of ${N_R}$, (1) if the global optimal $C_A^{ * {N_R}} > 0$, then $E_\pi ^D\left( {C_A^{ * {N_R}},{N_R}} \right) - E_\pi ^D\left( {C_A^{ * {N_R}} - 1,{N_R}} \right) > 0$ and $E_\pi ^D\left( {C_A^{ * {N_R}} + 1,{N_R}} \right) - E_\pi ^D\left( {C_A^{ * {N_R}}},\right. \left. {{N_R}} \right) < 0$, which are equivalent to

$$\begin{aligned}&E_\pi ^D\left( {C_A^{ * {N_R}},{N_R}} \right) - E_\pi ^D\left( {C_A^{ * {N_R}} - 1,{N_R}} \right) \nonumber \\&= \left( {1 - {\theta _O} + {\alpha _O} - \beta } \right) \, \left\{ {P\left( {{D_R} \le {N_R},{D_O}> N + C_A^{ * {N_R}} - 1 - {d_R}} \right) } \right. \nonumber \\&\qquad \left. { +\,P\left( {{D_R}> {N_R},{D_O}> N + C_A^{ * {N_R}} - 1 - {N_R}} \right) } \right\} - \gamma > 0 \end{aligned}$$

(A.18)

$$\begin{aligned}&E_\pi ^D\left( {C_A^{ * {N_R}} + 1,{N_R}} \right) - E_\pi ^D\left( {C_A^{ * {N_R}},{N_R}} \right) \nonumber \\&= \left( {1 - {\theta _O} + {\alpha _O} - \beta } \right) \, \left\{ {P\left( {{D_R} \le {N_R},{D_O}> N + C_A^{ * {N_R}} - {d_R}} \right) } \right. \nonumber \\&\qquad \left. { +\,P\left( {{D_R}> {N_R},{D_O} > N + C_A^{ * {N_R}} - {N_R}} \right) } \right\} - \gamma < 0 \end{aligned}$$

(A.19)

Therefore, the optimal conditions hold inequalities (A.20) and (A.21).

$$\begin{aligned}&P\left( {{D_R} \le {N_R},{D_O}> N + C_A^{ * {N_R}} - 1 - {d_R}} \right) \nonumber \\&\qquad +\,P\left( {{D_R}> {N_R},{D_O}> N + C_A^{ * {N_R}} - 1 - {N_R}} \right) \nonumber \\&\quad > {\gamma / {\left( {1 - {\theta _O} + {\alpha _O} - \beta } \right) }} \end{aligned}$$

(A.20)

$$\begin{aligned}&P\left( {{D_R} \le {N_R},{D_O}> N + C_A^{ * {N_R}} - {d_R}} \right) \nonumber \\&\qquad + P\left( {{D_R}> {N_R},{D_O} > N + C_A^{ * {N_R}} - {N_R}} \right) \nonumber \\&\quad < {\gamma / {\left( {1 - {\theta _O} + {\alpha _O} - \beta } \right) }} \end{aligned}$$

(A.21)

(2) if the global optimal $C_A^{ * {N_R}} = 0$, then $E_\pi ^D\left( {{1^{{N_R}}},{N_R}} \right) - E_\pi ^D\left( {{0^{{N_R}}},{N_R}} \right) < 0$, which is equivalent to inequality (A.21) by substituting $C_A^{ * {N_R}} = 0$.

8. Proof of Proposition 5

Denote $\varDelta \left( {{C_A},{N_R}} \right) = \varDelta E_O^D\left( {{C_A},{N_R}} \right) = \varDelta E_c^D\left( {{C_A},{N_R}} \right)$. According to the optimal conditions in Corollary 3, if $C_A^{ * {N_R}}$ is an optimal solution of additional capacity under a certain ${N_R}$,

(1) when $C_A^{ * {N_R}} > 0$, $\varDelta \left( {C_A^{ * {N_R}},{N_R}} \right) < {\gamma / {\left( {1 - {\theta _O} + {\alpha _O} - \beta } \right) }}$ and $\varDelta \left( {C_A^{ * {N_R}} - 1,}\right. \left. {N_R} \right) > {\gamma / {\left( {1 - {\theta _O} + {\alpha _O} - \beta } \right) }}$. Further, relationships also hold that

$$\begin{aligned}&\varDelta \left( {C_A^{ * {N_R}} + 1,{N_R} + 1} \right) - \varDelta \left( {C_A^{ * {N_R}},{N_R}} \right) \nonumber \\&= P\left( {{D_R} \le {N_R} + 1,{D_O}> N + C_A^{ * {N_R}} + 1 - {d_R}} \right) \nonumber \\&\quad +\,P\left( {{D_R}> {N_R} + 1,{D_O}> N + C_A^{ * {N_R}} - {N_R}} \right) \nonumber \\&\quad -\,P\left( {{D_R} \le {N_R},{D_O}> N + C_A^{ * {N_R}} - {d_R}} \right) \nonumber \\&\quad -\,P\left( {{D_R}> {N_R},{D_O} > N + C_A^{ * {N_R}} - {N_R}} \right) \nonumber \\&= - P\left( {{D_R} \le {N_R},{D_O} = N + C_A^{ * {N_R}} + 1 - {d_R}} \right) < 0\end{aligned}$$

(A.22)

$$\begin{aligned}&\varDelta \left( {C_A^{ * {N_R}} - 1,{N_R} + 1} \right) - \varDelta \left( {C_A^{ * {N_R}} - 1,{N_R}} \right) \nonumber \\&= P\left( {{D_R} \le {N_R} + 1,{D_O}> N + C_A^{ * {N_R}} - 1 - {d_R}} \right) \nonumber \\&\quad +\,P\left( {{D_R}> {N_R} + 1,{D_O}> N + C_A^{ * {N_R}} - {N_R} - 2} \right) \nonumber \\&\quad -\,P\left( {{D_R} \le {N_R},{D_O}> N + C_A^{ * {N_R}} - 1 - {d_R}} \right) \nonumber \\&\quad -\,P\left( {{D_R}> {N_R},{D_O}> N + C_A^{ * {N_R}} - 1 - {N_R}} \right) \nonumber \\&= P\left( {{D_R}> {N_R},{D_O} = N + C_A^{ * {N_R}} - {N_R} - 1} \right) > 0 \end{aligned}$$

(A.23)

$$\begin{aligned}&\varDelta \left( {C_A^{ * {N_R}},{N_R} + 1} \right) - \varDelta \left( {C_A^{ * {N_R}},{N_R}} \right) \nonumber \\&= P\left( {{D_R} \le {N_R} + 1,{D_O}> N + C_A^{ * {N_R}} - {d_R}} \right) \nonumber \\&\quad +\,P\left( {{D_R}> {N_R} + 1,{D_O}> N + C_A^{ * {N_R}} - {N_R} - 1} \right) \nonumber \\&\quad -\,P\left( {{D_R} \le {N_R},{D_O}> N + C_A^{ * {N_R}} - {d_R}} \right) \nonumber \\&\quad -\,P\left( {{D_R}> {N_R},{D_O}> N + C_A^{ * {N_R}} - {N_R}} \right) \nonumber \\&= P\left( {{D_R}> {N_R},{D_O} = N + C_A^{ * {N_R}} - {N_R}} \right) > 0 \end{aligned}$$

(A.24)

$$\begin{aligned}&\varDelta \left( {C_A^{ * {N_R}},{N_R} + 1} \right) - \varDelta \left( {C_A^{ * {N_R}} - 1,{N_R}} \right) \nonumber \\&= P\left( {{D_R} \le {N_R} + 1,{D_O}> N + C_A^{ * {N_R}} - {d_R}} \right) \nonumber \\&\quad +\,P\left( {{D_R}> {N_R} + 1,{D_O}> N + C_A^{ * {N_R}} - {N_R} - 1} \right) \nonumber \\&\quad -\,P\left( {{D_R} \le {N_R},{D_O}> N + C_A^{ * {N_R}} - {d_R} - 1} \right) \nonumber \\&\quad -\,P\left( {{D_R}> {N_R},{D_O} > N + C_A^{ * {N_R}} - {N_R} - 1} \right) \nonumber \\&= -\,P\left( {{D_R} \le {N_R},{D_O} = N + C_A^{ * {N_R}} - {d_R}} \right) < 0 \end{aligned}$$

(A.25)

Thus, we have

$$\begin{aligned}&\varDelta \left( {C_A^{ * {N_R}} + 1,{N_R} + 1} \right)< \varDelta \left( {C_A^{ * {N_R}},{N_R}} \right) < {\gamma / {\left( {1 - {\theta _O} + {\alpha _O} - \beta } \right) }} \end{aligned}$$

(A.26)

$$\begin{aligned}&\varDelta \left( {C_A^{ * {N_R}} - 1,{N_R} + 1} \right) \nonumber \\&\quad> \varDelta \left( {C_A^{ * {N_R}} - 1,{N_R}} \right) > {\gamma / {\left( {1 - {\theta _O} + {\alpha _O} - \beta } \right) }} \end{aligned}$$

(A.27)

$$\begin{aligned}&\varDelta \left( {C_A^{ * {N_R}},{N_R}} \right)< \varDelta \left( {C_A^{ * {N_R}},{N_R} + 1} \right) < \varDelta \left( {C_A^{ * {N_R}} - 1,{N_R}} \right) \end{aligned}$$

(A.28)

i) if $\varDelta \left( {C_A^{ * {N_R}},{N_R} + 1} \right) > {\gamma / {\left( {1 - {\theta _O} + {\alpha _O} - \beta } \right) }}$, together with inequality (A.26), the optimal additional capacity under ${N_R}$ holds that $C_A^{ * \left( {{N_R} + 1} \right) } = C_A^{ * {N_R}} + 1$, as Corollary 3;

ii) if $\varDelta \left( {C_A^{ * {N_R}},{N_R} + 1} \right) < {\gamma / {\left( {1 - {\theta _O} + {\alpha _O} - \beta } \right) }}$, together with inequality (A.27), $C_A^{ * \left( {{N_R} + 1} \right) } = C_A^{ * {N_R}}$.

(2) when $C_A^{ * {N_R}} = 0$, $\varDelta \left( {C_A^{ * {N_R}},{N_R}} \right) < {\gamma / {\left( {1 - {\theta _O} + {\alpha _O} - \beta } \right) }}$. It holds that $\varDelta \left( {C_A^{ * {N_R}},{N_R} + 1} \right) > \varDelta \left( {C_A^{ * {N_R}},{N_R}} \right)$, and $\varDelta \left( {C_A^{ * {N_R}} + 1,{N_R} + 1} \right)< \varDelta \left( {C_A^{ * {N_R}},}\right. \left. {{N_R}} \right) < {\gamma /{\left( {1 - {\theta _O} + {\alpha _O} - \beta } \right) }}$.

i) if $\varDelta \left( {C_A^{ * {N_R}},{N_R} + 1} \right) > {\gamma / {\left( {1 - {\theta _O} + {\alpha _O} - \beta } \right) }}$, together with inequality (A.26), $C_A^{ * \left( {{N_R} + 1} \right) } = C_A^{ * {N_R}} + 1$;

ii) if $\varDelta \left( {C_A^{ * {N_R}},{N_R} + 1} \right) < {\gamma / {\left( {1 - {\theta _O} + {\alpha _O} - \beta } \right) }}$, together with inequality (A.27), $C_A^{ * \left( {{N_R} + 1} \right) } = C_A^{ * {N_R}}$.

In summary, $C_A^{ * \left( {{N_R} + 1} \right) } \ge C_A^{ * {N_R}}$.