Admission Control Policies for Surgery Patients

Abstract

In China, day surgery has been promoted because its operation time and post-operative hospital stay are shorter than those of elective surgery. Day surgery can speed up the turnover of beds and operation rooms. Usually, the conditions of elective surgery patients are more complicated than those of day surgery patients. The development of the discipline, which means that the hospital has improved the skills of the doctors and the ability of doctors to cope with serious diseases and has increased the overall medical level of the hospital, requires surgeons to operate in some complicated elective surgeries. In the case of operating rooms and beds in short supply, there is a trade-off between the promotion of day surgery and the development of the discipline. Day surgery is relatively uncomplicated, but it requires more highly qualified surgeons. However, the development of the discipline requires surgeons to take on some complicated elective surgeris. Moreover, according to the notion of grading treatment, class-A tertiary hospitals are more suitable for patients with relatively complicated and serious conditions. In the emerging context of day surgery, highly qualified surgeons need to perform both day surgeries and elective surgeries. This paper studied how to control the admission of surgery patients. We take into account both day surgery promotion and discipline development in decision-making. A dynamic programming model was built for admission control, and a γ-adjust-threshold heuristic policy was proposed. We then compared the heuristic policy to three other policies through simulation. The results show that our heuristic policy outperforms the hospital’s target policy.

Appendix

Proposition 1

For each given t, R _t (n) is monotonically non-decreasing and concave in n;

Proof

• (1) For each given t, we prove R _t (n) is monotonically non-decreasing in n first.

When t = 0, R ₀(n) = 0, the conclusion is clear.

When t ≠ 0, we assume R _t − 1(n) is monotonically non-decreasing in n.

By Eq. (1) we have

$$ \begin{array}{l}{R}_t(n)-{R}_t\left(n-1\right)=\left(1-{\lambda}_t^{(1)}-{\lambda}_t^{(2)}\right)\left[{R}_{t-1}(n)-{R}_{t-1}\left(n-1\right)\right]+\sum_{i=1}^2{\lambda}_t^{(i)}\sum_kp\left({du}_t^{(i)}={d}_i^{(k)}\right)\hfill \\ {}\left[ \max \left\{{R}_{t-1}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right),{R}_{t-1}(n)-{c}_i\right\}- \max \left\{{R}_{t-1}\left(n-{d}_i^{(k)}-1\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right),{R}_{t-1}\left(n-1\right)-{c}_i\right\}\right]\hfill \\ {}=\left(1-{\lambda}_t^{(1)}-{\lambda}_t^{(2)}\right)\left[{R}_{t-1}(n)-{R}_{t-1}\left(n-1\right)\right]+\sum_{i=1}^2{\lambda}_t^{(i)}\sum_kp\left({du}_t^{(i)}={d}_i^{(k)}\right)\left[{H}_{t-1}(n)-{H}_{t-1}\left(n-1\right)\right]\hfill \end{array} $$

(4)

Let ${H}_{t-1}(n)-{H}_{t-1}\left(n-1\right)= \max \left\{{R}_{t-1}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right),{R}_{t-1}(n)-{c}_i\right\}- \max \left\{{R}_{t-1}\left(n-{d}_i^{(k)}-1\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right),{R}_{t-1}\left(n-1\right)-{c}_i\right\}$.

We discuss the values of H _t − 1(n) − H _t − 1(n − 1) as below:

• (a) ${R}_{t-1}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\ge {R}_{t-1}(n)-{c}_i$ and ${R}_{t-1}\left(n-{d}_i^{(k)}-1\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\ge {R}_{t-1}\left(n-1\right)-{c}_i$;

${H}_{t-1}(n)-{H}_{t-1}\left(n-1\right)={R}_{t-1}\left(n-{d}_i^{(k)}\right)-{R}_{t-1}\left(n-{d}_i^{(k)}-1\right)$;

• ∵ R _t − 1(n) is monotonically non-decreasing in n

• ∴${R}_{t-1}\left(n-{d}_i^{(k)}\right)-{R}_{t-1}\left(n-{d}_i^{(k)}\hbox{-} 1\right)\ge 0$ holds.

• ∴${H}_{t-1}(n)-{H}_{t-1}\left(n-1\right)={R}_{t-1}\left(n-{d}_i^{(k)}\right)-{R}_{t-1}\left(n-{d}_i^{(k)}-1\right)\ge 0$ holds.

• (b) ${R}_{t-1}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\ge {R}_{t-1}(n)-{c}_i$ and ${R}_{t-1}\left(n-{d}_i^{(k)}-1\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\le {R}_{t-1}\left(n-1\right)-{c}_i$;

• ∵R _t − 1(n) is monotonically non-decreasing in n,

• ∴R _t − 1(n) − c _i  ≥ R _t − 1(n − 1) − c _i holds;

• ∴${R}_{t-1}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\ge {R}_{t-1}(n)-{c}_i\ge {R}_{t-1}\left(n-1\right)-{c}_i\ge {R}_{t-1}\left(n-{d}_i^{(k)}-1\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)$ holds.

• ∴${H}_{t-1}(n)-{H}_{t-1}\left(n-1\right)={R}_{t-1}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)-\left[{R}_{t-1}\left(n-1\right)-{c}_i\right]\ge 0$ holds.

• (c) ${R}_{t-1}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\le {R}_{t-1}(n)-{c}_i$ and ${R}_{t-1}\left(n-{d}_i^{(k)}-1\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\ge {R}_{t-1}\left(n-1\right)-{c}_i$;

• ∵ R _t − 1(n) is monotonically non-decreasing in n,

• ∴${R}_{t-1}\left(n-{d}_i^{(k)}\right)-{R}_{t-1}\left(n-{d}_i^{(k)}-1\right)\ge 0$ holds.

• ∵${R}_{t-1}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\le {R}_{t-1}(n)-{c}_i$ and ${R}_{t-1}\left(n-{d}_i^{(k)}-1\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\ge {R}_{t-1}\left(n-1\right)-{c}_i$,

• ∴${R}_{t-1}(n)-{c}_i\ge {R}_{t-1}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\ge {R}_{t-1}\left(n-{d}_i^{(k)}-1\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\ge {R}_{t-1}\left(n-1\right)-{c}_i$ holds.

• ∴${H}_{t-1}(n)-{H}_{t-1}\left(n-1\right)={R}_{t-1}(n)-{c}_i-\left[{R}_{t-1}\left(n-{d}_i^{(k)}-1\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\right]\ge 0$ holds.

• (d) ${R}_{t-1}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\le {R}_{t-1}(n)-{c}_i$ and ${R}_{t-1}\left(n-{d}_i^{(k)}-1\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\le {R}_{t-1}\left(n-1\right)-{c}_i$;

• ∵R _t − 1(n) is monotonically non-decreasing in n

• ∴R _t − 1(n) − R _t − 1(n − 1) ≥ 0 holds.

• ∴H _t − 1(n) − H _t − 1(n − 1) = R _t − 1(n) − R _t − 1(n − 1) ≥ 0.

Based on the above four cases, H _t − 1(n) − H _t − 1(n − 1) ≥ 0 holds. So H _t (n) is monotonically non-decreasing in n, R _t (n) is monotonically non-decreasing in n.

• (2) For each given t, we prove R _t (n) is concave in n;

When t = 0, R ₀(n) − R ₀(n − 1) = R ₀(n − 1) − R ₀(n − 2). The conclusion is clear.

We assume R _t − 1(n) − R _t − 1(n − 1) ≤ R _t − 1(n − 1) − R _t − 1(n − 2) holds.

By Eq. (4), we have ${R}_t\left(n-1\right)-{R}_t\left(n-2\right)=\left(1-{\lambda}_t^{(1)}-{\lambda}_t^{(2)}\right)\left[{R}_{t-1}\left(n-1\right)-{R}_{t-1}\left(n-2\right)\right]+\sum_{i=1}^2{\lambda}_t^{(i)}\sum_kp\left({du}_t^{(i)}={d}_i^{(k)}\right)\left[{H}_{t-1}\left(n-1\right)-{H}_{t-1}\left(n-2\right)\right]$

$$ \begin{array}{l}{R}_t(n)-{R}_t\left(n-1\right)-\left[{R}_t\left(n-1\right)-{R}_t\left(n-2\right)\right]=\left(1-{\lambda}_t^{(1)}-{\lambda}_t^{(2)}\right)\left\{{R}_{t-1}(n)-{R}_{t-1}\left(n-1\right)-\left[{R}_{t-1}\left(n-1\right)-{R}_{t-1}\left(n-2\right)\right]\right\}\hfill \\ {}+\sum_{i=1}^2{\lambda}_t^{(i)}\sum_kp\left({du}_t^{(i)}={d}_i^{(k)}\right)\left\{{H}_{t-1}(n)-{H}_{t-1}\left(n-1\right)-\left[{H}_{t-1}\left(n-1\right)-{H}_{t-1}\left(n-2\right)\right]\right\}\hfill \end{array} $$

(5)

Respectively, corresponding to the above four cases of H _t − 1(n) − H _t − 1(n − 1):

• (a) ${R}_{t-1}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\ge {R}_{t-1}(n)-{c}_i$ and ${R}_{t-1}\left(n-{d}_i^{(k)}-1\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\ge {R}_{t-1}\left(n-1\right)-{c}_i$;

• ∵${H}_{t-1}(n)-{H}_{t-1}\left(n-1\right)={R}_{t-1}\left(n-{d}_i^{(k)}\right)-{R}_{t-1}\left(n-{d}_i^{(k)}-1\right)$,

• ∴${H}_{t-1}(n)-{H}_{t-1}\left(n-1\right)-\left[{H}_{t-1}\left(n-1\right)-{H}_{t-1}\left(n-2\right)\right]={R}_{t-1}\left(n-{d}_i^{(k)}\right)-{R}_{t-1}\left(n-{d}_i^{(k)}-1\right)-\left[{R}_{t-1}\left(n-{d}_i^{(k)}-1\right)-{R}_{t-1}\left(n-{d}_i^{(k)}-2\right)\right]$,

• ∴R _t − 1(n) − R _t − 1(n − 1) ≤ R _t − 1(n − 1) − R _t − 1(n − 2),

• ∴R _t − 1(n) − R _t − 1(n − 1) − [R _t − 1(n − 1) − R _t − 1(n − 2)]≤ 0,

• ∴${R}_{t-1}\left(n-{d}_i^{(k)}\right)-{R}_{t-1}\left(n-{d}_i^{(k)}-1\right)-\left[{R}_{t-1}\left(n-{d}_i^{(k)}-1\right)-{R}_{t-1}\left(n-{d}_i^{(k)}-2\right)\right]\le 0$.

• ∴Therefore, Eq. (5) R _t (n) − R _t (n − 1) − [R _t (n − 1) − R _t (n − 2)] ≤ 0 holds.

• (b) ${R}_{t-1}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\ge {R}_{t-1}(n)-{c}_i$ and ${R}_{t-1}\left(n-{d}_i^{(k)}-1\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\le {R}_{t-1}\left(n-1\right)-{c}_i$

• ∵${H}_{t-1}(n)-{H}_{t-1}\left(n-1\right)={R}_{t-1}\left(n-{d}_i^{(k)}\right)+{r}_i-\left[{R}_{t-1}\left(n-1\right)-{c}_i\right]$,

• ∴${H}_{t-1}(n)-{H}_{t-1}\left(n-1\right)-\left[{H}_{t-1}\left(n-1\right)-{H}_{t-1}\left(n-2\right)\right]={R}_{t-1}\left(n-{d}_i^{(k)}\right)-{R}_{t-1}\left(n-1\right)-\left[{R}_{t-1}\left(n-{d}_i^{(k)}-1\right)+{R}_{t-1}\left(n-2\right)\right]$,

• ∵R _t − 1(n) − R _t − 1(n − 1) ≤ R _t − 1(n − 1) − R _t − 1(n − 2),

• ∴R _t − 1(n) − R _t − 1(n − 1) − [R _t − 1(n − 1) − R _t − 1(n − 2)] ≤ 0 and ${R}_{t-1}\left(n-{d}_i^{(k)}\right)-{R}_{t-1}\left(n-1\right)-\left[{R}_{t-1}\left(n-{d}_i^{(k)}-1\right)+{R}_{t-1}\left(n-2\right)\right]\le 0$ holds.

Therefore, Eq. (5) R _t (n) − R _t (n − 1) − [R _t (n − 1) − R _t (n − 2)] ≤ 0 holds.

• (c) ${R}_{t-1}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\le {R}_{t-1}(n)-{c}_i$ and ${R}_{t-1}\left(n-{d}_i^{(k)}-1\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\ge {R}_{t-1}\left(n-1\right)-{c}_i$;

• ∵${H}_{t-1}(n)-{H}_{t-1}\left(n-1\right)={R}_{t-1}(n)-{c}_i-\left[{R}_{t-1}\left(n-{d}_i^{(k)}-1\right)+{r}_i\right]$

• ∴${H}_{t-1}(n)-{H}_{t-1}\left(n-1\right)-\left[{H}_{t-1}\left(n-1\right)-{H}_{t-1}\left(n-2\right)\right]={R}_{t-1}(n)-{R}_{t-1}\left(n-{d}_i^{(k)}-1\right)-\left[{R}_{t-1}\left(n-1\right)-{R}_{t-1}\left(n-{d}_i^{(k)}-2\right)\right]$

• ∵R _t − 1(n) − R _t − 1(n − 1) ≤ R _t − 1(n − 1) − R _t − 1(n − 2),

• ∴R _t − 1(n) − R _t − 1(n − 1) − [R _t − 1(n − 1) − R _t − 1(n − 2)] ≤ 0, and ${R}_{t-1}(n)-{R}_{t-1}\left(n-{d}_i^{(k)}-1\right)-\left[{R}_{t-1}\left(n-1\right)-{R}_{t-1}\left(n-{d}_i^{(k)}-2\right)\right]\le 0$ holds.

Therefore, Eq. (5) R _t (n) − R _t (n − 1) − [R _t (n − 1) − R _t (n − 2)] ≤ 0 holds.

• (d) ${R}_{t-1}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\le {R}_{t-1}(n)-{c}_i$ and ${R}_{t-1}\left(n-{d}_i^{(k)}-1\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\le {R}_{t-1}\left(n-1\right)-{c}_i$;

• ∵H _t − 1(n) − H _t − 1(n − 1) = R _t − 1(n) − R _t − 1(n − 1),

• ∴H _t − 1(n) − H _t − 1(n − 1) − [H _t − 1(n − 1) − H _t − 1(n − 2)] = R _t − 1(n) − R _t − 1(n − 1) − [R _t − 1(n − 1) − R _t − 1(n − 2)],

• ∵R _t − 1(n) − R _t − 1(n − 1) ≤ R _t − 1(n − 1) − R _t − 1(n − 2),

• ∴R _t − 1(n) − R _t − 1(n − 1) − [R _t − 1(n − 1) − R _t − 1(n − 2)] ≤ 0,

Therefore, Eq. (5) R _t (n) − R _t (n − 1) − [R _t (n − 1) − R _t (n − 2)] ≤ 0 holds.

Based on the four cases above, R _t (n) − R _t (n − 1) − [R _t (n − 1) − R _t (n − 2)] ≤ 0 holds. That is, R _t (n) − R _t (n − 1) ≤ R _t (n − 1) − R _t (n − 2) holds. So, for each given t, R _t (n) is concave in n.

From the proof (1) and (2), we get the conclusion that for each given t, R _t (n) is monotonically non-decreasing and concave in n.

Proposition 2

For each given n, R _t (n) is monotonically non-decreasing and concave in t and is super-modular in (t, n);

Proof

• (1) For each given n, we prove R _t (n) is monotonically non-decreasing in t;

When n = 0, R _t (0) = 0. The conclusion is clear.

When n ≠ 0, we assume for any n, R _t (n − 1) is monotonically non-decreasing in t

By Eq. (1), we have

$$ \begin{array}{l}{R}_t(n)-{R}_{t-1}(n)=\left(1-{\lambda}_t^{(1)}-{\lambda}_t^{(2)}\right)\left[{R}_{t-1}(n)-{R}_{t-2}(n)\right]+\sum_{i=1}^2{\lambda}_t^{(i)}\sum_kp\left({du}_t^{(i)}={d}_i^{(k)}\right) \max \left\{{R}_{t-1}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right),{R}_{t-1}(n)-{c}_i\right\}\hfill \\ {}\begin{array}{l}-\sum_{i=1}^2{\lambda}_t^{(i)}\sum_kp\left({du}_t^{(i)}={d}_i^{(k)}\right) \max \left\{{R}_{t-2}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right),{R}_{t-2}(n)-{c}_i\right\}=\left(1-{\lambda}_t^{(1)}-{\lambda}_t^{(2)}\right)\left[{R}_{t-1}(n)-{R}_{t-2}(n)\right]+\sum_{i=1}^2{\lambda}_t^{(i)}\sum_kp\left({du}_t^{(i)}={d}_i^{(k)}\right)\\ {}\left[{\mathrm{H}}_{t-1}(n)-{\mathrm{H}}_{t-2}(n)\right]\end{array}\hfill \end{array} $$

(6)

Let ${H}_{t-1}(n)-{H}_{t-2}(n)= \max \left\{{R}_{t-1}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right),{R}_{t-1}(n)-{c}_i\right\}- \max \left\{{R}_{t-2}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right),{R}_{t-2}(n)-{c}_i\right\}$.

We discuss H _t − 1(n) − H _t − 2(n) in four cases as below.

• (a) ${R}_{t-1}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\ge {R}_{t-1}(n)-{c}_i$ and ${R}_{t-2}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\ge {R}_{t-2}(n)-{c}_i$;

• ∵ R _t (n − 1) is monotonically non-decreasing in t

• ∴${R}_{t-1}\left(n-{d}_i^{(k)}\right)-{R}_{t-2}\left(n-{d}_i^{(k)}\right)\ge 0$ holds.

• ∴${H}_{t-1}(n)-{H}_{t-2}(n)={R}_{t-1}\left(n-{d}_i^{(k)}\right)-{R}_{t-2}\left(n-{d}_i^{(k)}\right)\ge 0$.

• (b) ${R}_{t-1}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\ge {R}_{t-1}(n)-{c}_i$ and ${R}_{t-2}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\le {R}_{t-2}(n)-{c}_i$;

• ∵ R _t (n − 1) is monotonically non-decreasing in t

• ∴R _t − 1(n) − c _i  ≥ R _t − 2(n) − c _i holds

• ∴${R}_{t-1}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\ge {R}_{t-1}(n)-{c}_i\ge {R}_{t-2}(n)-{c}_i\ge {R}_{t-2}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)$.

• ∴${H}_{t-1}(n)-{H}_{t-2}(n)={R}_{t-1}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)-\left[{R}_{t-2}(n)-{c}_i\right]\ge 0$.

• (c) ${R}_{t-1}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\le {R}_{t-1}(n)-{c}_i$ and ${R}_{t-2}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\ge {R}_{t-2}(n)-{c}_i$;

• ∵ R _t (n − 1) is monotonically non-decreasing in t

• ∴${R}_{t-1}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\ge {R}_{t-2}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)$ holds

• ∴${R}_{t-1}(n)-{c}_i\ge {R}_{t-1}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\ge {R}_{t-2}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\ge {R}_{t-2}(n)-{c}_i$ holds.

• ∴${H}_{t-1}(n)-{H}_{t-2}(n)={R}_{t-1}(n)-{c}_i-\left[{R}_{t-2}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\right]\ge 0$.

• (d) ${R}_{t-1}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\le {R}_{t-1}(n)-{c}_i$ and ${R}_{t-2}\left(n-{d}_i^{(k)}\right)+{r}_i+{f}_i\left({d}_i^{(k)}\right)\le {R}_{t-2}(n)-{c}_i$;

• ∵ R _t (n − 1) is monotonically non-decreasing in t,

• ∴ R _t − 1(n) − R _t − 2(n) ≥ 0 holds,

• ∴H _t − 1(n) − H _t − 2(n) = R _t − 1(n) − R _t − 2(n) ≥ 0;

Based on the four cases,R _t (n) − R _t (n − 1) − [R _t (n − 1) − R _t (n − 2)] ≤ 0 holds. Therefore, Eq. (6) is greater than or equal to 0. That is,R _t (n) − R _t − 1(n) ≥ 0. So R _t (n) is monotonically non-decreasing in t.

• (2) Then, we prove that for each given n, R _t (n) is super-modular in (t, n). According to definition 1, we should prove R _t (n) − R _t (n − 1) ≥ R _t − 1(n) − R _t − 1(n − 1).

By Eq. (2) ${R}_t(n)-{R}_{t-1}(n)=-{\lambda}_t^{(1)}{c}_1-{\lambda}_t^{(2)}{c}_2+\sum_{i=1}^2{\lambda}_t^{(i)}\sum_kp\left({du}_t^{(i)}={d}_i^{(k)}\right) \max \left\{{r}_i+{f}_i\left({d}_i^{(k)}\right)+{c}_i-\left[{R}_{t-1}(n)-{R}_{t-1}\left(n-{d}_i^{(k)}\right)\right],0\right\}$,

• ∴$\begin{array}{l}{R}_t(n)-{R}_{t-1}(n)-\left[{R}_t\left(n-1\right)-{R}_{t-1}\left(n-1\right)\right]=\sum_{i=1}^2{\lambda}_t^{(i)}\sum_kp\left({du}_t^{(i)}={d}_i^{(k)}\right) \max \left\{{r}_i+{f}_i\left({d}_i^{(k)}\right)+{c}_i-\left[{R}_{t-1}(n)-{R}_{t-1}\left(n-{d}_i^{(k)}\right)\right],0\right\}\hfill \\ {}-\sum_{i=1}^2{\lambda}_t^{(i)}\sum_kp\left({du}_t^{(i)}={d}_i^{(k)}\right) \max \left\{{r}_i+{f}_i\left({d}_i^{(k)}\right)+{c}_i-\left[{R}_{t-1}(n)-{R}_{t-1}\left(n-{d}_i^{(k)}\right)\right],0\right\}\hfill \end{array}$.

• ∵ R _t (n) is concave in n

• ∴${R}_{t-1}(n)-{R}_{t-1}\left(n-{d}_i^{(k)}\right)\le {R}_{t-1}\left(n-1\right)-{R}_{t-1}\left(n-{d}_i^{(k)}-1\right)$;

• ∴${r}_i+{f}_i\left({d}_i^{(k)}\right)+{c}_i-\left[{R}_{t-1}(n)-{R}_{t-1}\left(n-{d}_i^{(k)}\right)\right]\ge {r}_i+{f}_i\left({d}_i^{(k)}\right)+{c}_i-\left[{R}_{t-1}\left(n-1\right)-{R}_{t-1}\left(n-{d}_i^{(k)}-1\right)\right]$

• ∴R _t (n) − R _t − 1(n) − [R _t (n − 1) − R _t − 1(n − 1)] ≥ 0.

• ∴ R _t (n) is super-modular in (t, n).

  • (3) Last, we prove that for each given n, R _t (n) is concave in t. That is, we should prove R _t (n) − R _t − 1(n) ≤ R _t − 1(n) − R _t − 2(n).

• ∵ R _t (n) is super-modular in (t, n)and by Eq. (2)

$$ \begin{array}{l}\kern2em {R}_t(n)-{R}_{t-1}(n)=-{\lambda}_t^{(1)}{c}_1-{\lambda}_t^{(2)}{c}_2+\sum_{i=1}^2{\lambda}_t^{(i)}\sum_kp\left({du}_t^{(i)}={d}_i^{(k)}\right) \max \left\{{r}_i+{f}_i\left({d}_i^{(k)}\right)+{c}_i-\left[{R}_{t-1}(n)-{R}_{t-1}\left(n-{d}_i^{(k)}\right)\right],0\right\}\\ {}\le -{\lambda}_t^{(1)}{c}_1-{\lambda}_t^{(2)}{c}_2+\sum_{i=1}^2{\lambda}_t^{(i)}\sum_kp\left({du}_t^{(i)}={d}_i^{(k)}\right) \max \left\{{r}_i+{f}_i\left({d}_i^{(k)}\right)+{c}_i-\left[{R}_{t-2}(n)-{R}_{t-2}\left(n-{d}_i^{(k)}\right)\right],0\right\}={R}_{t-1}(n)-{R}_{t-2}(n)\end{array} $$

• ∴R _t (n) − R _t − 1(n) ≤ R _t − 1(n) − R _t − 2(n) holds

• ∴For each given n, R _t (n) is concave in t.

From the proof (1), (2) and (3), we can get the conclusion that for each given n, R _t (n) is monotonically non-decreasing and concave in t and is super-modular in (t, n);

Theorem 1

For any time t(t = T, T − 1,  … , 1), if the remaining capacity of the surgeon satisfies $n\ge {n}_t^{(i)}$, then the surgeon should accept the booking request of type i patients on the nearest main surgery day, but if the remaining capacity of the surgeon satisfies $n<{n}_t^{(i)}$, then the surgeon should reject the booking request of type i patients on the nearest main surgery day.

Proof

From Eq. (2), if and only when ${r}_i+{f}_i\left({d}_i^{(k)}\right)+{c}_i\ge {R}_{t-1}(n)-{R}_{t-1}\left(n-{d}_i^{(k)}\right)$, the new surgery request will be accepted.

Theorem 2

${n}_t^{(i)}$ is monotonically non-decreasing in t.

Proof

• ∵ R _t (n) is super-modular in (t, n), ${R}_{t-1}(n)-{R}_{t-1}\left(n-{d}_i^{(k)}\right)\ge {R}_{t-2}(n)-{R}_{t-2}\left(n-{d}_i^{(k)}\right)$ holds

• ∴${n}_t^{(i)}\ge {n}_{t-1}^{(i)}$, that is ${n}_t^{(i)}$ is monotonically non-decreasing in t.