Capacity allocation and revenue sharing in healthcare alliances

Abstract

This study investigates the operations of a healthcare alliance, where two types of independent hospitals [general hospitals (GHs) and community healthcare centers (CHCs)] collaborate in capacity allocation and revenue management to improve efficiency and revenue. Specifically, the GH allocates part of its capacity to establish a dedicated green channel for referral patients from the CHC based on a negotiated revenue sharing scheme. Generally, independent hospitals, like GHs and CHCs, in the healthcare delivery system provide complementary and substitute services due to the referral process and patients choice. Both service providers need each other, and their collaboration is very beneficial to their own market share. We propose a two-stage game-theoretic approach to study the optimal incentive and coordination mechanisms of the healthcare alliances. In the first-stage cooperative game, two providers negotiate fixed proportion rates to share the revenue from referral patients. In the second-stage non-cooperative game, the GH makes the capacity allocation decision and the CHC decides the treatment threshold to maximize their own revenues. We derive a revenue-sharing scheme that is efficient, in the sense that the decentralized system can achieve the same revenue as the situation where a a central planner manages the alliance. When the capacity of the CHC and the number of patients visiting the CHC for the first diagnosis is large enough, the CHC is willing to take more efforts to treat more patients with alliances. In our study, simulation is employed to analyze the characteristics of the healthcare alliances as a supplement of theoretical analyses and a verification of the revenue-sharing scheme’s feasibility and efficiency. The equilibrium decisions obtained through simulation provides suggestions of much practical value for the operation of healthcare alliances.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Notes

  1. 1.

    http://www.bjd.com.cn/zc/sjc/201610/27/t20161027_11042429.html.

References

  1. Eggleston K, Ling L, Qingyue M, Lindelow M, Wagstaff A (2008) Health service delivery in China: a literature review. Health Econ 17(2):149–165

    Article  Google Scholar 

  2. Fitzgerald L, Ferlie E, Addicott R, Baeza J, Buchanan D, McGivern G (2007) Service improvement in healthcare: understanding change capacity and change context. Clin Manag 15(2):61–74

    Google Scholar 

  3. Freeman M, Savva N, Scholtes S (2016) Gatekeepers at work: an empirical analysis of a maternity unit. Manag Sci 63(10):3147–3167

    Article  Google Scholar 

  4. Guo L, Wu X (2017) Capacity sharing between competitors. Manag Sci 64:3554–3573

    Article  Google Scholar 

  5. Hu X, Caldentey R, Vulcano G (2013) Revenue sharing in airline alliances. Manag Sci 59(5):1177–1195

    Article  Google Scholar 

  6. Lee HH, Pinker EJ, Shumsky RA (2012) Outsourcing a two-level service process. Manag Sci 58(8):1569–1584

    Article  Google Scholar 

  7. Li L, Zhang RQ (2015) Cooperation through capacity sharing between competing forwarders. Transp Res Part E Logist Transp Rev 75:115–131

    Article  Google Scholar 

  8. Liu GG, Vortherms SA, Hong X (2017) China’s health reform update. Annu Rev Public Health 38:431–448

    Article  Google Scholar 

  9. Meng Q, Liu X, Shi J (2000) Comparing the services and quality of private and public clinics in rural china. Health Policy Plan 15(4):349–356

    Article  Google Scholar 

  10. Organization WH et al (2016) Deepening health reform in china. World Bank, Washington

    Google Scholar 

  11. Parkin D (1993) International comparisons of health expenditure. J Public Health 15(1):114–115

    Article  Google Scholar 

  12. Richard MD, Allaway AW (1993) Service quality attributes and choice behaviour. J Serv Mark 7(1):59–68

    Article  Google Scholar 

  13. Roels G, Tang CS (2016) Win-win capacity allocation contracts in coproduction and codistribution alliances. Manag Sci. https://doi.org/10.1287/mnsc.2015.2358

    Article  Google Scholar 

  14. Shumsky RA, Pinker EJ (2003) Gatekeepers and referrals in services. Manag Sci 49(7):839–856

    Article  Google Scholar 

  15. Sloan FA (2000) Not-for-profit ownership and hospital behavior. Handb. Health Econ. 1:1141–1174

    Article  Google Scholar 

  16. Tan Y, Chiang IR, Mookerjee VS (2006) An economic analysis of interconnection arrangements between internet backbone providers. Oper Res 54(4):776–788. https://doi.org/10.1287/opre.1060.0288

    Article  MATH  Google Scholar 

  17. Tirole J (1988) The theory of industrial organization. MIT Press, Cambridge

    Google Scholar 

  18. Vanberkel PT, Boucherie RJ, Hans EW, Hurink JL, Litvak N (2012) Efficiency evaluation for pooling resources in health care. OR Spectr. 34(2):371–390

    MathSciNet  Article  Google Scholar 

  19. Wang X, Debo LG, Scheller-Wolf A, Smith SF (2010) Design and analysis of diagnostic service centers. Manag Sci 56(11):1873–1890

    Article  Google Scholar 

  20. Yip W, Hsiao WC (2008) The chinese health system at a crossroads. Health Aff 27(2):460–468

    Article  Google Scholar 

  21. Yu Y, Benjaafar S, Gerchak Y (2015) Capacity sharing and cost allocation among independent firms with congestion. Prod Oper Manag 24(8):1285–1310. https://doi.org/10.1111/poms.12322

    Article  Google Scholar 

Download references

Funding

This work was partially supported by National Science Foundation of China (NSFC) under Grant Nos. 71671005 and 71671006.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jie Song.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

Proof for Proposition 1

Proof

If the number of patients visiting the GH for the first diagnosis is less than half of the capacity of the green channel, i.e., \(a \le c/2\), the total number of patients visiting the CHC is smaller than the capacity of the CHC. Thus, the GHs profit:

$$\begin{aligned} \begin{aligned} \pi _{1}&=\frac{1-t}{2}+\frac{d(1-t)}{2[d+a(1-t)]}+\beta \frac{k+1}{2} \min \{(1-k)\min \left\{ c,\frac{c}{2}+at\right\} ,t\}\\&=\frac{1-t}{2}+\frac{d(1-t)}{2[d+a(1-t)]}+\beta \frac{k+1}{2} \min \left\{ (1-k)\frac{c}{2}+at\,t\right\} .\\ \end{aligned} \end{aligned}$$

Therefore, when the capacity of the green channel is less than the number of referral patients, i.e., \((1-k)\frac{c}{2}+at \ge t\), the first derivative of the GH’s profit \(\pi _{1}\) with respect to t is:

$$\begin{aligned} \begin{aligned} \frac{\partial 2\pi _{1}}{\partial t}&= \frac{\partial }{\partial t}\left( 1-t+ \frac{d(1-t)}{d+a(1-t)}\right) +\beta (1+k)t\\&= \beta (1+k)-1-\frac{d^2}{a^2}\left( \frac{1}{t-(1+\frac{d}{a})}\right) ^2.\\ \end{aligned} \end{aligned}$$

Thus, we \(\beta\) satisfies the relation that \(\beta \le \frac{1}{k+1}[1+ (\frac{d}{a+d})^2]\), the GH’s profits \(\pi _{1}\) decreases with the capacity of the green channel t. From \(k\in [0,1]\), we know \(\frac{1}{k+1}[1+(\frac{d}{a+d})^2]> \frac{1+\frac{d}{a+d})^2}{2}\). Thus, the GH would decides not to allocate capacity to the green channel when \(\beta \le \frac{(1+\frac{d}{a+d})^2}{2}\).

If the number of patients visiting the GH for the first diagnosis is larger than half of the capacity of the green channel, i.e., \(a > c/2\), the total number of patients visiting the CHC may probably exceed the capacity of the CHC. Thus, when \(a>c/2\), \((1-k)c \ge t > c/2a\), the GHs profit is:

$$\begin{aligned} \begin{aligned} 2\pi _{1}&= 1-t + \frac{d(1-t)}{d+a(1-t)} +\beta (1+k) \min \{(1-k)c,t\}\\&= -t+\frac{d(1-t)}{d+a(1-t)}+\beta (1+k)t + f(k,a,d).\\ \end{aligned} \end{aligned}$$

Therefore, the first derivative of the GH’s profit \(\pi _{1}\) with respect to t is:

$$\begin{aligned} \frac{\partial 2\pi _{1}}{\partial t}=[\beta (1+k)-1]-\frac{d^2}{a}\frac{1}{[at-(a+d)]^2} \end{aligned}$$

Thus, when \(\beta < \frac{1}{1+k}\), the GH’s profit decreases with t. Considering that \(k\in [0,1]\), when \(\beta < 1/2\), the GH is not willing to allocate capacity to facilitate the green channel. In addition, when \(t \le c/2a\), the GH’s profits is the same as in the situation where \(a \le c/2\).

In conclusion, \(\beta \le \frac{(1+\frac{d}{a+d})^2}{2}\) is a necessary condition for the formation of the alliance.

\(\square\)

Proof for Proposition 2

Proof

From Proposition 1, we have known that the GH would never allocate capacity for the green channel when \(\beta \le \frac{(1+\frac{d}{a+d})^2}{2}\). Therefore, only the circumstances where \(\beta > \frac{(1+\frac{d}{a+d})^2}{2}\) need to be considered.

When \(a\le \frac{c}{2}\) and \((1-k)(c/2+at)\ge t\), the first derivative of the GH’s profit \(\pi _{1}\) with respect to t is:

$$\begin{aligned} \begin{aligned} \frac{\partial 2\pi _{1}}{\partial t}&= \beta (1+k)-1-\left( \frac{d}{at-(a+d)}\right) ^2. \end{aligned} \end{aligned}$$

Thus, the zero point of the GH’s profit’s derivative is \(t_1 = 1+\frac{d}{a}-\frac{d}{a\sqrt{1-\beta a(1-k^2)}}\). The GH’s profit increases with t when \(t<t_1\) and decreases with t when \(t>t_1\). Thus the maximum point of the GH’s profit is

$$\begin{aligned} t^* = \min \left\{ 1+\frac{d}{a}-\frac{d}{a\sqrt{1-\beta a(1-k^2)}}, \frac{c(1-k)/2}{1-a(1-k)}\right\} \end{aligned}$$

Notably, \(t_1 > \frac{c(1-k)/2}{1-a(1-k)}\), i.e., \(t*= \frac{c(1-k)/2}{1-a(1-k)}\), when \(\beta < 1/a\).

When \(a\le \frac{c}{2}\) and \((1-k)(c/2+at) \le t\), the first derivative of the GH’s profit \(\pi _{1}\) with respect to t is:

$$\begin{aligned} \frac{\partial 2\pi _{1}}{\partial t} = \beta a(1-k)^2-1-\left( \frac{d}{at-(a+d)}\right) ^2. \end{aligned}$$

When \(\beta < 1/a\),

$$\begin{aligned} \frac{\partial 2\pi _{1}}{\partial t}< (1-k)^2-1-\left( \frac{d}{at-(a+d)}\right) ^2 < 0. \end{aligned}$$

Then the GH’s profit decreases when \(t>\frac{c(1-k)/2}{1-a(1-k)}\), i.e., \(t*= \frac{c(1-k)/2}{1-a(1-k)}\).

When \(a > \frac{c}{2}\), the discussion is similar with the situation where \(a\le \frac{c}{2}\). Therefore, it is derived that when \(\frac{1+(\frac{d}{d+a})^{2}}{2} < \frac{1}{a}\) and \(\beta \in (\frac{1+(\frac{d}{d+a})^{2}}{2}, \frac{1}{a})\), the capacity of the green channel decided by the GH would be equal to the number of referral patients from the CHC, i.e., \(t= (1-k)\min \{c,\frac{c}{2}+at\}\). \(\square\)

Proof for Proposition 3

Proof

When \(a\le \frac{c}{2}\) and \((1-k)(c/2+at)\ge t\), the first derivative of the CHC’s profit \(\pi _{2}\) with respect to k is:

$$\begin{aligned} \begin{aligned} \frac{\partial 2\pi _{2}}{\partial k}&= \frac{\partial }{\partial k}\left[ \left( \frac{c}{2}+at\right) (-k^3+k^2)+t(1-\beta )(1+k)\right] \\&= \left( \frac{c}{2}+at\right) \left[ -3k^2+2k+\frac{(1-\beta )t}{\frac{c}{2}+at}\right] .\\ \end{aligned} \end{aligned}$$

Thus, the zero point of the CHC’s profit’s derivative is \(k_1 = \frac{1+\sqrt{1+\frac{3(1-\beta )t}{c/3+at}}}{3}\). The CHC’s profit increases with k when \(k<k_1\) and decreases with t when \(k>k_1\). Thus the maximum point of the GH’s profit is

$$\begin{aligned} k^* = \min \left\{ \frac{1+\sqrt{1+\frac{3(1-\beta )t}{c/3+at}}}{3}, 1-\frac{t}{c/2+at}\right\} \end{aligned}$$

When \(a\le \frac{c}{2}\) and \((1-k)(c/2+at)\le t\), the first derivative of the CHC’s profit \(\pi _{2}\) with respect to k is:

$$\begin{aligned} \begin{aligned} \frac{\partial 2\pi _{2}}{\partial k}&= \frac{\partial 2}{\partial k}\left[ \left( \frac{c}{2}+at\right) (-k^3+k^2)+ t (1-\beta )(\frac{c}{2}+at)(1-k)^2\right] \\&=\left( \frac{c}{2}+at\right) \left[ -3k^2+2\beta k\right] .\\ \end{aligned} \end{aligned}$$

Thus, the zero point of the CHC’s profit’s derivative is \(k_2 = \frac{2}{3}\beta\). The CHC’s profit increases with k when \(k<k_2\) and decreases with t when \(k>k_2\). Thus the maximum point of the GH’s profit is

$$\begin{aligned} k^* = \max \left\{ \frac{2}{3}\beta , 1-\frac{t}{c/2+at}\right\} \end{aligned}$$

It is obvious that \(\frac{1+\sqrt{1+\frac{3(1-\beta )t}{c/3+at}}}{3} > \frac{2}{3}\), and \(\frac{2}{3}\beta \le \frac{2}{3}\). Thus, it is a sufficient condition of \(k^*>\frac{2}{3}\) that \(1-\frac{t}{c/2+at}\). It is equivalent to the condition that

$$\begin{aligned} t^* < \frac{c}{2(3-a)} \end{aligned}$$

When \(c+2a>6\), \(\frac{c}{2(3-a)}>1>t*\). Therefore, when \(c+2a>6\), the CHC would definitely exert effort to treat more patients, i.e., \(k*>2/3\).

When \(a > \frac{c}{2}\), the discussion is similar with the situation where \(a\le \frac{c}{2}\). Therefore, if \(c+2a>6\), the CHC will choose a higher threshold of treatment than that without alliances, i.e., \(k>2/3\). \(\square\)

Proof for Proposition 4

Proof

When \(a\le c/2\), the first derivative of the alliance’s total profit \(\pi\) with respect to t is:

$$\begin{aligned} \begin{aligned} \frac{\partial 2\pi }{\partial t}&= \frac{\partial }{\partial t}\left[ 1-t+\frac{d(1-t)}{d+a(1-t)} + \left( k^2-k^3\right) \left( \frac{c}{2}+at\right) +(1+k)t\right] \\&= k-\frac{d^2/a}{[at-(a+d)]^2}.\\ \end{aligned} \end{aligned}$$

Thus, the alliance total profit decreases with the capacity of the green channel t, i.e., \(t^*=0\) when \(k < d^2/[a(a+d)^2]\). Considering that \(k\in [0,1]\), when \(d^2/[a(a+d)^2]>1\), i.e., \(d>\frac{a}{\sqrt{a}-1}\), the GH should not allocate capacity for the green channel.

When \(a > \frac{c}{2}\), the discussion is similar with the situation where \(a\le \frac{c}{2}\). Therefore, to maximize the alliance’s profit, the GH should not allocate capacity for the green channel, i.e., \(t_0=0\) when \(d > \frac{a}{\sqrt{a}-1}\). \(\square\)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Gao, X., Wen, J. & Song, J. Capacity allocation and revenue sharing in healthcare alliances. Flex Serv Manuf J 32, 829–851 (2020). https://doi.org/10.1007/s10696-019-09356-5

Download citation

Keywords

  • Healthcare alliances
  • Revenue management
  • Contract design
  • Capacity allocation