Health care network design with multiple objectives and stakeholders

Abstract

This paper aims to address the problem of locating/upgrading health care facilities with appropriate service levels while considering the preferences of all stakeholders mapped across a hierarchical decision network. To this end, an extended network goal programming model is adapted to ensure that a mix of balance and optimisation is attained in order to offer the decision that best satisfies the objectives of the national health care system, which are: network coverage, service level, network cost, and social impact of health centres. In addition, due to the highly uncertain environment of the health care systems, the robust counterpart of this model using the budget-of-uncertainty approach is developed in order to analyse the health care network’s performance to deal with uncertain parameters such as the national allocated budget and social impact. Key parameters which indicate the level of non-compensation between objectives, level of non-compensation between stakeholders, and level of centralisation in the health network along with the uncertainty budget are utilised to analyse the dynamics of decision network. The effectiveness of the model is demonstrated through use of a case study. The best-worst method is used to select a number of appropriate potential projects that serve as an input for the proposed model. To highlight the practical implications, different parametric analyses under both deterministic and uncertain environments are conducted. In addition, these experiments explore the compensatory behaviour between objectives and stakeholders in the network. Finally, some managerial insights are provided arising from the analytical results.

Appendix

1.1 A: Robust Model

To present robust optimization framework, let us consider equations (21) to (23) presented below such that parameters $a^{1}$ and $b^{2}$ can be described with interval uncertainty.

$$\begin{aligned}&A^{1} x \le B^{1} \end{aligned}$$

(21)

$$\begin{aligned}&A^{2} x \le B^{2} \end{aligned}$$

(22)

$$\begin{aligned}&l \le x \le u \end{aligned}$$

(23)

It is assumed that each uncertain parameter takes value according to a symmetric distribution with mean equal to the nominal value such as $\left[ a_{i j}^{1}-a\prime _{i j}^{1}, a_{i j}^{1}+a\prime _{i j}^{1}\right]$ and $\left[ b_{i}^{2}-b\prime _{i}^{2}, b_{i}^{2}+b\prime _{i}^{2}\right]$. $a\prime _{i j}^{1}$ and $b\prime _{i}^{2}$ are shift values. $Zro_{i}^{1}$ and $Pro_{j}^{1}$ are the supplementary variables for the robust counterpart of the uncertainty of $a^{1}$. $Zro_{i}^{2}$ and $Pro_{j}^{2}$ are the supplementary variables for the robust counterpart of the uncertainty of $b^{2}$. The role of the parameter $\Gamma$, as an uncertainty budget, is to adjust the robustness of the proposed method against the level of conservatism of the solution. According to the approach proposed by Bertsimas and Sim (2004), the tractable robust counterpart of the aforementioned uncertain equations (21) to (23) is as follows:

$$\begin{aligned}&z r o^{0}+p r o_{j}^{0} \ge {\hat{c}}_{j} \cdot y_{j}^{0} \end{aligned}$$

(24)

$$\begin{aligned}&\sum _{j} a_{i j}^{1} x_{j}+\Gamma _{i}^{1} \cdot z r o_{i}^{1}+\sum _{j \in J_{1}} p r o_{j}^{1} \le b_{i}^{1} \end{aligned}$$

(25)

$$\begin{aligned}&z r o_{i}^{1}+p r o_{i j}^{1} \ge a\prime _{i j}^{1} \cdot y_{j}^{1} \end{aligned}$$

(26)

$$\begin{aligned}&\sum _{j} a_{i j}^{2} x_{j}+\Gamma _{i}^{2} \cdot z r o_{i}^{2}+\sum _{j \in J_{1}} p r o_{j}^{2} \le b_{i}^{2} \end{aligned}$$

(27)

$$\begin{aligned}&z r o_{i}^{2}+p r o_{j}^{2} \ge {\hat{b}}_{i}^{2} \end{aligned}$$

(28)

$$\begin{aligned}&-y_{j}^{0} \le x_{j} \le y_{j}^{0} \end{aligned}$$

(29)

$$\begin{aligned}&-y_{j}^{1} \le x_{j} \le y_{j}^{1} \end{aligned}$$

(30)

Uncertain parameters in the proposed model are $EI_{kj}$ and $AC_{0}$. $E I_{k j}^{\prime }$ and $AC_{0}^{\prime }$ are shift values in symmetric distributions of $\left[ E I_{k j}-E I_{k j}^{\prime }, E I_{k j}+E I_{k j}^{\prime }\right]$ and $\left[ AC_{0}-AC_{0}^{\prime }, AC_{0}+AC_{0}^{\prime }\right]$, respectively.

Constraints (31) to (36) are imposed to develop the robust counterpart optimization model by considering the specific auxiliary variables (i.e., first, second, and third) under interval uncertainty of potential social impact of each facility in each type at each region. Similarly, Constraints (37) to (42) are imposed to develop the robust counterpart optimization model under interval uncertainty of national available budget for health service providing.

$$\begin{aligned}&\left( \begin{array}{c} \sum _{k, j} E I_{k j} \cdot y_{k j}^{n}+\sum _{k, k^{\prime }, j} E I_{k j} \cdot y_{k k j}^{c}+n_{H}^{N}-p_{H}^{N} \\ +\Gamma _{1}^{1} \overline{zro}_{1}^{1}+\sum _{j \in J_{1}, k \in K_{1}} \overline{p r o}_{kj}^{1} \\ +\Gamma _{2}^{1} \overline{zro}_{2}^{1}+\sum _{j \in J_{1}, k^{\prime } \in K_{1}} \overline{p r o}_{k^{\prime }j}^{2} \end{array}\right) \le SI_{0} \end{aligned}$$

(31)

$$\begin{aligned}&\left( \begin{array}{c} -\sum _{k, j} E I_{k j} \cdot y_{k j}^{n}-\sum _{k, k^{\prime }, j} E I_{k j} \cdot y_{k k j}^{c}-n_{H}^{N}+p_{H}^{N} \\ +\Gamma _{1}^{1} \underline{zro}_{1}^{1}+\sum _{j \in J_{1}, k \in K_{1}} \underline{p r o}_{kj}^{1} \\ +\Gamma _{2}^{1} \underline{zro}_{2}^{1}+\sum _{j \in J_{1}, k^{\prime } \in K_{1}} \underline{p r o}_{k^{\prime }j}^{2} \end{array}\right) \le -SI_{0} \end{aligned}$$

(32)

$$\begin{aligned}&\overline{z r o}_{1}^{1}+\overline{p r o}_{k j}^{1} \ge E I_{k j}^{\prime } \cdot y_{k j}^{n} \end{aligned}$$

(33)

$$\begin{aligned}&\overline{z r o}_{2}^{1}+\overline{p r o}_{k j}^{2} \ge E I_{k j}^{\prime } \cdot y_{k j}^{n} \end{aligned}$$

(34)

$$\begin{aligned}&\underline{z r o}_{1}^{1}+\underline{p r o}_{k j}^{1} \ge -E I_{k j}^{\prime } \cdot y_{k j}^{n} \end{aligned}$$

(35)

$$\begin{aligned}&\underline{z r o}_{2}^{1}+\underline{p r o}_{k j}^{2} \ge -E I_{k j}^{\prime } \cdot y_{k j}^{n} \end{aligned}$$

(36)

$$\begin{aligned}&\left( \begin{array}{c} \sum _{k, j} I C_{k j} \cdot y_{k j}^{n}+\sum _{k, k^{\prime }, j} E C_{k k^{\prime } j} \cdot y_{k k^{\prime } j}^{c}+n_{c}^{N}-p_{c}^{N} \\ +\Gamma 1^{2} \cdot \overline{z r o}^{2}+\sum _{k \in K_{1} , j \in J_{1}} \overline{p r o}_{k, j}^{\prime 1} \\ +\Gamma 2^{2} \overline{z r o^{\prime }}^{2}+\sum _{k \in K_{1}, k^{\prime } \in K_{1}, j \in J_{1}} \overline{p r o}_{k k^{\prime } j}^{\prime 2} \end{array}\right) \le AC_{0} \end{aligned}$$

(37)

$$\begin{aligned}&\left( \begin{array}{c} -\sum _{k, j} I C_{k j} \cdot y_{k j}^{n}-\sum _{k, k^{\prime }, j} E C_{k k^{\prime } j} \cdot y_{k k^{\prime } j}^{c}-n_{c}^{N}+p_{c}^{N} \\ +\Gamma 1^{2} \cdot \underline{z r o}^{2}+\sum _{k \in K_{1} , j \in J_{1}} \underline{p r o}_{k, j}^{\prime 1} \\ +\Gamma 2^{2} \underline{z r o}^{\prime 2}+\sum _{k \in K_{1}, k^{\prime } \in K_{1}, j \in J_{1}} \underline{p r o}_{k k^{\prime } j}^{\prime 2} \end{array}\right) \le -AC_{0} \end{aligned}$$

(38)

$$\begin{aligned}&\overline{z r o}^{2}+\overline{p r o}_{k j}^{\prime 1} \ge AC_{0}^{\prime } \end{aligned}$$

(39)

$$\begin{aligned}&\overline{z r o}^{\prime 2}+\overline{p r o}_{k k^{\prime } j}^{\prime 2} \ge AC_{0}^{\prime } \end{aligned}$$

(40)

$$\begin{aligned}&\underline{z r o}^{2}+\underline{p r o}^{\prime 1}_{k j} \ge AC_{0}^{\prime } \end{aligned}$$

(41)

$$\begin{aligned}&\underline{zro}^{\prime 2}+\underline{p r o}^{\prime 2}_{k k^{\prime } j} \ge AC_{0}^{\prime } \end{aligned}$$

(42)

$$\begin{aligned}&\overline{z r o}_{1}, \overline{p r o}_{k, j}^{1}, \overline{z r_o}^{1}, \overline{p r o}_{k, j}^{2}, \underline{z r o}_{1}^{1}, \underline{p r o}_{k, j}^{1}, \underline{z r o}_{2}^{1}, \underline{p r o}_{k, j}^{2} \ge 0 \end{aligned}$$

(43)

$$\begin{aligned}&\overline{z r o}^{2}, \overline{p r o_{k j}^{\prime }}, \overline{z r o^{\prime }}^{2}, \overline{p r o_{k k^\prime j}^{\prime }}^{2}, \underline{z r o}^{2}, \underline{p r o}_{k j}^{\prime 1}, \underline{z r o}^{\prime 2}, \underline{p r o}_{k k^\prime j}^{\prime 2} \ge 0 \end{aligned}$$

(44)

1.2 B: Potential projects analysis using the best-worst fuzzy method

At first, a set of possible potential projects of health network development is ranked for each region. There are various potential choices at each region for health care network development such as new facility establishment at different levels and upgrading/downgrading the level of health services at the existing health centre facilities. Besides, in order to evaluate potential network development projects in each region, there may exist various qualitative and quantitative criteria. Often, the expert qualitative judgements include a certain level of ambiguity and intangibility. Hence, the crisp values of criteria may not be precise enough to model the real-life multi-criteria decision-making. That is the reason the best-worst method, one of the recent MCDM methods, under fuzzy environment is utilised in this study. This method introduced by Guo and Zhao (2017) extends the best-worst method to the fuzzy environment. After applying this method, only top-ranked projects are considered to be evaluated in the next stage, i.e. extended goal programming model.

These different criteria, $\{C_{1},\ldots ,C_{10}\}$, develop the decision criteria system which their values indicate the performance of considered alternatives associated with each specific criteria. This method aims at obtaining practical preference ranking for alternatives as well as greater comparison consistency. For this purpose, the best and the worst criterion presented as $C_{B}$ and $C_{W}$, respectively are determined based on the experts’ opinions. Then, the fuzzy reference comparisons using the linguistic terms of experts is deployed for the best criterion in two steps: (i) pairwise comparison between the best criterion and others criteria; and (ii) pairwise comparison between considered criteria and the worst one. Using transformation rules, these extracted fuzzy preferences are then transformed to triangular fuzzy numbers (TFNs) as follows: equally important $(EI) \sim (1, 1, 1)$, weakly important $(WI) \sim (2/3, 1,3/21)$, fairly important $(FI) \sim (3/2, 2, 5/2)$, very important $(VI) \sim (5/2, 3, 7/2)$, and absolutely important $(AI) \sim (7/2, 4, 9/2)$. Then, fuzzy Best-to-Others as well as Others-to-worst vector are obtained as $\tilde{A}_{B} = (\tilde{a}_{B1}, \tilde{a}_{B2}, \ldots , \tilde{a}_{Bn})$ and $\tilde{A}_{W} = (\tilde{a}_{1W}, \tilde{a}_{2W}, \ldots , \tilde{a}_{nW})$, respectively. Eventually, the optimal fuzzy weights, $(\tilde{w}_{1}, \tilde{w}_{2}, \ldots , \tilde{w}_{n})$, of considered criteria are determined by solving the following optimisation problem (Guo & Zhao, 2017).

$$\begin{aligned}{} & {} \min \max _{j}\left\{ \left| \frac{{\tilde{w}}_{B}}{{\tilde{w}}_{j}}-{\tilde{a}}_{B j}\right| ,\left| \frac{{\tilde{w}}_{j}}{{\tilde{w}}_{w}}-{\tilde{a}}_{j w}\right| \right\} \end{aligned}$$

(45)

$$\begin{aligned}{} & {} \sum _{j=1}^{n} R\left( {\tilde{w}}_{j}\right) =1 \end{aligned}$$

(46)

$$\begin{aligned}{} & {} l_{j}^{w} \le m_{j}^{w} \le u_{j}^{w} \quad \forall j=1, \ldots , n \end{aligned}$$

(47)

$$\begin{aligned}{} & {} l_{j}^{w} \ge 0 \quad \forall j=1, \ldots , n \end{aligned}$$

(48)

Objective function (45) could be transformed to a simpler one by using maximum definition. Equation (46) presents a weight normalisation. Equations (47-48) demonstrate the smallest likely, most probable, and largest possible values of any fuzzy event.