Choice-driven location-allocation model for healthcare facility location problem

Abstract

Improving access to care in medically under-served areas (MUA) is an important task for public health policy in many countries. A lack of access to care in MUAs is often addressed by establishing new service providers, which improves the accessibility in those regions in a geographic sense. However, the geographic accessibility is only a necessary condition for relieving the regions from being under-served; new service providers must be sufficiently attractive for care consumers to use so that the new establishment effectively translates to actual improvement in accessibility to care in the MUAs. In this paper, we first develop a location-allocation model for healthcare facilities that incorporates a choice model to represent care consumers’ preferences and choice decisions on the care facilities. Then we expand the model such that some of the attribute variables in the consumer utility function can be handled as decision variables in the location-allocation optimization. Thus, the proposed model determines the attributes variables of open facilities as well as their locations. Utility of the proposed model is demonstrated by using the Korea’s MUA support program for perinatal care.

Appendices

Appendix: A. Proof of linearization for choice constraints

We follow Aros-Vera et al. (2013) to linearize the original constraint on choice probability, (12) and (13). This linearization results in (14)–(16), and in this “Appendix”, we show that they are equivalent to the original constraints. For ease of reference, (12)–(16) are reproduced here.

$$\begin{aligned}&x_{ij} = \frac{e^{V_{ij}} y_{j}}{e^{V_{i0}}+\sum _{m \in J_i^\tau } e^{V_{im}} y_m},&\forall i \in I, \forall j \in J_i^\tau \qquad&(12)\\&x_{i0} = \frac{e^{V_{i0}}}{e^{V_{i0}}+\sum _{m \in J_i^\tau } e^{V_{im}} y_m},&\forall i \in I \qquad&(13)\\&\sum _{j\in J_i^\tau } x_{ij} + x_{i0} = 1,&\forall i \in I \qquad&(14)\\&e^{V_{i0}} x_{ij} \le e^{V_{ij}} x_{i0},&\forall i \in I, \forall j \in J_i^\tau \qquad&(15)\\&e^{V_{i0}} x_{ij} \ge e^{V_{ij}} (x_{i0}-(1-y_j)),&\forall i \in I, \forall j \in J_i^\tau \qquad&(16) \end{aligned}$$

Proof

For any two alternatives j and k, the ratio of the choice probabilities $x_{ij}$ and $x_{ik}$ is

$$\begin{aligned}\frac{x_{ij}}{x_{ik}} = \frac{e^{V_{ij}/ \sum _{m} e^{V_{im}}}}{e^{V_{ik}/\sum _{m} e^{V_{im}}}} = \frac{e^{V_{ij}}}{e^{V_{ik}}}=e^{V_{ij}-V_{ik}}\end{aligned}$$

This ratio is independent from other alternatives than j and k, and is said to be independent from irrelevant alternatives (IIA). A logit model naturally exhibits this IIA property. By definition of probability, we have

$$\begin{aligned} \sum _{j\in J} x_{ij} + x_{i0} = 1, \quad \forall i \in I \end{aligned}$$

(A.1)

With the IIA property a logit model, we can write choice probability $x_{ij}$ as

$$\begin{aligned}x_{ij} = e^{V_{ij}-V_{i0}} x_{i0}, \quad \forall i \in I,\ \forall j \in J \end{aligned}$$

(A.2)

Now, substitute (A.2) into (A.1), and we obtain the following:

$$\begin{aligned}&e^{V_{i1}-V_{i0}} x_{i0} + e^{V_{i2}-V_{i0}} x_{i0} +\cdots + x_{i0} = 1, \forall i \in I \nonumber \\&\leftrightarrow \quad e^{-V_{i0}} x_{i0} = \frac{1}{\sum _{m\in J} e^{V_{im}} + e^{V_{i0}}} \nonumber \\&\leftrightarrow \quad x_{i0} = \frac{e^{V_{i0}}}{\sum _{m\in J} e^{V_{im}} + e^{V_{i0}}} \end{aligned}$$

(A.3)

Then (A.2) can be rewritten as

$$\begin{aligned} x_{ij}&= e^{V_{ij}-V_{i0}} \frac{e^{V_{i0}}}{\sum _{m\in J} e^{V_{im}} + e^{V_{i0}}} \nonumber \\&= \frac{e^{V_{ij}}}{\sum _{m\in J} e^{V_{im}} + e^{V_{i0}}}, \quad \forall i \in I,\ \forall j \in J \end{aligned}$$

(A.4)

Note that the equality (A.2) can be split to upper and lower inequality:

$$\begin{aligned} x_{ij}\le e^{V_{ij}-V_{i0}} x_{i0}, \quad \forall i \in I,\ \forall j \in J \end{aligned}$$

(A.5)

$$\begin{aligned} x_{ij}\ge e^{V_{ij}-V_{i0}} x_{i0}, \quad \forall i \in I,\ \forall j \in J \end{aligned}$$

(A.6)

Thus, we have the following relationships:

(A.1), (A.2) $\leftrightarrow$ (A.1), (A.5) , (A.6) $\leftrightarrow$ (A.3), (A.4).

Now, we take into account $y_j$ in (12). Specifically, we need make sure that (1) if $y_{j}=0$ then $x_{ij}=0$, and (2) if $y_{j}=1$ then $x_{ij}=e^{V_{ij} }/ (\sum _{m\in J} e^{V_{im}} y_{m} + e^{V_{i0}})$.

In Constraint (11), we require $x_{ij} \le y_{j}$, and this ensures condition (1). For (2), it is evidently satisfied when $y_j$ = 1. Note that, since $e^{V_{ij}-V_{i0}} x_{i0}$ is always positive, if $x_{ij}=0$, (A.6) is invalid while constraint (A.5) is valid. To relieve this invalidity, we add the $-(y_{j}-1)$ term. This ensures that if $y_{j}=1$, constraint (16) is same as (A.6) and that if $y_{j}=0$, constraint (16) is valid. Finally,

Constraint (12) $\leftrightarrow$ Constraint (A.4),

Constraint (13) $\leftrightarrow$ Constraint (A.3),

Constraint (14) $\leftrightarrow$ Constraint (A.1),

Constraint (15) $\leftrightarrow$ Constraint (A.5),

Constraint (16) $\leftrightarrow$ Constraint (A.6).

Since constraint (A.1), (A.5), (A.6) $\leftrightarrow$ constraint (A.3), (A.4), constraint (14), (15), (16) $\leftrightarrow$ constraint (12), (13). For more detail, we refer readers to Aros-Vera et al. (2013). $\square$

Appendix: B. Full formulation of $P_{MUA}$

$$\begin{aligned} (P_{MUA}) \qquad max&\sum _{k\in K}{s_k}\\&\lambda _k^p = \sum _{i\in I_k} h_i\rho _i / \sum _{i\in I_k} \rho _i,&\forall k \in K \\&\lambda _k^r = \sum _{i\in I_k, j\in J_i^\tau } \rho _i x_{ij} / \sum _{i\in I_k} \rho _i,&\forall k \in K \\&s_k^p \ge \lambda _k^p - \varLambda ^{p},&\forall k \in K \\&s_k^p \le \lambda _k^p - \varLambda ^{p} +1 ,&\forall k \in K \\&s_k^r \ge \lambda _k^r - \varLambda ^{r},&\forall k \in K \\&s_k^r \le \lambda _k^r - \varLambda ^{r} +1,&\forall k \in K \\&s_k^p + s_k^r \ge s_k,&\forall k \in K \\&h_i \le \sum _{j\in J_i^\tau } y_{j},&\forall i \in I \\&x_{ij} \le y_{j},&\forall i \in I, \forall j \in J \\&\sum _{j\in J_i^\tau } x_{ij} + x_{i0} = 1,&\forall i \in I \\&\underline{z}_{ij}+\sum _l z_{ijl}=z_{ij},&\forall i \in I, \forall j \in J_i^\tau \cap J_c \\&z_{ijl} \le z_{ij(l-1)},&\forall i \in I, \forall j \in J_i^\tau \cap J_c, l=2,\ldots ,\hat{l}\\&e^{V_{i0}} x_{ij} \le e^{V_{ij}} x_{i0},&\forall i \in I, \forall j \in J_i^\tau \cap J_o \\&e^{V_{i0}} x_{ij} \ge e^{V_{ij}} (x_{i0}-(1-y_j)),&\forall i \in I, \forall j \in J_i^\tau \cap J_o \\&e^{V_{i0}} x_{ij} \le e^{V_{ij0}^\star }x_{i0}+\sum _{l=1}^{\hat{l}} \theta _{ijl} \phi _{ijl}^x,&\forall i \in I, \forall j \in J_i^\tau \cap J_c \\&e^{V_{i0}} x_{ij} \ge e^{V_{ij0}^\star }(x_{i0}-(1-y_j))\\&\qquad \qquad +\sum _{l=1}^{\hat{l}} \theta _{ijl} (\phi _{ijl}^x-(z_{ijl}-\phi _{ijl}^y)),&\forall i \in I, \forall j \in J_i^\tau \cap J_c \\&\phi _{ijl}^x \ge 0,&\forall i \in I, \forall j \in J_i^\tau \cap J_c, l=1,\ldots ,\hat{l} \\&\phi _{ijl}^x \le z_{ijl},&\forall i \in I, \forall j \in J_i^\tau \cap J_c, l=1,\ldots ,\hat{l} \\&\phi _{ijl}^x \le x_{i0},&\forall i \in I, \forall j \in J_i^\tau \cap J_c, l=1,\ldots ,\hat{l} \\&\phi _{ijl}^x \ge z_{ijl}+x_{i0}-1,&\forall i \in I, \forall j \in J_i^\tau \cap J_c, l=1,\ldots ,\hat{l} \\&\phi _{ijl}^y \ge 0,&\forall i \in I, \forall j \in J_i^\tau \cap J_c, l=1,\ldots ,\hat{l} \\&\phi _{ijl}^y \le z_{ijl},&\forall i \in I, \forall j \in J_i^\tau \cap J_c, l=1,\ldots ,\hat{l} \\&\phi _{ijl}^y \le y_{j},&\forall i \in I, \forall j \in J_i^\tau \cap J_c, l=1,\ldots ,\hat{l} \\&\phi _{ijl}^y \ge z_{ijl}+y_{j}-1,&\forall i \in I, \forall j \in J_i^\tau \cap J_c, l=1,\ldots ,\hat{l} \\&o_{j} = z_{ij},&\forall i \in I, \forall j \in J_c \\&o_{j} \ge y_j \underline{O},&\forall j \in J_c \\&o_{j} \le y_j \overline{O},&\forall j \in J_c \\&C_u\sum _{j\in J_c} y_j+C_p\sum _{j\in J_c}o_{j} \le B\\&z_{ijl} \in \{0,1\},&\forall i \in I, \forall j \in J_i^\tau \cap J_c, l=1,\ldots ,\hat{l} \\&0 \le x_{ij} \le 1, s_k, s_k^p, s_k^r, h_i, y_j \in \{0,1\},&\forall i \in I, \forall j \in J,\forall k \in K \end{aligned}$$