Insights into clients’ choice in preventive health care facility location planning

Abstract

In this contribution we build on the approach proposed by Zhang et al. (OR Spectrum 34:349–370, 2012) to consider clients’ choice in preventive health care facility location planning. The objective is to maximize the participation in a preventive health care program for early detection of breast cancer in women. In order to account for clients’ choice behavior the multinomial logit model is employed. In this paper, we show that instances up to 20 potential locations and 400 demand points can be easily solved (to optimality or at least close to optimality) by a commercial solver in reasonable time if the problem is modeled by an alternative formulation. We present an intelligible approach to derive a lower bound to the problem. Our paper provides interesting insights into the trade-off between minimum workload requirement (to ensure quality of care) and participation (and thus early diagnosis of disease). We present a general definition of clients’ utility (which allows for clients’ characteristics, for example) and discuss some fundamental issues (and pitfalls) concerning the specification of utility functions.

Appendix

1.1 Model proposed by Zhang et al. (2012)

Additionally to the definitions of Sect. 2.1 we denote the parameters

• $h_{i}$   fraction of clients at node $i$,

• $\lambda$   expected number of clients per period over the entire area (Poisson rate); the Poisson rate of node $i$ is $\lambda \cdot h_i$,

• $\overline{\lambda }_k$   maximum participation rate at a facility with $k$ servers; $\overline{\lambda }_0 = 0$,

• $\Delta \lambda _k$ $= \overline{\lambda }_k - \overline{\lambda }_{k-1}$,

• $M_1$   big number; = 1 for example Zhang et al. (2012),

• $M_2$   big number; = 1 for example Zhang et al. (2012),

as well as

• $z_{ijo}$ artificial continuous variable for avoiding non-linearity with $o \in J$; corresponds to the result of $x_{ij} w_{o1}$.

and the mathematical model (Model Z) corresponding to the original contribution of Zhang et al. (2012)

$$\begin{aligned} \max F^Z = \lambda \sum _{i \in I} h_i \sum _{j\in J} x_{ij} \end{aligned}$$

(28)

subject to

$$\begin{aligned} x_{ij} + \sum _{o\in J} e^{-\beta t_{io}} z_{ijo}&= e^{-\beta t_{ij}} w_{j1}&i \in I, j \in J \end{aligned}$$

(29)

$$\begin{aligned} z_{ijo}&\le x_{ij}&i \in I, j,o \in J \end{aligned}$$

(30)

$$\begin{aligned} z_{ijo}&\le M_1 w_{p1}&i \in I, j,o \in J \end{aligned}$$

(31)

$$\begin{aligned} z_{ijo}&\ge x_{ij} - M_2(1 - w_{p1})&i \in I, j,o \in J \end{aligned}$$

(32)

$$\begin{aligned} \sum _{j\in J} \sum _{k=1}^K w_{jk}&\le Q_{\max } \end{aligned}$$

(33)

$$\begin{aligned} w_{jk+1}&\le w_{jk}&j\in J, k = 1, 2,\ldots , K-1\end{aligned}$$

(34)

$$\begin{aligned} \lambda \sum _{i \in I} h_i x_{ij}&\ge R_{\min }w_{j1}&j\in J \end{aligned}$$

(35)

$$\begin{aligned} \lambda \sum _{i \in I} h_i x_{ij}&\le \sum _{k=1}^K \nabla \lambda _k w_{jk}&j\in J\end{aligned}$$

(36)

$$\begin{aligned} x_{ij}&\ge 0&i \in I, j \in J \end{aligned}$$

(37)

$$\begin{aligned} z_{ijo}&\ge 0&i \in I, j,o \in J \end{aligned}$$

(38)

$$\begin{aligned} w_{jk}&\in \{0,1\}&j \in J, k = 1,2,\ldots , K \end{aligned}$$

(39)

1.2 Basics of discrete choice analysis

The MNL is well known for analyzing discrete choice decisions of individuals (McFadden 1973, 2001). Let $N$ be the set of individuals (customers, clients etc.), $M$ the choice set (set of alternatives the individual chooses from), and $L$ the set of attributes or characteristics (attractiveness determinants). The choice set $M$ must be exhaustive and the alternatives have to be mutually exclusive. Roughly speaking, all alternatives the individuals face have to be included in the choice set. Individual $n \in N$ chooses exactly one alternative from choice set $M$. In the discrete choice modeling literature it is assumed that an individual $n \in N$ chooses alternative $j\in M$ that maximizes utility (see Train 2003, for example). That is, $n$ chooses $j$, iff

$$\begin{aligned} u_{nj} > u_{nm} \quad \forall \ m \in M, m \ne j. \end{aligned}$$

(40)

The utility $u_{nj}$ of alternative $j$ for individual $n$ consists of a deterministic component $v_{nj}$ and a stochastic component $\epsilon _{nj}$, i.e.

$$\begin{aligned} u_{nj} = v_{nj} + \epsilon _{nj}. \end{aligned}$$

(41)

Usually, the deterministic component is modeled as a linear function:

$$\begin{aligned} v_{nj} = \sum _{l\in L} \beta _{jl}c_{njl} , \end{aligned}$$

(42)

where $c_{njl}$ is the value of attribute $l$ concerning individual $n$ and alternative $j$, and the coefficient $\beta _{jl}$ is the utility contribution per unit of attribute $l$ related to alternative $j$ (Ben-Akiva and Lerman 1985). In applications, $\beta _{jl}$ have to be estimated (by maximum-likelihood) using choice data from empirical studies (Anderson et al. 1992; Ben-Akiva and Lerman 1985; Louviere et al. 2000; Street and Burgess 2007; Müller et al. 2008; and Train 2003). Since $u_{nj}$ of (41) is stochastic we can only make probabilistic statements about (40):

$$\begin{aligned} p_{nj} = \text {Prob}\left( u_{nj} > u_{nm} \ \forall \ m \in M, m \ne j\right) . \end{aligned}$$

(43)

Assuming that the stochastic component $\epsilon _{nj}$ is independent, identically extreme value distributed, the probability (43) that individual $n$ chooses alternative $j$ is determined by

$$\begin{aligned} p_{nj} = \frac{e^{v_{nj}}}{\sum _{m\in M}e^{v_{nm}} }, \end{aligned}$$

(44)

which is the well-known MNL (Ben-Akiva and Bierlaire , 2003). Having said this, it is obvious that the MNL of (44) exhibits utility maximization behavior of the choice makers. In other words, using MNL means to assume that clients choose the facility that maximizes their utility (i.e., clients choose the most attractive—“the optimal”—facility). Hence, the underlying choice rule is utility maximization. Note, if $\epsilon _{nj}=0 \ \forall \ n,j,$ then the choice problem of (40) becomes deterministic. Consider, for example, $u_{nj}=v_{nj}=-t_{nj}$ with $t_{nj}$ as the travel time of individual $n$ to location $j$. This means that clients wish to obtain services from the facility with the shortest travel time. Obviously, such a choice model is characterized to be deterministic. If we assume that all clients located in $i \in I$ exhibit the same observable characteristics, then (1) results from (42) and (2) results from (44).