Abstract
Pursuing the overarching goal of saving both lives and healthcare costs, we introduce an approach to increase the expected participation in a preventive healthcare program, e.g., breast cancer screening. In contrast to sick people who need urgent medical attention, the clients in preventive healthcare decide whether to go to a specific facility (if this maximizes their utility) or not to take part in the program. We consider clients’ utility functions to include decision variables denoting the waiting time for an appointment and the quality of care. Both variables are defined as functions of a facility’s utilization. We employ a segmentation approach to formulate a mixedinteger linear program. Applying GAMS/CPLEX, we optimally solved instances with up to 400 demand nodes and 15 candidate locations based on both artificial data as well as in the context of a case study based on empirical data within one hour. We found that using a Benders decomposition of our problem decreases computational effort by more than 50%. We observe a nonlinear relationship between participation and the number of established facilities. The sensitivity analysis of the utility weights provides evidence on the optimal participation given a specific application (data set, empirical findings).
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Appendices
Appendix 1: Tighter bounds on variables
We can improve the upper bound for \(X_{i,j,m}\) in (25): If it is certain that r facilities will exist, we do not need to do the calculation as if (j, m) was the only facility, but we can assume that \(r1\) other facilities will be established as well. We replace \(p_{i,j,m}\) and get
where \(\mathscr {J}_i^{\text {min}}\) is the set of the \(r1\) least attractive other facility locations for demand node i and \(\check{m}\) represents the globally least attractive mode. This change reduces solution times remarkably—a finding that is also discussed in Freire et al. (2016).
It is also helpful to determine bounds for the nochoice probability variables \(Z_i\). They can be easily computed by assuming worst case or best case scenarios. Let \(\mathscr {J}_i^{\text {worst}}\) and \(\mathscr {J}_i^{\text {best}}\) be the sets of the r least or most attractive alternatives j for demand node i, and \(\hat{m}\) the most attractive mode. Then we get upper bounds \(Z_i^{\text {UB}}\) and lower bounds \(Z_i^{\text {LB}}\):
If we also decided about the number of established facilities r with a variation of (30) to
our model is harder to solve and (34) and (35) would not be applicable.
Appendix 2: Modeling framework to derive a lower bound
Since it may take a long time to find a first integer solution within the solving process, it is expedient to determine a lower bound for our linear model PHCFLPP (23)–(33). We present three models that altogether yield a lower bound. This approach is basically introduced in Haase and Müller (2015). It relies on the constant part of individuals’ utility functions, which mainly depends on distances.
Step I
With the first auxiliary model the highest choice probabilities for each single demand node are chosen. I.e., the most attractive facilities for them are identified. We continue our nomenclature:
Additional variables
 \(\Upsilon _{i,j,m}\):

= 1 if demand node i is assigned to location j in mode m; 0, otherwise
 \(F^{\mathrm {I}}\):

Objective function value of Model Step I (cumulated r highest choice probabilities)
subject to
The objective function (38) maximizes the cumulated chosen choice probabilities subject to (39), which means that the at most r highest choice probabilities for each demand node i are selected. (40) ensures that a facility can only be demanded in exactly one mode m.
Step II
In the second auxiliary model, the r overall most attractive facilities are established. The specific mode is not of interest here and the best one is always chosen. The decision is based on the remaining influences, mainly the distance between demand and supply nodes. An additional parameter that makes use of the solution to Step I sums up for how many demand nodes a certain facility belongs to the most attractive ones. The more demand nodes assigned to a facility the higher its attraction by this definition. We further extend our nomenclature:
Additional parameter
 \(b_{j,m}\) :

Attractiveness value for each facility at location j in mode m with \(b_{j,m} = \sum _{i \in \mathscr {I}} {\Upsilon _{i,j,m}^{*}}\) where \({\Upsilon _{i,j,m}^{*}}\) is the optimal solution to Step I
Additional variables
 \(\tilde{Y}_{j,m}\):

= 1 if facility at location j is specified to offer healthcare service in mode m; 0, otherwise
 \(F^{\mathrm {II}}\):

Objective function value of Model Step II (attractiveness of located facilities)
subject to
The objective function (42) maximizes the cumulated overall attractiveness and thereby chooses the r most demanded facilities according to (43). The result represents a solution that depends on the individuals’ constant part of their deterministic utility function disregarding capacities.
Step III
The last auxiliary model is used to determine the final modes for the facility locations previously identified in Step II. We consider
Additional parameter
 \(\tilde{Y}_{j,m}^{*}\) :

optimal solution to Step II
Additional variable
 LB:

objective function value of Model Step III (a lower bound for PHCFLPP)
subject to (24)–(28), (31)–(33) as well as
The first restriction blocks are in fact PHCFLPP without two redundant constraints of which information is already included in Step I and Step II. The information is that only one mode per facility can be present as well as that r facilities are established. (45) maximizes the expected participation by now selecting the final modes of the predestined facility locations via (46). The objective function value is a lower bound for PHCFLPP as it is a feasible integer solution to it. It is allowed to add up the predefined locations on the right hand side of (46), because at most one value can equal 1 due to (40). Thus, (46) is a substitute for (29) and a linking constraint in addition. Those constraints can be left out in Step III, as well as (30), because of (43). So we can perform a MIP start with PHCFLPP after solving the Steps I, II and III in a row.
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Krohn, R., Müller, S. & Haase, K. Preventive healthcare facility location planning with qualityconscious clients. OR Spectrum (2020). https://doi.org/10.1007/s0029102000605w
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Keywords
 Discrete choice
 Healthcare
 Random utility
 Configurations of facilities
 Facility location
 Benders