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An integrated approach for planning a long-term care network with uncertainty, strategic policy and equity considerations
Accepted Manuscript
An integrated approach for planning a long-term care network with uncertainty, strategic policy and equity considerations Teresa Cardoso , Monica Duarte Oliveira , Ana Barbosa-Povoa , ´ ´ Stefan Nickel PII: DOI: Reference:
S0377-2217(15)00486-5 10.1016/j.ejor.2015.05.074 EOR 12998
To appear in:
European Journal of Operational Research
Received date: Revised date: Accepted date:
12 August 2014 14 April 2015 26 May 2015
Please cite this article as: Teresa Cardoso , Monica Duarte Oliveira , Ana Barbosa-Povoa , ´ ´ Stefan Nickel , An integrated approach for planning a long-term care network with uncertainty, strategic policy and equity considerations, European Journal of Operational Research (2015), doi: 10.1016/j.ejor.2015.05.074
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Highlights We propose an integrated approach for planning a multi-service network of LTC.
We develop a stochastic MILP model that considers uncertainty in demand and supply.
The model aims at minimizing expected costs while ensuring satisficing equity levels.
The model explores how strategic health policy options can influence LTC delivery.
The model is applied to a region of Portugal and equity-cost trade-offs are calculated.
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An integrated approach for planning a long-term care network with uncertainty, strategic policy and equity considerations Teresa Cardoso a *, Mónica Duarte Oliveira b, Ana Barbosa-Póvoa c, Stefan Nickel d Centre for Management Studies of Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais 1, 1049-001 a
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Lisbon, Portugal, [email protected]
Centre for Management Studies of Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais 1, 1049-001 Lisbon, Portugal, [email protected]
b
Centre for Management Studies of Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais 1, 1049-001 Lisbon, Portugal, [email protected]
c
Institute of Operations Research, Karlsruhe Institute of Technology (KIT), Kaiserstr. 12, 76131 Karlsruhe, Germany, d
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[email protected]
Corresponding author: [email protected] ; +351 218419959 *
Abstract: Considering key uncertainties and health policy options in the reorganization of a longterm care (LTC) network is crucial. This study proposes a stochastic mixed integer linear programming model for planning the delivery of LTC services within a network of care where such
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aspects are modeled in an integrated manner. The model assists health care planners on how to plan the delivery of the entire range of LTC services – institutional, home-based and ambulatory services
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– when the main policy objective is the minimization of expected costs and while respecting satisficing levels of equity. These equity levels are modeled as constraints, ensuring the attainment of equity of access, equity of utilization, socioeconomic equity and geographical equity. The proposed
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model provides planners with key information on: when and where to locate services and with which capacity, how to distribute this capacity across services and patient groups, and which changes to the
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network of care are needed over time. Model outputs take into account the uncertainty surrounding LTC demand, and consider strategic health policy options adopted by governments. The applicability
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of the model is demonstrated through the resolution of a case study in the Great Lisbon region in Portugal with estimates on the equity-cost trade-off for several equity dimensions being provided. Keywords: OR in health services; LTC planning; multi-service; uncertainty; equity 1.
Introduction
Long-term care (LTC) includes a broad range of health and social services designed for individuals suffering from chronic illness and/or disabilities (OECD, 2005). LTC is currently provided in many developed countries (Brodsky & Clarfield, 2009), but there are differences in the way they organize 2
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these services, either by: 1) providing care in a variety of settings, including short-term to longer institutionalizations, home-based and ambulatory services; 2) having the provision of care shared between the family, public and private sectors; and 3) establishing different boundaries between the health and social care services (European Commission, 2008; Kraus et al., 2010). For instance, in countries with a National Health Service (NHS), such as Portugal and England, health and social
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LTC services are delivered to the population under different schemes: while health-related services are nearly free at the point of use and financed by taxation, access to social care is usually meanstested (Barros, Machado & Simões, 2011; Boyle, 2011).
One of the most important challenges for European LTC systems is to be responsive to an increasing demand for care (Knapp & Somani, 2009). This is mainly due to the ageing phenomenon,
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to an increase in the prevalence of chronic diseases and to changes in family structures. An adequate supply of care is thus required to address this increasing demand, but most healthcare systems are still ill-equipped to meet that challenge (AARP Public Policy Institute, 2006). This low supply of LTC results in higher costs for the health system, mainly due to an inadequate utilization of acute care services by LTC patients (Mission Unit for Long-Term Care, 2010). Within this setting,
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currently investing in and planning LTC networks represent a health policy priority in many European countries. Additionally, in the context of a strong pressure to decrease public health spending, NHS-based countries (for instance, Portugal) are compelled to minimize the delivery costs,
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while equity becomes a secondary concern. The development of tools to support such decisions is indeed a challenge (Rais & Viana, 2010).
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Mathematical programming models have been widely used in the health care planning literature (Brailsford & Vissers, 2011), but several challenges still need to be addressed in this area. In
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particular, there is a lack of research on developing mathematical programming planning models in the LTC sector and comprehensively considering key features, such as i) the multi-service nature of
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health care services, ii) the effect of uncertainty and of strategic health policy decisions on planning decisions over time, and iii) the joint effect of cost and equity considerations. Our study aims to address this gap by proposing an integrated approach based on a two-stage stochastic mixed integer linear programming (MILP) model – the LTC2SS model, with 2SS standing for Two-Stage Stochastic model – to support the planning of strategic and tactical decisions in the LTC sector. The model is set for the context of an NHS-based country and takes into account cost and equity concerns. The main objective is the minimization of expected cost, while equity 3
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considerations – namely, equity of access, equity of utilization, socioeconomic equity and geographical equity – are modeled as constraints, in the form of satisficing equity levels that need to be respected in LTC delivery. The LTC2SS considers the impact of demand uncertainty and of strategic health policy options in the planning of an LTC network, as well as the multi-service nature of LTC by accounting for a wide range of institutional, home-based and ambulatory services. The
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LTC2SS provides planners (hereafter generically named as decision-makers, DMs) with detailed information regarding: i) where and when should multiple LTC services be located and with which capacities (in terms of beds and human resources); ii) how should this capacity be distributed across services and patient groups; and iii) which changes are needed in this network over time, including capacity adjustments and the opening/closure of services. The LTC2SS model is applied at the county
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level in the Great Lisbon region in Portugal for the 2014-2016 period.
Section 2 of this article presents a brief literature review on related methods. The problem under study is stated in Section 3. The mathematical formulation of the model is presented in Section 4, with the case study and key results being explored in Section 5. Some conclusions and lines for future research are included in Section 6. Literature review
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Mathematical programming models have been extensively used to support location selection and
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capacity planning decisions in a wide range of areas (Arabani & Farahani, 2011; Melo, Nickel & Saldanha-da-Gama, 2009), including in the health care planning literature (Brailsford & Vissers, 2011). In the context of our study, particular attention is given to multi-period and multi-service
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mathematical programming based approaches that account for the effect of uncertainty on planning decisions. Applications in the health care sector in general and in the LTC sector in particular are
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also reviewed.
Location selection and capacity planning under uncertainty
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Planning under uncertainty has been an important research topic in different areas of application, for instance, in general supply chain planning (Georgiadis, Tsiakis, Longinidis & Sofioglou, 2011), in petrochemical industry planning (Verderame, Elia, Li & Floudas, 2010) and in water resource planning (Verderame et al., 2010). Most studies focused on planning at tactical or strategic levels did not comprehensively address uncertainty (Santoso, Ahmed, Goetschalckx & Shapiro, 2005), with some exceptions, as for instance the work by Cardoso et al. (2013).
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When addressing uncertainty, sensitivity analysis has been the simplest approach in use (Owen & Daskin, 1998). Nevertheless, such an approach does not ‘incorporate uncertainty into models proactively’ (p. 424) (Owen & Daskin, 1998). Stochastic and robust approaches are alternative (proactive) approaches (Snyder, 2006; Verderame et al., 2010). While stochastic approaches usually attempt to minimize expected costs (see, for instance, Nickel et al. (2012)), robust approaches often aim at
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minimizing the worst-case performance or regret (see Aissi et al. (2009)). Notwithstanding, since robust approaches are considered to be too conservative (Daskin & Dean, 2004), stochastic models are the most common approach used to deal with uncertainty (Peidro, Mula, Poler & Lario, 2009), and we chose this approach in our study.
Stochastic approaches have been widely used for capacity planning and location selection, with
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particular emphasis devoted to two-stage stochastic approaches (Snyder, 2006; Verderame et al., 2010). Within this literature, location of facilities and capacity levels are typically used as first-stage decisions, whereas allocation decisions are usually defined in the second-stage (Snyder, 2006). Examples are the studies from Alonso-Ayuso et al. (2003) and Mete and Zabinsky (2010). These studies usually consider demand as a main source of uncertainty (Peidro et al., 2009) and propose
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stochastic approaches based on scenario planning, with a typically low number of scenarios being considered (Santoso, Ahmed, Goetschalckx & Shapiro, 2003). Our article uses a stochastic
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optimization approach that is based on some of the before-mentioned features. Location selection and capacity planning in the health sector
Specific features relevant for planning the delivery of health care services have been considered in
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previous mathematical programming models. Multi-objective approaches have been proposed in the last decade, but their number is still low (Mitropoulos, Mitropoulos, Giannikos & Sissouras, 2006;
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Oddoye, Tamiz, Jones, Foroughi & Schmidt, 2007; Smith, Harper & Potts, 2012). Within the objectives most widely used in the literature, equity plays a key role, being a key objective in NHS-
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based systems (Baker, 2000; Ministry of Health, 1990). Different equity concepts have been adopted, namely, equity of utilization (Oliveira & Bevan, 2006), equity of access (Mehrez, Sinuany-Stern, Arad-Geva & Binyamin, 1996; Mestre, Oliveira & Barbosa-Póvoa, 2012), geographical equity (Earnshaw, Hicks, Richter & Honeycutt, 2007) and socioeconomic equity (Drezner & Drezner, 2011). Nevertheless, although it is known that different definitions of equity may conflict with each other (Oliveira, 2003), most studies do not consider the joint effect of multiple equity concepts. In addition to equity, minimization of total costs has also been considered in some studies (Bretthauer 5
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& Coté, 1998; Rowse, Harper, Williams & Smithies, 2013; Stummer, Doerner, Focke & Heidenberger, 2004; Sun, DePuy & Evans, 2014). Few studies have explored the multi-service nature of health service delivery (see, for instance, Santibáñez et al. (2009) and Mestre et al. (2012; 2015)) despite being recognized that proper planning requires modeling this aspect (Hulshof, Kortbeek, Boucherie, Hans & Bakker, 2012). Multi-period
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approaches have also been rarely used for strategic and tactical health care planning, and, to our knowledge, only Santibáñez et al. (2009), Ghaderi and Jabalameli (2013) and Mestre et al. (2015) have developed research in this area. Accounting for the impact of strategic health policy options on planning decisions is also recognized as relevant, but very few studies have explicitly dealt with this issue (for example, Maenhout & Vanhoucke (2013)).
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Most planning studies in the health care area have used deterministic approaches that ‘are not able to adequately model the uncertainties inherent in making real-world strategic decisions’ (Owen & Daskin, 1998) (p. 424). To our knowledge, only Abdelaziz and Masmoudi (2012) and Mestre et al. (2015) have considered the effect of uncertainty in strategic and tactical health care planning decisions. In fact, when it comes to address uncertainty in health care planning, simulation has been the most
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commonly used method (see, for instance, Harper et al. (2005)).
Location selection and capacity planning in the LTC sector
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Few optimization approaches have been proposed for supporting the planning of LTC (Lin, Kong & Lawley, 2012). One of the first models was a single-objective mathematical programming model that
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analyzes the cost-effectiveness of home-based LTC (Greene, Ondrich & Laditka, 1998). Homebased LTC was also the focus of the study developed by Lin et al. (2012). Capacity planning was also addressed in the LTC sector (Lin et al., 2012; Zhang, Puterman, Nelson & Atkins, 2012). The
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location of LTC services has been researched by Cinnamon et al. (2009) and Kim and Kim (2010). Shroff et al. (1998) and Cardoso et al. (2015) have also studied the location of institutional LTC
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facilities and, to our knowledge, these authors have proposed the only multi-objective approaches in this area. None of these studies has, however, considered the multi-service nature of LTC services, as well as the impact of uncertainty and of adopting different health policy options in LTC delivery. Summing up, the main features considered as relevant for planning the delivery of health care services in general, and of LTC services in particular, are identified in Table 1. It can be observed that no study has addressed these features altogether, and so the LTC2SS model proposed in our study attempts to fill this gap in the literature. 6
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Table 1 Selected features of health (in general) and LTC studies using mathematical programming models
Oliveira & Bevan (2006) Santibáñez et al. (2009) Abdelaziz & Masmoudi (2012) Mestre et al. (2012) Ghaderi & Jabalameli (2013) Maenhout & Vanhoucke (2013) Mestre et al. (2015) Lin et al. (2012) Greene et al. (1998) Zhang et al. (2012) Cinnamon et al. (2009) Kim & Kim (2010) Shroff et al. (1998) Cardoso et al. (2015) LTC2SS
Multiperiod
Multiservice
Policy options’ impact
LTC sector applications
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3.
Multiple equity concepts
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Planning under uncertainty
LTC planning background
The present study develops a model to support planning decisions in the LTC sector at strategic and
LTC networks’ key features
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tactical levels with background information being presented in this section.
We consider an LTC network that is operating in the context of an NHS-based system. Such a system can be found in Portugal (Barros et al., 2011) and in England (Boyle, 2011), among other
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countries. We describe the key features of the LTC network currently operating in mainland Portugal, which we will use as a reference network. The National Network of Long-Term Care (Rede
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Nacional de Cuidados Continuados Integrados, RNCCI) was created in Portugal in 2006 so as to ensure the provision of health and social care services for the frail elderly, the highly disabled and/or the
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severely ill (Boyle, 2011; Mission Unit for Long-Term Care, 2009a). Our study is restricted to the health care component of LTC, for which, in theory, there should be universal coverage and nearly free access at the point of use. LTC is delivered by multidisciplinary teams of health professionals (such as physicians and nurses) within a mix of public and private organizations, which provide a wide range of institutional (IC), home-based (HBC) and ambulatory (AC) care services (Ministry of Health, 2006). IC comprises services characterized by different lengths of stay (LOS), integrating the delivery of convalescence 7
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care (CC), medium-term and rehabilitation care (MTRC), long-term and maintenance care (LTMC) and palliative care (PC) for individuals with a variety of conditions who cannot receive care at home. HBC is provided at patients’ residences to individuals whose conditions do not justify the admission into an IC service but are not autonomous to perform their daily activities. AC (such as day centers) is provided to patients whose conditions do not justify the provision of IC or HBC. Regarding the
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role of the state, on the one hand, it is mainly responsible for establishing contracts with IC providers (either public or private, but mainly private non-profit institutions), paying them according to the utilization of services (Mission Unit for Long-Term Care, 2009b). On the other hand, the state assumes a much more active role in providing HBC and AC within the scope of primary health care centers (PHCCs) (ERS, 2013; Ministry of Health, 2006). Individuals can have access to these services after being evaluated in an NHS hospital or by a primary care provider, being then allocated to the
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closest available service (Mission Unit for Long-Term Care, 2009a). Planning decisions
The following planning decisions are considered in this study:
Where and when should new IC services be located, both in existing and new locations, and
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with how much bed capacity? And which is the additional bed capacity in which one should invest over time? Location decisions are not considered for HBC and AC, as these services
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are provided by health professionals from PHCCs that are located in all small area units in European NHS-based countries (Barros et al., 2011; World Health Organization, 2013)
How many hours of care provided by different health professionals should be available for
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providing IC, HBC and AC over time? How should available resources (in particular, beds) be (re)distributed across services,
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locations and patient groups over time? Policy objectives
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According to policy statements and health policy literature, key policy objectives include ensuring universal LTC coverage, promoting equality of access to health care for all citizens, promoting equity in the distribution and utilization of resources, and ensuring access for the socioeconomic groups with lower levels of income (Baker, 2000; Barros et al., 2011). Nevertheless, in the current European context of severe budget cuts, there is a very high pressure to place cost considerations as a priority in the (re)organization of LTC delivery. Satisficing levels of equity in several dimensions should also be imposed, aiming to ensure: 8
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i.
Equity of access: patients should receive the care they need as close as possible to their place of residence, with travel time not exceeding a maximum level across regions (according to the DMs’ point of view);
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Equity of utilization: unsatisfied demand (usually named as unmet need in the health policy literature) for different services should not exceed a maximum level (according to the DMs).
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This corresponds to the imposition of a minimum service level across services, avoiding in this way that only the cheapest services are delivered (this is expected when cost minimization is pursued); iii.
Socioeconomic equity: unsatisfied demand for the population groups with lower income should not exceed a maximum level (according to the DMs), thus avoiding financial dependency situations;
Geographical equity: unsatisfied demand across geographical areas should not exceed a
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iv.
maximum level (according to the DMs), thus ensuring a minimum level of service across regions and, consequently, avoiding a total lack of service provision in some geographic areas.
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We term these levels as satisficing levels (a concept introduced by Simon (1956)), as in times of economic and financial crisis, the aim is often to minimize costs while respecting acceptable (non-
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optimal) levels of equity, rather than maximizing equity. Structuring policy decisions
Different strategic policy decisions in a health system may substantially impact the LTC sector. For
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instance, the Portuguese government has recognized the relevance of (Central Administration of the Health System, 2013; Ministry of Health, 2006): converting acute hospitals into LTC units,
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transferring resources (human [HR], material [MR] and financial [FR] resources) from the acute care to the LTC sector, and changing the LTC provision paradigm from an institutional towards a
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community-oriented paradigm (i.e., substituting IC by HBC). Hence, different combinations of these strategic decisions should be analyzed. Using a strategy generation table as designed by Kirkwood (1997), we have identified six meaningful combinations, denoted by policy strategies (PS) I to VI (Fig. 1).
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Fig. 1 Strategy generation table built for identifying policy strategies. Legend: PS – policy strategy; HR – human resources; MR – material resources; FR – financial resources
As an example, Fig. 1 should be read as follows: PS I (represented by a dark circle) results from combining the following decisions: 1) converting an acute hospital into an LTC unit, 2) transferring
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resources from the acute care to the LTC sector, and 3) following a community-based paradigm. Structuring uncertainty
Future LTC demand is difficult to foresee with confidence, depending on changes in epidemiological and demographic profiles and in the availability of informal care (Pickard, Jiménez-Martin, Sánches & Prieto, 2011). Our study addresses data uncertainty associated with future LTC demand in terms
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of a) the number of individuals requiring LTC in the future, and b) the amount of services required by those individuals, as captured by the LOS (note that this component is not strictly a demand
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parameter, being also influenced by supply). We assume that the probability distributions associated to these uncertain parameters are known, with a scenario tree approach (Birge & Louveaux, 1997) being selected to handle that uncertainty in the context of a stochastic programming model. A
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scenario tree with a finite number of scenarios is thus built, with the number of scenarios depending on the number of different realizations considered for each of the uncertain parameters (for more
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details see Cardoso et al. (2013)).
Within the group of mathematical programming methods based on scenario planning, a two-stage
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(MILP) stochastic approach (Birge & Louveaux, 1997) is explored, given that in the LTC context it is natural to consider as first-stage decisions the opening and closure of services (i.e., location decisions), and then allocation and reallocation of resources and individuals as second-stage (as recourse actions) (Snyder, 2006). First-stage decisions should also comprise the investment in new beds (similarly to what has been done in Nickel et al. (2012) in the context of a general supply chain network) because these decisions take time to implement, require significant resources and need to remain valid for longer periods of time (Ministry of Health, 2006). The objective of the LTC2SS 10
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model is thus to minimize the total expected cost over the uncertain demand scenarios, while ensuring the attainment of satisficing levels of equity. Fig. 2 shows the key features of the proposed LTC2SS model, being read as follows – departing from an initial network of care, one should consider: i) closing/opening services; ii) assigning patients to the required service (solid arrows represent patients receiving LTC in the associated service [for IC
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and AC], and dashed arrows depict the provision of LTC at patients’ homes [for HBC]); iii) determining how many beds and human resources are needed per service; and iv) reallocating beds
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across services and locations (dotted arrows). All these decisions are evaluated over time.
Fig. 2 Representation of how the LTC2SS model plan changes in an existing network of care. Legend: IC – institutional
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care; HBC – home-based care; AC – ambulatory care; CC – convalescence care; PC – palliative care
Building the LTC2SS model
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This section presents the mathematical details of the LTC2SS model. The notation used for the model
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formulation is presented in Tables 2-5. Table 2 List of indices
d g l, j r
s, p t, w n, k
Indices Demand points Socioeconomic groups Locations for services Human resources LTC services Time periods Scenario tree nodes
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Table 3 List of sets Sets
L LI LO
S S I SO A s, r : s S , r R F d , l : d D, l L
C k , n : k , n N Q t , n : t T , n N U s, l : s S , l L V s, l : s S , l L M s, l : s S , l L
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G G P GW
Set of demand points Set of human resources Set of time periods Set of scenario tree nodes Set of socioeconomic groups, divided into subsets GP (groups of individuals with priority as a result of having lower levels of income, gGPG) and GW (groups of individuals without priority as a result of having higher levels of income, gGWG). Set of locations for services, divided into subsets LI (subset of locations for IC services, l,jLIL) and LO (subset of locations for HBC and AC services, l,jLOL) Set of LTC services, divided into subsets SI (subset of IC services, s,pSIS) and SO (subset for HBC and AC services, s,pSOS) Set of LTC services s provided by human resources r Set of demand points d that can receive LTC in locations l Set of predecessors k of the scenario tree node n Set of time periods t associated to scenario tree nodes n Set of LTC services s that can be provided in locations l Set of services s provided in locations l at the beginning of the planning horizon Set of services s not provided in locations l at the beginning of the planning horizon
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Table 4 List of parameters
Parameters Number of inpatient beds available in IC service s (sSIS) located in l (lLIL) at t=0
bc s / bc s
Minimum/maximum bed capacity allowed per IC service s (sSIS)
hcrsl
Number of hours of care provided by human resource r in service s located in l at t=0 Number of individuals from demand point d and socioeconomic group g requiring service s at t in scenario tree node n Number of individuals belonging to the lower income groups (gGPG) requiring LTC at t in scenario tree node n Number of individuals from demand point d requiring LTC at t in scenario tree node n Number of individuals requiring service s at t in scenario tree node n
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ibsl
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idgstn
iLtn
iS stn ni s / ni s
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LOSsn
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iPdtn
rt
icst
AC
ocst
rDst / rSst
rs
s dl
tn
st
Minimum/maximum number of individuals allowed per HBC/AC service s (sSOS)
Average length of stay (LOS), as measured by the number of days in IC service s (sSIS) in scenario tree node n Inflation rate at t
Investment cost per new bed installed in IC service s (sSIS) at t Operational cost per service s per period t Cost of reallocating a bed to IC service s (sSIS) from a service delivered in a different location / from another service delivered in the same location at t Recommendation for the hours of care to be provided by human resource r for each individual requiring service s Utilization factor associated with the provision of service s Travel time between demand point d and service location l (in minutes) Maximum total travel time (in minutes) at t in scenario tree node n Maximum travel time allowed for LTC patients accessing institutional services (in minutes) Minimum level of demand that need to be satisfied per service s at t Number of days per time period 12
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High value auxiliary coefficient Probability of scenario tree node n Satisficing levels defined for equity of access (EA)/equity of utilization (EU)/geographical equity (GE)/socioeconomic equity (SE) at t (defined as proportions)
n SE tEA / tEU / GE t / t
Table 5 List of variables First-stage variables Equal to 1 if service s is located in l at t; 0 otherwise
X slt
Number of additional beds to invest in for IC service s (sSIS) located in l (lLIL) at t Second-stage variables Equal to 1 if at least one bed is reallocated to IC service s (sSIS) located in l (lLIL) at t in scenario tree node n; 0 otherwise Equal to 1 if there is need to invest in new beds for IC service s (sSIS) located in l (lLIL) at t; 0 otherwise Number of beds to be made available for individuals from demand point d belonging to socioeconomic group g and receiving IC service s (sSIS) located in l (lLIL) at t in scenario tree node n Number of beds reallocated to/from IC service s (sSIS) located in l (lLIL) from/to IC service p (pSIS) located in j (jLIL) at t in scenario tree node n Proportion of individuals from demand point d and socioeconomic group g receiving service s in location l at t in scenario tree node n Number of individuals belonging to the lower income groups receiving LTC at t in scenario tree node n Number of individuals from demand point d receiving LTC at t in scenario tree node n Number of individuals receiving service s at t in scenario tree node n Number of hours of care provided by human resource r for individuals from demand point d belonging to socioeconomic group g and receiving service s located in l at t in scenario tree node n Number of additional hours of care that need to be provided by human resource r in service s located in l at t in scenario tree node n Number of hours of care provided by human resource r that are no longer required in service s located in l at t in scenario tree node n
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ABslt Ysltn Z slt
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Bdgsltn BTslpjtn / BFslpjtn Rdgsltn
RLtn RPdtn
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RSstn HCrdgsltn
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AH rsltn EH rsltn
P Otn
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TItn / TOtn
Total travel time (in minutes) at t for scenario tree node n Penalty (in minutes) attributed to individuals not receiving institutional care at t in scenario tree node n Total investment/operational cost at t for scenario tree node n
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T Otn
Objective function
Equations (1-3) present the model objective function that considers the minimization of the
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expected cost associated with LTC provision for the entire planning period, including both investment and operational costs. Investment costs are related to the investment in new beds (first term in Equation (2)) and to the reallocation of beds between services in different locations (second term) and in the same location (third term). Operational costs include costs associated to the operation of beds in IC services (first term in Equation (3)) and to the provision of HBC and AC services (second term).
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Min
n TI tn TOtn n N t :(t , n)Q
(1) t , n Q (2)
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ic rDst rS TI tn ABslt st t BTslpjtn BTslpltn st t t I I I I I 1 rt pS jL 1 rt 1 rt sS l L pS l :( s, l )U j :( p, j )U p :( p, l )U j l oc oc TOtn Bdgsltn st t Rdgsltn idgstn st t 1 rt sS O lLO 1 rt dD gG sS I lLI l:( s,l )U l:( s,l )U l:( d ,l )F l:( d ,l )F
Constraints
t , n Q
(3)
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Several constraints are defined: i) opening and closure of services; ii) minimum level of demand satisfaction; iii) assignment of patients; iv) resources’ requirements; v) capacity; vi) resources reallocation; vii) equity satisficing constraints; and viii) variable domains. These constraints are formulated below.
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Opening and closure of services constraints
Constraints (4-5) state that the opening/closing of an IC service is not allowed after deciding upon closing/opening it in a previous time period. No openings or closures are considered for HBC and
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AC services as these services are provided within the scope of a primary care network, and is assumed that all the population has a PHCC structure in their area of residence (Constraint (6)). (4)
X slw X slt s S I , l LI : s, l U M , t , w T , w t
(5)
CE
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X slw X slt s S I , l LI : s, l U V , t , w T , w t
X slt 1 s S O , l LO : s, l U , t T
(6)
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Minimum level of demand satisfaction constraints Constraints (7-8) ensure that a minimum level of satisfied demand per service s (captured by the parameter st) is reached per time period t.
n:t , n Q
RSstn
RSstn st s S , t T iSstn
n
(7)
Rdgsltn idgstn s S , t , n Q d D g G l :s, l U l :d , l F
(8)
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Assignment of patients’ constraints The assignment of patients is defined by Constraint (9-11). Particularly, Constraint (9) states that individuals will receive service s in locations where that service is provided; and Constraints (10-11) ensure that individuals in each demand point d will receive LTC in the closest available service, given that in an NHS-based system services should be delivered as close as possible to where demand is
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located (Barros et al., 2011). According to Constraint (10), on the one hand, if service s is provided in location l, and location l is closer to demand point d than another location j, the patients from d should not receive LTC at j. On the other hand, Constraint (11) ensures that patients will not receive LTC in locations situated beyond a maximum travel time (). Notwithstanding, there may be situations in which these conditions should not apply. For instance, when considering the context of an NHS-based country where the primary care network is already established and where HBC and
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AC should be provided within the PHCC belonging to the place of residence of the populations (such as in Portugal), there is no need to apply these conditions for both HBC and AC (Barros et al., 2011; World Health Organization, 2013).
(9)
X slt Rdgsjtn 1 g G, s S , l , j L : s, l U , s, j U , d , l F , t , n Q, dl dj , l j
(10)
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Rdgsltn X slt g G, s S , l L : s, l U , d , l F , t , n Q
Rdgsltn 0 g G, s S , l L : s, l U , d , l F , t , n Q, dl
(11)
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Resources requirements constraints
Constraint (12) calculates the number of beds required at t for IC service s located in l for individuals
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from demand point d and socioeconomic group g for scenario tree node n; and Constraint (13) states that the total bed capacity that needs to be in place in each service and location cannot be less than
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the bed capacity required in each scenario (similarly to Birge and Louveaux (1997)). Similarly, Constraint (14) determines the number of hours of care that should be provided by human resource r; and Constraint (15) establishes a balance between the hours of care provided by existing human
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resources (first term), additional hours of care that need to be provided (second term) and hours of care that are no longer required (third term) for each type of human resource r. A utilization factor (s) is included in Constraints (12) and (14), since a full occupancy of services is not expected in the health care sector. This is in line with literature that has pointed out that health facilities operating with high occupancy rates (usually higher than 90%) are expected to face multiple problems, including regular bed crisis, thereby affecting ‘key measures of ‘poor performance’ such as the waiting time to find a bed, staff stress, dissatisfaction’ (Jones, 2011) (p. 243). In our model, this utilization factor takes the 15
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value of 1 when occupancy rates are expected to equal 100%, and lower values for other cases. This factor also varies depending on how the state pays providers, i.e., whether according to the utilization of services or to installed capacity. Bdgsltn
LOSsn
Rdgsltn idgstn
1
s
g G, s S I , l LI : s, l U , d , l F , t , n Q
(12)
ABslt
HCrdgsltn Rdgsltn i dgstn
HCrdgsltn
d :( d , l )F g G
g G, d , l F , s, l U , s, r A, t , n Q
(14)
s, l U , s, r A, t , n Q, t 1 hcrsl AH rsltn EH rsltn HC AH EH rdgsl t 1k rsltn rsltn s, l U , s, r A, t , n Q, t 1 (15) d :d , l F g G k :(n, k )C
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rs s
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(13)
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B ibsl BTslpjtn BFslpjtn s S I , l LI ; s, l U , t , n Q, t 1 dgsltn I I L d D g G pS j :( pj , j )U Bdgsltn Bdgsl (t 1)k BTslpjtn BFslpjtn s S I , l LI : s, l U , t , n Q, t 1 I k :( n, k )C j LI d :d , l F g G pS j :( p, j )U
Capacity constraints
Maximum and minimum capacity constraints are imposed for LTC services through Constraints (16-
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17). While Constraint (16) defines the maximum and minimum bed capacity per IC service, Constraint (17) defines the maximum and minimum number of individuals in need that should be
Bdgsltn
d :(d , l )F g G
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bc s X slt
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assigned per HBC and AC services.
ni s X slt
bc s X slt s S I , l LI : s, l U , t , n Q
Rdgsltn idgstn
d :(d , l )F g G
ni s X slt s S O , l LO : s, l U , t , n Q
(16) (17)
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Resources reallocation constraints Constraints (18-21) model the reallocation of beds within the network of IC services. According to Constraint (18), the number of beds reallocated to service s located in l from service p located in j must be equal to the number of beds removed from service p located in j to service s located in l (and this for each scenario tree node n). Constraints (19-20) define the maximum number of beds that can be reallocated from and to service s. Constraint (21) states that reallocating beds between services provided in different locations is only allowed in the first time period (note that reallocating beds 16
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across services in the same location is always allowed). This is essential to cope with an initial distribution of services that may be far from the optimal one. BTslpjtn BFpjsltn
s, p S I , l , j LI : s, l U , p, j U , (t , n) Q
(18)
pS
I
jL j : p, j U I
BTslpjtn
Bdgsltn
d :d , l F g G
s S I , l LI : s, l U , t , n Q
BTslpjtn 0 s, p S I , l , j LI : s, l U , p, j U , t , n Q, l j, t 1
(19)
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ib s S I , l LI : s, l U , t , n Q, t 1 sl BFslpjtn Bdgsl (t 1)k s S I , l LI : s, l U , t , n Q, t 1 p S I j LI d :d , l F g G k :n, k C j : p, j U
(20) (21)
Under certain circumstances, an additional set of reallocation conditions can be required in order to
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ensure the stability of the LTC system, so as to avoid excessive reallocations within the LTC network. For that purpose, Constraints (22-23) are used to avoid a removal of beds from a given service and location whenever a reallocation of beds takes place to that service and location (Ysltn=1); and Constraints (24-25) are used to avoid a removal of beds whenever additional beds are added in that service and location (Zslt=1). The need for using these conditions arises when the state finances
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only the effective delivery of services, not being responsible for supporting extra payments associated with the reallocation of beds. Under these particular circumstances, the parameters rDst
j LI j :( p, j )U
jL j :( p, j )U I
BFslpjtn 1 Ysltn s S I , l LI : s, l U , t , n Q
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pS
I
BTslpjwn Ysltn 0 s S I , l LI : s, l U , t , n Q, w, n Q, w t
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pS I
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and rSst in the objective function are set to zero, and the stability of the system is no longer ensured. (22)
(23)
(24)
BFslpjtn 1 Z slt s, p S I , l , j LI : s, l U , p, j U , t , n Q
(25)
AC
ABslw Z slt 0 s S I , l LI : s, l U , t , w T , w t
Equity satisficing constraints Four satisficing equity levels are considered in this study: equity of access (EA; Constraints (26-28)), equity of utilization (EU; Constraint (29)), socioeconomic equity (SE; Constraints (30-31)) and geographical equity (GE; Constraints (32-33)). The measures selected for defining EA, EU, SE and GE are expressed as follows:
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i.
EA: in order to ensure that LTC provision takes place as close as possible to the patients’ place of residence, it is defined that the expected travel time per patient accessing IC should not exceed the satisficing level defined for EA for each time period t (Constraint (26)). The use of the penalty OPtn (Constraint (28)) penalizes individuals without access to IC, by assuming that these individuals incur the maximum travel time;
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ii. EU: the expected proportion of individuals in need of each type of service and not receiving it should not exceed the satisficing level defined for EU for each time period t (Constraint (29)). This condition ensures a minimum service level (in comparison to need) for each LTC service, thus avoiding a total lack of provision of the most expensive services. Disregarding this objective can create high variations in the utilization of different types of LTC services; iii. SE: the expected proportion of lower income individuals not receiving LTC should not
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exceed the satisficing level defined for SE for each time period t (Constraint (30)). This will ensure a minimum level of service for those with lower ability to pay for care, thus avoiding situations of lack of provision due to poverty;
iv. GE: the expected proportion of individuals belonging to each geographical area not receiving LTC should not exceed the satisficing level defined for GE for each time period t (Constraint
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(32)). Accordingly, a minimum service level across geographical areas is ensured, avoiding in this way a total lack of provision in some regions.
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In summary, the satisficing equity levels correspond to levels that need to be respected in a certain period and in the context of very scarce resources. Such satisficing levels should be defined by the
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DMs (Simon, 1956). T
P
O Otn n tn tEA t T tn n:t , n Q
(26)
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d D g G sS I
AC
T Otn
l LI l :( s, l )U l :( d , l )F
Rdgsltn idgstn dl t , n Q
(27)
P Otn idgstn Rdgsltn idgstn t , n Q d D g G sS I l LI l :s, l U l ; d , l F
n:t , n Q
n 1
RSstn iS stn
tEU s S , t T
(28)
(29)
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n:t , n Q
RLtn iLtn
tSE t T
(30)
Rdgsltn idgstn t , n Q d D g G P sS l :s, l U l :d , l F
n:t , n Q
RPdtn
n 1
RPdtn iPdtn
(31)
GE d D, t T t
(32)
Rdgsltn idgstn d D, t , n Q g G sS l :s, l U l :d , l F
(33)
Variable domains constraints Finally, the variables domains are given by Constraints (34-46). X slt 0,1 s S , l L : s, l U , t T
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(34)
Ysltn 0,1 s S I , l LI ; s, l U , t , n Q Z slt 0,1 s S I , l LI : s, l U , t T s S I , l LI ; s, l U , t T
Bdgsltn
g G, s S I , l LI : s, l U , d , l F , t , n Q
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ABslt
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RLtn
n 1
(35) (36) (37) (38) (39)
RLtn t , n Q
(40)
RPdtn
d D, t , n Q
RSstn s S , t , n Q
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BTslpjtn , BFslpjtn s, p S I , l , j LI : s, l U , p, j U , t , n Q
(41) (42) (43)
AHrsltn , EHrsltn 0 s S , l L : s, l U , s, r A, t , n Q
(44)
CE
PT
g G, s S , l L : s, l U , d , l F , t , n Q
Rdgsltn 0
(45)
T P Otn , Otn , TI tn , TOtn 0 t , n Q
(46)
AC
HCrdgsltn 0 g G, s S , l L : s, l U , d , l F , s, r A, t , n Q
5.
Case study
In this section we apply the LTC2SS model to real data from a Portuguese region to illustrate how it can support planning decisions in the LTC sector. Specifically, the model is applied to the county level in the Great Lisbon region – to an LTC network that is under restructuring – in the 2014-2016 period. Notwithstanding, the model can be run for a longer time span.
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5.1.
Model implementation and dataset used
The LTC2SS model was implemented in the General Algebraic Modeling System (GAMS) 23.7 and was solved with CPLEX 12.0 on a Two Intel Xeon X5680, 3.33GHz computer with 12GB RAM. The dataset includes:
LTC supply at the end of 2013 in the Great Lisbon region (Mission Unit for Long-Term
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Care, 2013);
Operational and investment costs (Algarve Health Region, 2004; Central Administration of the Health System, 2013; Portuguese Observatory on Health Systems, 2011). Reallocation costs are not taken into account (i.e., rDst=rSst=0), since the Portuguese state finances mainly IC (Barros et al., 2011);
Travel time (in minutes) between each Great Lisbon county and each LTC service
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(ViaMichelin, 2013), and maximum travel time allowed per patient requiring IC (Health Regulation Authority, 2011);
Number of hours of care that need to be provided by physicians and nurses per individual receiving each type of LTC service (ERS, 2013);
Utilization factor associated with the provision of services with a score of 1, given that
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providers in Portugal are paid according to utilization (Mission Unit for Long-Term Care,
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2009b);
Satisficing equity levels defined by the Head of the Long-Term Care Coordination Team of the Lisbon and Tagus Valley Regional Health Authority (LTV RHA), who has participated in
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this study in the role of a real DM operating in the LTC sector (we hereafter refer to her as our DM). In particular, our DM defined that within three years (i.e., at the end of 2016 [t=3])
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patients should travel no more than 15 minutes to access the care they need (corresponding to EAt=3=0.5). Within the same time frame, our DM also defined that the level of unsatisfied
AC
demand should not exceed: a) 80% across LTC services (EUt=3=0.8), b) 25% for the lower income groups (SEt=3=0.25), and c) 50% across geographical areas (GEt=3=0.5). Smooth equity improvements were also imposed along the planning horizon, so as to ensure that changes to the network are gradual.
Uncertainty was estimated for two parameters, namely:
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i.
The number of individuals requiring IC, HBC and AC, disaggregated by socioeconomic groups (very low income [VLI] and not very low income [NVLI]), as predicted by the detailed simulation model developed by Cardoso et al. (2012);
ii.
The LOS associated with each type of IC service estimated with data from the Central Administration of the Health System (2013).
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A scenario tree with 81 scenarios is used to describe these uncertain parameters. The probability distributions associated with both parameters made use of data from the Central Administration of the Health System (2013) and Cardoso et al. (2012). Data was then converted into three scenarios using the extended Pearson-Tuckey method (Clemen & Reilly, 2003). The whole dataset is available from the authors, upon request. Selected results
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5.2.
Several analyses are performed in this study. In Planning Context I, the PS IV from Fig. 1 is selected for analysis (representing the baseline policy strategy) and results from running the LTC2SS model are compared with the results obtained for the deterministic case (i.e., when considering average values for the number of individuals in need of LTC and LOS). In Planning Context II, the model is run for
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each one of the policy strategies identified in Fig. 1, and the results are compared in terms of the costs that need to be incurred for reorganizing the network, and also in terms of key changes to be
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implemented in the network. In Planning Context III the cost of ensuring equity satisficing levels is estimated.
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5.2.1. Planning Context I
This analysis explores how the LTC network should evolve when there is uncertainty in the demand for LTC services, and when the PS IV from Fig. 1 is selected, corresponding to a planning context
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with no conversion of acute hospitals into LTC units, with transfer of financial resources allowed from the acute to the LTC sector, and with an institutional-based paradigm.
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According to the model results, LTMC and PC are the types of IC services with the highest and lowest bed capacity requirements throughout the planning horizon (detailed results available in Cardoso et al. (2013)), respectively. This can be explained by differences between their LOS when compared to the remaining IC services (Mission Unit for Long-Term Care, 2012). Regarding PC services, these are the type of IC services associated with lower levels of unsatisfied demand. Fig. 3 shows which changes are needed in the provision of PC services, in terms of service location, bed capacity and allocation of individuals in need to services, over the 2014-2016 period under Planning 21
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Context I (i.e., when uncertainty is accounted for; Fig. 3a)) and under a deterministic case (Fig. 3b)) (note that more details on the results obtained under a deterministic case can be found in Cardoso et al. (2013)). For simplification purposes, institutions located in each county are numbered – e.g., two
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institutions are located in Lisbon and are named Lisbon (1) and Lisbon (2).
Fig. 3 Changes in the provision of palliative care (PC) services within the LTC network, in terms of services’ location,
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bed capacity and allocation of individuals in need to services (given by arrows) under a) Planning Context I and b) deterministic case
Results show significant differences in the opening and closure of services when uncertainty is
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modeled. In particular, it can be seen that under a deterministic case a new PC service should be opened in Sintra (1) (Fig. 3b)) in 2016, whereas this opening should not take place under Planning
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Context I (Fig. 3a)). Differences on the allocation of patients to services and on the bed capacity used in each service are also observed from Fig. 3: these differences also vary with the scenario under analysis, with Fig. 3a) only reporting results for one specific scenario. For a more detailed analysis on the bed capacity in use in each IC service when uncertainty is modeled, see the unit in Lisbon (2) as an example: the average values shown in Fig. 4 correspond to the expected bed capacity in use, whereas extreme values correspond to the associated maximum and minimum values. As expected, these results show that the impact of uncertainty increases as time passes. 22
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Fig. 4 Bed capacity in use in convalescence care (CC), medium-term and rehabilitation care (MTRC), long-term and maintenance care (LTMC) and palliative care (PC) services provided in Lisbon (2) under Planning Context I
Additional information useful for planning is also generated by the model: Fig. 5 shows the distribution of PC bed capacity in Lisbon (2) across individuals belonging to different socioeconomic
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groups (VLI and NVLI groups). It can be read that the model is discriminating towards individuals
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with VLI (via the use of the satisficing SE level).
Fig. 5 Bed capacity in use in the palliative care (PC) service provided in Lisbon (2) for very low income (VLI) and not
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very low income (NVLI) individuals under Planning Context I
Although the capacity in use in each service varies across scenarios (as shown in Figs. 4 and 5), the
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number of new beds in which there is a need to invest for each service is defined in the first-stage, taking the same value across scenarios, as shown in Table 6.
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Table 6 Additional beds in which there is a need to invest under Planning Context I and under a deterministic case
CC MTRC LTMC PC Total additional bed capacity
2014 35 65 135 5
Planning Context I 2015 2016 269 212 535 232 325 680 84 84 2,661
2014 284 110 216 108
Deterministic case 2015 2016 122 22 347 397 151 515 11 29 2,312
Table 6 compares the number of new beds that are needed for all the types of IC services when uncertainty is modeled and for the deterministic case, showing that a greater investment in new beds 23
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is required when uncertainty is accounted for. These results suggest that the bed capacity should increase significantly so as to ensure that all the equity satisficing levels are achieved, and an even higher increase would be needed to ensure that all the demand for LTC is satisfied. These results indicate that current supply levels are far from answering the LTC demand in the Great Lisbon region.
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With respect to HBC and AC, Table 7 summarizes the total number of hours of care to be provided by physicians and nurses in each one of these services when uncertainty is accounted for. Similar results to the ones observed for IC provision were obtained for HBC and AC services: a higher amount of resources is required to cope with an increasing and uncertain demand over time.
Table 7 Evolution of the total number of hours of care (expected values) to be provided by physicians and nurses for 2014 Physicians Nurses
HBC 21,419 230,664
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home-based care (HBC) and ambulatory care (AC) provision under Planning Context I 2015
AC 312,199 3,362,142
5.2.2. Planning Context II
HBC 33,623 362,094
AC 404,939 4,360,879
HBC 47,906 515,913
2016 AC 526,210 5,666,879
Table 8 summarizes the results obtained when the LTC2SS model is run for all the policy strategies
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identified in Fig. 1.
Policy strategies
Investment costs
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Table 8 Key results obtained when different policy strategies (PS) are followed Costs Operational costs
Total costs
Amount to be transferred from the acute care to the LTC sector
€108M
€126M
€36M
€18M
PS II
€37M
€153M
€190M
€100M
PS III
€27M
€118M
€145M
€55M
PS IV
€53M
€170M
€223M
€133M
CE
PT
PS I
Key changes in the LTC network until 2016 10 new LTC services 9 LTC services closed 10 new LTC services 8 LTC services closed 9 new LTC services 8 LTC services closed 8 new LTC services 6 LTC services closed
AC
From Table 8 it can be seen that running the model under PS IV results in the highest investment and operational costs. This is an expected result since under this PS it is not possible to substitute IC by HBC, with this substitution being the key for saving some money – this is confirmed by the relatively lower operational costs obtained when running the model under PS III. Also, no hospital conversion is allowed, meaning that it is not possible to take advantage of existing structures with bed capacity installed for IC provision. Using existing structures with bed capacity installed in the acute care sector for IC provision is however allowed under PS I and II, thus resulting in lower 24
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investments (when compared to PS III and IV, respectively). This lower investment is related to the assumption that Maternidade Dr. Alfredo da Costa, which is an NHS hospital located in the Lisbon county, will be closed (note that the closure of this unit is currently under evaluation (Ministry of Finance, 2013)) and converted into an LTC unit. Accordingly, this location is a new potential location in the model, being able to deliver 209 beds for IC provision, with no extra investment
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being needed for this bed capacity. One should be aware that the total costs shown in Table 8 are significantly above the current budget available for operations and investments in the LTC sector in the Great Lisbon area – according to the Ministry of Labor and Social Solidarity (2011), the current budget is €5M and €25M for investments and operations per year, respectively, representing a total budget of €90M for a three-
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year period. Accordingly, the reorganization of the LTC network and the associated achievement of the satisficing equity levels defined by our DM are dependent on the transfer of financial resources from the acute care to the LTC sector – e.g., the reorganization of the network under PS I requires the transfer of €36M for the LTC sector.
PS V and VI are not included in Table 8 because it is not possible to reorganize the LTC network
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and achieve the satisficing equity levels defined by our DM under these policy strategies. This is because no transfer of financial resources is allowed under these strategies and the current budget satisficing equity levels.
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available for the LTC sector in the Great Lisbon area is far from the one required to achieve these
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5.2.3. Planning Context III
It is relevant for DMs in the LTC sector to know the costs associated with ensuring equity satisficing levels. For that purpose, the LTC2SS model was run for the baseline policy strategy (PS IV) and by
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relaxing each of the equity satisficing level constraints (cases a), b), c) and d)) and then by relaxing all the equity satisficing levels simultaneously (case e)). The cost of equity is then determined by
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computing the difference between the cost obtained when all the equity satisficing levels are imposed, and the cost obtained under cases a), b), c), d) and e). Table 9 presents the key results for these five equity costs, providing estimates of the cost required to ensure the achievement of the corresponding satisficing equity level.
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Table 9 Estimates for the cost of ensuring equity satisficing levels a) b) c) d) e)
Cost of equity Cost of ensuring equity of access satisficing level (EA) Cost of ensuring equity of utilization satisficing level (EU) Cost of ensuring socioeconomic equity satisficing level (SE) Cost of ensuring geographical equity satisficing level (GE) Cost of ensuring all the equity satisficing levels
€108M €1M €3M €3M €144M
As expected, the total cost of equity is significantly higher than the cost of ensuring each equity
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satisficing level individually. When considering each equity level individually, the highest cost is observed for equity of access (EA), and the lowest for equity of utilization (EU). The lower cost obtained for EU may be explained because the minimum level of demand satisfaction (imposed for all the cases) is approximate to the satisficing level for EU. Results show that if the DM in the LTC sector disregards the achievement of SE satisficing levels, a reduction of €3M in costs can be
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achieved; nevertheless, by neglecting SE, the model will not discriminate between locating services in areas with a higher proportion of individuals with lower income.
The computational results of the above cases are detailed in Appendix A, showing a significant growth in the number of variables and constraints when an extra location for IC provision is considered under Planning Context II (PS I and II). Summing up, analyses show that a significant
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reorganization and increase in installed capacity (both in terms of beds and human resources) in the LTC network of the Great Lisbon region needs to be carried out. Reorganization of the network and
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associated costs will depend on strategic policy options for the health care sector, there being naturally a cost for ensuring satisficing equity levels. Conclusion
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6.
The increasing need for LTC, along with an inadequate supply of LTC services and with strong
CE
budget constraints that many European countries currently face, make LTC planning a priority in the health policy agenda of many countries. This study proposes an integrated approach based on a
AC
stochastic mixed integer linear programming (MILP) model – the LTC2SS model – to assist the planning of networks of LTC services in the context of NHS-based countries, both at a strategic and tactical level. The model considers the impact of demand uncertainty and of different strategic policy options at the health care system level, providing detailed information for planning, both in terms of location of services and of planning capacity. The model aims at minimizing the expected costs, while ensuring a minimum level of demand satisfaction and considering that policy makers with planning responsibilities in the LTC sector demand that satisficing levels for different types of equity
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should be ensured. In particular, equity of access, equity of utilization, socioeconomic equity and geographical equity are modeled. This study adds to the literature by i) proposing a generic approach that informs planning decisions in the LTC sector, a health sector not widely studied in the literature, ii) considering the multi-service nature of LTC, iii) accounting for the impact of demand uncertainty and of different strategic health
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policy decisions while planning changes to an LTC network over an extended planning horizon, and iv) modeling multiple equity concepts relevant for planning in NHS-based systems, with the equitycost trade-offs being calculated.
Although it can be used for supporting the planning of an LTC network for a longer time span, the applicability of the LTC2SS model has been shown to the Great Lisbon region for the 2014-2016
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period. This case makes use of real data from the Portuguese health system and provides insights on how the current LTC provision in the area may evolve and on which information can support LTC decision-makers. Key results confirm that an inadequate supply of services currently exists in the area. The model results also show the relevance of considering uncertainty while planning the reorganization of an LTC network, since different decisions are to be taken when uncertainty is
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accounted for. Particularly, accounting for uncertainty implies a higher investment in beds to cope with future uncertainty. The impact of different strategic health policy decisions has also been shown
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to be relevant. In particular, the cost of reorganizing the LTC network is highly dependent on which health policy decisions are adopted, with higher costs being found when no substitution is allowed between institutional and home-based care services and when there is no conversion from the acute
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to the LTC sector. Results have also shown that a transfer of financial resources from other sectors (such as the acute care sector) to the LTC sector is crucial for this reorganization. In our model
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application, the highest cost is associated with achieving a satisficing level of equity in access. Although applied to a Portuguese case study, the developed model can assist planners in other
AC
jurisdictions (for instance, in other countries) with similar concerns. This will however require adapting the developed model so as to consider the specificities of the system under study. Such an adaptation will imply exploring, for instance: i) whether just a sub-set of LTC services needs to be modeled, ii) which equity-related objectives are to be pursued and with which satisficing levels, iii) which strategic health policy options are expected to impact on the delivery of LTC, and iv) which specific context features of LTC provision should be considered within the set of model constraints.
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As future work, several research topics are to be worth pursuing. First, it is relevant to include multiple objectives in the stochastic model. In particular, the maximization of equity (defined in several forms) and of health gains, as well as the minimization of costs, should be jointly addressed through a multi-objective approach. In order to deal with these multiple objectives, one can resort to a function that aggregates the different objectives into an overall objective function, with the weights
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assigned to each objective being defined together with DMs from the LTC sector (similarly to Cardoso et al. (2015)). Alternatively, one can determine the set of efficient solutions using, for instance, a constraint method. Alternative stochastic approaches should also be explored. For instance, it may be relevant to consider a two-stage stochastic approach where both location and allocation are considered as first-stage decisions. Such an approach would allow a more stable allocation of patients to be obtained across scenarios. Further work may develop tools to potentiate
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the interactive use of the model in real decision-making processes in Portugal. These tools should include the integration of the stochastic model with Geographic Information Systems, enabling a visualization of how LTC networks change. Acknowledgements
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The first author acknowledges financing from the Fundação para a Ciência e Tecnologia (Portugal) [(SFRH/BD/63966/2009);(SFRH/BPD/98270/2013)]. The authors acknowledge financing from Conselho de Reitores das Universidades Portuguesas (CRUP) and Deutscher Akademischer Austauschdienst
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(DAAD), in the scope of Acções Integradas Luso-Alemãs. The authors thank the support from the Institute of Operations Research, Karlsruhe Institute of Technology (KIT, Germany) and from the
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Centre for Management Studies of Instituto Superior Técnico (CEG-IST, Portugal). An earlier version of this article was published as a working paper of CEG-IST entitled ‘Planning a long-term care
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network with uncertainty, strategic policy and equity considerations: A stochastic planning approach’. The authors thank Dr. Regina Sequeira Carlos, the Head of the Long-Term Care Coordination Team
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of the Lisbon and Tagus Valley Health Authority, for her willingness to take part in this study. References
AARP Public Policy Institute. (2006). European experiences with long-term care: France, the Netherlands, Norway, and the United Kingdom. Washington, D.C. Abdelaziz, F. B., & Masmoudi, M. (2012). A multiobjective stochastic program for hospital bed planning. J Oper Res Soc, 63(4), 530-538. Aissi, H., Bazgan, C., & Vanderpooten, D. (2009). Min–max and min–max regret versions of combinatorial optimization problems: A survey. Eur J Oper Res, 197(2), 427-438.
28
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Algarve Health Region. (2004). Plano Director Regional da Saúde do Algarve [Regional plan for health in Algarve]. Algarve. Alonso-Ayuso, A., Escudero, L. F., Garín, A., Ortuño, M. T., & Pérez, G. (2003). An approach for strategic supply chain planning under uncertainty based on stochastic 0–1 programming. J Glob Optim, 26(1), 97–124. Arabani, A. B., & Farahani, R. Z. (2011). Facility Location Dynamics: An Overview of Classifications and Applications. Comput & Ind Eng. doi: 10.1016/j.cie.2011.09.018 Baker, M. (2000). Making sense of the NHS White Papers (Second ed.). Oxon: Radcliffe Medical Press.
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Barros, P. P., Machado, S. R., & Simões, J. A. (2011). Portugal: Health system review. Health systems in transition, 13(4), 1156.
Birge, J. R., & Louveaux, F. (1997). Introduction to stochastic programming New York: Springer Verlag.
Boyle, S. (2011). United Kingdom (England): Health system review. Health systems in transition, 13(1), 1-483.
Brailsford, S., & Vissers, J. (2011). OR in healthcare: A European perspective. Eur J Oper Res, 212(2), 223–234.
Bretthauer, K. M., & Coté, M. J. (1998). A model for planning resource requirements in health care organizations. Decis
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Sci, 29(1), 243–270.
Brodsky, J., & Clarfield, A. M. (2009). Organization of health services: long term care in health services. In G. Carrin, K. Buse, K. Heggenhougen & S. R. Quah (Eds.), Health systems policy, finance, and organization (1st ed., pp. 351-357). San Diego: Academic Press.
Cardoso, S. R., Barbosa-Póvoa, A. P. F. D., & Relvas, S. (2013). Design and planning of supply chains with integration of reverse logistics activities under demand uncertainty. Eur J Oper Res, 226(3), 436–451.
M
Cardoso, T., Oliveira, M., Barbosa-Póvoa, A., & Nickel, S. (2012). Modeling the demand for long-term care services under uncertain information. Health Care Manag Sci, 15(4), 385-412. Cardoso, T., Oliveira, M. D., Barbosa-Póvoa, A., & Nickel, S. (2013). Planning a long-term care network with
ED
uncertainty, strategic policy and equity considerations: A stochastic planning approach. Working Paper CEG-IST no. 4/2013, Instituto Superior Técnico.
Cardoso, T., Oliveira, M. D., Barbosa-Póvoa, A., & Nickel, S. (2015). Moving Towards an Equitable Long-Term Care
PT
Network: A Multi-Objective and Multi-Period Planning Approach. Omega (in Press). Central Administration of the Health System. (2013). Implementação e Monitorização da Rede Nacional de Cuidados
CE
Continuados Integrados - Relatório Final [Implementing and Monitoring the National Network of Long-Term Care Final Report]. Cuidados Continuados - Saúde e Apoio Social. Cinnamon, J., Schuurman, N., & Crooks, V. A. (2009). Assessing the suitability of host communities for secondary
AC
palliative care hubs: A location analysis model. Health & Place, 15(3), 822-830. Clemen, R. T., & Reilly, T. (2003). Making Hard Decisions with Decision Tools Suite Update 2004. Duxbury. Daskin, M. S., & Dean, L. K. (2004). Location of Health Care Facilities. In F. Sainfort, M. Brandeau & W. Pierskalla (Eds.), Handbook of OR/MS in Health Care: A Handbook of Methods and Applications (pp. 43-76): Kluwer. Drezner, T., & Drezner, Z. (2011). A note on equity across groups in facility location. Nav Res Logist, 58(7), 705-711. Earnshaw, S. R., Hicks, K., Richter, A., & Honeycutt, A. (2007). A linear programming model for allocating HIV prevention funds with state agencies: a pilot study. Health Care Manag Sci, 10(3), 239-252.
29
ACCEPTED MANUSCRIPT
ERS. (2013). Avaliação do acesso dos utentes aos cuidados continuados de saúde [Evaluation of the access of patients to long-term care]. Entidade Reguladora de Saúde. European Commission. (2008). Long-Term Care in the European Union. Brussels: Directorate-General for Employment, Social Affairs and Equal Opportunities. Georgiadis, M. C., Tsiakis, P., Longinidis, P., & Sofioglou, M. K. (2011). Optimal design of supply chain networks under uncertain transient demand variations. Omega, 39(3), 254–272. Ghaderi, A., & Jabalameli, M. S. (2013). Modeling the budget-constrained dynamic uncapacitated facility location– 57(3-4), 382-400.
CR IP T
network design problem and solving it via two efficient heuristics: A case study of health care. Math & Comput Model, Greene, V. L., Ondrich, J., & Laditka, S. (1998). Can home care services achieve cost savings in long-term care for older people? J Gerontol, 53B(4), S228-S238.
Harper, P., Shahani, A. K., Gallagher, J. E., & Bowie, C. (2005). Planning health services with explicit geographical considerations: a stochastic location-allocation approach. Omega, 33(2), 141-152.
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Health Regulation Authority. (2011). Estudo do acesso dos utentes aos cuidados continuados de saúde [Study on patients' access to long-term care]. Oporto.
Hulshof, P. J. H., Kortbeek, N., Boucherie, R. J., Hans, E. W., & Bakker, P. J. M. (2012). Taxonomic classification of planning decisions in health care: a structured review of the state of the art in OR/MS. Health Syst, 1, 129-175. Jones, R. (2011). Hospital bed occupancy demystified. Br J Health Care Manag, 17(6), 242-248. Kim, D.-G., & Kim, Y.-D. (2010). A branch and bound algorithm for determining locations of long-term care facilities.
M
Eur J Oper Res, 206(1), 168–177.
Kirkwood, C. W. (1997). Strategic Decision Making: Multiobjective Decision Analysis With Spreadsheets. Belmont, CA: Duxbury Press.
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Knapp, M., & Somani, A. (2009). Health financing: Long term care, organization and financing. In G. Carrin, K. Buse, K. Heggenhougen & S. R. Quah (Eds.), Health systems policy, finance, and organization (1st ed., pp. 250-259). San Diego: Academic Press.
PT
Kraus, M., Riedel, M., Mot, E., Willemé, P., Röhrling, G., & Czypionka, T. (2010). A Typology of Long-Term Care Systems in Europe ENEPRI Research Report Ner 91. European Network of Economic Policy Research Institutes.
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Lin, F., Kong, N., & Lawley, M. (2012). Capacity planning for publicly funded community based long-term care services. In M. P. Johnson (Ed.), Community-based operations research (Vol. 167, pp. 297-315). New York: Springer. Maenhout, B., & Vanhoucke, M. (2013). An integrated nurse staffing and scheduling analysis for longer-term nursing
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staff allocation problems. Omega, 41(2), 485-499. Mehrez, A., Sinuany-Stern, Z., Arad-Geva, T., & Binyamin, S. (1996). On the implementation of quantitative facility location models: The case of a hospital in a rural region. J Oper Res Soc, 47(5), 612-625. Melo, M. T., Nickel, S., & Saldanha-da-Gama, F. (2009). Facility location and supply chain management - A review. Eur J Oper Res, 196(2), 401-412. doi: 10.1016/j.ejor.2008.05.007 Mestre, A., Oliveira, M. D., & Barbosa-Póvoa, A. (2012). Organizing hospitals into networks: A hierarchical and multiservice model to define location, supply and referrals in planned hospital systems. OR Spectr, 34(2), 319-348.
30
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Mestre, A. M., Oliveira, M. D., & Barbosa-Póvoa, A. P. (2015). Location–allocation approaches for hospital network planning under uncertainty. Eur J Oper Res, 240, 791–806. Mete, H. O., & Zabinsky, Z. B. (2010). Stochastic optimization of medical supply location and distribution in disaster management. Int J Prod Econ, 126(1), 76-84. Ministry of Finance. (2013). Orçamento do Estado para 2014 [State Budget for 2014]. MF, Lisboa. Lei nº 48/90: Lei de bases da saúde [Fundamental principles of health] § I Série (1990). Ministry of Health. (2006). Decreto-lei nº 101/2006: Cria a Rede Nacional de Cuidados Continuados Integrados [Create
CR IP T
the National Network of Long-Term Care]. Diário da República: I Série-A, no 109 de 6 de Junho.
Ministry of Labor and Social Solidarity. (2011). Plano de implementação da RNCCI e orçamento RNCCI 2011 [RNCCI implementation plan and 2011 budget]. MTSS, Lisboa.
Mission Unit for Long-Term Care. (2009a). Modelo de referenciação de utentes - Rede Nacional de Cuidados Continuados [Referral model for patients - National Network of Long-Term Care]. Cuidados Continuados - Saúde e Apoio Social.
AN US
Mission Unit for Long-Term Care. (2009b). Rede Nacional para os Cuidados Continuados Integrados: Cuidados Continuados - Saúde e Apoio Social.
Mission Unit for Long-Term Care. (2010). Três anos de Rede Nacional de Cuidados Continuados Integrados [Three years of National Network of Long-Term Care]. Cuidados Continuados - Saúde e Apoio Social. Mission Unit for Long-Term Care. (2012). Relatório de monitorização do desenvolvimento e da actividade da Rede Nacional de Cuidados Continuados Integrados (RNCCI), 2011 [Report monitoring the development and the activity of
M
the National Network of Long-Term Care (RNCCI), 2011]. Cuidados Continuados - Saúde e Apoio Social. Mission Unit for Long-Term Care. (2013). Unidade de Missão para os Cuidados Continuados Integrados [Mission Unit for Long-Term Care] Retrieved September 2013, from URL: www.rncci.min-saude.pt
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Mitropoulos, P., Mitropoulos, I., Giannikos, I., & Sissouras, A. (2006). A biobjective model for the locational planning of hospitals and health centers. Health Care Manag Sci, 9(2), 171–179. Nickel, S., Saldanha-da-Gama, F., & Ziegler, H.-P. (2012). A multi-stage stochastic supply network design problem with
PT
financial decisions and risk management. Omega, 40(5), 511-524. Oddoye, J. P., Tamiz, M., Jones, D. F., Foroughi, A. A., & Schmidt, P. (2007). A multi-objective model to determine
CE
efficient resource levels in a medical assessment unit. J Oper Res Soc, 58(12), 1563-1573. OECD. (2005). Long-term Care for Older People. Paris: OECD. Oliveira, M. D. (2003). Achieving geographic equity in the Portuguese hospital financing system PhD Thesis. Operational
AC
Research Department, London School of Economics, University of London, London. Oliveira, M. D., & Bevan, G. (2006). Modelling the redistribution of hospital supply to achieve equity taking account of patient's behaviour. Health Care Manag Sci, 9(1), 19-30. Owen, S. H., & Daskin, M. S. (1998). Strategic facility location: A review. Eur J Oper Res, 111(3), 423-447. Peidro, D., Mula, J., Poler, R., & Lario, F.-C. (2009). Quantitative models for supply chain planning under uncertainty: a review. Int J Adv Manuf Technol, 43(3-4), 400-420. Pickard, L., Jiménez-Martin, S., Sánches, R. V., & Prieto, C. V. (2011). The supply of informal care in Europe ENEPRI Research Report Ner 94. European Network of Economic Policy Research Institutes.
31
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Portuguese Observatory on Health Systems. (2011). Relatório de Primavera 2011 [2011 Spring Report]. OPSS. Rais, A., & Viana, A. (2010). Operations Research in Healthcare: a survey. Int. Trans. Oper. Res., 18(1), 1-31. Rowse, E. L., Harper, P. R., Williams, J. E., & Smithies, M. (2013). Optimising the use of resources within the district nursing service: a case study. Health Syst, 2(1), 43-52. Santibáñez, P., Bekiou, G., & Yip, K. (2009). Fraser Health uses mathematical programming to plan its inpatient hospital network. Interfaces, 39(3), 196-208. Santoso, T., Ahmed, S., Goetschalckx, M., & Shapiro, A. (2003). A stochastic programming approach for supply chain
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network design under uncertainty. Eur J Oper Res, 167(1), 96–115.
Santoso, T., Ahmed, S., Goetschalckx, M., & Shapiro, A. (2005). A stochastic programming approach for supply chain network design under uncertainty. Eur J Oper Res, 167(1), 96–115.
Shroff, H. F. E., Gulledge, T. R., Haynes, K. E., & O'Neill, M. K. (1998). Siting efficiency of long-term health care facilities. Socioecon Plan Sci, 32(1), 25-43.
Simon, H. A. (1956). Rational Choice and the Structure of the Environment. Psychol Rev, 63(2), 129–138.
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Smith, H. K., Harper, P. R., & Potts, C. N. (2012). Bicriteria efficiency/equity hierarchical location models for public service application. J Oper Res Soc, 64(4), 500-512.
Snyder, L. V. (2006). Facility Location Under Uncertainty: A Review. IIE Trans, 38(7), 537-554. Stummer, C., Doerner, K., Focke, A., & Heidenberger, K. (2004). Determining location and size of medical departments in a hospital network: A multiobjective decision support approach. Health Care Manag Sci, 7(1), 63-71. Sun, L., DePuy, G. W., & Evans, G. W. (2014). Multi-objective optimization models for patient allocation during a
M
pandemic influenza outbreak. Comput & Oper Res, 51, 350-359.
Verderame, P. M., Elia, J. A., Li, J., & Floudas, C. A. (2010). Planning and Scheduling under Uncertainty: A Review Across Multiple Sectors. Ind Eng Chem Res, 49(9), 3993–4017. (2013).
ViaMichelin
http://www.viamichelin.co.uk/
Maps
and
ED
ViaMichelin.
Route
Planner
Retrieved
World Health Organization. (2013). Primary health care - Facts and figures
September,
2013,
from
Retrieved May, 2013, from URL:
PT
http://www.euro.who.int/en/what-we-do/health-topics/Health-systems/primary-health-care/facts-and-figures Zhang, Y., Puterman, M. L., Nelson, M., & Atkins, D. (2012). A simulation optimization approach to long-term care
AC
CE
capacity planning. Oper Res, 60(2), 249-261.
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Appendix A Table A.1 displays the computational results obtained when applying the LTC2SS model for all the cases under analysis. Table A.1 Computational results
Planning Context I & II
Integer variables
Total constraints
430,824
164,256
655,313
358,716 358,713 358,716 358,689 358,713 358,683
532,985 133,420
532,979 532,985 532,931 532,979 532,919
CPU (sec.)
Iterations 5,409,704 7,635,363 12,455,185 10,351,318 10,385,446 10,197,032 13,129,027 26,020,341 20,083,599
28,800
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Planning Context III
PS I PS II PS III PS IV a) b) c) d) e)
Total variables
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Gap (%) 5.01 5.03 2.59 2.78 2.75 3.41 3.82 2.78 0.48
Objective €126M €190M €145M €223M €115M €222M €221M €221M €79M
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Analysis
An integrated approach for planning a long-term care network with uncertainty, strategic policy and equity considerations Teresa Cardoso , Monica Duarte Oliveira , Ana Barbosa-Povoa , ´ ´ Stefan Nickel PII: DOI: Reference:
S0377-2217(15)00486-5 10.1016/j.ejor.2015.05.074 EOR 12998
To appear in:
European Journal of Operational Research
Received date: Revised date: Accepted date:
12 August 2014 14 April 2015 26 May 2015
Please cite this article as: Teresa Cardoso , Monica Duarte Oliveira , Ana Barbosa-Povoa , ´ ´ Stefan Nickel , An integrated approach for planning a long-term care network with uncertainty, strategic policy and equity considerations, European Journal of Operational Research (2015), doi: 10.1016/j.ejor.2015.05.074
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Highlights We propose an integrated approach for planning a multi-service network of LTC.
We develop a stochastic MILP model that considers uncertainty in demand and supply.
The model aims at minimizing expected costs while ensuring satisficing equity levels.
The model explores how strategic health policy options can influence LTC delivery.
The model is applied to a region of Portugal and equity-cost trade-offs are calculated.
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An integrated approach for planning a long-term care network with uncertainty, strategic policy and equity considerations Teresa Cardoso a *, Mónica Duarte Oliveira b, Ana Barbosa-Póvoa c, Stefan Nickel d Centre for Management Studies of Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais 1, 1049-001 a
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Lisbon, Portugal, [email protected]
Centre for Management Studies of Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais 1, 1049-001 Lisbon, Portugal, [email protected]
b
Centre for Management Studies of Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais 1, 1049-001 Lisbon, Portugal, [email protected]
c
Institute of Operations Research, Karlsruhe Institute of Technology (KIT), Kaiserstr. 12, 76131 Karlsruhe, Germany, d
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[email protected]
Corresponding author: [email protected] ; +351 218419959 *
Abstract: Considering key uncertainties and health policy options in the reorganization of a longterm care (LTC) network is crucial. This study proposes a stochastic mixed integer linear programming model for planning the delivery of LTC services within a network of care where such
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aspects are modeled in an integrated manner. The model assists health care planners on how to plan the delivery of the entire range of LTC services – institutional, home-based and ambulatory services
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– when the main policy objective is the minimization of expected costs and while respecting satisficing levels of equity. These equity levels are modeled as constraints, ensuring the attainment of equity of access, equity of utilization, socioeconomic equity and geographical equity. The proposed
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model provides planners with key information on: when and where to locate services and with which capacity, how to distribute this capacity across services and patient groups, and which changes to the
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network of care are needed over time. Model outputs take into account the uncertainty surrounding LTC demand, and consider strategic health policy options adopted by governments. The applicability
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of the model is demonstrated through the resolution of a case study in the Great Lisbon region in Portugal with estimates on the equity-cost trade-off for several equity dimensions being provided. Keywords: OR in health services; LTC planning; multi-service; uncertainty; equity 1.
Introduction
Long-term care (LTC) includes a broad range of health and social services designed for individuals suffering from chronic illness and/or disabilities (OECD, 2005). LTC is currently provided in many developed countries (Brodsky & Clarfield, 2009), but there are differences in the way they organize 2
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these services, either by: 1) providing care in a variety of settings, including short-term to longer institutionalizations, home-based and ambulatory services; 2) having the provision of care shared between the family, public and private sectors; and 3) establishing different boundaries between the health and social care services (European Commission, 2008; Kraus et al., 2010). For instance, in countries with a National Health Service (NHS), such as Portugal and England, health and social
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LTC services are delivered to the population under different schemes: while health-related services are nearly free at the point of use and financed by taxation, access to social care is usually meanstested (Barros, Machado & Simões, 2011; Boyle, 2011).
One of the most important challenges for European LTC systems is to be responsive to an increasing demand for care (Knapp & Somani, 2009). This is mainly due to the ageing phenomenon,
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to an increase in the prevalence of chronic diseases and to changes in family structures. An adequate supply of care is thus required to address this increasing demand, but most healthcare systems are still ill-equipped to meet that challenge (AARP Public Policy Institute, 2006). This low supply of LTC results in higher costs for the health system, mainly due to an inadequate utilization of acute care services by LTC patients (Mission Unit for Long-Term Care, 2010). Within this setting,
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currently investing in and planning LTC networks represent a health policy priority in many European countries. Additionally, in the context of a strong pressure to decrease public health spending, NHS-based countries (for instance, Portugal) are compelled to minimize the delivery costs,
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while equity becomes a secondary concern. The development of tools to support such decisions is indeed a challenge (Rais & Viana, 2010).
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Mathematical programming models have been widely used in the health care planning literature (Brailsford & Vissers, 2011), but several challenges still need to be addressed in this area. In
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particular, there is a lack of research on developing mathematical programming planning models in the LTC sector and comprehensively considering key features, such as i) the multi-service nature of
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health care services, ii) the effect of uncertainty and of strategic health policy decisions on planning decisions over time, and iii) the joint effect of cost and equity considerations. Our study aims to address this gap by proposing an integrated approach based on a two-stage stochastic mixed integer linear programming (MILP) model – the LTC2SS model, with 2SS standing for Two-Stage Stochastic model – to support the planning of strategic and tactical decisions in the LTC sector. The model is set for the context of an NHS-based country and takes into account cost and equity concerns. The main objective is the minimization of expected cost, while equity 3
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considerations – namely, equity of access, equity of utilization, socioeconomic equity and geographical equity – are modeled as constraints, in the form of satisficing equity levels that need to be respected in LTC delivery. The LTC2SS considers the impact of demand uncertainty and of strategic health policy options in the planning of an LTC network, as well as the multi-service nature of LTC by accounting for a wide range of institutional, home-based and ambulatory services. The
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LTC2SS provides planners (hereafter generically named as decision-makers, DMs) with detailed information regarding: i) where and when should multiple LTC services be located and with which capacities (in terms of beds and human resources); ii) how should this capacity be distributed across services and patient groups; and iii) which changes are needed in this network over time, including capacity adjustments and the opening/closure of services. The LTC2SS model is applied at the county
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level in the Great Lisbon region in Portugal for the 2014-2016 period.
Section 2 of this article presents a brief literature review on related methods. The problem under study is stated in Section 3. The mathematical formulation of the model is presented in Section 4, with the case study and key results being explored in Section 5. Some conclusions and lines for future research are included in Section 6. Literature review
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2.
Mathematical programming models have been extensively used to support location selection and
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capacity planning decisions in a wide range of areas (Arabani & Farahani, 2011; Melo, Nickel & Saldanha-da-Gama, 2009), including in the health care planning literature (Brailsford & Vissers, 2011). In the context of our study, particular attention is given to multi-period and multi-service
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mathematical programming based approaches that account for the effect of uncertainty on planning decisions. Applications in the health care sector in general and in the LTC sector in particular are
2.1.
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also reviewed.
Location selection and capacity planning under uncertainty
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Planning under uncertainty has been an important research topic in different areas of application, for instance, in general supply chain planning (Georgiadis, Tsiakis, Longinidis & Sofioglou, 2011), in petrochemical industry planning (Verderame, Elia, Li & Floudas, 2010) and in water resource planning (Verderame et al., 2010). Most studies focused on planning at tactical or strategic levels did not comprehensively address uncertainty (Santoso, Ahmed, Goetschalckx & Shapiro, 2005), with some exceptions, as for instance the work by Cardoso et al. (2013).
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When addressing uncertainty, sensitivity analysis has been the simplest approach in use (Owen & Daskin, 1998). Nevertheless, such an approach does not ‘incorporate uncertainty into models proactively’ (p. 424) (Owen & Daskin, 1998). Stochastic and robust approaches are alternative (proactive) approaches (Snyder, 2006; Verderame et al., 2010). While stochastic approaches usually attempt to minimize expected costs (see, for instance, Nickel et al. (2012)), robust approaches often aim at
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minimizing the worst-case performance or regret (see Aissi et al. (2009)). Notwithstanding, since robust approaches are considered to be too conservative (Daskin & Dean, 2004), stochastic models are the most common approach used to deal with uncertainty (Peidro, Mula, Poler & Lario, 2009), and we chose this approach in our study.
Stochastic approaches have been widely used for capacity planning and location selection, with
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particular emphasis devoted to two-stage stochastic approaches (Snyder, 2006; Verderame et al., 2010). Within this literature, location of facilities and capacity levels are typically used as first-stage decisions, whereas allocation decisions are usually defined in the second-stage (Snyder, 2006). Examples are the studies from Alonso-Ayuso et al. (2003) and Mete and Zabinsky (2010). These studies usually consider demand as a main source of uncertainty (Peidro et al., 2009) and propose
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stochastic approaches based on scenario planning, with a typically low number of scenarios being considered (Santoso, Ahmed, Goetschalckx & Shapiro, 2003). Our article uses a stochastic
2.2.
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optimization approach that is based on some of the before-mentioned features. Location selection and capacity planning in the health sector
Specific features relevant for planning the delivery of health care services have been considered in
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previous mathematical programming models. Multi-objective approaches have been proposed in the last decade, but their number is still low (Mitropoulos, Mitropoulos, Giannikos & Sissouras, 2006;
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Oddoye, Tamiz, Jones, Foroughi & Schmidt, 2007; Smith, Harper & Potts, 2012). Within the objectives most widely used in the literature, equity plays a key role, being a key objective in NHS-
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based systems (Baker, 2000; Ministry of Health, 1990). Different equity concepts have been adopted, namely, equity of utilization (Oliveira & Bevan, 2006), equity of access (Mehrez, Sinuany-Stern, Arad-Geva & Binyamin, 1996; Mestre, Oliveira & Barbosa-Póvoa, 2012), geographical equity (Earnshaw, Hicks, Richter & Honeycutt, 2007) and socioeconomic equity (Drezner & Drezner, 2011). Nevertheless, although it is known that different definitions of equity may conflict with each other (Oliveira, 2003), most studies do not consider the joint effect of multiple equity concepts. In addition to equity, minimization of total costs has also been considered in some studies (Bretthauer 5
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& Coté, 1998; Rowse, Harper, Williams & Smithies, 2013; Stummer, Doerner, Focke & Heidenberger, 2004; Sun, DePuy & Evans, 2014). Few studies have explored the multi-service nature of health service delivery (see, for instance, Santibáñez et al. (2009) and Mestre et al. (2012; 2015)) despite being recognized that proper planning requires modeling this aspect (Hulshof, Kortbeek, Boucherie, Hans & Bakker, 2012). Multi-period
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approaches have also been rarely used for strategic and tactical health care planning, and, to our knowledge, only Santibáñez et al. (2009), Ghaderi and Jabalameli (2013) and Mestre et al. (2015) have developed research in this area. Accounting for the impact of strategic health policy options on planning decisions is also recognized as relevant, but very few studies have explicitly dealt with this issue (for example, Maenhout & Vanhoucke (2013)).
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Most planning studies in the health care area have used deterministic approaches that ‘are not able to adequately model the uncertainties inherent in making real-world strategic decisions’ (Owen & Daskin, 1998) (p. 424). To our knowledge, only Abdelaziz and Masmoudi (2012) and Mestre et al. (2015) have considered the effect of uncertainty in strategic and tactical health care planning decisions. In fact, when it comes to address uncertainty in health care planning, simulation has been the most
2.3.
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commonly used method (see, for instance, Harper et al. (2005)).
Location selection and capacity planning in the LTC sector
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Few optimization approaches have been proposed for supporting the planning of LTC (Lin, Kong & Lawley, 2012). One of the first models was a single-objective mathematical programming model that
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analyzes the cost-effectiveness of home-based LTC (Greene, Ondrich & Laditka, 1998). Homebased LTC was also the focus of the study developed by Lin et al. (2012). Capacity planning was also addressed in the LTC sector (Lin et al., 2012; Zhang, Puterman, Nelson & Atkins, 2012). The
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location of LTC services has been researched by Cinnamon et al. (2009) and Kim and Kim (2010). Shroff et al. (1998) and Cardoso et al. (2015) have also studied the location of institutional LTC
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facilities and, to our knowledge, these authors have proposed the only multi-objective approaches in this area. None of these studies has, however, considered the multi-service nature of LTC services, as well as the impact of uncertainty and of adopting different health policy options in LTC delivery. Summing up, the main features considered as relevant for planning the delivery of health care services in general, and of LTC services in particular, are identified in Table 1. It can be observed that no study has addressed these features altogether, and so the LTC2SS model proposed in our study attempts to fill this gap in the literature. 6
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Table 1 Selected features of health (in general) and LTC studies using mathematical programming models
Oliveira & Bevan (2006) Santibáñez et al. (2009) Abdelaziz & Masmoudi (2012) Mestre et al. (2012) Ghaderi & Jabalameli (2013) Maenhout & Vanhoucke (2013) Mestre et al. (2015) Lin et al. (2012) Greene et al. (1998) Zhang et al. (2012) Cinnamon et al. (2009) Kim & Kim (2010) Shroff et al. (1998) Cardoso et al. (2015) LTC2SS
Multiperiod
Multiservice
Policy options’ impact
LTC sector applications
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3.
Multiple equity concepts
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Planning under uncertainty
LTC planning background
The present study develops a model to support planning decisions in the LTC sector at strategic and
LTC networks’ key features
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3.1.
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tactical levels with background information being presented in this section.
We consider an LTC network that is operating in the context of an NHS-based system. Such a system can be found in Portugal (Barros et al., 2011) and in England (Boyle, 2011), among other
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countries. We describe the key features of the LTC network currently operating in mainland Portugal, which we will use as a reference network. The National Network of Long-Term Care (Rede
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Nacional de Cuidados Continuados Integrados, RNCCI) was created in Portugal in 2006 so as to ensure the provision of health and social care services for the frail elderly, the highly disabled and/or the
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severely ill (Boyle, 2011; Mission Unit for Long-Term Care, 2009a). Our study is restricted to the health care component of LTC, for which, in theory, there should be universal coverage and nearly free access at the point of use. LTC is delivered by multidisciplinary teams of health professionals (such as physicians and nurses) within a mix of public and private organizations, which provide a wide range of institutional (IC), home-based (HBC) and ambulatory (AC) care services (Ministry of Health, 2006). IC comprises services characterized by different lengths of stay (LOS), integrating the delivery of convalescence 7
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care (CC), medium-term and rehabilitation care (MTRC), long-term and maintenance care (LTMC) and palliative care (PC) for individuals with a variety of conditions who cannot receive care at home. HBC is provided at patients’ residences to individuals whose conditions do not justify the admission into an IC service but are not autonomous to perform their daily activities. AC (such as day centers) is provided to patients whose conditions do not justify the provision of IC or HBC. Regarding the
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role of the state, on the one hand, it is mainly responsible for establishing contracts with IC providers (either public or private, but mainly private non-profit institutions), paying them according to the utilization of services (Mission Unit for Long-Term Care, 2009b). On the other hand, the state assumes a much more active role in providing HBC and AC within the scope of primary health care centers (PHCCs) (ERS, 2013; Ministry of Health, 2006). Individuals can have access to these services after being evaluated in an NHS hospital or by a primary care provider, being then allocated to the
3.2.
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closest available service (Mission Unit for Long-Term Care, 2009a). Planning decisions
The following planning decisions are considered in this study:
Where and when should new IC services be located, both in existing and new locations, and
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with how much bed capacity? And which is the additional bed capacity in which one should invest over time? Location decisions are not considered for HBC and AC, as these services
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are provided by health professionals from PHCCs that are located in all small area units in European NHS-based countries (Barros et al., 2011; World Health Organization, 2013)
How many hours of care provided by different health professionals should be available for
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providing IC, HBC and AC over time? How should available resources (in particular, beds) be (re)distributed across services,
3.3.
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locations and patient groups over time? Policy objectives
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According to policy statements and health policy literature, key policy objectives include ensuring universal LTC coverage, promoting equality of access to health care for all citizens, promoting equity in the distribution and utilization of resources, and ensuring access for the socioeconomic groups with lower levels of income (Baker, 2000; Barros et al., 2011). Nevertheless, in the current European context of severe budget cuts, there is a very high pressure to place cost considerations as a priority in the (re)organization of LTC delivery. Satisficing levels of equity in several dimensions should also be imposed, aiming to ensure: 8
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i.
Equity of access: patients should receive the care they need as close as possible to their place of residence, with travel time not exceeding a maximum level across regions (according to the DMs’ point of view);
ii.
Equity of utilization: unsatisfied demand (usually named as unmet need in the health policy literature) for different services should not exceed a maximum level (according to the DMs).
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This corresponds to the imposition of a minimum service level across services, avoiding in this way that only the cheapest services are delivered (this is expected when cost minimization is pursued); iii.
Socioeconomic equity: unsatisfied demand for the population groups with lower income should not exceed a maximum level (according to the DMs), thus avoiding financial dependency situations;
Geographical equity: unsatisfied demand across geographical areas should not exceed a
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iv.
maximum level (according to the DMs), thus ensuring a minimum level of service across regions and, consequently, avoiding a total lack of service provision in some geographic areas.
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We term these levels as satisficing levels (a concept introduced by Simon (1956)), as in times of economic and financial crisis, the aim is often to minimize costs while respecting acceptable (non-
3.4.
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optimal) levels of equity, rather than maximizing equity. Structuring policy decisions
Different strategic policy decisions in a health system may substantially impact the LTC sector. For
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instance, the Portuguese government has recognized the relevance of (Central Administration of the Health System, 2013; Ministry of Health, 2006): converting acute hospitals into LTC units,
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transferring resources (human [HR], material [MR] and financial [FR] resources) from the acute care to the LTC sector, and changing the LTC provision paradigm from an institutional towards a
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community-oriented paradigm (i.e., substituting IC by HBC). Hence, different combinations of these strategic decisions should be analyzed. Using a strategy generation table as designed by Kirkwood (1997), we have identified six meaningful combinations, denoted by policy strategies (PS) I to VI (Fig. 1).
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Fig. 1 Strategy generation table built for identifying policy strategies. Legend: PS – policy strategy; HR – human resources; MR – material resources; FR – financial resources
As an example, Fig. 1 should be read as follows: PS I (represented by a dark circle) results from combining the following decisions: 1) converting an acute hospital into an LTC unit, 2) transferring
3.5.
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resources from the acute care to the LTC sector, and 3) following a community-based paradigm. Structuring uncertainty
Future LTC demand is difficult to foresee with confidence, depending on changes in epidemiological and demographic profiles and in the availability of informal care (Pickard, Jiménez-Martin, Sánches & Prieto, 2011). Our study addresses data uncertainty associated with future LTC demand in terms
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of a) the number of individuals requiring LTC in the future, and b) the amount of services required by those individuals, as captured by the LOS (note that this component is not strictly a demand
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parameter, being also influenced by supply). We assume that the probability distributions associated to these uncertain parameters are known, with a scenario tree approach (Birge & Louveaux, 1997) being selected to handle that uncertainty in the context of a stochastic programming model. A
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scenario tree with a finite number of scenarios is thus built, with the number of scenarios depending on the number of different realizations considered for each of the uncertain parameters (for more
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details see Cardoso et al. (2013)).
Within the group of mathematical programming methods based on scenario planning, a two-stage
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(MILP) stochastic approach (Birge & Louveaux, 1997) is explored, given that in the LTC context it is natural to consider as first-stage decisions the opening and closure of services (i.e., location decisions), and then allocation and reallocation of resources and individuals as second-stage (as recourse actions) (Snyder, 2006). First-stage decisions should also comprise the investment in new beds (similarly to what has been done in Nickel et al. (2012) in the context of a general supply chain network) because these decisions take time to implement, require significant resources and need to remain valid for longer periods of time (Ministry of Health, 2006). The objective of the LTC2SS 10
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model is thus to minimize the total expected cost over the uncertain demand scenarios, while ensuring the attainment of satisficing levels of equity. Fig. 2 shows the key features of the proposed LTC2SS model, being read as follows – departing from an initial network of care, one should consider: i) closing/opening services; ii) assigning patients to the required service (solid arrows represent patients receiving LTC in the associated service [for IC
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and AC], and dashed arrows depict the provision of LTC at patients’ homes [for HBC]); iii) determining how many beds and human resources are needed per service; and iv) reallocating beds
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across services and locations (dotted arrows). All these decisions are evaluated over time.
Fig. 2 Representation of how the LTC2SS model plan changes in an existing network of care. Legend: IC – institutional
4.
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care; HBC – home-based care; AC – ambulatory care; CC – convalescence care; PC – palliative care
Building the LTC2SS model
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This section presents the mathematical details of the LTC2SS model. The notation used for the model
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formulation is presented in Tables 2-5. Table 2 List of indices
d g l, j r
s, p t, w n, k
Indices Demand points Socioeconomic groups Locations for services Human resources LTC services Time periods Scenario tree nodes
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Table 3 List of sets Sets
L LI LO
S S I SO A s, r : s S , r R F d , l : d D, l L
C k , n : k , n N Q t , n : t T , n N U s, l : s S , l L V s, l : s S , l L M s, l : s S , l L
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G G P GW
Set of demand points Set of human resources Set of time periods Set of scenario tree nodes Set of socioeconomic groups, divided into subsets GP (groups of individuals with priority as a result of having lower levels of income, gGPG) and GW (groups of individuals without priority as a result of having higher levels of income, gGWG). Set of locations for services, divided into subsets LI (subset of locations for IC services, l,jLIL) and LO (subset of locations for HBC and AC services, l,jLOL) Set of LTC services, divided into subsets SI (subset of IC services, s,pSIS) and SO (subset for HBC and AC services, s,pSOS) Set of LTC services s provided by human resources r Set of demand points d that can receive LTC in locations l Set of predecessors k of the scenario tree node n Set of time periods t associated to scenario tree nodes n Set of LTC services s that can be provided in locations l Set of services s provided in locations l at the beginning of the planning horizon Set of services s not provided in locations l at the beginning of the planning horizon
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D R T N
Table 4 List of parameters
Parameters Number of inpatient beds available in IC service s (sSIS) located in l (lLIL) at t=0
bc s / bc s
Minimum/maximum bed capacity allowed per IC service s (sSIS)
hcrsl
Number of hours of care provided by human resource r in service s located in l at t=0 Number of individuals from demand point d and socioeconomic group g requiring service s at t in scenario tree node n Number of individuals belonging to the lower income groups (gGPG) requiring LTC at t in scenario tree node n Number of individuals from demand point d requiring LTC at t in scenario tree node n Number of individuals requiring service s at t in scenario tree node n
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ibsl
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idgstn
iLtn
iS stn ni s / ni s
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LOSsn
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iPdtn
rt
icst
AC
ocst
rDst / rSst
rs
s dl
tn
st
Minimum/maximum number of individuals allowed per HBC/AC service s (sSOS)
Average length of stay (LOS), as measured by the number of days in IC service s (sSIS) in scenario tree node n Inflation rate at t
Investment cost per new bed installed in IC service s (sSIS) at t Operational cost per service s per period t Cost of reallocating a bed to IC service s (sSIS) from a service delivered in a different location / from another service delivered in the same location at t Recommendation for the hours of care to be provided by human resource r for each individual requiring service s Utilization factor associated with the provision of service s Travel time between demand point d and service location l (in minutes) Maximum total travel time (in minutes) at t in scenario tree node n Maximum travel time allowed for LTC patients accessing institutional services (in minutes) Minimum level of demand that need to be satisfied per service s at t Number of days per time period 12
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High value auxiliary coefficient Probability of scenario tree node n Satisficing levels defined for equity of access (EA)/equity of utilization (EU)/geographical equity (GE)/socioeconomic equity (SE) at t (defined as proportions)
n SE tEA / tEU / GE t / t
Table 5 List of variables First-stage variables Equal to 1 if service s is located in l at t; 0 otherwise
X slt
Number of additional beds to invest in for IC service s (sSIS) located in l (lLIL) at t Second-stage variables Equal to 1 if at least one bed is reallocated to IC service s (sSIS) located in l (lLIL) at t in scenario tree node n; 0 otherwise Equal to 1 if there is need to invest in new beds for IC service s (sSIS) located in l (lLIL) at t; 0 otherwise Number of beds to be made available for individuals from demand point d belonging to socioeconomic group g and receiving IC service s (sSIS) located in l (lLIL) at t in scenario tree node n Number of beds reallocated to/from IC service s (sSIS) located in l (lLIL) from/to IC service p (pSIS) located in j (jLIL) at t in scenario tree node n Proportion of individuals from demand point d and socioeconomic group g receiving service s in location l at t in scenario tree node n Number of individuals belonging to the lower income groups receiving LTC at t in scenario tree node n Number of individuals from demand point d receiving LTC at t in scenario tree node n Number of individuals receiving service s at t in scenario tree node n Number of hours of care provided by human resource r for individuals from demand point d belonging to socioeconomic group g and receiving service s located in l at t in scenario tree node n Number of additional hours of care that need to be provided by human resource r in service s located in l at t in scenario tree node n Number of hours of care provided by human resource r that are no longer required in service s located in l at t in scenario tree node n
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ABslt Ysltn Z slt
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Bdgsltn BTslpjtn / BFslpjtn Rdgsltn
RLtn RPdtn
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RSstn HCrdgsltn
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AH rsltn EH rsltn
P Otn
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TItn / TOtn
Total travel time (in minutes) at t for scenario tree node n Penalty (in minutes) attributed to individuals not receiving institutional care at t in scenario tree node n Total investment/operational cost at t for scenario tree node n
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T Otn
Objective function
Equations (1-3) present the model objective function that considers the minimization of the
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expected cost associated with LTC provision for the entire planning period, including both investment and operational costs. Investment costs are related to the investment in new beds (first term in Equation (2)) and to the reallocation of beds between services in different locations (second term) and in the same location (third term). Operational costs include costs associated to the operation of beds in IC services (first term in Equation (3)) and to the provision of HBC and AC services (second term).
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Min
n TI tn TOtn n N t :(t , n)Q
(1) t , n Q (2)
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ic rDst rS TI tn ABslt st t BTslpjtn BTslpltn st t t I I I I I 1 rt pS jL 1 rt 1 rt sS l L pS l :( s, l )U j :( p, j )U p :( p, l )U j l oc oc TOtn Bdgsltn st t Rdgsltn idgstn st t 1 rt sS O lLO 1 rt dD gG sS I lLI l:( s,l )U l:( s,l )U l:( d ,l )F l:( d ,l )F
Constraints
t , n Q
(3)
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Several constraints are defined: i) opening and closure of services; ii) minimum level of demand satisfaction; iii) assignment of patients; iv) resources’ requirements; v) capacity; vi) resources reallocation; vii) equity satisficing constraints; and viii) variable domains. These constraints are formulated below.
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Opening and closure of services constraints
Constraints (4-5) state that the opening/closing of an IC service is not allowed after deciding upon closing/opening it in a previous time period. No openings or closures are considered for HBC and
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AC services as these services are provided within the scope of a primary care network, and is assumed that all the population has a PHCC structure in their area of residence (Constraint (6)). (4)
X slw X slt s S I , l LI : s, l U M , t , w T , w t
(5)
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X slw X slt s S I , l LI : s, l U V , t , w T , w t
X slt 1 s S O , l LO : s, l U , t T
(6)
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Minimum level of demand satisfaction constraints Constraints (7-8) ensure that a minimum level of satisfied demand per service s (captured by the parameter st) is reached per time period t.
n:t , n Q
RSstn
RSstn st s S , t T iSstn
n
(7)
Rdgsltn idgstn s S , t , n Q d D g G l :s, l U l :d , l F
(8)
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Assignment of patients’ constraints The assignment of patients is defined by Constraint (9-11). Particularly, Constraint (9) states that individuals will receive service s in locations where that service is provided; and Constraints (10-11) ensure that individuals in each demand point d will receive LTC in the closest available service, given that in an NHS-based system services should be delivered as close as possible to where demand is
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located (Barros et al., 2011). According to Constraint (10), on the one hand, if service s is provided in location l, and location l is closer to demand point d than another location j, the patients from d should not receive LTC at j. On the other hand, Constraint (11) ensures that patients will not receive LTC in locations situated beyond a maximum travel time (). Notwithstanding, there may be situations in which these conditions should not apply. For instance, when considering the context of an NHS-based country where the primary care network is already established and where HBC and
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AC should be provided within the PHCC belonging to the place of residence of the populations (such as in Portugal), there is no need to apply these conditions for both HBC and AC (Barros et al., 2011; World Health Organization, 2013).
(9)
X slt Rdgsjtn 1 g G, s S , l , j L : s, l U , s, j U , d , l F , t , n Q, dl dj , l j
(10)
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Rdgsltn X slt g G, s S , l L : s, l U , d , l F , t , n Q
Rdgsltn 0 g G, s S , l L : s, l U , d , l F , t , n Q, dl
(11)
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Resources requirements constraints
Constraint (12) calculates the number of beds required at t for IC service s located in l for individuals
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from demand point d and socioeconomic group g for scenario tree node n; and Constraint (13) states that the total bed capacity that needs to be in place in each service and location cannot be less than
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the bed capacity required in each scenario (similarly to Birge and Louveaux (1997)). Similarly, Constraint (14) determines the number of hours of care that should be provided by human resource r; and Constraint (15) establishes a balance between the hours of care provided by existing human
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resources (first term), additional hours of care that need to be provided (second term) and hours of care that are no longer required (third term) for each type of human resource r. A utilization factor (s) is included in Constraints (12) and (14), since a full occupancy of services is not expected in the health care sector. This is in line with literature that has pointed out that health facilities operating with high occupancy rates (usually higher than 90%) are expected to face multiple problems, including regular bed crisis, thereby affecting ‘key measures of ‘poor performance’ such as the waiting time to find a bed, staff stress, dissatisfaction’ (Jones, 2011) (p. 243). In our model, this utilization factor takes the 15
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value of 1 when occupancy rates are expected to equal 100%, and lower values for other cases. This factor also varies depending on how the state pays providers, i.e., whether according to the utilization of services or to installed capacity. Bdgsltn
LOSsn
Rdgsltn idgstn
1
s
g G, s S I , l LI : s, l U , d , l F , t , n Q
(12)
ABslt
HCrdgsltn Rdgsltn i dgstn
HCrdgsltn
d :( d , l )F g G
g G, d , l F , s, l U , s, r A, t , n Q
(14)
s, l U , s, r A, t , n Q, t 1 hcrsl AH rsltn EH rsltn HC AH EH rdgsl t 1k rsltn rsltn s, l U , s, r A, t , n Q, t 1 (15) d :d , l F g G k :(n, k )C
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rs s
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(13)
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B ibsl BTslpjtn BFslpjtn s S I , l LI ; s, l U , t , n Q, t 1 dgsltn I I L d D g G pS j :( pj , j )U Bdgsltn Bdgsl (t 1)k BTslpjtn BFslpjtn s S I , l LI : s, l U , t , n Q, t 1 I k :( n, k )C j LI d :d , l F g G pS j :( p, j )U
Capacity constraints
Maximum and minimum capacity constraints are imposed for LTC services through Constraints (16-
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17). While Constraint (16) defines the maximum and minimum bed capacity per IC service, Constraint (17) defines the maximum and minimum number of individuals in need that should be
Bdgsltn
d :(d , l )F g G
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bc s X slt
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assigned per HBC and AC services.
ni s X slt
bc s X slt s S I , l LI : s, l U , t , n Q
Rdgsltn idgstn
d :(d , l )F g G
ni s X slt s S O , l LO : s, l U , t , n Q
(16) (17)
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Resources reallocation constraints Constraints (18-21) model the reallocation of beds within the network of IC services. According to Constraint (18), the number of beds reallocated to service s located in l from service p located in j must be equal to the number of beds removed from service p located in j to service s located in l (and this for each scenario tree node n). Constraints (19-20) define the maximum number of beds that can be reallocated from and to service s. Constraint (21) states that reallocating beds between services provided in different locations is only allowed in the first time period (note that reallocating beds 16
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across services in the same location is always allowed). This is essential to cope with an initial distribution of services that may be far from the optimal one. BTslpjtn BFpjsltn
s, p S I , l , j LI : s, l U , p, j U , (t , n) Q
(18)
pS
I
jL j : p, j U I
BTslpjtn
Bdgsltn
d :d , l F g G
s S I , l LI : s, l U , t , n Q
BTslpjtn 0 s, p S I , l , j LI : s, l U , p, j U , t , n Q, l j, t 1
(19)
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ib s S I , l LI : s, l U , t , n Q, t 1 sl BFslpjtn Bdgsl (t 1)k s S I , l LI : s, l U , t , n Q, t 1 p S I j LI d :d , l F g G k :n, k C j : p, j U
(20) (21)
Under certain circumstances, an additional set of reallocation conditions can be required in order to
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ensure the stability of the LTC system, so as to avoid excessive reallocations within the LTC network. For that purpose, Constraints (22-23) are used to avoid a removal of beds from a given service and location whenever a reallocation of beds takes place to that service and location (Ysltn=1); and Constraints (24-25) are used to avoid a removal of beds whenever additional beds are added in that service and location (Zslt=1). The need for using these conditions arises when the state finances
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only the effective delivery of services, not being responsible for supporting extra payments associated with the reallocation of beds. Under these particular circumstances, the parameters rDst
j LI j :( p, j )U
jL j :( p, j )U I
BFslpjtn 1 Ysltn s S I , l LI : s, l U , t , n Q
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pS
I
BTslpjwn Ysltn 0 s S I , l LI : s, l U , t , n Q, w, n Q, w t
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pS I
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and rSst in the objective function are set to zero, and the stability of the system is no longer ensured. (22)
(23)
(24)
BFslpjtn 1 Z slt s, p S I , l , j LI : s, l U , p, j U , t , n Q
(25)
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ABslw Z slt 0 s S I , l LI : s, l U , t , w T , w t
Equity satisficing constraints Four satisficing equity levels are considered in this study: equity of access (EA; Constraints (26-28)), equity of utilization (EU; Constraint (29)), socioeconomic equity (SE; Constraints (30-31)) and geographical equity (GE; Constraints (32-33)). The measures selected for defining EA, EU, SE and GE are expressed as follows:
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i.
EA: in order to ensure that LTC provision takes place as close as possible to the patients’ place of residence, it is defined that the expected travel time per patient accessing IC should not exceed the satisficing level defined for EA for each time period t (Constraint (26)). The use of the penalty OPtn (Constraint (28)) penalizes individuals without access to IC, by assuming that these individuals incur the maximum travel time;
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ii. EU: the expected proportion of individuals in need of each type of service and not receiving it should not exceed the satisficing level defined for EU for each time period t (Constraint (29)). This condition ensures a minimum service level (in comparison to need) for each LTC service, thus avoiding a total lack of provision of the most expensive services. Disregarding this objective can create high variations in the utilization of different types of LTC services; iii. SE: the expected proportion of lower income individuals not receiving LTC should not
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exceed the satisficing level defined for SE for each time period t (Constraint (30)). This will ensure a minimum level of service for those with lower ability to pay for care, thus avoiding situations of lack of provision due to poverty;
iv. GE: the expected proportion of individuals belonging to each geographical area not receiving LTC should not exceed the satisficing level defined for GE for each time period t (Constraint
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(32)). Accordingly, a minimum service level across geographical areas is ensured, avoiding in this way a total lack of provision in some regions.
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In summary, the satisficing equity levels correspond to levels that need to be respected in a certain period and in the context of very scarce resources. Such satisficing levels should be defined by the
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DMs (Simon, 1956). T
P
O Otn n tn tEA t T tn n:t , n Q
(26)
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d D g G sS I
AC
T Otn
l LI l :( s, l )U l :( d , l )F
Rdgsltn idgstn dl t , n Q
(27)
P Otn idgstn Rdgsltn idgstn t , n Q d D g G sS I l LI l :s, l U l ; d , l F
n:t , n Q
n 1
RSstn iS stn
tEU s S , t T
(28)
(29)
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n:t , n Q
RLtn iLtn
tSE t T
(30)
Rdgsltn idgstn t , n Q d D g G P sS l :s, l U l :d , l F
n:t , n Q
RPdtn
n 1
RPdtn iPdtn
(31)
GE d D, t T t
(32)
Rdgsltn idgstn d D, t , n Q g G sS l :s, l U l :d , l F
(33)
Variable domains constraints Finally, the variables domains are given by Constraints (34-46). X slt 0,1 s S , l L : s, l U , t T
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(34)
Ysltn 0,1 s S I , l LI ; s, l U , t , n Q Z slt 0,1 s S I , l LI : s, l U , t T s S I , l LI ; s, l U , t T
Bdgsltn
g G, s S I , l LI : s, l U , d , l F , t , n Q
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ABslt
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RLtn
n 1
(35) (36) (37) (38) (39)
RLtn t , n Q
(40)
RPdtn
d D, t , n Q
RSstn s S , t , n Q
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BTslpjtn , BFslpjtn s, p S I , l , j LI : s, l U , p, j U , t , n Q
(41) (42) (43)
AHrsltn , EHrsltn 0 s S , l L : s, l U , s, r A, t , n Q
(44)
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g G, s S , l L : s, l U , d , l F , t , n Q
Rdgsltn 0
(45)
T P Otn , Otn , TI tn , TOtn 0 t , n Q
(46)
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HCrdgsltn 0 g G, s S , l L : s, l U , d , l F , s, r A, t , n Q
5.
Case study
In this section we apply the LTC2SS model to real data from a Portuguese region to illustrate how it can support planning decisions in the LTC sector. Specifically, the model is applied to the county level in the Great Lisbon region – to an LTC network that is under restructuring – in the 2014-2016 period. Notwithstanding, the model can be run for a longer time span.
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5.1.
Model implementation and dataset used
The LTC2SS model was implemented in the General Algebraic Modeling System (GAMS) 23.7 and was solved with CPLEX 12.0 on a Two Intel Xeon X5680, 3.33GHz computer with 12GB RAM. The dataset includes:
LTC supply at the end of 2013 in the Great Lisbon region (Mission Unit for Long-Term
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Care, 2013);
Operational and investment costs (Algarve Health Region, 2004; Central Administration of the Health System, 2013; Portuguese Observatory on Health Systems, 2011). Reallocation costs are not taken into account (i.e., rDst=rSst=0), since the Portuguese state finances mainly IC (Barros et al., 2011);
Travel time (in minutes) between each Great Lisbon county and each LTC service
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(ViaMichelin, 2013), and maximum travel time allowed per patient requiring IC (Health Regulation Authority, 2011);
Number of hours of care that need to be provided by physicians and nurses per individual receiving each type of LTC service (ERS, 2013);
Utilization factor associated with the provision of services with a score of 1, given that
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providers in Portugal are paid according to utilization (Mission Unit for Long-Term Care,
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2009b);
Satisficing equity levels defined by the Head of the Long-Term Care Coordination Team of the Lisbon and Tagus Valley Regional Health Authority (LTV RHA), who has participated in
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this study in the role of a real DM operating in the LTC sector (we hereafter refer to her as our DM). In particular, our DM defined that within three years (i.e., at the end of 2016 [t=3])
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patients should travel no more than 15 minutes to access the care they need (corresponding to EAt=3=0.5). Within the same time frame, our DM also defined that the level of unsatisfied
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demand should not exceed: a) 80% across LTC services (EUt=3=0.8), b) 25% for the lower income groups (SEt=3=0.25), and c) 50% across geographical areas (GEt=3=0.5). Smooth equity improvements were also imposed along the planning horizon, so as to ensure that changes to the network are gradual.
Uncertainty was estimated for two parameters, namely:
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i.
The number of individuals requiring IC, HBC and AC, disaggregated by socioeconomic groups (very low income [VLI] and not very low income [NVLI]), as predicted by the detailed simulation model developed by Cardoso et al. (2012);
ii.
The LOS associated with each type of IC service estimated with data from the Central Administration of the Health System (2013).
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A scenario tree with 81 scenarios is used to describe these uncertain parameters. The probability distributions associated with both parameters made use of data from the Central Administration of the Health System (2013) and Cardoso et al. (2012). Data was then converted into three scenarios using the extended Pearson-Tuckey method (Clemen & Reilly, 2003). The whole dataset is available from the authors, upon request. Selected results
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5.2.
Several analyses are performed in this study. In Planning Context I, the PS IV from Fig. 1 is selected for analysis (representing the baseline policy strategy) and results from running the LTC2SS model are compared with the results obtained for the deterministic case (i.e., when considering average values for the number of individuals in need of LTC and LOS). In Planning Context II, the model is run for
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each one of the policy strategies identified in Fig. 1, and the results are compared in terms of the costs that need to be incurred for reorganizing the network, and also in terms of key changes to be
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implemented in the network. In Planning Context III the cost of ensuring equity satisficing levels is estimated.
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5.2.1. Planning Context I
This analysis explores how the LTC network should evolve when there is uncertainty in the demand for LTC services, and when the PS IV from Fig. 1 is selected, corresponding to a planning context
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with no conversion of acute hospitals into LTC units, with transfer of financial resources allowed from the acute to the LTC sector, and with an institutional-based paradigm.
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According to the model results, LTMC and PC are the types of IC services with the highest and lowest bed capacity requirements throughout the planning horizon (detailed results available in Cardoso et al. (2013)), respectively. This can be explained by differences between their LOS when compared to the remaining IC services (Mission Unit for Long-Term Care, 2012). Regarding PC services, these are the type of IC services associated with lower levels of unsatisfied demand. Fig. 3 shows which changes are needed in the provision of PC services, in terms of service location, bed capacity and allocation of individuals in need to services, over the 2014-2016 period under Planning 21
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Context I (i.e., when uncertainty is accounted for; Fig. 3a)) and under a deterministic case (Fig. 3b)) (note that more details on the results obtained under a deterministic case can be found in Cardoso et al. (2013)). For simplification purposes, institutions located in each county are numbered – e.g., two
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institutions are located in Lisbon and are named Lisbon (1) and Lisbon (2).
Fig. 3 Changes in the provision of palliative care (PC) services within the LTC network, in terms of services’ location,
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bed capacity and allocation of individuals in need to services (given by arrows) under a) Planning Context I and b) deterministic case
Results show significant differences in the opening and closure of services when uncertainty is
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modeled. In particular, it can be seen that under a deterministic case a new PC service should be opened in Sintra (1) (Fig. 3b)) in 2016, whereas this opening should not take place under Planning
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Context I (Fig. 3a)). Differences on the allocation of patients to services and on the bed capacity used in each service are also observed from Fig. 3: these differences also vary with the scenario under analysis, with Fig. 3a) only reporting results for one specific scenario. For a more detailed analysis on the bed capacity in use in each IC service when uncertainty is modeled, see the unit in Lisbon (2) as an example: the average values shown in Fig. 4 correspond to the expected bed capacity in use, whereas extreme values correspond to the associated maximum and minimum values. As expected, these results show that the impact of uncertainty increases as time passes. 22
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Fig. 4 Bed capacity in use in convalescence care (CC), medium-term and rehabilitation care (MTRC), long-term and maintenance care (LTMC) and palliative care (PC) services provided in Lisbon (2) under Planning Context I
Additional information useful for planning is also generated by the model: Fig. 5 shows the distribution of PC bed capacity in Lisbon (2) across individuals belonging to different socioeconomic
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groups (VLI and NVLI groups). It can be read that the model is discriminating towards individuals
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with VLI (via the use of the satisficing SE level).
Fig. 5 Bed capacity in use in the palliative care (PC) service provided in Lisbon (2) for very low income (VLI) and not
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very low income (NVLI) individuals under Planning Context I
Although the capacity in use in each service varies across scenarios (as shown in Figs. 4 and 5), the
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number of new beds in which there is a need to invest for each service is defined in the first-stage, taking the same value across scenarios, as shown in Table 6.
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Table 6 Additional beds in which there is a need to invest under Planning Context I and under a deterministic case
CC MTRC LTMC PC Total additional bed capacity
2014 35 65 135 5
Planning Context I 2015 2016 269 212 535 232 325 680 84 84 2,661
2014 284 110 216 108
Deterministic case 2015 2016 122 22 347 397 151 515 11 29 2,312
Table 6 compares the number of new beds that are needed for all the types of IC services when uncertainty is modeled and for the deterministic case, showing that a greater investment in new beds 23
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is required when uncertainty is accounted for. These results suggest that the bed capacity should increase significantly so as to ensure that all the equity satisficing levels are achieved, and an even higher increase would be needed to ensure that all the demand for LTC is satisfied. These results indicate that current supply levels are far from answering the LTC demand in the Great Lisbon region.
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With respect to HBC and AC, Table 7 summarizes the total number of hours of care to be provided by physicians and nurses in each one of these services when uncertainty is accounted for. Similar results to the ones observed for IC provision were obtained for HBC and AC services: a higher amount of resources is required to cope with an increasing and uncertain demand over time.
Table 7 Evolution of the total number of hours of care (expected values) to be provided by physicians and nurses for 2014 Physicians Nurses
HBC 21,419 230,664
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home-based care (HBC) and ambulatory care (AC) provision under Planning Context I 2015
AC 312,199 3,362,142
5.2.2. Planning Context II
HBC 33,623 362,094
AC 404,939 4,360,879
HBC 47,906 515,913
2016 AC 526,210 5,666,879
Table 8 summarizes the results obtained when the LTC2SS model is run for all the policy strategies
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identified in Fig. 1.
Policy strategies
Investment costs
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Table 8 Key results obtained when different policy strategies (PS) are followed Costs Operational costs
Total costs
Amount to be transferred from the acute care to the LTC sector
€108M
€126M
€36M
€18M
PS II
€37M
€153M
€190M
€100M
PS III
€27M
€118M
€145M
€55M
PS IV
€53M
€170M
€223M
€133M
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PS I
Key changes in the LTC network until 2016 10 new LTC services 9 LTC services closed 10 new LTC services 8 LTC services closed 9 new LTC services 8 LTC services closed 8 new LTC services 6 LTC services closed
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From Table 8 it can be seen that running the model under PS IV results in the highest investment and operational costs. This is an expected result since under this PS it is not possible to substitute IC by HBC, with this substitution being the key for saving some money – this is confirmed by the relatively lower operational costs obtained when running the model under PS III. Also, no hospital conversion is allowed, meaning that it is not possible to take advantage of existing structures with bed capacity installed for IC provision. Using existing structures with bed capacity installed in the acute care sector for IC provision is however allowed under PS I and II, thus resulting in lower 24
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investments (when compared to PS III and IV, respectively). This lower investment is related to the assumption that Maternidade Dr. Alfredo da Costa, which is an NHS hospital located in the Lisbon county, will be closed (note that the closure of this unit is currently under evaluation (Ministry of Finance, 2013)) and converted into an LTC unit. Accordingly, this location is a new potential location in the model, being able to deliver 209 beds for IC provision, with no extra investment
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being needed for this bed capacity. One should be aware that the total costs shown in Table 8 are significantly above the current budget available for operations and investments in the LTC sector in the Great Lisbon area – according to the Ministry of Labor and Social Solidarity (2011), the current budget is €5M and €25M for investments and operations per year, respectively, representing a total budget of €90M for a three-
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year period. Accordingly, the reorganization of the LTC network and the associated achievement of the satisficing equity levels defined by our DM are dependent on the transfer of financial resources from the acute care to the LTC sector – e.g., the reorganization of the network under PS I requires the transfer of €36M for the LTC sector.
PS V and VI are not included in Table 8 because it is not possible to reorganize the LTC network
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and achieve the satisficing equity levels defined by our DM under these policy strategies. This is because no transfer of financial resources is allowed under these strategies and the current budget satisficing equity levels.
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available for the LTC sector in the Great Lisbon area is far from the one required to achieve these
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5.2.3. Planning Context III
It is relevant for DMs in the LTC sector to know the costs associated with ensuring equity satisficing levels. For that purpose, the LTC2SS model was run for the baseline policy strategy (PS IV) and by
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relaxing each of the equity satisficing level constraints (cases a), b), c) and d)) and then by relaxing all the equity satisficing levels simultaneously (case e)). The cost of equity is then determined by
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computing the difference between the cost obtained when all the equity satisficing levels are imposed, and the cost obtained under cases a), b), c), d) and e). Table 9 presents the key results for these five equity costs, providing estimates of the cost required to ensure the achievement of the corresponding satisficing equity level.
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Table 9 Estimates for the cost of ensuring equity satisficing levels a) b) c) d) e)
Cost of equity Cost of ensuring equity of access satisficing level (EA) Cost of ensuring equity of utilization satisficing level (EU) Cost of ensuring socioeconomic equity satisficing level (SE) Cost of ensuring geographical equity satisficing level (GE) Cost of ensuring all the equity satisficing levels
€108M €1M €3M €3M €144M
As expected, the total cost of equity is significantly higher than the cost of ensuring each equity
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satisficing level individually. When considering each equity level individually, the highest cost is observed for equity of access (EA), and the lowest for equity of utilization (EU). The lower cost obtained for EU may be explained because the minimum level of demand satisfaction (imposed for all the cases) is approximate to the satisficing level for EU. Results show that if the DM in the LTC sector disregards the achievement of SE satisficing levels, a reduction of €3M in costs can be
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achieved; nevertheless, by neglecting SE, the model will not discriminate between locating services in areas with a higher proportion of individuals with lower income.
The computational results of the above cases are detailed in Appendix A, showing a significant growth in the number of variables and constraints when an extra location for IC provision is considered under Planning Context II (PS I and II). Summing up, analyses show that a significant
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reorganization and increase in installed capacity (both in terms of beds and human resources) in the LTC network of the Great Lisbon region needs to be carried out. Reorganization of the network and
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associated costs will depend on strategic policy options for the health care sector, there being naturally a cost for ensuring satisficing equity levels. Conclusion
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6.
The increasing need for LTC, along with an inadequate supply of LTC services and with strong
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budget constraints that many European countries currently face, make LTC planning a priority in the health policy agenda of many countries. This study proposes an integrated approach based on a
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stochastic mixed integer linear programming (MILP) model – the LTC2SS model – to assist the planning of networks of LTC services in the context of NHS-based countries, both at a strategic and tactical level. The model considers the impact of demand uncertainty and of different strategic policy options at the health care system level, providing detailed information for planning, both in terms of location of services and of planning capacity. The model aims at minimizing the expected costs, while ensuring a minimum level of demand satisfaction and considering that policy makers with planning responsibilities in the LTC sector demand that satisficing levels for different types of equity
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should be ensured. In particular, equity of access, equity of utilization, socioeconomic equity and geographical equity are modeled. This study adds to the literature by i) proposing a generic approach that informs planning decisions in the LTC sector, a health sector not widely studied in the literature, ii) considering the multi-service nature of LTC, iii) accounting for the impact of demand uncertainty and of different strategic health
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policy decisions while planning changes to an LTC network over an extended planning horizon, and iv) modeling multiple equity concepts relevant for planning in NHS-based systems, with the equitycost trade-offs being calculated.
Although it can be used for supporting the planning of an LTC network for a longer time span, the applicability of the LTC2SS model has been shown to the Great Lisbon region for the 2014-2016
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period. This case makes use of real data from the Portuguese health system and provides insights on how the current LTC provision in the area may evolve and on which information can support LTC decision-makers. Key results confirm that an inadequate supply of services currently exists in the area. The model results also show the relevance of considering uncertainty while planning the reorganization of an LTC network, since different decisions are to be taken when uncertainty is
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accounted for. Particularly, accounting for uncertainty implies a higher investment in beds to cope with future uncertainty. The impact of different strategic health policy decisions has also been shown
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to be relevant. In particular, the cost of reorganizing the LTC network is highly dependent on which health policy decisions are adopted, with higher costs being found when no substitution is allowed between institutional and home-based care services and when there is no conversion from the acute
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to the LTC sector. Results have also shown that a transfer of financial resources from other sectors (such as the acute care sector) to the LTC sector is crucial for this reorganization. In our model
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application, the highest cost is associated with achieving a satisficing level of equity in access. Although applied to a Portuguese case study, the developed model can assist planners in other
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jurisdictions (for instance, in other countries) with similar concerns. This will however require adapting the developed model so as to consider the specificities of the system under study. Such an adaptation will imply exploring, for instance: i) whether just a sub-set of LTC services needs to be modeled, ii) which equity-related objectives are to be pursued and with which satisficing levels, iii) which strategic health policy options are expected to impact on the delivery of LTC, and iv) which specific context features of LTC provision should be considered within the set of model constraints.
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As future work, several research topics are to be worth pursuing. First, it is relevant to include multiple objectives in the stochastic model. In particular, the maximization of equity (defined in several forms) and of health gains, as well as the minimization of costs, should be jointly addressed through a multi-objective approach. In order to deal with these multiple objectives, one can resort to a function that aggregates the different objectives into an overall objective function, with the weights
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assigned to each objective being defined together with DMs from the LTC sector (similarly to Cardoso et al. (2015)). Alternatively, one can determine the set of efficient solutions using, for instance, a constraint method. Alternative stochastic approaches should also be explored. For instance, it may be relevant to consider a two-stage stochastic approach where both location and allocation are considered as first-stage decisions. Such an approach would allow a more stable allocation of patients to be obtained across scenarios. Further work may develop tools to potentiate
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the interactive use of the model in real decision-making processes in Portugal. These tools should include the integration of the stochastic model with Geographic Information Systems, enabling a visualization of how LTC networks change. Acknowledgements
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The first author acknowledges financing from the Fundação para a Ciência e Tecnologia (Portugal) [(SFRH/BD/63966/2009);(SFRH/BPD/98270/2013)]. The authors acknowledge financing from Conselho de Reitores das Universidades Portuguesas (CRUP) and Deutscher Akademischer Austauschdienst
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(DAAD), in the scope of Acções Integradas Luso-Alemãs. The authors thank the support from the Institute of Operations Research, Karlsruhe Institute of Technology (KIT, Germany) and from the
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Centre for Management Studies of Instituto Superior Técnico (CEG-IST, Portugal). An earlier version of this article was published as a working paper of CEG-IST entitled ‘Planning a long-term care
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network with uncertainty, strategic policy and equity considerations: A stochastic planning approach’. The authors thank Dr. Regina Sequeira Carlos, the Head of the Long-Term Care Coordination Team
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of the Lisbon and Tagus Valley Health Authority, for her willingness to take part in this study. References
AARP Public Policy Institute. (2006). European experiences with long-term care: France, the Netherlands, Norway, and the United Kingdom. Washington, D.C. Abdelaziz, F. B., & Masmoudi, M. (2012). A multiobjective stochastic program for hospital bed planning. J Oper Res Soc, 63(4), 530-538. Aissi, H., Bazgan, C., & Vanderpooten, D. (2009). Min–max and min–max regret versions of combinatorial optimization problems: A survey. Eur J Oper Res, 197(2), 427-438.
28
ACCEPTED MANUSCRIPT
Algarve Health Region. (2004). Plano Director Regional da Saúde do Algarve [Regional plan for health in Algarve]. Algarve. Alonso-Ayuso, A., Escudero, L. F., Garín, A., Ortuño, M. T., & Pérez, G. (2003). An approach for strategic supply chain planning under uncertainty based on stochastic 0–1 programming. J Glob Optim, 26(1), 97–124. Arabani, A. B., & Farahani, R. Z. (2011). Facility Location Dynamics: An Overview of Classifications and Applications. Comput & Ind Eng. doi: 10.1016/j.cie.2011.09.018 Baker, M. (2000). Making sense of the NHS White Papers (Second ed.). Oxon: Radcliffe Medical Press.
CR IP T
Barros, P. P., Machado, S. R., & Simões, J. A. (2011). Portugal: Health system review. Health systems in transition, 13(4), 1156.
Birge, J. R., & Louveaux, F. (1997). Introduction to stochastic programming New York: Springer Verlag.
Boyle, S. (2011). United Kingdom (England): Health system review. Health systems in transition, 13(1), 1-483.
Brailsford, S., & Vissers, J. (2011). OR in healthcare: A European perspective. Eur J Oper Res, 212(2), 223–234.
Bretthauer, K. M., & Coté, M. J. (1998). A model for planning resource requirements in health care organizations. Decis
AN US
Sci, 29(1), 243–270.
Brodsky, J., & Clarfield, A. M. (2009). Organization of health services: long term care in health services. In G. Carrin, K. Buse, K. Heggenhougen & S. R. Quah (Eds.), Health systems policy, finance, and organization (1st ed., pp. 351-357). San Diego: Academic Press.
Cardoso, S. R., Barbosa-Póvoa, A. P. F. D., & Relvas, S. (2013). Design and planning of supply chains with integration of reverse logistics activities under demand uncertainty. Eur J Oper Res, 226(3), 436–451.
M
Cardoso, T., Oliveira, M., Barbosa-Póvoa, A., & Nickel, S. (2012). Modeling the demand for long-term care services under uncertain information. Health Care Manag Sci, 15(4), 385-412. Cardoso, T., Oliveira, M. D., Barbosa-Póvoa, A., & Nickel, S. (2013). Planning a long-term care network with
ED
uncertainty, strategic policy and equity considerations: A stochastic planning approach. Working Paper CEG-IST no. 4/2013, Instituto Superior Técnico.
Cardoso, T., Oliveira, M. D., Barbosa-Póvoa, A., & Nickel, S. (2015). Moving Towards an Equitable Long-Term Care
PT
Network: A Multi-Objective and Multi-Period Planning Approach. Omega (in Press). Central Administration of the Health System. (2013). Implementação e Monitorização da Rede Nacional de Cuidados
CE
Continuados Integrados - Relatório Final [Implementing and Monitoring the National Network of Long-Term Care Final Report]. Cuidados Continuados - Saúde e Apoio Social. Cinnamon, J., Schuurman, N., & Crooks, V. A. (2009). Assessing the suitability of host communities for secondary
AC
palliative care hubs: A location analysis model. Health & Place, 15(3), 822-830. Clemen, R. T., & Reilly, T. (2003). Making Hard Decisions with Decision Tools Suite Update 2004. Duxbury. Daskin, M. S., & Dean, L. K. (2004). Location of Health Care Facilities. In F. Sainfort, M. Brandeau & W. Pierskalla (Eds.), Handbook of OR/MS in Health Care: A Handbook of Methods and Applications (pp. 43-76): Kluwer. Drezner, T., & Drezner, Z. (2011). A note on equity across groups in facility location. Nav Res Logist, 58(7), 705-711. Earnshaw, S. R., Hicks, K., Richter, A., & Honeycutt, A. (2007). A linear programming model for allocating HIV prevention funds with state agencies: a pilot study. Health Care Manag Sci, 10(3), 239-252.
29
ACCEPTED MANUSCRIPT
ERS. (2013). Avaliação do acesso dos utentes aos cuidados continuados de saúde [Evaluation of the access of patients to long-term care]. Entidade Reguladora de Saúde. European Commission. (2008). Long-Term Care in the European Union. Brussels: Directorate-General for Employment, Social Affairs and Equal Opportunities. Georgiadis, M. C., Tsiakis, P., Longinidis, P., & Sofioglou, M. K. (2011). Optimal design of supply chain networks under uncertain transient demand variations. Omega, 39(3), 254–272. Ghaderi, A., & Jabalameli, M. S. (2013). Modeling the budget-constrained dynamic uncapacitated facility location– 57(3-4), 382-400.
CR IP T
network design problem and solving it via two efficient heuristics: A case study of health care. Math & Comput Model, Greene, V. L., Ondrich, J., & Laditka, S. (1998). Can home care services achieve cost savings in long-term care for older people? J Gerontol, 53B(4), S228-S238.
Harper, P., Shahani, A. K., Gallagher, J. E., & Bowie, C. (2005). Planning health services with explicit geographical considerations: a stochastic location-allocation approach. Omega, 33(2), 141-152.
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Health Regulation Authority. (2011). Estudo do acesso dos utentes aos cuidados continuados de saúde [Study on patients' access to long-term care]. Oporto.
Hulshof, P. J. H., Kortbeek, N., Boucherie, R. J., Hans, E. W., & Bakker, P. J. M. (2012). Taxonomic classification of planning decisions in health care: a structured review of the state of the art in OR/MS. Health Syst, 1, 129-175. Jones, R. (2011). Hospital bed occupancy demystified. Br J Health Care Manag, 17(6), 242-248. Kim, D.-G., & Kim, Y.-D. (2010). A branch and bound algorithm for determining locations of long-term care facilities.
M
Eur J Oper Res, 206(1), 168–177.
Kirkwood, C. W. (1997). Strategic Decision Making: Multiobjective Decision Analysis With Spreadsheets. Belmont, CA: Duxbury Press.
ED
Knapp, M., & Somani, A. (2009). Health financing: Long term care, organization and financing. In G. Carrin, K. Buse, K. Heggenhougen & S. R. Quah (Eds.), Health systems policy, finance, and organization (1st ed., pp. 250-259). San Diego: Academic Press.
PT
Kraus, M., Riedel, M., Mot, E., Willemé, P., Röhrling, G., & Czypionka, T. (2010). A Typology of Long-Term Care Systems in Europe ENEPRI Research Report Ner 91. European Network of Economic Policy Research Institutes.
CE
Lin, F., Kong, N., & Lawley, M. (2012). Capacity planning for publicly funded community based long-term care services. In M. P. Johnson (Ed.), Community-based operations research (Vol. 167, pp. 297-315). New York: Springer. Maenhout, B., & Vanhoucke, M. (2013). An integrated nurse staffing and scheduling analysis for longer-term nursing
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staff allocation problems. Omega, 41(2), 485-499. Mehrez, A., Sinuany-Stern, Z., Arad-Geva, T., & Binyamin, S. (1996). On the implementation of quantitative facility location models: The case of a hospital in a rural region. J Oper Res Soc, 47(5), 612-625. Melo, M. T., Nickel, S., & Saldanha-da-Gama, F. (2009). Facility location and supply chain management - A review. Eur J Oper Res, 196(2), 401-412. doi: 10.1016/j.ejor.2008.05.007 Mestre, A., Oliveira, M. D., & Barbosa-Póvoa, A. (2012). Organizing hospitals into networks: A hierarchical and multiservice model to define location, supply and referrals in planned hospital systems. OR Spectr, 34(2), 319-348.
30
ACCEPTED MANUSCRIPT
Mestre, A. M., Oliveira, M. D., & Barbosa-Póvoa, A. P. (2015). Location–allocation approaches for hospital network planning under uncertainty. Eur J Oper Res, 240, 791–806. Mete, H. O., & Zabinsky, Z. B. (2010). Stochastic optimization of medical supply location and distribution in disaster management. Int J Prod Econ, 126(1), 76-84. Ministry of Finance. (2013). Orçamento do Estado para 2014 [State Budget for 2014]. MF, Lisboa. Lei nº 48/90: Lei de bases da saúde [Fundamental principles of health] § I Série (1990). Ministry of Health. (2006). Decreto-lei nº 101/2006: Cria a Rede Nacional de Cuidados Continuados Integrados [Create
CR IP T
the National Network of Long-Term Care]. Diário da República: I Série-A, no 109 de 6 de Junho.
Ministry of Labor and Social Solidarity. (2011). Plano de implementação da RNCCI e orçamento RNCCI 2011 [RNCCI implementation plan and 2011 budget]. MTSS, Lisboa.
Mission Unit for Long-Term Care. (2009a). Modelo de referenciação de utentes - Rede Nacional de Cuidados Continuados [Referral model for patients - National Network of Long-Term Care]. Cuidados Continuados - Saúde e Apoio Social.
AN US
Mission Unit for Long-Term Care. (2009b). Rede Nacional para os Cuidados Continuados Integrados: Cuidados Continuados - Saúde e Apoio Social.
Mission Unit for Long-Term Care. (2010). Três anos de Rede Nacional de Cuidados Continuados Integrados [Three years of National Network of Long-Term Care]. Cuidados Continuados - Saúde e Apoio Social. Mission Unit for Long-Term Care. (2012). Relatório de monitorização do desenvolvimento e da actividade da Rede Nacional de Cuidados Continuados Integrados (RNCCI), 2011 [Report monitoring the development and the activity of
M
the National Network of Long-Term Care (RNCCI), 2011]. Cuidados Continuados - Saúde e Apoio Social. Mission Unit for Long-Term Care. (2013). Unidade de Missão para os Cuidados Continuados Integrados [Mission Unit for Long-Term Care] Retrieved September 2013, from URL: www.rncci.min-saude.pt
ED
Mitropoulos, P., Mitropoulos, I., Giannikos, I., & Sissouras, A. (2006). A biobjective model for the locational planning of hospitals and health centers. Health Care Manag Sci, 9(2), 171–179. Nickel, S., Saldanha-da-Gama, F., & Ziegler, H.-P. (2012). A multi-stage stochastic supply network design problem with
PT
financial decisions and risk management. Omega, 40(5), 511-524. Oddoye, J. P., Tamiz, M., Jones, D. F., Foroughi, A. A., & Schmidt, P. (2007). A multi-objective model to determine
CE
efficient resource levels in a medical assessment unit. J Oper Res Soc, 58(12), 1563-1573. OECD. (2005). Long-term Care for Older People. Paris: OECD. Oliveira, M. D. (2003). Achieving geographic equity in the Portuguese hospital financing system PhD Thesis. Operational
AC
Research Department, London School of Economics, University of London, London. Oliveira, M. D., & Bevan, G. (2006). Modelling the redistribution of hospital supply to achieve equity taking account of patient's behaviour. Health Care Manag Sci, 9(1), 19-30. Owen, S. H., & Daskin, M. S. (1998). Strategic facility location: A review. Eur J Oper Res, 111(3), 423-447. Peidro, D., Mula, J., Poler, R., & Lario, F.-C. (2009). Quantitative models for supply chain planning under uncertainty: a review. Int J Adv Manuf Technol, 43(3-4), 400-420. Pickard, L., Jiménez-Martin, S., Sánches, R. V., & Prieto, C. V. (2011). The supply of informal care in Europe ENEPRI Research Report Ner 94. European Network of Economic Policy Research Institutes.
31
ACCEPTED MANUSCRIPT
Portuguese Observatory on Health Systems. (2011). Relatório de Primavera 2011 [2011 Spring Report]. OPSS. Rais, A., & Viana, A. (2010). Operations Research in Healthcare: a survey. Int. Trans. Oper. Res., 18(1), 1-31. Rowse, E. L., Harper, P. R., Williams, J. E., & Smithies, M. (2013). Optimising the use of resources within the district nursing service: a case study. Health Syst, 2(1), 43-52. Santibáñez, P., Bekiou, G., & Yip, K. (2009). Fraser Health uses mathematical programming to plan its inpatient hospital network. Interfaces, 39(3), 196-208. Santoso, T., Ahmed, S., Goetschalckx, M., & Shapiro, A. (2003). A stochastic programming approach for supply chain
CR IP T
network design under uncertainty. Eur J Oper Res, 167(1), 96–115.
Santoso, T., Ahmed, S., Goetschalckx, M., & Shapiro, A. (2005). A stochastic programming approach for supply chain network design under uncertainty. Eur J Oper Res, 167(1), 96–115.
Shroff, H. F. E., Gulledge, T. R., Haynes, K. E., & O'Neill, M. K. (1998). Siting efficiency of long-term health care facilities. Socioecon Plan Sci, 32(1), 25-43.
Simon, H. A. (1956). Rational Choice and the Structure of the Environment. Psychol Rev, 63(2), 129–138.
AN US
Smith, H. K., Harper, P. R., & Potts, C. N. (2012). Bicriteria efficiency/equity hierarchical location models for public service application. J Oper Res Soc, 64(4), 500-512.
Snyder, L. V. (2006). Facility Location Under Uncertainty: A Review. IIE Trans, 38(7), 537-554. Stummer, C., Doerner, K., Focke, A., & Heidenberger, K. (2004). Determining location and size of medical departments in a hospital network: A multiobjective decision support approach. Health Care Manag Sci, 7(1), 63-71. Sun, L., DePuy, G. W., & Evans, G. W. (2014). Multi-objective optimization models for patient allocation during a
M
pandemic influenza outbreak. Comput & Oper Res, 51, 350-359.
Verderame, P. M., Elia, J. A., Li, J., & Floudas, C. A. (2010). Planning and Scheduling under Uncertainty: A Review Across Multiple Sectors. Ind Eng Chem Res, 49(9), 3993–4017. (2013).
ViaMichelin
http://www.viamichelin.co.uk/
Maps
and
ED
ViaMichelin.
Route
Planner
Retrieved
World Health Organization. (2013). Primary health care - Facts and figures
September,
2013,
from
Retrieved May, 2013, from URL:
PT
http://www.euro.who.int/en/what-we-do/health-topics/Health-systems/primary-health-care/facts-and-figures Zhang, Y., Puterman, M. L., Nelson, M., & Atkins, D. (2012). A simulation optimization approach to long-term care
AC
CE
capacity planning. Oper Res, 60(2), 249-261.
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Appendix A Table A.1 displays the computational results obtained when applying the LTC2SS model for all the cases under analysis. Table A.1 Computational results
Planning Context I & II
Integer variables
Total constraints
430,824
164,256
655,313
358,716 358,713 358,716 358,689 358,713 358,683
532,985 133,420
532,979 532,985 532,931 532,979 532,919
CPU (sec.)
Iterations 5,409,704 7,635,363 12,455,185 10,351,318 10,385,446 10,197,032 13,129,027 26,020,341 20,083,599
28,800
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Planning Context III
PS I PS II PS III PS IV a) b) c) d) e)
Total variables
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Gap (%) 5.01 5.03 2.59 2.78 2.75 3.41 3.82 2.78 0.48
Objective €126M €190M €145M €223M €115M €222M €221M €221M €79M
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Analysis