A stochastic location model for designing primary healthcare networks integrated with workforce cross-training

Journal Pre-proof A stochastic location model for designing primary healthcare networks integrated with workforce cross-training Amir Ahamdi-Javid, Nasrin Ramshe

PII: DOI: Reference:

S2211-6923(18)30089-4 https://doi.org/10.1016/j.orhc.2019.100226 ORHC 100226

To appear in:

Operations Research for Health Care

Received date : 5 July 2018 Revised date : 5 August 2019 Accepted date : 31 October 2019 Please cite this article as: A. Ahamdi-Javid and N. Ramshe, A stochastic location model for designing primary healthcare networks integrated with workforce cross-training, Operations Research for Health Care (2019), doi: https://doi.org/10.1016/j.orhc.2019.100226. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Published by Elsevier Ltd.

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A Stochastic Location Model for Designing Primary Healthcare

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Networks Integrated with Workforce Cross-Training

Amir Ahamdi-Javid1 and Nasrin Ramshe

Department of Industrial Engineering & Management Systems, Amirkabir University of Technology, Tehran, Iran

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Abstract. Health posts aim to elevate the level of public health by providing primary health services. The effectiveness of a health-post network depends on factors, such as the number and location of health posts. The service quality in such networks can be improved by controlling the amount of congestion at facilities. An important policy for decreasing operational cost of these networks is to consider a workforce mix of flexible and dedicated servers. This paper integrates the network design with workforce cross-training in the presence of congestion, where the queuing system at each health post is modeled by a set of multi-class M/G/m queues offering multiple service types. The problem is formulated as an integer nonlinear programming model, and a linearization method is used to solve it. A hypothetical case study illustrates how the model can be used and interesting managerial insights are presented.

Keywords: Primary healthcare; Service network design problem; Congested facility location; Flexibility and server mix; Integer nonlinear programming; Multi-class M/G/m queues.

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Founding: This research project is NOT founded by any research program or organization.

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Corresponding author’s email address: [email protected] 1

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1. Introduction

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Health system operates at three levels: macro (national), meso (district), and micro (individual). The meso level describes local or district health systems, which can be regarded as a subsystem of a national health system with certain characteristics, such as population, governance structure; and health services and resources (Gilson, 2012). A local health system comprises three health sub-systems: the public healthcare, Private-Not-For-Profit (PNFP), and Private-For-Profit (PFP). The public healthcare sub-system is composed of hospitals, health centers, health posts, and community health workers.

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UNICEF enumerates the four levels of healthcare for an efficient health system in the public sector. These levels are outreach services, health posts, health centers, and referral hospital (UNICEF, 2005). The focus of this paper is on the local health system and its public healthcare sub-system. The local health system has a network of health posts in a district, which is the lowest level of health facilities (Gilson, 2012; Zulu et al., 2015). Health posts are the first line of healthcare provision in urban areas and supervised by a community health center. Health posts provide free primary health services and play an important role in promoting the public health. Design of health post networks with their distinguishing features has not been studied in the literature, which is addressed in this paper.

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The multi-service nature of health posts implies the capability of adopting workforce cross-training policy. The policy has a significant effect on the service cost, and can reduce the total service cost and costly servers’ idle times. Servers of the health posts can be flexible. Flexibility here means that each server is capable of performing multiple services (serving different classes of clients) because most of these services are not specialized and need the similar equipment in many cases. Determining the appropriate mix of dedicated and flexible servers (i.e., server mix) in each center is one of the issues that need to be addressed in the design of health post networks.

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From the literature review presented in Section 2, one can see that workforce cross-training decisions have not yet be incorporated into any multi-service network design problem, while this integration has potential applications in healthcare systems. Moreover, those papers considering multi-service network design mostly ignored the congestion factor. The server-mix policy affects the congestion in the queuing systems. In fact, combining queues with multi-task servers (instead of parallel queues and multiple single-task servers) improves efficiency and congestion (Gupta, 2013, pp. 36-37).

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This paper for the first time integrates the network design with workforce-mix planning, while controlling the congestion at multi-service facilities to achieve targeted service levels. It is shown that the nonlinear model of this integration problem can be linearized and solved by existing optimization solvers. The results are finally applied to a hypothetical case study on a health post network design problem. The advantage of our integration approach is discussed, and it is shown that the sequential approach to determine the networkdesign decisions and workforce-mix yields either significantly inferior solutions or solutions that are infeasible to the service-quality and stability constraints. This paper uses queueing theory to incorporate congestion and service quality into the proposed healthcarenetwork design problem. For an introduction to this topic see Creemers et al. (2007) and Gupta (2013), and for a survey of the papers applying queuing theory in healthcare see Lakshmi and Iyer (2013). One can also

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refer to general review papers that survey queue systems in healthcare Ahmadi-Javid et al. (2017), RomeroConrado et al. (2017), Hu, et al. (2018), and Palmer et al. (2018).

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In healthcare systems, the type of health service can be used for classification of resources. Actually, in these systems, multiple types of services are provided, and multiple classes of clients are served where each class has its own characteristics such as arrival rates and service time requirements. Multi-class queuing systems are usually used to model these service systems. One of the issues that should be addressed in these systems is how to partition servers among different classes (Gautam, 2012). In other words, servers of these systems can be cross-trained to serve multiple service types at different service rates (Batta et al., 2007; Chakravarthy & Agnihothri, 2005).

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The congestion factor usually acts as a proxy of service quality in service systems. Waiting time for receiving primary health services has a significant effect on customer satisfaction. In the literature, there are two main service quality policies to deal with congestion. In one, the congestion factor is included in the objective function as a cost (Aboolian et al., 2008; Vidyarthi & Jayaswal, 2014; Vidyarthi & Kuzgunkaya, 2015), and in the other, a threshold constraint limits the congestion measures (Aboolian et al., 2012; Baron et al., 2008; Boffey et al., 2010; Silva & Serra, 2008). In this paper, we consider the second policy, which results in a nonlinear model for our problem, which is not the case for single-service network design problems.

2. Literature review

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The remainder of the paper is organized as follows. Section 2 presents the relevant literature review. Section 3 introduces the problem, presents the required preliminaries, and formulates the problem as a nonlinear mathematical programming model. This section also proposes a linearization method to optimally solve the model. Section 4 applies the model to a hypothetical case study in East Tehran health post network and provides interesting managerial insights. Section 5 concludes the paper and shortly discusses future research directions.

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A comprehensive review of general facility location problems with immobile servers, stochastic demand, and congestion in single-service networks is provided by Berman and Krass (2015). Recent advances can be found in Ahmadi-Javid and Hoseinpour (2107, 2019). Günes and Nickel (2015) and Ahmadi-Javid et al. (2017) recently reviewed studies that directly focus on healthcare facility location problems, which are closely related to location models with congestion.

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The literature on multi-service network design is limited to healthcare applications. Griffin et al. (2008) investigated the community health center network design. They proposed an optimization model to find the best location and service offerings. Stummer et al. (2004) provided a model for determining the location and sizes of medical departments in a hospital network. Galvao et al. (2006) studied the hierarchical location of maternal and perinatal healthcare facilities in Rio de Janeiro. Graber-Naidich et al. (2015) studied the primary care network development with different types of primary care facilities. Mestre et al. (2012) proposed a hierarchical and multi-service model to decide upon the location and structure of a hospital network. Mestre et al. (2015) proposed two location–allocation models for handling uncertainty in the strategic planning of multi-service hospital networks. Cardoso et al. (2015) developed a model by taking into consideration demand uncertainty, multiple services, and various forms of equity for planning a longterm care network. None of these papers considers the congestion at service facilities. 3

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The reader is referred to Hopp and Van Oyen (2004) for an extensive bibliography of cross-training and flexibility literature. Van den Bergh et al. (2013) reviewed the literature on personnel scheduling problems. De Bruecker et al. (2014) presented a review and classification of the literature regarding workforce planning problems incorporating skills. The papers that determine cross-training decisions are briefly reviewed in the following.

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Agnihothri et al. (2003), and Chakravarthy and Agnihothri (2005) presented a model to determine the best mix of dedicated and cross-trained servers in a field service system with two job types and a fixed number of servers. Agnihothri and Mishra (2004) studied a system with three job types and job mismatch. Bata et al. (2007) studied the problem of workforce planning for a service center with multiple groups and timedependent service demand. Simmons (2013) developed a simple model to determine the best mix of dedicated and cross-trained servers with two job types when the total delay cost is a nonlinear function of the mean delay time. Bard and Purnomo (2005) considered the nurse preference scheduling problem with different skill categories. Wright and Mahar (2013) investigated the scheduling of cross-trained nurses across multiple units in a hospital. Paul and MacDonald (2014) modeled the benefits of cross-training to address the nursing shortage. All the models developed in these papers are not integrated with any network design decisions.

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In a nutshell, in most of the location models with congestion developed for service network design, only one service type is considered. On the other hand, there are many papers published on designing multiservice networks, but only a few of them recently incorporated congestion; for more details see Ramshe and Ahmadi Javid (2019). To the best of our knowledge, server-mix (workforce cross-training) decisions have not yet been combined in any multi-service network design problem. Integrating server-mix decisions results in more challenging optimization models because of the complexity of multi-class queues. Hence, this paper for the first time contributes to the literature by incorporating both server mix and congestion into a multi-service location model. 3. Problem statement and formulation

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The focus of this paper is on public healthcare systems. Figure 1 visualizes the variety of public health providers in a district. A public healthcare system typically has a network of health posts, that provide a predetermined set of primary health services to all the people living in the district.

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Clients of the health posts network are mostly recognized based on their location and service types requested. Therefore, classify the clients into classes based on these two factors. We consider a set of demand zones in the district to distinguish the clients from a locational perspective. As a common assumption, we adopt a user-choice environment where the clients choose the closest open facility to obtain their services. This is reasonable because health posts do not provide many specialized services that are not available elsewhere (Griffin et al., 2008). The multi-service nature of health posts and simplicity of the service types provided by them allow using the workforce cross-training policy. In fact, health posts generally do not have many staffing requirements (UNICEF, 2005). Each health post has different units to provide a set of predetermined health service types. A unit in a health post can provide one or more types of services; if it is equipped with one or multiple parallel flexible or dedicated servers, respectively. A dedicate server only provides a single service type,

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while a flexible server can provide two or more service types. Therefore, each health post can have a mix of flexible and dedicated servers.

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We use server to represent either one person or a single team of people that provide one service type or multiple service types to clients. The server may use a set of equipment to perform the service delivery. Adding supporting staffs or more advance equipment can improve the server’s service rate, while incurring additional costs. Each server type has finite number of capacity levels (service rate options), one of which must be selected in the design. We also study the multi-server case, where each unit can be equipped with multiple flexible or dedicated servers. The problem is to determine the number and location of health posts, their server mixes, and the corresponding number and capacity of servers at each health post while considering the network operational cost, service accessibility, and service quality. Service accessibility is related to the total travel cost of clients to reach nearest health posts. The objective is to minimize the sum of operational and transportation costs. Service quality is incorporated by limiting the congestion at open health posts (the less congestion, the higher quality). A threshold on the expected waiting time in the queue for each class of clients at any health post is set to control the congestion.

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The arrival times are random and follow independent exponential distributions, which is common choice to model the interarrival times due to the memoryless property of exponential distributions (Lakshmi & Iyer, 2013). The service time for each service type follows a general distribution with finite mean and variance, which does not impose any modelling restriction in practice. In our service scheduling scheme, the clients are served according to FCFS (First Come, First Served) and none of the classes receive any preferential treatment, which is acceptable in non-emergency healthcare. Moreover, the switching times and costs, which are required for a flexible server to switch from serving a client class to serving another, are assumed to be negligible. As shown in Figure 2, the queue of each unit at facility can generally be modeled as a multi-class M/G/m queue (see Section 3.1).

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The notation used throughout the paper is summarized in Table 1. In the sequel, we first provide some preliminaries on queues used in our study in Section 3.1. Then, the problem is mathematically formulated as a nonlinear model in Section 3.2. Finally, the model is linearized in order to be solved in practice in Section 3.3.

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Public healthcare system

Hosp Hosp HC

HC

HC

HC

HC

HC

HP

HP

HP

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Hosp

HP

HP HP HP

HP

HP

HP

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HP

Multi-class M/G/1 queue

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Customers of service S1

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Fig. 1. Public healthcare providers in a health district (Hosp: hospital; HC: health center; HP: health post; O: Community health workers).

Unit with a single flexible server providing services S1 and S2

Customers of service S2

M/G/1 queue Unit with one dedicated server providing service S3

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Customers of service S3

Fig. 2. A facility offering three service types S1, S2, and S3, which includes a unit with one flexible server and a unit with one dedicated server; there are three classes of clients who want only one of three service types.

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Table 1. Notation: sets, parameters, and variables. Sets Set of (primary health) service types Set of potential facilities (health posts)

𝐽

Set of demand zones

R

Set of client classes Set of types of flexible and dedicated servers

𝐻 𝐹𝐻

Set of flexible server types, i.e., ℎ ∈ 𝐻| |𝑆 |

𝐷𝐻 𝐾

Set of dedicated server types, i.e., ℎ ∈ 𝐻| |𝑆 | Set of capacity levels for server type ℎ ∈ 𝐻

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𝑆 𝐼

𝑆

Set of service types that a server of type ℎ ∈ 𝐻 can serve

𝑆

Set of service types that a client of type 𝑟 ∈ 𝑅 requests for

𝑅

Set of client classes who can receive all or part of their services from a type-ℎ server, i.e., 𝑅

𝜑

Set of any possible number of servers of type ℎ ∈ 𝐻

Parameters Demand rate of client class 𝑟 ∈ 𝑅 in demand zone 𝑗 ∈ 𝐽



𝑟 ∈ 𝑅 |𝑆 ∩ 𝑆

∅

Service rate of a server of type ℎ ∈ 𝐻 with capacity level 𝑘 ∈ 𝐾 for client class 𝑟 ∈ 𝑅

𝜇 𝜎

Variance of service-time distribution for server of type ℎ ∈ 𝐻 with capacity level 𝑘 ∈ 𝐾 for client class 𝑟 ∈ 𝑅 Coefficient of variation of service-time distribution for server of type ℎ ∈ 𝐻 with capacity level 𝑘 ∈ 𝐾 for client class 𝑟 ∈ 𝑅 , i.e., 𝑐𝑣

𝑐𝑣 𝑑 𝑡𝑐

Distance from demand zone 𝑗 ∈ 𝐽 to facility 𝑖 ∈ 𝐼 Travel cost from demand zone 𝑗 ∈ 𝐽 to facility 𝑖 ∈ 𝐼

Fixed cost of establishing facility 𝑖 ∈ 𝐼 as a health post (amortized over the planning period)

𝑐

Service cost for server of type ℎ ∈ 𝐻 with capacity level 𝑘 ∈ 𝐾 at facility 𝑖 ∈ 𝐼 (amortized over the planning period)

𝑝

Cost premium for a flexible server of type ℎ ∈ 𝐹𝐻, which is in computing the service cost of the flexible server

𝜇

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𝑓

𝜎

𝑀𝑊

Maximum threshold set to control waiting time in queue (service level) at any open facility

𝑇𝑆

Total physical space of a facility 𝑖 ∈ 𝐼 for providing services

𝑀𝑆𝑅

Minimum space requirement for hosting a server of type ℎ ∈ ℎ with capacity level 𝑘 ∈ 𝐾

Decision variables 𝑥 𝑧

A binary variable that equals 1 if facility 𝑖 ∈ 𝐼 is established, and 0 otherwise

𝑦

A binary variable that equals 1 if 𝑚 servers of type ℎ ∈ 𝐻 with capacity level 𝑘 ∈ 𝐾 is used at facility 𝑖 ∈ 𝐼, and 0 otherwise

Abbreviations for measures

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A binary variable that equals 1 if demand zone 𝑗 ∈ 𝐽 is served by facility 𝑖 ∈ 𝐼, and 0 otherwise

Average waiting time in queue per client, i.e., ∑ ∈ ∑ queue of a server of type ℎ ∈ 𝐻 at facility 𝑖 ∈ 𝐼

AW TFC TSC TAC

Total network cost, i.e., TFC+TSC Total access (travel) cost, i.e., ∑ ∈ ∑

OF

Objective function, i.e., TNC+TAC

∈

∑

∈

∑

∈

∑

∈

𝜆 𝑧

𝑊

∑

∈

∑

∈

𝜆

, where 𝑊 is expected waiting time per client in

𝑦

𝑡𝑐 𝜆 𝑧

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TNC

Total establishment cost of facilities, i.e., ∑ ∈ 𝑓 𝑥 Total service cost, i.e., ∑ ∈ ∑ ∈ ∑ ∈ ∑ ∈ 𝑚𝑐

∈

3.1. M/G/m queue

In this paper, a congested network design problem is studied, in which the congestion is measured using queuing metrics. This subsection briefly provides some required preliminaries on the M/G/m queues. It also numerically compares two approximations and one upper bound for M/G/m queues.

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Consider an M/G/m queue, which is a queue with Poisson arrivals and general service times. In this queue, there is 𝑚 ( 1) parallel identical servers, and the service discipline is FCFS (First Come, First Served). The mean and variance of the service time are denoted by 1⁄𝜇 and 𝜎 , respectively. The customers arrive according a Poisson process, and the mean arrival rate is given by 𝜆. The traffic intensity 𝜌 is defined as 𝜆⁄𝑚𝜇 , and the queue is stable if 𝜌 1. The analysis of an M/G/m queue with multiple servers (𝑚 1) generally seems very difficult, and there is no known closed-form expression for its waiting-time expressions. However, there are approximations and

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upper bounds (Bolch et al., 1998; Kimura, 2010; Gautam, 2012). A number of these results that can be computationally tractable and manageable in our optimization procedure presented in Section 3.3. 3.1.1. Approximations

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An early and typical approximation for the expected waiting time for an M/G/m queue, denoted by 𝑊 M/G/𝑚 , is traced back to the work of Lee and Longton (1959) as 1 𝑐𝑣 𝑊 M/G/𝑚 𝑊 M/M/𝑚 (1) 2 where 𝑐𝑣 is the coefficient of variation of the service-time distribution, and 𝑊 M/M/𝑚 is the expected waiting time in the corresponding M/M/m queue, given by 𝑚𝜌 𝑊 M/M/𝑚 𝑃 𝑚! 1 𝜌 𝑚𝜇 with the empty probability 𝑃

𝑚𝜌 1 𝜌 𝑚!

𝑚𝜌 𝑖!

.

𝑊 M/M/𝑚

𝜌

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To find simpler approximation, it is possible to replace 𝑊 M/M/𝑚 with the following upper bound: 1 𝜌 𝑚𝜇

1

in formula (1) (Graves, 2005), which yields the following simpler formula: 𝜌 1

1 𝜌

𝑐𝑣 2

1 . 𝑚𝜇

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𝑊 M/G/𝑚

(2)

This approximation gives the exact values for the M/G/1, M/M/1, multi-class M/G/1 queues; and provides very close values for the M/M/m queue with heavy traffic, i.e., 𝜌 1.

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3.1.2. Upper bound

An upper bound for 𝑊 G/G/𝑚 is given in Bolch et al. (1998) using Kingman’s upper bound as 𝑊 G/G/𝑚

𝜆

𝑉𝑎𝑟 𝑇

𝑉𝑎𝑟 𝐵 𝑚 2 1

𝜌

𝑚

1 𝐸𝐵 𝑚

(3)

where 𝜌 𝜆⁄𝑚𝜇 is the traffic intensity, 𝑇 is the (random) inter-arrival time, and 𝐵 is the (random) service time. After substituting the appropriate values, the upper bound (3) can be used for M/G/m queue as follows:

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1 𝜆

𝑊 𝑀/𝐺/𝑚

𝜆

𝜎 𝑚 2 1

𝑚 1 𝜇 𝑚 . 𝜌

(4)

Unfortunately, this upper bound is tight only for M/M/1 queues. Even for M/G/1 queues, for which the two approximations (1) and (2) are exact, this upper bound is very conservative (see the next subsection).

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Table 2. A comparison of approximations and upper bound for M/G/m queue. Inputs

1

1

1.5

0.5

2

1

1.5

0.5

1

Approximation (1)

Approximation (2)

Upper bound (4)

0.03

0.3

2.6786

2.6786

24.3452

0.05

0.5

6.2500

6.2500

21.2500

0.07

0.7

14.5833

14.5833

26.7262

0.09

0.9

56.2500

56.2500

66.8056

0.03

0.3

4.2857

4.2857

25.9524

0.05

0.5

10.0000

10.0000

25.0000

0.07

0.7

23.3333

0.09

0.9

90.0000

0.03

0.3

4.0966

0.05

0.5

16.2500

0.07

0.7

37.9167

0.09

0.9

146.2500

0.06

0.3

0.6181

0.1

0.5

2.0833

0.14

0.7

6.0049

0.18

0.9

26.6447

0.06

0.3

0.9890

0.1

0.5

3.3333

0.14

0.7

9.6078

0.18

0.9

42.6316

0.06

0.3

1.6071

0.1

0.5

5.4167

0.14

0.7

15.6127

0.18

0.9

69.2763

0.09

0.3

0.2084

0.15

0.5

0.9868

0.21

0.7

0.27

0.9

0.09

0.3

0.15

0.5

0.21

0.7

0.27

0.9

35.4762

90.0000

100.5556

6.9643

28.6310

16.2500

31.2500

37.9167

50.0595

146.2500

156.8056

1.3393

13.5119

3.1250

13.7500

7.2917

20.6548

28.1250

61.5278

2.1429

15.1190

5.0000

17.5000

11.6667

29.4048

45.0000

95.2778

3.4821

17.7976

8.1250

23.7500

18.9583

43.9881

73.1250

151.5278

0.8929

9.9008

2.0833

11.2500

3.4191

4.8611

18.6310

17.0221

18.7500

59.7685

0.3335

1.4286

11.5079

1.5789

3.3333

15.0000

5.4705

7.7778

27.3810

27.2354

30.0000

93.5185

0.5419

2.3214

14.1865

5.4167

21.2500

0.09 1.5

0.3

23.3333

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𝜌

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0.5

λ

re-

𝑐𝑣

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𝑚

0.15

0.5

2.5658

0.21

0.7

8.8896

12.6389

41.9643

0.27

0.9

44.2575

48.7500

149.7685

3.1.3. A comparison of approximations and upper bound

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This subsection compares approximation formulas (1) and (2), and upper bound (4) for M/G/m queues by conducting a numerical study in Table 2. Formula (1) gives the exact values for 𝑚 1 (M/G/1 queues) and for 𝑐𝑣 1 (including M/M/m queues). The results show that upper bound (4) is very conservative in practice. Formula (2) is exact for M/G/1 queues and fairly close to (1) for 𝑚 1. Unfortunately, formula (1) is not simple enough that can be efficiently used in an integer programming model. Therefore, we apply approximation formula (2) for modeling our problem. 3.1.4. Multi-class M/G/m queueing system As we deal with a server-mix multi-service design problem, this subsection discusses how the results provided in the previous subsections can be adapted for multi-class M/G/m queues, in which multiple

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∈

𝐸𝐵

∈

1 𝜇

𝜎

1 𝜆

𝜆 𝐸𝐵 ∈

1 𝜆

𝜆

1

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classes of customers are served. The arrival process of each customer class 𝑟 ∈ 𝑅 is governed by a Poisson process with rate 𝜆 , which is independent of the other classes’ arrival processes. The service times are distributionally independent for all the customers, and they are identically distributed for any particular class of customers. Let 𝐵 denotes the (random) service time of a customer in class 𝑟 for which the mean and variance are 1⁄𝜇 and 𝜎 , receptively. Again, the FCFS service-scheduling scheme is considered, in which none of the customer classes receives any preferential treatment. Also assume there is no switchover time from a customer class to another. Therefore, all the arrival Poisson processes of the |𝑅| customer classes can be aggregated into a single Poisson process with rate 𝜆 𝜆 𝜆 ⋯ 𝜆| | . Let 𝐵 be the service time incurred by the server aggregated over all customers, which is distributed as 𝐵 with probability 𝜆 ⁄𝜆. Thus, we have 1 1 1 𝜆 𝐸𝐵 𝜆 𝐸𝐵 𝜇 𝜆 𝜆 𝜇 𝑐𝑣

𝜇

∈

𝑊

lP

re-

𝜎 𝜇 is the coefficient of variation of the service-time distribution for class-𝑟 customers. If where 𝑐𝑣 there is a single server, the expected waiting time in the queue for class 𝑟 (denoted by 𝑊 and 𝑊 , respectively) are given by (Gautam, 2012) 1 𝑐𝑣 ∑ ∈ 𝜆 𝜆𝐸 𝐵 𝜇 𝑊 𝑊 (5) 2 1 𝜌 2 1 𝜌 1 . 𝜇

𝑊

(6)

where 𝜌 is the overall traffic intensity defined as 𝜆𝐸 𝐵

𝜆 𝜇

𝜌 .

urn a

𝜌

∈

(7)

∈

A multi-class M/G/1 queue is stable if 𝜌 1. If there is 𝑚 ( 1) identical parallel servers in the system, the traffic intensity is redefined as 𝜌 𝜆𝐸 𝐵 ⁄𝑚, and the stability condition is similarly given by 𝜌 1. Approximation formula (2) and upper bound (4), can be naturally extended for multi-class M/G/m queues, after substituting the appropriate terms. Indeed, an analogue to (2) is given by

𝑊

1 𝑐𝑣 𝑚 𝜇 , 2 1 𝜌

∈

𝜆

Jo

∑

(8)

and upper bound (4) is modified as follows: ∑

𝑊

∈

𝜆

2 1

1

𝑐𝑣 𝑚𝜇 𝜌

1

𝜌 2𝜆

(9)

.

10

Journal Pre-proof

We could use the upper bound in (9) to get a linearized reformulation in Section 3.3, but it is not applied here because it retunes values that are significantly larger than the original values of 𝑊 (see Section 3.1.3 for a comparison) and because the associated linearized formulation becomes more complex. 3.2. Mathematical formulation

𝑓𝑥

min

𝑚𝑐 ∈

∈

∈

∈

𝑡𝑐  𝑧

𝑦

∈

∈

s.t. 𝑦 | ∈

∈

∈

∈

∈

∈

𝑧

∈

𝑦

1

1 𝑥

𝑧

𝑑 𝑧

𝑀

𝑑

𝑥

𝑀

𝑚𝑀𝑆𝑅 𝑦 ∈ ,

𝑀𝑊 , 𝑧 ∈ 0,1

𝑇𝑆 𝑥

urn a

𝑊 𝑥 ,𝑦

∈

lP

∈

∈

∈

(10)

∈

re-

∈

𝑥 𝜆 𝑧 𝑚𝜇

pro of

Now we can formulate our location problem, described earlier in this section, as the following Integer Nonlinear Programming (INLP) model:

𝑖 ∈ 𝐼, 𝑠 ∈ 𝑆

(11)

𝑖 ∈ 𝐼, ℎ ∈ 𝐻

(12)

𝑗∈𝐽

(13)

𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽

(14)

𝑖∈𝐼, 𝑗∈𝐽

(15)

𝑖∈𝐼

(16)

𝑖 ∈ 𝐼, 𝑟 ∈ 𝑅 𝑖 ∈ 𝐼, 𝑗 ∈ 𝐽 , ℎ ∈ 𝐻, 𝑘 ∈ 𝐾

(17) (18)

The first two terms of objective function (10) are related to the total network cost, which are the total establishment cost of facilities and their associated service units, and the total service-provision cost. The last term is the total access cost for the clients.

Jo

Constraint set (11) states that if a facility is open in the health post network, then it provides all service types. It also ensures that each service type can be served by only one server type. Constraint set (12) guarantees the stability of the queuing systems at each facility. Constraint set (13) enforces that each demand zone is served by exactly one facility. Constraint set (14) ensures that demand zones can only be served by open facilities. Constraint set (15) states that the clients of each demand zone patronize the open facility with the lowest travel distance. The parameter 𝑀 in constraint set (15) is a sufficiently large positive constant, which must be larger or equal to max ∈ , ∈ 𝑑 . Constraint set (16) ensures that the sum of the minimum spaces required for the servers included in each facility does not exceed its physical space. Constraint set (17) states that expected waiting time in the queue per class-r client at facility 𝑖 ∈ 𝐼, denoted by 𝑊 , , does not exceed the predetermined service level 𝑀𝑊 . This constraint represents the service quality policy, which controls the congestion level in the network. The threshold in (17) can also be 11

Journal Pre-proof

determined separately for each client class or facility. A client in some classes may receive different service types from more than one server types (when there is no flexible server providing all the requested service types), and subsequently, should wait in more than one queue. Therefore, the expected waiting time in the queue per client of class 𝑟 ∈ 𝑅 at facility 𝑖 ∈ 𝐼 is approximately given by 𝑊

,

𝑊

𝑖 ∈ 𝐼, 𝑟 ∈ 𝑅.

(19)

: ∈

pro of

where 𝑊 is the expected waiting time per client in the queue of server type ℎ ∈ 𝐻 at facility 𝑖 ∈ 𝐼. Then, we use the approximation in (2) to write 𝑊 in terms of variables 𝑦

∑

∈

∑

∈

𝜆 𝑧

𝑦

𝑊 ∈

2 1

∈

∑

∈

1 𝑐𝑣 𝑚 𝜇 ∑∈ 𝜆 𝑧 𝑚𝜇

and 𝑧 as follows:

(20)

.

re-

Constraint set (18) imposes binary restrictions on the decision variables. 3.3. Linear reformulation

∑

𝜆 𝑧

𝑎 ,𝑏

∑∈

𝑔

∑ ∈

∑ ∈

0 𝑎

𝑎 𝑦

urn a

∑ ∈

𝑦

∈

lP

The INLP model presented in Section 3.2 has two nonlinear constraint sets (12) and (17), which cannot be solved optimally using existing software packages. In this subsection, we reformulate it to Mixed-Integer Linear Programming (MILP) model, which can be solved optimally using MILP solvers such as CPLEX. ,𝑔 , and 𝑛 such that Let us define four sets of non-negative continuous variables 𝑎 , 𝑏

,𝑛

𝑔

𝑖𝑓 𝑦 𝑖𝑓 𝑦 𝑧

0 1 0 𝑔

if 𝑧 if 𝑧

0 1.

Using these variables, the problem can be linearized as follows: min

𝑓𝑥

𝑚𝑐

∈

∈

s.t.

∈

∈

∈

𝑡𝑐  𝑧

𝑦

∈

∈

∈

(21)

(11), (13)-(16), (18)

∈

𝑏

𝑎

Jo

𝜆 𝑧 ∈

∈

∈

𝑦

𝑏 𝑚𝜇

1

𝑀

12

𝑖 ∈ 𝐼, 𝑟 ∈ 𝑅

(22)

𝑖 ∈ 𝐼, ℎ ∈ 𝐻

(23)

𝑖 ∈ 𝐼, ℎ ∈ 𝐻, 𝑟 ∈ 𝑅 , 𝑚 ∈ 𝜑 , 𝑘 ∈ 𝐾

(24)

Journal Pre-proof

1

𝑀

𝑦

𝑏

𝑔 ∈

𝑀𝑊

∈

∈

∑

2 𝑔

∈

∈

𝑛

𝑧 𝑀

𝑔

1

𝑧

,𝑔

𝑖 ∈ 𝐼, 𝑟 ∈ 𝑅

(26)

𝜆 𝑛 𝑚𝜇

𝑖 ∈ 𝐼, ℎ ∈ 𝐻, 𝑚 ∈ 𝜑 , 𝑘 ∈ 𝐾

(27)

𝑛

𝑔

0

: ∈

max , ∈

, ∈

𝑖 ∈ 𝐼, ℎ ∈ 𝐻, 𝑚 ∈ 𝜑 , 𝑗 ∈ 𝐽, 𝑘 ∈ 𝐾

(29)

𝑖 ∈ 𝐼, ℎ ∈ 𝐻, 𝑟 ∈ 𝑅 , 𝑚 ∈ 𝜑 , 𝑗 ∈ 𝐽, 𝑘 ∈ 𝐾

(30)

𝑟∈𝑅

1 𝑐𝑣 𝑚 𝜇 𝜏 ⁄𝑚𝜇

ℎ ∈ 𝐻, 𝑚 ∈ 𝜑 , 𝑘 ∈ 𝐾

lP

∈

in which 𝜏 is 10 𝜆 , 𝜏 1.

(28)

𝑚𝜇

max 𝑚𝜇 𝜏 1 ∈ 2 1 max 𝑚𝜇

𝑀

𝑖 ∈ 𝐼, ℎ ∈ 𝐻, 𝑚 ∈ 𝜑 , 𝑗 ∈ 𝐽, 𝑘 ∈ 𝐾

are sufficiently large positive constants that can be set as follows:

min ∈

𝑀 ,𝑛

where 𝑀 and 𝑀 𝑀

(25)

1 𝑐𝑣 𝑚 𝜇

𝑏

𝑎 ,𝑏

𝑖 ∈ 𝐼, ℎ ∈ 𝐻, 𝑟 ∈ 𝑅 , 𝑚 ∈ 𝜑 , 𝑘 ∈ 𝐾

pro of

| ∈

𝑎

re-

𝑎

where 𝑛 is the largest number of digits after the decimal mark of all 𝜆 , 𝑗 ∈ 𝐽; for integer

4. Numerical analysis

urn a

We now present a hypothetical case study for our health post network design problem in district 14 of Tehran, the capital of Iran. This section first describes the case in Section 4.1, and then illustrates the results in Section 4.2. Managerial insights are discussed in Section 4.3. 4.1. East Tehran health post network

Health posts provide free primary health services and are the basic level of healthcare provision in East Tehran community healthcare network, which is under-supervision of Shahid Beheshti University of Medical Sciences, one of the three medical universities in Tehran.

Jo

Here we consider the network of health posts in district 14 of Tehran, which is a part of East Tehran community healthcare network. District 14 is divided into 21 neighborhoods, each of which represents a demand zone here. These 21 neighborhoods are depicted in Figure 3, in which 23 potential locations are also shown by black-filled circles. Each health post provides a set of nine primary health service types, i.e., |S| 9, which are (1) vaccination, (2) developmental screening, (3) growth monitoring, (4) family counseling, (5) iron and vitamin supplements, (6) obesity screening and nutrition counseling, (7) diabetes screening and counseling, (8) 13

Journal Pre-proof

re-

pro of

blood pressure screening and counseling, (9) pregnant women counseling, which are not specialized services. Each one of the 21 neighborhoods has a special demand rate for each of the nine health service types. To compute each average annual demand rate of each neighborhood for each service type, the number of people in the neighborhood that are likely to use the health posts are estimated by demographics, and then it is multiplied by the average number of annual referrals for that service type per person. Moreover, the average travel cost from each neighborhood to a potential health post is estimated based on the rectangular distance between them. Required data was collected through document studies, observations at several health posts and follow-up interviews with the relevant managers of Shahid Beheshti University of Medical Sciences, which supervises East Tehran community healthcare network.

2 ×

urn a

1 ×

Table 3. Service types offered by 21 potential server types in our case study. Server type 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 × × × × × × × × × × × × × × × × × × × × × × × × ×

Jo

Service type 1 2 3 4 5 6 7 8 9

lP

Fig. 3. Potential locations of health posts in district 14 of Tehran.

14

19 ×

20 × ×

× ×

21 × × ×

Journal Pre-proof

pro of

Health posts are not staffed by physicians; they are staffed by only health technicians such as family health technicians, midwife and nutrition technicians. These servers can be flexible and provide more than one service types. In addition, because of the simple nature of the service types, the cost premium for a flexible server is low, where the cost premium represents the additional training cost amortized over the tenure of the server (Simmons, 2013). The set of possible server types is given in Table 3. This table shows that what service types can be assigned to each server type. For example, the server type 20 can offer services of pregnant women counseling, developmental screening, and growth monitoring of children. Note that sever types 1 to 9 are dedicated, and the other ones are flexible. 4.2. Optimal network configuration

re-

For the case illustrated in Section 4.1, we solve the MILP model proposed in Section 3.3 to determine the optimal health-post network configuration. Here we assumed that only one sever of each type is used in 1 ). We also consider two capacity levels for each server type. The threshold parameter each unit (𝜑 𝑀𝑊 is set to 20 minutes. The optimal solution of the model is reported in Table 4, where the numbers related to open facilities or employed server types are only included. In this table, if a server type is used in an open facility, the notation L-1 or L-2 is used, where 1 and 2 determine the capacity level that is selected for the sever. The optimal network is composed of 15 health posts. Table 4. Optimal network configuration for our case study. Server type used in network

4

5

6

7

L-1 L-2 L-1 L-1 L-2

L-1

L-1

11

L-2

14

L-1

L-1

L-1

L-1

L-1

L-1

L-1

L-1

L-1

L-2

L-1

L-1

L-1

L-2

L-1

L-1

L-1

L-1

L-1

L-1

L-2

L-1

L-2

20

L-2

L-1

L-1

L-1

L-2

L-2

L-1

L-2

L-2

L-2

L-1

L-2

urn a

L-2

L-1

19

21

13

lP

L-1

9

L-1

L-1

L-1

L-1

L-1

L-2

L-2

L-2

L-2

L-1

L-1

L-1

L-1

L-1

L-1

L-1

L-1

L-1

L-1

L-1

L-1

L-1

L-2

L-1

L-2

L-1

L-2

L-2

L-2

Jo

Established health post’s location 1 3 4 5 9 11 12 14 16 17 18 20 21 22 23

15

L-1

L-1 L-1

Journal Pre-proof

4.3. Managerial insights From the results given in Table 4 one can see that mixes of flexible and dedicated servers are selected chosen in open health posts (see also Table 3). This means that the extreme cases where all servers are either flexible or dedicated are not optimal. In the following, other important managerial insights are presented.

pro of

4.3.1. Advantage of integration

Let us examine the importance of integrating server-mix decisions with our underlying multi-service network design problem. For this purpose, we compare our integrated approach with the sequential approach that decomposes the problem into two natural decision stages. In the first stage, the decisions on location of facilities and allocation of clients (𝑥 and 𝑧 ) are determined; and in the second stage, the server-mix decisions (𝑦 ) are determined.

lP

re-

First note that the solution of the first stage of the sequential approach may lead to the infeasibility of the second stage problem due the violation of service-level constraints (17). Next, even if these service-level constraints are removed, the integration approach may lead to better solutions. This can happen because the service-provision cost at facility 𝑖 may be different from facility 𝑖′, i.e., 𝑐 𝑐 . These differences may affect the location decisions of the integration model, while they are neglected in the sequential approach. Figure 5 shows that although the sum of establishment and access costs approach decreases by 3.7%, the total service cost increases by 13.3%, and the total cost (OF) increases by 3.5%. This means that integrating location-allocation and server-mix decisions significantly improve the overall quality of the network design.

10%

5%

urn a

Amount of change

15%

0%

-5%

TFC+TAC

TSC

OF

Fig. 4. Evaluation of solution obtained by sequential approach in comparison with one obtained by our integration approach.

4.3.2. Impact of cross-training policy

Jo

Let us investigate the impact of following the cross-training policy. In this regard, assume that all servers are dedicated, and no flexible server is available (i.e., 𝐻 𝐷𝐻). Under this assumption (no flexible server), Figure 5 shows that we have an increase of 36% in the total service cost of the network, and 21% increment in the total network cost. This clearly demonstrates the positive impact of the cross-training policy.

16

Journal Pre-proof

1100000 1000000

Currency

900000 800000 700000

500000 400000

TSC

pro of

600000

TNC

with cross-training policy

without cross-training policy

Fig. 5. Impact of considering cross-training decisions on TSC and TNC.

4.3.3. Impact of service quality policy

re-

In our model, the congestion level cannot go beyond a predetermined service level to ensure service quality. Therefore, we can investigate the effect of the service quality policy by removing the related constraint set from the model. This relaxation leads to a considerable change in the network configuration; the number of established facilities decreases to nine facilities. Moreover, this leads to a very high increase, about 800%, in the average waiting time (AW), while the total network cost decreases only by 33%. This confirms that controlling congestion is very influential in such systems.

lP

5. Conclusions

urn a

In this paper, we study the design of health post networks, motivated by the importance of primary healthcare. The aim of the new problem is to simultaneously determine the location of facilities, the workforce mix at each open facility, and appropriate number and capacity of them. We use multi-class M/G/m queues and approximately formulate the problem under a service quality policy, in which a threshold constraint controls the maximum waiting time in the queue. The resulting model is an integer nonlinear program, which can be reformulated as an MILP model.

Jo

Our results are applied to a hypothetical case study and several managerial insights are provided. We show that integrating multi-service network design with cross-training and quality consideration is significantly helpful and outperforms the traditional two-stage approach in which these network and server-mix decisions are made in separate phases. Moreover, our results demonstrate that cross-training can meaningfully reduce the total network cost. Our findings also show that if the service-level constraints are not considered to control congestion, the average waiting times at several open facilities may radically increase up to multiple times. Therefore, one may conclude that the configuration of a healthcare network that offering public primary care strongly depends on the policies on human resources and service quality. An open research area is extending our model to other different settings and under more realistic assumptions that can significantly affect network design. From a computational perspective, developing efficient solution algorithms that can solve such problems in larger scales seems very challenging because the basic models of such problems are non-convex integer programs.

17

Journal Pre-proof

References

[6].

[7]. [8]. [9]. [10]. [11]. [12]. [13]. [14].

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pro of

[5].

re-

[4].

lP

[3].

urn a

[2].

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19

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[44]. Vidyarthi, N., Jayaswal, S. (2014). Efficient solution of a class of location–allocation problems with stochastic demand and congestion. Computers & Operations Research, 48, 20-30. [45]. Vidyarthi, N., Kuzgunkaya, O. (2015). The impact of directed choice on the design of preventive healthcare facility network under congestion. Healthcare Management Science, 18(4), 459-474. [46]. Wright, P.D., Mahar,S. (2013). Centralized nurse scheduling to simultaneously improve schedule cost and nurse satisfaction. Omega, 41, 1042-1052. [47]. Zulu, J.M., Hurtig, A.K., Kinsman, J., Michelo, C. (2015). Innovation in health service delivery: integrating community health assistants into the health system at district level in Zambia. BMC Health Services Research, 15, 15-38.

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