Combining revenue and equity in capacity allocation of imaging facilities

Accepted Manuscript

Combining revenue and equity in capacity allocation of imaging facilities Liping Zhou , Na Geng , Zhibin Jiang , Xiuxian Wang PII: DOI: Reference:

S0377-2217(16)30486-6 10.1016/j.ejor.2016.06.046 EOR 13798

To appear in:

European Journal of Operational Research

Received date: Revised date: Accepted date:

19 December 2015 20 June 2016 22 June 2016

Please cite this article as: Liping Zhou , Na Geng , Zhibin Jiang , Xiuxian Wang , Combining revenue and equity in capacity allocation of imaging facilities, European Journal of Operational Research (2016), doi: 10.1016/j.ejor.2016.06.046

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Highlights Capacity allocation of imaging facilities for multiple patient types is addressed. The joint effect of revenue and equity is considered. A nonlinear mixed-integer model with chance constraint is proposed and solved. The examination queuing model is analyzed from a time-slot server perspective. Total revenue decreases as equity level increases or access time target decreases.

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Combining revenue and equity in capacity allocation of imaging facilities Liping Zhou, [email protected], Na Geng, [email protected], Zhibin Jiang*, [email protected], Xiuxian Wang, [email protected] School of Mechanical Engineering, Department of Industrial Engineering and Management, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, P.R. China *

Corresponding author.

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Abstract

Because of high procurement and operating costs, imaging facilities (e.g., magnetic resonance imaging (MRI)), are usually critical resources in hospitals. Hospital managers are under high pressure to pursue high utilization of the capacity, which leads to long waiting time for patients. However,

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different types of patients have different access time targets determined by their priorities according to the urgent levels and payments. The access time target is defined as the maximal amount of time between the appointment date and the examination date. For public hospitals, it is important to manage patient access to critical resources by considering the equity among different types of patients without sacrificing revenue. This paper proposes a nonlinear mixed-integer programming (NMIP)

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model for allocating the capacity of imaging facilities with the objective of maximizing revenue under

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the constraints of maintaining equity among different types of patients. The equity constraints are defined as the same access levels for different types of patients and the joint chance constraint for the

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same service levels in terms of waiting time. To solve this model, each time-slot, rather than the imaging facility, is considered as a server, which leads to an M/D/n queuing model. Based on an

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analysis of the M/D/n model, an approximation approach is proposed for the NMIP model, and CPLEX is used to solve the approximated model. Extensive numerical experiments based on real data

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from a large public hospital in Shanghai show the applicability and performance of the proposed model and investigate the impact of different parameters. Keywords

OR in health services; capacity allocation; equity; nonlinear mixed-integer programming; joint chance constraint 1. Introduction This paper is motivated by our collaboration with a large Chinese public hospital. Imaging facilities (e.g., magnetic resonance imaging (MRI)), are critical resources in the hospital because of high 2

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procurement and operating costs (Geng, Xie, & Jiang, 2011; Green, Savin, & Wang, 2006; United Kingdom, 2013). Hospital managers are under high pressure to pursue high utilization of such facilities. However, different types of patients have different access time targets (ATTs) determined by their priorities according to the urgent levels and payments. ATT is defined as the maximal amount of time between the appointment date and the examination date. In general, patients are classified into emergency patients (EPs), outpatients (OPs), and inpatients (IPs). EPs are usually in critical condition

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and require immediate treatment and diagnosis. OPs and IPs can be further subclassified as public (or regular) and private (or very important person (VIP)) depending on the payment. Public patients usually pay a lower fee and need to wait for longer periods of time. Private patients pay higher fees for which they receive higher service level treatment. In China, public hospitals are self-financing

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institutions, which have profit incentives (Hsiao, 2008; Reynolds & McKee, 2011; Wang, 2009). Essentially, public hospitals are the basis of the Chinese National Healthcare System (Zhang, Li, Ye, Ding, & Kang, 2015) and have responsibilities and obligations to pursue social equity and justice in the healthcare system (China, 2007; Liu & Griffiths, 2011; Wong, 2004). Therefore, public hospitals

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must maintain the equity among different types of patients while simultaneously maximizing hospital revenue as self-financing institutions. It is challenging for hospital managers to manage patients’

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access by considering the tradeoff between revenue and equity. One natural solution for this problem is to pool the capacity and by using the dynamic scheduling.

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However, scheduling multi-priority customers with the objective of balancing revenue and equity is a complex subject. In addition, the ATT requirements should be taken into account. To our knowledge,

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there is little research on scheduling that considers these issues. Pooling capacity is not always beneficial for customers with different capacities and service-level requirements (Joustra, Van der

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Sluis, & Van Dijk, 2010; Tekin, Hopp, & Van Oyen, 2009). Therefore, we propose capacity allocation for multiple types of patients to solve this problem by combining revenue and equity. According to the definitions from highly cited papers (Culyer & Wagstaff, 1993; Wagstaff & Van

Doorslaer, 2000) and World Health Organization (2000), equity in healthcare includes equity of health, equity of financing, responsiveness equity, equity of access, and equity of utilization. Among these dimensions, equity of access and responsiveness equity could be considered at the operational management level of a single hospital. Others should be considered from the perspective of multiple hospitals in a district or from a countrywide perspective. For instance, equity of financing can be 3

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realized through tax and insurance policies. From the government’s point of view, Cardoso, Oliveira, Barbosa-Póvoa, and Nickel (2016), Mestre, Oliveira, and Barbosa-Póvoa (2012) have used equity as planning objectives to handle the facility location and allocation problem. The government attempts primarily to maximize equity and secondarily to control the costs. Different from the above literature, this paper studies capacity allocation from a public hospital perspective. A public hospital must maximize its revenue to maintain normal operation as a self-financing institution. This paper proposes

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a nonlinear mixed-integer programming (NMIP) model for imaging facilities capacity allocation with the objective of maximizing revenue on the premises of satisfying two equity constraints, equity of access and responsiveness equity.

Some health policy studies (Gulliford et al., 2002; Le Grand, 1987) proposed that the opportunity to

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receive healthcare service is an important aspect of equity of access. Accordingly, in this paper, equity of access is described as controlling the acceptance ratio (which can be interpreted as the percentage of patients accepted for imaging examinations compared to the actual arrival number) of all types of patients with similar values. The World Health Organization (2000) proposed waiting time is one

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perspective for measuring responsiveness equity. In this paper, responsiveness equity is satisfied by ensuring the probability that each patient’s waiting time within his/her ATT will be above a given

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breaching probability, which is a joint chance constraint (JCC). To solve the NMIP model with JCC, we approximate the JCC by multiple single chance constraints concerning the waiting time service

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levels for each type of patients. To analyze patients’ waiting time distribution, each time-slot, instead of the imaging facility, is regarded as a server, which leads to an M/D/n model. A linearization based

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approximation approach is proposed to solve the NMIP model. Real data collected from our collaborating hospital is used in numerical experiments to show the applicability and performance of

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the proposed model.

Mathematical programming models have been widely used in healthcare capacity allocation

literature (Brailsford & Vissers, 2011). Different from the previous studies, this paper makes the following contributions: 1) it considers the joint effects of revenue and equity, 2) it considers the ATTs of different types of patients from an individual perspective, 3) it proposes an approximation solution approach to solve the NMIP model with JCC, and 4) it analyzes the queuing model from the perspective of a time-slot server rather than a machine server. The remaining of this paper is organized as follows. Section 2 presents a brief literature review on 4

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the related research. Section 3 describes the problem and the mathematical model. Section 4 proposes the solution approach. Section 5 presents numerical experiments, and the conclusions are given in Section 6. 2. Literature Mathematical programming models have been extensively used to support capacity allocation decisions in a wide range of fields including manufacturing (Klein & Kolb, 2015), service systems

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such as call centers (Koole & Van Der Sluis, 2003) and healthcare (Brailsford & Vissers, 2011). Because our study is in the context of healthcare, we first review the related literature on capacity allocation in healthcare. Then, we review relevant studies about capacity allocation by considering equity in other fields. Finally, the studies on capacity planning and allocation considering the chance

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constraints of service level requirements are reviewed. 2.1 Capacity allocation in the healthcare area

A considerable amount of studies have addressed healthcare capacity allocation problems in terms of hospital staff, rooms, beds, facilities, and so on. Kao and Tung (1981) proposed an approach based

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on queuing theory for periodically reallocating beds to services to minimize expected overflows. Lapierre, Goldsman, Cochran, and DuBow (1999) developed a time series model using hourly census

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data to make good decisions regarding the size of each unit. Blake and Carter (2002) presented a methodology that included two linear goal programming models for allocating resources in hospitals

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to achieve a preferred mix and volume of treatment types. Li, Beullens, Jones, and Tamiz (2009) proposed an integrated queuing and multi-objective bed allocation model to allocate limited beds to

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multiple departments by balancing patient admissions and profits among all departments. A dynamic programming approach was adopted in Ayvaz and Huh (2010) to allocate limited-capacity hospital

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resources to elective and emergency patients. To ensure a quick diagnosis, Geng, Xie, Augusto, and Jiang (2011) proposed an MRI reservation process for stroke patient examinations. A Monte Carlo approximation approach was combined with local search to determine the contract. The Markov Decision Process model was proposed to prove the structural properties of the optimal control policy. To solve the multi-objective medical resource allocation problem, Feng, Wu, and Chen (2015) developed a multi-objective simulation optimization algorithm by integrating a non-dominated sorting genetic algorithm with a multi-objective computing budget allocation. However, few studies have addressed equity objectives in the healthcare capacity allocation field. 5

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Using the Rawlsian definition of equity, Hooker

and Williams

(2012)

developed a

mixed-integer/linear programming model to allocate scarce healthcare resources by combining the objectives of equity and utilitarianism. Mestre et al. (2012) proposed a hierarchical multiservice mathematical programming model to determine the location and supply of hospital services for maximizing geographic equity of access. Oliveira and Bevan (2006), Cardoso et al. (2015, 2016) considered equity in the area of medical facility location and allocation.

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Additionally, few articles have studied the chance constraint of service level in the healthcare capacity planning and allocation area. Zhang, Puterman, Nelson, and Atkins (2012) described a discrete event simulation methodology for setting long term care capacity levels over a multiyear planning horizon to achieve target waiting time service levels.

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2.2 Capacity allocation by considering equity

Equity is one of the critical concerns when policy makers and managers allocate limited capacity to multiple demand streams. Unfortunately, to the best of our knowledge, capacity allocation that considers equity has received limited coverage in the literature. In many traditional industries such as

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manufacturing, airlines, and hotels, managers focus mainly on revenue maximization (or cost minimization) and pay little attention to equity. Few studies have considered equity in problems

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related to allocating capacity. Berger and Bechwati (2001) proposed a nonlinear programming model to optimize budget allocation between acquisition and retention spending to maximize customer

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equity. Sadegh, Mahjouri, and Kerachian (2010) proposed a crisp and fuzzy Shapley games methodology to allocate shared water resources in water transfer projects considering the equity

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criterion. Tian, Huang, Yang, and Gao (2012) investigated the efficiency and equity of freeway corridor capacity allocation when capacity is limited. Klein and Kolb (2015) considered capacity

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allocation in a firm serving heterogeneous customer segments with the objective of maximizing customer equity. 2.3 Capacity planning and allocation with the chance constraint of service level The chance constraint of service level has been described as the probability of satisfying the demand or the probability of keeping the waiting time below a certain threshold. Many articles have considered this constraint in call center studies. Koole and van der Sluis (2003) studied a shift-scheduling problem for call centers with an overall service level objective. Baron and Milner (2009) provided a square root safety staffing rule that is effective for solving horizon-based service 6

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level agreement. To meet a service level commitment, Robbins and Harrison (2010) formulated a mixed-integer two-stage stochastic programming model to address the uncertainty of customer arrivals and solved the model by an implicit enumeration (branch and bound) algorithm. Service level satisfaction has also been investigated in other research areas. Considering uncertain demand, travel time, and service time, Lin (2014) provided an analytical framework to tackle a three-level territory (including facility location, demand allocation, and resource capacity) planning

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problem of a logistic system to satisfy the response time service level.

Compared with the above literature, this paper investigates capacity allocation for multiple patient types at the hospital operation level by considering the equity of access and responsiveness equity. This paper maximizes revenue within the constraints of the equity requirement rather than the widely

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studied revenue/utilization maximization or cost/idle rate minimization. We develop the queuing model from the perspective of a time-slot server, whereas most studies investigated this issue from the machine perspective. An approximation approach based on queuing theory analysis is proposed to solve the model.

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3. Problem Description and Mathematical Programming Model 3.1 Problem Description

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This paper considers the capacity allocation of imaging facilities among multiple types of patients. Due to the self-financing feature of the hospital and the quasi-public nature of medical service, both

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revenue and equity are considered in this model. It is not easy to balance revenue and equity, especially when capacity is insufficient. In a market economy, it is reasonable to maximize the

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expected revenue while respecting acceptable levels of equity. In this paper, we divide the daily working time capacity of each facility into multiple time-slots

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with different durations. Each time-slot represents the correct examination duration for each type of patient. Each patient examination requires one corresponding time-slot. Different types of patients have different ATTs, and requests for examinations may be rejected due to insufficient capacity. In this manner, the issue of capacity allocation is transformed into a time-slots allocation problem. The decisions are the number of time-slots allocated to each patient type and the planned arrival rate (which can be interpreted as the expected number of patients accepted for imaging examinations) of each patient type. The objective is to maximize total revenue by accepting the patients. The unit revenue is patient type specific. Not that the rejection cost is not considered in this study, but it could 7

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be easily included by modifying the unit revenue. From a pure profit perspective, the hospital would first satisfy the private patients to maximize revenue; however, all types of patients should be served fairly. Therefore, we study the imaging facility capacity allocation problem with the objective of maximizing the revenue by considering two equity constraints: 1) Equity of access constraint: All types of patients should have the same opportunity to receive medical service. According to the definition of access from Penchansky and Thomas (1981), we

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utilize the dimension of “accessibility.” Furthermore, some studies of health policy (Gulliford et al., 2002; Le Grand, 1987) have proposed that the opportunity to receive healthcare service is an important aspect of equity of access. Equity of access in this paper is described as controlling the acceptance ratio of all types of patients with similar values.

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2) Responsiveness equity constraint: The World Health Organization (2000) first proposed the concept of responsiveness equity, and waiting time is one perspective. Therefore, responsiveness equity in this paper refers to equal satisfaction levels of patients within type-specific ATTs. 3.2 Mathematical Programming Model

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This section proposes an NMIP model to allocate imaging facilities capacity to various patient types by combining revenue and equity. The assumptions and notations of the mathematical

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programming model are formally stated as follows. The following assumptions are made throughout the paper.

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Assumption A1: There are several heterogeneous imaging facilities. The capacity of each facility is sliced into time-slots with different durations. Each slot is assigned to the patient type with

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corresponding examination duration. Assumption A2: Different types of patients arrive independently according to a Poisson distribution,

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and the arrival rates are stationary. Patients of the same type require the same examination durations, the same ATTs, and are examined based on the first-come-first-serve (FCFS) discipline. REMARK 1. Figure 1 presents each patient type’s weekly arrival pattern obtained from the data of our collaborating hospital, indicating that patients arrive smoothly. In practice, different types of patients have different expectations or degrees of emergency, so their ATTs could vary. For instance, EPs should be treated immediately, and private patients have higher expectations than public patients because of their more pressing needs and higher payments. Patients of the same type should have the same ATTs and be treated fairly based on the FCFS queuing discipline. 8

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Notations Indexes 1, 2,...,| I |

index of patient type;

j

1, 2,...,| J |

index of imaging facility;

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i

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Figure 1. Patient weekly arrival patterns over six months

Sets

set of patient types, indexed by i;

J

set of imaging facilities, indexed by j;

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I

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Parameters

daily regular capacity of facility j in minutes;

ti

examination duration of type-i patients in minutes; i

daily arrival rate of type-i patients; individual waiting time of type-i patients in days;

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wi

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Kj

ATTi access time target of type-i patients in days; minimal breaching probability of waiting time;

ri

unit revenue of type-i patients in RMB; maximal breaching value of workload balance; maximal breaching value of acceptance ratio deviation;

Variables

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xij

the number of time-slots assigned to type-i patients from facility j, (xij

yi i

0 if type-i patients can be examined by facility j, and xij

0 otherwise);

planned arrival rate of type-i patients; acceptance ratio of type-i patients.

Objective Equation (1) is the objective of maximizing the expected revenue. i I

ri yi

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max

Constraints

(1)

Capacity constraints: Each imaging facility has regular working time every day. Constraint (2) ensures that the assigned time-slots do not exceed the total capacity. xij ti

Kj

j

J

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i I

(2)

Equity of access constraints: Equity of access is ensured by controlling acceptance ratio deviations. Constraint (3) calculates the acceptance ratio of each patient type. Constraint (4) ensures equity of access because it prevents acceptance ratio deviations between each type’s acceptance ratio

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and the average value from exceeding the maximum level. Constraint (5) ensures that the planned

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arrival rate of type-i patients does not exceed their actual arrival rate.

i

yi

yi

i  I

i i

i I

0

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i 

|I| i

i

(3)

I

i I

(4) (5)

Responsiveness equity constraint: Different types of patients have their own ATTs according to

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the urgent levels and payments. Constraint (6) is a responsiveness equity constraint that ensures the probability that any patient’s waiting time is less than his/her ATT remains above a minimum level.

P(w1

ATT1 ,..., w|I |

ATT|I | )

(6)

Workload balance constraints: Based on the expected life of the facility and the employees’ sense of fairness, this study considers the workload balance of imaging facilities and the corresponding technicians. Constraint (7) balances the workload by ensuring that the deviations between each facility’s workload and the average value do not exceed a maximum level. 10

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xij ti i I

xij ti

Kj

Kj

j Ji I

|J|

j

J

(7)

Variable constraints: The constraints in (8) are variable constraints.

xij

;

i

, yi

i I, j

J

(8)

4. Solution Approach

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It is impossible to directly solve the proposed NMIP model due to JCC (6). JCCs are often non-convex and non-smooth, and are generally challenging to solve (Hu, Hong, & Zhang, 2013). In the following, we first break the JCC into multiple single chance constraints. Next we utilize queuing theory of the M/D/n model to obtain a performance measure which sets an analytical relationship between the capacity and the single chance constraints. Specifically, we approximate these single

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chance constraints to some nonlinear constraints by analyzing the waiting time distribution of M/D/n queuing model analytically. Then a linearization approach is proposed to approximate the model with nonlinear constraint. Finally, the original NMIP model is approximated by a mixed-integer linear programming (MILP) model, which can be efficiently solved by a number of well-researched

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algorithms and effective software solvers. Notations

M

discretization number (a large fixed positive integer); the discretized arrival rate of type-i patients;

im

yim

index of discretization number;

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0,1,..., M

{0,1}

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m

1 if planned arrival rate of type-i patients is

im

, 0 otherwise.

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4.1 JCC breaking

A number of general approaches have been proposed in the literature to solve JCC problems,

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including convex conservative approximations (Hong, Yang, & Zhang, 2011) and the scenario approach (Calafiore and Campi, 2006). In this subsection, we first break the JCC into multiple single chance constraints, as shown in (9).

P(wi

ATTi )

|I |

i I

(9)

Proposition 1: Equation (9) is the sufficient condition of equation (6). (Proof of Proposition 1 is given in Appendix A.) According to Proposition 1, we can obtain the subproblem of the original NMIP model by replacing 11

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constraint (6) with equation (9). Unsurprisingly, the solution of the new model is the local optimal solution. REMARK 2. Studies (Zhang et al., 2012) have used the Bonferroni approach (Miller, 1981) to break the joint chance constraint, and equation (6) will hold if equation (10) is satisfied. P( wi

ATTi ) 1

1 |I|

i

(10)

I

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Equation (10) is the sufficient condition of equation (9). Equation (11) can be proved by the mathematical induction approach. |I |

1

1 |I|

[0,1], I

N

(11)

Therefore, the approach of breaking the JCC into multiple single chance constraints in this paper is

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more accurate than that of the Bonferroni approach.

4.2 Waiting time distribution approximation for an M/D/n queuing system Equation (9) ensures the probability that type-i patients’ waiting time not exceed their ATTs is no less than a fixed value. To satisfy this constraint, there is one minimum capacity for each planned arrival rate. Let function 𝑓𝑖 (𝛼, |𝐼|, 𝑦𝑖 , 𝐴𝑇𝑇𝑖 ) be the minimum number of time-slots needed to meet

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constraint (9) with the planned arrival rate 𝑦𝑖 . Then, equation (12) can be used to replace constraint

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(9).

xij

fi ( ,| I |, yi , ATTi )

i

I

(12)

j J

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From the perspective of a traditional queuing system with facilities as servers and patients as customers, one straightforward model of such a system is represented by a multi-server, multi-class

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queuing model. It is too complex to analyze the patients’ waiting time distribution. Therefore, one solution is to use discrete event simulation to calculate the waiting time distribution. An analytical

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analysis has not been performed, which makes capacity allocation difficult and time consuming because of the black-box nature of discrete event simulation. In this paper, we propose a different modeling approach for the capacity allocation of imaging

facilities. The daily capacity of each facility is divided into basic time-slots that correspond to the durations of different examinations types. Each time-slot is considered a server, and each time-slot server serves one patient each day. Patients wait for their examination in a time-slot server queuing system. Therefore, if we know the number of time-slots allocated to each type, the queuing system can be modeled as an M/D/n queuing model with a constant service time of one day. The M/D/n 12

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queuing system has been widely investigated in the literature, and its properties have been thoroughly studied (Franx, 2001; Tijms, 2003, 2006). Although an accurate analytical expression of the waiting time distribution is not available, one well-known approximation was proposed by Franx (2001) and Tijms (2003). By using these equations (see Appendix B), we can calculate the corresponding 𝑓𝑖 (𝛼, |𝐼|, 𝑦𝑖 , 𝐴𝑇𝑇𝑖 ) for each 𝑦𝑖 . 4.3 Linearization of the NMIP model

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Due to the lack of a closed-form expression of the nonlinear function 𝑓𝑖 (𝛼, |𝐼|, 𝑦𝑖 , 𝐴𝑇𝑇𝑖 ), it is approximated by discretization, and then, the proposed NMIP model can be approximated by an MILP model that can be solved by standard solvers.

Function 𝑓𝑖 (𝛼, |𝐼|, 𝑦𝑖 , 𝐴𝑇𝑇𝑖 ) can be calculated with the approximation of the M/D/n queuing system

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in Section 4.2 if planned arrival rate 𝑦𝑖 is known. Therefore, we discretize the planned arrival rate using the following methods: M

yi

yim

im；

im =

m 0

M m i ； yim M m 0

1；yim

{0,1}

i

I , m {0,1,..., M }

(13)

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For each 𝑦𝑖𝑚 , let 𝑓𝑖𝑚 =𝑓𝑖 (𝛼, |𝐼|, 𝑦𝑖𝑚 , 𝐴𝑇𝑇𝑖 ). Then, we can calculate the corresponding 𝑓𝑖𝑚 . The discretization solution would be equivalent to its primary problem if M were large enough according

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to the central limit theorem. After the discretization transformation, the original NMIP model with JCC is transformed into a MILP model as follows: M

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max

ri yim

(14)

im

i I m 0

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S.t.

xij ti

Kj

j

J

(15)

i I

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M

yim

im

i

i m 0

i

i I

|I|

i

xij ti i I

xij ti

Kj

I

(16)

i

Kj

j J i I

i

|J|

I

(17)

j

J

(18)

M

yim

1

m 0

13

i

I

(19)

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M

xij

yim fim

j J

xij

;

yim {0,1};

i

(20)

I

m 0

i I, j

i

J , m {0,1,..., M }

(21)

The MILP model has been widely applied for more than twenty years, and many algorithms have been investigated to solve it. The most common are the linear programming based branch method and

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the bound method; branch-and-price and branch-and-cut methods have also been proposed to solve the MILP model (Grossmann, 2002). In addition to these algorithms, effective software programs can be used to solve the MILP models, including modeling systems such as GAMS, AMPL, and AIMMS and commercial solvers such as OSL, CPLEX and XPRESS (Johnson, Nemhauser, & Savelsbergh, 2000). Because the MILP solution algorithms have been thoroughly investigated, this paper does not

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commit to developing a new MILP solution method. The CPLEX Optimizer is used to solve the MILP model. 5. Numerical experiments

In this section, we collected real data from our collaborating hospital, which is a large public

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hospital in Shanghai, China to show the applicability and performance of the proposed model. There are five types of patients who need MRI examinations: EPs, public IPs, private IPs, public OPs, and

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private OPs. The solutions of the model are compared with an upper bound (UB). Sensitivity analyses were performed to investigate the effects of different parameters such as δ, α, ATT, revenue, and daily

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working time. All numerical experiments were performed on a computer with an Intel® Core™2 Duo CPU at 2.93GHz, and the MILP models were solved using the commercial solver CPLEX 12.6. The

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computation time for all numerical experiments was below 10 seconds; therefore, the computational time for the experiments are not presented.

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5.1 Data setting

This subsection presents a base case from which the numerical experiments will be defined.

According to the real data from the public hospital and the preceding analysis, there are five types of patients, including EPs, private and public IPs, and private and public OPs. These patients have different arrival rates, ATTs, and generate different revenues. There are four MRI scanners, each of which has a daily capacity of 600 minutes. Each scanner can examine all types of patients. The examination time-slot duration for all types of patients is 15 minutes, and the number of patients who

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arrive each day is assumed to follow a Poisson distribution. The time-horizon in this case-study is half a year. According to our field observations, the examination requests from Eps and IPs are limited and considerably smaller than the hospital’s total capacity, and they are not rejected. EPs are in emergency condition and require diagnosis and treatment immediately. Therefore, EPs should be examined on the same day their examinations are requested. Examinations of IPs are requested after admission to the

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hospital. Due to the bed occupancy and continuity of care, IPs cannot be rejected and should be examined as soon as possible. Therefore, the acceptance ratios of IPs and EPs are both equal to 1.0. We consider equity of access among public OPs and private OPs in the following experiments. Table 1 provides information on the patients’ daily arrival rate, unit revenue, and ATT.

Patient type

Daily arrival rate

Unit revenue (RMB)

ATT (days)

1

EPs

4.125

200.0

0

2

Private IPs

5.583

500.0

1

3

Public IPs

56.840

200.0

2

4

Private OPs

12.500

500.0

5

5

Public OPs

88.475

200.0

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Table 1 Patient information

Parameters θ, δ, and α are set to 0.05, 0.05, and 0.8, respectively. These values were defined by

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consultation with the vice president and the director of the management office, who are the hospital policy decision makers of our collaborating public hospital. The patient planned arrival rates are

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discretized by the number M=6000 because the objective value is not significantly improved when M is greater than 6000, which was ascertained by attempting various M values.

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The proposed NMIP model is solved by the approximation solution approach. To show the solution

quality of the approximated model and the accuracy performance of this approximation approach, we use a UB which is the objective value of the model by removing constraint (6) from the original NMIP model. We use “gap”, which is the deviation percentage of the approximated objective value (Ap_Obj) in the MILP model with the UB (𝑔𝑎𝑝 = (𝑈𝐵 − 𝐴𝑝_𝑂𝑏𝑗)⁄𝑈𝐵), to show the quality and the performance. The above base case is then modified to investigate the impact of parameters δ, α, ATT, revenue,

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and daily working time. The numerical cases considered are shown in Table 2. For example, Case 1 is the base case but with different service levels of δ with {0, 0.01, 0.03, 0.05, 0.07, 0.09, 0.1, 0.15} (impact of δ). Table 2 Information of sensitivity analysis numerical cases Modified parameter

Values

1

Service level δ

{0, 0.01, 0.03, 0.05, 0.07, 0.09, 0.1, 0.15}

2

Service level α

{0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9}

3

ATT (days)

4

Private patient unit revenue (RMB)

{50, 100, 200, 300, 400, 500, 600}

5

Daily working time (minutes)

{525, 540, 555, 570, 585, 600, 615, 630}

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Case

Multiples {0, 0.2, 0.5, 1, 2, 3, 4, 5, 10, 15} of

5.2 Base case study

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base case’s ATTs

This subsection presents the effectiveness of the proposed model, the capacity allocation decision, and the performance of decisions for the base case defined in Section 5.1. The proposed

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approximation approach produces a solution with a gap of 3.88% compared with the UB. Figure 2 shows the time-slots allocation solution 𝑥𝑖𝑗 . Table 3 presents the solution for time-slots allocation and

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the acceptance ratio of the base case, including each type’s optimal total time-slots allocation solution 𝑥𝑖 =∑4𝑗=1 𝑥𝑖𝑗 and acceptance ratios. In the base case, capacity is insufficient to accept all patients

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because the acceptance ratios of type 4 and type 5 are smaller than 1.0. Another observation is that the patient acceptance ratios satisfy the equity of access requirements because the constraint (4) is

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respected.

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Scanner 4 Scanner 3 Scanner 2 Scanner 1

Type 1 (EPs)

Type 2 (Private IPs)

Type 3 (Public IPs)

Type 4 (Private OPs)

Figure 2. Time-slots allocation solution 16

Type 5 (Public OPs)

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Table 3 Time-slots allocation and acceptance ratio for type-i patient per day Type i

1

2

3

4

5

𝑥𝑖

9

7

58

12

74

𝛾𝑖

1.0000

1.0000

1.0000

0.9300

0.8341

To illustrate the effectiveness of the planning approach and allocation results, we develop a discrete

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event system simulation procedure to simulate the MRI examination process. Each experiment is completed with an extremely long simulation of 10,000 days. The waiting time distribution of the simulation results based on the above allocation results are presented in Table 4. The grey color is used to highlight the value of the probability of the waiting time being less than or equal to the ATTs of the simulation results. From the cumulative probability distribution of waiting time (CPDWT), the

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probability of each patient type’s waiting time being smaller than the ATT is above the minimum level |𝐼|

of constraint (9) (𝑃(𝑤𝑖 ≤ 𝐴𝑇𝑇𝑖 ) ≥ √𝛼 =0.9564). (For instance, the first line of table 4 shows the CPDWT of type-1 patients. The probability of a type-1 patient’s waiting time being less than or equal to 0 days is 0.9962, and the probability of his/her waiting time less than or equal to 1, 2, 3, 4, 5, 10, or

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20 days is 1.0.)

CPDWT 0

1

0.9962

2

2

3

4

5

10

20

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

0.8319

0.9913

0.9995

1.0000

1.0000

1.0000

1.0000

1.0000

0.6672

0.9671

0.9979

1.0000

1.0000

1.0000

1.0000

1.0000

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3

Time (days)

1

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Type i

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Table 4 Waiting time distribution of the simulation results

0.3745

0.7087

0.8651

0.9366

0.9693

0.9835

1.0000

1.0000

5

0.1706

0.4079

0.5716

0.6916

0.7804

0.8472

0.9906

1.0000

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4

5.3 Sensitivity analysis This subsection addresses the sensitivity analysis of our approach with respect to five key

parameters: δ, α, ATT, revenue, and daily working time. Tables 5-9 show the expected total revenue (ETR), each type’s total allocated time-slots 𝑥𝑖 , the acceptance ratio of private OPs (A-Pri), the acceptance ratio of public OPs (A-Pub), and the solution gap with respective to the changes of different parameters. Sensitivity analysis with respect to (w.r.t.) equity level δ: This is relevant for hospital managers 17

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to know the costs associated with ensuring equity satisfaction levels. As shown in Table 5, the solution gaps are all below 5%. As parameter δ decreases, 𝑥4 , ETR and A-Pri also decrease, whereas 𝑥5 and A-Pub increase. When δ becomes smaller, the acceptance ratio difference between private and public OPs also becomes smaller. Because the total capacity is constant and insufficient, 𝑥4 and A-Pri decrease, and 𝑥5 and A-Pub increase as δ decreases, and additional revenue would be sacrificed. Table 5 MILP model solution and the comparison with the UB vs. δ ETR (RMB)

{𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 , 𝑥5 }

A_Pri

A_Pub

UB (RMB)

Gap (%)

0

35126.2

{9,7,58,11,75}

0.8412

0.8412

36635.6

4.20

0.01

35217.9

{9,7,58,11,75}

0.8558

0.8412

36760.6

4.20

0.03

35340.5

{9,7,58,12,74}

0.8943

0.8345

36984.3

4.45

0.05

35556.4

{9,7,58,12,74}

0.9300

0.8341

36990.0

3.88

0.07

35520.3

{9,7,58,12,74}

0.9458

0.8313

37034.3

4.09

0.09

35712.5

{9,7,58,13,73}

0.9987

0.8187

37034.3

3.57

0.10

35720.8

{9,7,58,13,73}

1.0000

0.8187

37034.3

3.55

0.15

35720.8

{9,7,58,13,73}

1.0000

0.8187

37034.3

3.55

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δ

Sensitivity analysis w.r.t. equity level α: As shown in Table 6, all the solution gaps are below 6%.

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The increasing α leads to an increasing solution gap, 𝑥1 and 𝑥2 , and decreasing 𝑥5 , ETR, A_Pri and A_Pub. The larger the parameter α is, the higher the requirements level of responsiveness equity is.

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Because all EPs and IPs are accepted, more capacity is needed to ensure their waiting time service level, which leads to less remaining capacity for private and public OPs. Consequently, more revenue

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will be sacrificed to ensure a higher level of responsiveness equity. Table 6 MILP model solution and the comparison with the UB vs. α

ETR (RMB)

{𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 , 𝑥5 }

A_Pri

A_Pub

UB (RMB)

Gap (%)

0.1

36495.3

{6,6,58,12,78}

0.9522

0.8793

36990.0

1.35

0.2

36070.8

{7,7,58,12,76}

0.9498

0.8562

36990.0

2.55

0.3

36050.5

{7,7,58,12,76}

0.9480

0.8557

36990.0

2.61

0.4

36029.2

{7,7,58,12,76}

0.9460

0.8552

36990.0

2.67

0.5

35803.0

{8,7,58,12,75}

0.9433

0.8433

36990.0

3.32

0.6

35784.9

{8,7,58,12,75}

0.9418

0.8428

36990.0

3.37

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α

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0.7

35757.5

{8,7,58,12,75}

0.9393

0.8422

36990.0

3.45

0.8

35556.4

{9,7,58,12,74}

0.9300

0.8341

36990.0

4.03

0.9

34909.2

{10,8,58,12,72}

0.9060

0.8060

36990.0

5.96

Sensitivity analysis w.r.t. ATT: As shown in Table 7, the increasing ATT leads to a decreasing solution gap. The gap is 25.84% when ATT=0. This large gap is partly due to the ineffective UB.

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When ATT decreases, the UB stays the same because it ignores the responsiveness equity constraint (6), which results in the larger gap between UB and the optimal objective of the NMIP model. The amount of capacity to serve EPs and IPs (𝑥2 , 𝑥3 ) decreases as ATT increases. That leaves more capacity left for OPs, so the indexes 𝑥5 , ETR, A_Pri and A_Pub first increase and then stay the same due to the diminishing benefit of an increased ATT.

Multiple of base ETR (RMB)

{𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 , 𝑥5 } A_Pri

A_Pub

UB (RMB)

Gap (%)

27430.4

{9,11,71,13,56}

0.5937

0.4937

36990.0

25.84

0.2

31101.9

{9,11,71,11,58}

0.7470

0.6470

36990.0

15.92

0.5

34138.9

{9,11,59,12,69}

0.8738

0.7738

36990.0

7.71

1.0

35556.4

{9,7,58,12,74}

0.9300

0.8341

36990.0

3.88

2.0

35576.9

{9,7,58,12,74}

0.9317

0.8347

36990.0

3.82

3.0

35625.8

{9,7,58,12,74}

0.9338

0.8367

36990.0

3.69

35751.2

{9,6,58,12,75}

0.9412

0.8412

36990.0

3.35

35751.2

{9,6,58,12,75}

0.9412

0.8412

36990.0

3.35

10.0

35751.2

{9,6,58,12,75}

0.9412

0.8412

36990.0

3.35

15.0

35751.2

{9,6,58,12,75}

0.9412

0.8412

36990.0

3.35

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5.0

PT

4.0

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0

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case’s ATT

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Table 7 MILP model solution and the comparison with the UB vs. ATT

Sensitivity analysis w.r.t. revenue: Table 8 shows that all the solution gaps are around 4%. The

increase of private patient unit revenue leads to increases in 𝑥4 , A_Pri and ETR but reductions in 𝑥5 and A_Pub. Unsurprisingly, more private OPs are accepted when their unit revenue increases, while 𝑥5 and A_Pub are reduced because of capacity limitations. However, when the private patient revenue is larger than 200, the acceptance ratio remains unchanged because of the insufficient capacity and the equity of access constraint limits the allocated solutions. In this situation, hospitals

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satisfy the private patients’ needs first, but the ceiling is confined by the equity constraint, regardless of how large the potential revenue is. Table 8 MILP model solution and the comparison with the UB vs. revenue ETR

revenue (RMB)

(RMB)

50

{𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 , 𝑥5 }

A_Pri

A_Pub

UB (RMB)

Gap (%)

28039.1

{9,7,58,10,76}

0.7758

0.8523

29222.0

4.05

100

28803.1

{9,7,58,10,76}

0.7758

200

30333.6

{9,7,58,11,75}

0.8558

300

32114.8

{9,7,58,12,74}

0.9300

400

33835.6

{9,7,58,12,74}

0.9300

500

35556.4

{9,7,58,12,74}

0.9300

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Private patient unit

600

37277.2

{9,7,58,12,74}

0.9300

30051.1

4.15

0.8412

31709.4

4.34

0.8341

33467.7

4.04

0.8341

35226.0

3.95

0.8341

36990.0

3.88

0.8341

38789.4

3.90

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0.8523

Sensitivity analysis w.r.t. daily working time: Table 9 shows that the solution gaps are all below 6%. All four performance indexes (𝑥𝑖 , ETR, A_Pri and A_Pub) increase as the daily working time

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increases. With an increase in daily working capacity, additional patients can be accepted and more revenue is obtained.

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Table 9 MILP model solution and the comparison with the UB vs. daily working time ETR

time (minutes)

(RMB)

{𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 , 𝑥5 }

A_Pri

A_Pub

UB (RMB)

Gap (%)

PT

Daily working

30647.0

{9,7,58,10,56}

0.7280

0.6280

32384.3

5.36

31720.5

{9,7,58,10,60}

0.7728

0.6728

33201.0

4.46

555

32535.5

{9,7,58,10,64}

0.7758

0.7178

34283.6

5.10

570

33600.2

{9,7,58,11,67}

0.8513

0.7513

35095.5

4.26

585

34424.6

{9,7,58,11,71}

0.8558

0.7963

36178.0

4.85

600

35556.4

{9,7,58,12,74}

0.9300

0.8341

36990.0

3.88

615

36313.6

{9,7,58,12,78}

0.9358

0.8748

37834.3

4.02

630

37310.4

{9,7,58,13,81}

1.0000

0.9085

38634.3

3.43

525

AC

CE

540

6. Conclusions and Future Work Imaging facilities are crucial and heavily used resources in hospitals. Hospital managers are under

20

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high pressure to allocate capacity to different types of patients with different access time targets to provide a tradeoff between revenue and equity. This study develops a NMIP model from a tactical level. Based on the time-slot server perspective, this paper proposes a solution technique that combines JCC approximation and linearization to solve the NMIP model. Numerical experiments show that the proposed approach is effective and applicable. The proposed solution is about 8% deviation from the UB in most cases. The sensitivity analysis results show that increasing equity level

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requirements or decreasing ATT could lead to reductions in expected total revenue.

This paper presents an investigation within the context of—but not limited to—hospital imaging facilities capacity allocation. It can be applied to similar settings such as government budget allocation, service system staffing planning, and manufacturing system machine allocation with

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tradeoffs between revenue/utility and equity. In addition, the proposed approach can also be used in the setting of continuous treatment with fixed service time using a time-slot server, for example, radiotherapy. The only difference between these two scenarios is the length of service time of the time-slot server in the M/D/n queuing analysis; both scenarios could be handled by our proposed

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approach.

Future research can be pursued in several directions. One direct extension is to consider uncertain

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service time for the time-slot server, for example, bed occupancy time. Another extension is to consider nonstationary arrival rates.

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Acknowledgements

We would like to thank the editors and the anonymous referees for their helpful comments and

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constructive suggestions, which greatly improved the presentation and the quality of this paper. The work described in this paper was supported by Research Grant from National Natural Science

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Foundation of China (71432006, 71471113). References

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Zhang, L., Li, M., Ye, F., Ding, T., & Kang, P. (2015). An investigation report on large public hospital reforms in China. Current Chinese economic report. Singapore: Springer Science+Business

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Media. doi:10.1007/978-981-10-0039-3. Zhang, Y., Puterman, M. L., Nelson, M., & Atkins, D. (2012). A simulation optimization approach to

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long-term care capacity planning. Operations Research, 60, 249-261. doi:10.1287/opre.1110.1026.

AC

Appendix A. Proofs of proposition 1 Because different types of patients arrive and are treated independently in our capacity allocation

problem, the joint probability distribution can be broken down as follows:

P(w1  ATT1 ,..., w|I |  ATT|I | )    P(w1  ATT1 )  ...  P(w|I |  ATT|I | )  

(A.1)

Let

P(wi  ATTi )   |I |

Then 25

i  I

(A.2)

ACCEPTED MANUSCRIPT

|I |

P( w1  ATT1 ,..., w|I |  ATT| I | )      |I |

(A.3)

i 1

Thus, P( wi

ATTi )

|I |

P( w1

ATT1 ,..., w|I |

(A.4)

ATT|I | )

Therefore, equation (9) is the sufficient condition of equation (6).

CR IP T

Appendix B. Approximation waiting time distribution equations of M/D/n queue Parameters: W

the customer waiting time;

x

a constant time duration; arrival rate; constant service time duration;

i, j , k

integers;

pi

the probability that there are i customers in the system;

qi

the probability that there are i customers in the queue;

Qi

the cumulative distribution of qi ;

G0

a big integer.

M

AN US

D

j

kc 1

x)

e

( kD x )

Qkc

ED

P(W

j 1

j 0

x

kD

(B.1)

j

Qj

PT CE AC

(kD x) j ; for (k -1) D j!

qi ,

j

0,1,....

(B.2)

i 0

C

q0

pi , qi

pi

C

i

1, 2,...

(B.3)

i 0

c

pi

pj j 0

( D)i e i! pj

i c D

pj j c 1

pG0

G0

G0 1

pi i 0

j

pG0

pG0 1

26

( D) i c j e (i c j )! j G0

1

；j

G0

D

; i

N

(B.4)

(B.5)

(B.6)