The therapist assignment problem in home healthcare structures

Expert Systems With Applications 62 (2016) 44–62

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The therapist assignment problem in home healthcare structures Meiyan Lin a, Kwai Sang Chin b,c,∗, Xianjia Wang d, Kwok Leung Tsui b a

Department of Systems Engineering and Engineering Management, City University of Hong Kong, Kowloon Tong, Kowloon, Hong Kong Department of Systems Engineering and Engineering Management and Centre of Systems Informatics Engineering, City University of Hong Kong, Kowloon, Hong Kong c School of Management, Wuhan University of Technology, Wuhan 430070, China d School of Economics and Management, Wuhan University, Wuhan 430072, China b

a r t i c l e

i n f o

Article history: Received 30 November 2015 Revised 5 April 2016 Accepted 8 June 2016 Available online 9 June 2016 Keywords: Therapist assignment problem Mixed-integer programming model Continuity of care Patient priority Preferred time periods

a b s t r a c t Staff planning in Home Health Care (HHC) context is challenging due to the complexity, such as, unavailability of resources, variation in patient health conditions, and diversity of continuity of care (COC) and patient’s priority (PP). This necessitates the implementation of adequately effective models and intelligent systems to improve the robustness of care plans that run with limited input from the support staff. The work proposed an effective, simple, compatible and extensible model for the therapist assignment problem (TAP). The model aims at maximizing the assignment rate of demand, subject to the constraints of workload capacity limitation and available time selection clash. It helps the HHC structure managers to make proper decisions through preferred time periods (PTPs) selections and weight allocations. The analysis of the PTPs claims that the HHC structures applying the TAP model should offer a selection of at most five PTPs to each patient for the sake of effectiveness and efficiency. Following this suggestion, optimal solutions for all instances can be provided within 0.4 s. The weight allocations depend on the various requirements for COC and PP. The analysis of results suggests that the HHC structures can adopt PP in the TAP model without hesitation. However, it also advises that they should pay attention on the adoption of COC, because it has a visible effect on the assignment rate of demand with the lower COC levels and the utilization rate of therapists, while slightly affecting the computational time of the TAP model and the total number of assigned demands. The work offers the HHC structures a demonstration of the core part of an effective planning system to help them make better decisions that satisfy patient demand, achieve high quality of service, and enhance efficiency. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction 1.1. Background Home health care (HHC) implies a wide range of care services, including nursing care; physical, occupational, and speechlanguage therapy; and medical, paramedical, and social services, which can be given to patients at home rather than in hospitals or nursing homes (Medicare.gov, 2016). These services help patients to improve their health condition and their living independence, assist patients in staying at home, and promote the optimal level of well-being of patients to avoid hospitalization or admission to long-term care institutions (Ellenbecker, Samia, Cushman, & Alster, 2008). HHC is economically attractive to an im-

∗

Corresponding author. E-mail addresses: [email protected] [email protected] (K.S. Chin), [email protected] [email protected] (K.L. Tsui). http://dx.doi.org/10.1016/j.eswa.2016.06.010 0957-4174/© 2016 Elsevier Ltd. All rights reserved.

(M. (X.

Lin), Wang),

poverished government because costs are lower than for nursing homes and residential care (Kok, Berden, & Sadiraj, 2015). For instance, the Department on Ageing of Illinois spends approximately US$650 monthly only for HHC, an amount is significantly lower than the US$3510 spent for people in nursing homes (Kielstra, 2009). HHC is a growing sector within the healthcare domain, influencing the global economy. In the United States, approximately 3.5 million Medicare beneficiaries received HHC from 12,613 agencies and cost roughly US$17.9 billion in 2013(MedPAC, 2015). The Hong Kong Government, for example, has increased the annual Elderly Health Care Voucher amount to HK$20 0 0 per elderly (aged 70 or above) to encourage older people to select private healthcare services (GovHK, 2015). Moreover, the demand for HHC is continuously increasing because of aging populations (Christensen, Doblhammer, Rau, & Vaupel, 2009; Lin, Chou, Liang, Peng, & Chen, 2010), dramatic changes in the needs of chronic diseases, development of innovative technologies for HHC (e.g. Ambient Assisted Living System (Botia, Villa, & Palma, 2012)), and a growing pressure on governments to contain healthcare costs (Stabile et al., 2013).

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Abbreviations PTP COC EVAP GAP HHC OT PP PT TAP FCOC PCOC NCOC ADL WA_I

WA_II WA_III WA_IV

WA_V

Preferred Time Periods Continuity of Care Evacuation Vehicle Assignment Problem Generalized Assignment Problem Home Health Care Occupational Therapist Patient Priority Physical Therapist Therapist Assignment Problem Full Continuity of Care Partial Continuity of Care No Continuity of Care Activities of Daily Living The weight allocation criteria relate to the case where the HHC structures consider neither COC nor PP. The weight allocation criteria refer to the case where the HHC structures only consider PP. The weight allocation criteria claim the case where the HHC structures only consider COC. The weight allocation criteria state the case where the HHC structures consider PP is more important than COC. The weight allocation criteria state the case where the HHC structures consider COC is more important than PP.

These trends force the HHC structures to explore means eagerly to reduce cost, improve service quality, and enhance productivity. A primary means of achieving these objectives is the optimal usage of available human resources. The main issues of HHC human resource planning are the human resource dimensioning (Benzarti, Sahin, & Dallery, 2013; Busby & Carter, 2006), the assignment of patients to operators (Carello & Lanzarone, 2014; Hertz & Lahrichi, 2009; Lanzarone, Matta, & Sahin, 2012; Yalcindag, Matta, & Sahin, 2012), and the scheduling and routing for each operator (Akjiratikarl, Yenradee, & Drake, 2007; Cappanera & Scutellà, 2013; Elbenani, Ferland, & Gascon, 2008; Kergosien, Lenté, & Billaut, 2009; Mankowska, Meisel, & Bierwirth, 2014; Nickel, Schröder, & Steeg, 2009; Rasmussen, Justesen, Dohn, & Larsen, 2012; Shao, Bard, & Jarrah, 2012; Trautsamwieser & Hirsch, 2011). Effective staff planning has become the essential means to avoid process inefficiencies, treatment delays, and quality deterioration to maintain profitability, as the HHC structures (i.e., service providers) typically have a large number of patients and have to deliver services to many different locations. In practice, the delivery of service and the feasibility of plans are usually affected by random events, such as variation in patient health conditions (Lanzarone, Matta, & Scaccabarozzi, 2010), unavailability of resources, and long durations of transportation between patients’ homes (Lanzarone & Matta, 2014). Moreover, the existence of certain constraints, such as continuity of care (COC) (Gulliford, Naithani, & Morgan, 2006) and burnout level of operators (Gandi, Wai, Karick, & Dagona, 2011), distinguishes HHC resource planning from the planning problems encountered in production and services systems. Despite the complexity required for planning in most HHC structures, effective staff planning is not supported by proper skills, methodologies, and tools needed for managing the logistics and organizational activities of care delivery. Therefore, the implementation of adequate, effective planning models and tools for the HHC structures is necessary to improve the robustness of plans with limited input from the support staff. The human resource assignment problem has attracted increasing

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attention from researchers and providers because it is of very importance to ensure efficiency and effectiveness, so that planning results can serve as a proper and solid base for the next step, which is obtaining optimal schedules and routes. This study analyzes and discusses the therapist assignment problem considering the different COC and patient priority (PP) levels, the total assigned demand, and the workload utilization of therapists. 1.2. Therapist assignment problem (TAP) This study addresses a particular segment of the HHC industry that provides physical, occupational, and other therapies for patients. The patient is provided with the suitable therapy services by capable therapists according to the estimation of his/her health condition. Usually, districts are considered independent of one another when assignments are planned; thus, a therapist is assigned to a patient based on the compatibility of his/her skills to the patient’s pathology as well as his/her geographical area (Lanzarone & Matta, 2014). A key issue in the assignment problem is the need to retain the COC for patients. A number of the HHC structures are eager to retain the COC to ensure that patients are satisfied with their services. However, other HHC structures do not adopt this concept, which means each visit is provided by any appropriate therapist who has sufficient available capacity in the required time period. Operators assignment under the different COC requirements consists of assigning each newly admitted patient to his/her preferred therapist from the capable ones based on the COC levels he/she requires (Carello & Lanzarone, 2014). In this work, the therapist assignment problem (TAP) is described from two perspectives, that is, patient and therapist or demand and supply. A patient demand is categorized on the basis of the different PP and COC levels, which are related to pathologies and that patient’s eagerness to the COC. A therapist supply is classified according to the main therapy skills of the therapist. Under the assumption that each therapist works as a full-time employee and refuses to work overtime, the total supply of therapists is set unless the HHC structure recruits new therapists. The limited supply of therapists along with the increasing rehabilitation service demand has overexerted the service provider’s ability to provide timely and high-quality treatment. Therefore, decisions should be made to enhance the utilization of therapists and improve patient satisfaction. Currently, those in the rehabilitation industry plan the assignments manually with one or two days to resolve the details. One goal of the TAP, from the perspective of supply, is to maximize the workload utilization rate of therapists under overtime bans. The concept of the utilization rate of each operator in each period was formulated and calculated by researchers in recent years (Lanzarone et al., 2012; Yalcindag et al., 2012). This maximization under overtime bans is important mainly to protect therapists from burnout, which is related to the care volume exceeding the contract capacity of each operator. As a prolonged response to chronic job-related stressors, burnout is a syndrome that can affect a broad range of professions (Gandi et al., 2011); it causes decreased job performance and reduced job commitment, resulting in stress-related health problems and low career satisfaction of workers. At the same time, such objectives guarantee that providers can satisfy as many patients as possible and deliver highquality service to each patient while following laws and regulations. Another goal of the TAP, from the perspective of demand, is to maximize the satisfaction rate of patients (Borsani, Matta, Beschi, & Sommaruga, 2006) by meeting patient requirements that involve the different COC and PP levels. A demand category with the higher COC or/and PP level/s is associated with a higher weight to be satisfied first in the TAP model. The flexibility of weight allocations can help the HHC structures to have enhanced knowledge on which strategy can fit their current situation.

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For the HHC structures, high efficiency is one of the most important criteria for the adoption of the TAP model in order to solve the real life problem. Good efficiency can be attained when minimal computational time for the assignment problem can be ensured. However, the computational time is related to the search space of optimal solutions, which is significantly related to the number of the preferred time periods (PTPs) of each patient. In practice, some HHC structures do not encourage patients to have any preference for time periods, and this variation assumes that all patients are available at any working time session. Conversely, some HHC structures offer patients to voice their PTPs. One extreme situation is that each patient prefers to be visited during one session only. In most situations, each patient selects at least one PTP to the HHC structures. Therefore, another goal of the TAP model is to figure out the proper number of the PTPs of each patient so to achieve minimal computational time and optimal solution.

allocation criteria, and Section 5 describes the conclusions, limitations and future works of our research. 2. Literature review The classic assignment problem and its many variations aim to find optimal pairings of agents and tasks (Pentico, 2007). The TAP can be classified as a multi-dimensional assignment problem that recognizes agent qualifications with side constraints. Tasks refer to the demands (patients, their required therapy services, and their PTPs), and agents refer to the supplies (therapists, their available working sessions). Constraints pertain to the time restrictions and the capacity limitations, and objectives correspond to the maximization of the integrated assignment rate of demand and the utilization rate of supply under considering the various requirements for the COC and the PP. 2.1. Priority

1.3. Contribution One of the main contributions is the proposed TAP model with the formulation of matching the patient demands and the therapist supplies optimally and considering time clash and capacity constraints. A patient demand can always be classified into a specific category despite its diversity; a therapist supply also can always figure out which category it belongs to despite its diversity. Moreover, the TAP model offers the HHC structures five cases of weight allocation criteria, from which to select the proper one for assignment. It works as the core part of an effective planning system, which includes data processing, TAP model, and outcomes explanation. Because of its simplicity, the model can easily be implemented and extended to fit different HHC structures when considering different variations of operational strategies in practice. The second main contribution is the discussion on the effect of the number of the PTPs. The analysis of the PTPs provides a suggestion for the HHC structures to make a better decision to attain high efficiency of the TAP model. With the increasing size of real instances, the computational time for providing an optimal solution exponentially increases and thus the efficiency of the TAP model drops, as the TAP is a NP-hard problem (Pentico, 2007). Adopting proper number of the PTPs is one of the keys to narrow down the search space of optimal solutions and make sure that the planning result can be provided within seconds. The third primary contribution is the analysis of the effect of the COC and the PP under the circumstance where the HHC structures offer a selection of proper number of the PTPs for each patient. We conduct the analysis of the experimental results and provide a better understanding of the effects of the COC and the PP from the perspective of HHC structure managers. The analysis claims significant tips for HHC structure mangers. Finally, the work offers the HHC structures an effective demonstration to handle their operational problems by patterning their problems into mathematical models from the perspective of matching demand and supply. The TAP model is validated through randomly generated instances and a real one from an HHC provider in Hong Kong and solved by a Gurobi Optimizer with Python interface (Robert, Edward, & Zonghao, 2016). 1.4. Structure of the following sections Section 2 presents a literature analysis of PP, COC and the assignment problem. Section 3 describes the problem statement of the TAP under the different COC and PP levels and formulates a mixed-integer program model under different weight allocation criteria. Section 4 discusses the results of the validation of the TAP model with instances in different groups under different weight

The PP is considered a critically important characteristic of demand in the TAP because it represents medical and social needs and indicates the relative position of patients in the waiting queue in practice. Whether in manufacturing systems or service industries, priority is an important real-life feature considered in modeling planning problems of many applications. The work by (Almeida, Correia, & Saldanha-da-Gama, 2016) indicated that the use of the activity priority rules to tackling project scheduling problems can be successfully adapted to the Multi-Skill Resource Constrained Project Scheduling Problem. Priority was associated with each block and defined the retrieval order of the block in the Blocks Relocation Problem (Expósito-Izquierdo, Melián-Batista, & Moreno-Vega, 2015). Two priority-based indicators to measure the benefits and costs are proposed to develop the priority-based constructive algorithms for scheduling agile earth observation satellites (Xu, Chen, Liang, & Wang, 2016). Priority can be formulated as either constraints or objective functions under different circumstances. Job priority constraints have been analyzed and discussed as one type of side constraint that appears both in the daily scheduling of nurses belonging to the float team and the availability list of a hospital (Caron, Hansen, & Jaumard, 1999; Volgenant, 2004). How the consideration of the PP affects the surgery scheduling policy has been presented in the surgery scheduling problem by Min and Yih (2010). They assumed that the priority assigned to each patient does not change until the patient is scheduled, and that priority is generated from the weighted sum of the numerical values of three clinical criteria. The difference is that the PP assigned to each patient in the TAP is updated if the evaluation of the patient updates and fits the corresponding criteria. Formulated as constraints in the equilibrium transit assignment, priority rules establish two useful propositions: “If not all passengers already on-board can obtain a seat, none of the passengers newly attempting to board will be able to obtain a seat.” and “If at least some passengers newly boarding can obtain a seat, all of the passengers with higher priority must have obtained a seat.” (Schmöcker, Fonzone, Shimamoto, Kurauchi, & Bell, 2011). The HHC structures also propose two similar propositions when only considering the PP: “If not all patients with higher priority can receive service during the upcoming week, none of the patients requiring same service with lower priority will be able to be visited.” and “If at least some patients with lower priority can be visited, all of the patients requiring same service with higher priority must have been assigned to be visited.” The priorities of vessel including handling priority and positioning priority are considered the weights in the objective function when balancing the costs of two vessels competing for the same berth in the mixed-integer programming model for the integrated

M. Lin et al. / Expert Systems With Applications 62 (2016) 44–62

berth allocation and quay crane assignment problem (Raa, Dullaert, & Schaeren, 2011). Their view on the priorities for the berth allocation problem is also found in Imai, Nishimura, and Papadimitriou (2003) “Any kind of weight/priority can be attached to individual vessels: After all, this formulation has the advantage that any kind of weight can be attached to individual ships. For instance, when a ship must be handled quickly for a certain reason such as an emergency, high priority may be realized in the resulting solution by adding a high value to it in the formulation.” Our view of priority is similar to these priority rules used in the berth allocation. We design the PP as the weight in the objective function because of the advantages. Firstly, it can avoid formulating the complex constraints related to the PP, thus, stop the probability of increasing the computational time due to these PP-related constraints. Secondly, it can still provide solutions that satisfy constraints and achieve optimal objective function value even though they cannot guarantee the two propositions mentioned above, which means the HHC structures need to compromise on the PP; Third, it works like a soft constraint that demand with the higher PP levels (associated with the higher weights) has a higher probability to be assigned. 2.2. COC and the assignment problems in HHC COC has been identified as an essential element of good primary care, and thus it is not an issue in hospitals. Its concept has changed over time since the 1950 s (Harper, 1958). Some multidimensional models reemerged in the 1990 s and described continuity as “the planning of care according to patient’s needs, providing an ongoing relationship with a care provider, communicating with patients and other care providers, and enabling patients to move orderly through services, having a broad range of services available, being able to move between services flexible and having easy access to care services.”(Uijen, Schers, Schellevis, & van den Bosch, 2012). The COC in most assignment problems in HHC is assumed as that patient is assigned to an operator without change during their whole treatment process. It plays a significant role in preventing potential information loss among operators and in helping to develop a good relationship between operators and patients (Freeman & Hughes, 2010). The TAP model considering the different COC levels is formulated and validated in this study. It adopts the concept of COC described in the aforementioned multidimensional models. For instance, a patient with the partial COC (PCOC) can be reassigned to other capable therapists. Communication among these therapists is necessary because information should be transferred from the responsible therapist to the reassigned therapist. Additional details on the COC in the assignment problems in the HHC context are presented in following. The assignment problem aims to assign personnel to patients in a fair manner with considering COC. This problem becomes an assignment of personnel to visits without considering COC. A body of literature discusses the assignment of a newly admitted patient to an operator under the assumption of the full COC (FCOC), whereas another body focuses on the assignment of a number of newly admitted and follow-up patients with or without considering COC. Under the consideration of COC, different models are applied to achieve workload balance among operators in the problem of assigning a newly-admitted patient to an operator. Lanzarone, Matta, and Jafari (2010) and Lanzarone and Matta (2012a) modeled the stochastic workload with a stationary triangular distribution and patient demand with a uniform distribution under the assumption of a cost quadratic in the number of extra visits. They proposed a cost assignment policy to ensure workload balance among operators and the lowest cost increment. A stochastic Markov chain model was proposed by Lanzarone et al. (2010) to handle uncertainty pertaining to the clinical and social conditions of the patient. The model predicted several major variables, such as the number

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of patients in the following period, duration of care, and amount of required visits. In these studies, a patient is assigned to one preferred operator without any changes throughout his/her whole treatment period regardless of what happens. However, in practice, the HHC structures prefer to solve the assignment problem either at a fixed frequency (e.g., week) or when a certain amount of newly admitted patients is reached. Therefore, many researchers focus on the assignment of a number of patients to a list of operators with regard to workload balance, operator/patient categories and COC in the HHC context. Borsani et al. (2006) proposed a mathematical support tool developed for the HHC structures, the first part of which is the assignment model with several features, including COC, patient satisfaction, outsourced visits, operator efficiency, workload balance, and burnout level. Lanzarone et al. (2012) formulated the specific features of HC services, such as deterministic or stochastic patient demand, COC, operator skills, and geographical areas, into their different assignment models. They also relaxed the complete COC constraint in two ways to obtain the partial or completelyneglected COC and formulated these relaxations into four assignment models. First, more than one reference operator of a category can be assigned to a patient (multi-operator assignment), and second, a change in the reference operator of a patient can be allowed over two consecutive periods. The COC considered in most studies has no levels except for the one from Carello and Lanzarone (2014), who modeled the problem of assigning a set of HHC patients to a set of nurses as a deterministic assignment model considering the different COC requirements and uncertainty in patient demand. The patients are classified into different sets according to their COC levels and their arrival status (follow-up or newly admitted). The different COC levels formulated in the proposed TAP model adopted both the relaxation of the COC mentioned by Lanzarone et al. (2012) and the different levels of the COC formulated by Carello and Lanzarone (2014). The difference is that the different COC levels are formulated as weights in the objective function of the TAP model. The advantages of this formulation are as follows. Firstly, the TAP model can solve the instances under different situations by adopting different weight allocation criteria. For instances, when completely-neglecting the COC, The TAP model applies the corresponding case of weight allocation criteria that treats the demand equally without discrimination. When partially-neglecting the COC, it applies a different case of weight allocation criteria that makes the weight associated with the demand of the FCOC level highest and these associated with the demand of the PCOC or no COC (NCOC) levels lower. Besides, this formulation ensures the flexibility of the TAP model to discuss the effect of the COC on performance such as in terms of the satisfaction rate of demand and the workload utilization of therapists. Several studies have considered patient classification and operator categories when formulating the assignment problems. Yalcindag et al. (2012) assumed that the assignment is held within a single category of operators who are divided into groups based on their main skills and geographical areas to serve. In each group, all operators have the same professional capabilities. Newly admitted patients are assigned to one of the operators at the beginning of the following week under the consideration of balanced workloads of operators and the satisfaction of a set of specific constraints. Similarly, the TAP in this study classifies therapists based on their main skills. Different from Yalcindag et al. (2012), the TAP considers different categories of therapists at the same time because it can be easily extended to model the situation where a “surplus” team of therapists can provide mixed therapies. For instance, these therapists can provide both physical and occupational therapies in this HHC structure. Hertz and Lahrichi (2009) classified the patients into five categories associated with different weights which represent the heaviness level of patient cases

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involved, and the nurses into three types particularly including the “surplus” nurse team. The patients are classified on the basis of their health conditions and requirements in the TAP model, such as their required therapy services, preferred therapists, and the estimated PP levels. The patient demands are classified into nonoverlapping clusters according to their similarity in terms of health conditions and requirements. Different demand clusters are associated with different weights that represent the importance level of demand to be satisfied under the assumption that the workload of each visit to any patient is the same for all therapists. Mathematical models are widely used to formulate the assignment problem in the literature with different tools and algorithms. Borsani et al. (2006) presented two linear programming models in their mathematical support tool and solved them in around 10 minutes by LINGO. Hertz and Lahrichi (2009) proposed a mixed-integer programming model solved by a Tabu search algorithm. Lanzarone and Matta (2012b) proposed a set of mathematical programming models and ran them using OPL 5.1. Yalcindag et al. (2012) analyzed two structural policies and one mixedinteger programming model at the assignment level. Lanzarone et al. (2010) also used a stochastic Markov chain to predict the number of patients in the future and the amount of required visits. Carello and Lanzarone (2014) applied the robust cardinalityconstrained approach proposed by Bertsimas and Sim (2004) to model their problem that can consider uncertainty in patient demand without assuming probability distributions. The model demonstrated enhanced performance in terms of overtime costs and fairness in nurses’ workload. Most of these mathematical models formulated the assignment problems from the perspectives of patients (e.g. related characteristics, constraints, and requirements) and operators (e.g. related skills, basic units, and capacities). Different from these models, the mixed-integer programming model proposed in the TAP is formulated from the perspective of demand and supply, which helps the HHC structures managers to implement and adapt the TAP model in their specific operational system. It can easily be implemented and extended to consider different variations in practice because of its simplification. The advantage is that this TAP model is relieved from the complexity of characteristics of patients and their requirements and the diversity of characteristics of therapists and their requirements. In this study, the strengths of the proposed TAP model come from two aspects: (1) the PP and the COC are formulated as the weights in the objective function of the TAP model instead of constraints; (2) The TAP model is formulated as an optimization match problem of the demands and the supplies in different categories. In summary, the above-mentioned strengths of the proposed TAP model are simplification, compatibility, flexibility and extensibility. 3. The TAP model The planning process begins with a list of patients who are to be treated at home during the upcoming week. Given a set of therapists with various skills, the objective is to develop a weekly assignment plan for each therapist that achieves full workload utilization, meets possibly maximum demand, and minimizes the total cost of providing rehabilitation services while adhering to a series of constraints and accommodating the patient/therapist preferences. A mixed-integer programming model is built based on the description of operational constraints, processes, and resources of the TAP. 3.1. Description of TAP Generally, most patients can be treated at any working session of any working day if they have no preference, but some may have restricted time windows. In practice, having preferred time

windows is more common than having none. The HHC structure faces the issue of determining the number of the PTPs each patient should have in order to maximize the workload utilization of therapists and the satisfaction of patients, and in the meantime, achieve high efficiency of the planning model. Overtime is not allowed for all therapists in this HHC structure because of the possibility of burnout and low service quality when working overtime. In the TAP, treatment duration is assumed to be 1 h, considering the transportation time. The therapist assignment in the HHC structure is completed at the end of a week and is then confirmed with the patients by the manager who is responsible for creating the weekly rosters for all therapists. Patient and therapist classifications are described in the following subsections. 3.1.1. Patient classification The different PP level is assigned to the patients according to their arrival status and health condition. The patients are classified into two categories based on their arrival status: (1) the newly admitted patients who will start their treatment during the upcoming weeks; (2) the follow-up patients who are already under treatment. The highest PP level (level 1, noted as L1) is assigned to the newly admitted patients because they should be visited within five working days. If the follow-up patients need care service more frequently than others, for example, once per week, the PP level of these patients is set to 2 (noted as L2). Level 3 (noted as L3) is graded to those who need care service once two weeks or more. The PP level assigned to the follow-up patients may be updated after treatments. When it’s short of therapists, the treatments of some patients with the lower PP levels may be delayed. The patients are also generally classified in one of three categories based on their eagerness for a preferred therapist: (1) patients who require the FCOC; (2) patients who require the PCOC; and (3) patients who require the NCOC. Each patient in the first category is adamant about being treated by the same preferred therapist during the whole treatment. Each patient in the second category can be reassigned to other capable therapists although he/she has a certain desire to be visited by the same preferred therapist. Each patient in the third category can be assigned to any capable therapist during their therapy process as he/she has no preference for therapists. According to the type of therapy service they need, the patients are distinguished into two different categories. One is the patients require physical therapy service provided by physical therapists (PTs), and the other is the patients require occupational therapy service provided by occupational therapists (OTs).Note that some patients who require both physical and occupational therapy services will be classified into both categories at the same time. Each of these patients cannot receive both therapies on the same working day in consideration of the effect of these treatments and his/her acceptability. Fig. 1 presents the patient clusters based on the above-mentioned classification criteria, namely, the types of therapy service, the PP levels, and the COC levels. 3.1.2. Therapist classification The rehabilitation services provided by the HHC structure are classified into two categories: one physical and the other occupational. The physical therapy consists of exercise therapy, mobility and walking training, chest physiotherapy, and pain relief, etc. The occupational therapy includes aids prescription/training, home assessment/modification, Activities of Daily Living (ADL) training, cognitive training, etc. Shao, Bard, and Jarrah (2012) summarized PTs “help restore function, improve mobility, relieve pain, and prevent or limit permanent physical disabilities of those suffering from injuries or disease. They work primarily with gross motor movements such as gait and disorders involving the spine and extremities”. OTs help patients to improve their ability to perform a job and common day-

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sions in {201, 202, 203, . . . , 212} refer to sessions on Tuesday (d = 2, h = 1, 2, . . . , 12), etc. In the TAP model, a patient in a category presents his/her demand of requesting the corresponding therapy service, the COC and the PP level. 3.2.2. Performance indicators The satisfaction rate Sk, q, r of demand under the (k, q, r) category over the upcoming week is used as the demand level indicator for patients, as shown in Eq.(1)



Fig. 1. The patient clusters.

to-day tasks. They work with individuals who suffer from a disabling condition induced by a mental, physical, developmental, or emotional impairment. The professions considered are PTs and OTs. All therapists are contracted to work full time (e.g. 48 hours per week), and they are typically scheduled six days a week between 9:00 am and 5:30 pm. 3.2. The mix-integer programming model (the TAP model) The TAP model is extracted on the basis of the description of operations from a rehabilitation center in Hong Kong. In general, each therapist can provide H hours of service to patients on each working day (D = six working days in each assumed planning horizon). Some key characteristics of this problem are summarized as follows: 1. The weekly assignment of patients to each therapist is completed at the end of the current week. It considers patients who need therapy next week and who are newly admitted this week. 2. Each therapist in this model is assumed to be a full-time worker who can provide H (in general, H equals eight) hours of service and work six days a week. Overtime is not allowed. 3. Each therapist visits only one patient during a working session. 4. Each patient requests to be served by one therapist during a working session. 5. Each patient requests no more than one session for therapy service of each type during the upcoming week. 6. Patients who require both physical and occupational therapy services cannot receive both of them on the same working day for the sake of their health and safety. 7. Patients who require to be served by the same therapist during their whole treatment are preferred not to be reassigned to other therapists. 8. Patients with the higher PP levels should be visited more frequently than those with the lower PP levels. 3.2.1. Notations of the TAP model Table 1 presents the list of parameters, sets, weights, performance indicators and decision variables in the TAP model. β The working sessions  are denoted  as T = {(d × 10 + h)|d ∈ 1, 0 < H < 10 [1, D], h ∈ [1, H ], β = , including all work2, 10 ≤ H < 24 ing sessions of a week (H × D hours). For instance, when H = 8, working sessions in {11, 12, . . . , 18} refer to sessions on Monday (d = 1, h = 1, 2, . . . , 8), and working sessions in {21, 22, . . . , 28} pertain to sessions on Tuesday (d = 2, h = 1, 2, . . . , 8), etc. When H = 12, for instance, working sessions in {101, 102, . . . , 112} refer to sessions on Monday(d = 1, h = 1, 2, . . . , 12), working ses-



i∈P k,q,r

S k,q,r =



j∈Oi,k

t∈T i,k

yi,j,t

(1)

Nk,q,r

The reassignment rate Rk, q, r of demand under the (k, q, r) category is also presented in the objective function of the TAP model, as the assignment of the demand is more welcome than the delay of the demand, as shown in Eq.(2) .





i∈P k,q,r

Rk,q,r =



j∈Oi,k

t∈T i,k

yi,j,t

(2)

Nk,q,r

As shown in Eq. (3), the assignment rate Ik, q, r of demand under the (k, q, r) category serves as a performance indicator.

Ik,q,r = S k,q,r + Rk,q,r

(3)

The workload utilization rate Uj of the j-th therapist over the upcoming week is presented as a performance indicator in terms of the supply level, as shown in Eq.(4).



Uj =



t∈Fj

(i,j,t)∈C Fj 

yi,j,t

,

∀ j∈O

(4)

3.2.3. Objective functions The TAP model aims at maximizing the assignment rate of demand, while meeting the requirements of a series of constraints and considering the patients’/therapists’ preferences. The assignment rate of demand consists of the satisfaction rate Sk, q, r of demand and the reassignment (the backup assignment) rate Rk, q, r of demand. The higher Sk, q, r and the lower Rk, q, r reflect the higher patients’ satisfaction of therapist assignment and the higher assignment rate of demand. By setting the weights associated with the satisfaction rate and the reassignment rate of demand under each category, we finally maximize the integrated objective function shown as the following Eq. (5), where f1 and f2 are shown in Eq. (6) and (7), respectively:

f = f1 + f2

(5)

Objective (6) explains the satisfaction rate considering the different COC and PP levels.

f1 =

K  L  C 







i∈P k,q,r

πk , q , r ×



j∈Oi,k

t∈T i,k

yi,j,t



Nk,q,r

k=1 q=1 r=1

(6)

Objective (7) explains the reassignment rate considering the different COC and PP levels.

f2 =

K  L  C 





k=1 q=1 r=1

δk,q,r ×



i∈P k,q,r



j∈Oi,k

t∈T i,k

Nk,q,r

yi,j,t



(7)

Using the performance indicators presented in Eq. (1) and (2), the formulation of the objective function can be modified as following Eq. (8) as the reassignment rates Rk, q, 3 and their associated weights δ k, q, 3 are always zero, because the demands without preference for therapists can be served by any capable therapist without reassignment:

f=

K  L  C 

πk,q,r × Sk,q,r + δk,q,r × Rk,q,r

k=1 q=1 r=1



(8)

50

M. Lin et al. / Expert Systems With Applications 62 (2016) 44–62

Table 1 Notations of the TAP model. Parameters and indices The total number of the types of therapy service provided by the HHC structure. The total number of the working days in the upcoming week. The total number of the working sessions per working day ruled by the HHC structure. The total number of the patients requiring to be served in the upcoming week. The total number of the therapists providing therapy service in the upcoming week. The total number of the PP levels set by the HHC structure. In this work, L = 3. The total number of the COC levels set by the HHC structure. In this work, C = 3. The index of the working sessions per working day. h ∈ [1, H]. The index of the working days per week. d ∈ [1, D]. The index of the types of therapy service. k ∈ [1, K]. The index of the patients. i ∈ [1, N]. The index of the therapists. j ∈ [1, M]. The index of the working sessions during the upcoming week. The index of the PP levels. q ∈ [1, L]. A lower value means a higher PP level (L1 > L2 > L3, 1 refers to L1, 2 to L2, 3 to L3). The index of the COC levels. r ∈ [1, C]. A lower value means a higher requirement of the COC (requirement relation follows FCOC > PCOC > NCOC, 1 refers to the FCOC, 2 to the PCOC, 3 to the NCOC). The total number of the patients belonging to the (k, q, r) category, which means that the patients require the k-th type of therapy service with the q-th PP level and the r-th COC level. Sets 1, 0 < H < 10 β ={d × 10 + h |h ∈ [1, H], β = { }, the set of working sessions in the d-th working day. 2, 10 ≤ H < 24 The set of the working sessions during the upcoming week. T = ∪d∈[1,D] Td . The set of the therapy services provided by the HHC structure. S = {1, 2, . . . , K}. The set of the patients who need to be served during the upcoming week. P = {1, 2, . . . , N}. The set of the therapists who are available to provide service during the coming week. O = {1, 2, . . . , M}. The set of the k-th type of therapists who can provide the k-th type of therapy service. k ∈ S. The set of the available working sessions of the j-th therapist during the upcoming week. j ∈ O. Fj ⊂T. The set of the k-th type of therapists whom the i-th patient prefers to be served by. Oi, k ⊂O. The set of the available PTPs which the i-th patient prefers to be served at, for the k-th type of therapy service. The set of the patients who require the k-th type of therapy service with the q-th PP level and the r-th COC level. ∪k∈[1,K ] ∪q∈ [1, L] ∪r∈ [1, C] Pk,q,r = P. The set of the patients who require the k-th type of therapy service. ∪q∈ [1, L] ∪r∈ [1, C] Pk,q,r = Pk . The backup set of capable therapists who can provide the k-th type of therapy service to the i-th patient inPk . Oi,k = Ok − Oi,k . Oi,k will be empty because Oi,k = Ok if the i-th patient is in Pk, q, 3 . ={(i, j, t): j ∈ Oi, k , t ∈ Ti, k |i ∈ Pk }. The set of all assignment tuples to satisfy the k-th type of therapy service demand of the i-th patient. ={ (i, j, t) : j ∈ Oi,k , t ∈ Ti,k |i ∈ Pk }. The set of all backup assignment tuples to reassign the k-th type of therapy service demand required by the i-th patient to capable therapists. = Ai,k ∪Bi,k . The set of all assignment tuples that makes sure the k-th type of therapy service demand required by the i-th patient to be served by capable therapists at his/her PTPs. =∪k∈[1,K ] ∪i∈Pk Ai,k . All assignment tuples to satisfy all therapy service demand required by all patients. =∪k∈[1,K ] ∪i∈Pk Bi,k . All backup assignment tuples (for reassignment) that mean reassignment of therapy service demand to capable therapists at the PTPs of patients. =A ∪B . The total assignment tuples that make sure the delivery of therapy service to patients at their PTPs. Weights The assignment weight of the demands from the patients in Pk, q, r . The satisfaction weight of the demands from the patients in Pk, q, r . The reassignment weight of the demands from the patients in Pk, q, r . Performance indicators The satisfaction rate of the demand under the (k, q, r) category. The reassignment rate of the demand under the (k, q, r) category. The assignment rate of the demand under the (k, q, r) category. The utilization rate of the j-th therapist. Variables 1, i f patient i is served by therapist j at working session t. ={ 0, otherwise.

K D H N M L C h d k i j t q r Nk, q, r

Td T S P O Ok Fj Oi, k Ti, k Pk, q, r Pk Oi,k Ai, k Bi, k Ci, k A B C

α k, q, r π k, q, r δ k, q, r Sk, q, r Rk, q, r Ik, q, r uj yi, j, t

3.2.4. Constraints Constraint (9) indicates that each available working session of each therapist can only be occupied at most once.



yi,j,t ≤ 1 ,

∀t ∈ Fj , ∀ j ∈ O

(9)

( i , j , t )∈ C



yi,j,t ≤ 1 ,

  yi,j,t ≤ Fj , ∀ j ∈ O

(10)

Constraint (11) specifies that each demand can be served by the preferred physical/occupational therapist, be reassigned to the

(11)

Constraint (12) shows that the demands required by the same patient should not be satisfied on the same working day.

⎝

 t∈Ti,p ∩

t ∈ F j ( i , j , t )∈ C

∀ i ∈ Pk , ∀ k ∈ [1, K]

( i , j , t )∈ C i , k

⎛

Constraint (10) shows that the total number of demands assigned to each therapist should not exceed his/her weekly capacity.

 

other capable physical/occupational therapists, or be delayed.

 Td

j∈Op

⎞ 



yi,j,t ⎠∗

t∈T i,k ∩

 Td

 yi,j,t

= 0,

∀i ∈ Pp ∩Pk ,

j∈Ok

∀ p, k ∈ [1, K], ∀ d ∈ [1, D]

(12)

Constraint (13) indicates the binary decision variable.

yi,j,t ∈ {0, 1}, (i, j, t) ∈ C

(13)

M. Lin et al. / Expert Systems With Applications 62 (2016) 44–62

3.2.5. Model and weight allocations analysis 3.2.5.1. Model analysis. As indicated, the demands in the TAP model are classified into K × L × C categories with Nk, q, r , ∀k ∈ [1, K], ∀q ∈ [1, L], ∀r ∈ [1, C]. If Nk,q,r = 0, ∃k ∈ [1, K], ∃q ∈ [1, L], ∃r ∈ [1, C], that is, there are no demands belonging to the (k, q, r) category, then the satisfaction rate Sk, q, r and the reassignment rate Rk, q, r equal zero. The satisfied assignment and reassignment make the satisfaction of Eq. (14), the left side of which relates to the total number of demands being served and the right side presents the total number of supplies being occupied. By maximizing the objective function (8) that considers the satisfaction of demand and allows few reassignments, the workload utilization rate of all therapists can achieve certain optimization. The reassignment strategy improves the utilization rate of therapists. K  L  C  k=1 q=1 r=1



  i∈P k,q,r j∈Ok t∈T i,k



yi,j,t

=

 

yi,j,t

(14)

j ∈ O t ∈ F j ( i , j , t )∈ C

Constraint (12) is the quadratic constraint that ensures patients who require more than one type of therapy service cannot be treated at the same time and on the same day. It increases the computational time of the TAP model with the increasing size of the Ti, k of each patient (namely, the number of the PTPs). 3.2.5.2. The general weight allocation criteria. The weight associated with the demand under each category should ensure the higher satisfaction rate of demand and the lower reassignment rate of demand so as to achieve the higher assignment rate of demand and the higher utilization rate of therapists without constraints violation. The weight allocation criteria imply that the demand under different categories has different levels of importance according to the various strategies of COC and PP. The first consideration focuses on the situation where both the COC and the PP have weights on each demand and where the importance of the COC than the PP is random. The more importance of the COC or the PP, the higher weights associated with the demand under the corresponding categories. The patients (demands) have been classified into K × L × C categories. The assignment weight α k, q, r associated with the demands from the patients in Pk, q, r satisfies the criteria (15a) to (15d). To achieve maximum assignment of the demands under each category, the model enables the reassignment with the small weights δ k, q, r . Then the satisfaction weights π k, q, r and the reassignment weights δ k, q, r associated with each demand category in the objective function should follow the criteria (15e) to (15k). K  L  C 

αk,q,r = 1

πk,1,r > πk,2,r > πk,3,r > . . . > πk,L,r > 0, ∀k ∈ [1, K], r ∈ [1, C] (15g)

πk,q,1 > πk,q,2 > πk,q,3 > . . . > πk,q,C > 0, ∀k ∈ [1, K], q ∈ [1, L] (15h)

δ1,q,r = δ2,q,r = . . . = δK,q,r , ∀q ∈ [1, L], r ∈ [1, C]

(15i)

0 < δk,1,r < δk,2,r < δk,3,r < . . . < δk,L,r , ∀k ∈ [1, K], r ∈ [1, C]

(15j)

0 < δk,q,1 < δk,q,2 < . . . < δk,q,C , ∀k ∈ [1, K], q ∈ [1, L]

(15k)

The criteria (15b), (15f) and (15i) indicate that no difference in the weights exists among the demand classified according to therapy needs. The criteria (15c) and (15d) (or (15g) and (15h)) state that the demands with the higher COC or/and PP levels are associated with higher assignment (or satisfaction) weights; while the criteria (15j) and (15k) claim that they are associated with lower reassignment weights. These four criteria ((15c), (15d), (15g), and (15h)) ensure that the demands with the higher COC and/or PP levels have a higher probability to be assigned and satisfied in the model. The weight allocation criteria in this project takes given values K = 2, L = 3, and C = 3. In following sections, discussion is conducted only based on these given values. 3.2.5.3. Weight allocation criteria without considering the COC. When there is no requirement on the COC for all patients, there will be no concept of backup assignments. So, the demands can be served by any capable therapist and hence the backup assignment sets are empty. The weights δ k, q, r associated with backup assignments will be meaningless and removed, so the weight π k, q, r satisfies the equation where αk,q,r = πk,q,r , δk,q,r = 0, ∀k ∈ [1, 2], q ∈ [1, 3], r ∈ [1, 3] according to Eq. (15e). If this model considers neither the COC nor the PP, the weights π k, q, r need to follow the criteria (16a) to (16d). 2  3  3 

πk , q , r = 1

(16a)

k=1 q=1 r=1

π1,q,r = π2,q,r > 0, ∀q ∈ [1, 3], r ∈ [1, 3]

(16b)

πk,q,1 = πk,q,2 = πk,q,3 > 0, ∀k ∈ [1, 2], q ∈ [1, 3]

(16c)

(15a)

πk,1,r = πk,2,r = πk,3,r > 0, ∀k ∈ [1, 2], r ∈ [1, 2]

(16d)

(15b)

π k, q, r associated with the demands in (k, q, r) category satisfies

k=1 q=1 r=1

α1,q,r = α2,q,r . . . = αK,q,r , ∀q ∈ [1, L], r ∈ [1, C]

51

When only considering the PP in the model, the weight

the criteria (16a) to (16c) and (16e) instead of (16d), which consider the PP levels but not the COC levels.

πk,1,r > πk,2,r > πk,3,r > 0, ∀k ∈ [1, 2], r ∈ [1, 3]

αk,1,r > αk,2,r > αk,3,r > . . . > αk,L,r > 0, ∀k ∈ [1, K], r ∈ [1, C] (15c)

αk,q,1 > αk,q,2 > αk,q,3 > . . . > αk,q,C > 0, ∀k ∈ [1, K], q ∈ [1, L] (15d)

αk,q,r = πk,q,r + δk,q,r , ∀k ∈ [1, K], ∀q ∈ [1, L], ∀r ∈ [1, C]

(15e)

π1,q,r = π2,q,r . . . = πK,q,r , ∀q ∈ [1, L], r ∈ [1, C]

(15f)

(16e)

3.2.5.4. Weight allocation criteria with considering the COC. When considering the COC, it is very important to meet all the requirements of the patients by assigning their demands to their preferred therapists instead of reassigning their demands to the other not-preferred but capable therapists. The assignment weight α k, q, r associated with the demands in (k, q, r) category should meet the criteria (15a), (15b) and (15d), which are revised as the criteria (17a), (17b) and (17c) when K = 2, L = 3, C = 3. The maximization of the assignment rate Ik,q,r (Ik,q,r = Sk,q,r + Rk,q,r ) can be achieved in the objective function by setting the satisfaction weight π k, q, r and the reassignment weight δ k, q, r following the criterion (17i) to make sure Sk, q, r Rk, q, r . Besides, π k, q, r and

52

M. Lin et al. / Expert Systems With Applications 62 (2016) 44–62

δ k, q, r should meet the criteria (15e), (15f), (15h), (15i) and (15k), which are revised as (17d) to (17h). With considering only the COC but not the PP, the weights π k, q, r , δ k, q, r and α k, q, r should also meet the following criteria (17j) to (17l). All these criteria (17a) to (17l) indicate that the demand with the higher COC levels is associated with a higher satisfaction weight and a lower reassignment weight. 2  3  3 

Table 2 Percentage of different types of demands. Demands

Categories

Percentage %

Physical or Occupational?

Physical Therapy Occupational Therapy Yes No New admitted Level 1(L1) Follow up Level 2 (L2) Level 3 (L3) FCOC PCOC NCOC

69 31 26 74 14 30 56 30 37 33

Belonging to same patient? The PP to be visited

αk,q,r = 1

(17a) the COC levels

k=1 q=1 r=1

α1,q,r = α2,q,r . . . = αK,q,r , ∀q ∈ [1, 3], r ∈ [1, 3]

(17b)

αk,q,1 > αk,q,2 > αk,q,3 > 0, ∀k ∈ [1, 2], q ∈ [1, 3]

(17c)

αk,q,r = πk,q,r + δk,q,r , ∀k ∈ [1, 2], ∀q ∈ [1, 3], ∀r ∈ [1, 3]

(17d)

π1,q,r = π2,q,r , ∀q ∈ [1, 3], r ∈ [1, 3]

(17e)

πk,q,1 > πk,q,2 > πk,q,3 > 0, ∀k ∈ [1, 2], q ∈ [1, 3]

(17f)

δ1,q,r = δ2,q,r , ∀q ∈ [1, 3], r ∈ [1, 2]

(17g)

0 < δk,q,1 < δk,q,2 , ∀k ∈ [1, 2], q ∈ [1, 3]

(17h)

0 < δk,q,r  πk,q,r , ∀k ∈ [1, 2], q ∈ [1, 3], r ∈ [1, 2]

(17i)

αk,1,r = αk,2,r = αk,3,r > 0, ∀k ∈ [1, 2], r ∈ [1, 3]

(17j)

πk,1,r = πk,2,r = πk,3,r > 0, ∀k ∈ [1, 2], r ∈ [1, 3]

(17k)

0 < δk,1,r = δk,2,r = δk,3,r , ∀k ∈ [1, 2], r ∈ [1, 2]

(17l)

If the HHC structure considers the PP and the COC at the same time, the assignment weight α k, q, r , the satisfaction weight π k, q, r , and the reassignment weight δ k, q, r in the problem should follow not only the criteria (17a) to (17i) but also the criteria (18a) to (18c). Two specific situations about the weight allocation are also considered in this study. If the PP means more important than the COC to patients, the weights meet not only the criteria (17a)-(17i) and the criteria (18a)-(18c), but also the following criteria (18d) to (18f), as patients would rather be visited by other capable therapists than be postponed to another week. On the contrary, the COC means more important to patients when they prefer to be visited by their preferred therapist and can wait for service if necessary. Then, the weights follow not only the rules (17a)-(17i) and the criteria (18a)-(18c), also the following criteria (18g) to (18i) .

αk,1,r > αk,2,r > αk,3,r > 0, ∀k ∈ [1, 2], r ∈ [1, 3]

(18a)

πk,1,r > πk,2,r > πk,3,r > 0, ∀k ∈ [1, 2], r ∈ [1, 3]

(18b)

αk,3,r > αk,1,r+1 , ∀k ∈ [1, 2], r ∈ [1, 2]

(18g)

πk,3,r > πk,1,r+1 , ∀k ∈ [1, 2], r ∈ [1, 2]

(18h)

δk,3,1 < δk,1,2 , ∀k ∈ [1, 2]

(18i)

In summary, five cases of weight allocation criteria are presented in Section 3.2.5.3 and Section 3.2.5.4. The first case (denoted as WA_I) relates to the case where the HHC structures consider neither COC nor PP. The second case (denoted as WA_II) refers to the case where the HHC structures only consider PP. The third case (denoted as WA_III) claims the case where the HHC structures only consider COC. The forth case (denoted as WA_IV) refers to the case where the HHC structures consider that PP is more important than COC. The final case (denoted as WA_V) refers to the case where the HHC structures consider that COC is more important than PP. 4. Computational tests Computational tests are conducted to evaluate the proposed model in the rehabilitation care services context. The aim is to evaluate the performance of the TAP model with regard to the maximum assignment rate while preserving the COC and the PP as patient preference. The TAP model is validated by different instances, each with different characteristics. In this model, the effect of the number of the PTPs are tested and discussed. Instances with all patients each selecting at most 48, 5, 4, 3, and 2 available sessions are randomly generated based on reality and are used to validate the model. In this section, we first describe the instance of a rehabilitation service provider in Hong Kong, followed by a set of random instances in Section 4.1. This service provider is one of the main rehabilitation service providers in a large district in Hong Kong, and it currently has four PTs, two OTs, and a number of assistants providing rehabilitation therapy service to patients. The computational results are presented in Section 4.2. 4.1. Data instances

0 < δk,1,r < δk,2,r < δk,3,r , ∀k ∈ [1, 2], r ∈ [1, 2]

(18c)

αk,q,3 > αk,q+1,1 , ∀k ∈ [1, 2], q ∈ [1, 2]

(18d)

πk,q,3 > πk,q+1,1 , ∀k ∈ [1, 2], q ∈ [1, 2]

(18e)

δk,q,2 < δk,q+1,1 , ∀k ∈ [1, 2], q ∈ [1, 2]

(18f)

The service provider offers two types of therapy services (K = 2 ), classifies the demands into three different levels (L = 3 ), and grades patients’ eagerness on the COC into three levels (C = 3 ). Table 2 and Fig. 2a-c show the percentage of the demands in the real-life instance from the service provider. Here, the demands are represented using the patients and their requiring therapies. The characteristics of each demand, such as the therapy details, the PP level, and the COC level, are identified after the evaluation of the corresponding patient’s health condition. Fig. 2d shows the percentage of the supplies and Table 3 presents

M. Lin et al. / Expert Systems With Applications 62 (2016) 44–62

53

Fig. 2. The percentage of demand and supple of real life instance. Table 3 Workforce details of each instance in each size group. Instance size

S

Therapist Details

No.

Capacity

No.

M Capacity

No.

Capacity

PTs OTs Total number of supplies

2 1 3

48 48 144

4 2 6

48 48 288

8 4 12

48 48 576

Table 4 Demand details of base instances. Instance size Patients

Demands

P1

PT

L1

L2

L3

OT

L1

L2

L3

The total

P1 ∩P2 P2 FCOC P1, 1, 1 PCOC P1, 1, 2 NCOC P1, 1, 3 FCOC P1, 2, 1 PCOC P1, 2, 2 NCOC P1, 2, 3 FCOC P1, 3, 1 PCOC P1, 3, 2 NCOC P1, 3, 3 FCOC P2, 1, 1 PCOC P2, 1, 2 NCOC P2, 1, 3 FCOC P2, 2, 1 PCOC P2, 2, 2 NCOC P2, 2, 3 FCOC P2, 3, 1 PCOC P2, 3, 2 NCOC P2, 3, 3 number of demands

S

M

L

110

219

442

20 50 6 5 4 12 15 9 15 21 23 3 3 2 3 6 3 9 10 11 160

41 100 14 6 11 16 30 23 33 47 39 6 4 4 7 12 8 20 20 19 319

83 201 14 21 21 31 45 47 90 97 76 6 14 15 20 28 22 32 33 31 643

the number of therapists and their workload capacity under each category. Based on Table 2, the base instances of different sizes are randomly generated, including instance of small size (2 PTs, 1 OT, and 140 patients) in the S group, real-life instance of medium size (4 PTs, 2 OTs, and 278 patients) in the M group, instance of large size (8 PTs, 4 OTs, and 560 patients) in the L group. The demand details of each base instance in each group are presented in Table 4. The demands are classified into the different categories according to the types of therapy service, the PP and COC levels. The set P1, 1, 1 consists of the patients having requirements for physical therapy, level 1 of the PP, and the FCOC level, namely the patients under the (PT, L1, FCOC) category and their demands under

L

the (1,1,1) category. P2, 2, 3 represents the patients under the (OT, L2, NCOC) category, namely, their demands in the (2,2,3) category; etc. The preferred therapist of a patient is also randomly generated according to the category he/she is classified into. For instance, if the demand from a patient falls into one of the (1, q, r) ∀q ∈ [1, 3], ∀r ∈ [1, 2] categories, his/her preferred therapist will be one physical therapist randomly selected from the O1 therapists set; instead, if the demand of the i-th patient is in any one of the (1, q, 3) ∀q ∈ [1, 3] categories, which means he/she has no preference for a specific therapist, the set of preferred therapists of the i-th patient Oi, 1 will consist of all physical therapists, namely Oi,1 = O1 . Table 3 and Table 4 show the undersupply exists in all instances. This undersupply leads to delays of demands because of the overtime ban. With the increasing size of instances, the delays of demands become more serious in this study (e.g. 16 delays for S group, 31 delays for M group, 67 delays for L group). The base instance in each group is used as base data for all tests in the project.

4.2. Computational results The instances in different groups are applied to validate the model by using the Gurobi Optimizer in a personal computer 2.2 GHz Intel Core, 8GB usable memory, OS X operating system). The percent gap between the optimal solution and the best node is 0.00%. The effect of the PTPs, the COC, and the PP are discussed in the following subsections. Different situations of the PTPs are discussed in this section to estimate the effect of the PTPs on the performance of the TAP model. The analysis of the PTPs provides the HHC structures suggestions on the PTPs. In practice, offering too many PTPs may cause the selection of proper PTPs difficult. Therefore, only seven situations are discussed. Seven instances in each size group are randomly generated by using the base instance in the same size group presented in Table 4. For example, one of the seven instances in the S (M/L) group falls into one of the (“PTPs= 48”, “PTPs ≤ 48”, “PTPs ≤ 5 ”, “PTPs ≤ 4 ”, “PTPs ≤ 3 ”, “PTPs ≤ 2 ”, “PTP = 1 ”)

54

M. Lin et al. / Expert Systems With Applications 62 (2016) 44–62 Table 5 Size of model obtained by Gurobi. Instances

Size

S

M

L

Categories

Rows Columns Nonzeros Rows Columns Nonzeros Rows Columns Nonzeros

PTPs = 48 Ins_A

≤ 48 Ins_B

≤5 Ins_C

≤4 Ins_D

≤3 Ins_E

≤2 Ins_F

=1 Ins_G

31,144 28,320 133,440 252,757 177,600 984,960 2,041,525 1,228,224 7,555,968

5397 9311 32,285 72,238 62,324 303,208 524,700 366,640 2,040,960

408 876 2143 1574 3884 10,088 7686 16,076 48,152

360 658 1522 1322 3504 7828 5686 13,232 37,504

290 514 1145 1120 2358 5954 5171 10,688 31,136

238 345 768 980 1772 4502 2903 7232 18,624

194 220 470 641 1088 2374 2054 4748 11,576

categories. If one instance falls into the (“PTPs = 48 ) category, it means that each patient in this instance can be served at any working session during the whole upcoming week. If the instance falls into the (“PTPs ≤ 5 ”) category, it means that each patient in this instance can only select at most five PTPs during the whole upcoming week, etc. For better description, we denote the instances under the (“PTPs= 48 ”) category, (“PTPs ≤ 48 ”) category, (“PTPs ≤ 5 ”) category, (“PTPs ≤ 4 ”) category, (“PTPs ≤ 3 ”) category, (“PTPs ≤ 2 ”) category and (“PTP = 1 ”) category as the Ins_A, the Ins_B, the Ins_C, the Ins_D, the Ins_E, the Ins_F, and the Ins_G respectively. Therefore, there are 21 instances (7instances per group × 3groups) generated randomly for the TAP model validation. To investigate the effect of the COC and the PP on the performance indicators, the TAP model is validated under the five cases of weight allocations criteria as discussed in Section 3.2.5.3 to 3.2.5.4: (1) the (neither the COC nor the PP) case, that is, the case WA_I; (2) the (only the PP) case, that is, the case WA_II; (3) the (only the COC) case, that is, the case WA_III; (4) the (the COC is more important than the PP) case, that is the case WA_IV; and (5) the (the PP is more important than the COC) case, namely, the case WA_V. The weights in the objective function of the TAP model are generated randomly based on these five cases of criteria. During the experiments, 10 runs are conducted for each instance under each PTPs category in each size group to test the TAP model under each case of weight allocation criteria, which means that experiments are conducted 1050(= 10 × 7 × 3 × 5) times. 4.2.1. Effect of the PTPs Table 5 shows the details of the TAP model when using the instances under the different PTPs categories in the different size groups, which are presented by three characteristics: the number of rows, the number of columns, and the number of nonzeros (i.e., “rows,” “columns,” and “nonzeros” in Table 5, respectively). The “PTPs” column represents the number of the PTPs of each patient. The size of the TAP model only depends on the size of instance and the number of the PTPs, as indicated in Table 5. The TAP model has more rows, columns and nonzeros when validating with instances of larger size (i.e. L > M > S). When using instances in the same group, the size of the TAP model booms with the growing number of the PTPs because of the rising number of potential assignments (i, j, t). When the size of the TAP model reaches more than one hundred millions of rows, sixty millions of columns and four hundred millions of nonzeros, the Gurobi optimizer falls to solve it. The computational time depends on the size of the TAP model, which depends on the size of each instance under each PTPs category. Table 6 indicates the multiple relation of the denominator, the computational time of the Ins_C in each row of Table A.1, and the numerator, the computational time of the instances under each PTPs category in the same row of Table A.1. When the value is larger than 1, the higher it is (or when the value is smaller than

1, the lower it is), the larger gap in the computational time exists between the instances under that category and the Ins_C. From Table 6, a very large gap in the computational time exists between the Ins_C and the Ins_A (Ins_B). For instance, the computational time of the Ins_A (Ins_B) is at least 18.3 times (7.01 times) longer than that of the Ins_C. A slight difference of the computational time exists between the Ins_C and the Ins_D. The computational time of the Ins_C is at most 1.81 (= 1/0.55) times (3.85 (= 1/0.26) times) longer than that of the Ins_D (Ins_E). When comparing the computational time of the Ins_C with that of the instances under the other two categories, which are, the Ins_F and the Ins_G, it is at most 4.34 (= 1/0.23) times and 11.11 (= 1/0.09) times longer than that of the Ins_F and the Ins_G respectively. The effect of the number of the PTPs on computational time is also illustrated in Fig. 3 The computational time of instances decreases sharply when the number of the PTPs decreases to 5 (see Fig. 3). The number of the PTPs has an obvious effect on the efficiency of the TAP model in terms of the computational time. As a conclusion, with the smaller number of the PTPs, the shorter computational time the model will run. It will be beneficial for the HHC structures to choose the model with the shorter computational time, so that the assignment plans of the upcoming week are obtained within the shorter running time of the whole planning system adopting the model. For the sake of the efficiency of the TAP model, the HHC structure should pay attention on the smaller number of the PTPs. Besides, the HHC structures are also concerned about the satisfaction of patient demand. Therefore, the number of assigned demands and the objective function value will be another two performance indicators for the HHC structures to choose the proper number of PTPs. Fig. 4 reveals a trend of dramatic decrease of the objective function value when the number of the PTPs decrease. The objective function value stays stable or slightly increases before the number of the PTPs decreases to 5 for all instances in all groups. From 5 PTPs, the objective function value had been dropping quickly till it reached the lowest point at 1 PTP. The number of assigned demands remains the same before the PTPs reaches 5. From 5 PTPs, the total number of assigned demands has been falling till it reached to the lowest point at 1 PTP (see Fig. 5). For the HHC structures, it is important to make every effort to satisfy the patient demand to succeed under strong competition in HHC markets, which means they should make sure to satisfy as much demand as possible. Therefore, the HHC structures that allow each admitted patient to select at most five PTPs with an improved chance of being visited, enables the model to obtain optimal results within 0.4 s for instances of S/M/L sizes (see Table A.1).

4.2.2. Effects of the COC and the PP The following section will help the HHC structures to deal with the decision making on the COC and the PP. In practices, most of

M. Lin et al. / Expert Systems With Applications 62 (2016) 44–62

Fig. 3. Run time of model.

Fig. 4. Objective function values of instances under different categories and groups.

Fig. 5. The number of assigned demands of instances in different groups.

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M. Lin et al. / Expert Systems With Applications 62 (2016) 44–62 Table 6 Computational time (s) of the TAP model validated by all instances. Instances

Size

S

M

L

Weights Setting

I II III IV V I II III IV V I II III IV V

PTPs B = 48 Ins_A

≤ 48 Ins_B

≤5 Ins_C

≤4 Ins_D

≤3 Ins_E

≤2 Ins_F

=1 Ins_G

18.3 31.9 32.5 29.2 35.4 26.2 25.8 25.4 23.8 33.3 67.1 70.1 57.7 62 57.4

10.5 13.5 14.8 14.4 14.7 8.06 8.04 7.01 7.29 10.5 25.7 25.2 17.5 20.1 18.6

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0.92 1.43 0.79 0.66 0.76 0.58 0.55 0.61 0.72 0.7 3.57 4.51 1.2 1.34 1.22

0.38 0.48 0.56 0.49 0.54 1.13 1.42 0.26 0.27 0.32 2.28 2.56 0.7 0.82 0.67

0.23 0.32 0.38 0.32 0.34 0.81 0.55 0.27 0.28 0.33 0.67 0.72 0.49 0.61 0.36

0.14 0.18 0.28 0.23 0.26 0.25 0.44 0.09 0.1 0.11 1.2 1.18 0.29 0.29 0.25

Fig. 6. The computational time of Ins_C under different weight settings. Fig. 7. The total assigned demands of Ins_C under different weight settings.

the HHC structures in HK have the M/L size of demand and supply (e.g. 300+ demands and 6 supplies, 500+ demands and 12 supplies) per week. Therefore, we assumed that the HHC structure manager have accepted to offer at most five PTPs for each patient with consideration of both the efficiency of the TAP model and the high satisfaction of patient demand. The analysis of the PP and the COC are based on the results of experiments conducted with the Ins_C in the M/L group. Firstly, the HHC structures concern the effect of the PP and the COC on performance indicators of higher level, which are, the computational time, the objective function value, and the total number of assigned demands. It will be discussed with Fig. 6 to Fig. 8 in following Section 4.2.2.1. The effects of the COC and the PP on the other performance indicators, including the satisfaction rate Sk, q, r of demand, the reassignment rate Rk, q, r of demand, the total assignment rate Ik, q, r of demand, and the utilization rate Uj of therapists are discussed in Section 4.2.2.2. Without losing generalizability, the discussion of the effects of the COC and the PP will focus on the results obtained with the Ins_C in the M group. The results of the Ins_C in the L group are presented in the Appendix. 4.2.2.1. Effects on computational time, assigned demands, and objective function value. The computational time of the Ins_C under the different cases of weight allocation criteria are compared, namely, the cases WA_III, WA_IV, and WA_V against the cases WA_ I and WA_II. The COC is considered in the former cases but not in latter cases. For the Ins_C in the M group, the value at the points III and IV is slightly larger than that at the points I and II, say, (0.114 − 0.099)/0.114 × 100% = 13.4% . However, the value at the point V is shorter than that at the points I and II, e.g. (0.099 − 0.087)/0.087 × 100% = 13.8% (see Fig. 6). For the Ins_C in the L group, the value at the points III, IV, and V is slightly larger than that at the points I and II (see Fig. A.1). In summary, the com-

putational time of the Ins_C in the M/L group that does not consider the COC is a bit shorter than that considers the COC. Therefore, considering the COC in the TAP problem can affect the computational time slightly. Fig. 6 shows no significant effect or trend of the PP on the computational time. For the Ins_C in the M/L group, the total number of assigned demands at the points III, IV, and V is smaller than that at the points I and II (see Fig. 7 and Fig. A.2). We can image that when patients require to be served by their preferred therapists, there is a probability of unavailability of their preferred therapists at their PTPs. The COC affects the total number of assigned demands slightly when there exists an over demand. The PP has no effect on the total number of assigned demands of the Ins_C in the M group without considering the COC, because it keeps the same at the points I and II (see Fig. 7). When considering the COC, the total number of assigned demands at the point IV is larger than that at the points III and V for the Ins_C in the M group, but it is not the case of the Ins_C in the L group. Therefore, the PP does not have observable influence on the number of total assigned demands. When there are enough therapists providing services, patients with the different PP levels can always be visited. When it is short of therapists, demands with the higher PP levels have occupied all of the available and capable therapists (see Fig. 9 and Fig. 11). The objective function value is smallest at the point III when the model only considers the COC (the case WA_III), as indicated at the bottom in Fig. 8. To explore the effect of the PP on the objective function value, comparisons are separately conducted among the results of the Ins_C under the cases WA_I and WA_II (without the COC), and among the results of instances under the cases WA_IV and WA_V (with the COC). The objective function value at the point IV is higher than that at the point V (see Fig. 8), where

M. Lin et al. / Expert Systems With Applications 62 (2016) 44–62

Fig. 8. The objective function value of Ins_C under different weight settings.

the point IV refers to the case WA_IV that the PP is more important than the COC and the point V refers to the case WA_V that the COC is more important than the PP. The objective function value at the point II is higher than that at the point I, where the former refers to the case that the PP is considered in the TAP model and the latter presents to the case that neither the PP nor the COC is considered. Therefore, the PP has a certain effect on the objective function value. 4.2.2.2. Effects on Sk, q, r , Rk, q, r , Ik, q, r and Uj . The satisfaction rate Sk, q, r , the reassignment rate Rk, q, r , the total assignment rate Ik, q, r of demand are presented by the values at the (k, q, r) points in Fig. 9 to Fig. 11. The utilization rate Uj of the j-th therapist is presented by the value at the j point in Fig. 12. The satisfaction rate Sk, q, r and the reassignment rate Rk, q, r of demand are shown as the curves III, IV, and V (considering the COC) and the curves I and II (without the COC) in Fig. 9 and Fig. 10. The two curves I and II in Fig. 9 and Fig. 10 overlap completely. Fig. 9 shows that the satisfaction rate S2, 3, 3 of the curves III, IV and V is smaller than the S2, 3, 3 of the curves I and II. It also reveals that the satisfaction rate S1, 3, 3 of the curve V is smaller than the S1, 3, 3 of the curves I and II. These facts indicates that the patients who have no preference still have the risk to be unsatisfied because of the undersupply of therapists, especially when their PP levels are lower than the other patients. The satisfaction rate S1, 2, 2 of the curves III, IV and V is smaller than the S1,2,2 (= 1 ) of the curves I and II (see Fig. 9) and the reassignment rate R1, 2, 2 of all curves equals to zero (see Fig. 10), which means the model cannot find backup assignments to have these unsatisfied demands in the (1, 2, 2) category assigned to capable physical therapists at their PTPs when considering the COC. The satisfaction rate S2, 3, 2 of the curves III and IV is smaller than the S2, 3, 2 of the curves I and II (see Fig. 9) but the corresponding reassignment rate R2, 3, 2 is non zero and equals 0.11 (see Fig. 10). It means that when considering the COC requirement of patients, there is a risk that some patients cannot always be served by their preferred therapists but reassigned to the other capable therapists in order to maximize the total assignment rate of demand and achieve the full utilization of therapists. Without considering the COC, the reassignment rate is always zero; while, when considering the COC, it is not zero for the demand under some categories, such as demands in the (1, 3, 2) category and the (2, 3, 1) category under the cases WA_III, WA_IV, and WA_V, and demands in the (2, 3, 2) category under the cases WA_III and WA_IV (see Fig. 10). The reassignment of these unsatisfied demands to capable therapists leads to the higher assignment rate I1, 3, 2 , I2, 3, 1 , and I2, 3, 2 under the cases WA_III, WA_IV and WA_V than under the cases WA_I and WA_II, as indicated in Fig. 11. Only I1, 2, 2 , I1, 3, 3 and I2, 3, 3 under the cases WA_III, WA_IV and WA_V are all smaller than under the cases WA_I and WA_II. The reason of smaller I1, 2, 2 under the cases WA_III, WA_IV and WA_V is the unavailability of capable therapists to visit patients at their PTPs with satisfying the operational constraints. The two

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main reasons of the smaller I1, 3, 3 and I2, 3, 3 are: (1) there exists an undersupply of the capable therapists; (2) the available capable therapists have been assigned to the patients who have the higher PP level and have preference for the capable therapists. On the premise that all operational constraints of the TAP model considering the COC are satisfied, the demand with the higher COC levels still has a risk to be unsatisfied and reassigned to achieve a higher total assignment rate, or be delayed because of unavailability of capable therapists at the PTPs. The demand with the lower COC levels has a higher probability to be delayed due to the undersupply of therapists. Without considering the COC in the TAP model, the PP has no effect on the satisfaction rate Sk, q, r , the reassignment rate Rk, q, r , and the total assignment rate Ik, q, r of demand, as the curves I and II overlap completely (see Fig. 9 to Fig. 11). However, it is pointed out that there exists a difference in rates of demand under the cases WA_III, WA_IV, and WA_V (when considering the COC), such as Sk, 3, 2 , Sk, 3, 3 , Rk, 3, 2 , Ik, 3, 2 and Ik, 3, 3 of the curves III, IV, and V in Fig. 9 to Fig. 11. The weight π k, 3, 2 is higher than the weight π k, 3, 3 under the case WA_V because the COC is more important than the PP. It leads to the higher satisfaction rate Sk, 3, 2 (the higher assignment rate Ik, 3, 2 ) than the Sk, 3, 3 (the Ik, 3, 3 ) of the curve V in Fig. 9 (Fig. 11). The satisfaction rate S2, 3, 2 (the assignment rate I2, 3, 2 ) of the curve IV is lower than the S2, 3, 2 (I2, 3, 2 ) of the curve III in Fig. 9 (Fig. 11), and the reassignment rate R2, 3, 2 of the curve IV is higher than that of the curve III. Hence, the demand with the lower PP levels has a higher probability to be reassigned and delayed when the PP is more important than the COC (the WA_IV). In summary, the PP has a limited effect on the satisfaction rate of demand Sk, q, r , the reassignment rate of demand Rk, q, r , and the total assignment rate of demand Ik, q, r of instances when considering the COC. The utilization rate of therapists under different cases of weight allocation criteria is shown in Fig. 12. There are 4 physical therapists and 2 occupational therapists in the Ins_C in the M group. Without considering the COC in the TAP model, 3 physical and 1 occupational therapists achieve full utilization of workload, 1 physical therapist achieves 97.9% of utilization of workload, and 1 occupational therapist achieves 97.9% of utilization of workload. When the TAP model considers the COC, only a physical therapist achieves full utilization of workload, the others can only achieve 97.9% to 99.3% of utilization of workload. The difference in utilization rate of therapists with or without the COC means that the COC has a certain effect on the utilization rate of therapists. The curves I and II indicates that the PP has no effect on the utilization rate of therapists without considering the COC. The curves III and IV are almost overlapping which indicates that the PP hardly has any effect on utilization rate of therapists when considering the COC. Therefore, the PP has no effect on the utilization rate of therapists no matter considering the COC or not. In summary, when considering the COC in the TAP model, the total number of assigned demands reduces slightly, e.g. four cases of instances in the M group and 2 cases of instances in the L group reduced; the computational time of TAP model becomes very slightly longer; the demand with the higher COC levels still bears the risk of being unsatisfied and reassigned to achieve higher assignment rate, or be delayed due to the operational constraints violation; the average utilization rate of therapists decreases by 2%. The PP has no or limited effects on these performance indicators except the objective function value. The summary can help the HHC structure managers to make decisions on adopting the concept of the PP without hesitation when there is an undersupply of therapists; however, the decision on adopting the concept of the COC should be made carefully by measuring the importance of goals that the HHC structures aim to achieve.

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M. Lin et al. / Expert Systems With Applications 62 (2016) 44–62

Fig. 9. The satisfaction rate of demand under different cases of weight allocation criteria.

Fig. 10. The reassignment rate of the demand under different cases of weight allocation criteria.

Fig. 11. The assignment rate of demand under different cases of weight allocation criteria.

Fig. 12. The utilization rate of therapist under different cases of weight allocation criteria.

5. Conclusion This study proposes a mixed-integer program model with the linear objective function and the quadratic constraints for the therapist assignment problem (TAP). The model provides five cases of weight allocation criteria for the HHC structures to choose from, given their current operational process. For instance, the HHC structures that do not provide any COC levels or PP levels to patients can adopt the model under the case WA_I to plan their assignment of care workers for the upcoming week. The HHC structures providing only the PP or the COC can use the TAP model under the case WA_II or the case WA_III. For those HHC structures providing both the PP and the COC can select the TAP model under the case WA_IV or the case WA_V. It can also be extended and adopted by other HHC structures by formulating their specific operational constraints from the perspectives of demand and supply, or adding demand categories according to patients’ requirements other than the PP and the COC.

The TAP model under the five cases of weight allocation criteria has been validated by the randomly generated instances of different sizes under different PTPs categories. The effect analysis claims that the number of the PTPs, the PP and the COC affect the performance indicators of optimal solutions, including the computational time, the total assigned demands, the objective function value, the satisfaction rate, the reassignment rate, and the assignment rate of demand and utilization rate of therapists. The number of the PTPs for each patient to select has a crucial effect on the computational time of the model and the total assigned demands. The results indicate that it should not be greater than five because the computational time and the performance indicators are all acceptable in the approximate optimal degree in this setting. The COC has an observable effect on the satisfaction rate, the reassignment rate, and the assignment rate (the sum of the satisfaction rate and the reassignment rate) of the demand, while it has a slight effect on the total number of assigned demands and the computational time. However, the COC has very limited effect on the objective function

M. Lin et al. / Expert Systems With Applications 62 (2016) 44–62

value. The PP scarcely has effect on these performance indicators except the objective function value. The analysis of the results can help the HHC structure managers to make better decisions on the adoption of the TAP model with proper consideration of the PTPs, the PP and the COC. To the best of our knowledge, this study is one of the few attempts to explore the effects of the PTPs, the COC, and the PP on patient/operator assignment problems in the HHC context. It is a novel optimization model under different weight allocation criteria for the TAP, which is the core of an effective planning system for the HHC structures. However, this study has a few limitations. First of all, the formulation of the COC and the PP into the objective function cannot guarantee the related propositions always be satisfied, because they do not be formulated as hard constraints in the model. Another limitation is that the instance for validating the TAP model is under certain percentage exacted from real-life. Instances under different situations, for example, under the situation where there are sufficient therapists and the percentage of each demand category is random, are not generated. Finally, our problem is mainly defined with respect to the actual setting of a local HHC provider. Some specific operational constraints are not considered in the TAP model, such as allowance for overtime, burnout levels of therapists, and more visits per patient. The future work will be conducted from several directions. Firstly, the validation of the TAP model should be conducted with different instances under different circumstances, not just these instances in the current work. Secondly, the TAP model should be extended by considering some common operational constraints mentioned in the assignment problem literature in HHC context, such as allowance for overtime, different work contracts of therapists, and more frequency of visits required by patient during the coming week. Finally, it is necessary to explore an effective algorithm to solve the problem with very large-scale instances within acceptable computational time that Gurobi cannot offer.

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allocation criteria obtained by 10 times running the TAP model implemented in GUROBI and its Python interface. Table A.1 presents the average computational time; the mean objective function value of the model with each instance is achieved and presented in Table A.2; the average number of assigned demands is calculated and presented in Table A.3. Fig. A.1–A.7.

Fig. A.1. The computational time of the Ins_C in the L group.

Fig. A.2. The assigned demands of the Ins_C in the L group.

Acknowledgement This work is partly supported by the Hong Kong RGC Grant No. T32-102/14 N and the NSFC Key Project Grant no. 71231007. Appendix. Results under different settings Table A.1 to Table A.3 present the results of each instance in each size group and each PTPs category under each case of weight

Fig. A.3. The objective function value of the Ins_C in the L group.

Table A.1 Computational time (s) of instances. Instances

Size

Weights Setting

PTPs B = 48 Ins_A

≤ 48 Ins_B

≤5 Ins_C

≤4 Ins_D

≤3 Ins_E

≤2 Ins_F

=1 Ins_G

S

I II III IV V

0.365 0.431 0.364 0.355 0.415

0.21 0.183 0.166 0.175 0.172

0.02 0.013 0.011 0.012 0.012

0.018 0.019 0.009 0.008 0.009

0.008 0.007 0.006 0.006 0.006

0.005 0.004 0.004 0.004 0.004

0.003 0.002 0.003 0.003 0.003

M

I II III IV V

2.591 2.596 2.892 2.599 2.89

0.796 0.809 0.799 0.795 0.91

0.099 0.101 0.114 0.109 0.087

0.057 0.056 0.07 0.078 0.061

0.111 0.143 0.03 0.03 0.028

0.08 0.056 0.031 0.031 0.029

0.025 0.044 0.011 0.01 0.01

L

I II III IV V

21.274 21.172 20.694 19.857 19.402

8.167 7.596 6.273 6.431 6.293

0.317 0.302 0.359 0.32 0.338

1.132 1.362 0.43 0.43 0.412

0.722 0.773 0.25 0.263 0.227

0.212 0.219 0.176 0.197 0.123

0.382 0.357 0.102 0.092 0.086

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M. Lin et al. / Expert Systems With Applications 62 (2016) 44–62 Table A.2 The objective function value of instances. Instances

Size

Weight setting

PTPs = 48 Ins_A

≤ 48 Ins_B

≤5 Ins_C

≤4 Ins_D

≤3 Ins_E

≤2 Ins_F

=1 Ins_G

S

I II III IV V

0.956 0.962 0.922 0.969 0.966

0.956 0.963 0.929 0.969 0.962

0.946 0.953 0.923 0.961 0.96

0.923 0.928 0.884 0.924 0.925

0.903 0.912 0.863 0.921 0.906

0.855 0.869 0.817 0.886 0.858

0.765 0.786 0.716 0.8 0.788

M

I II III IV V

0.957 0.963 0.927 0.967 0.949

0.957 0.963 0.928 0.961 0.957

0.953 0.96 0.907 0.959 0.95

0.951 0.957 0.934 0.954 0.946

0.929 0.938 0.884 0.946 0.928

0.911 0.922 0.854 0.92 0.898

0.842 0.859 0.775 0.841 0.805

L

I II III IV V

0.952 0.958 0.917 0.965 0.955

0.952 0.959 0.936 0.963 0.953

0.952 0.958 0.903 0.956 0.946

0.947 0.954 0.927 0.954 0.952

0.95 0.958 0.898 0.948 0.937

0.923 0.932 0.855 0.898 0.883

0.867 0.885 0.808 0.872 0.839

Table A.3 The total number of assigned demands of instances. Instances

Size

Weight setting

PTPs = 48 Ins_A

≤ 48 Ins_B

≤5 Ins_C

≤4 Ins_D

≤3 Ins_E

≤2 Ins_F

=1 Ins_G

S

I II III IV V

144 144 144 144 144

144 144 144 144 144

142 142 142 142 142

140 140 138.4 138.9 138

133 133 131.2 130.4 130

123 123 121 121.3 121

109 109 109 109 109

M

I II III IV V I II III IV V

288 288 288 288 288 576 576 576 576 576

288 288 288 288 288 576 576 576 576 576

286 286 285.1 285.2 282.1 576 576 574.2 574 574

286 286 282.7 283 282.3 574 574 574 574 574

277 277 276.1 275.9 275.4 575 575 572.8 571.7 570.8

271 271 266.1 266 264.8 561 560.8 550.8 550.7 546.4

236 236 235.7 235.1 236 511 511 510.7 510.3 508.1

L

Fig. A.4. The satisfaction rate of demand under different cases of weight allocation criteria.

Fig. A.5. The reassignment rate of demand under different cases of weight allocation criteria.

M. Lin et al. / Expert Systems With Applications 62 (2016) 44–62

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Fig. A.6. The assignment rate of demand under different cases of weight allocation criteria.

Fig. A.7. The utilization rate of therapists under different cases of weight allocation criteria.

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