142 Int. J. Services Operations and Informatics, Vol. 3, No. 2, 2008

Copyright © 2008 Inderscience Enterprises Ltd.

Computer simulation and swarm intelligence organisation into an emergency department: a balancing approach across ant colony optimisation

Fabio Fruggiero* and Alfredo Lambiase Department of Mechanical Engineering, University of Salerno, Via Ponte Don Melillo 1, 84084 Fisciano, SA, Italy E-mail: ffruggiero@unisa.it E-mail: lambiase@unisa.it *Corresponding author

Daithí Fallon Department of Manufacturing, Biomedical & Facilities Engineering, Cork Institute of Technology (CIT), Rossa Avenue, Bishopstown, Cork, Ireland E-mail: Daithi.Fallon@cit.ie

Abstract: Healthcare system must be sensitive to the needs of patient, financially viable and cost-effective. Emergency Department (ED) crowding and rising healthcare costs are perceived as significant issues that are getting worse. In order to respond to the growing number of incoming patients, hospital departments, including emergency rooms, have to re-evaluate their current facilities, procedures and practises from an operations management perspective. In a typical ED, it is important to minimise not only the patient’s waiting time but also the staff idle time while maintaining the high utilisation rate of medical facilities. Computer simulation is recognised as a powerful tool, for medical management, to enquire productivity trying to increase service level to patients.

Based on the analogy of a Job Shop Scheduling Problem (JSSP) and known patient scheduling methodologies, a metaheuristic Swarm Intelligence (SI) approach, focused on Ant System (AS) behaviour, was used in the balancing of an ED. The Ant Colony Optimisation (ACO) algorithm was implemented with the proposal to optimise patient scheduling under defined precedence, zoning and capacity constraints while balancing the workload between and within resource types. The ED of Cork University Hospital (CUH), Ireland, is the case in issue.

Keywords: healthcare; Emergency Department; ED; computer simulation; line balancing; Swarm Intelligence; SI; Ant System; AS; Ant Colony Optimisation; ACO.

Reference to this paper should be made as follows: Fruggiero, F., Lambiase, A. and Fallon, D. (2008) ‘Computer simulation and swarm intelligence organisation into an emergency department: a balancing approach across ant colony optimisation’, Int. J. Services Operations and Informatics, Vol. 3, No. 2, pp.142–161.

Computer simulation and SI organisation in ED 143

Biographical notes: Fabio Fruggiero is a Mechanical Engineer graduated from University of Salerno in 2004. Presently he is a post-PhD student at the Department of Mechanical Engineering, University of Salerno, Italy. His research interests are in intelligent agent system, optimisation techniques, industrial automation, factory planning and management, digital factory and virtual reality system.

Alfredo Lambiase is currently Full Professor of Design and Management of Industrial Plants at the Department of Mechanical Engineering, University of Salerno, Italy. He is author of a great deal of research studies published at national and international journals as well as conference proceeding. His research areas of interest include process simulation, factory planning and control in complex manufacturing environment, time cost and quality optimisation, digital factory and virtual reality.

Daithí Fallon is an Industrial Engineer. He graduated from the University College Galway with a BE in 1985 and subsequently an MEngSc in Computer Integrated Manufacturing in 1987. He worked as a Quality Engineer in Digital Equipment for five years before taking a lecturer position in Cork Institute of Technology in 1992, lecturing in industrial and quality engineering. His research interests include simulation of manufacturing systems and manufacturing process development. He is the Director of CAMMS (a large Continuing Professional Development unit within CIT) and is now Head of Department of Manufacturing Engineering at CIT.

1 Introduction

In today’s hyper-competitive environment, the natural goal of an organisation, whether it is producing goods or services, is to respond to demand in a timely fashion with efficacious operating costs. An organisation can be considered like the systematic arrangement of two or more entities who fulfil role and share common purpose. In relation to the nature of the process, every organisation has a distinct purpose or mission. Each organisation is looking for a constant search in new and better ways to manage the practice and measure the performance of implemented solutions. They are continually trying to improve the way in which they perform their tasks. The nature of the process and product, the awareness of uncertainty, the feedback of entities into the systems are the important factors to consider. Performance has to be defined in order to evaluate changes. A working knowledge of the process, as well as insight into process innovations and/or changes, can be achieved by the analysis of the system using a discrete event simulation model. Computer tools can perform a dynamic analysis of the process behaviour, manifesting problems (such as bottlenecks and/or resource availability and utilisation) and predicting the results of changes. Through a computer-realised model, it is possible to examine the process in a structured manner. Management considerations can be investigated using ‘what if’ questions.

Managers, researchers and analysts are beginning to uncover the potential for using simulation in the field of healthcare. Simulation can be used to test system’s behaviour and to study different configurations anticipating the potential performance of a system (Austin and Boxerman, 1995). Some reasons attributed to the move towards patient care include the demand for shorter hospital stay in prospect of costs or payments policy. In

144 F. Fruggiero, A. Lambiase and D. Fallon

the organisation of a hospital, a significant impact, in terms of patient population, is played by Emergency Department (ED). About the totality of the hospital’s inpatient population comes through ED. ED can be modelled (in abstract terms) as a set of tasks (i.e. treatments) in which the patients change state due to interactions with resources. It is the human factor element that makes determining a solution very tricky.

Throughout this paper, there will be references, as in industrial perspective, to patients as jobs, medical treatments as tasks and medical staff as resources or work-centres. The application of industrial engineering concepts and methods to healthcare has had direct and predictable positive effects. To provide high-quality healthcare for patients, reducing the time patients spend in ED is obviously a complex balancing problem (between patient’s waiting time and number/availability of resources). Predictive solutions are recognised as being useful in anticipating the potential performance of a system. A tool that integrates process simulation modelling and optimal scheduling would have obvious benefits in the health sector (Karvonen et al., 2004). Simulation can be used initially to analyse patient arrival rates and to study different system’s behaviours (Fetter et al., 1965). An optimising procedure can be adopted to maximise/minimise a defined goal (such as the number and utilisation of medical staff).

From an engineering point of view, ED treatments can be compared to a manufacturing Job Shop (JS) where the assignment of tasks becomes a complex scheduling problem. The process is considered as a group of related tasks (i.e. medical treatments) which are performed by hospital staff. The optimality consists of finding a feasible assignment of tasks to resources in such a way that the demand is met and the ‘output’ is optimised.

The goal is to assign a list of pending tasks to replicable, but not interchangeable, resources while minimising the throughput time and maximising the efficiency. In detail, it is described like an assembly line balancing problem (Baybars, 1986). The balancing policy would be able to produce a feasible sequence of operations (with properly defined constraints). The problem can be assimilated to an NP-hard instance (Sanieev, 1998) and a solution can be found in optimal criteria based on operation research fields.

Research from fields such as ethology has also contributed to scheduling and to more complex balancing problems (Bautista and Pareira, 2002). Through the use of animal-based evolutionary models (involving collaborative rather than hierarchical decision making) defined as ‘Swarm Intelligence’ (SI), it is possible to define a general method to solve this constrained form of complex problem. The present work implements an optimisation procedure called Ant Colony Optimisation (ACO): a metaheuristic approach applied to the scheduling and balancing problem of a real ED.

The paper starts from the implementation of a discrete event simulation model of ED of the Cork University Hospital (CUH), Ireland. After a brief introduction to healthcare management, the first section of this paper is focused on presenting the model on simulation perspective. Assessment and effect of time-dependent patterns of monthly, daily and hourly demand has been treated. Queuing networks was first used to assess the flows between units and establish target utilisations of bed unit. A simple procedure in layout validation/assessment was proposed. The outputs of this model, such as the utilisation of resources and patient’s waiting time, resulted in the impetus to define a more complex model querying the scheduling process and the utilisation of medical resources. The aim is to eliminate inefficiencies, maximise the utilisation of resources, reduce patients’ waiting times and minimise the maximum patient’s staying time in the department in question. These goals will be treated in the second part of the present work.

Computer simulation and SI organisation in ED 145

The problem is formulated as a line balancing problem (Vilarinho and Simaria, 2006). A short description of the procedures and flow within ED indicates the complexity of the problem. An Ant System (AS) procedure is utilised to find an optimal configuration of resources. Discrete-event simulation is then used to evaluate the flow through the balanced system. Model performances are presented and guidelines for future research are given.

2 Healthcare management perspective

In the healthcare service it is important to minimise staff’s idle time and patients’ waiting time, maintaining high utilisation rate of medical facilities and offering high quality of service to patients. Some hospitals have insufficient resources. Many hospitals have inefficient ways to use them. Because of the manifold increase in patient market and limited financial resources, the growing of healthcare efficiency is vital to foster healthy people. Moreover, interventions in this form of knowledge-based adaptive systems require careful consideration and planning perspectives, but of a different kind than in mechanistic systems. The approach must not ignore significant features of healthcare systems such as the ‘cultural’ role of hospitals. Hospitals have made the commitment to provide service and facilities to face up any form of disease.

Healthcare is one of the largest industries all over the world. Public dissatisfaction with such a system has been growing steadily and progressively from widely acknowledged problems beyond the current question about economic sustainability of such a system (Mintzberg, 1994; Mintzberg et al., 1998). Thus, healthcare management is a serious and impeding problem to face with. Expenditures are reaching costs and levels that put mangers and economists to quest the importance of using powerful methodologies to manage the healthcare services. According to national statistics (Organisation for Economics Co-operation and Development (OECD), 2007), the healthcare industry is a vital element of the economy, accounting for 7.5% of the Ireland Gross Domestic Product (GDP) (9% of Italian one). The US states are more likely to spend approximately 15.3% of GDP. This value becomes more remarkable when we learn that in the USA 14% of population has no access to care (i.e. they are without health insurance). With the aging of the population (life expectancy in Ireland and Italy stood approximately at 80 years) and the rising of costs, the situation of health costs has seen the public share increase nearly to 78%. Ireland and Italy have almost the same level in terms of health spending per capita. They spend about $2800 per capita in healthcare services. The situation is different into the USA which healthcare public expenditure covers just the 45% of the total $6400 per capita costs (see Figure 1). This amount is dwarfed in comparison with medical services offered to patients. The nation’s hospital inpatient beds capacity has declined over the past decade: the average, in Italy and Ireland, of 3.1 acute care hospital beds per 1000 population means there is a high possibility for patients to request resources that are still busy. In European countries, healthcare can be seen as a government service. It often has waiting list and other opportunity costs. The US metaphor for healthcare is as a commodity that can be bought and sold.

146 F. Fruggiero, A. Lambiase and D. Fallon

Figure 1 Total and public expenditure in healthcare service in relation to the national Gross Domestic Product in Milliards of local US$, year 2005

Source: OECD

Hospitals are considered extraordinarily complicated system. They are mainly bureaucratic organisations which use the principle of span of control very effectively. The management of the health system can be faced in different perspective. Some people manage primarily down, directly into the clinical operations (i.e. focusing on the treatment of patients and medical procedures). Others manage up, towards those who control the system and resource’s utilisation (i.e. state agencies, insurance companies etc.). The proposed approach, out of the operations (only a medical staff can face them), looks at the healthcare system from an up level. It is going to look forward to initiating and overseeing new programmes that will improve hospital’s performance, to predicting corrective action in response to unforeseen problems, to optimising the distribution human, physical and monetary resources. The main consideration is one of the key factors in hospital cost containment and revenue enhancement is effective and efficient resource utilisation, bed planning and capacity analysis. Enquiry in tasks, procedures and scheduling policies has to be investigated in order to improve the system capacity. An efficient perspective in management and control of goods, service, equipment and resource from acquisition to disposition has to be employed.

3 Computer simulation into healthcare frame

The absolute necessity to offer a constant high level of medical services to patients, in relation to surgical skills and medical facilities, has pushed healthcare managers to investigate into the process and utilisation of resources. The intent is to keep as little idle time for staff as possible, reducing costs, and as high utilisation rate of facilities as possible. This target, obviously, joins industrial environment and medical organisations. Healthcare is a ‘calling’. Most of healthcare systems worldwide are now using modelling to plan for clusters (Rotondi et al., 1997; Vissers et al., 2001). The use of engineering tools, such as discrete event simulation model, to investigate the process is recognised as a powerful management approach (Schneider et al., 1995). This work is settled in a particular environment but the main idea can be generalised as well. The development of a simulation model of ED of CUH, Ireland, has given the possibility to evaluate new ideas in the layout assessment, to reduce organisational friction and to fix the risks

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associated with the implementation of changes. A simulation model is able to depict the flow of patients, the resources’ utilisation, evaluating the ‘way’ and ‘time’ the patients are treated. The use of an ED simulation model is to test and validate the consideration of a balanced approach performing the new assessment into layout and resource organisation.

The consciousness that Ireland is ranked 25th out of 26th European states in term of healthcare availability to public and value for money offered to consumers in healthcare sectors (Health Costumer Powerhouse – 2006 Euro Health Consumer Index) pushed to implement a decision support system. It is going to provide how it is possible to review efficiently the scheduling of treatments and the number of resources (in terms of hospital staff). In an ED, because of its complex feature (i.e. queuing and various levels, pre-emptive priority among patients) traditional operating research techniques are of limited usefulness (Goitein, 1990; Mahachek, 1992). Through the implementation of a discrete event simulation model, we are able to predict the future step of the environment. Thus, we can obtain a dynamic view of the process and its performances. By means of an optimisation procedure, constructing the problem like an assembly line balancing model (Becker and Scholl, 2006), we are actually predicting the process behaviour according to the data collected.

3.1 The ED of CUH: a discrete event simulation model

An ED must provide medical service to anyone needing emergency care. ED was, generally defined as an urgicentre, an accident room or Accident and Emergency (A&E) department or accident ward under the management of a medical unit. It is open for emergencies, urgent care, and other routine services, and it is not primarily interested in patient’s ability to pay. It represents a walk-in medical clinic in most communities. Its duties are complementary to the General Practitioners’ ones. Student general practitioners (Senior House Officer – SHO), man specialist, surgeons, registrar, paediatrician, technicians are involved in this area. Typically, there are registered nurses to support the interns. When an illness strikes, ED became the natural place to seek care. Many patients have learned that their medical needs may be satisfied more quickly (not always true) in ED. Quality of care, access to care and the cost of care (apart from the fact that they spend public or private money) are the three main items to evaluate healthcare performance. Moreover, overcrowded ED and intolerable waiting list for treatment is the usual prospect of an emergency service. The objective of such a system is to provide facilities for physicians and their staff to treat patients by calling. Into ED facilities, a calling can have different priorities (i.e. 5 categories) in relation to revealed disease level. A FIFO system into level is adopted. The priority level is assigned across a Manchester code procedure (MackWay et al., 2005). ED patients commonly follow one of the four paths. Patients can arrive to the facility by ambulance (26%) or by car (74%). Differentiation between paediatric area and adult area is made by means of the patient’s age. An area for paediatric cure is assigned to child treatments (13% of arrivals). The patient may (1) be treated and sent home; (2) be treated/stabilised in ED, held over in the ED for observation in a holding room, and then sent home/hospitalised; (3) go directly to the operating room, if they require emergency surgery, and then to an inpatient unit; (4) be admitted directly to the inpatient unit.

148 F. Fruggiero, A. Lambiase and D. Fallon

Figure 2 Statistical data analysis of incoming patients. (a) Inner patient level; (b) Pathology percentage. Value modelled into the simulation model of ED (see online version for colours)

The physical facilities of ED in CUH consist of a registration desk (RR), a waiting room (WaitR), a triage room (T), an X-ray area, 12 treatment rooms for major patients (57% of arrival) and 4 beds for minors treatments (32% of arrivals are categorised as minor), 4 resuscitation areas (1.4% of incoming patients need its facilities) and separate stations for the unit secretary, nurses, nurses aides and physicians. Data modelled has been portrayed in Figure 2. The registration process must capture accurate patient identification. In addition to collecting patient data, registration systems should provide outpatient reports and forms. A triage process has to assess the patient’s pathology. The pathology’s rate (Figure 2(b)) was used to assess the arrival inpatient condition (according to hospital reports). The data of hospital tasks and treatment’s time, across ED facilities, had been present scattered and not available on computers, so those paths and values were obtained by means of staff and of consultant doctors. A nodal representation helps describing the processes and tasks into ED (Figure 3). The configuration of possible situations is too manifold to deny any other representation. It has recognised 50 different paths in relation to the same number of danger list (limb problems represent 39% of diseases). The patient highest priority is assigned, across a triage process, to critical patients (care type = 1–2). A resuscitation room should be, always, available for Priority 1 patient’s arrivals. The patient’s condition is assessed by a triage nurse (i.e. a very experienced nurse), who determinates the priority level of the scheduling policy, based on patient’s vital signs as well as symptoms. Patients with a non-life threatening condition (care type = 3, 4 and 5) go through the registration process next. Cubicles in major and minor areas are equipped for treatments. Afterwards, the patient is sent to waiting area and waits for medical cure. Medical staff could be a nurse or a doctor (SHO and/or Registrar). When a patient is called to bed, he/she is treated by a nurse and/or by a doctor (SHO) and receives adequate treatments. If a detailed examination is necessary, a specialist from the hospital is called for consulting. The specialist can order tests for the patient (e.g. Exams). There are specifically two kinds of tests ordered within ED, lab test and/or X-ray (includes Computerised Axial Tomography (CAT)). If an imaging unit is required, provision must be made for the consultation services of radiology technicians and radiologists. X-ray technician are eventually committed. Once a patient has been treated in ED, a decision is made on whether the patient should be hospitalised (admitted to other unit (39%) or other

Computer simulation and SI organisation in ED 149

clinic (23%)) or should be discharged (26%). Death (0.2%) and other cases (exit without treatment because of ‘excessive’ waiting time) occasionally occur.

Figure 3 Nodal representation of ED process

A scheduling policy should foresee the totality of tasks needed by the incoming patients. The aims of a simulation model in ED of CUH were to portray the situation in term of resource’s saturation, patient’s throughput time, plant utilisation, etc. Distribution of patients’ arrival, in a single work shift, can be assumed across historic data estimation.

The mean arrival rate can be fixed as one patient every 11 min. Difference between day and night shift occurs. Particular trends have been evaluated: growth of arrival during the bank holidays (an increment of 15%) was noticed. Daily and monthly trends have been implemented into the proposed discrete event simulation model. The daily arrival trend can be fixed across a Poisson function (equation (1)) distribution with pick during 9 a.m. to 13 p.m.

Pr (k arrival in [0,t]) = Pr(X(t) = k) = ( ) e .!

k ttk

δδ −

(1)

Here δ is the shape parameter which indicates the average number of events in the given slot [0,t]. Thus, the time between two consecutive arrivals Ai = ti – ti – 1 follows an exponential distribution with mean 1/δ. Different pathologies implemented with different δ values.

Performance of the existing ED (Figure 4) such as long waiting time for treatment and throughput time and unbalanced utilisation rate of resources highlighted that Irish patients have little to be happy with. The charts in Figure 4 highlight the patient’s throughput time across the major and minor treatments areas (these areas treat 89% of hospital’s incoming patients). The data are related to the simulation of a week.

150 F. Fruggiero, A. Lambiase and D. Fallon

Figure 4 Throughput time in the majors and minors areas of ED in CUH across a discrete event simulation model. Weekly time bucket (see online version for colours)

The implementation of a discrete event simulation model also portrayed the mean saturation level of the medical staff in the department: there is a discrepancy between 89% of doctor’s saturation and 45% of technicians. This value forced to the implementation of a balanced approach. Moreover, the number of bed care units is able to cover only 48% of the cumulative daily demand distribution. The mean throughput time in the hospital was nearly 350 min.

3.2 Improvement in layout

The layout of ED in CUH was investigated in order to improve system’s performance. The Quadratic Assignment Problem (QAP) (Koopman and Beckmann, 1957) represents the implemented mathematical formulation to validate the location of a set of treatment rooms. Cubicles, minors and majors, paediatric treatment room, X-ray and plaster room, intensive care units were switched, in location, considering flow paths and zoning constraints. A tabu list forbade the assignment to a particular location (e.g. the location of resuscitation close to ambulance’s entrance). Mathematically the problem is the allocation of a set of facilities to a set of locations. A fitness function related to distance, flow and user’s defined penalty value leads the assignment process (equation (2)).

( ) ( ) ( ), 1 1

min .n

n n

ij i j i iS i j if d bΦ Φ ΦΦ∈

= =

+∑ ∑ (2)

Where Sn is the set of permutation of {1,2,…,n} elements; fij represents the flow from facility i to facility j; dkl is the distance between location k and location l; bik is the cost (i.e. a function of assignment relevance according to expert analysis) of placing facility i in location k.

A Simulated Annealing (SA) procedure (Metropolis et al., 1953; Kirkpatrick et al., 1983) was implemented. Starting from an initial state, a perturbation to a new state in the neighbourhood is performed. Each state is associated to a particular energy level in the objective function. The solution is accepted if ∆E < 0. If ∆E > 0, the transformation is accepted with probability p(∆E) = exp(-∆E/(kbT)) where T represents the control parameter and kb the Boltzmann’s constant. This mechanism is able to guarantee the acceptance of small changes with a little growth of the objective function (equation (2)). The output of such an approach is the definition of emergency close areas. Which are the areas of ED that can be shifted to reduce Length of Stay (LOS) to overcome bottlenecks and to increase efficiency?

Computer simulation and SI organisation in ED 151

Paediatric area and its waiting hall can be displaced in the layout if there is a necessity to have more cubicles for major treatments. This was the first output in an improving system configuration.

4 From the simulation model to the balancing approach of an ED

The simulation model permits to evaluate the current process behaviour and predict the impact of changes. The line balancing approach detects fundamental problems (unnecessary and/or duplicated activities), evaluates the sequence of operation in the department in issue (analysing scheduling process), defines the optimal scheduling in relation to user targets (optimisation of patient’s scheduling, patient’s throughput time). Student general practitioners (SHO), registrar doctor, specialist, nurses (triage nurse and nurse), technician, desk’s clerk, stretcher and patient are the entities in the process and in the model. These entities, except patients, are assimilated to resources and represented like work-centres. Patients are assimilated to jobs that have to overcome a set of tasks to make a diagnosis. Each work-centre has to process patients with their treatments (tasks). The mean throughput time, after proposed improvements, was calculated across the implementation of the simulation model. Each patient (job) is classified across a priority level assigned by the triage nurse (from higher 1 to lower 5). A fuzzy controller was used like a filter of the flow paths (Kruse and Meyer, 1987). Besides, the way used to define the tasks is relatively general.

4.1 The patient’s scheduling problem

Traditionally the balancing problem is settled into a flow-oriented production system (Scholl, 1999). It consists of finding a feasible assignment of task to work-centres, minimising such an objective. For simplicity intent, in this work are investigated only medical and paramedical resources: eight different categories are contemplated. The ANTBAL procedure of Vilarinho and Simaria (2006) was modified in order to fulfil the healthcare frame.

A metaheuristic algorithm, contemplating agent-based research, inspired by natural ants behaviour, seems to assimilate the need of finding the right settlement of the large number of variables in matter (Dorigo et al., 1999; Dorigo et al., 2000). Little attention has been paid to the question of what is the value added by the addition and/or if the addition could be avoided by making the use of current resources efficient. It is an appealing outlook that re-evaluates the ED resources, from a management perspective, through the run-time scheduling of patients. Schematically, the problem of ED scheduling consists of finding a feasible assignment of patients (i.e. jobs: 1{ }n

i iJ J == ) to resources (i.e. work-centres: 1{ }m

k kW W == ) optimising an objective function. There is a great need in the healthcare industry to balance capabilities and capacities with patient demand to ensure that the patient flow remains smooth. Each patient is characterised by a Release-time (i.e. the arrival time into the ED according to equation (1)), a priority-level (i.e. the triage care type) and a medical-route (i.e. the investigations and treatment to attend the patient’s disease). They can be defined across historical data analysis and fuzzy analysis.

152 F. Fruggiero, A. Lambiase and D. Fallon

Each patient Ji can be ‘processed’ on interchangeable resources and requires a medical-route, i.e. a chain of tasks 1 2, , , ,

ii i imO O O… scheduled under constraints. Each

task of the medical-route is performed in a tik time. Oik is the treatment of patient Ji that has to be process on medical resource Wk for a treatment time .ikt

This description represents a classical formulation of Job Shop Scheduling Problem (JSSP). The output of scheduling is the minimisation of maximum completion time (equation (3)). The quality of scheduling is defined across the introduction of an overall objective function (equation (16)). Priority levels and constraints are taken in consideration. Across a balancing approach it is possible to define which resource must be activated to treat the patient. After the definition of the resource’s route, it is possible to evaluate the patient’s waiting and throughput time.

Despite the overall output, the main objective of the scheduling policies is to minimise the duration in which all operations for all jobs are completed, which is defined as min (makespan).

*max max( ) min( ( )) min{max [ ] : , }.i ik ik i kC t C t t s J J W W= = + ∀ ∈ ∀ ∈ (3)

Here 0iks ≥ represents the starting time of kth operation of ith job and t is the iteration value. This value can be normalised, and so added to the overall objective function, across the patients’ waiting time (Wtime).

N. ro Job N. ro Task

1 1( ) ( )

.N. ro Job

ii imi m

C t t tWtime = =

⎡ ⎤−⎢ ⎥

⎣ ⎦=∑ ∑

(4)

Reduction of waiting time (equation (4)) responds to patients’ perspectives. Reduction of the number of resources responds to managerial intent. The objective is to contemplate both the aspects in a complete problem formulation.

4.2 The SI optimisation procedure: the agent model architecture

The essential idea of an AS is that ‘optimal’ solutions are the result of an incremental output of good partial paths. The consciousness that natural ants are capable to establishing shortest route paths from their colony to feeding source is the inspiring source of this model. They rely on the phenomena of SI for survival. They make decisions that seemingly require high degree of cooperation smelling and following a chemical substance (i.e. pheromone) laid on the ground and proportional to goodness load that they carry on (i.e. in a scheduling approaches the goodness of objective function: reported to makespan in this applicative case). The same behaviour of natural ants can be overcome in an artificial system with an artificial communication strategy regarded as a direct metaphoric representation of natural evolution. Artificial ants live in a computer world and are able to memorise previous movement and to raise solution through a stigmergic approach (Grassè, 1959). Computationally, ACO (Van der Zwaan and Marques, 1999; Colorni et al., 1991; Blum et al., 2001) is a population-based approach found on stochastic solution construction procedures based on an autocatalytic positive feedback loop. The procedure of building a solution takes into account the following:

Computer simulation and SI organisation in ED 153

• heuristic information, on priority level of jobs

• pheromone amount, different from ant to ant and related to solution quality (which stores up and evaporates dynamically to reflect elapsed time factor).

The probabilistic approach focused on pheromone evaluation gives at the same time autocatalytic behaviour with a reinforcement strategy. Path’s attractiveness raises with path choice, and probability increases with the number of times that same path was chosen before. At the same time, the employment of heuristic information can guide the ants towards the most promising solutions.

Figure 5 Model optimiser architecture

The model starts by creating a sub-colony with N ants. Each ant in the sub-colony builds a feasible solution (i.e. the scheduling of all treatments, and consequentially patients access to medical resources). The initial feasible schedule is constructed by taking into account heuristic information (priority level as in the triage assignment). Between levels a visibility factor discriminates patient treatment policy (equation (6)). An array of arriving time (Release-time) brands the process of assignment. Each task (i.e. treatment of particular patient) is characterised by a probability value. Definition of patient’s arrival time, triage level, treatment sequence and mean treatment time is made across statistical analysis and fuzzy controller (Zadeh, 1965). If several routes are applicable, a self-created selection procedure chooses the task. The selection procedure is based on a Roulette Wheel Selection (RWS) mechanism. Path’s attractiveness, thought like change over duty i→j of medical resource, raises with path choices, and probability increases with the number of times that same path was chosen before (equation (5)). For each solution a measure of the quality of output, computed according to the problem’s overall objective function, is used to update the followed ant’s path (equation (16)). The procedure has to be repeated for each agent. The model of Figure 5 shows the optimiser architecture of the approach. A colony of agent is constructed and iterated until the counter (iterations) value. To apply AS we are needed to consider a useful representation of path choice (pathologies flow chart) and constraints. The basic idea is that a feasible and optimal solution of JSSP can be built from a permutation of task’s order defined

154 F. Fruggiero, A. Lambiase and D. Fallon

in a network representation. The representation is based on ‘disjunctive graph model’ G = (V, C, E) of Roy and Sussman (1964).

Each job–tasks pair (i,j) is to be processed on a specified machine M(i,j) for T(i,j) time units, so each node of graph is weighted with j operation’s processing time. This procedure makes available always legal schedules which do not violate hard constraints.

5 The ACO algorithm

Starting from a source node (i.e. a waiting room) every ant chooses the next node (i.e. treatment area) in its path according to a probability function and to a selection mechanism. According to the fuzzy controller, it is defined as the situation of the department at a time t. The patient arrival time and level and chain of treatments (with its tik) are defined. For each treatment, according to constraints, it is possible to choose the work-centre. According to the treatment’s constraints that bind the assignment of the treatment to particular range of resources, the definition of work-centre is made across an RWS approach considering the workload of each resource and the number of resource of the same type in ED. The intent is to balance and reduce the number of resources.

In a graph representation, each patient in the department can be represented by a group of nodes (i.e. treatments). At time *t St t= + (where St is the starting time of the shift in issue and *t kost= ) the number of patients in the department has to be updated.

Precedence constraints (which determine the sequence the patient has to be treated in the ED); zoning constraint (which forces the assignment of certain treatments to particular resources, i.e. the registration process and/or the triage visit and/or the calling of a specialist form the hospital etc.); capacity constraints (related to the saturation level of the resources) tighten the tabu list of resources.

The workload and the number of resources in ED define the slices (i.e. paths’ probabilities amount) of the RWS mechanism. A probability, 0 1,( )ant

ijP t≤ ≤ represents the trigger that at time t the generic agent, ant, makes ‘changing duty i→j’ as next routing path (equation (5)). At time t each ant chooses the next treatment, i.e. where it is going to be at time t+1. Sant(t) is the group of schedulable/feasible tasks for each agent (ant).

Likewise, for each ant the selected task can be treated on a group of work-centres (Work_optionsi). Pheromone trail is kept into a matrix, task x task (equation (8)).

To carry on search, we must select a task transforming the probability value in a fitness slice.

( )

otherwise

[ ( )] [ ( )]if ( )

( )

0

[ ( )] [ ( )] .ij ij

ant

ant

antij

ih ih ih S t

t tj S t

P t t t

α β

α β

τ η

τ η∈

∈=

⎧ ⎫⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭

∑ (5)

Here

• ( )ij

tτ represents the intensity of trail on connection ( , )i j at time t

• ( )ij

tη represents a heuristic value pointed a visibility factor:

Computer simulation and SI organisation in ED 155

1 1( ) .ijj ij

tTriageLevel t

η = ⋅ (6)

The basic idea is to connect the visibility factor to something that leads the explorer agent to choose the next task in relation to processing time (tij) value and according to triage’s priority level (TriageLevelj). Remember that, in this characterisation, Level 1 means patient needs immediate attention.

• The parameters α and β characterise the exploration and exploitation of the algorithm. They are user-defined values and tune the relative importance of the amount of pheromone versus the heuristic coefficient.

When all colony agents have constructed a complete solution (i.e. the sequence of feasible nodes), a pheromone update rule is applied (equation (7)). It represents the pheromone release strategy.

( 1) (1 ) ( 1) ( ).ij ij ijt t tτ λ τ τ+ = − + Δ (7)

Besides ants’ activity, pheromone trail evaporation is included through a coefficient ,λ representing pheromone vanishing during elapsing time. It imitates the natural decrease of pheromone trial intensity over time (0 1).λ≤ ≤ This step implements a useful form of forgetting. With evaporation procedures we consider a simple decay coefficient that works on total laid pheromone level between time t and t + n. From a practical point of view, evaporation is needed to avoid too rapid convergence of the model to a sub-optimal solution region. The pheromone is the trail of all colony agents (equation (8)). It is therefore the sum of single agent laid pheromone quantity:

1

( ) ( ).ants

antij ij

ant

t tτ τ=

Δ = Δ∑ (8)

At the beginning of the procedure, a user’s-defined initial amount of pheromone (τ0) is released in every path.

Once ant has built a solution, it evaluates the (partial) solution and deposits pheromone trails on the connections arcs (equation (9)), if used, proportionally to its objective function value (equation (16)), like in the natural behaviour.

-th followed edge ( , )if

otherwise

_.

( )

0 antant

ijant i jObj funcQ

tτ⎧ ⎫⎨ ⎬⎩ ⎭

⋅Δ = (9)

Q is the quantity of pheromone per unit time, defined as user-defined constant of pheromone updating and equal for all ants; Obj_funcant is the quality of solution function of ant agent. At the beginning of the procedure, an initial amount of pheromone is released on every path.

5.1 The balancing approach in an ED across ACO algorithm

The decision problem of optimally partitioning tasks among work-stations with respect to some objective is known as the line balancing problem (Akagi et al., 1983; Simaria and Vilarinho, 2001; Becker and Scholl, 2006). The main idea of our approach is to fit generic formulation of ANTBAL (Vilarinho and Simaria, 2006) on the ED environment.

156 F. Fruggiero, A. Lambiase and D. Fallon

During the present investigation, the traditional formulation of the assembly line balancing problem is continuously glanced at. The formulation of the balancing problem starts with the definition of the Cycle Time (Ct) of the line whose value will be assumed as the minimum replication time of the work-centre (i.e. medical staff). Ct is used like a trigger on the replication of resources. A work-centre is replicated if it has assigned a treatment with processing time higher than Ct. Planning horizon (Pl) is a work shift (8 hours). Generically, the resource (work-centres) into ED requires visiting over the planning horizon, L different types of patient care each with numerousness lD (defined across historic fuzzy data interpretation). Usually in ED there are five levels of patient care status according to the triage process. The cycle time of the line is formulated as follows (equation (10)).

1

.L

ll

PlCtD

=

=

∑ (10)

1

.ll L

ll

Dq

D=

=

∑ (11)

Besides, it is possible to evaluate the value of patients of a particular level in the system like the rate of level numerousness by total number of patients (equation (11)). During the scheduling of the patients in ED, the idea of a balancing model is to control the entity of the hospital staff into the line. The intent is to define the scheduling of the treatment, and consequentially the sequence of patient to visit, according to a balancing status inter- and intra-work-centres. It is a desirable output – the situation in which, according with the constraints of the problems, we minimise the number of the medical staff and maximise its saturation (reducing the patient’s throughput time and waiting time). The duplication of resource should be created only when it is necessary to satisfy the demand. Specialisation in task has been implemented according to fuzzy rules. Specialisation refers to the ways in which a hospital organises to identify specific task and to assign a job description to each employee. For example, a nurse’s aid has specific task to perform that is different from those of a physician, a registered nurse or a medical technologist. The scheduling sequence helps to process the patients in order to obtain the minimisation of the objective function. In order to define the capacity of a work-centre, it had to introduce a workload factor. The workload factor takes into consideration the sequence of previous assigned tasks, according to the scheduling sequence. The workload on work-centre k related to model l is evaluated across the sum of the processing time of the previous scheduled treatment plus the current assignment. If h is the candidate for assignment to work-centre k, the workload on this resource is formulated across the following statement (equation (12)).

.k

kl il hli T

W t t∈

= +∑ (12)

Here kT is the set of tasks (i.e. visits) already assigned to resource k. The definition of the objective function of the model is a difficult matter that should

be evaluated according to patients’ and hospital managers’ intents. The basic idea is to contemplate into the objective function, the patient’s throughput time. The definition of the objective function starts with consideration on the scheduling policy.

Computer simulation and SI organisation in ED 157

It is defined as follows:

• LL, the number of different resources/work-centres (settled to definition of the present instance to 8 and activated into the line with a threshold level fixed to 1)

• NN, the simultaneous number of resources (i.e. medical staff) in ED after the scheduling of the supposed activities process

• SSkl, the Idle Time of the resource k due to patient level l

• SSk, the average idle time of the resource k

• IT, the sum of SSk rated by the numerousness of k (i.e. the average idle time of the process).

A Weighted line Efficiency (WE) needs to be defined (equation (13)).

1

1

.

I

imLi

ll

tWE q

NN Ct=

=

⎡ ⎤⎢ ⎥⎢ ⎥=

⋅⎢ ⎥⎢ ⎥⎣ ⎦

∑∑ (13)

The minimisation of work-centres (NN) in a fixed Ct is obtained maximising the WE value.

Another important goal of the model is to obtain a balance (Bb) between work-centres (equation (14))

2

1

1 1

11 .1

L

klL LLl

b ll k

SSLLB q

LL IT LL=

= =

⎡ ⎤⎢ ⎥⎢ ⎥= − −

− ⎢ ⎥⎢ ⎥⎣ ⎦

∑∑ ∑ (14)

and a balance (Bw) of workloads within work-centres (equation (15)). 2

1

1 1

11 .( 1)

L

m klL LLl

wl k k

q SSLB

LL L SS L=

= =

⎡ ⎤⋅⎢ ⎥

⎢ ⎥= − −− ⎢ ⎥

⎢ ⎥⎣ ⎦

∑∑∑ (15)

According to the balancing formulation of the process, and the output of scheduling in terms of normalised value of patient’s throughput time, it is possible to define an overall objective function as the sum of two outputs (i.e. the scheduling and balancing approaches). The output of this function (Obj_func) is looped for each ant and at each iteration count. The objective is the maximisation of equation (16). The intent is to use some user-defined values ( , , , )γ χ ε φ to weight the addendum of the overall objective function. Task of the user (if you are checking the scheduling and resources in hospital manager perspective or patient one) is to define the relative importance of addendums.

*max

_ .b wWtimeObj func WE B BC

γ χ ε φ= ⋅ + ⋅ + ⋅ + ⋅ (16)

158 F. Fruggiero, A. Lambiase and D. Fallon

For each agent of the colony, a vector of the scheduling sequence is registered and a matrix of the work-centres employment is updated. The employment of work-centres is a constrained function of work-centre’s numerousness and its workload.

6 Results

Experimental analysis in ACO performance was run in order to depict the model tuning phase. Important is to fix a consideration: the number of agents influences the convergence rate (generally you can fix No. of ANTS ≈ No. of Jobs × No. of Work-Centres) and also the computational time solution. For the examined environment, makespan obtains the minimum value for 0.8 ≤ 1–λ ≤ 0.9. Moreover, it is important to have an empirical rule that establishes and regulates the value of weight of visibility vs. weight of pheromone. Across a detailed graphic analysis, it was stated the relative importance of visibility factor, in a range continuously varying in 0 < β < 10, and the relative importance of pheromone to small values 0 < α <1 (Fruggiero et al., 2005).

According to the convergence and exploration of the environment, the total number of iterations of the ACO algorithm and the settlement of its parameters were carried out. The implementation in issue was run on 1000 interactions different sub-colony agents (ants). The objective function output is portrayed in Figures 6(a)–(b). The relative importance of parameter in objective function value (equation (16)) was settled according to exploration and exploitation meanings. The main idea is to define , , ,γ χ ε φ in order to have no predominant paths after small cycle rate. The author implemented the approach in patient’s perspective (γ = 15). The ED configuration was with 60 patients of five different levels. The type of different resources (medical staff) was, initially, settled to eight. This is the minimum number of resources in the department. An activation value of the resource, in workload entity, is pointed to 1. The treatment’s time was evaluated in relation to historical data analysis and fuzzy logic control. Besides, the objective function has been maximised in order to reduce the number of resources (Figure 6(b)). The importance of small waiting time for treatment is another factor to take into consideration. As the scheduling police, the output of this model was 29 resources of different type with eight SHO doctors. For each type of resource it is possible to evaluate its numerousness. Needless to say, the reduction on the number of resources can be offset by a little growth of the patient’s waiting time for treatment and a non-proportional rise in the makespan of the scheduling. Finally, the average LOS for hospital inpatients also has declined from the initial system configuration. To test this new hospital assessment, experiments were run with the simulation model. Figure 7 shows the new system performance according to a new layout assessment with overcoming of bottleneck (increase from 12 to 16 major cubicles) and balanced resource assessment (i.e. system’s configuration like in Figure 6). From the results of this experiment, the system was more efficient as defined by the reduction in total time in the system for scheduled patients.

Computer simulation and SI organisation in ED 159

Figure 6 (a) Alternative system’s configuration with different sub-colonies entity. (b) Objective function value with different sub-colonies (see online version for colours)

Figure 7 Mean patient throughput time in ED according to the simulation model in balancing perspective (see online version for colours)

7 Conclusion and future perspective

This study has examined the improvement of ED functions through the aspects of capacity efficiency. Capacity efficiency was approached from different perspectives: resource efficiency, resource workload and resource availability. The intent is to optimise the patient’s scheduling and the total number of resources in order to reduce hospital management costs while boosting the patient satisfaction of the healthcare service. The implementation of a discrete event simulation model was able to depict the hypothetical alternative configuration of ED.

The main patient’s waiting time was checked. Additionally, the development of the scheduling theory should allow the review and analysis of existing procedures to ensure that they met the needs of physicians, patients and hospital staff in an effective manner.

160 F. Fruggiero, A. Lambiase and D. Fallon

The scheduling of inpatient episode was developed and tested in an agent-based optimal approach. A key feature will be based on striving to implement a real-time aspect of this approach. Supposing new layout configuration, evaluating the process outputs and implementing a simulation model is a preliminary step in process analysis. The introduction of an optimisation procedure, in term of resource and its allocation, should be the following step in yielding a wide range of benefits related to the level of service, quality, safety, job satisfaction and cost savings.

Acknowledgements

The authors are thankful to the staff of Cork University Hospital for their professional support, providing facilities and material for this research. Sincere thanks are also due to Dr. Nora McCarthy and Dr. Stephen Cusak for their active participation in the project and for believing in such a research.

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