Optimization of Nurse Numbers in Emergency Department of a District Hospital in a Developing Country Iran , 2014

Background: this study aims to optimize nurse numbers in emergency department of a district hospital in Iran using linear programming. Material and methods: through observation and checklist data about average patient arrival, delivery time of care services needed for patients and number of nurses were obtained. Using linear programming optimum number of nurse needed for right delivery of services was calculated. Results: optimum number of nurses was calculated as 12 nurses, but because of some issues in 3 shift, real number of nurses needed for a 24 hours period is equal to 16 (12+4) nurses (12 as minimum needed plus nurses who stay and don’t leave the 3 shift till tomorrow morning). Conclusion: Using LP models can be useful for estimating optimum number of nurses for different wards of hospitals, so they can reduce their costs and reach more productivity.

reach their full potential.Knowing that hospitals are greatest users of health resources makes it inevitable to design scientific and applicable model to optimize use of health resources in hospitals.Managers and policy makers like to know and control costs and financial status of every health system and especially hospitals.Among all costs of every hospital, it is said that Costs of human resources comprise more than 60% of resources of every hospital.So managing and optimization of human resources can optimize a large part of health resources.
In hospitals, different departments exist.Among them some have special complexities.Like emergency department.Shift scheduling and calculating true needed personnel of departments like emergency department is one of the most important concerns of managers and administrators (4).
Emergency department is one of the key departments of a hospital.When nurses are below than needed, the chance for overload and confusion of nurses are higher and undesirable outcomes for patients who need emergency services as its name it is and also hospital and its reputation seems to be inevitable.Since 1980s, concerns about time scheduling and appropriate number of personnel of emergency department have considered by researchers because of the importance of this issue in improvement of quality of services delivering to urgent patients and satisfaction of these group of patients.Determination of the minimum required number of nurses with Patient satisfaction of the service provided and also the number of nurses needed during each shift has been the common goal of all the studies that have been done in this area.
In Iran, however, managers of health care, especially in emergency department are often from medical staff rather than Analysts so they need easy-to-use tools to keep them flexible in different situations.It is several decades that researchers use different methods such as statistical methods, work measurement, queuing models and integer programming for scheduling problem for staff.
Isken & Hancock in 1991 developed a model for resource usage minimizing.In their study, resources were the nurses that must be allocated to various parts of hospital.They demonstrated that using a simple algorithm can have results like a simplex method.Their study showed some facts about shift scheduling of nurses in emergency department, but they failed to develop a comprehensive algorithm or model for shift scheduling and calculating number of nurses in a complicated environment like emergency department.
As said before, for optimization of resources, in past two decades the use of a simulation models as a tool for planning and decision making in the field of health care has increased considerably.Many simulation projects primarily in hospitals and emergency departments have been implemented .
Evans, Gor and Unger in 1996 used simulation for different studies in shift scheduling of nurses, technicians of emergency department and physicians for decreasing average length of stay (LOS) of patients in hospitals.They used the software ARENA for evaluation of shift scheduling of different employees in different parts of hospitals.In this model, 5 different time tables developed based on the time patient spends in hospital.Pitt in 1997 stated the PRISM(18), Presented use of simulation as a tool for resource planning.Cao & Tung in 1980 used a linear programming model to determine the need for regular nurses and overtime nurses.Ketabi and Monzavi barzaki in 2006 in Shahid Chamran hospital of Tehran calculated the optimum number of nurses for emergency department.
Considering the fact that data record of admission, service delivery and discharge of patients of emergency department is not well organized in Iran, using techniques of Operations Research (OR) like simulation and linear programming and so on is limited to determine the optimum number of personnel.
Despite this limitations and difficulties, still use of these techniques can be useful.In this study we calculated the average numbers of patients enter to emergency department in all day and night hours and also time needed for delivering nursing care services.Using results of these two steps, we could determine the optimum number of nurses for emergency department by use of linear programming.

Material and Methods
This is a descriptive, cross sectional study.Population of study, are patients and nurses of emergency department in summer of 2014.We used checklists for data collection.Validity of checklist was checked using expert validity.We used linear and integer programming method to determine optimum number of needed nurses in different shifts of emergency department.We used WinQSB software to solve the problem.This software is an easy-to-use and is common software in operations research.
Linear programming is a robust mathematical technique for finding optimum number in situations like allocation and combination of resources(7).This method is one of practical methods of operations research which used frequently by researchers for decision making about allocation of limited resources.A linear programming model consists of components listing below: 1-Decision Variables: variables with unknown amount which decision maker wants to find right number for them.
2-Objective function: an objective function is a mathematical phrase that indicates relations between decision variables and target of problem we want to solve.
3-benefit/cost coefficients: are coefficients of decision variables in objective function.These coefficients indicate the value of every decision variable in objective function.
4-Subjects (limitations): constraints indicate limitations of resources for implementation of a decision.Based on problem statement, constraints can be in three types that we show it by three symbols ≥, ≤ and =.
A linear programming model is useful for determining optimum number of nurses for every shift in emergency department in this study.
To determine the average patient arrival to the emergency department, like Ketabi and Monzavi Barzaki(1), we divided a night and day (24 hours) to 8 periods.Then we calculated average number of patients in these different periods during study.Based on surveys and previous studies(1) , in order to determine delivery time of care services needed for patients (medical and nursing services), we divided patients into 3 groups and we estimated average time needed for each group (Table 2).We investigated patients came to emergency department in 3 past months (summer 2014) and we found that 40% of patients needed low cares, 20% needed moderate cares and 40% needed intensive cares.
At the moment, shifts of Chamran hospital in Ferdows are 6 hour periods (Table 3).In order to make a linear programming model, first step is to identify shifts of emergency department.In this hospital, shifts are 6 hours periods (table-3).Shifts of this hospital can have nurses of different time periods were discussed in table 1.For example first shift-from 8 to 14-can have nurses from first-from 8 to 11-and second-from 11 to 14-time period.
We use an objective function to minimize number of nurses needed in emergency department of Chamran hospital.In this study, our linear programming model is as follows: Joustra et al in 2009 in a study tried to reduce access times for an endoscopy department by an iterative combination of computer simulation and Linear Programming.They showed that allocating current capacity in a more effective way, all procedures will be improved without extra resources like equipment and staff.Results of current study are in line with Joustra's.
In 1972, Warner et al declared that considering nurse shortage and cost of nursing, using a linear programming model will be necessary.In 2004 Matthews showed that Using linear programming is necessary for minimizing the cost of nursing.As we can see using linear programing for proper allocation of human resources is worldwide accepted and implemented.
In this study, although we used LP model for optimization of nurses in first model, results were not satisfying.Maybe this happened because of inappropriate shift scheduling of understudy hospital which results in more constraints.This (correcting shift scheduling of this hospital) is a solution for our objective (minimizing necessary nurses).In second situation, nurses can start working at the middle of current shifts, so same services for patients will be provided using fewer nurses.

Conclusion
Using LP, estimation of optimum number of nurses for emergency department can be achieved and also we can estimate number of necessary nurses for each shift.
According to results of this study, Chamran Hospital of Ferdows in current situation (with current shift scheduling) will need minimum of 16 nurses.It can be said that sing LP models can be useful for estimating optimum number of nurses for different wards of hospitals, so they can reduce their costs and reach more productivity.

Limitations
Like Ketabi and Monzavi, we considered 6 hours of every shift all net hours and if one wants to reach more practical results, the real net time of every nurse in each shift should be calculated and more precise results will be achieved.
Wider time interval for studying emergency department can help more in identifying real rate of patients' entrance to emergency department.Since we did this study in summer we may be biased toward some complaint happens more in summers like accidents of monocycles etc.
a-decision variables: X1: number of nurses starts working at 8 X2: number of nurses start working at 11 (zero) X3: number of nurses starts working at 14 X4: number of nurses start working at 17 (zero) X5: number of nurses starts working at 20 X6: number of nurses start working at 23 (zero) X7: number of nurses start working at 2 (zero) X8: number of nurses start working at 5 (zero) b-Objective Function: Min: z= x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 c-Constraints: C1.Time periods (demand): these constraints are the number of nurses in every time period as follows ≥ 2 X5 + X6 + X7 ≥ 1 nurses.They demonstrated that number of necessary nurses they estimated is fewer than what estimated by empirical methods.

Table 1 .
Entry average of Patients in the 8-periods(1)

Table 2 .
Needed care services based on patient group(1)

Table 3
Although 4 shifts are defined for emergency department, but in fact nurses which their work time starts at 20 (8 P.M.) are continuing till 8 (A.M.) and they work in 2 shifts consecutively.Considering patients arrival average in 3-hours periods mentioned in table 1, and also considering care time needed for every group of patients mentioned in table 2, needed care for patients in every time period calculated.Finally we calculated average nurses needed for delivery of services to patients per hour.