Set the Nurses Working Hours Using Graph Coloring Method and Simulated Annealing Algorithm

Set the Nurses Working Hours Using Graph Coloring Method and Simulated Annealing Algorithm Elham Photoohi Bafghi Department of Computer, Bafgh Branch, Islamic Azad University, Bafgh, Iran. Abstract Adjustment of nursing programs in the specified period as it can cover the requirements of hospitals, nurses and patients is a challenging task for nursing managers. The project aims to design intelligent system as a scheduler by a system of Simulated Annealing Algorithm to provide better services to patients and overcome traditional scheduling problems. Of course, the property of graph coloring is used to do this. The project has been applied to design programs based on the data derived from interviews and shifts of nurse managers in the hospital. Data analysis has been taken by converting the process of planning of the expert to the mathematical function of Simulated Annealing Algorithm using the C # programming language. Planning and scheduling of nursing shift work was designed by the system which has a high efficiency compared to individual and leads to increased efficiency of managers, increased job satisfaction of nurses, reduced problems of working with paper reports and observing the needs of the staff in addition to considering the hospital requirements. Finally, the performance of the proposed algorithm was compared with genetic algorithms. Regarding the number of groups made up in relation to the number of nurses in the conducted tests, Simulated Annealing Algorithm was capable for planning in all cases while the genetic algorithm programming is taken in 92% of the scheduling cases. Keywords: shift of nurses, genetic algorithm, algorithm of metals annealing. INTRODUCTION Health is one of the national strategic areas of development of information technology in the country and nursing care is an essential component of health care. Optimal regulation of nursing scheduler has a prominent effect on reducing hospital costs, increasing nurses job satisfaction, quality of care and increase of hospital entrance budget. So hospital can reduce programming problems in addition to increase of the performance by having the optimum regulation of nursing shifts by having better programming for the human resources and optimal use of working labors. Planning and scheduling of the sources in traditional ways and the respect to the individual excellence, staffs requirements and equitable distribution of advantages and favorable demographic shift demand for nursing managers, difficult and time consuming so. Intelligent and automated planning and business planning can be used for the calculation of working hours of nurses to provide better services to patients and nursing managers used to deal with problems. Planning and scheduling of the resources in traditional and the respect of staff superiority and equitable distribution of favorable division of shift is somehow difficult for nursing managers. Intelligent and automated planning can be used for the calculation of working hours and business planning of nurses to provide better services to patients and oppose nursing managers to deal with problems. In hospitals, nursing planning is taken in three personal methods, performed periodic and cyclical. Individual planning taken based on staff s demand, possesses large benefits such as reduced absence, increased organizational conscientiousness, induction of professional independence and saving managing times. However, there are some inherent problems, such as priority of the individual needs to the needs of health care organizations, non-compliance with proper distribution of ranks to provide better care and unfair distribution of desirable shifts. Private health centers that often have fixed sources, apply cyclic planning method, so that for a certain period of time, they assign a fixed program to their sources. But in this method less justice is observed. Non-cyclical program has high flexibility for working as the working hours and holidays of employees varies from week to week. In this method, in addition to the importance given to personal desires, needs of the sections are also considered. Since the current process of planning for nursing shifts at most of the hospitals, is taken in non-cyclic way and due to the diversity of the program, observation of demands of the section, planning for staffs requirements and the equitable distribution of sources in the traditional programs is more complicated and time-consuming. In this project, the problem is raised in this way and the solution is proposed. Aikline pointed out that since the 1970s, the viewpoint of solving problems using computer has been created by experts. He believes that his investigation about methods of optimization of planning follows a total overall goal of "how to find good possible solutions in the shortest possible time" [1]. The results show that for problem solving sessions for nurses appointments, the proposed algorithm, as a meta-heuristic method is the most applied method [2].Studies have shown that genetic algorithms, is the most evolutionary and most widely-known algorithm, possesses immense power of problem solving [3, 4]. GA, as a method of finding the optimal solution of a problem is defined as the conventional methods in artificial intelligence. Infrastructure components of the evolutionary process in genetic algorithm are included of the population genetic program, reproduction, mutation, competition and choice. In this way, by gradual removing the improper species and at the same time by optimal amplification of higher species, nature can continue to improve each generation by different characteristics. In fact, the natural evolution can be summarized as random search and survival of the fittest [2, 5]. The main purpose of this project is to design smart systems, nursing scheduler 8195

algorithm by Simulated Annealing Algorithm (annealing of metals). A method is implemented in this project to provide a consolidated program regarding the number of nurses in a hospital. RESEARCH BACKGROUND Chang and Sherman (2002), conducted a mathematical modeling in a two-stage study for a scheduling system according to the requirements of hospital management and government regulations of nurses and nurse's shift preferences. Researcher named Hofe, considered the purpose of planning, as a constrain for solving the nurses scheduling problem and have shown that other techniques can be also used for solving the nurses scheduling problems. Chang and Sherman (2002), mathematical modeling in a two-stage study for a scheduling system according to the requirements of hospital management and government regulations nurses and nurse's shift preferences. Researcher Hofe purpose of planning, problem solving scheduling restrictions for nurses and other techniques have shown that nurses are able to solve scheduling problems. Maier and Wolfe (2005) studied the Viena hospital and applied the ant colony optimization method to allocate nurses in hospitals based on some specific limitations. Rothstein and non-rotating schedule Warner had examined the non-rotating scheduling as the use of true method was related to operations of offices and hospitals house working. GRAPH COLORING METHODS Graph Coloring Problem (GCP) is about having a graph G and aim to determine the minimum required colors for coloring the vertices of the graph, so that no two adjacent vertices have the same color. The minimum number of colors required for this purpose is called the color number of graphs shown by χ (G). This is a very difficult problem in NP-Complete series [6]. Johnson et al., showed that none of the proposed deterministic algorithms, are able to color the even relatively small graph with 70 vertices and density of 0.5 [7]. The mean of graph density refers to the ratio of the number of graph edges to the edges of a complete graph with the same number of vertices. Solving the graph coloring problem is difficult even in approximation and it is proved that no polynomial algorithm is able to color a graph with colors lower than 2 χ (G) (Unless it is proven that P=N). GENETIC ALGORITHM This algorithm codes a potential solution to a specific problem on the data structure (such as chromosome) and applies combined operators on these structures to save the vital information. Board of problems apply Genetic algorithm is very extensive and it could be said that genetic algorithm is a universal search technology and a very convenient way to get the global optimal way. Genetic algorithm is type of a powerful search algorithms counts as the most popular and widely used evolutionary algorithms. Therefore, since these types of algorithms are based on biological evolution, they imitate the concepts of heritage, genetic and mutations. Considering the fact that the coloring is one of the oldest and most famous problems in graph theory and allocates various applications, it is known as NP-Hard problems for arbitrary graphs. While it can be solved for certain categories of graphs, including complete polynomial graphs. A lot of work have been dedicated to develop efficient algorithms for graph coloring problems, as an important part of these works can be devoted to intelligent design and exploration. So the genetic algorithm can be used for graph coloring problem to achieve the optimal solution [8]. THE STEPS OF GENETIC ALGORITHM The genetic algorithm steps include coding, evaluation, selection, cutting and mutations. Coding step is the hardest step to solve the problem by using genetic algorithms. In the standard genetic algorithms, string with limited length shows each chromosome. A string may include a series of binary, correct, or characters bits. Solution of the problem must be encoded in strings. In fact, at this stage, we build the possible chromosomal sequences for an answer. Coding scheme is important because it has a significant impact on the accuracy of the genetic algorithm. You may be able to improve genetic algorithm in a decent time by applying a proper simulation for the responses. After configuration for each possible response, the basic population is formed by assembly of these structures. In evaluation step, a fitness value is assigned to each chromosome. The genetic algorithm uses the quality of a chromosome to determine its compatibility, then applies it to determine the likelihood that the chromosome can stay alive in the next generation. Genetic algorithm produces offspring of a pair of chromosomes in the population. Chromosome with high fitness value will survive and chromosomes with low fitness value are destroyed. The next step is about selection. Parents are selected randomly with probability proportional to the fitness value attributed to them. Cutting is the most important step in genetic algorithm. Some parents chromosomes are directly simulated to the number of children. In other cases, pairs of parents are cut and resulting chromosomes are inserted into the population of children. Cutting is depended on the cutting rate. Mutation selects some of the genes of each chromosome randomly and varies them. The probability of this event is governed by mutation rate. Genetic Algorithm is evaluated by implementation of procedure step by step including: 1) Start algorithm with a population of N random individuals (chromosomes), 2) calculation of adjustment for each chromosome, 3) Selection of two parent chromosome, based on higher adjustment, 4) Apply of cutting regarding the rate for one of the children, 5) Put the created offspring into a complex as a new generation, 6) Replacement of the new generation in the basic population, 7) Go to step 2 (after consolidation of new generation, the algorithm returns once again to the evaluation stage.) [9]. RESEARCH METHODOLOGY In this project as a applicable one on the basis of data derived from nursing shifts scheduler, samples from two sections of the hospital with the highest complexity of planning were selected as default. The data collection tools included of 8196

programs of nurses shifts. Then the proposed system was designed based on data obtained by selected sections with non-random planning. Data analysis was taken by evaluation, pre-processing and post-processing of data and calculations of the price functions based on the trend of scheduling of the expert ones to the numerical function and programming based on the Simulated Annealing Algorithm, via C in Visual Studio environment. In the final step, the comparison was taken by genetic algorithms for evaluation of system performance. THE SAMPLE PROPOSED METHOD Evaluation of studies conducted in this area shows that there are several ways for solving the scheduling and timing of nurses programs, but most of them solve a simple model or are depended on a particular problem in a hospital. For example, in Chen and Wornell s studied the ratings were not considered. In a study conducted by Hancock, times of start of work were considered to be flexible instead of three constant times and ratings was not considered for nurses. In addition, it was allowed to have higher or lower staff relative to the number of required employees [,11].In this project, the purpose functions were determined based on the definition of the problem, while the purpose function in different studies were considered to be depended on the type of problem definition in a specific hospital. In this project lower number of limitations were considered in the problem, it is recommended to consider more limitations in future studies to get closer to the answer in the real world. In the present study, in the second part, based on the plans it was observed that program was written for 15 people instead of 20, practically, because one of the personnel did not have the physical presence because of leaving. So, in order to have a more natural response, the program was designed of 15 people in the system and obtained a more actual percentage in reduction of costs. Since data processing and adjustment of shift tables for the staff takes much time from the nursing managers and the applied model schedules the nursing affairs more rapidly, a 92% of save was occurred in the case of time scheduling and improvement of managers performance in relation to the application of genetic algorithms. In addition, program regulation time for each person was reduced. Difference of program regulation time in different studies was taken by using various methods and strategies. Providing people with nursing services in hospitals and health centers is considered as the person has the competency and skills for providing proper services. Also planning for the nurses is among the biggest challenges in the hospitals, because scheduling requires to provide hospitals while providing the individual requirements of the nurses which leads to a continues challenge. The system developed in this project has advantages over the existing internet-based systems, because in addition to valuing employees demands, it also cares about the regulations and requirements of hospital. Another benefit of the designed system is about the uniform distribution of forces that leads to opposition against any form of injustice in shifts distribution and due to the reduction of overtime work loads, employees will be more satisfied and will improve the quality of care. In addition, for evaluation o the approach presented by the designed system in the case of lack of personnel, a more justice program could be provided by homogenous distribution of working hours among other personnel in relation to the experts. While the approach of expert people is included of adding the working hours of some people and non-uniform distribution of working hours of absent people among the existed ones. So the designed system is able to have an equitable distribution of program while lack of personnel among the others, as the fairer program leads to higher satisfaction of the personnel and approximation of working hours to designed ones and reduction of excess load of some nurses. In general, in this project the case of regulation of working hours of nursing shifts was defined for the first time in local scale by means of Simulated Annealing Algorithm, as maximum number of 20 nurses were distributed in 3 shift appointments of morning, evening and night during a week uniformly. According to the results of this study, it can be concluded that since the present health organization is still prepared in the country in a manual and paper-based manner, this problem takes so much time and cost of excess paper for maintenance of programs and taking management decisions. Since there is the ability to automatically provide such information, the use of intelligent methods to provide optimum program seems to be necessary. Results showed the present condition of planned shift of nurses in hospitals often consists of a flexible noncyclic application of programming that converts individual planning to a satisfactory method for management of the complexities of nursing program. For creating a specific timing and setting appointment for each nurse in the specific time interval, some patterns and models should be provided including number of nurses, working hours and number of days determined for scheduling. Working appointments for a complete day is divided in 3 parts based on the scheduling program for each nurse which puts him or her in one of these three sections, as every day has a special schedule. Usually, for uniform distribution of working sources during a week, the first criteria is taken by conversion of individual s scheduling trend as the firs cost function. The number of nurses per shift varies according to different working hours in the day. Determination of the number of nurses per shift is taken by the administrator or supervisor first. To determine the working hours of nurses, some series of personal or business features are considered including profession and gender. These characteristics are very important in determining the appointment orders. This structure is called as the compatibility or non-compatibility of any nurse in relation to another one. In order to provide a mathematical model to determine the relationship between nurses for setting appointments, the adjacency matrix is used. The adjacency matrix is defined as a matrix with the equal number of rows and columns. Adjacency matrix has unique feature. It is known that all the elements of the main diagonal value are equal to zero. In an adjacency matrix, if the numerical value of the first row, second column is equal to one, for sure, it is clear that the numerical value of the second row, first column is also equal to one. In this part, number of each row and each column represents a person who is naturally considered to be a number. When the value of an element is equal to one, it means that two nurses have nothing to do with each other and 8197

cannot turn on a work plan or be connected simultaneously. Finally, the adjacency matrix is used for creating graphs. Then coloring is taken regarding the graph coloring properties and use of Simulated Annealing Algorithm. There is an unique adjacency matrix (alone) for each graph and there is not any adjacent for any two matrix. According to an adjacency matrix that shows the relationship between nurses actually one graph is achieved. Then the chromatic number obtains according to the graph coloring. The more accurate graph coloring, the more efficient number of nurses who are working in a same shift. Nurses are grouped here and the fewer number of colors used for coloring is preferred. This algorithm acts as a group to be neutral. The first group starts in this way and blank nodes take any color, respectively. The fitness function is used for optimization of graph coloring. The lower chromatic number the more optimized fitness function. Heuristic algorithms are both discrete and continuous. Discrete types are such as genetic algorithms, particle and of ant algorithms. In these types of algorithms first the answers are created and from the first in algorithm, the chromatic number is obtained and then multiplied by the number of errors existed in edges. Errors are so that for example if 3 1nd 1 nodes are connected in the graph, both of them in the same color (red), that is considered as an error. In the simulation, each nurse is considered as a node in the graph. Figure 1 shows how to paint nodes in the graph by the algorithm SA, with minimum 3 different colors. As shown in Table 1 after running the program, according to the number of nurses that has been determined in the first row, fitness value was determined for cooling algorithm (SA) and Genetic Algorithm (GA). If the target number is lower than fitness function, it is would be optimal. Once five nurses are considered, the fitness value of both will be the same for each algorithm. Of course, this case stands for a total of people, but when the number 20 is considered, the fitness value of Simulated Annealing Algorithm (SA) function is less than genetic algorithm (GA), and this trend also stands for higher categorization of nurses. In the case of nurses categorization, results are shown in Table 2 after running the program. Table 2: number of formed groups in relation to the number of nurses. Nurse SA GA 5 2 3 3 4 20 6 7 9 50 13 15 0 28 200 55 57 About the laws that included in routine programs, the nurses attended in the night shift, will not be involved completely in the next day shift. Also always the working conditions are not the same necessarily for all nurses who work in a hospital. Some of the nurses can be considered as alternative for replacement. In this part, some tests are taken on the working procedure and finally took the best of the simulated case will be shown. The number of nurses in the morning shift 5 people The number of nurses in the evening shift 4 people The number of nurses in the night shift 3 people The total number of considered nurses 20 people The program was run according to the above conditions and results are listed in Table 3. Table 3: Colors considered for each node (nurse) in SA algorithm Figure 1: Graph coloring with nodes by SA algorithm THE RESULTS OF THE SIMULATION Among the objectives concerned by the project, it is to minimize the error with lower chromatic number. Fitness function will be derived by multiplied chromatic number with the number of errors. After running the program, fitness value compared to the number of number of nurses was obtained as results can be seen in Table 1. Nurse SA GA Table 1: Fitness value in relation to the nurses 5 0.004 0.004 0.007 0.007 20 0.011 0.014 0.018 0.023 50 0.032 0.042 0 0.074 0.086 N. Nurse 1 2 3 45 6 7 8 9 11 12 13 14 15 16 1718 19 20 Color 16 16 12 42 16 16 16 2 0 4 4 4 16 4 0 2 0 12 2 Graph According to Table 3, each nurse represents a node of graph, representing a color by the specified number. Number of figures shows a specific color. G1=1, 2, 7, 14, G2=3, 19, G3=5, 9, 17, 20, G4=, 16, 18, G5=11, 12, 13, GB=8, 6, 15, 4, According to the above equation Gn represents the grouping of the colors, as each group has a special color. For example, in the group G1 nurses who hold the number 1, 2,7,14 have 16 colors. GB shows the neutral node. Neutral nodes will have the effect on the fitness function. GB can be in any color. The value of fitness function in this mode is equal to 0.0117 for 8198

cooling algorithm of metals. The number of errors is zero and its chromatic number is equal to 5. Weekly planning and appointments to any nurse is taken by SA as shown in the Table 4. Table 4: Nurses shift schedule during a week by GA Saturday N M E - N - M E M E - - M M - E N - - - Sunday O N M E O E M E M M - - N N E M O - - - Monday N O M E N E M E M M - - O O E M N - - - Tuesday O N M E O E M E M M - - N N E M O - - - Wednesday N O M E N E M E M M - - O O E M N - - - Thursday O N M E O E M E M M - - N N E M O - - - Friday N O M E N E M E M M - - O O E M N - - - As shown in Table 4, programming was taken for 20 nurses during a week. The letter N refers to night shift, the letter E means afternoon shift, the letter M stands for the morning shift and the letter O is also used to be the rest time for the mode nurse. Number of nurses was assigned from 1 to 20. As shown in the table, the nurse No.1 is shifted at night for Saturday, while Sunday is considered as the rest day. This programming is suggested by SA algorithm. This case was also done by genetic algorithms resulted in Tables 5 and 6. The value of fitting function in this case for the GA was 0.01404. Number of errors was zero while the chromatic No. was 7. Also grouping of the colours was as shown below: G1=1, 6, G2=2, 11, 12, 14, 18, G3=5, 15, G4=, 17, G5=16, 19, 20, GB=13, 7, 9, 8, Table 5: Colors considered for each node (nurse) in GA Color 17 0 2 13 6 0 17 13 2 0 13 6 2 17 2 0 5 2 6 5 Graph Table 6: Nurses appointments during a week by GA X N M - - E N E - E - M M N M E - - M - - As shown in Table 6, appointments of the 20 nurses were taken just in one day which is practically out of the program application trend. That happened because of the formed grouping by genetics algorithm. CONCLUSIONS This study was conducted to adjust the working hours of nurses using graph coloring and Simulated Annealing Algorithm. The project was applicable and aimed to design the Plan designed based on the data derived from shift programs and interviews with nurse managers in the hospital. Data analysis has been taken by converting the process of planning of the experts into the mathematical and programming function by Simulated Annealing Algorithm using the C # programming language. The results showed that problems of shifts and programming are especially proper to be solved by Simulated Annealing Algorithm. Results obtained in this project indicated a high-performance of SA algorithm in solving the nursing shifts problems compared to that of genetic algorithm in this respect. REFERENCES [1] Bahadorani, Solving the problem of graph coloring with genetic algorithm. First National Conference on application of intelligent systems in Science and Technology, Azad University, Quchan Branch, 2013. [2] A. Darvish. The design of intelligent systems of regulation of nursing schedules using genetic algorithms. Journal of Nursing and Midwifery, Tehran University of Medical Sciences (Haayat),Vol. 15, No. 2, pp. 23-27, 2003. [3] M. A. Bozorgzadh. The use of evolutionary algorithms with double chromosome structure for solving graph coloring problems, the first conference of fuzzy and intelligent systems, Ferdowsi University of Mashhad, 2007. [4] M. Shahsavar. The optimum design of genetic algorithms with statistical approaches, the National Conference of data security, Qazvin University, 2014. [5] H. Yarmohammadi. The new algorithm for metals cooling by dynamic type, for global optimization, The third National Conference of Metaheuristic and its application in Science and Engineering, Azad University of Fereydunkenar, 2014. [6] C. Maier, H. Wolfe. Cyclical Scheduling and Allocation of Nursing staff. Socio Economic Planning Sciences, vol. 7, pp. 471-487, 2005. [7] S. Siferd, W. Benton. Workforce Staffing and Scheduling Hospital Nursing Specific Models. European Journal of Operational Research, Vol. 60, No. 3, pp. 233-246, 20. [8] A. Neri. Application of Imperialist Competitive Algorithm in Online PI Controller. IEEE, Second International Conference on Intelligent Systems, Modelling and Simulation, Vol. 87, No. 2, pp. 152-163, 2011. [9] R. Dorne, J. Hao. A new genetic local search algorithm for graph coloring, in Parallel Problem Solving From Nature V. Amsterdam, The Netherlands: N. Holland, 1498, pp. 745 754.2012. [] W. M. Hancock. A heuristic approach to nurse scheduling in hospital units with non stationary, urgent demand, and a fixed staff size. J Soc Health Syst, Vol. 2, No. 2, pp. 24-41, 2003. [11] B. Chen, G. Wornell. Object-oriented implementation of heurestic search methods for graph coloring, maximum clique, and satisfiabilit", Proceedings of the 2nd DIMACS implementation challenge, Series in Discrete Mathematical and Theoretical Computer Science, vol. 26, pp. 619-652, 2001. 8199