A hybrid SCA–VNS meta-heuristic based on Iterated Hungarian algorithm for physicians and medical staff scheduling problem in outpatient department of large hospitals with multiple branches
Introduction
Personnel scheduling problem is one type of most concerned management problems in hospitals, especially in large ones [1], [2]. As we learned from the field survey in many big Chinese hospitals, some of them still use the manual way to schedule their staffs. Consequently, physician scheduling problem in fact cost hospital managers a lot of time considering the complexity of it. This paper is motivated by the practical problem, and particularly, given the background of typical Chinese hospitals.
Hospital personnel scheduling problem has been widely studied in the literature [3], [4], [5]. In this class of problem, nurse scheduling problem is the most studied one [6], [7], [8]. Nurse scheduling problem with different kind of nurse is discussed in [9]. Besides, another typical personnel scheduling problem, the physician scheduling problem (PSP) also has gained much attention from researchers [10], [11], [12]. In PSP, researches focus on personnel factor, while task constraints are more considered in nurse scheduling problem.
PSP is more complex than the nurse scheduling problem for considering a lot of constraints [13], such as, demand, preference, qualification and so on, and it has been proved to be NP-Hard [14], [15]. In [16], master physician assignment problem is studied as a tactical planning problem. The medical resident rotation and shift scheduling problem is solved in [17]. The PSP with flexible shifts considering physician preferences and fairness aspects is investigated in [18]. However, most papers just consider PSP for one hospital without branches, but a number of large hospitals in China have multiple branches as a group. Especially in outpatient department, the physicians are belonged to the hospital group and they need to work in hospital and all the branches. Hence, the PSP should be considered in the whole hospital group with all the branches. It means that the physicians should work in hospital and branches. The situation is described in Fig. 1. There is one larger hospital owning a group of physicians and m branches. In each branch, there are many medical staffs working in it and the physician belong to large hospital but also need to work in these branches.
This paper studies the scheduling problem of physicians and medical staff in the outpatient department of large hospital with multi-branches, where the hospital and the branches are considered equally. The duty of physicians is to diagnose the outpatients in the consulting room, and one medical staff is assigned to the physician to assist his/her work. Physicians going to different branches will incur corresponding cost, and they have a lot of preferences about the working time, working partners, and working places. It means that the schedules not only should satisfy the demands of each branch but also meet the preferences of physicians for the working conditions. The objective of this paper is to minimize the sum of the dissatisfaction degree of physicians regards the working time, medical staff, and place, the cost of physicians, the deviation of each physician allocated to each branch and the penalty of continuous shifts assignment and continuous branches assignment. In many PSP papers, the objectives include the personnel aspects [1], [10], such as the preference of physicians [12], the workload [19], and the fairness [20], and there are also researches which consider some or all of these aspects simultaneously [16], [18], [21].
PSP must satisfy a lot of limitations and constraints [22]. Some constraints may conflict with others or not easy to be satisfied. In some cases, constraints in PSP are divided into soft and hard constraints [13], [15], [23]. The soft violations are reflected in the objective function by adding the penalty function. Our work tries to minimize the number of continuous shifts assignment and continuous branches assignment by considering them as a penalty function in the objective. The hard constraints must be satisfied, including workload, resources limit, hospital requirements, etc. In this work, we consider the resources limit, medical staff working regulation, workload, and some ergonomic constraints as the compulsory constraints.
Many heuristic algorithms are proposed for solving the complexity optimization problems [24], [25], [26]. These heuristic algorithms are presented for which can solve intricate and large-scale problems in a valid time. There are many popular heuristic algorithms for optimization, such as Simulated Algorithm (SA) [27], Variable Neighborhood Search (VNS) [28], [29], [30], [31], [32], Particle Swarm Optimization (PSO) [33], [34], etc. In recent years, heuristic algorithms are widely used in the health care area, because the problems in health care have many constraints and intricate situations. For instances, Rahimian et al. [30] propose a novel hybrid integer programming and VNS to deal with a nurse rostering problem. They use the integer programming to improve solutions, in order to overcome the drawback of metaheuristic which cannot guarantee the optimality of solutions. Ceyhan and Özpeynirci [32] design two branch and price algorithms, which are basic branch and price and improve branch and price, to solve the pharmacy duty scheduling problem. The improve branch and price algorithm is embed several heuristics designed by the authors, in order to solve large-scale problems.
For solving the PSP problem, Niroumandrad and Lahrichi [11] present a stochastic tabu search algorithm to solve the physician scheduling problem in radiotherapy centers. The authors also provide a mathematical formulation for the problem, and apply their algorithm respectively in determine and stochastic cases. Rousseau et al. [35] propose a hybrid constraint programming and local search algorithm based on the concept of genetic algorithm. Moreover, genetic algorithm is also utilized to tackle the resident physician scheduling problem in [36]. Lo and Lin [34] apply the particle swarm optimization algorithm to solve the PSP in emergency department, in which an intelligent physician scheduling system is provided to assist the management of emergency department. What is more, Van Huele and Vanhoucke [5] propose six decomposition-based heuristics for the complexity problem which integrated PSP and surgery scheduling problem. In [5], three constructive heuristic and eighteen rules corresponding to the physician scheduling and surgery assignment are used to generate the solutions.
In this paper, we investigate the assignment problem of physician and medical staff in large hospitals, which is a NP-hard problem. We present the hybrid SCA–VNS meta-heuristic based on Iterated Hungarian algorithm to solve this assignment problem, which is a novel hybrid meta-heuristic motivated by the characteristics of VNS and SCA, and the features of the problem. VNS [37] proposed in 1997 is an efficient and flexible metaheuristic, which is to find better solution through changing the neighborhood structures and applying the operations of shaking and local search. Sine Cosine Algorithm (SCA) [38] presented in 2016 is a novel optimization algorithm based on the sine and cosine functions, which is verified that it can solve large-scale problem effectively. What is more, we find that the assignment problem of each clinic in a shift is an 0–1 assignment problem, and the Hungarian algorithm is a classical exact algorithm for solving 0–1 assignment problem. The main innovations and contributions of this paper have been summarized as following three aspects.
(1) A novel physicians and medical staff scheduling problem in outpatient department of large hospital with multiple branches is proposed in this paper. To the best of our knowledge, this is the first attempt to investigate such practical problem. This problem involves the collaboration of resources in each branch, such as the number of consultation rooms, and the number of medical staff in each branch, trying to optimize the preference and cost of physicians, and minimize the deviation of each physician allocated to different branch.
(2) Iterated Hungarian Algorithm (IHA) is used to deal with the personnel allocation in this problem for getting more exact solutions. We decompose the original problem to several single assignment problems, and use IHA to solve them, respectively.
(3) Due to the complexity and scale of the problem, we propose a hybrid SCA–VNS algorithm to solve it. We improve the SCA based on some specific features of our problem, design four neighborhood structures and utilize some local search operations to update solutions.
The remainder is organized as follows. The problem definition and modeling are presented in Section 2. In Section 3, the hybrid SCA–VNS algorithm and procedures are illustrated in detail. The computational experiment and comparison results are discussed in Section 4. Finally, we conclude the paper and give the prospect directions for future study in Section 5.
Section snippets
Problem definition and modeling
In this section, we give the notations and problem statement as below.
The proposed resolution strategy and hybrid algorithm
For solving the above collaborating scheduling problem, we propose a hybrid algorithm which corporates SCA and VNS
In this section, a hybrid SCA–VNS algorithm is put forward to solve this problem by dividing it into three stages: (i) generating the allocation table which specifies the number of different types of physician in each branch and shift, (ii) assigning the physicians to each shift and each branch according to the allocation table, (iii) assigning the medical staff to each assigned
Comparison experiments and discussion
In this section, the proposed algorithm SCA–VNS is compared with the original SCA [38], VNS [28], PSO [33], GA (Genetic Algorithm) [40], and SA [27]. We generate 22 instances with different number of physicians and branches based on the real hospital problem. We analyze the results and compare them using the Relative Percent Deviation (RPD) which is one of the most widely used measure method. The RPD is defined as the following equation:
Where Avg denotes the average
Conclusions
In this paper, a novel scheduling problem of physicians and medical staff in the outpatient department of large hospital with multi-branch has been studied. Taking into account the collaborative features of multiple branches, we use the IHA to assign the physicians and medical staff to get more exact results and reduce the complexity of calculation. To find better solutions optimally and faster, we propose a hybrid SCA–VNSalgorithm which combines Sine Cosine Algorithm and Variable Neighborhood
Declaration of Competing Interest
No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.asoc.2019.105813.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Nos. 71501058, 71601065, 71690235, and71690230), the Key research and development Projects in Anhui, China (1804b06020377), the Basic scientific research Projects in central colleges and Universities, China (JZ2018HGTB0232), and Innovative Research Groups of the National Natural Science Foundation of China (71521001).
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