# A Novel Hybrid Simulated Annealing for No-Wait Open-Shop Surgical Case Scheduling Problems
###### Abstract  
In this paper, the problem of finding an assignment of \"\(n\)\" surgeries to be presented in one of \"\(m\)\" identical operating rooms (ORs) or machines as the surgical case scheduling problem (SCSP) is proposed. Since ORs are among NP-hard optimization problems, mathematical and metaheuristic methods to address OR optimization problems are used. The job or surgical operation ordering in any OR is a permanent part of all sequencing and scheduling problems. The transportation times between ORs are defined based on the type of surgical operations and do not depend on distance, so there is no surgical operation waiting time for transferring. These problems are called no-wait open-shop scheduling problems (NWOSP) with transportation times. The transportation system for the problems is considered a multi-transportation system with no limitation on the number of transportation devices. Accordingly, this study modeled a novel combined no-wait open-shop surgical case scheduling problem (NWOSP-SCSP) with multi-transportation times for the first time to minimize the maximum percentile of makespan for OR as a single objective model. A mixed-integer linear program (MILP) with small-sized instances is solved. In addition to the small-sized model, a novel metaheuristic based on a hybrid simulated annealing (SA) algorithm to solve large-sized problems in an acceptable computational time is suggested, considering the comparison of the SA algorithm and a new recommended heuristic algorithm. Then, the proposed hybrid SA and SA algorithms are compared based on their performance measurement. After reaching the results with a numerical analysis in Nova Scotia health authority hospitals and health centers, the hybrid SA algorithm has generated significantly higher performance than the SA algorithm. hybrid meta-heuristic simulated annealing algorithms; operating rooms; no-wait open-shop surgical case scheduling problem; makespan; transportation time; mixed integer linear programming; heuristic algorithm +
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002330  
to the OR's time slot to optimize and level resource utilization. The tactical stage (cyclic timetable) output as training is applied for decision-making in operational problems. MSS is recognized as a cyclic schedule, and it is usually monthly or quarterly. MSS allocates OR time to specialties corresponding to their specific conditions. The last decision stage considered in this study, the operational stage, is recognized as the SCSP. In SCSP, scheduling resources and patients are usually planned, and surgical cases are scheduled for specific days and times. It is the shortest and involves resource allocation, surgical cases, and advanced scheduling decisions. Production scheduling is a key to structural productivity, which organizes a calendar for making components/products. The scheduling problems are categorized into single machine scheduling problems (SMSP), flow shop scheduling problems (FSSP), job shop scheduling problems (SSP), open shop scheduling problems (OSSP), and hybrid scheduling problems (HSP). In this paper, the OSSP problem is considered. The OSSP is also called moderated-JSSP between the FSSP and JSSP. The FSSP contains \"\(n\)\" jobs, each with \"\(m\)\" operations. The process sequences of these jobs are the same for this problem. The OSSP consists of \(n\) jobs, each with at most \(m\) operations. The operations of each job can be managed in any sequence. If a job consists of three operations of 1, 2, and 3, then this job can be controlled by employing any one of the following six sequences, which is n: Sequence 1: 1-2-3; Sequence 2: 1-3-2; Sequence 3: 2-1-3; Sequence 4: 2-3-1; Sequence 5: 3-1-2; and Sequence 6: 3-2-1. This study considers the OSSP with transportation times to minimize the makespan. The number of machines or ORs is described by the parameter \(m\), and the subscripts related to the ORs are \(i\) and \(l\), in such a way \(i\in(1,2,\ldots,m)\).