# A Novel Hybrid Simulated Annealing for No-Wait Open-Shop Surgical Case Scheduling Problems
## 2 Literature Review
Two solution methods were provided; a linear time algorithm for a problem case is expected. The transportation times between machines were less than the minimum tasks' processing times. Heuristic algorithms for a problem case in transportation times are process-dependent. They considered transportation times independent of the jobs but dependent on the route. It was believed that the transportation times depended on the distance between the shops and not on the size or weight of the jobs [52; 53; 54]. For the first time, transportation problems are divided numerically [55]:  
* Problems in which the number of transportation is more than or equal to the number of jobs. * Problems in which only one transportation can manage the jobs. They examined complexity and the results of scheduling problems in the flow shop and open shop (limited machine numbers) with transportation times; in their research, the most complex issue of the open shop was finally solved with two machines. In the multi-transportation system, there is no limitation on the number of transportation; in other words, there is no expected time or delay for transferring any halfway job from one machine to the next, and transportation is always available. In scheduling problems, based on the transportation times, there are two modes:  
* If transportation times depend only on distance (_JITT_). * If transportation times depend on the job sequences in addition to distance (_JDTT_). In this case, the job transmitted earlier affects the transportation time of the present job [56]. Since _JITT_ is a specific case state of the _JDTT_ problem, the complexity hierarchy rule demonstrates that by solving _JDTT_, we have also responded to the _JITT_ problem, so verifying the _JDTT_ case state is enough [57]. The transportation system in our research is based on [PERSON] et al. [58]. The transportation times are dependent on jobs. According to the problem's NP-hardness and the development of the above problem in this study, the present problem is also NP-hard. Optimization approaches [59; 60; 61; 62], including heuristic and meta-heuristic algorithms [63; 64; 65; 66; 67] and recently recommended well-known hybrid methods [68; 69; 70; 71; 72; 73], are used to solve the model. Many resources in OR are used in mathematical modeling. There are two kinds of patient arrival: elective and non-elective. Effective patients can be scheduled, while non-elective patients require surgery instantly. Considering inpatients admitted for an overnight stay, different resources, such as a post-anesthesia care unit (PHU), an OR, and intensive care units, could be applied. Measuring performance could expand to services other than an OR, such as ward, PHU, post-anesthesia care unit (PACU), and intensive care unit (ICU) [74; 75; 76; 77; 78]. Figure 2 shows a three-step surgery procedure with the following steps: (1) pre-operative step, in which the required resources are nurses and PHU; (2) intra-operative, which is the step in which surgery will be accomplished. Surgeons, anesthetists, nurses, and ORs are required in this step. (3) Post-operative step, patients are shifted to the PACU and ICU following surgery [78]. This paper considers a single-objective optimization OR problem. Thus, we want to show solution approaches for single-objective optimization OR problems. Figure 3 categorizes optimization approaches that have been applied to OR scheduling problems. For solving OR scheduling problems, exact, evolutionary, and intelligent algorithms (such as mathematical programming, simulation, heuristic, and meta-heuristic), known as NP-hard problems, are used [79]. For example, genetic algorithm (GA), non-dominated sorting GA II, the combination of greedy and novel meta-heuristics, SA, hybrid SA, a single-objective ant colony optimization (ACO), and a hybrid Pareto set ACO under deterministic conditions were previously applied to solve the problem.