## 4 Solution Approach
### New Heuristic Principles  
New decoding principles in this paper were built in order to meet the industrial requirements of the expected demand interval. The decision variables that make up the complete solution were grouped in the following four sets: configuration information (CI), station information (SI), operation information (OI), and buffer allocation (BA). The structural encoding method was introduced where each individual can be illustrated as code P in Figure 4.  
As shown in the figure, each code P consists of five vectors (P\({}_{1}\), P\({}_{2}\), P\({}_{3}\), P\({}_{4}\), and P\({}_{5}\)) and can be divided into three parts of information: station configuration, operation allocation and sequencing, and buffer allocation. During the first step of the decoding stage, the three parts are decoded into station configuration array (SI), operation information matrix (OI), and buffer allocation array (BA) through Principle I, Principle II, and Principle III based on the priority coding and constraints, where the three principles have ensured the  
Figure 4: Synthetic representation of code P for each individual. Figure 3: Logic diagram of the heuristic parts (the dashed frame in Figure 2). feasibility of SI, OI, and BA. Firstly, SI is decoded by Principle I, where the three constraints in Equations (12)-(14) are satisfied. Then, OI is obtained by Principle II together with SI, where Equations (15) and (16) are fulfilled. Afterwards, BA is calculated by Principle III, satisfying the buffer constraint. To handle the industrial requirement of the demand interval, new heuristic principles were proposed to decode the last element based on the demand interval and configuration information array (CI), which is the information used to decide the number of machine tools on each station of the production line. Considering the fact that the station configuration array (SI) has already been obtained, the number of stations (S), the fixturing device and part orientation, and the kind of machine tools of all stations are known. The investment cost of a single machine tool at station s is \(\text{MC}(\text{SI}(\text{s}))\), where \(\text{s}=1\ldots\text{S}\), while the availability of the type of machine used at station s, \(\text{A}(\text{SI}(\text{s}))\) can be obtained as the following equation:  
\[\text{A}(\text{SI}(\text{s}))=\frac{\text{MTTF}_{\text{SI}(\text{s})}}{\text{ MTTF}_{\text{SI}(\text{s})}+\text{MTTR}_{\text{SI}(\text{s})}}\]  
In this equation, \(\text{MTTF}_{\text{SI}(\text{s})}\) and \(\text{MTTR}_{\text{SI}(\text{s})}\) are the two reliability parameters of the machine tool with type SI(s), and stand for the Mean Time To Failure and the Mean Time To Repair, respectively. Similarly, the allocation and sequencing plan of the operations at all stations is also obtained from the operation information matrix (OI). The total machining time for each station \(\text{MTM}(\text{s})\), where \(\text{s}=1\ldots\text{S}\) can be obtained based on the known time of each operation. Assume that the annual demand interval is equal to \([\text{D}_{\text{min}}\,;\,\text{D}_{\text{max}}]\) parts per year. The annual working hours of the production line is \(\text{T}_{\text{A}}\) (hours/year) and can be calculated in Equation (1). Then, the production rate interval \([\text{PR}_{\text{min}};\,\,\text{PR}_{\text{max}}]\) can be obtained, where \(\text{PR}_{\text{min}}=\frac{\text{D}_{\text{min}}}{\text{T}_{\text{A}}}\) [parts/hour], \(\text{PR}_{\text{max}}=\frac{\text{D}_{\text{max}}}{\text{T}_{\text{A}}}\) [parts/hour]. Therefore, to meet demand, the production time (PT) for each part must be within the range of \([\text{PT}_{\text{min}};\,\,\text{PT}_{\text{max}}]\), where \(\text{PT}_{\text{min}}=\frac{3600}{\text{PR}_{\text{max}}}\) [s] and \(\text{PT}_{\text{max}}=\frac{3600}{\text{PR}_{\text{min}}}\) [s].