## 4 Solution Approach
### New Heuristic Principles
So, the constraint for the cycle time for each station \(\text{CTS}(\text{s})\), where \(\text{s}=1\ldots\text{S}\), can be defined as:  
\[\text{PT}_{\text{min}}\leq\text{CTS}(\text{s})=\frac{\text{MTM}(\text{s})}{ \text{CI}(\text{s})\times\text{A}(\text{s})}\leq\text{PT}_{\text{max}},\, \text{s}=1\ldots\text{S} \tag{20}\]  
Then, the feasible range of \(\text{CI}(\text{s})\) for each station can be calculated in Equation (21), where \(\text{CI}(\text{s})\) must be a positive integer:  
\[\left\lceil\frac{\text{MTM}(\text{s})}{\text{PT}_{\text{max}}\times\text{A}( \text{s})}\right\rceil\leq\text{CI}(\text{s})\leq\left\lfloor\frac{\text{MTM}( \text{s})}{\text{PT}_{\text{min}}\times\text{A}(\text{s})}\right\rfloor,\,\, \text{s}=1\ldots\text{S} \tag{21}\]  
In a real situation, there is also the possible constraint of a minimum (\(\text{CI}_{\text{min}}\)) and maximum (\(\text{CI}_{\text{max}}\)) number of machine tools per station. Therefore, the final range of \(\text{CI}(\text{s})\) for each station can be converted into Equation (22):  
\[\text{max}\bigg{(}\frac{\text{MTM}(\text{s})}{\text{PT}_{\text{max}}\times \text{A}(\text{s})},\text{CI}_{\text{min}}\bigg{)}\leq\text{CI}(\text{s})\leq \text{min}\bigg{(}\left\lfloor\frac{\text{MTM}(\text{s})}{\text{PT}_{\text{min} }\times\text{A}(\text{s})}\right\rfloor,\text{CI}_{\text{max}}\bigg{)},\,\, \text{s}=1\ldots\text{S} \tag{22}\]  
Since Equation (5) may provide multiple plans on the number of allocated machine tools at each station, the production line has more possible combinations for the configuration information CI. In order to further investigate these possible combinations for the current individual, all the possible CIs are generated. First of all, the total investment cost of each combination is calculated through Equation (23). Then, combinations that exceed the investment budget (Equation (24)) are all removed to fulfil the constraint of the maximum investment cost on the production line. \[\text{C}_{\text{total}}=\text{C}_{\text{machine}}+\text{C}_{\text{buffer}}= \sum_{\text{s}=1}^{\text{S}}\text{MC}(\text{SI}(\text{s}))\times\text{CI}( \text{s})+\sum_{\text{j}=1}^{\text{S}-1}\text{BC}\times\text{BA}(\text{j}) \tag{23}\]  
\[\text{C}_{\text{total}}\leq\text{C}_{\text{max}} \tag{24}\]  
Afterwards, in order to obtain the best of the remaining combinations, it is essential to find a proper function to evaluate them. As the considered problem is the combination of TLBP and BAP, we tried to introduce some common evaluating functions from TLBP to achieve the estimation of the solutions. Therefore, the two most typical objectives of line balancing were introduced: bottleneck time and deviation of allocation time. 1. The bottleneck time of a production line is the maximum cycle time among all the stations. It represents the cycle time of the whole production line. However, in this situation, the solutions may have different levels of investment cost, which may make them incomparable simply through the bottleneck time;
2. On the contrary, the deviation of allocation time has a more suitable mechanism in this situation. The deviation of allocation time stands for the deviation between the station cycle time and the mean cycle time at all stations of the production line. It evaluates the intrinsic quality of balancing for a single solution under the current investment cost of the manufacturing resources. Therefore, the sum of the squares of the time deviation (SQ) was built here to evaluate all the combinations. As \(\text{CTS}(\text{s})\), \(\text{s}=1\ldots\text{S}\) of each combination can be calculated through Equation (20), the mean \(\text{CTS}(\text{s})\) can be easily obtained as:  
\[\text{CT}_{\text{mean}}=\frac{\sum_{\text{s}=1}^{\text{S}}\text{CTS}(\text{ s})}{\text{S}} \tag{25}\]  
In addition, due to the requirement of the demand interval, the solutions whose \(\text{CT}_{\text{mean}}\) belong to the interval of \([\text{PT}_{\text{min}};\text{ PT}_{\text{max}}]\) apparently have more advantages than the others.