## 4 Genetic Algorithm
### Genetic Operators
#### 4.5.4 Tournament Selection
If \(o=0\), the selected gene \((j,s,o)\) can be moved only forward to an arbitral location before the location of gene \((j,s,o+1)\), whereas if \(o=O_{j}\), the gene can be moved only backward to a location after the location of the gene \((j,s,o-1)\). OGSM impacts both the machine assignment and operation sequencing as it alters the sequence of the genes, which is used to determine the operations assignment and sequencing in Stage-1 of the GA search by the greedy procedure described in Section 4.3.2. OSSM is applied on form-2 of Segment-2 (Figure 1d) to shift a location of an arbitrarily selected gene \((j,s,o,m)\) during Stage-2 of the search. It is applied in a similar manner as OGSM was applied during Stage-1. However, in Stage-2, since the genes directly encode the machine assignment, OSSM impacts only the operation sequencing but not the machine assignment of the operations. ROAM is a mutation operator responsible for altering the machine assignment in Stage-2 of the GA search. The operator arbitrarily selects a gene \((j,s,o,m)\), and changes the value of \(m\) to a different eligible machine \(m^{\prime}\) for operation \(o\) of job \(j\) (i.e., \(P_{o,jm^{\prime}}=1\)). IOAM intelligently changes a machine assignment in an attempt to lower the workload on a heavily loaded machine. This operator first identifies the machine with the largest workload because of the solution under consideration for a mutation (let that machine be donated as \(m^{*}\)). Then, it selects one of the operations assigned to \(m^{*}\) and relocates it to an eligible machine with the least load as long as the load transfer will not make the least loaded machine more loaded than \(m^{*}\) after the load transfer. ## 5 Numerical Studies

### Model Analysis

#### 5.1.1 Illustration of Objective Function Terms

This section attempts to illustrate the various objective function terms considered in this paper. For this purpose, the prototype problem presented in Section 4.1 was solved using the proposed algorithm, where makespan minimization was the only objective function, and other objective function terms were merely evaluated. The details of a typical



The values of the objective function terms related to sublot flowtime can be extracted from Table 4. The entry time to the shop floor of a sublot (\(e_{s,j}\)) is the setup begin (SB) time of the first operation if the setup is attached, or it is equal to the setup end (SE) time if the setup is detached. Here, it is important to note that, if the setup of the first operation is detached, the setup can begin and be completed before raw material is dispatched to the shop floor. The sublot departure time (\(d_{s,j}\)) is the process end (PE) time of the last operation. From the first row of column ten of Table 4, \(e_{1,1}=100\), because the job is dispatched after its setup is completed as the first operation of this sublot has a detached setup time. The departure time \(d_{1,1}=2587.5\). Hence, the flowtime \(f_{1,1}=2587.5-100=2487.5\). The flowtimes for the other sublots can be determined similarly and are summarized in Table 5. At the bottom of this table, the performance measures \(f_{max}\) and \(f_{total}\) are indicated as 2487.5 and 16,560.6, respectively. The job flowtime and sublot-finish-separation objective function terms can be evaluated from Table 5. The entry time \(\hat{e}_{1}\) and the departure time \(\hat{d}_{1}\) of the first job are the same as \(e_{1,1}\) and \(d_{1,1}\), respectively, of the first sublot since this job has only one sublot in the final solution. Hence, its flowtime \(\hat{f}_{1}=f_{1,1}=2487.5\). The second job has three sublots in the final solution. Thus, its entry time \(\hat{e}_{2}\) is the minimum of \(\{e_{1,2},\ e_{2,2},\ e_{3,2}\}=e_{3,2}=120.0\), and its departure time \(\hat{d}_{2}\) is the maximum of \(\{d_{1,2},\ d_{2,2},\ d_{3,2}\}=d_{3,2}=2599.8\). Therefore, the flowtime of job-2 is evaluated as \(\hat{f}_{2}=2599.8-120.0=2479.5\).