## 3 Proposed Approach
### Mc-Sim-GRASP: Construction Phase  
The purpose of the construction phase in GRASP and in the proposed algorithm is to construct a solution by the sequential aggregation of jobs to the solution from a set of possible jobs ranked through a greedy or fitness function. Even though the studied problem is a multicriteria objective, a pure strategy was selected for the construction phase, meaning that a unique objective function guides the entire construction [79]. Then, for this phase, an earliest due date (EDD) rule was selected to deal with the earliness/tardiness objective as a single greedy function. However, considering the other functions, a respective penalization was included for the customer and product importance functions regarding each job, each depending on customer importance or product importance, in comparison with the position of the job in the sequence. Table 3 shows the penalization for the customer importance criterion for an example of 10 jobs with five levels of customer importance. These penalization scores are based on the following criteria: A job that is not belated has a score of zero. If a job is delayed, the penalization is greater if the customer importance is high and lower if the customer importance is low. Then, the job penalization increases if the job is taking the place of a job that has greater importance. The case study for this paper defines five levels of customer importance, where Level 1 corresponds to the most important customers and Level 5 to the least important. For the instances tested in this research, the customer importance for each job was randomly assigned using the probabilities indicated in Table 4. Naturally, this scale for customer importance and the probability of the importance level were established here for testing purposes. In real scenarios, the assignment of customer importance will not be probabilistic, but rather deterministic according to the views of the decision-maker. A similar procedure was conducted to assign a product importance for each job and the corresponding penalization. The main contribution of the construction phase of the proposed approach is the alternation of three different greedy functions at each iteration of GRASP. For instance, the first iteration begins the construction phase with an EDD rule criterion; the second iteration continues using the customer importance; and the third iteration continues using the product importance. These iterations are repeated in the same order until the solution/schedule is completed. It is important to note that the EDD rule criterion is used to guide the constructions for both the earliness/tardiness mean and standard deviation objectives at the same time. Inside each step of construction, the restricted candidate list (RCL) is defined as the subset of jobs with the best value of 10% of the total range of the greedy function values obtained. A job is then randomly selected from the RCL to form part of the solution. The method continues iteratively until all jobs have been scheduled, included in the solution. After the sequence is completed, a Monte Carlo simulation is performed with as many runs as needed to obtain confidence intervals of at least 1% for the four objective functions (expected earliness/tardiness, standard deviation of earliness/tardiness, customer importance, and product importance). In the first iteration of the construction phase, this solution is included in the nondominated solutions (NDS) archive, which is initially empty. In the remaining iterations, this solution is evaluated using PAES to determine whether it is considered within the set of NDS. Figure 1 presents the flow diagram of the construction phase of the proposed approach. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Customer Importance** & 1 & 2 & 3 & 4 & 5 \\ \hline
**Probability of Occurrence** & 8\% & 12\% & 20\% & 28\% & 32\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Customer importance probability.