## 3 Problem Definition  
The job-shop scheduling problem with variable machine efficiency and maintenance activities (SSP-VME-MT) is a complex variant of the classic job-shop scheduling problem (JSSP). In the classical JSSP, a set of \(n\) jobs \(J=\{j_{1},j_{2},...,j_{n}\}\) must be processed on a set of \(k\) machines \(M=\{m_{1},m_{2},...,m_{k}\}\). Each job \(j_{i}\) has a processing time \(p_{i,j}\) on each machine \(m_{j}\), and the machines are arranged in a fixed sequence such that each job must visit each machine in the same order. The objective of the FSSP is to find a schedule that assigns each job to a machine such that each job is processed in the order prescribed by the machine sequence and the makespan, i.e., the time taken to complete all jobs is minimized. Formally, the problem can be stated as follows:  
Minimize the makespan, subject to:  
* Each job must be processed on exactly one machine. * Each machine can process only one job at a time. * Each job must visit each machine in the same order: \(m_{1},\ m_{2},\ \ldots,\ m_{k}\). * There is no considerable travel time between consecutive machines. The mixed-integer programming (MIP) model for the JSSP-VME-MT problem is presented as follows:  
Let \(J=\{j_{1},j_{2},\ldots,j_{n}\}\) represent the set of jobs and \(M=\{m_{1},m_{2},\ldots,m_{k}\}\) the set of machines. Let \(P_{ij}\) denote the processing time of job \(j\) on machine \(i\), and \(E_{i}(t)\) represent the efficiency of machine \(i\) at time \(t\), in Equation (1), modeled as:  
\[E_{i}(t)=max(E_{min},E_{max}-\alpha t) \tag{1}\]  
where \(E_{max}\) is the initial efficiency, \(E_{min}\) is the minimum efficiency, and \(\alpha\) is the decay rate. The objective is to minimize the makespan \(C_{max}\), in Equation (2), defined as:  
\[C_{max}=\max_{j\in J}C_{j} \tag{2}\]  
where \(C_{j}\) is the completion time of job \(j\). The constraints are:  
* Each job \(J\) must be processed in a specific order on machines \(m\), as shown in Equation (3): \[S_{ij}+P_{ij}\ /\ E_{i}(S_{ij})\ \leq S_{i+1,j}\,\ \forall\ i,j\] (3) where \(S_{ij}\) is the start time of job \(j\)  on machine \(i\). * Maintenance activities can restore efficiency but introduce idle times \(M_{i}\) machine \(i\), as presented in Equation (4): \[E_{i}(S_{ij}+M_{i})=E_{max},\ \forall i,j\] (4)  
The JSSP-VME-MT introduces additional layers of complexity by considering the variable efficiency of machines and the necessity of maintenance activities. Each machine has a variable efficiency rate that diminishes over time due to factors such as wear and tear. As a result, an operation postponed to be processed later may take longer than if it were processed earlier, given the declining efficiency of the machine. In this case, the study proposes a linear decay on the efficiency starting. In the proposed model, the machine efficiency decays linearly over time. This decay starts from the machine's maximum efficiency and decreases to a minimum efficiency level of 20% of the initial efficiency. Assuming no maintenance activities are undertaken, we calculated the decay slope to ensure that the machine reaches this minimum efficiency level by the time 2/3 of its totaloperating time has passed. The total operating time for a machine is defined as the sum of the processing times for all jobs assigned to that machine. Figure 1 shows the decaying efficiency of five different machines (assuming no maintenance activities are involved). The examples illustrated in Figure 1 are based on hypothetical, simulated scenarios designed to demonstrate the concept of decaying machine efficiency. The variation in machine efficiency is modeled using a linear decay function, starting at 100% efficiency and decreasing to a minimum of 20% over two-thirds of the total machine operating time. This model assumes a consistent and predictable decay pattern reflective of typical machine wear and tear. The user can perform voluntary maintenance activities to handle the decline in efficiency.