**3. Optimization Methodology**

The PMMS is a discrete problem, where the input variables are the maintenance activities, which are integer numbers between [0,4]. Also, the output variables are the PCR of pavement, which are integer numbers between [0,4]. Most of the metaheuristic optimization calculations are based on the real numbers, which are not reasonable for the PMMS issue. Thus, this paper displayed a modern discrete calculation called integer search algorithm (ISA) to optimize the PMMS problem.

#### *3.1. Integer Search Algorithm*

Most of the stochastic algorithms are based on the real numbers, which are used in all engineering applications. However, the variables of the PMMS problem are integer numbers, where these stochastic algorithms need adaptation to be suitable for discrete problems. Therefore, this paper has proposed a new discrete algorithm suitable for the discrete problem, which is called integer search algorithm (ISA). The ISA algorithm is proposed based on the random generation of integer numbers. The exploration and exploitation of the ISA algorithm are controlled using parameters a and A, as in (1) and (2). The ISA algorithm updates the integer variables *X*(*i*), as in (3). The main merits of the ISA algorithm are simplicity and speed.

$$a = round\left(b - b \times \frac{Iter}{Max\\_iter}\right) \tag{1}$$

$$A = \text{Integer } \text{Random} \in [-1, 1] \times a \tag{2}$$

$$X(i) = X\_{best}(i) + A \tag{3}$$

where *b* is a constant ≥2, *Xbest*(*i*) is the best position of variable *X*(*i*), *Iter* is the iteration number, and *Max\_iter* is the maximum number of iterations. the flowchart of the ISA algorithm is depicted in Figure 1.

**Figure 1.** Flowchart of the integer search algorithm.
