**1. Introduction**

Roadways are considered the arteries of economic and social activities at the national and local levels. Furthermore, they are one of the fundamental foundations of development countries. Pavement is an essential component of road infrastructure. Its surface plays an important role in providing a safe and comfortable environment for users. During the pavement design life, the pavement is exposed to increasing traffic volumes with heavy loads, environmental adverse impacts, and poor use, which lead to significant deterioration of the pavement. To make the pavement function better, regular maintenance should be done to repair the degradation of the pavement. Therefore, through regular maintenance, a sufficient budget will be spent to maintain the pavement at an adequate condition. Actually, limited budgets are the biggest difficulties and challenges facing the pavement maintenance. For this reason, the pavement maintenance activities should be managed within the available budget and resources [1]. A good pavement managemen<sup>t</sup> program will preserve the condition of all road sections at an adequate high performance with minimum cost without any reverse effects on traffic operation, environment, or social activity [2]. Therefore, decision-makers need to have the methodologies for maintenance managemen<sup>t</sup> to attain a sufficient level of service at a minimum cost [3]. In other words, optimizing the available budget allocation is needed for decision-makers to assist them in achieving the specified objectives of pavement performance under available budget and authority constraints [4].

Initially, the priority (rank) was used to determine pavement treatment plans. When the priority is applied, the maintenance activities (alternatives) are allocated using a ranking system based on parameters such as traffic volume, road class, quality index, etc. These

**Citation:** Alqaili, A.; Qais, M.; Al-Mansour, A. Integer Search Algorithm: A New Discrete Multi-Objective Algorithm for Pavement Maintenance Management Optimization. *Appl. Sci.* **2021**, *11*, 7170. https://doi.org/10.3390/ app11157170

Academic Editor: Peng-Yeng Yin

Received: 23 June 2021 Accepted: 29 July 2021 Published: 3 August 2021

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parameters are determined based on road condition data for the current year. Therefore, the best maintenance plan cannot be assured by using a prioritization approach, and multi-year planning will also be problematic. In the last few decades, the optimization approach has been a priority to create the best possible solution for pavement maintenance strategy. In early maintenance optimization, the decision-making process was based on singleobjective optimization. However, the decision-making process in pavement maintenance involves many objectives (e.g., cost, performance, etc.) that conflict with each other, and the solution resulting from single-objective optimization may be unacceptable to other objectives. Therefore, a reasonable solution to a multi-objective problem is to search for a set of solutions that satisfy the requirements of several objectives [5].

The programming methods, such as stochastic and deterministic, are utilized to satisfy the constraints related to the requirements of the road network for the proper maintenance plan. Stochastic techniques (e.g., Markov chain) are useful for the insufficient data of pavement conditions [6]. While, a set of deterministic techniques were used to resolve maintenance problems, such as dynamic programming [7], goal programming [8], quadratic programming framework [9], and nonlinear mathematical program [10].

Many methods are used for multi-objective optimizations, such as the weighted sum method, which is used widely. By this approach, several objective functions are summated with suitable weight for each objective function. The criteria are weighted, and the comparison of sorting orders is made for outcome reliability [11]. The priority process is divided into two stages: defining maintenance activities, and then analysis of trade-offs is executed to introduce many prioritized activities [12]. The multi-criteria are generated and weighted to specify the assets of prioritization [13].

By using the theory of multi-attribute utility, an axiomatizing mathematical process is suggested to quantify and analyze the alternatives, which include many competing results [14–16]. This method is a good trial to link the objectives of maintenance treatments on asphalt pavements and define the choices of decision-making. On the other hand, the analytic hierarchy process to estimate the weights of criteria set and alternatives are used [17–20]. In this regard, the other options are compared with related standards set by using pair-wise comparisons, then the criteria weight is determined. Thus, hierarchically, the alternatives and criteria are formed.

The genetic algorithm (GA) is an evolutionary computation technique, which is widely applied to solve different objectives of maintenance problems. The ability of a genetic algorithm in the optimization process is to deal with either simple issues (e.g., linear optimization) or the complex computational problems (e.g., multimodality) and support decision-making procedure with a reasonable solution in the maintenance plan. For this reason, GA has been used widely in maintenance optimization. GA technique provides a solution for allocation the available funds to achieve the objectives constrained of the resource of central and regional agencies [21]. Also, the GA is utilized to address the nature of combinatorial maintenance programing in the network-level of roads [22]. Besides, GA is applied to attain optimum maintenance plans at sufficient level for pavement sections [22]. For constrained optimization, the GA is used to optimize the maintenance of rural roads network by minimizing cost and maximizing the road performance [23–25]. An adaptive hybrid GA, which contains GA and local search, is applied for improving the effectiveness and efficiency of solution searching for the optimal maintenance plan [26].

Also, discrete particle swarm optimization (PSO) is applied to optimize a multiobjective problem for pavement maintenance optimization. In this method, random solutions are created, and optimal searching is made by updating populations [27,28]. Additionally, a parameter-free velocity term is introduced to the barebones algorithm, which can provide a feasible decision-making process [29].

Dominance-based rough set approach is mainly used in the selection of the best solution when a large number of the possible solutions has existed. In other words, this method enhances the decision-making process for optimizing various maintenance activities to accomplish predefined objectives and make optimal allocation of the available

budget. In this method, the rough set theory is used, making it possible to analyze the contradiction. The optimization is achieved by this method through two stages. In the first one, a set of solutions from the Pareto optimal is created. In the second one, through the decision-making process, the best solutions in the created set are indicated [30,31]. The interactive multi-objective optimization-dominance rough set approach is also used to support decision-making interaction to determine the optimal set of maintenance activities [32].

However, most of the metaheuristic optimization algorithms are based on the real random numbers, which are not suitable for the discrete problems. This paper presented a new discrete algorithm called integer search algorithm (ISA) to optimize the PMMS problem. To continue in searching for methods to find the optimal solution supporting the decision making of pavement maintenance, this paper presents a new discrete stochastic algorithm called integer search algorithm (ISA) for the pavement managemen<sup>t</sup> system. Also, the ISA and GA are applied to improve the pavement managemen<sup>t</sup> for the road networks in a developing country with a limited annual budget. Furthermore, the convergence curves of ISA and GA during the optimization process are compared.

#### **2. Pavement Maintenance Model**

#### *2.1. Pavement Condition Evaluation*

The state of the pavement plays a significant role in the decision-making process for pavement repair. The pavement repair activities are dependent on the current condition of the pavement. As a result, determining the best maintenance strategy necessitates assessing the state of the pavement. In order to describe the current pavement condition, the proposed study used the pavement condition rating (PCR) with the grading system from 0 (Failed) to 4 (Excellent) as presented in Table 1.


#### **Table 1.** Pavement condition rating for IRI

#### *2.2. Pavement Preservation Treatments*

Pavement treatments are selected based on the purpose of the maintenance strategies of the networks. For retarding the pavement deterioration and reducing the distresses, this study interested on preventive maintenance treatments. Five maintenance activities were used in this study to implement the pavement maintenance called do nothing, crack seal, slurry seal, thin asphalt overlay, and thick asphalt overlay. These activities are assigned codes which are M-00, M-01, M-02, M-03, and M-04, respectively.
