The recent decade has seen a rapid development of production planning along with different types of transport systems such as planes, trains and ships. In this With regard to Nasiri et al. [1] explored freight consolidation and containerization using a ship transport. Thus, Hajiaghaei-Keshteli et al. [2] first integrated production proposal planning and rail transport. Usually, the type of transport network plays a important role in an integrated enterprise network in terms of production planning problem [3-4]. In today's markets, having passed serial and custom production periods production, air transport provides a chance to reach customer growth satisfaction more effectively in both universities and industrial practitioners [5]. The importance of air transport in the scheduling of production systems has increased considerably [6]. Get closer to reality, the uncertainty of a set of keys settings in such systems makes this problem more practical [7-8]. These facts have motivated our attempts to contribute to a air transport and integrated production planning problem in a fuzzy approach. As pointed out by Chen [9] in a review article, coordinated decisions, especially in a procurement chain, have been increasingly motivated by academics and industrial practitioners in the following decades. From another point of view, in a coordinated way, managers seek to make sustainable decisions within the capacitated production scheduling context [10]. Usually, customers understand and accept small deviation of due date. This uncertainty refers to production problems such as damages to raw materials as well as machinery failures and problems such as delay delivery of aircraft, traffic problems, etc. In general, an insignificant distance from the agreed delivery date is considered to be delivery time window In the following, a set of recent and important studies is explained. In 2013, Fu et al. [11] studied the coordination of the production scheduling and delivery under two main restrictions: time windows and delivery capacity. In 2014, Low et al. [12] proposed a non-linear model to minimize the total costs, including transportation cost, vehicle arrangement cost, and penalty cost. They also assumed that goods delivery takes place in the time window. In 2015, Kang et al. [13] also modelled an integrated production and transportation problem by considering some suppositions to minimize the total production and transportation cost in each planned period. In a small-sized problem, MILP approach was used. Due to the difficulty of large-sized test problems, a Genetic Algorithm (GA) was presented. Similarly, Li et al. [14] addressed the integrated production on parallel batching machines and the delivery scheduling problem in order to maximize the revenue with a proposed heuristics. In 2016, Karaoğlan and Kesen [15] considered integration production and transportation of short-lived products and developed a branch-and-cut algorithm.

Recently, Tavakkoli-Moghaddam et al. [16] proposed an integrated air transportation and production scheduling problem. They applied GA and Particle Swarm Optimization (PSO) to tackle their introduced problem Zandieh and Molla-Alizadeh-Zavardehi [17] and Rostamian Delavar et al. [18] proposed some main mathematical models with different suppositions. They considered different types of capacities and solved their problem with two GA approaches. Afterwards, Mortazavi et al. [10] addressed the model of Rostamian Delavar et al. [18] with a new version of Imperialist Competitive Algorithm (ICA). Usually, the production and distribution-scheduling problem is considered as a problem that minimizes several costs such as production cost, transportation cost, and the earliness and tardiness penalty costs. Their problem was also considered as a non-deterministic polynomial hard problem [9]. To fill the aforementioned gaps and get closer to the real-world applications, this study formulates and solves an integrated capacitated air transportation and production-scheduling problem in a fuzzy environment. This paper also contributes to a number of recent natureinspired metaheuristics, which have been proposed recently, to solve complex and largescale test problems. The rest of this paper is summarized as follows. Section 2 explains the proposed problem exactly along with its characteristics in detail. Section 3 probes the encoding plan of algorithms’ representation and their details to tackle the problem. The outputs of experiments with different criteria are presented in Section 4. At the end, the conclusion and future works are presented in Section 5.

2- Mathematical Model

This section provides fundamental basics of the model. Herein, orders are allocated to existing capacities of air transportation and are sequenced within the site of production centers in order to minimize the total cost of the whole chain. The developed model is based on the study of Rostamian Delavar et al. [18]. The assumptions of the developed problem have been defined based on study of Rostamian Delavar et al. [18], too. Accordingly, a set of key parameters, including the capacity and quantity of order, is uncertain and formulated by fuzzy numbers. The indices, parameters, and decision variables of the problem presented are given in Tables 1 to 3, respectivelyAs noted earlier, the proposed model considers both of transportation and production centers. The first objective function aims to dedicate the orders that coordinate the ordinary flight. In addition, the second one stands for the orders that cannot satisfy the ordinary flight. Eventually, the allocated orders of each chapter are computed in the last term of objective. It should be noted that the main difference of this model with the mentioned works in the literature is the two types of delivery times (the earliest and latest delivery times are considered). The following equations are defined as follows function. Constraint (2) ensures that order i and ordinary flight f have the same destinations. Constraint (3) ensures that the allocated quantities to flight f are less than the capacity of flight f. Constraint (4) ensures that order i is fully allocated. A set of constraints (5) and (6) states the allocation constraints of a single machine. Constraint set (7) computes the completion time of jobs.

3- Solution Method

As can be seen in the literature, since this problem is NP-hard [10-18], a number of recent studies have focused mainly on developing efficient metaheuristic approaches to solve this problem. Similar to these studies, the main contribution of this study is to propose a number of nature-inspired algorithms to probe this problem more efficiently. In this regard, four metaheuristics are used in this paper: Genetic Algorithm (GA) as a famous evolutionary algorithm, Particle Swarm Optimization (PSO) as a well-known swarm intelligence, nature-inspired meta-heuristics Virus Colony Search (VCS), and Keshtel Algorithm (KA).

3-1. Encoding scheme

Similar to another metaheuristic solution planning when solving the discrete mathematical formulation, a representation plan of designing the encoding and decoding of algorithms is required [4-8]. Accordingly, the order of each comfortable ordinary flight should be assigned with the same destinations. The allocation matrix is divided into K submatrices. All of processes in algorithms are applied to each sub-matrix.

3-2. Keshtel algorithm (KA)

One of recent nature-inspired algorithms proposed in this study is Keshtel Algorithm

(KA). This metaheuristic developed by Hjiaghaei-Keshteli and Aminnayeri [19] is inspired by an amazing feeding behavior of a dabbling dock, namely Keshtel, in Anas family. To clarify the counterpart of the proposed algorithm, the user generates an initial population, called Keshtel, and divides them into three types (i.e., N1, N2, and N3). N1 includes some Keshtels that have found good food for the first time, called lucky Keshtels. In addition, N3 includes the worst solutions. The lucky Keshtels search for more food around them. When better food is found around a lucky Keshtel, a new lucky one is replaced; if not, the swirling will continue. For N2 population, they move between the two other Keshtels. In addition, for N3 population, they are regenerated randomly for each generation.

The steps of KA are detailed in Fig. 4.

Genetic Algorithm (GA) developed by Holland [21] is known as one of the well-known evolutionary algorithms. GA inspired by and Host Cells. The better ones are selected as viruses. Each virus in the diffusion process creates a new random individual. Then, each virus infects only one host cell. The algorithm is summarized by pseudo-code, as seen in Fig. genetics defines an array of variables named chromososme. Two operators change chromosomes: mutation and crossover [1]. Since this metaheuristic is well known and has been investigated by several earlier studies, the interested readers can refer to related studies in this regard [2-5].

3-5. Particle swarm optimization (PSO)

Eberhart and Kennedy [22] firstly proposed PSO. The social behavior of individuals or particle in nature, like flocks of birds or schools of fish, motivates the creators to develop the algorithm. In the PSO, any solution in a search space is a counterpart of a particle nature. Each particle selects a direction to move using a combination of its current location information, the best place where previously had, and the best experience of all the particles. This process is repeated until the termination criteria are met. Similar to GA, the interested readers can refer to [6-8] to see more illustrations and descriptions of this well-known metaheuristics.

4- Computational Results

Herein, first, the data have been generated by an approach benchmarked from the literature. Consequently, the presented metaheuristics are tuned by the Taguchi method to set the best set of algorithms’ parameters. Finally, a comparative study is adopted to assess the performance of metaheuristics in different criteria.

4-1. Generating data

To investigate the behavior of the solution approaches, a plan to generate the test data is shown in Table 4. To generate experimental problems, a dataset by considering J-F-K indices is benchmarked from [14]. The value of N is considered equal to 5×F for each problem. Then, nine problems with different sizes are generated for the experimental study. We show the total number of flights with the same destination by TFk. The corresponding flights are assigned to an ordinary flight number FNf, starting from 1 to TFk. Each flight’s departure time is then generated using uniform distribution from [24 * (FNf -1)/TFk, 24 * FNf /TFk]. It should be noted that some parameters, i.e., the capacity and quantity of order, are valued by fuzzy numbers, i.e., fuzzy triangle",