## 1 Introduction  
Transportation is a multimodal, multi-problem and multi-spectral system, as it involves different categories and activities, such as policy-making, planning, designing, infrastructure construction and development. Currently, considering the significant developments in technology, economy and society, an efficient transportation system plays a key role in passengers' satisfaction and the reduction of costs. Many people use public transportation systems to reach their destination; however, others employ personal vehicles. Passengers are more likely to use a transit service that is highly reliable. Travelers may switch to other transportation modes if a transit service does not provide the expected levels of service [1]. To prevent the increase of private transport entering city centers, effective alternative travel modes must be provided [2]. [PERSON] and [PERSON] [3] investigated methods of transit network modelling that can be implemented in transit decision-making support system tools to improve their performances, according to the latest innovations in information technology and telematics. In addition, a good public transportation system has been recognized as a potential means of reducing air pollution, decreasing energy consumption, increasing mobility and improving traffic congestion. In order to improve complicated public transportation systems, a well-integrated transit system in urban areas can play a crucial role in passengers' satisfaction and reduce operating costs. This system usually consists of integrated rail lines and a number of feeder routes connecting transfer stations. In general, previous approaches to tackle transit network problems can be divided into two major groups: analytic and network approaches. These approaches differ in their purposes and have different advantages and disadvantages. They should be considered as complementary rather than alternative [4]. Numerous studies have attempted to implement analytic models [5; 6; 7; 8; 9; 10]. On the other hand, some of the researchers adopted network approaches instead of analytical methods [2; 11; 12; 13; 14]. Many studies have been carried out to identify solutions using the aforementioned approaches. These can be categorized into four groups, namely mathematical, heuristic, metaheuristic and hybrid techniques [15]. [PERSON] and [PERSON] [6] presented a mathematical method for designing an optimal feeder bus network to access an existing rail line. Furthermore, [PERSON] and [PERSON] [10] developed a mathematical model to analyze the passenger waiting time in an intermodal station in which the intercity transit system was served by feeder buses. They presented the analytic model for quantifying the relationships of passenger waiting time to the reliability of feeder bus services and the capacity of intercity transit. A large number of research papers has been published in recent years utilizing heuristic methods due to their flexibility. [PERSON] and [PERSON] [12] developed a heuristic algorithm to integrate suburban stations and bus services, along with the optimization of coordinated schedules for feeder bus services using existing schedules for suburban trains. [PERSON] _et al._[16] proposed a stochastic network model for a multimodal transport network that considers auto, bus, underground and walking modes. [PERSON] [9] proposed a model for better coordination of the intermodal transit system. Furthermore, [PERSON] and [PERSON] [17] suggested the use of a specific feeder bus service to provide shuttle service between a recreation center and a major public transportation facility. They suggested an integrated methodology (i.e., analytical and numerical techniques) for the development and optimization of decision variables, including bus headway, vehicle size and route choice.