###### Abstract  
This paper presents a mixed-integer linear programming model for demand-responsive feeder transit services to assign vehicles located at different depots to pick up passengers at the demand points and transport them to the rail station. The proposed model features passengers' one or several preferred time windows for boarding vehicles at the demand point and their expected ride time. Moreover, passenger satisfaction that was related only to expected ride time is fully accounted for in the model. The objective is to simultaneously minimize the operation costs of total mileage and maximize passenger satisfaction. As the problem is an extension of the nondeterministic polynomial problem with integration of the vehicle route problem, this study further develops an improved bat algorithm to yield meta-optimal solutions for the model in a reasonable amount of time. When this was applied to a case study in Nanjing City, China, the mileage and satisfaction of the proposed model were reduced by 1.4 km and increased by 7.1%, respectively, compared with the traditional model. Sensitivity analyses were also performed to investigate the impact of the number of designed bus routes and weights of objective functions on the model performance. Finally, a comparison of Cplex, standard bat algorithm, and group search optimizer is analyzed to verify the validity of the proposed algorithm. D RCNN 2018 10 30 610390 10.3390/61003030 10.3390/61003030 10.  
designated place within one of the specified periods to pick them up. Similar to VRPs with multiple time windows [5; 6], DRTs with multiple time windows can save a greater amount of mileage compared with the single time window problem. Furthermore, it is very important for passengers to set a maximum and minimum expected ride time to meet personalized travel needs, which will encourage the drivers to take the shortest route distance in order to reduce the time in the vehicle for other passengers. However, DRTs with expected ride time have also not been examined previously. Another major motivation for this study was to address DRTs in terms of passenger satisfaction, which involves many factors, such as bus fares, congestion, environment, etc. Some studies have studied VRPs with consideration of customer satisfaction, although this has been considered less in the DRTs [7; 8]. Without loss of generality, it is only relevant in terms of expected ride time in feeder transit operation (i.e., the satisfaction is in the range of 0-1) when the ride time is between the minimum and maximum expected values. Obviously, greater passenger satisfaction and demand for more vehicles to provide direct services will increase the total mileage and operating expenses. Therefore, it is necessary to reveal the optimal relationship between passenger satisfaction and the total mileage in order to balance this with service quality and operating costs. The main objective of this research is to develop an optimization model for DRTs with passengers' multiple time windows and satisfaction in order to improve service quality and operating costs. The paper will focus on the following critical research tasks: (1) coordination of the passenger boarding time window guidance and feeder transit routing process to balance this with the total mileage and passenger satisfaction; and (2) development of a heuristic solution algorithm to efficiently yield the acceptable solution to the proposed mode. Finally, a numeric case study is used to illustrate the proposed methodology and apply the proposed model during the process of producing the optimal design fora DRT in the real world. The remainder of the paper is organized as follows. Section 2 reviews the related literature onDRTs. Section 3 analyses the framework of the proposed methodology and presents the formulation of the DRT model. Section 4 presents an improved bat heuristic algorithm for resolving the model. Section 5 displays a case study to illustrate the proposed model and algorithm. Some concluding remarks and possible future work are given in Section 6.