N with the line between the departures of trains j(l) and j(l – 1) and whose destinations are the other stations is as follows: qwait = j(l ),s =Nod1,s d j(l ),1 – d j(l -1),1 TR(ten)Overtaking occurs on the route and causes changes within the order in which trains arrive at every station. As a result, j(l) is utilized to denote the lth train that dwells at station j. three.two.two. Number of passengers Waiting for Trains That are Delayed at Every single Station Besides the Origin Station Unlike the passengers waiting at the origin station, the passengers waiting at station j(j = 1) consist of both the passengers who arrive through the existing time period and those that were unable to board the preceding train (i.e., train j(l – 1)) simply because its capacity was reached (i.e., delayed passengers). The total (��)-Leucine Endogenous Metabolite quantity of passengers waiting at station j is as follows: N odr,s qwait = qdet-1),j + d j(l ),j – d j(l -1),j (11) j(l ),j j(l TR s =r +1 The number of delayed passengers, which is, passengers who have been unable to board train j(l – 1) because the number of passengers waiting for train j(l – 1) exceeded its capacity is as follows: up qdet-1),j = qwait 1),j – n j(l -1),j (12) j(l j(l – The amount of passengers who board train j(l) at station j is determined by two variables: (1) the remaining capacity of train j(l) soon after passengers alighting at station j; and (two) the existing number of waiting passengers. If the number of waiting passengers is decrease than the remaining capacity from the train, then all the waiting passengers can board the train; otherwise, only a few of the waiting passengers can board the train, even though the rest ought to wait for the following train: n j(l ),j = min max D – n j(l ),j-1 + ndown , x j(l ),j qwait j(l ),j j(l ),jup(13)exactly where n j(l ),j-1 will be the number of passengers onboard train j(l) in the time when the train departs from station (j – 1), that is calculated employing the following equation: n j(l ),j-1 = n j(l ),j-2 – ndown-1 + n j(l ),j-1 j(l ),jup(14)exactly where ndown-1 may be the quantity of passengers who alight from train j(l) at station (j – 1), j(l ),j which can be calculated as follows:j -ndown-1 = j(l ),j 3.3. Objective Functionsr =odr,j-1 d j(l ),r – d j(l -1),r TR(15)Higher departure frequencies can minimize twait and thus ttotal but enhance Ctotal . The objectives of the optimization model established in this study are to decrease both ttotal and Ctotal .Appl. Sci. 2021, 11,eight of3.3.1. Total Dicaprylyl carbonate medchemexpress passenger Travel Time ttotal ttotal consists of twait , the time spent onboard trains tonboard , as well as the time spent transferring involving express and nearby trains ttran . 1. twait The typical twait is set to half on the interval involving the departure instances of two successive trains. For delayed passengers, their twait is set to the typical twait plus the whole interval between the departure instances of trains (i + 1) and i.wait ti = N -1 j =wait qi,j di,j – di-1,j det + qi,j (di+1,j – di,j )(16)2. tonboard and ttran ttotal contains tonboard and ttran . The sum of tonboard and ttran is often calculated by subtracting the departure time from the beginning station from the arrival time in the destination station:onboard ti + titran =k =2 j =N k -od j,k (di,j – di-1,j ) ( ai ,k – di,j ) TR(17)where i’ is the quantity of the train that the passenger takes to reach the destination station, and i could be the quantity of your train that the passenger requires at the starting station (when the passenger will not transfer in between trains, then I = i’; otherwise, I = i’). The ttotal in the course of the study period is as follows.