N in the line between the departures of trains j(l) and j(l – 1) and whose destinations will be the other stations is as follows: qwait = j(l ),s =Nod1,s d j(l ),1 – d j(l -1),1 TR(10)Overtaking occurs around the route and causes modifications within the order in which trains arrive at every station. As a result, j(l) is made use of to denote the lth train that dwells at station j. 3.two.two. Quantity of Passengers Waiting for Trains That are Delayed at Every Station Aside from the Origin Station Unlike the passengers waiting at the origin station, the passengers waiting at station j(j = 1) consist of each the passengers who arrive during the present time period and these that were PHGDH-inactive Technical Information unable to board the preceding train (i.e., train j(l – 1)) for the reason that its capacity was reached (i.e., delayed passengers). The total number 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 amount of delayed passengers, which is, passengers who were unable to board train j(l – 1) because the quantity 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 number of passengers who board train j(l) at station j depends upon two aspects: (1) the remaining capacity of train j(l) after passengers alighting at station j; and (two) the existing quantity of waiting passengers. In the event the number of waiting passengers is reduce than the remaining capacity on the train, then each of the waiting passengers can board the train; otherwise, only some of the waiting passengers can board the train, even though the rest ought to wait for the subsequent 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 is definitely the quantity of passengers onboard train j(l) at the time when the train departs from station (j – 1), that is calculated making use of the following equation: n j(l ),j-1 = n j(l ),j-2 – ndown-1 + n j(l ),j-1 j(l ),jup(14)where ndown-1 could be the quantity of passengers who alight from train j(l) at station (j – 1), j(l ),j which is calculated as follows:j -ndown-1 = j(l ),j three.three. Objective Functionsr =odr,j-1 d j(l ),r – d j(l -1),r TR(15)High departure frequencies can reduce twait and as a result ttotal but increase Ctotal . The objectives in the optimization model established within this study are to minimize both ttotal and Ctotal .Appl. Sci. 2021, 11,8 of3.three.1. Total Passenger Travel Time ttotal ttotal consists of twait , the time spent onboard trains tonboard , and also the time spent transferring involving express and nearby trains ttran . 1. twait The typical twait is set to half from the interval amongst the departure instances of two successive trains. For delayed passengers, their twait is set for the average twait plus the entire interval in between the departure times 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 consists of tonboard and ttran . The sum of tonboard and ttran may be calculated by subtracting the departure time from the beginning station from the arrival time at 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 definitely the quantity in the train that the passenger takes to reach the destination station, and i would be the quantity in the train that the passenger takes in the starting station (in the event the passenger does not transfer amongst trains, then I = i’; otherwise, I = i’). The ttotal through the study period is as follows.