A novel approach for travel time optimization in single track railway networks

A Novel Approach for Travel Time Optimization in Single-Track Railway Networks Nguyen Quang Thuan1(B) and Nguyen Duc Anh2 Vietnam National University Hanoi-International School, Hanoi, Vietnam nguyenquangthuan@vnu.edu.vn Hanoi University of Science and Technology, Hanoi, Vietnam Abstract Train scheduling plays an important role in the operation of railways systems This work focuses on a model of scheduling in which one minimizes the total travel time of trains in a single track railways network The model can be written in the form of a mixed 0–1 linear program which has the worst case exponential complexity to calculate the optimal solution In this paper, we propose a computationally efficient approach to solve the train scheduling problem Our approach is based on a so-called Difference of Convex functions Algorithm (DCA) to provide good feasible solutions with finite convergence The algorithm is tested on three different railway network topologies including one topology introduced in [18] and two practical topologies in Northern Vietnam The numerical results are encouraging and demonstrate the efficiency of the approach Keywords: Train scheduling · Penalty function · DC Algorithm Introduction Railways have advantage over the roadways in that they can carry a large number of passengers and large or heavy freight loads to long distances It becomes an essential pubic transport in most countries Among many problems arising in operating a railway system, train scheduling is critical to reduce costs, increase profits or improve service quality It generates train timetables to optimize total cost (time cost or financial cost) and satisfy some given conditions such as passenger demands, investment capital, time resource, etc Train scheduling is usually classified into two groups [18]: line planning and scheduling generation The former determines frequencies, routes, and scheduled times at each stop while the later finds the departure time and the arrival time of each train at sidings or stations Szpigel (1973) can be considered as a pioneer in studying train scheduling problems The author modeled a problem of minimizing the travel time of trains on single track line to a job shop scheduling problem then used a branchand-bound technique to solve it [31] Afterwards, many researchers focused on c Springer Nature Switzerland AG 2020 H A Le Thi et al (Eds.): ICCSAMA 2019, AISC 1121, pp 27–38, 2020 https://doi.org/10.1007/978-3-030-38364-0_3 28 N Q Thuan and N D Anh two aspects: modeling and solution methods for a given problem For modeling, practical problems were normally formulated into the form of a mathematical optimization problem such as an integer program [2–6] a mixed-integer program [1,7,10–15,18,25,26], a multi-objective linear programming [8,11] This kind of problems is NP-hard Finding a solution method (exact or heuristic) for them is a challenging mission Branch-and-bound techniques were usually used to get the optimal solution [14,16,31] However, it takes much time to get the optimal solution in the case of large-scale and complex instances Thus, heuristic approaches were often proposed to find feasible solutions, for instance, the priority-rulebased heuristics [1,9,19,28,28], backtracking search [1], look-ahead search [30], and meta-heuristic algorithms [14,15,17] Each solution method above furnishes different feasible solutions Its efficiency depends on the structure of formulations, the network topology, the size of instance, etc It is worthy to have a new solution method that may find a good feasible solution If this solution is not optimal, it is a good upper bound in the schema of branch-and-bound algorithms in order to accelerate the time of computing the optimal solution Our contribution is to propose a method finding such a good feasible solution It is based on DC (difference of two convex functions) programming and DCA (DC Algorithms) that has been efficiently applied to real world non convex programs in various fields [22,23] Obviously, we study a typical model that was proposed by Higgins et al [14]; and modified by Karoonsoontawong and Taptana [18] It is formulated in the form of a mixed 0–1 linear programming (MILP) By employing the exact penalty method, we first show that the MILP can be equivalently recast as a concave minimization problem Next, we reformulate the concave minimization problem in the form of a DC program then use DCA to solve We test the proposed algorithm on three different railway network topologies including one topology introduced in [18] and two practical topologies in Northern Vietnam The preliminary results demonstrate the efficiency of the proposed method The paper is structured as follows After the introduction, we describe the problem in Sect Section presents DC programming, DCA and show how to apply DCA to the problem In Sect 4, we provide some numerical experiments to evaluate the proposed approach The last section is dedicated to some conclusions Problem Description Among train scheduling problems in single-track railway line, the travel time optimization problem has specially attracted many researchers Higgins et al [14], Zhou and Zhong [32] introduced a mixed integer linear program for the aforementioned problem Basing on the formulation in [14], Karoonsoontawong and Taptana in 2017 proposed a modified formulation [18] This section presents the problem described in [18] with the following assumptions: Networks includes sidings or stations that divide railways into segments; Two trains or more are not allowed on any track segment; There are two tracks at sidings/stations and A Novel Approach for Train Scheduling Problems 29 each segment has the most tracks; and a pre-specified path is assigned for each train 2.1 Notation The sets, the parameters, the variables are denoted as follows Sets I = {1, 2, , nI } - set of trains, nI = |I|, nI ∈ N is total number of trains in the railway system; P = {1, 2, , nP } - set of rail segments, nP = |P |, nP ∈ N is total number of segments in the railway system; Q = {1, 2, , nQ } - set of stations or sidings nQ = |Q|, nQ ∈ N is the number of stations (or sidings) in the railway system; P (i) - ordered set of rail segments traversed by train i ∈ I; Q(i) - ordered set of stations (or sidings) traversed by train i ∈ I; P1 - set of single-track segments; P2 - set of double-track segments; P same d (i, j) - set of common rail segments for trains i and j, which traverse in the same direction; P1opp d (i, j) - set of common single-track segments for trains i and j, which traverse in opposite directions; D is set of segment directions: inbound or outbound Parameters q1 (p, d) - starting station (or siding) of segment p in direction d; q2 (p, d) - terminal station (or siding) of segment p in direction d; di,p - direction in which train i ∈ I traverses segment p ∈ P (i); hi,j 1,p - minimum headway between trains i, j ∈ I traversing in the same direction on p ∈ P ; hi,j 2,p - minimum headway between trains i, j ∈ I traversing in opposite directions on p ∈ P ; lp - length of segment p ∈ P ; i - earliest departure time of train i ∈ I; YdO i v p - minimum allowable average velocity of train i ∈ I on segment p ∈ P (i); v ip - maximum achievable average velocity of train i ∈ I on segment p ∈ P (i); Wi - weight showing the priority for train i ∈ I; Sqi - scheduled stop time for train i ∈ I at station q ∈ Q(i); M - sufficiently big constant Decision Variables Aijp equals to if train i ∈ I traverses track segment p ∈ P same d (i, j) before train j ∈ I when trains i and j traverse track segment p in the same direction; otherwise Bijp equals to if train i ∈ I traverses track segment p ∈ P1oppd (i, j) before train j ∈ I when trains i v` a j traverse track segment p in opposite directions; otherwise 30 N Q Thuan and N D Anh i Xa,q i Xd,q i XdO i XaD 2.2 is the arrival time of train i ∈ I at station/siding q ∈ Q(i) is the departure time of train i ∈ I from station/siding q ∈ Q(i) is the departure time of train i ∈ I from its origin station is the arrival time of train i ∈ I at its destination station Mathematical Model The problem can be formulated as follows: i i Wi (XaD − YdO ) , z = (1) i∈I subject to: j i ≥ Xa,q + hi,j M ∗ Aijp + Xd,q 1,p (p,di,p ) (p,dj,p ) ∀i = j ∈ I, p ∈ P same d (i, j) (2) j i M ∗ (1 − Aijp ) + Xd,q ≥ Xa,q + hi,j 1,p (p,di,p ) (p,dj,p ) j i M ∗ Bijp + Xd,q ≥ Xa,q + hi,j 2,p (p,di,p ) (p,dj,p ) ∀i = j ∈ I, p ∈ P same d (i, j) (3) opp d ∀i = j ∈ I, p ∈ P1 (i, j) (4) j i M ∗(1−Bijp )+Xd,q ≥ Xa,q +hi,j 2,p (p,di,p ) (p,dj,p ) ∀i = j ∈ I, p ∈ P1opp d (i, j) (5) lp lp i i ≤ Xa,q − Xd,q ≤ i (p,di,p ) (p,di,p ) vp v ip i i ≥ YdO XdO i Xa,q + Sqi ≤ i Xd,q ∀i ∈ I, p ∈ P (i) ∀i ∈ I ∀i ∈ I, q ∈ Q(i) (6) (7) (8) Aijp ∈ {0, 1} i, j ∈ I, p ∈ P same d (i, j) (9) Bijp ∈ {0, 1} i, j ∈ I, p ∈ P1opp d (i, j) (10) The objective function (1) minimizes the weighted sum of total train travel times Constraints (2) and (3) state that for any two trains i, j traversing the same segment p in same direction, Aijp equals to zero if and only if train j traverses segment p before train i, and train j must leave segment p for the period of hi,j 1,p before train i can enter it Constraints (4) and (5) also indicate that Bijp equals to zero if and only if train j traverses segment p before train i with the time headway not less than the minimum safety headway Constraints (6) ensure that the travel time of trains on any rail segment is in the range of the corresponding upper and lower limits Constraints (7) allow the train departure time from its origin station to be bigger than or equal to its earliest departure time Constraints (8) state that a train leaves a station siding after it arrives at this station and stops there for at least the scheduled stop time The problem above is a mixed 0–1 linear programming Finding a suitable method for solving this kind of problems is always challenging The challenge A Novel Approach for Train Scheduling Problems 31 does not only come from the binary variables but also the size of problems We propose here a method to solve the problem efficiently For this, we first use the theory of exact penalization in DC programming [24] to reformulate the MILP as that of minimizing a DC function over a polyhedral convex set The resulting problem is then handled by DCA which was introduced and extensively developed over the last decades [23] The mentioned approach has been applied successfully in several large scale problems (see [20–23,27] and reference therein) The details are provided in the following section 3.1 Solution Method DC Reformulation By using an exact penalty result, we reformulate the MILP in the form of a concave minimization program The exact penalty technique aims at tranforming the orignal MILP into a more tractable equivalent problem in the DC optimization framework Let S be the feasible set of the problem MILP (1)–(10) For notational simplicity, we group all arrival and departure time variables in a column vector nI T ] , where U = [ U11 , U12 , , U1nI , U21 , U22 , , U2nI , , Un1Q , , UnnQI , Un1Q +1 , , U2n Q i i i i T denotes the transpose operator; Uq = Xa,q and UnQ +q = Xd,q ∀i ∈ I, q ∈ Q In the same way, we group all the binary variables (includes Aijp and Bijp ) into a column vector C = [ c111 , c112 , , cnI nI nP , c(nI +1)nI nP , , c(2nI )nI nP ]T , where cijp = Aijp and c(nI +i)jp = Bijp ∀i, j ∈ I, p ∈ P We denote a new set K := {(U, c) ∈ S : c ∈ [0, 1]2nI nI nP } Assume that K is a nonempty, bounded polyhedral convex set in R2nI nQ × R2nI nI nP Therefore, the problem (1)–(10) can be expressed in the general form (Uopt , copt ) = argmin{z : (U, c) ∈ S, c ∈ {0, 1}2nI nI nP }, (11) i i where z = i∈I Wi (XaD − YdO ) Let us consider the function p defined by min{cijp , − cijp } p(U, c) := (12) i,j∈I;p∈P It is clear that p is concave and finite on K, p(U, c) ≥ ∀(U, c) ∈ K, and {(U, c) ∈ S : c ∈ {0, 1}2nI nI nP } = {(U, c) ∈ K : p ≤ 0} Hence, Problem (11) can be written as (Uopt , copt ) = argmin{z : (U, c) ∈ K, p(U, c) ≤ 0} (13) The following theorem is in order Theorem Let K be a nonempty bounded polyhedral convex set, f be a finite concave function on K and p be a finite nonnegative concave function on K Then there exists t˜0 ≥ such that for t˜ ≥ t˜0 the following problem has the same optimal value and the same optimal solution set: 32 N Q Thuan and N D Anh (Pt ) α(t) = {f (x) + t˜p(x) : x ∈ K} (P ) α = min{f (x) : x ∈ K, p(x) ≤ 0} Furthermore, • If the vertex set of K, denoted by V (K), is contained in x ∈ K : p(x) ≤ 0, then t˜0 = f (x) − α(0) • If p(x) > for some x ∈ V (K), then t˜0 = : x ∈ K, p(x) ≤ S0 , where S0 = min{p(x) : x ∈ V (K), p(x) > 0} Proof The proof for the general case can be found in [24] From Theorem we get, for a sufficiently large number t˜ (t˜ > t˜0 ), the equivalent concave minimization problem to (13) z + t˜p(U, c) : (U, c) ∈ K (14) which is a DC program of the form g(U, c) − h(U, c) (15) where g(U, c) = χK (U, c) and h(U, c) = −z − t˜ i,j∈I;p∈P min{cijp , − cijp } χK (U, c) = if (U, c) ∈ K, otherwise +∞ (the indicator function of K) We have successfully tranformed an optimization problem with integer variables into its equivalent form with continuous variables 3.2 DC Algorithm for (14) Now, we investigate a DC programming approach for solving (14) A DC program is that of the form: α := f (x) := g(x) − h(x) : x ∈ Rn (16) with g, h being lower semi-continuous proper convex function on Rn , and its dual problem is defined as α := h∗ (y) − g ∗ (y) : y ∈ Rn (17) where g ∗ (y) := max xT y − g(x) : x ∈ Rn is the conjugate function of g Based on local optimality conditions and duality in DC programming, the DCA consists in the construction of two sequences {xk } and {y k }, candidates to be optimal solutions of primal and dual programs respectively, in such a way that {g(xk ) − h(xk )} and {h∗ (y k ) − g ∗ (y k )} are decreasing and their limits points satisfy the local optimality conditions The idea of DCA is simple: each iteration of DCA approximates the concave part −h by its affine majorization (that corresponds to taking y k ∈ ∂h(xk )) and minimizes the resulting convex function A Novel Approach for Train Scheduling Problems 33 Generic DCA scheme: Initialization Let x0 ∈ Rn be a good guess, k ← 0; Repeat Calculate y k ∈ ∂h(xk ); (Pk ); Calculate xk+1 ∈ argmin g(x) − h(xk ) − x − xk , y k : x ∈ Rn k + ← k; Until Convergence of xk The convergence properties of DCA and its theoretical basis can be found in [23], for instant it is important to mention that: • DCA is a descent method without line search; • If the optimal value of problem (16) is finite and the sequence {xk } is bounded then every limit point x∗ of {xk } is a critical point of g − h; • DCA has a linear convergence for general DC programs; • DCA has a finite convergence for polyhedral DC programs ((16) is called polyhedral DC program if either g or h is polyhedral convex) We now describe the DCA applied to the DC program (14) By definition, a sub-gradient (v k , dk ) ∈ ∂h(U k , ck ) can be chosen as follows: • vqi = −Wi if q is destination station on travel path of train i, otherwise 0; • dijp = t˜ if cijp ≥ 0.5, otherwise dijp = −t˜ for all i, j ∈ I, p ∈ P By using (v k , dk ), we then compute (U k+1 , ck+1 ) by solving the linear program: (U k+1 , ck+1 ) = argmin χK (U, c) − (U, c) − (U k , ck ), (v k , dk ) : (U, c) ∈ K (18) or (U k+1 , ck+1 ) = argmin − (U, c), (v k , dk ) : (U, c) ∈ K (19) Thus, the DCA applied to (14) is as follows: Algorithm DCA: Let k = 0; er = 106 ; Choose a sufficiently small positive number ; Choose an initial point (U k , ck ); while er > Compute (v k , dk ) ∈ ∂h(U k , ck ); Solve − (U, c), (v k , dk ) : (U, c) ∈ K to obtain (U k+1 , ck+1 ); Compute error er =|| (U k+1 , ck+1 ) − (U k , ck ) ||; k = k + 1; endwhile Regarding the complexity of the proposed DCA, besides the computation of the sub-gradients which is trivial, the algorithm requires one linear program at each iteration and it has a finite convergence The linear program has polynomial complexity The convergence of Algorithm DCA can be summarized in the next theorem [29] 34 N Q Thuan and N D Anh Theorem (i) Algorithm DCA generates a sequence {(U k , ck )} contained in V (K) such that the sequence {g(U k , ck ) − h(U k , ck )} is decreasing (ii) If at iteration r, we have cr ∈ {0, 1}2nI nI nP , then ck ∈ {0, 1}2nI nI nP and f (U k+1 , ck+1 ) < f (U k , ck ) ∀k > r (iii) The sequence {(U k , ck )} converges to {(U ∗ , c∗ )} ∈ V (K) after a finite number of iterations The point (U ∗ , c∗ ) is critical point of the problem (14) Moreover such an (U ∗ , c∗ ) is almost always a strict local minimum of problem (14) Basing on the second affirmation of the theorem above, we should choose a feasible solution as an initial point of DCA This ensures that the solution obtained by DCA is feasible to the original problem although DCA works on the continuous domain The way we choose an initial point is as follows: We arrange the trains in the order of the earliest departure time at the original station of the itinerary With each train, we determine the schedule on the whole itinerary for that train The generated schedule of the next train is based on the schedule which was created by the previous trains to ensure that no conflict occurs The result is a feasible solution of the problem Computational Experiments In this section, we provide preliminary computational results of our approach We have coded the algorithm in C++ programming language and tested instances using PC Intel core i7 3770 3.4 GHz, 16 GB RAM The solver CPLEX 12.6.1 is used to solve the linear program in each iteration of DCA and get the optimal value of MILP We investigate the algorithm performance on three network topologies that are the topology shown in [18] (toy network) and two topologies in Northern Vietnam (HN-HP, HN-LC network) The toy network includes trains, segments, stations and the total length of segments is 297 km The HN-HP network is a single-track railway system connecting Hanoi capital and Hai Phong city, including trains (4 inbounds, outbounds), segments and stations The total length of HN-HP network is 102 km The HN-LC network is a line connecting Hanoi capital and Lao Cai province It is more complex as it Table The size of testing networks Toy network HN-HP network HN-LC network Trains 8 Segment 27 Stations/sidings 28 Continuous Variables 36 128 448 Binary variables 90 896 3456 216 1920 7360 Constraints A Novel Approach for Train Scheduling Problems 35 consists of trains (4 inbounds and outbounds), 27 segments, 28 stations and the total distance is 294 km The size of the networks are shown in Table In each network, 10 instances are generated by modifying the train parameters composed of the scheduled stop time at stations, the earliest departure time and the weight of trains The tested results for three networks are presented in Table 2, 3, and where ValDCA is the objective value obtained by the algorithm DCA; CPU is the computing time; OptVal is the optimal value; and GAP ValDCA − OptVal × 100% = OptVal Table The computation result for Toy network Instance ValDCA CPU OptVal GAP 2347.022 0.011 2190.053 7.17 % 2319.022 0.018 2174.437 6.65 % 2199.714 0.016 2199.714 0.00 % 2175.438 0.021 2143.438 1.49 % 2281.022 0.019 2136.438 6.77 % 2175.715 0.014 2175.715 0.00 % 2164.438 0.022 2136.438 1.31 % 2223.715 0.016 2223.715 0.00 % 4420.691 0.018 4420.691 0.00 % 10 4370.568 0.021 4224.106 3.47 % Average 2667.735 0.018 2602.475 2.69% From the tables of results, we can see that: – The computing time of DCA is good for all the instances Even for the big size (HN-LC network with 3456 binary variables and 7360 constraints), the time to get solution is still small – The GAP numbers are very small This means the solution quality is quite good For the Toy network, there are out of 10 instances in which GAP equals to zero, i.e DCA furnishes the optimal solution for 40% 36 N Q Thuan and N D Anh Table The computation result for HN-HP network Instance ValDCA CPU OptVal GAP 6755 0.077 6745 0.15 % 6749 0.061 6739 0.15 % 5493 0.075 5455 0.70 % 5482 0.076 5442 0.74 % 5447 0.067 5445 0.04 % 5283 0.081 5273 0.19 % 5340 0.068 5301 0.74 % 17014 0.071 16744 1.61 % 16722 0.063 16659 0.38 % 10 10087 0.077 10015 0.72 % Average 8437.2 0.072 8381.8 0.54 % Table The computation result for HN-LC network Instance ValDCA CPU OptVal GAP 4546 0.385 4470 1.70 % 4543 0.381 4505 0.84 % 4618 0.342 4528 1.99 % 4956 0.357 4687 5.74 % 4996 0.386 4683 6.68 % 4546 0.356 4470 1.70 % 7311 0.335 7150 2.25 % 11322 0.379 11137 1.66 % 6008 0.373 5914 1.59 % 10 6054 0.335 5969 1.42 % Average 5890.00 0.363 5751.30 2.56 % Conclusion In this paper, we have studied a model minimizing total travel time of trains in single track networks We have shown that the aforementioned problem can be formulated as a mixed-integer linear program Realizing the inherent difficulty in computing the optimal solution of MILPs, our main contribution was to propose a computationally efficient approach based on DCA The considered combinatorial optimization problem has been beforehand reformulated as a DC program with a natural choice of DC decomposition, and the resulting DCA then consists in solving a finite sequence of linear programs DCA is original because it gives an integer solution while it works in a continuous domain Preliminary A Novel Approach for Train Scheduling Problems 37 numerical results were encouraging and demonstrated the effectiveness of the proposed method The short computing time and the capacity for handling the large-scale instances make the proposed method valuable Moreover, notice that most problem formulations arising in train scheduling can be formulated as some sort of MILP problems, our proposed approach seems attractive and needs more investigation References Adenso-Diaz, B., Gonzalez, M.O., Gonzalez-Torre, P.: 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