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Transportation infrastructure network design in the presence of modal competition: computational complexity classification and a genetic algorithm

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Abstract

In this paper we analyze the computational complexity of transportation infrastructure network design problems, in the presence of a competing transportation mode. Some of these problems have previously been introduced in the literature. All problems studied have a common objective: the maximization of the number of travelers using the new network to be built. The differences between them are due to two factors. The first one is the constraints that the new network should satisfy: (1) budget constraint, (2) no-cycle constraint, (3) both constraints. The second factor is the topology of the network formed by the feasible links and stations: (1) a general network, (2) a forest. By combining these two factors, in total we analyze six problems, five of them are shown to be NP-hard, the sixth being trivial. Due to the NP-hardness of these problems, a genetic algorithm is proposed. Computational experiments show the applicability of this algorithm.

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References

  1. Balakrishnan A, Magnanti TL, Mirchandani P (1997) Network design. Annotated bibliography in combinatorial optimization. Wiley, New York

  2. Bussieck M, Winter T, Zimmermann U (1997) Discrete optimization in public rail transport. Math Prog 79(1–3):415–444

  3. Chakroborty P (2003) Genetic algorithms for optimal urban transit network design. Comput Aided Civ Infrastruct Eng 18:184–200

  4. Chakroborty P, Dwivedi T (2002) Optimal route network design for transit systems using genetic algorithms. Eng Optim 34(1):83–100

  5. Desaulniers G, Hickman MD (2007) Public transit. Transportation Handbooks in operations research and management science, vol 14. Elsevier, Amsterdam, pp 69–127

  6. García-Archilla B, Lozano AJ, Mesa JA, Perea F (2013) GRASP algorithms for the robust railway network design problem. J Heuristics 19(2):399–422

  7. Garey M, Johnson DS (1979) Computers and Intractability: a guide to the theory of NP-completeness. W.H Freeman and Company, US

  8. Goel G, Karande C, Tripathi P, Wanga L (2010) Approximability of combinatorial problems with multi-agent submodular cost functions. ACM SIGecom Exch 9(1):1–4

  9. Grötschel M, Lovász L, Schrijver A (1981) The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1:169–197

  10. Grötschel M, Lovász L, Schrijver A (1993) Geometric algorithms and combinatorial optimization, 2nd edn. Springer-Verlag, New York

  11. Guihaire V, Hao JK (2008) Transit network design and scheduling: a global review. Transp Res Part A 42:1251–1273

  12. Jegelka SS (2012) Combinatorial problems with sub-modular coupling in machine learning and computer vision. Thesis ETH Zurich

  13. Laporte G, Mesa JA, Ortega FA (1995) Assessing the efficiency of rapid transit configurations. Top 5:95–104

  14. Laporte G, Mesa JA, Perea F (2010) A game theoretic framework for the robust railway transit network design problem. Transp Res Part B 44:447–459

  15. Magnanti TL, Wong RT (1984) Network design and transportation planning: models and algorithms. Transp Sci 18:1–55

  16. Marín A, García-Ródenas R (2009) Location of infrastructure in urban railway networks. Comput Oper Res 36(5):1461–1477

  17. Mesa JA, Boffey BT (1996) A review of extensive facility location in networks. Eur J Oper Res 95:592–603

  18. Nayeem MA, Rahman MK, Rahman MS (2014) Transit network design by genetic algorithm with elitism. Transp Res Part C 46:30–45

  19. Nemhauser GL, Wolsey LA (1999) Integer and combinatorial optimization. Wiley, Amsterdam

  20. Perea F, Mesa JA, Laporte G (2014) Adding a new station and a road link to a road-rail network in the presence of modal competition. Transp Res Part B 68:1–16

  21. Puerto J, Ricca F, Scozzari A (2018) Extensive facility location problems on networks: an updated review. TOP 26(2):187–226

  22. Schmidt M, Schöbel A (2014) Location of speed-up subnetworks. Ann Oper Res 223(1):379–401

  23. Schöbel A (2012) Line planning in public transportation: models and methods. OR Spectr 34(3):491–510

  24. Sourirajan K, Ozsen L, Uzsoy R (2009) A genetic algorithm for a single product network design model with lead time and safety stock considerations. Eur J Oper Res 197(2):599–608

  25. Székely L, Wang H (2005) On subtrees of trees. Adv Appl Math 34:138–155

  26. Ukkusuri SV, Mathew TV, Waller ST (2007) Robust transportation network design under demand uncertainty. Comput Aided Civ Infrast Eng 22:6–18

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Acknowledgements

Mozart Menezes and Juan A. Mesa were partially supported by project MTM2015-67706-P (MINECO/FEDER,UE). Federico Perea was partially supported by the Spanish Ministry of Science, Innovation, and Universities, under projects “ OPTEP-Port Terminal Operations Optimization” (No. RTI2018-094940-B-I00) and MTM2016-74983, financed with FEDER funds, and by the Universitat Politècnica de València under grant SP20180164 of the program Primeros Proyectos de Investigaciòn (PAID-06-18), Vicerrectorado de Investigaciòn, Innovaciòn y Transferencia. All this support is gratefully acknowledged.

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Correspondence to Federico Perea.

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Appendix

Appendix

Proofs

We begin this appendix with the proof of Proposition 1.

Proof

We prove this result by reduction from the knapsack problem. Consider the knapsack problem with item values equal to \(b_i\), and item weights equal to \(w_i\), \(i=1,\dots ,n\), and capacity \(w_{\max }\), which is formulated as:

$$\begin{aligned} \begin{array}{rl} \max &\, \sum _{i=1}^n b_i x_i \\ \text {s.t.:} &\, \sum _{i=1}^n w_i x_i \le w_{\max } \\ &\, x_i \in \{0,1\}, i=1,\dots ,n. \\ \end{array} \end{aligned}$$
(26)

Consider the following input data: \(N=\{v_0,v_1,\dots ,v_n\}\), \(E = \left\{ \{v_0,v_i\}:i=1,\dots ,n\right\} \), \(W= \{(v_0,v_i):i=1,\dots ,n\}\), \(C_{\max } = w_{\max }\), \(c_{0,i} = 0\), \(c_i = w_i\), \(i=1,\dots ,n\); see Fig. 2. Choose \(g_{0,i}\), \(\ell _{0,i}\), and \(u^{ALT}_{0,i}\) so that \(g_{0,i} \varphi (\ell _{0,i} - u^{ALT}_{0,i}) = b_i\) for all \(i=1,...,n\). For this, choose \(u_{0,i}^{ALT}=u\) and \(\ell _{0,i}=\ell \) for all \(i=1,\dots ,n\) so that \(\varphi (\ell - u) >0\). Note that these values exist because \(\lim \varphi _{x \rightarrow +\infty }(x) = 1\). Let \(\varphi ^* = \varphi (\ell - u).\) Then define \(g_{0,i} = b_i / \varphi ^*\).

Fig. 2
figure2

A network to prove the NP-hardness of RND1.

Solving this instance of RND1 solves the knapsack problem (26). So, if there was an algorithm that could polynomially solve this RND1 instance, you would be able to solve the knapsack problem in polynomial time, which is a contradiction because the knapsack problem is NP-hard (see Garey and Johnson 1979). \(\square \)

The proof to Corollary 1 follows.

Proof

The reader may note that, with the input data in the proof of Proposition 1, the solution to problem RND1 is the same as the solution to problem RND2, with the same data (because the underlying chosen network (NE) is a star, and therefore any subnetwork of it is also a star). \(\square \)

MILP model

We now detail the MILP program we designed for solving the RND1 problem in the experiments, similar to the one introduced in García-Archilla et al. (2013). The following variables are needed:

  • \(y_i\) is a binary variable to decide whether or not node \(v_i\) is a station of the railway network.

  • \(x_{ij}\) is a binary variable to decide whether or not edge (ij) is a link of the railway network.

  • \(r_{pq}\) is a binary variable to decide whether or not there is a path for OD-pair (pq) in the railway network.

  • \(w_{pq}\) is a binary variable to decide whether or not the railway network has a better utility than the road network for OD-pair (pq).

  • \(f^{pq}_{ij}\) is a binary variable to decide whether or not OD-pair (pq) will use edge (ij) from \(v_i\) to \(v_j\) in their route on the railway network.

The objective of our model is to maximize the railway trip coverage:

$$\begin{aligned} \max \sum _{(p,q)\in W}g_{pq} w_{pq}. \end{aligned}$$
(27)

The constraints of our model have been grouped according to their aims:

  • Budget constraints,

    $$\begin{aligned} \sum _{(i,j) \in E} \, c_{ij}\,x_{ij}+ \sum _{n_i\in N} c_i y_i \le C_{\max } . \end{aligned}$$
    (28)
  • Edges can be used in both senses, and if a link is built, then its end nodes must be stations of the railway network:

    $$\begin{aligned} x_{ij}= & \, x_{ji}, \ (i,j)\in E , \end{aligned}$$
    (29)
    $$\begin{aligned} x_{ij}\le & \, y_i, \ (i,j) \in E, \end{aligned}$$
    (30)
    $$\begin{aligned} x_{ij}\le & \, y_j, \ (i,j) \in E. \end{aligned}$$
    (31)
  • Routing demand conservation constraints:

    $$\begin{aligned}&\sum _{i : (i,p)\in A}\,{f}_{ip}^{pq} = 0, \ (p,q)\in W , \end{aligned}$$
    (32)
    $$\begin{aligned}&\sum _{j : (p,j)\in A}\,{f}_{pj}^{pq} = r_{pq}, \ (p,q)\in W , \end{aligned}$$
    (33)
    $$\begin{aligned}&\sum _{i: (i,q)\in A}\,{f}_{iq}^{pq} = r_{pq},\ (p,q)\in W , \end{aligned}$$
    (34)
    $$\begin{aligned}&\sum _{j : (q,j)\in A}\,{f}_{qj}^{pq} = 0,\ (p,q)\in W , \end{aligned}$$
    (35)
    $$\begin{aligned}&\sum _{i : (i,k)\in A} {f}_{ik}^{pq}\,-\,\sum _{j : (k,j)\in A} {f}_{kj}^{pq}\,=\,0, \ \forall \ k\notin \{p,q\}, \ (p,q)\in W . \end{aligned}$$
    (36)
  • Location-allocation constraints:

    $$\begin{aligned} {f}_{ij}^{pq} +r_{pq}-1\,\le \, \, x_{ij}, \ (i,j)\in A,\ (p,q)\in W . \end{aligned}$$
    (37)
  • Disutility of the railway network

    $$\begin{aligned} u_{pq}\,= & \, \,\sum _{(i,j)\in A}d_{ij}{f}_{ij}^{pq}+ M(1 - r_{pq}) + t_s \left( \left( \sum _{(i,j) \in A} f^{pq}_{ij}\right) - 1\right) \nonumber \\&+ \gamma (|zone_p - zone_q| + 1) . \end{aligned}$$
    (38)
  • Splitting demand constraints:

    $$\begin{aligned} (u_{pq} - u^{ALT}_{pq}) - M(1-w_{pq}) \le 0, \end{aligned}$$
    (39)

    where M is a real number sufficiently large.

Constraint (28) states that construction costs cannot exceed the budget, \(C_{\max }\). Constraints (29) allow the constructed links to be used in both directions. Constraint (30) and (31) impose that, if a link is built, then its corresponding end nodes should have a station. Constraints (32)– (36) are flow conservation constraints for variables f. Note that if \(r_{pq} = 0\), there will be no flow from pq via the railway network. Constraints (37) force that demands are only allocated through public arcs if the corresponding edges are built. Constraints (38) define the utility of each OD-pair in the railway network, which depends on the riding time, the number of stops, and the journey price. In these experiments, we took \(t_s = 0.5\) and \(\gamma = 1\). This implies that the stop time at stations is 0.5, and that the ticket price is 1, 2, or 3, if the number of zone changes is 0, 1, or 2, respectively. Note that the utility of (pq) is a large enough constant M if there is no path from p to q in the railway network. Constraints (39) impose for each OD-pair (pq) that, if their utility using the road network is better than their utility using the railway network, then \(w_{pq} = 0\), and therefore this OD-pair is not covered.

Besides these constraints, we also added to our model the following cuts:

$$\begin{aligned} f^{pq}_{ij}\le & \, x_{ij}, \forall \ (p,q) \in W, \ (i,j) \in A. \\ w_{pq}\le & \, r_{pq}. \end{aligned}$$

The first one imposes that no edge can be used if its corresponding link is not built. The second one imposes that an OD-pair without a path in the railway network cannot be covered. Previous experience showed that these two cuts significantly reduced the computational times.

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Perea, F., Menezes, M.B.C., Mesa, J.A. et al. Transportation infrastructure network design in the presence of modal competition: computational complexity classification and a genetic algorithm. TOP (2020) doi:10.1007/s11750-019-00537-x

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Keywords

  • Networks/graphs
  • Transportation
  • Computational complexity
  • Genetic algorithms

Mathematics Subject Classification

  • 90B06