Global Optimization Methods for the Discrete Network Design Problem Dr. Shuaian Wang Lecturer University of Wollongong Dr. Qiang Meng Associate Professor National University of Singapore Dr. Hai Yang Chair Professor The Hong Kong University of Science and Technology

Network Design Problems Add capacity to existing links, or add new links Minimize total travel time Budget constraint Braess paradox 2

Network Design Problems Discrete network design problem (DNDP): finite candidate link capacities (e.g., integer number of lanes) Continuous network design problem (CNDP) Mixed network design problem (MNDP) 3

Literature on DNDP and MNDP Branch-and-bound: LeBlanc (1975), Poorzahedy and Turnquist (1982): Support function: Gao et al. (2005) Complementary constraint: Wang and Lo (2010) Variational inequality: Luathep et al. (2011) KKT condition: Farvaresh and Sepehri (2012) Single-level concave program (Li, Yang, Zhu and Meng 2012) Heuristics 4

Objective A DNDP with multiple capacity levels A new system-optimum relaxation based solution method A new user-equilibrium reduction based solution method 5

Notation Set of all feasible network design decisions 6

Notation We consider fixed demand Set of path flows: Set of link flows: 7

System-optimum (SO) relaxation If travelers' route choice behavior followed the SO principle, then the problem could be formulated as a mixed-integer convex optimization model: 8

User-equilibrium (UE) However, in reality, travelers' route choice behavior follows the user-equilibrium (UE) principle. For a given network design decision z, the link flow is the optimal solution to: 9

SO-Relaxation Based Method

Properties of the problem and formulations SO relaxation could provide a lower bound for the real total travel time at UE. An optimal network design solution under SO principle can be a good approximate solution under UE principle (unless there is the Braess paradox effect) The number of network design decisions is finite (discrete network design) 11

Idea of the SO-relaxation based algorithm Sort the solutions according to the total travel time at SO, and stop when the lower bound is larger than the upper bound. Consider two candidate links to expand, one lane may be added to each link. 12 Network design decision Total travel time at SOTotal travel time at UE Step 1: (1,1) Step 1: 100Step 2: 130 Step 3: (0,1) Step 3: 120Step 4: 125 Step 5: (1,0) Step 5: 140Step 6: 150 Step 7: (0,0) Step 7: 160Step 8: 165

Challenge: How to obtain the 2 nd (3 rd, 4 th etc.) best solution under SO? 13

Example: Exclude solutions 14 In the two-link example, the effect of the linear constraint is:

Steps of the SO-relaxation method Step 0: Define upper bound UB=0 Step 1: Solve SO-relaxation with constraints that exclude all previously generated solutions. The optimal objective value is LB. If LB>=UB, optimal solution, stop; Otherwise, evaluate the obtained provisional optimal solution by solving a UE problem, and update UB if this solution is better than all the previously generated ones. 15

UE-Reduction Based Method

Key observation The set of feasible link flows does not change with the network design decisions. If we construct a new link, we can assume that the link already exists, but its free-flow travel time is very large. 17

UE-reduction based method The UE link flow is the optimal solution to: Hence, for any, the following is a valid constraint: Luckily, this constraint is convex. 18

UE-reduction based method The link flows are obtained when evaluating the provisional solutions of SO-relaxation model Adding the following constraint to the SO-relaxation model strengthens the model: Convexity of the constraint ensures tractability. 19

Steps of the UE-reduction method Step 0: Define upper bound UB=0 Step 1: Solve SO-relaxation with constraints that exclude all previously generated solutions and with constraints that take advantage of the UE objective function. The optimal objective value is LB. If LB>=UB, optimal solution, stop; Otherwise, evaluate the obtained provisional optimal solution by solving a UE problem, save the resulting link flow, and update UB if this solution is better than all the previously generated ones. 20

Comparison of the two methods 21

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