Dynamic Network Design Problem Research Articles

Multi-period stochastic programming for relief delivery considering evolving transportation network and temporary facility relocation/closure

In this study, we address a dynamic network design problem pertaining to the distribution of essential supplies during the post-disaster response phase. We present a two-stage, stochastic multi-period model that minimizes both expected unmet demand and costs associated with operating emergency response facilities and distributing relief supplies, taking into account evolving road conditions as well as the closure and relocation of temporary relief facilities. To address the computational intractability inherent in large-scale mixed-integer linear programming (MILP), we implement an accelerated branch-and-Benders-cut algorithm, facilitating efficient problem-solving processes. We validate this approach through extensive numerical analysis and its application to a real-world scenario, demonstrating its superiority over deterministic methods, especially when considering the trade-off between nominal operational costs and out-of-sample reliability. Results show that dynamically adjusting the number of operating facilities and their locations may enhance cost efficiency while decreasing unmet demand during the relief delivery process as the transportation network changes with time. Additionally, we show that utilizing stochastic models for decision-making can substantially hedge against demand uncertainty, both in meeting demand and reducing operational costs. However, the benefits of stochastic solutions diminish as the disruption rate increases.

A dual approximation-based quantum-inspired genetic algorithm for the dynamic network design problem

In this paper, we formulate a dynamic transportation network design model in which traffic dynamics are modeled by the cell transmission model. In the formulation, transportation planners decide on the optimal capacity expansion policies of existing transportation network infrastructure with limited resources, while road users react to the capacity changes by selfishly choosing routes to maximize their own profit. Owing to the problem complexity, a majority of the research efforts have focused on tackling this problem using meta-heuristics. In this study, we incorporate a series of dual-variable approximation techniques into the paradigm of a quantum-inspired genetic algorithm (QIGA) and devise an efficient evaluation function based on these techniques. The proposed QIGA contains a series of enhancements compared to conventional genetic algorithms (GAs) and can be considered as a better alternative when solving problems with a complex solution space. The QIGA is applied to a synthetic network, a subnetwork of a real-world road network, and a realistic network to justify its theoretical and practical value. From the numerical results, it is found that in the same computational time, the QIGA outperforms the conventional GA by 3.86–5.63% in terms of the objective value, which can be significant, especially when network expansion of a large urban area is considered. Technical, computational, and practical issues involved in developing the QIGA are investigated and discussed.

A Distributionally Robust Joint Chance Constrained Optimization Model for the Dynamic Network Design Problem under Demand Uncertainty

This paper develops a distributionally robust joint chance constrained optimization model for a dynamic network design problem (NDP) under demand uncertainty. The major contribution of this paper is to propose an approach to approximate a joint chance-constrained Cell Transmission Model (CTM) based System Optimal Dynamic Network Design Problem with only partial distributional information of uncertain demand. The proposed approximation is tighter than two popular benchmark approximations, namely the Bonferroni’s inequality and second-order cone programming (SOCP) approximations. The resultant formulation is a semidefinite program which is computationally efficient. A numerical experiment is conducted to demonstrate that the proposed approximation approach is superior to the other two approximation approaches in terms of solution quality. The proposed approximation approach may provide useful insights and have broader applicability in traffic management and traffic planning problems under uncertainty.

Stochastic capacity planning and dynamic network design

Abstract The present paper proposes a mathematical model and a solution approach to the Stochastic Capacity Planning and Dynamic Network Design Problem. Here, strategic decisions usually comprise developing the necessary capacity — through either incrementing capacity on existing assets (facilities or logistics channels) or establishing new capacity in the form of new assets — in order to satisfy increasing demand. Hence, throughout the planning horizon, decisions on which new assets to establish and where to increment capacity must be taken at minimal cost and in a timely manner. However, when demand varies nonmonotonically, decisions on which assets to temporarily shut down in times of decreasing demand and which of those to reopen when market conditions improve must also be taken into account. We propose a multi-stage stochastic mixed-integer programming approach to the problem as well as a Lagrangian Heuristic procedure to attain reasonably well bounded feasible solutions. The proposed method is evaluated in a Global Mining Supply Chain context which, due to the inherently large capital expenses, could have the outcome of its strategic decision making process significantly improved.

A Dual Variable Approximation-Based Descent Method for a Bi-level Continuous Dynamic Network Design Problem

The transportation network design problem (NDP) considers modifying network topology or parameters, such as capacity, to optimize system performance by taking into account the selfish routing behavior of road users. The nature of the problem lends itself to a bi-level formulation of a problem that represents a static case of a Stackelberg game. The NDP is complex because users’ individual objectives do not necessarily align with system-wide objectives and it is difficult to determine the optimal allocation of limited resources. To solve the bi-level dynamic NDP, this study develops a dual variable approximation-based heuristic. This dual variable identifies the system-wide gradient as a descent direction and designs an iterative solution framework. Descent direction-based approaches designed to solve bi-level programming problems typically suffer from non-differentiability, which can hamper the solution process. The proposed method addresses this issue by approximating the descent direction with dual variables that correspond to cell transmission model constraints and using the constructed rational direction to iteratively decrease the upper-level objective while maintaining the feasibility of the lower-level program. The proposed method was empirically applied to three networks of various sizes. The results obtained from this empirical solution were compared with the results from an exact Kth-best algorithm and a genetic algorithm. The results demonstrate the efficacy and efficiency of the proposed descent method.

A genetic algorithm for bi-level linear programming dynamic network design problem

Solving the Bi-level Linear Programming Network Design Problem (BLPNDP), which is typically regarded as a strongly NP-complete problem, with traditional mathematical programming algorithms is computationally intractable. A majority of the research efforts have been focused on tackling this problem using meta-heuristics. However, the evaluation functions in most meta-heuristic approaches involve dynamic traffic simulation when evaluating feasible capacity expansion policies. This commonly used evaluation function becomes a major bottleneck constraining the applicability of meta-heuristics as it is computationally expensive and analytically complicated. To address this issue, we present a modified Genetic Algorithm (GA) to solve the BLPNDP with a novel evaluation function which reduces the number of dynamic traffic simulations needed. The heuristic exploits the underlying linear programming structure of the BLPNDP and employs dual variable approximation techniques to estimate the evaluation function instead of dynamic traffic simulation. Empirical analyses are conducted on three networks of varying sizes to examine the performance of the GA based heuristic. From the experiments conducted, the proposed GA based heuristic outperforms the traditional GA. Also the proposed method of estimating the evaluation function is flexible and can be used within the framework of many meta-heuristic approaches.

Modeling Departure Time Choice under Stochastic Networks Involved in Network Design

Different travel cost functions for modeling travelers’ departure time and route choice behavior under uncertainty are investigated; then the principal equivalency of the mean-variance and scheduling approaches in modeling departure time choice is proven analytically. Based on the equivalency, a general cost function is formulated for modeling travelers’ departure time and route choice behavior under uncertainty and applied to an analysis of the departure patterns under uncertainty and a reliability-based dynamic network design problem. The analytical investigation of the departure pattern considering travel time reliability in the individual's travel cost function is presented as an extension of Vickrey's deterministic bottleneck model toward stochastic capacities. The departure pattern is significantly different from that without reliability: travelers depart earlier to reduce travel time variability. This finding (corroborated by simulations) motivated the network design approach including travel time reliability in departure time and route choice behavior. A discrete dynamic network design problem is formulated subject to a reliability-based dynamic user equilibrium, with the cost function derived from the equivalency analysis among different cost functions. Total network costs–the costs of total travel (including travel time reliability) and construction–are minimized. The network design problem consists of optimally allocating link capacities by determining the optimal numbers of lanes for a subset or a full set of links of a potential network. A genetic algorithm with some intelligence is adopted to solve the discrete optimization problem. Use of a small hypothetical network indicates that different network designs may lead to diverse departure distributions among different origin-destination (O-D) pairs over time, directly affecting travel costs. Relatively dissimilar departure patterns among different O-D pairs over time give lower travel costs.

Linear Programming Models for the User and System Optimal Dynamic Network Design Problem: Formulations, Comparisons and Extensions

In this paper we formulate a network design model in which the traffic flows satisfy dynamic user equilibrium conditions for a single destination. The model presented here incorporates the Cell Transmission Model (CTM); a traffic flow model capable of capturing shockwaves and link spillovers. Comparisons are made between the properties of the Dynamic User equilibrium Network Design Problem (DUE NDP) and an existing Dynamic System Optimal (DSO) NDP formulation. Both network design models have different objective functions with similar constraint sets which are linear and convex. Numerical demonstrations are made on multiple networks to demonstrate the efficacy of the model and demonstrate important differences between the DUE and DSO NDP approaches. In addition, the flexibility of the approach is demonstrated by extending the formulation to account for demand uncertainty. This is formulated as a stochastic programming problem and initial test results are demonstrated on test networks. It is observed that not accounting for demand uncertainty explicitly, provides sub-optimal solution to the DUE NDP problem.