Sensitivity analysis of stochastic user equilibrium and its application to delivery services pricing

International audienceIn e-commerce the delivery of products is a crucial part for the success of a e-shop. An efficient delivery system should offer various services and predict customersbehavior. The latter are influenced by the price of a delivery service, but also byits quality (congestion effect induced by customers’ choices) in a higly competitveenvironment. In this study, we introduce a bilevel model to optimize a deliverysystem. At the upper level, the provider control services’ tariffs. At the lower level,users react by choosing their delivery service according to a utility function whichincorporates the provider tariff and the congestion effect. We model the customers’reaction using Stochastic User Equilibrium (SUE). We also present a sensitivityanalysis for the SUE that gives explicit expression of the derivatives of customersdistribution with respect to services’ tariffs. Based on a local search that exploit thederivatives information, a new heuristic algorithm for the bilevel delivery servicespricing problem is developed and compared to others existing approachs

A continuous network design model in stochastic user equilibrium based on sensitivity analysis

The continuous network design problem (CNDP) is known to be difficult to solve due to the intrinsic properties of non-convexity and nonlinearity. Such kinds of CNDP can be formulated as a bi-level programme, in which the upper level represents the designer's decisions and the lower level the travellers' responses. Formulations of this kind can be classified as either Stackelberg approaches or Nash ones according to the relationship between the upper level and the lower level parts. This paper formulates the CNDP for road expansion based on Stackelberg game where leader and follower exist, and allows for variety of travellers' behaviour in choosing their routes. In order to solve the problem by the Stackelberg approach, we need a relation between link flows and design parameters. For this purpose, we use a logit route choice model, which provides this in an explicit closed-form function. This model is applied to two example road networks to test and briefly compare the results between the Stackelberg and Nash approaches to explore the differences between them

Second best toll and capacity optimisation in network: solution algorithm and policy implications

This paper looks at the first and second-best jointly optimal toll and road capacity investment problems from both policy and technical oriented perspectives. On the technical side, the paper investigates the applicability of the constraint cutting algorithm for solving the second-best problem under elastic demand which is formulated as a bilevel programming problem. The approach is shown to perform well despite several problems encountered by our previous work in Shepherd and Sumalee (2004). The paper then applies the algorithm to a small sized network to investigate the policy implications of the first and second-best cases. This policy analysis demonstrates that the joint first best structure is to invest in the most direct routes while reducing capacities elsewhere. Whilst unrealistic this acts as a useful benchmark. The results also show that certain second best policies can achieve a high proportion of the first best benefits while in general generating a revenue surplus. We also show that unless costs of capacity are known to be low then second best tolls will be affected and so should be analysed in conjunction with investments in the network

Solving Signal Control Problems with Second-Order Sensitivity Information of Equilibrium Network Flows

The equilibrium network signal control problem is represented as a Stackelberg game. Due to the characteristics of a Stackelberg game, solving the upper-level problem and lower-level problem iteratively cannot be expected to converge to the solution. The reaction function of the lower-level problem is the key information to solve a Stackelberg game. Usually, the reaction function is approximated by the network sensitivity information. This paper firstly presents the general form of the second-order sensitivity formula for equilibrium network flows. The second-order sensitivity information can be applied to the second-order reaction function to solve the network signal control problem efficiently. Finally, this paper also demonstrates two numerical examples that show the computation of second-order sensitivity and the speed of convergence of the nonlinear approximation algorithm

The Network Improvement Problem for Equilibrium Routing

In routing games, agents pick their routes through a network to minimize their own delay. A primary concern for the network designer in routing games is the average agent delay at equilibrium. A number of methods to control this average delay have received substantial attention, including network tolls, Stackelberg routing, and edge removal. A related approach with arguably greater practical relevance is that of making investments in improvements to the edges of the network, so that, for a given investment budget, the average delay at equilibrium in the improved network is minimized. This problem has received considerable attention in the literature on transportation research and a number of different algorithms have been studied. To our knowledge, none of this work gives guarantees on the output quality of any polynomial-time algorithm. We study a model for this problem introduced in transportation research literature, and present both hardness results and algorithms that obtain nearly optimal performance guarantees. - We first show that a simple algorithm obtains good approximation guarantees for the problem. Despite its simplicity, we show that for affine delays the approximation ratio of 4/3 obtained by the algorithm cannot be improved. - To obtain better results, we then consider restricted topologies. For graphs consisting of parallel paths with affine delay functions we give an optimal algorithm. However, for graphs that consist of a series of parallel links, we show the problem is weakly NP-hard. - Finally, we consider the problem in series-parallel graphs, and give an FPTAS for this case. Our work thus formalizes the intuition held by transportation researchers that the network improvement problem is hard, and presents topology-dependent algorithms that have provably tight approximation guarantees.Comment: 27 pages (including abstract), 3 figure

On characterizing the relationship between route choice behavior and optimal traffic control solution space

Explicitly including the dynamics of users' route choice behaviour in optimal traffic control applications has been of interest for researchers in the last five decades. This has been recognized as a very challenging problem, due to the added layer of complexity and the considerable non-convexity of the resulting problem, even when dealing with simple static assignment and analytical link cost functions. In this work we establish a direct behavioural connection between the different shapes and structures emerging in the solution space of such problems and the underlying route choice behaviour. We specifically investigate how changes in the active equilibrium route set exert direct influence on the solution space's structure and behaviour. Based on this result, we then formulate and validate a constrained version of the original problem, yielding desirable properties in terms of solution space regularity. © 2017 The Authors

The chemical reaction optimization approach to solving the environmentally sustainable network design problem

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A Genetic Algorithm Approach for Optimising Traffic Control Signals Considering Routing

It is well known that coordinated, area-wide traffic signal control provides great potential for improvements in delays, safety, and environmental measures. However, an aspect of this problem that is commonly neglected in practice is the potentially confounding effect of drivers re-routing in response to changes in travel times on competing routes, brought about by the changes to the signal timings. This article considers the problem of optimizing signal green and cycle timings over an urban network, in such a way that the optimization anticipates the impact on traffic routing patterns. This is achieved by including a network equilibrium model as a constraint to the optimization. A Genetic Algorithm (GA) is devised for solving the resulting problem, using total travel time across the network as an illustrative fitness function, and with a widely used traffic simulation-assignment model providing the equilibrium flows. The procedure is applied to a case study of the city of Chester in the UK, and the performance of the algorithms is analyzed with respect to the parameters of the GA method. The results show a better performance of the signal timing as optimized by the GA method as compared to a method that does not consider rerouting. This improvement is found to be more significant with a more congested network whereas under a relatively mild congestion situation the improvement is not very clear

Robust Evaluation for Transportation Network Capacity under Demand Uncertainty

As more and more cities in worldwide are facing the problems of traffic jam, governments have been concerned about how to design transportation networks with adequate capacity to accommodate travel demands. To evaluate the capacity of a transportation system, the prescribed origin and destination (O-D) matrix for existing travel demand has been noticed to have a significant effect on the results of network capacity models. However, the exact data of the existing O-D demand are usually hard to be obtained in practice. Considering the fluctuation of the real travel demand in transportation networks, the existing travel demand is represented as uncertain parameters which are defined within a bounded set. Thus, a robust reserve network capacity (RRNC) model using min–max optimization is formulated based on the demand uncertainty. An effective heuristic approach utilizing cutting plane method and sensitivity analysis is proposed for the solution of the RRNC problem. Computational experiments and simulations are implemented to demonstrate the validity and performance of the proposed robust model. According to simulation experiments, it is showed that the link flow pattern from the robust solutions to network capacity problems can reveal the probability of high congestion for each link