09 ndp_throughsvd (tristan2001)

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Network Design Through Sensitivity Analysis and

Singular Value Decomposition

Guido Gentile and Natale Papola

Dipartimento di Idraulica, Trasporti e Strade

University of Rome La Sapienza, Italy

THE PROBLEM

The Network Design Problem (NDP) consists in the seeking of a transportation networksupply configuration and demand flow pattern which jointly maximize a given objective

function of a social type, while satisfying the demand-supply equilibrium constraint. It is a

rather complicated problem because:

The objective function is typically non convex; The equilibrium problem, to be introduced as a constraint, makes the general problem

become of the bi-level type;

The number of design variables is typically high; There are generally present, among others, budget constraints and external effects are

to be considered.

OVERVIEW OF THE MAIN APPROACHES USED IN THE LITERATURE

TheNDPapproaches can be grouped into two categories: discrete NDPand continuous NDP.

The first ones deal with the topology of the network, while the second ones, for any given

topology, assume as design variables the characteristics of the elements of the network (e.g.,

capacities of road links, frequencies of transit line links, link tolls and fares, signal setting atlink intersections).

A discrete NDP can be solved through combinatorial optimisation techniques, for example of

the Branch & Boundtype as in Magnanti e Wong (1984), yielding the exact solution. The

limit of this approach is the dimension of the network, though the use of the optimisation

probabilistic methods opens new perspectives.

Referring to the continuous NDP, the various approaches utilized could be grouped into the

following four categories:

Game theory approach, where a group of individuals make separate decisions, but the

solution depends on the iterative effect of all their decisions as, for example, in LeBlanc andAbdulaal (1984), where an Iterative Optimization Assignment, consisting in iteratively

solving an equilibrium assignment problem, for a given specification of the design variables,

and an optimization of the design variables for a given traffic flow pattern, yields a non

cooperative Cournot Nash equilibrium, when existing. To be underlined that such a solution is

not necessarily Pareto-efficient.

Normative approach, where, assuming as design variables, among others, the link tolls,

possibly a different toll for each link of the network, a system optimum can be achieved,

which is the same (Bellei, Gentile and Papola 2000) as stating in a normative way the network

flow pattern, as in Steenbrink (1974).

Variational inequality approach, which consists in formalizing the demand-supplyequilibrium problem, here necessarily of the deterministic type, through a system of


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variational inequalities to be considered as constraints of the objective function, then solving

the resulting problem using specific methods like: constraint accumulation (Marcotte, 1983),

cone projection (Clune, Smith and Xiang, 1999) and Minty parameterization (Patriksson and

Rockafellar, 2000).

Implicit approach, which consists in defining a function yielding the network equilibrium

flows connected with a given specification of the design variables and expressing through

these the flows appearing in the objective function, so yielding a classic mono-level problem

which can be solved through the solution procedures of the non linear programming

problems. To be noticed that the calculation of the objective function needs an equilibrium

problem to be solved. Consequently, descent methods without derivative can be used, like the

Hooke - Jeeves or the Powellalgorithm, as in Abdulaal e LeBlanc (1979), or, alternatively,the sensitivity analysis is to be used in order to calculate the derivatives of the assignment as

in Tobin and Friesz (1988), Davis (1994) and Yang (1997).

OUR APPROACH TO THE PROBLEM

In this paper we deal with the optimization of an existing network using the implicit approach,which is a classic continuous NDP, making use of the sensitivity analysis. On this regard we,

firstly, extends the current results in the literature to the multi-user and multi-modal context

with elastic demand. Specifically, by making reference to persons trips, the equilibrium will

be formalized as a fixed point problem, employing, on the demand side, behavioral models

with elastic demand based on random utility theory, and, on the supply side, congested

networks with non-separable arc cost functions.

Secondly, given that the main difficulties when facing real size NDP in this context, arise

from the high number of the independentsearch directions (one for each design link), we

overcome this difficulty applying, to the Jacobian of the network loading map, the Singular

Value Decomposition (Golub and Van Loan 1996). This allows us to individualize subsets ofefficient directions, corresponding to those interventions having greater impacts, by filteringthe noise arising from non-efficient directions, corresponding to those interventions havingirrelevant impacts. We have a reason to be confident that the use of this technique while

limiting the calculus of the sensitivity to the efficient directions, will drastically reduce the

calculation needed to individualize a local optimum: a first result obtained from a network test

shows that only 3 or 4 ortonormal directions, out of 100 independent descent directions, cause

relevant impacts.

Thirdly, when calculating the gradient of the objective function, the sensitivity analysis of the

network loading will be used, instead of that of the network equilibrium. Moreover, with

reference to a nested logitdemand model, a calculation procedure of the sensitivity will beformulated with path implicit enumeration.

Finally, being the problem typically non-convex, in order to individualize the global

optimum, the classical calculation procedure will be coordinated with an heuristic, like the

taboo search, to avoid insisting along descent directions already explored.

References

Abdulaal M. , LeBlanc L .J. (1979) Continuous Equilibrium Network Design Models.

Transpn. Res. 13B, 19-32.

Bellei G. , Gentile G. , Papola N. (2000) Ottimizzazione del Trasporto Urbano in Contesto

Multiutente e Multimodo Mediante lIntroduzione di Pedaggi. In Metodi e Tecnologie

dellIngegneria dei Trasporti, ed.s G. E. Cantarella, F. Russo, Franco Angeli s.r.l. , Milano,Italia.


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Clune A. , Smith M. , Xiang Y. (1999) A Theoretical Basis for Implementation of a

Quantitative Decision Support System Using Bilevel Optimisation. Proceedings of the 14th

International Symposium on Transportation and Traffic Flow Theory, Jerusalem, Israel.Davis G. A. (1994) Exact Local Solution of the Continuous Network Design Problem via

Stochastic User Equilibrium Assignment. Transpn. Res. 28B, 61-75.

Fisk C. S. (1984) Game Theory and Transportation System Modelling. Transpn. Res. 18B,301-313.

Golub G. H. , Van Loan C. F. (1996) Matrix Computation, 3th ed. The John HopkinsUniversity Press Ltd., London.

LeBlanc L. J. , Abdulaal M. (1984) A Comparison of the User-Optimum versus System-

Optimum Traffic Assignment in Transportation Network Design. Transpn. Res. 18B, 115-

121.

Magnanti T. L. , Wong R. T. (1984) Network Design and Transportation Planning: Models

and Algorithms. Transpn. Sci. 18, 1-55.

Marcotte P. (1983) Network Optimization with Continuous Control Parameters. Transpn. Sci.17, 181-197.

Patriksson M. and R.T. Rockaleller (2000) A Mathematical Model and Descent Algorithm forBilevel Traffic Management. In Proceedings of the8

thMeeting of the Euro Working Group

Transportation EWGT, ed.s M. Bielli, P. Carotenuto, Roma, Italia.

Steenbrink P. A. (1974) Optimization of Transportation Networks. John Wiley & Son,London.

Tobin R. L. , Friesz T. L. (1988) Sensitivity Analysis for the Equilibrium Network Flow.

Transpn. Sci. 22, 242-250.

Yang H. (1997) Sensitivity Analysis for the Elastic-Demand Network Equilibrium Problem

with Applications. Transpn. Res. 31B, 55-70.

09 ndp_throughsvd (tristan2001)

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