Numerical Approach of Network Problems in Optimal Mass Transportation

In this paper, we focus on the theoretical and numerical aspects of network problems. For an illustration, we consider the urban traffic problems. And our effort is concentrated on the numerical questions to locate the optimal network in a given domain (for example a town). Mainly, our aim is to find the network so as the distance between the population position and the network is minimized. Another problem that we are interested is to give an numerical approach of the Monge and Kantorovitch problems. In the literature, many formulations (see for example [1-4]) have not yet practical applications which deal with the permutation of points. Let us mention interesting numerical works due to E. Oudet begun since at least in 2002. He used genetic algorithms to identify optimal network (see [5]). In this paper we introduce a new reformulation of the problem by introducing permutations  . And some examples, based on realistic scenarios, are solved.


Introduction
In this paper we present some models of urban planning.These models are examples of applications in mass transportation theory.They describe how to optimize the design of urban structures and their management under realistic assumptions.The paper is organized as follows: in Section 2 we present at first some urban planning models and preliminaries.The Section 3 is devoted to the approximation of the models; and numerical simulations that are our main results.Finally, summary and conclusions are presented in Section 4.

Preliminaries and Mathematical Modeling
Given two distributions  and  on with equal total mass, the classical generic Monge transportation problem consists in finding among all the maps d verifying for any measurable set in , those which solve the minimization problem: These maps are said to be transportation maps; they transport a measure  (quantity) to a measure  .
For the existence of solutions, we recommend to see [6][7][8][9][10].We invite the reader to see the books written in this topic by Villani [11,12] for additional information.
In particularly Sudakov have studied in [13] the existence of optimal map transportation when   When the unknowns of the problem are the distributions  and  , the Monge-Kantorovich mass transportation problemcan be interpreted as an optimal urban design problem.When the unknown is the transportation network, we have an irrigation problem, that is an optimal design of public transportation networks.We also mention the dynamic formulation of mass transportation given in [4,14,15] and generalized in [16].In this framework, we consider that:  the space of measures acting is a time-space domain where the urban area  is a bounded Lipschitz open subset with outward normal vector n  ;  the mass density   , t x  at the position x and time t is a Borel measure supported on Q , i.e.

 
The Monge-Kantorovich mass transportation problem consists in solving the following optimization problem:  (1)   with the constraints: where is an integral functional on the  1 d   -valued measures defined on Q .Note that (2) is the continuity equation of our mass transportation model.

Optimal Urban Design
In the models of optimal design of an urban area we considered that  the urban area  is a well known regular compact subset of d  ;  the total population and the total production are fixed data of the problem;  only the density of residents  and the density of services  are unknowns data of the problem.The aim is to find the density of residents  and the density of services  minimizing the transportation cost.
Principally there are two models for studying the optimal urban design.The first one takes into account the following facts:  there is a transportation cost for moving from the residential areas to the services poles;  people do not desire to live in areas where the density of population is too high;  services need to be concentrated as much as possible, in order to increase efficiency and decrease management costs.
The transportation cost will be described through a Monge-Kantorovich mass transportation model.
In particularly, we will take it as the -Wasserstein distance defined by: Taking into account the total unhappiness of residents due to high density of population, we define a penalization functional of the form  So in the first model, the optimal urban design problem becomes the following optimization problem: In the second model the population transportation is considered as a flow, that is a vector field .The equilibrium condition is achieved when the emerging flow is the excess of the demand in 2 :   In order to take into account the congestion effects, we suppose that the transportation cost per resident at the point x depends on the traffic intensity at x , i.e.
Then, the transportation cost moving  to  is: The problem (1) with the constraints (2) allows both to take into account the congestion effects by an appropriated choice of the functionals and to widen the choice of unhappiness function and management cost function For more details, we refer the interested reader to the several recent papers on the subject (see for instance [4,[14][15][16][17]).

Network Problems Applied to the Urban Transportation
In the models of optimal design of an urban area we considered that  the urban area  is a well known regular compact subset of d  ;  the density of residents  and the density of services  are two well known positives measures with equal mass.
The irrigation problem consists to find among all feasible structures (or feasible network) those that minimize the transportation cost The particular irrigation problem of the average distance consists to find an optimal network opt for which the average distance for a citizen to reach the mos t nearby point of the network is minimal.
For every there exists an optimal network says for example a town.

pr ical Simulations the blem
rst approa rom this, we deduce a generawhere is the Hausdorff measure defined on .
1  Notice that there are other proposition of functionals to be minimized.
For more details, we refer the interested reader to the several recent papers on the subject (see for instance [1-3, 18,19]).
Our aim is to concentrate our effort on the numerical questions to locate the optimal network in a given do-

First Steps for the Formulation of Discrete Pro
In this subsection, we are going to propose a fi ch of discretization.F lized formulation but not the own possible formulation. Let For the simulation we are going to consider:  and of balls where and Therefore when , we obtain 0 Suppose that the map and the me In the next section, we show that it is quite possible to give a more general approximation .

New Reformulation Using Permutations
In the literature, many formulations (see for example [1][2][3][4]) have not yet practical applications which deal with th introducing permutations This is a theoretical formulation.And our aim is to apply d for pro-it to a practical urban transport network.As a first step, we decided to work on 2   with a reasonable number of points.
For a scenario in n  , if we consider m points: the number of pr e permutation of points.In this paper we introduce a new reformulation of the problem by ograms be solved beco s .We leave the reader to verify that for: Let us take a permutation  defined on   A scenario involving up to 18 points is use  l Experiments e models developed in programs, it is the reason we consider only some of these points for permutations in ( 2  ) and ( 3 ).

Numerica
This section shows how the thre the two previous Sections 3. ber of routes.At present, the total number of routes in the network is 289.First, the most important 18 sites of the network are identified.Thirty (30) permanent terminuses (terminals) and 810 bus stops are used (see Figure 1, where bus stops are not represented due to their size).The map in Figure 1 is obtained by using the software EMME [22].Table 1 gives the 18 sites, their latitude and longitude.
The data are based on the scenario of 3D; and the input data needed to use the models are the:  total length of the network kilometers;  number of points = 5902.62L k x (terminals and bus stops) (see Figure 1  latitude and longitude of points = 18 m ); k x representing the two terminals and bus stops.The total distance covered by all the buses from terminals to starting points of routes and from end points back to their terminals represents the total length of the network; and we have L = 5902.62kilometers (for the 18 sites).
The numerical experiments are executed:  on a computer: 2 × Intel(R) Core(TM)2 Duo CPU 2.00 GHz, 4.0 Gb of RAM, under UNIX system;  and by the software IPOPT (Interior Point OPTimization) 3.9 stable [23,24], running with linear solver ma27.
For the objective function, we have: tem) coordinates are calculated with Google map, and then transformed into coordinates on the plane with he formula: degree + (mi- , 0 x y x y x y x y e.: subroblems n:  nd we have 9  .There fore some constraints are p to solve.An illustratio The thr sub-prob for = y y  ee unconstrained lems are obtained ( (1), ( 2), ( 3)) . y coor three permutations ( All sub-problems constrai are reported i Table 2 1 and y m for optimal network Σ opt .

GPS dinates
y y 

6
(1,2,3) y y  10 (2,1,1) y y y y    2 y y  y y y y y y  27 (3,3,3) It is sufficient to and ince we , only com he quantities solve 1 2 3 4 5 6 7 11 14 , , , , , , , , P P P P P P P P P Finally, and is added to the value of the objecm computa results, we have obtained the same value for all 10 sub-problems.with

Conclusions
In this paper, we describe applications of mass transheory and n of urban netw programming problems scribed urban transpo of 3D applied to these three models.The results have shown that the optimal network is obtained with permutations including j uture works, we will and make a reformulation that solves a unique prog duced the obj portation t develop how to optimize the curve desig ork problems.Using the discrete formulations, we give three nonlinear wi iables, and have de th continue var rtation problem     , We would like to thank all DSI's (Division Système d'Information) members of 3D for their time and efforts for providing the data, and discussions related to the meaning of the data.


For studying the many cases where the Monge transportation problem doesn't give a solution, Kantorovich considered the relaxed version of the Monge problem.In this framework, the transportation problem consists in finding among all admissible measures : subset A of  .d The Monge-Kantorovich problem obtained, depends only on the two distributions  and  , and the cost which may be a function of the path connecting c x to .y points  we solve 27 programs  m = 4 points  we solve