﻿Numerical Approach of Network Problems in Optimal Mass Transportation

Applied Mathematics
Vol. 3  No. 5 (2012) , Article ID: 19057 , 10 pages DOI:10.4236/am.2012.35069

Numerical Approach of Network Problems in Optimal Mass Transportation

Lamine Ndiaye1, Babacar Mbaye Ndiaye1, Pierre Mendy1, Diaraf Seck1,2

1Laboratory of Mathematics of Decision and Numerical Analysis (LMDAN), FASEG, University of Cheikh Anta Diop, Dakar, Senegal

2Unité Mixte Internationale, UMMISCO, Institut de Recherche pour le Développement, Bondy, France

Email: babacar-mbaye.ndiaye@insa-rouen.fr

Received February 13, 2012; revised March 23, 2012; accepted March 31, 2012

Keywords: Optimal Mass Transportation; Network; Urban Traffic; Monge-Kantorovich Problem; Global Optimization

ABSTRACT

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.

1. 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.

2. 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 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 [6-10]. 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 and is absolutely continuous with respect to the Lebesgue measure.

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 having and as marginals, those which solve the minimization problem

The meaning of the following expressions and is explained respectively as follows:

and for any measurable subset of.

The Monge-Kantorovich problem obtained, depends only on the two distributions and, and the cost which may be a function of the path connecting to.

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;

·  the mass density at the position and time is a Borel measure supported on, i.e.;

·  the velocity field of a particle at is a Borel vectorial measures supported on;

·  the velocity field of the flow at is a Borel vectorial measures supported on and defined by .

The Monge-Kantorovich mass transportation problem consists in solving the following optimization problem:

(1)

with the constraints:

(2)

where is an integral functional on the -valued measures defined on. Note that (2) is the continuity equation of our mass transportation model.

2.1. 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;

·  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

where is the density of the population, is supposed to be convex, null at the origin and super linear (that is as). The increasing and diverging function represents the unhappiness to live in an area with population density.

Thus we define a functional which penalizes sparse services. This functional is of the form:

where is supposed to be concavenull with infinite slope at the origin (i.e. as ). Every single term in the sum above represents the cost for building and managing a service pole of size located at the point of. The Report is the productivity of pole of size.

So in the first model, the optimal urban design problem becomes the following optimization problem:

(3)

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, i.e.

In order to take into account the congestion effects, we suppose that the transportation cost per resident at the point depends on the traffic intensity at, i.e.

where is an increasing function.

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 of the first model to the local lower semicontinuous functionals on.

For more details, we refer the interested reader to the several recent papers on the subject (see for instance [4, 14-17]).

2.2. 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;

·  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 for which the average distance for a citizen to reach the mos t nearby point of the network is minimal.

Theorem 1 For every there exists an optimal network for the Optimization problem

where is the Hausdorff measure defined on.

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 domain says for example a town.

3. Approximation and Numerical Simulations

3.1. First Steps for the Formulation of the Discrete Problem

In this subsection, we are going to propose a first approach of discretization. From this, we deduce a generalized formulation but not the own possible formulation.

Let

For the simulation we are going to consider:

·  and;

·  such that;

·  a sequence of points and of balls such that

, with

, and.

and let and. We build an other map switching round cyclically the images of balls. Then satisfies

where and.

Then we have, i.e.

Therefore when, we obtain

(4)

Suppose that the map and the measure are regular, the Equation (4) leads

(5)

Then

is cyclically monotonous and.

At first, in problem () the objective is to find the points which minimize

subject to

In the next section, we show that it is quite possible to give a more general approximation .

3.2. New Reformulation Using Permutations

In the literature, many formulations (see for example [1-4]) have not yet practical applications which deal with the permutation of points. In this paper we introduce a new reformulation of the problem by introducing permutations.

Let us take a permutation defined on such that, we set:

and we solve the two following problems: ()

subject to

and problem ()

subject to

with the norm and

.

This is a theoretical formulation. And our aim is to apply it to a practical urban transport network. As a first step, we decided to work on with a reasonable number of points.

For a scenario in, if we consider points: the number of programs to be solved becomes. We leave the reader to verify that for:

·  m = 3 points we solve 27 programs

·  m = 4 points we solve 256 programs

·  etc.

A scenario involving up to 18 points is used for problem (). Using this scenario with problems () and () requires to solve programs, it is the reason we consider only some of these points for permutations in () and ().

3.3. Numerical Experiments

This section shows how the three models developed in the two previous Sections 3.1 and 3.2 are applied to real data of Dakar Dem Dikk (3D). Recall that 3D (see [20], [21]) is the main public urban transportation company in Dakar. This company manages a fleet of buses with different technical characteristics. Some of the buses can operate only in certain roads in the city center and the others can access in all over the network. Buses are parked overnight at Ouakam and Thiaroye terminals (see Figure 1).

To ensure network coverage, 3D manages its services by using 17 lines, with 11 from Ouakam terminal and 6 from Thiaroye terminal. Each line ensures a certain num-

Figure 1. Urban transportation network of 3D.

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 (terminals and bus stops) (see Figure 1);

·  latitude and longitude of points 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 = 5902.62 kilometers (for the 18 sites).

The GPS (Global Positioning System) coordinates are calculated with Google map, and then transformed into coordinates on the plane with he formula: degree + (minute/60) + (second/3600). Table 1 gives the coordinates of all points.

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:

with

Finally,

Table 1. Scenario of 3D.

and is added to the value of the objective function.

Constraint gives

.

Urban transportation network of 3D Thus, , i.e.:

And gives

i.e.:

with and.

We simply formulate the problem in AMPL [25] syntax, and solve the problem through the AMPL environment; with a total number of 38 variables for problem. The solution obtained is an optimal one (for each case) wherein the priority is assigned to the minimization of the distance between and. The IPOPT found an optimal point within desired tolerances; and we obtain the following results: the total number of iterations is 16 and for the optimal network we have

. The others obtained solutions, and are given in Table 2 with the GPS coordinates. The points and are represented in the network (see Figure 2).

Permutations make the resolution more complicated but can give better results. The number of sub-problems to solve depends on the number of points in the network. Thus, we choose points in 3D’s network.

Now, let us take the permutation which is the main idea of this work. For problem, such that, we have. Therefore some constraints are similar and we have 9 subproblems to solve. An illustration:

The three unconstrained sub-problems are obtained for with, i.e. in the three permutations (1,1,1), (2,2,2) and (3,3,3).

All sub-problems constraints are reported in Table 3.

Table 2. y1 and ym for optimal network Σopt.

Figure 2. Optimal network of 3D without permutation.

Table 3. All constraints with permutations.

It is sufficient to solve and; since we have, , , , , , , , and.

We only compute the quantities, and. For all sub-problems, the objective function is the same as problem with.

with

,

and

.

Finally,

and is added to the value of the objective function.

From computational results, we have obtained the same value for all 10 sub-problems. Thus, permutations do not influence the distance constraint on the curve of. The optimal value is for all with.

For problem, we consider and the number of possible permutation is. Recall that the number of sub-problems to solve depends on the number of points in the network. Also, we choose points in 3D’s network. Thus, for the considered permutation we have obtained a total of sub-problems to solve, see Table 4.

In order not to overload explanations, we develop only the sub-problem, the rest are left to the reader as an exercise.

In, the sub-problem is obtained for

with vectors and . For the objective function, we have

Table 4. Optimal values of αi with different permutations.

with

,

and

Finally,

and is added to the value of the objective function for all sub-problems

The constraint

gives

i.e.:

For the scenario of problems and, we choose x1 = Ouakam terminal, x2 = Thiaroye terminal and x3 = Leclerc (see Table 5).

We denote by and the optimal value and the optimal solution of sub-problem , respectively.

The results show that the following six sub-problems:, , , , , give the best value . The six solutions are different, i.e.:

with only.

The curve can be described by one of the points; see Figure 3 where

.

The points which define (with coordinates are given in Table 6, with.

The solutions giving the best possible permutations (optimum) are illustrated in Table 6 and include all permutations.

According to the simulations, we determine a set of optimal policy that can describe the optimal network. Finally: after comparison of the simulated models, we can deduce that the model for problem is better. It provides the best curve describing the optimal value, obtained with the permutations introduced in the objective function.

4. Conclusions

In this paper, we describe applications of mass transportation theory and develop how to optimize the curve design of urban network problems. Using the discrete formulations, we give three nonlinear programming problems with continue variables, and have described urban transportation problem of 3D applied to these three models. The results have shown that the optimal network is obtained with permutations including.

In future works, we will study an application in and make a reformulation that solves a unique program,

Figure 3. Optimal network Σopt.

Table 5. The network Σopt with permutations.

Table 6. The optimal solutions with permutations.

5. Acknowledgements

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.

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