OD-Matrix Estimation using Stable Dynamic Model

Transcription

1 OD-Matrix Estimation using Stable Dynamic Model Yuriy Dorn (Junior researcher) State University Higher School of Economics and PreMoLab MIPT Alexander Gasnikov State University Higher School of Economics and IITP RAS May 31, 2015 Best Paper by a Junior Researcher Prize Abstract In this work we propose a new equilibrium-based origin-destination matrix estimation model. We also propose a new equilibrium-based model to describe mutual dependence of network users and network developers. 1 Introduction In this work we consider two closely related problems. The first one is the origindestination matrix (OD-matrix, traffic matrix) estimation with respect to user equilibrium traffic assignment. The second one is the modeling of developers actions with respect to network users actions. The second problem is closely related to the first one. We assume that network users can change their ODpair and paths from origin to destination and that the payoff of the developers depends on the user network equilibrium. Usually one use simple iteration scheme to solve this problem. In this scheme at each iteration one evaluates a new OD-matrix estimation based on the cost matrix estimated on the previous iteration. After this the OD-matrix estimate is used futher to compute user equilibrium and the corresponding cost matrix. For OD-matrix estimation with a given cost matrix the gravity and the entropy models are used [9, 6]. The corresponding OD-matrix estimates can be considered as the stochastic equilibrium in a game, where agents (network users) can swap (with some probability) their origin or destination nodes with each other aiming to reduce their transportation costs. First author: The research results presented in the paper have been supported by a grant RFBR res.projects ofi-m, mol-a-ved. Second author: The research was conducted in IITP RAS and supported solely by the Russian Science Foundation grant (project ). 1

2 There are also a few models which allow to estimate OD-matrix and equilibrium flow as the solution of a single optimization problem [3, 10]. In all these models the OD-matrix estimate is obtained as a solution of the entropy maximization problem and can not be explicitly used to model of the developers actions. In this work we introduce a new equilibrium-based OD-matrix estimation model. In this model we does not compute the OD-matrix estimate directly. These model allows to include a developers actions in it. In our model the solution corresponds to the strict equilibrium, where users can change their OD-pair and paths in network, and developers can change the arc capacities. The underlying network assignment model is the Stable Dynamic model. To the best of our knowledge this is the first time, when the Stable Dynamic model is used for the OD-matrix estimation problem. In the real-life problems not only a OD-matrix depends from the user equilibrium, but also graph topology and arc properties, such as arc capacity and arc base travel cost. Usually to model these cases one use random graph models or closely related to them. This kind of models seems not to be realistic. In practice the developers try to predict the demands of customers and their preferences. In this work we propose a new equilibrium-type model, where the solution correspond to the best choice of the developers with respect to the user equilibrium. To the best of our knowledge this is the first time, when a developers equilibrium, a user equilibrium and a corresponding OD-matrix can be obtained as a solution of a single optimization problem. 2 OD-matrix estimation using stable dynamic model We consider the following problem. Let Γ(V, E) be a weighted oriented graph. Here V is the set of nodes, E is the set of arcs. The {d ij } is an unknown ODmatrix. Duplet (Γ(V, E), {d ij }) is called instance. Assume below that for every node i we know values of parametrs L i and W i : d ij = L i j d ji = W i j The triplet (Γ(V, E), L, W ), which correspond to instance (Γ(V, E), {d ij }), is called setup. Denote by f ({d ij }) (f R E ) the equilibrium traffic flow vector for instance (Γ(V, E), {d ij }) (whatever equilibrium model we use). Component f e is flow on arc e at equilibrium. Usually OD-matrix depends on equilibrium traffic flow vector {d ij } = {d ij }(f ). Denote by τ vector of arc weights (costs). Usually arc weight depend on arc flow τ e = τ e (f e ). 2

3 The standart problem is to find the agreed OD-matrix and the equilibrium flow (f, {d ij }), such that the following relations are hold: {d ij} = {d ij }(f ), f = f ({d ij }), d ij = L i, j d ji = W i. j (1) There are many solution concepts for this problem. Usually so called multistage (actually, 4-stage) transportation models are used [6, 2]. These models use simple iteration methods to solve (1). More precisely on the step k one compute {d ij } k = {d ij }(fk 1 ), after one follows model split step, and on the transportation assignment step one compute fk = f ({d ij } k ). After that we take k := k + 1 and start new iteration. To describe the dependence of the ODmatrix from the equilibrium flow vector f (more precisely, from corresponding to f cost matrix) gravitational or entropy models are used [6]. For the transportation assignment step one usually use Beckmann s model [1] or the Stable Dynamic model, proposed by Nesterov and de Palma [5]. In practice, the standart stopping criteria for multistage models is simple bound for number of iterations. One of drawbacks of multistage models is that we can not garantee it convergence. Hence any garantee sertificates for the solution can not be provided. There are many types of models describing the dependency of OD-matrix from current flow. Usually the gravity and the entropy models are used [9, 6]. The gravity model can be considered as a special case of the entropy model. The solution of the entropy model can be considered as the stochastic equilibrium in game, where agents (network users) can swap (with some probability) their origin or destination nodes with each other aiming to reduce their transportation costs. Now we introduce new model where estimated OD-matrix correspond to the (strict) equilibrium in some underlying congestion game. 2.1 Equilibrium approach to OD-matrix estimation We propose the following approach to estimation of the agreed OD-matrix and the equilibrium flow vector. Denote by s and t two dummy vertices (super-origin and super-destination respectively), and two sets of dummy arcs E s = {(s, i), i V } and E t = {(j, t), j V }. Denote by Γ ext (V {s} {t}, E E s E t ) extended graph. Denote by i L i = j W j = N. Denote proxy OD-matrix {d e ij } in the following fashion: { d e N, if (i, j) = (s, t), ij = 0 if (i, j) (s, t). 3

4 We propose to find the solution of (1) as the Wardrop equilibrium [8] for the instance (Γ ext, {d e ij }) with respect to the following constraints: { fij L i, if (i, j) = (s, i), = W j if (i, j) = (j, t). Note that we could use any good equilibrium transportation model. For our purpose good equilibrium model is such a model, that leads to the potential game [4] with convex potential function Φ(τ). For example, Beckmann s and the Stable Dynamic models are good. 2.2 Stable Dinamic model Here we give a brief description of the Stable Dynamic model [5]. This is the equilibrium traffic assinment model. Each arc e is decribed by two parameters: the free flow travel time τ e (or, the arc base cost) and the arc capacity f e. The pair (f, τ) is called admissible if the following conditions holds: τ e = τ e, if f e < f e, τ e τ e, if f e = f e. To say it simple, admissible pair must satisfy capacity constraints and only when arc capacity is used, the arc becomes congested. Denote by P ij the set of the available paths from the node i to the node j. Denote by C q (τ) the cost of the path q. We assume that cost of the path q is equal to the sum of the costs on the included arcs: C q (τ) = e E τ e δ eq, where δ eq is equal to 1 if the path q contain the arc e and equal to 0 otherwise. Denote by ˆP ij the set of paths for OD-pair (i, j), which is used. Denote by T ij (τ) = min q P ij C q (τ) the costs on the shortest path for the OD-pair (i, j). The Wardrop s equilibrium condition require the satisfaction of the following conditions: C q (τ) = T ij (τ), q ˆP ij, C q (τ) T ij (τ), q P \ ˆP ij, At the Stable Dynamic model the admissible pair (f, τ ) is called equilibrium if the Wardrop s equilibrium condition holds. 4

5 To say it simple, at a equilibrium the capacity constraints holds and not even one network user have the opportunity to reduce the transportation cost by switching to any other path. The underlying game for the Stable Dynamic model is so called Congestion Game [7]. Any congestion game can be veiwed as a potential game [4]. Any equilibrium in potential game can be found as a solution of the corresponding optimization problem. Same stay true for the equilibrium in the Stable Dynamic model. The equilibrium pair (f, τ ) in the Stable Dynamic model for the instance (Γ(V, E), {d ij }) can be found as a solution of the following optimization problem: (i,j) d ij T ij (τ) e E f e (τ e τ e ) max (2) s.t. τ τ (3) The equilibrium flow is f = f s, where s is the optimal Lagrange multipliers for the constraints τ τ. The Stable Dynamic model have one important property. Suppose that for some subset of the arcs Υ the equilibrium flow is known (f e = ˆf e for all e Υ). Then we can use this information explicitly. In particular, the problem (2-3) with additional information can be rewritten as: d ij T ij (τ) (i,j) e E\Υ f e (τ e ˆτ e ) e Υ ˆf e (τ e τ e ) max (4) s.t. τ e τ e, e E \ Υ (5) 2.3 Traffic matrix estimation via equilibrium assignment Now we can implement our idea for computing the OD-matrix using the Stable Dynamic model. An agreed pair ({d ij }, f ) for the setup (Γ(V, E), L, W ) is the solution of the following optimization problem. max{nt st (τ) e E f e (τ e τ e ) L i (τ (s,i) τ (s,i) ) W j (τ (j,t) τ (j,t) )} (s,i) E s (j,t) E t (6) s.t. τ e τ e, e E, (7) where s is the dual multiplier for the constraint τ τ at optimum. f e s e, e E, fe = L i, e = (s, i), W j, e = (j, t). 5

6 This is an example of the more general idea. Denote by Φ({d ij }) the value of the potential function at the equilibrium for the instance (Γ(V, E), {d ij }). Below is an example for the Stable Dynamic model. Φ({d ij }) = max { d ij T ij (τ) f e (τ e ˆτ e )} τ τ (i,j) e E Theorem 2.1 Let (Γ(V, E), L, W ) be a given setup and (Γ ext, N) is its extension. Let the transportation assignment model for the instance ({d ij }, Γ) have the potential function Φ({d ij }), and for every dummy arc e the delay τ e is the convex function on flow τ e (f e ). Then the unique agreed pair (f, {d ij }) exists and determined by the solution of the following optimization problem: where ({d ij}) = arg min Ψ({d ij}), (8) {d ij} 0 Ψ({d ij }) = Φ({d ij }) + (s,i) τ (s,i) ( j d ij ) + (j,t) τ (j,t) ( i d ij ) The problem (6-7) is dual for (8). In our approach the OD-matrix can not be computed explicitly. Nevertheless in many practical cases there is no need to compute it explicitly. This drawback is compensated by the fact, that we can explicitly compute dual multipliers (shadow prices) for dummy arcs. They represent cost correspond to particular vertice. For instance they can represent financial costs (parking tax, housing rents, average wages in area) or time costs (average parking time and so on). This model can also be used to modeling of developers actions. 3 Model with developers Previously a transportation network of a city were given. Now we introduce a new game-theoretic model which discribe the mutual influence of the network user equilibrium in a transportation network and developers. There are 2 n agents (developers). Each agent is assigned to the pair of indexes (i, j) from the set {(s, j) : j V } {(j, t) : j V }. Two agents can not be assigned to the same pair of indexes. Till now the parameters f, τ were fixed. Now we assume that there are new type of agents called developers. Each developer responsible for its own dummy arc e. We consider two cases. In the market base costs case developer of the arc e can choose the arc capacity f e and the arc base cost τ e. In the fixed base costs case developer of the arc e can choose only the arc capacity f e. In our model we assume that the topolody of the graph Γ(V, E) is fixed. The strategy of the developer (i, j) is the choice of the capacity f ij from the given closed and convex strategy set Ψ ij and the arc base cost τ ij from the given closed convex strategy set Φ ij. 6

7 Denote by Ψ = i,j Ψ ij. All elements of the set Ψ is not negative. Denote by Φ = i,j Ψ ij. All elements of the set Φ is not negative. Suppose that c e is the prime cost of the construction of one unit of the capacity for arc e. We assume that developers have no fixed costs. Then the choice f ij for the developer of the arc (i, j) leads to the total cost c ij f ij. After this the developer of the arc (i, j) can set the base cost (or price) τ ij. The total income for the developer will be τ ij f eq ij. Then the total payoff for the developer of the arc (i, j) will be F ij = τ ij f eq ij c ij f ij The aim of any developer is to maximize its own payoff. Note that the following problem is dual to the problem (2): min e f e τ e (9) h i = f f i O Ξh i = v i, h i 0, i O i O Here v i R m, v (j) i = d ij, v (i) i = j i d ij. Ξ R n m is the node-arc incidence matrix of the graph Γ. Ξ je = 1 if the arc e goes from the node j; Ξ je = 1 if the arc e goes to the node j; Ξ je = 0 in other case Market Base Costs Consider the case when the developer of arc (i, j) can choose not only the arc capacity f ij from the admissible set Ψ ij, but also the base arc cost τ ij from the admissible set Φ ij. We assume that Ψ ij and Φ ij is closed convex sets, all elements the sets is positive. Ψ = i,j Ψ ij, Φ = i,j Φ ij x 0, x Ψ, x 0, x Φ. Then the best strategy for the developer of the arc (i, j) is the solution of the following optimization problem: max τ ij f eq ij c ij f ij (10) f ij, τ ij f ij Ψ ij τ ij Φ ij Definition: The qwartet ( f, τ, f eq, τ eq ) is called equilibrium in the network equilibrium model with developers with market prices if (f eq, τ eq ) is the solution of the problem (6) and f as well as τ is the solution of the problem (10) with f = f eq. In other words there are no users or developers who can gain by switching their current strategy. 7

8 Theorem 3.1 The equilibrium in the network equilibrium model with developers for the instance (Γ(V, E), N) can be found as the solution of the following optimization problem: max f e, τ e:e E \E min f e E f e τ e 0 f f Ξf = v s f ij Ψ ij, τ ij Φ ij, e E \E (i, j) E \ E f e c e (11) and τ eq is the solution of a dual problem to (9). Proof: The problem (9) is dual to the problem (2) ([5]), so one can reconstruct from its solution the solution of the problem (2). Note that the solution of the problems (3.1) and (2) for variable f with fix f are equal because the additional term e E \E f e c e do not depend on f. The objective function for the problem (3.1) (after the minimization subproblem on variable f is solved) is equal to the sum of the payoffs of developers with respect to the user equilibrium. Note that the objective fuction for developers is separable. Then the solution of the problem (3.1) correspond to the solution of the problem (10). Thus the problem of finding equilibrium for the network equilibrium model with developers for the fixed prices case is reduced to a minimax saddle-point problem for a convex-concave objective function and can be solved by a standart optimization techniques for this kind of problems. 3.1 Fixed Base Costs There are some cases when the base arc costs is fixed. For example this case take place if a goverment plays the developer role and its price policy chase social or political goals. Then the best strategy for the developer of arc (i, j) is the solution of the following optimization problem: max τ ij f eq ij c ij f ij (12) f ij f ij Ψ ij Definition: The triplet ( f, f eq, τ eq ) is called equilibrium in the network equilibrium model with developers with fixed costs if (f eq, τ eq ) is the solution of the problem (6) and f is the solution of the problem (12) with f = f eq. 8

9 The equilibrium in the network equilibrium model with developers for the instance (Γ(V, E), N) can be found as the solution of the following optimization problem: max f e:e E \E min f f ij Ψ ij, e E f e τ e 0 f f Ξf = v s e E \E (i, j) E \ E f e c e (13) and τ eq is the solution of the dual problem to (9). Thus equilibrium can be found as the solution of the liniar programming optimization problem. 4 Concluding remarks In this work we present the equilibrium-based models for the OD-matrix estimation for instances with fixed and changing graph properties. Both models are very idealized and designed for a long-term modeling. On the other hand presented models are very flexible and allow to model the impact of many different socio-economic, city development and transportation policies in a uniform way. This include parking tolls polocy, number of storeys limitation and so on. References [1] Beckmann, M., B. McGuire, and C. Winsten. (1956). Studies in the Economics of Transportation. New Haven, CT: Yale University Press. [2] A. Gasnikov, Yu. Dorn, Yu. Nesterov, S. Shpirko. On the three-stage version of stable dynamic model // Matem. Mod., 26:6 (2014), [3] Fisk, C. S. On combining maximum entropy trip matrix estimation with user optimal assignment. Transportation Research Part B: Methodological 22.1 (1988): [4] D. Monderer and L. Shapley. Potential Games // Games and Economic Behavior. V. 14, P [5] A. de Palma, Yu. Nesterov, Stationary Dynamic Solutions in Congested Transportation Networks: sumary and Perspectives, Networks and Spatial Economics, 3, P , (2003) [6] Juan Ortuzar and Luis Willumsen. Modelling transport. John Wiley and Sons [7] Rosenthal, Robert W. A class of games possessing pure-strategy Nash equilibria. International Journal of Game Theory 2.1 (1973):

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