The Traffic Network Equilibrium Model: Its History and Relationship to the Kuhn-Tucker Conditions. David Boyce

Transcription

1 The Traffic Networ Equilibrium Model: Its History and Relationship to the Kuhn-Tucer Conditions David Boyce University of Illinois at Chicago, and Northwestern University 54 th Annual North American Meetings Regional Science Association International Savannah, November 8, 27 1

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3 Dramatis Personae, Harold Kuhn, 1925, born in California Ph.D. in Mathematics, Princeton University, 195 Lecturer in Mathematics, Princeton University, Professor of Mathematics, Princeton University, Albert Tucer, , born in Ontario, Canada Ph.D. in Mathematics, Princeton University, 1932 Professor of Mathematics, Princeton University,

4 Harold Kuhn, 1925-, USA Albert Tucer, , Canada 4

5 Importance of the Kuhn-Tucer Theorem Although the origins of constrained optimization are comple, etending bac to the formulation of Lagrange, Kuhn and Tucer (1951) Nonlinear Programming, decisively introduced this technique to economics, as well as operations research and fields of engineering, following WW II when the time was ripe for such methods; Kjeldsen (2) authored a fascinating historical account in Historia Mathematica; Taayama (1985) wrote in the Introduction to his tet, Mathematical Economics: nonlinear programming theory is probably the most important mathematical technique in modern economic theory. 5

6 Dramatis Personae, Martin Becmann, 1924, born in Germany Doctorate, Economics, University of Freiburg, Germany, 195 Research Associate, Cowles Commission, C. Bartlett McGuire, , born in Minnesota A. M., Economics, University of Chicago, 1952 Research Associate, Cowles Commission, Christopher B. Winsten, , born in England B. A., University of Cambridge, UK Research Associate, Cowles Commission, Tjalling C. Koopmans, , born in Netherlands Doctorate, Economics, University of Leiden, Netherlands, 1936 Research Director, Cowles Commission for Research in Economics, and Professor, University of Chicago, Later, Co-Recipient Nobel Prize in Economic Sciences in

7 Martin Becmann, 1924-, Germany 7

8 Bartlett McGuire, , USA 8

9 Christopher Winsten, , England 9

10 Tjalling Koopmans, , Netherlands 1

11 A Study of the Allocation of Resources During , the Cowles Commission for Research in Economics was the leading academic research center in mathematical economics and applications of mathematical programming to a broad range of problems in economics. Research was initiated in 1951 with support of the Rand Corporation on the Theory of Resources Allocation, with applications to transportation, location and population dispersal problems. Rand s main interest was railway capacity analysis, perhaps motivated by a desire to estimate the capacity of the USSR railway system. The research team wored on the application of activity analysis to transportation and location problems. A study of efficiency in road networs led to the discovery of 11 traffic networ equilibrium.

12 Becmann s Pioneering Achievement Represented a road networ with general lin cost functions and node conservation of flow constraints; Proposed a complementarity relationship for shortest route choices from an origin to a destination: if a route flow is positive, then its route cost must be a minimum, and if a route cost is not a minimum, its route flow must be zero (cf. Wardrop, 1952); Defined the general properties of a model of origindestination flow (demand) as a function of endogenously determined user-equilibrium route costs; Formulated a concave optimization problem whose solution incorporated both demand and route choice; Analyzed the properties of this formulation and the 12 related formulation for efficiency on road networs.

13 Significance of the Results Becmann was evidently the first economist to use the Kuhn- Tucer (1951) conditions as the basis for formulating an entirely new problem in economics; His formulation had major practical consequences: urban travel forecasting for road planning around the world; road pricing policy and design of congestion toll systems; Becmann may also have been the first to use the Kuhn-Tucer conditions to formulate a large-scale optimization problem in any field of engineering, maing a seminal contribution to the emerging field of operations research as well as engineering in general. 13

14 Objectives of This Tal Offer conjectures and insights into how Becmann achieved his result, described in Chapter 3 and 4, Studies in the Economics of Transportation; First, I review the Kuhn-Tucer (K-T) conditions; Then, I show how Becmann may have applied them to formulate his model. 14

15 Find The Kuhn-Tucer Conditions that maimizes ( ) g constrained by: f h ( ) h 1,..., m, = and. For to be a solution to the maimum problem, it is necessary that and someu satisfy conditions (1) and (2), and a certain constraint qualification: g m ( ) f ( ) i + h= 1 u h h i, i = 1,... n f h ( ), h = 1,... m (1) i g m ( ) f ( ) i + h= 1 u h h i =, i = 1,... n (2) u h [ ( )] f h =, h = 1,... m u 15

16 Setting up the Optimization Problem - 1 Let ( ) denote a directed lin connecting node i to node j, denotes the flow on lin ( ) terminating at node ; = ji = (, + j ) denotes the total flow on lin ( ) y ji y = denotes the cost of travel on lin ( ) (nondirectional) The definitions of lin flow and lin cost pertain to two-way roads. Node conservation of flow (based on Kirchhoff s Law): Let = ( ) i, j j, = j + j j, be the total flow from node i to node ; Or, flow out of node i = flow into node i + flow originating at node i 16

17 Setting up the Optimization Problem - 2 Choice of shortest routes (or least-cost route): If y i, is the least cost from node i to node, then y y + y j,, for all j. For every ( ) there is a unique value y i, such that y y j, y, with the equality holding for some j. Assuming travelers use shortest routes: if, >, then y y j, = y, and if y y j, < y, then, =, Therefore, ( y y y ) for all nodes j, and all node pairs., j, = For each pair ( ), then, all used routes have minimal and equal costs. Capacity or Lin Performance Function y = h be the capacity function, which is non-decreasing; Let ( ) where y is the travel cost on lin ( ) eperienced by each traveler. 17

18 Fied Demand Model Formulation Consider the following cost minimization problem: ma, s.t. 1 2 h ( ) (total cost of flow) ( ), j f, all ( ) (node conservation of flow) j where ( + ), for all ( ),, j and f i, is the fied flow from node i to node Define an auiliary function, now nown as the Lagrangean function: ( ) 1 ( ) ( ) L,, λ = h + λi ji f 2,,, j 18

19 If h ( ) (1) h ( ), we obtain the following K-T optimality conditions:,, [ h ( ) h ( ) + λ λ ] [ ( h ( ) h ( ) + λ λ )] j, j, = (2) λ, (, j ) j (, j ) j f f =, λ, The standard K-T interpretation follows: 1) if, >, λ, for all ( ) then = h ( ) + h ( ) 2) if, =, i, λ j, then h ( ) + h ( ) i, λ j, λ, for all ( ) λ as the cost of travel from node i to node j. where ( ) i, λ j, 19

20 This result shows an additional charge, or efficiency toll, must be placed on lin ( ) for the total cost of travel to minimized. Suppose we want the lin flows and costs without such tolls? This question must have confronted Becmann. To remove the unwanted term, he altered the objective function: ma, h ( ) d (, j ) fi, all ( ) j 1 2 s.t., where ( + ), all ( i j), j Defining the objective function as the integral over the lin flows yields the standard user-equilibrium result: 1) if, >, then = h ( ) 2) if, =, λ, for all ( ) i, λ j, then h ( ) i, λ j, λ, for all ( ) 2

21 Variable Demand Formulation Becmann also assumed variable origin-destination flow (demand) depends upon shortest route costs: = f ( y ): flow from i to is a function of the shortest route cost y, and independent of all other flows. For his formulation, Becmann needed the inverse demand function, which he stated as: y = g ( ), assumed to be strictly decreasing i To epand the fied demand formulation to include variable demand, we add the integral of the inverse demand function to the objective function, and change the node conservation of flows constraints to variable demand. ma (, ) i,, g 1 ( ) d h ( ) (, j ) i, all ( ) j i, s.t., 2 where +, all ( i j), j d 21

22 The corresponding K-T optimality conditions are: (1), [ g ( ) ] λi, [ g ( ) ] λi, = [ h ( ) + ] λi, λ j, ( h ( ) + λ λ ) [ ] ;, j, (2) = λ λ,, ( ), j j ( ), j j f f = From these K-T conditions, we have the following interpretations: 1) if >, 2) if =, λ, for all ( ), λ, for all ( ) ;,, for all (, ) j, j,, for all (, ) then = g ( ) = y then g ( ) = y, > then g ( ) ( ) ( ) i g j, j, = h then g ( ) g ( ) h ( ) 3) if, 4) if, =, 22

23 Conclusions The Kuhn-Tucer conditions were the basis for Becmann s formulation, including the representation of the user-equilibrium route choice conditions. The only prior application of the K-T conditions in economics found so far was by Dorfman (1951), who proved a nown result in a new way. One ey to Becmann s formulation was representing the shortest route assumption in the optimality conditions as a complementarity relationship. Then, he formulated an equivalent, or artificial, objective function that generated the desired results. 23

24 Later formulations of the problem changed to the lin-route representation of conservation of flow (Patrisson, 1994). Although this representation may be more intuitive, some insights may be lost. Similar formations involving the integral of a function are found in economics, and perhaps in theoretical mechanics, as Becmann hinted. Consumers surplus, discussed in Chapter 4, could be viewed in the same way, although a plausible interpretation is available. In such cases an interpretation of the objective function may not be possible, or it may need to be interpreted in a more comple framewor. An eample is the representative traveler interpretation proposed by Oppenheim (1995). 24