- Email: [email protected]
The use of large scale mathematical programming models in transportation systems
THE USE OF LARGE SCALE MATHEMATICAL PROGRAMMING MODELS IN TRANSPORTATION SYSTEMS LARRYJ. LEBLANC Department of Industrial Engineering and Operations Research, Southern Methodist University, Dallas, TX 75275, U.S.A.
LNTRODWIlON
The U.S. Bureau of Public Roads’ form of the travel time function is A,(x,) = oii + b,(x,)‘, and so the integration in (1) is easily performed. Problem (lH4) has a nonlinear, convex objective function with linear constraints. In LeBlanc (1973), LeBlanc, Morlok, Pier&alla (1975), and Nguyen (1973) the Frank-Wolfe algorithm is shown to be efficient for even very large equilibrium problems. In LeBlanc, Morlok, Pierskalla (1975),a problem with 1,824variables, 552 conservation of flow constraints and 1,824 nonnegativity constraints was solved in nine seconds. In Florian (1975), a successful application to a large problem in Winnipeg, Canada is described. In the equilibrium problem (l)-(4) above, the required amounts of traffic flow between each pair of nodes j and s, namely 9, were assumed to be constants. In the variable or elastic demand equilibrium problem, the quantities 0,. are taken to be functions of the travel time (at equilibrium) between nodes j and s : D,, = 4,(u,,),where u, is the travel time between nodes j and s. Beckmann (1956) has shown by Kuhn-Tucker analysis that the optimal flows for this problem are the equilibrium flows:
In recent years, a number of successful applications of nonlinear and integer programming models for transportation systems have been reported. Rigorous solution techniques have been used to find globally optimal solutions for extremely large nonlinear and integer optimization models. Notable examples are network equilibrium problems (with fixed or variable demands), urban network design problems, stochastic transportation problems, and aircraft routine problems. Efficient heuristics have also been employed to find very good, but not necessarily optimal, solutions. Heuristics have been successfully used to solve larger network design problems. THE NETWORK EQUILIBRIUM PROBLEM
network equilibrium problem is to determine how traffic will be distributed over the streets of an urban network. It is assumed that drivers will take the path with shortest driving time (or generalized time which includes other factors such as tolls, distance, etc.) between their origins and destinations. Because of trafhc congestion, the travel time on each street in the network is a nonlinear function of the traflic flow on the street. Because of the nonlinearity, the network equilibrium problem cannot be solved by finding the shortest path between each pair of nodes in the network and sending the required flow along each path. Beckmann (1956) has shown by examining Kuhn-Tucker conditions that the optimal flows for the following nonlinear network problem are the desired equilibrium flows: The
I
min C
0,
&j=zxfJ
+
c
xz
=
c t
xb
prcri, 0
subject to
‘u
vcrij 0
subject to
min C I=” A,(t)dt -c
A,(t)dt
(1)
allarcsij
(2)
all de&a~onss
(3)
allnodes j#s
xf z 0 all arcs ij, all destinations s
(4)
where xii = trtic
flow on arc ij. xii = t.dFic flow on arc ij with destination s.
Aij(x,,) = travel time per unit of flow on arc ij when the tKiflC flOW is Xij. 4. = required trafllc between nodes j and s.
Dj, r0
is
1” 0
W$(t)dr
(5) (6)
and also subject to (2x4). In (S), the second term is summed over all origin-destination pairs, j and s. In (5), the number of trips between nodes j and s, D,, is a decision variable; W,[D,,] is the inverse of the trip demand function 4, In (5), we are equivalently maximizing consumer surplus. The reader is referred to Beckmann (1956) for a further discussion. In (1974), Florian and Nguyen successfully applied generalized Benders decomposition (Geoffrin, 1972) to solve the elastic demand equilibrium problem. Generalized Benders decomposition is a technique for obtaining global solutions to an optimization problem by temporarily fixing certain complicating variables (the Di, in this case), solving the resulting problem, and then determining a better value for the complicating variables. This procedure is iterated until convergence to the optimal solution to the overall problem is obtained. The computational experience in Florian, Nguyen (1974)indicated that the variable demand equilibrium problem was only 2O-25% more difiicult to solve than the fixed demand problem.
419
420
L. J. TRE NETWORK DESIGN PROBLEM
The network design problem is a natural generalization of the network equilibrium problem in which we wish not only to predict tratlic flows, but also to determine improvements to a network to minimize traffic congestion. Improving a network includes increasing the capacity of existing streets by widening or resurfacing, and adding new streets or expressways. Ideally, the criterion for evaluating improvements to a network is the traffic congestion (or other measure) present in the equilibrium flows on the new network. A less desirable, but more computationally attractive, criterion is the congestion present in the system optimal flows. These are the flows that would result if some central controller could route all drivers between their origins and destinations in such a manner as to minimize total congestion in the network. However, this does not occur for automobile traffic: the equilibrium flows are intuitively a much closer predicter of actual traffic flows. In practice, the most common method of solving network design problems is probably to enumerate several possible networks, predict the traffic flows on each link, and choose the network with the least congestion. A more rigorous approach to the network design problem is to use piecewise linear approximations to the travel time functions. A linear programming model to choose flows and capacity increases is then solved. This type of model essentially replaces each nonlinear arc with two or more linear arcs with fixed capacities. The model routes the necessary amount of flow between origins and destinations and chooses capacity increases to minimize congestion, subject to conservation of flow, capacity and a budget constraint. An application of this model is described in Morlok et al. (1973). The optimal solution to the linear program above will specify the optimal network configuration so that congestion at the system optimal flow is minimized. This is true because the model chooses flows to minimize congestion. Integer programming models for determining improvements to a network to minimize congestion at equilibrium are described in LeBlanc (1975). Ochoa-Rosso (1968). and Bruynooghe (1972). Although efficient branching strategies have been proposed for integer programming network design models. heuristics must be used for truly large network design problems. A discussion of effective heuristic techniques is found in IMorlok and LeBlanc (1975) and Steenbrink (1973). Steenbrink describes an application of his technique to a large Dutch road network. THESTOCHASTlC TRANSPORTATION PROBLEM The stochastic transportation problem is to choose quantities to be shipped from supply points to demand points when the requirements at destinations are random variables rather than known constants. Since demands are not known, if a certain quantity of material is shipped to some destination, then an expected holding cost and an expecred shortage cost is incurred. In the stochastic transportation problem, we wish to choose amounts to be shipped from each supply point to each demand point in
LEBLANC
order to minimize shipping costs (which are deterministic) plus expected holding and shortage costs. The problem is shown to be a convex nonlinear programming problem in Hadley (1964). We consider m supply points, each with a known supply of S,, i = 1.2,. . . mj we are also given II demand points, each with demand Di, where D, is a mndom variable with density function di(u). We then wish to choose shipments x,,from supply points to demand points to
+P~J:(u-Y~)~~(U)~U] (7) subject to xii=0
i=1,2 ,..., m; Yj=g&j
j=l,2
,..., n
(8)
j=l,2,...,n i-1
$x,i=Si
i=l,2,...,m
(10)
where cii = unit shipping cost from supply point i to destination j. h, = unit holding cost at destination j. I#+(u) = probability density function for demand at destination j. pi = unit shortage cost at destination j. Si = supply at i. In (7), y, is the total amount shipped into destination j from all sources, and so J$(yi - r ) bi (u)du is the expected amount of material which must be held at destination j. Similarly, E (u - yi)4i(u)du is the expected shortage. Therefore the expected holding cost at destination j is h, Id; (y, - u)di(u)du and the expected shortage cost is pi Iy;i(u - y,Mu)du. Thus we are minimizing shipping costs plus expected
holding and shortage costs. Constraints (9) are definitional constraints relating y,, the total amount shipped into demand point j, to the shipments x,~ Constraints (10) are supply constraints for each supply point i. Computational results in Cooper, LeBlanc (1974) include the solution of 14 probiems with 5000 variables, 5000 constraints (8). 200 definitional constraints (9) and 25 constraints (4) in less than 50 seconds each on a CDC Cyber 70. The aircraft routing problem is to determine the routing of individual aircraft to meet a fixed schedule of departure and arrival times at minimum cost. This problem has been formulated as an ordinary linear assignment problem. Each scheduled arrival is treated as a “person” which can be assigned to any “job”, i.e. a scheduled departure. To prevent impossible assignments (such as assigning an aircraft which arrives at 5:OOo’clock in New York to a Right which departs New York at 4:OOo’clock) these costs are set equal to r. This model has been used at Braniff Airlines (PateI, 1975) for routing purposes. REFl3RENCF.S
Beckmann
M.,
McGuireC. B.
the Economics NW
of
and Winsten C. (1956) Studies in
Trmsporfafion.
Haven, Connecricur.
Yale
University
Press,
The use of large scale mathematical programming models in transportation Bruynooghe M. (1972) An Optimal Mefhod of Choice oj Inoesrmenrs in a Transporf Network. PTRC Proceedings. Cooper L. and LeBlanc L. J. (1974) An Eficienf Algorirhm for Large Scale Convex Network Problems. Technical Report No. CP-74007, Department of Industrial Engineering and Operations Research. Southern Methodist University, Dallas, Texas. Florian M. and Nguyen S. (1974) A method for computing network equilibrium with elastic demands. Transpn Sci. g(4). Florian M. and Nguyen S. (1976) An application and validation of equilibrium trip assignment methods. Transpn Sci. To be published. Geoffrion A. M. (1972) Generalized Benders decomposition. J. Optimization Theory and Application 10(4). Hadley G. (1964) Nonlinear and Dynamic Programming. AddisonWesley, Reading. MA. LeBlanc L. J. (1973) Mathematical Programming Algorifhms for Large Scale Network Equilibtium and Nefwork Design Problems. Ph.D. Dissertation, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois. LeBlanc L. J., Morlok E. K. and Pierskalia W. P. (1975) An eflicient approach to solving the road network equilibrium
systems
421
traffic assignment problem. Transpn Res. 9. 309-318. LeBlanc L. J. (1975) An algorithm for the discrete network design problem. Transpn Sci. 9, 183-199. Morlok E. K. et a/. (1973) Development and Application of a Highway Networli Design Model. Final Report: FHWA Contract Number DOT-FH-Il. Northwestern University. Morlok E. K. and LeBlanc L. J. (1975) A Marginal Analysis Technique for Determining improvements to an Urban Road Network. Presented at the ORSA/TIMS meeting, Las Vegas. Nguyen S. (1973) A Mathematical Programming Approach to Equilibrium Methods of Trafic Assignment with Fixed Demands, Publication No. 138, Department d’informatique, University of Montreal. Ochoa-Rosso F. (1968) Optimum project addition in urban transportation networks via descriptive trafhc assignment models. Search and Choice in Transporf Systems Planning, Vol. V. MIT Press. Cambridge, MA. Pate1 C. (1975) Manager, Operations Research. Braniff Intemational Airlines. private communication. Steenbrink P. A. (1974) Transport network optimization in the Dutch integral transportation study. Transpn Res. 8. 1l-27.
LNTRODWIlON
The U.S. Bureau of Public Roads’ form of the travel time function is A,(x,) = oii + b,(x,)‘, and so the integration in (1) is easily performed. Problem (lH4) has a nonlinear, convex objective function with linear constraints. In LeBlanc (1973), LeBlanc, Morlok, Pier&alla (1975), and Nguyen (1973) the Frank-Wolfe algorithm is shown to be efficient for even very large equilibrium problems. In LeBlanc, Morlok, Pierskalla (1975),a problem with 1,824variables, 552 conservation of flow constraints and 1,824 nonnegativity constraints was solved in nine seconds. In Florian (1975), a successful application to a large problem in Winnipeg, Canada is described. In the equilibrium problem (l)-(4) above, the required amounts of traffic flow between each pair of nodes j and s, namely 9, were assumed to be constants. In the variable or elastic demand equilibrium problem, the quantities 0,. are taken to be functions of the travel time (at equilibrium) between nodes j and s : D,, = 4,(u,,),where u, is the travel time between nodes j and s. Beckmann (1956) has shown by Kuhn-Tucker analysis that the optimal flows for this problem are the equilibrium flows:
In recent years, a number of successful applications of nonlinear and integer programming models for transportation systems have been reported. Rigorous solution techniques have been used to find globally optimal solutions for extremely large nonlinear and integer optimization models. Notable examples are network equilibrium problems (with fixed or variable demands), urban network design problems, stochastic transportation problems, and aircraft routine problems. Efficient heuristics have also been employed to find very good, but not necessarily optimal, solutions. Heuristics have been successfully used to solve larger network design problems. THE NETWORK EQUILIBRIUM PROBLEM
network equilibrium problem is to determine how traffic will be distributed over the streets of an urban network. It is assumed that drivers will take the path with shortest driving time (or generalized time which includes other factors such as tolls, distance, etc.) between their origins and destinations. Because of trafhc congestion, the travel time on each street in the network is a nonlinear function of the traflic flow on the street. Because of the nonlinearity, the network equilibrium problem cannot be solved by finding the shortest path between each pair of nodes in the network and sending the required flow along each path. Beckmann (1956) has shown by examining Kuhn-Tucker conditions that the optimal flows for the following nonlinear network problem are the desired equilibrium flows: The
I
min C
0,
&j=zxfJ
+
c
xz
=
c t
xb
prcri, 0
subject to
‘u
vcrij 0
subject to
min C I=” A,(t)dt -c
A,(t)dt
(1)
allarcsij
(2)
all de&a~onss
(3)
allnodes j#s
xf z 0 all arcs ij, all destinations s
(4)
where xii = trtic
flow on arc ij. xii = t.dFic flow on arc ij with destination s.
Aij(x,,) = travel time per unit of flow on arc ij when the tKiflC flOW is Xij. 4. = required trafllc between nodes j and s.
Dj, r0
is
1” 0
W$(t)dr
(5) (6)
and also subject to (2x4). In (S), the second term is summed over all origin-destination pairs, j and s. In (5), the number of trips between nodes j and s, D,, is a decision variable; W,[D,,] is the inverse of the trip demand function 4, In (5), we are equivalently maximizing consumer surplus. The reader is referred to Beckmann (1956) for a further discussion. In (1974), Florian and Nguyen successfully applied generalized Benders decomposition (Geoffrin, 1972) to solve the elastic demand equilibrium problem. Generalized Benders decomposition is a technique for obtaining global solutions to an optimization problem by temporarily fixing certain complicating variables (the Di, in this case), solving the resulting problem, and then determining a better value for the complicating variables. This procedure is iterated until convergence to the optimal solution to the overall problem is obtained. The computational experience in Florian, Nguyen (1974)indicated that the variable demand equilibrium problem was only 2O-25% more difiicult to solve than the fixed demand problem.
419
420
L. J. TRE NETWORK DESIGN PROBLEM
The network design problem is a natural generalization of the network equilibrium problem in which we wish not only to predict tratlic flows, but also to determine improvements to a network to minimize traffic congestion. Improving a network includes increasing the capacity of existing streets by widening or resurfacing, and adding new streets or expressways. Ideally, the criterion for evaluating improvements to a network is the traffic congestion (or other measure) present in the equilibrium flows on the new network. A less desirable, but more computationally attractive, criterion is the congestion present in the system optimal flows. These are the flows that would result if some central controller could route all drivers between their origins and destinations in such a manner as to minimize total congestion in the network. However, this does not occur for automobile traffic: the equilibrium flows are intuitively a much closer predicter of actual traffic flows. In practice, the most common method of solving network design problems is probably to enumerate several possible networks, predict the traffic flows on each link, and choose the network with the least congestion. A more rigorous approach to the network design problem is to use piecewise linear approximations to the travel time functions. A linear programming model to choose flows and capacity increases is then solved. This type of model essentially replaces each nonlinear arc with two or more linear arcs with fixed capacities. The model routes the necessary amount of flow between origins and destinations and chooses capacity increases to minimize congestion, subject to conservation of flow, capacity and a budget constraint. An application of this model is described in Morlok et al. (1973). The optimal solution to the linear program above will specify the optimal network configuration so that congestion at the system optimal flow is minimized. This is true because the model chooses flows to minimize congestion. Integer programming models for determining improvements to a network to minimize congestion at equilibrium are described in LeBlanc (1975). Ochoa-Rosso (1968). and Bruynooghe (1972). Although efficient branching strategies have been proposed for integer programming network design models. heuristics must be used for truly large network design problems. A discussion of effective heuristic techniques is found in IMorlok and LeBlanc (1975) and Steenbrink (1973). Steenbrink describes an application of his technique to a large Dutch road network. THESTOCHASTlC TRANSPORTATION PROBLEM The stochastic transportation problem is to choose quantities to be shipped from supply points to demand points when the requirements at destinations are random variables rather than known constants. Since demands are not known, if a certain quantity of material is shipped to some destination, then an expected holding cost and an expecred shortage cost is incurred. In the stochastic transportation problem, we wish to choose amounts to be shipped from each supply point to each demand point in
LEBLANC
order to minimize shipping costs (which are deterministic) plus expected holding and shortage costs. The problem is shown to be a convex nonlinear programming problem in Hadley (1964). We consider m supply points, each with a known supply of S,, i = 1.2,. . . mj we are also given II demand points, each with demand Di, where D, is a mndom variable with density function di(u). We then wish to choose shipments x,,from supply points to demand points to
+P~J:(u-Y~)~~(U)~U] (7) subject to xii=0
i=1,2 ,..., m; Yj=g&j
j=l,2
,..., n
(8)
j=l,2,...,n i-1
$x,i=Si
i=l,2,...,m
(10)
where cii = unit shipping cost from supply point i to destination j. h, = unit holding cost at destination j. I#+(u) = probability density function for demand at destination j. pi = unit shortage cost at destination j. Si = supply at i. In (7), y, is the total amount shipped into destination j from all sources, and so J$(yi - r ) bi (u)du is the expected amount of material which must be held at destination j. Similarly, E (u - yi)4i(u)du is the expected shortage. Therefore the expected holding cost at destination j is h, Id; (y, - u)di(u)du and the expected shortage cost is pi Iy;i(u - y,Mu)du. Thus we are minimizing shipping costs plus expected
holding and shortage costs. Constraints (9) are definitional constraints relating y,, the total amount shipped into demand point j, to the shipments x,~ Constraints (10) are supply constraints for each supply point i. Computational results in Cooper, LeBlanc (1974) include the solution of 14 probiems with 5000 variables, 5000 constraints (8). 200 definitional constraints (9) and 25 constraints (4) in less than 50 seconds each on a CDC Cyber 70. The aircraft routing problem is to determine the routing of individual aircraft to meet a fixed schedule of departure and arrival times at minimum cost. This problem has been formulated as an ordinary linear assignment problem. Each scheduled arrival is treated as a “person” which can be assigned to any “job”, i.e. a scheduled departure. To prevent impossible assignments (such as assigning an aircraft which arrives at 5:OOo’clock in New York to a Right which departs New York at 4:OOo’clock) these costs are set equal to r. This model has been used at Braniff Airlines (PateI, 1975) for routing purposes. REFl3RENCF.S
Beckmann
M.,
McGuireC. B.
the Economics NW
of
and Winsten C. (1956) Studies in
Trmsporfafion.
Haven, Connecricur.
Yale
University
Press,
The use of large scale mathematical programming models in transportation Bruynooghe M. (1972) An Optimal Mefhod of Choice oj Inoesrmenrs in a Transporf Network. PTRC Proceedings. Cooper L. and LeBlanc L. J. (1974) An Eficienf Algorirhm for Large Scale Convex Network Problems. Technical Report No. CP-74007, Department of Industrial Engineering and Operations Research. Southern Methodist University, Dallas, Texas. Florian M. and Nguyen S. (1974) A method for computing network equilibrium with elastic demands. Transpn Sci. g(4). Florian M. and Nguyen S. (1976) An application and validation of equilibrium trip assignment methods. Transpn Sci. To be published. Geoffrion A. M. (1972) Generalized Benders decomposition. J. Optimization Theory and Application 10(4). Hadley G. (1964) Nonlinear and Dynamic Programming. AddisonWesley, Reading. MA. LeBlanc L. J. (1973) Mathematical Programming Algorifhms for Large Scale Network Equilibtium and Nefwork Design Problems. Ph.D. Dissertation, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois. LeBlanc L. J., Morlok E. K. and Pierskalia W. P. (1975) An eflicient approach to solving the road network equilibrium
systems
421
traffic assignment problem. Transpn Res. 9. 309-318. LeBlanc L. J. (1975) An algorithm for the discrete network design problem. Transpn Sci. 9, 183-199. Morlok E. K. et a/. (1973) Development and Application of a Highway Networli Design Model. Final Report: FHWA Contract Number DOT-FH-Il. Northwestern University. Morlok E. K. and LeBlanc L. J. (1975) A Marginal Analysis Technique for Determining improvements to an Urban Road Network. Presented at the ORSA/TIMS meeting, Las Vegas. Nguyen S. (1973) A Mathematical Programming Approach to Equilibrium Methods of Trafic Assignment with Fixed Demands, Publication No. 138, Department d’informatique, University of Montreal. Ochoa-Rosso F. (1968) Optimum project addition in urban transportation networks via descriptive trafhc assignment models. Search and Choice in Transporf Systems Planning, Vol. V. MIT Press. Cambridge, MA. Pate1 C. (1975) Manager, Operations Research. Braniff Intemational Airlines. private communication. Steenbrink P. A. (1974) Transport network optimization in the Dutch integral transportation study. Transpn Res. 8. 1l-27.