1.225J
1.225J (ESD 205) Transportation Flow Systems
Lecture 6
Introduction to Optimization
Prof. Ismail Chabini and Prof. Amedeo R. Odoni
Lecture 6 Outline
‰ Mathematical programs (MPs)
‰ Formulation of shortest path problems as MPs
‰ Formulation of U.O. traffic assignment as an MP
‰ Relationship between U.O. and S.O. traffic assignment
‰ Solving S.O. traffic assignment by hand
‰ Lecture summary
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Lecture 6, Page 2
1
Optimization: Mathematical Programs
‰ General formulation (n variables, m constraints):
min z ( x1 , x2 ,..., xn )
Objective function
Subject to (s.t.): g1 ( x1 , x2 ,..., xn ) ≥ b1 

g 2 ( x1 , x2 ,..., xn ) ≥ b2 
Feasible set

M

g m ( x1 , x2 ,..., xn ) ≥ bm 
( x1 , x2 ,..., xn )
‰
g j ( x1 , x2 ,..., xn ) ≥ b j
‰
‰ Notes:
: decision variables
: A constraint
Max f ( x1 , x2 ,..., xn ) = Min z(x) = − f ( x1 , x2 ,..., xn )
g ( x1 , x2 ,..., xn ) ≤ b ⇔ − g ( x1 , x2 ,..., xn ) ≥ −b
g ( x1 , x2 ,..., xn ) = b ⇔ g ( x1, x2 ,..., xn ) ≥ b and − g ( x1 , x2 ,..., xn ) ≥ −b
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Types of Mathematical Programs (MPs)
‰ Linear programs (LPs): objective function is linear, and
constraints are linear
‰ Non-linear programs (NLPs): objective function is linear.
(constraints are usually linear. Otherwise, there might be more than
one optimal solution (finding such a solution can be a very time
consuming task))
‰ If decision variables are further constrained to take integer values, a
linear program is an integer program
‰ If decision variables are constrained to take 0/1 values: an integer
program is an 0/1 integer program
‰ If some, but not all, variables are constrained to take integer values: a
linear program is called a mixed integer program
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2
MPs Are Tools for Transportation Analysts
‰ Various models and quantitative analysis questions in transportation
can be formulated as minimization (or maximization) problems:
• Shortest path problems
• Traffic assignment models in congested networks
• Signal setting problems
• Ramp-metering optimization
‰ Examples of questions related to modeling:
• Formulate a model as a mathematical program (MP) (there might
be more than one model for the same modeling question)
• Study the properties of the model (i.e. does the model possess one
or multiple solutions? Is it easy to find a solution?
• Find a “solution” to the model (One may settle for an approximate
reasonable solution, as it is not always possible (desirable) to find
an optimal solution in a reasonable amount of time)
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How to Solve Mathematical Programs?
‰ Graphically: This gives you a feel of what is happening
‰ By hand using a systematic analytical method
‰ Use your software such as Xpress-MP, GAMES, LINDO, CPLEX,
Excel:
• Software tools are computer implementations of systematic methods
‰ There are also specialized software for some transportation applications
• Examples: TRANSCAD and EMME/2 for static traffic assignment
(We have licenses of these software systems in the CTS Computing
lab)
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3
Shortest Path Problems As An LP: Example
‰ We want to find shortest paths from all nodes to Node 5
‰ Decision variables:
1, if arc (i, j ) is used
xij = 
otherwise
 0,
‰ cij , (i, j ) ∈ A is the cost of each arc
1
2
2
4
1
1
1
1
8
2
3
1
4
4
5
4
6
3
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Lecture 6, Page 7
AllAll-toto-One Shortest Path Problem As An LP
‰ Formulation:
min 4 x12 + 3 x13 + x24 + 2 x25 +
8 x32 + 2 x34 + 6 x35 + 4 x45
s.t.
x12 + x13 = 1
x24 + x25 − x12 − x32 = 1
x32 + x34 + x35 − x13 = 1
x45 − x24 − x34 = 1
− x25 − x35 − x45 = −4
xij ≥ 0, (i, j ) ∈ A
‰ Note: constraints “the decision variables must be integers” should
have been added. However, it is known theoretically that the above
LP possesses an integer solution, and tools exist to it.
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OneOne-toto-One Shortest Path Problem As An LP
‰ Formulation:
min 4 x12 + 3 x13 + x24 + 2 x25 +
8 x32 + 2 x34 + 6 x35 + 4 x45
s.t.
x12 + x13 = 1
x24 + x25 − x12 − x32 = 0
x32 + x34 + x35 − x13 = 0
x45 − x24 − x34 = 0
− x25 − x35 − x45 = −1
xij ≥ 0, (i, j ) ∈ A
‰ Note: we could have added the fact that the decision variables are
either 0 or 1. However, it is known theoretically that the above LP
possesses a 0-1 solution, and tools exist that provide such solution
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Lecture 6, Page 9
SO Static Traffic Assignment as an MP: Example
min x1t1 (x1 ) + x2t2 (x2 ) + x3t3 (x3 )
f1ac = qac
s.t .
bc
1
f
+f
bc
2
Demand
= qbc
f1ac ≥ 0, f1bc ≥ 0, f 2bc ≥ 0
Non - negativity
x1 = f1ac + f 2bc
x2 = f1bc
x3 = f
Definition of link flows
bc
2
x1
x2
x3
0
0
0
‰min x1t1 ( x1 ) + x2t 2 (x2 ) + x3t3 (x3 ) = min ∫ m1 (x1 )dx1 + ∫ m2 ( x2 )dx2 + ∫ m3 (x3 )dx3
where m1 (x1 ) =
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d ( x1t1 (x1 ))
d (x2t2 ( x2 ))
d (x3t3 ( x3 ))
, m2 (x2 ) =
, m3 ( x3 ) =
dx1
dx2
dx3
Lecture 6, Page 10
5
Objective Function For U.O. Traffic Assignment
t 2 ( x2 ) = 1 + 2 x2
Link 2, x2
5
5
D
O
t1 ( x1 ) = 2 + x1
Link 1, x1
t2
t1
t2 ( x2 )
Minimum of ( ∫ t1 ( w) dw + ∫ t 2 ( w) dw)
x1
x2
0
0
t1 ( x1 )
x2
5
4
3
2
1
0
0
1
2
3
4
5
x1
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UO & SO Traffic Assignment As MPs: R7 Example
x1
x2
x3
x1
0
0
0
x2
x3
0
0
‰ min ∫ t1(x1 )dx1 + ∫ t2 (x2 )dx2 + ∫ t3 (x3 )dx3 ‰ min ∫ m1 ( x1 )dx1 + ∫ m2 ( x2 )dx2 + ∫ m3 (x3 )dx3
0
ac
1
s.t. f
= qac
f +f
bc
1
bc
2
s.t.
= qbc
ac
1
f
bc
1
f
= qac
+ f 2bc = qbc
f1ac ≥ 0, f1bc ≥ 0, f2bc ≥ 0
f1ac ≥ 0, f1bc ≥ 0, f 2bc ≥ 0
x1 = f1ac + f2bc
x1 = f1ac + f 2bc
x2 = f1bc
x2 = f1bc
x3 = f 2bc
x3 = f 2bc
‰ U.O.: All used paths have, between ‰ S.O.: All used paths have,
any O-D pair, equal and minimum
between any O-D pair, equal and
travel time
minimum marginal travel times
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6
Solving SO Static Traffic Assignment: Examples
Link 2, x2
5
D
O
5
Link 1, x1
1
a
3
b
t1 ( x1 ) = 2 + x1
t1 ( x1 ) = 10 + x1
c
2
t 2 ( x2 ) = 1 + 2 x2
t 2 (x2 ) = 90 + x2
t3 ( x3 ) = 0
(qac , qbc ) = (80,10)
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Lecture 6, Page 13
Solving SO Static Traffic Assignment: Examples
Link 2, x2
q
D
O
Link 1, x1
q
t2 ( x2 ) = 1 + 2 x2
t1 ( x1 ) = 2 + x1
‰ UO solutions for three values of q:
• q=1/8: x1=0, x2=q
• q=1/2: x1=0, x2=q
• q=5: x1=3, x2=2
‰ SO solutions for three values of q:
• m1(x1)=2+2x1; m2(x2)=1+4x2
• q=1/8: x1=0, x2=q
• q=1/4: x1=0, x2=q
• q=5: x1=(-1+4q)/6, x2=(1+2q)/6
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SO Static Traffic Assignment: Example
min x1t1 (x1 ) + x2t 2 (x2 ) + x3t3 (x3 )
s.t .
f1ac = qac
bc
1
f
+f
bc
2
Demand
= qbc
f1ac ≥ 0, f1bc ≥ 0, f 2bc ≥ 0
Non - negativity
x1 = f1ac + f 2bc
x2 = f1bc
x3 = f
(
Definition of link flows
bc
2
)
*
*
*
‰ S.O. solution: x1 , x2 , x3 = (80,10,0x )
x
x
‰ min x1t1 ( x1 ) + x2t 2 (x2 ) + x3t3 (x3 ) = min ∫ m1 (x1 )dx1 + ∫ m2 ( x2 )dx2 + ∫ m3 (x3 )dx3
1
2
3
0
0
0
d (x3t3 ( x3 ))
d (x2t 2 ( x2 ))
d ( x1t1 (x1 ))
, m2 ( x 2 ) =
, m3 ( x3 ) =
where m1 (x1 ) =
dx3
dx2
dx1
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Lecture 6, Page 15
Lecture 6 Outline
‰ Mathematical programs (MPs)
‰ Formulation of shortest path problems as MPs
‰ Formulation of U.O. traffic assignment as an MP
‰ Relationship between U.O. and S.O. traffic assignment
‰ Solving S.O. traffic assignment by hand
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Lecture 6, Page 16
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