bilinear approximation techniques to solve network flow...
TRANSCRIPT
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Bilinear Approximation Techniques toSolve Network Flow Problems with
Nonlinear Arc Cost Functions
Artyom Nahapetyan
University of FloridaDepartment of Industrial and Systems Engineering
July 18, 2006
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
1 Supply Chain ProblemsConcave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
2 Dynamic Traffic Assignment ProblemsDiscrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
3 Conclusion
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Description of the Problem
CPLNF Problem
Let
G (N,A) represent a network, and
fa(xa) denote a cost function of arc a.
minx
∑a∈A
fa(xa)
s.t. Bx = b
xa ∈ [λ0a, λ
naa ] ∀a ∈ A
B - node-arc incident matrix of the network Gfa(xa) - concave piecewise linear functions
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Arc Cost Function
Function fa(xa)
fa(xa) =
c1a xa + s1
a xa ∈ [λ0a, λ
1a)
c2a xa + s2
a xa ∈ [λ1a, λ
2a)
· · · · · ·cnaa xa + sna
a xa ∈ [λna−1a , λna
a ]
c1a > c2
a > · · · > cnaa
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Arc Cost Function
Function fa(xa)
fa(xa) =
c1a xa + 0 xa ∈ [0, λ1
a)c2a xa + s2
a xa ∈ [λ1a, λ
2a)
· · · · · ·cnaa xa + sna
a xa ∈ [λna−1a , λna
a ]
c1a > c2
a > · · · > cnaa
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Arc Cost Function
Function fa(xa)
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Arc Cost Function
Function fa(xa)
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Mixed Integer Formulation
CPLNF-IP Problem
minx ,y
∑a∈A
∑k∈Ka
cka xk
a +∑a∈A
∑k∈Ka
ska yk
a
Bx = b∑k∈Ka
xka = xa
∑k∈Ka
λk−1a yk
a ≤ xa ≤∑k∈Ka
λkayk
a ,∑k∈Ka
yka = 1
xka ≤ Myk
a , xka ≥ 0, yk
a ∈ 0, 1
where xka denotes the portion of the total flow that has cost
according to the linear function f ka (xa) = ck
a xa + ska
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Relaxation of the Binary Variables
CPLNF-R Problem
minx ,y
∑a∈A
∑k∈Ka
cka xk
a +∑a∈A
∑k∈Ka
ska yk
a
Bx = b∑k∈Ka
xka = xa
∑k∈Ka
λk−1a yk
a ≤ xa ≤∑k∈Ka
λkayk
a ,∑k∈Ka
yka = 1
xka = xay
ka , xk
a ≥ 0, yka ≥ 0
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Relaxation of the Binary Variables
CPLNF-R Problem
minx ,y
∑a∈A
∑k∈Ka
cka xk
a +∑a∈A
∑k∈Ka
ska yk
a
Bx = b
∑k∈Ka
λk−1a yk
a ≤ xa ≤∑k∈Ka
λkayk
a ,∑k∈Ka
yka = 1
xka = xay
ka , yk
a ≥ 0
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Relaxation of the Binary Variables
CPLNF-R Problem
minx ,y
∑a∈A
∑k∈Ka
cka yk
a
xa +∑a∈A
∑k∈Ka
ska yk
a
Bx = b
∑k∈Ka
λk−1a yk
a ≤ xa ≤∑k∈Ka
λkayk
a ,∑k∈Ka
yka = 1
yka ≥ 0
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Theoretical Results
Lemma
Any feasible vector of the CPLNF-IP problem is feasible to theCPLNF-R
Lemma
Any local optimum of the CPLNF-R problem is either feasible tothe CPLNF-IP or leads to a feasible vector of CPLNF-IP with thesame objective function value.
Theorem
A global optimum of the CPLNF-R problem is a solution or leadsto a solution of the CPLNF-IP .
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Theoretical Results
Lemma
Any feasible vector of the CPLNF-IP problem is feasible to theCPLNF-R
Lemma
Any local optimum of the CPLNF-R problem is either feasible tothe CPLNF-IP or leads to a feasible vector of CPLNF-IP with thesame objective function value.
Theorem
A global optimum of the CPLNF-R problem is a solution or leadsto a solution of the CPLNF-IP .
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Theoretical Results
Lemma
Any feasible vector of the CPLNF-IP problem is feasible to theCPLNF-R
Lemma
Any local optimum of the CPLNF-R problem is either feasible tothe CPLNF-IP or leads to a feasible vector of CPLNF-IP with thesame objective function value.
Theorem
A global optimum of the CPLNF-R problem is a solution or leadsto a solution of the CPLNF-IP .
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Concave Piecewise Linear Problem with a SeparableObjective Function
CPLPwSOF Problem
minx
n∑i=1
fi (xi )
x ∈ X ⊆ Rn
xi ∈ [λ0i , λ
nii ]
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Relaxation
CPLPwSOF-R Problem
minx ,y
n∑i=1
∑k∈Ki
f ki (xi )y
ki
x ∈ X ⊆ Rn∑k∈Ki
λk−1i yk
i ≤ xi ≤∑k∈Ki
λki yk
i
∑k∈Ki
yki = 1
yki ≥ 0,
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Theoretical Results
Theorem
A concave minimization problem with a separable piecewise linearobjective function, CPLPwSOF, is equivalent to a bilinear program,CPLPwSOF-R.
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Two Problems
CPLNF-R Problem
minx ,y
∑a∈A
∑k∈Ka
cka yk
a
xa +∑a∈A
∑k∈Ka
ska yk
a
Bx = b∑k∈Ka
λk−1a yk
a ≤ xa ≤∑k∈Ka
λkayk
a
∑k∈Ka
yka = 1
yka ≥ 0
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Two Problems
CPLNF-R Problem
minx ,y
∑a∈A
∑k∈Ka
cka yk
a
xa +∑a∈A
∑k∈Ka
ska yk
a
Bx = b∑k∈Ka
λk−1a yk
a ≤ xa ≤∑k∈Ka
λkayk
a
∑k∈Ka
yka = 1
yka ≥ 0
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Two Problems
LP(y)Problem (y is fixed)
minx
∑a∈A
∑k∈Ka
cka yk
a
xa
Bx = b
xa ∈ [0, λnaa ]
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Two Problems
CPLNF-R Problem
minx ,y
∑a∈A
∑k∈Ka
[cka xa + sk
a
]yka
Bx = b
∑k∈Ka
λk−1a yk
a ≤ xa ≤∑k∈Ka
λkayk
a
∑k∈Ka
yka = 1
yka ≥ 0
Artyom Nahapetyan Dissertation Defense
![Page 22: Bilinear Approximation Techniques to Solve Network Flow …plaza.ufl.edu/artyom/Presentations/DissertationDefense.pdf · Supply Chain Problems Dynamic Traffic Assignment Problems](https://reader036.vdocument.in/reader036/viewer/2022062604/5fb54826e09c9239202dce78/html5/thumbnails/22.jpg)
Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Two Problems
CPLNF-R Problem
minx ,y
∑a∈A
∑k∈Ka
[cka xa + sk
a
]yka
Bx = b
∑k∈Ka
λk−1a yk
a ≤ xa ≤∑k∈Ka
λkayk
a
∑k∈Ka
yka = 1
yka ≥ 0
Artyom Nahapetyan Dissertation Defense
![Page 23: Bilinear Approximation Techniques to Solve Network Flow …plaza.ufl.edu/artyom/Presentations/DissertationDefense.pdf · Supply Chain Problems Dynamic Traffic Assignment Problems](https://reader036.vdocument.in/reader036/viewer/2022062604/5fb54826e09c9239202dce78/html5/thumbnails/23.jpg)
Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Two Problems
LP(x) Problem (x is fixed)
miny
∑a∈A
∑k∈Ka
[cka xa + sk
a ]yka
∑k∈Ka
λk−1a yk
a ≤ xa ≤∑k∈Ka
λkayk
a
∑k∈Ka
yka = 1, yk
a ≥ 0
A binary variable that satisfies the inequality is a solution ofthe problem.
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Two Problems
LP(x) Problem (x is fixed)
miny
∑a∈A
∑k∈Ka
[cka xa + sk
a ]yka
∑k∈Ka
λk−1a yk
a ≤ xa ≤∑k∈Ka
λkayk
a
∑k∈Ka
yka = 1, yk
a ≥ 0
A binary variable that satisfies the inequality is a solution ofthe problem.
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Dynamic Cost Updating Procedure (DCUP)
DCUP: Iteratively Solves LP(x) and LP(y)
Step 1: Let y0 denote the initial vector of yk0a , where y10
a = 1 andyk0a = 0, ∀k ∈ Ka, k 6= 1. m← 1.
Step 2: Let xm = argminLP(ym−1), andym = argminLP(xm).
Step 3: If ym = ym−1 then stop. Otherwise, m← m + 1 and goto Step 2.
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Theoretical Results
Theorem
Given any initial binary vector y0, DCUP converges in a finitenumber of iterations.
Theorem
Let (x∗, y∗) be the solution returned by DCUP. If y∗ is a uniquesolution of the LP(x∗) problem then (x∗, y∗) is a local minimum ofCPLNF-R.
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Theoretical Results
Theorem
Given any initial binary vector y0, DCUP converges in a finitenumber of iterations.
Theorem
Let (x∗, y∗) be the solution returned by DCUP. If y∗ is a uniquesolution of the LP(x∗) problem then (x∗, y∗) is a local minimum ofCPLNF-R.
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Theoretical Results
y∗ is not unique only if ∃k ∈ K , k 6= 0, k 6= na, and a ∈ A,such that x∗a = λk
a .∑k∈Ka
λk−1a yk
a ≤ xa ≤∑k∈Ka
λkayk
a
It is highly unlikely that x∗a takes those values because it is asolution of the LP(y) problem.
minx
∑a∈A
∑k∈Ka
cka yk
a
xa
Bx = b, xa ∈ [0, λnaa ]
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Theoretical Results
y∗ is not unique only if ∃k ∈ K , k 6= 0, k 6= na, and a ∈ A,such that x∗a = λk
a .∑k∈Ka
λk−1a yk
a ≤ xa ≤∑k∈Ka
λkayk
a
It is highly unlikely that x∗a takes those values because it is asolution of the LP(y) problem.
minx
∑a∈A
∑k∈Ka
cka yk
a
xa
Bx = b, xa ∈ [0, λnaa ]
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
DSSP
Artyom Nahapetyan Dissertation Defense
![Page 31: Bilinear Approximation Techniques to Solve Network Flow …plaza.ufl.edu/artyom/Presentations/DissertationDefense.pdf · Supply Chain Problems Dynamic Traffic Assignment Problems](https://reader036.vdocument.in/reader036/viewer/2022062604/5fb54826e09c9239202dce78/html5/thumbnails/31.jpg)
Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
DSSP
Artyom Nahapetyan Dissertation Defense
![Page 32: Bilinear Approximation Techniques to Solve Network Flow …plaza.ufl.edu/artyom/Presentations/DissertationDefense.pdf · Supply Chain Problems Dynamic Traffic Assignment Problems](https://reader036.vdocument.in/reader036/viewer/2022062604/5fb54826e09c9239202dce78/html5/thumbnails/32.jpg)
Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
DSSP
Artyom Nahapetyan Dissertation Defense
![Page 33: Bilinear Approximation Techniques to Solve Network Flow …plaza.ufl.edu/artyom/Presentations/DissertationDefense.pdf · Supply Chain Problems Dynamic Traffic Assignment Problems](https://reader036.vdocument.in/reader036/viewer/2022062604/5fb54826e09c9239202dce78/html5/thumbnails/33.jpg)
Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
DSSP
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
DSSP
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
DSSP
Kim D., Pardalos P., “A Solution Approach to the FixedCharged Network Flow Problems Using a Dynamic SlopeScaling Procedure”, Operations Research Letters, 24, pp.195-203, 1999.
Kim D., Pardalos P., “Dynamic Slope Scaling and TrustInterval Techniques for Solving Concave Piecewise LinearNetwork Flow Problems”, Networks, 35(3), pp. 216-222,2000.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
An Alternative Formulation
NFPwFDCF Problem
minx
FT (x)x
s.t. Bx = b,
xa ∈ [0, λnaa ],
where F (x) is a vector function with components
Fa(xa) =
fa(xa)
xaxa > 0
M xa = 0=
c1a xa ∈ (0, λ1
a]
c2a + s2
axa
xa ∈ (λ1a, λ
2a]
· · · · · ·cnaa + sna
axa
xa ∈ (λna−1a , λna
a ]
M xa = 0
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Theoretical Results
Theorem
The NFPwFDCF problem is equivalent to the CPLNF problem.
Theorem
The solution of the DSSP is a user equilibrium solution of thenetwork flow problem with the flow dependent cost functionsFa(xa), i.e
“find feasible x∗ such that FT (x∗)(x − x∗) ≥ 0, ∀xa ∈ [0, λnaa ],
Bx = b”,
We need to find a system optimum solution of NFPwFDCF.
It is well known that system optimum and user equilibriumsolutions are not the same.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Theoretical Results
Theorem
The NFPwFDCF problem is equivalent to the CPLNF problem.
Theorem
The solution of the DSSP is a user equilibrium solution of thenetwork flow problem with the flow dependent cost functionsFa(xa), i.e
“find feasible x∗ such that FT (x∗)(x − x∗) ≥ 0, ∀xa ∈ [0, λnaa ],
Bx = b”,
We need to find a system optimum solution of NFPwFDCF.
It is well known that system optimum and user equilibriumsolutions are not the same.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Theoretical Results
Theorem
The NFPwFDCF problem is equivalent to the CPLNF problem.
Theorem
The solution of the DSSP is a user equilibrium solution of thenetwork flow problem with the flow dependent cost functionsFa(xa), i.e
“find feasible x∗ such that FT (x∗)(x − x∗) ≥ 0, ∀xa ∈ [0, λnaa ],
Bx = b”,
We need to find a system optimum solution of NFPwFDCF.
It is well known that system optimum and user equilibriumsolutions are not the same.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Theoretical Results
Theorem
The NFPwFDCF problem is equivalent to the CPLNF problem.
Theorem
The solution of the DSSP is a user equilibrium solution of thenetwork flow problem with the flow dependent cost functionsFa(xa), i.e
“find feasible x∗ such that FT (x∗)(x − x∗) ≥ 0, ∀xa ∈ [0, λnaa ],
Bx = b”,
We need to find a system optimum solution of NFPwFDCF.
It is well known that system optimum and user equilibriumsolutions are not the same.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Test Problems
Parameters of the Problems
Network size (nodes-arcs-supply/demand nodes): 12-35-2,20-100-3, 40-300-4, 100-2000-20, and 200-5000-50.
Demand: U[10,20], U[20,30], or U[30,40]
Number of linear pieces: 5 or 10.
Total number of problems sets is 30.
There are 30 problems per problem set. (In total 900problems)
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Test Problems
Computational Results
DCUP provides a better solution than DSSP in about 50% ofthe test problems.
Both algorithms provide the same solution in about 30% ofthe test problems.
DCUP spends 2-5 times less CPU time than DSSP.
DCUP converges using less number of iterations than DSSP.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Description of the Problem
FCNF Problem
minx
f (x) =∑a∈A
fa(xa)
s.t. Bx = b,
xa ∈ [0, λa],
where fa(xa) =
caxa + sa xa ∈ (0, λa]0 xa = 0
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
ε-Approximation
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
ε-Approximation
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
ε-Approximation
CPLNF(ε) Problem
minx
φε(x) =∑a∈A
φεaa (xa)
s.t. Bx = b,
xa ∈ [0, λa],
where ε is a vector of εa.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Theoretical Results
Let
x∗ = argmin(FCNF ),
xε = argmin (CPLNF (ε)), and
δ = minxva |xv ∈ V , a ∈ A, xv
a > 0, where V represents theset of vertices of the feasible region.
Theorem
For all ε such that εa ∈ (0, λa], ∀a ∈ A, φε(xε) ≤ f (x∗).
Theorem
For all ε such that εa ∈ (0, δ], ∀a ∈ A, φε(xε) = f (x∗).
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Theoretical Results
Let
x∗ = argmin(FCNF ),
xε = argmin (CPLNF (ε)), and
δ = minxva |xv ∈ V , a ∈ A, xv
a > 0, where V represents theset of vertices of the feasible region.
Theorem
For all ε such that εa ∈ (0, λa], ∀a ∈ A, φε(xε) ≤ f (x∗).
Theorem
For all ε such that εa ∈ (0, δ], ∀a ∈ A, φε(xε) = f (x∗).
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Theoretical Results
Let
x∗ = argmin(FCNF ),
xε = argmin (CPLNF (ε)), and
δ = minxva |xv ∈ V , a ∈ A, xv
a > 0, where V represents theset of vertices of the feasible region.
Theorem
For all ε such that εa ∈ (0, λa], ∀a ∈ A, φε(xε) ≤ f (x∗).
Theorem
For all ε such that εa ∈ (0, δ], ∀a ∈ A, φε(xε) = f (x∗).
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Adaptive Dynamic Cost Updating Procedure
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Adaptive Dynamic Cost Updating Procedure
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Adaptive Dynamic Cost Updating Procedure
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Test Problems
Parameters of the Problems
Network size (nodes-arcs-supply/demand nodes): 20-100-3,40-300-4, 100-1000-10, and 150-3000-15.
Setup cost: U[50,100], U[100,200], or U[200,400]
Slope: U[1,5], U[10,20], or U[30,40]
Demand: U[30,50]
Total number of problems sets is 36.
There are 30 problems per problem set. (In total 1080problems)
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Test Problems
Computational Results
ADCUP provides a better solution than DSSP
Small problems - 62%, andLarge problems - 98%.
Both algorithms provide the same solution
Small problems - 28%, andLarge problems - 0%.
ADCUP spends 2-8 times less CPU time than DSSP.
ADCUP converges using less number of iterations than DSSP.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Problem Description
Given Data:
Set of products.
Unit and setup costs.
Inventory costs.
Production capacities.
Demand is a function of the price.
Problem
Find a production and pricing policy, which maximizes the profit.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Problem Description
Given Data:
Set of products.
Unit and setup costs.
Inventory costs.
Production capacities.
Demand is a function of the price.
Problem
Find a production and pricing policy, which maximizes the profit.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Price-Demand and Profit-Demand Relationships
g(d) = f (d)d
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Formulation
CMDP Problem
maxx ,y ,d
∑p∈P
∑j∈∆
g(p,j)
∑i∈∆|i≤j
x(p,i ,j)
−
∑p∈P
∑i ,j∈∆|i≤j
[c in(p,i ,j) + cpr
(p,i)
]x(p,i ,j) −
∑p∈p
∑i∈∆
cst(p,i)y(p,i)
s.t.∑p∈P
∑j∈∆|i≤j
x(p,i ,j) ≤ Ci ,∑
j∈∆|i≤j
x(p,i ,j) ≤ Ciy(p,i),
x(p,i ,j) ≥ 0, y(p,i) ∈ 0, 1
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Approximation
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Approximate Formulation
ACMDP Problem
maxx ,y ,λ
∑p∈P
∑j∈∆
∑k∈K
gk(p,j)λ
k(p,j)
−∑p∈P
∑i ,j∈∆|i≤j
[c in(p,i ,j) + cpr
(p,i)
]x(p,i ,j) −
∑p∈p
∑i∈∆
cst(p,i)y(p,i),
s.t.∑p∈P
∑j∈∆|i≤j
x(p,i ,j) ≤ Ci ,∑
j∈∆|i≤j
x(p,i ,j) ≤ Ciy(p,i),
∑i∈∆|i≤j
x(p,i ,j) =∑k∈K
dk(p,j)λ
k(p,j),
N∑k=0
λk(p,j) = 1,
x(p,i ,j) ≥ 0, λk(p,j) ≥ 0, y(p,i) ∈ 0, 1.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Approximate Formulation
ACMDP Problem
maxx ,y
∑p∈P
∑i∈∆
∑j∈∆|i≤j
∑k∈K
qk(p,i ,j)x
k(p,i ,j) − cst
(p,i)y(p,i)
s.t.
∑p∈P
∑j∈∆|i≤j
∑k∈K
xk(p,i ,j) ≤ Ci ,
∑j∈∆|i≤j
∑k∈K
xk(p,i ,j) ≤ Ciy(p,i),
∑k∈K
∑i∈∆|i≤j
xk(p,i ,j)
dk(p,j)
≤ 1, xk(p,i ,j) ≥ 0, y(p,i) ∈ 0, 1,
where qk(p,i ,j) = f k
(p,j) − c in(p,i ,j) − cpr
(p,i)
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Objective Function
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Objective Function
Artyom Nahapetyan Dissertation Defense
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Bilinear Reduction
ACMDP-B Problem
maxx ,y
∑p∈P
∑i∈∆
∑j∈∆|i≤j
∑k∈K
qk(p,i ,j)x
k(p,i ,j) − cst
(p,i)
y(p,i) = ϕ(x , y)
x ∈ X and y ∈ Y ,
where Y = [0, 1]|P||∆| and
X = x |xk(p,i ,j) ≥ 0,
∑p,k,j |i≤j xk
(p,i ,j) ≤ Ci ,∑
k,i |i≤j
xk(p,i,j)
dk(p,j)
≤ 1
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Theoretical Results
Theorem
Any local maximum of the ACMDP-B problem is feasible or leadsto a feasible solution of the ACMDP problem with the sameobjective function value.
Theorem
A global maximum of the ACMDP-B problem is a solution or leadsto a solution of the ACMDP problem.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Theoretical Results
Theorem
Any local maximum of the ACMDP-B problem is feasible or leadsto a feasible solution of the ACMDP problem with the sameobjective function value.
Theorem
A global maximum of the ACMDP-B problem is a solution or leadsto a solution of the ACMDP problem.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Procedure 1
Tow LPs
LP(x) :
maxy∈Y
∑p∈P
∑i∈∆
[∑j∈∆|i≤j
∑k∈K qk
(p,i ,j)xk(p,i ,j) − cst
(p,i)
]y(p,i)
LP(y) :
maxx∈X
∑p∈P
∑i∈∆
∑j∈∆|i≤j
∑k∈K
[qk(p,i ,j)y(p,i)
]xk(p,i ,j)
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Procedure 1
Procedure 1
Step 1: Let y0 denote an initial binary vector, where y(p,i) = 1.m← 1.
Step 2: Let xm = argmaxLP(ym−1), andym = argmaxLP(xm).
Step 3: If ym = ym−1 then stop. Otherwise, m← m + 1 and goto Step 2.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Procedure 1
Disadvantage
The quality of the solution is not good enough.
Procedure 1 converges to a local maximum of the problem.
If ym(p,i) = 0 then in all following iterations xk
(p,i ,j) = 0, ∀j ∈ ∆,i ≤ j , and p ∈ P.
As a result, the local maximum can be far from being a globalone.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Approximation of the Objective Function
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Approximation of the Objective Function
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Approximation of the Objective Function
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Approximation of the Objective Function
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Procedure 2
Procedure 2
Step 1: Let ε(p,i) be a sufficiently large number, and y0 be suchthat y0
(p,i) = 1, ∀p ∈ P and i ∈ ∆. m← 0.
Step 2: Construct the approximation problem and run Procedure1 to find a local maximum of the problem, where ym is an initialbinary vector. Let (xm+1, ym+1) denote the local maximum.
Step 3: If ∃p ∈ P and i ∈ ∆ such that∑j∈∆|i≤j
∑k∈K qk
(p,i ,j)x(m+1)k(p,i ,j) − cst
(p,i) ≤ εm(p,i) and∑
j∈∆|i≤j
∑k∈K x
(m+1)k(p,i ,j) > 0 then ε← αε, m← m + 1 and go to
Step 2. Otherwise, stop.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
How Big is ε(p,i)?
Procedure 3
For all i ∈ ∆ and p ∈ P assign the available capacity first to thevariables with a higher value of qk
(p,i ,j)
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Test Problems
Parameters of the Problems
Number of products: |P| =5, 10, or 20.
Planning horizon: |∆| =12, or 52.
Capacity: Ci = |P|U[10, 100], |P|U[50, 150], |P|U[100, 200],or |P|U[150, 250].
Costs: cpr(p,i) = U[20, 40], cst
(p,i) = U[600, 1000], and
c in(p,i) = U[4, 8].
Maximum price/demand: U[70, 90]/U[500, 1000].
Profit per unit of investment: β ∈ [0.7, 1.3]
The value of the parameter α: 1/2, 2/3, 9/10.
Total number of problem sets is 24.
There are 10 problems per problem set. In total 240 problems.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Concave Piecewise Linear Network Flow ProblemFixed Charge Network Flow ProblemCapacitated Multi-Item Dynamic Pricing Problems
Test Problems
Computational Results
Quality of the solution:
Procedure 1 - 1-8.3%, andProcedure 2 - <1.2%.
CPU time:
Procedure 1 - 0.2-35 sec, andProcedure 2
α = 1/2 - 0.5-58 sec,α = 2/3 - 0.8-78 sec, andα = 9/10 - 1.8-257 sec.
A higher value of α provides a slightly better solution.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Description of the Problem
Given Data:
Network - G (N,A).
Travel time - φa(·).Set of OD pairs -C = (1, 4), (3, 4).Demand - hk(t).
Problem
Find traffic flows which minimize the total travel time of allvehicles during the time period [0,T ].
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Description of the Problem
Given Data:
Network - G (N,A).
Travel time - φa(·).Set of OD pairs -C = (1, 4), (3, 4).Demand - hk(t).
Problem
Find traffic flows which minimize the total travel time of allvehicles during the time period [0,T ].
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Description of the Problem
Variables:
xa(t) - number of cars on arc a at time t,
ua(t) - inflow rate into arc a at time t,
va(t) - outflow rate from arc a at time t.
Assumptions:
xa(0) = 0 and xa(T ) = 0,
If t is the time to enter arc a then t + φa(xa(t)) is the time toleave the arc.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Disadvantage
Requires the network to be empty at time 0 and T
It is not necessarily valid to ignore xa(0) even when it is small.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Cyclic Time Period
How to model DTA problem with positive xa(0) and xa(T )?
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Cyclic Time Period
How to model DTA problem with positive xa(0) and xa(T )?
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Cyclic Time Period
How to model DTA problem with positive xa(0) and xa(T )?
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Discrete-Time Formulation
[0,T ) =⇒ 0, 1, 2, . . . , (T − 1).
Γa = a set of possible discrete travel times on arc a.
Use time-expanded network
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Time-Expanded Network
T = 5,
∆ = 0, 1, 2, 3, 4,2 ≤ φa(·) < 5, and
Γa = 2, 3, 4
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Time-Expanded Network
T = 5,
∆ = 0, 1, 2, 3, 4,2 ≤ φa(·) < 5, and
Γa = 2, 3, 4
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Time-Expanded Network
T = 5,
∆ = 0, 1, 2, 3, 4,2 ≤ φa(·) < 5, and
Γa = 2, 3, 4
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Time-Expanded Network
T = 5,
∆ = 0, 1, 2, 3, 4,2 ≤ φa(·) < 5, and
Γa = 2, 3, 4
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Time-Expanded Network
T = 5,
∆ = 0, 1, 2, 3, 4,2 ≤ φa(·) < 5, and
Γa = 2, 3, 4
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Time-Expanded Network
T = 5,
∆ = 0, 1, 2, 3, 4,2 ≤ φa(·) < 5, and
Γa = 2, 3, 4
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Variables
Index a(t, s) uniquely identifies an arc in TE network.
Arc flow on TE network: Ya(t,s) =∑
k∈C yka(t,s)
za(t,s) =
1 if the arc is used0 o/w
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Variables
Index a(t, s) uniquely identifies an arc in TE network.
Arc flow on TE network: Ya(t,s) =∑
k∈C yka(t,s)
za(t,s) =
1 if the arc is used0 o/w
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Variables
Index a(t, s) uniquely identifies an arc in TE network.
Arc flow on TE network: Ya(t,s) =∑
k∈C yka(t,s)
za(t,s) =
1 if the arc is used0 o/w
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Variables
Index a(t, s) uniquely identifies an arc in TE network.
Arc flow on TE network: Ya(t,s) =∑
k∈C yka(t,s)
za(t,s) =
1 if the arc is used0 o/w
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Set Ωa(t)
It is necessary to compute xa(t)
Ωa(t) = (τ, s) : τ = [t − 1]T , · · · , [t − s]T , s ∈ Γa
xa(t) =∑
(τ,s)∈Ωa(t)Ya(τ,s)
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Set Ωa(t)
It is necessary to compute xa(t)
Ωa(t) = (τ, s) : τ = [t − 1]T , · · · , [t − s]T , s ∈ Γa
xa(t) =∑
(τ,s)∈Ωa(t)Ya(τ,s)
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Set Ωa(t)
It is necessary to compute xa(t)
Ωa(t) = (τ, s) : τ = [t − 1]T , · · · , [t − s]T , s ∈ Γa
xa(t) =∑
(τ,s)∈Ωa(t)Ya(τ,s)
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Discrete Formulation
Periodic DTA Problem
min(x ,y ,z,g)
∑t∈∆
∑a∈A
[φa(xa(t))
∑s∈Γa
Ya(t,s)
]= Φ(Y )TY
s.t. Byk + gk = bk ,∑t∈∆
gkd(k)t
=∑t∈∆
hkt
Ya(t,s) =∑
k∈C yka(t,s), xa(t) =
∑(τ,s)∈Ωa(t)
Ya(τ,s)∑s∈Γa
za(t,s) = 1∑s∈Γa
(s − δ)za(t,s) ≤ φa(xa(t)) ≤∑
s∈Γasza(t,s)
Ya(t,s) ≤ Maza(t,s)
yka(t,s) ≥ 0, gk
d(k)t≥ 0, xa(t) ≥ 0, za(t,s) ∈ 0, 1
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Linearization of the Constraint
Denote φ−1a (1) = 0
∑s∈Γa
(s − δ)za(t,s) ≤ φa(xa(t)) ≤∑
s∈Γasza(t,s)
m∑s∈Γa
φ−1a (s − δ)za(t,s) ≤ xa(t) ≤
∑s∈Γa
φ−1a (s)za(t,s)
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Bounds on the Problem
The upper bound on the objective
Φ(Y )TY ≤∑t∈∆
∑a∈A
[ ∑s∈Γa
sza(t,s)
] [ ∑s∈Γa
Ya(t,s)
]=
∑t∈∆
∑a∈A
∑s∈Γa
sYa(t,s) = qTu Y
The lower bound on the objectiveΦ(Y )TY ≥
∑t∈∆
∑a∈A
∑s∈Γa
(s − δ)Ya(t,s) = qTl Y
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Bounds on the Problem
The upper bound on the objective
Φ(Y )TY ≤∑t∈∆
∑a∈A
[ ∑s∈Γa
sza(t,s)
] [ ∑s∈Γa
Ya(t,s)
]=
∑t∈∆
∑a∈A
∑s∈Γa
sYa(t,s) = qTu Y
The lower bound on the objectiveΦ(Y )TY ≥
∑t∈∆
∑a∈A
∑s∈Γa
(s − δ)Ya(t,s) = qTl Y
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Three Problems
Let S(δ) represent the feasible region.
Upper Bounding Problem(Y u,Zu) = argminqT
u Y : (Y ,Z ) ∈ S(δ)
Original Problem(Y ∗,Z ∗) = argminΦ(Y )TY : (Y ,Z ) ∈ S(δ)
Lower Bounding Problem(Y l ,Z l) = argminqT
l Y : (Y ,Z ) ∈ S(δ)
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Theoretical Results
Lemma
For any δ > 0, qTl Y l ≤ Φ(Y ∗)TY ∗ ≤ qT
u Y u
Theorem
Given ε > 0, there exist δ > 0 such that
Φ(Y ∗)TY ∗ − qTl Y l ≤ ε, and
qTu Y u − Φ(Y ∗)TY ∗ ≤ ε
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Theoretical Results
Lemma
For any δ > 0, qTl Y l ≤ Φ(Y ∗)TY ∗ ≤ qT
u Y u
Theorem
Given ε > 0, there exist δ > 0 such that
Φ(Y ∗)TY ∗ − qTl Y l ≤ ε, and
qTu Y u − Φ(Y ∗)TY ∗ ≤ ε
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Solution Improvement
Refinement Problem
Find (Y u,Zu)⇓
Y u is the solution of the original problem with Z = Zu
⇓The refined solution is (Y u,Zu).
Corollary
Φ(Y ∗)TY ∗ ≤ Φ(Y u)T Y u ≤ Φ(Y u)TY u ≤ qTu Y u
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Solution Improvement
Refinement Problem
Find (Y u,Zu)⇓
Y u is the solution of the original problem with Z = Zu
⇓The refined solution is (Y u,Zu).
Corollary
Φ(Y ∗)TY ∗ ≤ Φ(Y u)T Y u ≤ Φ(Y u)TY u ≤ qTu Y u
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Test Problems
Parameters of the Problems
Time horizon - [0,T ) = [0, 10),
Travel time function
linear: φa(xa(t)) = 1.5 + 2.5(
xa(t)100
), or
quadratic: φa(xa(t)) = 1.5 + 2.5(
xa(t)100
)2
DemandTime
Traffic Intensity 0 1 2 3 4 5 6 7 8 9 TotalLow 20 25 30 35 40 40 35 30 25 20 300
Medium 30 35 40 45 50 50 45 40 35 30 400High 40 45 50 55 60 60 55 50 45 40 500
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Test Problems
Computational Results: Linear Travel Time Function
(Y∗, Z∗) (Y u , Zu) (Y l , Z l ) Rel. cpu
Traffic cpu∗ cpuu cpul Err Ratio
Intensity Delay (sec) Delay (sec) Delay (sec) (%)cpu∗cpuu
cpu∗
cpul
Low 1337.50 27.42 1385.00 2.57 1392.50 2.38 3.55 10.7 11.5Medium 1800.00 15.92 1866.30 2.66 1815.30 2.90 0.85 6.0 5.5
High 2290.00 95.02 2327.50 4.07 2315.00 1.25 1.09 23.3 76.0
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Test Problems
Computational Results: Quadratic Travel Time Function
(Y∗, Z∗) (Y u , Zu) (Y l , Z l ) Rel. cpu
Traffic cpu∗ cpuu cpul Err Ratio
Intensity Delay (sec) Delay (sec) Delay (sec) (%)cpu∗cpuu
cpu∗
cpul
Low 1054.50 0.88 1054.50 0.09 1054.50 0.08 0.00 9.8 11.0Medium 1543.80 6.62 1543.80 0.14 1543.80 0.34 0.00 47.3 19.5
High 2129.80 501.17 2128.60 0.10 2128.60 0.13 -0.06 5011.7 3855.2
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
The Upper Bounding Problem
DTDTA-U Problem
min(x ,y ,z,g)
qTu Y
s.t. Byk + gk = bk ,∑t∈∆
gkd(k)t
=∑t∈∆
hkt
Ya(t,s) =∑
k∈C yka(t,s), xa(t) =
∑(τ,s)∈Ωa(t)
Ya(τ,s)∑s∈Γa
za(t,s) = 1∑s∈Γa
φ−1a (s − δ)za(t,s) ≤ xa(t) ≤
∑s∈Γa
φ−1a (s)za(t,s)
Ya(t,s) ≤ Maza(t,s)
yka(t,s) ≥ 0, gk
d(k)t≥ 0, xa(t) ≥ 0, za(t,s) ∈ 0, 1
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Difficulties
For practical problems, general IP solvers (e.g., Cplex) are noteffective.
LP relaxation does not provide a good lower bound.
LP relaxation problem decomposes into shortest path problems.
The neighborhood search we implemented was slow and didnot produce a good solution.
Many infeasible neighbors.
Optimal solutions may be isolated.
Difficult to find feasible vectors z .
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Nonlinear Relaxation (DTDTA-R)
Replace za(t,s) byYa(t,s)P
r∈ΓaYa(t,r)
.
DTDTA-R Problem
min(x ,y ,z,g)
qTu Y
s.t. Byk + gk = bk ,∑t∈∆
gkd(k)t
=∑t∈∆
hkt
xa(t) =∑
(τ,s)∈Ωa(t)
∑k∈C yk
a(t,s)
∑s∈Γa
φ−1a (s − δ)Ya(t,s) ≤ xa(t)
∑s∈Γa
Ya(t,s) ≤∑s∈Γa
φ−1a (s)Ya(t,s)
yka(t,s) ≥ 0, gk
d(k)t≥ 0, xa(t) ≥ 0
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Theoretical Results
Theorem
DTDTA-R provides a tighter lower bound for the DTDTA-U thanthe LP relaxation.
Equivalent Objective Functions
Objective Function 1 Objective Function 2
min(x ,y ,z,g)
qTu Y min
(x ,y ,z,g)δeT x
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Heuristic Algorithm
Main Procedure
Step 1: Solve the DTDTA-R problem and let (yR , xR) denote thesolution of the problem.
Step 2: Find values of binary variables za(t,s) based on the
solution xRa(t,s).
Step 3: In the DTDTA-U problem, fix binary variables to thevalues of za(t,s) and solve the resulting LP.
Step 4: If the LP problem is feasible, stop and return the solution.Otherwise, run the UpSet procedure and go to Step 1.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Heuristic Algorithm
UpSet Procedure: Example 1
s = 1 2 3 4 5Ya(t,s)P
r∈ΓaYa(t,r)
= 0.8 0.0 0.0 0.0 0.2
Force Ya(t,5) to be 0
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Heuristic Algorithm
UpSet Procedure: Example 2
s = 1 2 3 4 5Ya(t,s)P
r∈ΓaYa(t,r)
= 0.0 0.3 0.7 0.0 0.0
Allow Ya(t,4) to be positive
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Heuristic Algorithm
UpSet Procedure: Example 3
s = 1 2 3 4 5Ya(t,s)P
r∈ΓaYa(t,r)
= 0.0 1.0 0.0 0.0 0.0
Allow Ya(t,3) to be positive
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Heuristic Algorithm
UpSet Procedure: Example 4∑r∈Γa
Ya(t,r) = 0
Allow Ya(t,s) to be positive ∀s ∈ Γa
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Heuristic Algorithm
UpSet procedure potentially cuts away the current localminima.
In the algorithm it is not require to find a global solution ofthe DTTDTA-R problem.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Test Problems
Parameters of the Problems
Two networks
Travel time, φa(xa(t)) = αa + βa(xa(t)/ca)γa
αa βa ca γa
U(1,2) U(2,8) U(200,300) random (1 or 2)
Demand=U(20,100)
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Test Problems
Computational Results: Separate Mode
Heuristic Approach Cplex Rel.A CPU Iter. CPU Err. B
(%) (sec.) (sec.) (%) (ratio)4-Nodes, [0,10), OBJ1 12% 0.81 10.23 0.84 4.09% 1.744-Nodes, [0,10), OBJ2 8% 1.03 10.35 1.00 4.13% 1.414-Nodes, [0,30), OBJ1 20% 19.76 34.47 376.58 3.92% 22.624-Nodes, [0,30), OBJ2 22% 27.05 38.64 396.51 3.32% 9.559-Nodes, [0,10), OBJ1 28% 633.00 40.58 1,515.44 4.16% 7.809-Nodes, [0,10), OBJ2 10% 111.92 9.98 1,758.65 2.09% 38.919-Nodes, [0,30), OBJ1 30% 1338.37 7.56 5,000.30 -3.33% 12.659-Nodes, [0,30), OBJ2 27% 1029.79 4.53 4,945.14 -4.22% 14.28
A - percentage of the problems not solved by the heuristic
B - CPUCplex/CPUheur.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Test Problems
Computational Results: Combined Mode
Heuristic Approach Cplex Rel.A CPU CPU Err. B
(%) (sec.) (sec.) (%) (ratio)4-Nodes, [0,10) 4% 0.59 0.97 3.63% 1.854-Nodes, [0,30) 10% 19.91 543.00 3.70% 24.229-Nodes, [0,10) 6% 166.86 1,359.16 2.50% 30.279-Nodes, [0,30) 14% 706.03 4,952.60 -3.62% 16.97
A - percentage of the problems not solved by the heuristicB - CPUCplex/CPUheur.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Toll Pricing Framework in the Static DTA
Problem Description: 4-Node Network
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Toll Pricing Framework in the Static DTA
Problem Description: User Equilibrium Solution
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Toll Pricing Framework in the Static DTA
Problem Description: System Optimum Solution
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Toll Pricing Framework in the Static DTA
Problem Description: Toll Pricing Problem
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Reduced TE Networks
(y , z) feasible
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Reduced TE Networks
RTE(z)
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Reduced TE Networks
(p, t)m
p(t) in RTE (y , x , z)
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Theoretical Results
SP(y , z)-A Problem
miny ,g
ΦT∑k∈C
yk
s.t. Bzyk + gk = bk ,
∑t∈∆
gkd(k)t
=∑t∈∆
hkt
yka(t,sa(t))
≥ 0, gkd(k)t
≥ 0
Theorem
A feasible solution (y , z) is an user equilibrium solution if and onlyif yk
a(t,sa(t))is an optimal solution of the corresponding SP(y , z)-A
problem.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Valid Toll Vector
Let β denote an arc toll vector.
Definition
β is a valid toll vector with respect to a feasible solution (y , z) if(y , z) is a solution of the tolled user equilibrium problem.
SP(y , z , β)-A Problem
miny ,g
(Φ + β)T∑k∈C
yk
s.t. Bzyk + gk = bk ,
∑t∈∆
gkd(k)t
=∑t∈∆
hkt
yka(t,sa(t))
≥ 0, gkd(k)t
≥ 0
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Theoretical Results
SP(y , z , β)-A Problem
miny ,g
(Φ + β)T∑k∈C
yk
s.t. Bzyk + gk = bk ,
∑t∈∆
gkd(k)t
=∑t∈∆
hkt
yka(t,sa(t))
≥ 0, gkd(k)t
≥ 0
Corollary
Given a feasible vector (y , z), vector β is a valid toll with respectto (y , z) if and only if y is an optimal solution of the correspondingSP(y , z , β)-A problem.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Dynamic Toll Set
Theorem
Given a feasible vector (y , z), β is a valid toll if and only if thereare vectors ρk and γ such that
∑k∈C
[(bk)T ρk + γk
∑t∈∆
hkt
]= (Φ + β)T
∑k∈C
yk
BTz ρk ≤ Φ + β
ρkd(k)t
≤ γk
β = −Φ is a valid toll vector given any feasible solution (y , z)
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Dynamic Toll Set
Theorem
Given a feasible vector (y , z), β is a valid toll if and only if thereare vectors ρk and γ such that
∑k∈C
[(bk)T ρk + γk
∑t∈∆
hkt
]= (Φ + β)T
∑k∈C
yk
BTz ρk ≤ Φ + β
ρkd(k)t
≤ γk
β = −Φ is a valid toll vector given any feasible solution (y , z)
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
What if (y , z) is a System Optimum Solution?
Let
(y∗, z∗) denote a system optimum solution,
L∗a(t) =∑
s∈Γaφ−1
a (s − δ)z∗a(t,s), and
U∗a(t) =
∑s∈Γa
φ−1a (s)z∗a(t,s)
DTDTA(z∗) Problem
min(y ,g)
ΦT (y)y
s.t. Bz∗yk + gk = bk ,
∑t∈∆
gkd(k)t
=∑t∈∆
hkt
L∗a(t) ≤∑
(τ,s∗a (τ))∈Θa(t)
∑k∈C
yka(τ,s∗a (τ)) ≤ U∗
a(t), yka(t,s∗a (t)) ≥ 0, gk
d(k)t≥ 0
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
What if (y , z) is a System Optimum Solution?
Theorem
Given a system optimum solution (y∗, g∗, z∗), vector βMSC withcomponents
βMSCa(t,s∗a (t)) =
∑r∈∆|(t,s∗a (t))∈Θa(r)
[∇xa(r)
φa(r)(x∗a(r))
∑k∈C
y∗ka(r ,s∗a (r))
]
+∑
r∈∆|(t,s∗a (t))∈Θa(r)
[λ∗a(r) − µ∗a(r)
]is a valid toll vector with respect to (y∗, z∗), where x∗a(t) representsthe optimal number of cars on arc a at time t, λ∗ and µ∗ are thedual multipliers of the bounding restrictions, and λ∗a(t)µ
∗a(t) = 0,
∀a ∈ A and t ∈ ∆.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Similarities with Marginal Cost.
βMSCa(t,s∗a (t)) =
∑r∈∆|(t,s∗a (t))∈Θa(r)
[∇xa(r)
φa(r)(x∗a(r))
∑k∈C
y∗ka(r ,s∗a (r))
]
+∑
r∈∆|(t,s∗a (t))∈Θa(r)
[λ∗a(r) − µ∗a(r)
]
The first component of the toll vector measures the totalbenefit that is experienced by drivers who enter the arc aftertime t
The second component of the toll vector measures the cost tomaintain the (optimal) network structure of RTE (z∗)
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Similarities with Marginal Cost.
βMSCa(t,s∗a (t)) =
∑r∈∆|(t,s∗a (t))∈Θa(r)
[∇xa(r)
φa(r)(x∗a(r))
∑k∈C
y∗ka(r ,s∗a (r))
]
+∑
r∈∆|(t,s∗a (t))∈Θa(r)
[λ∗a(r) − µ∗a(r)
]
The first component of the toll vector measures the totalbenefit that is experienced by drivers who enter the arc aftertime t
The second component of the toll vector measures the cost tomaintain the (optimal) network structure of RTE (z∗)
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Similarities with Marginal Cost.
βMSCa(t,s∗a (t)) =
∑r∈∆|(t,s∗a (t))∈Θa(r)
[∇xa(r)
φa(r)(x∗a(r))
∑k∈C
y∗ka(r ,s∗a (r))
]
+∑
r∈∆|(t,s∗a (t))∈Θa(r)
[λ∗a(r) − µ∗a(r)
]
The first component of the toll vector measures the totalbenefit that is experienced by drivers who enter the arc aftertime t
The second component of the toll vector measures the cost tomaintain the (optimal) network structure of RTE (z∗)
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Toll Pricing Problem
Let =(y∗, z∗) denote the set of valid tolls with respect to systemoptimum solution (y∗, z∗).
MinRev Problem
min(β,ρ,γ)∈=(y∗,g∗,z∗)
(y∗)Tβ
MinCost Problem
min(β,ρ,γ,κop ,κinf )
(cop)Tκop + (c inf )Tκinf
s.t. (β, ρ, γ) ∈ =(y∗, g∗, z∗)
|βa(t,s∗a (t))| ≤ Mκopa(t), κ
opa(t) ≤ κinf
a ,
κopa(t) and κinf
a ∈ 0, 1,
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Toll Pricing Problem
Let =(y∗, z∗) denote the set of valid tolls with respect to systemoptimum solution (y∗, z∗).
MinRev Problem
min(β,ρ,γ)∈=(y∗,g∗,z∗)
(y∗)Tβ
MinCost Problem
min(β,ρ,γ,κop ,κinf )
(cop)Tκop + (c inf )Tκinf
s.t. (β, ρ, γ) ∈ =(y∗, g∗, z∗)
|βa(t,s∗a (t))| ≤ Mκopa(t), κ
opa(t) ≤ κinf
a ,
κopa(t) and κinf
a ∈ 0, 1,
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Toll Pricing Problem
Let =(y∗, z∗) denote the set of valid tolls with respect to systemoptimum solution (y∗, z∗).
MinRev Problem
min(β,ρ,γ)∈=(y∗,g∗,z∗)
(y∗)Tβ
MinCost Problem
min(β,ρ,γ,κop ,κinf )
(cop)Tκop + (c inf )Tκinf
s.t. (β, ρ, γ) ∈ =(y∗, g∗, z∗)
|βa(t,s∗a (t))| ≤ Mκopa(t), κ
opa(t) ≤ κinf
a ,
κopa(t) and κinf
a ∈ 0, 1,
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Toll Pricing Problem
Additional Constraints
Name ConstraintNonnegativity β(a,t,s∗a (t)) ≥ 0, ∀t ∈ [0,T ]Maximum toll β(a,t,s∗a (t)) ≤ βmax
a ,∀t ∈ [0,T ]Maximum subsidy −βmin
a ≤ β(a,t,s∗a (t)), ∀t ∈ [0,T ]Road restriction β(a,t,s∗a (t)) = 0, ∀t ∈ [0,T ]Time restriction β(a,t,s∗a (t)) = 0, ∀t ∈ [t1
a , t2a ]
Variability |β(a,t,s∗a (t)) − β(a,t+δ,s∗a (t+δ))| ≤ εa, ∀t ∈ [0,T ]
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Illustrative Examples
Parameters of the Problems: Network
Time horizon= [0, 30)
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Illustrative Examples
Parameters of the Problems: Travel Time Function
φa(xa(t)) = Aa + Ba(xa(t)/Ca)Da ,
where parameters Aa, Ba, Ca and Da, are random numbers.
Aa Ba Ca Da
U[1,2] U[2,8] U[200,300] random (1 or 2)
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Illustrative Examples
Parameters of the Problems: Demand Function
hkt = 1.1αtζ,
where the value of αt depends on time t, and ζ is a numberuniformly generated from the interval [20, 30].
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Computational Results
Total Collected Toll and Total Cost
Total Collected Toll Total CostA εa = 1 εa = 0.5 εa = 0.1 A εa = 1 εa = 0.5 εa = 0.1
MinRev(ε) 3854 4015 5157 No Sol. 2670 3298 3870 No Sol.MinCost(ε) 5433 5563 6978 No Sol. 1249 1624 2418 No Sol.
Number of Toll Collecting Centers
Number of Toll Collecting CentersA εa = 1 εa = 0.5 εa = 0.1
MinRev(ε) 17 16 17 N/AMinCost(ε) 8 9 12 N/A.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Computational Results
Total Collected Toll and Total Cost
Total Collected Toll Total CostA εa = 1 εa = 0.5 εa = 0.1 A εa = 1 εa = 0.5 εa = 0.1
MinRev(ε) 3854 4015 5157 No Sol. 2670 3298 3870 No Sol.MinCost(ε) 5433 5563 6978 No Sol. 1249 1624 2418 No Sol.
Number of Toll Collecting Centers
Number of Toll Collecting CentersA εa = 1 εa = 0.5 εa = 0.1
MinRev(ε) 17 16 17 N/AMinCost(ε) 8 9 12 N/A.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Computational Results
Toll Vector for different values of εa: MinRev Problem
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Discrete-Time DTA with Periodic Planning HorizonA Heuristic Algorithm for DTDTA-UDynamic Toll Pricing Framework
Computational Results
Toll Vector for different values of εa: MinCost Problem
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Conclusion
Supply Chain Problems
MIP formulations are equivalent to continuous bilinearproblems.Bilinear reduction problems can be used in heuristic algorithms.By applying a cutting plane method to the bilinear problems itis possible to find an exact solution of the problem.
Dynamic Traffic Assignment Problems
Initial SO formulation belongs to the class of nonlinear mixedinteger problems.Theoretical results suggest solving upper bounding MIPproblem.Bilinear relaxation provides a tighter bound than the LPrelaxation and can be used in heuristic algorithms.A toll pricing framework can be constructed based on anapproximate or exact solution of the SO problem.
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Supply Chain ProblemsDynamic Traffic Assignment Problems
Conclusion
Questions?
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