An Improved Dynamic Programming Decomposition Approach for Network Revenue Management

(1)

An Improved Dynamic Programming

Decomposition Approach for

Network Revenue Management

Dan Zhang

Leeds School of Business

University of Colorado at Boulder

(2)

Outline

Background

Network revenue management formulation

Classical dynamic programming decomposition

An improved dynamic programming decomposition

Numerical results

(3)

Network RM Formulation (Gallego and van Ryzin, 1997; Gallego et. al., 2004; Liu

and van Ryzin, 2008)

m resources with capacity c (an m-vector)

Capacity for resource i is c

i

.

n products

N = {1, . . . , n}

Fare for product j is f

j

Product consumption matrix A = [a

ij

]

Finite time horizon with length τ

In each period, there is one customer arrival with probability λ, and

no customer arrival with probability 1 − λ.

Given a set of products S ⊆ N, a customer chooses product j with

probability P

j

(S ).

No-purchase probability P

0

(S) = 1 −

P

j ∈S

P

j

(S).

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Applications

Industry

Resources

Products

Airlines

Scheduled flights

O-D itineraries at certain fare levels

(5)

Dynamic Programming Formulation

DP optimality equations:

v

t

(x ) = max

S⊆N(x )

n

λ

X

j ∈S

P

j

(S )(f

j

+ v

t+1

(x − A

j

))

+ (λP

0 (S ) + 1 − λ)v

t+1

(x )

o

,

∀t, x,

v

τ +1

(x ) = 0,

∀x.

Notations

v

t

(x ): DP value function

A

j

: resource incidence vector of product j

N(x ): {j ∈ N : x ≥ A

j

}

Curse of dimensionality: state space grows exponentially with

the number of resources

(6)

Dynamic Programming Formulation

DP optimality equations:

v

t

(x ) = max

S⊆N(x )

n

λ

X

j ∈S

P

j

(S )(f

j

+ v

t+1

(x − A

j

))

+ (λP

0 (S ) + 1 − λ)v

t+1

(x )

o

,

∀t, x,

v

τ +1

(x ) = 0,

∀x.

Notations

v

t

(x ): DP value function

A

j

: resource incidence vector of product j

N(x ): {j ∈ N : x ≥ A

j

}

(7)

Choice-based Deterministic Linear Program (CDLP)

z

CDLP

= max

h

X

S⊆N

λR(S )h(S )

X

S⊆N

λQ(S )h(S ) ≤ c,

(Resource constraint)

X

S⊆N

h(S ) ≤ τ,

(Time constraint)

h(S ) ≥ 0,

∀S ⊆ N.

(Non-negativity)

Replace stochastic demand with deterministic fluid with rate λ

Given offer set S ⊆ N

Total time S is offered: h(S )

Revenue from unit demand: R(S ) =

P

j ∈S

f

j

P

j

(S )

Consumption of resource i from unit demand:

Q

i

(S ) =

P

j ∈S

a

ij

P

j

(S )

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CDLP (Gallego et. al, 2004; Liu and van Ryzin, 2008)

CDLP can by efficiently solved for certain class of choice

models.

The vector of dual values π

∗

associated with resource

constraints can be used as “bid-prices” for resources

z

CDLP

provides an upper bound on revenue

Some recent references:

Talluri (2010): Concave programming formulation

Gallego, Ratliff, Shebalov (2011): Efficient reformulation

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Classical Dynamic Programming Decomposition

For each i , approximate the DP value function with

v

t

(x ) ≈

v

t,i

(x

i

)

|

{z

}

Value of the

i -th resource

+

X

k6=i

π

∗k

x

k

,

|

{z

}

Value of all

other resources

∀t, x.

Using the approximation in DP recursion leads to

v

t,i

(x

i

) =

max

S⊆N(xi,c−i)

X

j ∈S

λP

j

(S )

f

j

−

X

k6=i

a

kj

π

k∗

|

{z

}

Fare proration

+v

t+1,i

(x

i

− a

ij

)

+ (λP

0

(S ) + 1 − λ)v

t+1

(x

i

),

∀t, x

i

.

Compute offer sets dynamically using the approximate value function

v

t

(x ) ≈

X

i

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Classical Dynamic Programming Decomposition

For each i , approximate the DP value function with

v

t

(x ) ≈

v

t,i

(x

i

)

|

{z

}

Value of the

i -th resource

+

X

k6=i

π

∗k

x

k

,

|

{z

}

Value of all

other resources

∀t, x.

Using the approximation in DP recursion leads to

v

t,i

(x

i

) =

max

S⊆N(xi,c−i)

X

j ∈S

λP

j

(S )

f

j

−

X

k6=i

a

kj

π

k∗

|

{z

}

Fare proration

+v

t+1,i

(x

i

− a

ij

)

+ (λP

0

(S ) + 1 − λ)v

t+1

(x

i

),

∀t, x

i

.

Compute offer sets dynamically using the approximate value function

v

t

(x ) ≈

X

i

(11)

Classical Dynamic Programming Decomposition

For each i , approximate the DP value function with

v

t

(x ) ≈

v

t,i

(x

i

)

|

{z

}

Value of the

i -th resource

+

X

k6=i

π

∗k

x

k

,

|

{z

}

Value of all

other resources

∀t, x.

Using the approximation in DP recursion leads to

v

t,i

(x

i

) =

max

S⊆N(xi,c−i)

X

j ∈S

λP

j

(S )

f

j

−

X

k6=i

a

kj

π

k∗

|

{z

}

Fare proration

+v

t+1,i

(x

i

− a

ij

)

+ (λP

0

(S ) + 1 − λ)v

t+1

(x

i

),

∀t, x

i

.

Compute offer sets dynamically using the approximate value function

v

t

(x ) ≈

X

i

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Classical Dynamic Programming Decomposition

A DP with m-dimensional state space is reduced to m

one-dimensional DPs, one for each resource.

101

4 states

(Assume 100 seats per flight)

⇒

4 × 101 states

Variants of the approach are widely used in practice.

Review: Talluri and van Ryzin (2004a)

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Classical Dynamic Programming Decomposition

A DP with m-dimensional state space is reduced to m

one-dimensional DPs, one for each resource.

101

4 states

(Assume 100 seats per flight)

⇒

4 × 101 states

Variants of the approach are widely used in practice.

Review: Talluri and van Ryzin (2004a)

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DP Decomposition Bounds

Proposition (Zhang and Adelman, 2009)

The following relationships hold:

(i)

v

t

(x ) ≤ min

l =1,...,m

n

v

t,l

(x

l

) +

P

k6=l

π

∗

k

x

k

o

≤

v

t,i

(x

i

) +

P

_k6=i

π

_k

∗

x

k

, ∀i , t, x ;

(ii)

v

1 (c) ≤ v

1,i

(c

i

) +

P

k6=i

π

∗

k

c

k

≤ z

CDLP

, ∀i .

Decomposition value for each leg provides an upper bound on

revenue

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Linear Programming Formulation of DP (Adelman, 2007)

min

{v

t

(·)}

∀t

v

1 (c)

v

t

(x )

≥

X

j ∈S

λP

j

(S )(f

j

+

v

t+1

(x − A

j

)

+ (λP

0 (S ) + 1 − λ)

v

t+1

(x )

,

∀t, x, S ⊆ N(x).

Huge number of decision variables and constraints

Functional approximation idea: use a parameterized

representation of the value function to reduce the number of

decision variables

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Linear Programming Formulation of DP (Adelman, 2007)

min

{v

t

(·)}

∀t

v

1 (c)

v

t

(x )

≥

X

j ∈S

λP

j

(S )(f

j

+

v

t+1

(x − A

j

)

+ (λP

0 (S ) + 1 − λ)

v

t+1

(x )

,

∀t, x, S ⊆ N(x).

Huge number of decision variables and constraints

Functional approximation idea: use a parameterized

representation of the value function to reduce the number of

decision variables

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Linear Programming Formulation of DP (Adelman, 2007)

min

{v

t

(·)}

∀t

v

1 (c)

v

t

(x )

≥

X

j ∈S

λP

j

(S )(f

j

+

v

t+1

(x − A

j

)

+ (λP

0 (S ) + 1 − λ)

v

t+1

(x )

,

∀t, x, S ⊆ N(x).

Huge number of decision variables and constraints

Functional approximation idea: use a parameterized

representation of the value function to reduce the number of

decision variables

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The Affine Functional Approximation (Zhang and Adelman, 2009)

Affine approximation is given by

v

t

(x )

≈

θ

t

+

X

i

V

t,i

x

i

,

∀t, x.

(1)

Using (1) in the linear programming formulation leads to

min

θ,V

θ

1

+

X

i

V

1,i

c

i

θ

t

+

X

i

V

t,i

x

i

≥

X

j ∈S

λP

j

(S )

f

j

+

θ

t+1

+

X

i

V

t+1,i

(x

i

− a

ij

)

!

+ (λP

0

(S ) + 1 − λ)

θ

t+1

+

X

i

V

t+1,i

x

i

!

,

∀t, x, S ⊆ N(x).

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The Affine Functional Approximation (Zhang and Adelman, 2009)

Affine approximation is given by

v

t

(x )

≈

θ

t

+

X

i

V

t,i

x

i

,

∀t, x.

(1)

Using (1) in the linear programming formulation leads to

min

θ,V

θ

1

+

X

i

V

1,i

c

i

θ

t

+

X

i

V

t,i

x

i

≥

X

j ∈S

λP

j

(S )

f

j

+

θ

t+1

+

X

i

V

t+1,i

(x

i

− a

ij

)

!

+ (λP

0

(S ) + 1 − λ)

θ

t+1

+

X

i

V

t+1,i

x

i

!

,

∀t, x, S ⊆ N(x).

(20)

The Affine Functional Approximation

The dual program is given by

zP1= max_Y X t,x ,S⊆N(x )   X j ∈S λPj(S)fj   Yt,x ,S X x ,S⊆N(x ) xiYt,x ,S= ( ci, if t = 1, P x ,S⊆N(x ) xi−Pj ∈SλPj(S)aij Yt−1,x ,S, ∀t = 2, . . . , τ ∀i , t, X x ,S⊆N(x ) Yt,x ,S= _1, _{if t = 1,} P x ,S⊆N(x )Yt−1,x ,S, ∀t = 2, . . . , τ. Y ≥ 0.

Due to the large number of columns, solving the linear program

above still requires considerable computational effort.

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The Affine Functional Approximation

The dual program is given by

zP1= max_Y X t,x ,S⊆N(x )   X j ∈S λPj(S)fj   Yt,x ,S X x ,S⊆N(x ) xiYt,x ,S= ( ci, if t = 1, P x ,S⊆N(x ) xi−Pj ∈SλPj(S)aij Yt−1,x ,S, ∀t = 2, . . . , τ ∀i , t, X x ,S⊆N(x ) Yt,x ,S= _1, _{if t = 1,} P x ,S⊆N(x )Yt−1,x ,S, ∀t = 2, . . . , τ. Y ≥ 0.

Due to the large number of columns, solving the linear program

above still requires considerable computational effort.

(22)

Functional Approximation Approaches for Network RM

Citation Choice Model Functional approximation Solution strategy Adelman (2007) Independent demand Affine Column generation Zhang and Adelman (2009) MNLD Affine Column generation

Zhang (2011) MNLD Nonlinear non-separable CDLP+Simultaneous DP

Liu and van Ryzin (2008) MNLD Separable (fare proration) CDLP+DP Decomposition Miranda Bront et. al. (2009) MNLO Separable (fare proration) CDLP+DP Decomposition Farias and Van Roy (2008) Independent demand Separable concave Constraint sampling Meissner and Strauss (2012) MNLD Separable concave Column generation Kunnumkal and MNLD Separable (fare proration) Convex programming

Topaloglu (2011) +DP Decomposition

Tong and Topaloglu (2011) Independent demand Affine Reduction

+ Constraint generation Vossen and Zhang Independent demand Affine Reduction

+ MNLD + Dynamic disaggregation

MNLD: Multinomial logit model with disjoint consideration sets MNLO: Multinomial logit model with overlapping consideration sets

(23)

Research Questions

Computational cost:

ADP (affine or separable concave approximation)

classical DP decomposition

How can we balance solution quality with solution time?

Can we improve the classical DP decomposition?

(24)

Research Questions

Computational cost:

ADP (affine or separable concave approximation)

classical DP decomposition

How can we balance solution quality with solution time?

Can we improve the classical DP decomposition?

(25)

A Strong Functional Approximation (Zhang, 2011)

v

t

(x )

≈

min

i







ˆ

v

t,i

(x

i

) +

X

k6=i

π

_k

∗

x

k







,

∀t, x.

Nonlinear

and

non-separable

functional approximation

Each value v

t

(x ) is approximated by a single value across legs

Motivated by the decomposition bounds (Zhang and

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A Nonlinear Optimization Problem

Using the new functional approximation leads to

z

NLP

=

min

ˆ vt,i(·)∀t,i

min

i







ˆ

v

1,i

(c

i

) +

X

k6=i

π

k∗

c

k







min

i







ˆ

v

t,i

(x

i

) +

X

k6=i

π

k∗

x

k







≥

X

j ∈S

λP

j

(S )



f

j

+

min

l







ˆ

v

t+1,l

(x

l

− a

lj

) +

X

k6=l

π

∗k

(x

k

− a

kj

)











+ (λP

0

(S ) + 1 − λ)

min

l







ˆ

v

t+1,l

(x

l

) +

X

k6=l

π

k∗

x

k







,

∀t, x, S ⊆ N(x).

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A Restricted Optimization Problem

Step 1: Writing each constraint as m equivalent constraints

Step 2: Restricting the constraints so that each constraint

only involves one resource

The restricted problem provides a relaxed bound:

Proposition

The objective value of the restricted program, z

d

NLP

, is bigger than

z

NLP

.

(28)

An Equivalent Simultaneous Dynamic Program

ˆ

v

t,i∗

(x

i

) =

max

S⊆N(xi,c−i)

X

j ∈S

λP

j

(S )

f

j

+ min

(

ˆ

v

t+1,i∗

(x

i

− a

ij

) −

X

k6=i

π

k∗

a

kj

,

min

l 6=i

max

0≤yl≤cl−alj

[

ˆ

v

t+1,l∗

(y

l

)

− y

l

π

∗l

] −

X

k

a

kj

π

k∗

+ π

∗ i

x

i

)!

+ (λP

0

(S ) + 1 − λ) min

ˆ

v

t+1,i∗

(x

i

), min

l 6=i

max

0≤yl≤cl

[

ˆ

v

t+1,l∗

(y

l

)

− π

∗ l

y

l

] + π

∗ i

x

i

∀i , t, x

i

.

DP recursion for resource i involves values from all other

resources

The dynamic program

is equivalent to the restricted nonlinear program

can be solved efficiently via a

simultaneous

dynamic

programming algorithm

(29)

An Equivalent Simultaneous Dynamic Program

ˆ

v

t,i∗

(x

i

) =

max

S⊆N(xi,c−i)

X

j ∈S

λP

j

(S )

f

j

+ min

(

ˆ

v

t+1,i∗

(x

i

− a

ij

) −

X

k6=i

π

k∗

a

kj

,

min

l 6=i

max

0≤yl≤cl−alj

[

ˆ

v

t+1,l∗

(y

l

)

− y

l

π

∗l

] −

X

k

a

kj

π

k∗

+ π

∗ i

x

i

)!

+ (λP

0

(S ) + 1 − λ) min

ˆ

v

t+1,i∗

(x

i

), min

l 6=i

max

0≤yl≤cl

[

ˆ

v

t+1,l∗

(y

l

)

− π

∗ l

y

l

] + π

∗ i

x

i

∀i , t, x

i

.

DP recursion for resource i involves values from all other

resources

The dynamic program

is equivalent to the restricted nonlinear program

can be solved efficiently via a

simultaneous

dynamic

programming algorithm

(30)

New Bounds

Proposition (Zhang, 2011)

Let {ˆ

v

_t,i

∗

(·)}

∀t,i ,x

i

be the optimal solution from the simultaneous

dynamic program. The following results hold:

(i)

v

ˆ

_t,i

∗

(x

i

) ≤ v

t,i

(x

i

), ∀i , x

i

;

(ii)

v

1 (c) ≤ z

NLP

≤ z

NLP

d

= min

i

n

ˆ

v

_1,i

∗

(c

i

) +

P

_k6=i

π

_k

∗

c

k

o

≤

min

i

n

v

1,i

(c

i

) +

P

_k6=i

π

∗

_k

c

k

o

≤ z

_CDLP

.

The simultaneous dynamic program provides tighter bounds

on revenue than the classical decomposition.

(31)

Recap

High dimensional dynamic program

m

Large scale linear program

⇓

Large scale nonlinear program with nonlinear constraints

⇓

Restricted nonlinear program with nonlinear constraints

m

(32)

Recap

High dimensional dynamic program

m

Large scale linear program

⇓

Large scale nonlinear program with nonlinear constraints

⇓

Restricted nonlinear program with nonlinear constraints

m

(33)

Recap

High dimensional dynamic program

m

Large scale linear program

⇓

Large scale nonlinear program with nonlinear constraints

⇓

Restricted nonlinear program with nonlinear constraints

m

(34)

Recap

High dimensional dynamic program

m

Large scale linear program

⇓

Large scale nonlinear program with nonlinear constraints

⇓

Restricted nonlinear program with nonlinear constraints

m

(35)

Recap

High dimensional dynamic program

m

Large scale linear program

⇓

Large scale nonlinear program with nonlinear constraints

⇓

Restricted nonlinear program with nonlinear constraints

m

(36)

Comparison: Classical vs. Improved Approaches

Classical dynamic programming decomposition:

Solve m single-leg DPs Solve CDLP Fare proration Static bid-prices Prorated fares

Network effects only captured through fare proration

Improved dynamic programming decomposition:

Solve one simultaneous DP

Solve CDLP Static bid-prices

(37)

Comparison: Classical vs. Improved Approaches

Classical dynamic programming decomposition:

Network effects only captured through fare proration

Improved dynamic programming decomposition:

(38)

Comparison: Classical vs. Improved Approaches

Classical dynamic programming decomposition:

Network effects only captured through fare proration

Improved dynamic programming decomposition:

(39)

Comparison: Classical vs. Improved Approaches

Classical dynamic programming decomposition:

Network effects only captured through fare proration

Improved dynamic programming decomposition:

(40)

Computational Study: Problem Instances

Randomly generated hub-and-spoke instances

Number of non-hub locations (flights) in the set {4, 8, 16, 24}

Number of periods in the set {100, 200, 400, 800}

Two products for each possible itinerary

Multinomial Logit Choice Model with Disjoint Consideration Sets

(MNLD)

Largest problem instance: 24 non-hub locations (flights), 336

products, 800 periods

(41)

Numerical Study: Policies

DCOMP1

: the new decomposition approach where the

approximation

v

t

(x ) ≈

m

X

i =1

ˆ

v

_t,i

∗

(x

i

), ∀t, x

is used to compute control policies.

DCOMP

: the classical dynamic programming decomposition

CDLP

: static bid-price policy based on the dual values of resource

constraints in CDLP

CDLP10

: A version of CDLP that resolves 10 times with equally

spaced resolving intervals

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Computational Time

Case # Parameters Capacity Load CPU seconds DCOMP1−DCOMP DCOMP per leg factor _CDLP _DCOMP _DCOMP1

A1 (100,4,4,16) 10 1.17 0.16 2.03 2.75 35.38% A2 (200,4,4,16) 20 1.27 0.23 7.89 10.88 37.82% A3 (400,4,4,16) 40 1.19 0.16 31.92 43.73 37.00% A4 (800,4,4,16) 80 1.28 0.20 127.66 174.48 36.68% A5 (100,8,8,48) 5 1.43 1.52 5.75 7.47 29.89% A6 (200,8,8,48) 10 1.36 0.72 22.83 29.58 29.57% A7 (400,8,8,48) 20 1.35 1.61 91.67 118.92 29.73% A8 (800,8,8,48) 40 1.21 0.72 362.84 471.73 30.01% A9 (100,16,16,160) 2 1.65 4.64 15.09 19.42 28.67% A10 (200,16,16,160) 5 1.45 4.69 75.84 96.97 27.85% A11 (400,16,16,160) 10 1.29 2.92 303.66 388.97 28.10% A12 (800,16,16,160) 20 1.40 3.64 1218.67 1560.19 28.02% A13 (100,24,24,336) 1 1.45 3.69 24.72 31.81 28.70% A14 (200,24,24,336) 2 1.35 4.59 98.52 127.36 29.28% A15 (400,24,24,336) 5 1.29 4.39 492.73 630.84 28.03% A16 (800,24,24,336) 10 1.38 4.23 1978.14 2532.20 28.01%

(43)

Bound Performance

Case # CDLP DCOMP DCOMP1 Bound improvement %-difference across legs bound bound bound _%-CDLP _%-DCOMP _DCOMP _DCOMP1 A1 24078.90 23985.56 22900.49 5.15% 4.74% 4.46% 0.00% A2 48367.58 48328.43 47588.56 1.64% 1.55% 1.87% 0.36% A3 89312.44 87576.49 86729.90 2.98% 0.98% 2.36% 0.00% A4 213102.50 211854.85 211087.37 0.95% 0.36% 0.58% 0.00% A5 32521.30 31029.90 30726.17 5.84% 0.99% 3.18% 0.05% A6 70541.63 68760.67 68617.41 2.80% 0.21% 2.18% 0.22% A7 107831.01 106339.36 106153.32 1.58% 0.18% 1.09% 0.00% A8 216080.83 212915.61 212848.05 1.52% 0.03% 1.49% 0.00% A9 26347.76 24953.24 24764.75 6.39% 0.76% 4.69% 0.00% A10 60629.35 58489.12 58118.33 4.32% 0.64% 2.95% 0.03% A11 101616.47 100069.27 99771.63 1.85% 0.30% 1.52% 0.01% A12 224780.69 222558.53 222231.72 1.15% 0.15% 0.94% 0.00% A13 13074.04 11845.73 10386.38 25.88% 14.05% 10.37% 0.00% A14 26296.19 24926.41 24373.33 7.89% 2.27% 5.50% 0.00% A15 74112.13 72089.14 71617.55 3.48% 0.66% 2.80% 0.03% A16 131457.79 129589.28 129273.91 1.69% 0.24% 1.44% 0.00%

(44)

Bounds from Individual Legs

132000 131000 131500 132000

al

Legs

130500 131000 131500 132000

dividual

Legs

129500 130000 130500 131000 131500 132000

s

fr

om

individual

Legs

DCOMP

DCOMP1

128500 129000 129500 130000 130500 131000 131500 132000

Bounds

fr

om

individual

Legs

DCOMP

DCOMP1

128000 128500 129000 129500 130000 130500 131000 131500 132000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Bounds

fr

om

individual

Legs

Leg

DCOMP

DCOMP1

128000 128500 129000 129500 130000 130500 131000 131500 132000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Bounds

fr

om

individual

Legs

Leg

DCOMP

DCOMP1

128000 128500 129000 129500 130000 130500 131000 131500 132000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Bounds

fr

om

individual

Legs

Leg

DCOMP

DCOMP1

(45)

Bounds from Individual Legs

132000 131000 131500 132000

al

Legs

130500 131000 131500 132000

dividual

Legs

129500 130000 130500 131000 131500 132000

s

fr

om

individual

Legs

DCOMP

DCOMP1

128500 129000 129500 130000 130500 131000 131500 132000

Bounds

fr

om

individual

Legs

DCOMP

DCOMP1

128000 128500 129000 129500 130000 130500 131000 131500 132000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Bounds

fr

om

individual

Legs

Leg

DCOMP

DCOMP1

128000 128500 129000 129500 130000 130500 131000 131500 132000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Bounds

fr

om

individual

Legs

Leg

DCOMP

DCOMP1

128000 128500 129000 129500 130000 130500 131000 131500 132000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Bounds

fr

om

individual

Legs

Leg

DCOMP

DCOMP1

(46)

A Hub-and-spoke Network with 2 Non-Hub Locations

Case # τ Load Capacity DCOMP1 OPT-GAP DCOMP1 Revenue Gains factor per leg REV _%-CDLP _%-CDLP10 _%-DCOMP B1 100 2.40 4 5775.47 -3.17% 182.90% 8.56% 2.74% B2 200 2.13 9 13262.92 -2.04% 209.79% 5.74% 4.11% B3 400 2.13 18 25456.41 -6.53% -1.00% -0.33% 0.03% B4 800 2.13 36 53946.59 -1.20% 212.87% 4.41% 7.17% B5 100 1.60 6 8034.27 -7.25% 48.88% 3.00% 0.09% B6 200 1.60 12 17318.40 -2.31% 13.42% 3.62% 5.77% B7 400 1.60 24 35472.06 -1.25% 311.96% 4.78% 7.97% B8 800 1.60 48 65618.95 -9.20% 42.16% -4.49% 0.00% B9 100 1.37 7 9269.56 -5.85% 13.27% 1.33% 1.94% B10 200 1.28 15 20521.01 -3.70% 15.05% 0.98% 4.65% B11 400 1.28 30 42471.91 -2.10% 15.46% 1.70% 8.14% B12 800 1.28 60 86841.58 -1.15% 15.47% 2.45% 5.52% B13 100 1.07 9 11107.51 -1.68% 2.89% 0.47% 0.34% B14 200 1.07 18 23268.75 -0.80% 4.84% 1.14% 0.11% B15 400 1.07 36 47824.97 -0.27% 22.11% 2.22% 0.04% B16 800 1.07 72 96993.79 -0.08% 7.92% 2.14% 0.01% B17 100 0.96 10 11854.02 -0.85% 23.82% 1.46% 0.18% B18 200 0.91 21 25259.70 -0.07% 2.61% 1.24% 0.04% B19 400 0.91 42 51593.10 0.03% 30.16% 2.48% 0.01%

(47)

DCOMP1 Percentage Revenue Gain vs. Load Factor

0.5

1

1.5

2

2.5 −5

0

5

10 Load factor

DCOMP1 percentage revenue gain

%−CDLP10

%−DCOMP

(48)

DCOMP1 Percentage Revenue Gain vs. Load Factor

0.5

1

1.5

2

2.5 −5

0

5

10 Load factor

DCOMP1 percentage revenue gain

%−CDLP10

%−DCOMP

(49)

DCOMP1 Percentage Revenue Gain vs. Number of Periods

0

100

200

300

400

500

600

700

800

900 −5

0

5

10 Number of periods

DCOMP1 percentage revenue gain

%−CDLP10

%−DCOMP

(50)

DCOMP1 Percentage Revenue Gain vs. Number of Periods

0

100

200

300

400

500

600

700

800

900 −5

0

5

10 Number of periods

DCOMP1 percentage revenue gain

%−CDLP10

%−DCOMP

(51)

A Hub-and-spoke Network with 4 Non-Hub Locations

Case # τ Load Capacity DCOMP1 OPT-GAP DCOMP1 Revenue Gains factor per leg REV _%-CDLP _%-CDLP10 _%-DCOMP C1 100 1.99 6 16795.66 -5.08% 14.60% 0.46% 0.10% C2 200 1.99 12 35028.96 -2.63% 52.79% 1.95% 1.32% C3 400 1.99 24 70163.84 -3.35% 12.35% -0.11% 0.03% C4 800 1.99 48 143921.85 -1.34% 52.31% 1.36% 0.66% C5 100 1.49 8 21860.34 -4.23% 44.57% 1.71% 1.96% C6 200 1.49 16 45171.83 -2.54% 14.18% 1.72% 2.67% C7 400 1.49 32 88532.95 -5.29% 29.01% -2.27% -0.81% C8 800 1.49 64 184410.85 -1.79% 1.02% 0.49% 0.18% C9 100 1.19 10 26270.03 -5.00% 4.98% 1.46% 2.67% C10 200 1.19 20 54509.68 -3.02% 4.18% 1.61% 4.11% C11 400 1.19 40 111520.14 -1.79% 3.44% 1.51% 4.43% C12 800 1.19 80 226059.91 -1.05% 57.07% 1.92% 3.08% C13 100 1.00 12 29208.18 -5.00% 1.30% -0.90% 0.21% C14 200 1.00 24 61175.75 -3.02% 2.18% 0.21% 0.08% C15 400 1.00 48 125854.79 -1.79% 6.38% 0.74% 0.00% C16 800 1.00 96 256236.11 -1.00% 2.66% 0.89% -0.06% C17 100 0.85 14 32057.27 -2.55% 2.50% -0.81% 0.44% C18 200 0.85 28 66527.87 -1.28% 3.05% -0.03% 0.20% C19 400 0.85 56 135897.71 -0.51% 3.24% 0.26% 0.05% C20 800 0.85 112 274817.89 -0.17% 3.21% 0.41% 0.02%