Yield Management in Freight Transportation - WordPress.com · Introduction Models Numerical Results...

IntroductionModels

Numerical Results

Yield Management in Freight Transportation

Long Gao

joint work with Michael F. Gorman

University of California, RiversideUniversity of Dayton

INFORMS, October, 2008

Long Gao Yield Management in Freight Transportation

IntroductionModels

Numerical Results

Outline

1 IntroductionMotivationsResearch Questions

2 ModelsFormulation and the Optimal PolicyComputational Feasibility

3 Numerical ResultsProfit PotentialsPolicy Robustness


IntroductionModels

Numerical Results

MotivationsResearch Questions

Outline





IntroductionModels

Numerical Results


Background: intermodal container transportation

Examples: CSX Intermodal, etc.


IntroductionModels

Numerical Results


Load tendering process: accept or reject?

Accept now or wait for the future more profitable orders?Long Gao Yield Management in Freight Transportation

IntroductionModels

Numerical Results


Motivations

Current: FCFSfairness, ’all business is good business’insufficient IT

Problems:limited capacity, unable to capture the most profitableordersservice suffers, especially for the important customersreal-time revenue management (RM) is essentiallynon-existent

Difference from airline RM models:fixed prices by contractscapacity availability spanned over time, returned or reroutedinventoriable capacity, intertemporal substitutionsimultaneous arrivals, not lower-before-highoperate in either batch mode or real-time mode


IntroductionModels

Numerical Results


Research Questions

What is the optimal (OPT) policy for load tendering infreight transportation?Is OPT computationally feasible for real world applications?What are the profit potentials of OPT and Airline RM(ARM) in freight transportation?How robust are these policies, if forecasts are biased?


IntroductionModels

Numerical Results

Formulation and the Optimal PolicyComputational Feasibility

Outline





IntroductionModels

Numerical Results


Model Assumptions

T-period load tendering system for an originPlanned capacity {Kt} (returned or rerouted) arrives at thebeginning of each periodCapacity are planned before execution horizon, noreplenishment within T periodsMultiple classes of orders bring in profit marginsr1 > · · · > rJ

Each order requires one unit of capacityNewly arrival orders Nt = {Ni

t : i ∈ I} with probability p(Nt)

Accepted orders xt = {xit} must be fulfilled within L periods

Operate in either batch mode, or real-time mode (no morethan 1 order per period)Objective: maximize the expected total profits over theexecution horizon T


IntroductionModels

Numerical Results


A Markov Decision Process Formulation

Vt(It, Nt) = maxxt∈At

{r · xt − ht(It + Kt − |xt|)+

∑Nt−1

p(Nt−1)× Vt−1 (It−1, Nt−1)

}, (1)

The action space At is defined by

0 ≤ xt ≤ Nt, Demand (2)|xt| ≤ It + [K]tt−L. Lead time supply (3)

Net inventory is updated by

It−1 = It + Kt − |xt|. (4)


IntroductionModels

Numerical Results


Optimal Policy for Batch Mode, Gao and Xu (2008)

Optimal order acceptance policyAccept in an increasing order of the indexReject class i if class i − 1 are not fully acceptedAccept class i until

1 all N it are accepted (demand)

2 cumulative leadtime capacity for i is exhausted (supply)3 the net capacity protection level is reached (inter-temporal

substitution)

Formally, for class i ∈ I, the optimal acceptance is

x̂it = min

Ni

t ,[It + [K]tt−L − [N]i−1

1

]+,[

It + Kt − [N]i−11 − ηi

t]+

. (5)


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Optimal Policy for Real-Time Mode

Protection level controlAccept an incoming class j order if

1 net inventory It ≥ ηjt;

2 lead time inventory It + [K]tt−L > 0

Bid price controlAccept an incoming class j order if

1 profit margin rj ≥ ∂Ht(I);2 lead time inventory It + [K]tt−L > 0

Bid price ∂Ht(I) := Ht(I)− Ht(I − 1), whereHt(I) = −ht(I) +

∑j pj

tVt−1(I)

Protection level ηjt := min { I : rj ≥ ∂Ht(I + Kt) }


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Numerical Results


Computational Feasibility

The policy characterization results in state independentthreshold ηi

tEase the “Curse of Dimensionality” for such multi-dim MDPO(T × I × J) v.s. O(T × I × NJ)

Challenges for large size problems: if N = 100, J = 50, forbatch mode, we need to evaluate 10100 times of Vt(It, Nt) tocompute

EVt(It, Nt) =N∑

N1t =0

· · ·N∑

NJt =0

p(N1t , N2

t , . . . , NJt )Vt(It, Nt)

Hybrid DP-simulation method: in each backward inductioniteration, evaluate EVt by Monte Carlo simulation.A 50-class problem can be solved within a minute on a PCwith 3G Hz CPU.


IntroductionModels

Numerical Results

Profit PotentialsPolicy Robustness

Outline





IntroductionModels

Numerical Results


Policy Comparison

OPT: Optimal policy, account for future supply process,and intertemporal substitutionARM: differentiate classes within a period, but ignore futuredemand and supply, no intertemporal substititionFCFS: accept as much as possible, ignore classdifferentiation, independent of forecastsCapacity Factor: imbalance of supply and demand,

ρ :=Total Capacity

Total Mean DemandControl: Bid-prices of OPT and ARM;Profit Improvement: percentage difference of total profits,e.g.,

∆VOPT :=VOPT − VFCFS

VFCFS × 100%


IntroductionModels

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How are ARM and OPT are different in bid price?

Figure: Origin CHWI: ρ = 0.65, J = 42

0 50 100 150 200 250 3000

1

2

3

4

5

6

7

Inventory Level: I

Bid

Price

:∂H

14(I

)

ARM

OPT

ARM and OPTcoincide for lowercapacity levelARM prices lower forhigher capacity levelARM does notaccount for futuresupply and demand,thus ignoresintertemporalsubstitution


IntroductionModels

Numerical Results


How will bid prices change w.r.t. supply condition?

Figure: Origin CHWI: ρ = 0.65, J = 42

0 50 100 150 200 250 3000

1

2

3

4

5

6

7

Inventory Level: I

Bid

Price

:∂H

14(I

)

ARM: 70%, 100%, 130%Kt

OPT: 70%Kt

OPT: 100%Kt

OPT: 130%Kt

130%Kt

70%Kt

100%Kt

70%, 100%, 130%Kt

ARM is insensitive tosupply processFor OPT, bid-pricedecreases as thefuture supplyincreasesLess likely to sell inthe futureCall for better supplyforecasts


IntroductionModels

Numerical Results


Data from June 7 to June 20, 2007

Capacity Mean # of Price RangeOrigin factor ρ Demand classes min max

1 0.09 6.29 9 0.04 3.532 0.14 2.61 3 0.07 5.303 0.62 3.33 6 0.06 5.714 0.65 1.32 6 0.01 0.735 0.72 0.89 3 0.07 8.846 0.74 129.87 51 0.05 6.527 0.78 13.86 9 0.08 8.458 0.79 32.08 21 0.03 4.799 0.79 7.13 3 0.03 2.86

10 0.84 1.52 3 0.01 0.7011 0.93 51.37 42 0.04 6.6812 1.04 45.49 24 0.04 5.7213 1.32 14.39 6 0.05 5.2614 1.38 29.41 27 0.04 7.6715 1.59 37.99 18 0.05 6.6916 1.59 8.34 6 0.04 7.4517 1.70 12.34 12 0.04 4.9018 1.71 61.40 33 0.05 8.5019 1.74 68.54 39 0.06 8.0320 1.77 5.13 12 0.02 1.9721 1.98 73.37 27 0.04 9.5422 2.00 19.97 12 0.04 5.5723 2.77 24.50 15 0.03 4.1224 5.21 2.32 3 0.04 7.01

14 days: 06/07 ∼06/20, 200724 major originsranked by capacityfactor ρ

11 shortagesMax 51 classesPoisson demand


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Supply and demand processes

2 4 6 8 10 12 14

50

100

150

200

250

300

350

400

450

500

550

Remaining Days: t

Supply

Pro

cess

:K

t

2 4 6 8 10 12 140

20

40

60

80

100

120

140

160

180

Remaining Days: t

Dem

and

Pro

cess

es:

µt


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Profit Improvements

0 5 10 15 20 250

5

10

15

20

25

30

Origin

Pro

fits

Impro

vem

ent

%

ARM

OPT+ρ < 1 ρ ≥ 1

ARM: 5 ∼ 30%improvement overFCFS for supplyshortageOPT: additional2 ∼ 20%improvement overARM for ρ < 1

11 origins havesignificant gains byusing RM andaccounting forinter-temporalsubstitution


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Profit Improvements for 70%, 50%Kt

Figure: Improvement for 70%Kt

0 5 10 15 20 250

10

20

30

40

50

60

Origin

Pro

fits

Impro

vem

ent

%

ARM

OPT+

ρ < 1 ρ ≥ 1

Figure: Improvement for 50%Kt

0 5 10 15 20 250

10

20

30

40

50

60

70

80

90

Origin

Pro

fits

Impro

vem

ent

%

ARM

OPT+

ρ ≥ 1ρ < 1


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How robust is the optimal policy?

What if the forecast is inaccurate?Is OPT still better than ARM and FCFS?Which error is more detrimental?


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Robustness for Systematic Forecast Errors

Forecast Error OPT dominance % Worst of ∆VOPT%

Type ∆µ% FCFS ARM FCFS ARMUnderestimate −40% 100.00 100.00 0.00 0.00

−20% 100.00 100.00 0.00 0.00Overestimate 20% 95.83 87.50 −0.03 −1.02

40% 87.50 70.83 −0.89 −4.11

Mean 95.83 89.58Worst Case −0.89 −4.11

OPT is robust for small to moderate forecast errors.Overestimation is more harmful due to over rejection,unutilized capacity


IntroductionModels

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Conclusions

Formulate the load tendering problem for freighttransportation that accounts for the supply process andinter-temporal substitution.Characterize the optimal policy for both bath mode andreal-time mode control.Develop hybrid DP-simulation algorithm for industrial sizeproblems.Using real data, we gain empirical understanding of theprofit potentials of RM techniques, and additional gainsfrom accounting for intermodal substitutionOptimal policy is robust for small to moderate forecasterrors.RM makes money for freight transportation!


IntroductionModels

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Future Research

Multiple types of capacity, multiple shipment datesEffects of short term forecasts on strategic or tacticalresources planningRandom supply processesOther Applications: Car rental, ATP manufacturing, etc.


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