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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
Long Gao Yield Management in Freight Transportation
IntroductionModels
Numerical Results
MotivationsResearch Questions
Outline
1 IntroductionMotivationsResearch Questions
2 ModelsFormulation and the Optimal PolicyComputational Feasibility
3 Numerical ResultsProfit PotentialsPolicy Robustness
Long Gao Yield Management in Freight Transportation
IntroductionModels
Numerical Results
MotivationsResearch Questions
Background: intermodal container transportation
Examples: CSX Intermodal, etc.
Long Gao Yield Management in Freight Transportation
IntroductionModels
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MotivationsResearch Questions
Load tendering process: accept or reject?
Accept now or wait for the future more profitable orders?Long Gao Yield Management in Freight Transportation
IntroductionModels
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MotivationsResearch Questions
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
Long Gao Yield Management in Freight Transportation
IntroductionModels
Numerical Results
MotivationsResearch Questions
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?
Long Gao Yield Management in Freight Transportation
IntroductionModels
Numerical Results
Formulation and the Optimal PolicyComputational Feasibility
Outline
1 IntroductionMotivationsResearch Questions
2 ModelsFormulation and the Optimal PolicyComputational Feasibility
3 Numerical ResultsProfit PotentialsPolicy Robustness
Long Gao Yield Management in Freight Transportation
IntroductionModels
Numerical Results
Formulation and the Optimal PolicyComputational Feasibility
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
Long Gao Yield Management in Freight Transportation
IntroductionModels
Numerical Results
Formulation and the Optimal PolicyComputational Feasibility
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)
Long Gao Yield Management in Freight Transportation
IntroductionModels
Numerical Results
Formulation and the Optimal PolicyComputational Feasibility
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)
Long Gao Yield Management in Freight Transportation
IntroductionModels
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Formulation and the Optimal PolicyComputational Feasibility
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) }
Long Gao Yield Management in Freight Transportation
IntroductionModels
Numerical Results
Formulation and the Optimal PolicyComputational Feasibility
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.
Long Gao Yield Management in Freight Transportation
IntroductionModels
Numerical Results
Profit PotentialsPolicy Robustness
Outline
1 IntroductionMotivationsResearch Questions
2 ModelsFormulation and the Optimal PolicyComputational Feasibility
3 Numerical ResultsProfit PotentialsPolicy Robustness
Long Gao Yield Management in Freight Transportation
IntroductionModels
Numerical Results
Profit PotentialsPolicy Robustness
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%
Long Gao Yield Management in Freight Transportation
IntroductionModels
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Profit PotentialsPolicy Robustness
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
Long Gao Yield Management in Freight Transportation
IntroductionModels
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Profit PotentialsPolicy Robustness
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
Long Gao Yield Management in Freight Transportation
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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
Long Gao Yield Management in Freight Transportation
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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
Long Gao Yield Management in Freight Transportation
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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
Long Gao Yield Management in Freight Transportation
IntroductionModels
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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
Long Gao Yield Management in Freight Transportation
IntroductionModels
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Profit PotentialsPolicy Robustness
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?
Long Gao Yield Management in Freight Transportation
IntroductionModels
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Profit PotentialsPolicy Robustness
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
Long Gao Yield Management in Freight Transportation
IntroductionModels
Numerical Results
Profit PotentialsPolicy Robustness
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!
Long Gao Yield Management in Freight Transportation
IntroductionModels
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Profit PotentialsPolicy Robustness
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.
Long Gao Yield Management in Freight Transportation