ewgt 2013 - bid price heuristics for unrestricted fare structures in cargo revenue management

BID-‐PRICE HEURISTICS FOR UNRESTRICTED FARE STRUCTURES IN CARGO REVENUE MANAGEMENT

L. Castelli 1, R. Pesen@ 2, D. Rigonat 1

1 Università degli Studi di Trieste, Italy

2 Università Ca’Foscari, Venezia, Italy

Revenue Management & Cargo

A collec'on of pricing, inventory, marke'ng techniques aimed at predic'ng consumer behaviour and op'mize product availability and price to maximize revenue growth.

In cargo: ¨  Total available capacity (= product inventory) may be uncertain; ¨  Bi-‐dimensional capacity (weight, volume); ¨  Tri-‐dimensional alloca@on; ¨  Product value increases over @me then drops to zero.

Product-‐oriented vs. Price-‐oriented Demand

RM forecasts shiY from restric@on/class-‐based to willingness to pay (wtp)-‐based

Product-‐oriented Price-‐oriented

Different fares for different products

A single type of product

Customer only interested in a specific fare/product

Customer purchase solely on price

Customer choice independent of the availability of cheaper services

Customer compare services from different carriers to get the cheapest fare

Objec@ve of our study

¨  Study effec@veness of capacity management algorithms based on willingness to pay;

¨  Develop policies that are suitable for cargo context;

¨  Explore different approaches: ¤ Dynamic Programming (DP) ¤ Bid-‐Prices (BP)

Roadmap

Defini@on of the scenario

General problem formula@on through Dynamic Programming (DP)

DP based algorithm

Bid-‐price (BP) approach: sta@c BP (SBP) and dynamic BP (DBP) algorithms

Tests on different shipment size /customer demand combina@on

Problem formula@on -‐ Assump@ons

¨  Cargo scenario (i.e. air cargo): bi-‐dimensional capacity (weight, volume);

¨  3-‐D alloca@on issues are ignored; ¨  Single class of customers; ¨  Customers are served one at a @me (1 customer = 1

shipment); ¨  A shipment can be accepted only if the flight has residual

capacity (volume and weight) to accommodate it; ¨  A shipment is paid for propor@onally to its weight (revenue =

weight * unitFare).

Problem formula@on -‐ Nota@on

Symbol Meaning

F={f1,…,fM} Set of increasing fares

f ∈ F Generic fare

pkm Willingness of k-‐th customer to pay fare m

{0,1,…,t,t+1,…,T} Time frame from reserva@on opening to closure

tk ∈ {0,…,T} Generic @me instant

φt probability of a customer showing up at @me t

Cw , Cv AircraY capacity for weight, volume

(w,v) Residual aircraY capacity for weight, volume

(ω,υ) Shipment size

qωυ Probability that a shipment has size (w,u)

Jm(t,w,v) Func@on that returns the expected op@mal revenues from @me t onwards assuming fare fm is displayed and there are residual capaci@es (w,v).

Problem formula@on (DP)

At @me T

No customer arrives

Revenues in T= revenues in T+1

A customer arrives

Shipment does not fit


Shipment fits

We offer fare f

Customer refuses f


Customer accepts f

Shipm. accepted Rev = f*w

Cap. w,v updated

We offer fare f+1

Calculate recursion for f+1

DP algorithm (DYM)

Assuming that n. of customers, arrival @mes, willingness to pay and shipment sizes are known in advance, DP is simplified into (2).

Jm(k,w,v) = maxj≥m{ pkj (fjω + Jj(k+1,w-ω,v-υ)) + (1- pkj)Jj(k+1,w,v)} (2) with final conditions Jm(k,0,v)= Jm(k,w,0)= Jm(K+1,w,v) = 0 for all 0 ≤ w ≤ Cw and 0 ≤ v ≤ Cv and k ≤ K

For each cust. arrival k

If shipment fits

Calculate (2) for all fares ≥ m

Check final condi@ons

Hence the algorithm:

Bid-‐price approach -‐I

Jm(t,w,v) – Jm(t,w-ω,v-υ) Represents the opportunity cost at fare fm for a shipment of size (ω,υ) appearing at @me t when capaci@es w,v are s@ll available.

Expected loss in future revenue from using the capacity now rather than reserving it for future use;

An op@mal policy, solu@on of DP, accepts a shipment iff generated revenues are ≥ to its opportunity cost.

Bid-‐price approach -‐ II

If Jm(t,w,v) is differen@able then ∂Jm(t,w,v)/∂w and ∂Jm(t,w,v)/∂v are the weight marginal opportunity cost and volume marginal opportunity cost, respec@vely.

We can calculate an approxima@on of the marginal opportunity cost (a bid-‐price)

We can design policies that accept a shipment only when its revenue is ≥ to the es@ma@on of the opportunity cost obtained through the BP

Bid-‐price approach -‐ III

Formally, the acceptance rule for a BP policy is:

Symbol Meaning

r Shipment revenue

πw(w,t) , πv(v,t) Weight and volume BP

πw(w,t)ω + πv(v,t)υ Opportunity cost

f Applied fare

r = fω ≥ πw(w,t)ω + πv(v,t)υ (3)

¨  Sta'c BP: fixed at the beginning of the booking period i.e., they do not change over @me and do not depend on the remaining capacity:

πw(w,t) = πw , πv(v,t) = πv.

¨  The chosen fare is unique for all customers: i.e. the min f s.t. rule (3) is respected by 1st accepted customer:

f = min{fj : fjωh ≥ πwωh + πvυh and phj = 1}

Sta@c BP Algorithm (SBP) -‐ I

Sta@c BP Algorithm (SBP) -‐ II

Calculates op@mal BP for each instance from a training set.

Training Phase SBP-‐A

Tes@ng Phase SBP-‐B

Checks acceptance rule (3) with avg. op@mal BP (πw , πv) obtained from training alg.

Op@mal BP per instance πw , πv

Dynamic BP (DBP)

Limita@ons of SBP: ¨  Revenue depends on willingness to pay of the first accepted customer: bad for inverse demand;

¨  BP do not change over @me but residual capacity value increases over @me.

Idea behind DBP: ¨  Dynamic BP are updated aYer each accepted customer by running SBP-‐A on the residual capacity;

¨  Fares are updated based on both users’ wtp and dyn. BP update.

Dynamic BP (DBP) -‐ Algorithm

For each cust. arr. k

If shipment fits AND

wtp current fare = 1

Update fare (4);

Accept k; Update res. capacity

Calculate new sta@c BP through

SBP-‐A

If both new st. BP are ≥ prev.

values, update dyn. BP

f= min in F={f1,…,fM}: fjωk ≥ Lkωk + Mkυk (4)

Fares are updated according to:

Where Lk, Mk are the dynamic BP for user k

Experimental test -‐ Setup

¨  Small shipments (SS): n = 750; between 2 and 45 Kg ¨  Large shipments (LS): n= 450; between 46 and 500 Kg

¨  Inverse Demand (ID): wtp decreases with customer arrivals ¨  Random Demand (RD): random wtp

SS-ID SS-RD

LS-ID LS-RD

Test Scenarios:

Experimental test – Results -‐ SR

85 85 85 85 113 100 100 100 100 100 93 96 97 93

0

Revenues Weight LF Volume LF Accepted requests (num.)

Running @me (sec.)

DYM

SBP

DBP

DYM SBP DBP Revenues 2,458,003 2,283,183 2,082,226 Weight LF 0.999 0.964 0.849 Volume LF 0.869 0.839 0.738 Accepted req. (num.) 178 166 152 Running @me (sec.) 552 0.5 624

Small Shipments, Random Demand

Experimental test – Results -‐ LR

100 100 100 100 100 98 100 99 96

1

93 94 94 94

3


Running @me (sec.)

DYM

SBP

DBP


Large Shipments, Random Demand

Experimental test – Results -‐ SI

100 100 100 100 100 76

100 100 101

0

78 100 100 102

21


Running @me (sec.)

DYM

SBP

DBP

DYM SBP DBP Revenues 2,712,138 2,053,969 2,121,717 Weight LF 0.999 1 1 Volume LF 0.87 0.87 0.87 Accepted req. (num.) 171 173 174 Running @me (sec.) 594 0.5 122

Small Shipments, Inverse Demand

Experimental test – Results -‐ LI

100 100 100 100 100 92 100 101 102

1

93 100 101 104

6


Running @me (sec.)

DYM

SBP

DBP


Large Shipments, Inverse Demand

Where we go from here..

We proved the reliability of wtp-‐based policies within the defined scenario (determinis@c demand, weight-‐based pricing etc.) Future work: ¨  Improvement to the proposed policies ¨  Comparison with policies developed by other authors

ewgt 2013 - bid price heuristics for unrestricted fare structures in cargo revenue management

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