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Markdown Optimization under Inventory Depletion Effect

Andrew Vakhutinsky, Alex Kushkuley, Manish Gupte

Oracle Retail GBU

Markdown Challenges for Large Retailers

• When should the Markdown occur and how deep

should the markdown be?

• How should a retailer balance the tradeoff between

sales volume and price?

• How can a retailer centrally manage millions of

combinations of Merchandise/Location/Candidate

Markdowns?

2

Last Year Actual

Sales

0

100

200

300

400

500

600

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Week

Sa

les

Un

its

0

100

200

300

400

500

600

Sale

s U

nit

s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Week

Last Year Sales Deconstructed

Markdown

Promotion

Seasonal

Lift

25% MD 50% MD

Circular BOGO

Christmas

Last Year’s Causal Factors

Natural Demand

We decompose historical sales patterns into specific demand drivers representing

different causal factors.

Demand Decomposition

Problem Formulation• Given: Initial product price p0, inventory level I0, and

markdown season length, T,

• Find: Sequence of non-increasing prices pt , t = 1,…,T, to

maximize the total revenue over the markdown season.

4

Ttpp

sII

Is

TtddppIds

sp

tt

ttt

tt

tttt

T

t

tt

,...,1

,...,1 ),...,,,...,,(

subject to

max

1

1

1

1110

1

Assumptions

• Single-product dynamic pricing without replenishment

• Merchandise items aggregated over size/color

• Some items’ stock is below presentation minimum resulting in

adverse inventory effect

• Myopic customers coming from infinite population

• Discrete price ladder

• Stochastic demand

1

6

11

16

21

26

31

36

41

46

51

0% 10% 20% 30% 40% 50% 60% 70% 80% 90%

Markdown %

Sa

les

Ac

ce

lera

tor

Markdown Events

Expected

High

Low

Estimating Price Elasticity

How much do sales change when prices change?

Observed Markdown Response Within a Subclass

Price Elasticity Model

0

1

2

3

4

5

6

7

8

9

5 10 15 20 25 30 35 40 45 50

% Off Full Price

Sa

les

Mu

ltip

lie

r

Girl's Dress Shoe Women's Sandal Women's Boot

Subclass Elasticities

Se

as

on

ali

ty In

de

x (

1 =

ave

rag

e w

ee

k)

Seasonality

Seasonality captures the ebb and flow of demand driven by holidays,

promotions and normal weather patterns.

0

1

2

3

4

5

6

7

8

03/15/03 05/24/03 08/02/03 10/11/03 12/20/03 02/28/04

0%

20%

40%

60%

80%

100%

0 10 20 30 40 50 60 70 80

Current OH as % of Max OH

Inv

en

tory

Eff

ec

t F

ac

tor

Inventory Effect

Sales Dampening vs. Inventory Sell-Through

Demand model

• Demand modeled as a function of: time, price, on-hand inventory.

d = d(t, p, I)

• Demand model components:

o Price Effect, dp(p): captures the sensitivity of demand to price

changes; modeled as isoelastic function of price p with constant

elasticity γ < –1

dp(p) = (p/pf)γ where pf is the full price of the item

o Inventory Effect (broken-assortment effect: willing-to-pay

customers cannot find their sizes/colors), dI(I): modeled as power

function of on-hand inventory I

dI(I) = (I/Ic)α where Ic is the critical inventory of the item

o Seasonality, s(t): seasonal variation of demand due to holidays

and seasons of the year; shared by similar items

Demand model and parameter fitting

• Demand is a product of the above-mentioned components:

d(t, p, I) = k dp(p) dI(I) s(t) δ(t) = k (p/pf)γ (I/Ic)

α s(t)

According to this model, the inventory must satisfy the following

differential equation:

- dI(t)/dt = k (p(t)/pf)γ (I(t)/Ic)

α s(t)

• Fitting the demand model:

• Estimate: base demand (k), price elasticity (γ) and inventory effect (α)

• Means: regression on multiple sales data points with known price,

inventory and seasonality.

Inventory effect at constant price: p(t) = p0

1 if ,1

1 if ,)(

)1(11))(()(

(1) where

1 if ,

1 if ,)(

)1(1)(

:are )( revenue cumulative and )( level inventory Then

),0()( and )(1

),( :Denote

0

0

2

1

/)(

00

1

1

0

0000

00

/)(

0

1

1

0

0

12

21

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cfItKt

t

t

eIp

I

tKtIp

tIIptR

I

I

p

pkK

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ttduustt

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Cumulative revenue: 50 weeks, various inventory effects

p0=$100; I0=1000; K=100; α=0.8,1,1.2 and α=0, no inventory effect

$0

$10,000

$20,000

$30,000

$40,000

$50,000

$60,000

$70,000

$80,000

$90,000

$100,000

0 10 20 30 40 50

α = 1

α = 0.8

α = 1.2

α = 0

weeks

Optimal Price Control:

continuous time and price

• MDO as a constrained variational problem

00

0

1

)0(,)0( :conditions initial with

)()()()(

subject to

)()()(

max

IIpp

tsp

tp

I

tIk

dt

tdI

dttsp

tp

I

tIkp

fc

T

fc

f

• Optimal solution:

(2) )()1(

and 1

where)(

)(1)( 0

1

1

0

k

θI τ

ttptp

Properties of the optimal solution for

continuous price and time

• Inventory level evolution under optimal price control:

• Optimal revenue when T = τ : R*(T) = p0I0 θ/(θ+1)

• Otherwise (if T ≠ t), the price control is not smooth:

• If T < τ, set p0 := p0 (τ/T)1/γ and re-apply optimal price control (2)

• If T > τ, keep price p(t) = p0 until time ts determined by:

then re-apply optimal price control (2) with new I0 = I(ts) where during

time period [0, ts] inventory level evolves at constant price according to (1)

1

0)(

)(1)(

ttItI

1 where

),()()1(

/),(1

1

01

θ

Ttt

KITtTt

ss

ss

Optimal Price Control:

discrete price ladder, periodic updates

• Approximate the optimal trajectory at each periodic price update

• At the price update time t compute optimal price:

• Compare p*(t) to the current price p(t):

• If p*(t) < p(t), then set the current price p(t) to the point in the price ladder,

which is closest to p*(t)

• If p*(t) ≥ p(t) and ts < tnext , lower p(t) one notch in the price ladder

• Performance can be further tuned up by selecting rounding

criteria based on update period, proximity of the switching time ts

to the next update time tnext , etc.

//1

*

)()(),(

)(

1)(

tI

I

tTkTt

tI

θ

θptp c

f

Example of optimal price control:

p0 = $100; I0 = 1000; K = 20; α = 0.8; γ = −2 → τ = 28 weeks

T = τ = 28 weeks: keep on the optimal trajectory (blue line)

T = 25 weeks: optimal trajectory starts at discounted price of $80 (green)

T = 31 weeks: price constant until it reaches optimal trajectory (red line)

0

20

40

60

80

100

120

0 10 20 30 40

T = 28

T = 31

T = 25

T=28, price ladder

weeks

Pricing Policy for Stochastic Demand

weekly price

adjustment

weekly

salesweekly

demand

Pricing Policy:

• Current price adjustment is a function of previous demand realizations

• Determined by several parameters tuned to optimize performance

Regret measure:

• Difference between revenue obtained by applying closed-loop control

and optimal solution obtained in hindsight, i.e. when the demand

realization is known.

Optimization Objective: Regret minimization

Computational Experiments

• Sales data from a large national fashion retailer

• ~100 merchandise x 500 locations ≈ 50,000 items

• Demand model parameters fitted via regression

• Closed loop simulated with multiple demand realizations

• Price policy parameters tuned-up to improve performance

• Regret measure: Mean relative regret

• Two price policies compared:

1. Closed-form analytical solution (fast computation)

2. Dynamic programming approach (slow)

0%

1%

2%

3%

4%

5%

6%

10 15 20 25 30

rela

tive

re

gre

t

markdown season length (weeks)

Relative regret vs. number of weeks in markdown season for two different policies

closed-form based policy

dp-search based policy

Effect of parameter estimation error

-0.5%

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

-25% -15% -5% 5% 15% 25%

relative revenue loss vs. parameter estimation error

stochastic demand

deterministic demand

References

•R. Lobel and G. Perakis. Dynamic Pricing Through Sampling

Based Optimization, Submitted 2010

•S. A. Smith and D. D. Achabal. Clearance pricing and

inventory policies for retail chains. Management Science,

44:285–300, 1998

•K.T. Talluri and G.J. Van Ryzin. Dynamic pricing. In The

Theory and Practice of Revenue Management, chapter 5.

Kluwer Academic Publishers, 2004

Summary

• Fast and scalable computational schema

• Adequately captures inventory effect

• Business rules friendly