Markdown Optimization under Inventory Depletion Effect
1. <Insert Picture Here>
Markdown Optimization under Inventory Depletion Effect
Andrew Vakhutinsky, Alex Kushkuley, Manish Gupte
Oracle Retail GBU
2. 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
4. 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
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1110
1
5. 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
6. 1
6
11
16
21
26
31
36
41
46
51
0% 10% 20% 30% 40% 50% 60% 70% 80% 90%
Markdown %
SalesAccelerator
Markdown Events
Expected
High
Low
Estimating Price Elasticity
How much do sales change when prices change?
Observed Markdown Response Within a Subclass
9. 0%
20%
40%
60%
80%
100%
0 10 20 30 40 50 60 70 80
Current OH as % of Max OH
InventoryEffectFactor
Inventory Effect
Sales Dampening vs. Inventory Sell-Through
10. 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
11. 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.
14. Optimal Price Control:
continuous time and price
• MDO as a constrained variational problem
00
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subject to
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• Optimal solution:
(2)
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and
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tt
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15. 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
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where
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16. 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.
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17. 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
18. Pricing Policy for Stochastic Demand
weekly price
adjustment
weekly
sales
weekly
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
19. 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)
20. 0%
1%
2%
3%
4%
5%
6%
10 15 20 25 30
relativeregret
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
22. 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
23. Summary
• Fast and scalable computational schema
• Adequately captures inventory effect
• Business rules friendly
