large-scale price optimization via network flow...๐ 5๐,๐=๐ 5 5๐ 5+๐ 5 6๐ 6+โฏ...
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ยฉ NEC Corporation 20171 NEC Group Internal Use Only
Large-Scale Price Optimization via Network Flow
ERATOๆ่ฌ็ฅญ SeasonIV 2017.8.3@NII
Shinji Ito, Ryohei Fujimaki
ยฉ NEC Corporation 20172 NEC Group Internal Use Only
Our goal: profit maximization by optimizing prices
โWhat is the best pricing strategy?
Profit
: $ 1.3 Price : $ 1.0
Sales quantity
Strategy 1 Strategy 2
? ?
ยฉ NEC Corporation 20173 NEC Group Internal Use Only
Our goal: profit maximization by optimizing prices
โWhat is the best pricing strategy?
Profit
: $ 1.3 Price : $ 1.0
Sales quantity
$ 5.2 $ 6.0
Strategy 1 Strategy 2
<
ร ร==
ยฉ NEC Corporation 20174 NEC Group Internal Use Only
Complicated structure in price optimization
โChanging the price of one product affects otherโs sales
Demand of
Demand of
Discount of
- Cannibalization:Sales
Growing the sales of
makes the sales of down
Price Quantity ProfitProduct 1 $1.3 รจ $1.0 +200 + $80Product 2 $1.2 -100 - $120
Gross profit - $40
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Predictive price optimization and its difficulty
โRecent advanced ML reveals relationship between prices and sales quantities
Predictive model:
Price Price Sales Sales โฏDay 1 $1.3 $1.0 2 2 โฏDay 2 $1.2 $1.0 4 2 โฏโฎ โฎ โฎ โฎ โฎ
Input:pos data
Machine learningsales = ๐(prices)
ยฉ NEC Corporation 20176 NEC Group Internal Use Only
Predictive price optimization and its difficulty
Predictive model:
Price Price Sales Sales โฏDay 1 $1.3 $1.0 2 2 โฏDay 2 $1.2 $1.0 4 2 โฏโฎ โฎ โฎ โฎ โฎ
Input:pos data
Machine learningsales = ๐(prices)Optimization (NP-hard) [This work]
Output:optimal prices
โRecent advanced ML reveals relationship between prices and sales quantities
ยฉ NEC Corporation 20177 NEC Group Internal Use Only
Our contribution
โScalable algorithm for price optimization
Based on:
1. Submodularity behind pricing
2. Network flow algorithm
3. Supermodular relaxation
ยฉ NEC Corporation 20178 NEC Group Internal Use Only
Our contribution
โScalable algorithm for price optimization
Based on:
1. Submodularity behind pricing
2. Network flow algorithm
3. Supermodular relaxation
Achieved:
- Can deal with thousands of products
- High accuracy for real-data problem
Table of contents
1. Introduction
2. Problem definition
3. Scalable price optimization algorithm
4. Experiments
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Objective function of price optimization
Gross profit:โ ๐,๐,โฆ ,๐ = (๐โ๐)๐price cost sales quantity
โWant to maximize is the gross profit โ
Unknown, but predictable
product id
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Predictive model for sales quantity
๐ ๐,๐= ๐ ๐ +๐ ๐ +โฏ+๐ ๐ +๐ ๐ +โฏpricesales
quantityweather calendar
Sat
price
$ $
โSales quantity ๐ is a function in prices ๐
Ex: ๐ ๐ =๐๐ +๐๐+๐ (polynomial model)๐ ๐ = exp(๐ผ ๐+๐ฝ ) , (generalized linear model)
โUse historical data to infer ๐, ๐
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๐ ๐,๐= ๐ ๐ +๐ ๐ +โฏ+๐ ๐ +๐ ๐ +โฏpricesales
quantityWeather calendar
Sat
price
$ $
โSales quantity ๐ is a function in prices ๐Predictive model for sales quantity
ใปใปใป
1: orange 2: apple
๐: milk
๐ด products
$
๐ : Price elasticity of demand
Price of orange
salesquantity
๐ : Cross price effect
Price of apple $
salesquantity
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โMany substitute goods in price optimization
Substitute goods in price optimization
๐ : Cross price effect
$
and : Substitute goods
Discounting apple makes
โdef
the sales of orange down(cannibalization)
๐ ๐,๐= ๐ ๐ +๐ ๐ +โฏ+๐ ๐ +๐ ๐ +โฏ
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Optimization problem
Maximize
Subject to ๐ โ {๐ ,๐ ,โฆ ,๐ }โฆDiscrete price candidates
โ ๐ = (๐โ๐)๐(๐)โOptimization is NP-hard
Gross profit
A commercial solver takes >24[h] for 50 products
Table of contents
1. Introduction
2. Problem definition
3. Scalable price optimization algorithm
4. Experiments
ยฉ NEC Corporation 201716 NEC Group Internal Use Only
Our contribution
โScalable algorithm for price optimization
Based on:
1. Submodularity behind pricing
2. Network flow algorithm
3. Supermodular relaxation
ยฉ NEC Corporation 201717 NEC Group Internal Use Only
Idea 1: Substitute goods and supermodular
Theorem [Substitute goods โ supermodular]
โ โ(๐) is a supermodular function
If all pairs of products are substitute goodsor independent
Demand of A
Demand of B
price of A
โConnection between substitute goods and submodular
Substitute goods:
Discounting โ less sales of
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Idea 1: Substitute goods and supermodular
Theorem [Substitute goods โ supermodular]
โ โ(๐) is a supermodular function
If all pairs of products are substitute goodsor independent
Demand of A
Demand of B
price of A
โConnection between substitute goods and submodular
Substitute goods:
Discounting โ less sales of
[Iwata, Fleischer, Fujishige (2001)]maximized in polynomial time
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Two issues in Key idea 1
โIf all products are substitute goods or independent, profit can be maximized in polynomial time
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Two issues in Key idea 1
Issue 1: General supermodular maximization is slow
Issue 2: Substitute goods assumption is too restrictive
โIf all products are substitute goods or independent, profit can be maximized in polynomial time
โThis approach is still impractical because of two issues
โผ ๐(๐)
ยฉ NEC Corporation 201721 NEC Group Internal Use Only
Two issues in Key idea 1
Issue 1: General supermodular maximization is slow
Issue 2: Substitute goods assumption is too restrictive
โIf all products are substitute goods or independent, profit can be maximized in polynomial time
โThis approach is still impractical because of two issues
Resolved by 2. network flow:
Resolved by 3. supermodular relaxation
โผ ๐(๐)๐๐ โผ ๐(๐)Applicable for non-substitute
ยฉ NEC Corporation 201722 NEC Group Internal Use Only
Two issues in Key idea 1
Issue 1: General supermodular maximization is slow
Issue 2: Substitute goods assumption is too restrictive
โIf all products are substitute goods or independent, profit can be maximized in polynomial time
โThis approach is still impractical because of two issues
Resolved by 2. network flow:
Resolved by 3. supermodular relaxation
โผ ๐(๐)๐๐ โผ ๐(๐)Applicable for non-substitute
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Substitute goods price optimization โMinimum Cut
Graph ๐บs
t
โMaximizing โ ๐ โ Finding minimum s-t cut:solved efficiently by network flow[Ford, Fulkerson (1956)], [Orlin (2013)]
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Substitute goods price optimization โMinimum Cut
Graph ๐บs
t
โMaximizing โ ๐ โ Finding minimum s-t cut:solved efficiently by network flow
๐=
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Substitute goods price optimization โMinimum Cut
Graph ๐บs
t
โMaximizing โ ๐ โ Finding minimum s-t cut:solved efficiently by network flow
๐=๐ =
ยฉ NEC Corporation 201726 NEC Group Internal Use Only
Substitute goods price optimization โMinimum Cut
Graph ๐บs
t
โMaximizing โ ๐ โ Finding minimum s-t cut:solved efficiently by network flow
๐=๐ =๐ =
ยฉ NEC Corporation 201727 NEC Group Internal Use Only
Substitute goods price optimization โMinimum Cut
Graph ๐บs
t
โMaximizing โ ๐ โ Finding minimum s-t cut:solved efficiently by network flow
๐=๐ =๐ =
โ ๐ =constant โ (capacity of st-cut of graph ๐บ)
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Two issues in Key idea 1
Issue 1: General supermodular maximization is slow
Issue 2: Substitute goods assumption is too restrictive
โIf all products are substitute goods or independent, profit can be maximized in polynomial time
โThis approach is still impractical because of two issues
Resolved by 2. network flow:
Resolved by 3. supermodular relaxation
๐(๐)๐๐ โผ ๐(๐)Applicable for non-substitute
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Non-SGP case: supermodular relaxation
modular
โ ๐ = โ ๐+ โ (๐)supermodular submodularโค โ ๐+ โ (๐)
supermodularโ maximized via network flow
โ (๐)โ (๐)โโ non-substitute goodsโ approximate โ by supemodular function
๐
๐โ
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supermodularsupermodular
Non-SGP case: supermodular relaxation
โ (๐)โ (๐)ฮ = 0 ฮ = 0.7 ฮ = 1Relaxation changes depending on ฮ โ 0,1 ร
โMany possibilities of relaxation functionBetter relaxation is chosen automatically
๐๐โ
๐๐โ
๐๐โ
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Simulation experiments
- Real-world retail data* of a supermarket*provided by KSP-SP Co., LTD.
- 1000 products- Compared with SDP, QPBO, QPBOI [Rother et.al (2007)]
Past data Proposed QPBO Others
Computing Time 36 [s] 964 [s] > 1day
Achieved Profit 1.40 M 1.88 M 1.25 M Nan
Upper bound 1.90 M 1.89 M Nan
Existing methods
โProposed approximation algorithm:fast and give solution with only <1% loss
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Conclusion
- Associating substitute goodswith supermodularity
Demand of A
Demand of B
price of A
- Using network flow andsupermodular relaxation
e.g., by robust optimization?
โConstructed a scalable price optimization algorithm by
โFuture work: how to cope with the errors in objective?
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