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Retail Pricing using Dynamic Optimization

October 2012

© 2012 Massachusetts Institute of Technology. All rights reserved.

Sabanci University

Fall 2012

Robert M. Freund

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A Retail Pricing Problem

The Coop is preparing for the Fall sales season of Power-Pro laptop bags for MBA students.

The “back-to-school” sales season runs from early August through late September, a total of 9 weeks.

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Retail Pricing Problem - Details

• 9-week sales period

• Set (or re-set) price each week

• Six different possible price levels

• Price cannot be increased from week to week

• The total stock level is 1,000 laptop bags

• Any bag remaining at the end of the season has a salvage value of $55

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Distribution of Demand

Stock Level 1,000Salvage Value $55Sales Season 9 weeks

Price cannot be increased from week to week.

Weekly Demand

Probability

Price 0.2 0.2 0.2 0.2 0.2

$70.00 220 225 230 235 240

$75.00 210 215 220 225 230

$80.00 165 170 175 180 185

$103.00 75 85 90 95 105

$112.00 60 65 70 75 80

$121.00 40 45 50 55 60

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Pricing Strategy

• What decision do we need to make in week 1?

• Is week 1 different than any other week?

• What decision(s) do we need to make in weeks 2, 3, …, 9 ?

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Pricing Strategy

• What will influence the pricing decision in week 1?

Starting Stock Optimal 1st Week Price600 ?500 ?400 ?300 ?

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Example of a Pricing Strategy

“Here is what you should do:

In weeks 1, 2, 3 set price to $112In weeks 4, 5 set price to $103In weeks 6, 7, 8 set price to $80In week 9 set price to $75 ”

• Could this be a description of an optimal strategy?

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Another Type of Pricing Strategy“Here is what you should do:

• Start with a price of $112

• In every week t , do the following:

• Given last week’s price Pt-1 and the average demand at price

Pt-1

• If:

(current stock level)/(weeks left in season) > (0.87) ,

then decrease the price to the next lowest value.”

Could this be a description of an optimal strategy?

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Yet Another Type of Pricing Strategy

“Here is what you should do:

• Start with a price of $112

• When the stock level falls to:

525 lower price to $103390 lower price to $80265 lower price to $75 ”

Could this be a description of an optimal strategy?

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A “Dynamic” Pricing Strategy• … • If we are in week 5 and stock level is 420 and last week’s price was $103,

then set this week’s price to $80

• If we are in week 5 and stock level is 415 and last week’s price was $103, then set this week’s price to $103

• … • If we are in week t and stock level is U and last week’s price was P, then set

this week’s price to ___ • …

Make these statements for all weeks, all stock levels, and all previous weeks’ prices.

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1-Period Model

Suppose it is the last week of the selling season

• It is the start of week 9

• 1 pricing period

• Week 9 Starting Stock Level = U

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1-Period Model, Stock Level = 100

For price of $103, Total EMV

= 103*(.2*75 + .2* 85 + .2*90 + .2*95 +.2*100) +55 * (.2*(100–75) + .2*(100-85) + .2*(100-90) + .2*(100-95) + .2 * (100-100))

= 9,167 + 605 = $9,772

For Stock Level = U = 100, and each price P, the expected total revenue is shown in the Total EMV column below:

Units Sold

  ProbabilityCurrent Week   Salvage Total

Price 0.2 0.2 0.2 0.2 0.2 Sales   Value EMV

$70.00 100 100 100 100 100 $7,000 + $0 $7,000

$75.00 100 100 100 100 100 $7,500 + $0 $7,500

$80.00 100 100 100 100 100 $8,000 + $0 $8,000

$103.00 75 85 90 95 100 $9,167 + $605 $9,772

$112.00 60 65 70 75 80 $7,840 + $1,650 $9,490

$121.00 40 45 50 55 60 $6,050 + $2,750 $8,800

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1-Period Model, Stock Level = 300

For price of $75, Total EMV

= 75 *(.2*210 + .2*215 + .2*220 + .2*225 + .2*230) +55 * (.2*(300–210) + .2*(300-215) + .2*(300-220) + .2*(300-225) + .2 * (300-230))

= 16,500 + 4,400 = $20,900

For Stock Level = U = 300, and each price P the expected total revenue is given in the Total EMV column below:

Units Sold

  ProbabilityCurrent Week   Salvage Total

Price 0.2 0.2 0.2 0.2 0.2 Sales   Value EMV

$70.00 220 225 230 235 240 $16,100 + $3,850 $19,950

$75.00 210 215 220 225 230 $16,500 + $4,400 $20,900

$80.00 165 170 175 180 185 $14,000 + $6,875 $20,875

$103.00 75 85 90 95 105 $9,270 + $11,550 $20,820

$112.00 60 65 70 75 80 $7,840 + $12,650 $20,490

$121.00 40 45 50 55 60 $6,050 + $13,750 $19,800

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Week-9 Table of Optimal EMV and Price

Stock Level

Last Week’s Price

(U) $70 $75 $80 $103 $112 $121

0

5

10

…

100 $7,000 ($70) $7,500 ($75) $8,000 ($80) $9,772 ($103) $9,772 ($103) $9,772 ($103)

…

300 $19,950 ($70) $20,900 ($75) $20,900 ($75) $20,900 ($75) $20,900 ($75) $20,900 ($75)

…

1,000

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Week-9 Model Interpretation

Define

J9(U, P) = Optimal EMV if we are in the last week (week 9 of a 9-week model) with a stock level of U units, and a price cap (i.e., previous week’s price) of P

J9(U, P) is precisely what is portrayed in the Week-9 table

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2-Period Model

• Stock level at beginning of week 8 = U

• 2 weeks, set price in each week

Now suppose it is the start of week 8 of the sales season:

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2-Period Model, Stock Level = 200

For price of $103, Total EMV= 103*(.2*75 + .2*85 + .2*90 + .2*95 + .2*105) +

.2*J9(200–75, 103) + .2*J9(200-85, 103) + .2*J9(200-90,103) + .2*J9(200-95, 103) + .2*J9(200-105, 103)

= 9,270 + 10,351 = $19,621

For Stock Level = U8 = 200, and each price P,the expected total revenue is given in the Total EMV column below:

Units Sold

  Probability Week 8   EMV after Total

Price 0.2 0.2 0.2 0.2 0.2 Sales   Week 8 EMV

$70.00 200 200 200 200 200 $14,000 + $0 $14,000

$75.00 200 200 200 200 200 $15,000 + $0 $15,000

$80.00 165 170 175 180 185 $14,000 + $2,000 $16,000

$103.00 75 85 90 95 105 $9,270 + $10,351 $19,621

$112.00 60 65 70 75 80 $7,840 + $11,470 $19,310

$121.00 40 45 50 55 60 $6,050 + $12,570 $18,620

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2-Period Model

I can do this for all stock levels

U = 0, 5, 10, … 1,000

and all prices

P = $70, $75, $80, $103, $112, $121

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Week-8 Table of Optimal EMV and Price

Stock Level

Last Week’s Price

(U) $70 $75 $80 $103 $112 $121

0

5

10

…

200 $14,000 ($70) $15,000 ($75) $16,000 ($80) $19,621 ($103) $19,621 ($103) $19,621 ($103)

…

…

…

1,000

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Week-8 Model Interpretation

Define

J8(U, P) = Optimal EMV if we are in the next-to-last week (week 8 of a 9-week model) with a stock level of U units, and a price cap of $P

J8(U, P) is precisely what is portrayed in the 2-period table

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9-Period Model

• 9 weeks, 9 prices to determine

• What is the optimal pricing strategy in Week 1?

Now suppose that we are in the first week of the full 9-week selling season.

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Week-1 Table of Optimal EMV and Price

Stock Level

Last Week’s Price

(U) $70 $75 $80 $103 $112 $121

0

5

10

…

200

…

…

…

1,000 $96,730 ($103)

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9-Period Model Interpretation

Define

J1(U, P) = Optimal EMV if we are in week 1 of a 9-week model with a stock level of U units, and a price cap of $P

J1(U, P) is precisely what is portrayed in the week-1 table

We wish to know J1(1000, $121) (for starters)

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The General Model

Let P denote the price decision for period t.

Let DP(i) denote the demand at price P, for i = 1, …, 5

Let Jt(Ut, Pt-1) denote the value function for period t, where

Ut is the stock level at the beginning of period t

Pt-1 is the price level of the previous period (t-1)

Jt(Ut, Pt-1) is the optimal expected value of revenues from

week t through the end of the season, given that it is now the beginning of week t, there are Ut bags in stock and we

cannot set the price higher than Pt-1 .

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The General Model

The value function for period t is essentially:

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111 )),(()(),(

1 ipttPi

PPttt PiDUJPiDqMaxPUJ

t

However, when the stock level falls to 0, there are no bags left to sell regardless of the demand.

Therefore, the equation is actually:

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111 ),)(,0(),(),(

1 iptttPi

PPttt PiDUMaxJPUiDMinqMaxPUJ

t

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Dynamic Optimization

• “Dynamic Optimization,” “Dynamic Programming,” “DP”

• State of the System

• Bellman Equation

• Backward Recursion

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Dynamic Optimization

• Dynamic optimization models capture complex interactions over time

• Dynamic optimization is a great conceptual tool, and very often is a good practical tool as well

• Dynamic optimization models are used in power system pricing, revenue management for airlines, hotels, car rental companies, internet retail pricing, military wargame models, etc.

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State of the System

• In this model, the state of the system is the stock level and price cap from the previous period: (Ut, Pt-1)

• In other models, the state of the system might be completely different

State of the system in period t

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State of the System, continued

• The “state of the system” in each period t is comprised of values of quantities that capture what is needed to determine the current status of the situation. In pricing, this might include:

• Current stock level Ut

• Last week’s price Pt-1

• Last week’s demand or average recent demand (indication of non-independent demand) Dt-1

• Number of markdowns already taken Nt-1

• others….

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Modeling Correlated (“learned”) Demand

Demand Level this Week

Low: 1 2 3 4 High: 5

Low: 1 0.50 0.30 0.20 0 0

2 0.25 0.50 0.15 0.10 0

3 0.10 0.15 0.50 0.15 0.10

4 0 0.10 0.15 0.50 0.25

High: 5 0 0 0.20 0.30 0.50

• Demand Probabilities after week 1:

Dem

and

Lev

el la

st

Wee

k

• This models demand being correlated from week to week

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Value Function

• For this model, the value function is Jt(Ut, Pt-1)

The value function is the optimal value of the system in period t for each possible state that the system might be in

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Backward Recursion

• We construct the value function recursively, starting with its value in the last time period, and working backwards to the first period.

• This is similar in concept to the way we construct EMV values at points in a decision tree, working from the end of the tree and working to the start of the tree.

• For our model, the backward recursion is:

• However, the exact nature of the recursion will differ from model to model

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111 ),)(,0(),(),(

1 iptttPi

PPttt PiDUMaxJPUiDMinqMaxPUJ

t

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Final Comments

• Dynamic optimization models compute the optimal decisions and associated EMV values

• Dynamic optimization models are very powerful

• Dynamic optimization models can be very computationally intensive

• When the models/systems become too complex, one needs to use “approximate dynamic optimization” methods to solve for very good (but necessarily optimal) pricing decisions