Apparel Business

Decision Models Lecture 4 1

Lecture 4

Non-Linear Models

Optimal Pricing in Retail

Modeling Pricing Competition


Nonlinear Programming

So far we have analyzed problems with objective functions and constraints that are linear – what happens if these are non-linear?

Where does non-linearity come from in the first place? Economies or diseconomies of scale - e.g. Marginal costs/benefits that

change as volume changes

Interactions effects – e.g. price effects demand, yet revenue is the product of price times demand, leading to nonlinearity

Threshold effects – e.g. the value of an option in the future is the maximum of i) the value if exercised, or ii) the value if not exercised.


Example: The optimal scale of business

Consider a business with the choice of two technologies:

The firm uses the lowest cost technology given its volume of output.

Annual Fixed Cost Variable Cost/Unit

Tech. 1 $0 $2.00

Tech. 2 $20,000 $0

}2,000,20min{)( xxc


Total Cost vs. Unit Volume

$0.00

$5,000.00

$10,000.00

$15,000.00

$20,000.00

$25,000.00

0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000

Unit Volume

To

tal C

ost

}2,000,20min{)( xxc


Market Price vs. Unit Volume

$0.00

$0.50

$1.00

$1.50

$2.00

$2.50

$3.00

$3.50

$4.00

0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000

Unit Volum e

Mar

ket

Pri

ce

xxp 01.03)(


Profit vs. Unit Volume

$0

$500

$1,000

$1,500

$2,000

$2,500

$3,000

0 2,000 4,000 6,000 8,000 10,000

Unit Volume

To

tal P

rofi

tWhat the problem looks like for small unit volumes ….

Maximum profit?

)()( Profit xcxxp(x)


Profit vs. Unit Volume

$0

$5,000

$10,000

$15,000

$20,000

$25,000

0 10,000 20,000 30,000 40,000 50,000 60,000 70,000

Unit Volume

To

tal P

rofi

t “Local” maximum – best solution within a limited “neighborhood”.

“Global” maximum – best solution overall

But for a wider range of unit volumes things look different ….

)()( xcxxp


Different starting points lead to different solutions

Starting solver at v=1,000 we get

x*=4,444 Local maximum Starting solver at v=10,000 we get

x*=40,000 Global maximum

Why? Solver checks for “local improvements” if marginal changes in volume do not improve profits, then it declares current value is “optimal”

For linear problems, local optima are always global optima and this logic works. But for non-linear problems, this is not always true.


Some practical decision-making consequences

Beware of the “small change” argument…

Argument: “Every time we’ve tried lowering prices a little, our profits have declined because the increased revenue doesn’t offset our increased costs. We’re best off staying with our current price points.”

Pitfall: If the price declines are large enough, the increased volume may justify a shift to a new technology that leads to MUCH higher profits (e.g. Wal-Mart).

Why cost-benefit analysis can be misleading:

Argument: If marginal benefits do not exceed marginal cost, the project is not worthwhile

Pitfall: Making larger changes (e.g.volume in our example) or making several changes simultaneously may lead to an improvement even if marginal changes do not.


Nonlinear Programming: Key Points

The solution returned by the optimizer may depend on the starting point: In general, optimizers are not guaranteed to give global optimal solutions to nonlinear programs. You may have to experiment with different starting points.

This behavior is due to the inherent limitations of marginal analysis, which has broader implications for decision making.

Also, nonlinear programs are less efficient numerically; they can take much more compute time to solve than linear programs.


Apparel Business

Product Assortment

Presentation

Advertising

Pricing

Supply Chain Management


Pricing in the Apparel Business

The price a customer is willing to pay for a particular product has little to do with how much it costs to produce it

Products are seasonal, if a product is not sold-out at the end of the season, the remainder is sold at a deep loss

There is a large number of products (styles, colors, sizes)

For some companies, there are many stores in different regions

Most product related decisions, including pricing, are made by “buyers”, whose main focus is selecting the “right” styles

Buyers are eager to get rid of the slow-selling merchandise by offering deep markdowns (“I just want it gone!”)

Recently, companies started realizing that a substantial amount is being left on the table by this mode of operation


Who does “Pricing Optimization”?


Example: Setting Retail Price

A retail firm is considering a pricing decision for a fashion item

According to marketing department estimates, if the price for the item is set at $P, the overall demand for the item over the 15-week selling season will be:

D = 7,725 - 97.5P

What price should a firm set to maximize its revenues over the selling season


Demand-Price Curve

Demand follows a linear pattern

PD 5.97725,7

D

P

7,725

79.24

1,875

60

Example: for P1=$60, demand is D=7,725-97.5x60=1,875 units


Revenues = R = P*D = P*(7,725 - 97.5P) = 7,725P - 97.5P2

R, $

P, $79.24

Non-linear function of P

39.62

153,014

0

Best price = $39.62


What real-life factors are ignored in this model?

Prices can be changed during the selling season

Supply constraints may limit the overall sales

Demand curve can be controlled by advertising

Retailers sell variety of products with inter-related demands

Retailers typically face competition

Etc. etc.


Ex: Multi-Period Revenue Management Problem

A retail company stores are stocked with 2,000 units of a single fashion item

The sales season consists of 15 weeks

No chance for re-stocking the item of reallocating among stores

Goal: maximize retailer’s profits over the selling season


Costs

All production and distribution costs have already been paid: they are sunk costs

Every unit kept in inventory at the end of each week incurs $1 in inventory and maintenance costs

All items in stores that are not sold at the end of 15 weeks are sold to discounters (“jobbers”) for $25 per unit (salvage value)


Demand-Price Curve

Demand in each week follows a linear pattern. Week 1:

11 5.6655 PD

D1

P1

655

100.8

265

60

Example: for P1=$60, demand is D1=655-6.5x60=265 units


Time-Dependent Demand Functions

We expect demand to decrease over time. Decreasing intercepts:

Each week, intercept drops by 20 units

1515

158

33

22

11

5.6375

...

,5.6515

...

,5.6615

,5.6635

,5.6655

PD

PD

PD

PD

PD


Time-Dependent Demand Functions

Demand

Price

655

375

100.857.7

Week 1

Week 15

Maximum demand in week 15 is about 60% lower than in week 1


Revenues

Revenue earned in Week 8 (assuming enough inventory):

R8

P879.2

88888 5.6515 PPPDR

Non-linear function of P8

39.6

10201.0


Decision Variables and Objective Function

Goal: maximize retailer’s profits over the selling season

Decision Variables:

Pt = Price in week t, t=1,…,15.

St = Sales in week t, t=1,…,15.

Objective Function:

Profits = Sales Revenue + Salvage Revenue – Inventory Cost

Additional (definitional) variables: Ending inventory and demand for each week:

It = Inventory in week t, t=1,…,15

Dt = Demand in week t, t=1,…,15


Revenues and Costs

Sales Revenue

S1P1+…+ S15P15=

Salvage Revenue

25I15

Inventory Cost

1xI1+…+ 1xI15=1x

Objective Function:

Profits=S1P1+…+ S15P15+25I15-(1xI1+…+ 1xI15)

15

1ttt PS

15

1ttI

15

115

15

1

125t

tt

tt IIPS


Problem Constraints

Constraints: For every week (t=1,…,15)

Sales in week tSt≤ Dt St≤ It-1

Inventory balance It= It-1-St I0=2000

Demand in week t

Non-negativitySt,Pt 0

It-1 It

St

Week t

1515

11

5.6375

...

,5.6655

PD

PD


Multi-period Revenue Management Problem: Complete Formulation

.0,

,2000

,15,...,1 ,

,15,...,1 ,5.6515

,15,...,1 ,

,15,...,1 ,

s.t.

125max

0

1

1

15

115

15

1

tt

ttt

tt

tt

tt

tt

ttt

SP

I

tSII

tPD

tIS

tDS

IIPS

Non-linear model: objective function is quadratic in price decision variables

Cannot check “Assume Linear Model” in Excel Solver options


Optimized Spreadsheet

123456789

1011121314151617181920212223242526272829303132

A B C D E FRetail.xls

Initial Inventory 2000Demand Slope 6.5Demand Intercept for Week 1 655Salvage Value 25.00$ Inventory Cost 1.00$ Intercept Trend 20

Week Price Intercept Demand Sales End Inv0 20001 65.99$ 655.0 226.1 226.1 1773.92 64.95$ 635.0 212.8 212.8 1561.13 63.91$ 615.0 199.6 199.6 1361.54 62.87$ 595.0 186.3 186.3 1175.25 61.83$ 575.0 173.1 173.1 1002.16 60.79$ 555.0 159.8 159.8 842.27 59.76$ 535.0 146.6 146.6 695.78 58.72$ 515.0 133.3 133.3 562.39 57.68$ 495.0 120.1 120.1 442.2

10 56.64$ 475.0 106.8 106.8 335.411 55.60$ 455.0 93.6 93.6 241.812 54.56$ 435.0 80.3 80.3 161.513 53.53$ 415.0 67.1 67.1 94.414 52.49$ 395.0 53.8 53.8 40.615 51.45$ 375.0 40.6 40.6 0.0

Revenue from sales $121,288.59Revenue from salvaged units $0.00Inventory Cost $10,290.00

Net Profit $110,998.59

=C12-$D$4*B12

=F11-E12

=D6*F26

=F28+F29-F30

=D7*SUM(F12:F26)


The optimal solution is to gradually decrease the price.

The total profit value is $110,999.

Optimal Prices

50.0

55.0

60.0

65.0

70.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Week

Pri

ce


Sales DECREASE over the course of the season as well. Why?

Optimal Sales

0.0

50.0

100.0

150.0

200.0

250.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Week

Sal

es


The LARGER is the inventory cost, the more FRONTLOADED sales are.

Optimal Sales (Inventory Cost = $10)

0.0

100.0

200.0

300.0

400.0

500.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Week

Sal

es


Modeling Pricing Competition

Two retailers compete by setting different prices for similar items

Demand for Company A depends on the prices of both Company A and Company B:

What should Company A’s price be, if Company B sets a price of $100?

3725+40*100 = 7,725

Company A should charge $39.62, if Company B charges $100

This is called the “Best Response” of Company A to Company B’s price

ABA PPD 5.97403725

BAB PPD 85352200


Company A’s Best Response Problem

Given competitor’s price , select own price to maximize revenues

Decision Variable:

Objective Function to be maximized:

Revenues =

Constraints:

Similar problem for Company B

Can be solved using Solver (see Retail.xls)

BP

ABAAA PPPDP 5.97403725

0,0 AA DP

AP

AP


Best Response

If Company B charges $100, the “Best Response” for Company A is to charge $39.62

What would Company B do in response?

Company B would charge $21.10, if Company A charges $39.62

If Company A knows how Company B chooses prices, would Company A set a price of $39.62?

100$62.39$ ABR

62.39$10.21$ BBR


Nash Equilibrium

If there is a set of prices such that nobody has an incentive to deviate unilaterally, that set is called a Nash Equilibrium

The Nash Equilibrium is a central concept in Game Theory, which analyzes competition

The Nash Equilibrium for our pricing model is:

BAA PBRP

ABB PBRP

72.22$AP 62.17$BP


Equilibrium: Prices, Sales, Revenues

Company A Company B

Price $ 22.72 $ 17.62

Sales 2,215 1,498

Revenues $ 50,314 $ 26,384

How would these change, if Company A acquires Company B?


Joint Optimization Problem

We can formulate this as a joint optimization problem: Select and to maximize total revenues.

Decision Variables:

Objective Function to be maximized:

Total Revenues =

Constraints:

See Retail.xls for spreadsheet implementation

0,0

0,0

BB

AA

DP

DP

BABABA

BBAA

PPPPPP

DPDP

853522005.97403725

BA PP ,

AP BP


Jointly Optimal Prices, Sales, Revenues

Company A Company B

Price $ 29.00 $ 25.74

Sales 1,927 1,028

Revenues $ 55,880 $ 26,443


Competition vs. Centralized Management

Competition CentralizedPercentage Difference

Average Price $ 20.66 $ 27.87 35 %

Total Sales 3,713 2,955 -20 %

Total Revenues $ 76,698 $ 82,323 7 %


Revenue Management Case Study: American Airlines & People Express

Airline Deregulation Act 1978 Price controls lifted Free entry and exit from markets

Rise of low-cost carriers PeopleExpress started 1981 1984 Results: $1B Rev., $60M profit

Major airlines like American were significantly affected, especially by loss of discretionary (leisure) travelers.


Crandall’s solution

Bob Crandall (VP Marketing at the time) recognized some essential facts: Many AA flights departed with empty seats The marginal cost of using these seat was very small AA could in fact compete on cost with the new entrants

using these “surplus seats”

But how? Created new restricted, discounted fares (“Super Saver”

and “Ultimate Super Saver” fares) Capacity-controlled the availability of these fares DINAMO – Dynamic Inventory Allocation and Maintenance

Optimizer: optimized the number of discount seats to sell on each and every flight departure (5000+ flights/day).


Results of the new strategy & capability

AMR shares initially plunged on first announcement of “Ultimate Super Saver” fares Jan. 1985 Analysts thought it was the start of a price war “AMR cannot operate profitably at these fares.”

But RM systems proved very effective AA revenues rose Competitors suffered: e.g. PeopleExpress

1984 $60M profit (all-time high)

1985 $160M loss

1986 Bankruptcy

Sold to Continental

DINAMO launched


“We were a vibrant, profitable company from 1981 to 1985, and then we tipped right over into losing $50 million a month. We were still the same company. What changed was American’s ability to do widespread Yield Management in every one of our markets. … That was the end of our run because they were able to under-price us at will and surreptitiously.”

“Obviously PeopleExpress failed . . .We did a lot of things right. But we didn’t get our hands around Yield Management and automation issues. [If I were to do it again . . . ] the number one priority on my list every day would be to see that my people got the best information technology tools. In my view, that’s what drives airline revenues today more than any other factor–more than service, more than planes, more than routes.”

Donald BurrCEO PeopleExpress


Pricing Optimization Software Companies

DemandTec www.demandtec.com

KhiMetrics www.khimetrics.com

ProfitLogic www.profitlogic.com

i2 www.i2.com

Zilliant www.zilliant.com


Summary

Non-linear optimization Price-dependent demands result in non-linear problems Beware of marginal analysis!

Modeling pricing competition Best Response Nash Equilibrium

Modeling inter-temporal dependencies Flow balance equations Surplus cash in the cash-flow matching problem vs. inventories in the

revenue management problem

Reference reading: P&B p. 200, p. 203-209

Apparel Business

Technology

Transcript of Apparel Business