Optimal Pricing and Optimal Pricing and Replenishment in an Inventory Replenishment in an Inventory

SystemSystem

Owen Wu

University of British Columbia

June 11, 2004

Joint work with Hong Chen and David Yao

2

Literature: Multiperiod Literature: Multiperiod Inventory Control ProblemInventory Control Problem

Deterministic demand

Stochastic Demand

Price-insensitive

EOQ Scarf (1960) Veinott (1966) … …

Price-sensitive

Whitin (1955)

Rajan et al. (1992)

Yano and Gilbert (2002)

… …

Zabel (1970) Thowsen (1975)

Federgruen and Heching (1999)

Thomas (1970)

Polatoglu and Sahin (2000)

Chen and Simchi-Levi (2003,2004)

Feng and Chen (2003)

3

QuestionsQuestions

What is the impact of demand variability on pricing and inventory replenishment decisions?

How to price dynamically within each replenishment cycle?

When is dynamic pricing significantly more profitable than static pricing?

4

0

5

10

15

20

25

0 5 10 15 20 25

Time

Arrivals

-8

-6

-4

-2

0

2

4

0 5 10 15 20 25

Time

Difference

Poisson

0

100

200

300

400

500

600

0 100 200 300 400 500 600

Time

Arrivals

-15

-10

-5

0

5

10

15

20

25

30

0 100 200 300 400 500 600

Time

Difference

Poisson

Demand Model: DiffusionDemand Model: Diffusion Unit Poisson

process:

Cumulative demand:

Brownian model can be viewed as an alternative model that approximates the real world.

5

Pricing and Inventory ControlPricing and Inventory ControlInventory X(t)

S

t0

Continuous review. Infinite horizon. Zero lead time.No backlog or lost sale.

Inventory policy: order up to S whenever inventory levelreaches zero.

Pricing strategy: single price per cycle, dynamic pricing.

Objective: To maximize the expected discounted/average profit.

6

replenishment cost c(S)

holding cost hX(t) per unit of time

cycle revenue: p S

Price p induces demand:

Long-run average profit under (S, ):

Additional holding cost per unit of time due to demand uncertainty

Single Price per Replenishment Single Price per Replenishment CycleCycle

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10 20 30 40 50 60250

300

350

400

450

500V

h 0.1h 0.5

h 2

10 20 30 40 50 600

50

100

150

200

250S

h 0.1

h 0.5

h 2

10 20 30 40 50 60

26

27

28

29p

h 0.1

h 0.5

h 2

Impact of Demand UncertaintyImpact of Demand Uncertainty

Example:c(S) = 100 + 5 S, (p) = 50 – p

8

Sequential optimization:Marketing:

Operations:

Joint optimization:

Joint vs. Sequential Joint vs. Sequential OptimizationOptimization

20 40 60 80 100 120 14020

22

24

26

28

p

Sequential

Joint

20 40 60 80 100 120 14066

68

70

72

74

76

S

Sequential

Joint

20 40 60 80 100 120 1400

100

200

300

400

V

Example: c(S) = 100 + 5 S, (p) = 50 – p, h = 1.

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Dynamic PricingDynamic Pricing

1 2 N–1 N

S

S(N–1)/NS(N–2)/N

S/N

0

p1p2

p3

pN

Inventory level

10

PropertiesProperties

V(, S) is pseudo-concave in

The marginal profit

or

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Impact of Demand Uncertainty Impact of Demand Uncertainty (Fixed S)(Fixed S)

5 10 15 200

10

20

30

40

50

p

p1

p2

p3

p4

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Impact of Demand UncertaintyImpact of Demand Uncertainty(Joint Optimization)(Joint Optimization)

Non-monotonicity and jumps (not very common)

p() = 10 – 10-3 +–1

c(S) = 50 + S2

h = 0.2

0.24 0.242 0.244 0.246 0.248 0.25

0

2

4

6

8

0.24 0.242 0.244 0.246 0.248 0.25

41

41.5

42

42.5

43

0.24 0.242 0.244 0.246 0.248 0.254.2

4.4

4.6

4.8

5

1*

2*

S*

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Profit Improvement over Single Profit Improvement over Single PricePrice

Quantify the advantage of dynamic pricing. When is the improvement significant?

(N, a, b, h, , K, c) (N, a – c, Khb, hb22)

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1 2 3 4 5 6 7 8 10437.4

437.5

437.6

V 1 2 3 4 5 6 7 8 10

27.5

28

28.5

29p

1 2 3 4 5 6 7 8 1066

66.1

66.2

66.3

66.4

S50

5050

c(S) = 100 + 5 S, (p) = 50 – p,

h = 1, = 10.

Number of PricesNumber of Prices

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Optimal Profit under Single Optimal Profit under Single PricePrice

010

20

30

40

50

h0

10

20

30

40

50

0

200

400

010

20

30

40

50

h

h

c(S) = 100 + S

(p) = 50 – p

16

Profit ImprovementProfit Improvement

010

20

3040

50

h0

10

20

30

40

50

0

2.5

5

7.5

10

010

20

3040

50

h

h

17

Percentage Profit Percentage Profit ImprovementImprovement

010

20

30

40

50

h0

10

20

30

40

50

0

2.5

5

7.5

10

010

20

30

40

50

h

h

0 10 20 30 40 50

h

0

10

20

30

40

50

00.000250.00050.000750.001

h

1% 2% 3%

180 10 20 30 40 50

h

0

10

20

30

40

50

00.000250.00050.000750.001

h

010

20

30

40

50

h

0

10

20

30

40

50

0

10

20

010

20

30

40

50

h

010

2030

40

50

h0

10

20

30

40

50

0

2.5

5

7.5

10

010

2030

40

50

h

010

20

30

40

50

h0

10

20

30

40

50

0

100

200

300

010

20

30

40

50

h

Percentage improvement under 8 prices (%)

Optimal average profit under single price

Profit improvement under 8 prices

h

h

h

c(S) = 100 + 10 S

(p) = 50 – p

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Upper Bound on Profit Upper Bound on Profit ImprovementImprovement

Theorem: Let be the optimal strategy, then

Heuristic Bound:

Lemma: For n > m,

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Upper Bound on Profit Upper Bound on Profit ImprovementImprovement

1020

3040

50h

10

20

30

40

50

0

5

10

15

1020

3040

50h

Heuristic Bound

h

21

Full Back-Order CaseFull Back-Order Case

SS(N–1)/N

S/N0

s/N

p1p2

pN

pN+1

Inventory level

s(N–1)/Ns

pN+M

(s, S) policy. s<0<S.

Properties:

If N=M,

22

Conclusion: Back to opening Conclusion: Back to opening questionsquestions

What is the impact of demand variability on pricing and inventory replenishment decisions?

How to price dynamically within each replenishment cycle?

When is dynamic pricing significantly more profitable than static pricing?

Most of the results hold under discounted objective.