optimal pricing and replenishment in an inventory system owen wu university of british columbia june...
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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
7
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.
9
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
11
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
12
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)
14
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
15
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
19
Upper Bound on Profit Upper Bound on Profit ImprovementImprovement
Theorem: Let be the optimal strategy, then
Heuristic Bound:
Lemma: For n > m,
20
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.
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