1. Optimal Pricing and Replenishment in an Inventory System Owen Wu University of British Columbia June 11, 2004 Joint work with Hong Chen and David Yao

2. 2 Literature: Multiperiod Inventory Control Problem Deterministic demand Stochastic Demand Price- insensitive EOQScarf (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. 3 Questions  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. 4 Poisson Demand Model: Diffusion  Unit Poisson process:  Cumulative demand:  Brownian model can be viewed as an alternative model that approximates the real world.

5. 5 Pricing and Inventory Control Inventory X(t) S t 0  Continuous review. Infinite horizon. Zero lead time. No backlog or lost sale.  Inventory policy:order up to S whenever inventory level reaches zero.  Pricing strategy:single price per cycle, dynamic pricing.  Objective: To maximize the expected discounted/average profit.

6. 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 Cycle

7. 7 Impact of Demand Uncertainty Example: c(S) = 100 + 5 S, (p) = 50 – p

8. 8  Sequential optimization: Marketing: Operations:  Joint optimization: Joint vs. Sequential Optimization Sequential Joint Sequential Joint Example: c(S) = 100 + 5 S, (p) = 50 – p, h = 1.

9. 9 Dynamic Pricing  1  2  N–1  N S S(N–1) / N S(N–2) / N S/N0S/N0 p1p1 p2p2 p3p3 pNpN Inventory level

10. 10 Properties  V( , S) is pseudo-concave in   The marginal profit  or

11. 11 Impact of Demand Uncertainty (Fixed S) 

12. 12 Impact of Demand Uncertainty (Joint Optimization)  Non-monotonicity and jumps (not very common) p( ) = 10 – 10 -3 + –1 c(S) = 50 + S 2 h = 0.2 1*1* 2*2* S*S*  

13. 13 Profit Improvement over Single Price  Quantify the advantage of dynamic pricing.  When is the improvement significant?  (N, a, b, h, , K, c)  (N, a – c, Khb, hb 2  2 )

14. 14 50 c(S) = 100 + 5 S, (p) = 50 – p, h = 1,  = 10. Number of Prices

15. 15 Optimal Profit under Single Price  h c(S) = 100 + S (p) = 50 – p

16. 16 Profit Improvement  h

17. 17 Percentage Profit Improvement  h  h 1% 2% 3%

18. 18  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. 19 Upper Bound on Profit Improvement  Theorem : Let be the optimal strategy, then  Heuristic Bound :  Lemma : For n > m,

20. 20 Upper Bound on Profit Improvement  Heuristic Bound h

21. 21 Full Back-Order Case S S(N–1) / N S/N0s/NS/N0s/N p1p1 p2p2 pNpN p N+1 Inventory level s(N–1) / N s pN+MpN+M  (s, S) policy. s<0<S.  Properties: If N=M,

22. 22 Conclusion: Back to opening questions  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.