Optimal Pricing and Replenishment in an Inventory System

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

Literature: Multiperiod Inventory Control Problem

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?

Poisson Poisson • Unit Poisson process: • Cumulative demand: Demand Model: Diffusion • Brownian model can be viewed as an alternative model that approximates the real world.

Inventory X(t) S t 0 Pricing and Inventory Control • 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.

holding cost hX(t) per unit of time cycle revenue: pS replenishment cost c(S)  • Long-run average profit under (S, ): Additional holding cost per unit of time due to demand uncertainty Single Price per Replenishment Cycle • Price p induces demand:

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

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

Inventory level p1 S S(N–1)/N S(N–2)/N p2 p3 pN S/N 0 1 2 N–1 N Dynamic Pricing

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

 Impact of Demand Uncertainty (Fixed S)

Non-monotonicity and jumps (not very common) 1* p()=10–10-3+–1 c(S)=50+S2 h=0.2 S*  2*   Impact of Demand Uncertainty(Joint Optimization)

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, hb22)

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

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

Profit Improvement  h

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

Percentage improvement under 8 prices (%)  h Optimal average profit under single price  h  Profit improvement under 8 prices h c(S)=100+10S (p)=50–p  h

Lemma: For n>m, • Theorem: Let be the optimal strategy, then • Heuristic Bound: Upper Bound on Profit Improvement

Heuristic Bound  h Upper Bound on Profit Improvement

Inventory level p1 S S(N–1)/N p2 pN S/N 0 s/N pN+1 pN+M s(N–1)/N s Full Back-Order Case • (s, S) policy. s<0<S. • Properties: If N=M,

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.

Optimal Pricing and Replenishment in an Inventory System