Dynamic Pricing for Non-Perishable Products with Demand Learning - PowerPoint PPT Presentation

Dynamic Pricing for Non-Perishable Products with Demand Learning Ren´ Victor F. Araman e A. Caldentey Stern School of Business New York University DIMACS Workshop on Yield Management and Dynamic Pricing Rutgers University August, 2005

Motivation Price Inventory 1 1 % Initial Inventory 0.8 0.8 % Initial Price Product 2 0.6 0.6 Product 2 Product 1 0.4 0.4 Product 1 0.2 0.2 0 0 0 10 20 30 40 0 10 20 30 40 Weeks Weeks Dynamic Pricing with Demand Learning 1

Motivation N 0 N 0 New Product New Product Product 1 Product 1 N’ 0 N’ 0 R R R Dynamic Pricing with Demand Learning 2

Motivation N 0 N 0 New Product New Product Product 1 Product 1 N’ 0 N’ 0 � R R Dynamic Pricing with Demand Learning 3

Motivation ◦ For many retail operations “capacity” is measured by store/shelf space. ◦ A key performance measure in the industry is Average Sales per Square Foot per Unit Time. ◦ Trade-off between short-term benefits and the opportunity cost of assets. Margin vs. Rotation. ◦ As opposed to the airline or hospitality industries, selling horizons are flexible. ◦ In general, most profitable/unprofitable products are new items for which there is little demand information. Dynamic Pricing with Demand Learning 4

Outline � Model Formulation. � Perfect Demand Information. � Incomplete Demand Information. - Inventory Clearance - Optimal Stopping (“ outlet option ”) � Conclusion. Dynamic Pricing with Demand Learning 5

Model Formulation I) Stochastic Setting: - A probability space (Ω , F , P ) . - A standard Poisson process D ( t ) under P and its filtration F t = σ ( D ( s ) : 0 ≤ s ≤ t ) . - A collection { P α : α > 0 } such that D ( t ) is a Poisson process with intensity α under P α . � t - For a process f t , we define I f ( t ) := 0 f s d s. Demand Intensity II) Demand Process: - Pricing strategy, a nonnegative (adapted) process p t . Exponential Demand Model - A price-sensitive unscaled demand intensity λ (p) = Λ exp(− α p) λ t := λ ( p t ) ⇐ ⇒ p t = p ( λ t ) . θλ (p) - A (possibly unknown) demand scale factor θ > 0 . Increasing θ - Cumulative demand process D ( I λ ( t )) under P θ . - Select λ ∈ A the set of admissible (adapted) policies λ t : R + → [0 , Λ] . Price (p) Dynamic Pricing with Demand Learning 6

Model Formulation III) Revenues: λ ∗ := arg max λ ∈ [0 , Λ] { c ( λ ) } , c ∗ := c ( λ ∗ ) . - Unscaled revenue rate c ( λ ) := λ p ( λ ) , - Terminal value (opportunity cost): R Discount factor: r - Normalization: c ∗ = r R . IV) Selling Horizon: - Inventory position: N t = N 0 − D ( I λ ( t )) . - τ 0 = inf { t ≥ 0 : N t = 0 } , T := {F t − stopping times τ such that τ ≤ τ 0 } V) Retailer’s Problem: �� � τ e − r t p ( λ t ) d D ( I λ ( t )) + e − r τ R max E θ λ ∈A , τ ∈T 0 N t = N 0 − D ( I λ ( t )) . subject to Dynamic Pricing with Demand Learning 7

Full Information Suppose θ > 0 is known at t = 0 and an inventory clearance strategy is used, i.e. , τ = τ 0 . Define the value function �� � τ 0 e − r t p ( λ t ) d D ( I λ ( t )) + e − r τ R W ( n ; θ ) = max E θ λ ∈A 0 N t = n − D ( I λ ( t )) τ 0 = inf { t ≥ 0 : N t = 0 } . subject to and r W ( n ; θ ) The solution satisfies the recursion = Ψ( W ( n − 1; θ ) − W ( n ; θ )) and W (0; θ ) = R, θ Ψ( z ) � max 0 ≤ λ ≤ Λ { λ z + c ( λ ) } . where Proposition. For every θ > 0 and R ≥ 0 there is a unique solution { W ( n ) : n ∈ N } . ◦ If θ ≥ 1 then the value function W is increasing and concave as a function of n . ◦ If θ ≤ 1 then the value function W is decreasing and convex as a function of n . ◦ lim n →∞ W ( n ) = θR . Dynamic Pricing with Demand Learning 8

Full Information 4.6 θ 1 R W(n; θ 1 ) θ 1 > 1 4 W(n;1) R 3.3 W(n; θ 2 ) θ 2 < 1 θ 2 R 0 5 10 15 20 Inventory Level (n) Value function for two values of θ and an exponential demand rate λ ( p ) = Λ exp( − α p ) . The data used is Λ = 10 , α = 1 , r = 1 , θ 1 = 1 . 2 , θ 2 = 0 . 8 , R = Λ exp( − 1) / ( α r ) ≈ 3 . 68 . Dynamic Pricing with Demand Learning 9

Full Information Optimal Demand Intensity Corollary. Suppose c ( λ ) is strictly concave. 4.5 The optimal sales intensity satisfies: λ ∗ ( n ; θ ) = arg max 0 ≤ λ ≤ Λ { λ ( W ( n − 1; θ ) − W ( n ; θ ))+ c ( λ ) } . θ 2 < 1 4 λ * (n) - If θ ≥ 1 then λ ∗ ( n ; θ ) ↑ n . λ * 3.5 - If θ ≤ 1 then λ ∗ ( n ; θ ) ↓ n . θ 1 > 1 - λ ∗ ( n ; θ ) ↓ θ . 3 5 10 15 20 25 Inventoty Level (n) - lim n →∞ λ ∗ ( n, θ ) = λ ∗ . Exponential Demand λ ( p ) = Λ exp( − α p ) . Λ = 10 , α = r = 1 , θ 1 = 1 . 2 , θ 2 = 0 . 8 , R = 3 . 68 . What about inventory turns (rotation)? Let s ( n, θ ) � θ λ ∗ ( n, θ ) be the optimal sales rate for a given θ and n . Proposition. d d λ ( λ p ′ ( λ )) ≤ 0 , If then s ( n, θ ) ↑ θ. Dynamic Pricing with Demand Learning 10

Full Information Summary: ◦ A tractable dynamic pricing formulation for the inventory clearance model. ◦ W ( n ; θ ) satisfies a simple recursion based on the Fenchel-Legendre transform of c ( λ ) . ◦ With full information products are divided in two groups: – High Demand Products with θ ≥ 1 : W ( n, θ ) and λ ∗ ( n ) increase with n . – Low Demand Products with θ ≤ 1 : W ( n, θ ) and λ ∗ ( n ) decrease with n . ◦ High Demand products are sold at a higher price and have a higher selling rate. ◦ If the retailer can stop selling the product at any time at no cost then: – If θ < 1 stop immediately ( τ = 0 ). – If θ > 1 never stop ( τ = τ 0 ). ◦ In practice, a retailer rarely knows the value of θ at t = 0 ! Dynamic Pricing with Demand Learning 11

Incomplete Information: Inventory Clearance Setting: - The retailer does not know θ at t = 0 but knows θ ∈ { θ L , θ H } with θ L ≤ 1 ≤ θ H . - The retailer has a prior belief q ∈ (0 , 1) that θ = θ L . - We introduce the probability measure P q = q P θL + (1 − q ) P θH . - We assume an inventory clearance model, i.e. , τ = τ 0 . Retailer’s Beliefs: 1 Define the belief process q t := P q [ θ | F t ] . 0.95 0.9 Proposition. q t is a P q -martingale that satisfies the SDE 0.85 q t 0.8 d q t = − η ( q t − ) [ d D t − λ t ¯ θ ( q t − ) d t ] , 0.75 0.7 ¯ where θ ( q ) := θ L q + θ H (1 − q ) 0.65 0.6 η ( q ) := q (1 − q ) ( θ H − θ L ) and θ L q + θ H (1 − q ) . 0.55 0.5 0 50 100 150 200 250 300 350 400 Time, t Dynamic Pricing with Demand Learning 12

Incomplete Information: Inventory Clearance Retailer’s Optimization: �� � τ 0 e − r t p ( λ t ) d D ( I λ ( s )) + e − r τ 0 R V ( N 0 , q ) = sup E q λ ∈A 0 � t subject to N t = N 0 − d D ( I λ ( s )) , 0 d q t = − η ( q t − ) [ d D t − λ t ¯ θ ( q t − ) d t ] , q 0 = q, τ 0 = inf { t ≥ 0 : N t = 0 } . HJB Equation: � � λ ¯ θ ( q )[ V ( n − 1 , q − η ( q )) − V ( n, q ) + η ( q ) V q ( n, q )] + ¯ rV ( n, q ) = max θ ( q ) c ( λ ) , 0 ≤ λ ≤ Λ with boundary condition V (0 , q ) = R , V ( n, 0) = W ( n ; θ H ) , and V ( n, 1) = W ( n ; θ L ) . Recursive Solution: � r V ( n, q ) � V (0 , q ) = R, V ( n, q ) + Φ − η ( q ) V q ( n, q ) = V ( n − 1 , q − η ( q )) . ¯ θ ( q ) Dynamic Pricing with Demand Learning 13

Incomplete Information: Inventory Clearance Proposition. -) The value function V ( n, q ) is a) monotonically decreasing and convex in q , b) bounded by W ( n ; θ L ) ≤ V ( n, q ) ≤ W ( n ; θ H ) , and Value Function 4.5 n=20 c) uniformly convergent as n ↑ ∞ , n= ∞ θ (q)R = [ θ L q+ θ H (1−q)]R n →∞ → R ¯ V ( n, q ) − θ ( q ) , uniformly in q. 4 n=10 V(n,q) -) The optimal demand intensity satisfies n=5 n=1 3.5 n →∞ λ ∗ ( n, q ) = λ ∗ . lim Conjecture: The optimal sales rate ¯ 3 θ ( q ) λ ∗ ( n, q ) ↓ q . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Belief (q) Dynamic Pricing with Demand Learning 14

Incomplete Information: Inventory Clearance Asymptotic Approximation: Since n →∞ V ( n, q ) = R ¯ lim θ ( q ) = lim n →∞ { q W ( n, θ L ) + (1 − q ) W ( n, θ H ) } , we propose the following approximation for V ( n, q ) � V ( n, q ) := q W ( n, θ L ) + (1 − q ) W ( n, θ H ) . Some Properties of � V ( n, q ) : - Linear approximation easy to compute. - Asymptotically optimal as n → ∞ . - Asymptotically optimal as q → 0 + or q → 1 − . - � V ( n, q ) = E q [ W ( n, θ )] � = W ( n, E q [ θ ]) =: V CE ( n, q ) = Certainty Equivalent. Dynamic Pricing with Demand Learning 15

Incomplete Information: Inventory Clearance Relative Error (%) := V approx ( n, q ) − V ( n, q ) × 100% . V ( n, q ) Value Function Relative Error(%) 9 40 n=5 CE V (n,q) 35 8 ~ 30 V(n,q) 7 CE V (n,q) (asymptotic) 25 6 20 5 n=5 15 ~ 4 10 V(n,q) V(n,q) 3 5 (optimal) 2 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Belief (q) Belief (q) Exponential Demand λ ( p ) = Λ exp( − α p ) : Inventory = 5, Λ = 10 , α = r = 1 , θ H = 5 . 0 , θ L = 0 . 5 . Dynamic Pricing with Demand Learning 16