Dynamic Pricing for Non-Perishable Products with Demand Learning

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

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

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

Motivation N 0 Product 1 N 0 New Product 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. - For a process f t, we define I f (t) := t 0 f s ds. II) Demand Process: - Pricing strategy, a nonnegative (adapted) process p t. - A price-sensitive unscaled demand intensity λ t := λ(p t ) p t = p(λ t ). Demand Intensity Exponential Demand Model λ(p) = Λ exp( α p) - A (possibly unknown) demand scale factor θ > 0. - Cumulative demand process D(I λ (t)) under P θ. θλ(p) Increasing 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: - Unscaled revenue rate c(λ) := λ p(λ), λ := argmax λ [0,Λ] {c(λ)}, c := c(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: max λ A, τ T subject to E θ [ τ 0 ] e r t p(λ t ) dd(i λ (t)) + e r τ R N t = N 0 D(I λ (t)). 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 W (n; θ) = max λ A E θ [ τ0 0 ] e r t p(λ t ) dd(i λ (t)) + e r τ R subject to N t = n D(I λ (t)) and τ 0 = inf{t 0 : N t = 0}. The solution satisfies the recursion r W (n; θ) θ = Ψ(W (n 1; θ) W (n; θ)) and W (0; θ) = R, where Ψ(z) max {λ z + c(λ)}. 0 λ Λ 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 4 W(n;θ 1 ) θ 1 > 1 R W(n;1) 3.3 W(n;θ 2 ) θ 2 < 1 θ 2 R Inventory Level (n) 0 5 10 15 20 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 Corollary. Suppose c(λ) is strictly concave. The optimal sales intensity satisfies: λ (n; θ) = argmax {λ (W (n 1; θ) W (n; θ))+c(λ)}. 0 λ Λ 4.5 4 Optimal Demand Intensity θ 2 < 1 - If θ 1 then λ (n; θ) n. λ * (n) λ * - If θ 1 then λ (n; θ) n. 3.5 θ 1 > 1 - λ (n; θ) θ. - lim n λ (n, θ) = λ. 3 5 10 15 20 25 Inventoty Level (n) 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)? Proposition. Let s(n, θ) θ λ (n, θ) be the optimal sales rate for a given θ and n. If d dλ (λ p (λ)) 0, 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: Define the belief process q t := P q [θ F t ]. Proposition. q t is a P q -martingale that satisfies the SDE dq t = η(q t ) [dd t λ t θ(qt )dt], 1 0.95 0.9 0.85 0.8 0.75 q t where θ(q) := θl q + θ H (1 q) and η(q) := q (1 q) (θ 0.6 H θ L ) θ L q + θ H (1 q). 0.55 0.5 0 50 100 150 200 250 300 350 400 Dynamic Pricing with Demand Learning 12 0.7 0.65 Time, t

Incomplete Information: Inventory Clearance Retailer s Optimization: HJB Equation: [ τ0 ] V (N 0, q) = sup E q e r t p(λ t ) dd(i λ (s)) + e r τ 0 R λ A 0 subject to N t = N 0 t 0 dd(i λ (s)), dq t = η(q t ) [dd t λ t θ(qt )dt], q 0 = q, τ 0 = inf{t 0 : N t = 0}. rv (n, q) = max 0 λ Λ [ λ θ(q)[v (n 1, q η(q)) V (n, q) + η(q)vq (n, q)] + θ(q) c(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 θ(q) η(q) V q (n, q) = V (n 1, q η(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 4.5 Value Function c) uniformly convergent as n, V (n, q) n R θ(q), uniformly in q. 4 n=10 n=20 n= θ(q)r = [θ L q+θ H (1 q)]r -) The optimal demand intensity satisfies lim n λ (n, q) = λ. V(n,q) 3.5 n=5 n=1 Conjecture: The optimal sales rate θ(q) λ (n, q) q. 3 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 lim V (n, q) = R θ(q) = lim {q W (n, θ L) + (1 q) W (n, θ H )}, n n we propose the following approximation for V (n, q) Ṽ (n, q) := q W (n, θ L ) + (1 q) W (n, θ H ). Some Properties of Ṽ (n, q): - Linear approximation easy to compute. - Asymptotically optimal as n. - Asymptotically optimal as q 0 + or q 1. - Ṽ (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) V (n, q) 100%. 9 8 7 6 5 4 3 Value Function n=5 CE V (n,q) V(n,q) (optimal) 2 0 0.2 0.4 0.6 0.8 1 Belief (q) ~ V(n,q) (asymptotic) 40 35 30 25 20 15 10 5 Relative Error(%) n=5 CE V (n,q) ~ V(n,q) 0 0 0.2 0.4 0.6 0.8 1 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

Incomplete Information: Inventory Clearance For any approximation V approx (n, q), define the corresponding demand intensity using the HJB λ approx (n, q) := arg max [λ θ(q)[v approx (n 1, q η(q)) V approx (n, q)]+λ κ(q)vq approx (n, q)+ θ(q) c(λ)]. 0 λ Λ Relative Price Error (%) := p(λ approx ) p(λ ) p(λ ) 100%. Asymptotic Approximation (%) Inventory (n) q 1 5 10 25 100 0.0 0.0 0.0 0.0 0.0 0.0 0.2 2.7-0.2-0.3-0.6-0.5 0.4 6.9 0.8-0.6-0.9-0.7 0.6 12.5 2.4-0.2-0.7-1.0 0.8 19.4 3.3 0.1-0.4-0.6 1.0 0.0 0.0 0.0 0.0 0.0 Certainty Equivalent (%) Inventory (n) q 1 5 10 25 100 0.0 0.0 0.0 0.0 0.0 0.0 0.2 5.3 2.6 2.7 2.4-0.4 0.4 14.4 11.6 12.0 10.1-0.5 0.6 29.9 28.2 28.0 17.6-1.0 0.8 54.6 46.2 37.4 11.1-0.7 1.0 0.0 0.0 0.0 0.0 0.0 Relative price error for the exponential demand model λ(p) = Λ exp( α p), with Λ = 20 and α = 1. Dynamic Pricing with Demand Learning 17

Incomplete Information: Inventory Clearance When should the retailer engage in selling a given product? When V (n, q) R. Using the asymptotic approximation q q(n) := Ṽ (n, q), this is equivalent to W (n; θ H ) R [W (n; θ H ) R] + [R W (n; θ L )]. 0.5 0.49 Non Profitable Products ~ q(n) 0.48 0.47 Profitable Products q(n) q := θ H 1 θ H θ L, as n. 0.46 1 5 10 15 20 25 30 Initial Inventory (n) Exponential demand rate λ(p) = Λ exp( α p). Data: Λ = 10, α = 1, r = 1, θ H = 1.2, θ L = 0.8. Dynamic Pricing with Demand Learning 18

Incomplete Information: Inventory Clearance Summary: Uncertainty in market size (θ) is captured by a new state variable q t (a jump process). V (n, q) can be computed using a recursive sequence of ODEs. Asymptotic approximation Ṽ (n, q) := E q[w (n, θ)] performs quite well. Linear approximation easy to compute. Value function: V (n, q) Ṽ (n, q). Pricing strategy: p (n, q) p(n, q). Products are divided in two groups as a function of (n, q): Profitable Products with q < q(n) and Non-profitable Products with q > q(n). The threshold q(n) increases with n, that is, the retailer is willing to take more risk for larger orders. Dynamic Pricing with Demand Learning 19

Incomplete Information: Optimal Stopping Setting: - Retailer does not know θ at t = 0 but knows θ {θ L, θ H } with θ L 1 θ H. - Retailer has the option of removing the product at any time, Outlet Option. Retailer s Optimization: U(N 0, q) = max λ A, τ T subject to E q [ τ N t = N 0 D(I λ (t)), 0 ] e r t p(λ t ) dd(i λ (t)) + e r τ R dq t = η(q t ) [dd(i λ (t)) λ t θ(qt )dt], q 0 = q. Optimality Conditions: { U(n, q) + Φ( r U(n,q) ) η(q) U θ(q) q (n, q) = U(n 1, q η(q)) if U R U(n, q) + Φ( r U(n,q) ) η(q) U θ(q) q (n, q) U(n 1, q η(q)) if U = R. Dynamic Pricing with Demand Learning 20

Incomplete Information: Optimal Stopping Proposition. a) There is a unique continuously differentiable solution U(n, ) defined on [0, 1] so that U(n, q) > R on [0, q n ) and U(n, q) = R on [q n, 1], where q n is the unique solution of ( ) r R R + Φ = U(n 1, q η(q)). θ(q) b) q n is increasing in n and satisfies θ H 1 θ H θ L q n n q Root {Φ ( ) r R θ(q) = η(q) q } (θ H 1) R < 1. c) The value function U(n, q) Is decreasing and convex in q on [0, 1] Increases in n for all q [0, 1] and satisfies max{r, V (n, q)} U(n, q) max{r, m(q)} for all q [0, 1], where m(q) := W (n, θ H ) (W (n, θ H) R) q n Converges uniformly (in q) to a continuously differentiable function, U (q). Dynamic Pricing with Demand Learning 21 q.

Incomplete Information: Optimal Stopping Relative Error (%) Critical Threshold (q * n ) 0.25 1 θ L θ L 0.62 Always Non Profitable 0.2 U(n,q) V(n,q) V(n,q) n=40 0.6 0.58 0.15 n=10 0.56 0.54 Profitable Under Stopping but Non Profitable Under Inventory Clearance 0.1 0.52 0.5 0.05 n=1 0.48 0.46 Always Profitable 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Belief (q) 0.44 5 10 15 20 25 30 Inventory (n) Exponential demand rate λ(p) = Λ exp( α p). Data: Λ = 10, α = 1, r = 1, θ H = 1.2, θ L = 0.8. Dynamic Pricing with Demand Learning 22

Incomplete Information: Optimal Stopping Approximation: Ũ(n, q) := max{r, W (n, θ H ) (W (n, θ H) R) q} q n ( ) r R where q n is the unique solution of R + Φ = Ũ(n 1, q η(q)). θ(q) 4.4 4.2 4 R U(n,q) (Optimal) Value Function ~ U(n,q) (Approximation) (%) 6 5 4 3 2 ~ Relative Error (%) U(n,q) U(n,q) 100% U(n,q) n=40 n=10 n=5 3.4 n=10 0 0.2 0.4 0.6 0.8 1 Belief (q) 1 n=1 0 0 0.2 0.4 0.6 0.8 1 Belief (q) Exponential demand rate λ(p) = Λ exp( α p). Data: Λ = 10, α = 1, r = 1, θ H = 1.2, θ L = 0.8. Dynamic Pricing with Demand Learning 23

Incomplete Information: Optimal Stopping Summary: U(n, q) can be computed using a recursive sequence of ODEs with free-boundary conditions. For every n there is a critical belief q n above which it is optimal to stop. Again, the sequence q n larger orders. is increasing with n, that is, the retailer is willing to take more risk for The sequence q n is bounded by θ H 1 θ H θ L q n ˆq := Root { ( ) r R Φ θ(q) = η(q) q } (θ H 1) R The outlet option increases significantly the expected profits and the range of products (n, q) that are profitable. 0 U(n, q) V (n, q) (1 θ L ) + R. A simple piece-wise linear approximation works well. Ũ(n, q) := max{r, W (n, θ H ) (W (n, θ H) R) q n q} Dynamic Pricing with Demand Learning 24

Concluding Remarks A simple dynamic pricing model for a retailer selling non-perishable products. Captures two common sources of uncertainty: Market size measured by θ {θ H, θ L }. Stochastic arrival process of price sensitive customers. Analysis gets simpler using the Fenchel-Legendre transform of c(λ) and its properties. We propose a simple approximation (linear and piecewise linear) for the value function and corresponding pricing policy. Some properties of the optimal solution are: Value functions V (n, q) and U(n, q) are decreasing and convex in q. The retailer is willing to take more risk ( q) for higher orders ( n). The optimal demand intensity λ (n, q) q and the optimal sales rate θ(q) λ (n, q) q. Extension: R(n) = R + ν n K 1(n > 0). Dynamic Pricing with Demand Learning 25