Stocking Retail Assortments Under Dynamic Consumer Substitution Siddharth Mahajan Garret van Ryzin Operations Research, May-June 2001 Presented by Felipe Caro 15.764 Seminar: Theory of OM April 15th, 2004 This summary presentation is based on: Mahajan, Siddharth, and Garrett van Ryzin. "Stocking Retail Assortments Under Dynamic Consumer Substitution." Operations Research 49, no. 3 (2001).Motivation RetailerSample path with T sequential customers Customer t with preferences Ut=(Ut0, Ut1,…, Utn) Category with n substitutable variants Mahajan and van Ryzin: “Stocking Retail Assortments Under Dynamic Consumer Substitution” RETAILERMotivation • Retail consumers might substitute if their initial choice is out of stock – Retailer’s inventory decisions should account for substitution effect • Consumers' final choice depends on what he/she sees available “on the shelf”. – In most previous models demand is independent of inventory levels. • Contribution of this paper: – Determination of initial inventory levels (single-period) taking into account dynamic substitution effects Mahajan and van Ryzin: “Stocking Retail Assortments Under Dynamic Consumer Substitution”Outline • Brief Literature Review •Model Formulation • Structural Properties – Component-wise Sales Function – Total Profit Function – Continuous Model • Optimizing Assortment Inventories •Numerical Experiments • Price and Scale Effects • Conclusions Mahajan and van Ryzin: “Stocking Retail Assortments Under Dynamic Consumer Substitution”Brief Literature Review •Demand models with static substitution: – Smith and Agrawal (2000) / van Ryzin and Mahajan (1999): 1. First choice is independent of stock levels 2. If first choice is out of stock, the sale is lost (no second choice) 3. Consequently, demand is independent of inventory levels • Papers that model dynamic effects of stock-outs on consumer behavior – Anupindi et al. (1997): concerned solely with estimation problems (no inventory decisions) – Noonan (1995): only one substitution attempt Mahajan and van Ryzin: “Stocking Retail Assortments Under Dynamic Consumer Substitution”Model Formulation • Notation: (See "Model Formulation" on page 336 of the Mahajan and Mahajan and van Ryzin: “Stocking Retail Assortments Under Dynamic Consumer Substitution” van Ryzin paper)Model Description •Assumptions: –Sequence of customers is finite w.p.1 –Each customer makes a unique choice w.p.1 • Some special cases: -Multinomial Logit (MNL): -Markovian Second Choice: -Universal Backup: all customers have an identical second choice -Lancaster demand: attribute space [0,1] and customer t has a random “ideal point” Lt, then Mahajan and van Ryzin: “Stocking Retail Assortments Under Dynamic Consumer Substitution” jjjttUuξ=+[1][2] [1] 0 [3] [ ]()(|)jjttkjk nttttjtt tqPU UPUUUU pUU U===== >>>…jjttUabLl=− −Model Formulation • Profit function: – hj(x,w) = sales of variant j given x and w. – Individual profit: pj(x ,w) = pj hj(x,w) – cjxj. – Total profit: p(x ,w) = Spj(x ,w). • Retailer’s objective: • Recursive formulation: – System function: –Sales-to-go: – Border conditions: Mahajan and van Ryzin: “Stocking Retail Assortments Under Dynamic Consumer Substitution” 0max [ ( , )]xExπω≥(, )1(, )ttdxUtt t tfxU x e x+=− =11(,) (, ) ( ,)jjj jtt t tt t txxfxU xηωηω++=− +11 1(,)0jTTxxxηω++==Structural Properties • Lemma 1: –x¥y ï xt¥yt for all sample paths. • Decreasing Differences: –h:SµQö√ satisfies decreasing differences in (z,q) if h(z’,q) – h(z,q) ¥ h(z’,q’) – h(z,q’) for all z’¥ z , q’¥q. – Lemma: if max{h(z,q): zœS} has at least one solution for every qœQ, then the largest maximizer z*(q) is nonincreasing in q. Mahajan and van Ryzin: “Stocking Retail Assortments Under Dynamic Consumer Substitution”Structural Properties • Theorem 1: The function hj(z·ej+ q,w) satisfies: (a) Concavity in z for all w. (b) Decreasing differences in (z,q) for all w. • Corollary 1: (a) A base-stock level is optimal for maximizing the component-wise profits. (b) The component-wise optimal base-stock level for j is nonincreasing in xi (i≠j). ï Usual newsboy problem Mahajan and van Ryzin: “Stocking Retail Assortments Under Dynamic Consumer Substitution”Structural Properties • Let: –T(x ,w)=∑ hj(x,w) = total sales –Hj(z ,w)= T(z·ej+ q,w) • If all variants have identical price and cost, then: p(x ,w) = p·T(x ,w) – c·∑ xj • Theorem 2: There exists initial inventory levels x and sample paths w for which: (a) T(x ,w) is not component-wise concave in x. (b) Hj(z ,w) does not satisfy decreasing differences in (z,q). Mahajan and van Ryzin: “Stocking Retail Assortments Under Dynamic Consumer Substitution”Structural Properties • Counterexample for (a): • Continuous model: –Customer t requires a quantity Qt of fluid with distribution Ft(·) –Redefine sample paths: w = {(Ut, Qt): t=1,…,T} • Theorem 3: There exist sample paths on which p(x ,w) is not quasi-concave Mahajan and van Ryzin: “Stocking Retail Assortments Under Dynamic Consumer Substitution” –(See the first two tables on page 339 of the Mahajan andvan Ryzin paper)Optimizing Assortment Inventories • Lemma 3: If the purchase quantities Qt are bounded continuous random variables then ∇E[h(x,w)] = E[∇h(x,w)] • Calculating ∇η(x,ω): Mahajan and van Ryzin: “Stocking Retail Assortments Under Dynamic Consumer Substitution” –(See the equations and explanation in section 4.1, page 341of the Mahajan and van Ryzin paper)Optimizing Assortment Inventories • Sample Path Gradient Algorithm • Theorem 4: If Ft(·) is Lipschitz for all t then any limit of yk is a stationary point Mahajan and van Ryzin: “Stocking Retail Assortments Under Dynamic Consumer Substitution” –(See the steps in section 4.2, page 341 of the Mahajan and vanRyzin paper)Numerical Experiments • Heuristic policies (with T~Poisson) Let qj(S) be the probability of choosing variant j from S 1. Independent Newsboy: demand for each variant is independent of stock on hand. 2. Pooled Newsboy: customers freely substitute among all available variants. Mahajan and van Ryzin: “Stocking Retail Assortments Under Dynamic Consumer Substitution” () 2 ()jj j jIxqSz qSjSλλ=+ ∈() () () () 2 ()()()()jjSjjPqS q S xS qS z