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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=(Ut
0, Ut1,…, Ut
n) Category with n substitutable
variants
Mahajan and van Ryzin: “Stocking Retail Assortments Under Dynamic Consumer Substitution”
RETAILER
Motivation
• 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”
j j jt tU u ξ= +
[1]
[2] [1] 0 [3] [ ]
( )
( | )
j jt t
k j k nt t t t j t t t
q P U U
P U U U U p U U U
= =
= = = > > >…
j jt tU a b L l= − −
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) = S pj(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 [ ( , )]
xE xπ ω
≥
( , )1( , ) t td x U
t t t tf x U x e x += − =
1 1( , ) ( , ) ( , )j j j jt t t t t t tx x f x U xη ω η ω+ += − +
1 1 1( , ) 0jT Tx x xη ω+ + = =
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 q œ Q, 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 ( )j j j jIx q S z q S j Sλ λ= + ∈
( ) ( ) ( ) ( ) 2 ( )
( )( )( )
j
j S
jjP
q S q S x S q S z q S
q Sx x Sq S
λ λ∈
= = +
=
∑
Numerical Experiments
• Assumptions: -T~Poisson(30).
-Qt~exp(1).
• Example 1: –Assume MNL utilities:
–Equal cost and prices.
–Result from van Ryzin and Mahajan: assume v1≥v2≥…≥v , then
optimal assortment is a set Ak={1,2,…,k} with k=1,…,n.
–Simulation: profit within ±1% with 95% confidence.
–Several starting points tested.
Mahajan and van Ryzin: “Stocking Retail Assortments Under Dynamic Consumer Substitution”
0
0
/
/
( ) ( max{ : })
0
j
jj j i
t t i
i S
uj
u
q S P U U i S
e j Swhere
e j
µ
µ
υυ υ
υ
∈
= = ∈ =+
⎧ ∈⎪= ⎨=⎪⎩
∑
Numerical Experiments
• Performance Comparison for Example 1. -Independent newsboy is biased: underestimates popular items, over estimates less popular items.
Mahajan and van Ryzin: “Stocking Retail Assortments Under Dynamic Consumer Substitution”
(See Table 2 and Figure 1 on pages 343-4 of the Mahajan and van Ryzin paper.)
Numerical Experiments
• Example 2: –Two variants: variant 1 less popular but with high margin, and variant 2 more popular but lower margin.
–Sample Path Gradient policy induces customers to “upgrade” to the high-margin variant.
Mahajan and van Ryzin: “Stocking Retail Assortments Under Dynamic Consumer Substitution”
(See Table 3 and Figure 4 on page 345 of the Mahajan and van Ryzin paper.)
Numerical Experiments
• Example 3: –Lancaster demand model:
–Lt~U[0,1], a=0.2, b=1
Mahajan and van Ryzin: “Stocking Retail Assortments Under Dynamic Consumer Substitution”
(See Table 4, Figures 5, and Figure 6 on pages 345-6 of the Mahajan and
van Ryzin paper.)
jt tU a b L l= − −
Price and Scale Effects
• Measure of “evenness”: – y is more “fashionable” than z, if z majorizes y:
• Observations: 1) If v is more fashionable than w, then w is more
profitable
2) Price or volume increase ï higher variety is offered
Mahajan and van Ryzin: “Stocking Retail Assortments Under Dynamic Consumer Substitution”
[ ] [ ]
1 1
[ ]
1 1[ ] 1, , 1
n ni i
i ik k
i
i i
y z
y i z k n
= =
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Conclusions
• Concluding remarks – General choice model. – Improvement upon the existing literature. – Interesting numerical experiments with valuable
insights.
• Comments – Assortment or inventory problem? – Assumes full information. – Single replenishment: price decision might be
relevant. – Industry evidence (field study).
Mahajan and van Ryzin: “Stocking Retail Assortments Under Dynamic Consumer Substitution”