Retail Inventories
and
Consumer Choice
__________________________
Siddharth Mahajan
Garrett J. van Ryzin
Overview
• Modeling Consumer Choice
– Attribute Models and Utility Models
– Binary Probit, Binary Logit, Multinomial Logit
• Inventory Management Under Static Choice
– Smith and Agrawal Model
– van Ryzin and Mahajan Models
•
Independent Population and Trend-Following Population
• Inventory Management Under Dynamic Choice
– Noonan Model
– Mahajan and van Ryzin Model
•
Sample Path Gradient Algorithm
• Maximum Likelihood Estimation of the MNL
Introduction
• Faced with limited choices, customers are often willing to
substitute rather than go home empty handed.
• Customers are heterogeneous in taste, and they are often
willing to pay a higher price for products with attributes closer
to their desired attributes.
• Therefore, a retailer has an incentive to offer a broad variety of
products to better cover the possible range of consumer tastes.
• There are few direct costs of variety for a retailer, but the
indirect costs of stockouts and overstocking impose an implicit
cost on variety. Trade-off: “breadth vs. depth” of assortment.
Choice Processes
• Given two alternatives, a choice corresponds to an expression
of preference of one alternative over another.
• Let X be a set of alternatives and let  be a binary relation on
the set X. For alternatives x, y  X, x  y denotes that y is
strictly preferred to x.
• Definition: A binary relation  on a set X is called a
preference relation if it is asymmetric and negatively transitive.
• Theorem: If X is a finite set, a binary relation  is a
preference relation if and only if there exists a function u: X 
 (called a utility function) such that
xy
iff
u(x) < u(y)
 Two approaches for modeling choice:
1.) Construct preference relations directly
2.) Construct utilities and then apply utility maximization
Lexicographic Model
• A product is made up of binary attributes. The consumer ranks
all attributes and then eliminates alternatives which do not
possess the most important attribute. If more than one
alternative remains, the next most important attribute is chosen
as a criterion for elimination of alternatives, and so on.
• This model implies attributes strictly dominate each other.
• Equivalent utility maximization model: There are n
alternatives that have m attributes (1 = highest, m = lowest).
Let ajk = 1 if alternative j possesses attribute k. Then a utility
satisfying Theorem 1 is the binary number
Uj = aj1aj2…ajm
Address Model
• We have n alternatives, each of which has m attributes that take
on real values. Therefore, alternatives can be represented as n
points, z1, …, zn, in m, which is called the attribute space.
• Each consumer has an ideal point (address) y  m, and
chooses the product closest to it in attribute space, where
distance is defined by a metric  on m m.
• These distances define a preference relation: i  j if and only
if (zi, y) > (zj, y).
• Equivalent utility maximization model:
Uj(y) = c - (zj, y)
Tversky Model
• Randomization of the deterministic lexicographic rule.
• Process:
1.) Delete attributes common to all alternatives.
2.)
Select one of the remaining attributes with probability
proportional to a specified (deterministic) utility value.
3.)
Eliminate alternatives not having the selected attribute.
4.)
If a single alternative remains, choose it. If several
alternatives remain, repeat the above process. If all
remaining alternatives have the same attributes, choose
one randomly.
Randomized Address Model
• Alternatives can be represented as elements z1, …, zn, of m,
and each individual has an ideal point (address) y  m.
• Let g(y), y  m, denote the density function for y.
• Uj(y) = c -  || zj – y ||, where c is the utility of y and 
represents the disutility due to deviations from y.
• The market space of variant j is given by
Mj = { y  m : Uj(y)  Ui(y), i = 1,…,n}
• The probability that a consumer buys variant j is given by
Pj =  g(y) dy
Mj
Random Utility Models
• Let the n alternatives be denoted j = 1,…,n. A consumer
associates a utility with alternative j, denoted Uj.
• This utility is decomposed into two parts: a representative
component uj that is deterministic and a random component j
with mean zero:
Uj = uj + j.
• The probability that an individual selects alternative j is
qj = P(Uj = max Ui)
i = 1,…,n
• This probability depends on the joint distribution of the random
components j. Common versions are the binary probit, binary
logit, and multinomial logit.
Binary Probit
• Two alternatives, and the error terms j, j = 1, 2 are iid N(0,2).
• Probability that Variant 1 is chosen:
q1
=
P(U1 = max {U1, U2})
=
P(U1  U2)
=
P(u1 + 1  u2 + 2)
=
P(2 – 1  u1 – u2)
=
u u 
 1 2 
 2 
• No closed form solution.
Binary Logit
• Two alternatives, and the error term  = 1 – 2 has a logistic
cumulative distribution, i.e.
F ( ) 
1
1 e

x

•  > 0 is a scale parameter and –  <  < 
• E() = 0 and Var() = (22)/3
• The logistic distribution provides a good approximation to the
normal distribution, but with “fatter tails.”
• Probability that Variant 1 is chosen:
q1
=
P(U1 = max {U1, U2})
=
P(U1  U2)
=
P(u1 + 1  u2 + 2)
=
P(2 – 1  u1 – u2)
u1
=
e
u1
u2
e  e 
Multinomial Logit (MNL)
• j are iid random variables with a Gumbel (or double
exponential) distribution, with cdf
F ( x)  P( j  x)  e
e
x
 (  )

•  is Euler’s constant (= 0.5772…) and  is a scale parameter,
E[j] = 0 and Var[j] = (22)/6
• The probability that an alternative j is chosen from a set
S  {1,2,…,n} that contains j is given by
uj
PS ( j ) 
e
ui
e
iS
Independence from Irrelevant Alternatives Property
• For all S  N, and T  N such that S  T, and for all i,j  S,
PS (i ) PT (i )

PS ( j ) PT ( j )
• The ratio of choice probabilities for i and j is independent of
the choice set containing these alternatives. This property is
not realistic if the choice set contains alternatives that can be
grouped such that alternatives within a group are more similar
than alternatives outside the group, because adding a new
alternative reduces the probability of choosing similar
alternatives more than dissimilar alternatives.
Blue bus/red bus paradox
• Suppose the individual selects to travel by car or by bus with
equal probability. Let the set S = {car, bus}. Then
PS(car) = PS(bus) = 1/2
• Introduce a new bus. Let the set T denote {car, blue bus, red
bus}. It can be shown that the MNL predicts
PT(car) = PT(blue bus) = PT(red bus) = 1/3
• However, a more natural outcome is
PT(car) = 1/2
PT(blue bus) = PT(red bus) = 1/4
Inventory Management Under Static Choice
• The retailer’s decision problem is to determine which subset S
from a universe of N possible variants should be included in the
assortment and how much to stock of each variant.
• Static choice models: Customer choices do not depend on the
transient inventory status of the variants in the assortment, but
only on S.
– Examples: catalog retailer, shoe store
Assumptions of Static Choice Models
(A1)
The initial choice of a variant is independent of the
inventory status of the variants in S.
(A2)
If a customer selects a variant in S and the store does not
have it in stock, the customer does not undertake a
second choice, and the sale is lost.
Notation
N
The set {1,2,…,n} of all variants available in the market
S
The subset of variants stocked by the retailer
xj
Initial inventory of variant j (decision variable)
x
Inventory level vector, {xj : j  N}
Y
Total number of customers, or store traffic (r.v.)

Mean of Y
Yj
Demand for the jth variant (a function of S, r.v.)
qj(S)
Probability that variant j is chosen by an arriving
customer
Smith and Agrawal Model
• Demand Model:
– Each customer has an initial preference for variant j  N
with probability fj.
– If j  S, the customer chooses to purchase variant j.
– If j  S, the customer will substitute with another variant
i  N with probability sji.
– A customer chooses not to substitute with probability
Lj = 1 -  sji
iN
• Probability that variant j is chosen from a set S:
q j (S )  f j  
iS ,i  j
fi sij
• Total store traffic: Y is modeled as a negative binomial r.v.
Let  represent the pmf for aggregate demand.
 k  y  1 k
 ( y)  
t (1  t ) y
 k 1 
where y = 0,1,2,… and k and t are parameters of the negative
binomial distribution, and
k (1  t )

t
• Given the total demand Y = y and the subset of variants S,
Yj has a binomial distribution with parameters y and qj(S).
Unconditioning over aggregate demand, Yj also has a negative
binomial distribution.
• Let j(yj,S) represent the pmf for the preference for each
variant j given S. Then
 k  y j  1 k
y
 j ( y j , S)  
r j (1  r j ) j
 k 1 
where
t
rj 
t  q j ( S )(1  t )
• Denote the cdf of Yj by
yj
 j ( y j , S )    j (d , S )
d 0
• Substitution Probability Models:
1.) Random Substitution
1 L 1 L 1 L

 0
3
3
3 
1  L
1 L 1 L
0


3
3 
 3
1 L
1  L 1  L
0
 3
3
3 
1  L 1  L 1  L

0
 3

3
3
2.) Adjacent Substitution
1 L
0
0 
 0
1 L
1  L
0
0 
 2

2

1 L
1 L
0
 0

2
2 

0
1 L
0 
 0
3.) One Variant Substitution
0
0

0
0

1  L 0 0
1  L 0 0

1  L 0 0
1  L 0 0
• Profit Function:
– Single-period, lost sales (newsvendor) model
– Notation
kj
Fixed cost to stock variant j
mj
Unit profit margin obtained from selling variant j
coj
Overage cost for variant j
cuj
Underage cost for variant j
– The profit obtained from stocking xj units of variant j, denoted
j(S,xj), is
xj
 j ( S , x j )  m j E[Y j ]  coj  ( x j  d ) j (d , S )
d 0
xj
 cuj  (d  x j ) j (d , S )  k j
d 0
– The total profit, denoted (S,x), is given by
 ( S , x)    j ( S , x j )
jS
• Assortment Optimization
– Smith and Agrawal propose a non-linear integer
programming formulation to solve for the optimal subset
S* and the optimal stocking quantities x*.
– Binary variable zj indicates whether item j is included in S
(zj = 1 if j  S).
– Possible constraints on x and S:
 t j z j x j  T1
jS
 z j  T2
jS
– From the static choice assumptions (A1) and (A2), given S,
the optimal stocking decision for each variant, xj*, is
determined by solving a simple, single-item newsvendor
problem:
 j ( x j  1, S ) 
*
cuj
coj  cuj
  j ( x*j , S )
– Substituting these values into the profit function yields a
discrete optimization problem with a nonlinear objective
function.
• Numerical Results:
1.)
The optimal profit for the No Substitution model is
always lower than the substitution model at any given
level of variety.
2.)
Random Substitution generally requires the largest
optimal assortment, Adjacent Substitution a somewhat
smaller optimal assortment, and One Variant Substitution
is often maximized with only one variant (the “universal
substitute”).
3.)
As L increases, the number of variants in the optimal
assortment increases.
van Ryzin and Mahajan Models
• MNL Choice Model:
– Each customer associates a random utility Uj with the
variants j  S.
– A no-purchase option, denoted j = 0 with associated utility
U0, is introduced.
– Customers choose the variant with the highest utility
among the set {Uj : j  S  {0}}.
– Each variant has an identical retail price p and has an
identical unit cost c. No fixed costs.
• Given S, the probability that variant j is selected is given by
q j ( S )  P(U j  max{U i : i  S  {0}}

vj
v0   vi
iS
where
uj
vj  e 
j  S  {0}
• The quantities vj are called preferences, because the values are
increasing in the systematic component of utility, uj.
Independent Population Model
• Each customer assigns utilities to the variants in the subset S
based on independent samples of the MNL model.
• The utility Uij that customer i assigns to variant j is given by
Uij = uj + ij
where the {ij, i  1} are iid random variables.
•  = mean number of customers arriving during the season
• The number of customers selecting variant j, Yj, is normally
distributed with mean  qj(S) and standard deviation ( qj(S)),
where  > 0 and 0   < 1.
• c = cost for each unit not sold (no salvage value)
• p – c = the opportunity cost for each unit short (loss in margin)
• The maximum expected profit given S and v is:
 I ( S , v)  max  E[ p min{ x j , Y j }  cx j ]
x 0 jS
• The optimal stocking level of variant j is given by
x*j
c
 q j ( S )  z (q j ( S )) , where z   (1  )
p

1
Trend-Following Population Model
• A fixed number  of customers visits the store. Each customer
has identical valuations of the utilities for the variants; these
utilities are determined by a single sample of the MNL model.
• {ij = 1j, i > 1}
• The common utility values are not observable to the retailer
prior to making assortment decisions.
• Demand for variant j, denoted Yj, is a scaled Bernoulli r.v.
y
q j ( S )

P(Y j  y )  1  q j ( S ) y  0
0
otherwise

• Expected profit from variant j given xj is E[p min{Yj, xj} – cxj].
• The optimal expected profit given S and v is:
 T ( S , v)   ( pq j ( S )  c)  
jS
• The optimal stocking level of variant j is given by
x*j
 0 if pq j ( S )  c

 if pq j ( S )  c
Assortment Optimization
• We can formulate the optimal assortment selection problem by
solving
max  ( S , v)
SN
• Let the variants be ordered as v1  v2 . …  vn.
• Theorem: Let Ai = {1, 2, … , i} for 1  i  n. Then for
each of the assortment problems defined above, there exists
an S*  {A1, … , An} that maximizes store profits.
 Choose the best i variants, where 1  i  n.
• Theorem: For all n > i  1,
a)
 (Ai+1, v) >  (Ai, v) for sufficiently high selling price p
b)
 (Ai+1, v) <  (Ai, v) for sufficiently low no-purchase
preference v0
c)
 (Ai+1, v) >  (Ai, v) for sufficiently high store volume 
(independent population case only)
Inventory Management Under Dynamic Choice
• Dynamic choice models: Customer choices do depend on the
transient inventory status of the variants in the assortment. A
consumer may have a preferred variant, but upon finding that
variant out of stock, he may decide to substitute a different
variant.
– Examples: grocery items, soft drinks
• Dynamic choice models are more realistic, but they are less
tractable so we must rely more heavily on computational
studies to understand the problem.
Noonan Model
• Single period inventory model of a merchandise category made
up of multiple product variants, each having a different unit
selling price and unit procurement cost.
• Customers have a first choice and a second choice and demand
is generated in two stages:
1.)
Primary demand is realized and satisfied as much as
possible with available inventory.
2.)
Unfilled demand is converted to secondary demand for
products based on deterministic proportions.
Notation
N
The set {1,2,…,n} of all variants available in the market
xj
Initial inventory of variant j (decision variable)
x
Inventory level vector, {xj : j  N}
cj
Unit procurement cost for variant j
pj
Unit selling price for variant j
sij
Fixed proportion of unfilled demand for variant i that is
transferred to variant j after stockouts
Definition of Demand Regions, n = 2
R
Region of demand where neither variant stocks out
R
Region of demand where both variants stock out on
original demand
R1
Variant 1 stocks out on original demand, but the
substitution demand is satisfied using Variant 2
R12
Variant 1 stocks out on original demand and its
substitution demand is large enough to stock out Variant 2
R2
Variant 2 stocks out on original demand, but the
substitution demand is satisfied using Variant 1
R21
Variant 2 stocks out on original demand and its
substitution demand is large enough to stock out Variant 1
Analysis of Demand Regions
• Let PA denote the probability that the demand vector lies in
region RA. The expression for expected profit can be written by
integrating over the different regions. Differentiating leads to
the following first-order necessary conditions:
c1  p1[ P  P12  P21  P1 ]  s12 p2 P1
c2  p2[ P  P12  P21  P2 ]  s21 p1P2
Mahajan and van Ryzin Model
• Dynamic choice version of the assortment problem using a
general random utility model.
• The merchandise category consists of n substitutable variants:
p j = selling price of variant j
c j = procurement cost of variant j
• Single-period (newsvendor-like) inventory model in which the
retailer’s only decision is the vector of initial inventory levels x
for each of the variants.
Sample Path Analysis
• T = number of customers on a sample path
• xt = (xt1, …, xtn) is the vector of inventory levels observed by
customer t (t = 1,…,T)
• x1 = x, the initial stocking decision
• S(y) = {j  {0} : yj > 0} for any real inventory vector y
– S(y) is the set of in-stock variants together with the no-purchase
option
– Customer t can only make a choice j  S(xt)
– S(xt+1)  S(xt), since inventory levels are nonincreasing over time
• Ut = (Ut0, Ut1, … , Utn) is the vector of utilities assigned by
customer t
• Based on xt and Ut, customer t makes the choice d(xt, Ut) that
maximizes his utility:
d ( xt ,U t )  arg max {U t j }
jS ( xt )
• Let  = {Ut : t = 1,…,T} denote a sample path from some
probability space (, , ).
– The retailer does not know the particular realization  but does
know the probability measure , so we think of  as
characterizing the retailer’s knowledge of future demand.
– The retailer’s objective is to choose x to maximize total expected
profit.
Total Profit
•  j(x, ) = the number of sales of variant j made on the sample
path  given initial inventory levels x
• (x, ) = (1(x, ), …,  n(x, ))
•  (x, ) = sample path profit
 (x, ) = pT (x, ) – cTx
• Retailer’s objective is to solve
max E[ ( x, )]
x0
A Fluid Model Relaxation
• Inventory is viewed as a fluid. Each customer t requires a
certain quantity of fluid, Qt, which could vary from customer
to customer and could be non-integral.
• The customer drains the inventory of the most preferred fluid.
If this fluid runs out, the customer drains the inventory of the
second most preferred fluid and so on.
• This process continues until either the customer’s requirement
is met or the inventory of all fluids valued higher than the nopurchase utility is exhausted.
• (x, ) is the sample path gradient.
Sample Path Gradient Algorithm
• Requires an initial starting inventory level y and a sequence of
step sizes {ak} with the following properties:

 ak  
k 0
•

and
2
 ak  
k 0
Sample Path Gradient Algorithm:
1. Initialize: k = 0 and y0 = y
2. At iteration k
i) Generate a new sample path k
ii) Calculate (yk, k) and ak
iii) Update the starting inventory level for the next iteration,
using the equation yk+1 = yk + ak (yk, k)
3. Set k = k + 1 and go to Step 2
Numerical Experiments – Assumptions
• The MNL model was used to generate the sequence of utilities
{Ut}. The systematic component of utility is broken down as
uj = y + a j – pj
and u0 = y + a0
where y stands for consumer income, aj is a quality index and
pj is the price for variant j.
• The examples have n = 10 variants, with linearly decreasing
quality indices
aj = 12.25 – 0.5(j – 1)
and a0 = 4.0.
j = 1,…,10
Numerical Experiments – Assumptions (cont’d)
• The error terms tj are iid, Gumbel distributed with parameter
 = 1.5, so the variance of tj is 1.18.
• The procurement cost for a unit is set at cj = 3.0 for all j, and
the price is set at pj = p ( j = 1,…,10), where p takes values in
the range 3–9. This simplification facilitates comparison with
other heuristic policies.
• The number of customers in the sequence, T, is a Poisson r.v.
with mean 30, and each customer demands exactly one unit of
product (Qt = 1 for all t).
Heuristic Policies
• Independent Newsvendor
• Pooled Newsvendor: The entire category is aggregated and
viewed as a single variant. An aggregate newsvendor
inventory level is then calculated for the category based on the
probability the category will be chosen over the no-purchase
option. This aggregate inventory is then allocated
proportionally to the preference values vj.
• Naïve Gradient: At each iteration, we decrease the inventory
level of all variants that are left in stock at the end of the
sample path and increase the inventory level of all variants that
are out of stock at the end of the sample path.
Comparison of Gross Profit
Price
5.0
8.0
Sample Path Gradient
47.82 98.15
Independent Newsvendor
46.31 95.40
Pooled Newsvendor
47.35 96.92
Naïve Gradient
42.48 92.74
Comparison of Stocking Levels, p = 8.0
Inventory Level
9
8
7
Sample Path Gradient
Independent Newsvendor
6
5
4
3
2
1
0
1
2
3
4
5
6
Variant
7
8
9
10
Explanation
• The demand for a given variant will be higher than the
independent newsvendor predicts because each variant receives
some additional substitute demand from other variants that are
out of stock. This effect increases the level of demand, which
provides an incentive to increase inventory.
• The unit underage cost is lower than the independent
newsvendor predicts because an underage in one variant does
not always result in a lost sale. This reduces the effective
underage cost, and creates an incentive to decrease inventory.
• The independent newsvendor is biased; it stocks too little of the
popular variants and too much of the less popular variants.
Comparison of Stocking Levels, p = 8.0
Inventory Level
9
8
7
Sample Path Gradient
Pooled Newsvendor
6
5
4
3
2
1
0
1
2
3
4
5
6
Variant
7
8
9
10
Inventory Level
Comparison of Stocking Levels, p = 8.0
10
9
8
7
6
5
4
3
2
1
0
Sample Path Gradient
Naïve Gradient
1
2
3
4
5
6
Variant
7
8
9
10
Explanation
• When inventory of a particular variant is left over at the end of
the sample path, both the naïve gradient and the sample path
gradient method choose a gradient direction which decreases
the inventory level of that variant.
• When a variant is sold out at the end of the sample path, the
naïve gradient always increases the inventory level of that
variant.
• However, the sample path gradient method does not always
increase the inventory level, because an incremental unit may
only result in a substitution rather than an additional sale.
• Therefore, the naïve gradient method overestimates the
inventory level.
Maximum Likelihood Estimation of the MNL
• We must estimate the systematic components of utility uj
required for computing the choice probabilities of the MNL.
Assume utilities are scaled so that  = 1.
• Linear in Attributes Model:
– Let each variant j = 1 ,…, n be associated with an m-vector of
attribute values, yj = (y1j,…, ymj).
– Let  = (1, … , m) be a vector of coefficients which determine
how attribute values are weighted to obtain uj.
– Then uj =  T yj, j = 1,…, n
Estimating 
• We will assume that we have data describing T customer choice
decisions. Let St denote the set of alternatives available to
Customer t and define Customer t’s decision by the values
zjt =
1 if alternative j is chosen by Customer t
0 otherwise
• The data consists of the values of {zjt : t = 1,…,T, j = 1,…,n}
together with the choice sets {St : t = 1,…,T}.
• Maximize the log-likelihood function, L() over .
T
 T

yi 


L(  )    z jt   y j  ln  e

t 1 jSt
i

S


t
T
Conclusions
• Consumer choice does affect inventory management, and
choice behavior has a significant impact on both stocking
decisions and profits.
• In particular, under dynamic substitution one needs to stock
relatively more of popular variants and relatively less of
unpopular variants.
• Both static and dynamic substitution models suggest narrower
assortments are optimal if there is a higher level of substitution
among variants in a category.