Capturing the Risk-Pooling Effect Through
Demand Reshape
Amit Eynan • Thierry Fouque
John M. Olin School of Business, Washington University, St. Louis, Missouri 63130
Department of Economics, Management, Mathematics, and Computer Sciences, University of Paris X,
92000 Nanterre Cedex, France
[email protected] • [email protected]
T
he risk-pooling effect has been documented to benefit inventory systems by reducing the
need for safety stock and consequently lowering costs such as inventory holding and
shortage penalty. In this paper, we propose a new approach, called “demand reshape,” to
take advantage of the risk-pooling effect. It is demonstrated that a company can improve
its profit by encouraging some of its customers, who intended to purchase one product to
switch to another. The effectiveness of this approach is evaluated in various scenarios and
found to be very promising.
(Demand Reshape; Risk Pooling; Consolidation; Multiproduct; Substitution)
1.
Introduction
Variability of products’ demand frequently attracts
managers’ attention because of its costly implications.
By carrying high inventory levels, firms may satisfy
customers’ orders and materialize high revenue in
periods when demand is high, but often end up with
excess inventory in periods when demand is low. On
the other hand, carrying low inventory levels to avoid
the costs of excess inventory may result in a poor
service level and loss of potential profit. To facilitate
this burden associated with demand variability, firms
attempt to employ various approaches to take advantage of the risk-pooling effect.
A company that sells a single product through
several outlets to satisfy demand in several markets
(locations) may consider instead establishing one central facility to satisfy demand from the various markets. Eppen (1979) demonstrated the cost benefits
resulted from using a centralized inventory system
to satisfy normally distributed demands generated
in several locations. Chen and Lin (1989) extended
Eppen’s (1979) results by incorporating concave holding and penalty cost functions. Mehrez and Stulman
(1984) replaced the penalty cost for stockouts with
Management Science © 2003 INFORMS
Vol. 49, No. 6, June 2003, pp. 704–717
a binding constraint on the maximum probability of
stockouts at each location and showed the superiority of centralized systems over decentralized systems. Stulman (1987) used a first-come, first-served
inventory allocation to determine the minimal starting inventory level subject to service-level constraints,
and demonstrated the reduction in inventory as a
result of centralization. Chen and Lin (1990) provided
a counterexample in which a centralized configuration requires a higher inventory level. Eynan (1999)
considered a profit-maximizing firm whose markets
may be characterized with different selling prices as
well as different shortage penalty costs, and showed
that: (1) centralization is recommended, and (2) managers should not fear “supply cannibalization,” where
customers from low-paying markets arrive early causing the firm to be unable to satisfy later demand from
high-paying markets.
When a company offers multiple products it can
capture the advantages of the risk-pooling effect by
using common components for different products and
employ an assemble-to-order strategy. Baker et al.
(1986) analyzed inventory changes resulting from the
use of commonality. Gerchak et al. (1988) argued
that the objective function should be minimization
0025-1909/03/4906/0704$05.00
1526-5501 electronic ISSN
EYNAN AND FOUQUE
Capturing the Risk-Pooling Effect Through Demand Reshape
of inventory value rather than the number of units.
Eynan and Rosenblatt (1996) suggested that a common component may be more expensive than each of
the components it replaces because of its wider functionality. However, even a more expensive component may still result in a cost reduction. Eynan (1996)
studied the effect of correlation of demand on the
benefits of commonality and showed that smaller correlations correspond with larger savings, and consequently more expensive common components can be
afforded. Rutten and Bertrand (1993) and Eynan and
Rosenblatt (1997) explored the employment of variable recipes (flexible design) in which products can
be made following several bills of material (recipes)
by allowing concurrent usage of the original (cheaper)
and the common (expensive) components.
In this paper, we consider a new approach that may
be used by multiproduct companies to take advantage
of the risk-pooling effect. The suggested approach is
called “demand reshape” as it takes a given aggregate (total) demand and changes its distribution (allocation) among the various products. This reshaping
is obtained by making an effort to persuade some of
the customers to purchase another (usually a substitute) item instead of the original item they had in
mind. The cost of such effort can vary widely, and
can be as minimal as displaying posters in the store
to make customers more aware of the available substitute item. Some Taco Bell drive-through facilities
employ another example: When placing their order,
customers are asked: “Would you like to try our new
Gordita?” The question makes some customers switch
from another item and apparently changes demands
for the two items. This practice increases the demand
mean and variability of one product while reducing
them for the other product. This paper shows that
demand reshape reduces the sum of demand variabilities and consequently increases total profit.
The topic of item substitution, where customers
may switch to an available item upon stockout of
another, has been previously explored. We mention
only a few examples: McGillvary and Silver (1978)
used substitution to reduce the total cost of holding and shortage. Parlar and Goyal (1984) considered profit maximization and assumed shortage cost
and salvage value to be zero. Pasternack and Drezner
(1991) extended these works suggesting that substiManagement Science/Vol. 49, No. 6, June 2003
tution results in a reduced (per-unit) revenue. Parlar
(1985) studied a special case of perishable items where
old items can substitute for fresh items and vice versa.
It should be noted that the substitution (switch) in
these works and this branch of research takes place
only when inventory of one item is exhausted, or, in
other words, when customers are “forced” to switch.
In this work, however, switches take place due to customers’ change of preference when inventory of their
originally intended item may be available.
In the next section, assumptions and definitions
are presented. In §3, the demand reshape model is
established and analyzed for a fixed switching rate
followed by a numerical example. Sections 5 and 6
generalize the model to include probabilistic switching. Section 7 extends the model to the multipleperiod case. Conclusions and suggestions for future
research appear in the summary.
2.
Assumptions and Definitions
The following definitions will be used throughout the
paper.
pi = selling price of product i i = 1 2.
ci = purchasing cost of product i.
vi = salvage value of product i.
To make the environment meaningful, we assume
pi > c i > v i .
qi = penalty for each unit short of product i.
i i = mean, standard deviation of the original
periodic demand for product i.
= correlation coefficient between products’
demands.
f x0 y0 = joint density function of the original
demand for the two products.
= switching parameter 0 ≤ ≤ 1—proportion of
customers who originally intended to purchase Product 1, however, because of the company’s effort, have
switched to Product 2.
Hence, any realization of an original demand
x0 y0 for Products 1 and 2, respectively, will be
transformed into x = 1 − x0 y = x0 + y0 . It should
be noted that occasionally reshaping efforts will lead
to steering customers from an available product to
an unavailable one. We “let” such scenarios take
place assuming that reshape efforts may not be suspended instantaneously, or even if paused, their
705
EYNAN AND FOUQUE
Capturing the Risk-Pooling Effect Through Demand Reshape
impacts remain unaffected. (Clearly, if reshape can be
paused or discontinued, the benefits of the suggested
approach will further increase.)
The effort to reshape demand (switch customers)
results in a linear transformation of variables:
Consequently, the parameters should be modified as
follows (the “ ˆ ” sign designates the respective parameters after reshaping):
ˆ 1 = 1 − 1 (1)
ˆ 2 = 1 + 2 (2)
ˆ 1 = 1 − 1 ˆ 2 = 2 12 + 21 2 + 22 (3)
(4)
Because the reshaped demand of both products
include a portion of the original demand of Product 1,
the new demands are correlated with the following
coefficient:
ˆ = 1 + 2
2 12 + 21 2 + 22
Furthermore, we define
fˆx y = joint density function of the reshaped
demand.
Si = initial inventory of product i.
For exposition purposes, the presentation of the
problem is in a single-period setting. However, it can
be converted to the multiperiod setting using a simple
transformation as described in §7.
3.
Demand Reshape with Fixed
Switching Rate
In this section, we express the firm’s profit function and analyze the effect of demand reshape on
its profit as well as the service level it provides to
customers. Given initial inventory levels S1 S2 , the
demand spectrum can be divided into four categories: X ≤ S1 Y ≤ S2 ; X ≤ S1 Y > S2 X > S1 Y ≤ S2 ;
X > S1 Y > S2 . Following these categories, the profit
function can be expressed as
=
S1
S2
x=0 y=0
p1 x + v1 S1 − x + p2 y + v2 S2 − y
× fˆx y dy dx
706
+
+
+
S1
x=0 y=S2
x=S1
y=0
S2
x=S1
p1 x + v1 S1 − x + p2 S2 − q2 y − S2 × fˆx y dy dx
p1 S1 − q1 x − S1 + p2 y + v2 S2 − y
× fˆx y dy dx
y=S2
p1 S1 − q1 x − S1 + p2 S2 − q2 y − S2 × fˆx y dy dx − c1 S1 − c2 S2 Property 1. The profit function, , can be expressed as the sum of two independent newsvendor
problems:
S1
p1 x + v1 S1 − xfˆ1 x dx
1 =
x=0
p1 S1 − q1 x − S1 fˆ1 x dx − c1 S1
+
x=S1
and
2 =
S2
y=0
+
where
fˆ1 x =
y=0
p2 y + v2 S2 − yfˆ2 y dy
y=S2
p2 S2 − q2 y − S2 fˆ2 y dy − c2 S2 fˆx y dy
and
fˆ2 y =
x=0
fˆx y dx
The proof appears in the appendix.
Consequently, the optimal solution can be obtained
by independently solving the two problems. These
solutions are, therefore, characterized by
p − c1 + q 1
p − c2 + q2
and F2 S2 = 2
F1 S1 = 1
p1 − v1 + q1
p 2 − v 2 + q2
Si
fˆi x dx. For the purpose of
(where Fi Si = x=−
analytical tractability (and without significant limitations), we will continue the analysis assuming a normally distributed demand. Hence, each one of these
c.d.f. levels corresponds to a z value which can be
used to express the optimal beginning inventory levels as S1 = ˆ 1 + z1 ˆ 1 and S2 = ˆ 2 + z2 ˆ 2 .
Property 2. The optimal z values are independent
of the reshape factor, , and the correlation coefficient, .
Proof. Fi Si is a function of only the price/cost
structure and not the demand distribution. Management Science/Vol. 49, No. 6, June 2003
EYNAN AND FOUQUE
Capturing the Risk-Pooling Effect Through Demand Reshape
Given the assumption of normally distributed
demand, each of the two independent problems can
be expressed as
Proof.
Lz =
=
i = pi − vi ˆ i − ci − vi Si − pi − vi + qi ˆ i Lzi See Silver et al. (1998, p. 388), where
x − zi fs x dx
Lzi =
x=zi
=
where F z =
is the loss function, and
1
2
fs x = √ e−x /2
2
is the standard normal distribution. Using Si = ˆ i +
zi ˆ i , the profit can be rewritten as
i = pi − ci ˆ i − ci − vi zi + pi − vi + qi Lzi ˆ i (5)
Using the expressions for ˆ 1 ˆ 2 ˆ 1 , and ˆ 2 , (1)–(4),
the profit function can be expressed in terms of the
original parameters as
= p1 −c1 1−
1 −c1 −v1 z1 +p1 −v1 +q1 Lz1 ×1−1 +p2 −c2 2 +
1 −c2 −v2 z2 +p2 −v2 +q2 Lz2 × 2 12 +21 2 +22 The practice of demand reshape affects the profit
in two ways: (1) Reduction in the sum of demand
variabilities, which reduces the need for safety stock
inventory, and (2) by choosing one product versus
another, there may be a change (increase or decrease)
in the revenue (due to different selling prices, i.e., p1
versus p2 or the cost (due to different unit cost, i.e.,
c1 versus c2 . To concentrate on the effect of demand
variability we have neutralized the effect of price
and cost by using a uniform price/cost structure (i.e.,
p1 = p2 = p c1 = c2 = c v1 = v2 = v q1 = q2 = q. Then,
the profit function becomes
= p − c
1 + 2 − c − vz + p − v + qLz (6)
where = 1 − 1 + 2 12 + 21 2 + 22 . This
expression can be used to demonstrate the shape and
direction of the function with respect to the switching
parameter, , as suggested in the following property.
Property 3. is increasing in , and concave.
Management Science/Vol. 49, No. 6, June 2003
z
x=z
x=z
x=z
x − zfs x dx
xfs x dx − z
x=z
fs x dx
xfs x dx − z1 − F z
f x dx.
x=− s
Because at optimality
p−c+q
Fi Si =
=F z
p−v+q
it follows that at optimality
c − vz + p − v + qLz = p − v + q
x=z
xfs x dx > 0
Therefore,
d
= −c − vz + p − v + qLz
d
12 + 1 2
× −1 + 2 12 + 21 2 + 22
= c − vz + p − v + qLz1
1 + 2
> 0
× 1− 2 12 + 21 2 + 22
d2 = −c − vz + p − v + qLz1
d2
× 1 2 12 + 21 2 + 22 − 1 + 2 12 + 1 2 ·
2 12 + 21 2 + 22
· 2 12 + 21 2 + 22 −1
= −c − vz + p − v − qLz1
× 1 2 12 + 21 2 + 22 − 2 13 − 212 2
− 2 1 22 · 2 12 + 21 2 + 22 −3/2
= −c − vz + p − v + qLz
2 2 1 − 2 × 2 2 1 2
< 0 1 + 21 2 + 22 3/2
This property implies that d/d decreases in ;
hence, most of the potential improvement resulting
from demand reshape can be obtained with a small
value of , as will be demonstrated later in a numerical example. This pattern is especially appealing
because the optimal value of is determined by the
707
EYNAN AND FOUQUE
Capturing the Risk-Pooling Effect Through Demand Reshape
optimal balance between the profit increase resulting
from demand reshape and the cost to convince customers to switch, where it may be argued that the
latter is convex increasing in .
The analysis so far assumed that customers switch
from Product 1 to Product 2. However, because no
specifics have been made regarding the products, it is
also the case that demand reshape is beneficial if customers switch from Product 2 to Product 1. The reason that profit increases even if customers switch from
a low variable-demand product to a high variabledemand product is due to the changes in demand
variability. The reduction in demand variability of
the product that customers switch from is linear in (see (3)), while the increase in demand variability of
the product customers switch to is smaller (because of
the square root in (4)). Even though applying demand
reshape is beneficial in both directions, one may wonder which is the preferred strategy. Should a company
make efforts to convince some Product-1 customers to
switch to Product 2, or should it attempt to convince
some Product-2 customers to switch to Product 1. As
claimed in the following property, it is more beneficial
to switch customers from the product whose demand
has a larger standard deviation to the product with
the smaller standard deviation of demand.
According to the definition of , a proportion of
Product-1 customers will switch to Product 2. Similarly, let us define to represent the symmetric case
where a company’s effort results in the same proportion of Product-2 customers to switch to Product 1.
Property 4. If 1 > 2 , then > .
Proof. Following a similar procedure to the one
resulted in (6), it can be determined that
= p − c
1 + 2 − c − vz + p − v + qLz where
=
12 + 21 2 + 2 22 + 1 − 2 In the following, it will be shown that ≤ :
t ≡ − = 12 + 21 2 + 2 22
+ 1 − 2 − 1 − 1 − 2 12 + 21 2 + 22
= 12 + 21 2 + 2 22 − 21 2 1 − 708
− 2 12 + 21 2 + 22 − 21 2 1 − + 1 − 2 − 1 =
1 + 2 2 − 21 2 1 − − 1 + 2 2 − 21 2 1 − − 1 + 2 − 1 + 2 Defining k = 1 + 2 2 , l = 1 + 2 2 , and m =
21 2 1 − , we get
√
√
√
√
t = k − m − l − m − k − l
Given that 1 ≥ 2 1 + 2 ≥ 1 + 2 or k ≥ l, thus,
dt
1
1
1
=
−√
≥0
√
dm 2
l−m
k−m
Because for m = 0, t = 0 it follows that for m > 0, and
especially m = 21 2 1 − , t > 0. As was demonstrated the practice of demand
reshape results in profit improvement. However,
firms may inquire how this practice would change the
service level they provide to customers. Using SL to
denote the service level, which is defined as the probability to satisfy all demands, it is suggested in the
next property that demand reshape not only improves
profit but service level as well.
Property 5. dSL/d > 0.
Proof. Given the assumption of normally distributed demand, the joint density function of the
original and reshaped demands are
x0 −
1 2
y0 −
2
x0 −
1
−2
f x0 y0 = exp −
1
1
2
y0 −
2 2 2 −1
· 21− +
2
·21 2 1−2 −1 x − ˆ 1 2
y − ˆ 2
x − ˆ 1
ˆ
f xy = exp −
−2ˆ
ˆ 1
ˆ 1
ˆ 2
2 y − ˆ 2
−1
· 21− ˆ 2 +
ˆ 2
·2 ˆ 1 ˆ 2 1− ˆ 2 −1 Management Science/Vol. 49, No. 6, June 2003
EYNAN AND FOUQUE
Capturing the Risk-Pooling Effect Through Demand Reshape
SL =
S1
S2
x=− y=−
z z
=
2
ˆ
× ex −2xy+y
z z
=
z
2 /21−ˆ 2 it implies that
dydx
1
2
2
2
e−x 1−ˆ /21−ˆ 2
y=− 2 1− ˆ
x=−
1 − 2 d ˆ
= 2 2 1 2
> 0
d 1 + 21 2 + 22 3/2
1
y=− 2 1− p̂ 2
x=−
=
Finally, as
fˆxydydx
x=−
×e
−y 2 −2xy+
ˆ
ˆ 2 x2 /21−ˆ 2 1
y=− 2 1− ˆ2
dSL dSL d ˆ
=
·
> 0
d
d ˆ d
dydx
It should be noted that products’ individual service levels (i.e., the probability to satisfy all demand
for a certain product) do not change with reshaping of demand because the z values (and the corresponding F values) do not change as was suggested
in Property 2.
z
2
2
2
ˆ /21−ˆ ×e−x /2 e−y−x
dydx
Define
y − x
ˆ
t=
1 − ˆ 2
We get
ˆ
y = t 1 − ˆ 2 + x
and dy =
1 − ˆ 2 dt
4.
Numerical Example
z2 /1+ˆ
1
dSL
e−u du
=− 2
d ˆ
u=
2 1 − ˆ
z2 /1+ˆ
1
−u =− −e 2 1 − ˆ 2
u=
1
2
=
e−z /1+ˆ > 0
2 1 − ˆ 2
A numerical example is provided to illustrate the
impact of demand reshape and the properties suggested earlier in this paper. Demand is normally
distributed with the following parameters: 1 =
1300 2 = 1000 1 = 400 2 = 300 = 0. Price and
costs are p = 12 c = 10 v = 6, and q = 4. As 1 > 2 ,
effort will be made to convince customers to switch
from Product 1 to Product 2.
Table 1 demonstrates the relationship among the
switching parameter, , the optimal inventory levels,
and the resulting profit and service level. As expected
from (1) and (2), the change in ˆ 1 and ˆ 2 is linear
with respect to ; ˆ 1 is linear as well, however ˆ 2 is
convex (it can be verified that d 2 ˆ 2 /d2 > 0. Consequently, S1 = ˆ 1 + z1 ˆ 1 decreases linearly (recall that
z1 is independent of , Property 2), and S2 = ˆ 2 + z2 ˆ 2
increases and is convex in . 1 decreases linearly,
and by reviewing (5) it can be seen that this is due
to a linear reduction of ˆ 1 and ˆ 1 at the same rate.
As ˆ 1 = 1 − 1 and ˆ 1 = 1 − 1 , it follows that
1 = 1−1=0 . 2 , on the other hand, increases and
is concave in because ˆ 2 is linear but ˆ 2 is convex. Finally, as = 1 + 2 it follows that is convex
in . The changes in these three profit functions with
respect to are illustrated in Figure 1.
The resulting profit improvement using demand
reshape is obtained by taking advantage of the riskpooling effect. The extent of the potential to which
Management Science/Vol. 49, No. 6, June 2003
709
Consequently,
SL =
dSL
=
d ˆ
z
√
z−x/
ˆ
x=− t=−
z
x=−
1 −x2 /2
e
2
1 −x2 /2 −t2 /2
e
e
dt dx
2
−x 1 − ˆ 2 + z − x
ˆ √ ˆ
2
1−ˆ 2
1−ˆ 2
1 − ˆ 2
2
ˆ /21−ˆ × e−z−x
dx
1 −x2 −2xz+z
x − z
ˆ
2 /21−ˆ 2 ˆ
=−
dx
e
2
1 − ˆ 3/2
x=− 2
z
Define
u=
ˆ + z2
x2 − 2xz
21 − ˆ 2 We get
du =
x − z
ˆ
dx
1 − ˆ 2
when x = z,
u=
and
ˆ 2 + z2
z2
z2 − 2z
=
21 − ˆ 2 1 + ˆ
EYNAN AND FOUQUE
Capturing the Risk-Pooling Effect Through Demand Reshape
Table 1
The Impact of Demand Reshape
ˆ 1
ˆ 2
ˆ1
ˆ2
1300
1170
1040
910
780
650
520
390
260
130
0
1000
1130
1260
1390
1520
1650
1780
1910
2040
2170
2300
40000
36000
32000
28000
24000
20000
16000
12000
8000
4000
0
30000
30265
31048
32311
34000
36056
38419
41037
43863
46861
50000
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0∗
ˆ
S1
000 140134
013 126120
026 112107
037 98094
047 84080
055 70067
062 56054
068 42040
073 28027
077 14013
NA
0
S2
1
2
SL %
107600
120668
133866
147186
160614
174135
187733
201396
215113
228872
242667
105463
94917
84370
73824
63278
52731
42185
31639
21093
10546
0
84097
109072
132047
153169
172644
190702
207572
223458
238537
252954
266832
189560
203988
216417
226993
235921
243434
249757
255097
256930
263500
266832
3600
3799
3992
4173
4340
4490
4622
4739
4841
4931
6000
Note. At = 1, the problem reduces to a single-product problem. Consequently, there is no ˆ value, and SL is calculated as in
a simple newsvendor problem.
this effect can improve profit depends on the correlation coefficient, . Figure 2 describes the relative profit
increase with respect to for three original levels of
correlation = −05 0 05. It can be seen that for
= −05, the improvement is very impressive even
for small values of : 11.8% for = 01 and 22.6%
for = 02. But even for a positive correlation with
coefficient = 05 the resulting profit increase is still
substantial: 3.7% for = 01 and 6.7% for = 02.
Finally, in the last column of Table 1, it can be
observed that the service level is also improving by
demand reshape.
This numerical example uses a coefficient of variation, i /
i , of 0.308 and 0.3 for Items 1 and 2, respectively. As demand reshape benefits from variability
reduction, it would be expected that larger demand
variation will imply more room for improvement
Figure 1
The Effect of on s
Figure 2
3,000
Relative Profit Improvement for Various Initial Correlation
Levels
I
2,500
80
70
60
50
40
30
20
10
0
I
2,000
1,500
1,000
I
500
0
0
0.2
0.4
0.6
D
710
through demand reshape. Maintaining the level of the
mean demands at 1 = 1300 and 2 = 1000, we have
defined ' = i /
i and evaluated the profit and profit
improvement for various coefficient of variation levels, ', and values. As can be observed from Table 2
the impact of the coefficient of variation is substantial; for any value, the larger the ', the smaller the
profit. This result is expected as more variability in the
system is costly. Moreover, the absolute improvement
because of demand reshape (i.e., compared with profit
levels obtained at = 0) is increasing in '. Consequently, the relative improvement because of demand
reshape is increasing in ' as well (see Figure 3). For
example, when = 025 the profit improvement is 3%
for ' = 01, 7% for ' = 02, 16% for ' = 03, 40% for
' = 04, and 336% for ' = 05.
To this point, a uniform price/cost structure has
been employed in an attempt to isolate all factors and
0.8
1
U = -0.5
U =0
U = 0.5
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
D
Management Science/Vol. 49, No. 6, June 2003
EYNAN AND FOUQUE
Capturing the Risk-Pooling Effect Through Demand Reshape
Table 2
The Impact of on Profit
0
0.25
0.5
0.75
1
0.1
0.2
0.3
0.4
0.5
3,711
3,817
3,888
3,935
3,966
2,823
3,034
3,176
3,270
3,333
1,934
2,251
2,464
2,605
2,699
1,046
1,468
1,752
1,939
2,065
157
685
1,040
1,274
1,432
focus only on the effect of demand reshape on the
variability of demand and its resulting improvement
with respect to profit and service level. Needless to
say, a firm may have products with different prices
and costs. To cover many possible price/cost scenarios while maintaining data manageable, we introduce
a ratio
p
c
v
q
(= 1 = 1 = 1 = 1
p2
c2
v2
q2
to indicate the price/cost difference between the
products (where ( = 1 represents the uniform price/
cost structure).
( < 1 represents cases where customers switch from
a low-price item to a high-price item (i.e., p1 < p2 .
In such cases, a profit improvement is expected due
to reduction in the sum of demand variabilities as
well as a higher revenue per unit. Of more interest are cases where ( > 1 where a trade-off exists.
On the one hand, the sum of demand variabilities
will dampen and increase profit as was suggested
earlier. On the other hand, customers switch to a
Figure 3
The Effect of Demand Variability on Profit Increase
lower-revenue item which implies a profit reduction.
Thus, it would be worthwhile to explore how much
cheaper a substitution may be before all the benefits
associated with demand variability become inferior to
the disadvantage of a lower per-unit revenue. To illustrate this trade-off, we used data for the demand of
both items, and price/cost parameters of Item 2 from
the last numerical example. Moreover, we have varied
the value of ( to obtain a wide range of p1 c1 v1 , and
q1 . The resulting profit with respect to ( is depicted in
Figure 4 (for three values of : 0, 0.5, and 1). The benefit of demand reshape can be observed by comparing
the profit level (for = 05 or = 1 with the reference profit level of = 0 (when reshape does not take
place). As one may expect, this benefit decreases with
( and is even negative for large values of ( implying that reshape should be avoided. The break-even
values are ( = 173 for = 1 and ( = 202 for = 05.
These values surprisingly suggest that customers may
switch to products that are even 42.2% and 50.5%
cheaper before the impact of lower revenue outgrows
the impact of lower demand variability for value of
1 and 0.5, respectively. This magnitude of low-price
tolerability is another indication for the potential benefits of demand reshape.
A more conservative (restrictive) setting to illustrate
price tolerability is to keep all cost parameters the
same for both products (i.e., c1 = c2 v1 = v2 q1 = q2 and differ only on the prices p1 = p2 . This ensures
that the two products have different absolute, as well
as relative, margins. Using the previous data (except
for p1 ), we determined for different values, the
Figure 4
900
D=1
800
The Effect of Uneven Price/Cost on the Benefit of Demand
Reshape
3,500
700
600
3,000
D =1
D=0.5
Profit
500
400
D=0.25
300
2,500
D =0.5
2,000
D =0
200
1,500
100
0
0.10
1,000
0.50
0.20
0.30
J
0.40
Management Science/Vol. 49, No. 6, June 2003
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
E
711
EYNAN AND FOUQUE
Capturing the Risk-Pooling Effect Through Demand Reshape
Figure 5
profit for two to five products. (To make the comparison simple, all products in this example correspond to the same data: i = 1300 i = 400 pi = 12
ci = 10 vi = 6, and qi = 4. Furthermore, initial correlation is zero.) It can be observed that the improvement
increases as more products are involved, making
demand reshape even more attractive.
Maximal Tolerable Price Difference
12%
(p1 – p2) / p1
10%
8%
6%
4%
5.
2%
0%
0
0.1
0.2
0.3
0.4
0.5
D
0.6
0.7
0.8
0.9
1
value of p1 that results in the same profit as the
case without reshape. Then, considering the difference
between p1 and p2 , we evaluated the extent to which
p2 can be cheaper than p1 while reshape still increasing profit, as described in Figure 5. For example, when
= 02, reshape is recommended as long as p2 is not
cheaper than p1 by more than 8.59%.
The benefits of demand reshape can also be
explored when more products are involved. For
example, with three products (1, 2, and 3), a company
may consider making an effort to shift a proportion of customers of Products 1 and 2 towards Product 3.
As expected, the profit associated with Products 1 and
2 will drop in , whereas the profit associated with
Product 3 will increase; the total profit however, will
increase. Similarly, in a four-product case, a shift of
demand can be made from three products toward a
fourth one. Figure 6 illustrates the improvement in
Figure 6
Relative Profit Improvement for Numerous Products
90
80
5 products
Demand Reshape with
Probabilistic Switching
So far, it was assumed that the switching rate is constant; for any realization of demand x0 exactly x0
customers switch. One may argue that a stochastic switching process where each Product-1 customer
switches to Product 2 with probability is more realistic. Such a model captures the individual’s decisionmaking pattern rather than aggregating all customers
together and applying a given ratio of switching.
Moreover, this probabilistic switching adds another
source of variability to the system. The purpose of this
section is to explore the benefits of demand reshape
in probabilistic switching cases and to assess whether
the promise of demand reshape as was demonstrated
earlier is still viable.
First, it can be noted that Properties 1 and 2 hold
for any reshaped joint probability function, fˆx y.
Consequently, these properties are also valid for the
probabilistic switching case.
To initiate this analysis, the new reshaped demand
parameters should be determined as is suggested in
the following property.
Property 6. If is the probability that a Product-1
customer switches to Product 2 (1 − is the probability to maintain choice), then the demand parameters
after reshaping are
70
% Profit Increase
ˆ 1 = 1 − 1 4 products
60
3 products
50
40
2 products
30
20
10
0
0
0.1
0.2
0.3
0.4
0.5
D
712
0.6
0.7
0.8
0.9
1
ˆ 2 = 1 + 2 ˆ 1 = 1 − 1 + 1 − 2 12 ˆ 2 = 1 − 1 + 2 12 + 21 2 + 22 The proof is provided in the appendix.
It should be mentioned that the new demands (after
reshaping) do not necessarily follow a normal distribution. However, given that a Poisson distribution
can be approximated using a normal distribution (see,
Management Science/Vol. 49, No. 6, June 2003
EYNAN AND FOUQUE
Capturing the Risk-Pooling Effect Through Demand Reshape
for example, Parzen 1960, p. 248) and that a random variable resulting from a probabilistic switching
(random selection) of a Poisson distribution follows
a Poisson distribution (Parzen 1962, p. 47), it implies
that the new demands can be approximated using
normal distribution, and that (5) and (6) are still
appropriate.
For the fixed switching model, it was shown that
demand reshape is beneficial in both directions (i.e.,
switching from Product 1 to Product 2 or vice versa).
Furthermore, Property 4 suggested that a firm will
be better off encouraging customers to switch from
the product with the larger standard deviation to the
one with the smaller standard deviation. In probabilistic switching, the additional source of variability
increases with the realized value of X0 . The mean
plays a role in the direction that demand reshape
should be practiced because this value relates not only
to 1 but also to 1 . Consequently, to make a general recommendation for the direction of reshape (i.e.,
without the need to solve for each direction and only
then to make a decision), the demand with smaller
standard deviation should have a larger mean as proposed in the following property.
Property 7. If 2 > 1 and 1 > 2 , then > .
Proof. As
= p − c
1 + 2 − c − vz + p − v + qLz
and
= p − c
1 + 2 − c − vz + p − v + qLz it is sufficient to show that > .
Defining A = 1 − 2 22 B = 2 22 + 21 2 + 12 ,
C = 1 − 2 12 , and D = 2 12 + 21 2 + 22 , we can
express
= A + 1 − 2 + B + 1 − 2
and
= C + 1 − 1 + D + 1 − 1 √
In
the
proof
of
Property
4,
it
was
shown
that
√
√
√
√A +
B
≥
C
+
D,
implying
that
D
is
bounded
by
D≤
√
√
√
A + B − C. Hence,
≤ -C ≡ C + 1 − 1
√
√
√ 2
+
A + B − C + 1 − 1 Management Science/Vol. 49, No. 6, June 2003
As
d-C
=0
dC
while
only at
d-C <0
dC C=0
and
√
√
C=
√
A+ B
2
d-C > 0
dC √C=√A+√B
it follows that -C is maximized at the extreme
points. Moreover, it can be verified
that A ≤ C ≤ B
A + 1 − 1 +
implying
that
-C
≤
-A
=
-B
=
B + 1 − 1 , and consequently, ≤ -C ≤
. 6.
Numerical Example
Focusing on the profit function for the fixed versus
probabilistic switching, one can observe that the only
difference is in the sum of the reshaped standard deviation as denoted by . As was mentioned earlier, the fixed-switching model is subject to only one
source of variability, variation of demand, whereas the
stochastic switching model is also subject to variability
because of switching uncertainty. To compare between
the two models, we have evaluated the values of of each model with respect to the standard deviation
of the original demand. We have used the same data
as in the previous numerical example with the exception of using 1 = 2 = . The values of with
respect to (for = 025 are depicted in Figure 7.
As can be observed, once is beyond a minimal level
(about 100; recall that 1 = 1300 2 = 1000; hence,
< 100 implies an extremely small coefficient of variation) there is a negligible difference between the two
Figure 7
Probabilistic vs. Fixed Switching with Respect to Variation
of Demand
250
200
150
100
Probabilistic
50
Fixed
0
0
20
40
60
V
80
100
120
713
EYNAN AND FOUQUE
Capturing the Risk-Pooling Effect Through Demand Reshape
curves. (Similar curves were obtained for other values. Furthermore, for = 0 and = 1 the two curves
are identical.) Hence, for any practical purpose, just a
small initial variation in demand is sufficient to override the variation due to switching. Consequently, the
analysis of the fixed-switching case is relevant to the
stochastic switching case as well.
7.
The Multiperiod Case
For ease of exposition, the presentation and analysis so far pertained to the single-period case. Clearly,
the benefits of demand reshape are valuable in multiperiod settings as well. By replacing the salvage cost,
vi , with holding cost, hi , to be charged for inventory
of product i left by the end of the period, using /
as the firm’s time preference for money, and adding
an index t to reflect period t, the profit function can
be established for the multiperiod case. Assuming no
backordering, 1 , for example, can be expressed as
t−1 t
1 = E
/ 1
t=1
=
/t−1
S1t
x=−
t=1
+
p1 x − h1 S1t − xfˆ1 x dx
x=S1t
p1 S1t − q1 x − S1t fˆ1 x dx
−c1 S1t − S1t−1 − Ezt−1
1 where zt1 is the number of units of item 1 which were
sold in period t.
Thus,
S1t
t
Ez1 =
fˆ1 x dx
xfˆ1 x dx + S1t
x=S1t
x=−
If demands are independent and identically distributed (throughout time), by rearranging, we get
S t
1
t−1
1 =
/
p1 − /c1 x − h1 S1t − x fˆ1 x dx
t=1
x=−
+
x=S1t
p1 − /c1 S1t − q1 x − S1t fˆ1 x dx
− c1 1 − /S1t
Because each element (period) of this summation is
independent (of other periods), it follows that it can
714
be solved separately. In other words, the myopic solution is optimal and all periods should start with the
same inventory level S1t = S1 ∀ t. Furthermore, comparing i of the multiperiod and the single-period
models it can be observed that the single-period
model can describe the multiperiod problem by substituting ci and pi with ci 1 − / and pi − /ci , respectively. Thus, the analysis and conclusions that were
made earlier for the single-period case are also valid
for the multiperiod case.
8.
Summary
In this paper, we proposed a new approach to reduce
the costly consequences associated with uncertainty
of demand. By making an effort that encourages some
customers of one product to switch to another, the
total uncertainty of demand (measured as the sum
of modified standard deviations) is reduced. Consequently, the company is able to improve its profit and,
at the same time, raise the service level it provides to
customers. It was (pleasantly) surprising to find out
that even a small value of (proportion of switching
customers) results in an impressive profit increase.
This result may also shed new light on the centralization problem. It was demonstrated in the literature
that using a centralized configuration by consolidating several decentralized points of sale into one is
beneficial (see, for example, Eppen 1979). However,
managers often are reluctant to adopt such a solution,
fearing that they will lose some customers who will
find the new location not as accessible as the location they used before it was consolidated. The results
of this work suggest that companies can enjoy most
of the benefits of centralization and still maintain a
decentralized configuration by attempting to encourage some customers who used to buy in one location
to approach another location. This can be done by distributing coupons or small gifts that will make one
location more favorable to some clients that originally
visited another location. At the same time, no sales
will be lost due to location inaccessibility.
Acknowledgments
The authors thank the associate editor and three anonymous
referees for their helpful comments. This work was partially supported by the French National Foundation for Corporate Management Studies (FNEGE).
Management Science/Vol. 49, No. 6, June 2003
EYNAN AND FOUQUE
Capturing the Risk-Pooling Effect Through Demand Reshape
Appendix
+
Proof of Property 1. As the original demand varies over the
domain X0 ≥ 0 Y0 ≥ 0, given a switching parameter, , the domain
of the reshaped demand is X ≥ 0 Y ≥ bX (where b = /1 − .
Consequently, the calculations of depend on the relationship
between b and S2 /S1 .
Case I. b < S2 /S1 .
=
S1
x=0
+
+
+
+
+
bS1
y=bx
S1
x=0
S1 x=0
x=S1
+
S2
y=bS1
p1 x + v1 S1 − x + p2 y + v2 S2 − yfˆx y dy dx
+
p1 x + v1 S1 − x + p2 S2 − q2 y − S2 fˆx y dy dx
=
S2
x=S1
p1 x + v1 S1 − x + p2 y + v2 S2 − yfˆx y dy dx
y=bx
S2 /b
y=S2
+
+
y=S2
S2 /b
+
x=S2 /b
p1 S1 − q1 x − S1 + p2 y + v2 S2 − yfˆx y dy dx
y=bx
+
p1 S1 − q1 x − S1 + p2 S2 − q2 y − S2 fˆx y dy dx
+
− c1 S1 − c2 S2
S1 bS1
=
p1 x + v1 S1 − xfˆx y dy dx
x=0
+
+
+
+
+
x=0
S1
x=0
S2 /b S2 /b
y=S2
x=S2 /b
=
p1 x + v1 S1 − xfˆx ydy dx
S2
p1 S1 − q1 x − S1 fˆx ydy dx
+
p1 S1 − q1 x − S1 fˆx y dy dx
y=bx
+
p1 S1 − q1 x − S1 fˆx y dy dx − c1 S1
+
The Regions of Integration (Case I)
S1
x=0
S1
y=S2
x=0
S2
y=bS1
y=S2
S1
p2 S2 − q2 y − S2 fˆx y dx dy
y/b
S2 /b
x=S1
p2 S2 − q2 y − S2 fˆx ydx dy
x=S2 /b
S2
y/b
y=0 x=0
y=S2
p2 S2 − q2 y − S2 fˆx y dx dy − c2 S2
p1 x + v1 S1 − xfˆx y dy dx
y=bx
y=bx
p2 y + v2 S2 − yfˆx y dx dy
y/b
x=S1
p2 y + v2 S2 − yfˆx y dx dy
x=S1
y=S2
S2 /b
p1 S1 − q1 x − S1 fˆx y dy dx − c1 S1
p2 y + v2 S2 − yfˆx y dx dy
y/b
x=0
S2
y=bx
S2 /b
x=0
S2 /b
x=0
p2 S2 − q2 y − S2 fˆx y dx dy − c2 S2
p1 x + v1 S1 − x + p2 y + v2 S2 − yfˆx y dy dx
bS1
y=S2
p1 x + v1 S1 − x + p2 S2 − q2 y − S2 fˆx y dy dx
y=bS1
S1
x=S2 /b
Figure A2
y = bx
Y
S2
p2 y + v2 S2 − yfˆx y dx dy
y=bS1
x=0
+
y/b
x=0
Case II. b > S2 /S1 .
p1 x + v1 S1 − xfˆx y dy dx
y=bx
x=S1
Figure A1
y=bS1
y=S2
x=S1
S2
y=0
= 1 + 2 y=bx
S1
bS1
x=0
+
p1 S1 − q1 x − S1 + p2 S2 − q2 y − S2 fˆx y dy dx
S1
x=S2 /b
bS1
y=bx
p1 x + v1 S1 − x + p2 S2 − q2 y − S2 fˆx y dy dx
p1 x + v1 S1 − x + p2 S2 − q2 y − S2 fˆx y dy dx
y=bS1
p1 x + v1 S1 − x + p2 S2 − q2 y − S2 fˆx y dy dx
The Regions of Integration (Case II)
Y
y = bx
bS 1
S2
S2
bS 1
S1
Management Science/Vol. 49, No. 6, June 2003
S2 /b
X
S2 /b
S1
X
715
EYNAN AND FOUQUE
Capturing the Risk-Pooling Effect Through Demand Reshape
+
x=S1
y=bx
p1 S1 − q1 x − S1 + p2 S2 − q2 y − S2 fˆx y dy dx
− c1 S1 − c2 S2
S2 /b S2
=
p1 x + v1 S1 − xfˆx y dy dx
x=0
+
+
+
+
+
+
+
+
+
+
+
=
+
+
S2 /b
x=0
S2 /b
x=0
bS1
y=S2
S1
x=S2 /b
y=bx
S2
y/b
y=0 x=0
bS1
y=S2
x=0
bS1
y/b
y=S2
x=S2 /b
y=bS1
y=bS1
S1
y/b
y=bx
S2
y/b
y=0 x=0
y=S2
p2 S2 − q2 y − S2 fˆx y dx dy − c2 S2
p1 S1 − q1 x − S1 fˆx y dy dx − c1 S1
p2 y + v2 S2 − yfˆx y dx dy
y/b
x=0
p2 S2 − q2 y − S2 fˆx y dx dy − c2 S2
= 1 + 2 Lemma. Let X be a random variable with mean and variance 2 ,
reflecting the number of customer arrivals. Each customer may choose
with probability to switch to another product. Let U be a random variable denoting the number of customers who switch. Then, U has mean
and variance 1 − + 2 2 .
Proof. Let gu x denote the probability density function of
the number of customers who switch u given the number of customers who originally arrive to buy Product 1 x. We can express
the cumulative and probability distribution functions of U as
FU u =
716
u=0
ufU u du =
x=0 u=0
u=0
u
x=0
gu xhx dx du
ugu x du hx dx =
x=0
EU xhx dx
Futhermore,
EU 2 =
=
u
x=0 t=0
gt x dt f x dx
u=0
u2 fU u du =
x=0 u=0
u=0
u2
x=0
gu xhx dx du
u2 gu x du hx dx =
x=0
EU 2 xhx dx
The second moment of any random variable with mean and standard deviation of and , respectively, can be expressed as 2 +
2 . Furthermore, given x the variance of a binomial process is
x1 − . Therefore,
EU 2 =
x=0
x1 − + x2 hx dx
= 1 − p2 S2 − q2 y − S2 fˆx y dx dy
p1 x + v1 S1 − xfˆx y dy dx
y=bx
where hx is the p.d.f. of x.
For a given value of x the mean of switching customer is x.
Hence,
EU =
xhx dx = EX = p2 S2 − q2 y − S2 fˆx y dx dy
x=S2 /b
x=S1
p2 S2 − q2 y − S2 fˆx y dx dy
S1
x=S1
p2 S2 − q2 y − S2 fˆx y dx dy
S2 /b
x=0
=
p2 y + v2 S2 − yfˆx y dx dy
y=bS1
EU =
dFU u =
gu xf x dx
du
x=0
x=0
p1 x + v1 S1 − xfˆx y dy dx
p1 S1 − q1 x − S1 fˆx y dy dx − c1 S1
S2 /b
p1 x + v1 S1 − xfˆx y dy dx
x=S1
p1 x + v1 S1 − xfˆx y dy dx
y=bS1
bS1
y=bx
S1
fU u =
p1 x + v1 S1 − xfˆx y dy dx
y=bS1
x=S2 /b
x=0
+
y=bx
and
x=0
xhx dx + 2
x=0
x2 hx dx
= 1 − + 2 2 + 2 Finally, VarU = EU 2 − E 2 U = 1 − + 2 2 + 2 − 2 =
1 − + 2 2 . Proof of Property 6. In our process, we have two original random variables X and Y , which are the demands for Products 1 and
2, respectively. Clearly, X and Y may be correlated. Furthermore, X
and Y may be defined as functions of three independent random
variables A, B, and C, such that X = A + B and Y = C + B (B is
included in both X and Y to introduce correlation).
To maintain the parameters of X and Y (i.e., 1 1 2 2 , and
), it can easily be verified that 1 = A + B 2 = B + C 12 =
A2 + B2 , and 22 = B2 + C2 . Furthermore,
CovX Y = CovA + B B + C
= CovA B + CovA C + CovB C + V B
= 0 + 0 + 0 + V B = V B
Hence,
=
V B
CovX Y =
1 2
1 2
and B2 = V B = 1 2 .
The decision not to switch to another product may be viewed as
is the sum of
a random selection with probability 1 − . Hence, X
two such random selections over A and B. Let A
and B denote
these random variables. Then, based on the lemma, it follows that
Management Science/Vol. 49, No. 6, June 2003
EYNAN AND FOUQUE
Capturing the Risk-Pooling Effect Through Demand Reshape
ˆ 12 = A2 + B2
= 1 − A + 1 − 2 A2 + 1 − B + 1 − 2 B2 =
1 − 1 + 1 − 2 12 .
, on the other hand, is the sum of four random variables: A
Y
(random selection with probability over A), B (random selection
with probability over B), B, and C. All of these variables, except
for B and B, are independent.
V B = V B − B = V B + V B − 2 CovB B or
CovB B = V B + V B − V B /2
Based on the lemma,
A2 = 1 − A + 2 A2 B2
= 1 − B + 2 B2 Hence,
CovB B = B2 + 1 − B + 2 B2 − 1 − B − 1 − 2 B2 /2
= B2 Consequently,
V B + B = V B + V B + 2 CovB B = 1 − B + 2 B2 + B2 + 2B2
= 1 − B + 1 + 2 B2 and finally,
= V A
+ V B + B + V C
ˆ 22 = V Y
= 1 − A + 2 A2 + 1 − B + 1 + 2 B2 + C2
= 1 − 1 + 2 12 − B2 + 1 + 2 B2 + 22 − B2
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