production
economics
ELSEVIER
Int. J. Production
Economics
44 (1996) 267.. 275
A two-item newsboy problem with substitutability
Moutaz
ADepartment
qJ‘MIS,‘M.
Khouja”**, Abraham
Mehrezb, Gad Rabinowitzb
The Belk College of Business Administration,
NC 28223,
‘Department
of Industrial
Engineering
and Management,
The Universi~
of .Vorth Carolina
at Charlotte,
Charlotte,
USA
Ben-Gurion
Universi&
of the Neger. Beer Shet>a, Israel
Received 28 June 1995; accepted 6 May 1996
Abstract
Previous
research on the newsboy problem is based on the assumption that in case of a shortage, unsatisfied demand is
lost. Such an assumption is inappropriate
for items that have a close substitute. In this paper, we formulate a two-item
newsboy problem with substitutability
(TINPS). Upper and lower bounds on the optimal order quantities of the two
items are derived. Since analytical solutions to the problem are difficult to obtain, a Monte Carlo simulation is used to
identify the optimal solution to the TINPS. Order quantities identified by the simulation provide higher expected profit
than would have been obtainable without considering substitutability.
Keywords:
Single-period
problem;
Substitutability;
Monte
1. Introduction
The classical newsboy
inventory
problem
is
to find product
order quantity
that maximizes
expected profit in a single period probabilistic
demand
framework.
The
problem
considers
a single perishable item and assumes that in case of
a shortage, unsatisfied demand is lost [l].
Several extensions to the newsboy problem have
been proposed and solved in the literature. Among
those extensions
are using alternative
objective
functions such as maximizing
the expected utility
instead of expected profit [Z, 31, maximizing
the
market value of the firm using the capital asset
*Corresponding
0925-5273/96/$15.00
SSDI
author.
Copyright
0925-5273(95)00059-X
Carlo simulation
pricing model framework [4], and maximizing the
probability of achieving a target profit level [2,3,5,
61. Other
extensions
consider
uncertainty
in
supplies [7], having more than one supplier [7],
price-dependent
demand
[S], quantity
discounts
offered by suppliers [9, lo], the availability
of an
emergency supply option [ll, 121, and the use of
progressive multiple discounts to sell excess inventory [13].
Early extensions
to multiple
items assume
demand for items to be independent
and solve
a profit maximization
problem under storage or
budget constraints
[l]. More recent extensions to
multiple items include a constrained
problem with
progressive
multiple discounts
[14], and a twoitem problem with independent
demand and an
objective of maximizing
the probability
of achieving a target profit level [15, 161. Similar to earlier
,c> 1996 Elsevier Science B.V. All rights reserved
268
M. Khouja et al. /ht. J. Production
models, these models assume that in case of a shortage, unsatisfied demand is lost.
A common situation that arises in practice is that
customers substitute a different item when the item
they desire is not in stock. Consider, for example,
grocery stores that stock two grades of a produce,
a regular grade and a premium grade at a higher
price. Customers
who are unable to satisfy their
demand for one of the grades due to a shortage may
substitute some quantity of the other grade rather
than trying to satisfy their demand at a different
store. A similar situation
may occur in a fresh
seafood market. The manager of the market may be
reluctant to order a large quantity of an expensive
item, since the cost of overestimating
demand is
quite large. The manager might be encouraged
to
do so by the fact that a number of customers who
are unable to purchase the expensive item due to
a shortage substitute some quantity of another item
that is less expensive to overstock.
In this paper, a two-item newsboy problem with
substitutability
(TINPS) is formulated. The expected
profit function for the new problem is derived. However, due to the difficulty of optimizing the expected
profit function, a Monte Carlo simulation is used to
identify the optimal solution to the TINPS. To limit
the number of combinations
of the order quantities
of the two items to be tested using the simulation,
upper and lower bounds on the optimal order
quantities of the two items are derived.
The rest of this paper is organized as follows. In
Section 2, the TINPS model is formulated and the
expected profit function is derived. A Monte Carlo
simulation
to solve the problem is developed in
Section 3 and upper and lower bounds on the
values of the optimal order quantities
are established. Some numerical examples are solved using
the simulation
in Section 4. Section 5 contains
a discussion and conclusion.
2. The model
Define the following
pi
si
notation
1,2, an item index,
sale price of one unit of item i,
salvage value of one unit of item i,
Economics
44 (I 9%) 267--275
Ci
%i
Bi
xi
.ZtxiI
4
(Xi)
f
(Xl,
X2)
t1
f2
Qi
QT
cost of one unit of item i,
Pi - Ci,
cost
of
underestimating
demand by one unit for item i,
CL - St, cost of overestimating
demand
by one unit for item i,
a random variable denoting demand for
item i,
a realization of Xi,
probability
density function of Xi,
cumulative probability
density function
Of Xi,
joint probability
density function
of
Xi and X,,
quantity of item 1 customers substitute
for one unit of item 2,
quantity of item 2 customers substitute
for one unit of item 1,
order quantity of item i, a decision variable,
optimal order quantity of item i.
The TINPS is to find Qr and Q2 that maximize the
expected profit, E(ZI(Qi, QJ) . The profit is given
by:
~(QI>Q2)=
I
it1
Cf’iXiiiI
C,Qr+ Si(Qi - X,)]
Cf’iQi- CiQJ
PiXi + PjQj -
~ CiQi
if Qi > Xi, i = 1, 2,
if Qi < Xi, i = 1, 2.
if Qi > Xi
(1)
i=l
+
PiMin{t;(Xj
- Qj), Qi - Xi}
\ +siMax{Qi-Xi-ti(Xj-Qj),O}
and Qj 6 Xj,
i,j=1,2and
i#j.
The complete derivation of E(L’(Qi, Q2)) is shown
in Appendix
A. As the
appendix
shows,
E(n(Qi, Q2)) is quite complex and optimizing
it
requires finding (Qr, QJ for which aE(n(Qr, Q2))/
8Q1 = aE(n(Qr,
Q2))/8Q2 = 0
which
presents
a high degree of difficulty. Actually, since no proof
of concavity of E(L’(Qi, Q2)) can be provided, the
problem may have many local optimal solutions.
Due to the difficulty of obtaining an optimal solution using optimization
methods, a Monte Carlo
269
M. Khouja et al. JInt. J. Production Economics 44 (1996) 267-275
E2:demand for each item is equal to or above the
order quantity (there will be no excess of either item
to be discounted).
If event 2 is realized, then the profit is given by
simulation is used to identify the optimal order
quantities. However, upper and lower limits on the
optimal order quantities are established so as to
limit the solution space to be tested by the simulation.
712 =
f',Q,
+ PZQZ- GQl - C2Q2,
which after simplification gives
3. The simulation
To develop a simulation of the TINPS, six possible events are identified and the resulting profit if
each occurs is defined. A graphical presentation of
the six events in the X1-X2 space is shown in Fig. 1.
QI~I+ QA.
~53 =
(QI<Xl and bW1 -QAG(Qz
=3=f'1Q1+PJ,-CIQ,
+
713 =
E2
=
-
C2Q2
QI)
which after some manipulation
+ SI(QI-XI) + SZ(QZ-X,1,
Xzg2
f’2t2(X,
-
+ S2(Q2-X2 - tz(X1- QJ,
+ P2X2-C,Q,-C2Q2
~1 = X,a, +
becomes
QI~I- P2Q2 +(a2 + B2HX2 + t2(X1-Qd,
becomes
(4)
-(Ql - XdP, - (Q2 -
E, = {QI <Xl, Q2>X2, and (2(X1-QJ
XJP2.
{Q1 < X1 and Q2 GX,}.
(2)
>(Q2 - X2,).
x2
E5 or
Qz
-X2)>.
quantity and the excess quantity of item 2 is sufficient to substitute for item 1 (there will be an excess
of item 2 that must be discounted).
If event 3 is realized, then the profit is given by
El:demand for each item is below the order quantity (there will be an excess of both items that must
be discounted).
If event 1 is realized, then the profit is given by
which after some manipulation
(3)
E,:demand for item 1 is greater than the order
E, = (Qi > X1 and Q2 > X,}.
ni =P,Xi
7~2 =
E6:
E2:
IQ, < X2 & tl(X2-Q2)
5 (Ql-Xl11
{Q, < X2 8, tl(XZ-QZl
> (a,-X,)1
or
_--__--------____
I________________
E
1
I
:
Eg or
E4:
I
{Q, ’ X1 & Q2 > X2)
;{Q,
< X1 & t2!Xl-Q1l
5 (a,-X2)>
;{Q,
< Xl & t,(X,-Q,)
> (Q2-X2)1
Ql
Fig. 1. A graphical
presentation
or
x1
of events of the TINPS.
270
M. Khouja et al. /Int. J. Production Economics
&: demand for item 1 is greater than the order
quantity and the excess quantity of item 2 is insufficient to substitute for item 1 (there will be no excess
of either item to be discounted).
If event 4 is realized, then the expression for profit
simplifies to (3).
44 (I 996) 267-275
where n(Qi,
Q2) is the sample
mean
and S(Qi, Q2) is the sample standard
S(Q1, Q2) =
& =
{Qz
<X2
and
~I(XZ
-
QJ
G
(QI
-
~5 =
PzQz
+
+ Pltl(X2
PIXI
C2Q2 - CIQI
-
- Q2) + SI(QI - XI - t,(X2 -
which after some manipulation
7~5 =
& =
>
QA
-
IQ1
(QI
BIQI
>X,,
-
+
Qz
(~1
+
<X2,
Qd.
becomes
Pd(X,
and
+
t1Gf2
tl(X2
-
-
Qzb
(5)
Q2)
d
Q1
<
Q?)
and
Q2(Qi
d
Qz
d
Qy).
The
re-
sulting profits depend on the realized events and
are computed
using (2)(5). For each Qi and
combination,
the
profits
~,(QI>
Q2h
Q2
and
w,(Q1,
Q2)
are independent
(02(Q1> Q2h . . . 3
random variables from a common distribution
with
mean E(17(Q1, Q2)) and variance
cr2(K7(Q1, Q2)).
For IZsufficiently large, the lOO(1 - x)% confidence
interval on E(L’(Q1, Q2)) is [17]
n(Qi,
j$i (aj(Qi,
- 1) o.5.
Qz) - n(Qi> QJ)‘l(k
[
Two factors are important
in determining
the
sample size n. The first factor is the desired level of
accuracy in estimating profit. The second factor is
the number of combinations
of Q1 and Q2 for which
profit is to be estimated. To limit the number of
combinations
of Q1 and Q2 for which expected
profit must be estimated, upper and lower bounds
on the optimal order quantities are established. The
lower bounds are developed in Lemma 1. Before
introducing
Lemma 1, define Property 1.
Property 1. Property 1 is said to koldfor item i if:
Profit per unit of item i > profit per
units of item j.
Or alternatively:
tj
XI,>.
E6: demand for item 2 is greater than the order
quantity and the excess quantity of item 1 is insufficient to substitute for item 2 (there will be no excess
of either item to be discounted).
If event 6 is realized, then the expression for profit
simplifies to (3).
A flow chart of the simulation
used to identify
the optimal solution to the TINPS is shown in
Fig. 2. Suppose Qk and Qk are lower bounds on
QT and Qz, and Q’: and Qy are upper bounds. As
Fig. 2 shows, a sample of y1realizations of demand
each
combination
of
generated
for
is
Q,(Q?
1
XdS.
E,: demand for item 2 is greater than the order
quantity and the excess quantity of item 1 is sufficient to substitute for item 2 (there will be an excess
of item 1 that must be discounted).
If event 5 is realized, then the profit is given by
deviation
Q2) - Za,2S(Q1, Q2Yn0.’ d E(~(QI,
d ~(QI, Q2) + Z,,,S(Q1,
Q2Yn0.5,
Q2))
pi >
tjr,j.
In other words, if Property 1 holds for item i then
it is more profitable to sell customers one unit of
item i than selling them the substitution
quantity of
item j. Otherwise, it is more profitable to only stock
item j. Define Qk and Q\ as F1(X1 = Qk) ~0 and
F,(X, = Q$ ~0, Lemma 1 establishes
the lower
bounds on QT and Qz.
Lemma 1. If‘ Property 1 holds for both items, then
any optimal solution to the TINPS
satisfies
QT
3
Q\
and
QZ
3
Qi.
Proof. The proof of Lemma
1 is in the Appendix.
Upper bounds on the optimal order quantities
under general conditions are developed in Lemma
upper
bounds
under
uncorrelated
2. Stricter
demands
with normal
or Poisson
distributions
are developed in Lemma 3. Define the following
random variables:
X, = X1 + tl(X2 - Q:)
(6)
271
M. Khouja et al./Int. J. Production Economics 44 (1996) 267-275
NO
;,;,,
Tl
lrYGsym7
L
Increase
Counter
by 1
Fig. 2. A flow diagram of the simulation model of the TINPS
and
x, = x* + f*(X1 - Qk,.
(7)
Define
Qy and
Qy as F3(X3 = Qy)z 1 and
F4(X4 = Qy)z 1, Lemma
2 establishes
Q’: and
Qy as upper bounds on QT and QT.
Stricter upper bounds on QT and QT can be
established for uncorrelated
X1 and X2 with normal or Poisson distributions.
The expected values
of X3 and X4 are
E(X,) = E(X,) + tl[E(X,)
-
Qkl,
E(Xd = EWd + t,[E(X,) - Qkl,
Lemma 2. Any optimal solution to the TZNPS satisjes QT < Q’: and QZ < Q’i’.
Proof. The proof is obtained using similar
ments used in the proof of Lemma 1 [18].
and if X1 and X2 are uncorrelated
Var(X,)
= Var(X,)
+ tf Var(XJ,
Var(X,)
= Var(XJ
+ tiVar(X,),
argu-
[19],
272
M. Khouja et al. lint. J. Production Economics
If Xi and X2 are normal, then X3 and X4 are
also normal and the same conclusion holds for
the Poisson distribution [20, p. 3191. Suppose Q’;”
solves the single item newsboy problem with demand X3 and Qy solves the single item newsboy
problem with demand X4. Then, Qy and Qy must
satisfy [ 11
MQ'I')
= dam + PI)
(8)
44 (1996) 267-275
Table 1
Data for numerical
examples
l-3
Item
Unit price, Pi
Unit salvage value, S,
Unit purchase cost, C,
Mean demand
Standard deviation
I
Item 2
$2.00
$1 .oo
$1.20
50
10
$7.50
$2.00
$5.00
30
6
and
UQ;) = 4bz + Pd.
(9)
Lemma 3 provides proof that QT and Q’;’are upper
bounds on QT and Q:.
Lemma 3. Any optimal solution to the TINPS satisfies QT < Q’;’ and Q2 < Q’;“.
Proof. The proof for Lemma 3 is provided in the
Appendix.
4. Numerical
examples
To gain some insights into the problem, we solve
six numerical examples of the TINPS using the
simulation. Table 1 shows the data for the first
three examples. In ail three examples, the costs of
underestimating
demand
by one unit are
x1 = $0.80 and CI~= $2.50 while the costs of over-
Table 2
Solutions
to numerical
examples
l-3 of the TINPS
Example
1
Item 1
t1
t2
Lower bound, QL
Upper bound, Q!’
UB under Property
QP
Q?
W'(Q:,QP))
99% C I on .E(n(Q:,
% increase in profit
Example
Item 2
1.00
1, QF
Q:))
estimating demand by one unit are /3i = $0.20 and
jz = $3.00. In other words, item 2 provides larger
profit per unit when sold than item 1 but also has
a higher loss if not sold. Table 2 shows the substitution patterns and the solutions to examples l-3. In
all three examples, customers substitute a quantity
of t2 = 0.267 units of item 2 for one unit of 1 (this
substitution results in the same expenditure that
would have been made without the shortage). Customers substitute quantities of tl = 1, 2 and 3 units
of item 1 for one unit of item 2 in examples 1,2 and
3 respectively (these substitutions result in expenditures of 0.267, 0.533, 0.80 of the expenditure that
would have been made without the shortage). Since
both items satisfy Property 1, lower and upper
bounds on the optimal order quantities (Qk and Qu,
i = 1,2) are established using 3 standard deviations
and are shown in Table 2. Also, with uncorrelated
demands for the two items, the upper bounds are
20
103
78
58
60
$99.1
$101.673-101.725
0~2.6
Item
2
1
Example
Item 2
20
133
100
58
69
99.1
$104.298-104.360
%5.3
Item 1
Item 2
3.00
2.00
0.267
12
58
38
29
29
3
0.261
12
58
38
29
26
20
166
121
58
110
$99.1
$108.89G108.895
%9.9
0.261
12
58
38
29
16
213
M. Khouja et al. JInt. J. Production Economics 44 (1996) 267-275
Table 3
Solutions
to numerical
examples
4-6 of the TINPS
Example
4
Item 1
t1
t2
Lower bound, QfUpper bound, Qu
UB under Property
Example
Item 2
1.00
1, Qy
QY
Q:
E(n(Q?>Q?))
99% C I on E(II(Q:, QF))
% increase in profit
20
103
78
58
49
Item
5
Example
I
Item 2
Item
2.00
0.267
12
58
41
32
34
$103.1
$105.61 l-105.707
%2.5
tightened by using (8), (9) and the definitions of
X3 and X4 to obtain (Q”, i = 1,2). Table 2 also
shows the optimal order quantities without
substitutability (Qp, i = 1,2) and the associated
expected profit E(ll(QF, Qp)). In all three examples,
as the table shows, considering substitutability
improves the profit of the TINPS. As f, increases,
Qi increases and Qz decreases. Such behavior can
be intuitively explained. As ti increases, the effective cost of underestimating demand of item 2
decreases because more of that demand is satisfied
by item 1 in case of a shortage. Subsequently, Qz
decreases which effectively increases the demand
for item 1 and hence QT increases.
Table 3 shows the solutions to examples 4-6.
These three examples are similar to examples 1-3
except that the salvage value of item 2 is increased
to S2 = $3.5 per unit. Since the cost of overestimating demand of item 2 is decreased to flZ = $1.50, it
is less profitable to substitute item 1 for it for any ti.
Hence, QT is smaller for every ti in examples 4-6
and Q: is larger.
6
I
Item 2
3.00
0.267
12
58
41
32
29
20
133
100
58
64
20
166
121
58
101
$103.1
$106.403-106.455
%3.2
0.267
12
58
41
32
20
$103.1
$109.017-109.035
%5.7
items using optimization methods is quite difficult,
simulation provides a useful tool for finding the
optimum. In order to reduce the number of combinations of order quantities for which to run the
simulation, lower and upper bounds on the optimal
order quantities are derived.
Numerical examples show that the solution to
the TINPS may be quite different from the classical
newsboy problem in which substitutability is ignored. Furthermore, the improvement in expected
profit due to the consideration of substitutability
can be substantial.
Future research may consider using prices as
means of changing substitution patterns. In other
words, prices become decision variables with the
goal of obtaining substitution patterns between
items that result in maximizing profit.
Appendix A
Development
ofE(IZ(Ql,
Q2))
Define
5. Discussion
and conclusion
This paper formulates a two-item newsboy problem in which customers substitute some quantity of
one item for the other in case of a shortage. While
finding global optimal order quantities of the two
gl(Ql,
Qz)= t Cf'iXi
- CiQi+ SdQi-X,)1,(A.11
i=l
g2(Ql,QA= i (Pi-Ci)Qi,
i=l
64.2)
274
M. Khouja et aLlInt. J. Production Economics
g3(Ql,
Q2)= PIXI + &_Q2 - i
CiQi
i=l
+
PI Min(tl(X2 - Q2), Q1 - Xl}
+ & MaxfQl- X1 - W2 - Q2k01,
64.3)
ga(Qls
Q2)=PzXZ + PIQI- C CiQi
i=l
+
P2Min{h(X1 - Qd Q2 - X2>
+ S2Max(Q2 - X2 - t2W1- Qd,01,
Letting
bl = b2 =0
(Qy = Qk and
Qz = Qk)
increases the profit by b,cx, + b2az.
2. For type 2 solutions,
one of the following
three events will occur:
2(a). Event EZ. The profit is given by (3) which
becomes
Letting b2 = 0 (QT = Q’;) increases profit by b2a2.
2(b). Event E5. The profit is given by (5) which
becomes
~5 =
(4.4)
The expected
functions:
profit is the sum of the following
four
QI Qz
h(Q,>Qd = j j sl(Qi,
QdfV,, X,)dX,dXz>
0 0
corn
QI 0~
MQI, Qz)=
jj
gs(Qi,
Qz)ufW~>
XddX, dXa
m QZ
&(QI,QJ =
jj
gdQ1,Q,)fV,,Xz)dx~dx,.
QI 0
Further analysis of hi, h2, h3, and h4 can be used to
further develop E(n(Qi, QJ) [lS].
(Qi-b&z + ~IXI+
z,t,(X,
-,%(QL;
+ h -XI -
~1x2
-
+
Q::+ b2)
t,Q::
-t~bz).
If b2 is decreased by 1 unit (Qz is increased by one
unit) and b1 is decreased by ti, the change in profit
isd =%2-t1a,.ByProperty1,d
>Oandthusthe
above change in bl and b2 leads to a superior
solution. The change in QT and QT is continued
until b2 = 0, which satisfies the lemma, or until
bl = 0, which reduces the solutions to a type 1 solution.
2(c). Event &. The profit is given by (3). Letting
b2 = 0 (Q: = Q$) increases the profit by b2a2.
3. For type 3 solutions, an argument similar to
the one made against type 2 solutions can be made.
Proof of Lemma 3. Suppose Q2 = Qi, the demand
for item 1 is X3 as given by (6) and hence the
optimal order quantity of item 1 is Q’;’defined in (8).
For any larger quantity
Q2 = Qi + b, b > 0, the
demand for item 1 is
x5 =
Proof of Lemma 1. Three types of solutions violate
Lemma 1:
1. QT = Qy - bl and Qt = Qi - b2, bl and b2 > 0,
2. QT = Qk + bl and Q: =Qi--b2,
bl 20 and
b, > 0, and
3. QT =Qkb, and QT =Qi+
b,, b, >O and
b2 > 0.
The profit of each of the three solution types is
increased by using solutions that satisfy the lemma.
1. For type 1 solutions, event E2 is realized as
long as bl and b2 > 0. The profit is given by (3)
which becomes
44 (1996) 267-275
For
if X2 <(Qi
X1 3
+ b), and
i X1 + ti (X, - Q$ - b) if X2 >(Q’; + b).
X, < (Q’; + b), X5-N(E(X,),
Var(X,))
and
X, - N(E(X,)
+ tr(E(X, I X2 d (Qi+ b))- Qi),
as
Var(X,) + ti Var(X, IX2 6 (Qk + b))). Hence,
Fig. 3 shows,
E(X,) < E(X,)
and
Var(X,) <
Var(X,), and the cumulative probability
functions
of X3 and X5 satisfy
F3(c) < F,(c) for any c > 0,
(A.5)
and subsequently
FC r(~J(~i
+ A)) > F; %&i
which with (8) implies
+ PA
Qy > QT.
(4.6)
M. Khouja et al. /ht. J. Production Economies 44 (1996) 267-275
Fig. 3. Probability
density
functions
For X2 > (Q2"
+ 61,X5 -NWXd
of X3 and X,.
+ tl(W2 IX2
> (Q\ + b)) - Qk - b), Var(X,) + ti Var(X, 1X2 >
>
(Qk+ b))) and XJ-N(E(X~)+~~(E(X~IX~
(Qk + b)) - Qk),
Var(X1) + t:
Var(Xz I X2 >
(Qk + b))). Since E(X,) < E(X,) and Var(X,) =
Var(X,) , the cumulative
probability
functions of
X3 and X, satisfy (A.6) which with (8) implies
Q’; > QT. A similar argument
can be made for
Qz and the same proof can be constructed
for
Poisson distributed
X1 and X2 [20, p. 3191.
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