Directory UMM :Data Elmu:jurnal:I:International Journal of Production Economics:Vol65.Issue2.Apr2000:
Int. J. Production Economics 65 (2000) 141}171
Optimal procurement policies under price-dependent demand
Hakan Polatoglu!,*, Izzet Sahin"
! Frank Sawyer School of Management, Suwolk University, Boston, MA 02108, USA
" School of Business Administration, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA
Received 19 June 1998; accepted 17 September 1998
Abstract
We study a periodic-review inventory model where, in addition to the procurement quantity, price is also a decision
variable. We develop a model where demand in each period is a random variable having a price- and, possibly,
period-dependent probability distribution, with the expected demand decreasing in price. The model includes price limits
and "xed ordering costs in addition to unit procurement holding and shortage costs. We study the optimal policies which
jointly maximize the discounted expected pro"t over a "nite planning horizon. We characterize the form of the optimal
procurement policy under a general price}demand relationship and give a su$cient condition for it to be (s , S ) type. We
n n
also discuss some special cases and extensions to the basic model, including the in"nite horizon problem. ( 2000
Elsevier Science B.V. All rights reserved.
Keywords: Pricing; Inventory; (s, S) policy
Notation
n
N
i
n
q
n
p
n
Pl
P
6
c
r
h
v
K
n
period index (n"1 corresponds to the last period)
total number of periods in the planning horizon
beginning inventory level before ordering in period n
beginning inventory level after ordering in period n
price in period n
price #oor
price ceiling
unit procurement cost
unit shortage cost
unit holding cost
unit salvage value at the end of the planning horizon
"xed ordering cost in period n
* Corresponding author. Protek Computer Systems Inc., Perpa Ticaret Merkezi, Elektrokent, Kat 13, No 1981, Okmeydani, 80270
Istanbul, Turkey. Tel:. #90 (212) 210-1730; fax: #90 (212) 210-1765.
E-mail: Hakan.Polatoglu@protek.com.tr (H. Polatoglu)
0925-5273/00/$ } see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 5 2 7 3 ( 9 8 ) 0 0 2 4 0 - 0
142
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
a
X(p)
XM (p)
¸(p)
;(p)
f (x; p)
F(x; p)
p (q )
n n
M (i , p , q )
n n n n
periodic discount factor
random demand when price is p
expected demand as a function of price ("E[X(p)])
lower bound on random demand, 0)¸(p))X(p)
upper bound on random demand, X(p));(p)(R
probability density function of X(p)
probability distribution of X(p)
optimal price when the inventory level is q in period n
n
n-period (periods n}1) pseudo-pro"t function (pro"t excluding the e!ect of the "xed ordering
cost)
M
M (i , p , q ) expected value of M (i , p , q )
n n n n
n n n n
n-period expected pseudo pro"t when the optimal value of p is implemented in period n, i.e.,
M
M w(i , q )
n n n
n
w
M (i , p (q ), q )
M
M (i , q ),M
n n n
n n n n n
PM (i , p )
n-period expected pro"t when the optimal value of q is implemented in period n
n n
n
PM w(i )
n-period expected pro"t when the optimal values of q and p are implemented in period n
n
n
n
d(z)
Heavyside function, d(z)"0 for z)0 and d(z)"1 for z'0
[z]`
positive value function, [z]`"zd(z)
1. Introduction
Under increased competition, inventory-based businesses are forced to better coordinate their procurement and marketing decisions, to avoid carrying excessive stock when sales are low or shortages when they
are high. An e!ective means of such coordination is to conduct the inventory control and pricing decisions
jointly. The main task in doing so is to determine the optimal inventory policy, given the price}demand
relationship that is expected to prevail in the market place in the short term.
In addressing the above issue, we are concerned in this paper with a single-item, periodic-review inventory
system where the vendor, who enjoys a degree of monopoly power in the market place, is in a position to
in#uence demand by its pricing decisions. It is thus confronted with simultaneous pricing and procurement
quantity decisions, which would jointly maximize the present value of expected pro"t over a planning horizon.
We will assume that the ordering policy does not change the demand pattern during the planning horizon.
Ordering cost, shortage cost and temporal increases in the procurement cost are among the important
factors that force the decision maker to carry inventories. Holding cost has the opposite e!ect. Thus, in the
absence of pricing, the main decision problem is to determine the optimal inventory levels to strike a balance
between these opposing factors, under a given demand forecast, cost structure, and a set of operating
conditions.
There are critical di!erences between a "xed-price, periodic-review inventory model [1] and a model that
also includes pricing decisions. The economical interpretations of these di!erences relate to the various ways
that price plays into the decision problem. First, price is a decision variable that determines the revenue per
unit sold. Second, price is a factor that in#uences the demand, thus the period-ending inventory levels. In
addition, when backlogging occurs, apart from the amount of backorders, the backlogging policy must also
account for the price that applies to backorders and the timing of revenue collection. For instance,
backorders could be sold at the current price in advance, or at a future price set at the time of delivery.
Therefore, when the decision maker confronts the pricing and inventory decisions simultaneously,
in addition to the two opposing cost-related e!ects that we noted above, price-related factors must also
be taken into account. Mainly, there will be a trade-o! between the high-price low-demand and low-price
high-demand scenarios, in terms of discounted total revenue. This revenue trade-o!, however, will not be
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
143
independent of the above described cost trade-o!, since the demand levels (i.e., inventory levels) will be
a!ected by the pricing decisions. Unfortunately, this complex relationship between the cost and revenue
trade-o!s does not allow the model to simplify into separate pricing and procurement decisions.
Various versions of the procurement-and-pricing model has been studied in the literature. In his pioneering work, Whitin [2] proposed a link between price theory and inventory control. He noted that such
a model would have stronger managerial implications, as compared to "xed-price models. Later, Mills [3,4]
and Karlin and Carr [5] established a conceptual framework for a general inventory model, where price is
a decision variable. Subsequent studies concentrated mostly on the characterization of the optimal solution
for some special cases of the general model [6}13]. Thomas [14] provided some numerical examples to
demonstrate the nature of the decision problem under the presence of "xed ordering costs. More recently,
Gallego and Van Ryzin [15] studied the continuous-time version of the problem, and Petruzzi [16]
investigated the approximate solutions under a learning approach. Also, under a dynamic model with
Bayesian learning about the demand distribution, Subrahmanyan and Shoemaker [17] studied a number of
numerical examples which provide insights about the sensitivity of optimal prices and inventory levels to
changes in the unit procurement cost, price elasticity of demand, form of the demand distribution and the
expected demand function.
The existing analytical models impose rather restrictive assumptions on the form of the expected demand
function (e.g., concavity assumptions in [9,11,12]), demand distribution (e.g., multiplicative demand in [9],
additive demand in [9,11]), or the cost structure (e.g., parameter restrictions and no "xed ordering cost in
[11,10]) in analyzing the optimal procurement policy. In this paper, by relaxing some of the more limiting
modeling assumptions, we seek to characterize the fundamental properties of the optimal procurement policy
and the resulting expected pro"t in a more general setting (i.e., general demand distribution, linear cost
structure with the addition of set-up cost). We also address the sensitivity of the optimal procurement policy
to the underlying price-dependent demand uncertainty.
In what follows, we describe the price}demand relationship in Section 2, develop the basic model in
Section 3, and characterize its solution in Section 4. The rest of the paper is devoted to special cases and
extensions. We consider the in"nite-horizon problem in Section 5, the model with no "xed ordering costs in
Section 6, the non-stationary extensions of the basic model in Section 7, and the special case with deterministic demand in Section 8. Section 9 is devoted to some concluding remarks. Sections 4 and 5 include
numerical examples. Proofs are given in the appendix.
2. Demand uncertainty
It has been a common practice in demand modeling to express random demand as a combination of an
expected demand function, which exhibits some form of price-dependency, and a random term, which is
price-independent. This approach conveniently isolates the e!ects of price and uncertainty, while retaining
mathematical tractibility. However, it has some shortcomings.
Under the additive model [3,5,9,11], we have X(p)"XM (p)#e where XM (p) is a decreasing function of p and
e is a random variable with E[e]"0. The additive model can also be interpreted as a homoscedastic
regression model where XM (p) represents the regression function and e is the error term. One implication of
this model is that while the expected demand is a function of the price, the demand variance is priceindependent (demand distribution shifts as price varies). Also, the model allows for negative demand, unless
the price values are bounded from above.
The multiplicative model [5,6,8] is another commonly used form where X(p)"eXM (p) with E[e]"1. This
model implies the restriction that the demand equals to the product of its expected value and a random term.
As a result, the coe$cient of variation of demand equals that of e, which is constant in price, and demand
variance decreases at a rate faster than the expected value and it approaches to zero at high prices.
144
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
In addition to these, there are models that represent random demand by a mixture of additive and
multiplicative terms [10,12,13], such as X(p)"XM (p)a (e)#a (e), or X(p)"a (p)#ea (p), where a and
1
2
1
2
1
a are di!erentiable functions.
2
Additional simpli"cation can be achieved by assuming that the price}demand relationship is perfectly
predictable. This leads to the deterministic model, X(p)"XM (p), which serves as a "rst-order approximation,
and which has been utilized as a benchmark in the literature [3,5,9,12]. It has been reported in these studies
that the form of demand uncertainty, whether it is additive or multiplicative, plays a critical role in
determining optimal prices. Optimal prices are lower than their deterministic counterparts under the additive
demand model, while they are found to be greater under the multiplicative demand model. Additional
"ndings about the form of demand uncertainty, as it impacts optimal prices and inventory levels, are reported
in [18] p. 617.
In this study, we represent the demand by a continuous random variable, distributed over the range
[¸(p), ;(p)] with a known density function f (x; p). ¸ and ; are di!erentiable functions which represent the
bounds on demand, where 0)¸(p))X(p));(p)(R for all p. For convenience, we "rst assume stationarity, whereby the demand density is given by f in all periods. We then show in Section 8 that the results we
obtain for the "nite-horizon problem can be extended to non-stationary (period-dependent) demand
distributions.
There is empirical evidence that consumers react to price in purchasing. Other things being equal, a lower
price facilitates demand in a `faira market. Therefore, it is natural to assume that the probability that demand
is less than a given level x, F(x; p)"P[X(p))x], is a non-decreasing function of price. That is,
LF(x; p)
*0 ∀x3(¸(p), ;(p)).
Lp
(1)
It follows from Eq. (1) that F(x; p ))F(x; p ) for p (p . Thus, price induces a stochastic ordering of
1
2
1
2
demand distributions.
The expected demand is given by
=
U(p)
xf (x; p) dx" [1!F(x; p)] dx.
XM (p)"
P
P
(2)
0
L(p)
It also follows from Eqs. (1) and (2) that XM (p) is a decreasing function of p:
=
LF(x; p)
LF(x; p)
dXM (p)
*0N
"!
dx(0.
Lp
Lp
dp
0
P
3. Mathematical model and assumptions
Consider a periodic-review inventory system with N periods where the decision variables are q and p . The
n
n
review periods are linked by period-ending inventory levels such that the leftovers are transferred fully to the
next period and shortages are lost. That is, i "[q !X(p )]` for 0)n)N!1, and i *0 is a given
n
n`1
n`1
N
parameter. At the beginning of a period, the vendor decides how much to order (q !i ) and what price to
n
n
charge until the next decision point. There is a "xed cost of ordering (K), but no cost is assumed for pricing.
Therefore, it is for the vendor's bene"t to reconsider pricing at each decision epoch.
We assume that the vendor has full information about the inventory and procurement costs, and the
demand distributions in all periods of the planning horizon. At the beginning of a review period, say period n,
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
145
given the inventory position, the vendor is to determine the procurement quantity and the price to maximize
the expected n-period pro"t which represents the expected value of the sum of the current period's pro"t and
the discounted optimal expected pro"t to be obtained during the remaining periods.
Let Pnl and Pn be the price #oor and the price ceiling, respectively, in period n, such that the interval
6
[Pnl, Pn ] represents the range of allowable prices. Then, the sequences MP1l , P2l ,2, PNl N and MP1, P2,2, PNN
6
6
6 6
establish limiting-price pro"les during the planning horizon. In regulated markets, these pro"les represent
the temporal rules of regulation. Also, in a case where a manufacturer might issue price limits for its dealers
(price-maintenance), or provide `suggested retail pricesa, the allowable price ranges could be represented by
the limiting-price pro"les.
In the basic model, without loss of generality, we assume uniform pro"les where the feasible price values in
any period are between Pl and P . If there is no price regulation or other constraint on prices, then the price
6
limits may be taken as 0 and p , respectively, where p , referred to as the `null pricea in the literature, is the
=
=
highest possible price level at which there will be no demand, that is, P[X(p))0]"1 for all p*p .
=
Allowing p "R leads to detailed technical considerations (see [8,9]) which we choose to avoid in this paper.
=
We assume that inventory costs are proportional to period-ending inventory levels. The cost and discountfactor parameters may be allowed to change from period to period. However, as in the case of the demand
distribution, we keep them time-invariant for convenience in developing the basic model (see Section 8 for
non-stationary extensions of the basic model). We take P 'c so that it is possible to make pro"t. In
6
addition, the leftovers at the end of the planning horizon are assumed to be salvaged at a discount price, that
is, v)c.
Under these assumptions, the n-period optimal expected pro"t, as a function of i , is obtained from
n
PM w(i )"maxMPM (i , p ): p 3[Pl, P ]N,
n n
n n n n
6
and
PM (i , p )"maxMM
M (i , p , q )!Kd(q !i ): q *i N,
n n n
n n n n
n
n n
n
where M
M is the expected value of the n-period pseudo-pro,t function which is de"ned for n*1 as
n
0)q )X(p ),
aPM w (0)# p q !r(X(p )!q ),
n~1
n n
n
n
n
n
M (i , p , q )"!c(q !i )#
w
n n n n
n
n
aPM
(q !X(p ))# p X(p )!h(q !X(p )), X(p ))q ,
n~1 n
n
n
n
n
n
n
n
with PM w(i )"v[i ]`. This leads to
0 0
0
M
M (i , p , q )"(p #r!c)q !rXM (p )! (p #r#h)H(p , q )#ci # aPM w (0)[1!F(q ; p )]
n n n n
n
n
n
n
n n
n
n~1
n n
qn~L(pn)
PM w (x) f (q !x; p ) dx,
(3)
#a
n~1
n
n
0
where H(p , q )"E[[q !X(p )]`] is the expected value of leftovers at the end of period n. It is seen that
n n
n
n
H(p , q ) is a convex increasing function in q and an increasing function in p ; its additional properties are
n n
n
n
given in [19].
Therefore, the overall decision problem is described by
G
P
M w(i , q )!Kd(q !i ): q *i N
(4)
PM w(i )"maxMM
n n
n n n
n
n n
n
M
M w(i , q )"maxMM
M (i , p , q ): p 3[Pl, P ]N"M
M (i , p (q ), q ),
(5)
n n n
n n n n n
6
n n n n n
for all n*1, where p (q ) represents the optimal price for a given q in period n. The objective is to determine
n n
n
the optimal decision variables pw and qw for all n, which jointly maximize PM for a given i . Note that if
n
n
N
N
w
PM (i ))0 then the optimal decision would be not to do business.
n N
146
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
4. Model solution
In order to determine the optimal procurement policy for period n, we need to characterize the form of
M w satis"es
M
M w(i , q ). First, we note from Eqs. (3) and (5) that M
n n n
n
M
M w(i , q )"MM w(0, q )#ci ,
n n n
n
n
n
M w(0, q ). For that
so that MM w is additively separable in i and q . Hence, it su$ces to characterize the form of M
n
n
n
n
n
purpose, we need to solve the maximization problem:
G
M
M w(0, q )"max (p #r!c)q !rXM (p )! (p #r#h)H(p , q )# aPM w (0)[1!F(q ; p )]
n
n
n
n
n
n
n n
n~1
n n
qn~L(pn)
PM w (x) f (q !x; p ) dx: p 3[Pl, P ] .
#a
(6)
n~1
n
n
n
6
0
It turns out that the solution methods that have been developed for the "xed price (i.e., Pl"P ) versions of
6
the model [20}23] are not directly applicable for the solution of the problem at hand. Instead, we will follow
a new solution approach which follows the inductive setting described in [21].
P
H
4.1. Form of the optimal procurement policy
The dynamic programming problem represented by Eq. (6) is worked out in the appendix. The results are
summarized below in a theorem and a corollary after the following de"nitions.
De5nition. Let G(i, p, q)"(p#r!c)q!rXM (p)!(p#r#h!ac)H(p, q)#ci, and Gw(0, q)"maxMG(0, p, q):
p3[Pl, P ]N: then, we de"ne the critical inventory levels 0)sm(s (Sl(sl(S (s6(S6 as shown in
g
g
g
g
g
g
g
6
Fig. 1.
Note that the generic function G represents the expected pro"t in a single-period, lost-sales model where
the unit salvage value is c (i.e., the leftovers are returned to the supplier at the original cost). It will play
a critical role in the characterization of MM w, and in establishing bounds on the optimal control parameters.
n
The following theorem describes the form of M
M w in general:
n
Theorem 1. ¹here exist an even integer k and critical inventory levels, s1(s2(2(skn~1(S (skn, de,ned by
n
n
n
n
n
n
S "argmaxMM
M w(0, q): 0)q(RN,
n
n
M w(0, S )!KN,
M w(0, q)*M
s1"minMq: M
n
n
n
n
M w(0, q)"M
sj "minMq: q'sj~1, M
M w(0, S )!KN for j"2, 3,2, k ,
n
n
n
n
n
n
such that if M
M w(0, q ) is unimodal, then for n*2:
1
1
(a) M
M w(0, q )"Gw(0, q )#aMM w (0, s1 ), ∀q 3[0, s1 ],
n~1
n~1
n
n
n
n~1
n
(b) M
M w(0, q ))Gw(0, q )#aMM w (0, s1 )#aK, ∀q 3[s1 ,R),
n~1
n~1
n
n
n
n~1
n
(c) M
M w(0, q )*Gw(0, q )#aM
M w (0, s1 ), ∀q 3[s1 , skn~1 ],
n~1 n~1
n~1
n
n
n
n~1
n
M w(0, q )*M
M w(0, q@ )!aK, for any q@ 3(q ,R),
(d) ∀q 3[S , skn~1 ], M
n
n
n
n
n
n
n
g n~1
(e) M
M w(0, 0)(R,
n
M
M w(0, q )"!R.
(f ) lim
n
qn?= n
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
147
Fig. 1. De"nitions of the critical inventory levels under Gw(0, q).
Corollary 1. For n*2, sm(s1, skn~1(sl, s6(skn, and Sl(S (S6; also, if s (s1, then s (s1.
n n
g
n
1
g
g
g
g g
n
g
n
Regarding the condition of unimodality of M
M w(0, q ) in Theorem 1, there are analytical di$culties
1
1
in proving this property, except for some special cases [3,5,8,12,19,24]. For instance, it is reported in
w
[19] that M
M (0, q ) is unimodal if (1) demand is deterministic, (2) demand is additive, e has a uniform
1
1
distribution and XM (p) is linear, or (3) demand is multiplicative, e has an exponential distribution and
XM (p) is linear. The discussions of unimodality and su$cient conditions can be found in the above-cited
references.
Since G is a special case of M
M (with v"c), under the unimodality assumption, Gw is also a unimodal
1
function with a maximizer at S and a reorder point at s ((S ). Thus, it follows from (a) and Corollary
g
g
g
w
1 (s1 (S ) that MM (0, q ) is an increasing function of q on [0, s1 ]. Furthermore, (a) implies that the
n~1
n
n
n
n~1
g
optimal pricing decision, p (q ), for q 3[0, s1 ] is given by argmaxMG(0, p, q ): p3[Pl, P ]N, the maximizing
n~1
n
n n
n
6
price under the generic single-period model. (Note that if q )s1 , there will be an order placed in period
n~1
n
n!1; hence, the inventory carried over to period n!1 would worth ac, and in this case the pricing decision
is based only on G.)
It is implied by (b) and (c) that MM w(0, q ) is bounded by two unimodal functions which are at most aK
n
n
apart on [s1 , skn~1 ]. Hence, (b) and (c) establish bounds on M
M w(0, q ) based on the value of the optimal
n~1 n~1
n
n
(n!1)-period pro"t, provided that an order takes place in period n!1. In this case, the contribution of the
w
w
current period's pro"t is limited to between G (0, q ) and G (0, q )!aK.
n
n
148
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
Fig. 2. A general n-period expected pseudo-pro"t function, MM w(0, q ) vs. q , for n*2 where k "6.
n
n
n
n
M w (0, s1 ); hence, properties (b) and (c) can be rewritten
It follows from (a) that M
M w(0, 0)"Gw(0, 0)#aM
n~1
n
n~1
w
independent of M
M
(0, s1 ) as
n~1
n~1
M w(0, 0))Gw(0, q )!Gw(0, 0)#aK, ∀q 3[s1 ,R),
(b@) M
M w(0, q )!M
(7)
n
n
n
n~1
n
n
w
w
w
w
M (0, 0)*G (0, q )!G (0, 0), ∀q 3[s1 , skn~1 ].
(c@) M
M (0, q )!M
n
n
n
n~1 n~1
n
n
Thus, under the optimal solution, the increase in expected n-period pro"t, upon raising the stock level from
0 to q , must be greater than or equal to the change in one-period expected pro"t (with v"c) under a similar
n
stock movement (i.e., Gw(0, q )!Gw(0, 0)). This increase, however, is limited by Gw(0, q )!Gw(0, 0)#aK.
n
n
Property (b) extends the upper bound function Gw(0, q )!aM
M w (0, s1 )#aK over [skn~1 , R). On the
n~1
n~1
n~1
n
w
other hand, it follows from (d) that M
M (0, q ) is aK-decreasing ([22]) over [S , skn~1 ], and the functional value
n
n
g n~1
w
w
of M
M , evaluated at any point beyond skn~1 , is strictly below the global maximum of M
M (0, S ).
n
n~1
n
n
A general expected n-period pseudo-pro"t function, as characterized by Theorem 1, is depicted in Fig. 2. It
is seen that there could be more than one order-up-to level, and for each order-up-to level there could be
more than one reorder interval. This characterization is weaker than that in the "xed-price model, where it
can be shown that in each period a single optimal reorder-point, order-up-to-level policy exists [21].
The possible presence of multiple order-up-to levels poses serious di$culties in characterizing and
computing the optimal procurement policies. These di$culties, however, can be circumvented if the
beginning inventory level before ordering in an arbitrary period n is less than or equal to skn. Then, since
n
M
M w(0, 0) is "nite and M
M w(0, q ) tends to decline as q tends to in"nity, the optimal procurement policy can be
n
n
n
n
characterized as follows provided that i )skn:
n
n
S if i 3O ,
w
n
n
q " n
(8)
n
i
if i 3OM ,
n
n
n
for all n, where O , the `reorder regiona, is the union of reorder intervals, given by
n
[0, s1]X[s2, s3]X2X[skn~2, skn~1] and OM is the complement of O with respect to [0, skn]. Note that under
n
n
n
n
n
n n
n
the unimodality condition imposed on M
M w, we have k "2 and qw is determined by an (s , S ) policy.
1
1
1
1 1
In implementing the above optimal procurement policy, we need to compute skn for each period, which can
n
be a computational burden. Instead, a stronger but more useful condition can be established by referring to
Corollary 1, where we have s6)skn. That is, we replace the condition of the above optimal policy (i )skn) by
n
n
n
g
G
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
149
i )s6 which involves the computation of s6 once and in advance by using the problem primitives. Also, it is
g
g
n
seen that the condition i )skn is satis"ed if S (s6 for all n and i )S .
g
N
N
n
n
n
The fundamental di!erence between the (s , S ) policies de"ned for the "xed-price models [20,21] and the
n n
policy we de"ne in Eq. (8) is that, at each decision epoch, the vendor has to make a pricing decision whether
or not it decides to place an order. Thus, the vendor needs to determine the optimal price, p (i ), at the
n n
beginning of each review period, given the observed value of i . This requires the preparation of price tables
n
for the vendor to read the optimal prices from. These tables can be prepared simultaneously during the
computation of reorder points and the order-up-to level in each period.
4.2. A suzcient condition for a single reorder point
Properties (b) and (c) in Theorem 1 indicate that M
M w(0, q ) is aK-increasing over [s1 , sl]. Theoretically,
n~1 g
n
n
this allows for more than one reorder point. However, operating a system under multiple reorder points
could be a burden in practice. Determination of the critical levels s1, s2,2, skn for all periods with su$cient
n
n n
accuracy could be di$cult. It is, therefore, important to know under what conditions there exists a single
reorder point (i.e., k "2).
n
It follows from property (a) in Theorem 1 that M
M w(0, q ) is increasing over [0, s1 ]; thus, if s1)s1 , that
n~1
n
n~1
n
n
w
w
is if M
M (0, s1 )*M
M (0, S )!K, then k "2 and s1 is the only reorder point. However, this is not
n~1
n
n
n
n
n
a convenient su$cient condition, since, under a general demand distribution, the value of s1 , or
n~1
M
M w(0, s1 ), is not analytically measurable with su$cient accuracy.
n
n~1
w
l
Another possibility is to show that M
M (0, q ) is increasing over [s1 , s ], implying k "2 under Theorem
n~1 g
n
n
n
1. (skn~1)sl for all n; see Corollary 1.) To this end, we assume that MM w (0, q ) is increasing over
g
n~1
n~1
n l
w
w
M (0, q ))MM (0, q@ ) for s1 )q
[s1 , s ], and investigate the su$cient conditions under which M
n~1
n
n~1 g
n
n
n
n
(q@ )sl with arbitrary q and q@ . This leads to the following result:
n
n
g
n
Corollary 2. MM w(0, q ) is non-decreasing in q over [s1 , sl] for n*1, that is there exists a single reorder point
n~1 g
n
n
n
(k "2), if sl)S and for all q 3[sm, sl]:
g g
g
1
n
n
p #r!c
n
, ∀p 3Mp : p#h!ac'0, Pl)p)P N.
F(q ; p ))
n
6
n n
p #r#h!ac
n
(9)
The RHS in Eq. (9) is a concave increasing function of p , and F(q ; p ) is assumed to be a non-decreasing
n
n n
function of p for all q (see Section 2).
n
n
In view of Eq. (9), we can make the following observations. If the vendor administers a higher price, he is
able to increase F(q ; p ), the likelihood that there will be no shortage at level q , to a desired level, while the
n n
n
relative weight of unit underage cost rises (relative weight of unit overage cost declines). Hence, the vendor is
inclined to increase the stock level to above q and this continues until a break-even point is reached. This is
n
veri"ed by the fact that LG(0, p, q)/Lq*0 under Eq. (9). Thus, based on the demand distribution and unit
costs that prevail in the market, Eq. (9) re#ects the vendor's capability to increase the expected current period
pro"t by increasing the stock level to above sl. In other words, Eq. (9) attributes a higher monopoly power to
g
the vendor than the vendor would have in its absence. In this interpretation, we utilized the myopic notion
that the higher the monopoly power, the less responsive is the demand to changes in price. That is,
probability of satisfying the demand fully, F(q ; p ), does not respond strongly to an increase in price, since
n n
the increase in F(q ; p ) is limited by the RHS in Eq. (9).
n n
Under Theorem 1 and Corollary 2, the optimal procurement policy is de"ned by
qw"i #(S !i )d(s1!i ),
n
n
n
n
n n
(10)
150
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
provided that i )s6 is satis"ed for n'1. It also follows that sl)S for n*1, such that sl is a lower bound
g
n
g
g
n
on order-up-to levels.
Note that in order to have k "2, M
M w(0, q ) need not be increasing over [s1 , sl]. There could be other
n~1 g
n
n
n
conditions leading to the same result. Our experience with a number of numerical examples suggests that
cases with k '2 would be rare in practice.
n
4.3. Suzcient conditions for the optimality of (s , S ) policies
n n
If, in property (d), the range for q were established as [S ,R), then M
M w(0, q ) would be characterized as
n
n
n
g
a K-decreasing function over [S ,R), S would be the only order-up-to level, and there would be no need for
g
n
the condition i )skn in Eq. (8) or Eq. (10). This is precisely the case in the development of the optimality of
n
n
(s , S ) policies under the "xed-price (Pl"P ) model [21].
n n
6
In the following corollary, we investigate the su$cient conditions under which MM w assumes a desirable
n
shape over [S ,R) to yield a single order-up-to level.
g
Corollary 3. M
M w(0, q )*MM w(0, q@ )!aK for all q and q@ with S )q (q@ , if s )S and
n
1
g
n
g
n
n
n
n
n
n
(p #r#h!ac)F(q ; p )*p #r!c, ∀p 3[Pl, P ], for n*1.
6
n
n n
n
n
(11)
Note that for large q values, F(q; p) tends to 1 (particularly, F(q; p)"1 when q*;(p)), and this complies
well with condition Eq. (11). That is, M
M w(0, q ) is K-decreasing in the limit as q tends to in"nity. Also, it is
n
n
n
intuitive that at considerably large inventory levels, the vendor would tend to reduce the price substantially,
even below the unit procurement cost, in order to deplete inventories. In this regard, it is seen in Eq. (11) that
the condition tends to hold when c and h get larger and r gets smaller (it holds at all q levels when
p#r!c)0)p#r#h!ac, that is, ac!h!r)p)c!r). In general, the condition tends to hold
when c and h get larger or r gets smaller.
Thus, under Theorem 1 and Corollary 3, the optimal procurement policy is characterized by
G
S if i 3O ,
n
n
qw" n
n
i
if i 3OM or i 'skn.
n
n
n
n
n
Also, under both Corollaries 2 and 3, it follows from Theorem 1 that k "2 for all n and qw is determined by
n
n
an (s , S ) policy: qw"i #(S !i )d(s1!i ).
n
n n
n
n
n
n n
4.4. Numerical example
To demonstrate our "ndings, we consider a 5-period problem de"ned by the base parameter set
c"0.5, r"1.5, h"0.4, v"0.1, K"5, a"0.95; the expected demand function XM (p)"ae~bp with a"150
and b"0.2; the price limits Pl"0.1 and P "10; and the additive-uniform demand distribution
6
F(x; p)"0.5(x!XM (p)#j)/j for !j)x!XM (p))j with j"20. (j"0 corresponds to the deterministic
demand case.) For sensitivity analyses, we used multiple parameter values as h"0.02, 0.1, 0.4;
K"1, 5, 10; b"0.05, 0.1, 0.2; and j"0, 5, 10, 20.
M w,2, M
M w over the
Using a computer program, we computed the expected pseudo-pro"t functions M
M w, M
1
2
5
q range of [0, 250] with a step size of 1. The resulting optimal values of the control parameters and the
n
critical functional values are shown in Tables 1 and 2. Also, the pseudo-pro"t functions computed for
j"0, K"10, the deterministic demand case, and j"10, K"10 are plotted in Figs. 3 and 4.
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
151
Table 1
Critical inventory levels, prices and pseudo-pro"t values under the 5-period example problem
K
j
b
h
n
s
n
S
n
M
M w(0, S )
n
n
p (S )
n n
10
20
0.2
0.4
10
20
0.1
0.4
10
20
0.05
0.4
10
20
0.2
0.1
10
20
0.2
0.02
5
20
0.2
0.4
1
20
0.2
0.4
10
10
0.2
0.4
10
5
0.2
0.4
10
0
0.2
0.4
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
49.6
51.1
51.1
51.1
51.1
64.2
65.3
65.2
65.2
65.2
100.0
100.9
100.9
100.9
100.9
50.8
52.4
53.2
52.4
53.1
51.1
56.2
56.8
55.6
55.5
54.2
55.7
55.7
55.7
55.7
60.5
62.1
62.1
62.1
62.1
43.0
43.7
43.7
43.7
43.7
39.7
40.0
40.0
40.0
40.0
36.5
36.5
36.5
36.5
36.5
65.7
67.5
67.5
67.5
67.5
72.5
73.6
73.6
73.6
73.6
108.3
109.3
109.3
109.3
109.3
67.1
110.4
125.8
125.4
125.0
67.5
119.9
161.4
196.3
133.0
65.7
67.5
67.5
67.5
67.5
65.7
67.5
67.5
67.5
67.5
57.8
58.7
58.7
58.7
58.7
53.9
54.2
54.2
54.2
54.2
49.9
49.9
49.9
49.9
49.9
235.3
455.8
665.2
864.1
1053.1
509.2
990.5
1447.7
1882.0
2294.6
849.3
1654.8
2420.1
3147.1
3837.7
240.3
466.1
681.5
885.3
1079.7
241.7
472.7
692.8
900.7
1098.1
235.3
460.5
674.4
877.7
1070.7
235.3
464.3
681.9
888.5
1084.9
242.5
466.7
679.7
882.0
1074.2
246.1
472.7
688.0
892.6
1086.9
249.7
477.3
693.6
899.1
1094.3
5.5
5.5
5.5
5.5
5.5
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.6
5.5
5.5
5.5
5.5
5.6
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
152
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
Table 2
Critical inventory levels that are obtained under the generic pseudo-pro"t function
K
j
b
h
sm
g
s
g
Sl
g
sl
g
S
g
s6
g
S6
g
Gw(0, S )
g
10
10
10
10
10
5
1
10
10
20
20
20
20
20
20
20
10
5
0.2
0.1
0.05
0.2
0.2
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.1
0.02
0.4
0.4
0.4
0.4
45.0
62.0
97.8
46.2
46.5
51.3
60.1
38.2
34.9
51.1
65.3
101.1
52.4
52.7
55.7
62.1
43.7
40.1
51.5
65.5
101.3
52.8
53.1
56.0
62.3
44.1
40.4
63.7
71.8
107.6
65.3
65.7
64.8
66.3
55.2
51.0
67.5
73.7
109.5
69.2
69.7
67.5
67.5
58.7
54.3
71.5
75.7
111.5
74.2
81.1
70.3
68.8
62.3
57.7
93.2
96.8
132.6
146.2
281.1
82.1
73.0
83.9
79.2
241.7
516.0
856.1
247.2
248.8
241.7
241.7
245.7
247.7
We observe that the optimal control values are quite sensitive to the price senstivity of demand (note that
b is the price elasticity multiplier of the expected demand). The more sensitive the expected demand to price,
the less is the expected pro"t. As for the holding cost, smaller h values tend to yield considerably higher
order-up-to levels.
Comparing the critical inventory levels listed in Tables 1 and 2, we see that the conditions given in
Corollary 1 hold in each case. We also note that S )s6 in all cases except h"0.1 and h"0.02, for which the
g
n
reorder-region order-up-to level policy will be optimal.
Table 1 shows that the optimal prices evaluated at respective order-up-to levels are the same in all periods.
Consequently, if an order is to be placed in each period, then the optimal prices in successive periods will be
constant. We also observe that demand uncertainty has considerable impact on the parameters of the
optimal procurement policy. As the demand variance is decreased (i.e., j is decreased), the order-up-to and
reorder levels both tend to be lower, and the vendor foresees higher expected pro"ts.
On the other hand, on comparing the deterministic and probabilistic pseudo-pro"t curves in Fig. 3, we
"nd that the deterministic values are greater than the corresponding probabilistic values, especially, about
the q mid-range, which includes respective order-up-to levels.
n
In Fig. 4, we plot M
M w(0, q )!MM w(0, 0) vs. q together with the boundary functions. It is seen that the
n
n
n
n
properties listed in Theorem 1 and Corollary 1 hold. The boundary functions established in Eq. (7) also
satisfy the conditions (b@) and (c@). We also observe that the pseudo-pro"t curves exhibit considerable slope
changes across the q mid-range.
n
The best price trajectories p (q ), evaluated for n"1, 2,2, 5, are plotted in Fig. 5. The "gure shows that
n n
the optimal pricing decision at a given inventory level is not necessarily trivial. We observe that p (q ) is
n n
generally decreasing in q , but it can exhibit mild increases, or stagnate (become `stickya) over various
n
q ranges where it does not respond to changes in stock levels. Also, on comparing the deterministic and
n
probabilistic optimal prices, we observe that it is not necessarily true that the deterministic prices are lower
than the probabilistic prices or vice versa (cf. [4,5]).
Intuitively, price should be lower at higher stock levels, so as to facilitate higher demand to deplete the
excess inventories. However, the vendor would still tend to increase the price for two basic reasons: to shrink
the demand so that there will be leftovers for future periods, or to realize a higher pro"t per unit sold. If the
"xed ordering cost in a future period is to be saved by satisfying that period's demand by carying a part of the
current inventory forward, then both of these reasons will prevail. Once the total cost of inventory holding
and the di!erential cost of procurement breaks even with the savings in the "xed ordering cost, however, it
would not be pro"table to consider carrying stock into future periods. As n gets larger, there are more periods
ahead to consider; thus, there could be a recurrence of this break-even behaviour at higher q levels. This can
n
be seen in Fig. 5 by comparing p (q ) with p (q ). The passage from one pricing regime to another, at
5 5
2 2
a break-even point, need not be smooth and there could be sudden changes in p (q ).
n n
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
153
Fig. 3. Expected pseudo-pro"t curves, MM w(0, q ) vs. q , that are obtained for the 5-period example problem in Section 5.4
n
n
n
(c"0.5, r"1.5, h"0.02, b"0.2, K"10, j"10.0). The thin curves represent the deterministic case where j"0.
154
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
Fig. 4. Combined representation of the curves in Fig. 3. The thin curves represent MM w(0, q )!MM w(0, 0) vs. q for n*1 and the thick
n
n
n
n
curves represent the boundary functions Gw(0, q )!Gw(0, 0) and Gw(0, q )!Gw(0, 0)#K.
n
n
5. In5nite-horizon model
In this section, we study the in"nite-horizon version of the model. We drop the period indices from the
notation and rewrite the expected pseudo-pro"t function from Eq. (6) as
q~L(p)
[MM w(0, x)!M
M w(0, s1)#a
M w(0, s1)]` f (q!x; p) dx: p3[Pl, P ]
M
M w(0, q)"max G(0, p, q)#aM
6
0
P
G
H
q~L(pq)
[M
M w(0, x)!M
M w(0, s1)]` f (q!x; p ) dx,
(12)
"G(0, p , q)#aM
M w(0, s1)#a
q
q
0
where p ,p(q), the maximizing price at q.
q
The solution for M
M w(0, q) involves the integral in Eq. (12) which represents the convolution of the density
function with a "ltering function based on M
M w(0, q) (i.e., the function MM w(0, q) is "ltered over the level
w
M
M (0, s1)). Because of this, we need to solve the integral Eq. (12) recursively. In each stage of the solution,
M
M w(0, q) is established over a non-overlapping interval, and the solution obtained is employed in the next
stage.
P
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
Fig. 5. Optimal price trajectories that are obtained for the 5-period example problem shown in Fig. 3.
155
156
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
It follows from Eq. (12) that for 0)q)s1 we have M
M w(0, q)"Gw(0, q)#aM
M w(0, s1), which is an
w
increasing function of q over [0, s1] where s1(S . Evaluating M
M (0, q) at s1, we obtain
g
M
M w(0, s1)"Gw(0, s1)/(1!a), which implies
a
M
M w(0, q)"Gw(0, q)#
Gw(0, s1),
I
1!a
(13)
where the index in roman numeral denotes the stage number. Successive stages I, II, III, 2 correspond to
intervals [0, s1], [s1, s2],[s2, s3],2, respectively.
In the next stage, we have s1)q)s2 for which the e!ective integration range starts at s1 in Eq. (12) and
we have
q~L(pq)
w
w
M w(0, s1)] f (q!x; p ) dx
[M
M w(0, x)!M
M
M (0, q)"G(0, p , q)#aMM (0, s1)# a
II
q
II
q
1
s
q~L(pq)
w
M
M w(0, x) f (q!x; p ) dx,
(14)
"G(0, p , q)#a[1!F(q!s1; p )]M
M (0, s1)#a
II
q
q
q
s1
which is a renewal equation [25] with solution
P
P
G
C
a
M
M w(0, q)"max G(0, p, q)#Gw(0, s1)
!R (q!s1; p)
II
a
1!a
D
H
q~L(p)
G(0, p, x) dR (q!x; p): p3[Pl, P ] ,
#
a
6
1
s
where R (x; p)"+= aiF(i)(x; p) and F(i) is the i-fold convolution of F with itself. (Derivation is given in the
i/1
a
appendix.)
Having established M
M w(0, q) over [0, s2], we check whether s6(s2 where s6 is a lower bound on sk (see
g
g
Corollary 1). If s6)s2, then k"2 and we stop. In this case, there will be one reorder point. If s2(s6,
g
g
however, then we need to consider the q range beyond s2 and pass to the next stage.
Evaluating Eq. (12) for s2)q)s3, we obtain
P
s2
P
M w(0, s1)] f (q!x; p )
M w(0, x)!M
M
M w (0, q)"G(0, p , q)#aMM w(0, s1)# a [M
III
q
II
q
1
s
s2
a
Gw(0, s1)[1#F(q!s2; p )!F(q!s1; p )]#a M
M w(0, x) f (q!x; p ) dx.
"G(0, p , q)#
q
II
q
q
q
1!a
1
s
Since M
M w inside the integral is obtained earlier, it can be substituted in the above to solve for M
M w.
II
III
The fourth stage will involve another renewal equation. For s3)q)s4, we have
P
s2
P
M w(0, s1)# a [M
M w(0, x)!MM w(0, s1)] f (q!x; p )
M
M w (0, q)"G(0, p , q)#aM
IV
q
II
q
1
s
q~L(pq)
#a
[M
M w (0, x)!M
M w(0, s1)] f (q!x; p )
q
IV
3
s
P
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
157
a
"G(0, p , q)#
Gw(0, s1)[1#F(q!s2; p )!F(q!s1; p )! F(q!s3; p )]
q
q
q
q
1!a
s2
q~L(pq)
#a M
M (0, x) f (q!x; p ) dx# a
M
M w (0, x) f (q!x; p ) dx,
II
q
IV
q
1
3
s
s
P
w
P
which is a renewal equation on MM w . The solution for this equation can be obtained by following a similar
IV
procedure to the one used in solving for M
M w(0, q).
II
M w in the next two
If s4)s6, then we stop. Otherwise, we follow the same solution process to establish M
g
stages, and check the stopping criterion.
This method works in principle. However, there is a computational di$culty due to the fact that critical
inventory levels s1, s2, s3,2 are unknown in advance. They have to be computed simultaneously as we
establish M
M w numerically.
One approach is to start with the assumption that s1"s . Based on this initial value of s1, M
M w can be
g
w
established over [0, s2] and the value of s2 can be found by incrementing the value of q until M
M (0, q) equals
II
M
M w(0, s1). Note that s2 will be the "rst point greater than s1 satisfying the above equality. Likewise, the
II
process can be continued to include s3 and s4 and so on. Once the stopping condition (s6)sk) is satis"ed,
g
using the current solution of M
M w, s1 can be updated. If it is tolerably close to its previous value, the procedure
w
can be terminated. Otherwise, based on its updated value, a new M
M will be established over [0, sk].
It is beyond the scope of this paper to investigate the convergence of the numerical method proposed
above. To demonstrate its merits, however, we use it to obtain the solution for an in"nite-horizon problem
de"ned by the parameter set c"0.5, r"0.25, h"0.3, v"0.1, K"8, a"0.7; the expected demand function XM (p)"150 exp(!0.5p); the price limits Pl"0.1 and P "4; and the multiplicative-exponential demand
6
distribution F(x; p)"1!exp(!x/XM (p)), x*0, under which the discounted renewal function is given by
R (x; p)"a[1!exp(!(1!a)x/XM (p))]/(1!a).
a
We obtained the solution in 10 iterations. The successive s1 values were 28.20, 35.25, 31.26, 33.40, 32.15,
32.87, 32.44, 32.70, 32.55 and 32.64 until we reached a tolerance of 0.1 units. In this example problem, it was
found that k"2; hence, it was not necessary to pass beyond the second stage in computations.
M w, we solved a 15-period problem under the
In order to demonstrate the transition from M
M w to M
1
same problem data, and compared the resultant expected pseudo-pro"t functions with the function
we obtained by solving the in"nite horizon problem. Fig. 6 depicts M
M w(0, q ) functions for n"1, 2,2, 15
n
n
w
and M
M , which become indistinguishable from the pseudo-pro"t functions for large n, as predicted by the
theory.
6. No-5xed-ordering-cost
In order to establish a link between this paper and the earlier studies reported in the literature, here we take
a brief look at the model with K"0.
Following a similar inductive approach to the one we used previously, we suppose that M
M w(0, q ) is
1
1
w
unimodal and PM
(i ) is determined by a single-parameter policy:
n~1 n~1
M
M w (0,
n~1
PM w (i )"ci #
n~1 n~1
n~1
M
M w (0,
n~1
G
S ), i )S ,
n~1
n~1
n~1
i ), S (i .
n~1
n~1
n~1
158
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
Fig. 6. Expected pseudo-pro"t curves, M
M w(0, q ) vs. q , that are evaluated for the 15-period problem speci"ed in Section 6. The curve at
n
n
n
the top represents the solution for the in"nite horizon model with the same parameter set. (It coincides with the expected pseudo-pro"t
curves at large n.)
Under this setting, it follows from Eq. (6) that
G
M
M w(0, q )"max G(0, p , q )#aM
M w (0, S )
n
n
n n
n~1
n~1
qn~L(pn)
M w (0, S )] f (q !x; p ) dx: p 3[Pl, P ] .
[M
M w (0, x)! M
(15)
#a
n~1
n~1
n~1
n
n
n
6
Sn~1
It can be seen that for q )S , the integral drops out and we have
n
n~1
M
M w(0, q )"Gw(0, q )#aM
M w (0, S ),
n
n
n~1
n~1
n
where Gw(0, q ) is a unimodal function. For S (q , however, we note that the integral in Eq. (15) is
n
n~1
n
negative-valued. Based on these observations, we establish M
M w(0, q ) in two alternative ways, as shown in
n
n
Fig. 7. Considering Eq. (15) and Fig. 7, we discover that
P
H
minMS , S N)S )S ,
(16)
g n~1
n
g
for n*2, and the con"guration shown in Fig. 7a can only occur for n"2.
The recursive ordering in Eq. (16) implies that if S )S , then S "S for all n*2. It is clear that if v"c,
g
1
n
g
then M
M ,G, and S "S by de"nition. But, it is more plausible, in general, that v(c, in which case S will
1
1
g
1
be di!erent from S .
g
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
159
Fig. 7. Construction of M
M w(0, q ) vs. q for K"0; dashed curves represent Gw(0, q ) for: (a) S (S , and (b) S (S .
n
n
n
n
g
n~1
n~1
g
Intuitively, when the price is "xed, S must be greater than S . Having the option of salvaging the leftovers
g
1
at a higher return, the decision maker will be encouraged to procure up to higher levels. But, under the
impact of pricing, there could be a trade-o! between small-quantity high-price and large-quantity low-price
scenarios. Demand uncertainty at a given level will then be contingent on the pricing decision. For this
reason, under a general price-demand relationship, it is theoretically possible that S (S .
g
1
Fig. 7b shows that the expected pro"t level will be higher if S
shifts towards S and the contribution of
n~1
g
G is more than the contribution of the negative valued integral in Eq. (15). Thus, overcoming the transient
behaviour represented by Eq. (16), there will be a propensity, under the optimal solution, for S to approach
n
S as the number of periods is increased.
g
Su$cient conditions for the optimality of a single-critical-number policy are given in [11]. In principle,
these conditions can be utilized to verify the form of the optimal procurement policy, but they are di$cult to
interpret in terms of their managerial implications. Under some restrictions, Thowsen [11] shows that the
single-parameter policy will be optimal if the demand is additive with a PF (Po& lya frequency function of type
2
2) distributed random term and a linear expected demand function. Also, su$cient conditions for the
existence and uniqueness of the optimal decision variables are given in [9,10]. The characterization
portrayed in these studies are based on taking derivatives. The drawback of that approach is that it can not
be immediately extended to the model with "xed ordering costs, where di!erentiability is restricted to
sub-regions.
160
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
7. Non-stationary extensions
In addition to being dependent on price, it is possible that the demand distribution exhibits periodic shifts
during the planning horizon due to seasonalities or changing market conditions. The natural way of
incorporating such time-variability in our model would be to represent the demand in period n by a random
variable X (p ) with distribution F (x; p ). The generic function, now denoted by G , and the critical
n n
n
n
n
parameters de"ned for Gw, sm (2(S6 , would also be period-dependent.
n gn
gn
By virtue of the fact that the demand distribution depends on price and the price can be set at the
beginning of each period, extensions to time-dependent demand are immediate. It can be seen, going through
the proofs in the appendix, that this does not alter the arguments and derivations, provided that Gw(0, q ) is
n
n
unimodal and the condition s6 )skn~1 holds for all n'1. This condition parallels the "xed-price version of
gn
n~1
the non-stationary demand model [21,23] where it is assumed that S )S
for n'1. It can also be seen
gn
gn~1
that if S )s1 , then k "2.
n~1
n
gn
Similarly, we can let the cost parameters and the discount factor depend on the period (i.e.,
c , r , h , K , a ), without any di$culty, provided that K *a K
(a "1), and M
M w is unimodal for
1
n n n
n n
n
n~1 n~1 0
w
n'1 (which implies unimodality of G ).
n
8. Deterministic demand
Under deterministic demand, X (p)"XM (p) for all n. For technical reasons, we will assume that X (p) is
n
n
n
o(1/p) as pPR and as pP0`. Under this assumption, the riskless revenue function R (p)"pX (p) starts
n
n
growing at 0 when p"0, reaches a maximum level, and declines down to 0 as p tends to in"nity. We further
assume that R (p) is a pseudoconcave function of p on (0,R) so that there is a unique breakeven point
n
between low-price-high-demand and high-price-low-demand situations in terms of revenue. It is shown in
[19] that if X (p) is a convex or concave decreasing function, then R (p) is pseudoconcave on (0,R).
n
n
A detailed discussion of the properties of the riskles revenue function and related functions can be seen in
[7,19].
Using X (p)"XM (p) in Eq. (3) we obtain
n
n
M
M (i , p , q )"G (i , p ,
Optimal procurement policies under price-dependent demand
Hakan Polatoglu!,*, Izzet Sahin"
! Frank Sawyer School of Management, Suwolk University, Boston, MA 02108, USA
" School of Business Administration, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA
Received 19 June 1998; accepted 17 September 1998
Abstract
We study a periodic-review inventory model where, in addition to the procurement quantity, price is also a decision
variable. We develop a model where demand in each period is a random variable having a price- and, possibly,
period-dependent probability distribution, with the expected demand decreasing in price. The model includes price limits
and "xed ordering costs in addition to unit procurement holding and shortage costs. We study the optimal policies which
jointly maximize the discounted expected pro"t over a "nite planning horizon. We characterize the form of the optimal
procurement policy under a general price}demand relationship and give a su$cient condition for it to be (s , S ) type. We
n n
also discuss some special cases and extensions to the basic model, including the in"nite horizon problem. ( 2000
Elsevier Science B.V. All rights reserved.
Keywords: Pricing; Inventory; (s, S) policy
Notation
n
N
i
n
q
n
p
n
Pl
P
6
c
r
h
v
K
n
period index (n"1 corresponds to the last period)
total number of periods in the planning horizon
beginning inventory level before ordering in period n
beginning inventory level after ordering in period n
price in period n
price #oor
price ceiling
unit procurement cost
unit shortage cost
unit holding cost
unit salvage value at the end of the planning horizon
"xed ordering cost in period n
* Corresponding author. Protek Computer Systems Inc., Perpa Ticaret Merkezi, Elektrokent, Kat 13, No 1981, Okmeydani, 80270
Istanbul, Turkey. Tel:. #90 (212) 210-1730; fax: #90 (212) 210-1765.
E-mail: Hakan.Polatoglu@protek.com.tr (H. Polatoglu)
0925-5273/00/$ } see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 5 2 7 3 ( 9 8 ) 0 0 2 4 0 - 0
142
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
a
X(p)
XM (p)
¸(p)
;(p)
f (x; p)
F(x; p)
p (q )
n n
M (i , p , q )
n n n n
periodic discount factor
random demand when price is p
expected demand as a function of price ("E[X(p)])
lower bound on random demand, 0)¸(p))X(p)
upper bound on random demand, X(p));(p)(R
probability density function of X(p)
probability distribution of X(p)
optimal price when the inventory level is q in period n
n
n-period (periods n}1) pseudo-pro"t function (pro"t excluding the e!ect of the "xed ordering
cost)
M
M (i , p , q ) expected value of M (i , p , q )
n n n n
n n n n
n-period expected pseudo pro"t when the optimal value of p is implemented in period n, i.e.,
M
M w(i , q )
n n n
n
w
M (i , p (q ), q )
M
M (i , q ),M
n n n
n n n n n
PM (i , p )
n-period expected pro"t when the optimal value of q is implemented in period n
n n
n
PM w(i )
n-period expected pro"t when the optimal values of q and p are implemented in period n
n
n
n
d(z)
Heavyside function, d(z)"0 for z)0 and d(z)"1 for z'0
[z]`
positive value function, [z]`"zd(z)
1. Introduction
Under increased competition, inventory-based businesses are forced to better coordinate their procurement and marketing decisions, to avoid carrying excessive stock when sales are low or shortages when they
are high. An e!ective means of such coordination is to conduct the inventory control and pricing decisions
jointly. The main task in doing so is to determine the optimal inventory policy, given the price}demand
relationship that is expected to prevail in the market place in the short term.
In addressing the above issue, we are concerned in this paper with a single-item, periodic-review inventory
system where the vendor, who enjoys a degree of monopoly power in the market place, is in a position to
in#uence demand by its pricing decisions. It is thus confronted with simultaneous pricing and procurement
quantity decisions, which would jointly maximize the present value of expected pro"t over a planning horizon.
We will assume that the ordering policy does not change the demand pattern during the planning horizon.
Ordering cost, shortage cost and temporal increases in the procurement cost are among the important
factors that force the decision maker to carry inventories. Holding cost has the opposite e!ect. Thus, in the
absence of pricing, the main decision problem is to determine the optimal inventory levels to strike a balance
between these opposing factors, under a given demand forecast, cost structure, and a set of operating
conditions.
There are critical di!erences between a "xed-price, periodic-review inventory model [1] and a model that
also includes pricing decisions. The economical interpretations of these di!erences relate to the various ways
that price plays into the decision problem. First, price is a decision variable that determines the revenue per
unit sold. Second, price is a factor that in#uences the demand, thus the period-ending inventory levels. In
addition, when backlogging occurs, apart from the amount of backorders, the backlogging policy must also
account for the price that applies to backorders and the timing of revenue collection. For instance,
backorders could be sold at the current price in advance, or at a future price set at the time of delivery.
Therefore, when the decision maker confronts the pricing and inventory decisions simultaneously,
in addition to the two opposing cost-related e!ects that we noted above, price-related factors must also
be taken into account. Mainly, there will be a trade-o! between the high-price low-demand and low-price
high-demand scenarios, in terms of discounted total revenue. This revenue trade-o!, however, will not be
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
143
independent of the above described cost trade-o!, since the demand levels (i.e., inventory levels) will be
a!ected by the pricing decisions. Unfortunately, this complex relationship between the cost and revenue
trade-o!s does not allow the model to simplify into separate pricing and procurement decisions.
Various versions of the procurement-and-pricing model has been studied in the literature. In his pioneering work, Whitin [2] proposed a link between price theory and inventory control. He noted that such
a model would have stronger managerial implications, as compared to "xed-price models. Later, Mills [3,4]
and Karlin and Carr [5] established a conceptual framework for a general inventory model, where price is
a decision variable. Subsequent studies concentrated mostly on the characterization of the optimal solution
for some special cases of the general model [6}13]. Thomas [14] provided some numerical examples to
demonstrate the nature of the decision problem under the presence of "xed ordering costs. More recently,
Gallego and Van Ryzin [15] studied the continuous-time version of the problem, and Petruzzi [16]
investigated the approximate solutions under a learning approach. Also, under a dynamic model with
Bayesian learning about the demand distribution, Subrahmanyan and Shoemaker [17] studied a number of
numerical examples which provide insights about the sensitivity of optimal prices and inventory levels to
changes in the unit procurement cost, price elasticity of demand, form of the demand distribution and the
expected demand function.
The existing analytical models impose rather restrictive assumptions on the form of the expected demand
function (e.g., concavity assumptions in [9,11,12]), demand distribution (e.g., multiplicative demand in [9],
additive demand in [9,11]), or the cost structure (e.g., parameter restrictions and no "xed ordering cost in
[11,10]) in analyzing the optimal procurement policy. In this paper, by relaxing some of the more limiting
modeling assumptions, we seek to characterize the fundamental properties of the optimal procurement policy
and the resulting expected pro"t in a more general setting (i.e., general demand distribution, linear cost
structure with the addition of set-up cost). We also address the sensitivity of the optimal procurement policy
to the underlying price-dependent demand uncertainty.
In what follows, we describe the price}demand relationship in Section 2, develop the basic model in
Section 3, and characterize its solution in Section 4. The rest of the paper is devoted to special cases and
extensions. We consider the in"nite-horizon problem in Section 5, the model with no "xed ordering costs in
Section 6, the non-stationary extensions of the basic model in Section 7, and the special case with deterministic demand in Section 8. Section 9 is devoted to some concluding remarks. Sections 4 and 5 include
numerical examples. Proofs are given in the appendix.
2. Demand uncertainty
It has been a common practice in demand modeling to express random demand as a combination of an
expected demand function, which exhibits some form of price-dependency, and a random term, which is
price-independent. This approach conveniently isolates the e!ects of price and uncertainty, while retaining
mathematical tractibility. However, it has some shortcomings.
Under the additive model [3,5,9,11], we have X(p)"XM (p)#e where XM (p) is a decreasing function of p and
e is a random variable with E[e]"0. The additive model can also be interpreted as a homoscedastic
regression model where XM (p) represents the regression function and e is the error term. One implication of
this model is that while the expected demand is a function of the price, the demand variance is priceindependent (demand distribution shifts as price varies). Also, the model allows for negative demand, unless
the price values are bounded from above.
The multiplicative model [5,6,8] is another commonly used form where X(p)"eXM (p) with E[e]"1. This
model implies the restriction that the demand equals to the product of its expected value and a random term.
As a result, the coe$cient of variation of demand equals that of e, which is constant in price, and demand
variance decreases at a rate faster than the expected value and it approaches to zero at high prices.
144
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
In addition to these, there are models that represent random demand by a mixture of additive and
multiplicative terms [10,12,13], such as X(p)"XM (p)a (e)#a (e), or X(p)"a (p)#ea (p), where a and
1
2
1
2
1
a are di!erentiable functions.
2
Additional simpli"cation can be achieved by assuming that the price}demand relationship is perfectly
predictable. This leads to the deterministic model, X(p)"XM (p), which serves as a "rst-order approximation,
and which has been utilized as a benchmark in the literature [3,5,9,12]. It has been reported in these studies
that the form of demand uncertainty, whether it is additive or multiplicative, plays a critical role in
determining optimal prices. Optimal prices are lower than their deterministic counterparts under the additive
demand model, while they are found to be greater under the multiplicative demand model. Additional
"ndings about the form of demand uncertainty, as it impacts optimal prices and inventory levels, are reported
in [18] p. 617.
In this study, we represent the demand by a continuous random variable, distributed over the range
[¸(p), ;(p)] with a known density function f (x; p). ¸ and ; are di!erentiable functions which represent the
bounds on demand, where 0)¸(p))X(p));(p)(R for all p. For convenience, we "rst assume stationarity, whereby the demand density is given by f in all periods. We then show in Section 8 that the results we
obtain for the "nite-horizon problem can be extended to non-stationary (period-dependent) demand
distributions.
There is empirical evidence that consumers react to price in purchasing. Other things being equal, a lower
price facilitates demand in a `faira market. Therefore, it is natural to assume that the probability that demand
is less than a given level x, F(x; p)"P[X(p))x], is a non-decreasing function of price. That is,
LF(x; p)
*0 ∀x3(¸(p), ;(p)).
Lp
(1)
It follows from Eq. (1) that F(x; p ))F(x; p ) for p (p . Thus, price induces a stochastic ordering of
1
2
1
2
demand distributions.
The expected demand is given by
=
U(p)
xf (x; p) dx" [1!F(x; p)] dx.
XM (p)"
P
P
(2)
0
L(p)
It also follows from Eqs. (1) and (2) that XM (p) is a decreasing function of p:
=
LF(x; p)
LF(x; p)
dXM (p)
*0N
"!
dx(0.
Lp
Lp
dp
0
P
3. Mathematical model and assumptions
Consider a periodic-review inventory system with N periods where the decision variables are q and p . The
n
n
review periods are linked by period-ending inventory levels such that the leftovers are transferred fully to the
next period and shortages are lost. That is, i "[q !X(p )]` for 0)n)N!1, and i *0 is a given
n
n`1
n`1
N
parameter. At the beginning of a period, the vendor decides how much to order (q !i ) and what price to
n
n
charge until the next decision point. There is a "xed cost of ordering (K), but no cost is assumed for pricing.
Therefore, it is for the vendor's bene"t to reconsider pricing at each decision epoch.
We assume that the vendor has full information about the inventory and procurement costs, and the
demand distributions in all periods of the planning horizon. At the beginning of a review period, say period n,
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
145
given the inventory position, the vendor is to determine the procurement quantity and the price to maximize
the expected n-period pro"t which represents the expected value of the sum of the current period's pro"t and
the discounted optimal expected pro"t to be obtained during the remaining periods.
Let Pnl and Pn be the price #oor and the price ceiling, respectively, in period n, such that the interval
6
[Pnl, Pn ] represents the range of allowable prices. Then, the sequences MP1l , P2l ,2, PNl N and MP1, P2,2, PNN
6
6
6 6
establish limiting-price pro"les during the planning horizon. In regulated markets, these pro"les represent
the temporal rules of regulation. Also, in a case where a manufacturer might issue price limits for its dealers
(price-maintenance), or provide `suggested retail pricesa, the allowable price ranges could be represented by
the limiting-price pro"les.
In the basic model, without loss of generality, we assume uniform pro"les where the feasible price values in
any period are between Pl and P . If there is no price regulation or other constraint on prices, then the price
6
limits may be taken as 0 and p , respectively, where p , referred to as the `null pricea in the literature, is the
=
=
highest possible price level at which there will be no demand, that is, P[X(p))0]"1 for all p*p .
=
Allowing p "R leads to detailed technical considerations (see [8,9]) which we choose to avoid in this paper.
=
We assume that inventory costs are proportional to period-ending inventory levels. The cost and discountfactor parameters may be allowed to change from period to period. However, as in the case of the demand
distribution, we keep them time-invariant for convenience in developing the basic model (see Section 8 for
non-stationary extensions of the basic model). We take P 'c so that it is possible to make pro"t. In
6
addition, the leftovers at the end of the planning horizon are assumed to be salvaged at a discount price, that
is, v)c.
Under these assumptions, the n-period optimal expected pro"t, as a function of i , is obtained from
n
PM w(i )"maxMPM (i , p ): p 3[Pl, P ]N,
n n
n n n n
6
and
PM (i , p )"maxMM
M (i , p , q )!Kd(q !i ): q *i N,
n n n
n n n n
n
n n
n
where M
M is the expected value of the n-period pseudo-pro,t function which is de"ned for n*1 as
n
0)q )X(p ),
aPM w (0)# p q !r(X(p )!q ),
n~1
n n
n
n
n
n
M (i , p , q )"!c(q !i )#
w
n n n n
n
n
aPM
(q !X(p ))# p X(p )!h(q !X(p )), X(p ))q ,
n~1 n
n
n
n
n
n
n
n
with PM w(i )"v[i ]`. This leads to
0 0
0
M
M (i , p , q )"(p #r!c)q !rXM (p )! (p #r#h)H(p , q )#ci # aPM w (0)[1!F(q ; p )]
n n n n
n
n
n
n
n n
n
n~1
n n
qn~L(pn)
PM w (x) f (q !x; p ) dx,
(3)
#a
n~1
n
n
0
where H(p , q )"E[[q !X(p )]`] is the expected value of leftovers at the end of period n. It is seen that
n n
n
n
H(p , q ) is a convex increasing function in q and an increasing function in p ; its additional properties are
n n
n
n
given in [19].
Therefore, the overall decision problem is described by
G
P
M w(i , q )!Kd(q !i ): q *i N
(4)
PM w(i )"maxMM
n n
n n n
n
n n
n
M
M w(i , q )"maxMM
M (i , p , q ): p 3[Pl, P ]N"M
M (i , p (q ), q ),
(5)
n n n
n n n n n
6
n n n n n
for all n*1, where p (q ) represents the optimal price for a given q in period n. The objective is to determine
n n
n
the optimal decision variables pw and qw for all n, which jointly maximize PM for a given i . Note that if
n
n
N
N
w
PM (i ))0 then the optimal decision would be not to do business.
n N
146
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
4. Model solution
In order to determine the optimal procurement policy for period n, we need to characterize the form of
M w satis"es
M
M w(i , q ). First, we note from Eqs. (3) and (5) that M
n n n
n
M
M w(i , q )"MM w(0, q )#ci ,
n n n
n
n
n
M w(0, q ). For that
so that MM w is additively separable in i and q . Hence, it su$ces to characterize the form of M
n
n
n
n
n
purpose, we need to solve the maximization problem:
G
M
M w(0, q )"max (p #r!c)q !rXM (p )! (p #r#h)H(p , q )# aPM w (0)[1!F(q ; p )]
n
n
n
n
n
n
n n
n~1
n n
qn~L(pn)
PM w (x) f (q !x; p ) dx: p 3[Pl, P ] .
#a
(6)
n~1
n
n
n
6
0
It turns out that the solution methods that have been developed for the "xed price (i.e., Pl"P ) versions of
6
the model [20}23] are not directly applicable for the solution of the problem at hand. Instead, we will follow
a new solution approach which follows the inductive setting described in [21].
P
H
4.1. Form of the optimal procurement policy
The dynamic programming problem represented by Eq. (6) is worked out in the appendix. The results are
summarized below in a theorem and a corollary after the following de"nitions.
De5nition. Let G(i, p, q)"(p#r!c)q!rXM (p)!(p#r#h!ac)H(p, q)#ci, and Gw(0, q)"maxMG(0, p, q):
p3[Pl, P ]N: then, we de"ne the critical inventory levels 0)sm(s (Sl(sl(S (s6(S6 as shown in
g
g
g
g
g
g
g
6
Fig. 1.
Note that the generic function G represents the expected pro"t in a single-period, lost-sales model where
the unit salvage value is c (i.e., the leftovers are returned to the supplier at the original cost). It will play
a critical role in the characterization of MM w, and in establishing bounds on the optimal control parameters.
n
The following theorem describes the form of M
M w in general:
n
Theorem 1. ¹here exist an even integer k and critical inventory levels, s1(s2(2(skn~1(S (skn, de,ned by
n
n
n
n
n
n
S "argmaxMM
M w(0, q): 0)q(RN,
n
n
M w(0, S )!KN,
M w(0, q)*M
s1"minMq: M
n
n
n
n
M w(0, q)"M
sj "minMq: q'sj~1, M
M w(0, S )!KN for j"2, 3,2, k ,
n
n
n
n
n
n
such that if M
M w(0, q ) is unimodal, then for n*2:
1
1
(a) M
M w(0, q )"Gw(0, q )#aMM w (0, s1 ), ∀q 3[0, s1 ],
n~1
n~1
n
n
n
n~1
n
(b) M
M w(0, q ))Gw(0, q )#aMM w (0, s1 )#aK, ∀q 3[s1 ,R),
n~1
n~1
n
n
n
n~1
n
(c) M
M w(0, q )*Gw(0, q )#aM
M w (0, s1 ), ∀q 3[s1 , skn~1 ],
n~1 n~1
n~1
n
n
n
n~1
n
M w(0, q )*M
M w(0, q@ )!aK, for any q@ 3(q ,R),
(d) ∀q 3[S , skn~1 ], M
n
n
n
n
n
n
n
g n~1
(e) M
M w(0, 0)(R,
n
M
M w(0, q )"!R.
(f ) lim
n
qn?= n
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
147
Fig. 1. De"nitions of the critical inventory levels under Gw(0, q).
Corollary 1. For n*2, sm(s1, skn~1(sl, s6(skn, and Sl(S (S6; also, if s (s1, then s (s1.
n n
g
n
1
g
g
g
g g
n
g
n
Regarding the condition of unimodality of M
M w(0, q ) in Theorem 1, there are analytical di$culties
1
1
in proving this property, except for some special cases [3,5,8,12,19,24]. For instance, it is reported in
w
[19] that M
M (0, q ) is unimodal if (1) demand is deterministic, (2) demand is additive, e has a uniform
1
1
distribution and XM (p) is linear, or (3) demand is multiplicative, e has an exponential distribution and
XM (p) is linear. The discussions of unimodality and su$cient conditions can be found in the above-cited
references.
Since G is a special case of M
M (with v"c), under the unimodality assumption, Gw is also a unimodal
1
function with a maximizer at S and a reorder point at s ((S ). Thus, it follows from (a) and Corollary
g
g
g
w
1 (s1 (S ) that MM (0, q ) is an increasing function of q on [0, s1 ]. Furthermore, (a) implies that the
n~1
n
n
n
n~1
g
optimal pricing decision, p (q ), for q 3[0, s1 ] is given by argmaxMG(0, p, q ): p3[Pl, P ]N, the maximizing
n~1
n
n n
n
6
price under the generic single-period model. (Note that if q )s1 , there will be an order placed in period
n~1
n
n!1; hence, the inventory carried over to period n!1 would worth ac, and in this case the pricing decision
is based only on G.)
It is implied by (b) and (c) that MM w(0, q ) is bounded by two unimodal functions which are at most aK
n
n
apart on [s1 , skn~1 ]. Hence, (b) and (c) establish bounds on M
M w(0, q ) based on the value of the optimal
n~1 n~1
n
n
(n!1)-period pro"t, provided that an order takes place in period n!1. In this case, the contribution of the
w
w
current period's pro"t is limited to between G (0, q ) and G (0, q )!aK.
n
n
148
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
Fig. 2. A general n-period expected pseudo-pro"t function, MM w(0, q ) vs. q , for n*2 where k "6.
n
n
n
n
M w (0, s1 ); hence, properties (b) and (c) can be rewritten
It follows from (a) that M
M w(0, 0)"Gw(0, 0)#aM
n~1
n
n~1
w
independent of M
M
(0, s1 ) as
n~1
n~1
M w(0, 0))Gw(0, q )!Gw(0, 0)#aK, ∀q 3[s1 ,R),
(b@) M
M w(0, q )!M
(7)
n
n
n
n~1
n
n
w
w
w
w
M (0, 0)*G (0, q )!G (0, 0), ∀q 3[s1 , skn~1 ].
(c@) M
M (0, q )!M
n
n
n
n~1 n~1
n
n
Thus, under the optimal solution, the increase in expected n-period pro"t, upon raising the stock level from
0 to q , must be greater than or equal to the change in one-period expected pro"t (with v"c) under a similar
n
stock movement (i.e., Gw(0, q )!Gw(0, 0)). This increase, however, is limited by Gw(0, q )!Gw(0, 0)#aK.
n
n
Property (b) extends the upper bound function Gw(0, q )!aM
M w (0, s1 )#aK over [skn~1 , R). On the
n~1
n~1
n~1
n
w
other hand, it follows from (d) that M
M (0, q ) is aK-decreasing ([22]) over [S , skn~1 ], and the functional value
n
n
g n~1
w
w
of M
M , evaluated at any point beyond skn~1 , is strictly below the global maximum of M
M (0, S ).
n
n~1
n
n
A general expected n-period pseudo-pro"t function, as characterized by Theorem 1, is depicted in Fig. 2. It
is seen that there could be more than one order-up-to level, and for each order-up-to level there could be
more than one reorder interval. This characterization is weaker than that in the "xed-price model, where it
can be shown that in each period a single optimal reorder-point, order-up-to-level policy exists [21].
The possible presence of multiple order-up-to levels poses serious di$culties in characterizing and
computing the optimal procurement policies. These di$culties, however, can be circumvented if the
beginning inventory level before ordering in an arbitrary period n is less than or equal to skn. Then, since
n
M
M w(0, 0) is "nite and M
M w(0, q ) tends to decline as q tends to in"nity, the optimal procurement policy can be
n
n
n
n
characterized as follows provided that i )skn:
n
n
S if i 3O ,
w
n
n
q " n
(8)
n
i
if i 3OM ,
n
n
n
for all n, where O , the `reorder regiona, is the union of reorder intervals, given by
n
[0, s1]X[s2, s3]X2X[skn~2, skn~1] and OM is the complement of O with respect to [0, skn]. Note that under
n
n
n
n
n
n n
n
the unimodality condition imposed on M
M w, we have k "2 and qw is determined by an (s , S ) policy.
1
1
1
1 1
In implementing the above optimal procurement policy, we need to compute skn for each period, which can
n
be a computational burden. Instead, a stronger but more useful condition can be established by referring to
Corollary 1, where we have s6)skn. That is, we replace the condition of the above optimal policy (i )skn) by
n
n
n
g
G
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
149
i )s6 which involves the computation of s6 once and in advance by using the problem primitives. Also, it is
g
g
n
seen that the condition i )skn is satis"ed if S (s6 for all n and i )S .
g
N
N
n
n
n
The fundamental di!erence between the (s , S ) policies de"ned for the "xed-price models [20,21] and the
n n
policy we de"ne in Eq. (8) is that, at each decision epoch, the vendor has to make a pricing decision whether
or not it decides to place an order. Thus, the vendor needs to determine the optimal price, p (i ), at the
n n
beginning of each review period, given the observed value of i . This requires the preparation of price tables
n
for the vendor to read the optimal prices from. These tables can be prepared simultaneously during the
computation of reorder points and the order-up-to level in each period.
4.2. A suzcient condition for a single reorder point
Properties (b) and (c) in Theorem 1 indicate that M
M w(0, q ) is aK-increasing over [s1 , sl]. Theoretically,
n~1 g
n
n
this allows for more than one reorder point. However, operating a system under multiple reorder points
could be a burden in practice. Determination of the critical levels s1, s2,2, skn for all periods with su$cient
n
n n
accuracy could be di$cult. It is, therefore, important to know under what conditions there exists a single
reorder point (i.e., k "2).
n
It follows from property (a) in Theorem 1 that M
M w(0, q ) is increasing over [0, s1 ]; thus, if s1)s1 , that
n~1
n
n~1
n
n
w
w
is if M
M (0, s1 )*M
M (0, S )!K, then k "2 and s1 is the only reorder point. However, this is not
n~1
n
n
n
n
n
a convenient su$cient condition, since, under a general demand distribution, the value of s1 , or
n~1
M
M w(0, s1 ), is not analytically measurable with su$cient accuracy.
n
n~1
w
l
Another possibility is to show that M
M (0, q ) is increasing over [s1 , s ], implying k "2 under Theorem
n~1 g
n
n
n
1. (skn~1)sl for all n; see Corollary 1.) To this end, we assume that MM w (0, q ) is increasing over
g
n~1
n~1
n l
w
w
M (0, q ))MM (0, q@ ) for s1 )q
[s1 , s ], and investigate the su$cient conditions under which M
n~1
n
n~1 g
n
n
n
n
(q@ )sl with arbitrary q and q@ . This leads to the following result:
n
n
g
n
Corollary 2. MM w(0, q ) is non-decreasing in q over [s1 , sl] for n*1, that is there exists a single reorder point
n~1 g
n
n
n
(k "2), if sl)S and for all q 3[sm, sl]:
g g
g
1
n
n
p #r!c
n
, ∀p 3Mp : p#h!ac'0, Pl)p)P N.
F(q ; p ))
n
6
n n
p #r#h!ac
n
(9)
The RHS in Eq. (9) is a concave increasing function of p , and F(q ; p ) is assumed to be a non-decreasing
n
n n
function of p for all q (see Section 2).
n
n
In view of Eq. (9), we can make the following observations. If the vendor administers a higher price, he is
able to increase F(q ; p ), the likelihood that there will be no shortage at level q , to a desired level, while the
n n
n
relative weight of unit underage cost rises (relative weight of unit overage cost declines). Hence, the vendor is
inclined to increase the stock level to above q and this continues until a break-even point is reached. This is
n
veri"ed by the fact that LG(0, p, q)/Lq*0 under Eq. (9). Thus, based on the demand distribution and unit
costs that prevail in the market, Eq. (9) re#ects the vendor's capability to increase the expected current period
pro"t by increasing the stock level to above sl. In other words, Eq. (9) attributes a higher monopoly power to
g
the vendor than the vendor would have in its absence. In this interpretation, we utilized the myopic notion
that the higher the monopoly power, the less responsive is the demand to changes in price. That is,
probability of satisfying the demand fully, F(q ; p ), does not respond strongly to an increase in price, since
n n
the increase in F(q ; p ) is limited by the RHS in Eq. (9).
n n
Under Theorem 1 and Corollary 2, the optimal procurement policy is de"ned by
qw"i #(S !i )d(s1!i ),
n
n
n
n
n n
(10)
150
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
provided that i )s6 is satis"ed for n'1. It also follows that sl)S for n*1, such that sl is a lower bound
g
n
g
g
n
on order-up-to levels.
Note that in order to have k "2, M
M w(0, q ) need not be increasing over [s1 , sl]. There could be other
n~1 g
n
n
n
conditions leading to the same result. Our experience with a number of numerical examples suggests that
cases with k '2 would be rare in practice.
n
4.3. Suzcient conditions for the optimality of (s , S ) policies
n n
If, in property (d), the range for q were established as [S ,R), then M
M w(0, q ) would be characterized as
n
n
n
g
a K-decreasing function over [S ,R), S would be the only order-up-to level, and there would be no need for
g
n
the condition i )skn in Eq. (8) or Eq. (10). This is precisely the case in the development of the optimality of
n
n
(s , S ) policies under the "xed-price (Pl"P ) model [21].
n n
6
In the following corollary, we investigate the su$cient conditions under which MM w assumes a desirable
n
shape over [S ,R) to yield a single order-up-to level.
g
Corollary 3. M
M w(0, q )*MM w(0, q@ )!aK for all q and q@ with S )q (q@ , if s )S and
n
1
g
n
g
n
n
n
n
n
n
(p #r#h!ac)F(q ; p )*p #r!c, ∀p 3[Pl, P ], for n*1.
6
n
n n
n
n
(11)
Note that for large q values, F(q; p) tends to 1 (particularly, F(q; p)"1 when q*;(p)), and this complies
well with condition Eq. (11). That is, M
M w(0, q ) is K-decreasing in the limit as q tends to in"nity. Also, it is
n
n
n
intuitive that at considerably large inventory levels, the vendor would tend to reduce the price substantially,
even below the unit procurement cost, in order to deplete inventories. In this regard, it is seen in Eq. (11) that
the condition tends to hold when c and h get larger and r gets smaller (it holds at all q levels when
p#r!c)0)p#r#h!ac, that is, ac!h!r)p)c!r). In general, the condition tends to hold
when c and h get larger or r gets smaller.
Thus, under Theorem 1 and Corollary 3, the optimal procurement policy is characterized by
G
S if i 3O ,
n
n
qw" n
n
i
if i 3OM or i 'skn.
n
n
n
n
n
Also, under both Corollaries 2 and 3, it follows from Theorem 1 that k "2 for all n and qw is determined by
n
n
an (s , S ) policy: qw"i #(S !i )d(s1!i ).
n
n n
n
n
n
n n
4.4. Numerical example
To demonstrate our "ndings, we consider a 5-period problem de"ned by the base parameter set
c"0.5, r"1.5, h"0.4, v"0.1, K"5, a"0.95; the expected demand function XM (p)"ae~bp with a"150
and b"0.2; the price limits Pl"0.1 and P "10; and the additive-uniform demand distribution
6
F(x; p)"0.5(x!XM (p)#j)/j for !j)x!XM (p))j with j"20. (j"0 corresponds to the deterministic
demand case.) For sensitivity analyses, we used multiple parameter values as h"0.02, 0.1, 0.4;
K"1, 5, 10; b"0.05, 0.1, 0.2; and j"0, 5, 10, 20.
M w,2, M
M w over the
Using a computer program, we computed the expected pseudo-pro"t functions M
M w, M
1
2
5
q range of [0, 250] with a step size of 1. The resulting optimal values of the control parameters and the
n
critical functional values are shown in Tables 1 and 2. Also, the pseudo-pro"t functions computed for
j"0, K"10, the deterministic demand case, and j"10, K"10 are plotted in Figs. 3 and 4.
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
151
Table 1
Critical inventory levels, prices and pseudo-pro"t values under the 5-period example problem
K
j
b
h
n
s
n
S
n
M
M w(0, S )
n
n
p (S )
n n
10
20
0.2
0.4
10
20
0.1
0.4
10
20
0.05
0.4
10
20
0.2
0.1
10
20
0.2
0.02
5
20
0.2
0.4
1
20
0.2
0.4
10
10
0.2
0.4
10
5
0.2
0.4
10
0
0.2
0.4
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
49.6
51.1
51.1
51.1
51.1
64.2
65.3
65.2
65.2
65.2
100.0
100.9
100.9
100.9
100.9
50.8
52.4
53.2
52.4
53.1
51.1
56.2
56.8
55.6
55.5
54.2
55.7
55.7
55.7
55.7
60.5
62.1
62.1
62.1
62.1
43.0
43.7
43.7
43.7
43.7
39.7
40.0
40.0
40.0
40.0
36.5
36.5
36.5
36.5
36.5
65.7
67.5
67.5
67.5
67.5
72.5
73.6
73.6
73.6
73.6
108.3
109.3
109.3
109.3
109.3
67.1
110.4
125.8
125.4
125.0
67.5
119.9
161.4
196.3
133.0
65.7
67.5
67.5
67.5
67.5
65.7
67.5
67.5
67.5
67.5
57.8
58.7
58.7
58.7
58.7
53.9
54.2
54.2
54.2
54.2
49.9
49.9
49.9
49.9
49.9
235.3
455.8
665.2
864.1
1053.1
509.2
990.5
1447.7
1882.0
2294.6
849.3
1654.8
2420.1
3147.1
3837.7
240.3
466.1
681.5
885.3
1079.7
241.7
472.7
692.8
900.7
1098.1
235.3
460.5
674.4
877.7
1070.7
235.3
464.3
681.9
888.5
1084.9
242.5
466.7
679.7
882.0
1074.2
246.1
472.7
688.0
892.6
1086.9
249.7
477.3
693.6
899.1
1094.3
5.5
5.5
5.5
5.5
5.5
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
10.0
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.6
5.5
5.5
5.5
5.5
5.6
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
5.5
152
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
Table 2
Critical inventory levels that are obtained under the generic pseudo-pro"t function
K
j
b
h
sm
g
s
g
Sl
g
sl
g
S
g
s6
g
S6
g
Gw(0, S )
g
10
10
10
10
10
5
1
10
10
20
20
20
20
20
20
20
10
5
0.2
0.1
0.05
0.2
0.2
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.1
0.02
0.4
0.4
0.4
0.4
45.0
62.0
97.8
46.2
46.5
51.3
60.1
38.2
34.9
51.1
65.3
101.1
52.4
52.7
55.7
62.1
43.7
40.1
51.5
65.5
101.3
52.8
53.1
56.0
62.3
44.1
40.4
63.7
71.8
107.6
65.3
65.7
64.8
66.3
55.2
51.0
67.5
73.7
109.5
69.2
69.7
67.5
67.5
58.7
54.3
71.5
75.7
111.5
74.2
81.1
70.3
68.8
62.3
57.7
93.2
96.8
132.6
146.2
281.1
82.1
73.0
83.9
79.2
241.7
516.0
856.1
247.2
248.8
241.7
241.7
245.7
247.7
We observe that the optimal control values are quite sensitive to the price senstivity of demand (note that
b is the price elasticity multiplier of the expected demand). The more sensitive the expected demand to price,
the less is the expected pro"t. As for the holding cost, smaller h values tend to yield considerably higher
order-up-to levels.
Comparing the critical inventory levels listed in Tables 1 and 2, we see that the conditions given in
Corollary 1 hold in each case. We also note that S )s6 in all cases except h"0.1 and h"0.02, for which the
g
n
reorder-region order-up-to level policy will be optimal.
Table 1 shows that the optimal prices evaluated at respective order-up-to levels are the same in all periods.
Consequently, if an order is to be placed in each period, then the optimal prices in successive periods will be
constant. We also observe that demand uncertainty has considerable impact on the parameters of the
optimal procurement policy. As the demand variance is decreased (i.e., j is decreased), the order-up-to and
reorder levels both tend to be lower, and the vendor foresees higher expected pro"ts.
On the other hand, on comparing the deterministic and probabilistic pseudo-pro"t curves in Fig. 3, we
"nd that the deterministic values are greater than the corresponding probabilistic values, especially, about
the q mid-range, which includes respective order-up-to levels.
n
In Fig. 4, we plot M
M w(0, q )!MM w(0, 0) vs. q together with the boundary functions. It is seen that the
n
n
n
n
properties listed in Theorem 1 and Corollary 1 hold. The boundary functions established in Eq. (7) also
satisfy the conditions (b@) and (c@). We also observe that the pseudo-pro"t curves exhibit considerable slope
changes across the q mid-range.
n
The best price trajectories p (q ), evaluated for n"1, 2,2, 5, are plotted in Fig. 5. The "gure shows that
n n
the optimal pricing decision at a given inventory level is not necessarily trivial. We observe that p (q ) is
n n
generally decreasing in q , but it can exhibit mild increases, or stagnate (become `stickya) over various
n
q ranges where it does not respond to changes in stock levels. Also, on comparing the deterministic and
n
probabilistic optimal prices, we observe that it is not necessarily true that the deterministic prices are lower
than the probabilistic prices or vice versa (cf. [4,5]).
Intuitively, price should be lower at higher stock levels, so as to facilitate higher demand to deplete the
excess inventories. However, the vendor would still tend to increase the price for two basic reasons: to shrink
the demand so that there will be leftovers for future periods, or to realize a higher pro"t per unit sold. If the
"xed ordering cost in a future period is to be saved by satisfying that period's demand by carying a part of the
current inventory forward, then both of these reasons will prevail. Once the total cost of inventory holding
and the di!erential cost of procurement breaks even with the savings in the "xed ordering cost, however, it
would not be pro"table to consider carrying stock into future periods. As n gets larger, there are more periods
ahead to consider; thus, there could be a recurrence of this break-even behaviour at higher q levels. This can
n
be seen in Fig. 5 by comparing p (q ) with p (q ). The passage from one pricing regime to another, at
5 5
2 2
a break-even point, need not be smooth and there could be sudden changes in p (q ).
n n
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
153
Fig. 3. Expected pseudo-pro"t curves, MM w(0, q ) vs. q , that are obtained for the 5-period example problem in Section 5.4
n
n
n
(c"0.5, r"1.5, h"0.02, b"0.2, K"10, j"10.0). The thin curves represent the deterministic case where j"0.
154
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
Fig. 4. Combined representation of the curves in Fig. 3. The thin curves represent MM w(0, q )!MM w(0, 0) vs. q for n*1 and the thick
n
n
n
n
curves represent the boundary functions Gw(0, q )!Gw(0, 0) and Gw(0, q )!Gw(0, 0)#K.
n
n
5. In5nite-horizon model
In this section, we study the in"nite-horizon version of the model. We drop the period indices from the
notation and rewrite the expected pseudo-pro"t function from Eq. (6) as
q~L(p)
[MM w(0, x)!M
M w(0, s1)#a
M w(0, s1)]` f (q!x; p) dx: p3[Pl, P ]
M
M w(0, q)"max G(0, p, q)#aM
6
0
P
G
H
q~L(pq)
[M
M w(0, x)!M
M w(0, s1)]` f (q!x; p ) dx,
(12)
"G(0, p , q)#aM
M w(0, s1)#a
q
q
0
where p ,p(q), the maximizing price at q.
q
The solution for M
M w(0, q) involves the integral in Eq. (12) which represents the convolution of the density
function with a "ltering function based on M
M w(0, q) (i.e., the function MM w(0, q) is "ltered over the level
w
M
M (0, s1)). Because of this, we need to solve the integral Eq. (12) recursively. In each stage of the solution,
M
M w(0, q) is established over a non-overlapping interval, and the solution obtained is employed in the next
stage.
P
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
Fig. 5. Optimal price trajectories that are obtained for the 5-period example problem shown in Fig. 3.
155
156
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
It follows from Eq. (12) that for 0)q)s1 we have M
M w(0, q)"Gw(0, q)#aM
M w(0, s1), which is an
w
increasing function of q over [0, s1] where s1(S . Evaluating M
M (0, q) at s1, we obtain
g
M
M w(0, s1)"Gw(0, s1)/(1!a), which implies
a
M
M w(0, q)"Gw(0, q)#
Gw(0, s1),
I
1!a
(13)
where the index in roman numeral denotes the stage number. Successive stages I, II, III, 2 correspond to
intervals [0, s1], [s1, s2],[s2, s3],2, respectively.
In the next stage, we have s1)q)s2 for which the e!ective integration range starts at s1 in Eq. (12) and
we have
q~L(pq)
w
w
M w(0, s1)] f (q!x; p ) dx
[M
M w(0, x)!M
M
M (0, q)"G(0, p , q)#aMM (0, s1)# a
II
q
II
q
1
s
q~L(pq)
w
M
M w(0, x) f (q!x; p ) dx,
(14)
"G(0, p , q)#a[1!F(q!s1; p )]M
M (0, s1)#a
II
q
q
q
s1
which is a renewal equation [25] with solution
P
P
G
C
a
M
M w(0, q)"max G(0, p, q)#Gw(0, s1)
!R (q!s1; p)
II
a
1!a
D
H
q~L(p)
G(0, p, x) dR (q!x; p): p3[Pl, P ] ,
#
a
6
1
s
where R (x; p)"+= aiF(i)(x; p) and F(i) is the i-fold convolution of F with itself. (Derivation is given in the
i/1
a
appendix.)
Having established M
M w(0, q) over [0, s2], we check whether s6(s2 where s6 is a lower bound on sk (see
g
g
Corollary 1). If s6)s2, then k"2 and we stop. In this case, there will be one reorder point. If s2(s6,
g
g
however, then we need to consider the q range beyond s2 and pass to the next stage.
Evaluating Eq. (12) for s2)q)s3, we obtain
P
s2
P
M w(0, s1)] f (q!x; p )
M w(0, x)!M
M
M w (0, q)"G(0, p , q)#aMM w(0, s1)# a [M
III
q
II
q
1
s
s2
a
Gw(0, s1)[1#F(q!s2; p )!F(q!s1; p )]#a M
M w(0, x) f (q!x; p ) dx.
"G(0, p , q)#
q
II
q
q
q
1!a
1
s
Since M
M w inside the integral is obtained earlier, it can be substituted in the above to solve for M
M w.
II
III
The fourth stage will involve another renewal equation. For s3)q)s4, we have
P
s2
P
M w(0, s1)# a [M
M w(0, x)!MM w(0, s1)] f (q!x; p )
M
M w (0, q)"G(0, p , q)#aM
IV
q
II
q
1
s
q~L(pq)
#a
[M
M w (0, x)!M
M w(0, s1)] f (q!x; p )
q
IV
3
s
P
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
157
a
"G(0, p , q)#
Gw(0, s1)[1#F(q!s2; p )!F(q!s1; p )! F(q!s3; p )]
q
q
q
q
1!a
s2
q~L(pq)
#a M
M (0, x) f (q!x; p ) dx# a
M
M w (0, x) f (q!x; p ) dx,
II
q
IV
q
1
3
s
s
P
w
P
which is a renewal equation on MM w . The solution for this equation can be obtained by following a similar
IV
procedure to the one used in solving for M
M w(0, q).
II
M w in the next two
If s4)s6, then we stop. Otherwise, we follow the same solution process to establish M
g
stages, and check the stopping criterion.
This method works in principle. However, there is a computational di$culty due to the fact that critical
inventory levels s1, s2, s3,2 are unknown in advance. They have to be computed simultaneously as we
establish M
M w numerically.
One approach is to start with the assumption that s1"s . Based on this initial value of s1, M
M w can be
g
w
established over [0, s2] and the value of s2 can be found by incrementing the value of q until M
M (0, q) equals
II
M
M w(0, s1). Note that s2 will be the "rst point greater than s1 satisfying the above equality. Likewise, the
II
process can be continued to include s3 and s4 and so on. Once the stopping condition (s6)sk) is satis"ed,
g
using the current solution of M
M w, s1 can be updated. If it is tolerably close to its previous value, the procedure
w
can be terminated. Otherwise, based on its updated value, a new M
M will be established over [0, sk].
It is beyond the scope of this paper to investigate the convergence of the numerical method proposed
above. To demonstrate its merits, however, we use it to obtain the solution for an in"nite-horizon problem
de"ned by the parameter set c"0.5, r"0.25, h"0.3, v"0.1, K"8, a"0.7; the expected demand function XM (p)"150 exp(!0.5p); the price limits Pl"0.1 and P "4; and the multiplicative-exponential demand
6
distribution F(x; p)"1!exp(!x/XM (p)), x*0, under which the discounted renewal function is given by
R (x; p)"a[1!exp(!(1!a)x/XM (p))]/(1!a).
a
We obtained the solution in 10 iterations. The successive s1 values were 28.20, 35.25, 31.26, 33.40, 32.15,
32.87, 32.44, 32.70, 32.55 and 32.64 until we reached a tolerance of 0.1 units. In this example problem, it was
found that k"2; hence, it was not necessary to pass beyond the second stage in computations.
M w, we solved a 15-period problem under the
In order to demonstrate the transition from M
M w to M
1
same problem data, and compared the resultant expected pseudo-pro"t functions with the function
we obtained by solving the in"nite horizon problem. Fig. 6 depicts M
M w(0, q ) functions for n"1, 2,2, 15
n
n
w
and M
M , which become indistinguishable from the pseudo-pro"t functions for large n, as predicted by the
theory.
6. No-5xed-ordering-cost
In order to establish a link between this paper and the earlier studies reported in the literature, here we take
a brief look at the model with K"0.
Following a similar inductive approach to the one we used previously, we suppose that M
M w(0, q ) is
1
1
w
unimodal and PM
(i ) is determined by a single-parameter policy:
n~1 n~1
M
M w (0,
n~1
PM w (i )"ci #
n~1 n~1
n~1
M
M w (0,
n~1
G
S ), i )S ,
n~1
n~1
n~1
i ), S (i .
n~1
n~1
n~1
158
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
Fig. 6. Expected pseudo-pro"t curves, M
M w(0, q ) vs. q , that are evaluated for the 15-period problem speci"ed in Section 6. The curve at
n
n
n
the top represents the solution for the in"nite horizon model with the same parameter set. (It coincides with the expected pseudo-pro"t
curves at large n.)
Under this setting, it follows from Eq. (6) that
G
M
M w(0, q )"max G(0, p , q )#aM
M w (0, S )
n
n
n n
n~1
n~1
qn~L(pn)
M w (0, S )] f (q !x; p ) dx: p 3[Pl, P ] .
[M
M w (0, x)! M
(15)
#a
n~1
n~1
n~1
n
n
n
6
Sn~1
It can be seen that for q )S , the integral drops out and we have
n
n~1
M
M w(0, q )"Gw(0, q )#aM
M w (0, S ),
n
n
n~1
n~1
n
where Gw(0, q ) is a unimodal function. For S (q , however, we note that the integral in Eq. (15) is
n
n~1
n
negative-valued. Based on these observations, we establish M
M w(0, q ) in two alternative ways, as shown in
n
n
Fig. 7. Considering Eq. (15) and Fig. 7, we discover that
P
H
minMS , S N)S )S ,
(16)
g n~1
n
g
for n*2, and the con"guration shown in Fig. 7a can only occur for n"2.
The recursive ordering in Eq. (16) implies that if S )S , then S "S for all n*2. It is clear that if v"c,
g
1
n
g
then M
M ,G, and S "S by de"nition. But, it is more plausible, in general, that v(c, in which case S will
1
1
g
1
be di!erent from S .
g
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
159
Fig. 7. Construction of M
M w(0, q ) vs. q for K"0; dashed curves represent Gw(0, q ) for: (a) S (S , and (b) S (S .
n
n
n
n
g
n~1
n~1
g
Intuitively, when the price is "xed, S must be greater than S . Having the option of salvaging the leftovers
g
1
at a higher return, the decision maker will be encouraged to procure up to higher levels. But, under the
impact of pricing, there could be a trade-o! between small-quantity high-price and large-quantity low-price
scenarios. Demand uncertainty at a given level will then be contingent on the pricing decision. For this
reason, under a general price-demand relationship, it is theoretically possible that S (S .
g
1
Fig. 7b shows that the expected pro"t level will be higher if S
shifts towards S and the contribution of
n~1
g
G is more than the contribution of the negative valued integral in Eq. (15). Thus, overcoming the transient
behaviour represented by Eq. (16), there will be a propensity, under the optimal solution, for S to approach
n
S as the number of periods is increased.
g
Su$cient conditions for the optimality of a single-critical-number policy are given in [11]. In principle,
these conditions can be utilized to verify the form of the optimal procurement policy, but they are di$cult to
interpret in terms of their managerial implications. Under some restrictions, Thowsen [11] shows that the
single-parameter policy will be optimal if the demand is additive with a PF (Po& lya frequency function of type
2
2) distributed random term and a linear expected demand function. Also, su$cient conditions for the
existence and uniqueness of the optimal decision variables are given in [9,10]. The characterization
portrayed in these studies are based on taking derivatives. The drawback of that approach is that it can not
be immediately extended to the model with "xed ordering costs, where di!erentiability is restricted to
sub-regions.
160
H. Polatoglu, I. Sahin/Int. J. Production Economics 65 (2000) 141}171
7. Non-stationary extensions
In addition to being dependent on price, it is possible that the demand distribution exhibits periodic shifts
during the planning horizon due to seasonalities or changing market conditions. The natural way of
incorporating such time-variability in our model would be to represent the demand in period n by a random
variable X (p ) with distribution F (x; p ). The generic function, now denoted by G , and the critical
n n
n
n
n
parameters de"ned for Gw, sm (2(S6 , would also be period-dependent.
n gn
gn
By virtue of the fact that the demand distribution depends on price and the price can be set at the
beginning of each period, extensions to time-dependent demand are immediate. It can be seen, going through
the proofs in the appendix, that this does not alter the arguments and derivations, provided that Gw(0, q ) is
n
n
unimodal and the condition s6 )skn~1 holds for all n'1. This condition parallels the "xed-price version of
gn
n~1
the non-stationary demand model [21,23] where it is assumed that S )S
for n'1. It can also be seen
gn
gn~1
that if S )s1 , then k "2.
n~1
n
gn
Similarly, we can let the cost parameters and the discount factor depend on the period (i.e.,
c , r , h , K , a ), without any di$culty, provided that K *a K
(a "1), and M
M w is unimodal for
1
n n n
n n
n
n~1 n~1 0
w
n'1 (which implies unimodality of G ).
n
8. Deterministic demand
Under deterministic demand, X (p)"XM (p) for all n. For technical reasons, we will assume that X (p) is
n
n
n
o(1/p) as pPR and as pP0`. Under this assumption, the riskless revenue function R (p)"pX (p) starts
n
n
growing at 0 when p"0, reaches a maximum level, and declines down to 0 as p tends to in"nity. We further
assume that R (p) is a pseudoconcave function of p on (0,R) so that there is a unique breakeven point
n
between low-price-high-demand and high-price-low-demand situations in terms of revenue. It is shown in
[19] that if X (p) is a convex or concave decreasing function, then R (p) is pseudoconcave on (0,R).
n
n
A detailed discussion of the properties of the riskles revenue function and related functions can be seen in
[7,19].
Using X (p)"XM (p) in Eq. (3) we obtain
n
n
M
M (i , p , q )"G (i , p ,