European Journal of Operational Research 148 (2003) 363–373
www.elsevier.com/locate/dsw
Production, Manufacturing and Logistics
A committed delivery strategy with fixed frequency
and quantity
Douglas J. Thomas
a
a,*
, Steven T. Hackman
b
Smeal College of Business Administration, The Pennsylvania State University, 509 Business Administration Building,
University Park, PA 16802-3005, USA
b
School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA
Received 20 February 2001; accepted 15 April 2002
Abstract
We analyze a supply chain environment in which a distributor facing price-sensitive demand has the opportunity to
contractually commit to a delivery quantity at regular intervals over a finite horizon in exchange for a per-unit cost
reduction for units acquired via committed delivery. Supplemental orders needed to meet demand are purchased at an
additional unit cost. For normally distributed demand, we use a simulation-based approximation to develop models
yielding closed-form solutions for the optimal order quantity and resell price for the distributor. Inventory, ordering
and pricing implications for this ‘‘committed delivery strategy’’ are investigated.
Ó 2002 Elsevier Science B.V. All rights reserved.
Keywords: Supply chain management; Inventory; Distribution
1. Introduction
Motivated by the supply chain of a distributor of industrial food products, we introduce a
committed delivery strategy as a vehicle for contractually shifting variability away from manufacturers and logistics providers to a distributor or
retailer in return for a reduction in unit acquisition
cost. We analyze a supply chain comprised of a
vendor, a distributor and a pool of potential customers. The customers in this case are actually
*
Corresponding author. Tel.: +1-814-863-7203; fax: +1-814863-7067.
E-mail addresses: [email protected] (D.J. Thomas), [email protected] (S.T. Hackman).
resellers that deliver food products to restaurants,
schools and hospitals. Each reseller has exogenous
random demand, unaffected by the transfer price
set by the distributor. A few distributors compete
for the business of these resellers. By lowering his
price a distributor will attract some but not all
resellers due to factors such as geographic location, quality of service and inertia. That is, demand is price-sensitive, but the market is not
purely price competitive.
In our framework, the distributor has the opportunity to commit to a delivery of a specific
quantity at regular intervals over some finite horizon in exchange for a per-unit discount in the
transportation and acquisition cost for committed
orders. Initially, we assume that the per-unit discount is independent of the commitment quantity.
0377-2217/03/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved.
doi:10.1016/S0377-2217(02)00398-3
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D.J. Thomas, S.T. Hackman / European Journal of Operational Research 148 (2003) 363–373
We show that for the case where the price is fixed,
a quantity dependent discount schedule can be
easily accommodated.
A commitment agreement might specify ‘‘one
full rail car of canned peaches each Monday
morning.’’ These commitment strategies are well
suited for the rail industry, due to the large fixed
cost component and historically low capacity utilization.
In our motivating case, the industrial food distributor ships products via rail. Since the volume
is low by rail company standards, they face long
and highly variable lead times. Our commitment
framework is based on the idea of many small
shippers forming a consortium and contracting
capacity on a periodic basis. That is, an entire train
would be reserved on a recurring basis, and individual firms would contract for capacity. Under
such an arrangement, members of the consortium
could commit to some substantial fraction of
available capacity and reserve the remaining capacity for supplemental orders. The precise behavior of the consortium is a topic for future work.
In this paper, we focus on the behavior of an individual firm.
We assume that all demand must be met within
each period. The distributor receives required
supplemental orders at unit cost at the end of each
period. We further ignore any fixed ordering cost,
as discussions with several distributors indicate
that a per-unit cost that includes a percentage for
ordering and transportation cost is a reasonable
proxy.
There are several implications of such commitment agreements. First, the distributor’s ability to
react to realized demand is limited. Under traditional stochastic inventory control models, realized demand triggers orders, whereas under a
commitment strategy, the committed supply will
arrive independently of the demand process.
Variation in demand will cause the distributor to
place supplemental orders; thus our commitment
strategy will have both inventory and ordering
implications. When demand is low, a pure control
policy adapts by simply placing fewer or smaller
orders. Under a commitment strategy inventory will build up if committed deliveries exceed
demand. Second, the distributor’s commitment
allows manufacturers and logistics providers,
particularly freight carriers, to improve utilization
of their critical assets (e.g., rail cars). Accordingly, in exchange for commitment the distributor
expects reductions in procurement and delivery
costs. Simply put, the distributor gives up some
flexibility and absorbs additional risk in exchange
for a cost reduction. The central question addressed in this paper is: What is the best commitment strategy? That is, how much should be
committed to up-front, and what price should be
charged? As we shall see any cost reduction will
give the distributor an incentive to lower price to
increase market share. Changes in price affect the
demand distribution, which in turn affects the
cost of inventory and supplemental orders as a
function of the commitment quantity. This is why
we initially explore the simplest kind of ordering
policy.
We emphasize that our model does not permit the distributor to pay for but not accept
committed deliveries. We assume that regular
shipments improve the supply chain cost structure,
which is why the supplier and/or carrier is willing
to offer a discount. This assumption is consistent
with our motivating case, where partially filling
the railcar results in greater product damage.
Furthermore, such a level ordering policy may
have production cost benefits for the manufacturer.
Our distributor’s flexibility to commit is more
restrictive than a quantity discount model, requiring a total commitment quantity, which has
been addressed by Anupindi and Bassok (1998),
Bassok et al. (1997) and Bassok and Anupindi
(1997). Weng (1995a,b, 1997) addresses a twoechelon, infinite horizon model with quantity
discounts and price-sensitive demand. His work
focuses on using quantity discount schedules as a
mechanism for channel coordination. Building on
the work of Ernst and Pyke (1993) and Yano and
Gerchak (1989), Henig et al. (1997) study a twoechelon system where a discount is offered for
transportation capacity commitment. For an infinite horizon, stationary demand system, they show
that for a given level of contracted transportation
capacity (which need not be used), two critical
numbers characterize the optimal ordering policy,
D.J. Thomas, S.T. Hackman / European Journal of Operational Research 148 (2003) 363–373
such that when on-hand inventory falls within a
certain range, exactly the contracted capacity is
used. Eppen and lyer (1997) examine a two-stage
stochastic inventory model motivated by backup
agreements common to the fashion industry.
Under such an agreement, a buyer chooses an
order quantity, and the vendor holds back a
fraction of the commitment. After observing initial
demand, the buyer can acquire up to the remainder of their commitment at the original price,
paying a penalty cost for committed units not
purchased. Tsay (1997) models a manufacturerretailer chain where the retailer gives a point estimate of demand. The two parties then agree on a
minimum purchase commitment, a maximum
quantity guaranteed to be available, and a transfer
price. He shows that without such contract structure, inefficiencies can results.
The organization of the paper is as follows. In
Section 2 we formalize the decision facing the
distributor and introduce the price-sensitive demand model. In Section 3 we analyze the inventory
and supplemental ordering dynamics associated
with a commitment strategy. The purpose of this
model is to accurately capture the true cost of inventory and supplemental orders. While analytical
expressions for such costs may be formulated, they
involve multi-fold convolutions that must be recalculated each time the price and/or commitment
level is changed, all but rendering standard analyses computationally intractable. When demand
each period is identically and normally distributed,
and period demands are independent, quadratic
approximations are shown to be a remarkably
close fit to the true ‘‘standardized’’ costs, the parameters of which are obtained via simulation.
These parameters are independent of the problem
cost parameters, and so these quadratic functions
may be used in the development of related models.
Through further approximation, we develop an
expression for the standardized commitment level
that depends only on the total horizon length and
not the number of sub-periods. In Section 4 we
introduce an approximation to the price-sensitive
demand model, namely that the mean is pricesensitive and the standard deviation is proportional to the mean. We discuss the conditions
under which this approximation would be accu-
365
rate. This approximation permits the development
of an effective all-units discount from which the
multi-period model can be reduced to an equivalent single-period model that is easily solved. Final
remarks and future research opportunities are offered in Section 5.
2. Distributor model
Our distributor faces price-sensitive demand
and has the opportunity to reduce transportation and procurement costs by agreeing to purchase and receive a fixed quantity in each of T
periods. The fixed quantity and frequency permit
the manufacturer and transportation provider to
efficiently utilize their resources, resulting in
overall system cost reduction. In exchange for
absorbing this variability, the distributor is offered
a percentage discount d for these committed deliveries only.
The distributor chooses total commitment
quantity Q over the planning horizon T as well as
selling price p to maximize expected profit. Demand is normally distributed with the mean and
standard deviation depending on the selling price
p. For a commitment level Q, Q=T units would
arrive each period. No backorders or lost sales are
permitted. Excess demand in each period is met via
a supplemental mode at unit cost c. In each period,
the distributor pays cð1 dÞ per unit for committed orders, c for orders acquired via the supplemental mode and h=T per unit held at the end of
the period. That is, h represents the cost of holding
a single unit for the entire horizon, T periods.
3. A multi-period model
In this section we develop accurate approximations to the cost of inventory and supplemental
ordering for the committed delivery strategy in a
multi-period setting.
Recall that our distributor commits to an aggregate quantity Q and receives Q=T units at the
beginning of each of T equal-length sub-periods.
Demand occurs throughout the sub-period. In
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D.J. Thomas, S.T. Hackman / European Journal of Operational Research 148 (2003) 363–373
order to meet demand supplemental orders are
placed at unit cost at the end of each sub-period, if
necessary. The opportunity to place supplemental
orders at unit cost with zero lead time could represent the opportunity to either buy product from
local competitors or to place additional orders via
an express carrier.
Let EST ðQ; pÞ, EIT ðQ; pÞ denote the expected
amount of supplemental orders and inventory over
the whole T-period horizon as functions of both
the total commitment quantity Q and the price p.
It is important to remember that the distributor’s
choice of p will affect the demand distribution. For
example, when T ¼ 2:
Z 1
ES2 ðQ; pÞ ¼
ðD1 Q=2Þf1 ðD1 Þ dD1
Q=2
þ F1 ðQ=2Þ
Z
1
ðD2 Q=2Þf2 ðD2 Þ dD2
Q=2
þ
Z
Q=2
0
Z
1
ðD1 þ D2 QÞ
QD1
f2 ðD2 Þ dD2 f1 ðD1 Þ dD1 ;
ð1Þ
where Dt is the demand in period t with pdf, cdf
ft ðÞ, Ft ðÞ, F t ðÞ ¼ 1 Ft ðÞ. The three terms in (1)
represent, respectively, the expected supplemental
orders in the first period, the expected supplemental orders in the second period if supplemental
orders were placed in the first period, and the expected supplemental orders in the second period if
inventory was carried from period one into period
two.
For moderate values of T the convolution in
(1) becomes computationally prohibitive. Furthermore, for any change in price p or commitment
quantity Q, the expression (1) (and its inventory
counterpart) must be re-evaluated, since a change
in price changes the demand distribution.
In what P
follows we assume that aggregate demand D ¼ Tt¼1 Dt is normally distributed with
mean lðpÞ and standard deviation rðpÞ, and that
the period demands fDt g are i.i.d. normal random
variables. When demands follow this stochastic
process we show in the next section how to circumvent the difficulty mentioned in the previous
paragraph.
3.1. A simulation-based approximation
We approximate the expectations of the inventory and supplemental ordering functions using
simulation, as follows. Define b
S ðzÞ, bI ðzÞ, to be
the standardized expected supplemental orders
and expected inventory functions so that ESðQ;
pÞ rðpÞ b
S ðzÞ, EIðQ; pÞ rðpÞbI ðzÞ, where z ¼ ðQ
lðpÞÞ=rðpÞ denotes the total standardized commitment level. For T ¼ 1; . . . ; 100 and z ¼ 3; . . . ;
3 in increments of 0.01, 100,000 sample paths were
generated. Parameters for quadratic functions for
were determined using regression. Quadratic forms
were chosen both for goodness of fit as well as ease
of use in the resulting profit function. Quadratic
forms provided a superior fit to logarithmic forms,
and the improvement in fit with higher order
polynomials was quite small. Regression was used
to determine the parameters of quadratic approximations to b
S ðzÞ and bI ðzÞ for T ¼ 10, 20, 30, 40,
50, 100:
b
S ðzÞ aS z2 þ bS z þ cS ;
ð2Þ
bI ðzÞ aI z2 þ bI z þ cI :
ð3Þ
(To simplify notation, we have suppressed the
functional dependence of the parameters aS , bS , cS ,
aI , bI , cI on T.) All r-squared and adjusted rsquared values were above 0.999, and the maximum absolute error was less than 0.06. It is a
remarkably close fit. Table 1 shows the regression
Table 1
Regression parameters
T
a
b
c
Orders
10
20
30
40
50
100
0.104545
0.102178
0.101350
0.100899
0.100632
0.100050
)0.500078
)0.500057
)0.500057
)0.500048
)0.500028
)0.500031
0.662116
0.704925
0.724966
0.737359
0.745929
0.767863
Inventory
10
20
30
40
50
100
0.041375
0.042274
0.042486
0.042581
0.042646
0.042715
0.224999
0.237507
0.241651
0.243722
0.244999
0.247482
0.358248
0.410866
0.434393
0.448471
0.458076
0.482085
D.J. Thomas, S.T. Hackman / European Journal of Operational Research 148 (2003) 363–373
coefficients obtained, and Figs. 1 and 2 show
the resulting approximations to the standardized
inventory and supplemental ordering functions
when T ¼ 20.
3.2. A multi-period optimization model
Since the sub-periods will be small it is reasonable to use the maximum inventory level for
costing purposes. Let h be the holding cost rate for
the entire horizon. The maximum will be achieved
at the beginning of each sub-period, when the
committed delivery of Q=T arrives. Expected in-
367
ventory (in unit-periods) will be Q=T þ rbI ðzÞ.
Recall that Q ¼ lðpÞ þ zrðpÞ. At the end of the
horizon, units may be salvaged for cð1 d sÞ
where s is the salvage penalty, representing the
fraction of the product value that is lost if the units
are held at the end of the horizon. Since we are not
dealing with perishable products, the units are
actually held over for the next planning horizon.
The salvage penalty s serves to discourage excessive ending or rollover inventories. Using the
simulation-based approximations, we now develop
the distributor’s profit as a function of price p and
standardized commitment level z.
The expected revenue generated is the sum of
sales and salvage:
plðpÞ þ cð1 d sÞ
½ðlðpÞ þ zrðpÞÞ þ rðpÞ b
S ðzÞ lðpÞ:
ð4Þ
Note that the expected number of units salvaged is
the difference between the expected number of
units acquired during the horizon, which equals
the commitment quantity plus all supplemental
orders, minus the expected demand. The expected
cost is the sum of 3 components, namely, the cost
to purchase the commitment quantity, the cost to
acquire all supplemental order and the cost to
carry inventory:
Fig. 1. Standardized inventory units for T ¼ 20 with quadratic
fit.
cð1 dÞðlðpÞ þ zrðpÞÞ þ crðpÞ b
S ðzÞ þ ch
½rðpÞbI ðzÞ þ ðlðpÞ þ zrðpÞÞ=T :
ð5Þ
The difference between (4) and (5) is the distributor’s expected profit pðp; zÞ, which he seeks to
maximize:
max pðp; zÞ ¼ ½p cð1 d h=T ÞlðpÞ crðpÞ
p;z
½zh=T þ sz þ ðd þ sÞ b
S ðzÞ þ hbI ðzÞ:
ð6Þ
For a fixed price p the profit function in (6) is a
quadratic function of z. That is, we seek to minimize the standardized cost as a function of z:
SCðzÞ ¼ sz þ ðd þ sÞðaS z2 þ bS z þ cS Þ þ h
ðaI z2 þ bI z þ cI Þ;
Fig. 2. Standardized supplemental orders for T ¼ 20 with
quadratic fit.
or rewriting in standard quadratic form,
ð7Þ
368
D.J. Thomas, S.T. Hackman / European Journal of Operational Research 148 (2003) 363–373
\a"
zfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflffl{
SCðzÞ ¼ ½ðd þ sÞaS þ haI z2
þ ½h=T þ s þ ðd þ sÞbS þ hbI z
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
\b"
\c"
zfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflffl{
þ ½ðd þ sÞcS þ hcI :
ð8Þ
SCðzÞ will have a unique cost minimizer when
\a" P 0. For this to be true, it is sufficient for aI ,
aS P 0, a condition satisfied here (see Table 1). The
optimal solution to Eq. (7) is
z1 ½h=T þ s þ ðd þ sÞbS þ hbI ;
2ðd þ sÞaS þ 2haI
ð9Þ
which is a linear fractional function of the cost
parameters d, h and s. (Keep in mind that the
coefficients of the quadratic form are functions of
T.) Since we are working with standardized demands, the optimal commitment fractile depends
only on the inventory, ordering and salvage costs
and not on the price. To interpret (9), first note
that bS < 0 and aS , aI , bI > 0 (see Table 1.) As
expected, the optimal commitment fractile decreases with increases in either the holding cost h
or the salvage penalty s and increases as the
commitment discount d increases.
Note for the regression parameters in Table 1
that aS 1=10, bS 1=2, aI 1=24, bI 1=4.
That is, for different values of T, these curves basically vary by a constant. In fact, as T increases,
the curves move upward. This is not surprising,
since for small T, the sub-periods are larger and
there is more opportunity for supply and demand
to balance.
The only place where T explicitly appears in (9)
is the h=T term in the numerator. The regression
parameters depend on T, but as noted above, the
parameters appearing in (9) do not vary greatly
with T. For moderate to large T, the h=T term may
be small relative to the other terms. By dropping
h=T and making the substitutions aS 1=10,
bS 1=2, aI 1=24, bI 1=4, we get a simplified
expression for the optimal commitment fractile
that does not depend on T:
z2 30d 30s 15h
:
12d þ 12s þ 5h
ð10Þ
Changes in the three cost parameters have the
same directional effect on z1 and z2 .
We have presented two expressions for the
commitment fractile, Eqs. (9) and (10). To evaluate the performance of these expressions, we use
the functions from the original simulation results
(rather than the quadratic fits) to estimate the actual standardized cost. Recall from Section 3.1
that we estimate via simulation the supplemental
order and inventory functions for values of z ¼
3; . . . ; 3 in increments of 0.01. For each of 1800
problem instances (T ¼ 10; 20; 30; 40; 50, a ¼ 0:05;
0:10; . . . ; 0:50, s, h ¼ 0:05; 0:10; . . . ; 0:30) we construct the cost function and find the best z. Table
2 shows the average percentage cost gap between
the best solution and the solutions generated by
our two expressions for z. Both expressions perform relatively well, with average gaps across all
problems of 0.90% and 0.83%. The difference in
performance between the two expressions is relatively small, supporting the use of the simple
expression.
Fig. 3 shows the percent deviation from our
simulation estimate of the optimal cost for the two
approximations for the 360 problems with T ¼ 20.
The jagged nature of this figure is due to the collection of many different sets of cost parameters.
Note that performance of either approximation is
relatively poor for very low values of z . In this
range, it is quite likely that the discount being
offered is not sufficient to incent the distributor
to commit at all.
Since we are minimizing a quadratic function,
we can easily accommodate a quantity-discount
schedule. One would follow the standard approach
Table 2
Average cost gap for a ¼ 0:05; 0:10; . . . ; 0:50, s, h ¼ 0:05; 0:10;
. . . ; 0:30
T
Gap with z1 (%)
Gap with z2 (%)
10
20
30
40
50
1.47
0.86
0.70
0.76
0.70
1.21
0.73
0.70
0.79
0.71
Average
0.90
0.83
D.J. Thomas, S.T. Hackman / European Journal of Operational Research 148 (2003) 363–373
369
It is obvious from Eq. (11) that kn2 is monotonically
decreasing in n, and that kn2 ¼ r20 =l20 when q ¼ 1.
In general, for high values of q or n, the aggregate
coefficient of variation will be close to constant.
For resellers of industrial food products, selling
the same product in the same region, one might
expect some positive correlation. Accordingly, we
now assume that the coefficient of variation of
aggregate demand is a constant k. As we will see in
the next section, this constant coefficient of variation assumption permits the development of an
effective all-units discount that does not depend
on the price.
Fig. 3. Deviation from optimal cost for T ¼ 20, a ¼ 0:05; . . . ;
0:50, s, h ¼ 0:05; . . . ; 0:30.
4.1. Effective discount
of finding z for each candidate d and checking the
appropriate limits.
Assuming a constant coefficient of variation for
market demand, and substituting the expression
for optimal quantity, we derive a constant effective
discount defined as:
4. Price-sensitive demand model
d^ ¼ d h=T k
In the previous section we derived expressions
for the optimal commitment fractile. This fractile
does not depend on the price, however, the price
will determine the mean and standard deviation,
thus the actual commitment quantity will depend
on price. In this section, we turn our attention to
the problem of selecting the selling price for the
distributor.
A few distributors compete for the business of
these resellers. By lowering his price a distributor
will attract some but not all resellers due to factors
such as geographic location, quality of service and
inertia. Thus, demand is price-sensitive and the
market is not purely price competitive.
Consider a pool of homogeneous customers
each with random demand Di , with identical mean
l0 and variance r20 . Let kn2 denote the squared coefficient
of variation for the aggregate demand
Pn
i¼1 Di of n customers, and let q denote the constant correlation coefficient between customers
(1 6 q 6 1). With these notations,
kn2
r20 1
1
¼ 2
þ 1
q :
n
l0 n
ð11Þ
½z h=T þ sz þ ðd þ sÞ b
S ðz Þ þ hbI ðz Þ:
ð12Þ
The distributor’s problem of setting the optimal
price now takes on the following simple form:
max pðp; d^Þ ¼ ½p cð1 d^ÞlðpÞ:
p
ð13Þ
Throughout this section we assume that lðpÞ is
non-negative, differentiable and strictly decreasing,
so that a lower price induces greater demand. We
furthermore assume that there exists a price that
leads to a positive profit, and that for a sufficiently
high price the function lðpÞ eventually becomes
and remains non-positive, so that the set of profitmaximizing prices to (13) is non-empty and
bounded.
4.2. Optimality conditions and pricing behavior
Each price p can be viewed as a percentage
markup m in the unit cost c so that p ¼ pðmÞ cð1 þ mÞ. The necessary and sufficient conditions
for a unique optimal price may be characterized in
terms of the markup m and the elasticity l ðpðmÞÞ.
Recall that the elasticity of demand approximately
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D.J. Thomas, S.T. Hackman / European Journal of Operational Research 148 (2003) 363–373
measures the percentage increase in demand for a
1% increase in price and is formally defined as:
l ðpÞ pl0 ðpÞ
:
lðpÞ
ð14Þ
Since a maximum on ðc; 1Þ exists, the first-order
necessary condition for optimality implies that
0 ¼ lðp Þ þ ðp cÞl0 ðp Þ, which is equivalent to
the condition that
1 þ m
¼ l ðpðm ÞÞ:
m
ð15Þ
It is clear from (15) that a necessary and sufficient
condition for a unique optimal price is that the
elasticity, viewed as a function of the markup,
crosses the function f ðmÞ ð1 þ mÞ=m only once.
In this paper we shall analyze the linear and
constant elasticity demand functions that frequently appear in economic research (see Shy,
1995):
lðpÞ ¼ d
ap
;
b
lðpÞ ¼ dp ;
a b 6 p 6 a;
ð16Þ
ð17Þ
p P 0; > 1:
For the linear form d is a scale value representing
the size of the market, and for p in the specified
range ða pÞ=b represents the distributor’s market
share. For a constant elasticity demand function
the right hand side of (15) always equals a constant
(hence, the name). The unique optimal prices,
markups and corresponding profits for the linear
and constant elasticity mean demand functions are
shown in Table 3.
For the two demand functions addressed here,
it is clear that a reduction in the unit cost c will
result in a reduction in the optimal price. As we
now demonstrate, a reduction in cost can never
d
ap
b
dp
p
aþc
2
c
1
m
aþc
1
2c
1
1
Theorem 1. pðc dÞ 6 pðcÞ for each c > d > 0.
Proof. It is sufficient to demonstrate that
pðpðcÞ; dÞ P pðp; dÞ for all p P pðcÞ. By definition
pðpðcÞ; 0Þ P pðp; 0Þ for all p. Since l is nondecreasing and d is positive dlðpðcÞÞ P dlðpÞ for
each p P pðcÞ. Thus,
pðpðcÞ; dÞ ¼ pðpðcÞ; 0Þ þ dlðpðcÞÞ P pðp; 0Þ þ dlðpÞ
¼ pðp; dÞ;
as required.
ð18Þ
When the coefficient of variation rðpÞ=lðpÞ is
increasing in p, the effective discount d^ðpÞ is decreasing in p. Under this condition, there cannot
be an optimal ‘‘commit’’ price greater than an
optimal ‘‘no commit’’ price.
Theorem 2. Let pN , pC be minimal and maximal
optimal solutions for the ‘‘no commit’’ profit function (with d^ ¼ 0), and the ‘‘commit’’ profit function,
with effective discount d^ðpÞ. If d^0 ðpÞ 6 0 then
pC 6 pN .
Proof. If d^ðpN Þ < 0 then d^ðpÞ < 0 for p P pN and
the distributor would not commit, so we may
assume that d^ðpN Þ P 0.
When the distributor commits his profit function is:
p^ðpÞ ¼ ðp cÞlðpÞ þ cd^ðpÞlðpÞ:
ð19Þ
The second term of this profit function is nonnegative and non-increasing. So for p P pN it follows that
Table 3
Optimal prices, markups and profits
lðpÞ
result in an increase in the optimal price. In what
follows, let P ðcÞ denote the set of all profit maximizing prices when the unit cost equals c, and
define pðcÞ maxfp : p 2 P ðcÞg, pðcÞ minfp :
p 2 P ðcÞg. (Both functions are well-defined, since
Pc is closed and therefore compact.)
ðpN cÞlðpN Þ þ cd^ðpN ÞlðpN Þ
p
2
ða cÞ
4b
d c 1
1
d
> ðp cÞlðpÞ þ cd^ðpÞlðpÞ;
ð20Þ
which implies that pC 6 pN , and the result follows. D.J. Thomas, S.T. Hackman / European Journal of Operational Research 148 (2003) 363–373
371
4.3. Sensitivity analysis
Now we analyze the market opportunities that
a commitment strategy offers. In what follows
we only consider the constant elasticity demand
function.
The objective function in each of our previous
models may be generically represented as:
pðp; d^Þ ½p cð1 d^ÞlðpÞ
ð21Þ
and the maximum expected profit as
Pðd^Þ max hðp; d^Þ:
p
ð22Þ
Fig. 4. Percentage profit increase.
Let pðd^Þ denote the unique optimal solution to (22)
so that Pðd^Þ ¼ pðpðd^Þ; d^Þ. Prior to the commitment opportunity the distributor uses his optimal
price
pð0Þ ¼
c
;
1
and receives his expected profit
d c 1
pðpð0Þ; 0Þ ¼
:
1
ð23Þ
ð24Þ
ð25Þ
and new expected profit of
1
Pðd^Þ ¼ Pðpð0ÞÞð1 d^Þ :
Hðd^; Þ ¼
Pðd^Þ
Pð0Þ
cost effect
# zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{
"
#
^
^
pðpðdÞ; dÞ
pðpð0Þ; d^Þ
¼
:
pðpð0Þ; 0Þ
pðpð0Þ; d^Þ
|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}
"
When the distributor commits he obtains the effective discount d^. Substituting cð1 d^Þ for c in
Eq. (23) gives the new optimal price
pðd^Þ ¼ pð0Þð1 d^Þ;
profit with the reduced unit cost at the optimal
price. Formally,
ð26Þ
Note that more elastic demand permits a greater
profit improvement. Fig. 4 shows the fractional
profit increase as a function of the effective discount.
Two effects contribute to the profit improvement: reduction in effective unit cost and improved
1
pricing. Let Hðd^; Þ Pðd^Þ=Pð0Þ ¼ ð1 d^Þ
denote the ratio of the optimal profit with commitment to the optimal profit without commitment.
To study these effects we decompose Hðd^; Þ into
its cost effect and price effect components. The
cost effect is the ratio of the profit at the original
price with the reduced unit cost to the original
profit. The price effect is the ratio of the profit at
the original price with the reduced unit cost to the
ð27Þ
price effect
Let Cðd^; Þ denote the percentage increase in profit
due to the cost reduction d^ with no change in price.
Let P ðd^; Þ denote the percentage increase in profit
obtained by choosing an optimal price after securing effective discount d^. In this case,
Cðd^; Þ ¼ d^ð 1Þ;
ð28Þ
1
ð1 d^Þ
P ðd^; Þ ¼
1:
1 þ d^ð 1Þ
ð29Þ
Fig. 5 shows the percentage increase in profit for
these two effects as a function of the effective discount for ¼ 2.
We now examine the relative impact on profit
of the price and cost effects. Assuming optimal
commitment and pricing behavior by the distributor, the fraction of profit improvement due to
cost reduction as a function of delta-hat and
epsilon is
372
D.J. Thomas, S.T. Hackman / European Journal of Operational Research 148 (2003) 363–373
Fig. 5. Price and cost effects for ¼ 2.
qðd^; Þ d^ð 1Þ
Cðd^; Þ
¼
:
1
Hðd^; Þ 1 ð1 d^Þ 1
Fig. 6. Fraction of profit increase due to cost reduction.
ð30Þ
First, we present Theorem 3, which states that for
very small discounts, the profit improvement is due
almost entirely to cost reduction. Furthermore, the
importance of pricing increases as both elasticity
and effective discount increase.
Theorem 3. For qðd^; Þ as defined in Eq. (30):
d^ð1 d^Þ1
oqðd^; Þ
¼
½1 þ ð 1Þ logð1 d^Þ
1
2
o
½ð1 d^Þ 1
ð1 d^Þ
1
:
ð33Þ
When > 1 and 0 6 d^ 6 1 the expression
d^ð1 d^Þ1 =½ð1 d^Þ1 12 is non-negative, so the
partial derivative will be non-negative if
1
gðd^; Þ ½1 þ ð 1Þ logð1 d^Þ ð1 d^Þ P 0:
ð34Þ
(1) limd^!0 qðd^; Þ ¼ 1,
oqðd^; Þ
(2)
6 0, 0 6 d^ 6 1,
od^
oqðd^; Þ
6 0, P 1.
(3)
o
For each d^ 2 ½0; 1 the function gðd^; Þ is maximized
at ¼ 1 with maximum value equal to zero, and
so the result follows. Proof. The proof of (1) is established by applying
l’H^
opital’s rule. To prove (2) it may be readily
verified that
Fig. 6 shows the fraction of profit increase due
to the cost reduction as a function of the per-unit
discount. As predicted by Theorem 3, the cost effect dominates small discounts.
oqðd^; Þ
1
½ð1 d^Þ ð1 d^Þ 1:
¼
od^
½ð1 d^Þ1 12
5. Final remarks
ð31Þ
Since > 1 the derivative will be negative when
1 d^=ð1 d^Þ 6 1, or equivalently when
f ðd^Þ 1 d^ ð1 d^Þ 6 0;
0 6 d^ 6 1:
ð32Þ
Since f0 ðd^Þ ¼ ½ð1 d^Þ 1 6 0 for 0 6 d^ 6 1 and
f ð0Þ 6 0, the function f is non-positive, as required. To prove (3) it may be readily verified that
The purpose of this work was to introduce
commitment delivery strategies and explore the
implications of such strategies to a single firm. We
have presented a model for a distributor facing
price-sensitive demand and given the opportunity
to commit to specific, periodic deliveries for a finite
horizon in exchange for a unit cost reduction. For
the case when demand is normally distributed,
approximating functions for inventory and sup-
D.J. Thomas, S.T. Hackman / European Journal of Operational Research 148 (2003) 363–373
plemental ordering effects are developed. These
approximating functions permit the development
of an expression for the optimal commitment level,
in terms of a standard normal fractile. This expression is both simple and robust, allowing firms
to estimate their appropriate commitment level
quickly. The simplicity of this expression will allow
future work to address the behavior of a consortium of shippers.
We also show that a firm reducing acquisition
costs via commitment will choose to lower their
price and increase market share. While the expression for the optimal fractile does not depend
on the demand distribution or price-sensitivity,
more complicated expressions must be evaluated
to determine price and actual quantity (rather than
fractile) decisions. We establish that the reduction
in unit cost constitutes most of the profit improvement, so firms can obtain most of the benefit
of commitment by using the simple expression for
the optimal fractile and even if they leave price
unchanged.
This work was motivated by the opportunity
for shippers to form a consortium and reap significant savings by committing to specific, periodic
quantities. This consortium was implicit in the
models considered here. Future work should address the behavior of the consortium. Questions
regarding consortium behavior include: How
much capacity should the consortium procure?
That is, if 10 shippers in a consortium are each
willing to commit to one railcar per week, should
the consortium commit to 10 per week or more to
accommodate supplemental orders. If the consortium commits to a level above the base level (10 in
this case), how should they determine which
member of the consortium gets to use the extra
capacity and at what cost?
373
Acknowledgements
The authors wish to thank two anonymous
referees for their helpful comments.
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