Customer lead time management when both demand and price are lead time sensitive

European Journal of Operational Research 153 (2004) 769–781

www.elsevier.com/locate/dsw

Stochastics and Statistics

Customer lead time management when both demandand price are lead time sensitive

Saibal Ray a, E.M. Jewkes b,*

a Faculty of Management, McGill University, 1001 Sherbrooke Street West, Montreal, Que., Canada H3A 1G5b Department of Management Sciences, University of Waterloo, Waterloo, Ont., Canada N2L 3G1

Received 25 October 2000; accepted 23 August 2002

Abstract

In this paper, we model an operating system consisting of a firm and its customers, where the mean demand rate is a

function of the guaranteed delivery time offered to the customers and of market price, where price itself is determined by

the length of the delivery time. Economies of scale are present. The firm�s objective is to maximize profits by selecting an

optimal guaranteed delivery time taking into account that (i) reducing delivery time will require investment, and (ii) the

firm must be able to satisfy a pre-specified service level. We show that it is imperative for managers to know whether

customers are price or lead time sensitive based on the simultaneous dependence of price and demand on delivery time

before selecting a time-based competitive strategy. We investigate the optimal policy and provide managerial insights

based on our analysis. Examples where our insights are consistent with actual practical situations are also provided. We

show that our model is different from those in the literature that assume price and delivery time to be independent

decision variables and present conditions under which ignoring the relation between price and delivery time can lead

managers to substantially sub-optimal decisions––with or without the presence of economies of scale.

� 2002 Elsevier B.V. All rights reserved.

Keywords: Delivery time strategy; Investment and capacity analysis; Economics of queues

1. Introduction

In the past decade, practitioners have focused

on speed as the basis of competitive advantage

(Stalk and Hout, 1990; Blackburn et al., 1992).

Companies use three main strategies to utilise

speed to attract customers: (i) to serve customers

as fast as possible, (ii) to encourage potential

* Corresponding author. Fax: +519-746-7252.

E-mail address: [email protected] (E.M.

Jewkes).

0377-2217/$ - see front matter � 2002 Elsevier B.V. All rights reserv

doi:10.1016/S0377-2217(02)00655-0

customers to get a delivery time ‘‘quote’’ prior to

ordering, and (iii) to guarantee a ‘‘uniform’’ de-livery lead time for all potential customers (So and

Song, 1998). Many companies, specifically in the

service and make-to-order manufacturing sectors,

are adopting the third strategy of advertising a

uniform delivery time for all customers within

which they guarantee to satisfy ‘‘most’’ orders

(refer to examples in So, 2000; So and Song, 1998;

Rao et al., 2000). While this strategy may attractmany customers, there is a risk that demand may

exceed the firms� capacity to respond. This can

lead to a penalty cost for the manufacturer or it

ed.

mail to: [email protected]

770 S. Ray, E.M. Jewkes / European Journal of Operational Research 153 (2004) 769–781

might lead to a decrease in repeat business. With

this strategy, it is important to have some internal

mechanism in place to ensure that the promised

delivery times are feasible and reliably met.

Since the late 1980s, a large volume of opera-tions management literature has recognized that

customer demand increases with lower delivery

times as well as with lower prices (Blackburn et al.,

1992; So and Song, 1998; So, 2000). Karmarkar

(1993) pointed out that lead times are most prob-

ably inversely related to market shares or price

premiums or both. So and Song (1998) noted that

shorter delivery times can allow a price premium,e.g., shipping costs from Amazon.com are more

than double when the guaranteed delivery time is

around two days than when it is around one week.

As well, recent industry practice suggests that

customers may be willing to pay a price premium

for shorter delivery times (Blackburn et al., 1992;

Weng, 1996; Magretta, 1998; Gupta and Weera-

wat, 2000).The potential for increased demand and/or a

price premium creates an incentive for firms to

reduce the length of the delivery time. One of the

ways that firms can reliably satisfy a short delivery

time is by investing in increasing capacity (So and

Song, 1998; Palaka et al., 1998). Firms then must

trade-off the potential for increased demand and

price against the costs of investment.It is well known that economies of scale can

bring down unit operating costs in manufacturing

facilities and even in service facilities (Scherer,

1980). Hence, operations management models have

sometimes assumed unit operating costs to be a

decreasing function of demand.

In this paper, we model an operating system of

a firm and its customers where demand is a func-tion of market price and a uniform guaranteed

delivery time and the market price is determined

by the length of the delivery time. We then expand

our initial model to incorporate economies of

scale. More specifically, this paper presents an

analytical approach for a firm to maximize its

profit by optimal selection of a guaranteed delivery

time. Mean demand rate is modeled as a decreas-ing function of price and delivery time while price

itself is a decreasing function of the guaranteed

delivery time. The model takes into account that

reducing delivery time by increasing capacity will

require investment, the firm must be able to satisfy

the guaranteed delivery time according to a pre-

specified reliability level, and higher demand can

reduce unit operating costs, i.e., economies of scaleexist. In Section 2, we summarise some of the

relevant literature while Section 3 presents our

initial analytical model assuming operating costs

per unit to be constant. In Section 4 we show how

our model differs from the existing models in the

literature which assume price to be independent of

delivery time. Section 5 extends the basic model to

include economies of scale. Our conclusions andfuture research opportunities are provided in Sec-

tion 6.

2. Literature review

Since the paper by Stalk and Hout (1990) on

time-based competition, there has been extensiveresearch on the effects of customer responsiveness

as a strategic competitive weapon (Hum and Sim,

1996). While much of this literature is qualitative,

there are a number of quantitative contributions.

Many of the quantitative models focus on the ef-

fects of lead time reduction on operational deci-

sions such as batch size and quality (for a detailed

review refer to Karmarkar, 1993) and demand istypically assumed to be an exogenous parameter.

Economics and marketing oriented research

recognizes that longer lead times might have a

negative impact on customer demand. Research in

this area typically focuses on internal pricing and

capacity selection issues for service facilities by

taking into account user�s delay costs and capacity

costs (Dewan and Mendelson, 1990). There alsoexists a stream of literature that investigates the

use of quoted customer lead times to explore the

impact of due-date setting on demand and profit-

ability (refer to Weng, 1999). Hill and Khosla

(1992) constructed a model where demand is a

function of actual delivery time and price and the

firm�s objective is to maximize profit by optimal

selection of price and delivery time, but theirmodel is totally deterministic.

Several authors have investigated the issue of

shorter delivery time in a game-theoretic frame-

S. Ray, E.M. Jewkes / European Journal of Operational Research 153 (2004) 769–781 771

work. For example, Lederer and Li (1997) studied

competition between firms serving delay-sensitive

customers and the resulting impact on price, pro-

duction rate and scheduling policies. Ha (1998)

derived pricing schemes that induce customers tochoose optimal service rates in a competitive

framework when services are jointly produced by

the customers and the facility. For other related

research based on game-theoretic models refer to

literature review in So and Song (1998).

While all the above lines of research are im-

portant, as So and Song (1998) point out, they

are basically different from the recently popularstrategy of committing to a ‘‘uniform’’ delivery

time guarantee for all customers, the focus of this

paper. In the case of a delivery time guarantee,

firms advertise a uniform delivery time for all

customers within which they guarantee to satisfy

most customer orders. The length of the delivery

time is a decision variable that directly affects

overall demand and a reliability constraint is usedto ensure a satisfactory service level once the de-

livery time is selected. In perhaps the first research

paper directly addressing the issue of uniform

guaranteed delivery time, So and Song (1998)

model the firm as a queuing system where the

mean customer demand has a log-linear relation-

ship with price and delivery time. The objective is

to maximize the profit per unit time by suitableselection of the decision variable values––length of

the guaranteed delivery time, price and capacity.

They characterize the optimal decision, perform

analytical comparative statics and provide useful

managerial insights on the effect of operating

characteristics on the optimal strategy of a firm. In

Palaka et al. (1998), the objective function, ca-

pacity costs and decision variables are similar tothat of So and Song albeit in a linear demand

framework. However, Palaka et al. explicitly take

into consideration work-in-process and penalty

costs whereas So and Song do not. So (2000) ex-

tended So and Song�s work by focusing on how

firms select the best price and guaranteed delivery

time in the presence of multiple-firm competition

and how different firm and market characteristicsaffect the optimal strategy. Rao et al. (2000) inte-

grate a uniform delivery time strategy with pro-

duction planning for a make-to-order firm with

time-dependent demand. The production schedule

for the firm is synchronised with the guaranteed

delivery time and the firm optimises on the deliv-

ery time to maximize the average expected profit

per period.While the papers based on uniform delivery

time framework assume demand per unit time to

be dependent on price and/or delivery time, they

do not consider the relationship between price and

delivery time. As mentioned earlier, customers

may be willing to pay a price premium for shorter

delivery times. We extend previous research by

explicitly modeling such a relationship betweenprice and delivery times. Numerical examples from

previous research also show that operating costs

play an important role in the optimal decision.

However, no previous work analytically models

the effect of demand on operating costs. We in-

clude economies of scale by modeling the unit

operating cost as a decreasing function of the

mean demand rate.

3. The analytical model

We consider a firm that announces a uniform

guaranteed delivery time (L), a decision variable,

for all its customer orders. Orders arrive according

to a Poisson process with mean rate k. The pro-cessing times of the orders are exponentially dis-

tributed with mean rate l. Customers are served in

a first-come-first-served fashion, and the arrival

rate depends on the market price of the service/

product (p) and the delivery time, L. We assume

that customers prefer shorter delivery times and

lower prices and that price is related to the length

of the guaranteed delivery time; specifically thatthe price, p, is higher for a shorter L. The firm has

established an internal target delivery time reli-

ability level, sR (0 < sR < 1), which is the proba-

bility that a random customer will have an actual

waiting time of L or less. As failure to satisfy an

arriving customer within the guaranteed lead time

L might have an adverse impact on repeat busi-

ness, the firm has set the internal target to be closeto 1.

The firm can invest in increasing the processing

rate, l, through for example, hiring extra workers


or acquiring improved equipment. In general, it is

reasonable to assume that successive investments

in increasing l by the same amount will cost equal

or more, i.e., the investment function, MðlÞ, is

increasing and linear/convex in l. In our initialmodel, we will assume that the investment function

takes a linear form, Al (So and Song, 1998; Palaka

et al., 1998), to obtain analytical results. We will

later discuss the extension to non-linear investment

functions. The firm has a unit operating cost of mper unit, initially assumed to be constant (i.e.,

economies of scale do not exist). Finally, the ob-

jective of the firm is to maximize its profit per unittime subject to satisfying the delivery reliability

constraint.

To further characterize the analytical model, we

now elaborate on the precise relationships between

k, p, and L. We assume that the mean demand rate

depends linearly on L and p, i.e.,

kðp; LÞ ¼ a� b1p � b2L; ð1Þ

where a denotes the mean demand rate when both

p and L are zero (a higher value of a represents a

higher overall potential for demand) and b1 and b2represent the price and lead time sensitivities of the

mean demand rate, respectively (a; b1; b2 > 0).

The linear demand function will help us to ob-

tain qualitative insights without much analyticalcomplexity. It also has the desirable properties

that the price and lead time elasticity of demand

are higher at higher prices and guaranteed delivery

times (refer to Palaka et al., 1998).

We explicitly model price premiums for shorter

delivery times by assuming that the price is deter-

mined by the length of the guaranteed delivery

time and that a shorter delivery time can commanda higher market price. For a given L, the market

price, p, is given by

p ¼ d � eL; ð2Þ

where d ¼ price when L ¼ 0, i.e., the maximum

price the market is willing to pay, and e ¼ deliverytime sensitivity of price (d; e > 0).

Combining (1) and (2) we can express k in terms

of L and system parameters as

kðLÞ ¼ ða� b1dÞ � ðb2 � b1eÞL ¼ a0 � b0L; ð3Þ

where

a0 ¼ ða� b1dÞ and b0 ¼ ðb2 � b1eÞ:

While the link between k, p and L has been used

before in the literature, the explicit dependence of pon L is an additional relationship that we introduce

(Gupta and Weerawat, 2000, also relates price and

lead time in a similar fashion but in a different

context). Though the new link reduces the number

of decision variables, it captures for managers a

relationship that exists in practice, and, as we will

show later in the paper, could lead to a decision

error if ignored. First, we will assume a0 > 0, since

otherwise when b0 is positive, k will be negative for

all L. If b0 > 0, k decreases with L, which is the case

to which most operations management literature

refers. This represents the situation when custom-

ers are ‘‘more lead time sensitive than price sensi-tive’’. We will henceforth refer to these customers

as lead-time-sensitive (LTS) customers. Some

thought shows that b0 6 0 (i.e., b1eP b2) also

makes sense when customers are ready to wait

longer to pay a lower price, in which case k in-

creases (or remains constant) with L. In this case

the customers are ‘‘more price sensitive than lead

time sensitive’’. We will refer to these customers asprice-sensitive (PS) customers. Note that the cus-

tomer preferences are based not only on b1 and b2but also on e. This type of price and lead time

sensitivity of customers has been referred to in the

literature before. Blackburn et al. (1992) and

Smith et al. (1999) pointed out that there are both

‘‘price sensitive’’ and ‘‘time-sensitive’’ customers in

the market. The former segment prefers a lowerprice even with longer delivery times while the

latter segment is ready to pay a price premium for

shorter delivery lead times.

Since the firm wishes to maximize profit per unit

time, its goal can be written as

ðP1Þ Maximize pðl;LÞ ¼ ½pðLÞ �m�kðLÞ �Al;

Subject to s¼ PðW < LÞ ¼ 1� e�ðl�kÞLP sR

ðdelivery reliability constraintÞ;l> k ðsystem stability constraintÞ;p > m> 0;L;k> 0

ðnon-negativity constraintsÞ;ð4Þ


where W is the steady state actual waiting time for

a random order, kðLÞ is given by (3), and pðLÞ by(2). For Poisson arrivals and exponential service

times assumptions, the form of the delivery reli-ability constraint is exact. However, note that for

high service levels, the tail of the waiting time

distribution is well approximated by the expo-

nential distribution even for a G=G=s queue (refer

to So and Song, 1998). Hence, our model is ap-

proximately valid for more general demand and

service time characteristics.

It is not difficult to show that P1 is decreasingconcave in l, and that at optimality, the delivery

reliability constraint must be binding (So and

Song, 1998; Palaka et al., 1998). The optimal l,lRðLÞ, is then

lRðLÞ ¼ � lnð1� sRÞL

þ kðLÞ: ð5Þ

Since the reliability constraint is binding, explicitly

modeling a penalty cost (per unit) in the objective

function is also straightforward (So, 2000).

Substituting lRðLÞ given by (5) for l, the profit

function in P1 can be expressed in terms of a single

variable L. Appendix A gives the details of deter-mining the profit-maximizing delivery time, L�, for

the firm. From the appendix it is clear that a mild

restriction on the parameter values can guarantee

a unique L�. If pðLÞ is positive for some feasible

region of L, the firm can announce L� and set its

processing rate based on (5) to satisfy the service

level. The optimal market price, p�, will be deter-

mined by (2). These p� and L� values will then in-duce the mean demand rate given by (3).

To illustrate, first let us provide two numeri-

cal examples with the following parameters: a ¼250, b1 ¼ 5, d ¼ 30, e ¼ 15, m ¼ 2, A ¼ 12 and

sR ¼ 0:99. The optimal solutions are provided in

Table 1.

Table 1

Comparison of L�, lðL�Þ, pðL�Þ and pðL�Þ for Examples 1 and 2

b0 L� p� lRðL�Þ kðL�Þ pðL�ÞExample 1 )55 0.23 26.48 132.54 112.90 1173.58

Example 2 25 0.18 27.35 121.62 95.58 963.23

Example 1 (b2 ¼ 20). We have b0 ¼ b2 � b1e ¼�55, and k ¼ 100þ 55L. In this example, custom-

ers are price sensitive.

Example 2 (b2 ¼ 100). We have b0 ¼ b2 � b1e ¼25 and k ¼ 100� 25L. In this example, customers

are lead time sensitive.

Recall that for b0 6 0, customers are price sen-

sitive––they are willing to wait longer if they can

pay a lower price. Comparison of Examples 1 and

2 shows the effect of the change in sign of b0 on theoptimal decision variable values – capacity, de-

mand and profit. For PS customers, it is intuitive

that L� is larger and p� is smaller than for LTS

customers. Note that the above examples are only

illustrative––depending on the parameter values,

the differences in L� and p� can be higher or lower.

Obviously as L� increases, p� will decrease.

From (5) we can also show that lRðLÞ is monotonedecreasing in L for LTS customers (b0 > 0), but maynot be monotone for PS customers (b0 6 0). For PS

customers, as L initially increases from a very low

value, the capacity costs decrease. However, as Lbecomes large, the price becomes very low. This

brings about high demand from PS customers and

then capacity cost again increases. It is for this

reason that l� for PS customers is greater than l�

for LTS customers in the examples.

3.1. Comparative statics and managerial insights

Now that we have seen how to determine the

optimal delivery time, it is useful from a manage-

rial viewpoint to understand how the behavior of

the optimal solution will be affected by changes insystem parameters. Table 2 illustrates the effects of

some firm and market characteristics on the opti-

mal delivery time, L� (see Appendix B for mathe-

matical proofs). While such analysis can be done

Table 2

Change in L� with increase in individual parameter values

Parameter Effect on L�

for b0 > 0

Effect on L� for b0 6 0

A Increases May increase or decrease

m Increases Decreases


for all the parameters (Ray, 2001), we focus on

two that generate seemingly counter-intuitive re-

sults. Specifically, we look at what happens to L�

as each of the parameter value increases.

The initial observation from Table 2 is that theeffect of the parameters on L� depends strongly on

sign of b0. Hence, managers must first ascertainwhether they are competing for primarily lead timesensitive (b0 > 0) or price sensitive (b0 6 0) custom-ers when deciding their delivery time strategy.

Impact of capacity expansion costs, A. A firm

serving LTS customers and mildly PS customers,

will naturally seek to increase L�, and thus reducedemand and hence the investment requirement.

For firms with highly PS customers, any increase

in L� will decrease price and so increase demand

and also the capacity investment requirement.

Hence, for such firms, as investment costs increase,

it is optimal to decrease L� (i.e., compete based ontime) despite the fact that they have PS customers.

This is less obvious, but understandable from thepoint of view that such an action will increase

price, decrease demand and hence their capacity

investment.

Impact of unit operating costs, m. Firms with

low unit operating costs (m) and PS customers

should guarantee a relatively long delivery time.

This will lead to a low market price, high demand

and ultimately maximized profits. Alternately,firms with low unit operating costs with LTS

customers should compete based on time in order

to capture the maximum price premium. For high

values of m, the profit-optimizing strategy for such

firms is the reverse.

Note that the implications provided by our re-

sults are consistent with actual industry practice.

For example, some courier services companies andthird-party logistics providers initially started by

serving PS customers. However, as demand in-

creased and capacity costs (Al) became an issue,

these companies resorted to time-based competi-

tion by guaranteeing a smaller delivery time and

charging a price premium. We can also relate the

insight provided by the effect of operating costs (m)in the following setting. Compare the strategy oflocal small computer ‘‘clone’’ assemblers to that of

a large company such as Dell both of which have

low operating costs, the former because of low

infrastructure costs and the latter because of effi-

ciency. Since Dell serves LTS customers, it com-

petes based on time and extracts a price premium

their customers are willing to pay. The local as-

semblers know that their customers are price sen-sitive in nature, and so they compete based on

price rather than time in order to maximize their

profits. Hence, companies with the same operatingcosts may choose to compete on a different basisif they are aware of their customer preferences.However, as we indicated before, firms with high

operating costs will have just the opposite strategy

for each type of firm.In the next section we show that our model is

quite different from models where price is assumed

to be an independent decision variable (as in So

and Song, 1998; Palaka et al., 1998).

4. Price as an independent decision variable

In this section, our goal is to see the effect on

optimal decision variable value(s) and profit if

managers fail to take into account the dependence

of price on the length of the delivery time (as in

(2)) and assume p to be a decision variable inde-

pendent of L.With p and L both as decision variables,

kðp; LÞ ¼ a� b1p � b2L (b1; b2 > 0). The firm�sprofit-maximization problem is similar to P1 with

pðl; p; LÞ ¼ kðp; LÞðp � mÞ � AlRðp; LÞ. Note that

the optimal capacity, lR, in this case depends

on both p and L, lRðp; LÞ ¼ ð� lnð1� sRÞ=LÞþkðp; LÞ. We will refer to this model as Model 1 and

its optimal price and delivery lead time guarantee

as p�1 and L�1 respectively. Our original model will

be referred to as Model 0.

Proposition 1. The optimal price for Model 1, p�1,can be found as follows:

p�1ðLÞ ¼Aþ m

2þ a2b1

� b22b1

L: ð6Þ

Assuming that the price will be determined optimallyfor all L, a mild condition can guarantee that thefirm will be able to ascertain its unique optimaldelivery time, L�

1, by solving a relatively simpleequation.


Proof. Refer to Appendix C. �

Comparing (2) and (6), it is immediately obvi-

ous that the optimal price is decreasing in deliverytime for both models, but that the rate of decrease

will depend on whether customers are price sensi-

tive or lead time sensitive. For PS customers, the

rate of decrease for Model 0 is greater than Model

1; for LTS customers, the result will depend on the

relative values of b1, b2 and e.Note that when the relationship between price

and delivery time is not considered, then the de-mand sensitivity of the customers is based only on

the relative values of b1 and b2. The sensitivity of

price itself towards delivery time (e) is not taken

into account and hence we do not have infor-

mation about the overall customer preferences.

Comparing the optimal delivery lead time values

for the models, we have the following proposition:

Proposition 2. L�1 6¼ L� except in the special case

when L�1 ¼ L� ¼ ðA2� A1Þ=ðB1� B2Þ, where A1 ¼

b0ðAþ m� dÞ � a0e, A2 ¼ ðb2=2ÞðAþ mÞ � ðb2a=2b1Þ, B1¼2eb0, B2¼ðb22=2b1Þ and C¼�Alnð1�sRÞ.Specifically, when A16A2, L�<L�

1.

Proof. Refer to Appendix D. �

Proposition 2 shows that taking into account

the relationship between price and delivery time

will, in general, give a different optimal solution

than assuming p is an independent decision vari-

able. The difference in optimal delivery times im-

plies that there will be a profit penalty for firms in

not taking into account the dependence of price on

delivery time.

Example 3. Using the same parameter values as in

Example 1, but assuming that p and L are inde-

pendent decision variables, we have p�1 ¼ 31:20and L�

1 ¼ 0:40. Note that L�1 is almost 74% more

than L�. Using L�1 in place of L� in Model 0 will

lead to p ¼ 24 and an associated profit of 1081.1, a

loss in profit of almost 8.6%.

While Example 3 is simply illustrative, other

computational work confirms that L�1 can be either

larger or smaller than L�, and that the loss in profit

can be substantial. Using L�1 in Model 0 can lead to

two types of ‘‘mistakes’’––(1) investing in lead time

reduction to guarantee a shorter delivery lead time

when customers are price sensitive, want lower

prices and are willing to wait longer, or (2) notproviding short enough lead times when the mar-

ket is willing to pay a price premium for shorter

lead times. These results are consistent with the

empirical work of Sterling and Lambert (1989)

who found that management frequently sets at-

tribute levels inconsistent with customer prefer-

ences, not realising that customers have different

needs than the seller.We have established the significance of our

model and the difference between it and those ex-

isting in the literature. From a managerial view-

point, it is important to understand under what

conditions will it be essential to explicitly account

for the interaction between price and delivery time,

i.e., when using L�1 in place of L� for Model 0 will

result in a substantial profit loss for the firm andwhen such a step will be valid. We list the condi-

tions below (refer to Appendix D):

• Large difference between L�1 and L� and hence

substantial profit loss by using L�1 for L� in

Model 0 can occur

– for large values of e, i.e., price is very sensi-

tive to L (L� � L�1),

– for large values of b1, i.e., demand is very

sensitive to price, and large values of d, i.e.,potentially expensive product (L� � L�

1),

– for large values of b1 and small values of d(L� � L�

1),

– for large values of b2, i.e., demand is very

sensitive to L (L� � L�1),

– for small values of A and/or m and large val-ues of b1 and/or e (L� � L�

1).

• L�1 � L� and profit loss by using L�

1 for L� in

Model 0 is small

– for large values of a (i.e., high potential for

demand),

– for small values of b0.

Based on these results, we observe that it isimportant to take the price and delivery time re-

lationship into account when the firm is serving

either (I) very LTS customers, or (II) very PS


customers, or (III) potentially very expensive

products, or (IV) when price is very delivery time

sensitive. On the other hand, when the potential

market is very large or when the customers are not

very sensitive towards either price or delivery time,ignoring the additional relationship will not be a

major cause of concern.

The above analysis showed the potential ad-

verse effects of using L�1 in place of L�. We now

focus on the effects of making an ‘‘error’’ in cal-

culation of L� itself. As in So and Song (1998), it is

possible to show that for both LTS and PS cus-

tomers, it is more harmful (in the sense of profitloss) to guarantee a shorter than optimal delivery

time compared to longer than optimal delivery

time provided the deviations are equal (Appendix

E). If managers are not sure about their parameter

values, it is better to err on the side of caution and

guarantee a ‘‘higher than optimal’’ delivery lead

time. We can even show that for any delivery time,

whether above or below the optimal (but close toit), the penalty for deviating from the optimal is

more for Model 0 than for Model 1, provided the

deviations are of equal amount (Appendix F).

Therefore, the loss of profit from making ‘‘errors’’

in calculating the optimal delivery time is greater

for Model 0 than for Model 1. From a managerial

viewpoint, this implies that the managers must bemore careful choosing the optimal delivery timewhen market price is governed by delivery time ra-ther than when price is a decision variable.

1 In this paper Zy represent the first derivative and Zyyrepresent the second derivative of Z with respect to y and !represents tends towards.

5. Incorporating economies of scale

Companies may be able to achieve economies of

scale by spreading fixed costs over a larger pro-duction volume (Scherer, 1980). For such opera-

tions, it is reasonable to assume that the unit

operating cost is a decreasing function of the de-

mand rate, at least within a certain volume range.

As Section 3 in this paper and numerical examples

of So and Song (1998), Palaka et al. (1998) and So

(2000) show, operating costs might have a signifi-

cant impact on the delivery time strategy of a firm.In this section, we analytically explore the impli-

cations of scale economies on the basic model in-

troduced in Section 3. While the exact nature of the

scale economies will depend on many factors, here

we analyse the case where the unit operating cost,

m ¼ ukð�vÞ, is decreasing convex with respect to the

mean demand rate. In our model, vP 1 (i.e., suf-

ficiently decreasing and convex) represents thesensitivity of unit operating cost to the mean de-

mand rate and u is a finite constant denoting the

operating cost for unit demand rate. The relation-

ships between k and L, p and L and form of in-

vestment function remain the same as in Section 3.

The optimisation problem can now be written

as

ðP2Þ Maximize pðl;LÞ ¼ ðp�mÞk�Al;

Subject to PðW < LÞPsR; i:e:; 1� e�ðl�kÞLPsR

ðdelivery reliability constraintÞ;l> k ðsystem stability constraintÞ;p>m> 0;L;k> 0

ðnon-negativity constraintsÞ;ð7Þ

where p ¼ d � eL, k ¼ a0 � b0L and hence m ¼uða0 � b0LÞð�vÞ

. It is clear that m is decreasing

convex with respect to k. When b0 6 0 (PS cus-

tomers), kL is non-negative and m is non-increasing

convex in L and when b0 > 0 (LTS customers), kL isnegative and m is increasing convex in L.

The reasoning in Section 3 showing that the

optimal l will be along lRðLÞ given by (5), is still

valid as the expression for lRðLÞ is independent ofm and so remains unchanged. Substituting lRðLÞfor l, the profit-maximization problem can again

be expressed in terms of a single decision variable

L. We will refer to this model as Model 2 and theoptimal price and guaranteed delivery time of this

model as p�2 and L�2 respectively.

Differentiating the profit function with re-

spect to L and analysing we can show that for

b0 6 0 there can be at most one feasible solution

to 1pL ¼ 0. If any feasible solution exists, it will be

the optimal delivery time, L�2, and if there is no

feasible solution, then L�2 is one of the feasible

limits of L (Appendix G). The profit function for

0.118

0.12

0.122

0.124

1 1.5 2 2.5 3v

0.23

0.24

0.25

0.26

1 1.5 2 2.5 3v

L*

L2*

L*

L2*

(a)

(b)

Fig. 1. Optimal delivery time versus v for Models 0 and 2: (a)

b0 > 0 and (b) b0 6 0.


b0 > 0 might not be unimodal. However, as has

been shown in Ray (2001), it is not difficult to

determine L�2 by a simple comparison of the profit

values at a finite number of possible alternatives.

Once L�2 is determined and if pðL�

2Þ > 0, the firmcan announce L�

2 and the optimal processing rate

and market price will be given by (5) and (2) re-

spectively. The resulting mean demand rate will

induce the operating cost, mðL�2Þ, and the firm�s

profit will be maximized.

In the following parts of this section, we assume

that there is a unique profit maximizing L�2 and

provide insights on the effects of economies ofscale on the optimal policy. We begin with the

particular case of v ¼ 1.

Proposition 3. For v ¼ 1, L�2 < L� when b0 > 0, and

L�2 P L� when b0 6 0.

Proof. Refer to Appendix H. �

For v > 1, analytical results are much more

difficult to obtain. However, based on numerical

experiments, we have been able to ascertain the

effects of increasing v. We observed that

• in general (i.e., even for v > 1), L�2 < L� when

b0 > 0 (LTS customers), and L�2 P L� when

b0 6 0 (PS customers);• as v increases, L�

2 changes in a convex–concave

manner for b0 > 0 (Fig. 1(a)) and in a con-

cave–convex manner for b0 6 0 (Fig. 1(b)).

The above observations are quite intuitive, but

have managerial importance. For LTS customers,

there is an incentive to guarantee a shorter delivery

time in order to attract more demand and therebydecrease m. Thus, for low values of v, as v in-

creases, L�2 decreases. Because of higher demand,

the firm must invest in increasing the processing

rate to satisfy the reliability constraint. Once vincreases to a point where the capacity investment

costs are offsetting the scale economies, L�2 starts to

increase with v. A similar, but reverse explanation

holds for PS customers. The important managerialinsight to note here is again the significant role thatcustomer characteristics play in setting the optimaldelivery time.

Example 4. In this example, we assume the same

parameters as in Examples 1 and 2 with u ¼ 115

and v ¼ 1:5 (to ensure that the operating cost with

economies of scale is comparable to m). We have

L�2 ¼ 0:25 for Example 1 (compared to L� ¼ 0:23)

and L�2 ¼ 0:17 for Example 2 (compared to L� ¼

0:18). While the optimal decision variable values

are different, the profit penalty of using L�2 for L

� in

Model 0 is not severe.

For a broad range of numerical experiments, we

observed results consistent with Example 4: that

incorporating economies of scale leads to a dif-ferent optimal delivery lead time than not model-

ing economies of scale, but the loss of profit is not

severe as long as the price delivery time relation-

ship captured, even for a comparatively large value

of v. This means that managers should determine therelation between market price and delivery timebefore investigating the impact of economies ofscale.


6. Conclusions and future research opportunities

In this paper, we model an operating system

consisting of a firm and its customers where themean demand rate is a function of market price

and guaranteed delivery time and the market price

is determined by the length of the delivery time.

The firm can invest in increasing capacity to

guarantee a shorter delivery time but must be able

to satisfy the guarantee according to a pre-speci-

fied reliability level. Our model accounts for whe-

ther customers are ‘‘price sensitive’’ or ‘‘lead timesensitive’’ by capturing the dependence of bothprice and demand on delivery time. We show how

the firm can determine the optimal length of the

guaranteed delivery time, analyse the optimal

policy and study the change in the behavior of the

policy as various system parameters change. We

provide examples where our insights are consis-

tent with actual practical situations. Our analy-sis clearly shows that the time-based competitive

strategy for firms whose customers are more sen-

sitive towards price than delivery time will be dif-

ferent from firms whose customers want shorter

delivery time and is ready to pay a price premium.

Hence, the primary concern for managers would beto understand customer characteristics––based onthe simultaneous dependence of price and demand ondelivery time––before deciding on a delivery timestrategy. We also show how our model is quite

different from those in the literature that assume

price and delivery time to be independent decision

variables. We specifically address under what

conditions ignoring the relation between price andthe delivery time can lead to substantial profit pen-alty for the firm. One of our more interesting re-sults is that when market price is dependent on

delivery time, managers need to take greater care

in choosing the optimal delivery time as compared

to when price is a decision variable.

We then extended our model by incorporating

economies of scale where the unit operating cost is

a decreasing function of the mean demand rate.

We show that for practicing managers it is im-portant not only to know the lead time sensitivity

of price, but also to take into account the effect of

economies of scale, when they are present. How-ever, our results indicated that the effect of delivery

time on price seems to be more important than thatof economies of scale.

The analytical results for this paper are based

on the assumption of a linear investment function.

We also investigated the impact of non-linear in-vestment functions. Analytical work and numeri-

cal experiments with the investment function Ali

(i > 1) show that most of the qualitative insights

from sensitivity results of Model 0 still hold. The

effect of increasing i will be similar to that of in-

creasing A, since for either case (i.e., an increase in

A or i) it will require more investment to attain a

particular processing rate. For LTS customers andmildly PS customers, as i increases, L� will increase

in order to reduce the investment cost. If the cus-

tomers are extremely price sensitive, L� could de-

crease with increase in i. When L� increases, it does

so in a concave manner for PS customers while for

LTS customers the increase in L� follows a S-curve

pattern. In our experiments we also noted that for

higher values of i, it becomes more important toinclude the additional price and delivery time re-

lationship while deciding on the optimal delivery

time (i.e. the profit penalty of using L�1 for L� in

Model 0 is more).

There are a number of further research oppor-

tunities for our model:

ii(i) Extending this work in a competitive frame-work (e.g., So, 2000), i.e., what managerial in-

sights can be provided in the presence of

multiple-firm competition and how different

firm and market characteristics would affect

the optimal delivery time strategies.

i(ii) Rather than assuming service level to be a

constraint, make it a decision variable (it will

perhaps model the small, repetitive customerswell).

(iii) Model the dependence of demand on price and

delivery time and price on delivery time in a

non-linear fashion (e.g., So and Song, 1998).

Acknowledgements

Helpful comments from an anonymous referee

and Prof. Yigal Gerchak are gratefully acknowl-

edged.


Appendix A

For b0 > 0, the feasible region for L is 0;ðmin a0

b0 ;d�me

� �Þ and for b0 6 0, the feasible region for

L is 0; d�me

� �� . As L tends towards the feasible

limits, the profits are negative––infinite at lowerlimit and finite at the upper feasible limit (say, LU).

Expressing the profit function in terms of a single

variable L, differentiating with respect to L and

simplifying we have

pLðLÞ ¼ b0ðAþ m� dÞ � a0eþ 2eb0L

� A lnð1� sRÞL2

; ðA:1Þ

pLLðLÞ ¼ 2eb0 þ 2A lnð1� sRÞL3

: ðA:2Þ

As L ! 0, the profit function is increasing concave

in L.

(a) If b0 6 0, pLL is non-positive which implies pðLÞis concave in L. If the solution to pL ¼ 0 is fea-sible then it will be L�, and if it is not, then

L� ¼ LU (since p will be increasing).

(b) If b0 > 0, pLL is non-positive till L ¼�2A lnð1�sRÞ

2b0e

h i1=3and then positive. This implies

that pðLÞ is concave for 0; �2A lnð1�sRÞ2b0e

h i1=3� �

and then convex. Therefore there can be at

most two solutions to pL ¼ 0. If both solutions

are feasible, then the smaller one will be L�

(since p is initially increasing); if only one solu-

tion is feasible, it will be L�; if none are feasi-

ble, then L� ¼ LU (since p will be increasing).

From (a) and (b) we can conclude that for any

b0, pLðLÞ < 0 for L ¼ LU is sufficient to guarantee

an interior profit-maximizing solution. Note that a

necessary condition for pLðLÞ < 0 at L ¼ LU to

hold is that a0 must be sufficiently positive.

Appendix B. Proof of comparative statics results

Assuming that pL ¼ 0 has a unique solution im-

plies that L� will be determined by that solution and

pLL ¼ 2b0eþ 2A lnð1� sRÞL�3 6 0:

Total differentiation of (A.1) with respect to Agives

oL�

oA¼

�b0 þ lnð1�sRÞL�2

2b0eþ 2A lnð1�sRÞL�3

:

If b0 > 0, ðoL�=oAÞ > 0. If b0 6 0, (oL�=oA) is

unrestricted in sign. However, if sR is high and b0

is not very negative, oL�=oA will be positive; if sR is

low and b0 � 0, oL�=oA can be negative.

Total differentiation of (A.1) wrt m gives

oL�

om¼ �b0

2b0eþ 2A lnð1�sRÞL�3

:

when b0 > 0, ðoL�=omÞ > 0. If b0 6 0, ðoL�=omÞ6 0.

Appendix C. Proof of Proposition 1

For this model, the profit function will be given

by

pðp; LÞ ¼ ½a� b1p � b2L�ðp � mÞ

� A� lnð1� sRÞ

L

�þ ½a� b1p � b2L�

�:

ðC:1Þ

Since ppp is negative, solving pp ¼ 0 we have p�1ðLÞof (6). Substituting p�1ðLÞ for p in (C.1) and dif-

ferentiating with respect to L, we have

pLðp�1ðLÞ; LÞ ¼ ðb2=2ÞðAþ mÞ � ðb2a=2b1Þ

þ ðb22=2b1ÞL� A lnð1� sRÞL2

ðC:2Þ

and

pLLðp�1ðLÞ;LÞ ¼ ðb22=2b1Þ þ2A lnð1� sRÞ

L3: ðC:3Þ

Following the same logic as in Appendix A (since

e ¼ 0, so b0 > 0), we can then convince ourselves

that as long as a is sufficiently high, L�1 will be

given by the unique solution to (C.2)¼ 0.


Appendix D. Proof of Proposition 2

With A1 ¼ b0ðAþ m� dÞ � a0e, A2 ¼ ðb2=2Þ�ðAþ mÞ � ðb2a=2b1Þ, B1 ¼ 2eb0, B2 ¼ ðb22=2b1Þ andC ¼ �A lnð1� sRÞ, we have

pLðLÞ ¼ A1þ B1�Lþ C;

pLðp�1ðLÞ;LÞ ¼ A2þ B2�Lþ C:

Note that A2 must be negative to have an interiorsolution ) A2 < 0, B2 > 0 and C > 0.

pL � pLðp�1ðLÞ;LÞ ¼ ðA1� A2Þ þ ðB1� B2ÞL;

pLL � pLLðp�1ðLÞ; LÞ ¼ ðB1� B2Þ:

B1� B2 ¼ �ð2b1e� b2Þ2

2b1< 0: ðD:1Þ

Since we are only comparing the optimal values,

the portion of interest is the increasing, concave

portion of the curve. Both pL and pLðp�1ðLÞ; LÞ will! þ1 as L ! 0 and will surely be decreasing upto their optimal values. Since ðB1� B2Þ is nega-

tive, it implies that pL will decrease faster than

pLðp�1ðLÞ;LÞ. If A16A2 (remember A2 is negative),

pL will start below pLðp�1ðLÞ; LÞ and also decrease

faster implying that L� < L�1. If A1 > A2, then until

L ¼ ðA2� A1Þ=ðB1� B2Þ, pL > pLðp�1ðLÞ; LÞ and

after that pL < pLðp�1ðLÞ; LÞ. So, the only way

L� ¼ L�1 is if both of them are equal to

ðA2� A1Þ=ðB1� B2Þ. Note that if A1 > A2, L� can

still be smaller than L�1 if ðB1� B2Þ is highly neg-

ative. If A1 is highly positive and ðB1� B2Þ is notnegative enough, then L� > L�

1.

Comparing the values of A1, A2, B1 and B2 it is

possible to deduce the conditions when L� and L�1

will differ significantly, or not. For example, if e islarge, both A1 and ðB1� B2Þ will be very negative.This implies that pL will start much below

pLðp�1ðLÞ;LÞ and also decrease much faster imply-

ing that L� � L�1

Appendix E

Let P ðLÞ ¼ ½�A lnð1� sRÞ�=L2 implying thatP ðLÞ is positive, decreasing convex in L. L� will be

determined by the solution to pL ¼ 0 which on

simplification yields

a0e� b0ðAþ m� dÞ � 2b0eL� ¼ P ðL�Þ:pLðL� þ dÞ ¼ �½P ðL�Þ � 2b0ed� P ðL� þ dÞ� and

pLðL� � dÞ ¼ �P ðL�Þ � 2b0edþ P ðL� � dÞ. Since,

P ðLÞ is positive, decreasing convex in L, so 0 <P ðL�þdÞ<P ðL��dÞ and 0<P ðL�Þ� P ðL�þdÞ<P ðL��dÞ�P ðL�Þ, implying that pLðL��dÞPjpLðL�þdÞj.

Appendix F

From (D.1), we already know that pLLðLÞ�pLLðp�1ðLÞ; LÞ < 0. Since both pLLðLÞ and pLLðp�1ðLÞ;LÞ are negative until the point they are concave

(which surely includes L� and L�1), we can conclude

that pðLÞ is more concave than pðp�1ðLÞ; LÞ in thevicinity of the optimal values. This implies that the

penalty for deviating from the optimal is more for

Model 0 than for Model 1, provided the deviations

are of equal amount.

Appendix G

Differentiating the profit function in terms of

the single decision variable L for Model 2 we have

pL ¼ ð�b0Þðp � mÞ � ekþ ð�b0Þuvkð�vÞ � AolR

oLðG:1Þ

and

pLL ¼ 2kLpL � 2kLmL � kmLL � Ao2lR

oL2: ðG:2Þ

On rearranging the terms of pL ¼ 0 we have

kLðp � mÞ þ kpL � kmL ¼ AolR

oL: ðG:3Þ

Differentiating both sides of (G.3) with respect to Lwe have

o

oLðLHSÞ ¼ 2kLpL þ ðuvÞðkLÞ2ðk�ðvþ1ÞÞð1� vÞ;

ðG:4Þ

o2

oL2ðLHSÞ ¼ �ðuvÞð1þ vÞð1� vÞðk�ðvþ2ÞÞðkLÞ3;

ðG:5Þ


o

oLðRHSÞ ¼ A

o2lRðLÞoL2

� �: ðG:6Þ

We can show that the RHS of (G.3) is increasing

and concave in L, negative until L ¼ffiffiffiffiffiffiffiffic

ð�b0Þ

qand

then positive (as L ! 0, the RHS ! �1). We can

also show that the LHS will be unrestricted in sign,

finite as L ! 0 and decreasing convex in L. Hence,

we can convince ourselves that there can be zero or

one feasible solution to pL ¼ 0. If the solution is

feasible, it must be L�2 (since as L ! 0, pL is in-

creasing concave). If there is no feasible solution,

one of the limits of L will be L�2 depending on

whether the profit function is increasing or de-

creasing.

Appendix H. Proof of Proposition 3

For v ¼ 1, pLL for Model 2 will be the same as

for Model 0. For pL, B1 and C will remain thesame. However, A1ðModel 2Þ ¼ A1ðModel 0Þ�b0m, implying that if b0 > 0, A1ðModel 2Þ <A1ðModel 0Þ and if b0 6 0, A1ðModel 2ÞPA1ðModel 0Þ. Hence for b0 > 0, pL for Model 2 will

intersect the L-axis earlier than for Model 1 and

for b0 6 0, pL for Model 2 will intersect the L-axislater than for Model 1 which proves the proposi-

tion.

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Documents

Customer lead time management when both demand and price are lead time sensitive