Reading Assignment Report -II

Supply chain focus dependent safety stock placement

Gangaraju Vanteddu Æ Ratna Babu Chinnam Æ Kai Yang Æ Oleg Gushikin

Published online: 18 April 2008

� Springer Science+Business Media, LLC 2008

Abstract Increasing globalization, growing product range diversity, and rising consumer awareness are making markets highly competitive, forcing supply chains

to adapt constantly to different stimuli. Growing competition between supply chains

(as well as players within them) is also warranting a priority for overall supply chain

performance over the goals of individual players. It is now well established in the

literature that, among the many order winners, both overall supply chain cost and

responsiveness (i.e., supply chain lead time) are the most significant determinants of

supply chain competitiveness. The literature, however, mostly focuses on supply

chain cost minimization with rather simplistic treatment of responsiveness. By

introducing the concept of a coefficient of inverse responsiveness (CIR), we facil-

itate efficient introduction of responsiveness related costs into the scheme of supply

chain (SC) performance evaluation and/or optimization. Thus, our model aids

supply chain managers in achieving better strategic fit between individual business

unit strategies and overall supply chain requirements in terms of cost efficiency and

responsiveness. In particular, it aids in strategic placement of safety stocks at dif-

ferent stages in the supply chain. Our model also offers managerial insights that help

improve our intuitions into supply chain dynamics. The model is more suited for

G. Vanteddu � R. B. Chinnam (&) � K. Yang Department of Industrial & Manufacturing Engineering, Wayne State University,

4815 Fourth Street, Detroit, MI 48202, USA

e-mail: [email protected]

G. Vanteddu

e-mail: [email protected]

K. Yang

e-mail: [email protected]

O. Gushikin

Ford Research and Advanced Engineering, 2101 Village Road, Dearborn, MI 48121, USA

e-mail: [email protected]

123

Int J Flex Manuf Syst (2007) 19:463–485

DOI 10.1007/s10696-008-9050-z

strategic SC alignment, for example, when dealing with product changeovers or

introduction of new product, rather than for operational control.

Keywords Supply chain strategy � Business strategy � Safety stock costs � Cost of responsiveness � Supply chain cycle time � Coefficient of inverse responsiveness

1 Introduction

In today’s dynamic global market environment, it has become necessary for

organizations to focus upon all the relevant performance metrics. Hill (1993)

advocates close attention to functional perspectives, customers’ views, and actual

orders while determining order winners and qualifiers. Among the possible order

winners, supply chain (SC) cost and responsiveness turn out to be more crucial than

others. Responsiveness here is the ability of the supply chain to respond quickly to

changing customer needs, preferences, and options through improved supply chain

cycle time (i.e., velocity), and not the higher service levels achievable through

increased distribution of channel inventory. The optimal supply chain strikes the

right balance between supply chain cost efficiency and responsiveness, depending

on the nature of product(s) supplied by it.

Figure 1 divides the cost-responsiveness spectrum of the SC into four basic

quadrants, and provides some indications of their appropriateness. In reality, the

treatment will not be that distinct, and depending on the type of products and

markets targeted by the SC, the focus could be anywhere on the spectrum. In this

context, Fisher (1997) emphasizes the importance of considering the nature of

product demand before devising the respective supply chains. More precisely, one

needs to account for both the nature of the product as well as uncertainty from both

demand and supply in designing the SC (Chopra and Meindl 2004). Similar

comments have been echoed in industry literature. Nazzal et al. (2006) present a

case study for Agere Systems, focusing upon the increase of market share and

profits by reducing lead times in the highly competitive semiconductor manufac-

turing industry. So is the example of Revlon whose supply chain includes more than

5,000 active finished good stock keeping units (SKUs) with product lifecycles

spanning less than 3 years. To meet aggressive inventory reduction targets and

achieving high customer service levels, Revlon is emphasizing reduction of

manufacturing and supplier lead times and the associated variability (Davis et al.

2005). Yang and Geunes (2007) emphasize that longer lead times, in addition to

reducing customer responsiveness, increase demand forecast error, since forecast

error generally increases as the forecast horizon increases. In addition, longer lead

times expose the supply chain to more in-process inventories, design changes,

degradation, accidents, changes in demand patterns, etc. (Felgate et al. 2007), which

in turn increase the supply chain costs. In summary, striking the right balance

between supply chain cost efficiency and responsiveness is critical, which can be

neglected at one’s own peril. On the contrary, existing literature overly focuses on

just one order winner: cost efficiency.

464 G. Vanteddu et al.

123

There are primarily four drivers influencing the performance of the SC, namely,

infrastructure, inventories, transportation, and information (Chopra and Meindl

2004). Given our assumption that the necessary network topology is already in

place, we do not need to include infrastructure related cost elements and

transportation related aspects explicitly in our model. These issues, however, are

addressed in an indirect fashion in our model. For example, cost added at a stage can

be considered to be a function of fixed costs associated with infrastructure such as

location, buildings, machinery, etc. and transportation to the immediate downstream

(D/S) stage. Even though we are developing the model assuming that all the stages

are involved in manufacturing, a stage purely dealing with transportation could be

easily accommodated. We are also assuming information symmetry at all the stages

and leave information asymmetry related issues for future research. Thus, we are

primarily considering the inventory cost driver in our model.

Among inventory cost elements, safety stock, maintained to account for the

internal and external variability in the supply chain, is vital in the sense that it

directly affects customer satisfaction and constitutes a significant portion of the cost

of goods sold (COGS).

In summary, in terms of supply chain strategy, one can choose to compete purely

on price (for a commodity product) or product velocity (fashion industry, for

example) or choose a third path, a niche strategy, a hybrid of the other two

(Cherukuri et al. 1995). Our research primarily focuses upon the third strategy,

where there are opportunities to be exploited with respect to both order winners.

The rest of this manuscript is organized as follows: Sect. 2 discusses relevant

literature; Sect. 3 deals with development of the overall cost expression for the

supply chain, which accounts for supply chain (SC) responsiveness; Sect. 4 offers

managerial insights into strategic safety stock placement in a serial supply chain;

Sect. 5 describes how one can use the model for achieving the strategic fit; finally,

H ig

h L

ow

C os

t

Low High

Responsiveness

Poor Strategy

Short Lifecycle Products

Mass Production

Ideal Strategy

Fig. 1 Supply chain responsiveness versus cost (source: Vanteddu et al. 2006)

Supply chain focus dependent safety stock placement 465

123

Sect. 6 offers conclusions and limitations that should be addressed in future

research.

2 Literature review

Gallego and Zipkin (1999) develop and analyze several heuristic methods to study

the problem of stock positioning in serial production-transportation systems and

offer a number of interesting insights into the nature of the optimal solution. A

stochastic service model as advocated by Graves and Willems (2003) addresses the

issue of strategic placement of safety stocks across a multi-echelon SC in the

presence of demand uncertainty. As with Graves and Willems (2003), the primary

emphasis of our research is to provide decision support for SC design, rather than

SC operation, and the intent is not to find inventory control policies as is the case

with much of the multi-echelon inventory literature. Another common feature is that

the local decisions (decentralized control as opposed to central control) are made at

each stage based on local information. It is worth noting at this point, as supported

by Graves and Willems (2003), that a SC subjected to decentralized control is not

equivalent to a SC being locally optimized. In an optimization context, the model

attempts to find the optimal parameter values, under the assumption of decentralized

control, that minimize the total SC cost. The primary purpose of their model is to

develop a multi-echelon model, and the relevant optimization algorithm is

specifically designed for optimizing the placement of safety stocks in a real-world

SC. Unlike Graves and Willems (2003), where inventory is the only lever to counter

demand and supply variability, our model has two levers, namely, echelon inventory

and responsiveness (cycle time).

The supply chain configuration problem, a special case of the supply chain design

problem, consists of the set of decisions that are made after the network design that

act to configure each stage in the network by taking into account certain relevant

parameters; see Graves and Willems (2003), who argue that the central question is

to determine what suppliers, parts, processes, transportation modes (called options)

to select at each stage in the supply chain. Our model is similar in spirit to their

optimization model (Graves and Willems 2003). The primary similarity is that both

models balance the increase in cost of goods sold (COGS) against the decrease in

inventory related cost, though the two models differ significantly with regard to the

methodology adopted and serve different end purposes.

For a stochastic service model (Graves and Willems 2003; Simchi-Levi and Zhao

2005; Lee and Billington 1993; Ettl et al. 2000), which we have adopted in our model,

we assume that the increase in cost at a stage depends on the opportunities that exist for

resource flexibility and model it as a continuous function of a novel dimensionless

parameter called the coefficient of inverse responsiveness (CIR), with a focus on: (a)

developing managerial insights with regard to strategic safety stock placement in a

serial supply chain and (b) achieving strategic fit between the supply chain strategy and

the individual business strategies for a given set of parameter values.

Assuming an installation, continuous time base stock policy for supply chains

with tree network structures, another interesting paper that is based on the stochastic

466 G. Vanteddu et al.

123

service model concept is that by Simchi-Levi and Zhao (2005) in which they derive

recursive equations for the back order delay (because of stock out) at all stages in

the supply chain, and, based on those recursive equations, dependencies of the back

order delays across different stages of the network are characterized and useful

insights with respect to the safety stock positioning are developed in various supply

chain topologies. The two other papers that use a stochastic service approach that

are relevant to our research are those by Lee and Billington (1993) and Ettl et al.

(2000). We share the ample production capacity assumption with Simchi-Levi and

Zhao (2005) and Graves and Willems (2003). For capacitated models using a

modified base stock policy, the reader can refer to Glasserman and Tayur (1995,

1996) and Kapuscinski and Tayur (1999). Simchi-Levi and Zhao (2005) enhance the

work of Gallego and Zipkin (1999) and provide stronger results. They also point out

through their computational study of the Bulldozer supply chain problem (Graves

and Willems 2003) the perils of ignoring lead time uncertainties, which is accounted

for in our model. In addition, Simchi-Levi and Zhao (2005) present a nice summary

of the literature for installation policies that are used in various network topologies:

multistage serial systems (Simpson 1958; Hanssmann 1959; Lee and Zipkin

1992) and distribution systems (Axsäter 1993; Graves 1985; Lee and Moinzadeh

1987a, b).

Given the size and complexity of supply chains such as those found in the

computer, electronics, and auto industries, a common problem for asset managers is

not knowing how to quantify the trade-off between service levels and the investment

in inventory required to support those service levels (Ettl et al. 2000). This problem

is compounded by the response related performance measures. Similar to Ettl et al.

(2000) model, our model is also intended to be a strategic model not caring much

about the operational details.

Eppen and Martin (1988) present a very nice critique of the standard procedure

using a (Q,r) model for setting safety stocks in the presence of stochastic lead time and demand. Chopra et al. (2004) build on the work of Eppen and Martin (1988) but

their focus is on the flaws in the managerial prescriptions implied by the normal

approximation. In particular they infer the following by using the exact demand

during the lead time instead of the normal approximation: For cycle service levels

above 50% but below a threshold, reducing lead time variability increases the

reorder point and safety stock, whereas reducing the lead time decreases the reorder

point and safety stock. Our research addresses the issue of reducing lead time

variability indirectly as a result of reducing the lead time because of the flexibility

options available at a stage.

For a thorough comparison of installation and echelon stock policies for

multilevel inventory control, the reader is referred to Axsäter and Rosling (1993),

who primarily consider serial and assembly systems and prove that for (Q,r) rules echelon stock policies are, in general, superior to installation stock policies.

Advocating the necessity of models that include both cost and responsiveness,

Moon and Choi (1998) suggest extending the lead time reduction concept to

different inventory models to justify the investment to reduce the lead times. Choi

(1994) used an expediting cost function to reduce the variance of suppliers’ lead

times.

Supply chain focus dependent safety stock placement 467

123

As opposed to network design models that focus on the trade-off between the

fixed costs of locating facilities and variable transportation costs between facilities

and customers, Sourirajan et al. (2007) present a model for single-product

distribution network design problem with lead times and service level requirements,

which enables them to capture the trade-off between lead times and inventory risk

pooling benefits. The objective is to locate distribution centers (DCs) in the network

such that the sum of the location and inventory (pipeline and safety stocks) is

minimized. Our CIR concept is similar in spirit to the replenishment lead time

calculation at a DC (Sourirajan et al. 2007), which depends upon the volume of flow

through the DC.

One very interesting (Q,r) model with stochastic lead times that could serve as a building block in supply chain management was proposed by Bookbinder and

Cakanyildirim (1999) as opposed to the constant lead time assumption in many

other studies. This paper introduces a term called an expediting factor for the lead

time, which is similar in spirit to the dimensionless quantity proposed in our model,

CIR. Ryu and Lee (2003) consider dual sourcing models with stochastic lead times

in which lead times are reduced at a cost that can be viewed as an investment. They

make use of the concept of expediting factors proposed by Bookbinder and

Cakanyildirim (1999) in their model. They analyze (Q,r) models with and without lead time reduction and compare the expected total cost per unit time for the two

models.

Even though we did not consider product mix flexibility related issues in our

model, the reader can refer to Upton (1997) to explore the relationship between

process range flexibility and structure, infrastructure and managerial policy at the

plant level. We assume information symmetry at all the stages in our model. The

effect of information sharing for the time series structure of the demand on safety

stocks is addressed in Gaur et al. (2005).

3 Model development

This section mostly develops the total cost expression, made up of safety stock costs

and responsiveness related costs, for a serial supply chain.

3.1 Expression for safety stock costs

We follow the building block model (Graves and Willems 2003) with installation

base stock policies and a common underlying review period for all stages. A

typical base stock policy works as follows. When the inventory position (i.e., on-

hand plus on-order minus back orders) at stage i falls below some specified base stock level Bi, the stage places a replenishment order, thereby keeping the inventory position constant. Simchi-Levi and Zhao (2005) attribute the popularity

of the base stock policy to the fact that it is simple, easily implementable, and has

been proven to be optimal or close to optimal in many cases. For example, in

serial supply chains with zero setup costs and without capacity constraints,

because the installation base stock policy is equivalent to an echelon base stock

468 G. Vanteddu et al.

123

policy under certain initial conditions (Axsäter and Rosling 1993), it is indeed

optimal in these cases (Clark and Scarf 1960). In serial systems, even a modified

base stock policy with capacity constraints is still close to optimal (Speck and van

der Wall 1991; van Houtum et al. 1996).

In an installation policy, each facility only needs the inputs from the immediate

upstream (U/S) and downstream (D/S) facilities and makes ordering decisions based

on its local order and inventory status (Simchi-Levi and Zhao 2005) as opposed to

an echelon base stock policy, which is a centralized control scheme that allows for a

central decision maker to coordinate and control the actions at all stages in the SC

(Graves and Willems 2003).

Even though our model assumes all stages to be manufacturing stages, without

loss of generality, a stage could be modeled as a DC as well. A pure transportation

function can also be modeled with the building block concept, wherein the transport

time is the lead time with pipeline inventories.

Orders are placed at discrete time intervals and each stage is considered as a

building block (Graves 1988) that generates a stochastic lead time. A building block

is typically a processor plus a stock keeping facility. Depending on the scope and

granularity of the analysis being performed, the stage could represent anything from

a single step in manufacturing or distribution process to a collection of such steps to

an entire assembly and test operation (Graves and Willems 2003). Demand is

assumed to be stationary and uncorrelated across nonoverlapping intervals and there

are no capacity constraints.

Our model is designed as a decentralized supply chain (Graves and Willems

2003; Lee and Billington 1993) to mimic reality more closely, with each stage

following a local base stock policy. Supporting the same view, Lee and Billington

(1993) state that organizational barriers and restricted information flows between

stages may result in complete centralized control of material flow in a supply chain

not being feasible or desirable.

The primary distinction between centralized and decentralized supply chain is

put in the following succinct form by Lee and Billington (1993) ‘‘Centralized

control means that decisions on how much and when to produce are made centrally,

based on material and demand status of the entire system. Decentralized control, on

the other hand, refers to cases where each individual unit in the supply chain makes

decisions based on local information’’.

Assuming that each building block operates independently using a simple

installation policy, one can first characterize various building blocks such as serial,

assembly, distribution, etc. and then identify the links among these building blocks

(Simchi-Levi and Zhao 2005). We have chosen series system for simplicity of

analysis and primarily to develop certain insights that are insensitive to the specific

supply chain topology. Also, other networks such as assembly system can be

reduced to an equivalent series system (Rosling 1989). Most of the features are

similar to the features of a serial system presented in Gallego and Zipkin (1999)

with some modifications.

There are several stages or stocking points arranged in series. The first stage

receives supplies from an external source. Demand occurs only at the last stage.

Demands that cannot be fulfilled are immediately backlogged. There is one product,

Supply chain focus dependent safety stock placement 469

123

or more precisely, one per stage. To move units to one stage from its predecessor,

the goods must pass through a supply system representing production or

warehousing activities. There is an inventory holding cost at each stage and our

model does not consider back-order penalty cost, which could easily be included.

The horizon is finite, all data are stationary, and the objective is to evaluate the

performance and optimize depending upon the supply chain focus on the cost–

responsiveness spectrum. Information is centralized but control is decentralized.

As in Gallego and Zipkin (1999), the numbering of the stages follows the flow of

goods; stage 1 is the first, and demand occurs at the last stage. The external source,

which supplies stage 1, has ample stock and it responds immediately to orders

(Fig. 2).

We have assumed that the service level targets required at each of the players

are exogenous, i.e., they are dictated by the immediate D/S player or the final

customer. Following Graves and Willems’ (2003) treatment of stochastic service

model in supply chains, let U(k1), …, U(kn) be the service levels for corresponding safety factors k1, …, kn where U(ki) represents the cumulative distribution function for a standard normal variable. kj is the multiplying factor for the variability term for stage j, which along with the average in process inventory would give the expected base stock to achieve a given service level. Let the

processing time at stage j be a random variable sj (measured in time periods) with mean Lj and variance r2j :

The stochastic service model (Graves and Willems 2003; Simchi-Levi and Zhao

2005; Lee and Billington 1993; Ettl et al. 2000) assumes that the delivery or service

time between stages varies based on the material availability at that stage and that

each stage in the supply chain maintains a base stock that is sufficient to meet its

service level target (Graves and Willems 2003). If Di is the random delay at the preceding stage i, then the replenishment cycle time at stage j equals

cj ¼ sj þ Di: ð1Þ

We have adopted the procedure for calculating this delay due to the stock out at the

preceding stage as presented in Graves and Willems (2003) and Ettl et al.(2000)

with a modification that takes into account the fact that there is only one player at

the preceding stage. Thus, we are assuming that the expected value of this delay is

simply equivalent to the probability of stock out at the preceding stage pi times its average processing time.

Therefore, expected replenishment cycle time at stage j is given by

E cj � � ¼ Lj þ piLi; ð2Þ

where

pi ¼ 1� UðkiÞ ð3Þ

Assuming that the demand is N(l,r2 ), where l is the mean demand during one time period and r2 is the demand variance during one time period, to satisfy average demand l, given the average replenishment cycle time from (2), the average cycle stock is given by

470 G. Vanteddu et al.

123

l� Lj þ 1� UðkiÞð ÞLi � �

: ð4Þ

Assuming the independence of processing times at a stage and between the stages,

we realize that

r2Di ¼ r 2 i ð1� UðkiÞÞ ð5Þ

r2cj ¼ r 2 j þ r2i ð1� UðkiÞÞ: ð6Þ

When the demands are uncorrelated between time periods, then the mean and

variance of the demand during replenishment period for stage j denoted by the continuous random variable WJ are obtained as follows by slightly modifying the equations to account for the portion of the lead time variability transferred from the

preceding stage (see, for example, Eppen and Martin 1988 and Feller 1960).

lWj ¼ lE½cj� r2Wj ¼ r

2E½cj� þ r2cjl 2:

ð7Þ

Now, we introduce another parameter cj, termed the coefficient of inverse respon- siveness (CIR), defined as the ratio of the average demand to the rate of production

(throughput) pj:

cj ¼ l=pj: ð8Þ

Assuming that there is enough capacity at all the stages to satisfy a given demand, cj B 1 at all times. When cj = 1, the expected replenishment cycle time is given by (2). We also assume that there is some upper bound above which the rate of production

cannot be increased further. The CIR at a stage is similar to the expediting factor

proposed by Bookbinder and Cakanyildirim (1999). They define the expediting

factor s as the constant of proportionality between the random variables ~T (the expedited lead time) and T(ordinary lead time). For expedited orders (s\ 1), shorter than average lead time can be obtained at a cost. Similarly, longer mean lead

time results in a rebate for the customer when (s[ 1). By considering a model with three decision variables (Q,r,s), they show that the expected cost per unit time is jointly convex in the decision variables and obtain the global best solution.

Keeping the average cycle stock constant (given in (4)), from Little’s law, at

expedited rates of production (cj\ 1), average replenishment cycle time for player j, when operating at a CIR level cj will be equivalent to

E cj � � ¼ ðLj þ ð1� UðkiÞÞLiÞcj; ð9Þ

r2cj ¼ ðr 2 j þ r2i ð1� UðkiÞÞÞcj: ð10Þ

As cj decreases, the average replenishment cycle time for player j decreases, which means that player j is becoming more responsive. Because of this inverse relationship between cj and the responsiveness at stage j, cj is termed the coefficient of inverse responsiveness at stage j.

In light of Eqs. 9 and 10, we assume that demand during replenishment period WJ for stage j is normally distributed as follows:

Supply chain focus dependent safety stock placement 471

123

lWj ¼ lE cj � �

rWj ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2E cj

� � þ ðr2cjÞl

2 q : ð11Þ

This equation considers both the demand variability and the replenishment cycle

time variability. Replenishment cycle time variability is made up of two

components:

1. The processing time variability at stage j. 2. The portion of the processing time variability at stage i that is transferred to

stage j.

We assume that base stock at stage j is given by Bj ¼ lE cj � � þ kjrWj ; where kj is

the safety factor to achieve the service level target U(kj) for that stage. After accounting for the average demand over the replenishment period, expected

on-hand inventory is given by

E Ij � � ¼ kjrWj : ð12Þ

By augmenting the above expression with the following term, which is the expected

number of back orders (Graves and Willems 2003; Ettl et al. 2000),

E BO½ � ¼ rWj Z1

z¼kj

ðz� kjÞ/ðzÞdz ð13Þ

we realize the following expression for the expected safety stock at stage j:

E SSj � �

¼ rWj kj þ Z1

z¼kj

ðz� kjÞ/ðzÞdz

0

B@

1

CA: ð14Þ

At stage j, let Cj be the nominal cumulative cost of the product realized when cj = 1, and let hj be the holding cost rate per period. The per-unit holding cost of safety stock at stage j per period equals Cj hj. The total safety stock holding cost at stage j per period is given by:

CSSCj ¼ CjhjE SSj � �

: ð15Þ

Total safety stock holding cost for the supply chain per period is given by:

CSSC ¼ Xn

j¼1 CSSCj : ð16Þ

3.2 Expression for responsiveness related costs

When players (1, …, n) operate with their respective CIRs, being (c1 , …, cn), stage j incurs two types of responsiveness related costs, as described below.

472 G. Vanteddu et al.

123

3.2.1 Direct responsiveness related costs

The difference in nominal cumulative costs of the product at two consecutive stages

j and i (i.e., Cj - Ci) will increase by a value that is assumed to be a function of (1 - cj). This is the cost that stage j will pay for operating at a higher processing speed that lowers the average replenishment cycle time at stage j.

The increase in cost is typically due to increase in investment in the so-called 5M

resources (i.e., manpower, machine, methods, material, and measurement). We are

addressing the issue of cycle time reduction due to the opportunities for flexibility

available at a stage which can be harnessed at a cost. This should not be confused

with cycle time reductions due to better operational efficiencies such as efficient

removal of wastes from the processes such as downtime, setup time, etc, which we

will consider in the extension of this framework. Flexibility related costs will

increase/decrease depending on how flexible these 5M resources are at a stage.

Direct response related costs at stage j per period are given by

DRCj ¼ ½f ð1� cjÞ�ðCj � CiÞl: ð17Þ

Even though we do not advocate any specific function type for modeling f(1 - cj), the cost of volume flexibility function (COF), we strongly recommend that it be

derived from the historical data. For example, for a specific average demand, from

the past data one could fit a regression model between (1 - cj) and the incremental cost at a stage. Moon and Choi (1998) advocate the use of a piecewise linear

crashing cost function that is widely used in project management in which the

duration of some activities can be reduced by assigning more resources to the

activities. They used a piecewise linear crashing cost function in their model. Ben-

Daya and Raouf (1994) could be a useful reference when considering using other

types of crashing cost functions.

Bookbinder and Cakanyildirim (1999) assume the expediting cost per unit

(because of technological investments or hiring extra workers etc.) time w(s) to be a decreasing convex function of the expediting factor with w(1) = 0 (additional cost when CIR = 1 is zero in our model as well). As opposed to our model, they allow

w(s) to take negative values for s[ 1, meaning that for longer lead times they assume that the manufacturer gives the buyer some rebate per unit time.

Proceeding from the research of Bookbinder and Cakanyildirim (1999), Ryu and

Lee (2003) chose their expediting cost functions w1 (s1) and w2(s2) to be decreasing convex functions of the expediting factors s1 and s2. They considered w1 (s1) = c1 (-1 + 1/s1) and w2(s2) = c2 (-1 + 1/s2), where c1 and c2 are positive coefficients. Our cost function, for example Eq. 17, looks similar in spirit to these cost functions.

Yang and Geunes (2007) expect the cost of reducing procurement time, because

of supplier’s investment in production processes or technologies, to increase at a

nondecreasing rate with the amount of lead time reduction and therefore employ a

convex function for lead time reduction. They consider a piecewise linear, convex

and decreasing form (as the production lead time increases) for the unit procurement

cost function but note that their analysis of this function applies to general piecewise

linear functions and convexity is therefore not required, although they expect this

function to be convex in practical context. For our numerical analysis, we consider a

Supply chain focus dependent safety stock placement 473

123

simple increasing convex function, although the results would not be different for

any nondecreasing function.

3.2.2 Indirect responsiveness related costs

In addition to the above mentioned direct responsiveness related costs, the stage

will experience an increase in safety stock costs when it operates at cj, for the reasons mentioned in Sect. 3.2.1. As a result, the difference in nominal

cumulative costs of the product at stages j and i will increase by a value, which is a function of (1 - cj). Therefore, indirect responsiveness related cost at stage j is given by:

IRCj ¼ ½f ð1� cjÞ�ðCj � CiÞhjE SSj � �

: ð18Þ

Therefore, the total responsiveness related cost at stage j is given by

TRCj ¼ DRCj þ IRCj: ð19Þ

Hence, total safety stock costs in the presence of increase in costs at stage j that account for increase in the responsiveness per period is given by

CTSCj ¼ CSSCj þ TRCj: ð20Þ

Hence, the total safety stock related cost for the whole supply chain per period is

given by

CTSC ¼ Xn

j¼1 CTSCj : ð21Þ

4 Managerial insights for strategic safety stock placement

In this section, we attempt to study the safety stock placement problem primarily

with respect to the key parameters such as replenishment cycle times, replenishment

cycle time variability, cost of responsiveness etc. as we travel from U/S to D/S and

offer some interesting insights.

4.1 Position of the bottleneck player

We consider a hypothetical serial supply chain with five stages to study the impact

of the position on the bottleneck player on supply chain performance. The different

parameters for this serial supply chain are illustrated in Table 1.

For the purpose of our discussion, we consider the player with no flexibility

whatsoever with respect to its processing time as the bottleneck player. The effect of

changing the position of the bottleneck player on the SC safety inventory costs is

depicted below. We have considered three cases, wherein the processing time

variability progressively increases (IV in Fig. 3), decreases (DV in Fig. 3), and

remains constant (EV in Fig. 3) as we move from the most U/S stage to the most D/

S stage. For the increasing and decreasing cases, we kept the total processing

474 G. Vanteddu et al.

123

variability constant and progressively increased/decreased the process variability as

we move from U/S to D/S in a symmetric fashion. When a particular stage is a

bottleneck, the CIR for that particular stage would be unity and for all other stages

the CIR is considered to be 0.9. Response related costs are not considered in the

analysis as our primary intention is to show the effect of reducing the lead time on

the safety stocks as the bottleneck player moves from U/S to D/S. By considering

the incremental total supply chain safety stock costs with respect to a benchmark

case, wherein all the players are operating at a CIR equivalent to 0.9 and the

processing variability is the same at all the stages (as with the EV case in the chart),

one can clearly see that the incremental total supply chain safety stock costs

progressively increase as the bottleneck player moves from U/S to D/S for all three

cases.

Key managerial insight #1: Safety stock cost depends on the placement of the bottleneck player in a serial supply chain and it is desirable for the bottleneck player to be located towards the U/S rather than the D/S.

4.2 Safety stock costs versus responsiveness costs

We illustrate the impact of CIR on the different cost curves in Fig. 4, using the cost

expressions developed earlier.

We know that, for any stage, safety stock related costs decrease (Eq. 15) and

response related costs increase (Eqs. 17 and 18) with decrease in cycle time, or in

Table 1 Parameters of the five- stage serial SC adopted for

numerical analysis

Stage/player

1 2 3 4 5

Nominal cumulative cost Cj 80 112 175 327 863

Average demand l 100 100 100 100 100

Demand variability r2 400 400 400 400 400

Mean processing time

(periods) Lj

1 1 1 1 1

Processing variability ðl2r2j Þ (in number of units

of product)

50 50 50 50 50

CIR cj 1 1 1 1 1

Safety coefficient kj 1.28 1.28 1.28 1.28 1.28

Average back order coefficient 0.1 0.1 0.1 0.1 0.1

Probability of stockout pj 0.1 0.1 0.1 0.1 0.1

Inventory holding cost hj 0.017 0.017 0.017 0.017 0.017

Service level U(kj) 0.9 0.9 0.9 0.9 0.9

External supplier Customer

Supply

Demand

h i j lk

Fig. 2 Schematic description of a serial supply chain

Supply chain focus dependent safety stock placement 475

123

other words, with decrease in CIR. Hence, without loss of generality, we can say

that the total cost function, which is a combination of safety stock and response

related costs, tends to be convex, the optimum CIR shifting either towards or away

from unity depending on whether responsiveness cost component or safety stock

cost component is dominating. A very interesting result relevant to the nature of the

total costs is due to Bookbinder and Cakanyildirim (1999), who show that the

expected cost per unit time is jointly convex in the decision variables (Q,r,s), s being the expediting factor for a (Q,r) inventory system with expedited orders and random lead times.

4.3 Effect of cycle times and processing time variability on total stage cost

Figure 5 illustrates the effect of increasing processing time variability at a particular

stage on the total cost while keeping the other parameters constant. For the sake of

numerical analysis, we have considered the parameter values for stage 3. PV in the

chart stands for processing time variability and DV stands for demand variability.

Increasing process variability tends to increase both the safety stock related costs

and responsiveness related costs at different rates, primarily depending on the

nominal cumulative cost Cj, the cost of volume flexibility function f(1 - cj), and the cost added at a stage (Cj - Ci). In the presence of increasing processing time variability, therefore, optimum CIR either moves away from or towards unity,

-10

0

10

20

30

40

1 2 3 4 5

Supply Chain Bottleneck StageI nc

re as

e in

T ot

al S

af et

y St

oc k

C os

t ( %

)

IV EV DV

Fig. 3 Incremental SC safety stock costs versus position of the bottleneck player

0

100

200

300

0.55 0.7

CIR

C os

t

SS cost Response Cost Total Cost

10.85

Fig. 4 CIR versus stage costs

476 G. Vanteddu et al.

123

depending on whether safety stock cost component/or the responsiveness cost

component is dominating. With increasing variability, the degree of departure of the

optimum CIR value either way depends on the three factors mentioned above for a

particular stage. To explain, to counter the excessive variability, a typical supply

chain manager tries to reduce the cycle time (by reducing CIR), which would

minimize the SS related costs. However, that action does not necessarily lead to

reduction of the overall costs. For example, if the cost added at a particular stage is

very high then the decrease in safety stock costs by cycle time reduction might not

be able to offset the increase in cycle time reduction (responsiveness) costs. In such

a scenario, moving CIR counterintuitively towards unity (i.e., increasing the cycle

time), which means in the direction that increases safety stock costs, could be a

better measure to reduce overall costs.

Key managerial insight #2: As the processing time variability increases for a stage, the optimum value of CIR moves away from unity if safety stock cost component dominates the responsiveness cost component and towards unity if the responsiveness cost component dominates the safety stock cost component.

In Fig. 5, optimum CIR is progressively moving away from unity as the PV/

DV ratio increases, because we have considered the safety stock costs component

to be larger than the responsiveness cost component for our numerical analysis.

The trend will be reversed if the response related costs dominate the safety stock

related costs. In Fig. 5, for the case of PV/DV = 0.25, the optimum CIR value is

0.919, for which the respective safety stock and responsiveness cost components

are $91.65 and $1.749 per period, respectively. For the case of PV/DV = 2,

wherein the process variability increases eightfold compared to PV/DV = 0.25, the

corresponding safety stock and responsiveness cost components are $138.6 and

$1.75, respectively. In this case, not only is the safety stock component larger but

also the effect of increasing process variability is much greater on safety stock

component compared with responsiveness cost component. Hence, the optimum

CIR value for the PV/DV = 2 case will move in the direction that will mitigate the

effect of increasing safety stock costs and to a point where the cost advantage in

terms of safety stock cost reduction by reducing the CIR is equalized by the

corresponding increase in responsiveness costs. The optimum value for the PV/

DV = 2 case occurs at a CIR value of 0.909 (moving away from unity), with

50

150

250

350

0.55 0.7 0.85

CIR

T ot

al C

os t

PV/DV:0.25 PV/DV:0.5 PV/DV:0.75 PV/DV:1 PV/DV:1.5 PV/DV:2

1

Fig. 5 CIR versus total stage costs

Supply chain focus dependent safety stock placement 477

123

corresponding safety stock and responsiveness cost components of $137.86 and

$2.48, respectively.

We extend the above analysis by performing a simple sensitivity analysis by

considering the changes in mean processing time and nominal cost added at stage 3,

while keeping other parameters constant. Table 2 presents the case with the mean

processing time and nominal cost added both being halved. Table 3 presents the

case with mean processing time and nominal cost added both being doubled.

For the sake of simplicity, we have just considered the cases PV/DV = 0.25 and

PV/DV = 2. For both of these cases, the optimum CIR for PV/DV = 2 occurs at a

slightly lower value than that for PV/DV = 0.25 for the reasons mentioned in the

preceding paragraph. The changes made in the parameter values are not able to

make the responsiveness related costs increase at a faster rate than the decrease in

safety stock related costs (with decreasing CIR), hence, the optimum CIR is still

moving away from unity, primarily due to the parameter values we have assumed

for our numerical analysis. We show in Sect. 4.4 that changes in the cost of

flexibility function (COF), that is f(1 - cj), in our model cause the responsiveness related costs to increase at a faster rate than the decrease in safety stock related costs

(with decreasing CIR), making the optimum CIR move towards unity. In a practical

context, it all depends on the actual values of the parameters, which make either of

the cost components dominating.

The optimum value of the CIR for a stage will primarily depend upon the

cumulative cost of the product, which in turn will depend upon the position of the

stage in the supply chain, the cost added, and the nature of cost of volume flexibility

function. Using the cycle time reduction as a lever may or may not be cost effective

depending on, for example, whether a windscreen wiper is being assembled or an

engine is being assembled at a stage. For the same cost of volume flexibility function,

it might be a good idea to reduce the cycle time if a windscreen wiper is being

assembled as opposed to an engine. Irrespective of the component that is being

assembled, this effect will be more pronounced D/S when compared to U/S because

the cumulative cost of the product will progressively increase as we move D/S.

Key managerial insight #3: To reduce the safety stock costs, reducing the cycle time is a better lever at downstream stages when compared to upstream stages for a given responsiveness cost, all other parameters being held constant.

Key managerial insight #4: All other parameters being constant, for a given cycle time reduction, the cost of flexibility function has to be milder for a stage with high cost addition when compared to a stage with low cost addition, to justify savings in safety stock costs.

Table 2 Sensitivity analysis 1 CIR PV/DV = 0.25 PV/DV = 2

1 69.869 103.747

0.9 68.432 100.405

0.8995 68.441 100.404

0.899 68.450 100.403

0.895 68.533 100.407

478 G. Vanteddu et al.

123

The insight from Ryu and Lee (2003) ‘‘that in order to attain greater savings from

the symmetric cost scheme for the two suppliers, the investment cost for the supplier

with the more unreliable lead time should be smaller’’ looks similar in essence to

our key managerial insight 2, albeit in a different context. One of the key insights

offered, that downstream lead times have a greater impact on system performance

than upstream ones (Gallego and Zipkin 1999), is also similar in spirit to our

research.

4.4 Effect of cost of volume flexibility on total stage cost

Figure 6 illustrates the effect of cost of volume flexibility at a particular stage on the

total cost while keeping the other parameters constant. For the sake of numerical

analysis, we have once again considered the parameter values for stage 3. COF in

the chart stands for cost of volume flexibility function mildness/steepness, when

compared to a randomly chosen increasing convex function (COF: 1).

In Fig. 6, as expected, optimum CIR is progressively moving towards unity as the

COF increases. As opposed to processing time variability, which affects both the

safety stock cost component and the responsiveness cost components, cost of

flexibility affects only the responsiveness cost component. Intuitively speaking,

given that the stage is not operating at optimal CIR, increasing the cost of flexibility

makes a typical supply chain manager operate close to nominal cycle times (CIR

close to unity), but going back to the example given earlier, if a windscreen wiper is

being assembled, particularly at a D/S stage, it is quite likely that there will be

significant savings in safety stock costs by reducing the cycle time (moving the CIR

away from unity), overtaking the extra cost of responsiveness because of a steeper

flexibility function. The opposite will be true, for example, if an engine is being

assembled at a stage.

Table 3 Sensitivity analysis 2 CIR PV/DV = 0.25 PV/DV = 2

1 132.232 202.513

0.95 129.834 198.290

0.93 129.952 197.664

0.92 129.952 197.664

0

500

1000

1500

2000

2500

0.55 0.7 0.85

CIR

T ot

al C

os t

COF:0.01 COF:0.1 COF:0.33 COF:1 COF:2 COF:5 COF:10

1

Fig. 6 CIR versus total stage costs

Supply chain focus dependent safety stock placement 479

123

Key managerial insight #5: Given that a stage is not operating at optimal CIR, as cost of volume flexibility function becomes steeper, the optimum value of CIR moves towards unity if the responsiveness costs segment dominates the safety stock cost segment, and vice versa.

Key managerial insight #6: All other parameters being constant, for the same cost addition at a stage, cost of flexibility function has to be milder at U/S than D/S for a given amount of savings in safety stock costs.

While our model assumes information symmetry, it is still applicable, including

the managerial insights, in supply chains with information asymmetry. For

example, in the presence of bullwhip effect attributable to information asymmetry

or other related issues, demand variability will progressively increase as we move

towards the upstream stages. It is still advisable that mitigating measures are put

in place to reduce the causative factors of the bullwhip effect. In the absence of

any mitigating measures, one could also consider the actual variability at different

stages and still use the model. The effect will be to accentuate both the safety

stock component and the responsiveness cost component as we proceed upstream.

If the increase in the cost of responsiveness at a stage is significantly larger

compared to the reduction in safety stock costs for unit processing time reduction,

the optimum CIR will move towards unity. The opposite trend will be witnessed

in the opposite case. The change in the optimum CIR value at downstream stages

may also be significant because of larger cumulative costs and typically large

amounts of cost addition, in spite of the effect of bullwhip effect being smaller

compared to upstream stages.

5 Model usability for achieving strategic fit

The model can be used primarily as a building block in any kind of supply chain

network both to evaluate the performance of individual stages with respect to the

key order winners, cost, and responsiveness, and to optimize the supply chain

depending upon the focus within the cost-responsiveness spectrum. The results can

be used to evaluate the gap between the individual business strategies of different

stages and the supply chain strategy, so that appropriate actions can be taken to

achieve the strategic fit. Once the supply chain design is completed and the network

is in place, the model will be particularly useful for making strategic decisions with

regard to the safety stock placement while considering a changeover or introducing

a new product.

5.1 Performance evaluation

Total safety stock related cost (safety stock cost and responsiveness cost)

expression for all the stages (using Eq. 20) can be easily evaluated given the

different parameter values and one can obtain the costs separately for safety stock

holding and cycle time reduction (if so desired by a stage) and the expected cycle

times assuming that all the stages operate without any coordination with respect to

the overall supply chain strategy (that is, there is no central decision maker). This

480 G. Vanteddu et al.

123

will allow stages to evaluate their performance with regard to the key order

winners of cost and responsiveness and facilitates their comparison to the internal

performance benchmarks. For example, a particular stage may realize that better

operations management practices that involve the reduction of waste (lean

practices) that had been planned for are ineffectual. Hence, they may want to

know what went wrong and implement appropriate corrective and preventive

actions such that cost and cycle time reductions can be achieved without any

major investments.

5.2 Optimization

Now, assuming that there is a central decision making mechanism, which acts to

coordinate the whole supply chain, the overall safety stock related cost expression

for the supply chain (Eq. 21) can be optimized, say for the optimal CIR values at all

the stages given the constraints on target supply chain cycle time, individual stages’

rates of production, service levels, inventories, cost of inventory holding, etc. The

optimal costs (safety stock inventory and responsiveness) can be compared with the

costs obtained assuming the absence of the central decision maker (as for

performance evaluation explained in Sect. 5.1) to initiate necessary actions.

We also offer a simple optimization example based on the parameter values

presented in Table 1 that lend support to the managerial insights developed in Sects.

4.3 and 4.4.

Figure 7 depicts the optimal CIR values under different scenarios considered in

Sects. 4.3 and 4.4. The only constraint considered was 0 B cj B 1. The total cost function was optimized under different scenarios.

SC: Standard cost of flexibility function for all stages and other parameter values

as per Table 1.

DC: Decreasing cost of flexibility function (i.e., the function becomes milder

compared to the SC case as we move from U/S to D/S); stage 3’s cost of flexibility

function remaining the same as in the SC case. Other parameters for all the other

stages remain as in the SC case.

IV: Progressively increasing process variability as we move from U/S to D/S. For

stage 3, process variability remains the same as in the SC case. Other parameters for

all the other stages remain as in the SC case.

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1 2 3 4 5

Player

C IR

SC

DC

IV

DV

Fig. 7 Player versus optimal CIR

Supply chain focus dependent safety stock placement 481

123

DV: Progressively decreasing process variability as we move from U/S to D/S.

For stage 3, process variability remains the same as in the SC case. Other parameters

for all the other stages remain as in the SC case.

The results, as can be seen from Fig. 7, clearly reinforce the insights developed in

the earlier sections. The shift in the optimum CIR value at stage 5 shows strong

variability owing to the fact that cost added constitutes a significant portion of the

cumulative cost at that stage, thereby increasing the cost of responsiveness

significantly for the same cost of flexibility function when compared to other stages.

The same is the case with stage 4, although on a reduced scale. Since stage 3 is the

transition stage under different scenarios retaining the same parameter values as

compared to the SC case, there is no perceptible change in the optimum CIR value.

5.3 Achieving strategic fit

If the overall costs at a stage are increasing as a result of optimization to achieve the

strategic supply chain objectives, when compared to the costs in the absence of a

central decision maker, i.e., by following its independent business strategy, then, by

devising a scheme for the sharing of the additional profits for the whole supply chain

in an equitable fashion, say, through changing the contract structure, discounts or

other incentives, efforts could be made to coordinate the individual business

strategies with the supply chain strategy to achieve the strategic fit.

One way of doing this is by considering the supply chain objectives at different

levels, say, strategic, tactical, and operational levels, and performing a gap analysis

Strategic Objective:Customer Satisfaction Tactical Objective:Total customer Cycle time Performance evaluation for a hypothetical supply chain:

Cost Estimate(Cip) $25,000 $100,000 $50,000 $0 $25,000 $0 Sum(Cip) $200,000 Time Required in months (Tip) 2 12 3 0 3 0 Max (Tip) 12

Tier-3 Supplier

Tier-2 Supplier

Tier-1 Supplier Mfg Unit D.C

Retail Unit

5

4

3

2

1

Cost Estimate(Cjp) $5,000 $15,000 $0 $0 $6,000 $0 Sum(Cjp) $26,000 Time Required in months (Tjp) 1 4 0 0 1 0 Max (Tjp) 4

5

4

3

2

1

Cost Estimate(Ckp) $5,000 $15,000 $2,000 $10,000 $6,000 $0 Sum(Ckp) $38,000 Time Required in months (Tkp) 2 2 1 1 1 0 Max (Tkp) 2

5

4

3

2 Overall cost= 1 Sum(Sum(Cip)+

…+Sum(Ckp)) $264,000

Max(T) : Max(max(Tip),max(Tjp),max(Tkp)) 12

NC product level

Throughput rate

Information symmetry

level

P E

R F

O R

M A

N C

E M

E T

R IC

S

Fig. 8 Performance comparison for different players in a supply chain (adopted from Vanteddu et al. 2006)

482 G. Vanteddu et al.

123

between the individual business strategies and the supply chain strategy in terms of

costs and responsiveness. Vanteddu et al. (2006) offers a simple Excel based model

to achieve strategic fit assuming that quantitative data is available to make use of the

tool, as illustrated in Fig. 8.

By simply modifying the performance metrics by placing emphasis on cost

efficiency and responsiveness, one can use the above model to achieve the strategic

fit. Actual costs related to cost and responsiveness can be obtained by using the

model proposed in this paper.

6 Conclusions

In this paper, in addition to offering several managerial insights with regard to

strategic safety stock placement in a supply chain, an attempt has been made to

address the problem of achieving compatibility between the supply chain strategy

and the individual stage’s business strategy by primarily considering safety stocks

and responsiveness related costs. We introduce a new parameter called the

coefficient of inverse responsiveness (CIR) to model response related costs at a

stage, which also enhances the scalability of the model. The developed cost function

for the building block could be easily extended for any type of supply chain

network. By knowing the values of the parameters of the model, one can know the

safety stock costs and responsiveness related costs at each stage and compare them

to an ideal supply chain strategy in terms of cost and responsiveness to make

informed decisions. The key managerial insights developed are generic in nature

and applicable to any supply chain irrespective of its placement on the cost-

responsiveness spectrum and topology.

To make the model more tractable we had to make certain simplifying

assumptions and following are some of the limitations related to those assumptions.

For example, we did not factor the bullwhip effect into our model. It would be an

interesting extension to our research if the bullwhip effect and other information

asymmetry related issues were factored into the model. We have not incorporated

order sizes into our model because the model is supposed to be largely a strategic

model. Research could be extended by incorporating this aspect into the model to

see how it affects the managerial insights that we offered. Relaxing the assumptions

on the stationary nature of the demand and stage base stock policy with periodic

review will make the model more realistic and robust to real-world situations.

Introducing contracts that affect the flexibility at a stage with financial ramifications

will also be a very fertile area to pursue that will make our model mimic reality

more closely. Another such area is the consideration of other types of network

topologies, particularly assembly and distribution; we are planning to consider

extension of the framework in this direction. It would also make more sense to

consider capacity constraints at certain stages. Product mix related flexibility is also

a crucial factor which is not addressed in our model. Addition of this feature would

truly make the research more in tune with reality, especially in mass customization

settings. Finally, we really would like to see this model used in some real-world

application so that insights presented in our model could be validated.

Supply chain focus dependent safety stock placement 483

123

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