Directory UMM :Data Elmu:jurnal:I:International Journal of Production Economics:Vol65.Issue3.May2000:

Int. J. Production Economics 65 (2000) 229}242

Determining the optimal target for a process with multiple
markets and variable holding costsq
Yuehjen E. Shao!,*, John W. Fowler", George C. Runger"
!Department of Statistics Fu Jen Catholic University, Taipei, Taiwan
"Arizona State University, Tempe, AZ 85287-5906, USA
Received 31 August 1998; accepted 2 March 1999

Abstract
The determination of the quality target for a manufacturing process represents an intricate and "scally vital decision.
This study examines methods for process target optimization in industries where several grades of consumer speci"cations (and hence several quality-grades of products) may be sold within the same market. In such situations, manufacturers may hold goods that have been rejected by one customer to sell the same goods to another consumer in the same
market at a later date. The expected pro"t function for such "rms must consider the holding costs as well as the pro"ts
associated with this sales strategy. This study provides a conceptual and mathematical overview of such situations.
A method for identifying the optimal process target that re#ects holding costs is presented and illustrated in the context of
the steel galvanization industry. ( 2000 Elsevier Science B.V. All rights reserved.
Keywords: Optimal target; Expected pro"t; Holding; Quality; Variability

1. Introduction
The determination of the optimal target, or set
point, for a manufacturing process has a tremendous impact on both a manufacturer's customer

satisfaction and on the "scal bottom line. Although
the quality characteristics of the "nished product
may satisfy consumer expectations when the process target is set high, the raw material and production costs necessary to maintain such high quality

q

This research was supported in part by the National Science
Council of the Republic of China, while the "rst author was
a Visiting Professor in Arizona State University.
* Corresponding author. Tel.: 011886-2-2903-1111, ext. 2647;
fax: 011886-2-2903-3753.

levels may prove prohibitively expensive [1]. Conversely, while the manufacturer may avoid excessive production costs by setting lower process
targets, the "nished product's quality characteristics may not meet the customer's speci"cations.
Depending on the industry and the market, such
unacceptable products may be reworked for later
sale (e.g., an over"lled or under"lled can of fruit
may be emptied and re"lled), sold in a secondary
market at a lower price, discarded, or, in some cases
`helda for later sale to another customer in the

primary market. Each of these methods of disposing of unacceptable products carries its own relative
costs and bene"ts. Therefore, setting the optimal
process target (OPT) is an integral and "nancially
signi"cant aspect in the design of any manufacturing process.

0925-5273/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 5 2 7 3 ( 9 9 ) 0 0 0 8 1 - X

230

Y.E. Shao et al. / Int. J. Production Economics 65 (2000) 229}242

Nomenclature
selling prices to the original in the
primary market
A
selling prices to the nonoriginal con2
sumers in the primary market
A
selling price in the secondary market

3
c
quality-speci"c production cost
(cost of maintaining speci"c quality
characteristic for one unit of "nished
product)
E[P(x); u] expected value of pro"t as a function of the process target when holding cost is "xed
E[P(x, t); u] expected value of pro"t as a function of the process target when holding cost is normally distributed
f (x)
probability density function (pdf) of
the random variable x which is normally distributed.
F(x)
cumulative distribution function
(CDF) of the random variable
x which is normally distributed
¸
customer's lower limit speci"cation
1
¸
plant tolerance limit speci"cation

2
P(x)
pro"t function when holding cost is
"xed
P(x, t)
pro"t function when holding cost is
normally distributed

A
1

Methods for determining an appropriate process
target have been studied under a variety of economic and industrial circumstances. Springer [2]
concentrated on the economic dimension of the
problem, determining the optimal target under the
assumption of a process with a net income function
with upper and lower speci"cation limits. Bettes
[3] took the optimal target and upper speci"cation
limit into account simultaneously. Hunter and
Kartha [4] proposed an approach which employed

a single (lower) speci"cation limit and assumed no
reworking of substandard output; instead, their approach assumed the sale of rejected products in
a secondary market at a "xed price. Although there
is no explicit solution for Hunter and Kartha's [4]
assumed conditions, Nelson [5] o!ered an approxi-

R

S

t
t
u
t
p
u
uH
x

basic production cost (cost of producing one unit of "nished product

independent of the speci"c quality
characteristic)
holding cost. Two cases are considered: (1) a "xed cost and (2) a normal random variable
holding time
average holding time
standard deviation of holding time
process mean
optimal process target
quality characteristic of the one unit
of "nished product (i.e., this study
assumes that x is normally distributed with mean u and standard deviation p)

Greek letters
a
"xed cost for the holding of the
0
product
a
holding cost per unit of time
1

p
process standard deviation
/(x)
pdf of the random variable x which
is standard normally distributed
W(x)
CDF of the random variable x which
is standard normally distributed

mated solution for their technique. Bisgard et al.
[6] modi"ed the assumptions of Hunter and
Kartha's study in their consideration of the `canning problema by assuming that an under"lled
canned product would be sold at a rate which is
proportional to the product's content.
Carlsson [7] applied Hunter and Kartha's [4]
"ndings to an investigation of the steel beam industry. Carlsson's method divides the producer's basic
costs into a "xed component and a variable component, then incorporates an additional premium
into the income function when the output displays
high quality and a deduction when the products
exhibit inferior quality. Golhar [8] investigated the

canning problem in circumstances in which rejected
canned products are emptied and re"lled for sale in

Y.E. Shao et al. / Int. J. Production Economics 65 (2000) 229}242

the primary market. Golhar and Pollock [9] indicated that it may prove more cost e$cient to
empty and re"ll an over"lled product (thus incurring additional production costs while recovering
the value of the excess product in the over"lled
cans) rather than simply selling the over"lled
products in the primary market at regular price
(thus avoiding the costs of re"lling the cans
while forfeiting the value of the excess contents).
Golhar and Pollock [9] suggested use of a one-way
table, and Golhar [10] provided a FORTRAN
computer program to determine settings for both
the optimal quality target and the upper speci"cation limit.
Arcelus and Rahim [11] introduced a targetsetting method which integrates the joint control of
both variable and attribute quality characteristics
of a product. Arcelus and Rahim [12] elaborated
upon their previous work and designed a statistical

experiment to gauge the e!ect of the model parameters. Schmidt and Pfeifer [13] extended Golhar
and Pollock's [9] work and provided a simple
closed-form and one-way table to solve the problems. Schmidt and Pfeifer's work may be applied in
determining the optimal upper limit and the optimal mean for the capacitated situation. Melloy
[14] proposed a technique which minimizes the
losses due to the over"ll of the packages to avoid
noncompliance, subject to an acceptable degree of
risk of noncompliance. Boucher and Jafari [15]
examined the relationship between use of a speci"c
sampling plan and the determination of the optimal
target of a "lling process.
Wilhelm [1] and Baxter et al. [16] examined
target optimization for the steel galvanization process. Wilhelm [1] suggested judging the variance of
output by observing the probability densities of the
output distribution. When the variance of the output is low, quality targets may be safely shifted
closer to the reference level (i.e., the customer's
speci"cations). When the variance is high, shifting
the target closer to the reference level may incur
a prohibitively high risk of delivering unacceptable
"nished products. Baxter et al. [16] provided

a more detailed analysis to buttress Wilhelm's "ndings. Signi"cantly, however, neither study speci"cally dwelled upon the economic dimension of the
issue.

231

The literature reviewed cumulatively suggests
that one reasonable strategy for determining the
OPT is to maximize expected net pro"ts with
respect to process variability and to "nancial considerations (i.e, production, distribution, and material costs). In some circumstances, this general
strategy may dictate the reworking of defective
goods; in others, it may prescribe the abandonment
of rejects or their relegation to a secondary market.
But how this general strategy should be applied in
situations where products that fail to meet one
customer's expectations may be acceptable to other
customers in the same primary market is not as
straightforward.
The present study shall consider strategies for
determining the OPT for industrial processes when
rejected goods may be held and sold to other customers in the same primary market at a later

date. If the level of quality is low enough, it may
be more e!ective to immediately sell it in a secondary market. In such industries, a manufacturer
may resolve to store the products declined by
the `originala intended customer in the hopes of
selling the same goods to a second or `nonoriginalacustomer for a comparable or slightly
lower price, rather than accept the lower pro"ts
to be fetched from the sale of the rejected goods
in a secondary market. However, the manufacturer
incurs additional holding costs while storing
the goods for later sale. Since determination of
the optimal quality target for an industrial process
is, in the end, a question of overall pro"t
maximization, the expected pro"t function must
be expressed to re#ect these holding costs. In
addition, although the case study presented in
Section 3 is a continuous process, the proposed
strategies are applicable to a batch production
process.
This paper is structured as follows. The next
section presents the mathematical models that address the need to maximize pro"ts. It also discusses
the models' assumptions and formulations, as well
as the techniques for deriving the OPT. In Section
3, the technique is illustrated in terms of the steel
galvanization industry. The sensitivity analysis is
performed and discussed in this section. The "nal
section of the paper assesses and summarizes the
"ndings in this research.

232

Y.E. Shao et al. / Int. J. Production Economics 65 (2000) 229}242

2. Model formulation for determining the optimal
process target
2.1. The model
The notation for this study is listed at the front of
this paper. To maximize the expected pro"t, the
expected pro"t function must be established "rst.
Consequently, the following situations should be
considered:
(i) If the quality characteristic of the process output is su$cient to equal or surpass the customer's limit speci"cation (¸ ), then the
1
original customer's expectations have been met
and the "nished product can be sold in the
primary market at a price of A .
1
(ii) If the quality characteri stic of the process
output does not meet the original customer's
speci"cations (¸ ) but clears the plant toler1
ance threshold ¸ (¸ '¸ and ¸ is close to
2 1
2
2
¸ ), then the "nished product may be held and
1
later sold to another customer in the primary
market at a price of A ). However, the "rm
2
incurs the holding costs (S) in this case.
(iii) If the output quality plunges below ¸ , then
2
the "nished product cannot be sold in the primary market. It may, however, be sold in the
secondary market at price A .
3
In these three cases, the pro"t would be:
(i) A !cx!R when x*¸ ,
1
1
(ii) A !cx!R!S when ¸ )x(¸
2
2
1
(iii) A !cx!R when x(¸ .
3
2
Therefore, the pro"t function P(x) would be

G

A !cx!R
when x*¸ ,
1
1
P(X)" A !cx!R!S when ¸ )x)¸ ,
2
2
1
A !cx!R
when x(¸ .
3
2
The next two subsections consider two di!erent
assumptions for the holding costs. First, the case
where the holding costs are "xed is discussed.
Then the case where these costs are represented by
a random variable is presented. In both cases, the
quality of the product is assumed to be a normal
random variable (x) with mean u and standard
deviation p.

2.1.1. Fixed holding cost
After estimating the variance of the independent
variable x, the expected pro"t function as a function of the process mean assumes the following
form:

P

=
(A !cx!R) f (x)dx
1
L1
L1
# (A !cx!R!S) f (x)dx
2
L2
L2
#
(A !cx!R) f (x)dx.
3
~=
Eq. (1) can be rewritten as follows:
E[P(x); u]"

P
P

P

=
(A !cx!R) f (x)dx
1
L1
L1
# (A !cx!R!S) f (x)dx
2
L2
L2
#
(A !cx!R) f (x)dx
3
~=
=
" (A !R) f (x)dx
1
L1
L1
# (A !R!S) f (x)dx
2
L2
L2
#
(A !R) f (x)dx!cu
3
~=
¸ !u
"(A !R) 1!W 1
1
p

E[P(x); u]"

P
P

P

P
P

C

A BD
CA
BBD
A B
¸ !u
1
p

#(A !R!S) W
2

A

!W

¸ !u
2
p

#(A !R)W
3

¸ !u
2
!cu
p

"(A !R!cu)
1

A
A

B
B

#(A !A !S)W
2
1

¸ !u
1
p

#(A !A #S)W
3
2

¸ !u
2
.
p

(1)

Y.E. Shao et al. / Int. J. Production Economics 65 (2000) 229}242

2.1.2. Variable holding cost
From a practical point of view, the holding cost
may be variable and for this reason, this section
assumes that the holding cost is a normal random
variable. This study assumes that
S"a #a t,
0
1
where S is the holding cost, a is the "xed cost for
0
the storage of the product, a is the holding cost per
1
unit of time, and t is the holding time which is
normally distributed with mean of t and standard
u
deviation of t .
a
The pro"t function p(x, t) then would be
P(x, t)"

G

A !cx!R when x*¸ ,
1
1
A !cx!R!S"A !cx!a !a t!R
2
2
0
1
when ¸ )x(¸ , 0)t(R,
2
1
A !cx!R when x(¸ .
3
2

Consequently, the expected pro"t function can be
derived to be (see Appendix A):
E[P(x, t); u]"(A !R!cu)
1

C
A

# (A !A !a !a B )
2
1
0
1 T

BD

¸ !u
]W 1
p

#(A !A #a #a B )
3
2
0
1 T
¸ !u
,
]W 2
p

B

A

where

C

D

t
/(D )
T #D and D " u.
B "t
T
T t
T
p W(D )
p
T
Notice that above equation would reduce to the
equation for "xed holding costs when a "0.
1
2.2. Solution
The way to determine the optimal process target
is discussed in this section. Again, the "xed holding
cost scenario will be discussed "rst; followed by the
variable holding cost scenario.

233

2.2.1. Fixed holding cost
Remember that A and A are both selling prices
1
2
in the primary market. A is the selling price in the
3
secondary market, and it is reasonable to assume
that both A and A are greater than A . This study
1
2
3
fu rther assumes that A is greater than or equal to
1
A since the quality characteristics of the "nished
2
product are acceptable to the original customer.
The optimal value of u may be found by solving
dE[P(x); u]
"0.
du

(2)

In addition, the second derivative should also
prove to be less than zero; that is,
d2E[P(x); u]
(0.
du2
Since
E[P(x); u]"(A !cu!R)
1

B
B

A
A

#(A !A !S)W
2
1

¸ !u
1
p

#(A !A #S)W
3
2

¸ !u
2
,
p

it follows that

A B

dE[P(x); u]
!1
"!c#(A !A !S)
2
1
du
p

B
A BA B

]/

A

¸ !u
1
#(A !A #S)
3
2
p

¸ !u
!1
/ 2
.
p
p

]

(3)

By letting (¸ !u)/p"m , then (¸ !u)/p"
1
1
2
(¸ !¸ )/p#m . Eq. (3) would then become
2
1
1
dE[P(x); u]
A !A !S
1
/(m )
"!c! 2
1
du
p

B

A

A
A

!

]/

B

A !A #S
3
2
p

B

¸ !¸
2
1#m .
1
p

(4)

The solution of uH may be obtained by setting Eq.
(4) equal to zero and solving for m . This results in
1

234

Y.E. Shao et al. / Int. J. Production Economics 65 (2000) 229}242

"nding uH by using (¸ !uH)/p"m . That is,
1
1
letting dE[P(x); u]/du"0, yields the following relationship:

A
A

!c!
]/

B
B

A

B

A !A #S
A !A !S
2
2
1
/(m )! 3
1
p
p

¸ !¸
2
1#m "0.
1
p

(5)

Given A , A , A , c, S, ¸ , ¸ , and p, let the solu1 2 3
1 2
tion for m be mH. Since (¸ !u)/p"m ,
1
1
1
1
(6)
uH"¸ !pmH.
1
1
Furthermore, in order to con"rm Eq. (6), it can
be easily shown that the second derivative of
E[P(x); uH] is less than zero. Note that Eq. (5) does
not yield a closed-form solution. However, numerical analysis approaches (e.g., Newton}Raphson
[17]) can be used to solve this equation.
2.2.2. Variable holding cost
In the case of variable holding cost, the optimal
value of u may be found by solving
dE[P(x, t); u]
"0.
du
Appendix B shows that
dE[P(x, t); u]
"!c#k /(m )#k /(m )"0. (7)
1
1
2
2
du
Given A , A , A , c, ¸ , ¸ , a , a , t , t , and p,
1 2 3
1 2 0 1 u p
uH can be solved and obtained. Again, Eq. (7) does
not yield a closed-form solution. Numerical analysis approaches (e.g., Newton}Raphson and Trapezoidal Approximation) are suggested to solve this
equation. In order to con"rm Eq. (7), it can be
shown that the second derivative of E[P(x, t); uH] is
less than zero.

3. An industrial case study
3.1. The steel galvanization process
The extraordinary expenses and potential hazards resulting from the corrosion of steel and other

metals are among the most prominent challenges
facing construction, manufacturing, and other
steel-related industries. One of the most e!ective,
a!ordable, and widely-used methods for preventing
the corrosion of steel is `hot-dipa galvanization.
Through the galvanization process, steel beams,
sheets, or components are coated with a thin layer
of zinc to insulate the steel against corrosive agents
in the environment. Industrial customers of steel
have increasingly turned to galvanized products to
avoid the dangers and costs associated with steel
corrosion.
From the customer's perspective, the most critical property of a galvanized steel product is the
thickness of the exterior coating of zinc. This coating weight is expressed in terms of the net weight of
the zinc coating in proportion to the surface area
of the sheet or beam. The intended use of the
galvanized steel product } ranging from galvanized
steel roo"ng nails to the structural members for
bridges and buildings } dictates the requisite
coating weight. This speci"ed coating weight,
in turn, provides the basis for the customer's expectations of the galvanized steel manufacturer's
products.
Given the customer's speci"cations, the manufacturer of galvanized steel must then set the target
point } in terms of coating weight } for the galvanization process. In general, the manufacturer
should strive to meet the customer's demands while
consuming the least amount of zinc possible. Since
the customer will, in most circumstances, not object
to a higher coating weight, the manufacturer may
opt to set a fairly high target for the process. However, the wasted zinc may prove a costly drain. Of
course, if the manufacturer shaves away the di!erence between the customer's speci"cations and the
actual process target, the risks of producing unacceptable products soar. Although buyers for steel
sheets and beams with unacceptably low coating
weights may often be found in a secondary market
for uncoated steel, the "rm must then settle for
a substantially lower price and forfeit the value of
the zinc coating.
Manufacturers of galvanized steel sheets and
beams often enjoy another alternative. Although
the coating weight for one batch of steel sheets may
fail to meet the speci"cations demanded by, say,

Y.E. Shao et al. / Int. J. Production Economics 65 (2000) 229}242

a speci"c industrial or municipal construction
project, the same steel sheets may satisfy the
requirements for another project. Rather than
dump the thinly coated steel sheets immediately
into the uncoated steel market, the "rm may hold
the rejected sheets in the hopes of selling the same
goods to a customer with lower coating-weight
demands. Although the "rm incurs the holding
costs associated with storing the steel, it can expect
to ultimately sell the steel at galvanized-steel prices.
The challenge for the manufacturer is to e!ectively
integrate consideration of the costs and bene"ts of
holding rejected steel products into a strategy for
setting process targets.

235

Table 1
The relationship among holding cost, process standard deviation, OPT, and the corresponding expected pro"ts
Holding Process OPT
cost
standard
deviation

Expected Expected Expected
pro"t at pro"t at bene"t
plant
OPT
from OPT
target
(i.e., 330)

40

1
3
5
7

313.0711
318.0691
322.4704
326.5798

423.3455
420.5446
418.0025
415.5964

415.0001
415.0001
414.9971
414.7513

8.3454
5.5445
3.0054
0.8451

80

1
3
5
7

313.1519
318.3447
322.9611
327.1097

423.3091
420.4232
417.7885
415.2945

415.0001
415.0001
414.9962
414.6814

8.3090
5.4231
2.7923
0.6131

120

1
3
5
7

313.2149
318.5584
323.3409
327.8025

423.2806
420.3286
417.6220
415.0574

415.0001
415.0001
414.9954
414.6116

8.2805
5.3285
2.6266
0.4458

160

1
3
5
7

313.2664
318.7323
323.6494
328.2431

423.2574
420.2513
417.4863
414.8632

415.0001
415.0001
414.9945
414.5417

8.2573
5.2512
2.4918
0.3215

200

1
3
5
7

313.31
318.8786
323.9085
328.6150

423.2376
420.1862
417.3718
414.6990

415.0001
415.0001
414.9937
414.4718

8.2375
5.1861
2.3781
0.2272

3.2. Fixed holding cost
To illustrate the proposed technique for setting
the OPT, actual galvanization process costs and
other data were "rst obtained from an industrial
collaborator. The examination of a sample of
data sets revealed that the process's standard
deviation (equal to 3 milliounces/foot2 or mpf2)
provided good data for testing this target optimization strategy. In addition, the actual process
target for the manufacturer in question has been
330 mpf2.
The OPT may be found by substituting the
following parameters' values in Eq. (5): A "600
1
(dollars), A "500 (dollars), A "220 (dollars),
2
3
c"0.5 (dollars), ¸ "310 (mpf2), ¸ "301 (mpf2),
1
2
and S ranges from 40 to 200 (dollars). Several
values of the standard deviation of the process (i.e.,
1, 3, 5, and 7 mpf2) were evaluated. The above
parameters' values supply general information
about the galvanized steel industry. Table 1 illustrates these OPT values and their corresponding
expected pro"ts under various conditions. For
example, consider the case where the holding
cost"200 and the process standard deviation"3
(mpf2), the value mH"!2.9595 may be derived
1
using the Newton}Raphson method. The OPT,
uH"318.8786, may then be found by applying
Eq. (5). In contrast to the plant's target (i.e.,
330 mpf2), this study's "ndings suggest that
setting a lower process target would enhance the
"rm's total pro"ts by curbing the consumption

of excess zinc needed to sustain such a high
coating weight. The expected pro"t is $420.1862
when the OPT is set at 318.8786 mpf2 and
the expected pro"t is $415.0001 when the plant sets
the target at 330 mpf2. Therefore, the expected
bene"t from OPT would be $5.1861 per unit of
the "nished product, which is almost 1.25%
savings. While this may seem like a modest
increase, it can be achieved with no additional
work and can certainly add up in mass-production
runs.
In order to understand the e!ects of various
parameters on pro"tability, Eq. (5) can be re-written in the following formula:
1!cH/(m )!cH/(c #m )"0,
2 3
1
1
1

236

Y.E. Shao et al. / Int. J. Production Economics 65 (2000) 229}242

Fig. 1. The e!ects of c and c on OPT.
1
2

Fig. 2. The e!ects of c and c on expected pro"t.
1
2

where
c "(A !(A !S))/(cHp),
1
1
2
c "((A !S)!A )/(cHp),
2
2
3
c "(¸ !¸ )/p.
3
1
2

Figs. 1 and 2 display the e!ects of determining
the OPT and the expected pro"t for di!erent combinations of c and c . c and c are chosen by using
1
2 1
2
the values of A , A and A and varying S as
1 2
3
described at the beginning of this section. c essen1
tially represents the di!erence between the revenue

Y.E. Shao et al. / Int. J. Production Economics 65 (2000) 229}242

237

Fig. 3. The e!ects of c /c on the loss.
1 2

Fig. 4. The savings in terms of the various "xed holding costs,
using process standard deviations of 1, 3, 5, and 7 mpf2.

from the original customer and the net revenue
(selling price minus holding cost) from nonoriginal
customer in the primary market. c essentially rep2
resents the di!erence between the revenue from
a nonoriginal customer in the primary market and
a customer in the secondary market. Fig. 1 provides
the obvious, but very important, observation that
the higher the process variability, the further away
the optimal target is from the customer's speci"cation limit. Fig. 2 conveys a useful fact that the
higher the process variability, the lower the pro"t.
Fig. 3 emphasizes the e!ects of c and c on the
1
2
pro"tability. It shows the relationship between
the parameter ratio, c /c , and the loss. In Fig. 3,
1 2
the loss is de"ned as the di!erence between the
expected pro"t for placing the plant's target setting
(i.e., 330 mpf2) and the expected pro"t from placing
of the OPT. The ratio, c /c is described as
1 2

increase the value of the numerator or decrease the
value of denominator in Eq. (8). Increasing the
numerator implies that the net revenue di!erence
between original and nonoriginal customers in the
primary market should be large. As the net revenue
di!erence increases, there is a corresponding
decrease of the loss. This seems reasonable because
the OPT would be further away from the customer's speci"cation limit. As a result, it leads to
greater chances of the acceptability of the products,
and it leads to a smaller loss. Decreasing the
denominator in Eq. (8) also reduces the loss. This is
because as the di!erence between net revenue in the
primary and secondary markets gets smaller, there
is no longer a bene"t in holding the product for sale
in the primary market.
Turning now to Fig. 4, we can see how it displays
the savings versus the various "xed holding costs,
considering four levels of process variability (i.e.,
1, 3, 5, and 7 mpf2, respectively). In this "gure, the
savings are expressed by taking the di!erence
between the expected pro"t from setting the OPT
and the expected pro"t from setting the plant's
target at 330 mpf2. Two conclusions can be drawn
from the results in Fig. 4. First, it is apparent that
greater savings can be obtained only when the
process variability is small. Second, it is also apparent that as process variability becomes smaller the
slope of the curve #attens. This #attening indicates
that the savings are insensitive to the holding cost.
For example, in a case with the process's standard
deviation of 1 mpf2, the savings are approximately
a constant value, no matter what the values of the

A !A #S
c
2
1" 1
.
A !A !S
c
2
3
2

(8)

Fig. 3 considers a case in which the process standard deviation is 3 mpf2. A process standard deviation of 3 mpf2 was chosen because the real plant
data reveals this is a logical choice. It was also
chosen because it has the same pattern as the other
cases (i.e., where the standard deviations are 1, 5,
and 7 mpf2.) Fig. 3 indicates that the larger the
c /c ratio, the smaller the loss. Obviously, this
1 2
study seeks to decrease the loss and for this reason,
examines ways to obtain a large c /c ratio. There
1 2
are two ways to achieve this. You can either

238

Y.E. Shao et al. / Int. J. Production Economics 65 (2000) 229}242

holding costs are. Similarly, in a case with the
process's standard deviation of 7 mpf2, the savings
decrease as the holding costs increase. This implies
that the smaller the process variability, the #atter
the slope, which leads to the process being more
insensitive to the holding cost. All of this means
that near constant savings can be achieved when
there is small process variability. Thus, one can
expect greater pro"t.

Table 2
The relationship among t , process standard deviation, OPT,
u
and their corresponding expected pro"ts
Process
standard
deviation

OPT

Expected
pro"t

Expected
pro"t at
plant?
target
(i.e., 330)

Expected
bene"t
from
OPT

3

1
3
5
7

313.0538
318.0099
322.3649
326.4346

423.3532
420.5706
418.0484
415.6606

415.0001
415.0001
415.0001
414.7639

8.3531
5.5705
3.0483
0.8967

6

1
3
5
7

313.0596
318.0298
322.4003
326.4833

423.3506
420.5619
418.033
415.6390

415.0001
415.0001
415.0001
414.7597

8.3505
5.5618
3.0329
0.8793

9

1
3
5
7

313.0653
318.0493
322.4352
326.5313

423.3481
420.5533
418.0178
415.6179

415.0001
415.0001
415.0001
414.7556

8.3480
5.5532
3.0177
0.8623

12

1
3
5
7

313.0710
318.0686
322.4695
326.5786

423.3456
420.5449
418.003
415.5970

415.0001
415.0001
415.0001
414.7514

8.3455
5.5448
3.0029
0.8456

t
u

3.3. Variable holding cost
In addition to the parameters discussed above,
some other parameters are needed to determine
the OPT from Eq. (7). These parameters are
arbitrarily selected as the values of the mean
holding time, t , ranging from 3 to 12 time
u
units, and the standard deviation of the holding
time, t , ranging from 1 to 5 time units. The
p
"xed cost, a , ranges from 10 to 50 units, and
0
the holding cost per unit time, a , ranges from
1
1 to 5.
By using Newton}Raphson and Trapezoidal
Approximation, the solution of Eq. (7) can be
computed. Our "ndings suggest that the various
combinations of a and a described above exhibit
0
1
the same basic pattern of e!ects on pro"tability.
Likewise, the implications are similar for various
combination of the t and t .
u
p
Table 2 shows the OPT values and their corresponding expected pro"t under various conditions.
In Table 2, the expected bene"t from OPT is shown
for average holding time 3}12 with 4 process standard deviations. t is equal to 1, a is equal to 30,
p
0
and a is equal to 1 in this table. For example, in the
1
case of t "3 time units and p"3 mpf2, the
u
expected pro"t is $420.5706 when OPT is
318.0099 mpf2 and the expected pro"t is $415.0001
when plant's target is set at 330 mpf2. This results
in a bene"t of $5.5705 for using OPT. During a
mass-production run, the total bene"t might be
substantial.
Fig. 5 displays the corresponding `plant-lossa
when various target settings (TS) are made, ranging
from 310 to 330 mpf2. The rest of the variables take
on the values described above. Here, the plant-loss
is de"ned as the di!erence between the expected
pro"t for placing the plant's TS (i.e., 330 mpf2)

Fig. 5. Display of plant-loss when target settings range from 310
to 330 mpf2.

and the expected pro"t from placing of the speci"ed
TS. Fig. 5 elicits three observations. The "rst observation evinces how the loss is greatest when the
TS is 310 mpf2 and the process standard deviation

Y.E. Shao et al. / Int. J. Production Economics 65 (2000) 229}242

is 7 mpf2. This is because, in this study, the customer's speci"cation is 310 mpf2. The process
outputs would fall below 310 with a higher probability when the process standard deviation is at
the largest value. If process outputs were to fall
below the customer speci"cation, then the products would either sell to the market with the
price A (with involvement of holding cost) or
2
sell to the secondary market with an even cheaper
price A . As a result, when process outputs fall
3
below the customer's speci"cation, the loss is the
greatest.
The second observation shows a case in which
there is no plant-loss (i.e., the plant-loss equals 0).
When no plant-loss is preferred, then the TS
should be a larger value as process standard
deviations increase. For example, in order to
have the zero loss, the TS should be approximately 311.6 and 323.3 mpf2 when standard deviations are 1 and 7 mpf2, respectively. Fig. 5 shows
how the slope is steeper when the TS is set at
311.6 mpf2 for p"1 mpf2 than it is when the TS
is set to 323.3 mpf2 (for p"7 mpf2). A steeper
slope results in dramatic #uctuations in the
plant loss if the mean value shifts. The third
observation deals with the fact that for all values of
the process standard deviation, the plant loss
approaches zero as the TS gets closer to 330 mpf2.
This implies that the plant loss may be insensitive to the process variability when TS becomes
large.
Fig. 6 presents the corresponding `true-lossa
when various target settings are made, ranging
from 310 mpf2 to 330 mpf2. True-loss is de"ned as
the di!erence between the expected pro"t from
setting OPT and the expected pro"t from setting
the speci"ed TS. The results of Fig. 6 are similar
to those of Fig. 5, with the one exception that
the true-loss will not be the same when the TS
increases. As the true-loss increases, the TS
also increases. For example, when the TS is
330 mpf2, the lower the process variability is, the
greater the true-loss is. Therefore, it seems reasonable to conclude that the determination of the
OPT is important as long as the process remains
in a stable state and lower process variability is
maintained. If the OPT is found, the loss can be
substantially reduced. In addition, the trend of

239

Fig. 6. Display of true-loss when target settings range from 310
to 330 mpf2.

Figs. 5 and 6 addresses that the plant and true
losses may be less sensitive to the process variability when the TS is large. This may explain the
reason why the plant would have set a target of
330 mpf2.
Table 3 sets out the trade-o! between product
rejection and the pro"tability. A lower TS has the
advantage of lower production costs but the corresponding expected pro"ts are even lower. Table
3 shows the expected pro"t, plant-loss, and trueloss, respectively, when the TS ranges from 301 to
309 mpf2. Table 3 presents this information under
four conditions of the process standard deviations
ranging from 1, 3, 5, and 7 mpf2. Certainly when
one sets these lower TS values there is a very high
chance of rejection of the product (i.e., the product
quality is below the acceptable level). Here we wish
only to demonstrate the potential savings this
process o!ers; but the potential saving is also a
potential loss. As a savings, you can save some of
production cost, for example raw materials, such as
zinc in the present application. But as a loss, if you
choose a TS that is too low, it will not produce
a high pro"t. For this reason, we call it a loss,
plant-loss, or true-loss. Table 3 provides several
conclusions. First, all of the plant- loss and trueloss values are positive. The loss will increase in any
condition as long as the TS falls between 301 and
309 mpf2. The loss is substantial. Second, when TS
is 303 and 305 mpf2, the expected pro"t increases as
the process variability decreases. This is rational
since the process variability is typically inversely
proportional to the expected pro"t. There are exceptions to this assumption when the values of TS

240

Y.E. Shao et al. / Int. J. Production Economics 65 (2000) 229}242

Table 3
The relationship among process standard deviation, expected
pro"t, plant-loss, and true-loss with target settings from 301 to
309
Target
setting

Process
standard
deviation

Expected
pro"t

Plant-loss

True-loss

301

1
3
5
7

173.857
173.9998
178.203
186.3022

241.1431
241.0003
236.7943
228.4643

249.4979
246.5764
239.8553
229.3721

303

1
3
5
7

292.0894
236.6036
222.2988
221.5644

122.9107
178.3965
192.6985
193.2021

131.2655
183.9726
195.7595
194.1099

305

1
3
5
7

296.2088
280.454
264.8691
257.1826

118.7913
134.5461
150.1282
157.5839

127.1461
140.1222
153.1892
158.4917

307

1
3
5
7

295.3569
310.2843
303.2319
291.1771

119.6432
104.7158
111.7654
123.5894

127.9980
110.2919
114.8264
124.4972

309

1
3
5
7

314.4090
341.5830
336.4958
321.9554

100.5911
73.4171
78.5015
92.8111

108.9459
78.9932
81.5625
93.7189

values of TS are 307 and 309 mpf2. For example,
the quality characteristic of the products would
have a higher probability to fall below 310 mpf2
when the process standard deviation is 1 mpf2, and
the quality characteristic of the products would
have a higher probability to fall over 310 mpf2
when the process standard deviation is larger. Indeed, this seems to suggest that when the process
variability is large, so is the pro"t margin. In summary, this study cautions against using values of TS
ranging from 301 to 309 mpf2 because even with
lower production costs there is still quite a high risk
of loss.

4. Summary

equal 301, 307, and 309 mpf2. In this study, the
plant's speci"cation (¸ ) is 301 mpf2 and the cus2
tomer's speci"cation (¸ ) is 310 mpf2. When the
1
quality characteristic of the product is lower than
the plant's speci"cation, then the selling price is A ,
3
which is the lowest selling price. When the quality
characteristic of the product is lower than the customer speci"cation, then the selling price is
A which is lower than the primary price A . There2
1
fore, considering the case of TS of 301 mpf2, the
quality characteristic of the products would have
a higher probability to fall below 301 mpf2 when
the process standard deviation is small. This results
in lower pro"t. The quality characteristic of the
products would have a higher probability to fall
over 301 mpf2 when the process standard deviation
is large. This could result in greater pro"t. Similarly, this feature works for the cases where the

This study has examined methods for determining the optimal target setting for industrial processes in order to maximize expected pro"ts with
respect to process variability and to production
costs, material costs, holding costs, and other "scal
considerations. Speci"c attention has been given to
the determination of the optimal target setting in
industries in which several grades of customer
expectations are possible within the same market.
In such circumstances, a manufacturer may hold
products that have been rejected by the intended
customer in the hopes of selling them to another
customer in the same market. Even though the
producer incurs a storage or holding cost, the net
pro"t may still be higher than when the rejected
goods are reworked, discarded, or sold in a secondary market.
This study has examined such circumstances
through mathematical modelling and through consideration of an actual case study. It presents
methods for determining the optimal process target
for an industrial process that considers the costs
and bene"ts associated with holding rejected goods
for later sale. It has also explored how these
methods may be applied in the steel galvanization
industry. Our research "ndings show that the
key issue for greater pro"tability is low process
variability.
The pivotal contribution of the present study is
the recognition that the determination of a process
target must re#ect a recognition of the costs as

Y.E. Shao et al. / Int. J. Production Economics 65 (2000) 229}242

well as the bene"ts of holding rejected products
for sale in the same market. Future research will
expand these e!orts to consider situations in
which holding costs are dependent on the distance
from target. For the moment, it su$ces to conclude
that it is reasonable for a manufacturer to set
the process target close to the customer's speci"cations when the process variability is low; a manufacturer should be more conservative when
confronting a higher process variability. Future
research will also allow for the distribution of the
holding time to be non-normal. Finally, future research will investigate how to extend the current
model to allow the value of ¸ to be a decision
2
variable.

!a
1

PC

"

L1

L2

AP

241

BD

=
t f (t)dt
0

f (x)dx

(A !cx!R!a )
2
0

!a
1

AP

BD

=
t f(t)dt
0

f (x)dx

"(A !R!a !a B )
2
0
1 T
!c

P

L1
f (x)dx
L2

P

L1
xf (x)dx.
L2

(A.2)

Substituting Eq. (A.2) into Eq. (A.1), then the following holds:
Appendix A. The derivation of E[P(x, t); u]

P

=
E[P(x, t); u]" (A !cx!R) f (x)dx
1
L1

"(A !R)
1

PP

#

L1 =
(A !cx!R!a
2
0
L2 0

!a t) f (t) f (x)dt dx
1

P

#

E[p(x, t); u]

(A !cx!R) f (x)dx. (A.1)
3
~=

P

AB

C

P CP
P CP
L2

"

D

L2

xf (x)dx#(A !R!a
2
0
~=

D

=

L2

0

P

P
C

!a B )
1 T

L1
L1
f (x)dx!c xf (x)dx
L2
L2

A

(A !cx!R!a ) f (t)dt
2
0

BD

¸ !u
1
p

¸ !u
2
p

B

#(A !R!a !a B )
2
0
1 T

=
(A !cx!R!a !a t) f (t)dt f (x)dx
2
0
1
0
L1

f (x)dx
~=

A

=
/(D )
T #D
t f (t)dt"t
p W(D )
T
0
T

=
xf (x)dx
L1

L2

#(A !R)W
3

where D "t /t . The element in Eq. (A.1), which is
T
u p
:L12[:=(A !cx!R!a !a t) f (t)dt] f (x)dx, can
L 0 2
0
1
be derived as
L1

L1

"(A !R) 1!W
1

1
e~(t~tu)2@2t2p and
f (t)"
t
J2nt W u
p t
p

P

P

f (x)dx!c

P

Now let

B "
T

=

#(A !R)
3
!c

L2

P

CA

] W

B A

¸ !u
¸ !u
1
!W 2
p
p

BD

!cu

"(A !R!cu)
1

A
A

B
B

#(!A #A !a !a B )W
1
2
0
1 T

¸ !u
1
p

#(!A #A #a #a B )W
2
3
0
1 T

¸ !u
2
.
p

242

Y.E. Shao et al. / Int. J. Production Economics 65 (2000) 229}242

Appendix B. The derivation of the 5rst derivative of
E[P(x, t); u]
By using the results in Appendix A, we can
obtain
dE[P(x, t); u]
du

A BA
A B

B

¸ !u
1
"(!A #A !a !a B ) ! / 1
1
2
0
1 T
p
p
1
#(!A #A #a #a B ) !
2
3
0
1 T
p
]/

A

B

¸ !u
2
!c.
p

(B.1)

Let m "(¸ !u)/p, and m "(¸ !u)/p. Then
1
1
2
2
Eq. (B.1) can be derived as
dE[P(x, t); u]
"k /(m )!k /(m )!c,
1
1
2
2
du
where
k "(1/p)(A !A #a #a B ),
1
1
2
0
1 T
k "(1/p)(A !A !a !a B ).
2
2
3
0
1 T
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