Full text of DOI:10.1111/j.1937-5956.2000.tb00333.x

PRODUCTION AND OPERATIONS MANAGEMENT
Vol. 9, No. 2, Summer 2000
Printed in U.S.A.
ECONOMIC PROCESS CONTROL UNDER
UNCERTAINTY*
FELICIEN KANYAMIBWA AND J. KEITH ORD
Prudential Insurance Company of America, Newark, New Jersey 07102, USA
McDonough School of Business, Georgetown University, Washington, DC 20057, USA
The future of the global industry lies in the continuous improvement of both products and
processes, a renewed commitment to competition, and an aggressive approach to satisfying custom-
ers’ needs in quality, quantity, and timing. In quality management, the degree of customer satisfaction
for a given product may be measured in the form of the loss to society. This loss is formulated as a
function of the deviation from the target for each of the product’s quality characteristics. The greater
the variability of uncontrolled factors during manufacturing or production the larger will be that loss.
In this paper, we develop a form of the loss function that takes into account the variability of a
production process, the decision loss, and the costs of sampling and inspection. Specifically, we
consider monitoring a production process, which may undergo continuous mean shift and variance
deterioration during a production run. We then examine decision rules for continuing production or
stopping and adjusting the production process.
(CONTINUOUS QUALITY IMPROVEMENT; QUALITY CONTROL; TAGUCHI LOSS; QUAL-
ITY COSTS)
1. Introduction
The revolution in quality control and improvement in the last two decades brought many
challenges to industry. That revolution came in response to customers’ increasingly high
expectations of product quality and to various answers to these challenges by quality experts
and firms. It is evident, now more than ever before, that the future of the global industry lies
in the continuous improvement of both products and processes, a renewed commitment to
competition, and an aggressive approach to satisfying customers’ needs. In quality manage-
ment, the degree of customer satisfaction for a given product may be expressed in the form
of the loss to society, defined as the loss caused by the quality of a product after it is shipped,
other than any loss caused by the product’s intrinsic quality functions (Logothetis and Wynn
1989). The loss is formulated as a function of the deviation from the target for each of the
product’s quality characteristics. The greater the variability of uncontrollable factors during
manufacturing or production the larger the loss to society. To deal with the uncontrollable
factors, economic models of quality control have been proposed in literature and applied in
industry.
Several researchers developed economic models to design quality control charts in
statistical process control (SPC). Most of these models were developed in the context of
* Received June 1997; revisions received January 1998 and November 1998; accepted March 1999.
184
1059-1478/00/0902/184$1.25
Copyright © 2000, Production and Operations Management Society

ECONOMIC PROCESS CONTROL UNDER UNCERTAINTY
185
specification limits and defects, and assume variability to be fixed (Montgomery 1980; Lee
and Rosenblatt 1987; Moinzadeh and Klastorin 1995). Sullivan (1984) notes that such
models may constitute a barrier to process improvement and other approaches to variability
reduction, such as those advocated by Deming (Deming 1982; Walton 1988) and Taguchi
and Wu (1985). The work by Moskowitz, Plante, and Chun (1994) addresses some of these
issues, but their work has two main limitations. First, while studying process variability, they
use the concept of control limits. Second, they overlook the importance of the tradeoffs
between sampling and inspecting and continuing the process, versus stopping and adjusting
the production process at any given time.
In this paper, we develop a loss function approach that enables us to determine optimal
sampling and inspection policies as well as to decide whether to adjust the process or to
continue production without adjustment. We allow for variability in the production process
and for the possibilities of both the continuous mean shift and variance deterioration within
a production run. We limit our analysis to the context of a quadratic loss function, consistent
with Taguchi’s approach (Logothetis and Wynn 1989).
The paper is organized as follows. In Section 2 we review relevant literature and revisit the
concepts of variability reduction. In Section 3 we develop the methodological framework and
formulate the model. We also establish preliminary results of our analysis. In Section 4 we
give a general decision model which allows for the possibility of sampling and inspection.
Specifically we derive an expression for selecting the optimal sample size and inspection
interval. In Section 5 we consider a case study and draw some analytical conclusions. We
give concluding remarks in Section 6.
2. Literature Review
Quality control relies on traditional methods such as SPC, control charts, reliability studies,
and process capability indices. The purpose of these on-line quality control tools is to
maintain the manufacturing process in control. But whereas these tools remain important,
they are not intended to and cannot deal with uncontrolled process variability.
To address such problems, Deming published his famous 14 points, which stress the
importance of quantitative methods in product and process design. Deming argues that
“Quality improvement is a pervasive, never-ending activity with a constancy of purpose” and
that it must be controlled during production, not usage (Deming 1982; Walton 1988). In the
same context, Taguchi (Taguchi and Wu 1985) proposed to move quality to the design and
product development stage. The concept of “off-line” design or design by design popularized
by Taguchi advocates the use of statistically designed experiments in engineering design to
control process variation. Taguchi argues that the indirect costs related to quality problems
such as loss of market share and increased marketing effort to ward off competition will
increase with variations in the product’s quality characteristics. He concludes that since
variations from socially desired functional specifications cause a loss of quality, then
minimizing variations would lead to reduced warranty and service costs and more satisfied
customers.
To achieve the objective of off-line design, Taguchi recommended that the purpose of the
actions taken on a regular basis during normal production be prevention and not rectification.
The out-of-control region in the control chart should never be allowed to be reached;
therefore, variability in the production process should be diagnosed and dealt with before it
becomes a concern.
Taguchi’s viewpoint may be formulated in the following way: a process that is in control
at the beginning of a production cycle can go “out of control” with a positive probability each
time an additional item in the lot is produced. Once out of control, the process remains so
until the next setup.
The concept of an out-of-control state in unreliable systems has been the subject of several

FELICIEN KANYAMIBWA AND J. KEITH ORD
186
studies in the recent literature. Lee and Rosenblatt (1987) considered the possibilities of
process deterioration and the existence of a constant rate of defective items in a production
lot. Addressing the problem of joint control of production cycles or manufacturing quantities
and maintenance by inspection, they showed that the optimal diagnostic intervals are equally
spaced.
Moinzadeh and Klastorin (1995) studied quality control and costs with a sampling plan in
an unreliable production process by making assumptions consistent with those in Lee and
Rosenblatt (1987) about the production of defective items. They found that in an unreliable
production process which produces only non-defective items when operating in control, but
produces defective items with fixed probability when the process has shifted to an out-of-
control state, an efficient sampling and testing of operations may significantly reduce
expected total costs. However, the important assumption made by Lee and Rosenblatt (1987)
and Moinzadeh and Klastorin (1995), of an out-of-control process, producing defective items
with a constant rate only from the time the process goes astray until the next adjustment,
assumes the concepts of fixed variability. The assumption ignores the concept of process
variability deterioration; that is, a situation in which the process variance increases with the
effective production run time between two process setups.
Several studies have stressed the need to consider variability deterioration; see Sullivan
(1984), Guo and Ord (1995), Moskowitz, Plante, and Chun (1994), and Wang and Gerchak
(1996). In this paper we address the impact of variability deterioration upon inspection and
sampling policy in unreliable production processes.
3. The Variability Loss Function Model
In the absence of an inspection plan, the decisions available at any given time are the
following:
●
Continue the process.
●
Sample and inspect with the possibility of adjusting the process in order to minimize the
loss.
In practice, decision makers follow an inspection plan, and the process is inspected
periodically by sampling a fixed number of observations. The decisions to continue or adjust
the production process are made only at inspection instants. Two decision variables are of
interest in this case:
●
The sample size, and
●
The inspection interval (time between samples).
In this paper, we define the loss as the sum of the cost due to variability and the cost of
sampling and inspection, given the sample size and the inspection interval. We assume that
a sample of fixed size is drawn at the end of the inspection interval. The sampling interval
(time taken to sample) is assumed to be small relative to the inspection interval so that the
observations in a particular sample may be taken to be independent and identically distrib-
uted; of course the density function may vary over time. The process variability may
deteriorate and its mean may continuously shift with time. Values of quality characteristics
are assumed to be normally distributed. We derive the results for the univariate case.
We adopt the following key notation:
Xt ϭ value of the quality characteristic X, at time t, with observed value xt;
␮ ϭ mean of the value of the quality characteristic at time t, after adjustment or setup at
t
time zero;
I ϭ inspection interval in time units;
k ϭ variability cost in $ per unit of variability measure;
m ϭ sample size in number of items;
n ϭ number of items produced during an inspection interval;
Th ϭ throughput rate in number of items per unit time;

ECONOMIC PROCESS CONTROL UNDER UNCERTAINTY
187
FIGURE 1. Tree for the Decision Making in Economic Process Control.
␾( x, T) ϭ k( x Ϫ T)2 ϭ variability cost function with target T in $;
Lc(t) ϭ loss, in $, of continuing the process at time t;
Lc(t, ␮t) ϭ loss, in $, of continuing the process at time t, when ␮ is known;
t
¯
¯
Lc(t, Xt) ϭ loss, in $, of continuing the process at time t, when Xt is observed;
Ls(t) ϭ loss, in $, of stopping the process at time t;
Ci ϭ cost of inspection in $ per unit;
Cr ϭ cost, in $, of adjusting the process or adjustment cost per setup;
f٪, g٪, h٪ ϭ probability density functions.
The results are derived using the following process probability distributions. We assume
Xt to be normally distributed with mean ␮ and variance ␻2. Also, we assume that the mean
t
␮ evolves as a random walk and the distribution for Xt given ␮0 is from a starting value ␮
t
0
at time zero with incremental variance ␴2. That is
2
Xt͉␮t ϳ ᏺ͑␮t, ␻ ͒,
2
␮ ͉␮0 ϳ ᏺ͑␮0, t␴ ͒;
t
2
Xt͉␮0 ϳ ᏺ͑␮0, ␻2 ϩ t␴ ͒.
(1)
The methodological approach followed in this paper is based on the principle of maximiza-
tion of expected utility and on the likelihood principle. Maximization of utility is expressed
in the form of minimization of a loss function. The likelihood principle requires that
inference be based on the observed data alone, and not on the data that could have been
observed but was not (Singpurwalla 1992). For this, we use a Bayesian decision analysis
approach. In Subsection 3.1, we develop the methodological approach implemented in the
paper. Singpurwalla (1992) provides a general discussion of the approach.
3.1. Decision Making Under Uncertainty
In Figure 1, we show a schematic description of the decision-making process using a flow
diagram. We represent the decision nodes by rectangles and the random nodes by circles.
Each decision may be associated with error due to uncertainty.
Let Ᏹ be a collection of sampling and inspection plans, with elements denoted e ϭ {m,
I}; Ᏸ be a collection of decisions, with only two elements, {c ϭ continue the process; s
ϭ stop the process}, indexed by d; ᏾ be a collection of possible quality characteristic
1
posterior mean observations, ␮t; and ᏾ be a collection of quality characteristic values, xt.
2
The decision-making process consists of two parts: at decision node Ᏸ1 we choose a
sampling and inspection plan, e, whereas at decision node Ᏸ2 we decide to continue or stop
the production process given the observed posterior mean of the quality characteristic value.
Uncertainty occurs at node ᏾1 , since the posterior mean is based on a random sample mean,

FELICIEN KANYAMIBWA AND J. KEITH ORD
188
and at node ᏾2, since the process variability, from which the loss is determined, is based on
a random variable, Xt. The revision of belief about the process mean is done in the light of
sampling information, through the posterior distribution, using a Bayesian approach.
From Figure 1, we describe the decision process as follows. Consider that we want to make
a decision at time t, for subsequent periods, t ϩ u, u Ͼ 0. At time (t ϩ u), we observe
quality characteristics Xu and we incur a loss that may be written as L(e, d, ␮t, xu); that is,
the loss is formulated as a function of decision d, sampling and inspection plan e, and the
posterior mean ␮ observed at time t. The expected loss may be written as
t
L͑e, d, ␮ ͒ ϭ
L͑e, d, ␮t, xu͒ p͑ xu͉␮t, d, e͒dxu,
t
͵
xu
Lc͑t, ␮ ͒ when d ϭ c
t
ϭ
(2)
ͭ
Ls͑t͒
when d ϭ s,
where p( xu͉␮t, d, e) is the posterior distribution of xu given ␮t, d, and e.
At node Ᏸ2 we minimize the expected loss over the two possible decisions: continuing or
adjusting the process. Hence, we find the decision that satisfies the function
L͑e, ␮ ͒ ϭ min ͓Lc͑t, ␮ ͒, Ls͑t͔͒.
(3)
t
t
At the random node ᏾ we compute the expected loss when we observe the sample mean,
1
¯
Xt or, equivalently, when we obtain the process posterior mean, ␮t. Given e, we assume that
␮ has the predictive density p(␮ ͉e). The expected loss is
t
t
L͑e, ␮ ͒ p͑␮ ͉e͒d␮t,
(4)
t
t
͵
␮
t
where p(␮ ͉e) is the predictive or the preposterior distribution of ␮t.
Finally, we need to find the optimal inspection and sampling plan. This optimal plan
t
minimizes the preceding expression in (4) and satisfies the function:
min
L͑e, ␮ ͒ p͑␮ ͉e͒d␮ .
(5)
t
t
t
͵
ͫ
ͬ
e
␮
t
The development of Subsections 3.2–3.4 follows the principle of maximization of expected
utility. We start at the terminus of the flow diagram and work our way back to get to Ᏸ2 in
Subsection 3.2, using (2) and (3), then to ᏾2 in Subsection 3.3, using (4), and finally to Ᏸ1 ,
in Subsection 3.4, using (5). In Section 4, we find the long run optimal sample size and
inspection interval, based on the loss function formulated in Subsection 3.4. The solution
procedure for finding the optimal sample size and interval involves two steps:
●
Given m and I, determine the steady-state conditions of continuing and adjusting the
process using E(loss) ϭ E(loss͉c)ءP(c) ϩ E(loss͉s)ءP(s);
Optimize expected loss over m and I.
●
3.2. Decision Rules When the Process Mean at Time t After Adjustment Is Known, and
the Inspection Interval Is Fixed
To invoke the maximization of expected utility principle, we start at the terminus of the
flow diagram. For this, the process mean and the inspection interval are assumed fixed. An
economic decision about the process implies balancing between the expected loss incurred if
the process is adjusted, which we define as E(Ls), and the expected loss incurred if the
process is continued, which we define as E(Lc). If E(Ls) Յ E(Lc) then the process is

ECONOMIC PROCESS CONTROL UNDER UNCERTAINTY
189
adjusted; otherwise, the process is continued. We use (2) and the quadratic loss ␾( xu, T) to
derive the results.
In (2), the formulated loss is per item and per unit time. When the inspection interval is
I, and the throughput rate is Th, the expected losses need to be integrated over the inspection
interval and to take into account the throughput rate. Moreover, the expected loss if we
continue production is evaluated over ␮tϩu. With fixed sample size and inspection interval,
the probability density function p( xu͉␮t, d, e) in (2) is no longer conditioned on d and e for
computing expected loss in either situation, so that we let f( xu͉␮t) ϭ p( xu͉␮t, e). Hence,
using (2), the expected loss with known process mean at time t is
I
E͑Lc͒ ϭ E͑Lc͑t, ␮ ͒͒ ϭ Th
Lc͑t, ␮ ͒du
t
t
͵
0
I
ϱ
ϭ Th
␾͑ xu, T͒ f͑ xu͉␮ ͒dxudu,
(6)
t
͵ ͵
0
Ϫϱ
where f( xu͉␮t) is the probability density function (PDF) of Xu given ␮ which may be
t
expressed as
ϱ
f͑ xu͉␮ ͒ ϭ
f͑ xu͉␮tϩu, ␻͒ g͑␮ ͉␮ ͒d␮
,
tϩu
t
tϩu
t
͵
Ϫϱ
where g is the PDF for ␮ ͉␮t, as introduced in (1). Then, we have
tϩu
I
ϱ
ϱ
E͑Lc͑t, ␮ ͒͒ ϭ Th
␾͑ xu, T͒ f͑ xu͉␮tϩu, ␻͒ g͑␮ ͉␮ ͒d␮tϩudxudu,
(7)
t
tϩu
t
͵ ͵ ͵
0
Ϫϱ Ϫϱ
where, from (1)
2
1
1 xu Ϫ ␮
tϩu
f͑ xu͉␮
exp Ϫ
, ␻͒ ϭ ͱ2␲␻
ͩ ͩ ͪ ͪ
tϩu
2
␻
2
1
1
␮
tϩu Ϫ ␮
t
͉␮ ͒ ϭ ͱ2␲u␴
g͑␮
exp Ϫ
ͩ ͩ ͪ ͪ
tϩu
t
2u
␴
If we stop the process to make an adjustment we incur the adjustment cost Cr. Hence, from
(2) the expected loss becomes
I
E͑Ls͒ ϭ E͑Ls͑t͒͒ ϭ Th
Ls͑t͒du
͵
0
I
ϱ
ϱ
ϭ Cr ϩ Th
␾͑ xu, T͒ f͑ xu͉␮u, ␻͒ g͑␮ ͉␮ ͒d␮udxudu, (8)
u
0
͵ ͵ ͵
0
Ϫϱ Ϫϱ
since the process mean is now reset at ␮ at time t. Therefore, using (1) and (6), we have
0

FELICIEN KANYAMIBWA AND J. KEITH ORD
190
2
I
ϱ
ϱ
1
1 xu Ϫ ␮
tϩu
2 ͱ2␲␻
E͑Lc͑t, ␮ ͒͒ ϭ Th
k͑ xu Ϫ T͒
exp Ϫ
ͩ ͩ ͪ ͪ
t
͵ ͵ ͵
2
␻
0
Ϫϱ Ϫϱ
2
1
1
␮tϩu Ϫ ␮
t
ϫ ͱ2␲u␴
exp Ϫ
d␮tϩudxudu;
ͩ ͩ ͪ ͪ
2u
␴
I
ϱ
ϱ
2
ϭ Th
k͑͑ xu Ϫ ␮ ͒2 ϩ ͑␮tϩu Ϫ T͒ ͒
tϩu
͵ ͵ ͵
0
Ϫϱ Ϫϱ
2
1
1 xu Ϫ ␮
tϩu
ϫ ͱ2␲␻
exp Ϫ
ͩ ͩ ͪ ͪ
2
␻
2
1
1
␮tϩu Ϫ ␮
t
exp Ϫ
d␮tϩudxudu;
ϫ ͱ2␲u␴
ͩ ͩ ͪ ͪ
2u
␴
I
ϱ
2
ϭ Th
k ␻2 ϩ
͑␮tϩu Ϫ ␮ ͒2 ϩ ͑␮t Ϫ T͒
t
͵ ͵
ͩ
ͪ
ͫ
0
Ϫϱ
2
1
1
␮
tϩu Ϫ ␮
t
exp Ϫ
d␮tϩudu;
ϫ ͱ2␲u␴
ͩ ͩ ͪ ͪ
ͬ
2u
␴
I
2
ϭ Thk
͑␻2 ϩ u␴2 ϩ ͑␮t Ϫ T͒ ͒du;
͵
0
1
ϭ T Ik ␻2 ϩ I␴2 ϩ ͑␮ Ϫ T͒
.
(9)
2
ͩ
ͪ
h
t
2
Following the same approach and using (8),
I
ϱ
2
E͑Ls͑t͒͒ ϭ Cr ϩ Thk
␻2 ϩ
͑͑␮u Ϫ ␮ ͒2 ϩ ͑␮0 Ϫ T͒ ͒
0
͵ ͵
ͫ
0
Ϫϱ
2
1
1
␮u Ϫ ␮
0
ϫ ͱ2␲t␴
exp Ϫ
d␮ du;
u
ͩ ͩ ͪ ͪ
ͬ
2u
␴
I
2
ϭ Cr ϩ Thk
͓␻2 ϩ u␴2 ϩ ͑␮0 Ϫ T͒ ͔du;
͵
0
1
ϭ C ϩ T Ik ␻2 ϩ I␴2 ϩ ͑␮ Ϫ T͒
.
(10)
2
ͩ
ͪ
r
h
0
2
Thus,
E͑Ls͑t͒͒ Յ E͑Lc͑t, ␮ ͒͒
2
if and only if
Cr Յ kITh͑͑␮t Ϫ T͒2 Ϫ ͑␮0 Ϫ T͒ ͒.
t
Hence we adjust the process whenever
2
Cr Յ kITh͑͑␮t Ϫ T͒2 Ϫ ͑␮0 Ϫ T͒ ͒,

ECONOMIC PROCESS CONTROL UNDER UNCERTAINTY
191
or
1/ 2
Cr
2
͉␮ Ϫ T͉ Ն
t
ϩ ͑␮ Ϫ T͒
,
(11)
ͫ
ͬ
0
kITh
i.e., when the deviation of the mean from the target exceeds a threshold.
3.3. Decision Rules When the Process Mean at Time t From the Last Adjustment Is
Unknown, and the Inspection Interval Is Fixed
In Subsection 3.2, we took ␮t, the mean of the process at t, the time of adjustment, to be
known; now working our way back on the flow diagram of Figure 1, let it be unknown and
¯
consider the posterior distribution of ␮t. Let Xt, m, and ␮ be respectively the sample mean,
t
the sample size, and the unknown process mean, at time t. Moreover, assume that based on
prior information and additional observation, the probability distribution of ␮ is determined
t
to be (Stuart and Ord, 1994, pp 304–6)
␮t ϳ ᏺ͑␪, Vt͒;
(12)
(13)
then
2
␮
͉␮t ϳ ᏺ͑␪, Vt ϩ u␴ ͒;
͑tϩu͒
2
␻
៮
X ͉␮ ϳ ᏺ ␮ ,
.
ͩ ͪ
t
t
t
m
Hence
2
␻
៮
X ͉␪ ϳ ᏺ ␪, ϩ V ;
(14)
(15)
(16)
ͩ
␪ ϩ
ͪ
t
t
m
2
␻
mVt
␻2Vt
៮
៮
Xt,
␮ ͉X ϳ ᏺ
;
ͩ
ͪ
t
t
␻2 ϩ mVt
␻2 ϩ mVt
␻2 ϩ mVt
2
␻
mVt
␻2 ϩ mVt
␻2Vt
␻2 ϩ mVt
2
៮
៮
Xt,
␮
͉X ϳ ᏺ
␪ ϩ
ϩ u␴ .
ͩ
ͪ
tϩu
t
␻2 ϩ mVt
Let
2
␻
mVt
␻2 ϩ mVt
៮
Xt;
E͑␮ ͒ ϭ
␪ ϩ
pt
␻2 ϩ mVt
␻2Vt
␻2 ϩ mVt
Var ͑␮ ͒ ϭ
.
(17)
pt
When the inspection interval and the sample size are respectively I and m, the inspection cost
is mCi. Furthermore, from (4), the expected loss for continuing the process is evaluated over
¯
␮
tϩu, given information at time t, Xt, and xu, whereas the expected loss for adjusting the
process is evaluated over ␮ and xu, following the process adjustment. Both expected losses
u
are integrated over the inspection interval. Following the same approach as in Subsection 3.2,
¯
we do not condition on ␮t, d, and e and, since the information available on ␮t is through Xt,
¯
we let f( xu͉Xt) ϭ p( xu͉␮t, d, e). If we adjust the process we incur the adjustment cost Cr.
Then, following the same approach as in Subsection 3.2, the expected losses with sampling
are, respectively, for continuing and adjusting the process

FELICIEN KANYAMIBWA AND J. KEITH ORD
192
I
ϱ
ϱ
៮
៮
E͑Lc͑t, Xt͒͒ ϭ mCi ϩ Th
␾͑ xu, T͒ f͑ xu͉␮tϩu, ␻͒ g͑␮ ͉Xt, ␴͒d͑␮ ͒dxudu;
tϩu
tϩu
͵ ͵ ͵
0
Ϫϱ Ϫϱ
I
ϱ
ϱ
E͑Ls͒ ϭ Cr ϩ mCi ϩ Th
␾͑ xu, T͒ f͑ xu͉␮u, ␻͒ g͑␮ ͉␮ ͒d͑␮ ͒dxudu. (18)
u
0
u
͵ ͵ ͵
0
Ϫϱ Ϫϱ
In this case
I
ϱ
៮
៮
E͑Lc͑t, Xt͒͒ ϭ mCi ϩ Thk
␻2 ϩ
͑␮tϩu Ϫ T͒2g͑␮ ͉Xt͒d͑␮ ͒ du;
tϩu
tϩu
͵ ͵
ͫ
ͬ
0
Ϫϱ
␻2Vt
I
ϭ mCi ϩ Thk
␻2 ϩ u␴2 ϩ
ͩ
ͪ
2
ͫ
␻2 ϩ mVt
͵
0
2
␻
mVt
␻2 ϩ mVt
៮
ϩ
␪ ϩ
X Ϫ T
du;
ͩ
ͪ
ͬ
t
␻2 ϩ mVt
I
␻2Vt
ϭ mC ϩ kIT ␻2 ϩ
␴2 ϩ
ͩ
ͩ
ͪ
ͩ
ͫ
i
h
2
␻2 ϩ mVt
2
2
␻
mVt
␻2 ϩ mVt
៮
ϩ
␪ ϩ
X Ϫ T
,
ͪ
ͬ
t
␻2 ϩ mVt
I
2
ϭ mC ϩ kIT ␻2 ϩ
␴2 ϩ Var ͑␮ ͒ ϩ ͑E͑␮ ͒ Ϫ T͒
,
(19)
ͪ
ͫ
ͬ
i
h
pt
pt
2
and the expected loss if we adjust the process corresponds to (10), plus the inspection cost:
1
2
E͑Ls͑t͒͒ ϭ Cr ϩ mCi ϩ kITh͑␻2 ϩ 2 I␴2 ϩ ͑␮0 Ϫ T͒ ͒.
Thus,
1
2
E͑L ͑t͒͒ Յ E͑L ͑t, X ͒͒ N C ϩ kIT ␻2 ϩ I␴2 ϩ ͑␮ Ϫ T͒
៮
ͩ
ͪ
s
c
t
r
h
0
2
2
2
I
␻2Vt
␻2 ϩ mVt
␻
mVt
␻2 ϩ mVt
Յ kIT ␻2 ϩ
␴2 ϩ
ϩ
␪ ϩ
(20)
៮
X Ϫ T
t
ͩ
ͪ ͩ
ͪ
ͬ
ͫ
h
2
␻2 ϩ mVt
Hence we adjust the process when
␻2Vt
␻2 ϩ mVt
C ϩ kIT ͑␮ Ϫ T͒2 Ϫ
ͫ
ͬ
r
h
0
1
2
2
៮
Յ
͓␻ ͑␪ Ϫ T͒ ϩ mVt͑Xt Ϫ T͔͒
2
kITh
͑␻2 ϩ mVt͒
or
1/ 2
Cr
͉E͑␮ ͒ Ϫ T͉ Ն
ϩ ͑␮ Ϫ T͒2 Ϫ Var ͑␮ ͒
.
(21)
ͭ
ͮ
pt
0
pt
kITh
This result is similar to the conclusion in (11). In fact, we adjust the process when the
deviation of the posterior mean from the target exceeds a threshold. The known mean ␮ in
t
(11) is replaced here by the (expected) posterior mean, E(␮pt). However, with the additional

ECONOMIC PROCESS CONTROL UNDER UNCERTAINTY
193
information from sampling, the precision increases so that the square of the threshold in (11)
is reduced by the variance of the posterior mean in (21).
3.4. Loss Function When the Sampling and Inspection Plan Are Fixed
In Subsection 3.3 we determined the decision rules when the process mean must be
estimated from sampling information. Following the same approach as in the preceding
subsections, we fold the flow diagram of Figure 1 to reach Ᏸ1. In this subsection, we
formulate the loss function that will serve as the basis in determining the optimal inspection
and sampling plan.
Consider a fixed sample size m and a known inspection interval I. At time t, two options
are possible. After inspecting the process, we may decide to adjust the process or to continue
¯
the process without adjustment. Let x¯t be the mean of the sample, and Lc(t, Xt) and Ls(t) be
respectively the loss of continuing and of adjusting the process after sampling. Then, the
minimum loss of making a decision with sampling is, from (3)
៮
L* ϭ min ͕Lc͑t, Xt͒, Ls͑t͖͒
(22)
Let H1 and H2 be respectively the lower and upper limits on the posterior mean for
continuing the process; from (21) these values are
1/ 2
Cr
H ϭ T Ϫ
ϩ ͑␮ Ϫ T͒2 Ϫ Var ͑␮ ͒
;
.
ͭ
ͭ
ͮ
ͮ
1
0
pt
kITh
1/ 2
Cr
H ϭ T ϩ
2
ϩ ͑␮ Ϫ T͒2 Ϫ Var ͑␮ ͒
(23)
0
pt
kITh
Let
2
៮
␻ ␪ ϩ mXtVt
␻2 ϩ mVt
␮
ϭ
;
then
pX t
C
pt ϭ ͕␮pt: H1 Յ ␮pt Յ H2͖,
␮
S
ϭ ͕␮pt: ␮pt Յ H1
or
␮pt Ն H2͖.
␮
pt
Therefore,
E͑L͒ ϭ
Lc͉␮pX dF͑␮ ͒ ϩ
Ls͉␮pX dF͑␮ ͒,
pt
pt
͵
t
͵
t
C
S
␮
pt
␮
pt
or, equivalently
2
2
2
␻2Vt
I␴
␻ ␪ ϩ mXtVt
␻2 ϩ mVt
៮
E͑L͒ ϭ
mC ϩ IkT ␻2 ϩ
ϩ
ϩ
Ϫ T
dF͑␮ ͒
ͩ
ͩ
ͪ ͪ
ͫ
ͬ
i
h
pt
␻2 ϩ mVt
2
͵
C
␮
pt
2
I␴
2
ϩ
C ϩ mC ϩ IkT ␻2 ϩ
ϩ ͑␮ Ϫ T͒
dF͑␮ ͒;
pt
ͩ
ͪ
ͬ
ͫ
r
i
h
0
͵
2
S
␮
pt
2
2
2
៮
I␴
␻2Vt
␻ ␪ ϩ mXtVt
ϭ mCi ϩ IkTh ␻2 ϩ
ϩ
ϩ
Ϫ T
dF͑␮ ͒
pt
␻2 ϩ mVt
2
͵
ͩ
ͩ
ͪ ͪ
ͫ
2
␻ ϩ mVt
C
␮
pt
Cr
2
ϩ
ϩ ͑␮0 Ϫ T͒
dF͑␮ ͒ .
pt
͵
ͩ
ͪ
ͬ
kITh
S
␮
pt

FELICIEN KANYAMIBWA AND J. KEITH ORD
194
Let
␮pt Ϫ E͑␮ ͒
pt
z ϭ
z1 ϭ
z2 ϭ
;
ͱ
Var ͑␮ ͒
pt
H1 Ϫ E͑␮ ͒
pt
;
ͱ
Var ͑␮ ͒
pt
H2 Ϫ E͑␮ ͒
pt
,
(24)
ͱ
Var ͑␮ ͒
pt
and let G( z) be a standard normal distribution with density function g( z). Then,
␻2Vt
d␮pt ϭ
dz;
ͱ
␻2 ϩ mVt
␻2Vt
z2
dF͑␮ ͒ ϭ
dG͑ z͒;
pt
ͱ
␻2 ϩ mVt
͵ ͵
C
z1
␮
pt
␻2Vt
ϭ
͕G͑ z2͒ Ϫ G͑ z1͖͒;
ͱ
␻2 ϩ mVt
␻2Vt
␻2 ϩ mVt
dF͑␮ ͒ ϭ 1 Ϫ
͕G͑ z2͒ Ϫ G͑ z1͖͒.
pt
ͱ
͵
S
␮
pt
Thus,
2
2
2
៮
I␴
␻2Vt
␻ ␪ ϩ mXtVt
␻2 ϩ mVt
E͑L͒ ϭ mC ϩ IkT ␻2 ϩ
ϩ
ϩ
Ϫ T
ͩ
ͩ
ͪ ͪ
2
ͫ
i
h
2
␻2 ϩ mVt
␻2Vt
␻2 ϩ mVt
Cr
ϫ
͑G͑ z ͒ Ϫ G͑z ͒͒ ϩ
ϩ ͑␮ Ϫ T͒
ͩ
ͪ
ͱ
2
1
0
kITh
␻2Vt
ϫ 1 Ϫ
͑G͑ z ͒ Ϫ G͑ z ͒͒
,
,
ͭ
ͮͬ
ͱ
2
1
␻2 ϩ mVt
2
2
2
␻2Vt
␻2Vt
␻ ␪ ϩ mXtVt
␻2 ϩ mVt
៮
I␴
ϭ mC ϩ IkT ␻2 ϩ
ϩ
ϩ
Ϫ T
ͩ
ͩ
ͪ
ͫ
ͭ
i
h
ͱ
2
␻2 ϩ mVt ␻2 ϩ mVt
Cr
Cr
2
2
Ϫ
Ϫ ͑␮ Ϫ T͒
ϫ ͕G͑ z ͒ Ϫ G͑ z ͖͒ ϩ
ϩ ͑␮ Ϫ T͒
ͪ
ͮ
ͩ
ͬ
0
2
1
0
kITh
kITh
2
I␴
ϭ mC ϩ IkT ␻2 ϩ
ϩ
ͱ
Var ͑␮ ͒ Var ͑␮ ͒ ϩ ͑E͑␮ ͒ Ϫ T͒
2
ͫ
ͭ
i
h
pt
pt
pt
2
Cr
Cr
2
2
Ϫ
Ϫ ͑␮ Ϫ T͒
ϫ ͕G͑ z ͒ Ϫ G͑ z ͖͒ ϩ
ϩ ͑␮ Ϫ T͒
.
(25)
ͪ
ͮ
ͬ
0
2
1
0
kITh
kITh
4. Optimal Sample Size and Inspection Interval, and Sensitivity Analysis
The key results in Section 3 assumed the knowledge of the optimal sample size and
inspection interval. In practice, economical design of a statistical process control involves

ECONOMIC PROCESS CONTROL UNDER UNCERTAINTY
195
determining an optimal sample size and inspection interval before fixing a sampling plan.
This decision corresponds to the node Ᏸ1 of Figure 1.
Before inspection, a decision maker will have only prior or pre-experimental information
about the loss associated with the two alternatives: inspecting and continuing the process or
inspecting and adjusting the process. This will be based on expectation about the loss due to
the two decisions. It is within this framework that we develop our results. We consider the
long-run or steady-state operating conditions for optimal inspection and sampling plan.
4.1. Optimal Sample Size and Inspection Interval
In the steady state, we assume that we have the initial conditions at time t ϭ 0 and we need
¯
to find the distribution of the posterior mean ␮ ͉X0, at the end of a diagnostic interval I in
I
order to estimate the expected loss over I. Thus, for the long-run operating conditions the
prior on ␮ is based on ␪ ϭ ␮ and Vt ϭ I␴2, so that, from (14)–(16)
pt
0
␮ ͉␮0 ϳ ᏺ͑␮0, VI͒;
I
2
␻
៮
X ͉␮ ϳ ᏺ ␮ ,
ͩ ͪ
0
0
0
m
2
␻
៮
␮ ͉X ϳ ᏺ ␮ ,
ϩ V .
(26)
ͩ
ͪ
pI
0
0
I
m
The process parameter distributions are completely determined a priori except for informa-
tion from the data yet to be observed. Thus, without other prior information on the process
behavior, decision-makers should be equally willing to accept adjusting or continuing the
process at the end of any inspection interval. This state of indifference may be expressed by
taking the prior probability of adjusting or continuing the process at the end of any inspection
1
2
interval to be uniform or noninformative, and likely equal to for either decision.
Hence, from (25) and (26),
2
I␴
1
2
␻2 ϩ mVI
1
Cr
2
2
E͑L͒ ϭ mC ϩ IkT ␻2 ϩ
ϩ
ϩ ͑␮ Ϫ T͒
ϩ
ϩ ͑␮ Ϫ T͒
;
ͫ
ͭ
ͮ ͭ
ͮͬ
i
h
0
0
2
m
2
kITh
2
I␴
1
2
Cr
␻2 ϩ mVI
ϭ mC ϩ IkT ␻2 ϩ
ϩ ͑␮0 Ϫ T͒2 ϩ
ϩ
.
(27)
ͫ
ͭ
ͮͬ
i
h
2
kITh
m
The expected loss per unit time is
2
E͑L͒ mCi
I␴
1
Cr
␻2 ϩ mVI
ϭ
ϩ kT ␻2 ϩ
ϩ ͑␮0 Ϫ T͒2 ϩ
ϩ
.
(28)
ͫ
ͭ
ͮͬ
h
I
I
2
2
kITh
m
The quantity of items produced over the inspection interval I equals Th ϫ I. We can only
sample m units with m less or equal to the number of items produced during the inspection
interval. In the long run, we would like to minimize the loss per unit time, E(L)/I. To find
the optimal sample size and inspection interval given information at time t we minimize
E(L)/I in (28) with respect to I and m and subject to m Յ ThI, or
E͑L͒
min
(29)
I
I,m
subject to:
m Յ ThI,
I Ն 0; m Ն 0.
(30)
(31)

FELICIEN KANYAMIBWA AND J. KEITH ORD
196
In general we have the following two possibilities:
1. m Ն ThI. This situation corresponds to 100% inspection, and reduces to m ϭ ThI since
the sample size cannot exceed the quantity of items produced over the inspection interval.
2. m Ͻ ThI. This case is consistent with the assumption in Section 3 of a small sampling
interval relative to the inspection interval.
Thus the constraint in (30) expresses the general conditions. The nonlinear programming
problem in (29)–(31) may be solved using the Kuhn-Tucker conditions (Chiang 1984), which
are satisfied and sufficient since the feasible region is a convex set formed by the linear
constraints (30) and (31) only.
From (28) to (31), we derive the following Kuhn-Tucker conditions:
2
2
Ѩᏸ Ci
␻
Ci
I
␻
ϭ
Ϫ kTh
Ѩᏸ
ϩ ␭ Ն 0,
mC ϩ
m Ն 0,
and
m
Ϫ kTh
ϩ ␭ ϭ 0;
ͩ
ͪ
Ѩm
I
2m2
2m2
1
I2
Cr
ϭ Ϫ
ϩ kT ␴2 Ϫ ␭T Ն 0,
I Ն 0,
and
ͩ ͪ
i
h
h
ѨI
2
1
Cr
I Ϫ
mC ϩ
ϩ kT ␴2 Ϫ ␭T ϭ 0;
ͩ ͪ
ͫ
ͬ
i
h
h
I2
2
Ѩᏸ
ϭ m Ϫ ThI Յ 0,
␭ Ն 0,
and
␭͑m Ϫ ThI͒ ϭ 0,
(32)
Ѩ␭
where ᏸ ϭ E(I)/I ϩ ␭(m Ϫ ThI) is the Lagrangian function, and ␭ is the Lagrangian
multiplier associated with constraint (30).
From (32), we have either m ϭ ThI, or ␭ ϭ 0 and m Ͻ ThI.
●
If m ϭ ThI then, from (28), we obtain a loss, E(LI), which is not a function of m, where
1
Cr
E͑L ͒ ϭ ThC ϩ kT ␻2 ϩ I␴2 ϩ ͑␮ Ϫ T͒2 ϩ
ϩ ␻
.
(33)
2
ͫ
ͭ ͮͬ
I
i
h
0
2Th I
k
We observe that limI3ϱ E(LI) ϭ limI30 E(LI) ϭ ϱ and that E(LI) is a continuous and
strictly convex function of I. Therefore minI E(LI) is reached when Ѩ{E(LI)}/ѨI ϭ 0,
which is equivalent to
k␻2 ϩ Cr
I ϭ
.
(34)
ͱ
2
2kTh␴
●
If m Ͻ ThI, then ␭ ϭ 0. Moreover, the Kuhn-Tucker conditions in (32) exclude the
cases where m ϭ 0 or I ϭ 0. Thus from (32), we have
2
Ci
I
␻
Ϫ kTh
ϭ 0;
(35)
(36)
2m2
1
I2
Cr
Ϫ
mC ϩ
ϩ kT ␴2 ϭ 0;
h
ͩ ͪ
i
2
which leads to the solution pair (m*, I*), where
␻2kTh
m* ϭ
I* ϭ
ͱ
I*
;
ͱ
2Ci
2m*Ci ϩ Cr
.
(37)
ͱ
2
2kTh␴
From (35) and (37), we obtain the following equations with one unknown decision
variable each:

ECONOMIC PROCESS CONTROL UNDER UNCERTAINTY
197
2
␻
␴
␻
kTh m* Cr
m*2 ϭ
I*2 ϭ
ϩ
;
ͩ ͪ
ͱ
Ci
4
Ci
Ci I*
Cr
ϩ
.
2
(38)
ͱ
2
␴
2kTh 2kTh␴
The Hessian determinant of (28) is
2
kTh␻
m3
Ci
I2
Ϫ
͉H͉ ϭ
;
Ci 2͑mCi ϩ Cr/ 2͒
Έ
Έ
Ϫ
I2
I3
3C2i
kTh␻2Cr
4Ci kTh␻
Ci
I
2
ϭ
ϩ
ϩ
Ϫ
.
(39)
ͩ
ͪ
3
͑mI͒
I4
I3
2m2
From (35), (kTh␻2/ 2m2) Ϫ (Ci/I) ϭ 0. Thus, the two successive minors of ͉H͉ are greater
than zero. Therefore H is positive definite at m* and I*.
The following properties may be established from (28) and are stated without proof:
1. For fixed I, E(L)/I is a strictly convex function of m and given I the optimal sample
size m is unique.
2. For fixed m, E(L)/I is a strictly convex function of I and given m the optimal inspection
interval I is unique.
3. limI3ϱ E(L)/I ϭ limI30 E(L)/I ϭ limm3ϱ E(L)/I ϭ limm30 E(L)/I ϭ ϱ. For other
values of I Ͼ 0, m Ͼ 0, E(L)/I is finite and continuous.
Hence, from the preceding properties, (32), and (39), we may conclude that for a given cost
structure there exits an optimal pair of diagnostic interval and sample size, that minimizes the
loss function. That pair is (m*, I*) in (37) and (38).
4.2. Sensitivity Analysis
In this subsection we examine the effects of the cost structure, composed of the inspection
cost, variability cost, and adjustment/repair cost, and the impact of process variability on
optimal operating conditions and variability loss. To get insight and more meaningful results,
we observe the effects of Cr/Ci, Cr/k, Ci/k, k, and ␻2/␴2, on E(L)/I, m, and I.
From (28) and (38), at optimality we observe the following relationships between the
optimal sample size m and the optimal inspection interval I, on the one hand, and the
inspection cost Ci, the variability cost k, the adjustment cost Cr, and ␻2/␴2 on the other hand:
(Ci, Cr, k, m): m increases as Ci/k decreases and Cr/Ci increases. m is sensitive to the
combined effects of Ci, Cr, and k. We observe that as Cr/Ci becomes smaller or Ci
becomes larger, the optimal sample size decreases. Hence, as the inspection cost
becomes larger, it is economically justified to take a smaller sample. However, as the
adjustment cost or the variability cost become larger, it becomes economical to take a
larger sample to avoid false alarms.
(␻2/␴, m): m increases as ␻2/␴ increases. This result indicates that the optimal sample
size is more sensitive to the initial variability of the process than to the variability due
to production run time.
(Ci/k, Cr/k, I): I increases as Ci/k decreases or as Cr/k increases, indicating that the
inspection interval is very sensitive to both Cr and Ci and should be reduced by frequent
inspections in these cases. Furthermore, as the adjustment cost becomes larger, it
becomes economical to wait longer between inspections and, at each inspection, take a
larger sample to avoid false alarms.
(␻/␴2, I): I increases as ␻/␴2 increases, and decreases with increasing rate as ␴ increases.

FELICIEN KANYAMIBWA AND J. KEITH ORD
198
This result indicates that if the variability of the process increases very fast with
production run time, it becomes economical to have more frequent inspections.
(k): I decreases as k increases. This result implies that as the variability cost increases,
frequent inspections may lead to a smaller loss.
The preceding propositions are based on the analytical results in (28), (36), and (38). As one
referee observed, learning would reduce Cr. The analytical results, especially (28), show that
adjustment cost reduction will necessarily reduce the loss. Also, a lower setup cost alone
makes the optimal inspection interval smaller. These findings confirm the results by Porteus
(1986) in the context of lot sizing, process quality improvement, and setup reduction. In
Section 5, we illustrate these results by an example from the automotive industry.
5. Case Study of a Painting Process in the Production of Steel Structures for
Trailers and Transportation Containers
5.1. Description and Optimal Operating Conditions
The data used to illustrate the results of Section 4 were collected by an international
company from audit files of its suppliers Societe de Transport International du Rwanda
(1989). We consider a supplier that produces steel structures for trailers and transportation
containers.
Before applying the paint on the steel, an anti-rust coating must be applied to the substrate.
Specifications of dry-film thickness are given before painting so that the operators can set a
wet-film thickness. The coat is applied by air-spray gun. To maintain satisfactory film
thickness, it is essential to adjust tip sizes and pump pressures as well as to select appropriate
types and quantities of thinners. It is then necessary periodically to adjust the air caps, fluid
caps, needles, spread-adjustment valves, fluid tips, and pumps to maintain the specified
dry-film thickness.
To decide whether the process should be stopped and the equipment cleaned and adjusted,
regular inspection of the film thickness is done using an elcometer thickness gauge. The
elcometer is based on the principle that magnetic attraction of the substrate is controlled by
the thickness of the coating. The elcometer, using the magnetic attraction recorded, indicates
the dry-film thickness on the dial in micro-millimeters (␮m).
The painting of structures was run for 36 hours without interruption to record process
performance in a pre-control state. The film thickness was measured using the elcometer and
recorded; during every hour, 31 measurements were randomly taken. Out of the 31 measure-
ments, one was randomly drawn and the remainder were used to compute the average recorded
film thickness. The one randomly selected measure was used as the observed film thickness at the
end of an hour. Data recorded during the 36-hour period are given in Table 1.
The target film thickness was 25 ␮m. On average, 18 square feet were coated every hour.
From the data we obtain the following:
●
The observed mean and standard deviation of Xt Ϫ ␮ are respectively Ϫ0.717 and
t
5.135, approximated respectively to 0 and 5 ␮m. [It should be noted that only the
average of the remaining 30 observations was available.]
●
The mean of the observed hourly average film thickness is 24.42, approximated to 24
␮m.
͌
●
The mean and the standard deviation of (␮ Ϫ 24)/ t are respectively Ϫ0.7422 and
t
3.480, respectively approximated to 0 and 3.5 ␮m.
Therefore, the following distributions were assumed:
Xt ϳ ᏺ͑␮t, 25.0͒;
(40)
(41)
␮t ϳ ᏺ͑24.0, 12.25t͒.
Also, the following parameters were estimated for the process from other information:

ECONOMIC PROCESS CONTROL UNDER UNCERTAINTY
199
TABLE 1
Observations of the Paint Film Thickness (in ␮m) in Trailer Steel Structure Painting
Time
Mean (M)
Xt
(M Ϫ T)/sqrt (t)
Xt-mean
OPT
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
14.0
15.8
12.0
18.2
13.1
17.5
20.2
35.0
22.2
13.1
25.2
16.4
7.2
12.5
18.7
17.3
10.1
21.0
35.3
23.7
18.2
28.5
28.6
27.5
26.0
16.8
43.8
24.0
16.3
48.7
11.6
20.0
40.9
61.8
51.3
46.5
9.4
17.5
10.0
16.9
11.9
17.6
25.3
28.2
26.9
10.9
33.1
7.0
Ϫ10.0
Ϫ5.8
Ϫ6.9
Ϫ2.9
Ϫ4.9
Ϫ2.7
Ϫ1.4
3.9
Ϫ4.6
1.7
Ϫ2.0
Ϫ1.3
Ϫ1.2
0.1
5.1
Ϫ6.8
4.7
14.4
22.4
31.8
20.1
22.0
28.6
22.0
25.7
25.1
24.0
9.2
35.8
18.4
30.0
25.3
29.8
19.4
16.1
38.5
29.2
18.2
22.5
21.9
28.7
24.2
20.0
12.5
22.2
19.6
22.6
26.9
33.7
21.3
21.8
15.0
19.9
Ϫ0.6
Ϫ3.5
0.4
10.0
11.0
12.0
13.0
14.0
15.0
16.0
17.0
18.0
19.0
20.0
21.0
22.0
23.0
24.0
25.0
26.0
27.0
28.0
29.0
30.0
31.0
32.0
33.0
34.0
35.0
36.0
Ϫ2.2
7.9
Ϫ2.2
Ϫ4.6
Ϫ3.1
Ϫ1.4
Ϫ1.7
Ϫ3.4
Ϫ0.7
2.6
Ϫ0.1
Ϫ1.3
1.0
1.0
0.7
0.4
Ϫ1.4
3.8
0.0
Ϫ1.4
4.5
Ϫ2.2
Ϫ0.7
2.9
6.5
4.6
Ϫ9.5
Ϫ1.1
Ϫ3.2
Ϫ7.9
1.7
6.1
9.3
10.8
19.0
10.7
22.2
48.7
16.3
14.4
35.0
25.9
32.9
23.7
18.7
39.8
30.4
15.5
48.2
8.2
0.6
1.2
13.4
Ϫ7.5
Ϫ3.9
6.5
Ϫ2.7
5.4
Ϫ2.2
1.9
Ϫ4.0
6.5
Ϫ0.8
Ϫ0.5
Ϫ3.3
Ϫ8.1
Ϫ4.0
3.5
11.9
37.0
65.3
42.5
46.1
Ϫ8.8
Ϫ0.4
3.7
Source: Societe de Transport International du Rwanda (1989)
Ci ϭ $1.0/sqft;
k ϭ $0.01/␮m;
Th ϭ 18 sqft/h;
Cr ϭ $10.
The inspection interval and the sample size are measured, respectively, in hours (h) and
square feet (sqft).
5.2. Current System Performance
We used our method to estimate the optimal operating conditions of the process. The
minimum loss using (28) and (36) was found at m* ϭ 2.00 sqft, I* ϭ 1.781 h, and the
expected loss per unit time ϭ $13.66. The computations were performed using routines from
Press et al. (1995). Since m is much less than ThI ϭ 32.04, the assumption that the sampling
time is much less than the production time between samples is justified.
To evaluate the performance of the process while operating under sub-optimal conditions
we used an additional set of 36 mean values provided by the company, computed based on
30 observations each. The data were collected as follows: measurement of the system

FELICIEN KANYAMIBWA AND J. KEITH ORD
200
FIGURE 2. Observed Dry Film Thickness With Precontrol.
performance were taken every 1.33 h (1 hour and 20 minutes of painting without interrup-
tion), with at the end of each interval an inspection of a sample of size 4.5 sqft. The process
was automatically adjusted every 1.33 h. Forty measurements were recorded per sample so
as to have 10 observations for each 20 minutes or equivalently 30 observations per hour of
painting. The average measurements using the optimal (OPT) policy are given in Table 1 under
OPT and are plotted in Figure 2. From Table 1, we observe the following:
●
When the system operates with automatic inspection and setup every 1.33 h and sample
of size 4.5 sqft, the observed average loss is $665.40 over 36 h, i.e., $18.48/h. The
expected loss under these conditions and using (28) is $15.25/h.
●
When the system operates without inspection and adjustment, the observed loss is
$1078.80 over 36 h, i.e., $29.94/h.
Table 2 shows the structure of savings when the painting process is improved with automatic
inspection and setup.
These findings are consistent with our estimates for optimal operating conditions. That is,
when we compare the performance with and without inspection, sampling, and process
adjustment, using data, respectively, under OPT and under Mean (M) in Table 1 and the
estimated parameters of the process, we observe the following:
●
The loss per unit time without inspection is higher than the loss per unit time with
inspection, sampling, and process adjustment.
●
Inspection, sampling, and process adjustment make OPT nearly stationary, in contrast to
the pre-control state as may be confirmed by the plots in Figure 2.
6. Conclusion and Directions for Future Research
In this paper we propose a loss function that takes into account both a continuous mean
shift and increases in production process variability. We show analytically how, given
TABLE 2
Variable Selection Results From Several Methods
Cost Item
Current System
Improved System
Savings
804.9
(121.5)
(270.0)
Variability cost in $
Inspection cost in $
Adjustment cost in $
Loss in $ (savings in %)
1078.8
0.0
0.0
1078.8
273.9
121.5
270.0
665.4
413.4 (38.2)

ECONOMIC PROCESS CONTROL UNDER UNCERTAINTY
201
information during a production run, an economic decision could be made regarding the
control of the process variability. Specifically, we propose an approach for determining an
optimal sampling and inspection plan. We develop a decision rule to choose between
continuing or stopping to adjust the process after inspection.
We find that the loss is very sensitive to the variability cost. We also show that there are
complex relationships between different cost parameters, so that it may be necessary to perform
a sensitivity analysis before deciding on a sampling plan. In general, we found that there is an
inverse effect on the sample size between the inspection cost on the one hand and the variability
and adjustment/repair cost on the other hand. Indeed, the combined increase of the variability and
adjustment/repair cost requires a higher sample size, whereas a higher inspection cost limits the
size of the sample. Moreover, we find that as the per unit variability cost becomes larger,
inspection should be more frequent and the sample size should be larger. We also perform a
sensitivity analysis on the variability of the process. The analysis shows that if the variability of
the process increases very quickly with production run time, it becomes economical to have more
frequent inspections, and that the optimal sample size is more sensitive to the initial variability of
the process than to the variability due to production run time.
Although the model used in this research is a quadratic loss function, it would be possible
to use any other model such as the inverted normal, linear, or polynomial form. Also, a
generalization to the multivariate case may improve the decision-making process, especially
when quality characteristics are correlated. Moreover, manufacturing consists in general of
many dependent stages, and the quality of a product at upstream stages may influence that at
downstream stages. This multistage effect on quality should be taken into account in
economic process control.
The approach appears to be applicable in various manufacturing industries where it is
necessary to control losses due to variability.1
1 The authors thank the area editor, two referees, and Dr. Duncan Fong for their helpful comments on earlier
versions of the manuscript.
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Felicien Kanyamibwa is Director of Management Performance and Information at the Prudential
Insurance Company of America. Dr. Kanyamibwa’s interests focus on statistical modeling with
particular reference to time-series analysis and response models. He is also interested in statistical
process control for time-dependent processes. Currently, Dr. Kanyamibwa is working on booking,
retention, and utilization models in the financial industry.
J. Keith Ord is Professor of Decision Sciences in the McDonough School of Business at George-
town University, Washington DC. Dr. Ord’s interests focus on statistical modeling with particular
reference to time-series analysis and spatial processes. He is also interested in statistical process control
for time-dependent processes. A related interest is the modeling of inventory and production processes
and the development of more effective control schemes. Dr. Ord is an elected member of the
International Statistical Institute and a Fellow of the American Statistical Association.