SPROCESS CONTROL:

A DYNAMIC PROGRAMMING APPROACH

by

Wm. Howard Beverly Jr.

Thesis submitted to the Graduate Faculty of the ·

Virginia Polytechnic Institute and State University

in partial Fulfillment of the requirements for the degree of

MASTER OF SCIENCE

in

Industrial Engineering and Operations Research

APPROVED:

G. Kemble Bennett, Chairman

Kenneth E. Case „; ig.W. Schmidt,

gg.

June l974

Blacksburg, Virginia

TABLE OF CONTENTS

@92 .ACKNONLEDGEMENTS ......................... ii

LIST OF FIGURES.......................... vi

LIST OF TABLES .......................... vii

Chapter

I INTRODUCTION....................... l °

Process Ouality Control................ T

Survey of Previous Research. . ............. 3

Purpose........................ 6

Outline of Succeeding Chapters ............ 8

II DEVELOPMENT OF THE DYNAMIC APPROACH ........... 9

General Description of the Model ........... 9

General Bayesian Decision Theory Approach....... l0

Prior and Posterior Probabilities ......... ll

Cost Assumptions ................... T4

List of Variables................... T6

Development of the Recursive Equation......... T8

Convergence of the Optimal Policy to a SteadyState Condition ................. 20

Summary........................ 30

III DEVELOPMENT OF THE GENERAL MULTI-STATE MODELS....... 3l

The Dynamic Model................... 3l

The Markovian Assumption.............. 3lThe Transition Probability Matrix ......... 35The Probability State Vector §........... 37The Probability Vector f; 40

iii

iv

· TABLE OF CONTENTS (continued)

Chapter Page

' Derivation of the Posterior State Vector Qt"'.... 41Derivation of the Predictive Distribution ..... 42The Explicit Form of the Recursive Equation .... 42Optimization of the Sampling Interval ....... 44Utilization of the Model.............. 46

The Steady State Y Control Chart Model ........ 47

General Cost Model.................48ProbabilityVector g_................ 50Probability Vector g_................ 51Probability Vector l_................ 55

Summary........................ 55

IV COMPUTATIONAL RESULTS .................. 57

Design of the Numerical Analysis ........... 57

Optimization of the Sampling Interval ....... 58Grid Values for pt and xt ............. 59

Example Problems ................... 60

Two State Processes ................ 60Example #l..................... 60Example #2...................., 66Example #3..................... 70Three State Process ................ 74Five State Process................. 78

Computational Efficiency ............... 82

Time Dependent Dynamic Model ............. 82

Summary........................ 83

V SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS ........ 84

Summary and Conclusions................ 84

Areas for Future Research............... 85

LIST OF REFERENCES ........................ 87

V

TABLE OF CONTENTS (continued)

Page

Appendix A DOCUMENTATION OF COMPUTER PROGRAM.......... 90

VITA ............................... I26

LIST 0F FIGURES

üsigß ÖBB E

2-l The Generalized Transition Probability Matrix ..... l3

3-l The Transition Probability Matrix for the Multi—stateDynamic Model .................... 38

4-l Steady State Cost Per Unit for Example #l ....... 63

4-2 Steady State Cost Per Unit for Example #2 ....... 68 _

4-3 Steady State Cost Per Unit for Example #3 ....... 72

4-4 Steady State Cost Per Unit for the Three State Process 76

4-5 Steady State Cost Per Unit for the Eive State Process . 80

vi

7LIST OF TABLES

@ Ess4-l Cost Terms and Process Parameters for Example Problems.

.76l

4-2 Example #l: Sampling Interval Results.......... 62

4-3 Optimal Steady State Control Policies: Example #l.... 65

4-4 Example #2: Sampling Interval Results.......... 67

4-5 Optimal Steady State Control Policies: Example #2.... 69‘

4-6 Example #3: Sampling Interval Results.......... 7l

4-7 Optimal Steady State Control Policies: Example #3.... 73

4-8 Three State Process: Sampling Interval Results ..... 75

4-9 Optimal Steady State Control Policies: Three Stateq Process ........................ 77

4-lO Five State Process: Sampling Interval Results...... 79

4-ll Optimal Steady State Control Policies: Five StateProcess ........................ 8l

vii

Chapter I

INTRODUCTION

Traditionally, the design of quality control systems has been based

on statistical criteria. Specifically, control charts have been used

extensively since their initial introduction by Walter A. Shewhart in

l924. Although tools such as these are statistically appealing, their I

economic aspects are often undesirable. In light of this fact, much

research and effort has been afforded to the economic design of quality

control systems. For the most part, this research has been divided

between the areas of process control and acceptance sampling.

with regard to process control, it is normally a manager‘s goal to

in some way minimize the cost of operating his process. Upon this

premise, this thesis will endeavor to develop an economic model describing

a typical process control situation. A Bayesian decision theory approach

will be undertaken and the model will be formulated using the techniques

of dynamic programming.

Process Quality ControlI

The function of a quality control procedure is to determine the

condition of the process. In most general terms, the process is either

in-control or out—of-control. But more specifically, it is often

desirable to know the degree, if any, of maladjustment. That is to say,

how far out-of—contr0l is the process? Because of the inherent

variation of all processes, no machine can be expected to manufacture

all items at the same level of quality. A machine may operate

l

2

automatically, but, in practice, the quality of its performance will .

deteriorate unless it is examined from time to time and repaired when it

is deemed necessary.

In general, any operating process is subject to two types of

variation. There are variations due to randomness or chance in the process

operation. Such deviations are uncontrollable, but are normally tolerable

since their impact is small. But secondly, variations due to assignable·

causes are of great concern to a process operator. Often such

variations plague a process and are responsible for the production of

both undesirable and unacceptable quality. In view of this latter cause,

a quality control procedure should specify regular inspections of the

process output to determine whether or not to stop and overhaul the

machine.

The use of control charts, first introduced by walter A. Shewhart

[29], has been a valuable statistical tool in process control. Undoubtedly

the most widely used chart is for controlling the average operating

level of a process, YZ Traditionally accepted control limits have been

positioned three standard deviations on either side of the center line

while sample sizes of either four or five items have been inspected.

Additionally, samples of the process output have been collected at

convenient intervals.

Because of the increased emphasis on economical operation in

industry over the past twenty-five years and the mushrooming of

_ competition, the expensive use of purely statistical quality control

procedures has, in some process control applications, become less and

3

less desirable. Conditions such as these initiated the interest in the

design of economically based quality control plans.

Survey of Previous Research

There are numerous authors in the area of designing cost-based or

economic quality control systems. One such author is Duncan [lO] [ll] 2

who proposed a procedure for determining the sample size, the interval

between samples, and the control limits for an Y°control chart which

maximize the average net income when a single assignable cause or out-of-

control state exists. His assumptions were that the process could shift

from the in-control state to a single out-of—control state at anytime

during the day and that the process possessed the Markov property. Goel

et al. [l5] then extended Duncan‘s model by developing an algorithm for

computing the optimal test parameters. Lave [23] considered the optimal

choice of process control plans based on a two-state Markov model and in

addition examined means in selecting the optimum parameters for the

respective plans.

More recent research by Duncan [12] and Knappenberger and Grandage

[20] addressed the problem of determining the optimal parameters for

an Y control chart when there are several assignable causes. The latter

authors assumed that the process parameter, U, was a continuous random

variable but that it could be satisfactorily approximatedbyia

discrete

random variable. One value, UO, of the discrete random variable was

associated with the single in-control state, and the remaining values

U], U2, ..., US were associated with out-of—control values of the process

2 parameter. Their model was then formulated based on three pertinent

4

costs: sampling, investigating and correcting the process, and producing

defective units. IAn even more recent article was published by Gabria [l3] who

again considered the economic design of an Y control chart to optimally

determine its parameters. This author introduced a new element into the

cost analysis defined as the worst cycle quality level (w.C.Q.L.). [This

w.C.Q.L. offered the chart user the option of placing a constraint on _

the permissible mean expected number of defectives produced within a

quality cycle time. Chiu and wetherill [ 8] proposed a semi-economic

scheme for the design of a control plan using an Y chart. These

researchers allow the Q.C. engineer to select a consumer's risk point on

the OC-curve and then proceed to outline a step-by-step procedure for

determining the optimal control chart design parameters.

In all cases, the previously discussed authors only addressed

their research for a process where a single measurable quality charac-

teristic was of interest. Contrary to this doctrine, Montgomery and

Klatt [24] formulated an economic process control model based on

dependent multiple characteristics and utilized the Hotelling T2 control

chart. Latimer et al. [22] have proposed a multicharacteristic model

with the dual purpose of controlling the lot mean and determining the

lot disposition.

The problem of establishing control policies has been undertaken by

only a few researchers using the Bayesian decision theory approach.

Such authors as Girshick and Rubin [14] inspected the problem of

minimizing the expected running and repair costs while operating a process

in an equilibrium condition. They assumed that the process level could

5

„occur in four possible states: two operating states and two repair

states. Bather [ 2] formulated a model using dynamic programming and

avoided the difficulties in assuming the existence of steady state

conditions. Bather developed optimal stopping rules which indicated

when the process should be stopped and overhauled and when it should be

allowed to continue production. His work was later extended by CarterI

[ 7] who included the determination of optimal sample sizes to collect

at each inspection and the optimal interval between sampling. Both'

Bather and Carter addressed their work to a process which has a single

quality characteristic of interest. Further, they assumed that the

quality characteristic was a continuous random variable and was normally

distributed. The process mean was also considered to be a continuous

random variable whose successive increments behaved as a random walk.

Additionally, Carter proved that the recursive equations of the dynamicIprogramming formulation converged to a limit as the time index on

operating the process grew without bound. Hence, a single set of

operating policies was shown to exist in the steady state environment.

Another application of dynamic programming was effected by Pruzan

[27] who considered a system of consecutive operations and determined

points of either no inspection or l00% inspection. Taylor [30] has shown

that inspecting a fixed number of items at a fixed interval of time is

non-optimal. Instead, decisions with respect to sampling should be

executed in a sequential manner based on current posterior probabilities.

Using this approach, Taylor [3l] developed an optimal control procedure

assuming that the process exists in only two states, in-control and

out—of—control.

6

The previously discussed background in the design of cost-based

process control plans is by no means intended to exhaust the body of

research. But rather, it is offered only as a general description of

the primary areas of concentration.

Purpose

The review of previous research has revealed a variety of approaches _

to the economical design of quality control systems. Generally, most

authors have assumed that the Markov property exists in the operation of

the process model and also that a steady state or equilibrium condition

has been achieved. Several authors have employed a Bayesian decision

theory approach to incorporate historical data with current sample

observations in making present operating decisions. _

Still the influence of the Y control chart has persuaded many

researchers to attempt to increase its salability. But one of the short-

comings of the Y chart is that explicit consideration of historical

decisions is not incorporated while making present operating decisions.

In order to overcome this malady,this thesis will employ the techniques

of Bayesian decision theory. It is next proposed that not all

processes operate in the steady state environment. For example,

consider a job shop operation which may require a production run of

several days. In following the operating policies of an Y control chart,

the situation may arise where the policy indicates an out—of—c0ntrol

condition and that an overhaul is warranted. But suppose only five to

six more hours of production may complete the run. Query--is it better,

economically, to initiate an expensive overhaul now, rather than turning

7

out a higher percent of defective products for the remainder of the

production run? Such a question is often not easily answered, but here

is where a dynamic programming approach can supply the necessary

guidelines. Accordingly, this thesis will first formulate the model in

a time dependent environment and employ the technique of dynamic

programming. The cost factors to be included in the model formulation

are for sampling, investigating and overhauling the process,

andproducingdefective items. The pertinent design parameters to be

determined are the sample size at each inspection, the interval between

successive inspections, and finally the values of the sample outcome

which will dictate the proper operating decisions.

The model will only accommodate process operations which have a

single quality characteristic of interest. It will be assumed that this

characteristic is a continuous random variable as is the process

parameter p, the mean operating level. From this point, this thesis

will follow the approach of Knappenberger and Grandage [20] in assuming

that the parameter p can be satisfactorily approximated by a discrete

random variable and that the time between successive shifts in the

operating level is dictated by the exponential distribution.

Finally, following the guidelines of Bather [ 2] and Carter [ 7]

the recursive equations will be shown to converge to a limit as the time

index grows without bound. Hence'an optimal set of design parameters

will be obtained to illustrate the steady state optimal control

policies. Upon accomplishing this, a comparison of the results obtained

will be made with the optimally designed Y control chart proposed by

8

Knappenberger and Grandage [20]. The evaluation will be based upon how

well each approach attains its premise of minimizing the operating

_ cost.

Outline of Succeeding Chapters

In Chapter II, a general development of the dynamic programming

formulation of the process control model is presented. Acursorydescription

of the techniques of Bayesian decision theory is provided.

After developing the recursive equation of the dynamic model, a steady

state optimal policy is shown to exist via an inductive proof of

convergence. V

In Chapter III, the dynamic model is presented in detail. Initially,

the Markovian assumption is used to derive the probabilities of shifts

in the state of the process mean. The transition probability matrix is

derived along with an explicit form of the dynamic recursive equation.

In addition, the steady state Y control chart model developed by

Knappenberger and Grandage [20] is presented to illustrate the similarities

in its formulation.

In Chapter IV, the numerical results for five example problems are

presented. For each example, the optimal steady state operating

policies are derived from both the dynamic model and the Y‘control chart

model. These policies are interpreted and a comparison of the optimal

costs obtained is effected.

Chapter V provides a sumnary of, and conclusions about, the findings

of the research. Also, areas for further research are discussed.

Chapter II

DEVELOPMENT DF THE DYNAMIC APPROACH

_ In this chapter a general description of the process model is pro-

vided along with assumptions concerning the pertinent costs incurred

while operating the process. A cursory discussion of the Bayesian

decision theory approach is included and the recursive equation of the

dynamic programming formulation is derived. Then employing some results

due to Carter [7 ], the optimal operating policies are shown to converge

to a steady state optimal policy as the time index on operating the

process grows without bound. ·

General Description of the Model

Consider a typical manufacturing process which fabricates units

that have a quality characteristic of interest which can be described in

terms of a single continuous random variable. This random variable, x,

is assumed to be dependent upon the process mean, p, which itself is a

continuous random variable. Adopting the approach of Knappenberger and

Grandage [20], it is assumed that the process mean, p, can be satis-

factorily approximated by a discrete random variable. 0ne value, u(O),

of the discrete random variable is associated with a single in-control

state of the process mean, and the remaining values p(—s), p(—s+l), ...,

p(-l) and p(l), u(2), ..., p(s) are all associated with the out-of-control

states of the process mean.

At any time t during the process operation, the quality charac-

teristic,xt,is assumed to be normally distributed with mean pt(i) and

9

10

variance 62, where 62 is constant for any time t. For the purpose of

the ana1ysis, the time index wi11 run backwards starting with T, then

T-1, T—2, ..., t+1, t, ..., 2, 1, O. Samp1es of size nt are co11ectedfrom the process output at each period t and the samp1e mean, xi, is

ca1cu1ated. It is assumed that xy is norma11y distributed with mean pt(i)

and variance 62/nt.

In order to further describe the stochastic nature of the process,

it is assumed that the time between successive shifts in the state of theI

process mean is exponentia11y distributed with parameter A. Any shift

from the in-contro1 state, u(O), to an out-of-contro1 state p(i)

(i = -s, -s+1, ..., -1, 1, 2, ..., s), is considered to be a deterioration

of the process; and once a shift does occur, the process can on1y be

returned to the in-contro1 state by an adjustment or an 0verhau1.

Genera1 Bayesian Decision Theory Approach

The main objective of a process operator is to monitor and contro1

his process so as to minimize the tota1 future costs of its operation

over some finite or possib1y infinite p1anning horizon. Since the

processing operation is inherent1y random, an operator cannot know with

certainty what costs he wi11 incur. According1y, he must make

inspections of the process from time to time and based on the resu1ts

obtained, effect operating decisions with the purpose of minimizing tota1

future expected costs.

ll

Prior and Posterior Probabilities

At any time t, the operating decision, either to overhaul or not to

overhaul, should be based on the value of the process mean. But the

. process mean is never known with certainty. Accordingly, the operator

must assess a prior probability for the state of the process (i.e., prior

to sampling). These prior beliefs can be based on past experience in

the form of historical data, or even upon subjective judgement. ·i

Define the ith element, at‘(i),of the probability state vector_ét',

to be the prior probability that the process mean is in state i at time

t (the prime denotes a prior belief). After observing a sample of size

nt at time t,the operator must make an operating decision. At this point,

he has two sources of information describing the process: the prior

state vector £t‘, and the sample result x,. These two sources may be

combined to depict the posterior condition of the process. Now define

the element at"(i) of the probability state vector_ét", to be the posterior

probability that process mean is in state i at time t (the double prime(

denotes a posterior belief). Having the posterior information about the

state of the process, the operator can effectively choose to either _

overhaul the process or allow it to continue production at its present

level.

In order to derive the posterior state vector, Bayes formula must

be employed. In general terms the formula is given by

P(A/B) = Ei§é%%&B$Al _ (2.0l)

l2

The posterior probabilities are dependent upon the prior probability

state vector Qt' and the likelihood function of the sample mean, denoted

by ß[§¥\pt(l),nt]. Hence the elements of the posterior state vector are

derived in the following manner:

§iß[Xtlut(l),nt]-at(i)

where i = -s, —s+l, ..., l, 0, l, ..., s.The prior beliefs about the state of the process must be explicitly

defined for each period t, t—l, and so on. Since the stochastic nature

of the process mean is described by a Markov process, the transition

probabilities of the process, that is, the probabilities of a shift in

the process mean from its present state to another state, must be

defined. These transition probabilities are depicted in matrix form in

Figure 2-l.

The row index of the matrix indicates the present state of the

process mean at time t, whereas the column index describes the state of

the process mean at the beginning of the following period, t-l. Each

element, bij, of the transition probability matrix éj represents the

probability of a shift in the process mean from state i to state j in a

single transition. It is also assumed that the transition probabilities

remain stationary from one period to the next.

with the transition probability matrix defined in general, the

procedure for obtaining the prior state vector at each period can be

defined. Since the posterior beliefs about the process at time t

incorporate the greatest amount of information, they will be used in

deriving the prior beliefs for period t-l. Hence, the prior state

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l4

vector, ét_]', is obtained by the matrix multiplication:

Each individual element of the prior state vector at t—l is then

¤t_]'(.i) = ¤t"(i)·bi’j (2·04)1

where i, j = —s, -s+l, ..., l, 0, l, ..., s. _

A detailed derivation of the prior and posterior state vectors as

well as the elements of the transition probability matrix is provided in

Chapter III.

In order to ease the notational burden in subsequent sections let

ut' and ut" denote the general random variables for the prior and

posterior means of the process, respectively, and let uo denote the state

of the process mean immediately after an overhaul is completed. .

Cost Assumptions

The costs incurred in operating and maintaining the repetitive

process generally arise from three sources.

l. Overhauling cost - This cost is incurred whenever the process

is halted to be adjusted or to be examined for assignable causes. In

general, this cost can be assumed to be a function of the state of the

process mean, but for most actual processes a fixed average cost for

maintenance is probably most realistic. Let k[pt(i)] be the general

cost expression for the overhauling cost where pt(i) is the mean of the

process at time t. After sampling at time t, the state of the process

l5

mean is still not known with certainty; thus, the overhaul cost must be

replaced by an expected cost. Let K(ut") be the posterior expected cost

of an overhaul where

K(ut") = _E H k[ut(l)] · (2-05)ut(l)/Ut

It is assumed that K(0) > O. That is to say, a cost will be incurred

regardless of the state of the process mean whenever an overhaul is

initiated.

2. Operating cost - This cost is incurred during the productive

cycle of the process. The cost will be solely attributed to the manu-

facture of defective units when the process is in a particular state pt(i).

The cost of operating for one period will be a function of the state of

process mean, the production rate, the length of the period, and the

cost per defective unit produced. (Let this cost be denoted by:

c[¤t(i)] = R·h·a4·f(i) (2 06)

where

R is the production rate,

h is the length of the period,2

a4 is the cost of a defective unit, and

f(i) is the conditional probability of producing a defective unit

given that the process mean is pt(i).

Since pt(i) is a random variable, the general cost expression,

c[pt(i)], must be replaced by an expected cost. Let C(ut“) be the

T6

posterior expected cost of operating the process for one period where

‘C(ut")= _E c[pt(i)] , (2.07)

¤t(i)/ut"

3. Sampling cost - This cost is incurred at each period when the

process output is inspected. In general, this cost will involve two

components. Let a] be the fixed cost of sampling for each period and

let a2 be the variable cost per unit sampled. Let the general cost(

expression be given by

S(nt) = a] + a2·nt .I

(2.08)

It will be assumed that the fixed portion of the sampling cost, a1, is .

incurred at each sampling point, even if the optimal size is zero. This

seems to be a reasonable assumption for many real processes.

List of Variables

t - time (stage) index; where t runs backwards from t = T,

t = T-l, ..., t = l, t = 0.

xt - continuous random variable representing the quality

characteristic of the product at time t.

pt(i) — the ith state of the process mean at time t; assumed to

be a discrete random variable where i = —s, —s-l, ...,

-l, 0, l, ..., s-l. s.

62- the process variance of the quality characteristic xt;

assumed to be constant for all t.

. 2 . . . .N[ut(i),6 ] — the distribution of xt at time t.

l7

xk - the sample mean observed at time t.

nt - the sample size at time t.

N[ut(i),¤2/nt] — the distribution of the sample mean, xé, at time t.

uti — the general expression for the prior mean of the ·

process at time t.

dt? - the prior probability mass function of the state of

the process at time t with elements at'(i).

ut" — the general expression for the posterior mean of the

process at time t.

Qt" - the posterior probability mass function of the state

of the process at time t.h

uo- the state of the process mean immediately after an

overhaul; known with certainty to be pt(O), the

in-control state.

a],a2 — the fixed and variable costs for sampling.

a4 — the unit cost of a defect. .

R - the production rate of the process.

h — the length of a time period from t to t-l.A-] — the mean time between successive shifts in the process

mean; assumed to be constant for all t.

f_ - the conditional probability mass function of producing

a defective unit given that the process is in state

ut(i); a row vector with elements f(i).

l8i

Development of the Recursive Eguation

After sampling at time t and computing the posterior state vector,

gi", for the process mean, the process operator must choose to either

overhaul or not overhaul the process. At each period, this decision

should not be based entirely on the present economic state of the system,

but should also include the minimization of the future expected costs of

operating the process. If an overhaul is initiated, the cost of the‘

overhaul plus the cost of operating the process for one period at the

adjusted level will be incurred. Otherwise, if no overhaul is performed,

the cost of operating the process at the current state of the process mean

will be contracted. Additionally, the decision now will effect the

costs of operating in future periods since the state of the process mean

is highly dependent upon when the process was last overhauled.

Consequently, after sampling at time t, the process operator should

choose, -

min , (2.09)”

C(“t") + Vt—i(“t—iI)

where Vt_](ut_]') represents the minimum future expected cost of having

prior beliefs, ut_]', with t—l periods left to operate. It should be

reemphasized that the values of Vt_](ut_]') included in each equation

are dependent upon the present operating decision, and consequently will

· not be identical.

Prior to sampling at time t, the process operator cannot know what

sample result he will see. But he does have a predictive distribution

19

for the sample mean, x], which is dependent upon the prior state of the

process and the sample size. This density function may be derived in

the following fashion:I

g(x£|ut',nt) = V;2[i£)pt(l),Ht]•gt‘(l) . (2.lO)

Taking the expected value of equation (209) for all values of xk, the

process operator may find the future expected costs before sampling, '

conditioned upon the prior state at time t and nt. Hence, the prior

future expected cost is given by ~

e [|<(ut") + C(u0) +Vt_](ut_]‘)__E

·min + S(nt) .

y Xtiut »"t C(ut") + Vt_](ut_]‘) (2.ll)

But at this point, the process operator should choose the sample

size nt, which will minimize his prior expected costs. Thus, the minimum

future expected cost given prior beliefs, ut', with t periods left to

operate is:

min __ E min + S(nt) .“1; {xtlutrnt Cwt") + V1;—1("t-i') }. (2.12) ,

This expression is the general form of the recursive equation for

a dynamic programming problem. It can be solved recursively each period

(t = l, 2, 3, ...) for the optimal policy [7]. The policies will

depict, for each set of prior probabilities (prior state vector Qt'),

what sample size to collect and, based on the sample result, what

operating decision to make.

20

Convergence of the Optimal Policy to a Steady State Condition

The model formulation, up to this point, has been in a time dependent

environment. Consequently, a process operator must always be cognizant

of the number of periods left to run whenever he chooses a sample size

and makes an operating decision. In many process control applications

the number of periods of operation is very large and consequently the

use of such an approach may tend to be impractical. But in such a ·

situation it is plausible to consider the existence of a steady state

optimal policy. That is to say, as the number of periods of operation

becomes large, do the optimal policies at each period converge to a

steady state policy? If such a policy exists, it would greatly simplify

the use of the control procedure since an operator would not have to be

cognizant of time while making operating decisions.

In order to show that a steady state optimal policy does exist, the

recursive equation of the dynamic programming formulation must be shown

to converge to a limit as the time index grows without bound. If such

a limit can be shown to exist, then the so—called infinite horizon

equation will be obtained.

In attempting to prove the existence of the infinite horizon

equation it is necessary to include a discount factor, a, in the recursive

equation to restrict the costs from becoming infinite as the time index

approaches infinity. Including this discount factor, the recursive

equation becomes

2l

Vt(ut') =

min E min + S(n )nt { ģ|ut',nt C(ut") + a Vt_](ut_]') t }

(2.l3)

In order to derive the infinite horizon equation, the sequence

{Vt(ut')} must initially be shown to be uniformly bounded, and secondly

be shown to be uniformly convergent. The following theorem is a slightly .

modified version of the derivation outlined by Carter [7]:

Theorem I: ‘

Assume,

i. k[pt(i)] is bounded for all possible states of the process

mean (i.e., k[ut(i)] j_K0 < w for all pt(i)).

ii. The cost of operating the process for one period after an

overhaul is bounded (i.e., C(u0) = CO < w).

iii. The cost of maintaining an inspection procedure each period

is a constant positive value when the optimal sample size

is zero (i.e., $(0) = $0 < w).

Then, Vt(ut') is uniformly bounded for all states of the process mean

and for all values of ut' and t.

Proof:

At each period t, the process operator has the opportunity of not

sampling, overhauling the process, and running for one period. The

cost of this policy when t = l is

1<(u]') + c(u0) + $(0) i KO + CO + $0 . (2.14)

22

Thus,

V](u]') j_K0 + CO + $0 . (2.15)

For t = 2 the same policy is again an alternative, so

V2(u2')_; K(u2‘) + C(uO) + $(0) + aV](u]') (2.16)

i KO + CO + S0 + a(1<0+c0+$0) . (2.17)

Then by induction, .

V (u‘)

< [K +C +S ] (l+a+a2+ +at”]) (2 18)t t —- 0 0 0 "' '

K + C + S”

, 0 0 0

The inequality holds for all t and all states of the prior mean ut';

hence, Vt(ut') is uniformly bounded.

Q.E.D.

Now, if it can be shown that the sequence {Vt(ut')} is monotonically

nondecreasing in t, then the sequence must be uniformly convergent since

it is uniformly bounded by Theorem I. Before uniform convergence is

proven,an important assumption must first be outlined. Because the

process mean is assumed to occur in a finite number of discrete states,

the possibility of considering too few states could arise in a practical

analysis. Suppose, for example, the cost of performing an overhaul is

extremely large and a single out—of—control state of the process mean is

chosen relatively close to the in—control state. In such a case, it may

be cheaper to let the process run even if it is out-of—control, rather

than incur the cost of an overhaul. Consequently, to overcome this

ma1ady,it will always be assumed that the number of out—of-control states

23

are chosen and positioned such that there exists one or more states of

the process mean which make the cost of running the process for one

period sufficiently greater than the overhaul cost. A convenient way

to state this assumption is that as pt(s) + w, C(ut") + w, where pt(s)

is the state of the process mean which is farthest out—of—control; hence,

a decision to overhaul will minimize the operating costs.

The following proof of uniform convergence is again a slightly

modified version of a theorem proposed by Carter [7].

Theorem II: „

Assume,

i. C(ut") + w as ut" + w (ut" + w implies pt(s) + w).

Then, the sequence {Vt(ut')} is uniformly convergent as t + w.

Proof:

Let E; be the value of nt which minimizes

u O IK(ut ) + C(u ) + aVt_1(ut_] )__ E min ‘ ‘

+ S(nt)>< lu ',n „.t”° t mi; ] + ""’1;—i(“t—i ] (2.20)

Define

K(u ") + C(u0) + aV (u ')t t-l t-lGt(u") = min

C(ut") + aVt_](ut_]') (2.2l)

Substituting and equation (2-2]) into equation (2.l3), one obtains

24

luEt+lt+l’

t+l

Vt(ut‘> =I

IUEIE

[Gt(¤")l + S(Ft) (2.23)t t

’t

·i —

E _ .[Gt(u")] +S(nt+]) (2.24)Xtlut *"t+i ‘

Observe that this inequality must hold since nk was defined to be the

minimizing value of nt. Hence, the value of Vt(ut') evaluated at n£+]

must at least be equal to or greater than the value of Vt(ut') evaluated

at n£. Subtracting equation (2.24) from equation (2.22) yields

Vt+](¤') — Vt(¤') ;;I5_E

[Gt+](¤")-Gt(u")l (2.25)’

t+l

Observe that the time index has been omitted from the expressions u', u"

and the parameter x] This change was effected since the expressions and

parameter take on the same set of values regardless of the time index.

Additionally, the time index would only be a source of confusion in the

remainder of the analysis.

If [Gt+](u")—Gt(u")] ;_0, for all possible values of u", then the

value of [Vt+](u')—Vt(u')] must also be greater than or equal to zero

by equation (2.25). The following inductive proof shows that

[Gt+1(u")—Gt(u")] ;_0 for all t and u".

It is first necessary to show that the inequality holds for t = l.

But first, it is desirable to distinguish between the values of u' when

25

an overhaul was performed in the preceding period and when no overhaul

was undertaken. If an overhaul did occur, the following notation will be

used:

u' = u* . · (2.26)

From equation (2.2l) one obtains

K(u") + C(u0)G](u") = min (2.27) ‘

C(uu)

and AK(u") + C(u0) + aV](u*)

G2(u") = min (2-28)C(u") + aV](u')

For simplicity, let OH denote a decision to overhaul and let R

denote a decision to run without an overhaul; then there are four

permutations of decisions that can be effected at periods one and two:

l) R-R 2) R-OH 3) OH—R 4) OH-OH .

The value of [G2(u")-G](u")] will now be examined for each case.

Case l:

G](u") = C(u") (R) (2.29)

G2(u") = C(u“) + aV](u') (R) (2.30)

G2(u") - G](u") = aV](u“) ;_0 . (2.3l)

The last inequality must hold since all costs are positive and

consequently V](u') must be positive.

26

Case 2:

G](u") = C(u") (R) (2.32)

G2(u") = K(u") + C(u0) + aV](u*) (OH) (2.33)

G2(u") - G](u") = K(u") + C(u0) — C(u“)+ aV](u*) (2.34)

Note that at period 1 the decision was made to run without an3

overhau1, which imp1ies that this decision had the 1east expected cost,

or that .

KQNU + C(u0)— C(u")= A ;_O (2-35)

Hence, .

G2(u") — G](u") = A + aV](u*) ;_O . (2.36)

Case 3:

G](u") = K(u") + C(u0) (OH) (2.37)

G2(u") = C(u") + aV](u') (R) (2.38)

G2(u") — G](u") = C(u") - K(u") - C(u0) + aV1(u') (2.39) 4

For t = 1 the process was overhau1ed which imp1ies that

C(u") - [K(u")+C(u0)] = A ;_0 . (2.40)

Hence, A

G2(u") — G](u") = A + aV1(u') ;_0 . (2.41)

Case 4:

(OH) (2-42)

G2(u") = K(u") + C(uO) + aV1(u*) (OH) (2.43)

G2(u") - = aV](u*) ;_0 . (2-44)

27

The preceding portion of the proof shows that for t = l,

[Gt+](u")-Gt(u")] ;_O for all values of u". Consequently, [V2(u')—V](u')]

g_0.

In effecting a proof by induction, the next step is to assume that

the inequality [Vt(u')-Vt_](u')] ;_O holds for period t and then demon-

strate that it holds for period t+l.

_ Assume that Vt(u') ;_Vt_](u'). Again from equation (2.2l) one'

obtains

K(u") + C(u0) + aVt(u*)Gt+1(u") = min (2.45)

C(u“)+ aVt(u')

and

K(u") + C(uO) + aVt_1(u*)Gt(u") = min (2.46)

C(u") + aVt_](u')

Once again consider the four permutations of decisions that can be

effected at periods t and t+l.

Case l:6t(u“)

= c(¤“) + aVt_](u‘) (R) (2.47)

Gt+](¤") = C(¤") + avt(¤') (R) (2.48)Gt+](u") - Gt(u") = a[Vt(u')—Vt_](u')] ;_O (2.49)

This follows from the assumption that Vt(u') ;_Vt_](u').

28

Case 2:

Gt(u") = C(u") + aVt_](u') (R) (2.50)Gt+](u") = K(u") + C(uO) + aVt(u*) (0H)(2.5l)

Gt+)(¤") · Gt(¤") = *<(¤") + C(¤°) - C(u")+ aVt(u*) — aVt_](u') (2.52)

= K(u") + C(u0) - C(u") + aVt(u*)

— aVt_](u') + [aVt_](¤*)—aVt_](u*)] (2.53) _= [K(u") + C(u0) + aVt_1(¤*)l i

— [C(¤")+öVt_](¤')] + ¤Lvt<¤*)—vt_]<¤*)J (2.54)

Observe that since the decision at period t was to run without an over-

haul, the sum of the terms in the first bracket must be greater than the

sum of the terms in the second bracket. Also, by assumption, the sum

of the terms in the third bracket is greater than or equal to zero.

Hence,

Gt+](u") — Gt(u") ;_0 . (2.55)

Case 3:

Gt(u") = K(u") + C(u0) + aVt_](u*) (OH)(2-56)

Gt+](u") = C(u") + aVt(u‘) (R) (2.57) .

IGt+](u“) — Gt(u") = C(u") — K(u") — C(u0)

+ aVt(u') — aVt_](u*) (2.58)

= C(u") - K(u") - C(u0) + aVt(u')

- aVt_](u*) + [aVt_](u')—avt_](u‘)] (2.59)

= [C(u")+aVt_](u')] - [K(¤")+C(¤0)+¤Vt_](u*)]+ a[Vt(u')-Vt_](u')] (2-60)

29

By the same argument as before, since the decision at period t was to

overhaul, the sum of the terms in the first bracket must be greater than

the sum of the terms in the second bracket. Also, by assumption, the

sum of the terms in the third bracket is greater than or equal to zero.

'Hence,

Gt+](u") - Gt(u") g_0. (2.6l)

Case 4:A

Gt(u") = K(u") + C(u0) + aVt_](u*) (OH) (2 62)Gt+](u") = K(u") + C(uO) + aVt(u*) (OH) (2.63)

Gt+](u") — Gt(u") = a[Vt(u*)-Vt_](u*)] i O (2.64)

This follows by assumption.

The preceding manipulations show that if it is assumed that

Vt(u') i Vt_](u'), then Gt+](u") ;_Gt(u"), for all values of t and u".

It was previously shown by equation (2.25) that,

Vt+](u') - Vt(u') g_YIUIEH

[Gt+1(u")-Gt(u")] (2.65)’

t+l

thus,

Vt+](u') - Vt(u') ; O (2-66)

for all values of u' and t.

Hence, Vt(u') is monotonically nondecreasing in t and is uniformly

bounded. Thus, the sequence {Vt(u‘)} is uniformly convergent as t

approaches infinity.

Q.E.D.

30

Now, because the sequence {Vt(u')} has been shown to converge to a

limit, it can be stated with certainty that, as the number of periods

of operation becomes large, the optimal control policies at each period

converge to a steady state optimal control policy. Consequently, a

process operator will make his decisions based on the prior beliefs about

the state of the process mean, and, assuming an equilibrium condition

has been achieved, will not have to be cognizant of time. u

Summary ”

In this chapter a generalized process control model has been

formulated using the techniques of dynamic programming. A number of

assumptions have been outlined and the pertinent costs have been

discussed. A cursory explanation of the Bayesian decision theory approach

was provided to serve as a preface to a more detailed development which

follows in Chapter III. And finally, the optimal control policies were

shown to converge to steady state policies which are independent of

time.

Chapter III

DEVELOPMENT OF THE GENERAL MULTI—STATE MODELS

In this chapter, two multi-state models are presented. The dynamic

model employs the technique described in Chapter II. All relations

essential to the utilization of the dynamic model are developed and the

procedure for optimizing the sampling interval is outlined. The steady _

state Y control chart model, proposed by Knappenberger and Grandage [20],

is presented to illustrate its similar formulation and as a preface to

its utilization in Chapter IV.

The Dynamic Model

Recall from Chapter II that the process being modeled fabricates

units which possess a single quality characteristic of interest. This

characteristic, xt, is assumed to be a continuous random variable which

is normally distributed with mean pt(i) and variance 62. Also,the process

mean, ot(i) is assumed to be a discrete random variable which can occur

in 2s + l different states. These states are denoted by pt(—s), pt(-s+l),

where ¤t(0) V€PV€$€Ot$ asingle in-control state and the remaining states are all considered to

represent an out—of-control condition of the process. Finally, the

process is assumed to possess the Markov property.

The Markovian Assumption

Since shifts in the state of the process mean are described by a

Markov process, the time between successive shifts is exponentially

3l

32

distributed. It will be assumed that the average amount of time the

process stays in the in—control state before a shift occurs, given byA-],

remains constant for all periods of operation. The probability

density function is then

f(t) (3.0l)for t > 0 and A > 0. IIf the length of a period of operation is h time units, then the

probability that the process mean remains in the in—control state for a

single period is

^ h -At —AhP = l —f ÄG dt = e . (3.02)00 0

Given that the process has a production rate R, P00 can be expressed as

^= —Ak/R

POO e (3.03)

where k is the number of units produced during a period of length h.

Now, letting A* = A/R one obtains,

(500 =éÄ*k. (3.04)

Observe that A*-1 represents the average number of units produced before

a shift occurs.

The probability that the process shifts out of the in—control state,

ut(0), during the production is k units is then l - éoo or l -e-4*k.

Since there are 2s states other than ut(0), this probability must be

allocated such that

33

s A _ ,kA 2 P . = l - e J

(3_05)J=·S JJ#0

where POJ represents the probability of a shift from state pt(0) to state

pt(j). One method for allocating this probability, proposed by Knappen-

berger and Grandage [20] and later modified by Latimer et al. [22], will

be employed. If it is assumed that PO _j = POJ, then equation (3-05) may

be expressed asJ

S ^ l —x*k_[ POJ = g·(l—a ) - _ (3.06)J=l

This probability may be distributed over each of the s states by utilizing

the binomial mass function which represents the probability of j successes

in s trials and is given by

¤<3> = (j) pl (l—p)S”J (3-07)

where j = 0, l, ..., s and 0 < p < l. To allocate this probability, some

fraction c, where 0 < c < l, is required such that

s A s.2 Poj = c *2 Q(j) . (3.08)j=l j=l

Summing equation (3.07)0ver the range of j gives

_2 0(J) = l (3-09)

J=0whichmay be written as

S

_2 0(J) = l - 0(0) (3-l0)J=l

_ s— 1 - (T-P) . (3.ll)

34

Substituting equations (3.06) and (3.11) into (3.08) and so1ving for c

yie1ds

c = —————————·—— . (3.12)1 — <i—p>S

Consequent1y, the probabi1ity of a shift from state pt(0) to state pt(3)

is given by

B . = B . = —ill§;äiäL— (ä) J (1- )S'j (313)°0,-3 O3 s 3p p '2[1—(1-P) 1

for j = 1, 2, ..., s [22].

Before deriving the remaining probabi1ities, Bij and Bii, i ¢ 0, it

is necessary to out1ine severa1 assumptions about the process. Initia11y,

any shift from state pt(0) is considered as a deterioration of the process;

and once the process goes out-of-contro1 it stays out-of-contr01 unti1

an overhau1 is performed. Thus, the process cannot correct itse1f.

Second1y, once the direction of deterioration away from pt(0) is

estab1ished, the process cannot improve but may get worse. That is to

say, if the process shifts from pt(i) to pt(j) (or pt(-i) to ut(-3)) where

|3| > li], then subsequent shifts can deteriorate on1y to ut(j+1), ...,

pt(s) (or ut(—j—1), ..., ut(-s)). Fina11y, it is assumed that on1y one

shift can occur during a sing1e period of operation.

Now, the remaining probabi1ities Bij and Bii, i ¢ 0, can be derived.

It seems p1ausib1e to assume that the probabi1ity of a deterioration

from state pt(i) to state ut(j), Bij, is proportiona1 to the probabi1ity

of a deterioration from state pt(0) to state pt(3) [22]. Let

35

é· .Pij — -§--—— (3.14)

P .J=1 OJ

215 .= —-Ä. (3-15)

1 ' Poo

Then, Pii is given by

. ä . °P., = 1 — P.. (3.16)11 j=1+] TJ

= P-- (3.17)J=1 U

2 i .=—T— Poj . (3.18)1 - POO 3-1

The Transition Probabi1ity Matrix

Reca11 the brief description of the transition probabi1ity matrix,

Q, in Chapter II. Each e1ement of 3, bij, represents the probabi1ity of

the process mean being in state 3 at period t-1, given that it was in

state i at period t. In order to define each e1ement, it is necessary

to consider the two p0ssib1e actions at time t. Name1y, the process can

either be overhau1ed or it can be a11owed to run without an overhau1.

First, suppose no overhau1 was performed at period t. Then the

e1ement boo, the probabi1ity of the process mean being in state 0 at

period t-1, given that it was in state 0 at period t, is simp1y the

probabi1ity of no shift. Hence, boo = PO0. Likewise, boj, the

probabi1ity of the process mean being in state 3 at period t-1, given

36

that it was in state 0 at period t, is the probability of a shift to

state j. Hence, boj = POJ (bo _j = By the same logic, the

elements bii, i f O, and bij, j > i and —j < -i, must be given by

bi, = Bii and bij = Bij. Since the process can only be corrected byan overhaul, the remaining elements of B must all be equal to zero since

each represents an improvement in, or a change in the direction of

deterioration of, the process mean. .

Now suppose that the process was overhauled at period t. In Chapter

II it was stated that whenever the process is adjusted, it is always

returned to the in-control state, pt(Ü). Thus, after an overhaul at

period t, the value of the element at"(O), of posterior state vector

Qt", will always be one; the remaining elements at"(i), i f 0, will be

equal to zero. Also recall from Chapter II that the prior state vector

at the next period, t—l, is obtained by the matrix multiplication:

It can then be observed, after performing this operation, that

at_]'(—S) = b0,_S (320)

(3.2))

A.

' =(0) b0O ( )

^· : ^

3.23am (S) bog . ( )

37

Each element, boj, is simply the probability of a shift in the process

mean from state 0 to state j during a single period of operation. Conse-

quently, boj = Foj. All the remaining elements of Q_need not be considered.

Note that the elements boj are identical for the two cases considered.Thus, a single transition probability matrix is now explicitly defined

in Figure 3-l. a

The Probability State Vector Q_The posterior state vector at period t, Qt", represents the

probability of the process mean being in each state immediately after

sampling. In order to determine the expected cost of operating the

process for a single period until the next sample is taken, the proba-

bility mass function for the state of the process at any point in time

during the t'th period is needed. Accordingly, define the i'th element

yt(i), of the probability state vector Qt, to be the probability that the

process mean is in state pt(i) at any point in time during the t'th

period.

Duncan [ll] has shown that, given a shift between the samples at t

and t-l, the average fraction of the interval that elapses before the

shift occurs is

A Z(l-e”" k) A

kwherek is the number of units produced between samples.

The following derivation of Qt parallels that given by Knappenberger

and Grandage [20]. Since the process is assumed to be unable to correct

itself, ;t(0) depends on the probability that the process mean is in

xxU1 U1 UI

xx Q Q Q . I I Q • I . •„ II ,—r•— II/'I (/1 (\I r—

I ^ ^ I I4.) O I- U1 In

I1

|'*‘ IQ FQ

I I I Q)I./1 r— r— I./1 U1 'O

xx Q Q Q I • . Q I I I • • II « Q OI- U1 UI (\I I- Z

I " '· I I-I-9 U') U') (J3

<Q_<Q_

<Q__<C.L -I—

E/'\ (U

C\I C\I CI I A

U1 ‘ N N U1 C)xx Q Q Q • • I Q I I I I . « Q Q

1- Ih I/7 (\I Q) .I

•~ •~ I -6-)4-, I- U1 IU

:1 <¤. <¤. <¤. -I->UI• • • „ . . . . I I I

. . I I I . I . • . 4.)

p-\-/ ® G • • • CD G r— • • • G G G,- <¤. <¤. (1)

I .C4.) +9

1L.z—x Q

G ® *~I—\-/ @ • • • C) C) Ö • ••1-

(D. XI ‘v—

4.) Lol :1 4-)I- FU

/\ I- E•——· I I-—·

I ^ I >>A- @ C] •

- ·r—

"C) • • • C) C) +9

I- I G '•—

I (Q. <D. I-·I-* ‘I—

3 .¤FO

. .' „ . I . I . . . _Q• • . • . . „ „ . I Q, . . I • I • I . I L

Q./·x Q](\I + C+ UI I\I OUI I + <\I 'v—

I•· U1 + +9

N- Q C\I ~ ·• I UI • • • E CD E ·r·

r—+ ^ I U7

I VI I-—^ C

+9 I I CD IG3 I1 <¤. IQ. LI-

Q r— I-I—+ + G)

+ U1 UI I-- .Cth I I +

r— I"I •· •~ U1 +-/ r·— C\I • • ¤ I UI Ü

·• •C)I-

+ + ··I

I U) UI r—"

YU-I-I I I I O I

:1 <¤. <C1. <I:1. IQ. PO

Q U1 U1 cuUI I I LI

•• •~ U1 3~/ I- (\.I I U) U')

r— r-' + +• • • •‘ I Ö • • • CD 'I'•

I U1 U1 I- •~ I-L-I-) I I I CD

3

(\_|rx"

+ +’°* (\I •—·

UI UI UI I- Q Q I I QE I I I • • • I @ r—

·•

·U1 I/I L/1

O L! AZ \./ x! \./ \-/ S.? \./ \./

L -I-* ·I-* ··I—* -I-* -I-* -I—* -I-* +-9 -I-*LI. 1 1 1 1 1 1 1 1 1

39

state ut(0) at the time the sample is taken (at time t) and remains there

until the next sample is taken, and the probability that the process

mean is in state pt(0) at the time of sampling and then Shifts to a

different state during the production of the next k units. Hence,

§0(o) = §t~(o) 500 + ;,~(o) (1-500) A . (3.25)

The remaining probabilities yt(i), i f 0, depend upon the probability _

that the process mean is in state pt(i) at the time a sample is inspected

and remains there during the production of the next k units, the

probability that the process mean is in state pt(0) after Sampling

(at time t) and Shifts to state pt_,(i) during the production of the

next k units, and the probability that the process mean is in Some lower

state (say, pt(m) where m < i) at the time a sample is taken and Shifts

to state ut_,(i) during the production of the next k units. Each yt(i)

also depends on the probability that the process mean is in state pt(i)

after Sampling (at time t) and deteriorates to Some higher state

(say, ut_,(n) where n > i) during the production of the next k units [20].

It is assumed that the fraction of time the process mean spends in a

lower state before it deteriorates to a higher state is on the average,

the same fraction, A, of the time the process mean spends in state pt(Ü),

given that a shift to Some other state has occurred [20]. Hence,

5,, + &,~(¤) 50, <i—A)

i—l 0 H 0+ mg] at (m) pm, (i—A)

0 S 0+ at"(i) A n=§+] P,n (3.26)

40

and

-1+1 A H „+ mg-] at (m) (1-A)

A —s A+ 6t"(—i) A [_ P_1 n (3·Z7)

n=—1-1’

for 1 = 1, 2, ..., s. Observe that the third term in each of the

equations (3-26) and (3.27) is zero when 1 = 1 and the 1ast term is zero

when i = s. That is to say, it is impossib1e to shift to state pt(1) or

pt(—1) from any other state except pt(0), and it is impossib1e to shift

from state pt(s) or pt(-s) to some higher state.

The Probabi1ity Vector f

In Chapter II it was assumed that the cost of operating the process

for each period is so1e1y attributed to the production of defective units.

Consequent1y, define the 1'th e1ement f(i), of the probab111ty vector E,

to be the conditiona1 probab111ty of producing a defect given that the

process mean is in state pt(i). Specifica11y, assume that SL and SU

represent the 1ower and upper specification 11m1ts, respective1y, for the

process qua1ity characteristic xt. whenever xt fa11s outside the

specification 11mits the unit is considered to be defective. Since xt

is assumed to be norma11y distributed with mean ut(i) and variance 62,

the 1'th e1ement of f_is given by

S -1%-1,11112^. U 1 1 2a2'F(1) = 1 GSL

/2n

41

Derivation of the Posterior State Vector ät"

The posterior probabilities describing the states of the process

mean are derived from the prior probabilities and the sample result.

From equation (2-02),the i'th element of Qt" is given by

° . ßlzglu (1)„11]Ä'(1)C,t··(,) = _ (3.29)

V) ß[xt(¤C(1)„nC1 ut (1) .

Also, the likelihood function for the sample mean is normally distributed

with mean ut(i) and variance 62/nt. Thus, the density function for x, is

— 1 ‘/E 2 22[xt|ut(i),nt] = ———- T- e O (3.30)

/211

where -w < ik < w and i = —s, ..., -1, 0, 1, ..., s. Therefore,

II

G out 1)C, ··(1) = _ (3.31)1 -11 tx — (1)l2t t **1;

1 ^ . .—j;·——— e at (1)gi /211 O

This expression can be reduced to

2 .20a"(i) = ———-—l————. (3.32)1 -1. [Z- (ni?

t t **1;2 „

20 atnü)Vi

42

It can be observed that

Z ¤t"<i) = T - (3.33)Vl

Derivation of the Predictive Distribution

As stated previously, the sampling distribution is normally distri-

buted with parameters pt(i) and 62/nt. The joint distribution of EZ and

pt(i), given nt, is given by the product

2[xt|pt(i),nt] «;t'(i) . . (3.34)

Then summing over all possible states of the process mean, one obtains

the marginal or predictive distribution for the sample mean conditioned

upon the prior beliefs about the process and the sample size. In

equation (2.lO), the predictive distribution is expressed as

9(?pIut'„nt> = Z_ßIYpI¤t(i)„¤t] Ät'(i) (3.35)1

which can be explicitly defined as

lpplip-¤t(i)l2

g(E lu ' n ) = Z Zig ·l e 202 Ä '(i) (3 36)ll li li vi ‘/EE 0 ll

The Explicit Form of the Recursive Eguation

Recall from Chapter II the general form of the recursive equation

given by

43

Vt(ut‘) =II 0 IK(ut ) + C(u ) + Vt_4(ut_4 )

min __ E min + $(nt)“1c(U ~) + v (U ·)

t t—l t—l(3.37)

Now, in order to explicitly define each of the cost terms, let the

posterior expected cost of an overhaul, K(ut“), be equal to a constant

average cost a3. Hence, ‘

K(ut") = a3 (3.38)

for all ut". ”

The posterior expected running cost is dependent upon the state of

the process mean throughout a single period of operation. The probability

that the process mean is in a particular state, Ut(i), at any point in

time during the t'th period, given by $(i), is needed. Then, utilizing

equation (2.07) in Chapter II, the posterior expected running cost,

C(ut"), when no overhaul is performed, is given by

c(Ut~) = R U 64 vg %(1) $(1) (3.39)1

. _ ^ ^t- R h a4 j_l_ (3.40)

where L 1S l tY‘äI‘\SpOS€. l

The posterior expected running cost, C(uO), given an overhaul was

just completed, is given by

C(u0) = R 11 64 if (3.4l)

44

where ij is derived from the posterior state vector, gt", whose 0'th

element at"(O) = l (ik is used to signify an overhaul was just completed).

The cost of sampling, S(nt), is simply a restatement of equation

(2.08):

S(nt) = a] + a2 nt . (3_42)

Then to summarize, the explicit recursive equation is given by

Vt(ut‘) =a3+Rha4il* +Vt-I(Ut—I)

min __|E minn x u

‘nt t t

’t ^ ^t .Rhaail +Vt—l(Ut-I)

· + 6] + 62 nt} (3.43)

where

at_]' is obtained from equations (3-20) through

(3.23) if an overhaul was performed at period t.ut_]' implies (

is obtained in general from equation (3.l9)

when no overhaul was performed at period t.

Optimization of the Sampling Interval

Two approaches, each proposed by Carter [7], for determining the

optimal interval between inspections will be discussed. In considering

h, the time between samples, it may be observed for certain kinds of

cost functions, the total expected cost per period may decrease as the

45

value of h decreases. But, on the other hand, the total cost for sampling

over the planning horizon may increase, especially if there is a fixed

component of the sampling cost. Thus, the effect of h during the

optimization procedure is not clear.

For a finite horizon problem, it would be possible to examine the

total expected cost for a fixed tgtal_time using different values for the

length of the sampling interval. If T is the total time and h is the

interval length, then T/h is the number of periods of operation that would

be considered. This procedure, however, has several limitations [7]:

l) There is no guarantee that the optimal policies derived

from a fixed interval length will yield a global optimum.

2) A search or enumeration technique would have to be employed

. and only a local optimum might be found.

3) Each time the total time of operation, T, changes, a new

» value for h would have to be determined.

The second approach, which will be employed in the optimization

procedure, is to consider the “gain" of the process. Recall from

Chapter II that Vt(ut') is the minimum future expected cost with t periods

left to operate. The “gain" of the process is then defined as

6 (3.44)

with the value of a in equation (2.l3)set equal to one. The gain is

interpreted as the rate per period at which the expected future total

costs decrease. For the infinite horizon problem, the steady state

46

optimal policy yields a minimum gain. (Conversely, the optimal policy

can be found by minimizing the gain) [7].

The optimization procedure would proceed as follows: a starting

value for h would have to be chosen. The recursive equation (3.43) would

be solved for each period t = l, 2, 3, ..., for all sets of prior

probabilities and the value of the gain, 6, would be computed. when t

increases such that 6 converges, then the steady state control policies I

for the given value of h would be obtained. The value of the gain would

then be divided by k, the number of units produced in that period, and

an average steady state expected cost per unit would be obtained. The

procedure would then be repeated by searching over values of h until the

optimal fixed interval is determined such that the average steady state

expected cost per unit is minimized.

Utilization of the Model

The application of the dynamic model would proceed as follows.

Initially, estimates of the costs for overhauling and sampling the process

would have to be determined. The cost of producing defective units as

well as the specification limits would have to be given. Secondly, the

possible discrete states of the process mean would have to be outlined.

An estimate of both the process variance and the average time between

shifts from the in-control state of the process mean would have to be

obtained. And lastly, the production rate would have to be determined.

Assuming that the process starts out in the in—control state, the

process operator would run the process for a single period. He would

then derive his prior beliefs about the process using equation (3.l9).

47

Given the new prior probabilities he would check the control policy to

determine what optimal sample size to inspect. After sampling he would

compute the posterior probabilities and again check the control policies

to determine the optimal operating decision. If the optimal sample size

is zero, then the operator immediately checks the control policies to V

determine the optimal operating decision based on his present prior

probabilities. The same procedure is repeated after operating the process

each period. The operator, by using the steady state control policy,

does not have to be cognizant of how many periods the process will be

operated.4

The Steady State Y Control Chart Model gOne of the main purposes of this thesis is to compare the results

obtained for the dynamic model with those of the optimally designed Y

control chart model proposed by Knappenberger and Grandage [20].

Accordingly, a cursory development of the steady state Y control chart

model is included. For a detailed derivation and description see [20].

It is the purpose of this model to derive the test parameters, which

include the control limits, the sample size, and the interval between

samples, such that the expected total cost per unit manufactured by the

process is minimized. It is assumed,likewise for the dynamic model,that

there is a single measurable quality characteristic, x, which is a

continuous random variable and is normally distributed. The process

parameter, u, is a continuous random variable which can be satisfactorily

approximated by a discrete random variable. One value, u(O), of the

, discrete random variable is associated with the in-control value of the

48

process mean and the remaining values p(l), p(2), ..., p(s) are

associated with out—of-control values of the process parameter. Here,

Knappenberger and Grandage develop their model assuming the process

mean only shifts in a positive direction, unlike the formulation of the

dynamic model. But since they assume that the control limits are

symmetric, the additional burden of developing a model which allows

shifts in either direction is avoided.

General Cost Model [20]

The expected total cost E(C), per unit of product, associated with

a quality control test procedure can be written as

E(c) (3.45)

where E(C]) is the expected cost per unit associated with carrying out

the test procedure, E(C2) is the expected cost per unit associated with

investigating and correcting the process when the test indicated the

process is out-of-control (that is, when the null hypothesis is rejected),

and E(C3) is the expected cost per unit associated with the production of

defective products.

The expected sampling and testing cost per unit is

al a2NE(C]) = E-+ ~?— (3.46)

where

N is the sample size,

a] is the fixed cost per sample,

a2 is the cost per unit of product sampled, and

k is the number of units produced between samples.

49

After each sample is collected, the null hypothesis, HO (the

assertion that the process is in-control), can be either accepted or

rejected. Let g_be the row vector of probabilities q(i), where q(i) is

the conditional probability of rejecting HO given U = U(i). Let Q_be

the row vector of probabilities a(i), where Q(i) is the probability that

U = U(i) at the time the test is performed. Then the expected cost of

rejecting the null hypothesis is given by

Ö3 S . .E(C2) = E- IE q(1) a(1) (3-47)1-0

= gg g Qt , (3.48)

where

Q) is the transpose of Q} and

a3 is the average cost of investigating and correcting the process.

If a4 is the cost associated with producing a defective unit, f_is the

row vector of probabilities f(i), where f(i) is the conditional proba-

bility of producing a defective unit given U = U(i), and I_is the row

vector of probabilities y(i), where y(i) is the probability that the

process mean is in state U(i), then the expected cost per unit associated}

with accepting the null hypothesis is

sE(C3) = a4 _) f(i) v(1) (3.49)

1=0

= aa jf . (3.60)

50

Combining equations (3.45), (3.46), (3.48), and (3.50), the total expected

cost per unit becomes

E(C)In

the above function, the a's are cost coefficients which are

independent of the test parameters. The vector f_depends only on the

nature of the process parameter, but the vectors g, Q, and I_are all .

functionally dependent upon the test parameter. The form of this

dependency is outlined in later sections.

Thus far, only two (that is, N and k) of the three test parameters

have been defined. In order to express the third test parameter (that is,

the critical region) as a single parameter, some restrictions must be

placed on the nature of the test. It will be assumed that the test

statistic, T, is normally distributed with mean u and variance 62/N, where

cz is constant, and that the critical region is symmetric and defined by

the critical region parameter L such that the null hypothesis is rejected

if

T > u(0) + LE- or if (3.52)„/N

T < MO) - E _ (3.53)I/N

Probability Vector g_[20]

Based on the assumption that T is normally distributed with mean u

and variance«¥/N, the probability of rejecting H0 when u = u(i) can be

written as

5l

- _ Lo Lo<i(i) — P(T > ¤(0> + —> + P(T < ¤(0) — —)JN JN

: /N-+ +‘|

_ ¢(M_%Q /,7- L) , (3.54)

where

l 2¢(d) = ( ———- e dz. (3.55) 'd „/E

When i = 0, the above reduces to

q(0) = 2 ¢(L) . (3.56)

Probability Vector Q [20]

The elements, Q(i), of the probability state vector Q_represent the

steady state probability that the process mean is in state i (that is,

p = u(i)) at the time a sample is inspected. To obtain these steady state

probabilities, the transition probability matrix, Q, is required. The

elements, bij of the matrix Q, represent the probability of the process

mean shifting from state u(i) to state u(j) during the production of k

units between samples.

It is assumed that the time between successive shifts in the state

of the process mean is exponentially distributed with parameter A and

whose probability density function is given by equation (3.0l). At this

point, the development proceeds in the same manner as for the dynamic

model, hence only the results will be presented. The probability that

the process mean remains in the in-control state during the production of

k units is

52

P00 =e‘^*"

. (3.57)

The probability of a shift from state b(O) to some other state u(j)

during the production of k units is _

= (j) pi Ü'P)S_j(3-58)

El-p(l-p) lfor j = l, 2, ..., s. Note that equation (3.58) is different from

equation (3.l3) derived for the dynamic model. The factor (ä) is omitted ‘

from equation (3.58) since shifts are only permitted in one direction.

It is assumed, likewise for the dynamic model, that when the process

goes out—0f·control it stays out-of-control until detected. Once an

overhaul has been performed the process mean is always returned to the

in—control state p(O). Also, shifts in the state of the process mean are

only allowed to proceed from one state p(i) to a higher state p(j), where

j > i. And only one shift is allowed in each testing period.V

On the basis of these assumptions, the elements of §_can now be

defined. when j is less than i, the probability, bij, that the process

mean is in state u(i) at some time t and in state u(j) at time t + k/R

(recall R is the production rate) is simply the probability of rejecting

HO (that is, q(i)) at time t multiplied by the probability Poj of shifting

to p(j) during the production of the next k units. Thus, for j < i,

For the case j > i, the probability of failing to reject H0 at time

t multiplied by the probability, Pij, of shifting directly from u(i) to

b(j) in the time required to produce k units must be added to the

53

above equation. Again, the development of Pqj proceeds in the same

fashion as for the dynamic model, thus the results are as follows:

J

iglP03

611 * @*65 (3-60)

where i f O, and

P .- OJp.. - —-—--- (3.61) _

iJ l — P00

where j > i.

Hence, for j > i, bij becomes ‘

[1-q(i)] PO-:

.bjjq(i) POJ + 6 _ P00 . (3-62)

For i # 0, the probability, bii, of being in state P(i) at time t

and also at time t + (k/R) is equal to the probability of rejecting H0 at

time t, multiplied by the probability of returning to P(i) during the

production of the next k units, plus the probability of failing to reject

H0 at time t, multiplied by the probability of remaining in state P(i)

during the production of the next k units. Thus, for i # 0, bii is

(3-63)

and, hence,J

[l—q(i)] X PO-q = qm P J, (3.64)ii Oi l —

P00 '

Finally, the probability of shifting from state P(O) at time t to

state P(j) at time t + (k/R) is simply Poj. Hence,

boj = Poj . (3.65)

54

It is shown in many texts on stochastic processes [26], [5] that the

above conditions define a transition matrix of an irreducible aperiodic

positive recurrent Markov chain. Thus, there exists a vector g_such that

g_§_= E-, (3.66)

where

a_= [a(0), a(l), ..., a(s)] and (3-67)

) am = 1. (3.68) ‘i=O

Furthermore, a(i) is the long-run (or steady state) unconditional‘proba-3

bility that the process mean is in state p(i) regardless of the initial

state of the process mean. The solution of equation (3-66) for the i'th

element of g_is

ag) : b;']+] (3.69)

where bF3]'s are the elements of B*-1; and

F. = .. (3.70 )

for i # j—l and j f s+l,

= 3.7l1 ( )

for j = s+l, and

#1 .= .. - (3.72)ba-l.3 ban 1

for j # s+l. For further details see [20].

55

Probability Vector l_[20]

In the previous section, the vector Q, where d(i) is the steadyV

state probability that the process mean is in state n(i) at the time a

sample is taken, was developed. In order to determine the cost of

producing defectives, as defined in equation (3.50), the steady state

probability, Y(i), of the process mean being in state n(i) at any point

in time is required. Again, at this point, the development of l_directlyI

parallels the development of i_ for the dynamic model; hence,only the

results will be reiterated. The steady state probability that the

process mean is in state u(0) at any point in time, y(0), is

Y(O) = o(Ü) POO + o(O) (l—P00) A (3.73)

where A is defined by equation (3-24). The remaining probabilities, y(i),

i f 0, are given by

v(i) = ¤(i) PO0 + ¤(¤) Poi (l-A)

= + d(m) P . (l-A)m=l

m)

s+ a(i) A X Pin (3-74)

n=i+l

where the third term is zero when i = l and the last term is zero when

i = s.

Sumary

In this chapter, two multi-state models have been presented. The

dynamic model was explicitly formulated along with all the relations

56

which are necessary for its use. Methods for the optimization of the

sampling interval are outlined and discussed. The steady state Y‘control

chart model, formulated by Knappenberger and Grandage [20], was presented‘

briefly to describe the technique of its design.

In Chapter IV, each model will be illustrated by the solution of

several example problems. A comparison of the results of each will then

be made based on how well each attains its premise of minimizing operating

costs.I

Chapter IV

COMPUTATIDNAL RESULTS

Utilization of the dynamic model is illustrated in this chapter by

a number of examples. Three of the example problems describe

hypothetical processes which possess a single out—of-control state, one

depicts a three state process, and one illustrates a five state process.

The results obtained for the dynamic model are then compared to those of

the steady state Y control chart model designed by Knapperberger and

Grandage [20].u

Design of the Numerical Analysis

Two computer programs were written in FDRTRAN IV and processed on

an IBM model 370/l58 computer to solve the dynamic model and the steady

state Y model.

The possible states of the process mean were determined arbitrarily

as follows:“

Two state process: u(l) = u(0) + 36 ;

Three state process: u(—l) = u(0) — 36

u(]) = ¤(Ü) + 3G ;

- Five state process: u(—2) = u(0) — 36

u(]) = u(0) + 1.50

u(2) = u(0) + 3¤ -

57

~ 58

In order to parallel the approach by Knappenberger and Grandage [20],

the lower and upper specification limits on the process quality

characteristic were positioned at

p(Ü) — 30

and

u(Ü) + 30 „

respectively.u

Optimization of the Sampling Intepyal U

Optimization of the sampling interval for the dynamic model was

effected in the manner described in Chapter III. The gain, 6, given by

equation (3.44), was evaluated at each period of the dynamic recursion

for a given value of the sampling interval, k units (recall k = h·R, where

h is the time (hours) between sampling and R is the production rate).

The gain per unit produced in the sampling interval was then found by the

division: 6/k. When the change in the gain per unit from one period to

the next was less than 0.0l, the criterion for convergence was satisfied.

In all problems, convergence was attained after three to five periods.

Since there is no assurance that the steady state cost per unit asla

function of k is convex, the computer program initially attempted to

establish bounds on the optimal value of k. Once this “interval of

uncertainty" was established, the internal points were enumerated.

59

Grid Values for Qt' and x,

In order to limit the magnitude of the optimization procedure,

appropriate grid values for the prior probabilities, at‘(i), and sample

mean, xk, were chosen. The values of at'(i) chosen for the two state

process example problems were as follows:

Q.; = Lé,·<¤>. _[1.0,0.0],

[ .9, .l], _

[ .8, .2],

[ .7, .3],A

[ .6, .4],

[ .5, .5],

[ .4, .6],

[ .3, .7],

[ .2, .8],

[ .l, .9], and

[0.0,1.0].

A greater number of different prior probabilities could have been

utilized, but the affect on the optimal control policies and the resulting

optimal cost per unit would probably not have been significant. On the

other hand, for an actual application of the dynamic model, more prior

probabilities should be considered to insure accuracy.I

Since the distribution of x, was assumed to be normal, onlysample means in the following interval were considered:

60

u(—S) — 30 30 .

The incremental values of xp in this interval were evaluated every 6/3

units.

Example Problems

For each of the examples the following process parameters were

utilized:62==

.36u(Ü) = Ü.O

A

SL = -l.8

SU = l.8

The remaining parameters and cost components are presented in Table 4-l.

Two State Processes

Example #lU

From Table 4-2, the optimal cost per unit is shown to be $.2702.

The associated optimal sampling interval, k, is 730 units. In solving

the steady state Y‘control chart model, the following results were

obtained:

E*(c> =$.3998K*

= 220‘

N* = l

L* = 2.32 .

61

Table 4-1

Cost Terms and Process Parametersfor Example Problems

1 S 25.00 S 6.00 S 200.00 S 3.00 .0625 200

2 $150.00 S15.00 S1000.00 S100.00 .10 100

3 S 50.00 $ 5.00 S 125.00 S 15.00 .10 100

62

Table 4-2

Example #l: Sampling Interval Results

_ Sampling——_* ExpectedU

Interval Cost Per(units) Unit

650 .27l4

670 .2708

690 .2704

7l0 .2703

730* .2702

750 .2704

770 .2706

790 .27l0

8l0 .2722

830.2728850

.2735

* denotes optimum sampling interval

GLO®

r-:¤=

®('7 GJG) •—

, Q.E

® füI- ><

Lu

fx LCJ VI QON 4-* *4-N ·r··E

4-*3 r— -1-O ~¤ I EN <l' EN 1-

fü CU L> S. cu

O S. 3 ¤.LÜ GJ UTN 4-* •r— 4-J

C LI. In"‘° OC) L.) .(*7 O')N C Q)

I- füG D. 4-*r·· E L/7N fü

V7 >~„'U

® füOW G.)LO 4-*

(./7

®NKD

GLO*.0

® LO (D LD (D LO G LO G<|' Ü') C') (U (\.| „ r—-· I- @ C)N N N N N N N N NC\| (\| (\1 C\1 (\| C\| (\| <\.| l\1

1}.Uf] Jöd 150:] DBILCJSÖXE]

64

Hence, the optimal testing procedure is to sample l unit every 220 units

produced and to reject H0 ifY”>

u(0) + 2.326 or if Y < u(O) - 2.326.

The optimal steady state control policies specified by the dynamic

model are listed in Table 4-3. For each prior probability of the process

mean being in the in—control state the corresponding optimal sample size

to collect is listed. Then based on the sample mean observed, the process

operator will select the optimal operating decision.„

Consider, = l.0. The process operator may observe that the

optimal sample, HQ is zero. Consequently, the optimal operating decision

whenn‘=

0 is to let the process run for another period,(R). The operator

may now obtain the prior probabilities for the next period by employing

equation (3.l9). (Qt" = Qt' since no sample was observed.)

Consider, 6{(O) = .80. The optimal sample size to collect is 3.

After sampling, if ik is found to be ;_l.20, then the optimal operating

decision is to let the process run. If Y; > l.2, then the process should

be overhauled.

After sampling, the posterior probabilities are then derived from

equation (3.32). If the process is allowed to run, the prior probabilities

for the next period are obtained from equation (3.l9). If the process

is overhauled dt" becomes (l.0,0.0) and then equation (3.l9) is used to

obtain 6t_]".

Results obtained in optimizing the sampling interval are listed

in Table 4-2 and illustrated in Figure 4-l. Observe that the expected

unit cost as a function of k appears to be convex. Also note that

expected unit cost is extremely insensitive to large changes in the

65

Table 4-3

Optimal Steady State Control Policies:Example #l

Range of thePrior sample mean xt for Optimal

Probability Optimal which "Run without Operating ·of State O Samplg_Size Overhaul" is Optimal Detjsion if

&£(O) n Operating Decision n = O

l.00 0 · — R

.90 2 j_l.20

:80 3 j_.95.70

3 j_ .95

.60 3 i .95

.50 3 i .95

.40 3 j_ .70

.30 3 i .70

.20 3 i .70

.l0 l i-.30

0.00 0 - OH

66

sampling interval. That is to say, the difference in the expected cost

per unit for k = 650 and k = 850 is a mere $.002l.

Now consider how well each model attains its premise of minimizing

the expected cost per unit for operating the respective quality control

procedure. Observe that by employing the control policies dictated by

the dynamic model,a 32.4% savings in the expected cost per unit may be

achieved over the Y control chart method.

Example #2

The cost terms in Table 4-l were employed in optimizing both the

dynamic model and the X steady state model. The results for the dynamic

model yielded a minimum steady state expected cost per unit of $.6493

for a sampling interval of 800 units. These results are listed in

Table 4-4. The optimal expected cost per unit using the steady state Y

model was found to be:

E*(C) = $1.049 .

The optimal testing procedure is to sample 3 units every 5l2 units

produced and reject H0 is x”< 0(0) — 3.l2¤ or if x°> p(0) + 3.l2¤.

Observe in Figure 4-2 that the expected unit cost as a function of k 4

appears to be convex as in Example l. But note that the sensitivity

of the function is more significant than in Example l. Still a large

change in the sampling interval, k, from 500 to 800 only effects a

change of approximately $.04 in the expected cost per unit.

The optimal operating policies dictated by the dynamic model are

illustrated in Table 4-5.

67

Table 4-4

Example #2: Sampling Interval Results

Sampling ExpectedInterval Cost Per(units) Unit A

500 .6883

550 .6723

600 .6626

650 .6559

700 .6518

750 .6497

. 800* .6493

850 .6507

900 .6532

950 .6567

1000 .6610

* denotes optimal sampling interval

C\I=¤:

G GJ '

G Q.I- E

IG><

G L1.!LOON L

' O

O umG 4-* 4-*CN ·r—· ·•—

C E3 G

C) GLO (\1 LG 1- I Q)

FU Q O.>

G L CU 4-*O Cu L thG 4-* 3 O

‘ E U) LJ•—« ·«—-G LI- 'ULO U'! Cl)I\ C 4-*

·r·· Ur—· GJG Q. Q.G E X|\ IG Lu

U)GJ

G 4-*LO IGkO 4-*

L/7

G >~•G 'URD IG

O.)-I-*

G (/7LOLO

GGUT

G 0 0 0 0 0 0 ~ 0 0 0G LD 0 LO 0 1.0 0 1.0 0 1::G 0 0 1\ 1\ go ao 1.0 LD <rRQ KO no ao ao ao go LO ao no

].IU[] Jöd 1SOj pa10adx3

69

Table 4-5

Optimal Steady State Control Policies:Example #2

Range of thePrior sample mean xi for Optimal

Probability Optimal which "Run Without Operating ‘of State 0 Sample Size 0verhaul" is Optimal Decjsion if

atKO) n° Operating Decision n = 0

1.00 0 R -R.904 j_.95 Q

.80 6 i .95

.70 E 6 i .95

.60 6 . j_ .95

.50 6 i .70

.40 6 i .70

.30 6 i .70

.20 6 i .70

.l0 6 i

.700.000 — 011 T

70

Again consider how well each model attains its premise of

minimizing expected unit costs. Again, a significant savings, 38.l%,

in the expected cost per unit to operate the quality control procedure

can be effected by employing the operating policies of the dynamic model.

Example #3

The resultant optimal expected cost per unit derived from the

dynamic model was $l.067 for a sampling interval of l6O units. TheseA

results are listed in Table 4-6. The optimal expected cost per unit

derived from the Y control chart model was:

E*(C) = $1.791 .

The corresponding test procedure is to sample 2 units every 78 units

manufactured, and overhaul the process if x-< p(0) — 2.5lo or if

Y> „(o) + 2.5lo.

Again the sensitivity of the expected unit cost as a function of k

is depicted in Figure 4-3. Observe that for this example,the expected

cost per unit is extremely sensitive to even small changes in the sampling

interval.

The optimal steady state operating policies are presented in

Table 4-7.

Again, comparing the procedures outlined by each model, a 40.4%

savings is attained by following the doctrines of the dynamic model

rather than those of the steady state Y control chart model.

7l

Table 4-6

Example #3: Sampling Interval Results

Sampling ExpectedInterval Cost Per(units) Unit

l00 l.l89

l20 l.l27

l40 l.085

l60*l.067l8O

l.l79

200 l.l8l

220 l.9l2

240 l.208

260 l.230

280 l.l96i

300 l.285

* denotes optimal sampling interval

OO

-

O (*7CO

:~‘*=.

(\1G)

_ r—_ O D.

LO EC\J fü

><¢·~ |,__|__|O thÜ 4-* LC\I •r— O

C '4—3

4.I(\I P') -:—C\I r·—- I C

fü Ü O>

O L CU LO GJ L G)C\| 4-* 3 Q.

_ C C7v—• -:- 4-*L1. L/1

U) O•— C L)

r- GJ(3 Q. 4-*a¤ E v¤•— fü 4-*

V7 L/7

O >“•Ü 'Ur- fü

GJ4-*‘ O V)

(\Ir-

OO•-

O O O O O O O O O O OÜ

€\JO RO Ü C\| O LD Ü

C\] €\] (\l r— r- •—r-*- r- O O O

r- r— 1— :— e-- r— 1- •—— •—- r- r—·

1I.U['] Jöd 1SOj paqoadxg

73

Table 4-7

Optimal Steady State Control Policies:Example #3

Range of thePrior Optimal sample mean xt for Optimal

Probability Samplg_Size which "Run without Operating ‘

of State O n 0verhaul" is Optimal Detjsion ifa€(O) Operating Decision n = O

l.00 O ‘- R

.90 3 $ .95

.80 3 $ .95

.70 3 $_.95

· .60 3 i•95

.50 6 $ .95

.40 6 $ .95

.30 6 $ .95

.20 6 $ .95

.l0 6 $ .70

0.00 0 - OH

74

Three State Process

The cost components and process parameters used in Example #l

were also used in the analysis of the three state process. This process

possesses two out-of-control states which are located symmetrically

about the single in-control state. Since the process mean can be in

three states, a greater number of combinations of prior probabilities

had to be considered in the analysis. These combinations are listed inA

Table 4-9.

In solving the dynamic model, the minimum expected cost per unit

was found to be $.2826 for a sampling interval of 7lO units. The same

control procedure as in Example #l was derived for the Y control chart

model. Recall that E*(C) = $.3998. The results of the optimization

procedure are listed in Table 4-8 and the sensitivity of the expected

unit cost as a function of k is illustrated in Figure 4-4. The optimal

steady state operating policies are outlined in Table 4-9.

Note that the results obtained for the three state process differ

only slightly from those of the two state:

Expected Cost SamplingPer Unit Interval

Two state: $.2702 730

Three state: $.2826 7l0

In utilizing the three state process dynamic model a savings of

29.3% can be achieved over the Y-control chart model.

75

Table 4-8

Three State Process: Sampling Interval Results

Sampling ExpectedInterval Cost Per(units) Unit

650 .2833

670 .2829

690 .2827

7l0* .2826 _

730 .2828

750 .283l

770 .2834

790 .2840

8l0 .2847

830 .2854

850 .2862

* denotes optimal sampling interval

U)V7 .GJm 8co LQ.

G‘“° I3.

OO •'¤

4-*

Q)co:/1 GJ4-* L

3 s: I-|\ 3

sz Q)

IG <t'rt} I

'\ > <' LL O

B E“*‘

C 3 4-*·v—w- E

U) LI. GE

gw- L-Q)

" E ¤.ä 4-*E vv 3'\

L.)

0 ¤* .¤= tz?LO 4-*U)

G

~ äko •‘¤

BGLO U)LO

LO 0 :.0 0 :.0 CD :.0 0 LO CJRO KO :.0 LKÜ <r

<?‘ M M N N¤O OO 00 oo oo OO oo oo oo~ ooQ Q N N N Q N N N N

Z],I.Uf] Jöd 1].50:) p31I)8dX]

77

Table 4-9

Optimal Steady State Control Policies!Three State Process

Range of thePrior^State Vector Optimal sample mean xt for Optimal

a{(i) Sample which "Run without OperatingStge Overhaul" is Optimal Dectßion when ·

—l 0 +l n Operating Decision n = O

0.00 l.00 0.00 0 — R

.05 ..90 .05 2 -.90 j_x£ t_l.2

.lO .80 .lO 2 -.90 ;_xt i l.2

.l5 .70 .l5 4 -.90 ;_xt ;_l.2

.20 .60 .20 4 -.60 ;_x£ ;_.90

.25 .50 .25 4 -.60 ;_x£ ;_.90

.30 .40 .30 4 -.60 ;_x£ ;_.90

.33 .34 .33 4 -.60 ix-ti .90

.70 .2 .lO 4 -.60 ;_x£ ;_.90

.lO .2 .70 4 -.60 ixt j_ .90

0.00 .l .90 2 ;_—.30

.90 .l 0.00 l g_ .60

l.00 0.0 0.00 O - OH

0.00 0.0 l.00 O — OH

78 ·

Five State Process

Again the cost components of Example #1 are used in the analysis of

the five state process. Like the three state process, a greater number

of combinations of prior probabilities were needed in the analysis.

These combinations are listed in Table 4-ll.

The minimum expected cost per unit obtained from the dynamic model

was found to be $.2357 for a sampling interval of 1150 units. The

optimal expected cost per unit found for the Y model was:

E*(c) = $.3478 . '

The corresponding test procedure is to sample 2 units every 318 units

produced and overhaul the process (reject H0) ifx‘<

u(0) - 2.256 or if

§°> „(0) + 2.268.

In performing the analysis, the binomial parameter, p, in equation x

(3.13) was equal to .597 (value used in [20]). The results of the

optimization procedure for the dynamic model are listed in Table 4-10.

Figure 4-5 illustrates that when the sampling interval is greater than

1150 the increase in the expected cost per unit is very significant.

Note, however, that the results for the five state process model

are significantly different from those of the two and three state process

models. The expected cost per unit has decreased approximately $.035

and the optimal sampling interval has increased from 710 to 1150.

Observe in Table 4-ll that in order to compensate for a longer sampling

interval, the optimal sample sizes have increased from 3 and 4 to 7.

79

Table 4-]0

Five State Process: Sampling Interva] Results

Sampling ExpectedInterva] Cost Per(units) Unit

V l050 .2369

l070 .2366

l090 .2366 T

lll0 .2365

ll30 .236]

ll50* .2357 .

l]70 .2440

ll90 .244]

* denotes optimal sampling interval

thU)GJ

G UOW O1-- L '•··- O.

GJ-0-)G fü

N -0-)u- V)r—

. ‘ GJ>

G V) LLLO -0-)r— ·•— G)r·- C .C

3 -0-)xx

LC) L, G

r—0 O

C') fü Ü‘·0—

r- >r—· L CU -0-)(1) L ·r-

-0-) 3 CCU')Q

•·-• ·r—,— LL Le— C7) O)•-· E CL·

·r-r— -6-)DV. th

C) E OON fü L)G tf)r— GJ

-0-)fü-0-)

G tf)NG >$r— °O

füGJ-0-)

G U')LOG

G G G G G G Gr·· IN LD LOÜ Ü C') C') V C') C') C")(\| (\0 <\| C\| f.\l (\1 (\l

8l

Table 4-ll

Optimal Steady State Control Policies:Five State Process

Range of thePrior State Vector Optimal sample mean xt for Optimal

a{(i) Sample which "Run without OperatingStze Overhaul" is Optimal Detision if ·

-2 -l 0 +l +2 n Operating Decision n = 0

0.00 0.00 l.00 0.00 0.00 0 — R .

0.00 .05 .90 .05 0.00 2 -l.20 ;_x£ j_l.50

0.00 .l0 .80 .l0 0.00 7 -.30 j_x£ j_ .60

.05 .l0 .70 .l0 .05 7 -.30 ;_x£_j_ .60

.05 .l5 .60 .l5 .05 7 -.30 j_x£ j_ .60

.05 .20 .50 .20 .05 7 -.30 j_x£ j_ .60

.05 .25 .40 .25 .05 7 -.30 j_x£ j_ .60

0.00 0.00 .33 .34 .33 7 j_ .60

.33 .34 .33 0.00 0.00 7 -.30 i

.05 .30 .30 .30 .05 7 -.30 j_x£ j_ .60

.20 .20 .20 .20 .20 7 -.30 j_x£ j_ .60

- - * — - 0 - OH

* For all observed values of < .20, the optimal sample sizewas found to be 0.

82

Upon comparing the expected unit cost derived from the two models,

a 32.2% savings may be achieved hy utilizing the steady state optimal

operating policies derived from the dynamic model.

Computational Efficiency

The execution time on the IBM model 370/158 computer for optimizing

the dynamic model for each of the two state example problems ranged .

between 4 and 7 minutes, depending on how fast the gain, 6, converged.

More specifically, to solve a single period of the dynamic recursive

equation required approximately 8 seconds.

In attempting to solve the three state example problem, approximately

9 minutes of execution time was required. In solving the five state

process dynamic model approximately 30 minutes of execution time was

required.

Hence, as the size of the problem increases,the corresponding

execution time required for optimization increases drastically. Also

bear in mind that only a coarse grid on the prior probabilities was

considered. Consequently, in an actual application of the dynamic model

the amount of computer time required would be considerable.

On the other hand, the optimization procedure for the steady state

Y control chart model for the two, three, and five state process example

problems only required at most 6 seconds of execution time.

Time Dependent Dynamic Model

Recall that in order to show the existence of steady state optimal

policies the dynamic model was initially formulated in a time

83

dependent environment. Then after establishing the convergence of the

gain, 6, the steady state policies were obtained.

In observing the numerical results for each of the example

problems, the gain has been shown to converge in three to five periods.

Therefore, any application of the time dependent model would only be

reasonable for operations of only one or two periods. For operations of

any greater length of time, the steady state policies would be applicable. l

Consequently, it seems reasonable to assume that the use of a time

dependent set of operating policies is not practical.

Summary

In this chapter the numerical results of five example problems have

been described, tabularized, and illustrated. It has been shown for each

example problem that by utilizing the steady state optimal control

policies derived from the dynamic model,a savings of from 29% to 40%

can be achieved in the expected cost per unit over the use of the control

policies derived from the Y control chart model. Additionally, it

was shown that, in most cases, the expected cost per unit was

relatively insensitive to substantial changes in the sampling interval.

Chapter V

SUMMARY, CONCLUSIONS, AND RECDMMENDATIONS

In this chapter, conclusions are made on the numerical results

presented in Chapter IV and areas for future research are discussed.

Summary and Conclusions

The conclusions to be drawn from the results of Chapter IV are

primarily in three areas. Initially, it has been shown in all cases

considered, that there is a substantial cost savings available by

employing the optimal steady state operating policies derived from the

dynamic model rather than utilizing the operating policies of an Y

control chart. In process control situations where quality costs are

relatively high,the dynamic approach can yield significant results.

However, the simplicity in using this approach in a practical

application is not comparable to Y control chart policies.“ Since anY

increased amount of computations are required in the use of the dynamic

model (i.e., deriving posterior probabilities and prior probabilities for

future periods), the use of a computer terminal or high-speed calculator

would most likely be needed. Dtherwise, decision tables and/or

conversion tables would have to be utilized. And for a process con-

sisting of more than two or three’states,this method would probably be

extremely cumbersome. Consequently, it is suggested that the dynamic

model be utilized in controling a process whose quality costs are high

and in a moderately sophisticated environment.

84

85

Secondly, a great deal of accuracy in deriving an optimal set of

operation policies is not warranted. Judging from most of the example

problems, the optimal unit cost appears to be relatively insensitive over

a wide range of sampling intervals. Like many quality control cost

models, a necessary condition is only to operate in a region near an

optimal solution.

Finally, the efficiency of the optimization procedure is considered I

to be very poor. The application of the dynamic model to a process of any

considerable size may require many hours of execution time to derive an

optimal set of steady state operating policies. For a process of

considerable size, it is suggested that this factor be considered.

Perhaps in such a case, the application of the Y control chart technique

would suffice.

Areas for Future ResearchI

Probably the most important follow-up to this research is the

examination of the sensitivity of the optimal operating policies to the

accuracy designed into the optimization procedure. That is to say, a

much finer grid of values for the prior state vector and possible sample

means would need to be considered.

Extensions to the dynamic model could include the consideration of

an additional random variable in the overhaul cost term. The average

cost of an overhaul could be replaced by random variable which describes

the cost of lost profit due to machine down time. In addition, the

dynamic model could be reformulated in order to describe multiple

quality characteristics.

. 86”

In following the research of Carter [7], a logical extension to

his work would be effected through the development of a multi-

characteristic model. The process characteristic could be described in

terms of a multivariate normal random variable. Then, assuming multi-

variate normal sampling, the conjugate prior relationship could be

utilized to simplify model analysis.

LIST OF REFERENCES ·

l. Barnard, G. A. "Control Charts and Stochastic Processes," Journalof Royal Statistical Society, Sereis B, XXI, No. 2 (l9595,239-27l.

2. Bather, J. A. "Control Charts and the Minimization of Costs," Journalof the Royal Statistical Society, Series B, XXV, No. l (l963§,49-80.

3. Bellman, R. E. and S. E. Dreyfus. Applied Dynamic Programming,Princeton, New Jersey: Princeton University Press, l962. _

4. Bellman, R. E. Dynamic Programming, Princeton, New Jersey: PrincetonUniversity Press, l957.

5. Bhat, U. N. Elements of Applied Stochastic Processes, New York,N. Y.: John wiley and Sons, l972.

6. Carter, P. L. "A Bayesian Approach to Quality Control," ManagementScience, XVIII, No. ll (l972), 647-655.

7. Carter, P. L. "A Bayesian Decision Theory Approach to Process Control,"Unpublished Doctoral Dissertation, Indiana University (l970),57-l29. _

8. Chiu, N. K. and G. B. Netherill. "A Simplified Scheme for theEconomic Design of X-Charts," Journal of Qualit Technolo , VI,No. 2 (l974), 63-69.

9. Comer, J. P. "Application of Stochastic Approximation to ProcessControl," Journal of Ro al Statistical Societ , Series B, XXVII,No. 2 (l965 , 32l-33l.

l0. Duncan, A. J. Quality Control and Industrial Statistics, Thirdedition, Homewood, Illinois: Richard D. Irwin, Inc., l965.

ll. Duncan, A. J. "The Economic Design of X Charts to Maintain CurrentControl of a Process," Journal of American StatisticalAssociation, LI, No. 274 (l956S, 228-242.

l2. Duncan, A. J. "The Economic Design of X Charts when There is aMultiplicity of Assignable Causes," Journal of the AmericanStatistical Association, LXVI (l97l).

l3. Gabrial_I. N. "Economically Optimal Determination of the Parametersof X-Control Chart," Management Science, XVII, No. 9 (l97l),635-646.

87

88

14. Girshick, M. A. and H. Rubin. "A Bayes Approach to a Quality ControlModel," Annals of Mathematical Statistics, XXIII (1952), 114-125.

15. Goel, A. L., S. C. Jain and S. M. Wu. "An Algorithm for theDetermination of the Economic Design of X Charts Based onDuncan's Model," Journal of American Statistical Association,LXIII, No. 321 (1968), 304-320.

16. Hastings, N. A. J. Dynamic Programming With Management Applications,London: Butterworth and Co. Ltd., 1973.

17. Hoel, P. G. Introduction to Mathematical Statistics, Fourth edition,New York, N. Y.: John Wiley and Sons, 1971.

18. Howard, R. A. Dynamic Programming and Markov Processes, Cambridge,° Massachusetts: The M.I.T. Press, 1960.

19. Jedamus, P. and R. Frame. Business Decision Theory, New York, N. Y.:McGraw-Hill Book Company, 1969.

20. Knappenberger, H. A. and A. H. E. Grandage. "Minimum Cost QualityControl Tests," AIIE Transactions, I, No. 1 (1969), 24-32.

21. Ladany, S. P. "0ptimal Use of Control Charts for Controlling CurrentProduction," Management Science, XIX, No. 7 (1973), 763-772.

22. Latimer, B. A., G. K. Bennett, and J. W. Schmidt. "The EconomicDesign of a Dual Purpose Multicharacteristic Quality ControlSystem," AIIE Transactions, V, No. 3 (1973), 214-222.

23. Lave, R. E. "A Markov Model for Quality Control Plan Selection,"AIIE Transactions, I, No. 2 (1969), 139-145.

24. Montgomery, D. C. and P. J. Klatt. "Minimum Cost MultivariateQuality Control Tests," AIIE Transactions, IV, No. 2 (1972),103-110.

25. Nemhauser, G. L. Introduction to Dynamic Programming, New York, N. Y.:John Wiley and Sons, 1966.

26. Parzen. S. Stochastic Processes, San Francisco, California:Holden-Day, Inc., 1962. V

27. Pruzan, P. M. "A Dynamic Programming Application in Production LineInspection," Technometrics, IX, No. 1 (1967), 73-81.

28. Raiffa, H. and R. Schlaifer. Applied Statistical Decision Theory,Boston: Harvard Graduate School of Business Administration,Division of Research, 1961.

89

29. Shewhart, w. A. Statistical Method from the Viewpoint of QualityControl, The Department of Agriculture, washington, D. C., 1939.

30. Taylor, H. M. "Markovian Sequential Replacement Processes," Annalsof Mathematical Statistics, XXXVI (1965), 1677-1694.

31. Taylor, H. M. "Statistical Control of a Gaussian Process,"Technometrics, IX, No. 1 (1967), 29-41.

32. Tummala, V. M. R. Decision Analysis with Business Applications,New York, N. Y.: Intext Educational Publishers, 1973.

33. wagner, H. M. Principles of Operations Research with Applications toManagerial Decisions, Englewood Cliffs, New Jersey: Prentice—Hall, °Inc., 1969.

Appendix A

DOCUMENTATION OF COMPUTER PROGRAM

90

A9l

General Information

This appendix contains a documented listing of the computer program

used to optimize the dynamic model. The program is written in FORTRAN IV

and was run on an IBM 370/l58 system.

EDLE

In order to utilize the program,a variable amount of input data is”

necessary. The necessary data cards are described as follows:

Qapg_ fjjigi Variable4Description

l 215 S Specifies the number of out-of-controlstates on one side of u(0). (e.g., threestates, s = l.) s = 0, if two stateprocess is to be analyzed.

NPS Specifies the number of different priorstate vectors, pi', to consider.

2 6Fl0.0 Al The fixed cost of sampling.

A2 The variable cost of sampling.

A3 The average cost of an overhaul.

A4 The cost per unit of a defect.

3 6Fl0.0 LAM The exponential parameter, A.

P1 The binomial parameter, p.

SIG The process variance, 62.

SL Lower specification limit.

SU Upper specification limit.

R Production rate.

4 6Fl0.0 MU(1) The possible states of the process mean;listed as u(-s), u(—s+l), ..., p(0), u(l),...,p(S).

I

92

Card Field Variable Description

5 to 6Fl0.0 PRIOR(I,J) Prior state vector, Q_'; listed as Ät'(-s),

NPS+5 6Fl0.0 H Starting value of the sampling intervalwhere h is in time units.

XBARL Minimum value of x.

XBARU Maximum value of xÄ

b XBINT Incremental steps in the value of x-to .be evaluated between XBARL and XBARU.

DELTA Convergence factor for the unit gain, 6/k.

NPS+6 215 NSMAX Maximum sample size to consider.

ITMAX Maximum number of iterations to establishgain,

NPS+7 6Fl0.0 KSTEP Step size for the sampling intervalbounding procedure (in terms of unitsproduced). The "interval of uncertainty"is equal to 2*KSTEP.

_ KENUM Incremental values of k, the samplinginterval, to enumerate within the "intervalof uncertainty.“

NPS+8 215 IPNS = l if output is to include a listing ofall sample sizes considered for each pxat each period of the dynamic recursionalso with the corresponding information.

IPR = l if output is to include the operatingpolicies found in the initial boundingprocedure on the sampling interval.

Output

The output includes a listing and description of all input

parameters. In addition, for each period of the dynamic recursion for a

given value of the sampling interval, each prior state vector is listed

93

along with the corresponding optimal operating policies (i.e., the

optimal sample size, the minimum future expected cost, and the description

of optimal operating decisions). Also a gain analysis is provided after

each period of the dynamic recursion.

Program Restrictions

Due to the dimension of variables within the program,a maximum of _

five states of the process mean can be analyzed and no more than 30

different prior state vectors, gt', can be considered.

94

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PROCESS CONTROL:

A DYNAMIC PROGRAMMING APPROACH

by

Nm. Howard Beverly Jr.

(ABSTRACT)

In this thesis, a cost based process control model is formulated. _

A dynamic programming approach is used and along with the techniques

of Bayesian decision theory, an optimal set of steady state control

policies are shown to exist which are dependent upon prior beliefs about

the condition of the process.

It is the objective of this thesis to compare the results obtained

from this approach to those of an X control chart approach. The model

proposed by Knappenberger and Grandage [20] is used as a basis for

comparison. Numerical examples are used to illustrate each procedure.

The results obtained illustrate that by using the operating policies

specified by the dynamic approach,a savings of from 29% to 40% in the

optimal cost per unit to operate the quality control procedure can be

achieved rather than utilizing the policies of the X control chart model.