The effect of the covariance factor on the Procurement Problem Variance of net leadtime demand

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1988-09

The effect of the covariance factor on the
Procurement Problem Variance of net
leadtime demand

Adams, Keith T.

http://ndl.handle.net/10945/23190

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ath
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NAVAL POSTGRADUATE SCHOOL
Monterey , California

THESIS

THE EFFECT OF THE COVARIANCE FACTOR ON THE
PROCUREMENT PROBLEM VARIANCE OF NET
LEADTIME DEMAND
by
Keith T. Adams

september 1988

Thesis Advisor: Alan W. McMasters

Approved for public release; distribution is unlimited

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|. TITLE (include Security Classification) THE EFFECT OF THE COVARIANCE FACTOR ON THE
PROCUREMENT PROBLEM VARIANCE OF NET LEADTIME DEMAND

WORK UNIT
ACCESSION NO.

|. PERSONAL AUTHOR(S)
| ADAMS, Keith T.

Ja. TYPE OF REPORT) 13b TIME COVERED 14. DATE OF_REPORT (Year, Month, Day) [15 PAGE COUNT
Master's Thesis FROM TO 1988 September 44

.. SUPPLEMENTARY NOTATION The views expressed in this thesis are those of the autho

Bnd do not reflect the official policy or position of the Department of

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| COSATI CODES 18 SUBJECT TERMS (Continue on reverse if necessary and identify by block number)
FIELD Covariance, Variance, Standard deviation,

: fT tC CLeadtime, Demand, Repairables, inventory models,
|. AP ICP's, Inventory Control Points, Wholesale

. ABSTRACT (Continue on reverse if necessary and identify by block number)
| An analysis is made of the formulae used by the Navy's Inventory
Control Points in calculating the variance of Net Leadtime Demand of
repairable items. A new formula is then derived, which makes use of actual
Icalculations of covariance between regenerations and demands. The resulting
variance values derived from the new formula are compared with the variance
Ivalues resident on the Navy's Ships Parts Control Center data base and are
Ishown to produce lower variances. The new formula is also compared to the
Joption path formula to determine which formula produces the smallest
Ivariance. The comparison suggests an under-estimation of variance results
when the option path with its estimate of the covariance is used. The
thesis concludes with recommendations for implementation of the new formula.

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Prof. Alan W. McMasters 408-646-2678 54Mg
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Approved for public release; distribution is unlimited

The Effect of the Covariance Factor
on the Procurement Problem Variance of Net Leadtime Demand

by
Keith T. Adams
Lieutenant Commander, Supply Corps, United States Navy

B.S., Purdue University, 1973
M.Ed., University of Missouri, 1975

Submitted in partial fulfillment of the
requirements for the degree of

MASTER OF SCIENCE IN OPERATIONS RESEARCH
from the

NAVAL POSTGRADUATE SCHOOL

September 1988

ABSTRACT

An analysis is made of the formulae used by the Navy’s Inventory
Control Points in calculating the variance of Net Leadtime Demand of repairable
items. A new formula is then derived, which makes use of actual calculations of
covariance between regenerations and demands. The resulting variance values
derived from the new formula are compared with the variance values resident on
the Navy’s Ships Parts Control Center data base and are shown to produce lower
variances. The new formula is also compared to the option path formula to
determine which formula produces the smallest variance. The comparison suggests
an under-estimation of variance results when the option path with its estimate of
the covariance is used. The thesis concludes with recommendations for

implementation of the new formula.

11

THESIS DISCLAIMER

The views and judgements presented in this thesis are those solely of
the author. They do not necessarily reflect official positions held by the Naval
Postgraduate School, the Department of the Navy, the Department of Defense, or
any other US government agency or organization. No citation of this work may

include references or attributions to any official US government source.

1V

TABLE OF CONTENTS

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MOURNS cig cccieyscecccveacccccesecscccosecseccssectsccsecsscessecssesssecsseessens ene 3
DICT, 3
Be cc eeccicceccecsassccsccssssessssessecssecssesseessesssessonssosssessees 3

I. FORMULA DEVELOPMENT.......cccccsscccsscccsscsssccsccseccsseessessecssesssessessreessesssesseess 4
PUMA IONAL, CAVEAT: ..n.c.ccccscccccccccscccsecoccccescssccosecssccsssssssssessssesseen 4
B. PROCUREMENT PROBLEM VARIABLE...ee.ccsscsssssssesssesosessessseeosee. 4
C. UICP VARIANCE FORMULA .o..cocccssccssessssssesssessesstesstsssesssesstessessece: 5
BAU VOR MMOL As cscscccscvcsccecssccccecoseccccsssecsscsuccscsssseccncsssccsucsseccsesees 7
E. FORMULA COMPARISONS. .eecccscsssscsscsssessecsssessesssessecsvesseesensevesseesees 9

II]. FORMULA COMPARISON METHODOLOGY .0.....::-ssssscscesssssssecessssssessessssvcce 12
AA, YATES AVCOTCMSIER (GIN. eae 12
B. FORMULA COMPARISON PROCEDURES. ..-.cccccccssssssessecssesssecesene 13
©y AMAT FORMATE, c.cccccc.-ccccccsccsccecssecsscoscsasscssecssecssecssessseessevsee 15

po SB GUUIEINS) a rc 16

V. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS. .....scccsocssscssseees 21
(CUI CS Se 21
BO ONC WU STON GS, cic iccccsscccssccccsscccssecscscsvecssecssecseccoscsseccsscssscaneceeseen 21
C. RECOMMENDATIONS .ovcceccssscssssssscosecssessscssecssessesssessscarsessessessessseese 22
D. RELATED FURTHER STUDY .oeccccccccscsssccosssssssssscssecessesesssessstsnecsasce: 23

MMMSNO)E REFERENCES, ......cccc ccc cccccccceccccssscssescsocesscsessessssousecsecsssssussssessestsensevsres. 24

V

APPENDIX A - DEN/NOMECLATURE Of DATA ELEMENTS 20
APPENDIN B - FORTRAN PROGRAM FOR VARIANCE CALCULATIONS.... 26
APPENDIX CG - DEVAILET) OUD PU als CIN ye 34

INITIAL DISTRIBUTION Ores iiss ccessecs ce 35

V1

TABLE OF ABBREVIATIONS

Variables

Z - Procurement Problem Variable (PPV)

Z - procurement problem random variable

V - Procurement Problem Variance

D - mean demand per quarter

d - quarterly demand random variable
Var(d)- variance of quarterly demand

L - mean procurement leadtime

] - procurement leadtime random variable
L, - net acquisition time

Var(])- variance of leadtime

B - mean regenerations per quarter (CR)(SR)
b - quarterly regeneration random variable
T - mean procurement problem turn around time

(or mean repair cycle time)
t - repair cycle time random variable
Var(t)- variance of repair cycle time

Abbreviations

A/O - Application/Operation

ASO - Aviation Supply Office, Philadelphia, PA

COG - Cognizant Activity

DEN - Data Element Number

1G - Inventory Control Point

ICPDAT - Inventory Control Point Data

IHF - Inventory History File

FMSO - Fleet Material Support Office, Mechanicsburg, PA
MAD - Mean Absolute Deviation

NICN - Navy Identification Code Number

NIIN - National Item Identification Number

NPS - Naval Postgraduate School, Monterey, CA

NSF - Navy Stock Fund

OPTION - Option path formula for variance calculations at ICPs
PVAR - Mathematically correct formula for variance calculation
SAS - Statistical Analysis System

SIG - Selective Item Generator

SPCC - Ships Parts Control Center, Mechanicsburg, PA

UICP - Uniform Inventory Control Point

Vil

I. INTRODUCTION

A. BACKGROUND

In the U.S. Navy there are approximately 228,800 items classified as
repairables. The responsibility for managing these items is shared between the
Navy’s two inventory control points (ICPs), the Aviation Supply Office (ASO) in
Philadelphia, PA., and Ships Parts Control Center (SPCC) in Mechanicsburg, PA.
The tota] dollar value of these items is in excess of $28 billion with an annual
Navy Stock Fund (NSF) budget for procurement of just under $2 billion [Ref 1].
To manage the inventories of these high dollar value items, the ICPs use a
complex mathematical model which incorporates formulae for the calculation of
means and variances of attrition demand over a net leadtime of procurement for
specific items.

The mean net leadtime demand calculated is called the Procurement Problem
Variable (Z) and the variance of that demand is called the Procurement Problem
Variance (V). These two parameters are key elements in determining the
procurement quantity that 1s necessary to maintain a repairable item inventory at
prescribed protection levels. Specifically, the mean is the quantity that should be
available to meet the average demand over the net leadtime. Additionally, a
percentage of the square root of the variance (standard deviation) could be
purchased to meet any additional demand that may be experienced. This is
essentially a safety level [Ref 2]. The sum of the mean and safety level is the
procurement réorder point used by the ICPs.

If, in the calculation of the variance, an error is made resulting in too large a
value, more safety stock than necessary may be held. This would tie up money
in unnecessary stock and prevent it from being used elsewhere. If the variance
calculation was too small, not enough material would be available, resulting in
the chance of a ‘stock-out’ being higher than desired.

In the late seventies, the ICPs recognized that the variance model being used,
generally calculated variance values that were too high. Two attempts to correct

this situation were then incorporated into the model. One was a result of a

iL

study completed by Fleet Material Support Office (FMSQO) in 1977 [Ref 3]. This
study hypothesized that the large variances were a result of ignoring a dependent
relationship between the quarterly demand for an item and the quarterly
regeneration of carcasses that were returned for repair. The dependent
relationship manifests itself as a covariance between these two random variables.
This was ignored in the original model when calculating the variance of the net
leadtime demand. As a result of this study, an estimate of the covariance
between regeneration and demand rates was incorporated into the computerized
Levels program (UICP A/O D01) by the ICPs. This estimate was provided as an
option path in the Levels program [Ref 4].

The second attempt to reduce variance was done by SPCC in a study
completed in the same year [Ref 5]. To prevent excessively large safety levels
from being created, a "patch" was added to the Levels program which performs a
variance to mean ratio check for each item. If this ratio exceeds an ICP selected
parameter, it modifies the program to recompute the variances of the net
leadtime demand using a power rule formula [Ref 6].

The variance to mean ratio check, the power rule formula and the estimate of
covariance are included in DO1, but the use of the covariance term is only an
option. This option path is currently not being used at SPCC [Ref 7]. The only
definitive reason for not using it was that the ICPs felt that the variance values
that were obtained did not provide sufficient safety stock (.e., too small a
variance). Thus, the large variances (that are not recalculated by the power rule
because they do not exceed the ICP parameter) which precipitated the initial
studies, appear to remain on file at SPCC.

This thesis will look at possible reasons for the large variances mentioned
above and will attempt to offer a method for estimating the value of the

variances more accurately.

B. OBJECTIVES

There are two main objectives of this thesis. The first is to develop a
theoretically correct variance formula for the net leadtime demand which will use
the expected values of demand and regeneration rates to calculate the covariance.
The second is to compare the theoretically correct formula with the actual
variance values on file from SPCC'’s data base and the option path variance
formula of DO1l. By the comparison with the latter, the degree to which the
estimate of the covariance agrees with the theoretically correct formula for

covariance can also be obtained.

C. SCOPE

The comparisons made to satisfy the second objective were limited to using a
5% sample of items resident on SPCC's files. No ASO data was examined. No
attempt was made to actually calculate safety level or determine actual changes
in costs of stock which would result from different variance calculations.
However, it follows that any reduction in variance, with all other factors
remaining constant, would reduce the amount of safety stock required to provide

a given level of protection.

D. PREVIEW

In Chapter II, the two alternatives to be used in this thesis for computing the
procurement problem variance will be presented. In particular, the theoretically
correct variance formula will be derived and the difference between it and the
option path formula will be discussed. Chapter III contains a short discussion on
how the data was acquired and the procedures used in the comparison of the
three alternatives. In Chapter IV, the results of the comparisons are shown and
discussed. Chapter V summarizes the previous chapters, presents conclusions
from the analysis, and makes recommendations for further testing and

implementation.

IH. FORMULA DEVELOPMENT

This chapter begins with a notational caveat and then discusses the concept of
the procurement problem variable as the mean demand for an item over a net
leadtime. It continues with an explanation of the variance formula used by the
ICPs which includes the covariance estimate and variance to mean ratio check
that is used to reduce the variance values. The fourth section presents the
derivation of a theoretically correct variance formula which will be called "PVAR’.
The chapter concludes with comparisons of the correct formula with the formulae

that are currently being used at SPCC.

A. NOTATIONAL CAVEAT

Capita] letters are used to denote the mean values of the variables that they
represent. Occassionally, there will be a need to distinguish between these mean
values and the distributed random variable from whence they came. This will be
accomplished by adopting the expediancy of using the lower case version of the

symbol to represent the random variable. Alj time is measured in quarters.

B. PROCUREMENT PROBLEM VARIABLE

The Procurement Problem Variable (known as PPV and denoted by Z) is the
expected demand over an "average acquisition time". The term "variable", in this
case, is a misnomer. It is a mean of the distribution of the procurement problem
random variable, not a random variable itself. However, the term has been
accepted by convention, to represent the expected demand over a net leadtime.

To develop this net leadtime, let B represent the average number of items
regenerated per quarter and let D be the average number of items demanded per
quarter. The ratio of B/D then represents the average proportion of demands
that are satisfied by regenerations and 1 - B/D is the average proportion of
demands that are not, and thus have to be procured. Next, let L represent the

mean procurement leadtime and T represent the mean repair turn-around time.

The average of the net acquisition time, L,, can then be represented by the

following formula:
(1) L, = (1 - B/D)L + (B/D)T.

Multiplying this formula by the average quarterly demand, D, will produce the

average demand over L.,.
(2) itve= Ol).- BlL+ BT =e

Equation (2) is the formula used by the ICPs for computing Z, the mean of the
net leadtime demand [Ref 4].

C. UICP VARIANCE FORMULA

The variance formula that was used in the middle 1970's was:

(3) V =(L- T)[Varid) + Var(b)] + TVar(d) + D2Var(t)
+ (D - BP[Var(]) + Vari(t)].

The above equation was pieced together from a Fleet Material Support Office
(FMSO) Working Memorandum [Ref 3] and the current computerized Levels
program documentation (UICP A/O DO1) [Ref 4]. The memorandum, which was a
summary of a study completed in 1977, suggested changes to the above equation
(3) that would reduce the variance of net leadtime demand of repairables. The
problem of observed large variances at the Inventory Control Points (ICP) in the
mid seventies was important to them because of increasing funding restrictions
and budgetary. hmitations that were being imposed upon the supply system at
that time. They recognized that a reduction in variance values would reduce the
amount of money needed to fund safety stock. To accomplish this reduction, the

ICPs incorporated the changes that were recommended by the study.

The major change that was incorporated was an estimate of the covariance
between the demand rate and regeneration rate of a repairable item. From the

FMSO study the estimate had the form of:

(4) Var(d)B/D

The ICPs programmed the above covariance estimate into the variance

equation as an option path. The option path has the following form:

(5) OPTION = (L - T)[Var(d) + Var(b) -2Var(d)B/D] + TVar(d) + D?Var(t)
+ (D - BP{Var(]) + Var(t)}.

The above equation (5) is the same formula that is documented in the current
Levels program. However, the option path, according to SPCC’s Operations
Analysis Division [Ref 7], is not being used. The only variance reduction
technique that is currently being used is a variance to mean ratio check and
subsequent power rule recalculation of variance.

The variance to mean ratio check and the power rule were implemented as a
result of a study completed by SPCC in 1977 [Ref 5] which was also motivated by
the excessively large variances of net leadtime demand that were on file. To
prevent large safety levels from occurring, a "patch" was added to the Levels
program which compared the variance of net leadtime demand, calculated from
equation (3), with the mean of net leadtime demand, calculated from equation (2).
If this ratio exceeded a preset ICP parameter (SPCC = 150, ASO = 450), the
variance calculated by equation (3) was recalculated using the following formula

(power rule):

(6) VieewalZie

where a and b are preset parameters.

The above parameters (a,b) are currently set at SPCC as 4.849 and 1.502,
respectively, and at ASO as 27.458 and 1.559, respectively [Ref 10]. These
parameters are reviewed approximately every three years by FMSO.

In summary, the current variance calculations at the ICPs are obtained by
using equation (3) and the variance to mean ratio check with the power rule.
The actual variance values on file at SPCC will be referred to as "V" throughout
the rest of this paper. Note that even though equation (3) and equation (5) are
calculations for the variance of net leadtime demand, V, to prevent confusion, the
results of equation (5) will be referred to as "OPTION’. OPTION, equation (5), is
only programmed as an option path and, as previously mentioned, is not being

used.

D. PVAR FORMULA

The procurement problem variable, as shown in formula (2) can be derived in
another way as follows. Let | be the number of quarters required for
procurement of a new item. Let t be the repair turn-around time necessary to
repair a carcass of the same item. The mean net number of items to buy to meet

demand over | can be described by the regression function [Ref 9] as follows:
(8) Elz ll] = 1D - (1-t)B

This equation has the following interpretation. The first term, ID = IlE[d], is
the expected number of items demanded given the procurement leadtime, 1. This
value must be offset by the mean number of carcasses expected to be returned to
inventory in ‘ready for issue” condition (RFI) over |. For the first t periods of the
given | periods a number of carcasses are being repaired. The number of such
carcasses is the consequence of the number of items returned to supply for repair
prior to our time origin. After t such items can be used to fill demands. The
term (1 - t)B = (1-t)E(b) represent a conditional expectation of those regenerations
after our time origin. This is the reason for the negative term in (8). Using the

basic rule of iterated expectations,

(9) Elz] = ElE|z |1.t]].

It then follows:

(10) 7 PL = 1B.

which can be rewritten to show that it is identical to formula (2):

Z= DL - BL + BT.

To develop the variance of net leadtime demand, the regression function (8) can
be used. Rewriting the regression function of net demand (z) on leadtime (1) and

repair cycle time (t) provides the following:

(11) Elz {1.t] = d - t)(D - B) + tD,

and the conditional variance of z given | and t is:

(12) Variz]1t) = ( - t)Var(d - b) + tVar(d),

because we are summing (] - t) independent observations of (d - b) and adding it
back to independent observations of d.

Using the Lemma stated and proved by FMSO [Ref 10] (i.e., the unconditional
variance is the mean of the conditional variance plus the variance of the

regression function) results in:

(13) Var(z) = (L - T)Vari(d - b) + TVar(d) + Var((D - B) + tB),
= (L - T)Var(d - b) + TVar(d) + (D - B)?Var(1) + B?Var(t),

because procurement leadtime and repair cycle time are independent variables.
Since current repairables inventory management procedures [Ref 11] require a

return of a carcass concurrently with a requisition for another unit of the

repairable (i.e., a one for one exchange), this creates a dependent relationship
between the number of carcasses returned to supply for repair and the demand
for the same item. Accounting for this dependent relationship, twice the
covariance between demand and regeneration (because each is dependent on the

other) is subtracted from [Var(d) + Var(b)]. The following formula results:

(14) Variz) = (L - T)/Var(d) + Var(b) - 2Cov(d,b)] + TVar(d) + B?Var(t)
+ [(D - B)?Var(1)].

The covariance term from the above equation (14) can be derived using

expectations [Ref 14]:

(15) Covid,b) = Ef(d - D)(b - B)],
E[db] - DB.

When (15) is inserted in (14) the resulting equation, which will be called

PVAR, for calculating the variance of demand over a net acquisition leadtime is:

(16) PVAR = (L - T)iVari(d) + Var(b) - 2(E[db] - DB)] + TVar(d) + B2Var(t)
+ [(D - B)?Var(1)].

E. FORMULA COMPARISONS
If PVAR, equation (16), is subtracted from V, equation (3), the difference is:

(17) V - PVAR = 2DVar(t)(D-B) + (L - T)2Covid,b).

Adding PVAR to both sides and expanding terms results in an expression relating
V and PVAR:

(18) V = PVAR +(D?2 - B2)Varit) + (D - B)*Var(t) + (L - T)2Cov(d,b).
Collecting terms and simplifying:

(19) V = PVAR + 2DVar(t)(D-B) + (L - T)2Cov(d,b).

It is interesting to note when PVAR would equal V. If we assume that L > T,
then V = PVAR when the following is true:

(20) DVar(t)D - B) = -(L - T)Cov(d,b).

A special case of the above would occur when both terms are zero. That results
from any one term (on both sides) being zero. This is not an uncommon event
(i.e., Cov(d,b) and Var(t) equal to zero) as will be shown in the following chapters.
Also note that if the covariance term was negative (1.e., E[db] > DB) and any
term on the left side of equation (20) was zero (1.e., Var(t) = 0), then PVAR would
be greater than V. Mathematically it is possible for the covariance term to be
negative, but conceptually it is not since a probability of a regeneration will exists
when a demand occurs and the regeneration rate can never be negative. The
negative covariance term is not an uncommon event when working with the data
and may suggest problems with the data on file. This investigation is left for
further study.

The same procedures as above can be used to compare PVAR and OPTION.
For simplicity, let the estimate of covariance, equation (4), be represented by Cov’
and let the calculation of covariance, equation (15) be represented by Cov. This

comparison results in:

(20) OPTION = PVAR - 2[Cov’(d.b) - Cov(d,b)] + (D2? - B2)Var(t)
+(D - B?Varit).

As discussed above, if Var(t) = 0, then the difference between OPTION and PVAR

reduces to:

(21) OPTION = PVAR -2[Cov(d.b) - Cov(d,b)].

Then PVAR and OPTION will be equal when:

10

(22) Cov(d,b) = Covid,b),

and PVAR will be less than OPTION when:

(23) Cov(d,b) > Covid,b).

This last situation, equation (23), will be discussed in depth in Chapter IV.

tl

Il. FORMULA COMPARISON METHODOLOGY

This chapter begins with an explanation of how the data was obtained from
the files of SPCC and loaded to the Naval Postgraduate School’s (NPS)
mainframe computer. It then explains the process used to compare the variance,
V, on file at SPCC, with the option path formula for variance, OPTION, and the

theoretically correct variance formula, PVAR.

A. DATA ACQUISITION

The data used to compare the three models was taken from SPCC’s data files
on the Univac 494 computer. The data consisted of all repairable items with a
cognizant activity code (COG) of 7H, 7I, and 7G. These COGs indicate that the
items are specically managed by SPCC. The data elements necessary to calculate
the variances were downloaded to tape via the ICPDAT (inventory control point
data) network using the computer resources of the Operations Research
Department (Code 93) at FMSO. The specific Data Element Number (DEN) and
nomenclature of each data element are presented in Appendix A. It was
necessary to access two different files to obtain all the data elements. The SIG
(selective item generator) file was used for most of the data and the IHF
(inventory history file) was accessed for specific data necessary to calculate
expected values (for Covid,b)). Once the data was acquired, it was translated into
IBM format for storage in National Item Identification Number (NIIN) sequence
on the new IBM 3090 mainframe at SPCC. A mainframe data analysis package,
SAS, was used to eliminate any NIIN which had blanks or data missing from any
DEN. An example would be a NIIN that had data on the SIG file but no IHF
entries and vice versa. For the purpose of this data selection, zero was
considered a valid data entry, but blanks were not. Finally, a tape was obtained
of the remaining data. This tape was taken to the Naval Postgraduate School
(NPS) where it was uploaded on the IBM 370/3033AP mainframe and stored in a
batch data file. Due to the size of the data (in excess of 47,000 line items or

12

NIINs), a 5% sample was taken from the batched data set and loaded to a
private disk (B-disk). The private disk allowed interactive programming, which
was not available if kept on the batch file. The 5% size was the largest sample
size that could be loaded and stored on a private disk (1672K bytes of disk
space). The resulting sample had a total sample size of 2,345 observations. Each
observation consisted of a NIIN and all data elements pertaining to that NIIN
that were needed for computing the variances being compared.

Since the batch file was arranged in NIIN sequence, the sequential sampling
technique [Ref 12] was used to ensure a continuous, representative sample across
all NIINs. To obtain the 5% sample, the data was sequentially subdivided into
blocks of 20 items. A number between 1 and 20 was selected at random to
determine which item from each block would be sampled. The 5% sample,

therefore, consisted of one item from each block.

B. FORMULA COMPARISON PROCEDURES

The V, PVAR and OPTION formulae were programmed on the NPS mainframe
computer using FORTRAN. The actual code is presented in Appendix B. The
resulting variances from each of these equations were compared to the
corresponding variance obtained directly off SPCC’s file, V1. The file variance
value is denoted by V1 to distinguish it from the programmed UICP variance
formula, V. Vl was used as the comparison value because it is the actual
variance used in the calculation of inventory levels. V was used only to compare
it to V1 to see if the variance on file could be duplicated by a simple formula. If
Vi could not be duplicated then some method other than direct calculation of the
variance was used by SPCC. It is assumed that the power rules were used to
estimate the variances of the components within the variance formula. A recent
study by FMSO [Ref 8] indicates that the mean absolute deviations (MAD) that
are used to compute the variances of several of the variables in the calculation of
the variance are estimated by power rules similar to the one discussed above.
The affect of the power rules and the resulting variance values is. left to further

study.

13

As discussed under SCOPE, no direct comparison of OPTION and PVAR will
be done with V. Thus, the comparison of variance values will be done between
V1 (the values on file at SPCC) and V (the UICP variance formula), and between
Vi and OPTION (the UICP option formula for calculating variance) and PVAR
(the theoretically correct variance formula).

A series of data checks were built into the program to remove any item with
data that resulted in calculations of a negative Z, a leadtime demand of zero or
less or D (mean quarterly demands) that were equal to zero. The last check was
done to prevent division by zero when using the OPTION equation.

The values of the three variances were tabulated in a series of output files.
The output files were then divided into specific categories of demand for several
reasons. It was important to reduce the size of the comparison groups to make
data analysis easier in GRAFSTAT. When the data set is too large, the graphic
output exceeds its capacity. Another reason is that the ICPs use certain mean
quarterly demand values as a criteria for determining underlying probability
distributions for demand during net leadtime. It was also considered important
to separate the high demand items from the lower demand items since they are
managed more intensely. A series of demand groupings were therefore defined.
Costs associated with a stock-out are higher if the safety levels for these high
demand items are inaccurate.

The "Low Low Demand" items had mean quarterly demands of less than one
unit. The "Medium Low Demand” items had mean quarterly demands equal to
or greater than 1 but less than 2 units. Items with mean demands equal to or
greater than 2 but less than 5 were grouped into the "High Low Demand’
category. The "Medium Demand’ category contained items with mean demands
equal to or greater than 5 but less than 20 and the "High Demand’ items were
those with mean quarterly demands of 20 units or more.

In addition to V1, PVAR, OPTION, and V, the output files contained an
identification number for a specific NIIN (I), the PPV (Z) value and various other
data elements. Finally, the standard deviation or square root of each variance

(except V because this was not in the comparison) and the ratio (V/Z) were

14

included. This ratio was used to look at how many of the samples exceeded the
variance to mean ratio parameter at SPCC of 150.

The output files were input to an NPS mainframe statistical analysis package,
GRAFSTAT, for graphical analysis. The output from this package did not
integrate well into a microcomputer word processor and thus was used only to
find trends between the variance calculations. Once trends were observed, the
original FORTRAN program was modified to produce summary data of the
results. These results were then fed into a microcomputer. Using the
microcomputer and Harvard Graphics, graphs of the summary data were then

prepared and imported to WordPerfect 5.0 for use in this thesis.

C. DATA FILE OUTPUT

A total of 1,261 items (53.8%) passed through all the data checks. A cursory
look at the items not passing the check showed that most of the items had mean
demands that were less than one per quarter and many of the data elements had
zero values. A large majority of these items were identified as new items
because they were coded with Navy Item Code Numbers (temporary NICN’s
appeared instead of NIIN’s) for which little or no historical data was available.
Most of these items should have been screened from the data set during initial
download at SPCC, but were not because of the presence of zeros in the data
fields instead of blanks. It could not be determined why the zeros were entered
in the DENs. However, zeros allowed them to pass through the initial screening
but then caused them to fail the final data checks built into the calculation
programs. In addition, some of these items were identified as having gone
through a Cognizant Activity change (i.e., COG migration) which is a change of
activity responsible for the supply management of the particular item or
reclassification from an item having been identified by a NICN to an item which
is now identified by a NIIN. This would cause a "disconnect" between data on
the IHF (Inventory History File) which was associated with a NICN and the
same item on the SIG (Selective Item Generator) File which is now identified by
a NIIN. This normally would have produced blanks and would have been

screened out initially but the presence of unexplainable zeros prevented it.

15

IV. RESULTS

The data was run through the different variance calculations and the results
were divided into demand groups as mentioned in Chapter III. Figure 1 shows
the distribution of the items among the different demand groups.

Number of Items per Category

800

600

400

200

Low Low Med Low High Low Medium High
Demand Category :

MM No.per dem. category

Figure 1 - Distribution of the 1261 items by quarterly
demand category.

A sample of the detailed output file for high demand items are presented in
Appendix C.

16

As can be seen from Figure 1, most of the items were in the low low demand

group. Those items that the ICP consider for intense management are in the

medium and high demand group. Even though they are only a-.small percent of

the total items in the sample. they reflect the relative percentages for the entire

population.

Figure 2 shows the percentage of items in each demand category that had a

reduction in variance values (over V) as a result of the PVAR calculations and

OPTION calculations.

% Decrease in Variance

_— ce,

ee ips
ZEEE:

Demand Group

Low Low MedLow HighLlow Medium

Ma PVAR Formula OPTION Formula

High Total

Figure 2 - Percent of items within each demand group
that showed a reduction of variance values by PVAR

amma, OPTION.

Note that in every demand category, the OPTION formula reduced the variance

by a larger percentage than did the PVAR formula.

The main reason for this is

that a large number of items, when using the PVAR formula, had demand-

regeneration covariances equal to zero. ‘This was caused by regeneration data

17

equal to zero. This did not occur when using the OPTION formula because it
used the mean regeneration value that was on file while PVAR used the raw
data to calculate mean regeneration. This suggests that mean regeneration
values are being calculated at SPCC by some other method and not from data on
file. The investigation of this point is left for further study.

As can be seen in Figure 2, for the high demand items, PVAR reduced the
variance for only 57.7% of the items. This was the lowest improvement shown by
PVAR. Those items that did not have their variances reduced, fell into two
categories. They were either items that had covariances equal to zero (in the
PVAR formula) because of regeneration data equal to zero or the variance to
mean ratios (as shown by V/Z) were greater than the variance to mean ratio
check parameter of 150. In the latter case, V1 was computed using the power
rule while PVAR was calculated as programmed (the use of the power rule in
calculating V1 was verified by hand).

Table I shows typical items in these categories. Item number 313 had a zero
covariance when PVAR was used to calculate variance. Item number 1287 had

V1 recomputed using the power rule. Finally, item number 560 fell into both

categories.
TABLE I
ITEM Z Vi PVAR V ratio
BLS 96.75 2671.48 a) lea Dial 59.40
1287 93.43 AAD? 21 15911.4 15922.9 170.43
560 99.08 4824.57 15793.5 15793.5 159.40

A small quantity of the items (7 items with high demand and 14 total) from
the output file had variance to mean ratios greater than 150 (V/Z > 150). Of
these 14 items 4 had PVAR values that would not have passed the variance to
mean ratio check. This suggests, in this particular case, the cut-off parameter of
150 may be too severe. If this situation is true then not enough safety stock is

being held to meet the required protection level.

18

For the rest of the demand categomes, Figure 2 shows that PVAR is only
slightly less effective than OPTION, in reducing the variances of the sample.
The main reason given that SPCC has not used the OPTION formula is that it
calculates variances which have been shown to be too small to provide sufficient
safety stock. If PVAR were implemented, then quite possibly the same would
hold true. However, the discussions so far have been limited to the number of
items for which variances were reduced, not the degree of reduction. To
determine the degree of reduction, the differences in standard deviations (square
root of the variances) were plotted for all items where V1 (the variance on file)
was less than PVAR and PVAR was less than OPTION. From the plots,
summary data was gathered and is shown in histogram form in Figure 3. This
figure accounts for 94% of the items sampled. The other 6% of the sample that
is not included are items where the PVAR formula calculated a zero covariance or
where the variance to mean ratio exceeded 150. These items were discussed
above. |

In Figure 3 the data is grouped by the difference in number of items. Option
shows a decrease in standard deviation over PVAR by a median value of 3.
PVAR shows a decrease in standard deviation over V1 by a median value of 1.
From the difference between V1 and PVAR it appears that for the same level of
protection, ‘on the average’, less safety stock would be required if the PVAR
formula was used. From the differences between OPTION and PVAR, the
OPTION formula, “on the average", provides even less safety stock than PVAR.

According to SPCC, the OPTION formula is not used because it reduces the
variance of net leadtime demand too much and thus does not provide enough
safety stock. On the other hand the variances on file (V1) are apparently too
large and have been the object of a number of studies and program modifications
to reduce their values. The PVAR formula, as presented in this thesis, reduces
the variance, as compared to V1. However, PVAR does not reduce it to the level
of OPTION. Thus, PVAR might be the solution to this dilemma.

19

% of total in each group

Median of 8.d.PVAR-s.d.OPTION = 3

Median of s.d Vi- 8.d.PVAR = 1

difference in std. dev.

[ s.d.PVAR-s.d.Option s.d.V1-s.d.PVAR

a

Figure 3 - Difference in standard deviations between
Vaiiancecewermul ac.

V. SUMMARY, CONCLUSIONS
AND RECOMMENDATIONS

A. SUMMARY

This thesis has compared different formulae that are or could be used to
calculate the variances of the net leadtime demand for SPCC managed repairable
items. PVAR, the theoretically correct variance formula, was derived directly
calculate the covariance between quarterly demands and quarterly regenerations.
The differences between the OPTION formula (documented in UICP A/O DOI),
the PVAR formula (derived in Chapter II), and the variance formula used to
compute the current values listed on SPCC’s data files were discussed. The
variance values listed on SPCC’s data files were then compared with the variance
values calculated ‘by both the OPTION formula and the PVAR formula. Finally,
an analysis of the results from the comparisons of the different variance formula

was presented.

B. CONCLUSIONS

It is a well known fact that a large variance in net leadtime demand resident
on SPCC’s file can result in unusually large safety stock. In the past, various
changes to the UICP programs have been implemented which reduce the variance
to acceptable levels to prevent large sums of money from being tied up in possibly
unused and unnecessary safety stock. The current procedure is to make a
variance to mean ratio check and to recalculate the variance of net leadtime
demand if it exceeds a predetermined threshold. An alternative available
estimates the covariance factor and uses the option path for computing the
variance. This approach was designed to reduce the variance to acceptable levels
by accounting for covariance between the dependent variables of demand and
regeneration. The option path, if it were used, apparently underestimates the
variance of net leadtime demand and would excessively reduce the amount of

safety stock required. While this would reduce, considerably, the amount of

dollars necessary to procure and maintain the safety level, it could also reduce
the levels of operational availability of various weapon systems by not providing
enough safety stock.

The PVAR model, when used with complete and current data, reduces the
variance of over 95% of the repairable items sampled. It also does not estimate
the covariance of regeneration and demand, but calculates it directly and thus
gives a more theoretically correct variance output. In addition, it does not reduce
the variances to the levels calculated by the OPTION formula. By using the
PVAR model, SPCC could reduce the amount of money tied up in unnecessary
safety stock for those items which had large variances on file and redistribute
some of the money to items which may require, for what ever reason, an increase
in protection level G.e., more safety stock). This would possibly allow an increase
in operational availability of weapon systems and at the same time could reduce
the amount of money necessary for spares support. It would allow the ICP to do

its job cheaper and smarter.

C. RECOMMENDATIONS

The results indicated here, should not imply that the PVAR model is a
panacea for a restrictive funding environment. The model should be thoroughly
tested and verified through simulation and under actual operating conditions
prior to any consideration being given to incorporating it into levels setting.

In particular, PVAR should continue to be tested using data obtained from
ASO to see if similar results (as obtained in this study) apply to aviation
material. In addition, simulation and actual field testing of PVAR should be done
to see if the variance values that are calculated by PVAR provide for enough
safety stock.

FMSO has recently completed a new Functional Description (PD-80) [Ref 13]
for a UICP program system design to forecast leadtime and repair turn-around
time. The documentation and program are to be incorporated into the software
modifications being made as part of the ICP modernization project. The
procedures described in PD-80 include many significant improvements over D0O1,

but the basic formula for calculating the variance of demand over net leadtime is

22

still similar to equation (5) in Chapter Il. When the PVAR model passes testing,
corrections can easily be made to PD-80 and then implemented without delay.
By correcting only the variance formula, and maintaining the other significant
improvements of PD-80, the ICPs would not only operate more economically but

also provide the necessary spares support for the fleet.

D. RELATED FURTHER STUDY

Further study should be directed toward the policy governing the use of the
power rules for estimating the mean absolute deviation of the components of the
variance formula. If the reason for estimating these MADs is due to lack of data,
then this lack of data needs to be investigated as well. Blank data fields were
screened out of this study. These blanks will affect the new Levels program (PD-
80) that does not use MADs but instead calculates directly the variances of the

individual components of the formula for variance of net leadtime demand.

23

LIST OF REFERENCES

a ogo ne Naval Supply Systems Command, Command Presentation for
2. Tersine, Richard J. Princtples of Inveutory and Matertals Managenient, 2d
ed., pg 126, North-Holland, 1982.

3. Navy Fleet. Materia! Sapa Office, ALRAND Worlung Memorandum 292,
Calculation of Procurement Problem Variance, 2 March 1977.

4. Navy Fleet Material Support Office, System Design Documentation (FD-
DO1), Levels, appendix O, by Richard S. Jackson, 31 March 1984.

Oo. Ships Parts Control Center letter 790C/EE/140_ to Commander, Naval Sung
Systeins Command, Subj: “Lnplementation of Variance-to-Mean Ratio Check in
UICP A/O DO1", 9 February 1977.

6. Navy Fleet Material Support Office, ACLRAND Working Memorandum 357,
Update to Power Rule Parameters, 30: May 1980.

7. — Interview with Mr. John Boyarski, Operations Analysis Division, SPCC,
Mechaniesbure, PA, 16 November 1987.

8. Navy Fleet Material Support Ofitcee, AL RAND Working Memorandum 535,
Update to Porweer Rule Parameters, 15 November 1987.

OF oe Morris UL. Probability aud Statistics, 2™ ed., pg 604, Addison-
Wesley, 1986

10. Navy Fleet) Material Support Office, ALRAND Report 50B, Stattstical

framing Manual, Vol Lil, pe V22 by 13. 110 Bigs miei te cnet

It. Navy Repairables Management Manual ONAVMATINST 4400.14B), pg V-12,
17 Peluso .

12.) Duncan, Acheson J. Qualtty Control and Industrial Statistics, Sth ed., pg
201, Irwin, LOXG.

13. Navy Fleet Material Ce a Office, Uniform Inventory Control Prograni
Systent Desien Documentation Forecastug LUEITAT Requtrements Model, pg M-43,
December 1987.

APPENDIA A

DEN/Nomenclature of Data Elements

AQ19 - Observed Quarterly Demand MAD
BO11B - Procurement Leadtime
AO19B - Observed Quarterly Careass Return MAD
BOILA - Procurement Leadtinie Forecast .
BOI2ZF - Average Procurement Turn-Around Time for repair
BO12B - Average Carcass Return Rate
BOLSA SVariince om Pile at SPCC
BOZ3C - Demand over Procurement Leadtime |
BO23I]5 - Regenerations over Procurement Leadtime.
BO23G - Regenerations during repair turn-around time
BO74 - Average Quarterly Svsten: Demand Forecast
BO32C - Observed Leadtune Demanc
3074 A . Neuen RET Regenerations

C001 /CO02 - National Item Identification Number

C003 - Cognizant Activity

C005 - Unit of Issue |

F020 - FO20G - Depot completions reported for the last 8 quarters
OO - Repair Survival Rate

FOO9A - Repair Survival Rate MAD

HOM HOMA HOLIC

- HO2Z1, HOZIA ,HOZLC- Total quarterly demand reported for the last 8 quarters

APPENDIX B
FORTRAN PROGRAM FOR VARIANCE CALCULATION

tedededededetedesetedetoletedste dv tedtedetetetsieseheteteteioteteiedciohdicekiceiddssdcedicichiciviciciciggihkecledvivietvdetcdlRdict
*THIS PROGRAM READS THE DATA FROM THE DATA FILE "STGIHF" AND PUTS IT %*
“IN COLUMN VECTORS FOR FURTHER ANALYSIS. THE DATA SET IS IN CHARACTER *

“FORMAT WITH A LRCL = 236. THE OUTPUT IS “RATIODAT LISTING A’ ¥
*COMPILE THE PROGRAM USING FORTVS AND USE THESIS EXEC TO RUN ¥
*THE VARIABLES ARE: %

CNIIN - NATIONAL ITEM IDENTIFICATION NUMBER (CO01E/C002) ¥
* COG - COGNIZANT ACTIVITY (C003) *
* DEMMAD - OBSERVED DEMAND MAD (A019) ve
* PLTMAD - PROCUREMENT LEADTIME MAD (BO11B) ¥
* PLTFC - PROCUREMENT LEADTIME FORCAST (BO0114) ve
* DEM - AVERAGE QUARTERLY SYSTEM DEMAND FORCAST (B074) ve
* LTDEM - OBSERVED LEADTIME DEMAND (BO023C) ve
** RATIO1 - VARIANCE TO MEAN RATIO FROM FILE (V1/PPV) 7%
* RATIO2 - NEW VARIANCE TO MEAN RATIO CALCULATED (PVAR/PPV) ¥
* RATIO3 - VARIANCE TO MEAN RATIO CALC FROM (OPTION/PPV) ¥
** RATIOS - VARIANCE TO MEAN RATIO WITHOUT COVARIANCE DO1 (V/PPV) ve

RATDIF - DIFFERENCE BETWEEN CALCULATED VAR/MEAN AND FILE VAR/MEAN *
** CRMAD - OBSERVED CARCASS RETURN MAD (A019B) ¥
* PTAT - AVERAGE PROCUREMENT TURN-AROUND TIME FOR REPAIR (B012F) ve
* NTTMAD - NAVY (NON-REPROTING) REPAIR TURN-AROUND TIME (B012B) ve
** AVGCR - AVERAGE CARCASS RETURN RATE (BO22B) ve
* LREGEN - RFI REGENERATIONS DURING LEADTIME (BO23E) ¥

** TREGEN - RFI REGENERATIONS DURING PTAT (B023G) ‘
** QREGEN - QUARTERLY RFI REGENERATIONS (BO74A) zs

* RSRMAD - REPAIR SURVIVAL RATE MAD (FOO9A) *
* QTRIRP THRU QTR8RP - DEPOT COMPLETIONS REPORTED FOR THE Gas? S Gis
2 (PO20 THRU BO ZveT x
* RSR - REPAIR SURVIVAL RATE (F009) we
% CUl= Nee ISSUE Cele *
* QTRIDM THRU QTR8DM - TOTAL QTRLY DEMAND REPORTED FOR THE LAST 8 QTR *
= (HO14+HO14A+HO14C THRU HO21+HO21A+H021C) *
“© OPTION - CALCULATED VARIANCE BY THIS PROGRAHN WITH COVARIANCE COV1 "
* V1 - VARIANCE OF PPV ON SPCC'S FILE(BO19A) *
ve 6 Vo = VARIANCE FROM DO1 WITH OUT COVARIANCE cf
* COV1 - EST OF COVARIANCE FACTOR USED AT THE ICPS %
** =COV - COVARIANCE FACTOR CALCULATED BY EXPECTED VALUES %
* PPV - PROCUREMENT PROBLEH YARIABLE (BO23¢G=202 02 5G) ¥
* PVAR - CALCULATED PROCUREMENT PROBLEM VARIANCE WITH COVARIANCE COV *
* BDATA - COUNTER FOR BAD DATA WHICH WILL NOT BE USED IN ANALYSIS *
%* GDATA - COUNTER FOR GOOD DATA WHICH WILL BE USED IN ANALYSIS ¥

* POSDIF - COUNTER FOR POSITIVE IMPROVENENT IN VARIANCE WITH PROGRAM *
* NEGDIF - COUNTER FOR NEGATIVE IMPROVEMENT IN VARIANCE WITH PROGRAM *

*% UNCHNG - TOTAL QTY OF NIINS WITH VARIANCE UNCHANGED BY PROGRAM *
* VDIF - DIFFERENCE BETWEEN VARIANCE ON FILE AND CALCULATED VARIANCE *
* COUNT1-5 - COUNTER FOR VAR EXCEEDING SPCC PARAMETER FOR RATIO *
* DELTA - DIFFERENCE IN STANDARD DEVIATION %
* NUM - NUMBER OF ITEMS USED 1Os@Ake Ui ice: %
*% J - SETS THE NUMBER OF DATA LINES (NIINS) TO BE READ/USED ¥

26

eee - oelG PRESET PARAMETER

* § - CONSTANT FOR THE ESTIMATION OF VARIANCE FROM THE MAD

low -SIPeiive Cr Vi
pee vARSD - STD DEV OF PVAR
feb) = STD DEV OF OPTION

Pees. = DIFFERENCE eee ee AND OPTION S.D. ( PVARSD- ON SIe

Seseseveves esksestrledevestcvesesevestedevestestevesistc ve

* DECLARE VARIABLES ,
PARAMETER (J=2345,

s\e stes'e 77 ves'e

ses tose slcs'es tesles's \s

svsevesicsles

sevesicsiesicsles'c

sesvvesesveseseseseveses

‘SET PARAMETERS, DIMENSION ARRAYS
= eZee 150 )

Poe deed yy ORMMADC J), DEM(J), LIDEM(J), V1iCJ), CRMAD(CJ),
CPTATI( J) ,NITMAD( J), AVGCR(J), LREGEN( J), TREGEN( J), QREGEN(J),

CRSRMAD(J), RSR(J),
CRATIO2(J), RATIO3(J),

PYani@ee, COV (J),
AB TUE (rs es

PPV(J),
RATDIF(J), COV1(J),

NCJ), RATION gy
RATDEL(J)

C,RATIO4(J), OPTION(J), DELTA(J), TOTDEL,V1SD(J) ,PVARSD(J),
COPSD( J) ,DIFF(J)
INTEGER PLTMAD(J), QTRIRP(J), QTR2RP(J), NEGDIF, POSDIF, NEGDEL

Ceeeeowrt, UNDEL, COUNTI,

COUNT2,

COUNT3, COUNT4, COUNTS,

OUirerro?), QPRakP( J), OTRSRP( J), QTRORP(J), QTR7RP(J), UNCHNG,
OtEZUM J) ,
CQTR5DM( J), QTR6DM(J), QTR/DM(C J), QTR8DM( J), BDATA, GDATA,

COTR8RP(J), OQTRIDN(J),

erOrr 1 ,

Pee Lis

NTE i,

NDELI1,

OTR). OTRSDMCT) ,

ee Nee. POE 2 NDELZ ,

POLE.

eee eee, NDELS, PDIf4, NDIF4, PDEL4, NDEL4, PDIF5, NDIF5,

Pee. NUE, NUM ,DEL,ONEF ,ONEL, IWOF ,TWOL, THREEF , THREEL,
evi! VPI VEL, TEN ,TENL,GIEN?F ,GIENL
CHARACTER*9 CNIINCJ)

CHARACTER*2 UIC J) ,COG(J)

BDATA=0
GDATA=0

NEGDIF=
POSDIF=

PDIF1=0

0
0

TOTDEL=0. 0

27

9’ ?
a
¢

ay

ils:

NUM=0
DEL=0
ONEF=0
ONEL=0
TWOF=0
TWOL=0
THREEF=0
THREEL=0
FIVEF=0
FIVEL=0
TENF=0
TENL=0
GTENF=0
GTENL=0

WRITECGR 3 — | ae' V1 ',' PVAR ',* OPTION
Ce V es Z i) V1 SD ',' PVAR SD ',
C’ OPTION sie Vi

READ DATA FILE AND CREATE DATA VECTORS

DO 10) ieee
READ (1,15) CNIIN(I), COG(I), DEMMAD(I), PLTMAD(I), PLTFC(I),
DEM(I), LTDEM(I), V1(1), CRMAD(I), PTAT(I), NTTMAD(I),
AVGCR(1), LREGEN(I), TREGEN(I), QREGEN(1I), RSRMAD(I),
QTRIRP(I), QTR2RP(I), QTR3RP(1), QTR4RP(I), QTRSRP(I),
QTR6RP(I), QTR7RP(I), QTR8RP(I), RSR(I), UI(I),
QTRIDM(I), QTR2DM(I), QTR3DM(I), QTR4DM(I), QTRSDM(I),
QTR6DM( I), QTR7DM(I), QTR8DM(I)
FORMAT (A9, A2, F10.4, 13, 2(F9.2), F10.2, 2(F10.4), F4.2,
F3,1, Fl10.2, 2(F9.1), F9.2, F3.2, 8(15), F3.2, AZ. S@leue

Be 5 Fl SP 1 a Jy oa 1 ea a og

* CALCULATE COV, COV1, V, PPV AND PVAR

COV(I)= (((QTRIDM(1I)*QTR1IRP( 1) )+(QTR2DM( I )*QTR2RP(I))
C+( QTR3DM(1)*QTR3RP( 1) )+( QTR4DM( 1)*QTR4RP( 1) )+( QTRSDM( 1)
C*QTR5RP( 1) )+(QTR6DM( I )**QTR6RP( I) )+( QTR7DM( I )*QTR7RP(I))
C+( QTR8DM( I )**QTR8RP( 1) ))/8)-C( (QTRIDM( I )+QTR2DM( 1)+QTR3DM( I)
C+QTR4DM( 1)+QTR5DM(1)+QTR6ODM(1)+QTR7DM(I)+QTR8DM(1))/8)
C**( (QTRIRP( I)+QTR2RP( 1)+QTR3RP( 1)+QTR4RP( 1)+QTRSRP(1)+
CQTR6RP( I)+QTR7RP(1)+QTR8RP(1))/8))

IF(DEM( 1). LE.0) THEN
BDATA=BDATA + 1
GOTO 10

END

COVICTI)= CCRSRCI))*CAVGCRCI))*( (S*DEMMAD( 1) )**2)) /DEM(C I)

VCI)= (CPLTFCC(I)-PTATCI))*( CC S*DEMMAD(I))**2) + CRSRCI)**2)*
CCC S*CRMAD( 1) )**2)+( AVGCR(I)**2)*((S*RSRMAD( I) )**2) +
CCC S*CRMADC I) )**2)*( (S*RSRMAD(I))**2)) + (PTATCI)*((S*DEMMAD( 1) )**2
C))+ CCDEMCI)**2)%*( (S*NTIMADCI))**2)) + ((DEMCI) -QREGEN( I) )**2)*
CCC CS*PLTMAD( I) )**2)+¢C CS*NTTMAD( I) )****2))

OPTION(I)= ( PLTFC(1)-PTAT(1))*(((S*DEMMAD(1))**2) + (RSR(1)**2)*

28

ees Oh lee 2+ AVGCR( 1)*-2 )*( (S*RSRMAD( 1) )**2) - 2*COVICL) +
Ses enh (ie 2 )*((S*RSRMAD(1))**2)) + (PTATCI)*((S*DEMMAD( I) )**2
Cy) CODEM( 1) *2)*( (S*NITMADC I) )**2)) + (CDEM( I) -QREGEN( 1) )**2)*
CGC Ge een) =? )+( (S*NTIMAD( 1) )**2))

PVAR(I)= (PLTFC(1)-PTAT( I) )*( ((S*DEMMAD(I))**2) + (RSR(1)%*2)*
6(( S*#CRMAD( 1) )*"2)+( AVGCR( 1)7*2)e*( (S*RSRMAD(1))**2) - 2*COV(I) +
C(( S*CRMAD( 1) )2**2)**( (S*RSRMAD(1))**2)) + (PTAT(1)*((S*DEMMAD(1))**2
C))+ ((QREGEN( I )***2)*( (S*NTTMAD(1))***2)) + (((DEM(1)-QREGEN( I) )**
C2)*((S*PLTMAD( 1) )***2))

PPV( 1I)=LTDEM( 1) -LREGEN( I)+TREGEN( I)
* DATA CHECK AND SCRUB FOR BAD OR ERRONEOUS DATA ELEMENTS

PCE COE Gia me be. 0}, THEN
BDATA = BDATA + 1
GO TO 10

tpi VAR( 1). bi. 07 OR, PPV(1). LT. 0.OR. V(1).LT.0) THEN
BDATA = BDATA + 1
ce 10 10

ELSE IF (V1(1). LT. 0. OR. OPTION(1I).LT.0) THEN
BDATA = BDATA + 1
GO TO 10

END IF
GDATA = GDATA + 1

* CALCULATE VARIANCE TO MEAN RATIOS
RATIOI(1)=V1(1)/PPV(1)
RATIO2( 1)=PVAR(1)/PPV(I)
RATIO3(1)=OPTION(1)/PPV(1)
RATIO4(1)=V(1)/PPV(I)
RATDEL(I) = RATIOI(I) - RATIO3(1)
RAMDIE( 1) = RATIO1(1) - RATIO2(1)
IF (RATDIF(1).LT.0.) THEN
NEGDIF = NEGDIF + 1
ELSE IF(RATDIF(1).GT. 0.) THEN
POSDIF = POSDIF + 1
END IF

Wee RSeOEn( 1). LT. 0.) THEN
NEGDEL = NEGDEL + 1

Pion ir CRATDEL( 1).GT. 0.) THEN
POSDEL = POSBEL + i

ENDe IF

PaGe sib avr) ). LT. 0) THEN
DEL=DEL+1
END ar

* CALCULATE STANDARD DEVIATION

AG.

V1ISD(I)=V1(1)**. 5
PVARSD( I1)=PVAR(1)***. 5
OPSD( 1)=OPTION(1)***. 5

* REPORT WRITER AND DATA OUTPUT

* DATA OUTPUT FOR IMPROVEMENT CALCULATION
IF(PVAR( 1). GT. V1(1). OR. OPTION(I).GT. PVAR(I)) THEN
GO TO 100

END IF

DIFF(I)=PVARSD(1)-OPSD(T)

DELTA(I)=(V1(1)**. 5)-( PVARCI)**. 5)

TOTDEL=TOTDEL + DELTA(T)

NUM=NUM+1

WRITE(3,95) I, DELTA(1),DIFF(I)

95 FORMAT(' -' ,I15,2X, ‘DELTA = ‘,F10.3,2X, DIFF = ‘,F10.3)

IF(DIFF(I). LE. 1) THEN
ONEF=ONEF+1

ELSE IF(DIFF(1). LE. 2. AND. DIFF(I).GT. 1) THEN
TWOF=TWOF+1

ELSE IF(DIFF(I). LE. 3. AND. DIFF(I). GT. 2) THEN
THREEF=THREEF+1

ELSE IF(DIFF(I). LE. 5. AND. DIFF(I). GT. 3) THEN
FIVEF=FIVEF+1

ELSE IF(DIFF(i). LE. 10. AND. DIFF(I).GT.5) THEN
TENF=TENF+1

ELSE IF(DIFF(1I).GT.10) THEN
GTENF=GTENF+1

END IF

IF(DELTA(I). LE. 1) THEN
ONEL=ONEL+1

ELSE IF(DELTA(1). LE. 2. AND. DELTA( I). GT. 1) THEN
TWOL=TWOL+1

ELSE IF(DELTA(1). LE. 3. AND. DELTA( I). GT. 2) THEN
THREEL=THREEL+1

ELSE IF(DELTA(I). LE. 5. AND. DELTA( I). GT. 3) THEN
FIVEL=FIVEL+1

ELSE IF(DELTA(I). LE. 10. AND. DELTA(1).GT.5) THEN
TENL=TENL+1

ELSE IF(DELTA(I).GT.10) THEN
GTENL=GTENL+1

END IF

* SPLIT DATA INTO LLOW, MLOW, HLOW, MED AND HIGH DEM ITEMS FOR ANALYSIS

100 LES CDEMC HT) Li. 1) THEN

GO TO 114

ELSE IF (DEM(1). LT. 2. AND. DEMC1 ). GE=d) Sie
GO TO 154

ELSE IF (DEM(1). LT. 5. AND) DEMC1! ) Giz ina
GO TO 164

ELSE IF (DEM(1). LT. 20. AND. DEM( 1). GE. 5) aaa
GO TO 124

ELSE IP (CBEMC 1). GE. 20) SEN
GO TO 134

END IF

51)

*LOW LOW DEMAND OUTPUT

114 Pownce LoL. 0. ) THEN
NDIF1 = NDIF1 + 1
ELSE IF(RATDIF(I).GE. 0.) THEN
PUTRI = POIF1 + 1
END IF

PoeCRet DE bCi). Ll. 0. jimainiN
NDEL1 = NDEL1 + 1

EUSE DTFCRATDEL( 1). GE. 0. ) THEN
PDEL1 = PDEL1 + 1

BND IF

IF(RATIO4(1).GE.P) THEN
COUNT1 = COUNT1 + 1
END IF

WRITE(10,115) I, V1(1), PVARCI), OPTION(I), V(I), PPV(I),
CDIFF(I),V1SD(I), PVARSD(I), OPSD(I), RATIO4(I)
115 FORMAT ('-' ,15,9(F10. 3),F10. 3)
So) “dol ae

*MED LOW DEMAND FILE OUTPUT

154 IF (RATDIF(1I). LT. 0.) THEN
NE? = NDIFZ +t
bok LPCRATDIE( 1).GE. 0.) THEN
Pte 2 = RbIk2 +1
RIND eke

Pe GRADER LC Toh, 0.) THEN
NDEL2 = NDEL2 + 1

Por UCR TDEL( 1),CGE.0. ) THEN
POEL2 = PDELZ s+ 1

END ear

PiCRAT LOA jeGE. P) THEN
COUNT2 = COUNT2 + 1
END IF

WRITE(11,155) I, V1(1), PVAR(I), OPTION(1I), V(1I), PPV(I),
CDIFF(I),V1SD(I), PVARSD(I), OPSD(I), RATIO4(1)
155 FORMAT ('-' ,15,9(F10. 3),F10. 3)

GO TO 10

*HIGH LOW DEMAND OUTPUT

164 IF (RATDIF(1I).LT.0.) THEN
NPWESe— NDIF3 +1
Poobelra@rcsdpih(1).GE.0.) THEN
Polror— PDIFS + 1
END IF

Balk

IF (RAIDER 1). LT.O. ) THEN
NDEL3 = NDEL3 + 1

ELSE IFCRATDEL(I).GE.0. ) THEN
PDEL3 = PDEL3 + 1

ENDeah

IFCRATIOS(1).GE.P) THEN
COUNT3 = COUNT3 + 1
oN) Ble he

WRITE( 7,165) I, V1C1), PVARC1), OPTIONC Iie eee Cle
CDIFF(1I),V1SD(1), PVARSD(I), OPSD(I), RATIO4(I)
165 FORMAT (=o, 9( F10.s) ems)

GO TO 10
*MED DEMAND OUTPUT

124 IF CRAQDIF( 1). LT. 0. ) GaEN
NDIF4 = NDIF4 + 1
ELSE IFCRATDIF(1).GE.0. ) THEN
PDIF4 = PDIF4 + 1
END ek

IF (RATDEL(1 ). LT. GepeiEN
NDEL4 = NDEL4 + 1

ELSE TFCRATDER 1) Che” ji THEN
PDEL4 = PDEL4 + 1

END IF

IFCRATIO4(1).GE.P) THEN
COUNTS = COUNT4 + 1
ENE

WRITE(8,125) I, V1(1), PVARC1), OPTIONGD Sev Gl Sere Gn
CDIFF(1),V1iSD(I), PVARSD(I), OPSD(I), RATIO4(1)
125 FORMAT ( = 915, 9¢P1G.c er teen

GOERS
*HIGH DEMAND OUTPUT

134 IF (RATDIF(1).LT.0.) THEN
NDIS = NOUPSs ee
ELSE JFCRATDIFC] GE, OC”) Hien
EDIEe = Dis + 1
END IF

IF (RATDEL(I).LT.0.) THEN
NDELS = NDELS + 1

ELSE IF(RATDEL(1).GT. 0.) THEN
PDEL5 = PDELS + 1

END IF

IFCRATIO4(1).GE.P) THEN
COUNTS = COUNTS + 1

32

END IF

WRITE(9,135) I, V1(1I), PVAR(I), OPTIONCI), VC1I), PPV(I),
CDIFF(I),V1SDC(I), PVARSD(I), OPSD(I),RATIO4(1)
ie FORMAT ( = ,15,9(F10.3),F10. 3)

10 CONTINUE

UNCHNG = I - (POSDIF + NEGDIF)
Nites 1 = (POSDEL + NEGDEL)

*TOTAL SUMMARY DATA OUTPUT

WRITE (3,145) BDATA, GDATA, NEGDIF, POSDIF, UNCHNG, NEGDEL, POSDEL
C, UNDEL, TOTDEL, NUM, DEL

145 FORMAT ('-'/'0O BDATA = ‘',1I5/'0 GDATA = ',I5/'ONEGDIF = ',15/
C'OPOSDIF = ',15/'OTOTAL VARIANCE UNCHANGED = ',15/'ONEGDEL = ',7
CI5/'OPOSDEL = ',I5/'OTOTAL VARIANCE UNCHANGED = ',15/
G'OTOTAL DELTA OF S.D. = ',F10.3/'ONUMBER OF ITEMS = ',I5/
C'ONUMBER OF ITEMS WHEN V<V1 = ',F10. 3)

WRITE(3,300) ONEF,TWOF , THREEF ,FIVEF,TENF ,GTEN,ONEL,TWOL, THREEL,
CFIVEL,TENL,GTENL

ogee FORMAT ('-'/ 12(15))

“SUMMARY DATA OUTPUT BY DEMAND

WRITE (3,215) NDIF1, PDIF1, NDEL1, PDEL1, COUNT1
WRITE (10,215) NDIF1, PDIF1, NDEL1, PDEL1, COUNT1

215 FORMAT ('-'/'OLOW DEMAND SAMPLES'/'ONDIF1 = ',I5/'OPDIF1 = ',15/
oer = .15/ OPDEL1 = ,15/'OCOUNT1 = ‘,I5)

WRITE (3,255) NDIF2, PDIF2, NDEL2, PDEL2, COUNT2
Meme (11,255) NDIF2, PDIF2, NDEL2, PDEL2, COUNT2

255 FORMAT ('-'/'OMED LOW DEMAND SAMPLE'/'ONDIF2 = ',15/'OPDIF2 = ',I5
own? = —§15/ OPDEL2 = ,15/'OCOUNT2 = ',I5)

WRITE (3,265) NDIF3, PDIF3, NDEL3, PDEL3,COUNT3
WRITE (7,265) NDIF3, PDIF3, NDEL3, PDEL3, COUNT3

265 FORMAT ('-'/'OHIGH LOW DEM SAMPLES'/'ONDIF3 = ',15/'OPDIF3 = ',IS/
eee) = 9l5/ OPDELS = ',15/ OCOUNT3 = ',I5)

WRITE (3,225) NDIF4, PDIF4, NDEL4, PDEL4, COUNT4
WRITE (8,225) NDIF4, PDIF4, NDEL4, PDEL4,COUNT4

225 FORMAT (‘'-'/'OMEDIUM DEMAND SAMPLES'/'ONDIF4 = ',15/'OPDIF4 = ',I5
me Ege —) .15/ OPDEL4 = ,15/' OCOUNT4 = ' ,15)

WRITE (3,235) NDIFS, PDIF5, NDELS, PDELS, COUNTS
WRITE (9,235) NDIFS5, PDIF5, NDELS, PDELS, COUNTS

235 FORMAT ( -'/'OHIGH DEMAND SAMPLES'/'ONDIF5 = ',I5/'OPDIFS = ',15/
© eee —) 9157 OPDELS = ,15/ OCOUNTS = ',15)

STOP
END

52

APPENDIX C
DETAILED OUTPUT LISTING

I - Item Number
PVAR -. Vets colcul eee
- Variance calculate
OPTION - Variance calculated i OPTION
- Variance calculated by UICP formula
- Mean net leadtime demand (PPV)
4 sd - difference in between PVAR_ standard deviation and OPTION
standard deviation (PVAR s.d. - OPTION s.d.)
V1 sd - Square Root of V1 (standard deviation)
PVAR sd - Square Root of PVAR (standard deviation)

1

Square Root_of OPTION (standard deviation)

OPTION sd
V/Z Variance to Mean ratio

1

] V1 PVAR OPTION Vv Z 4 3D V15D PVAR SD OPTION SD V/Z

36 5023.379 1482.285 254.502 1482.285 105.620 ce25G7 70.876 38.500 15.9 so 14.034

97 13199.719 18595.754 2413.862 18595.756 193.650 0.000 114.890 136.366 49.131 96.028

113° 5271.066 2987.308 274.486 3106.028 64.920 38.089 72.602 54.656 16.568 47.844

241 2942.332 6507, 905 465.541 650.705 717250 a. 7 36 54.245 2oeol 5 cl. 576 9.135

290 4100.773 2518.112 566.498 2589.786 51.450 26.380 64.037 50.181 23.801 50.336

292 6695.910 1497.289 728.998 1997-269 123.220 11.704 81.829 38.695 26.991 12.15)

S13 2671.982 S797. 102 SIS7 628657 aie 96077 50 0.000 51.686 75.810 “5. S20 59.402

387 29102.812 11955-1176 260779955 11455 e176 188.350 55.961 lp. col 107.029 a), 06's 60.819

560 482%. 566 157935.531 25783528 515795555) 99.080 0.000 69.459 125. Ga2 S07 779 159.402
879 - 369670 2I eco 2.097 926.285 350k? 82.890 26.950 60.795 37 wo 20.455 39.942
9120 -27E4.769 122469.891 1604.28) 1°351.262 68.710 0.000 Sear) 110.679 40.053 179.760

L123 Sesle.9357 S959 .59e 198) 2 e567 See eee eo 103.700 COL cNG ao 140.759 182.7

1286 11429.637 23712.079 2579.768 22867.633 Ty S710 0.000 106.910 153.987 SOR 179.996
WB? 44072, 21) Pool e76 (1715-97 Wo%ce. 93.4930 0.000 66.500 126.140 41.394 ° 170.426
134) 2603/7 3.969.300 bes0 26c6209e) 5497 2erU 306.980 7.146 162.401 58.381 by ere Lt ll .o3
1365 «67726.999 17683.113 193525038 21a2 ee eA 0.000 87.903 132.978 44.182 158 454
1366 19959. 168 601073.) 36787 055039976, 000 274, 750 0.000 141.277 Zoo. s0 196.949 392.170
1370 87616 .5002319494.875 3624.6%8269876 .062 729.660 0.000 296.001 481.087 euler [1 2) 369.865
1646592673 .500467730.000467309.250467730.000 6558.996 0.308 769.853 683.908 683.600 7h
1663 4712.086 12041.215 2381.762 12041.215 97 2 2eU 0.000 68.645 109.732 “8.805 123.474
1730 15199.637 22515255) S779 28 Fee 2751 Ss. 55) Clee eu 0.000 123.287 1G9 5377 69.137 104.896

1791)» =93207.099 1352.95] 373.564 1998.95] 60.%60 17.455 56.631 36.782 19.328 23.965
1890 15284.000 2287.788 587.166 2326.068 169.690 23.599 123.628 47.851 249 22 5c 13.708
189] 3$5834.844 30795.516 174955.398 37147 .898 438.280 43.368 189.501 175.487 132.119 84.758
1959 5019.367 4016.4]? 375.707 4163.137 65.600 a4 70.847 63.375 19.3835 63.462

2320 10893.074 436%.207 1129.218 4364.207 191.110 52.496 104.370 66.062 33.604 22.856

q1

10.

et.

Pere eer 1 OUTION LIST

No. of Copies

Defense Technical Infonmation Center Z
Cameron Station — |
Alexandria, Virginia 22314

pumech

Defense Logistics Studies Information Exchange
U. 8. Anny Logistics Management Center
Fort Lee, Virginia 23501

Library, Code 0142 Z
Naval Postgraduate School
Monterey, Calfornia 93943-5002

Professor Rohert R. Read

Code 55Re, Department of Operations Research
Naval Posteraduate School

Monterey, California 93943-5000

Professor Alan W. McMasters _ D
Code 54Mg, Departinent of Adiinistrative Sciences
Nava] Postgraduate School

Monterey, California 93943-5000

Commanding Officer an
Navy Fleet: Material Support Office

Attn: Code 93 (LCDR Ik. Adams)

Mechanicsburg, Pennsylvania 17055-0787

Commanding Officer 3
Naval Supply Svstems Conimmand

Attn: SUP 042 (CDR M. Mitchell

Washington, 1D. C. 20376-5000

Commanding Officer 3
Navy Ships Parts Control Center

Attn: Code 0-412

Mechanicsburg, Pennsylvamia = 17055

Commanding Officer _ 3
Navy Aviation Supply Ofce

Attn: Code SDB4-A

Philadelphia, Pennsylvania 19111

Professor Peter Purdue ]
Departinent. of Operations Research

Naval Postgraduate Schoo!

Monterey, California 93943-5000

Chief of Naval Operations 1
Navy Hepartment

Attn: OP31

Washington, DC 20350-2000

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