STOCKAGE POLICIES FOR MEDIUM AND LOW-COST PARTS

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP81-01043R002300090004-9
• Release Decision: RIPPUB
• Original Classification: K
• Document Page Count: 83
• Document Creation Date: December 27, 2016
• Document Release Date: October 25, 2013
• Sequence Number: 4
• Publication Date: April 18, 1958
• Content Type: MEMO

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,.
4
U.S. AIR FORCE
PROJECT RAND
RESEARCH MEMORANDUM
I
WOCKAGE POLICIES FOR MEDIUM AND LOW-COST PARTS
Allen R. Ferguson
Lawrence Fisher
RM-1962
18 April 1958
STAT
Assigned to
This is a working paper. It may be expanded, modified, or with-
drawn at any time. The views, conclusions, and recommendations
expressed herein do not necessarily reflect the official views or
policies of the United States Air Force.
STAT
74 P-fl n D emAstettitut
1700 MAIN ST. ? SANTA MONICA 6 CALIFORNIA
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This memorandum describes a set of practical, systematic procedures for estab-
lishing efficient base and depot stock levels for low priced Air Force parts,
Categories II and III. The use of such procedures should make it possible to
reduce the work load on supply personnel and to increase supply effectiveness
by cutting reorder requirements and supply shortages. The procedures can be
used in manual data processing at base level, and also with integrated data
processing and centrally controlled resupply.
Under the procedures for base stocking outlined in Air Force Supply Manual 67-1
the Order Quantity is based on demand rate and cost category; and the Reorder
Level is determined by taking into account demand rate and pipeline time.
The procedures outlined. here take into account several additional factors in
an effort to reduce overall provisioning costs. The factors include expected
mean demand for each item, variability of demand, unit value of item, vari-
ability of demand, cost of reorder, cost of holding Operating Stocks, cost
of obsolescence in event of program termination, expected shortage cost, and
resupply and procurement pipeline times.
Equations are developed for determining efficient stock levels, taking these
factors into account. Tables devised from these equations can be used by
clerical personnel to determine stock levels without complicated procedures
and decision processes; the equations may be used in an integrated data
processing system.
The suggested procedures have been compared with those of Supply Manual 67-1
through tests by regular Air Force supply personnel in the RAND Logistics
Laboratory. Results bear out the improved efficiency of the new practices
suggested here. (Actually, provisioning has for some time been undergoing
changes not reflected in "67-1" and to some extent already take into account
some of the features of the suggested system.)
Briefings on this study have been presented to AMC and to members of the Air
Staff during recent months.
741anD
e4tot4T4ti54(
1700 Main St. ? Santa Konica ? California
STAT
STAT
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SUMMARY
This Research Memorandum presents for Air Force consideration some sim?
ple and practical proposals for stocking ? that is, provisioning and distrib?
uting ? those items for which detailed intensive management is not appropriate,
for example, Cost Category II and III items. These policies are complementary
to deferred procurement policies for some Category / items.' While either
could be implemented alone, together they would provide increased supply ef?
fectiveness and economy through buying fewer costly parts and investing some
of the saving in base stocks of cheap items. They would entail more manage?
ment and closer control of the expensive parts, but decreased materiel move?
ment, paper processing and priority resupply of the low?cost parts.
The research underlying this Memorandum leads to the following con?
elusions:
A. Stockage rules should consider:
1. Expected mean demand for each item
2. Variability of demand
3. Unit value
4. Cost of incurring a reorder
5. Cost of holding the Operating Stocks
6. Cost of expected terminal obsolescence (termination of the pro?
gram being supported)
7. Expected shortage cost
8. Resupply and procurement pipeline times.
1
J. W. Petersen, Savings from Procurement Deferral with Interim Con?
tractor Support: The Case of High Value Airframe Spares, The RAND Corpor?
ation Research Memorandum RM-2085, 10 January 1958,
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B. The dynamics of weapon-system or other program phase-in or phase-out
can be taken into account effectively by
1. Limited depot stockage during the early part of a program, and
subsequently stocking the depot to its full Stock Control Level
2. Gathering and analyzing consumption data intensively early in
the phase-in and reacting to that information
3. Using a "final buy" calculation during the later stages of the
program and,for some of the least costly low-demand items early
in the program; and
46 Using a "terminal buy" calculation at the time when an item is
expected to go out of production.
C. With an integrated data processing system, these results can be
largely achieved by using the equations developed in the mathematical appall-.
dix. With a manual data processing system and local determination of levels,
tables based upon the formulas can be used by clerical personnel to set the
appropriate levels.
ID. The Air Force can increase supply effectiveness, decrease personnel
pressures in supply and achieve dollar economies by adopting the policies
described and proposed in this Memorandum.
E. Because the proposed policies permit reduced management per line
item, their use should free management to manage the more costly and critical
items batter, or, alternatively, it might permit reducing somewhat base-level
manning where (as in hardened missile installations) there is a premium on
personnel space.
F. Further research is needed at RAND to extend the scope of the study,
and further developmental studies are required, particularly in the Air Force,
to derive adequate estimates of cost and other parameters.
G. The rapid development of an integrated data processing system will
improve the application of these as well as other supply policies.
4.1
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This Memorandum is concerned with aircraft parts and other technical
items) not necessarily with quartermaster or bulk items. Inasmuch as INT
do not yet have formal solutions for certain of the reparable items) the
policies described cover nonreparables and, when applied to base stockage)
parts repaired at the depot. Some significant error would result if parts
repaired on the base were stocked at the base in accordance with these rules.
Similarly, for determining depot stock levels, parts repaired at the depot
should not be stocked as described below. Hence, this Memorandum applies
without modification to virtually all Category III items and to the non-
recoverable Category 11 items.
The study covers the question of how much to stock of line items for
which the decision has already been made that they be brought into the in-
ventory. It covers, then, the depth of stockage, but not the breadth or
range of items that should be stocked. This range problem is an important
one, on which work is under way.
In studying this problem, the aim has been to develop policies which
would result in near-minimum system cost and still be practical. The ap-
proach is to consider each item as independent of the others in the inven-
tory. First the base levels, then the depot levels are determined; for each)
the Order Quantity (Q), which establishes the amount of the Operating Stocks)
then the Reorder Point (R), which establishes the amount of the Safety and
Pipeline Stocks, are determined. Together Q plus RI of course, constitute
the Stock Control Level (SCL).
For any given annual demand rate, the size of Q fixes the average fre-
quency with which orders must be placed as well as the size of the Operating
Stocks. The more frequent the reorders, the greater the supply workload and
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the greater its cost. Since Category II and III parts account for as much as
99 percent of the depot-to-base issues, it is desirable to have infrequent
orders for each line item. This requires a relatively large Order Quantity,
and a relatively large investment in Operating Stocks. Therefore, the high-
er the price of the item the smaller Q should be. The proposed policies bal-
ance the cost of holding an item against, the cost of frequent reorders.
The Reorder Point establishes the level of the Safety and Pipeline Stocks
which serve to provide protection during the resupply cycle. In deter-
mining R, account is taken not only of the demand rate but also of the var-
iability in demand, considering whatever is known about the probability as-
pects of demand for groups of parts. Pipeline time is also considered the
longer it is, the larger the Safety and Pipeline Stocks should be because of
the greater risk of a shortage during the pipeline time. Given the demand
rate, the Order Quantity determines how frequently the Safety and Pipeline
Stock is dipped into and, therefore, how frequently the risk of a shortage
is run.
In addition, the shortage cost, i.e., the expected cost to the Air Force
of overcoming a shortage, is corisidered. The minimum cost of a shortage ap-
pears to be the cost of priority actions. To this minimum should be-added,
in some circumstances, such costs as that of local manufacture, the cost of
keeping a higher assembly on hand, or even the cost of maintaining an extra
aircraft (or missile) to substitute for an AOCP (or MOCP) caused by a short-
age.
The higher the demand, the more frequent the orders, the longer the
pipeline time, and the greater the shortage cost, the higher the economical
Reorder Point. Against this must be balanced the investment cost. The higher
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the unit cost and the higher the cost of keeping items, the lower the Reorder
Point should be.
These, then, are the factors considered in determining the Order Quantity
and the Reorder Point:
Order Quantity Reorder Point
Equations have
Demand Rate
Unit Cost
Holding Cost
Reorder Cost
Demand Rate
Variation in Demand
Unit Cost
Keeping Cost
Pipeline Time
Order Quantity
been developed to take these factors into account with a good
approximation of a theoretically ideal solution. From these equations, tables
such as that in Figure 1 are developed. Such tables are proposed for use at
base level. With them, complex procedures could be avoided, but full advan-
tage could be taken of the relatively complete and efficient decision rules
underlying them.
In Figure 1 the demand rate is shown across the top, increasing from
left to right. The price is shown increasing from top to bottom. As demand
increases, Reorder Points and Order Quantities increase. On the other hand,
as the price increases both fall. For the very high-priced items in Category II,
the Reorder Points and Order Quantities are very small. Some large Stock
Control Levels appear in the upper left corner of the chart. These figures
can readily be rounded to dozens or dollars' worth, to unit pack, or any other
convenient unit.
In stocking a base initially for a new weapon, the Stock Control Level
would be placed at the base. By stocking several years' supply of the cheap
low-demand parts, the danger of a shortage is nearly eliminated, and the
chance of having to incur the reorder cost during the program is made small.
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RM-1962
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4-18-58
Tables could be prepared at some central point, using either manual
or computer methods, and distributed to the bases. Alternatively, such
tables could be used by, say, a Weapon System Supply Manager to set the
Order Quantities and Reorder Points directly. With integrated data processing,
the levels could be determined by the Data Processing Center, following the
equations upon which such tables are based.
To be practical, these policies must handle phase-in and phase-out.
For common parts the phase of life of any particular weapon is relatively
unimportant, but stockage of parts peculiar to a weapon should, of course,
reflect its program. For individual bases, where the transition to a new
weapon is relatively abrupt, no change in the decision rules is required.
As the end of a program approaches at any base, the Order Quantity should
be adjusted to support the expected remaining life of the program; a down-
ward adjustment is typically called for.
Given the base stockage rules which handle stock distribution, the
depot-stockage and, hence, requirements rules are developed. The same
factors are used as in base stockage; but some of the terms have somewhat
different meanings in the depot case. For a long stable program the
principles of base and depot stockage are very similar, but the dynamics
of phase-in and phase-out have much greater impact upon the depot than upon
the base.
During the phase-in, the depot must support a growing program with
little demand experience for individual parts. For many items the bases
will have several years' stocks from the beginning. If the phase-in is
gradual and the procurement leadtime is short, it may be possible for the
depot to take advantage of this situation and stock little more than its
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Reorder Points, waiting until it has had demands from the bases and experience
with the program before bringing its stocks up to their Stock Control Levels.
Computations have been developed, adjusting depot Order Quantities and Re-
order Points as the phase-out approaches, to take account of the termination
of production of peculiar parts and the declining demand for them.
The proposed policies appear to provide close to optimal solutions to
the system stockage problems. They constitute consistent distribution and
requirements rules which are practical within the present data processing
system and organization and which can be applied with little modification
in an integrated data processing system. They appear to take into account
all of the major relevant factors. But, what evidence is there that they
will work in fact?
In the first experiment in the Logistics System Laboratory (LP-1) the
results of applying these policies were compared with the results of applying
a set of policies which Air Force members of the Laboratory staff developed
as an approximation to the current best practice in the Air Force. In
general, the proposed policies for low and medium value parts compared
extremely favorably. This comparison took account of complete base and depot
stockage policies, i.e., distribution and requirements in a dynamic situation
in which readiness to fight a simulated war was demanded of both systems.
The proposed policies resulted in far fewer AOCPts, slightly fewer
ANFEls, and less than 10 per cent as many priority actions as did the alter-
native system. Further, they cost less in terms of supply workload.
To conclude this summary statement, this Memorandum presents policies
for stocking nonrecoverable items at base and depot level. Their application
promises substantial economies in dollars and in personnel and, particularlyt
large improvement in the effectiveness of the supply support of the combat
forces.
RM -1962
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FOREWORD
This Research Memorandum covers the setting of base and depot levels for
items which the activity in question does not repair. The setting of levels
for items which the activity in question does repair includes much of the
formulation discussed here but requires the application of some additional
principles.
The methods described here were developed principally by A. J. Clark and
the authors and are extensions, corrections, and simplifications of the pro-
cedures described by E. B. Berman and Clark in An Optimal Inventory Policy for
a Military Organization (The RAND Corporation Paper P-647, 30 March 1955),
and by Clark in A Technique for Optimal Distribution of Available Stocks to
Bases (The RAND Corporation Research Memorandum RM-1621, 30 January 1956).
The extension to the short-program case has been simplified by comparison
with the results of unpublished studies by H. W. Karr, using a dynamic pro-
graming model for procurement decisions.
These policies have formed a major portion of the first project in the
RAND Logistics Systems Laboratory (LP-1).
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CONTENTS
SUMMARY
FOREWORD
I. INTRODUCTION
A. Scope and Approach
B. The Problem
Page No.
iii
xi
1
1
3
II.
BASE LEVELS UNDER STABLE CONDITIONS
15
A. Approximation Formulas
16
1. Base Order Quantity
17
2. Base Reorder Point
25
B. Resultant Stock Levels
33
1. The Base Stockage Table
33
2. Sample Computation
36
3. Effects of Pipeline Time and Shortage Cost
?
?
40
C. Resultant Effectiveness and Costs
44
D. Summary
53
III.
DYNAMIC ADJUSTMENTS TO BASE LEVELS
55
A. Quantities for Initial Base Support
55
B. The Phase?Out or Short Program
57
IV.
DEPOT OR STORAGE?SITE STOCKAGE ?
67
A. The Equations
68
B. Depot Stockage Under Dynamic Conditions
71
C. Depot Order Quantities
73
D. Depot Reorder Points
83
E. Summary of Depot Stockage Policies
88
V.
DATA REQUIREMENTS
95
A. Sensitivity
96
B. Description of Data
102
1. Identifying and Program Information
102
2. Demand Data
102
3. Pipeline Time
105
4. & 5. Holding Costs and Keeping Costs
106
6. Reorder Costs
107
7. Shortage Cost
108
VI.
CONCLUSION
113
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Appendix
I. MATHEMATICAL APPENDIX
A. Fundamental Base Cost Equation with a
Stable Program
B. Economical Base Order Quantity
C. Economic Base Reorder Level
D. Determination of the Holding Cost Rate and
Keeping Cost Rate
E. Adjusting Q for Short Programs
II. SAMPLE BASE TABLES
Table 11.1 - Base Stock Levels
Page No.
121
122
123
126
128
129
133
134
Table 11.2 - Base Stockage Tables 135
III. RELATIONSHIP BETWEEN STOCK-LIST PRICE AND NUMBERS OF
UNITS ISSUED FOR B-47 AND F-86H SPARE PARTS
141
fABLES
1. Demand Experience for B-47 Aircraft Parts over
1300 Aircraft-Months
2. Some Statistical Characteristics of F-86H Parts.
3. Base Stock Levels
4. Effect of Pipeline Time and Shortage Cost Upon
The Reorder Point
5. Effect of Pipeline Time and Shortage Cost Upon
The Reorder Point
6. Effectiveness and Cost Under "67-1" and The
Proposed Policies - Illustrative Comparison -
Base Level 45
7. Effectiveness and Cost Under "67-1" and The
Proposed Policies - Illustrative Comparison -
Base Level
8. Effectiveness and Cost Under "67-1" and The Proposed
Policies - Illustrative Comparison - Base Level -
9. Effectiveness and Cost Under "67-1" and The Proposed
Policies - Illustrative Comparison - Base Level - ??
10. Sample Table of Economical Base Ord..1. Quantities:
Long Program
11. Sample Table of Economical Base Order Quantities:
5-Year Program
12. Sample Table of Economical Base Order Quantities:
1-Year Program
13. Initial Depot Stockags
14. Depot Reaction to a Base Order
15. Terminal Buy
16. Sensitivity Table (using a Five-Year Program and
a Correct Yearly Demand of 10) 98
17. Sensitivity Table (using a Five-Year Program and
a Correct Yearly Demand of 10) 99
18. Sensitivity Table (using a Five-Year Program and
a Correct Yearly Demand of 84) 100
19. Sensitivity Table (using a Five-Year Program and
a Correct Yearly Demand of 84) 101
20. Shortage Costs Implied By "67-1" Policies. ??? 111
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Page No.
5
6
34
41
42
48
51
52
59
61
62
91
92
93
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FIGURES
1. Base Stock Levels
2. Some Definitions
3. Base Level Demand Pattern for a Gasket
4. Pattern of Base Stocks on Hand and Due-In For Same.
Gasket (see Fig. 3)
5. Annual Distribution Costs
6. Annual Cost of Reordering
7. Reorder Cost, Holding Cost and Total Cost ? ? ? ? ? ?
8. How a Higher Unit Price Reduces The Economical Order
Quantity
9. Determinants of the Protection Provided By the Rth
Unit in the Reorder Point
10. Determination of the Economical Reorder Point . . .
11. Supply Cost Implications of Proposed Versus "1956"
Policies
12. Supply Effectiveness Implications of Proposed Versus
"1956" Policies
Page No.
viii
8
9
11
19
20
22
24
27
31
117
118
I. INTROYUCTION
A. Scope and Approach
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There is at present a great deal of interest in the Air Force and else-
where in the defense establishment in improving the management of the costly
inventories of spare parts. Hi-Valu and Lo-Valu programs are now under way
in the Air Materiel Command, and Headquarters USAF has initiated a Supply
Improvement Program. RAND has done research in this general area for some
years.1 The present study, dealing with those items for which detailed line-
item management is not appropriate, is complemented by the work on deferred
procurement of Hi-Valu items.2
Policies for the provisioning and distribution of support for the major
missiles must be firmed up in the near future. Similarly, the development
of electronic Data Processing Centers in the Air Materiel Command may present
an especially good opportunity for introducing new policies over the next few
years. Specifically, within the next year the ELECTRO LOGS Project at Okla-
homa City Air Materiel Area may be in a position to introduce some of these
policies after the completion of operational testing and evaluation of the
Inventory-Control phase.
This paper presents simple and practical provisioning and distribution
calculations for items which do not justify detailed management, such as
Category III and nonrecoverable Category II parts,
1
R. B. McNeill, E. B. Berman, A. J. Clark, H. W. Nelson, A Proposal for
a New Air Force Supply Procedure, The RAND Corporation Research Memorandum
RM-1417, 28 January 1955.
J. W. Petersen, Savings from Procurement Deferral with Interim Contractor
Support: The Case of High Value Airframe Spares, The RAND Corporation Research
Memorandum 1114-20851 10 January 1958.
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The characteristics of a practical method are:
1. It can deal with the uncertain and erratic demands, characteristic
of the supply system.
2. It has reasonable data requirements.
3. It is compatible with the present manual and punched-card
data-processing practice with local determination of stock
levels, but it is also compatible with integrated data-
processing and centrally-controlled resupply; it should
also provide rules from which the necessary machine appli-
cations for an integrated system can be developed.
4. It must provide effective stockage rules under the dynamic
conditions of weapons' phasing-in and -out.
Simplicity of computation is achieved by using, at least for the cheap-
est parts, approximate, rather than rigorously optimal, formulas for deter-
mining the appropriate stock levels. Simplicity in operation can be achieved
by printing and distributing a small number of tables to each base for the
use of supply personnel.
The methods appear to be applicable to the bulk of the parts in Cate-
gories II and III; however, there are some exceptions. No attempt is made
to treat problems associated with the long service life of some parts, or
the special problems associated with dated items. The former may be important
in the case of many Hi-Valu parts, but it does not appear to be of major sig-
nificance for most lower-cost ones. Further, the methods do not apply to
quartermaster, bulk or local-manufacture items.
The general approach is to find economical basel stock levels on an
item-by-item basis. After we find the economical base levels, depot or
storage-site levels are set which will provide effective support for the
bases.
1
For purposes of setting stock levels, parts-repair depots and IRAN
depots are considered to be the sane as bases; they might have different
pipeline times and, of course, different demand rates, but they are con-
suming activities and, in that sense, are like other bases.
?.
????
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The computational methods discussed in this paper may be used to set
base and depot stock levels for nonrecoverable line items with a long and
stable base program or during the phase-in or the phase-out of a weapon or
other end item. The methods are also applicable to reparable items at
activities which do not repair the items in question.
The paper is subject to two important limitations. First, it is con-
fined to the question: Given the affirmative decision to stock an item at
an activity, how much should be stocked? The determination of what to
stock is outside the scope of this paper. Second, the paper does not deal
with the stockage of all reparable parts, but only with those parts which
are not normally reparable at the activity in question. Thus it deals only-
with stocking those parts which are consumed when they are used or which, if
reparable, go off base for repair. In both these case, demand for a part
can be satisfied only with a part from some outside source. In the other
case where a part can be repaired on base, a demand for a part can be met
either from an off-base or an on-base, i.e., maintenance, source. The
ambiguity in the latter case introduces problems which have not been com-
pletely worked out.
B. The Problem
The major characteristics of the Air Force supply system are widely
known, but it may be well to sununarize the relevant ones briefly as a con-
text for the discussion to follow.
The vast majority of the parts in the Air Force supply system are low-
cost parts, and the great bulk of the supply activity measured in terms of
the numbers of parts issued or consumed is similarly concentrated in the
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low-cost items.1 Since there are many hundreds of thousands of such parts
in the Air Force inventory,2 it is not
to devote a great deal of management to
other hand, these parts must be managed
experience of the Air Force that a very
are caused by inexpensive parts.3 Most
possible - certainly not economical -
the control of each item. On the
effectively, since it has been the
large fraction of supply difficulties
supply activity is accounted for by
the cheap parts. On the other hand, the bulk of the investment is in the
mall fraction of expensive parts.
These characteristics of the supply system are reflected in Tables 1
and 20 which show the base level experience for two quite different aircraft
in two widely separated periods. Table 1 shows very detailed B-47 experience
at March and McDill Air Force Bases during 1953 and 1954. Table 2 shows
F-86D experience at Clovis Air Force Base in 1956. Both tables show base
level supply activity for roughly the equivalent of a base-year, broken down
by price and issue-rate groupings. Each shows the percent of total active
line itens? the percent of total transactions and the percent of the value
of transactions accounted for by each price-demand group. While the tables
differ in detail, their general characteristics are much the same. Both
clearly indicate the concentration of line items in the very low-price, very
low-demand groups, the concentration of supply activity - as reflected in
1 Bernice B. Brown, Characteristics of Demand for Aircraft Spare Parts,
The RAND Corporation Report R-292, July 1956.
2 M. A. Geisler and A. R. Mirkovich, Analysis of Worldwide Data on Air-
craft Spare Parts as to Unit Cost, Quantity and Value Issued, and Inventory
Value, The RAND Corporation Research Memorandum RM-1481, 6 May 1955.
3
H. W. Karr, Analysis of 13-47 AOCP Experience, The RAND Corporation
Research Memorandum RH-1340, 14 September 1954.
?
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Table 1.
Demand Experience for B-47 Aircraft-Parts over 1300 Aircraft-Months1
Unit Cost
(Dollars)
Under
Demand per 100 Aircraft-Months
0.1
0.1-0.9
1.0-9.9
10.0 and Over
600 and
over
% Line Items
% Quantity
Demanded
% Dollar Value
2.7
0.01
0.5
0.1
0.3
0.4
0.02
0.2
Demanded
0.4
9.2
39.2
36.7
% Line Items
16.4
2.5
1.1
0.1
10 - 600
% Quantity
Demanded
0.1
0.5
1.6
1.3
% Dollar Value
Demanded
0.2
1.7
7.0
4.8
Line Items
58.9
7.7
6.5
3.0
Under 10
% Quantity
Demanded
0.1
1.7
12.8
81.3
% Dollar Value
Demanded
0.0
0.1
0.3
0.5
Total Line Items:
26,445, from Master Spare Parts List
Total Quantity Demanded: 635,334 Parts
Total Value Demanded: 20.4 million dollars
1The data were collected at March, MacDill and Fairford Air Force Bases
in 1953 and were previously published in M. A. Geisler's Analysis of Base
Stockage Policies, The RAND Corporation Research Memorandum 1431, 17 February
1955, p.4.
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ACTERISTICS OF F-86H PARTS
SOME STATISTICAL
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r-1 1-()
Or-40
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RM-1962
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transactions - in the low-price, high-demand group and the concentration of
dollar value of transactions in the very few high-value, relatively high-
demand cells. All the data available at RAND indicate that these charac-
teristics are typical.1
Almost any supply system would function effectively if there were a
smooth flow of issues as in Figure 2.2 Issues would be made until the
Reorder Point was reached, then additional stock would be ordered up to
the Stock Control Level. Issues would continue during the pipeline time.
By the time the new order arrived, the stock on hand would be down to the
safety level, which would be needed only if pipelines were interrupted.
With no uncertainties, it might be possible by trial and error to estab-
lish adequate Reorder Points and Order Quantities. However, there is uncer-
tainty. Pipeline times vary, so there is no assurance that stocks will ar-
rive when they are expected. More important, demands do not occur in neat
order, one per week, or one every month, or anything of the sort. Figure 3
is a more typical representation of an actual sequence of base demand over a
period of 35 weeks for a particular item. This is, if anything, a less errat-
ic demand pattern than is typical; yet, in spite of the fact that this part
had a mean demand of 1.7 units per week, there were many weeks when it was
1
Bernice B. Brown and M. A. Geisler, Analysis of the Demand Patterns
for B-47 Airframe Parts at Air Base Level, The RAND Corporation Research
Memorandum RM-l2971 27 July 1954; M. A. Geisler and A. R. Mirkovich, Analysis
of the Flyinr,,. Activity and Spare Parts Demand of F-86D Aircraft at Perrin
Air Force Base, 1 September 1953 - 28 February 1954, The RAND Corporation
Research Memorandum M-1456, 4 June 1955; idem, Analysis of Worldwide Data
on Aircraft Spare Parts as to Unit Cost, Quantity and Value Issued, and In-
ventory Value, The RAND Corporation Research Memorandum RM-14811 6 May 1955.
2
The figure also serves to define some of the key terms which will be
used throughout the discussion.
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6-170006000?Z001?1701-0-1-8dC1I-V10 SZ/01-/? -1A-09 ? eseeiej -104 panaiddv Ado paz!l!ueS u! PeWsseloaCI
6-170006000?Z001?1701-0-1-8dC1I-V10 SZ/01-/? -1A-09 ? eseeiej -104 panaiddv Ado paz!l!ueS u! PeWsseloaCI
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not demanded at all, and several weeks in which there were demands as high
as 9.
If the rule, "stock 90 days' stock plus the expected number of demands
during the pipeline time plus a 15-day safety level," were rigidly applied
to the 23-cent gasket, the solid line in Figure 4 would show how the stock
on hand at a base would fluctuate over a period of 35 weeks. There are some
stockouts. (Stockouts for the cheap items, such as this 23-cent item, have
been a major cause of criticism of Air Force stockage policy.)
Further, if that rule were followed, three reorders would have been
placed while some 68 parts were demanded. If it costs as little as $5 to
process an order, $15 would have been spent on reorders to support $15.64
worth of consumption.
What do these facts mean for stockage policy? To the extent that good
management should be directed toward dollar economy, it is desirable to con-
centrate it on the small fraction of high-value items. It is desirable to
buy them in minimal quantities and to control them closely. Since they are
few, this may be entirely practical and is, of course, the objective of the
Hi-Valu program. However; in view of the great number of less expensive
parts, it is impractical to attempt to control them in that fashion, and be-
cause so little is invested in them, in relative terms, there is no economic
justification for doing so. Further, since a large proportion of the supply
activity is accounted for by the cheap parts, it is only by curtailing the
administration and management of them that management resources will be
freed to control properly the higher-cost items. The low-value parts, how-
ever, can and do ground weapons and stop maintenance lines, so they can by
no means be ignored.
plus due-ins
'flu,
4-4-
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(f) 4.)
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O0
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a_
t.-
LI-
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Sound management, therefore, appears to call for striking a balance
among the costs of managing the different kinds of items, the consequence
of shortages of them and the costs of investing in and holding them. The
study reported here does that. The methods proposed are sufficiently simple
to permit their use in the management of the masses of lower-cost items,
with considerable improvement in the effectiveness of supply support of the
operational units. The policies call for large base stocks and infrequent
requisitions of the cheapest parts so as to reduce greatly the risks of
shortages of the cheap parts and the costs of processing them. On the other
hand, to reduce the investment in the higher-cost items, much =inner stocks
and more frequent reorders are appropriate.
The paper is organized as follows:
This introductory chapter describes the organization and scope of the
paper and defines the problem of setting efficient stock levels.
In Chapter II, the principles of efficient stockage are stated and used
to compute a sample table of economical base stock levels under a stable
program. These stock levels are then compared with the base levels which
would result from rigid application of the general rules stated in the Air
Force Supply Manual.1
Chapter III takes up the effect of some dynamic considerations on base
stock levels: the setting of quantities for initial-support tables and the
lAir Force Manual 67-1. This manual is referred to as u67-1fl. By the
?67-1 system'? is meant the rules calling for a reorder of the number of units
expected, on the average, to be issued in a period equal to 15 days, plus
the length of the routine pipeline time and an operating period of 60 days
for Cost Category II items and 90 days for Cost Category III items.
In many areas, the Air Force applies stodkage policies which are dif-
ferent from and improvements over the u67-1? system, but the ?67-1fl system
provides a ready bench mark with which the proposed rules may be compared,
and, further, for purposes of comparison our set of general rules can best
be related to some other set of general rules.
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setting of levels as the phase-out of a weapon or program approaches.
Chapter IV, ',Depot or Storage-Site Stock Levels", parallels the dis-
cussion in Chapters 11 and III, first taking up the stable-program case
and setting levels so that efficient support may be given to bases; and
then, the dynnmic aspects of determining depot Order Quantities and Reorder
Points.
Chapter V describes the Data Requirements of the proposed policies
and discusses their sensitivity to inaccuracies in the data.
Chapter VI lists the conclusions which may be drawn from this study.
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II. BASE LEVELS UNDER STABLE CONDITIONS
The present chapter is basic to the whole discussion. It develops the
fundamental principles which will be elaborated and modified later on to take
account of dynamic and system stockage problems. Specifically, this chapter
develops formulas proposed for the computation of base Reorder Points (for
determining when to resupply) and base Order Quantities (how much to ship at
a time) for a stable program.
Two sets of equations are developed. A rather rigorous set is derived
in Appendix I, but most of the discussion in the text of this and the
succeeding chapters relates to some approximations to the rigorous equations.
The approximations are quite adequate for determining levels for the lower-
priced items in Category . II and for Category III and are very easy for
manual computation. The more exact equations can be used where electronic
computing equipment is available. For some of the higher-cost items they
give significantly better results than do the approximate equations.1 These
equations still permit automatic computation of levels and, hence, are
appropriate for the more expensive parts (including Category I items) in
cases where it is decided not to give individual attention (as required, for
example, in the deferred-procurement program) to the management of particular
line items. As will be shown later, it would also be practical to use tables
based on either the approximate or exact equations for very simple manual
computation. The choice among these three procedures for actual implementa-
tion requires the exercise of judgment based upon the cost and importance of
the items in question, the accuracy of the input data available and the com-
puting capacity at hand.
1See Chapter V for a discussion of the sensitivity of system costa to
the approximation.
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?
After the formulas are presented, some reasonable assumptions will be
made about the environment in which the supply system operates, and the
approximation formulas will be used to compute a sample table of basel
Reorder Points and Order Quantities. The sample table is for illustrative
purposes only and is not proposed for actual use. We shall see the way
Reorder Points differ for different resupply pipeline times and for differ
ent costs associated with a shortage, and we shall compare the sample
stock levels with "67-1" levels. Finally, this section will present a
very rough comparison of the supply effectiveness and costs of the proposed
policies and the "67-1" policies.
A. Approximation Formulas
The definitions to be used are as follows (See Figure 2 ):
Stock Control Level has the same meaning as the Stock Control Level
under the present system.
Reorder Point is the level for determining when to reorder, i.e.,
when "stock-on-hand-plus-due-in/8 minus due-out's" is equal to or less
than the Reorder Point, an additional order is to be placed.
Order Quantity represents the difference between the Stock Control
Level and the Reorder Point.
Operating Stock refers to the "stock-on-hand-or-due-in" over and
above the Reorder Point.
Safety and Pipeline Stock is that "stock-on-hand-or-due-in" up to
the Reorder Point.
1"Base" As used throughout this Memorandum includes the base support
activity of a depot.'
17,
1. Base
In
we treat
Figure 2
19 62
4-18-58
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Order Quantity
explaining how to set base levels for any particular line item,
first the Order Quantity, then the Reorder Point. A glance at
will show that determining these two quantities at each activity
for each item solves the stockage problem.
Turning first to the Order Quantity, it is obvious that, if there
were no cost or inconvenience associated with ordering again and again, it
would be cheapest and simplest to order one unit at a time, thus avoiding
the costs of holding the Operating Stocks. On the other hand, if it cost
nothing (in money, trouble, or material resources) to hold stock, there
would be an incentive to order each item once and for all and to keep vast
quantities on hand -- thus avoiding the problems and costs of reordering.
Neither extreme situation is true, of course, but looking at both brings
out the fact that for any given demand rate the desirable size of order
depends on the costs of reordering and the costs of holding the Operating
Stocks and, in the approximation, upon nothing else.1
If each shipment
Is small, shipments are frequent, and the costs of placing an order are
incurred frequently, but only a small amount of Operating Stock is allocated
to the base. If each shipment is large, the number of orders -- and hence
the yearly reorder costs -- are reduced, but the average Operating Stock
is large and so is the cost of holding it. To find the economical Order
Quantity for each demand rate, we balance holding cost against reorder
costs.
1In the rigorous formulation, explicit account is taken of the fact
that the Operating Stocks also provide some protection against shortages.
See Appendix I.
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a. To define reorder costs, compare what happens if a base were to
receive two small shipments of an item with what would happen if it were
to receive only one large shipment of the same number of units of the
same item.. Since at both base and depot level some order costs are
independent of the number of units on an order, it costs more to make two
small than one large shipment of the same total amount. As illustrated
in Figure 5, we call that difference in cost the reorder cost, i.e., the
extra distribution cost incurred by each additional order for the base
during the year.1 The reorder costs are those costs which are incurred
in the placing and filling of an order, and which do not depend on the
number of units ordered or shipped. They consist largely of paperwork
and communications costsand some fraction of the stock-picking, packing,
transportation and receiving costs.
Figure 6 shows that, given the demand rate at a base, annual2 cost
of reordering declines as the size of each shipment increases. Note
that annual cost of reordering declines rapidly as the size of shipment
increases fram one unit to two units (since the number of shipments is
reduced by half), but the cost declines less rapidly for larger shipments,
1For further discussion, see Chapter V, Section B-6.
2Costs and demand are referred to as pertaining to a year -- any
convenient period would do equally well.
Reorder
cost
Reorder cost
One large order
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Two small
orders a year
Fig.5 Annual distribution costs
(Given total number of units shipped)
Three small
orders a year
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(siollop) EoupapJoai Jo 'SOO ionuuv
Order Quantity
CT
I-.
"Cf
0
demand rate
RM-1962
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e.g., if the Order Quantity is increased from nine units to ten, the number
of shipments --and hence annual costs - -is reduced only 10 per cent.
b. The Operating Stocks must be held until used. The Annual cost to
hold a unit of stock consists of the unit value of the item multiplied by
the unit holding cost. The latter is (1) the physical cost of storing a
unit of the item at a base for a year, plus (2) a charge for the capital
invested in a unit of the stock.1 The unit holding cost is expressed as
a per cent of the value of the item.
The larger the Order Quantity, the larger the stock held and the greater
the costs of holding it. Thus, two kinds of cost are involved: annual cost
to reorder, which declines as Q increases, and annual cost to hold, which
increases as Q increases, given the unit,price of the item and the unit
holding cost.
Figure 7 represents annual cost to hold, annual cost of reorders and
the sum of the two. Note that annual reorder costs decline rapidly at
first as Order Quantity increases; total cost decreases at first and then,
as reorder cost levels off, total cost increases. Obviously, it is desirable
to select an Order Quantity which minimizes total cost. The economical
Order Quantity is that size of order associated with the smallest total of
_annual cost to reorder plus annual holding cost.
1This is a charge for the risk of obsolescence or modification due to
engineering changes in the part itself or in the end item or higher assembly
of which it is a member, plus a capital charge reflecting the value of money
to the Air Force, i.e., the cost of using money to buy this item rather than
spending it in some other way. See Chapter V, Section B-4 and Appendix I.
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4.*
L. 411;
to
O ? o c
o 0
E w E E
41)
6 .0
C
a, .2
o
Economical Order Quantity
Fig.7?Reorder cost, holding cost and total
i5
0
0
0
0
LU
RM-1962
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The demand rate and the unit cost of the item also influence the
economical value of Q. If the demand rate for one item is greater
than for another, the annual cost to reorder associated with any given
size of shipment will also be larger (Figure 7). Thus, the economical
Order Quantity will be larger, the greater the demand rate. The higher
the unit cost, the more it costs to hold any given number of units; hence,
the economical Order Quantity decreases, as illustrated in Figure a.
In order to use these principles in computing levels we must have a
formula;1 it is
Economical
(1) Order
Quantity, Q
2(Reorder Cost) (Annual Demand Rate) 4. .
(Holding Cost) (Unit Cost
For all practical purposes, this is the equation we would have obtained
if, instead of plotting the curves in the last several figures, we had
written down their equations and found the mathematical solution for the
minimum cost point. The derivation is developed more fully in Appendix I.
Equation 1 shows that the higher the demand rate, the larger the
economical Order Quantity, but doubling the demand rate does not double Q;
the higher the reorder cost, the larger the economical Order Quantity, but,
again, doubling this less than doubles Q; the higher the holding cost or
the unit cost, the smaller the economical Order Quantity, but, again,
'In the equation, the "ln under the radical is a constant factor which
results primarily from the fact that fractional shipments cannot be made.
In other respects this equation can be derived from the equation for the
optimal number of orders per year which appears in the Air Force Supply
Management Handbook, AFM 67-10, Page 149 (March 1, 1956). For the
derivation of Equation 1, see the Mathematical Appendix.
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doubling the holding cost does not cut Q in half. In fact, Q changes as
the square root of these factors: a doubling of Q would occur if demand
or reorder cost were quadrupled or if unit price or holding cost were
quartered. These effects can be seen clearly in Figure 1or in Table 3
below.
2. Base Reorder Point
The next step is to determine when shipments should be made to a base,
i.e., what the basets Reorder Point should be. The purpose of the Reorder
Point is to indicate at what inventory level an order should be placed. It
marks the amount to be held as Safety and Pipeline Stocks (Figure 2) to
avoid shortages while the base is waiting for a shipment. If resupply were
instantaneous and the pipeline were never interrupted, or if there were no
cost or inconvenience in being without a part while awaiting a shipment,
there would be no justification for investing in Safety and Pipeline Stocks.
They would be zero. Resupply takes time, however, and it is costly to be
without parts when they are demanded. Furthermore, demand is uncertain, and
there is no way of predicting with accuracy how many parts will be demanded
during the routine pipeline time when the base is awaiting a delivery.
Therefore, there are good reasons for having Safety and Pipeline Stocks;
the larger those stocks (all else equal), the less is the risk that a shortage
will occur. Since it does cost something to keep stocks, the problem in
setting the Reorder Point is to stock just enough so that the protection
provided against shortages is worth what it costs, and so that keeping addi-
tional stocks would cost more than the additional protection would be worth.
This is not an easy problem.
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To determine the appropriate Reorder Point, we need to know three
things:
a. how many shortages a year are avoided by each additions.1 unit in
the Safety and Pipeline Stock;
b. how much it is worth to avoid each of these shortages;
c. what the cost is of keeping an additional unit of Safety and
Pipeline Stock.
Since stock levels are set before, not after, the fact, only expected values
of these quantities can be used.
a. The number of shortages a year avoided by having an additional unit
(call it the Rth'unit) in the Safety and Pipeline Stocks depends on how
frequently these stocks are exposed to use and on the probable number of
demands during eadh period of exposure. Figure 9 shows diagrammatically the
factors determining how much protection is provided by additional Safety and
Pipeline Stocks.
The frequency with which these stocks are exposed to use depends upon
the size of the Order Quantity, given the annual demand rate, for they are
subject to use whenever the base has exhausted its Operating Stocks and is
(presumably) awaiting an order. The expected number of orders1 a year, of
course, equals the expected annual demand rate divided by the Order Quantity,
e.g., if the demand rate is 100 per year and 25 are shipped at a time, there
will be an average of four shipments a year; but if 50 are shipped at a time,
there will be only two shipments a year, etc. The proballe amber of demands during
1The number of orders per year is the reciprocal of the operating period
(measured in years). Operating period equals Order Quantity 4 Annual Demand.
The number of orders per year equals Annual Demand 4 Order Quantity.
O's
O .-
4- C
0
..c
CC
o
>,
.0.0
C
V
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? >
0
a)
V
s-
0
Economical
RM-1962
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the pipeline time--which is the period of exposure--depends upon the length
of the pipeline time, the annual demand rate and the variability of demand.
All else equal, the longer the pipeline time, the greater the number of
demands expected while the base is awaiting shipment, and hence the greater
the chance that the Rth unit will avoid a shortage.
The larger the annual demand rate, the greater the chance that the
additional unit will avoid a shortage, because the more demands one can expect
during any pipeline period. Usually the greater the variability of demand,
i.e., the more erratic is demand, the greater the probability that the Rth
unit will avoid a shortage.
Variation in the pipeline time has much the same effect as does variation
in demand. It can be treated in the same way since we are interested in the
probable number of demands during the period while the base is awaiting
resupply.
To get some "feel" for how an additional unit in the Safety and Pipeline
Stocks avoids ashortage, refer again to Figure 4 showing the stock of the
gasket on-hand and due-in. In this case, the Reorder Point is 11 units, and
the Order Quantity is 22. At the first reorder, since a total of 14 units
were demanded within one routine pipeline time, each of the 11 units in the
Safety and Pipeline Stock avoided one shortage because, had the Reorder Point
been 10 instead of 11, there would have been three shortages instead of two,
if the Reorder Point had been 9 units, there would have been four shortages,
etc. On the next order, each unit in the Safety and Pipeline Stock avoided
a shortage for similar reasons. But, at the third reorder the seventh,
eighth, ninth, tenth, and eleventh units of Safety and Pipeline Stocks avoided
no shortages because after the reorder was placed five units were still on
RM-1962
4-18-58
-29-
hand and there were no demands during that particular pipeline time. The
eleventh unit avoided two shortages in this example and on the average
avoided two-thirds of a shortage in each of the three periods of exposure.
Thus, the expected number of shortages avoided during a year by the Rth
unit is equal to the probability that R or more units will be demanded during
a pipeline time (for that probability is the fraction of pipeline times in
which R or more units will be demanded times the number of reorders per
year.)
b. The second quantity which must be known in order to determine the
economical Reorder Point for a base is how much it is worth to avoid a
shortage.1 If a shortage would result in priority action, the cost of that
action is a lower limit on the amount it is worth to avoid a shortage. In
addition, there is the loss in operational capability caused by the shortage.
This latter cost depends upon the immediacy of the need, the mission
effect of a shortage and the cost of possible compensatory action. If the
item is needed immediately, the shortage cost will be higher than if the
need is discovered during, say, a periodic inspection when the item could
as well be installed several days later. Secondly, a shortage of one (of
two) landing light would have far less effect on operational capability
than would the shortage of a nose wheel. If the effect of the shortage
can
be reduced temporarily or permanently by local manufacture or cannibaliza-
tion, the actual shortage cost is likely to be less than if the aircraft is
forced to remain without the item. If a shortage can be compensated by
using an inexpensive higher assembly, of which there are many stocked on the
base anyway, it is far less serious and less costly than would be the case
'The shortage cost is discussed more fully in Chapter V, Section B-q.
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RM-1962
4-18-58
%I. the next higher available assembly were the aircraft itself. It appears
that the pro-rated cost of the aircraft or other end item sets an upper
limit on the shortage cost.
c. The third quantity that must be known to set the economical
Reorder Point is the cost of keeping each additional unit of Safety and
Pipeline Stock on hand. In general, we can expect that, after the first
unit of an item is stocked, the cost of keeping the Safety and Pipeline
Stock is increased by roughly the same amount by each additional unit in
the Safety and Pipeline Stock. To determine the cost of keeping a unit for
Safety and Pipeline Stock, all of the factors used for the determination of
unit holding cost for operating stock are needed, viz., physical-storage
costs, capital costs, and engineering-obsolescence costs. It was expected
that the Operating Stocks would be used up periodically, but there is a
good chance that an additional unit of Safety and Pipeline Stock will
eventually have to be salvaged or otherwise disposed of at less than the
purchase price; for, by having enough stock to achieve a high degree of
protection, there is a very good chance that some of the Safety and Pipeline
Stock will be on hand at the end of the program. Thus, terminal obsolescence
must also be taken into account in determining keeping cost.1
Figure 10 illustrates how the three factors are used to determine the
Economic Reorder Point. Compare the expected saving of'shortage costs
from each additional unit of stock with the cost of having that unit. For
instance, the expected saving due to the first unit (the first bar in
Figure 10) of the Safety and Pipeline Stock is the probability of one or
1Terminal obsolescence is defined as that obsolescence of parts caused
by the phasing-out of the program to which they apply. Keeping cost is
discussed more fully belwin Chapter V., Section B-5.
?
Declassified in Part - Sanitized Copy Approved for Release @ 50 -Yr 2013/10/25 ? C
s z+ d
s('+8 T.(1,
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7
Units in the Safety and Pipeline Stocks
of orders a year
or more, 2 or
RM-1962
4-18-58
-31-
14+
0
C
0
2
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a
0
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Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/10/25: CIA-RDP81-01043R002300090004-9
RM-1962
4-18-58
-32-
more demands during the routine pipeline time, (P1), multiplied by the
number of orders expected each year, (+), multiplied in turn by the
shortage cost per unit (s). Then compare this figure with the cost of
keeping an additional unit on hand (kv). If the savings are greater than
the cost, hold at least one unit in the Safety and Pipeline Stock, make
the Reorder Point at least one. Then, look at the expected saving from the
second unit. It is the same as the expected saving from the first unit,
except that the probability of two or more demands during the routine pipe-
line time must be used instead of the probability of
this, of course, is less than the probability of one
probability that exactly one unit will be demanded),
one or more demands;
or more demands (by the
so the net savings are
less. If, however, the savings still are'greater than the cost of keeping
the additional item, it pays to have at least two units. Do this for each
successive unit until you come to the Rth unit; the expected savings from
the Rth unit are greater than the cost of keeping the Rth unit, so it pays
to stock the Rth unit, but the expected savings from the (R+1)5t unit, i.e.,
one more unit, are less than the cost of keeping a unit, so stop at R units.
This process is summarized by Equation 2:1
(2) s.S1 P > kv >BA- P
-R+1 where terms are defined as in Figure 10.
R
In words:
A particular Reorder Point, R, is the economical Reorder Point if
the expected savings in shortage costs from the last unit in R (the
Rth unit) are greater than the cost of keeping the Rth unit, but
the additional cost of keeping one more unit, (the R+1)5t unit,
is greater than the expected additional savings from that unit.
1For the derivation of Equation 2 see Appendix I.
RM-1962
4-18-58
-33-
If the savings from the (R+1) st unit are greater than the keeping cost,
then the economical Reorder Point is greater than R. If the savings from
the Rth unit are less than the keeping cost, then the economical Reorder
Point is less than R.
Notice, the lower the cost of keeping an additional unit (k) or the
lower the unit price (v), the greater the economical Reorder Point tends
to be. The greater the shortage cost, the greater the relative benefits
from each unit are; hence, the greater the economical Reorder Point would
be. For higher-demand items and for longer pipeline times, the probability
of R or more demands would be greater during the pipeline time, and so the
economical Reorder Point would be greater. Since an increase in reorder
cost increases the Order Quantity and redUces the annual number of orders,
the greater the reorder cost, the smaller the Reorder Point, but this is a
rather indirect and small effect.
Now we have seen in principle how economical Order Quantities and
Reorder Points may be computed for a base. What sort of stock levels result
when we apply these principles?
B. Resultant Stock Levels
To evaluate these policies one must know what sort of base stock levels
are likely to result from the use of Equation 1 (to compute Order Quantities)
and Equation 2 (to compute Reorder Points) and how the levels are affected
by changes in the parameters used in their computation.
1. The Base Stockage Table
Table 3 is a sample table of base levels under a particular set of
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Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/10/25: CIA-RDP81-01043R002300090004-9
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RM-1962
4-18-58
-35-
hypothetical conditions,' viz., a 30-day pipeline time,2 an expected
shortage cost of $50 per unit, a reorder cost of $5 per order, annual
holding costs of 20 per cent of unit price (say, 10 per cent storage cost
plus 10 per cent capital cost), a five-year program, and negative-binomial
demand probability distributions with the variance equal to four times the
mean.3 It shows in a form which can be readily used by clerks, Reorder
Points and Order Quantities which might result from the application of the
proposed policies for those items which it has been determined will be
stocked at the base with unit prices of one cent to $500 and expected
annual demand rates from less than 2 up to 506 (monthly demand rates from
less than 1/6 to 42).
Note that the annual demand classes used for the table (0-2, 2-6,
6-12, ... 380-420, 420-462, etc.) are not equal in absolute or relative
width and that (except for the second) each price class is about twice
the width of the preceding class ($0.12-0.24, $0.25-.0.49, $0.50-1.00, etc.).
This choice of class intervals resulted from an analysis of the effects of
'Sample tables of base levels under some other conditions are
presented in Appendix II.
2B1 pipeline time we mean the time from the occurrence of that issue
at the base which reduces the stock-on-hand-and-due-in to the Reorder
Point until the resulting shipment arrives and is available for issue.
3The cost and probability assumptions are believed to be reasonable
guesses for many items in the Air Force supply system. But they are guesses
and not estimates. Therefore, Table 3 is presented for illustrative purposes
only. It is not necessarily recommended for use in either service-testing
or implementing the proposed policies.
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RM-1962
4-18-58
-36-
errors on system performance.1 It is designed to minimize the effects of
these errors. Sensitivity to error is discussed in Chapter V. Among other
things, the discussion shows that it is better to overstate the demand rate
than to understate it. Hence, the Reorder Points (R) and Order Quantities
(Q) are not computed for the midpoint of each cell but, instead, are computed
for a point near the lower end of the price class and near the upper end of
the demand class.
2. Sample Computation
For example, the levels for the cell "Demand, 72-90, Price, $4.00-7.99"
was computed for an item with a demand of 84 per annum (72 x 1/3 + 90 x 2/3)
and a price of $4.80 ($4.00 x 4/5 + $8.00, x 1/5).2
then carried out by solving Equations 1 and 2.
(1) Q-\/ 1+ 2rd
hv
assuming
r $5.00
d 84
h 0.20
v - $4.80
Hence,
The computation was
1If a cell computed on the basis of a demand rate of 10 is used for an
item with a demand rate of 8, an "error" of 2 units is made in using the
formulas.
2In the $0.01-0.12 price class $0.075 was used. In the cells to the
left of the heavy line the adjustment for a short base program which will
be described in the next section was used.
Q 1 !)2((:3 (41.32)
(3)
J876 = 29.60
RM-1962
4-18-58
-37-
We then enter 30 as the Order Quantity in the cell "Demand, 72-90;
Price, $4.00 - 7.99.? The next step is to find the Reorder Point, Rs
from Equation 2.
(2)
sd sd
Q EH) icy
Equation 2 may be rewritten:
(4) PR *7 lira-9" PR+1
For these values we have:
k = 0.351
V = $4.80
Q =. 30 (derived above)
s = $50
d = 84
So that Equation (4) becomes
( PR (0.3 )5s) .(EV;. (30)
PR4-1
( 6)
Then
pR 7 0.012143 Prol
1For an explanation of how to derive k for different expected program
lengths see Appendix I.
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S.
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/10/25: CIA-RDP81-01043R002300090004-9
(7)
RM-1962
4-18-58
-38-
The Reorder Point, R, is then found by comparing 0.012143 with the
values in a table showing the probability of j or more demands per pipe-
line time. Demands have been assumed to have a negative binomial distribu-
tion with variance four times the mean, expected annual demands(d) are
84 and the pipeline time (t) is 0.08 years. Hence, the table is used for
a mean of (84)(608) = 6.92 and variance of (4)(6.92) = 27.68.
Such tables can be computed by an electronic computer. The relevant
part of the particular table needed shows:
(Possible Reorder Probability of Exactly
Point) j Demands
p = 1 -
P(x)
Probability of j or
More Demands
0
0.044811
1.000000
1
.075283
.955189
2
.091468
.879906
?
?
?
20
.005539
.026200
21
.004400
.020660
22
.003486
.016261
23
.002755
.012775
24
.002173
.010020
25
.001711
.007846
By inspection of the probability table we find that:
P23 IT. 0.012775 7 0.012143 > P24 .1.= 0.010020
therefore, the Reorder Point, Ri, is 23 units. Thus, we have found the
entries (R 23, Q= 30) for the cell "Ahnual Demand: 72-90, Unit Price:
)4.00-7.99" in Table 3 .
RM-1962
4-18-58
-39-
Table 3 shows that the higher the demand rate, the greater the Reorder
Point and the Order Quantity. It also shows that the lower the price, the
greater the Reorder Point and the Order Quantity; however, there is no simple
relationship between the two. For a demand of 6 to 12 a year (1/2 to 1 a
month) the Reorder Point is 11 units for the cheapest item (about one or
two years' supply) and the Order Quantity is 70 units (six to thirteen
years' supply) -- the Order Quantity is seven times the Reorder Point. For
8-16-dollar items, again those with demands of about 10 per annum, however,
the Reorder Point is 4 (four to eight months' supply) and the Order Quantity
is 7 (seven to fourteen months' supply) -- the Order Quantity is less than
twice as great as the Reorder Point.
For items with a base demand rate of 110 to 132 annually, the relation-
ship is quite different. The Reorder Point drops from 39 units (about four
months' supply) for the cheapest items to 19 units (or two months' supply)
for the $250-$500 cell and the Order Quantity drops from over two years
to about two weeks supply.
With or without integrated data processing, the formulas from which
these Reorder Points and Order Quantities were derived can be used to
produce tables like Table 3 for use by bases.1 A moderate number of tables
should be sufficient for any one base. It should be noted that total cost
is barely affected if the Order Quantity for low-cost Category III items
is rounded to some standard unit, e.g., dozens, 25's, gross, etc., or, to
the nearest unit pack.
1If tables are centrally computed, the more rigorous process described
in Appendix I may be more desirable than the approximations provided by
Equations 1 and 2.
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Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/10/25: CIA-RDP81-01043R002300090004-9
RM-1962
4-18-58
-40-
In the high price ranges the errors introduced by using the approxi-
mation equations are relatively large. If computing facilities are avail-
able, it would be better to use the more rigorous formulation of Appendix I
for parts costing, say, more than $50 each. If tables are prepared centrally,
with EDPE, of course, the more rigorous form should be used throughout.
3. Effect of Pipeline Time and Shortage Cost
Table 3 is based upon a 30-day pipeline and a 50-dollar shortage cost.
What happens with other pipeline times and other shortage costs? As stated
in discussing Equation (2), the larger the shortage cost and the longer the
pipeline: the larger the economical Reorder Point. The Order Quantity, of
course, is not affected. Appendix II provides five additional tables
similar to Table 3 showing all combinations of 4-day pipelines (such as
might be realized in the ZI with highly effective air-electronic resupply)
and 30-day pipelines, and $50, $500, MOO shortage costs (which probably
bracket the relevant ranges of shortage costs for technical items).
Tables 4 and 5 give particular examples of the effect of different
pipeline times and shortage costs upon the Reorder Level. An examination
of these summary tables or tables in Appendix II will indicate the sensitivity
of the Reorder Level to the pipeline time and its relative insensitivity to
the value of the shortage cost. An increase in the pipeline time from 4 to
30 days causes increases in the Reorder Point of several months' stock in
most cases. In Table 4, for example, the increases range from about five
months' to nearly eleven months worth of stock (ignoring the upper right-
hand cell). In Table 5, the higher demand example, the increases are less
dramatic but are still large.
0
E-4 ??????
E r9
H
CD 0 N
1:1
5
0 A
tx1 e-I
E-1
0
0
$300.00 (Q = 1)
4 days 30 days
H
0
?0
04
P
?Co
H
r4
tl'is
2
$5.00 (Q = 10)
4 days 30 days
IA
rl
UN
0
Cq
1
H
r4
M
O-...
m
.....?
Unit Price: $ .25 (Q = 41)
Pipeline Time: 4 days 30 Days I
CN
m
y3
H
CN
01
01
tr?
H
H
H
m
M
Shortage Cost
s = $50
8
I
to
?
UN
44
11
to
c
H
1
t-
'0
=
c
H
I
:0
RM-1962
4-18-58
-41-
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Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/10/25: CIA-RDP81-01043R002300090004-9
RM-1962
4-18-58
-42-
fr.4
0
2
$300.00 (Q = 4)
4 days 30 days
0
H
t?-?
....?
N
......0
$ 5.00 (Q 30)
4 days 30 days
to
0
01
'5*
CV
Unit Price: $ .25 (Q 11e) ]
Pipeline Time: 4 days 30 days
tO
CV
0
CV
.......
0
CV
......
Shortage Cost
a = $50
8
N
0
0
Cd
,!.1
1
'0
t
"',.:4"
1
VD
'0
t
....,
sl?
RM-1962
4-18-58
-43-
In contrast, the Reorder Point is relatively insensitive to the value
of the shortage cost. A one-hundred-fold range in shortage cost is shown
in the Appendix and in these brief tables. Yet the largest increase (ignoring
the increase from zero in the fifth column) in either Table 4 or 5 is thir-
teen-fold. In the Appendix tables the same general situation is shown
throughout. In low demand items, however, which account for the bulk of
the items,1 the increase is proportionately greater than in the cases of
high annual demand.
The fact that the Reorder Point is not very sensitive to the value of
the shortage cost does not mean that the shortage cost can be ignored.
What it does mean is that taking it into account at all covers a large range
of possibilities reasonably well. In conttast, the 1167-111 rules, which take
no explicit account of shortage cost, provide entirely different /border
points. These (and the n67-1? Order Quantitites) are shown for comparison
in Tables 4 and 5.
Notice that the Order Quantity, which is shown for reference in Tables
4 and 5, is not affected by the differences in pipeline or shortage costs.
1See for example the Tables in Appendix III.
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RM-1962
4-18-58
-44-
C. Resultant Effectiveness and Costs
In Section A of this chapter, the stockage equations which provide effec-
tive support at or near least cost were derived. In Section B, the resultant
f.?
stock levels were described. Of primary interest, however, is the impact of
these policies on supply effectiveness and, second, the cost of achieving
that effectiveness. Four tables follow which compare base stockage under
the "67-1? rulesi and under the proposed /saes 'trith vailowshortage costs
and ,pipeline times. The comparisons are of limited significance, being
based on a single sample of issue data, viz., issue experience for B-47
2
aircraft spares at March and McDill Air Force Bases in 1953 and 1954.
It is assumed that each of the 7,000 items issued during that period, and
on which we have unit prices, is stocked at the base and that no other
parts are stocked. The "67-1" stock levels and the proposed stock levels
are then found, using the assumptions underlying Table 3 (except for pipe-
line time and shortage cost).
Table 6 shows the comparison with a 30-day pipeline and a 50-dollar
shortage cost. The 30-day pipeline is approximately the present routine
pipeline in the continental United States. The 50-dollar shortage cost is
chosen as an extreme on the low side. This represents a rough guess of the cost of
'The "67-1" policies used in computing these tables are the general
rules in effect 1 September 1957, viz., stock 15-day safety level plus pipe-
line stocks plus, for Category III items, 90 days' 'Operating Stock and for
Category II items 60 days' Operating Stock. These rules have been modified
since September 1957 and, in any event, were never applied blindly. They
are used here merely as a bench-mark.
2Laboratory Problem I in the LogistiCs,SyStemo Laboratory compared these
policies with more realistic current policiekand.'cleiseIy confirmed the
results reported here. This is discussed mi fully below.
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Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/10/25: CIA-RDP81-01043R002300090004-9
RM-1962
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-46-
priority action, which is the minimum action necessary in the event of a
shortage of a technical item and makes no allowance for the loss of military
effectiveness which might result from a shortage, for example, by having an
aircraft, missile, or an essential piece of ground support equipment out of
commission for parts.
Under the assumed costs, the proposed policies require holding sub-
stantially larger base-level inventories of Cost Categories II and III items
than does the "67-1" system. With the cost of holding the Operating Stocks
computed at 20 per cent per annum and the annual cost of keeping Safety and
Pipeline Stock taken as 35 per cent, the economical policies also require a
greater charge for stock than the theoretical "67-1" policy, i.e., $60,000
a year to pay for Operating Stocks versus $20,000 under "67-1," and about
$126,000 a year for Safety and Pipeline Stocks versus $87,000 under '167-1,n
a total of $186,000 a year versus $107,000.
What is the return from these additional expenditures? Rather than
the 24,000 routine requisitions needed each year if these 7,000 items are
stocked under the theoretical "67-1" policy, only 7,000 routine requisitions
are needed under the proposed policies. At $5 per requisition, the 24,000
routine reorders under "67-1" would cost $120,000; the 7,000 routine.requisi-
tions under the economical policy would cost only $35,000. A large part of
the savings would be in the form of reduced workload requirements for
management of Categories II and III in base and depot supply. Thus, the
out-of-pocket costs with the economical policy are less: $2210000 a year
as against $2270000. This difference is, of course, insignificant. What
is important is that the extra investment is roughly offset by the reduced
cost of supply operations.
RM-1962
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-47-
More important, the proposed policies increase supply effectiveness.
There is a striking difference in the shortages to be expected under the
two systems. Given the assumed probability distribution, we would expect
4000 shortages a year under the "67-1" policies but only 840 under the pro-
posed policies. This difference in the number of shortages is, of course,
of major importance in itself. It is achieved at no increase in out-of-
pocket costs. Further, if only the costs of priority action (assumed to
be $50) are considered, the reduction in shortages is worth $160,000. Thue,
the over-all cost of using the proposed policies is almost 40 percent less
than the cost using "67-1," strictly applied.
To summarize Table 6: With a 50-dollar shortage cost and a 30-day
pipeline, although the proposed policies call for larger stocks of Category
II and III items at base level and, hence, a larger annual cost for carrying
this investment, the reduction in routine reorders alone just about makes
up for this additional investment cost. In addition, there is a very
large increase in the supply effectiveness. The number of shortages is cut
by nearly 80 percent, with the over-all support costs for these items reduced
by nearly 40 percent. Procurement costs are increased, but the cost of the
requisitions, both routine and priority, which follow is greatly reduced.
Now, instead of the 50-dollar shortage cost, which takes
cost of priority action into account and ignores the effects
on base performance, Table 7 reflects a 5000-dollar shortage
is large enough to take account of a substantial chance that
only the
of shortages
cost, which
a shortage
has a considerable impact on operational capability. The cost of holding
the Operating Stocks is not affected by the shortage cost and is again
$20,000 for the theoretical "67-1" system, and about $60,000 a year for
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U)
RM-1962
4-18-58
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RM-1962
4-18-58
?49?
the economical system. The amount of Safety and Pipeline Stock does not
change for the 1167-1fl system with a change in the expected shortage cost)
because the 7167-1? formulas make no adjustment for shortage costs. Hence,
the average value of stock-on-hand-or-due-in to the base, with a 30-day
pipeline, is still $3500000 for the "67-1? system and has an annual keeping
cost of about $110,000 a year. The proposed policies do take shortage cost
into account. With a shortage cost of $5000, instead of $50, the proposed
policies increase the Safety and Pipeline Stocks by an amount large enough
to reduce the expected annual number of shortages from 840 to 8.40. Such
a reduction in shortages requires a large increase in base inventories of
Category II and III parts: $1,700,000 with a 5000-dollar shortage cost
yersus $350,000 under the theoretical n67-1" system. Just as there is no
change in the Operating Stocks, so there is no change in the number of
routine reorders--since shortage cost does not enter the equation for Q.
However, compared with the u67-1? system, the relative cost advantage of the
economical policies is even greater with a 5000-dollar shortage cost than
it is with a 50-dollar shortage cost. This is as one would expect, since
it explicitly adjusts to the shortage cost.
For $1.4 million in additional inventories, expected shortages are cut
from 4000 a year, under ?67-1,11 to less than 10. If there is a 10 percent
chance that a shortage will cause an AOCP and that on the average each
AOCP will last four days, 4000 shortages implies 1600 AOCP days, (.10)(4000)(4)=1600.
In terms of the number of aircraft available for operations, a reduction in
expected shortages from 4000 to 8.4 would be equivalent to adding 4.4
aircraft to the base. This can be thought of roughly as being equivalent
to making the administrative support aircraft operationally available.
This additional effectiveness is bought for only $1.4 million, or
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Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/10/25: CIA-RDP81-01043R002300090004-9
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4-18-58
-50-
$1,400,000 = $3200000 per aircraft. This provides a clue as to the upper
4.4
limit on the shortage cost. If, in the judgment of the managers responsible
for operations and logistics support, this is too high a price to pay for
an additional operational aircraft, the shortage cost (under the assumptions
stated above in this paragraph) should be less than $5000. It is important
to repeat two things: first, these calculations are presented solely for
purposes of illustration, and, second, the Air Force does not now in fact
follow u67-1? policies blindly, so the differences between the proposed
policies and present practice might well be smaller than these calculations
indicate, but they would certainly be in the direction indicated.1
Tables 8 and 9 show the effect of a routine pipeline time of 4 days
instead of 30 days. Table 8 is similar to Table 6, except that the pipe-
line is reduced from 30 days to 4 days. Under both sets of policies, the
value of stockrequired is reduced: Under the u67-1fl system it is reduced
from about $350,000 to only $2000000; under the economical policy, it is
reduced from $6600000 to about $350,000, a 40-to-50 percent reduction in
each case. The Operating Stocks are unchanged by this change in pipeline;
the number of routine reorders a year are still 24,000 for ?67-1? and 7000
for the economical system. With the 50-dollar shortage cost and the 4-day
pipeline, the ?67-1? policies (if the 15-day safety level were maintained
with the 4-day pipeline) would show slightly fewer shortages than the pro-
posed system.2 Because of the large reduction in the number of routine
'Base-Depot Model Studies (The RAND Corporation Research Memorandum
RM-1803, 1 January 1957), indicates that bases, in fact, hold more stock
than the H67-111 system calls for; hence, adoption of the proposed policies
would, result in a smaller increase in effectiveness than is shown in the
tables but would require a smaller addition to base inventories.
2The character of the shortage is drastically changed; however, under
"67-1? most of the shortages of items carried in stock would be for Category
III items; under the proposed policies almost all would be for Category II
items.
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RM-1962
4-18-58
-51-
Total Costs
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Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/10/25: CIA-RDP81-01043R002300090004-9
RM-1962
4-18-58
-52-
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RM-1962
4-18-58
. -53-
reorders, however, the cost of operating under the proposed system is still
about one-third less than the cost of operating under the u67-111 system.
It is desirable, under either the '167-1" system or the economical policies,
to pay something to reduce the pipeline time. It would be easier to reduce
the pipeline times with the proposed policies in use than with the '167-1"
rules because the former call for a much smaller number of requisitions.
With a 5000-dollar shortage cost and a 4-day pipeline, shown in Table 9,
the ?67-lu policy would again have lower costs than with a 30-day pipeline)
but, since the proposed policies take shortage costs into account and
reduce shortages to an expected 8.4 per year, they would allow savings
of over $3.8 million a year, or over 90 per cent. Again, these savings
can be thought of as being equivalent to increasing the capability of the
base.
D. Summary
This chapter has developed the approximation formulas for economical
stockage: Given the decision to stock an item at a base, reorder costs
are balanced against holding costs to find the economical Order Quantity
for any given line item; and the expected gains from the shortages avoided
by having an added unit of Safety and Pipeline Stock are compared with
the cost of keeping that unit to determine the Reorder Point. The types
of stock levels which result under a long and stable base program from the
applicatim of these principles have been examined, and the theoretical
effectiveness and costs of operating under the proposed policies have been
compared with those of the It67-ln policy under stable conditions. Appli-
cation of the proposed policies for Cost Category II and III items would
increase base supply effectiveness markedly. Such an application would
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Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/10/25: CIA-RDP81-01043R002300090004-9
RM-1962
4-18-58
apparently require an increase in the value of base inventories in these
Cost Categories,1 but once it was installed and working it would also
greatly reduce the workload on base supply and on supply personnel at
depots or storage sites by reducing the number and frequency of both
routine and priority requisitions.
The next chapter will examine how the proposed system operates when
account is taken of some of the dynamic aspects of base operation.
1As is developed in Chapter VI, it is not clear that there would be A
net increase in the total value of system stocks of Categories II and III.
RM-1962
4-18-58
-55-
III. DYNAMIC ADJUSTMENTS TO BASE LEVELS
The preceding Chapter assumed first that base programs (i.e., expected
mean demands' for the items at a base) would be stable for a long time into
the future. But, in fact, weapons phase into and out of the Air Force and
individual bases.2 A particular model or series of aircraft may be assigned
to a base for only two or three years (or even less) and there are always
some bases which are nearing the phase-out of some weapon.
Before implementing the proposed policies, it is necessary to look
into the effect that the phasing-in and -out of a weapon system may have
upon the proper levels of base stocks for parts peculiar to that weapon.3
That is the subject of this Chapter. Spebifically, it will outline how
quantities for initial-support tables are found and how an approaching
phase-out of a base, or the short life of a program at a base, will alter
somewhat the economical Reorder Points and Order Quantities.
A. Quantities for Initial Base Support
In this section it will be assumed that bases convert to new weapons
at once -- or over very short spans of time. This appears to be realistic,
and for the lower-cost parts moderate deviations from this assumption will
cause only inconsequential over-stockage in the early months.4
10f course, the erratic deviations from the expected mean demand
were considered.
23tockfisch, J. A., Logistics Support During Phase-In of the F-102,(U)
The RAND Corporation, Research Memorandum RM-2166, May 2, 1958, p.l. (Secret)
3For parts common to many weapons, the program for a particular weapon
has little effect. Parts used on only two, or a few, weapons constitute a
borderline case requiring further examination.
4For the depot, of course, the corresponding assumption cannot be made;
in Chapter IV a method of taking account of the fact that demand increases
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RM-1962
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To stock the first base the best available estimate of mean demand for
each item should be used and the base be stocked up to its Stock Control
Level for each. The equations or the tables should be used in determining
the levels. As will be discussed in the next section, some modification of
the formulation in the preceding chapter may be called for for very low-
demand, low-price parts.
The problem of how to get a usable estimate of the demand rates for
individual line items is a serious one; and it is beyond the scope of this
Research Memorandum. When the initial support tables for the first base
have to be computed, only sketchy information is available from limited
test experience, engineering estimates and experience with similar parts
in other applications. Such estimates are subject to wide error. For low-
cost items it is desirable to keep the probability of stockouts very small,
but for high-cost items it is appropriate to run a greater risk of shortage
Therefore, in determining the initial-support tables for the first base,
judgment should be applied to whatever engineering or other estimates are
available, "leaning toward the high side" for the cheapest parts and, if
anything, tending to underestimate demand for the more expensive items.
As a weapon program grows, after the activiation of the first units,
every effort should be made to adjust the initial demand estimate on the
basis of actual experience. As subsequent units or bases convert to the
new weapon, their initial support tables should be computed, taking account
during the early stages is explained. In the event that at any base the
major weapon is expected to phase in slowly, or to be slow in building up
to its normal level of activity, that method could be used for base stocking
during the phase-in. For most of the Category III parts, however, such re-
finement would probably be unnecessary.
-_-_-_-_-_?-_,,---?-?-?-?,'.-?4
Declassified in Part - Sanitized Copy Approved for Release
mm-1962
4-18-58
-57-
of the demand experience at the first few bases converted. The errors in
the first demand estimates can be corrected in time to influence provisioning
expenditures only if the procurement leadtime is considerably shorter than
the period of phase-in of the weapon. If the leadtime is short enough to
permit adjustments in procurement to be made on the basis of improved
information, moderate over- or under-estimates of demand early in the phase-
in will not be very serious.
B. The Phase-,Out or Short Program
The formulation (Equation 2) for the Reorder Point takes account of
the dynamic aspects of weapon phasing, because terminal obsolescence is
1
explicitly included in the keeping cost..
On the other hand, Equation 1, for the Order Quantity, does not take
account of the dynamics, since it is based on the assumption that the base
program will last indefinitely. This assumption is satisfactory whenever
the base can be expected to order the item several times in the future, in
which case no change is required. It is not accurate, however, if the base
can be expected to order the item only once or twice during the remainder
of the program.
For the lowest-price low-demand parts, the economical initial stock
may be so large that no reorder is expected during the life of the program.2
Thus, for these items the initial stockage should take account of the length
of the program. Further, as the end of the program approaches, the routine
reorders for more and more parts will be such as to cover, or nearly cover,
lAppmdix I, D.
2See, for example, the upper left cells in Table 3.
50-Yr 2013/10/25: CIA-RDP81-01043R002300090004-9
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RM-1962
4-18-58
?58?
the needs through the expected remainder of the program. In this case,
too, the program life should be considered in determining the appropriate
Order Quantity.
The consideration of program life has two effects: First, it is desir-
able to reduce some Order Quantities to avoid excessive stockage for the
(remaining) short program. Second, it is desirable in certain other cases
to increase the Order Quantity to reduce the chance that additional orders
will need to be placed during the remainder of the program.
These points can best be explained with reference to Table 10. In the
upper left portion of the table, there are many Order Quantities which equal
several years? expected consumption. For example, the third cell in the
second column calls for an Order Quantity of from 5 to 14 years' supply.
For a one- or two-year program this is excessive. On the other hand, the
fifth cell in the seventh column calls for an Order Quantity approximately
equal to one year's supply. It may be economical, if an order is needed
one year from the end of the program, to order a few more than the 46
units shown so as to reduce the risk of incurring an additional reorder
cost, still nearer to the end of the program. Thus, in some cases, as
the end of a program approaches, the Order Quantity should be reduced to
avoid excessive stockage and excessive terminal obsolescence, and in some
other cases increased to avoid the risk of excessive reorders.
Looking at the matter a little more formally: Since demand is typi-
cally erratic, a final shipment is the last shipment only with some degree
of confidence, not with certainty. The larger the shipment, the greater
the probability that it will be the last shipment, and the greater the
number of units likely to be left over at the end of the program.
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Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/10/25: CIA-RDP81-01043R002300090004-9
Rm-1962
4-18-58
?60?
In finding the "final,' Order Quantity, essentially the same sort of
problem must be faced as in setting the economical Reorder Point: each
additional unit in the "final" Order Quantity yields some expected savings
in future reorder costs by reducing the chance that a reorder will be needed,
but each additional unit must be bought and stored at the base, probably
until the end of the program.
The savings expected from each additional unit must be compared with
the expected cost of adding that unit to base stocks. If savings are less
than costs, the unit should not be shipped.1
The effects of making the computation of the "final" Order Quantity
for items with various unit prices and demand rates are shown on Tables ll
and 12. These tables are all based on the cost assumptions of Table 3.
Table 10 might apply to a very long program or to common parts, e.g.,
commercial hardware. However, given the general uncertainty of Air Force
programs, there is little advantage in using a table for more than, say, a
five-year program. In that case, Table 10 should be adjusted to read as
Table 112 which shows the long-program Order Quantities crossed out and
replaced by those appropriate for a five-year program. Note that in Table
11 some of the Order Quantities have been adjusted upward and some downward.
3
With a one-year program, as in Table 12, the adjustments apply to
nearly half of the area shown on these tables. As the phase-out of the
1For a more detailed statement, see Appendix I.E.
2
Table 11 has the same Order Quantities for the range of demands covered
as does Table 3 which is also a five-year table.
j
Table 12 summarizes the adjustments. Each long-program Order Quantity
appears in the upper left-hand corner of a cell. "Final" Order Quantities
11
RM-1962
4.-18-58
?
a.
N
to
o
N
54
o
H
4
t.--
N
tr.
V-
M
..0
N
W
H
M
H
0,
C.-
. _
a (i
,i...
80
0 4
-.1N
M
rk.
so
...
H
w
m
H
w
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as
0..
,
m
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N
A
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NO
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H
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0.
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4
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0,
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sl cv
LAW
M
a
0
N
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0
LA,
I
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M LI-%0
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a
iq
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m
n
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'r-i I-1
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n
4
0
N
1
N
0
H
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C.-
...0
M
fit > rIPQF
F-1
where: QF = the economical Final Order Quantity;
2
'Whereas the relevant comparison for the base is Q hi > (si + ri)PQB
the discounted depot shortage cost taking account
of the fact that the shortage would last throughout
the post-termination leadtime.
the Final Buy Quantity
the probability of QBor more demands between now and
In interpreting eouation (4) the reorder cost should always be taken
as the post-termination reorder cost since QB would normally be exhausted
only very late in the program. The unit cost, used in determining hi
(Appendix I), should be the pre-termination price if the Final Buy is in fant
1
small error is introduced in this formulation since the shortage cost
would be incurred only if the demands in the rest of the program exceed QB
while the reorder costs are incurred if demand exceeds
BM-1962
4-18-58
-83-
to be made before the termination of production. Otherwise, it should, of
course, be the post-termination price.
D. Depot Reorder Points
Now that the determination of the depot Order Quantity under various
conditions has been described, we can discuss the depot Reorder Point com-
putation. The general policy to be applied in the determination of depot
Reorder Points is to use the equation (2) used for the determination of
base Reorder Points, with those modifications necessary to make it fit the
depot problem. The differences in the interpretation of the factors in the
equation have been outlined above. There remain only three central problems,
all related to the dynamics of the program of the end items being supported.
They are: (1) the determination of what demand rate to use in the computa-
tion of R; (2) the effect of the termination of production of the item upon
Ri and (3) the appropriate R to use in the event that, at any point in the
program, bases have life-of-type Operating Stocks.
1. Determining the Appropriate Demand Rate: As in the case of computing
Q, there is a problem of deciding what demand rate to use in computing R.
The size of R depends upon the demand rate used; in a growing program, it
will be smaller if demand expected in the first year should be used than
if demand for the third or fourth year
demand rate at the peak of the program
demand near the end of the phase-out.
were used. Later in the program,
would call for a larger R than would
Of course, the demand rate to use is
the expected demand rate at the time when the depot is expected to reach its
Reorder Point.
a. For those parts where the depot will stock up to the Stock Control Level
at the beginning of the program, the important date is that date at which
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R14-1962
/4 -i-55
-84-
the depot Order Quantity is expected to be exhausted, as that is the time
when the Safety and Pipeline Stocks probably will be needed. In this case
the depot operating period determines the appropriate system demand rate
for determining the depot Reorder Point. In other words, using the program
information for the date at which the depot Order Quantity is expected to
be consumed, estimate the system demand rate and from it derive the depot
Reorder Point.
For those parts which the depot will not stock up to its Stock Contro)
Level early in the phase-in, the relevant date in determining R is that at
which demands on the depot may first be expected from the bases.
taking the program level expected at the time when the first base
phased inlis expected to reach its Reorder Point, estimate system
Therefore,
to be
demand
for the item in question. Compute the depot Reorder Point on the basis of
that demand estimate. In this way, the depot Reorder Point is made consis..
tent with the expected system demand at the time when orders will probably
be placed against the depot.
For some items this procedure will result in depot Reorder Points
based upon expected demand rates late in the program.2 Since, at the time
of initial stockage, demand late in the program is most uncertain, this
proposal must be examined more closely. Taking two extreme situations:
1If for any reason some operating base other than the first is expected
to use up its Operating Stocks earlier than the first, the relevant date
is, of course, the date at which that base is expected to hit its Reorder
Point. This does not appear to be a case of major practical importance.
2The natural tendency to provide ample base stockage for a new weapon
will cause this to be true in more cases than are indicated in the Tables
or in,Eauation (1), because this tendency has the same effect as an over-
estimate of base demand rates.
RM-1962
4-18,58
-85-
(1) Suppose, first, that the first base is expected to require resupply
at the peak of the program. Then, the depot Reorder Point would be com-
puted on the basis of the highest system demand expected for the entire
program. At first glance this may seem to be a source of serious over-
stockage. More careful examination indicates that, unless demand per program
element is badly overestimated in determining R, this is not apt to be the
case for three reasons. First this situation typically will arise only
for the low-value, low-demand items, so any overstockage is not likely to
be serious in dollar terms. Second, for such items the Reorder Point
usually constitutes only a small fraction of the Stock Control Level. Third,
if there is a demand against the depot at that time, part of the (large)
Safety and Pipeline Stocks will be used to fill the base order or orders
and a new Order Quantity and Reorder Point will be computed.
The real problems arise only if demand per program element has been
seriously overestimated. During the phase-in, it should be possible to
adjust the Reorder Point to reflect the demand actually experienced during
the early part of the phase-in. With the present manual data processing
system this could be done only imperfectly but it is not impossible. It
appears to be highly desirable to increase the data-gathering at the first
bases issuing any new weapon and, absent even that, it might be possible to
catch the worst errors from the Stock Balance and Consumption Reports.
For a program phasing in over three years, the occurrence of the first
demand against the depot at the program peak could arise (under the assump-
tions of Table 3) for items with an annual value of issues of $5.50. This
group of parts accounts for only a small fraction of the dollars invested
in inventory but for a modest proportion of the line items.
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RM-1962
4-18-58
-86-
(2) Take as the second extreme the case where bases are initially
stocked with life-of-type stocks; this procedure (of setting R by rising
the demand expected when a base first orders on the depot) would, of course,
result logically in zero depot Reorder Points -- the bases would be expected
to reach their Reorder Points when system demand should be zero.
Early in the program, when demand estimates are very unreliable and
when there are few bases from which the part might be drawn if demand were
far higher than estimated, it is clearly risky to operate with no depot
stocks. On the other hand, there is at least as good a chance that future
demands will have been overestimated as underestimated, so there is some
advantage in limiting depot stocks. There are many ways of meeting this
situation (which, of course, will arise only in the case of very inexpensive
items1). One simple way is to brImg into the depot at the beginning of the
program the full amount of the Stock Control Level for one or two of the
bases which will receive the weapon late in the phase-in. In this way, the
system has some degree of protection should the demands at the first few bases
prove so much greater than expected that they must be resupplied early. At
the same time, should the demand prediction turn out to be roughly correct or
overestimated, the depot can ship out to the later bases the stocks brought
in for them and no extra procurement need be incurred. Further, if informa-
tion that demand has been greatly overestimated is made available to the
appropriate manager early in the phase-in, he may be able to curtail the
1Cf. Table 3.
RM-1962
4-18-58
-87-
orders outstanding and use the depot stocks, which were originally earmarked
for, say, the last two bases to be phased in, to support several bases earlier
in the phase-in.
There remains the problem of determining the Reorder Point later in
the program. Once demands are made against the depot by the bases, the
depot should be stocked up to the Stock Control Level, and the earlier
description of how the depot Reorder Point should be determine4pusing
expected system demand at the end of the depot operating period, applies
except that the termination of production should be taken into account.
2. Termination of Production: Once the item goes out of production,
the Reorder Point should be high enough to provide adequate protection
through the longer procurement leadtime, any increase in price will have a
(probably slight) offsetting tendency. The increase in reorder cost will
have only an indirect effect through increasing Q, and hence slightly
decreasing R. One procurement leadtime before the termination of productiaa
of the item, the Reorder Point should be computed, using the projected post-
production leadtime, price and Q determined with the post-procurement values.
The appropriate demand rate is the rate expected to be in effect one operating
period after the termination of production. R and Q computed in this way
provide the Stock Control Level for the period after the end of production.
If the stocks on hand in the depot one procurement leadtime before the end
of production are less than the new Reotder Point (plus expected depot issues
prior to the termination of production), a buy equal to the difference between
the new Stock Control Level and the expected depot assets at the end of
production should be made atthe in-production prices.
3. Depot R's and Base QpIa: One contingency remains to be discussed.
What should be the depot R after the initial phase-in if the bases are holding
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1114-1962
4-18-58
-88-
life-of-type Operating Stocks? The logic of the argument of this chapter
calls for a zero Reorder Point under these conditions and, possibly, no
depot Operating Stocks either. This is too risky. Early in the program
the system could be adequately protected as discussed above by the stocks
held at the depot for bases phasing in late. During the phase-out, some
assets could be returned to the depot to constitute a system reserve. There
remains the problem of the mid-range of a program. Here it appears that
the appropriate policy might be to hold back from some of the bases their
full life-of-type Operating Stocks or to rely on redistribution between
bases to cover individual base shortages. This is an area in which manage-
ment will have to exercise judgment until a practical and reasonably rigorous
solution is worked out.
E. Summary of Depot Stockage Policies
In general, the stockage rules developed in Chapters II and III for
base stockage are taken over and modified to develop depot stockage rules.
Some redefinition of terms is needed to make the formulations fit the depot
situation, but the chief problems of developing depot rules are associated
with the dynamic aspects of the growth and decline of weapon system and
other end-item programs.
Because of the dominance of the dynamic aspects of depot stockage, no
simple static comparison between the present and proposed policies such as
was presented for the base situation in Tables 6-9 would be of value. The
overall effect of applying the proposed policies at the depot level would
appear to involve reduced stocks of the relatively high-cost, high-demand
items, because of the weight given to the cost of holding and keeping them
in the system.
RM.-1962
4-18-58
-89-
Early in the program, at least, there should be a similar reduction
of depot stocks of the great bulk of low-cost, low-demand items. This fol-
lows from the fact that the proposed policies explicitly take account of
the fact that the bases should be stocked with up to several years, expected
consumption of such items. Early in the program the depot stocks of such
items should be very small and, assuming that it is feasible to have direct
shipment to the bases, should remain small for many of these items during
most of the phase-in.
Later in the program the depot stpcks of such items would theoretically
be greater than in the present system and the frequency of reordering them
from the manufacturers would be correspondingly decreased. As individual
items go out of productiopp a terminal buy is called for and its net effects
as compared to the terminal buys in the present practice are impossible to
determine at this point.
In order for the system to operate at maximum effectiveness, there is
a real need for better information coming from the earliest operational
locations of a new weapon or other end item to the managers, because much
of the economy of depot stockage depends upon being able to react in a
period of a few months.
The stockage of common parts at the depot levels presents no great
problems because of the fact that the dynamics of a particular program are not
apt to influence significantly the consumption of common items. All that
is required in this case is that the equation developed for base stockage
be interpreted to reflect the depot situation. Thus, the reorder cost is
the procurement reorder cost, the relevant pipeline time is the rrocurement
leadtime, the shortage cost is the cost of overcoming a shortage at the depot
and the pertinent demand rate is the system consumption or condemnation rate.
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RM-1962
4-18-58
-90-
For purposes of illustration, the following three tables (13-15) trace
out without elaboration or qualification the major steps, questions and
decisions for depot stockage of items peculiar to particular end items.
The appropriate initial depot stockage (Table 13) for an item will
depend upon the depth to which the bases are initially stockedvand second-
arily upon the size of the depot Order Quantity and the total expected con-
sumption of the item during the program. If the stocks at the first bases
are expected to last for a relatively long period, say in excess of six
months, the depot need only stock up to its Reorder Point plus some addi-
tional protection such as the Stock Control Levels of one or two bases.
If the base stocks are initially so small that the depot can expect to have
demands placed against it in the first few months of the program, the depot
should stock up to its Stock Control Level initially. In the unlikely event
that the bases might have short operating periods but the depot Order Quan-
tity is large relative to the expected issues throughout the program, the
depot should stock only a Final Buy Quantity.
Later in the program when the base receives an order or request to
ship to a base, Table 14, it should normally make the issue unless doing
so will cause a shortage at the depot. If the issue carries the depot's
stocks to or below the Reorder Point, the depot should compute its own Order
Quantity using the system demand expected in the relatively near future.
If the item can still be obtained at the pre-termination price and reorder
cost, Q should be computed using those values of v and r;if not,the post-
termination values should be used. Once Q is computed, a determination
should be made as to whether to make a Final Buy computation. If so, that
is the amount to be stocked. It should be noted that the depot Final Buy
G
Get estimates
of post-termination
r and t if
available
Compute Q
Stock QB 1
Eyes?
Table 13
Initial Depot Stockade
Areearly base Ws 7/
6 mo. base demand?
no
? Yee ??
Compute Q using system
demand at procurement
leadtime plus one year
for Cat III items pills
6 mos. for other items
Compute R using Q and
demand as described above
Stock R plus Q
RM-1962
4-18-58
-91-
At what date is first
depot demand expected
from base?
Compute R using the
operating period de-
rived above and system
demand at the end of
that seriod
Stock at depot R as
computed plus SCL for
two bases expected to
phase in late in the
program.
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RM-1962
4-18-58
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Make ad hoc
decision as
to how to handle
the order'
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Table 14
Depot Reaction to a Base Order
_____yes
Can item still be bought at
pre-termination values of
yes ro
Compute QB
with post-
term. r and
pre-term. v
Will filling the
order carry stocks
to or below R?
Compute QB with
post-term.
values of r & v
Stock QB
Yr
Will it carry stock
to below zero?
Compute Q with post-?c
termination values
of v and r
Is Q t.d?
rc)
Make
issue
Make
issue
Determine expected
system demand 1 yr.
or 6 mo. plus lead-
time in the future,
0 Can item still be
bought at in-production
v and r?
yLs
At the end of the
operating period, can
item be bought at pre-
term, values of v & r?
no
Compute R using the
system demand rate at
the time Q is expected
to be exhausted and
using post-term. values
of r, t and v.
Compute Q with those
values of v and r
Compute R using the
system demand rate
yes-at the time Q is
expected to be ex-
hausted and using
the pre-term. values
of r, t and v.
Stock Q plus R
3-Falls outside the scope of this Research Memorandum
No
Action
yes
Terminal Buy
Upon receipt of information that
r, v and t will increase at a
date one procurement leadtime
in the future
LCompute Q using pre-termination
rice & post-termination reorder cost.
1
2
Is
Determine expected system
demand rate when Q is ex-
acted to be exhausted.
Compute R using that
demand rate and the post-
termination values of
r, v and t.
Is stock on hand plus
due-in Q R?
n
RM-1962
4-18-58
-93-
Compute QB using post-
term. values of r, v & t.
Is stock or
hand plus
due-in
QB?
no
uy up to Q
yes
No
Action
1Criteria for disposal are outside the scope of this Research Memorandum.
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Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/10/25: CIA-RDP81-01043R002300090004-9
RM-1962
4-18-58
-94-
computation takes account of the risk of incurring a depot shortage. If
a Final Buy is not appropriate, a new Reorder Point is to be computed with
the appropriate values of r, t and v and the depot should bring its stock
on hand and due-in up to the new Stock Control Level.
At least for the higher value Category II items, but if possible for
all items, it is desirable to recompute the levels shortly before the item
is scheduled to go out of production so that the Air Force may buy at the
pre-termination prices and reorder costs. The post-termination values
of r, t and v should be used in computing R and Q and, if appropriate, in
computing Q. The depot stocks should be brought up to the sum of R and Q
or to QB as appropriate; Table 15 illustrates the steps involved.
V. DATA REQUIREMENTS
RM-1962
4-18-58
-95-
Any inventory policy requires data if it is to operate. The purpose
of this chapter is to describe the data needs of the proposed policies
and to discuss the sensitivity of the policies to inaccuracies in the data.
By understanding the latter a reasonable judgement can be made about the
effort that should be devoted to improving the accuracy of each kind of
data.
It is not intended to discuss systematically here how to make the
required estimates, nor will the sources of the data in the Air Force,
the information flows or the data manipulations be treated. At present,
joint efforts to determine some of the parameter values are under way by
personnel from RAND, from the AMC Directorates of Plans and Programs and
of Supply with some assistance from Comptroller personnel. It appears
that adequate data and estimates can be obtained without excessive
cost or delay. The policies will operate far more effectively with an
integrated electronic data-processing system than with the present
predominantly manual and mechanical system. However, even with present
data processing the proposed policies should result in considerably more
effective and economical stockage than do the present practices.
For the purposes of applying the equations, the data requirements
may be classified as follows:
1. Identifying and program information 6. Reorder cost
2. Demand Data 7. Shortage cost
3. Pipeline Time
4. Holding cost
5. Keeping cost
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R11-1962
4-18-58
-96-
These are the parameters required for solving Equations 1 and 2 and
for setting the "final" Order Quantity. Explicit information on demand
and on pipeline times is required by the u67-1? system also. The ?67-1?
policy parameters carry implicit assumptions about the other variables.
A. Sensitivity
Before discussing further the kinds of data required, it is worth
considering the accuracy with which the various parameters need be de-
termined. None of them can be determined with perfect accuracy. Demand
and shortage cost are particularly hard to estimate. Even unit cost,
which at first glance seems very straightforward, is not, for the cost
that we would like to have is the value of the item to the Air Force at the
time a stockage decision is being made. The cost information available
is usually the price at which the item was last purchased. Since the price
may have risen or fallen since then, the last price is only an estimate of
the cost of purchasing the item now. Further, for items in short supply
the actual value to the Air Force may be a great deal more than the
current purchase price, and for items on which disposal action is appropri-
ate the value to the Air Force and on the market may be much less than the
purchase cost new. This illustration is presented here merely to show that
at best any value for the parameters will be an estimate.
Also, it is relatively easy to get good estimates of some parameters
and much harder to get good ones of others. One of the common problems
in applying improved management techniques is, in fact, that the costs of
getting the necessary information may exceed the value of the improvements.
One should not spend more effort and money in estimating any particular
parameter than it is worth.
s.
RM-1962
4-18-58
-97-
Tables 16-19 summarize sensitivity information to provide some guidance
on the importance of accuracy of the different variables. Tables 16 and 18
use a 30-day pipeline time and Tables 17 and 19 a 4-day pipeline time.
Tables 16 and 17 cover a moderately low demand case: an annual demand of ten
units per base year. The tables are based upon the distribution of demands
and prices shown in Table I of Appendix III. They show the percentage
increase in cost resulting from using a wrong estimate of each parameter in
setting base levels. In computing the tables values were assumed as true
values for all the parameters in equations (1) and (2); then R, Q and the
total system cost were determined with the approximate equations, given those
values. The results and the percentage increase in cost over the exact
equations appear at the top and bottom respectively of each table. R and Q
were then determined, with one parameter "estimated" at twice or half its
true value. The costs of using this R and Q in the face of the "true"
meter, value were determined and compared with the cost realized with R and 4
computed using the true value. The percentage deviation of the "erroneous"
cost from the true cost is shown in each table.1
The upper portion of each table shows those errors which tend to increase
stockage above what it should be, and the lower portions show those which tend
to reduce stockage. Thus, an over-estimate of demand and an under-estimate
of price are both in the upper portions. In nearly all the cases shown,
errors causing over-stockage are less costly than are the corresponding
errors resulting in under-stockage.
3-Since the comparison is made using the approximate formulas in a few
cases, where the error in the estimate results in levels closer to those which
would' have been obtained with the accurate equations, the use of an erroneous
value actually decreases cost.
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RM-1962
4-18-58
-98- Table 16. Sensitivity Table (using a Five Year
Program and a Correct Yearly Demand of 10)
30 Day Pipeline
The Correct
Input Data
Unit Value = v
Cost Per Year
,
The Solution of the Approximation Formulas
using Correct Input Data.
$ .30 $ 4.80 $300.00
$ 3.74 $23.79 $245.87
Erroneous
Input Data
Yearly Demands = 2d
Reorder Cost = 2r
Unit Value = iv
Shortage Cost = 2s
The Solution with Input Errors Which Cause
an -Increase iii_the Average Stock Level.
g Increase in the Cost PacIgaz
8.6 8.7 9.8
6.7 -4.1 -14.6
7.8 -1.1 - 3.9
1.6 2.9 2.4
Erroneous
Input Data
Yearly Demands = id
Reorder Cost = ir
Unit Value = 2v
Shortage Cost - iS
Poisson Dist.
The Solution with Input Errors Which Cause
a Decrease in the Average Stock Level
/ Increase in the Cost per Year
20.1 30.6 9.8
6.7 1.3 0
9.1 2.1 0
2.4 2.6 9.8
25.4 1.4 0
% Increase Over Exact
Formula
_
.5 4.9 36.6
Note:
d = 10
r= 5
S = 50
The correct distribution is a negative
binomial distribution with a mean of 10
and a variance of 40.
1,1
?
Table 17. Sensitivity Table (Using a Five Year
Program and a Correct Yearly Demand of 10)
4 Day Pipeline
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-99-
The Correct
Input Data
Unit Value = v
Cost Per Year
The Solution of the Approximation Formulas
using Correct Input Data.
$ .30 $ 4.80 $300.00
$ 3.06 $14.72 $ 95.00
Erroneous
Input Data
Yearly Demands = 2d
Reorder Cost 2r
Unit Value- 1
- ilI
Shortage Cost .., 2s
The Solution with Input Errors Which Cause
an Increase in the Average Stock Level.
cZ, Increase in the Cost per Year
24.8 13.4 - 1.8
9.5 - 2.6 - 1.8
10.1 .- 1.0 - 1.8
2.3 4.3 0
Erroneous
Input Data
Yearly Demands .. Ad
Reorder Cost = ir
Unit Value = 2v
Shortage Cost = is
Poisson Dist.
The Solution with Input Errors Which Cause
a Decrease in the Average Stock Level.
/ Increase in the Cost p2/: Year
11.1 23.2 0
7.5 12.8 0
11.4 23.2 0
1.3 3.8 0
6.2 0 0
,
% Increase Over
Exact Formula
.3 10.1 1.8
Note:
d = 10
r = 5
s = 50
The Correct Distribution is a negative binomial
distribution with a mean of 10 and a variance
of 40.
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Table 18. Sensitivity Table (Using a Five-
Year Program and a Correct Yearly Demand of 84)
aQ Du Pipeline
The Correct
Input Data
Unit Value = v
Cost per Year
The Solution of the Approximation Equations
using Correct Input Data.
$ .30 $ 4.80 $ 150.00
10.56 74.35 1235.65
Erroneous
Input Data
Yearly Demand = 2d
Reorder Cost = 2r
Unit Value = iv
Shortage Cost = 2s
The Solution with Input Errors Which Cause
an Increase in the Average Stock Level.
/ Increase in the cost per Year
15.1 19.8 34.7
2.7 - .7 - 7.7
3.5 .9 - 3.4
.7 2.3 4.3
Erroneous
Input Data
Yearly Demand = id
Reorder Cost = ir
Unit Value = 2v
Shortage Cost = is
Poisson Dist.
The Solution with Input Errors Which Cause
a Decrease in the Average Stock Level.
/ Increase in the Cost 221.- Year
47.9 78.0 47.1
2.7 5.5 4.3
6.5 9.2 8.6
.9 2.4 4.1
55.1 63.1 15.0
% Increase Over
Exact Formula
,
0 1,0 13.7
Note:
d = 84
r= 5
s = 50
The Correct distribution is negative binomial
with a mean of 84 and a variance of 336.
Table 19. Sensitivity Table (Using a Five-
Year Program and a Correct Yearly Demand of 84)
pa pe1ine
RM-1962
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-101-
,
The Correct
Input Data
Unit Value = v
Cost Per Year
,
The Solution of the Approximation Equations
Using Correct Input Data.
$ .30 $ 4.80 $150.00
8.75 47.77 518.40
Erroneous
The Solution with Input Errors Which Cause
Input Data
an Increase in the Average Stock Level.
/ Increase in the Cost pa Year
Yearly Demand = 2d
7.0 8.5
12.5
Reorder Cost = 2r
3.7 .2
-10.0
Unit Value = iv
4.1 1.2
- 7.0
Shortage Cost = 2s
.5 1.7
7.3
,
Erroneous
The Solution with Input Errors Which Cause
Input Data
a Decrease in the Average Stock Level,
(/ Increase in the Cost per Year
Yearly Demand = id
17.5 25.8
49.8
Reorder Cost = ir
6.1 7.3
7.4
_Unit Value = 2v
7.4 11.2
13.4
Shortage Cost = is
1.5 1.0
9.0
Poisson Dist.
49.9 38.0
9.0
% Increase Over
Exact Formula
.1 .7
14.2
Note: d = 84
r = 5
s = 50
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The Correct distribution is negative binomial
with a mean of 84 and a variance of 336.
ft
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B. Description of Data
The significance of sensitivity will be discussed further in con-
nection with the individual kinds of data. The data will be taken up in
the order indicated in the enumeration on Page 95.
1. Identifying and Program Information
Unit cost, stock number, noun and other descriptive information,
applicability and program information for each item are all necessary if
the proposed policies are to be implemented fully. Cost, program and
identifying information are required for any reasonable policies. Applicability
and program information are desirable both to improve the demand estimation
and, in particular, to make the "final order" and other "dynamic" computations.
The policies can be used without these data, but to the extent that it is
practical to get the information the policies can be used more effectively.
Applicability information is not systematically available on all items
now, but in connection with other efforts to continue improving the supply
system it is being gathered and disseminated increasingly. The other kinds
of information are currently available in usable form.
2. Demand Data
Any system of inventory management is based upon some kind of pre-
diction of demand. Tables 16-19 show that errors in the estimation of
demand can be very costly, especially if demand is underestimated. In the
examples, estimating demand at half its true rate would result in
inventory costs of up to 78 percent greater than they should be. Conse-
quently, it is worthwhile to expend considerable effort in making as good
estimates of demand as possible.
J.
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Demand, however, is one of the most difficult parameters to estimate.
Errors of many hundred percent are not uncommon. Early in a program, at
the time of initial provisioning, for example, accurate prediction is im-
possible. As the program develops every effort should be made to improve
the accuracy of the estimates from experience. This is undoubtedly one of
the major payoffs to be gained from an integrated data processing system.
As was discussed earlier (Chapter IV), early in a program the depot
should not be stocked up to its full Stock Control Level. In this way
it is possible to hedge against over-stocking the system. The idea is
to bring in depot stocks only up to the Reorder Point (with exceptions
discussed above), and when additional information about the actual demand
for the part is available to bring the rest of the depot stocks into the
system or to reduce the planned stockage for bases coming in later. Unless
demand data are accumulated, properly identified to exclude one-time demands
and other peculiarities and made available to the appropriate manager early
in the program, this way of economizing on initial provisioning cannot
be exploited.
At the base level, at present, it is possible to get information on
the recent issues, and, using this plus class knowledge and knowledge
of program changes, usable estimates of future demands can be made. However,
these, too, are crude as is indicated by the fact that typically a very large
proportion of the demands upon base supply are for items which are not stocked
at that base.1
1
A substantial amount of research has been undertaken at RAND dealing
with the nature of demand. B. Brown, Characteristics of Demand of Aircraft
Spare Parts, The RAND Corporation Report R-292, July 1956; T. Goldman,
Relationships Between Program Elements and System Demand for Airframe Spare
Parts, The RAND Corporation Research Memorandum RM-1858, 24 January 1957;
T. Goldman, A Priori Demand Prediction - A Case Study of B-52 Airframe Parts,
The RAND Corporation Research Memorandum RM-2088, 10 January 1958.
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An error in the mean demand rate changes both the Order Quantity and
the Reorder Point and changes them in the same direction. The sensitivity
of these levels to the demand rate is greatest for high-cost, high-demand
items and least for low-cost, low-demand items. Good estimates of the
demand are more easily made for high-demand items than for low-demand items.
In implementing the proposed policies, therefore, data should be gathered
more carefully and in greater detail for the higher-cost items than for the
lower-cost items. A practice of analyzing demand (and re-computing levels
if necessary) each time the stock-on-hand-or-due-in falls to within one unit
of the Reorder Point would automatically accomplish this objective; under
the proposed policies, the average number of orders a year is nearly propor-
tional to the square root of the dollar value of annual consumption. For
example, if $200 worth of an item is consumed annually at a base, the base
will reorder twice a year; if $800 worth is consumed, the base will reorder
four times a year. Thus, given the demand rate, the higher the unit cost,
the more often would levels be revised. An alternative is to compute demand
and levels whenever demands exceed or fall short of expectations by more
than some predetermined amount.
RM-1962
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-105-
Since probability considerations influence the Reorder Point, the type
of demand distribution which exists for an item is important. Very few
items have sufficiently high-demand rates to permit a reliable estimate of
demand distribution to be made on an individual item basis. Probably the
best way to estimate demand distributions is to study the relatively high-
demand items in a particular property class or functional group and to find
the kind of demand distribution which fits the data best. In this way it
May be possible to find probability distributions applicable to large groups
of parts.
Notice that in Table 17 the use of the Poisson distribution, if the trUe
distribution were negative binomial with a variance equal to four times the
mean, would introduce only moderate errors except in the lowest price ranges.
For the higher demand rate shown in Table 19 the cost is far more sensitive
to error in selecting the probability distribution. The greater sensitivity
in the case of the high-demand parts is at least partially offset by the fact
that it is easier to get a good estimate of demand characteristics for high-
than for low-demand parts.
3. Pipeline Time
Both the current and proposed systems require pipeline times; they are
nearly as important as demand data in setting Reorder Points which are based
on the average demand per pipeline time. The variance in pipeline time is
also needed to specify the probability distribution of demands per pipeline
time, but computationally this variance can be incorporated into the demand
variance, and the comments above on demand variance apply equally to pipe-
line yariance.
For the depot the relevant pipeline time is, of course, the contractor-
depot pipeline time including both Air Force and manufacturer administrative
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leadtime, materials purchase, fabrication, assembly and set-up cost as appro-
priate, and the transportation, communication and similar times.
Data on pipeline times should be relatively easy to collect.
4. and 5. Holding Costs and Keeping Costs
Holding cost is the sum of the physical holding cost, engineering obso-
lescence and a capital charge. Keeping cost also includes these same
elements. An error in any of these elements will have exactly the same
effect as an error of the same proportion in unit cost.' The physical
holding cost consists of the cost of storing the item in a warehouse, main-
taining its static records, inventorying and inspecting it and complying
with the non-engineering technical orders pertaining to it. These elements
of holding cost should not be particularly difficult to ascertain with
satisfactory accuracy.
Engineering obsolescence is the expected cost that will be incurred per
year in modifying the item. Statistical estimates of this cost are possible
with little difficulty.
The capital charge is another matter. It includes the interest rate
but is really the opportunity cost of capital, that is, a measure of the
value lost to the Air Force by not spending the price of the item in ques-
tion in some other way. This is a difficult concept theoretically and fur-
ther research on it will be needed before completely satisfactory estimates
can be obtained.
Since the holding cost is really an expression of the value of money to
the Air Force, it is one of the elements in the equations which can be mani-
pulated to apply certain kinds of policy constraints systematically through-
'Hence, holding cost and keeping cost errors are not shown separately
in the Tables.
y
- r - "nr - ?
RM-1962
4-18-58
-107-
out the system. For example, in the event that the budget available at some
particular time should not permit stocking the system up to the levels called
for by these policies and that it should, consequently, be necessary to re-
duce stockage, it is not easy to see how one might best do so. By raising
the capital charge, Stock Control Levels and Q and R would be reduced system-
atically in a manner which should cause minimum disruption. Such a step
would be less disruptive than an arbitrary decision to reduce all stocks by
some specified percent for example.
Keeping cost includes holding cost and also depends on the length of
time the item will be kept at the base or depot respectively, and on the net
salvage price at the end of the base or system program. An error in either
of the last two factors will cause an error of the opposite sign in keeping
cost per unit. For example, if a program is expected to last for four years
but it lasts for five, the keeping cost will be somewhat larger and the Re-
order Point somewhat lower than they should be.
Errors in holding costs affect the Order Quantity, and through keeping
cost, the Reorder Point. For most items, errors in the obsolescence charge
in the keeping cost have only moderate effect on the Reorder Point and none
on the Order Quantity. Hence, system performance is less adversely affected
by an error in holding or keeping cost than by one in demand rate or pipeline
time.
6. Reorder Costs
Reorder costs determine the frequency with *hich items are to be ordered
and have less effect on Reorder*Points than do the other factors discussed so
far; but they affect the Order Quantity and the total cost significantly, up
to 13 % in one case in Table 17.
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''????? ? .
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The costs to be taken into account are all those costs which are incurred
whenever a reorder takes place, but which are independent of the size of the
reorder. For depot-base resupply they include the paperwork costs at base and
depot; the costs of stockpicking, handling, packing and shipping the minimum
order (one unit, one unit pack, etc.) at depot; the corresponding costs of
receiving, handling and warehousing at base; communications and accounting
costs. For the procurement reorder cost, the relevant costs are, again, those
incurred whenever a reorder is placed regardless of the size of the order.
They include the Air Force and contractor administrative costs, depot receiv-
ing, handling, etc. - or in the event of a direct shipment to base, the cor-
responding base cost - manufacturer's set-up cost if the item is not in pro-
duction, and the cost of packing and shipping the minimum order. These should
be relatively easy to estimate and because of their limited effect on total
cost the effort required to determine them adequately should not be very great.
7. Shortage Cost
The shortage cost is the most difficult of the parameters to estimate.
Fortunately, system performance, as measured by cost, is fairly insensitive
even to errors as large as those shown in Tables 16 - 19. Given the insen-
sitivity of total costs to the shortage cost and the difficulty in obtaining
accurate estimates, perhaps a desirable goal might be to find shortage costs
within a factor of two or three, e. g., an estimate of $1,000 might be sat-
isfactory so long as the true shortage cost lay between $500 and $2,000.
As mentioned above, the fact that a shortage cost must be taken into
account does not imply that one need estimate the cost of losing a war or of
not having a critical weapon available in the critical phase of the war or
gnything of the sort. It implies that it is reasonable to get some estimate
of the average cost of overcoming a shortage. The concept is very similar
?
t-
"t'''''???""a015M-SrFlIM.
.%7
RM-1962
4-18-58
?109?
to the concept of having a specified level of confidence that a demand will,
in fact, be met.
One can look at the shortage cost in any one of three ways. First, it
can be thought of, as a minimum, as the average cost of the supply and other
work needed to overcome a shortage if one occurs. At the base these costs
would consist of the cost of searching for the item in maintenance, receiving
and elsewhere on the basej the extra cost of premium communication to and
transportation from the depot and perhaps most important, the cost of dis-
rupting routine activities. This would provide a minimum estimate of the
shortage cost, but in spite of expedited action there is some risk that a
weapon would be out of commission because of some part shortages. The short-
age cost, therefore, should be greater than the costs of the expediting
action. One can think of an upper limit to the shortage cost in the follow-
ing way. In the worst case if an item is short, a missile (or aircraft) will
be MOCP (or AOCP). (In some cases a maintenance facility could be idled.)
Obviously, one way to avoid the consequences of such a shortage is to pro-
vide the operational unit with an additional weapon to stand-in for the one
out of commission. The maximum cost of an expected part shortage would be
the cost of providing stand-by weapons as a means of overcoming the effect
of a shortage. Of course, providing a component or a black box rather than
a stand-by weapon might be a considerably less costly way of overcoming a
shortage of most parts. In such cases the cost of keeping on hand the next
higher assembly would constitute the shortage penalty for the item in
question.
The purpose of the above paragraphs is not to indicate how to estimate
the shortage cost but only to show that it is not an impossible concept.
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Further research is under way at RAND and in AMC to get usable estimates of
shortage cost.
There is a third way of looking at shortage cost. It can be considered
not as an objective cost) but as a device whereby a policy decision can be
introduced systematically throughout the system or in a part of it, the
policy decision being to increase or decrease the risk of a shortage. A
decision to decrease the chances of shortages throughout the system, for
example, could, if funds were available, be affected by computing stock
levels with an increased shortage cost. Similarly, greater protection for
high-precedence units than for low-precedence ones could be assured by using
differential shortage costs in computing their levels.
Since the shortage cost is not a well-understood concept, let us look
it it another way. The establishment of an inventory at all implies that
there is some shortage cost. From the level of that inventory, one may est-
imate'the implicit shortage cost? Table 3, showing stock levels, was de-
rived by assuming that demands during a routine pipeline time had negative
binomial distributions with variance four times the mean, 20 percent-per-
annum holding costs, and a five-year program (35 percent keeping costs.)
Given the same parameters, the "67-1" policies imply the shortage costa
shown in Table 20. The implied shortage costs vary from three cents to over
$1,300. What is important about Table 20 is not the absolute levels of the
implicit shortage costs, which depend on the demand distribution used and
on the holding-cost rate, but rather the ratio of the shortage cost for the
price on the one hand and the relationship of the shortage cost to the demand
rate on the other. With a 30-day pipeline time the implied shortage cost is
1
An implicit shortage cost is the shortage cost which must prevail if
the levels are correct.
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-111-
SHORTAGE COSTS IMPLIED BY "67-1" POLICIES *
Annual
Demand
Rate
Unit
30-day:Pipeline
Cost
Reorder
Point
Order
Quantity
Shortage Cost
-
11/3
4 .10
0
1
$ 0.03 - $ 0.56
1 1/3
5.00
0
1
1.31 - 27.81
1 1/3
250.00
0
1
65.62 -1,390.62
10
.10
1
3
.03- .06
10
5.00
1
3
1.70 - 3.07
Io
250.00
1
1
28.31 - 51.21
103 1/3
.10
13
26
.04 - .05
103 1/3
5.00
13
26
2.22 - 2.69
103 1/3
250.00
13
9
38.36 - 46.48
* Note: The parameter values are the same as those used
in Table 3, except for the shortage cost.
Table 20
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'4
3
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nearly 30 times as great for a 250-dollar item when the annual demand is
1-1/3 as when that rate is 103-1/3. There is no reason to assume that a
single shortage is that much more important for a high-demand item than for
a low-demand one.
An inventory system which sets the Reorder Point without considering
unit cost has the implicit assumption that shortage cost is proportionate
to price, that is, a shortage of a 5-dollar item is treated as being 50
times as serious as the shortage of a 10-cent item, and the shortage of a
250-dollar item is treated as about 25 times as serious as the shortage of
a 10-dollar item and 5 times as serious as the shortage of a 50-dollar iteni,
if all three items are stocked. It may well be true that shortages of
higher-cost items are more serious than shortages of low-cost items, but
the implication that shortage cost is proportional to price appears to be
altogether unreasonable.
=
'44:1
'
RM-1962
4-18-58
-113-
VI. CONCLUSION
This Memorandum has described a simple and practical method for de-
termining base and depot stock levels in the face of the dynamic and un-
certain environment with which the Air Force supply system has to deal.
The method takes account of the major variables which must be considered
to achieve economical and effective provisioning and distribution under
such circumstances.
The logic of the basic method is straightforward. The factors which
determine the costs and effectiveness of the supply system are identified
and the manner in which they affect cost -- after converting effectiveness
into a cost-equivalent through the use of the shortage cost concept -- has
been spelled out mathematically. This statement, equation (IA.) of Ap-
pendix I, is the keystone of the whole discussion. It has been "solved"
to show those values of the Order Quantity and Reorder Point which would
provide the minimum system cost. The least cost solution for the Reorder
Point is equation (2). Equation (1) for the Order Quantity is an approxi-
mation of the rigorous solution in the Appendix (I.5).
The least cost statements which are the subject matter of Chapter II
and of the first Section of Chapter IV provide rules for provisioning and
distribution where dynamic elements are not of major significance. In
depot stockage and where the remaining life of the program at base is short
the dynamic elements must be taken into consideration. At the base this
means that the approaching termination of the program must be taken into
account in determining the Reorder Point through the gradual increase in
the keeping cost. This is incorporated in equation (2). For the Order
Quantity, adjustments in the equation are required; the expected costs
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of holding additional operating stocks must be balanced against the expected
cost of incurring an additional reorder near the end of the program. This
is the subject of Chapter III.
Depot stockage is dominated by its dynamic aspects. Early in a weap-
onts life the depot can take advantage of the fact that most of the items
in base supply will be stocked adequately for relatively long periods and
the depot need not stock up to its Stock Control Level until it, in fact,
faces demands from the bases. Such a practice would allow the depot stocks
to be kept reasonably small and would reduce the risk of gross overproaure-
ment because of serious overestimation of the demand rate.
Depot stockage also must reflect the fact that the costs and lead time
involved in procuring items increase markedly when the item goes out of
production. This and the problem e phasing-out the weapon from the system
as a whole are taken into account through a terminal buy computation.
The corrections for the dynamic aspects of the problem are not rigor-
ously correct; but they do take account of the major factors in the problem,
they are reasonably simple to implement and they do not have excessive data
requirements. Further they provide close approximations to more elaborate
dynamic programming computations which have been developed at RAND.
Consequently the proposed policies appear logically sound; but without
actually implementing them, there is no certainty as to how well they will
really operate. Some effort has been made to get an empirical evaluation
of the policies. The simple comparison in Chapter II with the rigid appli-
cation of ?67-1it policies, indicates that the proposed policies promise
considerable improvement. However, since the policies in 67-1 are in fact
applied with a good deal of judgment and since those policies have been
RM-1962
4-18-58
-115-
changed in recent months, that comparison is not particularly important.
Further, it covers only the impact of applying the proposed policies at base.
A much more meaningful comparison has been made in the Logistics
Systems Laboratory Project I (LP-1). The full details of that expetiment
will be developed in other publications. However, because of the fact that
it provided a far superior semi-empirical appraisal of these policies than
is available in any other wey,some major features of the experiment and
some relevant results are summarized here.
The proposed policies were a part of a proposed Logistics System
called the 1'1960 System" which also incorporated deferred provisioning and
data processing innovations. That system was compared over the simulated
life of a weapon with the u1956 System". The latter system was developed
by Air Force personnel, who were
present current best practice in
were compared not with the rigid
serious approximation of current
members of the Laboratory staff, to re-
the Air Force. Thus the proposed policies
application of the manuals but with a
Air Force practice.
The LP-1 comparison had the further advantage of being a system com-
parison. Bases, parts repair depots, IRAN facilities and storage sites
were all stocked in accordance with the policies of the two systems.
Maintenance and operational activities were controlled in such away that
the comparisons between the supply systems would not be vitiated by off-
setting adjustments in those areas. Both logistics systems were further
constrained to stand ready to "fight" a war at any time during the program.
In short, every effort was made to make the comparison scientifically sound
and operationally significant.
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RM-1962
4-18-58
-116-
The general results of the comparison are shown in the following charts.
Figure 11 compares some measures of the costs of the two systems. The
system in-and-out movements provide some measure of the routine supply
workload and the priority requisitions, of the expediting workload of the
two sets of policies. The proposed Category II and III policies were only
39 and 25 per cent respectively as costly as the "1956" policies. The
investment in parts was negligible (less than 1 per cent) greater with the
proposed than with the "1956" policies. This is true in spite of the fact
that under the proposed policies considerably more, especially of the cheap
parts were stocked at the base level.
Figure 12 provides the more important comparison of the policies) on
the basis of their supply effectiveness. The "1956" policies resulted in
2.6 times as many Category II and III AOCP-days as did the proposed policies;
they showed 15 per cent more ANFE's and more than double the number of
stockout days of the proposed system.
These results do not provide any final "proof" of the superiority of
the proposed policies, and certainly they do not prove that these are the
best of all possible policies. They are, however, evidence that the
policies are superior to present policies.
There is a great deal of interest at all levels in the Air Force in
implementing policies such as these. Work is underway with Headquarters
AMC and Headquarters USAF on such implementation. In the process of
development and implementation in the real environment, a great deal more
will be learned. Modifications and perhaps further simplifications of the
decision rules will suggest themselves.
Total system in-and-out Priority requisitions Cost of spares purchased
(thousands of dollars)
movements
Fig.!! ?Supply cost implications of proposed versus
LP-I experience for categories 11 and III (fourteen quarters)
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Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/10/25: CIA-RDP81-01043R002300090004-9
Fig.I2 Supply effectiveness implications
of proposed versus"1956"policies
LP-I experience for categories 'Iran(' m (fourteen quarters)
RFI-1962
4-18-58
-119-
To summarize, the research underlying this Memorandum leads to the
following conclusions:
A. Stockage rules should consider:
1. Expected mean demand for each item,
2. Variability of demand,
3. Unit value,
4. Cost of incurring a reorder,
5. Cost of holding the Operating Stocks,
6. Cost of expected terminal obsolescence (termination
of the program being supported),
7. Expected shortage cost,
8. Resupply and procurement pipeline times.
B. The dynamics of weapon-system or other program phase-in or phase-
out can be taken into account effectively by
1. Limited depot stockage during the early part of a
program, subsequently stocking the depot to its
full Stock Control Level;
2. Gathering and analyzing consumption data intensively
early in the phase-in and reacting to that information;
3. Using a "final buy" calculation during the later
strizes of the program and for some of the least costly
low-demand items early in the program; and
4. Using a "terminal buy" calculation at the time when
an item is expected to go out of production.
C. With an integrated data processing system, these results can be
largely achieved by using the equations developed in the mathematical
appendix. With a manual data processing system and local determination
of levels, tables based upon the formulas can be used by clerical personnel
Lo set the appropriate levels.
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Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/10/25: CIA-RDP81-01043R002300090004-9
RM-1962
4-18-58
-120-
D. The Air Force can increase supply effectiveness, decrease personnel
pressures in supply and achieve dollar economies by adopting the policies
described and proposed in this Memorandum.
E. Because the proposed policies permit reduced management per line
item, their use should free management to manage the more costly and
critical items better, or, alternatively, it might permit reducing somewhat
base-level manning where -- as in hardened-missile installations -- there
is a premium on personnel space.
F. Further research is needed at RAND to extend the scope of the study,
and further developmental studies are required, particularly in the Air
Force, to derive adequate estimates of cost and other parameters.
G. The rapid development of an integrated data processing system
will improve the application of these as well as other supply policies.
RM-1962
4-18-58
-121-
Appendix I
MAT LW TICAL APPENDIX
In Section A of this Appendix the base cost equation for a stable
program is stated. From it, the Order Quantity equation and its approxi-
mation are derived and the Reorder Point equation is derived: Sections
D and C. Section D is a short statement of the derivation of the holding
cost, and Section Edertves and explains the final Order Quantity
calculation.
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Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/10/25: CIA-RDP81-01043R002300090004-9
1
RM-1962
4-18-58
-122-
A. Fundamental Base Cost Equation with a Stable Program
Let
= Reorder level for an item;
Q = Order quantity;
c(R,Q) = Annual variable cost of operating with a
particular R and Q;
= Average annual demand rate;
= Unit price;
m Annual unit holding cost as a fraction of
unit price;
m Annual unit keeping cost as a fraction of
unit price;
= Reorder cost;
m Unit shortage cost;
OD
ZR = E (X-R)P(X) = Expected number of shortages
XmR+1 per order placed if the reorder
level is R, where
P(X) = probability of exactly X demands
during a routine pipeline time.
Then, to a first approximation,
1.1) c(RQ)hv(i) + + + :c11 Z
2 Q Q R.
RM-1962
4-18-58
-123-
B. Economical Base Order Quantity
To find the particular Q associated with minimum cost,
given R, set
ac/?= 0
1.2) aciaQ hv d(r+sZR)
2
1.6)
1.7)
0 hv d(r+sZR)
=
2
42 2d(r+sZR)
hv
If R is chosen so that ZRCs.1 1.5 becomes
The expected annual number of orders,
d
2dr
hv
dhv
2r
1/See AFM 67-10, p. 149.
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Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/10/25: CIA-RDP81-01043R002300090004-9
RM-1962
4-18-58
-124-
This solution (1.8) assumes that C(RA) is a continuous
function. However, fractional shipments are impossible; there-
fore, different equations must be used.
Subtract 1.1 from
1.9) c(R,Q-1) h_vc1 kvR + dr + ds
2 c1=1 Q-1 R
which yields
1.10) c(R,Q-1) - c(R,Q)_ hv
+ d(r+sZR) _ _ hv +
Q-1 Q
2_
I.11a)
1.12)
1.13)
Define
_ hv + d(r+sZR)
2
If R is the lowest cost Reorder Level
2e +hv d(r+sZR) 0
2
2.44 d(r+sZR) 0
2 2E +hv
1.14) Q . 1 4, A. 2d(r+sZR) 1 + 1
2 2 +hv 2 4 hv
2d(r+sZR)
Similarly,
1.15) c(R,Q) - e(R,Q+1) _ hv = d(r+sZR)
2 Q244,
0
1.16) hv-2e/ _ d(r+sZR)
" 0
2
(1'4,1Q
L.17)
1.18)
1.19)
1.20)
1.21)
1)
Hence:
Q -2 +
2
2d(r+sZR)
0PQF
where QF = the economical "final" order quantity,
r' = the discounted cost of a shipment of one unit
at the end of the program;
h' = the discounted cost of holding one unit from
now until the end of the program;
PQF = probability of QF or more demands from now
until the end of the program.
= (r+v)e-in
Also, if the salvage price is zero,
111 = vg+ 1)(i-e17
?
Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2013/10/25 ? CIA-RDP81-01043R002300090004-9
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/10/25: CIA-RDP81-01043R002300090004-9
RM-1962
4-18-58
-133-
Appendix II
SAMPLE BASE TABLES
This Appendix consists of six base tables reflecting different pipeline
times and shortage costs, but otherwise based on the same assumptions as
Table 3 in the text. By comparing them the reader can get a detailed im-
pression of the effect of differences in these values over the whole array
of prices and demand rates. Of course, only the Reorder Point is affected
by shortage cost and pipeline time in the approximations upon which these
tables are based. Consequently, tables of just Reorder Points rather than
tables including the Order Quantity would have been adequate for just this
purpose. It was deemed better to show the full tables to permit other com-
parisons and to show how the relationship between R and Q is affected by
different values of the shortage cost and pipeline time.
Comparisons of the effect of differences in the values of the other
parameters (other than the frequency distribution and the variance) can be
made visually. Thus, using any one table, the effect of different values
of d and v can be readily determined by comparing rows or columns. The
effect of differences in h and k can be made in the same way, since price
is multiplied by h and v. Thus, doubling h has exactly the same effect
upon Q as does doubling v. Similarly, a change in k (with h constant) has
the same effect upon R as does a change of the same proportion in v. Changes
In h have a more complex effect upon R and cannot be read easily from the
tables.
Lastly, the Order Quantity increases as the square root of the reorder
cost. So the effect of differences in the Reorder Point can be determined
by inspection. The effect of changes in r upon R are indirect and, over
most of the cells in any of the tables, small.
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Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/10/25: CIA-RDP81-01043R002300090004-9
Appendix III
RM-1962
4-18-58
-141-
The following tables show the relationship between stock-list price
and numbers of units issued for B-47 and F-8611 spare parts.
The data are classified according to the cells of Table 3, with number
of units issued converted to annual rates.
The B-47 data were obtained at March and McDill Air Force Bases during
1953 and 1954. They represent approximately 18 base months! experience.
The F-86H data were obtained from Clovis Air Force Base during 1956
and represent about six base months' experience.
In both cases, items with no issues, on which prices were not available,
or which had units of issue other than 'teach,' (e.g., pounds, feet) were
excluded.
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2013/10/25 : CIA-RDP81-01043R002300090004-9
Appendix Table 111.1
RELATIONSHIP BETWEEN PRICE AND ISSUE RATES OF B-47 SPARE PARTS
Annual Demand Rate
......"1.-...
THig.1,
at leas but
'
2/3
1-1 3
5-1 3
11-1/3
19-1/3
29-1/3
41-1/3
55-1/3
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89-1/3
109-1/3
131-1/3
155-1/
181-1/3
209-1/
239-1/3
271-1/
30546
341-1 3
379-
419-1 3
461-1/3
Total
Price less than
IMIIIMMINTMEAlma
241
330
266
219
109
173
55-1/3
76
71-1/3
144
89-1/3
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2 9-1 ",
271-1 *,
0 -1/111MIEMENIME
co
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14
32
9
8
5
12
1
5
3
2
3
1
1
4
628
1.00- 1.99
207
132
48
38
19
18
9
7
6
4
2
1
2
2
1
2
1
1
500
2.00- 3.99
216
124
40
31
24
17
6
8
5
1
3
2
1
1
1
1
1
485
4.00- 7.99
179
118
33
33
14
11
11
5
1
2
2
1
1
6
1
1
1
420
8.00- 15.99
158
86
31
25
11 -
6
1
2
2
1
4
1
328
16.00- 31.99
131
91
30
11
9
4
2
4
1
3
1
2
1
290
32.00- 63.99
118
66
23
14
8
10
1
1
1
2
2
1
247
64.00- 124.99
118
79
20
19
8
6
4
1
2
2
1
1
1
1
1
1
265
125.00- 249.99
84
60
26
11
11
10
3
2
1
1
1
1
211
250.00- 499.99
68
38
21
22
3
2
5
3
1
2
1
166
500.00- 999.99
35
43
6
17
2
6
2
3
2
1
1
118
000.00- 1 999.99
28
13
6
8
4
3
1
1
2
1
67
2 000.00- 3 999.99
24
16
13
7
4
1
1
1
1
68
4 000.00- 7 999.99
6
7
5
2
1
1
22
8,000.00-15,999.99
2
2
3
2
1
1
11
16 000.00-31 999.99
1
2
1
1
1
6
32 000.00 and over
Total
2160
1562
783
603
1
169
2
117
108
90
140
6
IIITUIMIIIIIIMIMIEMINIIWMI
a
-
Source:
Demand
data collected
at March
and Ma
MINEMIEW1
1 .
Appendix Table 11/.2
RELATIONSHIP BETWEEN PRICE AND ISSUE RATES OF F-866 SPARE PARTS
Annual Demand Rat
111.1
, more than but
0
2
6
12
20
30
42
56
72
90
110
12/
156
162
210
240
272
306
342
360
421
412
Total
Un:i.t.,.. ,
3.:-.. ''......?,1.ess th
2
6
12
20
30
42
56
72
90
110
128
151
182
210
240
272
306
342
360
420
462
top
..> - .01- .1
15
11
19
18
21
6
16
14
7
26
10
5
15
2
8
10
9
7
17
15
337
.13-
4
6
3
6
9
6
5
2
3
8
2
2
1
2
71
.25- .4
7
5
9
7
6
2
4
3
6
2
4
1
4
1
1
21
67
.50- .99
11
10
22
8
11
2
5
2
2
4
1
1
.1
1
1
86
1.00- 1.9
16
21
10
10
8
7
6
2
1
2
1
2
1
1
2
1
2.00- 3.99
37
35
14
8
14
6
3
4
1
4
1
1
1
1
4.00- 7.99
60
8
42
19
12
5
1
1
1
1
8.00- 15.99
31
23
16
10
4
3
2
1
1.1.1
1
1
1
93
16.00- 31.99
39
49
41
18
9
4
M
1
161
32.00- 63.99
24
12
4
1
3
3
2
1
1
1
64.00- 124.99
19
6
10
8
2
1
1
47
125.00- 219.99
10
22
12
8
1
1
1
1
1
a
250.00- 499.99
8
7
4
1
1
1
1
1
2
500.00- 999.99
11
6
3
2
2
3
1
1
2
1
1,000.00- 1,999.99
2
6
1
1
1
1
1
2
1
17
2,000.00- 3,999.99
1
2
1
2
1
1
1
9
4 000.00- 7 999.99
8 000.00-15 999.99
1
1
16 000.00-31 999.99
1
32 000.00-63 999.99
o
64,000.00 and over
Total
296
259
210
126
102
48
55
6
2
o
22
11
1
26
MIIMMIMIIIMMII
Source:
Demand
data ceillected
.,.t Cann
n Air F
rce Base
Clovi
New Mexico
February
to June
19 6.
IM
111.11.1.1.1111
MIIIIIMIHM
1.1111111111
Mill
MIMI.
Mil
IMIMII
MIMI
MM.
IMI
=MM..
MI
1111.M.
MIIIMEIM
I.
Mil
Z: 7 rg
11111MMME
6 c3"
Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2013/10/25: CIA-RDP81-01043R002300090004-9
S.
? S.
The RAND Corporation, Santa Monica, Calif
USAF Project RAND
111-1962
STUMM POLICIES FOR MEDIUM- AND LOW-CCT PARTS by A. R. Ferguson and
L. Fisher; Research Memorandum E14-196, April 18, 1958, Unclassified.
158 pp. incl. illus.
ASTIA Document NO. AD 156001.
A method for determining brae and depot stock levels in the face of the
dynamic and uncertain environment with ehith the Air Force supply system
has to deal. The method examines the major variables which must be
considered to achieve economical and effective provisioning and distribu-
tion under such circumstances. The study indicates what stockage rules
should include, how the dynamics of weapon-system or other program pima/t-
in or phase-out can be taken into account, how to free management to manage
the more costly and critical it. better, what further rose/m.6h is needed,
and hal/the application of these as well as other supply policies can be
improved.
?
The RAND Corporation, Santa Monica, Calif.
USAF Project RAND
114 -1962
STOCKADE POLICIES FOR MEDIUM- AND LOW-COST PARTS by A. R. Ferguson and
L. Fisher; Research Memorandum 114-1962, April 18, 1958, Unclassified.
158 pp. incl. illus.
ASTIA Document NO. Al) 156001.
A method for determining base and depot stock levels in the face of the
dynamic and uncertain environment with which the Air /bine supply system
Dm to deal. The method examines the major variables which must be
considered to achieve economical and effective provisioning and distribu-
tion under suth circumstances. The study indicates what stockage rules
should include, how the dynamics of weapon-system or other program phase-
in or phase-out can be taken into account, botrto free management to manage
the more costly and critical items better, what further research is needed,
and hovrtbe application of these as well as other supply policies can be
improved.
?
The RAND Corporation, Santa Monica, Calif
USAF Project RAND
ret-1962
3T0CKAGE POLICIES FOR MEDIUM- AND LOW-COST PARIS by A. R. Ferguson and
I. Fisher; Research Monorsohme 111-1962, April 18, 1958, Unclassified.
158 pp. incl. illus.
MIA Document $0o. AD 156001.
A method for determining base and depot stock levels in the face of the
dynamic and uncertain environment with which the Air Force supply system
has to deal. The method examines the major variables which rust be
considered to achieve economical and effective provisioning and distribu-
tion under such circumstances. The study indicates what stockage rules
should include, how the dynamics of weapon-system or other program phase-
in or phase-out can be taken into account, how to free management to manage
the more costly and critical it. better, what further romerth is needed,
and hal/the application of these as well as other supply, policies can be
improved.
?
The RAND Corporation, Santa Monica, Calif.
USAF Project RAND
am-1962
STOMCADE P0LSCIE3 FOR MEDIUM- AND Low-cosr PARTS by A. R. Ferguson and
L. Fisher; Reeser& Memorandum 114-1962, April 18, 1958, Unclassified.
158 pp. incl. illus.
AST/A Document No. AD 156001.
A method for determining base and depot stock levels in the face of the
dynamic and uncertain environment with which the Air Force supply system
has to deal. The method examines the major variables which must he
considered to achieve economical and effective provisioning and distribu-
tion under such circumstances. The study indicates what stockage rules
should include, hey the d,rmaics of weapon-system or other program ph....
in or phase-out can be taken into account, haw to free mansgmeent to manage
the more costly and critical items better, what further research is needed,
and how the application of these as well as other sway policies OAR be
improved.
?
Declassified in Part - Sanitized Copy Approved for Release @ 50-Yr 2013/10/25: CIA-RDP81-01043R002300090004-9