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A-A2OS 743 DEVELOPING ANd TIN D;RY NOD L FOR THE--rOatN AIR-FOKtE- 12-REPAIRABLE ITEM INVENTORYCU) NAVAL POSTGRADUATE SCHOOL

1 UNCLASSIFIED MONTEREY CA B G PARK DEC 88 F/ 55 U

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NAVAL POSTGRADUATE SCHOOLMonterey, California

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" ZLECTEI

THESISDEVELOPING AN INVENTORY MODEL FOR THE

KOREAN AIR FORCEREPAIRABLE ITEM INVENTORY

by

Park. Bveoung Gone

December 1988

Thesis Advisor Thomas P. Moore

Approved for public release; distribution is unlimited.

co 2 28 Q0

DISCLAIMER NOTICE

THIS DOCUMENT IS BEST QUALITYPRACTICABLE. THE COPY FURNISHEDTO DTIC CONTAINED A SIGNIFICANTNUMBER OF PAGES WHICH DO NOTREPRODUCE LEGIBLY.

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Approved for public release; distribution is unlimited.

Developing an Inventory Model for the Korean Air ForceRepairable Item Inventory

by

Park, Byeoung GoneCaptain, Republic of Korea Air Force

B.S., Korean Air Force Academy, Seoul 1983

Submitted in partial fulfillment of therequirements for the degree of

MASTER OF SCIENCE IN MANAGEMENT

from the

NAVAL POSTGRADUATE SCHOOLDecember 1988

Author:

Park, Bveoung Gone

Approved by: @-P PZ666L,

Thomas P. Moore, Thesis Advisor

rietsch, Second Reader

David R. Whipple, Chairman,Department of Administrative Sciences

Dean of Information and P Scien

Unclassified

RIKPOR I DOCUMENTAT ION' 1"AGEL_I co:7 Seu:>Csu~: nclas,.tficd b ct~~sakin,,s

a 'c7: Cs~s A j*.wT \ 3 D,,tribution Avallabilit% of Repc rt- eas;c:o osrd:Scriecu~e Approved for public release: distribution is unlimited.

-Perfor==2 Orsar.:zatior, Report Nunberis) 5 MIonitoring Organization Report Number(s)Nanne Of Pc;fo 2. O 1aiza::on1 61) Office Svrnbo! 7 a \nae of monitoring Organization

Naval Postizaduate School (ifapplca'c 54 Naval Postirraduate School7c Address (cnv siar. and ZIP code) 7b Address (city, stare. and ZIP code)Monterev. CA 93943-5000 Monterey, CA 93943-5000, a Nam-ne oFund.nzg Sponsorinog Organization Sb Oflice Sxmibol 9 Procurement Instrument Identification Number

c Address (t.stare. ar-d ZIP code,, 10 Source of Funding Numbers

Program Element No IProject No jTask No IWXork Unit Acccssion No

Tie t i.'audc se urcv:n sla- DEVELOPING AN INVENTORY MODEL FOR THlE KOREAN AIR FORCL RE-PA\IRABILE ITENI INVENORY

!2 Perso na: Authors1) Park. Bveousi- Gone* a Io"o Reror7 I 3b I tine Cosered 1- aeo eort (ya.r~~,di) 15 Pac-e Count

NlsersT Fom To December 19-SS 1 24it) Supplementary Notation The viewvs expressed in this thesis are those of the author and do not reflect the official policy or po-s ition of the Department of Defense or the U.S. Government.17 Cosat: Coc%?s IS Subiect Terms (cownrP:-~ oii re~erse yj necessary and identify by biock numiber)

F~eld Ir.u I Subgroup Inventoryv Theory, Repairable Itemn Management, Queueing Theoryv

19 Abstract (conrinae on reverse tf necessary and idenu'v by block number)The Korean Air Force has determined that repairables management is one of the areas in which attention could be ex-

pected to lead to substantial improvement in the efficient management of defense resources and in maintaining an adequatelevel of force effiectiveness. This thesis reviews various inventory models for the management of repairable items and developsan inv entory model for the Korean Air Force. It discusses the characteristics of each model, and, identifies and explains thedifferences, in each model with respect to assumptions, objectives, constraints, and optimization methods. Also this researchevaluates the proposed model using sample data sets.

DDI0\ 43s5%.I 3 A\PR ed:tion mnaY be used until exhausted security classification of ti-is pageAll other editions are obsolete

U.nclassified

ABSTRACT

The Korean Air Force has determined that repairables management is one of the

areas in which attention could be expected to lead to substantial improvement in the

efficient management of defense resources and in maintaining an adequate level of force

effectiveness. This thesis reviews various inventory models for the management of re-

pairable items and develops an inventory model for the Korean Air Force. It discusses

the characteristics of each model, and, identifies and explains the differences in each

model with respect to assumptions, objectives, constraints, and optimization methods.

Also this research evaluates the proposed model using sample data sets.

K r

- a3

iil

TABLE OF CONTENTS

1. INTRODUCTION ............................................. 1

A, BACKGROUND............................................1

B. PURPOSE OF RESEARCH.....................................2

C. PREVIEW................................................2

11. REPAIRABLES AND THE KOREAN AIR FORCE LOGISTICS SYSTEM . 4

A. INTRODUCTION...........................................4

B. THE KOREAN AIR FORCE LOGISTICS COMMAND ORGANIZATION 4

C. REPAIRABLE ITEMS MANAGEMENT SYSTEM.................. 6

1. The General System OvervieN ................................. 6

2. Operating Characteristics of Repairables Management...............9

a. Material Repair Schedule (MRS)...........................9

b. Material Repair Return List (MRRL)....................... 9c. The Measure of Performance............................. 10

D. MATHEMATICAL BACKGROUND OF REPAIRABLES MANAGE-

MENT........................................ ............. 1I1

1. DDR (Daily Demand Rate).................................11

2. DRP (Depot Repair Percent)................................ 11

. RCT (Repair Cycle Time)..................................11I

4. RCQ (Repair Cycle Quantity)...............................12I

5. OLQ (Operational Level Quantity)............................ 12

6. OST (Order and Shipping Time).............................. 12

7. OSTQ (Order and Shipping Time Quantity) ..................... 12

8. SLQ (Safety Level Quantity).................................139. RO (Requisition Objective)..................................13

10. History of RO Computation................................15

a. Before 1973..........................................15

b. Between 1973 and 1976.................................15

c. Between 1976 and 1979................................. 15

d. Between 1979 and 1984................................. 16

e. After 1984..........................................16

iv

E. SUMNMARY AND PROBLEM STATEMENT...................... 16

111. OVERVIEW OF REPAIRABLE MIANAGEMENT MODEL............ Is

A. INTRODUCTION ........................................... 18S

B. DETERMINISTIC INVENTORY MODEL........................ 18

1. Aiiaivsis of The Continuous Supplement Policy Model..............19

a. Notation...........................................19

b. Model Formulation....................................20

2. The Substitution Policy Model............................... 24

a. Notation...........................................24

b. Model Formulation....................................25

3.Summary and Evaluation..................................2. 8C. STOCHASTIC INVENTORY MODELS.......................... 2S

I. M*vodel Analysis .......................................... 29

a. Analysis of M ETRI C MAodel............................. 29

b. Mod-METRIC Model..................................3 4

c. Dyna-METRIC Model................................. 39

d. Mathematical Models Used In The Navy...................44

e. Availability Centered Inventory Mode].....................49

D. SUMMARY ............................................... 53

IV. MODEL PROPOSAL FOR THE KOREAN AIR FORCE .............. 54

A. INTRODUCTION..........................................54

B. THEORETICAL BASE FOR THE MODEL PROPOSAL ............. 54

1. Notation...............................................54

2. The M,MlIK!K Queueing System............................54

3.The MM. c,!K;K Queueing System........................... 55

C. DEVELOPING A MODEL FOR THE KOREAN AIR FORCE......... 551. Application Conditions....................................55

2. Model Formulation.......................................57

a. Step I..............................................58

b. Step 2.............................................60

c. Step 3.............................................61

D. SUMMARY ............................................... 61

V

V. COMPARING THE COMPUTATION RESULTS OF THIE MODEL ...... 62

A . IN TRO D U CTION .......................................... 62

B. MEASURE OF EFFECTIVENESS (MOE) ........................ 62

1. F ill R ate ...............................................62

2. Operational Rate ....................................... 62

3. 'Mean Supply Response Time (MSRT) ........................ 63

4. The Average Number of NORS Aircraft ....................... 63

5. C onstraints ............................................. 63

C. COMPARING THE MODEL RESULTS ......................... 64

1. D ata Characteristic ....................................... 64

2. Com paring the Results .................................... 65

3. Description of the Computer Program ......................... 65

4. Summ ary of the Results ................................... 70

D . SU M M A RY .............................................. 71

VI. SUMMARY AND CONCLUSION ............................... 72

APPENDIX A. SAMPLE DATA LIST 1 .............................. 73

A. SAMPLE DATA DEMAND DISTRIBUTION LIST ................ 73

APPENDIX B. SAMPLE DATA LIST 2 .............................. 78

A. SAMPLE DATA DISPOSAL RATE DISTRIBUTION LIST .......... 78

B. SAMPLE DATA ANALYSIS .................................. 83

APPENDIX C. COMPUTER PROGRAMS AND RESULTS .............. 85

LIST OF REFERENCES .......................................... 110

INITIAL DISTRIBUTION LIST ................................... 113

vi

LIST OF TABLES

Table 1. SUPPLY RESPONSE TIME REQUIREMENTS ................. 10

Table 2. DESCRIPTION OF NORS CODE ........................... 10Table 3. P(SUCCESSFUL REPAIR) COMPUTED FROM SAMPLE DATA .. 56Table 4. M EAN REPAIR TIM E .................................... 64

Table 5. DEMAND DISTRIBUTION LIST ........................... 73




Table 9. DEMAND DISTRIBUTION LIST ........................... 77Table 10. DISPOSAL RATE DISTRIBUTION LIST ..................... 78

Table 11. DISPOSAL RATE DISTRIBUTION LIST ..................... 79

Table 12. DISPOSAL RATE DISTRIBUTION LIST ..................... SOTable 13. DISPOSAL RATE DISTRIBUTION LIST ..................... 81

Table 14. DISPOSAL RATE DISTRIBUTION LIST ..................... S2

Table 15. DEMAND DISTRIBUTION ANALYSIS LIST ................. 83

Table 16. DEMAND DISTRIBUTION ANALYSIS LIST ................. 84

vii

LIST OF FIGURES

Figure 1. The Organization of the Korean Air Force Logistics Command ....... 5

Figure 2. The Korean Air Force Repairable Management Cycle .............. 7

Figure 3. The Korean Air Force Repairables Inventory Periodic Review ....... 14

Figure 4. Repair Cycle and System for "Continuous Supplement" Model ....... 21

Figure 5. Repair Cycles and System for "Substitution" Model ............... 26

Figure 6. Queueing System with One Repairman (MM 1 KK) ............. 55

Figure 7. Queueing System with c Repairman (M, Mic, K/K) ............... 56

Figure S. Difference of Total Stock Level .............................. 66

Figure 9. Difference in Safety Levels Between the Two Models .............. 67

Figure 10. Difference of Optimal Procurement Quantity . ................... 6S

Figure 11. Computer programs for the KAF Model ....................... S5Figure 12. Computer programs for the KAF Model ....................... 86

Figure 13. Computer programs for the KAF Model ....................... 87

Figure 14. Computer programs for the KAF Model ....................... 8S

Figure 15. Computer programs for the proposed .Model .................... 89

Figure 16. Computer programs for the proposed Model .................... 90


Figure IS. Computer programs for the proposed Model .................... 92


Figure 20. Computational Results for the KAF Model ..................... 94

Figure 21. Computational Results for the KA Model ..................... 95





Figure 26. Computational Results for the KAF Model .................... 100

Figure 27. Computational Results for the KAF Model .................... 101

Figure 28. Computational Results for the proposed Model (OA = 0.90) ........ 102

Figure 29. Computational Results for the proposed Model (OA= 0.90) ........ 103


Figure 31. Computational Results for the proposed Model (OA=0.91) ........ 105

v'iii


Iiture 33. CoIIputational Results for the proposed Model (OA=0.92) ........ 107

Figure '-I. Computational Results for the proposed Model (O.-N = 0.93) ........ .108

Figure 35. Computational Results for the proposed Model (OA=0.93) ........ 109

ix

ACKNOWLEDGEMENT

I am truly grateful to my God and the Korean Air Force that allowed me to study

at the Naval Post2raduate School. I owe much to Dr. Thomas P. Moore, thesis adviser.

and Dr. Dan Trietsch, the second reader. Their guidance and thoroughness tremen-

dously assisted t. completion of this research. They comprised the basis for this re-

search. Also. thanks to Dr. Kang and Capt. Shin for their advice for this research.

I'Maliv. I extend for my thanks to family. my wife Young-hwa and my son Jae-wook,

who not only endured my long periods of absence, but also gave me the encouragement

to complete this thesis.

x

1. INTRODUCTION

A. BACKGROUND

The objective of the Korean Air Force supply system is to ensure the desired level

of operational availability for its population of repairable equipment and maintain suf-

ficient stock levels for replacement components and repair parts to support the mainte-

nance of existing weapon systems and equipment. Also, the Korean Air Force supply

system is responsible for stocking a number of items which are failure prone and have

been designated as repairable. Because of engineering and economic considerations, at-

tempts are made by Korean Air Force repair activities to restore these items to service-

able condition whenever they fail.

The repairable inventory system differs from that of the consumable inventory sys-

teni in two ways. First. the repairable system contains two distinct inventories, one of

which has items that are in a Ready-For-Issue (RFI) state and another which has items

that are in a Non-Ready-For-Issue (NRFI) state. The first inventory contains items

which are usable and the second contains items that must be repaired before they can

be used. Second, the RFI inventory is made up of a mixture of new items and items that

have been used. failed, repaired, and are ready to be used again. This is one reason why

repairable inventor- management is a complicated process. [Ref. 1: p. 2]

The Korean Air Force Logistics Command (KAFLC) tries to determine an appro-

priate stock level for repairables based on a primitive mathematical model. This model

estimates stock levels using rules of thumb based on expert opinion. However, it is very

diflIcult to determine an optimal stock level for a repairable without using a more so-

phisticated inventorv management model. The result of the lack of such a model is that

the Korean Air Force supply system tends to maintain relatively high stock levels for

repairable components. Although, repairable items account for only a small percentage

of the quantity of items stocked throughout the Korean Air Force, they account for a

large portion of dollars invested in inventor-v. The characteristics of high cost and high

essentiality that these items typically possess make efficient inventor. control difficult

and extremely critical.

Historically, most inventory models have been developed for the private sector

where the profit motive is important. Such models consider the various average annual

variable costs associated with managing inventories and strive to minimize the sum of

these costs. The typical relevant costs in most inventory models are ordering cost,

holding cost and stockout cost. Certainly. these cost parameters are very important in

the common concept of inventory management. [Ref. 2: p. 1]

Ilowever. in military supply systems, it is hard to estimate the costs associated with

stockouts. In addition, the military supply system is not interested in profit

maximization or cost minimization. Instead. such an organization is usually interested

in maximizing the ability of its forces to respond to any threat. An objective which

maximizes some measure of readiness with given resources is therefore appropriate.

B. PURPOSE OF RESEARCH

The effective accomplishment of each flying mission is dependent upon having suf-

ficient numbers of aircraft ready to fly and to perform at their fullest capability. Tosupport this goal, the Korean Air Force emphasizes an extensive system of supply and

nmintenance capabilities. The Korean Air Force also tries to maximize the availability

of aircraft by quickly identifying malfunctioning parts, removing them. and rapidly in-

stalling replacements which have been positioned at the base supply department.

The purpose of this thesis is the development of an inventory model for the man-agement of repairable items at the wholesale level by the Korean Air Force Logistics

Command (KAFLC). This will be accomplished by the analysis of various repairable

inventor.y management models which have been developed and applied to the manage-

ment of inventories of repairable items in the United States military. And also. a nu-

merical analysis will be accomplished based on a proposed model. Finally the inventory

system parameters, and management concept will be studied.

C. PREVIEWChapter II of this thesis presents an overview of the current Korean Air Force Lo-

gistics Command (KAFLC) repairable items management process. This overview willexplain the organization of the Korean Air Force Logistics Command, describe the

present general methodology used for repairable items management and the current

overall approach to the problem.

Chapter III gives a detailed description of the specific mathematical inventory

models in use today for the management of repairables. Some explanation of their

mathematical approach in solving the inventory problem is given.

In Chapter IV, the development of a model for the Korean Air Force repairables

inventory management system is described. Model formulation and solution techniques

are provided.

2

Chapter V will show numerical results from a computer program for comparing the

Frn',o :dc1 -nd describe ma1,jor components based on the findings in Chapters II I

aniJ IV.

inx.Chapter VI contains a sunu~irvn and recoimmendations for future study

based on the proposed repairabics inventory management model.

II. REPAIRABLES AND THE KOREAN AIR FORCE LOGISTICSSYSTENI

A. INTRODUCTION

The Korean Air Force didn't use any kind of repairables inventory management

model until the late 1960's. In the beginning of the 1970's, the first inventory manage-

ment concept was introduced by the U.S Air Force. The economic inventory manage-

ment model was used during this period.

The model adopted in the 1970's has been revised systematically several times since

its adoption by the Korean Air Force. However, the usefulness of this inventory model

has decreased as the weapon systems used in the Korean Air Force have become more

sophisticated, and the overall size of the Korean Air Force has increased. The annual

budget for repairables takes approximately 60-70 percent ofthe total annual stock fund

budget of the Korean Air Force. however the essentiality' of high quality repairables in-

ventory management was not realized. [Ref. 1: p. 1]

In this chapter, the Korean Air Force Logistics system will be described. The

chapter specifically deals with repairables management. organization, sy'stem parame-

ters. and mathematical models related to repairable item management. This chapter will

provide the basis for the analysis and suggested improvements to the Korean Air Force

repairables inventory management system which will be described later.

B. THE KOREAN AIR FORCE LOGISTICS COMMAND ORGANIZATION

The Korean Air Force Logistics Command (KAFLC) is responsible for managing

the allocation of the logistics resources to support maximum weapon system operational

avaiiability. It functions as an intermediate echelon command among the tactical units

of the Korean Air Force under the policies and planning of Air Force Headquarters.

Figure 1 shows the organization of the KAFLC. The following paragraphs will explain

the basic missions and responsibilities of the six major divisions of the command.

The DMM (Directorate of Materials Management) is the focal point of material

management for the Korean Air Force. It procures all materials according to its esti-

mates of require, ents, distributes the material to all of the tactical units and the sup-

porting units under the policies of the KAFLC. Thus, the DMM is the equivalent of a

wholesale level Inventory Control Point (ICP). The DMM manages approximately

210.0o0 items which are allocated among the hundreds of item managers. To manage

4

MMDER

DL4~1 DSTDIC DA

-mg vc 1. The Organization of the Korean Air Force Log(,istics Command

eflecivlv 1Nl NI relies heavily oin thc Electronic D)ata P~rocessing Centcr' s (LDP,(i)

comiputer servies to access historical transaction data, compute demand forecasts, cur-

rnt asset data. etc.

The DST (Depot of' Storage and Transportation) is the centralized warehouse (6r

Lhe Korean Air Force where all procured and repaired materials, inciluding con1SUmab11le's.

are -stored. Thley control the physical invenitory of thc items needed by the ultimiate USCIr.

T*:-.c DST also deals with the disposal of salvage, scrap, excess and Obsolete nlatcri~ll.

TLie other I-ota Ole of' the D)ST is transporting, all materials to arrive at thle tacticalt

The DOW1 1 (lDcpo of MIaintenance and Equipment) is the ili-houLSe de1M~t leVel

ImainwrialnCc organizati on For the Korean Air Force. 11e primry task or the mi Nii is

thei Overhaul Of aircraft onl a scheduled basis (prvenive maintenance). It perlorI-0-1s the

maintenance or aviation end items which are beyond the capabilities of' the the intermec-

dliate and oruanizational maintenance level fIacilities. These items would be the major

end items of an aircraft such as the engne, fuselage, gearbox and the Wdirc ground

support equipment f-or the oper-ation of the aircraft.

The DMIEC (Depot of Maintenance, Electronics and Communication) performs the

maintenance of radar equipment and its subassemblies, ground cormnunication equip-

ment and weather forecasting equipment. The DMEC also performs preventive main-

tenance.

The DMA (Depot of Maintenance and Ammunition) performs the maintenance of

the precision measurement equipment (PM E), airborne equipment and armament of the

aircraft, such as laser guidance system.

The EDPC (Electronic Data Processing Center) provides logistics software devel-

opment and support, establishes job processsing standardization, collection of data re-

lated to preventive and corrective maintenance, provides analysis and guidance for each

of the other major divisions. The major objective of the EDPC is to assist each division

in performing their specialized logistics missions within the Korean Air Force I oci tics

Command.

C. REPAIRABLE ITEMS MANAGENENT SYSTEN

1. The General System Overview

In the Korean Air Force an item of supply is designated as a repairable if it can

be repaired faster and less expensively than it can be procured. Weapon systems in-

stalled in aircraft have become increasingly sophisticated and complex. Hence, many

weapon systems are made up of a number of subsystems which in turn are comprised

of several repairable modules. Often the complexity of these individual modules are such

that the personnel and equipment are not available at the end user level to repair failed

units. Consequently, these modules are designated as repairables and failed units are

returned to designated repair activities for repair. Therefore, repairables management

has become an essential part of the Korean Air Force supply system in terms of decision

making. Figure 2 shows the current Korean Air Force repairables management cycle.

The Korean Air Force has two kinds of maintenance operations for repairables,.

the base maintenance (organizational and intermediate level) and the depot level main-

tenance (DME, DMEC, DMA). It should be noted that not all depot level repairables

are repaired at the Korean Air Force facilities. Because of the complexity, some repairs

are done by the USAF under Foreign Military Sales (FMS) using commercial contrac-

tors. As depicted in Figure 2, three kinds of repairables flows are shown. The first is the

flow of carcasses to base level maintenance. The second is the flow of carcasses to depot

level maintenance. The third level is the flow of carcasses from the DST to the USAF

(the activities that coordinate with USAFLC for repair and procurement follow FMS

6

LL vi n~

if!i

0u

-Y-z

Cl 0

rigiie 2. Thc EreanAir orce epai~th~ci\Iaagcn itCyi

procedures). Note that the third flow implies the exchange of a carcass for a new item

from the USAFLC.When a failure of a system or subsystem occurs, the equipment is inspected to

determine the cause of the failure. After isolation of the failure to a component or major

assembly, further inspection is done to determine if base level repair is possible. In caseof the failure of a major end item. an RFI (Ready-for-Issue) item from the base supply

is used to replace the failed item and the aircraft returns to operational status. This in-sures the operational availability of an aircraft. If possible, the failed item is repairedby the Base Maintenance Squadron (organization or intermediate level of repair). Aftersuch a repair, the item is released to the Base Supply Squadron in serviceable condition.

For the determination of the maintenance level, the Korean Air Force has aspecial code assigned to each replaceable component of a major end item. Thus, thelevel of maintenance for each component is predetermined by this code. The code is as

foflows. [Ref. 3: p. 71

First code position

X expendable item

N repairable item

Second code position

D depot level maintenance item

F field level maintenance item (intermediate)

B base level maintenance item

Third code position

1 depot level maintenance

2 intermediate level maintenance

3 no maintenance action

After the Base Supply Squadron has issued a serviceable item from its stocks

and when base level maintenance has been determined to be impossible, the Base SupplySquadron sends the failed item to the DST. Of course, all these actions are reported to

and directed centrally bv the DMM. The failed items from each base are placed in the

DST facility and await repair. These carcasses are turned over to the depot maintenancefacility (DML. DMEC, DMA) in batches under the approval of the DMM. In other

I I • I

words, all carcasses from each base are stocked in the DST facility during the quarter.

At the end of each quarter, all carcasses are sent to the depot maintenance facility

(DME. DMEC. DMA). However, at this time, item managers of the DMM consider

the capacity limit of the depot maintenance facility.1

2. Operating Characteristics of Repairables Management

As mentioned earlier, there are several depot maintenance organizations in the

Korean Air Force. However. their concept of operation and relations with the DMM

are identical for the DME, DMEC and DMA. So, only the operation of DME will be

covered here. The maintenance organization of the Korean Air Force is restricted in

terms of its capacity to process incoming repairs and in terms of the level of technology

to deal with the repair of complex systems. Thus, the MRS (Material Repair Schedule)

and MRRL (Material Repair Return List) are established to manage these restrictions.

a. Material Repair Schedule (IRS)

The MRS is the repair schedule for the depot maintenance facility. It de-

scribes the type and quantity of repairable material which can be repaired during the

next fiscal year by repair depots operated by the Korean Air Force. In September of the

fiscal year, all item managers estimate the demand for each repairable item for the next

year, and the total repair quantity, based on data from the last several years. The inte-

grated results for each item are provided to the DME, DMEC. and DMA. Then. the

DME. DMEC and DMA set up their repair schedules after discussion and coordination

with DMM. The forecasted maintenance requirements which are not covered by the

MRS are turned over to the MRRL. The repair quantity for the MRS is set up on a

quarterly basis and carcasses are issued fiom the DST based on capacity of the DME.

b. Material Repair Return List (MRRI.)

This is the list of repairables for which the Korean Air Force does not have

depot level maintenance capability, either due to technology or capacity restrictions. In

the case of an MRRL repair, the DST sends the carcasses to a USAF depot facility.

Upon the receipt of the carcass from the Korean Air Force, the USAF returns a serv-

iceable unit to the DST. Serviceable units from both the USAF and Korean Air Force

depot facility are integrated at the DST to make up the serviceable stocks which are

available to support for the tactical unit.

1 This capacity limit established based on the MRS, see paragaph 2.a. above.

9

c. The Measure of Petformance

The Korean Air Force supply system (wholesale level) uses two measures

of performance. The first one is fill rate. It is also used by the base level supply system.

The second measure is supply response time. It measures the length of time elasped for

base backorders to be satisfied by the stock from the DST. In other words, it is the

length of time from the placement of an order with the DMMI to the receipt of the or-

dered item. Notice that the DMM uses this more as a priority rule for issuing the stock

than as a gencral measure of performance for management of the repairable items.

Currently, the Korean Air Force assumes that any backorders for spares result in an

aircraft being "not operationally ready because of supply" (NORS). Thus. the Korean

Air Force has developed codes which indicate the NORS condition. Each code provides

the maximum supply response time requirement to fill backorders. Table 1 summarizes

the supply response time requirements for each demand with priority and Table 2 de-

scribes the meaning of each NORS code. [Ref. 4: p. 81]

Table 1. SUPPLY RESPONSE TIME REQUIREMENTS

Priority Type of order Response time (day) Circumstances03 Express 3 G. K NORS

06 Semi-express 14 A, F NORS13 Routine 30 Routine

Table 2. DESCRIPTION OF NORS CODENORS Code Description

G Aircraft is in a Totally inoperational condition

K Radar malfunctioning

F Operational, incapable of missionA NORS is anticipated

10

D. MATHEMATICAL BACKGROUND OF REPAIRABLES MANAGEMENT

1. DDR (Daily Demand Rate)

This is the average number of demands per day and it is computed following

formula:2

365DDR = - D'

L 4 365

where:

D, = demand on the ith day of the year

2. DRP (Depot Repair Percent)

It is computed in the following manner for each repairable item from the last

three years of historical data:

DRP = TT x 100RRTS + .R TS + COND

where:

RTS = repair this station

NRTS = non-repair this station

COND = condemned

RTS refers to the number of units repaired at the Korean Air Force depot.

Conversely NRTS refers to the number of units repaired by the commercial contractors

or by the USAF under FMS policies. Thus, NDRP (Non-Depot Repair Percentage) is

the complement of DRP.

3. RCT (Repair Cycle Time)

The Korean Air Force has two kinds of repair cycle time. The one is the repair

cycle time associated with base maintenance and the other is associated with the depot

level maintenance. Usually RCT refers to the time allowed for the depot level mainte-

nance only. This time allowance is the policy of the Korean Air Force. So, RCT is

constrained 3 to be not less than 30 days and not more than 120 days.

2 This is computed by the end of year. The time period of this computation starts from thefirst day of the year to the end of the year.

3 This is the policy of the Korean Air Force, not a value which is computed from actualobservations which are recorded in the inventory management database.

11

4. RCQ (Repair Cycle Quantity)This refers to the quantity of the repairable item which must be stocked to mcet

demands during the repair cycle time. RCQ is applied to MRS items only and computedby the following formula:

RCQ = DDR x DRP x RCT

where:

DDR = daily demand rate

DRP = depot repair percentage

RCT = repair cycle time (days)

5. OLQ (Operational Level Quantity)This is the stockage quantity which is required to cover forecasted demand

during the time an MRRL item is in the hands of the USAF for repair. OLQ is related

to MRRL as RCQ is to an MRS item. The formula for OLQ is as follow:

OLO = DDR x 60 day

where:


6. OST (Order and Shipping Time)OST is defined as the time elasped from the initiation of a procurement or re-

pair order, to the receipt of the order. In the case of a procurement, the Korean AirForce constrains the OST to be not less than 120 days and not greater than 365 days.

For MRRL repairables. OST is constrained to be not less than 220 days and not greaterthan 465 days. In both cases, the upper and lower intervals are adopted to avoid an

inventory stockout. The 100 days increment in the OST is due to the additional trans-portation time for MRRL items from Korea to the continental U.S.A.

7. OSTQ (Order and Shipping Time Quantity)OSTQ refers to the quantity of repairable items needed to meet the demand

during the order and shipping time. The formula for OSTQ is as follows:

for MRS items, OSTQ = DDR x NDRP x OST

for MRRL items, OSTQ = DDR x OST

where:

12


NDRP = non-depot repair percentage

OST = order and shipping time

8. SLQ (Safety Level Quantity)

This refers to the quantity of repairables stocked to meet demand during delays

in shipping. delays in maintenance, or unexpected increases in demand. The formula for

SLQ is as follows:

for MRS items, SLQ < x (RCQ + OSTQ)

for MRRL items, SLO = \3 x OSTQ

where:

RCQ = repair cycle quantity

OSTQ = order and shipping time quantity

9. RO (Requisition Objective)

Even with the assistance of the existing computer system, it is hard for the item

managers to manage the more than 2.000 items (per item manager) that are assigned to

them. Thus. the DMYM designates items that have two or more requisitions per 'ear as

requisition objective items. The item managers will monitor these items more closely.

Currently. the DMIM applies a periodic inventory review model which is based

on a policy of reviewing and repairing at fixed regular intervals to bring inventory levels

back up to the Requisition Objective (RO). Repair orders are placed at predetermined

intervals- (the expected demand between intervals plus some allowance for the variabil-

ity of demand must be considered).

The requisitioning objective consists of three terms. These are the order and

shipping time quantity (OSTQ), the repair cycle quantity (RCQ) and safety level quan-

tity (SLQ). The requisitioning objective is the sum of these components. In Figure 3.

the repair order quantity is simply the difference between RO and the on hand inventory

at time of review.

Since the system parameters such as OST, RCT, DDR, DRP and NDRP differ

at each review period, the level of the RO may also be different at the time of each re-

4 This predetermined interval is the repair induction interval on the MRS. The current lengthof the intcrval is 2 months.

13

nO

0t L

st,

n +-2 ri4n+G H~on t h

-eur 3. The l'orcaii Air Force Repairahb'cs Iiivezlitory Periodic -eie

v~cv..- invcntory revICwAs tlrc USLUI 11 donc cvcrv 2 months. Thec formula for R() i, showNv

buIowx:

for MR11S items. RO SLQ ±RCO + OS 1k)

wi crc:

SLQ ,!' x (R"CQ + OSlIQ)

RCQ DDR x DRP x RCT

OSTQ DDR x NDRP x OST

for NMRRL items, RC) SLO + OLQ + OSTO

Xvhcr-c:

SLQ / \3 x Q.VIQ0

OLQ =DDR x 60 days

OSTQ DI)R x OST

14

10. History of RO Computation

ThIIP section Shows the history of the R() Computation formula. 'Ihle K orean

Air Fcrc,(e tries, to Jeternine optimal stock level fhor the repairables iI]'%cntorV fluflaeI-c-

mercIt SV "emI. HIowever. the results of'JaniMng11 thle RO forltion1 without adequatecnideration of the systemn parameters may result in excess stock for someiteso

s~ ockout, [or some item".

a. Bcf14e 19-3

RO = OLQ +SLQ + OSTQ

where:

01[Q = IDR x 7

SLQ = IDR X 15

OS FQ =DIR x '(

h. Bertween 19-3 and 1976

For %IRS. RO = DDR x 120)

For NIRRL. RO = R('Q, + SLQ + OSTQ

R('Q = IDR x 15

S LQ= . 3 RCQ + OSI1Q)

()SlQ =DI)R x 84

.Between 19-6 and 1979

RO = RCQ + SLO + .CO + OSTQ

xvh rc:


SLQ =S-(R(-Q + OSTQ 4- NCQ)

NCQ = LDDR x \DRP x \"cr

(JSTQ = DDR x NDRI' x OST

NCT = RIS Condemn Timnei

III ViN(I [) I', a ti:nc, spenlt for deLisioninge a condemnation among NRTS items.

d. Between 1979 and 1984

RO = RCQ + SLQ + 0S1Q

where:


SLQ = (DDR x NDRP x POST) + RCQ

OSTQ = DDR x NDRP x ROST

e. After 1984

For MRS, RO = RCQ + SLQ + OSTQ

where:

RCQ =DDR x DRP x RCT

SLQ = .3 x (RCQ + OSTQ)

OSTQ = DDR x NDRP x OST

For MRRL. RO = OLQ + SLQ + OSTQ

where:

OLQ =DDR x 60

SLQ = ,3OSTQ

OSTQ = DDR x OST

E. SUMMARY AND PROBLEM STATEMENT

As mentioned earlier, militar inventory managers tend to want to increase their

operational availability rather than minimize inventory costs. However, in the case of

the Korean Air Force repairables management process, the mathematical model is in-

sufficient to adequately model the stochastic characteristics of the inventory manage-

ment problem. In the RO (Requisition Objective) computation it does not specify the

wear-out rate or the regeneration rate, which are important parameters in dealing with

repairables inventory management.

The DDR (Daily Demand Rate) computation is the mean of a stochastic process.

However, the current inventoR model treats demand as if it were deterministic. In the

formula for SLQ (Safety Stock Quantity), they considered the stochastic nature of de-

mand but the actual probability distribution is not considered at all. As evidence of the

16

model's shortcomi nes, the Korean Air Force repairables management tends to stocklarce quantities of' spare parts and components in order to avoid stockout situations.

17

III. OVERVIEW OF REPAIRABLE MANAGEMENT MODEL

A. INTRODUCTION

Inventory systems such as the one described in the previous chapter can be classified

rather broadly as a multiechelon system with repair. Analytical studies of multiechelon

systems have shown the computation in such models to be hard, so either simplifyingassumptions or approximations are necessary. The introduction of the repair aspect intothe system certainly complicates the structure of a model even at single echelon levels.

Suppose a repairable system, consisting of one Inventory Control Point (ICP), one

stock point and one overhaul and repair activity controls the inventory of a single item.

Demands from various tactical units are placed only at the stock point. The system hascontinuous updating of records, i.e., transaction reporting. When items wear out or fail.

the tactical unit can either scrap the item or return it to the depot repair facility (DME.

DMA. DMEC). After inspection, the depot can either scrap the item or repair and re-

turn it to the RFI inventory.

In this chapter, we will discuss various repairable inventory models based on twodistinguishing characteristics. One is a deterministic model, the other is a stochastic

model. In case of deterministic model, we will describe two models, "continuous sup-

plement" model and "substitution" model. For the stochastic case. we will describe fivemodels based on current usage in U.S military. Each description will include model

formulation and solution techniques.

B. DETERMINISTIC INVENTORY MODELThese deterministic inventory models were developed by David A. Schrady and

others between 1966 and 1967. These deterministic inventory models are described in aresearch report for the U.S Navy repairable inventory management system [Ref. 5, 6,

7]. The first model calls for the repair activity to induct a batch of carcasses when theNRFI inventory level reaches a certain point. With a deterministic system, this rule in-

sures regularly spaced repair inductions of fixed batch size, which is a simple process for

the repair activity to manage. Repair batch sizes can be maintained if there is contin-

uous supplementing of repaired RFI items with new procurement. The procurementorder will be issued with enough lead time so that when the on-hand RFI stock drops

to zero, a procurement quantity will arrive. The two inventories, RFI and NRFI. will

have time histories as shown in Fi2ure 4. Note that in this, the "continuous supplement"

is

model, the procurement trigger is in the RFI inventory and the repair trigger is in the

NRIFI inventory. Ief. 6, 7: pp. 9-10. pp. 392-393j

The second modcl is suggested by noticing that there is a trade-off between stock

held in RFI condition and stock held in NRFI condition. NRFI carcasses are a legiti-

mate resource because they rcquirc only repair to be rertored to full usefulness. But the

value of this resource is less than the value of the RFI resource by at least the cost of

labor and replacement parts. Thus. if inventory is to be held in the system, it would be

slightly cheaper to hold it in NRFI condition than in R1 condition. [Ref. 6. 7: p. 10,

p. 3931

Notice that the "substitution" policy minimizes the RFI inventory while the "con-

tinuous supplement" policy ninimizes the NRFI inventory [Ref. 6: p. 13]. The next

section describes the mathematics of these two models.

1. Analysis of The Continuous Supplement Policy Model

a. ;'otation

Under the assumption of deterministic demand and lead times, it is not

necessary to maintain a saflety stock. Whenever the RFI inventory reaches zero, a pro-

curement quantity will arrive. Thus, the system is never out of stock. Between pro-

curement arrivals, depleting RFI stock is replenished with repaired items. Also, arepetitive system will be established regardless of the initial provisioning policy, so that

it is suficcient to analyze only one cycle to determine system characteristics. Figure 4,

shows basic characteristics of the "continuous supplement" policy model. The basic

notation as follows [Ref. 5: pp. 12-22]:

D = annual demand rate

T = procurement cycle

T, = repair cycle

t, = procurement lead time

r, - repair lead time

ro = field recovery rate

ra = overhaul and repair recovery rate

Q, = procured lot size

Q, = repaired lot size

/, = holding cost for RFI

h2 = holding cost for NRFI

X' = reorder point for procurement

19

X, =reorder point for repair

n = number of' repair cycles per procurement cycle. n = 7- / T2

= fixed procurement cost per one cycle

A,2 = fixed repair cost per induction

b. MIodel Formulation

The objective of the model is to determine an optimal procurement quan-

tity Q, and an optimal repair quantity Q; that minimize the average annual cost. These

optimal values, coupled with respective reorder points, X, and X, constitute an optimal

operating doctrine. To determine the average annual variable cost, the costs per repair

reorder cycle must be investigated. The product of per cycle costs and the number of

cycles per year will then yield the average annual cost. The total cost for any given cycle

T, is the sum of procurement, repair and holding costs.

Since there is only one procurement per reorder cycle, the variable order

cost is the actual cost of the items ordered and can be expressed as Q1 C. The fixed

procurement cost for one cycle is A, as mentioned. To determine the repair costs per

cycle it is necessary to compute the number of repair cycles per procurement cycle

i = T, / 7,. n is an integer). In Figure 4:

T2 = t1 + t2

where:

t,=KT2 , O<K<I

t2=(1I- A)T

The cost of items repaired per procurement cycle is C2Q, and the fixed repair cost is

A,n, where A, is defined to be the fixed repair cost per induction. The holding costs per

procurement cycle will be hAr1 + h2A 2,. where AT, is the area under RFI curve and

At2 is the area under the NRFI curve. To compute the area under RFI curve, consider

Figure 5 showing RFI inventory for one procurement cycle. Since D is constant and

known, the area of U, A,, is given by:

Au = t, (Q1 + a)

20

iZ L ' -- ______Po-cimrOK~ z

Arsr or r RrL 0.)

S TOCIk D~mnNOtD

ro'NT

j.D I o or a, i urks

.~I., -r rtip~rL THK'l- IN __________R

rt~fMv ~

P.1

/t 1-1

r. D~

P T'IA

II V *

Figure 4. Repair Cyclc and System for "Continuous Supplenlielit" 1\ lodel

21

where:i, = K T

a = 01 - D t1

The formula will reduce to:

2AU = (tlQj - D 1 ) / 2

Since n is an integer, the area of V is the sum of n-I trapezoids each having

one side of length T. To determine the area of each trapezoid, a recursive relation is

developed. Let:

A= area of the ith trapezoid in V (see Figure 4)= .v, + A,2 . ..... +Av_

where:

A T2[2 Q 2 + 2 (Qj- rD) -DT]

= [4Q + 2(Q, - tD)- 3DT]

therefore, the general formula for the area of V:

At.. 7 2 T2 [2iQ 2 + 2(Q 1 - r1 D) - (2i-I)DT2] i= 1,2,....,n- 1.

"-I

So the area of V is. A,,= E-AV:

-I n-I

A v - T02Q2 i + 2(n- 1)(Ql - rD) - DT 2Z(2i- 1)]

1 T2[Q 2n (n- 1) + 2(n- I)Q, - 2(n- l)tD - DT 2n(n- 1) + DT 2(n-1)]

- 2 T12(n - 1)(Q. - t.D) + (n2 - n)Q 2 - (n - 1)2DT2]

Finally, area of W in Figure 4, A, is easily calculated as follow:

'w -2 2

22

where:

b = nQ 2 + (Q, - t1D) - (n - 1)DT2

Thus. the area of W:

I T T2[nQ2 + Q1 - tID - (n - I)DT2]

The total area under RFI curve is:

Ar- Au + A, + Aw

-jQI - I Dti + 1 T2{2(n - 1)(Q) - D t + (n2 - n)Q2 - (n2 - n)DT2}

Since n is an integer and the build up rate of NRFI items, r.D, is constant,

the area under the NRFI curve is simply n times the area under the repair cycle curve

is TQ 2 r,. So the NRFI holding cost per procurement cycle is nT2Qh2 / 2r2. The totalcost per procurement cycle becomes:

+j2 2 T 1_ +__7_ h2Q 2 T1KT = Q1 C 1 + A+ 72 + T2 2 r2

+ hAT,

The total average annual cost is then:

KT _ Q +C1 ,41 ,A2 C2Q2 h2Q2 hAT

-, r + -- ± + + +

Simplifying the total average annual cost:

S .41RD h2Q2 h1Q1 hQ 2K- + C1RD + A2r2r°D + + - hjkQ2 + 2Q,2r 2 2 2

where:

T2 = Q2 I ror 2D

T,/T 2 = n, n = 1,2,3.

t= k T2 = kQ 2 / ror2D for, 0<k<l

t1+ t2 = T 2

T Q I - rOr2D = Q, / RD

23

R = I - r2

Note that the terms C1RD and CrrorD are independent of Q, and Q2. and hence do not

affect the operating doctrine. Therefore it is appropriate to redefine the average annual

cost of ordering, repairing and holding as follow:

A RD + 42rr 2D + h2Q2 h C1 hIQ 2Q Q2 + 2 l2 + 2

From differential calculus the optimal Q and Q will be those that satisfy the equations,

8K OK 08Q1 ' Q2

Taking the partial derivatives results in:

IK AJRD +h

2

-Q1 Q 2

C',K A2rorz2D + h2 _k +hi

Solving the equation to get the optimal procurement and repair quant swies,

2.4IRD 2 A2 r C D

Q, Q /1 = h2 +r2hl 2k)

2. The Substitution Policy Model

a. Notation

Notice that this model minimizes the RFI inventory. Figure 5 shows basic

characteristics of the "substitution " policy model. The basic notation as follows [Ref.

6, 7: pp. 13-17. pp. 393-396]:

QP = procurement quantity

QR = repair batch size

d = demand rate

(1 - r) = scrap rate, r is the recoverv rate measured

t. tR = procurement and repair lead times

,tp = fixed procurement cost per order

24

A4 = fixed repair batch induction cost per batch

= RFI holding cost

h, = NRFI holding cost

T = system cycle time

b. Model Formulation

Referring to Figure 5, define T as the time period during which inductionsare suspended and the repair is simply accumulating NRFI carcasses. It may be deter-

mined that:

T, = (Qp + QR)/d

Next. let n be the number of inductions per cycle. In general, it will not be possible to

insure that the last induction before T, begins will reduce the NRFI stock to zero. The

residual NRFI stock when T. begins will be some fraction of the net loss in NRFI items

per repair cycle, where the repair cycle is the time between regular consecutive in-

ductions. The net loss of NRFI inventory over the period between successive in-

ductions:

rd(QRd -OR= -QR(l-r)

Thus, the residual when T. begins will be some fraction of QR(I - r), call it fQA(I - r)

where 0 < /? < 1. While the ,# factor is unavoidable due to the requirement that there be

an integral number of repair batches in each cycle, we shall set fl = 0 in subsequent de-

velopments.

Now the first batch after inductions are resumed takes the amount QR fromthe NRFI inventory. Subsequent inductions cause a net reduction of only QR(1 - r)

items. Thus, the amount of NRFI items available for the (n-I) inductions are

rdTo - QR or QR(r - 1) + rQp. Dividing the last expression by Q,(I - r) yield:

(n-l) QR(r- I) + rQp rQR(I - r) + -r )OR

or

-r QR

25

tIEPOMIlu( STOCK

wiv.4olry ofRFI I

r d Derrxid flats, d

a n

CoI incjtor

ToT

T.-; IT 5. Repair Cycles anid Systeni for "Substituition" M~odel

26

The system cycle time is given by:

"n2R + Q_ QPT

where:

n r )QPQR*

From the Figure 7, the area under the RFI curve, over one complete cycle A,, is then:

A= 1 r1- 2

Al -2d ( - r )[Q RQ , + (rr )3]

Referring again to Figure 7 the area the NRFI curve:

2 r

A 2- )[QRQP + .uI

Now we get total cost per unit times expression:

TOd( I - r) .4Rrd h12r-r

IC _A d( + + T[QR + r )Q-] + h2r[Qp + QR]T ()P QRr

where:

Qp + nQR Q ,d d(l - r)

The total cost under substitution model:

hL r [ Q h2 r 2TC = Ap + nA4R + 2dPQ +- r P1 + d 1-r

2d Il-rQ Q± ~ Q] 2d lr QPQR + PJ

The optimal order quantities are obtained by setting the partial deriatives,

with respect to Q, and QR:

8TC T=0 TC 0CQl, ' CQR

Next we get new formula for optimal order quantities:

27

.TC _ Al4(1 - r) hI( - r) h~r

J~p p

8 JC __ R__11_ 1 1, r ~

-- + +

jQR -- _

The optimal order quantities for repair and procurement:

2Ad(lI - r) Q2 " 12h(l-r) + h2r ' QR = h + h2

3. Sumnmary and Evaluation

An important result of deterministic inventory model analysis is the realization

that economic order quantities do not vary directly w ith demand but as the square roct

of demand. This nonlineai relationship could explain why many organizations experi-ence inventory problems. Intuitive inventory policies are undermined by difficult-to-

grasp nonlinearities. [Ref. S: p. 141]

Although the models described in this section are evidently applicable to re-pairables management, they" are fraught with many limitations. The reasonableness of

model assumptions and their sensitivity to actual conditions determines the utility of anyparticular model. For example, the two models assume that the demand for a repairable

item is known with certainty. Also stockouts either are not permitted to exist or are

backordered and satisfied when replenishments are received.In terms of the Korean Air Force repairable management system, the models

described here are difficult to adapt to the Korean Air Force system. This is because

of demand variability. relatively long procurement lead time. repair capacity and capa-

bilitv, a number of backorder situations, and current system management concept. The

other important point is that the Korean Air Force often encounters NORS conditions

which results in decreased operational availability. However, these two deterministicmodels do not deal with stockouts. These reasons didn't allow to use for the Korean

Air Force repairable management system.

C. STOCHASTIC INVENTORY MODELS

As the weapon systems installed in modern aircraft become increasingly sophisti-

cated and complex. repairable items represents an important subset of the total inven-

tory of items which are managed by the military supply system.

2S

In general. repairable items are supported by a two-echelon inventory and repair

ss:em. When a repairable item fails at the base level, it is returned to base supply and

a new serviceable unit is issued from extra RFI stock of the base. If possible, the failed

item is then repaired by the base maintenance organization and then stored in RFI

conditions at base supply. Sometimes. however, the failed item must be returned to the

repair depot where more sophisticated equipment and specialized skills are available to

repair it.

When we consider the condemnation of repairables, there should be an inflow of

newly manufactured items to depot level supply should look at the material flow of all

echelons, and then. place a replenishment order to resupply the condemned repairables.

At times. forward base locations will be supplied from another closely located base. This

is called lateral resupply. For other items, a manulacturcr may provide both the source

of procurement for new assets and the source of repair for failed items.

The decisions concerning which maintenance levels will repair the failed items sub-

sequently affects the supply support provided at the organizational level. If an item is

repaired at the organizational level, repair equipment and maintenance personnel must

be made available at that level. However, if the item is not repairable at that level, then

the question is whether the item can be replaced at the organizational level. Organiza-

tionall level repairables will normally be transferred to the next higher echelon of repair

if the repairs cannot be accomplished at the organizational level. Similarly. repairables

which cannot be repaired by the intermediate level are usually sent to the depot level forrepair.

The difference between stochastic models and deterministic models is that if de-

mands or future requirements are uncertain, then regardless of the stockage policy

adopted there is generally some probability that available stock levels will be insuflicient

to meet demand [Ref. 9: pp. 187-ISS]. In this section, we will describe five stochastic

models based on current usage.

1. Model Analysis

a. Analysis of METRIC Alodel

(1/ Background. The METRIC model was developed over a period of

years by a research group at the RAND corporation and presented in the literature by

Sherbrooke (1968) and extened by Muckstadt (1973). METRIC was developed with the

ultimate goal of implementation and a slightly modified version of METRIC was actu-

ally implemented by the USAF. [Ref 10. 11. 121

29

METRIC is a mathematical model of a base-depot supply system in

which item demand is assumed to be compound Poisson with a mean value estimated

by a Ba\ esian procedure. The model is also capable of determining base and depot stock

levels for stock levels of a group of recoverable items. Its governing purpose is to opti-

mize system performance for specified levels of system investment. It is designed for

application at the weapon-system level, where a particular line item may be demanded

at several bases and the bases are supported by one central depot. The support depot

may vary by item as in the item-manager system or it may be fixed as in the weapon-

system storage site concept. The general purposes of this model are optimization, re-

distribution and evaluation. [Ref. 13: p. 2]

The basic METRIC model considers a two-echelon system in which

independent bases (lower echelon) are supported by a repair depot (upper echelon). At

the occurrence of a failure (it could be more than one), the failed item is either replaced

by available base stock or back ordered if the base stock is not available. The item is

inspected to determine the extent of repair required. If the repair can be made at the

base. the unrepaired item enters base repair. If the item cannot be repaired at the base

level, it is shipped to the depot. Upon shipping the item to the depot, the base places

an order with the depot for a replacement, so that the inventory position for item i at

base j can be maintained.

(22 ) Mathematical Assumptions

1. System Objective of Minimizing the Expected Number of Backorders: The objectivewill be to minimize the sum of expected backorders on all recoverable items at allbases pertinent to a specific weapon system. Thus, unless all bases are identical,the expected number of backorders will vary by base. Expected backorders take afixed period of time and add together the number of days on which any unit of anyitem at any base is backordered. In order to minimize the expected number ofbackorders one divides this number by the length of the period and taking the ex-pected value. This yields a number that is independent of period length. This isthe value we seek to minimize. [Ref. 14: pp. 126-131]

2. Compound Poisson Demand: We assume that demand for each item is described bya logarithmic Poisson process. Compound Poisson processes are generalizationsof Poisson processes: the compound processes allow the flexibility of incorporatingmore parameters, yet retain the simple analytical properties of the Poisson. Thelogarithmtic Poisson is obtained by considering batches of demands where thenumber of batches follows a Poisson process and the number of demands per batchhas a logarithmic distribution.

3. Demand is Stationary over the Prediction Period: It is assumed that the distributionof- demand over some future period of interest is stationary.

30

4. Repair Decision Deponds on the Complexity:. The assumption is that the decision torepair a unit at base level or send it to the depot is a function only of the type ofmalfunction and the base maintenance capability.

5. Lateral Resupplv is Ignored: When a unit is shipped from base to depot for repair.a serviceable repiacenent will be resupplied irom the depot if-possible. If the depot

has no unit on the shelf" the base must wait until a unit emerges from depot repair.

6. S 'stem is Conservative.: Consider a particular stock item demanded at base j. Weassume that a unit of' stock has a probability r, of being repairable at base j. witha repair time drawn at random from a base repair distribution with mean A,; aprobability (1 - r.) of being depot repairable, with an order and shipping-timedistribution having mean 0, and a depot repair distribution having mean D. Thisimplies that there are no condemnations or that the system is conservative, as thename "recoverable item" suggests. A higher condemnation rate usually indicatesthat the item should be redesigned. The condemnation rate must be considered forprocurement purposes. but the procurement process is not considered in theNMLTRIC optinization.

7. The Depot Does Not Batch Units of Recoverable Item for Repair. The model as-sumes that depot repair begins when the repairable base turn-in arrives at the de-pot. This appears to be a reasonable approximation to current depot schedulingpractice. Since METRIC economizes by buying fewer high-cost items, these arethe items which are most likely to be in short supply.

S. Demand Data from DiJferent Bases Can Be Pooled. We assume that demand fromthe several bases can be pooled in some manner so that a composite initial estimateof demand per flying hour can be obtained. The pooled estimate can be obtainedby a simple averaging technique or a more sophisticated procedure such as expo-nential smoothing. METRIC multiplies this number by the flying hours per monthfor each item.

(3) Model Formulation. By mathematical assumptions, demand in this

system follows a compound Poisson process. A compound Poisson process may be

thought of as a series of customers who arrive following a Poisson process. each of

whom can demand an amount that is independently distributed according to a com-

pounding distribution. Assume that item i is stocked at each of base j. and the cus-

tomers who place demand for the item at each base have a known mean arrival rate of

,, j= 1.2.3 ........ J. When a customer arrives at base to place one or several demands, he

turns in an equal number of carcasses. These carcasses can be repaired at base level with

probability of r,., while (1 - r,,) is the probability that they must be repaired at the de-

pot. The arrival of carcasses from base j at the depot is described by a Poisson process

whose mean is (1 - r,,) times the mean of the Poisson customer arrival process at the

base j. Therefore, the total demand at the depot for item i is compound Poisson, with

mean customer arrival rate:

31

-I

1=1

where:

= mean arrival rate fot item i at base j

Letf, be the mean demand per customer at base j. Then, the mean

depot demand rate for unit i is:

1 .1

0 = ,)= ZOij(i - r,)j= 1 j=1

where:

0,. = the mean rate for item i at basej

In the special case of the logarithmic Poisson process, the probability

that x customer demands are in the repair supply process is negative-binomial with pa-

rameters q and K (Note: K =T/ In q where . is mean customer arrival rate and T is

avcrage resupply time).

(q - 1)2

P(x .ijTij) = (K + x - 1)! - 1)x+(K - l,)!x!qX+

x = 0,1,2 ........ q 1, K>0

where:

q = the variance to mean ratio of P(x I),";

K= ;.,,TEf/(q - 1)

T= average resupply time for a demand for item i at base j.

A,) = base repair cycle time

D = depot repair cycle time

0,, = the order and shipping time

6(So) = average depot delay

Then:

T . = rjj,4j + (1 - rtj){O + 6(So)}

32

Since it takes an average of D time units for an arrival to complete

the repair process, the probability distribution of the number of units in the depot repair

cycle is compound Poisson with mean of A).D. Hence, the expected number of units back

ordered at the depot is:

BO(So ID) = (x - SO) P (x D)

x> SO

As mentioned in the beginning, the objective of METRIC is to min-imize the sum of backorders for all item i and for all bases j within a budget constraint.

Thus, the METRIC problem will be represented as follows:

I ,

minimize I Z BOij(sio Sij)

i=1 j=1

I J

subject to V cii:! Ci=1 j=1

S~ ,l<i<Il O<_j<J

where:

S, = the decision variables, stock for item i at base j

C = the total amount of budget available

C,= the cost of item i

S,o = the depot stock for item i

(4, Solution Technique. The METRIC problem is solved by using the

generalized Lagrange Multiplier method suggested by Fox and Landi [Ref. 15: pp.

258-261]. Let (D be a Lagrange Multiplier associated with the budget constraint. The

Lagrange function is written:

I J I J

ZYBO(SI iyT,) - DE ClS1=1 j=1 i=1 J=1

The auxiliary problem attempts to minimize this equation. By trial

and error, we try to find the value of CI' which satisfies a given constraint. Therefore.

33

we need to solve the above equation for several values of (1I, and choose that value of

(D for which the required resources are closest to the budget limit. Fox and Landi sug-

gest a binary search procedure. Their computational experience was that at most six

bisections were required to obtain budget allocations that were within one half of 1 "oof the original budget C.

The objective function is separable in the items. Dropping the sub-script i in the original problem allows us to rewrite the equation as:

I

min2BIS I ;.~)- OCSj- CIo

L1

Since BOJ(S, I /,T) is discretely convex for a given S, the optimum

base level is obtained by simply finding the smallest non-negative integer satisfying:

BOj(Si + I I ;jj) - BO(Sj I >_ J) > ( 1 S o

(5) Model Evaluation. The current Korean Air Force Logistics system's

objective is maximizing operational availability subject to a budget constraint. This issimilar to the METRIC model. The objective of minimizing the expected number of

backorders is related to the objectives of the Korean Air Force Logistics system. How-

ever, two assumptions are distinctly violated. One is batch size repair policy is not used,

the other is lateral resupply is ignored. Currently, these assumptions are not used in the

Korean Air Force because of relatively long procurement leadtimes and repair capability

limitations. It may be possible for the Korean Air Force computer resources to handlethe data requirements for the METRIC model, but it is doubtful for actual usage.

However, the METRIC model is obviously related to the Korean Air Force in terms of

its basic concept and system objective.

b. MIod-METRIC Model

(I) Background. Mod-METRIC model was developed by Muckstadt to

deal with problems which the METRIC approach did not consider. Mod-METRIC

considers the relations in parts hierarchy and tries to solve this multi-indenture level

problem. Mod-METRIC was implemented by the U.S. Air Force as the method for

computing repairable stock levels for the F-15 weapon system. [Ref. 16: p. 472]

Most repairables contain subassemblies or other componznts whichare also repairable. For example an aircraft engine, it may have modules for intake,

combustion and exhaust. If an engine fails, it is replaced by a serviceable engine from

34

base stock. The failed component is then repaired either at the base or depot level de-

pending upon the complexity of repair. A serviceable module from the base stock, if

available, will replace the failed module, and the repaired engine is placed in base engine

stock.

/2) Mathematical Assumptions. The assumptions stated in METRIC are

also applicable to Mod-METRIC except that Mod-METRIC assumes that the demand

process is the simple Poisson process. [Ref. 16 pp. 475-4761

(3,1 Model Formulation. In METRIC, the objective is to minimize ex-

pected backorders for all items subject to an investment constraint. The Mod-METRIC

objective, however, is to minimize the backorders for the end item, subject to an invest-

ment constraint on the total dollars allocated to the end item and its components [Ref.

10. 17: p. 475. p. 391. This difference is caused by the hierarchical maintenance re-

lationship bctxxecn the module and the end item. As an example, an engine backorder

indicates that an aircraft is missing an engine and is unavailable to perform its flying

mission. Modules, on the other hand. are used only to repair engines. A backorder for

a module only delays the repair of an engine. The impact of module backorders and

engine backorders is clearly not the same.

Let T, denote the average engine resupply time at a base. By the na-

ture of repairables. T, depends on several factors. When it is repaired at base level, T,

would be the time it takes to flow through the base maintenance system. If it is repaired

at the depot, T, consists of the time to place the depot order for a serviceable part and

to receive the part from the depot, assuming a serviceable asset is on hand at the depot.

However, when there are no serviceable assets on hand at the depot. an additional delay

is included in the resupply time:

T = rIB I + (1 - r1){A t + 6(SoD)}

where:

= the probability an engine will be repaired at base i

B, = the average resupply time, given an engine is repaired at base i

A, = the average order and ship time for an engine at base i

S, = the depot engine stock

D = the average depot repair time

5SoD = the delay days per demand

35

6SoD can be derived in the following manner. The expected number

of engines back ordered at the depot is:

BO(SOI.D) = Z (X - So)P(XAI.D)

X> S,

where:

,A = X(1 -rp.

, = the daily engine removal rate at base i

In other words, this expression is the expected number of units on

which delay is being incurred at a random point in time. Dividing this expression by the

expected number of demands per day yields a statistic which has the dimension of delay

days per demand:

6 SD = B(So I. .D)/.

= expected backorders: expected daily demand

Further,. B, can be divided into two components. That is, B, is equal

to the average remove and replace time, given that the necessary module is available,

plus the expected delay due to the unavailability of the module which is required to re-

pair the engine. Then:

B, = Ri + A,

where:

R, = the average repair time at base i if modules are available

A,= the average delay in base engine repair due to unavailability of a needed module

Further, assume that the engine should be repaired by the failure of

module j. Then, AU is the expected delay in engine base repair time due to a back order

on module j at base i. Thus:

A = (X - SU)P(XUI ; f TO / xj

X,> S",

where:

36

= average number of daily removals of module j at base i

S,, = stock level of module j at base i

7, = average resupply time for module j at base i

Then:

T• = r,,B, + (1 - r.)(A, + 6jD,)

where:

r,'= the probability that a failure isolated to module j will be repaired at base level

B,. average base repair time for module j at base i

A,, = average order and ship time for module j at base i

D,,= average depot repair time for module j

6,- E (X - S,.)P(X OjD )/ O,D;

S,; = the stock level of module j at the depotJ

0r = Z).j(1- r,.)

Consequently, the expected delay in engine repair at base i due to

module unavailability is:

J

A! I / ri) .AY

j=1

Thus, expression T represents all the system components including

the depot engine and modules stock level, and the base module stock level. By using this

relationship, the engine and module stock levels can be derived. Finally, the math-

ematical statement of the problem is:

I

mi E X, - S1)P(XJI)T,)i=1 X> Si

I J J

subject to Z(CSi +±~ j~j + ZCIS'j + CESO Ci=1 j=1 J=1

where:

S, = stock level of base i

37

CE = unit cost of an engine (end itam)

c'= unit cost of module j

C = dollar budget limit

,4j Solution Technique. Unfortunately, the equation above is not sepa-

rable. because T, is a complex function of the S,. Thus, Muckstadt recommended that

it should be broken down into two parts; the component subproblem and the end item

subproblem [Ref. 16: p. 477]. Even after the break-down of the problems, however, it

requires the solution of many subproblems each of which corresponds to a particular

division of the available budget between components and end items. The solution pro-

cedure is as follows.

First allocate the budget C into C, and C2 for components and end

items repectively. Allocate C, among components so as to minimize the expected end

items repair delays sunmed over all bases subject to the budget constraint C,. This

problem is mathematically represented as:

Jrain E 1J).j

j=1

J I

subject to Z(CjSoj + Zcjsi)<_ C,

j=1 j=1

This problem can be solved using the METRIC technique. Given the

result from this step, compute the average resupply time T for the components of each

base. Then, allocate the remaining budget C2 so as to minimize the expected end item

base backorders. The METRIC budget allocation procedure may again be used. The

steps provide a set of proposed stock levels upon a given allocation of the budget among

the end items and components. These steps then are repeated several times using new

values for C, and C2 to establish the best allocation.

(5) Mfodel Evaluation. Basically, the Mod-METRIC model has a similar

structure to METRIC. All assumptions used in Mod-METRIC model are applicable to

METRIC except for the compound Poisson demand. In other words, this model is the

same as METRIC for the Korean Air Force repairable management system. The only

disadvantage is that the Mod-METRIC model was developed for new weapon sy'stems

such as the F-15 aircraft. However, the Korean Air Force has a lot of non-standard

38

weapon systems. These non-standard weapon systems are causing problems such as

long procurement leadtinics.

c. Dyna-METRIC Model

(I, Background. D'na-METRIC was developed by the RAND Corpo-

ration to provide an analytic method for studying the transient behaviour of

component-repair and inventorv systems under time-dependent operational demands

and logistics decisions like those that might be experienced in wartime [Ref. 14, 18].

Note that the past work regarding the repairable item stockage prior to Dyna-METRIC

only dealt with a steady state inventory system with constant average demand and ser-vice rates. These steady state assumptions may provide a good approximation during

peace time operations. But in wartime demands for components may suddenly increase

very much relative to the previous peacetime operation and then may decrease gradually

or. in some case, drastically due to attrition in the system. [Ref. 14: pp. 1-6]

A key characteristic of Dvna-METRIC is its ability to deal with the

dynamic or transient demands placed on component repair and inventory support

caused by time variables in a scenario that includes sortie rates, mission changes, phased

arrival of component repair resources, interruptions of transportation, and the like, all

of which would be experienced in wartime. It computes how given resource levels and

process times would contribute to the wartime capability. By exploiting the mathemat-

ical structures of its underlying equations, Dyna-METRIC suggests the alternative cost

effective repair or stockage resource purchases which would achieve a target aircraft

availability goal throughout the wartime scenario.

Dyna-METRIC considers a three echelon inventory repair system.

Each base has an in-house repair facility which may have various test and repair capa-

bilities. This base repair facility may be supported by several Centralized Intermediate

Repair Facilities (CIRFs). Each operating base is capable of doing only linited types

of maintenance, usually limited to simple removal and replacement operations at the

flight line. It should be noticed that some of the bases have a CIRF while others do not

have direct flows of parts to the depot. A depot is represented as existing outside of the

model. It is seen from the model's point of view, as an infinite source of supply located

some order and ship time away.

2) Mathematical Assumptions. The major assumptions of

Dyna-METRIC which distinguish it most from the others are shown below.

I. Demand for items are generated by a nonhomogeneous Poisson process with in-tensity function m(t), and mean value function r(t)= fni(s) ds. The functions,

39

m(t) and r(t). will be defined later. The t denotes an arbitrary time. Thus, the de-mand process is dynamic.

2. The repair process is independent of the arrival process and has slack repair ca-pacity so that each repairable item demanding repair immediately receives serviceswith average service time based on the function F(s.t). F(s.t) will be defined later.

3. No more than one subcomponent can fail or be demanded in the repair of eachassembly.

(3) Time Dependent Pipeline Equations. Since the major objective of the

system is to avoid the loss of aircraft mission capability due to a shortage of correctlyfunctioning components on the aircraft, it is necessary to compute the number of com-ponents awaiting repair, being repaired, being on the way to and from another echelon

of repair, and partially repaired but awaiting :pare parts. Each state is a pipeline seg-

ment is which characterized by a delay time that arriving components must spend in thepipeline before exiting the segment. The model expands each component's expectedpipeline size into a complete probability distribution for the number of components

currently undergoing repair and on order, so the probability distribution for all compo-

nents can be combined to estimate aircraft availability and sorties. [Ref. 14: pp. 7-23]

Under the assumptions that the probability distribution of repairtime is independent of the failure process. the average number of components in the re-

pair pipeline will be:

ss dT

where:

d = average daily failure rate

7 = average repair time

With the further assumption that demand has a Poisson probability distribution, theprobability that there are K components in the pipeline at any point in time would be:

-K - a.

P( K in pipeline) = .se K e K!

However, in Dyna-METRIC the demand is a function of time so that:

40

d(t) = (failure per flying hour)

x(flving hour sorties at time t)

x(number of sorties day per aircraft at time t)

x(quantity of the component on the aircraft)

x(percentage of aircraft with the component)

Also, in place of a constant average repair time, T, the dynamic model uses the proba-

bility that a repair started at time s is not completed at time t. That is:

F(t.s)= Probability(component entering at s is still in repair at t)

- Probability(repair time > t-s when started at s)

The average number of components in the pipeline is derived by combining those two

functions. Consider only those components that arrived in an interval of time. As. cen-

tered at time s. Then:

A.(t,s) = d(s) F(t,s)As

where:

Al.(t.s) = expected number of components in the repair pipeline

d(s) = daily failure rate at time s

F(t,s) = Probability that a component is not out of repair by time t

As = interval of time centered at s

If we assume that the number of failures arriving in the interval As is independent of the

number of failures arriving in similar intervals centered at other time other than s and

F(t) is independent of the probability distribution generating the demand rate, then:

;.(t) = Zd(s) F(t,s) As

s<t

Further assume that As is very small, so that:

).(t) = d(s) F(t,s) ds0

41

With the additional assumption that the component failure probabil-

itv distribution is Poisson. ;.(i) is the mean of a time varying (nonhomogeneous) Poisson

process. That is, the probability of K components in repair at time t is:

PK) = )(e - (r) / K!

where:

Z(tf= d(s) F(t.s) ds0

14) Time Dependent Component Peiformance Measure. The component

measures typically computed by the Dvna-ME FRIC model are [Ref 14: pp. 24-291:

R(t)= ready rate at time t

FR(t) fill rate at time t

EB(t)= expected back orders

VBD(t) = variance of the backorders

DT(t)= average cumulative demands by time t

S(t) = supply level at time t

The ready rate is given by:

S(I)

R(t) = P{(KIK=O

Since the definition of the fill rate is the probability that a component

will be available when a demand is placed, it is therefore the probability that demands

have left at least one component available, that is, the sum of the probabilities of de-

mands less than the stock level. Expected backorders are given by:

EB(t) = Z {K - s(r)}P{K/ ).(t)}K>S(t)

S

1.(t) - s(t) + Z(t){s(t) - K}P{KI l.t))K=0

42

For K greater than s(t), there will be backorders of (K - s(t)l. The

probability of any demand level, tnat is K. is P{Ki .it)}, and the expected value of the

backorders is merely the product of the various values the backorders can take on times

the probability of a demand at that given value. The variance in backorders is given b\:

'B(t) = ( K - s(t)) P fK I(z)) - EB(t)K>S(,:)

<5, Time Dependent Optimal Determination of Spare Parts. The fact

that pipelines have time-dependent probability distributions means that the optimal mix

of spare components at one point in time may not be the optimal mix at another. Thus,

the approach to take is to compute. for each time period of interest, the marginal in-

crease in spare parts to achieve a given capability over those already input or determined

for a previous time. In determining the supply level, the model attempts to provideenough spare parts to give the desired confidence at the lowest cost at each point in time

of interest. Thus. the objective function is to minimize the total cost of spare parts.

Let:

S, = the spare parts level for component i

C, = the unit cost of component I

a. = the desired confidence level

K = the non-mission capable rate iot to be exceeded

P(K.S) = the probability that the non-miission capable rate is less than K,given a stock level S

I = types of repairables required on each aircraft

l hen, the problem to solve is:

I

minimize CiS!

subject to P(K,.S) ., S,> S1,

where:

S= the input stock level or previous time optimization stock level for component i

43

Assuming complete cannibalization. P(K,.S) equals:

I

P(K ,S) = EIPi(IQK,)i= 1

where:-51 - Q,Kn

P(QK) = ( PkKP,(K) = the KrObability of exactly K failures of component i

The necessary condition for the performance constraint to be met is to have.

P,(Q,K) > o., for each i. Then marginal analysis is used to determine the best mix of

additional components to achieve the desired goal. This process proceeds by investing

in one additional component at a time which is selected by finding the component that

gives the largest increase in the logarithms of the confidence level at the lowest cost.

That is. we determine:

AjIn P ( KS) / C1

where:

A, In P (K,.S) = In{P(K&, S) I P(K,,S))

S, = (s, S. ,s ..... )

The component for which supply is increased one unit is the one whose index solves:

max A In P(K.S) / C,. This process continues until the given confidence level is achieved.

At this point, the resulting value of S is the efficient solution of the base stockage

problem.

/6) Model Evaluation. The Dyna-METRIC model is relatively hard to

adapt for the Korean Air Force. Especially since this model assumes that each repair-

able item needing repair immediately receives service. This does not occur in the Korean

Air Force. Currently, the Korean Air Force applies a batch repair policy for repairables,

except in case of NORS condition aircraft. This is a serious violation of the assumption

of Dyna-METRIC. So, the Dyna-METRIC model is difficult to use for the Korean Air

Force repairable management system without using a continuous repair policy.

d. Mathematical Mlodels Used In The Navy

(1 Background. The Uniform Inventory Control Program (UICP) was

developed in August 1965 to provide a standard system to be used at all U.S Naval

44

Supply Systems Conunand (NAVSUP) ICPs. The Fleet Material Support Office

(FNISO) under the direction of NAVSUP is responsible for the system design. ADP

analysis, programming and documentation of UICP. The development of the UICP

formulas for inventory levels follows the approach used by Hadley and Whitin. [Ref.

13. 19]

2 ) Mathematical Assunptions

1. It is a continuous review system. Wholesale inventor levels requirements and as-sets are known by the Inventory Control Point (ICP) at all times. [Ref. 13, 19: pp.162-165, Chapter 3 Appedix A]

2. It is a steady state environment. The key characteristics of the items managed bythe ICP are assumed to be constant over the forecast period. These characteristicsinclude the forecasted average values and variances of the random variables of therate of customer demand, procurement leadtime. production leadtime. depot repairtimes, depot repair survival rate and the rate of carcass returns.

3. An order for procurement or repair is placed when the assets reach the reorder levelor the repair level.

4. Customer demands and carcass returns do not occur in more than one unit pertransaction.

5. The unit procurement cost or repair cost of an item is independent of the magni-tude of order quantity or repair quantity.

6. The cost of a backorder and the time-weighted cost of a backorder can be accu-rately quantified.

7. The reorder level and repair level are always non-negative.

S. The cosz to hold one unit of stock in the inventory is proportional to the unit costof the item.

9. No interaction exists among families of items or individual nonfarnily items orboth. Each family or nonfanfilv item's inventory levels requirements are calculatedindependently of those of families or nonfamilv items.

10. The optimal inventory levels are determined by minimizing an average annualvariable costs equation composed of order cost plus holding costs plus shortagecosts.

(3) UICP Depot Level Repairables (DLRs) Procurement fodel. The

repairables procurement model starts with a total variable cost (TVC) equation which

is to be minimized. A notable difference from the model for consumables is the inclusion

of receipt of RFI assets from a repair process in the DLR model.

(a) Ordering Cost

Let:

A = administrative cost to order

45

D = forecasted quarterly recurring demand

B forecasted quarterly regenerations

Q = economic order quantity

Then. 4(D - B)IQ is the expected number of procurements per year. Thus, the annual

ordering cost is given by 4(D - B)A/Q.

(b) Holding Cost

Let:

I = percent cost per dollar of inventory held annually

C = item cost (replacement)

L = procurement leadtime

T = repair turnaround time

R = repair level

F(x) = the function of probability distribution of leadtime

demand

The mean demand during a procurement leadtime is given by:

DT + (D - B) x (L - 7)

Thus. holding cost is given by:

0fIC--" + R - {(D - B) x L + B x 7) + f (x - R)F(x) dr]

x>R

let:

= shortage cost per requisition short

E = military essentiality weight

F = quarterly requisition rate

Then:

(.E){4(D - B) Q}F

DJ[>_,(x - R) F(x) dx

46

It equals the cost of the expected number of backorders in a year. The TVC equation

is symbolized by:

TIC = (D - B)A / Q

+ IC[Q/2 + R - {(D - B)xL + B xT} + f (x- R)F(x) dx]xa R

+ (/.E)[4(D - B)/Q] F/ D f (x - R)F(x) dxx?_R

Then, by setting dTVC ; dQ = 0:

Q = , s(D B) (A + X.>EF RO3 x- R)F(x)t Iq

Also, by setting dTVC ; dR = 0, we find:

QICD

f - R)F(x)dx- {QICD + 4).EF(D- B)}x>!R

Because the expression f ,R(x - R)F(x) dx is the cumulative

distribution for leadtime demand, this is the quantity defined as RISK. Note that since

Q and R are related, the reorder quantity Q cannot be solved independently. Thus,

UICP approximates Q by using a variation F the economic order quantity formula:

Q 18(D - B)A

IC

Then, OICD / {QICD + 4.EF(D - B)} can be computed for RISK determination.

14} UICP Depot Level Repairables Repair Model. The repair model also

starts with a total variable cost equation viewed largely independently of the procure-

ment problem. Note that its time horizon is the depot level repair turn around time.

The total variable cost equation for the repair model is the sum of the order cost, holding

cost and backorder cost.

47

Let:

0, =economidc repair quantity,

A,2 repair administrative order cost

C,2 repair price

R,=repair level

F2,(x) = probability distribution of demand during during repair turnaround time

z,- repair shortage cost

Then:

TI"C = 4 min(D,A).42 / Q2

Q2+ 1C2 {j R2 - (D XT) + f (x- - R 2)F2(x-)dX}

+ A,E{4 tnin(D,B) / Q,}F / Df x 2(x - R 2)F2(X)dX

B,: setting teT1C I Q2 =0-

Q2 n8 in_(D,B)A2 ± 2 EF/ DJ (x - R2)F2 (x)d~x iic 2x R2

Also. by setting ('TVC / 8'R2 =0:

Jf F2(X)dx= Q21C2D / Q21C2D + 4; 2EF min(D,B)x>o

Again, Q2 and R2 are related. The UICP model approximates Q2 as follows:

Q2= .,/8 nzin (D,B)A42 / 'C2

Then, Q21C2D I Q21CD + 4; 2EFB can be computed by using the above results.

48

"5; Intgrated Repairabks Model. As stated earlier, the requirements

computed by the procurement and repair models are accomplished independently of

each other. As a result, this leads to a carcass constrained situation. That is, the com-

puted procurement inventor level for an item does not provide sufficient carcasses to

allow repairs at the computed repair inventory level. To solve this problem. Naval

Supply System Conmmand (NAVSUP) has made some changes. Under the model inte-

2ration. there is only one RISK formula:

RISK = IC 3 D / IC 3 D + ;.FE

where:

C3 = (B/D)(C2) + (1 B

Also, rather than using a procurement leadtime or a depot level re-

pair turnaround time. it is uses an average acquisition time as the horizon for computing

the safety level. The average acquisition time (L,) is defined by:

L, = (1 - B/D)L + (B/D)T

It is clear that L2 is the weighted average of the procurement leadtime

and the repair turnaround time because D-B represents the quantity to be procured and

B represents the quantity from the regeneration.

S6, Model Evaluation. For the UICP model, three major problems exist.

The first problem is that UICP models is a continuous review inventory system. The

Korean Air Force repairable management system applies a periodic review. The second

problem is that UICP assumes that no interaction exists among failures of items.

Finally. the optimal inventory level for UICP model is determined by minimizing annual

variable cost (order costs plus handling costs plus shortage costs). However, the Korean

Air Force does not have the same objective.

e. Availability Centered Inventory MVlodel

I1 Background. ACIM was developed by CACI Inc. under the spon-

sorship of the Ship Support Improvement Project, Naval Sea System Command.

PNS-306. and aproved in March 1981 by the Chief of Naval Operations for use in de-

ternuning consumer level stockage quantities for selected equipment [Ref 20: pp. 1-2].

llowever, this model was initially used as a part of a larger model (Logistics Support

49

Economic Evaluation Model) which provided necessary inputs and enabled comparisons

with other Navy stockage policies. After this. successive refinements for the simplifi-

cation of the solution procedure and associated computer programs have been made for

several years.

This model was originally designed to calculate inventory levels for

all items in the parts breakdown of an equipment and at all stockage facilities in a

multi-echelon support system. Thus, ACIM is capable of computing levels for all ships,

intermediate maintenance activities, and depots that use or support the equipment.

However, this model is mainly used in the provisioning process to compute shipboard

allowances to achieve specified weapon system readiness levels which have not been

achieved with the standard protection level models.

2; Availabliiy Measure. ACIM recognizes that the purpose of a supply

system is to provide sufficient support so that a weapon system is operational when it

is needed. The terminology used to described this goal is operational availability (A).

ACIM defines A, by the following formula [Ref. 21: p. 5]:

,4 = up time (up time + down time)

= MTBF (MTBF + MTTR + MSRT)

A, may also be interpreted as the probability that the equipment is in an operable con-

dition at a random point in time. Among the three factors of A,, the MTBF and MTTR

are system parameters outside the control of the supply system and are viewed as con-

straints. MSRT is the only term which depends on stockage postures. it is therefore the

one that the ACIM model focuses on to achieve a given value for A,.

(3/ .1atheinatical Assumptions

1. All parts are organized in terms of equipment with a top-down breakdown that canbe represented as an arborescent network. Any part may be totally consumable,totally repairable, or any mix thereof. [Ref. 21: pp. 10-111

2. Stockage and maintenance facilities are organized in a hierarchical structure ac-cording to supply! maintenance flows which can be represented as an arborescentnetwork as illustrated below. Each facility has a colocated maintenance and supplycapability. The facility at the top of the structure is assumed to have an infinitesupply of all items.

3. External demands upon supply are stationary and compound-Poisson distributed.

4. All stockaze locations use a continuous review, (S-1,S) ordering policy.

50

5. Mean Time to Repair (MTTR) is defined to include all equipment downtimes that

are not supply related.

6. There is no lateral resupply (transshipment) among bases.

7. Items repaired at any location are assumed to be returned to collocated stocks forissue.

S. If, for a given facility in the network, the stocks are physically distributed in severalplaces. it is assumed that the resupply time for direct customers (next lower ech-elon) is independent of the location.

9. The order policy assumption precludes consideration of economies of scale for re-supply. That is, all ordering is on a one for one basis.

(4) Model formulation. The goal of ACIM is to maximize the opera-

tional availability (A) of a weapon system subject to a given inventory budget. With the

definition given above of A0 and with the view that MSRT is the only term which is af-

fected b the stockage decision, the developers of ACIM argue that the allocation which

maximizes A, is equivalent to the allocation which minimizes MSRT. ACIM therefore

actually attempts to minimize MSRT subject to given constraints.

Let i be an arbitrary item in equipment e (which may be e itself). Letu= 0 represent an arbitrary facility in the support system and u= 1,2.3,...., represent fa-

cilities at the next lower level, i.e, those facilities that submit items for repair directly to

or obtain resupply from facility o. Then, the objective can be explicitly stated as follows:

Find values for S, which minimize D,, for all user locations u:

subject to ECkSv = Bk,v

where:

C, = unit cost of item k

B = given budget for spares procurement

D,= expected delay per demand upon inventory for for item i at location u

S,= the level of inventory for item k at location v

D,, is one of the components of MW which is defined as the mean time to return a failed

unit of item i at location u to a serviceable condition. M, is given by:

A, = Di. + Ti ,

51

where:

D_ = expected delay per demand upon inventory for for item i at location u

T,. = mean time to repair item i at user location u

D,, in turn is given by:

Dju = (1/-.) Z (X - Su) P(X: ;.iuT.u)X2 S.~

where:

S,, = stock level of item i at location u

= expected number of demands upon inventory for item i at location u

P(X : ,.,j = probability of X units of stock reduction for item i at location u

T, = mean resupply time for item i at location u

T, is given by:

Ti = (Liu + Liu) + (1 - ru)(RI. + Ri,)

where:

= probability that a demand for item i upon inventory at location u

L,, = average resupply leadtime assuming stock is is available

L' = additional resupply leadtime due to expected shortage at the resupply source

R, = average shop repair cycle assuming availability of spares for items within i

R, = additional shop repair cycle due to expected shortage of spares items within i

Since ACIM assumes that MTBF and MTTR are independent of the

stockage policy, minimizing D,, is assumed to maximize A,, (operational availability of

a equipment at location u).

(5, Solution Technique. Assume that initial values for S,, are given for

all items and locations. These may all be zero or some minimum value given by policy

or current assets. Then compute the MSRTs for all items.

ACIM calculates what the new MSRT would be if one additionalunit of stock were placed against that particular item. Subtracting the new MSRT from

the old MSRT and then dividing by the unit cost of the item provides the model with a

value which is multiplied by the item's demand to determine a selection-rank. After the

selection number is calculated for each item, one unit of stock is added to the candidate

52

with the highest selection rank number. The operational availability (A0) is calculated

and if the target A, has not been achieved, the model continues the same procedure.

6/ A1odel Evaluation. As with the UICP model, three major violations

of'the basic assumptions of the ACIM model exist. The first is that ACIM assumes that

you have a continuous review system. The second is that ACIM assumes there is no

lateral resupply. Finally, ACIM assumes that the stocks are physically distributed atseveral locations. Obviously. the Korean Air Force supply system does not require se-

veral physical locations, even though the system objective is to maximize operational

availability.

D. SUMMARY

Throughout Section B and C of this chapter, we have reviewed the various inven-tory models and theories for repairable items which are currently in use in the military

services, or in the process of being evaluated. All of the assumptions and the math-

ematical formulations of the models were stated and discussed.

53

IV. !ODEL PROPOSAL FOR THE KOREAN AIR FORCE

A. INTRODUCTION

In Chapter II, we summarized the current Korean Air Force repairables manage-

ment system and repair process. The general structure of repairables management in the

Korean Air Force is similar to the structure assumed by several of the models described

in Chapter III.

In this Chapter, we describe a proposed inventory control process for the Korean

Air Force. This process is based partly on queueing modelF, and partly the Wilson-

larris economic order quantity model.

B. THEORETICAL BASE FOR THE MODEL PROPOSAL

I. Notation

c = number of identical repair channels

a, = average repair rate per repair channel

= average failure rate of an individual unit of a repairable item

X = random variable describing the steadystate number of NRFI units of item 1

P,= steady state probability that there are n NRFI units

2. The N/M//K/K Queueing System

This model is a limited source queueing model in which there are only K cus-

tomers. It is variously called the machine repair model, the machine interference model.

or the cyclic queueing model. It is one of the most useful of all queueing theory models.

One way to view this model is shown in Figure 6. [Ref. 22: pp. 186-1901

The population of potential customers for this queueing system consists of K

identical devices, each of which has an operating time of 0 time units between break-

downs. 0 having an exponential distribution with average value I / . The one

repairman repairs the machines at an exponential rate with an average repair time of

1 /,u time units. The operating machines are outside the queueing system (outlined by

the dashed lines) and enter the system only when they break down. The queueing system

always reaches a steady state because there can be no more than K customers in the

system (one machine being repaired and K-1 waiting for repairs).

54

Down time0 1 Oueueing time Repair time

Figure 6. Q ucucMichS ilc wIthOeRnimi MM1K

\\ en oftheMachines ar on(o ocaigthnKnozhe r p

atin~~ and the timc until thumbnrxt machinebrk s down h iiu fKnietia

exponential distributions with parameter (K-n) 0'..

3. The M/////Queticing System

This qUCucinel1_ systemi is slimillar to thc machine repair model considered In the

sctLionI abOVC CXCCpt dtat We have c rather than one repairman, wherec K, as sh o'wn

in 1Fi-tire 7. [Ref. 22: pp, 190)- 192]

C. DEVELOPING A MODEL FOR THE KOREAN AIR FORCE

I. Application Conditions

1.Supporting 11) Items. i.e.. we want 1.) items to be available at all timecs. In this case1.D rep~resents thei number of' aircraft which the K orean Air F'orce wants to haveavaliable at ahi timecs.

2. P(SUCCeSsful RepirII) is Close to 1. Whenl P(successftll Repair) = I, we have a pureiiwa.hine repair quIcuCing' Sy'Stcm1. WnVC P(SL1CCeSShIlI Repir1l) W 0.e ave a Coll-

sieitemn whose inanacment is best accomplished using, standard miethio o-oL1eie frontl invenito:.% theory,,. The Samle1 daita colClece fr-on the K orean Air I orocfor, thlis Study indiCa ted that P( S tICCcOSfI1 RepirII) = 0. 9S or- ItIIII 1 he- '11 r all repa I Nitemls examiincd. Uor tis reCason,. the proposed model Is based pairtlY oil the 1it:I-cine~l repir q uLCuIc,1! mo1del. I he data baise Used 1 0or the St UdV a,'s o1)H id I ro il

55

Operating time Repair time

Number of machinel down

Firture 7. Qiteuiing System )lith c Repairman (l\1/i\1/e/K/K)

tcK LCdurinu the 19S4-1988. -able 3 shows thc nican probabilitY of, suc(.-c cs s l 11 repair bascd on thc sample data. 6

-'-1) 1 3. SC C E S S 17(LREPAIR) COMlPUTED FROM SAMPLE DATAYear S4 S5 S6 87 8sAcr~

I .5 )M8 0.898 7 (0.986 0.9S 84 juou. .98 7 (19 0.976) S 1.

.CondJition1S close to the M, NI I K: K quuc Ngssteml hold. Becaulse. there a i-C anumer f rpaame inreairdept.it igh b reasonable at first to assumec that

th NI.~c NI' K iijodel applies. I lowever, most repairs rq~uire speciali/edk sKillshldl by few people lor a .:ivcn rcpairable part. FurtherCI- tlIe repair dcpo0t IS stal*\2d

adscheduled SO thalt there- IS little overl-l exceSS capaityV. AS aI rCslt. ti CI cI"littleC Chan,11e in the recpair process oultput rate wNVII theC Second. thirld. 'ourIth. dCt.unit backs up inI the repair ll LCLIC.

4. Steady state1 has beenl achieved.

6 Tic sainple data ]list appears in Appenldix A and B

56

2. Model Formulation

This proposed model consists of 3 steps. The first step computes a minimumpopulation size K for the ith type of repairable item. Population refers in this case to

the installed material plus the stored RFI material plus the NRFI material awaiting re-pair. In other words, all material, both good and broken.

Once the minimum population size K is chosen, an attrition buy safety levelS, is chosen. This safety level is used to insure that the actual population size usuallystays above K during the period when the system is awaiting the arrival of an attrition

buy.The last step is to compute the size of an attrition buy itself. This Q7 is com-

puted using the Wilson-ttarris economic order quantity model.

In step 1. described on the next page. the \1 M I K K queueing system willbe used to obtain probability expressions of the form P(N, > K - D,). These expressionsare obtained as follows:

K,

P(Vi > K - D) = Z P(, = n)n= K, - D, +1

where:

N. = Discrete random variable representing the numberof units of item i which are broken

K = Decision variable representing planned minimumnumber of units of the item to be owned, including thoseinstalled in the repairable equipment

D = Number of units of item i which the Korean AirForce wants to have available at all times

-he riLht hand side of the last expression is equal to:

K, - D,

1- Z P(Al = n)fl=C)

using the MM 1 I K K queue. Allen [Ref. 22: p. 192] gives the following formula forP(.\ = a):

. K-"

(k -4

57

where:

K,

, Z K,-k)! (-k=O

In step 2. described in the following section, the expression P(X, > S,) is also

used. It is obtained using the Poisson distribution:

(DaiLi)lK e- (Do,L,)

P(X1 = A) Dii

where:

XF = discrete random variable representing attritiondemand for item i during a procurement leadtime

D, = forecasted attrition demand for item I

L= forecasted procurement leadtime for item 1

We will want S, such that P(.. > S.) is less than a risk value, b,,, which is pre-specified

by the user, where P(A, > S). is given by:

0o

P(X > Si) = 3 P(Xi = k)k= S, +1

' (DaLke- (Dt.)

k= S, +1

S D kL) e- (D,L)

k=O

a. Step I

There are several different objective functions which might be used to help

choose the size of the population fot the ith item. The following four subsections de-

scribe four different math programs for choosing K,.

1) Ahernative I

Choose K, by the following:

58

mi n K,'

subject to P(.V > (Ki - Di)) _ bli

where:

D, = Number of units of item i which the Korean AirForce wants to have available at all times

b, = Parameter chosen by the decision maker

This first alternative formulation contains no explicit budget constraint. However.

minimizing K, for all items is equivalent to minimizing the cost )f that portion of the

population which is owned to meet the availability goals. Unfortunately. this formu-

lation doesn't identify solutions which exceed the available budget. The next math

programming formulation accomplishes this.

(2) Alternative II

rain E Ki

i=1

subject to P(Xi > K, - D) < b1l, i = I .......

and ZhiKi + Zai[max(O. Ki - K 1)I B

where:

h = annual holding cost per unit held

B = total budget for one year

m = number of repairable items being considered

K, = previous planned minimum number of units owned

a. = unit cost of item i

(3) lternative IllI

Maximize customer service subject to investment constraint:

59

ni n Z = 'P(N >(Ki - Di))i=1

m "I

subject to YhiI + Za-[max(O. K - K0,)] < B

'4 Alternative IV

Maximize customer service subject to investment constraint (another form):

nun Z = max {P(.\.> (Ki- O))

subject to ZiKi + ZajmaxO, K o, - IM B

i=l i=1

The choice between these alternate formulations depends on the way the Korean Air

Force feels most comfortable with resolving the conflict between budget limitations and

customer service. For the numerical computation of this study, we chose alternative I.

This choice balanced the need for realism against the time constraints imposed on this

research.

b. Step 2

Attrition buy reorder point, R, = /, + S,, where S, is obtained from the

following:

min Si

subject to P(A;. > Si) < b2i

where:

X = discrete random variable representing attritiondemand for item i during a procurement leadtime

b2, = parameter chosen by the decision makerfor item i

Using the reorder point R,, place an attrition buy when the total number of unattrited

units (working and failed. installed and uninstalled) reaches or drops below R,.

60

c. Step 3Use the Wilson-Harris EOQ model to choose procurement quantity:

* 2"4Da,Qi I lCi

where:

A = administration and ordering cost for item i

D= attrition demand rate for item i

,= annual holding cost rate for item i

C,= unit cost for item I

So the initial buy quantity formula is R, + Q: = K, + S, + Q:.

This proposed repairable item inventory control process works as follows.To start the system, purchase K, + S, + Q: units. As installed items fails, repair them

as soon as possible (i.e. do not use batch repair). When attritions in the repair process

reduce the actual population size to K + S,, place an attrition buy.

D. SUMMARY

In this chapter, we developed a new inventory model based on the M,:MI,K K

queueing theory and the current repairables management system of the Korean Air

Force. In the next chapter, wve will compare the current inventor model with the pro-

posed model, in terms of the numerical results of each, using computer computations.

61

V. COMPARING THE COMPUTATION RESULTS OF THE MODEL

A. INTRODUCTION

In this chapter. we will discuss a specific result for the two models, one is a existing

model of the Korean Air Force, the other is a proposed model. The results show only

summary of the computer computation and a comparison of the results. The whole

computation results are shown in Appendix C.The purpose of this chapter is to provide a numerical analysis between the proposed

model and the current Korean Air Force model. The data used in the computer com-

putation consists of a selection of 8 line items form the data on 50 line items which was

obtained from the Korean Air Force. Due to the limited time available for this research.

only 8 items were selected for use in the comparison of the models. A selection of high

demand and low demand items were chosen. The complete data set is described in Sec-

tion C of this chapter.

B. MEASURE OF EFFECTIVENESS (MOE)

The following MOEs are all reasonable methodologies for evaluating repairable item

inventory models for the Korean Air Force. However, we will concentrate on one MOE

for model comparison such as operational availability.

I. Fill Rate

It is computed by taking the total number of units demanded at a base over a

fixed period time and dividin2 that number into the total number of units issued at the

time they were are demanded. Thus, fill rate is the percentage of demands that are filled

immediatelv.2. Operational Rate

Operational rate is the probability that, at any given point in time, there will be

no stockouts from base supply (backorders). Operational rate is computed by counting

up the length of time (in days) during a year that no backorders existed and dividing this

number by 365. This gives us the percentage of time during the year that no backorders

were in existence.

Operational rate has an advantage over both the fill rate and mean number of

backorders in that it may be directly related to the supply system's effect on operations.

However, it has a disadvantage over both the fill rate and mean backorders in that it has

a rather bothersome all-or-nothing character.

62

3. Mean Supply Response Time (MSRT)

.ISRT is the mean time it takes for the supply system to respond to the demandfor a replacement part or component. It is obtained by taking each stockout that is es-

tablished during a period of' time. observing how many days it takes to satisfy the

backorders. adding up all these numbers and dividing the sum by the total number of

demands during that period. It is the average time weighted unit short (TWUS).

MSRT is important by itself as an indicator of success of the supply system in

meeting response time goals. As a measure, it considers both the likelihood of satisfying

demands from stock on hand and the length of the delay in satisfying demands when the

system runs out of stock.

4. The Average Number of NORS Aircraft

NORS stands for "not operationally ready because of supply". The averagenumber of NORS aircraft is computed b% counting for each aircraft the number of

NORS days during the course of a year, adding these numbers up for each aircraft, and

finally dividing by 365. Considering that the purpose of a spare parts supply system for

the Korean Air Force is to maintain the operational readiness of aircraft, the average

number of NORS aircraft would certainly seem to be a reasonable measure of effective-

ness. A stockage model attempting to optimize with respect to a NORS measure would

require more restrictive assumptions than those optimizing with respect to fill rate, the

average backorders. MSRT. or the operational rate. Above all, its lack of"separability"

is a cause of major mathematical problems.

Something to note is that the operational sector of the KAF has consistently

suggested that the Korean Air Force supply system should be managed in operational

terms, not by supply terms. That is, their imminent question has been "How manyspares are needed to provide an operational readiness of X percent of the aircraft?".

5. ConstraintsThe budget available for the investment in spares is either a constraint in all of

the models or the objective is to achieve specified performance at the minimum budget.

The essence of any inventory control problem is the trade-off between cost and systemperformance. As mentioned in the beginning of Chapter II, the budget constraint is the

major resource constraint for the Korean Air Force.

63

C. COIPARING THE MODEL RESULTS

1. Data Characteristic

The data used for the model comparison are 8 line items of the 50 line items in

the sample data. The whole sample data is shown in Appendices A and B. This data

consists of: a) quarterly demand data for 19 quarters; and b) quarterly attrition data for

19 quarters for each item. The demand data was analyzed using Minitab. The mean,

standard deviation, variance and squared coefficient of variation were calculated for each

item. Since data concerning the failure process for each installed item wasn't available,

we estimated the mean item failure rate. a, by computing mean demand over 5 years,

divided by number of aircraft. We estimated the annual item repair rate. p,, by dividing

actual working hours per year by the average repair time in hours for the item. The

current average repair time for each of the S items in the sample was obtained from the

Korean Air Force [Ref. 23]. These values are shown in the table below. For the pur-

poses of this comparison we assumed that the random variables associated with the

failure process and the repair process had exponential distributions.

Table 4. MEAN REPAIR TIME:This table shows the meanactive repair times for the 8sample items

Item number Mean RepairTime (hours)

2.9

3.94 2.1

5_ _7.5

6 6.1

7 1.5

S 1.2

The actual data for an item consists of 19 quarterly demand observations during

1984-1988. Items were selected from three different kinds of aircraft, the F-4D E.

F-5A B and F-16A B. Appendix B shows the Minitab results for all 50 items in the

sample data set.

64

The data used in this research also included quarterly attrition quantities for 19 quarters.

However. for the model comparison, we selected 8 items. 7 The items which were chosen

had a variety of annual demands.

2. Comparing the Results

The complete computation results for both models are shown in Appendix C.

A summarv of these results is shown on the following pages. For the Korean Air Force

model, the target operational availability is 90 percent. In the Korean Air Force, this

is a fixed level of availability.

During the computational steps, the actual operational availability allows an

increase from 90 percent to 93 percent, however the proposed model didn't meet the

budget constraint for a 94 percent operational availability. Figure 8 shows the difference

of K values: a comparison of the current K values of the Korean Air Force with min K

friom the proposed model. Figure 9 shows the difference in safety levels for the proposed

model and the Korean Air Force model. In the comparison, the attrition protection

levels of 0.98 is fixed. Appendix C shows the results based on several attrition protection

levels between 0.90 and 0.98. SLQ is a safety level for the Korean Air Force model. min

S is the attrition safety level for the proposed model. Figure 10 shows the difference in

procurement quantities. For the Korean Air Force model, RO is a procurement level,

optimal Q is for the proposed model. In the actual difference for the two models, the

proposed model has slightly lower procurement quantities than the Korean Air Force

modcl.

3. Description of the Computer Program

For the Korean Air Force repairable item inventory model, we first computed

a demand forecast using a trend value to guide our choice of alpha value to be used in

an exponential smoothing forecast for the next quarter's demand. When the trend value

fell between 0.9 and 1.1. we chose an alpha value of 0.2, otherwise we used an alpha of

0.4. Then the demand forecast was used to calculate the DDR, DRP, RCQ, OSTQ. SLQ

and RO. The computational procedure for this current Korean Air Force model is

briefly described in the following paragraph. The complete computational results for the

Korean Air Force model are shown in Figures 20 to 27, and the whole mathematical

" Item numbers for the 8 items chosen from the 50 items are 10, 16, 20, 27, 30, 33, 37 and46.

8 Attrition protection level is the probability that attrition will not reduce the population ofitem i below A, when an attrition reorder is placed when the population drops to the K, + S, level.

65

Operational Availability 0.91

_._[ _-_ 4 '5 U ,iHM n I, 9$ I 15. 2...24 1.8r3. 1il. . -" ' ;,, i..L U9

Curr K 99 110 195 183 110 199 19, 12{.

Difference -% +5 +39 -3 -8 +4 u r$ -1 1

Op_erational AvaiIabilit_. .91

_ 2 4 5 7 7 _

Mir I'. .3 115 235 10-J!_) 1Q2 239 Jl 109

Curr K 99 110 195 183 11-I( 199 19) 120

Difference -6 +5 +41J -3 -8 +40 4.7 -11

Operational Availability =.92

1 2 2 ,= 5 (5 7 __

Min ?. 93 I 16S 236 181 10J 24) 202 I I'Curr K 99 110 195 183 I 10 199 194 12U

Difference -6 +6 +41 -2 -7 +41 '-9 -I1

Operational Availability = ,) .93

1 2 3 4 5 5 7M.in K 93 1 1 23 7 1.1 10J 241 2-3 1iCurr K' 99 110U 195 183 1 10.) 199 19,1 12

Difference -tUp +6 +',12 - -7 4-42 +9 -9

Figure 8. DilfrecICC of Total Stock Level

66

Operational Availability = o..9() Protection Level = 9V

1 _ _ 3 4 '5 6 "Mll S I 1 2 , 4- Z >

iS L 1 1 1S L r 2 I 2 I 1 1 3 2

Di f ference -1 0 -1 +I I r -1 41

Operational Availability = 0.91 Protection Level =.93

1 2 " 4 5 t5 7 uH i S 1 1 1 2 2 I 2SLQ 2 1 2 I 1 I 3 2Difference -1 0 -1 +1 -1 C) -I +I

Qperational Availability = Q.92 Protection Level = 0.353

1 2 3 4 5 5 -7 3lin S 1 I I 2 2 1SLO 2 I 2 I 1 1 2Difference -1 0 -1 1 +1 Q -1 +1

_peritional Availabilit = .,1) - Protection Level =.97

1 2 3 4 5 6 7MinS I 1 I 2 2 1 2 3S 2 1 ", 1 I I 3Difference -1 0 -I +1 41 ' --1 +1

Figure 9. Difference in Safety Levels Bctwecn the Two Models

67

Operational Availability =

1 2 3 4 5 5 7 8Optimal Q 1 I 1 1 I I 2 2RU 3 1 2 1 2 1 4 3Difference -2 0 -1 0 -1 0 -2 - I

Operational Availability = U.91

__ 2_ 4 5 6 i

Uptimal U I I 1 I 2R0 3 1 2 1 2 1 4 3

Difference -2 0 -1 0 -1 I -2 -l

Operational Availability = o.92

1 2__ 4 '5 (3 7 _

Opt i ma , 1 1 I I 1 I 2RU 3 1 2 1 2 1 4 3Difference -2 0 - 0 -1 0 -2 -1

Operational Availability = .93

_ 2 3 " 5 (3- 7 'Optimal Q I I I 1 1 I 2 2RU 3 1 2 1 2 I 4 3Difference -2 0 -1 0 -I -2 -

Figuire 10. Dilffrelice of Optimal Procurement Quantity

6S

process for the Korean Air Force repairable item inventory model is described in Chap-

ter II.

For the DDR. we computed the total demand for each quarter, divided by 365

days. The DRP equals the RTS divided by the sum of RTS. NRTS and COND. The

RCQ was co ,puted by multiplying DDR times DRP times RCT. This RCT refers to

the repair cycle time allowed for depot level maintenance. For these computations, we

used 120 days for RCT because this is the maximum allowed by the Korean Air Force.

The OSTQ is computed by multiplying DDR times NDRP times OST. For OST, we

used 365 days which is the maximum allowed by the Korean Air Force. The SLQ is

computed by finding the square root of 3 times RCQ plus OSTQ. The RO is computed

by adding SLQ. RCQ and OSTQ.

For the model comparison, we used two system parameters (SLQ and RO) and

the item's current K value. Using the historical demand data obtain from the Korean

Air Force, and their current repairable item inventory as described above, the requisition

objective (RO) and safety level quantity (SLQ) were computed for each calendar quarter

in 19SS (note that the equivalent historical information was not available from the

Korean Air Force). The current K value in Figure 8 was obtained by taking the sum

of RFI, NRFI and the currently installed population, as of November 1988 (previous

historical data was unavailable). The SLQ for model comparison in Figure 9 is the av-

erage safety level in 198S. Also, the RO in Figure 10 is the average RO level in 19SS.

For the proposed model, we first need to choose b, for the first constraint set

for each item. For b ,, one minus the current target operational availability in the

Korean Air Force was used. To obtain the constraining total budget value B, we com-

puted the sum of the current K,, value for each item times its unit cost plus the holding

cost for each item, over all 8 items. The rho value (-) for an item was derived from

the average failure rate (a.) of that repairable item, divided by its average repair rate (p).

For the model comparison, a value for the budget constraint had to be obtained.

We estimated a value for this variable by sumnming the following products over all items:

K., times unit cost plus annual holding cost for item i (reminder: K, is the number of

units of item i currently owned by the Korean Air Force).

In step 1, the min K values are computed based on two constraints. First, we

calculate the probability that ., is greater than K, - D, for an item. The starting value

used for K is D, (the size of the currently installed population). For the min K in the

computer program, we first compute P0 of the M,:MI K'K queueing system. And then.

we compute P(.\ = i) using P0. In the P(., = n) computation, n equals K, - D,. If the

69

constraint on P(. > K, - D,) isn't satisfied by choosing K = D, then we increased K, by

1. These steps are continued, until PiN > K, - D) is less than 0.10. At this time. we

took the current n value, and then added D, to get the min K value. After obtaining the

min K values from the first constraint set for each of the 8 items, the budget constraint

is checked. If these A, values don't satisfy the budget constraint, we go back to the first

constraint set and increase the b, values. In this case, a method for choosing an in-

creased set of b1, 's must be chosen. Two ways exist to do this. The first method is to

increase the b1, values simultaneously for all values. The other way of solving this

problem is to apply different operational availabilities for each item. In other words. the

operational availability chosen would depend on the item characteristics. The first

-nethod described above (simultaneous increase of every b,,) was used in this work. If

the nm'n K satisfies the budget constraint, this is min k used the model comparison, in

Figure S. As it turns out, the first set of min K. values met the budget constraint. So,

the min K values from the first constraint set gave a feasible solution in the budget

constraint. Actually, the budget constraint allowed up to 93 percent operational avail-

abilitY for all S items.

For step 2. we use the Poisson probability distribution to get minimum attrition

sa'ety levels. Suppose that risk b2, = 0.1. This would mean that the attrition protection

level is 0.90. Therefore, 1 - b,, equals the protection level. For the min S,, we first cal-

culate probability of , = 0, and next calculate probability X, = 0 and so on. We sum

these individual probabilities up until the cumulative value meets or exceeds the pro-

tection level (for the model comparison, we use 0.98). This is the nuin S, value for the

model comparison in Figure 9.

For step 3, the Wilson-Flarris EOQ formula is used. For this formula, a per

order ordering cost of 10 percent of the unit cost was used. This is the value which the

Korean Air Force currentiv believes to be in effect [Ref. 1: p. 65]. Similarly. a holding

cost rate of 10 percent per year of unit cost was used. The resulting optimal procure-

ment quantities for the model comparison are shown Figure 10. The complete compu-

tational results for the proposed model are shown in Figures 28 to 35.

4. Summary of the Results

The proposed model allows for a 93 percent operational availability within the

budget constraint. This is an improvement of 3 percent in operational availability with

the same budget that was used with the current Korean Air Force model to obtain 90

percent operational availability.

70

D. SUMMARY

Inr this chapter, we compared two models based on computer computations using a

samile data set. The proposed model has shown an improvement over the current

model. Ilowever. it doesn't mean the proposed model is better for the Korean Air Force.

A tremendous effort would be required to actually implement this model in the Korean

Air 1orce. Furthermore. additional research should be done to explore the performance

of the proposed model.

71

VI. SUMMARY AND CONCLUSION

For this research, there were three major objectives. The first was to review the lit-

erature to understand the current Korean Air Force model and other models, such as

METRIC, Mod-METRIC, for controlling the stockage decisions in multi-item, multi-

echelon inventory systems involving repairable items. The second objective was to de-

velop a new model for the Korean Air Force inventory management system. The last

objective was to evaluate this new model by comparing the current Korean Air Force

model with this proposed model.

For the first objective, we studied the METRIC family model, ACIM model and

U.S Navy model. We then reviewed their basic assumptions and mathematical ap-

proach. For the second objective, we reviewed the Korean Air Force inventory man-

agement system concept and conditions, and developed a proposed model. In order to

accomplish the last objective, sample data from the KAF was uscd in comparing the

Korean Air Force model and proposed model.

Analysis of the sample data revealed that the proposed model yielded some im-

provement in operational availability and safety level with the same budget. As men-

tioned in Chapter V, the proposed model requires further research. Currently the

research is hard to do because of time constraints and data available from the Korean

Air Force. So. additional research should be done to eliminate these shortcomings in the

proposed model. Also, for the implementation. we need more data points to obtain

better distributions for the demand process and repair time process.

72

APPENDIX A. SAMPLE DATA LIST 1

A. SAMPLE DATA DEMAND DISTRIBUTION LIST

Table 5. DEMAND DISTRIBUTION LIST

QTR I 2 3 4 5 6 7 8 9 10

84-1 0 0 0 0 0 0 0 0 0 0

84-2 0 0 0 0 0 0 0 0 0 0

84-3 0 0 0 0 0 0 0 0 0 2

S4-4 0 0 0 0 0 0 0 0 0 2

1984 0 0 0 0 0 0 0 0 0 4Total

85-1 0 0 0 0 0 0 0 0 0 1

85-_ 0 0 0 0 0 0 0 0 0 2

85-3 0 0 0 0 0 0 0 0 085-4 0 0 0 2

1985 0 0 0 0 0 0 0 0 0 5Total

86-1 0 0 0 2 0 0 0 0 0 086-2 0 0 0 0 0 0 0 0 0 2

86 -3 0 0 0 1 0 1 0 0 1 086-4 0 0 0 0 0 1 0 0 21 3

1986 0 0 0 3 0 2 0 0 3 5Total

87-1 0 2 0 2 1 0 0 0 5 2

87-2 2 1 0 4 0 1 1 1 7 0

87-3 0 0 0 0 0 0 0 1 2 0

87-4 1 0 1 3 0 2 0 1 2 4

1987 3 3 1 9 1 3 1 3 16 6Total

88-1 0 2 1 1 0 0 2 0 4 3

88-2 0 1 0 3 0 3 0 1 6 0

88-3 1 0 0 5 1 0 0 1 4 2

1988 1 3 1 9 1 3 2 2 14 5Total

73


QTR 11 12 13 4 15 16 17 is 19 2()

S4-1 0 0 1 0 6 1 2 1 0

8-4-2 ( 0 U 2 1 1 0 4 7 3

84-3 0 0 0 2 0 0 0 0 2 0

S4-4 0 0 0 2 0 2 0 2 5 2

1l)8- 0 0 0 7 1 9 1 8 15 5Total

85-1 1 0 4 0 1 1 0 8 4 2

5- 2 ( U 0 5 0 1 0 0 3 U

-. . . . .U ( 2 3 0 0 1 2

85-4 0 1 0 3 1 0 U 1 3

19S5 1 1 4 2 5 0 8 9 7Total

S6-1 0 1 0 4 0 0 2 2 3 4

86-2 () 0 0 1 1 2 0 5 3 0

86-3 0 0 0 3 0 0 1 5 386-4 0 0 0 2 1 1 0 1 1

19S6 0 1 0 10 5 3 2 9 12 9Total

S7-1 4 0 0 5 2 1 0 I 0 0

S7-2 I 1 0 4 5 0 0 2 4 0

5 1 1 3 1 0 0 1 1

1987 12 1 1 13 12 3 0 3 3Toral

8S-1 1 . 1 4 3 0 1 0 2 088-2 2 1 0 2 2 1 0 1 3 U

88-3 1 0 0 4 0 0 0 0 1 2

19SS 4 1 1 10 5 1 1 1 6 2Total

74


QTR 21 22 23 24 25 26 27 28 29 30

S4-1 0 2 0 1 9 4 36 S4 4 4

S4-2 1 0 0 1 5 5 44 66 7 0

84-3 1 0 0 0 11 4 52 43 1 0

84-4 3 2 4 3 1 0 55 32 7 3

19S4 5 4 4 5 26 13 1S7 225 19 7Total

s5-1 1 4 1 10 0 2 101 71 2 3

85-2 I 1 0 0 () 2 67 j 59 0 185-3 1 1 0 5 5 5 28 5o 5 0

85-4 0 1 1 4 4 1 42 25 2 2

1985 3 7 2 19 19 10 238 205 9 5Total

S6-1 2 5 4 5 7 6 61 40 0 3

86-2. 4 3 3 1 8 1 17 39 4 1

86-3 0 2 1 1 S 1 95 12 5 )

86-4 1 4 0 4 4 3 47 73 0

1986 7 14 S 11 27 11 220 164 9 9Total

87-1 5 6 2 2 10 2 19 50 11 387-2 4 1 0 2 7 7 24 37 7 0

s7-3 1 3 0 0 3 0 25 21 4 0

7-4 1 I 7 0 0 4 70 24 13 4

1987 10 11 9 4 20 13 138 132 35 7Total I88-1 0 5 1 5 9 10 42 31 5 2

88-2 6 2 0 6 9 1 21 18 9 0

88-3 0 3 2 2 5 3 Is 23 3 4

1988 6 10 3 13 23 14 81 72 17 6Total

75


QTR 31 32 33 13- 35 36 37 3S 39 40

84-1 0 Q 0 9 1 2 2 0 12 1

S4-2 5 0 1 7 7 1 3 0 21 7

84-3 4 0 2 9 0 1 0 0 6 3

84-4 0 0 1 18 2 9 3 5 2 1

19S4 9 0 4 43 10 13 8 5 41 12Total

85-1 12 0 1 17 10 4 2 1 17 2

85-2 20 3 1 2 26 3 0 1 9 0

85-3 1 0 3 2 32 1 1 0 13 585-4 14 0 0 7 25 3 2 2 5 1

1985 47 3 5 28 93 1 4 44 8Total

86-1 17 1 0 4 30 5 2 4 10 086-2 25 0 1 2 11 0 5 0 0 13

86-3 1 0 9 17 0 5 0 5 7

86-4 21 2 1 0 4 6 0 3 6 6

19S6 63 4 2 15 62 11 12 7 21 26Total

S7-1 5 0 0 4 9 3 1 1 2 5

87-2 5 1 2 5 5 3 2 1 2 5

8-3 0 0 2 5 2 1 0 0 787-4 4 0 0 0 17 7 3 0 4 3

1987 14 1 4 14 33 14 6 2 14 18Total

88-1 10 0 0 0 0 1 0 0 0 0

8S-2 3 0 1 0 10 1 0 0 4 3

88-3 7 0 1 5 4 2 4 0 5 0

1988 20 0 2 5 14 4 4 0 9 3Total

76


QTR 41 42 43 44 45 46 47 4S 49 5()

S4-1 0 0 1 4 7 14 28 5 3 13

8-1-2 () () (1 9 7 ( 30 6 3 9

84-3 1 0 0 0 9 9 0 17 11 9 10

84-4 0 0 U 10 2 11 29 12 3 17

1984 0 0 1 32 25 25 104 34 18 49Total ___ __

TO-1 S 1 2 5 1 5 23 S 5 15

85-2 S 1 2 5 1 10 23 8 5 15

S5-3 1 0 8 4 Q 13 13 . S

85-4 3 3 4 5 2 6 11 7 4 8

I 9S5 20 12 S 23 8 21 70 36 14 46lot al

86-1 4 0 10 0 9 7 12 2086-2 7 4 1 13 11 0 19 10 3 0

86-S 1 1 1 7 9 0 1 5 7 6S6-4 4 2 3 5 5 11 6 16 0 !6

19S6 16 7 5 35 25 20 50 ,43 12 2

T o tal1 0 5 1) 6 9 5 9 s

8., 1 1 1 6 7 9 10 1

S 7 - ) 1 0 9 2 0 8 3 3 9

8-4 2 4 13 8 11 0 151987 2 7 1 17 2 29 30 33 9 33

To tal

S-I 0 0 5 7 0 4

8-_ 0 4 1 3 2 0 5 4 5 7

S8-3 1 0 0 3 1 12 14 7 1 6

1986 1 4 1 6 8 12 26 22 6 17Total

77

APPENDIX B. SAMPLE DATA LIST 2

A. SAMPLE DATA DISPOSAL RATE DISTRIBUTION LIST

Table 10. DISPOSAL RATE DISTRIBUTION LIST

QTR 1 2 3 4 5 6 7 8 9 10S-1 0 0 0 0 0 0 0 0 0 0

S4-2 0 0 0 0 0 0 0 0 0 0

S4-3 0 0 0 0 0 0

8-1-4 0 0 0 0 0 0 0 0 0 0

S5-1 0 0 0 0 0 0 0 0 0 0

85-2 0 0 0 0 0 085 0 0 0 0 0 0 0 0 0 1

85-4 0 0 0 0 0 0 0 0 0 0

86-1 0 0 0 0 0 0 0 0 0 086-2 0 0 0 0 0 0 0 0

86-3 0 0 0 0 0 0 0 0 0 0

86-4 0 0 0 0 0 0 0 0 0

87-1 0 0 0 0 0 0 0 0 0 0

87 -2 0 0 0 0 0 0 0 0 0 087-3 0 0 0 0 0 0 0 0 0 0

87-4 0 0 0 0 0 0 0 0 0 1

88-1 0 0 0 0 0 0 0 0 0 0

s8-2 0 0 0 0 0 0 0 0 0 088-3 0 0 0 0 0 0 0 0 0 0

P(successfu 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.92repair)

78


QTR 11 12 13 14 15 16 17 1S 19 20

8--1 0 0 0 0 0 0 0 0 0 0S4-2 0 0 0 0 0 2 0 0 0 0

8--3 0 0 0 0 0 0 0 0 0 0

84-4 0 0 0 0 ( 0 0 0 1 0

85-1 0 0 0 0 0 0 0 0 0 085-2 0 0 0 0 0 0 0 0 0 085-3 0 0 0 0 0 0 0 0 0 085-3 0 0 0 1 0 0 0 0 0 0

S6-1 0 0 0 1 0 0 0 0 0 0

86-2 0 0 0 0 0 0 0 0 0

S6-' I) ) 0 0 0 0 0 0 1

86-4 0 0 0 0 0 0 0 0 0 0

T. I 0 0 0 0 0 0 0 0 0 0

S7-2 0 0 0 0 0 0 0 0 0 0

8 7-3 0 0 0 0 0

87-4 0 0 0 1 0 0 0 0 0 1

S,- 1 0 0 0 1 0 0 0 0 0 0

$8-2 0 ) 0 0 0 0 0 0 1 0

s8-3 0 0 0 0 0 0 0 0 0 0


79


QI R 1 22 23 24 25 26 27 28 29 Y3

S-li1 o 1 Of 0 0 0o

84-2 0 0 0 0 1 0 0 2 0 0

S4-3 0 0 0 0 0 0 0 0 0

84-4 0 U 0 0 0 0 0 0 0 0

55-1 0 0 0 0 0 0 2 0 1 085-2 0 0 0 3 0 0 0 1 0 U

85-3 0 U 0 0 1 1 0 0 0 0

85-4 U U 0 0 0 0 0 0

86-1 U 0 0 4 0 0 0 U U U

86-2 U U U U U 0 0 0 U

S6-3 0 0 U 0 0 0 1 0 1 U

86-4 U 0 0 2 0 0 0 0 0 0

87- 1 0 0 0 0 U 0 0 0 0 0

S-;7-2 2 0 1 0 0 1 0 0

S7.3 0 0 0 0 0 0 0 0 1 0

87-4 0 0 1 1 0 0 0 0 0 0

8 -1 0) U 0 1 0 1 1 1 0 U

S8-2 0 0 0 0 0 0 0 1 )

S -) 0 U } 1 0 0 0 0 (.) U

PrsucceUl 0.94 1.00 0.96 0.97 0.97 0.97 0.99 0.99 0.96 1.00repair8

80


Q-IR 31 32 33 34 35 36 37 38 39 4)

84-1 0 0 0 0 0 0 3 0 0 0

84-2 0 0 0 0 0 0 0 2 )

84-3 0 0 0 0 0 0 1 0 0 (

84-4 040 0 0 0 0

85-1 ) 0) C 0 0 0 0 1 4

85-2 1 0 0 0 1 0 0 H

85-3 0 0 1 0 0 0 0 0 0 185-4 4 (4 (4 4) (I (4 (4 44 (4 (

S6-1 0 ] ( 0 0 1 0 1 (4 ( )

S6-2 1 ( 0 ) 0 0 1 4) (4

86-4 o) C) o) C) 4 ) ( ) 4

7T- ) 4) 1 0 0 0 0 0

0 4) Q) 4) 0 0 0 0 1 1S7-.4 0 0 0 0 0 0 0 0 0

sS- 1 1 0 0 0 (Q 0 0 0 (.4

88-2 t )0 0 4 ) 0 0

SS-3 C) 0 4 0 0 0 0 1 0

P(successfu{ 0.9S 1.00 0.99 1.00 0.99 1.00 0.99 1.00 0.96 0.94repair)_

81


QTR 41 42 43 44 45 46 47 4s 49 50

S4-1 0 0 0 0 0 2 0 0 0 1

8-4-2 0 0 1 0 1 0

S4-3 0 0 0 0 0 0 0 0

84-4 0 0 0 1 0 0 0 1 0 0

85-1 0 0 0 0 1 0 0 1 0

85-2 0 0 0 0 0 0 2 0 0 0

85-S 0 1 0 0 0 0 0 1 0 0

85-- 0 0 0 0 0 0 0 0 1 0

S6-1 0 0 0 0 0 0 0 0 0 1

86-2 0 0 0 1 0 0 0 0 0

S6-3 0 0 0 2 0 0 0 0 0 0

86-4 0 0 0 0 1 0 0

S1-1 0 0 0 0 0 1 0 2 0 0

87-2 0 0 0 0 1 0 0 2 0

-- 0 0 0 0 0 0 0 0 0

87-4 0 0 0 0 0 0 1 0 0 1

ss- 1 0 0 0 0 0 0 1 0 0

8S-2 0 0 0 0 0 0 0 0

S,_ 0 0 0 () 0 0 0 0 0 0


82

B. SAMPLE DATA ANALYSIS

Table 15. DEMAND DISTRIBUTION ANALYSIS LIST

No Mean Mean 2 S.D Var 01

1 0.364 0.133 0.674 0.454 3.432 0.545 0.297 0.820 0.672 2.26

3 0.182 0.033 0.405 0.164 4.97

4 1.909 3.644 1.710 2.890 0.79

1.909 3.644 1.700 2.890 0.79

6 0.727 0.529 1.009 1.018 1.92

7 0.273 0.075 0.647 0.419 5.59

8 0.455 0.207 0.522 0.275 1.32

9 3.091 9.554 2.343 5.489 0.57

10 1.,16 1.73- 2) 2 1.293 2.940 0.98

11 0.,95 0.801 1.449 2.099 2.62

12 0.211 0.044 0.419 0.176 3.99

13 0.316 0.099 0.946 0.895 8.97

14 2.63 2 6.927 1.422 2.022 0.29

15 1.316 1.732 1.176 2.055 1.19

16 0.842 0.709 0.834 0.696 0.98

17 0.211 0.045 0.535 0.286 6.36

18 1.526 2.329 2.118 4.486 1.93

19 2.579 6.651 1.805 3.258 0.49

20 1.367 1.869 1.342 1.837 0.981I 1.632 1

.663 1.862 3.467 1.30

22 2.421 5.861 1.742 3.035 0.52

23 1.444 2.0S5 1.947 3.791 1.82

24 2.737 7.491 2.642 6.967 0.93

25 5.362 31.719 3.609 13.025 0.41

S3

Table 16. DEMAND DISTRIBUTION ANALYSIS LIST

No Melcn Nlean2 S.D Var Q.

26 3.211 10.311 2.594 6.279 0.65

45.470 2067.521 24.8. 619.014 0.30

28 42.000 1764.000 20.700 428.490 0.24

29 4.684 21.939 3.667 13.447 0.6130 1.7S9 3.201 1.7S2 3.168 0.99

31 8.050 64.810 7.920 62.726 0.97

32 0.421 0.177 0.83S 0.702 5.650.89)5 .801 0.875 0.766 0.)6

34 5.530 30.581 5.230 27.353 0.8911. 160 124.550 10.420 108.576 0.87

36 2 .7S89 -. 779 2.463 6.066 0.75

37 1.52 6 1.467 2.236 0.96I8 0.947 0.897 1.508 2.274 2.5439 6.790 46.104 5.700 32.490 0.770

40 3.684 13.572 3.284 10.785 0.7941 2.053 4.215 2.838 8.054 1.91

42 1.579 2.4 9 3 2.0()63 4.256 1.7143 0.842 0.709 1.167 1.362 1.92

44 6.053 36.639 3.353 12.496 0.34

45 4.632 21.455 3.077 9.468 0.44

-16 5.630 -31.697 5.380 30.112 0.9547 14.890 221.712 8 720} 76.s.38 .34

8 8.842 78.181 3.4()4 11.587 0.15

-49 3. 10i 9.6414 2.622 6,8759 (). 1

.2L1 9.420 88.736 5.370 28S7 0.5

84

A1

AD--A205 -71r DVELOPING AN INVENVORY MODEL FOR- Tg~E-KOREAN AIR FORCE '2/2REPAIRABLEAITEM INVE:TORY(J) NAVAL POSTGRADUATE SCHOOLMONTEREy CA 8 G PARE DEC 88

UNCLASS IFIlED F/G 15/5 U

SEEN

6- U---

I.2.-

"1.0 *- 'till- 11,321

111112.

-. 2 il 5 -11. .

APPENDIX C. COM"NPUTER PROGRAMS AND RESULTS

rro~lr&m ROKAF ( input outPwit)tyr',tnl,1O = rrny or.91..]a real;tob2fl = rroy (I.. 'I] of roal;

tb =arrt'y [or01..91..] real;

clik :boolvati;

DDRI..tnh tolfl:

OSTC)..nb toblO;SI.Q-..tmb tali 10:110'nb tnl, I10All' _ ob Lnti,'O

IT.. t n 1 1. n b",MEXP-Lob tnb I UDI ',tnb tit) 10:TEIIP...tnb tab,3t);

PT)OCCEDURL title(kt:integer);

I irrI. a oI it 1 *10 IT HiIMf AllIALYS I S *i*)z itr I ta Ii ( 11 Ex o I I I. I uT I Al1, SIlOOTlI I 11IG t~

I1 war ILe li( Is ir I to Iit C t I t 1,C 1. * +

0 iirlItolIts( OrTq ~7 uri tte I n t- SIC]*1tt)0 trlt i ( 111 110 t*~.)

lid etid ( c it s It

UN()CEIMEI) re ( I It Ili tegr ;var TEIIP-tab : tnb3U

I TIIPtf.~bk It, ]: ltl-tob (I .J

3 111)1 talid t I *j I DOI..til, I)

5 Tl~t'-Lohli :0 IICO] Lb E ij]6 TI III 'tobfkIt I j I OSrI...tnb (1,3]7 : iCIII'_L6 I~c I..j I 5Ifl~tab I Ij ;0 lEIll'-tnbil, jJ 0 10..tab [ i .j j

end (canse k)end;

L uuu~ Co.101tl -iol-li ol h oe

85

H* z IAI III VIMO ItAll 4 * I t

begin

> )> DATA IIPIUT lulrtlIE <<<<ft

uriteln ( I)EIIAIII) DATA H!For I := I to 19 dobe, in

ori., In F Orll Ii , ilor .j : I L, LI dorend (itnb(i,.j

end ;

uritel, (ISPUSAI. RATE E !for I := I to 19 dober, It

wr i t-'! (I " FOR 1) 1,. . i ;

for j := I to 0 (Iorend (DIStab ( i,j1);

end;

(* ) ) ''TRE'I1 ANAIYS'-IS (c<<( )

bn,,ror I := I to 0 do

'llf_Lnb [i 1:=((Jill nl) JO. i] l[ 11Lnl.,[ 19,iJ )k2)/( III-tnbf 16, i~tlll_Lnb[ 17, iJfblt t ih[i O i

I f 1. WL=,,[ 19.,i1 )) ;i1 I' I ID.. Lnh ilI > 4 0 . 9) n iI ( 10 11) t nhI1,i[I < = I . ) .hell

ALP-Lab li U.2

ALP_t. nhIj 0.'I:

all

(€ > > EX110111I':1 V I Il 51100'J1111 I( (((0't

for i := I Lo 13 doYEX[_Lsnb [I .i]j :- = I _.tLn b [l I j;

for = 2 to 19 do

begninfor I 1a 3 do

EXP_tab Cii (Al,.Ptnb I. t hIl.tab (i,.]) + (( I - ALPLnb [I])* XI'_.Lnb Li-l.i]);

for i : lo 0 doFCSTLob [i] := (AIP-tnb (iJ * IIIL tab [19.i]) 4 ((1 - ALl' _tb [])

I E'XP tld L [ (19,.il)';

(*At> > DAILY DEIIAIIU FIATHI' ( < *)

for I := I to 19 do

bagin,for j I to 0 do

DDR tab [ i..j : Ill_Lab [ i,.j] / 91;end;

Figure 12. Computer programs for the KAF Model

S6

(' Dl1"PO1 IEPAIR rLIICmI*tlr )

for I :r1 to 19 dobo, Xin

i UISnb11 0jU.0 then 0tP..tnb C i j j .

b'eL i'Iiif r 111 tnb ( 1 j U thorn Ditr-.tab i j 1 (1.0

D)[L'..j.nb ( ij j LdS-tab ( i j I IIL-tab C(i j];

it' 1) LS...L.b C(.jJ 1)= acid (1L..Iab (ij 1) theii DIIP..tnji, j jj U 00

>. riq u1n;11 SI'tAI h ( (jn I L o I U (',

for j: I to 0 doicrr!iiiIiCO.n 1)i DDi 11 Uft.n b i j I UI~.a HP _L j ]b 120:U!U -... t 111 L .iJ iI D t _L.t 11.1] J I 1 - DU It i L,- 1 J) ,5;;I.lLnt i . .j j r7- r I.( 3 It (hCOILtab f i ,i j 1 ~ l 0- I -i *i )

11OlJ.,il I i .j I SIt.tcl(~J 1C-f.n(i.jj + USTO..tnb (i.,j;

r.,rc I t Lo 1 3 0 r

bow I it o Ac

hr Ic I itcw r iii' I it

r I~t I itsI 11

to i 1. I t ic I; twr i nIII T

'rIt, )* I*,)

Vr ir I t r 19 cI

racq ( it . c.i~i n,wri t (1 iT~l..tbk s

uor i I tIi4i1

riguic~~~(! 13 iuctcprgasorteKA l\dI

rec I TFI11'..L n7

if k I LlI irbegt inwr ito I ;wr teInwrite ( ALI'):for j : I to ,1 doutile (" ",ALP-tab CU]);writLeln;

end ;

if k 2 1U¢,ubet! i n

wriLoln;writeln;write ( FC'r );for j I to 4 dowrite (C .FCSTtnb[j]);

write)n;

end ;

wr Li I iwr i Le I: uritoln ( QlTI 56 7

0);-rit-l ( -.

I'r i - I to 19 dobelin

writ.e D '.i);for .i 5 to 0 do

hea!! ij I

re f h .. I h tI".Im .n )

Ur1Il.o( " ", h'l'F l i. nlij k. . I .1 ):

i f k 1 l ihu.nbet'.in

ur iL I ii;W" v tL I itwriI.e( A[X');

r,,r *, i 5 to 0 dowriLe( ,AIX,'_..nb[.i J )

Ur i L i;

if It 2 H.i r,

w r i te hiwri te Ji;write(* FCT );

for j := 5 to 13 doari te( ' ', FCS'r.tab CJ I)

writor,;ed ;

end;end.

Figure 14. Computer programs for the KAF Model

8s

prog~ram test:

nri-ny-.) = orrny ( B.0 of rcl1:sr-yK=nrrny [I . .0]1 of iLfr!cr

arrny.x = rray (1-9.1 of real;vor

nrr-n vryn

nrr-jKc: -irrnyY,;arrj) :nrriy-.K;nrr~rho :ncray-n:arrj)L: arrny.a:arrjc arr,-y-.x;arrOn tr-ynnrr Q rynnrr-S array-K;

i In I '1 .: ~ uIn .i' iIc rIacL-t. SLIM. P(. r.xp .),r;I., Prot- lovpl1 renl;

IC =0.1:OIA 0.119;

(*')>'> STEP I < < <:

(uniction gipt. F(J( Kc:t- ,c:ro c- 'o

k in: wr

faIct 1:

Jrr n:= () I.o Ke fir)

for i:= iKc nlr'uno iKc-h' 1 do

IwritcJm,( * ) :Fad . : = I;

AcL-Ot := I /aur;

Figure 15. Computer progranis for the proposed Mlodel

S9

pro' ,'Iurc iii itinln(vtir i -n:zDrrpiy..n);

be, i i

otirrij 3] 5:354 .10 1. 1

nrr-n 1 1 07tI.I

nrr-.nLOj 51043 .7ItI. t;end:

r-roc'diro dir~jilny.:uiiK(iirrK array-K);

ui!'Ir( mill 0

wr iLn Ii w( mill K 3 .*mrr K 13 );u r i Lr!I i ( mill FA .nrr-rf'I I);uri 1 Inr( iiiii K5 .nrrK1 5]);w r i 1c II I( miiilt~ K nr 1fl)siritrJi( ' mill K.7 .nrrKf.71 ):oir i LpIli( miii V.O = .nrrK(Oj);

tiiloti nih vcwqur([Ij, r!nJ ;pouwrr :iiutcoer): reol;berc iti

nilh powcr li, i Lh rowcr(I., pcur- 1riml:

fiinc iii

[tic. I(illiti I

Vii un r rnl

bt-rli1l

Iiid:ji

i 90

procc-Juro ifel.-)L(vtur nr-Li.UI..s)begin

arr-01 .4621 :0.0526;nrr.r)Il J 0. IF51

err-)I,(J 1 . 31 ;

prormduro displziy..S(arrS :nrray_K);

wr it(,( .ril1))

"IriLc( , r.S f)

... i L( ,n rr-SII35

w ri Ir? .nrr-517 1);write In( .urr.SLIO1)

bri

nrr..Kr[2 111)

Yj rr. Kc 15 1 110;:

n r r._.K': 1f 1 20 1;

nrr.I)I I Ul ;1.

urr DI 3 1 ::17!);t'r.1) 11 175 L,

nrr-DIU G I 1730

nrr..-tl 1 (30 ;.IIi

nr h I ). 0.i101M;

nrr..rltn7 rvlll7G

Figure 17. C0w1lputer' PI-OfV-lnS for tile proposed Model

91

nrr..xtlj U.IJU;nrr...x" U.91:nrr-x[3] 0.92;

arx[41] 0.93;

nrr-x[l] 0.95;nrr..x(71 0.96;err-.x[Ul 0.97;arr..x[9J 0.90;

end;

bcr~i iilli ti I -(nr,);geL.init(nrr..Kc.nrrI1,avrrlio);

rrperntd~o --irnin :=~~'jUtrmgio 0;

cxip-cosL U01;

item ,no ji.rmrro + J

rnct 1;

sim 0;

for j:= nrd~t~mr cowirLo vrr..Kc[iLemno]-n#f do

fnct := Cat,11; (iieccs.ry to coirnrL)

sum:= ruin + fat.;wrjtelii( * str"fact 1;

wr i~n rh( *rjn K = '-r 1) i Lirn-tiot r-i);

until iteni-io =0;

do-s ii := tIlruec;utitil not do-jpgi.ii;

writelnt( 'pro;...r~vc] Si S2 SO S-1 S5 SO S7 SO');w.riLci( - -- - - - -- - - - - - - - - - - -for x :Ito 0 dob'?r in

Figure IS. Comiputer programs for the proposed Model

92

u r iLe( irr .x jx J

gr~tDL( arr_)1,)repent

i Leijo :=iLcm-nio *

utitil itemnno =0;

end;

(*))> STET' III ((<()

n r rj-nl L U.102:a r r-Da12 J 0.2;arr_.Unt31 0(1.2;

n r t- Da 5 (J. 2;o r riOn j 0. 2;ar rDa[7 1 U. 6:arrl-n[0J 0.6;wriLclii:

belii;

or x Ito 3 d o

nrr-O Ix] m'rL ((*'! (03.1 a ~r1x/.)i rr-..a [x])/(IC *(rjx].IM);

Write C i- x);H'iLclIr( .nrr_(4Ilx])

ur jic In;

ur i l.r!u In K G 4 Q TOTAL')*11 I., I --- --

for x := I Lo 0 dorIbci'.iii

c.ii1 c r ~ I: 2 onti.5 x0 -fn q=..Icn

it nxrll, I e .rlhcn :=I-

writc('

ifc=1 Iien wi~a C

wri~e(' .qtm;

writc1n;

ri~lll- 19. Coinpu91tcr prOgramlils for- tho imrOposed Model

93

QrRf 1 2 3 4

DI 4 .cUUUIilnuflflii(i n 2. Oouifuoiiciltr f 0.0muOUCJOU0iE 00 0 .000UUO0if1tt I U

n23 0. flttitittlltuuEttF YUOUOUU 0. U1luU~n~st .1HUMOULOUUE '00 0. (tUl)trUattIE,

04l U. O ttittmiitmMu3 utiC u , no)I.UfljIt~tj:1UI U 2MTIiU(JOU11.1LJIMIJU 2. VOUU(tUtlUE '00

D5t3 :.tttiirttriiir "M Inn~uuc" .ltitlltlgttI .tUUnuttUttntjnt 2,oliouinhliiirn,

U I C 4 uh otIU rii Yuit 't n .0 01 1 LI U O tttiltiUiti~ Ill .01 LIHU U I. 00 ' t 2.0110 ttittt UJOIj l 1 1

in 2SI.00iuHIAtIOUtiun:' Ut 2U UnUnnUU Ut U 2. OUUUUUU UU I t N.t~UOU111.111111l

A011 43. u~tinionhEo I 2V. thi)UoolontillC I .ouUUUUULooUUU- .UJnnn~ -a

Ca) 1 U 7 f.lmo l~o l. i) ! I M O O I 0 U ( O M11( 1r I

U. tltiitiiliiiilC fll . liltt~ttltttii Ut I . UUMiIIOOIUn E IU (1 .lJU 0tlo ttltlt I fil

OHt I0. 1iill (1fliiiit i) U lllIEiiO O I UU 2.OU UUUE" ~U JU JJUUttL 'U I .u;:ntitt~jtn"" v

C)I lo 11 0 y1 l t " 1 17a 1 0 .0 0 0 1 0 " U D M tite" ' 0 O U I ~ o 1F 'u u n o ~ n INDIG 4 .a t~(IIIr o itil 10' 3. MM OL)odIe! .uuuun m 0 F L"QILU117

01 Z 00f(t 11(M,1 0 mm nu~"E u .ununo~up n y uno94n m

U ~PoUMIFI iAL. fliOUllil11103 tqr it .1 24

02 2.0OM M 1Uftm i:.U" 2. 2000HIOUC * (I 0.000) 'ooloonn900 O0UlJ(uUUUflOE -01I03 1 . 411(1111i11h:1LI It . 7 lr, ilIt n u: u0 0. 0000000000ILE -0 1 4 .OO000hh1Oh IF I I114 2.(1II;*lI i)M" i.*00 2 IIJIIIIIUCIIO I . 21I11J0II1TIDUF, O 7. fl otIH(umunut-0

v5 2. onn ll, : 101 2. MM~i~I .llJ0l~l UOH I . 1IntiiICIUi.JKIOl .0 /?o' l:*I)1; 1 I )1 - I I IIIfII . 1j I . L-tiW 0I11 0 1 j,5uunuIUOucltlF'0 0.21 70000n'10001IN7 U3 .7I 1 i4U0O" - U 1 . ' II 10H1 11; 91)11 9. O'4.13 O UIIUF-fl I 1.13/4 hI1I0r1-11131111OPn I ;uiiIIiilF.l ,32/iifrlhr .34020000101:.I 1. ii5o2c040001.noP1 . imsi r'uu111 I .UI;UU?144UU~iIJI 0 .U4 I72fldOII-n 1 I.913741I 2mr-mi)'

PP I.MIMU9r9lummHElc 1.A:1:5)7NjE1011~l 1 .2112VU00IEU1,111I i.56539mvisormn'fPI h~:'~mI-~ 2 .w;'6135721PEOUn1 7 .6951JAZ20000E-O1 1 .0573143 1131-117,11

P12 2.5 13i:.;MAMOU 9.'ll:r:111 1.6I7113210lI01) .r1n51ll34I.toi 7l1305MODl . 2'9lI~Il:1t ,0L79i4HIMI-fI 1.5009:1MUC13'.3F''

1111 F /IJ,104=71"MIu 1.619167lUOSF, 1 *07021247CP9Y0 I .711.13U4I!302IIllP! 13 1.I3Ij3"1,:I%2,:-UI 1 2qnr33JG71GQUuH 0.4t10274OUIGE-0I 163113071117WOMuPIG F 2 i221'l:n 1.;:t?69;373111HO 1A.9UI51349I7011171.300 ,13.1~PH .15/~fSIFlI 1 .11111113141I1ITIOU 2 .392093flRA5IF.00 I or;4f45lH347F..ti'

I~ ?lif21nII I tj7Immfl uv':Iijn .43137J33U~i:.0 .51716061 111-HIDO' .JUIGIMMMUMC1 I .3117G6ii71SQLUU 1.615ULIUu2EO.0 1 .001372013539000U

F'ur i.Oiruuj7tvui:puu 1.US-12ItCUSC40 1.706610013E'0U 1.261135fl203E-(00

0 11 It 6 7 nl

h0 1'~ui 0.: (0 " IM II 1 I . nrlu116UM Ii;,11 lIIlIfl: I OU In) I0)O rIOU 0. 7 1313MOtOICl 1 11p7 .I PC131.l~tF i 1 1l.IIIi11 '0 . 770960u:.'I 0011J 7. 021)(10000011 4 11

* I ~ '~'.1t'l' I I IMEiI-III I 1 ? 57; lko J 2 .~I.IIOlU:)I:1 II: 1119.011mnl4!u2I'F, nO) I t!; H0.7 1(r'~i'lmI APU.II l12U13I100U 21. 9P) Ii00I1UlIF9 t U 4 961 10111'. M)11, 1 J 1. 111F I 'i I 'l-IM I .t17 1Cr II10110Il.Vl1 0 U 2.01113l~lE11 I1 D 137 fl111/I-11I~1I'1: *I . I,5,'.,l;;.ljhlVV 9.41HOU"VINIO-(U 2 3 3502313451ITU . 51341MUQUOI~l~

P1i 1 (1 1,:1Ptlftlill:i11I0 1l . /~. 13G:111 H 1 .3 r:0103 7 01 0011U/F 1 6. 21321lfI JO k911

I'll ! .11 I'I1lil 1' 7. 37.I7lflI;Ill.-rul 3.001031:III .3/l~1492flI00011

Il. /112,!,I01 11u.1 )-I 7 W,:J'. 15W7391 I:-0 3. I'll9 117 In15 E, I W 5 . ll1) 2 412 11)l 00Il

& ~ ~ ~ ~ ~ ,nuviOKU ,....512WH .I .:~ainIR~~sb lcIA1iIcc

95

Will.

((Tn 1 4

1) 1 1 . 0 9 5iir 41(1i lE -()2 5.479J45205461:-03 0.UU0(J U((J(.0 0.UoUOO(JUOUEI001) 7 . 0110t10m)imIO17 100 0.7121 700?27 ,- 0:1 0. U0000000OOo, an0 0. 2 19 17nf0027I1 0 3

DI I[. (Iif 1i1111i ' , i t' I III 1 . 000Io(U3(II~ I (ill (1.. 471. S fr n3 03 (1. 0f(1 )UIIMO 11I OffD11 i 7. ,0 1 irI ((1 2130,20(101413 -03 5. 4ou/! nou4 b2 (JO- 5. 441 ,1(5-1 I I 0131)0 " . ;'1 Mll .:i'I:-U1J ,. 4 19-1 7jI1- 03 2. 717(1234 Fl-03 5.471 7OJ7 (03

(Il 0 U u ilOli iU iO '( 0 1 uumm3'l nulr 1/3 00 .4 7 1 -157 0 1V0 0 0.2U1010 0 0 F -03UU1)1.

72 0. (Ii Ui(021- 2. 7 3 9/ 7 6027 11 -03 5 .UO19152.05.FJO 50 .1 01 0(E (i30(4 fl .4 (10IiiIEi: 01. 5. 1 1 1154013 r.- 03 . !0020540n0F3- 010 .2 1117 000(E U Oh r 1

01 0 r i (nu.i( 19 131 (3F! : J 5. 479 ii 0!iO10 1:E- rig 0 .U U0 Q iljii0 0ii0 E+10 21. 0 .730 1 1h2' 1 ( .- 0

Il (1 0. ''1(M11.( I -i2 I. 3c"110 1TIEi1- 02 U. 0050O11101: 10 0.21!11710022E-0l3DI ? I .( 'MOWPI tlIhhi:i U 0(j~UL03i.iiiiJii31( 0,210170(1021-03 5.4 0 570i(Ih(i5((fr.,03D13 (3.. (h19 1 1000271(IhI:_03i 2(.7391"!60ii21r-03 0. 13791i10(i-MtE0(3 0. UUioiLI(J0i1331 00(U11 0. 01u0LouOOM I 1013-0 5. 170000004030 S.47940b00 0. 00100000M,41)i

01 Q.iiiiiilihiiJ1i 0.i .7:1 0627'1-03 2n .3 372601Ii1)0 3.31121[1307.4,- 03

131 5 .4 U. 01,1001::-(111).001i- 1i(if)iili3.2i19 U. (00000ii300 0 0. (i(i ii0m 1t M(F (If''I f 1./Jn1fl0/.k. 2 1- /!'501 0(C127'-13 .2u1oro13-u3 t 0l00 ll(i30)(1113i 02I

D5 1 0. :ifiiil(imim11: -00 2. :ll ?1021'( F3-03 2 .730 2G027413-03 3 .36S 3 SI 37210 E(*30D21 2. 731i1,0 /(13 2. 731i3720lI774 FI113 0.3764-3(. 000MMOi000001 00 .. 02307101 5.- 0. 7(1(7311ii223-i( 11. I~(0I 2 ll)(: 1. 1101.1006000E71002 0 00110011001.11iii , (10

F' 0 -'i~~JiJ11,1 11,! 70 U. bA L1 E- 3 G. 1 3U 173(U 137 1. F-0U2 2.4U13 6 9n:1424701 o

P1 2.. 791 !1!/ i ll2.1 -13 Z. 7111i.I., 4 1i.l1I 2 . 3 7 0.!'7 4 E - 3 0 1. UliGl9 0 l(3 li i 137 :-;p1 .0 :12.iihl1'' 1 1 1: - 0: 1 2. 7J 3!O1U2()1J2 (04E-0 2. (1 . i121;3)214 13-03 0U .7((03972f;07 7,1 -02i

P72 n .7 till 7 110 >(3 1). " !I13 [0 11 ~i2,'l1) -0. I . G 9 0 I )13 - 02 .11 0000 01310. I Oi033 0. Oifilli)(1iI: Iii MI .(11MI1iJ21., 1 (11) 1. W170'1 1ibiil1-03 62.(35753422 u2 U?_fI

0 9S U . 111 I 11 10 [u' I f 03 .U 1 1)1 I77 0 t I 11 12 (11 31. 3I(1 (3 0 03 13 V ~ - .4 1; 5 1 :I 4 2 4 71 -H 2 1, 110~''' on 2.: - ). VIJ:;1/.:-il E 0( . *73 7 060U0074Fi113 U. 310000M III

1) 1 1 (fHl (I K ( U .i (3(31.! 001 OUn lI2ii 3 . 739 S0 0 V E03/ -0U 13. MliIUMMOU001iI I ill)

D17 2 1ih:r(? 13i:!l 09U 2. h107i;.1-: E' .- bsoou- 2 3U.111 ii,9r.O0i1 F- Ill

I35 !1. Ill1( 0i/(3-( 2i .0(i'OI .6)"I l. (Ii o.'17015uumo'1013-on u.20113712311013-13

1-i-me3I 22. Corn;,,z11,i oiial Re~sults for the KA F M\odel

23; 1 *lfrIMMffII(IIfII:i. M? I OfffOOOfff!IOHKf1 .aourimonvoJIs~u IUffflff~ff:fr

Ill. I *. (-I Imfff: 1 ff1 1.tOlfffftfltfffffI nuf I 00000HUMM~~ffff~U 7.Ifff'f~uffffff-o2

""f I 1 0 01 Iff I IIO 1 I . 00f1,100 001(J 1 Of~ffffff ifl'i 5.LI1.uff LIfIJIUr-.1ff M1.0fffifffl0f1fl 'fl!

*) 11 1 . fIMO Irfit' Ml~ fI: 1~ 00 1 ffff'ffffftflffffl 1 ff0 1. f0 (1u f)Iuu fJ(lUo I. (13) 3f 31 (3 ~f ")011 1 . 111101100110011fff: 1 it ( If . twIffIII)O( l. 1 o (1.1 J 10 f0011"OHIJUKIUU (I i . 10U 110 IfIfffff1)1 1, I uOfI'(.I IfIf?: . f1) 1 . Ufff,(ffifltffl I)1 (I J.0OlfOuOUoJU1fEsu1o ff~ffftff~:IP]) I, I , 1 0 0 1 f 1 I,10 01111. M )of I . J n I 'f f f I I 'l L) . 0 0 0ffl it)f 0 f0 0 0)E I, (13) I . 0 0 10 (1 0 0 0Vff ff f 3 1 (1f 1Ili I I .(1000W, IIToffr OU I) 1 flO10 fif30t1 E , 'M fflff)1co"oUUIJ(JIHO( 1ft~lfffftfE'pl; I . fIM1,0IIIJ0if)f1 (III I . I If~fff f lu i00 1.n u o u uE UffotfJfOffOf' ') I rffrff~fff: 1f

I .220',ujfIhMI;UM:I I .~oWuntiMMOU IW mu.0000000ffU U 1.006UUOINfIL 'Uf

If1 ODMM1I1j I 0i)II.Wunoulu uu~mm

Ii Iff (IfflHfI ffI iff IO M(fffUffu 1:) , 313 I .fffhfJDUflfIM :OR 1 321Lflffffffffffl 11.1P I I f I)(I~ffIroffffljf;Ifu00O00E t .(fffffffffIflu Iflffffff''(f 012.000II Ifffffffff:fI1)11 M, lffit l:ffffff ill'.. ;Ii I .fiffffffffffllEfff Iit) 2 . UU 1.1 .1oflfffpft'flf 2. u l-lffff o uftfffil

fI I . f M i tIil t IfffIIl. 13 J11 1 . (1111 flfjffffff fr I off) I . O IUlIflffff11 ff 9. ItlfOl 0ffff' -07 1 I 0 1 .111 itIM Ifll: 1 1 .fff ffffffffI frt I .Iltlfffflffflf IfiU 1. Otlf(flfffffffjffr III)

PD5 m,I Ifflllff f llI 1.flffflffff(Ilffff) I.(I0000110009flff. 1fll I ffflfffflfffflffffIFfffOOff) I fw!ilfffimlfff Ilft I .*fflffffffffffffff fllf I 0 0 1 0E.Ifffoffo fff U Iff) 1.Uffff Olffffff f it)f''

lidf to I . (III MOII'~;If~ ff'ff I MO1'f1flflf I .(Iffffffff 11Fl ~f fl'f) 1.otIffff lfffffff: lftf

:2) i; I(.I rll I fffff'11 f''fff0jfr: 4 Of) . UffffrlffAOfffff f ir) I . OMMOMIlOllif -f1)!1f 1 (01 1 If IM 1W1;flf 1fff I *ffffifffff~ff (tt .Uf000110fff10(107.ff21 I .1(ifffff1ffffIEflfW]1) " I fffr Iflf I tillIff I .1111)(101f1f0f1ffflff i l 1.Huo ut~inlomfI mff EI .(flflfffffflif9ff

D17 1* .fff I' UfI Off I .ffff1ufffffffff :Ilitf) I .00f0Ut MOfOf.. Ihf) I *tfffflIf'M~lflff

I)1 1 . (Plf) II)f~J)OD it . ODM1fII,100 1 UutffMuUOtJOfJ II DO i .000000000E * 0 I 00f~lf fil'ff

97

n r n 23 4

1)L .1 1,510GOV I- 11. #(71 0.- 575:34 24(:1'-I 0U. 1100000cU15. 100) 0. ou0u0001002. 0D;!' 01. (illn) 12)12: 2(HMI 2 ID 9.) Iln:2 1:3CM!2303-0 I U . UujtJUOUUUUU2r 210 3. Zn76711 -.211 t0)3 If. (1000MJIIUMIt~: I O(3 0. (IOU(I'I1UIU) E 1 00 0. 57531216f, /1--0 1 0. UUOUUOUUU)Jl. sOi1 .19. U ; 3 1! 3 U q )G1.- 1 11 . ulu:I2 31i'J"GE1-() I 6. 57534241151'r-01 6. 57534 21))57r-U 1Ds (I . 01;3f) I 3r;!n2)(;1- (I I G. !,7 5:2,.!4W2')G/2--0 I 3.207071 2320IH-U I 6. 5753-I 24657r-O I01 11 U. (l))l1)i0llom))1 1 i) U . 011111IIIIUIC (11) 3.27077 ?12- 1 0. UI)UUUUU000I' (31l),7 0. M1J)I))1UQH)11):: I O J . 2 11107 1 V:2L- U I 0. UUU0U0U2U[700 0. 57 5:14 65 17 -01

2)~~~ 0 . 2I5 I:I G. 0 744070 6753424(;7E0 9 0.1 I1000-0

F 1) P 0 . 0C 3 ) 1:16.9)( G C-u I U. (1:I.25110l .00U00O0UU00:i(2 3.2(107712329r.I'lI011 ) 3. 217 13712 3,7 .j1 l-u 0 U. U1,1000(20001: I UO 0 .57531246571-01 0. 00000010000C .01)021 0. 0UI)J)1(()U 31 1. (1-1:111356) 1 ), . (10 0. OUOuOCOO)UuF'o 00 U. 061) 1 waw9(Mt- 0022 1 64 (J.)!)(; I (-I C (II) U. UMJ1UIUMAU'U2.,l 0j .0133 230!1E-11.1 6. 5'153-1 2I,57r.-(V I

Ii 3 01113(1 )iU)0-U I 3.20-77.323Jl-0 1 G. N1534 24 657r- 0 1 0. 110000)11: Oll11,1,1I. 1; It" 0)1u1)111 1 2:OJ . '() (I. 57534' 2'I4(3-: 0 1 0. U000Il000UE iu7i u UlUUOU0l7 ("I1)12 !,u. utionoomIl(J)'P 0(2 u. tioJ() fl(l)(M10 F, .00 0. 000000tMlIF I'00 3.270707 12 3290 (11) 1; I .3I50;UfiJ3 11l:()( . . t)MHJ)1;E-(1 1 531M 3 r. fmll G.57534?24 1;571: -011 1 ;. !, :,:I .I-;l57l: 1.' U. (IOJUU1IUOOU 01 O 9.0(13(3 13U0011;-0 1 0. 0)ll)0(11101 ( 01 1 rie U. WeIMuOUUI 101) 9. oUI;o 1369o-u ul~010"- 1 J00i(l011 I 0 0. 0)IIJO)U0))Ut'0 '0')

2) 1 1 3. 2.a07 1223 291- 01 13.0(IUOIJUOUUU(3U G. 5534240j71-U1 6 . 515.'34 2-1f;5 7 F-0 1

12it S 13 *1 0

1) 1 U. I)I(2UM)l01j)JlI- 1n 11)3. 2.0767? 132.: -02 3. 2076;7 173291--0 1 4 .602.73:27211.01, O1) 7 3. 2.07137 2:12191- 11 :5. 21 7 r;7 1737!)E -101 3.20715712321W-Ill U. UUUOUIJIJIIf, M1)11)3 6.5,15 34 2-I0!OF/I I I U. 1.1011011LUM'2301) 3. UUUUUUUUUUE '00 U. UUJU(1U0 (11)

D5 3 . 2071137 12:1291.,:-0 1 3. "2/ 6712329 1'.-0 1 3. 2777 1232910'f(1) 1. 654.11 15G61316!0 1DD 11 3. 2r3/16 7 1I "!295-F- r2 3 . 22313R(73 ;::211 V- (I 0. 131(MOOJ(3Ur I~ ou) 3.27,,7 1732,11 #(i))2)7 (1. 10,10 I~1I0 12 J.. "11611 ;'2!0 (1I 3. 2071;7 1222-0 1 0. uUUOUQU3.' 013

lin . l)llJ2112)JIJfill. (2. lnnlolim uol:'ul()I 1 .315000J493 ls#l) I . 912U02?71970 1- (JO)

03U 0. O2)f0001)Jt)))) '1) 0 . fill)2u1l)11 Iu 1.IM 32135r)164 01100 2. 951:1-211 MICEnI 003r) 10 3., n~f'; r I I ":121., -I I (2. !', 3-2 ;:-Ir;7 *- i1 3 32 V1/ r7 11 2 U F- 01 fl. rMMOODO1)2113? I3111

1) 11 U. 6)IIIJI iU)11201 I))) U . U0111111011 J11, 1211)1 3. 20 11; I1232.91-201 0. .U01 1))JIlllll2- I0It

F 15 .f'/'2!0-11 31. 2)17W 1122 ;0 1 1 . 3 11UGOV431I0E013 3. 20701 1 73;1 F -(11D3 a.13 MIMW.t)l(()~l3 I2211 3 . 7.1I'767 122: 111 3 .20767 1 : 2-0 1 3.2'? 20 7 12'.l0 -(11I

1) 11 :. 2fl'1/ 1 ;'(:it0 I U. 11(0110001 1 (10 G22 . !,5131 21057r-0 2.(1 30nI:00 1

3)31 03 51 :25/1: -1, t :3 .07 071 :.29 00-21 (. (000000112(22:4 (00 U. (IMUounOwn: oil1

PI' V 3 .2 1 ? :G'I I I: -u I 3 I I. ;111 *1117 1 V -os I 1 . U ' I2; U273 U *1E10 0 0 .LU~3U U(IfI'~t ('11)3 3. *07 rI1 23j2! 1: 11 1U . U 0. 11 UU 1)lE- (31 0. 5753,121 05E-o 1 3.1) 95705419 11'1)

Figure 2-4. Computational Results for the KAF iMoldcl

o~lo 1 23 4

c) 3 o. 0II00l)1:111011:o o 1) . o10Ji)(0000(Ir 00 0 . OOIJ0000000)E* 00 U. 0001.1)OIJOO(l r. I U0. C). O1iLIO~iI 0LI I. 0 ~IIIIIi)) 1 0(1U Ui 00I))I00E. . UUI)OLII))tIl L I)

EI~ II II(IL'II~lIJII~iI)IJI IOU 0. U l0IOUllII0(l) lU 0. U1J00OU000.0 . IJ00)IU)JI 0U(3 0.M OOII1IOO)1I: LI)0t00))01 00 0.000000000001t 00 0.OOO000IIO E .1011 01 . IU ) 11) (l 01111111 .111(1 >1 ) 01 0000 (;(E 00 0 .000001100 00 0. U000001I0F 11

P!,) 0. t 00LI1I1.1001.10 lt) 0J . ())1)1 10110E 10 0. 0)JUOOOOOU(E .00 0. IJULI00)0)L1)'E 'I UOG U 0

1,1011110.1000E4 00) a 0I(tI0JJI)EJ+U(O 1.0U0E0 . 000000000I) 0 .OOOfUIE * o

11 . 1HII)tlflI 701 -0 000 00I]JL 1)I1O,0 0.000O LI OU OE 0 0.0000000000f,1;.#1)0I'll1 0.0 0(0I :0,1, I 1). 0 U. U0(1110)0( f *O 0 0. 00101.10000 E # 00 0. 010000000111-: 1.11)

12 0. jIqiji~flIILII1IIIE .00 II. O0()OIIJOII1EI 11 . OUO)OOOOOE#.00 0 .Ot)OOI1IIIOI i

V) I 1 0. 0lOf J11IIIIIIE: ()1 u. wi0ll II)00 00 wt #a 0. IJOOU000000COU0 0. 1000I00OFI(f (00V 1240. I11111001M11I: .100 0. 0001100000FL1U. I 0f 0. 000fr0O000I0 .00 0. O0iC11110WE ITfl)

01 0 1). )jI~II WO 1 0 oOo(oOUOO. to 0. 00Lll)1000FA00 0J. 001),00110I00E I O)

0 3I).Ijo 01)tiliflk'l: ~I M0) o. 0I1III)0I00(E+00 0. UI11lI~r)0000F + 00 0 . 1)1)L)11)I)111'F .100D i 111)I.U1IMi rn)0 0. 011 1111J0000E,00 0. !)00100OEO * (0AU (t)011)OJIF.)

1) ''10 .)I~Ijlr11:IL 0.t1011111111101:.Io1 I.000hI1Or~II)or~OE O o. oull111111111 III

1)2 0. 0u1I)0I I: U 0 0l011000 I) HI':00 1. 0000011OF I0 0.l 0,010000(1"ir M1)

PJ II. 01100(111U1J10T1WE It 0. '11r)1L11:00 i.001011000C00~o 0. 0n0000000il)A I 0)QD)12 L1) (1')11 ''fJl; 1):));1 11 ,1 ) 1 (0 IlI 0 , 110)1111) ul .01 0 ,I' U . 001 1l0JOE 1) 1.00001J'U +U 0 1011()E.IO003 0. OffoODIlilnrlIC II 4)11 01J11t~).l U(31 (I. OOOOjOL3jrIoa *00 0. 001)1111111)I1E #UiII'* (I. I)JlI)Jirgifil i lj: III (. tl000)fj)irI :' i~ .1)i) U. 010001111 1,10EL 0 U I. U0I0f.0000J1)1 #E .10

D5o U.0000)1,1101111)(11 I 1) . 0000001'IIIOI: ill) 0. 00100001))1OE400 0.M000"(1)1)00l ''3I) ) ). I~L~~t))(lI,,I 11 0. (I01'.1~JJ)j11 )1100 U 1 ) 1JOOUfIJt)0E 010 0. OT111J1)001)1)F 111

-,)7I 0. 1'llllll Illi: M11 2. 110000011111l 1t00o 4I . 0 10 oiil~rOil UL' -) 00L00 )1100OW : I Oiltit(2 (. Jimll)11 Offi11 . 1) .I. fJ (iUIliU IIrI Oi,( 0. 1.000000)001'E 100 0. OJI)0ul111(101 a.0

P 9 0 . I I II II I11111111r1 : ;l 1) .l-1,V(I1111111OEI 0011 U. 00000001110E 1 )0 0U . 0001000M(1)Jv Onn

U10 I (I. (11)IlmI)l)Il~IIi ) OfJ111)111)IF IIF 0. (10000)O.0 .0i~)OOI, 11 0. i111101:~l 1,00 0. M:1)1U'OJ)i t ( 0. 9000001000PJf 00] 0. 0000000L1,110

1)! 0 L. 0I1.1 jl11M I'll: Ilfll 0I. t1IIW11111111001 I fill 0. 11000001110iE I(Oil 0. 00110n)1 :M 'I fil

111 0~ n. 0r:1'l11)I111ru:I I 1(1) 0.. O.0IJIIOIF .00 0. O0OOOOOOOUE .00 0. O00011lflLr O .01

0IJ 0. HUOUOUOUOUE!. f00 U00OOOU OOOOE00 0. 00000010001E,.00 0. 00000M00111 t 'III

fluc.5 C1 uatoa Reslt WIMJIfo 101e KA. oi\t(1)MOE Ito0odeltUMII

5 .0 ; : l 1;ul: 11 , U)M10 011 0 O. UU D)IU fU U JO OO99O r

010' 1 2 3 4

1)1 . nG(3 I 137!r' (Ii l U137Cr313.I ' 0.000OUI.OU*0 0.000oooo oor . 'J1)2 0 .nouooKu .720G15372.1.ul 0. OOuOUOUI:*'3 2.l)13245..fD2 0.O~jQC OM,0I~( I: F.u 0 VU. oUOIOUOUOtOE~o , .1 1Al4937664E+00 a .aOOO0UofloE'0001 I.'2041.1327ZE1,00 1. 72011C572EWO1I 1.4U44923/6GlE+UO I.404493761-EIOOD5~ 1. 7?.I'lr,!72FiU0 j.4j.vL17If3r,1FOO 9.9312/'362,62E-01 1.40449376154E400Do u . UI0v1inJ :no 0 .o'umIOIooon :10 Ia' .13572,105,1iE,0 Una. UOOOOaE t OD)7 a0.0 U 0 oOUr#.10 9. 93 1 27/066 3E-0 I 0.0000OO00000K.00 1I. 4014 937613.E # 0(1V8I 1. 4 937664 f DO 1.'lL11937ti541l1 I.4114497334E+UU I .72.01.165372ER(oo09 1?. 0lX*0 14 : I ,f II 104! 3 7 fy -E F U0 0. 01J000000(1Z4 00 2 . I G 0 1 10 9 74 .I & (0DIU0 9. 93 17 2/10 ; ' -f I t0 . 0 0 0 U0I 1!10 0 E00 I .40449371364E a0 a. .000000000E 4 001)1 1 J. UU,!j:!HI)0[RIEIM 2 . M2111 fl0206# 00 0,.0)000000001>00 I .720146537?K.')00)12 2. Oi0.10 0 l-'0 0. f0I0UJoOOI:4 00 t . 72U014f::172 F 1,00 1 .40-1,193766.1 E.I0(MD1 1 . 721f ::' , 11 I77F111 1 9*12 oj:if ~:21 1)1 1 .40449'76S1Q'O 0.OOOOUoloqqnrJF.')L014 0. flUI((10Il1' (10 1I. 4 0419: 97 (;34S 00 0. UUOUOUUOOOE0 U. 0)0000000K.IE I)015 0. JO(OUOOO' *, U:)0 fit rII)Hi) lt iOUUKf 10 0.O 00000000 '110 9.931 270)63? E-011)1(; 1 .YI(2P 9131) ")( I .70Id57E0lI.021(>0 1 *44973l1(u

017 1. *043cn:! F 1) 1, . 137E Oouloo'K*) 1.902514l326E 00 1.0. 49311rO0l IFI017 0. 1)(1 UI0000:muc*UlI.21 1.13/2 *00 0.72(1Uu11107E 00 0 . 0 f 01 l J1))O u10(1

* 019 3. 1001109743iL.:ut 0.UUUOOO000UE*00UO 1.40449:37G64E09 1.404 492' 76 GIK1''.0

0 I*I1 5 6 7 a

01 U.011001110111)uE: 11') U.9 1' I2711f022K-Ol1 9.9231270662-01 3.7 1514 122.0FltD 12 9. 0:11 " 2:-I 0j. !13 12/h;r:l-o I q.93127066.I2E-U1 0. (oLnflonl(1010.I032 1 . 4u.I.I'J:17 r ; . I I Q1 U. 0(jt011011110'~jUE ' 00 0.00000000001>1U 0 000000001: 'OlD)4 !).9:12/ooIs;:).-I t . 4o WI.3I.0 I .72n l453721 s(III S. 50fl1015361: 4 00

I i 17 U.'1 11 1;1' 1 20 v K- I I U. 93 1 17'j.I- :i . 14bU.5,1555 11 2.22U6991330 31: '01IIG .. 12b~2:I-j .J17!lu::.,.1 0.IIIOU(10001)K*01) U 2. 14054.15355F '0OrP 7 1. 72116',0:1 7:1't110 1.;.13 1':;, ;!45 51'f 10 :J. (,036'5 11I160 Es00 (). OIOU OOO0 1'$ 00

D9 0. W1IMIi(IIIIV (il U. 01(11IllIIlIK '0)1 2. 2205996306F. WO 2. 97:10111 M1'*(M0 .i I 3 ! I 1. fIW ).". T_. -:1, I 1 . -'~ 7 ; l I: I 9. 9:J I 2'70~12 *-(I 1 0. rIFFlw,1)11 r1 ,: F Oft

oil 1 ). ()lIIl0rlIIIlI0V '00 I J OV . 01Mll0ll)IIJII: f 11(0 9. 9:3I27U6mGJ2.I-U I 0. 011111.0001 .11111

S2 UJ. 9312I(W,v132K.-oI 9. U3 1271.1632%K-01 .1 .917254 1326E#lJ0 5.5GG5J40t;3Gi002) 03 . 000000000K.4 (ll0 9. 91:1 1270MME2-11 I I . 90105 /2.1054Et10U 4 . 9,1U299 1l11:L ['0

Di 11I . (it)2b0,. U.1 0. )(JMMO)20000 00 1 .4 4'1937664 0' Do 2. 627.,6722 '379.10(1C 15 1 .4 0419:171131 V 4 0'0 9.93127066321:2-U-t 1 0. 000'lO fOOK '(1 0. OCI)'O1)OIIOIIL * ()1) 1 G 0. [i MIMM'10 ) 9. .9: 12"7 01(3 2 1:- 0 1. U. 0 00()00 0 0 00 2 .5 00)17 05 60 U1 F 00017 0). U0 1 IV () I11)1 01: K10 U0). U00'1 (IU0 1100 K' . 0 0 2. 221399320GE + 0r 0. (rl0l)f(f.

211 9.:11 "ur /1)W. - 01 9. 9:312 O10:!12K-01 2 .'326b45622EtO00. a00O0000001: 'r019 9.931270062-OL 0.utoououUooi',0o 1.4044937664E+00 3.4402930745F-.'')[

Fi-urC 26. Comuputat(ional Results foi- the KAI iodel

100

1IA 2 34

D! 3.3013222,33'Il 2.0620230121F'0 O.UOOOUUOOOE#00 0.00000001111E'00002 0.o mo(300l iJr:.no 2.7al34i171071E'l10 0.00U01.10OUOE'UO 4.97 19V360 1: f0003 0. 0fl100I0VII0 U. UO101OrJ(UOUE '1.3 2.06202001 29EtO O .00000011000K'004 2 .700.1479U1 IE:U0 2.7063447!)1171r. UO 2.O';202130129E'OO 2 .O62020(1211E,00

05 1-70G447J07If E JO 2.rJ32U211(0129FUU 1.321OU41096E.OO 2.062020J2IE&O03t)!l 0.Ot1oti1i1l11flir o0 0. fUOUOrUO03i+n 3. 32"3395206F~+00 0. OO00000LJUE # OIJ0/I U.1000(.1113 0h*IO I 1.:J2I1 (39G M + O U. UU3J1)U0UU0E. f 0 2. 0G2020U 1,9E # OflC 9 2 . 06 2 0 2F)1 1.1: 0 0 2.0WU2 0 1 '1! 1 Fi s 0au 2 .06202130129K.00 2. 7L)14479071FllfD9 2 . 7 0 ; 14 7 9 01 - ' . . Qi~i~t G 129 lr' (IV. O(fUOOUOU0E I UU 6. 400D71019 MrK# 00U I f3 I .R 111.3 1 (19'J0 '01 UO . l1ioIJUoIIIUE +00 2. 0132.0Z0U 129F +001 0. UtJoO)OUUUO'J. # 13111)1 U. UU01u)IUMllI100J 3. 0Fi353.52470UL) U. 0UU)J03)00E#00 2. 7014790 / I F. &fir)1 - 3. 0#. or).31K 3 1). 10f)001 UIJOE + 031 2 .7013447110711K 00 2. 010202nl(1 -i 'q3 il')D13 2. 711 14/791)7 1EF (OU 1 . .11 1 I94 10961H +110 2.06202001 2.C110 0. UO110001J11u)Ill 0

0,I) 01111IMIO1~Utir) +00. 2. on 2107110 L 29E 0 a . O(1UU001UUU 00O 0. 0UoooouoUOUtI: '101 , U . Uo''l)OM101l: '00 U. IIl1113113.3 fill U. 01IOUUOUIJUC). (it) 1.3z 1094 1 09lnF 0 nf)

11P, 3.3.i 'M1* 00 27 1- /flu; E1 01 3 .J01 I3221312')3-:100 2. 01,20110 1.I211: 11)) 17 2.. 1JfiZIIJI1 11. 1I) U,0. 10JI(il'E I J11 2.70611790173 '11) 0.1)01113W100(' I 1(0 1 U. Ouuiiuu0101iuo: 0111 2. 7011.7911? 7 11 toi U. ULJUOIIOU1JI: f00 U. UU013UII.3)1)IK * (3)

0 19 6.'3UUt1/U0'JiU /61. U. t)UIJUUUUUOUL-.+ U 2 .ur20280 12901:+UU 2.0132(120(.129JE-00

(fill 6 7

1)0. 0000000~1JU, f (10 1 . 32 103941 I096EK ' 0 1 .321094 10913K +00 8. 3 1 G09197 1 10

0iz I . :s2 1 fl'jl 1 our). I:, 1 1. :13094 10(96r#300 1 . I132180 U10JE' (JO 0. 01-10(00000017 4 101P 3 2 .0 (31 : ')1o11. 0U. [111011(IrJ00lOfl 1' .131 n.31. 00000000000+110 0. O0001300000F* t MD' 1 .:31 fT I W33l Ii I'l '! :(Jl23320Il3'II 011 I19Ui/IiE1I U I .511930I1* IulV l) I .3210 U) 1 n 3f:3l qK I II 1. J;! I 189 3I.i: oil 13. 4 22l1.4 7 ; 1 3 1U J. 861335 2,

71) 's 0 )

0' ; 1 .3 2 01 l.lW3 t3 1 1 1)1. :1? 1 I34 139 1: + O U.OUUO00000000+03J 6. 4211211*7 601 F, 00

0/ 2. nh';4.llj 4 9,7)1 19 7 1,1f; (11 K +Or 7. 1324 lUZ3113K+31 U. U3JJ)II)10&111)(1 33)II U.IIll: 1 0it .t 1lIjllrt 11 I 01-,rI(1 3.31 3 13 2 2 6 2 5 I 1t1) U. 4 .40 It)2 b7 10; 01: I1)(IDI1 U. U0000003,111111 11U.' U. 0IU0UtI30III)()r. 'JO 3 . Ot3-15324 711' .00 5. o3311:015; 1113 U 1 3 2 111111 1);: W1111.,Itl 2. O1;20211l I 2S1K. 110) 1 . 32309111 W116E i,) U. 17109~r~110 11 0 '1 IN

3131 0. 3lIIl01111110O l*: 1 U. 00OUL1h1I.J) 31 I .32 169~4 1U'JCE31 0 . UOOIJ()IU)1110l1 0311P1)2 1. A321011,1 N,1:iIIlU I1. 3210911 1 WI363it OU 31. 3(J32262fZ(OE#003 I . b09iAu 177F UID133 U. 1100000013)OIIIK'll0l 1 . 3"!131 31 IIGKfHl 0 3. 325:1395211t.'UU I1. 332739/I111131)1PH3. 3. 3;'!,:l]U5?U3;E 3003Lt. 001J0I1130E,111 2 .3.of; 21lJn1 I .11K. 00) 4 .9209)317 3110 lullFO

01! 2 2. OW312123 I NI: I;Oil I . 32 1811J I1396C10 0ll .001100)IUO(U001: 100 1. 1101JI)IJ JI)0111 1 (11V)36 0. UOIIf 3 31100 1 1 . 3"' 139. 119 1 1#*(3 0. UOUJ(UUUUK+03 7. 15,1743 1 '4 1 MIP 17 1.3. u(111,!PIJII)M3K 00) U. UI0010111 I OF00 13 3.004 5:3524 7UK4: 011 U. 0130I1)'l1111)3 ()1Vin I. 323 0:! 1l1W!; I 1 I. 32 1 14 1 WME-I, OQ 4 .,10525730200')I jj 0. O0OOOF0JJ111I:. nlC19 1.321103136KUL U.0U0UU0UUUEi11U 2.060200129I.400 7. 30 54 U 355330K'11

Figur~e 27. Comuputational Results for the KAF Mlodel

101

Ps 3. 10 0J~~3I37 us 3.070925113:4>ol:.. 0-'lnn7fl1Ir.-O1 n z 30 U., 3 3. 3 151,s n 70 b-o I

2 un 7.lUUJ9233F-UI n - 10 sus 3 .2 1 U C-2 52~1C 01nz3 zu 0 .32320-15071E-01 n c41 sul - .2327943102E-01

S isun : .053703032GE-01 n It 12 auI 4.5521303039F-01n.Ssun Ii 9.3530974000E-01 n 2 43 nun s 4.197500-13603Z-01

I1 V. 93 n z 44 5u .1995627.00OZ-ni

PO I l._UlIl15P.UloE-02 n - 45 :un 5S.52342701flr,01

n U.PIhu~:o n =46 su 5.14 15 10 70 0 1 E- 0

n 1un 391 2 -1It 2E.- U Z u 47 sumn .11606004921tE-01M2 7S 5..15 U 9 V5 2:- F -11 n 240 'un . .47U I PICOIE-0I

n :3 C., 7.75,1415'2.IIF.-(2 n , 40 *%,a 2 G.77001700lIE01

m = 4 V1i, s .U3262205i)IE-()I n 50 su., : 7.09390083E-0

M 5 SIM 2 1.35690G591F.-01 n 51 Sl iu it 7.339007451.OE-01n =0 -,in -1.624017330('E-01 n 52 a sU. 7.60315033229!_oi

n 2 7 Sit" 2 1.95916000li1E-OL n 553 sum 7.853OG29005E-01

R - 0 suJ'z 2.) 73003U4311L-QI n 5 4 s A 8.00000rfl!3OF-0IM z 9 It 2.69,7Z31307 CE-01 I 5 55 oun . 0a.307544 j5 29C0 1

n T 10 uns3,09113020M3E-0: n = 56 sun 2 8.5;00751S27F"]In - I 1u 3.511331 .- IIE - 0 n 57 sun s 0.6 0150001SE-0

n 2 sn 3.T13 0 6 567c 5 K-0 I n = 50 sun 2 OA659006930F-01n 13 uS 4,3111005003.-QI Ml - 59 sum z 9.019613G777E-01n I I u 4.01353701P0r.-01 min K z 234

M r 1n 5.252190550jr.-0I TO 1 3,3UL77911OOE-01n % I Gr u S.603500755GE-Ut n % 0 sun z 2.301770II02r.-01

n z 17 V1n 0. U0Z2VO237IS-01 nl I su~n T5 53 7113 7 fOr-0 1n . II sun G.5:00470201E-01 n 52 z:: . 7.0420?01000f.-0ln = I~ su::n 0.A11133237r.-0l 0 3 sun T 8.051107 30~jE-n~n It70 sun. 7.2VI1,006SEl-0 n 4 su.. x 0.72250O01t-Ol

2 s 1 Sun s 7 G!9S11V!I7E-0I n n5 Cun 2.1672DGgnOIC-01

27 i 7,900!Hi316nEI-ol .i.K 1302.n 1 0. 2I5201?431Il-01 P 0 51.0264 10500tt-Ot

n2 1 zu a 8. 5 194 :0 7 0 0 1 0 1 it 0 sns100iSOt0

25-u 0.4) 103f'-o3 n % 1 su .0214 5055 C-0l

it- u1 :.' -u 8. 9 P 1'fI 179 E -0 1 n z 2 sun 2.9100520371K-01

M . u., % 9.1 Li2.IU7739E-01 n = 3 2u 3.011(3503177F-01

sinzK ll5 n z 4 un 4.61107793001-01

.0 j .00 10 .7 5 1:- I) n z 5 sn=5.31715932O.'i-01

11 Sn s iOf170i- n %6 su 0 .0I76274l00I-0I

n T 2.1 i4 v 15 3 1 "I I E- 0 n a S , . % 7.141724397.-fil

I~ I n il : u

n = C s . 5 . 1. I J ' 2 I 03 7 !0 2.0 1 0 651"0 1 5 9 a 4n s u . .6 t0 3 .11E0 E f

n I flu .1 7.f11J0356'/t3 1:-) n 1 0 sn e. 015159C-1

n S 2 5.01,57l1352C01.13 n z It zu .300551.

n 13" =us 9. 1.1 -0IrU f 2 3 .10311 .,.7_I

n 0z, U: Iu I / 9i7 ' ' .1V -11. n 3 FU 2.n I 77.g )203E-IE

n 1S 5u I G G I 56S ) 0 r. - sun 3 s 2.05 J04'19I.-Uis

.' %u I I.I0!'1,31 779q~ 0C n II nu 2 I.6I)UI25.-oI

n 1 - . ' .1 .114 r 14UF 02o' n 13 1.n 77-IC j23iZIi.-n I

n 5 0 i 2. l9503Z5!M14r-02 n 7sun z .7713011921rt-OI

q n 2, ,u 3 7 1172 .07 5 1:- 0 nt 0 sun : .qon7002703V.-OA

n 2 20 Sun = 3. l:111UU1SUOUI-02 n s9 oun : 1.179710053OF-03

211 Zu 3.7jr.r110007E-02 n 10 1u l.53 1 91,51 -93

nz22 itn 4 ..17129507frr3L-02 n i 2 mI zu ;l.O77l2;1'30I:I ()3

n 527 u 5.7135131316E-02 n T I2 .. 257225 ;. I E-')

n 2 1u 6 S. I ( 37 2 3 93 1)6IU- 02 ni = 13 snS3.20?2'5S~nlIC-73T2.5 7u 7. fI (l 731-f17.W 11 =I su1 4. 039 107,017. -illn20 ,C 0 . " 3'9 2 0 /C.IFr- 0 " n 2 15 s,.n 5 .0 C1 .1~3 19 1 FI -11

n s27 ns 9.50) 21151r0'3E-02 n m I0 run 6 . 31It2 32 5 ( II W9-II.

ns20 Ot' s . 3117Ur07r-~ 3i 1 u

n 2 2q '. I . 2-f50)55r0l 71F -il1 to ns 9.64 0 77-1 111':11 -0 3* 1 30 s 1 1.4 140-191J3 251-0I1 n x 19 1u IP 102 1 GI .1-

n 1 =uI I I . 6 1)0fn3 G17 1 .1.- 01 n = 20 sn51414'12F0

n 32 ,u 0 I.0 07 8 5 .11G 1*- 01 n r21 DUN 1. l.5Illi70.1)ll:02

n l: 33 snS2.01707007U06'01 n t 27go vu Z. 115 Inn V91-0.mv31 2', 2240 CA0 ()13 3 E-0I1 n t 23 sun. 2.51SG109011412

n . 35 un.2.415172/'7M-01 n z 24 si' x 3.0370440001I-02

n 2 3litn 2.Th51153fIiF.-01 n 2 25 sum : 3.611752590tIE-07

rigui-e 23. Compulationil Rcsults for thc proposed M-odel (OA = 0.90)

ni 2= 4n . 27374713753E-C02 n - 7 sun 4 01 7 5720'1E-0In T27 2 5.03190458357E-02 n x e Sum . 4.5610236511E-01n - 73 2u S.130q5177I1iC-02 n - 13 su 5.010 70523:21-Ot

n2211 xu 0. 0721002019E-02 n -10 5u 5.519$21542CE-01n 0Sm%7.97210')0333E-02 n I I ,u .01523119SE-0t

n 1 s31.2026(134373E:02 n 2sms645IZ741672CE-0:: 32 sm ',.0571020231E 01 .3, g 1m s .173U90 017E-0l

n 33 1i. Z .0a3GS39734E-OL ni 14 7u 7.2600193071E-01*34 1 .37452133tOE-Ot n 15 All.n 7 0t164017f001F-0i

n -35 ii. 155622VCG94E-0I n 170gu 7.9450025O0621:0I39) *u * 1 31330317-0I ii 17 . 1.12 30M170C 0

537 ,,i 1 I9013i209300E-0I n 103 2im a,*507941402E0i

40 s.z2.6950(JII73ZE-01 n t 21 sum 9.1I330300691E-01n=4 1 22.96610613267E-0 I mI n K 2101'42 s:: 3.219 172310COE-OI min EI =93

n 4 un 3.5443033121E-01 mIi n 22 115M 4 t .=3.840175750 0 0E-0I : Imni63 =2 34

nT4Gsm24.479457416G7-01 min. K 5 10247 51u.n 4.0019430472E-01 mlin X 0 230

n 10I sum S .126412542CF-01 nm 97 2200m149 u z 5. 50011132"IF-02 mli 613a 109n ' SOsn 5.7771221072E-01 Pnot...Imsm SI S2 S3 54 55 SG S7 5')

SI 5n 1 1 .0906103707E-01 - ----------- - - - - - - - - - - -ni 52 3.1 13.401027.14Zr-01 9.00000O00006-o 1I 0 I I I 0 I 2ni 53 'u. .704053370CE-0: 9.100000000"E-01 I0III0I2

5 1 .997fl373333E-aL 9.20000000001-0I n 0 I2n 55 si. 7.27fl60"GI14E-01. 9.3000000000-ll 1 0 1 I 1 0 I 2n 50 7. 7. 5 4 ; C150 E -01 13.40O02000006-0I 1 0 1 I 1 0 I 2mz57 su ?.n002113I22113-o 9.50,'1000000()E-01 I I 1 2 I I 1 2

n 5 80313'314134E-01 9.6000000006-01 I 1 1 2 I I 2 259 su s 1.26I17301I4F-O1 9.7ooooooflfAF-01 i 1 1 2 1 1 z zri*G 0u 1345712157013e-oI 9.0O00U00O0-0II 1 1 2 2 1 2 3

n 2 tu 8.8305 963517E-01n 5=3 1.0078145520C-0I t, OPTIMAL PROCUR.I1r."1T QUAhNTITY (i

m:6 1 sum 9.1294 102426E-0alqI 6.3245553203E-01-in. K %233 02 =6.3Z45553203E-0Iro 4.0774309057E-OZ 03 =6.3245553203E-01

M=0 sun I.077lt009(257E-02 Ill O.gi42719I00E0I1n I slim = 13. 2G20G722rG-U2 0 5 = 6. 3245553203E-01

n : 2 slim 2 1.253137200,E-01 (16 z 5.3245553203E.-01n23 -1us, 1.G073137(IGI-01 (27 = I.095415I150E-0fl

n 4 =u 2.12500704174r-01 Q0 r I.0954451150E-00n = 5 sn= 2.SG(;40223001-OIn 56 slim 3.D(27Q73Gn03E3I

m 7=3. l14520705SE-01 K S Q 1071 At.

23.01'70055G I -U I - -- --- ---sum 4 .3052050137E-01 1 93 1 I "5

n 04.7225U05190GCr-UI 2 1t5 I I I

n I 5.900a 2736 Gl0 1 5 2 1:11 =65 .6U06'144llE-UI 7 200 2 2 704

ii = IS Gu 6.9355,?'25'IE-01 0I lm t 17 N. 7.21334939 1!E- 01ni = 0 s u: 7.531570G002E-0l1n : 19 sun 7.7999160033E-01n z 20 %U41 T0.04033690105E-0I

ii 51 Su S1.270935IO9E-01

iis22 slms13.4062r,24417E-01us - 23 sun z 0.6706Cu7G4S0E-0I*i 2 2.1 gigs = C.S8I090r02954E-01rk 25 sum = 9.003000SLU2E-01

Pin K 200PO T 1.39I41303GH3-0Z

M20 un 4.SqI'4l303G331-02ni - 1Iu 9.39035794io-J2ni z 2 slim 2 .43GI*416347E-0Iii T 3 cu. 2 1.917Gr35i074E-01it T 4 sun = 2.4UCG30221C-13-us z 5 sun z 2.9950003120E-01

n s un = 3.5222469132E-01

Fi-urc 29. Computational Results for the proposed Model (OA 0.90)

103

P " 3 51)1 V~IG763 36-UII' VsL~ 0902t5413 'r- -I3I3 'Q n 0s.. 3S ' 10.11 E 3-0 n i 3 n1. 3 331 0'6.057E-.

3 5.3'I~~1fln7,-? , ! - ~ 4u,, 3023373n 2 7 .13'J3!l ~ Lo n .) ,_

n /9Ji33L - it 42 u m. 4.552.,03'.(.-0n 2 5 :u~ 9.35.3007310U-Cj3 n I 1105' 1OC0

33'U I 501175311arr-02 I' 1i, 5. 12 11 W3-0in 1 , .5 1 1 r V Ci r r-f 0 'I = 5.Q 4 15 -1 3371 -3f*(3-31

n t In' 1.33617219r.7r..u7. , , 47 6 1 0 6.! U 1) 1 q 2 S -0n5.A15 v 7 z :1:E-1 F .-U n 4 si 6 4701910013E;-01

.1. , 'I. 7r,0 431G0527.:tI U- 0z n I q 11 in .7 7 0 0 17 CI I:! .-niz. 3.03707235 31E-01 I' 50 al.1 i 7.0 039 3 11).1 : I r-0 1

in'~. F . 3.3053-01 sul x1 7.33190U14516L-016 ~ ~ ...- GI F 310F 3'3i-0 1 ii 52 s. 7. 603CO332COEK-03

n : I. AVI ,UO 3011:J5-0 I n 51 a-n 2 0530023)35-03n =s" 2 2.317 Uruiio-01 n z 54 ri.. T .U033006353'6-01

nt9 s- = 2.0972337GEr-Ut ni' 55 s:... - = .07335 IE3 0 si. 3.0963 U20092E.- 0 fn 56 a.5100 55?-023m 3 5u n= 3.53 33372404E-0L 57 8*1 861050;39E-11

.2 2 3.9 6 6G3005E-0 n 50 5,. G1 590030-03it % in. 4 .374,0051203f r-03 n 59 sun. = .0396;30777E-01I-Iisn .01 3035-03 n 60 su 2 9. 177 5507E-01

- ' .. =5'55U 7'25,GF-03 (P0 = .301779110JE-01t i 6. 1 Y333L,0 37E-n1 m 0 .. 3.301779tio:r-oi

n 06.5mi903.'33-0L n I All. 2 5.5374132*Ir3ji-Q3n 11C .

3 j3 3 37 E - 0 1 n 2 " 2 703'i''1323i'lrf.Q,

n . s 7. Z'3'W i3qfr5 r. .n n 3 ::in.' O.U 1 07

1031.-(

flnr- 1 5ln 6.2011I-at.2T7.9'3230r-3 I 9.107239900K-073T 0. 2 15 2 -W V13 113C- Q . I1 K * 100

n 731 si, 0.53 343'123J.-03 10 1 02043050U30-0ln 15 a. 11 t, 13 1 1 C I 19 n 0 in 6 1 .030 1 9591333: -0

n f. 95 9 11 V11 11. iiO1 r. - ( I1. I 2 u .0?j313 5 nflL-n i 9I t131 Q3;7i3:.V.-3,1 n t 2 Mn.52 30sf73;-3

ni. K n =: 3 'i.. 3.VjiCG50jfl7'1.-0iPO T 0 I 9U3I3/!j, 5!!c -5 n I Isi .fl3 111 7 11 io1 0 1

n l. V 1) V ,1,, 5. n1 5.331,/5937n35-nlnu~ 2.11 39,jil '1o. 6.0327131!0

1~~07. I40 ; -1 7~ii!1ll 11 71 7.I"~ 3- 1)

or,~l 3 1'h1 3',037T133 ns 20 3SO. . 0019 1221MU3J0! -031T G.. I. 113 V

1J5f,771:(1 in t I3 I t- 0.377333-15-5

1.. 15 . (513 1113,Er 0 3 12 =.r 33 Ii, Ij 0 i531 r I n .-r

33 1 U! I:I' -: l n 1 1 SUn m 9.1303333 9E-U 3

I3 s: 5.,I PU.337 =o 2.f333375U1511E-0Sn 3 1. , C..7 7 3 1 Q3 ns r 0 2 . a MI( 1.3l5J59 1 .-11

1 0.i~ 1 . 3J~93P 3 0 i 3slim 6. 3 01it3,3,.3in 3 i.S 1 3.l34. 13J141,413 n j-l I. so. 13,391, 11 ; 1*/:-il

n 2 IS ".. 1 2.2 1 11i~'3-32. 11 3 1' im t I.774r,3 1.3I:_c IGi 31 1o.= 3 15137 V-7:191:-112 I = 4 no.s 2. 59023303G.0311-131

ns : Ill*Oi 2. 1953Ji72V.U41 ;02 ns =3 6in . T 5 .0 7. C0 -1 7203n in 3')~I 1' (37 ? 7.5I n 57 son. 07 13 0333.

in : 291c., 3. 35 30 511-,'ZE0-332 ns. 0 sun O830/067 3q.11-03R 21 1o~ .71111;1 0tJ1'?332 9 sun z I 3. 197.i70u,35I;03

ni, m 24.4?0t'507r303:-32 Ii30 1i .35 3 1 jjf .2'3 9C- fn.2 : 2 rE-l7 n I33lm 39733I21I I '31-03

n3 0.t 3 u3 !13'.331::- 0 is n 2 2o. .57 2.5 07; 3 0 3

n = 25nn 7. o00 n o i 3r,:un 2 it 33 ali. 3. 207.5533(33I0r-13320. 23'70351732 3 f9Z*3 ,(1 1 t 1 ?t1.1 1 4.3T)310 2507 7 r- u

ns 77 0. 9.50137.13131'-t33 it 15 nli, 5 01111.3311[K-33nis 20 1i. U 3 U: 1U3U330670.-o3 ns 16 s::n 3 2137 1 uti it 1v- n 3

~29 Sn 3. S0 .- u3 n 371m7 0 5(;;19 IE3-03mn 73 10, 3.4 I,1-13fl3'J3J53-333 ns 10 S.it . 9.640127133 '330-03in : 33 no.. I G O)il I , rJ'33 . -J 0 33 ftsi9n.. 13.103211551,11.-07

n :j ?-.o: E, 10 0al 15 416 ) F- 0 n m201 sli 1.131101f523 -02n 3 -in J n 2.0170/0''77090-3 n = 21 son.: I.7, 1113701I E-037

n 2 3 14 sun. % 2 1 2AUGO3)U 331: -(it n. 22 s1131 2.AI 335 Ti339jqf- an 35 s",. % 2.4951271723:-uj n 23 us.S2.530io.33743-fl

n 6 3r ~ n.i 2. 75505:1111 57E-3 1 n 2.1 in 3 .0 3 7040559rC-02

':igtvc 30J. Comutationial Results for Ili,- pr-oposed i lodel (OA =0.9 1)

104

n 25 S.. 317 r, Z 1 O K- 02 5 2 9950IM 0 17-01714 273 7 4 70 ,3 1'- 7. n 23 39 2 IfU 1 3 -

n 7 5 03130t4"(15"'E-IZ n 1 4.015/3132 UOr.-U29 7i1 5.0537.323E7 43 4.5OO23C5 11E-3L

n 21J 8.1,7210.71)1917-Wl n 9 ,, .J3'933-330 7.07730 3033.-U? n 1 5 .5311 34 5 1 .CC0 0I

31.1 0317731!-02 n 330.0 5Z30'].39E-0

n 21US7102023K1F-0 I. 32 u.. .451274G77OE-03

n 15 1 . S G7.2 131 S0 1 E- Of n 15 go. 7 . 11, 0 )0 7 0 : -0 10 G 1 3 .1 13 3 -0 11 n :6 7 7.0915 .1, 0'

1) 2i- (I

fl*37 1 90 G 12 '9 3IfF -o3 n 37 0. O.121U170E-139 i. 2 1 117 13 7 -1, -33 Of I30 0 0. 5 07 Of4 19 (17 r -0 1

W7 2 2.69,11732E-fli n 20 ,. 5045''-3M 1 2. 99030032317F.-01 n 230 qu. 9. 133030111094V-01

n3.?419117230O~r.-o I in K 30, 43 i. 3.S-31439777t.-0 min KIL 93

n : -1 .1 3 . a ',7 5 - _ C 0u -o03 pin F. 2 1 15n z 45 t.. 4 . 10 IOU 561:1 C 0 1 3 :i K3 2 235nl z 40 2* 1 4 .4714574r,7K-0I **In K41 :00

M 371 n:. 4 .01119431.172E-0I pn K 5 302n 1 0 5.12324425420E-O: pin K16 z 239

19 II , 5. 497j3137gznOF3.-03 :in K17 x201Su = 9777-j7r,11771E-03 min KS0 = 109

2 A= 0 .33'J19 33tJ , 1 I. -U prot...3 ftI St S2 SJ S4 ss5 s SO 7 SO12 z 6. B.1) It21 ,11GA 1.-0 I - - - - -- --- - - - - - - - - -

n 93 *.. 0.70 035331131 lI 9 3100001000rlE-01 1 0 1 1 1 0 1 251 91 ;. 6 6.7 337 32,31 r-031 9.2300000003-01 1 0 3 3 3 0 3, 2

T5 7.. 72 7 a C 0 C1 -1 . 03 9. 00000000,3r-03 3 0 1 , , 9 2M 9 ~ i. 75971233 ( 9.301: 1000GOUL1.U I 1 0 3 1 3 0 1 2n 7 7.. 7. M1073r 7337 133 t 03 3. 47311t73 0 10 1 a I I 1 0 3 2

511 17 11.J 1 1 I 3 q 310373jo31333313-01 1 1 1 2 1 1 2 201) -1. 0 , *3.2 4 57 If13 03r,- 9.710000001.01 -03 I 1 1 2 1 1 2 2

n 3 ,, 3.3'52.32771-1 q.OoUOOO3UOE-OjI I 1 2 2 1 2 3

n G5I I. I j. f:J9'1n34 3rY; /.1 1 1101! I7. 1 3 . 11 1,7'43'7 c E - uI33lAL rfc rciti IIIt

I V(I tH 0,4303530J 3 -1 47 72 99 1 *r3 1:-

73 ~ ~ ~ ~ ~ ~ ~ ~ O .33/.t3731 .- 3 3 .37415572073=-0n . U1 .1 1, r. -*~' D*,"~ 1 3. 0.1 4 7 I I :L 'I

2 ~ ~ ~~ fill I ;:. :111.:1 3 .O374'j95311fo. onU

n 3 Q3 7-)~ 11. 3 : 10371 3 3)7 - 1) 13154357 3

n ." 2. till)n7~r~'3 IE11 63 I. 0U34 Ior A 1

1, i3~ .= *.ju52nu/.ir-ui3 - -- --- --------

n 15 . ,72. 1 1;133;,lIIl-O 3 7 35 1 1 "J7

*... 5.9u33'2:3:f';3733 5 133 U Z 1

I" . 6 1) 0 G7 11,1j 1: - U 6 239 1 2133M r3 6. .1 3i 3217J3 -11 7 231 7 2 705

n r* 77.7 1 4"31 0ISE -0: 0 NO0 3 2 1314

n 0 7.5315", .u37r.-3I31 ~ Im ,, '!'3'3U12-3f

i 7 . 7 r 39 4 9~ V 9 33

23 n, 3. 4 1117174 117-r.- 31

2I I7. 3. U 4 It q f ,7.U 54 E - U

7 '3CI . 01.7 407"53'3-0 1

P0 .9. 9,4345133 333:-. -1).9 391 7 9 1 It r. - U .-33

n 3 9 . 3 47 r,31,54 U3)13E- U I

I i-twe 31. Coniputational Results foi- tCle proposed Model (OA =0.9 1)

105

ro : 33.016" [;11 -It7'Iii72 : 1 30 niG 5: 2 l / 5 3 01571 '1

'7 G 9 7i') 11 9. -532140i 1U ml :7 sm sa 2n952 7L r. -

j if .11 10 tIS Itrl'36-0 II !t.m IN . t94502701,1: -0tIt 1 ! .. 3fl t01lj'Cr01:- itII45it 4g .2 34 t2 7 ;2F: 01n

n 2 rim . .5.452 OU10 5 .I-iI n 417 111: of 18.20112009 a _O

at X v iiit! ., l 2 1 : .. i 2 - i n 421 J 4u .: 1 54 7 Ut 1 2 1'.0 3fl . - fI

SOifli n ztT7251Ft1 4-1 tun 5 .1100515 270- In 11 '5 11 .9 r2St,2;~ t:-I :ut 45 - 50 2imm 7 .O l1JIt tlnI~r -01

0 i3 I 62m0103s0,EIftan. 151nni" e r.a300 n7P :i-Fu0l

4 t i I ti I. 09 I ! i I I. 52tI n o .i 7 60. 771LT 'UtPIil;-1t

ti' 1 : 2.II rJ I:lI II 591Kit 1 1 53) @i: 7l 701; IM 310PI1 )r.-01

n:01 2 11 0101 n 54 7m 5 1

ni :2 3 19.o j1 1.-1 o. 70 51 12 071,3F.-O I

I 9 23n. 2 1) fl7 1 !, 1: -1 n 54 sin .7l001 OIV131-0

3i: I 1; 74 152 V("12 f t 6 sum 55 9.460153 r-t

35 5i113379JnttfE- nt a

n . of' '2 t400 f7Jost:E -I 1.2)1 tO15flITtE-Ot

n r.z ao 9u'5 .01102: I-

n : G.i~ 19 11 n~n 75 t:t 11 3 Yi -II011'0 =0.30 779 103 -0

It 'I -., - (2'27 /IIm I .i 7 sli r flit0052tt fl

n? mi 4 I'. r%231 f suz 0.32?0740702 'JI0

i7 5 -ii j 71 USI 0 1 !-sU I 'z r 2 .0t Gf;: .517

m 25 0 9 .11352 n' I ~ Oum 0.5 t1f3InSz i

Z UI ti 3ifl 2 19 392 1ijCuI:- to Tu ± os I .1r.1 li ,t I22.2 n

Ii It t I -"i 2 3 2 9f2,2 i. 2 07ijI) liC 1

it ()0ri: 5 3 Lr 1,tfl ;, tIi M so p l: f 3.11114 155H 7 i t

Itrm L46tntt~ 2 ru S.2t51 5L 2

it 7 22.l g, z r 1u o u.Joolr2 s 2ur 16.7 no;lrt - I

ItI " 29 .7 24 232 -'2fi QI n 8 --im 8. I!217 1 270 : .0

it ± 22 4i~ 21 i:.!' "11, 1 I: 22)I'' 1 'mu1 I. t/11, 0 2,.3EiSn '-1

it -'ti 7 truuli'J-lt tn au . u 193 001 1'it~~~~i a 123 IUi± sI 114070,tSU 1t nu t.Is7 1 uI

is T 71) 2i 1 ~ 9 2t:r2 22 ' 3 notgs 93 3.Z .::U20S I 0It n : 1N.2115672 Ut 14 ni,1 n 4 0t7 I. I I)1 13

rum : 7n, , 1 It 4t22 93 i2 i i) C -1 I 15 sum 5 7 2 4 3t;22 03 1 1

ftt m. I 1) 2 . r 1!1254 12' It 11 i I7 1um ml 7.7G 2 loll0

of. 5 2.2422209223.1.is21 rup 2s . t03I2t62V52W12F22 0'n 9 15 uim UP 2. It17)711'E-02 it n 29o, su 1 . 32910 211 27 12'-

iili 2. Cll pu . of I;aa rot~t .or t7 prpoe M1e (G = 1 0U Ufirm 15 n, 1 42)99 11C2-11F i

n zI U1135 10;' (I;! of 5 .1, 3 G 1-12311156

S; 21 "*'"." ti ";',inl;::ii:,,gR.-i,, I 4J: '.5''JlI3i 1,:irn .'

"::sr :,2 ,11'i;RI I I,:-,. I .,: : 9 3 '1 j i RI l 0',

/ / " i llil !'.ii2, ' E

C JI;'i " , +: 7',IIq jjp ,i 0. t 17 .11 !, 1I G 3 1:S: ', '/,:: : 0. 4-0'

I .w5'~r''~ui iirr:U.r0 1;~ :i

,t i.Jtl ",, : "I I ! '9 ru-in . ' i:u' ll I.IIl

j I: - I f 1 0 i i I1 9 'RIi'i r n: II: r.:, : : 1 *;' 'l ; I: II : r 0 1: -tit

QaMW, " . ilIi.'.i"I (RI -1 = :1 7 452-'--* F II- :, i i .ll ,l l!iii ''Re :41 .1,,1"0 F~ll 11 1 lZ * ~ 'l -: i 11 niZ l I I1 SI 519 '3 ni IWi 3

1 illi"i I PI II I+i.E -i R I i -IRi,-ri i i 7 R'bI I, " ' I,t - P , m : :'"I~ l ' Ii'ilsll..IilsIt:I I ri Im I+ n l ~ ~~ +I +I .

II "S i ' I 'J -1;:illsRlililI,.|iI III RI II I 0 l l lI .+J l -~- "U i :''l ':i lrlR, i l i '1 issRIsRlsii i Ii: 1 . I t) . I: 11I1 + I l

' - I ,. i i ,~ : i1 i|;I I illi5RR I . Ri I I ll I 2 I I lf l ,+ l'. l-I

,

",. - , , ' I ,',i 11 , T. I , i ii Y, sIliiis s i

I 3 R i i 5R,'jR n isI I , Ii 4 I IIl

-" ii "/' iii"Iiii.4 ii'I .II ,,' 9RRRIRAI IIRRRi~ lRI RIRi

, Il :::: : K7 2;T ;;2::lrq~l :,+, -' H :l. 'Ktl ? "

g"" l - }fl 1 ' 3lll] l li, Ir , I+'1 I ' i R Ii, I

Fi - I * 'sri : is II' "l5/i rli;4 . li R*I I II .43i) i iiIK Ii

2 . '- , , : I 1 1 ' : + + 1 . : 1 , -Sll -:' - -I. -

ri [Ii-, . 'R= I ;'/ |iiu - I RU i Ii = i

I t ii'

-I " r 1 I(I

nl:A 7 . Ili I ii

I 's~r ,';Hr - I .i ', ' i'.'.' i t i l l lllllfl'. 3 I / i t' I

' ± , I :: ± I i nI RnII; ";1r'I: I I Is1

n A } j ,iU i Ii s rii i , , Iii . l . llll lllll l -I I t I t) ,ti PG+ -,,,i" II. ii':/ cI ~lll c I. I l 'IlIII I Il)I - | l I I ,Ii 27 l "i. +' Ii. .i~ lI4'l *.l llll' ll +' HiIIIIII(II:'l

n107

S I ., I 'It I;,II ;7 I II II IIA il l} Illll . l I I I t

r I.: f, 15. 1 1 "Ill I filllll +ll

+ ',. "I ll , ; l ,ll'l I I C Ill l * G : .,l , b ~ l J l

2: n+"' " ; , ~14 l I,14 III (1' V t- t I1 + I+b .+/l Z I .I" b!

33.' "op tto a Res lt foIrI;',I111 l hI pro ose M o e (OA 0I' ,+: I . 92)

- r,',i~ : IIIII :IIIII 1 0 7 I I . I.'.' I I , I T I +

C'o- 3. I- l(317151L-OI nsi Itis. 2 755(0530157E-01i 0 - 3 .1. C3 iI 1 o17 s 3 ON 1/113:L-0I

0. - -. .0 '7:111-1777 1:- 111' 3 3 5oC 11c , 5/r -hI)is 2 3 'I IC':11:-. 111.is. 3 61 U 7I t.~,, r. - i

So. - U. 4 3 It -1 -I I I )LI 3 ~:io U 9 Il01K- 1I ~, 41.7. 303U37F-U

;U1 n :K 03 n 3 I-,1 4 07L.0.I23;03F-0l

2 isiss.l?'1,:-l 4s . 19 (17 7 7 1;',''' , 11 :

Us 11.. I.ii~J 11isit1:: -U ! . is.1 11 3')i 141/ .- 17 1

72-. 3. l 'jlC16522017:l-01 -to n. is. 0 6 1 * I 11 G I It -U 11

Isi r, 'i7 -1 1 .?f!l 1:-j 1 $10 3i.. 0 0 11/ 17$ , n- I

IC ISsii " I I 1:nn. . *. - 111 .iI-is.. 7l m) mms I '1 11 / .1 -oitJ1 Is. ~ l7Is l'"1 -7 Isi is V~ s..i 1113rI -III 705.11

10 !W.10sssi 7. nIiz1IJ.II0l5iJII'-lfl A 2

is i1 . 1*.11si 00 .... 1-11 01 I'l 6 V.01.9 11U.10 3-0l0I' 1. . 1 11.lr1 : 0 I is r. - n1 On.' 3 007l F"-re' 1

is1 n1 -'. 7l..'nSI-n-I ,r1- s I O..' S53l I3ir'r - I

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I'i"Utre 35S. Cwiiput. aiwmal Results for the jproposcd Model (OA =0.93)

109

LIST OF REFERENCES

1. Lee. K. S., Scientifc .Mlanagement for the Korean Air Force Logistics System, Korean

Air Force Logistics Command Research Report, 1986.

2. Apple, C. L.. A System Approach to Inventory Management of Repairables in the

.a'v, Master's Thesis, Naval Postgraduate School, Monterey, California, March

19S2.

. . l1am. S. P.. Sqfciv Level of the Base Repair Cycle Item, Master's Thesis. I lanyang

University, Seoul, Korea. December 1986.

.4. Korean Air Force Manual 67-1, Supply Support, Korean Air Force Publication

Center. 197s.

5. Flatchett, J. WX., McNall, P. F., Schrady, D. A. and Zenha, P. W., A Repairable Item

Invemorv Miodel, Naval Postgraduate School Technical Report Research Paper

No.71, November 1966.

6. Schrady. I).. .. fathematical .1odels of the Repairable Item Inventory System, Na-

val Postgraduate School. Monterey. California, August 1967.

7. Schrady, D. A., A Deterministic Inventory Model for Repairable Items, Naval Re-

search Logistics Quarterly. V.14, No.3, pp.3 91-398, 1967.

S. Tersine. R. J., Principles of Inventory and Materials Management. North-Holland,

19S8.

9. Scarf, H-1. E., Gilford, D. H. and Shell,, M. W., Muhistage Inventory models and

Techniques, Stanford University Press. 1963.

10. Sherbrooke, C. C., M[ETRIC, A M1fulti-Echelon Technique for Recoverable Item

Control, Operations Research, V.16. No.1, pp.122-141. 1968.

110

11. Sherbrooke, C. C., METRIC, A Multi-Echelon Technique for Recoverable Item

Control. Rand Cooperation Report, RM-507S-PR, November 1966.

12. Sherbrooke, C. C.. A management Perspective on METRIC, Multi-Echelon Tech-

nique for Recoverable Item Control, Rand Cooperation Report, RM-5078, I-PR,

January 1968.

13. Hadly, G. and Whitin, 1. M., Analysis of Inventory system, Prentice-Hall, Inc.,

1963.

14. Ilillestad. R. J., Rand Corporation Report, R-2785-AF, Dvna-fETRIC, Dynamic

Multi Echelon Technique for Repairable Item Control, Rand Cooperation Report.

R-27S5-AF, July 1982.

15. Fox, B. L. and Landi, D. M., Searching for the Multiplier in One-Contriant Oph-

mization problem, Operations Research, V. 18, No.2, pp.253-262, 1970.

16. Muckstadt, J. A.. A Model fo;- a .1fulti-Item, Multi-Echelon, .1fulti-Indenture Inven-

tory sy'stem, Management Science, V.20, No.4, pp.472-4S1, 1973.

17. Navy Fleet Material Support Office, Overview of Multi-Echelon Models, Report 160,

November 1984.

18. Hillestad, R. J. and Carrillo, M. J., Models and Techniques for Recoverable Item

Stockage When Demand and the Repair Process are Nonstationary, Rand Cooper-

ation Report N-1482-AF. May 1980.

19. Naval Supply Systems Command Publication 553, Inventory Management, A Basic

Guide to Requirements Determination in the Navy, 1984.

20. CACI, Inc., User's manual, An Availability Centerd Inventory Model, by CACI,

Inc.-Federal Operations and Management Analysis Department, 1982.

21. CACI, Inc.. Mathematical Description, Optimal Operational Availability Inventory

Model, by CACI, Inc.-Federal Systems and Logistics Division, 1978.

111

22. Allen. A. 0.. Probabilitv, Statistics and Queueing Theory, Academic Press, Inc.,197S.

2.Telephone Conversation between Oh, Dock Soo, Korean Air Force LogisticsCommand and Park, Byeoung Gone, November, 1988.

112

INITIAL DISTRIBUTION LIST

No. Copies

Defense Technical Information Center 2Cameron StationAlexandria, VA 22304-6145

2. Library. Code 0142 2Naval Postgraduate SchoolMonterey, CA 93943-5002

3. Defense Logistics Studies Information Exchange IUS Army Logistics Management CollegeFort Lee, VA 23801-6043

4. Professor T. P. MooreDepartment of Administrative Science, Code 54.MrNaval Postgraduate SchoolMonterey, CA 93943-5008

5. Professor Dan Trietsch IDepartment of Administrative Science, Code 54TrNaval Postgraduate SchoolMonterey, CA 93943-5008

6. Air Force Central Library 2Postal Code 151-00Sindaebang Dong, Kwanak Ku, SeoulRepublic of Korea

7. Library of Air Force Academy 2Postal Code 370-72Chongwon Gun. Chung Cheong Bug Do,Republic of Korea

S. Park. Byeoung Gone 2Taegu, Dong-Gu, Sin-Gi Dong541-14. 6Tong-6BanRepublic of Korea

9. LCDR. Mike Hardee 1Carrier Air Wing TwoUSS RangerF.P.O. San Francisco CA 96601

113

eelhhehhi - dtic · the repairable inventory system differs from that of the consumable inventory...

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