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NAVALPOSTGRADUATESCHOOLMonterey,California

T H E S I SANNUAL SCHEDULING OF ATLANTIC FLEET

NAVAL COMBATANTS

by

Clarke E. Goodman J r .

March 1985

T h e s i s A d v i s o r : R. Kevin Wood

Approved f o r p u b li c r e le a s e ; d i s t r i b u t i o n i s u n l i m i t e d .


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SECURITy CLASSIFICATION OF THIS PAGE (When Data Bntered)

REPORT DOCUMENTATION PAGE READ INSTRUCTIONSBEFORE COMPLETING FORM1. REPORT NUMBER r'GOVT ACCESSION NO. 3. RECIPIENT'S CATAL.OG NUMBER4. TITL.E (and Subtitle) 5. TYPE O F R EP ORT a. "'ERIOO COVEREO

Annual Schedul ing o f Atlan t i c F l e e t Naval Maste r ' s Thesis

Combatants t-1arch 19856. "'ERFORMING ORG. REPORT NUMBER

7. AUTHOR(e) 8. CONTRACT OR GRANT NUMBER(e)

Clarke Elmer Goodman J r .,

PERFORMING ORGANIZATION NAME ANO ADDR E SS 10. PROGRAM ELEMENT. PROJECT. TASKAREA a. WORK UNIT NUMBERS

Naval postgraduate SchoolMonterey, Cal i fo rn ia 93943

11. CONTROL.L.ING OFFICE NAME AN D ADDRESS 12. REPORT OATE

Naval postgraduate School March. 198513. NUMBER OF "'AGESMonterey, Cal i fo rn ia 93943 10 3

14. M O NI TO R IN G A GE N CY N AM E a. AOORESS(II dll /erentirom Controlllnll Olllce) 15. SECURITY CLASS. (o f t i l l . report)

ISa. OECLASSIFICATION/OOWNGRAOINGSCH EOUL.E

16. OISTRIBUTION STATEMENT (o f thle Report)

Approved fo r p ub lic re lease ; d i s t r i b u t i o n i s un l imi t ed .

17. OISTRIBUTION STATEMENT (01 th e ab.tract entered In Block 20, II dll ierentirom Report)

18. SUPPLEMENTARY NOTES

19. K E y WOROS (ContInue on reverse . Ide I f nece er y en d I d en t ll y b y block number)

employment schedul ing, i n t e g e r programming, s e t cover ing , math programming,column genera t ion

20.ABSTRACT (Continue on rever. e . Ide

I fnece .ae ry e n d I d en t it y b y b lo ck nUlllberJ

Employment Scheduling i s th e t a sk o f ass igning sh ips to f u l l f i l u. S.Navy commitments a t home an d abroad. Commitments a re even t s , with f ixeds t a r t and completion da tes , t h a t r equ i re spec i f i ed sh ip r e sources . Theob jec t ive o f the employment schedule i s to s a t i s f y a l l even t requirementswhi le prov id ing an equ i t ab le r o t a t i o n o f sh ips an d an even d i s t r i b u t i o nof workload.

This s tudy provides a mathematical programming model to a s s i s t employ-

DO FORMI JA N 73 1473 EOITION OF 1 N OV 65 IS OBSOLETES/ N 0102- LF014-6601 1

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ment schedul ing. A s e t cover ing formula t ion o f th e schedul ing problemm i n i m i z e s dev ia t ions from an " idea l " schedule , developed in terms o f navyschedul ing p o l i c y, while s a t i s f y i n g e v en t r eq u ir e m en ts . An e f f i c i e n tcolumn genera t ion program, using p r o b l e m ~ s p e c i f i ccolumn r educ t ion t e c hn iques , produces a moderate s i z e d problem which i s then solved as an i n t e g e rprogram.

The model i s t e s t e d using da ta from the 1983 Atlan t i c F l e e t schedule fo rc a r r i e r s and su r face combatants . The da ta i nvo lv ing I I I s h i p s , 19 majoreven t s , 73 sepa ra te sh ip - type requirements , an d 44 fo rce weapon systemc a p a b i l i t y requirements yie lds a s e t cover ing problem with 10,723 v a r i a b l e sand 22 8 c o n s t r a i n t s . T hi s p ro bl em i s solved on an IBM 3033 AP in 84seconds o f CPU t ime.

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Approved for public release, distribution unlimited

Annual Scheduling of Atlant ic Fleet

Naval Comba tan t s

by

Clarke E. $(oodman Jr .Lieutenant, Ulllted States Navy

B.S., Miami University, 1977

Submitted in partial fulfillment of th erequirements for th e degree of

MASTER OF SCIENCE IN OPERATIONS RESEARCH

from th e

NAVAL POSTGRADUATE SCHOOLMarch, 1985

Author:

Approved by:

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ABSTRACT

Employment scheduling is the task of assigning ships to fulfill U. S. Navy

commitments at home a nd a broa d. Commitments are events, with fixed start

an d completion dates, that require specified ship resources. Th e objective of the

employment schedule is to satisfy all event requir ements while providing an

equitable rotation of ships an d an even distribution of workload.

This study provides a mathematica l p rogramming model to assist

employment scheduling. A se t covering formulation of the scheduling problem

minimizes devia tions from an "ideal" schedule, developed in terms of navy

schedu ling policy, while satisfying event requ irements . An efficient column

generation program, usmg problem-specific column reduction techniques,

produces a moderate-sized problem which is then solved as an integer program.

The model is tested using data from th e 1983 Atlantic Fleet schedule for

carriers an d surface combatants. Th e data involving 111 ships, 19 major events,

73 separate ship-type requirements, an d 44 force weapon s ys te m capability

requirements yields a se t covering pro blem w ith 10,723 variables an d 228

constraints. This problem is solved on an IBM 3033 AP in 84 seconds of CPU

time.

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.W>

TABLE OF CONTENTS

1. INTRODUCTION 8-

A. CURRENT PROCEDURES 8

B. PROBLEM SCOPE 10

C. THE NEED FO R COMPUTER ASSISTANCE 12

D. CPSKED SOLUTION STRATEGy.................................. 13

E. THESIS OUTLINE 16

II. PROBLEM DEFINITION 18

A. NAVY MANAGEMENT AND PLANNING CONCEPTS 18

B. TH E CPSKED PROBLEM 23

C. MEASURES OF EFFECTIVENESS (MOEs) 24

III. MODEL DESCRIPTION 28

A. MODEL SELECTION 28

B. TH E SET COVERING MODEL 29

C. CPSKED PROBLEM FORMULATION 31

D. SCHEDULE COSTS 34

E. PENALTIES 40

IV. PROBLEM GENERATION 44

A. GENERATING FEASIBLE SCHEDULES 44

B. COLUMN REDUCTION 46

V. IMPLEMENTATION AND RESULTS 48

A. COMPUTER PROGRAMS 48

B. TEST DATA 49

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INITIAL DISTRIBUTION LIST

APPENDIX E SHIP SCHEDULE REPORT .

APPENDIX F SOLVER OUTPUT R E P O RT .

LIST OF REFERENCES

VII. CONCLUSIONS AND RECOMMENDATIONS ..

APPENDIX A EVENT INPUT DATA .

APPENDIX B SHIP INPUT DATA .

APPENDIX C SHIP STATISTICS R E P O RT .

APPENDIX D EVENT REPORT

C.

D.

SCHEDULE QUALITY

COMPUTATIONAL RESULTS

53

56

59

61

66

69

73

79

91

101

102

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ACKNOWLEDGEMENT

I would like to thank th e scheduling staff at CINCLANTFLT for providing

data and insight into th e Navy employment schedul ing problem. I am grateful to

Professor Gerald G. Brown whose assistance with th e X system was crucial to the

development of this model. I am deeply indebted to Captain Dan Bausch,

USMC, whose previous research at th e Naval Postgraduate School contributed

substantially to this study. I thank my wife, Linda, and two sons, Eric and

Bobby, for their patience, understanding, an d assistance during many long days

an d nights devoted to this study. Lastly, I express my most sincere appreciation

to Professor Kevin Wood without whose support and professional assistance this

project could not have been undertaken.

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1. INTRODUCTION

In i ts broades t sense, fleet readiness is th e degree to which th e force is ready tocarry ou t its mission to wage p ro mp t a nd sustained combat at sea. Supportingmilitary strategy involves not only having units properly m an ne d, t ra in ed ,equipped, an d supported, bu t also deployed to pos it ions f rom which they may beable to best support U.S. interests an d rapidly engage potential enemies....Aproperly ba lanced employment schedule is essent ial to attain high states ofreadiness, because th e individual requirements for maintenance, training, an dmorale ar e frequently in competition with each other. [Ref. 1]

Employment scheduling is th e process whereby U. S. Navy ships,

submarines, aircraft an d other units are assigned to major operations, exercises,maintenance periods, inspections an d other events. Th e effectiveness of th e

employment scheduling process directly influences overall fleet combat readiness.

Currently, thi s process is largely manual requiring several full time scheduling

officers an d additional personnel at various levels of management. This study

develops an d implements an optimization model that automates a substantial

part of th e employment scheduling problem. The model is formulated as a

generalized se t covering problem an d ma y be applied to a number of independent

subsets of th e employment scheduling problem. For explanatory purposes, th e

model is applied to the annual planning schedule for naval c om bat an ts o f t he

Atlantic Fleet.

A. CURRENT PROCEDURES

Th e Atlantic Fleet Employment Schedule details th e day-to-day operations

of th e 700 to 750 units that comprise th e Atlantic Fleet. The schedule is one of

t he p ri ma ry methods for managing these fleet assets. Requests for fleet units to

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--

participate i n event s, referred to as event r eques ts in this study, originate from

several sources, e.g., Secretary of Defense, Chief of Naval Operations, Type

Commanders, F leet Commanders, Group Commanders , Squadron Commanders,

an d individual unit commanders. Fleet assets are always in short supply relative

to th e demands r esu lt ing f rom all event request s. Flee t schedu le rs a re faced with

th e problem of selecting which event requests will be scheduled and how to most

efficiently schedule those events. The size an d complexity of this schedul ing

p ro blem d em an ds th e resources of numerous management personnel , e.g.,

operation and planning staffs, at all levels in th e command structure.

Current Navy employment schedules ar e pr odu ced with little computer

assistance. Th e Commander in Chief Atlantic Fleet (CINCLANTFLT) convenes

a schedul ing conference each quarter. This conference is th e culmination of th e

employment scheduling process an d resul ts in publication of a detailed quarterly

employment schedule with annual schedule projections. CINCLANTFLT's

conference is preceded by Type Commander scheduling conferences. The Type

Commander conferences are th e working conferences where schedules are

developed. At these conferences, rough schedules are proposed, reviewed,

discussed, conflicts resolved, and bargains made until a final schedule is selected

for submission to CINCLANTFLT. In th e overall process, compute rs a re only

used to store an d retrieve schedule data; they are not used to assist decision-

making.

CINCLANTFLT IS th e overall schedule coordinator. Fleet assets are

managed by th e Type Commanders who, in turn, delegate part of their

management responsibilities to group, squadron and unit commanders.

CINCLANTFLT an d t he ope ra tional fleet commander (OPFLT) are primari ly

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concerned with meeting major operational commitments while Type Commanders

an d lower levels of command are pr inc ipal ly concerned w it h m ai nt en an ce ,

inspections, an d training.

B. PROBLEM SCOPE

Th e entire employment scheduling problem IS formidable. However,

because of its structure, th e problem can be divided in to in depe nde nt

subproblems of manageable size. This s tudy develops a model for the Combatant

Primary Event Schedule (CPS KED) problem. Th e derivation of this problem

from th e overall employment scheduling problem is discussed in this section; th e

resulting CPSKED problem is defined in de ta il in Chapter II.

CINCLANTFLT has operational commitments in th e home fleet (Second

Fleet) an d abroad. These commitments result from event requests that have

been approved for inclusion in the fleet schedule an d are referred to in this study

as primary events. Primary events include all extended operations an d major

exercises. These events are th e most i mpor tan t a nd t he most demanding events

in th e fleet schedule. Other events are classified as either major maintenanceevents or secondary events and may be viewed as events necessary to support th e

successful conduct of pr imary events.

This study focuses on scheduling ships to the CINCLANTFLT primary

events. I t is a ssumed that (1) all primary events ar e fixed in start time an d

duration, an d (2) all primary events are uniformly more important than

supporting events. Assumption 1 effectively separates th e process of the timing

of primary events from th e problem of schedul ing (assigning) ships to these

events. This is a good approximation of c ur re nt Naval pract ice since most

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commitments are made years in advance without detai led knowledge of future

fleet assets, an d also because of long-term fixed commitments. Assumption 2

allows assignment of ships to primary events without requiring concomitant

schedul ing of secondary events, a lt ho ug h ti me must be set aside in a ship's

primary event schedule to allow for subsequent scheduling of secondary events.

Thus, with th e above assumptions th e problems of determining which events to

schedule an d when to schedule them ar e presumed solved. Th e remaining

problem is to determine which fleet assets should be used to satisfy th e primary

event requirements while distributing th e workload equi tably among th e ships.'

Fleet assets may be divided into th e following functional categories: navalcombatant units, amphibious units, marine units, support units , submarine units,

an d aviation units. Within a functional category, unit operational capabilities

are similar an d units are employed in similar missions. Hence, substitutions

within a funct ional category ma y be allowed bu t substitutions across category

b ou nd s a re no t allowed. Primary events may require assets from one or more of

these functional categories; however, since substi tutions are confined to functional

categories, an individual asset requirement for a primary event is dependent on

only one functional category. ConsequentlY,the CPSKED model can be

developed to generate annual planning schedules for assigning assets from one

functional category, naval combatants in this study, to primary events without

regard to other functional categories. Primary event scheduling problems

considering other funct ional categories, e.g., amphibious uni ts , aviation units,

submarines, o r s up po rt ships, ca n be f ormulated in a manner analogous to th e

methods presented in this study.

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An instance of t he C PS KE D p ro bl em consis ts of 19 primary events an d 111

ships based on 1983 historical data. The primary events ma y require assets by

ship-type and/or weapon system capability. Th e 19 primary events translate to

73 event/ship-type requirements an d 44 force weapon system capability

constraints. Th e goal is to select th e bes t an nu al pla nn in g schedule from all

possible candidate schedules.

C. TH E NEED FOR COMPUTER ASSISTANCE

Scheduling decisions directly affect fleet readiness an d fleet operational

performance.

The optimized peacet ime employment schedule which has as i ts objec tive maximizing combat readiness should always be th e goal and guide. [Ref. 1]

Unfortunately, readiness is a vague measure which c an no t b e directly optimized.

However, computers can be used effectively as management tools to assure that

the employment schedule provides th e best opportunity t o m ai nt ai n readiness at

th e highest level possible.

The opportunity to maintain readiness ca n be measured in te rm s o f efficient

utilization of fleet assets. Th e unnecessary over-employment of fleet assets

adversely affects personnel morale an d reduces th e opportunities for maintenance

an d t ra in ing. While over-employment is considered more detrimental to fleet

readiness, under-employment results in deficiencies in operational experience with

a consequent reduction in overall readiness. Thus, th e effect of either over-

employment or under-employment of fleet assets is a reduction in fleet readiness.

In addition, assignment of a suboptimal mix of forces an d capabilities to perform

an operational mission o r ma jo r exercise will result in degraded performance and,

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in the extreme, ma y r esult in failure to achieve th e objectives of th e mission or

exercise.

N av y e mp lo ym en t schedules have been successfully produced for years

without th e assistance o f c om put ers o r c om put er models. Furthermore, because

of the unpredictable nature of ships an d navy operations, it is unlikely that

computer models will ever be sufficiently sophisticated to replace fleet schedulers.

Computer models can, however, become valuable tools to assist fleet schedulers.

Computer models may be used to speed up th e process of generating a schedule

an d conduct "what-if ' analysis on a schedule proposal. Additionally, an

optimization model ca n provide a method of measuring th e relative merit ofdifferent schedule proposals.

Currently, there exist no concrete methods for- judging th e acceptabili ty of a

proposed schedule. Exper ienced schedu ler s have a n i nt ui ti ve feeling, based on

Navy policy an d guidelines, about the merit of a proposed schedule. The

mathematical modeling process requires that th e scheduler's in tui tion be replaced

by concrete rules an d measurable criteria, yielding an analytic f ramework for

comparing proposed schedules. Thus, the modeling process provides additional

insight into th e scheduling problem an d results in a s tandardized method for

evaluating a proposed schedule. The ability to critically evaluate and compare

alternative proposals is potentially th e greates t management tool to be gained

from a ut om at in g t he scheduling process through th e use of an optimization

model.

D. CPSKED SOLUTION STRATEGY

CPSKED is an optimization scheduling tool developed an d implemented as

a se t covering model. "Optimization" increases the model 's power as a decision

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support system and "set covering" provi des model flexibility and precision.

Large-scale se t covering models, despite their advantages, are generally

considered difficult or impossible to solve. This sec tion provides th e rationale,

based on modeling concepts an d experience gained from prior research, for

selecting this approach'to solve the primary event scheduling problem.

Because of thei r combinator ial nature, scheduling problems are difficult to

solve optimally. Consequently, many suboptimal, heuristic techniques have been

developed for attacking schedul ing problems. However, optimization should be

preferred over suboptimal techniques because op tima l solut ions p rovide a proper

reference for judging th e acceptabi li ty of all alternative schedules. Geoffrion an d

Powers [Ref. 2] have stated ,the need for optimization:

The problem is not that optimization- capabilrty is needed to cope with thestaggering number of alternatives...although this is impor tant . I t is not that optimization capability is needed to resolve the cost trade-offs inherent in planning, al though this too is important . I t is not even that managers would ratherhave the best answers possible from their planning support systems, althoughcertainly this is compelling. Rather, the crux of the matter is that optimizationcapability is needed to permit reliable comparisons between different runs of themodel.

Therefore, th e goal of this s tudy is to deve lop a model and solution techniques

that reliably provide optimal solutions to the CPSKED problem.

Scheduling problems ca n fre qu ent ly be viewed as selection models, e.g.,

route selection, crew selection, etc. In th e CPSKED problem, a set of individual

ship schedules must be selected such that demands for ship-types and weapon

system capabilities required by different events are sat is fi ed . Select ion problems

may be formula ted as se t covering or set partitioning problems. In terms of a

scheduling problem, the objective of a se t covering model is to select a minimum

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satisfied or "covered." When th e problem constraints are stated as equalities, Le.,

demands must be satisfied exactly, th e problem is referred to as a "set

partitioning" or "equal ity constrained set covering" problem. The CPSKED

problem is formulated here as a se t covering problem with certain generalizations.In th e CPSKED problem, variables correspond to individual ship schedules

an d constraints correspond t o p ri ma ry event requirements. Event requirements

are stated in terms of force composition (ship-types) and force weapon system

capabilities. Th e basic development of th e model is discussed in Chapter III.

Set covering problems represent a class of integer programming problems

which is simple in concept. Unfortunately, like most integer programming

problems, se t covering problems are quite difficult to solve. However, r ecent

advances in solution techniques have made possible th e solution of large

problems. (See Bausch [Ref. 3] for a survey of these computational advances.)

Brown, Graves, an d Ronen [Ref. 4] have applied th e se t partitioning model to a

crude oil ocean tanker scheduling problem. Their large-scale problems (74

constraints an d greater than 7,000 binary var iables) were typical ly solved in less

than one minute of IBM 3033 CP U time. Their success is based on th e X System[Ref. 5] which is an advanced general purpose op timizat ion sys tem. Since th is

system is available at th e Naval Postgraduate School, it is employed as th e solver

for th e CPSKED set covering problem.

The set covering approach allows many of th e real-world modeling

constraints to be included in problem generator versus th e problem solver. This

allows for flexible an d precise modeling. Essentially, th e problem generator

generates columns of th e integer programming constraint matrix, each of which

corresponds to a feasible schedule. As Bausch [Ref. 3] states "The art of

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formula ting pract ical set covering problems lies in th e schemes used for column

generation." Details of th e problem generation scheme are given in Chapter IV.

The development of this model requires a method for evaluating each ship

schedule in terms of th e employment schedul ing objectives. Commercia l ship

scheduling and/or routing models typ ical ly address d is tances , speeds, p rof its ,

capacities, etc. T he C PS KE D problem is concerned with more abstract military

objectives, readiness and operational effectiveness. A pp ro pr ia te s ur ro ga te

objectives that provide th e best oppor tuni ty to maximize th e real, bu t abstract,

objective ar e often used both in mode ling an d in rea li ty. CINCLANTFLT's

scheduling policy guidelines are in fac t surrogate objectives designed to provide

each unit th e best opportuni ty to maximize the real objective, combat readiness.

Precedence for th e use of surrogate objectives in modeling also exist. Sibre

[Ref. 5] employs surrogate object ives in place of abstract military objectives in his

study of a U . S. Coast Guard ship scheduling problem. In that study, Sibre used

a quadratic assignment model with morale-related objectives in terms of "away

from home port time", "ba lanced workload" , an d "maximum s ingle cruise

duration." In this study, military objectives are developed in t er ms o f scheduling

policies as they relate to fleet readiness (see Chapter II) an d are implemented

using techniques b as ed o n goal programming methods [Ref. 6].

E. THESIS OUTLINE

This study presents a set covermg optimization model for solving th e

CPSKED problem. In Chapter II, the problem is defined in de ta il an d measures

of effectiveness ar e developed. Chapter I II develops th e set covering solut ion

method. Th e method used to generate th e problem is descr ibed in Chapter IV.

In Chapter V, th e model is implemented using data from th e 1983

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CINCLANTFLT schedule; th e resul ts are then compared with th e actual 1983

CINCLANTFLT schedule. Conclusions an d recommendations are summarized in

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II. PROBLEM DEFINITION

Indus tr ia l p roduct ion problems are often concerned with maximizing

productivity subject to constrained resourc.es. The ~ avy employment scheduling

problem closely parallels th e industrial problem, i.e., the Navy is concerned with

maximizing national defense subject to constrained fleet resources. Analysis of an

industrial production problem requires a working knowledge of th e company's

management goals an d procedures; similarly, analysis of th e Navy employment

scheduling problem requires a knowledge of th e Navy's management goals and

procedures. This chapter provides a brief background in Navy management and

planning concepts. Th e insight provided by this background information is used

to isolate a moderate- sized, independent subproblem (CPSKED) from th e overall

scheduling problem an d to develop specific measures of effectiveness for this

subproblem.

A. NAVY MANAGEMENT AND PLANNING CONCEPTS

Navy management an d planning concepts a re con ta ined in NWP-l [Ref. 1],

NWP-7 [Ref. 7], an d A tla nti c Fle et Regulations [Ref. 8]. Th e background

provided in this section is divided in to th e following four areas: management and

control of operating forces; employment schedule events; fleet assets an d

employment cycles; an d planning policy.

1. Management and Control of Operating Forces

Navy organization distinguishes between two types of control for its

operating forces: administrative con trol (ADCON) an d operational control

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(OPCON). Administrative control is concerned with tra ining, maintenance, an d

readiness while operational control is concerned with conduct ing naval operations

an d exercises. All Navy operating forces are assigned to either th e Atlantic or

Pacific Fleet Commanders for administrative control. Th e Fleet Commanders

normally delegate administrative control to Typ e C om ma nd er s. Operational

control is exercised by Unified Commanders (CINCs) and is normally delegated

through Naval C om po ne nt Co mma nd er s ( FLTC IN Cs ) to Operational Fleet

Commanders (OPFLTs) or Type Commanders.

Operational control of an Atlantic Fleet unit is transferred to

CINCUSNAVEUR or CINCPACFLT when the unit is operating away from th e

home fleet. Operational' control of units operat ing in th e home fleet is normally

delegated t o C OM SE CO ND FL EE T, OPFLT in th e Atlantic. Adminis trat ive

control of Atlantic Fleet units is delegated to the Type Commanders:

COMN AVSURFLANT for surface ships, COMNAVAIRLANT for aircraft

carriers and ai r squadrons, COMSUBLANT for submarines, an d FMFLANT for

marine units.

Th e Atlantic Fleet Employment Schedule provides detailed

information on the utilization and status of naval forces. Th e schedule is

published quarterly an d consists of a deta il ed quarterly schedule and an annual

planning schedule. Th e detailed quarterly schedule contains all tasks an d

activities to be conducted by fleet units an d is directive in nature. The annual

planning schedule contains only major activities an d is informative in nature.

Th e quarterly schedule must account for every day in th e quarter for each unit ;

th e annual planning schedule need no t account for each da y in th e year.

Th e Employment Schedule is a primary management tool for both

planning and con trol o f fleet units. As administrative commanders, th e Type

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Commanders develop th e major por tions of these schedules. CINCLANTFL T

coordinates, approves, and promulgates th e schedule. This study is concerned

with the annua l planning schedule.

2. Employment Schedule Events

Th e tasks an d activities contained I n th e Employment Schedule ar e

broken down into 27 categories which ar e further s ub di vi de d i nt o specific

e mp lo ym en t t er ms ( EM PT ER Ms ). A comple te desc rip tion of categories and

terms ma y be found in NWP-7 [Ref. 12]. In this s tu dy, t he term "event" is used

to refer to a collection o f E MP TE RM s related to th e same task. Eve nts a re

categorized as e ithe r primary events, major maintenance events, or secondary

e,vents.

Primary events consist of extended operations an d major exercises.

These events are th e backbone of th e schedule. Primary events result f rom fleet

operational commitments, e.g., commitments to deploy a battle group to the

Indian Ocean, or commitments to participate in a specific NATO exercise. These

events are fixed in t ime, i.e., they have fixed start an d completion times.

Major maintenance events, e.g., construction, conversion, overhaul,

etc., are d ep end en t on s hip ya rd availability an d ship cycles. These events ar e

generally scheduled independently of all other events . Unit s scheduled for major

maintenance events a re no t considered available for primary events.

Secondary events include th e rem aining events associated with

maintenance, training, inspections an d other individual unit events . Secondary

events ma y be viewed as preparation and support for th e primary events. These

events are generally scheduled no t to conflict w it h t he p ri ma ry events.

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This study develops an d implements a method for producing annual

planning schedules for th e primary events. Non-operational periods resul ting

from major maintenance events are presumed known an d sufficient time is se t

aside in the primary event schedule to permit subsequent scheduling of secondary

events.

3. Fleet Assets and Employment Cycles

Fleet assets are classified in functional categories as combatant ships,

amphibious ships , service ships , submarines, a ircraf t units, fleet mar ine units,

training units, an d shore support units. Within each functional category, unit

operational requirements are s imilar; consequent ly, units ma y exchange roles

within cer ta in limits, e.g., a f rigate may be able to fill th e a requirement for a

destroyer. Capabilitie s o f units in different funct ional categor ies are raeJically

different with respect to primary events an d unit substitution across functional

boundaries is no t acceptable.

Fleet assets are further classified as either COR (Command

Operat iona lly Ready) or C NO R ( Co mm an d Not Operational ly Ready). CO R

units are capable of participating in " ...operat ional tasks which contr ibute to the

effective accomplishment of the FLTCINC's responsibilities. Commands that are

CNOR are assigned to the OPCON of th e Type Commander who is responsible

for conducting t he t ra in ing an d maintenance required for the unit to attain COR

status." Only CO R assets ca n be assigned to primary events. A fleet unit's

status is primarily dependent on its employment cycle. Th e ship employment

cycle is defined in NWP-1 [Ref. 1] an d consists of th e following phases: th e new

construction or overhaul phase , the operational phase, and th e refit phase. A

new cycle begins each time t he ship enters overhaul. A ship is CO R only during

th e operational phase.

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-

B. THE CPSKED PROBLEM

Th e overall employment scheduling problem involves scheduling primary,

secondary, an d major maintenance events for all operating forces in th e Atlantic

Fleet. Several independent subproblems may be identified in the overall

employment scheduling problem. Divisions ca n be made in terms of f leet assets

an d event types. Th e C PS KE D p ro bl em is an example of one of th e possible

independent subproblems.

1. Division by Fleet Assets

As mentioned earlier in this chapter, fleet a sse ts inc lude a wide variety

of units performing very different funct ions . With respect to primary eventscheduling, each o f th es e functional categories is independent since a unit in on e

functional category cannot perform th e mission of a unit in a different category.

In primary event scheduling, mission capability is th e primary consideration and

th e primary event schedul ing problem ca n be divide d into subproblems by

functional category.

2. Division by Event Types

Major maintenance events are dependent on a unit's employment cycle

an d a re scheduled based on shipyard availability an d optimum maintenance

cycles. Major maintenance schedules ar e developed prior to scheduling other

events. Units scheduled for major maintenance become non-operational (CNOR)

assets; thus, th e effect of scheduling major maintenance is to limit t he quan ti ty of

available operational assets for subsequent primary event employment scheduling.

Primary events are the "end p ro du cts " o f all fleet a cti vity duringpeacetime an d receive th e highest priority when schedul ing operational fleet

assets. Primary event requirements cannot be satisfied by CNOR assets.

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th e fact that readiness is a difficult entity to measure , Navy policy defines th e

following as significant factors supporting fleet readiness: effective deployment of

forces, maintenance, training, an d personnel morale.

In th e broad sense, effective deployment of forces means satisfying th e

primary event requirements . Decisions to commit forces to operations and

exercises at home an d abroad are made at th e highest levels with careful

consideration for their contribution to overall military readiness. Thus, effective

deployment of forces IS accomplished by prescribing th e primary event

requirements, in terms of force compos ition an d capability, which are then

converted to problem const ra in ts . These constraints must be satisfied at th e

sacrifice of th e remaining factors.

Th e remaining three major factors, maintenance, training, an d personnel

morale, are difficult to measure directly, an d hence, more concrete MOEs that

provide th e opportunity to achieve these criteria are sought. Th e

CINCLANTFLT scheduling policies described earlier in this chapter are

guidelines or goals designed to maximize the opportunity for each unit to achieve

th e highest degree of readiness in maintenance, training, an d personnel morale.

During th e operational phase , CINCLANTFLT policy states that 20

working days per quarter should be assigned for maintenance upkeep. Fo r th e

CPSKED problem, this implies that at leas t one third of th e home fleet time

should be reserved for in-port upkeep.

To maintain training readiness, CINCLANTFLT policy states that te n days

per quarter should be provided for each s hip to conduct individual ship training

(ISE). ISE periods are considered secondary events and not scheduled in

CPSKED; however, th e CPSKED solut ion should reserve sufficient home fleet

at-sea time to satisfy this requirement.

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Th e major factors affecting personnel morale are family separation and crew

liberty. To ensure family separation is not excessive an d crew liberty is

adequate , CINCLANTFLT policy establishes th e following guidelines: no more

than one third of th e time between overhauls should be deployed time;

deployments will be followed by a post deployment leave period; an d ships in th e

operational phase should be scheduled for no more than 30 days at-sea per

quarter.

A schedule that provides the o pt im al a mo un t of home port time for

training, morale, an d maintenance, the opt im al a mo un t of home fleet underway

exercise time for t ra ining, an d an equitable deployment rotation of ships will

provide th e best opportunity to achieve th e CINCLANTFLT goals for readiness.

Based on this observation , a measurab le MO E ca n be constructed from th e

CINCLANTFLT policy guidelines.

Th e approach is a goal - programming technique. Policy statements are used

to derive ideal target t imes, or goals, for deployment t ime, home fleet at-sea time,

an d deployment rotation time. Home fleet time consists of th e operational phase

time less deployment t ime. Home p or t t im e is th e home fleet time less the home

fleet at-sea time. Assuming all constraints can be satisfied, th e objective

becomes: minimize th e deviations from th e ideal target times an d th e single MO E

is a function of th e deviations from t he t arge t times. If some of the constraints

cannot be satisfied, constraint violation penalties, discussed in th e nex t chapter,

ar e included in the objective.

This objective captures t he i nt en t of the CINCLANTFLT policy guidelines;

however, it cannot measure many of th e intangible factors that must be

considered when developing an employment schedule. Neither ca n the intangible

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facto rs always be included as problem const ra in ts . On th e other hand, a human

scheduler cannot possibly evaluate all schedul ing alternatives to determine an

optimum schedule. A human scheduler is required to ensure "al l" criteria are

satisfied; th e optimization model is required to ensure th e resul ting schedule is

t he " be st " schedule in terms of th e established criteria.

The CPSKED problem may now be stated as follows: Gene rat e a n a nnu al

planning schedule for all carriers and surface combatants that minimizes th e

deviations from th e fleet 's ideal schedule (specified by target deployment time,

home fleet at-sea time, an d deployment rotation time) while sa tisfying, as best

possible, all primary event requirements.

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III. MODEL DESCRIPTION

This chapter presents th e rationale for modeling t he C PS KE D problem as ase t covering problem. Th e se t covering model an d generalizations are discussed

and the CPSKED model is developed as an elastic se t covering model. Th e

objective function costs an d penalties ar e developed in terms of th e

CINCLANTFLT policy described in th e previous chapter.

A. MODEL SELECTION

M an y ty pes of scheduling problems may be solved as se t covering or se t

partitioning problems. The basic formulation is straightforward; however, for

practical problems, these formulations typically r es ul t i n t ho us an ds of variables

an d are considered difficult to solve optimally. For this reason, approximate

heuristic methods have been used extensively in solving these problems.

Fortunately, a sophisticated large-scale mixed integer linear programming solver,

th e X System [Ref. 9], permits the efficient solution o f m an y large-scale problems.

Bausch [Ref. 3] employed th e X System on test problems consist ing of several

hundred constraints and thousands of var iables in his survey of computational

techniques for solving large-scale se t covering problems; th e results were qui te

favorable. The crude oil tanker problem, Brown, Graves, an d Ronen [Ref. 4], in

. which columns represent possible ship rou tes and the object is to select th e least

cost set of rou tes, con ta ined thousands of variables an d was solved using th e X

System in less than one minute of IBM 3033 CPU time.

Official Navy policy states that "The opt imized peacetime employment

schedule which ha s as its objective maximizing combat readiness should always

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be the goal and guide." [Ref. 1] Thus schedule optimization is a Navy goal. The

exis tence of a sophist icated, proven, large-scale solver allows formulation of th e

CPSKED problem as a set covering problem with high expectation of achieving

optimal solutions.

B. THE SE T COVERING MODEL

Set covering models formulated as integer l inear programs have been known

an d proposed fo r pr actical applications for many years. Th e standard

formulation is:

J

mm ~ c i x ii=1

J

s.t. ~ aii x i ~ bii=1

i = l , . . . , I

Xi E {O,l} j = l , . . . , J

where

aii E {O,l}, and b >0 and integer.

In this formulation, a minimum cost set of columns from t he c on st ra in t m at ri x

must be chosen such that that each constraint is satisfied, Le., "covered" at least

bi times.

In many practical applications, th e columns ma y be partitioned into sets

where only one column per set is allowed. Fo r example , a se t ma y consis t of all

possible schedules for a single ship an d exact ly one of t h ~schedules in the set

must be selected. If there are K such set s, S 11 . , SK , th e model ma y be

generalized to admit only one column pe r set in th e final solution. This is

accomplished by adding th e following constraints:

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1 k = 1, ... , K

where

_ { 1 if j E Sk8 kj - 0 otherwise.

The standard formulation ma y be general ized to admit ranges on th e

const ra in ts . Thi s generalized se t covering problem is:

minJ

~ C j X ;j= 1

J

s.t. E 8 k j X j;=11 k=1, ... , K

Jb - :::;; ~ a i j Xj :::;; b + i = 1, ... , I

j= 1

where

Note that equality c on st ra in ed s et covermg problems, Le., se t partitioning

problems, ca n be fo rmu la ted this way by setting bi + = bi - for a ll i .Efficient, reliable solution of se t covering models is difficult. Th e X Sys tem

is an advanced genera l purpose large-scale op ti miz atio n s ys te m w it h special

features for solving integer and mixed-integer models. This system employs

"elastic" programming techniques [Ref. 10]. Elastic programming assumes that

all constraints ma y be violated at a cost. This t echn ique allows the feasible

region to be "stretched," subject to penalty costs, an d general ly result s in more

rapid convergence to an optimal integer solution. In an elastic formulation,

feasible solutions always exist; th e ,objective, then, is to find a feasible solution

that minimizes both th e original objective and t he sum of t he elastic penalties.

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Th e elastic formulation of the generalized set-covering problem is:

J K 1min ~ C j X j+ ~ ( P k - S k - + P k + S k + )+ ~ ( P j - S j - + P j + S j + )

j= l k=l j = l

Js. t. l - s k - : : : ; ~ 8 k j X j:::; l + s k+ k = I , ... ,K .

j= l

Jb j - - s j -:::; ~ a j j x j:::; b j + + s J i = I , ... , I .

j= l

Xj E {0,1} j = 1, .. . , J

Sk - ? 0, S k + ? 0, and integer k = I , ... , K

Sj - ? 0, Sj+? 0, and integer. i = I , ... , I

where

Pk +, pj + = upper constraint violat.ion penalties

Pk , P j - = lower constraint violation penalties

b + = upper constraint limit

b - = lower constraint limit.

C. CPSKED PROBLEM FORMULATION

Th e C PS KE D p ro bl em is formulated as a generalized elastic se t covering

problem using th e following notation:

Indicies:

k = 1, ... , K

t = 1, ... , I= 1, .. . , L

(rows) constraints requiring that oneschedule column be selectedfor each ship,

(rows) event/ship-type requirements,

(rows) event/ weapon systemcapability requirements,

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J = 1, . . . , J

p = 1, .. . , Pq = 1, .. . ,Qw = 1, . . . , W

R,

v,

Data:

8 A:j' j E SA:, k = 1, . . . , K

(columns) each representing anindividual ship schedule,

primary schedule events,

ship-types,

weapon system types,

index se t identifying all schedulecolumns belonging to ship k,

index se t identifying allevent/ship-type requirementsbelonging t o e ve nt p ,

index set ident ifying allevent/ship-type requirementsrequiring ship-type q ,

index se t identifying allweapon system capabilityrequirements belonging for event p

index se t identifying allweapon system capabi li tyrequirements requiringweapon sys tem type w .

cost of schedule j for shipk.

1 if schedule j is for ship k;o otherwise,

1 if schedule j assigns ship kto event p as ship-type q;o otherwise,

1 if ship k has weaponsystem w;o otherwise,

minimum number of ships oftype q required for event p ,

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Pk - , k = 1, . . . , K

Pk +, k = 1, . . . , K

Decision Variable:

Logical Variables:

Sk +, k = 1, . . . , K

Sk - , k = 1, . . . , K

Si +, i = 1, . . . , I

maximum number of ships oftype q allowed for event p ,

minimum number of weaponsystems of type w required

for event p ,maximum number of weaponsystems of type wallowedfor event p ,

penalty for no t scheduling ship k ,

penalty for assigning morethan one schedule to ship k ,

pe r unit penalty for assigningtoo few ships of type q to event P ,

pe r unit penalty for assigningtoo many ships of type q to event P ,

p er u ni t penalty for assigningtoo few weapon systems o f t yp e wto event p ,

pe r unit penalty for assigningtoo many weapon systems o f t yp e wto event p .

1 if schedule j is selected;o otherwise.

greater than 1 if more than oneschedule is selected for ship k;o otherwise,

1 if no schedule is selected forship k;

o otherwise,amount by which b +is violated,

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Si - , i = 1, . . . , I

SI +, l = 1, . .. , L

S/ - , l = I , ... , L

Formulation:

amount by which h iis violated,

amount by which bl +is violated,

amount by which blis violated.

J K I L

mm E Cj Xj + E (P k - Sk - + P k +Sk +) + E (P i - Si - + P i +Si +) + E (P I - SI - + PI +81+)j = l k = l i = l 1=1

J

s. t. l - s k - ~ EOkjXj ~ l+ s k+j= l

k = I , . . . , K

J

b - - 8 - ~ ~ a x ~ b ++8 + " ~ 1 I, 1 '" L.J 11 1 '" i i - , . . . ,j = l

J

b l - - s l - ~ E A l j X j ~ bl + + 8 1 + l = I , ... , Lj= l

Xj E {O,l} j = l , .. . , J

In words, th e model is interpreted as: "Choose the min imum cost se t of ship

schedules such that one schedule per ship is inc luded in th e se t and most (ideally

all) event requirements are satisfied." To produce meaningful planning schedules,

th e costs an d penalty structures are critical to the model. The se topics ar e th e

subject of th e n ext two sections.

D. SCHEDULE COSTS

This section details th e computation of the costs for individual ship

schedules, Le., th e Cj values of the CPSKED model. The objective for th e

CPSKED modelis

to satisfy the event requirements while providing an equitablerotation of th e ships between deployed and home fleet status an d providing an

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even distribution of th e home fleet workload. Thi s objective is decomposed into

th e following three components based on CINCLANTFLT policy goals:

1. Achieve an ideal time between successive deployments for an individualship.

2 . . Maintain an ideal ratio of a ship's deployment tim e t o between overhaultime.

3. Maintain an ideal ratio of a ship's home fleet sea time to home fleettotal time.

Th e first two objectives replace th e "equitable rotation" requirement while th e

third replaces th e "even workload" requirement.

Under th e model assumptions, an employment schedule that satisfies th e

event r equirements while achieving th e ideal times an d ratios specified is

considered an ideal sc}ledule. Given the ship assets an d event requirements for

t he Atlan ti c fleet, th e likelihood of achieving an ideal schedule is extremely small .

To o bta in a schedule as close as possible to th e ideal, a cost structure measuring

th e deviations from the ideal schedule is imposed on th e problem. Th e following

targets are established for all ships:

T I time (in days) between deployments,

T2 target ratio of deployed days to between overhaul days,

T 2 deployment time (in days) required to achieve ratio T 2,

T 3 target ratio of home fleet se a days t o t ot al home fleet days,

T 3 home fleet sea time (in days) required to achieve ratio T 3

Costs C I j, C 2 j , an d C 3 j with respect to a particular schedule j are then

defined in terms of t he targe ts as follows:

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~ {0.1 x ( deviation above T 1 )

C l j 1.0 X ( deviation below T 1 ) ,

~{.O x ( deviat ion above T 2 )C 2j 0.1 x ( deviation below T 2 ) ,~ {1.0 x ( deviation a b o ~ eT 3 )C 3 j 0.25 X. ( deviation below T 3 ) .

The costs h e r ~ar e functions of deviation in days from th e target. In terms

of CINCLANTFLT policy, it is more costly to over-employ a unit rather than

under-employ a unit. Consequently, costs are relatively reduced when they

reflect under-employment of a unit, Le., more time between deployments , less

deployed time, or less home fleet sea time.

Th e linear cost of a schedu le j is defined to be th e sum of th e three cost

functions:

This column cost is intuitively appealing since it can be viewed as a measure of

t he t ot al deviation in days from an ideal schedule for a particular ship. The su m

of the linear costs over all ships indicates a measure of th e dev ia tion in days for

the fleet employment schedule from an ideal schedule.

Frequently there will be insufficient assets of a given ship-type to satisfy th e

event requirements. When this occurs, ships of a different type are generally

substituted to satisfy th e shortfall. Th e acceptability of ship subst itut ions

depends on th e mission requi rements for th e given event. In this model,

substitutions are allowed at an increased cost. Acceptable substitutions are partof the event input data, e.g., for a given event it may be acceptable to substitute

an FFG for a DDG with an acceptability factor of 0.8. The acceptability factor

is a measure of how well th e substituting ship can perform th e duties of th e

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required ship for t he pa rt ic ular event an d lies in th e range (0,1]. The

acceptabili ty factor is used to adjust the linear cost of a schedule column

containing substitution assignments. I f there are no substitutions, th e

acceptability factor is considered to be 1.0 an d th e l inear cost for th e column is as

described above. I f there are subst itu tions , t he n t he l inear column cost is divided

by th e average of th e accep tabi li ty factors for th e events contained in the

schedule column. Then, for two similar columns, one with subst itu tions an d one

without, th e costs of th e column with substitutions will be greater with the

amount of th e difference a function of the acceptability of th e substitution. This

procedure allows th e model to discriminate between substitutions and primary

. assets and keeps substitutions to a minimum level.

Though appealing, the linear cost, adjusted for subst itu tions , may resul t in

poor decis ions i f used directly I n the model. Th e problem is i ll us tr at ed by the

following example:

Suppose ships A and B have schedules with costs of 50 and 50 respectively. I fship A and B also have schedule columns with costs 0 and 100 respectivelywhich satisfy th e same combined se t of event requirements, then the model will

no t different iate a preference between the first cover (cost 100) or th e secondcover (cost 100). Part of th e scheduling objective is to d ist ribu te th e workloadequitab ly over al l fleet asse ts , hence, when costs are equal, t he model shou ld becapable of selecting th e cover that distr ibutes the costs over th e most ships.

To avoid this problem, th e squares of th e adjusted linear costs are used in th e

model. This cost allows th e model to resolve ties by spreading th e cost over th e

greater number of ships.

All components of a ship's column cost are computed with regard to the

ship's current employment cycle. This requires a knowledge of the following

historical information for each ship:

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D 1 total da ys in th e current operational phase,D 2 total deployed days in th e current operational phase,D 3 t ot al h om e fleet sea days in th e current operational phase,D 4 last deployment completion date.

Th e cutoff date for thi s info rmat ionis

th e last day priorto

th e modelplanning

period. If a ship has no t deployed since beginning i ts operational phase , i ts last

deployment completion date is se t to the date th e ship last completed overhaul or

was commissioned.

. Col um n cost computation is descr ibed in th e following equations:

Terms:Cj

C jC ljC 2 jC 3 jai j

Ci j

t lt 2i

t 3i

T 1T 2T 3

model column cost,l inear column cost,time between deployment cost,deployment cost,home fleet sea cost,substitution acceptability factor,column' average acceptability factor,time between deployments,deploy time for event i ,

home fleet s ea t im e for event i ,

time between deployment target,deploy time target,home fleet s ea t im e t arge t,deploy time target ratio,home fleet s ea t im e target ratio,starting total days in operational phase ,starting total deployed days,star t ing total home fleet sea days,last deployment completion date,

d 1 total days in operational phase ,d 2 total deployed days,d 3 total home fleet sea days,d 4 last deployment completion date,

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N ship non-operational days for th eplanning period (generally overhaulperiods),

X slack operating days for ship-type .training an d individual ship exercises.

Counters:

d 1 = D 1 + 365 - N

Targets:

T 1 = 360 (may be varied for each ship-type)

Cost formulas:

0.1( T 1 - t 1)

t e T 1

o

T 3 - d 3

0.25( d 3 - T 3)

if T 1 ~ t 1 ,

if T 1


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E. PENALTIES

In th e elastic formulation of the model, penalties can be categor ized as

either model disruption penalties or goal violation penalties. When violation of

an inelastic constraint has no physical interpretation, th e penalty for violating

th e constraint is a model disruption penalty; these penalties should be sufficiently

small to allow reasonable relaxat ion of th e feasible region, yet great enough to

enforce the constra int in th e final solution. When a constra int can be violated at

a cost in th e final solution, th e constraint is actually a goal an d th e penalty is a

goal violation penalty.

In th e CPSKED problem, th e first se t of constraints require that exactly

one schedu le column be selected for each ship. The second se t of constraints

requires that th e correct force composition be ass igned for each event . Th e third

set of constraints requires that correct se t of weapon system capabilities IS

assigned for each event. Th e associated ranges an d penalties for these sets of

constraints ar e assessed separately.

1. Ship-Schedule Constraints

Since exactly one schedule is desired for each ship, the upper and

lower ranges on th e ship-schedule const ra in ts a re both se t to one. Vio la tion of

th e upper range implies that a ship would receive more than one schedule for th e

planning period. A ship cannot be employed I n different locations

simultaneously; hence, th e upper range must no t be v io la ted in th e final solution.

Th e penalty then is a model disrupt ion penalty that increases problem elasticity

while enforcing the upper range on th e constraint. A ship schedule cost is

measured in terms of days deviation from an ideal ship schedule . Schedule costs

beyond a certain limit, typically 200-300 days deviation, would be

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counterproductive to maintaining a high state of combat readiness. An upper

bound on this limit of 1000 days deviation is used to establish th e model

disruption penalty. The CPSKED objective function is in terms of days

deviation squared, thus th e penalties must also be in days dev ia tion squared and

th e resulting penalty is l.Ox 10 6 days deviation squared. Any combination of

columns for a part icular ship will cost more, including penalty, than an y single

column for th e ship an d consequently, multiple schedules will not be selected in

an y optimal solution.

Violation of th e lower range on a ship-schedule constraint corresponds

to no t scheduling that ship. Th e lower penalty, then, should be th e price at

which it is acceptable to allow th e ship to remain idle throughout th e planning

period. In th e CPSKED model, th e "idle" price is computed for each ship, this

p rice is equiva lent to th e column cost for a "d o nothing" column. Th e "idle"

price squared is then used as th e penalty for violating th e lower range of the

ship-schedule constraints.

2. Event Requirement Constraints

In the CPSKED model, th e events are CINCLANTFLT commitments

and the event requirement constraints can be interpreted as goals to meet those

commitments. I t may no t be possible to meet these goals at any reasonable cost.

Th e penalties associated with these constraints are goal violation penalties.

Th e lower range b - on an event requirement constraint corresponds to

th e m in im um n um be r o f ships of a particular type required for th e event. Event

values are assumed to be related to th e event duration an d deployment status.Generally, short home fleet se a events ar e more easily canceled or rescheduled

than long duration deployed events an d consequently receive a lesser value in th e

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schedule planning process. In th e CPSKED model the event value s are defined

to be the durat ion of th e h ome fleet sea days and/or th e deployed days contained

in th e event. Th e lower penalty P i - i s a price above which th e cost of

committing additional assets to the event exceeds th e value of th e contribution of

those assets. I n t his model, th e lower penalty is a func tion o f th e event value an d

may be adjusted within th e program.

Situa tions arise where a sh ip would be under-employed if all minimum

event requirements hi-are me t exactly. Under these circumstances, it may be

desirable to schedule th e ship for some events in excess of minimum event

requirements in order t o m ai nt ai n training and proficiency for th e ship. To allow

for this possibility, th e upper range bi + for all event requirements ma y be se t

above th e minimum requirement. In most instanc es, sh ip assets will be in short

supply an d th e lower range will be binding. Th e upper penalty Pi +, in effect

when the u pper range is exceeded, is a function of th e event value an d may be

adjusted within th e program.

3. Force Weapon System Capabil i ty Constra ints

Frequently, primary events ma y require a specified set of force weapon

system capabilities. Weapon sys tem capabilities are no t necessarily unique to

s hi p t yp es an d hence, th e force system capabi l ity requirements ma y be satisfied

by var ious mixes of ships. Penalties for viola ting these constra ints are related to

th e additional value a particular weapon sys tem contr ibutes to an event's mission

an d consequently should be input data under th e scheduler's c ontrol. These

penalties should be high enough to enforce th e constraints but less than

event/ship-type pe nalties since a w ea pon system contr ibutes less than an entire

unit to th e event's mission. I n t hi s p ro to ty pi c i mp le men ta ti on , these penalties

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were all se t to 1,000 (lower) an d 0 (upper). These penalties worked well in th e

model; however, a more thorough knowledge of mission requirements an d system

contributions would enable improvements.

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Rule 1. A u ni t m us t be of the proper type, or an allowable substitute, tosatisfy an event requi rement const ra in t.

Rule 2. A u ni t m ay n ot p ar ti ci pat e in primary event s when th e u nit is ina non-operational status.

Rule 3. A u ni t c an no t be participate in more than one primary event atan y given time.

These rules ar e used to generate all feasible ship schedule as follows:

For each ship k perform th e following steps:

Step 1. Determine th e ship-type q and, using rule 1, select all eventrequirement constraints that d em an d t yp e q units or allow typeq units as s ub st it ut io ns . T hi s "potential ship-event list" is th elist of events that ship k could potentially participate in .

. Step 2. Determine th e ship non-operational periods from input data and,using rule 2, compute the time intersection of each event in th e"potential ship-event list" with the non-operational periods. Ifth e time intersection is not nu ll , d el ete the event from th e"potential ship-event list." The resulting list is th e "ship-eventlist."

Step 3. Construct a schedule network as follows: Define a starting node,s , an d connect this node to all events in th e ship-event list.Using rule 3, connect addi t ional arcs between event pairs if thetime intersection of th e events in th e p ai r is nul l; th e direction ofth e arc is from th e earlier event to the later event.

Step 4. Le t v correspond to an event in a schedule, th e se t of all directeds - v paths for all v in th e network corresponds to th e se t of allfeasible schedules for the ship. Enumerate each s - v path j an dse t column coefficients: (a) ai; = 1 if i is on th e s - v path; (b)b k; = 1 ; (c) )./; = 1 if ship k satisfies part of event /weapon systemcapability requirement l ; an d (d) 0 otherwise.

In the CPSKED column generation program, event requirement i np ut s m ay

be specified by ei ther ship- type or s hi p h ul l number. When a scheduler knows a

prwrt that a ship must par ti cipa te in a certain event, th e requirement should be

input by hull number. The column generator then forces all columns for that

hull number to contain th e event. Additionally, if a type requirement demands

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n units and all n units are specified by hull number, then only those units will

contain th e event in their ship-event network, i.e., only those units will

be considered for sa tisfying the event /ship- type requirement. Thus column

reductions will occur if all units for a specific type requirement are specified by

hull number. This is equivalent to fixing assignments in th e schedule. Ev en ts i n

progress at th e beginning of th e planning period should be fixed in thi s manner.

Also, any requirements that must be satisfied by a particular unit should be fixed

to ensure th e desired results an d to reduce th e size of th e problem.

The CPSKED column generator allows ship-type subst itutions to be

specified, at a cost, for each type/ requirement. If there are n of th e required

ship-type an d m of th e substitution ship-type, then there will be n + m

candidates available to satisfy th e requirement, an d a consequent ia l inc rease in

th e number of columns. Allowable substitutions should be used sparingly and

only where t ac ti ca lly feasible, e.g., a car ri er would never substitute for a. frigate

an d a f riga te would probably never substitute for a c ru ise r. Substitution strategy

may have a dramatic effect on the number of feasible columns generated.

B. COLUMN REDUCTION

T he nu mbe r of columns produced by th e method described above is much

less than th e 21 - 1 combinations which would be pr oduced by a naive generator.

Nevertheless, the number o f columns ca n grow very large. Many of these

columns ma y correspond to unit schedules that are unacceptable because of

excessive cost. Excessive cost corresponds to severe over-employment of th e unit

an d is counterproductive to the maintenance of high fleet readiness.

After each schedule column is generated, a cost for that column is

computed. The cost represents a measure of th e deviation from th e ideal

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ind iv idua l sh ip schedule. Fo r each of th e component costs, limits may be

established beyond which th e schedule is deemed completely unacceptable. If th e

cost of a column exceeds these limits, it is no t included in th e problem. The

CPSKED column generation program accepts th e following limiting parameters

by ship-type:

Maximum home fleet sea cost,Maximum deployment cost,Maximum time between deployment cost,Maximum column cost.

If an event requires a specific ship by hul l number, then that event becomes

mandatoryfor th e ship; th e cost l imits

ar eignored for

th ecolumn that contains

only mandatory events. Significant reductions in t he n um be r of columns sent to

th e solver are possible using this cost l imit ing approach.

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V. IMPLEMENTATION AND RESULTS

The CPSKED model has been implemented at th e Naval Postgraduate

SchooL Input data for testing this implementation has been extracted from th e

Atlantic Fleet projected annual schedu le for calendar year 1983. Th e testing

results indicate that high qua lity schedules a re p roduc ed efficiently. Schedule

quality IS based on comparisons of th e CPSKED schedule and the

CINCLANTFLT schedule. Model efficiency IS discussed I n terms of

computational experience based on four model runs.

A. COMPUTER PROGRAMS

The CPSKED model ha s been implemented on an IBM 3033 AP computer

sys tem under th e CMS operating system. CPSKED consists of thr ee pa rt s; t he

column generator, th e solver, and th e report writer.

1. Problem Generator

The CPSKED problem generator is written in ANSI standard

FORTRAN 77 an d compiled by IBM VS FORTRAN. The program uses a Ship

Data file an d an Event Data file for input. Th e program produces an

unformatted file which is read directly by th e solution driver; this file represents

th e CPSKED p rob le m in a compact data format suggested by Bausch [Ref. 3].

2. Problem Solver

Th e solver consists of a problem driver, XSCOVC, an d severalsubroutines. Th e X System solver rou tines are wri tt en in Level 66 FORTRAN

an d compiled by th e IBM FORTRAN IV H (Extended) compiler. Th e solver

employs many advanced featu res inc luding hypersparse data representation,

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complete constructive degeneracy resolution, basis'factorization, and elastic range

constraints. Th e X System may be tailored to specific models to form th e

computa tional foundat ion for ' specialized application packages. In this

development, th e CPSKED programs have not been integ ra ted with th e solver to

take full advantage of th e solver's capabilities. In th e CPSKED implementation,

th e solver generates a compact data file representing th e CPSKED solution; this

file is used as an input file to th e CPSKED report writer. Th e driver, XSCOVC,

also produces a condensed output repor t containing solution characteristics an d

computational statistics for th e problem solution.

3.Report WriterTh e C PS KE D r ep or t w ri te r is written in ANSI standard FORTRAN

77. The program uses th e Ship Data file, Event Data file, an d schedule. solution

file as inputs an d produces th e following reports:

Ship Statistic Report;Ship Schedules Report;Event Force AssignmeIl;t Report.

Samples of the input data files and the C PS KE D re po rt s are included m th e

Appendices.

B. TEST DATA

The model has been tested usmg actual data from th e Atlantic Fleet for

calendar year 1983. Model input consists of a ship data input file and a n e vent

data input file. Sample input data files are included as Appendices A and B.

Scheduling parameters , or goals, are set within th e column generation program.

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1. Ship Data

Th e Atlantic Fleet carrIe r an d surface combatant assets for th e

calendar year 1983 consisted of th e ships l is ted in Table 1.

TABLE 1

1983 Atlantic Fleet Combatants

Aircraft CarriersGuided Missile CruisersGuided Missile DestroyersDestroyersGuided Missile FrigatesFrigates

Total

Type

CVjCVNCGjCGNDD GDDFFG

FF

Number9

14

2317

1929

111

Non-operational periods, overhaul etc., an d historical data for these

assets a re kn own an d included in th e ship input data file. Th e requirement to

select exactly. one schedule for each ships results in 111 schedule selection

constraints.

2. Event Data

All extended operations and major exercises involving surface

combatant units were extracted from t he CINCLANTFLT annual schedule

resulting in th e event lis t displayed in Table 2.

A primary event is composed of a collection of sub-events; each of

these su b events corresponds to an employment term (EMPTERM) used in the

Atlantic Fleet Schedule. Each sub-event is designated as deployed time, homefleet s ea t im e, or home fleet inport time. Th e primary event, MED 2-83, is used

as an example, see Table 3.

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TABLE 2.

1983 Primary Event List

Extended OperationsMED 1-8310 1-83ME F 1-83ME F 2-83SNFL 1-8310 2-83MED 2-83ME F 3-83SNFL 2-83UNITASME F 4-83MED 1-84

Major ExercisesCOMPTUEX 2-83SOLID SHIELD 83OCEAN SAFARI 83COMPTUEX 3-83COMPTUEX 4-83COMPTUEX 1-84

(listed in order of event start time)

TABLE 3.

MED 2-83 Sub-events

Primary event:

Sub-events:

EMPTERM

MED 2-83

EXER (Readex 1-83)POMEN R (Transit)OPCONEN R (Transit)LVUPK (Stand down)

START

069

069093123134316326

END355

092122133315325355

CODE

SI

D

DD

I

codes: D - Deployed time,I - Home fleet in port time,

S - Home fleet sea time.

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A primary event requIres a specific force composi tion, with possible

allowance for substitution of assets. These requirements result in th e

event/ship-type c on st ra in ts . Typ ic al requirements, based on th e MED 2-83

example, ar e l isted in Table 4.

TABLE 4.

MED 2-83 Ship-Type Requirements

Type Hull Substitution NumberCV/CVN 69 none 1CG/CGN any DDG, a = 0.7 2DD G any DD, a = 0.8 2DD any none 2FFG an y FF , a = 0.7 3FF any none 3

Force weapon system c apa bility re quir em en ts are ba sed on current

.requirements for forces deploying to the Mediterranean, Middle E as t, an d I ndi an

Ocean. Typical requirements using th e MED 2-83 example a re l is ted in Table 5.

TABLE 5.

MED 2-83 Capability Requirements

System Number

AAW Missile (SM-l/ER) 2AAW Missile (SM-l/MR) 4AA W Radar (SPS-48) 3Data Link (NTDS) 4Passive Sonar (TASS/TACTAS) 3AS W Helicopter (LAMPS) 3Guns (5in/54) 4

Th e 1983 primary events result in a total of 73 event/ship-type

constraints. Force weapon system capabil ity requirements resul t in 44 add it iona l

constraints.

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3. Scheduling Parameters

The parameters li sted in Table 6 represent the scheduling policy goals

an d cost limits used in th e model runs. These parameters may be modified in th e

problem generator.

TABLE 6.

Scheduling Parameters

CV/CVN CG/CGN DD G DD FF G FFT 1 360 360 360 360 360 360

1" 2 .33 .33 .33 .33 .33 .331"3 .33 .33 .33 .33 .33 .33Ma x C 1 120 120 120 120 120 120.Ma x C 2 180 180 120 120 90 90Max C 3 90 90 60 60 45 45Ma x C j 300 180 150 150 120 120

C. SCHEDULE QUALITY

CPSKED c ap tu re s t he essence of CINCLANTFLT scheduling policy an d

provides an optimum schedule with respect to that policy. Th e objective costs,

including penalty costs, indicate th e overall quali ty of a schedule, e.g. a schedule

with a total objective value of zero is one that satisfies all requirements an d

exactly achieves all o f t he CINCLANTFLT policy goals.

Th e CIN C LA NT FLT a nn ua l schedule did no t contain projec ted ship

assignments for all projected primary events, e.g., UNITAS an d several exercises

were scheduled with ship assignments indicated "DTMD" for "t o be determined."

To place th e CINCLANTFLT schedule on an equal basis with CPSKED for

conducting comparisons, all known CINCLANTFLT ship assignments were fixed

and CPSKED was run to optimize th e remaining part of th e schedule. Th e

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TABLE 8.

CPSKED Results

Run 1 Ru n 2 Run 3 Run 4

CPSKED(NS) CPSKED(SU) CPSKED(S) CPSKED(CLF)

-Characteristics -Total Ships: 111 111 111 111

Operational Ships: 105 105 105 105

Total Events: 19 19 19 19

Allowed Subs: no yes yes yes

Cost Limits: yes no yes yes

-Objectives -cost: 395,200 427,000 446,700 1,472,500

-Penalties -

Schedule Selection: 0 0 0 0Event/Ship-type: 4,144,000 0 0 0

Weapon Capability: 10,000 7,000 9,000 11,000

Total: 4,549,200 434,000 455,700 1,483,500

-Problem Size-Rows: 228 228 228 228

Columns: 4,109 15,193 10,723 3,984Non-Zeros: 19,019 84,247 55,404 19,092

-R un Times-

-(in cpu seconds)-

Generator: 2.3 8.3 6.2 2.4Solver: 23.0 172.8 113.0 22.6

Reports: 0.6 0.7 0.7 0.7

Substitutions dramatically increase the problem size as indicated by a

comparison of runs 1 an d 3; however, the event /ship- type penal ties observed in

run 1 indicate that all requirements could not be satisfied without substitutions.

Commitments must be met, an d consequently, substitutions are necessary to

avoid event/ship-type penalties.

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Fixing schedule assignments that are known a priori will significantly

decrease th e problem size; however, fixing assignments ca n be expected to

increase th e costs an d ma y Increase the number of goal viola tion penal ties.

Compare runs 3 an d 4.

T he n um be r of event/ship-type constraints will i n f l u e n c ~th e number

of columns generated because more events are added to th e event list used to

generate th e columns. However, th e addition of weapon capabi l ity constra ints

only increases the number o f rows in th e problem.

Th e inclusion of cost limits in th e problem generator results in a

problem size reduction of approximately 30% with little degradation in th e cost

objective, compare runs 2 an d 3.

3. Execution Times

Total execution time for model runs consists of generation time,

solution time, and reporting time. To effectively employ CPSKED as a decision

support system requires rapid execution. Generation an d r ep or ti ng t im e are

relatively insignificant when co mp ared to solution time. Solution time is

influenced by th e problem size, problem penalties, and th e techniques employed

by the solver. Th e solut ion t imes observed in this study compare very favorably

with solution t imes for other large-scale set covering problems. [Refs. 3,4, an d 11J

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VI. CONCLUSIONS AND RECOMMENDATIONS

This study has demonstrated that optimization techniques can produce high

quality annual fleet employment schedules efficiently. Response t imes are short

enough to permit using this model in an interactive schedule planning system.

Ref inements in th e implementation of this model ca n f ur th er reduce solution

times.

Th e CIN CLANTFL T versus CPSKED schedule comparisons indicate there

is room for improving fleet employment schedules. Optimization models similar

to C PS KE D c an become powerful management tools for developing, refining, an d

maintaining employment schedules.

An optimization model provides a means for considering "al l" alternatives

to determine th e "best" schedule subject to the constraints supplied to the model.

This schedule ma y then be used as a reference for comparing alternate schedules

that ma y include additional c ri te ri a not evident in th e initial model run. Because

of the relat ively fas t response t imes , this process may be conducted iteratively

until a final acceptable annual schedule is developed. Th e optimization model

ensures that costs are minimized. Th e scheduler, or decision maker, must decide

whether th e add it iona l c ri te ri a a re justifiable in terms of th e resulting increased

costs. Thus, the model provides th e decision maker with th e capability -of

producing high quali ty optimum schedules that satisfy, or at least consider, all

scheduling criteria.

In its present state of development, th e CPSKED implementat ion is not an

end-user product. It does not possess a user-friendly front end and has no t been

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fully i nt eg ra te d w it h th e solver. Input data requirements are extensive an d

presently require fixed formatted files. An end-user implementation should

include an interactive front end for genera ting event requirement input . Th e

fro nt-en d s ho uld in corpo rate " can ne d" requirements that ca n be edited for

recurring events. Th e model should also have access to a data base for extracting

an d updat ing the ship i np ut d at a. Integration of th e problem generator and the

solver can reduce file handling and exploit more of the X System's capabilities to

reduce overall execution time.

Th e model development i n t hi s study has focused on scheduling combatants

to primary events. Th e model ma y be applied to other pr imary event scheduling

problems, e.g., amphibious forces, service forces, etc., by changing t he i np ut data

files an d scheduling parameters ..

Navy doctrine states that "The optimized peacetime employment schedule

that ha s as i ts object ive maximizing combat readiness should always be th e goal

an d guide." C PS KE D, o r a similar optimizing decision support system, can, an d

should, be used to assist schedulers in achieving that objective.

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APPENDIX A

S A M PL E E VE N T INPUT DATA

Card CardType Columns Data Description

All 1 Card typeM = Major event cardS = Sub-event cardH = Hull requirement cardT = Ship-type requirement cardW = Weapon system requirement cardN = Hull specification card

M 2-4Event number

5-26 Event name27-30 Julian start date31-34 Julian completion date

S 2 Event codeE = Major employmentC = Concurrent employment

3 Status codeS = Home fleet at-sea operationsD = Deployed operations

4-12 Employment t erm (EMPTERM)13-28 Location term .29-43 Supplemental information

44-63 Remarks64-67 Julian start date68-71 Julian completion date

H 2-14 Six 2 digit codes indicating th e number of shipsof t ypes 1 thru 6 required by hull number.Fo r each non-zero field an N card is required.

T 2-13 Six 2 digit codes indicating th e number of shipsof t ypes 1 thru 6 required by ship-type.Does no t include ships required on H card.

W 2-19 Up to nine 2 digit codes indicating th e number o fweapon systems of ty pes 1 thru 9 required.

N 2-3 2 digit code indicating ship type

4-39 Up to nine 4 d ig it ship hul l numbers , t he numberof fields used must correspond with thenumber indicated on th e H card for th ecorresponding ship type.

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ship types

weapon systems

l = C V / C V N

2 = CG/CGN3 = DD G4 = DD5 = FFG6 = FF

1 = AA W missile systems SMI-ER2 = AA W missile systems SMI-MR3 = AA W Radar system SPS-484 = Data link system NTDS Link- l l5 = Passive sonar system TASS/TACTAS6 = Helo capab il it y LAMPS7 = 5"/54 Gu n system

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F I L E : EVENTLG DATA A l

31663365

31073303"

31403365

30953112

31403170317131803 1 7 1 3 3 t : 53161335533553365

310731363137314631373273314731633264327332743303

31193131

30693092309331223123313331233325313333153316332533263355

31483157315831683180318831893198

31523168

31213151315233443152334433453365

316631953196320531963355320633453346335533563365

MED 2-83

MEF 3-83

SOLI 0 SHI H O 83

SNFL 2-83

UNITED EFFORTOCEAN SAFARI 83BALTOPS 83

UNIT AS - WATC

MEF 4 - 8 3

C INCUSNAVElR

COMIOEASTFQR

STANAVFORlAfIT

COMIDEASTFOR

3095311202-83

3152316803-83

31523365

COMSOLANT

SESREAOEX C A ~ I B B E A NSEA 01-83SI: POMSE ENR ROTASCDOEPLOYS f CPCONSE ENR C O ~ U SSE LVUPKHOI0COOOOOOOOT CO C 202 0203 03W C2 C'I03 0'103 0304X000027383557NOI0069MCOSCOMPTlJ: X 2-83Si:SCOMPTUEXH00 00000000 00TCOCZ02020303M009MEF 3-S3SE POMSE ENP. ROTASCDOEPLOYSE CPCCNSE EN? C01USS f L VUFKHCO CCOOOOOOOOTeO COOOOI000 1WOO COOOOI000 lJ OX00004S006800MOI0S0LID S H I ~ L r83 31193131S E:S EXO::P. ATLANTIC CCEANH 00 OCOOOOOO 00TOlCZ02020202X000027383648XOO.C000006859MOllS/IFL .2-83SE P:JMSE ENRSCDOEPLOYS : OPCON~ E ENR C Q ~ U SH CO CCOCOOOOOO" '00C(01000000XOOOOOO 393738M012CCEAN S A ! = ~ R I83 31483198S J:SE XE=R A TLANTIC C C E ~ NS ES EXER NOHH ATUr-ITICSESEXER NOHH ATL,ANTICSESENR C O ~ U SHOI 0000000000T CO (201 0202 00X000027383647NOI0067M013COMPTUEX 3-83SESCOMPTUEXHOO 00000000 00TCOC202C20303MOI4UN::TASSE POMSE OPCONSCDDEPLOYSE LVUPKH CO COOOCOOOOOTOO COOI0100 01XOOOC49396948" ' 0 1 5 ~ : = F4-83SE POMSE ENR ROTASCDDEPLOYSE OPCONSE ENR C O ~ U SSE L VUPKHCOCOOOOOOOOOTOOOOOOOI0001

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FILE: eVENTLG DATA Al

\01000000010001 )0XOO CC48C068 00M 0 1 6 C O M P T U ~ X4-83SESCOMPTUEXHCOCOOOCOOOOOT000202020303MOl7 10 1-84SE' POMSE ENR ROTASCDDEPLOYSE OPCONH 00 0000 0000 00TCO 01 01 0000 01WOl C20002020200XOOC0003 769 00MOl8MED 1-84SESREADEXSE POMS E ENFl ROTASCDOEPLOYS = OPCONHOI0000000000TCOC202C20303\oI020403040303J4XOO 0027383547XOOOC00006900NOI0C62MC19COMPiUEX 1 - 8 45ES CO"1PTUEXHCOCCOOCOOOOOTOO 0202020303

322132380 4 - 8 3

32413365

CINCPACFLT

324433650 2 - 8 3

CINCUSNAVEUR

3334334901-84

65

10 1 - 8 4

MED 1-84

32213238

32413270327132813271336532823365

3244326332043293329433033294336533043365

33343349


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APPENDIX B

S AM P LE S HI P I N P U T DATA

Card Data DescriptionColumn1-2 Ship-type code (1 thru 6)3-6 Ship-type designation (CV /CVN, CG/CGN, DDG, DD, FFG, FF)7-10 Ship hull number11-33 Ship name34-37 Overhaul or precom s ta r t date (current planning period)38-41 Overhaul or precom completion date42-45 Non operational start date (except overhaul)45-48 Non operational completion period

49-52 Total days since last overhaul or commissioning thruth e start of the current planning period.

53-56 Total deployed days since last overhaul thru thestart of the current planning period.

57-60 Total home fleet opera tional days since las t overhaulthru start of current planning period.

61-64 Total home fleet a t-sea days since las t overhaulthru start of current planning period.

65-68 Date last deployment completedor last da y before planning period is ship is deployedor overhaul completion dateor commissioning date.

69-78 weapon system indicators l=installed, O=not installed.ship types 1 = CV/CVN

2

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