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NPS ARCHIVE1967MCCORVEY, D


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UNIVERSITY OF CALIFORNIALos Angeles

Mathematical Modeling of FacilityMaintenance Planning

A thesis submitted in partial satisfaction of therequirements for the degree Master of Science

in Engineering

by

Donald Laney McCorvey, Jr.

Committee in chargeProfessor J. Morley English, ChairmanProfessor Moshe F. RubinsteinProfessor Robert B. Andrews

1967

DUDLEYKNOX U6RAKVS^TGRADUATE SCHOOL.MONTOH* ';#.


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C


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LIBRARYNAVAL POSTGRADUATE SCHOOLMONTEREY, CALIF. 93940

University of California, Los Angeles

1967


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TABLE OF CONTENTS

ACKNOWLEDGEMENTSABSTRACT OF THE DISSERTATIONCHAPTER

I INTRODUCTION1.1 Objectives .

II

1.2 Outline of the Facility MaintenancePlanning Problem.1 . The Approach ....1.4 Related Work Done by Others1.5 Order of Presentation.FACILITIES MAINTENANCE PLANNINGDEFINITIONS SYSTEM ORIENTATION ANDA COMPARISON OF EXPECTED MISSIONAND FACILITY LIFE ....2.1 Terminology ....2.2 A Systems Orientation for FacilitiesMaintenance ....2.3 Expected Mission and FacilitiesLire. . . . . .

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ChapterIII

IV

VI

DEVELOPMENT OF A VALUE SYSTEM FORFACILITIES MAINTENANCE PLANNING3.1 The General Value System DesignProcess ....3.2 Facilities Maintenance PlanningValue SystemOPTIMAL VALUE SEEKING MATHEMATICALTECHNIQUES FOR CAPITAL BUDGETINGPROBLEMS .....4.1 Linear Programming Basic Formula4.2 Integer Programming .4.3 Dynamic ProgrammingAPPLICATION OF DYNAMIC PROGRAMMINGTO FACILITIES MAINTENANCE PLANNING5.1 Transformation of the Value SystemObjective Function to a DynamicProgramming Recursion Equation5.2 Practical Application of theModel ....5.3 Computer Program5.4 Results of the ComputerEvaluationsCONCLUSIONS AND RECOMMENDATIONSFOR FURTHER RESEARCH6.1 Conclusions6.2 Recommended Future Research

BIBLIOGRAPHYAPPENDIX

A PRINCIPAL COMPUTER PROGRAM TERMSB PROGRAM LISTING "REPAIR PROJECTSALLOCATION PLAN

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ACKNOWLEDGMENTS

The author wishes to thank the members of his com-mittee, Professor Robert B. Andrews, Professor Moshe F.Rubinstein and Professor J. Morley English. Particularappreciation is expressed to Professor J. Morley English,chairman, for his guidance and encouragement.

The support provided by the United States Navy PostGraduate Program for Civil Engineer Corps Officers isgratefully acknowledged.

Finally the author wishes to thank his wife, Helen,for her encouragement and understanding during thislast year of graduate study, and for the numerous draftsand the final preparation of the manuscript.


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ABSTRACT OF THE THESIS

Mathematical Modeling of FacilityMaintenance Planning

byDonald Laney McCorvey, Jr.

Master of Science in EngineeringUniversity of California, Los Angeles, 1967

Professor J. Morley English, Chairman

The planning of maintenance for large facilitycomplexes as found in the military and in large corpora-tions requires a rational decision making process forefficiently allocating resources for maintenance. Engin-eering economics has provided a basis for choosing be-tween competing projects where all values are reducibleto economic values. When the number of alternativeprojects becomes large, the search for optimal combina-tions of projects becomes difficult. Decisions mustalso consider value parameters, intangibles, not repre-sented in economic evaluation. It is the purpose ofthis thesis investigation to contribute to the develop-ment of a mathematical model for planning the maintenanceof large facility complexes.


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In order to accomplish the foregoing purpose, therelation of facility maintenance planning to the largersystem it serves was analyzed. A general planning modelwas developed, based upon a value system design process.The general planning model was found to require extensiveadditional research with respect to the definition andanalysis of values in facilities maintenance other thanthose having economic interpretations. A simplifiedplanning model based on economic value only, is defined.Mathematical techniques including linear, integer anddynamic programming used in capital budgeting were in-vestigated for applicability to the planning problem.

A computer solution to the planning model objectivefunction employing dynamic programming was evaluated.The dynamic programming model was found to be tractablefor large sets of projects when a decomposition techni-que was employed, but impractical because of the lackof constraints on the solution, making it incompatiblewith current financial management practices.


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CHAPTER I

INTRODUCTION

I.I Objectives

The overall objective of this thesis is to developa mathematical model for planning the maintenance offacilities for organizations operating large scale faci-lity complexes. Of special concern is the military basesystem. The specific objectives are: to define theproblem of planning facility maintenance for large scalefacility systems and its relationships with the object-ives of the overall system; to develop a planning sys-tem model by applying systems engineering value designmethodology; to report on mathematical techniques usedfor analyzing capital investments showing their relationto the facility problem; and to develop and brieflyevaluate an algorithmic solution to a simplified main-tenance planning model.1.2. Outline of the Facility Maintenance Planning Problem

The management of facilities maintenance may bedefined to have two principle elements. First, it mustdecide what must be done , and second how the tasks canbe accomplished in the most efficient manner. The latterproblem is one involving: organization of the work force,planning work schedules for efficient production, and


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supervising the performance of the work. This task isextremely important and is generally treated under suchcategories as production planning and control. Greatimprovements have been made in this field aided by moreeffective building products and production equipment.

The task of deciding what must be done remains inthe domain of the top level decision maker, aided by aplanning organization. He is faced with the task ofdeciding specifically what must be maintained, and towhat extent it is to be maintained. The simple answeris to maintain everything in a "like new" condition.Unfortunately, the resources available seldom permitsuch decisions.

The simple solution Of maintaining everything ina new condition would not be seriously in error, if theoriginal requirements for the facilities remained con-stant (also assuming the new condition was the minimumacceptable condition). The real world condition is,However, one of constant change. New requirements forfacilities are generated and old ones eliminated. Itis essential that maintenance of a facility be accom-plished to the extent that the facility will continueto be required.

Another form of the problem is deciding which ofseveral deserving maintenance tasks will be performedfirst, in a limited budget where each is equivalent


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in cost. The question of relative values has now beenintroduced. The maintenance planner must decide whichproject, if performed, will add the greatest benefitto the overall system which the set of facilities serves.

These decisions are difficult to make and oftensufficient information is not at hand to properly judgebetween competing maintenance projects. The same pro-blem faces the maintenance planner whether he is respon-sible for a single facility, a military base or the totalsystem of military bases.

There is a need for a maintenance planning systemwhich permits the pi anner at each echelon to contributethe information he is best qualified to provide, con-sidering his vantage point. The system should permitall of the values pertinent to good resource investmentdecisions to be expressed; it should permit the main-tenance planner to apply mathematical techniques to searchfor the optimal plan for investing the limited resourcesavailable to the set of maintenance requirements; andfinally the planning system's introduction should begradual and compatible with the existing method of solvingthe planning problem.1.3 The Approach

A mathematical model for planning the maintenanceof facilities should consist of these principal parts:


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1. Definition of the facilities maintenance problemand its relation to the overall system it serves.

2. Development of a value system to be used to judgethe merits of all proposed allocation plans.

3. Selection of an algorithmic technique to searchthe set of feasible solutions for optimal solu-tions.

Facilities maintenance planning has been definedas a management subsystem serving the total set of mis-sion subsystems oriented toward accomplishment of theoverall system objectives. The national defense systemhas been used as the specific case under consideration.A general value system is developed which has sufficientdimension to include all the pertinent value laden para-meters of the general planning problem.

Because of the lack of practical measures of valuesother than economic, a simplified value system basedupon the economic theory of value is then derived whichmay be suitable as an initial step in applying the moregeneral value system. Finally, a dynamic programmingformulation is presented which may be used to search foroptimal solutions to the resource allocation problem.A computer program was developed to evaluate its feasi-bility.

1.4 Related Work Done by Others


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While no specific works were found in the litera-ture of facility maintenance planning, the followingworks provide the background upon which the developmentsin this thesis are based. The general field of engin-eering economics has been the source of decision rulesapplied in selecting mutually exclusive plans where econo-mic measures dominate. Grant and Ireson (9 ) provideone of the basic works in general engineering economics,while Barish (1 ) and Morris (15) are more current worksattempting to introduce probabilistic considerationsinto economic decisions.

The economic theories of decision making have beenlimited in their ability to measure all the pertinentvalue parameters in a decision making situation; and ithas been necessary to develop broader measures of valueand to find means to compare dissimilar value measures.Systems engineering has been a source of value systemdesign. Hall (10) presents a treatment of value mea-surement including both the economic and the psycholo-gical theories of value. Fields (7) and Fox (8) pro-vide treatments on cost effectiveness. A general systemdesign value model is developed by Lifson. His workis the basis for the general planning model proposedin Chapter 111. The methodology of systems engineeringdesign has been employed in this thesis to structurethe maintenance planning problem, because it is systems


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oriented and requires a value model for comparing solu-tions analogous to the comparison of alternative designconcepts

Capital budgeting treats the problem of choosingbetween multiple courses of action to provide the great-est economic rewards for the allocation of resources.Models have been developed based upon the economic theoryof value in the fields of securities investments (13),capital improvement project selection (14), and equip-ment replacement (20). Capital budgeting problems areresolved to mathematical models requiring optimizationof an objective function subject to constraint. Themethods used in capital budgeting have served as aguide to the selection of an algorithm for resolvingthe objective function developed in Chapter III.

The mathematical methods of linear, integer anddynamic programming have been developed in capital bud-geting models. Linear programming advanced by Dantzig(5) was applied to the capital budgeting problem byWeingartner (22). Dynamic programming advanced firstby Bellman was applied to the capital budgeting problemby Weingartner (21) and Cord (4).1.5 Order of Presentation

Chapter I provides an introduction to the problembackground, relative to the general development of this


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thesisChapter II provides basic definitions to be used

and defines further the relation of facility maintenanceto the larger system in which it is imbedded. The mis-sion versus facility life relation is also treated.

Chapter III provides a development of a generalvalue system design process and its application to themaintenance planning problem. A simplified planningmodel based upon economic theory only is also developed.

Chapter IV reviews the mathematical techniques oflinear, integer and dynamic programming as applied tocapital budgeting problems.

Chapter V presents the development of a dynamicprogramming algorithm for solution of the objectivefunction of the economic planning model developed inChapter III and reports the results of computer runsemploying the algorithm.

Chapter VI presents conclusions.


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CHAPTER IIFACILITIES MAINTENANCE PLANNINGDEFINITIONS , SYSTEM

ORIENTATION AND A COMPARISON OFEXPECTED MISSION AND FACILITY LIFE

This thesis is concerned with the planning of faci-lities which are elements of large complexes of facili-ties designed to satisfy dynamic sets of mission require-ments. Of special interest is the problem of planningthe maintenance for the set of military bases which forma part of our national defense system. While the emphasiswill be placed upon the military problem, the presentationis applicable to other organizations, especially govern-mental, which operate facility maintenance programs in-dependent of programs for capital improvements to theirrespective physical plants.

In this chapter terms to be used will be defined.A system orientation of the facilities planning problemwill be presented, and the relation between mission andfacility life, as it affects the planning problem willbe discussed. The purpose of this chapter is to pre-pare for the development of a mathematical model of avalue oriented planning system to be presented in Chap-ter III, and the investigation of techniques for opti-mizing the objective function to be derived as presentedin Chapters IV and V.


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2.1 TerminologyFacility

A facility is a structure or ground structure in-cluding all of the attachments and equipment that serve tocreate a desired environmental state for the accomplish-ment of a mission or set of missions. It excludes equip-ment that can be detached or removed from the facilitywhich is productive in nature, e.g., a machine tool maybe detached from a building, a locomotive may be removedfrom a section of railroad track.System

A system is a set of objects with relationshipsbetween the objects and their attributes . Objects aresimply the parts or components of a system. Attributesare properties of objects. For example, springs exhibitspring tension and displacement. Relationships tie thesystem together.Subsystems

A system which is an element or object within a largersystem is called a subsystem. This concept gives riseto the hierarchial order of systems, wherein all subsystemsmay be defined as being an object in successively largersystems. The largest system is defined as the universe.New Construction

The provision of a facility or expansion of an exist-ing facility for the purpose of meeting the requirements


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of a new mission, or expanded mission or set of missions.It includes the rebuilding of a facility for a new mission.Missions and Objectives

A general statement of need is a problem situationfor which a system is to be designed and operated. Thestatement of missions and objectives is the basis for thedesign of a value system for the evaluation of possiblesolutions to the problem.Maintenance

Maintenance is defined as all work necessary to keepa facility in an operable condition for the satisfactoryperformance of its assigned missions. Maintenance isdefined to include two subdivisions:

1. Routine maintenanceThe frequent or continuous work performed ona facility to keep it at a satisfactory opera-tional level of condition. Routine has the con-notation of being sets of small independent tasksperformed repeatedly which do not require signi-ficant replacement of component parts. (In Chap-ter III this definition will be modified to in-clude operational expenses incurred because ofthe deterioration of the facility.)

2. RepairRepair is the infrequent work performed on afacility to return a facility to a satisfactory


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operational level of condition. This work in-cludes the replacement of constituent parts ofthe facility. It includes no expansion of thecapacity of a facility and is not performed forthe purpose of rebuilding an unused and deteriora-ted facility for the satisfaction of a new mission,

2.2 A Systems Orientation For Facilities Maintenance

The definitions of systems and subsystems permit anysystem to be defined as a subsystem of some larger, encom-passing system. A key consideration of systems designis the optimization of the objectives of the system. Caremust be exercised in optimization efforts, for the rela-tions between the system under consideration and adjacentsubsystems in its encompassing system, may result in asuboptimization in the larger system. It is importantto know how the system under consideration affects itsenvironment. Hitch and McKean ( 12) , advise that theeffect of subsystem optimization on at least one levelhigher in the hierarchy of systems should be examined toinsure that the subsystem optimization is desirable. Thehierarchy of systems in which the maintenance planning formilitary facilities is imbedded will be reviewed.

The largest system which can reasonably be visualizedas defining the objectives of military facilities main-tenance is the national government. The policies esta-


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blished at this level state the need for the next lowersystem, the national defense system, represented by theDepartment of Defense. It is within the defense systemthat the objectives of the facilities maintenance sub-system are defined. A recent address by the AssistantSecretary of Defense (Comptroller), Robert N. Anthony,outlined the current set of systems employed by the De-partment of Defense to define its missions and objectivesThe largest set of subsystems are called Major Programs .Figure 1 lists the set of major subsystems.

FIGURE 1

MAJOR DEFENSE PROGRAMS

1 Strategic Forces2. General Purpose Forces3. Specialized Activities (includes MAP )4. Airlift and Sealift5. Guard and Reserve Forces6. Research and Development7. Logistics8. Personnel Support9. Administration

The major problems are subdivided into Program Ele-ments . For example, B-52 Squadrons and Base Operations(Offensive) are two elements under the major programstrategic forces. Both of the above subdivisions may be


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described as mission oriented, that is, they defineactivities which must be performed to satisy the principleobjectives of the defense system--to counter possiblehostile action against the nation.

Figure 2 is a list of Functional Categories which aredefined as the elements of the new program elements definedabove. At this level a new systems orientation is intro-ducedthe management system. Each of the functionalcategories is common to a degree to each of the programelements (mission oriented). This permits or requiresthat a management system be designed which will providethe services of the specific function to all of the majorprograms and program elements. The system of functionalCategories may be considered a shadow system designed tooptimize the performance of each functional category, ac-cording to a set of management objectives under a set ofconstraints defined by the major programs and programelements

Figure 3 illustrates the relationship between themajor programs and the two functional categories which arerelevent to this thesis: Operations and Maintenance ofUtilities; and Maintenance of Real Property Facilities.A maintenance planning system will be developed to opti-mize the objectives of the management system of the twofunctional categories within the constraints imposed bythe operational system. The two functional categories


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FUNCTIONAL CATEGORIESOF THE MILITARY MISSIONS SYSTEM

1. Mission Operations2. Supply Operations3. Maintenance of Materiel4. Modernization5. Transportation6 Comtnuni cat ions7. Medical Operations8. Food Service9. Personnel Housing Operations10. Overseas Dependent Education11. Other Personnel Support12. Base Services13. Operation and Maintenance of Utilities14. Maintenance of Real Property Facilities15. Minor Construction16. Administration

FIGURE 2


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Major Programs (Mission Systems)1 2 3 . . j n

Facilities' Operation and MSubsystem aintenance

CO &Q) 0)H 4->U wO >,60 CO94-> 4Jcd CO CDBr-l (1)Ctf 60C ctjO CoC

m

INTERRELATIONS BETWEEN THE MILITARYMISSIONS SYSTEMS, MAJOR PROGRAMS AND MANAGEMENT

SYSTEM, FUNCTIONAL CATEGORIES

Figure 3


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will be referred to jointly as, facilities maintenance.2.3 Expected Mission and Facilities Life

By limiting the problem to facilities maintenance,the planning of new facilities is placed beyond the scopeof this thesis. However, realistic maintenance planningmust reflect the dynamic character of the set of missionsand objectives of the defense system, and in this respect,the problem is similar to that of planning new construc-tion. Maintenance resources must not be applied to faci-lities which have no future mission assigned or foreseen.The more difficult case is to decide how much maintenanceis justified for a facility with a limited estimate ofmission life.

A problem inherent in facilities p lining is therequirement to make an initial investment in a facilitywhich can not be consumed by an originally assigned mis-sion. This is not to say that all facilities are subjectto the assignment of short duration missions, but thatsome facilities are, and optimal planning requires thatthe problem be considered. If successive missions couldbe defined by the mission oriented planning system, theproblem would be resolved. The life as a measure of thetotal requirement for a facility would be defined, andmaintenance plans prepared accordingly. In the absenceof long range mission projections which may be taken asdeterministic, the life of a facility must be taken to be


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a function of the expressed mission Life, and the char-acteristics of the facility which indicate that it maybe expected to serve future undefined missions.

Two methods of examining future missions probabilitiespresent themselves. The first is based upon an evaluationof individual facilities. For example, an office buildingis usually of such a general design that it can be expect-ed to be usable for its designed purpose for the durationof successive missions until its economic life is reached.

The extreme of this case is the special purposefacility designed for the performance of a single researchexperiment. Such a facility may be so specialized thatit has no future use. Both of these cases are simple toanalyze, as stated; but add to the first case a consider-ation such as remoteness of location, and to the second,an extension of mission life to include a series of experi-ments of unknown duration and the problems become complex.

The key consideration in both of the above cases isthe desired longevity of the maintenance work performed;not whether or not the facility should be maintained forsome currently defined mission. This first method, then,is an evaluation of each individual facility, attemptingto determine the probability that it will have someestimated life beyond the currently planned major programlife.

The second evaluation is of complete base complexes.


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We have recognized that when missions are defined, faci-lities are constructed to meet these requirements, andthat because of the dynamic nature of the defense sys-tem, requirements are often eliminated, leaving an excessof facilities. The total military base complex systemmust be analyzed to eliminate excess capacity. Regard-less of the value of an individual, facility, it may bedeclared excess as a part of an entire military base.

To evaluate the probability of the life of a faci-lity beyond the currently defined major program life,requires an evaluation of the individual facility forconvertibility to meet future mission requirements, andan evaluation of the total set of military base complexes.The estimation of facility life is an important elementin maintenance planning. In the following chapters itwill be assumed that a reasonable estimate can be made.

Footnotes1. Definition paraphrased from Hall, page 60.2. Address given October 24, 1966 before the fin-ancial Management Roundtable, a Department ofDefense Conference.3. MAP is the Military Assistance Program for foreigncountries


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CHAPTER IIIDEVELOPMENT OF A VALUE SYSTEM FORFACILITIES MAINTENANCE PLANNING

In Chapter II the relations between facilities,their maintenance, and the higher order systems in whichthey are imbedded as a subsystem were established. Itwas also established that management of the maintenanceof facilities is a subsystem of a total management sys-tem for functional elements of major programs orientedtoward accomplishment of missions. In this chapter aplanning concept for maintenance of a complete systemof facilities will be developed by application of valuesystem design methodology to the facilities problem.The value system design process will be described ini-tially, and then the steps of the process will be appliedto the problem.

The purpose of using the value system design processis to provide a well defined model to which refinementsmay be added as additional definitions of value in theproblem are developed through research. This presup-poses the hypothesis that management and planning fora facilities system is analogous to: establishment ofa value system; and by some process synthesizing andevaluating all feasible courses of action to select thatcourse of action which will produce the highest value


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to the system. Application of the general value systemplanning model is limited by the practical limits ondefining elements having value, referred to as intan-gibles, and the inability to mathematically solve theproblem even were all values completely defined. Assump-tions will be made which reduce the problem to a moretractable form.3.1 The General Value System Design Process

The design of a system, whether it be a piece ofhardware, such as an airplane, or a management process,must have some criteria by which the goodness of thedesign may be judged. Therefore, a value system designprocess is essential to systems engineering methodology.Hall (10) distinguishes two value theories: the economictheory, and the psychological theory. The economic theoryis the type of engineering economics presented by Grantand Ireson (9 }, Barish (1), and others. The psycho-logical theory is defined to contain such value measuringtechniques as the Von Neumann and Morgenstern UtilitiesTheory, and the Churchman and Ackoff (3 ) Order Scales,among others.

The combination of the two general theories is anobjective in systems engineering and operations researchvalue systems. Cost effectiveness is an example. Lifson(23) presents a general value system design process, amodified form of which will be presented here. The steps


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of the process are shown in Figure 4 , and will be treat-ed in the following paragraphs.Problem Formulation

To initiate the process, a general statement of theproblem must be formulated. In a product design process,an initial statement of the problem is provided by thecustomer, but in the case of designing a system withinan organization, higher level management must providethe initial statement of the problem.Estimate of Needs and Objectives! Nl

In this step specific needs, missions or objectiveswhich are to be fulfilled by the system are defined.English (6) describes design as an iterative processand indicates that even the initial statement of needsmay not be accurately stated, but will be improved asthe process evolves. N is the set of needs goals andobjectives which are to be accomplished by the system.Estimate of Resources JR

The resources pertinent and available to the systemfor fulfilling the needs N must be defined. Quantitativerepresentations are desirable. The statement of resourceswill be used as a set of constraints on the solutionof the system. R is the set of vectors which describeavailable resources.Estimate of the Environment

The system must exist in an environment which is


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FIGURE 4THE VALUE SYSTEM DESIGN PROCESS

Problem FormulationEstimate of theNeedsResourcesEnvironment

""

Design of valueSystem\f

SynthesizeAlternativeDesign conceptsand configurations

Includingy, design parameteru(y) , utility functionsPM, performance modelswj , weighting factorsO.F., the objective functionSR, the stopping rule


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some larger system. For the purposes of design, thestatement of environment may be limited to those elementswhich are pertinent to the system. For example, environ-mental elements may be present which have little or noeffect on the system. Such elements may be omitted.In the general case each element of the set of pertinentenvironments may consist of a vector of subordinateenvironmental elements. An assumption in the definitionof the environment is that the decisions made in thesystem are of an order of magnitude sufficiently smallthat the state of the environment is not affected.E is the set of pertinent environments, each of whichis a vector of environmental elements not significantlyaffected by the decisions within the system.Design Parameters jy }

A design parameter is a criterion which is consideredto have an influence on the degree to which the needs Njof the system are satisfied. An example of a designparameter in facilities may be the number of square feetof usable space in a building, y . is the set of designparameters which are selected to represent the system.Utility Functions

Utility functions are the relations which convertthe quantitative measures of a design parameter,into measures which express value to the satisfactionof the system needs. Schlaifer (l -) presents methods

yj


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of constructing utility functions. Utility functionsto be used in the value system are normalized utilities.That is, they represent the segments of the scale of autility function defined by the aspiration points andthe point of indifference, where the indifference pointis assigned the value of zero utiles and the aspirationpoints may be assigned the value of plus and minus one.The negative aspirations may not exist. Normalizedutilities have the properties of linear functions.Performance Models

Performance models are the relationships whichtransfer the characteristics or properties of a givendesign alternative into the quantitative measure of adesign parameter , jy .. For example a design parametermay be the floor loading capacity of a building. . .

Vf f M E e|.| sMwtUsMD iJ 3-1Where I =

S =

Wml

D, =

the set of moments of inertia ofbeams and columns in the structurethe set of moduli of elasticity forthe beams and columns in the struc-turethe set of beam and column lengthsin the structurethe set of possible wind conditionsthe set of possible seismic loadsthe set of dead loads on the struc-ture


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In this example the performance model is a structuralanalysis of the building, where wind and seismic loadsare given in the pertinent environmental vector, E^rDesign Concepts

A design concept is a possible solution to thesystem which is to be evaluated by the value system. Itis described by a set of vectors X, for the kth designconcept. f(y^) j,. is a set of explicit probabilisticeffectiveness estimates associated with a configuration,X^, of a design concept. Each f(y-j) ^ is a probabilitydensity function of design parameter y-, given that X^is implemented in the Lth environment. The probabilityof the occurrance of the environmental states is definedby the set E of discrete probabilities of occurancesof the Lth environmental state.The Utility Matrix

The elements of the value system may be visualizedas a three dimension array as shown in figure 5. Eachof the u-k .functions is defined to be a normalized utilityva lueProbability Matrix

The probability matrix for a given design configura-tion shows the probability functions associated withgiven design parameters and environmental conditions(figure 6). Each fj^jis the probability that for agiven design parameter, y., and a given state, L, a


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Environmental vectors and probabilities

COuCD

CD

%U


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Environmental Vecotrs and Probabilities

MATRIX OF PROBABILITY DENSITY FUNCTIONSDESIGN CONCEPT x^ FOR VALUESDESIGN PARAMETERS, y -kLGIVEN

Figure 6


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value, y, of the design parameter, Yj^L* m occur.Weighting Factors Wj

Weighting factors are a measure of the importanceattached to a given design parameter to the satisfactionof the set of needs, N. Weighting factors may be devel-oped using Churchman, Ackoff and Arnoff order scales (3).The Objective Function

The value of elements of alternative design conceptscan be evaluated by the parameters, yj, presented above.It is necessary to develop a relation which describesthe total value of the system. One method is to developone single measure of the system by operating on theelemental values of a design concept represented by thedesign parameter utilities. The general expression maybe stated as :

Objective Function = f(wj, u^, f^L ?]_) 3-2Where j = 1,2,3. . .m and L = 1,2,3. . . n

The value of the objective function of the kth designconcept is a function of the weighting factor, Wj, fordesign parameter, j, the normalized utility, u^, of de-sign parameter, j, for environmental state, L, the proba-bility, f-jkL, that the value, yj^L* f the design parameterj, will occur, given environmental state L, and the proba-bility that the environmental state L will occur for theset of all combinations as j takes on values from 1 to m


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and L takes on values from 1 to n. The functional re-lationship is not defined in the general case, but anadditive function which assumes linearity, permittedby the normalized utility functions is assumed as a sim-plication. Equation (3-2) may then be restated as:

m nO.F. = y X (wJUJLfjLP L> 3 " 3j=l 1=1

The above form of the objective function assumes thatall elements of the value system can be translated intoa single unit of measure. If this can not be done, amultidimensional objective function results. The costeffectiveness model is a two dimensional value systemused in systems design.Synthesis of Alternative Designs

The general value system has been established andmay be used to evaluate a set of competing design con-cepts to be synthesized in this step. Design of theperformance model usually must follow the synthesis ofalternative designs, because the translation of designproperties to the dimensions of the design parametersmay be unique for each design.Evaluation of Each Design and the Stopping Rule

Application of the value system to each design con-cept will result in an objective function value for eachdesign. A decision must then be made to accept or reject


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the design having the highest objective function value.If the decision to reject is made, a review of each ele-ment of the value system, as well as the set of designconcepts is in order. During the design process, addi-tional information may have been acquired which maychange one or perhaps all of the value system relations.After revisions have been made, the value system is ap-

iplied, repeatedly, until an acceptable design is selected.3.2 Facilities Maintenance Planning Value System

The steps of the value system design process willbe applied to facilities maintenance, to define the plan-ning problem, and to provide a model in which the psycho-logical theory of value may be applied when measuresof these values are developed. A simplified objectivefunction based upon the economic theory of value isdeveloped which will be used in Chapters IV and V in theapplication of techniques for finding optimal designconcepts, S^.3.21 Statement of Needs N

The primary objective of the facility maintenancesystem is the satisfaction of requirements for facilitiesestablished by the missions assigned by the major programsystem. Maintenance represents the work necessary tokeep a facility functioning, but a characteristic ofsome facility maintenance is its postponability. A


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facility may function for a period of time with no main-tenance, but a point is reached where work must be ac-complished. Inherent in this characteristic is thequestion of degree of mission satisfaction. Assume thatthere is a range of facility conditions which will satisfya mission requirement. At the highest level of condi-tion, an optimal value of mission satisfaction may bedefined and at the lowest level of condition missionperformance may be marginal, or the effects are notmeasurable quantitatively. An objective of facilitiesmaintenance must be to maintain a level of facility con-dition for all facilities in the system, which producesthe optimal value to the combined missions oriented system,and facilities maintenance system. Smith (is) uses theterm concrescence, meaning growing together, to definethe process where the conflicts of two intersecting valuesystems are resolved by adopting a single value systemencompassing both of the original systems. The principleof concrescences requires that the maintenance planningsystem have as an objective the continuous provision ofthe facilities optimally required by the mission orientedsystem. A second objective of the maintenance system isthat a minimum of resources be expended. The conflict mustbe resolved by defining negative values to non-attainmentof the optimal facility conditions. The cost of opera-ting a primary mission due to less than optimal mainten-


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ance may be taken as a cost of postponement of maintenance.3.22 Statement of Resources R

The set of resources at the disposal of the main-tenance planner include money, manpower, and material.Money represents the dominant resource under conditionsof stable national economy. The amount of money thatcan be made available is a decision made at higher levelswithin the defense system and should reflect the rela-tive need for facilities, compared to other elements ofthe functional categories. The constraints on money arepresented in the form of a fiscal budget.Manpower is defined as the "in house" capability toperform work. It consists of the military and civilianpersonnel who are assigned to facility maintenance tasks.Manpower planning is an additional tool in controllingthe expenditure of money and therefore must be regardedas a constrained resource in most cases. Man powerplanning serves two purposes: first, it fosters stabilityof employment, thus, serving national objectives abovethe national defense system; and second, it forms aready tool for managing, in the event that money failsas a planning tool for allocation of national resources.Material is a resource that is usually constrained bythe timing of its availability. When the economy isin a relatively stable state, material is unconstrained,but during a military emergency, this resource becomes


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heavily constrained as normal production processes con-vert to military production. A maintenance planningmodel should consider constraints on the above resources,3.23 Environments N

j

Two general statements of environmental statesmay apply to military facility maintenance planning:

1. four states--peace, preparing for war, war,post war, or

2. the set of all feasible combinations of missionsfor all combinations of facilities.

Both must be considered approximations of reality, butneither are practical. The first is indefinite, andthe second can not be defined. If it were, the solution

would be intractable.The assumption to be made in the elementary mathe-

matical model is that only a single environment existswhich is the best estimate available of the future.This assumption can be made for all of the characteristicsof the problem. The probabilistic representation ofsome elements such as life of requirement may be per-mitted and still retain a tractable solution in smallproblems. In making this assumption about a dynamicsystem, it is necessary to apply static analysis fre-quently or as often as new projects and resource condi-tions present themselves.3.24 Design Parameters X


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The set of design parameters must measure how wella maintenance plan achieves the two primary objectives:optimal condition of facilities for the set of assignedmissions, and minimum consumption of resources. Toreflect reality the design parameters must measure bothpsychological and economic values inherent in each de-sign concept. This has been and remains a principlelimitation to effective modelling of reality, for littlework has been done from a facilities viewpoint to esta-blish a relation between the two value systems. Theapproach has been to assign economic value to all recog-nizable quantities of value and to treat the remainingpsychological values as non quantifiable intangiblestreated subjectively along with the economic analysiswhen making decisions. The result often seems to be anarbitrary decision, when the decision maker, influencedby the intangible factors decides counter to the economicfacts. It would be more accurate to say, assuming thatthe decision maker is correct, that the economic analy-sis failed to reflect the reality of the case.

The development of a comprehensive set of pertinentdesign parameters for the complete evaluation of thefacilities plan is beyond the scope of this thesis.It will be suggested that the following are some of thevalue laden parameters which require definition andquantification for inclusion in the facility maintenance


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planning value system:1. Habitability of personnel support facilities.2. Esthetic qualities of interiors and exteriors

of facilities and adjacent landscapes.3. Conveniences created for personnel who operate

the mission, but have no contracted claim forconveniences improved or provided by highermaintenance of a facility.

4. Safety.5. Maintenance of facilities for possible future

missions.This list is far from exhaustive, but serves only toillustrate the nature of values which are not treatedby the economic value theory. Items 4 and 5 do haveeconomic interpretations, if values can be assigned tofuture requirements for the facilities. In item 4 theloss of a human life or the cost of crippling a human

3being, must be determined. English treated an analo-gous case concerning the seismic design of buildings.An economic value of an earthquake caused failure wastaken to be a function of the probability of failure ofthe building for conditions of load and failure causingload. The economic value, then, is the expected valuederived from the sum of the probabilities of failureand the associated cost of failure for each value of load.

For the mathematical model treated in Chapter V,


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a simplication will be made by resorting to economicvalue theory and the engineering economics approach.Traditional engineering economics as presented by Grantand Ireson (9 ) and Barish (1) use the parameters ofnet present worth of future cost streams, equivalentannual cost and internal rate of return as measures ofvalues in engineering projects. The three may be brieflydescribed, as follows: n

1. Net present worth, p i = / ci / i + \i=l

- V c 7TT-rr 3-4Where c$_ = the net annual cost for time period i,

r = a discount rate for the decreasedvalues of future money,tt? \ i_- the discount factor,

and n = the number of time periods duringthe life of the project.2. Equivalent annual cost, R

R " P (1+rP-l 3"3, Internal rate of return

The internal rate of return is found by solvingequation (3-4) for the value of r, given thatthe present worth is equal to zero.

If all of the attributes affecting the conditionof a facility can be translated into annual costs, oneof the above parameters can be used to measure the valueof a facility maintenance plan. For the purposes of


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this thesis, net present worth money saved, S, will beused for measuring the positive effects of investingresources in facilities maintenance. A second non addi-tive design parameter, present worth money spent, Q,will be used for measuring the negative value of inves-ting resources in facilities maintenance.Objective Function

The cost effectiveness type two dimensional object-ive function will be used. Effectiveness is defined tobe the net present worth money saved, S, and cost, Q,the net present worth of money spent to create the ef-fectiveness condition.

0. F. = f(S,Q) 3-6

3.25 Alternative Design Concepts X^The alternative courses of action in the planning

sense represent the set of different feasible combinationsfor allocating resources to the total facility system.It is postulated that the state of the system is to bemeasured against a set of optimal facility conditions.Several additional assumptions will now be made to pro-vide a total measure of system condition:

1. If the system missions remain constant, thereis some cost, which if allocated, will maintainthe system at its current state of conditionindefinitely. (This condition is not necessar-ily the optimal.)


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2. The constant condition cost can be divided intotwo general types of cost: Routine Maintenance,(Definition from Chapter II modified to include:all operational inconveniences which have aneconomic interpretation or which reflect theextent of variance from the optimal conditionsdefined at the beginning of the assumed constantcondition period), and repairs (Chapter II de-finition.)

3. That repair projects can be defined which esti-mate the routine maintenance costs of a facilityfor its expected mission life.

4. That routine maintenance costs will change whenwhen a facility is repaired, providing reducedroutine maintenance cost.

5. That the set of all projects for the repair ofthe total set of facilities represents the pos-sible future condition states of the facilitysystem as a function of the quantity of resourcesto be applied.

6. That the maintenance plan will only includerepair projects which influence, but do not con-trol, the management of the routine maintenancebudget. (No attempt will be made to manageroutine maintenance in this planning system.)


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The alternative design concepts X^ , courses of ac-tion, consist of the total possible combinations forallocating resources to the set of repair projects overthe total budget planning horizon for k = 1,2,3. . . o.3.26 Performance Models

First Design Parameter. S.

a. Sk = f(Rlt Clj , c^j, rt ) 3-7

b. sk =m LjlX cij nfep* - 2 ( Ci j } nfcp- + Ri n+n*

n

Where i = project identification number, i=l,2,3. . ,mk = the number of the design concept,k=l,2,3. . ,oR^= the cost to repair project i

c- j= the cost of routine maintenance ofproject i in the jth time periodbefore the repair is funded, j=l,2,3. . .n

n = the total number of budget periodsin the life of each project or somearbitrary limit in which all costsbeyond the limit are discountedback to the final included time period,n. (Permits an infinite future.)

c'j j=the cost of maintenance for the ithproject after the accomplishment ofthe repair project.

r^ = the rate of disutility for value


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/ + \ j = the time discount factor of presentr *- worth value for the ith project forvalues in the jth time period.L^= the time period in which the repairfor the ith project is made.

The net present worth return, S, for the courseof action X. consists of the sum of the returns fromthe set of repair projects, i, through m, where the returnfrom each project is the sum of the discounted futurecost stream for the life of the project, minus the totaldiscounted revised cost stream of the project if therepair were funded in year L=l.

The revised cost stream is the set of originalperiod costs before the repair is made, plus the costof the repair, plus the set of periodic costs whichoccur after the repair project is funded to the end ofthe facility life.

Figure 7a shows an example of a cost stream beforea repair has been made for a single project. Figure 7bshows the revised cost stream which will occur if theproject were funded immediately. Figure 7c shows a com-bination of Figures a and b for the case whece L is theyear that the project is funded.

The two arbitrary limits introduced into the pro-blem are the budget horizon, and an arbitrary time periodduring which costs are considered. The budget horizonbecomes a practical limit over which the maintenance


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a.

b.

c.

CO


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plan is to be valid. The effect of this horizon on theselection of projects is one of the elements investigatedin Chapter V. The arbitrary maximum life of a projectof n years presents difficulties where the actual lifeexceeds this period. The total cost stream, before therepair can be represented as follows:

ooc la l + c2a 2 + cj aj- - cnan+anV Cfa f

f=lik

Where the last expression on the right representsthe discounted sum of all future costs beyondthe time period discounted to time period nand then to the present, aj is the time dis-count factor for period jThe difficulty occurs in projecting revised cost streamsbeyond period n because the final term of a revised coststream depends upon the time of funding the repair.This limitation may be reduced by making n large toreduce the significance of errors.Discrete Budget Values

A final practical feature of the representation ofthe cost stream is the discrete representation of periodiccosts in lieu of a continuous fromulation. This permitsthe project planner to represent cost estimates in theformat of fiscal budgets and to express expected jumpsin the cost stream caused to large routine maintenancecosts that occur infrequently. The latter comment iscounter to the defined meaning of repair and routine


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maintenance previously stated. Therefore, the defini-tion of routine maintenance will be extended to includerepairs which do not exceed some prescribed level ofcost. All repairs exceeding this amount must be includedin the form of repair projects.3.26 Second Design Parameter. Q

The second design parameter, Q, present worth costof resources isn

Qk = I RiL TT7rTTL 3 - 9i=lWhere L is the year each repair Rj_ is funded.

3.27 Evaluation of Alternative Design Concepts (Coursesof Action)

The final step in the process is to evaluate eachfeasible course of action and select the one which hasthe most desirable objective function values. The twodimensional objective function has been defined as acost effectiveness model, because it relates an effect-iveness defined as a net present worth savings, S^to a present worth cost of resources, Q^ f The methodof evaluating the objective function is the subject ofChapters IV and V. The problem to this point has beendefined in a form analogous to capital budgeting models.In Chapter IV the characteristics of mathematical pro-gramming techniques for capital budgeting models will


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be reviewed to find a suitable model for evaluating thefacilities maintenance planning system objective function.In Chapter V a dynamic programming algorithm is developedfor the solution of the objective function and evalua-tions of its use pertinent to the practical applicationof the algorithm are described.

Footnotes1. Flow chart and symbology suggested by Dr. R.B . Andrews2. See Lifson for a description of normalized uti-lities.3. Unpublished paper "Economics of Structural Safetyin Seismic Design" by J. Morley English.


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CHAPTER IVOPTIMAL VALUE SEEKING MATHEMATICAL TECHNIQUES

FOR CAPITAL BUDGETING PROBLEMSIn Chapter III a value system for planning the main-

tenance of facilities was developed. The value systemestablishes the set of all possible courses of actionwhich must be evaluated by application of the objectivefunction, equation (3-6). In this chapter mathematicaltechniques which have been applied to other capitalinvestment problems are reviewed. The purpose of thisreview is to show the advantages and disadvantages ofthe several techniques based on the work done by others,and to show the reasons for the choice of a dynamicprogramming technique for solution of equation (3-6).The three general methods are linear programming, inte-ger programming and dynamic programming.4.1 Linear Programming Basic Formulation

Dantzig ( 5) has been the principle contributerto the extension of linear programming to a wide varietyof problems suitable for computer solutions. Includedis a large body on allocation models. Weinga rtner (22)has contributed extensively to its application to thecapital budgeting problem. The capital budgeting prob-lem may be described simply as finding a set of invest-ments (l g ) from a larger set I, I g I, such that the


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return from the allocation of a sum of money over a num-ber of time periods is maximized. The problem is con-strained to spend no more than the amount allocated toeach time period. This may be stated:

na. 0. F. = Maximum > ^jX. ^~^

j=ln

b. subject to "S c bj ^ Ct t = 1,2,3, . . ,Tj=l

c. * Xj * 1 j = 1,2,3, . . ,nbj = the net present value of the jth projectdiscounted to the presentct .= the outlay required for the jth projectin the ith time periodCt

= the maximum permissible outlay in thetth period

xj = the fraction of the jth project accepted

The objective function for equation (4-la) representsthe sum of the returns expected when the set of optimalfractions x of the total set of projects is found bysolution of the linear programming algorithm. The setof equations (4-lb) constrains the outlay in each ofthe budget periods t to the set of constant values Ct .When each project may be funded only once, equation(4-lc) constrains the solution to cases when the maximuminvestment on any project is the value of its maximumor total outlay. The basic linear programming model


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has these characteristics:1. It permits constraints on budgets to be set in

each of a number of budget periods.2. It permits the incremental funding of a project

over a set of budget periods, not always con-secutive.

3. Fractional projects may occur.4. The optimal time for initiating the funding of

a project is not considered.4.11 Additional Feasible Constraints

The limitation of the basic linear programmingformulation and of most traditional forms f multipleanalysis is the assumption of complete independencebetween projects (21), This assumption was also madefor the repair projects defined in Chapter III, Linearprogramming permits the consideration of three idealizedtypes of interdependencies between projects: mutualexclusion, contingency relations, and forced acceptance.Mutually Exclusive Projects

In equipment replacement problems, mutually exclusiveprojects may be illustrated by the following example.A new machine tool is to be purchased, where there areseveral competing models available. Each model has anassociated return value and a resource cost. The ac-ceptance of one model, excludes the acceptance of any


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of the other models available. In the context of faci-lity repairs, two or more alternative repair methodsmay serve to solve a facility deficiency. Each methodmay vs-ry in its investment cost, expected durability,or other value oriented characteristic. Acceptance ofone method excludes the others. The linear programmingconstraint which provides this condition may be stated:

S xj 1 4 - 2jeyWhere j is the set of mutually exclusive projects

Equation (4-2) requires that no more than one of theprojects in the mutually exclusive set of j be accepted.A limitation to the linear programming algorithm is thepossibility of accepting more than one fractional projectwhose sum is less than one.4.12 Contingent Projects

It may be desirable in the facility repair problemto divide a repair undertaking for a large facility intoseveral increments, each of which serves a useful pur-pose, independent of the undertaking of the others, butaccomplishment of certain minor projects should be fundedafter the major ones. This condition defines the minorprojects as projects whose funding is contingent uponthe prior or concurrent funding of the major projects.


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Another form of this example is the case where the secondproject is of no value unless a specified initial pro-ject is funded. These examples imply a time relation.An example which can be included in the linear program-ming model is illustrated by a case where the desira-bility of purchasing a machine tool attachment is con-tingent upon the purchase of the basic machine. In thelinear programming model consider the case where projecta must be funded before project b. This may be insuredby requiring the following additional constraints:

a. xb * xa 4-3

b. xa * 1

If xa = 1, that is, project a is accepted, then projectb may be accepted or rejected. The non-negativity con-constraint of equation (4-lc) requires x^ to be at leastzero. The addition of equations (4-3 a and b) requiresproject b to have a value between zero and one. Againthe problem of fractional projects prohibits a completecontingency constraint.4.13 Mutually Exclusive and Contingent Projects

The conditions of the previous two forms of inter-dependencies may be combined. Consider the facilitiesrepair case, where either one of two types of roof re-pairs must be performed on a building. If one of the


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roof repair projects is accomplished, the remodellingof the interior may proceed. Either project a or bmust be funded, before project c. The linear programmingconstraint may be stated:

a - xa + xb " l 4" 4

b. xc xa + xb4.14 Contingent Project Chains

The basic contingent constraint may be extended torequire a specific sequence of funding projects:

a. xa ^ 1 4-5

b - xb " xac. xc * xb

Thus, project a must be funded before project b, andproject b must be funded before project c.4.15 Forced Acceptance of Projects

The forced acceptance of projects may be treatedas external to the application of linear programming bymerely deceeasing the quantities, Ct , available for outlaysin the respective budget periods by the amount of theone or more projects for which funding is demanded.4.2 Integer Programming


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The integer programming formulation is stated inthe same manner as the linear programming model with theadditional constraint that all x^ be integral. This ad-ditional constraint eliminates fractional projects whichmay occur in the linear programming model. The limita-tion on the model is the performance of the availableinteger programming algorithm. Weingartner (21) reportedthat in a problem of this type using three constraintsand 10 projects, conversion was not obtained within5000 iterations.4.3 Dynamic Programming

4.31 The Basic Recursion Relation

The basic allocation problem can be formulated inthe recursion relation of dynamic programming. Thelinear programming problem of equation (4-1) is analo-gous to the knapsack problem which Bellman and Dreyfustreat using dynamic programming ( 2) # Instead of con-straints on the total volume and total weight, the usualcase in the knapsack problem, these constraints arereplaced by constraints on outlays in budget periods.Theoretically, the number of constraints of this naturemay be greater than two, but computational limitationsplace limits on the total number of constraints feasible.Dynamic programming is based upon the "Principle ofOptimality) stated by Bellman (2).


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"The Principle of Optimality . An optimal policy-has the property that whatever the initial stateand initial decisions are, the remaining deci-sions must consittute an optimal policy with regardto the state resulting from the first decision."A proof of this principle is also presented by Bellman(2 ). The Principle Of Optimality permits the statementof a general recursion relation as follows:

fn(q) = [Maximum gn(qn ) + ^-1^"%)] 4" 6O^q^Q 0^qn ^q

Where fn (q) = maximum return feasible when aquantity of resource q, the statevariable is allocated to the nstages of the problem.

The right side of equation (4-6) provides that fn willbe maximized for the investment of a quantity of resourceq when the allocation of q is divided between the returngn from the nth stage and fn.^ all previous stages, suchthat their sum is maximized. This required that thereturn from gn + fn_i be evaluated for all values of qas it varies over all feasible values from zero to q.The Principle of Optimality prohibits negative resourceallocations. The projects in the allocation or capitalbudgeting problem are treated as stages in the recursionrelation, equation (4-6) but no particular order foraddi'ng projects is necessary.4.32 Multiple State Variables and Dimensionality


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Equation (4-6) may be extended to the multiple budgetcase by adding for each budget time period t one addi-

tional state variable. The linear programming formulation,equation 4-1 may be converted to the recursion relation:

a. fn (C, l> C '2 C, 3 ' C 't ) 4" 7

= Max[bxn+fn. 1 (C 1-cixi , C' 2 -c2Xi , . ,C t-ctn^i = 1 ,2, . . , nt = 1,2, . . , t

subject to:b. C' t - cti =

c. f (C) =Where C'jC^ . . . C are JfafjBPfKS&o&F the

i = identifies the project (all n projectshave been considered.)fn (C'^,C*2, . , C' t )^the maximum returnfrom an optimal allocation of fundsfrom the set of C' budget periodsto the n projects

Each C' is a state varaable of the dynamic programmingformulation. Inthe solution to the above type problemwhere only the discrete valuse zero or one are permit=ted for the fractional projects, x^ , funded, all of thefeasible combinations of values of the set of ct foreach stage of the problem must be evaluated and the setsof decisions made, recorded, or stored in a computer


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solution.While the computational time using a computer is

important to obtaining tractable solutions, the criticalconsideration is currently the demands made upon rapidmemory capacity for storing the decisions from the pre-vious stages. Nemhauser (16 ) provides the followingequation for estimating computer emeory capacity:

Rapid Memory Storage Units Required = 3NK^ 4-8Where N = the number of stages (projects)

P = the number of state variables perstageand K = the number of feasible values perproject

As an example and comparison of current computercapacity, consider an allocation problem where ten pro-jects which may have two feasible values each, eitherzero or one and the number of budget periods, that isstate variables, is ten. Then N = 10, K = 2, and P = 10,and rapid memory storage requirements is approximately30,000 words of computer memory. By comparison the IBM7094 computer has a 32,000 word rapid access memory.Even for this small problem, this model computer approachesfull capacity. Emphasis is placed upon computer rapidmemory, because the speed of solution of the extensiveenumerations required for discrete value problems isprimarily a function of data access time. If external


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memory units, such as magnetic tape and discs are used,the access time is on the order of hundreds or thousandsof times slower than the main rapid access memory of thecomputer.

In equation (4-8) the increase in the number ofstate variables causes a power increase in the storagecapacity requirements. Bellman terms this problem the"Curse of Dimensionality." As stated above, the dynamicprogramming formulation of capital budgeting problemsof equation (4-1) present no theoretical problem, butits algorithmic solution for th e discrete value case islimited by the capacity of the computing equipment avail-able.4.33 Interdependence of Projects

The types of interdependence of projects describedunder linear programming may also be included in thedynamic programming model. However, each equation inthe constraint formulation must be treated as a statevariable in the recursion relationship. The dimensiona-lity of the problem creates a practical limit to thenumber of dependent relationships which can be treated.4.34 Application to Capital Budgeting

Both Cord ( 4) and Weingartener (21) have reportedapplication of dynamic programming to capital budgeting.Cord used a two state variable formulation using the


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Lagrange multiplier, suggested by Bellman ( 2 ) to reducedimensionality. His recursion relation was of the form:

fn(l) = Max[pnxn - qwnXn + 1^(1' - 1^)] 4-90^1 '=1 0=xn^l

Where I = funds available for allocation tocapital projectsI.= the funds required by the ith capitalproject where i = 1,2, . . , N.

The I^'s are constantsP^= the expected annual income over thelife of the investment foom the ithcapital investmentq = the Lagrange multiplierXj= a variable constrained to one of twovalues: zero, if the ith investment

is not included in the budget; one,if the ith investment is included inthe budget.

v^= the variance of the expected interestrate of return on the i th capitalproject.w. = (I Vj/I) The ith variance weighted1 by the ratio of the ith investmentto the total funds available.

Cord's formulation includes a constraint on the totalvariance assigned to the selected set of projects, inaddition to the constraint on total funds available.The function of the variance constraint is to limit therisk to the investor. Hillier ( ll) presents probabilis-tic relations for the investment problem, and Markowitz 0-3)English ( 6 ) and Morris d5) describe the use of variance


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for the measurement of risk in investments as used byCord. Using the IBM 7070 computer, Cord solved a 25project problem in 12 minutes. Weingartener questionsthe validity of the argument that the introduction ofa probability distribution can automatically be regardedas treating the problems of uncertainty. His complaintis not with the derivation of the probabilistic infor-mation as presented by Hillier, but the means by whichthe data is obtained.

Weingar tner (21) reports on the development ofa dynamic programming computer program using a programlanguage similar to FORTRAN with the exception that theresults of the strategies are stored and computed in

binary form. This permits the program to test 2000 stra-tegies at each stage with 10 separate constraints.For Cord's problem, 63 seconds were required for solutionon an IBM 7094 computer, compared to 12 minutes by Cordon an IBM 7070. Weingart ner s solution also producedan exact solution, where Cord's, using the Lagrange multi-plier formulation did not produce the optimum. Cordattributed the less than optimal solution to the coarse-ness of the iterations on the Lagrange multiplier, butWeingartner concludes from a comparison with the exactsolution and a finer evaluation of the Lagrange multi-plier formulation, that the exact solution can not befound using the Lagrange multiplier formulation.


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4.4 Conclusions on Existing Capital Budgeting Formulations

The capital budgeting models discussed in thischapter were investigated for the purposes of findinga suitable model for selecting the optimal course of actionin the facility maintenance planning problem. In eachof the above models, the timing of the funding is deter-mined in advance, and the question answered is whichprojects should be funded. In the facilities maintenanceplanning problem, the solution must also indicate whenfunding of a project should be accomplished.

The linear and integer programming models providedthe option of several approaches to expressing depen-dence between projects. This is an advantage over thedynamic programming model which rapidly becomes dimen-sionally intractable when constraint conditions are con-sidered. In Chapter V a dynamic programming algorithmis developed which treats the time of initial fundingproblem inherent in the planning problem; and reasonsfor the selection of dynamic programming will be pre-sented.


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CHAPTER VAPPLICATION OF DYNAMIC PROGRAMMING TO

FACILITIES MAINTENANCE PLANNINGChapter II introduced the systems orientation of

facility maintenance planning for a system of militarybases. In Chapter III a value system for planning wasformulated using a value system design process as amodel. Chapter IV provided a review of mathematicalprogramming techniques used in capital budgeting andresource allocation problems. In this chapter the ob-jective function of equation (3-7) is transformed intoa one state variable dynamic programming recursion equa-tion. This formulation permits the evaluation of thefeasible combinations for funding a set of repair pro-jects, considering the funding of each project in everyyear of a budget horizon. Practical limitations areintroduced and methods for reducing the effects of theselimitations are described. The results of a computersolution to a sample problem are provided.5.1 Transformation of the Value System Objective Functionto a Dynamic Programming; Recursion Equation

5.11 The Project Return Function, ^(q)The dynamic programming recursion equation (4-6)

requires that for every feasible value of resource as-


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signed to an activity, there must be defined some returnfunction. The general form of equation (3-7) consideredfor a single project has this characteristic. Theresource required is the present worth of the cost ofthe repair when it is funded in year k. Where k = 1,2,3, . . , p, as k varies, the net present worth returnvaries discretely, thus, a return function is generated.Given project i:

n k na. 8l (q) = c. jaj -][ cjaj + Rak + c^a-j 5-1

j = l j=l j=k1 = k = p

b. q = Ra,i * i


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for evaluating the objective function equation (3-6)has the following desirable characteristics:

1. The optimal time of funding each project in aset of projects may be determined for a rangeof resource allocations.

2. The single constraint on the present worth valueof future annual budgets defines an optimalset of annual budgets.

3 g The optimal time of funding each project inde-pendently is evaluated when the quantity ofresource available, Q, is a set at a level whichwould permit immediate funding of all projects.

4. A set of solutions are generated which establisha cost/effectiveness comparison for a range offeasible resource investment values.

The assumption of one state variable, Q, and indepen-dence of projects creates the following disadvantages:

1. Dependence of relations between projects areexcluded.

2. The model is deterministic.3. Large sets of projects make the solution intract-

able.4. Annual budgets imposed by higher authority can

not be modelled.All of the above shortcomings of the basic equation arecaused by the rapid increase in dimensionality when more


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than one state variable is specified.For the large military facility system, the number

of repair projects to consider each year can be recog-nized to be in the hundreds, if each base submits onlya single project.. Consideration of all projects overa ten year planning horizon raises this to thousandsof projects. A practical solution requires some methodof reducing the dimensionality of the problem. Twomethods have been suggested in the literature: "coarsegrid" search ( 2 ) ; and decomposition (5 ). A formof each technique has been applied in the computer pro-gram developed utilizing equation (5-2).5.21 Coarse Grid Search

In using a coarse grid search the number of feasiblevalues of the state variable, q, searched as q variesfrom zero to Q is reduced by sampling the feasible valuesat equal intervals. A coarse grid problem solutioncan be used to define a local neighborhood of feasiblevalues to be searched exhaustively. Bellman recommendsthe method for return functions, gn (q) , not subjectto sharp peaks which may be bracketed by the grid interval5.22 Decomposition

The technique of decomposition was suggested byDantzig for linear programming problems. In cases whereproject independence is assumed, and a single state


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variable is used, its use in dynamic programiing appearsfeasible. Decomposition refers to decomposing a largeset of projects into a set of subsets of projects tobe solved independently. After each subset has beensolved by application of dynamic programming, the solu-tions of each subset are treated as return functionsin an aggregation application of dynamic programming.5.23 Reduction in Number of Feasible Values

A third method of reducing dimensionality is toreducethe total number of feasible values in the returnfunction gn(qn ) of equation (5-1). This may be doneby shortening the planning horizon. In the fiscal senseonly the current budget year is important, for plansfor future years can and will be changed as the environ-ment changes. Therefore, if a shortened planning hori-zon provides an equivalent funding plan in terms of totalresources to be expended, and total savings generated,it may be assumed that the shortened horizon search isequivalent to the i_ong horizon search.5.3 Computer Program

A computer program was written in FORTRAN IV foruse on the IBM 360 model 15 computer. The program wasdeveloped to evaluate the methods of increasing thefeasible problem size discussed above. The comparisonsmade were


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1. A comparison of a coarse grid search on the statevariable with a search of all feasible values.

2. A comparison of a coarse grid search in thestate variable with a search of all feasiblevalues on subsets of projects aggregated bydynamic programming. A coarse grid s amplingof the solution of the subsets is used as areturn function g^(q).

3. A comparison of three value s of the budget hori-zon ranging from five to twenty years

5.31 DataThe data for the program consists of a set of fifty

projects. Figures 8 and 9 represent the original annualcost streams c^j and revised annual cost stream c\afor a basic set of ten projects. Discrete annual costvalues were interpolated from these continuous repre-sentations. Figure 10a lists the set of repair costs,R^ , for the basic ten projects, and Figure 10b is theset of discount factors r. for the set of fifty projects.The annual costs and repair costs are repeated five timesto construct the fifty project set. Each project hasa unique discount rate, r^ , which gives it a uniqueret um function.5.32 General Description of the Program

The program consists of these major elements:1. Return function generator.


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o

2


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M-lrHJ4->coBd>rl

cr


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a. Project Repair Costs, R^ t for 10 Basic Projects

Project Number 12 3 4 5Repair Cost, in $ 270 200 400 1600 1900

Project Number 6 7 8 9 10

Repair Cost, in $ 250 800 1200 1180 2500

b. Discount Rates, r 1 , for Each of 50 Projects

Projects 123456789 10110 .06 .07 .10 .15 .05 .08 .20 .25 .22 .301120 .04 .06 .08 .13 .08 .13 .15 .20 .16 .3521 30 .02 .05 .07 .10 .06 .12 .10 .18 .10 .253140 .01 .04 .04 .08 .04 .10 .08 .15 .08 .1041--50 .05 .03 .02 .03 .02 .08 .04 .10 .04 .05

PROJECT PARAMETERS

Figure 10


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2. Dynamic programming subroutine.3. Data storage and output section.

The principal features of these elements will be out-lined. The computer listing and an interpretation ofthe program symbols is included in the appendix.Return Function Generator

The return function generator operates on equation(5-1). The sets of annual costs, c-^, and c 1 . .; thelength of the planning horizon, p; the maximum project

t

life, n; and the set of repair costs, R. , are providedas inputs. Using this data, the stage return functiongi(q) | is computed for all feasible values. There isone feasible value for each period in the budget horizon,A penalty cost is included in the program to be assignedwhen the allocation of the resource is proposed afterthe project has terminated. This is detected when avalue of zero is assigned to c^..-. If an intermittentannual cost of zero exists in a project, the assignmentof a nominal cost other than zero will avoid the penaltycost. The set of all return functions g^(q) is storedfor use by the dynamic programming subroutine.Dynamic Programming Subroutine

This subroutine operates on equation (5-2) fordiscrete values of q. Three features are of interest:

1. The state variable is incremented over all feasi-


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ble values by first incrementing over the feasi-ble values qn of the stage return function gn (qn )and when these are exhausted, adding successivelyto the highest value of qn , the values of q formthe previous stage. This permits every feasiblevalue from the lowest feasible value of the pre-sent return function, qn , to a value which permitsall projects considered to be funded in the yeargenerating their independent optimal returns.

2. The optimum values of qn and (q - qn ) are foundby varying qn ovef all feasible values of thereturn function, Sn (qn ) or to the limit of thestate variable.

3. When a coarse grid is applied to the state vari-ble, the only modification to the program isto increment the first values of qn by an in-terval greater than one. The values of q fromthe previous stage applied afterward, alreadyexhibit the greater interval.

Data Storage and Output SectionThe principal item of interest in this section of

the program is the method of generating a syntheticreturn function for the application of the subroutineto the aggregation step in the decomposition method.The method used is coarse grid sampling. The resultsof each solution of a subset is a set of decisions


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defining the funding year of each project for a set ofsavings and resource costs for each feasible incrementof resource allocation, q. The technique used is tosample the savings, fn (q) , and cost increments, q, withan interval size that will permit the total set of in-crements to be represented by twenty values. For theselected increments, the cost, q, and savings, fn(q),values and the funding year decisions for the projectsassociate d with these values are stored to be retrievedif that increment is used in the final aggregated solu-tion.5.4 Resu

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