NBS BUILDING SCIENCE SERIES 129

Cost Estimation

and Cost Variability

in Residential Rehabilitation

4,;

U.S. DEPARTMENT OF COMMERCE • NATIONAL BUREAU OF STANDARDS °' .^^^
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NBS BUILDING SCIENCE SERIES 129

Cost Estimation and Cost Variability

in Residential Rehabilitation

Robert E. Chapman

Hatloial Buf«au of StaoriiriJ' Lftrsry, E-Ol Admin. BW; DEC 1 5 1980

/lot ClU

Center for Building Technology

National Engineering Laboratory ,

' '

' National Bureau of Standards ^ Washington, DC 20234 ' "^i

U.S. DEPARTMENT OF COMMERCE, Philip M. Klutznick, Secretary

Luther H. Hodges, Jr., Deputy Secretary

Jordan J. Baruch, Assistant Secretary for Productivity, Technology and Innovation

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued November 1980
Library of Congress Catalog Card Number: 80-600174

National Bureau of Standards Building Science Series 129

Nat. Bur. Stand. (U.S.), Bldg. Sci. Ser. 129, 120 pages (Nov. 1980)

CODEN: BSSNBV

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1980

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

Price $4.50

(Add 25 percent for other than U.S. mailing)
PREFACE

This research was conducted under the sponsorship of the Rehabilitation

Technology Group, Building Economics and Regulatory Technology Division,

Center for Building Technology, National Engineering Laboratory, National

Bureau of Standards. This paper analyzes four methods for estimating the

costs of residential rehabilitation. A theoretical procedure based on an approach which integrates the performance concept with established

engineering economics techniques is also developed.

The author would like to extend appreciation to those individuals who

generously provided information on all aspects of the residential rehabilitation process. Special appreciation is extended to Dr. Harold E. Marshall,

Applied Economics Group, Building Economics and Regulatory Technology

Division, who provided many useful comments at the formative stages of

this study while the author was a member of the Applied Economics Group.

(The author is currently with the Operations Research Division, Center for

Applied Mathematics, National Engineering Laboratory, National Bureau of

Standards.) Special appreciation is also extended to Dr. Joseph G. Kowalski,

formerly with the Building Economics and Regulatory Technology Division,

who prepared a working draft of several portions of chapter 3, and to

Mr. William G. Hall, Operations Research Division, whose programming

efforts in the area of fire safety in health care facilities has greatly

affected the formulation of an approach for applying the performance

concept to the residential rehabilitation problem.

Cover: Residential rehabilitation

activities have increased dramatically in recent years. In some cases these activities have resulted

in substantial gains to investors in

inner city housing. In other cases

the uncertainty surrounding renovation costs have had a significant

and adverse effect on the demand for

houses in older neighborhoods.

iii
ABSTRACT

This study analyzes four methods of estimating the costs of residential

rehabilitation. Each method is critiqued with regard to its treatment of

changes in the size of the renovation project, the productivity of labor,

and the contractor's markup for overhead and profit. Cost comparisons and

a discussion of the way in which the inherent riskiness of renovation

activities may be asessed are also presented. A theoretical approach for

dealing with cost variability which integrates the performance concept

with established engineering economics techniques is also developed.

Key words: Applied economics; building codes; building economics; cost

estimation; economic analysis; engineering economics; housing;

mathematical programming; optimization; rehabilitation;

renovation

iv
TABLE OF CONTENTS

Page

PREFACE ill

ABSTRACT iv

LIST OF FIGURES vi

LIST OF TABLES vii

LIST OF EXHIBITS vlii

SI CONVERSION UNITS ix

1. INTRODUCTION 1

1.1 Background 1 1.2 Purpose 2 1.3 Scope and Approach 3

2. ALTERNATIVE METHODS OF ESTIMATING REHABILITATION COSTS .... 5

2.1 Average Costs and Costs Indices 5 2.2 Parametric Cost Estimating Procedures 9 2.3 Probabilistic Cost Estimating Procedures 17

2.4 Cost Functions 26

3. CRITIQUE OF ALTERNATIVE METHODS OF ESTIMATING

REHABILITATION COSTS 35

3.1 Impact of Changing Job Size 37

3.2 Treatment of Productivity 42

3.3 Markup and Its Determinants 45

3.4 Assessment of Reliability 46

3.5 Treatment of Risk 48

4. AN APPROACH FOR DEALING WITH COST VARIABILITY 57

4.1 Cost Functions and the Performance Concept: An

Overview 58

4.2 Achieving Code Compliance in the Least Costly Manner:

A Case Study 75

4.3 Potential Applications of an Approach Integrating

Cost Functions and the Performance Concept 91

V
TABLE OF CONTENTS (Cont'd)

5. SUMMARY AND RECOMMENDATIONS FOR FURTHER RESEARCH 97

5.1 Summary 97

5.2 Recommendations for Further Research 98

6. REFERENCES 100

APPENDIX A Derivation of a Cost Function for Residential

Rehabilitation 103

LIST OF FIGURES

Page

Figure 2.1 Fitting a Model to Data 13

Figure 2.2 Fitting a Confidence Band About a Parametric Cost

Estimate 16

Figure 2.3 Implications of Extrapolation With a Linear Cost Model. 16

Figure 2.4 Commonly Used Probability Distributions for Key

Construction Factors 19

Figure 2.5 Cost Profile for a Renovation Project 25

Figure 3.1 Average Total Cost as a Function of Job Size 41

Figure 3.2 An Average Total Cost Curve With a Constant Cost Per

Unit Segment 41

Figure 4.1 Optimal Solution: Graphical Method 70

Figure 4.2 Alternative Solution: Graphical Method 72

Figure 4.3 Delphi Method 77

Figure 4.4 Floorplan for the Patient Room Floors 85

Figure 4.5 Cost Comparison 87

Figure 4.6 Alternative Retrofit Packages for the Case Application. 90

vi
LIST OF TABLES

Page

Table 2.1 A Description of Selected Cost Manuals 7

Table 2.2 Density Functions for Commonly Used Probability

Distributions 20

Table 2.3 Input Requirements for a Standard Probabilistic Cost

Estimating Procedure 23

Table 2.4 Estimated Coefficients for the Cement-Coated Fiberglass

and Gypsum-Impregnated Jute Fabric Cost Model 33

Table 3.1 An Illustration of the "Horizontal" and "Vertical"

Dimensions of Cost Estimation 38

Table 3.2 A Comparison of Estimated and Actual Direct Costs for Two

Surface Refinishing Tasks 49

Table 4.1 Building Component/State Score Matrix 61

Table 4.2 Score Assessment Matrix 61

Table 4.3 Code Requirements Matrix 64

Table 4 .4 Equivalency Test Matrix 64

Table 4.5 Cost Matrix Used in Linear Programming Approach .... 66

Table 4.6 Retrofit Packages for the Case Application 89

vii
LIST OF EXHIBITS

Page

Exhibit 4.1 Occupancy Risk Factor Calculation Worksheet 78

Exhibit 4.2 Worksheet for Determining the Value of the Safety

Parameters 79

Exhibit 4.3 Worksheet for Calculating Containment Safety,

Extinguishment Safety, People Movement Safety, and

General Safety 80

Exhibit 4.4 Worksheet for Evaluating Fire Zone Safety Equivalency • 81

viii
Most Common SI Units and their Equivalent Values in Customary Units

QUANTITY INTERNATIONAL (SI) UNIT U.S. CUSTOMARY UNIT APPROXIMATE CONVERSION

LENGTH meter (m) millimeter (mm) foot (ft) inch (in)

1 m

1 mm

- 3.2808 ft = 0.0394 in

AREA square meter (m^

)

2 square yard (yd ) 1 m2

1 mm

- 1.1960 yd^

- 10.764 ft^

= 1.5500 X 10"-'ln'^ 2 square millimeter (mm ) square foot (ft^)

2 square inch (in )

vULUrllj 2 cubic meter (m )

cubic millimeter (mm )

cubic yard (yd^) cubic foot (ft^) cubic inch (in )

1

1 mm''

- 1.3080 yd^

= 25.315 ft^

= 61.024 X lO'^ln-"

CAPACITY liter (L) mUllliter (mL) gallon (gal) fluid ounce (fl oz)

1 L

1 mL

= 0.2642 gal

= 0.0338 fl oz

VELOCITY, SPEED meter per second (m/s) kilometer per hour (km/h)

foot per second (ft/s or f.p.s.)

mile per hour (mile/h or m.p.h.)

1 m/s

1 km/h

= 3.2808 ft/s

= 0.6214 mlle/h

ACCELERATION meter per second squared (m/s^) foot per second squared (ft/s^) 1 m/s^ = 3.2808 ft/s^

MASS metric ton (t) [1000 kg) kilogram (kg) gram (g)short ton [2000 lb] pound (lb) ounce (oz)1 t

1 kg

1 g

= 1.1023 ton

= 2.2046 lb

= 0.0353 oz

DENSITY metric ton per cubic meter (t/m^) kilogram per cubic meter (kg/m )

ton per cubic yard (ton/yd^) pound per cubic foot (lb/ ft )

1 t/m3

1 kg/m^

= 0.8428 ton/yd^

= 0.0624 lb/ft

kilonewton (kN)

newton (N)

ton~force ( tonf

)

kip [1000 Ibf] pound-force (Ibf)

1 kN

1 kN

1 N

= 0.1124 tonf

= 0.2248 kip

= 0.2248 Ibf

MOMENT OF FORCE, TORQUE kilonewton meter (kN*m) newton meter (N*m) ton-force foot (tonf'ft) pound-force inch (lbf*in)

1 kN'm

1 N'm

= 0.3688 tonfft

= 8.8508 Ibf'in

PRESSURE, STRESS megapascal (MPa)

kilopascal (kPa)

ton-force per square inch (tonf/in^) ton-force per square foot (tonf/ft^) pound-force per square inch (Ibf/ln ) pound-force per square foot (Ibf/ft^)

1 MPa

1 MPa

1 kPa

1 kPa

= 0.0725 tonf/in^

= 10.443 tonf/ft^

- 0.1450 lbf/ln2

= 20.885 blf/ft^

WORK, ENERGY, QUANTITY OF HEAT megajoule (MJ) kllojoule (kJ) joule (J)

kilowatthour (kWh) British thermal unit (Btu)

foot pound-force (ft'lbf)

1 MJ

1 kJ

1 J

= 0.2778 kWh

= 0.9478 Btu

= 0.7376 fflbf

POWER, HEAT FLOW

RATE kilowatt (kW) watt (W) horsepower (hp) British thermal unit per hour (Btu/h)

foot pound-force per second (ft*lbf/s)

1 kW

1 W

1 W

= 1.3410 hp

= 3.4121 Btu/h

= 0.7376 fflbf/s

COEFFICIENT OF HEAT watt per square meter kelvin Btu per square foot hour degree Fahrenheit (Btu/f t^'h'°F)

1 W/m^'K = 0.1761 Btu/ft^'h'°F

TRANSFER [U-value] (W/m^-K) [=(W/m2'°C))

THERMAL CONDUCTIVITY watt per meter kelvin (W/m'K) Btu per square foot degree Fahrenheit 1 W/m'K = 0.5778 Btu/ft^'°F

[k-value] [=(W/m'°C)] (Btu/ft^'°F)

NOTES: (1) The above conversion factors are shown to three or four places of decimals.

(2) Unprefixed SI units are underlined. (The kilogram, although prefixed, is an SI base

unit)

.

REFERENCES: NBS Guidelines for the Use of the Metric System, LC1056, Revised August 1977; The Metric System of Measurement, Federal Register Notice of October 26, 1977,

LC 1078, Revised November 1977; NBS Special Publication 330, "The International System of Units (SI)," 1977 Edition;

NBS Technical NOte 938, "Recommended Practice for the use of Metric (SI) Units in Building Design and Construction," Revised edition June 1977; ASTM Standard E621-78, "Standard Practice for the Use of Metric (SI) Units in Building Design and Construction," (based on NBS TN 938), March 1978; ANSI Z210 .1-1976, "American National Standard for Metric Practice;" also Issued as ASTM E380-76'^, or IEEE Std. 268-1976.

ix
Facing page:

The rapidly vis-ing costs of new housing

ooupted with the looationat advantages of

the oentval city have stimulated rehabilitation activities in many urban areas. Unfortunately this trend is mitigated by

the extreme variability of residential

rehabilitation costs.

X
1 . INTRODUCTION

1 .1 BACKGROUND

Between 1972 and 1976 the median cost of a new single-family housing unit

increased at an annual rate of 12.5 percent. •'^ Such rapidly rising prices

have forced many prospective homeowners to look elsewhere in order to

satisfy their housing demands. This trend is likely to continue over the

next 5 to 10 years. Current forecasts indicate that between 22 and 30

million additional housing units will be needed by 1988.^ A significant

increase in the number of housing units being renovated is therefore likely

U.S. Department of Housing and Urban Development, Final Report of the Task

Force on Housing Costs, May 1978.

"Millions of Housing Units Needed by 1988," Engineering News Record , April 27, 1978.

1
if the nation's housing needs are to be satisfied. Since costs play a vital role in all investment decisions, any significant change in current

housing investment trends calls for a reduction in the uncertainty associated with the extreme cost variability of residential rehabilitation

activities

.

The housing renovation industry has received little research attention in

the past. Due to its rapid growth, however, many research questions are now being raised. One important area where knowledge is lacking relates

to the nature of the cost structure of the housing renovation industry.

Focusing more narrowly, how reliable is cost estimation in the area of

housing renovation? What methods are available for homeowners and/or

investors, contractors, subcontractors^ and public officials for estimating

renovation costs? How reliable are such methods? Need they be improved?

Accurate cost estimation is important to all the major participants in the

building process. Reasonably accurate cost information is essential to the

housing owner as an input into the investment decision. Unreliable cost

information renders housing renovation investments riskier than they would

be if accurate cost estimates existed. To contractors and subcontractors

the reliability of cost estimation procedures also affects the riskiness of

doing business. Finally, accurate cost estimation is needed by Federal,

State, and local program managers for numerous reasons. For example, a local public housing official may be faced with a decision about choosing

a contractor to upgrade publicly owned housing units. Accurate cost estimates would be a major concern to the HUD official engaged in program

budgeting for an Urban Homesteading Demonstration Program.

The primary sources of cost data for housing renovation are cost estimation

guides or manuals, which subdivide tasks into fine levels of detail. These

guides can be put into two categories. The first includes the guides whose

primary market is for new building construction contractors, subcontractors,

architects, engineers, and cost estimators. These guides are often used to

prepare budgetary estimates of construction costs. They provide reference

information on unit costs. More detailed or final estimates in new construction, however, usually rely directly on a construction firm's cost

estimator. These estimates would be based on detailed quantity takeoff

s

coupled with unit prices drawn from the firm's past experiences with similar construction projects and current prices in the local market for labor

and materials. The second category of guides are those directly aimed at

the housing renovation contractor or subcontractor. These guides can also

be used to prepare preliminary or budgetary estimates for housing

renovation costs.

1 .2 PURPOSE

The purpose of this report is to present an analysis of four basic methods

for estimating housing renovation costs. Particular emphasis is placed on how these methods permit (or prohibit) the introduction of the inherent

riskiness of the renovation process into the housing investment decision.

This report also serves to develop a conceptual procedure, based on

2
engineering economics principles, which will permit risk to be explicitly

incorporated into the preparation of a pre-bid cost estimate. Guidelines

are given to show how this procedure could eventually be used by participants in the renovation process. This report is intended to serve as a reference document for researchers; architectural and engineering consultants; and Federal, State, and local program managers concerned with the

problems of estimating the costs of housing renovation.

1 .3 SCOPE AND APPROACH

The focus of this study is on analyzing the ways in which cost calculations

are performed and cost variability is treated in several alternative procedures for estimating the costs of housing renovation. For the most part,

the emphasis of this study is on theoretical considerations. Actual cost

data are introduced in several instances, however, to illustrate how these

considerations could be treated in practice. It is not the intent of this

study to downgrade a particular method of estimation. Each method has its

strengths and weaknesses. However, if a particular method is deficient in

its treament of risk, it would seem advisable, given the importance of

this factor in the renovation process, that this deficiency be called to

the attention of potential users.

The basic format of this study consists of a description of the four

primary methods for estimating housing renovation costs, a critique of

each method, the development of a "theoretical" approach for dealing with

cost variability, and the implications that the use of such an approach

would have on the purchases of renovation services

.

Specifically, chapter 2 describes each of the four basic methods for

estimating renovation costs, what their data requirements are, how they

may be applied to the problem, and a measure of their flexibility in

dealing with varying technical and economic conditions.

Chapter 3 focuses on a critique of the per unit cost method of estimation

with regard to such crucial factors as the impact of changing job size;

the treatment of productivity and contractor markup; and the treatment of

risk.

Chapter 4 develops a theoretical approach for dealing with cost variability

based on the economic theory of cost functions. A framework for introducing the performance concept is also developed. The cost saving potential

of this approach (based on past studies in the area of fire safety) is

then discussed. The chapter concludes with a discussion of how such an approach could be of benefit to individual homeowners and/or investors,

financial institutions^ and governmental program managers.

Chapter 5 contains a short summary of the research findings and

recommendations for future research, including an econometric analysis of

actual housing renovation projects.

3
The report also includes a technical appendix which shows how, given an underlying set of technical relationships, a cost function suitable for

estimating residential rehabilitation costs may be derived.

Facing page:

The extreme variability of residential

rehabilitation costs is a reflection of

constraints on the construction process

due to the condition of the building as well as economic factors determined in

the market place. Accounting for the

interactions among technical and economic

variables can only be achieved by increasing the complexity of the cost estimating

procedure.
2. ALTERNATIVE METHODS OF ESTIMATING REHABILITATION COSTS

2.1 AVERAGE COSTS AND COSTS INDICES

Cost estimation is usually based on past experiences projected to include

anticipated increases in wage and material costs. A premium is, thus,

placed on historical data as a basis for estimating future costs.

Guidebooks and manuals are frequently the only source of cost information

.5
for prospective homeovmers and/or investors. They also form an important

supplement to a contractor's own past cost experience. In general, cost

guides or manuals rely upon cost reports submitted, under a variety of

conditions, by contractors who report their own historical experience.

Thus, subtask cost figures reported in a cost manual are based on the,

average cost experiences of the reporting contractors. Furthermore, the

guidebooks rely on the judgment of construction experts to modify, if

necessary, their collected cost figures to ensure that the reported

figures are representative of cost behavior under typical or average

conditions

.

The first three publications listed in table 2.1 are primarily directed

to the estimation of new construction costs. ^ The last two publications

focus on the estimation of home repair, remodeling and renovation costs.

There are, of course, other manuals available which address slightly

different aspects of the building construction process. Table 2.1

should therefore be looked at as a representative sample of the manuals

available rather than an exhaustive listing.

In every case the particular figures that the guidebooks report are average figures per unit, e.g., per square foot or linear foot. The common approach taken is to report average costs per unit as equal to

the sum of average material cost per unit and average labor cost per

unit where each of the latter have been factored upwards by the

contractor's markup for overhead and profit.

More specifically the average total cost of task i may be expressed

mathematically by either equation 2.1, 2.2 or 2.3.

ATCi = AMCi + ALCi

ATC-j^ = k(ADMCi) + k(ADLCi) = kCADMC^ + ADLCi)

ATCi = ^(^m + Wi/PRODi)

2.1

2.2

2.3

where

ATC^

AMC^

ALC^

ADMC^

ADLC^

m

Average Total Cost of the i subtask

Average Material Cost of the i^ subtask

Average Labor Cost of the i^^ subtask

Average Direct Material Cost (excludes markup) of the 1

subtask

Average Direct Labor Cost (excludes markup) of the i*"^

subtask

Markup factor for contractor's (or subcontractor's)

overhead and profit.

Average gross material price per unit.

th

Iaii three guides referred to above contain provisions which would allow the

cost of certain types of renovation processes to be estimated.

6
Table 2.1 A Description of Selected Cost Manuals

Data Approximate Recorded Nature and Title Number of for Each Source of Overhead Regional Statistical Publisher Subltems Listed Subitem Data and Profit Variation AsseBSibl 1 Ity

Dodge Manual for over 10,000 1 Description Average Data Includes Adjustment No standard Building Construc- of task f iguresj subcontractor Indices for errors published.

tion Pricing and 2 Crew com- compiled overhead and 152 cities Reliability not Scheduling position from profit but are reported. determinable • 3 . Output per records not general 24 separate

day of Wood & contractor '

s

trade & sub- McGraw-Hill Unit Tower overhead & trade indiInf ormation definition Incor- profit

.

cles are Systems 5 Per unit porated . reported

Company material for each

cost city

.

6 Per unit labor costs

7 Total unit costs

Building Construe- over 16,000 1 Description Average Overhead Over 108 No standard

tion Cost Data of task figures profit city cost errors published.

2 Crew compiled included indices Reliability not Robert Snow Means, composition from In cost can be determinable

.

Company 3 Output per actual figures used to day job costs which adjust

k Unit defini- reported can be per unit

tion on indus- subcon- material

,

5 Per unit trial and tracted; labor, and material commercial bare costs total cost

cost buildings reported in 16 cate6 Per unit costing for items gories for labor costs $150,000 that are each city.

7 Total unit and up or not subconcosts large tracted . housing

projects

Building Cost over 7,000 1 Specification Data based Overhead Regional No standard

File and descrip- on cost and profit editions errors

tion of task records of included and city published . Construction 2 Unit HcKee

,

and specified cost Reliability

Publishing definition Berger

,

markups are indices not Company 3 Per unit and obtainable

.

are determinable

.

material Mansueto

.

reported

.

cost

k Per unit labor cost

5 Per unit

total cost

Home-Tech Estimator over 9,000 1 Specification Data based Overhead Local area No standard Vol. II Manager's and descrip- on "actual and profit cost modifi- errors Manual tion of task time and not included cation index publ ished

.

2 . Per unit material in cost aval lable Reliability

cost of studies ." figures . for over 100 not material cities . determinable

.

3 . Per unit cost

of labor

4 . Total per unit cost

(no markup

included)

5 Per unit price (cost times markup)

National Repair over 2,000 1 Descrip- Developed Overhead Geographical No standard

and Remodeling tion of by author and profit wage modi- errors Estimator task through not Included fication publ ished . 2 Crew consulta- in cost factors Reliability Craftsman Book composi- tion with figures

.

for 16 not Company tion experts

.

crafts determinable

.

3 Ouput per Derivation and 17 day of figures cities are

4 . Unit defini- explicitly reported

.

tion described in 5. Per unit many cases, material

6. Per unit

labor costs

7 . Total unit costs

7
Wj[ = Average wage rate per hour of the crew required to perform

task i, and

PRODj^ = Number of units of output per hour of the crew required to

do task i.

Equations 2.1 through 2.3 are very useful because they explicitly

identify the major items which must be verified by the cost estimator to ensure that a cost figure from a guide will be applicable to the specific

job under consideration. First, is the markup factor appropriate? This

is because the figure in equations 2,2 and 2.3 will vary from subcontractor to subcontractor and depends upon market conditions as well as those factors which affect general overhead and project specific overhead

costs. Second, are the material prices assumed in equations 2.1 through

2.3 appropriate to that location at that particular time? Third, are local wage rates consistent with the average hourly rate assumed in the

guidebook? Fourth, is the crew mix required to accomplish a specific

task consistent with a given firm's practices? Finally, and probably

most critically, is the productivity figure appropriate to the situation

at hand?

The above questions indicate the potential sources of error which may

render estimates based on the figures reported in a guide inaccurate in

any particular construction application. The authors and editors of

these manuals are fully aware of such variations, and for this reason,

they include methods for adjusting or modifying estimates for major

sources of variations. Assumptions for making such adjustments are usually given in the preface.

"The unit costs presented in the DODGE MANUAL represent average

prices. Users of construction costs recognize that such costs are

not exact for a broad range of building projects. Implicit in the

development of these unit costs are a number of assumptions which

include purchase of materials in quantities normal for most building

projects, no volume or special discounts, no labor cost premiums

due to trade shortages, no unusual weather conditions, standard

"good workmanship" and standard grade materials .. .Prices contained

in the DODGE I4ANUAL are those that would be incurred by a general

contractor who subcontracted for all Items of Work, with the exception of the mechanical and electrical items (Divisions 15 & 16)

which do not include the subcontractor's overhead and profit. The

general contractor's overhead and markup are not included in any of

the prices. No two projects are identical nor are the Items of

Work that make up two different projects. When using unit costs,

the user must consider special project conditions such as weather,

site, etc., and must reflect the effect of simplicity or complexity

on the Items of Work."-*^

1979 Dodge Manual for Building Construction Pricing and Scheduling , McGraw-Hill Information Systems Company, New York, 1978, p. vi.

8
2.2 PARAMETRIC COST ESTIMATING PROCEDURES

The second category of cost estimating procedures Is parametric or regression based. These procedures make use of historical cost data

just as do the average cost procedures. The major difference is that

parametric procedures use a set of key factors (parameters) which are weighted to reflect their importance in order to estimate a response

variable (usually the cost of a particular subtask) . Parametric procedures are superior to average cost procedures because they permit specific

factors to (differentially) register their impact on average total cost

(or on total cost for that matter). Thus, they may remove to some extent

the reliance on judgmental decisions which can neither be confirmed nor denied. Their application, however, is more complicated. In addition,

they require more data inputs in order to develop a satisfactory procedure.

The equation used to estimate the response in a parametric procedure is

frequently referred to as a cost model. The process of developing such

a cost model may be long and involved. It always begins with the postulation of a theoretical model. Ideally, information on the renovation

process would be used to postulate the theoretical model. One should be

cautioned against postulating a model without knowledge of the process.

Using a trial and error approach to "discover" the key factors may produce a model which requires information as input which is not available

before the fact.

One commonly used cost model, where cost is in dollars, is of the form m

ATC = 3o + 2 3iX^ + e 2.4

j=l

where ATC = average total cost, the response variable,

3q = the intercept term,

3j = the coefficients of the explanatory variables (weighting

factors)

,

Xj = the explanatory variables (key factors), and

e = the error term.

Models having a structure^ similar to equation 2.4 are referred to as linear. This may be seen by noting that for the case where m is equal

to 1 the above model reduces to:

ATC = 3o + 3iXi + e

^The intercept term is included since the assumption that the response is zero when all explanatory variables are zero is a rather strong and

usually unjustified assumption.

^The term structure is used to denote the basic relationship between the

variables which affect average total cost.

9
which is the equation

greater than or equal

m-dimensional space.

Models of a nonlinear form may also be included under the category of

parametric cost estimating procedures. An example of a nonlinear cost

model would be :

ao ai a2 a3

ATC = Bo(T) (L) (M) (E) e 2.5

where T = the construction technology factor,

the labor inputs,

the materials inputs,

the equipment inputs,

the intercept term,

the weighting factors, and

the error term.

As one might suspect the estimation of the 3 and a terms (weighting factors)

for nonlinear models is much more complicated than for the general linear

case. The flexibility of the nonlinear model is, however, greater than

that of the general linear model. Cost models having a structure similar

to equation 2.5 are frequently referred to as cost estimating relationships

or cost functions. Since the emphasis of this section is on the more

commonly used linear cost models, a thorough discussion of cost functions

will be postponed until section 2.4.

The methodology through which the underlying structure and determinants of

average total cost may be analyzed involves a four stage iterative process

of (1) selecting a model, (2) estimating the coefficients, 3j, (3)

testing the validity of the underlying assumptions, and (4) testing the

adequacy of the model. Once estimated, the finalized model can then be

used to predict average total costs under a wide variety of conditions.

If desired, a similar approach may also be used to estimate the markup

ratio

.

The actual estimation of the 3's in equation 2.4 would normally be

accomplished through application of the econometric technique known as ordinary least squares. For example, suppose we have n observations on average total cost, where n is greater than m + 1. Then the i^^

observation may be expressed as:

m

ATCi = go + E 3^Xi^ + ei

of a straight line. In higher dimensions (m

to 2) the equation defines a plane or hyperplane in

10
All n observations may be expressed in matrix form by

ATCi ' 1 Xii Xi^

ATCnJ JXni

or more compactly

ATC = X 6 + e

1

Po

3l +

"nm ! &m

2.6

where the line under ATC, X, 3 and e indicates that they are matrices

order to apply ordinary least squares to equation 2.6, the following

assumptions should be satisfied:

In

(1) the explanatory variables, X-[ j, are fixed,

(2) the expected value of the error term,

E(ei) , is zero

,

(3) the variance of the error terra,

ECe^ei), is constant, and

(4) the covariance among the i^^ and k*^^ error terms, E(ej[e^), is zero.

Under these conditions ordinary least squares will produce best linear

unbiased estimators of the 3's. The estimates, 3, are unbiased because

the expected values of 3j is equal to 3j for each j (i.e., E (3j) = 3j). The estimates are linear because they are linear combinations of

the experimental observations. That is, using the matrix notation of

equation 2.6, _3 may be expressed as:

_3 = (X'X)"1 X' ATC

The ordinary least squares estimates are best because no other linear

estimator of the 3's has a smaller variance. This attribute is very

useful because it permits more precise statements to be made about the

impact of changes in the value of one or more of the key factors on average

total cost.

As will be shown in chapter 3, economists and engineers generally agree

that average total cost may be expressed as a function of the number of

units of output (say square feet of floor area renovated). This usually

implies that average total cost falls over a certain range and then remains

11
constant over a fairly wide range. (Eventually, for very large projects,

average total costs will begin to rise again.) Suppose now that only the

square feet of floor area renovated is allowed to vary (i.e., all other

factors are held constant). Figure 2.1 illustrates how the ordinary least

squares procedure operates. In this case average total cost, ATC, is

shown along the vertical axis whereas the number of units of output, Q, is

shown along the horizontal axis. (The dots shown on the figure represent

(ATCjQ) combinations.) The straight line shown in the figure represents

the "best fit" for the given data. Note that the straight line is declining, indicating that the average total cost decreases as the number of

square feet of floor area increases.

It is important to point out that the straight line is not extended beyond

the range of observation. This is because ordinary least squares estimates

are rather "data specific" and extrapolating beyond the range of observation may lead to gross inaccuracies in the predicted value of the response.

In order to better illustrate the ordinary least squares approach, we shall refer to the particular combination (ATC-j^,Q-[) in figure 2.1. In

this case ATC-j[ is the response and Qj^ is the number of square feet of

floor area renovated. The predicted response, ATC-^, lies on the straight

line. The difference between ATC-^ and ATC-j^, dj^, is denoted by a darkened

line connecting the two points.

The rationale behind any estimation procedure, including ordinary least

squares, is to keep as small as possible the distance between the response

and the predicted response. An examination of figure 2.1 would reveal that

some of the (ATC,Q) combinations lie above the line and some lie below it.

Consequently, any summation of just the d^ would probably end up being

zero. In order to get around this problem ordinary least squares suras the

squared deviations (i.e., d?) . Since the square of a number is always

greater than zero unless the number is zero (i.e., ATC-j^ = ATC-j^), the sum

of their squares will have to be greater than or equal to zero. The concept of "closeness" that ordinary least squares uses is to minimize the sum of the squared deviations. Although it is conceptually "nice", ordinary least squares may give too much weight to outliers (points way above

or way below the line). Consequently, the stability of the estimates of

the 3's may be very sensitive to the exclusion of a particular

observation

As mentioned earlier, a cost model combines one or more explanatory

variables in order that an estimate of a response variable (direct cost,

material cost, output per hour, or markup) is produced. Although these

Another more theoretical issue relates to the stability of the estimates

over time. For an authorative source on this problem see R.L. Brown,

J. Durbin, and J. M. Evans, "Techniques for Testing and Constancy of

Regression Relationships Over Time," Journal of the Royal Statistical

Society , Series B, Vol. 37, pp. 149-192, 1975.

12
13
models can be shown to be best (minimum variance) linear unbiased estimators

of the desired response variable, they represent point estimates rather

than interval estimates. Thus, there may be situations in which other

technical approaches might have costs very near to the one revealed as the

least costly. This would imply that the least costly approach can not be

unambiguously identified.

Even though in most cases there will be a wide enough variation in costs

between technical approaches to identify the one which is least costly,

there may be instances where confidence intervals about each estimated

response are desired. In addition, confidence intervals tell us how good

or how bad our cost model is for that particular combination of inputs.

Other things being equal small confidence intervals are preferred to large

confidence intervals. The purpose of this discussion is to illustrate

how a confidence interval may be fitted about the response. First, we must define precisely what we mean by a confidence interval. The 100(1 -

e) percent confidence Interval associated with a particular cost model

is def ined as

C' 3 ± (tn_k^i_,/2) (C'(X'X)-lc)

where

_3 = the vector of estimated coefficients (weighting factors);

the vector of explanatory variables (key factors);

an estimator of the standard deviation of the predicted

value , C^' 3 and

'^n-k l-e/2 ~ ^ ~ value of the t statistic with n - k degrees

of freedom. •'^

It is important to point out that the output of the cost model is the

point estimate C'_3

Suppose we wish to fit a 90 percent confidence interval about the estimate

resulting from covering over 510 square feet of wall area in a dwelling

unit with plywood paneling. Prior information on wage rates and prices

for a four by eight foot sheet of plywood paneling indicate that the

average wage rate is $10.00 per hour and the cost per square foot of

plywood paneling is $0.45.

The predicted value, ^'_3, which results is $1.20. That is, the average

cost of installing plywood paneling is $1.20 per square foot in this case.

_C =

4(C'(X'X)C~1)^ =

The number of degrees of freedom is based on information used in estimating

the cost model; in particular, n is the number of observations and k is

the number of explanatory variables.

14
We now wish to compute the width of the confidence interval, w, where:

^ = ^8, 0.95 (C'42(X'X)"^C)^

A statistical table may be used to show that the appropriate value for the

t distribution with eight degrees of freedom is 1.86. This permits the

above statement to be reduced to:

w = 1.86 (C'42(x'X)-lc)^

Performing the indicated matrix multiplication, ^ C^i^ (X'X)~^^, and taking

the square root of the resulting scalar yields:

(C'42(X'X)-lc)^ = 0.06

Thus, the width of the confidence interval is 11 cents (w = (0.06)x

(1.86)). The 90 percent confidence interval, Io.90> ^bout the predicted

value for pljwood paneling in this case is thus:

l0.90 = (1.09, 1.31) or

^0.90 = 1 .20 + 0.11 = ^'_3 + w.

The above discussion has shown how a confidence interval would be fitted

about the estimate for a particular set of inputs. If a whole set of

inputs were considered, one would get a confidence band about the best

fit straight line. An example of such a confidence band is shown in figure 2.2. Note that the band becomes wider at the ends. This is because

the cost model is most accurate in the area where the largest number of

observations lie (i.e., in the central region).

It was mentioned earlier that nonlinear cost models tended to be more

flexible than linear cost models. This stems from the fact that real

world considerations are more frequently nonlinear than linear.

Figure 2.3 serves to illustrate this point. As in the earlier cases,

average total cost is shown along the vertical axis and Q along the horizontal axis. The theoretical average total cost curve is labeled ATC in

the figure. By contrast the linear approximation of the desired portion

The 6^(X X)~^ matrix is usually printed as a part of the standard output

of the ordinary least squares package. The matrix actually used in this

exercise is available from the author upon written request.

There are many applications where the assumption of linearity is quite

appropriate (see, Robert E. Chapman and Joseph G. Kowalski, Lead Paint

Abatement Costs: Some Technical and Theoretical Considerations , National

Bureau of Standards, Technical Note 979, February 1979). In these cases,

however, it is often true that the key factors affecting cost are only

varying over a small range. In such cases, even if the process were

nonlinear, a linear cost model could be extremely accurate.

15
gure 2.2 Fitting a Confidence Band About a Parametric Cost Estimate

Figure 2.3 Implications of Extrapolation With a Linear Cost Model

16
of the theoretical average total cost curve based on actual observations

is labeled ate. Note that ate consists of three distinct segments. The

two segments which appear at the ends are drawn as solid lines.

In these regions the predicted response for average total costs, ate, would

be too low. Furthermore, if the line ate were extended in either direction

the magnitude of the error would increase. This illustrates why one should

not extrapolate beyond the range of observation. The segment in the middle

of the line ate is dashed. In this region the predicted response would be

too high. It is worth noting, however, that even though the cost estimates

lie above the ATC curve, they may be quite accurate. This is due to the

fact that the ATC curve is the theoretical average total cost curve. Consequently it is not possible for actual average total costs to lie below

it whereas due to inefficient (less than optimal) utilization of labor,

material, and equipment, it is possible to lie above it.-'-

2.3 PROBABILISTIC COST ESTIMATING PROCEDURES

One form of construction cost estimation which has grown rapidly as low-cost computer software packages have become available is probabilistic

cost estimation. The term probabilistic is a reflection of the fact that

a probability distribution can be associated with each key factor in the

renovation process. Once these probability distributions have been specified for each factor, it becomes possible to perform a monte carlo simulation of the cost estimation process. The usual output of such a simulation

The above statement implies that the theoretical average total cost curve will never be observed in its entirety. To illustrate why this is so, a discussion on technical and allocative inefficiency similar to that given

by Farrell (M. J. Farrell, "The Measurement of Productive Efficiency,"

Journal of the Royal Statistical Society , 1957) will be presented. For a two factor constant returns to scale production technology, the two types

of inefficiency may be defined in a straightforward manner. Technical

inefficiency (i.e., producing using inputs in the right proportions but

wrong quantities) is measured by the ratio of the inputs required to produce a given output as a percent of those actually used. Allocative

inefficiency (i.e., producing using inputs in the wrong proportions but

using quantities that are technically correct) is measured, for a given

input price ratio, by constructing a line tangent to the desired isoquant

(a curve giving all the technically feasible ways of producing a given

level of output) and extending it until it intersects (denoted as point

A) the line connecting the origin and the point on the desired isoquant

corresponding to the combination of inputs actually used (denoted as point B) . The measure of allocative inefficiency is then defined as the

distance OA divided by OB, In both cases, the total cost of production

is higher due to inefficiency. Since firms may not be flexible enough to

always use the right quantities (technical efficiency) or right proportions (allocative efficiency) observed cost will lie above the envelope

curve defined by the theoretical average total cost function.

17
is what is known as a cost profile. A cost profile may be defined as a graphical or tabular portrayal, for the given values of the data input, of

the probability of overrunning any given budget estimate. Probabilistic

cost estimating procedures are quite attractive because they can be applied

to either an average cost method or a parametric cost estimating procedure.

Since the application of probabilistic procedures to average costs is

simpler than for a whole series of parameters, the discussion in this

section will focus upon average cost methods.

Although no theoretical limitation exists on which probability distribution

can be used in the simulation of the cost estimating process, most actual

applications rely on four basic distributions. These distributions are:

(1) the uniform; (2) the normal; (3) the log normal; and (4) the triangular.

These four probability distributions are plotted graphically in figure 2.4.

In each case the value of the random variable (average total cost for a particular subtask) is plotted along the horizontal (x) axis. The value

taken on by the density function, f(x), is shown along the vertical axis

on figure 2.4. The mathematical form of each density function is given in

table 2.2. A nonmathematical interpretation of the density function is

that it provides a measure of the frequency with which a certain event'

will take place for a given "small" interval along the x axis. Note that

in figure 2.4 the uniform and the triangular distribution both have well

defined starting and stopping points. These distributions might be appropriate if average total cost was known to be at least $a but no more than

$c. It is also important to point out that the log normal (of necessity)

and the triangular (by construction) distributions can be skewed. That

is, the tails of the distribution are of unequal length. For the distributions as drawn, it reflects the possibility that an extremely high value

of the random variable (average total cost) can occur with non-zero probability. Such cost patterns are a common occurrence in the construction

industry, where the most likely cost (the point where f(x) is a maximum)

for a particular subtask may be rather low but due to the riskiness of the

process, costs may take on a very large value with non-zero probability.

The crucial step which must be taken prior to the application of any

probabilistic cost estimation procedure is to determine which distribution

is appropriate for the case at hand. The first step is to collect cost

data from similar projects or contact local building material suppliers

and contractors. The actual cost of each subitem should then be recorded.

Once all data has been collected it will then be necessary to group them

into intervals. (A general rule of thumb states that at least five data

points should be in each interval .)

The number of times average total cost occurred within the interval may

then be used to construct a histogram. The histogram may then be compared

to sample standard histograms such as appear in the text by Hastings and

Peacock.-'- Based on these comparisons, a class of distributions which

N. Hastings and S.B. Peacock, Statistical Distributions , John Wiley and Sons

New York, 1975.

18
UNIFORM

LOG NORMAL

TRIANGULAR

Figure 2.4 Commonly Used Probability Distributions

for Key Construction Factors

19
Table 2.2 Density Functions for Commonly Used Probability Distributions^

UNIFORM

f(x) = 1 in the interval [a, c]

c - a

0 otherwise

NORMAL

1 exp (-(x-m)^/2o^) - «> < x < «> f(x) = a(2n)

LOG NORMAL

f (x) =0 X < a

1 exp (-( Jln(x-a)-m)2/2a2) x > a (x - a)a(2n)

TRIANGULAR

f (x) = 2 . X

(c - a) (b - a)

(c - a) (c - b)

0

in the interval [a, b]

in the interval [b, c]

otherwise

^Source: Marek Fisz, Probability Theory and Mathematical Statistics , Third

Edition, John Wiley and Sons, New York, 1963.

20
might fit the data can be hypothesized. The next step is to develop

maximum likelihood estimators for the parameters of the distribution

(e.g., the mean and variance) under the assumption that the hypothesized

distribution is correct.-^ The final step is to perform a goodness-of-f it

test on the data. If the fit is unacceptable it will be necessary to

hypothesize a new class of distributions and reestimate the parameters

of the distribution. Several points along these lines are worth noting.

First, there are two goodness-of-f it tests which may be used. They are:

(1) the Chi-square test; and (2) the Kolmogorov-Smlrnov test.^ Second,

it is possible to have the goodness-of-f it test accept more than one distribution as appropriate. This unfortunate circumstance is rather common

with the Chi-square test. However, since the Chi-square test is by far

the easiest to apply, such a potential problem may not be deemed serious.

Should the situation result where two distributions are "accepted" by

the goodness-of-f it test, it is advisable to carefully examine the tails

of the two distributions. If one tail is significantly more important than

the other, then choose the distribution which better fits that tail.

Finally, it is important to recognize that the Chi-square test is an asymptotic result whereas the Kolmogorov-Smirnov test is exact for any number

of data points. Thus, if the sample size is small the emphasis should be on the result of the Kolomogorov-Smirnov test rather than the Chi-square

test.

The focus of the previous paragraph on fitting distributions to data was

not to overpower the reader with a whole set of statistical procedures

which must be followed prior to any meaningful analysis. The focus was rather on recognizing that if probability distributions are used to estimate renovation costs, the underlying assumptions implicit in that application must be understood. It is crucial to recognize that the distributions

are ideal theoretical constructions whereas the data are from the "real

world" and hence can not be "ideal." In some cases it may in fact be

impossible to obtain a sample of actual average total cost figures for a particular subtask. In this case expert judgment can be used to choose

the distribution. In such cases past researchers have recommended the use of the triangular distribution because it explicitly allows for lowprobability high-cost events. Furthermore, it may be completely defined

by only three points: (1) the minimum point; (2) the maximum point; and

(3) the mode or most likely point. (If one believed that costs were equally

likely to be distributed throughout the interval then the uniform distribution would be appropriate. The uniform distribution may be completely

defined by only two points: (1) the minimum point; and (2) the maximum

point .)

The likelihood function of a sample, given a parameter, is the product of the

density function with respect to the parameter at each sample point. For an excellent discussion see John Freund, Mathematical Statistics , Second Edition,

McGraw-Hill, New York, 1970.

•Both tests are discussed in L. Breiman, Statistics : With a View Toward

Applications

,

Haughton Mifflin, Boston, 1973.

21
The previous discussion has touched upon how the distributions would be

applied in an actual monte carlo simulation to estimate the costs of

residential rehabilitation. Although only two of the four distributions

were discussed it can be easily shown that the normal and log normal distributions may be applied almost as simply in actual cost studies. The

basic difference between the normal—log normal and uniform—triangular

distributions is that the upper tail of the normal and log normal distributions extends to infinity. This is not true for both lower tails since

the log normal distribution has a minimum point at a (a is greater than or equal to zero) . The lower tail on the normal distribution has no minimum

point however. Thus, in applying probabilistic cost estimating techniques

it is useful to specify three sets of numbers when a normal or log normal

distribution enters the process. Each set of three numbers may be either

estimated from actual data or based on the opinions of construction experts.

The first number needed is the "optimistic" estimate of average total

cost. This estimate is called optimistic because there is only a 10 percent

(subjective) probability that average total cost will fall below it.

Mathematically, the optimistic estimate is the 10th percentile point of

the distribution. The second number needed is the "middle ground" estimate

of average total cost. The term middle ground is used because in 50 percent

of the cases average total cost will fall below it and in 50 percent of

the cases average total cost will fall above it. Mathematically, the

middle ground estimate is the median of the distribution. The third number

needed is the "pessimistic" estimate of average total cost. This estimate

is called pessimistic because there is only a 10 percent (subjective)

probability that average total cost will fall above it. Mathematically,

the pessimistic estimate is the 90th percentile point of the distribution.

In essence, once the distribution is specified the user need enter (at

most) three numbers for each subtask into the computer software package.

These numbers are summarized in table 2.3.

Assuming that the user has correctly specified the probability distribution

and correctly input the information identified in table 2.3, the software

package will perform a monte carlo simulation. The term monte carlo is used

to Indicate that the process is patterned after several popular games of

chance. Basically what the computer program does is estimate the cost of

each subtask sequentially. This is done by generating a random number.

Each random number will correspond to a value between 0 and 1. The random

number is then associated with the parent probability distribution (i.e.,

the average total cost distribution for that subtask) . Since each random

number is between 0 and 1 it can be interpreted as the probability that

the value of the random variable will be less than or equal to a specified

amount. If we denote the random number as R, the random variable as x ^^'^ the specified amount as X then the expression may be written formally as

Pr (x < X) = R.

22
Table 2.3 Input Requirements for a Standard Probabilistic

Cost Estimating Procedure

DISTRIBUTION NUMERICAL INPUTS

UNIFORM 1. MINIMWl COST

2. MAXIMUM COST

TRIANGULAR

1. MINIMUM COST

2. MAXIMUM COST

3. MODAL OR MOST LIKELY COST

NORMAL AND

LOG NORMAL

OPTIMISTIC: 10 th PERCENTILE

th MIDDLE GROUND: SC-" PERCENTILE

PESSIMISTIC: 90^^ PERCENTILE

23
Since we are concerned with the average total cost of the subtask, the

relevant cost is X. Thus, if there are n (say 100) subtasks we will get n random numbers and n separate X's. Suppose there are N (say 1000) iterations.

Then for the first iteration we get

Pr (x < Xij) = Rij j = 1, . . . n

The estimated total cost for the overall job on the first iteration, TCj^,

is thus

n TCi = Z Xi^

j=l

Similarly, the estimated total cost for the overall job on the second

iteration, TC2, would be

n TC2 = 2 X2i

j=l

and for the i^^ iteration

n TCi = 2 Xi^

j=l

The computer will generate N estimates of total cost for the overall job.

These estimates are then ranked from least costly to most costly. For

example, the least costly estimate (i.e., the first order statistic) is

denoted as TCq) . The parentheses are used to distinguish the first

order statistic from the estimated cost on the first iteration. Since the

total costs are ranked from least to most costly, it is possible to compute

the probability that total cost will be less than or equal to a specified

dollar amount. For example,

Pr (tc < TC(k)) = k/N,

where tc = the random variable total cost

TC^j^^ = the k^^ order statistic for total cost; and

N = the number of iterations.

Conversely, the probability of exceeding TC(}^), the projected budget, may

be expressed as

Pr (tc ^ TC(i^)) = 1 - k/N

The process described above is most easily understood through reference to

a cost profile. An example of a cost profile is shown in figure 2.5.

24
15 -

10 -

5 -

I I I I I I I I I I 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1. PROBABILITY OF OVER RUN

Figure 2.5 Cost Profile for a Renovation Project

25
Note that the total cost of the job is shown along the vertical axis. In

this case, the probability of overrunning a specified budget is shown

along the horizontal axis. For example, the probability of overrunning a $35,000 budget is only 15 percent whereas the probability of overrunning a $30,000 budget is 30 percent. Thus, the perspective investor can specify

a given level of risk (in terms of probability of overrun) and then choose

a budget which will satisfy this constraint. In addition to dealing with

risk, the use of probabilistic cost estimation permits the investor to

more effectively manage any funds held for contingencies. That is, the

investor may proceed with a basic contract which permits contingencies in

terms of better quality products (say floor coverings) to be installed

should the cost of the job fall below some agreed upon figure.

It is important to point out that the term monte carlo is quite appropriate

since there is still some element of chance remaining. In essence there

are no absolutes. Total costs will either exceed or fall below the projected

budget. The measure of risk is only approximate . In particular, different

sequences of random numbers can yield different estimates. Thus, one should

be cautioned against demanding a one percent risk. On one simulation the

figure might be $45,000 and on another $55,000. This is because each simulation (N iterations) yields a single estimate of the cost profile. Therefore, in order to get a more meaningful measure of the true risk being

assumed, one should replicate the simulation using a different starting

random number (seed). Should the investor be using the simulation to

choose between two alternative methods of renovation, replications should

certainly be performed. For an excellent discussion of this topic the

interested reader is referred to the article by Law.-*^

2.4 COST FUNCTIONS

The discussion so far has focused on engineering based procedures for

estimating rehabilitation costs. The purpose of this section is to show how

the requirements of having a sound cost engineering approach can be integrated with the economic theory of cost functions. Combining both aspects

of the problem results in a cost estimating procedure which is sensitive to

the technical considerations of the renovation process as well as local

market conditions. Furthermore, the development of such a procedure facilitates the treatment of risk through the use of probabilistic methods. (A

theoretical structure which allows performance concepts to be introduced into

the rehabilitation investment decision will be developed in chapter 4.)

The cost function approach is highly desirable due to the duality relationship

between the cost function and the production function associated with the

physical process. (The term production function as used in this study refers

to an explicit relationship between a set of inputs (i.e., labor, materials,

and capital) and technological factors which taken together produce a given

output, e.g., square feet of floor area renovated.)

Averill M. Law, "Confidence Intervals in Discrete Event Simulation: A

Comparison of Replication and Batch Means," Naval Research Logistics

Quarterly , Vol. 24, 1977.

26
This attribute has been documented in numerous economic and engineering

economics articles.^ >2 Through reference to the duality relationship, it

is possible to assert that the cost function tells us the least-cost way

of renovating Q square feet of floor area. In an actual empirical study,

the cost function associated with the underlying process would be derived

from the production function by solving a constrained optimization problem.

In order to illustrate this concept more fully, a mathematical discussion

followed by a numerical example will be presented.

The derivation of the cost function will proceed on the assumption that the

production function underlying the construction process is Cobb-Douglas.

This assumption is made both because it is consistent with previous studies

of the construction industry-^ and for ease of exposition. In the mathematical discussion which follows, it is assumed that the prices of all inputs

are independent. Under this assumption the quantity of output, Q, the

square feet of floor area renovated, and the cost of output, C, may be

expressed as

:

m n a-j Q = do n Ti(ki))( n (X.) ) 2.7

i=l j=l

n

C = E XjPj 2.8

j=l

where

Q = total square feet of floor area to be renovated;

basic construction technology factor;

construction technology factor associated with i^^ building

system or subsystem (e.g., plumbing, mechanical, electrical);

t" v» measure of the condition of the i system or subsystem (e.g.,

1 = excellent; 2 = sound; 3 = some deterioration . . . );

^Eugene Silberberg, The Structure Of Economics: A Mathematical Analysis , McGraw-Hill Publishing Company, New York, 1978.

^Gerald L. Musgrave and Robert H. Rasche, "Estimation of Cost Functions,"

The Engineering Economist , Vol. 22, No. 3, 1973.

^For an authoritative source on this subject see John S. McConnaughey

,

Production Functions in Contract Construction for the United States , 1972 (unpublished), Ph.D. dissertation, Michigan State University, 1976.

27
number of systems or subsystems considered;

quantity of the j input (materials, labor, equipment)

required to perform the job;

the percentage change in output associated with a one percent

change in the utilization of the j^'^ input; '^

number of inputs considered;

the total cost (expected bid price) of the job; and

the unit cost of the j*-'^ input.

Note also that the summation sign, E, and the product sign, II, are used

in equations 2.7 and 2.8. They are defined as:

n

E XjPj = XiPi + X2P2 + + XnPn

j=l

and

n a a2

n X.j = (Xj) (X2) (X^)

j=l

Letting n equal 2 would thus imply that

n 2

E XjPj = E XjPj = XiPi + X2P2

j=l j=l

and

n a-j 2 a-j a2

n (X.) = n (Xj) = ((xi) )((X2) ) j=i j=i

The constrained optimization problem may then be solved through application

of the method of La Grange multipliers.^

Based on the constrained optimization problem stated above, it is possible

to assert that the cost minimizing demand curve for each input, X*, reduces

m =

In economics is referred to as an output elasticity.

In order to avoid a rather mathematical discussion at this point the

derivation of the cost function will not be presented here. Those

readers wishing an in-depth discussion are encouraged to turn directly

to appendix A.

28
1/R m -1/R n -aJR -aJR

n where R = E a^; and

j=l

The minimum cost solution, C*, for renovating Q square feet of floor area is

then obtained by substituting equation 2.9 into equation 2.8. This substitution yields :

n

C* = E x'^P.,

j=l ^

or equivalently

:

1/R m -1/R n aJR n -a^R

c* = (Q) (To n Ti(ki)) ( n (P.) ) (R) ( n (a.) ) 2.10

i=l j=l j=l

A closer examination of equation 2.10 reveals that it consists of three

distinct factors: (1) a technology factor; (2) a size factor; and (3) a market factor. The technology factor is defined by the underlying

construction/renovation process. That is, certain basic construction

techniques (technologies) interact with the condition of the building's

system and subsystems in defining an approach which is feasible in the

engineering sense. The technology factor, TF, in equation 2.10 is given as:

m -1/R

TF = (Tq n Ti(ki))

i=l

The size factor may be expressed as the product of the number of structures

being renovated, N, and the average number of square feet per structure

renovated, q. This division is important because it permits the existence

or lack of existence, of economies of scale to be tested. The size factor,

SF, in equation 2.10 is given as:

SF = q1/R = (n1/R) (ql/R).

The market factor reflects the influence that supply and demand conditions

in the local construction market for key labor, material, and equipment

inputs have on the overall cost of the job. The market factor, MF, in

equation 2.10 is given as:

29
n a-j/R n a-j/R

MF = ( n (Pj) ) (R) ( n (a-j) )..

j=l j=l

Equation 2.10 may thus be written as:

C* = (TF)x(SF)x(MF) . 2.11

Thus far no mention has been made about how such a cost function would

be estimated. In its present state neither equation 2.10 or 2.11 is in a form which is directly estimable. Taking the natural logarithm of either

equation 2.10 or 2.11 would, however, produce an equation which was directly estimable. Stated in its most simple form the natural logarithm

of C* would be

ln(C*) = bo + biiln(TFa) (hi^niSF^) + JinCMF^) +

where a, a=l, ... A, denotes the contract package of one or more dwelling

units and is a normally distributed random variable, error term, with

mean 0 (E(e^) = 0 for all a), variance a (E(e^, e^) = a for all a) and

covariance 0 (E(e£, e^) = 0 for all i^j) . Taking the natural logarithm

of equation 2.10, letting t^ (k^) = S^k^^ , and collecting all constant

terms into bQ yields

n m

in(cl) =

3o + 2 e-j^n(Pj^) + E (3^+i)to(k.^)

j=l -^-^1=1

+ (ei„+n+l)^^(Qa) + ^a- 2.12

The method of ordinary least squares may now be applied to equation 2.12.

Equation 2.12 in its estimated form would be given as

j=l i=l

+ (^+^+i)lniq^) 2.13

In equation 2.13 the 3 , t • • •

» m+n+l, are the ordinary least

squares estimates of the 3^»

It is important to point out that the lef^ hand side of equation 2.13 is an estimate o| the natural logarithm of C . Therefore, in order get an estimate of , it is necessary to take the antilogarithm of iln(C^). This

may be accomplished simply by making use of the relationship = exp( 5-n(C^) ) .

30
Example

In order to illustrate how the actual estimation of a cost function would

be accomplished, the methodology developed earlier will be applied to the

problem of estimating the costs of surface refinishing work. Examples of

surface refinishing include covering over walls on which the paint may be

in a cracked or peeling state, may be of an undersirable color or texture,

or may contain lead pigments which are potentially dangerous to small

children.

One method of surface refinishing which is currently used is the

application of a flexible wall covering. Flexible wall coverings include

cement coated fiberglass and gypsum impregnated jute fabric. Both products

are applied in the same manner as wall paper, usually with a water base

adhesive. Their strength and durability, however, greatly exceed that of

most commercially available wallpaper. Due to their similarity, it is

possible to use one cost function for both products. The discussion

given here is patterned after results presented in a report by Chapman

and Kowalski on methods for eliminating the lead-based paint hazard

from housing. The reader interested in further details on surface refinishing techniques is referred to the NBS report Lead Paint Abatement

Costs: Some Technical and Theoretical Considerations .

The products under study, cement-coated fiberglass and gypsum impregnated

jute fabric, consist of a woven glass or jute fabric impregnated with

Portland cement. After application, the portland cement absorbs moisture

from the environment and hardens becoming a fairly rigid and penetrationresistant material. Whenever the material is to be used in a wet area,

such as a bathroom or a kitchen, a protective coating should also be

applied. Based on a previous study, the method of application is known

to be affected by the surface condition and the occupancy status. The

primary labor cost is the wage of the painter/wallpaper hanger who installs

the product. The only material cost which is significant is the delivered

cost per square yard of the product itself.

The production function for cement-coated fiberglass and gypsum impregnated

jute fabric may thus be hypothesized as:

3l 32 «! ^2

Q = To(ki) (k2) (L) (M)

Robert E. Chapman and Joseph G. Kowalski, Lead Faint Abatement Costs;

Some Technical and Theoretical Considerations , National Bureau of

Standards, Technical Note 979, February 1979.

31
where :

Q = the total number of square feet of wall area covered;

Tq = the basic technology factor;

ki = the occupancy status (1 = unoccupied; 2 = occupied);

k2 = the condition of the wall surface;

^2 ~ the "complexity" factors;

L = the amount of painter/wallpaper hanger labor required to do

the job;

M = the square feet of cement-coated fiberglass or gypsum

impregnated jute fabric required; and

ai, 0L2 = the output elasticities.

Using equation 2.10 as a guide, it is possible to write the cost function

as

1/R -1/R 2 2 a./R «

C* = (Q) (Tq) ( n (k.) 1 ) ( n (P.) J

) (D ) , 2.14

1=1 j=l

where

R = aj^ + 0.2',

Pj = the hourly wage rate of a painter/wallpaper hanger;

P2 = the delivered price per square yard for cement-coated

fiberglass or gypsum impregnated jute fabric; and

Dg = a dummy variable associated with the use of gypsum jute

(1 = gypsum jute not used; 2 = gypsum jute used) .

Using the general form given in equation 2.12 to put equation 2.14 into

a form which is readily estimable yields:

£n(C*) = + 3i^n(Q.) + g2^n(kj^) + Q^Zn(k2^)

2.15 + 34Jin(PLi) + 35iln(Pi4i) + 36^n(Dgi) +

Twenty three separate observations were then used to estimate 3q through 35

in equation 2.15. The estimates for 3© through 3^ are given in table 2.4.

32
Table 2.4 Estimated Coefficients for the Cement-Coated Fiberglass

and Gypsum Impregnated Jute Fabric Cost Model

FACTOR COEFFICIENT ESTIMATE

Intercept

Size 1

2

3

4

5

6

o -1.63

0.97

-0.24

0.16

0.52

0.46

-0.42

Occupancy

Condition

Wage Rate

Material Price

Jute Dummy

An examination of table 2.4 reveals the following. First, the coefficient

of the size factor is slightly less than one, indicating that (other things

being equal) the size of the job can be doubled without doubling the cost.

Second, the coefficient of the occupancy factor is negative, indicating that

occupied units are relatively less expensive to treat. This rather sur- prising result stems from the fact that although the presence of furniture

and people in the dwelling does affect the installation of the product,

these added costs are more than compensated for by being able to store the

materials in the dwelling until the job is completed.-*- Third, the coefficients

associated with condition, the average wage rate and the material price per

square yard were all positive, indicating that a rise in the value of those

factors would result in higher installation cost. Finally, the coefficient

associated with the gypsum jute dummy variable is negative indicating that

(others things being equal) it is relatively less expensive to install this

product than cement-coated fiberglass. This difference is due to less

material wastage and a greater ease in cutting the product as well as minor differences in the method of applying the material.

An illustration of the model is now in order. Suppose we wished to estimate

the direct cost of using cement-coated fiberglass to refinish 800 square

feet of wall area in an unoccupied dwelling unit. An inspection of the unit

revealed the walls to be in a sound but peeling condition. The average wage

rate and delivered material price in the locale are $10.00 per hour and

$3.60 per square yard, respectively. The estimated cost thus becomes

Zn C* = -1.63 + 0.97 iln(800) - 0.24 iln(l)

+ 0.16 in(2) + 0.51 Jln(lO.OO)

+ 0.45 ilnC3.60) - 0.42 iln(l),

which reduces to

in C* = -1.63 + 6.49 - 0 + 0.11 + 1.18 + 0.58 - 0,

Theft and vandalism were prevalent in the neighborhoods containing the

dwellings making up the sample.

33
or

In C* = 6.73.

Recall that in order to get an estimate of C*, the direct cost of the job,

it was necessary to make use of the formula C* = exp(ilnC*), which implies,

in this case, that

C* =

or

C* ^ $840.

Chapter Summary

The purpose of this chapter was to provide an introduction to each of four

basic methods for estimating rehabilitation costs. The presentation of the

methods stressed both theoretical and empirical considerations with particular emphasis being placed on statistical techniques. The development of the

theoretical discussion was arranged to highlight the increasing order of

complexity of the cost models in terms of data requirements and ease of

application. The reason for this approach was to demonstrate why increased

flexibility in dealing with varying technical and economic conditions could

only be achieved through a greater emphasis on detail in the data inputs or sophistication on the part of model builders and/or users.

Facing page:

Although many aspects of residential

rehabilitation are similar to those of

new construction^ differences in job size^

productivityJ and contractor markup cause

the application of average cost methods to

systematically underestimate the true costs

of building renovation.

34
3. CRITIQUE OF ALTERNATIVE METHODS OF ESTIMATING REHABILITATION COSTS

The data reported in cost guides on per unit costs is essentially of two

kinds. First the data can be based on averages. Average figures are com- puted by compiling cost reports submitted by many independent reporting

units. Second, the data reported in these manuals may be nonstatistical

,

not derived from samples, and thus to some extent, judgmental. In this

case, per unit cost figures are based on opinions of experts who understand

the nature of the subtask, how it is best done, and how long it would take

to do it. Such figures represent the expert's judgment as to what is

typical (or average) behavior in completing a specific subtask. The repair

and alteration cost guides appear to lean to expert opinion as their sources

for their reported subtask figures.

35
None of the cost guides, either of the new construction genre or of the

repair and remodeling genre, describe the degree of variability in their

data. Such information would be useful in order to assess the reliability

of estimates based on such data.

In this chapter we shall discuss in more detail some of the difficulties

associated with judgmental or expert estimates. The focus will be on the

notion of the "reliability" of the estimates and how to determine it.

Most guidebooks also contain detailed cost adjustment indices which differ

by region and labor categories. They are usually applied to the average

figures in order to attain figures which are more appropriate to local

construction market conditions. Although cost indices serve to control

for systematic differences in the structure of construction costs, they

may promote a false sense of security. '^ This stems from three factors:

(1) the construction project mix; (2) local construction practices; and

(3) local building code requirements. Understanding the reasoning behind

the previous statement may be seen by recognizing that cost indices are really little more than a market basket of construction labor, materials,

and sometimes equipment. The weights used in computing the index are based on specific types of construction. Consequently, should the type of

construction being put in place for a particular region differ from that

used to compute the weights for the index, it is quite likely that the

index would introduce a bias into any cost to which it was applied.

Unfortunately, there is no prior information which can be used to identify

the direction and size of such a bias. This claim is supported by recent

empirical studies which have shown that the use of construction cost

indices for military construction projects is likely to introduce systematic

biases into the decisionmaking process.^

Local construction practices relating to crew size and composition and

local building code requirements may also introduce biases which will

render meaningless the weights used in a construction cost index.

In addition to these factors, the impact of changing job size, the treatment

of productivity, and variations in contractor's markup ratios, will all

affect the reliability of the cost estimates. Each of these topics will

be examined in turn prior to an assessment of the reliability of the most

commonly used techniques for introducing risk into the process of estimating

Lawrence Jaquith, "The Cost Index: Working Tool or Trap," Architectural

Record , February 1969.

James M. Johannes, Paul D. Koch, and Robert H. Rasche, An Investigation of

Factors Affecting Geograpic Cost Differentials on Military Construction

Projects , National Bureau of Standards, NBS-GCR-80-197 , February 1980.

'For an authorative source on assessing the cost implications of a building

code or code provision, see John S. McConnaughey , Jr., An Economic Analysis

of Building Code Impacts: A Suggested Approach , National Bureau of

Standards, NBSIR 78-1528, October 1978.

36
rehabilitation costs. The chapter will conclude with a discussion of how

and why the calculation of risk can be rigorously introduced into the

cost estimation process.

3.1 IMPACT OF CHANGING JOB SIZE

The term job size really has two meanings, only one of which is purely

economic. Throughout this paper the meaning, unless otherwise stated,

will be the one which is purely economic in nature. The economic definition

of job size may be thought of as an aggregate measure of the gross square

feet of floor/wall area treated, the number of building systems renovated,

and the number of items installed. This economic concept of job size is

closely related to a more pragmatic concept of job size, namely the dollar

value of a particular construction contract or subcontract. The economic

meaning will be used because it makes more sense to talk about costs varying

as a function of scale rather than as a function of costs.

A cost estimation procedure is basically a summation approach. A construction

project is subdivided into hundreds of subtasks . Ideally the cost estimation effort would therefore also be subdivided into hundreds of subtasks.

Separate cost estimates are then made for each subtask and summed to obtain

overall cost figures. Usually the estimation process follows, as closely

as possible, the order of the construction process. It may start with

items involving excavation or earth work and end with fixture installation.

Job size affects the estimation process along a horizontal dimension, where

the "horizontal dimension" refers to the relationship between job size and

how many square feet, linear feet or item counts are specified in any one

of the particular subtasks. Job size also affects the "vertical dimension"

of a cost estimate. The number of subtasks included in the summation

process is the vertical dimension of a cost estimate. Both dimensions

will be positively related to the total size and price of the job.

Table 3.1 illustrates the two dimensions of cost estimation. We shall

consider each dimension in turn.

The "Horizontal" Impact of Job Size

The data base in the major cost estimating guides is drawn from cost reports

of contractors whose new construction projects had total project costs in excess of $150,000. (This minimum figure varies from manual to manual but

in all cases it is large.) Average housing renovation costs would most

likely fall in the $20,000 to $30,000 range, however.

Bureau of Building Marketing Research, Homeowners Remodeling/Modernization

Study , November 1975.

"HUD Launches a New Rehab Program," Engineering News Record , November 25,

1976.

37
•u CO

X o o H O v£) O -H

03OH

CO LTi

4-1 CO

X

OD

O

S-i o o

)-l

COoo oH

<:Ho a

•H

!-i CJ

CO o6

cu

4-1

Pio

01

4-) M o

38
Because the descriptive subtask of the work to be done in housing renovation

can be made to coincide with the descriptive subtasks reported in new con- struction manuals, however, cost estimates for residential rehabilitation

can be gleaned from these manuals. For example, both a new construction

and renovation subtask may require painting of an interior wall (one primer

coat and one finish coat). Thus, since the per square foot cost from a manual "can" be specified in terms compatible with the guidebooks' descriptions, they have often been used in putting together a renovation cost

estimate. That this may lead to substantial errors may be seen by examining

an economic construct known as a cost function. For purposes of illustration

in this section we shall focus on the average cost curve (function) .

Economists generally think of average total costs as depending on units of

output. Average costs fall as the output increases, reach a minimum at some

size of job, and begin to rise as output level continues to increase.

An Illustration of the typical behavior of average total costs as a function

of job size is shown in figure 3.1.^ Note that for any particular output

level, whether it be Qj^ or Q2 , (see figure 3.1), it will always be true

that the accounting definitions of equations 2.1 through 2.3 will be met.

Thus, the guidebooks could be thought as reporting average cost figures

from one such point along a given cost curve. Most economists and engineers

agree, however, that average total cost remains constant over some fairly

wide range. This concept of average total cost is illustrated in figure 3.2.

If costs do in fact behave in this way, little error will be contained in

a cost figure which ignores the impact of job size. The above being true

as long as the number of units involved (e.g., gross square feet) exceeds

some critical minimum. Therefore in order to ensure that average total

costs are approximately constant, most major guides give a minimum total

cost figure in their preface.

Given that there may be two orders of magnitude in the difference in job

size between housing renovations and "major" new construction projects,

the applicability of the constant cost assumption is probably not valid.

The practical meaning, or source, of such differences in cost as a function

of job size is illustrated by the following quote from R. L. Peurifoys

book. Estimating Construction Costs:

The economic principles behind this claim are discussed in detail in

Walter Nicholson, Microeconomic Theory; Basic Principles and Extensions , The Dryden Press, Inc., Hinsdale, Illinois, 1972.

Recall that average total cost is equal to total cost divided by Q, where

Q is output (the economic measure of job size). Although average total

cost may in fact be falling, it does not follow that total cost will also

be falling.

39
"All experienced construction men know that the productivity

of labor is usually low during the early stages of construction.

As the organization becomes more efficient, its productivity rates

will improve, then as the construction job enters the final stages,

there will usually be a reduction in productivity rates. This is

important to an estimator. For a small job it is possible that

labor will never reach its most efficient rate of production because

there will not be sufficient time. If a job is of such a type that

laborers must frequently be transferred from one operation to another

or if there are frequent interruptions, the productivity rates will be

lower than when the laborers remain on one operation for long periods

of time without interruption ." '•

The importance of the role of job size on cost behavior for renovation

activities is supported by empirical work in the area of lead-based paint

hazard abatement. In the lead paint report aluded to above, cost data were collected and analyzed for using barrier materials to cover over the

lead-based paint hazard (e.g., gypsum wallboard, and pl3rwood paneling) on existing walls. These tasks would easily be subitems in a room remodeling

or renovation contract. In every case where barrier materials were applied,

it was found that average direct costs-^ were significantly and negatively

related to job size (square feet of wall area covered).^ This suggests

that using the average cost from new construction activities in housing

renovaton might lead to persistent errors in (per square foot) average

direct cost estimates which would lead to significant errors in the estimate

of the total cost of the job. The differences in job size between major

new construction projects and housing renovation are so great that it is

reasonable to assume that the new construction average costs will be

significantly lower than those experienced in renovation activities (see

figure 3.2).

R.L. Peurifoy, Estimating Construction Costs , McGraw-Hill Book Co., New

York, 1958.

Robert E. Chapman and Joseph G. Kowalski, Guidelines for Cost-Ef f ective

Lead Paint Abatement , National Bureau of Standards, Technical Note 971,

January 1979.

'Recall that direct costs do not include a markup for the contractor's

overhead and profit.

A negative relationship between job size and average direct cost means

that as the job size increased average direct cost decreased.

40
41
The "Vertical" Impact of Job Size

Although errors may occur at the horizontal level of detail in a construction

cost estimate, some cancellation may in fact occur (i.e., positive errors may be compensated for by negative errors) . If for any particular task the

direction and size of error is random, and if each subitem comprises a small

proportion of total project costs, then violations of the constant cost

assumptions will likely be canceling in effect because of the cumulative

nature of cost estimation.^ Since a cost estimate for a new construction

job is quite detailed in the vertical direction (hundreds of subtasks may

be involved), the statistical laws involved in the summation approach will

tend to promote a minimization of the aggregated error. Thus, there is a built-in safeguard against large errors of estimation for the job as a whole which is due to the additive nature of cost estimation. Again job

size is important, but now in the vertical sense. ^ Here increasing job

size will be associated with an increasing number of estimates for the

required subtasks. Since job sizes are smaller in housing renovation

activities, the potential number of subitems will also be smaller. This means that the chances for the estimation error of any one subtask to

cancel an error in another subtask is reduced. This implies that estimating the costs of housing renovation may be inherently less accurate than

for new construction activities. Unfortunately, it is not clear how many

subtasks are needed in order to achieve a satisfactory, much less optimal

level, of protection against estimation errors.

3.2 TREATMENT OF PRODUCTIVITY

The data reported in cost guides can be adjusted relatively easily for

differences in wage rates and material prices (see equation 2.3). As we have seen in the earlier parts of this paper differences in average costs

because of differences in job size may be less amenable to judgmental modification in transfering new construction cost experiences to housing

renovation applications.

The statistical laws of large numbers provide a theoretical rationale for

this claim.

Harrison D. Weed in his report, A Study of Variability of Construction

Cost Estimates , U.S. Army Armament Command, Rock Island, Illinois, July

1976, found that as job size increases, the coefficient of variation of

bids submitted fell. (See page 11 of the above cited report.) His

finding is consistent with the discussion above.

42
Inherent in the arithmetic of cost accounting equations (equations 2.1

through 2.3) was that job size will be reflected in changing values of the

productivity figures associated with the average labor costs. Job size,

however, is not the only factor which affects productivity

Repair, remodeling, and renovation contracting is conducted in a very

different construction environment than new building construction. Productivity figures for renovation activities (beyond the modification due

to job size differences) may differ from productivity in new construction

because of factors completely foreign to typical new construction activity.

Several factors readily suggest themselves.

First, productivity of labor will undoubtedly be affected by the occupancy

status of the dwelling unit. The productivity of labor will be affected

by the accessability of building elements requiring treatment. Such access may be significantly reduced due to interference from the occupants.

It seems likely that productivity will be affected by the size and composition

of the crew carrying out the work; the construction practices which govern

how these workers carry out their tasks; and the relative skill levels of

the labor force engaged in housing renovation. Whether labor is union or non-union may have a consistent impact on average productivity. It shouldn't

be assumed that union labor is more productive in the renovation environment.

The contrary may be true because renovation activities may best be served

by labor with breadth of skills rather than depth of skills. For example,

one writer argues that:

"More skill is required for remodeling work than for new construction.

Most people working on new homes are highly specialized; one carpenter

will do nothing but hang doors, while another installs trim and another

lays floors. This kind of specialization is not practicable in

remodeling, and each worker must have a variety of skills."^

Past empirical studies indicate that technical as well as economic

considerations are of crucial importance in estimating the level of productivity. See Robert E. Chapman and Joseph G. Kowalski; Guidelines for

Cost-Ef f ective Lead Paint Abatement , National Bureau of Standards,

Technical Note 971, January 1979.

R.M. Burns, How to Buy and Fix Up an Old House , Home-Tech Publication,

Bethesda, Maryland, 1976.

43
If higher paid union workers are proportionately more productive than

non-union workers, the total wage bill may be lower under union than

non-union contracts. The fact that union workers are accepting reduced

scales in order to get major rehab jobs in several large urban areas,

however, seems to indicate that the total wage bill is higher for union

contracts .

•'^ »^

Another factor affecting labor productivity will be reflected in differences,

if any exist, in the amount of mechanical equipment and specialized tools

that workers have available to them to work with. Since housing renovation

contractors tend to be small relative to new construction contractors and

subcontractors, it seems reasonable to assume that less "capital" equipment

will be available to their workers.

One other important factor bearing on productivity is managerial skill.

Given that new construction contractors often have fairly sophisticated

management systems and techniques, it implies that their average productivities will be higher than in the area of housing renovation where the

physical process is more complicated and less expertise is available.

Although none of the above factors have been explicitly quantified in terms

of their impacts on the levels of productivity in housing renovation, they

do suggest that differences in the working environment are large enough to

cast doubt on the applicability of new construction productivity figures

for renovation activity.

Another question of crucial importance is whether the production function

for new (housing) construction is the same as for housing renovation.

Since the technical options open to a renovation contractor are much more

constrained than for a new housing contractor, it is likely that the physical

process will exhibit systematic differences in the utlization of labor and

materials.^ The levels of productivity are therefore likely to differ

significantly. Past empirical studies'^ have shown that production functions

for different classes of construction activities may be significantly

"Unions and Contractors Try to Get More Rehab Work," Engineering News

Record , July 22, 1976.

"HUD Launches a New Rehab Program," Engineering News Record , November 25, 1976.

The mathematical concepts underlying this statement are based on the Le

Chateller principle. For an indepth discussion of the Le Chatelier

principle the interested reader is referred to Eugene Silberberg, The

Structure of Economics; A Mathematical Analysis , McGraw-Hill Book Company,

New York, 1978.

For an authortative source, see John S. McConnaughey , Jr., Production

Functions in Contract Construction For the United States, 1972 (unpublished)

Ph.D. dissertation, Michigan State University, 1976.

44
different. This would imply that marked differences in the marginal

physical product of the factors of production (labor, materials and

equipment) would quite likely exist.

3.3 MARKUP AND ITS DETER^IINANTS

The average material cost and average labor cost for each subtask must be

multiplied by a markup factor to arrive at an average total cost estimate

for that subtask. By definition, the markup figure is equal to the sura of

the total costs for a job divided by the variable costs for that job. By

definition, average total cost is equal to average variable cost plus

average fixed cost. Fixed costs are composed of general overhead, project

specific overhead, and a target profit figure. General overhead represents

those costs which represent payments for general office expense, rental for

space, payments for such services as telephones and advertising, and the

portion of the salaries of management which is not assignable to any specific

job. Project specific overhead is a cost which would not be incurred if

the project were not undertaken but which can not be assigned to, or tied

to, any particular subtask' s output level. Profit, although included for

cost estimation purposes as a targeted dollar amount, is in reality the

residual which remains after total costs have been subtracted from the

gross revenue of the project.

In practice, the estimating manuals and guidebooks treat the markup factor

as a constant across the "horizontal" items of the cost estimate. Such a practice will be generally accurate when a firm is involved in a project

which is large enough so that it is producing in the constant cost range

(see figure 3.2) of its average total cost curve. If a particular job is

small, i.e., if the firm is operating at output levels which do not demonstrate constant costs, the markup factor will be inversely related to the

job size. This is because it is easier to lose money on a small job than

a big one due both to unanticipated technical problems and relatively

higher average costs (see figure 3.2 where it is assumed that renovation

activities are carried out at a level below Qj^) . In addition, markup

factors will be higher for smaller sized jobs because fixed costs are

spread over a smaller base. Both of these effects suggest that housing

renovation markup rates should be higher than new (housing) construction

rates. One further point is that we would expect markup rates for the

successful firm to demonstrate greater variability across jobs. This is

due both to variations in job size and in the contractor's assessment of

the building's condition.

'•In an accounting sense, project specific overhead is often a horizontal

or line item in a cost estimate.

45
The above, although quite speculative and requiring empirical confirmation,

suggests that new (housing) construction markup factors are not transferable

to housing renovation. And that doing so could lead to gross underestimates

of the true costs of renovating the unit.

3.4 ASSESSMENT OF RELIABILITY

The primary difficulty with (subtask) cost per unit estimates based on

"expert" opinion is that the user of such estimates, if not an "expert,"

may improperly apply those estimates. Experts in subfields of construction

will have some shared scenarios in mind when referring to a subtask like

removing and replacing a sink in the same location of a 40 year old

house. The steps involved in such a task will be well understood by such

experts. Thus, variations between the actual case in hand and a reported

case in a cost guide may be clear to an expert cost estimator in housing

renovation construction. However, the cost guides cannot detail the sce- narios which lie behind any particular subtask figure. Space considerations

alone make such detailing impossible.

The irony in all this is that cost figures reported in cost guides are most

usable, from the viewpoint of accuracy, by those who need to refer to them

the least. This implies that data in cost guides become less and less

reliable as the user becomes less and less expert, (Incorrect estimates

based on guidebook data can be explained away by asserting that the guidebook

user did not correctly modify the reported figures.) Thus, a homeowner

(the potential guide user most in need of assistance) attempting to use a repair and alteration guide will be least capable of judgmentally modifying

the reported figures.

If the data were statistically derived it would be possible to define an interval estimate which could be believed with a specified degree of confidence. Data of this type, if reported, would enable a user to estimate a range which would probably contain the expected costs. However, data of

this kind is considerably more expensive to collect. For example, the

repair and alteration manuals break out figures for over 500 separate

items. If the data were to be statistically based, samples of at least 20

observations would probably have to be collected for each subitem for each

major region of rehab activity. This means that at least 10,000 separate

figures would have to be collected and processed for each region. Such

data collecting, processing and evaluating requirements are probably beyond

the present size of the market for this kind of information.

Cost estimation is a form of prediction. Reliability refers to the accuracy

of predictions; more intuitively, the difference between predicted costs

and actual costs. Esoteric discussions exist on how to evaluate predictive

methods. Unfortunately, there is no basis for assessing, in these terms,

the reliability of the data contained in cost manuals. •'^

The above statement applies with equal force to both the new construction

manuals and the repair and alteration manuals.

46
In order to test the reliability of the figures in any manual, a systematic

matching or comparison of subtask predictions and actual experiences would

have to be undertaken. Also total job estimates and actual figures would

have to be compared. Errors of prediction could be measured, and explicitly

quantified statements could then be made about the accuracy of the cost

estimates based on guidebook data.

A mechanism does not exist for such an undertaking for a variety of reasons. Subcontractors and contractors usually do not rely on guides for their final

estimates. They use them primarily as references. Architects and engineers

may use such reference data in preliminary stages of design work but have

little incentive to compare preliminary estimates with actual figures.

Government contracting officers can only compare their predicted bids (when

based on guidebook data) with actual bids. Such a comparison is not between

predicted costs and actual costs, but between predicted costs and predicted

costs (to the extent that a bid price represents a contractor's own prediction)

.

The above discussion implies that given currently available information it

is impossible to assess the reliability of cost figures reported in such

manuals. Statistically derived data at least is conceptually assessable.

Based on existing sample data, standard errors for each factor in the cost

estimation "equation" could be computed as well as standard errors of

prediction and confidence intervals Judgmentally derived estimates may

be of such a nature that in theory there may be no way of demonstrating

that they could be wrong

.

In order to illustrate the notion of reliability we can compare cost data

given in an NBS report^ on the actual cost of refinishing interior walls

to cost estimates derived from repair and alteration manuals. Per unit

material and labor costs and their sum, per unit direct costs, are reported

in the NBS report for gypsum wallboard and plaster and metal lath. It is

important to point out that these costs were actually experienced by con- tractors for surface refinishing activities in a number of separate dwelling

units. Twelve separate observations are reported in the NBS report on the

costs of installing and painting gypsum wallboard on interior walls. Nine

separate observations of the costs of plaster and lath are also observed.

Three series of estimates are derived from three different cost manuals.

They are the per unit direct costs of: (1) labor; (2) materials; and (3)

Those readers who are interested in the mechanics of "fitting" confidence

intervals about the estimated cost for a particular subtask are referred

to Robert E. Chapman and Joseph G. Kowalskl, Lead Paint Abatement Costs:

Some Technical and Theoretical Considerations, National Bureau of Standards,

Technical Note 979, February 1979.

Robert E. Chapman, Economic Analysis of Experiment Lead Paint Abatement

Methods: Phase I , National Bureau of Standards, Technical Note 922,

September 1976.

47
the total. A comparison of the observed average direct cost experiences

with the cost estimates permits us to compute a percentage error of prediction for each series of manual estimates. Table 3.2 contains the

results of this exercise.

Upon examination of table 3.2 it can be seen that the manuals underestimate

both labor and material costs. In general, however, the underestimates for

labor costs are more serious than those for material costs. The dollar

errors, the difference between actual and estimated costs, range from six

cents to 42 cents per square foot. For even a modest job, say refinishing

a single room (approximately 500 square feet of wall and ceiling area)

these errors could exceed $200. In terms of percentage errors, the amount

actual costs exceeded estimated costs, the manuals ranged from 4.8 percent

to 80.3 percent. It may come as a surprise that the widely used technique

of installing gypsum wallboard had relatively higher percentage errors

than for plaster with metal lath.

Data similar to that illustrated in table 3.2 would be needed on a subtask

by subtask basis in order to assess the overall reliability of the figures

reported in the cost manuals. To do this adequately would require a com- prehensive research project. In the absence of such a research project, a quantitative assessment of the reliability of the data in the cost manuals

cannot be made.

Two comments on the data in table 3.2 are worth reiterating. First, the

direct cost estimates are in all cases underestimates of the actual average

experiences reported in table 3.2. Although it would not be appropriate

to generalize from two small subtasks to the population of subtasks as a whole, it is of interest that the examples chosen do underestimate housing

renovation subtask costs. This is consistent with the observations in

sections 3.1 and 3.2 of this paper. Second, note that the repair and

alteration manual's estimates are not appreciably better than the new construction manual's estimates.

3 .5 Treatment of Risk

Although it was possible to give only a qualitative assessment of the

reliability of the data in the cost manuals, it is possible to discuss in more detail the means through which risk can be treated in these manuals

.

As in the previous section, the discussion will be qualitative rather than

quantitative

.

There are three basic methods of treating risk in construction cost

estimation. These three methods are: (1) judgmental modifications;

(2) defining a confidence interval about the estimate; and (3) probabilistic

(monte carlo simulation) concepts. Each method will be treated in turn.

In the discussion which follows, unless explicitly stated, it will be

assumed that the cost estimating technique being discussed is based on average total costs.

48
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49
Judgmental Mof If Ications

By far the most popular method of risk asessment is to make use of judgmental

modifications. Unfortunately, the use of such a method can only be viewed

as a placebo since no a priori grounds can be established which would

enable the user to associate percentage changes in risk due to a particular

judgmental modification. In reality, judgmental modifications are more a means for adjusting the cost estimate to reflect a change in some technical

attribute rather than a measure of the interaction of technical attributes

and economic forces in the market place. Thus, experts applying judgmental

modification methods are, in a sense, superimposing a parametric cost

estimating procedure on top of an average total cost procedure. Although

this concept may seem appropriate, it is important to point out that parametric or regression based techniques obey certain rules of the real numbers.

This is because most cost estimating relationships are fitted using a continuum of data. An expert who modifies the average total cost procedure,

however, is usually only capable of establishing a relative order or rank

among different scenarios. Such an approach makes use of what is known as ordinal level data.^

Unfortunately, ordinal level data need not obey all the rules of the real

number system which a regression based procedure obeys by construction.

Since the estimator of the rehab project is concerned with the absolute

rather than the relative cost of the project, it does not appear that any

level of significance can be attached to the "reduction in risk" associated

with the judgmental modification method.

Defining a Confidence Interval About the Estimate

The second method for treating risk involves fitting a confidence interval

about the estimate. In section 2.2 where the mechanics of fitting a con- fidence interval were discussed, it was shown that an estimate of the

standard deviation of the predicted value must be known in order to define

the upper and lower limits on the confidence interval. Since none of the

guidebooks cited in table 2.1 provide measures of variability associated

with their average total cost estimates, it is not possible to fit a

Statisticians have defined four levels of measurement in data. The first

level, nominal, makes no assumption about the values being assigned to the

data. The second level, ordinal, assumes that it is possible to rank-order

all categories but that any numeric values assigned to the categories does

not imply that any other properties of the real numbers follow. The third

level, interval, implies that the distance between categories are defined

in terms of fixed and equal units. The fourth level, ratio, implies that

a zero point is inherently defined by the measurement scheme. In the context of the previous discussion, parametric procedures are usually estimated

at the interval or ratio levels. For an excellent discussion on the levels

of measurement see Norman H. Nie, C. Hadlai Hull, Jean G. Jenkins, Karin

Steinbrenner , and Dale Bent, SPSS; Statistical Package for the Social

Sciences , Second Edition, McGraw-Hill Book Company, New York, 1975.

50
confidence Interval about the estimate. Even if the guidebooks published

such information, there is some question about the effects of aggregation

on the variance of the sample. This criticism stems from the fact that

observations are drawn from cities across the nation. Consequently, the

structure of costs in St. Louis might so differ from Boston that the

variances about the means would not be homogeneous. In this case, pooling

Information about the variance of the sample would not be justified.

The thrust of this criticism is that in order to fit confidence intervals

about the average total cost figures in the guidebooks, it would be

necessary to have sample variances for each city.

Although the use of confidence intervals helps in assessing the impacts of

risk on the rehab decision, they are lacking in some respects. For example,

a 95 percent confidence interval implies that if samples were taken over and

over under identical circumstances from the same population, then 95 out of

100 intervals would contain the true mean of the population. If the sample

is not random or the desired estimate is for an atypical case, then the

estimate of the mean may be biased. This implies that the concept of a confidence interval loses some of its meaning. The previous statement is

reinforced by the fact that no discussions of sampling and nonsampling

errors are given in the guidebooks. Consequently, it is not possible to

quantify the bias, if any, associated with these average total cost

estimates. Under these circumstances fitting a confidence interval must

be viewed as a rather academic exercise.

Closely related to the concept of a confidence interval is a technique known

as sensitivity analysis. This technique derives its name from measuring the

cost sensitivity to a change in one or more factors involved in the process.

That is, it permits one to determine how "sensitive" the average total cost

estimate is to a change in one or more of the key factors. Sensitivity

analysis is extremely difficult to apply to average total cost procedures

due to the aggregation of all key factors affecting average total cost.

Parametric procedures are quite easily adapted to sensitivity analysis,

however. More succinctly, the sensitivity of the estimate to a change in

a factor may be defined by simply differentiating the cost estimation

equation. In the simplest case, where the relationship is linear and cost

depends on only two factors, we see that

C = 3o + 3iXi + 32X2

which upon differentiation (partially) with respect to reduces to

8C = BiOXi)

That is, the change in cost, AC, associated with a change in X]^, AX]^, is

given as

AC = ei (AXi)

51
If both factors were likely to change, it would be necessary to compute the

total derivative, which is given as (using the A notation)

AC = 3l(AXi) + 32(AX2)

The change in C would then be computed by plugging in the AX' s . Delta C

would then be added to the estimate of C. A similar equation can be written

for a linear relationship containing n factors. Now if due to the riskiness

of the process, one felt AX^, would be -1 with a probabilijty of 0.1 and

+5 with a probability of 0.9, the expected value of AX]^, AXj^, would be

AXi = (0.1)(-1) + (0.9)(5) = -0.1 + 4.5 = 4.4

If probabilities (either objective or subjective) for the other X's could

be computed, then the expected change in average total cost due to the

riskiness of the process would be

AC = 3i(AXi) + 62(^2)

or in general terms

n _

AC = E 3i(AXi) 3.1

i=l

As in the previous case, AC would then be added to the estimate of C to

determine if a revision of the cost estimate is needed.

In the previous example where the relationship was linear, the rate of

change of the cost estimate was independent of the level of any given

factor. Furthermore, by virtue of linearity, a change in one factor had

no effect on any other factor. That is, all factors are assumed to be

independent and to cause no interaction. In some cases these assumptions

may not be justified. Based on the information presented in section 2.4,

it can readily be seen that cost functions do not suffer from these

problems.-^ This is one of the many reasons why the cost function approach

will be stressed in chapter 4. Another reason is the relative ease with

which probabilistic methods can be introduced into its framework.

Following the notation of equation 3.1, C may be expressed as n

AC = E OC/8Xi)(Mi)

1=1 '

52
Proballlistic (Monte Carlo Simulation) Concepts

The third and most comprehensive method for treating risk draws upon

probabilistic concepts. Probabilistic concepts are the most appropriate for

treating risk because they allow the estimator to "fit" a distribution about

each factor and then run through a whole series of "what if" questions.

Although it is possible to manually fit a confidence interval about an average

total cost estimate, most probabilistic methods rely on computer software

for support. The major advantages of probabilistic methods are: (1) they

do explicitly treat risk, and (2) they do not suffer from some of the criticisms voiced about confidence intervals. In particular, if one has strong a priori beliefs about the structure of rehab costs, it is still possible to

apply probabilistic methods even if the average total cost estimate in the

guide is biased. More pragmatically, it is possible to adjust for the bias

through a judicious choice of a distribution so that the behavior of the

distribution parallels the critical steps in the process.

As was shown in section 2,3, probabilistic methods can be applied to both

average cost and parametric procedures. If one wishes to use probabilistic

methods in conjunction with a parametric procedure, however, it would be

necessary to fit a probability distribution about each factor rather than

just about the average total cost estimate. It is fairly easy to see that

this approach is superior to the average total cost approach since costs are "sensitive" to a change in any factor.^ The "degree" of sensitivity is,

however, dependent upon the relative weight of that factor in determining

the renovation cost for that task. In the case where the relationship is

linear, the application of probabilistic methods is rather simple and

straightforward. The computer software package will first perform a monte

carlo simulation on the cost estimating equation (relationship). As mentioned in section 2.3, this process will be repeated until a cost profile

for the task results (see figure 2.5 in section 2.3). The user can then

choose a particular level of risk which he is willing to accept. In order

to compute a confidence interval about the desired level, the user must

replicate the simulation a certain number of times (the number of replications

is dependent on the desired "tightness" of the confidence interval).

Probabilistic methods may also be applied to cost functions. To properly

apply these methods to a cost function is relatively more difficult, however.

This can be seen by noting that the basic form of the cost function, as given by equation 2.10, was shown to be nonlinear. From the discussion in

Many of the major advantages associated with the use of probabilistic methods

have already been discussed in section 2.3. Consequently, this discussion

will focus on how these methods would be used in conjunction with procedures

based on average total costs, parametric relationships, and cost functions.

If costs are not sensitive to a particular factor, then the estimate of its

coefficient is probably not statistically significant. In such an event,

that factor could probably be eliminated from the cost estimation equation.

53
section 2.4, equation 2.10 is known to be log linear. That is, if one takes

the natural logarithm of equation 2.10, the resulting equation is linear in

the logarithms of the factors. In order to determine the appropriate distribution, it will be necessary for the user to fit a probability distribution

to the logarithms of each key factor. Guidelines for doing this were provided in section 2.3. Monte carlo simulations can then be performed just

as in the previous cases. Probabilistic methods can also be used to perform

a type of sensitivity analysis. The type of sensitivity analysis may be

either qualitative, in the sense that it is based solely on judgmental modification, or quantitative. For example, one might not be sure about the

true distribution of a particular factor. (Perhaps the Chi-square test

accepted two distributions.) The monte carlo simulation could then be

repeated under the assumption that the factor was distributed differently.

By the same token, uncertainty about the condition of the building can be

incorporated by either shifting the entire distribution upward or by

requiring the distribution to be more skewed. Such an approach could

thus complement expert judgment about the physical process. In any event

by using the baseline estimate as a reference point, it will be possible

to attach a percentage change in the risk being borne by the investor due

to a change in a particular factor or group of factors.

Chapter Summary

The emphasis of this chapter has been on the assessment of per unit cost

estimation techniques relating to the impact of changing job size, the

treatment of productivity and contractor markup. Two examples were also

presented which introduced the importance of assessing the reliability of

the model and its ability to explicitly deal with risk. Although each of

the four techniques is capable of generating per unit cost estimates, the

focus was on the cost index/average total cost technique used in most

costruction cost manuals. The inflexibility of this technique was shown

to exist across the horizontal measure of job size, the treatment of productivity and contractor markup. Since the vertical measure of job size

depends on the number of subtasks, no one technique can be claimed to be

best. However, in the event that a technique systematically under or overestimates the cost of the job, increasing the vertical dimension will

increase the overall error due to the additive nature of the error term.

Theoretical and empirical evidence was also presented which suggested the

technical constraints imposed on rehabilitation activities would lead to

systematic differences in the utilization of construction labor and

materials which could profoundly affect the productivity of the factor

inputs. Qualitative evidence was advanced that showed why the use of

markup factors had to be more closely studied and that transferring new

construction markup rates to rehabilitation projects was probably not

justified. An example, based on actual cost data was presented to show

that the reliability of the cost estimates from guidebooks specializing in

repair and alteration are probably no better than those for new construction.

Both types of guidebooks resulted in substantial underestimates of the

actual costs experienced.

54
Of the three methods for dealing with risk, judgmental modifications,

confidence intervals, and probablistic concepts, only the probabilistic

approach is capable of assessing the risk being borne by the investor as a function of changes in one or more factors. In addition, this method was revealed to be less sensitive to any biases in the average total cost

figures presented in the guidebooks or in technical reports, Probalistic

methods were also shown to complement expert judgment since they are flexible enough to allow for judgmental modifications in cost figures due

to a priori beliefs about the system or process.

55
Facing page:

The application of the pevfovmanoe

ooneept is a key ingvedient in reduoing

the costs of residential vehahilitation.

If this approach were combined with a technically sound cost estimation procedure, the risks of undertaking a particular residential rehabilitation

Qoh could he reduced significantly

.

56
4. AN APPROACH FOR DEALING WITH COST VARIABILITY

The focus of this chapter is on combining the advantages of the use of cost

functions with the cost reducing potential of the performance concept in

developing a means for dealing comprehensively with cost variability.

Although the discussion in this chapter will be of a theoretical nature,

empirical results will be presented which support the claim that this

method of solution will permit substantial cost reductions in some types

of rehab work.

57
4.1 COST FUNCTIONS AND THE PERFORMANCE CONCEPT: AN OVERVIEW

In section 2.4 it was shown that the use of cost functions permitted the

requirements for a sound cost engineering approach to be integrated with

traditional economic theory. More specifically, the duality relationship

between cost and production functions permitted:

(1) local market conditions; and

(2) technical considerations of the renovation process

to register their impact on residential rehabilitation costs.

In section 3.5 it was shown that the use of probabilistic methods permitted

the effects of risks to be explicitly introduced into the estimation of

rehab costs. When these advantages are combined with the cost reducing

potential of the performance concept, an approach which reduces both cost

and cost variability begins to emerge.

There are two major advantages of the performance concept which make it

highly desirable for application to the rehab process:

(1) its cost reducing potential; and

(2) the fact that it provides decisionmakers with greater latitude in

making choices.

The major disadvantage is that performance is difficult to measure unless

it can be tied to some prescriptive solution. In order to get around this

obstacle, previous researchers have focused on the development of a hybrid

system known as an equivalence methodology.-*- This system separates the

levels of performance into a series of discrete steps, or states, each of

which can be tied to a prescriptive solution. Futhermore, the equivalency

methodology approach is sufficiently general that it is possible to define

a distinct level of performance (or performance score) associated with

each code or code attribute under consideration. Thus, the use of an equivalent methodology:

(1) provides a mechanism for demonstrating code compliance; and

(2) promotes greater latitude in making choices by permitting

substitutions among building components.

It is the second reason which lends support to the claim that this approach

will reduce cost variability. (A detailed discussion of the cost reducing

potential of this system is given in section 4.2.) This relationship may

be seen more clearly by noting that greater freedom in making retrofit

Harold E. Nelson and A. J. Shibe, A System for Fire Safety Evaluation of

Health Care Facilities , National Bureau of Standards, NBSIR 78-1555,

November 1978.

58
choices will permit the investor to avoid some of those situations which

are more risk prone. This would imply that the variance about any estimate

where a substitution was made could be reduced. Consequently, the end

result would be a reduction in the variance of the budget estimate.

The discussion which follows will be divided into four parts. The first

deals with the structure of the equivalence methodology. The second outlines

how cost functions and probabilistic methods can be used to develop a cost

structure for the problem. The third provides a general mathematical

formulation of the problem as well as a graphical means for selecting the

least-cost rehabilitation (retrofit) strategy and for defining an alternative class of retrofit strategies. The final part provides guidelines for

selecting the best retrofit strategy from those provided by a computerized

version of the "equivalency methodology."

STRUCTURE OF THE EQUIVALENCE METHODOLOGY

This subsection will develop a series of tables (matrices) which reveal the

basic structure for a generalized equivalency methodology. The discussion

is of necessity abstract in order to highlight the flexibility of such a generalized methodology.

Prior to the development of the generalized methodology, however, it is

necessary to define several terms and state the assumptions upon which the

analysis rests. The following terms will be used throughout the discussion.

(1) Building Component - any portion of the building or a building

system for which a prescriptive solution is or can be defined in

the code(s) under consideration.

(2) State - a discrete level of performance for a particular building

component

.

(3) State Value - a numeric score associated with the level of

performance of that state.

(4) State Variable - a variable which takes on a value of 1 if the

building component is in that state and a value of 0 if it is

not

.

(5) Code Requirement - a numeric score associated with the level of

performance required by each code under consideration.

(6) Score Assessment - a numeric score associated with the level of

performance provided within the building for each code under

consideration

.

(7) Retrofit Cost - the cost of moving from any given or initial

state to any other state.

59
The following assumptions will be made in the discussion which follows

(1) All code requirements must be satisfied simultaneously in order

for the building to be deemed in compliance.

(2) The score assessment is the sum of all state values for those

building components which affect the code under consideration.

(3) Each building component can be in one and only one state.

(4) The state of a building component is defined by the worst case condition of that component.

(5) The number of building components is much larger than the number

of codes considered.

(6) It is possible to stay in a given or initial state at no cost.

(7) Moving to a lower state is impossible.

As one might suspect, the structure of the generalized equivalency methodology

will be rather complex. Fortunately, previous efforts in the area of fire

safety have demonstrated that such a methodology can be easily adapted to

computer optimization. The specific method which will be used in this

exposition is the mathematical programming technique known as linear programming. In its usual context, linear programming deals with the problem

of allocating limited resources among competing activities in an optimal

way. At the foundation of any linear programming problem is a mathematical

model which describes the problem of concern. The term "linear" refers to

the requirement that all mathematical functions in the model are linear.'^

The term "program" is used in the general sense in that it refers to a plan rather than a computer program perse. (The basic reason why all

mathematical functions involved in the program are linear may be explained

through reference to tables 4.1 and 4.2.)

Assumptions 4 and 7 are not necessary from a computational viewpoint.

They are included here to promote a more consistent approach to the

engineering problem.

A linear function is defined as a function of the form

f(X) = aQ + a^xj + . . . + ajXj + . . . + a^Xn

where a^ are coefficients not all zero and the xj are variables. The

geometrical representation of a linear function is a straight line, a plane, or a hyperplane. For example, f(x) = a + bx, a straight line, is a linear function where g(x) = c + dx^, a parabola, is not.

60
Table 4.1 Building Component/State Score Matrix

Building

Comoonent

1

Level of Performance

State and Value

2 3 m

1 Vii V,2 Vl3 Vim

2 V21 V22 V23 V2tn

3 hi y32 ^33

• V3m

•

n Vnl V„2 V„3 Vnm

lable 4.2 Score Assessment Matrix

Building Code

Component 1 2 3 P

1

2

3

n

Total Pi- P2 = P3 =
The foundation of the linear programming model is shown in table 4.1, the

Component/State Score Matrix. An examination of table 4.1 reveals that

there are n building components and m states associated with each building

component.^ The state value associated with the i^^ component and j^^ state

is denoted V-j^j. The state values, V-[^j, ascend in score as one moves from

left to right. That is, Vn < < • • • < ^im* ^ general point of

departure, it will be assumed that negative state values represent undesirable circumstances whereas positive state values represent desirable

circumstances. A state value of 0 is assumed to be neutral.

Based on assumption 2 and the previous discussion it can be seen that a movement to a higher state represents a potential retrofit.^

Thus far, the mechanics of actually calculating the level of performance

provided within the building for a particular code have not been discussed.

Table 4.2 provides the format for performing these calculations. Notice

that table 4.2 is a matrix with n rows, one for each building component,

and p columns, one for each code or code attribute under consideration.

An examination of table 4.2 would reveal that some cells of the matrix are shaded. This is due to the fact that some building components have no effect on the level of performance required by the code. Thus, to include

them in the score assessment for that code would be redundant. In order

to perform the score assessment calculation, one would take the state

values from table 4.1 and enter them in the appropriate spaces in table

4.2. Once this task has been completed, it is only necessary to sum each

column and enter the resultant score in the appropriate space in the row

labeled "Total" in table 4.2. Each of these scores are labeled P^, P2>

. . . Pp, indicating the level of performance provided within the building

for each of the p codes under consideration.

In reality there need not be an equal number of states for each building

component. This stronger assumption is used only for purposes of expository

clarity. It will be shown in the subsequent discussion that an unequal number

of states can easily be accommodated in the mathematical formulation of the

problem.

It is of course possible that a movement to a higher state in one building

component will affect the performance of another building component. This

"row interdependence" can easily be handled through the use of dummy state

variables representing dummy states. See exhibit 4.2 in section 4,2 for an example of a "row interdependence."

Rigorously speaking, each cell shaded in table 4.2 is assigned a weight of

0 and hence contributes 0 points to the score assessment for that code,

whereas each cell not shaded in table 4.2 is assigned a weight of 1. It is

of course possible to have any weight w-j^j, but for purposes of expository

clarity, is assumed to be either 0 or 1

.

62
The code requirements are given in table 4.3. These numeric values are determined by identifying the state in table 4.1 which includes or is the

prescriptive solution defined in the code. These values are then entered

in the appropriate spaces in table 4.2 and summed in order to get the code

requirements, Rj, R2, . . . , Rp • Note that the equivalency methodology

is sufficiently general to accomodate variations in the level of performance

required by different code jurisdictions.

The final step, determining if the building is in compliance with each code,

is outlined in table 4.4. This step is taken by transferring the first

set of scores P]^ , ?2> ••• table 4.4, from table 4.2 to the boxes

labeled , P2> ••• Pp, in table 4.4. The second set of scores

^1 > ^2 » '^p' then transferred from table 4.3 to the boxes labeled

R]^ , R2 , . . . Rp . Code equivalence is tested by determining if the

differences between the first set of numbers, P]^ , P2 , ... Pp, and the

second set of numbers, Rj^ , ... Rp, are all greater than or equal to zero

.

In the event that one or more of the differences is negative, the building

is deemed not to be in compliance with the code. At this point it becomes

necessary to define a plan of correction or, more simply, a retrofit

strategy. Although retrofits can be defined based on score improvement

alone, such an approach overlooks the potential of the linear programming

procedure alluded to earlier.

COST STRUCTURE

Given the basic framework of the equivalency methodology discussed in the

previous subsection, it becomes possible to use cost functions and probabilistic methods to develop a cost structure for the problem. The

structure of costs is, however, more complicated than that discussed in

section 2.4. The nature of these complications is twofold. First, through

reference to table 4.1 it can be seen that there are nm (n rows by m columns) -1 possible states The likelihood of one cost function being able to

treat all of these cases is quite remote. Thus, in the discussion which

follows, it will be assumed that there are N distinct cost functions, 1 < N < nm. The second complication is of a more subtle nature.

From the discussion in section 2.4 we know that each cost function contains

a set of key factors. Since we now have N cost functions, it is likely

that some of them will contain factors in common. This implies that greater

care must be exercised in applying probabilistic methods. For example,

several building components might be associated with certain categories

of plumbing system retrofits all of which require a skilled plumber. We

shall assume that the same plumbing contractor would perform whichever

tasks are selected by the Investor and would use the same staff in carrying

The number of possible retrofits is at most nm-n and, in general, will

depend upon the initial state of each factor.

63
Table 4.3 Code Requirements Matrix

Code

1 2 3 p

Score R2 R3 Rp

Table 4.4 Equivalency Test Matrix

Code P - R = > 0

Yes No

1 Pi - Ri =

2 P2 - R2 =

P Pp — Rp =

64
them out. Now if each retrofit were treated independently using

probabilistic methods, it would be possible to get differing wages from

the same plumber doing different tasks. Such a state of affairs does not

accurately reflect the way in which construction services are contracted.

Fortunately, the solution to the problem is rather simple. To see this,

denote the key factors contained in the first cost function as K][, in the

second cost function K2, and so on. The universe of key factors, K , is

thus the union of all key factors in the N cost functions. This relationship may be expressed mathematically as

As in the discussion in section 2.4, each element of K has associated with

it an estimated mean k and a probability distribution. -'-

Once all elements of K have been identified and the relevant information

about their probability distributions have been put into the software

package, it becomes possible to generate a meaningful set of cost estimates.

The cost estimates desired for this problem are shown in table 4.5. From

the table it can be seen that a cost estimate is needed for each building

component/state pair. Based on assumption 6, however, we know that it is

possible to stay in the initial or given state at zero cost. This implies

that costs of staying in the n input states of the nm possible states are

0. From assumption 7 we know that it is impossible to move to a lower

state. This may be assured by making the cost of a lower state arbitrarily

high. For example, n times the highest single retrofit cost. Thus, although costs for all nm entries of table 4.5 will eventually have to be

computed, the only costs which are of importance at this stage are the

costs associated with the potential retrofits. Since the goal of this

operation is to reduce both cost and variability, probabilistic methods

must be used in conjunction with the linear programming procedure.

In order to apply probabilistic methods to the problem at hand, it is first

necessary to select a random number for each factor, refer to the probability

distribution of that factor, and select the appropriate value of the random

variable. Once all L random variables have been determined, it is necessary

to plug them into the appropriate cost function in order to get an estimated

cost for each potential retrofit. Just as in the discussion in section

2.4, this process is reiterated until a cost profile for each potential

retrofit is defined (see figure 2.5). The next step is for the user to

In the event that k is a fixed constant, say square feet of floor area,

then the probability distribution associated with k is the point

distribution.

n

K = U Ka

a=l

65
Table 4.5 Cost Matrix Used in Linear Programming Approach

RETROFIT COST

STATE

BUILDING

COMPONENT 1 2 3

• • • m

1 ^12 ^13

• • •

2 C22 ^23

• • • C2m

3 ^31 ^32 ^33

• • • ^3m

•

••

••

•

••

•

••

•

••

•

n Cnl Cn2 Cn3

• • • ^nm

66
define a risk level, say 20 percent. The software package will then survey

each potential retrofit and pick off the retrofit cost which has a 20 percent chance of being overrun. The program would then automatically select

the maximum value, multiply it by some constant, say n, and use it as the

"arbitrarily high" cost for all retrofits corresponding to a lower state

than the initial or given state. The result of this exercise is that

table 4.5 is now filled with all the appropriate cost figures. These

values can now be used as inputs for the linear programming procedure.

What is particularly attractive about this is that the linear programming

algorithm will produce the least-cost solution for achieving compliance to

all p codes. However, by construction this solution has only a 20 percent

chance of being overrun. Thus, we have obtained the least-cost solution

for a given level of risk.

PROBLQl FORMULATIOnI

The subject of the next two subsections is the formulation of the linear

prograramiag problem and a discussion of how such a computer algorithm

would select the optimal solution and generate a set of alternatives.

Certain portions of the discussion which follows are more technical than

previous sections. Therefore, the reader who is primarily interested in

obtaining a conceptual understanding of linear programming and how it may

be applied to the problem of residential rehabilitation is directed to the

subsection entitled "Graphical Formulation".

MATHEMATICAL FORMULATION

To formulate the mathematical model (linear program) for this problem, we shall refer to information in table 4.1. Table 4.1 may be thought of as a matrix with n rows, one for each of the n building components, and m columns

The variable X-j^^ is used to identify the state of each building component.

By definition the state variable X^j is equal to one if the i^ factor

is in the j^^ state and is zero otherwise. (Recall that the score for

each factor is determined by the worst case condition.) The variable V^j

is the score, state value, associated with the i^^ building component

and j^^ state (entry in table 4.1). These values are taken directly from

table 4.1. For example, if the i^^ building component is in the j'-" state

the value, or score, associated with the i*" row is V^jX^^^, since X^^. is

equal to zero for j' unequal to j . Score improvements are thus possible

as a result of retrofits. If the i^^ building component is currently in

the j^^ state, X^. would be the pre retrofit or current state, jc, and

Xj^^ would be the post retrofit or new state. It is of course required

The discussion of the mathematical and graphical formulation of the problem

is a modified and abbreviated version of a writeup prepared by

Mr. William G. Hall and which appears in Robert E. Chapman, William G. Hall

and Phillip T. Chen, A Computerized Approach for Identifying Cost-Ef f ective

Fire Safety Retrofits in Health Care Facilities , National Bureau of Standar

NBSIR 80-1926, January 1980.

67
that j > jc otherwise no improvement has been made. In actual practice

(see section 4.2) the score associated with X^j, V-^ j

, would be greater

than the score associates with Xj^j^,. It Is now possible to show how

costs may be introduced in the mathematical model. The variable C-j^j Is

used to record the cost of upgrading the 1^^ building component from the

Input state, j^, to the j*"^ state. For each possible retrofit being

coasLdered, C-j^j is calculated through use of a cost function.

Keeping in mind that the objective is to match or exceed the scores for each

of the p code requirements in the least costly manner, it is now possible

to formulate the. problem mathematically. The problem is to choose X-j^j,

j^<j<li;> where is the current state of the 1^^ factor and — —-^m' -^c 1 m

is the maximal state of factor 1^ so as the minimize total retrofit

costs

n

E I . ^ij^lj ^ij = 0 or 1

i=l j=jj

subject to

(1) parameter value constraints for each building component given as:

ji

-JmE _ Xi-j = 1 for 1 = 1, ... , n

• 1

(2) total value requirements

2 ^2 VijXij - Yi = Ri

ieS^ j=jj

i I Z VijXij - Y2 = R2

jm

Z E VijXij - Yp = Rp

ieSp j=jj

The superscript on j, j-'", indicates that the maximal state, j^, may vary

and hence the total number of states, as a function of the building

component under consideration.

68
where

j = the state index (column of table 4.1),

V = the score associated with the i b

and j^^ state (entry in table 4.1),

building component

the total retrofit cost of going from the

the j^^ state, for the i^^ building component,

state to

S-^y S2, , Sp = the set of building components not shaded in table 4.2,

R]^, R2, ••• ) ~ mandatory safety requirements from table 4.3, and

^1> » ^p ~ nonnegative surplus variables representing a requirement

excess

.

GRAPHICAL FORMULATION

The mathematical formulation of the problem presented in the previous

discussion is of necessity rather terse. It is also possible to show how

an optimal combination of retrofit options can be selected via a graphical

procedure

.

In order to illustrate graphically how such a computerized procedure would

operate, it would be necessary to use an example in which it is assumed

that the code level is a function of the performance of two techniques,

"A" and "B." A graphical solution is the least-cost means of achieving

compliance to the code is shown in figure 4.1. In the figure, any movement

up the vertical axis indicates higher levels of performance for technique

"A." Similarly, any movement along the horizontal axis indicates higher

levels of performance for technique "B." Consequently, if one were to

construct a straight line which passed through the origin, any movement

along that line (i.e., a move in the north-easterly direction) would

indicate a movement to higher code levels.

In part A of figure 4.1 the lightly shaded region represents all the

combinations of the performance of techniques "A" and "B" which match or exceed the requirements of the code. The lightly shaded region in part A

of figure 4.1 is referred to as the feasible region since all points which

lie along its boundary or within are technically feasible. The boundary

is all those combinations of the two techniques which exactly satisfy the

requirements of the code.

In part B of figure 4.1 a series of equal cost lines are shown. An equal

cost line shows all the combinations of technique "A" and technique "B"

which cost the same. It is drawn based on the assumption that the unit

prices for technique "A" and technique "B" are constant. Higher equal

cost lines imply greater costs are being incurred. By referring to part B

of figure 4.1, it can be seen that the first equal cost line does not

touch the shaded region, implying that not enough funds are being allocated

69
Part A

LEVEL

OF A

LEVEL OF B

Part B

LEVEL OF B

Figure 4.1 Optimal Solution: Graphical Method
to attain the level of "performance" required by the code. The second equal

cost curve just touches a vertex of the shaded feasible region. Since no other equal cost line which is lower also touches the feasible region,

Lt can be asserted that the point of contact is the least-cost combination

of the two factors satisfying the code. Figure 4.1 also illustrates that

if more money were spent it would still be possible to achieve the same level of performance as shown by the vertex of the feasible region labeled

"b." However, from the graph it can be seen that it would be more cost

effective to use those dollars to achieve a higher level of performance

than required by the code. Such a strategy would result in a point lying

entirely within the feasible region such as "c."

One way in which alternative solutions may be generated by such a computeriz

procedure is shown in figure 4.2.

In real world applications, the procedure actually used would probably

generate two classes of alternative solutions. Only the first class is

illustrated in figure 4.2. The optimal solution derived earlier is denoted

by "a" in figure 4.2.

For example, suppose X-L-j appears in the optimal solution as a retrofit

(i.e., Xj^j = 1), then the cost, C-j^ j

, of going from Xj^j^,, the pre retrofit

state, to X-j^j , the post retrofit state is made arbitrarily high. This

step guarantees that Xj[j can no longer be in the optimal solution.

Graphically this would correspond to a rotation of the equal cost line.

The new solution would thus be the point at which the lowest of the new

class of equal cost curves just touches the feasible region. Such a point

is designated by "b" in figure 4.2. Exactly analagous is the case where

X-j^j did not appear in the optimal solution (i.e., X-^a =0). In this

case, the cost of going from X-j^j^ to Xj^j , , is held fixed and

all CjlI^, j , are made arbitrarily high. This step guarantees that

X^-j will appear in the optimal solution as a retrofit. The number of

solutions generated in this class is equal to the number of variables in

the original linear programming problem minus the number of retrofit states

which cannot be attained (those within a row having a lesser value than

the input). In the simple case illustrated graphically, up to six alternative solutions could be generated. The second class of alternatives is

determined by the rows of the table. For example, suppose Xj^j appears

in the optimal solution as a retrofit, then the cost of going from Xj^j^

to X..' for all j', j„ < j' < ji:, where ji; is the maximal state in row

i, is made arbitrarily high. This step guarantees that no potential retrofits associated wU;h t\3.^. buLlding component can appear in the optimal

solution. The number of solutions generated is equal to the number of

building components (rows in table 4.5) which moved to a higher state

(were retrofitted) in the optimal solution.

71
LEVEL

OF A

LEVEL OF B

Figure 4.2 Alternative Solution: Graphical Method

72
The two classes of alternative solutions described above are very interesting

because they are "close" to the optimal solution. Close is put in quotation

marks because through reference to figure 4.2 it can be seen that the alternative solutions are adjacent vertices of the boundary of the feasible

region. '^ Although these solutions will be more costly than the optimal

solution, some of them will probably be close (in dollar terms) to the

least-cost combination of retrofits.

All of the discussion has proceeded as if the continuous solution to the

optimization were actually the integer solution. In general, this is not

true; however, there are several strong arguments for the approach used.

The first is the "near integer" property of the solutions; the second, the

judicious selection of the family of alternative solutions; and the third,

the computational cost, efficiency, and reliability of the algorithm.

Each of these is discussed in turn.

The near integer property occurs because of the structure of the constraint

matrix. A solution will have exactly n + m variables, where n was assumed

to be much greater than m, since n constraints corresponding to the building

component in table 4.1 have no variables in common and each of the constraints

has a right hand side of 1. Therefore, there are n mutually exclusive

subsets of the n + m variables which must sum to 1. Since it requires at

least two fractions to sum to 1, there must be at least n-m variables

which are exactly one; n-m or more integers among the n variables of interest

is defined as near integer. Furthermore, this is the worst case since

some of the variables may be zero and/ or some of the surplus variables

(the Y's) may be nonzero. Should either condition occur, there are fewer

than n + m variables to be included in the n subsets and more than n-m of

the variables may be integer.

Regardless of the integer characteristics of the optimal solution, it

represents a lower bound on the total retrofit cost. Should this approach

be adopted a heuristic integerization procedure could be applied to each

solution generated. This would produce an integer solution by making a minimal change in the basic continuous solution.

An additional reason for advocating the design of the alternate solution

family discussed earlier is that any variable appearing as a fraction in

the optimal solution must appear in at least one of the alternates as one and be supressed in at least one of the alternates. Another, more pedestrian,

criterion is that the alternates be amenable to generation via a systematic

but comprehensive specification, that they meet user needs, and that the

number of alternates be adequate but not burdensome.

In n dimensional space, the alternative solutions correspond to the

vertices of the polyhedron (feasible region) in the neighborhood of the

optimal basis (optimal combination of retrofits).

73
SELECTING THE "BEST" RETROFIT STRATEGY

The output of the computer program discussed in the previous subsection

will include :

(1) the optimal retrofit strategy; and

(2) a series of alternative retrofit strategies.

It is important to point out that all retrofit strategies are ranked on the

basis of construction cost only. As such, they do not include any recurring

or nonrecurring costs which will take place in the future. Differences in

the length of the renovation period are only treated through their effect

on construction cost. In reality, longer renovation periods will result

in lost revenues or increased housing expenses if the dwelling is to be

occupied by the investor. By the same token, computer printouts would not

include a measure of aesthetic quality or professional design judgment,

unless these attributes were reflected in the retrofit cost of going to a particular state. For these reasons, it is important that the user of the

linear programming model carefully review the complete set of retrofit

strategies with respect to any of the additional objectives outlined above

in order to select the one which is optimal for the case at hand. For

example, if one of the attributes under consideration was related to energy

performance, one might opt for a higher initial cost in order to increase

savings on future energy bills. Similar statements can be made about

building aesthetics.

The previous discussion was not intended to leave the reader with a feeling

that the final selection was an arbitrary one. On the contrary, different

investors have different objectives which are not always reflected in

renovation costs. In some cases, the additional constraints placed on the

problem by the potential investor are purely subjective, and hence, not

amenable to mathematical optimization. In other cases the costs (and

benefits) of a particular retrofit strategy can be quantified. It is

these cases where a clear-cut economic rationale can be applied. Stated

more simply, if future costs (e.g., fire insurance, energy, water) are unlikely to vary across retrofit strategies, then other things being equal,

the potential investor would be advised to select the least-cost solution.

Any differences in amenities provided by one retrofit strategy over any

other can be evaluated by the investor's willingness to pay for that amenity.

On the other hand, if future costs are likely to vary across retrofit

strategies, then a well proven building investment tool such a life-cycle

costing should be used. In this case the user would assess all costs

occurring over the investment horizon and select that retrofit strategy

which minimizes life-cycle costs. Furthermore, if any differences in

amenities exist between the retrofit strategy which minimizes life-cycle

costs and any other, the economic viability of that strategy can be assessed

by the user's willingness to pay for that amenity or set of amenities.

74
4.2 ACHIEVING CODE COMPLIANCE IN THE LEAST COSTLY MANNER: A CASE STUDY

This section provides a case illustration of how the performance approach,

developed in the previous section, may permit the costs of code compliance

to be drastically reduced. It is important to point out that although

this case application is concerned with the problem of fire safety in

health care facilities, it exhibits many of those attributes discussed in

the previous section. More precisely, the methodology presented in this

section includes:

(1) the concept of code equivalence; and

(2) the cost reduction potential of the performance approach over

that of strict compliance to the code.

Although the methodology is concerned with compliance to only one code, it

explicitly identifies four code attributes all of which must be satisfied

simultaneously in order for the facility to be deemed "up to code." If

each of these attributes is thought of as a code, then the problem becomes

quite similar to the one discussed in the previous section.

THE FIRE SAFETY EVALUATION SYSTEM

The Fire Safety Evaluation System is a quantitative evaluation system for

grading fire safety in health care facilities. The primary use of the

system is to determine how combinations of widely accepted fire safety

equipment and building construction features may provide a level of safety

equivalent to that required by the 1973 Life Safety Code-*-. Three major

concepts basic to the Fire Safety Evaluation System are:^

(1) Occupancy Risk ; the number of people affected by a given fire, the

level of fire they are likely to encounter, and their ability to

protect themselves.

(2) Building Safety Features : the ability of the building and its fire

protection systems to provide measures of safety commensurate with

the risk.

(3) Safety Redundancy : in-depth protection, through the simultaneous

use of alternative safety methodologies such as Containment,

Extinguishment, and People Movement. The design of the complete

fire safety system is intended to ensure that the failure of a single protection device or method will not result in a major

failure of the entire system.

Code for Safety to Life from Fire in Buildings and Structures , National

Fire Protection Association, NFPA 101-1973 (Chapter 10: Institutional

Occupancies)

Definitions are taken from the report by H.E. Nelson, and A. J. Shibe

,

A System for Fire Safety Evaluation of Health Care Facilities , National

Bureau of Standards NBSIR 78-1555, November 1978.

75
The concept of safety redundancy in the Fire Safety Evaluation System is

directly related to Provision 2-1111 of the 1973 Life Safety Code.

The task of ensuring that the Fire Safety Evaluation System satisfied

Provision 2-1111 of the Life Safety Code was the responsibility of a panel

of fire safety experts. The goal of the panel was to reach consensus on all relevant fire safety issues. To facilitate this process, a management

tool known as the Delphi Method was used. The Delphi Method, as used in

developing the Fire Safety Evaluation System, consisted of four steps.

These steps are illustrated in figure 4.3. Note that the steps illustrated

in the figure form a closed loop. This is because a certain amount of

recycling of ideas was needed in order to achieve consensus. The first

step in the process was to select a set of key safety factors. These

factors were related to the concepts of Occupancy Risk and Building Safety

Features mentioned earlier. (For example, Occupancy Risk includes as a factor Patient Mobility whereas Building Safety Features includes as a factor Interior Finishes in the Corridor and Exits.) The second step

focused upon the identification of a set of parameters associated with

each factor. (For example. Patient Mobility includes as parameters Mobile,

Limited Mobility, Not Mobile, and Not Movable. Interior Finishes in the

Corridor and Exits includes Class A, Class B, and Class C flame spread

ratings as parameters.) The third and most critical step was to assign a weight to each parameter which best reflected the relative degree of risk

or safety associated with that parameter. More specifically, negative

values reflected greater risks whereas positive values contributed toward

a higher level of safety within the fire zone. (The term fire zone is

defined as a space separated from all other spaces by floors, horizontal

exists, or smoke barriers.) The system treats a value of zero as "safety"

neutral. (The values for the three parameters associated with the safety

factor Interior Finishes in the Corridor and Exits are: Class C, -5 points;

Class B, 0 points; and Class A, 3 points.) In the fourth step, the value

of each safety parameter assigned by the panel was reassessed for adequacy

and consistency. In the event that the system which emerged from the panel

was shown to be inadequate or inconsistent, the entire four-stage process

was repeated.

The end result of the panel's work was a series of worksheets which permitted

the relative merits of each fire safety parameter to be carefully assessed.

These worksheets are presented as exhibits 4.1 through 4.4. In addition,

since each of the 13 Building Safety Features has a unique parameter which

corresponds to strict compliance, it was possible to compute the score, or level of safety, provided by the Life Safety Code for Extinguishment,

Containment and People Movement Safety. These values were then used as a base which any alternative to strict compliance to the Life Safety Code

had to match or exceed.

76
SELECT

SAFETY

FACTORS

IDENTIFY

REASSESS SAFETY

PARAMETERS

ASSIGN

WEIGHTS TO

PARAMETERS

Figure A. 3 Delphi Method

77
Exhibit 4.1 Occupancy Risk Factor Calculation Worksheet

Table 1. OCCUPANCY RISK PARAMETER FACTORS

RISK PARAMETERS

1. PATIENT

MOBILITY [Ml

RISK FACTOR VALUES

MOBILITY

STATUS

MOBILE

LIMITED MOBILITY

NOT

MOBILE

NOT

MOVABLE

RISK FACTOR 1.0 1.6 3.2 4.5

2. PATIENT PATIENT 1-5 6-10 11-30 >30

DENSITY (D] RISK FACTOR 1.0 1.2 1.5 2.0

3. ZONE FLOOR 1ST 2ND OR 3RD

4TH TO 6TH7TH AND ABOVEBASE- MENTS

LOCATION (L) RISK FACTOR 1.1 1.2 1.4 1.6 1.6

4. RATIO OF PATIENTS 11 11 yo >]] ONE OR*

MORE

PATIENTS TO ATTENDANT 1 1 1 1 NONE

ATTENDANTS (T) RISK FACTOR 1.0 1.1 1.2 1.5 4.0

5. PATIENT

AVERAGE

AGE [A]

1 AGE UNDER 65 YEARS

AND OVER HEAR

65 YEARS & OVER

1 YEAR & YOUNGER

Frisk factor 1.0 1.2

* RISK FACTOR OF 4.0 IS CHARGED TO ANY ZONE THAT HOUSES

PATIENTS WITHOUT ANY STAFF IN IMMEDIATE ATTENDANCE

[able 2. OCCUPANCY RISK FACTOR CALCULATION

M D T A F

1 ^

Table 3A. (NEW BUILDINGS] Table 3B. [EXISTING BUILDINGS]

F R

78
Exhibit 4.2 Worksheet for Determining the Value of the Safety Parameters

Table 4. SAFETY PARAMETERS VALUES

PARAMETERS PARAMETERS VALUES

1. CONSTRUCTION

COMBUSTIBLE

NON-COMBUSTIBLE

FLOOR OF ZONE UNPROTECTED PROTECTED

FIRST -2

SECOND -7

THIRD -9

4TH & ABOVE -13

UNPROTECTED PROTECTED

-4-9

UNPROTECTED PROTECTED FIRE RESIST

-7 -7

2. INTERIOR FINISH

(Corr. & Exit)

CLASS C -5

3. INTERIOR FINISH [Roomsj

CLASS C CLASS A -3

4. CORRIDOR

PARTITIONS/WALLS

NONE OR

INCOMPLETE

=1/3 HR >1/3<1.0 HR -10

5. DOORS TO CORRIDOR

>20 MIN. FR & AUTO CLOS.

2 (0|*"

6. ZONE DIMENSIONS

DEAD END MORE THAN 100' DEAD END

30 -100' NO DEAD ENDS>30' & ZONE LENGTH IS: >150'

-6 101" -4

7. VERTICAL OPENINGS

OPEN 4 OR MORE

FLOORS OPEN 2 OR 3 FLOORS

ENCLOSED WITH INDICATED FIRE RESIST.

-14 -10

>1HR e2 HR.

DOUBLE DEFICIENCY SINGLE DEFICIENCY

}. HAZARDOUS AREAS

-11

OUTSIDE ZONE

-5 -6

IN ADJACENT ZONE

NO DEFICIENCIES

-2

SMOKE PARTITION MECH. ASSISTED SYSTEMS

9. SMOKE CONTROL

BY CORRIDOR

10. EMERGENCY MOVEMENT

ROUTES

MULTIPLE ROUTES

DEFICIENT CAPACITY -2

W/O HORIZONTAL

EXITlsl HORIZONTAL EXITlsl DIRECT EXITlsl

11. MANUAL FIRE ALARM

NO MANUAL FIRE ALARM MANUAL FIRE ALARM W/O F.D. CONN. W/F.D. CONN.

12. SMOKE DETECTION & ALARM

CORRIDOR ONLY ROOMS ONLY CORRIDOR & HABIT SPACE

13. AUTOMATIC

SPRINKLERS

CORRIDOR &

HABIT. SPACE

10

NOTE; *Use 101 when item 5 **Use 101 when item 1 ***Use |01 in zone with

in existing buildings.

is -10.

0 is -8.

less than 31 patients

'Use 10] when item 1 is based on first floor zone or on an unprotected type of construction. 'Use 10] when item 1 is based on an unprotected type of construction •Use 101 when item 4 is -10.

79
Exhibit 4.3 Worksheet for Calculating Containment Safety, Extinguishment

Safety, People Movement Safety, and General Safety

Table 5. INDIVIDUAL SAFETY EVALUATIONS

80
Exhibit 4.4 Worksheet for Evaluating Fire Zone Safety Equivalency

Table 6. MANDATORY SAFETY REQUIREMENTS

CONTAINMENT

Sa

EXTINGUISHMENT

Sb

PEOPLE MOVEMENT

Sc

ZONE LGCATION New Exist. New Exist. New Exist.

FIRST FLOOR 9.0 4.0 6.0 3.0 6.0 1.0

ABOVE FIRST FLOOR 14.0 8.0 8.0 5.0 9.0 3.0

Table 7. ZONE SAFETY EQUIVALENCY EVALUATION YES NO

CONTAINMENT MANDATORY

SAFETY |Si) '^^^ CONTAINMENT (Sa)-=Si Sa C

EXTINGUISHMENT . MANDATORY

SAFETY (S2) EXTINGUISHMENT (Sb)

" -=S2 Sb E

PEOPLE MANDATORY

MOVEMENT,. , less PEOPLE >0

SAFETY 1^3) MOVEMENT (Sc)

S3 Sc P - =

GENERAL , OCCUPANCY

SAFETY (Sq) RISK (R)

Sg R G - =

81
The first worksheet, exhibit 4.1, consists of a brief description of the

fire zone and the means for calculating occupancy risk. Occupancy risk is

calculated by selecting the appropriate value for each of the five factors

shown in table 1 and multiplied together to get an "unadjusted" occupancy

risk factor. The resultant is then entered either in table 3A if the

building is new, or in table 3B if the building was in existence at the

time of the adoption of the 1973 Life Safety Code. The occupancy risk

factor for the fire zone is then calculated by taking the product of the

weighting factor and the "unadjusted" occupancy risk factor. The occupancy

risk factor is then used to establish the minimum level of General Safety

which must be provided by the 13 Building Safety Features in order to be

deemed in compliance to the Life Safety Code.

The second worksheet, table 4 shown as exhibit 4.2, provides the foundation

for the computerized procedure. Table 4 gives the scores associated with

the parameters for each of the 13 Building Safety Features. These scores

are used in evaluating the fire zone. To evaluate the fire zone it is

first necessary to identify the appropriate parameter and score associated

with each of the 13 Building Safety Features. The existing state of the

fire zone is then defined by recording (circling, checking or marking in some other manner) all of these parameters and their associ'.ated scores in

table 4. In performing the evaluation, it is important to point out that

for each Building Safety Feature which has parameters which have a higher

score, those parameters represent potential retrofits. For example, if in

the current state the flame spread rating on Interior Finishes in the

Corridor and Exits was Class C, then both Class B and Class A flame spread

ratings would be potential retrofits. More importantly, by systematically

combining improvements in score with anticipated retrofit costs, it becomes

possible to establish a means for upgrading the level of fire safety within

the fire zone in the most cost-effective manner.

The third worksheet, table 5, is shown in exhibit 4.3. Table 5 provides

the means for calculating th<^; . core associated with each of the four safety

redundancy requirements. (The four safety redundancy requirements are:

(1) Extinguishment Safety, (2) Containment Safety, (3) People Movement

Safety, and (4) General Safety.) In order to calculate the score for each

of the safety redundancy requirements, it is necessary to enter the score associated with the parameter identified in table 4 as corresponding to

the existing state of the Building Safety Feature into the appropriate

spaces in the coded rows of table 5. (No values are entered in the shaded

spaces of table 5.) Each of the four columns is then summed to get an overall score. These scores are labeled S^, S2, S3, and Sq in table 5.

The fourth worksheet provides the means for determining if the fire zone possesses a level of fire safety equivalent to that of the 1973 Life Safety

Code. Basically this is done by taking the four scores calculated in

table 5 and entering them in the boxes labeled Sj^, S2, S3, and Sq in

table 7. The user then selects the values from table 6 for Containment

Safety, Extinguishment Safety, and People Movement Safety for the appropriate building type and fire zone location. These values are entered in

82
the boxes labeled S^, S\^, and S^, in table 7. The occupancy risk factor

calculated on the first worksheet is then entered in the box labeled R.

Based on these two sets of numbers it is possible to test if the fire zone provides a level of safety equivalent to the 1973 Life Safety Code. This

test is performed by determining if the differences between the first set

of numbers, S][, S2, S3, and Sq, and the second set of numbers, S^, S^,,

S(,, and R, in table 7 are greater than or equal to zero.

A COMPUTERIZED VERSION OF THE FIRE SAFETY EVALUATION SYSTEM

In the event that the fire zone fails to pass the equivalency test, it will

be necessary to select a retrofit strategy which will ensure that the

Building Safety Features produce scores which match or exceed each of the

four redundancy requirements. A systematic means for doing this which

explicitly introduces relative costs into the retrofit decision is the

computerized version of the Fire Safety Evaluation System developed jointly

by the Center for Building Technology and the Center for Applied Mathematics

at the National Bureau of Standards

This computerized procedure is particularly attractive since it uses information collected during the fire safety evaluation as its primary

input. This information is used not only to define the current state of

fire safety in the fire zone, but also to calculate the expected retrofit

costs^ for that fire zone.

Using linear programming, the computerized procedure utilizes information

on the current state of the fire zone, the minimum passing "score" needed

to achieve compliance, and the anticipated cost of each retrofit measure

to identify the least-cost or optimal combination of retrofits. The com- puterized procedure then systematically analyzes other retrofit combinations

to see if alternatives might exist which are close to the one identified

as least costly. The least costly combination of retrofits and any

alteratives which the program produces, usually between 10 and 20, are then summarized in tabular form so that they can be ranked from least

costly to most costly. By using this approach, health care facility

decisionmakers have greater flexibility in choosing among retrofit com- binations. In particular, by providing alternatives, the decisionmaker

has the opportunity to assess the relative costs of the alternative retrofit

strategies, to assess the effects of nonconstruction costs, and to match

Robert E. Chapman, Phillip T. Chen and William G. Hall, Economic Aspects

of Fire Safety in Health Care Facilities: Guidelines for Cost-Ef f ective

Retrofits , National Bureau of Standards, NBSIR 79-1902, November 1979.

Unless otherwise noted, the term retrofit cost refers to initial cost of a given retrofit measure.

83
common retrofit packages across the fire zones. The computerized

procedure also contains a series of user options which make it possible

to alter the cost of any retrofit, preclude a retrofit, force a retrofit

to be included, or demand a higher level of safety than required by the

Life Safety Code.^

CASE APPLICATION

The focus of this section is on the results of the application of the

computerized procedure to a "typical" general hospital. This approach was taken in order to introduce as many engineering and architectural design

considerations as possible into the analysis. The formulation of the

typical hospital design was based on an extensive review of actual hospital

layouts and working drawings. The review was also useful in defining an expected level for each of the 13 Building Safety Features.

The design used in the case application was a seven story "T" shaped

structure capable of housing approximately 300 patients. The facility

also houses an outpatient clinic and an emergency room. The facility,

assumed to have been built around 1960, is constructed with structural

steel framing protected by a fire resistive concrete covering, reinforced

concrete floors, fixed windows and masonry exterior walls. In all cases,

but one, the fire zone under examination was the entire floor.

Through reference to floor plans for the hospital, it was possible to

establish fire hazard scenarios for each fire zone. All fire hazard sce- narios showed the expected level of each Building Safety Feature as well

as potential retrofit courses of action should the fire zone be deficient.

Figure 4.4 shows the plan for the patient room floors. (The patient room

floors are floors three through seven.) Through reference to the figure,

one can identify such important factors as the location of the means of

egress, hazardous areas and a dead end corridor as well as information on zone dimensions and the expected number of patients being housed.

Additional information was then used to establish a series of critical

element counts which uniquely defined the scope of the retrofit. They

include information on such topics as the number of "No Door" charges

which must be removed or the square feet of floor space requiring sprinkler

protection in order to move to a higher state. Recall that the level of

each Building Safety Feature is determined by the worst case condition

within the fire zone. Consequently, there may be several retrofit options

possible for a particular Building Safety Feature. (This may be seen by

examining exhibit 4.2.)

Those readers interested in a detailed description of the computerized

procedure are referred to the report by R. E. Chapman, W. G. Hall and

P. T. Chen, A Computer Approach for Identifying Cost-Effective Retrofits

in Health Care Facilities , National Bureau of Standards, NBSIR 80-1929,

January 1980.

The first floor, which houses the emergency room, consists of two fire

zones

.

84
85
Application of the Fire Safety Evaluation System to the selected hospital

design indicated that all fire zones in areas of patient use, with the

exception of the fire zone containing the emergency room, would require

some degree of renovation in order to meet the requirements of the 1973

Life Safety Code. Computer analyses were then conducted for each floor to

identify retrofit options which would satisfy Provision 1-3118 of the Life

Safety Code. These analyses produced anywhere from seven to 21 retrofit

strategies which were less costly than strict compliance to the prescriptive

provisions of the Life Safety Code. In most cases, the cost of the least-cost

solution was about half the cost of the prescriptive solution. Since the

retrofit strategies mentioned earlier were for a particular floor, it was necessary to use information on all fire zones to synthesize a comprehensive

retrofit strategy. In reality, synthesizing a comprehensive retrofit

strategy could be a major problem. Due to the built in procedure for

generating alternative retrofits, however, this problem is reduced to a minimum. More succinctly, a large number of retrofit strategies for each

fire zone is desirable from the viewpoint of design flexibility because

they provide an opportunity to match common retrofits across fire zones.

In the case application discussed here, the retrofit strategy identified

as the least costly on an individual fire zone basis exhibited a marked

similarity across fire zones. Thus, the least-cost solution for the entire

building is composed of the least-cost solutions for all fire zones within

the building. As in actual practice, the synthesis of comprehensive retrofit

strategies was based on technical considerations as well as anticipated

retrofit costs.

The cost saving potential of the Fire Safety Evaluation System is shown in

figure 4.5. It displays graphically the anticipated costs of retrofitting

the entire hospital under two courses of action. The first course of action

involves the selection of the least-cost solution based on the Fire Safety

Evaluation System. This case is labeled "Performance" in figure 4.5. The

second course of action implies that strict compliance to the prescriptive

provisions of the Life Safety Code is adhered to. This case is labeled

"Prescriptive" in figure 4.3. Even a brief examination of figure 4.5 would

reveal that the performance based Fire Safety Evaluation System reduces

overall retrofit costs by more than 50 percent. The estimated retrofit

costs for the two courses of action are $115,000 and $250,000, respectively.

Even though the cost saving potential of the Fire Safety Evaluation System

is substantial in this case, it is possible that the overall least-cost

solution may vary considerably from one fire zone to another. The use of

this solution may, thus, be unwarranted from a design viewpoint. In order

to address this point, engineering judgment was used to establish a series

of alternative retrofit strategies which matched retrofit packages across

fire zones. The retrofit costs associated with the least-cost solution.

86
COST COMPARISON

300

250

= 200

H

(0O

O 150

lZ

2 100

LU

50

Performance Prescriptive

RETROFIT PACKAGE

Figure 4.5 Cost Comparison

87
five alternative retrofit strategies and the prescriptive solution are shown graphically in figure 4,4. A brief description of each package is

given in table 4.6. A more detailed description of each retrofit package

is given in the NBS report by Chapman et al.-*-

A close examination of table 4.6 and figure 4.4 reveals that at least three

alternative retrofit strategies can be defined which are close in cost to

the least-cost solution. Additional retrofit strategies could also be

defined which both require a specific retrofit package on a particular

floor and are fairly close to the least-cost solution. Through the use of

this procedure, health care facility decisionmakers will have a great deal

of freedom in tailoring the retrofit strategy to their own particular

needs and objectives.

Two additional retrofit strategies based on the Fire Safety Evaluation

System are also shown in figure 4.4. These are total sprinklering of the

entire facility (package E) and the removal of the dead end corridor

through the installation of an exit stairwell (package F) . They are included to show how the Fire Safety Evaluation System can promote substantial cost savings even if a relatively expensive retrofit strategy,

such as totally sprinklering each fire zone, is required. In particular,

it was possible to totally sprinkler the entire facility for only two

thirds the cost of strict compliance.

The inherent flexibility of the Fire Safety Evaluation System is more

evident, however, when retrofit package F is compared to the prescriptive

package G. This stems from the fact that a significant portion of the

cost of strict compliance to the Life Safety Code was due to the installation of an exit stairwell to remove the dead end corridor. Consequently,

any requirement that the exit stairwell be included in the retrofit package

would provide a critical test of the Fire Safety Evaluation System's

cost-saving capability. Since the computer program generates two classes

of alternative solutions, one of which includes the installation of the

exit stairwell as a retrofit, it is possible to formulate a comprehensive

retrofit package from the standard computer output.^ Based on this information, it was possible to develop a retrofit strategy which met the

requirements of the Life Safety Code at a cost saving of over 25 percent.

This is due to the fact that the Building Safety Features which are not

constrained are upgraded in the most cost-effective manner.

R. E. Chapman, P. T. Chen, and W. G. Hall, Economic Aspects of Fire Safety

In Health Care Facilities; Guidelines for Cost-Ef f ective Retrofits , National Bureau of Standards, NBSIR 79-1902, November 1979.

Users of the computer program could also utilize the option CHANGE in

putting together a comprehensive retrofit package. For a programmer-oriented

description, see R. E. Chapman, W. G. Hall, and P. T. Chen, A Computerized

Approach for Identifying Cost-Ef f ective Retrofits in Health Care Facilities , National Bureau of Standards, NBSIR 80-1929, January 1980.

88
Table 4.6 Retrofit Packages for the Case Application

RETROFIT APPROXIMATE

PACKAGE DESCRIPTOR RETROFIT COST

A Least-Cost $115,000

B Fire Department Connection on Manual $121,000

Fire Alarms

C Sprinklers in the Corridor $131,000

D Smoke Detection and Alarm System in $133,000

the Corridor

E Total Sprinkler $158,000

F Removal of the Dead End Corridor $184,000

G Prescriptive $250,000

89
ALTERNATIVE SOLUTIONS

Figure 4.6 Alternative Retrofit Packages for the Case Application

90
In addition to making it easier to formulate a comprehensive retrofit

strategy, the use of alternative retrofits has a second advantage in that

it allows health care facility decisionmakers to introduce explicitly

nonconstruction costs into the process of retrofit selection and design

development. Nonconstruction costs include not only the potential for

lost revenues and a reduction in the level of services being provided, but

insurance differentials and changes in operating and maintenance costs as well. Since these costs may be significant, the use of alternatives permits

the decisionmaker to choose an option which may h^ve slightly higher con- struction costs but which is less costly when other cost considerations

are introduced. Furthermore, some retrofit decisions may affect the costs

of owning, operating and maintaining the facility in future periods of

time. Should this be the case, it would be worthwhile to base the retrofit

decision on an established building investment technique such as life-cycle

costing which explicitly weighs all costs and savings over a set period of

time

.

The previous discussion has shown how the Fire Safety Evaluation System can

be used to achieve significant savings in the costs of compliance to fire

safety codes without compromising the safety and well being of the persons

being housed in health care facilities. Although the Fire Safety Evaluation

System is the only comprehensive equivalence methodology in use at this

time and its application is clearly nonresidential, the basic concept shows

much promise for applications in other code areas and for other occupancy

classifications. At this time, the nonfire related code areas where a full fledged equivalence methodology seems appropriate, relate to health

and sanitation and security and egress, these areas offer a definite potential because alternatives do not exist and the interdependence among

potential alternatives does not appear to be as serious as in other areas such as strength and stability.

4 .3 POTENTIAL APPLICATIONS OF AN APPROACH INTEGRATING COST FUNCTIONS AND

THE PERFORMANCE CONCEPT

The discussion, thus far, has focused primarily on the mechanics of a combined cost f unction/perf ormance concept approach. That such an approach

could both reduce costs and cost variability was examined in section 4.1.

Empirical evidence which documents the cost saving capability of the perfor- mance concept was presented in the previous section. In terms of the real

world application of this procedure, however, a natural question is "who

really benefits from it?". The answer to this question is quite basic.

The primary benefactors of such an approach are the purchasers of rehabilitation services, because this approach permits them to purchase a known

quality of housing services (mandated by the code(s)) at the least cost

for a known level of risk acceptance. It can also be argued that this

approach will benefit rnnfractors preparing bids for residential

91
rehabilitation projects.-'- However, since the major burden of the costs of

residential rehabilitation must be borne by the purchaser a contractor's

services, the emphasis in this section will be on how this approach helps

these individuals.

There are three types of "purchasers" who will benefit from the approach

outlined in section 4.1. They are:

(1) individual homeowners and/or investors;

(2) financial institutions; and

(3) government

To some extent, all three groups of "purchasers" are interrelated, however,

the group which would be most affected by this approach is the first.

At the present time, the individual homeowner or investor shoulders the

greatest burden in undertaking a residential rehabilitation project. To

be sure the use of the approach proposed in this report will not shift the

responsibility from the owner to someone else, it must be assumed that

people undertake rehabilitation because, in the end, they believe it will

be profitable. Current rehabilitation efforts, however, tend to be overly

risky and may promote speculation which often adversely affects the

inhabitants of the area being "developed." As rehabilitation activities

become less risk prone, they will attract a larger segment of society. In some cases it may even be possible for low-income homeowners to undertake

remodeling efforts which were previously beyond their means. By the same

token, renters may suffer lower rent increases due to rehabilitation efforts

because the absolute cost of renovation is lower

For an excellent discussion of this topic the interested reader is referred

to the following articles: Marvin Gates, "Bidding Strategies and

Probabilities," Journal of the Construction Division , American Society of

Civil Engineers, Vol. 93, March 1967; and Michael Curran, "A Scientific

Approach to Bidding: Range Estimating," Constructor , January 1975.

As Anthony Downs points out (Anthony Downs, "Investing in Rehabilitation

Can Be Successful," Real Estate Review , Vol. 6, No. 2, Summer 1976.) any

renovation activity must be accompanied with an increase in the rental

rate for the property. Reducing the cost of renovation will, however,

reduce the absolute increase in the rental rate. It may also promote a more rapid upgrading of lower quality housing. For an extensive discussion

of this topic which includes a model of housing supply, the interested

reader is referred to the study by Valenza (Joseph J. Valenza, Residential

Rehabilitation , unpublished Ph.D. dissertation, George Washington

University, 1978).

92
The crucial issue for either the homeowner or the investor is of course the

profitability of residential rehabilitation. In terms of profitability,

one must differentiate between expected return and expected risk. Most

people are to some extent risk averse. Thus, maximizing the expected return

on an investment regardless of the risk is an unlikely investment strategy

for most people. Economic studies have shown that individuals tend to

prefer a balance between risk and rate of return.^ It is uncertain what

effect the widespread use of the approach discussed in section 4.1 would

have on the rate of return of a rehabilitation project.

It is certain, however, that the inherent riskiness of residential rehabilitation will be reduced somewhat by better cost estimates and known levels

of risk acceptance.

The second group of "purchasers," financial institutions, would also be

affected by a reduction in the costs and riskiness of rehabilitation activities. In certain instances, financial institutions practice a subtle

form of redlining, thus precluding any investor, no matter how qualified,

from receiving a loan for undertaking a rehabilitation effort. Many of

these activities are defended on the basis of "poor past performance," or "high risks." As the trend toward rehabilitation has grown, many financial

institutions have formed consortiums or other risk-sharing arrangements.^*^

Even the widespread use of an approach which provides accurate cost estimate

that can be tied to a measure of risk will not remove the need for financial

risk-sharing arrangements. In all cases one would expect residential rehabilitation to be more risky than new housing construction. What this

approach would do, which is not done now, is provide an objective means

for assessing the merits of a particular rehabilitation loan application.

J. Tobin, "Liquidity Preference as Behavior Towards Risk," The Review of

Economic Studies , Vol. XXVI, No. 1, February 1958.

This may be seen by noting that more people will probably be drawn into

rehabilitation activities if the costs begin to fall. As more people enter

and the market interactions become more frequent, the likelihood of individuals behaving as "free riders" and hence appropriating a portion of

the investors profit for themselves will be increased. For an excellent

discussion of this topic see the report by Colwell (Peter F. Colwell, An

Economic Analysis of Residential Abandonment and Rehabilitation , National

Bureau of Standards, NBSIR 76-1043, May 1976).

'David Gressel, Financing Techniques for Local Rehabilitation Programs , National Association of Housing and Redevelopment Officials, 1976.

^J. Thomas Black, Allan Bornt, and Robert Dubinsky, Private Market Housing

Renovation in Older Urban Areas, The Urban Land Institute, 1977.

93
Furthermore, if financial institutions are aware that the likelihood of a given budget being overrun is very small, they may be more willing to make

loans in neighborhoods where little or no rehabilitation is being carried

out

.

The last "purchaser" of rehabilitation services to benefit is government.

As in many public policy situations, the government recognizes that private

market rehabilitation may, if left to itself, create serious problems.

Thus, the government may find itself in a position of advocating rehabilitation in order to get houses into a higher tax paying status while coping

with the problem of providing adequate shelter for those families displaced

by upward filtering of the housing stock due to rehabilitation. The problem

is further complicated since government agents (Federal, State, and local)

all operate under a budget constraint. The fact that the emphasis in the

past has been on urban renewal reflects this claim (throughout most of the

50 's and 60' s renewal efforts were thought to be less expensive than re- habilitation efforts). Unfortunately, renewal activities generated many

of the same problems of dislocation that widespread private market rehabilitation activities generate. The current government emphasis on rehabilitation seems to reflect a desire to save what we already have. Housing

conservation in the future, however, can be more than just saving. It may

even be possible to pursue a moderately aggressive program with lower

budgets. More precisely reducing the costs of renovation may be a valuable

tool for government planners in providing adequate housing for low-income

families. It may make code enforcement activities more effective and

promote the goal of providing standard (higher quality) housing for lower

income households.-'- If costs can be reduced substantially, government

sponsored rehabilitation may be used in conjunction with private market

rehabilitation so that dislocation problems are minimized while assuring

that increased tax revenues will be forthcoming due to private market

activities

.

Unfortunately, the rehabilitation of inner city housing is not a panacea.

There are many problems facing the nation's cities. The fact that rehabilitation activities cannot be divorced from these other problems is a direct

result of the economic Infrastructure of the modern urban community. On

the positive side. Improving the housing stock may make other problems

seem somewhat more tractable. On the other side, the uncertainty associated

with the efficacy of treating some of the major problems facing the nation's

cities is so grave that it may prevent the inner city renalassance so desired

from coming to fruition.

Daniel R. Mandelker and Roger Montgomery, "Housing Codes and Housing Code

Enforcement—Saving What We Have," Housing in America: Problems and

Perspectives , Bobbs-Merrill Company, Inc., Indianapolis, Indiana, 1973.

94
Chapter Summary

The purpose of this chapter was to develop a theoretical approach for

dealing with cost variability. This approach combined the technical and

economic advantages of cost functions with the cost-reducing potential of

the performance concept. It was shown that through the use of monte

carlo simulation this approach could identify the least-cost solution for

achieving compliance to a given set of codes for a specified level of risk

acceptance. Consequently, the generalized equivalence methodology outlined

in this chapter provides a means for dealing comprehensively with cost

variability

.

As a means of illustrating the cost-saving potential of the generalized

methodology, an example from the area of fire safety in health care facilities was presented. Although the method presented was concerned

with only one code, the Life Safety Code, it contained four code attributes

which had to be satisfied simultaneously in order to be deemed in compliance

If each of these attributes is thought of as a code, the analogy to the

residential rehabilitation problem comes into sharp focus. The equivalence

methodology was explained through a series of worksheets and applied to a prototypical hospital. The results of this application revealed a cost-saving potential of over 50 percent when compared to the costs of

prescriptive compliance. In addition, it was shown that several retrofit

strategies could be identified which were significantly less expensive

than prescriptive compliance. These alternatives were a means through

which nonconstruction costs could be introduced into the design selection

process

.

The chapter also included a discussion of how the generalized methodology

could benefit individual homeowners and/or investors, financial institutions

and government. Basic concepts which highlighted the interdependence

among the three purchasers of rehabilitation services were also presented.

95
Facing page:

The gains from res-tdentiat rehabilitation

ai>e both financial and aesthetic. Work

outlined in this study has shown that a potential for reducing the extreme

voj'iability of rehabilitation costs

may both promote private market activities and permit the upgrading of

lower-income housing units by government

authorities

.

96
5. SUMMARY AND RECOMMENDATIONS FOR FURTHER RESEARCH

5 .1 SUMMARY

Accurate cost estimation is important to all major participants in the

rehabilitation process. Reasonably accurate cost information is essential

to the housing owner as an input to the investment decision and to financial

institutions in assessing the merits of each investment decision. Unreliable

cost information renders residential rehabilitation investments riskier than

they would be if accurate cost estimates existed. To contractors and

subcontractors, the reliability of cost estimation procedures has a substantial

effect on the riskiness of doing business. Finally, accurate cost estimation

is needed by Federal, State and local government agents in developing a coordinated approach which promotes private market activity without escalating

the housing problems faced by lower income residents.

97
Four methods for estimating the costs of residential rehabilitation were analyzed in this study. The first method, average total cost, is the most

widely used and least flexible of the four methods studied. The second

method, parametric procedures, is more flexible but suffers from the assumed

linear relationships between the key factors in the procedure. The third

method, probabilistic cost estimation, is extremely flexible; it may be

used with either average total cost or parametric procedures. The fourth

method, cost functions, is the most consistent from an engineering economics

viewpoint. In addition, probabilistic concepts are easily incorporated

into the cost function approach.

Due to the extreme variability of residential rehabilitation costs, it is

necessary that any totally acceptable procedure be able to explicitly

treat risk in developing a cost estimate. Three means of risk assessment

were studied for each of the four methods. Based on this analysis, judgmental modifications were shown to be inappropriate, since it is qualitative

rather than quantitative, and the use of confidence intervals might be

misleading if the estimated value for either average total cost or sample

variance was biased. On the other hand, probabilistic concepts were shown

to be highly desirable since they permitted one to define percentage changes

in the level of risk being borne by the investor as a function of changes

in one or more factors in the rehabilitation process.

The engineering economics construct known as the cost function was then

adapted to a general framework based on the performance concept. The

performance concept was then used to define a generalized equivalence

methodology for residential rehabilitation. This theoretical methodology

permitted both compliance to the code(s) under consideration and a mathematical structure suitable for optimization techniques to be identified.

Probabilistic methods were then developed which would permit a potential

investor to identify an optimal retrofit strategy for a given level of

risk acceptance. Past empirical studies in the area of fire safety in

health care facilities were used to illustrate how such an approach might

reduce retrofit costs by as much as 50 percent.

5.2 RECOMMENDATIONS FOR FURTHER RESEARCH

In order to expand the concepts developed in this study, future research in

several areas is needed.

This study has concluded that the engineering economics construct known as

a cost function is the most consistent technique for estimating residential

rehabilitation costs. Unfortunately, the development of a series of cost

functions for residential rehabilitation can only be accomplished through

the application of econometric techniques to actual housing renovation

cost data. Past empirical studies have defined the type and nature of

such cost data. It would be helpful if these guidelines could be followed

in collecting a complete set of residential rehabilitation cost data.

98
The use of probabilistic methods is an important tool for assessing the

riskiness of a potential residential rehabilitation project. At the

present time, little empirical evidence exists on the likely candidates

for the probability distributions associated with a given factor in the

rehabilitation process. Further work in this area has two basic merits.

First, it explicitly ties the problem of risk in the physical process to

the cost estimation process. Second, since cost functions are more com- plicated than the other methods of estimating rehabilitation costs,

defining the most likely probability distributions for each factor would

simplify their use in the field.

This study has presented empirical evidence which shows that cost

reductions of 50 percent were possible in other areas of building renovation. If these savings can be achieved in residential rehabilitation

activities, the level of rehabilitation may increase substantially. The

theoretical model developed in this paper is of necessity abstract. However

previous experience in other areas has facilitated the development of the

generalized methodology presented in this paper. The actual process of

selecting each building component, defining each state and assigning each

state value will be extremely difficult. If this effort is balanced against

the cost reducing capability of such a procedure it may seem less formidable

however. The fact that the generalized equivalence methodology can actually

identify the least-cost solution for a given level of risk implies that a comprehensive treatment of costs in this area is theoretically feasible.

The productivity of labor in residential rehabilitation activities is one

of the major topics which affects rehabilitation costs. At this time

there is a widespread belief that productivity levels in all sectors of

the construction industry are declining. If this is true and can be

measured, one could show directly how rehabilitation costs will be changing

as a result of the decline in productivity. The sparsity of empirical

data on this crucial topic, however, prevents one from giving a definitive

answer as to the existence or severity of the decline in productivity.

Along similar lines, it would be beneficial to know how the level of

productivity in residential rehabilitation compares to that in new con- struction. If answers to this question could be obtained, one might be

able to identify the circumstances under which rehabilitation would have a distinct cost advantage over new construction. Guidelines for making

these decisions would be of value to potential investors whether they

contract for renovation services or do the job themselves.

99
6 . REFERENCES

Black, J. Thomas, Allan Bornt, and Robert Dubinsky, Private-Market Housing

Renovation in Older Urban Areas , The Urban Land Institute, 1977.

Breiman, L., Statistics: With a View Toward Applications , Houghton Mifflin

Boston, 1973.

Brown, R. L., J. Durbin, and J. M. Evans, "Techniques for Testing and

Constancy of Regression Relationships Over Time," Journal of the Royal

Statistical Society , Series B, Volume 37, 1975.

Bureau of Building Marketing Research, Homeowners Remodeling/Modernization

Study , November 1975.

Burns, R. M., How to Buy and Fix Up an Old House , Home-Tech Publication,

Bethesda, Maryland, 1976.

Chapman, Robert E., Economic Analysis of Experiment Lead Paint Abatement

Methods: Phase I , National Bureau of Standards, Technical Note 922,

September 1976.

Chapman, Robert E., Phillip T. Chen, and William G. Hall, Economic Aspects

of Fire Safety in Health Care Facilities; Guidelines for Cost-Ef f ective

Retrofits, National Bureau of Standards, NBSIR 79-1902, November 1979.

Chapman, Robert E., William G. Hall, and Phillip T. Chen, A Computerized

Approach for Identifying Cost-Ef f ective Retrofits in Health Care Facilities

National Bureau of Standards, NBSIR 80-1929, January 1980.

Chapman, Robert E. and Joseph G. Kowalski, Guidelines for Cost-Ef f ective

Lead Paint Abatement , National Bureau of Standards, Technical Note 971,

January 1979.

Chapman, Robert E. and Joseph G. Kowalski, Lead Paint Abatement Costs;

Some Technical and Theoretical Considerations , National Bureau of Standards

Technical Note 979, February 1979.

Code for Safety to Life from Fire in Buildings and Structures , National

Fire Protection Association, NFPA 101-1973.

Colwell, Peter F., An Economic Analysis of Residential Abandonment and

Rehabilitation , National Bureau of Standards, NBSIR 76-1043, May 1976.

Curran, Michael, "A Scientific Approach to Bidding: Range Estimating,"

Constructor , January 1975.

Downs, Anthony, "Investing in Rehabilitation Can Be Successful,"

Real Estate Review , Vol. 6, No. 2, Summer 1976.

100
Farrell, M. J., "The Measurement of Productive Efficiency," Journal of the

Royal Statistical Society, 1957.

Fisz, Marek, Probability Theory and Mathematical Statistics , Third Edition,

John Wiley and Sons, New York, 1963.

Freund, John, Mathematical Statistics , Second Edition, McGraw-Hill,

New York, 1970.

Gates, Marvin, "Bidding Strategies and Probabilities," Journal of the

Construction Division , American Society of Civil Engineers, Vol. 93,

March 1967.

Gressel, David, Financing Techniques for Social Rehabilitation Programs , National Association of Housing and Redevelopment Officials, 1976.

Hastings, N. and J. B. Peacock, Statistical Distributions , John Wiley and

Sons, New York, 1975.

"HUD Launches a New Rehab Program," Engineering News Record , November 25,

1976.

Jaquith, Lawrence, "The Cost Index: Working Tool or Trap," Architectural

Record , February 1969.

Johannes, James M., Paul D. Koch, and Robert H. Rasche, An Investigation of

Factors Affecting Geographic Cost Differentials on Military Construction

Projects , National Bureau of Standards, NBS-GCR-80-197 , February 1980.

Law, Averill M., "Confidence Intervals in Discrete Event Simulation: A

Comparison of Replication and Batch Means," Naval Research Logistics

Quarterly , Vol. 24, 1977.

Mandelker, Daniel R. and Roger Montgomery, "Housing Codes and Housing Code

Enforcement—Saving What We Have," Housing in America: Problems and

Perspectives , Bobbs-Merrill Company, Inc., Indianapolis, Indiana, 1973.

McConnaughey , John S., Production Functions in Contract Construction for the

United States, 1973 , unpublished Ph.D. dissertation, Michigan State

University, 1976.

McConnaughy, John S., An Economic Analysis of Building Code Impacts

:

A Suggested Approach , National Bureau of Standards, NBSIR 78-1528,

October 1978.

101
"Millions of Housing Units Needed by 1988/' Engineering News Record , April 27, 1978.

Musgrave, Gerald L. and Robert H. Rasche, "Estimation of Cost Functions,"

The Engineering Economist , Vol. 22, No. 3, 1973.

Nelson, Harold E. and A. J. Shibe, A System for Fire Safety Evaluation of

Health Care Facilities , National Bureau of Standards, NBSIR 78-1555,

November 1978.

Nicholson, Walter, Microeconomic Theory; Basic Principles and Extensions , The Dryden Press, Inc., Hinsdale, Illinois, 1972.

Nie, Norman H., C. Hadlai Hull, Jean G. Jenkins, Karin Steinbrenner , and

Dale Bent, SPSS; Statistical Package for the Social Sciences , Second

Edition, McGraw-Hill Book Company, New York, 1975.

1979 Dodge Manual for Building Construction Pricing and Scheduling , McGraw-Hill Information Systems Company, New York, 1978.

Peurifoy, R. L., Estimating Construction Costs , McGraw-Hill Book Company,

New York, 1958.

Silberberg, Eugene, The Structure of Economics; A Mathematical Analysis , McGraw-Hill Book Compaay, New York, 1978.

Tobin, J., "Liquidity Preference as Behavior Towards Risk," The Review of

Economic Studies , Vol XXVI, No. 1, February 1958.

"Unions and Contractors Try to Get More Rehab Work," Engineering News Record , July 22, 1976.

U.S. Department of Housing and Urban Development, Final Report of the Task

Force On Housing Costs , May 1978.

Valenza, Joseph J., Residential Rehabilitation , unpublished Ph.D.

dissertation, George Washington University, 1978.

Weed, Harrison D., A Study on Variability of Construction Cost Estimates , U.S. Armament Command, Rock Island, Illinois, July 1976.

102
APPENDIX A

DERIVATION OF A COST FUNCTION FOR RESIDENTIAL REHABILITATION

This appendix developes a form of economic model known as a cost function.

The derivation of the cost function will proceed on the assumption that

the production function underlying the construction process is Cobb-Douglas.

This assumption is made both because it is consistent with previous studies

of the construction industry-'- and for ease of exposition. The term production function as used in this study refers to an explicit relationship

between a set of inputs (labor, materials, and capital) and technological

factors which taken together produce a given output (square feet of floor

area renovated) . The cost function associated with the underlying process

may then be derived from the production function by solving a constrained

optimization problem. A non-mathematical statement of the problem is to

minimize the cost of renovation subject to the requirement that Q square

feet of floor area are renovated. Since portions of the discussion which

follows are of a rather technical nature, readers interested primarily in

the implications of such an approach are encouraged to go directly to

sections 2.4 and 4.1.

In the mathematical discussion which follows, it is assumed that the prices

of all inputs are independent. Under this assumption the quantity of

output, Q, the square feet of floor area renovated, and the cost of output,

C, may be expressed as

m n a

,

Q = n T.(k^) n (X.) j

i=l j=l A.l

n

C = E X^P^ A.

2

where

Q = total square feet of floor area to be renovated;

Tq = basic construction technolory factor;

T. = construction technology factor associated with the i building

1 system or subsystem (e.g., plumbing, structural, electrical);

-'-For an authoritative source on this subject see John S. McConnaughey , Jr.

Production Functions in Contract Construction for the United States, 1972

(unpublished), Ph.D. dissertation, Michigan State University, 1976.

103
kj^ = measure of the condition of the i^'^ system or subsystem (e.g.,

1 = excellent, 2 = sound, 3 = some deterioration, e..);^

m = the number of building systems or subsystems;

Xj = quantity of the j input (materials, labor, equipment) required

to perform the job;

Oj = the percentage change in output associated with a one percent

change in the utilization of the j*-^ input; ^

n = the number of inputs;

C = the total cost (expected bid price) of the job; and

Pj = the unit cost of the j input.

Note also that the summation sign, Z, and the product sign, IT, are used in

equations A.l and A. 2. They are defined as

n

S XjP-j = XiPi + X2P2 + . . . + X^Pn

and

n aj ai 0L2 n (X.) = (Xi) (X2) . . .(x^)

Letting n equal 2 would thus imply that

n 2

I XjPj = E XjPj = Xi?i + X2P2

j=l j=l

and

n a-j 2 a-! (xi 0.2 I (X.) = 1 (X.) = (Xi) (X2)

j=l j=l

The constrained optimization problem may then be solved through application

of the method of La Grange multipliers. This method consists of the formation of an auxiliary function subject to a constraint condition

<}) (X]^, Xfi) = 0.

In terms of the four levels of measurement discussed in section 3.5, the

niaraerical values assigned to the condition variable should be defined

over the interval or ratio levels.

I In economics aj is referred to as an output elasticity.

104
Since the problem may be written as

/. — C + X(}) (X]^, Xj^)

and <J) (X]^, X^) is equal to zero, minimizing minimizes C (equation A. 2)

subject to the constraint that Q square feet of floor area are renovated

(equation A.l). More succinctly, the auxiliary function may be written as

n m n a, L= l XjPj + X (Q - JI T^dc^) n (Xj) h

j=l i=l j=l

Differentiating partially equation A. 3 with repect to each input and

equating the resultant to zero, produces the following system of equations

9Xj

a m n a. = Pi - ^ 3^ (-^Oin^Tidci) n (Xj) ) = 0

1

• a m n a- |i- = Ph - (T n Ti(ki) n (Xj)-^) = 0

9X^ X^ 1-1 j-1 -J am n 1^ = ^xl(Vri^'^i>jJi"3>

• ttj^ m n

This system consists of n equations in n + 1 unknowns, X-^, X^ and

Thus it is only possible to express the level of each input, say X^, as a function of one of the n inputs, say X\^, For example

P P P P

i Z = = h h

m n a. m n a.

«Ji (t .n T.(kp .n (X.) ) «h(T^.n T (k^) .n (X.) ) °i=i ^ ^ j=i J "i=i ^ ^ j=i J

which implies

a Pv, _i __ % P^

X^ = — = 1, . . . , n A.

4

i

105
The cost minimizing demand curve for the h^^ input-'- may then be obtained by

substituting equation A, 4 in equation A.l and solving for X^^. This is

expressed mathematically as m r n a.ir n a-l '"n a- Q = n (T.)(k.) I n ((a^/ia^))^ i _n n (Xj)J

1=1 j=i

or equlvalently

n (X^) J = Xf = Q To n T^Cki)

j=l ^ 1=1

L

n n

n ((aj)/(ah))J n ((Ph)/(Pj))'j"l

^ A.

5

where

R = S aj.

j=l

The cost minimizing demand curve for the h input, X^, is then obtained by

taking the one over R^^ equation A. 5. The cost minimizing demand

curve for the h*"^ input, Xi^, is therefore

^h = Q ^o ^ -^i(ki))

1=1

"1 "1

/ n a^\^

n ((a.)/(cxj^))J

/ n a. ^ R

I n ((p^)/(Pj))J i

i

A.

6

Substituting equation A. 6 into equation A. 4 permits the cost minimizing demand

curves for the remaining n-1 factor Inputs to be derived. These demand curves

are given as 1 "1

R m ^ n -a./R n -a-/R

Q T^ n T^Cki)) n ((aj)/(at,)) J n ((p^)/(Pj)) J {o^)){{v^)K? {))

1-1 J j=l

£= 1, n, ^ h.

'-The cost minimizing demand curve for the h*^^ input gives the level of the h^'^

input needed in order to minimize equation A. 2.

106
The above equations raay be simplified by noting that

and

n -a./R

n ((a.)/(aj^)) J = (ecu)

(l-(aj^/R))

j=l

-((a.)/R)

which implies

OLn n -a^R n -((a^)/R)

n ((a.)/(at^)) J = a n (a.^ ^

Applying the same logic one gets

n -((a.)/R) n -((a.)/R)

a„ n (a.) J = n ((a.)/(oi{)) ^ ^ j=l j=l

Through a similar set of manipulations it is possible to show that

Pu n , -((a.)/R) n , -((a.)/R)

n ((Ph)/(Pj)) J = n ((pp/(Pj)) J

The cost minimizing demand curve for each input is thus

* 1/R m -1/R n -((aJ/R) -((a.)/R)

Xo = Q do n T.(k.)) n((a.)/(a^)) J ((P;,)/PJ) J A.

7

i=l j=l

where Jl = 1 , . . . , n.

107
The minimiim cost solution for renovating Q square feet of floor area if

then gotten by substituting equation A. A into equation A. 2. This

substitution yields

n J. C* = E X Pn

il=l ^

^

or equivalently

1/R m -1/R n n -((a.)/R) -((a.)/R)

C* = Q (Tq n T.(k.)) E n((ap) J ((Pji)/P.)) J A.

i=l 1=1 j=l

Fortunately, equation A. 8 can be simplified by noting that

^ X,, , ^ , ((a.)/R)

n ((p^)/(p.)) J = n (p.) J j=i j=i

which implies

1/R m -1/R n ((a.)/R) n n -((a.)/R

C* = Q (Tq n Tj^(k^)) n p. J E n ((a.)/(aj^)) J i=l j=l ^^1 j=l

The last term in brackets in the previous equation can also be simplified

by noting that

n , -((aJ/R) n -((aJ/R)

n ((aj)/(a^)) J =

«jl n (aj) ^

which implies

n n , -((a.)/R) n -((aJ/R) n

S n ((a.)/(a3^)) J = n (a.) ^ E a^^ i=l j=l j=l ^Fl

n where R = E aj^

1=1

108
Equation A. 8 may thus be rewritten as

1/R/ m \ -1/R n ((a.)/R) n -((a.)/R)

c* = Q n T^Ck.) n Pj J (R) n ((a^) J a.9

V i=l / j=l j=l

A comparison of equation A, 9 to equation 2.10 in section 2.4 reveals the

two to be identical in form. The empirical estimation of the theoretical

cost function (either equation A.9 or 2.10) could then proceed in a manner

similar to that outlined in section 2.4.

109
BIBLIOGRAPHIC DATA

SHEET (See instructions)

1 PlIRI irATIDN OR

REPORT NO. NBS BSS 129

7 P^rfnrmino 1 CI luiiiiiiig Orp?*n \_/ 1 g cii 1 • Rpnrtrt r\ c|i^iw> 1 L Kin 1 >i • 3. Publication Date November 1980

4. TITLE AND SUBTITLE

Cost Estimation and Cost Variability in Residential Rehabilitation

5. AUTHOR(S)

Robert E. Chapman

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No,

8. Type of Report & Period Covered

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

Same as item 6.

10. SUPPLEMENTARY NOTES

Library of Congress Catalog Card Number: 80-600174

I I Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11, ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

This study analyzes four methods of estimating the costs of residential

rehabilitation. Each method is critiqued with regard to its treatment of changes

in the size of the renovation project, the productivity of labor, and the contractor's

markup for overhead and profit. Cost comparisons and a discussion of the way in

which the inherent riskiness of renovation activities may be assessed are also

presented. A theoretical approach for dealing with cost variability which

integrates the performance concept with established engineering economics techniques

is also developed.

12. KEY WORDS (S/x to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Applied economics; building codes; building economics; cost estimation; economic

analysis; engineering economics; housing; mathematical programming; optimization;

rphab-flifflfinn; renovation.

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