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ISIBSIR 81-2416

MBS

publications

Application of an Equivalency

Methodology to Building

Rehabilitation: A Pilot Study

NBSIR 81-2416

NATIONAL BUKEAL

ON BTANDAXPS

UBIAIT

MAR 1 6 1982 .

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APPLICATION OF AN EQUIVALENCY

METHODOLOGY TO BUILDING

REHABILITATION: A PILOT STUDY

James H. Pielert

Center for Building Technology

and

Robert E. Chapman and William G. Hall

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Applied Mathematics

Washington, DC 20234

January 1 982

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS. Ernest Ambler, Director
HO

S$?f % i ISA**
ABSTRACT

With increased emphasis on the re-use of existing buildings, new approaches

must be developed to assist regulators in making code-related decisions. The

application of performance criteria to building rehabilitation provides flexibility in the use of technically sound design alternatives in lieu of prescriptive provisions which may be restrictive. This report presents the results of

a pilot study on the application of an equivalency methodology in achieving

regulatory compliance. The use of such a methodology is particularly attractive

in this area because prescriptive type provisions have been shown to constrain

rehabilitation activities and, in some cases, may be mutually contradictory.

Regulatory requirements were chosen so as to explicitly incorporate conflicting

requirements as affecting the design of windows and doors — illumination,

ventilation, egress and security. The methodology is computerized to allow the

selection of least-cost means of achieving compliance with these requirements.

A prototypical townhouse is evaluated using the pilot equivalency methodology

and optimal compliance strategies are identified and compared with the cost of

prescriptive compliance. The results of the study produced potential savings

ranging from 20 to 30 percent depending on the initial conditions of the

building.

Key Words: applied economics; building codes; health and safety; housing;

mathematical programming; rehabilitation; renovation.

iii
PREFACE

This research was conducted under the joint sponsorship of the Building

Economics and Regulatory Technology Division of the Center for Building Technology (CBT) and the Operations Research Division of the Center for Applied

Mathematics (CAM), National Engineering Laboratory, National Bureau of Standards (NBS). The work was initiated as part of the building rehabilitation

technology program in CBT. It is an extension of the work being conducted

within NBS by CAM and CBT on the Fire Safety Evaluation System developed by

the Center for Fire Research.

The application of prescriptive code provisions to existing buildings has been

shown to constrain the rehabilitation process. These constraints, coupled with

the rapidly rising costs of new construction, have aggravated the Nation's

building problems. This report develops a pilot equivalency methodology which

includes a computerized procedure for identifying the least-cost means of

achieving compliance with selected regulatory attributes. The focus of the

study is on how these regulatory attributes affect and are affected by various

window and door compliance strategies in buildings being rehabilitated. An

objective of the study was to demonstrate the feasibility of applying equivalency methodologies to regulatory areas where conflicting requirements may be

present (e.g., window hardware may improve security performance but hinder

egress). Application of the computerized procedure was tested on a prototypical

townhouse residence. The results of the computer study indicate that potential

cost savings associated with the use of such an approach to code compliance

are substantial when compared to the more traditional prescriptive

approach.

Special appreciation is extended to Messrs. Patrick W. Cooke and Phillip T. Chen

who provided support in the formulation of the prototypical building and its

associated retrofit costs, as well as, in the development of code requirements

and related performance values. Special appreciation is also extended to

James G. Gross, Building Economics and Regulatory Technology Division, and

Christoph J. Witzgall, Operations Research Division, whose stimulating discussions provided guidance and encouragement throughout this effort. The significant contributions of Mary Chaney, secretary to the Rehabilitation Technology

Group, and Mary L. Ramsburg and Lisa Gray of the CBT Word Processing Center in

the typing of this report are gratefully acknowledged.

iv
TABLE OF CONTENTS

Page

LIST OF TABLES vi

LIST OF FIGURES vii

LIST OF EXHIBITS viii

EXECUTIVE SUMMARY ix

SI CONVERSION UNITS xi

1. INTRODUCTION 1

1 . 1 Background 1 1.2 Purpose 4

1.3 Scope and Approach 5

2. APPLICATION OF THE EQUIVALENCY APPROACH TO THE BUILDING

REHABILITATION PROCESS: SOME BASIC GUIDELINES AND EXTENSIONS 7 2.1 The Model Development Process: An Application of the Delphi

Method 8

2.1.1 Evaluative Models 8

2.1.2 Optimization Models 10

2.1.3 Application of Delphi Method 11

2.2 Structure of a Generalized Equivalency Methodology 14

2.3 Optimal Compliance Strategies: A Graphical Approach 21

3. CASE APPLICATION OF THE EQUIVALENCY METHODOLOGY TO RESIDENTIAL

REHABILITATION 29

3.1 Selection of Regulatory Requirements 29

3.1.1 Applicable Building Regulatory Requirements 29

3.1.2 Sample Worksheets Developed for the Pilot Study 31

3.2 Selection and Assessment of a Prototypical Design 39

3.3 Some Preliminary Results 44

4. SUMMARY AND RECOMMENDATIONS FOR FURTHER RESEARCH 49

4.1 Summary 49

4.2 Recommendations for Further Research 49

REFERENCES 52

APPENDIX A - COST ESTIMATION AND COST VARIABILITY: AN APPROACH BASED

ON COST FUNCTIONS AND PROBABILISTIC CONCEPTS 55

APPENDIX B - A MATHEMATICAL PROGRAMMING APPROACH TO EOUIVALENCY-BASED

REGULATIONS 71

v
LIST OF TABLES

Page

2.1 Building Component/ State Score Matrix 16

2.2 Calculating the Score for a Particular Regulatory Requirement 18

2.3 Score Assessment Matrix 19

2.4 Regulatory Requirement Matrix 20

2.5 Equivalency Test Matrix 20

3.1 Windov; Characteristics and Related Regulatory Requirements 32

3.2 Exterior Door Characteristics and Related Regulatory Requirements .. 33

3.3 Comparison of Initial, Prescriptive and a Near Optimal Cost

Solution for the Prototypical Building 46

A.l Density Functions for Commonly Used Probability Distributions 59

A. 2 Input Requirements for a Standard Probabilistic Cost Estimating

Procedure 61

A. 3 Cost Matrix Used in Linear Programming Approach 69

B. l Definitions of the Variables Used in the Computer Program 73

vi
LIST OF FIGURES

Page

2.1 Stages in the Development of an Equivalency Methodology 13

2.2 A Graphical Representation of the Constraints Associated with the

Four Regulatory Requirements 22

2.3 The Feasible Region Associated with the Four Regulatory

Requirements 22

2.4 The Selection of the Optimal Compliance Strategy 24

2.5 The Generation of an Alternate Compliance Strategy 24

2.6 The Selection of Compliance Strategies When Performance Levels

Vary Discretely 27

3.1 Relationship of Regulatory Requirements and Building Components .... 30

3.2 Typical Buildings Inspected in St. Louis, Missouri 40

3.3 Typical Buildings Inspected in Baltimore, Maryland 41

3.4 Floor Plan for Prototypical Building 42

3.5 Front and Rear Elevations of Prototypical Building 43

3.6 Typical Window and Door Retrofits 45

A.l Commonly Used Probability Distributions for Key Construction

Factors 58

A. 2 Cost Profile for a Renovation Project 64

vii
LIST OF EXHIBITS

Page

3.1 Sample Worksheet for Evaluating Security Requirements 34

3.2 Sample Worksheet for Evaluating Egress Requirements 35

3.3 Sample Worksheet for Evaluating Ventilation Requirements 36

3.4 Sample Worksheet for Evaluating Illumination Requirements 37

viii
EXECUTIVE SUMMARY

Throughout the United States, increasing concern is being expressed for the

need to more efficiently utilize the existing building stock. Studies of the

impact of the current regulatory process on existing buildings have concluded

that application of prescriptive code requirements developed for new

construction to such buildings presents a constraint to rehabilitation.

The purpose of this report is to demonstrate how an equivalent performance

approach can be adapted, in a fairly general but rigorous manner, to the

problem of identifying and selecting cost effective code compliance strategies

for rehabilitating existing buildings. The primary objective is to determine

the feasibility of applying equivalency methodologies to regulatory areas which

adversely impact rehabilitation and to highlight the advantages of such an approach to code compliance over the prescriptive approach. The major disadvantage of the performance approach 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 an equivalency methodology for fire safety known as the Fire Safety Evaluation System

(FSES). 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.

An equivalency methodology is constructed which is sufficiently general that

it is possible to define a distinct level of performance (or performance score)

associated with each code or code requirement under consideration.

A pilot study was formulated to extend the previous work on the FSES to

regulatory areas other than fire safety. The procedure identifies the leastcost means of achieving compliance with selected regulatory requirements which

affect windows and doors in residential buildings being rehabilitated. The

specific regulatory areas considered are security, egress, ventilation, and

illumination

.

Equivalency methodologies can be easily adapted to computer optimization. The

specific method may vary according to the problem. For purpose of exposition

however, it is sufficient to focus on one technique known as linear programming

which is used in this pilot study. 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 per se

.

A case application was designed to test the equivalency methodology and to

demonstrate how the procedure would be applied in practice. This includes the

selection of regulatory requirements, development of sample worksheets, and the

formulation of a prototypical design. Criteria for the design of the case study

included the following: (1) regulatory requirements were selected such that an improvement in one area could result in a negative effect in another area;

(2) regulatory requirements were selected such that an improvement in one area

could affect another area in the same way; and (3) building components were

ix
selected such that their design would be affected by the regulatory requirements.

Egress and security would be affected in a conflicting way by improvement in

window and door security hardware (criterion 1); ventilation and illumination

would be similarly affected by changes in window size (criterion 2); and window

and doors can contribute to the level of compliance of all four regulatory

requirements (criterion 3).

Worksheets capable of defining a level of performance were developed for each

set of regulatory requirements. This task was accomplished by subdividing each

building component into a set of states which cover all possible situations

expected to be encountered in practice. For example, window type includes slider,

casement, single hung, double hung and fixed as states. Associated with each

component/ state pair is a value which represents the contribution of this pair

to meeting the overall objective; i.e., attaining the level of performance

required by the regulatory requirement under consideration. The state values

in the worksheets were assigned by the project team based on their knowledge

of the subject areas. There was no attempt made to utilize broader technical

input in setting these values since the purpose of the study was to evaluate

the applicability of equivalency methodologies for the regulatory areas considered and was not intended to be a comprehensive technical treatment.

A prototypical building was synthesized in a manner which provided a test of

the equivalency methodology formulated in the case study. A typical inner city

residential townhouse was selected with various initial window and door conditions. "In-place prices" for accomplishing these various retrofits were identified. The computer program which was developed based on the model equivalency

methodology was applied to this building with the various initial window and

door conditions (type, size, number, location, etc.).

Potential savings associated with the use of an equivalency based approach to

compliance versus the more traditional prescriptive approach were then estimated

for eight buildings with different initial conditions. Generally, the cost of

code compliance based on the equivalency approach is about three-quarters of

those associated with strict prescriptive compliance. If the number of regulatory requirements and building components are increased to reflect the actualities of building rehabilitation activities, this percentage may either increase

or decrease. However, based on the studies in the fire safety area, fairly

substantial savings, i.e., 20 percent or more, would be expected.

The study demonstrated the feasibility of applying equivalency techniques to

regulatory requirements which may have conflicting impacts on building

rehabilitation. A mathematical optimization model can be used to identify

least-cost means of achieving compliance with significant cost savings.

x
SI CONVERSION UNITS

The conversion factors and units contained in this report are in accordance

with the International System of Units (abbreviated SI for Systeme International d' Unites). The SI was defined and given official status by the 11th

General Conference on Weights and Measures which met in Paris, France in

October 1960. For assistance in converting U.S. customary units to SI units,

see ASTM E 380, ASTM Standard Metric Practice Guide, available from the American Society for Testing and Materials, 1916 Race Street, Philadelphia, PA

19103. The conversion factors for the units found in this Standard are are follows

Length

1 in = 0.0254* meter

1 ft = 0.3048* meter

1 mil = 0.001* in

1 yd = 0.9144* meter

Area

1 in2 = 6.4516* x 10“^meter2

1 ft 2 = 0.0929 meter2

1 yd 2 = 0.836 meter 2

Volume

1 in2 = 1.639 x 10“2 meter2

1 liter = 1.00* x 10-2meter2

1 gallon = 3.785 liters

Temperature

°C = 5/9 (Temperature °F-32)

* Exactly

xi

1 . INTRODUCTION

1.1 BACKGROUND

Throughout the United States, increasing concern is being expressed for the

need to utilize more fully the existing building stock. The re-use of old

buildings has grown far beyond the preservation movement which spawned it with

all existing structures now viewed as assets that can be reclaimed. Soaring

construction costs and Federal incentives are creating a positive environment

for building rehabilitation. During the late 1970' s, national policy has promoted a program to rehabilitate older city buildings instead of tearing them

down and erecting new ones, a process that had destroyed many inner city neigh

borhoods. The Department of Commerce reported that in the third quarter of

1980, seasonally adjusted annual expenditures for residential alteration and

repair reached a new high of almost $50 billion. 1 The Architectural Record

reported in October of 1979 that expenditures for nonresidential additions,

alterations, and major replacements are expected to increase from $15 billion

in 1978 to as much as $30 billion annually by the mid 1980' s. z

Anderson3 gives the following reasons for the recycling boom: (1) from start to

finish a recycling project may take only half as long to complete as a similarly

sized new development, (2) recycling costs are competitive with new construction, (3) recycled space is much in demand, and (4) attitudes of city officials

and lenders are changing. Because of these reasons, more and more decaying

and abandoned houses, warehouses, schools, railroad stations, and other build

ings once slated for demolition are being turned into money-making properties

while having positive impacts on the neighborhood.

Several studies of the impact of the current regulatory process on building

rehabilitation have concluded that application of prescriptive code provisions

for new construction to existing buildings presents a constraint to rehabilita1 U.S. Department of Commerce Construction Reports, "Residential Alterations

and Repairs, Annual 1979,” issued April 1980.

2 Philip E. Kidd, Value of Nonresidential Rehabilitation Will Double by

Mid-1980'

s

, Architectural Record (pg. 61), October 1979.

3 Timothy Anderson, Recycling: Old Buildings Big New Opportunity , Professional

Builder (pp. 197-199), January 1979.

1
tion. 1 >2 >3 ,4 ,5 This is manifested in two ways: (1) most codes contain administrative provisions which state that a building's conformance with the requirements of the code for new construction should increase in relation to the dollar

amount of rehabilitation planned; and (2) when the occupancy classification of an existing building changes the building must be made to conform to the code

requirements for the new occupancy. Applying codes for new construction to

existing buildings, with their generally prescriptive format, presents difficulties since: (1) they may not address the types of construction present in many

older buildings; (2) innovative solutions are limited since building officials

feel they do not have a legal or technical basis for approving code deviations;

and (3) they are structured to follow the design process associated with new

construction as contrasted to the analytical procedure required in building

rehabilitation.

Existing buildings constructed to some code or standard that reflect the

state-of-the-art at the time will probably not comply with current codes.

However, if they have performed their intended use without exposing the public

to any undue hazard, one could conclude that they should be considered for re-use. This philosophy has become increasingly accepted over the past several

years with several significant rehabilitation code and guideline development

activities taking place.

A. Code Provisions for Existing Buildings by the State of Massachusetts

Massachusetts has published a new Article 22 of the State Building Code

entitled "Repair, Alterations, Additions and Changes in Use of Existing

Buildings. "6 This article allows rehabilitation of existing buildings

1 Saundra A. Berry, Ed., Proceedings of the National Conference on Regulatory

Aspects of Building Rehabilitation , National Bureau of Standards, Special

Publication 549, August 1979.

2 James G. Gross, James H. Pielert and Patrick W. Cooke, Impact of Building

Regulations on Rehabilitation — Status and Technical Needs , National Bureau

of Standards, Technical Note 998, May 1979.

3 Impact of Building Codes on Housing Rehabilitation , Hearings Before the

Committee on Banking, Housing and Urban Affairs — United States Senate,

March 24, 1978.

^ N. John Habraken, et al . , An Investigation of Regulatory Barriers to the

Re-Use of Existing Buildings , National Bureau of Standards, GCR 78-139,

January 1978.

3 Robert J. Kapsch, Building Codes: Preservation and Rehabilitation , National

Bureau of Standards, Special Publication 473 (pp. 437-452), June 1977.

6 State of Massachusetts, Article 22 - Repair, Alteration, Additions and Changes

in Use of Existing Buildings , Massachusetts State Building Codes Commission,

October 1980.

2
without necessarily meeting all new construction code requirements, as long as, minimum health and safety provisions are met. The following rules

apply: (1) ordinary repairs (paint and wallpaper) require no permit; (2)

repairs (maintenance) made with similar materials are allowed; (3) alterations are allowed with similar materials, but any new system must comply

with the code for new construction (e.g., egress, fire protection, plumbing,

electrical, mechanical, energy conservation, and building enclosure system);

and (4) additions must conform to the code for new construction including

height and area limits. A recent NBS report documents the development of

Article 22 and provides four case studies of its application.!

Complete or partial changes in use are regulated according to the change in

hazard index which depends on the use group of the building occupancy.

Appendix T of of the Massachusetts State Building Code contains a detailed

listing of occupancies and related hazard indexes. The change in the

hazard index from the existing occupancy to the proposed occupancy determines the level of code compliance under the following requirements:

(a) Hazard index stays the same or decreases (Article 22 - Section 2203.0) - Eight items of code compliance are mandatory including floor loads,

structural loads (excluding seismic), number of exits, capacity of

exits, exit signs and lights, means of egress lighting, fire alarm

system, and smoke enclosure of stairways.

(b) Hazard index increases by one (Article 22 - Section 2204.0) - Changein-use must conform to the code for new construction with the following exceptions: fire limits, area and height limits, accessibility for

physically handicapped, earthquake resistance and soil liquefaction,

mortar, and certain requirements related to fire and party walls.

( c) Hazard index increases by two or more (Article 22 - Section 2205.0) - change-in-use must conform to the code for new construction.

Section 2206.1 of Article 22 provides that "where compliance with the

provisions of the code for new construction, required by this article, is

impractical because of structural or construction difficulties or regulatory conflicts, compliance alternatives may be accepted by the building

official." Appendix T of Article 22 contains some acceptable compliance

1 James H. Pielert, Removing Regulatory Restraints to Building Rehabilitation:

The Massachusetts Experience , National Bureau of Standards, Special

Publication 623, October 1981.

3
alternatives. The development and implementation of Article 22 has been

discussed in several publications . 1 >2 ,3

B . Rehabilitation Guidelines by Department of Housing and Urban Development

The U.S. Department of Housing and Urban Development (HUD) has developed

Rehabilitation Guidelines to help communities resolve building code problems encountered in implementing rehabilitation programs. ^ The Rehabilitation Guidelines are divided into eight volumes:

Volume 1 Volume 2 Volume 3 Volume 4

Volume 5 Volume 6

Volume 7 Volume 8

- Setting and Adopting Standards for Building Rehabilitation

- Municipal Approval of Building Rehabilitation

- Statutory Guideline for Building Rehabilitation

- Managing Official Liability Associated With Building

Rehabilitation

- Egress Guideline for Residential Rehabilitation

- Electrical Guideline for Residential Rehabilitation

- Plumbing DWV Guideline for Residential Rehabilitation

- Fire Ratings of Archaic Materials and Assemblies

The information contained in Volumes 1 through 4 is most closely related to

the subject of this report. Potential problems caused by application of the

"25-50 percent rule" and the change-in-occupancy triggering mechanism are recognized and guidance is provided on a more realistic approach. Locally

developed rehabilitation codes which no longer use these triggering mechanisms

are described in the appendices of Volume 1 , including codes for Massachusetts, San Francisco, Denver, Washington, D.C., Los Angeles and Detroit.

Such performance oriented approaches make use of the technical guidelines in

Volumes 5 through 8.

1.2 PURPOSE

The purpose of this pilot study is to demonstrate how the equivalent performance

approach can be adapted, in a fairly general but rigorous manner, to the problem

of identifying and selecting cost effective code compliance strategies when

rehabilitating existing buildings. The primary objective is to determine the

feasibility of applying equivalency methodologies to regulatory areas which

adversely impact rehabilitation and to highlight the advantages of such an approach to code compliance over the prescriptive approach* The pilot study

1 James G. Gross, James H. Pielert, and Patrick W. Cooke, op. cit.

2 Sandra A. Berry, Ed., op. cit.

2 Proceedings of the Conference on Building Rehabilitation Research and

Technology for the 1980 8 s , National Conference of States on Building Codes

and Standards, August 1980.

^ Rehabilitation Guidelines , U.S. Department of Housing and Urban

Development, October 1980.

4
considers the design of windows and doors in buildings being rehabilitated as affected by the regulatory areas of security, egress, ventilation, and

illumination.

1.3 SCOPE AND APPROACH

The general plan of this report is to describe briefly previous work on the

application of equivalency methodologies to regulatory applications; to discuss the mathematical concepts and a computer program based on the equivalency

methodology; and to identify and discuss the relevant engineering considerations which must be exercised in order to use this approach in solving a particular problem. Specifically, this report is organized as follows:

Chapter 2 describes the general application of the equivalency approach

to the building rehabilitation process including: (1) development of

equivalence methodologies based on a management tool known as the Delphi

Method; and (2) the structures of such equivalency methodologies including

the technical bases. The advantages of this approach to code compliance

over the more restrictive prescriptive approach are highlighted. An optimization model which minimizes construction and materials costs is utilized. The chapter concludes with the presentation of a graphical approach

for obtaining optimal compliance strategies.

Chapter 3 presents a case application of the equivalency methodology to a residential rehabilitation. The rationale for the selection of regulatory

requirements in the area of health and safety is discussed and specific

requirements applied to window and door retrofits are presented. Specific

worksheets are included which define the state values (level of performance)

for each regulatory requirement associated building components. Details

of the prototypical residential building are then presented. The chapter

concludes with some preliminary results of the study where a computer program was exercised to identify the least-cost means of achieving compliance to the selected regulatory requirements for the prototypical building.

The cost of strict compliance, as well as the costs of the alternative

solutions, are then compared in order to demonstrate the flexibility

available to decision makers who use the equivalency methodology.

Chapter 4 summarizes the major findings of the study and sets forth several

recommendations for continued development of the methodology including

technical and mathematical considerations.

Appendix A presents a discussion incorporating a cost engineering approach

with the economic theory of cost functions in arriving at a cost estimating procedure which is sensitive to the technical considerations of the

rehabilitation process as well as local market conditions. Appendix B

presents the mathematical formulation of the equivalency methodology for

the prototypical building discussed in Chapter 3.

5
It should be pointed out that the pilot study on the use of equivalency

methodologies discussed in the report was conducted to assess the feasibility

of the approach as applied to regulatory areas other than fire safety in the

building rehabilitation process. There was no attempt made to subject the

worksheets shown in Chapter 3 to a comprehensive Delphi-type review because of

funding and time constraints. However, it is believed that the engineering and

mathematical treatment was of sufficient rigor to have demonstrated the

feasibility of the approach and to justify continued development.

6
2.0 APPLICATION OF THE EQUIVALENCY APPROACH TO THE BUILDING REHABILITATION

PROCESS: SOME BASIC GUIDELINES AND EXTENSIONS

The purpose of this chapter is to demonstrate how the equivalent performance

approach can be applied, in a fairly general but rigorous manner, to the problem of identifying and/or selecting efficient investment strategies for rehabilitating existing buildings. The objective of this exercise is to highlight

the advantages of such an approach to code compliance over the more restrictive

prescriptive approach. A previous study 1 has shown that there are major advantages of the equivalent performance approach which make it highly desirable for

application to the rehabilitation process: it provides decision makers with

greater latitude in making choices, has cost reducing potential, and permits

innovation

.

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 system known as an equivalency methodology. 2 >3 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. Furthermore, the equivalency methodology approach is sufficiently general that it is possible to define a distinct quantitative level

of performance (or performance score) associated with each regulatory requirement under consideration. Thus, the use of an equivalency 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 both cost and cost variability. This relationship may be seen more clearly by noting that greater freedom in making retrofit choices will permit

the investor to avoid not only one or more "expensive" retrofits but also 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.

1 Robert E. Chapman, Cost Estimation and Cost Variability in Residential

Rehabilitation , National Bureau of Standards, Building Science Series 129,

November 1980.

3 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.

3 Robert E. Chapman, Phillip T. Chen and William G„ Hall, Economic Aspects of

Fire Safety in Health Care Facilities: Guidelines for Cost-Effective

Retrofits , National Bureau of Standards, NBSIR 79-1902, November 1979.

7
Consequently, the end result would be a reduction in both the budget estimate

and its variability.

The discussion which follows is divided into two parts. The first deals with

the development of equivalency methodologies based on a management tool known

as the Delphi Method and the second part develops the structure of a generalized

equivalency methodology.

2.1 THE MODEL DEVELOPMENT PROCESS: AN APPLICATION OF THE DELPHI METHOD

In essence, an equivalency methodology or model is a formal mathematical

representation of the states( levels of performance) associated with building

rehabilitation. As with a conceptual model of a physical process or process

control situation, there are both internal constraints and external trade-offs

involved in the formulation. Resolution of the trade-off problem is strongly

dependent upon a complete understanding of the way in which the model is to be

utilized. The model must be useful and in a form which is usable.

The internal constraints involve items such as validity (Does the model reflect,

with sufficient accuracy, the physical situation?); generality (Does the validity hold over the expected range of physical situations? Is an inordinate

volume of input required?); calibration (Can the required parameter or coefficients be determined?); and technological state-of-the-art issues (Are variables

of measurable dimensions and quantities? Can they be quantified?). Computational tractability or feasibility is also an important internal factor; it is,

however, of lesser importance here than in the external trade-off decision.

The use envisioned for the model impacts upon all of the model's internal

attributes. Use, in turn, depends primarily on the user class, where the user

class is actually a target market for a very specialized product. The user

class (institutionally and individually) must be understood and characterized

in terms of its expectations, orientations, motivations and capabilities.

2.1.1 Evaluative Models

There are two general types of equivalency model formulations. The first and

simplest of these is an evaluative model . Its only output is an assessment of

the expected performance of a given configuration of components and/or subsystems, or possibly a given sequence of configurations. The performance is in

terms of desirable or mandatory functions of the building. Any considerations

other than performance, such as cost and aesthetics, must be accommodated

external to the model. The required calculations may be done by hand, by

calculator, or by a small computer and the user may be relatively

unsophisticated in computer science.

The extent to which the use and user can dominate the model formulation will

be illustrated by several scenarios for an evaluative model. These are not

intended to be exhaustive but they are moderately realistic.

8
As a first scenario, consider a model whose purpose is to determine if a particular combination of materials, components, and/or subsystems is adequate

with respect to some law, regulation, or standard. The primary users are regulatory agencies who use the model to produce a basis for acceptance or rejection of a proposed design. For these uses and users, the desirable model

characteristics are:

1) Validity must be very good; a mistake in the accept-reject decision

may have costly consequences.

2) Generality should apply to all building types over which the agency

has authority to allow for broad application. It may be attained by

a family of models or by parameterization of a single model. A

large volume of perhaps rather detailed data is acceptable.

3) Calibration is not a particular problem. The components and subsystems

are homogeneous; most of the information required is standard

engineering data.

4) Technological state-of-the-art issues do not pose problems.

5) Computational tractability is not important for this use. A tremendous

amount of computation for the evaluation of a single case can be

tolerated.

For the second scenario, consider an evaluative model used by an architectural/

engineering (A/E) firm as an aid to evaluate a fairly large number of conceptual

designs, say several hundred, from among the millions of potential configurations. For this use and user, the desirable model characteristics are somewhat

different. They are:

1) Validity requirements can be relaxed considerably. The user applies

the model as a filter to select a smaller group of candidate configurations for further analysis. Accuracy to perhaps as little as one significant figure should be adequate.

2) Generality is much the same as in the first scenario except that large

volumes of data (per case) should be avoided.

3) Calibration is again no problem.

4) Technological state-of-the-art is of even less importance than in the

first scenario.

5) Computational tractability is a moderately critical factor. Long,

involved calculations are burdensome and conducive to error , and there

are two orders of magnitude more cases to evaluate.

A third scenario for an evaluative model deals with an existing facility.

Depending upon the user, the model characteristics of validity, generality, and

9
computational tractability are as those in scenarios one and two. The other

model characteristics differ:

1) Calibration becomes considerably more complicated. It may be difficult

or even impossible to estimate calibration parameters either empirically or theoretically. Obsolete technologies, heterogeneous components, and performance degradation due to age, use, or abuse may be

encountered

.

2) The technological state-of-the-art may pose problems of measurement of

the component or subsystem properties. These difficulties are similar

to those expected in calibration.

2.1.2 Optimization Models

The second general type of model is an optimization or optimum seeking model . Functionally, it differs from the evaluative model in that it determines the

component/subsystem configuration which is acceptable and optimizes some objective. The most obvious example is to determine the least-cost configuration

which maintains the functional integrity of the building. Other objectives are possible, the requirement being that they are quantifiable in commensurate

units, but considered here is only the case in which the objective is to minimize costs. The optimization model must include the evaluative model, additional explicit constraints, a costing submodel, an optimization procedure,

and a procedure for producing a class of alternate solutions all integrated

into a single model. It is more complex, more expensive to develop and exercise, and requires much more mathematical and computer science sophistication.

As with the evaluative model, the use and user class impact upon the model

formulation are the same as described in scenarios one through three. However,

there are some more stringent requirements upon the developer.

A costing sub-model is essential to quantify the\ costs of various retrofit

options. The complexity of the costing model may vary from a mechanism for

direct input to rather sophisticated costing functions (see appendix A for a detailed discussion of this issue).

The evaluation portion of the model must be augmented by formal statements of

certain requirements relating to such things as aesthetics and engineering judgment which are not explicit in the performance requirements. Otherwise, the

optimum configuration with respect to the performance criteria may be nonfeasible with respect to some implicit construction, architectural, engineering,

or aesthetic requirements. To some extent, if the non-performance requirements

are difficult to quantify, these relations may be replaced by augmentation via

a set of near-optimum solutions. In effect this strategy evades the formal

statement of these requirements at the expense of producing extra sets of

solutions for critical examination.

In addition to the optimum solution, a class of near optimum or "good" solutions

should be generated. These are required to answer many "what if" questions of

value to the user; they should provide a basis for accommodation of non- construction costs and whatever requirements are not explicit in the model.

10
Ideally, the alternate solutions should contain all configurations reasonably

close to the optimum with respect to the value of the objective; each candidate

subsystem should appear in at least one solution. The class of solutions identified combined with engineering judgment should provide for the selection of

the best overall configuration.

The optimization model is much more costly to develop and exercise than an evaluative model. However, in principle, most of the extra burden can be

absorbed by the model developer. Consequently, it is practical to produce software which is "user-friendly" to the extent that the additional requirements

placed upon the user are primarily additional computational resources. While

some user-supplied information is required, it does not require any change in user orientation, skills, or capability.

The procedures required can be characterized as cost-effective constrained

optimization models. The obvious candidate methodologies are mathematical programming procedures. These include linear programming, integer programming,

and various forms of non-linear programming . 1 The procedure to be used is

dependent upon the form of the relations between the costs, the subsystem

characteristics, and the performance requirements. The task of developing an optimization model could range from almost trivial to impossible depending on

the form and structure of these relations.

2.1.3 Application of the Delphi Method

The model development process outlined earlier is the output of a consensus generating technique known as the Delphi Method. The Delphi Method in its

present sense and as applied to the development of an equivalency methodology,

consists of a panel of experts charged with achieving each of six objectives

relative to the application of any resulting model. These objectives are safety, flexibility, workability, economy, choice, and accountability.

The first objective, safety, is the primary reason for regulations and

consequently any alternative to prescriptive compliance must ensure that the

health and well-being of the individuals occupying the building are not compromised. The second objective, flexibility, is designed to provide options

which can be categorized according to some rule (e.g., cost, aesthetics, function, mission, etc.). The third objective, workability, implies that the system can in fact be used (e.g., no new education of inspectors or designers is

needed since the system uses present well-understood principles.) The fourth

objective, economy, explicitly recognizes that the introduction of alternatives

to prescriptive compliance permits the intent of the regulation to be met at some savings in cost to the building owner. Greater economy in design and con- duction should also result because an equivalency methodology permits innovations in the building process to be introduced more easily. The fifth objective,

choice, is the building owners option; unlike the prescriptive compliance

approach, the equivalency methodology produces classes of solutions which

1 An introduction to these techniques is given in Saul Gass, Linear Programming:

Methods and Applications, Fourth Edition, McGraw-Hill, New York, 1975

.

11
satisfy the regulation( s) under consideration. Building owners may have

a variety of reasons for undertaking building renovation; by providing a class

of solutions all of which satisfy the regulation( s) , the building owner can select the retrofit alternative which best satisfies his needs. The sixth

objective, accountability, is a requirement of the authority having jurisdiction. Since an equivalent solution is an alternative to prescriptive compliance, the building official(s) responsible for enforcing the regulation( s)

must be assured that the solutions provided by the model are based on sound

technical considerations and satisfy all of the requirements associated with

the prescriptive solution. Another area of accountability concerns the granting

of waivers. Traditionally, waivers have been granted when prescriptive compliance can be achieved only at undue cost to the building owner. It is believed

that waivers will be granted on a more systematic basis under an equivalency

methodology than in the past.

The task of achieving the six objectives just described is illustrated in

figure 2.1. The figure shows the stages in the development of an equivalency

methodology. It is designed to highlight the needs associated with analysis of

the complex interactions expected in the model development process. The initial

step, which is essential for success, involves the postulation of a model which

governs the way in which the various regulatory requirements interact (i.e.,

are they independent, complementary or conflicting?). The next step is to

select a universe of key components which span all regulatory requirements

(e.g., window size and glazing type). Once all components have been identified,

it is then necessary to subdivide each one into a set of conditions or states

which span all likely possibilities. At this stage it is important to point

out that the set of parameters must be large enough to include the component/

state pair which corresponds to prescriptive compliance for each regulatory

requirement under consideration. The next step is to assign weights to each

component/state pair which best reflects the relative degree of risk or safety

associated with that state. More specifically, negative values should be used

to reflect greater risks whereas positive values should be used to reflect con- tributions toward a higher level of safety within the area under consideration.

Ideally, the system would treat a value of zero as "safety neutral". In order

to reduce the complexity of this step in the process, it is recommended that

separate weights be assigned to each regulatory requirement under consideration.

The postulation of a model in a way in which the requirements interact as a first step ensures that such an approach is technically valid. The final step

is to reassess the weight assigned to each state for consistency with the interaction model, the list of components and the list of states. In the event that

one or more of the previously mentioned attributes is found to be lacking or inconsistent, the sequence of steps following the inconsistency up to and

including the entire process should be repeated. The level of performance

associated with each regulatory requirement is calculated by first identifying

a unique state for each component which corresponds to prescriptive compliance.

This process is repeated for each regulatory requirement. The score, or weight,

associated with this state is then recorded on a worksheet. The interaction

model is then used to evaluate the level of performance associated with

prescriptive compliance. The set of scores which results provides constraints

which must be met or exceeded in order for the solution to satisfy the intent

12
DELPHI

METHOD

13

Figure

2.1

Stages

in

the

Development

of

an

Equivalency

Methodology
of the regulatory requirements. Upon achievement of consensus on the entire

process it will be beneficial for the members of the Delphi group to act as liaison among the various groups which formulate and promulgate building regulations. This step is necessary to demonstrate that an equivalency methodology

is based on a sound technical basis which is broad enough to avoid problems in

implementation.

2.2 STRUCTURE OF A GENERALIZED EQUIVALENCY METHODOLOGY

This section contains a series of tables (matrices) which reveal the basic

structure for a generalized equivalency methodology. Prior to the development

of this generalized methodology, however, it is necessary to define several

items 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 for the regulatory

requirements 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

a particular state for a given requirement.

(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. It may take on positive integer values if the state admits multiples (e.g., the number

of doors).

(5) Regulatory Requirement - the level of performance required by each

requirement under consideration. It is assigned a numeric score which

later becomes a constraint.

(6) Score Assessment - a numeric score associated with the level of performance

provided within the building for each requirement under consideration. It

is the sum of all state value/state variable products.

(7) Retrofit Co st - the cost of moving from any given or initial state to any

other stated

The following assumptions will be made in the discussion which follows:

Cl) All regulatory requirements must be satisfied simultaneously in order for

the building to be deemed in compliance.

1 Appendix A outlines how the engineering economics concept of a cost function

coupled with probabilistic methods can be used to develop a cost structure

for the problem.

i 14
(2) The score assessment is the sum of all state values for those building

components which affect the regulatory requirements under consideration.

(3) With respect to a given requirement, each building component can be in one

and only one state. (This assumption can be relaxed to accomodate design

considerations .

)

(4) The state of a building component is governed by a worst case condition

for that component. (This assumption can be relaxed without loss of

generality.

)

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 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. 2 The term "program" is used

in the general sense in that it refers to a plan rather than a computer program

per se .

The foundation of the linear programming model is shown in table 2.1, the

building component/ state score matrix. An examination of table 2.1 reveals

that there are n building components, m states (levels of performance) associated with each building component 2 and p regulatory requirements associated

with each component/ state pair. The state value associated with the i

fc h com- ponent, j state and k

tn code is denoted (see the cell removed from the

building component/ state score matrix in table 2.1). As a general point of

departure, it will be assumed that negative state values represent undesirable

circumstances whereas positive state values represent desirable circumstances

.

1 The requirement that all functions be linear is relaxed in appendix B where

the general mathematical structure of the problem is presented.

2 A linear function is defined as a function of the form

f(X) = aQ + a^x^ + ... + ajxj + ... + anxn where a-j are coefficents 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 whereas g(x) =c + dx2 , a parabola, is not.

2 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 appendix B that an unequal number of states

can easily be accommodated in the mathematical formulation of the problem.

15
Table

2.1

Building

Component/State

Score

Beatrix

16
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 from the initial state,

j c , to some other state j, represents a potential retrofit if Vjjk is greater

than Vijck-‘

Thus far, the mechanics of actually calculating the level of performance

provided within the building for a particular code have not been discussed.

Tables 2.2 and 2.3 provide the format for performing these calculations.

Table 2.2 is a vertical slice of the building component/state score matrix.

Note that table 2.2 is a matrix with n rows, one for each building component,

and m columns, one for each possible state. An examination of table 2.2 would

reveal that some cells of the matrix are shaded. This is due to the fact that

each building component can be in one and only one state. In view of this fact

and for expository clarity we can, without loss of generality, replace Vnk

with S-^, V

2mk with 82^., and so on through V

R ^ with S n

^. In order to

perform the score assessment calculation, one ’would record the state values,

Sik, in the appropriate spaces in table 2.3. 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 2.3. These scores are labeled , P 2 , ... Pp , indicating the level of performance provided within

the building building for each of the p regulatory requirements under

consideration.

The regulatory requirements are given in table 2.4. These numeric values are determined by identifying the state in table 2.2 which includes or is the prescriptive solution defined in each requirement. These values are then entered

in the appropriate spaces in table 2.3 and summed in order to get the regulatory

^ It is of course possible that a movement to a higher state in one building

component will affect the performance of another building component with

respect to the same regulatory requirement. "Component interdependence" can usually be handled through the use of dummy state variables representing

dummy states. A more important issue however, is "requirement interdependence"

whereby a retrofit in one area may represent a retrograde in another. This

relationship (if and when it exits) serves to constrain the feasible region

to the class of bounded polyhedrons.

o At a more rigorous level, may be defined as follows:

m

S ik = ^ Vijk * xij

3=1

V» Vi where V.^^. = the state value associated with the i

u component, the j state

and the k^ regulatory requirement; and

Xij = the state variable

Since Xjj can only take on values of 0 or 1, S^k is uniquely determined by

the value of j, j', for which Xjj takes on a value of 1.

17
Table

2.2

Calculating

the

Score

for

a

Particular

Regulatory

Requirement

E

CM

18

STATE
Table

2.3

Score

Assessment

Matrix

Q_ Q. CL. % a= OQ_ CM CO a a a a II GO GO GO c to Q_ GO Q.

*-* 03E

03

P-1

R Ql

CO

Ql

CM

GO

a a.CO

GO

aa

a oLa GO 0 CL. GO

II l QL

O"

03

a

a

B

a

a OC a a a a # a a a a a a a a a a a o

bi)

03 CM

CM CM

CM CM

CO

a

aa CM

CM

II C£ GO GO GO GO GO CM

D_

a

a

a --

II GO GO GO

GO GO

oT

CM CO

a

a

a i COo

luauoduiOQ Suping +

GO++

19

Pk- X

S,kSik +

S2k
Table 2.4 Regulatory Requirement Matrix

Regulatory requirement

i 2 3 P

Score Ri R2 R3 a a a Rp

Table 2.5 Equivalency Test Matrix

> 0

Regulatory

requirement

P - R = Yes No

i Pi - Ri

i II

2 P2 - R2

~~ a •

II

• 1•

P Pp - R

p

1 II

20
requirements, , R2 , ..., R

p

. Note that the equivalency methodology is sufficiently general to accommodate variations in the level of performance required

by different regulatory jurisdictions.

The final step, determining if the building is in compliance with each

requirement, is outlined in table 2.5. This step is taken by transferring the

first set of scores P]_, P2 , ... P

p

from table 2.3 to the boxes labeled P]^ , P 2 , ... P

p

, in table 2.5. The second set of scores R]_, R2 , ... R

p

, is then transferred from table 2.4 to the boxes labeled R^, R2 » ••• R

p

« Regulatory equivalency is tested by determining if the differences between the first set of

numbers, P]_, P2 , • •• P

p

, and the second set of numbers, R]_, ... R

p

, 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 regulatory requirement . At this point

it becomes necessary to define a plan of correction or, more simply, a com- pliance 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. In order to illustrate how such a procedure

would be applied in practice, a scaled down version of the generalized equivalency methodology was applied to a prototypical building. The results of this

case study are the subject of Chapter 3; a mathematical statement of the

optimization model is given in appendix B.

2.3 OPTIMAL COMPLIANCE STRATEGIES: A GRAPHICAL APPROACH

This section draws upon the mathematical formulation of the problem discussed

in the previous section. A way in which alternative solutions are generated

will also be presented. In order to focus on the main concepts, a geometrical,

rather than an algebraic, approach will be used.

In order to illustrate how the mathematical programming approach used in the

pilot study operates, an example is used in which it is assumed that the performance for each of the four regulatory requirements discussed in section 3.1

is a function of the level (e.g., size, hardware), of two techniques (e.g.

building components). Limiting the discussion to two dimensions permits the

basic concepts to be illustrated graphically. Problems like the generalized

equivalency methodology just described would, however, be many dimensional.

In the discussion which follows, the level of each technique is denoted by

Lj and L2 respectively. In all cases, the level of technique "1" will be

shown along the horizontal axis and the level of technique "2" along the vertical axis. More precisely, any movement out along the horizontal axis indicates

higher levels for technique "1". Similarly, any movement up along the vertical

axis indicates higher levels for technique "2”. Consequently, if one were to

construct a straight line which passed through the origin, any movement outward

along that line (i.e., a move in the north easterly direction) would indicate a movement to higher levels for both techniques.

Turning now to figure 2.2, it can be seen that four sets of lines have been

constructed. Each of these lines represents a regulatory requirement. All

points on a given line have equal performance. The points of intersections of

21
LEVEL OF TECHNIQUE 1 (LI)

Figure 2.2 A Graphical Representation of the Constraints Associated

with the Four Regulatory Requirements

LEVEL OF TECHNIQUE 1 (LI)

Figure 2.3 The Feasible Region Associated with the Four Regulatory Requirements

22
the lines are denoted as A, B, C and D. These points also serve to define a quadrilateral. Notice that associated with each line is a series of arrows.

These arrows indicate the portion of the quadrant (all possible combinations of

techniques "1" and "2" where Lj , L 2 >_ 0) within which that regulatory requirement is met or exceeded. A closer examination of figure 2.2 reveals that one

of the sets of arrows points away from the other three sets. This is a graphical representation of a conflicting requirement. As discussed in section 3.1,

the scores associated with security tended to conflict with some of the other

regulatory requirements. If one takes the intersection of each set of potential solutions, the quadrilateral ABCD and its interior results. In mathematical terminology ABCD is referred to as a convex set since for each pair of

points in the entire collection, the entire line segment joining these two

points is also in the collection.

This intersection is shown in figure 2.3 by the lightly shaded region

representing all the combinations of the various levels of techniques "1" and

”2” which match or exceed each of the four regulatory attributes discussed earlier. The lightly shaded region in figure 2.3 is referred to as the feasible

region since all points which lie along its boundary or within are technically

feasible. The boundry excluding the vertices represents those combinations of

the two techniques which exactly satisfy one of the regulatory requirements,

while the vertices represent those combinations which exactly satisfy two

regulatory requirements.

A graphical solution to the least-cost means of achieving compliance is shown

in figure 2.4. The solution is illustrated via a series of equal cost lines.

An equal cost line shows all the combinations of technique "1" and technique

"2" which cost the same. It is drawn based on the assumption that the unit

prices for technique "1” and technique "2" are constant. Higher equal cost

lines imply greater costs are being incurred. By referring to figure 2.4, it can be seen that the first equal cost line, CQ , does not touch the shaded

region, implying that it is not possible to achieve compliance with the funds

being allocated. The second equal cost line, , just touches a vertex of

the feasible region. Since no other equal cost line, which is lower also

touches the feasible region, it can be asserted that the point of contact is

the least-cost combination of the two techniques which complies with all four

regulatory requirements. Figure 2.4 also illustrates that if slightly more

money were spent it would still be possible to hold the level of performance

for one regulatory requirement constant (i.e., the requirement associated with

line segment AB as illustrated in figure 2.2) and increase the other (i.e.,

the requirement associated with line segment BC.) Such a solution would lie

on the boundary of the feasible region (i.e., where C 2 intersects the line

segment AB). However, from the figure, it can be seen that it may be preferable to use those dollars to achieve a higher-level of performance for both

requirements. Such a strategy would result in a point lying entirely within

the feasible region such as

,, B

, ". Points like "B 1 " may have other desirable

requirements, for example, they may refer to a standard product rating such

as size or weight.

Notice that the move from point B to B' resulted in an increase in the overall

performance for three of the four regulatory requirements. This is because the

23
LEVEL OF TECHNIQUE 1 |L1|

Figure 2.4 The Selection of the Optimal Compliance Strategy

LEVEL OF TECHNIQUE 1 (LI]

Figure 2.5 The Generation of an Alternate Compliance Strategy

24
constraint gradients associated with line segments AB, BC and CD (see figure 2.2)

form acute angles with the cost gradient. The score associated with the fourth

regulatory requirement, line segment AD, is reduced however. This is due to

the conflict between security and the other three regulatory requirements (i.e.,

egress, ventilation and illumination). Conflicting requirements serve to

constrain the size and shape of the feasible region and may, in some cases,

reduce the potential savings associated with the use of an equivalency methodology. Generally speaking, these problems are most serious when building com- ponents are only available in a small number of fixed sizes or types. The

implications of this additional constraint will be explored shortly.

As mentioned earlier, optimization models should also produce a set of alternate

solutions which contain some configurations reasonably close to the optimum

with respect to costs but differ from the optimum solution in terms of the post

retrofit states. One way in which alternate solutions may be generated is

shown graphically in figure 2.5. This procedure which generates two classes of

alternate solutions was used in a previous study*. Only the first class is

illustrated in figure 2.5. The optimal solution derived earlier is denoted by

"B" in the figure. Let us define i as representing the technique index ( i = 1,2) and j as representing the various states (j = 1, ..., j where j 1

-.,- is the maximal state of the i technique). Now if we define Xjj to be a state

variable and it appears in the optimal solution, it is equal to one, otherwise

it is equal to zero. Suppose Xjj appears in the optimal solution as a retrofit,

then the cost, C-jj , of going from X-^^ c , the preretrofit state, to Xjj , the postretrofit state is made arbitrarily high.^ This step guarantees that X^ 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 lines just touches the feasible

region. Such a point is designed as "C" in figure 2.5. Exactly analagous is

the case where X-j_j

did not appear in the optimal solution (i.e., X-jj =0). In

this case, the cost of going from Xjj C to Xij , Cjj , is held fixed and all C^k,

k^j , are made arbitrarily high. This step guarantees that Xij 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 problem minus the

number of retrofit states which are excluded for technical or other related

reasons

.

The second class of alternatives is determined by the number of techniques.

For example, suppose X^j appears in the optimal solution as a retrofit, then

the cost of going from X^ to X.j , for all j

' , j c < j

' <_ j 1 max , is made arbitrarily high. This step guarantees that no potential retrofits associated with

that technique can appear in the optimal solution. The number of solutions

generated is equal to the number of techniques which moved to a higher state

(were retrofitted) in the optimal solutions.

* Robert E. Chapman, Phillip T. Chen and William G. Hall, op. cit.

2 An operational definition of an arbitrarily high retrofit cost could be one

for which its cost is n + 1 times the highest individual retrofit cost,

where n is the number of building components.

25
The two classes of alternative solutions described above are useful because

they are "close" to the optimal solution. "Close" is put within quotation

marks because through reference to figure 2.5 it can be seen that the alternative solutions are adjacent vertices of the feasible region. Thus "close" means a small change in the basis ( component/ state pair combinations) rather

than a small change in the objective function (retrofit costs). Although

these solutions will be more costly than the optimal solution, some of them

will be extremely close (in dollar terms) to the least-cost combination of

retrofits

.

All of the discussion so far has proceeded as if the levels of the techniques were continuously variable. In some cases this may not be correct since building materials often come in fixed sizes, weights, thickness, etc. Thus what

must be solved is an integer program rather than a continuous program. There

are, of course, classes of problems for which the continuous solution is actually the integer solution. In general, this is not true; figure 2.6 illustrates a case in which the solution to the continuous problem is not the solution to the integer problem. In figure 2.6 the grid points which satisfy all

four regulatory requirements are shown as dots within the quadrilateral ARCD.

Note that points "A” and "B" are not shown as dots since they are not grid

points (i.e., both techniques do not take on integer values). Now if the cost

of technique "1" with respect to technique "2" were quite high, producing an equal cost line steeper than the line segment AB, the optimal solution for the

continuous problem would be point "A". Since point "A" is not a grid point,

we may seek to round it up with respect to one technique, as shown by point "a",

or with respect to both techniques, as shown by point "a f ". Unfortunately, due

to the conflicting requirements, neither of these points lies within the

feasible region. If we now choose the nearest grid point to "A", point "A’",

it is fairly easy to see (by sweeping an equal cost line rightward) that a point like "A"" would be less costly. If the optimal solution were at point

"B", an ad hoc integerization would probably produce a feasible solution such

as "B

,M which may or may not be the optimal integer solution. Regardless of

the integer characteristics of the optimal solution, it represents a lower

bound on the total retrofit cost. It is obvious that the introduction of an additional constraint (i.e., each value for each variable must be integer)

cannot improve the cost.

Fortunately, several arguments can be advanced which permit this general class

of problems to be solved as continuous linear programs. The reason that a linear programming approach is attractive is due to: (1) the "near integer"

property of the solutions; (2) the judicious selection of the family of alternative solutions; and (3) the computational cost, efficiency, and reliability

of the algorithm. The near integer property occurs because of the structure of

the constraint matrix. One criterion used in the design of the alternate solution family was that any variable appearing as a fraction in the optimal solution must appear in at least one of the alternates as a one and be suppressed

in at least one of the alternates. Other criteria were that the alternates

26
LEVEL

OF

TECHNIQUE

2

(L2)

LEVEL OF TECHNIQUE 1 (LI)

Figure 2.6 The Selection of Compliance Strategies when

Performance Levels Vary Discretely

27
should be amenable to generation via a systematic but comprehensive

specification, that they meet (at least our conception of) user needs, and that

the number of alternates be adequate but not burdensome. Parametric linear programming techniques provide a convenient vehicle for achieving these objectives.

To solve the optimization problem as an integer model is actually possible.

The reasons for selecting a continuous model are completely pragmatic. Computation time for an integer solution can be expected to be an order of magnitude

higher, however.

Thus the solutions presented, while not guaranteed to include the optimum

integer solution, do contain:

(1) a bounding value on the retrofit cost;

(2) costs corresponding to the alternates most likely to be considered by

the user; and

(3) one or more costs close to the integer optimum.

28
3.0

CASE APPLICATION OF THE EQUIVALENCY METHODOLOGY TO RESIDENTIAL REHABILTATION

It was necessary to design a case application which would test the innovative

equivalency methodology formulated in this report. This included the selection

of regulatory requirements and a prototypical building which could be evaluated

to determine optimal compliance strategies based on cost considerations.

3.1

SELECTION OF REGULATORY REQUIREMENTS

Since this was a pilot study with time and funding constraints, it was necessary

to select a narrow range of regulatory requirements and building components.

This selection was based on an NBS study which provided a matrix showing the

relation between these factors. ^ Figure 3.1 reproduces this matrix except that

building components under "Enclosure and Space" have been specifically

identified.

Criteria for the design of this pilot study included the following:

1 . Regulatory requirements would be selected such that an improvement in

one area could result in a negative effect in another area.

2. Regulatory requirements would be selected such than an improvement in

one area could affect another area in the same way.

3. Building components would be selected such that they could contribute

to compliance with the regulatory requirements.

As shown by the shaded areas in figure 3.1, the code requirements selected are ventilation, illumination, egress and security and the building components are doors and windows. Egress and security would be affected in a conflicting way

by improvement in window and door security hardware (criterion 1); ventilation

and illumination would be similarly affected by changes in window size (criterion 2); and window and doors can contribute to the level of compliance of

all four regulatory requirements (criterion 3).

3.1.1

Applicable Building Regulatory Requirements

The selection of regulatory requirements which affect the design of windows and

doors in existing buildings was based on a recent NBS study by Cooke^. The

study was undertaken to identify and compare the levels of selected code provisions contained in seven model codified documents which deal with the occupancy,

maintenance, and rehabilitation of existing residential buildings. The code

1 James G. Gross, James H. Pielert and Patrick W. Cooke, op. cit.

2 Patrick W. Cooke, Comparison of Selected Codes and Standards Relating to

Existing Residential Buildings , National Bureau of Standards, NBSIR 80-2081,

July 1980.

29
Figure

3.1.

Relationship

of

Regulatory

Requirements

and

Building

Components

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30

NOTE:

Shaded

areas

represent

regulatory

requirements

and

building

components

treated

in

the

pilot

study
provisions are compared and analyzed in sixteen major code areas. Areas

considered which are relevant to this case study are windows and doors, means

of egress /exits , light and illumination, and ventilation. Hardware and physical security are covered in the section on "windows and doors."

A conclusion of the report is that windows and doors do not receive extensive

coverage in most of the codes studied with regard to such factors as exterior

protection, energy conservation, maintenance of glazing, and physical security.

Practically all the codes directly integrate their ventilation requirements

with those provisions for light, since, in many cases, the natural ventilation

requirements are a function of the window area requirements for natural lighting. Except for a very few instances, however, no specific illumination levels

are provided. The study finds that regulatory requirements for providing and

maintaining means of egress from existing buildings are in some cases expressed

in such generalized terminology as to be vague and, in other cases, the provisions are very specific (e.g., clearances, headroom). However, requirements

are not consistent among the codes and several significant differences exist

regarding number of required exits and clearances for emergency egress.

Because of the inconsistency of the coverage of the regulatory requirements

being considered in this case study as shown by the Cooke report, it was decided

to develop a hybrid list of requirements for windows and doors which could be

used to evaluate the equivalency methodology x^hich was formulated for illumination, egress, ventilation and security. An additional resource in establishing

these requirements were those used by the City of Baltimore in rehabilitating

abandoned city owned housing.

1

Table 3.1 shows the window characteristics and related regulatory requirements

which have been selected for the prototypical building. The characteristics

are of two general categories: (1) configuration/construction (size, type,

hardware, glazing, screen type, and percent of floor area); and (2) vertical

location in the exterior envelope (height above grade and height of sill).

The right column of the table indicates the prescriptive regulatory level for

each of the characteristics.

Table 3.2 shows similar information for exterior doors. Configuration/

construction characteristics include size, type of glass, hardware, composition,

number of exterior doors, and the presence of storm doors. The prescriptive

regulatory levels were selected in the same way as described above for windows.

3.1.2 Sample Worksheets Developed for the Pilot Study

The regulatory requirements highlighted in figure 3.1 and outlined in the

previous section are presented in greater detail in exhibits 3.1 through 3.4.

Each exhibit focuses on a single regulatory requirement; 3.1 is concerned with

security, 3.2 with egress, 3.3 with ventilation, and 3.4 with illumination.

1 Housing Authority of Baltimore City - Construction and Buildings Inspection,

Construction Standards for Housing Renewal Program, August 1978.

31
Table 3.1 Window Characteristics and Related Regulatory Requirements

CHARACTERISTICS PRESCRIPTIVE LEVEL

<20 in. Height or < 20 in. Width

Size 20-40 in. Height and > 20 in. Width

40-60 in. Height and > 20 in. Width*

>60 in. Height and > 20 in. Width

° at least one window per habitable room

° minimum 5.7 sq. ft. net clear opening

0 20 in. minimum width of opening

slider

casement

Type single hung

double hung*

fixed

° double hung windows shall be used

° units shall be fully weatherstripped and

counter balanced for easy operation

latch

Hardware locking latch*

dead bolt

° hardware to be openable from inside

° units shall be easily opened and held in

position by window hardware

single

double (1 frame )*

Glazing double (2 frames)

triple (2 frames)

° provide double glazing or equivalent

% <5%

Floor 5-10%*

Area >10%

° illumination: aggregate glazing area of

10% of floor area

° clear ventilation area of not less than

5% of floor area

<0

Above 0-5 ft.*

Grade 5 ft. -8 ft.

>8 ft.

° dwellings below fourth floor shall be provided with exterior door or window of such

dimension to be used as means of emergency

egress

° habitable rooms below grade to satisfy

same standards as for rooms below gradeparticularly with regard to light and

ventilation

Height 0-24 in.

of 24 in. -44 in.*

Sill >44 in.

° minimum sill height of 24 in.

° maximum sill height of 44 in.

Yes*

Screen No

° provide screens that will effectively

prevent the entrance of insects

* Prescriptive Regulatory Level Selected for Prototypical Building

32
Table 3.2 Exterior Door Characteristics and Related Regulatory Requirements

CHARACTERISTICS PRESCRIPTIVE LEVEL

Size

(Area)

<18 ft. 2 18-21 ft. 2 * 21-25 ft. 2 >25 ft. 2

Entrance doors shall be 1 3/4 in. x 3 ft. - 0x7 ft. 0

unless existing openings are smaller and conditions

prevent the increase of opening.

Glass

1 lite

3 lite*

6 lite

9 lite none

Rear and side doors shall have three horizontal

lites at the top section.

Hardware

lock set*

dead bolt(l key)

dead bolt(2 key)

Exit doors to be easily opened from inside without

a key.

Composition

hollow core solid wood*

metal

Exterior doors shall be of solid core or solid

lumber construction.

Number

12*

>3

Two exits to the outside per dwelling for egress.

Storm

Door

Yes*

No

All entrance doors shall have installed a combination aluminum storm and screen door.

* Prescriptive Regulatory Level Selected for Prototypical Building

33
Exhibit 3.1 Sample Worksheet for Evaluating Security Requirements

COMPONENT SECURITY

1 WINDOW SIZE

< 20”W

or < 20”H

»20”W

20”-40”H

=* 20”W

40”-60”H

>20”W

>60”H 1 7 2 0 -2

2 WINDOW TYPE

SLIDER CASEMENT S. HUNG D. HUNG FIXED

-3 -2 0 0 10

3 WINDOW

HARDWARE

LATCH LOCKING LATCH DEAD BOLT

0 5

4 WINDOW

GLAZING

SINGLE DOUBLE(IF) D0UBLE(2F] TRIPLE (2F)

0 0 3 5

5 WINDOW SCREEN

YES NO

0 -1

6 WINDOW HEIGHT

ABOVE GRADE

5 O' 0’-5’ 5-8’ > 8’ -3 0 2 5

7 WINDOWSILL

HEIGHT

0”-24” 24”-44” >44”

0 0 0

8 DOOR SIZE

< 18 18-21 21-25 > 25

1 0 -2 -3

9 GLASS IN DOOR

NONE 1 LITE 3 LITES 6 LITES 9 LITES

6 1 0 “2 "4

10 DOOR HARDWARE

LOCKSET 1 KEY (DB) 2 KEY (DB)

0 2 6

11 DOOR

COMPOSITION

HOLLOW CORE SOLID WOOD METAL

-2 0 2

12 SECOND DOOR

YES NO

0 -5

34
Exhibit 3.2 Sample Worksheet for Evaluating Egress Requirements

COMPONENT EGRESS

1 WINDOW SIZE

< 20”W

or< 20”H

>20”W

20”-40”H

>20”W

40”-60”H

>20”W

> 60”H

-10 -1 0 2

2 WINDOW TYPE

SLIDER CASEMENT S. HUNG D. HUNG FIXED

2 -4 0 0 -10

3 WINDOW

HARDWARE

LATCH LOCKING LATCH DEAD BOLT

5 0 -4

4 WINDOW

GLAZING

SINGLE DOUBLE(IF) D0UBLEI2F) TRIPLE (2F)

0 0 -2 -5

5 WINDOW SCREEN

YES NO

0 3

6 WINDOW HEIGHT

ABOVE GRADE

5 0’ 0-5’ 5’-8’ >8’

-10 0 -1 -5

7 WINDOWSILL

HEIGHT

0”-24” 24”-44” >44”

4 0 -5

8 DOOR SIZE

< 18 18-21 21-25 > 25

-1 0 1 2

9 GLASS IN DOOR

NONE 1 LITE 3 LITES 6 LITES 9 LITES

0 0 0 0 0

10 DOOR HARDWARE

LOCKSET 1 KEY (DB) 2 KEY (DB)

0 -2 -6

11 DOOR

COMPOSITION

HOLLOW CORE SOLID WOOD METAL

0 0 0

12 SECOND DOOR

YES NO

0 5

35
Exnibit 3.3 Sample Worksheet for Evaluating Ventilation Requirements

COMPONENT VENTILATION

1 WINDOW SIZE

< 20”W

or « 20”H

>20”W

20 ”-40 ” H

> 20”W

40-60”H

>20”W

*>60”H

-2 -1 0 2

2 WINDOW TYPE

SLIDER CASEMENT S. HUNG D. HUNG FIXED

5 -1 0 -10

3 WINDOW

HARDWARE

LATCH LOCKING LATCH DEAD BOLT

2 0 “3

4 WINDOW

GLAZING

SINGLE DOUBLED F) DOU BLE(2 F j TRIPLE (2F)

0 0 0 ”2

5 WINDOW SCREEN

YES NO

0 -5

6 WINDOW HEIGHT

ABOVE GRADE

SO’ 0-5’ 5-8’ > 8’ -5 0 1 2

7 WINDOW SILL

HEIGHT

0”-24” ^3" esj >44”

=-1 . 0 1

8 DOOR SIZE

< 18 18-21 21-25 > 25

-1 0 1 2

9 GLASS IN DOOR

NONE 1 LITE 3 LITES 6 LUES 9 LITES

0 0 0 0 0

10 DOOR HARDWARE

LOCKSET 1 KEY (DB) 2 KEY (DB)

0 0 0

11 DOOR

COMPOSITION

HOLLOW CORE SOLID WOOD METAL

0 0 0

12 SECOND DOOR

YES NO

0 -4

36
Exhibit 3.4 Sample Worksheet for Evaluating Illumination Requirement

COMPONENT ILLUMINATION

1 WINDOW SIZE

< 20”W

or < 20”H » 20”W

20”-40”H

>20”W

40”-60”H

>20”W

>60”H

"3 -1 0 4

2 WINDOW TYPE

SLIDER CASEMENT S. HUNG D. HUNG FIXED

1 0 0 0 0

3 WINDOW

HARDWARE

LATCH LOCKING LATCH DEAD BOLT

0 0 0

4 WINDOW

GLAZING

SINGLE DOUBLED F| D0UBLE(2F) TRIPLE (2F)

0 0 -1 -2

5 WINDOW SCREEN

YES NO

0 1

6 WINDOW HEIGHT

ABOVE GRADE

SO’ 0-5’ 5-8’ > 8’ ”5 0 1 2

7 WINDOWSILL

HEIGHT

0”-24” 24”-44” >44”

0 0 0

8 DOOR SIZE

< 18 18-21 21-25 > 25

0 0 0 0

9 GLASS IN DOOR

NONE 1 LITE 3 LSTES 6 LITES 9 LITES

-2 -1 0 +2 +4

10 DOOR HARDWARE

LOCKSET 1 KEY (DB) 2 KEY (DB)

0 0 0

11 DOOR

COMPOSITION

HOLLOW CORE SOLID WOOD METAL

0 0 0

12 SECOND DOOR

YES NO

0 0

37
The four exhibits are designed as worksheets capable of defining a level of

performance for each regulatory requirement. This task is accomplished by

subdividing each building component into a set of states which cover all possible situations expected to be encountered in practice. For example, window

type includes slider, casement, single hung, double hung and fixed as states.

Associated with each component/state pair is a value which represents the con- tribution of this pair to meeting the overall objective; i.e., attaining the

level of performance required by the regulatory requirement under consideration.

1

In the context of the pilot study, negative state values detract from the performance goal whereas positive state values contribute towards the attainment

of the performance goal. A state value of zero is goal neutral. It should be

recognized that a zero value can occur both for a particular component/state

pair or for all states within a component. An example of the second issue can

be seen in exhibit 3.4 where all classes of window hardware and door hardware

are goal neutral with respect to illumination requirements.

The level of performance provided within the building for each regulatory

requirement is calculated by first recording the number of units occurring

within each component/state pair. The state values for all units within each

component are then summed. The level of performance for the regulatory requirement under consideration is then computed by summing across all components. As

shown in appendix B, the unit counts are subject to a set of constraints which

require the total number of units to be equal across components 1 through 7 (window components) and across components 8 through 12 (door components). These

constraints are designed to impose a set of consistency requirements. Compliance with each regulatory requirement is then checked by determining if the

performance score associated with all component/ state pairs matches or exceeds

the performance score associated with prescriptive compliance. It is important

to point out that a satisfactory retrofit strategy is one in which all perfor- mance scores match or exceed that which results from prescriptive compliance.

An examination of exhibits 3.1 through 3.4 reveals that improvements in

security often exert a negative impact on egress, ventilation or illumination

(e.g., window size and glazing). This conflicting requirement is thought to

be fairly common in rehabilitation activities and was discussed earlier as a driving force in the design of the pilot study. Another issue relates to the

close score relationship between ventilation and illumination, where the same general trend in state values is seen across most components. The final issue

relates to the basic structure of the system. As pointed out earlier, each

component was selected so that it contributed to goal attainment. (More precisely, the system was constructed so that each component contributed in an additive or linear fashion.) In order to illustrate the flexibility of the

equivalency approach, however, it was necessary to embed the pure prescriptive

1 The state values shown in exhibits 3.1 through 3.4 were assigned by the

project team based on their knowledge of the subject areas. There was no attempt made to utilize a formal Delphi technique in setting these values.

The purpose of the study was to evaluate the applicability of equivalency

methodologies for the regulatory areas considered and was not intended to

be a comprehensive technical treatment.

38
solution into the system. This was done by insuring that for each component

one or more states bracketed the state corresponding to prescriptive compliance.

In this manner some state values exceeded and some were exceeded by the prescriptive state's value. Combining this feature with anticipated rehabilitation

costs permits one to identify optimal compliance strategies. An illustration

of this feature was given in section 2.3. A detailed mathematical treatment is

given in appendix B.

3.2 SELECTION AND ASSESSMENT OF A PROTOTYPICAL DESIGN

A prototypical design was selected to demonstrate the usefulness of the

equivalency methodology developed. Specific aspects of the design include:

1. applicable building regulatory requirements (discussed earlier);

2. configuration of the prototypical building to be rehabilitated; and

3. typical window and door retrofits and related costs.

Typical inner city residential buildings in several major cities in the United

States were visited and inspected in order to identify the relevant engineering

and architectural design information required for the synthesis of a prototypical building. Figures 3.2 and 3.3 shows some of the typical buildings inspected

in St. Louis and Baltimore. Photographs and sketches of candidate buildings were collected and analyzed and discussions were held with building officials

and contractors involved with the rehabilitation of such buildings. Staff of

the Construction and Building Inspection Department of the Housing Authority

of Baltimore City were particularly helpful in allowing visits to housing

undergoing rehabilitation in the city.

A prototypical building was then synthesized in a manner which provided a test

of the equivalency methodology formulated in this case study. Figure 3.4 shows

the basement plan, first floor plan and second floor plan of the prototypical

single family townhouse, or rowhouse with residential units on both sides. The

first floor contains a living room, dining room, kitchen and den; the second

floor has three bedrooms and a bath; and the basement contains a utility room

and storage area. There is a deck extending in the rear of the house at the

first floor level. The building is assumed to be at least 50 years old, all

exterior walls are brick bearing wall construction, and roof, floor and interior

wall construction is timber. The occupancy of the building is assumed to

comply with housing code requirements for structures of that size.

Figure 3.5 illustrates the front and rear elevations of the dwelling including

the locations of windows and doors. All windows are of the steel casement

type of the sizes and types indicated; doors are wood of the sizes indicated.

All windows and doors are assumed to be in a severely deteriorated condition

(e.g., not secure, lack of weathertightness) requiring replacement including

all frames and jambs. Costs are estimated on this basis and include necessary

masonry work in the exterior walls. Since this study is concerned with only

the window and door components, the type and condition of the other building

structural and mechanical systems need not be identified.

39
Figure 3.2 Typical Buildings Inspected in St. Louis, Missouri

40
Figure 3.3 Typical Buildings Inspected in Baltimore, Maryland

.s.-s/fZ's m mm m

41
Figure

3.4

Floor

Plans

for

Prototypical

Building

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Figure

3.5

Front

and

Rear

Elevations

of

Prototypical

Building

co

CM

CO X

X

CO

1 o1 CO 2s£ CM

11VM N0WN03

OUJo

11VM N0MHN00

43

EXISTING

CONSTRUCTION:

FLAT

ROOF

BRICK

EXTERIOR

STEEL

CASEMENT

WINDOWS

WOOD

DOOR
It is necessary to identify potential window and exterior door retrofits and

associated costs in testing the equivalency methodology. Since the case study

was designed to evaluate the methodology and is not intended for use in actual

rehabilitation projects, cost data were developed in-house by cost engineers.

Sources of these data include the City of Baltimore Construction and Building

Inspection Department, Building Construction Cost Data 198C) 1 published by

Robert S. Means Company, Inc., and discussions with designers and contractors

involved with residential rehabilitation. Since very little cost data on actual rehabilitation projects were obtained, the resulting cost data should

not be considered as representing current rehabilitation experience. It is

felt that data used were developed in a consistent manner and that the trends

and comparisons presented are valid.

Window and exterior door retrofits are classified into the following five

categories as shown on figure 3.6

1. Adding a component where there is no existing opening,

2. blocking up a component opening including removal of existing component

and closing of the opening in a manner compatible with the exterior

facade

,

3. replacing an existing component with one of the same size (component

type may or may not be the same),

4. replacing an existing component with one of a smaller size (component

type may or may not be the same) , and

5. replacing an existing component with one of a larger size (component

type may or may not be the same).

"In-place prices" for accomplishing these retrofits include the following

factors; removal of existing component, addition or enlargement of an opening,

waste removal, building materials, labor, installation, and finishing,

refinishing an adjacent area, and mark-up for contractor overhead and profit.

3.3 SOME PRELIMINARY RESULTS

The computer program outlined in appendix B was applied to the prototypical

building described in section 3.2. The purpose of this section is to summarize

in a qualitative manner those costs associated with prescriptive compliance and

the optimal cost solution.

Table 3.3 summarizes the window and door details for the initial prototypical

building, the retrofitted building based on a prescriptive solution, and the

configuration based on a near optimal cost solution. The costs associated with

the optimal cost solution are about 29 percent less than those for the

1 Building Construction Cost Data 1980 , Robert Snow Means Company, Inc., New

York, 1979.

44
Figure 3.6 Typical Uindow and Door Retrofits

BEFORE AFTER

ADDING A

COMPONENT

BLOCKING UP

COMPONENT

OPENING

REPLACING EXISTING

COMPONENT WITH

ONE OF EQUAL SIZE

REPLACING EXISTING

COMPONENT WITH

ONE OF SMALLER SIZE

REPLACING EXISTING

COMPONENT WITH

ONE OF LARGER SIZE

i i i i

' ' r-r

35^ i ii' i i a

dr

xir

32 i

I5r

i l i. i

1

1 ji 1 r

I I .'ITT

Original Opening

45
Table

3.3

Comparison

of

Initial,

Prescriptive

anda

Near

Optimal

Solution

for

the

Prototypical

Building^'

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CO

4-> CD TJO<4-1

LT) CO

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(I) i43

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i-i bO

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CN

46 - ’

Enlargement

of

two

rear

window

openings

required.

4/

cost

of

near

optimal

solutions

is

29

percent

less

than

the

prescriptive

solution.
prescriptive solution. By referring to table 3.3, and comparing these two

solutions, it can be seen that these savings are possible because the near optimal soution:

1. selects windows which conform to existing openings, thereby, avoiding

the costly process of enlarging openings to accept windows which meet

the minimum prescriptive level,

2. does not require upgraded security hardware on second floor windows

over that normally provided,

3. upgrades the main exterior doors to metal construction with no glass

and dead bolt (1 key) hardware, and

4. requires no second doors.

It is not possible to specifically identify the impact of these retrofit items

on each of the regulatory areas under consideration because of the complex

interactions in the optimization model. However, the state value excess over requirement (P - Rp shown in table 2.5) for the near optimal solution are 25

for security, 15 to egress, and zero for ventilation and illumination.

Other initial building configurations were analyzed with the computerized

equivalency approach and similar cost savings in the 25 to 30 percent range were obtained. It should be noted that these figures represent only the four

regulatory requirements; security, egress, ventilation and illumination. If

the number of requirements and building components are increased to reflect the

actualities of building rehabilitation activities this percentage may either

increase or decrease. Also, broad based technical input was not possible in

developing components states and state values shown on the worksheets. Application of Delphi techniques in formulating such data may effect results. Based

on studies in the fire area, however, fairly substantial savings would still be

expected

.

Computer generated solutions such as those just discussed are designed to

produce: (1) an optimal retrofit strategy; and (2) a series of alternative

retrofit strategies.

Note that optimal as used here implies the minimum retrofit cost based on the

original configuration. However, if the investor opts for a change in configuration, the new minimum retrofit cost solution is not neccessarily similar to

the original configuration. Consequently, the solution corresponding to the

minimum percentage of prescriptive compliance costs may deviate from the original configuration. By superimposing constraints on the rehabilitation process,

the user can substantially increase the costs of prescriptive compliance.

Since the computer generated retrofits address these user-imposed constraints

in the most cost-effective manner, it is possible that the costs as a percent

of prescriptive compliance may actually decline.

47
Furthermore, if all retrofit strategies are ranked on the basis of construction

cost only, 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 rented or 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 upgrading to a particular state. For these reasons, it is important

that the user of such computer programs 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 as life-cycle costing should be used. 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.

48
4. SUMMARY AND RECOMMENDATIONS FOR FURTHER RESEARCH

4 . 1 SUMMARY

The application of prescriptive type code provisions designed for new construction has been identified as a constraint when applied to the rehabilitation of existing buildings. The study which has been presented in this report

addresses the need for a systematic procedure to evaluate the impact of a wide

range of regulatory requirements on a building being considered for rehabilitation. This research is an extension of Fire Safety Evaluation System developed

by the NBS Center for Fire Research (CFR); the Center for Applied Mathematics

and the Center for Building Technology were a source of technical support to

CFR in implementing the Fire Safety Evaluation System.

A computer based pilot equivalency methodology has been developed for

identifying the least-cost means of achieving compliance with selected regulatory requirements applied to existing buildings. It has been shown that the

use of a mathematical optimization model along with the Delphi Method to generate consensus on technical issues can be used to consider regulatory requirements which have conflicting impacts on building component design (e.g., window

hardware criteria have a different effect on egress performance as compared to

security performance). A case application of an equivalency methodology to

the design of window and door retrofits in a residential rehabilitation project

has been presented. This included the selection of the prototypical design,

cost data for various retrofits, and regulatory requirements related to health

and safety concerns including egress, security, ventilation and illumination.

A computer program developed as part of the study permits the effects of these

specific regulatory requirements on various window and door compliance strategies to be identified and assessed. Comparing the costs of optimal compliance

strategies with those of prescriptive compliance indicated potential saving

ranging from 20 to 30 percent depending on the initial condition of the building.

4.2 RECOMMENDATIONS FOR FURTHER RESEARCH

The pilot study was conducted to assess the feasibility of applying an equivalency methodology to regulatory areas other than fire safety in the building rehabilitation process. The study was carried out with technical expertise

available within the project team with no attempt made to use a broad base

Delphi group of technical experts. With this in mind, the following recommendations are presented for broadening the scope of the pilot equivalency methodology

and improving the technical bases.

A. Develop the Equivalency Methodology to Include a Broader Range of

Regulatory Requirements and Building Components

This study has demonstrated the feasibility of the equivalency

approach presented and the need for continued development. A comprehensive Delphi approach should be applied in selecting additional

regulatory requirements and in providing technical expertise in defining performance levels. Interested members of the building community

49
might be informed of the research and their support sought in providing

technical assistance and implementation of results.

B . Develop Revised Classification Schemes for Building Component/

Regulatory Requirement Relationships

The classifications shown in Exhibits 3.1 through 3.4 are based

primarily on dimensional and hardware considerations which may not

truly represent component performance for the regulatory requirement

and may cause problems from a computational perspective. For example,

the window size classification shown in figure 3.1 for security is

based on the width and height of the component. A better approach may

be similar to that used to classify flame spread ratings of interior

finish which is designated either Class A, B or C based on results of

tests conducted using standard procedures. A result of the Delphi

activity would be to provide technical guidance in presenting

performance classifications in a more appropriate manner.

C. Illustrate Ways in Which Mathematical Models Can be Used as Instruments of Measurement

Engineering principles occupy a central role in the successful

development of equivalency methodologies. A related but more fundamental concept is the use of mathematical models as instruments of

measurement. More precisely, the performance of a system is a function of the levels of its subsystems and components. The axioms of

logic and mathematics which provide a basis for the building of models

impart a structure which ensures that the measurement of performance

is carried out in a consistent manner. This structure also permits

one to measure attainment or nonattainment of an engineering goal

(e.g., compliance or noncompliance with a standard). Furthermore,

since known goals can be embedded in the model (e.g., prescriptive

compliance) a given model is capable of measuring attainment against

a wide variety of goals. Thus given the model one need only specify

the status quo, use the model to determine the level of performance

associated with the status quo, and then set the goal at that level

of performance. In this sense the model is acting as an instrument

of measurement, one which is capable of comparing alternatives in an objective manner.

D

.

Use Econometric Techniques to Estimate a Set of Cost and Production

Functions for Residential Rehabilitation Activities

The duality relationship between cost functions and production

functions is an important criterion in choosing a technically sound

and logically consistent method for estimating building rehabilitation

costs. Unfortunately, the development of a series of cost functions

for rehabilitation activities can only be accomplished through the

application of econometric techniques to actual rehabilitation 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

50
in collecting a complete set of rehabilitation cost data. The use of

probabilistic methods is an important tool for assessing the riskiness

of a potential 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. It would be useful if such a study were undertaken in con- junction with the analysis of rehabilitation cost data. The use of

equivalency methodologies is expected to have a substantial effect on

the cost structure of the firm carrying out rehabilitation activities.

However, due to the duality relationship mentioned earlier it can be

asserted that equivalancy methodologies will also affect the production

choices of the firm via the production function. More precisely, on theoretical grounds it can be shown that the regulatory structure

facing the firm affects the ease with which inputs can be substituted

for one another. Empirical studies relating the elasticity of substitution of the firm's production function to the regulatory structure

facing the firm would thus be an important step towards understanding

how regulations affect productivity.

51
REFERENCES

1. Anderson, T. , Recycling; Old Buildings Big New Opportunity , Professional

Builder (pp. 197-199), January 1979.

2. Berry, S.A., Editor, Proceedings of the National Conference on Regulatory

Aspects of Building Rehabilitation , National Bureau of Standards, Special

Publication 549, August 1979.

3. Building Construction Cost Data 1980, Robert Snow Means Company, Inc., New

York, 1979.

4. Chapman, R.E., Cost Estimation and Cost Variability in Residential

Rehabilitation , National Bureau of Standards, Building Science Series 129,

November 1980.

5. Chapman, R.E., Chen, P.T., and Hall, W.G., Economic Aspects of Fire Safety

in Health Care Facilities: Guidelines for Cost-Effective Retrofits , National

Bureau of Standards, NBSIR 79-1902, November 1979.

6. Chen, P.T., and Chapman, R.E., "Budget Estimates for Replacement of Plant

and Facility Equipment at the National Bureau of Standards," ASHRAE Transactions 1981 , Vol. 87, Pt. 1, pp. 1243-59.

7. Cohan, J.L., Multiobjective Programming and Planning, Academic Press, New

York, 1978.

8. Construction Standards for Housing Renewal Program , Housing Authority of

Baltimore City-Construction and Buildings Inspection, August 1978.

9. Cooke, P.W., Comparison of Selected Codes and Standards Relating to Existing

Residential Buildings, National Bureau of Standards, NBSIR 80-2081, July

1980.

10. Curron, M. , "A Scientific Approach to Bidding: Range Estimating,"

Construction , January 1975, pp. 27-33.

11. Fisz, M. , Probability Theory and Mathematical Statistics , Third Edition,

John Wiley and Sons, New York, 1963.

12. Freund, J., Mathematical Statistics, Second Edition, McGraw-Hill, New

York, 1970.

13. Gass, S., Linear Programming: Methods and Applications , Fourth Edition,

McGraw-Hill, New York, 1975.

14. Gross, J.G., Pielert, J.H., and Cooke, P.W., Impact of Building Regulations

on Rehabilaitation-Status and Technical Needs , National Bureau of Standards,

Technical Note 998, May 1979.

52
15 . Habraken, N.J., et al., An Investigation of Regulatory Barriers to the

Re-Use of Existing Buildings , National Bureau of Standards, GCR-78-139,

January 1978.

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

and Sons, New York, 1975.

17. Ignazio, J.P., Goal Programming and Extensions , D.C. Health, Lexington,

Mass. 1976.

18. Impact of Building Codes on Housing Rehabilitation, Hearings Before the

Committee on Banking, Housing and Urban Affairs —- United States Senate,

March 24, 1978.

19. Kapsch, R.J., Building Codes: Preservation and Rehabilitation , National

Bureau of Standards, Special Publication 473 (pp. 437-452), June 1977.

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

Comparison of Replication and Batch Means," Naval Research Logistics

Quarterly , Vol. 24, 1977.

21. McConnaughey , J.S., Production Functions in Contract Construction for the

United States, 1972 (unpublished), Ph.D. dissertation, Michigan State

University, 1976.

22. Musgrave, G.L. and Rasche, R.H., "Estimation of Cost Functions," The

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

23. Nelson, H.E., and Shibe , A.J., A System for Fire Safety Evaluation of

Health Care Facilities , National Bureau of Standards, NBSIR 78-1555,

November 1978.

24. Nicholson, W., Microeconomic Theory: Basic Principles and Extensions , The

Dryden Press, Inc., Hinsdale, Illinois, 1972.

25. Nie, N.H., Hull, C.H., Jenkins, J.G., Steinbrenner , K. , and Bent, D.,

SPSS: Statistical Package for the Social Sciences , Second Edition McGrawHill Book Company, New York, 1975.

26. Ostwald, P.F., Cost Estimating for Engineering and Management , PrenticeHall Inc., Englewood Cliffs, New Jersey, 1974.

27. Pielert, J.H., Removing Regulatory Restraints to Building Rehabilitation:

The Massachusetts Experience , National Bureau of Standards, Special

Publication 623, October 1981.

28 . Proceedings of the Conference on Building Rehabilitation Research and

Technology for the 1980 "s . National Conference of States on Building Codes

and Standards, August 1980.

29. Silberberg, E., The Structure of Economics: A Mathematical Analysis , McGraw-Hill Publishing Company, New York, 1978.

53
30. State of Massachusetts, Article 22-Repair, Alteration, Additions and

Changes on Use of Exiting Buildings , Massachusetts State Building Code,

October 1980.

54
APPENDIX A. COST ESTIMATION AND COST VARIABILITY: AN APPROACH BASED ON COST

• FUNCTIONS AND PROBABILISTIC CONCEPTS

The purpose of this appendix 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.

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.)

This attribute has been documented in numerous economic and engineering

economics articles.^»2»3 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 the discussion which follows, it is assumed that the prices of all inputs

are independent. Under this assumption the quantity of output, 0, the square

feet of floor area renovated, and the cost of output, C, may be expressed as:

0

= 9(X|, ..., X^, t q , t ^, ...,

T

ra

) A.l

n

C = I X i P i A.

2

j-1

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

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

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

3 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.

55
where

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

q = the basic functional relationship;

til Xj = quantity of the j input (materials, labor, equipment) required to

perform the job;

n = number of inputs considered;

T 0 = basic construction technology factor;

T ^ ^ = construction technology factor associated with i building system or subsystem (e.g., plumbing, mechanical, electrical);

m = number of systems or subsystems considered;

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

Pj = the unit cost of the j

1-*1 input.

The cost function associated with equation A. 1 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 systems and subsystems in defining an approach which is feasible in the engineering sense.

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 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 cost function based on these attributes may thus be expressed as:

C* = c(TF, SF, MF) (A. 3)

where C* = the cost minimizing solution

c = the basic functional relationship

TF = the technology factor;

SF = the size factor; and

MF = the market factor

56
Statistical and/or econometric techniques would then be used to estimate the

parameters associated with the basic functional form c.

Another related 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. 2 The usual output of such a simulation 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 A.l. 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 A.l. The

mathematical form of each density function is given in table A.l. 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 A.l 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.

1 Michael Curran, "A Scientific Approach to Bidding: Range Estimating,"

Constructor , January 1975, pp. 27-33.

2 Phillip F. Ostwald, Cost Estimating for Engineering and Management , Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1974.

57
f(x)

1/c-a

o

UNIFORM

a c

Figure A.l Commonly used Probability Distributions

for Key Construction Factors

58
Table A.l Density Functions for Commonly Used Probability Distributions3

UNIFORM

f(x) = 1/ (c-a)

0

in the interval [a, c]

otherwise

NORMAL

f(x) = exp (-(x-m) 2

/ 2 a

2 )/a( 2 TT)l /2 - 0° < x < °°

LOG NORMAL

f(x) = 0 x < a

exp (-(£n(x-a)-m)2/2cj2)/(x-a)a(2Tl) x > a

TRIANGULAR

f(x) = 2x/ (c-a)(b-a) in the interval [a, b]

2x/(c-a)(c-b) in the interval [b, c]

0 othewise

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

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

59
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 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. 1 Based

on these comparisons, a class of distributions which 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.

It is important to recognize 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 low-probability 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 distributions may

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

maximum point.)

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

New York, 1975,

- 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.

3 Phillip T. Chen and Robert E. Chapman, "Budget Estimates for Placement of

Plant and Facility Equipment at the National Bureau of Standards," ASHRAE

Transactions 1981, Vol . 87, Pt . 1, pp. 1243-59.

60
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 constructon

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

cost. This estimate is called optimistic because there is onlv 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 A. 2.

Table A. 2 Input Requirements for a Standard Probabilistic

Cost Estimating Procedure

Distribution Numerical Inputs

Uniform 1. Minimum Cost

2. Maximum Cost

1. Minimum Cost

Triangular 2. Maximum Cost

3. Modal or Most Likely Cost

Normal and 1. Optimistic: 10th Percentile

Log Normal 2. Middle Ground: 50*-h Percentile

3. Pessimistic: 90 fc h Percentile

61
Assuming that the user has correctly specified the probability distribution and

correctly input the information identified in table A. 2, 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 and the specified amount as X then the expression can be

written formally as

Pr ( x < X) = R.

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 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, TCi, is

thus

n TCi = E X;m • j-1

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

TC?, would be

n TC 2 = E X2j

•

j=l

and for the ith iteration

n TCi = E Xij . 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,

62
where

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

tc = the random variable total cost

TC(k) = the ^

th order statistic for the total cost; and

N = the number of iterations.

Conversely, the probability of exceeding TC( k ), the projected budget, may be

expressed as

Pr (tc > TC( k )) = 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 A. 2.,

Note that the total cost of the job is shown along with 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 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. 1

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

Comparison of Replication and Batch Means," Naval Research Logistics Ouarterlv,

Vol. 24, 1977.

63
* Nl 1S0Q

4-' c;0)

•r-) O

CU

COou

CNJ <1

o>u

3

60

•H

fc

64

PROBABILITY

OF

OVER

RUN
Before proceeding with a discussion of the mechanics of the cost model, it is

useful to review the ways in which the inherent riskiness of the building

renovation process may be assessed. 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.

Judgmental Modifications

By far the most popular method of risk assessment 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.

1

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.

1 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. Hadlia 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.

65
Defining a Confidence Interval About the Estimate

The second method for treating risk involves fitting a confidence interval

about the estimate. To use this method however, 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 the construction cost guidebooks,

the traditional source of cost data, do not provide measures of variability

associated with their average total cost estimates, it is not possible to fit a 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 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 desirable 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 should 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. More succinctly, the sensitivity of the estimate to a change in a factor may be defined by simply

differentiating the cost estimation equation.

Probabilistic (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

66
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.

If one wishes to combine parametric and probabilistic methods, 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 rather simplistic average total cost

approach since costs are "sensitive" to a change in several factors. The

"degree" of sensitivity is, however, dependent upon the relative weight of

that factor in determining the renovation cost for the specific task. In the case where the relationship is linear (or log linear), the application of

probabilistic methods is rather simple and straightforward. The computer

software package would first perform a monte carlo simulation on the cost

estimating equation (relationship) until a cost profile for the task results.

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 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. 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.

Given the basic framework of the equivalency methodology discussed in the main

body of the text, it becomes possible to use cost functions and probabilistic

methods to develop a cost structure for the problem. The structure of costs

is, however, complicated by two factors. First, through reference to table 2.1

it can be seen that there are nm (n rows and m columns) possible states. Since

the installation of a new door for security purposes would cost the same as the

installation of that door for egress purposes, we can assert that the retrofit

cost associated with a particular component/state transition is independent of

the regulatory requirement under consideration. Even with this simplification,

however, 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.

67
From the previous discussion 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 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 , 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

n K = U Ka a=l

Now if there are L distinct key factors in K, then we may express K

mathematically as

K = { ki, k2, . . . kL }

As in the earlier discussion, each element of K has associated with it an estimated mean k^ and 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 A. 3. From the table it can be seen that a cost estimate is needed for each building component/state

pair. Since the goal of this operation is to reduce both cost and variability,

probabilistic methods should 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 previous discussion, this process

is reiterated until a cost profile for each potential retrofit is defined (see

figure A. 2). The next step is for the user to define a risk level, say 20 percent. The software package will then survey each potential retrofit and pick

1 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.

68
Table

A.3

Cost

Matrix

Used

in

Linear

Programming

Approach E

E

O

E

CM O

E

CO O

9 9 9 1

o

• 9 9 9

•

o

•

e

9

9

9

9

9

9 |

State eo

CO

O

CO

CM o

CO

CO O

0 9 9 CO

cO

CSI

CM

O

CM

CM o

CM

CO O

9 9 9 CM

o

- o

CM oCO O

9 9 9 O

Building

Component - CM CO e 9 9 e

69
off the retrofit cost which has a 20 percent change of being overrun. The

program would then automatically select the maximum value, multiply it by some constant, say n+1 , and use it as the "arbitrarily high" cost for all retrofits

which are deemed to be non-f easible . The result of this exercise is that

table A. 3 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 approach is that the linear

programming algorithm will produce the least-cost solution for achieving com- pliance to all p regulatory requirements. 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.

70
APPENDIX B. A MATHEMATICAL PROGRAMMING APPROACH TO EOUIVALENCY-BASED REGULATIONS

This appendix is designed to supplement the discussions in chapters 2 and 3 on identifying and pursuing optimal compliance strategies. Its focus is on how

the worksheets presented in chapter 3 can be used as a basis for the development of a linear programming procedure which ranks compliance strategies according to retrofit cost. A secondary focus is on some of the technical challenges

associated with implementing a generalized equivalency methodology. Since the

worksheets occupy a central role in the cost-optimization model, a moderately

detailed description of them is required for exposition of the model. The primary use of such a set of worksheets is to determine how combinations of various levels, or states, of several widely known building components can be used

to provide a level of performance equivalent to that prescribed in a set of

building codes or standards. Four major regulatory areas are basic to the

evaluation system developed in the pilot study. They are:

the ability of the building and its associated

components to resist entry of individuals intent upon

theft or violence.

the ability of the building and its associated

components to facilitate an orderly evacuation of the

building should such a need arise.

the ability of the building and its associated

components to facilitate the natural expulsion of stale

air and the circulation of fresh air.

the ability of the building and its associated

components to promote the use of daylighting.

\ It is important to point out that the design of the evaluation system developed

in the pilot study is intended to ensure that the failure of a single device or method will not result in a major failure of the entire system. This concept

of system redundancy is an integral part of several major codes and hence was

deemed appropriate for inclusion in the pilot study.

In order to promote a more complete understanding of the cost-optimization

model, the framework will first be developed from a descriptive viewpoint.

Each of the components of the model will then be reintroduced and developed

more fully. An explicit algebraic statement of the model will be given at the

end of the appendix.

( 1 ) Security:

(2) Egress

:

(3) Ventilation:

(4) Illumination:

71
The problem actually solved may be stated as:

minimize the total retrofit cost subject to the following constraints^:

(1) conservation of units equations;

(2) accessory matching equations; and

(3) performance requirements.

The Objective Function

The problem as stated has a single objective^, namely the minimization of total

retrofit cost. The total retrofit cost is merely the sum of all the individual

retrofit costs. Recall that retrofit cost was defined in chapter 2 as the cost

of moving from any given or initial state to any other state. Unfortunately,

some minor modifications to this definition had to be made due to a cost allocation problem. More precisely, window size, window type and window glazing were

each identified as building components in the evaluation system. Since replacement windows usually come as units (i.e., the size, type, and glazing characteristics are determined simultaneously) it was impossible to allocate the retrofit

cost in any meaningful manner to the individual size-type-glazing components.

Similar cost allocation problems resulted from door systems. This complication

required the redefinition of several variables which had the effect of increasing

the number of variables so as to accomodate all possible combinations. Since

these variables are used throughout the remainder of this appendix, they are numbered individually and defined in table B.l according to system (i.e., windows

or doors), preretrofit state (input) and postretrofit state (output). Table B.l

therefore provides a ready reference for each variable's code number used in the

algebraic statement of the problem.

The Constraints

The Conservation of units equations are five identities which define the

maximum number of windows and doors as well as the number of windows and doors

of various sizes that already exist within the dwelling unit.

The accessory matching equations are four identities which contain the

allocation of accessories (window hardware, window screens, door hardware, and

second doors). The equations are designed to ensure that each accessory is

counted once and only once.

1 An additional set of constraints which is also important for this problem are

the nonnegativity requirements. This set of constraints requires each and

every state variable to be greater than or equal to zero. These constraints

are noted for completeness in exposition but will not be discussed in any

detail

.

^ The problem may also be extended and formulated as a multiobjective program,

where there are a combination of objectives to be minimized or maximized.

For the reader interested in this approach to the problem, the texts by Cohon

and Ignizio are strongly recommended.

72
Xi

1234

5

6

7

89

10

1112

1314

15

16

1718

1920

212223

24

25

2627

2829

30

31323334

3536

3738

3940

4142

43

44

4546

47

4849

50

5152

53

54

5556

5758

59

60

6162

63

6465

66

676869

70

71

Table B.l Definitions of the Variables Used in The Computer Program

Input State Output State Comments

Window Blank Window 20-40" H Window 40-60" H Window Blank Window Blank Window Blank Window Blank Window Blank Window Blank Window Blank Window Blank Window 20-40" H Window 20-40" H Window 20-40" H Window 20-40" H Window 20-40" H Window 20-40” H Window 20-40" H Window 20-40" H Window 40-60" H Window 40-60" H Window 40-60" H Window 40-60" H Window 40-60" H Window 40-60" H Window 20-60" H Window 40-60" H Window Blank Window Blank Window Blank Window Blank Window Blank Window Blank Window Blank Window Blank Window 20-40" H Window 20-40" H Window 20-40" H Window 20-40" H Window 20-40" H Window 20-40" H Window 20-40” H Window 20-40" H Window 40-60" H Window 40-60" H Window 40-60" H Window 40-60" H Window 40-60” H Window 40-60" H Window 40-60" H Window 40-60” H

Mo Window Hardware

No Window Hardware

No Window Hardware

Window Blank Window Blank Window Blank

20-40" H Slider Single

20-40" H Slider Double

20-40" H Casement Single

20-40" H Casement Double

20-40" H Casement Triple

20-40" H D. Hung Single

20-40" H D. Hung Triple

20-40" H D. Hung Triple

20-40" H Slider Single

20-40" H Slider Double

20-40" H Casement Single

20-40" H Casement Double

20-40" H Casement Triple

20-40" H D. Hung Single

20-40" H D. Hung Double

20-40" H D. Hung Triple

20-40" H Slider Single

20-40" H Slider Double

20-40" H Casement Single

20-40" H Casement Double

20-40" H Casement Triple

20-40" H D. Hung Single

20-40" H D. Hung Double

20-40” H D. Hung Triple

40-60" H Slider Single

40-60" H Slider Double

40-60" H Casement Single

40-60" H Casement Double

40-60" H Casement Triple

40-60" H D. Hung Single

40-60" H D. Hung Double

40-60" H D. Hung Triple

40-60" H Slider Single

40-60" H Slider Double

40-60" H Casement Single

40-60" H Casement Double

40-60" H Casement Triple

40-60" H D. Hung Single

40-60" H D. Hung Double

40-60" H D. Hung Triple

40-60" H Slider Single

40-60" H Slider Double

40-60" H Casement Single

40-60" H Casement Double

40-60" H Casement Triple

40-60“ H D. Hung Single

40-60" H D. Hung Double

40-60" H D. Hung Triple

Latch Locking Latch Maximum Window Hardware

Window Place Holder

"Brick Up" Option

"Brick Up” Option

Add A New Window

Add A New Window

Add A New Window

Add A New Window

Add A New Window

Add A New Window

Add A New Window

Add A New Window

Replace with Same Size Replace with Same Size Replace with Same Size Replace with Same Size Replace with Same Size Replace with Same Size Replace with Same Size Replace with Same Size Replace with Smaller Size Replace with Smaller Size Replace with Smaller Size Replace with Smaller Size Replace with Smaller Size Replace with Smaller Size Replace with Smaller Size Replace with Smaller Size

Add A New Window

Add A New Window

Add A New Window

Add A New Window

Add A New Window

Add A New Window

Add A New Window

Add A New Window

Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Same Size Replace With Same Size Replace With Same Size Replace With Same Size Replace With Same Size Replace With Same Size Replace With Same Size Replace With Same Size Hardware Place Holder Hardware Installation

Hardware Installation

No Window Screen No Window Screen Screeni Placeholder

No Window Screen Install Window Screen Screeni Installation

Door Blank Door Blank Door Placeholder

Door 18-21 ft 2 Door Blank "Brick: Up" Option

Door Blank 18-21 ft^ Hollow 0 Lite Add A New Door

Door Blank 18-21 ft 2 Hollow 1 Lite Add A New Door

Door Blank 18-21 ft 2 Hollow 3 Lite Add A New Door

Door Blank 18-21 ft 2 Hollow 6 Lite Add A New Door

Door Blank 18-21 ft 2 Hollow 9 Lite Add A New Door

Door Blank 18-21 ft 2 Solid 0 Lite Add A New Door

Door Blank 18-21 ft 2 Solid 1 Lite Add A New Door

Door Blank 18-21 ft 2 Solid 3 Lite Add A New Door

Door Blank 18-21 ft 2 Solid 6 Lite Add A New Door

Door Blank 18-21 ft 2 Solid 9 Lite Add A New Door

Door Blank 18-21 ft 2 Metal 0 Lite Add A New Door

Door Blank 18-21 ft 2 Metal 1 Lite Add A New Poor

Door Blank 18-21 ft 2 Metal 3 Lite Add A New Door

Door Blank 18-21 ft 2 Metal 6 Lite Add A New Door

Door Blank 18-21 ft 2 Metal 9 Lite Add A New Door

Door 18-21 ft 2 18-21 ft 2 Hollow 0 Lite Replace With Same Size

Door 18-21 ft 2 18-21 ft 2 Hollow 1 Lite Replace With Same Size Door 18-21 ft 2 18-21 ft 2 Hollow 3 Lite Replace With Same Size

Door 18-21 ft 2 18-21 ft 2 Hollow 6 Lite Replace With Same Size

Door 18-21 ft 2 18-21 ft 2 Hollow 9 Lite Replace With Same Size

Door 18-21 ft 2 18-21 ft 2 Solid 0 Lite Replace With Same Size

Door 18-21 ft 2 18-21 ft 2 Solid 1 Lite Replace With Same Size

73
*i

8182

83

8485

86

8788

8990

9192

9394

95

96

9798

99

100

101 102

103 104 105

106

107 108

109 110

111 112

113

114

115

116

117 118

119 120

121 122

123

124

125

126

127 128

129

130

131 132

133 134

135

136

137 138

139 140

141 142

143

144

145

146

147 148

149

150

151

152 153

154

155

156 157

158 159

160

161

Table B.l (cont.)

Input State Output State

Door 18-21 ft 2 Door 18-21 ft 2 Door 18-21 ft2 Door 18-21 ft 2 Door 18-21 ft2 Door 18-21 ft2 Door 18-21 ft 2 Door 18-21 ft2 Door Blank

Door Blank

Door Blank

Door Blank

Door Blank

Door Blank

Door Blank

Door Blank Door Blank

Door Blank

Door Blank

Door Blank

Door Blank

Door Blank

Door Blank

Door 18-21 ft2 Door 18-21 ft2 Door 18-21 ft2 Door 18-21 ft 2 Door 18-21 ft 2 Door 18-21 ft2 Door 18-21 ft2 Door 18-21 ft2 Door 18-21 ft2 Door 18-21 ft 2 Door 18-21 ft2 Door 18-21 ft 2 Door 18-21 ft 2 Door 18-21 ft2 Door 18-21 ft2 Door Blank

Door Blank Door Blank

Door Blank

Door Blank

Door Blank

Door Blank

Door Blank Door Blank

Door Blank Door Blank

Door Blank

Door Blank

Door Blank Door Blank

Door 18-21 ft2 Door 18-21 ft2 Door 18-21 ft2 Door 18-21 ft2 Door 18-21 ft2 Door 18-21 ft2 Door 18-21 ft2 Door 18-21 ft2 Door 18-21 ft2 Door 18—21 ft2 Door 18-21 ft2 Door 18-21 ft2 Door 18—21 ft2 Door 18-21 ft2 Door 18—21 ft2 No Door Hardware

No Door Hardware

No Door Hardware

No Second Door

No Second Door

# of Rooms

# of Doors Egress Coefficient

Security Coefficient Illumination Coefficient Ventilation Coefficient

18-21 ft 2 Solid 3 Lite

18-21 ft 2 Solid 6 Lite

18-21 ft 2 Solid 9 Lite

18-21 ft 2 Metal 0 Lite

18-21 ft 2 Metal 1 Lite

18-21 ft 2 Metal 3 Lite

18-21 ft 2 Metal 6 Lite

18-21 ft 2 Metal 9 Lite

21-25 ft 2 Hollow 0 Lite

21-25 ft 2 Hollow 1 Lite

21-25 ft 2 Hollow 3 Lite

21-25 ft 2 Hollow 6 Lite

21-25 ft 2 Hollow 9 Lite

21-25 ft 2 Solid 0 Lite

21-25 ft 2 Solid 1 Lite

21-25 ft 2 Solid 3 Lite

21-25 ft 2 Solid 6 Lite

21-25 ft 2 Solid 9 Lite

21-25 ft 2 Metal 0 Lite

21-25 ft 2 Metal 1 Lite

21-25 ft2 Metal 3 Lite

21-25 ft2 Metal 6 Lite

21-25 ft 2 Metal 9 Lite

21-25 ft 2 Hollow i0 Lite

21-25 ft 2 Hollow 1 Lite

21-25 ft 2 Hollow 3 Lite

21-25 ft2 Hollow i5 Lite

21-25 ft2 Hollow 19 Lite

21-25 ft2 Solid 0 Lite

21-25 ft2 Solid 1 Lite

21-25 ft 2 Solid 3 Lite

21-25 ft 2 Solid 6 Lite

21-25 ft 2 Solid 9 Lite

21-25 ft2 Metal 0 Lite

21-25 ft2 Metal 1 Lite

21-25 ft2 Metal 3 Lite

21-25 ft2 Metal 6 Lite

21-25 ft2 Metal 9 Lite

>25 ft 2 Hollow 0 Lite

>25 ft 2 Hollow 1 Lite

>25 ft2 Hollow 3 Lite

>25 ft 2 Hollow 6 Lite

>25 ft2 Hollow 9 Lite

>25 ft 2 Solid 0 Lite

>25 ft 2 Solid 1 Lite

>25 ft 2 Solid 3 Lite

>25 ft2 Solid 6 Lite

>25 ft 2 Solid 9 Lite

>25 ft2 Metal 0 Lite

>25 ft2 Metal 1 Lite

>25 ft2 Metal 3 Lite

>25 ft2 Metal 6 Lite

>25 ft2 Metal 9 Lite

>25 ft2 Hollow 0 Lite

>25 ft 2 Hollow 1 Lite

>25 ft2 Hollow 3 Lite

>25 ft2 Hollow 6 Lite

>25 ft2 Hollow 9 Lite

>25 ft 2 Solid 0 Lite

>25 ft2 Solid 1 Lite

>25 ft2 Solid 3 Lite

>25 ft2 Solid 6 Lite

>25 ft 2 Solid 9 Lite

>25 ft2 Metal 0 Lite

>25 ft2 Metal 1 Lite

>25 ft 2 Metal 3 Lite

>25 ft2 Metal 6 Lite

>25 ft2 Metal 9 Lite Ordinary Lock

Dead Bolt 1 Key

Dead Bolt 2 Keys Install Second Door

No Second Door

It of Rooms

It of Doors Egress Coefficient Security Coefficient Illumination Coefficient Ventilation Coefficient

Percent Illumiination Percent Illumination

Percent Ventilation Percent Ventilation

Comments

Replace With Same Size Replace With Same Size Replace With Same Size Replace With Same Size Replace With Same Size Replace With Same Size Replace With Same Size Replace With Same Size Add A New Door

Add A New Door

Add A New Door

Add A New Door

Add A New Door

Add A New Door

Add A New Door

Add A New Door

Add A New Door Add A New Door Add A New Door

Add A New Door Add A New Door Add A New Door Add A New Door Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size

Add A New Door

Add A New Door Add A New Door

Add A New Door

Add A New Door Add A New Door

Add A New Door

Add A New Door Add A New Door

Add A New Door Add A New Door

Add A New Door

Add A New Door

Add A New Door Add A New Door Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Replace With Larger Size Hardware Placeholder

Hardware Installation

Hardware Installation

Door Installation

Door Placeholder Room Placeholder

Door Placeholder

Surplus Value

Surplus Value

Surplus Value

Surplus Value

Surplus Value

Surplus Value

74
The last eight constraints define the performance requirements . The first two

constraints define the minimum number of doors and windows permissible. The

third and forth constraints define the total window and door area. These

figures are used to ensure that the ratio of window area to floor area exceeds

the code requirements for ventilation and illumination. The levels for each

regulatory requirement are calculated by first identifying the component/state

pair which corresponds to prescriptive compliance, recording the state value

and summing across all components.

The discussion which follows provides an explicit algebraic statement of the

model.

The Objective Function

Minimize the total retrofit costs

161

cTOTAL =

Z cixi

i=l

where c-^ = the cost of retrofitting from

the initial state to state i

The Constraints

A. Conservation of Units Equations

11 35

A. 1 X 1 + £ xi + Z xi = 20

i=4 i=28

Constraint A.l requires that the number of windows not created (i.e., a transition from a blank (no window) to a blank) plus the number of new windows

(i.e., a transition from a blank to a window 20 to 40 inches high or 40 to 60

inches high) remain fixed. The right hand side variable (20) is a limit on

the number of new windows; it is conceptually, an arbitrarily large number.

73 103 133

A. 2 X57 + E X± + z X± + E Xi = 20

i=59 i=89 i=119

Constraint A. 2 requires that the number of doors not created plus the number of new doors remain fixed. The right hand side variable is a limit on the number

of new doors.

19 43

A. 3 X2 + E Xi + z X± = W2 i=12 i=36

This constraint conserves the number of windows 20 to 40 inches high. W2 is

the number of such windows existing; it is input for each problem.

75
27 51

A. 4 X 3 + Z Xi + E Xi = W3

i=20 i=44

This constraint conserves the number of windows 40 to 60 inches high. W3 , the

number of such windows, is an input.

88 118 148

A. 5 X 58 + E Xi + Z Xi + Z Xi = D X i=74 i=104 i=134

This constraint conserves the number of doors which are 18 to 21 square feet in

size. Dj , the number of such doors, is an input.

B. Accessory Matching Equations

51 54

B.l Z Xi - Z Xi = 0

i=4 i=52

Constraint B.l requires that the number of window hardware systems (latch,

locking latch, maximal locking hardware) equals the number of windows.

51 56

B. 2 Z X t - Z Xi = 0

i=4 i=55

Constraint B.2 requires that the number of window screen systems (the number

with and the number without) equals the number of windows.

148 151

B. 3 Z Xi - Z Xi = 0

i=59 i=l49

Constraint B.3 requires that the number of door hardware systems (lockset,

deadbolt (1 key), deadbolt (2 key) equals the number of doors.

148 153

B.4 Z Xi - Z Xi = 0

i=59 i=152

This constraint requires that the number of second door systems (the number

with and the number without) equals the number of doors.

C. Performance Requirements

C.l X 152 + X i5 3 - X!55 =

Constraint C.l requires a minimum of Dm

doors. is input for each problem;

X 155 is an explicit surplus variable.

76
51

C.2 2 - X 154 = Wm

i=4

Constraint C.2 requires a minimum of Wm

windows. Wm is input for each problem;

X 154 is an explicit surplus value.

C.3

159

2 Al

i=l

X

i - *161 " FV

This constraint requires the effective ventilation area of all doors and windows

to meet or exceed a minimum area requirement. aY is the effective ventilation

area in square feet per item in state i. The variable X]^i is an explicit

surplus and Fv is the required area in square feet. Fv is equal to a prescribed

fraction of floor area (see table 3.1).

159

C.4 E A’ Xi - X160 - F

1 i=l

This constraint requires the effective illumination area of all doors and

windows to meet or exceed a minimum area requirement. aJ is the effective

illumination area in square feet per item in state i. Tne variable X^q is an

explicit surplus and F^- is the required area in square feet. F* is equal to a prescribed fraction of floor area (see table 3.1).

161

C .5 2 s| X

± - X157 = R

S

i=l

Constraint C.5 defines the level of performance necessary to attain the security

requirement, R

E

. The variable X^y is an explicit surplus; the S? terms in

the equation represent state values indexed by code attribute, S is for

security.

161

C .6 2 sf Xt - X

156 = R

E

i=l

Constraint C .6 defines the level of performance necessary to attain the egress

requirement, R

E

. The variable X^^ is an explicit surplus; the SE terms in the

equation represent state values indexed by code attribute, E is for egress.

161

C .7 2 sjx.- X 159 = R

V

Constraint C.7 defines the level of performance

ventilation requirement, R

V

. The variable X^9

sY terms in the equation represent state values

V is for ventilation.

necessary to attain the

is an explicit surplus; the

indexed by code attribute,

77
161

C .8 Z s\ X ± - X 158 = R

1 i=l

This constraint defines the level of performance necessary to attain the

illumination requirement, R

1

. The variable X^g is an explicit surplus; the

terms in the equation represent state values indexed by code attribute, I is for illumination.

Some Comments on a Generalized Equivalency Methodology

The generalization of the pilot equivalency methodology outlined in this report

poses technical challenges both in the computational aspects of the problem and

in several theoretical issues in operations research. The computational aspects

of a generalized equivalency methodology involve: (1) fixed charge problems;

(2) large integer programming formulations; and (3) nonlinear constraints. The

theoretical aspects are associated with the task of systematically generating

alternate solutions for large integer programming problems.

As more emphasis is placed on the accuracy of the cost estimates, it is likely

that fixed charge problems will be encountered. For example, a contractor may

quote a figure of $1,000 plus $350 for each replacement window. In this case,

$1,000 is the fixed charge. Although the algorithm used in this report can not

handle the fixed charge problem directly, algorithms do exist which can handle

it as well as explicitly model economies of scale. (The existence of economies

of scale implies that the objective function is nonlinear. A piecewise linear

approximation to the objective function is possible however.)

The structure of the problem solved in this report was such that a linear

programming formulation could be used, and almost all variables which appeared

in the solution took on integer values. Depending on the structure of the

generalized equivalency methodology this result may or may not occur. If it

does not occur, it will be necessary to rely on an integer programming

formulation.

The problem can, of course, be solved using an integer programming formulation

but only at the expense of greatly increased computational costs. The last

computational problem is associated with nonlinear constraints. In the general

formulation of the problem it may occur that some building components interact

in a nonadditive manner. The mixture of additive and multiplicative interactions implies that the regulatory requirement constraints are nonlinear. In some cases, a transformation of variables or an ingenious redefinition of the

regulatory requirement constraints can induce linearity. Under such circumstances a linear programming formulation would still be appropriate. However,

it may not be satisfactory from an engineering point of view to redefine,

transform or rely on pricewise linear approximations to the regulatory requirement constraints. In these cases it would be necessary to use a nonlinear

programming formulation. Such a decision would again be at the expense of

greatly increased computational costs.

78
The generation of alternate solutions to large integer programming problems is an important theoretical issue in operations research. At the present time,

most integer programming codes use an approach known as the branch and bound

technique. In essence this technique constructs a tree, selects a node to

branch on, and places a bound on it through a process known as fathoming. If

the code is written such that a branch is only partially fathomed and the nodes

representing potential solutions are stored, it may be possible to generate a class of "good" solutions. Unfortunately, there seems to be no priori reason

for believing that this set of solutions would be as systematic as the one generated with the linear programming formulation. Furthermore, this approach

may terminate before the optimal solution is found.

79
NBS-114A (REV. 2-80

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO. NBSIR 81-2416

2. Performing Organ. Report No. 3. Publication Date

March 1982

4. TITLE AND SUBTITLE

APPLICATION OF AN EQUIVALENCY METHODOLOGY TO BUILDING REHABILITATION - A

PILOT STUDY v

5. AUTHOR(S)

James H. Pielert , Robert E. Chapman and William G. Hall

6. PERFORMING ORGANIZATION (If joint or other than N BS. see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City. State, ZIP)

National Bureau of Standards

Department of Commerce

Washington, D.C. 20234

10, SUPPLEMENTARY NOTES

j 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

bi bliography or literature survey, mention it here) With increased emphasis on the re-use of existing buildings, new approaches must be

developed to assist regulators in making code related decisions. The application

of performance criteria to building rehabilitation provides flexibility in the use

of technically sound design alternatives in lieu of prescriptive provisions which

may be restrictive. This report presents the results of a pilot study on the

application of an equivalency methodology in achieving regulatory compliance. The use of such a methodology is particularly attractive in this area because prescriptive type provisions have been shown to constrain rehabilitation activities, and in some cases, may be mutually contradictory. Regulatory requirements were chosen so as to explicitly incorporate conflicting requirements as affecting the design of

windows and doors — illumination, ventilation, egress and security. The methodology is computerized to allow the selection of least-cost means of achieving

compliance with these requirements. A prototypical townhouse is evaluated using

the pilot equivalency methodology and optimal compliance strategies are identified

and compared with the cost of prescriptive compliance. The results of the study

produced potential savings ranging from 20 to 35 percent depending on the initial

conditions of the building.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Applied economics; building codes; health and safety; housing; mathematical

programming; rehabilitation; renovation.

13. AVAILABILITY 14. NO. OF

PRINTED PAGES

[X] Unlimited

j | For Official Distribution. Do Not Release to NTIS 91

1 Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.

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!/^ j Order From National Technical Information Service (NTIS), Springfield, VA. 22161 $10.50

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