NBSIR 73-182

Analyses for Decision in the

Office of Flammable Fabrics:

The Level of the Standard

Technical Analysis Division

National Bureau of Standards

Washington, D. C. 20234

April 1973

Status Report

Prepared for Office of Flammable Fabrics

Fire Technology Division

National Bureau of Standards

Washington, D. C. 20234

NBSIR 73-182

ANALYSES FOR DECISION IN THE

OFFICE OF FLAMMABLE FABRICS:

THE LEVEL OF THE STANDARD

Alexander R. Craw

Technical Analysis Division

National Bureau of Standards

Washington, D. C. 20234

April 1973

Status Report

Prepared for Office of Flammable Fabrics

Fire Technology Division

National Bureau of Standards

Washington, D. C. 20234

U. S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts. Director
ACKNOWLEDGEMENT

Particular thanks are due Dr. Joseph E. Clark, who as

Chief of the Office of Flammable Fabrics and later on Acting

Chief of the Fire Technology Division, gave strong support

and encouragement throughout the effort represented here.
TABLE OF CONTENTS

Page

Executive Summary 1

List of Symbols 5

1. INTRODUCTION 7

1.1 Purpose 7

1.2 Caution 8

2. THE SINGLE STAGE DECISION PROBLEM 9

3. THE LEVEL OF THE STANDARD 17

4. OUTCOMES 19

5. MEASUREMENT OF WORTH (RISK) OF OUTCOME 23

5.1 Number of Burns 23

5.2 Disutility 25

5.3 Assessment of Disutility (loss) -The Cost Tree 26

5.4 The Disutility to the Manufacturer 29

5.5 Benefit/Cost Analysis 30

5.6 Factors to be Included in "Cost"

(Disutility) Assessment 31

6. PROBABILITIES OF OUTCOMES 33

6.1 Extending the Conversation 33

6.2 Example: Prob (u|a^) 38

6.3 Suggested Detail for Initial Probability

Tree for Outcomes 40

7. EXAMPLE OF DETAIL FOR A DECISION TREE FOR

LEVEL OF STANDARD 40

7.1 Several Probability Trees 40

7.2 Alternatives (a) 42

7.3 Technology (T) 43

7.4 Nightwear (N) 4 3

7.5 Additional Costs to Consumer (C) 44

7.6 Use (U) or Non-Use (u) 50

7.7 Hazard (H) 50

7.8 Exposure (E) 50

7.9 Ignition (I) 54

7.10 Burn (B) and Burn Serverity (Bj_) 57

7.11 Survival fS) or Death (D) 61

7.12 Body Image (1^) 62
8. SUMMARY OF CURRENT NEEDS 62

8.1 Probability Trees for Outcomes 62

8.2 Measures of Worth or Risk 64

8.3 Manufacturers Cost Analysis 65

8.4 Intangibles 65

9. CONCLUSIONS 66

Appendix A: Summary of Background Papers 67

Appendix B: The Assessment of Multi-Attributed Utility

Functions 77
List of Illustrations

Page

Figure 1: A Decision Tree for K=3, m=4 10

Figure 2: Probabilities on Decision Tree 12

Figure 3: The Probability Matrix for Three Actions

With Four Outcomes 13

Figure 4: Tree Diagram for Analysis of Decision

between Three Alternatives with Four

Possible Outcomes 15

Figure 5: Utility or "fay-Off" Matrix 16

Figure 6: "Extending the Conversation" for Outcomes 22

Figure 7: Probability Tree for Prob(B|ai

) 35

Figure 8: Extending the Tree for Outcomes 36

Figure Al : The Probability Tree for Prob(B|Sj_) 70

Figure A2'i Decision Analysis Diagram for McDonald's

Problem 7 4

EXECUTIVE SUMMARY

This paper serves as an expansion of the brief exposition

in Myron Tribus 1 paper: Decision Analyses Approach to Satisfying the Requirement of the Flammable Fabrics Act. It starts

with a decription of the rudiments of a single stage decision

problem and how this problem is solved in principle by the

formalism of decision analysis. For this one needs to identify explicity what alternatives form the basis of decision,

what outcomes result from those alternatives, and what is the

probability of each outcome if a particular alternative were

in force. For each of the possible outcomes, one has to

assign a value or utility to that outcome, and then for each

alternative determine the expectation of the utility

(disutility) of the outcome. The decision rule then is:

Choose that alternative for which the expectation of the utility (disutility) is a maximum (minimum) . All of the above is

discussed in some detail in the light of a particular example,

the level of the standard for children's sleepwear.

For the particular example, in place of having simply the

outcomes burn (B) and no-burn (b or Bo), one allows the outcomes to include several degrees of burn injury B 1 ,B 2 ,B3.

Also included is whether or not the injured person survives

(S) or dies (D) , and if he survives, whether or not his bodyimage area (I d

) is affected or not.

Section 5 is a discourse on several of the possible measures of risk (disutility) that might be chosen. Some are

1
relatively crude, others more sophisticated. The ultimate

choice of outcomes may depend on the choice of the criteria

to be used in the evaluation part of the decision process.

Note that if one uses the minimum expected number of

burns as the measure of risk of the outcomes , this automatically implies that one should set the standard as stringently

as possible, and we remark that in this case one need not make

any explicit calculation . In addition to discussing what is

to be included in the disutility measurement, a discussion

is given of the essentials of benefit/cost analysis. If one

is unable to complete a formal decision analysis, much of the

information generated might be used to perform a conventional

benefit/cost study, for example, or to adopt some other method.

Besides assigning a measure of disutility to the outcomes,

one must assign the conditional probabilities of arriving at

each outcome given that a particular alternative is in effect.

It is usually extremely difficult to do this in a straightforward way. One then resorts to an insertion of intermediate

events so that the sequence of conditional probabilities will

combine to yield the desired probabilities of specifiec outcomes. This procedure is sometimes called "extending

the conversation". One attempts to choose a set of intermediate events which make it possible to calculate each

of the intermediate conditional probabilities. Section 6

concludes with a listing of the intermediate events suggested

for the probability tree for the level of the standard for

2
children's sleepwear. The details for the probability tree

are

:

1. Six probability trees with variables for each age

(2) and income level (3) (3x2=6)

2. Alternatives (a)

3. Technology (T)

4 . Nightwear (N)

5. Additional cost to the consumer (C)

6. Use (U) or non-use (u)

7. Existence of a hazard (H)

8. Exposure (E)

9. Ignition (I)

10. Burn (B) and burn-severity (B^

11. Survival (S) or death (D)

12. Body Image (1^)

Section 7 provides discussion of the bases on which one

might assign first estimates for each of the conditional

probabilities needed. The probabilities that are known are

stated. Suggestions are made as to the sources of information

that would assist in assignment of probabilities for each

item. Unsurprisingly, much of the needed information is not

available, especially that dealing with the social and behavioral aspects of the problem. Section 7.8 contains a discussion of the concept of "exposure" to an ignition hazard for

which an operational definition does not exist. One suggestion, in the nature of a trial-balloon, is to use a concept

3
of critical distance to define an operationally measurable

"exposure "

.

The paper concludes with a summary of current informational needs necessary to carry out a decision-making analysis.

It is determined that the considerable information voids

preclude an immediate application of the decision analysis

technique to the problem of the level of the standard.

To make the report reasonably self-sufficient, we have

summarized the work of M. Tribus and of K. MacDonald, as well

as some of the recent work on the assessment of multi-attriu

buted utility functions. These are included as appendices.

4
List of Symbols: (in order of appearance)

a = alternative (here a particular level of standard)

0 = outcome

p = probability

p. . = prob (0 . I a . ) =conditional probability of outcome 0. i] j

' i c -1 j

given that alternative a. is in effect

DM = decision-maker

Z = summation symbol

u = utility (also used to denote non-use)

u.. = u (0 .I a . ) = u (a . , 0 ) j

1 l l ' j' = the utility of outcome 0 . conditional on the alter3 native a^

= expectation operator

OFF = Office of Flammable Fabrics

FF = Flammable Fabrics

FFACTS = Flammable Fabrics Accident Case and Testing System

B = Burn

b or B = no-burn

o

B^ = less than a "major" burn -\ = a major burn that is not extensive \ Burn severity

B

3 = extensive burn J

Levels

S - survival

D = death

1 = ignition or ignite

i = not ignite

1^ = an individual's "body image" area is involved in

burn
= an individual's "body image" area is not involved

in burn

n = number of burns per unit time

N = number in the population under discussion

d = disutility

p . . = Prob ( 0 . la.)

HEW = Department of Health, Education, and Welfare

HEW-1 = the first annual report of the Secretary of HEW

under the FF Act to the President and The Congress

Shriner's= data from Shriner's Burn Institute

Rice = "Cost of Illness" by Mrs. Dorothy Rice

C = cost (additional) to consumer

U = use

u = non-use

E = exposed

e = not exposed

= perceived cost

Y = family invome

G = geographical area

W = washability

F = feel (or 'hand')

N = nightwear garment type

T = available technology

H = existence of a hazard

Q(c) = the demand function for quantity Q at price c.

6
1. Introduction

1.1 Purpose

This memorandum is intended to serve several purposes.

The first is to introduce the rudiments of the formal discipline of Decision Analysis to personnel working in the area

of flammable fabrics research and development. It extends

in more elaborate detail the relatively brief exposition given

by Myron Tribus in his paper: Decision Analysis Approach to

Satisfying the Requirements of the Flammable Fabrics Act. An

important contribution of that paper was the identification

of certain gross information needs necessary for a rational

approach (through the mechanism of Decision Analysis) to the

problem of determining the level at which a particular standard for flammable fabrics should be set.

A second purpose of this note is to present this detail

in the setting of a familiar, specific problem, and one for

which some abundance of information is available. The problem

of children's sleepwear is especially appropriate since, in

addition to information availability, it was the subject of

Tribus 1 original paper; the existing standard will likely be

subject to review and/or revision in the future; and the

problem is typical (in the sense of mathematical structure

and needs) of standard setting for many other flammable fabrics

The third purpose of this note is to start a formal,

detailed decision analysis for the children's sleepwear problem. In this analysis we would hope to specifically indicate

7
our present state of knowledge of the conditional probabilities

necessary for the many branches of the probability tree.

This involves identifying what estimates of probabilities

are known and possible sources of improvement for those probability estimates which are not available; indications are

given of methods for establishing them and the likely sources

of information.

Even if a complete decision analysis cannot be performed,

much information will have been generated to serve usefully in

other methods of decision making, e.g., benefit/cost analysis.

1. 2 Caution

The paper is in the nature of a status report and

reports on work that is ongoing but incomplete. Emphasis is

placed on the logical structure of the analysis and attempts

are made to identify those variables that are helpful or necessary to the analysis and for which information (data) need be

obtained. While attempt will be made to indicate how the

various pieces should be put together, some of the operational

methods for doing this need to be tested.

We will not discuss other decision problems in the management area of flammable fabrics. In particular, we will not

discourse on the decision problem of determining the priorities

for standard setting nor the decision problem on the allocation

of research and development in the flammable fabrics area.

8
2 . The Single Stage Decision Problem

In this section we will develop the essentials of the

solution of a single stage decision problem. The basic singlestage decision problem can be described generally as one in

which the decision maker (DM) must choose a course of action

from a set of k alternative actions, call them a-^, . . - , a^.

which have associated a set of m possible outcomes, call them

0-^, C^/.-.f 0 . This situation may be depicted graphically

through a tree diagram, called a decision tree. Such a tree

for three alternative actions and four outcomes is illustrated

in Figure 1.*

If action a^ is chosen (see Figure 1) the outcome may be

either 0^ or 0^ or 0^ or 0^ since, by assumption, the outcomes are mutually exclusive and exhaustive. The decision

maker (DM) usually has some information on the nature of the

possible outcomes and part of the procedure of decision theory

is to have the DM assign his (subjective) estimate of the probability of each possible outcome; this assignment is made on

the basis of the information available to him at the time of

assignment. This subjective probability is a measure of the

DM's belief that if action a^ is chosen that the particular

outcome will result.

_For the reader who may want to visualize a concrete rather

than an abstract example, we suggest that the alternatives

may be viewed as different levels of a standard for flammable fabrics (FF) , and the outcomes are different severities

of burns. As is done in sections 3 and 4, one may take

0-, = B no burn, 0~ = B, = less than a major burn, 0~ = B„ = a

1 o 2 1

J 3 2

major burn that is not extensive, 0^ = = an extensive burn.

9

In our illustrative example we denote the subjective conditional probabilities that the outcomes will be CK (j=l,2,3,4)

if action a^ is chosen by p^.. = Prob(0j|a^). Since the outcomes are assumed mutually exclusive and exhaustive the probabilities add to unity, i.e.,

.JPij

= Pll + p 12

+ P12 + P i3

+ P 14

= 1 -

It is convenient to indicate the probabilities on the

outcome branches and to identify the outcomes at the tips of

the branches (Figure 2)

It is the assignment of probabilities to outcomes that

characterizes decision theory as decision making under uncertainty.

We remark that the information displayed in Figure 2 may

also be displayed in matrix form (see Figure 3)

One of the difficult procedure problems in the application

of decision theory is that of getting the DM's assignment of

probabilities of outcomes. There are several operational ways

of doing this, but for the time being we will defer discussion

of these until we get to the concrete problem to be studied in

detail

In order to choose between the various actions or alternatives the DM must have some way of placing a value, worth

or preference for each of the outcomes. If the decision

problem is in a business context, monetary values of the outcomes are usually used to make the decision. If monetary value

is used, there is a question of whether or not the DM

11
Figure 2. Probabilities on Decision Tree

12
v Outcome

Action\

°1 °2 °3 °4 I

a

i Pn P12 Pl3 Pl4 1

P21 P22 P23 P24 1

a

3 P31 P32 P33 P34 1

Figure 3. Trie Probability Matrix for Three Actions

With Four Outcomes

13
is a risk taker or a risk averter and this brings up the

question of the utility of money to the DM rather than its

face value. We will not pursue this point at this time.

In social problems, like that of consumer protection, the

valuation will depend on a number of factors or attributes

and recourse usually is made to the application of multiattributed utility theory. This theory is based on a number

of postulates that most rational persons are willing to accept,

such as: if outcome A is preferred to outcome B, and outcome

B is preferred to outcome C, then outcome A is preferred to

outcome C. For the present discussion we will skip over the

difficult problem of assessing a multi-attributed utility

function and assume that the valuations have been made. We

denote the utility of the outcome CK conditional on the

alternative a^ as u(a^,0j) or simply as Uj_ j . In the graphical display of the decision problem it is convenient to write

the valuations of the outcomes alongside the outcomes (see

Figure 4). Correspondingly, we may display the utilities in

a utility or "payoff" matrix (Figure 5)

.

Once the probabilities and values of outcomes have been

assigned and determined, the DM is in a position to act.

The usual decision rule is: Choose that alternative for which

the expected utility of the outcomes is an optimum . The optimum is either a maximum or a minimum depending on whether the

valuation is considered as a utility or disutility. The

expected utility of the outcomes for a given act is obtained

14
15
xQutcomes

Actions.

°1 °2 °3 °4

a

i "11 "12 "13 "14

*2 u

21 "22 "23 "24

a

3

' U

31 U

32 ^3 "34

Figure 5. Utility or "Pay-Off" Matrix

16
by multiplying together the corresponding probabilities and

utilities and adding over the set of outcomes. ^ u(0\a±

) = Ep±i u(a if 0j) = zpj_j u

ii

j j

In outline form the decision process may be summarized:

a) Determine the set of alternatives, A = (a^ , . . . , a^}

b) Determine the outcome set, 0 =

' {0 , 02,...,0m >

c) Assign the DM's subjective probabilities for the

outcomes, i.e., p^j = Prob (Oj | aj_)

d) Evaluate the utility of the outcomes: u (a^ , 0j ) =u^

j

e) For each act, determine the expected utility of

the outcomes, i.e. ,t> u(0|ai) for all i from 1 to k

f) Choose that act for which the expected utility of

the outcomes is an optimum.

3 . The Level of the Standard

In this section we recognize that the problem of the level

at which to set the standard for a flammable fabric, in

particular that for children's sleepwear, is a single stage

decision problem, so that in principle, at least, we know how

to go about the solution of the problem.

In principle the problem is solved once we have determined

the valuations of the outcomes and the conditional probabilities of each possible outcome, i.e., the u^j's and the Pij's.

Our first problem is to delineate the set of possible

alternative standards. In order to keep the problem within

bounds some preliminary elimination of alternatives is essential. This process of elimination is sometimes colorfully

17
referred to as "pruning the tree" to avoid a bushy mess.

As a start, the following three standards representing

different levels of stringency, have been suggested:

a^ = Present Commercial Standard CS191-53. In this test

specimens are at an angle of 45° and the measured

burning distance is 5 inches. A small flame impinges

on the surface of the fabric for one (1) second.

The fabric must not ignite and burn a 5-inch length

in less than four (4) seconds.

a

2 = The so-called Blanket Flammability Test. This is a

resistance to ignition test. The test specimen is

2h Inches in diameter. A small flame impinges on

the center of the specimen for one second. The

sample must not ignite, i.e., allow the establishment

of a self-supporting flame.

a

3 = Vertical Test DOC FF3-71. Specimens are 3h x 10

inches and are hung vertically in the test chamber.

A flame source impinges on the bottom edge of the

specimen for 3 seconds. Pass-fail criteria depend

on char length of specimen and time of burning of

drips or other fragments on the base of the cabinet.

The alternative a^ is considered to be a minimal test,

and is one which must be satisfied by all fabrics used for

clothing in the United States. a^ may be considered as a new

baseline, while a^ is very stringent. a^ is the current

standard for children's sleepwear.

18
4 . Outcomes

The stated objective of the OFF is to prevent death and

reduce injuries due to FF. Here injury is interpreted as a burn injury. Tribus , in his expository paper, used burn (B)

or no-burn (BQ )

, and suggested that in a more elaborate

analysis one might want to take into accouat the level or

severity of the burn. This seems to be a worthwhile suggestion.

The degree of burn injury is a continuous variable, but we

will discretize this to a number of levels devoted by B , o

Bj... r B^ in increasing level of severity. One might think

of these as midpoints (or other representative points) of

some class interval and that we are representing each entire

class by a single member.

Several injury classification schemes were considered.

After reviewing the burn-injury literature and discussions

with physicians who treat burned individuals, and thinking

ahead to the valuation of the outcomes, it seems advisable

to adopt a classification scheme based on the depth of the

burn and the total body area subjected to second and third

degree burns. Sometimes this is referred to as Total Extent

of Burn and is written (3°+2°). Most important, such data are

available in the Flammable Fabrics Accident Case and Testing

Systems (FFACTS) , so there is hope of quantitative assessment.

The depth of burns is divided into first, second, and third

degrees. First degree burns have a redness; second degree,

blisters; and third degree, full thickness injury. First

19
second degree burns are partial thickness injury and will heal

spontaneously, if infection does not supervene. Third degree

is brown or gray, has considerable resiliency and is dry and

anesthetic. The second degree burn, however, is red, wet soft

and pliable: and sensation is usually present. (Larson,

Nebraska Med. J., October 1969).

At the Shriner's Burn Institute at Galveston, Texas and

at many other hospitals an estimate of the areal extent of burn

injury is quickly evaluated by using the "rule of nines"

(see diagram for age 15 years)

For younger children, the ratio of the sum of head and leg

areas to the total body area is larger. For children of other

ages the rule has been modified as indicated:

/

(Total is 100)

36

Note that 4 percent

added to the head

each time, whereas

2 percent is subtracted from the

leg. (Larson)

36 36

16

5 years

20
The term "major" burn is used if more than 15 percent of

body area has second and third degree injury, while the term

"extensive" burn is used if 35 percent or more of body area

is involved. (Ref: Quindlen and Abram, Southern Med. J.,

1969) . This suggests, as a start, that we adopt the following

categories and symbols for burns:

B = burn

B (or b) = no-burn

o

B^ = total extent of burn area less than or equal to 15

percent = less than a "major" burn

B^ = total extent of burn area more than 15 percent but

less than 35 percent = a major burn that is not extensive

B^ = extensive burn = 3 5% or more of body area covered

with second and third degree burns.

The economic "cost of illness" depends on whether or not

the injured person survives or dies, and data is available on

which to estimate the probability or survival (or death) by

age, race, of persons who have suffered a given body area

burn. We would like to allow for the outcomes the joint

event of survival (S) for a person suffering a burn of severity

level B.. We will write this joint event as SB. or B.S, the

order of symbols being unimportant here. Also let D denote

the event of death.

21
Figure 6. "Extending the Conversation" for Outcomes

22
If we are to consider the psychic "cost of illness",

psychiatrists advise that special psychic problems are

encountered by those who suffer burns to the head, face and

upper chest area, which effectively account for the individuals "body image" .. .this corresponds to the concept of "me,

as a person," and that it appears to radiate outward; to a

lesser extent the hands have importance. While the symbol

I will be used to denote ignition (following Tribus ' notation),

it seems appropriate to use the symbol 1^ if the person's

body image area is involved in the burn injury, and to use

the symbol i^ if it is not.

The ultimate choice of the outcomes to be considered will

depend upon the choice of the criteria to be used in the

evaluation part of the decision process. We take up this

matter next.

5 . Measurement of Worth (Risk) of Outcome

It is possible to provide a number of measures for the

worth (risk) of outcomes, some very crude while others may

possess a high degree of sophistication depending on the level

of detail to be included.

5 . 1 Number of Burns

If we restrict the outcome level to that of burn (B)

or no-burn (b) , we can use the number (n) of burns per unit

of time as a crude measure of worth (risk) , and we could pick

that level of standard, a^ for which the expected number of

burns was a minimum. Our problem would then be resolved if

23
if we could (1) determine the number N of the population

at stake per unit of time and (2) determine the conditional

probability of a burn if alternative a^ is chosen, since

£ (n| ai ) = N p(B| ai ) i.e. , the expected number of burns conditional on the adoption of ai equals the number in the population times the

probability of a burn if alternative a^ is chosen. Tribus

discusses a number of factors affecting the calculation of

the probability of a burn, and we shall go into this matter

in some detail in the Section 6.

We remark here that (n|aj_) and p (6 | ai) are on one-to-one

correspondence above and we could take p(B|ai) as a measure

of the "risk" involved (following the approach of Chauncey

Starr*) in adopting the alternative a^. The difficulty here

is that we do not know whether the level of risk as measured

by p(B|a-j_) is acceptable or not. We do, however, have the

option of comparing these probabilities with those of other

acts for which the public is at risk.

Note that if one uses the minimum expected number of burns

as the measure of risk of the outcomes, this automatically

says that one should set the standard as stringently as possible, and we remark that in this case one need not make any

explicit calculation .

*Starr, Chauncey (1969) : "Social Benefit versus Technological

Risk," Symposium on Public Safety, National Academy of Engineering, May 1, 1969. (See also, SCIENCE, 19 Septerber 1969,

"Vol. 165, No. 3899, p. 1232-1238.

24
Another option mentioned by Tribus is to determine the

curve of expected numbers of burns vs. expected increased cost

to produce (this being the increased dollar cost to produce

a garment) but indicates that this still begs the question

of where is it reasonable to put the standard.

5.2 Disutility

If we proceed to consider outcomes that include

severity levels of burn injury such as B

Q , B]_, B2, B3 we

could use the probability tree approach to calculate for each

alternative each of the probabilities p(B0

|a.j_), p(B]Jai),

p(B2|aj_) and p(B3|aj_) and the expected number of burns of each

severity level n(BQ |ai), n(B-]Jai), n(B2|ai) and n(B3|a-j_).

Now, however, we are reluctant to add up these numbers and

use the total number of burns as a measure of risk. Since

we are considering classes of burn-injuries as outcomes it

seems natural to consider these outcomes as undesirable

events and to associate a degree of disutility or undesirableness with the outcomes. Since the symbol u is customarily

used to denote utility, we choose here to designate the

disutility of an outcome by d(0) or d(0|a^) if the outcome

is conditional. By the mere fact of introducing severity into

the problem we recognize disutility varying with severity.

We want to weight the severity levels in some way, i.e., we

want to assess the disutility associated with each outcome.

If we consider the joint events BjS and BjD as outcomes,

in effect we are recognizing that the events have different

25
disutilities and want to take this into consideration. By

further breakdown of the outcomes into compound events,

such as BjSI^ or BjSi^, we are taking into account that the

disutilities of these events differ from that associated with

BjS.

By reverse token, if one cannot assess the difference in

the disutilities between the events B-S and B-SId , there is

j J no advantage to include the higher order outcome in the

decision tree.

No matter what level of detail is considered for the outcome it is possible to use the probability tree approach to

determine each p(0j|aj_) = Pij , i.e., the probability of the

outcome given that alternative a^ is in effect and also the

corresponding expected number n(0.|ai), items of interest in

their own right.

The above probability tree might be referred to as an

"Outcome Tree" or as a "Result Tree". A separate but similar

"Cost Tree" might be constructed to examine the costs (in

whatever coin) that would have to be paid to gain the increased

protection. Here as in many public decisions the benefit

and costs accrue to different people or classes of people.

5 . 3 Assessment of Disutility (loss) - The Cost Tree

One must decide on what factors (attributes) are to

be considered in the disutility (or loss) function and to

whom the disutility accrues. For example, McDonald's loss

function represents the average financial loss to a single

26
individual as a consequence of Oj , given action a^. The

value of the function is the sum of (1) the individual's loss

of earnings (mortality or morbidity) , (2) hospital costs (3)

costs of physicians services, and (4) the cost to implement

the action a^. For mortality losses McDonald used the present

value of lifetime earnings, discounted at a 6% rate.

We remark here that these are economic losses and losses

to a single individual. They are quantifiable. But they are

not necessarily exhaustive. Practioners of benefit/cost

analysis in the health and safety area theorize on what should

be included but often do not give any practical methods for

quantifying these. They also refer to losses due to intangibles , primarily pain and suffering associated with injury, but

usually do not evaluate these in the analysis. (Mishan,

Wiederkehr)

.

We choose to consider the disutility to an individual in

the population who might be subject to burns from use or nonuse of children's sleepwear, in some specified age group,

say 5 years or younger. Other age groupings can be cared for

in the analysis as well. Actually, while we talk of an

individual, we will have in mind the members of the individuals immediate family, particularly the parents or guardians

(who are the decision or rule makers for the young children)

.

We will refrain from considering a personal individual, such

as your child, but will consider a statistical or representative individual from the population. Although, for any

27
particular property such as psychological effect, there is

a distribution of individuals within the particular age class,

when one views these from the point of view of the population

of all people, the individuals in one age class are viewed as

fairly homogeneous, especially if we consider such factors as

the loss of potential earnings, treatment costs, etc.

With regard to intangibles, physical pain in an individual

is an extremely subjective phenomenon, but is usually restricted

to some relatively short period of time, the treatment period,

in comparison to the time of psychological effects. We are

currently examining the literature on psychological effects

associated with burn injuries in the hope of at least identifying qualitatively the major effects. We have found that in

addition to the individual's immediate family; there are

considerable psychological problems encountered by those who

are involved in the medical care of the burned children, in

particular nurses, doctors, and physical therapists.

These are the other intangibles that have to be presented

to the decision maker in the prose which accompanies the

quantification, and we would like to include some allowance

for the disutilities of the intangibles, even if only crudely.

We have looked at several possible approaches for doing some

quantification of these factors which others have classed as

intangibles. Discussion of these is contained in Appendix B.

In particular we give there a discussion of some of the recent

results in the methods of the assessment of multi-attribute

utility functions. 2 fi
We will not include in the discourse other losses to

society as a result of outcomes.

5 . 4 The Disutility to the Manufacturer

To this point we have included the consumer and the

public in the measurement of disutility of the outcomes. In

some way or other we would like to take into account the effect

on the manufacturing industry (manufacturers and their employees) of the act of setting the standard at a particular level.

Any change in the standard level will undoubtedly cause

a change in the cost of manufacture of fabrics, which eventually will result in a change in cost to the consumer who

purchases the fabric (or garment)

The law requires that the level of the standard set should

be technically feasible and reasonable. We assume here that

in order that an alternative standard (a^) be under consideration that (a) there exist a technological means for the manufacturing industry to meet the standard, (b) that there is a

way of estimating the cost to the manufacturer to comply with

the standards and (c) that there is a way of transforming this

cost into a cost to the consumer for the unit of time under

consideration in the valuation of outcomes.

There are then at least two possible ways to proceed in

selecting the level of the standard:

Introduce a cost variable in the probability tree for the

calculation of the probability of each possible outcome and

(1) use the expected disutility of the outcomes to the consumer

29
public as the decision rule for the selection of the level of

the standards, or (2) use the probability tree to calculate

the expected numbers of lives saved and injuries (of different

severities) avoided, treat them as benefits, and then calculate

a benefit to cost ratio for each alternative standard and perform a conventional benefit/cost analysis.

The first alternative has been the topic of discussion of

this note to the present. The theoretical discussion of this

methodology will be complete when we discuss the detailed makeup of the structure of the probability tree in Section 6.

5 . 5 Benefit/Cost Analysis

The benefit/cost ratio is simply the present value of

all benefits expected to be realized over a selected period of

time attributable to a given standard divided by the present

value of all costs expected to be incurred over the same time

period and attributable to the same standard. It is traditionally assumed that money (dollars) can serve as a common unit

of measurement of all benefit variables as well as of all

costs. The costs reflect actual losses to society resulting

from the accidental events.

Costs are measured by (1) expenditures on the part of

government to develop, promulgate and enforce standards programs and (2) expenditures on the part of consumers with

respect to the purchase and use of flammable fabrics and garments, in this problem we may treat governmental expenditure

as fixed, i.e., federal expenditures do not vary significantly

with the output of rule making actions.

30
The consumer cost component includes not only the manufacturer and/or suppliers cost but also reflects any additional

expenditures on the part of the user and any additional costs

that are borne by society as a whole that are caused by and

attributable to the rule making action, e.g., the need to take

special precautions in the laundering or cleaning of the

fabrics, or the loss of sectors of employment due to displacement of workers due to change in technologies.

The benefit/cost ratio may be used to rank the alternatives,

i.e., the greater the ratio the higher the ranking. Caution

should be exercised in the use of the ratio alone as a measure

of program worth for in certain instances it may lead to a

choice that does not maximize the net benefit (i.e., benefits

minus costs) to society.

5 . 6 Factors to be Included in "Cost" (Disutility)

Assessment

We list the factors that we will consider in the

assessment of disutility, and indicate possible sources of

information for data as well as some comments on each of the

items. HEW-1 will be used to designate the first annual

report of the Secretary of Health, Education, and Welfare

under the Flammable Fabrics Act to the President and the

Congress: Studies of Death, Injuries, and Economic Losses

Note: This discussion of benefit/cost analysis reflects

the summary of this topic by Dr. Harris Hordon in

an unpublished working paper for the National Highway Traffic Safety Administration (NHTSA)

31
Resulting from Accidental Burning of Products, Fabrics or

Related Materials (Sept. 1965-Feb. 1969) ; (undated)

.

Shriners will designate data available from the Shriners

Burn Institutes. Rice, is used to designate "the HEW Report",

"The Cost of Illness" by Mrs. Dorothy Rice.

1. Hospital Costs (HEW-1, pp. 48-49 and data from

Shriners Burn Institute.

2. Costs of physicians services (HEW-1, Shriners).

3. Individual's loss of earnings - mortality (Rice, p. 93).

4. Individual's loss of earnings - morbidity (HEW-1, p.. 56 1.

Note: This item will most likely be omitted for

children.

5. Transportation of victims or family or both to and

from hospital (Shriners)

.

Note: Data may be difficult to obtain; not in HEW-1.

6. Costs of subsequent hospital visits. (Shriners)

Note: This will vary with severity levels. Primarily

elective surgery, skin grafting, prosthetic

devices, psychiatric treatment.

7. Cost to implement action (government agencies).

Note: This may be considered as fixed across alternatives. Some estimates should be made to

ascertain magnitude.

8. Costs to manufacturer.

Note: This will probably require an independent study.

Needed to Determine "Reasonablenes " and also

additional costs to individual.

32
9. Purchase cost to individual.

Note: Also needed for calculation of probabilities

of outcomes.

10. Intangibles:

Note: Here we feel the primary intangible is the

psychic effect.

We feel that physical suffering ("pain") has been measured

to a large extent in part of the economic valuation, particularly in hospital costs and costs of physician's services.

We will try to at least discuss the psychic effects qualitatively. We will investigate the possibility of use of newly

developed techniques of assessment of multi-attributed utility

functions (see Appendix B)

.

6 . Probabilities of Outcomes

6 . 1 Extending the Conversation

In Section 2 we denoted the probability of the outcome

Oj conditional on the selection of alternative aj_ by p^j = Prob (Qjfai). A major portion of the work in decision analysis

is over once we have arrived at estimates for the national

values of the j

1 s that are acceptable to the decision maker.

But arriving at these estimates directly is almost an impossible task, and we look around for means of accomplishing our

goal.

This is done by introducing into the problem auxiliary

events for which- the estimation of the probabilities of the

auxiliary events is more easily accomplished, and from which,

33
by using the laws for the combinations of probabilities, it

is possible to calculate the p^j's. This process is sometimes

referred to as "Extending the Conversation".

In his previous work on this problem Tribus introduced

the auxiliary variables of additional costs (Cj), use (U) or

non-use (u) , exposure (E) and ignition (I) to calculate the

probability of a burn (B) conditional upon the adoption of

the level of standard a^ by

p(B|ai ) = Zp(B|UEICjai) p(l|UECjai) p(E|uCjai) p(u|Cjai ) p(Cj |aj.)

sum over

all j

+ Zp(B|uEICjai) pUluECja^ p(E|uCjai) p(uCjai) pCj|ai)

sum over

all j

This somewhat formidable collecting of symbols can be depicted

graphically by a"probability tree" (see Figure 6) . The terms

following the summation signs represent the probability of a

burn following a particular path through the tree from aj_ to

B; the total probability is the sum of the probabilities over

all the paths from a^ to B. Tribus also gave an indication

of the interpretation of each term and how it might be determined. For example, "the term p (I | UEa^Cj ) represents the

probability that a fabric will ignite when exposed to a flame.

This probability is reckoned conditional on the truth of UEa^

(Cj is immaterial here) , and a study of the various conditions

of exposure as gleaned from field studies correlated with

laboratory tests will enable an assignment of this term"

.

34
'35
Figure 8. Extending the Tree for Outcomes

36
In Section 4 we chose to include several categories of

outcomes by severity burn levels, which we denoted by Bj_, B2,

and B3. We can determine each of the probabilities Prob (S | aiBj

)

by extending the probability tree from each B to Bj(j=l, 2, 3)

and then extending to S or D from each B j

.

Other variables can be introduced to facilitate the calculation of the probability, Prob(Ok |ai) = Pj^.

For example, in the case of children's sleepwear, we might

ask the question of how the manufacturers might possibly meet

a standard by using different materials and/or techniques.

For convenience let us call these technologies , and let T^,

T2,...,T.£ denote the totality of technologies available to

the manufacturers to comply with the candidate standards

a^,a2,a3» As an example, T]_ might denote the use of cotton

fabric alone; T2 might denote the use of cotton treated in

some particular way; etc. We then ask the question: If standard ai is chosen, what fraction of (or what is the probability

that) the children's sleepwear will be made using each of the

technologies? It may well turn out that, for example, if

standard a^ is chosen the probability of compliance with

is zero, the fraction of garments using T2 is .30, etc. The

fractions T-^, T2,...,Tt may differ for different levels of

standards. For convenience, let

ProbCTjJai) = tki

denote the probability of a garment manufactured using Tj,

given that standard a^ is adopted. If we know what particular

37
T^ is being considered, we may be in a better position to

estimate the probability of outcome 0^.

It is suggested that the T variable be inserted between

aj_ and C in the tree. We must remember that our aim is to

facilitate the calculation of the p^

j

1 s and that we are at

liberty to do this in many ways. The more intermediate

variables we introduce into the discussion the greater is the

number of intermediate conditional probabilities that have to

be estimated. So it behooves us to introduce as few intermediate variables as possible and to choose those variables which

lead to easy estimation of the intermediate conditional probatilities

.

6 . 2 Example; Prob(ula-j)

In attempting to estimate the probability of use

(by a consumer and nationally) of garments of children's

sleepwear manufactured to conform to a given level of standard

a± [call this Prob(U|a^)], if we can do this well without too

much attention to detail - all the better. However, use will

depend on a variety of things : what the garment is made of

and behavioral characteristics of the user. The latter may

depend on perceived cost (actual cost plus "cost" of discomfort, etc.) (r )

, and these in turn may depend on family

income (Y) and local geographical area (G) , washability (W)

,

and feel (F) or hand of the garment.

We would like to throw in as many variables as needed to

make the estimation of Prob(u|ai) easier, since it is easy to

38
combine probabilities of lower order events. On the other

hand, if a variable does not enter the probabilities signifi

cantly, one shouldn't bring it in. If the decision-maker

(DM) knows, or feels that, say, geographical location doesn

effect use vary much, then we would not want to include this

variable in the chain of calculation. We should have some

basis (like data) for including or excluding some variable

from discussion.

Use might depend on garment type (N) [for nightwear] and

this might include pajamas, nightgowns, robes, single or mul

tiple layers and combinations of them, such as robes (single

layer) robes (multiple layers) , etc.

Information on family income by geographical area is

available from the latest census. Data on sales of garmant

type by geographical area should be available from the manufacturers and distributors. Behavioral scientists might be

called upon to furnish information on the behavioral characteristics relating to use of non-use of garment type by vari

ous sections of the population of potential users.

It is conjectured that information on perceived cost (^)

may be difficult to come by but here again one might appeal

the question to behavioral scientists.

39
6 . 3 Suggested Detail for Initial Probability

Tree for Outcomes

For separate age class 0-5 years and 6-12 years, by

income class:

alternative .(a)

available technologies (T)

additional cost to consumer (C)

use or non-use (U) or (u)

existence of a hazard (H)

exposure (E)

ignition (I)

burn (B)

burn-severity (Bj_)

survival or death (S) or (D)

body image (1^)

7 . Example of Detail for a Decision Tree for Level of Standard

7 . 1 Several Probability Trees with Variables for Each

For the present, and guided by members of OFF, we

suggest the following level of detail for the decision trees.

For the time being we will ignore geographical area (G)

,

washability (W) , feel or hand (F) . In order to bring in socioeconomic conditions and their effect on the behavioral characteristics of individuals with respect to "use" and "exposure"

and noting that the morbidity-mortality effects may vary by

40
age* and to differentiate the effects by age, we suggest

that six (6) identically structured probability trees be

developed: age (2), income level (3); 2x3= 6. The age

groups are: 0-5 years, 6-12 years.

The income level classes for families are:

= less than or equal to the poverty level r

Y2 = above the poverty level and below the median level,

Y3 = above median level.

Data on the numbers of children in the various age classes,

by race and income level can be obtained from the Tables 491

and 499 of the Statistical Abstract of the United States and

from the Bureau of Census Current Population Reports. These

data on the number of children in each group are needed to

calculate the expected numbers of burns by severity and the

expected numbers of burn deaths for each of the trees . From

these probability trees one can obtain first estimates of the

expected numbers of each of the outcomes. These could serve

as crude measures of disutility as discussed in sections 5.1

and 5.2. Further, by choosing some alternative, say a^, as

a reference level, one could then have estimates of the number

of lives saved and injuries avoided, items which furnish the

*Rittenbury et al present data to show that there is a racial

difference in mortality for children in the 0 to 14 age group,

but further indicate that there is insufficient data to determine why the racial difference exists. However, the lower

socioeconomic status of the Negro in the population from which

the patients studied were taken may explain this difference.

41
basis for a conventional benefit/cost analysis. In the latter

the net benefits (benefits minus costs) could serve as a measure of the expected utility for purposes of decision making

(see section 5)

.

Purposefully, we will try to steer some sort of middleground in the detail to be incorporated in the decision trees,

in order to have some means of possibly collasping (at some

later time) the details of the tree, or on the other hand, of

pointing to the necessity for greater detail should this be

borne out in the analysis.

Each tree will have the following sequence of branches

:

Alternatives (a) , Technology (T) , Nightwear (N) , Cost (C)

,

Use (U) , Hazard (H) , Exposure (E) , Ignition (I) , Burn (B)

,

Survival (S) , Body Image (I^) . Hazard is used here as a

short term for "existence of a hazardous condition, in particular, the presence (in the home) of ignition sources". We

will elaborate on the definition and operational evaluation of

"exposure" when we take up the detail of this variable.

The suggested detail for each of the variables is given

below along with some elaborating discussion.

7 . 2 Alternatives (a)

a-^ = Present Commercial Standards CS 191-53

a2 = Blanket Flammability Test

a3 = Vertical Test DOC-FF-3-71

The detail of these alternatives have been discussed in section

3. In the real life decision process these will have to be
expanded in number, probably with several levels between a^

and

7 . 3 Technology (T) = Cottons

T2 = Man-mades (synthetics)

L. J. Sharman has suggested the following probability

(fraction of market) matrix:

a

l a

2 a3

0.90 0.60 0.30

T2 0.10 0.40 0.70

1.00 1.00 1.00

7.4 Nightwear (N)

N, = Loose-fitting (nightgowns)

N2 = Semi-loose (pajamas)

N3 = Snug-fitting (knit sleepwear)

N

4 = Robe

Many classifications of technologies are possible. For

example , Lyman Fourt of Gillette Research Institute has used

four principal groups: (a) cotton, (b) polyester/cellulose

blends from 35 to 50% cellulose, (c) melt-shrink thermoplastics, and (d) wool.

43
For each of the alternatives aj_ (i = 1, 2, 3) we need to

develop a matrix of the market split:

T.

Nfc ll t 12

N.

13 ; 14 1.00

•23 24 1.00

Here, t-j, = Prob (N^ | a^Tj) = Fraction of cotton made nightwear that are devoted to nightgowns under condition that a^

is in effect.

t^ = Prob (N2 ^iT^) = Fraction of cotton nightwear that

are devoted to pajamas under condithat a^ is in effect.

7.5 Additional Costs to Consumer (C)

Here we mean the relative additional dollar costs to

the purchaser, with the reference cost taken as that to comply

with the weakest alternative (a^) ; i.e. A^/cCa-^) where

= c(aj,) - c(a^) and c(a^) is the cost under alternative a^.

Tentative levels of fractional additional cost are:

Cq = Essentially no additional cost,

Cj_ = Up to 10 per cent additional cost,

C 2 = Between 10 and 2 0 per cent additional cost,

44
= Between 20 and 30 per cent additional cost,

= Above 30 per cent additional.

Under the assumptions that:

. The typical garment contains 1.5 yards of fabic

. The increased cost per yard of the fabric is 30 cents

. The demand for the product is inelastic

. The manufacturing process will not change with a change

in fabric

. The fire retardant chemical increases the fabric

weight by 22 per cent

the Ernst and Ernst* study for the Office of Textiles, Department of Commerce estimates the (average) increased retail price

is $0.89 to $0.92 per garment. This applied under alternative

a

3

. We will now proceed to demonstrate how to get some estimates for the probabilities like Prob (C^|a^T,N^). For this

particular probability we will show that Prob (C^ | a^T-^N^ ) -Fraction of Robes made of treated cotton orginally (under a^)

retailing for more than $9.00.

Proof: Let Q = Q(c) denote the demand function for quantity Q at Price C. We may plot a graph of Q versus c.

*

°- b c

Ernst and Ernst (April 1970) : A Study to Estimate the Additional Cost of Children's sleepwear under Flammability Standards, COM-71-00847

.

4~5
The total demand for a quantity with cost between a and b is

given by Q(a,b) =

J Q(c)dc.

a

We now assume that the demand curves for treated and untreated materials are the same, i.e., they depend on cost

alone and not on any other factors such as wear, feel, and

washability

.

To determine limits of integration for assessing Q (a ,b)

we assume that A-^c) = $0.90 under alternative a.2*

C± is in effect if A3 (c) /c (a-j_>_< 0.10; or c (a^) ^3/. 10=$9 .00

C2 is in effect if 0 . 10 < Aj/c (ai) _< 0 . 20 ; or $4 . 50_< c (a^ < $9 . 00

C 3 is in effect if 0 . 20 < A

3 /c (a x ) _<0.30; or $3 . 00_< c (a^ $4 . 50

C 4 is in effect if 0.30 < A

3 /c(a1

) or c(ax

) < $3.00

Let On denote the total quantity of sales of untreated

cotton robes corresponding to condition Cj_. In particular,

Qll = f° Q(c)dc.

^"9.00

Let Q31 denote the quantity of sales of treated cotton

robes corresponding to Cj_ . In particular,

Q3 1 = ' Q(c)dc,

§ .90

and the change in quantity due to the increased cost of treatment is

Q31 " Qll = ^--Q

t

or

,9.90

Q 31 = Qxl -

j Q(c)dc

F. 00

This states that the total quantity of sales of treated cotton

robes (a3TjN4 ) corresponding to C

1 equals the total quantity
of sales of untreated cotton robes (a^T^N^) originally selling

for $9.00 or more diminished by the amount of sales of

untreated robes originally selling between $9.00 and $9.90.

Now Prob (C 1 |a 3 T1N4) = Number of sales of treated cotton robes

corresponding to condition divided by

the total number of sales of treated

cotton robes.

f9.90 = £31 = £ll _i 9.00 Q(c)dc

f 00 where Q3 =^

Q(c)dc

- Qll , since the second term is small

- Qll since * Q n =J Q(c)dc

57 0

Thus

,

Prob (C1 |a

3

T

1

N

4

)

25 Fraction or Robes made of treated

cottons originally (under a^)

retailing for more than $9.00.

This completes the proof for the estimation of the approximate

value for Prob (C^ | a

3

T-j_N4) .

This first approximation for the probability may be refined

-9.90

by taking into account the term - Q(c) dc/Q-, to obtain

9.00

47
the approximation. These estimates may be adjusted further

by taking into account factors other than price in determining

the demand function for sales

.

In like manner,

f9.90 A9.00 /9.90 /5.40

Q32 =1 Q(c)dc = + -\ Q(c)dc

^5.40 J4.50 79.00 74.50

'5.40 /9.90

^J4.50 J9.OO

= Q

12 "I \ -\ Q(c)dc

$32 Ql2 Ql2

63 Q 3 Q x Thus, Prob (C2 I

a

3

T

1N4

) - Fraction of Robes made of treated cotton originally (under a^) retailing

between $4*50 and $9.00.

Similarly, Prob (C^la^T^N^) ~ Fraction of robes made of treated

cottons originally retailing

between $3.00 and $4.00

and

Prob (C4 I ^T^N^) * Fraction of robes of treated cottons

originally (under a^) retailing for less

than $3.00

while

Prob (^33^4) = 0

Note that if we are satisfied to use an average A3 = $0.90, we

would need information on retail pricing of nightwear and the

original market splits to estimate the various probabilities

above

.

48
We remark that we automatically have

Prob(CQ

|a 1 T1N4

) = 1.00

Prob(Cj |a 1 T1N4

) = 0 for j = 1,2,3,4.

Using a different demand curve for pajamas as a function

of cost and making similar assumptions about the invariance of

the demand curve with respect to factors other than cost,

one can approximate Prob (C^ | the fraction or pajamas

made of treated synthetics retailing (under a^) for more than

$9.00, etcetera. Also

Prob(C

Q

|a

1

T

2

N ) =1.00

Prob(Cj |a

1

T

2

N

2

) = 0 for j = 1,2,3,4.

This takes care of the estimation of the conditional probabilities of additional costs for alternatives a^ and a^

To estimate the conditional probabilities Prob (Cj | a2TjNjJ

under alternative a2 , one would need to determine the value of

A2 = c(a2) - c(ai) along the lines in the Ernst and Ernst

report for A3. Then one could apply the same type reasoning

as done in the case for a3 to estimate Prob (Cj_ | ^TjN^) * To complete the numerical evaluation of the conditional

cost probabilities one needs two things:

(a) a market study to determine the demand curves for each

type nightwear under the assumption a^ was in effect.

This was the state of affairs before a^ was promulgated.

(b) A costing study to determine the additional cost per

treated garment to meed the standard a2 , i.e., determine A 2

49
7.6 Use (U) or Non-Use (u)

To determine quantities like Prob (u| aj_TjNk

C£ ) will

to require some sort of behavioral and/or market study.

7.7 Hazard (H)

Examination of FFACTS Quick Query for August 7,

1972 reveals that the following were the principal hazardous

conditions that led to ignition (scorch) and burn while

children (0-12 years) were wearing sleepwear:

- Matches/lighters

H2 = Open flame/hearth/fireplace/candle

H3 = Space heater/gas heat/stove

H4 = Ranges, gas and electric

H5 = Miscellaneous hazards other than H^, H2 , H3 , H4.

Let HQ

= No-Hazard.

The information on the existence of hazardous conditions can

be obtained from a survey, but undoubtedly some excellent

estimates should be available from home economists.

The probability of the existence of a particular hazardous

condition would appear to be independent of the level of the

standard.

7.8 Exposure (E)

As yet, the meaning of "exposure" has not been defined.

We would like to let Ej denote the event of "exposure to

hazardous condition Hj " and let E

Q (or e) denote the event

of not being "exposed". Intuitively, one would not like to

say that he is "exposed" to a hazard such as a fireplace, just
by being in the room containing the fireplace. Some idea of

closeness in space and length of time is involved. We need

a definition of "exposure" that is operationally measureable,

so that (a) one can determine the probability of "exposure"

to hazard H:

Prob(E) = Prob(H) Prob(E |H)

and equally important that (b) one can calculate the probability of ignition given exposure to hazard H.

Prob(I) = Prob(l|E) Prob(E)

One must look ahead to see what information is available to

calculate the Prob(l|E). The research program recommended

by the Government, Industry Research Committee on Flammable

Fabrics (GIRCFF) has been aimed at calculating Prob(l|E)

when "exposure" is given by some set of factors.

Before proceeding to an operational definition of "exposure"

one can indicate what one would call the probability of

"exposure" to hazard Hj . Time enters in several ways. One

important time is the time the individual spends in nightwear.

One might choose to call this the wearing time (t^) . This

wearing time will be dependent on age and socio-economic

conditions, family habits and doctrine. A behavioral study

could determine representative wearing times. One would like

to exclude the sleeping time (t s

) as a time under which the

child is "exposed" to hazards so that the total time available

(tA

) for "exposure" to hazards while in nightwear is the

difference between wearing time and sleeping time. Thus, one

51
could define the probability of exposure to hazard Hj as the

fraction of the wearing time less sleeping time that the

individual is "exposed" to Hj . In symbols,

Prob(Ej iHjlll) = fjtA/tA

= fj

where tA

= t^ - t

s So far, exposed (or exposure ) to hazard Hj is still

undefined. Experience and experiments tell us that for a

specific Hj and material some combination of critical

values of factors that represent the properties and environment of the ignition source, of the fabric and of distance

and time, like distance, intensity of heat, time, RH, etc.,

is needed before "ignition" is possible. Call this multidimensional space of ignition factors the factor space for

the given specific situation and materials. The factor space

can be separated into two complementary subsets (Hj ) and

F2 (Hj ) . The first subset is one in which ignition is impossible (one is just too far away or the heat intensity is not

enough,...). The complementary set F2 (Hj ), which we shall

call the critical factor set for Hj , is that collection of

factor values for which "ignition" is possible. The critical

factor set for H^ implicity defines the exposure to H.. , which

we have previously labeled E j . When one is outside the

critical factor set for H., i.e., in F 1 (Hj), we would like to

say the person is not exposed to the Hazard Hj . The above is

merely a refined way of saying that a person is exposed to a

hazard if something detrimental (ignition) can happen. If a

12
detrimental effect cannot occur, a person would not be

"exposed" to a hazard.

Unfortunately, the critical factor set thus defined is

not reasonably good for operational evaluation of exposure to

a hazard, for each candidate fabric would have a separate critical factor set and we would have to know all the possible

combinations of factors for which ignition were possible for

the fabric in order to define the set. We have to inject some

degree of arbitrariness into the definition and slack off a

little.

If we concentrate on a particular hazard, say an open

flame, hearth or fireplace (H2

) one has some inherent representative characterization of the flame (which will eventually

have to be determined) . Then the remaining factors are

essentially distance, time, and properties of the materials.

For hazard H2 there is some critical distance , which when

exceeded would lead to no ignition, independent of the length

of time any kind of candidate fabric (T) is used in the vicinity of the particular hazard. This critical distance for a

particular hazard depends on the alternative (to define the

class of candidate fabrics) but not on a particular fabric

nor on time.

In place of using the critical factor set to define the

exposure to a hazard we suggest the use of critical distance . The ratio of the sum of the times spent within the hazard's

critical distance to the time available (tA) for exposure to

53
hazards will be taken as the definition of the probability of

exposure to the hazard H j

, given the existence of the hazard

Hj , i.e., Prob (Ej |Hj . . . ) • Note that this definition of

probability of exposure allows for a frequency of residence

occasions within the critical distance associated with the

particular hazard and is independent of the fabric (T)

Laboratory experiments should lead one to a determination

of the critical distance for each type of hazard, Hj . The

evaluation of the Prob (Ej | Hj . . . ) can only be evaluated through

a knowledge of the behavioral patterns of individuals in the

presence of a hazard.

To this point we have used the properties of the hazard

(ignition source) the properties of the materials (T) to

define a critical distance, and the total residence time and

wearing and sleeping times to define the Prob(Ej|Hj).

To relate exposure to the next variable (I) in the

probability tree we need a concept of exposure time. By

shall mean the time (for that occasion) spent within the

critical distance set for Hj . The determination of represen

tative exposure times is again dependent on the behavioral

characteristics of individuals.

7 . 9 Ignition (I)

Information of Prob (I | Ej . . . ) will have to come from

the experimental program, both in-house and that done under

contract at the recommendation of GIRCFF.
Ignition time (t^) appears to be a central element in the

determination of the probability of ignition under fixed

conditions of exposure. A major question is whether (under

fixed conditions of exposure, labelled X here) ignition time

is a random of a deterministic variable. The ignition time

is deterministic if under each specified set of conditions,

there is a unique value for the ignition time. If the

ignition time, under repeated application of the same conditions, exhibits a range of times with an associated probability

(frequency) distribution, then the ignition time is a random

variable . In the deterministic case, if the time of exposure (t e ) is

less than the ignition time, i.e., T e <Tj[, then the

Prob ( 1 1 x e

<xj_ ; X) = 0 and ignition is not possible; however,

if the exposure time is greater than or equal to the ignition

time, i.e., T e >ji, then Prob (I | t s

>_t i ;X) = 1, i.e., ignition

is certain.

In the random variable case, the best that can be said

is something like this: if for given fixed conditions X, the

exposure time is below the range of ignition times, then the

probability of ignition is zero; if the exposure time is

within the range of ignition time there is a non-zero

probability of ignition. This non-zero probability can be

determined provided the frequency distribution for ignition

times is known.

55
The work done at Georgia Institute of Technology (GT)

for the case of radiative transfer indicates that the

"ignition time depends strongly and deterministically on fabric

properties". (pages 11, 61 Final Report 1971). However, a

fourth phase of their work for radiative heating is to determine whether or not the measured ignition time is a unique

function of heating intensity for fixed fabric properties and

conditions. Thus, at the present this question is not conclusively settled, but fortunately, the radiative case applies

only to one part (electric ranges) of hazard . The predominant hazards leading to burn injuries in

children using nightwear are convective in nature (flaming

heat sources) and here the evidence indicates strongly that

the ignition time is a random variable.

Work at Gillette Research Institute* on materials held

at fixed distances above a gas stove and in the flame provide

a considerable body of data that can be used to determine the

probability of ignition for different exposures, certainly for

the gas range part of hazard , if not for the other flame

heat sources. Results are presented for GIRCFF Fabrics using

the classification of (a) Cotton, (b) Polyester/Cellulose,

(c) Melt-Shrink, (d) Wool.

Lyman Fourth (Dec. 31, 1971) : Study of Ignition and Exposure,

Gillette Research Institute, Harris Research Laboratories.

56
Some of the above data may help define the critical

distances defined in the previous section. Before these

data can be applied one must have some behavioral information

on the distances and exposure times actually involved with

children in the proximity of flame sources.

7.10 Burn (B) and Burn Serverity -(B^ ) At this level we find it convenient to extend the

conversation by expanding the outcomes from burn (B) and

no burn (Bq or b) to include severity levels of burns

(B^) , survival (S) or death (D) , and body image area (1^)

.

This is indicated in Figure 8 (page 36)

.

As described in Chapter 4, we suggest adopting for

severity levels of burn inquiry:

Bq = no-burn

B-^ = second and third degree burn area less than or

equal to 15 percent Ethan a "major" burn.

t B2 = second and third degree burn area more than 15

percent, but less than 35 percent = a "major"

burn that is not "extensive".

B^ = second and third degree burn area greater than 35

percent e an "extensive" burn.

One needs to determine the probability of a burn given

that ignition has occurred while the child was wearing

nightwear of a particular kind satisfying the requirements

of a designated alternative, i.e., Prob (B | a^TjUEe

I )

.

57
As pointed out by the Factory Mutual Research Corporation*:

Clearly, the behavior of people enters into

the problem. If a person is able to sense the heat

source (by means of pain mechanism in skin tissues)

before ignition occurs, he may draw away from the

source and not be injured. However, if the fabric

worn by the person is heavy and opaque, (or if

the person is inebriated or is a very small child)

ignition may precede detection by pain and the

chances os a severe burn injury are greatly increased.

We remark also that the behavior of children subsequent

to recognition of ignition of their nightwear will effect the

severity of the burn injury.

However, if we assume that a burn has occurred, we may

turn to available accident data on burns such as that contained

in the FFACTS system to get some preliminary estimates of

Prob (BjJb...). We have prepared such a preliminary table

from the FFACTS Quick Query Output dated August 7, 1972 to

illustrate the procedure. (See Table 7.1)

For Children from 0-5 years old, ignoring the technology,

fabric and income groups, but for alternative a^ and wearing

Robes (Nj : Prob (B

1 |BIa

1

N

4

UE

1

) = 2/3

Prob (B

2 |BIa

1

N

4

UE

1

) = 0

Prob (B

3 |BIa

1

N

4

UE

1

) = 1/3

A. S. Kaleekar and H. C. Kung (October 1, 1971) : A Study

of Preignition Heat Transfer Through a Fabric-Skin System

Subjected to a Heat Source, Factory Mutual Research Corporation .

58
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59
while for children aged 0-5, wearing pajamas (N

2

)

Prob (B 1 |BIa

1

N

2

UE

1

)' >_ 26/42

Prob (B 2 |BIa1N2 UE 1 ) >_ 4/42

Prob (B 3 |BIa1N2 UE 1

) > 7/42, etc.

In the table Unk denotes that the severity of burn level is

unknown. If the case of Pajamas (N 2

) for ignition (I) due to

exposure to matches/lighters (E-^) there are 7 such entries.

These 7 cases could be allocated to the severity levels Bi,

B2 and B3 in a variety of ways. It is for the reason that

we have indicated >_ in the second example.

Similar tables could (in principle) be constructed for

each of the various technologies and income groups. However,

one should expect even smaller values for the entries in the

tables

.

7.11 Survival1 (S) or Death (D)

For preliminary estimates we suggest the use of the

clinical burn data for the Medical College of Virginia (MCV)

published in Rittenbury et al* (Table 1, p. 124). This is

based on percent of total area burned. We will use as

preliminary estimates the age grouping 0-4 and 5-14 for the

0-5 and 6-12 respectively. From Table 1 of that article we

Rittenbury M.S. et al (July 1966): Probit Analysis of Burn

Mortality in 1,831 Patients, Annuals of Surgery, Vol. 144,

123-138.

60
aggregate cases and otbain the following table:

MCV Clinical Burn Data 1949-1962

Burn

Severity

Class

% Total 5-4 Vr. 'r.

Area

Burned

No.

PtS.

No.

Deaths

%

Mort.

No.

Pts

No.

Deathi

%

s Mort,

Bl < 15 339 3 0.8 145 0 0

B 2 15-34 110 15 13.6 80 14 17.5

B

3 > 35 44 37 84.1 50 35 71.4

Totals 493 55 11.1 275 49 17.7

This will provide Prob(D

Prob (S |Bi. . . ) = l-Prob(D

Bj_»...).

Bi. . . )

7 . 12 Body Image (1^)

We suggest that preliminary estimates of

Prob (1^ | SBj_ . . . ) be obtained from an examination of the details

of the case histories of accidents in FFACTS.

8 . Summary of Current Needs

8 . 1 Probability Trees for Outcomes

In the previous sections (7.1 through 7.12) we have

discussed in some detail what needs to be done in order to

calculate the probabilities of the outcomes associated with

the alternative levels of standard for flammable fabrics for

children's nightwear. The solution of this problem depends

on the availability of information on social and economic

factors, on the behavioral patterns of individuals as well as

on engineering type data . We will indicate a "shopping list"

61
of current informational needs, and will try to place them

in some semblance of priority.

a) In order to have all of the branches of the tree fit

together there is need to have some agreed upon operational

definition of "exposure" to a hazard. We have discussed this

at some length in section 7.8 and have suggested the use of

a critical distance to define this exposure. If this is an

acceptable concept there is need to determine a critical

distance for each of the hazards under consideration. This

will probably entail a professional engineering judgment based

on existing engineering data . Once this distance has been

defined for each hazard, the probability of exposure to a

particular hazard can be determined from a study of the

behavior of individuals in the presence of the hazard.

The behavioral study should, at the same time, determine the

distribution of representative exposure times; these are

needed in order to estimate the probability of ignition given

exposure

b) There is need to summarize all of the experimental

work done at the recommendation of the GIRCFF to determine

Prob (I | E . . . ) and the availability of other data on this topic

for the hazards under consideration. From these data, professional engineering judgment based on an "up and down" procedure* probably could finish preliminary estimates of probabilities of ignition for the remaining cases.

By an "up and down" procedure we mean an adjustment upward or

downward from some known reference level, with the amount of

adjustment estimated by professional judgement.

62
c) There is need to determine the probability of a burn

given that ignition has occurred. This can be addressed best

through a behavioral study. (See Section 7.10)

d) The FFACTS can generate reasonable information on

burn severity levels only for fabrics satisfying alternative

a]_. The extent of what detailed information, on burn severity

can be extracted from this system should be determined. Then

one probably could utilize the "up and down" procedure to

provide the remaining preliminary estimates of probabilities.

(See Section 7.10)

e) Social and behavioral studies are needed to determine

probabilities of use and the existence of hazards.

f) Economic studies are needed to determine the market

splits for the various technologies used to satisfy the

different alternate levels of standards. The present study

by Ernst and Ernst is applicable only to the case of treated

cottons for alternative a 3

. (See also Section 8.3 below)

8 . 2 Measures of Worth or Risk

From this probability tree one could then obtain first

estimates of the expected numbers of each of the outcomes.

These could serve as crude measures of disutility. Further,

by choosing some alternative as a reference level, one could

then have estimates of the number of lives saved and injuries

avoided, items which furnish the basis for a conventional

benefit/cost analysis. In the latter the net benefits

(benefits minus costs) could serve as a measure of the expected

utility for purposes of decision making.

63
8 . 3 Manufacturers Cost Analysis

In order to carry out a benefit/cost analyses or to

calculate an expected net disutility, it will be necessary

to have an analysis performed on the costs to the manufacturing

industry to comply with the requirements of each alternative

standard. This cost analysis is also essential to the determination of the probabilities of the levels of increased cost

to the consumer in the probability tree for outcomes. Part

of the cost analysis will involve identification of the technologies (fabrics and processes) (T^) needed to comply with

each of the prospective standards, and will be most helpful in

determining the Prob (Tk | aj_) . The analysis should also provide

evidence of the "reasonableness" of the requirements imposed

on the industry. This cost analysis appears to play a central

role in the decision making problem at hand.

8 . 4 Intangibles

We consider the primary intangible in this problem to

be the psychic effects, which accrue not only to the individual

and his immediate family but also to those who are involved

in the burn victims therapy and rehabilitation. We have been

reviewing the literature on the psychic effects of burn victims

and have had discussions with physicians, nursing staff,

and psychiatrists who treat these personnel. It is our

intention to write a separate paper which will review and

summarize what is published in this area. This will permit

the decision makers in this problem area to have at least

64
qualitative information on this factor. It is recommended

that we try to do more, particularly in attempting to

evaluate multi-attributed disutility functions which contain

psychic effects as a component of a vector factor, following

the procedures recommended by Keeney and others.

9 . Conclusion

We have looked at the details of the problem of applying

decision analysis to determine the level of the standard for

a flammable fabric. We find that there are considerable information voids that preclude immediate application of this technique and have listed these in detail in Section 8. The solution of this problem depends on the availability of information

on social and economic factors, on the behavioral patterns of

individuals as well as on engineering type data. Much of the

behavioral and social information needs may be very difficult

to obtain. Survey techniques, a tool used often by social

scientists, will undoubtedly be needed. The results of surveys will have to be supplemented by the judgements of social

and behavioral specialists in exactly the same way that the

engineering data on ignition will have to abetted by the judgments of physical scientists and engineers.

The most important need at this time is a conceptual one,

that of defining what is meant by "exposure" to a hazard. In

addition, there is need for an operational way of assessing

the probability of exposure, once "exposure" has been defined.

Suggestions for these have been given in Section 7.8.

65
Most of the information needed for a decision analysis is

equally necessary to conduct other methods of decision making,

whatever they may be. We feel that the examination of this

problem from the point of view of decision analysis has been

benefical in identifying those areas where additional information is needed.

66
Appendix A

Summaries of Background Papers

A.l M. Tribus: Decision Analysis Approach to Satisfying the

Requirements of the Flammable Fabrics Act.

(ASTM Standardization News, pp. 22-27,

February 1973)

This paper was presented by the then Assistant Secretary

of Commerce for Science ant Technology at the meeting of the

Textiles and Needle Trades Division of the American Society

for Quality Control at Greensboro, North Carolina, February 12,

1970. The problem addressed was that of determining the level

at which a standard for flammable fabrics should be set.

After discussing the requirements placed on the Secretary

of Commerce by the Flammable Fabrics Act and reviewing some of

the beginning work of Dr. Chauncey Starr of the University of

Califormia at Los Angeles relative to the appropriate risk

levels which might be used by public officials in setting

standards for the protection of consumers, the author lays

down the elements of a probability tree to calculate the probability of a burn, if a given level of flammable fabrics standard for children's sleepwear is adopted. The raw data required

were not immediately available. However, he did feel that the

analysis provided a conceptual framework required to guide both

the data gathering and laboratory experimentation.

Tribus depicted a probability tree concerned with five (5)

variables, given that the decision* maker had selected a

67
particular standard (S^) . These variables were: (1) additional cost (C) to the user to comply with the standard; (2)

use (U) or non-use (u) by the consumer of materials satisfying

the standard; (3) exposure (E) or non-exposure (e) of the

wearer of the flammable fabrics to ignition sources; (4) ignition (I) or non-ignition (i) of the exposed fabric and finally

the outcome variables burn (B) or non-burn (b) . This probability tree may be depicted as in Figure Al.

The total probability of a burn, contingent on a choice of

p(B|Si) = Zp(B|UEICjSi )p(l|UEC.S i )p(E|UCjS i )p(u|CjS i )p(Cj |Si)

sum over

all j

+ Ip(B|uEICjSi)p(l|uECjSi )p(E|uCjSi )p(u|CjS i )p(Cj |Si)

sum? over

all j

Each product of terms following the summation signs represents the probability of a burn by proceeding along a particular path through the branches of the tree starting at and

ending at B. The summation over all j merely adds up the

probabilities of all ways of going from to the outcome B.

The author gives an interpretation of each term and how

it might be determined in practice. For example: "p (B | UEICj Si)

is the probability of a burn, given use, exposure to an ignition, cost and a certain standard. ... (It) can be determined

from laboratory tests on various flame retardant materials.

(It turns out that given UEIS^ true, Cj is immaterial)".

The analyses provide a means for computing the expected

number of burn cases at various levels of expenditure for protection. 6 8
C0 U E I _ B

I u e i b

4

Figure Al. The Probability Tree for Frcb (EfS^)

69
I

The author then discusses a possible way at arriving at the

level of the standard by considering possible trade-offs

through non-use of the garments meeting the standard by consumers of the disutility associated with increasing costs accompanying the increase in stringency of standards.

Tribus points out that the above trade-off considerations

also provide a basis for considering the effects on the number

of burns by programs complementary to standard setting such

as publicity and education. {This latter type of analysis was

the subject of Kathryn McDonald's masters thesis at Cornell

University in 1970; the thesis is summarized in the next

section of this appendix.

]

No attempt at actual analysis was made in this concept

paper.

A. 2 Kathryn McDonald: Consumer Protection: The Case of

Flammable Fabrics, Master's Thesis,

Cornell University, September 1970.

and K. McDonald, B.F. Smith, and R. Dardis : Flammable Fabrics

and The Cost of Consumer Protection,

Textile Research Journal, Vol. 41, No.

6

492-500, June 1971.

The McDonald thesis is in two parts, the first on an investigation of burn injuries in general with a review of the

literature and a sample survey and analysis of burn injuries

in the city of Syracuse, New York for the four year period

1966 through 1969. The second part of the thesis is devoted

70
to the decision analysis and it is this part of the thesis that

appears in published form.

For the Children's Sleepwear problem the following points

from the first part appear important. There are cultural and

sociological influences on the incidence of accidental burns:

age, sex, culture, geographical location and whether are factors. Wilkinson is quoted that a prominent factor was not

bad living accommodations but rather "domestic habits of the

family." Another study points out that most burn accidents

occurred in the three lowest (out of five) social groups (based

on occupation of head of household). An early study (1950' s)

by Mayor indicated that death rates from burn injuries were

highest in the East South Central States and lowest in MidAtlantic States, that non-whites were more succeptible to burn

fatalities than whites and that (at that time at least) in

the South and Southwest there was a predominant use of movable,

open-flame heaters. For Children under 5 years in Birmingham,

England one study showed a male: female ratio of 3:2 in home

accidents. Source of ignition, such as open solid-fuel fires,

electric and gas heaters, floor heaters are important. A

probit analysis of burn cases indicates that mortality was

related to age, depth of burn and extent (size of burn)

.

We now comment on the second part of the thesis dealing

with the decision analysis. We also draw out a decision-tree

picturization rather than present the matrix representation

of the problem. In what follows page numbers are those in the

published paper.

71
K. McDonald's Decision Analysis Problem: choose between

four actions a^ , a2 , a 3 , a4 each with three possible outcomes,

Q-^t &21 e 3» Definition of symbols:

a± = do nothing (status quo) = ...potentially fatal

a2 = strengthen standard alone 62 = ...potentially injurious

p493

a3 = educational program alone 63 = ...neither 6^ nor 6 2

a

4 = do both p>493

v(6 a±) represents average financial loss to a single

individual as a consequence of 6j, given action a^, and is the

sum of

(1) individual's loss of earnings (mortality or morbidity)

(2) hospital costs

(3) cost of physician services

(4) cost to implement action aj_.

Loss caused by property (structural) damage was not included.

No allowance was made for the pain and grief suffered by the

injured person or his family and the author notes (and quotes

D. Rice) that the loss is intangible and difficult to measure.

Two different decision criteria are employed: (a) minimize

expected "loss" (disutility) and (b) minimize the maximum

expected disutility.

As pointed out previously, this problem is different from

that addressed by M. Tribus, who was concerned about the level

at which action a2 = (strengthen standard alone) should be set.

However, the McDonald paper does address the level at which

a standard could be cost-effective . This is done through the

7'2
= prob(e..fa.)

3

For matrix form

see Table I, p. 495

Action Outcome

Value

of outcome (Evaluated as a "a loss" dollar cost)

Figure A2: Decision Analysis Diagram for McDonald's Problem

73
introduction of a parameter 3 (called b in the original thesis)

(see p. 493) , which is meant to be a measure of the effectiveness of the f ire-retardant material.

The assumption is made that protective clothing would never

cause more loss than was anticipated when no protective clothing was used. Thus, 0 <_ $ <_ 1. When 3 = 1 the treated apparel

provided no additional protection to the individuals and when

3=o, 100% burn protection was provided [Note: The writer

would think of 3 as a measure of ineffectiveness of the fireretardant material].

This 3 factor is used in a study of cost-effectiveness

with the cost to implement action a2 , labelled Ca , in the

following way.

Let E(a^) denote the expected loss from action a^ , and

let E(ai) = S; then E(a2) = 3S = Ca (p. 496). The expected

losses for actions ai and a.2 are equal when

D(a±) = E(a2) or when S = 3S + Ca or S(l-3) = Ca . By studying this relationship one can determine the relation

between program cost and effectiveness (protective clothing)

This is done in the published paper on pp. 496-498.

There are several weaknesses in the paper and the authors

seem to be aware of them. These weaknesses have to do with

the value of loss functions v(6j_, aj

) . These are treated as

cost functions and are expressed in dollars and all costs are

additive. "No allowance was made for "psychic" losses, that

is, pain and grief suffered by the injured person and his
family.... In addition, it was not possible to take into

account the loss due to non-hospitalized injuries. Thus, the

cost of burn injuries is likely to have been underestimated

in the present model" (p. 500) . This latter point could

affect the detail of much of the cost-effectiveness arguments.

For example, in Figure 1, p. 496, the a3 line may be displaced

upward and intersect the a% line, and Figure 2, p. 497 may

well turn out to be the actual rather than the hypothetical

case

.

Also, the arguments may well turn around, if the loss to

the individual is "expressed in utility rather than monetary

terms" (p. 499)

.

The authors' conclude:

"Because the present cost to treat apparel for fire

retardance exceeds the expected loss from apparel burn

injuries, the expected-loss criterion directed the decision

maker to purchase treated apparel. The minimax criterion,

on the other hand, led to a decision in favor of protective

clothing .

"The authors believe that therein may lie a solution

to the problem of flammable apparel - let the consumer

decide, on an individual basis, which criterion he wishes

to employ and then select the appropriate course of action.

In such instances, education is essential."

75
Appendix B

The Assessment of Multi-Attributed Utility Functions

In the evaluation of the outcomes one leans heavily to the

use of modern utility theory as developed by von Neumann and

Morgenstern, Luce, Raiffa, Peter Fishburn and others. A

utility function, u, is a function which assigns a real value

to every outcome (consequence) in a manner such that u(b)>u(c)

if and only if consequence b is preferred to consequence c. The importance of utility functions is that they may be used

as a guide to rational behavior. However, a major operational

problem is that of assessing the DM's utility function for the

outcomes. The technique of assessment of utility functions is

primarily that of asking appropriate questions relative to

trade-offs. Here the concept of a "lottery" (or its equivalent)

and of a "certainty equivalent" is useful. The reader is

referred to Chapter 4 of the delightful paperback by Raiffa

for a convenient discussion, with example, of the assessment

of a single attribute utility function, and to pages 246-255

of the same book for an excellent introduction to the multiattribute problem. The latter terminates with a discussion of

a medical treatment problem, in which the weakest link in the

chain turned out to be the treatment of the atility structure.

The task in that problem was to assess a utility function

involving 7 attributes.

For the case of a multi-attributed utility function much

of the operationally useful techniques for its assessment have

76
been based on theoretical developments due to Raiffa and his

colleagues and students, notably R. L. Keeney. Central in

this work is the concept of utility independence of one

attribute Y on another, Z. It is possible for Y to be utility

independent of Z, but for Z not to be utility independent of

Y. An example of this phenomenon is given at the end of

this analysis. However, in the case the attributes Y and Z

are mutually utility independent, it can be shown that the

utility function u(y,z) can be evaluated from the quasiadditive functional form

u(y, z) = u(yQ , z) + u(y, z D ) + ku(y0 ,z) u(y,zQ ) where k is an empirically evaluated constant. The notation

here indicates that y and z denote amounts of the attributes

Y and Z and yQ and z Q are fixed values of y and z . It should

be noted that u(yQ , z) and u(y, z c ) are one-attribute conditional utility functions.

In some problems it may be more convenient to assess an

iso-preference curve than a conditional utility function. An

iso-preference curve (also called an indifference curve) is

a curve along which the DM has a fixed value of utility, i.e.,

he is indifferent to combinations of attributes that make up

the curve. Thus, if (y^, z^) and (y2 , Z2) are two arbitrary

points on an iso-preference curve, the utility of each combination of attributes is the same, i.e., u(yi, Z]_) = u(y2# Z2) =

a constant.

Under the same consitions of mutual utility independence

•n
of the attributes Y and Z, an iso-preference curve may be substituted for either a conditional utility function for Y or Z

provided it covers the same range; details are not indicated

in this note. We mention, in passing, that there are operational ways of determining when Z is utility independent of

Y, and there are ways of assessing u(y,-z) if Z is utility

independent of Y, but Y is not utility independent of Z.

In his dissertation Keeney outlines a procedures for

assessing multi-dimensional utility functions in five steps:

1. Introducing the terminology and ideas of DA and

utility theory to the DM.

After these discussions about the definitions and

the DM's understanding of them one should make clear to

the DM that the preferences one is interested in are

his, that there are no objectively correct preferences,

that the preferences of importance represent the subjective feelings of the DM and that he is free at any

time to indicate a change in his feelings about any of

the preferences stated. One of the purposes of a DA

is to require i DM to reflect on his preferences and

hopefully straighten them out in his own mind.

2. Identifying the applicable utility independence assumptions .

If, for example, there are two attributes Y and z under discussion one tries to determine if Y is utility

independent of Z and whether Z is utility independent
of Y. The independence properties will determine the mathematical functional form of the multi-attributed utility function,

and how to proceed further in the assessment of that utility

function.

3. Assessing conditional utility functions.

One assesses each of the required conditional utility functions on an arbitrary scale.

In this part of the procedure, for example, one

assesses say the utility of the attribute Z for a

fixed amount of attribute Y, say yQ , and then one

assesses the utility of the attribute Y keeping the

amount zQ of the attribute Z fixed in the process.

Recall that in certain cases it may be easier to

assess an iso-preference curve than a conditional utility function.

4. Scaling the conditional utility functions.

In this step one scales the conditional utility

functions so that they have a common origin and unit

of measure.

After this stage of the procedure one puts the

scaled conditional utility functions in the appropriate mathematically functional form and one then has

a utility function for the DM.

5. Checking the consistency of the utility function.

This is accomplished by further questioning of the

DM about his preferences. If any inconsistencies occur

79
one repeats some or all of the steps 1 through 5 until

one becomes satisfied that the utility function represents the DM's true preferences.

The systematic assessment of multi-dimensional utility

functions is probably the most difficult part of the procedure

of applying DA and may be a limitation on the DA approach.

The following discussion, due to Keeney, indicates the

possibility of a one-way utility independence between two

variables without the existence of a mutual utility independence:

Suppose one has a serious disorder and must undergo a

serious operation. Also more than one type of operation

exists for controlling this particular disorder. A choice

must be made as to what type of operation will be proformed.

The consequences are to be evaluated in terms of the probability of death and patient expense.

One might assume the probability of death is utility

independent of patient expense by arguing that the relative

preferences for death should be the same regardless of cost.

However, the patient expense might not be utility indepen- dent of the probability of death . One might have a completely different preference structure for expense, given

that he is almost sure to live, than he would given the

operation is likely to be fatal.

80
References (for Appendix B)

Fishburn, P. C. : Independence in Utility Theory with Whole

Product Sets, Operations Research, Vol. 13, 28-45, 1965.

Fishburn, P. C. : Utility Theory, Management Science, Vol. 14,

No. 5, p. 335-378, January 1968.

Fishburn, P. C. : Utility Theory for Decision Making, Publications in Operations Research No. 18, Wiley, N.Y. 1970.

Keeney, R. L. : Multidimensional Utility Functions: Theory,

Assessment and Application Technical Report No. 43, Operations

Research Center, Massachusetts Institute of Technology,

October 196 9.

Keeney, R. L. : Multiplicative Utility Functions Technical

Report No. 70, Operations Research Center Massachusetts

Institute of Technology, March 1972.

Keeney, R. L. : Utility Independence and Preferences for Multiattributed Consequences, Operations Research, Vol. 19, pp.

875-893, 1971.

Keeney, R. L. : Utility Functions for Multiattributed Consequences, Management Sciences, Vol. 18, pp. 276-287, 1972.

Pollak, R. A.: Additive von Neumann-Morgenstern Utility

Functions Econometrica, Vol. 35, No. 3-4 (July-October 1967),

pp. 485-494.

Raiffa, Howard: Decision Analysis: Introductory Lectures on

Choices Under Uncertainty, Addison-Wesley , Reading, Mass., 1968.

Raiffa, Howard: Preferences for Multi-attributed Alternatives,

The RAND Corp. Memo. RM-586 8-DOT/RC, April 1969.

Ting, H. M. : Aggregation of Attributes for Multiattributed

Utility Assessment, Technical Report No. 66, Operations

Research Center, Massachusetts Institute of Technology, August

1971.

81
References

Decision Analysis

Howard, Ronald A.: The Foundations of Decision Analysis, IEEE

Transactions on Systems Science and Cybernetics, Vol. 55C-4,

No. 3, pp. 211-219, September 1968.

North, D.W. : A Tutorial Introduction to Decision Theory, IEEE

Transaction on Systems Science and Cybernetics, Vol. 55C-4, , No. 3, pp. 200-210, September 1968.

Raiffa, Howard: Decision Analysis: Introductory Lectures on

Choice Under Uncertainty, Addi son-Wesley, Reading, Mass., 1968.

Tribus, Myron: Rational Descriptions, Decisions and Designs,

Pergamon Press, 1969.

Decision Analysis for Flammable Fabrics

McDonald, Kathryn: Consumer Protection: The Case of Flammable

Fabrics, Master's Thesis, Cornell University, September 1970.

McDonald, K. , Smith, B. F. , and Dardis, R. : Flammable Fabrics

and the Cost of Consumer Protection, Textile Research Journal

Vol. 41, pp. 492-500, June 1971.

Tribus, Myron: Decision Analysis Approach to Satifying the

Requirements of the Flammable Fabrics Act, ASTM Standardization News, pp. 22-27, February 1973.

Burn Information

Frye, R. E. , Katz, R. G., and Tovey, H.: Protocol for Processing Flammable Fabrics Case Histories, National Bureau of

Standards Report 10530, January 15, 1971.

Holter, J. C. : Etiology and Management of Severely Burned

Children, Amer.J. of Diseases for Children, Vol. 118, pp. 680-

686, 1969.

Larson, D. L. : Shriners Burn Institute - Early Care of the

Acutely Burned Child, Nebraska Med. J., Vol. 54, pp. 672-676

and 733-736, 1969.

Pruitt, B. A. Jr. et al : Mortality in 1,100 Consecutive Burns

Treated at a Burns Unit, Annals of Surgery, Vol. 159, pp. 396-

401, 1964.

82
Quindlen, E. A. and Abram, H. S.: Psychosis in the Burned

Patient: A Neglected Area of Research Southern Medical

Journal, Vol. 62, pp. 1463-1466, December 1969.

Rittenbury, M. S. et al: Factors Significantly Affecting

Mortality in the Burned Patient, Journal of Trauma, Vol. 5,

pp. 587-600, 1965.

Rittenbury, M. S. et al: Probit Analysis of Burn Mortality in

18 31 Patients: Comparison with Other Large Series, Annals of

Surgery, Vol. pp. 123-138, July 1966.

The Secretary of Health, Education and Welfare: Studies of

Death, Injuries, and Economic Losses Resulting from Accidental

Burning of Products, Fabric or Related Materials: (September

1955-February 1969) , A Report to The President and The Congress

(undated)

.

Tovey, H . : Burn Injury Data for Proposed Children's Sleepwear

Flammability Standard, Information Council on Fabric Flammability, Proceedings of the Fourth Annual Meeting, New York City,

pp. 116-128, December 3, 1970.

Tovey, H. and Giroux, R. : Ignition Sources in Accident Cases

Involving Flammable Fabrics, National Bureau of Standards

Report 10629, November 11, 1971.

Benefit, Cost, Risk

Hordon, H. E.: A Guide to Benefit-Cost Analyses of Motor

Vehicle and Highway Safety Programs (DRAFT of Internal Working

Paper); Office of Systems Analysis, National Highway Traffic

Safety Administration, August.

Mishan, E. J.: Evaluation of Life and Limb: A Theoretical

Approach, Journal of Political Economy, Vol. 79, No. 4. pp.

687-705, July/August 1971.

Mushkin, S. J.: Health as an Investment, Journal of Political

Economy, Vol. LXX (Supplement)^, October 1962.

Prest, A. R. and Turvey, R. : Cost-Benefit Analysis: A Survey,

The Economic Journal , Vol. LXXV, pp. 684-7 35, December 1965.

Rice, Dorothy P.: Estimating the Cost of Illness, U.S. HEW

PHS Health Economic Series No. 6, PHS Ser. Pub. 947-6, May

1966.

Rice, D. P. and Cooper, B. S.: The Economic Value of Human

Life, American Journal of Public Health, Vol. 57, 1954-1966,

November 1967.

83
Schelling, T. C. : The Life You Save May be Your Own, in Chase,

S. B., Jr. (ed.): Problems in Public Expenditure Analysis,

The Brookings Institution, 1968, pp. 127-176.

Starr, Chauncey: Social Benefit versus Technological Risk,

Science, Vol. 165, pp. 1232-1238, September 1969.

Wiederkehr, R. : A Methodology for Programming Highway Safety

Improvement Projects, in Analysis for Planning, Programming

and Budgeting [M. Alfondary-Alexander , (ed) ] , Washington

Operations Research Council, 1968.

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U.S. Department of Transportation, National Highway Traffic

Safety Administration, April 1972.

84
FORM NBS-114A (1.71)

1. PUBLICATION OR REPORT NO.

NBSIR 73-182

2. Gov't Accession No.

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Analyses for Decision in the Office of Flammable

Fabrics; The Level of the Standard

5. Publication Date

6. Performing Organisation Code

8. Performing Organization

NBSIR" 73-182

10. Project/Task/Work Unit No. 4900200

7. AUTHOR(S)

Alexander R. Craw

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

11. Contract/Grant No.

12. Sponsoring Organization Name and Address

Office of Flammable Fabrics

Fire Technology Division

National Bureau of Standards

Washington, D.C. 20234

15. SUPPLEMENTARY NOTES

13. Type of Report & Period

Covered Status Report

to April 1973

14. Sponsoring Agency Code

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.) The fundamentals of a single-stage decision problem are discussed and

illustrated in the problem: The Level of the Standard for Children's

Sleepwear, originally discussed by M. Tribus. Outcomes are identified,

and various potential measures of disutility are discussed.

Given a particular alternative is in effect one must assign the conditional probabilities of arriving at each outcome. This process is aided

by introducing intermediate events (extending the conversation),

the childrens' sleepwear problem this is done by considering for

two age groups and three income levels the probability tree with

alternatives (a), technology (T) , nightwear (N) , additional cost

consumer (C) , Use (U) or non-use (u) ; existence of a hazard (H)

,

sure (E) , ignition (I), burn (B) and burn-severity (B^) , survival

or death (D) and body image (1^).

For

each of

branchejs

to the

expo-

(S)

Attention is given to the preliminary assignment of each of the conditional probabilities needed. Suggestions are made as to sources of

information. Much of the needed information is not available, especially

that dealing with the social and behavioral aspects of the problem. The

concept of exposure to an ignition hazard, for which an operational definition does not exist, is discussed.

17. KEY WORDS (Alphabetical order, separated by semicolons)

Children's Sleepwear; decision analysis; flammable fabrics; probability

assessment; standards; utility theory

18. AVAILABILITY STATEMENT

UNLIMITED.

I I FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE

TO NTIS.

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