D I- ILLGA~

ARI Research Note 87-42

0

00

INFORMATION PROCESSING ORGANIZATIONS WITH

00 ACYCLICAL INFORMATION STRUCTURES 00

Alexander H. Levis, I.M. Tomovic, and P.H. Cothier

Massachusetts Institute of Technology

< TI

for

Contracting Officer' s Representative 1 M

George.H. Lawrence NOV

r D

BASIC RESEARCH LABORATORY

Michael Kaplan, Director

U. S. Army

Research Institute for the Behavioral and Social Sciences

October 1987

ADoroved for Dublic release; distribution unlimited.

87 11 13 05
U. S. ARMY RESEARCH INSTITUTE

FOR THE BEHAVIORAL AND SOCIAL SCIENCES

A Field Operating Agency under the Jurisdiction of the

Deputy Chief of Staff for Personnel

* WM. DARRYL HENDERSON

EDGAR M. JOHNSON COL, [N

Technical Director Commanding

Research accomplished under contract

for the Department of the Army

Massachusetts Institute of Technology

Technical review by

[-~I~ For k

Dan Ragland 7 +77

.. . . .. . . .

OTIC

copyt

INSPECTED

IThis report. as submitted by the contractor. has been cleared fee release to Defense Technical information Ctet I(OTICI to comply with requilorV requirements. It hat been given no primary distribution other then to OTIC

and will be available only thremugh OTIC or other reference services such as the National Technocal information IService (NTIS). The vitws. c-nit-na. and/or findings containted in this report art those of the authorisi "n

should not be consirus,% at an off ocia; Cmepaitmenr of the Army position, Policy. or decision. unless so designated

by other official documeintaion.

0z
UNCLASSIFIED

SECURITY CLASSIFICATION OF THIS PAGE (*%en Date Entered)

REPORT DOCUMENTATION PAGE READ INSTRUCTIONS

BEFORE COMPLETING FORM

1. REPORT NUMBER 2. GOVT ACCESSION No. 3. RECIPIENT'S CATALOG NUMBER

4. TITLE (end Subtitle) 5. TYPE OF REPORT & PERIOD COVERED

Information Processing Organizations with Final Report

Acyclical Information Structures April 1983 - October 1986

6. PERFORMING ORG. REPORT NUMBER

LIDS-FR-1703

7. AUTHOR(a) 8. CONTRACT OR GRANT NUMBER(&)

Alexander H. Levis, M.M. Tomovic and P.H. Cothier MDA903-83-C-196

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASK

Laboratory for Information and Decision Systems AREA & WORK UNIT NUMBERS

Massachusetts Institute of Technology 2Q161102B74F

Cambridge, Massachusetts, 02139

tI. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

U.S. Army Research Institute for the Behavioral October 1987

and Social Sciences, 5001 Eisenhower Avenue, 13. NUMBEROF PAGES

Alexandria, VA 22-33-5600 65

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Unclassified

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Approved for public release; distribution unlimited.

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IS. SUPPLEMENTARY NOTES

George H. Lawrence, contracting officer's representative

19. KEY WORDS (Continue on reverse side if necery and Identify by block number)

Decisionmaking

Organization Theory

Measures of Effectiveness

20 AwrRACT (CirntJae - -- de If newmr-awy a Identify by block number)

Information theory models were used to analyze the effects of different

perceptions of uncertainty and the relative value of tasks held by individual

decision makers on organizational performance. The definition and evaluation

timeliness as a measure of the effectiveness of command and control systems

was also examined.

FOrD 1473 EInTOW OF I NOV 6S IS OBSOLETE

DX I JAN 73 UNCLASSIFIED

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I .p-l 79. IW7 i I
TABLE OF CONTENTS

Page

LIST OF FIGURES ....................................................... 3

EXECUTIVE SUMMARY ....................................................... 5

1. INTRODUCTION ........................................................ 6

2. THE INTERACTING DECISIONMAKER ..................................... 8

2.1 Petri Nets ............ ........ ............ 8

2.2 Information Theory .......................................... 9

2.3 Task Model ................................................... 11

2.4 The Decisionmaker Model ...................................... 14

2.5 Workload ..................................................... 16

2.6 Measures ofPerf'ormance .................. %...................18s

2.7 Performance-Workload Locus .................. 19

3. THE DESIGNER'S AND THE DECISIONMAKER'S POINTS OF VIEW ............. 25

3.1 The Organization Model ... .................. ... 25

3.2 Workload and Performance ..................................... 26

3.3 The Performance-Workload Locus ............... 28

3.4 Two Points of View ............. ........... 30

3.5 Two Three-Person Organizations . .......... 31

3.6 Results . ................................. 35

3.7 Conclusion ........ ...................................... . . 39

4. MEASURES OF EFFECTIVENESS IN COMMAND AND CONTROL .................. 40

4.2 System and Mission ............. ............ ................. 43

4 .3 System and Mission Models ................. .. .... .......... 46

4.4 System and Mission Loci .................................... 51

.5 Measures of Effectiveness ............. ... o. ............ 53

4.6 Effectiveness Analysis and Comparison of Doctrines ........... 55 , 4.7 Conclusion .......................... 58

* 5. CONCLUSIONS ................. ..... ................. *............... 60

5.1 Organizations ......................... 60

35.2 Measures of Effectiveness .... ..................... ... ....... 61

5.3 P blications ...... o.................................... .... 61

6. REFERENCES ....... o.........................................,. ..... 63
LIST OF FIGURES

2.1 A Simple Petri Net ............................................... 9

2.2 Information Structures for Organizations ......................... 13

2.3 Two-Stage Model ................................................. 14

2.4 The Interacting Decisionmaker with Memory ........................ 14

2.5 The Memoryless Interacting Decisionmaker Model ................... 15

2.6 The SA and the RS Subnets ........................................ 16

2.7 Performance Evaluation of an Organization ........................ 18

2.8 The Petri Net for the Example .................................... 21

2.9 Strategy Space for Example ....................................... 22

2.10 Performance-Workload Locus for Example ........................... 22

2.11 Performance-Workload Locus for the Case of Three Pure Strategies . 23

2.12 Strategy Space for the Case of Three Pure Strategies ............. 24

3.1 Three Sectors (Parallel Organization) ............................ 32

3.2 Organization A: Parallel Structure .............................. 32

3.3 Two Sectors (Hierarchical Organization) .......................... 33

3.4 Organization B: Hierarchical Structure ....................... 34

3.5 Consistency Measure of Q for Organization A: p(x) ............... 36

3.6 Consistency Measure of Q for Organization A: q(x) ............... 36

3.7 Consistency Measure of Q for Organization B: p(x) ............... 37

3.8 Consistency Measure of Q for Organization B: q(x) ............... 37

3
4.1 System, Environment and Context .................................. 42

4.2 Fire Support System Structure .................................... 44

4.3 Scenario ............. ........................................... 45

4.4 Geometric Relations Between System and Threat .................... 47

4.5 Time Profile of the System Response ............................. 48

4.6 Single Shot Kill Probability as a Function of Impact Time ........ 49

4.7 Tree Representing System's Operational States .................. 50

4.8 Mapping of the System Parameters into the System MOPs ........... 52

4.9 Doctrine 1 (LSSS) System and Mission Loci ........................ 56

4.10 Doctrine 2 (LSLS) System and Mission Loci ........................ 56

4.11 Option 1 (immediate fire without coordination)

System and Mission Loci ........................................ 57

4.12 Option 2 (wait and coordinate) System and Mission Loci ........... 58

4

T4
EXECUTIVE SUMMARY

The goal of the research was the development of a mathematical theory of

task-oriented organizations, where the tasks are primarily information

processing and decision making.

Information theoretic models were used to analyze the effect of different

perceptions of uncertainty and relative value of tasks by individual decision

makers on organizational performance. The definition and evaluation of

timeliness as a measure of effectiveness of command and control systems was

also addressed.

A measure of organizational effectiveness was developed for the case in

which systems designer and decision maker differ in their perception of the

uncertainty associated with inputs or tasks. This measure was defined as the

ratio of strategies that yield satisficing performance to the total admissible

strategies. A hierarchical and a parallel three-person organization were used

to illustrate the results. The complexities of assessing the timeliness of C3

• systems were ad4dressed through the extension of the system effectiveness

methodology to consider "nt only the structure and protocol of a C3 system,

but also the effect of doctrine on measures of effectiveness. Again, the

theoretical development were illustrated through application to a generic C,

system.

The overall research effort will contribute to methodologies for the

analysis, design, and evaluation of command and control systems that support

tactical Army organizations.

02

'-I v -,, '',-',- < " " , I , , " .. .
1. INTRODUCTION

When information processing and decision making tasks, which have to be

performed, exceed the capabilities of a single decision maker, organizations

are formed. The functions that each organization member is allocated, the

interactions among the members, and the protocols that govern these

interactions constitute the organizational form or structure. This structure,

in turn, affects the individual member's workload, as well as the performance

of the organization as a whole.

The point of view taken in this report is that of the organization

designer. Given a set of tasks, the designer develops an organizational

structure and then evaluates it to determine if it meets a set of performance

requirements. If it does, then the next step is to assess the sensitivity of

the design to the assumptions. To carry out the assessments, it is necessary

that a methodology be developed and that measures of effectiveness,

appropriate for the problem at hand, be developed. p.

Both of these steps are addressed in this report. They are imbedded in a

broader framework, that of the development of an analytical methodology for

the design and evaluation of command and control organizations. Consequently,

for completeness, elements of that methodology, which form the basis for the

specific results obtained in this investigation, will be presented. First, in

Chapter 2, the basic model of the interacting decisionmaker, developed by

Boettcher and Levis [1], [2], [3] will be introduced and its properties

described.

In the following chapter, the model of organizations performing demanding

tasks under time contraints is presented and the methodology for evaluating

them is given in some detail, since it is the focus of the research carried

out under this contract. In developing a design, the designer assumes that he

knows the uncertainty that characterizes the tasks to be performed by the

organization and that he also knows the value of each task. He also assumes

that the organization members, in processing the information and making

decisions, have identical knowledge of task uncertainty and perception of task

value. In general, it is very difficult to assess the probability

distribution of the tasks; it is also quite unlikely that the designer and the

actual members of the organization have the same perception of the tasks'

probability distribution. In this chapter, this assumption is relaxed: it is

assumed that the designer knows the tasks' real probability distribution,

while the organization members have a different perception of this

distribution. The second assumptions about common perception of task value is

also improbable: the decisionmakers in the organization may assign different

value to the various tasks from the ones the designer assumed. The relaxation

of this assumption is also presented in Chapter 3.

In order to pose both problems properly, two additional assumptions must

be made. The first is trivially true in practice, but needs to be made

explicit in the mathematical analysis: there is no communication between the

designer and the organization members. The second assuption is that the

6
designer knows both the true task uncertainty and the uncertainty as perceived

by each of the organization members, as well as both the true value of the

tasks and the value assigned to the tasks by each decisionmaker. This

information is needed in orcer to carry out sensitivity analyses. The

theoretical development is then applied to two three-person organizations, a

hierarchical and a parallel one, in order to illustrate the sensitivity

analysis and its use in selecting organizational designs.

The same effectiveness analysis methodology, but with different measures

of performance, can be applied to the evaluation of the architecture of

command and control systems. Such an application to a model abstracted from

an actual US Army system is presented in Chapter 4.

Finally, in the concluding chapter, the results are summarized and

suggestions for further research are made.

7
7

2. THE INTERACTING DECISIONMAKER MODEL

The first step in modeling an organizational structure is the modeling of

the tasks to be performed by the organization. The second step is to develop

an appropriate mathematical model of the organization member. Specifically,

*: this model must incorporate provisions for the variety of interactions that

can exist among decisionmakers in an organization. These two steps are

discussed in this chapter. In addition, the necessary analytical tools are

introduced, namely, Petri Nets and N-dimensional information theory. The

former is used to describe, rather precisely, the architecture of the

*. decisionmaking model, and of the organizations, while the latter is used to

model the cognitive workload of the individual decisionmakers.

2.1 PETRI NETS

In this work, only the basic properties of Petri Nets are needed to

describe the models. In related work for the Office of Naval Research and for

the Technical Panel on C3 of the Joint Directors of Laboratories, several

measures of performance (MOPs) of organizations have been obtained using some

more advanced concepts from Petri Net theory [4,5]. For an introductory

* treatment of Petri Nets as modeling tools, the text by Peterson [6] is

recommended.

Petri Nets are bipartite directed graphs reprsented by a quadruple

(P,T,IO). By convetion, P is the set of one type of nodes, called places or

circle nodes, and T is the set of the second type of nodes, called transitions

or bar nodes. Places can depict the presence of signals or represent

conditions; transitions can depict processes or events. Consequently, the

arcs that connect the nodes that form the graph can only go from one type of

node to another - either from a place to transitions, or from a transition to

places. The mapping I corresponds to the set of directed arcs from places to

transitions, i.e., it defines the input places of the transitions, while the

mapping 0 corresponds to the set of directed arcs from transitions to places;

i.e., it defines the output places of each transition. For ordinary Petri

Nets - the only type considered here - the mappings I and 0 take values from

the closed set (0,1); 1 denotes the presence of a link between two nodes,

*while 0 denotes the absence.

A Petri Net consisting of four transitions and five places is shown in

Figure 2.1. Tokens, denoted by dots in places or circle nodes, control the

execution of a Petri Net. A marking of a Petri Net is a mapping which assigns

a non-negative integer number of tokens to each place of the net. Since the

number of tokens in a place, in general, is not bounded, there can be an

*infinite number of markings associated with each net. A Petri Net is said to

execute when a transition fires. A transition can fire, only if it is

enabled. For a transition to be enabled, all its input places must contain at

least one token each. When a transition fires, it removes one token from each

input place and creates a new token in each of the output places of that

transition. One can envision a sequence of firings in the Petri Net of

Figure 2.1: Let the initial marking consist of a token in tne first (left

8
most) place. Then the first transition is enabled and it fires. The token in

the first place is removed and a token appears in the second place. Now the

second transition is enabled: it fires and the token is removed from the

second place; a new one appears in the third place, and so on. The execution

halts when the fourth transition fires and a token appears on the fifth place.

Figure 2.1. A Simple Petri Net

A transition may have more than one output places. When it fires, a

token is generated in each output place. However, to model decision making,

it is convenient to introduce a special transition, a decision switch, in

which the output places represent alternatives. When the decision switch

fires, a token is generated in only one of the output places. A decision rule

associated with this special transition determines the place in which the

* token is generated. The rule can be deterministic or stochastic; it can be

independent of the attributes of the tokens in the input places or it may

depend on them.

A subnet of a Petri Net PN is a Petri Net PNs with places P. that are a

subset of the places P of the original net and transitions T. that are a

subset of the transitions T of the original net. The input and output

mappings, Is and Os , are restricted to the arcs between the subsets T. and Ps.

The use of subnets simplifies the graphical representation of complex

organizations and allows the depiction of the decisionmaker model at a level

of detail appropriate to the problem being solved.

2.2 INFORMATION THEORY

Information theory was first developed as an application in communication

theory [7]. But. as Khinchin [8] showed, it is also a valid mathematical

theory in its own right, and it is useful for applications in many

disciplines, including the modeling of simple human decisionmaking processes

6 and the analysis of information-processing systems.

There are two quantities of primary interest in information theory. The

first of these is entropy: given a variable x, which is an element of the

alphabet X, and occurs with probability p(x), the entropy of x, H(x), is ~defined to be

H(x) * - p(x) log p(x) (2.1)

x

and is measured in bits when the base of the algorithm is two. The other

04 9
a.

quantity of inte2rest is average mutual information or transmission: given two

variables x and y, elements of the alphabets X and Y, and given p(x). p(y) ,

and p(xly) (the conditional probability of x, given the value of y), the

transmission between x and y, T(x:y) is defined to be

T(x:y) H(x) - H yx) (2.2)

where

Hy(x) = - 2 p(y) 2 p(xly) log p(xly) (2.3)

y x

is the conditional uncertainty in the variable x, given full knowledge of the

value of the variable y.

*@ McGill [9] generalized this basic two-variable input-output theory to N

dimensions by extending Eq. (2.2):

N

T(x :x,:... :x) H(xi) - H(x ,x2 ... x N ) (2.4)

~i=.

For the modeling of memory and of sequential inputs which are dependent

on each other, the use of the entropy rate, H(x), which describes the average

entropy of x per unit time, is appropriate:

H(x) lim I H[x(t), x(t+1),..,x(t+m-1)] (2.5) ' ' m- - m

The transmission rate, T(x:y), is defined exactly like transmission, but using

entropy rate in the definition rather than entropy.

The Partition Law of Information [10] is defined for a system with N-i

internal variables, w, through WN_ ,, and an output variable, y, also called

wN. The law states

N

H(w ) = T(x:y) + T (x:ww * + T(w :...:w :y) i- y N-i N': i=1

+ Hx (w,,w ,...,w N_1,y) (2.6)

010
and is easily derived using information theoretic identities. The left-hand

side of Eq. (2.6) refers to the total activity of the system, also designated

by G. Each of the quantities on the right-hano side has its own

interpretation. The first term, T(x:y), is called throughput and is

designated Gt. It measures the amount by which the output of the system is

* related to the input. The second quantity,

• Ty(x:wW , ..,wN_) N- = T(x:w ,w 2' .... w n-l',y) - T(x:y) (2.7)

is called blockage and is designated Gb. Blockage may be thought of as the

amount of information in the input to the system that is not included in the

output. The third term, T(wi:wz:...:wN_,:y) is called coordination and is

designated Gc. It is the N-dimensional transmission of the system, i.e., the

amount by which all of the internal variables in the system constrain each

other. The last term, Hx(W±,WSO.... , designated by Gn represents the

uncertainty that remains in the system variables when the input is completely

known. This noise should not be construed to be necessarily undesirable, as

it is in communication theory: it may also be thought of as internallygenerated information supqled by the system to supplement the input and

facilitate the decisionmaking process. The partition law may be abbreviated:

G=Gt + Gb + G +G (2.8)

A statement completely analogous to (2.8) can be made about information

rates by substituting entropy rate and transmission rates in Eq. (2.6).

2.3 TASK MODEL

The organization, interacts with its environment; it receives signals or

messages in various forms that contain information relevant to the

organization's tasks. These messages must be identified, analyzed, and

transmitted to their appropriate destinations within the organization. From

this perspective, the organization acts as an information user.

Let the organization receive data from one or more sources (N') external

to it. Every vn units of time on the average, each source n generates

symbols, signals, or messages xni from its associated alphabet Xni with

probability Pni' i.e.,

S-P(X=X) Xni Xn i = 1,2 ....,y (2.9)

*1 ix
Yn

= 1 n= 1,2 N' (2.10)

where In is the dimension of xn . Therefore, 1/.n is the mean frequency of

symbol generation from source n.

The organization's task is defined as the processing of the input symbols

xn to produce output symbols. This definition implies that the organization

designer knows a priori the set of desired responses Y and, furthermore, has a

function or table L(x n) that associates a desired response or a set of desired

responses, elements of Y, to each input Xni e C Xn

It is assumed that a specific complex task that must be performed can be

modeled by N' sources of data. Rather than considering these sources

separately, one supersource, composed of these N' sources, is created. The

input symbol x' may be represented by an N'-dimensional vector with each

source corresponding to a component of this vector, i.e.,

xv' (x,,x, .... X) ; x' a X 2.11)

To determine the probability that symbol x' is generated, the

independence between components must be considered. --If all components are

mutually independent, then pj is the product of the probabilities that each

component of x' takes on its respective value from its associated alphabet.

If two or more components are probabilistically dependent on each other, but

as a group are mutually independent from all other components of the

N'

pj FJpnj (2.12)

input vector, then these dependent components can be treated as one

supercomponent, with a new alphabet. Then a new input vector, x, is defined,

composed of the mutually independent components and these super-components.

This model of the sources implies synchronization between the generation

of the individual source elements so that they may, in fact, be treated as one

input symbol. Specifically, it is assumed that the mean interarrival time Tn

for each component is equal to T. It is also assumed that the generation of a

particular input vector, xj, is independent of the symbols generated prior to

or after it.

The last assumption can be weakened, if the source is a discrete

stationary ergodic one with constant interarrival time v that could be

approximated by a Markov source. Then the information theoretic framework can

be retained [11).

12

.. . . . + .. . ...
The vector output of the source is partitioned into groups of components

that are assigned to different organization members. The j-th partition Is

denoted by xJ and is derived from the corresponding partition matrix _J which

has dimension nj x N and rank nj, i.e.,

x = RJ x. (2.13)

Each column of nJ has at most one non-zero element. The resulting vectors xj

may have some, all, or no components in common.

The set of partitioning matrices [n{1 ,7r ...,1n} shown in Figure 2.2

specify the components of the input vector received by each member of the

subset of decisionmakers that interact directly with the organization's

environment. These assignments can be time invariant or time varying. In the

latter case, the partition matrix can be expressed as

*J for t eT

iJ(t) = (2.14)

0 fort (TI

The times [T) at which a decisionmaker receives inputs for processing can be

obtained either through a deterministic (e.g., periodic) or a stochastic rule.

The question of how to select the set of partition matrices, i.e., design the

information structure between the environment and the organization, has been

addressed by Stabile [12,13].

ORGANIZATION

0.

• . Figure 2.2. Information Structures for Organizations

13
2.4 THE DECISIONMAKER MODEL

The basic model of the memoryless decisionmaker with bounded rationality

is based on the hypothesis of F. C. Donders [14] that information processing

is done in stages. Specifically, it is assumed that the two stages are (a)

situation assessment (SA), and (b) response selection (RS), which correspond

to March and Simon's [15] two stage process of discovery and selection. The

structure of this model, shown in Figure 2.3, has been extended to include

interactions with other organization members, as well as memory. The extended

model is shown in Figure 2.4.

. v SAZ -- p RS --

Figure 2.3 Two-Stage Model

ZV

Figure 2.4 The Interacting Decisionmaker with Memory

~The DM receives singals x £ X froL. the environment with interarrival time

r. A string of signals may be stored first in a buffer so that they can be *1

I~ 74

" ",; 9,!]E ;'',w r r dj Do! DI'+
processed together in the situation assessment (SA) stage. The SA stage

contains algorithms that process the incoming signals to obtain the assessed

situation z. The SA stage may access the memory or internal data base to

obtain a set of values do . The assessed situation z may be shared with other

organization members; concurrently, the DM may receive the supplementary

situation assessment z' from other parts of the organization; the two sets z

and z' are combined in the information fusion (IF) processing stage to obtain

2. Some of the data (dI ) from the IF process may be stored in memory.

The possibility of receiving commands from other organization members is

modeled by the variable v'. A command interpretation (CI) stage of processing

is necessary to combine the situation assessment 2 and v' to arrive at the

choice V of the appropriate strategy to use in the response selection (RS)

stage. The RS stage contains algorithms that produce outputs y in response to

the situation assessment 2 and the command inputs. The RS stage may access

data from, or store data in memory [11].

In this report, only the memoryless case is considered. Consequently,

the general model reduces to the one shown Figure 2.5, where the Petri Net

formalism has been used.

x SA IF CI PS

Z'b d 6V.

Figure 2.5. The Memoryless Interacting Decisionmaker Model

A more detailed description of the model is obtained, if the internal

structure of the SA and RS stages is considered. The situation assessment

stage consists of a set of U algorithms (deterministic or not) that are

capable of producing some situation assessment z. The choice of algorithms is

achieved through specification of the internal variable u in accordance with

the situation assessment strategy p(u), or p(ulx), if a decision aid (e.g., a

preprocessor) is present. A second internal decision is the selection of the

algorithm in the RS stage according to the response selection strategy

p(VIZ,v'). The two strategies, when taken together, constitute the internal

decision strategy of the decisionmaker.

The subnets representing the SA and the RS stages are shown in Figure

2.6. Note the presence of decision switches in place of the regular

transitions to indicate that only one of the output places can receive a token

at each firing.

15
Z ) V

SA IF CI RS

0

Figure 2.6. The SA and the RS Subnets

2.5 WORKLOAD

The analytical framework presented in Section 2.2, when applied to the

single interacting decisionmaker with deterministic algorithms in the SA and

RS stages, yields the four aggregate quantities that characterize the

information processing and decisionmaking activity within the DM:

Throughput:

Gt = T(x,z',v':z,y) (2.15)

Blockage:

Gb = H(x,z',v') - Gt (2.16)

Internally generated information:

Gn = H(u) - H..(v) (2.17)

Coordination:

U

G= pig'(p(x)) + ai H(pi) + H(z) + gI (p(z.z')) + gC (P(ZV,)

@,'a i=1

V

+ p g (p(ZIV - J)) + aj H(p) + H(y) + H(z) + H(M) + H(IoV)

j=1

+ T (xl:z') + T (xlz':v') (2.18)

16
The expression for Gn shows that it depends on the two internal

strategies p(u) and p(vI!) even though a command input may exist. This

implies that the command input v' modifies the DM's internal decision after

p(vI2) has been determined.

In the expressions defining the system coordination, Pi is the

probability that algorithm fi has been selected for processing the input x and

Pj is the probability that algorithm h has been selected, i.e., u = i and

V=J. The quantities g. represent the internal coordinations of the

corresponding algorithms and depend on the probability distribution of their

respective inputs; the quantities ai , aj are the number of internal variables

of the algorithms fi and hj, respectively. Finally, the quantity H is the

entropy of a binary random' variable that takes one of its two values with

probability p.

H(p) = - p log 2 p - (1 - p)log 2 (1-p) (2.19)

I

Equations (2.15) to (2.18).determine the total activity G of the decisionmaker

according to the partition law of information, Eq. (2.6). The activity G can

Ube evaluated alternatively as the sum of the marginal uncertainties of each

system variable. For any given internal decision strategy. G and its component

parts can be computed.

Since the quantity G may be interpreted as the total information

*" processing activity of the system, it can serve as a surrogate for the

*: workload of the organization member in carrying out his decisionmaking task.

The qualitative notion that the rationality of a human decisionmaker is

not perfect, but is bounded [16], has been modeled as a constraint on the

total activity G. The specific form for the constraint has been suggested by

the empirical relation

t - c + cG t

* where t is the average reaction time, i.e., the time between the arrival of

* the input and the generation of an output y. It is assumed that the

decisionmaker must process his inputs at a rate that is at least equal to the

rate with which inputs arrive. The latter has been modeled by v, the mean

symbol interarrival time:

t-c +cGt <-

or

17
1 j

- t=- +G <-

c = c - c2

The modeling assumptions in this paper are that

c

2 Gb + G

and that c2 does not depend on p(x). Then, the bounded rationality constraint

takes the form

G=G +G + G G _ = F (2.20) t b n c c 2

where F can be considered as a rate of total activity and is measured in bits

per second. Inequality (2.20) represents a mathematical expression of only

one aspect of bounded rationality. Many other formulations are possible.

Weakening the assumption that the algorithms are deterministic changes

the numerical values of Gn and of the coordination term Ga [17]. If memory is

present in the model, then additional terms appear in the expressions for the

coordination rate and for the internally generated information rate [11].

2.6 MEASURES OF PERFORMANCE

As stated in Section 2.3, it is assumed that the designer knows a priori

the set of desired responses Y to the input set X. One measure of performance

(MOP) of the organization that reflects the degree to which the actual

response matches the desired response can be computed as shown in Figure 2.7.

X ORGANIZATION

Figure 2.7. Performance Evaluation of an Organization

18
The decisionmaker's actual response y can be compared to the desired

response y' and a cost is assigned using the cost function d(y,y'). If this : function is a binary one, i.e.,

/( 0 if y =y'

d(y,y') = (2.21)

( 1 if y #y

then the expected value of this cost denotes the probability that the wrong

decision is made, i.e., it is the probability of error.

In general, however, there is a cost cij associated with selecting Yi e

Y when the desired response is yj a Y':

C = d(yy) (2.22)

so that

J- p(xj) ciip(yilx ) (2.23)

I i

where y is the desired response to task xj. This measure of performance can

be inte-preted as a measure of the accuracy of the response, to the extent

that a cost is associated with the degree with which the actual decision

deviates from the desired one.

This class of performance measures, described generically by (2.23), is

'o not the only one that has been considered. In related work [18], measures of

performance that address time have been modeled and analyzed.

2.7 PERFORMANCE-WORKLOAD LOCUS

A useful way of describing the properties of the decisionmaker model,

which is generalizable to the properties of an organization, is through the

performance workload locus. In the case of a single performance measure, the

accuracy measure J, and a single decisionmaker with workload G, a two

dimensional space is defined with ordinate J and abscissa G. The locus is

constructed by considering the functional dependence of J and G on the

internal decision strategies of the single decisionmaker.

Let an internal strategy for a given decisionmaker be defined as pure, if

both the situation assessment strategy p(u) and the response selection

strategy p(vII) are pure, i.e., an algorithm f is selected with probability

one and an algorithm h is selected also with probability one when the

situation is assessed as being r:

w E " Hil Il l iii il l i liI
Dk = (p(u=i) = 1 ; p(v=JI2=) = 11 (2.24)

for some i, some J, and for each 2 element of the alphabet Z. There are n

possible pure internal strategies,

n = UVM (2.25)

where U is the number of f algorithms in the SA stage, V the number of h

algorithm in the RS stage and M the dimension of the set Z. All other

internal strategies are mixed [1] and are obtained as convex combinations of

pure strategies:

n

* D(pk) = Pk Dk (2.26)

k=l

where the weighting coefficients are probabilities.

Corresponding to each D(pk) is a point in the simplex

n

Pk 1, pk > 0 Yk (2.27)

k=1

* The possible strategies for an individual DM are elements of a closed convex

polyhedron of dimension n-1 whose vertices are the unit vectors corresponding

to pure strategies.

The total activity G, the surrogate for the cognitive workload, is a

convex function of the decision strategy, i.e.,

n

G (D(pk)) Pk Gk (2.28)

k=1

where Gk is the workload that results when the pure strategy Dk, given by Eq.

(2.24), is used.

The accuracy measure J can be related to the decision strategies in a

similar manner. Corresponding to each pure strategy Dk is a value of the

performance measure, denoted by Jk" Since each strategy is a convex

ni
combination of pure strategies, the value of J for an arbitrary D(Pk) is given

as a convex combination of the values of Jk' i.e.,

n

J(D(p ) = p k k (2.29)

k=l

The two expressions, (2.28) and (2.29) can be used now to characterize the

locus of points in the (J,G) space that describe the decisionmaker.

Example: Consider first the case of two pure strategies, D. and D2 . This

would correspond to the case where the decisionmaker can choose only between

two different algorithms f in the SA stage, as shown in Figure 2.8. The

-strategy space for this case can be parametrized as follows: Any strategy, D,

*. can be expressed as

D = P1 D +p P D 2 (2.30)

where p, +p 1 =1

in accordance with (2.26) and (2.27). Let

p 1 -8 and p1 :6

and let

Then, (2.30) can be rewritten as

D = (1-6) D + 6D (2.31)

f 2 EF CI RS Y AS

Figure 2.8. The Petri Net for the Example

21
The strategy space can be described by the parameter 6: it is the line

segment [0,1], as shown in Figure 2.9. with the point 0 corresponding to pure

strategy D,, point 1 to pure strategy D., and all points in between to all the

mixed strategies.

D D2

0 1

Figure 2.9. Strategy Space for Example

Then, it follows from (2.28) and (2.29) that

G(D(pk)) = G(D(6)) (1-6)G + 6 G (2.32)

and

J(D(Pk)) = J(D(5)) = (1-6)J + & (2.33)

Equations (2.32) and (2.33) are parametric in 6 and result in the locus shown

in Figure 2.10. The relative position of the end points (J,,G,) and (J,,G,)

is problem specific; it is not true that smaller workload leads to worse

performance, as Figure 2.10 indicates.

,%i

• ' J28=1

S 2

* 0

U.

u j =0

'4G G2 1 G2

WORKLOAD

Figure 2.10. Performance-Workload Locus for Example

22
-

In the general case, there are n pure strategies, as given by Eq. (2.25).

Then, the P-W locus is constructed as follows:

First, the values of (Ji,Gi) for the n pure strategies are determined.

This corresponds to evaluating the performance and the workload for the values

" of Pk' Eq. (2.27), that correspond to the vertices of the strategy space. The

result is a set of n points in the two-dimensional P-W space.

Then, the binary variations between each possible pair of pure strategies

are considered. This corresponds to the mapping of the edges of the strategy

space. For example, consider pure strategies Di and Dk1 then

D = (1-6)D i + 6Dk

for all combinations (i,k) where i=,...,n and k=l,...,n and for which i~k.

" By varying 6 from 0 to 1. the loci (Jik(b)' Gik(b)) are obtained. These are

convex lines joining the two boundary points, as shown in Figure 2.10. These

binary loci are quite useful, since they define the minimum workload locus for

any feasible value of J.

The third step consists of considering, successively, the binary

variation between all possible binary strategies until all mixed strategies

are accounted for. The result is a locus such as the one shown in Figure 2.11

for the case when there are three pure strategies. The corresponding strategy

space, for this case, is shown in Figure 2.12.

W

z J

2 Ic

G G G 1 2 3

WORKLOAD

Figure 2.11. Performance-Workload Locus for the Case of Three Pure Strategies

23

ICLM
r .2

P 3

Figure 2.12. Strategy Space for the Case of Three Pure Strategies

Thus, the decisionmaker model can be considered as a system that maps the

strategy locus, the simplex defined by Eq. (2.27), into the PerformanceWorkload Locus (J,G). Any change in the algorithms f or h, or the functions

in IF and CI, or the input x will affect the mapping.

. Any specific instantiation of the decisionmaker model can be analyzed by

considering the strategy space and the corresponding P-W locus. The detailed

procedure for doing both a qualitative and a quantitative evaluation will be

presented in the next chapter.

2

.524
3. THE DESIGNER'S AND THE DECISIONMAKERS'S POINTS OF VIEW

When the tasks which have to be performed exceed the capabilities of a

single person, organizations are formed, consisting of individuals interacting

with each other in specific ways. The most important characteristic of the

organizations considered here is that their task involves information

" processing and decisionmaking.

In earlier work [1-3], several simplifying assymptions has been made in

the design and analysis of decisionmaking organizations: (a) designers and

decisionmakers have identical knowledge of the tasks' uncertainty, i.e., the

probability distribution of the tasks and (b) identical perception of the

value of each task. These assumptions are very restrictive and often

unjustifiable. In general, it is very difficult to assess the probability

distribution of the tasks; it is also unlikely that the designer and the

decisionmakers have the same perception of the tasks' probability

distribution. In this chapter, this assumption is relaxed.

It is assumed that the designer knows the tasks' real probability

distribution, while the decisionmaker's peception of this distribution is

different. The second assumption, that all tasks are of equal value, is also

improbable; usually, different tasks have different utilities, i.e., different

,.. weights are assigned to them by the designer. Therefore, the second

assumption must be weakened, so that each task can be weighted differently.

In order to pose these problems properly, two additional assumptions must be

introduced: (a) there is no communication between the designer and the

decisionmakers and (b) the designer knows the tasks' uncertainty as perceived

by each decisionmaker in the organization, as well as the relative weights

assigned by them to each task.

3.1 THE ORGANIZATION MODEL

The task consists of receiving data (signals), processing that data, and

producing an output in the form of actions or signals. The input data can

originate from a single source or from many different sources as described in

the previous chapter. The data may be a single element or a set of elements.

In general, it is modeled as a vector, x, generated by a single source. This

vector signal is partitioned by partitioning matrices x and allocated to the

appropriate decisionmakers. A task can be specified fully by its finite

scheme, which consist of the task's alphabet and its probability distribution,

i.e.,

X = (X, P) = I P* .. m (3.1)

*This section is based on the work of M. M. Tomovic [181; the computational

results have been revised recently by J. Azzola.

25
The basic model of the memoryless decisionmaker introduced in the

previous chapter will be used. It is assumed that the overall task x is

partitioned by the matrix nn and that only the appropriate elements of x are

allocated to the n-th decisionmaker. The decisionmaker processes the data

using the algorithms in the SA stage in order to assess the situation. The

assessed situation, 2n, is then processed in the RS stage, where the decision

of an appropriate action or response, yn, is made. Which ones of the SA and

RS algorithms a decisionmaker will use depends on his choice of internal

decision strategy, Dn. For the situation assessment stage, it is assumed that

the strategy un is independent of the input xn, whereas in the response

selection stage vn depends on the value of the assessed situation Mn. THe

assumption that the choice of un is not dependent on xn has been relaxed by

Chyen and Levis [17].

In order to make communication and interaction between decisionmakers

possible, the information fusion (IF), and the command interpretation (CI)

elements have been included. The IF process allows sharing of the information

,II on the state of environment between decisionmakers. This functional element

associates information on the assessed situation obtained by the n-th

decisionmaker, zn, and the corresponding information sent to him from the rest

of organization, zon, and gives the cumulative updated information on the

state of environment 2n. It is also possible for the n-tb decisionmaker to

communicate his knowledge on the state of environment zno to other members of

the organization, who accept and fuse that information with their own in the

corresponding IF stage. Commands von received by the n-th decisionmaker from

the rest of the organization can modify or even override his own decision vn.

3.2 WORKLOAD AND PERFORMANCE

The concepts of decision stategies, workload, and performance introduced

in Chapter 2 for the single decisionmaker need to be generalized to apply to

organizations consisting of a number of decisionmakers operating as a team of

accomplish the given set of tasks. As before, the two fundamental quantities

are the workload or the activity level of each individual DM and the

performance index of the organization. Both of them are functions of the

input, the decisionmaker's internal structures, the organization's protocol or

standard operating procedures and the decision strategies. For a specified

input, a protocol and an internal structure (set of algorithms) both the

performance and the workload depend parametrically on the organizational

behavioral strategy which is defined as

{'p) DN(PN ) - .NpN(3.2)

A -iP ,.., p I" A&'..1N ~k- PKN k k

where Dn.pn) is the n-th decisionmaker's mixed decision strategy, and

AkX, .... denotes a pure strategy of the organization (see Levis and

Boettcher (2]).

26

. N I +" "j
The surrogate workload for each decisionmaker is defined as the activity

necessary to reduce uncertainty and arrive at a decision. For each DM it is

defined as

Gn = G (A) - ~ nlo ~ in (3.3)

• i j

where Win is the J-th value of the i-th internal variable of the n-th

decisionmaker. When the tasks' finite scheme X, organizational structure

(algorithms and protocols) and organization's behavioral strategy A, Eq.

(3.2), are specified, it is then possible to evaluate the activity of each

decisionmaker within the organization.

In has been shown in Chapter 2 that the total activity G is a convex

function of the decision strategy A in the sense that

G(A) 2 G(Ak ...kN) Pk'" PkN (3.4)

kN

where G(Ak1...kN) is the workload corresponding to the pure strategy.

The bounded rationality of each decisionmaker is modeled as a constraint

on his total activity

Gn (A) S Fn. (3.5)

where Fn is the maximum rate at which the n-th decisionmaker can process

information and v is the mean interarrival time of tasks.

The organization's performance is defined as

*J = E~d(y,y')) = p(x) i d(yiy?) P(Yjlxi) (3.6)

-%i j

where yj is the actual output of the organization as a whole in response to

the input xi and where yj is the designer. As in the case of a single

decisionmaker, the comparison function d(y,y') can take any form appropriate

to the particular problem; in its simplest form it is defined as

27

' "",IN
0 y y= d(y~y') =(3.7)

In that case, the performance index is reduced to the probability of producing

an incorrect response, i.e., the probability of making an error.

The performance measure has meaning only for the organization as a whole

and can be expressed as

J(A) J JL N k I .. p NN (3.8)

k' kN

to show its dependence on the choice of an organizational strategy. The

designer is the one who assigns the value of the performance threshold 0J)

which the decisionmaking organization has to meet, i.e.,

J (A) < J (3.9)

This condition determines the set of strategies that yield satisficing

performance.

3.3 THE PERFORMANCE-WORKLOAD LOCUS

For each decision strategy, the organization's performance and the

workload of each of the N decisionmakers can be computed using the algorithms

first developed by Boettcher [20], revised by M. Tomovic [19] and then

completely revised by Andreadakis [18] as part of the Computer-Aided

Evaluation of System Architectures (CEASAR) workstation. The performanceworkload locus,S, is Ndefined in the (N+1)-dimensional space S as the set of

* all points (J.G1,?..,GN) that correspond to the set of all admissible decision

strategies. Let I be the set of all admissible strategies of the

organization; then H = ~ ~(3.10)

where In is the set of admissible strategies of the n-th decisionmaker.

The bounded rationality contrainst for each DM, Eq. (3.5) is expressed

simply as a bounding hyperplane in the performance-workload space; the

satisficing constraint, Eq. (3.9), is also a bounding hyperplane that

intersects the J axis at J.

The resulting performance-workload locus can be used to compare

28

04

'M ,*v7
qualitatively two different organizational structures or designs that are to

perform the same task. It can also be used to compare the effectiveness of

the same organization in dealing with different degrees of uncertainty, i.e.,

for different probability distributions of the input alphabet X, the set of

tasks. This qualitative analysis is based on how well the locus So meets the

requirements of Eqs. (3.5) and (3.9).

A quantitative approach to the comparison and effectiveness analysis of

organizational forms is based on comparisons not in the performance-worklaod

space, but in the strategy space. Bounded rationality contraints, Eq. (3.5),

and performance requirements , Eq. (3.9), partition the space of organization

strategies into a set of strategies that lead the points in the performanceworkload locus that satisfy constraints Eqs. (3.5) and (3.9), and a set that

does not. The set of feasible strategies is defined as follows:

2 N N. = [AJ(A) < J, G'(A) < F T,...,GN(A) _ F-r} (3.11)

Then, a measure of effectiveness, Q, called the consistency measure, can be

defined [3]:

Q = V(y,)/V() (3.12)

where V is a volume in the N-dimensional strategy space. Therefore, Q denotes

the fraction of all strategies that are feasible. The higher it is, the more

consistent the decisions of the organization will be in the design

specifications. A second interpretation of the measure is the following: if

all organizational decision strategies are equally probable, then Q is the

probability that the organization will make a decision that satisfies the

individual bounded rationality constraints and leads to satisficing

performance. Hence, Q is a bounded non-decreasing function of the performance

threshold J and individual decisionmaker's activity threshold Fn ,

(n=l,...,N).

0 < Q(J,F'%, F2 .... FNr) < 1 (3.13)

The consistency measure Q is equal to zero, if there is no decision strategy

which meets the specifications of the task. Its value is equal to unity, if

all admissible decision strategies satisfy the requirements. It is evident

that the higher value of the mutual consistency measure Q, the better

* performance can be expected from that system. Therefore, the designer can

compare systems with respect to Q and select the one with the highest value of

Q for a given J and v.

A more useful representation of Q is in terms of the performance

threshold J and the mean interarrival time v. The resulting expression

29
0 5 Q(J,') 5 1 (3.14)

can be depicted in a three-dimensional space regardless of the size of the organization.

3.4 TWO POINTS OF VIEW

The basic premise in earlier work is that the designer knows the finite

scheme, Eq. (3.1), and the performance index and that he assumes the

decisionmakers in the organization will have the same knowledge. On that

basis, he can design an organizational form and evaluate it. However, the

question arises as to how robust the design is, i.e., how sentitive is the

value of Q to the assumption that the decisionmakers have indeed the same

knowledge. Suppose, that they have different perceptions of the tasks'

uncertainty and they assign different values to the inidividual tasks. The

inidivudal decisionmakers who only receive partial information about the

organization's tasks may have a different perception of the probability

distribution p(x). Furthermore, their local objectives may distort the values

of the weights assigned to each task by the designer who maintains a global

perspective.

The designer can adopt two points of view in order to study the

robustness of his design. First, he can assume that the DMs will operate on

the basis of his perception of task uncertainty, p(x), and objective function

J. Or, he can assume that the DMs will operate on the basis of a different

perception of task uncertainty, e.g., q(x) and that they will assign

different values, c(x), to the various tasks xI .

These two points of view lead to the formualtion of four problems that

the designers must analyze.

Problem 1: (Basic Problem) The DMs know the objective probability

distribution of the tasks, p(x), and the weighting coefficients for the

4various tasks. For simplicity, it is assumed that all the coefficients c(x)

are equal to unity for the base case. 0

7 Problem 2: (Task Uncertainty) The DMs have their own perception of the

probability distribution of the input, q(x), instead of p(x), but assign the

same values to the tasks as the designer (c(x) = 1).

Problem 3: (Task Value) The DMs know the objective probability distribution

to the tasks, p(x), but assign different values of the various tasks, i.e.,

their c(x) differs from the designer's (c(x) # 1).

Problem 4: (General Problem) The DMs have their own perception of the

probability distribution of the input, q(x), and the value of the tasks

(c(x)#i).

} 30
Problem 1 is the one that has been analyzed in detail in [1],[2], and [3]

while Problem 4 is the general case: Problems 1, 2, and 3 are really special

cases of 4. In this study, Problem 2 is analyzed because it addresses the

complex interrelationship between uncertainty and workload.

It is possible to develop the analytical expressions for the workload as

a function of p(x) and of q(x). It is also possible to introduce weighted

entropy [21], and derive the expressions for the difference in the workload Gn

due to the difference in the perception of the task uncertainty by the

designer and the decisionmakers. However, the analytical expressions have not

yielded any particular insight to the problem. Consequently, the sensitivity

analysis will be described in terms of an example.

3.5 TWO THREE-PERSON ORGANIZATIONS

"4 Two three-person organizations have been used to illustrate the analysis

and comparison of alternative organizational forms. This simple example has

been based on the problem of designing the organization of batteries of

4- surface-to-air missiles.

Let the trajectory of a target be defined by an ordered pair of points

located in a rectangle that represents a two-dimensional (flat) sector of

airspace. From the ordered pair, the speed and direction of flight of the

target can be determined. On the basis of that information, the organization

should respond by firing either a slow or a fast surface-to-air missile or by

not firing at all. It is assumed that the size of that sector, the frequency

of the arrival of targets, and the response time of the weapons systems are

such that three units are needed, i.e., it is necessary to design a three-

, member organization to accomplish the task. Two such organizational

structures are considered.

The first structure, Organization A, is a parallel structure with lateral

links, and is defined as follows. The rectangular sector is divided into

.. three equal subsectors and a decisionmaker is assigned to each one, see Figure

3.1. Each DM is capable of observing only the points that appear in his

" subsector. He can assess the situation, i.e., estimate the trajectory, and

select the response, i.e., which weapons to fire, for targets with

trajectories totally within his subsector. This is the case when two points

that define the target are within his subsector. Since it is possible for

trajectories to *straddle* the subsector boundaries, it is necessary that SA

information be shared. Thus, DM' and DM' share information that relates to

their common boundary. Similarly, DM' and DM' share information that relates

to targets that cross their common boundary. To keep the example

computationally simple, the situation assessment stages of DM1 and D4 are

assumed to contain a single algorithm fn; that of DM2 contains two algorithms

f 1 and f. In contrast, the response selection stage of DM' contains a single

algorlthm h', while the RS stages of DM' and DM' contain two algorithms hn and

,, n - 1,3. Therefore, the internal decision strategies are p(us), p(v11 ),

and p(v'sI'). The detailed structure of this organization is shown in Figure

3.2. Note that the interactions defined by the information structure exhibit

I .31
Sector C

Sector B

Sector AI

Figure 3.1. Three Sectors (Parallel Organization)

-SJ

Figure 3 .2. Organization A: Parallel Structure

32
no loops or cycles. In particular, consider the SA information shared between

DMI and DM 2. The variables zI and z1 2 are generated as a result of DMI's

processing of x'. Similarly, zz and z2' are produced by DM2 . Once DM2 has

forwarded x1 to DM', the final assessment z can be determined using the

information fusion (IF) stage of D.4. DM2 determines his value of E2 after

receiving z ', z32 and after producing z2 . While precise timing is not

explicitly required, it is implicit in that a particular stage of processing

cannot be completed until all the requisite input variables are received.

Furthermore, it is assumed that the overall input-output processing of the

organization can be accomplished on the average within the mean interarrival

Wtime v.

The second organizational structure, Organization B, incorporates a

decisionmaker who has a supervisory role. It is defined as follows: the

rectangular sector is divided into two equal subsectors for which DM and DM

are responsible for assessing the threat; however, data from the area adjacent

to the boundary betweem DM" and DM are transmitted to the coordinator or

supervisor, DM , who resolves conflicts and assigns targets either to DM' or

to DM' as appropriate (Fig. 3.3). This is accomplished through command inputs

v 21vand v" from the coordinator to the two commanders. They in turn exercise

their response yI and y3, respectively. Again, for computational simplicity,

it is assumed that DMI and DM have a single algorithm fn' for their SA stage

two algorithms h nand and h for the RS stage. The coordinator, DM2, has an

algorithm IF2 for processing the assessed situations z'1 and z'2 and two

algorithms, h3 and h., in the RS stage. The internal decision strategies are

p(v 'I1), p(v 112) and p(v'1 2). The structure of this organization is shown

in Figure 3.4. Again, note that the information structure is indeed

acyclical.

Sector C

Sector A

Figure 3.3. Two Sectors (Hierarchical Organization)

% In order to avoid complex protocols, it is assumed that no more than one

threat can be in each sector and there can be only one threat that crosses

sectors at a time. These assumptions avoid two conflicts: (a) one DM having

to decide about two threats at the same time and (b) the center DM receiving

J1.- data from the other two DMs and having to assess two threats simultaneously.

Actually, the second assumption can be relaxed so that two threats that cross

boundaries can be present provided they do so on different boundaries and the

net result is such that a single DM does not have to respond to both of them.

33

* " ".- * . .. " -
q -

-- -- --- ---- -

">0-1

(P

C-A\

Figure 3.4. Organization B: Hierarchical Structure

Fourteen different threat configurations are possible. For the

evaluation of the performance-workload locus, fifty six distinct threat sets

were used where sets were distinguished by whether the threat was fast or slow

and by what the direction of the threat was. The objective probability distribution p(x) was taken to be non-uniform. In order to provide some

contrast and also represent a plausible situation, the probability

distribution q(x) perceived by the three decisionmakers was based on the

assumption that the DMs do not take into account that threats can cross sectors; they assign zero probability to such events. On the other hand, they

assume a uniform probability distribution for the various configurations that

do not include threats crossing sectors.

34
The level of activity, measured by whether a decisionmaker expects to

respond to a threat or not is shown in Table 1. The active status denotes

either autonomous operation -the DM receives a threat in his sector and

responds to it -- ot interaction with another DM by sharing situation

assessment information.

TABLE 1. Expected Activity by DMs for p(x) and q(x)

DM STATUS p(x) q(x)

ORG A ORG B ORG A ORG B

1 Active 64% 82% 50% 65%

Inactive 36% 18% 50% 35%

2 Active 79% 50% 30% 309

Inactive 21% 50% 70% 70%

3 Active 64% 82% 50% 65%

- Inactive- ,36% 18% 50% 35%

3.6 RESULTS

The consistency measure Q for the parallel organization A and for task

-. probability distributions p(x) and q(x) is shown in Figures 3.5 and 3.6, while

Figures 3.7 and 3.8 are the corresponding plots for a hierarchical

Organization B.

Figure 3.5 shows that there are no strategies that the organization

A" members can use that will not lead to overload, if the interarrival time v is

less than 29 units of time. Furthermore, if x is more than 43 units, then no

strategy will overload the DMs. The fraction of strategies that satisfy the

bounded rationality constraints is a non-decreasing function of T, as shown by

the gradual, step like increases in Q. Similarly, there are no strategies

that will yield a cost that is less than 3.2 units, while if the constraint is

that the cost be no more than 3.4 units, then all strategies are feasible.

This is indicated in Figure 3.S by the steepness of the slope in the J

direction. Now consider Figure 3 .6. This shows the effect of underestimating

the task requirements by the decisionmakers. By neglecting to take into

consideration the common occurence of threats that cross over from one sector

to another and the resulting need for communication and coordination, the

decisionmakers perceive that they can handle threats arriving every 14 units

of time on the average. All strategies are feasible, if the threats arrive

every 28 or more units of time. On that basis, they can choose strategies

they believe to be feasible, but which in reality will cause overload and,

consequently, a degradation of performance, possibly a severe one.

35,
Figure 3.5. Consistency Measure Q for Organization A: p(x) 

Figure 3.6. Consistency Measure Q for Organization A: q(x) 

36 
Figure 3.7. Consistency Measure Q for Organization B' p(xl 

0 

Figure 3.8. consistency Measure Q for Organization B' q(xl 

37 
Under this assumption, it is also clear that performance will degrade.

Indeed, there are no strategies that will yield a cost of less than 3.5 units.

Superposition of Figure 3.6 on Figure 3.5 shows clearly the effect of q(x).

The DMs assume that the task is simpler than it really is and consequently,

select inappropriate strategies that lead to high workload and degraded

performance.

Very similar observations can be made by comparing the Q loci of

Organization B, Figures 3.7 and 3.8. In that case, too, the Q locus for the

objective probability is farther from the origin than the locus resulting when

the perceived uncertainty by the decisionmakers is used. The decisionmakers

have underestimated the need for coordination to handle threats that occur

near or on the boundary between the two sectors.

The assumption that a decisionmaker may select any of the admissible, but

.4 no necessarily feasible, strategies with equal probability is not a realistic

one. If the DM thinks that a certain subset 1; of the decision strategy space

3' will meet the performance and workload constraints, then he will choose

strategies from that set only and ignore the rest. This can be modeled by

assuming a uniform probability distribution for the strategies in and

assigning zero probability to the strategies not in 3 If a decisionmaker's perception of the task's probability distribution q( such that the

null set, then a uniform probability distribution is assumed for selecting a

V strategy from the set 7.

If the decisionmaker's q(x) is the same as the designer's, the objective

probability distribution p(x), then the feasible strategy set is denoted by

7.. The relative position, of the two strategy sets, the one due to q(x),

and the one due to p(x), I., may be such that the two sets are disjoint,

overlapping, or identically the same. Therefore, in selecting a strategy from

5s" the decisionmaker may be selecting a strategy that meets the

specifications of the task as seen.by thedesigner or he may not. In order to

analyze the relationship between is and 1, the consistency measure introduced

in Eq. (3.12) can be modified as follows:

@ V(_o)/V(D) for V(1') = 0

h VV(37) V(s

in r s ts0 for VW') (3.15) -s

1for V(Z:) V(1)

where V(*) is again the volume in the strategy space. The quantity Qcan be

interpreted as the probability that a decisionmaker, who perceives the

probability distribution of the tasks are _being q(x), will select a decision

strategy that satisfies the requirements (J,v).

38
3.7 CONCLUSION

In this chapter, an approach to the evaluation of alternative

organizational designs has been presented. The issues that arise when the

designer's and the decisionmakers' knowledge of the tasks and of their

relative value differ have been discussed. A measure of organizational

effectiveness has been introduced, the consistency measure Q, for the case

when the designer and the decisionmakers differ in their perception of the

uncertainty associated with the inputs or tasks. Two three-person

organziations, one with parallel structure and one with a hierarchical one,

have been used to illustrate the approach.

One weakness of the procedure is that the analytical formulation of the

workload does not yield much insight to the differences between the two points

of view. Alternatively, the computational approach yields results that are

specific to each case and problem that are being studied. Thus, upon

completion of this study, a major effort was initiated to develop efficient

computational algorithms and implement them on a workstation so that many

eases can be studied with ease. Such a workstation has now been developed and

will be used for further studies and analyses.

39
4. MEASURES OF EFFECTIVENESS IN COMMAND AND CONTROL

In the previous chapter, a methodology was presented for analyzing and

comparing alternative organizational forms. In the language of the theories

for system evaluation, two measures of performance (MOPs), accuracy and

workload, were introduced and the MOP locus, the performance-workload locus,

was defined. Then, a measure of effectiveness (MOE), the measure of

consistency Q, was defined and evaluated on the basis of the MOP locus. These

concepts are quite general and translate easily to the evaluation or

assessment of C' systems. One of the tasks in the research effort was to

approach the problem of evaluation from both sides - from the organization

theory perspective and from the system design perspective. The long term

objective has been to develop a unified approach.

In this chapter, a methodology for measuring the effectiveness of C"

systems is described and is then applied to an abstraction of a C' system of

the US Army.

4.1. INTRODUCTION

Improvements in weapon system technology, and higher capacity and speed

in data transmission, combined with an increasing complexity of the

battlefield, impose severe time constraints on hardware, software, and human

decisionmakers. The purpose of this task was to extend the methodology

presented in Bouthonnier and Levis [23] to consider measures of performance

(MOPs) and measures of effectiveness (MOEs) that include time.

Time has always been of crucial importance in combat and, consequently,

to command and control [24]. As Lawson [25] relates, win a typical discussion

of Command and Control, it is taken as axiomatic that the information

presented to the commander must be 'timely' as well as accurate, complete,

etc. Little or nothing is said about how timely is timely enough; nor is any

yardstick given by which to measure 'timeliness'. Rather, the clear

implication is that all would be well if only communications and computers

were 'faster'. In addition, this attention to rates (e.g., information

processing rates, rate of fire, etc.) in which time only appears in the

denominator, has led to a preoccupation with the performance characteristics

of the component parts of a C3 system. It does not provide any means of

comparing the effect of an increase in one 'rate' with that of an increase in

some othe rate.ff

The methodological framework is the one developed by Bouthonnier and

Levis [23]. The aspects that time can take in a warfare environment are

numerous. The ones that this assessment methodology consideres are:

*This chapter is based on the work of Phillipe H. Cothier [28].

40

Jil
System response time. It characterizes the time delay between the moment

when the C' system receives a stimulus and the moment it can deliver a

response. It is the sum of all the time delays at every level of the process.

Tempo of operations. In most military situations, rates are used to

express the important quantities, e.g., rounds per minute, miles per hour.

The term in commom usage for the operating rate of a C' system is its Itempow.

Lawson [25] defines it as the number of actions per unit of time which the

system is executing and states, further, that Nthe tempo tells us how complex

an environment the system can handle (i.e., its bandwidth) while the response

time tells us when it responds in time (i.e., the phase delay in the system),.

These notions depend on what is actually taking place, i.e., the

scenario. The event that stimulates the C" system is only the partial

perception by the system of a global scenario. Different scenarios can be

perceived through identical events, and the system is confronted with

uncertainty. Once the scenario is identified with enough certainty, then an

option must be selected. It appears that any assessment of a CS system must

consider the crucial role of the scenario and the tempo of operations it

implies: to each scenario corresponds an evaluation of the effectiveness of

the system. These partial evaluations can be merged into an overall measure

of effectiveness for a given range of possible scenarios.

Timeliness is defined here as a system's ability to respond within an

allotted time. Allotted time, in this context, is the time interval over

which our forces, which include a C2 system being assessed, can affect the

environment. This allotted time is described by a window of opportunity whose

parameters are determined by the system and the mission. Two quantities are

needed to specify the window of opportunit : one choice is the lower and the

upper bound of the time interval, t and t *, respectively; another one is one

of the bounds and the length of the interval, e.g., t and At. Such a time

interval is not sufficient to yield by itself a measure of effectiveness; one

must also consider the way this time is employed, i.e., one must consider the

doctrine that is in effect, was well as the tactics that are feasible in such

an interval.

The methodology used in this paper is based on six concepts: system,

environment, context, parameters, measures of performance, and measures of

effectiveness. The first three are used to pose the problem, while the last

three define the key quantities in the analytical formulation of the problem.

The analytical aspects of the methodology address mainly the relationships

between hardware characteristics, system structure, and standard operating

procedures (SOPs) to system performance.

The system consists of components, their interconnections, and a set of

operating procedures. A boundary can be drawn that defines what is included

within the system whose effectiveness is to be assessed. What is included

depends on the analysis at hand. The environment consists of our own forces

and the adversary's forces upon which our forces can act and which can act

upon ours. For example, the C3 system is used to direct forces and monitor

41

P .. . . 1.-". ..- ,
(sense) the environment. An engagement between two forces in an urban area or

at a mountain pass define typical environments. A Cs SUDsystem of a fire

support system or a communication network are typical systems. The context

denotes the set of conditions and assumptions within which the system and the

environment exist. The relationship between system, environment, and context

are shown in Figure 4.1.

ENVIRONMENT

SYSTEM

OWN FORCES

CONTEXT

Figure 4.1. System, Environment and Context

Parameters are the independent quantities used to specify the system and

the mission requirements. For example, in the case of a fire support system,

system parameters may include quantities that describe the detection

equipment, computational time delays, kill radius of the munition, and failure

probabilities associated with the components, to name but a few. Parameters

of the mission may be the tempo of operations, as described by the speed of

the threats, and the size of the engagement.

Measures of Performance are quantities that describe system properties or

mission requirements. MOPs for a command and control system may include

reliability, survivability, cost, and probability to kill. The mission

requirements should be expressed by the same quantities as the system MOPs,

e.g., minimum reliability or survivability, maximum cost, or minimum

probability to kill. System parameters are defined within the system

boundary; MOPs may be defined within the boundary or they may include aspects

of the environment.

* Measures of Effectiveness (MOEs) are quantities that result from the

comparison of the system MOPs to the mission requirements. They reflect the

extent to which the system meets the requirements. To evaluate the MOEs, it

is necessary to go outside the boundary and consider the environment. Only

then could the effect of the system on the mission outcome be evaluated.

These definitions of parameters, MOPs, and MOEs are consistent with those

developed at a recent MORS Workshop on the subject [26].

42

Z:
In this methodology for assessing effectiveness, the system MOPs and

mission requirements must be modeled and analyzed independently, but in a

common context. The system capabilities should be determined independently of

the mission requirements, and the mission requirements should be derived

without considering the system to be assessed. Otherwise, the assessment is

biased. Since the steps of the methodology have been presented in [23], they

and the modeling of timeliness will be described here through application to

an idealized tactical fire direction system. This problem is formulated in

Section 4.2; in Section 4.3, the system and the mission are modeled and their

respective loci defined. In Section 4.4, a class of measures of effectiveness

is defined, while in Section 4.5 the results are presented and discussed.

4.2 SYSTEM AND MISSION

The concepts discussed in Section 4.1 can be illustrated by applying the

methodology for System Effectiveness Analysis to an 'electronically integrated

command and control information system that also processes fire missionsff

[27].

One can isolate three main elements in a fire support system at the

battalion level: the forward observer, the battalion fire direction center,

and the field artillery cannon battery. The system can include several

forward observers and several batteries connected to the same central

battalion computer.

The Forward Observer (FO) receives the initial stimulus by detecting an

enemy threat. The FO is equipped with vehicle position determining equipment

and a laser rangefinder. These allow the FO to locate accurately area targets

for fire and to conduct one-round adjustments. The FO is also equipped with

the Digital Message Device (DMD), a portable, battery-powered device that

transmits and receives digital bursts. The FO uses the DMD to communicate

estimates of the position and velocity of the target, and requests for fire to

*the battalion computer.

The battalion Fire Direction Center (BN FDC) is provided with a central

computer. Digital communication over any standard communication means (radio

or wire) provides for input of data into the computer center and for the

return of the results. Forward observers and firing batteries are provided

* with remote terminal equipment to obtain data from the central computer.

The Battery Display Unit (BDU) is the cannon battery's link with the C'

system. Each battery has one BDU. The BDU assists execution of fire plans by

receiving and printing firing data for each target that the battery will fire.

This receive-only unit initiates an automatic acknowledgement that is sent

back to the FDC computer.

W . While this is the basic configuration, additional equipment is maintained

4 in parallel to augment the basic system. Voice communication links can be

added in parallel with the digital links, for instance, between the battalion

fire direction center and the cannon battery. Voice communication is slower,

more vulnerable, but still very useful, if the digital link fails. If the

643

-. k. Nl~i Ai a" L " "gfiAMM-
fire support system computer fails at the battalion level, the battery has the

capacity to do the firing computations locally. This alternative is slower,

though.

A schematic representation of the system that will be analyzed is shown

in Figure 4.2. Seven stages are shown. In this model, nodes are not subject

to failure; only links are. Alternative implementations of each stage are

shown by parallel links; e.g., a voice link is in parallel with the digital

link between the battalion fire direction center and battery B.

computer digital link voice

a OMD ~digitallink x N FDC .

6.r dgl.1linkBDU OD battery

relay voice link FADAC

2 3 4 5 6 7

Figure 4.2. Fire Support System Structure

If the BN FDC computer does not work, the target estimates from the FO

can be sent to battery B through voice communication (the BN FDC acts as a

simple relay). The battery crew can then compute the firing data manually

- with the Field Artillery Digital Automatic Computer (FADAC). In the case

where the firing data are computed at the BN FDC level and transmitted by

voice communication to battery B, neither the BDU nor the manual technique

have to be used. The voice communication of the firing data reaches directly

the firing platform of the battery. In order to assess properly the

effectiveness of this system, it is necessary to specify the environment and

context in which it operates, i.e., define the scenario.

The idealized scenario that will be considered is shown in Figure 4.3.

Some vital node of our forces' C' system is situated at the end of a valley.

A road along this valley leads to this node. The topography of the area is

perfectly known by our forces, and the road is the only access to the node. A

fire support battalion including one forward observer (FO), one battalion fire

direction center (BN FDC), and two batteries B. and B. has been positioned to

protect this access. This battalion is equipped with the fire support system

defined in Figure 4.2. The batteries cannot see the road: they shoot

according to the firing directions that are computed on the basis of the

observer's estimates.

An enemy tank (threat) appears in the field of vision of the forward

observer. It is moving on the road towards our forces with hostile

intentions. The mission is to defend the node, i.e., to prevent the tank from

attacking the node.

44
Road FRIENDLY

NODE

ForwardI

".5.

Battalion

cantore cto

Battery *2

Battery *1

Figure 4.3. Scenario

It is assumed that the threat cannot attack the fire support battalion

directly; the only countermeasure that will be considered is the jamming of

communications by the enemy. It is also assumed that the threat will pursue

its attack, even after it is fired upon. It will try to carry out its own

offensive mission, as if it encountered no reaction from our forces. This is

a highly idealized situation. It is used to illustrate the methodology,

rather than represent current practice in terms of doctrine or tactics.

The problem of interest is to determine how effective this particular

command and control system is. But, as indicated earlier, to assess

effectiveness, one has to go outside the boundaries of the C s system itself

and consider the mission and the environment. In this case, it is postulated

that the system to be assessed consists of the forward observer, the BN FDC.

the BDU, the FADAC, and the associated communication links. The batteries

themselves, as well as the threat and the node to be defended, constitute the

environment; the terrain constitutes the context. Note that in applying this

45

" 41"
methodology, a choice has to be made each time as to what is included within

the boundaries of the system. In this case, the sensors are included, as are

some parts of the weapons system. In another application, the whole fire

support system could be considered as a integrated weapons system.

The system's performance will be characterized by three MOPs Two of

them define the window of opportunity, namely, the upper bound t* and the

width of the window At. The third one is the overall probability of kill

(OPK) that measures the ability of the total fire support system to stop the

threat.

For each value of the selected system parameters, a value for the triplet

(t* *, At, OPK) is obtained. As the parameters are allowed to vary over their

respective range of values, a locus is constructed in the space of MOPs. This

is defined as the system locus, Ls .

The mission requirements in this case are expressed simply as the minimum

acceptable probability that the node will be defended successfully, i.e., that

the threat will be prevented from firing at the node. This requirement can

also be expressed as a locus in the MOP space; it is the mission locus Lr.

Effectiveness for this system and mission is defined by how well the overall

kill probability meets the mission requirement.

In the next section, the mathematical models of the system and the

mission will be developed and the corresponding loci constructed.

4.3 SYSTEM AND MISSION MODELS

Each node and each link of the C1 system is assumed to have a probability

of failure, independently of the countermeasures taken by the enemy. These

component failure probabilities determine the system's reliability. Since the

system is operating in a hostile environment, the communication links are

subject to jamming by the enemy. The probability of failure due to Jamming

determine the system's survivability. Although the two concepts of

reliability and survivability are distinct because the underlying

probabilities of failure have different causes, in this analysis they will be

merged to reduce the dimensionality of the problem. A single vector of

probabilities, p. will be used as a parameter that describes the failure

characteristics of the system's components. In a full scale analysis, the two

will be separate.

The parameter that determines the tempo of operations is, in this case,

the speed w of the threat. A whole range of different versions of the

scenario can be investigated by varying the speed w.

*It is assumed that the main source of uncertainty is the estimate of the

threat's state by the forward observer. If the FO uses two sightings to

determine the threat's state, then an appropriate system parameter can be, for

example, the angle A that separates the two sightings. Intuitively, the

larger the angle p the more accurate the speed estimate, but the longer the

response time.

46
Since the commander has a choice as to the resources he may use, two

cases will be considered: (a) one battery (B.), and (b) two batteries (B. and

B2). In the second case, two different tactics can be analyzed: coordinated

fire and uncoordinated fire.

Response Time and Window of Opportunity

The geometric relations for this scenario are shown in Figure 4.4.

Y

T

threat M \

trajectory -- o

-'(road ) WF

02max

S~2

01@ max YT

01.mi BN FDC

" I mile

r X

0 Imile

Figure 4.4. Geometric Relations between System and Threat

To begin the process of quantifying the timeliness characteristics of the

system through the determination of the window of opportunity, consider Figure

4.5, which shows the chronological sequence of the response process. The time

that a round fired by a battery hits the ground, timpact, is given by:

.3

timpact t obs + i (4.1)

i =1

A-ci is computed from the geometry in Figure 4.4. It is a function of the

speed w, the angle P, and the observation time (281.

AT, - A Pt(W,P,tobs) (4.2)

47
Estimation of the transmission of the estimates Flight time

position and velocity: of the target by FO computation and transmission of the of the firing data projectile

-Setting -up cf the battery B

=t

tobs End of t o timpac t

observation estimation firing time time

Figure 4.5. Time Profile of the System Response

A sensitivity analysis has shown that AT, can be taken as constant for

this geometry and characteristics of the weapon system. For the values chosen

for this example, this constant is 36 seconds.

Let the lower bound of the window of opportunity, t*, be equal to the

minimum impact time. For a given angle P and a given target velocity w, the

earliest impact time is determined by the earliest possible observation time,

i.e., tobs = 0, and by the minimal time delay (Ax,)min between the end of the

estimation and the actual firing of the battery. Thus:

t = ATI(w.,O) + (AT)min + AT3 (4.3)

This earliest impact time may be considered as the response time of the

system. Note that it depends on the scenario, the sensor characteristics, and

the positioning of the forward observer and the batteries with respect to the

threat.

Let M. be the point on the trajectory where the threat leaves the area

covered by battery B± (see Fig. 4.4) in the single battery case, or the area

covered by either batter *in the two battery case. Let the upper bound of the

window of opportunity, t , be the time that the threat moves past point Mi.

This bounds from above the admissible impact times, i.e., only impact times

occuring within the window of opportunity are allowed:

i(timpact )max _ t (4.4)

where

t K/w (4.5)

and K is the distance to point M. and depends on the geometry of the

situation.

48

4
The resulting time interval, At, is the system window of opportunity: the

system can deliver a response to the stimulus at any time t impact between t

and t (for t < t ). The window of opportunity is completely characterized

by the ordered pair (t , At), where At = t - t . Changes in the values of

the parameters w, and A lead to changes in the window of opportunity.

Performance

The single shot kill probability SSPK associated with an impact time is

easily computed by taking into account the uncertainty in the speed estimate,

and the kill radius of the munition. For fixed values of w and tobs, the

shape of the variations of SSPK with t is given in Figure 4.6; the latter also

shows an important trade-off. As P increases, the width of the window of

opportunity decreases because it takes a longer time for the FO to obtain his

estimate. But at the same time, a large A yields a more accurate estimate of

the speed of the target. Therefore, the kill probability is increased. The

upper limit t** is unaffected by changes in .

SSPK </33

* Figure 4.6. Single Shot Kill Probability as a Function of Impact Time

:. Also, as time goes by, if there are no updates of the threat's state, the

3"- uncertainty of its exact position increases and, therefore, SSPK decreases

tre ha epesnsthesstmtttemiure47

pO The seven-stage structure of the C' system has been presented in Figure

. reveals that out of the ten possible paths, six paths do not lead to the

- trnsmisionof information from the forward observer to the batteries.

" For each of the four paths that indicate successful communcation, the

~following quantities are defined:

49

014
q(i): probability that the path #i is operational; i = 1,2,3,4.

y(i) = (At1 )min(i) + Ats, i.e., y(i) is the minimum time delay between

the estimates by the FO and the impact time.

6(i): minimum time delay necessary to recompute new firing data based

on the initial estimates, to transmit them and to set up the

battery accordingly. If the system recomputes the firing data

immediately after each shot and fires in sequence, then 6(i)

represents the minimum time delay between two shots.

the

DMD

does not work

U the digital link does nct work the voice link the FADAC

FO • d% does ,ot work does not work

P. the

works the BN F1C the voice liK - the FADAC

1- / computer P works works

doe:, not work

. l- the voice link

the digital P5 does iK)t work

link works the digital

1,nk li doe s 4,

1 - 4 not w ork

the BN FDC 5 the voice

computer link works

works

1-P the vLice link '2/does .it

the digital -P6 the BD)U or, link works. - does not N

"ework- ' . the vOice

link wures

-P6 (2)

the b)U

workb

0

*Figure 4.7. Tree Representing System's Operational States

The earliest response time to the stimulus is t*. The system can use the

remaining time within the window of opportunity to deliver other responses,

e.g., to fire again, therefore increasing the overall kill probability. This

can be done in many different ways. This analysis focuses on two of them,

which are classical military doctrines, known as 'LOOK-SHOOT-SHOOTSHOOT... [LSS] and "LOOK-SHOOT-LOT... [LS].

Doctrine 1: "LOOK-SHOOT-SHOOT-SHOOT... The observer initially makes

estimates of the speed and position of the threat, and then the battery keeps

on shooting at the target, recomputing each new firing data on the basis of

these initial estimates.

50

C
The observation time tobs is the same for each shot , since there is no

updating of the estimates. The time delay between two shots is thus the

interval B. The battery fires as many shots as possible within the window of

opportunity, since there is no feedback from the observer.

Doctrine 2: OLOOK-SHOOT-LOOK-SHOOT... N~ After each shot, if the threat is

neither destroyed nor incapacitated, the observer makes new estimates of its

speed and position, new firing data are computed on the basis of these updated

estimates, the battery (or batteries) shoots according to these new firing

data, and so on until the upper limit of the window of opportunity is reached.

The overall probability of kill (OPK) can be computed from the single

shot probability of kill, as follows:

n

OPK = 1 - SSPK (timpactfn) (4.6)

Pp n=1

* where n* is the number of shots possible for a given doctrine and a given

* window of opportunity. The single shot probability is given by:

SSPK(t ) =~ (4.7) impact#n t imatn- t bn

where the function depends on the kill radius of the munition, the speed of

the target, and the accuracy of the observation. Similar expressions have

been derived for Doctrine 2 as well as for the two battery case with

* ,coordinated and uncoordinated fire. For details, see Cothier [28].

Sl

In calculating OPK, it is assumed that the C 3system operates in the same

mode (i.e., the same path is used) throughout that engagement. Therefore,

%four different values of OPK can be computed, one for each possible path.

4.4 SYSTEM AND MISSION LOCI

The three system MOPs (t**, At. OPK) can be derived for each path i.

However, it is preferable to obtain an overall probabilistic description of

these MOPs. For any of the 6 paths (Fig. 4.7) that fail to transmit the

information from the FO to the batteries, At is zero, and so is the OPK.

For paths #1 to #4, Ati and OPKi vary according to what doctrine is

chosen. The MOPs to consider are thus the expected values of these

quantities.

51

OP SP 46
4

At = E[At = q Ati (4.8)

i-i

4

OPK = E{OPKi } = q(i) OPKi (4.9)

i=1

where q(i) is the probability that the i-th path is used. The values of q(i)

. is obtained by using structure functions and the failure probabilities

(Cothier [28]). From now on, only the expected values of At and OPK will be

considered.

The dependence of the system MOPs on the system parameters is shown in

Figure 4.8. Note that OPK depends directly on the parameters w, A, and p, on

the two other MOPs, t and At (i.e., the window of opportunity), and on the

doctrine used. In other words, the parameters are mapped twice in the third

MOP, directly and indirectly.

Parameters MOPS

w * Doctrine

p OPK

Figure 4.8. Mapping of the System Parameters into the System MOPs

To each value of the parameter set (wj,p) corresponds a point in the MOP

space (t* , At, OPK). Now consider all the allowable values that the

parameters make take:

I

Wmin _W w _<w

O min A P a. (4.10)

~Pmin -( P - P..

14 52
If the parameters are allowed to vary over their admissible ranges, then the

variations define a locus in the MOP space. This is the system locus Ls .

Mission Locus

The analysis of the mission in this case is quite simple: the mission

requirements can be expressed directly at the MOP level. The mission

objective reduces to a single requirement, a condition on the third attribute

OPK. If X is the minimum probability that the threat will not be able to

attack the node being defended, then the mission locus, Lr , is the region in

the MOP space (t , At, OPK) that satisfies the inequality:

I S OPK < 1 (4.11) N

In general, this would not be the case. Substantial analysis would be

required to convert mission objectives into requirements and then express the

latter as a locus in the MOP space.

In this section the system locus and the mission locus have been derived.

The timeliness of the system has been described in terms of the window of

opportunity, which, together with the performance measure OPK, have led to the

system locus. In the next section, the two loci will be used to define and

evaluate MOEs.

4.5. MEASURES OF EFFECTIVENESS

The next step consists of comparing quantitatively the system MOPs to the

mission requirements, using the geometric relationship between the two loci,

Ls and Lr , in the MOP space. If the two loci do not overlap, i.e., if

L 3( L = (4.12)

then the system MOPs do not satisfy the mission requirements for any operating

state of the system. Consequently, effectiveness should be zero, regardless

of which measure is used. On the other hand, if the two loci coincide, i.e.,

if

L. n Lr = Ls = Lr (4.13)

then the system and the mission are perfectly matched, and effectiveness

should be equal to one. Three cases can be distinguished when the two loci

have points in common.

(a) Neither locus is included in the other,

53
L. nTL # and L r) L < L and L n' L < L (4.14)

In this case, only some of the values that the system MOPs may take

satisfy the mission requirements. Many different measures can be used to

* described the extent to which the system meets the requirements. Each of

these measures can be considered as an MOE which, if normalized, takes values

in the open interval (0,1). Let V be such a measure. Then one MOE is defined

4- by

E = V(L s L )/V(Ls ) (4.15)

while another is

E = V(Ls (- L r)/V(L ) (4.16)

(b) The system locus is included in the mission locus:

L (-' L = L (4.17)

The measure E. is equal to unity - all operating conditions of the

system meet the requirements. But E. is less than one; the system can attain

only certain values of the requirements.

(c) The mission locus is included in the system locus:

L S s L r = L r (4.18)

that is, the system meets all the requirements, but also has operating

conditions that do not meet the requirements. Then E. is less than one, and

E. is equal to unity.

These measures are only two of the many that can be defined; they can be

thought as partial MOEs, since they give only partial information about the

relationship between the two loci. These partial measures can be combined

into a single measure through the use of a utility function:

E = u(E ,E2 ... ,Ek ) (4.19)

The subjective judgements of the system developers and the users can be

incorporated into the effectiveness analysis through the selection of the

partial measures and the utility function.

54
,..

4.6 EFFECTIVENESS ANALYSIS AND COMPARISON OF DOCTRINES

For this example, only the first partial MOE of Eq. (4.15), E., will be

used. The mission locus is such that E. is very small and does not

discriminate between different cases.

The One-Battery Case: Comparison of Two Doctrines

Figures 4.9 and 4.10 show the system locus and its intersection (shaded

region) with the mission locus for each doctrine. The ratio of the shaded

volume over the total volume of the system locus is larger for doctrine 1 and

for doctrine 2:

E.(1 battery, doctrine 1) = 0.55

E,(1 battery, doctrine 2) = 0.50

When the tempo of operations is fast (the threat moves rapidly), the window of

opportunity is small: it is better to make an accurate measurement of its

speed once and then fire rapidly in sequence without taking time to make new

estimates, rather than to make an estimate, shoot, make a new measurement, and

so on. Therefore, the LSS doctrine is more effective than the LSLS doctrine.

The first doctrine provides a more timely response - while the initial

response time of both doctrines is the same, doctrine 1 allows for a higher

rate of firing than doctrine 2.

The Two Battery Case

a," When the two batteries B. and B, are considered, it appears from Fig.

4.4 that their areas of coverage overlap. The threat moves first on a part of

the road that is covered by battery B., then on a part that is covered by both

B1 and B ,and then 3n the part that is covered by B. alone. It has been

assumed that the tank has to go beyond the range of B. in order to be able to

fire at the Cs node. Therefore, the probability of kill varies with time,

suddenly increasing, then decreasing. Assuming a "LOOK-SHOOT-SHOOT-SHOOT...

doctrine, two different options for the fire support commander will be

* considered:

Option 1: The two batteries shoot at the threat independently, each one using

its window of opportunity at the maximum. There is no coordination between

the two batteries.

Option 2: Battery B, starts firing only when the threat enters the area

covered by both batteries. In other words, the commander decides not to fire

a. immediately with B1 , but to wait until coordinated fire can be achieved, i.e.,

both batteries B. and B. shooting so that their projectiles hit the target

trajectory at the same time. The system's window of opportunity is thus

reduced to that of battery B.. The time interval during which B. holds its

fire can be used to keep the observer's estimate updated.

S5S
OPK

9-

.8

75 -

.6

1%.5 -

.4- 7 02

- ,351

J...

*0 100 200 300 400 500 6 t

Figure 4.9. Doctrine 1 (LSSS) System and Mission Loci

OPK

.75 _ _

.7-

.6-

.5-

702

351

0 100 200 300 400 500

Figure 4.10. Doctrine 2 (LSLS) System and Mission Lci

56
Figure 4.11 and 4.12 show the system locus and its intersection (shaded

region) with the mission locus for both Options 1 and 2. The evaluation of

the effectiveness of the system for both options, measured by the ratio of the

shaded volume over the total volume of the system locus, yields the following

results. Let E,(1) be the MOE when Option 1 is used and E,(2) when Option 2

is used.

Then

E (1) = E (2) 2 0.6

Therefore, both options result in approximately the same value for the

effectiveness of the system. In Option 2, fewer shots are fired than in

Option 1. In this case, coordination reduces costs for the same kill

probability. However, this is not reflected in the MOE. Additional MOPs,

such as cost, must be introduced, if one would like to discriminate between

the two options.

It is important to note that Option 2, coordinated fire, is a better

choice than Option 1, although its window of opportunity is much narrower. In

* fact, the time available is better managed: it is more effective to wait in

order to implement a better option. This example shows that a larger size of

the window of opportunity does not necessarily lead to higher effectiveness.

OPK --

.9

.8.

75

.6

.5

861

702

0 t00 200 500 400 500 600 l

Figure 4.11 Option 1 (immediate fire without coordination)

System and Mission Loci

57
OPK

.75 -. ...

.6-

dt

.58

4- 702

0 100 200 30::0 480 500 600 ZAt

Figure 4.12. Option 2 (wait and coordinate) System and Mission Loci

,..6

i 4.7 CONCLUSIONS

" In this chapter, the notion of timeliness has been integrated in the

analysis of a system's effectiveness by treating the temporal characteristics

-]of the system as performance measures. The resulting measures of

effectiveness have embedded in them the tme-related notions of response time

and tempo of operations through their impact on the window of opportunity.

These MOEs allow also the quantitative comparison of different doctrines:

Some doctrines are shown to make better use of the available time than others.

Therefore, this methodology for effectiveness analysis can be used to evaluate

g doctrines appropriate to a given situation.

~A second point has been illustrated by considering the relationship

~between system components. While the speed of processing and transmission of

~data can be improved, the effectiveness Of the system may not change if, for

~instance, the reliability and survivability of the system's components are not

~improved also. Faster does not necessarily mean better; it can even mean

~worse, if the increase in speed is gained at the expenses of the system's

survivability. The methodology allows one to relate a change in one part of

the system to a change in another part: the variations in the features. of a

given system are not considered separately, but jointly. This yields Useful

iperspectives for the design of future C* systems. The influence of any

modification either in the components, or the structure, or the doctrine, can

- be evaluated, Using the proposed measures of effectiveness of the system.

L&

m.

i $61
A third point refers to the window of opportunity. The size of the

window of opportunity is not a sufficient determinant of a system's

timeliness. The location of the window, determined by the response time, must

also be considered.

The methodology is flexible enough to be adapted to many kinds of

systems: evolving systems (29], automotive [30]. or manufacturing [31.

Current efforts are directed toward developing the properties of several

classes of MOEs, developing graphics software for the efficient construction

and analysis of the system and mission loci, and applying the methodology to

- more complex problems such as evaluating the effectiveness of a large C'

system for doing indirect identification of friend, foe, or neutral.

.5.9
4m,

5. CONCLUSIONS

The goals of the research reported in this document were: (a) the

further development of a quantitative methodology for the design analysis, and

evaluation of command and control organizations, and (b) the development of a

methodology for the assessment of C3 systems that is consistent with the

methodology for the evaluation of the decisionmaking organizations these

systems support.

5.1 ORGANIZATIONS

The specific objective designed to meet the first goal was the

consideration of the effect that different subjective probabilities and

perceptions of the value of the tasks by individual decisionmakers can have on

the organization's performance. The original approach included the

development of the counterparts to N-dimensional information theory and the

Partition Law of Information when weighted entropy is used in the place of

classical entropy. While this development was carried out, the formal results

did not yield any insight into the research questions. The resulting

expressions were too complicated to be amenable to a physical interpretation.

Consequently, the research effort was redirected toward using the conventional

formulation of the Partition Law of Information, but complementing these

results with efficient computational procedures for the study of the phenomena

of interest.

The problem of qualitative differences in the tasks of an organziation

has been approached by distinguishing four cases:

Case 1: The organization designer assumes that the decisionmakers know

the objective probabilities of the tasks to be performed and all tasks

are of equal importance. In this case, the weighting coefficients in the

entropy expressions and in the objective function are unity. This is the

base case that has been analyzed fully in earlier work.

Case 2: The organization designer considers that the decisionmakers do

not know the correct probability distribution for the various tasks. In

this case, weighted entropy is used to estimate workload, but the

objective function has unity weighting coefficients, as in Case 1.

Case 3: The designer assumes that the decisionmakers do know the actual

task probabilities, but have a different perception of the relative

importance of tasks. In this case, the computation of workload is as in

Case 1, but the objective function is different.

Case 4: This is the most general case. The designer assumes that the

decisionmakers do not know the actual task probabilities and give

different relative weights to the various tasks. This case subsumes the

other three cases.

60
Detailed analytical results for the difference in workload under the

correct probability distribution and the perceived ones were obtained for the

interacting decisionmaker with bounded rationality. The expressions for the

individual components of workload, i.e., throughput, blockage, coordination,

and noise were also obtained. However, the complexity of the resulting

expressions made it difficult to obtain general results. Therefore, a

previously developed algorithm for three member organizations was modified and

expanded to handle cases, 2, 3, and 4. Computational results for two

alternative three person organizations were obtained using a new

implementation of the algorithm. This implementation was on an IBM PC/AT and

uses graphics extensively.

In order to evaluate the effect that differences in the perception of

task uncertainty (Case 2) have on organizational performance, a consistency

measure was introduced. This measures the effect of selecting strategies from

a perceived feasible strategy space as opposed to selecting one from the

actual feasible strategy space.

5.2 MEASURES OF EFFECTIVENESS

The basis for the analysis and evaluation of alternative organizational

designs was the performance-workload locus, which is a graphical depiction of

the values the measures of performances of a system can take, when all the

possible operating points are considered. This locus represents, therefore,

the system capabilities. The concept can be generalized when instead of

performance and workload, any measures of performance (MOPs) are considered.

The locus then becomes the system locus. Similarly, the design requirements

can be expressed in the performance space. In the case of organizations,

performance as measured by accuracy and workload were the pertinent MOPs.

Consequently, the corresponding requirements were for satisficing performance

and for workload less than the bounded rationality constraint. In the case of

C systems, the requirements define another locus, the requirements locus Lr.

Measures of effectiveness (MOEs) are then quantities obtained by

determining how well a system satisfies the requirements. In the case of

organizations, the consistency measure Q was used as an MOE.

To illustrate the methodology, a generic version of the C3 system for

surface-to-air missile batteries was used. One novel aspect of the research

was the modeling and analysis of timeliness as it pertains to C3 systems.

5.3 PUBLICATIONS

The research results have been documented in full in thesis reports and

technical papers. The pertinent publications are listed below:

Bejjani, G. J., (1985). Information storage and access in decision making

organizations. M.S. Thesis, LIDS-TH-1434, Laboratory for Information and

Decision Systems, Massachusetts Institute of Technology, Cambridge,

Massachusetts.

61
Bejjani, G. J., and Levis, A. H. (1985). Information storage and access in

decision making organizations. Proceedings Eighth MIT/ONR Workshop on C3

Systems. Cambridge, Massachusetts.

' Cothier, P.H. (1984). Assessment of timeliness in command and control. M.S. • Thesis, LIDS-TH-1391, Laboratory for Information and Decision Systems,

Massachusetts Institute of Technology, Cambridge, Massachusetts.

Cothier, P.H., and Levis, A. H. (1985). Assessment of timeliness in command

and control. Proceedings Eighth MIT/ONR Workshop on C3 Systems. Cambridge,

Massachusetts.

Tomovic, M.M., and Levis, A. H. (1984). On the design of organizational

structures for command and control. Proceedings Seventh MIT/ONR Workshop on

C3 Systems. Cambridge, Massachusetts.

ARCHIVAL PUBLICATIONS:

Levis, A. H. (1984). Information processing and decision making

organizations: A mathematical description. Journal of Large Scale Systems,

No. 7.

Cothier, P.H., and Levis, A. H. (1986). Timeliness and measures of

effectiveness in command and control. IEEE Trans. on Systems, Man, and

Cybernetics, Vol. SMC-16, No. 6, November/December.

6

,

62

4Z
6. REFERENCES

[I] K. L. Boettcher and A. H. Levis, (1982). "Modeling the Interacting

Decisionmaker with Bounded Rationality," IEEE Transactions on Systems,

Man, and Cybernetics, SMC-12, No. 3.

[2] A. H. Levis and K. L. Boettcher, (1983). wDecisionmaking Organizations

with Acyclical Information Structures," IEEE Transactions on Systems,

Man, and Cybernetics, SMC-13, No. 3.

[3] K. L. Boettcher and A. H. Levis, (1983). "Modeling and Analysis of

teams of Interacting Decisionmakers with Bounded Rationality,"

Automatica, Vol. 19, No. 5.

[4] H. P. Hillion and A. H. Levis, (1987). "Timed Event-Graph and

Performance Evaluation of Systems,0 Proc. 8th European Workshop on

Applications and Theory of Petri Nets, Zaragoza, Spain.

[5] H. P. Hillion, (1986). "Performance Evaluation of Decisionmaking

Organizations Using Timed Petri Nets," S.M. Thesis, LIDS-TH-1590,

Laboratory for Information and Decision Systems, MIT, Cambridge, MA.

[6] J. L. Peterson, (1981). Petri Net theory and the modeling of systems,

Prentice-Hall, Englewood, NJ.

[7] C. E. Shannon and W. Weaver, (1949). "The Mathmetical Theory of Communication," University of Illinois, Urbana, IL.

[8] A. I. Khinchin, (1957). "Mathematical Foundations of Information

Theory," Dover, New York.

[9] W. J. McGill, (1954). "Multivariable Information Transmission,"

Psychometrika, Vol. 19, No. 2.

[10] R. C. Conant, (1976). "Laws of Information Which Govern Systems," IEEE

Trans. on Systems, Man, and Cybernetics, SMC-6, No. 4.

[11] S. A. Hall and A. H. Levis, (1983). "Information Theoretic Models of

Memory in Human Decisionmaking Models," Proc. of 6th MIT/ONR Workshop on

C Systems, LIDS-R-1354, MIT, Cambridge, MA.

[12] D. A. Stabile and A. H. Levis, (1984). "The Design of Information

Structures: Basic Allocation Strategies for Organizations," Large Scale Systems, Vol. 7.

[13] D. S. Stabile, (1981). "The Design of Information Structures: Basic

Allocation Strategies for Organizations," S.M. Thesis, LIDS-TH-1098,

Laboratory for Information and Decision Systems, MIT, Cambridge, MA.

63

In =&4UdK~fdh

aa"0 Um
[14] F. E. Donders, (1983). "Die Schnelligkeit Psychischer Processe," Archiv

Anatomie und Physiologie.

[15] J. G. March and H. A. Simon, (1958). Organizations, John Wiley and

Sons, New York.

[16] J. G. March, (1978). "Bounded Rationality, Ambiguity, and the

Engineering of Choice," Bell J. Economics, Vol. 9.

[17] G. H.-L. Chyen and A. H. Levis, (1985). "Analysis of Preprocessors and

Decision Aids in Organizations," Proc. IFAC/IFIP/IFORS/IEA Conference on

Analysis, Design and Evaluation of Man-Machine Systems, Varese, Italy.

[18] S. K. Andreadakis, and A. H. Levis, (1987). "Accuracy and Timeliness in

Decisionmaking Organizations," LIDS-P-1650, Proc. 10th IFAC World

Congress, Pergamon Press, Newy York.

[19] M. M. Tomovic and A. H. Levis, (1984). 0On the Design of Organizational

Structures for Command and Control," Proc. 7th MIT/ONR Workshop on C'

Systems, LIDS-R-1437, Laboratory for Information and Decision Systems,

MIT, Cambridge, MA.

[20] K. L. Boettcher, (1981). "An Information Theoretic Model of the

Decision Maker," M.S. Thesis, LIDS-TH-1096, Laboratory for Information

and Decision Systems, MIT, Cambridge, MA.

[21] S. Guiasu, (1977). Information Theory with Applications, McGraw-Hill,

London.

[22] A. H. Levis, (1984). "Information Processing and Decisionmaking Organizations: A Mathematical Description," Large Scale Systems, Vol. 7.

[23] V. Bouthonnier and A. H. Levis, (1984). "Effectiveness Analysis of Cs

Systems," IEEE Trans. on Systems, Man, and Cybernetics, Vol. SMC-14.

[24] M. Van Greveld, (1985). Command in War, Harvard University Press,

Cambridge, MA, and London, UK.

[25] J. S. Lawson, Jr., "The Role of Time in a Command Control System," Proc.

4th MIT/ONR Wokshop on Distributed Information and Decision Systems

motivated by Command-Control-Communication (C° ) Problems, Vol. 4, C'

Theory, LIDS-R-1159, MIT, Cambridge, MA.

[26] Command and Control Evaluation Workshop (1985). R. Sweet, M. Metersky,

and M. Sovereign, Eds. MORS.

[27] TACFIRE Operations, (1979). Field Artillery Fire Direction Systems,

Dept. of the Army, Field Manual FM-6-1, Washington DC.

[28] P. H. Cothier, (1984). "Assessment of Timelines in Command and

64
Control," M.S. Thesis, LIDS-TH-1391, Laboratory for Information and

Decison Systems, MIT, Cambridge, MA.

[29] J. G. Karam, and A. H. Levis, (1984). 'Effectiveness Assessment of the

METANET Demonstration,' Proc. 7th MIT/ONR Workshop on C' Systems, LIDSR-1437, MIT, Cambridge, MA .

[30] A. H. Levis, P.K. Houpt, and S. K. Andreadakis, (1986). ' Effectiveness

Analysis of Automotive Systems," Proc. 3rd IAVD Congress on Vehile

Design and Components.

[31] L. A. Washington and A. H. Levis, (1986). 'Effectiveness Analysis of

Flexible Manufacturing Systems,' Proc. 1986 IEEE Int. Conference

Robotics and Automation.

.4

A.

n

6

A.

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