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WHS-542 auD Copy 4/ of 1/ 

11 Sheets 

28 November 1967 

Notes on Selection of Targets for Viewing 

References: 1. WHS-480, "The Effect of Alternate Targets upon 

Primary Path Selection," dtd 9 October 67. 

2. P. Perkins" Summary of Effective Worth of Primary," 

dtd 25 October 67. 

I. Problem Statement 

The requirement has been established that after taking a primary or one 

of its alternates, as the case may be, the main optics must then be able 

to acquire the next primary. This requirement geometrically imposes 

a latest possible decision time upon each alternate. In other words, for 

any alternate under consideration the decision to take that alternate must 

be made on or before a certain time or the main optics will not be able 

to acquire the next primary. This constitutes one limitation. If there 

are relatively few alternates for each primary, this may be the only 

limitation that need be considered. In that case multiplexing or other 

techniques may be used to define various sets of alternates which may be 

viewed by the ATS up to certain key decision times. The equations of 

Reference 1 and 2 may then be used to determine the average score for 

each of these sets and the maximum case selected therefrom. 

On the other hand, if there are many alternate targets, the constraint 

of having to be able to acquire the next primary with the main optics, 

may not sufficiently limit the number of alternates for any specific 

decision time. There may be more alternates still available for viewing 

through the ATS than time permits. The question of which of these 

alternates should be selected for viewing may not be answered by taking 

all possible combinations (multiplexing) because of excessive computation time. 
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The operational flight problem may be stated as follows. Given a target 

complex of M total targets consisting of one primary and a group of 

possible alternates. Only P (generally P < M) targets may be viewed 

by the man and each observed to be either active or inactive in his 

opinion or cloud covered. One and only one target will then be selected 

to be taken on the basis of greatest probable worth. 

The problem at hand is to determine before flight which P targets should 

be selected for viewing on the basis of greatest average expected score. 

Reference 1 and 2 present the exact closed form solution for the average 

expected score once it has been decided which Ptargets are to be viewed. 

Manipulation of this solution is used herein to solve the problem of 

deciding which targets should be selected for viewing. 

II. Method of Solution 

The solution of Reference 1 and 2 for average effective worth of a complex required that the component worth entries of observed active, observed 

inactive, or unobserved targets be entered in a single list in a fixed order 

or hierarchy. That identical hierarchy will be preserved throughout this 

solution. 

The equation for average total score shows clearly that, no matter which 

targets are viewed, the contribution of any entry is greatly influenced by 

all entries ahead of it on the final list and that it in turn effects the contributions to average expected score of all entries below it. Therefore, 

it is presumed that there is no simple closed form equation to tell which 

targets should be selected for viewing. Basically, the procedure used 

herein will be to first compose a trial list of P targets with entries 

nearest the head of the master list. The next target from the master 

list is now added as a trial value. Successively from top to bottom the 

effect on total score of deleting each single target including the one just 

added is determined and the one the deletion of which results in the 

highest total score is permanently discarded. Then the next target from 

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the master list is added and the procedure repeated until the master list 

is exhausted. Depending upon the magnitude of the numbers involved, a 

second pass through the target list may be required. The targets then 

remaining on the partial list are the ones to be viewed. 

Actually the implementation of this procedure is more complex than the 

foregoing explanation may indicate, because continuously added as the 

lowest entry on each trial partial list must be the greatest unobserved 

effective worth selected from the remaining targets on the master list. 

Also some targets will have two instead of one entry on the trial list as 

it is being developed. Further, these two entries may have between 

them the entries of other targets. 

The general outline of the following parts of this report is to first present 

the solution of Reference 1 and 2, then analyze the effect of each target 

upon the possible value of each other target in the process of selecting 

targets to be observed and, finally, outline a possible procedure for 

solution. 

Symbols 

The following symbols are the same as those of Reference 1 and 2, 

though the contractor has used other symbols when he programmed 

this solution for the average expected gain. 

WA Basic Active Worth 

WI 

Basic Inactive Worth 

P

A Probability of Activity 

V 

Probability of Visibility 

P

R 

Probability of Recognizing Activity 

P

F Probability of False Alarm 

PA 

Probability Man will Designate Target as Active 

PI Probability Man will Designate Target as Inactive 

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WA Effective Active Worth of Target which has been 

Designated as Active 

CSI Effective Inactive Worth of Target which has been 

Designated as Inactive 

WA= -d A pA = The part of the Effective Worth of a Target, 

which is due to the possibility that it might 

be Designated as Active 

WI = p

I = The Part of the Effective Worth of a Target, 

which is due to the possibility that it might be 

Designated as Inactive 

= WA +WI = [PA WA + (1- PA) WI]P

V = Effective Worth of a 

Target which the man will not attempt to observe. 

EFF W = Effective Average Worth of Complex or Primary 

HA. Computation of Average Worth of a Defined Group 

The Appendix attached hereto is the solution of Reference 2 for the 

effective average worth of a group of targets when the selection of targets 

for viewing from within that group has been pre-ordained. 

The final equation for this solution may be written in abbreviated notation 

as follows. Both the Table of Symbols and the Appendix must be read 

to obtain the meaning of the symbols. 

N j-1 

EFF W = +E ( p 

P j=1 i 

j=2 i=1 

This equation may be written in expanded form as follows: 

7.) 717: 'IT: 

Ji' 

, .--- 

11AftNiji;;;1?' 
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EFF W = (4) 1 

+ (1-71) w2 

+ (1-71) (1-P2) 3 

(1-71.) (1-72) (1-7) W4 

+ (1-1f) (1- /-3-) (1--73

) 6-74) 4) 1 2 5 

+ etc. 

Each line on the right hand side of the equation represents an entry from 

the list of a s as explained in the Appendix. Some targets may have two 

such entries, not necessarily consecutively. The value of the entry 

represented by the third line in the above equation has been degraded by 

the factor, 

(

1-7r) (1- 2

< 1. 0 

1 

Always O<T)< 1.0 

The amount of this degradation depends upon which targets are selected 

for entries one and two. All entries after the third are degraded by the 

factor, 

3 

Thus, the net contribution to total average worth of adding the k'th entry 

may be precomputed as, 

NET Wk= Ll Wk- Pk Ed W. 

i=k+1 

j=i-1 

where QW. cd. 3i1 (1-70 

As shown by this equation for NET Wk, adding another entry into the 

list, diminishes the net contribution of every other entry beneath 
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the one being added and generally lowers the contribution of entries 

preceding it. Profitable additions to the list may render negative the 

net contribution of some other target, in which case the latter should 

be removed from the list. 

III. Computational Procedure 

1. For each target there will be given a 

WWPPP P A' I' A' V' R' F 

2. Compute for each target, 

PA = [ PA PR (1- PA) PF] PV 

P

I = P - PA V 

WA = [ PA PR WA + (1- PA PF WI PV 

WI = [ PA(1- PR) WA "F (1- PA)(1- PF) WI] PV 

[PA WA + (1- PA) -WI] PV 

WA 

WA pA 

- - 

p1 

3. List all W, WA, and wI, in order of descending magnitude 

without regard to subscript or superscript. Without changing 

the order, replace all wA and 71 in this list by the [.CIAand WI' 

respectively, for the corresponding targets. This is the master 

list containing three entries, WA, (Jrand w, for each target. 

Throughout the solution various entries will be extracted from 

this master list and transferred to an ever changing shorter 

(partial) list in which the same relative order of entries will 

be maintained. 
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Generally, the three entries for any target will fall in the order WA, 

6), WI, although it is possible that in some cases the reverse order will 

result. The relative magnitude of these three entries will not necessarily 

follow any particular order. Of course, between these three entries 

may fall some of the entries for other targets. 

All three entries for each target must be subscripted or superscripted 

in some way to indicate to which target they apply. Those with the .A 

or I subscript are referred to as "observed" values, or entries, and 

those without it as "unobserved". The second observed entry for each 

target is flagged as a second entry. Whenever an observed entry for a 

target is transferred from the master list to the trial list or vice versa, 

the other observed entry for that same target is also transferred. The 

terminology "next observed target on the list" means the target with an 

observed entry nearest the top of the list. 

The following steps constitute a possible solution: 

1. Transfer the P first-entry observed values from the head of 

the master list to a trial target list and also transfer the second 

entry observed values for these same targets. In other words, 

transfer the first P observed targets from the master list to 

the trial list. 

2. Transfer to the trial list the two entries for the next observed 

target on the master list. 

3. Temporarily remove from the trial list the observed target 

at the head of the list. Transfer from the master list to the 

trial list the greatest unobserved entry selected from among 

those targets not represented on the trial list and label it as 

the Nth entry. Compute the score of the complex represented 

on the trial list by means of the equation presented in the 

Appendix. Notice that the computation stops with the Nth entry. 

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Return the Nth entry to the master list and restore to the trial 

list the target that was temporarily removed. 

4, Temporarily remove from the trial list the second observed 

target and repeat the process as outlined in Step 3. Continue 

to repeat this process until a score has been obtained for the 

removal of each individual target from the trial list. Permanently remove that target, the removal of which results in 

the highest score. 

5. Repeat the entire process starting with Step 2 and continue to 

repeat it until all of the observed targets on the master list 

have been transferred to the trial list. The solution is now 

probably complete and the trial list should be a near optimum 

selection. If, however, the relative magnitudes of the numbers 

involved do not render an adequate solution, a single iteration 

should suffice. 

To select P targets from a total of M by the foregoing method requires 

that the average effective worth (score) be computed (P+I) (M-P) times. 

This may be reduced slightly by rearranging the procedure, though for 

computer usage it is probably not worth the bother. 

In the foregoing procedure, total average expected score was computed 

periodically. It is possible to compute only incremental scores caused 

by the various additions and deletions of entries to the trial list. However, 

this may result in extra complication if more than one entry per target 

is actually involved in the computation. If the magnitudes are such that 

only a single observed value for each target enters into the computation, 

then incremental scores are sufficient. 

/21, 

F. Perkins 

FP:dmm 

C.11175) 

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Appendix 

As sumption. s 

1. Certain targets in a group consisting of a primary and several 

alternates will be pre-selected for viewing by the Man. This 

analysis does not determine how to select those targets which 

should be viewed. It determines an average score for the group 

once the selection has been made. (A crude rule of thumb for 

selecting which alternates should be viewed if there is not time 

to view all of them, might be to take those with the higher WA. ) 

2. Operationally only one target will ultimately be taken. 

Computation 

1. For each target there will be a 

WA

' WI' PA' PV' PR' PF (see Table of Symbols) 

2. Compute for each target selected for viewing, 

PA PA PR -F. (1-1°A) PE] PV 

P = p _ P V A 

[ PA PR WA + (1- PA) PF WI] PV 

WI = [PA ( PR) WA + ( PA) (1- PF) WI] PV 

(2) 'Tr_ A 

A 

PA 6.) 

- I p

1 

3. Compute for each target not selected for viewing 

w= [ PA WA + (1- PA) WI] PV 

MYNA A.,711 
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4. From among the targets not selected for viewing, choose the 

one with the highest w and designate this single value as 6) 

without a subscript. 

5. List all 7518 in order of descending magnitude without regard 

to subscripts A, I, or the one value with no subscript. 

6. Remove all entries below the single non-subscripted W. 

7. Some targets will have two entries, both an WAand an tali 

remaining in this reduced list. For these targets flag the 

second (lower) entry for later reference. 

8. The effective average worth of the complex is as follows. 

Subscripts A or I are not shown in this equation because they vary. 

wj=1 N i-1 (4). 

1- P ) S. P. 

1J J 

where: i or j is the number of the entry in order 

of descending mixed TY 

N is the total number of entries 

— — 

W. is either WA' WI' or W whichever applies 

to the jth entry 

P. has the same subscript as GI. P is unity 

for the final entry. 

P is PAi orchwhichever has the correct 1 

subscript for the ith entry, except when 

the repetition flag is on in which case Pis Pv. 

EFF W = w. P. +E P j=1 j=1 

j=2 
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Theory 

See memo of 9 October 1967 for more detail. The reason the W's are 

ordered according to descending magnitude without regard to subscript 

is that in the operational case the target actually taken will be the 

first one on this list which the man finds is both visible and has an . 

activity in agreement with its subscript A or I. If none of those viewed 

by the man qualify, then the final entry (the non- subscripted one) would 

definitely be taken because it has a higher effective worth than the lower 

items previously deleted from the list. The probability that this last 

item is invisible is included in its effective worth. 

In the final equation for EFF W the terms (1 . P) occur. 

J J 

is the algebraically simpler term W. as indicated. 

This product 

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