on choosing combinations of weapons

ON CHOOSING COMBINATIONS OF WEAPONS*

Samuel Karlin Stanford university+

William E. Pruitt University of Minnesota

William G. Hadow Stanford Research Institute

INTRODUCTION The purpose of this paper is to develop a model for evaluating the advantages of various

weapons mixes for battle engagements and to consider the choice of weapons mixes for various purposes.1

Various alternative weapons systems are available, each possessing individual merits. For example, some weapons are more effective for distance firing, but are severely limited in their capacity for accurate rapid fire. Other weapons may be desirable for battles typified by closein fighting. Some weapons require greater maintenance capabilities, use of more manpower, and the like. The objective is to determine the best weapons combination that optimizes some criterion subject to the natural constraints of the problem.

A s a preliminary, the problem of optimum firing policy for a single weapon of known accuracy is considered.

The effectiveness in firing a single type of weapon is described by a Poisson process with variable rate x(s) where s represents the distance of the enemy from our position. We postulate that in any engagement the enemy is approaching at a uniform rate. We decompose A(s) into two factors p ( s ) and a (s) such that A (s) = p ( s ) a (s). The interpretation is as follows: a (s) is the infinitesimal accuracy function as a function of the distance s for the weapon being considered, i.e., the probability of hitting the enemy at the distance from s i h to s when engaging the weapons system at unit rate is a (s) h + o (h). By its very meaning, a (s) is naturally assumed to be decreasing and could possibly vanish for s 2 So.

The function p (s) is associated with the firing policy and usually signifies the rate of firing. There are natural constraints like 0 5 p (s) 5 M ( s ) which express limitations of the rate of firing. The restriction

p (s) ds 5 6 r 0

indicates a constraint on the quantity of total firepower available for that weapons system. Broadly speaking, a (s) reflects the influence of the enemy and other physical consid-

erations on the effectiveness of the weapon, while p ( s ) refers to the policy at our disposal to

*Preparation of the paper was pr imar i ly supported by Stanford Research Institute. t P a r t i a l support was given by the Office of Naval Research (Nonr-225-28). IAlthough motivated by the problem of weapons mixes, we believe that the models formulated and the techniques developed here in a r e useful in the analysis of other models occurring in mili tary situations and general managem,ent problems.

95

96 S. KARLIN, W. E. PRUITT AND W. G . MADOW

adopt in using the weapon. The accuracy and rate of fire may be affected by how close the enemy is. This is accounted for in the variable nature of a (s) and p (s) as functions of s. Of course, i f p ( s ) = 0 on any interval, then the firing policy calls for no firing in this interval.

Suppose we adopt the firing policy p (9). Since a (s) is assumed fixed by the specifica- tion of the weapon system, the choice of p ( s ) determines A(s). Let PA ( 6 ) be the probability that the enemy survives up to the distance s (we assume the enemy begins at a and approaches 0, our position). Then, in a standard way,

P,(s) = PA(s + h) [l - A(s) h] + O(h).

It follows, as h - 0, that

Solving and using P (a) = 1, we have

The probability density of the location of kill is

dPx(s) = A(s) e 6 ds . From (l),

means zero probability of the enemies' survival to our position;

for s > 0 means non-zero probability of survival up to distance s. Usually both are assumed. For a single weapons system, only p ( s ) (the firing policy) is considered as a free vari-

able to be judiciously determined. A criterion for determining p is as follows: Choose po to maximize

where p obeys the constraints specified earlier and g (s) is interpreted as the gain derived in destroying the enemy at distance s. We assume g(0) = 0 and g(s) [l - P,(s)] -+ 0 as s ..+ 0 and a. (This assumption will be justified for many of the interesting cases.) Integrating by

ON CHOOSING COMBINATIONS OF WEAPONS 97

This is a concave functional of p(s ) . A method of solution of (2) is described in Ref. [2] (vol. II, chap. 8). For a single weapon system we always determine p according to (2) for an appropriate g(s).

Superposing k units of a single weapons system and assuming the use of each is governed by the same policy and also postulating that the weapons act independently, means effectively that we replace the Poisson parameter A (s) by kA (s). To each weapons system corresponds a combined policy-accuracy function ferent weapons systems, one corresponding to A (s) and the other to

binations of these systems to assign to our force when we specify n of type A (s), and N-n of type i (s) . The kill rate will be

(6 ) . Suppose there are available two dif-

Assume now that our facilities are such that we must select a total of N of any com- (s).

(3) A(n, s) = nA(s) + (N-n) x(s) . For the weapons mix (3)) we have the cumulative survival probability distribution of kill.

P(s,n) = e

Plotting P(s, n) for each n gives a profile graph of the probability of kill for each combination of weapons mix.

Let a3 Q)

H(n) = 5, g(s) dsP (s,n) = g'(s)[l - p (s,n) 1 ds *

Given the functions A (6 ) and Since H (n) is concave but not necessarily monotone, the optimal n will not, in general,

be either 0 or N. If H is monotone, then of course n = 0, N according as H is decreasing or increasing.

More generally, suppose available r different types of weapons systems with associated accuracy functions

(s), we may seek n to maximize H (n), 0 5 n 5 N.

A s a first approximation we assume that the choice of firing policy to be used for any weapon is that one which optimizes (2). Therefore, the survival probability of the enemy against one unit of weapon system i is

Pi(s) = e 9 hi (0 = ai ( 5 ) ~ ~ ( 5 ) (i = 1, ... , r).

If we employ ni units of weapon systems i (i = 1, . . . , r), the probability of survival becomes

2Karlin, S., Mathemat ica l Methods and T h e o r y in G a m e s , P r o g r a m m i n g , and Economics .- (Addi- son Wesley Publishing Go., 1959).

98 S. KARLIN, W. E. P R U I T T AND W. G. MADOW

(4)

r - Js Zl ni xi (5) d l

U(nl, n2, . . . , nr, s) = e

The function (4) is a convex function of nl, n2, . . . , nr . The constraints on ni may be of the following sor t

a ln l + a2n2 + ... + arnr = y ,

where, for example, o i is the amount of manpower needed in handling one unit of weapons i , and y is the total manpower available. Another type of constraint is

n1 + n2 + ... + n = N r

which means that exactly N weapons units are available for the given force. Other constraints may reflect inventory and replacement opportunities, fixed costs, and so forth. In any case, we have a set of linear constraints satisfied by nl, n2, . , . , nr and we seek to determine ni maxi- mizing the concave functional

g (s) - u (n1, "2, * , nr, s>3 d s *

0

There are standard computation techniques for this concave programming Another point of view is to solve for X (s) which maximizes

f g (s) dP, (s ) 0

set up.

under some suitable constraints. Denoting the solution by X * (s), we then determine the weapons mix which most closely approximates that solution.

We summarize briefly the contents of this paper. In the second section (The Optimal Firing Policy for One Weapon), the technique of de-

termining the optimal firing policy is elaborated in the case of a single weapon. The method is a standard variant of the calculus of variations where the functional maximized is concave. The solution is composed of successive intervals on which the usual calculus of variations solutions is valid followed by an interval where the constraint restrictions dominate. In the present context several typical cases of the method are developed. We do not exhaust all of the possibilities arising from different kinds of assumptions imposed on the type of weapon, the gain function, and the like.

In the third section (Examples-Optimal Policy for One Weapon Systems), we apply the results of the second section to various examples and interpret their contrasting features.

The fourth section (The Optimal Firing Policy for a Mixture of Weapons) is devoted to an analysis of the optimal firing policy for a mixture of weapons. This displays the method of solution for the case of several types of weapons with mutually interacting effects.

ON CHOOSING COMBINATIONS OF WEAPONS 99

The last section (General Formulation) is devoted to a brief discussion of various generalizations of the models previously introduced.

THE OPTIMAL FIRING POLICY FOR ONE WEAPON

follows. The enemy is assumed to approach at a constant rate. The accuracy of the weapon is characterized by a function a (s) in that the probability of destroying the enemy when the enemy is in the interval (s, s + h) is a (s) h + o (h) assuming the weapon fired a t unit intensity. The firing policy is specified by a function p (s) which prescribes the intensity of firing the weapon as a function of s. The natural constraints satisfied by p ( s ) a,- e

The essential elements of the problem for determining the optimal firing policy is as

(5) 0 5 p ( s ) 5 M , p ( s ) d s 5 6 .

The interpretation of these conditions a re discussed in the Introduction. Our objective is to determine the optimal firing policy, i.e., the firing policy satisfying

the constraints (5) which maximizes expected gain. To this end, we postulate a gain function g(s) which represents the value attained if the

enemy is destroyed at a distance s. We assume that g(s) is bounded increasing and absolutely continuous with a bounded derivative and g (0) = 0. The expected gain effected by a firing policy p ( s ) is computed by the formula

where P (s) denotes the probability that the enemy survives to a distance s. (We assume throughout that a and g are continuously differentiable. These require-

ments a r e far more stringent than necessary but simplify considerably the exposition, thereby avoiding technical complexities of a tedious nature not decisive as to the results that follow.)

For the firing policy p (s), the probability P (s) is

(see Introduction). Then

Let po (s) denote an optimal policy. The existence of po (s) under suitable conditions will be proved by explicitly exhibiting po (s). The method employed is a standard variant of the

100 S. KARLIN, W . E. PRUITT AND W. G. MADOW

calculus of variations with the special feature that the given functional is c ~ n c a v e . ~ The procedure is to linearize the problem which puts it in a form to which the classical Neyman-Pearson lemma is applicable. To this end, consider

A d s (0 5 A 5 1)

where p ( s ) represents another policy satisfying (5). Then

and

Thus I (A) is a strictly concave function of A. Using this fact, we infer that po is the optimal firing policy i f and only i f I' (1) 2 0 for all choices of p, i.e., i f and only if

for all p satisfying

0 5 p(u) 5 M and r p ( u ) du 5 6 . 0

3This method is exemplified in considerable de ta i l in chap. 8 of Ref. [ Z ] . Also, s e e Refs. [4-71 for other examples of the use of this method.

4Bellman. R. , I. Glicksberg and 0. Gross , "On Some Variational P rob lems Occurring in the Theory of Dynamic Programming," Rendiconti de l Circolo Mathematico di- P a l e r m o (1954).

5Danskin, J., "Mathematical Trea tment of a Stockpiling Problem," NRLQ, 2:99-109 (1955). 6Arrow, K. and S. Karlin. Studies in the Mathematical Theory of Inventory and Production

(chap. 4) , edited by Arrow, Karlin. and Scarf (Stanford University Press, Stanford, California, 1958).

7Koopmans, B. O., "The Theory of Search , I , Kinematic Bases," Operations Research.4:324-346 (1956).

pp. 1-35.

ON CHOOSING COMBINATIONS O F WEAPONS 101

Interchanging orders of integration, we see that (7) is equivalent to the inequality

where

(9)

Since H(s) 2 0 (recall that g'(u) 2 0 a.e), it follows that i f

for all admissible p(s) , then (7) holds and indeed po ( 6 ) is an optimal firing policy. The diffi- culty in determining po (s) from (10) is that H (6) itself depends on the unknown po (s) and, therefore, the problem is not truly linear as it seems to appear. Nevertheless, we proceed for the moment regarding H (s) as known. In this event, the classical Neyman-Pearson lemma applies and, thereby, provides a constant c such that

M i f H ( s ) a(s) > c

i f H (5) a(s) < c

arbitrary i f H ( s ) a(s) = c . Of course, po (5) is required to be admissible (i.e., satisfies (5)). Now suppose it is possible to prescribe an admissible function po (s), and calculate H (s) so that the relations (11) are verified for some constant c; then po (s) is necessarily an optimal firing policy. In fact, suppose po (5 ) exists satisfying (11) and (5). Then, according to the Neyman-Pearson criteria the inequality (10) is assured and consequently po ( 8 ) is an optimal policy.

obtained following the method just described. We will now list several cases of interest where the explicit optimal firing policy was

THEOREM 1. Suppose a(0) = Q), a(s) decreases to zero at 00, and define

S. KARLIN, W. E. PRUITT AND W. G. MADOW 102

If f (s) > 0 for all s > 0, then

where so is determined by the condition f(s) ds = 6 , provided f(s) 5 M (0 < s 5 so). 6" REMARK 1. The expression (12) can be motivated as follows. If H (s) a (s) 2 c on an

interval, then twice differentiation and further obvious manipulation leads to the identity po (s) = f (s) on this interval.

PROOF. Let PO (s) be defined according to (13). Then a direct calculation produces

By setting

the consistency requirements (11) will be satisfied, referring to the REMARK 1, i f it can be shown that H(s ) < c/a(s) for s > so. Let h ( s ) = H ( s ) - c/a(s). Then h'(s) = g'(s) + c[a'(s))/a (s)] for s > so. Also 2

and consequently

or equivalently h '(s) < 0 for s > so. Thus, h ( s ) < 0 for s > so since h(so) = 0.

103 ON CHOOSING COMBINATIONS O F WEAPONS

We now show that so can be chosen so that f(u) du = 6. Indeed, we will prove 8;" co

f(u) du = co for any X o > 0.

If we suppose the contrary, then A0

S co f(u) du 5 \ f(u) du 5 A < co (A = a cor.stant)

0

It follows as in (14) since a(u) is uniformly bounded for u in [x0p) that

and A' is a suitable constant. Integrating the relation - (a'(s))/(a2(s)) 5 Cg'(s) we obtain

But the conditions that g is bounded and a(s) 1 0 (s - co) a re clearly untenable in the presence of the inequality (15).

Since f (s) > 0 entails that the integral

diverges it follows that for any 6 > 0, a point so can be determined satisfying

f(u) du = 6 . 0

The proof of Theorem 1 is thus concluded.

proofs. The next two theorems are proved by the same methods. We will merely sketch the

THEOREM 2. Let f (s) be defined by (12). Suppose there exists so such that

d - [a(s) g(s)J = k(s) > 0 ds

for s < So and kb0) = 0; and

f(s) > 0 for s > S O .

104 S. KARLIN, W. E. PRUITT AND W. G. MADOW

Then the optimal firing policy has the form

where sl is determined by the condition

f(u) du = 6 , 80

provided f(s) 5 M for so < s < sl.

PROOF. Let po (8 ) be specified a8 in (16). An immediate calculation yields H (8 ) = A * g(s) for s < so where A isaconstant. Explicitly,

Define

Since H(s) a(s) = A g(s) a(s) is increasing for 0 < s < so, we have

with the last identity resulting from the fact that k(so) = 0. This establishes the consistency relations on the interval (0, so).

H (so) a(so) = c. The verification of the consistency properties for s > sl is accomplished as in the proof of THEOREM 1.

A slight generalization of THEOREM 2 is as follows: Let so be the maximum point of a (s) g (8). Suppose f (s) 2 0 for s 2 so. The optimal policy has the form (16).

Another specific example of THEOREM 2 is embodied in the following theorem.

On the interval (so, sl), we have H (8 ) a (6) = c by the definition of p o (6) and c, since

THEOREM 3. Let a(s) = l/s, and g(s) have the property: g"(s) > 0 for s < so; and g"(s) < 0 for s > so. Then


Io 0 S S l S 0

where so is the maximum point of g(s)/s, and s1 is determined by the condition

provided that po(s) 5 M for so 5 s 5 sl. Since g(0) = 0, the convexity of g(s) on s < xo implies that g(s)/s is increasing for

S ' % .

REMARK 2. We could enumerate other types of solution by imposing different sets of conditions on a (s) and g (s). The number of possibilities that can arise is very large. It seems impossible to present an exhaustive accounting of these solutions. THEOREMS 1 - 3 are typical of the cases where the solutions involve firing policies confined to a single interval. More complicated optimal policies involving several intervals of firing appear when the functions a'(s) and g'(s) oscillate.

REMARK 3. Since the functional I (A) defined in (6a) is strictly concave, we conclude that the optimal firing policy, i f it exists, is unique. This leads to the interesting analytical fact that the hypotheses of THEOREMS 1 and 2 are non-overlapping in view of the contrasting char- acter of the optimal policy. A direct proof can also be given that the properties f (8) > 0 for s > 0, a(0) = a, compared with the condition that k(s) > 0 for 0 5 s < so cannot hold simultaneously.

REMARK 4. Several qualitative results pertaining to the general nature of the optimal policy can be developed. We list two such properties which are valid under slight smoothness conditions without entering into their proofs:

1. If g' (6) is bounded for 0 5 s < so, then the optimal policy never calls for firing at the maximum rate M in the neighborhood of s = 0.

2. If a(s) is integrable at s = 0, and g'(s) is bounded for 0 5 s 5 so, the optimal policy requires no firing in a neighborhood of s = 0.

EXAMPLES-OPTIMAL POLICY FOR ONE WEAPONS SYSTEMS In this section, we consider the problem of comparing different weapons. The point of

view is to consider a weapon, characterized by its accuracy function a(s), and determine the optimal firing policy for that weapon. Then the probability of survival P ( 6 ) (see (1)) is computed and plotted as a function of 8. These curves may then be compared for a number of different weapons to see which have various desirable characteristics.

106 S. KARLIN, W . E. PRUITT AND W . G. MADOW

We also examine different choices of the gain functions g (s) in combination with various weapons. On the basis of these curves and formulae, the reader could easily interpret for himself the relative merits of the different choices of g ( s ) and weapons a (s).

For simplicity we assume throughout our discussion of these examples that M = 1 and 6 = 1.

EXAMPLE 1. A-. Let

The first weapon considered will have accuracy function al ( s ) I

bl (a l - s ) 0 i s 5 al

s > a l . (19) a1b) =

It is assumed that a1 5 d and (albl + 4)2 5 2a:bl. Then the optimal firing policy is, by THEOREM 2,

s < s1

s > t l ,

where s1 = a1/2, t l = a l - 2al/(albl + 4). The probability of survival of the enemy is then

This will be plotted below with the other probability of survival functions.


B. For the next weapon consider a2 (s) = l/s and g (s) defined in (18).

S

This provides an example which fits none of the categories included in THEOREMS 1 - 3. In fact, in this case the function (12) is identically zero. Nevertheless, we asser t that the optimal firing policy has the form

s < s2 f o

where s2 = - 6 + i n , t2 = 1/62,d2. The probability of survival is given by

s2 5 s < t2 -1 c1 s > t 2 .

We briefly indicate the proof that (21) is the optimal policy. A direct calculation yields

s 5 52 s2

t2 H(s ) a(s) E c = k -

H ( s ) a(s) > c s2 < s < t2

H ( s ) a(s) < c s > t 2 .

These relations permit immediate verification that the consistency relations (11) hold for the policy (21).

policy still has the form (21). Then s2 and t2 depend on 6 , d, and b2. A slightly more elaborate argument will show that i f a(s) = b2/s then the optimal

108 S. KARLIN, W. E. PRUITT AND W . G. MADOW

C. Let a3(s) = b3/sci, (Y > 1. Then i f 6 0 5 [(6ab3)/(ci - l)]l/cy , the optimal firing policy in accordance with THEOREM 1 is given by

S > S 3 '

where s3 = [(6cib3)/(cr- l)ll/". The restriction p3 (s) 5 1 is guaranteed by virtue of the con-

dition [(6cib3)/(ci - 1)11/" 2 6 0 . The probability of survival of the enemy is

s > s3

-a4s D. Consider a4 (s) = b4e . Then i f a4[(e + b46)fi4 5 1, the optimal firing policy

is given by THEOREM 2 as

where s4 = l/a4 , t4 = l/a4 log (e + b46). The corresponding probability of survival is

-a4 04 - s4) s < s4 ie s > t4 I'

These probability of survival functions will now be plotted for 6 = 1, and some particular curves. The values chosen were al = 6, bl = 1/2, d = 7, b2 = 1, b3 = 2, ci = 2, b4 = 6, and a4 = 2/3. The accuracy functions, and probability of survival curves a re plotted in Figure 1 for these values of the parameters.

It is interesting to notice here that the accuracy function al (s) gives rise to a probability of survival curve that is almost uniformly better than that corresponding to the accuracy function a4 (s) although the latter accuracy function is considerably greater for small values of s. The extreme differences in the survival curves P2(s) and P3(s) corresponding to the accuracy functions l/s, 2/s is somewhat surprising. 2

ON CHOOSING COMBINATIONS O F WEAPONS

ACCURACY FUNCTJONS

- I I

0 I 2 3 4 5 6 7 B

PROBABILITY OF SURVIVAL

I - P p PI (s)

0.75 -

I I I I 0 1 2 3 4 5 6 7 B

Figure 1

EXAMPLE 2: Let

109

I k s > d

This gain function was motivated by the probability of survival function P3 (9) just obtained, the thought being that our gain should be proportional to our probability of survival, assuming the enemy is using a similar weapon system. Notice that when P = 1 this is the previous case.

The same accuracy functions will be considered as in the previous example. A. For al (s), it is assumed that al 5 d and there will be an assumption to the effect

that p(s) 5 1 for sl 5 s 5 tl although this will not be as easily stated as it was in the previous case. Here, we have

110 S. KARLIN, W. E. PRUITT AND W. G. MADOW

s < s1

where s1 = (p/l+p)al and tl is the solution of

The probability of survival is then

S > t l '

B. We consider next the accuracy function a3 (s) = b3/sff for a > 1, and (Y > /3. In

this case i f a6 5 [(o6b3)/(a - S ) ] l / a , then

0 5 S 5 S 3

S > S 3 '

P 3 b ) =

where s3 = [(ab36)/(0 -B) ] ' / " . The probability of survival function is then

P (s) = [ (;)- s < s 3 3

c' s > s 3 .

C. For the final accuracy function a4 (s) = b4e-a4s, the optimal firing policy is given


f o s < s4

s > t4

where s4 = p/a4 and t4 is determined by

p4 (tq) cr 1. The probability of survival function is then

p4 ( 5 ) d( = 6. This is provided, as always, that P 54

l1 s > t4

THE OPTIMAL FIRING POLICY FOR A MIXTURE OF WEAPONS In this section, some results will be obtained on the optimal firing policy for a mixture

of weapons. It is assumed that there is available a system of weapons with accuracy functions al (s), . . . , ak (s). It is not assumed that all of the ai (s) are different. There is a gain function g (s), as before, and the desire is to maximize expected gain, which is expressed by

A s before, it is assumed that the firing policies, pi ( 5 ) are restricted by

This problem can be attacked in much the same way as the problem of finding the optimal firing policy for one weapon, although there are , of course, many more possibilities that arise.

following: If a number of different8 kinds of weapons are in the system, whenever two o r more types a r e being fired simultaneously all but one type must necessarily be fired at maximum

A couple of general observations can be made immediately. The most important is the

8For the purpose of this statement, weapons with accu racy functions a ( s ) and a (s) will be 1 2 considered of the same type if a ( s ) = c a ( s ) on any in te rva l and both a r e nonzero there . 1 2

693-440 0 - 63 - 2

112 S . KARLIN, W. E. PRUITT AND W. G. MADOW

intensity. The tendency, in fact, will be to fire different types of weapons at different times. For example, some type will be best for long range fire, another for intermediate range, etc., and one must use the long range weapons to their full potential when they are the most effective. The formal proof of the assertion of this paragraph goes as follows. By the methods expounded next, we deduce that whenever weapon I is used at non-maximum capacity on an interval I, then H (s) ai (s) = ci holds for s in I where ci is an appropriate constant. It follows that two weapons i and j used at less than maximum capacity lead to the relation ai (s) = c a. (5) on that interval and therefore weapons i and j are not different as presupposed.

should be fired when they are the only weapons in the system. If the restrictions on one weapon of this type are M and 6 , the procedure is to solve the optimization problem with restrictions nM and n6 obtaining a firing policy pl (t). Then the policy for each of the individual weapons is taken to be l/n p1 (t). It should be pointed out that this means that where weapons of a given type are added, there will be a tendency to fire each one at a lower intensity, but for a longer period of time.

We will now give a class of examples where explicit answers can be obtained. It will first be assumed that there is only one weapon of each type involved. Then the problem of finding the optimal number of weapons of each type to use as well as the optimal firing policy for each will be discussed afterward.

J

The other general observation is the manner in which n weapons of the same type

and that

let

0 1

Then if f,({) 5 Mi on the interval (si, ti) to be defined later, the optimal firing policyp. ( t ) con- s is ts of firingthe ithweapon with intensityf.(t) on the interval (si, ti) where { ~ i > ~ = ~ , {tJf& are determined recursively by the relations

k 1


and s1 = 0, where

k These relations determine the {siliZl, {ti$l uniquely. The expected gain will be fk(tk).

REMARK

fire the total complement of Now suppose that we consider using ni weapons with accuracy function a. (s) and we

1

weapons according to their optimal firing policy. Then the following question is of fundamental importance: Subject to the restraint Ccini 5 C, what choice of {ni}fZl will result in the maximum expected gain.

If the weapons are of the type described in the previous THEOREM, this problem can be solved by the method of concave programming.

The proof of the THEOREM and of the statement of concavity will be given next. It should be pointed out that these cases include a wide range of accuracy functions. In particular

the interesting cases of accuracy function of the form ai(s) = bi/s ', bi > 0, oi > 1 are included. The PROOFS given will also give an indication of the method that may be used to attack the problem for accuracy functions that do not meet the requirements of the THEOREM.

0.

PROOFS. Consider the function

This is easily seen to be a concave functional of the Ai, and will have a maximum over the unit cube at (1, . . . , 1) i f and only i f

Expanding the differentiation and interchanging orders of the integration, we see that (25) is equivalent to

"co

114 S. KARLIN, W . E. PRUITT AND W. G. MADOW

for all i where

The solution is again given by the Neyman-Pearson LEMMA, i.e., there is a constant ci such that

i f H ( s ) ai(s) > ci

H ( s ) ai(s) = ci

H ( s ) a i ( s ) < ci .

Mi

arbitrarybetween 0 and M.

0

1

k In order to complete the proof of the theorem, it is enough to establish that for {py ( s ) } ~ = ~ a s prescribed, the relations (27) a r e satisfied (see section entitled "The Optimal Firing Policy for One Weapon"). For si < s < t i ,

so that we want

Then we will have H (s) = ci/ai ( s ) on this range provided

'i H(s i ) = - . ai (si)

This will be shown by induction. It is clearly true for i = 1. Now


since H (ti-1) = ci-l/[ai-l (ti-l)]. But inserting and dividing an exponential factor, we have

since the last exponential is equal to

2 - ai (si) g' (si)

Therefore, H (si) = ci/ai (si) from the relations determining si. This shows that i f the si and ti a re determined a s stated in the THEOREM we will have H (s) = ci/ai (s) for si < s < ti . To see that fi (si) = f i - l (ti-1) has a solution, first notice that

116 S. KARLIN, W. E. PRUITT AND W. G. W O W

and so the solution if it exists is unique. Now, fi (ti-1) < fi- l (ti-1) by the assumption (-a; (s)/ ai (s) is increasing in i) and as s - m,

- - and so lim fi (s) = lim g ( s ) and obviously fi(s) < g(s ) so that fi(s) > fi-l(ti-l) infinitely

s --+ co s - Q , often. Thus a solution for si exists when timl is given. It was already noticed earlier (see THEOREM 1) that

1

so that ti is uniquely determined when si is known.

ci/ai(s). For ti-l < s < si, The remaining consistency relations a r e checked as follows: Let hi ( 6 ) = H (S) -

.. 1

But

is monotone implies h;(s) > 0. Thus H(s) - c/ai (s) < H (si) - ci/ai (si) = 0 for this range. A similar argument applies to H (s) - ci-l/ai-l ( 6 ) on the same range. Finally, notice that an integration of the assumption on a; (s)/ai (s) shows that ci/ai (s) and c./a. (6 ) can intersect at most once. Since

I 1

it follows that ci/ai (9) and ci-l/ai-l (8 ) intersect in the interval (ti-l, si). The fact that they cannot intersect elsewhere gives the desired consistency relations everywhere.


The expected gain is

We now present a proof that the expected gain as a function of {ni>fzl is concave. Be- cause weapons of a single type can be combined, the constraint set on the case of multiple weapons becomes

where pi ( 5 ) represents the cumulative firing capacity of the ith type weapon. Regarding all copies of a single weapon type as indistinguishable, we may reduce the conditions (30) to that where each single weapon of type i obeys the restrictions

0 5 pi(t) 5 Mi (i = 1, 2, . . . , k) 0

with the understanding that identical firing policy is adopted for identical copies of a given weapon. The expected gain for a specified policy satisfying (31) is computed by the formula

If we fix pl (t), p2 (t), . . . , pk(t) satisfying (31), it is clear from the form of the expression (32) that cp(nl, n2, . . . , nk; pl, . . . , p j ) is a concave function of {nJiZl. It can be shown by an elaborate argument that

k

(33)

is also a concave function. Some slight reflections show that one can regard the problem of determining the optimal

{p& in (33) as merely the problem of THEOREM 4 where ai ( 5 ) is replaced by niai ( 5 ) . This is clear after comparing the expression (32) with that of (29).

The optimal policy {py, p i , . . . , p;} as a function of {nJf=l has been explicitly characterized in THEOREM 4. Moreover,

(34) cp* (nl, " 2 9 ' * ' t "k) = fk ctk)

118 S . KARLIN, W . E. PRUITT A N D W. G . M A D O W

according to (29). Therefore, fk( ) is a concave function of {nl, n2, . . . , nk}. In order to determine the optimal choice of {ni ? which maximizes (34) subject to the restriction C nici 5 C we may apply the standard [9], [lo] algorithms of concave programming.

GENERAL FORMULATION In this last section, we suggest some generalizations of the previous models. The model of the section entitled "The Optimal Firing Policy for One Weapon," formu-

lated the battle engagement and firing policy in terms of a Poisson process in which the time parameter corresponded to the distance. The occurrence of an event in the Poisson process signified a lethal strike of the enemy. Our first extension proposes that the damage done on a hit be a random variable with various degrees of lethality. We may assume a s a first approximation that the damage done on successive hits are cumulate. Thus the second event, third event, and further events of the Poisson process have a bearing on the battle.

of all, it is easy to derive (and indeed is hown) that the probability density of the kth event occurring in the interval s, s + d s is

Various alternative mathematical formulations of these ideas suggest themselves. First

This can be substituted for (1) in evaluating the gain accrued from each successive hit. More generally, corresponding to the kth hit is associated a different gain function and these are to be appropriately cumulated.

The conditional distribution of the extent of damage done on each hit may depend on the number and types of weapons. For example, suppose n units of weapon h(s) and N-n units of weapon i ( s ) are utilized. The expression

nh(s) + x(s) (N-n)

is the probability that a hit at time s is due to the weapons of type h(s). Similarly,

(N-n) x(s) nA(s) + (N-n) X(s)

is the conditional probability that a hit sustained at time s comes from weapon type (s). This kind of probability information could be useful in deciding on the correct weapon apportionment.

There is no overwhelming reason for regarding the battle as a Poisson process although the previous discussion offers some plausibility. It is also reasonable to consider the battle in terms of a general continuous time Markov chain with a drift to the right. Thus, the process is

9Frank , M. and P. Wolfe, "An Algorithm for Quadratic Programming," Naval R e s e a r c h Logis- t i c s Quar te r ly , 3 : 95-1 10 (1956).

IOVajda, S., Mathematical P rogramming (Addison-Wesley Publishing Co., 1961).


in state k at time s, i f the enemy has sustained exactly k hits up to time s. Moreover, the waiting time in state k is assumed to follow a general distribution law which may depend on k.

An example of this is the Yule process. This is a pure birth process. If k hits oc- curred by time s, then kA(s) ds is the probability of a hit in the next interval s, s + ds. In other words, the more hits sustained by the enemy, the more vulnerable he becomes to further hits.

in a sense analogous to the way h (s) determines the Poisson process. Some suitable optimization cri teria may be:

1. To choose the weapon system which minimizes the expected time of reaching state k. 2. To maximize the probability of getting to state k in a prescribed time duration. The general formulation of a battle should embody the following concepts: 1. A battle is represented by a semi-Markov process in continuous time whose states

a r e identified with the non-negative integers with the following possible interpretations: 2. (a) States may represent the number of hits on the enemy;

(b) States may represent the damage level sustained by the enemy. The waiting time until the next transition is a random variable depending on the state of the system at the time of the last hit. The waiting times in the various states are determined by the weapons mix. Examples are the Generalized Poisson Processes, the Yule Process, and the Branching Process.

The structure of the weapon system determines the parameters of the stochastic process

For each state there is a gain function which is a function of time = distance of the enemy. We may compute expected gain and optimize appropriately.

* * *

on choosing combinations of weapons

Documents