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MEMORANDUM REPORT NO. 1509SEPTEMBER 1963

A DYNAMIC PROGRAMMING SOLUTION OF A

MISSILE ALLOCATION PROBLEM

William Sacco

RDT & E Project No. I 01O5O1AOO3and 1W232O1A098

BALLISTIC RESEARCH LABORATORIES

ABERDEEN PROVING GROUND, MARYLAND

DDC AVAILABILITY NOTICE

Qualified requesters may obtain copies of this report from DDC.

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The findings in this report are not to be construedas an official Department of the Army position.

BALLI ST IC RE SEARC1 LAB ORAT0R IES

MEMORANDUM REPCRT NO. 1509

SEPTEMBER 1965,,

A DYNAMIC PROGRAMMING SOLUTION OF AMISSILE ALLOCATION PROBLEM

William Saf~co

Computing Laboratory

PJ)T & E Project No. 1MO105OIA005 and JMO25201AO98

A B E R D E E N P R O V I N G G :1 O UN D, M A R Y L A N

BALLISTIC RESEARCCH LABORATORIES

1"N0RANDUM REPORT NO. 1509

WSacco/rhgAberdeen Proving Ground, Mi.September 1965

A DYNAMIC PROGRAMMING SOLUTION OF A

MISSILE ALLOATLION PIXOBLEMA-

ABSTRACT

A generalization of a missile allocation problem proposed by

Piccariello is formulated using dynamic programming techniques. The

formulation leads tc an efficient algorithm for computing integer solutions

to the discrete problem.

5

INTRODUCTION

H. PiccarieiLo1 has considered the following interesting missile alloca-

tion problem. Consider a target complex consisting of N missile launch

sites and M control centers. It is assumed that if one (or more) control

center survives an attack, then all surviving missiles can be launched and

if all control centers are destroyed chen none of the surviving missiles can

be launched. Furthermore the assumption is made that each of the N + M missile

sites and control centers presents a separate target to an enemy missilu

attack. hiis is equivalent to the assumption that a:n enemy missile can destroy

no target other than the one agaiast which it is directed. The problem is to

find an allocation of T identical missiles that when attacking the target

complex will minimize tne expected nwrber of' missiles capable of being launched.

Piccariello gives a solution to the problem when it is considered in its

continuous form (allocations are not restricted to integer values). He also

investigates the discrete form of the problem and shows by example, tnat, in

general, the solution in the continuous case is not a solution for the discrete

case. He then derives a necessary condition for the existence of a solution

for the discrete case to differ from the continuous case.

In this paper E generalization of Piccarie.lo's problem is formulated

using dynamic programming techniques., The formulation leads to an efficient

algorithm for computing integer solutions to the discrete problem.

THE FORMUlATION OF THE PROBLEM

We shal) use the following notations and definitions:

P i probability of survival of missile site i; i = 1,2,..when attacked by a single incoming missile,

P. probability of survlval of control center j; j = 1,2,...MJ when attacked by a single incoming missile,

a.: number of missiles at missile site i,1

T: total number of attacking missiles;

E(X,Y): expected number of surviving missiles after an attack byT missiles, where Y = T - X missiles are allocated to thecontrol centers and X missiles are allocuted to +he

missile sites.

The following expression for E(X,Y) ca1 be given:-

N [ M _Y j)

E(XY) a i 7T (1- ) ()1i L j=l

where x is the number of missiles allocated to the ith launcher site, yj is

the number uf missiles allocated to the jth control center and

N

Sxi =X (a)

\~y..Y=T - X. (0)

j=1

The objectiv" is to find the minimum values of E(X,Y) subject to the

constraints a) and b) where each xi and each yj is an integer.

The above problem reduces to the Piccariello problem when al=a2 ... =N=3,PI=P2.... =P,, and P =P =....P

TYNAMIC PROGRAMMUNG FORMUIATION

An examination of equation (f.), subject to the constraints (a) and (b)

indicates that we are faced with an N t M dimensional problem. In order to

circumvent thin dimensional difficulty., we will refconulate the problem as a

multi-stage problem and apply the techniques of dynamic programming to obtain

a feasible computational schemu. lt us defin-,

9k(X) = the minimum expected number of surviving missiles

for an allocation of X attacking missiles to k missile

sites,

and f (Y) = the minimnun value of the probability of the survival of at

least one control center for an allocation of Y attacking missiiefs

to I control centers,

Wi,

Then we have

mn E(XY) = Min m (x)fM(y) (2)

subject to the constraint X + Y = T, where

N x.

M, M i P i X1 + x 2 (3)(x 1=1

and

M v.fM(Y) = Min 1 - TT ( i -') bubject to the condition

(yj j - -l

Yl + Y 2 +'".+YM = Y"

Using the Principle of Optimality 2 we can express the g1 recursively as

F= M a P xk +g, k_ = 2 ,3,...,N, where Xk is permittedgk() M"'k tkk +gl k (5)to vary over the set 0,I,2,...XI. For k - 1 we have

gl(x) = a1 1 (6)

To obtain fM(Y) we first observe that fm(Y) = 1 - hM(Y), where (Y)

Max M (1 - PJ).jT=l - j)

P.ploying the Optimality Principle once again, we obtain

h,(y) = M.• 1 - Pj) h•_ (Y-Y;) , £ 2 ,5,...,M where yE

O,1, 2 ,...,Y) and (7)

h1 (Y) (1 - IY) (8)

The value Y is also permitted to range over the set of integers[O.I,2,...,T].

COMPUTATIONAL PROCEDURE

The dynamic programming formulation imbeds the original problem within

a family of analogous problems in whLch the basic parameters N, N, and T

assume sets of values which permit u2 to obuain, in the course of the compu-

tation, the solution to a variety of sub-problems. Because of the structure

of the process we are able to use the information obtained from sub-problems

of the originial problem to obtain th! solution of the original problem. The

input information that is required is the knowledge of N, M, and T and the

values of the PiIs, P.'s, and a 's.

We begin the computatiun by ubtaining the sequences (gk(X') and [Xk(X)]

from f-qustions (5) and (6), and the sequences [h,(Y)j and (y,(Y)) from

equations (7) and (8). Given this information, we are then prepared to use

equati.ou (2) to compute Min E(X,!').

NUMERICAL EXAMPLE

Lot N=, M= 2, T 5, P 1 ý .50, P2 = .30, P5 = .60, a1 =10, a2 = 8 ,

a. = 12, P1 = .60, P2 = .50. From equations 5, 6, 7, and 8, we obtain the

relations:

gý(X) a •,11

g2(X) M M P + gl(X-x{) +

g:(X) Min {aPx5 + g,(X-x,)}

hl(Y) = (1 - Y),

and Y y2h2 (Y) = Ma (1 - •2) hl(yy 2 )J.

Y"8

Using the previous relations we obtain the values,

gl(O) = 10

g1(l) = 5

91 (2) 2.5

g1(3) = 1.25

91(4)= .625

g1 (5) = .3125

g2(0) = a2 + g1(0) =8 + lo 18

8'.3010 + gl U

g9 i) = Min - 12.4

( + gl(O) 2.4 + 10 = 12.4

8 + g1 (2) = 1.0.5

g2 (2) = Min 2 = g1(1) = 7.4 Y 7.4

L.72 + g1(0) = 10.721

(8 + g1 (5) = 9.25

9 )= M 2.4 + gl(2) = 4.9 L72 + gl(1) = .72

,216 + gl(O) = 10.216

"8 + gl(4) = 8.625

2.4 + g1 (3) = 5.65

g2 (4) : Min .72 + gl( 2 ) = 5.52 ) ).52

.216 + gl(1) 5.216

.o648 + (o) n 10.0618

9

91(g5)

2.4 + g1(4) = 5.025

.72 + g1 (3) = 1.97 .97

.216 + 4.(2) = 2.716

,.064,8 + g1.(1) = 5.064-8

S.01944 + g 1 (o) = 10.01944.

g3(0) = [12 + 20-] 30

9 () =Min 12 + g2 (1) --24.4• 24.

g3(l 7.2 + g2(0) =25.21I 2).= m } i

1A-7.= 19.4+

g3(2) = Min 7.2 + 12.4 9.6 19.4i

4.32 + 18 =22.32

12 + 4.9 - 16.9

7.2 + 7.h = 14+.6

(5) Min 7 14.634.32 + 12.4 = 16.72

2.592 + 18 20.592

12 + 5.52 15.52

g93(4) t$Min 4.E2 + 7.4 1172 = 11.72

2.592 1 12.4 14.992

1.5552 + 18 19.5552

10

"12 + 1.97 15.97

7.2 + 5.52 = 10.72

4.32 + 4.9 = 9.222.992 + 7.4 = 9.992

1.5552 ý 12.4 = 13.9552

-.93r5±1 + 18 = 18.9'j12

h1 (O) = 0

h1 (1) = .40

h1 (2) = .64

h1 (3) = .784

h1 (4) = .8804

1h(5) = .92224

h 2 (0) 0 =

h,(1) = Max 4 0 =

h2 (2) = Ma{(.5o)(.4o) = .20j = .200

(.5o)(.64) = .32

h2(3) = Max 3 .32

(.75)(.40) .30

0

0

(.50)(.784) = .392h 2(I) = Max (.75)(.64) = .48 = .48

1 (.875)(.4o) = .35

1,o

'-0(-50)(,8804) = .440o2

(15' (.75)(.7810 = .588h2 (87564 = .56 .588

(.9s5)(.14o) =.575

We are finally ready to make use of~ eq. (2)

Min E(Y,X) =Min g,,(X) f (ý

where

=1-hM(Y),

and

X + Y =T.

Fo-r tuI example N 5, Nv = 2 so that

imi E(Y/X) Min g3i(X) mm(Y) i.e.,tx,Y)

g3 (0/) f2,(5) =(30)(.412) 12.;56

9.(1) f2(4) = (24.4))(.42) = 12.688

g.(2) f(3) = (19.4)(.68) = 13.192

Mi n .(Y,X) =Mi n -,9x).22

li E(X Y ,X Mii 8(5) f2(2) = (14.6)(.830) = 11.68 ~ 2txY 11y72

g i f,)(O) (-.22)(l) = 9.22

The minimum vrt1iip~ is 9.22. The allocation of missiles which yieldsthe minimum value is = Y, Y = 0 What remains is the determination of the

optimal values of the x;1 I = 1,2,5:

S= M x(O) 2,

= x2 (X-x.) = ',C(3) 1

xI =xl(X-~xj -2','• - I(2) : 2

Thtref ort thu optLmJil allocatioui po] jy Is giwv n by

(x 1! x. yI' y•) = (2,1,2,0,0).

DISCUSSION

The author is grateful to Mr. fi. L. Morritt for many helpful discussions

during the formulation and solution of this problem. Mr. Me-rrtt hbs pointed

out to the author •evei-al problem areas of interest in which this procedure

would have direct application. For instance it would be possible by utilizing

these methods to decide upon an optimum ratio of vuutrol ctn(:!r L, missile

sites. It would Aso be possible to use this methodology to ,ietermine whether

it is desirablL to harden ICBM siteS, o- to utilize the additional money

planned to be spent on active or passive defense by building more imdefended

launching situs.

WILLIAM SACC"

p.,

BIBLIOGIjAPHY

1. Piccariello, H. A Missile Allocation Problem, Operations Research,

Vol. 10, No. 6. November - December 1962, p. 795.

2. Bellman, R. Dynamic Programming. PrintuLon University Press: 1957.

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