37
2017
8
8
Vol.37, No.8
Systems Engineering — Theory & Practice
: E917
doi: 10.12011/1000-6788(2017)08-2200-09
Aug., 2017
: A
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(defense effectiveness,
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Defense effectiveness analysis of multi-mode shooting to
multi-type targets
LI Longyue, LIU Fuxian
(Air and Missile Defense College, Air Force Engineering University, Xi’an 710051, China)
Abstract A lot of military confrontation scenarios can be viewed as a shooting battle problem for
the defense weapons (Red) to shoot the offense target (Blue). We consider two kinds of blue targets,
homogeneous and heterogeneous. Firstly, for shoot-shoot strategy, interceptors’ allocation method to gain
maximum defense effectiveness (DE) is proposed for given number of red interceptors and blue targets.
Secondly, for salvo-look-salvo strategy, in consideration of the factors such as shooting times and salvo size,
the iterative recursive generation method for DE of red salvo shootings. Thirdly, the general calculation
method of two important factors (interceptors requirements and available shooting times) affecting DE is
given. Simulation examples validate the methods of this paper.
Keywords shooting mode; salvo-look-salvo; defense effectiveness; generating function
1
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(1988–), ,
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, E-mail: lilong yue@126.com;
(1962–), ,
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, E-mail: liuxqh@126.com.
:
; 2014
Foundation item: Military Graduate Student Foundation of Army; Foundation for Excellent Doctoral Dissertation of AFEU
:
,
.
[J].
, 2017, 37(8): 2200–2208.
: Li L Y, Liu F X. Defense effectiveness analysis of multi-mode shooting to multi-type targets[J]. Systems Engineering — Theory & Practice, 2017, 37(8): 2200–2208.
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M
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pi ,
E = 1 − (1 − p)N .
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,",
M i F
n1
n2
E DE K
n
E = (1 − (1 − p1 ) ) (1 − (1 − p2 ) ) · · · (1 − (1 − pk ) k )
(1)
O ni , i = 1, 2, · · · , k
G%M i , n1 + n2 + · · · + nk = N .
Nguyen Miah GP&@ (1) L NL, BMH [10] .
&Q (1) -./I, J
J n = [n1, n2, · · · , nk], n1 + n2 + · · · + nk = f (n), KRNOOSPL, -.
∂f
∂E
=λ
.
∂n
∂n
QTGP, -.
− ln(1 − pi )(1 − pi )ni (1 − (1 − pi )ni )
O i = 1, 2, · · · , k, ⎧ 3
(2)
(a) R (b) -.
Q, -. −1
E = λ.
−1
n1
n1
⎪
⎨ − ln(1 − p1 )(1 − p1 ) (1 − (1 − p1 ) ) E = λ
−1
− ln(1 − p2 )(1 − p2 )n2 (1 − (1 − p2 )n2 ) E = λ
⎪
⎩
n −1
− ln(1 − p3 )(1 − p3 )n3 (1 − (1 − p3 ) 3 ) E = λ
(a)
(b)
(c)
(2)
−1
ln(1 − p1 )(1 − p1 )n1 (1 − (1 − p1 )n1 )
n
n
−1 = 1
(3)
ln(1 − p2 )(1 − p2 ) 2 (1 − (1 − p2 ) 2 )
R (3) - n2
1
ln
n2 =
ln (1 − p2 )
n
ln (1 − p1 ) (1 − p1 ) 1
n
(ln ((1 − p1 ) / (1 − p2 )) (1 − p1 ) 1 + ln(1 − p2 ))
(4)
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T HIS n2, T (2) (c) P
−1
N −n1 −n2
− ln(1 − p3 )(1 − p3 )N −n1 −n2 1 − (1 − p3 )
E=λ
(2) (a) R (5) -
(5)
n
N −n1 −n2
N −n1 −n2
n
− ln (1 − p3 ) (1 − p3 )
F (n1 ) = ln (1 − p1 ) (1 − p1 ) 1 1 − (1 − p3 )
(1 − (1 − p1 ) 1 ) = 0
(6)
n U (4) V-, E (6) WNO n , U (6) !XU-.
2
(6)
1
n1
N −n1 −n2 ,···,−nk−1
F (n1 ) = ln (1 − p1 ) (1 − p1 )
1 − (1 − pk )
ln (1 − pk ) (1 − pk )
N −n1 −n2 ,···,−nk−1
V 0 < pi < 1, i = 1, 2, · · · , k H N > 0 Y
n
−
(1 − (1 − p1 ) 1 ) = 0.
F (0) = ln (1 − p1 ) (1 − pk )nk < 0
N
F (N ) = − ln (1 − pk ) (1 − p1 ) > 0
;=;<33,>@
37
?F SS n WX@ 0 N AE, HO , QFY 0 BZEL (GD!V: [
RZEEB x = N/2 T [0, N ] G [0, x] [x, N ] ZE. O", V F (0)F (x) = 0, E x ;
V F (0)F (x) < 0, E [0, x] A; V F (0)F (x) > 0, E [x, N ] A. \P SZEZF]
[ , ^ ZE\ _ ZE]. IK%&VW ε, ET`UXY log (N/ε) "(GP, -.
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1
2
3
3.1
2. - 34 - 2.
56789
^Y Bourn [11] , T(Z _c
q
p
=
q!
.
p!(q − p)!
bin(k, q)(i) =
k
i
q i pk−i
! i " P, i
d`!, O q = 1 − p. &M 2 \!ZL, J
- $+ )" DE E(n, k, s) _c, O n, k, s GL V_ a] e[
". @! - $+ - ! , J P (j |i , k, s) _c k
XY s " i P,
[3,12]
j
! . f^ ^
γ = i/k, _c^"
a.&f^ γ = i/k ! . Ub&' γ = i/k , X, J k = k + kγ − i
!! f^ γ,
e[ k = i − kγ
!! f^ γ, Oc_ GL_c"@d/X/@d
"X. V NKL, E⎧
! - $+ - ! )" P (j |i , k, s)
_c
k
P (j |i , k, s) =
V-!
-
bin(k, q γ )(j), γ ∈ N
⎪
⎪
⎨
z=min(j,k)
⎪
⎪
⎩
(bin (k, q γ ) (z)) bin k, q γ (j − z)
z=j−min(j,k)
$+ - ! )"/ DE,⎧ e`fX# abgFc ⎫ [3,12]
E(n, k, s) = max
i
O i = 0, 1, · · · , n, Qdhe
⎨ k
⎩
j=0
⎬
P (j |i , k, s)E(n − i, j, s − 1)
⎭
⎧
⎪
⎨ E(n, k = 0, s) = 1
⎪
⎩ E(n, k, s = 0) =
(9)
(7)
Qdhe SS, V
" 0, E DE 0. (7) γ
0,
E
1, if k = 0
0, else
(8)
(9)
\ 1 (100%); V
, 0,
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DE
Æ,
I,
^"
! γsalvo &, E ns γsalvo = N, γsalvo ≥ 0, ns ! ".
IK /
" S, 1 ≤ s ≤ S, &]e[ S " fg,
k
`!Gg
P (0 ≤ x ≤ k), ?fXiL
GFS (ns , k) =
_c, Oe[
1,
S=0
(RS )min(ns ,k) = (RS−1 qsalvo + psalvo )min(ns ,k) , S > 0
(10)
"fXIL [13] &]
Rs =
V P (x = j) _c j
1,
s = 0,
Rs−1 qsalvo + psalvo , s > 0.
!, E
k
GFS (ns , k) = P (0 ≤ x ≤ k) =
P (x = j) = 1
j=0
(11)
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8
^YM 2 \jh, [RAB
fgO
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GF1 (ns , k) = (qsalvo + psalvo )min(ns ,k) .
$ DE
AB
1
2203
⎧
⎪
k = 0,
⎨ 1,
E(ns , k, 1) =
0,
0 ≤ ns < k,
⎪
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psalvo , ns ≥ k.
fg
2
" (S = 2), H ns > k Y, M 2 "! iL
k
GF2 (ns , k) =
min(ns −k,α1 )
(qsalvo + psalvo )
bin(k, qsalvo )(α1 )
(12)
α1 =0
d`, h,i min(ns − k, α1) Ta.M 2 "! . AB
fg 3 " (S = 3), H ns > k Y, M 3 "! iL
(12)
M 1 "! P, α
1
min(ns −k,α1 )
k
GF3 (ns , k) =
bin(k, qsalvo )(α1 )
α1 =0
min
qsalvo +
psalvo
α2 =0
ns −k−
min(ns −k,α1 ),α2
bin(α1 , qsalvo )(α2 ) (13)
(12) , (13) M 2 "! P, α
d`, h,i min(n − k − min(n − k, α ),
α )
Ta.M 3 "! . _AB fg S ", H n > k Y, M S "! iL
2
s
2
1
s
s
β1
k
GFS (ns , k) =
β2
bin(k, qsalvo )(α1 ) ·
α1 =0
bin(β1 , qsalvo )(α2 ) ·
α2 =0
bin (β2 , qsalvo ) (α3 )·
α3 =0
···
(14)
βS−2
(qsalvo + psalvo )βS−1 bin(βS−2 , qsalvo )(αS−1 )
αS−1 =0
O
β0 = k < ns , β1 = min(ns − k, α1 ), β2 = min (ns − k − min (ns − k, α1 ) , α2 ) ,
S−2
· · · , βS−1 = min ns − k −
βχ , αS−1
χ=1
?@ (14) βS−1
β
(qsalvo + psalvo ) S−1 =
bin(βS−1 , qsalvo )(αS ),
αS=0
E (14) P
GFS (ns , k) =
S βS−1
bin(βs−1 , qsalvo )(αs )
WT., Va
αs−1 = βs−1 , 1 < s ≤ S,
E_P ` aj k/"! . VSiLP, TOl i p
j, -. M s " DE
O cij $Z .
:;<56
E (n, k, s) = max
iA ,iB ⎩
jA =0,jB =0
Z
(16)
i,j
,bF, IK A, B F
"M 2 \, AB
" s⎧A, sB , E e`fX# -.
⎨k=kA ,k=kB k
salvo
i
cij qsalvo
pjsalvo |k=j
E(ns , k, s) = GFs (ns , k) |k=j =
3.2
(15)
s=1 αs =0
,
GL qA, qB _c, F
−
→ → −
→
P (jA |i A , kA , sA )P (jB |i B , kB , sB )E n − (iA + iB ), j , −
s − 1
⎫
⎬
⎭
(17)
;=;<33,>@
2204
37
i = 0, 1, · · · , n, i = 0, 1, · · · , n, i + i ≤ n, −→j = [j , j ], −→s = [s , s ]. IK s ≤ s
(s ≥ s jhc), EQdhe
(17)
A
A
B
A
B
A
B
A
B
A
B
B
kA
→
−
k
E (n, k , [1, sB ]T ) = max 1 − q γ
(1 − q γ ) A E (iB , kB , sB ) .
iA
O γ = iA/kA, kA = kA + kAγ − iA, kA = iA − kA γ. &@&
&]fXiL
" 3.1 \c, ns
DE, ^Y 3.1
GFSA ,SB (ns , kA , kB ) = GFSA (ns , kA ) · GFSB (ns , kB )
(18)
! ". e` (18)
(A)
β0
GFSA ,SB (ns , kA , kB ) =
(A)
α1
(B)
β0
(A)
(A)
bin(β0 , qsalvo )(α1 )
(B)
=0
α1
(A)
β1
(A)
(A)
(A)
α2
(A)
βS −1
A
(A)
(B)
(A)
=0
(19)
(B)
βS −1
A
(B)
bin(βSA −1 , qsalvo )(αSA )
(A)
(B)
bin(βSA −1 , qsalvo )(αSA )
(A)
αS =0
αS =0
A
GFSB −SA
(B)
bin(β1 , qsalvo )(α2 ) · · ·
(B)
=0
(B)
=0
(B)
β1
bin(β1 , qsalvo )(α2 )
α2
(B)
bin(β0 , qsalvo )(α1 )
B
ns −
SA −1 (B)
βχ(A) + βχ(B) , αSA
χ=0
O
(A)
β0
(B)
= min (ns , kA ) , β0
βs(A)
= min ns −
A
(A)
= min ns − β0 , kB ,
sA −1 βχ(A) + βχ(B) , α(A)
sA
,
χ=0
βs(B)
= min ns −
B
sA −1 βχ(A) + βχ(B) − βs(A)
, α(B)
sA
A
.
χ=0
FGd SS,
VSiLP,
DE:
HR
"Cm (
"CmnnT`UoeC/). k
1 ≤ sA ≤ SA , 1 ≤ sB ≤ SB , TOl i psalvo pksalvo Zj,
-.
A
i
cij qsalvo
pjsalvo |j=kA +kB ,
E(ns , kA , kb , sA , sB ) = GFSA ,SB (ns , kA , kB ) |k=kA +kB =
i,j
O cij $Z .
4
=>?@AB
M 2 \M 3 \GD #kI: , %SQ IUHI D.
4.1
CDEFGH
V
W #, [R
B
, T =
; (
YE T ". $\l@
Ek τ (fgmYE) pq
rl − r0
.
vt
O rl
h no; r0 = max{rd, rmin }, rd F&
mf^r.
l no; vt
pRno, rmin
-.&, A: @:>;<@Æ67989?:
8
r
r
0
vi t1
㓒ᯩ
ሴᕩ
rmax
r
vt (W V )
vt t1
t W V t1
t1
2205
rl
㬍ᯩ
t W V
0ⴞḷ
t
I JKLMI
1 rl
s 1, . h no t = 0 Yq, E τ + σ YEPML S, O σ
pqn$YE. V t1 YEPML ij,
rmax
,
vi
t1 =
O vi
vt
rl = vi t1 + vt (t1 + τ + σ) = rmax 1 +
+ vt (τ + σ).
vi
mf^r, E
T
N = int W
τ
AB VV
,
W
τ
= int
rl − r0
vt
.
DE ,"@ Ea , M 2 \ E ≥ Ea, J Nmin %& DE mV
Ea = 1 − (1 − p)Nmin
Nmin =
(20)
4.2
%S&%& DE mV
CDNEOPQ
log Ea
log(1 − p)
(20)
HI D.
&HI
", [R%S
ijYE t1 no R1 ,
t1 =
ijYEBnoHI D
[14]
@["
.
rl − (τ + σ)vt
, R1 = vi t1
vi + vt
(21)
l@ - $+ - )", M 2 " t1 + τ + ξ YEP i, O ξ $+lkr?CY
E, T (21) R1 oRP, M 2 " ijYE
t2 no R2
t2 =
T R2 oRP, M 3 "
t3 =
R2 − (τ + ξ)vt
=
vi + vt
3
rl − (τ + σ)vt
vi + vt
rl − (τ + σ)vt
vi + vt
vi
vi + vt
5
RSTU
/
(τ + ξ)vt
vi + vt
ij
YE t no R
s
vi
vi + vt
s−1
max ts ≥
".
−
1+
, R2 = vi t2 .
vi
vi + vt
+
vi
vi + vt
2
,
R3 = vi t3 .
2
(τ + ξ)vt
vi + vt
−
3
O R1, R2 , · · · , Rs ≥ max{rd, rmin }, HWlta
e` (22) IS
vi
vi + vt
ijYE
t no R
", U,poRfX, M s "
ts =
rl − (τ + σ)vt
vi + vt
R1 − (τ + ξ)vt
=
vi + vt
s
−
(τ + ξ)vt
vi + vt
max{rd , rmin }
vi
s
s−1
i=0
vi
vi + vt
i
, Rs = vi ts .
(22)
$\qYum%S$^nsV.
)", s 2 DE 1 o
ÆÆ)&', pQ: Q 1(qs) J 6 L, (D 2 ,
2
!GL 0.6 0.4 rQ 2(rs) J 8 L, (D 2 ,
2
!GL 0.4 0.6. s 2 Q 1, n1=2.55 n2=3.45 Y DE /, Ht /I, bXPsV
;=;<33,>@
37
J2
GL 3 L; s 2 Q 2, n =4.85 n =3.15 Y DE /, Ht /I, bXP
sV JM 1
5 L, JM 2
3 L. s 3
F 1 oÆ
Æ)&', qsrss 2 Ic, ?s SS F sfsL, GLV n =2.55 n =4.85
Y F = 0, s 2 sVc, E tG&M 2 \D@VNLut.
! - $+ - ! )", V NKLY, IK
3 ,
6 "! fg, n = 6,
" S = 3. ? (14) 2206
1
2
1
1
s
3
2
GF1 (6, 3) = (qsalvo + psalvo )3 = qsalvo
+ 3qsalvo
psalvo + 3qsalvo p2salvo + p3salvo .
Ee` (15)
E(6, 3, 1) = p3salvo ,
? (12)(13) (15) -
2
2
GF2 (6, 3) = p3salvo + 3(psalvo + qsalvo )qsalvo p2salvo + 3(psalvo + qsalvo )2 qsalvo
psalvo + (psalvo + qsalvo )3 qsalvo
,
E(6, 3, 2) = p3salvo (1 + qsalvo )3 ,
2
psalvo
3qsalvo
GF3 (6, 3) = p3salvo + 3qsalvo p2salvo (psalvo +qsalvo (psalvo + qsalvo )) +
2
3
2
psalvo +2psalvo qsalvo (psalvo + qsalvo ) +qsalvo
(psalvo + qsalvo ) +qsalvo
(psalvo + qsalvo )3 ,
2
2
+ 6qsalvo
)3 .
E(6, 3, 3) = p3salvo (1 + 3qsalvo + 6qsalvo
0.8
0.9
DE
0.7
DE
0.8
N=6
K=2
p1=0.6
p2=0.4
0.6
N=8
K=2
p1=0.4
p2=0.6
0.7
0.6
0.5
防御效率
防御效率
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
1
2
I
0
3
射击目标1的射弹数量
0.6
4
VWX&Y ZW[\]^_`Z_a
5
2 DE
6
0
1
2
3
4
射击目标1的射弹数量
5
6
7
1
1
F
F
0.4
0.6
N=6
K=2
p1=0.6
p2=0.4
0
0.4
-0.2
F
F
N=8
K=2
p1=0.4
p2=0.6
0.8
0.2
0.2
-0.4
0
-0.6
-0.2
-0.8
-0.4
-1
0
1
2
8
I VWX&Y ZW[\]^_`Z_a
3
射击目标1的射弹数量
4
3 F
5
-0.6
6
1
0
1
2
3
4
5
射击目标1的射弹数量
6
7
8
s 4 GL ! - $+ - ! )", NKLY
! v ! Y !
! "! uAE# ^nsV. V
F Y, IK
2 ,O
1
( ), B
1
( vv),
4 "! fg, ns = 4,
A
"GL SA = 1, SB = 2 (rh/@vv, E
""@vv). ", ^
Ys 4 %S! - $+ - ! )", ,"YY
! v ! Y !
! "! uAE# ^nsV, ws 5. SS, s 4 5 UCs 3,
(!
)- $+ (! ) )"
u sfsL, wU! " !
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1
0.9
0.8
0.8
0.7
0.7
0.6
0.6
防御效率
1
0.9
0.5
2207
第1次双发齐射
第2次双发齐射
第3次双发齐射
0.5
0.4
0.4
0.3
0.3
第1次单发射击
第2次单发射击
第3次单发射击
0.2
0.2
0.1
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
红方对蓝方目标的单发杀伤概率
0.7
0.8
0.9
0
0
1
0.1
0.2
0.3
0.4
0.5
0.6
红方对蓝方目标的单发杀伤概率
0.7
0.8
0.9
1
1
0.9
第1次三发齐射
第2次三发齐射
第3次三发齐射
0.8
0.7
防御效率
0.6
0.5
0.4
0.3
0.2
0.1
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0
0
0.1
4
0.2
0.3
0.4
0.5
0.6
红方对蓝方目标的单发杀伤概率
-
0.8
0.9
1
-
0.9
0.8
0.8
0.7
0.7
0.6
0.6
防御效率
0.9
0.5
0.5
0.4
0.4
0.3
0.3
第1次对A单发射击, 第2次对B单发射击
第1次对A双发齐射, 第2次对B双发齐射
第1次对A三发齐射, 第2次对B三发齐射
0.2
第1次对A单发射击, 第1次对B单发射击
第1次对A双发齐射, 第1次对B双发齐射
第1次对A三发齐射, 第1次对B三发齐射
0.2
0.1
0.1
0.1
0.2
0
0
I bW cd bWefg&Yopqjklmn
0.3
0.4
0.5
0.6
红方射击蓝方目标的单发杀伤概率
5
0.7
0.8
0.9
-
1
0.1
0.2
0.3
0.4
0.5
0.6
红方对蓝方目标的单发杀伤概率
-
4
Ea=0.80
Ea=0.85
Ea=0.90
Ea=0.95
3.5
3
最少所需射弹数
0
0
0.7
1
1
防御效率
防御效率
8
2.5
2
1.5
1
I op r gstuvW[\Z_awx
0.5
0.55
6
0.6
0.65
p
0.7
Ea
0.75
单发杀伤概率
0.8
0.85
0.9
0.95
0.7
0.8
0.9
1
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