Research Journal of Applied Sciences, Engineering and Technology 5(20): 4890-4894, 2013
ISSN: 2040-7459; e-ISSN: 2040-7467
© Maxwell Scientific Organization, 2013
Submitted: September 27, 2012
Accepted: December 13, 2012
Published: May 15, 2013
Based on the Force Deployment Model of Unascertained Expectation
Jianli Chen
Armed Police Force Engineering University College of Science, Xi'an 710086, China
Abstract: In this study, we utilize the unascertained mathematics method to give the unascertained number of
countermeasure of anti-terrorism strategic force deployment and unknown event. It has been defined the situation
sets of force deployment, condition density and mathematical expectation of density model. It has been given the
unascertained parameters Cij which decide and direct the force deployment. Find out the condition density matrix of
force deployment, further get the conditional density of single target force deployment, using the maximum density
mathematical expectation in order to get the optimal mathematical model of multiple target force deployment.
Analyzing the coefficient of model and provide two kinds of discussed computing method. The model overcomes
the limitation of past deterministic thinking method which study the force deployment and provide the decision
maker a relative substantial theory evidence.
Keywords: Conditional density, force deployment, mathematical expectation, model, situation, unascertained
number
INTRODUCTION
The problem of forces ‘deployment is a
fundamental problem in military decision-making, in
the past decision we can usually meet optimal result in
a positive background and a result in stochastic
condition .the practical anti-terrorism strategic forces’
deployment problem is usually anfractuosity., because
it’s not a doubtless problem but not a Stochastic
Process. It has a preordained fate relationship, it also
has some complicated indeterminacy factors. In
practice the system of anti-terrorism strategic forces’
deployment is a complicated "indeterminacy system.
Thus it’s practical for us to use the Unascertained
Mathematical ideal method to study the problem forces’
deployment. Bi and Yan (2010) study the research of
military strategy thought in the countemporaryera. Li
(2005) study the military strategy learns lectures. Shen
et al. (2005) study the targets being misty to make
policy troops an allotment model. Ren et al. (2007)
have a research of the troops of the rate deployment
model. Liu et al. (2010) analyze the troop disposition
assessment Reconnaissance-Attack system.
In this study, we utilize the unascertained
mathematics method to give the unascertained number
of countermeasure of anti-terrorism strategic force
deployment and unknown event. It has been defined the
situation sets of force deployment, condition density
and mathematical expectation of density model. It has
been given the unascertained parameters Cij which
decide and direct the force deployment. Find out the
condition density matrix of force deployment; further
get the conditional density of single target force
deployment, using the maximum density mathematical
expectation in order to get the optimal mathematical
model of multiple target force deployment. Analysing
the coefficient of model and provide two kinds of
discussed computing method. The model overcomes the
limitation of past deterministic thinking method which
study the force deployment and provide the decision
maker a relative substantial theory evidence.
DESCRIPTION OF THE PROBLEM
At present, the research of forces’deployment
model consist of ascertain model and stochastic model.
the problem of forces’optimized deployment in Military
Operations Research is practically a ascertain
designating problem. The model is expressed 0-1
planning:
m
max p( s ) = ∑ aij xij s.t.
i =1
m
∑
i =1
n
xij = 1 ∑ xij = 1
j =1
x ij = 0.1; (i = 1, 2, …, m; j =1, 2, …, n)
The 0-1 variable x ij is defined as:
when A i is disposed B j , x ij = 1
when A i isn’t disposed B j x ij = 0
Although we can use parallelism method in
parallelism to solve problem in order to get the result of
forces ‘deployment (it is a 0-1 matrix in practice) to
guide the military decision-making. Actually, it is based
on the premise that the above-mentioned model is
specific ascertain assumption. But in the practical
application, the eruption of hostilities, even in the
forces’ deployment process is full of many uncertain
factors, thus many scholars use the probability method
to study forces’ deployment so as to make forces’
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Res. J. Appl. Sci. Eng. Technol., 5(20): 4890-4894, 2013
deployment model transform studying
problem to study stochastic problem:
ascertain
(b )
ij m×n
n
=
p( B j ) p( S / B j )
∑
max p ( S )
=
m
∑ pij p( Aij B j )
=j 1 =i 1
There, S is represented where we gain the victory
For the problem of studying probability is a stochastic
process at the same time we need a larger specimen
space， the eruption of hostilities, even in the forces’
deployment process are not only a stochastic progress.
It has a preordained fate relationship ,it also has some
complicated indeterminacy factors. Thus it’s practical
for us to use the Unascertained Mathematical ideal
method to study the problem forces’ deployment.
Definition1: All troops countermeasures of deployment
problem research of troops are called countermeasures
collection. Called A = (A 1 , A 2 , …, A m ), Among them,
A i (i = 1, 2, …, m) is the i-th deployment
countermeasure troops, If mean the x i grow the troops
number of troops deployment countermeasures (when
𝑥𝑥
i<j, have x i <x j ）, if mean φ(x i ) = ∑𝑚𝑚 𝑖𝑖 ,
𝑖𝑖=1 𝑥𝑥 𝑖𝑖
Then call [[x 1 , x m ], φ(x)] for the unascertained
number of countermeasures.
R
R
Definition 4: For situation (A i , B j ), Use C ij to mean the
reliability of a defence B j for countermeasures A i , then
we called C ij as the unascertained density of situation.
Definition 5: For situation (A i , B j ), Use b ij to mean the
reliability of performance the task and obtaining a
victorious while using countermeasure A i when event
B j occur. Then we called b ij as the unascertained factor
density of situation. Then we called:
b12
b22
...
bm 2
... b1n 
... b2 n 
... ... 

... bmn 
as the unascertained factor density array.
Definition 6: Set [[a, b], φ(x)] as an unascertained
number, φ(x) is an factor density, Then we called E =
𝑏𝑏
∫𝑎𝑎 𝑥𝑥𝜑𝜑 (𝑥𝑥)𝑑𝑑𝑑𝑑 as the unascertained density expectation.If
[[x 1 , x m ], φ(x)] is an unascertained racional,then E =
∑𝑚𝑚
1 𝑥𝑥𝑖𝑖 𝜑𝜑(𝑥𝑥𝑖𝑖 ).
Definition 7: Suppose there are R experts analyse the
density of situation (A i , B j ). order b ij (k) to be density b j
generalized residual result which the k th expert deal
(1) (2)
(𝑅𝑅)
with situation (A i , B j )，thus (𝑏𝑏𝑗𝑗 , 𝑏𝑏𝑗𝑗 , … , 𝑏𝑏𝑗𝑗 ) are b j
generalized residual vector. If the weight of k th expert
is w k 0<w k ≤ 𝑤𝑤𝑘𝑘 ≤ 1 (k = 1, 2, …, R) that
∑𝑅𝑅𝑘𝑘=1 𝑤𝑤𝑘𝑘 𝑏𝑏𝑗𝑗(𝑘𝑘) is the generalized residual vector 𝑏𝑏𝑗𝑗(∗) of b j
(∗)
which is density of (A i , B j ) 𝑏𝑏𝑗𝑗
∑𝑅𝑅𝑘𝑘=1 𝑤𝑤𝑘𝑘 𝑏𝑏𝑗𝑗(𝑘𝑘) .
Definition 8: Order b ij (k) be density b j generalized
residual result which the k th expert deal with situation
(A i , B j ), thus:
Definition 2: All events that may occur of deployment
problem of troops are called events collection. Called B
= (B 1 , B 2 , …, B n ), among them, B j (j = 1, 2, …, n) is
the j-th event. The b j is the possibility size of the
occurrence of the event B j when x = j, h(x) = b j ,
We called [[1,n], h(x)] as the unascertained number
of the event.
Definition 3: Cartesian product of countermeasures
collection A = (A 1 , A 2 , …, A m ) and events collection B
= (B 1 , B 2 , …, B n ), A×B = {(A i , B j )�Ai ∈ A, Bj ∈ B} are
called the situation in collection, called S = A×B. For
arbitrarily A i∈ A, B j∈ B, we called (A i , B j ) as situation.
Make S ij = (A i , B j ), Then event B j occur and A i deploy
troops and carry out a task with B j .
 b11
b
=  21
 ....

bm 2
(b )
(k )
ij
m×n
 b11( k )
 (k )
b
=  21
 ....
 (k )
bm 2
b12 ( k )
b22 ( k )
...
bm 2 ( k )
... b1n ( k ) 

... b2 n ( k ) 
...
... 

... bmn ( k ) 
The generalized residual matrix of b ij which is
made by the k the expert (i = 1, 2… m, j = 1, 2 …n) If
the weight of the k th expert is w k 0< w k ≤ 1(k = 1, 2,
(k)
…, R) thus ∑Rk=1 wk bij .
Is be density b ij (*) generalized residual result which
is the unascertained condition density b ij of situation
(k)
(A i , B j ), thus b ij (*)∑Rk=1 wk bij
The troops deployment problem can describe for
our armies have the troops of total amount M,
Constitute m battle groups A 1 , A 2 , …, A m of
independence according to the amount organic
combination of the weapon material, true war technique
and the personnel's amount, probably produce the point
region of the riot activity of the terror raid as B 1 , B 2 , …,
Bn.
At some a particular period, the region of B j takes
place the possibility of terrible activity as b j and regard
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as war the group A i to set up defense at the B j , setback
terror assault riot activity of the credibility is b ij , how to
set up defense from the overall situation, namely with
how of the situation set up defense formation situation
to gather, or choice which c ij to make the possibility of
setback enemy biggest for our army, so we can obtain
warlike victory.
deployed in B j . So then(c 1j , c 2j ,…,c mj ) means
respectively the possibility size that battling group A 1 ,
A 2 ,…,A m is organized a defiance to B j .
The density model of troops deployment: Make
T
c1 j , c2 j , ... cmj  b1 j , b2 j , =
... bmj 
m
=
bc
∑
i =1
ij ij
g(b j )
(4)
THE MODEL OF TROOPS DEPLOYED
The uncertain number of troops deployed: Suppose
that independent battle group number is m that is as A 1 ,
A 2 , …, A m ，the independent battle group A i consist
of x i members, Suppose x 1 <x 2 <…<x m . The main
defiance areas to have possible terrorism war are B 1 ,
B 2 , …, B n. The possibility that B i take place terrible riot
activity is b i , Then we can get following 2 uncertain
number:
𝑥𝑥 𝑖𝑖
[[x 1 , x m ], φ(x)] where φ(x i ) = ∑𝑚𝑚
1 𝑥𝑥 𝑖𝑖
where h(y = j) = b j
[[1, n], h(y)]
(1)
The condition density of troops deployment:
(2)
=
(i 1,=
2,...t ; j 1, 2,...n)
0 ≤ bij ≤ 1
b ij is defined to the condition density of troops
deployment, The b ij means the credibility to obtain
overall victory, which happen in B j region, when A i
troops is organized to defiance B j region to carry out
the task of battling .
so get the uncertain condition density matrix of troops
deployment:
(b )
ij
m×n
 b11
b
=  21
 ....

bm 2
b12
b22
...
bm 2
... b1n 
... b2 n 
... ... 

... bmn 
 bp , bq  , g (b) 


Define:
n
The b j means that the possibility size of the war to
happen in B j region.
Suppose
=
bij b=
xi )
j ( x, x
The g(b j ) means the credibility that A 1 , A 2 ,…,A m
defeated of enemy war that happen in B j region, when
B j being defended with A 1 , A 2 ,…, A m b p = min{b j }b q =
-max{b j }(j = 1, 2, …, n) Combine to line up a preface
according to the size of b j , Get a condition density type
uncertain number:
(3)
E (b) = ∑ b jg(b j )
(5)
1
as uncertain mathematics expectation of density type.
The E(b) means the uncertain mathematics
expectation that our army acquires overall victory. The
different troops deployment gets the different
mathematics expectation. Comprehensive analysis on
the above to know: troops deployment problem of the
counter-terrorism strategic is look for reasonable troops
deployment, To obtain reasonable cij (i = 1, 2, …, t; j =
1, 2, …, n)
Make the E(b) maximize, namely:
n
n
m
max E (b) = ∑ b j g (b j ) = ∑ b j ∑ bij cij
j =1
(6)
=j 1 =i 1
That means under Guiding of the target to look for
reasonable uncertain density distribution, make the
biggest possibility obtain the victory of war, this c ij is a
comprehensive information characteristic that mean the
credibility when A i is deployed to B j . All of b j and b ij
are uncertain condition density, which relies on politics,
culture, diplomacy and military...etc of nation and
region of terrible influence place, they are analyzed and
calculated by the various factors that induct war.
The model (6) is also abstract into a problem as
follows:
n m
The jth list (b 1j , b 2j , …, b mj )T of in the matrix show
f
x
=
aij xij
max
(
)
∑∑
the credibility distribution that when battle group A 1 ,
=j 1 =i 1
A 2 , …, A m is organized the defence to B j , respectively,
m
A i defeated the enemy to get overall victory. The c ij
xij = 1 , 0 ≤ xij ≤ 1
∑
i =1
Mean the credibility of the battling task to carry out
(i 1,=
2,...t ; j 1, 2,...n)
when war happens in B j region , the A i troops is =
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(7)
Res. J. Appl. Sci. Eng. Technol., 5(20): 4890-4894, 2013
Solve the problem and then we can get a set of decision
variables x ij (i = 1, 2, …, t; j = 1, 2, …, n), which can
guide troops deployment.
The weight vectors of K experts is [w 1 , w 2 , …w R ],
thus get the 𝑓𝑓 (∗)
which is the prior estimate value of f ij :
𝑖𝑖𝑖𝑖
THE ANALYSIS AND ASCERTAINMENT OF
MODEL COEFFICIENT =
f ( ∗)
ij
The coefficient a ij in the model (7) and coefficient
b j b ij in the model (6) are the promise of optimizing
deployment. Next, we carry on analysis and discussion.
Both b j and b ij are unknown condition density and
depending on the comprehensive situation which
constituted by the politics, culture, diplomacy, military
strength of terrible nation and region.
Suppose these comprehensive situations can be divided
into s kind of state z 1 , z 2 ,…, z s and form into
comprehensive situation SZ:
zi∩ 𝑧𝑧𝑧𝑧 = 𝛷𝛷, (i≠ 𝑗𝑗)
Method one: The result has been given through the
unascertained measure of expert estimation, under the
predisposing of F i that caused the event occurs in war
situation B j , this constitute the predisposing factors
matrix:
 ( B1 , F1 ) ( B1 , F2 )
( B , F ) ( B , F )
2 2
= 2 1
 ....
...

( Bn , F1 ) ( Bn , F2 )
k =1
k
f ij ( k )
=
(i 1,=
2,...t; j 1, 2,...n)
It is the same for the b ij which is able to be find out
through the past experience and expert forecast method.
Method two: Use the probability, because z 1 , z 2 ,…,z s
and it has became general trend SZ.
zi∩ 𝑧𝑧𝑧𝑧 = 𝛷𝛷, (i≠ 𝑗𝑗)
p( Fi ) = ∑ p( Fi / Z k ) p( Z k ), (i = 1, 2,...t )
s
k =1
thus: b j = p(B j ) should be comprehend as probability:
t
b j = ∑ p( B j / Fi )
i =1
while:
P( B j Fi )
p(=
B j / Fi ) =
P( Fi )
p( Fi / B j ) P( B j )
s
∑ p( F / Z ) p( Z )
k =1
i
k
k
=
(i 1,=
2,...m, j 1, 2,...n)
here:
... ( B1 , Ft ) 
... ( B2 , Ft ) 
...
... 

... ( Bn , Ft ) 
p( B j ) = ∑ p( B j / Z k ) p( Z k ), ( j = 1,2,...n)
s
k =1
DISCUSSION
The reason that a war occurs is able to be caused by
one factor, or also can be caused by some factors at
mean time; here we only discuss the circumstance of
single factor independence that caused the war.
Suppose there are R experts, the K experts think
under the predisposing of 𝐹𝐹𝑖𝑖 , the possibility which a war
occurs in the place of B j is 𝑓𝑓𝑖𝑖𝑖𝑖 (𝑘𝑘)
, the evaluation set is
𝑖𝑖𝑖𝑖
consist of predisposing which caused the war from R
experts:
bij (1) , bij (2) , ... bij ( R ) 
R
∑w
thus:
Regard SB as the appearance space of war
occurrence. Obviously the SB has two elements. (Occur
war or don't). Regard B as the event of war occurrence.
and suppose an induct of riot a factor for F 1 , F 1 ,…, F t ,
Only being causing the inducement appear just may
cause war. Make (𝐵𝐵𝑗𝑗 , 𝐹𝐹𝑖𝑖 ) as war situation caused by
remote cause 𝐹𝐹𝑖𝑖 . 𝑓𝑓𝑖𝑖𝑖𝑖 is under 𝐹𝐹𝑖𝑖 remote cause, cause B j
war inducement of possibility size, so b j ∑𝑡𝑡𝑖𝑖=1 𝑓𝑓𝑖𝑖𝑖𝑖 = 1,
here 𝑓𝑓𝑖𝑖𝑖𝑖 have two methods to ascertainment.
( B j , Fi ) 
n×t
 ( bij (1) ) 


 ( bij (2) ) 
w1 w2 ... wR ] 

[=
 ... 
 (R) 
( bij )  R×1
In fact, Still return to produce one some problems
which stay at the further study to comprehension or
useful property from the research of the b j and the b ij ,
Thus produce the model canning and solve method to be
advantageous to analysis more and solve. If we can
understand ∑𝑚𝑚
𝑖𝑖=1 𝑏𝑏𝑖𝑖𝑖𝑖 ≤ 1, under the some condition or
not. Here"=" establish to mean inevitable occurrence
war feeling, "<" establish to express uncertain
occurrence war feeling:
T
m
n
i =1
j =1
0 ≤ ∑ bij ≤ 1, (i = 1, 2,...m) 0 ≤ ∑ bij ≤ 1, ( j = 1, 2,...n)
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Res. J. Appl. Sci. Eng. Technol., 5(20): 4890-4894, 2013
m
∑
i =1
bi ≤ 1,
n
∑
j =1
b j ≤ 1,
Moreover for c ij to speak, when area B j has war
feeling, -1 battle group must defend the war feeling.
Namely ∑𝑚𝑚
𝑖𝑖=1 𝑐𝑐𝑖𝑖𝑖𝑖 = 1, (j = 1, 2, …, n), But for any
deployment, each war area's hasing war feeling is an
indetermination affairs, namely:
n
∑
j =1
cij ≤ 1, . (i = 1, 2,...m)
Then we can give model's (6) stipulation condition
of increment then get new model thus.
CONCLUSION
According to unascertained condition density
deployment model of troops, use unascertained thought
method to analysis deploy environment, lead to deploy
into the troops of unascertained number and condition
density matrix. We can get the best deployment model
of troops though the unascertained number expect of
biggest worthy, avoided using the limit of the dispose
troops for battle of the thought method of assurance
before. The key of this model is indeed to make sure that
3 unascertained numbers and condition density matrix.
Along with the development of war, the
deployment model of troops by all means faces a new
topic and the deployment model of troops also needs a
further research.
REFERENCES
Bi, W. and G. Yan, 2010. The Research of Military
Strategy Thought in the Countemporaryera.
Military Science Publisher, Beijing, pp: 513-520,
(In Chinese).
Li, C.S., 2005. The Military Strategy Learns Lectures.
Military Science Publisher, Beijing.
Liu, S. and Z. Li, C. Zhang and H. Ren, 2010. Troop
disposition
assessment
reconnaissance-attack
system. Acta Armament., pp: 9-10, DOI: CNKI:
SUN:BIGO.0.2010-S2-038, (In Chinese).
Ren, J.Y. et al., 2007. According to all the troops of the
rate deployment model. Air Force Eng. Univ. Coll.
J., pp: 5.
Shen, M.X. et al., 2005. According to Many Targets be
Misty to Make Policy Troops an Allotment Model.
Engineering Science and Technology College,
College J. Xian, pp: 2166-168.
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