1
http://www.Lohmander.com/mil/ORFAU1.pdf
Operations Research for Army Units
- Methodology for optimal combat decisions
Peter Lohmander
http://www.lohmander.com/
Version 2010-02-10
CASE 1.
Figure 1.1.
2
A1
A2
A1
A2
X1
X2
B1
Y1
C11
0
B2
Y2
0
C22
X1
X2
B1
Y1
2
0
B2
Y2
0
1
c11 0 
 2 0

0 c 
0 1 



22 
max E
s.t.
E  c11 x1  0 x2
E  0 x1  c22 x2
1  x1  x2
0  x1
0  x2
x2  1  x1
 if B1 
 if B2 
3
max E
s.t.
E  c11 x1
 if
E  c22 (1  x1 )
max E
B1 
 if
B2 
 if
 if
B1 
s.t.
E  c11 x1
E  c22  c22 x1
B2 
max E
s.t.
E  2 x1
E  1  1x1
 if
 if
B1 
B2 
4
Figure 1.2.
2 x1  1  x1
3 x1  1
x1 
1
 0.33
3
1 2
x2  1  x1  1    0.67
3 3
2
E  2 x1   0.67
3
5
c11 0 
 2 0

0 c 
0 1 



22 
min E
s.t.
E  c11 y1  0 y2
E  0 y1  c22 y2
 if
 if
A1 
A2 
1  y1  y2
0  y1
0  y2
y2  1  y1
min E
s.t.
E  c11 y1
E  c22 (1  y1 )
 if
 if
A1 
 if
 if
A1 
A2 
min E
s.t.
E  c11 y1
E  c22  c22 y1
A2 
6
min E
s.t.
E  2 y1
E  1  y1
 if
 if
A1 
A2 
7
Figure 1.3.
Figure 1.4.
8
CASE 2.
Figure 2.1.
 p11
0

0

 0
0
0
p22
0
0
p33
0
0
0   0.8 0
0
0 

0   0 0.4 0
0


0  0
0 0.2 0 
 

p44   0
0
0 0.9 
9
max E
s.t.
E  p11 x1
(if B1 )
E  p22 x2
(if B2 )
E  p33 x3
(if B3 )
E  p44 x4
(if B4 )
1  x1  x2  x3  x4
x1  0; x2  0; x3  0; x4  0
Assumption: All x’s are strictly positive. (The usual case.)
E  p11 x1
E  p22 x2
E  p33 x3
E  p44 x4
E  p11 x1  p22 x2  p33 x3  p44 x4
10
x1 
E
p11
x2 
E
p22
x3 
E
p33
x4 
E
p44
x1  x2  x3  x4  1
E
E
E
E



1
p11 p22 p33 p44
1
1
1
1
1




p11 p22 p33 p44 E
p22 p33 p44  p11 p33 p44  p11 p22 p44  p11 p22 p33 1

p11 p22 p33 p44
E
0.4  0.2  0.9  0.8  0.2  0.9  0.8  0.4  0.9  0.8  0.4  0.2 1

0.8  0.4  0.2  0.9
E
11
E
0.8  0.4  0.2  0.9
0.4  0.2  0.9  0.8  0.2  0.9  0.8  0.4  0.9  0.8  0.4  0.2
E  0.10140845
E  10%
x1 
E
E

 12.68%
p11 0.8
x2 
E
E

 25.35%
p22 0.4
x3 
E
E

 50.70 %
p33 0.2
x4 
E
E

 11.27 %
p44 0.9
Figure 2.2.
12
 p11
0

0

 0
0
0
p22
0
0
p33
0
0
0   0.8 0
0
0 

0   0 0.4 0
0


0  0
0 0.2 0 
 

p44   0
0
0 0.9 
min E
s.t.
E  p11 y1
(if A1 )
E  p22 y2
(if A2 )
E  p33 y3
(if A3 )
E  p44 y4
(if A4 )
1  y1  y2  y3  y4
y1  0; y2  0; y3  0; y4  0
Assumption: All y’s are strictly positive. (The usual case.)
E  p11 y1
E  p22 y2
E  p33 y3
E  p44 y4
E  p11 y1  p22 y2  p33 y3  p44 y4
13
y1 
E
 x1
p11
y2 
E
 x2
p22
y3 
E
 x3
p33
y4 
E
 x4
p44
y1  y2  y3  y4  1
E
E
E
E



1
p11 p22 p33 p44
1
1
1
1
1




p11 p22 p33 p44 E
p22 p33 p44  p11 p33 p44  p11 p22 p44  p11 p22 p33 1

p11 p22 p33 p44
E
0.4  0.2  0.9  0.8  0.2  0.9  0.8  0.4  0.9  0.8  0.4  0.2 1

0.8  0.4  0.2  0.9
E
14
E
0.8  0.4  0.2  0.9
0.4  0.2  0.9  0.8  0.2  0.9  0.8  0.4  0.9  0.8  0.4  0.2
E  0.10140845
E  10%
y1 
E
E

 12.68%
p11 0.8
y2 
E
E

 25.35%
p22 0.4
y3 
E
E

 50.70 %
p33 0.2
y4 
E
E

 11.27 %
p44 0.9
y1  x1
y 2  x2
y3  x3
y 4  x4
15
Figure 2.3.
Test:
Z  x1 y1 p11  x2 y2 p22  x3 y3 p33  x4 y4 p44
Z  0.10140845  E
16
Figure 2.4.
17
Case 3.
Complete example (in Swedish) with general software:
http://www.lohmander.com/mil/2pzsg/Taktik_PL_1.htm
Figure 3.1.
18
 a11 a12   (9  1) (1  2)   8 1

a



 21 a22  (5  6) (5  3)   1 2 
max E
s.t.
E  8 x1  1x2
E  1x1  2 x2
1  x1  x2
x1  0; x2  0
x2  1  x1
max E
s.t.
E  8 x1  1(1  x1 )
E  1x1  2(1  x1 )
max E
s.t.
E  9 x1  1
E  2  3 x1
E  9 x1  1  2  3x1
12 x1  3
19
3 1
  0.25  25%
12 4
x2  1  x1  75%
x1 
E  9 x1  1 
9 4 5
   1.25
4 4 4
Figure 3.2.
 a11 a12   8 1

a


 21 a22   1 2 
20
min E
s.t.
E  8 y1  1 y2
E  1 y1  2 y2
1  y1  y2
y1  0; y2  0
y2  1  y1
min E
s.t.
E  8 y1  1(1  y1 )
E  1 y1  2(1  y1 )
min E
s.t.
E  9 y1  1
E  2  3 y1
E  9 y1  1  2  3 y1
12 y1  3
y1 
3 1
  0.25  25%
12 4
y2  1  y1  75%
21
E  9 y1  1 
Figure 3.3.
9 1 5
   1.25
4 4 4
22
Figure 3.4.
23
Case 4.
Complete example (in Swedish) with general software:
http://www.lohmander.com/mil/ResFStri.html
Figure 4.1.
24
Figure 4.2.
Figure 4.3.
25
Table 1.
Summary of results reported here:
http://www.lohmander.com/mil/ResFStri.html
(Value of one
Probability
object)/(Value of that attack
one unit) = w
with one unit
per object is
optimal
Probability that a
concentrated
attack against
only one object is
optimal
Probability
that the
defence should
use one unit
per object
Probability that a
concentrated
defence of only
one object is
optimal
2
63%
37%
50%
50%
4
45%
55%
78%
22%
10
32%
68%
97%
3%
Table 2. (Swedish version of Table 1.)
Sammanfattning av resultat som rapporteras här:
http://www.lohmander.com/mil/ResFStri.html
(Värdet av ett
Sannolikhet att
objekt)/(värdet
A bör anfalla
av en grupp) = w med en grupp
mot varje
objekt
Sannolikhet att
A bör anfalla
kraftsamlat med
2 grupper mot
ett objekt
Sannolikhet att
B bör skydda
varje objekt
med en grupp
per objekt
Sannolikhet att
B bör
kraftsamla 2
grupper till
skydd av ett
objekt
2
63%
37%
50%
50%
4
45%
55%
78%
22%
10
32%
68%
97%
3%