The spectral analysis of simulated N-team of interacting decision makers with bounded rationality constraints of Oladejo, which assumes triangular probability density function of command inputs is hereby restructured and analysed, to have hierarchical command inputs that are predicated on order statistics distributions. The results give optimal distributions.
The Double-Team of decision Maker Model that Boettcher and Levis developed [
The Model: The model of N-Team interacting decision makers with bounded rationality constraints was decomposed into coherent component strategies which were then used to analyze the system of specified C3I.
SymbolsXi is the input signal having exponential tempo/inter arrival times of the ith team.
p is the portioning algorithm of inputs to respective DM.
u is the internal decision.
fi is the algorithm for process ui to obtain the battle scenario.
qi is the ith team input regulator.
zi is the initial situational assessment.
z ¯ i is the final situational assessment.
g C A i , g C B i are the internal coordination strategies using SITREP z’s and vi, the command input to the ith team.
z' is situation assessment from other organizational members (SITREP or Feedback).
v is the final choice of decision.
v' is the command input.
v ¯ i is the output of DM i.
v ¯ is the final choice or response.
Y is the output or desired result.
hj is the processing algorithm for the final choice leading to Y.
hi is the algorithm for process vi.
P ( v | z ) is response selection strategy that maps z to Y in the absence of v' and determines choices of hj.
P ( v ¯ | z ¯ , v ′ ) is response selection strategy that maps z and v' to v, and it also determines choices of hj.
H ( x ) = − ∑ x p ( x ) log 2 p ( x ) is entropy of inputs, where p(x) is probability or uncertainty associated with N random variables, X.
T ( x , z ′ , v ′ , Y ) is mutual information or transmission or throughput between inputs x, z', v' and output Y.
Gt is the throughput.
Gb is the blockage.
Gn is the noise.
Gc is the internal coordination.
ℍ is the entropy of a random variable with probability p for taking one or two values, which also gives the amount of information transmitted.
A is the set of situational assessment options or strategies.
B is the set of available choices of tactics, modus operandi to remedy problem.
y is the convolution of strategies which is the assumed combined effects of process activities.
where
j 11 = 1 , 2 , ⋯ , U 1 , j 21 = 1 , 2 , ⋯ , V 1 , j 12 = 1 , 2 , ⋯ , U 2 , j 22 = 1 , 2 , ⋯ , V 2 , ⋯ , j 1 N = 1 , 2 , ⋯ , U N , ⋯ , j 2 N = 1 , 2 , ⋯ , V N
This is the structure of the generalized developed model as shown below:
The generalized developed model of
D M t ( t = 1 , 2 , ⋯ , N )
G t = T ( X t , Z t ( N − 1 ) , ⋯ , Z t 1 ; V ( N − 1 ) 1 Z 1 t ; ⋯ , Z ( N − 1 ) t ; Y t ) + H ( X t ; Z t ( N − 1 ) ; Z t 1 ; V ( N − 1 ) ′ ) − G t t + H ( u t ) + H z ( v t ) + ∑ i = 1 U t [ p i g c i ( p X t ) + α i ℍ ( p i ) ] + H ( Z t , Z 1 t , ⋯ , Z ( N − 1 ) t ) + g c A t ( p ( Z t ) , p ( Z t 1 ) , ⋯ , p ( Z t ( N − 1 ) ) ) + g c B t ( p ( Z t ) , p ( V ( N − 1 ) ′ ) ) + ∑ j = 1 V t [ p j g c j ( p ( Z t / V t ) ) + α j ℍ ( p j ) ] + H ( Y t ) + H ( Z t ) + H ( Z ¯ t ) + H ( Z t , V t ) + T ( X t : Z t 1 ; ⋯ , Z t ( N − 1 ) ) + T Z − t ( X t ; Z t 1 , ⋯ , Z t ( N − 1 ) : V ( N − 1 ) ′ ) (1)
Strategies which are probability density functions (pdfs) are given as follows:
U i ( x i ) ~ exponential inter arrival times or tempo of operations: 1 θ e − t / θ , t , θ > 0
f i ( u i ) ~ normal because of a defined goal (i.e. goal is focus):
1 2 π σ e − 1 2 ( x − μ ) 2 / σ 2 , − ∞ < x , μ < ∞ , σ 2 > 0 (2)
A ( z i ) ~ uniform distribution: 1 N or x b − a , x = 0 , 1 , 2 , ⋯ , N or a < x < b , respectively (3)
v ′ ~ triangular V ¯ C , 0 < V C , F = R = v − 2 2 C | 0 c , v − 1 = 2 R , ( i . e for v − 1 = c ) (4)
g C k is internal coordination strategy of corresponding algorithm which depends on the distribution of their respective inputs.
g c A 1 ~ Bernoull: pxq1-x, x = 0, 1 (appropriate strategies are employed. Hence, åxi is binomial and all samples are appropriate. These are the internal coordination strategies among situational assessments represented by set A1).
g c B 1 ~ geometric: pqx, x = 1 , 2 , ⋯ , depending on x (the strategies would result in a success hence åx1 is negative binomial that all samples are successful.
These are the internal coordination strategies among final choices and command inputs represented by set B1).
h ( v ′ ) ~ Weibull (due to reliability of subsystem before reaching final stage: α β X β − 1 e − α X β , X > 0 , α , β > 0 ).
μ ( X ) ~ expo (this is due to random occurrences).
F ( u ) ~ normal (i.e. geared towards a goal).
A ( z ) ~ uniform (simple random sample , equally likely samples).
B ( z , v ¯ | v ¯ ′ ) ~ conditional jpdf {(uniform × triangular) × ( g C A g C B ) yielding interaction due to z’s to determine choices of v depending on v-1 from superior command}.
h ( v ¯ ) ~ Weibull (output Y depends on survivability or reliability of other systems components).
v ′ ~ order Statistic-Hierarchical command inputs.
Now if command inputs v−1 has pdf Ordered statistics: f ( y n ) , f ( y 1 ) , f ( y K ) , f ( y i , y j )
g n ( y n ) = n [ F ( y n ) ] n − 1 f ( y n ) , a < y n < b (5)
g 1 ( y 1 ) = n [ 1 − F ( y 1 ) ] n − 1 f ( y 1 ) , a < y 1 < b (6)
Let g ( y i ) = i be hierarchical transformations of signal inputs
g k ( y k ) = n ! ( k − 1 ) ( n − k ) ! [ F ( y k ) ] k − 1 [ 1 − F ( y k ) ] n − k f ( y k ) , a < y k < b (7)
g i j ( y i , y j ) = n ! ( i − 1 ) ! ( j − i − 1 ) ! ( n − j ) ! [ F ( y i ) ] i − 1 [ F ( y j ) − F ( y i ) ] j − i − 1 × [ 1 − F ( y j ) ] n − j f ( y i ) f ( y j ) , a < y k < b (8)
and zero elsewhere.
An analytical approach was used to derive the optimal distributions of the convoluted strategies. Since the various events are independent the convolution of strategies was obtained by their product. The derivative of this convoluted strategy was equated to zero then solved.
B ( z ¯ , v ¯ | v ¯ ′ ) ~ discrete conditional jpdfs is given by (uniform × order statistic) × ( g C A ⋅ g C B ) .
B ( z ¯ , v ¯ | v ¯ ′ ) of first/initial command input.
B ( z ¯ , v ¯ | v ¯ ′ ) = 1 N ⋅ N [ 1 − F ( y 1 ) ] N − 1 f ( y 1 ) ( N x ) P x q N − x ( N − 1 x − 1 ) P x q N − x (9)
= [ 1 − F ( y 1 ) ] N − 1 p [ ( N x ) P x q N − x ] 2 = [ 1 − F ( y 1 ) ] N − 1 p [ ( N y 1 ) P y 1 q N − y 1 ] 2 ⋅ y 1 N (10)
p ≡ f ( y 1 ) , y 1 = x , 1 < y 1 < N (11)
F ( y ) = ∫ f ( y ) d y or (12)
F ( y ) = ∑ f ( y ) , q = 1 − p (13)
B ( z ¯ , v ¯ | v ¯ ′ ) of final/last command input.
B ( z ¯ , v ¯ | v ¯ ′ ) = 1 N ⋅ N [ 1 − F ( y n ) ] N − 1 f ( y n ) ( N y n ) P y n q N − y n ( N − 1 y n − 1 ) P y n q N − y n = [ 1 − F ( y n ) ] N − 1 p [ ( N y n ) P y n q N − y n ] 2 = y n N , p ≡ f ( y n ) , a < y n < b (14)
B ( z ¯ , v ¯ | v ¯ ′ ) of intermediate kth command input.
B ( z ¯ , v ¯ | v ¯ ′ ) = 1 N ⋅ ( N y k ) [ F ( y k ) ] k − 1 [ 1 − F ( y k ) ] N − k ⋅ p [ ( N y k ) P y k q N − y k ] 2 y k N 1 N ⋅ ( N y k ) [ F ( y k ) ] k − 1 [ 1 − F ( y k ) ] N − k ⋅ p [ ( N y k ) P y k q N − y k ] 2 ⋅ y k N , p ≡ f ( y k ) , a < y k < b (15)
B ( z ¯ , v ¯ | v ¯ ′ ) of interacting ith and jth command input.
B ( z ¯ , v ¯ | v ¯ ′ ) = N ! ( i − 1 ) ! ( j − i − 1 ) ! ( N − j ) ! [ F ( y i ) ] i − 1 [ F ( y j ) − F ( y i ) ] j − 1 ⋅ [ 1 − F ( y j ) ] N − j f ( y i ) f ( y j ) ⋅ ( N y i ) P y i q N − y i ( N − y i y j − 1 ) P y j q N − y j − y i = N ! ( i − 1 ) ! ( j − i − 1 ) ! ( N − j ) ! [ F ( y i ) ] i − 1 [ F ( y j ) − F ( y i ) ] j − i − 1 ⋅ [ 1 − F ( y j ) ] N − j p i p j [ ( N y i ) p i y i q i N − y i ] 2 ⋅ y i N ⋅ [ ( N − y i y j − 1 ) p i y i q i N − y j − y i ] 2 ⋅ y j N − y i
p i ≡ f ( y i ) , p j ≡ f ( y j ) , a < y i < y j < b (16)
d d p B ( z ¯ , v ¯ | v ¯ ′ ) = 0 to get optimal prob of command inputs. These derivatives yield:
( N − 1 ) [ 1 − F ( y 1 ) ] N − 2 ( − p ) ( p ) [ ( N y i ) p y 1 q N − y 1 ] 2 y 1 N + [ 1 − F ( y 1 ) ] N − 1 ⋅ [ ( N y i ) p y 1 q N − y 1 ] 2 y 1 N + 2 p [ 1 − F ( y 1 ) ] N − 1 ⋅ y 1 N [ ( N y i ) p y 1 q N − y 1 ] [ ( N y 1 ) y 1 p y 1 − 1 q N − y 1 − ( N − y 1 ) p y 1 q N − y 1 − 1 ] = 0 (17)
− p 2 ( N − 1 ) 1 − F ( y 1 ) y 1 N + y 1 N + 2 p [ ( N y 1 ) p y 1 q N − y 1 ] − 1 ⋅ [ ( N y i ) y 1 p y 1 − 1 q N − y 1 − ( N − y 1 ) p y 1 q N − y 1 − 1 ] y 1 N = 0
1 − p 2 ( N − 1 ) 1 − F ( y 1 ) + 2 p [ ( N y 1 ) p y 1 q N − y 1 ] − 1 ⋅ [ ( N y i ) y 1 p y 1 − 1 q N − y 1 − ( N − y 1 ) p y 1 q N − y 1 − 1 ] = 0 (18)
Simplifying Equation (18) to get
1 − p 2 ( N − 1 ) 1 − F ( y 1 ) + 2 p ( y 1 p − N − y 1 q ) = 0 (19)
1 + 2 p ( y 1 p − N − y 1 q ) = p 2 ( N − 1 ) 1 − F ( y 1 ) (20)
1 + 2 p p q ( q y 1 − p ( N − y 1 ) ) = p 2 ( N − 1 ) 1 − F ( y 1 ) (21)
q + 2 ( q y 1 − p ( N − y 1 ) ) = p 2 q ( N − 1 ) 1 − F ( y 1 ) = q + 2 ( y 1 ( q + p ) − N p ) (22)
1 − F ( y 1 ) = p 2 q ( N − 1 ) q + 2 ( y 1 ( q + p ) − N p ) (23)
∴ F ( y 1 ) = 1 − ( N − 1 ) p 2 q q + 2 ( y 1 ( q + p ) − N p ) = 1 − ( N − 1 ) p 2 q q + 2 ( y 1 − N p ) (24)
Similarly
F ( y N ) = 1 − ( N − 1 ) p 2 q q + 2 ( y N ( q + p ) − N p ) = 1 − ( N − 1 ) p 2 q q + 2 ( y N − N p ) (25)
B ( z ¯ , v ¯ / v ′ ) for intermediate kth command inputs
B ( z ¯ , v ¯ / v ′ ) = 1 N ( N y i ) [ F ( y k ) ] k − 1 [ 1 − F ( y k ) ] N − k p [ ( N y i ) p y k q N − y k ] 2 y k N
d d p B ( z ¯ , v ¯ / v ′ ) = 0 (26)
d d p B ( z ¯ , v ¯ / v ′ ) = y k N ( N y i ) { ( k − 1 ) [ F ( y k ) ] k − 2 ( p ) p [ 1 − F ( y k ) ] N − k [ ( N y k ) p y k q N − y k ] 2 + ( N − k ) [ F ( y k ) ] k − 1 [ 1 − F ( y k ) ] N − k − 1 ( − p ) p [ ( N y i ) p y k q N − y k ] 2 + 2 p [ F ( y k ) ] k − 1 [ 1 − F ( y k ) ] N − k [ ( N y i ) y k p y k − 1 q N − k − ( N − y k ) p y k q N − y k − 1 ] ⋅ [ ( N y i ) p y k q N − y k ] } = 0 (27)
Simplifying Equation (27) to get
1 + ( k − 1 ) p 2 F ( y k ) − ( N − k ) p 2 1 − F ( y k ) + 2 p ( y k p − N − y k q ) = 0 (28)
F ( y k ) [ 1 − F ( y k ) ] + ( k − 1 ) [ 1 − F ( y k ) ] p 2 − ( N − k ) F ( y k ) p 2 F ( y k ) [ 1 − F ( y k ) ] + 2 q ( q y k − ( N − y k ) p ) = 0 (29)
F ( y k ) [ 1 − F ( y k ) ] + p 2 ( k − 1 ) [ 1 − F ( y k ) ] − p 2 − ( N − k ) F ( y k ) = − 2 q ( y k ( q + p ) − N p ) F ( y k ) [ 1 − F ( y k ) ] (30)
F ( y k ) − [ F ( y k ) ] 2 + p 2 ( k − 1 ) − p 2 ( k − 1 ) F ( y k ) + p 2 F ( y k ) − p 2 ( N − k ) F ( y k ) = − 2 q [ y k ( q + p ) − N p ] F ( y k ) + 2 q [ y k ( q + p ) − N p ] [ F ( y k ) ] 2 (31)
F ( y k ) − [ F ( y k ) ] 2 + p 2 ( k − 1 ) − k p 2 F ( y k ) + p 2 F ( y k ) − N p 2 F ( y k ) + k p 2 F ( y k ) = − 2 q [ y k ( q + p ) − N p ] F ( y k ) − 2 q [ y k ( q + p ) − N p ] [ F ( y k ) ] 2 (32)
{ 1 − 2 q [ y k ( q + p ) − N p ] } [ F ( y k ) ] 2 + { − 1 + p 2 ( k − 1 ) + p 2 ( N − k ) − 2 q [ y k ( q + p ) − N p ] } F ( y k ) − p 2 ( k − 1 ) = 0 (33)
{ 1 − 2 q [ y k ( q + p ) − N p ] } [ F ( y k ) ] 2 + { ( N − 1 ) p 2 − 1 − 2 q [ y k ( q + p ) − N p ] } F ( y k ) − p 2 ( k − 1 ) = 0 (34)
By quadratic equation: a x 2 + b x + c = 0 , solution: ( x − α ) ( x − β ) = 0 , ( α , β ) = − b ± b 2 − 4 a c 2 a
Let
F ( y i ) = x (35)
∴ F ( y k ) = 1 − ( N − 1 ) p 2 + 2 q [ y k ( q + p ) − N p ] 2 { 1 − 2 q [ y k ( q + p ) − N p ] } ± { 1 − ( N − 1 ) p 2 + 2 q [ y k ( q + p ) − N p ] } 2 + 4 { 1 − 2 q [ y k ( q + p ) − N p ] ( k − 1 ) p 2 } 2 { 1 − 2 q [ y k ( q + p ) − N p ] } (36)
F ( y i j ) = N ! ( i − 1 ) ! ( j − i − 1 ) ! ( N − j ) ! [ F ( y i ) ] i − 1 p i [ ( N y i ) p i y i q N − y i ] 2 y i N ⋅ [ F ( y j ) − F ( y i ) ] j − i − 1 [ 1 − F ( y j ) ] N − j p j [ ( N y j ) p j y j q N − y j − y i ] 2 y j N − y i (37)
d d p B ( z ¯ , v ¯ / v ′ ) = 0
Let
ϕ 1 = [ F ( y i ) ] i − 1 p i [ ( N y i ) p i y i q i N − y i ] 2 y i N (38a)
ϕ 2 = [ F ( y j ) − F ( y i ) ] j − i − 1 (38b)
ϕ 3 = [ 1 − F ( y j ) ] N − j p j [ ( N − y i y j − 1 ) p j y j q j N − y j − y i ] 2 y j N − y i (38c)
d d p B ( z ¯ , v ¯ / v ′ ) = ϕ ′ 1 ϕ 2 ϕ 3 + ϕ 1 ϕ ′ 2 ϕ 3 + ϕ 1 ϕ 2 ϕ ′ 3 = 0 (39)
ϕ ′ 1 = ( i − 1 ) [ F ( y i ) ] i − 2 p i 2 [ ( N y i ) p i y i q N − y i ] 2 + [ F ( y i ) ] i − 1 [ ( N y i ) p i y i q N − y i ] 2 + 2 p i [ F ( y i ) ] i − 1 [ ( N y i ) p y i q N − y i ] [ ( N − y i ) p y i q N − y i − 1 ] y i N = y i N ⋅ { ( i − 1 ) p i 2 F ( y i ) + 1 + 2 p i [ y i p i − ( N − y i ) q i ] } [ F ( y i ) ] i − 1 [ ( N y i ) p i y i q i N − y i ] 2 y i N = y i N { ( i − 1 ) [ F ( y i ) ] − 1 p i 2 + 1 + 2 q i [ y i ( q i + p ) − N p i ] } [ F ( y i ) ] i − 1 [ ( N y i ) p i y i q N − y i ] 2 (40)
ϕ ′ 2 = ( j − i − 1 ) [ F ( y j ) − F ( y i ) ] j − i − 2 ( p j − p i ) (41)
ϕ ′ 3 = ( N − j ) [ 1 − F ( y i ) ] N − j − 1 ( − p j ) p j [ ( N − y i y j − 1 ) p i y i q j N − y j − y i ] 2 + [ 1 − F ( y j ) ] N − j [ ( N − y i y j − 1 ) p j y j q j N − y j − y i ] 2 + 2 [ 1 − F ( y j ) ] N − j p j [ ( N − y i y j − 1 ) p j y j q j N − y j − y i ] ⋅ [ ( N − y i y j − 1 ) ( y j p j y j q j N − y j − y i − ( N − y j − y i ) p j y j q j N − y j − y i ) ] ⋅ y j N − y i (42)
= { − ( N − j ) p j 2 1 − F ( y j ) + 1 + 2 p j [ y j p j − N − y j − y i q j ] } ⋅ [ 1 − F ( y j ) ] N − j [ ( N − y i y j − 1 ) p j y j q j N − y j − y i ] 2 ⋅ y j N − y i = y j N − y i { − ( N − j ) p j 2 [ 1 − F ( y j ) ] − 1 + 1 + 2 q j [ y j ( q j + p j ) − N p j ] } ⋅ [ 1 − F ( y j ) ] N − j [ ( N − y i y j − 1 ) p j y j q j N − y j − y i ] 2 (43)
d B d p ( z ¯ , v ¯ \ v ′ ) = { ( i − 1 ) p i 2 F ( y i ) + 1 + 2 p j [ y j ( q i + p i ) − N p i ] } B ( z ¯ , v ¯ / v ′ ) + ( j − i − 1 ) ( p j − p i ) [ F ( y j ) − F ( y i ) ] − 1 B ( z ¯ , v ¯ / v ′ ) + { − ( N − j ) p j 2 [ 1 − F ( y j ) ] − 1 + 1 − 2 q j [ y j ( q j + p j ) − N p j ] } B ( z ¯ , v ¯ / v ′ )
= ( i − 1 ) p i 2 [ F ( y i ) ] − 1 + 1 − 2 q i [ y i ( q i + p i ) − N p i ] + ( j − i − 1 ) ( p j + p i ) [ F ( y j ) − F ( y i ) ] − 1 − ( N − j ) p j 2 [ 1 − F ( y j ) ] − 1 + 1 − 2 q j [ y j ( q j + p j ) − N p j ] = 0 (44)
Simplifying Equation (44) to get
( i − 1 ) p i 2 [ F ( y j ) − F ( y i ) ] [ 1 − F ( y i ) ] + { 1 + 2 q i [ y i ( q i + p i ) − N p i ] } F ( y i ) [ F ( y j ) − F ( y i ) ] [ 1 − F ( y j ) ] + ( j − i − 1 ) ( p j − p i ) F ( y i ) [ 1 − F ( y j ) ] − ( N − j ) p j 2 F ( y i ) [ F ( y j ) − F ( y i ) ] + { 1 − 2 q j [ y j ( q j + p j ) − N p j ] } [ F ( y i ) − F ( y j ) − F ( y i ) ] = 0 (45)
( i − 1 ) p i 2 [ F ( y j ) − F ( y j ) 2 − F ( y i ) + F ( y i ) F ( y j ) ] + { 1 − 2 q i [ y i ( q i + p i ) − N p i ] } F ( y i ) [ F ( y j ) − F ( y i ) F ( y j ) 2 ] [ 1 − F ( y j ) ] + ( j − i − 1 ) ( p j − p i ) F ( y i ) [ 1 − F ( y j ) ] − ( N − j ) p j 2 F ( y i ) [ F ( y j ) − F ( y i ) ] + { 1 − 2 q j [ y j ( q j + p j ) − N p j ] } [ F ( y i ) − F ( y j ) − F ( y i ) F ( y j ) 2 ] ⋅ [ F ( y i ) 2 + F ( y j ) F ( y i ) 2 ] = 0 (46)
{ ( N − j ) p j 2 − 1 + 2 q j [ y j ( q j + p j ) − N p j ] − 1 − 2 q j [ y i ( q i + p i ) − N p i ] } ( F ( y i ) ) 2 + [ ( j − i − 1 ) ( p j − p i ) − ( i − 1 ) p i 2 ] F ( y i ) − [ ( i − 1 ) p i 2 ( F ( y j ) ) 2 − F ( y i ) ] + [ ( i − 1 ) p i 2 ] ( F ( y j ) ) 2 − F ( y j ) + { 1 + 2 q i [ y i ( q i + p i ) − N p i ] } ⋅ [ F ( y j ) ( F ( y i ) ) 2 − [ F ( y i ) ( F ( y j ) ) 2 ] + { ( i − 1 ) p i 2 + 1 + 2 q i [ y i ( q i + p i ) − N p i ] − [ ( j − i − 1 ) ( p j − p i ) − ( N − j ) p j 2 − 1 − 2 q j [ y j ( q j + p j ) − N p j ] ] } F ( y i ) F ( y j ) ] = 0 (47)
Then from Equation (35)
{ ( N − j ) p j 2 − 1 + 2 q j [ y j ( q j + p j ) − N p j ] − 1 − 2 q i [ y i ( q i + p i ) − N p i ] } ( F ( y i ) ) 2 + [ ( j − i − 1 ) ( p j − p i ) − ( i − 1 ) p i 2 ] F ( y i ) + 1 = 0 (48)
∴ F ( y i ) = − ( j − i − 1 ) ( p j − p i ) − ( i − 1 ) p i 2 ± [ ( j − i − 1 ) ( p j − p i ) − ( i − 1 ) p i 2 ] 2 − 4 { ( N − j ) p j 2 − 1 } 2 { ( N − j ) p j 2 − 1 + 2 q j [ y j ( q j + p j ) − N p j ] − 1 − 2 q i [ y i ( q i + p i ) − N p i ] } + 2 q j [ y j ( q j + p j ) − N p j ] − 1 − 2 q i [ y i ( q i + p i ) − N p i ] [ ( i − 1 ) p i 2 ] [ ( F ( y j ) ) 2 − F ( y j ) ] − 1 2 { ( N − j ) p j 2 − 1 + 2 q j [ y j ( q j + p j ) − N p j ] − 1 − 2 q i [ y i ( q i + p i ) − N p i ] } = 0 (49)
Simplifying the above quadratic equations to get
− [ ( i − 1 ) p i 2 ] [ ( F ( y j ) ) 2 − F ( y j ) ] − 1 = 0 (50)
⇒ ( i − 1 ) p i 2 [ ( F ( y j ) ) 2 − F ( y j ) ] + 1 = 0 (51)
∴ F ( y j ) = ( i − 1 ) p i 2 ± ( i − 1 ) p i 2 − 4 ( i − 1 ) p i 2 2 ( i − 1 ) p i 2 (52)
But
F ( y j ) ( F ( y i ) ) 2 − F ( y i ) ( F ( y j ) ) 2 = F ( y i ) F ( y j ) [ F ( y i ) − F ( y j ) ] (53)
Then
{ 1 + 2 q i [ y i ( q i + p i ) − N p i ] } [ F ( y i ) − F ( y j ) ] = − ( i − 1 ) p i 2 + 2 q i [ y i ( q i + p i ) − N p i ] − ( j − i − 1 ) ( p j − p i ) − ( N − j ) p j 2 − 1 − 2 q j [ y j ( q j + p j ) − N p j ] (54)
∴ F ( y j ) − F ( y i ) = ( i − 1 ) p i 2 + 1 + 2 q i [ y i ( q i + p i ) − N p i ] − ( j − i − 1 ) ( p j − p i ) 1 + 2 q i [ y i ( q i + p i ) − N p i ] + − ( N − j ) p j 2 − 1 − 2 q j [ y j ( q j + p j ) − N p j ] 1 + 2 q i [ y i ( q i + p i ) − N p i ] = 1 + ( i − 1 ) p i 2 − ( j − i − 1 ) ( p j − p i ) − ( N − j ) p j 2 − 1 − 2 q j [ y j ( q j + p j ) − N p j ] 1 + 2 q i [ y i ( q i + p i ) − N p i ] (55)
F ( y 1 ) = 1 − ( 10 − 1 ) ( 0.3 ) 2 ( 0.7 ) 0.7 + 2 ( 1 − 3 ) = 1 − 0.567 − 3.3 = 1 + 0.171818 = 1.172 unrealistic
F ( y 3 ) = 1 − 9 ( 0.3 ) 2 ( 0.7 ) 0.7 = 1 − 0.81 = 0.19
F ( y 10 ) = 1 − ( 10 − 1 ) ( 0.3 ) 2 ( 0.7 ) 0.7 + 2 ( 10 − 3 ) = 1 − 0.567 14.7 = 1 − 0.0386 = 0.9614
These results indicate:
1) F ( y 1 ) not feasible due to no (or low) command input(s).
2) F ( y 3 ) feasible.
3) F ( y 10 ) feasible. As the hierarchy increases correspondingly the distribution increases. This can be taken as workload.
Example 1: Given p k = 0.3 , k = 5 , N = 10 , y k = k
F ( y k ) = F ( y 5 ) = 1 − 9 ( 0.09 ) + 2 0.7 ( 5 − 10 ( 0.3 ) ) 2 { 1 + 2 0.7 ( 5 − 10 ( 0.3 ) } ± { − 1 + 9 ( 0.09 ) − 2 0.7 ( 5 − 10 ( 0.3 ) } 2 + 4 ( 1 + 2 0.7 ( 5 − 10 ( 0.3 ) ) 4 ( 0.09 ) ) 2 { 1 + 2 0.7 ( 5 − 10 ( 0.3 ) }
= 1 − 0.81 + 4 0.7 ± ( − 0.19 + 4 0.7 ) 2 + 4 ( 1 + 4 0.7 ( 0.36 ) ) 2 { 1 + 4 0.7 } = 0.12 + 5.7143 ± ( 5.5243 ) 2 + 4 ( 3.0572 ) 2 ( 6.7143 ) = 5.8343 ± 30.5179 + 12.288 13.4286 = 5.8343 ± 42.7467 13.4286 = 5.8343 ± 6.5381 13.4286
12.3724 13.4286 , − 0.7467 13.4286 = 0.92135 , − 0.05241 ( unrealistic )
Example 2: Given p i = 0.3 , p j = 0.5 , i = 4 , j = 7 , k = 5 , N = 10 , y k = K
F ( y 4 ) = F ( y 4 ) = − ( 2 ( 0.2 ) − 3 ( 0.09 ) ) 2 { 3 ( 0.25 ) − 1 + 2 0.5 [ 7 − 5 ] − 1 − 2 0.7 [ 4 − 3 ] } ± [ 2 ( 0.2 ) − 3 ( 0.09 ) ] 2 + 4 { 3 ( 0.25 ) − 1 + 2 0.5 [ 7 − 5 ] − 1 − 2 0.7 [ 4 − 3 ] } 2 { 3 ( 0.25 ) − 1 + 2 0.5 [ 7 − 5 ] − 1 − 2 0.7 [ 4 − 3 ] } = − ( 0.4 − 0.27 ) ± ( 0.4 − 0.27 ) 2 + 4 { 0.75 − 1 + 4 ( 2 ) − 1 − 2.8571 } 2 { 0.75 − 1 + 4 ( 2 ) − 1 − 2.8571 } = − 0.13 ± 15.5885 7.7858 = − 0.13 ± 3.9482 7.7858 = 3.8182 7.7858 = 0.49041 , − 4.0782 7.7858 ( unrealistic )
F ( y j ) = F ( y 7 ) = 3 ( 0.09 ) ± ( 3 ( 0.09 ) ) 2 + 4 [ 3 ( 0.09 ) ] 2 ( 3 ( 0.09 ) = 0.27 ± 0.0729 + 1.08 0.54 = 0.27 ± 0.1529 0.54 = 0.27 ± 1.073732 0.54 = 1.343732 0.54 = 2.4884 , − 0.803732 0.54 = − 1.4884 ( both unrealistic ) ≈ 0 certainly unrealistic
F ( y j ) = F ( y i ) = 3 ( 0.09 ) + 1 + 2 0.7 [ 4 − 3 ] − 2 ( 0.2 ) − 3 ( 0.25 ) − 1 − 2 0.5 [ 7 − 5 ] 1 + 2 0.7 − 2 ( 0.2 ) − 1 − 2 0.5 ( 2 ) = − 6.0229 − 5.5429 = 1.0866 ≈ 0 certainly unrealistic
The distribution of events at various stages of interactions were convolutions to obtain system distribution. Thereafter, analytical approach was used at various stages to obtain optimal distribution of the command inputs.
Optimal distribution values obtained can be taken as system efficiency of the model, which can be used for system control.
The authors declare no conflicts of interest regarding the publication of this paper.
Oladejo, M.O., Okafor, U.L., Chinyio, D.T. and Uwa, C.O. (2019) Optimal Distribution of N-Team Interacting Decision Makers with Hierarchical Command Inputs That Are Predicated on Order Statistics. Open Journal of Optimization, 8, 1-14. https://doi.org/10.4236/ojop.2019.81001