Optimal Distribution of N-Team Interacting Decision Makers with Hierarchical Command Inputs That Are Predicated on Order Statistics

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 ana-lysed, to have hierarchical command inputs that are predicated on order statistics distributions. The results give optimal distributions.


Introduction
The Double-Team of decision Maker Model that Boettcher and Levis developed [1] was extended and generalized by Oladejo [2] to N-Teams of Decision Makers with N n Bounded Rationality Constraints. A Spectral Analysis of the model was provided by Oladejo [3], where command inputs were based on uniform distribution. This work considered the same model with modified command inputs that were hierarchical and whose distribution was predicated on order statistics. In this new work, the command inputs were categorized according to the superiority of the commander who was next higher in rank to the officer making input. The analytical procedure of optimizing convoluted strategies was used to derive the optimal distribution functions of the hierarchical command inputs. In the previous work there was a mixture of vertical and horizontal signal commu-How to cite this paper: Oladejo π is the portioning algorithm of inputs to respective DM. u is the internal decision. f i is the algorithm for process u i to obtain the battle scenario. q i is the i th team input regulator. z i is the initial situational assessment. Y is the output or desired result. h j is the processing algorithm for the final choice leading to Y. h i is the algorithm for process v i .
( ) | P v z is response selection strategy that maps z to Y in the absence of v' and determines choices of h j .

| ,
P v z v′ is response selection strategy that maps z and v' to v, and it also determines choices of h j .
is entropy of inputs, where p(x) is probability or uncertainty associated with N random variables, X.

( )
This is the structure of the generalized developed model as shown below: The generalized developed model of Figure 1 is as shown:

Analysis
Strategies which are probability density functions (pdfs) are given as follows: g is internal coordination strategy of corresponding algorithm which depends on the distribution of their respective inputs. ( ) h v′ ∼ Weibull (due to reliability of subsystem before reaching final stage: ∼ expo (this is due to random occurrences).
and zero elsewhere.

Methodology
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) ( ) , | B z v v′ of first/initial command input.

Derivation of pdf
B z v v′ of interacting i th and j th command input.

Optimal Probabilities
to get optimal prob of command inputs. These derivatives yield: Simplifying Equation (27) to get Simplifying the above quadratic equations to get

Discussion
The distribution of events at various stages of interactions were convolutions to obtain system distribution. Thereafter, analytical approach was used at various Open Journal of Optimization stages to obtain optimal distribution of the command inputs.

Conclusion
Optimal distribution values obtained can be taken as system efficiency of the model, which can be used for system control.