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\title{Notes on the Application of Algorithmic Principles to Organizational Analysis}
\author{John Conover}
\date{\today}
\maketitle

\begin{abstract}

    Professional managers have observed that organizational
    performance under certain circumstances has exhibited non-linear
    characteristics as a function of increasing task complexities.
    Certain organizational activities are characterized by a search
    for a solution to a problem.  If these search activities could be
    analyzed using algorithmic analysis, in much the same way that
    algorithmic performance of a computer program is analyzed, then
    some qualitative, and possibly quantitative, insight could be
    derived into the way that organizations solve problems.  This in
    turn could lead to the optimization of the parametric issues
    confronting working organizations.  A heuristic proof will be
    offered that indicates that these non-linearities as a function of
    problem complexity do indeed exist, are unavoidable, and can
    severely impact the organization's efficiency and performance, if
    not appropriately optimized.  A general model, using algorithmic
    principles, is derived for organizational problem solving, and
    several optimizations presented.  It is the general conclusion of
    this presentation that problem solving organizations do indeed
    exhibit non-linear time/resource requirements in response to
    complexity issues, and that time and/or resources, may be
    optimized to provide the best organizational design. Further, it
    is shown that ignoring these non-linear issues, and operating
    non-optimally, can lead to organizational dysfunction.  Finally,
    it is shown that the interface coordination between organizational
    members is, singularly, the limiting issue in organizational
    performance and as such is the most important issue in
    organizational optimization.  Throughout the text, metric methods
    are suggested, for measurement of parameters, that effect
    organizational performance and optimization.  It is the basic
    premiss that if it is assumed that problem solving is
    algorithmical, then problem solving must obey the rules of
    algorithmical dynamics, and if problem solving obeys the rules of
    algorithmical dynamics, then problem solving can be analyzed and
    is therefore predictable.  Any other premiss essentially states
    that problem solving is not a systematic procedure.  This
    presentation explores the nature of organizational problem solving
    with algorithmic analysis.

\end{abstract}
\pagenumbering{roman}

    \chapter*{Preface}
        \addcontentsline{toc}{chapter}{Preface}

        The concepts presented herein were developed over a period of
        25 years and are not innovational.  That organizations doing
        problem solving must have disproportionately more resource
        allocation to handle more complex problems is not a new
        concept-military theorist have known that for millennia.  That
        interface coordination in the group is important is a concept
        as old as humanity itself.  What is new here is that these
        issues can be quantified and subjected to numerical methods,
        or metrics, to arrive at an optimized operating situation for
        the organization, and that qualitative issues of
        organizational design can be addressed analytically using only
        remedial engineering optimization techniques, in a
        deterministic fashion.  Further, in this age of computer
        automation of the organization, it can, within reasonable
        accuracies, be done automatically.

        It is not a difficult task, particularly with today's system
        accounting techniques that are available in contemporary
        operating systems, to write a program that can tell how many,
        for how long, and for what, commands were issued in a given
        interval of a project task.  The analysis of these activities
        determine the parametric values, which can be optimized, using
        the techniques presented herein.  Indeed, the issue of what
        complexity really means is probably better addressed by the
        amount of computer resources dispensed to a project, than the
        subjective evaluation of those responsible for the project.

        Is such collection of data an ethical issue?  I always took
        the courtesy of informing people that they were being
        monitored, and most agreed, particularly if they could
        participate in the analysis of the data.  But the ethical
        issue probably is still on the table.  Most were intrigued
        when told that the interface coordination issue, which as it
        will be shown, is centrally the most important issue
        confronting the organization-and not only that, the issue is
        endemic to the problem solving process of the organization
        itself.\smallskip

        \def\thismonth % convert numeric month to month name
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                    \or January\or February\or March\or April\or May\or June
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                \fi
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        \begin{flushright}
            ---J. C. C.\\
        \end{flushright}

        \begin{flushleft}
            Campbell, California \\
            \thismonth, \number\year \\
        \end{flushleft}
        \clearpage

\tableofcontents
    \clearpage

\pagenumbering{arabic}
\pagestyle{headings}

    \chapter{General Analysis of Organizations}
        \label{general}

        \section{Introduction}

            When confronted with an organization that takes
            disproportionately more time and resources to solve
            problems of increasing complexity, (ie., the
            organization's time and resource commitment to solve a
            problem is a non-linear function of the problem's
            complexity,) managers often, erroneously, jump to the
            conclusion that the organization is non-productive,
            inefficient, or mismanaged.  It is the premiss of this
            presentation that these organizational non-linearities not
            only happen, but are the very essence of the way
            organizations operate. On the one hand, the assumption
            that these non-linearities exist complicates the
            understanding of how to operate an organization, however
            on the other hand, since these complexities are inherently
            non-linear in nature, they lend well to optimization
            techniques that can be used to increase organizational
            efficiency, productivity, and performance. It will be
            shown in this chapter that, in at least complex
            organizations, it is not the completion of tasks that is
            an issue, but the process by which the tasks are completed
            that determines how effective an organization is at
            solving problems. Throughout this presentation, the
            emphasis will be on analyzing operational organizations
            that must perform services on a continuing basis, as
            opposed to project management.

            An algorithm is a method for solving a problem.  The
            formal foundations for algorithmic analysis were
            established during the axiomatization of the theory of
            mathematical operators and set theory developed during the
            first half of this century as mathematicians established
            the theoretical foundations of mathematics itself. In lay
            terms, an algorithm is a procedure that searches for the
            solution to a problem.  Algorithmic analysis is the study
            of the performance and methodologies of these procedures,
            and is a well established science, dating back to the
            early numerical analysis techniques used by Newton.
            Although the concept of the algorithm is generally
            associated with computer technology, the application of
            the concept to computer science is relatively recent. This
            discussion is about the application of algorithmic
            analysis to the problem of organizational performance.

        \section{The Organizational Performance Problem}

            Consider an organization, composed of many people, with
            many tasks to do.  If the organization's objective is to
            search for a solution to some problem, then algorithmic
            analysis may be used to analyze the organization's
            operations\footnote{Note that not all organizations fall
            into this category. Organizations that process things in
            an administrative or mechanically predetermined fashion
            may, but not necessarily.  The key issue is the words
            ``searching for a solution''.}.  A good example of this
            type of organization is a marketing organization that is
            searching for a solution on how to introduce a new product
            into the marketplace, or searching for a solution on how
            to increase sales. Another example is an engineering
            development organization that is searching for a
            technological solution in new product development. Since
            these organizations are searching for a solution to a
            problem, algorithmic analysis dictates that as a function
            of the problem's complexity, the problem's solution will
            probably require an exponential solution time.  In
            technical parlance, the ``search for solution'' problem is
            NP in nature, (and probably NP-hard in the examples given
            above.) The organization's attempt at solving the problem
            is therefore constrained by the characteristic nature of
            how NP problems lend there self to solution.

            Software organizations are familiar with the non-linear
            characteristics of development time as a function of
            increasing task complexities, and the phenomena is well
            documented.  More importantly, these non-linearities
            create software cost and time overruns which are well
            known in the industry.  The complexity of software is an
            essential property, not an accidental one. Software
            systems have orders-of-magnitude more states than
            computers do.  It is not surprising that much effort in
            the industry is focused on automating the complexity, ie.,
            the development process, with such things as CASE and
            concurrent engineering.  As mentioned above, complexity
            issues must be addressed by organizations other than
            software developers.  Software organizations, however,
            have been the subject of many published empirical studies
            on these complexity issues, which will be referenced
            throughout this analysis, and are therefore of interest
            when studying how organizations, in general, handle
            complexity issues.

        \section{Derivation of the Organizational Model}

            For tutorial purposes, the organization will be modeled,
            starting with a single person organization, then one
            person will be added to the organization, with the tasks
            divided equally between the two persons.  Then
            compensation for coordination issues between the two
            persons will be added. Finally, the general case will be
            analyzed.

            \subsection{Variables used in Derivations}

                \begin{description}

                    \item [$c$] is the linear factor of complexity.
                    If $c$ is divided by two, the task is exactly half
                    as complex. This variable should also reflect the
                    ability of the organization to handle complex
                    tasks. An organization that is twice as efficient
                    as another organization would have half the $c$.
                    (It may be appropriate to add a factor of
                    $\frac{1}{s}$ to $c$, where $s$ represents an
                    organization's skill level which may change do to
                    learning.)

                    \item [$e$] is 2.718\dots

                    \item [$k$] is a constant of proportionality with
                    the same dimension as $t$ below.

                    \item [$n$] is the number of persons in the
                    organization.

                    \item [$p$] is the probability, multiplied by a
                    constant of proportionality, that a coordination
                    issue will increase the time required to solve the
                    problem.  In some sense, $p$ reflects the quality
                    of the coordination process.  A $p$ of zero means
                    that there are no coordination issues. The
                    constant of proportionality reflects how sensitive
                    the system is to problems created by coordination
                    issues.

                    \item [$t$] is the execution time required for an
                    organization of $n$ many persons to solve a
                    problem of complexity, $c$. It would generally
                    have the dimension of time.

                    \item [$r$] is the total time allocation for an
                    organization of $n$ many persons to solve a
                    problem of complexity, $c$. It would generally
                    have the dimension of time, and is more pertinent
                    as an indicator of the total resources allocated
                    to solve the problem, ie., man hours.

                \end{description}

            \subsection{Derivation of the One Person Organization
            Model}

                Assuming that the time, $t_1$, required for completion
                of the search for solution to a problem is exponential
                on complexity

                \begin{equation}
                    t\left(1\right) = ke^c
                \end{equation}

                \noindent where $k$ is a constant of proportionality,
                and $c$ is proportional to the complexity of the
                problem. Likewise, in this simple case

                \begin{equation}
                    r\left(1\right) = ke^c
                \end{equation}

            \subsection{Derivation of the Two Person Organization Model}

                Assuming that the task can be divided equally between
                the two persons, and each has the same technology and
                skills in searching for a solution to the problem

                \begin{equation}
                    t\left(2\right) = ke^\frac{c}{2}
                \end{equation}

                \noindent and likewise for the resources required

                \begin{equation}
                    r\left(2\right) = ke^\frac{c}{2} + ke^\frac{c}{2} = 2ke^\frac{c}{2}
                \end{equation}

                Note that the task was divided equally, with each
                person addressing half of the complexities.  However,
                a new situation exists in that the work done by one
                person may not coordinate with the work done by the
                other, and visa versa, perhaps requiring a rework of
                some tasks, etc. If we assume that searching for, and
                solving, the problem created by the coordination issue
                is itself NP, then the previous equation must be
                multiplied by a factor of $e^p$ where $p$ is the
                probability that a coordination issue will occur
                multiplied by a constant that represents the affect
                that the issue has on the organization's
                efficiency\footnote{Note that the objective here is
                consistent with other research which presents the
                following equation

                \begin{displaymath}
                    \mbox{Actual Productivity} = \mbox{Potential Productivity} - \mbox{Losses Due to Faulty Process}
                \end{displaymath}

                \noindent The ``Actual Productivity'' in this case would be
                $\frac{c}{2}~-~p$.}. Therefore,

                \begin{equation}
                    t\left(2\right) = ke^\frac{c}{2}e^p = ke^{\frac{c}{2} + p}
                \end{equation}

                \noindent and likewise for the resources required

                \begin{equation}
                    r\left(2\right) = 2ke^\frac{c}{2}e^p = 2ke^{\frac{c}{2} + p}
                \end{equation}

            \subsection{Derivation of the n Person Organizational
            Model}

                The general solution for the problem's solution time can be
                implied

                \begin{equation}
                    t\left(n\right) = ke^{\frac{c}{n} + \left(n - 1\right) p}
                    \label{t_n}
                \end{equation}

                \noindent and likewise for the resources required

                \begin{equation}
                    r\left(n\right) = nke^{\frac{c}{n} + \left(n - 1\right) p}
                    \label{r_n}
                \end{equation}

                The reason for the $\left(n~-~1\right)$ multiplier in
                the exponent is because the probability that a
                coordination issue will happen is proportional to the
                number of interfaces that can occur.  It is assumed
                that each person interfaces with all others in the
                organization, and each interface has an equal
                probability of creating a coordination issue.

        \section{Summary}

            Equation~\ref{t_n} and~\ref{r_n} express the time and
            committed resources, respectively, for an organization to
            solve a problem, where $n$ is the number of persons in the
            organization, $c$ represents the complexity of the
            problem, $p$ represents the quality of the organization's
            interface coordination, and $k$ is a constant of
            proportionality. The metrics on $p$ can be obtained, for
            comparative studies, by solving equations~\ref{t_n}
            and~\ref{r_n} with $p~=~0$, which would be an idealized
            organization with no interface issues-naturally, a real
            organization must operate less efficiently, but how close
            the organization is to an idealized organization can be
            expressed numerically.  The metrics on problem complexity,
            $c$ is a more difficult proposition because complexity is
            somewhat subjective.  However, it can be measured, post
            facto, by knowing $p$, $n$, and the time required to solve
            the problem.  As it turns out, if the organization is
            large and it is desired to operate the organization
            optimally, these parameters can be ascertained indirectly.

    \chapter{The Flat Organization} \label{flat}

        \section{Introduction}

            The flat organization is an organization without division
            of task responsibility, and is therefore the simplest
            organization structure to analyze. Although performance
            comparisons will be made with structured organizations,
            the comparisons are presented in the more general sense
            that the predictions, derived in this section, are a
            pessimistic theoretical limit. An optimistic theoretical
            limit will be derived in Chapter~\ref{struct} which
            analyzes the performance of structured organizations. The
            performance of real organizations lay between these two
            theoretical limits since no organization is completely
            flat, or completely structured. The optimizations derived
            in this section are useful when analyzing the performance
            of the various flat organizations that are the constituent
            parts of a structured organization.

            \subsection{Variables used in Derivations}

                \begin{description}

                    \item [$c$] as before, is the linear factor of
                    complexity. If $c$ is divided by two, the task is
                    exactly half as complex. This variable should also
                    reflect the ability of the organization to handle
                    complex tasks. An organization that is twice as
                    efficient as another organization would have half
                    the $c$.  (It may be appropriate to add a factor
                    of $\frac{1}{s}$ to $c$, where $s$ represents an
                    organization's skill level which may change do to
                    learning.)

                    \item [$e$] as before, is 2.718\dots

                    \item [$k$] as before, is a constant of
                    proportionality with the same dimension as $t$
                    below.

                    \item [$n$] as before, is the number of persons in
                    the organization.

                    \item [$n_f|_{opt}$] is the number of persons in
                    the flat organization that will be required to
                    achieve the economic optimum operating situation
                    for the organization.

                    \item [$n_f|_{t=\min}$] is the number of persons
                    in the flat organization that will be required to
                    achieve the minimum solution time for a problem.

                    \item [$n_f|_{r=\min}$] is the number of persons
                    in the flat organization that will be required to
                    achieve the minimum resource allocation to solve a
                    problem.

                    \item [$p$] as before, is the probability,
                    multiplied by a constant of proportionality, that
                    a coordination issue will increase the time
                    required to solve the problem.  In some sense, $p$
                    reflects the quality of the coordination process.
                    A $p$ of zero means that there are no coordination
                    issues. The constant of proportionality reflects
                    how sensitive the system is to problems created by
                    coordination issues.

                    \item [$t_f$] is the execution time required for a
                    flat organization of $n$ many persons to solve a
                    problem of complexity, $c$.  It would generally
                    have the dimension of time.

                    \item [$r_f$] is the total time allocation for an
                    flat organization of $n$ many persons to solve a
                    problem of complexity, $c$.  It would generally
                    have the dimension of time, and is more pertinent
                    as an indicator of the total resources allocated
                    to solve the problem, ie., man hours.

                \end{description}

        \section{The Case to Increase Organizational Size}

            The question can be posed, when can a person, or persons,
            be added to an organization, presumably to decrease the
            time required to find a solution to a problem.  By
            analyzing the ratio of the time required to solve a
            problem with an additional person to the time required to
            solve the same problem without the additional person, some
            insight can be obtained on whether it is an advisable
            proposition, or from equation~\ref{t_n}

            \begin{equation}
                \frac{t_f\left(n + 1\right)}{t_f\left(n\right)} = \frac{ke^{\frac{c}{n + 1} + np}}{ke^{\frac{c}{n} + \left(n - 1\right) p}}
            \end{equation}

            \noindent and reducing

            \begin{equation}
                \frac{t_f\left(n + 1\right)}{t_f\left(n\right)} = e^{\frac{c}{n + 1} - \frac{c}{n} + p}
            \end{equation}

            \noindent Note that for a decrease in the time to solve
            the problem to occur

            \begin{equation}
                p + c\left(\frac{1}{n + 1} - \frac{1}{n}\right)
                \label{p_a}
            \end{equation}

            \noindent must be less than zero. If it is equal to zero,
            ie.,

            \begin{equation}
                p = c\left(\frac{1}{n} - \frac{1}{n + 1}\right)
                \label{p}
            \end{equation}

            \noindent then adding a person will have no effect on the
            time to solve the problem. Likewise, if equation~\ref{p_a}
            is greater than zero, then the situation exists where
            adding a person can actually increase the time required to
            solve a problem.  (Perhaps this is the mathematical
            definition of bureaucracy.)  Combining terms in
            equation~\ref{p}

            \begin{equation}
                p = c\left(\frac{n + 1 - n}{n\left(n + 1\right)}\right) = c\left(\frac{1}{n\left(n + 1\right)}\right) = \frac{c}{n^2 + n}
            \end{equation}

            \noindent then if $n~>>~1$, ie., the organization is large, and
            solving for $n$

            \begin{equation}
                n_f|_{opt} \approx \sqrt{\frac{c}{p}}
                \label{approx_1}
            \end{equation}

            \subsection{A Possible Economic Optimum}

                In some sense, it is apparent that adding persons to
                an organization without altering $p$ will result in
                diminishing returns after a certain point, and to
                maintain $p$ small will probably require automating
                the coordination interface.  It is interesting to note
                that this is one of the basic premises of concurrent
                engineering.  If it is determined that
                equation~\ref{p} really is the onset of diminishing
                returns, then this may possibly constitute the optimum
                economic operating point for the organization.

        \section{Possible Optimum Staffing Levels}

            The possible economic optimum operating point for the
            organization suggests that an optimum staffing level
            exists that will minimize the time to solve a problem, or
            alternately, minimize the resources that must be committed
            to solve the problem.

            \subsection{Optimum Staffing Level for Minimum Time}

                Taking the derivative of equation~\ref{t_n} with
                respect to the number of persons working on solving a
                problem will possibly yield the minimum solution time
                to solve the problem

                \begin{equation}
                    \frac{dt_f\left(n\right)}{dn} = ke^{\frac{c}{n} + \left(n - 1\right)p}\left(p - \frac{c}{n^2}\right)
                    \label{t_min_a}
                \end{equation}

                \noindent and by equating equation~\ref{t_min_a} to
                zero, the minima occurs when

                \begin{equation}
                    \frac{c}{n^2} = p
                    \label{t_min_b}
                \end{equation}

                \noindent or solving for $n$

                \begin{equation}
                    n_f|_{t=\min} = \sqrt{\frac{c}{p}}
                    \label{t_min}
                \end{equation}

                It is not clear whether this minima is of any
                practical interest, but it agrees nicely with
                equation~\ref{approx_1}, at least for large
                organizations.

            \subsection{Optimum Staffing Level for Minimum Resources}

                Taking the derivative of equation~\ref{r_n} with
                respect to the number of persons working on solving a
                problem will possibly yield the minimum resource
                solution to solve the problem

                \begin{equation}
                    \frac{dr_f\left(n\right)}{dn} = nke^{\frac{c}{n} + \left(n - 1\right)p}\left(p - \frac{c}{n^2}\right) + ke^{\frac{c}{n} + \left(n - 1\right)p}
                    \label{r_min_a}
                \end{equation}

                \noindent and by equating equation~\ref{r_min_a} to
                zero, the minima occurs when

                \begin{equation}
                    n\left(p - \frac{c}{n^2}\right) + 1 = 0
                \end{equation}

                \noindent or

                \begin{equation}
                    np - \frac{c}{n} + 1 = 0
                \end{equation}

                \noindent and multiplying by n

                \begin{equation}
                    pn^2 + n - c = 0
                \end{equation}

                \noindent then by the quadratic equation

                \begin{equation}
                    n = \frac{\sqrt{1 + 4pc} - 1}{2p}
                    \label{r_min}
                \end{equation}

                Again, it is not clear whether this minima is of any
                practical interest. If $n~>>~1$, ie., the organization
                is large ($\sqrt{4pc}~>>~2p$), then

                \begin{equation}
                    n_f|_{r=\min} \approx \frac{\sqrt{4pc}}{2p}
                \end{equation}

                \noindent or

                \begin{equation}
                    n_f|_{r=\min} \approx \sqrt{\frac{c}{p}}
                    \label{approx_2}
                \end{equation}

                This agrees nicely with equation~\ref{t_min}, at least
                for large organizations, which was the optimization by
                which equations~\ref{approx_a} and~\ref{approx_b} were
                derived.

        \section{Discussion}

            Severable multiple optimization alternatives exist. The
            first is the minimum time optimum. The next is the minimum
            time with optimal resources, and finally, the minimum time
            and optimal resources.

            \subsection{Minimum Time Optimum}

                Substituting $p$ from equation~\ref{t_min_b} into
                equation~\ref{t_n} yields

                \begin{equation}
                    t_f\left(n\right)|_{\min} = ke^{\frac{c}{n} + \left(n - 1\right)\frac{c}{n^2}}
                \end{equation}

                \noindent and combining terms

                \begin{equation}
                    t_f\left(n\right)|_{\min} = ke^{\frac{c}{n} + \frac{c}{n} - \frac{c}{n^2}} = ke^{\frac{c}{n}\left(2 - \frac{1}{n}\right)}
                \end{equation}

                \noindent then if $n~>>~1$, ie., the organization is large,

                \begin{equation}
                    t_f\left(n\right)|_{\min} \approx ke^{2\frac{c}{n}}
                    \label{approx_a}
                \end{equation}

                \noindent or by substituting equation~\ref{t_min}

                \begin{equation}
                    t_f\left(n\right)|_{\min} \approx ke^{2\sqrt{pc}}
                    \label{approx_a1}
                \end{equation}

                Interestingly, comparing equation~\ref{approx_a} to
                equation~\ref{t_n} with $p~=~0$, (ie., a perfect
                organization that has no interface coordination
                issues,) states that if the organization is operating
                optimally, that only 50 percent of the organization's
                efforts are directed at problem solving, and the
                remaining 50 percent is directed at interface
                coordination issues.  Possibly, this could account for
                the time slippages of 50 percent cited in the
                literature.  Although the references assumed a
                constant linear time slippage of 100 percent in the
                PERT elements, this analysis would indicate a 100
                percent increase in the amount of work that had to be
                accomplished, which would not necessarily be
                reflected, unless extrapolated, in a linear time
                slippage, (assuming that the organizations were
                operating optimally.) This may well be a question of
                semantics.

            \subsection{Minimum Time Optimum Resources}

                Likewise, substituting $p$ from equation~\ref{t_min_b}
                into equation~\ref{r_n} yields

                \begin{equation}
                    r_f\left(n\right)|_{t=\min} = nke^{\frac{c}{n} + \left(n - 1\right)\frac{c}{n^2}}
                \end{equation}

                \noindent and again combining terms

                \begin{equation}
                    r_f\left(n\right)|_{t=\min} = nke^{\frac{c}{n} + \frac{c}{n} - \frac{c}{n^2}} = nke^{\frac{c}{n}\left(2 - \frac{1}{n}\right)}
                    \label{r_n_t}
                \end{equation}

                \noindent then if $n~>>~1$, ie., the organization is
                large,

                \begin{equation}
                    r_f\left(n\right)|_{t=\min} \approx nke^{2\frac{c}{n}}
                    \label{approx_b}
                \end{equation}

                \noindent or by substituting equation~\ref{t_min}

                \begin{equation}
                    r_f\left(n\right)|_{t=\min} \approx nke^{2\sqrt{pc}}
                    \label{approx_b1}
                \end{equation}

                If it is assumed that the organizations studied in the
                literature are operating optimally and that the
                project complexity is proportional to the number of
                lines of instructions coded.

                This presentation yields a pessimistic prediction of
                these organization's performance.  The significance is
                that the performance of structured organizations are
                being compared to the predicted performance of a flat
                organization. (Chapter~\ref{struct} will analyze the
                structured organization, which will predict a
                performance that is too optimistic-the performance of
                real organizations will be between these two
                theoretical limits.)

                Taking the derivative of equation~\ref{r_n_t} with
                respect to the number of persons working on solving a
                problem will possibly yield the minimum solution
                resources to solve the problem

                \begin{equation}
                    \frac{dr_f\left(n\right)|_{t=\min}}{dn} = ke^{2\frac{c}{n} - \frac{c}{n^2}} + nke^{2\frac{c}{n} - 2\frac{c}{n^2}}\left(-2\frac{c}{n^2} + 2\frac{c}{n^3}\right)
                    \label{r_n_t_a}
                \end{equation}

                \noindent and by equating equation~\ref{r_n_t_a} to
                zero, the minima occurs when

                \begin{equation}
                    0 = 1 + n\left(2\frac{c}{n^3} - 2\frac{c}{n^2}\right) = 1 + \left(\frac{2c}{n^2} - \frac{2c}{n}\right)
                \end{equation}

                \noindent and multiplying by $n^2$

                \begin{equation}
                    0 = n^2 - 2cn + 2c
                \end{equation}

                \noindent then by the quadratic equation

                \begin{equation}
                    n = \frac{2c + \sqrt{4c^2 - 8c}}{2} = c + \sqrt{c^2 - 2c} = c + \sqrt{c}\sqrt{c - 1}
                \end{equation}

                \noindent then if $n~>>~1$, ie., the organization is large,
                and solving for $n$

                \begin{equation}
                    n_f|_{t=\min,r=\min} \approx 2c
                    \label{approx_c}
                \end{equation}

            \subsection{Minimum Time and Optimum Resources}

                Substituting equation~\ref{approx_c} into
                equation~\ref{approx_a} yields

                \begin{equation}
                    t_f\left(n\right)|_{t=\min,r=\min} \approx ke^{\frac{2c}{2c}} \approx ke
                \end{equation}

                \noindent and into equation~\ref{approx_b} yields

                \begin{equation}
                    r_f\left(n\right)|_{t=\min,r=\min} \approx nke^{\frac{2c}{2c}} \approx nke \approx 2cke
                    \label{r_n_b}
                \end{equation}

                These optimal solutions exist when, by substituting
                equation~\ref{approx_c} into equation~\ref{t_min_b}

                \begin{equation}
                    p = \frac{c}{\left(2c\right)^2} = \frac{c}{4c^2} = \frac{1}{4c} =\frac{1}{2n}
                \end{equation}

                To reiterate, with the variables $p$, $c$, and $n$

                \begin{equation}
                    n_f|_{t=\min,r=\min} = \frac{1}{2p}
                \end{equation}

                \begin{equation}
                    c = \frac{1}{4p}
                \end{equation}

                \begin{equation}
                    t_f\left(n\right)|_{t=\min,r=\min} = ke
                \end{equation}

                \begin{equation}
                    r_f\left(n\right)|_{t=\min,r=\min} = \frac{1}{2p}ke
                \end{equation}

            \subsection{Importance of Interface Coordination}

                Of particular interest is the fact that as the
                organization grows, the coordination interface quality
                must improve, ie., $p$ must get smaller, if adding
                additional persons is to impact the time to solve a
                problem.  Consider equation~\ref{p} for a given size
                of organization, then calculate what $p$ has to be for
                an organization that is 10 times as large, ie., $p$
                must decrease, approximately, with $\frac{1}{n^2}$ for
                large organizations. In point of fact, it may be wise
                to do metrics on the value of $p$ by some management
                scheme as a qualitative indicator of how well the
                organization's coordination interface is implemented.

        \section{Summary}

            It was suggested in this chapter that equation~\ref{p} is
            a mathematical definition of the onset of organizational
            bureaucracy and this is a possible economic optimum
            operating point for the organization.
            Equation~\ref{t_min} suggests that the quality of the
            organization's interface coordination process must improve
            (ie., $p$ decreases) with the square of the number of
            persons in the organization as the organization expands.
            Equation~\ref{t_min} further suggests that an organization
            can operate optimally, as far as time to problem solution
            is concerned.  Additionally,~\ref{r_min} further suggests
            that an organization can operate optimally, as far as
            minimization of resources for a problem solution is
            concerned.

            If the organization is large, then from
            equation~\ref{approx_1},~\ref{t_min},~\ref{approx_2},
            and~\ref{approx_c}, the organization's optimum operating
            situation is when

            \begin{displaymath}
                n_f|_{t=\min,r=\min} \approx \sqrt{\frac{c}{p}} \approx 2c
            \end{displaymath}

            Under these circumstances, from
            equations~\ref{approx_a},~\ref{approx_a1},~\ref{approx_b},~\ref{approx_b1},
            and~\ref{r_n_b}, the time to complete a project, and the
            resources required will be, respectively,

                \begin{displaymath}
                    t_f\left(n\right)|_{t=\min,r=\min} \approx ke^{2\frac{c}{n}} \approx ke^{2\sqrt{pc}} \approx ke
                \end{displaymath}

            \noindent and

                \begin{displaymath}
                    r_f\left(n\right)|_{t=\min,r=\min} \approx nke^{2\frac{c}{n}} \approx nke^{2\sqrt{pc}} \approx nke
                \end{displaymath}

            The organization will then be operating at the economic
            optimum, minimal time to solution optimum, and minimal
            resources for solution optimum, ie., at operational
            optimum\footnote{Perhaps this is an organizational minimax
            optimum. If the organization's product environment is
            stochastic in nature, the project would have required the
            minimum commitment of resources to complete in the minimum
            amount of time, allowing quick market introduction to
            maximize revenues with the least amount of risk of
            resource commitment.}.  This agrees favorably, but
            pessimistically, with historical presentations in the
            literature.

            The interpretation of the operational optimum is that the
            organization must operate at the point of diminishing
            returns, which allows the use of conventional
            economic/operational metrics to evaluate organizational
            dynamics.  The point of diminishing returns can be
            measured dynamically by increasing the number of persons,
            or complexity issues in the organization, and observing
            the organizational output. If the organizational output is
            inversely effected, then the point of diminishing returns
            has been attained. Otherwise, it hasn't.  If the point of
            diminishing returns has been attained, the only
            alternative is to affect the interface coordination
            process of the organization.  The interface coordination
            process, or quality thereof, can be optimized, by
            instituting changes and then observing if the point of
            diminishing returns has been altered, for better or worse.

    \chapter{The Structured Organization}
        \label{struct}

        \section{Introduction}

            This analysis will consider the hierarchical
            organization. The hierarchical organization will be
            considered to be made up, at the lowest level, of $n$ many
            persons, each reporting into a team leader. The team
            leader is a member of another organization which, for
            simplicity, consists of $n$ many persons. This
            organization reports into a team leader that is a member
            of yet another higher level organization that also
            consists of $n$ many persons, and so on.  It is assumed
            that organizational interface is vertically, through the
            team leaders to the individual teams, and not laterally
            between members of different organizations.

            \subsection{Variables used in Derivations}

                \begin{description}

                    \item [$c$] as before, is the linear factor of
                    complexity that each organization in the hierarchy
                    must address. If $c$ is divided by two, the task
                    is exactly half as complex.  This variable should
                    also reflect the ability of the organization to
                    handle complex tasks. An organization that is
                    twice as efficient as another organization would
                    have half the $c$. (As before, it may be
                    appropriate to add a factor of $\frac{1}{s}$ to
                    $c$, where $s$ represents an organization's skill
                    level which may change do to learning.

                    \item [$C$] is the linear factor of complexity
                    that the entire hierarchical organization must
                    address. If $C$ is divided by two, the task is
                    exactly half as complex.  This variable should
                    also reflect the ability of the organization to
                    handle complex tasks. An organization that is
                    twice as efficient as another organization would
                    have half the $C$.

                    \item [$e$] as before, is 2.718\dots

                    \item [$k$] as before, is a constant of
                    proportionality with the same dimension as $t$
                    below.

                    \item [$n$] as before, is the number of persons in
                    each organization of the hierarchy.

                    \item [$n_h|_{opt}$] is the number of persons in
                    the hierarchical organization that will be
                    required to achieve the economic optimum operating
                    situation for the organization.

                    \item [$n_h|_{t=\min}$] is the number of persons
                    in the hierarchical organization that will be
                    required to achieve the minimum solution time for
                    a problem.

                    \item [$n_h|_{r=\min}$] is the number of persons
                    in the hierarchical organization that will be
                    required to achieve the minimum resource
                    allocation to solve a problem.

                    \item [$N$] is the number of persons in the entire
                    hierarchical organization.

                    \item [$O$] is the number of organizations in the
                    hierarchy.

                    \item [$p$] as before, is the probability,
                    multiplied by a constant of proportionality, that
                    a coordination issue will increase the time
                    required to solve the problem.  In some sense, $p$
                    reflects the quality of the coordination process.
                    A $p$ of zero means that there are no coordination
                    issues.  The constant of proportionality reflects
                    how sensitive the system is to problems created by
                    coordination issues.

                    \item [$t_h$] is the execution time required for
                    an organization in the hierarchy, of $n$ many
                    persons to solve a problem of complexity, $c$.  It
                    would generally have the dimension of time. If all
                    organizations in the hierarchy solve problems
                    equally fast (ie., there is no wasted time) then
                    $t$ is also the execution time required for the
                    entire hierarchical organization consisting of $N$
                    many persons, $T$, to solve a problem of
                    complexity, $C$.

                    \item [$T$] see $t$ above.

                    \item [$r_h$] is the total time allocation for an
                    organization in the hierarchy, of $n$ many persons
                    to solve a problem of complexity, $c$.  It would
                    generally have the dimension of time, and is more
                    pertinent as an indicator of the total resources
                    allocated to solve the problem, ie., man hours.

                    \item [$R$] is the total time allocation for the
                    entire hierarchical organization, of $N$ many
                    persons to solve a problem of complexity, $C$.  It
                    would generally have the dimension of time, and is
                    more pertinent as an indicator of the total
                    resources allocated to solve the problem, ie., man
                    hours.

                \end{description}

            \subsection{Derivation of the Hierarchical Organizational Model}

                For an organization in the middle of the hierarchy,
                each team member has to interface with the other
                $n~-~1$ team members in the organization, and $n$
                persons in the reporting organization, plus one person
                in the organization that this organization reports to,
                for a total of $2n$ interfaces. It is assumed that
                inter-organizational interface issues, between
                organizations in the hierarchy, are handled by the by
                the next higher level organization in the
                hierarchy. Then with reasoning similar to that used in
                the derivation of equation~\ref{t_n}, a problem's
                solution time can be implied

                \begin{equation}
                    t_h\left(n\right) = T = ke^{\frac{c}{n} + 2np}
                    \label{t_nh_1}
                \end{equation}

                \noindent and with likewise similar reasoning to that
                used in the derivation of equation~\ref{r_n}, the
                resources required

                \begin{equation}
                    r_h\left(n\right) = nke^{\frac{c}{n} + 2np}
                    \label{r_nh_1}
                \end{equation}

                Additionally, the number of organizations in the
                hierarchy is the total number of persons in the entire
                hierarchical organization divided by the number of
                persons in each organization in the hierarchy

                \begin{equation}
                    O = \frac{N}{n}
                    \label{O}
                \end{equation}

                \noindent and the complexity that each organization in
                the hierarchy must address is the total complexity
                divided by the number of organizations in the
                hierarchy

                \begin{equation}
                    c = \frac{C}{O}
                    \label{c_1}
                \end{equation}

                \noindent Solving equation~\ref{c} for $O$ and
                equating to equation~\ref{O}

                \begin{equation}
                    \frac{C}{c} = \frac{N}{n}
                    \label{cc}
                \end{equation}

                \noindent and solving for $c$

                \begin{equation}
                    c = \frac{Cn}{N}
                    \label{c}
                \end{equation}

                \noindent which is the complexity that each
                organization in the hierarchy must
                address. Substituting equation~\ref{c} into
                equation~\ref{t_nh_1}

                \begin{equation}
                    T = ke^{\frac{C}{N} + 2np}
                    \label{T_N}
                \end{equation}

                \noindent and likewise, substituting into
                equation~\ref{r_nh_1}

                \begin{equation}
                    r_h\left(n\right) = nke^{\frac{C}{N} + 2np}
                    \label{r_nh}
                \end{equation}

                Since all organizations are assumed to operate in
                parallel to solve a problem, the total resources for
                the entire hierarchical organization would be the
                resources expended by each organization in the
                hierarchy multiplied by the number of organizations in
                the hierarchy

                \begin{equation}
                    R = Onke^{\frac{C}{N} + 2np}
                \end{equation}

                \noindent and substituting equation~\ref{O}

                \begin{equation}
                    R = Nke^{\frac{C}{N} + 2np}
                    \label{R_N}
                \end{equation}

        \section{The Case to Increase Organizational Size}

            By analyzing the ratio of the time required to solve a
            problem with an additional person to the time required to
            solve the same problem without the additional person, some
            insight can be obtained on whether it is an advisable
            proposition to increase the size of an organization in the
            hierarchy, or from equation~\ref{t_nh_1}

            \begin{equation}
                \frac{t_h\left(n + 1\right)}{t_h\left(n\right)} = \frac{ke^{\frac{c}{n + 1} + 2\left(n + 1\right)p}}{ke^{\frac{c}{n} + 2np}}
            \end{equation}

            \noindent and reducing

            \begin{equation}
                \frac{t_h\left(n + 1\right)}{t_h\left(n\right)} = e^{\frac{c}{n + 1} - \frac{c}{n} + 2p}
            \end{equation}

            \noindent Note that for a decrease in the time to solve
            the problem to occur

            \begin{equation}
                2p + c\left(\frac{1}{n + 1} - \frac{1}{n}\right)
                \label{p_ah}
            \end{equation}

            \noindent must be less than zero. If it is equal to zero,
            ie.,

            \begin{equation}
                2p = c\left(\frac{1}{n} - \frac{1}{n + 1}\right)
                \label{ph}
            \end{equation}

            \noindent then adding a person will have no effect on the
            time to solve the problem. Likewise, if
            equation~\ref{p_ah} is greater than zero, then the
            situation exists where adding a person can actually
            increase the time required to solve a problem.  (Perhaps
            this is the mathematical definition of bureaucracy.)
            Combining terms in equation~\ref{ph}

            \begin{equation}
                p = \frac{c}{2}\left(\frac{n + 1 - n}{n\left(n + 1\right)}\right) = \frac{c}{2}\left(\frac{1}{n\left(n + 1\right)}\right) = \frac{\frac{c}{2}}{n^2 + n}
            \end{equation}

            \noindent then if $n~>>~1$, ie., the organization is
            large, and solving for $n$

            \begin{equation}
                n_h|_{opt} \approx \sqrt{\frac{c}{2p}}
            \end{equation}

            \subsection{A Possible Economic Optimum}

                If it is determined that equation~\ref{ph} really is
                the onset of diminishing returns (similar to
                equation~\ref{p}), then this may possibly constitute
                the optimum economic operating point for the
                organization. It is probably important to note that in
                adding one person to an organization in the middle of
                the hierarchy is equivalent to adding an organization
                to the next level down in the hierarchy, and such
                organizational designs are subject to diminishing
                returns.

        \section{Possible Optimum Staffing Levels}

            Using reasoning similar to that used in chapter~\ref{flat}
            to optimize the number of persons in the hierarchical
            organization for minimum time and resources will provide
            the optimum operating situation for the hierarchical
            organization.

            \subsection{Optimum Staffing Level for Minimum Time}

                Taking the derivative of equation~\ref{T_N} with
                respect to the number of persons in each hierarchical
                organization will possibly yield the minimum solution
                time to solve the problem

                \begin{equation}
                    \frac{dT\left(n\right)}{dn} = ke^{\frac{C}{N} + 2np}\left(2p - \frac{c}{n^2}\right)
                    \label{T_min_a}
                \end{equation}

                \noindent and by equating equation~\ref{T_min_a} to
                zero, the minima occurs when

                \begin{equation}
                    \frac{c}{n^2} = 2p
                    \label{T_min_b}
                \end{equation}

                \noindent or solving for $n$

                \begin{equation}
                    n_h|_{t=\min} = \sqrt{\frac{c}{2p}}
                    \label{T_min}
                \end{equation}

            \subsection{Optimum Staffing Level for Minimum Resources}

                Taking the derivative of equation~\ref{R_N} with
                respect to the number of persons in each hierarchical
                organization will possibly yield the minimum resource
                solution to solve a problem

                \begin{equation}
                    \frac{dR\left(n\right)}{dn} = nke^{\frac{c}{n} + 2p}\left(2p - \frac{c}{n^2}\right) + ke^{\frac{c}{n} + \left(n - 1\right)p}
                    \label{R_min_a}
                \end{equation}

                \noindent and by equating equation~\ref{R_min_a} to
                zero, the minima occurs when

                \begin{equation}
                    n\left(2p - \frac{c}{n^2}\right) + 1 = 0
                \end{equation}

                \noindent or

                \begin{equation}
                    2np - \frac{c}{n} + 1 = 0
                \end{equation}

                \noindent and multiplying by n

                \begin{equation}
                    2pn^2 + n - c = 0
                \end{equation}

                \noindent then by the quadratic equation

                \begin{equation}
                    n = \frac{\sqrt{1 + 8pc} - 1}{4p}
                \end{equation}

                Again, it is not clear whether this minima is of any
                practical interest. If $n~>>~1$, ie., the organization
                is large ($\sqrt{4pc}~>>~2p$), then

                \begin{equation}
                    n_h|_{r=\min} \approx \sqrt{\frac{c}{2p}}
                \end{equation}

                \noindent which agrees nicely with
                equation~\ref{T_min}.

        \section{Discussion}

            The total resources for the entire organization can be
            derived by solving equation~\ref{T_min_b} for $2np$ and
            substituting in equation~\ref{R_N}

            \begin{equation}
                R = Nke^{\frac{C}{N} + \frac{c}{n}}
            \end{equation}

            \noindent then solving~\ref{cc} for $\frac{c}{n}$ and
            substituting

            \begin{equation}
                R\left(N\right) = Nke^{2\frac{C}{N}}
                \label{R_N_H}
            \end{equation}

            Taking the derivative of equation~\ref{R_N_H} with respect
            to the total number of persons in the organization will
            possibly yield the minimum resource solution to solve a
            problem

            \begin{equation}
                \frac{dR\left(N\right)}{dN} = ke^{2\frac{C}{N}} + kNe^{2\frac{C}{N}}\left(-2\frac{C}{N^2}\right)
                \label{R_N_min_a}
            \end{equation}

            \noindent and by equating equation~\ref{R_N_min_a} to
            zero, the minima occurs when

            \begin{equation}
                \frac{C}{N} = \frac{1}{2}
                \label{R_N_min}
            \end{equation}

            Substituting equation~\ref{R_N_min} into
            equation~\ref{R_N_H} yields

            \begin{equation}
                R|_{\min} = Nke = 2Cke
            \end{equation}

            Solving equation~\ref{T_min_b} for $2np$ and substituting
            into equation~\ref{T_N} yields

            \begin{equation}
                T = ke^{\frac{C}{N} + \frac{c}{n}}
            \end{equation}

            \noindent and solving equation~\ref{c} for $\frac{c}{n}$
            and substituting

            \begin{equation}
                T = ke^{2\frac{C}{N}}
            \end{equation}

            \noindent and substituting equation~\ref{R_N_min}

            \begin{equation}
                T|_{R=\min} = ke
            \end{equation}

            Solving equation~\ref{T_min_b} for $2np$ and substituting
            into equation~\ref{r_nh}

            \begin{equation}
                r_h = nke^{\frac{C}{N} + \frac{c}{n}}
            \end{equation}

            \noindent and solving equation~\ref{c} for $\frac{c}{n}$
            and substituting

            \begin{equation}
                r_h = kne^{2\frac{C}{N}}
            \end{equation}

            \noindent and substituting equation~\ref{R_N_min}

            \begin{equation}
                r_h|_{R=\min} = kne
                \label{R_min}
            \end{equation}

            \subsection{Minimum Time Optimum}

                Substituting $2np$ from equation~\ref{T_min_b} into
                equation~\ref{T_N} yields

                \begin{equation}
                    T = ke^{\frac{C}{N} + 2n\frac{c}{2n^2}} = ke^{\frac{C}{N} + \frac{c}{n}}
                \end{equation}

                \noindent and solving~\ref{cc} for $\frac{c}{n}$ and
                substituting

                \begin{equation}
                    T|_{\min} = ke^{2\frac{C}{N}}
                \end{equation}

            \subsection{Minimum Time Optimum Resources}

                Likewise, substituting $p$ from equation~\ref{T_min_b}
                into equation~\ref{R_N} yields

                \begin{equation}
                    R|_{t=\min} = nke^{\frac{C}{N} + 2n\frac{c}{2n^2}} = nke^{\frac{C}{N} + \frac{c}{n}}
                \end{equation}

                \noindent and solving~\ref{cc} for $\frac{c}{n}$ and
                substituting

                \begin{equation}
                    R|_{t=\min} = nke^{2\frac{C}{N}}
                \end{equation}

                Substituting $n$ from equation~\ref{T_min} and $N$
                from equation~\ref{R_N_min} into equation~\ref{O}
                yields

                \begin{equation}
                    O = \frac{\frac{2}{C}}{\sqrt{\frac{c}{2p}}}
                \end{equation}

                \noindent and substituting $c$ from equation~\ref{c_1}

                \begin{equation}
                    O = \frac{2C}{\sqrt{\frac{C}{2Op}}} = 2\sqrt{2COp}
                \end{equation}

                \noindent and squaring both sides

                \begin{equation}
                    O^2 = 8COp
                \end{equation}

                \noindent or

                \begin{equation}
                    O = 8Cp
                    \label{O_opt}
                \end{equation}

                \noindent which is the optimum number of organizations
                to solve a problem of complexity $C$. Interestingly,
                it is a linear relationship on
                complexity. Substituting $c$ from equation~\ref{c_1}
                into equation~\ref{T_min} yields

                \begin{equation}
                    n_h|_{R=\min,T=\min} = \sqrt{\frac{C}{2Op}}
                \end{equation}

                \noindent and substituting $O$ from
                equation~\ref{O_opt}

                \begin{equation}
                    n_h|_{R=\min,T=\min} = \sqrt{\frac{C}{16Cp^2}} = \sqrt{\frac{1}{16p^2}} = \frac{1}{4p}
                    \label{n_R_T_min}
                \end{equation}

                \noindent which is the optimum number of persons in
                each organization of the hierarchy.

                Equating~\ref{T_min} and~\ref{n_R_T_min}

                \begin{equation}
                    \frac{1}{4p} = \sqrt{\frac{c}{2p}}
                \end{equation}

                \noindent and squaring

                \begin{equation}
                    \frac{1}{16p^2} = \frac{c}{2p}
                \end{equation}

                \noindent or

                \begin{equation}
                    p = \frac{1}{8c}
                    \label{p_c}
                \end{equation}

                \noindent and substituting into
                equation~\ref{n_R_T_min}

                \begin{equation}
                    n = \frac{1}{4\frac{1}{8c}} = 2c
                    \label{n_c}
                \end{equation}

                \noindent and substituting into equation~\ref{R_min}

                \begin{equation}
                    r = 2cke
                \end{equation}

                \noindent and solving equation~\ref{p_c} for $c$ and
                substituting

                \begin{equation}
                    r = 2\frac{1}{8p}ke = \frac{1}{4p}ke
                \end{equation}

                Solving equation~\ref{p_c} for $c$ and substituting
                into equation~\ref{n_c} yields

                \begin{equation}
                    n = 2\frac{1}{8p} = \frac{1}{4p}
                \end{equation}

                \noindent and solving~\ref{p_c} for $c$

                \begin{equation}
                    c = \frac{1}{8p}
                \end{equation}

                The last three equations represent the optimum
                organization size in the hierarchical structure, with
                only $p$ as a parameter.

    \chapter{The Polynomial Organization}
        \label{polynomial}

        \section{Introduction}

            The flat organization with exponential responses to
            complexity was analyzed in chapter~\ref{flat}, and was
            found to lead to pessimistic predictions of organizational
            performance. The flat organization with simple polynomial,
            ie. $c^2$, response to complexity will be analyzed in this
            section and comparisons made to the derivations in
            chapter~\ref{flat}. The outline of the derivations will
            follow those presented in chapters~\ref{general}
            and~\ref{flat}.

            \subsection{Variables used in Derivations}

                \begin{description}

                    \item [$c$] is the linear factor of complexity.
                    If $c$ is divided by two, the task is exactly half
                    as complex. This variable should also reflect the
                    ability of the organization to handle complex
                    tasks. An organization that is twice as efficient
                    as another organization would have half the $c$.
                    (It may be appropriate to add a factor of
                    $\frac{1}{s}$ to $c$, where $s$ represents an
                    organization's skill level which may change do to
                    learning.)

                    \item [$e$] is 2.718\dots

                    \item [$k$] is a constant of proportionality with
                    the same dimension as $t$ below.

                    \item [$n$] is the number of persons in the
                    organization.

                    \item [$n_p|_{opt}$] is the number of persons in
                    the polynomial organization that will be required
                    to achieve the economic optimum operating
                    situation for the organization.

                    \item [$n_p|_{t=\min}$] is the number of persons
                    in the polynomial organization that will be
                    required to achieve the minimum solution time for
                    a problem.

                    \item [$n_p|_{r=\min}$] is the number of persons
                    in the polynomial organization that will be
                    required to achieve the minimum resource
                    allocation to solve a problem.

                    \item [$p$] is the probability, multiplied by a
                    constant of proportionality, that a coordination
                    issue will increase the time required to solve the
                    problem.  In some sense, $p$ reflects the quality
                    of the coordination process.  A $p$ of zero means
                    that there are no coordination issues. The
                    constant of proportionality reflects how sensitive
                    the system is to problems created by coordination
                    issues.

                    \item [$t_p$] is the execution time required for
                    an organization of $n$ many persons to solve a
                    problem of complexity, $c$.  It would generally
                    have the dimension of time.

                    \item [$r_p$] is the total time allocation for an
                    organization of $n$ many persons to solve a
                    problem of complexity, $c$.  It would generally
                    have the dimension of time, and is more pertinent
                    as an indicator of the total resources allocated
                    to solve the problem, ie., man hours.

                \end{description}

            \subsection{Derivation of the One Person Organization Model}

                Assuming that the time, $t\left(1\right)$, required
                for completion of the search for solution to a problem
                is a simple polynomial on complexity

                \begin{equation}
                    t_p\left(1\right) = kc^2
                \end{equation}

                \noindent where $k$ is a constant of proportionality,
                and $c$ is proportional to the complexity of the
                problem.  Likewise, in this simple case

                \begin{equation}
                    r_p\left(1\right) = kc^2
                \end{equation}

            \subsection{Derivation of the Two Person Organization Model}

                Assuming that the task can be divided equally between
                the two persons, and each has the same technology and
                skills in searching for a solution to the problem

                \begin{equation}
                    t_p\left(2\right) = k\left(\frac{c}{2}\right)^2
                \end{equation}

                \noindent and likewise for the resources required

                \begin{equation}
                    r_p\left(2\right) = k\left(\frac{c}{2}\right)^2 + k\left(\frac{c}{2}\right)^2 = 2k\left(\frac{c}{2}\right)^2
                \end{equation}

                Note that the task was divided equally, with each
                person addressing half of the complexities.  However,
                a new situation exists in that the work done by one
                person may not coordinate with the work done by the
                other, and visa versa, perhaps requiring a rework of
                some tasks, etc.  If we assume that searching for, and
                solving, the problem created by the coordination issue
                is itself polynomial, then the previous equation must
                have an additional value of $p$ added, where $p$ is
                the probability that a coordination issue will occur
                multiplied by a constant that represents the affect
                that the issue has on the organization's
                efficiency. Therefore,

                \begin{equation}
                    t_p\left(2\right) = k\left(\frac{c}{2} + p\right)^2
                \end{equation}

                \noindent and likewise for the resources required

                \begin{equation}
                    r_p\left(2\right) = 2k\left(\frac{c}{2} + p\right)^2
                \end{equation}

            \subsection{Derivation of the n Person Organizational Model}

                The general solution for the problem's solution time
                can be implied

                \begin{equation}
                    t_p\left(n\right) = k\left(\frac{c}{n} + \left(n - 1\right) p\right)^2
                    \label{t_np}
                \end{equation}

                \noindent and likewise for the resources required

                \begin{equation}
                    r_p\left(n\right) = nk\left(\frac{c}{n} + \left(n - 1\right) p\right)^2
                    \label{r_np}
                \end{equation}

                The reason for the $\left(n~-~1\right)$ multiplier in
                the exponent is because the probability that a
                coordination issue will happen is proportional to the
                number of interfaces that can occur.  It is assumed
                that each person interfaces with all others in the
                organization, and each interface has an equal
                probability of creating a coordination issue.

        \section{The Case to Increase Organizational Size}

            The question can be posed, when can a person, or persons,
            be added to an organization, presumably to decrease the
            time required to find a solution to a problem.  By
            analyzing the ratio of the time required to solve a
            problem with an additional person to the time required to
            solve the same problem without the additional person, some
            insight can be obtained on whether it is an advisable
            proposition, or from equation~\ref{t_np}

            \begin{equation}
                \frac{t_p\left(n + 1\right)}{t_p\left(n\right)} = \frac{k\left(\frac{c}{n + 1} + pn\right)^2}{k\left(\frac{c}{n} + p\left(n - 1\right)\right)^2}
            \end{equation}

            \noindent and for convenience taking the square root of
            both sides and reducing

            \begin{equation}
                \sqrt{\frac{t_p\left(n + 1\right)}{t_p\left(n\right)}} = \frac{\frac{c}{n + 1} + pn}{\frac{c}{n} + p\left(n - 1\right)}
                \label{p_ap}
            \end{equation}

            \noindent Note that for a decrease in the time to solve
            the problem to occur, the numerator of equation~\ref{p_ap}
            must be less than the denominator. If they are equal, ie.,

            \begin{equation}
                \frac{c}{n + 1} + pn = \frac{c}{n} + p\left(n - 1\right)
            \end{equation}

            \noindent or

            \begin{equation}
                \frac{c}{n + 1} + pn = \frac{c}{n} + pn - p
            \end{equation}

            \noindent and reducing

            \begin{equation}
                \frac{c}{n + 1} = \frac{c}{n} - p
            \end{equation}

            \noindent and combining terms of $c$

            \begin{equation}
                c\left(\frac{1}{n + 1} - \frac{1}{n}\right) = - p
            \end{equation}

            \noindent and finding the lowest common denominator

            \begin{equation}
                c\left(\frac{n - \left(n + 1\right)}{n\left(n + 1\right)}\right) = - p
            \end{equation}

            \noindent and reducing

            \begin{equation}
                c\left(\frac{1}{n\left(n + 1\right)}\right) = p
            \end{equation}

            \noindent and expanding

            \begin{equation}
                c\left(\frac{1}{n^2 + n}\right) = p
            \end{equation}

            \noindent then if $n~>>~1$, ie., the organization is
            large, and solving for $n$

            \begin{equation}
                n_p|_{opt} \approx \sqrt{\frac{c}{p}}
                \label{approx_1p}
            \end{equation}

            \noindent which is in agreement with
            equation~\ref{approx_1}.

            \subsection{A Possible Economic Optimum}

                In some sense, it is apparent that adding persons to
                an organization without altering $p$ will result in
                diminishing returns after a certain point. If it is
                determined that this really is the onset of
                diminishing returns, then this may possibly constitute
                the optimum economic operating point for the
                organization.

        \section{Possible Optimum Staffing Levels}

            The possible economic optimum operating point for the
            organization suggests that an optimum staffing level
            exists that will minimize the time to solve a problem, or
            alternately, minimize the resources that must be committed
            to solve the problem.

            \subsection{Optimum Staffing Level for Minimum Time}

                Taking the derivative of equation~\ref{t_np} with
                respect to the number of persons working on solving a
                problem will possibly yield the minimum solution time
                to solve the problem

                \begin{equation}
                    \frac{dt_p\left(n\right)}{dn} = 2k\left(\frac{c}{n} + \left(n - 1\right)p\right)\left(p - \frac{c}{n^2}\right)
                    \label{t_min_ap}
                \end{equation}

                \noindent and by equating equation~\ref{t_min_ap} to
                zero, the minima occurs when

                \begin{equation}
                    \frac{c}{n^2} = p
                    \label{t_min_bp}
                \end{equation}

                \noindent or solving for $n$

                \begin{equation}
                    n_p|_{t=\min} = \sqrt{\frac{c}{p}}
                    \label{t_minp}
                \end{equation}

                It is not clear whether this minima is of any
                practical interest, but it agrees nicely with
                equation~\ref{approx_1p}, at least for large
                organizations.

            \subsection{Optimum Staffing Level for Minimum Resources}

                Taking the derivative of equation~\ref{r_np} with
                respect to the number of persons working on solving a
                problem will possibly yield the minimum resource
                solution to solve the problem

                \begin{equation}
                    \frac{dr_p\left(n\right)}{dn} = k\left(\frac{c}{n} + \left(n - 1\right)p\right)^2 + 2nk\left(\frac{c}{n} + \left(n - 1\right)p\right)\left(p - \frac{c}{n^2}\right)
                    \label{r_min_ap}
                \end{equation}

                \noindent and by equating equation~\ref{r_min_ap} to
                zero, the minima occurs when

                \begin{equation}
                    \frac{c}{n} + p\left(n - 1\right) + 2n\left(p - \frac{c}{n^2}\right)= 0
                \end{equation}

                \noindent or expanding and combining terms

                \begin{equation}
                    \frac{c}{n} + pn - p -\frac{2nc}{n^2} + 2np = 0
                \end{equation}

                \noindent and combining terms

                \begin{equation}
                    \frac{c}{n} + 3pn - p - \frac{2c}{n} = -\frac{c}{n} + 3pn -p = 0
                \end{equation}

                \noindent and multiplying by n

                \begin{equation}
                    3pn^2 + -pn - c = 0
                \end{equation}

                \noindent then by the quadratic equation

                \begin{equation}
                    n = \frac{p + \sqrt{p^2 + 12pc}}{6p}
                \end{equation}

                \noindent or by multiplying the numerator and
                denominator by $p$

                \begin{equation}
                    n = \frac{1 + \sqrt{1 + \frac{12p}{c}}}{6} = \frac{1}{6} + \sqrt{\frac{1}{36} + \frac{c}{3p}}
                \end{equation}

                Again, it is not clear whether this minima is of any
                practical interest. If $n~>>~1$, ie., the organization
                is large, then

                \begin{equation}
                    n_p|_{r=\min} \approx \sqrt{\frac{c}{3p}}
                    \label{tr_minp}
                \end{equation}

                And substituting $n$ from equation~\ref{tr_minp} into
                equation~\ref{t_np} yields

                \begin{equation}
                    t_p\left(n\right)|_{r = \min} = k\left(\frac{c}{\sqrt{\frac{c}{3p}}} + p\left(\sqrt{c}{3p} - 1\right)\right)^2
                \end{equation}

                \noindent and combining terms

                \begin{equation}
                    t_p\left(n\right)|_{r = \min} = k\left(\sqrt{3cp} + \sqrt{\frac{cp}{3}} - p\right)^2 = k\left(2\sqrt{cp} - p\right)^2
                \end{equation}

                \noindent and factoring

                \begin{equation}
                    t_p\left(n\right)|_{r = \min} = k\left(\left(\sqrt{3} + \sqrt{\frac{1}{3}}\right)\sqrt{pc} - p\right)^2
                \end{equation}

                \noindent then if $n~>>~1$, ie., the organization is
                large, then

                \begin{equation}
                    t_p\left(n\right)|_{r = \min} \approx k\left(\left(\sqrt{3} + \sqrt{\frac{1}{3}}\right)\sqrt{pc}\right)^2 \approx 5.333kpc
                \end{equation}

        \section{Discussion}

            \subsection{Minimum Time Optimum}

                Substituting $p$ from equation~\ref{t_min_bp} into
                equation~\ref{t_np} yields

                \begin{equation}
                    t_p\left(n\right)|_{\min} = k\left(\frac{c}{n} + \left(n - 1\right)\frac{c}{n^2}\right)^2
                \end{equation}

                \noindent and combining terms

                \begin{equation}
                    t_p\left(n\right)|_{\min} = k\left(\frac{c}{n} + \frac{c}{n} - \frac{c}{n^2}\right)^2 = k\left(\frac{c}{n}\left(2 - \frac{1}{n}\right)\right)^2
                \end{equation}

                \noindent then if $n~>>~1$, ie., the organization is large,

                \begin{equation}
                    t_p\left(n\right)|_{\min} \approx k\left(2\frac{c}{n}\right)^2
                    \label{approx_ap}
                \end{equation}

                Interestingly, comparing equation~\ref{approx_ap} to
                equation~\ref{t_n} with $p~=~0$, (ie., a perfect
                organization that has no interface coordination
                issues,) states that if the organization is operating
                optimally, that only 50 percent of the organization's
                efforts are directed at problem solving, and the
                remaining 50 percent is directed at interface
                coordination issues.

                Alternately, substituting $n$ from
                equation~\ref{t_minp} into equation~\ref{t_np} yields

                \begin{equation}
                    t_p\left(n\right)|_{\min} = k\left(\frac{c}{\sqrt{\frac{c}{p}}} + p\left(\sqrt{c}{p} - 1\right)\right)^2
                \end{equation}

                \noindent and combining terms

                \begin{equation}
                    t_p\left(n\right)|_{\min} = k\left(\sqrt{cp} + \sqrt{cp} - p\right)^2 = k\left(2\sqrt{cp} - p\right)^2
                \end{equation}

                 \noindent then if $n~>>~1$, ie., the organization is
                 large, then

                 \begin{equation}
                    t_p\left(n\right)|_{\min} \approx k\left(2\sqrt{cp}\right)^2 \approx 4kcp
                 \end{equation}

            \subsection{Minimum Time Optimum Resources}

                Likewise, substituting $p$ from
                equation~\ref{t_min_bp} into equation~\ref{r_np}

                \begin{equation}
                    r_p\left(n\right)|_{t=\min} = nk\left(\frac{c}{n} + \left(n - 1\right)\frac{c}{n^2}\right)^2
                \end{equation}

                \noindent and again combining terms

                \begin{equation}
                    r_p\left(n\right)|_{t=\min} = nk\left(\frac{c}{n} + \frac{c}{n} - \frac{c}{n^2}\right)^2 = nk\left(\frac{c}{n}\left(2 - \frac{1}{n}\right)\right)^2
                \end{equation}

                \noindent then if $n~>>~1$, ie., the organization is large,

                \begin{equation}
                    r_p\left(n\right)|_{t=\min} \approx nk\left(2\frac{c}{n}\right)^2
                \end{equation}

        \section{Summary}

                Interestingly, in contrast with the optimizations
                derived in chapter~\ref{flat}, the optimized time to
                solve a problem is linear on complexity. Additionally,
                an option exists on whether to allocate minimum
                resources to solve the problem, or solve the problem
                in minimum time. If the problem is solved with minimum
                resources, it will take approximately $\frac{1}{3}$
                longer, than if the problem is solved in minimum time,
                but require about $\frac{1}{3}$ fewer
                resources. Interestingly, the optimizations still
                exhibit diminishing returns at the economic optimum,
                and any other operating situation will exhibit a
                non-linear solution time and resource requirement.

\end{document}
