Manageability of complex organisational systems – system-theoretical confines of control

Organisation is all about improving efficiency of production by implementing division of work (Smith 1776). Long since, Taylor (1911) laid the foundation of this concept by introducing the idea of perfect construction of work packages where the effort of perfect determination tends to infinity. The common structural approach to developing appropriately matching work packages is based on hierarchical tree structures using reduced but nonetheless non-ambivalent interfaces to superior as well as sub-ordered structural elements (Schulte-Zurhausen 2002; Kerzner 2003; Eber and Zimmermann 2018). However, furthermore, in particular, horizontal interactions are clearly ignored and replaced by an attempt to determine and require perfect outcomes of each package so that interdependencies automatically become fulfilled. Not only recent development of understanding implies that this strict concept is bound to fail principally. Meanwhile, alternative approaches like self-deterministic management (Malik 2003; Schelle et al. 2005), agile management (Beck et al. 2001; Schwaber and Sutherland 2020) and to some degree concepts of lean management (Koskela 2000; Koskela et al. 2002; VDI 2019), where definitions are less strict and interfaces more cooperative, are gaining more and more importance.

Understanding the principle of incomplete contracts (Ebeling et al. 1998; Picot et al. 2008; Liening 2017; Hoffmann and Körkemeyer 2018) already states that no result nor consumption of resources can be perfectly determined in advance which would be absolutely required by a hierarchical approach to division of work (Caldarelli and Vespignani 2007). This is valid for horizontal interdependencies where independently produced packages need to perfectly match as well as for vertical relationships.

The well-known principal-agent problem (Picot et al. 2008) points out that a perfect description of work and respective supervision can only be gained at a price of an at least equivalent effort as if not outsourcing the package. Thus, the goal of organisation is finding the second-best solution instead of hoping for a never occurring first-best solution. The second-best solution is then represented by the balance between control and tolerance (Coase 1937).

The known principle of balancing transaction cost (Picot et al. 2008) only considered the coordination effort versus the gained production advantage at an optimal degree of division of work. Therewith, perfect definition of outcome and interfacing is preconditioned while in no way taking tolerance into account.

This problem occurs not only with real division of work distributing to different people or companies but also with a single person taking up different roles within a project. Even then, the perfect transfer of knowledge and understanding from one role to another cannot be gained without significant losses.

Doubtless, the implementation of control mechanisms to adjust processes to required results is part of the concept of coordination. However, even then, the definition of perfect results to compare with is required. Thus, coordination not only focusses on the agents’ side in order to fulfil the given requirements but also needs to deal with the principals’ inability to formulate the perfect determination of work package as a solution to a given problem. This situation resembles the V-model (Schelle et al. 2005) controlling not only the quality of a solution, measured as deviation from then given plan, but also the applicability of the result to solving the given problem appropriately. The therewith demanded extension of the principle of control, being absolutely compatible with the traditional approach of governing, however, requires not only a significantly higher provision of controlling resources but also higher controlling delays as well since the goal comparison procedure resides on a much more abstract level than just comparing facts to a plan.

On this background, we state that perfect determination, e.g. based on strict hierarchical approaches, is principally not possible, neither on the agents’ side nor on the principals’ side (Wassermann and Faust 1994; Strogatz 2001; White et al. 2004; Winch 2006; Eber 2020). Therefore, the following question is to be brought up: to which degree within an inevitably inconsistent system local contradictions can be ruled out by respective controlling mechanisms and where perfection reaches its limits given by fundamental rules?

Recent approaches, e.g. lean management VDI 2553 (VDI 2019) or the SCRUM manifest (Beck et al. 2001; Schwaber and Sutherland 2020), emphasize a strategical turn towards non-hierarchical management where self-determination plays the fundamental role rather than the execution of perfectly determined tasks. These ideas are mainly based on the experience of a lack of perfection in predetermination of sub-elements, execution of these or the ability of controlling instances to ensure perfection. However, these concepts seem to be missing a mathematical foundation and are, hence, introduced on a more heuristic basis.

This paper represents an extension to ‘System-theoretical Approach to Fundamental Limits of Controllability in Complex Organization Networks’ (Eber 2021), where the fundamentals and principal limits of controlling precision based on harmonic control are discussed. This understanding is again presented in Sections 2–7 while a simulational experiment is added in Section 7, validating the resulting boundaries. The novelty of this present paper is first on the development of a set of scenarios of different stochastic deviations of input parameters providing limited indetermination and, hence, the requirement of a particularly determined set of resources in need to be held ready to face these (Section 8). Secondly, on the background of understanding harmonic control as dynamic and dissipative contributions to the behaviour, a principal limit of stable behaviour is proposed and discussed (Section 9).

In order to understand the behaviour of an organisation (Booch et al. 2007), modelling it as a system is required. Based on a hierarchical work breakdown structure (WBS), the overall task is separated into a large number of non-overlapping subtasks. The WBS as a graph-theoretical tree structure is bound to ignore all horizontal interactions between subtasks and branches solely according to a singular specific criterion, e.g., the physical structure of the product. Only after identifying the entirety of the product elements, the (horizontal) interfaces between these are reintroduced and form the system as a set of elements and respective (systemic) interactions (Bertalaffy 1969; Haken 1983; Luhmann 2001; Newman 2003; Zimmermann and Eber 2017). This comprises not only physical elements but also all imaginable elements to be processed in order to complete the project. Broken down to the utmost level of detail, each element is represented by a single variable q_i forming the state vector Q⇀ \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over Q} , while all interactions are given by linear differential equations as the second term of a respective Taylor series. The constant terms can be eliminated by linear transformation of the reference system to the state of equilibrium q₀ = 0 ∂Q⇀/∂t=0 \partial \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over Q} /\partial t = 0 . Higher order terms are neglected for this approach; linear superposition property of impact is assumed. (1) Q⇀t+dt=Q⇀t+F⇀(Q⇀)→∂Q⇀∂t=Q⇀0+AQ⇀+…(higher orders) {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over Q}_{t + dt}} = {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over Q}_t} + \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over F}\left({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over Q}} \right) \to {{\partial \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over Q}} \over {\partial t}} = {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over Q}_0} + A\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over Q} + \ldots \left({{\rm{higher}}\,{\rm{orders}}} \right) where A is the adjacency matrix and Q⇀ \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over Q} the state vector (Figure 1).

Fig. 1

Written as components we obtain the following: (2) ∂qj∂t=∑cj,iqi {{\partial {q_j}} \over {\partial t}} = \sum {{c_{j,i}}{q_i}}

As long as the system is not contradictory (Zimmermann and Eber 2017), the differential equations lead to the trivial solution and the system stabilizes at a state Q⇀=0⇀ \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over Q} = \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over 0} (zero vector). Inconsistency is given if multiple influencing nodes are driving a common target towards different values. Formally, within a linear system, the differential equations are superposed. However, if the influencing parameters c_i,j are strong and the impact is not performed simultaneously but e.g. alternatingly, a nonlinear system is built and instabilities show up on the time axis.

Example (Principal-Agent Problem) (Picot et al. 2008): Two partners (1 and 2) are not agreeing on a certain value q_j. The principal (1) expects and formulates a certain quality q_j for a sub-product, while the agent (2) provides only a different value, for whatever reason. He probably knows better due to his expertise or cannot do better due to the given circumstances. Then, over time, this absolute value is determined by each of the participants according to their personal (valid, however inconsistent) definition somehow alternatingly and induces the respective consequences to the adjacent elements (Figures 2 and 3).

Fig. 2

Fig. 3

In this context, the temporary q_j(t) value is simply set by the locally determining party. Since the differential equation refers to only a modification of the previously set value, this needs to be understood as a very strong and very fast adaption to the given value (i.e., notifying the inadequate value and immediately setting it).

Remark: If the adjustment process is slow compared to the alternating determination of the contradicting elements, the resulting value represents the average opinion. This situation is equivalent to both parties accepting the other side's value to some degree over time.

Such inconsistencies are represented by the average resulting fuzziness of system variables. As long as factually inconsistencies are existing, the system models the behaviour correctly. This includes averaging as well as oscillating or even escalating development.

Obviously, the two determining elements (players) maintain no direct interaction, namely nothing in order to clarify the situation. Only if one of the players is officially declared to be wrong or both agree on a common value, the discrepancy is solved. Such a procedure, however, is based on both players to realize each other's position or at least the resulting value and start an attempt to solve, which means establish a corrective interaction.

If means of controlling are established, one or both of the disagreeing parties or possibly a third party is observing the value in question and comparing this to an expected value. Based on this knowledge, forces are implied to bring the value back to expectation. In terms of systems theory, this is the introduction of a loop, where some impact is derived from an observation of a value (Wiener 1992) (Figure 4).

Fig. 4

In case of self-organised consolidation of inconsistent expectations, both the impacting elements are observing the state of the respective element and are taking the current situation – that is the influence of the disturbing element – into account when deciding for their reacting strength.

Considered as classical controlling, this process turns out to be equivalent. The development of a value, probably as the result of some production process, is compared with a predefined value of quality, per contract or plan, and brought back on value reacting to the observed discrepancies.

The mathematical representation of any short (one member only, the controller returns only proportional reaction) controlling loop is given by the simplest structure (Figure 5):

Fig. 5

The general solution of this kind of equations is either oscillating, exponentially escalating or damped, which is the preferred controlling behaviour (Zimmermann and Eber 2014; Eber 2019a). (3) ∂qi∂t=ciqi⇒qi∼ecit {{\partial {q_i}} \over {\partial t}} = {c_i}{q_i} \Rightarrow {q_i} \sim {e^{{c_i}t}}

The most general approach to understand the time-related behaviour of a starkly simplified subsystem of this kind is by the differential equations for the harmonic oscillator. Any deviation of a value Q is answered by an automatised force leading back − β to the desired value 0 working against the inertness μ towards any change and additionally taking some retarding forces ρ proportional to the rate of change into account. (4) μ∂2Q∂t2=−βQ−ρ∂Q∂tresp.∂2Q∂t2=−βμQ−ρμ∂Q∂t→∂Q∂t−ω2Q−k∂Q∂t. \matrix{{\mu {{{\partial^2}Q} \over {\partial {t^2}}} = - \beta Q - \rho {{\partial Q} \over {\partial t}}{\rm{resp}}.} \hfill \cr {{{{\partial^2}Q} \over {\partial {t^2}}} = - {\beta \over \mu}Q - {\rho \over \mu}{{\partial Q} \over {\partial t}} \to {{\partial Q} \over {\partial t}} - {\omega^2}Q - k{{\partial Q} \over {\partial t}}.} \hfill \cr} using the damping factor k = ρ/μ and the frequency ω=β/μ \omega = \sqrt {\beta /\mu} .

The general solution is given by an approach of damped oscillations as a complex function (5) Q(t)=Q0⋅exp(λt) Q\left(t \right) = {Q_0} \cdot \exp \left({\lambda t} \right) with ∂Q(t)∂t=Q0⋅λexp(λt) {{\partial Q\left(t \right)} \over {\partial t}} = {Q_0} \cdot \lambda \exp \left({\lambda t} \right) and ∂2Q(t)∂t2=Q0⋅λ2exp(λt) {{{\partial^2}Q\left(t \right)} \over {\partial {t^2}}} = {Q_0} \cdot {\lambda^2}\exp \left({\lambda t} \right) leading to Q0⋅λ2exp(λt)+βμQ0⋅exp(λt)+ρμQ0⋅λexp(λt)=0 {Q_0} \cdot {\lambda^2}\exp \left({\lambda t} \right) + {\beta \over \mu}{Q_0} \cdot \exp \left({\lambda t} \right) + {\rho \over \mu}{Q_0} \cdot \lambda \exp \left({\lambda t} \right) = 0 λ2+βμ+ρμ⋅λ=0→λ2+βμ+ρμ⋅λ+(ρ2μ)2=+(ρ2μ)2→(λ+ρ2μ)2=−βμ+(ρ2μ)2 \matrix{{{\lambda^2} + {\beta \over \mu} + {\rho \over \mu} \cdot \lambda = 0 \to {\lambda^2} + {\beta \over \mu} + {\rho \over \mu} \cdot \lambda + {{\left({{\rho \over {2\mu}}} \right)}^2} = + {{\left({{\rho \over {2\mu}}} \right)}^2}} \hfill \cr {\to {{\left({\lambda + {\rho \over {2\mu}}} \right)}^2} = - {\beta \over \mu} + {{\left({{\rho \over {2\mu}}} \right)}^2}} \hfill \cr} and finally (6) λ=−ρ2μ±(ρ2μ)2−βμ:=−k2±(k2)2−ω2 \lambda = - {\rho \over {2\mu}} \pm \sqrt {{{\left({{\rho \over {2\mu}}} \right)}^2} - {\beta \over \mu}} : = - {k \over 2} \pm \sqrt {{{\left({{k \over 2}} \right)}^2} - {\omega^2}}

Depending on the relationship between the damping factor k = ρ/μ and the frequency ω=β/μ \omega = \sqrt {\beta /\mu} , this system is capable of performing damped or escalating oscillations, as well as exponential approximating characteristics (Figure 6).

Fig. 6

The situation of weakly damped oscillations is determined by k/2 ≪ ω where the root becomes negative and the solution therefore complex-valued. (7) Q(t)=Q0⋅exp(−k2t)exp(±i⋅tω2−(k2)2) Q\left(t \right) = {Q_0} \cdot \exp \left({- {k \over 2}t} \right)\exp \left({\pm i \cdot t\sqrt {{\omega^2} - {{\left({{k \over 2}} \right)}^2}}} \right)

The relevant real part of this is representing exponentially damped oscillations: (8) Q(t)=Q0⋅exp(−k2t)cos(±tω2−(k2)2) Q\left(t \right) = {Q_0} \cdot \exp \left({- {k \over 2}t} \right)\cos \left({\pm t\sqrt {{\omega^2} - {{\left({{k \over 2}} \right)}^2}}} \right)

The frequency is different from the undamped frequency ωd2=ω2−k2/4 \omega_d^2 = {\omega^2} - {k^2}/4 , while the relaxation time is τ_R = 2/k = 2μ/ρ.

Critical dampening refers to the situation where the root term vanishes as no oscillations occur k/2=ω→ρ/2μ=β/μ k/2 = \omega \to \rho /2\mu = \sqrt {\beta /\mu} . The solution reduces to a single exponential descent with an identical relaxation time constant τ_R = 2/k = 2μ/ρ: (9) Q(t)=Q0⋅exp(−k2t) Q\left(t \right) = {Q_0} \cdot \exp \left({- {k \over 2}t} \right)

Finally, the overdamped case is given if the root yields a real solution, i.e., k / 2 ≫ ω (10) Q(t)=Q0⋅exp(−k2t±t(k2)2−ω2) Q\left(t \right) = {Q_0} \cdot \exp \left({- {k \over 2}t \pm t\sqrt {{{\left({{k \over 2}} \right)}^2} - {\omega^2}}} \right)

The characteristic of the time development is also exponential; however, the time constant of relaxation is somewhat different. (11) τR=[k2∓(k2)2−ω2]−1 {\tau_R} = {\left[{{k \over 2} \mp \sqrt {{{\left({{k \over 2}} \right)}^2} - {\omega^2}}} \right]^{- 1}}

The primary solution (negative sign) indicates the damping factor k/2 reduced by a term depending on the relationship between frequency and damping factor and, thus, an increased relaxation time τ_R.

Excursion: In order to approximate a strongly over-damped situation which moves slow enough that the originally dominant inertia term would cease to play a role (μ → 0), the differential equation needs to be slightly modified: (12) μ∂2Q∂t2=−βQ−ρ∂Q∂t→0=−βQ−ρ∂Q∂t→∂Q∂t=−βρQ \mu {{{\partial^2}Q} \over {\partial {t^2}}} = - \beta Q - \rho {{\partial Q} \over {\partial t}} \to 0 = - \beta Q - \rho {{\partial Q} \over {\partial t}} \to {{\partial Q} \over {\partial t}} = - {\beta \over \rho}Q

The solution of the remaining purely integral controller is given by Q(t) = Q₀ · (−βt/ρ) where the relaxation time constant is τ_R = ρ/β.

In the end, this is the reason why the primary solution out of the two possible signs is correctly chosen: Obviously, only then the development of τ_R for large values of τ_C is sensibly approximating the inertia-free approach.

These theoretical results can easily be applied on a singular presumably linear production process. Let the rate of production, corresponding the invested production resources, be R_Prod = dQ/dt, leading to the product determined as quality Q_Fin after the production duration t_Fin (Figure 9).

Fig. 9

Based on this principal dependency, the required controlling resources can be investigated from the quality deviations to be faced. However, this strongly relies on the character of deviation considered. Several scenarios are to be discussed in order to provide some framework for dimensioning sensible control mechanisms.

8.1

Maximum response scenario

Quality deviation incidents resulting from sources outside the production process are expected to occur unexpectedly and randomly (Figure 10). In order to settle these, the resources R_Contr = k_CδQ_Contr corresponding to the initial (maximum) controlling response need to be held available all over the process time t_Fin. While settling, only a fraction will be required due to the exponential decay of the remaining δQ_Contr. However, no storing of R_Contr in order to average the demand is feasible without violating the guaranteed controlling reaction time Δt.

Fig. 10

This ‘Maximum Response Scenario’ requires controlling ressources R˜Contrmax=δQ˜max/0.74δτ˜ \tilde R_{Contr}^{\max} = \delta {\tilde Q^{\max}}/0.74\delta \tilde \tau held ready where δQ˜max \delta {\tilde Q^{\max}} is given as the maximum amplitude of every possible incident as the fraction of the final product Q_Fin.

Remark: The shorter the time constant, the faster this incident can be settled. Thus, the shorter the time constant, the the higher the gradient dQ/dt at the incident: (39) ddte−t/τ|t=0=−1τe−t/τ|t=0=−1τ {\left. {{d \over {dt}}{e^{- t/\tau}}} \right|_{t = 0}}{\left. {= - {1 \over \tau}{e^{- t/\tau}}} \right|_{t = 0}} = - {1 \over \tau}

8.4

Simplified estimation scenario

Starkly simplifying, we might assume the (uncontrolled) production process to be halted after t_Fin and then using R_Prod during a subsequent period with the duration of the time tolerance δτ producing further and therewith repairing for the incomplete quality. Then, we obtain the manageable δQContrEst=0,74⋅δτ⋅RProd \delta Q_{Contr}^{Est} = 0,74 \cdot \delta \tau \cdot {R_{\Pr od}} where the time tolerance is needed as well as the provision of the production ressources over this time period. This approach also provides a measure for the manageable quality deviation, using R_Prod instead of R_Contr. (Figure 11) (45) δQContrEstQFin=0.74⋅δτ⋅RProdQFin=0.74⋅δτ⋅QFinQFintFin=0.74⋅δτtFin⇒δQ˜ContrEst=0.74⋅δτ˜ and R˜ContrEst=1 \matrix{{{{\delta Q_{Contr}^{Est}} \over {{Q_{Fin}}}} = {{0.74 \cdot \delta \tau \cdot {R_{\Pr od}}} \over {{Q_{Fin}}}} = {{0.74 \cdot \delta \tau \cdot {Q_{Fin}}} \over {{Q_{Fin}}{t_{Fin}}}} = 0.74 \cdot {{\delta \tau} \over {{t_{Fin}}}}} \cr {\Rightarrow \delta \tilde Q_{Contr}^{Est} = 0.74 \cdot \delta \tilde \tau \,{\rm{and}}\,\tilde R_{Contr}^{Est} = 1} \cr}

Fig. 11

Besides the attempts to quantify resources taken from the optimal controlling mechanisms on the basis of limited deviation scenarios (Section 8), a principal limit to control within a system may be described based on a much more fundamental argument:

In fluid dynamics, the ratio of dynamic forces and frictional forces is used to identify turbulent behaviour dominated by dynamics in contrast to linear flow which is driven by retarding forces (Reynolds number, conceptually introduced by Stokes (Stokes 1851)). Within the given context, a similar situation is at hand and, hence, a comparable criterion to distinguish between linear manageable and nonlinear escalating behaviour is proposed:

Based on the description of delayed controlling units, the differential equation is (see above, second-order approach) written as follows: (46) ∂2Q∂t2=−βμQ−ρμ∂Q∂t {{{\partial^2}Q} \over {\partial {t^2}}} = - {\beta \over \mu}Q - {\rho \over \mu}{{\partial Q} \over {\partial t}}

Here, the dynamical impact is represented by the linear response of the partial system to any deviation βQ/μ and can be understood as the force leading back to the target value Q = 0 proportional to the deviation Q.

The retarding forces are represented by the second term ρQ˙/μ \rho \dot Q/\mu which is proportional to the first derivative of the deviation ∂Q / ∂t; hence, the speed of change inducing the frictional term.

The ratio of these two components which determines the overall behaviour of the system yields the following: (47) S=Dynamic TermRetarding Term=βμμρQQ˙=βρQQ˙ S = {{Dynamic\,Term} \over {{\mathop{\rm Re}\nolimits} tarding\,Term}} = {\beta \over \mu}{\mu \over \rho}{Q \over {\dot Q}} = {\beta \over \rho}{Q \over {\dot Q}}

Using the equivalents to dynamical and frictional terms taken from the delayed controlling approach (second order) provides: (48) μ=Δt2τC β=2τC ρ=2(1−ΔtτC)S=βρQQ˙=2⋅tχτC⋅2(1−ΔtτC)QQ˙=1τC−ΔtQQ˙ \matrix{{\mu = {{\Delta {t^2}} \over {{\tau_C}}}\,\beta = {2 \over {{\tau_C}}}\,\rho = 2\left({1 - {{\Delta t} \over {{\tau_C}}}} \right)} \cr {S = {\beta \over \rho}{Q \over {\dot Q}} = {{2 \cdot {t_\chi}} \over {{\tau_C} \cdot 2\left({1 - {{\Delta t} \over {{\tau_C}}}} \right)}}{Q \over {\dot Q}} = {1 \over {{\tau_C} - \Delta t}}{Q \over {\dot Q}}} \cr}

It turns out that the characteristic of this development lies mainly with the denominator, while the undetermined factor Q/Q˙ Q/\dot Q only scales the behaviour linearly (Figure 12):

Fig. 12

Clearly, the overall behaviour is dominated by dynamic forces if the time constant of the implemented control τ_c approaches the time delay Δt. The temporal development will be non-linear, likely to be unpredictable and, hence, uncontrollable. Only if the controllers are operating much slower than the time delay, stability can be expected. Then, linear behaviour allows for predictability and, thus, controllability.

Understanding Q = Q_Contr as the manageable deviation and Q˙=RContr \dot Q = {R_{Contr}} as the resources available for control, we obtain for stable situations S < 1: (49) 1>1τCΔtQQ˙ respectively (τC−Δt)Q˙Contr>δQContrδQContr<(τC−Δt)RContr ∀τC>ΔtδQContrRProdtFin<(τC−Δt)RContrRProdtFinδQ˜Contr<τC−ΔttFinR˜Contr=(τ˜C−Δ˜t)R˜Contr \matrix{{1 > {1 \over {{\tau_C}\Delta t}}{Q \over {\dot Q}}\,{\rm{respectively}}\,\left({{\tau_C} - \Delta t} \right){{\dot Q}_{Contr}} > \delta {Q_{Contr}}} \cr {\delta {Q_{Contr}} < \left({{\tau_C} - \Delta t} \right){R_{Contr}}\,\forall {\tau_C} > \Delta t} \cr {{{\delta {Q_{Contr}}} \over {{R_{\Pr od}}{t_{Fin}}}} < \left({{\tau_C} - \Delta t} \right){{{R_{Contr}}} \over {{R_{\Pr od}}{t_{Fin}}}}} \cr {\delta {{\tilde Q}_{Contr}} < {{{\tau_C} - \Delta t} \over {{t_{Fin}}}}{{\tilde R}_{Contr}} = \left({{{\tilde \tau}_C} - \tilde \Delta t} \right){{\tilde R}_{Contr}}} \cr}

Remark: Using the first-order approach yields δQ˜Contr<=τ˜CR˜Contr \delta {\tilde Q_{Contr}} < = {\tilde \tau_C}{\tilde R_{Contr}} , implying very short time delays.

Hence, in order to achieve stable systems, the deviation of quality needs to be less than the available controlling resources scaled by a factor given by the difference between the relative controlling time constant and the relative control delay, i.e., the more effective control is expected to be, the slower control is required to keep the system stable.

This completely different understanding of a confinement to stability and therewith for controllability is plotted into Figure 13. The proportionality of manageable deviation δQ˜Contr \delta {\tilde Q_{Contr}} (understood as percentage of the scheduled quality) and the controlling resources on hold _Contr (also understood as percentage of the productive resources) remain for obvious reasons while the scaling factor τ˜C−Δ˜t {\tilde \tau_C} - \tilde \Delta t becomes meaningful. Since this factor is the difference of two durations, furthermore denoted as percentage of the total duration, the unit is t_Fin. Besides the criterion of stability, this value also represents the induced time tolerance given by this approach (formally not to be confused with the time tolerance derived on the basis of optimised control, but providing an equivalent idea).

Fig. 13

This picture clearly points out that the control ratio needs to exceed unity by far R˜Contr/δQ˜Control≫1 {\tilde R_{Contr}}/\delta {\tilde Q_{Control}} \gg 1 in order to provide sensible time tolerance values.

Example: A given quality deviation of e.g. δQ˜Control=5% \delta {\tilde Q_{Control}} = 5\% is expected to be compensated by additonal controlling resources of also R˜Contr=5% {\tilde R_{Contr}} = 5\% . Then, the timing constant of the controller needs to be equal to the production time t_Fin on top of the time delay which cannot be zero and, hence, is at least t_Fin, giving an idea of the magnitude of a time tolerance. Vice versa, a reasonable time tolerance of, e.g., τ˜C−Δ˜t≃5%tFin {\tilde \tau_C} - \tilde \Delta t \simeq 5\% {t_{Fin}} which is to be shared by the controller time constant and the controlling delay, requires the ratio of controlling resources to quality deviation to be at least R˜Contr/δQ˜Control>0.05−1=20 {\tilde R_{Contr}}/\delta {\tilde Q_{Control}} > {0.05^{- 1}} = 20 . Thus, controlling resources of R˜Contr=5% {\tilde R_{Contr}} = 5\% allow only for compensation of deviation values δQ˜Control=5%/20=0.25% \delta {\tilde Q_{Control}} = 5\% /20 = 0.25\% .

Hence, the various previously discussed scenarios offer limited deviation situations based on a number of different arguments limiting the absolute values of expected quality deviation. As a consequence, the resulting normalised resources required ready for control sit somewhere in the range of slightly below unity. However, the stabililty criterion which is clearly not as sophisticated but much more fundamental overrides this outcome by far and demands at least by one magnitude higher control resources in order to guarantee stability. Thus, in particular, this limit turns out to be crucial to designing controllable systems.

Based on these theoretical considerations, we state that any organisational structure can principally not be set up consistently and therefore will be in fundamental need of concepts of controlling not only overhead but also tolerance. Any inconsistent system, i.e., not only far off the equilibrium state but also where the equilibrium state is only dynamically determined, cannot be stabilised totally.

If ruling mechanisms are at hand which principally allow inducing forces strong enough to compensate for inconsistencies, the system may become controllable. The optimal time constant of the balancing process as the central controlling parameter is widely independent of the degree of inconsistency or controlling force. The main parameter turns out to be the controlling delay, i.e., the reaction time of the controlling mechanism which cannot be assumed to vanish. Therewith, a principal minimum balancing time is given. At any attempt to reduce this by employing stronger controlling mechanisms, the system becomes increasingly unstable and develops escalating behaviour.

The maximum deviation to be handled is clearly an undetermined parameter and, however, can be limited considering some practical scenarios. The dependency of the resources required to correct for the given deviations develops nevertheless proportional. Considering a balanced situation where the time tolerance of control also needs to be kept within reasonable limits, this ratio is forced to at least a magnitude beyond unity, hence principally limiting the capabilities of control with mutual restrictions on time tolerance as well as available control resources and manageable quality deviation values.

Less hierarchical and therewith more self-determining organisational systems, e.g. concepts of lean management VDI 2553 (VDI 2019) or the SCRUM manifest (Beck et al. 2001; Schwaber and Sutherland 2020), are in fact taking these principal limitations of control into account proposing flexible structures on the basis of short-range collaboration. Therewith, based on originally hierarchical structures, substantial decentral control is manifested and, however, equipped with a well-established set of resources ready to correct for unavoidable planning deviations and inconsistencies. Probably unaware of the conceptual implication, finally, the need of a significant number of resources including the therewith connected time tolerance is accepted and readily provided.

Hence, as a practical implication of this paper, a very clear and inevitable understanding of controlling confines in any organisational system is pointed out. This applies to any kind of project and, however, in particular to construction projects, focusing on unique production assemblies bound to very tight time frames and cost frames with no options to reconsider decisions. Typically, strictly hierarchical approaches are maintained predetermining all details in advance and then relying on controlling mechanisms to ensure the required specifications. Time and cost reserves are then integrated on a heuristic basis to a degree required by the markets rather than on a substantiated background. Based on the findings described in this paper, a fundamental minimal time tolerance is given through the well-known time delay of the implemented controlling mechanisms. Furthermore, the resources required to correct for the expected misalignment can be expressed directly in terms of a percentage of the creation of value of the respective process, in dependence of the intended controlling time constant and delay.

Therewith, the required controlling effort in terms of explicit cost is no more subjected to personal experience with more or less substantial statistical significance. Instead, this most decisive knowledge can be explicitly derived from available project parameters as are the planned time tolerance respectively the tolerance regarding the final value of the product.