This page is a sub-page of our page on Knowledge Algebra.
///////
Milestone dependency matrices:
A business model, for example, can be considered as a plan for how to achieve a consecutive sequence of (sets of) results at certain moments in time (“deadlines”). Each of the results achieved at moment \, t \, will depend on some of the results achieved at moment \, t-1 . A milestone dependency matrix is a formal adjacency matrix that links the results achieved at time t to the results that they have depended on from time \, t-1 .
Milestone dependency chains:
As discussed above, a business model can be considered as a kind of map of how to get from a “starting state” (at time \, t = 0 \, ) consisting of the \, p \, resources that are available to us at the start, and a “stopping state” (at time \, t = n \, ) consisting of the \, q \, goals that we want to achieve. We can then construct a (most probably very large) matrix \, \left< \, R \, || \, G \, \right> \, that can express all the possible relations between our initial resources, represented by a column-vector \, R_i \triangleq \left< R_{esources(0)} \right| \, of length \, p , and the “fulfillment state”, represented by a row-vector G^f \triangleq \left| G_{oals(n)} \right> \, of length \, q , when the goals of the business plan are fulfilled and our business is “up and running”. Hence we can write:
The start and stop of this development process can be naturally thought of as separated by \, n-1 \, intermediate “milestone moments” \, i = 1, 2, … , n-1 . These milestone moments occur when certain combinations of intermediate results have been reached. Such milestones correspond to the completion of combinations of key tasks, and they are often connected with different forms of deliverables. Moreover, the dependencies between the milestones are often represented visually, e.g., in the form of a Gantt chart.
///////
\, R \, M^1 \, M_1 \, M^2 \, M_2 \, \cdots \, M_{n-1} \, G^n \, \, R \, O^1 \, O_1 \, M^1 \, M_1 \, O^2 \, O_2 \, M^2 \, M_2 \, \cdots \, M_{n-1} \, O^n \, O_n \, G^n \, \, R \, H_{ow} O^1 \, W_{hy} O_1 \, M^1 \, M_1 \, H_{ow} O^2 \, W_{hy} O_2 \, M^2 \, M_2 \, \cdots \, M_{n-1} \, H_{ow} O^n \, W_{hy} O_n \, G^n \, \, R \, (H_{ow} O^1) \, (W_{hy} O_1) \, M^1 \, M_1 \, (H_{ow} O^2) \, (W_{hy} O_2) \, M^2 \, M_2 \, \cdots \, M_{n-1} \, (H_{ow} O^n) \, (W_{hy} O_n) \, G^n \, \, \begin{pmatrix} R_1 \\ R_2 \\ \vdots \end{pmatrix} \begin{pmatrix} H_{ow} O_{11} \, H_{ow} O_{12} \cdots \end{pmatrix} \begin{pmatrix} W_{hy} O_{11} \\ W_{hy} O_{12} \\ \vdots \end{pmatrix} \begin{pmatrix} M_{11} \, M_{12} \cdots \end{pmatrix} \begin{pmatrix} M_{11} \\ M_{12} \\ \vdots \end{pmatrix} \begin{pmatrix} H_{ow} O_{21} \, H_{ow} O_{22} \cdots \end{pmatrix} \begin{pmatrix} W_{hy} O_{21} \\ W_{hy} O_{22} \\ \vdots \end{pmatrix} \begin{pmatrix} M_{21} \, M_{22} \cdots \end{pmatrix} \, \cdots \,The product of the first four matrices (to the left) is equal to
\, \begin{pmatrix} R_1 \\ R_2 \\ \vdots \end{pmatrix} (H_{ow} O_{11} W_{hy} O_{11} + H_{ow} O_{12} W_{hy} O_{12} + \cdots) \begin{pmatrix} M_{11} \, M_{12} \cdots \end{pmatrix} \, \, \begin{pmatrix} R_1 (H_{ow} O_{11} W_{hy} O_{11} + H_{ow} O_{12} W_{hy} O_{12} + \cdots) M_{11} \,\, R_1 (H_{ow} O_{11} W_{hy} O_{11} + H_{ow} O_{12} W_{hy} O_{12} + \cdots) M_{12} \, \cdots \\ R_2 (H_{ow} O_{11} W_{hy} O_{11} + H_{ow} O_{12} W_{hy} O_{12} + \cdots) M_{11} \,\, R_2 (H_{ow} O_{11} W_{hy} O_{11} + H_{ow} O_{12} W_{hy} O_{12} + \cdots) M_{12} \cdots \\ \vdots \end{pmatrix} \, \, \begin{pmatrix} R_1 (H_{ow} O_{11} W_{hy} O_{11}) M_{11} + R_1 (H_{ow} O_{12} W_{hy} O_{12}) M_{11} + \cdots \,\, R_1 (H_{ow} O_{11} W_{hy} O_{11}) M_{12} + R_1 (H_{ow} O_{12} W_{hy} O_{12}) M_{12} + \cdots \\ R_2 (H_{ow} O_{11} W_{hy} O_{11}) M_{11} + R_2 (H_{ow} O_{12} W_{hy} O_{12}) M_{11} + \cdots \,\, R_2 (H_{ow} O_{11} W_{hy} O_{11}) M_{12} + R_2 (H_{ow} O_{12} W_{hy} O_{12}) M_{12} + \cdots \\ \vdots \end{pmatrix} \,///////
\, H_{ow} (R_1 O_{12} M_{12}) \, \, W_{hy} (R_1 O_{12} M_{12}) \,///////
Placificational options:
For \, i = 0, … , n-1 , the expression \, S_{pace(i)} S^{pace(i+1)} \, represents the “intentional adjacency matrix” between \, M_{ileStone(i)} \, and \, M_{ileStone(i+1)} . Focusing on the options for placification (= deployment = realization = implementation) that connect these adjacent intentions, we can write:
S_{pace(i)} S^{pace(i+1)} = S_{pace(i)} O_{ptions(\bullet)} O^{ptions(\bullet)} S^{pace(i+1)} == S_{pace(i)} P_{lace(\bullet)} P^{lace(\bullet)} S^{pace(i+1)} .
Hence we can write the “placificational options” of the intentional chain (of milestones) of the business model in the following form:
S_{pace(0)} P_{lace(\bullet)} P^{lace(\bullet)} S^{pace(1)} S_{pace(1)} P_{lace(\bullet)} P^{lace(\bullet)} S^{pace(2)} \cdots S_{pace(n-1)} P_{lace(\bullet)} P^{lace(\bullet)} S^{pace(n)}This matrix product captures the available choices (= options) involved in realizing (= placifying) the intentions of the business model in the form of a DAG (= Directed Acyclic Graph) of dependency-related milestone results.
Milestones of abilities development:
We can also consider a business plan/model as a plan for the development of abilities. An advantage with this view is that it opens up a broader perspective that includes all kinds of learning processes.
Remembering the relationship between the tensor notation and the bra-ket notation
\, A_{bilities(t)} \equiv \left< A_{bilities(t)} \right| \,
A^{bilities(t+1)} \equiv \left| A_{bilities(t+1)} \right> \, ,
we can write the bra-ket matrix product \, \left< A_{bilities(t)} \right| \left| A_{bilities(t+1)} \right> \, in tensor notation as:
\, A_{bilities(t)}A^{bilities(t+1)} \, .
Let \, O_j \triangleq {O_{ption}}_{j(t)} \in O_{ptions(t)} \, .
Here \, O_j \, denotes an option that is available at time \, t \, . It is an element of the set all available options \, O_{ptions(t)} \, at that time. In the \, O_j \, notation, for reasons of simplicity, we have suppressed the dependence of the index parameter \, j \, on time.
As discussed in the TELL ME deliverable on Business Algebra [TELL ME D1.3 appendix (version 2)], we define the option spread at time \, t \, as
\, O_i O^j \triangleq {O_{ption}}_{i(t)} O^{{ption_{j(t)}}} \, .
This matrix describes the (binary) relationships between all options available at time \, t \, . Hence we can introduce the abilities development formal adjacency matrix:
\, A_{bilities(t)}^{bilities(t+1)} \triangleq A_{bilities(t)} O_i O^j A^{bilities(t+1)}\,It represents the available choices/options when moving from the “abilities milestone” at time \, t \, to the abilities milestone at time \, t + 1 \, . The contextual and the symbolic groupings of the right hand side are given by
\, (A_{bilities(t)} O_i) (O^j A^{bilities(t+1)}) \, , and
\, A_{bilities(t)} (O_i O^j) A^{bilities(t+1)} \, .
Their interplay ensures the emergence of relevance regarding the options available for realizing the step from the abilities milestone at time \, t \, to the abilities milestone at time \, t + 1 \, .
Evaluation of an option spread with respect to effectiveness and efficiency
Effectiveness/efficiency evaluation applies an effective/efficient assessment metric to each option of an option spread \, O_i O^j \, . For example, in order to evaluate the performance of the abilities development plan above, for each of the available options, we should ask the two questions:
Effectiveness: Why are we pursuing option \, O^j \, ?
Efficiency: How well are we pursuing option \, O_i \, ?
Making the definition:
\, E_{val-How}^{val-Why} \left( A_{bilities(t)}^{bilities(t+1)} \right) \triangleq \triangleq A_{bilities(t)} O^i E_{val-How(O_i)} E^{val-Why(O_j)} O_j A^{bilities(t+1)} \,and introducing the notation
\, H_{ow(i)?} \triangleq E_{val-How(O_i)} \, and \, W^{hy(j)?} \triangleq E^{val-Why(O_j)} \,
we can conclude that:
\, E_{val-How}^{val-Why} \left( A_{bilities(t)}^{bilities(t+1)} \right) \equiv A_{bilities(t)} O^i H_{ow(i)?} W^{hy(j)?} O_j A^{bilities(t+1)} \, .
/////