Selection of the initial vectors

For any FCM network with N concepts, one can choose 2N _{activating vectors that include} all the possible combinations of the initial values of the concepts when initial values are

only limited to 0 or 1. This means that for an imaginary FCM with 40 concepts initial states

could be defined in more than one trillion (1.995e+12) ways. Feeding this massive amount

of combinations into the model requires excessive computing resources and can lead to

outputs that are difficult to interpret. Moreover, because FCMs have meta-rules, input

variations do not always lead to variations in outputs, so that a lot of the results would be

redundant. Accordingly, it is important to strategically select the right subset of initial

values (Jan H. Kwakkel, Haasnoot, and Walker 2015; J. Kwakkel and Haasnoot 2015).

There are fundamentally two strategies for achieving this: One strategy is to randomly

select a subset of initial vectors from all the possible permutations. This assumes that this

smaller set of vectors will allow me to observe patterns that are similar to an observation

of all permutations. A second strategy is to select input vectors based on plausible

managerial strategies. A manager would likely not attempt to change a large number of

largely different variables at a time but focus efforts on coherent strategies, such as “focus

on human resources”, “reorganize departments”, or “implement open innovation

principles”. However, without involving managers in this study it is difficult to formulate

such plausible strategies. I therefore focused on the first strategy and only did a limited test

of the second strategy by running the model with an input vector that represents a focus on

open innovation.

83 There are 371 concepts in the model that can either be activated (+ 1 or -1) or off (0). (I

chose to not consider “in-between” activation levels in the interval of [-1, 0, 1]). A random

assignment of these values means that, on average, half of all concepts (185.5) would be

activated to their full extend regardless. This is likely unrealistic in a real-world setting,

where companies cannot do all and fewer variables can be expected to be active at the same

time. It is also problematic because 50% of the concepts (i.e. all activated concepts) would

be clamped, effectively rendering large parts of the model inactive. Moreover, exploratory

and exploitative innovations have a 25% probability to both be activated at the same time.

Thus, ambidexterity would be high in a quarter of all cases, regardless of other concepts.

To resolve the issue, I therefore assigned a probability for every given concept to be

activated at p=0.05. This means that I studied the impact of initial vectors that activate an

average of 18.5 concepts (np=371*0.05=18.5). Figure 12 shows the distribution: the x-axis

shows the number of concepts that were activated in each class, the y-axis shows the

frequency (i.e. the number of vectors in each class). The minimum number of concepts that

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Figure 12- Distribution of sum of 1000 randomly selected initial vectors (P=0.05)

This set of 1000 randomly generated initial vectors was used in simulation and results is

explained ahead. Table 4 provides a glimpse of processing time and memory needed for

generating different size of initial vector sets.

Table 4- Time and memory used for generating initial vector permutations of the FCM with 371 concepts

# of Initial vectors Process time Memory size

1000 1 S 728 kb

100,000 25 S 70.8 Mb

1000,000 250 S 708 Mb

Hardware used for simulations above: HP Workstation, Core i7 processor, and 64Gb RAM.

4.4.1.2 Initial vector to represent managerial intervention

A comprehensive model of ambidexterity can be used to test hypotheses that are proposed

in the literature. It can also provide a “sandbox” for managers to examine their ideas for

achieving ambidexterity through computational experiments. In both cases, an initial

85 hypothesis or the planned managerial intervention. There is no limit to the number of

hypotheses or managerial ideas that could be tested with the model. However, such

experiments are only meaningful if they are carefully constructed. I therefore focused my

attention on only one case.

For this case, the literature of open innovation was used to examine the behavior of the

system when a set of practices, which are suggested in the literature, are represented as an

initial vector and fed into the system. Please refer to section 5.6 for the details of simulation

and results.

4.4.2 Generation of the adjacency matrix

Please refer to section 4.2.3 for details.

4.4.3 Running the FCM for all the acceptable initial values

Furthermore, as explained in 3.1, I analyzed the behavior of the landscapes resulting from

all the plausible permutations of initial vectors and numerous adjacency matrices.

Plausible permutations of initial vectors only includes concepts that have impacts on others

in a network. Therefore receiver-only concept analysis (ROCA) as described in Appendix

III was used to exclude concepts that have only inward links—such as outcome variables—

and the concepts that only have outward link to this group of concepts. Thus 107 concepts

were identified, as depicted in Figure 13, with no impact on the value of concepts of the

interest such as exploitative innovation, exploratory innovation and ambidexterity. This

step also eliminated a large set of unnecessary calculations and shortened the simulations

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Figure 13- Concepts excluded from the initial vector permutations. Concepts with no outward connections (red) and predecessor concepts only causing the first group (magenta)

The number of possible permutations of the initial combinations could be calculated as

180,352,320 (all the combinations of 4 activated concepts out of 258 concepts). For the list

of these concepts refer to

Appendix E- Domain and collective cognitive maps memo. This step alongside identifying

the initial values which lead to the landscapes of interest –using mathematical filtering or

visualization techniques—are the two typical steps per (Jan H. Kwakkel, Auping, and Pruyt

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4.4.4 Visualization of the results

The key question of visualization is to identify meta-rules (Dickerson Kosko) that govern

the behavior of the system. Specifically, I am interested to see a) how many scenarios

balance exploration and exploitation (i.e. achieve ambidexterity), b) how may scenarios

perform well with regard to one aspect but at the expense of the other (i.e. low

exploration/high exploitation and vice versa), c) how many scenarios result in low

performance in both aspects, and d) what theoretically possible positions in the scenario

space are not populated. Answering these questions makes it possible to contribute to

ambidexterity theory.

I used five type of visualization techniques to answer the questions above:

1) Scatter plot: I generated scatter plots using the R.Plotly package to visualize the

outcomes of my simulation runs. I plotted each simulation result against two axes, i.e. the

amount of exploitation and the amount of exploration. (See Figure 42).

2) Cluster map: I generated cluster maps, using the R.Plotly package, for visualizing

different groups of scenarios that contained similar scenarios and were distinctly different

from other scenario groups. This visualization also gave me information about the

frequency of each scenario type. (See Figure 52).

3) Heat map: I generated heat maps, using the R.ggplot2 package for visualizing the

density of scenarios in an area that covers all possible combinations of exploration and

exploitation. (See Figure 49).

4) Topology map: It was generated by combining the capabilities of R.ggplot2 and

88 elevation layers for all the scenarios with different value of exploitative and exploratory

innovation. (See Figure 48).

In addition to these visualizations, which directly contribute to answering the questions

posed above, I also developed:

5) Pulse diagram: A pulse diagram shows the activation levels of different concepts for

each iteration of the simulation. I developed this visualization using R, to study the system

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