# Revisiting Problem-Solution Co-Evolution in the Context of Team Conceptual Design Activity
## 3 Results
### Transitions Between the Problem and Solution Space  
The final set of hypotheses concerns the interaction between the problem and the solution space; that is, whether the synthesis of new problem and solution entities directly follows problem or solution space exploration. The procedure for testing the hypotheses again relies on the probabilities of specific state transition sequences appearing. First, all instances of sequences that result in the synthesis of design entities were singled out (e.g., the two transitions highlighted in the example illustrated in Figure 2). Second, these separated sequences were split into two groups: sequences that result with new problem entities and sequences that result with new solution entities. Third, the two groups were again divided into sequences that start within the problem space, and sequences that start within the solution space. Fourth, the counted instances of state-transition sequences were normalised across the groups to calculate their probabilities. Thus, one can analyse whether it is more probable that new problem entities (requirements, needs, or constraints) are a direct result of problem space exploration (PRO\(\rightarrow\)PS) or solution space exploration (SOL\(\rightarrow\)PS). The same analysis can also be performed for the new solution entities (PRO\(\rightarrow\)SS vs. SOL\(\rightarrow\)SS). The probabilities across all activity fragments and all teams are summarised using the boxplots in Figure 7, whereas all normalised results are shown in Table A3 in Appendix A.  
The boxplots show that both the new problem entities and new solution entities are more probable to follow after state transitions related to solution space exploration. This likelihood difference is more pronounced in the case of new solutions. Nevertheless, both differences were shown as statistically significant. The Wilcoxon signed-rank test was again used to compare the sequence pairs plotted in Figure 6 since the Shapiro-Wilk test rejected the normality assumption for the probabilities of sequences resulting in problem synthesis. The Wilcoxon test (paired per protocol fragment) revealed a significant difference at the \(\alpha\) = 0.05 level between both the PRO\(\rightarrow\)PS and SOL\(\rightarrow\)PS (\(p\) \(<\) 0.05) probability pair, and the PRO\(\rightarrow\)SS and SOL\(\rightarrow\)SS (\(p\) \(<\) 0.001) probability pair.  
Figure 7: Boxplots of proportions of synthesis of problems (left) and solutions (right) based on the space explored during the previous state transition.  
It must, however, be noted that the significant difference in the case of problem synthesis very much depends on the type of activity. Namely, during ideation activity, it was more probable that problem entities will follow after state transitions related to problem space exploration. In contrast, during concept review, the results are the same as on the overall level. Hence, while there exists empirical support for hypothesis H3a, that the identification of new problem space entities (e.g., new requirements) is more probable to be a result of solution space exploration, it is valid solely on the overall conceptual design process level, where no distinction is made between ideation and concept review. In addition, the results show the identification of solution entities was more probable to be a result of the solution space, rather than the problem space exploration (both overall and for each activity separately). Hence, no empirical support was found for the hypothesis H3b.