Optimization and Computational Fluid Dynamics - Department of ...

286 J. Hämäläinen, T. Hämäläinen, E. Madetoja, H. Ruotsalainen
Table 9.2 Six optimal compromise solutions
Tensile strength Formation Basis weight Dry solids
ratio (g/m 2 ) (g/m 2 ) content (%)
Desired values 3.00 0.30 . ..0.35 54.00 92.00
Compr.1 2.80 0.43 54.03 92.48
Compr.2 3.00 0.40 54.00 92.14
Compr.3 2.34 0.39 54.33 92.30
Compr.4 4.91 0.35 54.35 92.27
Compr.5 3.90 0.38 53.74 92.27
Compr.6 3.38 0.41 53.92 92.12
9.4.3.2 Example 2
A more complicated optimization example of papermaking targets was studied
next. In this example, there were four conflicting process and quality
targets: tensile strength ratio, formation, basis weight and dry solids content.
Basis weight describes the mass of the paper per square meter, and dry solids
content is measured from finished paper. All four papermaking targets were
given the desired values and the deviation from these desired values were
to be minimized. Table 9.2 presents the desired values of the optimization
targets.
We used a multi-objective optimization method combined with virtual
papermaking line to search the optimal solution, i.e., those control variable
values that correspond to the optimal process and quality target values. Six
Pareto-optimal compromise solutions were calculated and they are shown in
Table 9.2. When comparing the solutions, it was apparent that the chosen
targets were really conflicting, and hence all the targets could not reach the
optimum simultaneously. For example, in Compromise 2, we obtained the
best values for tensile strength ratio and basis weight (in bold face in Table
9.2), but, at the same time, formation and dry solids content were not so
good. In addition, we can see that formation attained its best value in Compromise
4 and dry solids content was the best in Compromise 6, but, in these
solutions, tensile strength ratio and basis weight were unfavorable. Nevertheless,
all the solutions found were mathematically equally good Pareto optimal
solutions, and the most satisfying final solution could be selected from these
solutions using the papermaking expert’s knowledge.
9.5 Towards Decision Support Systems
The virtual papermaking line integrates mathematical modeling with multiobjective
optimization, and thus provides a basis for the development of an