LION COMMUNITY USAGE CASE

Welded beam design (structural steel engineering): how to tradeoff multiple objectives.

NOTE: to reproduce the example you must first install a free version of Scilab on your computer (available from http://www.scilab.org/), and save the following Scilab file weldedBeam.sci in your local files, so that it can then be loaded into the LIONoso workbench.

Welded beam: problem definition

The problem is a simplified example of many complex desing issues arising in structural engineering, dealing with designing the form of steel beams and with connecting them to form complex stucture like bridges, buildings, etc. It has been used by many experts as a benchmark problem of multi-objective optimization.

The problem of designing an optimal welded beam consists of dimensioning a welded steel beam and the welding length so as to minimize its cost subject to constraints on shear stress, bending stress in the beam, the buckling load on the bar, the end the deflection of the beam, and side constraints. There are four design variables: which are shown with the letters (h,l,t,b) in the figure.

Structural analysis of this beam leads to two nonlinear objective functions subject to five nonlinear and two linear inequality constraints. The objectives are: the fabrication cost and the end deflection of the beam.

Because the (conflicting) objectives are two, one is dealing with a Multiple-Objectives Optimization Problem. There is not a single optimal solution, but more Pareto-optimal solutions. A solution is Pareto-optimal if none of its components can be improved without worsening at least one of the others. In our case, if one cannot reduce fabrication cost without causing a higher deflection. Deciding the preferred solution among the Pareto-optimal set requires the intelligent participation of the designer, to decide about the proper trade-off between cost and quality (deflection). LIONoso helps the designer to become aware of the different posibilities and focus on his preferred solutions.

Setting up the LIONoso tools:

The constraints are transformed into a penalty function which sums the absolute values of the violations of the constraints plus a large constant (estimated to be larger than the maximum value achieved by the functions in the input space). In this case, the order of magnitude of the two functions is very different: the end deflection ranges approximately from 0.001 to 0.01, while the cost ranges approximately from 2.0 to 40.0, as the following figure shows.

Unless the two functions are scaled, the effect of deflection in the weighted sum will tend to be negligible (too small to matter), and most Pareto-optimal points will be in the area corresponding to the lowest cost. We therefore divided each function by the estimated maximum value of each function in the input range. Note that only a very approximated estimate of the maximum value is sufficient.

LIONoso snapshots

It is interesting to consider all generated solutions and the Pareto-optimal ones. This can be obtained by inserting a filter and eliminating all rows with Pareto=0, as in the figure.

The dashboard configuration is as follows.

By associating a trellis plot to the results table, one observes that the welding length l and depth h are inversely proportional, the shorter the welding length, the larger the depth has to be, and that height t tends to be close to its maximum allowed value.

These observations can inspire a problem simplification, by fixing the height to its maximum value and by expressing the length as a function of depth, therefore eliminating two variables from consideration in the future focused explorations of this design issue.

References

The problem is listed in: Rekliatis G.V., Ravindrab A., and Ragsdell K.M. Engineering Optimisation Methods and Applications. 1983, New York: Wiley.
Image by Dr. Dervis Karaboga, published in scholarpedia.

NOTE: to reproduce the example you must first install a free version of Scilab on your computer (available from http://www.scilab.org/), and save the following Scilab file weldedBeam.sci in your local files, so that it can then be loaded into the LIONoso workbench.