## Dissertations

Dissertation

Spring 1988

#### Degree Name

Doctor of Engineering Science in Mechanical Engineering

#### Department

Mechanical Engineering

Michael Pappas

Harry Herman

Bernard Koplik

R. S. Sodhi

Rajesh N. Dave

#### Abstract

The Feasible Direction Finding Problem (DFP) of Zoutendijk is adapted to create a general Mathematical Programming (MP) algorithm for treating optimization problems with multiple objective functions. Classically such problems are reduced to standard MP form by converting them to single objective function problems by the use of weighting functions. Unfortunately not all practical problems can be so reduced. Consider the problem of maximizing the strength of a structure. Typically there are several, or even many, failure modes. All active failure modes must be included in the optimal search in such a way that resistance to one active mode can not be increased at the expense of another. Thus this problem can not be treated by reduction. The search must seek to increase resistance to all active modes.

The DFP formulation seeks to improve the objective function by including said function as a constraint in the DFP Linear Programming problem. Multiple objective functions can be treated by simply including each such function as a constraint in the DFP. Thus the solution to such a DFP improves all the objective functions considered. There is no need to resort to reduction to a single objective function. An algorithm based on the DFP is described. This procedure locates a variable set where, at least locally, no further improvement in all objective functions is available (a Parato Optimum). A general multiple objective formulation is developed defining a wide range of optimization problems. It is shown that this formulation also includes the problem of locating the feasible region, either from an infeasible starting point, or for feasibility restoration during the search. Thus the method is of value in single objective function optimization.

The procedure is applied to a six variable problem with eleven constraints where the objective is to separate the two lowest natural frequencies of a stiffened thin shell. Four active frequencies are considered. Several two-variable, constrained and unconstrained, problems are also treated. The procedure was found to efficiently locate Parato Optima and was effective in feasible region location and restoration.

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