Solution of stochastic multi-objective system design problems

A methodology is presented to solve multiple objective system design problems with some (or all) stochastic objectives. For these problems, the objective is to determine the maximum system reliability, but at a minimum cost and weight without explicit constraint limits. The reliability and cost objectives may not be deterministic due to estimation uncertainties and cost fluctuations, respectively. Objective function variance measures are explicitly included in the formulation as additional objectives to be minimized for risk-averse decision-makers. The described approach does not require explicit objective function numerical weights, but it does provide prioritized Pareto optimal solutions based on a non-numerical ranking of the importance of scaled objectives. The rankings are based on Monte-Carlo simulation and randomly selected and prioritized objective weights. A tabu search meta-heuristic approach is used to initially find Pareto optimal solutions to the multi-objective optimization problems. Multi-objective problems are often solved by modifying them into equivalent single objective problems using user-defined weights or utility functions. Then, the multi-objective problem is solved similar to single objective problems. Complicating the problem, many real life situations have uncertainty in their objectives or constraints. Unlike deterministic problems, in which all parameters are fixed, stochastic problems have parameters that either varies within a certain range or has a probability distribution. Hence, when both factors are combined, the resulting problem is very difficult. A new method is being developed to tackle the issue of multiple objectives with uncertainty. This method has been tested on the redundancy allocation problem with very encouraging results. The purpose of this study is to create a bridge between Pareto optimality and single solution approaches. Methods like utility theory, and weighted sum method return a single solution by combining all the objective functions into a single objective. These methods prove to be problematic because assigning appropriate numerical values to an objective function can be challenging for many practitioners. Alternatively, methods like genetic algorithms and tabu search yield a list of non-dominated Pareto optimal solutions. In such cases, it can be difficult to select the single best compromise solution out of hundreds (or thousands) of solutions. The new method is a logical evolution, and tries to combine both approaches by generating a list of prioritized solutions from among the Pareto optimal solutions. The first step is to obtain a Pareto optimal set using tabu search. The objective functions are then normalized and prioritized (ranked) according to the decision makers needs. Random objective function weight assignments are then repetitively selected from a joint probability density function that captures the previously determined sequence of ranked objectives. The objective functions are then multiplied by 'ordered random weights' and combined. In every iteration, the multi-objective function is transformed into a single objective function, and the best solution obtained is observed and recorded. This procedure is repeated many times, and the end result is a group of solutions that appear as the "best" solution most often. These solutions are further put into priority groups as an aid to the decision maker. This approach was tested on the redundancy allocation problem, where the most important objective is to maximize reliability of a series-parallel system through redundancy allocation, but there are also objectives to minimize cost and weight. Furthermore, there is a strong desire to select a solution with low reliability estimation variability. The method indicates that the Pareto optimal set of over 1,000 solutions can be filtered to recommend the strongest candidates based on the prioritized objective function ranks.