A light and reliable aircraft has been the major goal of aircraft designers. It is imperative to design the aircraft wing skins as efficiently as possible since the wing skins comprise more than fifty percent of the structural weight of the aircraft wing. The aircraft wing skin consists of many different types of material and thickness configurations at various locations. Selecting a thickness for each location is perhaps the most significant design task. In this paper, we formulate discrete mathematical programming models to determine the optimal thicknesses for three different criteria: maximize reliability, minimize weight, and achieve a trade-off between maximizing reliability and minimizing weight. These three model formulations are generalized discrete resource-allocation problems, which lend themselves well to the dynamic programming approach. Consequently, we use the dynamic programming method to solve these model formulations. To illustrate our approach, an example is solved in which dynamic programming yields a minimum weight design as well as a trade-off curve for weight versus reliability for an aircraft wing with thirty locations (or panels) and fourteen thickness choices for each location.

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In this paper, a multiple stage wastewater treatment system (WTS) is solved for the selection of technological options at each stage to minimize (economic cost, size, odour emissions) and to maximize (nutrient recovery, robustness, global desirability). Stages in the wastewater treatment system are the levels of treatment. There are 17 levels of treatment, where the first 11 levels are for the liquid treatment and the last 6 levels are for the solid treatment. This results in a 20-dimensional, continuous-state, 17-stage, 6-objective, stochastic optimization problem. The resulting multiple stage, multiple objective (MSMO) WTS is solved using the three-phase methodology in conjunction with the multiple objective version of highdimensional, continuous-state, stochastic dynamic programming (SDP). The three-phase methodology comprises the input phase, the matrix generation phase and the weighting phase. The primary goal of three-phase methodology is to obtain weight vectors at each stage of the WTS utilizing expert's opinions in the input phase, computing pairwise comparison matrices at each stage using the geometric mean-based methods in the matrix generation phase, and then calculating weight vectors at each stage using the eigenvector method in the weighting phase. The weight vectors are then used to scalarize the vector optimization problem, which is solved using the high-dimensional, continuous-state SDP augmented for handling multiple objectives at each stage.The results obtained are practical as evidenced by the selection of new technologies in levels 1 and 5 thereby validating expert's decision to include them in the evaluation process. In addition to encouraging reviews from WTS experts, the implementation results satisfy a set of external constraints in the form of interstage dependencies between technological options in the WTS. Furthermore, the solution technique presented here utilizes expert's opinions in the solution development process, and is quite general in its application to a variety of large-scale MSMO problems.

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