The D-optimal design problems in polynomial regression models with a one-dimensional control variable and k-dimensional response variable Y=(Y_1,...,Y_k) where there are some common unknown parameters are discussed. The approximate D-optimal designs are shown to be independent of the covariance structure between the k responses when the degrees of the k responses are of the same order. Then, the exact n-point D-optimal designs are also discussed. Krafft and Schaefer (1992) and Imhof (2000) are useful in obtaining our results. We extend the proof of symmetric cases for k>= 2.

The purpose of this study was to compare response rates and resistance to extinction in single-task and multiple-task phases. Research was conducted with thirty undergraduate college students in a controlled experimental setting. Each Participant was exposed to 4 treatment phases: single-task, fixed-ratio of one (ST-FR1), multiple-task fixed-ratio of one (MT-FR1), single-task fixed-ratio of 5 (ST-FR5) and multiple-task fixed-ratio of 5 (MT-FR5) all beginning with a baseline phase and reverting back to baseline after the first two conditions were presented. Half of the Participants received the single-task phase first, and the other half received the multiple-task phase first, in order to observe the behavior in transition. A trials-to-criterion measure was used to determine how long it took each Participant’s behavior to adapt to the new contingency in the next phase, which was presented without any signal. The data reveal that regardless of the order of phase presentation it took Participants more than twice as long to reach the criterion in the single-task phases, than in the multiple-task phases.

The focus of this dissertation is on improving decision-maker trade-offs and the development of a new constrained methodology for multiple response surface optimization. There are three key components of the research: development of the necessary conditions and assumptions associated with constrained multiple response surface optimization methodologies; development of a new constrained multiple response surface methodology; and demonstration of the new method. The necessary conditions for and assumptions associated with constrained multiple response surface optimization methods were identified and found to be less restrictive than requirements previously described in the literature. The conditions and assumptions required for a constrained method to find the most preferred non-dominated solution are to generate non-dominated solutions and to generate solutions consistent with decision-maker preferences among the response objectives. Additionally, if a Lagrangian constrained method is used, the preservation of convexity is required in order to be able to generate all non-dominated solutions. The conditions required for constrained methods are significantly fewer than those required for combined methods. Most of the existing constrained methodologies do not incorporate any provision for a decision-maker to explicitly determine the relative importance of the multiple objectives. Research into the larger area of multi-criteria decision-making identified the interactive surrogate worth trade-off algorithm as a potential methodology that would provide that capability in multiple response surface optimization problems. The ISWT algorithm uses an ε-constraint formulation to guarantee a non-dominated solution, and then interacts with the decision-maker after each iteration to determine the preference of the decision-maker in trading-off the value of the primary response for an increase in value of a secondary response. The current research modified the ISWT algorithm to develop a new constrained multiple response surface methodology that explicitly accounts for decision-maker preferences. The new Modified ISWT (MISWT) method maintains the essence of the original method while taking advantage of the specific properties of multiple response surface problems to simplify the application of the method. The MISWT is an accessible computer-based implementation of the ISWT. Five test problems from the multiple response surface optimization literature were used to demonstrate the new methodology. It was shown that this methodology can handle a variety of types and numbers of responses and independent variables. Furthermore, it was demonstrated that the methodology can be successful using a priori information from the decision-maker about bounds or targets or can use the extreme values obtained from the region of operability. In all cases, the methodology explicitly considered decision-maker preferences and provided non-dominated solutions. The contribution of this method is the removal of implicit assumptions and includes the decision-maker in explicit trade-offs among multiple objectives or responses.

Nitrogen-fixing bacteria in soybean nodules convert atmospheric nitrogen to plant-available forms in exchange for carbon from the plant, but other non-nitrogen-fixing bacteria also reside in nodules, and their role in the nodule is not well understood. This study was conducted to determine the effect of three non-nitrogen-fixing Pseudomonas spp. strains isolated from nodules on soybean, and we hypothesized these strains benefit soybean. A greenhouse study in which two cultivars of soybean (Asgrow AG46X6 and Pioneer P48A60X) were treated with three fluorescent Pseudomonas spp. strains (referred to in this study as Bullseye, Pancake, and Starfish) and an uninoculated control. Soybeans were harvested at two time points: the R2/R3 growth stage and the R6 growth stage. Following each harvest, measures of growth, yield, and nitrogen fixation were taken, and data were analyzed using two non-parametric, multivariate analyses: multiple response permutation procedure (MRPP) and permutational multivariate analysis of variance (PERMANOVA). Both analyses showed soybeans of both cultivars treated with Pancake differed from controls following the first harvest but not the second. When analyzed individually, most metrics for growth, yield, and nitrogen fixation following the first harvest were not significantly different between Pancake and control treatments, but Pancake treatment means were still generally higher than controls. If metrics are considered collectively in conjunction with the results of the multivariate analyses, the results show Pancake generally increased soybean growth and nitrogen fixation. These findings support the hypothesis that non-nitrogen-fixing bacteria from nodules benefit plants, and such bacteria have the potential to serve as biofertilizers. / Master of Science in Life Sciences / Soybeans are one of the most commonly grown crops in the world, and nitrogen-fixing bacteria colonize the roots of soybeans and initiate the formation of spherical nodules attached to the roots. Inside the nodules, these bacteria convert atmospheric nitrogen to plant-available forms in exchange for sugar from the plant, and such bacteria reduce the need to add nitrogen fertilizer to agricultural fields. Other non-nitrogen-fixing bacteria also reside in nodules, but their role in the nodule is not well understood. If these bacteria benefit soybeans, they have the potential to serve as biofertilizers (microbial inoculants that promote plant growth). This study was conducted to determine whether non-nitrogen-fixing bacteria isolated from nodules benefit soybean. A greenhouse study in which two cultivars of soybean (Asgrow AG46X6 and Pioneer P48A60X) were grown in soil and were either left uninoculated or were inoculated with one of three strains of bacteria from the genus, Pseudomonas (referred to in this study as Bullseye, Pancake, and Starfish). Following harvest, measures of growth, yield, and nitrogen fixation were taken, and data showed the bacteria generally benefited the soybean plants. Although, these results showed the bacteria benefitted the plants, field trials and further testing in the greenhouse should be conducted before using these bacteria as commercial biofertilizers. Additionally, the effects of other non-nitrogen-fixing nodule bacteria on soybeans should also be tested to identify other beneficial strains, and the cost of production should be compared to the potential gains of using such bacteria before they are developed into biofertilizers.