No / The marginal cost approach for the analysis of repair/replacement models was introduced by Berg in 1980 and has since been applied to many maintenance policies of various complexity. All models hitherto analysed in the literature by the marginal cost approach have one single decision variable only, this being, typically, the age of the current item at the time of ordering or replacement. This paper is concerned with the extension of the marginal cost technique to maintenance policies with several decision variables. After addressing the general framework appropriate for the multi-parameter case, we exemplify the workings of the technique by analysing a two-variable maintenance model involving replacement and minimal repair. We demonstrate that the marginal cost approach is an attractive and intuitively appealing technique also for models with several decision variables. Just as in the single-parameter situation, the approach is amenable to economic interpretation, a welcome feature for users of maintenance models with a prime interest in its economic (rather than its mathematical) aspects. As an added bonus of the marginal cost approach, in our example, some otherwise necessary tools from the theory of stochastic processes are dispensable.

Green manure crops are critical to maintaining soil fertility in organic cropping systems. However, little research has been conducted to address their contribution to weed control. Indianhead black lentil (Lens culinaris Medikus), AC Green Fix chickling vetch (Lathyrus sativus L.), and Trapper field pea (Pisum sativum L.) are legumes developed for use as annual green manure crops in the Northern Great Plains. Currently, no plant population density recommendations exist for these three species when grown as green manure crops under weedy conditions. The objective of this research was to determine the yield-density response of these three species under weedy and weedy-free conditions and to develop plant population density recommendations for use as annual green manure crops. Each species was grown at five plant population densities (10, 24, 64, 160, and 400 plants m-2) with weedy and weed-free treatments. Wild oat (Avena fatua L.) and wild mustard (Brassica kaber (D.C.) L.C. Wheeler) were planted in weedy treatments to supplement the natural weed community. Biomass samples and soil moisture measurements were taken at early bud and full bloom to simulate when these crops would be terminated. Biomass samples from the early bud stage were analysed for total nitrogen content. Green manure biomass production for all species was lower under weedy conditions. Weed biomass in weedy treatments decreased with increasing green manure plant population density for all species. Trapper field pea was the most competitive crop while Indianhead black lentil was the least competitive. Although total plot biomass differed among species and green manure crop density, changes in soil moisture levels were not greatly affected. No significant difference in total nitrogen concentration was found among green manure species. Differences in total nitrogen accumulation occurred because of differences in biomass production. Marginal cost analysis based on green manure seed costs and their nitrogen contribution to the value of subsequent wheat crop yield were used to determine optimum plant population densities. Under weedy conditions field pea and black lentil should be planted at densities of 49-78 and 223-300 plants m-2, respectively. Under weed-free conditions plant population densities for field pea and black lentil could be reduced to 45-60 and 184-223 plants m-2, respectively. No profitable plant population density was determined for chickling vetch when assuming a lower nitrogen contribution under both weedy and weed-free conditions. However, when assuming a higher nitrogen contribution, a profitable plant population density for chickling vetch of 24 plants m-2 was determined under weedy conditions and 32 plants m-2 under weed-free conditions.

Green manure crops are critical to maintaining soil fertility in organic cropping systems. However, little research has been conducted to address their contribution to weed control. Indianhead black lentil (Lens culinaris Medikus), AC Green Fix chickling vetch (Lathyrus sativus L.), and Trapper field pea (Pisum sativum L.) are legumes developed for use as annual green manure crops in the Northern Great Plains. Currently, no plant population density recommendations exist for these three species when grown as green manure crops under weedy conditions. The objective of this research was to determine the yield-density response of these three species under weedy and weedy-free conditions and to develop plant population density recommendations for use as annual green manure crops. Each species was grown at five plant population densities (10, 24, 64, 160, and 400 plants m-2) with weedy and weed-free treatments. Wild oat (Avena fatua L.) and wild mustard (Brassica kaber (D.C.) L.C. Wheeler) were planted in weedy treatments to supplement the natural weed community. Biomass samples and soil moisture measurements were taken at early bud and full bloom to simulate when these crops would be terminated. Biomass samples from the early bud stage were analysed for total nitrogen content. Green manure biomass production for all species was lower under weedy conditions. Weed biomass in weedy treatments decreased with increasing green manure plant population density for all species. Trapper field pea was the most competitive crop while Indianhead black lentil was the least competitive. Although total plot biomass differed among species and green manure crop density, changes in soil moisture levels were not greatly affected. No significant difference in total nitrogen concentration was found among green manure species. Differences in total nitrogen accumulation occurred because of differences in biomass production. Marginal cost analysis based on green manure seed costs and their nitrogen contribution to the value of subsequent wheat crop yield were used to determine optimum plant population densities. Under weedy conditions field pea and black lentil should be planted at densities of 49-78 and 223-300 plants m-2, respectively. Under weed-free conditions plant population densities for field pea and black lentil could be reduced to 45-60 and 184-223 plants m-2, respectively. No profitable plant population density was determined for chickling vetch when assuming a lower nitrogen contribution under both weedy and weed-free conditions. However, when assuming a higher nitrogen contribution, a profitable plant population density for chickling vetch of 24 plants m-2 was determined under weedy conditions and 32 plants m-2 under weed-free conditions.

This dissertation addresses five fundamental resource allocation problems on networks, all of which have applications to support Homeland Security or industry challenges. In the first application, we model and solve the strategic problem of minimizing the expected loss inflicted by a hostile terrorist organization. An appropriate allocation of certain capability-related, intent-related, vulnerability-related, and consequence-related resources is used to reduce the probabilities of success in the respective attack-related actions, and to ameliorate losses in case of a successful attack. Given the disparate nature of prioritizing capital and material investments by federal, state, local, and private agencies to combat terrorism, our model and accompanying solution procedure represent an innovative, comprehensive, and quantitative approach to coordinate resource allocations from various agencies across the breadth of domains that deal with preventing attacks and mitigating their consequences. Adopting a nested event tree optimization framework, we present a novel formulation for the problem as a specially structured nonconvex factorable program, and develop two branch-and-bound schemes based respectively on utilizing a convex nonlinear relaxation and a linear outer-approximation, both of which are proven to converge to a global optimal solution. We also investigate a fundamental special-case variant for each of these schemes, and design an alternative direct mixed-integer programming model representation for this scenario. Several range reduction, partitioning, and branching strategies are proposed, and extensive computational results are presented to study the efficacy of different compositions of these algorithmic ingredients, including comparisons with the commercial software BARON. The developed set of algorithmic implementation strategies and enhancements are shown to outperform BARON over a set of simulated test instances, where the best proposed methodology produces an average optimality gap of 0.35% (compared to 4.29% for BARON) and reduces the required computational effort by a factor of 33. A sensitivity analysis is also conducted to explore the effect of certain key model parameters, whereupon we demonstrate that the prescribed algorithm can attain significantly tighter optimality gaps with only a near-linear corresponding increase in computational effort. In addition to enabling effective comprehensive resource allocations, this research permits coordinating agencies to conduct quantitative what-if studies on the impact of alternative resourcing priorities. The second application is motivated by the author's experience with the U.S. Army during a tour in Iraq, during which combined operations involving U.S. Army, Iraqi Army, and Iraqi Police forces sought to interdict the transport of selected materials used for the manufacture of specialized types of Improvised Explosive Devices, as well as to interdict the distribution of assembled devices to operatives in the field. In this application, we model and solve the problem of minimizing the maximum flow through a network from a given source node to a terminus node, integrating different forms of superadditive synergy with respect to the effect of resources applied to the arcs in the network. Herein, the superadditive synergy reflects the additional effectiveness of forces conducting combined operations, vis-à-vis unilateral efforts. We examine linear, concave, and general nonconcave superadditive synergistic relationships between resources, and accordingly develop and test effective solution procedures for the underlying nonlinear programs. For the linear case, we formulate an alternative model representation via Fourier-Motzkin elimination that reduces average computational effort by over 40% on a set of randomly generated test instances. This test is followed by extensive analyses of instance parameters to determine their effect on the levels of synergy attained using different specified metrics. For the case of concave synergy relationships, which yields a convex program, we design an inner-linearization procedure that attains solutions on average within 3% of optimality with a reduction in computational effort by a factor of 18 in comparison with the commercial codes SBB and BARON for small- and medium-sized problems; and outperforms these softwares on large-sized problems, where both solvers failed to attain an optimal solution (and often failed to detect a feasible solution) within 1800 CPU seconds. Examining a general nonlinear synergy relationship, we develop solution methods based on outer-linearizations, inner-linearizations, and mixed-integer approximations, and compare these against the commercial software BARON. Considering increased granularities for the outer-linearization and mixed-integer approximations, as well as different implementation variants for both these approaches, we conduct extensive computational experiments to reveal that, whereas both these techniques perform comparably with respect to BARON on small-sized problems, they significantly improve upon the performance for medium- and large-sized problems. Our superlative procedure reduces the computational effort by a factor of 461 for the subset of test problems for which the commercial global optimization software BARON could identify a feasible solution, while also achieving solutions of objective value 0.20% better than BARON. The third application is likewise motivated by the author's military experience in Iraq, both from several instances involving coalition forces attempting to interdict the transport of a kidnapping victim by a sectarian militia as well as, from the opposite perspective, instances involving coalition forces transporting detainees between interment facilities. For this application, we examine the network interdiction problem of minimizing the maximum probability of evasion by an entity traversing a network from a given source to a designated terminus, while incorporating novel forms of superadditive synergy between resources applied to arcs in the network. Our formulations examine either linear or concave (nonlinear) synergy relationships. Conformant with military strategies that frequently involve a combination of overt and covert operations to achieve an operational objective, we also propose an alternative model for sequential overt and covert deployment of subsets of interdiction resources, and conduct theoretical as well as empirical comparative analyses between models for purely overt (with or without synergy) and composite overt-covert strategies to provide insights into absolute and relative threshold criteria for recommended resource utilization. In contrast to existing static models, in a fourth application, we present a novel dynamic network interdiction model that improves realism by accounting for interactions between an interdictor deploying resources on arcs in a digraph and an evader traversing the network from a designated source to a known terminus, wherein the agents may modify strategies in selected subsequent periods according to respective decision and implementation cycles. We further enhance the realism of our model by considering a multi-component objective function, wherein the interdictor seeks to minimize the maximum value of a regret function that consists of the evader's net flow from the source to the terminus; the interdictor's procurement, deployment, and redeployment costs; and penalties incurred by the evader for misperceptions as to the interdicted state of the network. For the resulting minimax model, we use duality to develop a reformulation that facilitates a direct solution procedure using the commercial software BARON, and examine certain related stability and convergence issues. We demonstrate cases for convergence to a stable equilibrium of strategies for problem structures having a unique solution to minimize the maximum evader flow, as well as convergence to a region of bounded oscillation for structures yielding alternative interdictor strategies that minimize the maximum evader flow. We also provide insights into the computational performance of BARON for these two problem structures, yielding useful guidelines for other research involving similar non-convex optimization problems. For the fifth application, we examine the problem of apportioning railcars to car manufacturers and railroads participating in a pooling agreement for shipping automobiles, given a dynamically determined total fleet size. This study is motivated by the existence of such a consortium of automobile manufacturers and railroads, for which the collaborative fleet sizing and efforts to equitably allocate railcars amongst the participants are currently orchestrated by the \textit{TTX Company} in Chicago, Illinois. In our study, we first demonstrate potential inequities in the industry standard resulting either from failing to address disconnected transportation network components separately, or from utilizing the current manufacturer allocation technique that is based on average nodal empty transit time estimates. We next propose and illustrate four alternative schemes to apportion railcars to manufacturers, respectively based on total transit time that accounts for queuing; two marginal cost-induced methods; and a Shapley value approach. We also provide a game-theoretic insight into the existing procedure for apportioning railcars to railroads, and develop an alternative railroad allocation scheme based on capital plus operating costs. Extensive computational results are presented for the ten combinations of current and proposed allocation techniques for automobile manufacturers and railroads, using realistic instances derived from representative data of the current business environment. We conclude with recommendations for adopting an appropriate apportionment methodology for implementation by the industry. / Ph. D.