Multi-objective optimization algorithms can generate large sets of Pareto optimal (non-dominated) solutions. Identifying the best solutions across a very large number of Pareto optimal solutions can be a challenge. Therefore it is useful for the decision-maker to be able to obtain a small set of preferred Pareto optimal solutions. This paper analyzes a discrete optimization problem introduced to obtain optimal subsets of solutions from large sets of Pareto optimal solutions. This discrete optimization problem is proven to be NP-hard. Two exact algorithms and five heuristics are presented to address this problem. Five test problems are used to compare the performances of these algorithms and heuristics. The results suggest that preferred subset of Pareto optimal solutions can be efficiently obtained using the heuristics, while for smaller problems, exact algorithms can be applied.

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The globalization of today's supply chains (e.g., information and communication technologies, military systems, etc.) has created an emerging security threat that could degrade the integrity and availability of sensitive and critical government data, control systems, and infrastructures. Commercial-off-theshelf (COTS) and even government-off-the-self (GOTS) products often are designed, developed, and manufactured overseas. Counterfeit items, from individual chips to entire systems, have been found in commercial and government sectors. Supply chain attacks can be initiated at any point during the product or system lifecycle, and can have detrimental effects to mission success. To date, there is a lack of analytics and decision support tools used to analyze supply chain security holistically, and to perform tradeoff analyses to determine how to invest in or deploy possible mitigation options for supply chain security such that the return on investment is optimal with respect to cost, efficiency, and security. This paper discusses the development of a supply chain decision analytics framework that will assist decision makers and stakeholders in performing risk-based cost-benefit prioritization of security investments to manage supply chain risk. Key aspects of our framework include the hierarchical supply chain representation, vulnerability and mitigation modeling, risk assessment and optimization. This work is a part of a long term research effort on supply chain decision analytics for trusted systems and communications research challenge.

The paper proposes a new exact approach, based on a Branch, Bound, and Remember (BB&R) algorithm that uses the Cyclic Best First Search (CBFS) strategy, for the 1|r i | U i scheduling problem, a single machine scheduling problem, where the objective is to find a schedule with the minimum number of tardy jobs. The search space is reduced using new and improved dominance properties and tighter upper bounds, based on a new dynamic programming algorithm. Computational results establish the effectiveness of the BB&R algorithm with CBFS for a broad spectrum of problem instances. In particular, this algorithm was able to solve all problems instances, up to 300 jobs, while existing best known algorithms only solve problems instances up to 200 jobs. Furthermore, the BB&R algorithm with CBFS runs one to two orders of magnitude faster than the current best known algorithm on comparable instances.