Robust Combinatorial Optimization with Locally Budgeted Uncertainty

Budgeted uncertainty sets have been established as a major influence on uncertainty modeling for robust optimization problems. A drawback of such sets is that the budget constraint only restricts the global amount of cost increase that can be distributed by an adversary. Local restrictions, while being important for many applications, cannot be modeled this way. We introduce new variant of budgeted uncertainty sets, called locally budgeted uncertainty. In this setting, the uncertain parameters become partitioned, such that a classic budgeted uncertainty set applies to each partition, called region. In a theoretical analysis, we show that the robust counterpart of such problems for a constant number of regions remains solvable in polynomial time, if the underlying nominal problem can be solved in polynomial time as well. If the number of regions is unbounded, we show that the robust selection problem remains solvable in polynomial time, while also providing hardness results for other combinatorial problems. In computational experiments using both random and real-world data, we show that using locally budgeted uncertainty sets can have considerable advantages over classic budgeted uncertainty sets

Investigating the Recoverable Robust Single Machine Scheduling Problem Under Interval Uncertainty

We investigate the recoverable robust single machine scheduling problem under interval uncertainty. In this setting, jobs have first-stage processing times p and second-stage processing times q and we aim to find a first-stage and second-stage schedule with a minimum combined sum of completion times, such that at least \Delta jobs share the same position in both schedules. We provide positive complexity results for some important special cases of this problem, as well as derive a 2-approximation algorithm to the full problem. Computational experiments examine the performance of an exact mixed-integer programming formulation and the approximation algorithm, and demonstrate the strength of a proposed polynomial time greedy heuristic

Robust Assignments via Ear Decompositions and Randomized Rounding

Many real-life planning problems require making a priori decisions before all parameters of the problem have been revealed. An important special case of such problem arises in scheduling problems, where a set of tasks needs to be assigned to the available set of machines or personnel (resources), in a way that all tasks have assigned resources, and no two tasks share the same resource. In its nominal form, the resulting computational problem becomes the \emph{assignment problem} on general bipartite graphs. This paper deals with a robust variant of the assignment problem modeling situations where certain edges in the corresponding graph are \emph{vulnerable} and may become unavailable after a solution has been chosen. The goal is to choose a minimum-cost collection of edges such that if any vulnerable edge becomes unavailable, the remaining part of the solution contains an assignment of all tasks. We present approximation results and hardness proofs for this type of problems, and establish several connections to well-known concepts from matching theory, robust optimization and LP-based techniques.Comment: Full version of ICALP 2016 pape

Accuracy of Time Phasing Aircraft Development using the Continuous Distribution Function

Early research on time phasing primarily focuses on the theoretical foundation for applying the continuous distribution function, or S-curve, to model the distribution of development expenditures. Minimal methodology is provided for estimating the S-curve\u27s parameter values. Brown, White, and Gallagher (2002) resolve this shortcoming through regression analysis, but their methodology has not been widely adopted by aircraft cost analysts, as it is judged as overly broad and not specific to aircraft. Instead, analysts commonly apply the 60/40 rule of thumb to aircraft development, assuming 60 percent expenditures at 50 percent schedule. It is currently unknown if the 60/40 heuristic accurately describes contemporary aircraft development programs. Therefore, using a sample of 26 DoD aircraft programs, we first test the accuracy of 60/40, discovering that, as a heuristic, the 60/40 cannot account for differences between new start and upgrade programs. Next, we improve upon prior research by using program characteristics to construct an aircraft-specific methodology for estimating parameters. Finally, we conclude our research by comparing the accuracy of our Rayleigh, Weibull, and Beta S-curve models. Our Weibull model explains 74.6 percent of total variation in annual budget, improving the estimation of budgets by 6.5 percent, on average, over the baseline 60/40 model