Discrete Optimization On accuracy, robustness and tolerances in vector Boolean optimization Y. Nikulin ⇑ , O. Karelkina, M.M. Mäkelä University of Turku, Department of Mathematics and Statistics, FI-20014 Turku, Finland article info Article history: Received 24 June 2011 Accepted 11 September 2012 Available online 18 September 2012 Keywords: Pareto optimum Robust measure Worst-case relative regret Accuracy function Tolerances abstract A Boolean programming problem with a finite number of alternatives where initial coefficients (costs) of linear payoff functions are subject to perturbations is considered. We define robust solution as a feasible solution which for a given set of realizations of uncertain parameters guarantees the minimum value of the worst-case relative regret among all feasible solutions. For the Pareto optimality principle, an appro- priate definition of the worst-case relative regret is specified. It is shown that this definition is closely related to the concept of accuracy function being recently intensively studied in the literature. We also present the concept of robustness tolerances of a single cost vector. The tolerance is defined as the max- imum level of perturbation of the cost vector which does not destroy the solution robustness. We present formulae allowing the calculation of the robustness tolerance obtained for some initial costs. The results are illustrated with several numerical examples. Ó 2012 Elsevier B.V. All rights reserved. 1. Introduction While solving practical optimization problems, it is necessary to take into account various kinds of uncertainty due to lack of input data, inadequacy of mathematical models to real processes, round- ing off, calculating errors, etc. It is known that in many cases initial data as a link between a reality and a model cannot be defined explicitly. The initial data is defined with a certain error, generally depend on many parameters and require to be specified during the problem solving process. In practice any problem cannot be prop- erly posed and solved without at least implicit use of the results of stability analysis and related issues of parametric analysis. There- fore widespread use of discrete optimization models in the last decades inspired many specialists to investigate various aspects of ill-posed problems theory and, in particular, the stability issues. The implications of enhanced optimization methods have in some areas been lead to the situation that optimal or near-optimal solutions have become ‘‘too good’’. For example, in design of a communication network, a network configuration can now be made so good (with respect to the original objective optimization) that there is hardly any possibility left in the network to accommo- date for potential disruptions and possible contingency in terms of e.g. routing delays. Similar problems are faced nowadays in many other areas where deterministic models do not properly reflect possible uncertainty of input parameters. In practice, it usually leads to undesirable situations where optimality (sometimes even feasibility) of solutions is very sensitive to some possible realiza- tions of problem parameters. Thus, chasing for solution optimality, we lose its robustness and vice versa. As a consequence, two lines of research within the operations research and mathematical optimization community have been initiated:  Post-optimal and parametric analysis investigate how an optimal solution found behave in response to initial data (problem parameters) changes. A general sensitivity and sta- bility analysis methodology is used based on analyzing the properties of the point-to-set mapping which specifies the optimality principle of the problem. Such research methods have been studied in great detail and covered e.g. in the liter- ature on optimization problems with a continuous set of fea- sible solutions. Numerous articles are devoted to analysis of conditions when a problem solution possesses a certain prop- erty of invariance to the problem parameters perturbations (see, e.g. [10,31,37,38]).  Robust optimization – instead of producing an optimal solu- tion for a normal situation, which is described by determinis- tic models but rarely occurs in practice, and where recovery to optimality can be complicated, the aim is to produce solu- tions that optimize an additionally constructed objective. The objective must assure that the optimal solution will remain feasible under worst case realization of uncertain problem input parameters. Worst-case optimization is also known as robust optimization, and optimal solutions of worst case opti- mization are often referred to as robust solutions (see e.g. [15]). 0377-2217/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ejor.2012.09.018 ⇑ Corresponding author. E-mail addresses: yurnik@utu.fi (Y. Nikulin), volkar@utu.fi (O. Karelkina), makela@utu.fi (M.M. Mäkelä). European Journal of Operational Research 224 (2013) 449–457 Contents lists available at SciVerse ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor