Research Project: Çok Amaçlı Eniyileme Problemleri İçin Temsili Çözüm Yöntemleri ve Veri Sınıflandırma
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Contributors
Funders
ID
TB.00109
Authors
Sayın, Serpil
Faculty Member
Publications
Generating representative sets for multiobjective discrete optimization problems with specified coverage errors
(Springer, 2024) Sayın, Serpil; Kirlik, Gokhan; Department of Business Administration; Yes; College of Administrative Sciences and Economics
We present a new approach to generate representations with a coverage error quality guarantee for multiobjective discrete optimization problems with any number of objectives. Our method is based on an earlier exact algorithm that finds the entire nondominated set using epsilon\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}-constraint scalarizations. The representation adaptation requires the search to be conducted over a p-dimensional parameter space instead of the (p-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p-1)$$\end{document}-dimensional one of the exact version. The algorithm uses rectangles as search elements and for each rectangle, two-stage mathematical programs are solved to obtain efficient solutions. The representation algorithm implements a modified search procedure and is designed to eliminate a rectangle if it can be verified that it is not of interest given a particular coverage error requirement. Since computing the coverage error is a computationally demanding task, we propose a method to compute an upper bound on this quantity in polynomial time. The algorithm is tested on multiobjective knapsack and assignment problem instances with different error tolerance levels. We observe that our representation algorithm provides significant savings in computational effort even with relatively low levels of coverage error tolerance values for problems with three objective functions. Moreover, computational effort decreases almost linearly when coverage error tolerance increases. This makes it possible to obtain good quality representations for larger problem instances. An analysis of anytime performance on two selected problem instances demonstrates that the algorithm puts together a diverse representation starting from the early iterations.
Computing the nadir point for multiobjective discrete optimization problems
(Kluwer Academic Publishers, 2015) Kirlik, Gökhan; Sayın, Serpil; N/A; Department of Business Administration; Yes; College of Administrative Sciences and Economics
We investigate the problem of finding the nadir point for multiobjective discrete optimization problems (MODO). The nadir point is constructed from the worst objective values over the efficient set of a multiobjective optimization problem. We present a new algorithm to compute nadir values for MODO with objective functions. The proposed algorithm is based on an exhaustive search of the -dimensional space for each component of the nadir point. We compare our algorithm with two earlier studies from the literature. We give numerical results for all algorithms on multiobjective knapsack, assignment and integer linear programming problems. Our algorithm is able to obtain the nadir point for relatively large problem instances with up to five-objectives.
Bilevel programming for generating discrete representations in multiobjective optimization
(Springer, 2018) Sayın, Serpil; Kirlik, Gökhan; Department of Business Administration; Yes; College of Administrative Sciences and Economics
The solution to a multiobjective optimization problem consists of the nondominated set that portrays all relevant trade-off information. The ultimate goal is to identify a Decision Maker's most preferred solution without generating the entire set of nondominated solutions. We propose a bilevel programming formulation that can be used to this end. The bilevel program is capable of delivering an efficient solution that maps into a given set, provided that one exits. If the Decision Maker's preferences are known a priori, they can be used to specify the given set. Alternatively, we propose a method to obtain a representation of the nondominated set when the Decision Maker's preferences are not available. This requires a thorough search of the outcome space. The search can be facilitated by a partitioning scheme similar to the ones used in global optimization. Since the bilevel programming formulation either finds a nondominated solution in a given partition element or determines that there is none, a representation with a specified coverage error level can be found in a finite number of iterations. While building a discrete representation, the algorithm also generates an approximation of the nondominated set within the specified error factor. We illustrate the algorithm on the multiobjective linear programming problem.
