Abstract
In this paper, two new sandwich algorithms for the convex curve approximation are introduced. The proofs of the linear convergence property of the first method and the quadratic convergence property of the second method are given. The methods are applied to approximate the efficient frontier of the stochastic minimum cost flow problem with the moment bicriterion. Two numerical examples including the comparison of the proposed algorithms with two other literature derivative free methods are given.
Highlights
The network cost flow problems which describe a lot of real-life problems have been studied recently in many Operation Research papers
There exist exact computation methods for finding the analytic solution sets of bicriteria linear and quadratic cost flow problems, Ruhe [3] and Zadeh [4] have shown that the determination of these sets may be very perplexing, because there exists the possibility of the exponential number of extreme nondominated objecttive vectors on the efficient frontier of the considered problems
We need only three given points on the efficient frontier to start the first method called the Simple Triangle Method (STM) and two points to start the second method called the Trapezium Method (TM), the described methodologies work for any number r of initial points, which may be obtained by solving scalarization problems corresponding to problem (3), i.e
Summary
The network cost flow problems which describe a lot of real-life problems have been studied recently in many Operation Research papers. The fact that efficient frontiers of bicriteria linear and quadratic cost flow problems are the convex curves in R2 allows to apply the sandwich methods for a convex curve approximation in this field of optimization A derivative free method was introduced first by Yang and Goh in [8], who applied it to bicriteria quadratic minimum cost flow problems. The efficient frontiers of these problems are approximated by two piecewise linear functions called further approximation bounds, which construction requires solving of a number of one dimensional minimum cost flow problems. Proofs of lemmas and Theorem 2 are given in Appendix
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
