On accuracy of solutions for discrete optimization problems with perturbed coefficientsof the objective function

M. Libura

Annals of Operations Research, 1999, vol. 86, issue 0, 53-62

Abstract: An approach to the sensitivity analysis for discrete optimization problems with perturbedobjective function is presented. The problem is stated in the following general form:minΣ e∈Y c(e) : Y ∈ F, where c=(c(e), e ∈ E) is a vector of weights for some finite set E,and F ⫅ 2 E is a given family of feasible subsets. It is assumed that the set of feasiblesolutions F is fixed, but the coefficients of the vector c may vary. The main problem consideredin this paper concerns the following question: How does the maximum relative errorof a given feasible solution depend on inaccuracies of the problem data? Two particularperturbations of the vector c are considered:(i) it is assumed that c is in a closed ball with radius δ≥0 and center c o ≥0 or(ii) the relative deviation of c(e) from the value c o (e) is not greater than δ for any e ∈ E, i.e.,|c(e) ‐ c o (e)| ≤δ c o (e), e ∈ E.For a given feasible solution X ∈ F , two functions of the parameter δ are introduced: thesensitivity function s(X, δ) and the accuracy function a(X, δ). The values of s(X, δ) anda(X, δ) are equal to the maximum relative error of the solution X when the perturbations ofproblem data are of the type (i) or (ii), respectively. Some general, but computationallyinefficient formulae for calculating these functions are derived and their properties arestudied. It is shown that s(X, δ) and a(X, δ) are nondecreasing, convex functions with alimited number of breakpoints. A practically efficient method of calculating upper and lowerenvelopes for the accuracy function is presented. This method is based on the notion of k‐bestsolutions of the problem. It gives an interval to which the maximum relative error ofthe solution X must belong when the coefficients of vector c are given with accuracy δ. Theapproach is illustrated with an example of the symmetric traveling salesman problem. Copyright Kluwer Academic Publishers 1999

Date: 1999
References: Add references at CitEc
Citations: View citations in EconPapers (3)

Downloads: (external link)
http://hdl.handle.net/10.1023/A:1018991826728 (text/html)
Access to full text is restricted to subscribers.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:annopr:v:86:y:1999:i:0:p:53-62:10.1023/a:1018991826728

Ordering information: This journal article can be ordered from
http://www.springer.com/journal/10479

DOI: 10.1023/A:1018991826728

Access Statistics for this article

Annals of Operations Research is currently edited by Endre Boros

More articles in Annals of Operations Research from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().