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Confidence in heuristic solutions?
Dalarna University, School of Technology and Business Studies, Statistics.ORCID iD: 0000-0003-2317-9157
Dalarna University, School of Technology and Business Studies, Statistics.ORCID iD: 0000-0003-2970-9622
2015 (English)In: Journal of Global Optimization, ISSN 0925-5001, E-ISSN 1573-2916, Vol. 63, no 2, 381-399 p.Article in journal (Refereed) Published
Abstract [en]

Solutions to combinatorial optimization problems frequently rely on heuristics to minimize an objective function. The optimum is sought iteratively and pre-setting the number of iterations dominates in operations research applications, which implies that the quality of the solution cannot be ascertained. Deterministic bounds offer a mean of ascertaining the quality, but such bounds are available for only a limited number of heuristics and the length of the interval may be difficult to control in an application. A small, almost dormant, branch of the literature suggests using statistical principles to derive statistical bounds for the optimum. We discuss alternative approaches to derive statistical bounds. We also assess their performance by testing them on 40 test p-median problems on facility location, taken from Beasley’s OR-library, for which the optimum is known. We consider three popular heuristics for solving such location problems; simulated annealing, vertex substitution, and Lagrangian relaxation where only the last offers deterministic bounds. Moreover, we illustrate statistical bounds in the location of 71 regional delivery points of the Swedish Post. We find statistical bounds reliable and much more efficient than deterministic bounds provided that the heuristic solutions are sampled close to the optimum. Statistical bounds are also found computationally affordable.

Place, publisher, year, edition, pages
2015. Vol. 63, no 2, 381-399 p.
Keyword [en]
p-median problem, deterministic bounds, statistical bounds, jackknife, discrete optimization, extreme value theory
National Category
Probability Theory and Statistics Computational Mathematics
Research subject
Komplexa system - mikrodataanalys, General Microdata Analysis - methods
Identifiers
URN: urn:nbn:se:du-17153DOI: 10.1007/s10898-015-0293-4ISI: 000361485700009OAI: oai:DiVA.org:du-17153DiVA: diva2:796036
Available from: 2015-03-17 Created: 2015-03-17 Last updated: 2015-10-12Bibliographically approved
In thesis
1. Optimization heuristic solutions, how good can they be?: With empirical applications in location problems
Open this publication in new window or tab >>Optimization heuristic solutions, how good can they be?: With empirical applications in location problems
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Combinatorial optimization problems, are one of the most important types of problems in operational research. Heuristic and metaheuristics algorithms are widely applied to find a good solution. However, a common problem is that these algorithms do not guarantee that the solution will coincide with the optimum and, hence, many solutions to real world OR-problems are afflicted with an uncertainty about the quality of the solution. The main aim of this thesis is to investigate the usability of statistical bounds to evaluate the quality of heuristic solutions applied to large combinatorial problems. The contributions of this thesis are both methodological and empirical. From a methodological point of view, the usefulness of statistical bounds on p-median problems is thoroughly investigated. The statistical bounds have good performance in providing informative quality assessment under appropriate parameter settings. Also, they outperform the commonly used Lagrangian bounds. It is demonstrated that the statistical bounds are shown to be comparable with the deterministic bounds in quadratic assignment problems. As to empirical research, environment pollution has become a worldwide problem, and transportation can cause a great amount of pollution. A new method for calculating and comparing the CO2-emissions of online and brick-and-mortar retailing is proposed. It leads to the conclusion that online retailing has significantly lesser CO2-emissions. Another problem is that the Swedish regional division is under revision and the border effect to public service accessibility is concerned of both residents and politicians. After analysis, it is shown that borders hinder the optimal location of public services and consequently the highest achievable economic and social utility may not be attained.

Abstract [sv]

Kombinatoriska optimeringsproblem, är en av de viktigaste typerna av problem i operationsanalys (OR). Heuristiska och metaheuristiska algoritmer tillämpas allmänt för att hitta lösningar med hög kvalitet. Ett vanligt problem är dock att dessa algoritmer inte garanterar optimala lösningar och sålunda kan det finnas osäkerhet i kvaliteten på lösningar på tillämpade operationsanalytiska problem. Huvudsyftet med denna avhandling är att undersöka användbarheten av statistiska konfidensintervall för att utvärdera kvaliteten på heuristiska lösningar då de tillämpas på stora kombinatoriska optimeringsproblem. Bidragen från denna avhandling är både metodologiska och empiriska. Ur metodologisk synvinkel har nyttan av statistiska konfidensintervall för ett lokaliseringsproblem (p-median problemet) undersökts. Statistiska konfidensintervall fungerar väl för att tillhandahålla information om lösningens kvalitet vid rätt implementering av problemen. Statistiska konfidensintervall överträffar även intervallen som erhålls vid den vanligen använda Lagrange-relaxation. I avhandlingen visas även på att metoden med statistiska konfidensintervall är fungerar väl jämfört med många andra deterministiska intervall i ett mer komplexa optimeringsproblem som det kvadratiska tilldelningsproblemet. P-median problemet och de statistiska konfidensintervallen har implementerats empiriskt för att beräkna och jämföra e-handelns och traditionell handels CO2-utsläpp från transporter, vilken visar att ehandel medför betydligt mindre CO2-utsläpp. Ett annat lokaliseringsproblem som analyserats empiriskt är hur förändringar av den regionala administrativa indelningen av Sverige, vilket är en aktuell och pågående samhällsdiskussion, påverkar medborgarnas tillgänglighet till offentlig service. Analysen visar att regionala administrativa iv gränserna lett till en suboptimal placering av offentliga tjänster. Därmed finns en risk att den samhällsekonomiska nyttan av dessa tjänster är suboptimerad.

Place, publisher, year, edition, pages
Borlänge: Högskolan Dalarna, 2015. 200 p.
Series
Dalarna Doctoral Dissertations, 2
National Category
Computational Mathematics
Research subject
Komplexa system - mikrodataanalys
Identifiers
urn:nbn:se:du-17353 (URN)978-91-89020-94-8 (ISBN)
Public defence
2015-04-23, Clas Ohlson, Borlänge, 10:00 (English)
Opponent
Available from: 2015-05-07 Created: 2015-05-06 Last updated: 2015-05-18Bibliographically approved

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