A LAGRANGEAN RELAXATION-BASED SOLUTION APPROACH FOR MULTICOMMODITY NETWORK DESIGN PROBLEM WITH CAPACITY VIOLATIONS

Year 2021, Volume: 17 Issue: 2, 241 - 263, 08.11.2021

Levent Erişkin

Abstract

In this study, we formulate and compare two different Lagrangean relaxation-based decompositions for multicommodity network problems with penalized constraints. These problems are different versions of capacitated multicommodity network problems where capacity constraints can be violated for additional penalty costs. These costs are reflected as nonlinear terms in the objective function; hence, these problems turn out to be nonlinear mixed integer optimization problems. To the best of our knowledge, there is no exact solution algorithm for this type of problem. We propose two kinds of Lagrangean relaxation-based decompositions and solve these problems with the subgradient algorithm. The resulting subproblems are easy to solve and the proposed algorithms can reach reasonable solutions where CPLEX solver cannot even find a solution. In the study, we also conduct a computational analysis where we compare two relaxations over various performance measures. Even though two relaxations present similar performances in terms of computation times and the number of iterations, we observed that Relaxation 1 statistically outperforms Relaxation 2.

Keywords

Multicommodity Network Design Problem, Lagrangean Relaxation, Subgradient Algorithm, Decomposition

KAPASİTE İHLALLİ ÇOKLU MAL ŞEBEKE DİZAYN PROBLEMİ İÇİN LAGRANGEAN GEVŞETMESİ TABANLI BİR ÇÖZÜM YAKLAŞIMI

Abstract

Bu çalışmada, cezalandırıcı kısıtlara sahip çoklu mal şebeke problemi için Lagrangean gevşetmesi tabanlı iki farklı ayrıştırma yaklaşımı formüle edilmekte ve karşılaştırılmaktadır. Bu problemler kapasite kısıtlarının ilave bir ceza maliyeti ile ihlal edilebilediği kapasite kısıtlı çoklu mal şebeke problemlerinin farklı versiyonlarıdır. Bu maliyetler amaç fonksiyonuna doğrusal olmayan terimler olarak yansıtılmakta, bu kapsamda bu problemler doğrusal olmayan karışık tam sayılı eniyileme problemlerine dönüşmektedir. Bilgimiz dahilinde, bu tip problemlerin çözümü için herhangi bir kesin çözüm algoritması bulunmamaktadır. Bu problemler için iki farklı Lagrangean gevşetmesi tabanlı ayrıştırma teklif etmekte ve gradyant altı algoritması ile çözmekteyiz. Ortaya çıkan alt-problemler kolaylıkla çözülebilmekte ve önerilen algoritmalar CPLEX çözücünün herhangi bir çözüm bile bulamadığı durumlar için makul sonuçlar elde etmektedir. Çalışmada ayrıca bu iki gevşetmenin farklı performans metrikleri bazında karşılaştırmasının yapıldığı bir hesaplamalı analiz de yapmaktayız. Her ne kadar iki gevşetme de çözüm süresi ve iterasyon adedi açısından benzer performanslar gösterse de Gevşetme 1’nin istatistiksel olarak Gevşetme 2’den daha üstün olduğunu gözlemledik.

Keywords

Çoklu Mal Şebeke Dizayn Problemi, Lagrangean Gevşetmesi, Gradyan Altı Algoritması, Ayrıştırma

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