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Optimum Solution of Resource Leveling Problem by Evaluating the Whole Search Domain

Year 2022, Volume: 34 Issue: 2, 779 - 794, 30.09.2022

Önder Halis Bettemir Tuğba Erzurum

https://doi.org/10.35234/fumbd.1105174

Abstract

The endeavor to minimize the fluctuations of the resource assignments without postponing the project deadline is named as resource leveling problem. This problem is solved by heuristic methods, consist of simple rules, meta-heuristic methods, explore the search domain systematically but randomly, and analytical methods. Nevertheless, aforementioned approaches come to be insufficient when the number of activity of the project increases which oversize the search domain of the resource leveling problem. In this study, the number of the feasible solutions is determined by detecting the number of feasible schedules obtained by delaying the noncritical activities without violating the restrictions on the activities. The entire search domain is evaluated and the optimum feasible solution is obtained in a guaranteed manner. Suitability of the resource distribution is evaluated by minimum moment metric. A spreadsheet application is developed and macro is written by Visual Basic programming language to implement the developed method. In the literature survey, twenty seven resource leveling problems are collected and optimum solutions of them are obtained by the developed method. Definition of the parameters and the evaluation of the search domain of the resource leveling problem by the spreadsheet application are explained in detail on two of the solved case study problems so that the researchers can implement the developed method. Moreover the developed method is coded on C++ in order to expedite the computations. Test problems are also solved by genetic algorithm and the computation duration and the obtained results are compared with the proposed method. The comparison reveals that the developed method can be implemented for optimizing the resource distribution. Moreover, this research contributes to the literature by providing a methodology for the determination of the search domain of the resource leveling problem.

Keywords

Resource leveling problem activity-on-arrow optimization

References

  • Rieck J, Zimmermann J. Handbook on Project Management and Scheduling. Exact methods for resource leveling problems, Springer International Publishing: 2015; 1, 361-387.
  • Demeulemeester E, Herroelen W. Project Scheduling: A Research Handbook, Kluwer Academic Publishers, Boston, USA, (2002).
  • Harris RB. Resource and arrow networking techniques for construction, Wiley, New York, 1978.
  • Harris RB. Packing method for resource leveling (PACK), J. Constr. Eng. Manage., 1990; 116(2), 331-350.
  • Hiyassat MAS. Modification of minimum moment approach in resource leveling, J. Constr. Eng. Manage., 2000; 126(4), 278-284.
  • Hiyassat MAS. Applying modified minimum moment method to multiple resource leveling, J. Constr. Eng. Manage., 2001; 127(3), 192-198.
  • Hegazy T. "Optimization of resource allocation and leveling using genetic algorithms, J. Constr. Eng. Manage., 1999; 125(3), 167-175.
  • Hegazy T, Kassab M. Resource optimization using combined simulation and genetic algorithms, J. Constr. Eng. Manage., 2003; 129(6): 698-705.
  • Hossein HD, SeifiSA, Shariat SY. Efficient hybrid genetic algorithm for resource leveling via activity splitting, J. Constr. Eng. Manage., 2010; 137(2): 137-146.
  • Ponz-Tienda JL, Yepes V, Pellicer E, Moreno-Flores J. "The resource leveling problem with multiple resources using an adaptive genetic algorithm", Automation in Construction, 2013; 29: 161-172.
  • Zheng DX, Ng, ST, Kumaraswamy MM. "GA-based multiobjective technique for multi-resource leveling", Bridges, 2003; 10(40671): 29.
  • Leu SS, Yang CH, Huang JC. Resource leveling in construction by genetic algorithm-based optimization and its decision support system application, Automation in construction, 2000; 10(1): 27-41.
  • Karaköse, E. Sürü İnsansız Hava Araçlarının Görev Paylaşımı için Genetik Algoritma Tabanlı Bir Yaklaşım. Fırat Üniversitesi Mühendislik Bilimleri Dergisi, 2022; 34(1), 351-360.
  • Li Z, Wuliang P, Zhongliang Z. An ant colony system for solving resource leveling problem, In IEEE International Conference on Intelligent Computation Technology and Automation (ICICTA), 2010; 1: 489-492.
  • Geng JQ, Weng LP, Liu SH. An improved ant colony optimization algorithm for nonlinear resource-leveling problems, Computers & Mathematics with Applications, 2011; 61(8), 2300-2305.
  • Wang Q, Qi JX. Research on resource leveling problem under resource constrained condition, IEEE International Conference on Machine Learning and Cybernetics, 2009; 2: 901-906.
  • Li H, Dong X. "Multi-mode resource leveling in projects with mode-dependent generalized precedence relations", Expert Systems with Applications, 2018; 97, 193-204.
  • Son J, Skibniewski MJ. Multiheuristic approach for resource leveling problem in construction engineering: Hybrid approach, J. Constr. Eng. Manage., 1999; 125(1), 23-31.
  • Qi, JX, Wang Q, Guo XZ. Improved particle swarm optimization for resource leveling problem, IEEE International Conference on Machine Learning and Cybernetics, 2007; 2: 896-901.
  • Tanyıldızı, E., ve Demir, G. Nümerik Optimizasyon için Kaotik Altın Sinüs Algoritması. Fırat Üniversitesi Mühendislik Bilimleri Dergisi, 2019; 31(1), 91-97. Akyol, S. Global Optimizasyon için Yeni Bir Hibrit Yöntem: Kaya Kartalı Optimizasyonu-Tanjant Arama Algoritması. Fırat Üniversitesi Mühendislik Bilimleri Dergisi, 2021; 33(2), 721-733.
  • Yetiş, H., ve Karaköse, M. Kuantum Uyarlamalı Genetik Algoritmalar için Çözüm Kalitesini Artıracak Yeni Bir Yaklaşım. Fırat Üniversitesi Mühendislik Bilimleri Dergisi, 2021; 33(1), 71-79.
  • Aydemir, S. B. Küresel Optimizasyon için Gauss Kaotik Haritası ile Kartal Optimizasyonu. Fırat Üniversitesi Mühendislik Bilimleri Dergisi, 2022; 34(1), 85-104.
  • Demeulemeester E, Herroelen W. A branch-and-bound procedure for the multiple resource-constrained project scheduling problem, Management science, 1992; 38(12): 1803-1818.
  • Neumann K, Zimmermann J. Procedures for resource leveling and net present value problems in project scheduling with general temporal and resource constraints, European Journal of Operational Research, 2000; 127(2): 425-443.
  • Nübel H. The resource renting problem subject to temporal constraints, OR-Spektrum, 2001; 23(3): 359-381.
  • Easa SM. Resource leveling in construction by optimization, J. Constr. Eng. Manage., 1989; 115(2): 302-316.
  • Hariga M, El-Sayegh SM. Cost optimization model for the multiresource leveling problem with allowed activity splitting, J. Constr. Eng. Manage., 2010; 137(1): 56-64.
  • Karaa FA, Nasr AY. Resource management in construction", J. Constr. Eng. Manage., 1986; 112(3), 346-357.
  • Gather T, Zimmermann J, Bartels JH. Exact methods for the resource levelling problem, Journal of Scheduling, 2011; 14(6): 557-569.
  • Mattila KG, Abraham DM. Resource leveling of linear schedules using integer linear programming, J. Constr. Eng. Manage., 1998; 124(3): 232-244.
  • Özcan, H. Comparison of particle swarm and differential evolution optimization algorithms considering various benchmark functions, Politeknik Dergisi, 2017; 20(4): 899-905.
  • Erzurum, T., ve Bettemir, Ö. H. Kaynak Dengeleme Probleminin Arama Uzayını Paralel Programlama ile Tarayarak Kesin Çözümü, Teknik Dergi, 2021; 32(3): 10767 – 10805.
  • Bettemir ÖH, Erzurum T. Comparison of resource distribution metrics on multi-resource projects, Journal of Construction Engineering, Management & Innovation, 2019; 2(2): 93-102, ().
  • Bettemir ÖH, Erzurum T. Comparison of Resource Distribution Metrics on Small Projects, International Civil Engineering and Architecture Conference, Trabzon, Turkey, (2019).
  • Erzurum T. Kaynak Dengeleme Probleminin Optimum veya Yakın Optimum Çözülmesi, Yüksek Lisans Tezi, T.C. İnönü Üniversitesi, Malatya, Türkiye, (2019).
  • Bandelloni M, Tucci M, Rinaldi R. Optimal resource leveling using non-serial dyanamic programming, European Journal of Operational Research, 1994; 78(2): 162-177.
  • Erzurum T. Bettemir, ÖH. Kaynak Dengeleme Problemlerinin Arama Uzayının Belirlenmesi Determination of Search Domain of Resource Leveling Problem, Uluslararası Katılımlı 7. İnşaat Yönetimi Kongresi, 437-453, Samsun, Türkiye, (2017).
  • Erzurum T, Bettemir ÖH. Optimum or Near-Optimum Resolution of Resource Leveling Problems with Spreadsheet Application, 5th International Project and Construction Management Conference (IPCMC 2018):1285-1299, Northern Cyprus, (2018).
  • Gordon, J., ve Tulip, A. Resource scheduling, Int. J. Proj. Manage., 1997; 15(6): 359-370.
  • Rui, L., and Xiao-ya, W., Using elitist particle swarm optimization to facilitate resources leveling optimization analysis, In IEEE 3rd IEEE Conference on Industrial Electronics and Applications, ICIEA: 90-95, (2008).
  • Abeyasinghe, M. C. L., Greenwood, D. J., ve Johansen, D. E., An efficient method for scheduling construction projects with resource constraints, International Journal of Project Management, 2001; 19(1), 29-45.
  • Younis, M. A., ve Saad, B., Optimal resource leveling of multi-resource projects, Computers and industrial engineering, 1996; 31(1): 1-4.
  • Mutlu, M. Ç. A branch and bound algorithm for resource leveling problem, MSc Thesis, METU, (2010).
  • Newitt, J. S. Construction Scheduling: Principles and Practices, Pearson Prentice Hall, NJ, (2004).
  • Hinze, J. W. “Construction Planning and Scheduling, 3rd Edition, Pearson Prentice Hall, Upper Saddle River, NJ, (2006).
  • Mubarak, S. A., Construction Project Scheduling and Control, 2nd Ed. Wiley, India (2010).
  • Akpan, E. O. P., Resource smoothing: a cost minimization approach, Production Planning & Control, 2000; 11(8): 775 – 780.
  • Stevens, J.D., Techniques for Construction Network Scheduling, McGraw-Hill, New York, (1990).
  • El-Rayes, K., Jun, D.H. Optimizing resource leveling in construction projects, J. Constr. Eng. Manage., 2009; 135(11): 1172-1180.

Tüm Arama Uzayı Taranarak Kaynak Dengeleme Probleminin Optimum Çözülmesi

Year 2022, Volume: 34 Issue: 2, 779 - 794, 30.09.2022

Önder Halis Bettemir Tuğba Erzurum

https://doi.org/10.35234/fumbd.1105174

Abstract

İnşaat süresi boyunca kaynak kullanımında gözlemlenen dalgalanmaların proje süresinde gecikme olmadan asgari düzeye indirilmesi kaynak dengeleme problemi olarak tanımlanır. Bu problem basit kurallar içeren sezgisel, sistematik fakat rassal biçimde arama uzayını tarayan üst-sezgisel ve analitik yöntemlerle çözülmektedir. Ancak aktivite sayısının artması ile arama uzayı çok büyüyen kaynak dengeleme probleminin çözümünde oluşan sorunlara karşı belirtilen yöntemler yetersiz kalmaktadır. Bu çalışmada, aktiviteler arasındaki kısıtlar ihlal edilmeden kritik olmayan aktivitelerin ertelenmesi ile kaç farklı şekilde uygulanabilir çözüm elde edilebileceği hesaplanarak kaynak dengeleme probleminin arama uzayının büyüklüğü belirlenmiştir. Belirlenen arama uzayının tamamı taranarak uygulanabilir en iyi çözüm garantili biçimde elde edilmiştir. Kaynak dağılımının uygunluğu minimum moment ölçeği ile incelenmiştir. Yöntemin uygulanabilmesi için bir hesap tablosu uygulaması oluşturularak Visual Basic programlama dilinde makro yazılmıştır. Literatürden derlenen 27 kaynak dengeleme probleminin geliştirilen yöntemle optimum çözümü elde edilmiştir. Hesap cetveline kaynak dengeleme probleminin nasıl tanıtılacağı çözülen problemler arasından seçilen 2 örnek problem üzerinde detaylı biçimde anlatılarak yöntemin tüm araştırmacılar tarafından uygulanabilmesi sağlanmıştır. Ayrıca geliştirilen yöntem çözüm süresini kısaltmak için C++ dilinde de kodlanmıştır. Test problemleri genetik algoritmayla çözülmüş, hesaplama süreleri ve sunduğu sonuçlar önerilen yöntemin çıktıları ile karşılaştırılmıştır. Çözüm süresi ve çözüm iyiliğinin karşılaştırılması sonucunda geliştirilen yöntemin kaynak dengeleme probleminin kesin çözümünde uygulanabilir olduğu belirlenmiştir. Ayrıca çalışma kaynak dengeleme probleminin arama uzayını belirleyen bir yöntem geliştirerek literatüre katkı sağlamaktadır.

Keywords

Kaynak dengeleme problemi ok diyagramı optimizasyon

References

  • Rieck J, Zimmermann J. Handbook on Project Management and Scheduling. Exact methods for resource leveling problems, Springer International Publishing: 2015; 1, 361-387.
  • Demeulemeester E, Herroelen W. Project Scheduling: A Research Handbook, Kluwer Academic Publishers, Boston, USA, (2002).
  • Harris RB. Resource and arrow networking techniques for construction, Wiley, New York, 1978.
  • Harris RB. Packing method for resource leveling (PACK), J. Constr. Eng. Manage., 1990; 116(2), 331-350.
  • Hiyassat MAS. Modification of minimum moment approach in resource leveling, J. Constr. Eng. Manage., 2000; 126(4), 278-284.
  • Hiyassat MAS. Applying modified minimum moment method to multiple resource leveling, J. Constr. Eng. Manage., 2001; 127(3), 192-198.
  • Hegazy T. "Optimization of resource allocation and leveling using genetic algorithms, J. Constr. Eng. Manage., 1999; 125(3), 167-175.
  • Hegazy T, Kassab M. Resource optimization using combined simulation and genetic algorithms, J. Constr. Eng. Manage., 2003; 129(6): 698-705.
  • Hossein HD, SeifiSA, Shariat SY. Efficient hybrid genetic algorithm for resource leveling via activity splitting, J. Constr. Eng. Manage., 2010; 137(2): 137-146.
  • Ponz-Tienda JL, Yepes V, Pellicer E, Moreno-Flores J. "The resource leveling problem with multiple resources using an adaptive genetic algorithm", Automation in Construction, 2013; 29: 161-172.
  • Zheng DX, Ng, ST, Kumaraswamy MM. "GA-based multiobjective technique for multi-resource leveling", Bridges, 2003; 10(40671): 29.
  • Leu SS, Yang CH, Huang JC. Resource leveling in construction by genetic algorithm-based optimization and its decision support system application, Automation in construction, 2000; 10(1): 27-41.
  • Karaköse, E. Sürü İnsansız Hava Araçlarının Görev Paylaşımı için Genetik Algoritma Tabanlı Bir Yaklaşım. Fırat Üniversitesi Mühendislik Bilimleri Dergisi, 2022; 34(1), 351-360.
  • Li Z, Wuliang P, Zhongliang Z. An ant colony system for solving resource leveling problem, In IEEE International Conference on Intelligent Computation Technology and Automation (ICICTA), 2010; 1: 489-492.
  • Geng JQ, Weng LP, Liu SH. An improved ant colony optimization algorithm for nonlinear resource-leveling problems, Computers & Mathematics with Applications, 2011; 61(8), 2300-2305.
  • Wang Q, Qi JX. Research on resource leveling problem under resource constrained condition, IEEE International Conference on Machine Learning and Cybernetics, 2009; 2: 901-906.
  • Li H, Dong X. "Multi-mode resource leveling in projects with mode-dependent generalized precedence relations", Expert Systems with Applications, 2018; 97, 193-204.
  • Son J, Skibniewski MJ. Multiheuristic approach for resource leveling problem in construction engineering: Hybrid approach, J. Constr. Eng. Manage., 1999; 125(1), 23-31.
  • Qi, JX, Wang Q, Guo XZ. Improved particle swarm optimization for resource leveling problem, IEEE International Conference on Machine Learning and Cybernetics, 2007; 2: 896-901.
  • Tanyıldızı, E., ve Demir, G. Nümerik Optimizasyon için Kaotik Altın Sinüs Algoritması. Fırat Üniversitesi Mühendislik Bilimleri Dergisi, 2019; 31(1), 91-97. Akyol, S. Global Optimizasyon için Yeni Bir Hibrit Yöntem: Kaya Kartalı Optimizasyonu-Tanjant Arama Algoritması. Fırat Üniversitesi Mühendislik Bilimleri Dergisi, 2021; 33(2), 721-733.
  • Yetiş, H., ve Karaköse, M. Kuantum Uyarlamalı Genetik Algoritmalar için Çözüm Kalitesini Artıracak Yeni Bir Yaklaşım. Fırat Üniversitesi Mühendislik Bilimleri Dergisi, 2021; 33(1), 71-79.
  • Aydemir, S. B. Küresel Optimizasyon için Gauss Kaotik Haritası ile Kartal Optimizasyonu. Fırat Üniversitesi Mühendislik Bilimleri Dergisi, 2022; 34(1), 85-104.
  • Demeulemeester E, Herroelen W. A branch-and-bound procedure for the multiple resource-constrained project scheduling problem, Management science, 1992; 38(12): 1803-1818.
  • Neumann K, Zimmermann J. Procedures for resource leveling and net present value problems in project scheduling with general temporal and resource constraints, European Journal of Operational Research, 2000; 127(2): 425-443.
  • Nübel H. The resource renting problem subject to temporal constraints, OR-Spektrum, 2001; 23(3): 359-381.
  • Easa SM. Resource leveling in construction by optimization, J. Constr. Eng. Manage., 1989; 115(2): 302-316.
  • Hariga M, El-Sayegh SM. Cost optimization model for the multiresource leveling problem with allowed activity splitting, J. Constr. Eng. Manage., 2010; 137(1): 56-64.
  • Karaa FA, Nasr AY. Resource management in construction", J. Constr. Eng. Manage., 1986; 112(3), 346-357.
  • Gather T, Zimmermann J, Bartels JH. Exact methods for the resource levelling problem, Journal of Scheduling, 2011; 14(6): 557-569.
  • Mattila KG, Abraham DM. Resource leveling of linear schedules using integer linear programming, J. Constr. Eng. Manage., 1998; 124(3): 232-244.
  • Özcan, H. Comparison of particle swarm and differential evolution optimization algorithms considering various benchmark functions, Politeknik Dergisi, 2017; 20(4): 899-905.
  • Erzurum, T., ve Bettemir, Ö. H. Kaynak Dengeleme Probleminin Arama Uzayını Paralel Programlama ile Tarayarak Kesin Çözümü, Teknik Dergi, 2021; 32(3): 10767 – 10805.
  • Bettemir ÖH, Erzurum T. Comparison of resource distribution metrics on multi-resource projects, Journal of Construction Engineering, Management & Innovation, 2019; 2(2): 93-102, ().
  • Bettemir ÖH, Erzurum T. Comparison of Resource Distribution Metrics on Small Projects, International Civil Engineering and Architecture Conference, Trabzon, Turkey, (2019).
  • Erzurum T. Kaynak Dengeleme Probleminin Optimum veya Yakın Optimum Çözülmesi, Yüksek Lisans Tezi, T.C. İnönü Üniversitesi, Malatya, Türkiye, (2019).
  • Bandelloni M, Tucci M, Rinaldi R. Optimal resource leveling using non-serial dyanamic programming, European Journal of Operational Research, 1994; 78(2): 162-177.
  • Erzurum T. Bettemir, ÖH. Kaynak Dengeleme Problemlerinin Arama Uzayının Belirlenmesi Determination of Search Domain of Resource Leveling Problem, Uluslararası Katılımlı 7. İnşaat Yönetimi Kongresi, 437-453, Samsun, Türkiye, (2017).
  • Erzurum T, Bettemir ÖH. Optimum or Near-Optimum Resolution of Resource Leveling Problems with Spreadsheet Application, 5th International Project and Construction Management Conference (IPCMC 2018):1285-1299, Northern Cyprus, (2018).
  • Gordon, J., ve Tulip, A. Resource scheduling, Int. J. Proj. Manage., 1997; 15(6): 359-370.
  • Rui, L., and Xiao-ya, W., Using elitist particle swarm optimization to facilitate resources leveling optimization analysis, In IEEE 3rd IEEE Conference on Industrial Electronics and Applications, ICIEA: 90-95, (2008).
  • Abeyasinghe, M. C. L., Greenwood, D. J., ve Johansen, D. E., An efficient method for scheduling construction projects with resource constraints, International Journal of Project Management, 2001; 19(1), 29-45.
  • Younis, M. A., ve Saad, B., Optimal resource leveling of multi-resource projects, Computers and industrial engineering, 1996; 31(1): 1-4.
  • Mutlu, M. Ç. A branch and bound algorithm for resource leveling problem, MSc Thesis, METU, (2010).
  • Newitt, J. S. Construction Scheduling: Principles and Practices, Pearson Prentice Hall, NJ, (2004).
  • Hinze, J. W. “Construction Planning and Scheduling, 3rd Edition, Pearson Prentice Hall, Upper Saddle River, NJ, (2006).
  • Mubarak, S. A., Construction Project Scheduling and Control, 2nd Ed. Wiley, India (2010).
  • Akpan, E. O. P., Resource smoothing: a cost minimization approach, Production Planning & Control, 2000; 11(8): 775 – 780.
  • Stevens, J.D., Techniques for Construction Network Scheduling, McGraw-Hill, New York, (1990).
  • El-Rayes, K., Jun, D.H. Optimizing resource leveling in construction projects, J. Constr. Eng. Manage., 2009; 135(11): 1172-1180.
There are 49 citations in total.

Details

Primary Language Turkish
Subjects Engineering
Journal Section MBD
Authors

Önder Halis Bettemir 0000-0002-5692-7708

Tuğba Erzurum 0000-0003-4788-6999

Publication Date September 30, 2022
Submission Date April 20, 2022
Published in Issue Year 2022 Volume: 34 Issue: 2

Cite

APA Bettemir, Ö. H., & Erzurum, T. (2022). Tüm Arama Uzayı Taranarak Kaynak Dengeleme Probleminin Optimum Çözülmesi. Fırat Üniversitesi Mühendislik Bilimleri Dergisi, 34(2), 779-794. https://doi.org/10.35234/fumbd.1105174
AMA Bettemir ÖH, Erzurum T. Tüm Arama Uzayı Taranarak Kaynak Dengeleme Probleminin Optimum Çözülmesi. Fırat Üniversitesi Mühendislik Bilimleri Dergisi. September 2022;34(2):779-794. doi:10.35234/fumbd.1105174
Chicago Bettemir, Önder Halis, and Tuğba Erzurum. “Tüm Arama Uzayı Taranarak Kaynak Dengeleme Probleminin Optimum Çözülmesi”. Fırat Üniversitesi Mühendislik Bilimleri Dergisi 34, no. 2 (September 2022): 779-94. https://doi.org/10.35234/fumbd.1105174.
EndNote Bettemir ÖH, Erzurum T (September 1, 2022) Tüm Arama Uzayı Taranarak Kaynak Dengeleme Probleminin Optimum Çözülmesi. Fırat Üniversitesi Mühendislik Bilimleri Dergisi 34 2 779–794.
IEEE Ö. H. Bettemir and T. Erzurum, “Tüm Arama Uzayı Taranarak Kaynak Dengeleme Probleminin Optimum Çözülmesi”, Fırat Üniversitesi Mühendislik Bilimleri Dergisi, vol. 34, no. 2, pp. 779–794, 2022, doi: 10.35234/fumbd.1105174.
ISNAD Bettemir, Önder Halis - Erzurum, Tuğba. “Tüm Arama Uzayı Taranarak Kaynak Dengeleme Probleminin Optimum Çözülmesi”. Fırat Üniversitesi Mühendislik Bilimleri Dergisi 34/2 (September 2022), 779-794. https://doi.org/10.35234/fumbd.1105174.
JAMA Bettemir ÖH, Erzurum T. Tüm Arama Uzayı Taranarak Kaynak Dengeleme Probleminin Optimum Çözülmesi. Fırat Üniversitesi Mühendislik Bilimleri Dergisi. 2022;34:779–794.
MLA Bettemir, Önder Halis and Tuğba Erzurum. “Tüm Arama Uzayı Taranarak Kaynak Dengeleme Probleminin Optimum Çözülmesi”. Fırat Üniversitesi Mühendislik Bilimleri Dergisi, vol. 34, no. 2, 2022, pp. 779-94, doi:10.35234/fumbd.1105174.
Vancouver Bettemir ÖH, Erzurum T. Tüm Arama Uzayı Taranarak Kaynak Dengeleme Probleminin Optimum Çözülmesi. Fırat Üniversitesi Mühendislik Bilimleri Dergisi. 2022;34(2):779-94.
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