IDEAS home Printed from https://ideas.repec.org/a/eee/ejores/v243y2015i3p723-730.html
   My bibliography  Save this article

A practicable branch and bound algorithm for sum of linear ratios problem

Author

Listed:
  • Jiao, Hong-Wei
  • Liu, San-Yang

Abstract

This article presents a practicable algorithm for globally solving sum of linear ratios problem (SLR). The algorithm works by globally solving a bilinear programming problem (EQ) that is equivalent to the problem (SLR). In the algorithm, by utilizing convex envelope and concave envelope of bilinear function, the initial nonconvex programming problem is reduced to a sequence of linear relaxation programming problems. In order to improve the computational efficiency of the algorithm, a new accelerating technique is introduced, which provides a theoretical possibility to delete a large part of the investigated region in which there exists no global optimal solution of the (EQ). By combining this innovative technique with branch and bound operations, a global optimization algorithm is designed for solving the problem (SLR). Finally, numerical experimental results show the feasibility and efficiency of the proposed algorithm.

Suggested Citation

  • Jiao, Hong-Wei & Liu, San-Yang, 2015. "A practicable branch and bound algorithm for sum of linear ratios problem," European Journal of Operational Research, Elsevier, vol. 243(3), pages 723-730.
    • Handle: RePEc:eee:ejores:v:243:y:2015:i:3:p:723-730
    DOI: 10.1016/j.ejor.2015.01.039
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0377221715000594
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.ejor.2015.01.039?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Billionnet, Alain, 2013. "Mathematical optimization ideas for biodiversity conservation," European Journal of Operational Research, Elsevier, vol. 231(3), pages 514-534.
    2. Du, Juan & Cook, Wade D. & Liang, Liang & Zhu, Joe, 2014. "Fixed cost and resource allocation based on DEA cross-efficiency," European Journal of Operational Research, Elsevier, vol. 235(1), pages 206-214.
    3. Benson, Harold P., 2007. "A simplicial branch and bound duality-bounds algorithm for the linear sum-of-ratios problem," European Journal of Operational Research, Elsevier, vol. 182(2), pages 597-611, October.
    4. Kao, Chiang, 2014. "Network data envelopment analysis: A review," European Journal of Operational Research, Elsevier, vol. 239(1), pages 1-16.
    5. Jeyakumar, V. & Li, G.Y. & Srisatkunarajah, S., 2013. "Strong duality for robust minimax fractional programming problems," European Journal of Operational Research, Elsevier, vol. 228(2), pages 331-336.
    6. Costa, Joao Paulo, 2007. "Computing non-dominated solutions in MOLFP," European Journal of Operational Research, Elsevier, vol. 181(3), pages 1464-1475, September.
    7. Schaible, Siegfried, 1981. "Fractional programming: Applications and algorithms," European Journal of Operational Research, Elsevier, vol. 7(2), pages 111-120, June.
    8. Yang, Min & Li, Yongjun & Chen, Ya & Liang, Liang, 2014. "An equilibrium efficiency frontier data envelopment analysis approach for evaluating decision-making units with fixed-sum outputs," European Journal of Operational Research, Elsevier, vol. 239(2), pages 479-489.
    9. Lim, Sungmook & Zhu, Joe, 2013. "Integrated data envelopment analysis: Global vs. local optimum," European Journal of Operational Research, Elsevier, vol. 229(1), pages 276-278.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Dreyfuss, Michael & Giat, Yahel, 2017. "Optimal spares allocation to an exchangeable-item repair system with tolerable wait," European Journal of Operational Research, Elsevier, vol. 261(2), pages 584-594.
    2. Hongwei Jiao & Binbin Li & Wenqiang Yang, 2024. "A criterion-space branch-reduction-bound algorithm for solving generalized multiplicative problems," Journal of Global Optimization, Springer, vol. 89(3), pages 597-632, July.
    3. Jiao, Hongwei & Ma, Junqiao, 2022. "An efficient algorithm and complexity result for solving the sum of general affine ratios problem," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    4. S. M. Mirhadi & S. A. MirHassani, 2022. "A solution approach for cardinality minimization problem based on fractional programming," Journal of Combinatorial Optimization, Springer, vol. 44(1), pages 583-602, August.
    5. Li, Yifu & Qi, Xiangtong, 2022. "A geometric branch-and-bound algorithm for the service bundle design problem," European Journal of Operational Research, Elsevier, vol. 303(3), pages 1044-1056.
    6. Dreyfuss, Michael & Giat, Yahel, 2019. "Allocating spares to maximize the window fill rate in a periodic review inventory system," International Journal of Production Economics, Elsevier, vol. 214(C), pages 151-162.
    7. Bo Zhang & YueLin Gao & Xia Liu & XiaoLi Huang, 2022. "An Outcome-Space-Based Branch-and-Bound Algorithm for a Class of Sum-of-Fractions Problems," Journal of Optimization Theory and Applications, Springer, vol. 192(3), pages 830-855, March.
    8. Shen, Peiping & Zhu, Zeyi & Chen, Xiao, 2019. "A practicable contraction approach for the sum of the generalized polynomial ratios problem," European Journal of Operational Research, Elsevier, vol. 278(1), pages 36-48.
    9. Hou, Zhisong & Liu, Sanyang, 2023. "A spatial branch-reduction-bound algorithm for solving generalized linear fractional problems globally," Chaos, Solitons & Fractals, Elsevier, vol. 176(C).
    10. Mahdiloo, Mahdi & Toloo, Mehdi & Duong, Thach-Thao & Farzipoor Saen, Reza & Tatham, Peter, 2018. "Integrated data envelopment analysis: Linear vs. nonlinear model," European Journal of Operational Research, Elsevier, vol. 268(1), pages 255-267.
    11. Huang, Bingdi & Shen, Peiping, 2024. "An efficient branch and bound reduction algorithm for globally solving linear fractional programming problems," Chaos, Solitons & Fractals, Elsevier, vol. 182(C).
    12. Chen, Kun & Zhu, Joe, 2020. "Additive slacks-based measure: Computational strategy and extension to network DEA," Omega, Elsevier, vol. 91(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Shen, Peiping & Zhu, Zeyi & Chen, Xiao, 2019. "A practicable contraction approach for the sum of the generalized polynomial ratios problem," European Journal of Operational Research, Elsevier, vol. 278(1), pages 36-48.
    2. Phung, Manh-Trung & Cheng, Cheng-Ping & Guo, Chuanyin & Kao, Chen-Yu, 2020. "Mixed Network DEA with Shared Resources: A Case of Measuring Performance for Banking Industry," Operations Research Perspectives, Elsevier, vol. 7(C).
    3. Zhang, Linyan & Chen, Kun, 2019. "Hierarchical network systems: An application to high-technology industry in China," Omega, Elsevier, vol. 82(C), pages 118-131.
    4. Khoveyni, Mohammad & Fukuyama, Hirofumi & Eslami, Robabeh & Yang, Guo-liang, 2019. "Variations effect of intermediate products on the second stage in two-stage processes," Omega, Elsevier, vol. 85(C), pages 35-48.
    5. João Costa & Maria Alves, 2013. "Enhancing computations of nondominated solutions in MOLFP via reference points," Journal of Global Optimization, Springer, vol. 57(3), pages 617-631, November.
    6. Chen, Ya & Li, Yongjun & Liang, Liang & Salo, Ahti & Wu, Huaqing, 2016. "Frontier projection and efficiency decomposition in two-stage processes with slacks-based measures," European Journal of Operational Research, Elsevier, vol. 250(2), pages 543-554.
    7. Yin, Pengzhen & Sun, Jiasen & Chu, Junfei & Liang, Liang, 2016. "Evaluating the environmental efficiency of a two-stage system with undesired outputs by a DEA approach: An interest preference perspectiveAuthor-Name: Wu, Jie," European Journal of Operational Research, Elsevier, vol. 254(3), pages 1047-1062.
    8. Li, Yongjun & Liu, Jin & Ang, Sheng & Yang, Feng, 2021. "Performance evaluation of two-stage network structures with fixed-sum outputs: An application to the 2018winter Olympic Games," Omega, Elsevier, vol. 102(C).
    9. Wu, Jie & Chu, Junfei & Sun, Jiasen & Zhu, Qingyuan, 2016. "DEA cross-efficiency evaluation based on Pareto improvement," European Journal of Operational Research, Elsevier, vol. 248(2), pages 571-579.
    10. Meng, Fanyong & Xiong, Beibei, 2021. "Logical efficiency decomposition for general two-stage systems in view of cross efficiency," European Journal of Operational Research, Elsevier, vol. 294(2), pages 622-632.
    11. Yang, Min & Li, Yong Jun & Liang, Liang, 2015. "A generalized equilibrium efficient frontier data envelopment analysis approach for evaluating DMUs with fixed-sum outputs," European Journal of Operational Research, Elsevier, vol. 246(1), pages 209-217.
    12. Ashtiani, Alireza M. & Ferreira, Paulo A.V., 2015. "A branch-and-cut algorithm for a class of sum-of-ratios problems," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 596-608.
    13. Qingyou Yan & Fei Zhao & Xu Wang & Guoliang Yang & Tomas Baležentis & Dalia Streimikiene, 2019. "The network data envelopment analysis models for non-homogenous decision making units based on the sun network structure," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 27(4), pages 1221-1244, December.
    14. Chergui, M. E-A & Moulai, M., 2007. "An exact method for a discrete multiobjective linear fractional optimization," MPRA Paper 12097, University Library of Munich, Germany, revised 09 Jan 2008.
    15. Liu, Hui-hui & Song, Yao-yao & Liu, Xiao-xiao & Yang, Guo-liang, 2020. "Aggregating the DEA prospect cross-efficiency with an application to state key laboratories in China," Socio-Economic Planning Sciences, Elsevier, vol. 71(C).
    16. Hongwei Jiao & Binbin Li & Wenqiang Yang, 2024. "A criterion-space branch-reduction-bound algorithm for solving generalized multiplicative problems," Journal of Global Optimization, Springer, vol. 89(3), pages 597-632, July.
    17. Qingyou Yan & Fei Zhao & Xu Wang & Tomas Balezentis, 2021. "The Environmental Efficiency Analysis Based on the Three-Step Method for Two-Stage Data Envelopment Analysis," Energies, MDPI, vol. 14(21), pages 1-14, October.
    18. Yu, Shasha & Lei, Ming & Deng, Honghui, 2023. "Evaluation to fixed-sum-outputs DMUs by non-oriented equilibrium efficient frontier DEA approach with Nash bargaining-based selection," Omega, Elsevier, vol. 115(C).
    19. Xiao, Huijuan & Wang, Daoping & Qi, Yu & Shao, Shuai & Zhou, Ya & Shan, Yuli, 2021. "The governance-production nexus of eco-efficiency in Chinese resource-based cities: A two-stage network DEA approach," Energy Economics, Elsevier, vol. 101(C).
    20. Li, Feng & Zhu, Qingyuan & Chen, Zhi, 2019. "Allocating a fixed cost across the decision making units with two-stage network structures," Omega, Elsevier, vol. 83(C), pages 139-154.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:ejores:v:243:y:2015:i:3:p:723-730. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/eor .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.