Estimating the Solution of the Best-Worst Method Non-Linear Programming Model by solving the Mixed Integer Linear Programming Model Solutions

Subject Areas : Statistics

Mohammad Reza Dehghani ¹ , Mehdi Abbasi ²

1 - Department of Industrial Engineering, Shiraz Branch, Islamic Azad University, Shiraz,,, Iran.
2 - Department of Industrial Engineering, Shiraz Branch, Islamic Azad University, Shiraz,, Iran.

Received: 2020-09-06 Accepted : 2021-04-27 Published : 2022-08-23

Keywords: مجموعه‌های منظم خاص (SOS), تقریب تکه‌ای خطی (PLA), روش بهترین-بدترین (BWM), مدل برنامه‌ریزی خطی مختلط (MILPM),

Abstract :

The Best-Worst Method (BWM) is one of the latest methods in Multiple Attribute Decision Making (MADM) problems. The mentioned method determines the optimal solution by forming and solving a Non-Linear Programming Model (NLPM). In respect to the difficulties of solving the relevant NLPM, some attempts have been made to provide Linear Programming Models (LPM) or Equivalent Mixed Linear Programming Models. But there are some deficiencies in each of the proposed models. In this paper, by removing the mentioned deficiencies, an algorithm was proposed to estimate the solution of BWM NLPM with an acceptable error by forming and solving Mixed Integer Linear Programming Models (MILPMs). In the suggested algorithm, first, the equivalent model of NLPM (ENLPM) was formed. Then by Piecewise Linear Approximation (PLA) using the SOS2 method, the non-linear sentences of ENLPM were approximated and the first MILP model was formed and solved. If the error of solution is not acceptable, improving the PLA of the ENLPM nonlinear sentences, the forming of new MILPMs, and its solving continues until a solution with acceptable error obtained. To investigate the reliability of the algorithm, a new method for generating samples was proposed. Then using this method, 128 samples with three and five attributes were generated. The results of implementing the proposed algorithm to solve generated samples showed the performance and efficiency of the proposed algorithm. In this regard by forming and solving a maximum of three MILPMs of the samples, an estimation of the solution with 1% error was obtained.

References:

• اصغرپور، محمدجواد. تصمیم‌گیری‌های چند معیاره، چاپ هفدهم، انتشارات دانشگاه تهران. (1398).


• Rezaei, J. (2015). Best-worst multi-criteria decision-making method. Omega, 53, 49-57.


• Rezaei, J. (2016). Best-worst multi-criteria decision-making method: Some properties and a linear model. Omega, 64, 126-130.


• Beemsterboer, D. J. C., Hendrix, E. M. T., & Claassen, G. D. H. (2018). On solving the best-worst method in multi-criteria decision-making. IFAC-PapersOnLine, 51(11), 1660-1665.


• Beale, E. M. L., & Tomlin, J. A. (1970). Special facilities in a general mathematical programming system for non-convex problems using ordered sets of variables. OR, 69(447-454), 99.


• Aboutorab, H., Saberi, M., Asadabadi, M. R., Hussain, O., & Chang, E. (2018). ZBWM: The Z-number extension of Best Worst Method and its application for supplier development. Expert Systems with Applications, 107, 115-125.


• Liang, F., Brunelli, M., & Rezaei, J. (2019). Consistency issues in the best worst method: Measurements and thresholds. Omega, 102175.


• Mohammadi, M., & Rezaei, J. (2019). Bayesian best-worst method: A probabilistic group decision making model. Omega, 102075


• AMIRI, M., & EMAMAT, M. S. M. M. (2020). A Goal Programming Model for BWM. INFORMATICA, 31(1), 21-34.


• Vafadarnikjoo, A., Tavana, M., Botelho, T., & Chalvatzis, K. (2020). A neutrosophic enhanced best–worst method for considering decision-makers’ confidence in the best and worst criteria. Annals of Operations Research, 1-28.


• Leyffer, S., Sartenaer, A., & Wanufelle, E. (2008). Branch-and-refine for mixed-integer nonconvex global optimization. Preprint ANL/MCS-P1547-0908, Mathematics and Computer Science Division, Argonne National Laboratory, 39, 40-78.


• انصاری, محمدرضا, حسنی‌فرد, فاطمه. (1396). حل یک مسئله بهینه‌سازی غیرخطی، عدد صحیح و غیرمحدب با استفاده از روش‌های محدب‌سازی مبتنی بر مجموعه منظم خاص. فصلنامه سیستم‌های مختلط و غیرخطی, 1(1), 71-85.


• Kang, C., Guo, M., & Wang, J. (2017). Short-term hydrothermal scheduling using a two-stage linear programming with special ordered sets method. Water Resources Management, 31(11), 3329-3341.


• Hua, H., Hovestadt, L., Tang, P., & Li, B. (2019). Integer programming for urban design. European Journal of Operational Research, 274(3), 1125-1137.


• Huchette, J., & Vielma, J. P. (2019). A combinatorial approach for small and strong formulations of disjunctive constraints. Mathematics of Operations Research, 44(3), 793-820.


• Epelle, E. I., & Gerogiorgis, D. I. (2020). A Computational Performance Comparison of MILP vs. MINLP Formulations for Oil Production Computers & Chemical Engineering, 106903.


• Akbari-Dibavar, A.,Mohammadi-Ivatloo, B., & Zare, K. (2020). Optimal stochastic bilevel scheduling of pumped hydro storage systems in a pay-as-bid energy market environment. Journal of Energy Storage, 31, 101608.


• MirHassani, S. A., &Hooshmand, F. (2019). Methods and Models in Mathematical Programming. Springer International Publishing.