پیوندهای مفید
اخبار و اعلانات
آمار
| تعداد نشریات | 41 |
| تعداد شمارهها | 1,102 |
| تعداد مقالات | 9,447 |
| تعداد مشاهده مقاله | 17,031,973 |
| تعداد دریافت فایل اصل مقاله | 11,942,973 |
Applying Duality Results to Solve the Linear Programming Problems with Grey Parameters | ||
| Control and Optimization in Applied Mathematics | ||
| دوره 5، شماره 1، فروردین 2020، صفحه 15-28 اصل مقاله (375.78 K) | ||
| نوع مقاله: Research Article | ||
| شناسه دیجیتال (DOI): 10.30473/coam.2021.56072.1152 | ||
| نویسندگان | ||
| Farid Pourofoghi* ؛ Davood Darvishi Salokolaei | ||
| Department of Mathematics, Payame Noor University (PNU), Tehran, Iran. | ||
| چکیده | ||
| Linear programming problems have exact parameters. In most real-world, we are dealing with situations in which accurate data and complete information are not available. Uncertainty approaches such as fuzzy and random can be used to deal with uncertainties in real-life. Fuzzy and stochastic theories cannot be used if the number of experts and the level of experience is so low that it is impossible to extract membership functions or the number of samples is small. To solve these problems, the grey system theory is proposed. In this paper, a linear programming problem in a grey environment with resources in interval grey numbers is considered. Most of the proposed methods for solving grey linear programming problems become common linear programming problems. However, we seek to solve the problem directly without turning it into a standard linear programming problem for the purpose of maintaining uncertainty in the original problem data in the final solution. For this purpose, we present a method based on the duality theory for solving the grey linear programming problems. This method is more straightforward and less complicated than previous methods. We emphasize that the concept presented is beneficial for real and practical conditions in management and planning problems. Therefore, we shall illustrate our method with some examples in different situations. | ||
| کلیدواژهها | ||
| Grey number؛ Grey linear programming؛ Duality theory؛ Uncertainty | ||
| عنوان مقاله [English] | ||
| کاربرد نتایج دوالیتی برای حل مساله برنامهریزی خطی با پارامترهای خاکستری | ||
| نویسندگان [English] | ||
| فرید پورافقی؛ داود درویشی سلوکلایی | ||
| گروه ریاضی، دانشگاه پیام نور، تهران، ایران | ||
| چکیده [English] | ||
| مسائل برنامهریزی خطی دارای پارامترهای دقیق هستند. در دنیای واقعی، ما با شرایطی روبرو هستیم که اطلاعات دقیق و کامل در دسترس نیست. در این شرایط میتوان از رویکردهای عدم قطعیت مانند فازی و تصادفی برای مقابله با عدم اطمینان در زندگی واقعی استفاده کرد. اگر تعداد متخصصان و سطح تجربه به قدری کم باشد که استخراج توابع عضویت غیرممکن باشد یا تعداد نمونه ها کم باشد، نمیتوان از نظریههای فازی و تصادفی استفاده کرد. برای حل این مشکل، نظریه سیستم خاکستری ارائه شد. در این مقاله ، یک مساله برنامهریزی خطی در یک محیط خاکستری با منابعی به صورت اعداد خاکستری بازهای در نظر گرفته شده است. بیشتر روشهای پیشنهادی برای حل مسائل برنامهریزی خطی خاکستری بر مبنای تبدیل آن به یک مساله برنامهریزی خطی معمولی میباشند. با این حال، ما به دنبال حل مستقیم مساله و بدون تبدیل آن به یک مساله برنامهریزی خطی استاندارد به منظور حفظ عدمقطعیت دادههای ورودی در جوابهای بدست آمده نهایی هستیم. برای این منظور، ما یک روش، بر اساس دوال مساله برنامهریزی خطی با منابع خاکستری ارائه میدهیم. روش پیشنهادی سادهتر از روشهای قبلی بوده و از پیچیدگی کمتری نسبت به آنها برخوردار است. تأکید میشود که روش ارائه شده برای شرایط واقعی و عملی در مسائل مدیریت و برنامهریزی می تواند مفید باشد. بنابراین، کارایی روش پیشنهادی با ارایه چند مثال در شرایط مختلف نشان داده شده است. | ||
| کلیدواژهها [English] | ||
| عدد خاکستری, برنامهریزی خطی خاکستری, نظریه دوگان, عدم قطعیت | ||
| مراجع | ||
|
[1] Azar A. 2007. “Operation research”, Payame Noor University Press, Tehran.
[2] Bai Y., Wang P., and Xie J. 2014. “Optimization of urban water supply schemes based on grey system theory”, International Journal of Control and Automation, 7 (9), 239-246.
[3] Baidya A., Bera U.K. and Maiti M. 2016. “The grey linear programming approach and its application to multi-objective multi-stage solid transportation problem”, Research, 53 (3), 500–522.
[4] Bevilacqua M., Ciarapica F. E. and Giacchetta G. 2006. “A fuzzy-QFD approach to supplier selection”. Journal of Purchasing and Supply Management, 12, 14-27.
[5] Bhattacharyya R. 2015. “A grey theory-based multiple attribute approach for R&D project portfolio selection”, Fuzzy Information and Engineering, 7, 211-225.
[6] Cheng F.J., Hui, S.H., and Chen, Y.C. 2002. “Reservoir operation using grey fuzzy stochastic dynamic programming”, Hydrological Processes, 16, 2395-2408.
[7] Darvishi D. 2019. “Some duality results in grey linear programming problem”, Journal of Operational Research and Its Applications, 16(3), 55-68.
[8] Darvishi D., Forrest J., and Liu, S. 2019. “A comparative analysis of grey ranking approaches”, Grey Systems: Theory and Application, 9(4), 472-487.
[9] Darvishi D., Liu S., and Nasseri S.H. 2018. “A new approach in animal diet using grey system theory”, Grey Systems: Theory and Application, 8(2), 167-180.
[10] Deng J.L. 1989. “Introduction to grey system theory”, The Journal of Grey Systems, 1(1), 1-24.
[11] Honghua W., and Yong M. 2013. “Emerging technologies for information systems”, Computing and Management Lecture Notes in Electrical Engineering, 236, 97-106.
[12] Kumar M. 2014. “Applying weakest t-norm based approximate intuitionistic fuzzy arithmetic operations on different types of intuitionistic fuzzy numbers to evaluate the reliability of PCBA fault”, Applied Soft Computing, 23, 387–406.
[13] Li Q.X., and Lin Y. 2014. “A briefing to grey systems theory”, Journal of Systems Science and Information, 2(2), 178-192.
[14] Li Q.X., Liu S.F., and Wang N.A. 2014. “Covered solution for a grey linear program based on a general formula for the inverse of a grey matrix”, Grey Systems: Theory and Application, 4(1), 72-94.
[15] Liu S.F., Dang Y., and Forrest J. 2009. “On positioned solution of linear programming with grey parameters”, International Conference on Systems, Man and Cybernetics, USA, 751-756.
[16] Liu S.F., and Lin Y. 2006. “Grey Information, theory and Practical Applications”, Springer.
[17] Liu S.F., and Xie N. 2013. “Grey systems theory and its applications”. The Science Press of China, Beijing.
[18] Luhandjula M.K., and Rangoaga M.J. 2014. “An approach for solving a fuzzy multi-objective programming problem”, European Journal of Operational Research, 232, 249–255.
[19] Mahmoudi A., Feylizadeh M.R., and Darvishi D. 2018. “A note on “a multi-objective programming approach to solve grey linear programming”, Grey Systems: Theory and Application, 8(1), 35-45.
[20] Michael C.F., Olvi L.M., and Stephen J.W. 2007. “Linear programming with MATLAB”, MOS-SIAM Series on Optimization.
[21] Moore R.E., Kearfott R.B., and Cloud M.J. 2009. “Introduction to interval analysis”, Society for Industrial and Applied Mathematics Press, Philadelphia.
[22] Mujumdar P., and Karmakar S. 2008. “Grey fuzzy multi-objective optimization”. In Fuzzy Multi-Criteria Decision Making, Springer.
[23] Nasseri S.H. and Darvishi D. 2018. “Duality results on grey linear programming problems”, Journal of Grey Systems, 30(3), 127-142.
[24] Nasseri S.H., Yazdani A., and Darvishi D. 2016. “A primal simplex algorithm for solving linear programming problem with grey cost coeffcients”, Journal of New Researches in Mathematics, 1(4), 115-135.
[25] Ramesh G., and Ganesan K. 2012. “Duality theory interval linear programming”, International Organization of Scientific Research Journal of Mathematics, 4(4), 39-47.
[26] Ramik J. 2006. “Duality in fuzzy linear programming with possibility and necessity relations”, Fuzzy Sets and Systems, 157, 1283–1302.
[27] Razavi Hajiagha S.H., Akrami H. and Hashemi S.S. 2012. “A multi-objective programming approach to solve grey linear programming”, Grey Systems: Theory and Application, 2 (2), 259-271.
[28] Rodder W., and Zimmermann H.J. 1980. “Duality in fuzzy linear programming”, International Symposium on External Methods and Systems Analysis, University of Texas at Austin, 415– 427.
[29] Rohn J. 1980. “Duality in interval linear programming”, International Symposium on Interval Mathematics, Academic Press, New York, 521–529.
[30] Soyster A.L. 1979. “Inexact linear programming with generalized resource sets”, European Journal Operations Research, 3, 316– 321. [31] Singh V., Yadav S.P., and Singh S.K. 2019. “Duality theory in Atanassov’s intuitionistic fuzzy mathematical programming problems: Optimistic, pessimistic and mixed approaches”, Annals of Operations Research, 275 (2).
[32] Thuente D.J. 1980. “Duality theory for generalized linear programs with computational methods”, Operations Research. 28(4), 1005–1011.
[33] Voskoglou M.G. 2018. “Solving linear programming problems with grey data”, Oriental Journal of Physical Sciences, 3 (1), 17-23.
[34] Wei J., Zhou L., Wang F., and Wu D. 2015. “Work safety evaluation in Mainland China using grey theory”, Applied Mathematical Modelling, 39, 924–933.
[35] Xia D.C., Li B.T., Wang J.P., and Shi G.P. 2014. “Research on general data envelopment analysis model based on the grey linear programming”, Applied Mechanics and Materials, 577, 828-831.
[36] Xiao X.P. 1998. “New study on the solution of grey linear programming”, The Journal of Grey System, 10 (2), 133-140.
[37] Yang Y. and Liu S.F. 2011. “Reliability of operations of grey numbers using kernels”, Grey Systems: Theory and Application, 1(1), 57–71.
[38] Yan X., and Song Z. 2014. “The portfolio models of contained grey profit under uncertainty”, Grey Systems: Theory and Application, 4(3), 487-494. | ||
|
آمار تعداد مشاهده مقاله: 342 تعداد دریافت فایل اصل مقاله: 231 |
||