بهینه‌سازی مدل برنامه‌ریزی تولید بلند مدت با در نظر گرفتن عدم قطعیت عیار با روش آزادسازی لاگرانژی - الگوریتم‌ خفاش

نوع مقاله : مقاله پژوهشی

نویسندگان

1 گروه مهندسی نفت و معدن، واحد تهران جنوب، دانشگاه آزاد اسلامی

2 گروه مهندسی معدن، دانشگاه کاشان

10.29252/anm.2020.12767.1420

چکیده

یکی از ارکان اصلی برنامه‌ریزی معدن، برنامه‌ریزی تولید بلند مدت است که یک امر بسیار مهم در تحقیقات نظری استخراج معادن روباز بوده و توزیع جریان نقدینگی را در سراسر عمر معدن مشخص می‌نماید. در واقع هدف برنامه‌ریزی، بیشینه‌کردن ارزش خالص فعلی از مجموع سودهایی است که در آینده تولید می‌شوند. برای رسیدن به این هدف باید همه محدودیت‌های عملیاتی از قبیل شیب، آمیختن عیارهای مختلف، تولید ماده معدنی و ظرفیت استخراج راضی‌کننده باشند. طرح‌های تولید بلند مدت بهینه در برابر عدم قطعیت‌های مربوطه با داده‌های مدل بهینه‌سازی دارای حساسیت هستند. در میان عدم قطعیت‌ها، عدم قطعیت عیار، سهم عمده‌ای را در دقت برنامه‌ریزی تولید بلند مدت خواهد داشت. در این مقاله مدل ترکیبی به وسیله روش آزادسازی لاگرانژی و الگوریتم خفاش برای حل مساله برنامه‌ریزی تولید بلند مدت معادن روباز ارائه شده که در آن عدم قطعیت عیار نیز لحاظ گردیده است. رویکردهای جدید پیشنهاد شده براساس بهینه سازی ضرایب لاگرانژ و مقایسه آن با روش سنتی است. الگوریتم خفاش برای به روزرسانی ضرایب لاگرانژ مورد استفاده قرار گرفته شده است. برای حل و اعتبارسنجی مدل به دست‌آمده، معدن سنگ آهن چادرملو به عنوان مورد مطالعاتی مناسب، در نظرگرفته شده است. نتایج حاصل از مطالعه موردی نشان می‌دهد که استراتژی‌ ترکیبی می‌تواند راه‌حل قابل قبولی را نسبت به روش تقریبی سنتی ارائه کند؛ به طوری ‌که، در طول یک دوره معین ارزش خالص فعلی با استفاده از روش ترکیبی پیشنهادی 69/6 درصد بیشتر از روش سنتی موجود است.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

Optimization of the Long-Term Production Scheduling Model by Considering the Grade Uncertainty by the Lagrangian Relaxation Method - Bat Algorithm

نویسندگان [English]

  • Kamyar Tolouei 1
  • Ehsan Moosavi 1
  • Amirhossein Bangian Tabrizi 1
  • Peyman Afzal 1
  • Abbas Aghajani Bazazi 2
1 Dept. of Petroleum and Mining Engineering, South Tehran Branch, Islamic Azad University, Tehran, Iran
2 Dept. of Mining Engineering, University of Kashan, Kashan, Iran
چکیده [English]

Summary
One of the main problems of mine planning is long-term production scheduling, which is very important in the theoretical research of open-pit mining and determines the distribution of cash flow throughout the life of the mine. In fact, the purpose is to maximize the net present value of the future profits generated. Among the uncertainties, grade uncertainty will play a major role in the accuracy of long-term production scheduling. In this paper, a hybrid model is presented by the Lagrangian relaxation method and bat algorithm to solve the problem of long-term production of open-pit mines, in which the uncertainty of the grade is also considered. The new approaches proposed are based on optimizing Lagrange coefficients and comparing them with the traditional method. The bat algorithm is used to update the Lagrange coefficients. The results of the case study show that the hybrid strategy can provide an acceptable solution compared to the traditional approximation method so that over a given period the net present value using the proposed hybrid method is 6.69% higher than the traditional one.
 
Introduction
In recent years, a new approach to cheaper computational algorithms, such as meta-heuristic techniques, has attracted more attention from researchers to solve production scheduling problems. Although these techniques do not guarantee optimization as a final solution for production, they can provide suitable solutions for production at a lower computational cost.
 
Methodology and Approaches
In this paper, an optimal hybrid model by the Lagrangian relaxation method and bat algorithm is presented to solve the problem of long-term production of open-pit mines, where the uncertainty of the grade is also considered. The newly proposed approach is based on optimizing Lagrange coefficients and comparing it with the traditional method. The results of the proposed approach are also compared with the combined approach based on the Lagrangian relaxation method and genetic algorithm. The bat algorithm is used to update the Lagrange coefficients.
 
Results and Conclusions
The results of a case study show that the Lagrangian relaxation method can provide a suitable solution to the main problem and the combined strategy can produce a more effective solution than the traditional approximation method. It was also found that the proposed method has advantages, such as stable convergence property and prevention of early convergence. Over a given period, the net present value using the LR-BA hybrid method is 6.69% higher than the traditional method and also 5.58% higher than the LR-GA method.

کلیدواژه‌ها [English]

  • Open pit mine
  • Long-term production scheduling
  • Grade uncertainty
  • Lagrangian relaxation
  • Bat algorithm

اصل مقاله

در عملیات معدنکاری در ابتدا از طریق پی‌جوئی و اکتشاف، محدوده‌ای که پتانسیل معدن شدن را دارد شناسایی شده و در سراسر منطقه کارهای اکتشافی از قبیل: حفاری، نقشه‌برداری و تفسیر زمین‌شناسی صورت می‌گیرد. در صورتی که روش روباز برای استخراج تأیید شود آنگاه مدل ماده معدنی به وسیله روش‌های مدلسازی عددی بسط داده می‌شود و ذخیره معدنی به زون‌هایی تقسیم‌بندی می‌شود که سرانجام این زون‌ها از طریق بلوک‌های مستطیلی شکل سه بعدی بلوک‌بندی می‌شوند. در هر بلوک عیار و عنصر کانسار و دیگر اطلاعات لازم از طریق مغزه‌گیری گمانه‌های حفاری مشخص می‌شود که در ارزیابی اقتصادی ارزش بلوک‌ها از آنها استفاده می‌شود. سپس از مدل توده معدنی که شامل توزیع کانسار و ارزش اقتصادی مورد انتظار آن می‌باشد در تعیین محدوده نهایی پیت مورد استفاده قرار می‌گیرند، که محدوده اقتصادی معدن را مشخص می‌نماید. به بیان دیگر، برنامه‌ریزی تولید سالانه یک مساله تصمیم‌گیری است که در آن بلوک‌ها در درون محدوده پیت باید به گونه‌ای و در زمانی استخراج شوند که بیشترین ارزش خالص فعلی[i](NPV) با در نظرگرفتن محدودیت‌های موجود حاصل شود. بلوک‌ها در داخل حجم‌های بزرگ‌تری که فاز استخراجی نامیده می‌شوند، جمع می‌شوند که آنها را می‌توان در طول زمان مشخص استخراج نمود. مشکل در یافتن مناسب­ترین راه ممکن برای جمع آوری بلوک‌ها در یک حجم که بتواند بیشترین مقدار ارزش خالص فعلی را بدهد، است.


[i] Net present value

مراجع

[1]           Johnson, T.B. (1969) Optimum Production Scheduling. Proceedings, 8th International Symposium on Computers and Operations research, Salt Lake City, 539-562.
[2]           Williams, C.E. (1974) Computerized Year-by-Year Open Pit Mine Scheduling. Society of Mining Engineers, AIME, Transactions, 256, 45-52.
[3]           Gershon, M.E. (1983). Optimal mine production scheduling: evaluation of large scale mathematical programming approaches. International journal of mining engineering, 1(4), 315-329.
[4]           Dagdelen K. and Johnson, T.B. (1986). Optimum Open Pit Mine Production Scheduling by Lagrangian Parametrization. Proceeding of the 19th International Symposium on the Application of Computers and Operations Research in the Mineral Industry, Pennsylvania State University, University Park, Pennsylvania, 13, 127-142.
[5]           Elevli, B. (1995). Open pit mine design and extraction sequencing by use of OR and AI concept. International Journal of surface mining, reclamation and environmental, 9, 149-153
[6]           Rovenscroft, P.J. (1992). Risk analysis for mine planning by conditional simulation. Trans. Instn Min. Metall. (Sec.A: Min. Industry), 101, 82-88.
[7]           Tolwinski, B. (1994). Scheduling production for open pit mines. Proceedings of APCOM’98, 19-23.
[8]           Lerchs, H. and Grossman, F. (1965). Optimum design of open-pit mines. Trans. CIM, 58, 47-54.
[9]           Akaike, A. and Dagdelen, K. (1999). A strategic production scheduling method for an open pit mine. Proceedings of the 28th Application of Computers and Operation Research in the Mineral Industry, 729–738.
[10]         Whittle, D. (2000). Proteus Environment: Sensitivity Analysis Made Easy presented at Whittle North American Strategic Mine Planning Conference, Colorado.
[11]         Johnson T.B., Dagdelen K. and Ramazan S. (2002). Open pit mine scheduling based on fundamental tree algorithm, Proceeding of the 30th International Symposium on the Application of Computers and Operations Research in the Mineral Industry, (SME: Littleton), 147-159.
[12]         Dimitrakopoulos, R. and Ramazan, S. (2004). Uncertainty based production scheduling in open pit mining. SME Transactions, 316, 106-112.
[13]         Ramazan, S. and Dimitrakopoulos, R. (2004). Recent application of operations research in open pit mining, SME Transactions, 316, 73-78.
[14]         Ramazan, S. and Dimitrakopoulos, R. (2007). Stochastic optimization of long term production scheduling for open pit mines with a new integer programming formulation. Orebody Modelling and Strategic Mine Planning, The Australasian Institute of Mining and Metallurgy, Spectrum Series, 14(2), 385-392.
[15]         Boland, N., Dumitrescu, I., Froyland, G., and Gleixner, A.M. (2009). LP-based disaggregation approaches to solving the open pit mining production scheduling problem with block processing selectivity. Computers & Operations Research, 36(4), 1064-1089.
[16]         Bley, A., Boland, N., Fricke, C., and Froyland, G. (2010). A strengthened formulation and cutting planes for the open pit mine production scheduling problem. Computers & Operations Research, 37(9), 1641-1647.
[17]         Lamghari, A. and Dimitrakopoulos, R. (2012). A diversified Tabu search approach for the open pit mine production scheduling problem with metal uncertainty. European Journal of Operational Research, 222(3), 642-652.
[18]         Sattarvand, J. and Niemann-Delius, C. (2013). A new metaheuristic algorithm for long-term open pit production planning. Archives of Mining Sciences, 58(1), 107-118.
[19]         Dimitrakopoulos, R. and Jewbali, A. (2013). Joint stochastic optimization of short and long term mine production planning: Method and application in a large operating gold mine. IMM Transactions, Mining Technology, 122(2), 110-123.
[20]         Soleymani Shishvan, M. and Sattarvand, J. (2015). Long term production planning of open pit mines by ant colony optimization. European Journal of Operational Research, 240(3), 825-836.
[21]         Mokhtarian, M. and Sattarvand, J. (2016). An Imperialist Competitive Algorithm for Solving the Production Scheduling Problem in Open Pit Mine, Int. J. Min. & Geo-Eng. 50(1), 131-143.
[22]         Mokhtarian, M. and Sattarvand, J. (2016). Commodity price uncertainty propagation in open-pit mine production planning by Latin hypercube sampling method. Journal of Mining & Environment, 7(2), 215-227.
[23]         Lamghari, A. and Dimitrakopoulos, R. (2016). Progressive hedging applied as a metaheuristic to schedule production in open pit mines accounting for reserve uncertainty. European Journal of Operational Research, 253(3), 843-855.
[24]         Lamghari, A. and Dimitrakopoulos, R. (2016). Network-flow based algorithms for scheduling production in multi-processor open pit mines accounting for metal uncertainty. European Journal of Operational Research, 250(1), 273-290.
[25]         Sayyadi, A., Fathianpour, N. and Mousavi, A. (1390). "Performance comparison of production planning of Esfordi phosphate mine using linear and dynamic programming." Journal of Analytical and Numerical Methods in Mining Engineering, 1(2): 1-8 (In Persian).
[26]         Sayyadi, A., Fathianpour, N. and Mousavi, A. (1391). "Application of Conditional Sequential Gaussian Simulation in Uncertainty Assessment of the Estimated Blocks Grade in Esfordi Phosphate Mine." Journal of Analytical and Numerical Methods in Mining Engineering, 2 (3): 44-52 (In Persian).
[27]         Zare Naghadehi, M., Dehghani, H. and Naderipour, R. (1396). "The Probabilistic Analysis of Block Economic Value (BEV) in Open-Pit Mines Considering the Effect of Uncertainties in Metal Price and Operational Costs." Journal of Analytical and Numerical Methods in Mining Engineering, 7 (13): 15-26 (In Persian).
[28]         Goovaerts, P. (1997). Geostatistics for Natural Resources Evaluation Oxford University Press.
[29]         Deutsch, C.V. and Journel, A.G. (1997).  GSLIB: Geostatistical Software Library and User's Guide (Applied Geostatistics Series).
[30]         Menabde, M., Froyland, G., Stone, P. and Yeates, G. (2004).  Mining schedule optimization for conditionally simulated orebodies. Proceedings of the international symposium on orebody modelling and strategic mine planning: uncertainty and risk management. 347-352.
[31]         Gholamnejad, J., Osanloo, M. and Khorram, E. (2008).  A chance constrained integer programming model for open pit long-term production planning [J]. IJE Transactions A: Basics, 21(4), 307-318.
[32]         Denby, B. and Schofield, D. (1995). Inclusion of risk assessment in open pit design and planning. Trans. Instn Min. Metall. (Sec.A: Min. Industry), 104, 67-71.
[33]         Dowd, P.A. (1994). Risk assessment in reserve estimation and open pit planning, Trans. Instn Min. Metall. (Sec.A: Min. Industry), 103, 148-154.
[34]         Dimitrakopoulos, R., Farrelly, C.T. and Godoy, M. (2002). Moving forward from traditional optimization: grade uncertainty and risk effects in open pit design. Transactions of the Institutions of Mining and Metallurgy, Section A: Mining Technology, 111, 82-88.
[35]         Godoy, M. and Dimitrakopoulos, R. (2004). Managing risk and waste mining in long-term production scheduling of open pit mines. SME Transactions, 316, 43-50.
[36]         Ramazan, S. and Dimitrakopoulos, R. (2004). Traditional and new MIP models for production planning with in-situ grade variability, International Journal of Surface Mining, Reclamation and Environment, 18(2), 85-98.
[37]         Gholamnejad J., Osanloo, M. and Karimi, B. (2006). A chance-constrained programming approach for open pit long-term production scheduling in stochastic environments. The Journal of the South African Institute of Mining and Metallurgy, 106, 105-114.
[38]         Gholamnejad, J. and Osanloo, M. (2007). A chance constrained integer programming model for open pit long-term production planning. Proceedings of the sixteenth international symposium on mine planning and equipment selection (MPES 2007), 359-372.
[39]         Gholamnejad, J. and Moosavi, E. (2012).  A new mathematical programming model for long-term production scheduling considering geological uncertainty. The Journal of the Southern African Institute of Mining and Metallurgy, 112(2), 77-81.
[40]         Goodfellow, R., and Dimitrakopoulos, R. (2013). Algorithmic integration of geological un-certainty in push back designs for complex multi-process open-pit mines. Mining Technology, 122(2), 67-77.
[41]         Leite, A. and Dimitrakopoulos, R. (2014). Stochastic optimization of mine production scheduling with uncertain ore/metal/waste supply. Int J Min Sci Technol.
[42]         Koushavand, B., Askari-Nasab, H. and Deutsch, C.V. (2014). A linear programming model for long-term mine planning in the presence of grade uncertainty and a stockpile. International Journal of Mining Science and Technology, 24, 451-459.
[43]         Goodfellow, R. and Dimitrakopoulos, R. (2016). Global optimization of open pit mining complexes with uncertainty, Applied Soft Computing, 40, 292-304.
[44]         Khan, A. (2018). Long-term production scheduling of open pit mines using particle swarm and bat algorithms under grade uncertainty. Journal of the Southern African Institute of Mining and Metallurgy, 118, 361-368.
[45]         Fisher, M.L. (1981). The Lagrangian relaxation method for solving integer programming problems. Management Science Journal, 27(1), 1−18.
[46]         Yang, X.S. (2010). A new metaheuristic bat-inspired algorithm. Nature inspired cooperative strategies for optimization (NICSO 2010), Springer Berlin Heidelberg, 65- 74.
[47]         Mirjalili, S.A., Mirjalili, S.M. and Yang, X.S. (2014). Binary bat algorithm, Neural Computing and Applications 25, 663-681.
[48]         Yang, X. S. and He, X. (2013). Bat algorithm: literature review and applications, International Journal of Bio-Inspired Computation, 5(3), 141-149.
[49]         Yang, X.S. (2011). Bat algorithm for multi-objective optimization. International Journal of Bio-Inspired Computation, 3(5), 267-274.
روش های تحلیلی و عددی در مهندسی معدن
دوره 10، شماره 24
آبان 1399
صفحه 13-26

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سابقه مقاله

  • تاریخ دریافت: 16 مرداد 1398
  • تاریخ بازنگری: 07 دی 1398
  • تاریخ پذیرش: 27 فروردین 1399

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