Optimization of the Inflationary Inventory Control Model under Stochastic Conditions with Simpson Approximation: Particle Swarm Optimization Approach

Document Type : Review article


1 Faculty of Industrial Engineering, Islamic Azad University, Science and Research Branch, Saveh, Iran

2 Faculty of Industrial Engineering, Kharazmi University, Tehran, Iran

3 Faculty of Industrial Engineering, Islamic Azad University, Karaj Branch, Iran

4 Department of Information Technology of Sufi Razi, Zanjan, Iran


In this study, we considered an inflationary inventory control model under non-deterministic conditions. We assumed the inflation rate as a normal distribution, with any arbitrary probability density function (pdf). The objective function was to minimize the total discount cost of the inventory system. We used two methods to solve this problem. One was the classic numerical approach which turned out to be prohibitively difficult. The other was a proposed combination method which used Simpson approximation and particle swarm optimization (PSO). To illustrate the theoretical results, we have provided numerical examples.


Main Subjects

Article Title [Persian]

بهینه‌سازی مدل کنترل موجودی تورمی تحت شرایط احتمالی با رویکرد تقریب سیمپسون : بهینه‌سازی ازدحام ذرات

Authors [Persian]

  • سید مصطفی اورند 1
  • ابوالفضل میرزازاده 2
  • فرزانه احمدزاده 3
  • فرید طالب لو 4
1 دانشکدة مهندسی صنایع، دانشگاه آزاد اسلامی، واحد علوم و تحقیقات، ساوه، ایران
2 دانشکدة مهندسی صنایع، دانشگاه خوارزمی، تهران، ایران
3 دانشکدة مهندسی صنایع، دانشگاه آزاد اسلامی، واحد کرج، ایران
4 گروه فناوری اطلاعات، دانشگاه صوفی رازی، زنجان، ایران
Abstract [Persian]

این مقاله یک مدل کنترل موجودی تورمی تحت شرایط غیرقطعی در نظر می‌گیرد. فرض می‌کنیم نرخ تورم از تابع توزیع نرمال، با هر تابع چگالی احتمال دلخواه (PDF) پیروی می‌کند. تابع هدف به حداقل رساندن هزینة تخفیف کل سیستم موجودی است. از دو روش جهت حل این مسئله استفاده شد: رویکرد کلاسیک عددی که معلوم می‌شود روش دشواری است؛ و روش ترکیبی پیشنهادی با استفاده از تقریب سیمپسون و بهینه‌سازی ازدحام ذرات (PSO). مثال‌های عددی جهت نشان دادن نتایج نظری ارائه شده است.

Keywords [Persian]

  • احتمالی
  • ازدحام ذرات
  • بهینه‌سازی تقریب سیمپسون
  • تورم
  • سیستم موجودی
Kennedy, J. & Eberhart, R. (1995). Particle swarm optimization (Vol. 4). Piscataway, NJ: IEEE Press.
Parsopoulos, K.; Skouri, K., & Vrahatis, M. (2008). Particle Swarm Optimization for Tackling Continuous Review Inventory Models. M. Giacobini et al. (Eds.): EvoWorkshops, 103–112.
A.Lubik, T. & Teo, W. (2012). “Inventories, inflation dynamics and the New Keynesian”. European Economic Review. 56(3), 327–346.
Blum, C., & Merkle, D. (2011). Swarm Intelligence: Introduction and Applications. Springer-Verlag.
Buzacott, J. (1975). “Economic Order Quantities with Inflation”. Journal of the Operational Research Society, (26), 553-558.
Chapra, S. (2005). Applied Numerical Methods W/MATLAB: For Engineers & Scientists (3rd ed., Vol. 1). New York, USA: McGraw-Hill Companies, Incorporated.
Chung, K. (3003). “An Algorithm for an Inventory Model with Inventory-Level-Dependent Demand Rate”. Computers and Operations Research(30), 1311-1317.
Datta, T., & Pal, A. (1991). “Effects of Inflation and Time-Value of Money on an Inventory Model with Linear Time-Dependent Demand Rate and Shortages”. Journal of operational Research, (52), 326-333.
Domoto, E.; Okuhara, K.; Ueno, N. & Ishii, H. (2007). “Target Inventory Strategy in Multistage Supply Chain by Particle Swarm Optimization”. Asia Pacific Management Review, 12(2), 117-122.
Fatih Taşgetiren, M. & Liang, Y.C. (2003). “A Binary Particle Swarm Optimization Algorithm for Lot Sizing Problem”. Journal of Economic and Social Research, 5(2), 1-20.
Hariga, M., & Ben-Daya, M. (1996). “Optimal Time Varying Lot-Sizing Models under Inflationary Conditions”. European Journal of operational Research, (89), 313-325.
Horowitz, I. (2000). “EOQ and Inflation Uncertainty”, International Journal of Production Economics, (65), 217-224.
Hsu, C.H.; Tsou, C.S. & Yu, F.J. (2009). “Multicriteria Tradeoffs In Inventory Control Using Memetic Particle Swarm Optimization”. International Journal of Innovative Computing, Information and Control, 5(11), 3755–3768.
Jeffrey, A. & Dai, H. H. (1995). Handbook of Mathematical Formulas and Integrals (4 ed., Vol. 1). (L. S. Yuhasz, Ed.) San Diego, California, USA: Academic Press.
Kennedy, J.; Eberhart, R. & Shi, Y. (2001). Swarm Intelligence (Vol. 1). San Francisco, CA, USA: Morgan Kaufmann.
Li, R.; Lan, H. & Mawhinney, J. (2010). “A Review on Deteriorating Inventory Study”. J. Service Science & Management, 3, 117-129.
Mirzazadeh, A. (2007). Effects of Uncertain Inflationary Conditions on Inventory Models Using the Average Annual Cost and the Discounted Cost. Bangkok: AIMS.
Mirzazadeh, A. & Sarfaraz, A. (1997). Constrained Multiple Items Optimal Order Policy under Stochastic Inflationary Conditions. USA, San Diego.
Misra, R. (1979). “A Note on Optimal Inventory Management under Inflation”. Naval Research Logistics Quarterly, (26), 161-165.
NEETU, & Tomer, A. (2012). A Deteriorating Inventory Model Under Variable Inflation When Supplier Credits Linked to Order Quantity. Procedia Engineering.
Orand, S.; Mirzazadeh, A. & Ahmadzadeh, F. (2012). Application of Particle Swarm Optimization Approach in the Inflationary Inventory Model under Stochastic Conditions. Lulea.
Rao, S. (2009). “Engineering optimization: Theory and practice”. In Engineering optimization: Theory and practice (Vol. 1, pp. 708-714). Hoboken, New Jersey, USA: John Wiley and Sons.
Sarker, B. & Pan, H. (1994). “Effects of Inflation and the Time Value of Money on Order Quantity and Allowable Shortage”. International Journal of Production Economics, (34), 65-72.
Sedighizadeh, D. & Masehian, E. (2009). “Particle swarm optimization methods, taxonomy and applications”. International Journal of Computer Theory and Engineering (IJCTE), 1(5), 486-502.
Shi, Y. & Eberhart, R.C. (1998). Parameter Selection in Particle Swarm Optimization. Berlin: Springer-Verlag.
Sue-Ann, G.; Ponnambalam, S. & Jawahar, N. (2012, October). “Evolutionary algorithms for optimal operating parameters of vendor managed inventory systems in a two-echelon supply chain”. Advances in Engineering Software, 52, 47–54.
Tsai, C.Y. & Yeh, S.W. (2008). “A multiple objective particle swarm optimization approach for inventory classification”. International Journal of Production Economics, 656–666.
Varga, T.; Király, A. & Abonyi, J. (2013). “Improvement of PSO Algorithm by Memory-Based Gradient Search—Application in Inventory Management”. Swarm Intelligence and Bio-inspired Computation Theory and Applications. 403–422.
Vrat, P. & Padmanabhan, G. (1990). “An Inventory Model under Inflation for Stock Dependent Consumption Rate Items”. Engineering Costs and Production Economics, (19), 379-383.
Wikipedia. (2012, December 28). Simpson's Rule. Retrieved from Wikipedia: http://en.wikipedia.org/wiki/Simpson's_rule.
Wong, J.T.; Chen, K.H. & Su, C.T. (2009). “Replenishment Decision Support System Based on Modified Particle Swarm Optimization in a VMI Supply Chain”. International Journal of Industrial Engineering: Theory, Applications and Practice, 16(1), 1-12.
Yang, H.L. (2012). Two-warehouse partial backlogging inventory models with three-parameter Weibull distribution deterioration under inflation. Int. J. Production Economics.
Yang, H.L. & Chang, C.T. (3013). “A two-warehouse partial backlogging inventory model for deteriorating items with permissible delay in payment under inflation”. Applied Mathematical Modelling, 37(5), 2717–2726.