A Lot-Sizing Model for Non-Instantaneous Deteriorating Products Under Advance Payment and Non-Linear Partial Backlogging

Document Type : Research Paper


1 School of Industrial and Systems Engineering, College of Engineering, University of Tehran, Tehran, Iran

2 Department of Industrial Engineering, KHATAM University, Tehran, Iran


In real life conditions, the buyers sometimes pay all or a percentage of the product price before receiving it, and the wholesaler sometimes allows them to prepay it at equal intervals. The present study develops a new mathematical model for products with non-instantaneous deteriorating rates by considering consecutive advance payments. In the proposed inventory model, the shortage is consisting of lost sales along with backorders simultaneously. In addition, the model considers the backlogging as totally dependent on the waiting time for the further cycle. In addition, the appropriate conditions to achieve the optimal solutions have been developed, and numerical instances have been provided to verify and evaluate the results and solution method. The useful methods to effectively reduce the annual total cost are provided according to the results of the sensitivity analysis.


Main Subjects

Article Title [Persian]

مدل تعیین اندازه دسته تولید برای محصولات زوال‌پذیر غیر آنی تحت پیش‌پرداخت و پشتیبانی جزئی غیر خطی

Authors [Persian]

  • سارا توسلی 1
  • ندا معنوی زاده 2
  • آیدا رضائی 1
  • مسعود ربانی 1
1 دانشکده مهندسی صنایع و سیستم‌ها، دانشکده فنی، دانشگاه تهران، تهران، ایران
2 گروه مهندسی صنایع، دانشگاه خاتم، تهران، ایران
Abstract [Persian]

در واقعیت، گاهی خریداران تمام یا درصدی از قیمت محصول را قبل از دریافت آن می‌پردازند و گاهی عمده‌فروش به آن‌ها اجازه می‌دهد تا در بازه‌های زمانی مساوی پیش‌پرداخت را انجام دهند. مطالعه حاضر با در نظر گرفتن پیش‌پرداخت‌های متوالی، یک مدل ریاضی جدید برای محصولات با نرخ زوال غیر آنی ارائه می‌کند. در مدل موجودی پیشنهادی، کمبود بصورت ترکیبی از فروش از دست‌رفته و پس‌افت بطور همزمان می‌باشد. همچنین، مدل، پشتیبانی را کاملاً وابسته به زمان انتظار برای چرخه بعدی می‌داند. علاوه بر این، شرایط مناسب برای دستیابی به راه حل‌های بهینه ایجاد و مثال‌های عددی برای تصدیق و ارزیابی نتایج و روش حل ارائه شده است. راهکارهای موثر برای کاهش هزینه کل سالانه بر اساس نتایج تحلیل حساسیت ارائه شده است.

Keywords [Persian]

  • مقدار سفارش اقتصادی
  • پشتیبانی جزئی غیر خطی
  • زوال‌پذیری غیر آنی
  • پیش‌پرداخت
  • اقلام زوال‌پذیر
Ai, X. Y., Zhang, J. L., & Wang, L. (2017). Optimal joint replenishment policy for multiple non-instantaneous deteriorating items. International Journal of Production Research55(16), 4625-4642.‏ https://doi.org/10.1080/00207543.2016.1276306
Bakker, M., Riezebos, J., & Teunter, R. H. (2012). Review of inventory systems with deterioration since 2001. European Journal of Operational Research221(2), 275-284.‏ https://doi.org/10.1016/j.ejor.2012.03.004
Bishi, B., & Sahu, S. K. (2018). An inventory model for deteriorating items with quadratic demand and partial backlogging. Journal of Computer and Mathematical Sciences9(12), 2188-2198.‏
Cambini, A., & Martein, L. (2008). Generalized convexity and optimization: Theory and applications. Springer Science & Business Media.‏
Chakraborty, D., Jana, D. K., & Roy, T. K. (2020). Multi-warehouse partial backlogging inventory system with inflation for non-instantaneous deteriorating multi-item under imprecise environment. Soft Computing24(19), 14471-14490.‏
Das, S. C., Manna, A. K., Rahman, M. S., Shaikh, A. A., & Bhunia, A. K. (2021). An inventory model for non-instantaneous deteriorating items with preservation technology and multiple credit periods-based trade credit financing via particle swarm optimization. Soft Computing,  , 5365-5384.‏
Diabat, A., Taleizadeh, A. A., & Lashgari, M. (2017). A lot sizing model with partial downstream delayed payment, partial upstream advance payment, and partial backordering for deteriorating items. Journal of Manufacturing Systems45, 322-342.‏ https://doi.org/10.1016/j.jmsy.2017.04.005
Dutta, D., & Kumar, P. (2015). A partial backlogging inventory model for deteriorating items with time-varying demand and holding cost. International Journal of Mathematics in Operational Research7(3), 281-296.‏ https://doi.org/10.1504/IJMOR.2015.069144
Dye, C. Y. (2007). Joint pricing and ordering policy for a deteriorating inventory with partial backlogging. Omega35(2), 184-189.‏ https://doi.org/10.1016/j.omega.2005.05.002
Educba. (2020). Mathematica vs Matlab. Accessed from https://www.educba.com/mathematica-vs-matlab/
Geetha, K. V., & Udayakumar, R. (2016). Optimal lot sizing policy for non-instantaneous deteriorating items with price and advertisement dependent demand under partial backlogging. International Journal of Applied and Computational Mathematics2(2), 171-193.‏
Ghoreishi, M., Weber, G. W., & Mirzazadeh, A. (2015). An inventory model for non-instantaneous deteriorating items with partial backlogging, permissible delay in payments, inflation-and selling price-dependent demand and customer returns. Annals of Operations Research226(1), 221-238.‏
Goyal, S. K., & Giri, B. C. (2001). Recent trends in modeling of deteriorating inventory. European Journal of Operational Research134(1), 1-16.‏ https://doi.org/10.1016/S0377-2217(00)00248-4
Gupta, R. K., Bhunia, A. K., & Goyal, S. K. (2009). An application of Genetic Algorithm in solving an inventory model with advance payment and interval valued inventory costs. Mathematical and Computer Modelling49(5-6), 893-905.‏ https://doi.org/10.1016/j.mcm.2008.09.015
Halim, M. A., Paul, A., Mahmoud, M., Alshahrani, B., Alazzawi, A. Y., & Ismail, G. M. (2021). An overtime production inventory model for deteriorating items with nonlinear price and stock dependent demand. Alexandria Engineering Journal60(3), 2779-2786.‏ https://doi.org/10.1016/j.aej.2021.01.019
Hung, K. C. (2011). An inventory model with generalized type demand, deterioration and backorder rates. European Journal of Operational Research208(3), 239-242.‏ https://doi.org/10.1016/j.ejor.2010.08.026
Jaggi, C. K., Tiwari, S., & Goel, S. K. (2017). Credit financing in economic ordering policies for non-instantaneous deteriorating items with price dependent demand and two storage facilities. Annals of Operations Research248(1-2), 253-280.
‏Khakzad, A., & Gholamian, M. R. (2020). The effect of inspection on deterioration rate: An inventory model for deteriorating items with advanced payment. Journal of Cleaner Production254, 120117.‏ https://doi.org/10.1016/j.jclepro.2020.120117
Khan, M. A. A., Shaikh, A. A., Konstantaras, I., Bhunia, A. K., & Cárdenas-Barrón, L. E. (2020). Inventory models for perishable items with advanced payment, linearly time-dependent holding cost and demand dependent on advertisement and selling price. International Journal of Production Economics230, 107804.‏ https://doi.org/10.1016/j.ijpe.2020.107804
Khan, M. A. A., Shaikh, A. A., Panda, G. C., Bhunia, A. K., & Konstantaras, I. (2020). Non-instantaneous deterioration effect in ordering decisions for a two-warehouse inventory system under advance payment and backlogging. Annals of Operations Research, 289, 243-275.
Khan, M. A. A., Shaikh, A. A., Panda, G. C., & Konstantaras, I. (2019). Two-warehouse inventory model for deteriorating items with partial backlogging and advance payment scheme. RAIRO-Operations Research53(5), 1691-1708.‏ https://doi.org/10.1051/ro/2018093
Kumar, P., & Keerthika, P. S. (2018). An inventory model with variable holding cost and partial backlogging under interval uncertainty: Global criteria method. International Journal of Mechanical Engineering and Technology9(11), 1567-1578.‏
Lashgari, M., Taleizadeh, A. A., & Sadjadi, S. J. (2018). Ordering policies for non-instantaneous deteriorating items under hybrid partial prepayment, partial trade credit and partial backordering. Journal of the Operational Research Society69(8), 1167-1196.‏ https://doi.org/10.1080/01605682.2017.1390524
Lee, Y. P., & Dye, C. Y. (2012). An inventory model for deteriorating items under stock-dependent demand and controllable deterioration rate. Computers & Industrial Engineering63(2), 474-482.‏ https://doi.org/10.1016/j.cie.2012.04.006
Liao, J. J., Huang, K. N., Chung, K. J., Lin, S. D., Chuang, S. T., & Srivastava, H. M. (2020). Optimal ordering policy in an economic order quantity (EOQ) model for non-instantaneous deteriorating items with defective quality and permissible delay in payments. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas114(1), 1-26.‏
Mahato, C., & Mahata, G. C. (2021). Optimal inventory policies for deteriorating items with expiration date and dynamic demand under two-level trade credit. OPSEARCH,  , 1-24.‏
Maihami, R., Ghelehkhondabi, I., & Ahmadi, E. (2021). Pricing and inventory planning for non-instantaneous deteriorating products with greening investment: A case study in beef industry. Journal of Cleaner Production, 126368.‏ https://doi.org/10.1016/j.jclepro.2021.126368
Maiti, A. K., Maiti, M. K., & Maiti, M. (2009). Inventory model with stochastic lead-time and price dependent demand incorporating advance payment. Applied Mathematical Modelling33(5), 2433-2443.‏ https://doi.org/10.1016/j.apm.2008.07.024
Mashud, A. H. M., Hasan, M. R., Wee, H. M., & Daryanto, Y. (2019). Non-instantaneous deteriorating inventory model under the joined effect of trade-credit, preservation technology and advertisement policy. Kybernetes, 49(6), 1645-1674. https://doi.org/10.1108/K-05-2019-0357
Mashud, A. H. M., Roy, D., Daryanto, Y., & Wee, H. M. (2021). Joint pricing deteriorating inventory model considering product life cycle and advance payment with a discount facility. RAIRO-Operations Research55, 1069-1088.‏ https://doi.org/10.1051/ro/2020106
Mashud, A. H. M., Wee, H. M., Sarkar, B., & Li, Y. H. C. (2020). A sustainable inventory system with the advanced payment policy and trade-credit strategy for a two-warehouse inventory system. Kybernetes,  .‏ https://doi.org/10.1108/K-01-2020-0052
Mishra, U., Wu, J. Z., & Sarkar, B. (2021). Optimum sustainable inventory management with backorder and deterioration under controllable carbon emissions. Journal of Cleaner Production279, 123699.‏ https://doi.org/10.1016/j.jclepro.2020.123699
Musa, A., & Sani, B. (2012). Inventory ordering policies of delayed deteriorating items under permissible delay in payments. International Journal of Production Economics136(1), 75-83.‏ https://doi.org/10.1016/j.ijpe.2011.09.013
Papachristos, S., & Skouri, K. (2000). An optimal replenishment policy for deteriorating items with time-varying demand and partial–exponential type–backlogging. Operations Research Letters27(4), 175-184.‏ https://doi.org/10.1016/S0167-6377(00)00044-4
Rahman, M. S., Khan, M. A. A., Halim, M. A., Nofal, T. A., Shaikh, A. A., & Mahmoud, E. E. (2021). Hybrid price and stock dependent inventory model for perishable goods with advance payment related discount facilities under preservation technology. Alexandria Engineering Journal60(3), 3455-3465.‏ https://doi.org/10.1016/j.aej.2021.01.045
Rajan, R. S., & Uthayakumar, R. (2017). EOQ model for time dependent demand and exponentially increasing holding cost under permissible delay in payment with complete backlogging. International Journal of Applied and Computational Mathematics3(2), 471-487.‏
Rezagholifam, M., Sadjadi, S. J., Heydari, M., & Karimi, M. (2020). Optimal pricing and ordering strategy for non-instantaneous deteriorating items with price and stock sensitive demand and capacity constraint. International Journal of Systems Science: Operations & Logistics,  , 1-12.‏ https://doi.org/10.1080/23302674.2020.1833259
Sen, N., Saha, S., & Nath, B. (2017). Inventory model with price dependent demand under permissible delay in payment. Asian Journal of Mathematics and Computer Research20(1), 1-12.‏
Shah, N. H., & Naik, M. K. (2020). Inventory policies for deteriorating items with time-price backlog dependent demand. International Journal of Systems Science: Operations & Logistics7(1), 76-89.‏ https://doi.org/10.1080/23302674.2018.1506062
Shaikh, A. A., Das, S. C., Bhunia, A. K., Panda, G. C., & Khan, M. A. A. (2019). A two-warehouse EOQ model with interval-valued inventory cost and advance payment for deteriorating item under particle swarm optimization. Soft Computing23(24), 13531-13546.‏
Skouri, K., Konstantaras, I., Papachristos, S., & Teng, J. T. (2011). Supply chain models for deteriorating products with ramp type demand rate under permissible delay in payments. Expert Systems with Applications38(12), 14861-14869.‏ https://doi.org/10.1016/j.eswa.2011.05.061
Soni, H. N., & Suthar, D. N. (2020). Optimal pricing and replenishment policy for non-instantaneous deteriorating items with varying rate of demand and partial backlogging. OPSEARCH, 57, 986-1021.
Sundararajan, R., & Uthayakumar, R. (2018). Optimal pricing and replenishment policies for instantaneous deteriorating items with backlogging and permissible delay in payment under inflation. American Journal of Mathematical and Management Sciences37(4), 307-323.‏ https://doi.org/10.1080/01966324.2017.1422202
Taleizadeh, A. A. (2014a). An economic order quantity model for deteriorating item in a purchasing system with multiple prepayments. Applied Mathematical Modelling38(23), 5357-5366.‏ https://doi.org/10.1016/j.apm.2014.02.014
Taleizadeh, A. A. (2014b). An EOQ model with partial backordering and advance payments for an evaporating item. International Journal of Production Economics155, 185-193.‏ https://doi.org/10.1016/j.ijpe.2014.01.023
Taleizadeh, A. A., & Nematollahi, M. (2014). An inventory control problem for deteriorating items with back-ordering and financial considerations. Applied Mathematical Modelling38(1), 93-109. https://doi.org/10.1016/j.apm.2013.05.065
Taleizadeh, A. A., Niaki, S. T. A., & Barzinpour, F. (2011). Multiple-buyer multiple-vendor multi-product multi-constraint supply chain problem with stochastic demand and variable lead-time: A harmony search algorithm. Applied Mathematics and Computation217(22), 9234-9253.‏ https://doi.org/10.1016/j.amc.2011.04.001
Taleizadeh, A. A., Pentico, D. W., Jabalameli, M. S., & Aryanezhad, M. (2013). An economic order quantity model with multiple partial prepayments and partial backordering. Mathematical and Computer Modelling57(3-4), 311-323.‏ https://doi.org/10.1016/j.mcm.2012.07.002
Taleizadeh, A. A., Tavakoli, S., & San-José, L. A. (2018). A lot sizing model with advance payment and planned backordering. Annals of Operations Research271(2), 1001-1022.‏
Teng, J. T., Cárdenas-Barrón, L. E., Chang, H. J., Wu, J., & Hu, Y. (2016). Inventory lot-size policies for deteriorating items with expiration dates and advance payments. Applied Mathematical Modelling40(19-20), 8605-8616.‏ https://doi.org/10.1016/j.apm.2016.05.022
Udayakumar, R., Geetha, K. V., & Sana, S. S. (2020). Economic ordering policy for non‐instantaneous deteriorating items with price and advertisement dependent demand and permissible delay in payment under inflation. Mathematical Methods in the Applied Sciences.‏ https://doi.org/10.1002/mma.6594
Viji, G., & Karthikeyan, K. (2018). An economic production quantity model for three levels of production with Weibull distribution deterioration and shortage. Ain Shams Engineering Journal9(4), 1481-1487.‏ https://doi.org/10.1016/j.asej.2016.10.006
Wang, W.C., Teng, J.T. & Lou, K.R. (2014). Seller’s optimal credit period and cycle time in a supply chain for deteriorating items with maximum lifetime. European Journal of Operational Research, 232(2), 315-321. https://doi.org/10.1016/j.ejor.2013.06.027
Wu, J., Skouri, K., Teng, J. T., & Ouyang, L. Y. (2014). A note on “optimal replenishment policies for non-instantaneous deteriorating items with price and stock sensitive demand under permissible delay in payment.” International Journal of Production Economics155, 324-329.‏ https://doi.org/10.1016/j.ijpe.2013.12.017
Wu, J., Teng, J. T., & Chan, Y. L. (2018). Inventory policies for perishable products with expiration dates and advance-cash-credit payment schemes. International Journal of Systems Science: Operations & Logistics5(4), 310-326.‏
Wu, K. S., Ouyang, L. Y., & Yang, C. T. (2006). An optimal replenishment policy for non-instantaneous deteriorating items with stock-dependent demand and partial backlogging. International Journal of Production Economics101(2), 369-384.‏ https://doi.org/10.1016/j.ijpe.2005.01.010
Zhang, Q., Tsao, Y. C., & Chen, T. H. (2014). Economic order quantity under advance payment. Applied Mathematical Modelling38(24), 5910-5921. https://doi.org/10.1016/j.apm.2014.04.040
Zia, N. P., & Taleizadeh, A. A. (2015). A lot-sizing model with backordering under hybrid linked-to-order multiple advance payments and delayed payment. Transportation Research Part E: Logistics and Transportation Review82, 19-37.‏ https://doi.org/10.1016/j.tre.2015.07.008