# American Institute of Mathematical Sciences

• Previous Article
Pseudo-polynomial time algorithms for combinatorial food mixture packing problems
• JIMO Home
• This Issue
• Next Article
An inventory control problem for deteriorating items with back-ordering and financial considerations under two levels of trade credit linked to order quantity
July  2016, 12(3): 1075-1089. doi: 10.3934/jimo.2016.12.1075

## Production planning in a three-stock reverse-logistics system with deteriorating items under a periodic review policy

 1 Industrial Engineering Department, College of Engineering, King Saud University, P.O Box 800, Riyadh 11421 2 King Saud University, Computer Engineering Department, College of Computer and Information Sciences, Riyadh 11421, Saudi Arabia 3 Fairleigh Dickinson University, Silberman College of Business, Department of Information Systems and Decision Sciences, 842 Cambie Street, Vancouver, BC, V6B 2P6

Received  December 2014 Revised  May 2015 Published  September 2015

In this paper, we study a reverse supply chain with three stocks for a firm implementing the periodic-review as policy. The first stock consists of newly manufactured items, while the second one is devoted to remanufactured items. The third stock contains all the items returned from the market. We made the following assumptions: items are not as-good-as-new, deterioration and dynamic demands affect new and remanufactured items, and dynamic customer return rate. Using optimal control theory, explicit expressions of the optimal rates in all three stocks have been provided: those for manufacturing, remanufacturing, disposal, and inventory levels. In the case where items deterioration rates are assumed to be unknown, self-tuning optimal control has also been implemented using the recursive least-squares method. Numerical examples and sensitivity analyses illustrate the results.
Citation: Abdelghani Bouras, Ramdane Hedjar, Lotfi Tadj. Production planning in a three-stock reverse-logistics system with deteriorating items under a periodic review policy. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1075-1089. doi: 10.3934/jimo.2016.12.1075
##### References:
 [1] S. S. Ahiska and E. Kurtul, Modeling and analysis of a product substitution strategy for a stochastic manufacturing/remanufacturing system,, Computers & Industrial Engineering, 72 (2014), 1. doi: 10.1016/j.cie.2014.02.015. [2] N. Aras, R. Gullu and S. Yurulmez, Optimal inventory and pricing policies for remanufacturable leased products,, International Journal of Production Economics, 133 (2011), 262. doi: 10.1016/j.ijpe.2010.01.024. [3] R. U. Ayres, Metals recycling: economic and environmental implications,, Resources, 21 (1997), 145. doi: 10.1016/S0921-3449(97)00033-5. [4] M. Bakker, J. Riezebos and R. H. Teunter, Review of inventory systems with deterioration since 2001,, European Journal of Operational Research, 221 (2012), 275. doi: 10.1016/j.ejor.2012.03.004. [5] A. Bouras and L. Tadj, Production planning in a three-stock reverse-logistics system with deteriorating items under a continuous review policy,, Journal of Industrial and Management Optimization, 11 (2015), 1041. doi: 10.3934/jimo.2015.11.1041. [6] A. E. Bryson and Y.-C. Ho, Applied Optimal Control,, Halsted Press, (1975). [7] X. Cai, M. Lai, X. Li, Y. Li and X. Wu, Optimal acquisition and production policy in a hybrid manufacturing/ remanufacturing system with core acquisition at different quality levels,, European Journal of Operational Research, 233 (2014), 374. doi: 10.1016/j.ejor.2013.07.017. [8] H. K. Chan, S. Yin and F. T. Chan, Implementing just-in-time philosophy to reverse logistics systems: A review,, International Journal of Production Research, 48 (2010), 6293. doi: 10.1080/00207540903225213. [9] P. Chanintrakul, A. E. Coronado Mondragon, C. Lalwani and C. Y. Wong, Reverse logistics network design: A state-of-the-art literature review,, International Journal of Business Performance and Supply Chain Modelling, 1 (2009), 61. [10] A. L. Craighill and J. C. Powell, Lifecycle assessment and economic evaluation of recycling: A case study,, Resources, 17 (1997), 75. doi: 10.1016/0921-3449(96)01105-6. [11] O. S. S. Filho and I. R. Salviano, Choosing an optimal return rate and a reverse logistics policy from the solution of a constrained LQG problem,, 19th World Congress, (2014). [12] M. Fleischmann, J. M. Bloemhof-Ruwaard, R. Dekker, E. van der Laan, J. A. E. E. van Nunen and L. N. Van Wassenhove, Quantitative models for reverse logistics: A review,, European Journal of Operational Research, 103 (1997), 1. doi: 10.1016/S0377-2217(97)00230-0. [13] A. Foul, S. Djemili and L. Tadj, Optimal and self-tuning optimal control of a periodic-review hybrid production inventory system,, Nonlinear Analysis: Hybrid Systems, 1 (2007), 68. doi: 10.1016/j.nahs.2006.05.001. [14] M. S. Garcia-Alvarado, M. Paquet and A. Chaabane, An inventory model with recovery and environment considerations,, Interuniversity Research Centre on Entreprise Networks, 3 (2014). [15] P. Goodall, E. Rosamond and J. Harding, A review of the state of the art in tools and techniques used to evaluate remanufacturing feasibility,, Journal of Cleaner Production, 81 (2014), 1. doi: 10.1016/j.jclepro.2014.06.014. [16] K. Govindan, H. Soleimani and D. Kannan, Reverse logistics and closed-loop supply chain: A comprehensive review to explore the future,, European Journal of Operational Research, 240 (2015), 603. doi: 10.1016/j.ejor.2014.07.012. [17] A. Gungor and S. M. Gupta, Issues in environmentally conscious manufacturing and product recovery: A survey,, Computers and Industrial Engineering, 36 (1999), 811. doi: 10.1016/S0360-8352(99)00167-9. [18] S. Guo, G. Aydin and G. C. Souza, Dismantle or remanufacture?,, European Journal of Operational Research, 233 (2014), 580. doi: 10.1016/j.ejor.2013.09.042. [19] R. Haijema, Optimal ordering, issuance and disposal policies for inventory management of perishable products,, International Journal of Production Economics, 157 (2014), 158. doi: 10.1016/j.ijpe.2014.06.014. [20] R. Hedjar, M. Bounkhel and L. Tadj, Receding horizon control of a hybrid production system with deteriorating items,, Nonlinear Analysis, 63 (2005), 405. doi: 10.1016/j.na.2005.05.027. [21] M. A. Ilgin and S. M. Gupta, Environmentally conscious manufacturing and product recovery (ECMPRO): A review of the state of the art,, Journal of Environmental Management, 91 (2010), 563. doi: 10.1016/j.jenvman.2009.09.037. [22] X. Li, Operations management of logistics and supply chain: Issues and directions,, Discrete Dynamics in Nature and Society, 2014 (2014). doi: 10.1155/2014/701938. [23] A. D. Lokhande, R. L. Shrivastava and R. R. Shrivastava, A review on critical success factors of remanufacturing,, International Journal International Journal of Entrepreneurship & Business Environment Perspectives, 3 (2014), 982. [24] M. Mahmoudzadeh, S. J. Sadjadi and S. Mansour, Robust optimal dynamic production/pricing policies in a closed-loop system,, Applied Mathematical Modelling, 37 (2013), 8141. doi: 10.1016/j.apm.2013.03.008. [25] J. Pahl and S. Voß, Integrating deterioration and lifetime constraints in production and supply chain planning: A survey,, European Journal of Operational Research, 238 (2014), 654. doi: 10.1016/j.ejor.2014.01.060. [26] J. Pahl, S. Voß and D. L. Woodruff, Production planning with deterioration constraints: A survey,, 19th International Conference on Production Research, (2007). [27] S. Pokharel and A. Mutha, Perspectives in reverse logistics: A review,, Resources, 53 (2009), 175. doi: 10.1016/j.resconrec.2008.11.006. [28] M. Pourakbary, E. van der Laanz and R. Dekker, End-of-life inventory problem with phase-out returns,, Production and Operations Management, 23 (2014), 1561. [29] H. Sarimveis, P. Patrinos, C. D. Tarantilis and C. T. Kiranoudis, Dynamic modeling and control of supply chain systems: A review,, Computers & Operations Research, 35 (2008), 3530. doi: 10.1016/j.cor.2007.01.017. [30] P. Sasikumar and G. Kannan, Issues in reverse supply chains, part II: Reverse distribution issues-An overview,, International Journal of Sustainable Engineering, 1 (2008), 234. doi: 10.1080/19397030802509974. [31] P. Sasikumar and G. Kannan, Issues in reverse supply chains, part I: End-of-life product recovery and inventory management-An overview,, International Journal of Sustainable Engineering, 1 (2008), 154. doi: 10.1080/19397030802433860. [32] P. Sasikumar and G. Kannan, Issues in reverse supply chain, part III: Classification and simple analysis,, International Journal of Sustainable Engineering, 2 (2009), 2. doi: 10.1080/19397030802673374. [33] C. S. Tang and S. Zhou, Research advances in environmentally and socially sustainable operations,, European Journal of Operational Research, 223 (2012), 585. doi: 10.1016/j.ejor.2012.07.030. [34] Z. Tao and S. X. Zhou, Approximation balancing policies for inventory systems with remanufacturing,, Mathematics of Operations Research, 39 (2014), 1179. doi: 10.1287/moor.2014.0647. [35] D. Wang, O. Tang and L. Zhang, A periodic review lot sizing problem with random yields, disruptions and inventory capacity,, International Journal of Production Economics, 155 (2014), 330. doi: 10.1016/j.ijpe.2014.02.007. [36] T. Ye and Y. Zhenhua, Reverse logistics network: A literature review,, Journal of Chemical and Pharmaceutical Research, 6 (2014), 1916. [37] S. Zhang, C. K. M. Lee, H. K. Chan, K. L. Choy and Z. Wu, Swarm intelligence applied in green logistics: A literature review,, Engineering Applications of Artificial Intelligence, 37 (2015), 154. doi: 10.1016/j.engappai.2014.09.007.

show all references

##### References:
 [1] S. S. Ahiska and E. Kurtul, Modeling and analysis of a product substitution strategy for a stochastic manufacturing/remanufacturing system,, Computers & Industrial Engineering, 72 (2014), 1. doi: 10.1016/j.cie.2014.02.015. [2] N. Aras, R. Gullu and S. Yurulmez, Optimal inventory and pricing policies for remanufacturable leased products,, International Journal of Production Economics, 133 (2011), 262. doi: 10.1016/j.ijpe.2010.01.024. [3] R. U. Ayres, Metals recycling: economic and environmental implications,, Resources, 21 (1997), 145. doi: 10.1016/S0921-3449(97)00033-5. [4] M. Bakker, J. Riezebos and R. H. Teunter, Review of inventory systems with deterioration since 2001,, European Journal of Operational Research, 221 (2012), 275. doi: 10.1016/j.ejor.2012.03.004. [5] A. Bouras and L. Tadj, Production planning in a three-stock reverse-logistics system with deteriorating items under a continuous review policy,, Journal of Industrial and Management Optimization, 11 (2015), 1041. doi: 10.3934/jimo.2015.11.1041. [6] A. E. Bryson and Y.-C. Ho, Applied Optimal Control,, Halsted Press, (1975). [7] X. Cai, M. Lai, X. Li, Y. Li and X. Wu, Optimal acquisition and production policy in a hybrid manufacturing/ remanufacturing system with core acquisition at different quality levels,, European Journal of Operational Research, 233 (2014), 374. doi: 10.1016/j.ejor.2013.07.017. [8] H. K. Chan, S. Yin and F. T. Chan, Implementing just-in-time philosophy to reverse logistics systems: A review,, International Journal of Production Research, 48 (2010), 6293. doi: 10.1080/00207540903225213. [9] P. Chanintrakul, A. E. Coronado Mondragon, C. Lalwani and C. Y. Wong, Reverse logistics network design: A state-of-the-art literature review,, International Journal of Business Performance and Supply Chain Modelling, 1 (2009), 61. [10] A. L. Craighill and J. C. Powell, Lifecycle assessment and economic evaluation of recycling: A case study,, Resources, 17 (1997), 75. doi: 10.1016/0921-3449(96)01105-6. [11] O. S. S. Filho and I. R. Salviano, Choosing an optimal return rate and a reverse logistics policy from the solution of a constrained LQG problem,, 19th World Congress, (2014). [12] M. Fleischmann, J. M. Bloemhof-Ruwaard, R. Dekker, E. van der Laan, J. A. E. E. van Nunen and L. N. Van Wassenhove, Quantitative models for reverse logistics: A review,, European Journal of Operational Research, 103 (1997), 1. doi: 10.1016/S0377-2217(97)00230-0. [13] A. Foul, S. Djemili and L. Tadj, Optimal and self-tuning optimal control of a periodic-review hybrid production inventory system,, Nonlinear Analysis: Hybrid Systems, 1 (2007), 68. doi: 10.1016/j.nahs.2006.05.001. [14] M. S. Garcia-Alvarado, M. Paquet and A. Chaabane, An inventory model with recovery and environment considerations,, Interuniversity Research Centre on Entreprise Networks, 3 (2014). [15] P. Goodall, E. Rosamond and J. Harding, A review of the state of the art in tools and techniques used to evaluate remanufacturing feasibility,, Journal of Cleaner Production, 81 (2014), 1. doi: 10.1016/j.jclepro.2014.06.014. [16] K. Govindan, H. Soleimani and D. Kannan, Reverse logistics and closed-loop supply chain: A comprehensive review to explore the future,, European Journal of Operational Research, 240 (2015), 603. doi: 10.1016/j.ejor.2014.07.012. [17] A. Gungor and S. M. Gupta, Issues in environmentally conscious manufacturing and product recovery: A survey,, Computers and Industrial Engineering, 36 (1999), 811. doi: 10.1016/S0360-8352(99)00167-9. [18] S. Guo, G. Aydin and G. C. Souza, Dismantle or remanufacture?,, European Journal of Operational Research, 233 (2014), 580. doi: 10.1016/j.ejor.2013.09.042. [19] R. Haijema, Optimal ordering, issuance and disposal policies for inventory management of perishable products,, International Journal of Production Economics, 157 (2014), 158. doi: 10.1016/j.ijpe.2014.06.014. [20] R. Hedjar, M. Bounkhel and L. Tadj, Receding horizon control of a hybrid production system with deteriorating items,, Nonlinear Analysis, 63 (2005), 405. doi: 10.1016/j.na.2005.05.027. [21] M. A. Ilgin and S. M. Gupta, Environmentally conscious manufacturing and product recovery (ECMPRO): A review of the state of the art,, Journal of Environmental Management, 91 (2010), 563. doi: 10.1016/j.jenvman.2009.09.037. [22] X. Li, Operations management of logistics and supply chain: Issues and directions,, Discrete Dynamics in Nature and Society, 2014 (2014). doi: 10.1155/2014/701938. [23] A. D. Lokhande, R. L. Shrivastava and R. R. Shrivastava, A review on critical success factors of remanufacturing,, International Journal International Journal of Entrepreneurship & Business Environment Perspectives, 3 (2014), 982. [24] M. Mahmoudzadeh, S. J. Sadjadi and S. Mansour, Robust optimal dynamic production/pricing policies in a closed-loop system,, Applied Mathematical Modelling, 37 (2013), 8141. doi: 10.1016/j.apm.2013.03.008. [25] J. Pahl and S. Voß, Integrating deterioration and lifetime constraints in production and supply chain planning: A survey,, European Journal of Operational Research, 238 (2014), 654. doi: 10.1016/j.ejor.2014.01.060. [26] J. Pahl, S. Voß and D. L. Woodruff, Production planning with deterioration constraints: A survey,, 19th International Conference on Production Research, (2007). [27] S. Pokharel and A. Mutha, Perspectives in reverse logistics: A review,, Resources, 53 (2009), 175. doi: 10.1016/j.resconrec.2008.11.006. [28] M. Pourakbary, E. van der Laanz and R. Dekker, End-of-life inventory problem with phase-out returns,, Production and Operations Management, 23 (2014), 1561. [29] H. Sarimveis, P. Patrinos, C. D. Tarantilis and C. T. Kiranoudis, Dynamic modeling and control of supply chain systems: A review,, Computers & Operations Research, 35 (2008), 3530. doi: 10.1016/j.cor.2007.01.017. [30] P. Sasikumar and G. Kannan, Issues in reverse supply chains, part II: Reverse distribution issues-An overview,, International Journal of Sustainable Engineering, 1 (2008), 234. doi: 10.1080/19397030802509974. [31] P. Sasikumar and G. Kannan, Issues in reverse supply chains, part I: End-of-life product recovery and inventory management-An overview,, International Journal of Sustainable Engineering, 1 (2008), 154. doi: 10.1080/19397030802433860. [32] P. Sasikumar and G. Kannan, Issues in reverse supply chain, part III: Classification and simple analysis,, International Journal of Sustainable Engineering, 2 (2009), 2. doi: 10.1080/19397030802673374. [33] C. S. Tang and S. Zhou, Research advances in environmentally and socially sustainable operations,, European Journal of Operational Research, 223 (2012), 585. doi: 10.1016/j.ejor.2012.07.030. [34] Z. Tao and S. X. Zhou, Approximation balancing policies for inventory systems with remanufacturing,, Mathematics of Operations Research, 39 (2014), 1179. doi: 10.1287/moor.2014.0647. [35] D. Wang, O. Tang and L. Zhang, A periodic review lot sizing problem with random yields, disruptions and inventory capacity,, International Journal of Production Economics, 155 (2014), 330. doi: 10.1016/j.ijpe.2014.02.007. [36] T. Ye and Y. Zhenhua, Reverse logistics network: A literature review,, Journal of Chemical and Pharmaceutical Research, 6 (2014), 1916. [37] S. Zhang, C. K. M. Lee, H. K. Chan, K. L. Choy and Z. Wu, Swarm intelligence applied in green logistics: A literature review,, Engineering Applications of Artificial Intelligence, 37 (2015), 154. doi: 10.1016/j.engappai.2014.09.007.
 [1] Xiaohui Liu, Jinhua Wang, Dianhua Wu. Two new classes of binary sequence pairs with three-level cross-correlation. Advances in Mathematics of Communications, 2015, 9 (1) : 117-128. doi: 10.3934/amc.2015.9.117 [2] Semu Mitiku Kassa. Three-level global resource allocation model for HIV control: A hierarchical decision system approach. Mathematical Biosciences & Engineering, 2018, 15 (1) : 255-273. doi: 10.3934/mbe.2018011 [3] Pedro Piñeyro, Omar Viera. Inventory policies for the economic lot-sizing problem with remanufacturing and final disposal options. Journal of Industrial & Management Optimization, 2009, 5 (2) : 217-238. doi: 10.3934/jimo.2009.5.217 [4] Ernan Haruvy, Ashutosh Prasad, Suresh Sethi, Rong Zhang. Competition with open source as a public good. Journal of Industrial & Management Optimization, 2008, 4 (1) : 199-211. doi: 10.3934/jimo.2008.4.199 [5] Carlos Tomei. The Toda lattice, old and new. Journal of Geometric Mechanics, 2013, 5 (4) : 511-530. doi: 10.3934/jgm.2013.5.511 [6] Tanja Eisner, Jakub Konieczny. Automatic sequences as good weights for ergodic theorems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4087-4115. doi: 10.3934/dcds.2018178 [7] Shengtian Yang, Thomas Honold. Good random matrices over finite fields. Advances in Mathematics of Communications, 2012, 6 (2) : 203-227. doi: 10.3934/amc.2012.6.203 [8] Ethan Akin. Good strategies for the Iterated Prisoner's Dilemma: Smale vs. Markov. Journal of Dynamics & Games, 2017, 4 (3) : 217-253. doi: 10.3934/jdg.2017014 [9] Anke D. Pohl. Symbolic dynamics for the geodesic flow on two-dimensional hyperbolic good orbifolds. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2173-2241. doi: 10.3934/dcds.2014.34.2173 [10] Aihua Fan, Shilei Fan, Lingmin Liao, Yuefei Wang. Minimality of p-adic rational maps with good reduction. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3161-3182. doi: 10.3934/dcds.2017135 [11] Alexander J. Zaslavski. Good programs in the RSS model without concavity of a utility function. Journal of Industrial & Management Optimization, 2006, 2 (4) : 399-423. doi: 10.3934/jimo.2006.2.399 [12] Zalman Balanov, Yakov Krasnov. On good deformations of $A_m$-singularities. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1851-1866. doi: 10.3934/dcdss.2019122 [13] Jui-Jung Liao, Wei-Chun Lee, Kuo-Nan Huang, Yung-Fu Huang. Optimal ordering policy for a two-warehouse inventory model use of two-level trade credit. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1661-1683. doi: 10.3934/jimo.2017012 [14] Roberta Ghezzi, Benedetto Piccoli. Optimal control of a multi-level dynamic model for biofuel production. Mathematical Control & Related Fields, 2017, 7 (2) : 235-257. doi: 10.3934/mcrf.2017008 [15] Shuren Liu, Qiying Hu, Yifan Xu. Optimal inventory control with fixed ordering cost for selling by internet auctions. Journal of Industrial & Management Optimization, 2012, 8 (1) : 19-40. doi: 10.3934/jimo.2012.8.19 [16] Yanqing Hu, Zaiming Liu, Jinbiao Wu. Optimal impulse control of a mean-reverting inventory with quadratic costs. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1685-1700. doi: 10.3934/jimo.2018027 [17] A.V. Borisov, I.S. Mamaev, A.A. Kilin. New periodic solutions for three or four identical vortices on a plane and a sphere. Conference Publications, 2005, 2005 (Special) : 110-120. doi: 10.3934/proc.2005.2005.110 [18] Ellina Grigorieva, Evgenii Khailov. A nonlinear controlled system of differential equations describing the process of production and sales of a consumer good. Conference Publications, 2003, 2003 (Special) : 359-364. doi: 10.3934/proc.2003.2003.359 [19] Samuel T. Blake, Thomas E. Hall, Andrew Z. Tirkel. Arrays over roots of unity with perfect autocorrelation and good ZCZ cross-correlation. Advances in Mathematics of Communications, 2013, 7 (3) : 231-242. doi: 10.3934/amc.2013.7.231 [20] Yujing Wang, Changjun Yu, Kok Lay Teo. A new computational strategy for optimal control problem with a cost on changing control. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 339-364. doi: 10.3934/naco.2016016

2017 Impact Factor: 0.994