# American Institute of Mathematical Sciences

October  2015, 11(4): 1041-1058. doi: 10.3934/jimo.2015.11.1041

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

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

Received  November 2013 Revised  August 2014 Published  March 2015

We consider in this paper a reverse supply chain with three stocks. Newly manufactured items are stored in the first stock. The second stock is reserved for remanufactured items, while the third stock contains the items that are returned from the market. One of our main assumptions is that remanufactured items are not as-good-as-new. We also assume that new and remanufactured items are subject to deterioration and to dynamic demands, that customer return rate is also dynamic, and that the firm adopts a continuous-review policy. Using optimal control theory, we obtain the explicit expressions of the optimal manufacturing rate, remanufacturing rate, disposal rate, and inventory levels in all three stocks. Numerical examples and sensitivity analyses illustrate the results obtained.
Citation: Abdelghani Bouras, Lotfi Tadj. Production planning in a three-stock reverse-logistics system with deteriorating items under a continuous review policy. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1041-1058. doi: 10.3934/jimo.2015.11.1041
##### References:
 [1] J. Aitken and A. Harrison, Supply governance structures for reverse logistics systems,, International Journal of Operations & Production Management, 33 (2013), 745. Google Scholar [2] A. A. Al-Babtain, Optimal control of reverse logistics model with logistic demand and return rates,, Applied Mathematical Sciences, 4 (2010), 2561. Google Scholar [3] A. Alshamrani, Adaptive control of a reverse logistic inventory model with uncertain deteriorations and disposal rates,, Advances in Operations Research, (): 1. Google Scholar [4] W. Bei and S. Linyan, A review of reverse logistics,, Applied Sciences (APPS), 7 (2005), 16. Google Scholar [5] D. F. Blumberg, Introduction to Management of Reverse Logistics and Closed Loop Supply Chain Processes,, CRC Press, (2004). Google Scholar [6] T. Clottey, W. C. Benton and R. Srivastava, Forecasting product returns for remanufacturing operations,, Decision Sciences, 43 (2012), 589. Google Scholar [7] M. P. De Brito, Managing Reverse Logistics or Reversing Logistics Management?,, ERIM PhD Series; EPS-2004-035-LIS, (2004), 2004. Google Scholar [8] M. P. De Brito, S. D. P. Flapper and R. Dekker, Reverse logistics,, Econometric Institute Research Papers., (). Google Scholar [9] R. Dekker, M. Fleischmann, K. Inderfurth and L. N. Van Wassenhove (eds.), Reverse Logistics. Quantitative Models in Closed-Loop Supply Chains,, Springer-Verlag, (2004). Google Scholar [10] I. Dobos, Optimal production-inventory strategies for a HMMS-type reverse logistics system,, International Journal of Production Economics, 81 (2003), 351. Google Scholar [11] I. Dobos, Készletmodellek a visszutas logisztikában= Inventory Models in Reverse Logistics,, PhD thesis, (2007). Google Scholar [12] H. Dyckhoff, R. Lackes and J. Reese, Supply Chain Management and Reverse Logistics,, Springer, (2004). Google Scholar [13] A. M. A. El Saadany, Inventory Management in Reverse Logistics with Imperfect Production, Learning, Lost Sales, Subassemblies, and Price/Quality Considerations,, PhD thesis, (2009). Google Scholar [14] L. Feng, J. Zhang and W. Tang, Optimal control of production and remanufacturing for a recovery system with perishable items,, International Journal of Production Research, 51 (2013), 3177. Google Scholar [15] O. S. S. Filho, Suboptimal production planning policies for closed-loop system with uncertain levels of demand and return,, in Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, (2011). Google Scholar [16] S. D. P. Flapper and T. S. Spengler (eds.), Product recovery,, Special issue of OR Spectrum, 28 (2006), 53. Google Scholar [17] S. D. P. Flapper, J. E. E. Van Nunen and L. N. Van Wassenhove, Managing Closed-Loop Supply Chain,, Springer, (2005). Google Scholar [18] M. Fleischmann, Quantitative Models for Reverse Logistics,, PhD thesis, (2000). Google Scholar [19] A. Foul, An advertising inventory system with product recovery,, International Mathematical Forum, 5 (2010), 125. Google Scholar [20] A. Foul, A. Al-Gohary and A. Essayed, Optimal control of HMMS-model with deteriorating items,, International Mathematical Forum, 4 (2009), 1515. Google Scholar [21] A. Foul and L. Tadj, Optimal control of a hybrid periodic-review production inventory system with disposal,, International Journal of Operational Research, 2 (2007), 481. doi: 10.1504/IJOR.2007.014175. Google Scholar [22] A. Gharbi, R. Pellerin and J. Sadr, Production rate control for stochastic remanufacturing systems,, International Journal of Production Economics, 112 (2008), 37. Google Scholar [23] M. Gharote, Inventory Models for Reverse Logistics,, Master's thesis, (2004). Google Scholar [24] V. D. R. Guide and L. N. Van Wassenhove, Business Aspects of Closed-Loop Supply Chains, vol. 2,, Carnegie Mellon University Press Pittsburgh, (2003). Google Scholar [25] V. D. R. Guide and L. N. Van Wassenhove, Or forum-the evolution of closed-loop supply chain research,, Operations Research, 57 (2009), 10. Google Scholar [26] V. D. R. Guide, T. P. Harrison and L. N. Van Wassenhove, The challenge of closed-loop supply chains,, Interfaces, 33 (2003), 3. Google Scholar [27] A. Gunasekaran and T. C. E. Cheng (eds.), New perspectives and challenges,, Special Issue of Omega, 36 (). Google Scholar [28] R. Hedjar, M. Bounkhel and L. Tadj, Receding horizon control of a hybrid production system with deteriorating items,, Nonlinear Analysis: Theory, 63 (2005), 405. doi: 10.1016/j.na.2005.05.027. Google Scholar [29] M. Y. Jaber and A. El Saadany, The production, remanufacture and waste disposal model with lost sales,, International Journal of Production Economics, 120 (2009), 115. Google Scholar [30] V. Jain (ed.), Sustainable supply chain management and reverse logistics,, Special Issue of International Journal of Production Research, 50 (). Google Scholar [31] P. Jie, Production Planning and Inventory Control of Two-Product Recovery System in Reverse Logistics,, PhD thesis, (2010). Google Scholar [32] Jonrinaldi, Modelling of Coordinating Production and Inventory Cycles in a Manufacturing Supply Chain Involving Reverse Logistics,, PhD thesis, (2012). Google Scholar [33] M. L. Junior and M. G. Filho, Production planning and control for remanufacturing: Literature review and analysis,, Production Planning & Control, 23 (2012), 419. Google Scholar [34] D. Kannan, A. Diabat, M. Alrefaei, K. Govindand and G. Yonge, A carbon footprint based reverse logistics network design model,, Resources, 67 (2012), 75. Google Scholar [35] J.-P. Kenné, P. Dejax and A. Gharbi, Production planning of a hybrid manufacturing-remanufacturing system under uncertainty within a closed-loop supply chain,, International Journal of Production Economics, 135 (2012), 81. Google Scholar [36] W. Kerr and C. Ryan, Eco-efficiency gains from remanufacturing. a case study of photocopier remanufacturing at fuji xerox australia,, Journal of Cleaner Production, 9 (2001), 75. Google Scholar [37] G. P. Kiesmüller, A new approach for controlling a hybrid stochastic manufacturing/remanufacturing system with inventories and different leadtimes,, European Journal of Operational Research, 147 (2003), 62. Google Scholar [38] G. P. Kiesmüller, Optimal control of a one product recovery system with leadtimes,, International Journal of Production Economics, 81 (2003), 333. Google Scholar [39] G. P. Kiesmüller and C. W. Scherer, Computational issues in a stochastic finite horizon one product recovery inventory model,, European Journal of Operational Research, 146 (2003), 553. doi: 10.1016/S0377-2217(02)00249-7. Google Scholar [40] G. P. Kiesmüller, S. Minner and R. Kleber, Optimal control of a one product recovery system with backlogging,, IMA Journal of applied mathematics in business and industry, 11 (2000), 189. Google Scholar [41] M. Klausner, W. M. Grimm and C. Hendrickson, Reuse of electric motors in consumer products,, Journal of Industrial Ecology, 2 (1998), 89. Google Scholar [42] R. Kleber, The integral decision on production/remanufacturing technology and investment time in product recovery,, OR spectrum, 28 (2006), 21. doi: 10.1007/s00291-005-0011-6. Google Scholar [43] R. Kleber, S. Minner and G. P. Kiesmüller, A continuous time inventory model for a product recovery system with multiple options,, International Journal of Production Economics, 79 (2002), 121. Google Scholar [44] A. Langevin and D. Riopel (eds.), Logistics Systems: Design and Optimization,, Springer, (2005). doi: 10.1007/b106452. Google Scholar [45] Y. Li, J. Zhang, J. Chen and X. Cai, Optimal solution structure for multi-period production planning with returned products remanufacturing,, Asia-Pacific Journal of Operational Research, 27 (2010), 629. doi: 10.1142/S0217595910002910. Google Scholar [46] L. L. Lim, Optimal Control of a Hybrid Production/Remanufacturing System Using one Shared Resource,, Master's thesis, (2008). Google Scholar [47] F. Liu, Reverse Logistics Matters: Impact of Resource Commitment on Reverse Logistics and Customer Loyalty,, Master's thesis, (2010). Google Scholar [48] Y. Lu, Optimal Acquisition and Sorting Policies for Remanufacturing Over Single and Multiple Periods,, Master's thesis, (2009). Google Scholar [49] S. Minner and G. P. Kiesmüller, Dynamic product acquisition in closed loop supply chains,, International Journal of Production Research, 50 (2012), 2836. Google Scholar [50] S. Minner and R. Kleber, Optimal control of production and remanufacturing in a simple recovery model with linear cost functions,, OR Spektrum, 23 (2001), 3. doi: 10.1007/PL00013344. Google Scholar [51] R. Montemanni (ed.), Optimization methods in logistics,, Special Issue of Journal of Applied Operational Research., (). Google Scholar [52] S. D. Morgan and R. J. Gagnon, A systematic literature review of remanufacturing scheduling,, International Journal of Production Research, 51 (2013), 4853. Google Scholar [53] K. Nakashima, H. Arimitsu, T. Nose and S. Kuriyama, Optimal control of a remanufacturing system,, International Journal of Production Research, 42 (2004), 3619. Google Scholar [54] N. A. Nik Ab Halim, Y. Sabariah and A. Haim Hilman, Reverse logistics adoption among malaysian manufacturers,, in International Conference on Management, (2011). Google Scholar [55] M. Olovsson and L. Khalil, Reverse Logistics Study at Volvo CE CST Europe,, Master's thesis, (2008). Google Scholar [56] R. Pellerin, J. Sadr, A. Gharbi and R. Malhamé, A production rate control policy for stochastic repair and remanufacturing systems,, International Journal of Production Economics, 121 (2009), 39. Google Scholar [57] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, John Wiley & Sons, (1962). Google Scholar [58] C. Prahinski and C. Kocabasoglu, Empirical research opportunities in reverse supply chains,, Omega, 34 (2006), 519. Google Scholar [59] M. Préjean, Half-soles, kettles and cures: Milestones in the tire retreading and repairing industry,, American Retreaders' Association, (). Google Scholar [60] D. S. Rogers, R. S. Tibben-Lembke and Council and R. L. Executive, Going Backwards: Reverse Logistics Trends and Practices, vol. 2,, Reverse Logistics Executive Council Pittsburgh, (1999). Google Scholar [61] B. S. Sahay (ed.), 3PL, 4PL and Reverse Logistics, Part 2,, Special issues on in the International Journal of Physical Distribution and Logistics Management, 36 (). Google Scholar [62] B. S. Sahay (ed.), 3PL, 4PL and Reverse Logistics, Part 1,, Special issues on in the International Journal of Physical Distribution and Logistics Management, 36 (). Google Scholar [63] J. Sarkis (ed.), Greening the Supply Chain,, Springer, (2006). Google Scholar [64] P. S. Sethi and G. L. Thompson, Optimal Control Theory: Applications to Management Science and Economics, 2nd ed.,, Kluwer Academic Publishers, (2000). Google Scholar [65] D. W. Steeneck and S. C. Sarin, Pricing and production planning for reverse supply chain: A review,, International Journal of Production Research, 51 (2013), 6972. Google Scholar [66] L. Van Stek, Using Reverse Logistics for Failed Spare Parts: A New Opportunity for Vanderlande Industries,, Master's thesis, (2012). Google Scholar [67] V. Verter and T. Boyaci (eds.), Reverse logistics,, Special Issue of Computers & Operations Research, 34 (). Google Scholar [68] J. Wan, Roles of Postponement in Supply Chain Strategies,, PhD thesis, (2009). Google Scholar [69] Z. Y. Wang, Integration of Reverse Logistics within New Product Design: The Cases study in Chinese Electric and Electronic Equipment (EEE) Industry,, Master's thesis, (2008). Google Scholar [70] H. Zerhouni, Intégration des Flux Inverses dans la Gestion des Stocks et de la Production},, PhD thesis, (2009). Google Scholar

show all references

##### References:
 [1] J. Aitken and A. Harrison, Supply governance structures for reverse logistics systems,, International Journal of Operations & Production Management, 33 (2013), 745. Google Scholar [2] A. A. Al-Babtain, Optimal control of reverse logistics model with logistic demand and return rates,, Applied Mathematical Sciences, 4 (2010), 2561. Google Scholar [3] A. Alshamrani, Adaptive control of a reverse logistic inventory model with uncertain deteriorations and disposal rates,, Advances in Operations Research, (): 1. Google Scholar [4] W. Bei and S. Linyan, A review of reverse logistics,, Applied Sciences (APPS), 7 (2005), 16. Google Scholar [5] D. F. Blumberg, Introduction to Management of Reverse Logistics and Closed Loop Supply Chain Processes,, CRC Press, (2004). Google Scholar [6] T. Clottey, W. C. Benton and R. Srivastava, Forecasting product returns for remanufacturing operations,, Decision Sciences, 43 (2012), 589. Google Scholar [7] M. P. De Brito, Managing Reverse Logistics or Reversing Logistics Management?,, ERIM PhD Series; EPS-2004-035-LIS, (2004), 2004. Google Scholar [8] M. P. De Brito, S. D. P. Flapper and R. Dekker, Reverse logistics,, Econometric Institute Research Papers., (). Google Scholar [9] R. Dekker, M. Fleischmann, K. Inderfurth and L. N. Van Wassenhove (eds.), Reverse Logistics. Quantitative Models in Closed-Loop Supply Chains,, Springer-Verlag, (2004). Google Scholar [10] I. Dobos, Optimal production-inventory strategies for a HMMS-type reverse logistics system,, International Journal of Production Economics, 81 (2003), 351. Google Scholar [11] I. Dobos, Készletmodellek a visszutas logisztikában= Inventory Models in Reverse Logistics,, PhD thesis, (2007). Google Scholar [12] H. Dyckhoff, R. Lackes and J. Reese, Supply Chain Management and Reverse Logistics,, Springer, (2004). Google Scholar [13] A. M. A. El Saadany, Inventory Management in Reverse Logistics with Imperfect Production, Learning, Lost Sales, Subassemblies, and Price/Quality Considerations,, PhD thesis, (2009). Google Scholar [14] L. Feng, J. Zhang and W. Tang, Optimal control of production and remanufacturing for a recovery system with perishable items,, International Journal of Production Research, 51 (2013), 3177. Google Scholar [15] O. S. S. Filho, Suboptimal production planning policies for closed-loop system with uncertain levels of demand and return,, in Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, (2011). Google Scholar [16] S. D. P. Flapper and T. S. Spengler (eds.), Product recovery,, Special issue of OR Spectrum, 28 (2006), 53. Google Scholar [17] S. D. P. Flapper, J. E. E. Van Nunen and L. N. Van Wassenhove, Managing Closed-Loop Supply Chain,, Springer, (2005). Google Scholar [18] M. Fleischmann, Quantitative Models for Reverse Logistics,, PhD thesis, (2000). Google Scholar [19] A. Foul, An advertising inventory system with product recovery,, International Mathematical Forum, 5 (2010), 125. Google Scholar [20] A. Foul, A. Al-Gohary and A. Essayed, Optimal control of HMMS-model with deteriorating items,, International Mathematical Forum, 4 (2009), 1515. Google Scholar [21] A. Foul and L. Tadj, Optimal control of a hybrid periodic-review production inventory system with disposal,, International Journal of Operational Research, 2 (2007), 481. doi: 10.1504/IJOR.2007.014175. Google Scholar [22] A. Gharbi, R. Pellerin and J. Sadr, Production rate control for stochastic remanufacturing systems,, International Journal of Production Economics, 112 (2008), 37. Google Scholar [23] M. Gharote, Inventory Models for Reverse Logistics,, Master's thesis, (2004). Google Scholar [24] V. D. R. Guide and L. N. Van Wassenhove, Business Aspects of Closed-Loop Supply Chains, vol. 2,, Carnegie Mellon University Press Pittsburgh, (2003). Google Scholar [25] V. D. R. Guide and L. N. Van Wassenhove, Or forum-the evolution of closed-loop supply chain research,, Operations Research, 57 (2009), 10. Google Scholar [26] V. D. R. Guide, T. P. Harrison and L. N. Van Wassenhove, The challenge of closed-loop supply chains,, Interfaces, 33 (2003), 3. Google Scholar [27] A. Gunasekaran and T. C. E. Cheng (eds.), New perspectives and challenges,, Special Issue of Omega, 36 (). Google Scholar [28] R. Hedjar, M. Bounkhel and L. Tadj, Receding horizon control of a hybrid production system with deteriorating items,, Nonlinear Analysis: Theory, 63 (2005), 405. doi: 10.1016/j.na.2005.05.027. Google Scholar [29] M. Y. Jaber and A. El Saadany, The production, remanufacture and waste disposal model with lost sales,, International Journal of Production Economics, 120 (2009), 115. Google Scholar [30] V. Jain (ed.), Sustainable supply chain management and reverse logistics,, Special Issue of International Journal of Production Research, 50 (). Google Scholar [31] P. Jie, Production Planning and Inventory Control of Two-Product Recovery System in Reverse Logistics,, PhD thesis, (2010). Google Scholar [32] Jonrinaldi, Modelling of Coordinating Production and Inventory Cycles in a Manufacturing Supply Chain Involving Reverse Logistics,, PhD thesis, (2012). Google Scholar [33] M. L. Junior and M. G. Filho, Production planning and control for remanufacturing: Literature review and analysis,, Production Planning & Control, 23 (2012), 419. Google Scholar [34] D. Kannan, A. Diabat, M. Alrefaei, K. Govindand and G. Yonge, A carbon footprint based reverse logistics network design model,, Resources, 67 (2012), 75. Google Scholar [35] J.-P. Kenné, P. Dejax and A. Gharbi, Production planning of a hybrid manufacturing-remanufacturing system under uncertainty within a closed-loop supply chain,, International Journal of Production Economics, 135 (2012), 81. Google Scholar [36] W. Kerr and C. Ryan, Eco-efficiency gains from remanufacturing. a case study of photocopier remanufacturing at fuji xerox australia,, Journal of Cleaner Production, 9 (2001), 75. Google Scholar [37] G. P. Kiesmüller, A new approach for controlling a hybrid stochastic manufacturing/remanufacturing system with inventories and different leadtimes,, European Journal of Operational Research, 147 (2003), 62. Google Scholar [38] G. P. Kiesmüller, Optimal control of a one product recovery system with leadtimes,, International Journal of Production Economics, 81 (2003), 333. Google Scholar [39] G. P. Kiesmüller and C. W. Scherer, Computational issues in a stochastic finite horizon one product recovery inventory model,, European Journal of Operational Research, 146 (2003), 553. doi: 10.1016/S0377-2217(02)00249-7. Google Scholar [40] G. P. Kiesmüller, S. Minner and R. Kleber, Optimal control of a one product recovery system with backlogging,, IMA Journal of applied mathematics in business and industry, 11 (2000), 189. Google Scholar [41] M. Klausner, W. M. Grimm and C. Hendrickson, Reuse of electric motors in consumer products,, Journal of Industrial Ecology, 2 (1998), 89. Google Scholar [42] R. Kleber, The integral decision on production/remanufacturing technology and investment time in product recovery,, OR spectrum, 28 (2006), 21. doi: 10.1007/s00291-005-0011-6. Google Scholar [43] R. Kleber, S. Minner and G. P. Kiesmüller, A continuous time inventory model for a product recovery system with multiple options,, International Journal of Production Economics, 79 (2002), 121. Google Scholar [44] A. Langevin and D. Riopel (eds.), Logistics Systems: Design and Optimization,, Springer, (2005). doi: 10.1007/b106452. Google Scholar [45] Y. Li, J. Zhang, J. Chen and X. Cai, Optimal solution structure for multi-period production planning with returned products remanufacturing,, Asia-Pacific Journal of Operational Research, 27 (2010), 629. doi: 10.1142/S0217595910002910. Google Scholar [46] L. L. Lim, Optimal Control of a Hybrid Production/Remanufacturing System Using one Shared Resource,, Master's thesis, (2008). Google Scholar [47] F. Liu, Reverse Logistics Matters: Impact of Resource Commitment on Reverse Logistics and Customer Loyalty,, Master's thesis, (2010). Google Scholar [48] Y. Lu, Optimal Acquisition and Sorting Policies for Remanufacturing Over Single and Multiple Periods,, Master's thesis, (2009). Google Scholar [49] S. Minner and G. P. Kiesmüller, Dynamic product acquisition in closed loop supply chains,, International Journal of Production Research, 50 (2012), 2836. Google Scholar [50] S. Minner and R. Kleber, Optimal control of production and remanufacturing in a simple recovery model with linear cost functions,, OR Spektrum, 23 (2001), 3. doi: 10.1007/PL00013344. Google Scholar [51] R. Montemanni (ed.), Optimization methods in logistics,, Special Issue of Journal of Applied Operational Research., (). Google Scholar [52] S. D. Morgan and R. J. Gagnon, A systematic literature review of remanufacturing scheduling,, International Journal of Production Research, 51 (2013), 4853. Google Scholar [53] K. Nakashima, H. Arimitsu, T. Nose and S. Kuriyama, Optimal control of a remanufacturing system,, International Journal of Production Research, 42 (2004), 3619. Google Scholar [54] N. A. Nik Ab Halim, Y. Sabariah and A. Haim Hilman, Reverse logistics adoption among malaysian manufacturers,, in International Conference on Management, (2011). Google Scholar [55] M. Olovsson and L. Khalil, Reverse Logistics Study at Volvo CE CST Europe,, Master's thesis, (2008). Google Scholar [56] R. Pellerin, J. Sadr, A. Gharbi and R. Malhamé, A production rate control policy for stochastic repair and remanufacturing systems,, International Journal of Production Economics, 121 (2009), 39. Google Scholar [57] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, John Wiley & Sons, (1962). Google Scholar [58] C. Prahinski and C. Kocabasoglu, Empirical research opportunities in reverse supply chains,, Omega, 34 (2006), 519. Google Scholar [59] M. Préjean, Half-soles, kettles and cures: Milestones in the tire retreading and repairing industry,, American Retreaders' Association, (). Google Scholar [60] D. S. Rogers, R. S. Tibben-Lembke and Council and R. L. Executive, Going Backwards: Reverse Logistics Trends and Practices, vol. 2,, Reverse Logistics Executive Council Pittsburgh, (1999). Google Scholar [61] B. S. Sahay (ed.), 3PL, 4PL and Reverse Logistics, Part 2,, Special issues on in the International Journal of Physical Distribution and Logistics Management, 36 (). Google Scholar [62] B. S. Sahay (ed.), 3PL, 4PL and Reverse Logistics, Part 1,, Special issues on in the International Journal of Physical Distribution and Logistics Management, 36 (). Google Scholar [63] J. Sarkis (ed.), Greening the Supply Chain,, Springer, (2006). Google Scholar [64] P. S. Sethi and G. L. Thompson, Optimal Control Theory: Applications to Management Science and Economics, 2nd ed.,, Kluwer Academic Publishers, (2000). Google Scholar [65] D. W. Steeneck and S. C. Sarin, Pricing and production planning for reverse supply chain: A review,, International Journal of Production Research, 51 (2013), 6972. Google Scholar [66] L. Van Stek, Using Reverse Logistics for Failed Spare Parts: A New Opportunity for Vanderlande Industries,, Master's thesis, (2012). Google Scholar [67] V. Verter and T. Boyaci (eds.), Reverse logistics,, Special Issue of Computers & Operations Research, 34 (). Google Scholar [68] J. Wan, Roles of Postponement in Supply Chain Strategies,, PhD thesis, (2009). Google Scholar [69] Z. Y. Wang, Integration of Reverse Logistics within New Product Design: The Cases study in Chinese Electric and Electronic Equipment (EEE) Industry,, Master's thesis, (2008). Google Scholar [70] H. Zerhouni, Intégration des Flux Inverses dans la Gestion des Stocks et de la Production},, PhD thesis, (2009). Google Scholar
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