Document Type : Article

**Authors**

Faculty of Industrial and Systems Engineering, Tarbiat Modares University, Tehran, P.O. Box 14115-111, Iran.

**Abstract**

In this paper, we consider a manufacturer that produces products in a finite horizon time and sells products with non-renewing free replacement warranty policy. The manufacturer is responsible to provide spare parts for failed products, whether the products are under or out of warranty. Previous research on warranty optimization has focused on maximizing manufacturer profit without considering the spare part market for out-of-warranty products. This study proposes a novel nonlinear model that maximizes manufacturer profit by optimization of price, warranty length and spare part inventory for under- and out-of-warranty products in a manufacturing/remanufacturing system. Due to the model’s unique structure, we propose a new two-stage approach that combines metaheuristic and an exact method, in which the first stage is to determine product’s prices and warranty length with metaheuristic algorithm and in the second stage the remaining inventory related problem is transferred to a Minimum Cost Network Flow Problem which is solved for spare part inventory control. To illustrate the effectiveness of the suggested method, the model is solved for a case study of Iranian SANAM electronic company with two different metaheuristic algorithms and a sensitivity analysis is conducted to study the effect of various parameters on the optimal solution.

**Keywords**

- Non-renewing free replacement warranty
- Dynamic pricing
- Spare parts inventory control
- Remanufacturing

**Main Subjects**

References:

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29. Kashan, A.H., "A new metaheuristic for optimization: optics inspired optimization (OIO)". Comput. Oper. Res., 55, pp. 99-125 (2015).

30. Jalili, S. and Husseinzadeh Kashan, A. "Optimum discrete design of steel tower structures using optics inspired optimization method". The Str. Des. of Tall and Spe.Build., 27(9): pp. e1466 (2018).

31. Özdemi̇r, M.T. and Öztürk, D. "Comparative Performance Analysis of Optimal PID Parameters Tuning Based on the Optics Inspired Optimization Methods for Automatic Generation Control". Energies, 10(12), pp. 2134 (2017).

32. ÖZDEMİR, M.T. and ÖZTÜRK, D. "Optimal PID Tuning for Load Frequency Control using Optics Inspired Optimization Algorithm", ICNES (2016).

33. Kashan, A.H., "An effective algorithm for constrained optimization based on optics inspired optimization (OIO)". Comput. Aided Design., 63, pp. 52-71 (2015).

34. Mohammadi, M., Musa, S.N. "Bahreininejad, Optimization of economic lot scheduling problem with backordering and shelf-life considerations using calibrated metaheuristic algorithms". Comput. Aided. Design., 251(Supplement C), pp. 404-422. (2015).

35. Kuo, R.J., et al., "Solving bi-level linear programming problem through hybrid of immune genetic algorithm and particle swarm optimization algorithm". Appl. Math. Comput., 266(Supplement C), pp. 1013-1026 (2015).

36. Yang, M.-F. and Lin, Y. "Applying the linear particle swarm optimization to a serial multi-echelon inventory model". Expert. Syst. Appl., 37(3), pp. 2599-2608 (2010).

37. Mousavi, S.M., et al., "A modified particle swarm optimization for solving the integrated location and inventory control problems in a two-echelon supply chain network". J. INTELL. MANUF., 28(1), pp. 191-206 (2017).

38. Majumder, P. Bera, U.K. and Maiti, M. "An EPQ model for two-warehouse in unremitting release pattern with two-level trade credit period concerning both supplier and retailer". Appl. Math. Comput., 274(Supplement C), pp. 430-458 (2016).

39. Bhunia, A.K. and Shaikh, A.A. "An application of PSO in a two-warehouse inventory model for deteriorating item under permissible delay in payment with different inventory policies". Appl. Math. Comput., 256(Supplement C), pp. 831-850 (2015).

40. Jiang, Y., et al., "An improved particle swarm optimization algorithm". Appl. Math. Comput., 193(1): pp. 231-239. (2007).

41. Kennedy, J. and Eberhart, R. "Particle swarm optimization (PSO)". in Proc. IEEE International Conference on Neural Networks, Perth, Australia. (1995).

42. Huang, W. Kulkarni, V. and Swaminathan, J.M. "Coordinated inventory planning for new and old products under warranty". Probab. Eng. Inform. SC., 21(02): pp. 261-287 (2007).

43. Yeo, W.M. and Yuan, X.-M. "Optimal Inventory Policy For Products With Warranty Agreements. in Industrial Electronics", ISIE 2007. IEEE International Symposium on. IEEE (2007).

44. Khawam, J., Hausman, W.H. and D.W. Cheng, "Warranty Inventory Optimization for Hitachi Global Storage Technologies", Inc. Interfaces, 37(5), pp. 455-471. (2007).

45. Darghouth, M.N. Ait-kadi, D. and Chelbi, A. "Joint optimization of design, warranty and price for products sold with maintenance service contracts". Reliab. Eng. Syst. Safe., 165, pp. 197-208 (2017).

46. Shi, Y. and Eberhart, R.C. "Empirical study of particle swarm optimization. in Evolutionary Computation", CEC 99. Proceedings of the 1999 Congress on. IEEE (1999).

2. Nguyen, D. and Murthy, D.P. "Optimal burn-in time to minimize cost for products sold under warranty". IIE. Trans., 14(3), pp. 167-174 (1982).

3. Murthy, D.N.P. "Product warranty and reliability". Ann. Oper. Res., 143(1): p. 133-146 (2006).

4. Zhou, Z. Li, Y. and Tang, K. "Dynamic pricing and warranty policies for products with fixed lifetime". Eur. J. Oper. Res., 196(3), pp. 940-948 (2009).

5. Lin, P.-C. Wang, J. and Chin, S.-S. "Dynamic optimisation of price, warranty length and production rate", Int. J. Syst. Sci,. 40(4), p. 411-420 (2009).

6. Wu, C.-C. Lin, P.-C. and Chou, C.-Y. "Determination of price and warranty length for a Gamma lifetime distributed product". J. of Inf. and Opt. Sci., 28(3), pp. 335-355 (2007).

7. Wu, C.-C. Lin, P.-C. and Chou, C.-Y. "Determination of price and warranty length for a normal lifetime distributed product". Int. J. Prod. Econ., 102(1), pp. 95-107 (2006).

8. Glickman, T.S. and Berger, P.D. "Optimal price and protection period decisions for a product under warranty". Manage. Sci., 22(12), pp. 1381-1390 (1976).

9. nasrollahi, m. and asgharizadeh, E. "Pro-Rata Warranty pricing model with risk-averse buyers". The Modares. J. of Manage. Res. in Iran, 20(1), pp. 131-154 (2016).

10. Lin, P.-C. and Shue, L.-Y. "Application of optimal control theory to product pricing and warranty with free replacement under the influence of basic lifetime distributions". Comput. Ind. Eng, 48(1), pp. 69-82 (2005).

11. Huang, H.-Z. Liu, Z.-J. and Murthy, D. "Optimal reliability, warranty and price for new products". Iie. Trans., 39(8), pp. 819-827 (2007).

12. Manna, D.K. "Price-warranty length decision with Glickman-Berger model". Int. J. of Rel. and Saf., 2(3), pp. 221-233 (2008).

13. Wu, C.-C. Chou, C.-Y. and Huang, C. "Optimal price, warranty length and production rate for free replacement policy in the static demand market". Omega, 37(1), pp. 29-39 (2009).

14. Fang, C.-C. and Huang, Y.-S. "A study on decisions of warranty, pricing, and production with insufficient information". Comput. Ind. Eng., 59(2), pp. 241-250 (2010).

15. Shafiee, M. and Chukova,S. "Optimal upgrade strategy, warranty policy and sale price for second‐hand products". Appl. Stoch. Model. Bus., 29(2), pp. 157-169 (2013).

16. Faridimehr, S. and Niaki, S. "Optimal strategies for price, warranty length, and production rate of a new product with learning production cost". Scientia Iranica. Transaction E, Industrial Engineering, 20(6), pp. 2247 (2013).

17. Mahmoudi, A. and Shavandi, H. "Analyzing price, warranty length, and service capacity under a fuzzy environment: Genetic algorithm and fuzzy system". Scientia Iranica, 20(3), pp. 975-982 (2013).

18. Tsao, Y.-C., et al., "Pricing and inventory policies for Hi-tech products under replacement warranty". INT. J. SYST. SCI., 45(6), pp. 1255-1267 (2014).

19. Wei, J., J. Zhao, and Y. Li, "Price and warranty period decisions for complementary products with horizontal firms' cooperation/noncooperation strategies". J. CLEAN. PROD., 105: pp. 86-102 (2015).

20. Yazdian, S.A., K. Shahanaghi, and A. Makui, "Joint optimisation of price, warranty and recovery planning in remanufacturing of used products under linear and non-linear demand, return and cost functions". Int. J. Syst. Sci., 47(5), pp. 1155-1175 (2016).

21. Murthy, D., O. Solem, and T. Roren, "Product warranty logistics: Issues and challenges". Eur. J. Oper. Res., 156(1), pp. 110-126 (2004).

22. Gallagher, T., Mitchke, M.D. and Rogers, M.C. "Profiting from spare parts". The McKinsey Quarterly, 2: pp. 1-4 (2005).

23. Kim, B. and Park, S. "Optimal pricing, EOL (end of life) warranty, and spare parts manufacturing strategy amid product transition". Eur. J. Oper. Res., 188(3), pp. 723-745 (2008).

24. Chari, N., et al., "Modeling and analysis of a warranty policy using new and reconditioned parts". Appl. Stoch. Model. Bus., 32(4), pp. 539-553 (2016).

25. Agrawal, S. Singh, R.K. and Murtaza, Q. "A literature review and perspectives in reverse logistics". Resour. Conserv. Recy., 97, pp. 76-92 (2015).

26. Pokharel, S. and Mutha, A. "Perspectives in reverse logistics: a review". Resour. Conserv. Recy., 53(4), pp. 175-182 (2009).

27. Wang, K.-H. and C.-T. Tung, "Construction of a model towards EOQ and pricing strategy for gradually obsolescent products". Appl. Math. Comput., 217(16), pp. 6926-6933 (2011).

28. Fulkerson, D.R., "An out-of-kilter method for minimal-cost flow problems". J. of the Soc. for Ind. and Ap. Math., 9(1), pp. 18-27 (1961).

29. Kashan, A.H., "A new metaheuristic for optimization: optics inspired optimization (OIO)". Comput. Oper. Res., 55, pp. 99-125 (2015).

30. Jalili, S. and Husseinzadeh Kashan, A. "Optimum discrete design of steel tower structures using optics inspired optimization method". The Str. Des. of Tall and Spe.Build., 27(9): pp. e1466 (2018).

31. Özdemi̇r, M.T. and Öztürk, D. "Comparative Performance Analysis of Optimal PID Parameters Tuning Based on the Optics Inspired Optimization Methods for Automatic Generation Control". Energies, 10(12), pp. 2134 (2017).

32. ÖZDEMİR, M.T. and ÖZTÜRK, D. "Optimal PID Tuning for Load Frequency Control using Optics Inspired Optimization Algorithm", ICNES (2016).

33. Kashan, A.H., "An effective algorithm for constrained optimization based on optics inspired optimization (OIO)". Comput. Aided Design., 63, pp. 52-71 (2015).

34. Mohammadi, M., Musa, S.N. "Bahreininejad, Optimization of economic lot scheduling problem with backordering and shelf-life considerations using calibrated metaheuristic algorithms". Comput. Aided. Design., 251(Supplement C), pp. 404-422. (2015).

35. Kuo, R.J., et al., "Solving bi-level linear programming problem through hybrid of immune genetic algorithm and particle swarm optimization algorithm". Appl. Math. Comput., 266(Supplement C), pp. 1013-1026 (2015).

36. Yang, M.-F. and Lin, Y. "Applying the linear particle swarm optimization to a serial multi-echelon inventory model". Expert. Syst. Appl., 37(3), pp. 2599-2608 (2010).

37. Mousavi, S.M., et al., "A modified particle swarm optimization for solving the integrated location and inventory control problems in a two-echelon supply chain network". J. INTELL. MANUF., 28(1), pp. 191-206 (2017).

38. Majumder, P. Bera, U.K. and Maiti, M. "An EPQ model for two-warehouse in unremitting release pattern with two-level trade credit period concerning both supplier and retailer". Appl. Math. Comput., 274(Supplement C), pp. 430-458 (2016).

39. Bhunia, A.K. and Shaikh, A.A. "An application of PSO in a two-warehouse inventory model for deteriorating item under permissible delay in payment with different inventory policies". Appl. Math. Comput., 256(Supplement C), pp. 831-850 (2015).

40. Jiang, Y., et al., "An improved particle swarm optimization algorithm". Appl. Math. Comput., 193(1): pp. 231-239. (2007).

41. Kennedy, J. and Eberhart, R. "Particle swarm optimization (PSO)". in Proc. IEEE International Conference on Neural Networks, Perth, Australia. (1995).

42. Huang, W. Kulkarni, V. and Swaminathan, J.M. "Coordinated inventory planning for new and old products under warranty". Probab. Eng. Inform. SC., 21(02): pp. 261-287 (2007).

43. Yeo, W.M. and Yuan, X.-M. "Optimal Inventory Policy For Products With Warranty Agreements. in Industrial Electronics", ISIE 2007. IEEE International Symposium on. IEEE (2007).

44. Khawam, J., Hausman, W.H. and D.W. Cheng, "Warranty Inventory Optimization for Hitachi Global Storage Technologies", Inc. Interfaces, 37(5), pp. 455-471. (2007).

45. Darghouth, M.N. Ait-kadi, D. and Chelbi, A. "Joint optimization of design, warranty and price for products sold with maintenance service contracts". Reliab. Eng. Syst. Safe., 165, pp. 197-208 (2017).

46. Shi, Y. and Eberhart, R.C. "Empirical study of particle swarm optimization. in Evolutionary Computation", CEC 99. Proceedings of the 1999 Congress on. IEEE (1999).

Transactions on Industrial Engineering (E)

January and February 2020Pages 494-515