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January 2018, 14(1): 267-282. doi: 10.3934/jimo.2017046

## Optimal production run time and inspection errors in an imperfect production system with warranty

 1 Department of Industrial & Management Engineering, Hanyang University, Ansan Gyeonggi-do, 15588, South Korea 2 Department of Mathematics, Hooghly Mohsin College, Chinsurah, Hooghly-712 101, West Bengal, India

* Corresponding author:ss.sumonsarkar@gmail.com (Sumon Sarkar)

Received  March 2016 Revised  September 2016 Published  June 2017

This paper considers an imperfect production system to obtain the optimal production run time and inspection policy. Contrary to the existing literature this model considerers that product inspection performs at any arbitrary time of the production cycle and after the inspection, all defective products produced until the end of the production run are fully reworked. Due to some misclassification during inspection, from the inspector's side two types of inspection errors as Type Ⅰ and Type Ⅱ are considered to make the model more realistic rather than existing models. Defective items, found by the inspector, are salvaged at some cost before being shipped. Non-inspected defective items are passed to customers with free minimal repair warranty. The model gives three special cases, where it is found that this model converges over the exiting literature. Some numerical examples along with graphical representations are provided to illustrate the proposed model with comparison with the existing models. Sensitivity analysis of the optimal solution with respect to key parameters of the model has been carried out and the implications are discussed.

Citation: Biswajit Sarkar, Bimal Kumar Sett, Sumon Sarkar. Optimal production run time and inspection errors in an imperfect production system with warranty. Journal of Industrial & Management Optimization, 2018, 14 (1) : 267-282. doi: 10.3934/jimo.2017046
##### References:
 [1] L. E. C$\acute{a}$rdenas-Barr$\acute{o}$n, On optimal batch sizing in a multi-stage production system with rework consideration, European Journal of Operational Research, 196 (2009), 1238-1244. doi: 10.1016/j.ejor.2008.04.015. [2] C. J. Chung, G. A. Widyadana and H. M. Wee, Economic production quantity model for deteriorating inventory with random machine unavailability and shortage, International Journal of Production Research, 49 (2010), 883-902. doi: 10.1080/00207540903460232. [3] G. Chryssolouris and S. Patel, In Process Control for Quality Assurance, in Industrial High Technology for Manufacturing (eds. K. McKee and D. Tijunelis), Marcel Dekker, Inc., New York, 1987,609-643. [4] M. A. Darwish and M. Ben-Daya, Effect of inspection errors and preventive maintenance on a two-stage production inventory system, International Journal of Production Economics, 107 (2007), 301-313. doi: 10.1016/j.ijpe.2006.09.008. [5] S. O. Duffuaa and A. El-G$\acute{a}$aly, A multi-objective mathematical optimization model for process targeting using 100$\%$ inspection policy, Applied Mathematical Modelling, 37 (2013), 1545-1552. doi: 10.1016/j.apm.2012.04.008. [6] S. O. Duffuaa and M. Khan, An optimal repeat inspection plan with several classifications, Journal of Operational Research Society, 53 (2002), 1016-1026. doi: 10.1057/palgrave.jors.2601392. [7] S. O. Duffuaa and A. El-Ga'aly, Impact of inspection errors on the formulation of a multi-objective optimization process targeting model under inspection sampling plan, Computers and Industrial Engineering, 80 (2015), 254-260. doi: 10.1016/j.cie.2014.07.025. [8] S. K. Goyal and L. E. C$\acute{a}$rdenas-Barr$\acute{o}$n, Economic production quantity with imperfect production system, Industrial Engineering Journal, 34 (2005), 33-36. [9] M. Harriga and M. Ben-Daya, Economic manufacturing lot sizing problem with imperfect manufacturing process: bounds and optimal solutions, Naval Research Logistics, 45 (1998), 423-433. [10] J. T. Hsu and L. E. Hsu, An integrated vendor-buyer inventory model with imperfect items and planned back orders, International Journal of Advance Manufacturing Technology, 68 (2013), 2121-2132. doi: 10.1007/s00170-013-4810-7. [11] F. Hu and Q. Zong, Optimal production run time for a deteriorating production system under an extended inspection policy, European Journal of Operational Research, 196 (2009), 979-986. doi: 10.1016/j.ejor.2008.05.008. [12] T. H. Lee, Optimal production run length and maintenance schedule for a deteriorating production system, International Journal of Advance Manufacturing Technology, 43 (2009), 959-963. doi: 10.1007/s00170-008-1773-1. [13] B. Pal, S. Sana and K. Chaudhuri, A mathematical model on EPQ for stochastic demand in an imperfect production system, Journal of Manufacturing System, 32 (2013), 260-270. doi: 10.1016/j.jmsy.2012.11.009. [14] E. L. Porteus, Optimal lot sizing, process quality improvement and setup cost reduction, Operations Research, 34 (1986), 137-144. doi: 10.1287/opre.34.1.137. [15] A. Raouf, J. K. Jain and P. T. Sathe, A cost-minimization model for multi characteristic component inspection, IIE Transaction, 15 (1983), 187-194. [16] M. J. Rosenblatt and H. L. Lee, Economic production cycles with imperfect production processes, IIE Transaction, 18 (1984), 48-55. doi: 10.1080/07408178608975329. [17] M. K. Salameh and M. Y. Jaber, Economic production quantity model for items with imperfect quality, International Journal of Production Economics, 64 (2000), 59-64. doi: 10.1016/S0925-5273(99)00044-4. [18] S. Sana, S. K. Goyal and K. S. Chaudhuri, On a volume flexible inventory model for items with an imperfect production system, International Journal of Operational Research, 2 (2007), 64-80. [19] S. Sana, S. K. Goyal and K. S. Chaudhuri, On an imperfect production process in a volume flexible inventory model, International Journal of Production Economics, 105 (2007), 548-559. [20] S. Sana, A production-inventory model of imperfect quality products in a three-layer supply chain, Decision Support System, 50 (2011), 539-547. doi: 10.1016/j.dss.2010.11.012. [21] S. Sana, Preventive maintenance and optimal buffer inventory for products sold with warranty in an imperfect production system, International Journal of Production Research, 50 (2012), 6763-6774. doi: 10.1080/00207543.2011.623838. [22] B. Sarkar, An inventory model with reliability in an imperfect production process, Applied Mathematics and Computation, 218 (2012), 4881-4891. doi: 10.1016/j.amc.2011.10.053. [23] B. Sarkar, Supply chain coordination with variable backorder, inspections, and discount policy for fixed lifetime products Mathematical Problems in Engineering (2016), Art. ID 6318737, 14 pp. doi: 10.1155/2016/6318737. [24] B. Sarkar, B. Mondal and S. Sarkar, Quality improvement and backorder price discount under controllable lead time in an inventory model, Journal of Manufacturing Systems, 35 (2015), 26-36. doi: 10.1016/j.jmsy.2014.11.012. [25] B. Sarkar and I. Moon, An EPQ model with inflation in an imperfect production system, Applied Mathematics and Computation, 217 (2011), 6159-6167. doi: 10.1016/j.amc.2010.12.098. [26] B. Sarkar and I. Moon, Improved quality, setup cost reduction, and variable backorder costs in an imperfect production process, International Journal of Production Economics, 155 (2014), 204-213. doi: 10.1016/j.ijpe.2013.11.014. [27] B. Sarkar, K. S. Chaudhuri and I. Moon, Manufacturing setup cost reduction and quality improvement for the distribution free continuous-review inventory model with a service level, Journal of Manufacturing Systems, 34 (2015), 74-82. doi: 10.1016/j.jmsy.2014.11.003. [28] B. Sarkar, M. Sarkar, L. E. C$\acute{a}$rdenas-Barr$\acute{o}$n and M. L. Singgih, An economic production quantity model with random defective rate, rework process and backorders for a single stage production system, Journal of Manufacturing Systems, 33 (2014), 423-435. doi: 10.1016/j.jmsy.2014.02.001. [29] B. Sarkar and S. Saren, Product inspection policy for an imperfect production system with inspection errors and warranty cost, European Journal of Operational Research, 248 (2016), 263-271. [30] B. Sarkar, S. S. Sana and K. S. Chaudhuri, An imperfect production process for time varying demand with inflation and time value of money --An EMQ model, Expert Systems with Applications, 38 (2011), 13543-13548. [31] B. Sarkar, S. S. Sana and K. S. Chaudhuri, An Economic production quantity model with stochastic demand in an imperfect production system, International Journal of Services and Operations Management, 9 (2011), 259-283. [32] C. H. Wang, Economic off-line quality control strategy with two types inspection errors, European Journal of Operational Research, 179 (2007), 132-147. doi: 10.1016/j.ejor.2006.03.024. [33] C. H. Wang and S. H. Sheu, Simultaneous determination of the optimal production-inventory and product inspection policies for a deteriorating production system, Computers} & Operations Research, 28 (2001), 1093-1110. doi: 10.1016/S0305-0548(00)00030-7. [34] C. H. Wang and F. C. Meng, Optimal lot size and offline inspection policy, Computers and Mathematics with Applications, 58 (2009), 1921-1929. doi: 10.1016/j.camwa.2009.07.089. [35] C. H. Wang, Integrated production and product inspection policy for a deteriorating production system, International Journal Production Economics, 95 (2005), 123-134. doi: 10.1016/j.ijpe.2003.11.012. [36] C. H. Wang, T. Dohi and W. C. Tsai, Coordinated procurement/inspection and production model under inspection errors, Computers and Industrial Engineering, 59 (2010), 473-478. doi: 10.1016/j.cie.2010.06.008.

show all references

##### References:
 [1] L. E. C$\acute{a}$rdenas-Barr$\acute{o}$n, On optimal batch sizing in a multi-stage production system with rework consideration, European Journal of Operational Research, 196 (2009), 1238-1244. doi: 10.1016/j.ejor.2008.04.015. [2] C. J. Chung, G. A. Widyadana and H. M. Wee, Economic production quantity model for deteriorating inventory with random machine unavailability and shortage, International Journal of Production Research, 49 (2010), 883-902. doi: 10.1080/00207540903460232. [3] G. Chryssolouris and S. Patel, In Process Control for Quality Assurance, in Industrial High Technology for Manufacturing (eds. K. McKee and D. Tijunelis), Marcel Dekker, Inc., New York, 1987,609-643. [4] M. A. Darwish and M. Ben-Daya, Effect of inspection errors and preventive maintenance on a two-stage production inventory system, International Journal of Production Economics, 107 (2007), 301-313. doi: 10.1016/j.ijpe.2006.09.008. [5] S. O. Duffuaa and A. El-G$\acute{a}$aly, A multi-objective mathematical optimization model for process targeting using 100$\%$ inspection policy, Applied Mathematical Modelling, 37 (2013), 1545-1552. doi: 10.1016/j.apm.2012.04.008. [6] S. O. Duffuaa and M. Khan, An optimal repeat inspection plan with several classifications, Journal of Operational Research Society, 53 (2002), 1016-1026. doi: 10.1057/palgrave.jors.2601392. [7] S. O. Duffuaa and A. El-Ga'aly, Impact of inspection errors on the formulation of a multi-objective optimization process targeting model under inspection sampling plan, Computers and Industrial Engineering, 80 (2015), 254-260. doi: 10.1016/j.cie.2014.07.025. [8] S. K. Goyal and L. E. C$\acute{a}$rdenas-Barr$\acute{o}$n, Economic production quantity with imperfect production system, Industrial Engineering Journal, 34 (2005), 33-36. [9] M. Harriga and M. Ben-Daya, Economic manufacturing lot sizing problem with imperfect manufacturing process: bounds and optimal solutions, Naval Research Logistics, 45 (1998), 423-433. [10] J. T. Hsu and L. E. Hsu, An integrated vendor-buyer inventory model with imperfect items and planned back orders, International Journal of Advance Manufacturing Technology, 68 (2013), 2121-2132. doi: 10.1007/s00170-013-4810-7. [11] F. Hu and Q. Zong, Optimal production run time for a deteriorating production system under an extended inspection policy, European Journal of Operational Research, 196 (2009), 979-986. doi: 10.1016/j.ejor.2008.05.008. [12] T. H. Lee, Optimal production run length and maintenance schedule for a deteriorating production system, International Journal of Advance Manufacturing Technology, 43 (2009), 959-963. doi: 10.1007/s00170-008-1773-1. [13] B. Pal, S. Sana and K. Chaudhuri, A mathematical model on EPQ for stochastic demand in an imperfect production system, Journal of Manufacturing System, 32 (2013), 260-270. doi: 10.1016/j.jmsy.2012.11.009. [14] E. L. Porteus, Optimal lot sizing, process quality improvement and setup cost reduction, Operations Research, 34 (1986), 137-144. doi: 10.1287/opre.34.1.137. [15] A. Raouf, J. K. Jain and P. T. Sathe, A cost-minimization model for multi characteristic component inspection, IIE Transaction, 15 (1983), 187-194. [16] M. J. Rosenblatt and H. L. Lee, Economic production cycles with imperfect production processes, IIE Transaction, 18 (1984), 48-55. doi: 10.1080/07408178608975329. [17] M. K. Salameh and M. Y. Jaber, Economic production quantity model for items with imperfect quality, International Journal of Production Economics, 64 (2000), 59-64. doi: 10.1016/S0925-5273(99)00044-4. [18] S. Sana, S. K. Goyal and K. S. Chaudhuri, On a volume flexible inventory model for items with an imperfect production system, International Journal of Operational Research, 2 (2007), 64-80. [19] S. Sana, S. K. Goyal and K. S. Chaudhuri, On an imperfect production process in a volume flexible inventory model, International Journal of Production Economics, 105 (2007), 548-559. [20] S. Sana, A production-inventory model of imperfect quality products in a three-layer supply chain, Decision Support System, 50 (2011), 539-547. doi: 10.1016/j.dss.2010.11.012. [21] S. Sana, Preventive maintenance and optimal buffer inventory for products sold with warranty in an imperfect production system, International Journal of Production Research, 50 (2012), 6763-6774. doi: 10.1080/00207543.2011.623838. [22] B. Sarkar, An inventory model with reliability in an imperfect production process, Applied Mathematics and Computation, 218 (2012), 4881-4891. doi: 10.1016/j.amc.2011.10.053. [23] B. Sarkar, Supply chain coordination with variable backorder, inspections, and discount policy for fixed lifetime products Mathematical Problems in Engineering (2016), Art. ID 6318737, 14 pp. doi: 10.1155/2016/6318737. [24] B. Sarkar, B. Mondal and S. Sarkar, Quality improvement and backorder price discount under controllable lead time in an inventory model, Journal of Manufacturing Systems, 35 (2015), 26-36. doi: 10.1016/j.jmsy.2014.11.012. [25] B. Sarkar and I. Moon, An EPQ model with inflation in an imperfect production system, Applied Mathematics and Computation, 217 (2011), 6159-6167. doi: 10.1016/j.amc.2010.12.098. [26] B. Sarkar and I. Moon, Improved quality, setup cost reduction, and variable backorder costs in an imperfect production process, International Journal of Production Economics, 155 (2014), 204-213. doi: 10.1016/j.ijpe.2013.11.014. [27] B. Sarkar, K. S. Chaudhuri and I. Moon, Manufacturing setup cost reduction and quality improvement for the distribution free continuous-review inventory model with a service level, Journal of Manufacturing Systems, 34 (2015), 74-82. doi: 10.1016/j.jmsy.2014.11.003. [28] B. Sarkar, M. Sarkar, L. E. C$\acute{a}$rdenas-Barr$\acute{o}$n and M. L. Singgih, An economic production quantity model with random defective rate, rework process and backorders for a single stage production system, Journal of Manufacturing Systems, 33 (2014), 423-435. doi: 10.1016/j.jmsy.2014.02.001. [29] B. Sarkar and S. Saren, Product inspection policy for an imperfect production system with inspection errors and warranty cost, European Journal of Operational Research, 248 (2016), 263-271. [30] B. Sarkar, S. S. Sana and K. S. Chaudhuri, An imperfect production process for time varying demand with inflation and time value of money --An EMQ model, Expert Systems with Applications, 38 (2011), 13543-13548. [31] B. Sarkar, S. S. Sana and K. S. Chaudhuri, An Economic production quantity model with stochastic demand in an imperfect production system, International Journal of Services and Operations Management, 9 (2011), 259-283. [32] C. H. Wang, Economic off-line quality control strategy with two types inspection errors, European Journal of Operational Research, 179 (2007), 132-147. doi: 10.1016/j.ejor.2006.03.024. [33] C. H. Wang and S. H. Sheu, Simultaneous determination of the optimal production-inventory and product inspection policies for a deteriorating production system, Computers} & Operations Research, 28 (2001), 1093-1110. doi: 10.1016/S0305-0548(00)00030-7. [34] C. H. Wang and F. C. Meng, Optimal lot size and offline inspection policy, Computers and Mathematics with Applications, 58 (2009), 1921-1929. doi: 10.1016/j.camwa.2009.07.089. [35] C. H. Wang, Integrated production and product inspection policy for a deteriorating production system, International Journal Production Economics, 95 (2005), 123-134. doi: 10.1016/j.ijpe.2003.11.012. [36] C. H. Wang, T. Dohi and W. C. Tsai, Coordinated procurement/inspection and production model under inspection errors, Computers and Industrial Engineering, 59 (2010), 473-478. doi: 10.1016/j.cie.2010.06.008.
Plot of expected total cost $C (u_{1}, u_{2}| t^{*}\; = \;2.19)$
Plot of expected total cost $C (t|u_{1}^{*}\; = \;0.00132, u_{2}^{*}\;=\;0.01023)$
Impact of holding cost $(C_h)$ on expected total cost $C(t*, u_{1}*, u_{2}*)$
Impact of salvage cost $(C_s)$ on expected total cost $C (t*, u_{1}*, u_{2}*)$
Impact of warranty cost $(C_w)$ on expected total cost $C (t*, u_{1}*, u_{2}*)$
Impact of rework cost $(C_r)$ on expected total cost $C (t*, u_{1}*, u_{2}*)$
Impact of inspection cost $(C_I)$ on expected total cost $C (t*, u_{1}*, u_{2}*)$
Impact of restoration cost (r) on expected total cost $C (t*, u_{1}*, u_{2}*)$
Summary of numerical results
 This model $(t^*, u_{1}^*, u_{2}^*)$$C(t^*, u_{1}^*, u_{2}^*) Sarkar and Saren[29] (t^*, u^*) C(t^*, u^*) Hu and Zong [11] (t^*, u_{1}^*, u_{2}^*) C(t^*, u_{1}^*, u_{2}^*) Wang [35] (t^*, u^*) C(t^*, u^*) (2.04, 0.058, 0.251) {\}8.48 (1.85, 0.064) {\}8.90 (2.05, 0.062, 0.239) {\}8.49 (1.83, 0.069) {\}8.97  This model (t^*, u_{1}^*, u_{2}^*)$$C(t^*, u_{1}^*, u_{2}^*)$ Sarkar and Saren[29] $(t^*, u^*)$ $C(t^*, u^*)$ Hu and Zong [11] $(t^*, u_{1}^*, u_{2}^*)$ $C(t^*, u_{1}^*, u_{2}^*)$ Wang [35] $(t^*, u^*)$ $C(t^*, u^*)$ (2.04, 0.058, 0.251) ${\$}$8.48 (1.85, 0.064)${\$}$8.90 (2.05, 0.062, 0.239) ${\$}$8.49 (1.83, 0.069)${\$}$8.97
Impact of key parameters on optimal solution
 $C_{h}$ $t^*$ $u_{1}^*$ $u_{2}^*$ $C(\cdot)$ $C_{r}$ $t^*$ $u_{1}^*$ $u_{2}^*$ $C(\cdot)$ 0.1 4.56 0.026 0.112 7.12 2.4 2.14 0.0552 0.15 8.00 0.3 2.64 0.045 0.194 7.92 2.6 2.11 0.0560 0.17 8.16 0.5 2.04 0.058 0.251 8.48 2.8 2.08 0.0568 0.21 8.32 0.7 1.72 0.068 0.296 8.93 3.0 2.04 0.0579 0.25 8.48 0.9 1.52 0.078 0.336 9.32 3.2 1.10 0.0592 0.30 8.62 1.1 1.37 0.086 0.372 9.68 3.4 1.94 0.0608 0.36 8.76 $C_{s}$ $t^*$ $u_{1}^*$ $u_{2}^*$ $C(\cdot)$ 3.6 1.87 0.0631 0.49 8.88 2.0 1.98 0.032 0.382 8.41 $C_{i}$ $t^*$ $u_{1}^*$ $u_{2}^*$ $C(\cdot)$ 2.2 2.01 0.042 0.317 8.44 1.0 2.007 0.003 0.017 8.437 2.4 2.03 0.050 0.278 8.46 1.2 2.041 0.058 0.251 8.479 2.6 2.04 0.058 0.251 8.48 1.4 2.068 0.084 0.214 8.511 2.8 2.05 0.065 0.229 8.49 1.6 2.085 0.105 0.181 8.531 3.0 2.06 0.072 0.211 8.51 1.8 2.093 0.124 0.151 8.542 3.2 2.07 0.079 0.196 8.52 2.0 2.094 0.141 0.120 8.543 $C_{w}$ $t^*$ $u_{1}^*$ $u_{2}^*$ $C(\cdot)$ $r$ $t^*$ $u_{1}^*$ $u_{2}^*$ $C(\cdot)$ 6.4 2.014 0.123 0.254 8.446 1200 1.94 0.061 0.264 8.35 6.8 2.024 0.105 0.253 8.457 1300 1.97 0.060 0.259 8.40 7.2 2.031 0.089 0.252 8.466 1400 2.01 0.059 0.255 8.44 7.6 2.037 0.073 0.251 8.473 1500 2.04 0.058 0.251 8.48 8.0 2.042 0.058 0.251 8.479 1600 2.07 0.057 0.246 8.52 8.4 2.045 0.041 0.250 8.483 1700 2.11 0.056 0.243 8.56 8.8 2.047 0.018 0.249 8.486 1800 2.14 0.055 0.239 8.60
 $C_{h}$ $t^*$ $u_{1}^*$ $u_{2}^*$ $C(\cdot)$ $C_{r}$ $t^*$ $u_{1}^*$ $u_{2}^*$ $C(\cdot)$ 0.1 4.56 0.026 0.112 7.12 2.4 2.14 0.0552 0.15 8.00 0.3 2.64 0.045 0.194 7.92 2.6 2.11 0.0560 0.17 8.16 0.5 2.04 0.058 0.251 8.48 2.8 2.08 0.0568 0.21 8.32 0.7 1.72 0.068 0.296 8.93 3.0 2.04 0.0579 0.25 8.48 0.9 1.52 0.078 0.336 9.32 3.2 1.10 0.0592 0.30 8.62 1.1 1.37 0.086 0.372 9.68 3.4 1.94 0.0608 0.36 8.76 $C_{s}$ $t^*$ $u_{1}^*$ $u_{2}^*$ $C(\cdot)$ 3.6 1.87 0.0631 0.49 8.88 2.0 1.98 0.032 0.382 8.41 $C_{i}$ $t^*$ $u_{1}^*$ $u_{2}^*$ $C(\cdot)$ 2.2 2.01 0.042 0.317 8.44 1.0 2.007 0.003 0.017 8.437 2.4 2.03 0.050 0.278 8.46 1.2 2.041 0.058 0.251 8.479 2.6 2.04 0.058 0.251 8.48 1.4 2.068 0.084 0.214 8.511 2.8 2.05 0.065 0.229 8.49 1.6 2.085 0.105 0.181 8.531 3.0 2.06 0.072 0.211 8.51 1.8 2.093 0.124 0.151 8.542 3.2 2.07 0.079 0.196 8.52 2.0 2.094 0.141 0.120 8.543 $C_{w}$ $t^*$ $u_{1}^*$ $u_{2}^*$ $C(\cdot)$ $r$ $t^*$ $u_{1}^*$ $u_{2}^*$ $C(\cdot)$ 6.4 2.014 0.123 0.254 8.446 1200 1.94 0.061 0.264 8.35 6.8 2.024 0.105 0.253 8.457 1300 1.97 0.060 0.259 8.40 7.2 2.031 0.089 0.252 8.466 1400 2.01 0.059 0.255 8.44 7.6 2.037 0.073 0.251 8.473 1500 2.04 0.058 0.251 8.48 8.0 2.042 0.058 0.251 8.479 1600 2.07 0.057 0.246 8.52 8.4 2.045 0.041 0.250 8.483 1700 2.11 0.056 0.243 8.56 8.8 2.047 0.018 0.249 8.486 1800 2.14 0.055 0.239 8.60