Mitigation of High-Tech Products with Probabilistic Deterioration and Inflations
Affiliation(s)
1 Department of Industrial & Management Engineering, Hanyang University, Seoul, South Korea. 2 Department of Mathematics, Hooghly Mohsin College, Hooghly, India. 3 Department of Mathematics, Indian Institute of Technology, Kharagpur, India. 4 Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, India.
1 Department of Industrial & Management Engineering, Hanyang University, Seoul, South Korea. 2 Department of Mathematics, Hooghly Mohsin College, Hooghly, India. 3 Department of Mathematics, Indian Institute of Technology, Kharagpur, India. 4 Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, India.
ABSTRACT
This paper describes a deteriorating inventory model with ramp-type demand pattern under stock-dependent consumption rate. The deterioration of the product is considered as probabilistic to make the research a more realistic one. The proposed model assumes partially backorder rate which follows a negative exponential with the waiting time. The effect of inflation and time value of money are incorporated into the model. The purpose of this study is to develop an optimal replenishment policy so that the total profit is maximized. We provide a simple solution procedure to obtain the optimal solutions. Numerical examples along with graphical representations are provided to illustrate the model. Sensitivity analysis of the optimal solution with respect to key parameters of the model has been carried out and the implications are discussed.
This paper describes a deteriorating inventory model with ramp-type demand pattern under stock-dependent consumption rate. The deterioration of the product is considered as probabilistic to make the research a more realistic one. The proposed model assumes partially backorder rate which follows a negative exponential with the waiting time. The effect of inflation and time value of money are incorporated into the model. The purpose of this study is to develop an optimal replenishment policy so that the total profit is maximized. We provide a simple solution procedure to obtain the optimal solutions. Numerical examples along with graphical representations are provided to illustrate the model. Sensitivity analysis of the optimal solution with respect to key parameters of the model has been carried out and the implications are discussed.
KEYWORDS
Ramp-Type Demand, Stock-Dependent Consumption Rate, Partial Backorder, Inflation, Probabilistic Deterioration
Ramp-Type Demand, Stock-Dependent Consumption Rate, Partial Backorder, Inflation, Probabilistic Deterioration
Cite this paper
Sarkar, B. , Sett, B. , Goswami, A. and Sarkar, S. (2015) Mitigation of High-Tech Products with Probabilistic Deterioration and Inflations. American Journal of Industrial and Business Management, 5, 73-89. doi: 10.4236/ajibm.2015.53009.
Sarkar, B. , Sett, B. , Goswami, A. and Sarkar, S. (2015) Mitigation of High-Tech Products with Probabilistic Deterioration and Inflations. American Journal of Industrial and Business Management, 5, 73-89. doi: 10.4236/ajibm.2015.53009.
References
[1] Covert, R.P. and Philip, G.C. (1973) An EOQ Model for Items with Weibull Distribution Deterioration. AIIE Transactions, 5, 323-326.
http://dx.doi.org/10.1080/05695557308974918 [2] Dye, C.Y., Chang, H.J. and Teng, J.T. (2006) A Deteriorating Inventory Model with Time-Varying Demand and Shortage-Dependent Partial Backlogging. European Journal of Operational Research, 172, 417-429.
http://dx.doi.org/10.1016/j.ejor.2004.10.025 [3] Chung, C.J. and Wee, H.M. (2008) An Integrated Production-Inventory Deteriorating Model for Pricing Policy Considering Imperfect Production, Inspection Planning and Warranty-Period and Stock-Level-Dependant Demand. International Journal of System Science, 39,823-837.
http://dx.doi.org/10.1080/00207720801902598 [4] Sana, S.S. (2010) Demand Influenced by Enterprises Initiatives—A Multi-Item EOQ Model for Deteriorating and Ameliorating Items. Mathematical and Computer Modelling, 52, 284-302.
http://dx.doi.org/10.1016/j.mcm.2010.02.045 [5] Wee, H.M., Lee, M.C., Yu, J.C.P. and Wang, C.E. (2011) Optimal Replenishment Policy for a Deteriorating Green Product: Lifecycle Costing Analysis. International Journal of Production Economics, 133, 603-611.
http://dx.doi.org/10.1016/j.ijpe.2011.05.001 [6] Sett, B.K., Sarkar, B. and Goswami, A. (2012) A Two-Warehouse Inventory Model with Increasing Demand and Time Varying Deterioration. Scientia Iranica: E, 19, 1969-1977.
http://dx.doi.org/10.1016/j.scient.2012.10.040 [7] Sarkar, B. (2013) A Production-Inventory Model with Probabilistic Deterioration in Two-Echelon Supply Chain Management. Applied Mathematical Modelling, 37, 3138-3151.
http://dx.doi.org/10.1016/j.apm.2012.07.026 [8] Sarkar, M. and Sarkar, B. (2013) An Economic Manufacturing Quantity Model with Probabilistic Deterioration in a Production System. Economic Modelling, 31, 245-252.
http://dx.doi.org/10.1016/j.econmod.2012.11.019 [9] Sarkar, B. and Sarkar, S. (2013) Variable Deterioration and Demand—An Inventory Model. Economic Modelling, 31, 548-556.
http://dx.doi.org/10.1016/j.econmod.2012.11.045 [10] Sarkar, B., Saren, S. and Wee, H.M. (2013) An Inventory Model with Variable Demand, Component Cost and Selling Price for Deteriorating Items. Economic Modelling, 30, 306-310.
http://dx.doi.org/10.1016/j.econmod.2012.09.002 [11] Cárdenas-Barrón, L.E., Sarkar, B. and Treviño-Garza, G. (2013) An Improved Solution to the Replenishment Policy for the EMQ Model with Rework and Multiple Shipments. Applied Mathematical Modelling, 37, 5549-5554.
http://dx.doi.org/10.1016/j.apm.2012.10.017 [12] Sarkar, B. and Saren, S. (2014) Partial Trade-Credit Policy of Retailer with Exponentially Deteriorating Items. International Journal of Applied and Computational Mathematics.
http://dx.doi.org/10.1007/s40819-014-0019-1 [13] Sarkar, B., Saren, S. and Cárdenas-Barrón, L.E. (2014) An Inventory Model with Trade-Credit Policy and Variable Deterioration for Fixed Lifetime Products. Annals of Operations Research.
http://dx.doi.org/10.1007/s10479-014-1745-9 [14] Deb, M. and Chaudhuri, K.S. (1987) A Note on the Heuristic for Replenishment of Trended Inventories Considering Shortages. Journal of Operational Research Society, 38, 459-463. [15] McDonald, J.J. (1979) Inventory Replenishment Policies-Computational Solutions. Journal of Operational Research Society, 30, 933-936.
http://dx.doi.org/10.2307/3009548 [16] Chang, H.J. and Dye, C.Y. (1999) An EOQ Model for Deteriorating Items with Time Varying Demand and Partial Backlogging. Journal of Operational Research Society, 50, 1176-1182.
http://dx.doi.org/10.2307/3010088 [17] Cárdenas-Barrón, L.E. (2011) The Economic Production Quantity (EPQ) with Shortage Derived Algebraically. International Journal of Production Economics, 70, 289-292.
http://dx.doi.org/10.1016/S0925-5273(00)00068-2 [18] Teng, J.T., Chang, H.J., Dye, C.Y. and Hung, C.H. (2002) An Optimal Replenishment Policy for Deteriorating Items with Time-Varying Demand and Partial Back-Logging. Operations Research Letters, 30, 387-393.
http://dx.doi.org/10.1016/S0167-6377(02)00150-5 [19] Cárdenas-Barrón, L.E. (2009) Economic Production Quantity with Rework Process at a Single-Stage Manufacturing System with Planned Backorders. Computer & Industrial Engineering, 57, 1105-1113.
http://dx.doi.org/10.1016/j.cie.2009.04.020 [20] Sarkar, B., Sana, S. and Chaudhuri, K. (2010) Optimal Reliability, Production Lotsize and Safety Stock: An Economic Manufacturing Quantity Model. International Journal of Management Science and Engineering Management, 5, 192-202. [21] Sarkar, B., Gupta, H., Chaudhuri, K. and Goyal, S.K. (2014) An Integrated Inventory Model with Variable Lead Time, Defective Units and Delay in Payments. Applied Mathematics and Computation, 237, 650-658.
http://dx.doi.org/10.1016/j.amc.2014.03.061 [22] Sarkar, B. and Majumder, A. (2013) Integrated Vendor Buyer Supply Chain Model with Vendor’s Setup Cost Reduction. Applied Mathematics and Computation, 224, 362-371.
http://dx.doi.org/10.1016/j.amc.2013.08.072 [23] Sarkar, B. and Sarkar, S. (2013) An Improved Inventory Model with Partial Backlogging, Time Varying Deterioration and Stock-Dependent Demand. Economic Modelling, 30, 924-932.
http://dx.doi.org/10.1016/j.econmod.2012.09.049 [24] Sarkar, B., Cárdenas-Barrón, L.E., Sarkar, M. and Singgih, M.L. (2014) An Economic Production Quantity Model with Random Defective Rate, Rework Process and Backorders for a Single Stage Production System. Journal of Manufacturing System, 33, 423-435.
http://dx.doi.org/10.1016/j.jmsy.2014.02.001 [25] Sarkar, B., Mandal, B. and Sarkar, S. (2015) Quality Improvement and Backorder Price Discount under Controllable Lead Time in an Inventory Model. Journal of Manufacturing Systems, 35, 26-36.
http://dx.doi.org/10.1016/j.jmsy.2014.11.012 [26] Mandal, B. and Pal, A.K. (1998) Order Level Inventory System with Ramp Type Demand Rate for Deteriorating Items. Journal of Interdisciplinary Mathematics, 1, 49-66.
http://dx.doi.org/10.1080/09720502.1998.10700243 [27] Wu, K.S. (2001) An EOQ Inventory Model for Items with Weibull Distribution Deterioration, Ramp-Type Demand Rate and Partial Backlogging. Production Planning & Control, 12, 787-793.
http://dx.doi.org/10.1080/09537280110051819 [28] Giri, B.C., Jalan, A.K. and Chaudhuri, K.S. (2003) Economic Order Quantity Model with Weibull Deterioration Distribution, Shortage and Ramp-Type Demand. International Journal of System Science, 34, 237-243.
http://dx.doi.org/10.1080/0020772131000158500 [29] Skouri, K., Konstantaras, I., Papachristos, S. and Ganas, I. (2009) Inventory Models with Ramp Type Demand Rate, Partial Backlogging and Weibull Deterioration Rate. European Journal of Operational Research, 192, 79-92.
http://dx.doi.org/10.1016/j.ejor.2007.09.003 [30] Sana, S.S. (2010) Optimal Selling Price and Lotsize with Time Varying Deterioration and Partial Backlogging. Applied Mathematics and Computation, 217, 185-194.
http://dx.doi.org/10.1016/j.amc.2010.05.040 [31] Cárdenas-Barrón, L.E., Sarkar, B. and Treviño-Garza, G. (2013) Easy and Improved Algorithms to Joint Determination of the Replenishment Lot Size and Number of Shipments for an EPQ Model with Rework. Mathematical and Computational Applications, 18, 132-138. [32] Sarkar, B., Chaudhuri, K. and Moon, I. (2015) Manufacturing Setup Cost Reduction and Quality Improvement for the Distribution Free Continuous-Review Inventory Model with a Service Level Constraint. Journal of Manufacturing Systems, 34, 74-82.
http://dx.doi.org/10.1016/j.jmsy.2014.11.003 [33] Buzacott, J.A. (1975) Economic Order Quantities with Inflation. Operational Research Quarterly, 26, 553-558.
http://dx.doi.org/10.1057/jors.1975.113 [34] Datta, T.K. and Pal, A.K. (1991) Effects of Inflation and Time-Value of Money on an Inventory Model with Linear Time-Dependent Demand Rate and Shortages. European Journal of Operational Research, 52, 326-333.
http://dx.doi.org/10.1016/0377-2217(91)90167-T [35] Jaggi, C.K., Aggarwal, K.K. and Goel, S.K. (2006) Optimal Order Policy for Deteriorating Items with Inflation Induced Demand. International Journal of Production Economic, 103, 707-714.
http://dx.doi.org/10.1016/j.ijpe.2006.01.004 [36] Sarkar, B. and Moon, I. (2011) An EPQ Model with Inflation in an Imperfect Production System. Applied Mathematics and Computation, 217, 6159-6167.
http://dx.doi.org/10.1016/j.amc.2010.12.098 [37] Sarkar, B., Sana, S.S. and Chaudhuri, K.S. (2011) An Imperfect Production Process for Time Varying Demand with Inflation and Time Value of Money: An EMQ Model. Expert System with Application, 38, 13543-13548. [38] Sarkar, B., Mandal, P. and Sarkar, S. (2014) An EMQ Model with Price and Time Dependent Demand under the Effect of Reliability and Inflation. Applied Mathematics and Computation, 231, 414-421.
http://dx.doi.org/10.1016/j.amc.2014.01.004
[1] Covert, R.P. and Philip, G.C. (1973) An EOQ Model for Items with Weibull Distribution Deterioration. AIIE Transactions, 5, 323-326.
http://dx.doi.org/10.1080/05695557308974918 [2] Dye, C.Y., Chang, H.J. and Teng, J.T. (2006) A Deteriorating Inventory Model with Time-Varying Demand and Shortage-Dependent Partial Backlogging. European Journal of Operational Research, 172, 417-429.
http://dx.doi.org/10.1016/j.ejor.2004.10.025 [3] Chung, C.J. and Wee, H.M. (2008) An Integrated Production-Inventory Deteriorating Model for Pricing Policy Considering Imperfect Production, Inspection Planning and Warranty-Period and Stock-Level-Dependant Demand. International Journal of System Science, 39,823-837.
http://dx.doi.org/10.1080/00207720801902598 [4] Sana, S.S. (2010) Demand Influenced by Enterprises Initiatives—A Multi-Item EOQ Model for Deteriorating and Ameliorating Items. Mathematical and Computer Modelling, 52, 284-302.
http://dx.doi.org/10.1016/j.mcm.2010.02.045 [5] Wee, H.M., Lee, M.C., Yu, J.C.P. and Wang, C.E. (2011) Optimal Replenishment Policy for a Deteriorating Green Product: Lifecycle Costing Analysis. International Journal of Production Economics, 133, 603-611.
http://dx.doi.org/10.1016/j.ijpe.2011.05.001 [6] Sett, B.K., Sarkar, B. and Goswami, A. (2012) A Two-Warehouse Inventory Model with Increasing Demand and Time Varying Deterioration. Scientia Iranica: E, 19, 1969-1977.
http://dx.doi.org/10.1016/j.scient.2012.10.040 [7] Sarkar, B. (2013) A Production-Inventory Model with Probabilistic Deterioration in Two-Echelon Supply Chain Management. Applied Mathematical Modelling, 37, 3138-3151.
http://dx.doi.org/10.1016/j.apm.2012.07.026 [8] Sarkar, M. and Sarkar, B. (2013) An Economic Manufacturing Quantity Model with Probabilistic Deterioration in a Production System. Economic Modelling, 31, 245-252.
http://dx.doi.org/10.1016/j.econmod.2012.11.019 [9] Sarkar, B. and Sarkar, S. (2013) Variable Deterioration and Demand—An Inventory Model. Economic Modelling, 31, 548-556.
http://dx.doi.org/10.1016/j.econmod.2012.11.045 [10] Sarkar, B., Saren, S. and Wee, H.M. (2013) An Inventory Model with Variable Demand, Component Cost and Selling Price for Deteriorating Items. Economic Modelling, 30, 306-310.
http://dx.doi.org/10.1016/j.econmod.2012.09.002 [11] Cárdenas-Barrón, L.E., Sarkar, B. and Treviño-Garza, G. (2013) An Improved Solution to the Replenishment Policy for the EMQ Model with Rework and Multiple Shipments. Applied Mathematical Modelling, 37, 5549-5554.
http://dx.doi.org/10.1016/j.apm.2012.10.017 [12] Sarkar, B. and Saren, S. (2014) Partial Trade-Credit Policy of Retailer with Exponentially Deteriorating Items. International Journal of Applied and Computational Mathematics.
http://dx.doi.org/10.1007/s40819-014-0019-1 [13] Sarkar, B., Saren, S. and Cárdenas-Barrón, L.E. (2014) An Inventory Model with Trade-Credit Policy and Variable Deterioration for Fixed Lifetime Products. Annals of Operations Research.
http://dx.doi.org/10.1007/s10479-014-1745-9 [14] Deb, M. and Chaudhuri, K.S. (1987) A Note on the Heuristic for Replenishment of Trended Inventories Considering Shortages. Journal of Operational Research Society, 38, 459-463. [15] McDonald, J.J. (1979) Inventory Replenishment Policies-Computational Solutions. Journal of Operational Research Society, 30, 933-936.
http://dx.doi.org/10.2307/3009548 [16] Chang, H.J. and Dye, C.Y. (1999) An EOQ Model for Deteriorating Items with Time Varying Demand and Partial Backlogging. Journal of Operational Research Society, 50, 1176-1182.
http://dx.doi.org/10.2307/3010088 [17] Cárdenas-Barrón, L.E. (2011) The Economic Production Quantity (EPQ) with Shortage Derived Algebraically. International Journal of Production Economics, 70, 289-292.
http://dx.doi.org/10.1016/S0925-5273(00)00068-2 [18] Teng, J.T., Chang, H.J., Dye, C.Y. and Hung, C.H. (2002) An Optimal Replenishment Policy for Deteriorating Items with Time-Varying Demand and Partial Back-Logging. Operations Research Letters, 30, 387-393.
http://dx.doi.org/10.1016/S0167-6377(02)00150-5 [19] Cárdenas-Barrón, L.E. (2009) Economic Production Quantity with Rework Process at a Single-Stage Manufacturing System with Planned Backorders. Computer & Industrial Engineering, 57, 1105-1113.
http://dx.doi.org/10.1016/j.cie.2009.04.020 [20] Sarkar, B., Sana, S. and Chaudhuri, K. (2010) Optimal Reliability, Production Lotsize and Safety Stock: An Economic Manufacturing Quantity Model. International Journal of Management Science and Engineering Management, 5, 192-202. [21] Sarkar, B., Gupta, H., Chaudhuri, K. and Goyal, S.K. (2014) An Integrated Inventory Model with Variable Lead Time, Defective Units and Delay in Payments. Applied Mathematics and Computation, 237, 650-658.
http://dx.doi.org/10.1016/j.amc.2014.03.061 [22] Sarkar, B. and Majumder, A. (2013) Integrated Vendor Buyer Supply Chain Model with Vendor’s Setup Cost Reduction. Applied Mathematics and Computation, 224, 362-371.
http://dx.doi.org/10.1016/j.amc.2013.08.072 [23] Sarkar, B. and Sarkar, S. (2013) An Improved Inventory Model with Partial Backlogging, Time Varying Deterioration and Stock-Dependent Demand. Economic Modelling, 30, 924-932.
http://dx.doi.org/10.1016/j.econmod.2012.09.049 [24] Sarkar, B., Cárdenas-Barrón, L.E., Sarkar, M. and Singgih, M.L. (2014) An Economic Production Quantity Model with Random Defective Rate, Rework Process and Backorders for a Single Stage Production System. Journal of Manufacturing System, 33, 423-435.
http://dx.doi.org/10.1016/j.jmsy.2014.02.001 [25] Sarkar, B., Mandal, B. and Sarkar, S. (2015) Quality Improvement and Backorder Price Discount under Controllable Lead Time in an Inventory Model. Journal of Manufacturing Systems, 35, 26-36.
http://dx.doi.org/10.1016/j.jmsy.2014.11.012 [26] Mandal, B. and Pal, A.K. (1998) Order Level Inventory System with Ramp Type Demand Rate for Deteriorating Items. Journal of Interdisciplinary Mathematics, 1, 49-66.
http://dx.doi.org/10.1080/09720502.1998.10700243 [27] Wu, K.S. (2001) An EOQ Inventory Model for Items with Weibull Distribution Deterioration, Ramp-Type Demand Rate and Partial Backlogging. Production Planning & Control, 12, 787-793.
http://dx.doi.org/10.1080/09537280110051819 [28] Giri, B.C., Jalan, A.K. and Chaudhuri, K.S. (2003) Economic Order Quantity Model with Weibull Deterioration Distribution, Shortage and Ramp-Type Demand. International Journal of System Science, 34, 237-243.
http://dx.doi.org/10.1080/0020772131000158500 [29] Skouri, K., Konstantaras, I., Papachristos, S. and Ganas, I. (2009) Inventory Models with Ramp Type Demand Rate, Partial Backlogging and Weibull Deterioration Rate. European Journal of Operational Research, 192, 79-92.
http://dx.doi.org/10.1016/j.ejor.2007.09.003 [30] Sana, S.S. (2010) Optimal Selling Price and Lotsize with Time Varying Deterioration and Partial Backlogging. Applied Mathematics and Computation, 217, 185-194.
http://dx.doi.org/10.1016/j.amc.2010.05.040 [31] Cárdenas-Barrón, L.E., Sarkar, B. and Treviño-Garza, G. (2013) Easy and Improved Algorithms to Joint Determination of the Replenishment Lot Size and Number of Shipments for an EPQ Model with Rework. Mathematical and Computational Applications, 18, 132-138. [32] Sarkar, B., Chaudhuri, K. and Moon, I. (2015) Manufacturing Setup Cost Reduction and Quality Improvement for the Distribution Free Continuous-Review Inventory Model with a Service Level Constraint. Journal of Manufacturing Systems, 34, 74-82.
http://dx.doi.org/10.1016/j.jmsy.2014.11.003 [33] Buzacott, J.A. (1975) Economic Order Quantities with Inflation. Operational Research Quarterly, 26, 553-558.
http://dx.doi.org/10.1057/jors.1975.113 [34] Datta, T.K. and Pal, A.K. (1991) Effects of Inflation and Time-Value of Money on an Inventory Model with Linear Time-Dependent Demand Rate and Shortages. European Journal of Operational Research, 52, 326-333.
http://dx.doi.org/10.1016/0377-2217(91)90167-T [35] Jaggi, C.K., Aggarwal, K.K. and Goel, S.K. (2006) Optimal Order Policy for Deteriorating Items with Inflation Induced Demand. International Journal of Production Economic, 103, 707-714.
http://dx.doi.org/10.1016/j.ijpe.2006.01.004 [36] Sarkar, B. and Moon, I. (2011) An EPQ Model with Inflation in an Imperfect Production System. Applied Mathematics and Computation, 217, 6159-6167.
http://dx.doi.org/10.1016/j.amc.2010.12.098 [37] Sarkar, B., Sana, S.S. and Chaudhuri, K.S. (2011) An Imperfect Production Process for Time Varying Demand with Inflation and Time Value of Money: An EMQ Model. Expert System with Application, 38, 13543-13548. [38] Sarkar, B., Mandal, P. and Sarkar, S. (2014) An EMQ Model with Price and Time Dependent Demand under the Effect of Reliability and Inflation. Applied Mathematics and Computation, 231, 414-421.
http://dx.doi.org/10.1016/j.amc.2014.01.004