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An alternative tree method for calibration of the local volatility
Effect of reliability on varying demand and holding cost on inventory system incorporating probabilistic deterioration
Department of Mathematics, National Institute of Technology Puducherry, Karaikal-609609, India |
This paper presents a mathematical framework to derive an inventory model for time, reliability, and advertisement dependent demand. This paper considers the demand rate is high initially, and then the demand rate reduces later stage, which reflects the situation related to cash in hand. The uncertain deterioration of the product presents through Uniform, Triangular, and Double Triangular probability distributions. The holding cost of the proposed inventory system is dependent on the reliability of the item to make this study a more realistic one. This proposed inventory system allows the situation of shortage and partially backlogged at a fixed rate. Numerical examples, along with managerial implications and sensitivity analysis of the inventory parameters, discuss to examine the effect of changes on the optimal total inventory cost.
References:
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B. Ahmad and L. Benkherouf,
Economic-order-type inventory models for
non-instantaneous deteriorating items and backlogging, RAIRO - Operations Research, 52 (2018), 895-901.
doi: 10.1051/ro/2018010. |
[2] |
H. K. Alfares and A. M. Ghaithan, Inventory and pricing model with price-dependent demand, time-varying holding cost, and quantity discounts, Computers & Industrial Engineering, 94 (2016), 170-177. Google Scholar |
[3] |
Z. T. Balkhi and L. Benkherouf,
On an inventory model
for deteriorating items with stock dependent and time-varying demand rates, Computers and Operations Research, 31 (2004), 223-240.
doi: 10.1016/S0305-0548(02)00182-X. |
[4] |
H. Barman, M. Pervin, S. K. Roy and G. W. Weber, Back-ordered inventory model with inflation in a cloudy-fuzzy environment, Journal of Industrial and Management Optimization, 2020.
doi: 10.3934/jimo.2020052. |
[5] |
S. Barzegar, M. Seifbarghy, S.H. Pasandideh and M. Arjmand, Development of a joint economic lot size model with stochastic demand within non-equal shipments, Scientia Iranica, 23 (2016), 3026-3034. Google Scholar |
[6] |
C. K. Chan, W. H. Wong, A. Langevin and Y. C. E. Lee, An integrated production-inventory model for deteriorating items with consideration of optimal production rate and deterioration during delivery, International Journal of Production Economics, 189 (2017), 1-13. Google Scholar |
[7] |
H. J. Chang and C. Y. Dye, An EOQ model for deteriorating items with time varying demand and partial backlogging, Journal of Operational Research Society, 50 (1999), 1176-1182. Google Scholar |
[8] |
C.T. Chang,
Inventory model with stock-dependent demand and nonlinear holding costs for deteriorating items, Asia-Pacific Journal of Operational Research, 21 (2004), 435-446.
doi: 10.1142/S0217595904000321. |
[9] |
U. Chanda and A. Kumar, Optimization of fuzzy EOQ model for advertising and price sensitive demand model under dynamic ceiling on potential adoption, International Journal of Systems Science, 4 (2016), 145-165. Google Scholar |
[10] |
R. R. Chowdhury, S. K. Ghosh and K.S. Chaudhuri, An order-level inventory model for a deteriorating item with time-quadratic demand and time-dependent partial backlogging with shortages in all cycles, American Journal of Mathematical and Management Sciences, 33 (2014), 75-97. Google Scholar |
[11] |
R. R. Chowdhury, S. K. Ghosh and K. S. Chaudhuri,
An
inventory model for perishable items with stock and advertisement sensitive
demand, Int. J. Appl. Comput. Math., 1 (2015), 187-201.
doi: 10.1007/s40819-014-0011-9. |
[12] |
P. S. Deng, R. Lin and P. P. Chu,
A note on the inventory
models for deteriorating items with ramp type demand rate, European Journal of Operational Research, 178 (2007), 112-120.
doi: 10.1016/j.ejor.2006.01.028. |
[13] |
B. K. Dey, B. Sarkar, M. Sarkar and S. Pareek,
An integrated inventory
model involving discrete setup cost reduction, variable safety factor,
selling price dependent demand, and investment, RAIRO -
Operations Research, 53 (2019), 39-57.
doi: 10.1051/ro/2018009. |
[14] |
T. K. Datta and A. K. Pal, 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 (1991), 326-333. Google Scholar |
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K. V. Geetha and R. Udayakumar, Optimal replenishment policy for deteriorating items with time sensitive demand under trade credit financing., American Journal of Mathematical and Management Sciences, 34 (2015), 197-212. Google Scholar |
[16] |
M. Ghoreishi, G. W. Weber and A. Mirzazadeh,
An inventory
model for non-instantaneous deteriorating items with partial backlogging,
permissible delay in payments, inflation and selling price-dependent demand and customer returns, Annals of Operations Research, 226 (2014), 221-238.
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R. Haji and H. Tayebi, Comparing four ordering policies in a lost sales inventory model with Poisson demand and zero ordering cost, Scientia Iranica, 22 (2015), 1294-1298. Google Scholar |
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M. Hemmati, S. M. T. Fatemi Ghomi and M. S. Sajadieh, Inventory of complementary products with stock-dependent demand under vendor-managed inventory with consignment policy, Scientia Iranica, 25 (2018), 2347-2360. Google Scholar |
[20] |
M. R. A. Jokar and M. S. Sajadieh, Optimizing a joint economic lot sizing problem with price-sensitive demand, Scientia Iranica, 16 (2009), 159-164. Google Scholar |
[21] |
B. C. Giri and K. S. Chaudhuri, Deterministic models of perishable inventory with stock-dependent demand rate and nonlinear holding cost, European Journal of Operational Research, 105 (1998), 467-474. Google Scholar |
[22] |
A. Goli, H. K. Zare, R. Sadeghieh and A. Tavakkoli-Moghaddam, Multiobjective fuzzy mathematical model for a financially constrained closed-loop supply chain with labor employment, Computational Intelligence, 36 (2020), 4-34. Google Scholar |
[23] |
A. Goli, H. K. Zare, R. Tavakkoli-Moghaddam and A. Sadeghieh, Hybrid artificial intelligence and robust optimization for a multi-objective product portfolio problem Case study: The dairy products industry, Computers & Industrial Engineering, 37 (2019), 106090. Google Scholar |
[24] |
A. Goli and S. M. R. Davoodi, Coordination policy for production and delivery scheduling in the closed loop supply chain, Production Engineering, 12 (2018), 621-631. Google Scholar |
[25] |
S. Khanra, S. K. Ghosh and K. S. Chaudhuri,
An EOQ model
for a deteriorating item with time dependent quadratic demand rate under
permissible delay in payment, Applied Mathematics and Computation, 218 (2011), 1-9.
doi: 10.1016/j.amc.2011.04.062. |
[26] |
I.
P. Krommyda, K. Skouri and I. Konstantaras,
Optimal
ordering quantities for substitutable products with stock-dependent demand, Applied Mathematical Modelling, 39 (2015), 147-164.
doi: 10.1016/j.apm.2014.05.016. |
[27] |
S. Kumar and U. S. Rajput, A probabilistic inventory model for deteriorating items with ramp type demand rate under inflation, American Journal of Operational Research, 6 (2016), 16-31. Google Scholar |
[28] |
R. Lotfi, G. W. Weber, S. M. Sajadifar and N. Mardani,
Interdependent demand in the two-period newsvendor problem, Journal of Industrial & Management Optimization, 16 (2018), 117-140.
doi: 10.3934/jimo.2018143. |
[29] |
G. S. Mahapatra, S. Adak and K. Kaladhar, A fuzzy inventory model with three parameter Weibull deterioration with reliant holding cost and demand incorporating reliability, Journal of Intelligent and Fuzzy Systems, 36 (2019), 5731-5744. Google Scholar |
[30] |
K. Maity and M. Maiti,
Inventory of deteriorating
complementary and substitute items with stock dependent demand, American Journal of Mathematical and Management Sciences, 25 (2005), 83-96.
doi: 10.1080/01966324.2005.10737644. |
[31] |
U. Mishra, L. E. Cárdenas-Barrón, S. Tiwari, A. A. Shaikh and G. Trevi
ño-Garza,
An inventory model under price and stock
dependent demand for controllable deterioration rate with shortages and preservation technology investment, Annals of Operations
Research, 254 (2017), 165-190.
doi: 10.1007/s10479-017-2419-1. |
[32] |
N.
M. Modak and P. Kelle,
Managing a dual-channel supply
chain under price and delivery-time dependent stochastic demand, European Journal of Operational Research, 272 (2019), 147-161.
doi: 10.1016/j.ejor.2018.05.067. |
[33] |
H. Mokhtari, A. Naimi-Sadigh and A. Salmasnia, A computational approach to economic production quantity model for perishable products with backordering shortage and stock-dependent demand, Scientia Iranica, 24 (2017), 2138-2151. Google Scholar |
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show all references
References:
[1] |
B. Ahmad and L. Benkherouf,
Economic-order-type inventory models for
non-instantaneous deteriorating items and backlogging, RAIRO - Operations Research, 52 (2018), 895-901.
doi: 10.1051/ro/2018010. |
[2] |
H. K. Alfares and A. M. Ghaithan, Inventory and pricing model with price-dependent demand, time-varying holding cost, and quantity discounts, Computers & Industrial Engineering, 94 (2016), 170-177. Google Scholar |
[3] |
Z. T. Balkhi and L. Benkherouf,
On an inventory model
for deteriorating items with stock dependent and time-varying demand rates, Computers and Operations Research, 31 (2004), 223-240.
doi: 10.1016/S0305-0548(02)00182-X. |
[4] |
H. Barman, M. Pervin, S. K. Roy and G. W. Weber, Back-ordered inventory model with inflation in a cloudy-fuzzy environment, Journal of Industrial and Management Optimization, 2020.
doi: 10.3934/jimo.2020052. |
[5] |
S. Barzegar, M. Seifbarghy, S.H. Pasandideh and M. Arjmand, Development of a joint economic lot size model with stochastic demand within non-equal shipments, Scientia Iranica, 23 (2016), 3026-3034. Google Scholar |
[6] |
C. K. Chan, W. H. Wong, A. Langevin and Y. C. E. Lee, An integrated production-inventory model for deteriorating items with consideration of optimal production rate and deterioration during delivery, International Journal of Production Economics, 189 (2017), 1-13. Google Scholar |
[7] |
H. J. Chang and C. Y. Dye, An EOQ model for deteriorating items with time varying demand and partial backlogging, Journal of Operational Research Society, 50 (1999), 1176-1182. Google Scholar |
[8] |
C.T. Chang,
Inventory model with stock-dependent demand and nonlinear holding costs for deteriorating items, Asia-Pacific Journal of Operational Research, 21 (2004), 435-446.
doi: 10.1142/S0217595904000321. |
[9] |
U. Chanda and A. Kumar, Optimization of fuzzy EOQ model for advertising and price sensitive demand model under dynamic ceiling on potential adoption, International Journal of Systems Science, 4 (2016), 145-165. Google Scholar |
[10] |
R. R. Chowdhury, S. K. Ghosh and K.S. Chaudhuri, An order-level inventory model for a deteriorating item with time-quadratic demand and time-dependent partial backlogging with shortages in all cycles, American Journal of Mathematical and Management Sciences, 33 (2014), 75-97. Google Scholar |
[11] |
R. R. Chowdhury, S. K. Ghosh and K. S. Chaudhuri,
An
inventory model for perishable items with stock and advertisement sensitive
demand, Int. J. Appl. Comput. Math., 1 (2015), 187-201.
doi: 10.1007/s40819-014-0011-9. |
[12] |
P. S. Deng, R. Lin and P. P. Chu,
A note on the inventory
models for deteriorating items with ramp type demand rate, European Journal of Operational Research, 178 (2007), 112-120.
doi: 10.1016/j.ejor.2006.01.028. |
[13] |
B. K. Dey, B. Sarkar, M. Sarkar and S. Pareek,
An integrated inventory
model involving discrete setup cost reduction, variable safety factor,
selling price dependent demand, and investment, RAIRO -
Operations Research, 53 (2019), 39-57.
doi: 10.1051/ro/2018009. |
[14] |
T. K. Datta and A. K. Pal, 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 (1991), 326-333. Google Scholar |
[15] |
K. V. Geetha and R. Udayakumar, Optimal replenishment policy for deteriorating items with time sensitive demand under trade credit financing., American Journal of Mathematical and Management Sciences, 34 (2015), 197-212. Google Scholar |
[16] |
M. Ghoreishi, G. W. Weber and A. Mirzazadeh,
An inventory
model for non-instantaneous deteriorating items with partial backlogging,
permissible delay in payments, inflation and selling price-dependent demand and customer returns, Annals of Operations Research, 226 (2014), 221-238.
doi: 10.1007/s10479-014-1739-7. |
[17] |
S. K. Ghosh, T. Sarkar and K. Chaudhuri, A multi-item inventory model for deteriorating items in limited storage space with stock-dependent demand, American Journal of Mathematical and Management Sciences, 34 (2015), 147-161. Google Scholar |
[18] |
R. Haji and H. Tayebi, Comparing four ordering policies in a lost sales inventory model with Poisson demand and zero ordering cost, Scientia Iranica, 22 (2015), 1294-1298. Google Scholar |
[19] |
M. Hemmati, S. M. T. Fatemi Ghomi and M. S. Sajadieh, Inventory of complementary products with stock-dependent demand under vendor-managed inventory with consignment policy, Scientia Iranica, 25 (2018), 2347-2360. Google Scholar |
[20] |
M. R. A. Jokar and M. S. Sajadieh, Optimizing a joint economic lot sizing problem with price-sensitive demand, Scientia Iranica, 16 (2009), 159-164. Google Scholar |
[21] |
B. C. Giri and K. S. Chaudhuri, Deterministic models of perishable inventory with stock-dependent demand rate and nonlinear holding cost, European Journal of Operational Research, 105 (1998), 467-474. Google Scholar |
[22] |
A. Goli, H. K. Zare, R. Sadeghieh and A. Tavakkoli-Moghaddam, Multiobjective fuzzy mathematical model for a financially constrained closed-loop supply chain with labor employment, Computational Intelligence, 36 (2020), 4-34. Google Scholar |
[23] |
A. Goli, H. K. Zare, R. Tavakkoli-Moghaddam and A. Sadeghieh, Hybrid artificial intelligence and robust optimization for a multi-objective product portfolio problem Case study: The dairy products industry, Computers & Industrial Engineering, 37 (2019), 106090. Google Scholar |
[24] |
A. Goli and S. M. R. Davoodi, Coordination policy for production and delivery scheduling in the closed loop supply chain, Production Engineering, 12 (2018), 621-631. Google Scholar |
[25] |
S. Khanra, S. K. Ghosh and K. S. Chaudhuri,
An EOQ model
for a deteriorating item with time dependent quadratic demand rate under
permissible delay in payment, Applied Mathematics and Computation, 218 (2011), 1-9.
doi: 10.1016/j.amc.2011.04.062. |
[26] |
I.
P. Krommyda, K. Skouri and I. Konstantaras,
Optimal
ordering quantities for substitutable products with stock-dependent demand, Applied Mathematical Modelling, 39 (2015), 147-164.
doi: 10.1016/j.apm.2014.05.016. |
[27] |
S. Kumar and U. S. Rajput, A probabilistic inventory model for deteriorating items with ramp type demand rate under inflation, American Journal of Operational Research, 6 (2016), 16-31. Google Scholar |
[28] |
R. Lotfi, G. W. Weber, S. M. Sajadifar and N. Mardani,
Interdependent demand in the two-period newsvendor problem, Journal of Industrial & Management Optimization, 16 (2018), 117-140.
doi: 10.3934/jimo.2018143. |
[29] |
G. S. Mahapatra, S. Adak and K. Kaladhar, A fuzzy inventory model with three parameter Weibull deterioration with reliant holding cost and demand incorporating reliability, Journal of Intelligent and Fuzzy Systems, 36 (2019), 5731-5744. Google Scholar |
[30] |
K. Maity and M. Maiti,
Inventory of deteriorating
complementary and substitute items with stock dependent demand, American Journal of Mathematical and Management Sciences, 25 (2005), 83-96.
doi: 10.1080/01966324.2005.10737644. |
[31] |
U. Mishra, L. E. Cárdenas-Barrón, S. Tiwari, A. A. Shaikh and G. Trevi
ño-Garza,
An inventory model under price and stock
dependent demand for controllable deterioration rate with shortages and preservation technology investment, Annals of Operations
Research, 254 (2017), 165-190.
doi: 10.1007/s10479-017-2419-1. |
[32] |
N.
M. Modak and P. Kelle,
Managing a dual-channel supply
chain under price and delivery-time dependent stochastic demand, European Journal of Operational Research, 272 (2019), 147-161.
doi: 10.1016/j.ejor.2018.05.067. |
[33] |
H. Mokhtari, A. Naimi-Sadigh and A. Salmasnia, A computational approach to economic production quantity model for perishable products with backordering shortage and stock-dependent demand, Scientia Iranica, 24 (2017), 2138-2151. Google Scholar |
[34] |
U. Mishra, J. Tijerina-Aguilera, S. Tiwari and L. E. C árdenas-Barrón, Retailer's joint ordering, pricing, and preservation technology investment policies for a deteriorating item under permissible delay in payments, Mathematical Problems in Engineering, 2018 (2018), Art. ID 6962417, 14 pp.
doi: 10.1155/2018/6962417. |
[35] |
S. M. H. Molana, H. Davoudpour and S. Minner, An (r, nQ) inventory model for packaged deteriorating products with compound Poisson demand, Journal of the Operational Research Society, 63 (2012), 1499-1507. Google Scholar |
[36] |
S. Pal and G. S. Mahapatra,
A manufacturing-oriented supply chain model
for imperfect quality with inspection errors, stochastic demand under rework
and shortages, Computers & Industrial Engineering, 106 (2017), 299-314.
doi: 10.1016/j.cie.2017.02.003. |
[37] |
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Author's | Cash in hand | Demand depend on | Holding Cost depend on | Deterioration | Backlog |
Giri & Chaudhuri (1998) | NA | stock | Non linear | NA | No |
Chang (2004) | NA | stock | Non linear | constant | No |
Skouri et al. (2009) | NA | ramp type | NA | Weibull | Yes |
Sana (2010) | NA | stock | NA | probabilistic | No |
Sett et al. (2012) | NA | time demand | NA | NA | No |
Sarkar & Sarkar (2013) | NA | time | NA | probabilistic | No |
Chowdhury et al. (2014) | NA | time-quadratic | NA | time demand | Yes |
Ghoreishi et al. (2014) | NA | price & time | NA | non-instantaneous | Yes |
Ghosh et al. (2015) | NA | stock | Constant | constant | No |
Wu & Zhao (2015) | NA | inventory & time | NA | constant | No |
Bhunia et al. (2015) | NA | time, advertisement | NA | constant | Yes |
Alfares & Ghaithan (2016) | NA | price | time | NA | No |
Chanda & Kumar (2016) | NA | advertising & price | NA | NA | No |
Sanni & Chukwu (2016) | NA | deterministic | NA | Weibull | Yes |
Shah & Vaghela (2016) | NA | time & advertisement | NA | constant | No |
Mahapatra et al. (2017) | NA | time & reliability | NA | constant | Yes |
Mokhtari et al. (2017) | NA | stochastic | NA | constant | Yes |
Pervin et al. (2018) | NA | time | time | Weibull | Yes |
Lotfi et al. (2018) | NA | interdependent | NA | NA | Yes |
Dey et al. (2019) | NA | selling price | NA | NA | Yes |
Pervin et al. (2019) | NA | price and stock | purchasing cost | constant | Yes |
Pervin et al. (2020) | NA | time & price | NA | constant | Yes |
Roy et al. (2020) | NA | probabilistic | Constant | Weibull | Yes |
This paper | Consider | time, reliability, advertisement | reliability | probabilistic | Yes |
Author's | Cash in hand | Demand depend on | Holding Cost depend on | Deterioration | Backlog |
Giri & Chaudhuri (1998) | NA | stock | Non linear | NA | No |
Chang (2004) | NA | stock | Non linear | constant | No |
Skouri et al. (2009) | NA | ramp type | NA | Weibull | Yes |
Sana (2010) | NA | stock | NA | probabilistic | No |
Sett et al. (2012) | NA | time demand | NA | NA | No |
Sarkar & Sarkar (2013) | NA | time | NA | probabilistic | No |
Chowdhury et al. (2014) | NA | time-quadratic | NA | time demand | Yes |
Ghoreishi et al. (2014) | NA | price & time | NA | non-instantaneous | Yes |
Ghosh et al. (2015) | NA | stock | Constant | constant | No |
Wu & Zhao (2015) | NA | inventory & time | NA | constant | No |
Bhunia et al. (2015) | NA | time, advertisement | NA | constant | Yes |
Alfares & Ghaithan (2016) | NA | price | time | NA | No |
Chanda & Kumar (2016) | NA | advertising & price | NA | NA | No |
Sanni & Chukwu (2016) | NA | deterministic | NA | Weibull | Yes |
Shah & Vaghela (2016) | NA | time & advertisement | NA | constant | No |
Mahapatra et al. (2017) | NA | time & reliability | NA | constant | Yes |
Mokhtari et al. (2017) | NA | stochastic | NA | constant | Yes |
Pervin et al. (2018) | NA | time | time | Weibull | Yes |
Lotfi et al. (2018) | NA | interdependent | NA | NA | Yes |
Dey et al. (2019) | NA | selling price | NA | NA | Yes |
Pervin et al. (2019) | NA | price and stock | purchasing cost | constant | Yes |
Pervin et al. (2020) | NA | time & price | NA | constant | Yes |
Roy et al. (2020) | NA | probabilistic | Constant | Weibull | Yes |
This paper | Consider | time, reliability, advertisement | reliability | probabilistic | Yes |
Range of |
Distribution | ||||
Uniform | |||||
Triangular | |||||
Double Triangular | |||||
Uniform | |||||
Triangular | |||||
Double Triangular | |||||
Uniform | |||||
Triangular | |||||
Double Triangular |
Range of |
Distribution | ||||
Uniform | |||||
Triangular | |||||
Double Triangular | |||||
Uniform | |||||
Triangular | |||||
Double Triangular | |||||
Uniform | |||||
Triangular | |||||
Double Triangular |
Parameter | Change(%) | Change |
|||
Parameter | Change(%) | Change |
|||
Parameter | Change(%) | Change |
|||
Parameter | Change(%) | Change |
|||
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