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March  2020, 16(2): 553-578. doi: 10.3934/jimo.2018167

A two-warehouse probabilistic model with price discount on backorders under two levels of trade-credit policy

Sankar Kumar Roy 1,, , Magfura Pervin 1, and  Gerhard Wilhelm Weber 2,

1. 

Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore-721102, West Bengal, India
2. 

Faculty of Engineering Management, Chair of Marketing and Economic Engineering, Poznan University of Technology, ul. Strzelecka 11, 60-965 Poznan, Poland

* Corresponding author: sankroy2006@gmail.com

The research of Sankar Kumar Roy is partially supported by the Portuguese Foundation for Science and Technology ("FCT-Fundação para a Ciência e a Tecnologia"), through the CIDMA - Center for Research and Development in Mathematics and Applications, University of Aveiro, Portugal, within project UID/MAT/04106/2013.
The author, Magfura Pervin is very much thankful to University Grants Commission (UGC) of India for providing financial support to continue this research work under [MANF(UGC)] scheme: Sanctioned letter number [F1-17.1/2012-13/MANF-2012-13-MUS-WES-19170 /(SA-Ⅲ/Website)] dated 28/02/2013.
The research of Gerhard-Wilhelm Weber (Institute of Applied Mathematics, Middle East Technical University, 06800, Ankara, Turkey) is partially supported by the Portuguese Foundation for Science and Technology ("FCT-Fundação para a Ciência e a Tecnologia"), through the CIDMA - Center for Research and Development in Mathematics and Applications.

Received  January 2018 Revised  July 2018 Published  March 2020 Early access  October 2018

It is impossible in this competitive era to assess the demand for items in advance. So, it is essential to refer to a stochastic demand function. In this paper, a probabilistic inventory model for deteriorating items is unfolded. Here, the supplier as well as the retailer adopt the trade-credit policy for their customers with the aim of promoting the market competition. Shortages are included into the model, and when stock on hand is zero, the retailer offers a price discount to those customers who are willing to back-order their demands. We consider two different warehouses in which the first one is an Own Warehouse (OW) where the deterioration is constant over time and the other is a Rented Warehouse (RW), and where the deterioration rate follows a Weibull distribution. An algorithm is provided for finding the solutions of the formulated model.Global convexity of the cost function is established which shows that our proposed model is very helpful for any supplier or retailer to finalize the optimal ordering policy. Beside of this, we target to increase the total profit for retailer by reducing the corresponding total inventory cost. The theoretical concept is justified with the help of some numerical examples. A sensitivity analysis of the optimal solution with respect to the major parameters is also provided in order to stabilize our model. We finalize the paper through a conclusion and a preview onto possible future studies.

Keywords: Probabilistic inventory model, two-level trade credit, warehouse, deterioration, shortage, price discount on backorder, optimization.
Mathematics Subject Classification: Primary: 90B05, 90B36; Secondary: 90C26.
Citation: Sankar Kumar Roy, Magfura Pervin, Gerhard Wilhelm Weber. A two-warehouse probabilistic model with price discount on backorders under two levels of trade-credit policy. Journal of Industrial and Management Optimization, 2020, 16 (2) : 553-578. doi: 10.3934/jimo.2018167
References:
[1]

S. P. Aggarwal and C. K. Jaggi, Ordering policies of deteriorating items under permissible delay in payments, The Journal of the Operational Research Society, 46 (1995), 658-662. 

[2]

B. Bank, J. Guddat, B. Kummer and K. Tammer, Non-Linear Parametric Optimization, Birkhäuser, Basel, 2014. doi: 10.1007/978-3-0348-6328-5.

[3]

L. Benkherouf, A deterministic order level inventory model for deteriorating items with two storage facilities, International Journal of Production Economics, 48 (1997), 167-175.  doi: 10.1016/S0925-5273(96)00070-9.

[4]

A. K. Bhunia, C. K. Jaggi, A. Sharma and R. Sharma, A two-warehouse inventory model for deteriorating items under permissible delay in payment with partial backlogging, Applied Mathematics and Computation, 232 (2014), 1125-1137.  doi: 10.1016/j.amc.2014.01.115.

[5]

A. K. Bhunia and M. Maiti, A two-warehouse inventory model for a linear trend in demand, Opsearch, 31 (1994), 318-329. 

[6]

T. Chakrabarty, B. C. Giri and K. S. Chaudhuri, An EOQ model for items with Weibull distribution deterioration, shortages and trended demand: An extension of Philip's model, Computers & Operations Research, 25 (1998), 649-657.  doi: 10.1016/S0305-0548(97)00081-6.

[7]

K. Chung and T. Huang, The optimal retailer's ordering policies for deteriorating items with limited storage capacity under trade credit financing, International Journal of Production Economics, 106 (2007), 127-145.  doi: 10.1016/j.ijpe.2006.05.008.

[8]

K. J. Chung and J. J. Liao, Lot-sizing decisions under trade credit depending on the ordering quantity, Computers and Operations Research, 31 (2004), 909-928.  doi: 10.1016/S0305-0548(03)00043-1.

[9]

R. P. Covert and G. S. Philip, An EOQ model for items with Weibull distribution deterioration, AIIE Transactions, 5 (1973), 323-326.  doi: 10.1080/05695557308974918.

[10]

D. Das, M. B. Kar, A. Roy and S. Kar, Two-warehouse production model for deteriorating inventory items with stock-dependent demand under inflation over a random planning horizon, Central European Journal of Operations Research, 20 (2012), 251-280.  doi: 10.1007/s10100-010-0165-4.

[11]

T. K. Datta and A. K. Pal, Order level inventory system with power demand pattern for items with variable rate of deterioration, Indian Journal of Pure and Applied Mathematics, 19 (1988), 1043-1053. 

[12]

L. N. De and A. Goswami, Probabilistic EOQ model for deteriorating items under trade credit financing, International Journal of System Science, 40 (2009), 335-346.  doi: 10.1080/00207720802435663.

[13]

P. M. Ghare and G. P. Schrader, A model for an exponentially decaying inventory, Journal of Industrial Engineering, 14 (1963), 238-243. 

[14]

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 (2015), 221-238.  doi: 10.1007/s10479-014-1739-7.

[15]

A. Goswami and K. S. Chaudhuri, An economic order quantity model for items with two levels of storage for a linear trend in demand, Journal of the Operational Research Society, 43 (1992), 157-167. 

[16]

S. K. Goyal, Economic order quantity under conditions of permissible delay in payments, Journal of Operational Research Society, 36 (1985), 335-338. 

[17]

V. R. Hartely, Operations Research - A Managerial Emphasis, Chapter 12, Good Year, Santa Monica, CA, (1976), 315-317. 

[18]

C. K. Jaggi, L. E. Cárdenas-Barrón, S. Tiwari and A. A. Shafi, Two-warehouse inventory model for deteriorating items with imperfect quality under the conditions of permissible delay in payments, Scientia Iranica E, 24 (2017), 390-412.  doi: 10.24200/sci.2017.4042.

[19]

H. Th. Jongen and G. W. Weber, On parametric nonlinear programming, Annals of Operations Research, 27 (1990), 253-284.  doi: 10.1007/BF02055198.

[20]

N. K. Kaliraman, R. Raj, S. Chandra and H. Chaudhary, Two warehouse inventory model for deteriorating item with exponential demand rate and permissible delay in payment, Yugoslav Journal of Operations Research, 27 (2017), 109-124.  doi: 10.2298/YJOR150404007K.

[21]

N. A. Kurdhi, J. Prasetyo and S. S. Handajani, An inventory model involving back-order price discount when the amount received is uncertain, International Journal of Systems Science, 47 (2016), 662-671.  doi: 10.1080/00207721.2014.900136.

[22]

M. Lashgari, A. A. Taleizadeh and A. Ahmadi, Partial up-stream advanced payment and partial down-stream delayed payment in a three-level supply chain, Annals of Operations Research, 238 (2016), 329-354.  doi: 10.1007/s10479-015-2100-5.

[23]

L. Y. Ouyang, C. T. Chang and P. Shum, The inter-dependent reductions of lead time and ordering cost in periodic review inventory model with backorder price discount, International Journal of Information and Management Sciences(3), 18 (2007), 195-208. 

[24]

M. Palanivel, R. Sundararajan and R. Uthayakumar, Two-warehouse inventory model with non-instantaneously deteriorating items, stock-dependent demand, shortages and inflation, Journal of Management Analytics, 3 (2016), 152-173.  doi: 10.1080/23270012.2016.1145078.

[25]

M. Pervin, G. C. Mahata and S. K. Roy, An inventory model with demand declining market for deteriorating items under trade credit policy, International Journal of Management Science and Engineering Management, 11 (2016), 243-251.  doi: 10.1080/17509653.2015.1081082.

[26]

M. Pervin, S. K. Roy and G. W. Weber, Analysis of inventory control model with shortage under time-dependent demand and time-varying holding cost including stochastic deterioration, Annals of Operations Research, 260 (2018), 437-460.  doi: 10.1007/s10479-016-2355-5.

[27]

M. Pervin, S. K. Roy and G. W. Weber, An integrated inventory model with variable holding cost under two levels of trade-credit policy, Numerical Algebra, Control and Optimization, 8 (2018), 169-191.  doi: 10.3934/naco.2018010.

[28]

M. Pervin, S. K. Roy and G. W. Weber, Multi-item deteriorating two-echelon inventory model with price- and stock-dependent demand: A trade-credit policy, Journal of Industrial and Management Optimization, (2018). doi: 10.3934/jimo.2018098.

[29]

M. Pervin, S. K. Roy and G. W. Weber, A Two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items, Numerical Algebra, Control and Optimization, 7 (2017), 21-50.  doi: 10.3934/naco.2017002.

[30]

J. Ray and K. S. Chaudhuri, An EOQ model with stock-dependent demand, shortage, inflation and time discounting, International Journal of Production Economics, 53 (1997), 171-180.  doi: 10.1016/S0925-5273(97)00112-6.

[31]

B. Sarkar, B. Mandal 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.

[32]

B. Sarkar, B. Mandal and S. Sarkar, Preservation of deteriorating seasonal products with stock-dependent consumption rate and shortages, Journal of Industrial and Management Optimization, 13 (2017), 187-206.  doi: 10.3934/jimo.2016011.

[33]

K. V. S. Sarma, A deterministic inventory model with two level of storage and an optimum release rule, Opsearch, 20 (1983), 175-180. 

[34]

S. Shabani, A. Mirzazadeh and E. Sharifi, A two-warehouse inventory model with fuzzy deterioratinn rate and fuzzy demand rate under conditionally permissible delay in payment, Journal of Industrial and Production Engineering(2), 33 (2016), 134-142. 

[35]

N. H. Shah, Probabilistic order level system with lead time when delay in payments are permissible, TOP, 5 (1997), 297-305.  doi: 10.1007/BF02568555.

[36]

N. H. Shah and Y. K. Shah, A discrete-in-time probabilistic inventory model for deteriorating items under conditions of permissible delay in payments, International Journal of System Science, 29 (1998), 121-125.  doi: 10.1080/00207729808929504.

[37]

S. Singh, J. Sharma and S. Singh, Profit maximizing probabilistic inventory model under the effect of permissible delay, International Multi Conference of Engineers and Computer Scientists, 3 (2010), 17-19. 

[38]

A. A. Taleizadeh and M. Noori-daryan, Pricing, manufacturing and inventory policies for raw material in a three-level supply chain, International Journal of Systems Science, 47 (2016), 919-931.  doi: 10.1080/00207721.2014.909544.

[39]

G. W. Weber, On the topology of parametric optimal control, Journal of the Australian Mathematical Society, Series B, 39 (1997), 1-35.  doi: 10.1017/S033427000000775X.

[40]

H. Yang, Two-warehouse inventory models for deteriorating items with shortages under inflation, European Journal of Operational Research, 157 (2006), 344-356.  doi: 10.1016/S0377-2217(03)00221-2.

[41]

H. L. Yang and C. T. Chang, A two-warehouse partial backlogging inventory model for deteriorating items with permissible delay in payment under inflation, Applied Mathematical Modelling, 37 (2013), 2717-2726.  doi: 10.1016/j.apm.2012.05.008.

show all references

References:
[1]

S. P. Aggarwal and C. K. Jaggi, Ordering policies of deteriorating items under permissible delay in payments, The Journal of the Operational Research Society, 46 (1995), 658-662. 

[2]

B. Bank, J. Guddat, B. Kummer and K. Tammer, Non-Linear Parametric Optimization, Birkhäuser, Basel, 2014. doi: 10.1007/978-3-0348-6328-5.

[3]

L. Benkherouf, A deterministic order level inventory model for deteriorating items with two storage facilities, International Journal of Production Economics, 48 (1997), 167-175.  doi: 10.1016/S0925-5273(96)00070-9.

[4]

A. K. Bhunia, C. K. Jaggi, A. Sharma and R. Sharma, A two-warehouse inventory model for deteriorating items under permissible delay in payment with partial backlogging, Applied Mathematics and Computation, 232 (2014), 1125-1137.  doi: 10.1016/j.amc.2014.01.115.

[5]

A. K. Bhunia and M. Maiti, A two-warehouse inventory model for a linear trend in demand, Opsearch, 31 (1994), 318-329. 

[6]

T. Chakrabarty, B. C. Giri and K. S. Chaudhuri, An EOQ model for items with Weibull distribution deterioration, shortages and trended demand: An extension of Philip's model, Computers & Operations Research, 25 (1998), 649-657.  doi: 10.1016/S0305-0548(97)00081-6.

[7]

K. Chung and T. Huang, The optimal retailer's ordering policies for deteriorating items with limited storage capacity under trade credit financing, International Journal of Production Economics, 106 (2007), 127-145.  doi: 10.1016/j.ijpe.2006.05.008.

[8]

K. J. Chung and J. J. Liao, Lot-sizing decisions under trade credit depending on the ordering quantity, Computers and Operations Research, 31 (2004), 909-928.  doi: 10.1016/S0305-0548(03)00043-1.

[9]

R. P. Covert and G. S. Philip, An EOQ model for items with Weibull distribution deterioration, AIIE Transactions, 5 (1973), 323-326.  doi: 10.1080/05695557308974918.

[10]

D. Das, M. B. Kar, A. Roy and S. Kar, Two-warehouse production model for deteriorating inventory items with stock-dependent demand under inflation over a random planning horizon, Central European Journal of Operations Research, 20 (2012), 251-280.  doi: 10.1007/s10100-010-0165-4.

[11]

T. K. Datta and A. K. Pal, Order level inventory system with power demand pattern for items with variable rate of deterioration, Indian Journal of Pure and Applied Mathematics, 19 (1988), 1043-1053. 

[12]

L. N. De and A. Goswami, Probabilistic EOQ model for deteriorating items under trade credit financing, International Journal of System Science, 40 (2009), 335-346.  doi: 10.1080/00207720802435663.

[13]

P. M. Ghare and G. P. Schrader, A model for an exponentially decaying inventory, Journal of Industrial Engineering, 14 (1963), 238-243. 

[14]

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 (2015), 221-238.  doi: 10.1007/s10479-014-1739-7.

[15]

A. Goswami and K. S. Chaudhuri, An economic order quantity model for items with two levels of storage for a linear trend in demand, Journal of the Operational Research Society, 43 (1992), 157-167. 

[16]

S. K. Goyal, Economic order quantity under conditions of permissible delay in payments, Journal of Operational Research Society, 36 (1985), 335-338. 

[17]

V. R. Hartely, Operations Research - A Managerial Emphasis, Chapter 12, Good Year, Santa Monica, CA, (1976), 315-317. 

[18]

C. K. Jaggi, L. E. Cárdenas-Barrón, S. Tiwari and A. A. Shafi, Two-warehouse inventory model for deteriorating items with imperfect quality under the conditions of permissible delay in payments, Scientia Iranica E, 24 (2017), 390-412.  doi: 10.24200/sci.2017.4042.

[19]

H. Th. Jongen and G. W. Weber, On parametric nonlinear programming, Annals of Operations Research, 27 (1990), 253-284.  doi: 10.1007/BF02055198.

[20]

N. K. Kaliraman, R. Raj, S. Chandra and H. Chaudhary, Two warehouse inventory model for deteriorating item with exponential demand rate and permissible delay in payment, Yugoslav Journal of Operations Research, 27 (2017), 109-124.  doi: 10.2298/YJOR150404007K.

[21]

N. A. Kurdhi, J. Prasetyo and S. S. Handajani, An inventory model involving back-order price discount when the amount received is uncertain, International Journal of Systems Science, 47 (2016), 662-671.  doi: 10.1080/00207721.2014.900136.

[22]

M. Lashgari, A. A. Taleizadeh and A. Ahmadi, Partial up-stream advanced payment and partial down-stream delayed payment in a three-level supply chain, Annals of Operations Research, 238 (2016), 329-354.  doi: 10.1007/s10479-015-2100-5.

[23]

L. Y. Ouyang, C. T. Chang and P. Shum, The inter-dependent reductions of lead time and ordering cost in periodic review inventory model with backorder price discount, International Journal of Information and Management Sciences(3), 18 (2007), 195-208. 

[24]

M. Palanivel, R. Sundararajan and R. Uthayakumar, Two-warehouse inventory model with non-instantaneously deteriorating items, stock-dependent demand, shortages and inflation, Journal of Management Analytics, 3 (2016), 152-173.  doi: 10.1080/23270012.2016.1145078.

[25]

M. Pervin, G. C. Mahata and S. K. Roy, An inventory model with demand declining market for deteriorating items under trade credit policy, International Journal of Management Science and Engineering Management, 11 (2016), 243-251.  doi: 10.1080/17509653.2015.1081082.

[26]

M. Pervin, S. K. Roy and G. W. Weber, Analysis of inventory control model with shortage under time-dependent demand and time-varying holding cost including stochastic deterioration, Annals of Operations Research, 260 (2018), 437-460.  doi: 10.1007/s10479-016-2355-5.

[27]

M. Pervin, S. K. Roy and G. W. Weber, An integrated inventory model with variable holding cost under two levels of trade-credit policy, Numerical Algebra, Control and Optimization, 8 (2018), 169-191.  doi: 10.3934/naco.2018010.

[28]

M. Pervin, S. K. Roy and G. W. Weber, Multi-item deteriorating two-echelon inventory model with price- and stock-dependent demand: A trade-credit policy, Journal of Industrial and Management Optimization, (2018). doi: 10.3934/jimo.2018098.

[29]

M. Pervin, S. K. Roy and G. W. Weber, A Two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items, Numerical Algebra, Control and Optimization, 7 (2017), 21-50.  doi: 10.3934/naco.2017002.

[30]

J. Ray and K. S. Chaudhuri, An EOQ model with stock-dependent demand, shortage, inflation and time discounting, International Journal of Production Economics, 53 (1997), 171-180.  doi: 10.1016/S0925-5273(97)00112-6.

[31]

B. Sarkar, B. Mandal 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.

[32]

B. Sarkar, B. Mandal and S. Sarkar, Preservation of deteriorating seasonal products with stock-dependent consumption rate and shortages, Journal of Industrial and Management Optimization, 13 (2017), 187-206.  doi: 10.3934/jimo.2016011.

[33]

K. V. S. Sarma, A deterministic inventory model with two level of storage and an optimum release rule, Opsearch, 20 (1983), 175-180. 

[34]

S. Shabani, A. Mirzazadeh and E. Sharifi, A two-warehouse inventory model with fuzzy deterioratinn rate and fuzzy demand rate under conditionally permissible delay in payment, Journal of Industrial and Production Engineering(2), 33 (2016), 134-142. 

[35]

N. H. Shah, Probabilistic order level system with lead time when delay in payments are permissible, TOP, 5 (1997), 297-305.  doi: 10.1007/BF02568555.

[36]

N. H. Shah and Y. K. Shah, A discrete-in-time probabilistic inventory model for deteriorating items under conditions of permissible delay in payments, International Journal of System Science, 29 (1998), 121-125.  doi: 10.1080/00207729808929504.

[37]

S. Singh, J. Sharma and S. Singh, Profit maximizing probabilistic inventory model under the effect of permissible delay, International Multi Conference of Engineers and Computer Scientists, 3 (2010), 17-19. 

[38]

A. A. Taleizadeh and M. Noori-daryan, Pricing, manufacturing and inventory policies for raw material in a three-level supply chain, International Journal of Systems Science, 47 (2016), 919-931.  doi: 10.1080/00207721.2014.909544.

[39]

G. W. Weber, On the topology of parametric optimal control, Journal of the Australian Mathematical Society, Series B, 39 (1997), 1-35.  doi: 10.1017/S033427000000775X.

[40]

H. Yang, Two-warehouse inventory models for deteriorating items with shortages under inflation, European Journal of Operational Research, 157 (2006), 344-356.  doi: 10.1016/S0377-2217(03)00221-2.

[41]

H. L. Yang and C. T. Chang, A two-warehouse partial backlogging inventory model for deteriorating items with permissible delay in payment under inflation, Applied Mathematical Modelling, 37 (2013), 2717-2726.  doi: 10.1016/j.apm.2012.05.008.

Figure 1.  Characteristic path of the OW
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Figure 2.  Characteristic path of the RW
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Figure 3.  Convexity of the function of total cost
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Figure 4.  Convexity nature of total cost in case of joint effect
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Figure 5.  Convexity nature of total cost in case of joint effect
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Figure 6.  Variation of total cost $TC$ with respect to demand factor $x$
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Figure 7.  Variation of total cost $TC$ with respect to shape parameter $\alpha$
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Figure 8.  Variation of total cost $TC$ with respect to ordering cost $A$
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Figure 9.  Variation of total cost $TC$ with respect to OW capacity $W$
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Table 1.  Previous works of different authors in this field including our work
Author(s) Two warehouse Probabilistic demand Trade credit Deterio-rations Shortage Price discount
Datta and Pal (1988) $\surd$ $\surd$
Bhunia and Maiti (1994) $\surd$
Shah and Shah (1998) $\surd$ $\surd$
Shah (1997) $\surd$ $\surd$
Palanivel et al. (2016) $\surd$ $\surd$
Jaggi et al. (2017) $\surd$ $\surd$
Benkherouf (1997) $\surd$ $\surd$ $\surd$
Singh et al. (2010) $\surd$ $\surd$
Kaliraman et al. (2017) $\surd$ $\surd$ $\surd$
Bhunia et al. (2014) $\surd$ $\surd$ $\surd$
Chung and Liao (2004) $\surd$
Yang (2006) $\surd$ $\surd$ $\surd$
De and Goswami (2009) $\surd$ $\surd$ $\surd$
Chung and Huang (2007) $\surd$ $\surd$ $\surd$
Kurdhi et al. (2015) $\surd$ $\surd$
Pervin et al. (2018) $\surd$ $\surd$
Pervin et al. (2017) $\surd$ $\surd$ $\surd$ $\surd$
Goyal (1985) $\surd$ $\surd$ $\surd$
Yang and Chang (2013) $\surd$ $\surd$ $\surd$
Sarkar et al. (2015) $\surd$ $\surd$
Ray and Chaudhuri (1997) $\surd$ $\surd$
Hariga (1995) $\surd$ $\surd$
Our paper $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Table Options

Download as excel

Author(s) Two warehouse Probabilistic demand Trade credit Deterio-rations Shortage Price discount
Datta and Pal (1988) $\surd$ $\surd$
Bhunia and Maiti (1994) $\surd$
Shah and Shah (1998) $\surd$ $\surd$
Shah (1997) $\surd$ $\surd$
Palanivel et al. (2016) $\surd$ $\surd$
Jaggi et al. (2017) $\surd$ $\surd$
Benkherouf (1997) $\surd$ $\surd$ $\surd$
Singh et al. (2010) $\surd$ $\surd$
Kaliraman et al. (2017) $\surd$ $\surd$ $\surd$
Bhunia et al. (2014) $\surd$ $\surd$ $\surd$
Chung and Liao (2004) $\surd$
Yang (2006) $\surd$ $\surd$ $\surd$
De and Goswami (2009) $\surd$ $\surd$ $\surd$
Chung and Huang (2007) $\surd$ $\surd$ $\surd$
Kurdhi et al. (2015) $\surd$ $\surd$
Pervin et al. (2018) $\surd$ $\surd$
Pervin et al. (2017) $\surd$ $\surd$ $\surd$ $\surd$
Goyal (1985) $\surd$ $\surd$ $\surd$
Yang and Chang (2013) $\surd$ $\surd$ $\surd$
Sarkar et al. (2015) $\surd$ $\surd$
Ray and Chaudhuri (1997) $\surd$ $\surd$
Hariga (1995) $\surd$ $\surd$
Our paper $\surd$ $\surd$ $\surd$ $\surd$ $\surd$ $\surd$
Table 2.  Effect of change in capacity of OW and trade-credit period
$W$ $M$ $N$ $T^*$ $b(T^*)$ $TC$ Case
0.5 0.4 0.088 521 1923.41 $M\le T$
800 0.9 0.8 0.097 560 1979.05 $N\le T <M$
0.6 0.9 0.109 582 1992.16 $0 <T <N$
0.5 0.4 0.0896 559 2075.04 $M\le T$
1000 0.9 0.8 0.0927 570 2152.00 $N\le T <M$
0.6 0.9 0.0989 590 2203.12 $0 <T <N$
0.5 0.4 0.0735 601 2214.07 $M\le T$
1500 0.9 0.8 0.0857 647 2539.11 $N\le T <M$
0.6 0.9 0.0875 685 2886.00 $0 <T <N$
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$W$ $M$ $N$ $T^*$ $b(T^*)$ $TC$ Case
0.5 0.4 0.088 521 1923.41 $M\le T$
800 0.9 0.8 0.097 560 1979.05 $N\le T <M$
0.6 0.9 0.109 582 1992.16 $0 <T <N$
0.5 0.4 0.0896 559 2075.04 $M\le T$
1000 0.9 0.8 0.0927 570 2152.00 $N\le T <M$
0.6 0.9 0.0989 590 2203.12 $0 <T <N$
0.5 0.4 0.0735 601 2214.07 $M\le T$
1500 0.9 0.8 0.0857 647 2539.11 $N\le T <M$
0.6 0.9 0.0875 685 2886.00 $0 <T <N$
Table 3.  Sensitivity analysis for different parameters involved in Example 1
Parameter $\%$ change value $T_1$ $T_2$ $T_3$ $T_4$ $T_5$ $T^*$ $b(T^*)$ $TC$ $\%$ change of TC
+50 600 1 1.5 ... ... ... 0.274 594 2284.24 +36.64
$A$ +20 480 1 1.5 2 ... ... 0.240 573 2170.33 +28.41
-20 320 1 1.5 2 2.5 ... 0.208 557 2001.16 +9.71
-50 200 1 1.5 2 2.3 3 0.173 529 1981.07 -0.85
+50 90 1 1.5 ... ... ... 0.198 468 2478.21 +27.53
$h_o$ +20 72 1 1.5 2 ... ... 0.183 450 2356.34 +21.23
-20 48 1 1.5 2 2.5 ... 0.175 438 2213.08 +18.73
-50 30 1 1.5 2 2.5 3 0.166 426 2087.60 +9.39
+50 105 1 1.5 ... ... ... 0.098 537 3798.26 +47.07
$h_r$ +20 84 1 1.5 2 ... ... 0.082 522 3523.65 +31.67
-20 56 1 1.5 2 2.5 ... 0.076 517 3247.43 +23.81
-50 35 1 1.5 2 2.5 3 0.680 510 3068.11 -1.27
+50 1.2 1 1.5 ... ... ... 0.176 526 2109.87 +28.45
$\theta$ +20 0.96 1 1.5 2 ... ... 0.248 538 2084.21 +17.39
-20 0.64 1 1.5 2 2.5 ... 0.273 559 1985.06 -1.87
-50 0.4 1 1.5 2 2.5 3 0.296 570 1867.30 -2.64
+50 0.075 1 1.5 ... ... ... 0.211 523 2075.32 +12.37
$\alpha$ +20 0.06 1 1.5 2 ... ... 0.250 538 1924.00 +10.22
-20 0.04 1 1.5 2 2.5 ... 0.310 550 1775.71 -0.79
-50 0.025 1 1.5 2 2.5 3 0.352 567 1528.66 -2.85
+50 4.5 1 1.5 ... ... ... 0.560 413 1968.20 +25.75
$\beta$ +20 3.6 1 1.5 2 ... ... 0.581 399 1876.11 +10.29
-20 2.4 1 1.5 2 2.5 ... 0.615 307 1718.53 -2.20
-50 1.5 1 1.5 2 2.5 3 0.672 279 1528.04 -5.43
+50 0.12 1 1.5 ... ... ... 0.736 578 1727.34 +24.74
$\delta$ +20 0.096 1 1.5 2 ... ... 0.703 530 1783.05 +18.42
-20 0.064 1 1.5 2 2.5 ... 0.682 492 1816.11 +10.53
-50 0.04 1 1.5 2 2.5 3 0.644 454 1874.20 -8.64
+50 0.725 1 1.5 ... ... ... 0.675 649 2665.93 +43.25
$M$ +20 0.6 1 1.5 2 ... ... 0.510 687 2682.75 +21.36
-20 0.4 1 1.5 2 2.5 ... 0.509 760 2789.77 +13.85
-50 0.25 1 1.5 2 2.5 3 0.588 781 2895.34 +7.04
+50 0.6 1 1.5 ... ... ... 0.322 300 2541.11 +40.52
$N$ +20 0.48 1 1.5 2 ... ... 0.379 349 2562.47 +37.06
-20 0.32 1 1.5 2 2.5 ... 0.401 373 2580.63 -7.43
-50 0.2 1 1.5 2 2.5 3 0.419 388 2558.47 -8.65
+50 15 1 1.5 ... ... ... 0.411 644 1563.72 +15.27
$R$ +20 12 1 1.5 2 ... ... 0.458 541 1571.08 +9.04
-20 8 1 1.5 2 2.5 ... 0.392 520 1584.60 -10.11
-50 5 1 1.5 2 2.5 3 0.450 501 1599.01 -5.14
+50 75 1 1.5 ... ... ... 0.749 385 1932.84 +29.27
$c$ +20 60 1 1.5 2 ... ... 0.755 337 1920.03 +21.43
-20 40 1 1.5 2 2.5 ... 0.759 249 1907.32 +37.19
-50 25 1 1.5 2 2.5 3 0.780 277 1871.92 +22.48
+50 15 1 1.5 ... ... ... 0.753 495 1884.67 +23.42
$s$ +20 12 1 1.5 2 ... ... 0.734 327 1940.59 +17.99
-20 8 1 1.5 2 2.5 ... 0.690 224 1982.47 -2.33
-50 5 1 1.5 2 2.5 3 0.638 200 2027.59 -7.21
+50 1200 1 1.5 ... ... ... 0.922 540 1825.49 +29.36
$W$ +20 960 1 1.5 2 ... ... 0.870 511 1871.52 +22.15
-20 640 1 1.5 2 2.5 ... 0.761 487 1905.14 -5.22
-50 400 1 1.5 2 2.5 3 0.739 475 1917.26 -9.46
+50 75 1 1.5 ... ... ... 0.875 610 1932.34 +31.06
$x$ +20 60 1 1.5 2 ... ... 0.852 587 1956.07 +26.17
-20 40 1 1.5 2 2.5 ... 0.830 551 1988.23 +13.50
-50 25 1 1.5 2 2.5 3 0.781 513 2130.54 +4.21
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Parameter $\%$ change value $T_1$ $T_2$ $T_3$ $T_4$ $T_5$ $T^*$ $b(T^*)$ $TC$ $\%$ change of TC
+50 600 1 1.5 ... ... ... 0.274 594 2284.24 +36.64
$A$ +20 480 1 1.5 2 ... ... 0.240 573 2170.33 +28.41
-20 320 1 1.5 2 2.5 ... 0.208 557 2001.16 +9.71
-50 200 1 1.5 2 2.3 3 0.173 529 1981.07 -0.85
+50 90 1 1.5 ... ... ... 0.198 468 2478.21 +27.53
$h_o$ +20 72 1 1.5 2 ... ... 0.183 450 2356.34 +21.23
-20 48 1 1.5 2 2.5 ... 0.175 438 2213.08 +18.73
-50 30 1 1.5 2 2.5 3 0.166 426 2087.60 +9.39
+50 105 1 1.5 ... ... ... 0.098 537 3798.26 +47.07
$h_r$ +20 84 1 1.5 2 ... ... 0.082 522 3523.65 +31.67
-20 56 1 1.5 2 2.5 ... 0.076 517 3247.43 +23.81
-50 35 1 1.5 2 2.5 3 0.680 510 3068.11 -1.27
+50 1.2 1 1.5 ... ... ... 0.176 526 2109.87 +28.45
$\theta$ +20 0.96 1 1.5 2 ... ... 0.248 538 2084.21 +17.39
-20 0.64 1 1.5 2 2.5 ... 0.273 559 1985.06 -1.87
-50 0.4 1 1.5 2 2.5 3 0.296 570 1867.30 -2.64
+50 0.075 1 1.5 ... ... ... 0.211 523 2075.32 +12.37
$\alpha$ +20 0.06 1 1.5 2 ... ... 0.250 538 1924.00 +10.22
-20 0.04 1 1.5 2 2.5 ... 0.310 550 1775.71 -0.79
-50 0.025 1 1.5 2 2.5 3 0.352 567 1528.66 -2.85
+50 4.5 1 1.5 ... ... ... 0.560 413 1968.20 +25.75
$\beta$ +20 3.6 1 1.5 2 ... ... 0.581 399 1876.11 +10.29
-20 2.4 1 1.5 2 2.5 ... 0.615 307 1718.53 -2.20
-50 1.5 1 1.5 2 2.5 3 0.672 279 1528.04 -5.43
+50 0.12 1 1.5 ... ... ... 0.736 578 1727.34 +24.74
$\delta$ +20 0.096 1 1.5 2 ... ... 0.703 530 1783.05 +18.42
-20 0.064 1 1.5 2 2.5 ... 0.682 492 1816.11 +10.53
-50 0.04 1 1.5 2 2.5 3 0.644 454 1874.20 -8.64
+50 0.725 1 1.5 ... ... ... 0.675 649 2665.93 +43.25
$M$ +20 0.6 1 1.5 2 ... ... 0.510 687 2682.75 +21.36
-20 0.4 1 1.5 2 2.5 ... 0.509 760 2789.77 +13.85
-50 0.25 1 1.5 2 2.5 3 0.588 781 2895.34 +7.04
+50 0.6 1 1.5 ... ... ... 0.322 300 2541.11 +40.52
$N$ +20 0.48 1 1.5 2 ... ... 0.379 349 2562.47 +37.06
-20 0.32 1 1.5 2 2.5 ... 0.401 373 2580.63 -7.43
-50 0.2 1 1.5 2 2.5 3 0.419 388 2558.47 -8.65
+50 15 1 1.5 ... ... ... 0.411 644 1563.72 +15.27
$R$ +20 12 1 1.5 2 ... ... 0.458 541 1571.08 +9.04
-20 8 1 1.5 2 2.5 ... 0.392 520 1584.60 -10.11
-50 5 1 1.5 2 2.5 3 0.450 501 1599.01 -5.14
+50 75 1 1.5 ... ... ... 0.749 385 1932.84 +29.27
$c$ +20 60 1 1.5 2 ... ... 0.755 337 1920.03 +21.43
-20 40 1 1.5 2 2.5 ... 0.759 249 1907.32 +37.19
-50 25 1 1.5 2 2.5 3 0.780 277 1871.92 +22.48
+50 15 1 1.5 ... ... ... 0.753 495 1884.67 +23.42
$s$ +20 12 1 1.5 2 ... ... 0.734 327 1940.59 +17.99
-20 8 1 1.5 2 2.5 ... 0.690 224 1982.47 -2.33
-50 5 1 1.5 2 2.5 3 0.638 200 2027.59 -7.21
+50 1200 1 1.5 ... ... ... 0.922 540 1825.49 +29.36
$W$ +20 960 1 1.5 2 ... ... 0.870 511 1871.52 +22.15
-20 640 1 1.5 2 2.5 ... 0.761 487 1905.14 -5.22
-50 400 1 1.5 2 2.5 3 0.739 475 1917.26 -9.46
+50 75 1 1.5 ... ... ... 0.875 610 1932.34 +31.06
$x$ +20 60 1 1.5 2 ... ... 0.852 587 1956.07 +26.17
-20 40 1 1.5 2 2.5 ... 0.830 551 1988.23 +13.50
-50 25 1 1.5 2 2.5 3 0.781 513 2130.54 +4.21
Table 4.  Sensitivity analysis for the combined effect of total cost involved in Example 1
$\%$ change $x$ $M$ $N$ $\delta$ $T^*$ $b(T^*)$ $TC$ $\%$ change of TC
+50 75 0.725 0.6 0.12 0.165 610 2685.00 +25.16
+40 70 0.70 0.56 0.112 0.137 589 2576.27 +32.00
+30 65 0.65 0.52 0.104 0.114 576 2450.08 +23.76
+20 60 0.6 0.48 0.096 0.105 558 2329.18 +21.34
+10 55 0.55 0.44 0.088 0.098 543 2249.71 +19.47
0 50 0.5 0.4 0.08 0.089 522 1923.46 ...
-10 45 0.45 0.36 0.072 0.068 516 1879.34 +13.32
-20 40 0.4 0.32 0.064 0.062 511 1794.11 -11.06
-30 35 0.35 0.28 0.056 0.058 504 1720.57 +5.48
-40 30 0.30 0.24 0.048 0.051 497 1685.00 -2.21
-50 25 0.25 0.2 0.04 0.048 483 1649.27 -0.23
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$\%$ change $x$ $M$ $N$ $\delta$ $T^*$ $b(T^*)$ $TC$ $\%$ change of TC
+50 75 0.725 0.6 0.12 0.165 610 2685.00 +25.16
+40 70 0.70 0.56 0.112 0.137 589 2576.27 +32.00
+30 65 0.65 0.52 0.104 0.114 576 2450.08 +23.76
+20 60 0.6 0.48 0.096 0.105 558 2329.18 +21.34
+10 55 0.55 0.44 0.088 0.098 543 2249.71 +19.47
0 50 0.5 0.4 0.08 0.089 522 1923.46 ...
-10 45 0.45 0.36 0.072 0.068 516 1879.34 +13.32
-20 40 0.4 0.32 0.064 0.062 511 1794.11 -11.06
-30 35 0.35 0.28 0.056 0.058 504 1720.57 +5.48
-40 30 0.30 0.24 0.048 0.051 497 1685.00 -2.21
-50 25 0.25 0.2 0.04 0.048 483 1649.27 -0.23
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