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March 2017, 7(1): 21-50. doi: 10.3934/naco.2017002

A two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items

1. 

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

2. 

Institute of Applied Mathematics, Middle East Technical University, 06800, Ankara, Turkey

* Corresponding author: sankroy2006@gmail.com.

Received  September 2016 Published  February 2017

In this study, we develop an inventory model for deteriorating items with stock dependent demand rate. Shortages are allowed to this model and when stock on hand is zero, then the retailer offers a price discount to customers who are willing to back-order their demands. Here, the supplier as well as the retailer adopt the trade credit policy for their customers in order to promote the market competition. The retailer can earn revenue and interest after the customer pays for the amount of purchasing cost to the retailer until the end of the trade credit period offered by the supplier. Besides this, we consider variable holding cost due to increase the stock of deteriorating items. Thereafter, we present an easy analytical closed-form solution to find the optimal order quantity so that the total cost per unit time is minimized. The results are discussed with the help of numerical examples to validate the proposed model. A sensitivity analysis of the optimal solutions for the parameters is also provided in order to stabilize our model. The paper ends with a conclusion and an outlook to possible future studies.

Citation: Magfura Pervin, Sankar Kumar Roy, Gerhard Wilhelm Weber. A two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 21-50. doi: 10.3934/naco.2017002
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]

K. Annadurai and R. Uthayakumar, Decaying inventory model with stock-dependent demand and shortages under two-level trade credit, International Journal of Advanced Manufacturing Technology, 77 (2015), 525-543.

[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.

[4]

S. Chand and J. Ward, A note on economic order quantity under conditions of permissible delay in payments, Journal of the Operational Research Society, 38 (1987), 83-84.

[5]

B. R. ChuangL.Y. Ouyang and Y. J. Lin, A minimax distribution free procedure for mixed inventory model with backorder discounts and variable lead time, Journal of Statistics and Management Systems, 7 (2004), 65-76.

[6]

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

[7]

T. K. Datta and K. Paul, An inventory system with stock-dependent, price-sensitive demand rate, Production Planning and Control, 12 (2001), 13-20.

[8]

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

[9]

M. GhoreishiA. Mirzazadeh and G. W. Weber, Optimal pricing and ordering policy for non-instantaneous deteriorating items under inflation and customer returns, Optimization, 63 (2014), 1785-1804.

[10]

M. GhoreishiG. 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.

[11]

B. C. GiriA. Goswami and K. S. Chaudhuri, An EOQ model for deteriorating items with time-varying demand and costs, The Journal of the Operational Research Society, 47 (1996), 1398-1405.

[12]

M. Goh, EOQ models with general demand and holding costs functions, European Journal of Operational Research, 73 (1994), 50-54.

[13]

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

[14]

F. W. Harris, Operations and Cost A. W. Shaw Company, Chicago, 1915.

[15]

R. I. Levin, C. P. McLaughlin, R. P. Lamone and J. F. Kottas, Productions Operations Management: Contemporary Policy for Managing Operating System McGraw-Hill Series in Management, New York, 1972, 373.

[16]

H. C. LiaoC. H. Tsai and C. T. Su, An inventory model with deteriorating items under inflation when a delay in payment is permissible, International Journal of Production Economics, 63 (2000), 207-214.

[17]

J. J. Liao, An EOQ model with non instantaneous receipt and exponentially deteriorating items under two-level trade credit, International Journal of Production Economics, 113 (2008), 852-861.

[18]

G. C. Mahata, An EPQ-based inventory model for exponentially deteriorating items under retailer partial trade credit policy in supply chain, Expert Systems with Applications, 39 (2012), 3537-3550.

[19]

S. K. Manna and K. S. Chaudhuri, An economic order quantity model for deteriorating items with time-dependent deterioration rate, demand rate, unit production cost and shortages, International Journal of System Science, 32 (2001), 1003-1009.

[20]

J. MinY. W. Zhou and J. Zhao, An inventory model for deteriorating items under stock-dependent demand and two-level trade credit, Applied Mathematical Modelling, 34 (2010), 3273-3285.

[21]

V. K. Mishra, L. S. Singh and R. Kumar, An inventory model for deteriorating items with time-dependent demand and time-varying holding cost under partial backlogging Journal of Industrial Engineering International 2013, DOI: http://jiei-tsb.com/content/9/1/4.

[22]

L. Y. OuyangB. R. Chuang and Y. J. Lin, 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, 18 (2007), 195-208.

[23]

M. Pal and S. Chandra, A periodic review inventory model with stock dependent demand, permissible delay in payment and price discount on backorders, Yugoslav Journal of Operations Research, 24 (2014), 99-110.

[24]

C. H. J. Pan and Y. C. Hsiao, Inventory models with back-order discounts and variable lead time, \International Journal of System Science, 32 (2001), 925-929.

[25]

M. PervinG. 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(4), 11 (2016), 243-251.

[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 (2016), DOI: 10.1007/s10479-016-2355-5.

[27]

A. Roy, An inventory model for deteriorating items with price dependent demand and time-varying holding cost, Advanced Modelling and Optimization, 10 (2008), 25-37.

[28]

S. W. ShinnH. P. Hwang and S. Sung, Joint price and lot size determination under conditions of permissible delay in payments and quantity discounts for freight cost, European Journal of Operational Research, 91 (1996), 528-542.

[29]

A. SwamiS. R. SinghS. Pareek and A. S. Yadav, Inventory policies for deteriorating item with stock dependent demand and variable holding costs under permissible delay in payment, International Journal of Application or Innovation in Engineering and Management, 4 (2015), 89-99.

[30]

R. P. Tripathi and H. S. Pandey, An EOQ model for deteriorating item with Weibull time dependent demand rate under trade credits, International Journal of Information and Management Sciences, 24 (2013), 329-347.

[31]

R. P. Tripathi, Inventory model with stock-level dependent demand rate and shortages under trade credits, International Journal of Modern Mathematical Sciences, 13 (2015), 122-136.

[32]

H. M. Wee, A deterministic lot-size inventory model for deteriorating items with shortages and a declining market, Computers and Operations Research, 22 (1995), 345-356.

[33]

T. M. Whitin, The theory of Inventory Management 2nd edition, Princeton University Press, Princeton, 1957.

[34]

R. H. Wilson, A scientific routine for stock control, Harvard Business Review, 13 (1934), 116–128.

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]

K. Annadurai and R. Uthayakumar, Decaying inventory model with stock-dependent demand and shortages under two-level trade credit, International Journal of Advanced Manufacturing Technology, 77 (2015), 525-543.

[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.

[4]

S. Chand and J. Ward, A note on economic order quantity under conditions of permissible delay in payments, Journal of the Operational Research Society, 38 (1987), 83-84.

[5]

B. R. ChuangL.Y. Ouyang and Y. J. Lin, A minimax distribution free procedure for mixed inventory model with backorder discounts and variable lead time, Journal of Statistics and Management Systems, 7 (2004), 65-76.

[6]

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

[7]

T. K. Datta and K. Paul, An inventory system with stock-dependent, price-sensitive demand rate, Production Planning and Control, 12 (2001), 13-20.

[8]

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

[9]

M. GhoreishiA. Mirzazadeh and G. W. Weber, Optimal pricing and ordering policy for non-instantaneous deteriorating items under inflation and customer returns, Optimization, 63 (2014), 1785-1804.

[10]

M. GhoreishiG. 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.

[11]

B. C. GiriA. Goswami and K. S. Chaudhuri, An EOQ model for deteriorating items with time-varying demand and costs, The Journal of the Operational Research Society, 47 (1996), 1398-1405.

[12]

M. Goh, EOQ models with general demand and holding costs functions, European Journal of Operational Research, 73 (1994), 50-54.

[13]

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

[14]

F. W. Harris, Operations and Cost A. W. Shaw Company, Chicago, 1915.

[15]

R. I. Levin, C. P. McLaughlin, R. P. Lamone and J. F. Kottas, Productions Operations Management: Contemporary Policy for Managing Operating System McGraw-Hill Series in Management, New York, 1972, 373.

[16]

H. C. LiaoC. H. Tsai and C. T. Su, An inventory model with deteriorating items under inflation when a delay in payment is permissible, International Journal of Production Economics, 63 (2000), 207-214.

[17]

J. J. Liao, An EOQ model with non instantaneous receipt and exponentially deteriorating items under two-level trade credit, International Journal of Production Economics, 113 (2008), 852-861.

[18]

G. C. Mahata, An EPQ-based inventory model for exponentially deteriorating items under retailer partial trade credit policy in supply chain, Expert Systems with Applications, 39 (2012), 3537-3550.

[19]

S. K. Manna and K. S. Chaudhuri, An economic order quantity model for deteriorating items with time-dependent deterioration rate, demand rate, unit production cost and shortages, International Journal of System Science, 32 (2001), 1003-1009.

[20]

J. MinY. W. Zhou and J. Zhao, An inventory model for deteriorating items under stock-dependent demand and two-level trade credit, Applied Mathematical Modelling, 34 (2010), 3273-3285.

[21]

V. K. Mishra, L. S. Singh and R. Kumar, An inventory model for deteriorating items with time-dependent demand and time-varying holding cost under partial backlogging Journal of Industrial Engineering International 2013, DOI: http://jiei-tsb.com/content/9/1/4.

[22]

L. Y. OuyangB. R. Chuang and Y. J. Lin, 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, 18 (2007), 195-208.

[23]

M. Pal and S. Chandra, A periodic review inventory model with stock dependent demand, permissible delay in payment and price discount on backorders, Yugoslav Journal of Operations Research, 24 (2014), 99-110.

[24]

C. H. J. Pan and Y. C. Hsiao, Inventory models with back-order discounts and variable lead time, \International Journal of System Science, 32 (2001), 925-929.

[25]

M. PervinG. 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(4), 11 (2016), 243-251.

[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 (2016), DOI: 10.1007/s10479-016-2355-5.

[27]

A. Roy, An inventory model for deteriorating items with price dependent demand and time-varying holding cost, Advanced Modelling and Optimization, 10 (2008), 25-37.

[28]

S. W. ShinnH. P. Hwang and S. Sung, Joint price and lot size determination under conditions of permissible delay in payments and quantity discounts for freight cost, European Journal of Operational Research, 91 (1996), 528-542.

[29]

A. SwamiS. R. SinghS. Pareek and A. S. Yadav, Inventory policies for deteriorating item with stock dependent demand and variable holding costs under permissible delay in payment, International Journal of Application or Innovation in Engineering and Management, 4 (2015), 89-99.

[30]

R. P. Tripathi and H. S. Pandey, An EOQ model for deteriorating item with Weibull time dependent demand rate under trade credits, International Journal of Information and Management Sciences, 24 (2013), 329-347.

[31]

R. P. Tripathi, Inventory model with stock-level dependent demand rate and shortages under trade credits, International Journal of Modern Mathematical Sciences, 13 (2015), 122-136.

[32]

H. M. Wee, A deterministic lot-size inventory model for deteriorating items with shortages and a declining market, Computers and Operations Research, 22 (1995), 345-356.

[33]

T. M. Whitin, The theory of Inventory Management 2nd edition, Princeton University Press, Princeton, 1957.

[34]

R. H. Wilson, A scientific routine for stock control, Harvard Business Review, 13 (1934), 116–128.

Figure 1.  Graphical representation of our proposed Inventory control model
Figure 2.  Flowchart of the solution procedure
Figure 3.  Graphical representation to show the convexity of total cost. The figure represents $T$, $t_1$ and the total cost $\Pi(T)$, along the x-axis, the y-axis and the z-axis, respectively
Figure 4.  Graphical representation to show the convexity of total cost. The figure represents T, t1 and the total cost Π(T), along the x-axis, the y-axis and the z-axis, respectively
Figure 5.  Change of total cost with respect to ordering cost, A, of our proposed model
Figure 6.  Change of total cost with respect to parameter α of our proposed model
Figure 7.  Change of total cost with respect to holding cost, h, of our proposed model
Figure 8.  Change of total cost with respect to deteriorating cost, θ, of our proposed model
Table 1.  Research works of various authors related to this area.
Author(s)ShortagesTrade credit policyStock dependent demandPrice discount on backordersDeterio-rationsTime varying costs
Ghare and Scharder (1963)
Giri et al. (1996)
Manna and Chaudhuri (2001)
Roy (2008)
Min et al. (2010)
Mishra et al. (2013)
Tripathi and Pandey (2013)
Tripathi (2015)
Annadurai and Uthayakumar (2015)
Pervin et al. (2015)
Swami et al. (2015)
Our paper
Author(s)ShortagesTrade credit policyStock dependent demandPrice discount on backordersDeterio-rationsTime varying costs
Ghare and Scharder (1963)
Giri et al. (1996)
Manna and Chaudhuri (2001)
Roy (2008)
Min et al. (2010)
Mishra et al. (2013)
Tripathi and Pandey (2013)
Tripathi (2015)
Annadurai and Uthayakumar (2015)
Pervin et al. (2015)
Swami et al. (2015)
Our paper
Table 2.  Sensitivity Analysis for different Parameters involved in Example 1.
Parameter% changevalueTt1t2TC
A+504503.98590.19780.1865225.4871
+253753.97010.19750.1841225.4991
+103303.96210.19680.1820225.5123
-102703.95280.10420.1792225.5472
-252253.95190.10400.1763225.5612
-501503.94820.10370.1730225.5860
s+50303.98830.10540.1579225.6102
+25253.98650.10490.1556225.6372
+10223.98510.10460.1534225.6819
-10183.98400.10420.1518225.6960
-25153.98320.10390.1475225.7542
-50103.98180.10340.1455225.7620
c+50903.98640.10430.1618226.0171
+25753.98470.10390.1632226.1261
+10663.98400.10440.1659226.4189
-10543.98340.11030.1671226.4703
-25453.98210.11110.1690226.5100
-50303.98110.12350.1724226.5275
a+501.054.08540.13520.1858225.8906
+250.8753.99810.14320.1822225.8940
+100.773.97020.16570.1805225.8976
-100.633.94530.17230.1778225.9121
-250.5253.93210.18050.1751225.9407
-500.353.91610.18230.1736225.9522
b+501.23.90930.18340.1780228.3131
+251.03.9240.18740.1799228.4309
+100.883.93980.19160.1827228.4971
-100.723.9830.13960.1848228.5102
-250.604.08060.16680.1864228.6524
-500.403.15030.16250.1882228.7601
Parameter% changevalueTt1t2TC
A+504503.98590.19780.1865225.4871
+253753.97010.19750.1841225.4991
+103303.96210.19680.1820225.5123
-102703.95280.10420.1792225.5472
-252253.95190.10400.1763225.5612
-501503.94820.10370.1730225.5860
s+50303.98830.10540.1579225.6102
+25253.98650.10490.1556225.6372
+10223.98510.10460.1534225.6819
-10183.98400.10420.1518225.6960
-25153.98320.10390.1475225.7542
-50103.98180.10340.1455225.7620
c+50903.98640.10430.1618226.0171
+25753.98470.10390.1632226.1261
+10663.98400.10440.1659226.4189
-10543.98340.11030.1671226.4703
-25453.98210.11110.1690226.5100
-50303.98110.12350.1724226.5275
a+501.054.08540.13520.1858225.8906
+250.8753.99810.14320.1822225.8940
+100.773.97020.16570.1805225.8976
-100.633.94530.17230.1778225.9121
-250.5253.93210.18050.1751225.9407
-500.353.91610.18230.1736225.9522
b+501.23.90930.18340.1780228.3131
+251.03.9240.18740.1799228.4309
+100.883.93980.19160.1827228.4971
-100.723.9830.13960.1848228.5102
-250.604.08060.16680.1864228.6524
-500.403.15030.16250.1882228.7601
Table 3.  Sensitivity Analysis for different Parameters which are involved in Example 1.
Parameter% changevalueTt1t2TC
M+500.453.93610.17900.1570227.1092
+250.3753.94730.17960.1589227.4121
+100.333.95280.18530.1603227.6708
-100.273.95990.18540.1639227.8211
-250.2253.96470.18610.1672227.8355
-500.153.97200.18640.1700227.8708
N+500.753.94710.15970.1968226.9858
+250.6253.95100.15940.1940226.9987
+100.553.95400.16590.1903227.6891
-100.453.95930.16680.1881227.6988
-250.3753.95490.16780.1854227.7408
-500.253.96010.16820.1826227.1923
θ+500.093.98400.16040.1725227.83
+250.0753.98440.16290.1756227.71
+100.0663.98470.16360.1791227.16
-100.0543.98300.16460.1824226.88
-250.0453.98710.16530.1847226.41
-500.033.98890.16640.1879226.30
δ+500.0753.98400.16870.1763225.74
+250.06253.98440.16950.1735225.63
+100.0553.98490.17360.1712225.16
-100.0453.98500.17410.1675224.98
-250.03753.98670.17500.1633224.47
-500.0253.98930.17690.1600224.39
Parameter% changevalueTt1t2TC
M+500.453.93610.17900.1570227.1092
+250.3753.94730.17960.1589227.4121
+100.333.95280.18530.1603227.6708
-100.273.95990.18540.1639227.8211
-250.2253.96470.18610.1672227.8355
-500.153.97200.18640.1700227.8708
N+500.753.94710.15970.1968226.9858
+250.6253.95100.15940.1940226.9987
+100.553.95400.16590.1903227.6891
-100.453.95930.16680.1881227.6988
-250.3753.95490.16780.1854227.7408
-500.253.96010.16820.1826227.1923
θ+500.093.98400.16040.1725227.83
+250.0753.98440.16290.1756227.71
+100.0663.98470.16360.1791227.16
-100.0543.98300.16460.1824226.88
-250.0453.98710.16530.1847226.41
-500.033.98890.16640.1879226.30
δ+500.0753.98400.16870.1763225.74
+250.06253.98440.16950.1735225.63
+100.0553.98490.17360.1712225.16
-100.0453.98500.17410.1675224.98
-250.03753.98670.17500.1633224.47
-500.0253.98930.17690.1600224.39
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