American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Pricing and hedging catastrophe equity put options under a Markov-modulated jump diffusion model
  • JIMO Home
  • This Issue
  • Next Article
    Optimal asset control of a geometric Brownian motion with the transaction costs and bankruptcy permission
April  2015, 11(2): 479-492. doi: 10.3934/jimo.2015.11.479

The EPQ model with deteriorating items under two levels of trade credit in a supply chain system

Pin-Shou Ting 1,

1. 

Department of International Business Management, Shih Chien University, Taipei, Taiwan

Received  October 2013 Revised  March 2014 Published  September 2014

There are two kinds of two levels of trade credit policies addressed by Huang [15] and Teng [29], respectively. Liao [24] presents the EPQ model with deteriorating items under two-level trade credit policy from the viewpoint of Huang [15], while Chang, Teng and Chern [2] discuss the model from the viewpoint of Teng [29]. However, errors are found in the process of the modeling in Chang et al. [2]. The main purpose of this study is to improve the model of Chang et al. [2] and develop a more general EPQ inventory model. A theorem is also provided for practitioners to make right decisions.
Keywords: permissible delay in payments., trade credit, EPQ, Inventory, deteriorating items.
Mathematics Subject Classification: Primary: 90B05; Secondary: 65H0.
Citation: Pin-Shou Ting. The EPQ model with deteriorating items under two levels of trade credit in a supply chain system. Journal of Industrial and Management Optimization, 2015, 11 (2) : 479-492. doi: 10.3934/jimo.2015.11.479
References:
[1]

F. J. Arcelus, N. H. Shah and G. Srinivasan, Retailer's pricing, credit and inventory policies for deteriorating items in response to temporary price/credit incentives, International Journal of Production Economics, 81-82 (2003), 153-162. doi: 10.1016/S0925-5273(02)00269-4.

[2]

C. T. Chang, J. T. Teng and M. S. Chern, Optimal manufacturer's replenishment policies for deteriorating items in a supply chain with up-stream and down-stream trade credits, International Journal of Production Economics, 127 (2010), 197-202. doi: 10.1016/j.ijpe.2010.05.014.

[3]

C. K. Chen and Y. X. Liao, A deteriorating inventory model for an intermediary firm under return on inventory investment maximization, Journal of Industrial and Management Optimization, 10 (2014), 989-1000. doi: 10.3934/jimo.2014.10.989.

[4]

L. H. Chen and F. S. Kang, Coordination between vendor and buyer considering trade credit and items of imperfect quality, International Journal of Production Economics, 123 (2010), 52-61. doi: 10.1016/j.ijpe.2009.06.043.

[5]

S. C. Chen, J. T. Teng and K. Skouri, Economic production quantity models for deteriorating items with up-stream full trade credit and down-stream partial trade credit, International Journal of Production Economics, 155 (2014), 302-309. doi: 10.1016/j.ijpe.2013.07.024.

[6]

C. Y. Chiu, M. F. Yang, C. J. Tang and Y. Lin, Integrated imperfect production inventory model under permissible delay in payments depending on the order quantity, Journal of Industrial and Management Optimization, 9 (2013), 945-965. doi: 10.3934/jimo.2013.9.945.

[7]

P. Chu, K. J. Chung and S. P. Lan, Economic order quantity of deteriorating items under permissible delay in payments, Computers and Operations Research, 25 (1998), 817-824. doi: 10.1016/S0305-0548(98)00006-9.

[8]

K. J. Chung, Comments on the EOQ model under retailer partial trade credit policy in the supply chain, International Journal of Production Economics, 114 (2008), 308-312. doi: 10.1016/j.ijpe.2008.02.010.

[9]

K. J. Chung and T. S. 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.

[10]

K. J. Chung and Y. F. Huang, The optimal cycle time for EPQ inventory model under permissible delay in payments, International Journal of Production Economics, 84 (2003), 307-318. doi: 10.1016/S0925-5273(02)00465-6.

[11]

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.

[12]

K. J. Chung and J. J. Liao, The optimal ordering policy in a DCF analysis for deteriorating items when trade credit depends on the order quantity, International Journal of Production Economics, 100 (2006), 116-130. doi: 10.1016/j.ijpe.2004.10.011.

[13]

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

[14]

S. K. Goyal, Economic order quantity under conditions of permissible delay in payments, Journal of Operational Research Society, 36 (1985), 335-338. doi: 10.2307/2582421.

[15]

Y. F. Huang, Optimal retailer's ordering policies in the EOQ model under trade credit financing, Journal of Operational Research Society, 54 (2003), 1011-1015. doi: 10.1057/palgrave.jors.2601588.

[16]

Y. F. Huang, An inventory model under two levels of trade credit and limited storage space derived without derivatives, Applied Mathematical Modelling, 30 (2006), 418-436. doi: 10.1016/j.apm.2005.05.009.

[17]

Y. F. Huang, Optimal retailer's replenishment decisions in the EPQ model under two levels of trade credit policy, European Journal of Operational Research, 176 (2007), 1577-1591. doi: 10.1016/j.ejor.2005.10.035.

[18]

Y. F. Huang and K. H. Hsu, An EOQ model under retailer partial trade credit policy in supply chain, International Journal of Production Economics, 112 (2008), 655-664. doi: 10.1016/j.ijpe.2007.05.014.

[19]

C. K. Jaggi and S. P. Aggarwal, Credit financing in economics ordering policies of deteriorating items, International Journal of Production Economics, 34 (1994), 151-155. doi: 10.1016/0925-5273(94)90031-0.

[20]

C. K. Jaggi, S. K. Goyal and S. K. Goel, Retailer's optimal replenishment decisions with credit-linked demand under permissible delay in payments, European Journal of Operational Research, 190 (2008), 130-135. doi: 10.1016/j.ejor.2007.05.042.

[21]

V. B. Kreng and S. J. Tan, The optimal replenishment decisions under two levels of trade credit policy depending on the order quantity, Expert Systems with Applications, 37 (2010), 5514-5522. doi: 10.1016/j.eswa.2009.12.014.

[22]

J. J. Liao, On an EPQ model for deteriorating items under permissible delay in payments, Applied Mathematical Modelling, 31 (2007), 393-403. doi: 10.1016/j.apm.2005.11.016.

[23]

J. J. Liao, A note on an EOQ model for deteriorating items under supplier credit linked to ordering quantity, Applied Mathematical Modelling, 31 (2007), 1690-1699. doi: 10.1016/j.apm.2006.05.003.

[24]

J. J. Liao, An EOQ model with noninstantaneous receipt and exponentially deteriorating items under two-level trade credit, International Journal of Production Economics, 113 (2008), 852-861. doi: 10.1016/j.ijpe.2007.09.006.

[25]

J. J. Liao and K. J. Chung, An EOQ model for deterioration items under trade credit policy in a supply chain system, Journal of the Operations Research Society of Japan, 52 (2009), 46-57.

[26]

J. Min, Y. 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. doi: 10.1016/j.apm.2010.02.019.

[27]

K. Skouri and I. Konstantara, Two-warehouse inventory models for deteriorating products with ramp type demand rate, Journal of Industrial and Management Optimization, 9 (2013), 855-883. doi: 10.3934/jimo.2013.9.855.

[28]

H. Soni and N. H. Shah, Optimal ordering policy for stock-dependent demand under progressive payment scheme, European Journal of Operational Research, 184 (2008), 91-100. doi: 10.1016/j.ejor.2006.10.048.

[29]

J. T. Teng, Optimal ordering policies for a retailer who offers distinct trade credits to its good and bad credit customers, International Journal of Production Economics, 119 (2009), 415-423. doi: 10.1016/j.ijpe.2009.04.004.

[30]

J. T. Teng and C. T. Chang, Optimal manufacturer's replenishment policies in the EPQ model under two levels of trade credit policy, European Journal of Operational Research, 195 (2009), 358-363. doi: 10.1016/j.ejor.2008.02.001.

[31]

J. T. Teng, C. T. Cheng, M. S. Chern and Y. L. Chan, Retailer's optimal ordering policies with trade credit financing, International Journal of Systems Science, 38 (2007), 269-278. doi: 10.1080/00207720601158060.

[32]

J. T. Teng and S. K. Goyal, Optimal ordering policies for a retailer in a supply chain with up-stream and down-stream trade credit, Journal of the Operational Research Society, 58 (2007), 1252-1255.

[33]

J. T. Teng, J. Chen and S. K. Goyal, A comprehensive note on: An inventory model under two levels of trade credit and limited storage space derived without derivatives, Applied Mathematical Modelling, 33 (2009), 4388-4396. doi: 10.1016/j.apm.2009.03.010.

[34]

A. Thangam and R. Uthayakumar, Two-echelon trade credit financing for perishable items in a supply chain when demand depends on both selling price and credit period, Computers and Industrial Engineering, 57 (2009), 773-786. doi: 10.1016/j.cie.2009.02.005.

[35]

Y. C. Tsao and G. J. Shreen, Dynamic pricing, promotion and replenishment policies for a deteriorating item under permissible delay in payments, Computers and Operations Research, 35 (2008), 3562-3580. doi: 10.1016/j.cor.2007.01.024.

[36]

D. E. Varberg, J. Purcell and S. E. Rigdon,, Calculus, $9^{th}$ edition, Pearson Education, New Jersey, 2007.

[37]

C. T. Yang, L. Y. Ouyang, H. F. Yen and K. L. Lee, Joint pricing and ordering policies for deteriorating item with retail price-dependent demand in response to announced supply price increase, Journal of Industrial and Management Optimization, 9 (2013), 437-454. doi: 10.3934/jimo.2013.9.437.

show all references

References:
[1]

F. J. Arcelus, N. H. Shah and G. Srinivasan, Retailer's pricing, credit and inventory policies for deteriorating items in response to temporary price/credit incentives, International Journal of Production Economics, 81-82 (2003), 153-162. doi: 10.1016/S0925-5273(02)00269-4.

[2]

C. T. Chang, J. T. Teng and M. S. Chern, Optimal manufacturer's replenishment policies for deteriorating items in a supply chain with up-stream and down-stream trade credits, International Journal of Production Economics, 127 (2010), 197-202. doi: 10.1016/j.ijpe.2010.05.014.

[3]

C. K. Chen and Y. X. Liao, A deteriorating inventory model for an intermediary firm under return on inventory investment maximization, Journal of Industrial and Management Optimization, 10 (2014), 989-1000. doi: 10.3934/jimo.2014.10.989.

[4]

L. H. Chen and F. S. Kang, Coordination between vendor and buyer considering trade credit and items of imperfect quality, International Journal of Production Economics, 123 (2010), 52-61. doi: 10.1016/j.ijpe.2009.06.043.

[5]

S. C. Chen, J. T. Teng and K. Skouri, Economic production quantity models for deteriorating items with up-stream full trade credit and down-stream partial trade credit, International Journal of Production Economics, 155 (2014), 302-309. doi: 10.1016/j.ijpe.2013.07.024.

[6]

C. Y. Chiu, M. F. Yang, C. J. Tang and Y. Lin, Integrated imperfect production inventory model under permissible delay in payments depending on the order quantity, Journal of Industrial and Management Optimization, 9 (2013), 945-965. doi: 10.3934/jimo.2013.9.945.

[7]

P. Chu, K. J. Chung and S. P. Lan, Economic order quantity of deteriorating items under permissible delay in payments, Computers and Operations Research, 25 (1998), 817-824. doi: 10.1016/S0305-0548(98)00006-9.

[8]

K. J. Chung, Comments on the EOQ model under retailer partial trade credit policy in the supply chain, International Journal of Production Economics, 114 (2008), 308-312. doi: 10.1016/j.ijpe.2008.02.010.

[9]

K. J. Chung and T. S. 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.

[10]

K. J. Chung and Y. F. Huang, The optimal cycle time for EPQ inventory model under permissible delay in payments, International Journal of Production Economics, 84 (2003), 307-318. doi: 10.1016/S0925-5273(02)00465-6.

[11]

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.

[12]

K. J. Chung and J. J. Liao, The optimal ordering policy in a DCF analysis for deteriorating items when trade credit depends on the order quantity, International Journal of Production Economics, 100 (2006), 116-130. doi: 10.1016/j.ijpe.2004.10.011.

[13]

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

[14]

S. K. Goyal, Economic order quantity under conditions of permissible delay in payments, Journal of Operational Research Society, 36 (1985), 335-338. doi: 10.2307/2582421.

[15]

Y. F. Huang, Optimal retailer's ordering policies in the EOQ model under trade credit financing, Journal of Operational Research Society, 54 (2003), 1011-1015. doi: 10.1057/palgrave.jors.2601588.

[16]

Y. F. Huang, An inventory model under two levels of trade credit and limited storage space derived without derivatives, Applied Mathematical Modelling, 30 (2006), 418-436. doi: 10.1016/j.apm.2005.05.009.

[17]

Y. F. Huang, Optimal retailer's replenishment decisions in the EPQ model under two levels of trade credit policy, European Journal of Operational Research, 176 (2007), 1577-1591. doi: 10.1016/j.ejor.2005.10.035.

[18]

Y. F. Huang and K. H. Hsu, An EOQ model under retailer partial trade credit policy in supply chain, International Journal of Production Economics, 112 (2008), 655-664. doi: 10.1016/j.ijpe.2007.05.014.

[19]

C. K. Jaggi and S. P. Aggarwal, Credit financing in economics ordering policies of deteriorating items, International Journal of Production Economics, 34 (1994), 151-155. doi: 10.1016/0925-5273(94)90031-0.

[20]

C. K. Jaggi, S. K. Goyal and S. K. Goel, Retailer's optimal replenishment decisions with credit-linked demand under permissible delay in payments, European Journal of Operational Research, 190 (2008), 130-135. doi: 10.1016/j.ejor.2007.05.042.

[21]

V. B. Kreng and S. J. Tan, The optimal replenishment decisions under two levels of trade credit policy depending on the order quantity, Expert Systems with Applications, 37 (2010), 5514-5522. doi: 10.1016/j.eswa.2009.12.014.

[22]

J. J. Liao, On an EPQ model for deteriorating items under permissible delay in payments, Applied Mathematical Modelling, 31 (2007), 393-403. doi: 10.1016/j.apm.2005.11.016.

[23]

J. J. Liao, A note on an EOQ model for deteriorating items under supplier credit linked to ordering quantity, Applied Mathematical Modelling, 31 (2007), 1690-1699. doi: 10.1016/j.apm.2006.05.003.

[24]

J. J. Liao, An EOQ model with noninstantaneous receipt and exponentially deteriorating items under two-level trade credit, International Journal of Production Economics, 113 (2008), 852-861. doi: 10.1016/j.ijpe.2007.09.006.

[25]

J. J. Liao and K. J. Chung, An EOQ model for deterioration items under trade credit policy in a supply chain system, Journal of the Operations Research Society of Japan, 52 (2009), 46-57.

[26]

J. Min, Y. 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. doi: 10.1016/j.apm.2010.02.019.

[27]

K. Skouri and I. Konstantara, Two-warehouse inventory models for deteriorating products with ramp type demand rate, Journal of Industrial and Management Optimization, 9 (2013), 855-883. doi: 10.3934/jimo.2013.9.855.

[28]

H. Soni and N. H. Shah, Optimal ordering policy for stock-dependent demand under progressive payment scheme, European Journal of Operational Research, 184 (2008), 91-100. doi: 10.1016/j.ejor.2006.10.048.

[29]

J. T. Teng, Optimal ordering policies for a retailer who offers distinct trade credits to its good and bad credit customers, International Journal of Production Economics, 119 (2009), 415-423. doi: 10.1016/j.ijpe.2009.04.004.

[30]

J. T. Teng and C. T. Chang, Optimal manufacturer's replenishment policies in the EPQ model under two levels of trade credit policy, European Journal of Operational Research, 195 (2009), 358-363. doi: 10.1016/j.ejor.2008.02.001.

[31]

J. T. Teng, C. T. Cheng, M. S. Chern and Y. L. Chan, Retailer's optimal ordering policies with trade credit financing, International Journal of Systems Science, 38 (2007), 269-278. doi: 10.1080/00207720601158060.

[32]

J. T. Teng and S. K. Goyal, Optimal ordering policies for a retailer in a supply chain with up-stream and down-stream trade credit, Journal of the Operational Research Society, 58 (2007), 1252-1255.

[33]

J. T. Teng, J. Chen and S. K. Goyal, A comprehensive note on: An inventory model under two levels of trade credit and limited storage space derived without derivatives, Applied Mathematical Modelling, 33 (2009), 4388-4396. doi: 10.1016/j.apm.2009.03.010.

[34]

A. Thangam and R. Uthayakumar, Two-echelon trade credit financing for perishable items in a supply chain when demand depends on both selling price and credit period, Computers and Industrial Engineering, 57 (2009), 773-786. doi: 10.1016/j.cie.2009.02.005.

[35]

Y. C. Tsao and G. J. Shreen, Dynamic pricing, promotion and replenishment policies for a deteriorating item under permissible delay in payments, Computers and Operations Research, 35 (2008), 3562-3580. doi: 10.1016/j.cor.2007.01.024.

[36]

D. E. Varberg, J. Purcell and S. E. Rigdon,, Calculus, $9^{th}$ edition, Pearson Education, New Jersey, 2007.

[37]

C. T. Yang, L. Y. Ouyang, H. F. Yen and K. L. Lee, Joint pricing and ordering policies for deteriorating item with retail price-dependent demand in response to announced supply price increase, Journal of Industrial and Management Optimization, 9 (2013), 437-454. doi: 10.3934/jimo.2013.9.437.

[1]

Prasenjit Pramanik, Sarama Malik Das, Manas Kumar Maiti. Note on : Supply chain inventory model for deteriorating items with maximum lifetime and partial trade credit to credit risk customers. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1289-1315. doi: 10.3934/jimo.2018096
[2]

Mohsen Lashgari, Ata Allah Taleizadeh, Shib Sankar Sana. An inventory control problem for deteriorating items with back-ordering and financial considerations under two levels of trade credit linked to order quantity. Journal of Industrial and Management Optimization, 2016, 12 (3) : 1091-1119. doi: 10.3934/jimo.2016.12.1091
[3]

Chui-Yu Chiu, Ming-Feng Yang, Chung-Jung Tang, Yi Lin. Integrated imperfect production inventory model under permissible delay in payments depending on the order quantity. Journal of Industrial and Management Optimization, 2013, 9 (4) : 945-965. doi: 10.3934/jimo.2013.9.945
[4]

Puspita Mahata, Gour Chandra Mahata. Two-echelon trade credit with default risk in an EOQ model for deteriorating items under dynamic demand. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3659-3684. doi: 10.3934/jimo.2020138
[5]

Sankar Kumar Roy, Magfura Pervin, Gerhard Wilhelm Weber. Imperfection with inspection policy and variable demand under trade-credit: A deteriorating inventory model. Numerical Algebra, Control and Optimization, 2020, 10 (1) : 45-74. doi: 10.3934/naco.2019032
[6]

Chandan Mahato, Gour Chandra Mahata. Optimal replenishment, pricing and preservation technology investment policies for non-instantaneous deteriorating items under two-level trade credit policy. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021123
[7]

Magfura Pervin, Sankar Kumar Roy, Gerhard Wilhelm Weber. Multi-item deteriorating two-echelon inventory model with price- and stock-dependent demand: A trade-credit policy. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1345-1373. doi: 10.3934/jimo.2018098
[8]

Deepak Kumar Nayak, Sudhansu Sekhar Routray, Susanta Kumar Paikray, Hemen Dutta. A fuzzy inventory model for Weibull deteriorating items under completely backlogged shortages. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2435-2453. doi: 10.3934/dcdss.2020401
[9]

Yu-Chung Tsao, Hanifa-Astofa Fauziah, Thuy-Linh Vu, Nur Aini Masruroh. Optimal pricing, ordering, and credit period policies for deteriorating products under order-linked trade credit. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021152
[10]

Jui-Jung Liao, Wei-Chun Lee, Kuo-Nan Huang, Yung-Fu Huang. Optimal ordering policy for a two-warehouse inventory model use of two-level trade credit. Journal of Industrial and Management Optimization, 2017, 13 (4) : 1661-1683. doi: 10.3934/jimo.2017012
[11]

Magfura Pervin, Sankar Kumar Roy, Gerhard Wilhelm Weber. An integrated inventory model with variable holding cost under two levels of trade-credit policy. Numerical Algebra, Control and Optimization, 2018, 8 (2) : 169-191. doi: 10.3934/naco.2018010
[12]

Kun-Jen Chung, Pin-Shou Ting. The inventory model under supplier's partial trade credit policy in a supply chain system. Journal of Industrial and Management Optimization, 2015, 11 (4) : 1175-1183. doi: 10.3934/jimo.2015.11.1175
[13]

Vincent Choudri, Mathiyazhgan Venkatachalam, Sethuraman Panayappan. Production inventory model with deteriorating items, two rates of production cost and taking account of time value of money. Journal of Industrial and Management Optimization, 2016, 12 (3) : 1153-1172. doi: 10.3934/jimo.2016.12.1153
[14]

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 and Optimization, 2017, 7 (1) : 21-50. doi: 10.3934/naco.2017002
[15]

Mahdi Karimi, Seyed Jafar Sadjadi. Optimization of a Multi-Item Inventory model for deteriorating items with capacity constraint using dynamic programming. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1145-1160. doi: 10.3934/jimo.2021013
[16]

Shuhua Zhang, Longzhou Cao, Zuliang Lu. An EOQ inventory model for deteriorating items with controllable deterioration rate under stock-dependent demand rate and non-linear holding cost. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021156
[17]

Jun Li, Hairong Feng, Mingchao Wang. A replenishment policy with defective products, backlog and delay of payments. Journal of Industrial and Management Optimization, 2009, 5 (4) : 867-880. doi: 10.3934/jimo.2009.5.867
[18]

Honglin Yang, Heping Dai, Hong Wan, Lingling Chu. Optimal credit periods under two-level trade credit. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1753-1767. doi: 10.3934/jimo.2019027
[19]

Lianxia Zhao, Hui Qiao, Qi An. Optimal pre-sale policy for deteriorating items. Numerical Algebra, Control and Optimization, 2022, 12 (1) : 109-120. doi: 10.3934/naco.2021054
[20]

Cheng-Kang Chen, Yi-Xiang Liao. A deteriorating inventory model for an intermediary firm under return on inventory investment maximization. Journal of Industrial and Management Optimization, 2014, 10 (4) : 989-1000. doi: 10.3934/jimo.2014.10.989

2021 Impact Factor: 1.411

Tools

Metrics

  • PDF downloads (106)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy