American Institute of Mathematical Sciences

Advanced
October  2016, 12(4): 1333-1347. doi: 10.3934/jimo.2016.12.1333

An inventory model for items with imperfect quality and quantity discounts under adjusted screening rate and earned interest

Tien-Yu Lin 1, , Ming-Te Chen 2, and  Kuo-Lung Hou 3,

1. 

Department of Marketing and Supply Chain Management, Overseas Chinese University, Taichung 40721, Taiwan
2. 

Department of Business Administration, National Chung Cheng University, Chia-Yi 621, Taiwan
3. 

Department of Industrial Engineering and Management, Overseas Chinese University, Taichung 40721, Taiwan

Received  October 2014 Revised  October 2015 Published  January 2016

In this paper we develop a new inventory model for items with imperfect quality and quantity discounts under adjusted screening rate and earned interest. Three highlights are included in this new model: (1) interest is earned by depositing the sales revenue from the perfect and imperfect items into an interest-bearing account (2) the screening rate may not be given but is a decision variable (3) the supplier offers quantity discounts to trigger the retailer into ordering greater lot sizes. This scenario has not been discussed in previous EOQ models with imperfect quality. Our model could determine two decision variables, order quantity and screening rate, to maximize retailer profit. The expected total profit function is derived with two special cases explored to validate the proposed model. An algorithm is developed to help the manager determine the optimal order quantity and screening rate. A numerical example is given to illustrate the proposed model and algorithm. Sensitivity analyses are carried out to investigate the model parameters effects on the optimal solution. Managerial insights are also included.
Keywords: imperfect quality, overlapping order policy., Inventory, quantity discounts.
Mathematics Subject Classification: Primary: 591B24, 93C15; Secondary: 90B3.
Citation: Tien-Yu Lin, Ming-Te Chen, Kuo-Lung Hou. An inventory model for items with imperfect quality and quantity discounts under adjusted screening rate and earned interest. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1333-1347. doi: 10.3934/jimo.2016.12.1333
References:
[1]

W. C. Benton and S. Park, A classification of literature on determining the lot size under quantity discounts,, Eur. J. Oper. Res., 92 (1996), 219.  doi: 10.1016/0377-2217(95)00315-0.  Google Scholar

[2]

T. H. Burwell, D. S. Dave, K. E. Fitzpatrick and M. R. Roy, Economic lot size model for price-depend demand under quantity and freight discounts,, Int. J. Prod. Econ., 48 (1997), 141.   Google Scholar

[3]

L. E. Cárdenas-Barrón, Observation on: Economic production quantity model for items with imperfect quality,, Int. J. Prod. Econ., 67 (2000).   Google Scholar

[4]

L. E. Cárdenas-Barrón, Economic production quantity with rework process at a single-stage manufacturing system with planned backorders,, Comput. Ind. Eng., 57 (2009), 1105.   Google Scholar

[5]

W.-M. Chan, R. N. Ibrahim and P. B. Lochert, A new EPQ model: Integrating lower pricing rework and reject situations,, Prod. Plan. Control, 14 (2003), 588.  doi: 10.1080/09537280310001626179.  Google Scholar

[6]

H. -C. Chang, An application of fuzzy sets theory to the EOQ model with imperfect quality,, Comput. Oper. Res., 31 (2004), 2079.  doi: 10.1016/S0305-0548(03)00166-7.  Google Scholar

[7]

H.-C. Chang and C.-H. Ho, Exact closed-form solutions for optimal inventory model for items with imperfect quality and shortage backordering,, Omega, 38 (2010), 233.  doi: 10.1016/j.omega.2009.09.006.  Google Scholar

[8]

A. Eroglu and A. Ozdemir, An economic order quantity model with defective items and shortages,, Int. J. Prod. Econ., 106 (2007), 544.  doi: 10.1016/j.ijpe.2006.06.015.  Google Scholar

[9]

S. K. Goyal and L. E. Cárdenas-Barrón, Note on: Economic production quantity model for items with imperfect quality-a practical approach,, Int. J. Prod. Econ., 77 (2002), 85.  doi: 10.1016/S0925-5273(01)00203-1.  Google Scholar

[10]

J. T. Hsu and L. F. Hsu, An EOQ model with imperfect quality items, inspection errors, shortage backordering, and sales returns,, Int. J. Prod. Econ., 143 (2013), 162.   Google Scholar

[11]

W. K. Hsu and H. F. Yu, An EOQ model with imperfective quality items under an announced price increase,, J. Chinese Ins. Ind. Eng., 28 (2011), 34.  doi: 10.1080/10170669.2010.532347.  Google Scholar

[12]

C.-K. Huang, An optimal policy for a single-vendor single buyer integrated production-inventory problem with process unreliability consideration,, Int. J. Prod. Econ., 91 (2004), 91.  doi: 10.1016/S0925-5273(03)00220-2.  Google Scholar

[13]

M. Y. Jaber, S. K. Goyal and M. Imran, Economic production quantity model for items with imperfect quality subject to learning effects,, Int. J. Prod. Econ., 115 (2008), 143.  doi: 10.1016/j.ijpe.2008.05.007.  Google Scholar

[14]

M. Y. Jaber, S. Zanoni and L. E. Zavanella, An entropic economic order quantity (EnEOQ) for items with imperfect quality,, Appl. Math. Modelling, 37 (2013), 3982.  doi: 10.1016/j.apm.2012.07.046.  Google Scholar

[15]

X. Ji and Z. Shao, Model and algorithm for bilevel newsboy problem with fuzzy demands and discounts,, Applied Math. Comput., 172 (2006), 163.  doi: 10.1016/j.amc.2005.01.139.  Google Scholar

[16]

M. Khan, M. Y. Jaber, A. L. Guiffrida and S. Zolfaghari, A review of the extensions of a modified EOQ model for imperfect quality items,, Int. J. Prod. Econ., 132 (2011), 1.  doi: 10.1016/j.ijpe.2011.03.009.  Google Scholar

[17]

M. Khan, M. Jaber and M. Wahab, Economic order quantity model for items with imperfect quality with learning in inspection,, Int. J. Prod. Econ., 124 (2010), 87.  doi: 10.1016/j.ijpe.2009.10.011.  Google Scholar

[18]

T. -Y. Lin and K. -L. Hou, An imperfect quality economic order quantity with advanced receiving,, TOP, 23 (2015), 535.  doi: 10.1007/s11750-014-0352-x.  Google Scholar

[19]

T. -Y. Lin, An economic order quantity with imperfect quality and quantity discounts,, Applied Math. Model., 34 (2010), 3158.  doi: 10.1016/j.apm.2010.02.004.  Google Scholar

[20]

B. Maddah and M. Y. Jaber, Economic order quantity for items with imperfect quantity: Revisited,, Int. J. Prod. Econ., 112 (2008), 808.   Google Scholar

[21]

B. Maddah, M. K. Salameh and C. H. Karame, Lot sizing with random yield and different qualities,, Appl. Math. Modelling, 33 (2010), 1997.  doi: 10.1016/j.apm.2008.05.009.  Google Scholar

[22]

B. Maddah, M. K. Salameh and L. Moussawi-Haidar, Order overlapping: A practical approach for preventing shortages during screening,, Comput. Ind. Eng., 58 (2010), 691.  doi: 10.1016/j.cie.2010.01.014.  Google Scholar

[23]

G. C. Mahata and A. Goswami, Fuzzy inventory models for items with imperfect quality and shortage backordering under crisp and fuzzy decision variables,, Comput. Ind. Eng., 64 (2013), 190.  doi: 10.1016/j.cie.2012.09.003.  Google Scholar

[24]

A. Mendoza and J. A. Ventura, Incorporating quantity discounts to the EOQ model with transportation costs,, Int. J. Prod. Econ., 113 (2008), 754.  doi: 10.1016/j.ijpe.2007.10.010.  Google Scholar

[25]

J. P. Monahan, A quantity discount pricing model to increase vendor profits,, Manage. Sci., 30 (1984), 720.  doi: 10.1287/mnsc.30.6.720.  Google Scholar

[26]

J. Paknejad, F. Nasri and J. F. Affisco, Quality improvement in an inventory model with finite-range stochastic lead times,, Int. J. Prod. Res., 33 (2005), 2767.  doi: 10.1155/JAMDS.2005.177.  Google Scholar

[27]

S. Papachristos and I. Konstantaras, Economic ordering quantity models for items with imperfect quality,, Int. J. Prod. Econ., 100 (2006), 148.  doi: 10.1016/j.ijpe.2004.11.004.  Google Scholar

[28]

H. Qin, M. Luo, X. Gao and A. Lim, The freight allocation problem with all-units quantity-based discount: A heuristic algorithm,, Omega, 40 (2012), 415.  doi: 10.1016/j.omega.2011.05.005.  Google Scholar

[29]

S. M. Ross, Introduction to Probability Models,, Academic Press, (1993).   Google Scholar

[30]

M. K. Salameh and M. Y. Jaber, Economic production quantity model for items with imperfect quality,, Int. J. Prod. Econ., 64 (2000), 59.  doi: 10.1016/S0925-5273(99)00044-4.  Google Scholar

[31]

W. Shih, Optimal inventory policies when stockouts result from defective products,, Int. J. Prod. Res., 18 (1980), 677.  doi: 10.1080/00207548008919699.  Google Scholar

[32]

E. A. Silver, Establishing the reorder quantity when amount received is uncertain,, INFOR, 14 (1976), 32.   Google Scholar

[33]

J. F. Tsai, An optimization approach for supply chain management models with quantity discount policy,, Eur. J. Oper. Res., 177 (2007), 982.  doi: 10.1016/j.ejor.2006.01.034.  Google Scholar

[34]

J. Vörös, Economic order and production quantity models without constraint on the percentage of defective items,, Cent. Eur. J. Oper. Res., 21 (2013), 867.  doi: 10.1007/s10100-012-0277-0.  Google Scholar

[35]

M. I. M. Wahab and M. Y. Jaber, Economic order quantity model for items with imperfect quality, different holding costs, and learning effects: A note,, Comput. Ind. Eng., 58 (2010), 186.  doi: 10.1016/j.cie.2009.07.007.  Google Scholar

[36]

H.-M. Wee, J. Yu and M.-C. Chen, Optimal inventory model for items with imperfect quality and shortage backordering,, Omega, 35 (2007), 7.  doi: 10.1016/j.omega.2005.01.019.  Google Scholar

[37]

H. F. Yu, W. K. Hsu and W. J. Chang, EOQ model where a portion of the defectives can be used as perfect quality,, Int. J. Syst. Sci., 43 (2012), 1689.  doi: 10.1080/00207721.2010.549593.  Google Scholar

show all references

References:
[1]

W. C. Benton and S. Park, A classification of literature on determining the lot size under quantity discounts,, Eur. J. Oper. Res., 92 (1996), 219.  doi: 10.1016/0377-2217(95)00315-0.  Google Scholar

[2]

T. H. Burwell, D. S. Dave, K. E. Fitzpatrick and M. R. Roy, Economic lot size model for price-depend demand under quantity and freight discounts,, Int. J. Prod. Econ., 48 (1997), 141.   Google Scholar

[3]

L. E. Cárdenas-Barrón, Observation on: Economic production quantity model for items with imperfect quality,, Int. J. Prod. Econ., 67 (2000).   Google Scholar

[4]

L. E. Cárdenas-Barrón, Economic production quantity with rework process at a single-stage manufacturing system with planned backorders,, Comput. Ind. Eng., 57 (2009), 1105.   Google Scholar

[5]

W.-M. Chan, R. N. Ibrahim and P. B. Lochert, A new EPQ model: Integrating lower pricing rework and reject situations,, Prod. Plan. Control, 14 (2003), 588.  doi: 10.1080/09537280310001626179.  Google Scholar

[6]

H. -C. Chang, An application of fuzzy sets theory to the EOQ model with imperfect quality,, Comput. Oper. Res., 31 (2004), 2079.  doi: 10.1016/S0305-0548(03)00166-7.  Google Scholar

[7]

H.-C. Chang and C.-H. Ho, Exact closed-form solutions for optimal inventory model for items with imperfect quality and shortage backordering,, Omega, 38 (2010), 233.  doi: 10.1016/j.omega.2009.09.006.  Google Scholar

[8]

A. Eroglu and A. Ozdemir, An economic order quantity model with defective items and shortages,, Int. J. Prod. Econ., 106 (2007), 544.  doi: 10.1016/j.ijpe.2006.06.015.  Google Scholar

[9]

S. K. Goyal and L. E. Cárdenas-Barrón, Note on: Economic production quantity model for items with imperfect quality-a practical approach,, Int. J. Prod. Econ., 77 (2002), 85.  doi: 10.1016/S0925-5273(01)00203-1.  Google Scholar

[10]

J. T. Hsu and L. F. Hsu, An EOQ model with imperfect quality items, inspection errors, shortage backordering, and sales returns,, Int. J. Prod. Econ., 143 (2013), 162.   Google Scholar

[11]

W. K. Hsu and H. F. Yu, An EOQ model with imperfective quality items under an announced price increase,, J. Chinese Ins. Ind. Eng., 28 (2011), 34.  doi: 10.1080/10170669.2010.532347.  Google Scholar

[12]

C.-K. Huang, An optimal policy for a single-vendor single buyer integrated production-inventory problem with process unreliability consideration,, Int. J. Prod. Econ., 91 (2004), 91.  doi: 10.1016/S0925-5273(03)00220-2.  Google Scholar

[13]

M. Y. Jaber, S. K. Goyal and M. Imran, Economic production quantity model for items with imperfect quality subject to learning effects,, Int. J. Prod. Econ., 115 (2008), 143.  doi: 10.1016/j.ijpe.2008.05.007.  Google Scholar

[14]

M. Y. Jaber, S. Zanoni and L. E. Zavanella, An entropic economic order quantity (EnEOQ) for items with imperfect quality,, Appl. Math. Modelling, 37 (2013), 3982.  doi: 10.1016/j.apm.2012.07.046.  Google Scholar

[15]

X. Ji and Z. Shao, Model and algorithm for bilevel newsboy problem with fuzzy demands and discounts,, Applied Math. Comput., 172 (2006), 163.  doi: 10.1016/j.amc.2005.01.139.  Google Scholar

[16]

M. Khan, M. Y. Jaber, A. L. Guiffrida and S. Zolfaghari, A review of the extensions of a modified EOQ model for imperfect quality items,, Int. J. Prod. Econ., 132 (2011), 1.  doi: 10.1016/j.ijpe.2011.03.009.  Google Scholar

[17]

M. Khan, M. Jaber and M. Wahab, Economic order quantity model for items with imperfect quality with learning in inspection,, Int. J. Prod. Econ., 124 (2010), 87.  doi: 10.1016/j.ijpe.2009.10.011.  Google Scholar

[18]

T. -Y. Lin and K. -L. Hou, An imperfect quality economic order quantity with advanced receiving,, TOP, 23 (2015), 535.  doi: 10.1007/s11750-014-0352-x.  Google Scholar

[19]

T. -Y. Lin, An economic order quantity with imperfect quality and quantity discounts,, Applied Math. Model., 34 (2010), 3158.  doi: 10.1016/j.apm.2010.02.004.  Google Scholar

[20]

B. Maddah and M. Y. Jaber, Economic order quantity for items with imperfect quantity: Revisited,, Int. J. Prod. Econ., 112 (2008), 808.   Google Scholar

[21]

B. Maddah, M. K. Salameh and C. H. Karame, Lot sizing with random yield and different qualities,, Appl. Math. Modelling, 33 (2010), 1997.  doi: 10.1016/j.apm.2008.05.009.  Google Scholar

[22]

B. Maddah, M. K. Salameh and L. Moussawi-Haidar, Order overlapping: A practical approach for preventing shortages during screening,, Comput. Ind. Eng., 58 (2010), 691.  doi: 10.1016/j.cie.2010.01.014.  Google Scholar

[23]

G. C. Mahata and A. Goswami, Fuzzy inventory models for items with imperfect quality and shortage backordering under crisp and fuzzy decision variables,, Comput. Ind. Eng., 64 (2013), 190.  doi: 10.1016/j.cie.2012.09.003.  Google Scholar

[24]

A. Mendoza and J. A. Ventura, Incorporating quantity discounts to the EOQ model with transportation costs,, Int. J. Prod. Econ., 113 (2008), 754.  doi: 10.1016/j.ijpe.2007.10.010.  Google Scholar

[25]

J. P. Monahan, A quantity discount pricing model to increase vendor profits,, Manage. Sci., 30 (1984), 720.  doi: 10.1287/mnsc.30.6.720.  Google Scholar

[26]

J. Paknejad, F. Nasri and J. F. Affisco, Quality improvement in an inventory model with finite-range stochastic lead times,, Int. J. Prod. Res., 33 (2005), 2767.  doi: 10.1155/JAMDS.2005.177.  Google Scholar

[27]

S. Papachristos and I. Konstantaras, Economic ordering quantity models for items with imperfect quality,, Int. J. Prod. Econ., 100 (2006), 148.  doi: 10.1016/j.ijpe.2004.11.004.  Google Scholar

[28]

H. Qin, M. Luo, X. Gao and A. Lim, The freight allocation problem with all-units quantity-based discount: A heuristic algorithm,, Omega, 40 (2012), 415.  doi: 10.1016/j.omega.2011.05.005.  Google Scholar

[29]

S. M. Ross, Introduction to Probability Models,, Academic Press, (1993).   Google Scholar

[30]

M. K. Salameh and M. Y. Jaber, Economic production quantity model for items with imperfect quality,, Int. J. Prod. Econ., 64 (2000), 59.  doi: 10.1016/S0925-5273(99)00044-4.  Google Scholar

[31]

W. Shih, Optimal inventory policies when stockouts result from defective products,, Int. J. Prod. Res., 18 (1980), 677.  doi: 10.1080/00207548008919699.  Google Scholar

[32]

E. A. Silver, Establishing the reorder quantity when amount received is uncertain,, INFOR, 14 (1976), 32.   Google Scholar

[33]

J. F. Tsai, An optimization approach for supply chain management models with quantity discount policy,, Eur. J. Oper. Res., 177 (2007), 982.  doi: 10.1016/j.ejor.2006.01.034.  Google Scholar

[34]

J. Vörös, Economic order and production quantity models without constraint on the percentage of defective items,, Cent. Eur. J. Oper. Res., 21 (2013), 867.  doi: 10.1007/s10100-012-0277-0.  Google Scholar

[35]

M. I. M. Wahab and M. Y. Jaber, Economic order quantity model for items with imperfect quality, different holding costs, and learning effects: A note,, Comput. Ind. Eng., 58 (2010), 186.  doi: 10.1016/j.cie.2009.07.007.  Google Scholar

[36]

H.-M. Wee, J. Yu and M.-C. Chen, Optimal inventory model for items with imperfect quality and shortage backordering,, Omega, 35 (2007), 7.  doi: 10.1016/j.omega.2005.01.019.  Google Scholar

[37]

H. F. Yu, W. K. Hsu and W. J. Chang, EOQ model where a portion of the defectives can be used as perfect quality,, Int. J. Syst. Sci., 43 (2012), 1689.  doi: 10.1080/00207721.2010.549593.  Google Scholar

[1]

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 & Management Optimization, 2013, 9 (4) : 945-965. doi: 10.3934/jimo.2013.9.945
[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 & Management Optimization, 2016, 12 (3) : 1091-1119. doi: 10.3934/jimo.2016.12.1091
[3]

Tien-Yu Lin. Effect of warranty and quantity discounts on a deteriorating production system with a Markovian production process and allowable shortages. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2020013
[4]

Ata Allah Taleizadeh, Hadi Samimi, Biswajit Sarkar, Babak Mohammadi. Stochastic machine breakdown and discrete delivery in an imperfect inventory-production system. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1511-1535. doi: 10.3934/jimo.2017005
[5]

Zhenwei Luo, Jinting Wang. The optimal price discount, order quantity and minimum quantity in newsvendor model with group purchase. Journal of Industrial & Management Optimization, 2015, 11 (1) : 1-11. doi: 10.3934/jimo.2015.11.1
[6]

Mahdi Karimi, Seyed Jafar Sadjadi, Alireza Ghasemi Bijaghini. An economic order quantity for deteriorating items with allowable rework of deteriorated products. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1857-1879. doi: 10.3934/jimo.2018126
[7]

Alexander O. Brown, Christopher S. Tang. The impact of alternative performance measures on single-period inventory policy. Journal of Industrial & Management Optimization, 2006, 2 (3) : 297-318. doi: 10.3934/jimo.2006.2.297
[8]

Javad Taheri-Tolgari, Mohammad Mohammadi, Bahman Naderi, Alireza Arshadi-Khamseh, Abolfazl Mirzazadeh. An inventory model with imperfect item, inspection errors, preventive maintenance and partial backlogging in uncertainty environment. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1317-1344. doi: 10.3934/jimo.2018097
[9]

Ata Allah Taleizadeh, Biswajit Sarkar, Mohammad Hasani. Delayed payment policy in multi-product single-machine economic production quantity model with repair failure and partial backordering. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-24. doi: 10.3934/jimo.2019002
[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 & Management Optimization, 2017, 13 (4) : 1661-1683. doi: 10.3934/jimo.2017012
[11]

Jing Shi, Tiaojun Xiao. Service investment and consumer returns policy in a vendor-managed inventory supply chain. Journal of Industrial & Management Optimization, 2015, 11 (2) : 439-459. doi: 10.3934/jimo.2015.11.439
[12]

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 & Optimization, 2018, 8 (2) : 169-191. doi: 10.3934/naco.2018010
[13]

Kun-Jen Chung, Pin-Shou Ting. The inventory model under supplier's partial trade credit policy in a supply chain system. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1175-1183. doi: 10.3934/jimo.2015.11.1175
[14]

Qi Feng, Suresh P. Sethi, Houmin Yan, Hanqin Zhang. Optimality and nonoptimality of the base-stock policy in inventory problems with multiple delivery modes. Journal of Industrial & Management Optimization, 2006, 2 (1) : 19-42. doi: 10.3934/jimo.2006.2.19
[15]

Yiju Wang, Wei Xing, Hengxia Gao. Optimal ordering policy for inventory mechanism with a stochastic short-term price discount. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-16. doi: 10.3934/jimo.2018199
[16]

Jianbin Li, Mengcheng Guan, Zhiyuan Chen. Optimal inventory policy for fast-moving consumer goods under e-commerce environment. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-13. doi: 10.3934/jimo.2019028
[17]

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 & Optimization, 2020, 10 (1) : 45-74. doi: 10.3934/naco.2019032
[18]

Sung-Seok Ko. A nonhomogeneous quasi-birth-death process approach for an $ (s, S) $ policy for a perishable inventory system with retrial demands. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-19. doi: 10.3934/jimo.2019009
[19]

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 & Management Optimization, 2019, 15 (3) : 1345-1373. doi: 10.3934/jimo.2018098
[20]

Daijun Jiang, Hui Feng, Jun Zou. Overlapping domain decomposition methods for linear inverse problems. Inverse Problems & Imaging, 2015, 9 (1) : 163-188. doi: 10.3934/ipi.2015.9.163

2018 Impact Factor: 1.025

Tools

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences