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July  2020, 16(4): 1753-1767. doi: 10.3934/jimo.2019027

Optimal credit periods under two-level trade credit

1. 

School of Business Administration, Hunan University, Changsha, Hunan Province 410082, China

2. 

School of Business, State University of New York at Oswego, Oswego, NY 13126, USA

* Corresponding author: ottoyang@126.com(HonglinYang)

Received  January 2018 Revised  September 2018 Published  May 2019

Fund Project: This work was supported by the National Natural Science Foundation of China under Grant Nos. 71571065, 71790593 and 71521061

In a two-echelon single-supplier and single-retailer supply chain with permissible delay in payment, we investigate the two-level trade credit policy in which the supplier offers the retailer with limited capital a credit period and in turn the retailer also provides a credit period to customers. The demand rate is sensitive to both retail price and the customerso credit period. By using the backward induction method, we analytically derive the unique equilibrium of both credit periods in the Stackelberg game to determine the retaileros pricing strategy. We find that the optimal retail price is not always decreasing in the credit period offered by the supplier to the retailer. In addition, we characterize the conditions under which the retailer is willing to voluntarily provide customers a credit period. Numerical examples and sensitivity analysis of key parameters are presented to illustrate the theoretical results and managerial insights.

Citation: Honglin Yang, Heping Dai, Hong Wan, Lingling Chu. Optimal credit periods under two-level trade credit. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1753-1767. doi: 10.3934/jimo.2019027
References:
[1]

P. L. Abad and C. K. Jaggi, A joint approach for setting unit price and the length of the credit period for a seller when end demand is price sensitive, International Journal of Production Economics, 83 (2003), 115-122.   Google Scholar

[2]

C. T. Chang, An EOQ model with deteriorating items under inflation when supplier credits linked to order quantity, International Journal of Production Economics, 88 (2004), 307-316.   Google Scholar

[3]

L. H. Chen and F. S. Kang, Integrated inventory models considering the two-level trade credit policy and a price-negotiation scheme, European Journal of Operational Research, 205 (2010), 47-58.   Google Scholar

[4]

S. C. ChenL. E. Cárdenas-Barrón and J. T. Teng, Retailer's economic order quantity when the supplier offers conditionally permissible delay in payments link to order quantity, International Journal of Production Economics, 155 (2014), 284-291.   Google Scholar

[5]

M. S. ChernQ. PanJ. T. TengY. L. Chan and S. C. Chen, Stackelberg solution in a vendor-buyer supply chain model with permissible delay in payments, International Journal of Production Economics, 144 (2013), 397-404.   Google Scholar

[6]

Y. FengY. MuB. Hu and A. Kumar, Commodity options purchasing and credit financing under capital constraint, International Journal of Production Economics, 153 (2014), 230-237.   Google Scholar

[7]

Y. Ge and J. Qiu, Financial development, bank discrimination and trade credit, Journal of Banking & Finance, 31 (2007), 513-530.   Google Scholar

[8]

B. C. Giri and T. Maiti, Trade credit competition between two retailers in a supply chain under credit-linked retail price and market demand, Optimization Letters, 8 (2014), 2065-2085.  doi: 10.1007/s11590-013-0702-x.  Google Scholar

[9]

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

[10]

C. H. Ho, The optimal integrated inventory policy with price-and-credit-linked demand under two-level trade credit, Computers & Industrial Engineering, 60 (2011), 117-126.   Google Scholar

[11]

Y. F. Huang, Optimal retailer's ordering policies in the EOQ model under trade credit financing, Journal of the Operational Research Society, 54 (2003), 1011-1015.   Google Scholar

[12]

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.   Google Scholar

[13]

Y. F. Huang, Economic order quantity under conditionally permissible delay in payments, European Journal of Operational Research, 176 (2007), 911-924.   Google Scholar

[14]

C. K. JaggiS. 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.   Google Scholar

[15]

A. M. M. JamalB. R. Sarker and S. Wang, An ordering policy for deteriorating items with allowable shortage and permissible delay in payment, Journal of the Operational Research Society, 48 (1997), 826-833.   Google Scholar

[16]

S. KhanraB. Mandal and B. Sarkar, An inventory model with time dependent demand and shortages under trade credit policy, Economic Modelling, 35 (2013), 349-355.   Google Scholar

[17]

M. LashgariA. A. Taleizadeh and S. S. 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, 12 (2016), 1091-1119.  doi: 10.3934/jimo.2016.12.1091.  Google Scholar

[18]

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.   Google Scholar

[19]

J. J. LiaoW. C. LeeK. N. Huang and Y. F. Huang, Optimal ordering policy for a two-warehouse inventory model use of two-level trade credit, Journal of Industrial & Management Optimization, 13 (2017), 1661-1683.  doi: 10.3934/jimo.2017012.  Google Scholar

[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.  doi: 10.1016/j.apm.2010.02.019.  Google Scholar

[21]

S. M. MousaviA. BahreininejdS. Nurmaya and F. Yusof, A Modified Particle Swarm Optimization for solving the integrated location and inventory control problems in a two-echelon supply chain network, Journal of Intelligent Manufacturing, 28 (2017), 191-206.   Google Scholar

[22]

S. M.MousaviS. T. A. NiakiA. Bahreininejad and S. Nurmaya, Optimizing a location allocation-inventory problem in a two-echelon supply chain network: A modified Fruit Fly optimization algorithm, Computers & Industrial Engineering, 87 (2015), 543-560.   Google Scholar

[23]

L. Y. Ouyang, C. H. Ho and C. H. Su, An optimization approach for joint pricing and ordering problem in an integrated inventory system with order-size dependent trade credit, Computers & Industrial Engineering, 57 (2009) 920-930. Google Scholar

[24]

N. PakkiraM. K. Maiti and M. Maiti, Uncertain multi-item supply chain with two level trade credit under promotional cost sharing, Computers & Industrial Engineering, 118 (2018), 451-463.   Google Scholar

[25]

J. SadeghiS. M. MousaviS. T. A. Niaki and S. Sadeghi, Optimizing a multi-vendor multi-retailer vendor managed inventory problem: two tuned meta-heuristic algorithms, Knowledge-Based Systems, 50 (2013), 159-170.   Google Scholar

[26]

J. SadeghiS. M. MousaviS. T. A. Niaki and S. Sadeghi, Optimizing a bi-objective inventory model of a three-echelon supply chain using a tuned hybrid bat algorithm, Transportation Research Part E: Logistics and Transportation Review, 70 (2014), 274-292.   Google Scholar

[27]

B. SarkarS. Saren and L. E. Cárdenas-Barrón, An inventory model with trade-credit policy and variable deterioration for fixed lifetime products, Annals of Operations Research, 229 (2015), 667-702.  doi: 10.1007/s10479-014-1745-9.  Google Scholar

[28]

D. SeifertR. W. Seifert and M. Protopappa-Sieke, A review of trade credit literature: Opportunities for research in operations, European Journal of Operational Research, 231 (2013), 245-256.   Google Scholar

[29]

N. H. Shah, Manufacturer-retailer inventory model for deteriorating items with price-sensitive credit-linked demand under two-level trade credit financing and profit sharing contract, Cogent Engineering, 2 (2015), 1-14.   Google Scholar

[30]

N. H. ShahD. G. Patel and D. B. Shah, Optimal pricing and ordering policies for inventory system with two-level trade credits under price-sensitive trended demand, International Journal of Applied and Computational Mathematics, 1 (2015), 101-110.  doi: 10.1007/s40819-014-0003-9.  Google Scholar

[31]

C. H. SuL. Y. OuyangC. H. Ho and C. T. Chang, Retailer's inventory policy and supplier's delivery policy under two-level trade credit strategy, Asia-Pacific Journal of Operational Research, 24 (2007), 613-630.  doi: 10.1142/S0217595907001413.  Google Scholar

[32]

J. T. TengJ. Min and Q. Pan, Retailer's inventory policy and supplier's delivery policy under two-level trade credit strategy, Omega, 40 (2012), 328-335.   Google Scholar

[33]

J. T. TengK. R. Lou and L. Wang, Optimal trade credit and lot size policies in economic production quantity models with learning curve production costs, International Journal of Production Economics, 155 (2014), 318-323.   Google Scholar

[34]

Y. C. Tsao, Ordering policy for non-instantaneously deteriorating products under price adjustment and trade credits, J. Ind. Manag. Optim., 13 (2017), 327-345.  doi: 10.3934/jimo.2016020.  Google Scholar

[35]

Y. C. Tsao, Channel coordination under two-level trade credits and demand uncertainty, Applied Mathematical Modelling, 52 (2017), 160-173.  doi: 10.1016/j.apm.2017.07.046.  Google Scholar

[36]

J. WuL. Y. OuyangL. E. Cárdenas-Barrón and S. K. Goyal, Optimal credit period and lot size for deteriorating items with expiration dates under two-level trade credit financing, European Journal of Operational Research, 237 (2014), 898-908.  doi: 10.1016/j.ejor.2014.03.009.  Google Scholar

[37]

C. T. YangC. Y. Dye and J. F. Ding, Optimal dynamic trade credit and preservation technology allocation for a deteriorating inventory model, Computers & Industrial Engineering, 87 (2015), 356-369.   Google Scholar

[38]

H. YangH. Dai and W. Zhuo, Permissible delay period and pricing decisions in a two-echelon supply chain, Applied Economics Letters, 24 (2017), 820-825.   Google Scholar

[39]

H. YangW. ZhuoY. Zha and H. Wan, Two-period supply chain with flexible trade credit contract, Expert Systems With Applications, 66 (2016), 95-105.   Google Scholar

[40]

Y. ZhangY. Lu and X. Jiang, Research on dynamic pricing of supply chain products based on channel advantages, Kybernetes, 41 (2012), 1377-1385.   Google Scholar

[41]

Y. W. ZhouY. G. Zhong and M. I. M. Wahab, How to make the replenishment and payment strategy under flexible two-part trade credit, Computers & Operations Research, 40 (2013), 1328-1338.  doi: 10.1016/j.cor.2012.12.013.  Google Scholar

show all references

References:
[1]

P. L. Abad and C. K. Jaggi, A joint approach for setting unit price and the length of the credit period for a seller when end demand is price sensitive, International Journal of Production Economics, 83 (2003), 115-122.   Google Scholar

[2]

C. T. Chang, An EOQ model with deteriorating items under inflation when supplier credits linked to order quantity, International Journal of Production Economics, 88 (2004), 307-316.   Google Scholar

[3]

L. H. Chen and F. S. Kang, Integrated inventory models considering the two-level trade credit policy and a price-negotiation scheme, European Journal of Operational Research, 205 (2010), 47-58.   Google Scholar

[4]

S. C. ChenL. E. Cárdenas-Barrón and J. T. Teng, Retailer's economic order quantity when the supplier offers conditionally permissible delay in payments link to order quantity, International Journal of Production Economics, 155 (2014), 284-291.   Google Scholar

[5]

M. S. ChernQ. PanJ. T. TengY. L. Chan and S. C. Chen, Stackelberg solution in a vendor-buyer supply chain model with permissible delay in payments, International Journal of Production Economics, 144 (2013), 397-404.   Google Scholar

[6]

Y. FengY. MuB. Hu and A. Kumar, Commodity options purchasing and credit financing under capital constraint, International Journal of Production Economics, 153 (2014), 230-237.   Google Scholar

[7]

Y. Ge and J. Qiu, Financial development, bank discrimination and trade credit, Journal of Banking & Finance, 31 (2007), 513-530.   Google Scholar

[8]

B. C. Giri and T. Maiti, Trade credit competition between two retailers in a supply chain under credit-linked retail price and market demand, Optimization Letters, 8 (2014), 2065-2085.  doi: 10.1007/s11590-013-0702-x.  Google Scholar

[9]

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

[10]

C. H. Ho, The optimal integrated inventory policy with price-and-credit-linked demand under two-level trade credit, Computers & Industrial Engineering, 60 (2011), 117-126.   Google Scholar

[11]

Y. F. Huang, Optimal retailer's ordering policies in the EOQ model under trade credit financing, Journal of the Operational Research Society, 54 (2003), 1011-1015.   Google Scholar

[12]

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.   Google Scholar

[13]

Y. F. Huang, Economic order quantity under conditionally permissible delay in payments, European Journal of Operational Research, 176 (2007), 911-924.   Google Scholar

[14]

C. K. JaggiS. 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.   Google Scholar

[15]

A. M. M. JamalB. R. Sarker and S. Wang, An ordering policy for deteriorating items with allowable shortage and permissible delay in payment, Journal of the Operational Research Society, 48 (1997), 826-833.   Google Scholar

[16]

S. KhanraB. Mandal and B. Sarkar, An inventory model with time dependent demand and shortages under trade credit policy, Economic Modelling, 35 (2013), 349-355.   Google Scholar

[17]

M. LashgariA. A. Taleizadeh and S. S. 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, 12 (2016), 1091-1119.  doi: 10.3934/jimo.2016.12.1091.  Google Scholar

[18]

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.   Google Scholar

[19]

J. J. LiaoW. C. LeeK. N. Huang and Y. F. Huang, Optimal ordering policy for a two-warehouse inventory model use of two-level trade credit, Journal of Industrial & Management Optimization, 13 (2017), 1661-1683.  doi: 10.3934/jimo.2017012.  Google Scholar

[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.  doi: 10.1016/j.apm.2010.02.019.  Google Scholar

[21]

S. M. MousaviA. BahreininejdS. Nurmaya and F. Yusof, A Modified Particle Swarm Optimization for solving the integrated location and inventory control problems in a two-echelon supply chain network, Journal of Intelligent Manufacturing, 28 (2017), 191-206.   Google Scholar

[22]

S. M.MousaviS. T. A. NiakiA. Bahreininejad and S. Nurmaya, Optimizing a location allocation-inventory problem in a two-echelon supply chain network: A modified Fruit Fly optimization algorithm, Computers & Industrial Engineering, 87 (2015), 543-560.   Google Scholar

[23]

L. Y. Ouyang, C. H. Ho and C. H. Su, An optimization approach for joint pricing and ordering problem in an integrated inventory system with order-size dependent trade credit, Computers & Industrial Engineering, 57 (2009) 920-930. Google Scholar

[24]

N. PakkiraM. K. Maiti and M. Maiti, Uncertain multi-item supply chain with two level trade credit under promotional cost sharing, Computers & Industrial Engineering, 118 (2018), 451-463.   Google Scholar

[25]

J. SadeghiS. M. MousaviS. T. A. Niaki and S. Sadeghi, Optimizing a multi-vendor multi-retailer vendor managed inventory problem: two tuned meta-heuristic algorithms, Knowledge-Based Systems, 50 (2013), 159-170.   Google Scholar

[26]

J. SadeghiS. M. MousaviS. T. A. Niaki and S. Sadeghi, Optimizing a bi-objective inventory model of a three-echelon supply chain using a tuned hybrid bat algorithm, Transportation Research Part E: Logistics and Transportation Review, 70 (2014), 274-292.   Google Scholar

[27]

B. SarkarS. Saren and L. E. Cárdenas-Barrón, An inventory model with trade-credit policy and variable deterioration for fixed lifetime products, Annals of Operations Research, 229 (2015), 667-702.  doi: 10.1007/s10479-014-1745-9.  Google Scholar

[28]

D. SeifertR. W. Seifert and M. Protopappa-Sieke, A review of trade credit literature: Opportunities for research in operations, European Journal of Operational Research, 231 (2013), 245-256.   Google Scholar

[29]

N. H. Shah, Manufacturer-retailer inventory model for deteriorating items with price-sensitive credit-linked demand under two-level trade credit financing and profit sharing contract, Cogent Engineering, 2 (2015), 1-14.   Google Scholar

[30]

N. H. ShahD. G. Patel and D. B. Shah, Optimal pricing and ordering policies for inventory system with two-level trade credits under price-sensitive trended demand, International Journal of Applied and Computational Mathematics, 1 (2015), 101-110.  doi: 10.1007/s40819-014-0003-9.  Google Scholar

[31]

C. H. SuL. Y. OuyangC. H. Ho and C. T. Chang, Retailer's inventory policy and supplier's delivery policy under two-level trade credit strategy, Asia-Pacific Journal of Operational Research, 24 (2007), 613-630.  doi: 10.1142/S0217595907001413.  Google Scholar

[32]

J. T. TengJ. Min and Q. Pan, Retailer's inventory policy and supplier's delivery policy under two-level trade credit strategy, Omega, 40 (2012), 328-335.   Google Scholar

[33]

J. T. TengK. R. Lou and L. Wang, Optimal trade credit and lot size policies in economic production quantity models with learning curve production costs, International Journal of Production Economics, 155 (2014), 318-323.   Google Scholar

[34]

Y. C. Tsao, Ordering policy for non-instantaneously deteriorating products under price adjustment and trade credits, J. Ind. Manag. Optim., 13 (2017), 327-345.  doi: 10.3934/jimo.2016020.  Google Scholar

[35]

Y. C. Tsao, Channel coordination under two-level trade credits and demand uncertainty, Applied Mathematical Modelling, 52 (2017), 160-173.  doi: 10.1016/j.apm.2017.07.046.  Google Scholar

[36]

J. WuL. Y. OuyangL. E. Cárdenas-Barrón and S. K. Goyal, Optimal credit period and lot size for deteriorating items with expiration dates under two-level trade credit financing, European Journal of Operational Research, 237 (2014), 898-908.  doi: 10.1016/j.ejor.2014.03.009.  Google Scholar

[37]

C. T. YangC. Y. Dye and J. F. Ding, Optimal dynamic trade credit and preservation technology allocation for a deteriorating inventory model, Computers & Industrial Engineering, 87 (2015), 356-369.   Google Scholar

[38]

H. YangH. Dai and W. Zhuo, Permissible delay period and pricing decisions in a two-echelon supply chain, Applied Economics Letters, 24 (2017), 820-825.   Google Scholar

[39]

H. YangW. ZhuoY. Zha and H. Wan, Two-period supply chain with flexible trade credit contract, Expert Systems With Applications, 66 (2016), 95-105.   Google Scholar

[40]

Y. ZhangY. Lu and X. Jiang, Research on dynamic pricing of supply chain products based on channel advantages, Kybernetes, 41 (2012), 1377-1385.   Google Scholar

[41]

Y. W. ZhouY. G. Zhong and M. I. M. Wahab, How to make the replenishment and payment strategy under flexible two-part trade credit, Computers & Operations Research, 40 (2013), 1328-1338.  doi: 10.1016/j.cor.2012.12.013.  Google Scholar

Figure 1.  The case of two-level trade credit
Figure 2.  $ T+N\leq M $
Figure 3.  $ M<T+N $
Figure 4.  $ M, N $ and $ p $ in case 1
Figure 5.  $ N, D $ and $ TPR $ in case 1
Figure 6.  $ M, TPR $ and $ TPS $ in case 1
Figure 7.  M, N and p in case 1
Figure 8.  N, D and TPR in case 2
Figure 9.  M, TPR and TPS in case 2
Figure 10.  M, N and p in case 3
Figure 11.  N, D and TPR in case 2
Figure 12.  The case of two-level trade credit
Table 1.  The comparisons of three models with permissible delay in payment
Trade credit policy Demand Decision variable Optimal delay period
one-level price-and-time $ M $ obtain the analytic solution of
(Yangos et al., 2017) dependent optimal $ M $
two-level price-and-credit constant give a solution algorithm
(Shahos et al., 2015) dependent of optimal
two-level price-and-credit $ M,N $ obtain the analytic solutions
(our paper) dependent of optimal $ M $ and $ N $
Note:M represents the credit period offered by the supplier to the retailer, and N represents the credit period provided by the retailer to customers.
Trade credit policy Demand Decision variable Optimal delay period
one-level price-and-time $ M $ obtain the analytic solution of
(Yangos et al., 2017) dependent optimal $ M $
two-level price-and-credit constant give a solution algorithm
(Shahos et al., 2015) dependent of optimal
two-level price-and-credit $ M,N $ obtain the analytic solutions
(our paper) dependent of optimal $ M $ and $ N $
Note:M represents the credit period offered by the supplier to the retailer, and N represents the credit period provided by the retailer to customers.
Table 2.  Notations and explanations
Notation Definition
$ w $ wholesale price per unit.
$ p $ retail price per unit (decision variable).
$ M $ credit period offered by the supplier (decision variable).
$ N $ credit period provided by the retailer (decision variable)
$ T $ ordering cycle.
$ D(p,N) $ customers$ ' $ annual demand rate depending on $ p $ and $ N $.
$ I_c $ retailer$ ' $s interest charged per dollar per year.
$ I_e $ retailer$ ' $s interest earned per dollar per year.
$ c $ supplier$ ' $s procurement cost per unit.
$ h $ retailer$ ' $s holding cost per unit.
$ A $ ordering cost per order.
$ TPS $ supplier$ ' $s annual total profit.
$ TPR $ retailer$ ' $s annual total profit.
Notation Definition
$ w $ wholesale price per unit.
$ p $ retail price per unit (decision variable).
$ M $ credit period offered by the supplier (decision variable).
$ N $ credit period provided by the retailer (decision variable)
$ T $ ordering cycle.
$ D(p,N) $ customers$ ' $ annual demand rate depending on $ p $ and $ N $.
$ I_c $ retailer$ ' $s interest charged per dollar per year.
$ I_e $ retailer$ ' $s interest earned per dollar per year.
$ c $ supplier$ ' $s procurement cost per unit.
$ h $ retailer$ ' $s holding cost per unit.
$ A $ ordering cost per order.
$ TPS $ supplier$ ' $s annual total profit.
$ TPR $ retailer$ ' $s annual total profit.
Table 3.  Sensitivity analysis with respect to γ
Parameter Value $ M^* $ $ N^* $ $ p^* $ $ \pi_r^* $ $ \pi_s^* $
$ \gamma $ 4500 0.5938 0.0000 2185.6 2295832 4595664
5000 0.5938 0.0000 2185.6 2295832 4595664
5500 0.5938 0.0000 2185.6 2295832 4595664
6000 0.5938 0.0000 2185.6 2295832 4595664
6500 0.7252 0.3502 2221.8 2500038 4619147
7000 0.8333 0.5833 2248.3 2720222 4666667
7500 0.9229 0.7979 2276.4 2957288 4734861
8000 0.9974 0.9974 2306.0 3212169 4821253
8500 1.0613 1.0613 2316.5 3481700 4918165
9000 1.1181 1.1181 2326.9 3762082 5018776
Parameter Value $ M^* $ $ N^* $ $ p^* $ $ \pi_r^* $ $ \pi_s^* $
$ \gamma $ 4500 0.5938 0.0000 2185.6 2295832 4595664
5000 0.5938 0.0000 2185.6 2295832 4595664
5500 0.5938 0.0000 2185.6 2295832 4595664
6000 0.5938 0.0000 2185.6 2295832 4595664
6500 0.7252 0.3502 2221.8 2500038 4619147
7000 0.8333 0.5833 2248.3 2720222 4666667
7500 0.9229 0.7979 2276.4 2957288 4734861
8000 0.9974 0.9974 2306.0 3212169 4821253
8500 1.0613 1.0613 2316.5 3481700 4918165
9000 1.1181 1.1181 2326.9 3762082 5018776
Table 4.  Sensitivity analysis with respect to Ie and Ic
Parameter Value $ M^* $ $ N^* $ $ p^* $ $ \pi_r^* $ $ \pi_s^* $
$ I_e $ 0.050 1.2530 1.0863 2289.8 3437846 4914066
0.055 1.0245 0.8245 2269.0 3035272 4772856
0.060 0.8333 0.5833 2248.3 2720222 4666667
0.065 0.6701 0.3368 2224.8 2469392 4589728
0.070 0.5268 0.0000 2186.9 2269113 4542227
$ I_c $ 0.07 0.8244 0.8244 2281.5 2734825 4691700
0.075 0.8304 0.6637 2259.4 2725086 4675004
0.080 0.8333 0.5833 2248.3 2720222 4666667
0.085 0.8351 0.5351 2241.7 2717306 4661668
0.090 0.8363 0.5030 2237.3 2715363 4658337
Parameter Value $ M^* $ $ N^* $ $ p^* $ $ \pi_r^* $ $ \pi_s^* $
$ I_e $ 0.050 1.2530 1.0863 2289.8 3437846 4914066
0.055 1.0245 0.8245 2269.0 3035272 4772856
0.060 0.8333 0.5833 2248.3 2720222 4666667
0.065 0.6701 0.3368 2224.8 2469392 4589728
0.070 0.5268 0.0000 2186.9 2269113 4542227
$ I_c $ 0.07 0.8244 0.8244 2281.5 2734825 4691700
0.075 0.8304 0.6637 2259.4 2725086 4675004
0.080 0.8333 0.5833 2248.3 2720222 4666667
0.085 0.8351 0.5351 2241.7 2717306 4661668
0.090 0.8363 0.5030 2237.3 2715363 4658337
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