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July  2019, 15(3): 1345-1373. doi: 10.3934/jimo.2018098

Multi-item deteriorating two-echelon inventory model with price- and stock-dependent demand: A trade-credit policy

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

1. 

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

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

* Corresponding author: sankroy2006@gmail.com

The author, Magfura Pervin is very much thankful to University Grants Commission (UGC) of India for providing financial support to continue this research work under [MANF(UGC)] scheme: Sanctioned letter number [F1-17.1/2012-13/MANF-2012-13-MUS-WES-19170 /(SA-Ⅲ/Website)] dated 28/02/2013

Received  September 2017 Revised  December 2017 Published  July 2018

Fund Project: The research of Gerhard-Wilhelm Weber (Institute of Applied Mathematics, Middle East Technical University, 06800, Ankara, Turkey) is partially supported by the Portuguese Foundation for Science and Technology ("FCT-Fundação para a Ciência e a Tecnologia"), through the CIDMA - Center for Research and Development in Mathematics and Applications.

This article is concerned with a multi-item inventory model for deteriorating items. The model is formed on the basis of a two-level supply chain policy, i.e., based on manufacturer's and retailer's perspective. The deterioration rate is considered as constant. The demand factor of any items suffer from a large amount of stock level; so, we consider stock-dependent demand function. The demand of any item is also dependent on its selling price; thus, a price-dependent demand function is introduced here. The retailer adopts the trade-credit policy for his customers in order to promote market competitiveness. He can earn revenue and interest after the customer pays the amount of purchasing cost to the retailer until the end of the trade-credit period, offered by the supplier. Shortages are allowed in the retailer's model as it is a very realistic item, too. A price discount on backordered commodities is offered for those customers who are willing to backorder their demand. Thereafter, we present an easy analytical solution procedure to find the total profit for both manufacturer and retailer. We also use the classical game theory and Nash equilibrium approach to find an optimal solution of the joint profit. The results are discussed with several numerical examples to illustrate our model and to provide some managerial insights related to the model. Furthermore, a parametric sensitivity analysis of the optimal solutions is provided and a concavity figure of our profit function is supplied to stabilize our model. The paper ends with a conclusion and an outlook to future research projects.

Keywords: Multi-item inventory, price- and stock-dependent demand, trade-credit, deterioration, price discount on backorders, Nash equilibrium, optimization.
Mathematics Subject Classification: Primary: 90B05, 91A10; Secondary: 90C26.
Citation: 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
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.   Google Scholar

[2]

M. Ben-daya and A. Raouf, On the constrained multi-item single period inventory problem, International Journal of Production Management, 13 (1993), 104-112.  doi: 10.1108/01443579310046472.  Google Scholar

[3]

L. Benkherouf and M. Omar, Optimal integrated policies for a single-vendor single-buyer time-varying demand model, Computers and Mathematics with Applications, 60 (2010), 2066-2077.  doi: 10.1016/j.camwa.2010.07.047.  Google Scholar

[4]

D. K. Bhattacharya, Production, manufacturing and logistics on multi-item inventory, European Journal of Operational Research, 162 (2005), 786-791.  doi: 10.1016/j.ejor.2003.02.004.  Google Scholar

[5]

J. Chen and P. C. Bell, Coordinating a decentralized supply chain with customer returns and price-dependent stochastic demand using a buyback policy, European Journal of Operational Research, 212 (2011), 293-300.  doi: 10.1016/j.ejor.2011.01.036.  Google Scholar

[6]

V. Choudri, M. Venkatachalam and S. 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, 12 (2016), 1153-1172.  doi: 10.3934/jimo.2016.12.1153.  Google Scholar

[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.  doi: 10.1080/09537280150203933.  Google Scholar

[8]

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

[9]

A. Goswami and K. S. Chaudhuri, An EOQ model for deteriorating items with shortages and a linear trend in demand, Journal of the Operational Research Society, 42 (1991), 1105-1110.   Google Scholar

[10]

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

[11]

S. Kar, T. Roy and M. Maiti, Multi-item inventory model with probabilistic price dependent damand and imprecise goal and constraints, Yugoslav Journal of Operations Research, 11 (2001), 93-103.   Google Scholar

[12]

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

[13]

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. Google Scholar

[14]

T. Y. Lin, M. T. Chen and K. L. Hou, An inventory model for items with imperfect quality and quantity discounts under adjusted screening rate and earned interest, Journal of Industrial and Management Optimization, 12 (2016), 1333-1347.  doi: 10.3934/jimo.2016.12.1333.  Google Scholar

[15]

M. Mohammadzadeh, A. A. Khamseh and M. Mohammadi, A multi-objective integrated model for closed-loop supply chain configuration and supplier selection considering uncertain demand and different performance levels, Journal of Industrial and Management Optimization, 13 (2017), 1041-1064.  doi: 10.3934/jimo.2016061.  Google Scholar

[16]

L. Y. Ouyang, B. R. Chuang and Y. J. Lin, Impact of backorder discounts on periodic review inventory model, International Journal of Information and Management Sciences, 14 (2003), 1-13.   Google Scholar

[17]

S. Pal, G. S. Mahapatra and G. P. Samanta, An inventory model of price and stock dependent demand rate with deterioration under inflation and delay in payment, International Journal of System Assurance Engineering and Management, 5 (2014), 591-601.  doi: 10.1007/s13198-013-0209-y.  Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

M. J. Rosenblatt, Multi-item inventory system with budgetary constraint: A comparison between the Lagrangian and fixed cycle approach, International Journal of Production Research, 19 (1981), 331-339.  doi: 10.1080/00207548108956661.  Google Scholar

[23]

B. Sarkar, B. Mandal and S. Sarkar, Quality improvement and backorder price discount under controllable lead time in an inventory model, Journal of Manufacturing Systems, 35 (2015), 26-36.  doi: 10.1016/j.jmsy.2014.11.012.  Google Scholar

[24]

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

[25]

B. Sarkar, A. Majumder, M. Sarkar, B. K. Dey and G. Roy, Two-echelon supply chain model with manufacturing quality improvement and setup cost reduction, Journal of Industrial and Management Optimization, 13 (2017), 1085-1104.  doi: 10.3934/jimo.2016063.  Google Scholar

[26]

J. T. Teng and C. T. Chang, Economic production quantity models for deteriorating items with price- and stock-dependent demand, Computers and Operations Research, 32 (2005), 297-308.  doi: 10.1016/S0305-0548(03)00237-5.  Google Scholar

[27]

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

[28]

Y. C. Tsao, Ordering policy for non-instantaneously deteriorating products under price adjustment and trade credits, Journal of Industrial and Management Optimization, 13 (2017), 327-345.  doi: 10.3934/jimo.2016020.  Google Scholar

[29]

Q. Wang and M. Parlar, A three-person game theory model arising in stochastic inventory control theory, European Journal of Operational Research, 76 (1994), 83-97.  doi: 10.1016/0377-2217(94)90008-6.  Google Scholar

[30]

J. Wu, L. Y. Ouyang, L. 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

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.   Google Scholar

[2]

M. Ben-daya and A. Raouf, On the constrained multi-item single period inventory problem, International Journal of Production Management, 13 (1993), 104-112.  doi: 10.1108/01443579310046472.  Google Scholar

[3]

L. Benkherouf and M. Omar, Optimal integrated policies for a single-vendor single-buyer time-varying demand model, Computers and Mathematics with Applications, 60 (2010), 2066-2077.  doi: 10.1016/j.camwa.2010.07.047.  Google Scholar

[4]

D. K. Bhattacharya, Production, manufacturing and logistics on multi-item inventory, European Journal of Operational Research, 162 (2005), 786-791.  doi: 10.1016/j.ejor.2003.02.004.  Google Scholar

[5]

J. Chen and P. C. Bell, Coordinating a decentralized supply chain with customer returns and price-dependent stochastic demand using a buyback policy, European Journal of Operational Research, 212 (2011), 293-300.  doi: 10.1016/j.ejor.2011.01.036.  Google Scholar

[6]

V. Choudri, M. Venkatachalam and S. 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, 12 (2016), 1153-1172.  doi: 10.3934/jimo.2016.12.1153.  Google Scholar

[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.  doi: 10.1080/09537280150203933.  Google Scholar

[8]

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

[9]

A. Goswami and K. S. Chaudhuri, An EOQ model for deteriorating items with shortages and a linear trend in demand, Journal of the Operational Research Society, 42 (1991), 1105-1110.   Google Scholar

[10]

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

[11]

S. Kar, T. Roy and M. Maiti, Multi-item inventory model with probabilistic price dependent damand and imprecise goal and constraints, Yugoslav Journal of Operations Research, 11 (2001), 93-103.   Google Scholar

[12]

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

[13]

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. Google Scholar

[14]

T. Y. Lin, M. T. Chen and K. L. Hou, An inventory model for items with imperfect quality and quantity discounts under adjusted screening rate and earned interest, Journal of Industrial and Management Optimization, 12 (2016), 1333-1347.  doi: 10.3934/jimo.2016.12.1333.  Google Scholar

[15]

M. Mohammadzadeh, A. A. Khamseh and M. Mohammadi, A multi-objective integrated model for closed-loop supply chain configuration and supplier selection considering uncertain demand and different performance levels, Journal of Industrial and Management Optimization, 13 (2017), 1041-1064.  doi: 10.3934/jimo.2016061.  Google Scholar

[16]

L. Y. Ouyang, B. R. Chuang and Y. J. Lin, Impact of backorder discounts on periodic review inventory model, International Journal of Information and Management Sciences, 14 (2003), 1-13.   Google Scholar

[17]

S. Pal, G. S. Mahapatra and G. P. Samanta, An inventory model of price and stock dependent demand rate with deterioration under inflation and delay in payment, International Journal of System Assurance Engineering and Management, 5 (2014), 591-601.  doi: 10.1007/s13198-013-0209-y.  Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

M. J. Rosenblatt, Multi-item inventory system with budgetary constraint: A comparison between the Lagrangian and fixed cycle approach, International Journal of Production Research, 19 (1981), 331-339.  doi: 10.1080/00207548108956661.  Google Scholar

[23]

B. Sarkar, B. Mandal and S. Sarkar, Quality improvement and backorder price discount under controllable lead time in an inventory model, Journal of Manufacturing Systems, 35 (2015), 26-36.  doi: 10.1016/j.jmsy.2014.11.012.  Google Scholar

[24]

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

[25]

B. Sarkar, A. Majumder, M. Sarkar, B. K. Dey and G. Roy, Two-echelon supply chain model with manufacturing quality improvement and setup cost reduction, Journal of Industrial and Management Optimization, 13 (2017), 1085-1104.  doi: 10.3934/jimo.2016063.  Google Scholar

[26]

J. T. Teng and C. T. Chang, Economic production quantity models for deteriorating items with price- and stock-dependent demand, Computers and Operations Research, 32 (2005), 297-308.  doi: 10.1016/S0305-0548(03)00237-5.  Google Scholar

[27]

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

[28]

Y. C. Tsao, Ordering policy for non-instantaneously deteriorating products under price adjustment and trade credits, Journal of Industrial and Management Optimization, 13 (2017), 327-345.  doi: 10.3934/jimo.2016020.  Google Scholar

[29]

Q. Wang and M. Parlar, A three-person game theory model arising in stochastic inventory control theory, European Journal of Operational Research, 76 (1994), 83-97.  doi: 10.1016/0377-2217(94)90008-6.  Google Scholar

[30]

J. Wu, L. Y. Ouyang, L. 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

Figure 1.  Proposed inventory control model for manufacturer
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Figure 2.  Proposed inventory control model for retailer
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Figure 4.  Profit function in respect of deterioration
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Figure 3.  The concavity of the total profit. Included are $t_1$, $T$ and the total profit $M_1(T, t_1)$, along the x-axis, the y-axis and the z-axis, respectively
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Table 1.  Contributions of some authors related to inventory model
Authors Multi items Two-echelon model Stock- and Price- dependent demand Deterio- rations Trade-credit policy Price discount on backorders
Lenard and Roy (1995) √
Kar et al. (2001) √ √
Ben-daya and Raouf (1993) √
Chen and Bell (2011) √
Datta and Paul (2001) √
Pal et al. (2014) √ √
Teng and Chang (2005) √ √
Thangam and Uthayakumar (2009) √ √ √
Pervin et al. (2016) √ √ √
Goswami and Chaudhuri (1991) √ √
Wu et al. (2014) √ √
Sarkar et al. (2017) √ √
Pervin et al. (2017) √ √
Ouyang et al. (2015) √
This paper √ √ √ √ √ √
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Authors Multi items Two-echelon model Stock- and Price- dependent demand Deterio- rations Trade-credit policy Price discount on backorders
Lenard and Roy (1995) √
Kar et al. (2001) √ √
Ben-daya and Raouf (1993) √
Chen and Bell (2011) √
Datta and Paul (2001) √
Pal et al. (2014) √ √
Teng and Chang (2005) √ √
Thangam and Uthayakumar (2009) √ √ √
Pervin et al. (2016) √ √ √
Goswami and Chaudhuri (1991) √ √
Wu et al. (2014) √ √
Sarkar et al. (2017) √ √
Pervin et al. (2017) √ √
Ouyang et al. (2015) √
This paper √ √ √ √ √ √
Table 2.  Computational parametric configuration for 3 items
Parameters i=1 i=2 i=3 Joint effects of all 3 items
$A_i$ 30 35 40 105
$p_i$ 10 15 20 45
$k_i$ 500 600 700 1800
$\theta_i$ 0.05 0.07 0.09 0.21
$c_i$ 10 14 18 42
$q_i$ 5 10 15 30
$s_i$ 0.12 0.18 0.23 0.53
$l_i$ 0.20 0.25 0.30 0.75
$a_i$ 0.50 0.55 0.60 1.65
$b_i$ 0.30 0.35 0.40 1.05
$\delta_i$ 0.7 0.8 0.9 2.4
$\alpha_i$ 0.2 0.3 0.4 0.9
$\beta_i$ 0.5 0.6 0.7 1.8
$\gamma_i$ 0.4 0.6 0.8 1.8
$M$ 1.2 1.6 2.0 4.8
$I_e$ 0.8 1.0 1.2 3.0
$I_c$ 0.5 0.7 0.9 2.1
$t_1$ 10.62 17.55 29.37 32.76
$t_2$ 9.20 15.72 22.18 29.18
$t_3$ 15.11 19.83 26.07 36.30
$t_4$ 13.57 22.15 34.49 35.02
$M_r(T, t_1)$ 26.6214 42.1753 63.8068 91.4128
$R_{1r}(T, M)$ 95.1506 117.3394 182.7139 220.6210
$R_{2r}(T, M)$ 165.3726 199.5379 274.2680 326.1660
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Parameters i=1 i=2 i=3 Joint effects of all 3 items
$A_i$ 30 35 40 105
$p_i$ 10 15 20 45
$k_i$ 500 600 700 1800
$\theta_i$ 0.05 0.07 0.09 0.21
$c_i$ 10 14 18 42
$q_i$ 5 10 15 30
$s_i$ 0.12 0.18 0.23 0.53
$l_i$ 0.20 0.25 0.30 0.75
$a_i$ 0.50 0.55 0.60 1.65
$b_i$ 0.30 0.35 0.40 1.05
$\delta_i$ 0.7 0.8 0.9 2.4
$\alpha_i$ 0.2 0.3 0.4 0.9
$\beta_i$ 0.5 0.6 0.7 1.8
$\gamma_i$ 0.4 0.6 0.8 1.8
$M$ 1.2 1.6 2.0 4.8
$I_e$ 0.8 1.0 1.2 3.0
$I_c$ 0.5 0.7 0.9 2.1
$t_1$ 10.62 17.55 29.37 32.76
$t_2$ 9.20 15.72 22.18 29.18
$t_3$ 15.11 19.83 26.07 36.30
$t_4$ 13.57 22.15 34.49 35.02
$M_r(T, t_1)$ 26.6214 42.1753 63.8068 91.4128
$R_{1r}(T, M)$ 95.1506 117.3394 182.7139 220.6210
$R_{2r}(T, M)$ 165.3726 199.5379 274.2680 326.1660
Table 3.  Sensitivity analysis for the parameters $\theta_i (i = 1, 2, 3)$. $JE$ represents the joint effects, respectively
Parameter $-30\%$ $-20\%$ $-10\%$ $10\%$ $20\%$ $30\%$
$\theta_1$ 0.035 0.04 0.045 0.055 0.06 0.065
$M_1(T, t_1)$ 28.6725 28.4218 28.2240 27.9031 27.7633 27.5502
$R_{11}(T, M)$ 96.1162 95.9631 95.8812 95.7243 95.5708 95.3276
$R_{21}(T, M)$ 166.502 166.372 166.127 165.872 165.609 165.421
$\theta_2$ 0.049 0.056 0.063 0.077 0.084 0.091
$M_1(T, t_1)$ 43.3213 43.1045 42.8871 42.7490 42.5289 42.3407
$R_{11}(T, M)$ 118.6731 118.4830 118.2571 117.9013 117.7856 117.5126
$R_{21}(T, M)$ 199.410 199.107 198.884 198.619 198.470 198.159
$\theta_3$ 0.063 0.072 0.081 0.099 0.108 0.117
$M_1(T, t_1)$ 65.1173 64.8243 64.5740 64.2581 63.8056 63.5731
$R_{11}(T, M)$ 183.9920 183.7526 183.4911 183.2354 182.8744 182.8530
$R_{21}(T, M)$ 273.576 273.291 272.876 272.651 272.304 272.118
$JE((M_i (T, t_1))_{i=1, 2, 3})$ 324.13 307.47 299.53 271.84 256.77 231.64
$JE((R_{1i}(T, M))_{i=1, 2, 3})$ 573.38 529.13 478.46 431.47 390.52 358.30
$JE((R_{2i}(T, M))_{i=1, 2, 3})$ 642.10 618.34 583.57 549.73 499.05 452.17
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Parameter $-30\%$ $-20\%$ $-10\%$ $10\%$ $20\%$ $30\%$
$\theta_1$ 0.035 0.04 0.045 0.055 0.06 0.065
$M_1(T, t_1)$ 28.6725 28.4218 28.2240 27.9031 27.7633 27.5502
$R_{11}(T, M)$ 96.1162 95.9631 95.8812 95.7243 95.5708 95.3276
$R_{21}(T, M)$ 166.502 166.372 166.127 165.872 165.609 165.421
$\theta_2$ 0.049 0.056 0.063 0.077 0.084 0.091
$M_1(T, t_1)$ 43.3213 43.1045 42.8871 42.7490 42.5289 42.3407
$R_{11}(T, M)$ 118.6731 118.4830 118.2571 117.9013 117.7856 117.5126
$R_{21}(T, M)$ 199.410 199.107 198.884 198.619 198.470 198.159
$\theta_3$ 0.063 0.072 0.081 0.099 0.108 0.117
$M_1(T, t_1)$ 65.1173 64.8243 64.5740 64.2581 63.8056 63.5731
$R_{11}(T, M)$ 183.9920 183.7526 183.4911 183.2354 182.8744 182.8530
$R_{21}(T, M)$ 273.576 273.291 272.876 272.651 272.304 272.118
$JE((M_i (T, t_1))_{i=1, 2, 3})$ 324.13 307.47 299.53 271.84 256.77 231.64
$JE((R_{1i}(T, M))_{i=1, 2, 3})$ 573.38 529.13 478.46 431.47 390.52 358.30
$JE((R_{2i}(T, M))_{i=1, 2, 3})$ 642.10 618.34 583.57 549.73 499.05 452.17
Table 4.  Sensitivity analysis for the parameters $b_i (i = 1, 2, 3)$. $JE$ represents the joint effects, respectively
Parameter $-30\%$ $-20\%$ $-10\%$ $10\%$ $20\%$ $30\%$
$b_1$ 0.21 0.24 0.27 0.33 0.36 0.39
$M_1(T, t_1)$ 27.6820 27.8921 28.1675 28.4691 28.8722 29.3225
$R_{11}(T, M)$ 96.3546 96.6704 96.9124 97.3510 97.6021 97.9146
$R_{21}(T, M)$ 165.318 165.702 165.993 166.247 166.583 166.875
$b_2$ 0.245 0.28 0.315 0.385 0.420 0.455
$M_1(T, t_1)$ 42.6813 42.8904 43.3612 43.6675 43.9130 44.4362
$R_{11}(T, M)$ 117.1325 117.4076 117.7451 118.2169 118.5306 118.8539
$R_{21}(T, M)$ 198.478 198.825 199.350 199.718 199.958 237.574
$b_3$ 0.28 0.32 0.36 0.44 0.48 0.52
$M_1(T, t_1)$ 66.3753 66.5206 66.8347 67.2541 67.6088 67.9346
$R_{11}(T, M)$ 182.7090 182.9364 183.2674 183.5104 183.7984 184.2576
$R_{21}(T, M)$ 273.157 273.409 273.749 273.986 274.370 274.875
$JE((M_i (T, t_1))_{i=1, 2, 3})$ 142.39 187.52 224.78 279.70 330.87 367.42
$JE((R_{1i}(T, M))_{i=1, 2, 3})$ 413.49 437.61 475.23 493.27 534.17 562.19
$JE((R_{2i}(T, M))_{i=1, 2, 3})$ 650.14 688.07 723.47 767.33 784.35 820.69
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Parameter $-30\%$ $-20\%$ $-10\%$ $10\%$ $20\%$ $30\%$
$b_1$ 0.21 0.24 0.27 0.33 0.36 0.39
$M_1(T, t_1)$ 27.6820 27.8921 28.1675 28.4691 28.8722 29.3225
$R_{11}(T, M)$ 96.3546 96.6704 96.9124 97.3510 97.6021 97.9146
$R_{21}(T, M)$ 165.318 165.702 165.993 166.247 166.583 166.875
$b_2$ 0.245 0.28 0.315 0.385 0.420 0.455
$M_1(T, t_1)$ 42.6813 42.8904 43.3612 43.6675 43.9130 44.4362
$R_{11}(T, M)$ 117.1325 117.4076 117.7451 118.2169 118.5306 118.8539
$R_{21}(T, M)$ 198.478 198.825 199.350 199.718 199.958 237.574
$b_3$ 0.28 0.32 0.36 0.44 0.48 0.52
$M_1(T, t_1)$ 66.3753 66.5206 66.8347 67.2541 67.6088 67.9346
$R_{11}(T, M)$ 182.7090 182.9364 183.2674 183.5104 183.7984 184.2576
$R_{21}(T, M)$ 273.157 273.409 273.749 273.986 274.370 274.875
$JE((M_i (T, t_1))_{i=1, 2, 3})$ 142.39 187.52 224.78 279.70 330.87 367.42
$JE((R_{1i}(T, M))_{i=1, 2, 3})$ 413.49 437.61 475.23 493.27 534.17 562.19
$JE((R_{2i}(T, M))_{i=1, 2, 3})$ 650.14 688.07 723.47 767.33 784.35 820.69
Table 5.  Sensitivity analysis for the parameters $\beta_i(i = 1, 2, 3)$. $JE$ represents the joint effects, respectively
Parameter $-30\%$ $-20\%$ $-10\%$ $10\%$ $20\%$ $30\%$
$\beta_1$ 0.35 0.40 0.45 0.55 0.60 0.65
$M_1(T, t_1)$ 30.2670 30.8305 31.4758 32.0165 32.7736 33.3814
$R_{11}(T, M)$ 98.4236 98.9470 99.5024 100.0631 100.5861 101.1425
$R_{21}(T, M)$ 167.264 167.752 168.369 168.804 169.362 169.972
$\beta_2$ 0.42 0.48 0.54 0.66 0.72 0.78
$M_1(T, t_1)$ 45.6149 46.1543 46.5783 47.1462 47.6388 48.3592
$R_{11}(T, M)$ 118.4268 118.9005 119.2457 119.8319 120.3670 120.7591
$R_{21}(T, M)$ 210.375 210.763 211.136 211.545 211.987 212.307
$\beta_3$ 0.49 0.56 0.63 0.77 0.84 0.91
$M_1(T, t_1)$ 68.3871 68.8414 69.4206 70.9126 70.4756 70.8225
$R_{11}(T, M)$ 185.2477 185.6470 186.0853 186.4800 186.8175 187.2052
$R_{21}(T, M)$ 275.8145 276.335 276.857 277.3670 277.8950 277.4352
$JE((M_i (T, t_1))_{i=1, 2, 3})$ 150.26 163.73 175.44 189.71 213.46 252.04
$JE((R_{1i}(T, M))_{i=1, 2, 3})$ 409.24 422.19 435.07 442.86 453.20 467.15
$JE((R_{2i}(T, M))_{i=1, 2, 3})$ 660.14 671.42 684.35 697.23 713.75 729.06
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Parameter $-30\%$ $-20\%$ $-10\%$ $10\%$ $20\%$ $30\%$
$\beta_1$ 0.35 0.40 0.45 0.55 0.60 0.65
$M_1(T, t_1)$ 30.2670 30.8305 31.4758 32.0165 32.7736 33.3814
$R_{11}(T, M)$ 98.4236 98.9470 99.5024 100.0631 100.5861 101.1425
$R_{21}(T, M)$ 167.264 167.752 168.369 168.804 169.362 169.972
$\beta_2$ 0.42 0.48 0.54 0.66 0.72 0.78
$M_1(T, t_1)$ 45.6149 46.1543 46.5783 47.1462 47.6388 48.3592
$R_{11}(T, M)$ 118.4268 118.9005 119.2457 119.8319 120.3670 120.7591
$R_{21}(T, M)$ 210.375 210.763 211.136 211.545 211.987 212.307
$\beta_3$ 0.49 0.56 0.63 0.77 0.84 0.91
$M_1(T, t_1)$ 68.3871 68.8414 69.4206 70.9126 70.4756 70.8225
$R_{11}(T, M)$ 185.2477 185.6470 186.0853 186.4800 186.8175 187.2052
$R_{21}(T, M)$ 275.8145 276.335 276.857 277.3670 277.8950 277.4352
$JE((M_i (T, t_1))_{i=1, 2, 3})$ 150.26 163.73 175.44 189.71 213.46 252.04
$JE((R_{1i}(T, M))_{i=1, 2, 3})$ 409.24 422.19 435.07 442.86 453.20 467.15
$JE((R_{2i}(T, M))_{i=1, 2, 3})$ 660.14 671.42 684.35 697.23 713.75 729.06
Table 6.  Sensitivity analysis for the parameters $\gamma_i(i = 1, 2, 3)$. $JE$ represents the joint effects, respectively
Parameter $-30\%$ $-20\%$ $-10\%$ $10\%$ $20\%$ $30\%$
$\gamma_1$ 0.28 0.32 0.36 0.44 0.48 0.52
$M_1(T, t_1)$ 28.2743 28.6183 29.0179 29.3746 29.6524 29.9032
$R_{11}(T, M)$ 97.3176 97.6480 97.9208 98.4635 98.7705 99.1432
$R_{21}(T, M)$ 165.418 165.727 166.284 166.546 166.920 167.335
$\gamma_2$ 0.42 0.48 0.54 0.66 0.72 0.78
$M_1(T, t_1)$ 44.1267 44.4725 44.8213 45.2018 45.5537 45.8715
$R_{11}(T, M)$ 116.4503 116.6931 116.9917 117.4361 117.8543 118.5174
$R_{21}(T, M)$ 206.421 206.885 207.367 207.753 208.290 208.668
$\gamma_3$ 0.56 0.64 0.72 0.88 0.96 1.04
$M_1(T, t_1)$ 67.3417 67.8864 68.4011 68.9172 79.5140 79.8711
$R_{11}(T, M)$ 189.1106 189.6587 190.2275 190.6340 191.3719 191.5603
$R_{21}(T, M)$ 279.9902 280.4275 280.8670 281.2552 282.7732 283.5112
$JE((M_i(T, t_1))_{i=1, 2, 3})$ 140.31 153.10 161.49 175.25 183.39 199.37
$JE((R_{1i}(T, M))_{i=1, 2, 3})$ 408.72 417.28 425.01 437.26 445.49 467.00
$JE((R_{2i}(T, M))_{i=1, 2, 3})$ 657.29 669.81 672.31 685.01 693.33 707.46
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Parameter $-30\%$ $-20\%$ $-10\%$ $10\%$ $20\%$ $30\%$
$\gamma_1$ 0.28 0.32 0.36 0.44 0.48 0.52
$M_1(T, t_1)$ 28.2743 28.6183 29.0179 29.3746 29.6524 29.9032
$R_{11}(T, M)$ 97.3176 97.6480 97.9208 98.4635 98.7705 99.1432
$R_{21}(T, M)$ 165.418 165.727 166.284 166.546 166.920 167.335
$\gamma_2$ 0.42 0.48 0.54 0.66 0.72 0.78
$M_1(T, t_1)$ 44.1267 44.4725 44.8213 45.2018 45.5537 45.8715
$R_{11}(T, M)$ 116.4503 116.6931 116.9917 117.4361 117.8543 118.5174
$R_{21}(T, M)$ 206.421 206.885 207.367 207.753 208.290 208.668
$\gamma_3$ 0.56 0.64 0.72 0.88 0.96 1.04
$M_1(T, t_1)$ 67.3417 67.8864 68.4011 68.9172 79.5140 79.8711
$R_{11}(T, M)$ 189.1106 189.6587 190.2275 190.6340 191.3719 191.5603
$R_{21}(T, M)$ 279.9902 280.4275 280.8670 281.2552 282.7732 283.5112
$JE((M_i(T, t_1))_{i=1, 2, 3})$ 140.31 153.10 161.49 175.25 183.39 199.37
$JE((R_{1i}(T, M))_{i=1, 2, 3})$ 408.72 417.28 425.01 437.26 445.49 467.00
$JE((R_{2i}(T, M))_{i=1, 2, 3})$ 657.29 669.81 672.31 685.01 693.33 707.46
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