doi: 10.3934/jimo.2019020

A collaborative EPQ inventory model for a three-echelon supply chain with multiple products considering the effect of marketing effort on demand

1. 

Department of Industrial Management, Agroindustry and Operations, Universidad de la Costa, Colombia

2. 

Department of Industrial Engineering, Universidad Tecnologica de Bolivar, Colombia

3. 

Grupo de Energia y Termodinamica, Universidad Pontificia Bolivariana, Colombia

4. 

Department of Industrial and Systems Engineering, School of Engineering and Sciences, Tecnológico de Monterrey, Monterrey, México

5. 

Principal, Kishore Bharati Bhagini Nivedita College, 148 Ramkrishna Sarani, Behala, Kolkata-700060, India

* Corresponding author: SHIB SANKAR SANA

Received  July 2018 Revised  October 2018 Published  March 2019

This paper presents an inventory model for a three-echelon supply chain with multiple products and multiple members considering the demand as an increasing function of the marketing effort. In the proposed inventory model, a collaborative approach is studied and an analytical method is applied to obtain the optimal production lot size and the optimal marketing effort in order to achieve the maximum profits. Some numerical examples are illustrated to justify the model. Moreover, a sensitivity analysis is well done in order to analysis the effect of the changes of key parameters of inventory model on the the maximum benefits of all members of the chain.

Citation: Katherinne Salas Navarro, Jaime Acevedo Chedid, Whady F. Florez, Holman Ospina Mateus, Leopoldo Eduardo Cárdenas-Barrón, Shib Sankar Sana. A collaborative EPQ inventory model for a three-echelon supply chain with multiple products considering the effect of marketing effort on demand. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019020
References:
[1]

H. M. Abdelsalam and M. M. Elassal, Joint economic lot sizing problem for a three-Layer supply chain with stochastic demand, International Journal of Production Economics, 155 (2014), 272-283.   Google Scholar

[2]

A. Ahmadi-Javid and P. Hoseinpour, A location-inventory-pricing model in a supply chain distribution network with price-sensitive demands and inventory-capacity constraints, Transportation Research Part E: Logistics and Transportation Review, 82 (2015), 238-255.   Google Scholar

[3]

F. Alawneh and G. Zhang, Dual-channel warehouse and inventory management with stochastic demand, Transportation Research Part E: Logistics and Transportation Review, 112 (2018), 84-106.   Google Scholar

[4]

M. Ben-DayaR. Asád and M. Seliaman, An integrated production inventory model with raw material replenishment considerations in a three layer supply chain, International Journal of Production Economics, 143 (2013), 53-61.   Google Scholar

[5]

L. E. Cárdenas-Barrón and S. S. Sana, A production-inventory model for a two-echelon supply chain when demand is dependent on sales teams' initiatives, International Journal of Production Economics, 155 (2014), 249-258.   Google Scholar

[6]

L. E. Cárdenas-Barrón and S. S. Sana, Multi-item EOQ inventory model in a two-layer supply chain while demand varies with promotional effort, Applied Mathematical Modelling, 39 (2015), 6725-6737.  doi: 10.1016/j.apm.2015.02.004.  Google Scholar

[7]

L. E. Cárdenas-BarrónJ. T. TengG. Trevino-GarzaH. M. Wee and K. R. Lou, An improved algorithm and solution on an integrated production-inventory model in a three-layer supply chain, International Journal of Production Economics, 136 (2012), 384-388.   Google Scholar

[8]

Z. DaiF. Aqlan and K. Gao, Optimizing multi-echelon inventory with three types of demand in supply chain, Transportation Research Part E: Logistics and Transportation Review, 107 (2017), 141-177.   Google Scholar

[9]

S. K. De and S. S. Sana, The (p, q, r, l) mode l fo r stochastic demand unde r Intuitionistic fuzzy agg regation with Bonferroni mean, Journal of Intelligent Manufacturing, check (2016), 1-19.   Google Scholar

[10]

J. P. Dube, Multiple discreteness and product differentiation: Demand for carbonated soft drinks, Marketing Science, 23 (2004), 66-81.   Google Scholar

[11]

B. C. Giri and S. Bardhan, Sub-supply chain coordination in a three-layer chain under demand uncertainty and random yield in production, International, Journal of Production Economics, 191 (2017), 66-73.   Google Scholar

[12]

S. K. Goyal and C. T. Chang, Optimal ordering and transfer policy for an inventory with stock dependent demand, European Journal of Operational Research, 196 (2009), 177-185.   Google Scholar

[13]

S. K. Goyal and A. Gunasekaran, An integrated production-inventory-marketing model for deteriorating items, Computers & Industrial Engineering, 28 (1995), 755-762.   Google Scholar

[14]

K. L. Hou and L. C. Lin, An EOQ model for deteriorating items with price-and stock-dependent selling rate under inflation and time value of money, International Journal of Systems Science, 37 (2006), 1131-1139.  doi: 10.1080/00207720601014206.  Google Scholar

[15]

M. JohariS. M. Hosseini-MotlaghM. NematollahiM. Goh and J. Ignatius, Bi-level credit period coordination for periodic review inventory system with price-credit dependent demand under time value of money, Transportation Research Part E: Logistics and Transportation Review, 114 (2018), 270-291.   Google Scholar

[16]

S. Karray, Periodicity of pricing and marketing efforts in a distribution channel, European Journal of Operational Research, 228 (2013), 635-647.  doi: 10.1016/j.ejor.2013.02.012.  Google Scholar

[17]

S. Kim and D. I. Gilliland, Working more or working less? Contingent allocation of reseller effort in distribution channels, Industrial Marketing Management, 64 (2017), 44-56.   Google Scholar

[18]

H. KrishnanR. Kapusciniski and D. A. Butz, Coordinating contracts for decentralized supply chain with retailer promotional effect, Management Science, 50 (2004), 48-63.   Google Scholar

[19]

W. LeeS. P. Wang and W. C. Chen, Forward and backward stocking policies for a two-level supply chain with consignment stock agreement and stock-dependent demand, European Journal of Operational Research, 256 (2017), 830-840.  doi: 10.1016/j.ejor.2016.06.060.  Google Scholar

[20]

P. MaH. Wnag and J. Shang, Supply chain chanel strategies with quality and marketing effort-dependent demand, International Journal of Production Economics, 144 (2013), 572-581.   Google Scholar

[21]

P. MaH. Wang and J. Shang, Contract design for two-stage supply chain coordination: Integrating manufacturer-quality and retailer-marketing efforts, International Journal of Production Economics, 146 (2013), 745-755.   Google Scholar

[22]

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

[23]

N. M. ModakS. Panda and S. S. Sana, Pricing policy and coordination for a two-layer supply chain of duopolistic retailers and socially responsible manufacturer, International Journal of Logistic: Research and Applications, 19 (2015), 487-508.   Google Scholar

[24]

N. M. ModakS. Panda and S. S. Sana, Two-echelon supply chain coordination among manufacturer and duopolies retailers with recycling facility, International Journal of Advanced Manufacturing Technology, 87 (2016), 1531-1546.   Google Scholar

[25]

P. A. NaikK. Raman and R. S. Winer, Planning marketing-mix strategies in the presence of interaction effects, Marketing Science, 24 (2005), 25-34.   Google Scholar

[26]

M. G. Nagler, An exploratory analysis of the determinants of cooperative advertising participation rates, Marketing Letters, 17 (2006), 91-102.   Google Scholar

[27]

T. PaksoyN. Y. Pehlivan and E. Ozceylan, A new tradeoff model for fuzzy supply chain network design and optimization, Human and Ecological Risk Assessment: An International Journal, 19 (2013), 492-514.   Google Scholar

[28]

T. PaksoyE. Ozceylan and G. W. Weber, Profit oriented supply chain network optimization, Central European Journal of Operations Research, 21 (2013), 455-478.  doi: 10.1007/s10100-012-0240-0.  Google Scholar

[29]

B. PalS. S. Sana and K. Chaudhuri, Three-layer supply chain – A production-inventory model for reworkable items, Applied Mathematics and Computation, 219 (2012), 530-543.  doi: 10.1016/j.amc.2012.06.038.  Google Scholar

[30]

S. PandaN. M. ModakM. Basu and S. K. Goyal, Channel coordination and profit distribution in a social responsible three-layer supply chain, International Journal of Production Economics, 168 (2015), 224-233.   Google Scholar

[31]

S. PandaS. Saha and S. K. Goyal, Dilema of rented warehouse and shelf for inventory systems with displayed stock level dependent demand, Economic Modelling, 32 (2013), 452-462.   Google Scholar

[32]

S. PandaN. M. ModakS. S. Sana and M. Basu, Pricing and replenishment policies in dual-channel supply chain under continuous unit cost decrease, Applied Mathematics and Computation, 256 (2015), 913-929.  doi: 10.1016/j.amc.2015.01.081.  Google Scholar

[33]

M. PervinS. 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

[34]

M. PervinS. 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 & Optimization, 7 (2017), 21-50.  doi: 10.3934/naco.2017002.  Google Scholar

[35]

M. PervinS. K. Roy and G. W. Weber, An integrated inventory model with variable holding cost under two levels of trade-credit policy, Numerical Algebra, Control & Optimization, 8 (2018), 169-191.  doi: 10.3934/naco.2018010.  Google Scholar

[36]

M. Pervin, S. K. Roy and G. W. Weber, Multi-item deteriorating two-echelon inventory model with price-and stock-dependent demand: A trade-credit policy, Journal of Industrial & Management Optimization, (2018). Google Scholar

[37]

X. PuL. Gong and G. Han, A feasible incentive contract between a manufacturer and his fairness-sensitive retailer engaged in strategic marketing efforts, Journal of Intelligent Manufacturing, 30 (2019), 193-206.  doi: 10.1007/s10845-016-1239-5.  Google Scholar

[38]

A. RoyS. S. Sana and K. Chaudhuri, A joint venturing of single supplier and single retailer under variable price, promotional effort and service level, Pacific Science Review B: Humanities and Social Sciences, 1 (2015), 8-14.   Google Scholar

[39]

A. RoyS. S. Sana and K. Chaudhuri, Optimal replenishment order for uncertain demand in three layer supply chain, Economic Modelling, 29 (2012), 2274-2282.   Google Scholar

[40]

K. Salas-NavarroJ. Acevedo-ChedidN. Mercado-Caruso and S. S. Sana, An inventory model of three-layer supply chain of wood and furniture industry in the Caribbean region of Colombia, Internarional Journal of Systems Science: Operations and Logistics, 5 (2018), 69-86.   Google Scholar

[41]

S. S. Sana, An EOQ model for salesmen's initiatives, stock and price sensitive demand of similar products – A dynamical system, Applied Mathematics and Computation, 218 (2011), 3277-3288.  doi: 10.1016/j.amc.2011.08.067.  Google Scholar

[42]

S. S. Sana, Optimal contract strategies for two stage supply chain, Economic Modelling, 30 (2013), 253-260.   Google Scholar

[43]

S. S. SanaJ. Acevedo-Chedid and K. Salas-Navarro, A three layer supply chain model with multiple suppliers, manufacturers and retailers for multiple items, Applied Mathematics and Computation, 229 (2014), 139-150.  doi: 10.1016/j.amc.2013.12.006.  Google Scholar

[44]

S. S. Sana, A production-inventory model of imperfect quality products in a three-layer supply chain, Decision Support Systems, 50 (2011), 539-547.   Google Scholar

[45]

S. S. Sana, Sales team's initiatives and stock sensitive demand – A production control policy, Economic Modelling, 31 (2013), 783-788.   Google Scholar

[46]

J. SongF. LiD. D. WuL. Liang and A. Dolgui, Supply chain coordination through integration of innovation effort and advertising support, Applied Mathematical Modelling, 49 (2017), 108-123.  doi: 10.1016/j.apm.2017.04.041.  Google Scholar

[47]

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.  Google Scholar

[48]

Y. C. Tsao and G. J. Sheen, Effects of promotion cost sharing policy with the sales learning curve on supply chain coordination, Computers & Operations Research, 39 (2012), 1872-1878.  doi: 10.1016/j.cor.2011.07.009.  Google Scholar

[49]

S. Zhao and Q. Zhu, A risk-averse marketing strategy and its effect on coordination activities in a remanufacturing supply chain under market fluctuation, Journal of Cleaner Production, 171 (2018), 1290-1299.   Google Scholar

show all references

References:
[1]

H. M. Abdelsalam and M. M. Elassal, Joint economic lot sizing problem for a three-Layer supply chain with stochastic demand, International Journal of Production Economics, 155 (2014), 272-283.   Google Scholar

[2]

A. Ahmadi-Javid and P. Hoseinpour, A location-inventory-pricing model in a supply chain distribution network with price-sensitive demands and inventory-capacity constraints, Transportation Research Part E: Logistics and Transportation Review, 82 (2015), 238-255.   Google Scholar

[3]

F. Alawneh and G. Zhang, Dual-channel warehouse and inventory management with stochastic demand, Transportation Research Part E: Logistics and Transportation Review, 112 (2018), 84-106.   Google Scholar

[4]

M. Ben-DayaR. Asád and M. Seliaman, An integrated production inventory model with raw material replenishment considerations in a three layer supply chain, International Journal of Production Economics, 143 (2013), 53-61.   Google Scholar

[5]

L. E. Cárdenas-Barrón and S. S. Sana, A production-inventory model for a two-echelon supply chain when demand is dependent on sales teams' initiatives, International Journal of Production Economics, 155 (2014), 249-258.   Google Scholar

[6]

L. E. Cárdenas-Barrón and S. S. Sana, Multi-item EOQ inventory model in a two-layer supply chain while demand varies with promotional effort, Applied Mathematical Modelling, 39 (2015), 6725-6737.  doi: 10.1016/j.apm.2015.02.004.  Google Scholar

[7]

L. E. Cárdenas-BarrónJ. T. TengG. Trevino-GarzaH. M. Wee and K. R. Lou, An improved algorithm and solution on an integrated production-inventory model in a three-layer supply chain, International Journal of Production Economics, 136 (2012), 384-388.   Google Scholar

[8]

Z. DaiF. Aqlan and K. Gao, Optimizing multi-echelon inventory with three types of demand in supply chain, Transportation Research Part E: Logistics and Transportation Review, 107 (2017), 141-177.   Google Scholar

[9]

S. K. De and S. S. Sana, The (p, q, r, l) mode l fo r stochastic demand unde r Intuitionistic fuzzy agg regation with Bonferroni mean, Journal of Intelligent Manufacturing, check (2016), 1-19.   Google Scholar

[10]

J. P. Dube, Multiple discreteness and product differentiation: Demand for carbonated soft drinks, Marketing Science, 23 (2004), 66-81.   Google Scholar

[11]

B. C. Giri and S. Bardhan, Sub-supply chain coordination in a three-layer chain under demand uncertainty and random yield in production, International, Journal of Production Economics, 191 (2017), 66-73.   Google Scholar

[12]

S. K. Goyal and C. T. Chang, Optimal ordering and transfer policy for an inventory with stock dependent demand, European Journal of Operational Research, 196 (2009), 177-185.   Google Scholar

[13]

S. K. Goyal and A. Gunasekaran, An integrated production-inventory-marketing model for deteriorating items, Computers & Industrial Engineering, 28 (1995), 755-762.   Google Scholar

[14]

K. L. Hou and L. C. Lin, An EOQ model for deteriorating items with price-and stock-dependent selling rate under inflation and time value of money, International Journal of Systems Science, 37 (2006), 1131-1139.  doi: 10.1080/00207720601014206.  Google Scholar

[15]

M. JohariS. M. Hosseini-MotlaghM. NematollahiM. Goh and J. Ignatius, Bi-level credit period coordination for periodic review inventory system with price-credit dependent demand under time value of money, Transportation Research Part E: Logistics and Transportation Review, 114 (2018), 270-291.   Google Scholar

[16]

S. Karray, Periodicity of pricing and marketing efforts in a distribution channel, European Journal of Operational Research, 228 (2013), 635-647.  doi: 10.1016/j.ejor.2013.02.012.  Google Scholar

[17]

S. Kim and D. I. Gilliland, Working more or working less? Contingent allocation of reseller effort in distribution channels, Industrial Marketing Management, 64 (2017), 44-56.   Google Scholar

[18]

H. KrishnanR. Kapusciniski and D. A. Butz, Coordinating contracts for decentralized supply chain with retailer promotional effect, Management Science, 50 (2004), 48-63.   Google Scholar

[19]

W. LeeS. P. Wang and W. C. Chen, Forward and backward stocking policies for a two-level supply chain with consignment stock agreement and stock-dependent demand, European Journal of Operational Research, 256 (2017), 830-840.  doi: 10.1016/j.ejor.2016.06.060.  Google Scholar

[20]

P. MaH. Wnag and J. Shang, Supply chain chanel strategies with quality and marketing effort-dependent demand, International Journal of Production Economics, 144 (2013), 572-581.   Google Scholar

[21]

P. MaH. Wang and J. Shang, Contract design for two-stage supply chain coordination: Integrating manufacturer-quality and retailer-marketing efforts, International Journal of Production Economics, 146 (2013), 745-755.   Google Scholar

[22]

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

[23]

N. M. ModakS. Panda and S. S. Sana, Pricing policy and coordination for a two-layer supply chain of duopolistic retailers and socially responsible manufacturer, International Journal of Logistic: Research and Applications, 19 (2015), 487-508.   Google Scholar

[24]

N. M. ModakS. Panda and S. S. Sana, Two-echelon supply chain coordination among manufacturer and duopolies retailers with recycling facility, International Journal of Advanced Manufacturing Technology, 87 (2016), 1531-1546.   Google Scholar

[25]

P. A. NaikK. Raman and R. S. Winer, Planning marketing-mix strategies in the presence of interaction effects, Marketing Science, 24 (2005), 25-34.   Google Scholar

[26]

M. G. Nagler, An exploratory analysis of the determinants of cooperative advertising participation rates, Marketing Letters, 17 (2006), 91-102.   Google Scholar

[27]

T. PaksoyN. Y. Pehlivan and E. Ozceylan, A new tradeoff model for fuzzy supply chain network design and optimization, Human and Ecological Risk Assessment: An International Journal, 19 (2013), 492-514.   Google Scholar

[28]

T. PaksoyE. Ozceylan and G. W. Weber, Profit oriented supply chain network optimization, Central European Journal of Operations Research, 21 (2013), 455-478.  doi: 10.1007/s10100-012-0240-0.  Google Scholar

[29]

B. PalS. S. Sana and K. Chaudhuri, Three-layer supply chain – A production-inventory model for reworkable items, Applied Mathematics and Computation, 219 (2012), 530-543.  doi: 10.1016/j.amc.2012.06.038.  Google Scholar

[30]

S. PandaN. M. ModakM. Basu and S. K. Goyal, Channel coordination and profit distribution in a social responsible three-layer supply chain, International Journal of Production Economics, 168 (2015), 224-233.   Google Scholar

[31]

S. PandaS. Saha and S. K. Goyal, Dilema of rented warehouse and shelf for inventory systems with displayed stock level dependent demand, Economic Modelling, 32 (2013), 452-462.   Google Scholar

[32]

S. PandaN. M. ModakS. S. Sana and M. Basu, Pricing and replenishment policies in dual-channel supply chain under continuous unit cost decrease, Applied Mathematics and Computation, 256 (2015), 913-929.  doi: 10.1016/j.amc.2015.01.081.  Google Scholar

[33]

M. PervinS. 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

[34]

M. PervinS. 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 & Optimization, 7 (2017), 21-50.  doi: 10.3934/naco.2017002.  Google Scholar

[35]

M. PervinS. K. Roy and G. W. Weber, An integrated inventory model with variable holding cost under two levels of trade-credit policy, Numerical Algebra, Control & Optimization, 8 (2018), 169-191.  doi: 10.3934/naco.2018010.  Google Scholar

[36]

M. Pervin, S. K. Roy and G. W. Weber, Multi-item deteriorating two-echelon inventory model with price-and stock-dependent demand: A trade-credit policy, Journal of Industrial & Management Optimization, (2018). Google Scholar

[37]

X. PuL. Gong and G. Han, A feasible incentive contract between a manufacturer and his fairness-sensitive retailer engaged in strategic marketing efforts, Journal of Intelligent Manufacturing, 30 (2019), 193-206.  doi: 10.1007/s10845-016-1239-5.  Google Scholar

[38]

A. RoyS. S. Sana and K. Chaudhuri, A joint venturing of single supplier and single retailer under variable price, promotional effort and service level, Pacific Science Review B: Humanities and Social Sciences, 1 (2015), 8-14.   Google Scholar

[39]

A. RoyS. S. Sana and K. Chaudhuri, Optimal replenishment order for uncertain demand in three layer supply chain, Economic Modelling, 29 (2012), 2274-2282.   Google Scholar

[40]

K. Salas-NavarroJ. Acevedo-ChedidN. Mercado-Caruso and S. S. Sana, An inventory model of three-layer supply chain of wood and furniture industry in the Caribbean region of Colombia, Internarional Journal of Systems Science: Operations and Logistics, 5 (2018), 69-86.   Google Scholar

[41]

S. S. Sana, An EOQ model for salesmen's initiatives, stock and price sensitive demand of similar products – A dynamical system, Applied Mathematics and Computation, 218 (2011), 3277-3288.  doi: 10.1016/j.amc.2011.08.067.  Google Scholar

[42]

S. S. Sana, Optimal contract strategies for two stage supply chain, Economic Modelling, 30 (2013), 253-260.   Google Scholar

[43]

S. S. SanaJ. Acevedo-Chedid and K. Salas-Navarro, A three layer supply chain model with multiple suppliers, manufacturers and retailers for multiple items, Applied Mathematics and Computation, 229 (2014), 139-150.  doi: 10.1016/j.amc.2013.12.006.  Google Scholar

[44]

S. S. Sana, A production-inventory model of imperfect quality products in a three-layer supply chain, Decision Support Systems, 50 (2011), 539-547.   Google Scholar

[45]

S. S. Sana, Sales team's initiatives and stock sensitive demand – A production control policy, Economic Modelling, 31 (2013), 783-788.   Google Scholar

[46]

J. SongF. LiD. D. WuL. Liang and A. Dolgui, Supply chain coordination through integration of innovation effort and advertising support, Applied Mathematical Modelling, 49 (2017), 108-123.  doi: 10.1016/j.apm.2017.04.041.  Google Scholar

[47]

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.  Google Scholar

[48]

Y. C. Tsao and G. J. Sheen, Effects of promotion cost sharing policy with the sales learning curve on supply chain coordination, Computers & Operations Research, 39 (2012), 1872-1878.  doi: 10.1016/j.cor.2011.07.009.  Google Scholar

[49]

S. Zhao and Q. Zhu, A risk-averse marketing strategy and its effect on coordination activities in a remanufacturing supply chain under market fluctuation, Journal of Cleaner Production, 171 (2018), 1290-1299.   Google Scholar

Figure 1.  Diagram of inventory level of the supply chain
Table 1.  Contributions of some authors related to EPQ model
Authors Multiple Three layer Marketing effort Defective Collaborative Different
items model sensitive demand items approach cycle time
Sana [44] $\checkmark$ $\checkmark$ $\checkmark$
Pal et al. [29] $\checkmark$ $\checkmark$
Ben-Daya et al.[4] $\checkmark$ $\checkmark$
Sana et al.[43] $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
Roy et al. [39] $\checkmark$ $\checkmark$
Abdelsalam and Elassal [1] $\checkmark$
Giri and Bardhan [11] $\checkmark$ $\checkmark$
Dai et al.[8] $\checkmark$
Sana [45] $\checkmark$
Cárdenas-Barrón and Sana [6] $\checkmark$ $\checkmark$ $\checkmark$
Tsao and Sheen [48] $\checkmark$ $\checkmark$
Roy et al.[38] $\checkmark$ $\checkmark$
Cárdenas-Barrón and Sana [5] $\checkmark$ $\checkmark$
Our present paper $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
Authors Multiple Three layer Marketing effort Defective Collaborative Different
items model sensitive demand items approach cycle time
Sana [44] $\checkmark$ $\checkmark$ $\checkmark$
Pal et al. [29] $\checkmark$ $\checkmark$
Ben-Daya et al.[4] $\checkmark$ $\checkmark$
Sana et al.[43] $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
Roy et al. [39] $\checkmark$ $\checkmark$
Abdelsalam and Elassal [1] $\checkmark$
Giri and Bardhan [11] $\checkmark$ $\checkmark$
Dai et al.[8] $\checkmark$
Sana [45] $\checkmark$
Cárdenas-Barrón and Sana [6] $\checkmark$ $\checkmark$ $\checkmark$
Tsao and Sheen [48] $\checkmark$ $\checkmark$
Roy et al.[38] $\checkmark$ $\checkmark$
Cárdenas-Barrón and Sana [5] $\checkmark$ $\checkmark$
Our present paper $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
Table 2.  Values of parameters for manufacturers
Parameters of Product 1 Product 2
Manufacturer 1 $p_{11} = 700$ units/year, $\alpha_{11} = 0.01$ $p_{12} = 740$ units/year, $\alpha_{12} = 0.03$,
$c_{11} = 15$$/units/year, $w_{11}^{M} = 55$$/unit, $\epsilon = 1$ $c_{12} = 17$$/ unit, $w_{12}^{M} = 75$$/unit, $\epsilon = 1$
$L_{11} = 2000$$ /production cycle $L_{12} = 1900$$ /production,
$AM_{11} = 200$$/setup, $k_{11} = 5.1$$/unit $AM_{12} = 208$$/setup, $k_{12} = 4.5$$/unit
$h_{11}^{P} = 2.75$$/unit/year, $\beta_{11} = 15$% $h_{12}^{P} = 3.75$$/unit/year, $\beta_{12} = 15$%
Manufacturer 2 $p_{21} = 900$ units/year, $\alpha_{21} = 0.04$ $p_{22} = 1050$ units/year, $\alpha_{22} = 0.03$
$c_{21} = 16$$/units/year, $w_{21}^{M} = 65$$/unit, $\epsilon = 1$ $c_{22} = 21$$/ unit, $w_{22}^{M} = 80$$/unit, $\epsilon = 1$
$L_{21} = 1700$$ /production cycle $L_{22} = 1750$$ /production cycle
$AM_{21} = 205$$/setup, $k_{21} = 6.0$$/unit $AM_{22} = 210$$/setup, $k_{22} = 8.0$$/unit
$h_{21}^{P} = 3.25$$/unit/year, $\beta_{21} = 10$% $h_{22}^{P} = 4.0$$/unit/year, $\beta_{22} = 10$%,
Parameters of Product 1 Product 2
Manufacturer 1 $p_{11} = 700$ units/year, $\alpha_{11} = 0.01$ $p_{12} = 740$ units/year, $\alpha_{12} = 0.03$,
$c_{11} = 15$$/units/year, $w_{11}^{M} = 55$$/unit, $\epsilon = 1$ $c_{12} = 17$$/ unit, $w_{12}^{M} = 75$$/unit, $\epsilon = 1$
$L_{11} = 2000$$ /production cycle $L_{12} = 1900$$ /production,
$AM_{11} = 200$$/setup, $k_{11} = 5.1$$/unit $AM_{12} = 208$$/setup, $k_{12} = 4.5$$/unit
$h_{11}^{P} = 2.75$$/unit/year, $\beta_{11} = 15$% $h_{12}^{P} = 3.75$$/unit/year, $\beta_{12} = 15$%
Manufacturer 2 $p_{21} = 900$ units/year, $\alpha_{21} = 0.04$ $p_{22} = 1050$ units/year, $\alpha_{22} = 0.03$
$c_{21} = 16$$/units/year, $w_{21}^{M} = 65$$/unit, $\epsilon = 1$ $c_{22} = 21$$/ unit, $w_{22}^{M} = 80$$/unit, $\epsilon = 1$
$L_{21} = 1700$$ /production cycle $L_{22} = 1750$$ /production cycle
$AM_{21} = 205$$/setup, $k_{21} = 6.0$$/unit $AM_{22} = 210$$/setup, $k_{22} = 8.0$$/unit
$h_{21}^{P} = 3.25$$/unit/year, $\beta_{21} = 10$% $h_{22}^{P} = 4.0$$/unit/year, $\beta_{22} = 10$%,
Table 3.  Values of parameters for distribution centers
Parameters of Product 1 Product 2
Distribution Centre 1 $AD_{11} = 190$ $/setup, $w_{11}^{D} = 75$$/units $AD_{12} = 200$ $/setup, $w_{12}^{D} = 95$$/units
$DEM_{11}^{D} = 170 $ units/year $DEM_{12}^{D} = 180 $ units/year
$h_{11}^{D} = 3.75$$/unit/year $h_{12}^{D} = 4.75$$/unit/year
$k_{11} = 2.25$$/unit, $\beta_{11} = 10%$ $k_{12} = 2.85$$/unit, $\beta_{12} = 10%$
Distribution Centre 2 $AD_{21} = 180$ $/setup, $w_{21}^{D} = 85$$/units $AD_{22} = 195$ $/setup, $w_{22}^{D} = 100$$/units
$DEM_{21}^{D} = 250 $ units/year $DEM_{22}^{D} = 290 $ units/year
$h_{21}^{D} = 4.25$$/unit/year $h_{22}^{D} = 5.0$$/unit/year
$k_{21} = 2.55$$/unit, $\beta_{21} = 5%$ $k_{22} = 3.0$$/unit, $\beta_{22} = 5%$
Parameters of Product 1 Product 2
Distribution Centre 1 $AD_{11} = 190$ $/setup, $w_{11}^{D} = 75$$/units $AD_{12} = 200$ $/setup, $w_{12}^{D} = 95$$/units
$DEM_{11}^{D} = 170 $ units/year $DEM_{12}^{D} = 180 $ units/year
$h_{11}^{D} = 3.75$$/unit/year $h_{12}^{D} = 4.75$$/unit/year
$k_{11} = 2.25$$/unit, $\beta_{11} = 10%$ $k_{12} = 2.85$$/unit, $\beta_{12} = 10%$
Distribution Centre 2 $AD_{21} = 180$ $/setup, $w_{21}^{D} = 85$$/units $AD_{22} = 195$ $/setup, $w_{22}^{D} = 100$$/units
$DEM_{21}^{D} = 250 $ units/year $DEM_{22}^{D} = 290 $ units/year
$h_{21}^{D} = 4.25$$/unit/year $h_{22}^{D} = 5.0$$/unit/year
$k_{21} = 2.55$$/unit, $\beta_{21} = 5%$ $k_{22} = 3.0$$/unit, $\beta_{22} = 5%$
Table 4.  Values of parameters for retailers
Parameters of Product 1 Product 2
Retailer 1 $w_{11}^{R} = 115$ $/unit, $k_{11} = 6.6$, $\beta_{1} = 20$% $w_{12}^{R} = 135$ $/unit, $k_{12} = 6.9$, $\beta_{1} = 20%$
$AR_{11} = 130 $ $/setup, $h_{11 }^{R} = 5.75$$/unit/year $AR_{12} = 140 $ $/setup, $h_{12 }^{R} = 6.75$$/unit/year
$DEM_{11}^{R} = 220$units/year, $\tau_{11} = 44$units/year $DEM_{12}^{R} = 230$units/year, $\tau_{12} = 46$units/year
Retailer 2 $w_{21}^{R} = 125$ $/unit, $k_{21} = 9.0$, $\beta_{2} = 25$% $w_{22}^{R} = 140$ $/unit, $k_{22} = 10.2$, $\beta_{2} = 25$%
$AR_{21} = 115 $ $/setup, $h_{21 }^{R} = 6.25$$/unit/year $AR_{22} = 118 $ $/setup, $h_{22 }^{R} = 7.0$$/unit/year
$DEM_{21}^{R} = 300$units/year, $\tau_{21} = 60$units/year $DEM_{22}^{R} = 340$units/year, $\tau_{22} = 68$units/year
Parameters of Product 1 Product 2
Retailer 1 $w_{11}^{R} = 115$ $/unit, $k_{11} = 6.6$, $\beta_{1} = 20$% $w_{12}^{R} = 135$ $/unit, $k_{12} = 6.9$, $\beta_{1} = 20%$
$AR_{11} = 130 $ $/setup, $h_{11 }^{R} = 5.75$$/unit/year $AR_{12} = 140 $ $/setup, $h_{12 }^{R} = 6.75$$/unit/year
$DEM_{11}^{R} = 220$units/year, $\tau_{11} = 44$units/year $DEM_{12}^{R} = 230$units/year, $\tau_{12} = 46$units/year
Retailer 2 $w_{21}^{R} = 125$ $/unit, $k_{21} = 9.0$, $\beta_{2} = 25$% $w_{22}^{R} = 140$ $/unit, $k_{22} = 10.2$, $\beta_{2} = 25$%
$AR_{21} = 115 $ $/setup, $h_{21 }^{R} = 6.25$$/unit/year $AR_{22} = 118 $ $/setup, $h_{22 }^{R} = 7.0$$/unit/year
$DEM_{21}^{R} = 300$units/year, $\tau_{21} = 60$units/year $DEM_{22}^{R} = 340$units/year, $\tau_{22} = 68$units/year
Table 5.  Optimal solution for Example 1
$\rho$ APM APD APR $\mu_{j}$ $q_{11}^{M}$ $q_{12}^{M}$ $q_{21}^{M}$ $q_{22}^{M}$
0 60051.71 21632.00 43070.27 124753.98 264.86 246.98 304.93 305.42
1 59879.90 21860.12 57593.64 139333.66 320.20 292.53 366.26 359.89
2 59784.84 21956.74 62462.60 144204.18 347.51 314.00 396.29 385.45
3 59724.75 22010.88 64902.89 146638.52 363.97 326.60 414.30 400.41
4 59683.00 22045.53 66367.86 148096.40 375.01 334.90 426.33 410.25
5 59652.02 22069.56 67343.81 149065.39 382.93 340.79 434.95 417.22
6 59627.88 22087.13 68039.75 149754.76 388.90 345.19 441.43 422.42
7 59608.36 22100.49 68560.39 150269.24 393.56 348.60 446.48 426.45
45 59390.16 22159.61 71340.32 152890.10 424.16 370.47 479.48 452.19
46 59386.77 22159.26 71346.60 152892.63 424.31 370.57 479.65 452.32
47 59383.40 22158.88 71352.30 152894.58 424.46 370.68 479.80 452.44
48 59380.05 22158.49 71357.47 152896.01 424.60 370.78 479.95 452.56
49 59376.72 22158.08 71362.12 152896.92 424.74 370.87 480.10 452.67
50 59373.40 22157.66 71366.31 152897.37 424.87 370.96 480.24 452.77
51 59370.11 22157.23 71370.04 152897.37 424.99 371.05 480.37 452.88
52 59366.83 22156.78 71373.35 152896.96 425.11 371.13 480.50 452.98
$\rho$ APM APD APR $\mu_{j}$ $q_{11}^{M}$ $q_{12}^{M}$ $q_{21}^{M}$ $q_{22}^{M}$
0 60051.71 21632.00 43070.27 124753.98 264.86 246.98 304.93 305.42
1 59879.90 21860.12 57593.64 139333.66 320.20 292.53 366.26 359.89
2 59784.84 21956.74 62462.60 144204.18 347.51 314.00 396.29 385.45
3 59724.75 22010.88 64902.89 146638.52 363.97 326.60 414.30 400.41
4 59683.00 22045.53 66367.86 148096.40 375.01 334.90 426.33 410.25
5 59652.02 22069.56 67343.81 149065.39 382.93 340.79 434.95 417.22
6 59627.88 22087.13 68039.75 149754.76 388.90 345.19 441.43 422.42
7 59608.36 22100.49 68560.39 150269.24 393.56 348.60 446.48 426.45
45 59390.16 22159.61 71340.32 152890.10 424.16 370.47 479.48 452.19
46 59386.77 22159.26 71346.60 152892.63 424.31 370.57 479.65 452.32
47 59383.40 22158.88 71352.30 152894.58 424.46 370.68 479.80 452.44
48 59380.05 22158.49 71357.47 152896.01 424.60 370.78 479.95 452.56
49 59376.72 22158.08 71362.12 152896.92 424.74 370.87 480.10 452.67
50 59373.40 22157.66 71366.31 152897.37 424.87 370.96 480.24 452.77
51 59370.11 22157.23 71370.04 152897.37 424.99 371.05 480.37 452.88
52 59366.83 22156.78 71373.35 152896.96 425.11 371.13 480.50 452.98
Table 6.  Values of parameters for manufacturers
Parameters of Product 1 Product 2
Manufacturer 1 $p_{11} = 500$ units/year, $\alpha_{11} = 0.02$ $p_{12} = 560$ units/year, $\alpha_{12} = 0.01$,
$c_{11} = 12$$/units/year, $w_{11}^{M} = 43$$/unit, $\epsilon = 1$ $c_{12} = 10$$/ unit, $w_{12}^{M} = 62$$/unit, $\epsilon = 1$
$L_{11} = 1500$$ /production cycle $L_{12} = 1450$$ /production,
$AM_{11} = 160$$/setup, $k_{11} = 4.3$$/unit $AM_{12} = 176$$/setup, $k_{12} = 3.9$$/unit
$h_{11}^{P} = 2.05$$/unit/year, $\beta_{11} = 8$% $h_{12}^{P} = 2.94$$/unit/year, $\beta_{12} = 8$%
Manufacturer 2 $p_{21} = 870$ units/year, $\alpha_{21} = 0.022$ $p_{22} = 920$ units/year, $\alpha_{22} = 0.015$
$c_{21} = 18$$/units/year, $w_{21}^{M} = 53$$/unit, $\epsilon = 1$ $c_{22} = 19$$/ unit, $w_{22}^{M} = 66$$/unit, $\epsilon = 1$
$L_{21} = 1530$$ /production cycle $L_{22} = 1550$$ /production cycle
$AM_{21} = 185$$/setup, $k_{21} = 4.9$$/unit $AM_{22} = 190$$/setup, $k_{22} = 6.8$$/unit
$h_{21}^{P} = 3.05$$/unit/year, $\beta_{21} = 5$% $h_{22}^{P} = 3.2$$/unit/year, $\beta_{22} = 5$%,
Parameters of Product 1 Product 2
Manufacturer 1 $p_{11} = 500$ units/year, $\alpha_{11} = 0.02$ $p_{12} = 560$ units/year, $\alpha_{12} = 0.01$,
$c_{11} = 12$$/units/year, $w_{11}^{M} = 43$$/unit, $\epsilon = 1$ $c_{12} = 10$$/ unit, $w_{12}^{M} = 62$$/unit, $\epsilon = 1$
$L_{11} = 1500$$ /production cycle $L_{12} = 1450$$ /production,
$AM_{11} = 160$$/setup, $k_{11} = 4.3$$/unit $AM_{12} = 176$$/setup, $k_{12} = 3.9$$/unit
$h_{11}^{P} = 2.05$$/unit/year, $\beta_{11} = 8$% $h_{12}^{P} = 2.94$$/unit/year, $\beta_{12} = 8$%
Manufacturer 2 $p_{21} = 870$ units/year, $\alpha_{21} = 0.022$ $p_{22} = 920$ units/year, $\alpha_{22} = 0.015$
$c_{21} = 18$$/units/year, $w_{21}^{M} = 53$$/unit, $\epsilon = 1$ $c_{22} = 19$$/ unit, $w_{22}^{M} = 66$$/unit, $\epsilon = 1$
$L_{21} = 1530$$ /production cycle $L_{22} = 1550$$ /production cycle
$AM_{21} = 185$$/setup, $k_{21} = 4.9$$/unit $AM_{22} = 190$$/setup, $k_{22} = 6.8$$/unit
$h_{21}^{P} = 3.05$$/unit/year, $\beta_{21} = 5$% $h_{22}^{P} = 3.2$$/unit/year, $\beta_{22} = 5$%,
Table 7.  Values of parameters for distribution centers
Parameters of Product 1 Product 2
Distribution Centre 1 $AD_{11} = 150$ $/setup, $w_{11}^{D} = 65$$/units $AD_{12} = 1650$ $/setup, $w_{12}^{D} = 84$$/units
$DEM_{11}^{D} = 150 $ units/year $DEM_{12}^{D} = 164 $ units/year
$h_{11}^{D} = 2.95$$/unit/year $h_{12}^{D} = 3.5$$/unit/year
$k_{11} = 1.83$$/unit, $\beta_{11} = 5%$ $k_{12} = 2.05$$/unit, $\beta_{12} = 5%$
Distribution Centre 2 $AD_{21} = 174$ $/setup, $w_{21}^{D} = 72$$/units $AD_{22} = 182$ $/setup, $w_{22}^{D} = 84$$/units
$DEM_{21}^{D} = 236 $ units/year $DEM_{22}^{D} = 279 $ units/year
$h_{21}^{D} = 3.10$$/unit/year $h_{22}^{D} = 3.90$$/unit/year
$k_{21} = 2.10$$/unit, $\beta_{21} = 2%$ $k_{22} = 2.75$$/unit, $\beta_{22} = 2%$
Parameters of Product 1 Product 2
Distribution Centre 1 $AD_{11} = 150$ $/setup, $w_{11}^{D} = 65$$/units $AD_{12} = 1650$ $/setup, $w_{12}^{D} = 84$$/units
$DEM_{11}^{D} = 150 $ units/year $DEM_{12}^{D} = 164 $ units/year
$h_{11}^{D} = 2.95$$/unit/year $h_{12}^{D} = 3.5$$/unit/year
$k_{11} = 1.83$$/unit, $\beta_{11} = 5%$ $k_{12} = 2.05$$/unit, $\beta_{12} = 5%$
Distribution Centre 2 $AD_{21} = 174$ $/setup, $w_{21}^{D} = 72$$/units $AD_{22} = 182$ $/setup, $w_{22}^{D} = 84$$/units
$DEM_{21}^{D} = 236 $ units/year $DEM_{22}^{D} = 279 $ units/year
$h_{21}^{D} = 3.10$$/unit/year $h_{22}^{D} = 3.90$$/unit/year
$k_{21} = 2.10$$/unit, $\beta_{21} = 2%$ $k_{22} = 2.75$$/unit, $\beta_{22} = 2%$
Table 8.  Values of parameters for retailers
Parameters of Product 1 Product 2
Retailer 1 $w_{11}^{R} = 94$ $/unit, $k_{11} = 5.10$, $\beta_{1} = 10$% $w_{12}^{R} = 119$ $/unit, $k_{12} = 5.5$, $\beta_{1} = 10$%
$AR_{11} = 110 $ $/setup, $h_{11 }^{R} = 3.8$$/unit/year $AR_{12} = 115 $ $/setup, $h_{12 }^{R} = 5.5$$/unit/year
$DEM_{11}^{R} = 180$units/year, $\tau_{11} = 35$units/year $DEM_{12}^{R} = 190$units/year, $\tau_{12} = 37$units/year
Retailer 2 $w_{21}^{R} = 107$ $/unit, $k_{21} = 7.9$, $\beta_{2} = 15$% $w_{22}^{R} = 126$ $/unit, $k_{22} = 8.6$, $\beta_{2} = 15$%
$AR_{21} = 102 $ $/setup, $h_{21 }^{R} = 5.2$$/unit/year $AR_{22} = 105 $ $/setup, $h_{22 }^{R} = 6.1$$/unit/year
$DEM_{21}^{R} = 265$units/year, $\tau_{21} = 49$units/year $DEM_{22}^{R} = 310$units/year, $\tau_{22} = 56$units/year
Parameters of Product 1 Product 2
Retailer 1 $w_{11}^{R} = 94$ $/unit, $k_{11} = 5.10$, $\beta_{1} = 10$% $w_{12}^{R} = 119$ $/unit, $k_{12} = 5.5$, $\beta_{1} = 10$%
$AR_{11} = 110 $ $/setup, $h_{11 }^{R} = 3.8$$/unit/year $AR_{12} = 115 $ $/setup, $h_{12 }^{R} = 5.5$$/unit/year
$DEM_{11}^{R} = 180$units/year, $\tau_{11} = 35$units/year $DEM_{12}^{R} = 190$units/year, $\tau_{12} = 37$units/year
Retailer 2 $w_{21}^{R} = 107$ $/unit, $k_{21} = 7.9$, $\beta_{2} = 15$% $w_{22}^{R} = 126$ $/unit, $k_{22} = 8.6$, $\beta_{2} = 15$%
$AR_{21} = 102 $ $/setup, $h_{21 }^{R} = 5.2$$/unit/year $AR_{22} = 105 $ $/setup, $h_{22 }^{R} = 6.1$$/unit/year
$DEM_{21}^{R} = 265$units/year, $\tau_{21} = 49$units/year $DEM_{22}^{R} = 310$units/year, $\tau_{22} = 56$units/year
Table 9.  Optimal solution for Example 2
$\rho$ APM APD APR $\mu$ $q_{11}^{M}$ $q_{12}^{M}$ $q_{21}^{M}$ $q_{22}^{M}$
0 45832.63 18356.66 33700.57 97889.86 222.13 209.85 262.20 297.45
1 45763.73 18467.95 43950.02 108181.70 253.32 239.95 294.82 342.42
2 45727.88 18510.28 47378.86 111617.02 266.67 252.99 308.54 362.42
3 45706.07 18532.68 49095.31 113334.07 274.11 260.32 316.12 373.8
4 45691.26 18546.52 50125.18 114362.96 278.87 265.02 320.94 381.14
5 45680.40 18555.88 50811.22 115047.50 282.17 268.29 324.27 386.28
6 45671.99 18562.61 51300.58 115535.17 284.60 270.70 326.71 390.08
7 45665.20 18567.64 51666.90 115899.74 286.46 272.55 328.58 393.01
57 45566.23 18586.60 53725.82 117878.66 298.47 284.52 340.54 412.12
58 45564.88 18586.41 53728.36 117879.65 298.50 284.56 340.57 412.18
59 45563.53 18586.21 53730.69 117880.44 298.54 284.59 340.60 412.23
60 45562.19 18586.00 53732.84 117881.03 298.57 284.62 340.64 412.28
61 45560.85 18585.80 53734.80 117881.45 298.60 284.66 340.67 412.33
62 45559.51 18585.59 53736.58 117881.69 298.63 284.69 340.70 412.38
63 45558.18 18585.38 53738.20 117881.76 298.66 284.72 340.73 412.43
64 45556.85 18585.17 53739.66 117881.68 298.69 284.74 340.75 412.48
$\rho$ APM APD APR $\mu$ $q_{11}^{M}$ $q_{12}^{M}$ $q_{21}^{M}$ $q_{22}^{M}$
0 45832.63 18356.66 33700.57 97889.86 222.13 209.85 262.20 297.45
1 45763.73 18467.95 43950.02 108181.70 253.32 239.95 294.82 342.42
2 45727.88 18510.28 47378.86 111617.02 266.67 252.99 308.54 362.42
3 45706.07 18532.68 49095.31 113334.07 274.11 260.32 316.12 373.8
4 45691.26 18546.52 50125.18 114362.96 278.87 265.02 320.94 381.14
5 45680.40 18555.88 50811.22 115047.50 282.17 268.29 324.27 386.28
6 45671.99 18562.61 51300.58 115535.17 284.60 270.70 326.71 390.08
7 45665.20 18567.64 51666.90 115899.74 286.46 272.55 328.58 393.01
57 45566.23 18586.60 53725.82 117878.66 298.47 284.52 340.54 412.12
58 45564.88 18586.41 53728.36 117879.65 298.50 284.56 340.57 412.18
59 45563.53 18586.21 53730.69 117880.44 298.54 284.59 340.60 412.23
60 45562.19 18586.00 53732.84 117881.03 298.57 284.62 340.64 412.28
61 45560.85 18585.80 53734.80 117881.45 298.60 284.66 340.67 412.33
62 45559.51 18585.59 53736.58 117881.69 298.63 284.69 340.70 412.38
63 45558.18 18585.38 53738.20 117881.76 298.66 284.72 340.73 412.43
64 45556.85 18585.17 53739.66 117881.68 298.69 284.74 340.75 412.48
Table 10.  Sensitivity analysis of numerical example 1
Change $\rho$ $\mu$ APM APD APR $q_{11}^{M}$ $q_{12}^{M}$ $q_{21}^{M}$ $q_{22}^{M}$
-60% 51 139800.21 59899.63 22282.85 57617.73 320.20 292.53 366.26 1058.53
-40% 51 153314.43 58598.34 22828.59 71887.49 424.99 371.05 480.37 1758.29
-20% 51 153048.94 59343.71 22268.86 71436.37 424.99 371.05 480.37 620.22
$h_{22}^{P}$ = 4.0 51 152897.37 59370.11 22157.23 71370.04 424.99 371.05 480.37 452.88
20% 50 152778.80 59350.99 22092.75 71335.05 424.87 370.96 480.24 373.87
40% 50 152678.01 59316.37 22045.70 71315.95 424.87 370.96 480.24 325.64
60% 50 152588.82 59278.03 22008.07 71302.73 424.87 370.96 480.24 292.27
-60% 50 152984.89 59546.27 22100.25 71338.37 424.87 370.96 480.24 382.25
-40% 50 152954.03 59484.25 22121.56 71348.22 424.87 370.96 480.24 407.12
-20% 50 152924.95 59426.92 22140.53 71357.50 424.87 370.96 480.24 430.55
$AM_{22}$ = 210 51 152897.37 59370.11 22157.23 71370.04 424.99 371.05 480.37 452.88
20% 51 152871.09 59319.76 22172.90 71378.44 424.99 371.05 480.37 474.06
40% 51 152845.94 59272.08 22187.38 71386.47 424.99 371.05 480.37 494.34
60% 51 152821.78 59226.72 22200.86 71394.20 424.99 371.05 480.37 513.82
-60% 51 155297.37 45450.11 38477.23 71370.04 424.99 371.05 480.37 452.88
-40% 51 154497.37 50090.11 33037.23 71370.04 424.99 371.05 480.37 452.88
-20% 51 153697.37 54730.11 27597.23 71370.04 424.99 371.05 480.37 452.88
$w_{22}^{M}$ = 80 51 152897.37 59370.11 22157.23 71370.04 424.99 371.05 480.37 452.88
20% 51 152097.37 64010.11 16717.23 71370.04 424.99 371.05 480.37 452.88
40% 51 151297.37 68650.11 11277.23 71370.04 424.99 371.05 480.37 452.88
60% 51 150497.37 73290.11 5837.23 71370.04 424.99 371.05 480.37 452.88
-60% 52 152921.92 59391.79 22156.78 71373.35 425.11 371.13 480.50 452.98
-40% 51 152913.69 59386.43 22157.23 71370.04 424.99 371.05 480.37 452.88
-20% 51 152905.53 59378.27 22157.23 71370.04 424.99 371.05 480.37 452.88
$k_{22}$ = 8.0 51 152897.37 59370.11 22157.23 71370.04 424.99 371.05 480.37 452.88
20% 50 152889.37 59365.40 22157.66 71366.31 424.87 370.96 480.24 452.77
40% 50 152881.37 59357.40 22157.66 71366.31 424.87 370.96 480.24 452.77
60% 49 152873.40 59353.20 22158.08 71362.12 424.74 370.87 480.10 452.67
-60% 51 144762.14 59159.30 22857.50 62745.34 347.51 314.00 396.29 2147.77
-40% 51 153112.57 59319.01 22321.38 71472.18 424.99 371.05 480.37 710.55
-20% 51 152969.73 59374.85 22200.75 71394.13 424.99 371.05 480.37 513.66
$p_{22}$ = 1050 51 152897.37 59370.11 22157.23 71370.04 424.99 371.05 480.37 452.88
20% 50 152851.11 59362.66 22134.13 71354.32 424.87 370.96 480.24 422.51
40% 50 152817.83 59351.56 22119.19 71347.09 424.87 370.96 480.24 404.27
60% 50 152792.05 59341.01 22108.80 71342.25 424.87 370.96 480.24 392.04
-60% 50 152768.88 59461.52 21953.66 71353.71 424.87 288.55 480.24 452.77
-40% 50 152808.30 59440.42 22010.95 71356.93 424.87 309.65 480.24 452.77
-20% 50 152850.83 59412.31 22077.53 71360.99 424.87 336.18 480.24 452.77
$h_{12}^{D}$ = 4.75 51 152897.37 59370.11 22157.23 71370.04 424.99 371.05 480.37 452.88
20% 51 152949.33 59313.00 22258.88 71377.46 424.99 419.51 480.37 452.88
40% 51 153009.19 59220.99 22399.36 71388.84 424.99 493.82 480.37 452.88
60% 51 153082.24 59044.14 22628.41 71409.70 424.99 630.01 480.37 452.88
-60% 50 152979.22 59446.72 22176.51 71355.99 424.87 303.47 480.24 452.77
-40% 50 152950.06 59421.66 22168.73 71359.66 424.87 327.52 480.24 452.77
-20% 50 152922.90 59397.20 22162.61 71363.09 424.87 349.91 480.24 452.77
$AD_{12}$ = 200 51 152897.37 59370.11 22157.23 71370.04 424.99 371.05 480.37 452.88
20% 51 152873.23 59346.99 22153.15 71373.09 424.99 390.97 480.37 452.88
40% 51 152850.25 59324.52 22149.74 71375.99 424.99 409.92 480.37 452.88
60% 51 152828.3 59302.68 22146.85 71378.77 424.99 428.04 480.37 452.88
-60% 51 152897.37 59370.11 9047.23 84480.04 424.99 371.05 480.37 452.
-40% 51 152897.37 59370.11 13417.23 80110.04 424.99 371.05 480.37 452.88
-20% 51 152897.37 59370.11 17787.23 75740.04 424.99 371.05 480.37 452.88
$w_{12}^{D}$ = 95 51 152897.37 59370.11 22157.23 71370.04 424.99 371.05 480.37 452.88
20% 51 152897.37 59370.11 26527.23 67000.04 424.99 371.05 480.37 452.88
40% 51 152897.37 59370.11 30897.23 62630.04 424.99 371.05 480.37 452.88
60% 51 152897.37 59370.11 35267.23 58260.04 424.99 371.05 480.37 452.88
-60% 51 152906.10 59370.11 22165.95 71370.04 424.99 371.05 480.37 452.88
-40% 51 152903.19 59370.11 22163.04 71370.04 424.99 371.05 480.37 452.88
-20% 51 152900.28 59370.11 22160.13 71370.04 424.99 371.05 480.37 452.88
$k_{12}$ = 2.85 51 152897.37 59370.11 22157.23 71370.04 424.99 371.05 480.37 452.88
20% 50 152894.52 59373.40 22154.81 71366.31 424.87 370.96 480.24 452.77
40% 50 152891.67 59373.40 22151.96 71366.31 424.87 370.96 480.24 452.77
60% 50 152888.82 59373.40 22149.11 71366.31 424.87 370.96 480.24 452.77
-60% 50 152734.53 59481.70 22071.37 71181.46 424.87 370.96 480.24 351.04
-40% 50 152784.43 59456.07 22095.57 71232.80 424.87 370.96 480.24 377.00
-20% 50 152838.33 59421.60 22123.72 71293.02 424.87 370.96 480.24 409.72
$h_{22}^{R}$ = 7.0 51 152897.37 59370.11 22157.23 71370.04 424.99 371.05 480.37 452.88
20% 51 152963.44 59298.54 22200.46 71464.44 424.99 371.05 480.37 513.24
40% 51 153039.80 59181.53 22260.64 71597.62 424.99 371.05 480.37 606.70
60% 51 153133.59 58950.93 22360.44 71822.22 424.99 371.05 480.37 781.50
-60% 51 152982.69 59273.13 22214.42 71495.15 424.99 371.05 480.37 533.98
-40% 51 152952.65 59311.79 22192.94 71447.93 424.99 371.05 480.37 502.31
-20% 51 152924.30 59343.55 22174.07 71406.68 424.99 371.05 480.37 475.68
$AR_{22}$ = 118 51 152897.37 59370.11 22157.23 71370.04 424.99 371.05 480.37 452.88
20% 50 152871.68 59395.91 22142.43 71333.34 424.87 370.96 480.24 432.98
40% 50 152847.07 59415.22 22128.52 71303.34 424.87 370.96 480.24 415.57
60% 50 152823.41 59431.94 22115.69 71275.79 424.87 370.96 480.24 400.11
-60% 45 118742.28 59390.16 22159.61 37192.50 424.16 370.47 479.48 452.19
-40% 47 130125.92 59383.40 22158.88 48583.64 424.46 370.68 479.80 452.44
-20% 49 141511.00 59376.72 22158.08 59976.20 424.74 370.87 480.10 452.67
$w_{22}^{R}$ = 140 51 152897.37 59370.11 22157.23 71370.04 424.99 371.05 480.37 452.88
20% 52 164285.03 59366.83 22156.78 82761.42 425.11 371.13 480.50 452.98
40% 54 175673.72 59360.32 22155.84 94157.55 425.34 371.29 480.75 453.16
60% 55 187063.41 59357.09 22155.36 105550.96 425.45 371.37 480.87 453.25
-60% 54 152977.57 59360.32 22155.84 71461.41 425.34 371.29 480.75 453.16
-40% 53 152950.20 59363.57 22156.32 71430.32 425.23 371.21 480.63 453.07
-20% 52 152923.48 59366.83 22156.78 71399.87 425.11 371.13 480.50 452.98
$k_{22}$ = 10.2 51 152897.37 59370.11 22157.23 71370.04 424.99 371.05 480.37 452.88
20% 49 152871.93 59376.72 22158.08 71337.13 424.74 370.87 480.10 452.67
40% 48 152847.05 59380.05 22158.49 71308.51 424.60 370.78 479.95 452.56
60% 47 152822.67 59383.40 22158.88 71280.39 424.46 370.68 479.80 452.44
Change $\rho$ $\mu$ APM APD APR $q_{11}^{M}$ $q_{12}^{M}$ $q_{21}^{M}$ $q_{22}^{M}$
-60% 51 139800.21 59899.63 22282.85 57617.73 320.20 292.53 366.26 1058.53
-40% 51 153314.43 58598.34 22828.59 71887.49 424.99 371.05 480.37 1758.29
-20% 51 153048.94 59343.71 22268.86 71436.37 424.99 371.05 480.37 620.22
$h_{22}^{P}$ = 4.0 51 152897.37 59370.11 22157.23 71370.04 424.99 371.05 480.37 452.88
20% 50 152778.80 59350.99 22092.75 71335.05 424.87 370.96 480.24 373.87
40% 50 152678.01 59316.37 22045.70 71315.95 424.87 370.96 480.24 325.64
60% 50 152588.82 59278.03 22008.07 71302.73 424.87 370.96 480.24 292.27
-60% 50 152984.89 59546.27 22100.25 71338.37 424.87 370.96 480.24 382.25
-40% 50 152954.03 59484.25 22121.56 71348.22 424.87 370.96 480.24 407.12
-20% 50 152924.95 59426.92 22140.53 71357.50 424.87 370.96 480.24 430.55
$AM_{22}$ = 210 51 152897.37 59370.11 22157.23 71370.04 424.99 371.05 480.37 452.88
20% 51 152871.09 59319.76 22172.90 71378.44 424.99 371.05 480.37 474.06
40% 51 152845.94 59272.08 22187.38 71386.47 424.99 371.05 480.37 494.34
60% 51 152821.78 59226.72 22200.86 71394.20 424.99 371.05 480.37 513.82
-60% 51 155297.37 45450.11 38477.23 71370.04 424.99 371.05 480.37 452.88
-40% 51 154497.37 50090.11 33037.23 71370.04 424.99 371.05 480.37 452.88
-20% 51 153697.37 54730.11 27597.23 71370.04 424.99 371.05 480.37 452.88
$w_{22}^{M}$ = 80 51 152897.37 59370.11 22157.23 71370.04 424.99 371.05 480.37 452.88
20% 51 152097.37 64010.11 16717.23 71370.04 424.99 371.05 480.37 452.88
40% 51 151297.37 68650.11 11277.23 71370.04 424.99 371.05 480.37 452.88
60% 51 150497.37 73290.11 5837.23 71370.04 424.99 371.05 480.37 452.88
-60% 52 152921.92 59391.79 22156.78 71373.35 425.11 371.13 480.50 452.98
-40% 51 152913.69 59386.43 22157.23 71370.04 424.99 371.05 480.37 452.88
-20% 51 152905.53 59378.27 22157.23 71370.04 424.99 371.05 480.37 452.88
$k_{22}$ = 8.0 51 152897.37 59370.11 22157.23 71370.04 424.99 371.05 480.37 452.88
20% 50 152889.37 59365.40 22157.66 71366.31 424.87 370.96 480.24 452.77
40% 50 152881.37 59357.40 22157.66 71366.31 424.87 370.96 480.24 452.77
60% 49 152873.40 59353.20 22158.08 71362.12 424.74 370.87 480.10 452.67
-60% 51 144762.14 59159.30 22857.50 62745.34 347.51 314.00 396.29 2147.77
-40% 51 153112.57 59319.01 22321.38 71472.18 424.99 371.05 480.37 710.55
-20% 51 152969.73 59374.85 22200.75 71394.13 424.99 371.05 480.37 513.66
$p_{22}$ = 1050 51 152897.37 59370.11 22157.23 71370.04 424.99 371.05 480.37 452.88
20% 50 152851.11 59362.66 22134.13 71354.32 424.87 370.96 480.24 422.51
40% 50 152817.83 59351.56 22119.19 71347.09 424.87 370.96 480.24 404.27
60% 50 152792.05 59341.01 22108.80 71342.25 424.87 370.96 480.24 392.04
-60% 50 152768.88 59461.52 21953.66 71353.71 424.87 288.55 480.24 452.77
-40% 50 152808.30 59440.42 22010.95 71356.93 424.87 309.65 480.24 452.77
-20% 50 152850.83 59412.31 22077.53 71360.99 424.87 336.18 480.24 452.77
$h_{12}^{D}$ = 4.75 51 152897.37 59370.11 22157.23 71370.04 424.99 371.05 480.37 452.88
20% 51 152949.33 59313.00 22258.88 71377.46 424.99 419.51 480.37 452.88
40% 51 153009.19 59220.99 22399.36 71388.84 424.99 493.82 480.37 452.88
60% 51 153082.24 59044.14 22628.41 71409.70 424.99 630.01 480.37 452.88
-60% 50 152979.22 59446.72 22176.51 71355.99 424.87 303.47 480.24 452.77
-40% 50 152950.06 59421.66 22168.73 71359.66 424.87 327.52 480.24 452.77
-20% 50 152922.90 59397.20 22162.61 71363.09 424.87 349.91 480.24 452.77
$AD_{12}$ = 200 51 152897.37 59370.11 22157.23 71370.04 424.99 371.05 480.37 452.88
20% 51 152873.23 59346.99 22153.15 71373.09 424.99 390.97 480.37 452.88
40% 51 152850.25 59324.52 22149.74 71375.99 424.99 409.92 480.37 452.88
60% 51 152828.3 59302.68 22146.85 71378.77 424.99 428.04 480.37 452.88
-60% 51 152897.37 59370.11 9047.23 84480.04 424.99 371.05 480.37 452.
-40% 51 152897.37 59370.11 13417.23 80110.04 424.99 371.05 480.37 452.88
-20% 51 152897.37 59370.11 17787.23 75740.04 424.99 371.05 480.37 452.88
$w_{12}^{D}$ = 95 51 152897.37 59370.11 22157.23 71370.04 424.99 371.05 480.37 452.88
20% 51 152897.37 59370.11 26527.23 67000.04 424.99 371.05 480.37 452.88
40% 51 152897.37 59370.11 30897.23 62630.04 424.99 371.05 480.37 452.88
60% 51 152897.37 59370.11 35267.23 58260.04 424.99 371.05 480.37 452.88
-60% 51 152906.10 59370.11 22165.95 71370.04 424.99 371.05 480.37 452.88
-40% 51 152903.19 59370.11 22163.04 71370.04 424.99 371.05 480.37 452.88
-20% 51 152900.28 59370.11 22160.13 71370.04 424.99 371.05 480.37 452.88
$k_{12}$ = 2.85 51 152897.37 59370.11 22157.23 71370.04 424.99 371.05 480.37 452.88
20% 50 152894.52 59373.40 22154.81 71366.31 424.87 370.96 480.24 452.77
40% 50 152891.67 59373.40 22151.96 71366.31 424.87 370.96 480.24 452.77
60% 50 152888.82 59373.40 22149.11 71366.31 424.87 370.96 480.24 452.77
-60% 50 152734.53 59481.70 22071.37 71181.46 424.87 370.96 480.24 351.04
-40% 50 152784.43 59456.07 22095.57 71232.80 424.87 370.96 480.24 377.00
-20% 50 152838.33 59421.60 22123.72 71293.02 424.87 370.96 480.24 409.72
$h_{22}^{R}$ = 7.0 51 152897.37 59370.11 22157.23 71370.04 424.99 371.05 480.37 452.88
20% 51 152963.44 59298.54 22200.46 71464.44 424.99 371.05 480.37 513.24
40% 51 153039.80 59181.53 22260.64 71597.62 424.99 371.05 480.37 606.70
60% 51 153133.59 58950.93 22360.44 71822.22 424.99 371.05 480.37 781.50
-60% 51 152982.69 59273.13 22214.42 71495.15 424.99 371.05 480.37 533.98
-40% 51 152952.65 59311.79 22192.94 71447.93 424.99 371.05 480.37 502.31
-20% 51 152924.30 59343.55 22174.07 71406.68 424.99 371.05 480.37 475.68
$AR_{22}$ = 118 51 152897.37 59370.11 22157.23 71370.04 424.99 371.05 480.37 452.88
20% 50 152871.68 59395.91 22142.43 71333.34 424.87 370.96 480.24 432.98
40% 50 152847.07 59415.22 22128.52 71303.34 424.87 370.96 480.24 415.57
60% 50 152823.41 59431.94 22115.69 71275.79 424.87 370.96 480.24 400.11
-60% 45 118742.28 59390.16 22159.61 37192.50 424.16 370.47 479.48 452.19
-40% 47 130125.92 59383.40 22158.88 48583.64 424.46 370.68 479.80 452.44
-20% 49 141511.00 59376.72 22158.08 59976.20 424.74 370.87 480.10 452.67
$w_{22}^{R}$ = 140 51 152897.37 59370.11 22157.23 71370.04 424.99 371.05 480.37 452.88
20% 52 164285.03 59366.83 22156.78 82761.42 425.11 371.13 480.50 452.98
40% 54 175673.72 59360.32 22155.84 94157.55 425.34 371.29 480.75 453.16
60% 55 187063.41 59357.09 22155.36 105550.96 425.45 371.37 480.87 453.25
-60% 54 152977.57 59360.32 22155.84 71461.41 425.34 371.29 480.75 453.16
-40% 53 152950.20 59363.57 22156.32 71430.32 425.23 371.21 480.63 453.07
-20% 52 152923.48 59366.83 22156.78 71399.87 425.11 371.13 480.50 452.98
$k_{22}$ = 10.2 51 152897.37 59370.11 22157.23 71370.04 424.99 371.05 480.37 452.88
20% 49 152871.93 59376.72 22158.08 71337.13 424.74 370.87 480.10 452.67
40% 48 152847.05 59380.05 22158.49 71308.51 424.60 370.78 479.95 452.56
60% 47 152822.67 59383.40 22158.88 71280.39 424.46 370.68 479.80 452.44
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