American Institute of Mathematical Sciences

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doi: 10.3934/jimo.2021155
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Joint emission reduction dynamic optimization and coordination in the supply chain considering fairness concern and reference low-carbon effect

Ziyuan Zhang and  Liying Yu ,

School of Management, Shanghai University, Shanghai, China

* Corresponding author: Liying Yu

Received  April 2021 Revised  May 2021 Early access September 2021

Fund Project: The first author is supported by the National Natural Science Foundation of China (NSFC) under Grants 12071280, 11671250

In the context of low-carbon economy, in order to explore the impact of the fairness concern and reference low-carbon effect on supply chain members' balanced emission reduction decisions and profits, supply chain joint emission reduction dynamic optimization models under four different scenarios are built, in which the manufacturer's optimal emission reduction strategy, the retailer's optimal low-carbon promotion strategy and other equilibrium solutions are solved by differential game theory. On the basis of analysis, a contract is designed to achieve the coordination of the supply chain when members are fairness concern. Some findings are as follows. First, when consumers' purchasing behavior is significantly affected by the reference low-carbon effect, and they have higher expectations for the product's emission reduction level, consumers' reference low-carbon effect will discourage the manufacturer's enthusiasm to reduce emissions, and do harm to the profits of the manufacturer and the retailer. Second, the fairness concern behavior of both parties will aggravate the adverse effects of reference low-carbon effect, bring a detrimental effect on the performance of the supply chain, aggravate the double marginal effect of the supply chain, and cause continuous negative social influence. Third, the bilateral cost-sharing contract can encourage the manufacturer to increase emission reduction investment, the retailer to increase low-carbon promotion investment, and can achieve a Pareto improvement of both parties' profits and utilities. In addition, the two cost-sharing ratios are only proportional to the marginal revenue and fairness concern intensity of both parties. Finally, when the two cost-sharing ratios and the revenue-sharing coefficient meet a certain relationship and are within a reasonable range, the bilateral cost sharing-revenue sharing hybrid contract can reduce the double marginal effect and achieve supply chain coordination.

Keywords: Reference low-carbon effect, fairness concern, joint emission reduction, supply chain coordination, differential game, dynamic optimization.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Ziyuan Zhang, Liying Yu. Joint emission reduction dynamic optimization and coordination in the supply chain considering fairness concern and reference low-carbon effect. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021155
References:
[1]

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Y. Y. Wang, R. J. Fan, L. Shen and M. Z. Jin, Decisions and coordination of green e-commerce supply chain considering green manufacturer's fairness concerns, International J. Production Research, 58 (2020), 7471-7489.   Google Scholar

[26]

Z. R. Wang, A. E. I. Brownlee and Q. H. Wu, Production and joint emission reduction decisions based on two-way cost-sharing contract under cap-and-trade regulation, Computers & Industrial Engineering, 146 (2020), 106549.   Google Scholar

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C. X. Wang, W. Wang and R. B. Huang, Supply chain enterprise operations and government carbon tax decisions considering carbon emissions, J. Cleaner Production, 152 (2017), 271-280.   Google Scholar

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J. Wang, X. X. Cheng, X. Y. Wang, H. T. Yang and S. H. Zhang, Myopic versus farsighted behaviors in a low-carbon supply chain with reference emission effects, Complexity, 2019 (2019), 3123572.   Google Scholar

[29]

L. J. Xia, Y. W. Bai, S. Ghose and J. J. Qin, Differential game analysis of carbon emissions reduction and promotion in a sustainable supply chain considering social preferences, Annals of Operations Research, 2020. doi: 10.1007/s10479-020-03838-8.  Google Scholar

[30]

L. J. Xia, T. T. Guo, J. J. Qin, X. Yue and Z. Ning, Carbon emission reduction and pricing policies of a supply chain considering reciprocal preferences in cap-and-trade system, Ann. Oper. Res., 268 (2018), 149-175.  doi: 10.1007/s10479-017-2657-2.  Google Scholar

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J. Xu and N. Liu, Erratum to: Research on closed loop supply chain with reference price effect, J. Intelligent Manufacturing, 28 (2017), 51-64.   Google Scholar

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B. Q. Yu, J. Wang, X. M. Lu and H. T. Yang, Collaboration in a low-carbon supply chain with reference emission and cost learning effects: Cost sharing versus revenue sharing strategies, Journal of Cleaner Production, 250 (2020), 119460.   Google Scholar

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S. Yu and Q. Hou, Supply chain investment in carbon emission-reducing technology based on stochasticity and low-carbon preferences, Complexity, 2021 (2021), 1-18.   Google Scholar

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H. Q. Zhang, P. Li, H. Zheng and Y. X. Zhang, Impact of carbon tax on enterprise operation and production strategy for low-carbon products in a co-opetition supply chain, J. Cleaner Production, 287 (2021), 125058.   Google Scholar

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L. H. Zhang, J. G. Wang and J. X. You, Consumer environmental awareness and channel coordination with two substitutable products, European J. Oper. Res., 241 (2015), 63-73.  doi: 10.1016/j.ejor.2014.07.043.  Google Scholar

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Z. C. Zhang, Q. Zhang, Z. Liu and X. X. Zheng, Static and dynamic pricing strategies in a closed-loop supply chain with reference quality effects, Sustainability, 10 (2018), 157.   Google Scholar

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Z. Y. Zhang, D. X. Fu and Q. Zhou, Optimal decisions of a green supply chain under the joint action of fairness preference and subsidy to the manufacturer, Discrete Dyn. Nat. Soc., 2020 (2020), 1-18.  doi: 10.1155/2020/9610503.  Google Scholar

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Z. Y. Zhang and L. Y. Yu, Dynamic optimization and coordination of cooperative emission reduction in a dual-channel supply chain considering reference low-carbon effect and low-carbon goodwill, International Journal of Environmental Research and Public Health, 18 (2021), 539.   Google Scholar

[39]

J. Zhang, W. Y. K. Chiang and L. Liang, Strategic pricing with reference effects in a competitive supply chain, Omega-International Journal of Management Science, 44 (2014), 126-135.   Google Scholar

[40]

J. Zhang, Q. L. Gou, L. Liang and Z. Huang, Supply chain coordination through cooperative advertising with reference price effect, Omega-International Journal of Management Science, 41 (2013), 345-353.   Google Scholar

[41]

L. H. Zhang, B. W. Xue and X. Y. Liu, Carbon emission reduction with regard to retailer's fairness concern and subsidies, Sustainability, 10 (2018), 1209.   Google Scholar

[42]

Y. Zhang, J. Y. Li and B. Xu, Designing buy-online-and-pick-up-in-store (bops) contract of dual-channel low-carbon supply chain considering consumers' low-carbon preference, Math. Probl. Eng., 2020 (2020), 15pp. doi: 10.1155/2020/7476019.  Google Scholar

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X. D. Zhou, B. Xu, F. Xie and L. Yu, Research on quality decisions and coordination with reference effect in dual-channel supply chain, Sustainability, 12 (2020), 2296.   Google Scholar

[44]

Y. J. Zhou, M. J. Bao, X. H. Chen and X. H. Xu, Co-op advertising and emission reduction cost sharing contracts and coordination in low-carbon supply chain based on fairness concerns, J. Cleaner Production, 133 (2016), 402-413.   Google Scholar

[45]

H. Zou, J. Qin and B. Dai, Optimal pricing decisions for a low-carbon supply chain considering fairness concern under carbon quota policy, International Journal of Environmental Research and Public Health, 18 (2021), 556.   Google Scholar

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Y. Zu and L. Chen, Myopic versus far-sighted behaviors in dynamic supply chain coordination through advertising with reference price effect, Discrete Dyn. Nat. Soc., 2017 (2017), 1-15.  doi: 10.1155/2017/9759561.  Google Scholar

show all references

References:
[1]

C. K. Anderson, H. Rasmussen and L. Macdonald, Competitive pricing with dynamic asymmetric price effects, Int. Trans. Oper. Res., 12 (2005), 509-525.  doi: 10.1111/j.1475-3995.2005.00522.x.  Google Scholar

[2]

E. Bendoly, K. Donohue and K. L. Schultz, Behavior in operations management: Assessing recent findings and revisiting old assumptions, J. Operations Management, 24 (2006), 737-752.   Google Scholar

[3]

S. Benjaafar, Y. Z. Li and M. Daskin, Carbon footprint and the management of supply chains: Insights from simple models, IEEE Transactions on Automation Science and Engineering, 10 (2013), 99-116.   Google Scholar

[4]

K. Y. Cao, B. Xu, Y. He and Q. Y. Xu, Optimal carbon reduction level and ordering quantity under financial constraints, Int. Trans. Oper. Res., 27 (2020), 2270-2293.  doi: 10.1111/itor.12606.  Google Scholar

[5]

C. Y. Dye, C. T. Yang and C. C. Wu, Joint dynamic pricing and preservation technology investment for an integrated supply chain with reference price effects, J. Operational Research Society, 69 (2018), 811-824.   Google Scholar

[6]

S. F. Du, L. Hu and L. Wang, Low-carbon supply policies and supply chain performance with carbon concerned demand, Ann. Oper. Res., 255 (2017), 569-590.  doi: 10.1007/s10479-015-1988-0.  Google Scholar

[7]

S. F. Du, C. Du and L. Liang, Supply chain coordination considering fairness concerns, Journal of Management Sciences in China, 13 (2010), 41-48.   Google Scholar

[8]

G. Fibich, A. Gavious and O. Lowengart, Explicit solutions of optimization models and differential games with nonsmooth (asymmetric) reference-price effects, Oper. Res., 51 (2003), 721-734.  doi: 10.1287/opre.51.5.721.16758.  Google Scholar

[9]

A. Gavious and O. Lowengart, Price-quality relationship in the presence of asymmetric dynamic reference quality effects, Marketing Letters, 23 (2012), 137-161.   Google Scholar

[10]

D. Ghosh and J. Shah, Supply chain analysis under green sensitive consumer demand and cost sharing contract, International Journal of Production Economics, 164 (2015), 319-329.   Google Scholar

[11]

H. Huang, J. Zhang, X. Ren and X. Zhou, Greenness and pricing decisions of cooperative supply chains considering altruistic preferences, Int. Journal of Environmental Research and Public Health, 16 (2019), 51.   Google Scholar

[12]

Q. Han, Y. Y. Wang, L. Shen and Y. Wang, Decision and coordination of low-carbon e-commerce supply chain with government carbon subsidies and fairness concerns, Complexity, 2020 (2020), 1-19.   Google Scholar

[13]

W. Jiang and X. Chen, Optimal strategies for low carbon supply chain with strategic customer behavior and green technology investment, Discrete Dyn. Nat. Soc., 2016 (2016), 1-13.  doi: 10.1155/2016/9645087.  Google Scholar

[14]

P. K. Kopalle and R. S. Winer, A dynamic model of reference price and expected quality, Marketing Letters, 7 (1996), 41-52.   Google Scholar

[15]

G. W. Liu, S. P. Sethi and J. X. Zhang, Myopic vs. far-sighted behaviours in a revenue-sharing supply chain with reference quality effects, International J. Production Research, 54 (2018), 1334-1357.   Google Scholar

[16]

Z. Liu, L. L. Lang, b. Hu, L. H. Shi, B. T. Huang and Y. J. Zhao, Emission reduction decision of agricultural supply chain considering carbon tax and investment cooperation, J. Cleaner Production, 294 (2021), 126305.   Google Scholar

[17]

L. Liu and F. T. Li, Differential game modelling of joint carbon reduction strategy and contract coordination based on low-carbon reference of consumers, J. Cleaner Production, 277 (2020), 123798.   Google Scholar

[18]

Z. Liu, X. X. Zheng, B. G. Gong and G. Y. Miao, Joint decision-making and the coordination of a sustainable supply chain in the context of carbon tax regulation and fairness concerns, International Journal of Environmental Research and Public Health, 14 (2017), 1464.   Google Scholar

[19]

C. H. Loch and Y. Z. Wu, Social preferences and supply chain performance: An experimental study, Management Science, 54 (2008), 1835-1849.   Google Scholar

[20]

L. H. Lu, Q. L. Gou, W. S. Tang and J. X. Zhang, Joint pricing and advertising strategy with reference price effect, International Journal of Production Research, 54 (2016), 5250-5270.   Google Scholar

[21]

Z. B. Lin, Price promotion with reference price effects in supply chain, Transportation Research Part E-Logistics and Transportation Review, 85 (2016), 52-68.   Google Scholar

[22]

Q. Q. Li, T. J. Xiao and Y. Z. Qiu, Price and carbon emission reduction decisions and revenue-sharing contract considering fairness concerns, J. Cleaner Production, 190 (2018), 303-314.   Google Scholar

[23]

X. J. Pu, Z. P. Song and G. H. Han, Competition among supply chains and governmental policy: Considering consumers' low-carbon preference, International Journal of Environmental Research and Public Health, 15 (2018), 1985.   Google Scholar

[24]

X. H. Qian, F. T. S. Chan, J. H. Zhang, M. Yin and Q. Zhang, Channel coordination of a two-echelon sustainable supply chain with a fair-minded retailer under cap-and-trade regulation, J. Cleaner Production, 244 (2020), 118715.   Google Scholar

[25]

Y. Y. Wang, R. J. Fan, L. Shen and M. Z. Jin, Decisions and coordination of green e-commerce supply chain considering green manufacturer's fairness concerns, International J. Production Research, 58 (2020), 7471-7489.   Google Scholar

[26]

Z. R. Wang, A. E. I. Brownlee and Q. H. Wu, Production and joint emission reduction decisions based on two-way cost-sharing contract under cap-and-trade regulation, Computers & Industrial Engineering, 146 (2020), 106549.   Google Scholar

[27]

C. X. Wang, W. Wang and R. B. Huang, Supply chain enterprise operations and government carbon tax decisions considering carbon emissions, J. Cleaner Production, 152 (2017), 271-280.   Google Scholar

[28]

J. Wang, X. X. Cheng, X. Y. Wang, H. T. Yang and S. H. Zhang, Myopic versus farsighted behaviors in a low-carbon supply chain with reference emission effects, Complexity, 2019 (2019), 3123572.   Google Scholar

[29]

L. J. Xia, Y. W. Bai, S. Ghose and J. J. Qin, Differential game analysis of carbon emissions reduction and promotion in a sustainable supply chain considering social preferences, Annals of Operations Research, 2020. doi: 10.1007/s10479-020-03838-8.  Google Scholar

[30]

L. J. Xia, T. T. Guo, J. J. Qin, X. Yue and Z. Ning, Carbon emission reduction and pricing policies of a supply chain considering reciprocal preferences in cap-and-trade system, Ann. Oper. Res., 268 (2018), 149-175.  doi: 10.1007/s10479-017-2657-2.  Google Scholar

[31]

J. Xu and N. Liu, Erratum to: Research on closed loop supply chain with reference price effect, J. Intelligent Manufacturing, 28 (2017), 51-64.   Google Scholar

[32]

B. Q. Yu, J. Wang, X. M. Lu and H. T. Yang, Collaboration in a low-carbon supply chain with reference emission and cost learning effects: Cost sharing versus revenue sharing strategies, Journal of Cleaner Production, 250 (2020), 119460.   Google Scholar

[33]

S. Yu and Q. Hou, Supply chain investment in carbon emission-reducing technology based on stochasticity and low-carbon preferences, Complexity, 2021 (2021), 1-18.   Google Scholar

[34]

H. Q. Zhang, P. Li, H. Zheng and Y. X. Zhang, Impact of carbon tax on enterprise operation and production strategy for low-carbon products in a co-opetition supply chain, J. Cleaner Production, 287 (2021), 125058.   Google Scholar

[35]

L. H. Zhang, J. G. Wang and J. X. You, Consumer environmental awareness and channel coordination with two substitutable products, European J. Oper. Res., 241 (2015), 63-73.  doi: 10.1016/j.ejor.2014.07.043.  Google Scholar

[36]

Z. C. Zhang, Q. Zhang, Z. Liu and X. X. Zheng, Static and dynamic pricing strategies in a closed-loop supply chain with reference quality effects, Sustainability, 10 (2018), 157.   Google Scholar

[37]

Z. Y. Zhang, D. X. Fu and Q. Zhou, Optimal decisions of a green supply chain under the joint action of fairness preference and subsidy to the manufacturer, Discrete Dyn. Nat. Soc., 2020 (2020), 1-18.  doi: 10.1155/2020/9610503.  Google Scholar

[38]

Z. Y. Zhang and L. Y. Yu, Dynamic optimization and coordination of cooperative emission reduction in a dual-channel supply chain considering reference low-carbon effect and low-carbon goodwill, International Journal of Environmental Research and Public Health, 18 (2021), 539.   Google Scholar

[39]

J. Zhang, W. Y. K. Chiang and L. Liang, Strategic pricing with reference effects in a competitive supply chain, Omega-International Journal of Management Science, 44 (2014), 126-135.   Google Scholar

[40]

J. Zhang, Q. L. Gou, L. Liang and Z. Huang, Supply chain coordination through cooperative advertising with reference price effect, Omega-International Journal of Management Science, 41 (2013), 345-353.   Google Scholar

[41]

L. H. Zhang, B. W. Xue and X. Y. Liu, Carbon emission reduction with regard to retailer's fairness concern and subsidies, Sustainability, 10 (2018), 1209.   Google Scholar

[42]

Y. Zhang, J. Y. Li and B. Xu, Designing buy-online-and-pick-up-in-store (bops) contract of dual-channel low-carbon supply chain considering consumers' low-carbon preference, Math. Probl. Eng., 2020 (2020), 15pp. doi: 10.1155/2020/7476019.  Google Scholar

[43]

X. D. Zhou, B. Xu, F. Xie and L. Yu, Research on quality decisions and coordination with reference effect in dual-channel supply chain, Sustainability, 12 (2020), 2296.   Google Scholar

[44]

Y. J. Zhou, M. J. Bao, X. H. Chen and X. H. Xu, Co-op advertising and emission reduction cost sharing contracts and coordination in low-carbon supply chain based on fairness concerns, J. Cleaner Production, 133 (2016), 402-413.   Google Scholar

[45]

H. Zou, J. Qin and B. Dai, Optimal pricing decisions for a low-carbon supply chain considering fairness concern under carbon quota policy, International Journal of Environmental Research and Public Health, 18 (2021), 556.   Google Scholar

[46]

Y. Zu and L. Chen, Myopic versus far-sighted behaviors in dynamic supply chain coordination through advertising with reference price effect, Discrete Dyn. Nat. Soc., 2017 (2017), 1-15.  doi: 10.1155/2017/9759561.  Google Scholar

Figure 1.  The impact of the bilateral cost sharing-revenue sharing hybrid contract.(a)the effect of $ \tau_m $ and $ \tau_r $ on $ \theta $.(b)the effect of $ \theta $ on both parties' profits
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Figure 2.  (a) The evolution path of low-carbon goodwill $ G(t) $.(b)The evolution path of reference low-carbon level $ R(t) $
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Figure 3.  (a) The evolution path of the manufacturer's profit $ \pi_m(t) $.(b)The evolution path of the retailer's profit $ \pi_r(t) $.(c) The evolution path of overall supply chain profit $ \pi_{sc}(t) $.(d) The evolution path of the manufacturer's utility $ U_m(t) $.(e) The evolution path of the retailer's utility $ U_r(t) $.(f) The evolution path of the utility difference $ \Delta U(t) $
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Figure 4.  (a) The impact of $ \tau_m $ and $ \tau_r $ on the low-carbon promotion cost-sharing ratio.(b)The impact of $ \tau_m $ and $ \tau_r $ on the emission reduction cost-sharing ratio.(c) The impact of $ \tau_m $ and $ \tau_r $ on the manufacturer's emission reduction investment.(d) The impact of $ \tau_m $ and $ \tau_r $ on the retailer's low-carbon promotion investment.(e) The impact of $ \tau_m $ and $ \tau_r $ on the manufacturer's profit $ \pi_m^f $. (f) The impact of $ \tau_m $ and $ \tau_r $ on the retailer's profit $ \pi_r^f $
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Figure 5.  (a) The impact of $ \epsilon $ on the manufacturer's emission reduction investment.(b)The impact of $ \alpha $ and $ \epsilon $ on the manufacturer's emission reduction investment.(c) The impact of $ \tau_m $ and $ \epsilon $ on the manufacturer's emission reduction investment.(d) The impact of $ \tau_m $ and $ \alpha $ on the manufacturer's emission reduction investment.(e) The impact of $ \alpha $ and $ \epsilon $ on the manufacturer's profit $ \pi_m^f $. (f) The impact of $ \alpha $ and $ \epsilon $ on the retailer's profit $ \pi_r^f $.(g) The impact of initial reference low-carbon level $ R_0 $ on the manufacturer's profit.(h) The impact of initial reference low-carbon level $ R_0 $ on the retailer's profit
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Table 1.  Some literature most relevant to this paper
Author Reference Low Carbon on Effect Fairness Concern Low Carbon Goodwill Supply Chain Coordination
Wang et al. [28] $\surd$
Liu et al. [18] $\surd$ $\surd$
Yu et al. [32] $\surd$
Zhang et al. [41] $\surd$
Zou et al. [45] $\surd$
This paper $\surd$ $\surd$ $\surd$ $\surd$
Table Options

Download as excel

Author Reference Low Carbon on Effect Fairness Concern Low Carbon Goodwill Supply Chain Coordination
Wang et al. [28] $\surd$
Liu et al. [18] $\surd$ $\surd$
Yu et al. [32] $\surd$
Zhang et al. [41] $\surd$
Zou et al. [45] $\surd$
This paper $\surd$ $\surd$ $\surd$ $\surd$
Table 2.  Related parameters and decision variables
Decision variables Definition
$ E(t) $ Manufacturer's emission reduction investment at time $ t $
$ A(t) $ Retailer's low-carbon promotion investment at time $ t $
Parameters Definition
$ \alpha $ Influence coefficient of reference low-carbon effect on the demand
$ \beta $ Influence coefficient of manufacturer's emission reduction on the demand
$ k $ Influence coefficient of retailer's low-carbon promotion on the demand
$ \rho $ Discount rate
$ \mu $ Influence coefficient of low-carbon goodwill on the demand
$ \pi_1 $ Manufacturer's marginal revenue
$ \pi_2 $ Retailer's marginal revenue
$ G(t) $ Low-carbon goodwill level at time $ t $
$ R(t) $ Consumer's reference low-carbon level at time $ t $
$ \lambda $ Influence coefficient of the low-carbon promotion on the low-carbon goodwill
$ \sigma $ Natural decay rate of low-carbon goodwill
$ \epsilon $ Memory parameter
$ \phi $ Low-carbon promotion Cost-sharing ratio
$ \gamma $ Emission reduction cost-sharing ratio
$ \theta $ Revenue-sharing coefficient under the supply chain coordination
$ \tau_m $ Manufacturer's fairness concern intensity
$ \tau_r $ Retailer's fairness concern intensity
$ \eta_m $, $ \eta_r $ Emission reduction cost coefficient and low-carbon promotion cost coefficient.
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Decision variables Definition
$ E(t) $ Manufacturer's emission reduction investment at time $ t $
$ A(t) $ Retailer's low-carbon promotion investment at time $ t $
Parameters Definition
$ \alpha $ Influence coefficient of reference low-carbon effect on the demand
$ \beta $ Influence coefficient of manufacturer's emission reduction on the demand
$ k $ Influence coefficient of retailer's low-carbon promotion on the demand
$ \rho $ Discount rate
$ \mu $ Influence coefficient of low-carbon goodwill on the demand
$ \pi_1 $ Manufacturer's marginal revenue
$ \pi_2 $ Retailer's marginal revenue
$ G(t) $ Low-carbon goodwill level at time $ t $
$ R(t) $ Consumer's reference low-carbon level at time $ t $
$ \lambda $ Influence coefficient of the low-carbon promotion on the low-carbon goodwill
$ \sigma $ Natural decay rate of low-carbon goodwill
$ \epsilon $ Memory parameter
$ \phi $ Low-carbon promotion Cost-sharing ratio
$ \gamma $ Emission reduction cost-sharing ratio
$ \theta $ Revenue-sharing coefficient under the supply chain coordination
$ \tau_m $ Manufacturer's fairness concern intensity
$ \tau_r $ Retailer's fairness concern intensity
$ \eta_m $, $ \eta_r $ Emission reduction cost coefficient and low-carbon promotion cost coefficient.
Table 3.  The relationship between related parameters
$ \pi_1 $ $ \pi_2 $ $ \tau_m $ $ \tau_r $ $ \eta_m $ $ \eta_r $ $ \alpha $ $ \beta $ $ k $ $ \epsilon $ $ \sigma $ $ \lambda $ $ \mu $ $ \rho $
$ E $ $ + $ $ - $ $ - $ $ \times $ $ - $ $ \times $ $ + $ $ + $ $ \times $ $ - $ $ \times $ $ \times $ $ \times $ $ + $
$ A $ $ - $ $ + $ $ \times $ $ - $ $ \times $ $ - $ $ \times $ $ \times $ $ + $ $ \times $ $ - $ $ + $ $ + $ $ - $
$ G_{\infty} $ $ - $ $ + $ $ \times $ $ - $ $ \times $ $ - $ $ \times $ $ \times $ $ + $ $ \times $ $ - $ $ + $ $ + $ $ - $
$ R_{\infty} $ $ + $ $ - $ $ - $ $ \times $ $ - $ $ \times $ $ + $ $ + $ $ \times $ $ - $ $ \times $ $ \times $ $ \times $ $ + $
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$ \pi_1 $ $ \pi_2 $ $ \tau_m $ $ \tau_r $ $ \eta_m $ $ \eta_r $ $ \alpha $ $ \beta $ $ k $ $ \epsilon $ $ \sigma $ $ \lambda $ $ \mu $ $ \rho $
$ E $ $ + $ $ - $ $ - $ $ \times $ $ - $ $ \times $ $ + $ $ + $ $ \times $ $ - $ $ \times $ $ \times $ $ \times $ $ + $
$ A $ $ - $ $ + $ $ \times $ $ - $ $ \times $ $ - $ $ \times $ $ \times $ $ + $ $ \times $ $ - $ $ + $ $ + $ $ - $
$ G_{\infty} $ $ - $ $ + $ $ \times $ $ - $ $ \times $ $ - $ $ \times $ $ \times $ $ + $ $ \times $ $ - $ $ + $ $ + $ $ - $
$ R_{\infty} $ $ + $ $ - $ $ - $ $ \times $ $ - $ $ \times $ $ + $ $ + $ $ \times $ $ - $ $ \times $ $ \times $ $ \times $ $ + $
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