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doi: 10.3934/jimo.2022042
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Trade credit and information leakage in a supply chain with competing retailers

College of Economics and Trade, Hunan University, Changsha, 410079, China

*Corresponding author: Man Yu

"World Bank urged to lift trade credit finance", Financial Times, November 11, 2008.

Received  June 2021 Revised  November 2021 Early access March 2022

This paper investigates the issue of information leakage in a supply consisting of one manufacturer, one incumbent retailer endowed with superior demand information, and a capital-constrained entrant retailer financed by the manufacturer's trade credit. The incumbent retailer faces the situation that the manufacturer may leak the incumbent retailer's order information to the uninformed entrant retailer. We first examine whether trade credit can prevent information leakage. It shows that the manufacturer always leaks information in the non-bankruptcy scenario where the capital-constrained retailer does not go bankrupt in the low-demand state. In contrast, in the bankruptcy scenario, the manufacturer does not leak information since it could get more transfer revenue from the bankrupt entrant retailer than the revenue obtained from the incumbent retailer. We also explore the impacts of information leakage on each member's optimal decision and profit. Interestingly, the findings show that the informed incumbent retailer, taking the advantage of being the first-mover, is able to actively place a larger order to induce the manufacturer to leak information and benefits from information leakage under certain conditions.

Citation: Man Yu, Erbao Cao. Trade credit and information leakage in a supply chain with competing retailers. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022042
References:
[1]

K. S. Anand and M. Goyal, Strategic information management under leakage in a supply chain, Management Science, 55 (2009), 438-452. 

[2]

C. Atanasova and N. Wilson, Bank borrowing constraints and the demand for trade credit: Evidence from panel data, Managerial & Decision Economics, 24 (2003), 503-514.  doi: 10.1002/mde.1134.

[3]

Y. Aviv, A time-series framework for supply-chain inventory management, Oper. Res., 51 (2003), 210-227.  doi: 10.1287/opre.51.2.210.12780.

[4]

V. Babich, Independence of capacity ordering and financial subsidies to risky suppliers, Manufacturing & Service Operations Management, 12 (2010), 583-607.  doi: 10.1287/msom.1090.0284.

[5]

V. Babich and C. S. Tang, Managing opportunistic supplier product adulteration: Deferred payments, inspection, and combined mechanisms, Manufacturing & Service Operations Management, 14 (2012), 301-314.  doi: 10.1287/msom.1110.0366.

[6]

F. Boissay and R. Gropp, Payment defaults and inter-firm liquidity provision, Review of Finance, 17 (2013), 1853-1894. 

[7]

G. CaiX. Chen and Z. Xiao, The roles of bank and trade credits: Theoretical analysis and empirical evidence, Production and Operations Management, 23 (2014), 583-598.  doi: 10.1111/poms.12035.

[8]

J.-A. Chevalier, Capital structure and product-market competition: Empirical evidence from the supermarket industry, American Economic Review, 85 (1995), 415. 

[9]

J. Chod, Inventory, risk shifting, and trade credit, Management Science, 63 (2017), 3207-3225.  doi: 10.1287/mnsc.2016.2515.

[10]

J. Chod, N. Trichakis and G. Tsoukalas, A signaling theory of in-kind finance, Working Paper, Boston College, 2016.

[11]

L. Y. ChuN. Shamir and H. Shin, Strategic Communication for Capacity Alignment with Pricing in a Supply Chain, Management Science, 63 (2017), 4366-4388. 

[12]

F. DelbonoL. Lambertini and R. Bertrand, Cournot and supply function equilibria in oligopoly, Energy Economics, 60 (2016), 73-78. 

[13]

S. DengK. FuJ. Xu and K. Zhu, The supply chain effects of trade credit under uncertain demands, Omega, 98 (2021), 102-113.  doi: 10.1016/j.omega.2019.102113.

[14]

Y. Ding, Y. Jiang, L. Wu and Z. Zhou, Two-echelon supply chain network design with trade credit, Comput. Oper. Res., 131 (2021), Paper No. 105270, 13 pp. doi: 10.1016/j.cor.2021.105270.

[15]

R. DominguezS. CannellaA. P. Barbosa-Póvoa and J. M. Framina, Information sharing in supply chains with heterogeneous retailers, Omega, 79 (2018), 116-132.  doi: 10.1016/j.omega.2017.08.005.

[16]

P. Fontaine and S. Zhao, Suppliers as financial intermediaries: Trade credit for undervalued firms, Journal of Bank and Finance, 124 (2021), 106-143.  doi: 10.1016/j.jbankfin.2021.106043.

[17]

M. GiannettiM. Burkart and T. Ellingsen, What you sell is what you lend? Explaining trade credit contracts, Review of Financial, Studies, 24 (2011), 1261-1298. 

[18]

S. HuangX. Guan and Y. J. Chen, Retailer information sharing with supplier encroachment, Production and Operations Management, 27 (2018), 1133-1147. 

[19]

C. K. JaggiG. MamtaK. Amrina and T. Sunil, Inventory and credit decisions for deteriorating items with displayed stock dependent demand in two-echelon supply chain using stackelberg and nash equilibrium solution, Ann. Oper. Res, 274 (2019), 309-329.  doi: 10.1007/s10479-018-2925-9.

[20]

A. JainS. Seshadri and M. Sohoni, Differential pricing for information sharing under competition, Production and Operations Management, 20 (2011), 235-252.  doi: 10.1111/j.1937-5956.2010.01161.x.

[21]

G. KongS. Rajagopalan and H. Zhang, Revenue sharing and information leakage in a supply chain, Management Science, 59 (2013), 556-572. 

[22]

P. Kouvelis and W. Zhao, Financing the newsvendor: Supplier vs. bank, and the structure of optimal trade credit contracts, Oper. Res., 60 (2012), 566-580.  doi: 10.1287/opre.1120.1040.

[23]

P. Kouvelis and W. Zhao, Who should finance the supply chain? impact of credit ratings on supply chain decisions, Manufacturing & Service Operations Management, 20 (2018), 19-35.  doi: 10.1287/msom.2017.0669.

[24]

Lenovo Group Limited, 2019–20 Financial Year Annual Results Announcement, 2020, Cninfo. Available from: http://www.cninfo.com.cn/.

[25]

T. Murphy, Protection in question. Survey: Detroit compromises intellectual property, Wards Auto World, (2007).

[26]

N. Shamir, Cartel formation through strategic information leakage in a distribution channel, Marketing Science, 36 (2017), 70-88. 

[27]

N. Shamir and H. Shin, Public forecast information sharing in a market with competing supply chains, Management Science, 62 (2016), 2994-3022. 

[28]

W. ShangA. Ha and S. Tong, Information sharing in a supply chain with a common retailer, Management Science, 62 (2016), 245-263.  doi: 10.1287/mnsc.2014.2127.

[29]

P. Vandenberg, Adapting to the financial landscape: Evidence from small firms in Nairobi, World Development, 31 (2003), 1829-1843. 

[30]

Y. WangW. Tang and R. Zhao, Information sharing and information concealment in the presence of a dominant retailer, Computers & Industrial Engineering, 121 (2018), 36-50.  doi: 10.1016/j.cie.2018.04.039.

[31]

N. A. Yan and X. L. He, Optimal trade credit with deferred payment and multiple decision attributes in supply chain finance, Computers & Industrial Engineering, 147 (2020), 106627.  doi: 10.1016/j.cie.2020.106627.

[32]

S. A. Yang and J. R. Birge, Trade credit, risk sharing, and inventory financing portfolios, Management Science, 64 (2018), 3667-3689. 

[33]

S. A. YangJ. R. Birge and R. P. Parker, The supply chain effects of bankruptcy, Management Science, 61 (2015), 2320-2338.  doi: 10.1287/mnsc.2014.2079.

[34]

H. Zhang, Vertical information exchange in a supply chain with duopoly retailers, Production and Operations Management, 11 (2002), 531-546.  doi: 10.1111/j.1937-5956.2002.tb00476.x.

[35]

J. ZhangS. LiS. Zhang and R. Dai, Manufacturer encroachment with quality decision under asymmetric demand information, European J. Oper. Res., 273 (2019), 217-236.  doi: 10.1016/j.ejor.2018.08.002.

[36]

D. ZhaoM. Chen and Y. Gong, Strategic information sharing under revenue sharing contract: Explicit vs. tacit collusion in retailers, Computers & Industrial Engineering, 131 (2019), 99-114.  doi: 10.1016/j.cie.2019.03.035.

show all references

References:
[1]

K. S. Anand and M. Goyal, Strategic information management under leakage in a supply chain, Management Science, 55 (2009), 438-452. 

[2]

C. Atanasova and N. Wilson, Bank borrowing constraints and the demand for trade credit: Evidence from panel data, Managerial & Decision Economics, 24 (2003), 503-514.  doi: 10.1002/mde.1134.

[3]

Y. Aviv, A time-series framework for supply-chain inventory management, Oper. Res., 51 (2003), 210-227.  doi: 10.1287/opre.51.2.210.12780.

[4]

V. Babich, Independence of capacity ordering and financial subsidies to risky suppliers, Manufacturing & Service Operations Management, 12 (2010), 583-607.  doi: 10.1287/msom.1090.0284.

[5]

V. Babich and C. S. Tang, Managing opportunistic supplier product adulteration: Deferred payments, inspection, and combined mechanisms, Manufacturing & Service Operations Management, 14 (2012), 301-314.  doi: 10.1287/msom.1110.0366.

[6]

F. Boissay and R. Gropp, Payment defaults and inter-firm liquidity provision, Review of Finance, 17 (2013), 1853-1894. 

[7]

G. CaiX. Chen and Z. Xiao, The roles of bank and trade credits: Theoretical analysis and empirical evidence, Production and Operations Management, 23 (2014), 583-598.  doi: 10.1111/poms.12035.

[8]

J.-A. Chevalier, Capital structure and product-market competition: Empirical evidence from the supermarket industry, American Economic Review, 85 (1995), 415. 

[9]

J. Chod, Inventory, risk shifting, and trade credit, Management Science, 63 (2017), 3207-3225.  doi: 10.1287/mnsc.2016.2515.

[10]

J. Chod, N. Trichakis and G. Tsoukalas, A signaling theory of in-kind finance, Working Paper, Boston College, 2016.

[11]

L. Y. ChuN. Shamir and H. Shin, Strategic Communication for Capacity Alignment with Pricing in a Supply Chain, Management Science, 63 (2017), 4366-4388. 

[12]

F. DelbonoL. Lambertini and R. Bertrand, Cournot and supply function equilibria in oligopoly, Energy Economics, 60 (2016), 73-78. 

[13]

S. DengK. FuJ. Xu and K. Zhu, The supply chain effects of trade credit under uncertain demands, Omega, 98 (2021), 102-113.  doi: 10.1016/j.omega.2019.102113.

[14]

Y. Ding, Y. Jiang, L. Wu and Z. Zhou, Two-echelon supply chain network design with trade credit, Comput. Oper. Res., 131 (2021), Paper No. 105270, 13 pp. doi: 10.1016/j.cor.2021.105270.

[15]

R. DominguezS. CannellaA. P. Barbosa-Póvoa and J. M. Framina, Information sharing in supply chains with heterogeneous retailers, Omega, 79 (2018), 116-132.  doi: 10.1016/j.omega.2017.08.005.

[16]

P. Fontaine and S. Zhao, Suppliers as financial intermediaries: Trade credit for undervalued firms, Journal of Bank and Finance, 124 (2021), 106-143.  doi: 10.1016/j.jbankfin.2021.106043.

[17]

M. GiannettiM. Burkart and T. Ellingsen, What you sell is what you lend? Explaining trade credit contracts, Review of Financial, Studies, 24 (2011), 1261-1298. 

[18]

S. HuangX. Guan and Y. J. Chen, Retailer information sharing with supplier encroachment, Production and Operations Management, 27 (2018), 1133-1147. 

[19]

C. K. JaggiG. MamtaK. Amrina and T. Sunil, Inventory and credit decisions for deteriorating items with displayed stock dependent demand in two-echelon supply chain using stackelberg and nash equilibrium solution, Ann. Oper. Res, 274 (2019), 309-329.  doi: 10.1007/s10479-018-2925-9.

[20]

A. JainS. Seshadri and M. Sohoni, Differential pricing for information sharing under competition, Production and Operations Management, 20 (2011), 235-252.  doi: 10.1111/j.1937-5956.2010.01161.x.

[21]

G. KongS. Rajagopalan and H. Zhang, Revenue sharing and information leakage in a supply chain, Management Science, 59 (2013), 556-572. 

[22]

P. Kouvelis and W. Zhao, Financing the newsvendor: Supplier vs. bank, and the structure of optimal trade credit contracts, Oper. Res., 60 (2012), 566-580.  doi: 10.1287/opre.1120.1040.

[23]

P. Kouvelis and W. Zhao, Who should finance the supply chain? impact of credit ratings on supply chain decisions, Manufacturing & Service Operations Management, 20 (2018), 19-35.  doi: 10.1287/msom.2017.0669.

[24]

Lenovo Group Limited, 2019–20 Financial Year Annual Results Announcement, 2020, Cninfo. Available from: http://www.cninfo.com.cn/.

[25]

T. Murphy, Protection in question. Survey: Detroit compromises intellectual property, Wards Auto World, (2007).

[26]

N. Shamir, Cartel formation through strategic information leakage in a distribution channel, Marketing Science, 36 (2017), 70-88. 

[27]

N. Shamir and H. Shin, Public forecast information sharing in a market with competing supply chains, Management Science, 62 (2016), 2994-3022. 

[28]

W. ShangA. Ha and S. Tong, Information sharing in a supply chain with a common retailer, Management Science, 62 (2016), 245-263.  doi: 10.1287/mnsc.2014.2127.

[29]

P. Vandenberg, Adapting to the financial landscape: Evidence from small firms in Nairobi, World Development, 31 (2003), 1829-1843. 

[30]

Y. WangW. Tang and R. Zhao, Information sharing and information concealment in the presence of a dominant retailer, Computers & Industrial Engineering, 121 (2018), 36-50.  doi: 10.1016/j.cie.2018.04.039.

[31]

N. A. Yan and X. L. He, Optimal trade credit with deferred payment and multiple decision attributes in supply chain finance, Computers & Industrial Engineering, 147 (2020), 106627.  doi: 10.1016/j.cie.2020.106627.

[32]

S. A. Yang and J. R. Birge, Trade credit, risk sharing, and inventory financing portfolios, Management Science, 64 (2018), 3667-3689. 

[33]

S. A. YangJ. R. Birge and R. P. Parker, The supply chain effects of bankruptcy, Management Science, 61 (2015), 2320-2338.  doi: 10.1287/mnsc.2014.2079.

[34]

H. Zhang, Vertical information exchange in a supply chain with duopoly retailers, Production and Operations Management, 11 (2002), 531-546.  doi: 10.1111/j.1937-5956.2002.tb00476.x.

[35]

J. ZhangS. LiS. Zhang and R. Dai, Manufacturer encroachment with quality decision under asymmetric demand information, European J. Oper. Res., 273 (2019), 217-236.  doi: 10.1016/j.ejor.2018.08.002.

[36]

D. ZhaoM. Chen and Y. Gong, Strategic information sharing under revenue sharing contract: Explicit vs. tacit collusion in retailers, Computers & Industrial Engineering, 131 (2019), 99-114.  doi: 10.1016/j.cie.2019.03.035.

Figure 1.  Sequence of events
Figure 2.  The manufacturer's non-leakage region in the bankruptcy scenario
Figure 3.  The manufacturer's non-leakage region in the non-bankruptcy scenario
Figure 4.  Wholesale price w with respect to $ R $ and $ r $
Figure 5.  Impacts of $ r $ on retailers
Figure 6.  Impacts of $ r $ on the manufacturer and the supply chain
Table 1.  Optimal order quantities given different wholesale prices
Non-leakage equilibrium without bankruptcy $ q_{i L}^{E N 1^{*}}=\dfrac{3 A_{L}-u+2 w_{e}(1+R)-4 w_{i}}{6} $
$ q_{i H}^{E N 1^{*}}=\dfrac{3 A_{H}-u+2 w_{e}(1+R)-4 w_{i}}{6} $
$ q_{e}^{E N 1^{*}}=\dfrac{u+w_{i}-2 w_{e}(1+R)}{3} $
Non-leakage equilibrium with bankruptcy $ q_{i L}^{E N 2^{*}}=\dfrac{3 A_{L}-A_{H}+2 w_{e}(1+R)-4 w_{i}}{6} $
$ q_{i H}^{E N 2^{*}}=\dfrac{A_{H}+w_{e}(1+R)-2 w_{i}}{3} $
$ q_{e}^{E N 2^{*}}=\dfrac{A_{H}-2 w_{e}(1+R)+w_{i}}{3} $
Separating equilibrium $ \phi^{\prime} \geq 3 $ $q_{i L}^{E S 1^{*}}=\dfrac{A_{L}-2 w_{i}+w_{e}(1+R)}{2}$
$q_{e L}^{E S 1^{*}}=\dfrac{A_{L}-3 w_{e}(1+R)+2 w_{i}}{4}$
$q_{i H}^{E S 1^{*}}=\dfrac{A_{H}-2 w_{i}+w_{e}(1+R)}{2}$
$q_{e H}^{E S 1^{*}}=\dfrac{A_{H}-3 w_{e}(1+R)+2 w_{i}}{4}$
Separating equilibrium $\phi^{\prime}<3$ $\begin{array}{c} q_{i L}^{E S 2^{*}}=\dfrac{1}{2}\left[2 A_{H}-A_{L}+w_{e}(1+R)-2 w_{i}\right]-\\\dfrac{1}{2} \sqrt{\left(A_{H}-A_{L}\right)\left[3 A_{H}-A_{L}-2 w_{e}(1+R)+4 w_{i}\right]} \end{array}$
$\begin{array}{c} q_{e L}^{E S 2^{*}}=\dfrac{1}{4}\left[3 A_{L}-2 A_{H}-3 w_{e}(1+R)+2 w_{i}\right]+\\ \dfrac{1}{4} \sqrt{\left(A_{H}-A_{L}\right)\left[3 A_{H}-A_{L}-2 w_{e}(1+R)+4 w_{i}\right]} \end{array}$
Pooling equilibrium $q_{i}^{E P^{*}}=\dfrac{2 A_{L}-u+w_{e}(1+R)-2 w_{i}}{2}$
$q_{e}^{E P^{*}}=\dfrac{3 u-2 A_{L}-3 w_{e}(1+R)+2 w_{i}}{4}$
Non-leakage equilibrium without bankruptcy $ q_{i L}^{E N 1^{*}}=\dfrac{3 A_{L}-u+2 w_{e}(1+R)-4 w_{i}}{6} $
$ q_{i H}^{E N 1^{*}}=\dfrac{3 A_{H}-u+2 w_{e}(1+R)-4 w_{i}}{6} $
$ q_{e}^{E N 1^{*}}=\dfrac{u+w_{i}-2 w_{e}(1+R)}{3} $
Non-leakage equilibrium with bankruptcy $ q_{i L}^{E N 2^{*}}=\dfrac{3 A_{L}-A_{H}+2 w_{e}(1+R)-4 w_{i}}{6} $
$ q_{i H}^{E N 2^{*}}=\dfrac{A_{H}+w_{e}(1+R)-2 w_{i}}{3} $
$ q_{e}^{E N 2^{*}}=\dfrac{A_{H}-2 w_{e}(1+R)+w_{i}}{3} $
Separating equilibrium $ \phi^{\prime} \geq 3 $ $q_{i L}^{E S 1^{*}}=\dfrac{A_{L}-2 w_{i}+w_{e}(1+R)}{2}$
$q_{e L}^{E S 1^{*}}=\dfrac{A_{L}-3 w_{e}(1+R)+2 w_{i}}{4}$
$q_{i H}^{E S 1^{*}}=\dfrac{A_{H}-2 w_{i}+w_{e}(1+R)}{2}$
$q_{e H}^{E S 1^{*}}=\dfrac{A_{H}-3 w_{e}(1+R)+2 w_{i}}{4}$
Separating equilibrium $\phi^{\prime}<3$ $\begin{array}{c} q_{i L}^{E S 2^{*}}=\dfrac{1}{2}\left[2 A_{H}-A_{L}+w_{e}(1+R)-2 w_{i}\right]-\\\dfrac{1}{2} \sqrt{\left(A_{H}-A_{L}\right)\left[3 A_{H}-A_{L}-2 w_{e}(1+R)+4 w_{i}\right]} \end{array}$
$\begin{array}{c} q_{e L}^{E S 2^{*}}=\dfrac{1}{4}\left[3 A_{L}-2 A_{H}-3 w_{e}(1+R)+2 w_{i}\right]+\\ \dfrac{1}{4} \sqrt{\left(A_{H}-A_{L}\right)\left[3 A_{H}-A_{L}-2 w_{e}(1+R)+4 w_{i}\right]} \end{array}$
Pooling equilibrium $q_{i}^{E P^{*}}=\dfrac{2 A_{L}-u+w_{e}(1+R)-2 w_{i}}{2}$
$q_{e}^{E P^{*}}=\dfrac{3 u-2 A_{L}-3 w_{e}(1+R)+2 w_{i}}{4}$
Table 2.  Optimal order decisions
Non-leakage equilibrium without bankruptcy $q_{i L}^{I N 1^{*}}=\dfrac{3 A_{L}-u-2 w(1+R)}{6}$
$q_{i H}^{I N 1^{*}}=\dfrac{3 A_{H}-u-2 w(1+R)}{6}$
$q_{e}^{I N 1^{*}}=\dfrac{u-w(1+R)}{3}$
Non-leakage equilibrium with bankruptcy $q_{i L}^{I N 2^{*}}=\dfrac{3 A_{L}-A_{H}-2 w(1+R)}{6}$
$q_{i H}^{I N 2^{*}}=\dfrac{A_{H}-w(1+R)}{3}$
$q_{e}^{I N 2^{*}}=\dfrac{A_{H}-w(1+R)}{3}$
Separating equilibrium $\phi^{I} \geq 3$ $q_{i L}^{I S 1^{*}}=\dfrac{A_{L}-w(1+R)}{2}$
$q_{e L}^{I S 1^{*}}=\dfrac{A_{L}-w(1+R)}{4}$
$q_{i H}^{I S 1^{*}}=\dfrac{A_{H}-w(1+R)}{2}$
$q_{e L}^{I S 1^{*}}=\dfrac{A_{H}-w(1+R)}{4}$
Separating equilibrium $\phi^{I}<3$ $\begin{array}{c} q_{i L}^{I S 2^{*}}=\dfrac{1}{2}\left(2 A_{H}-A_{L}-w-w R\right)\\ -\dfrac{1}{2} \sqrt{\left(A_{H}-A_{L}\right)\left(3 A_{H}-A_{L}-2 w-2 w R\right)} \end{array}$
$\begin{array}{c} q_{e L}^{I S 2^{*}}=\dfrac{1}{4}\left(3 A_{L}-2 A_{H}-w-R w\right)\\ +\dfrac{1}{4} \sqrt{\left(A_{H}-A_{L}\right)\left(3 A_{H}-A_{L}-2 w-2 w R\right)} \end{array}$
Non-leakage equilibrium without bankruptcy $q_{i L}^{I N 1^{*}}=\dfrac{3 A_{L}-u-2 w(1+R)}{6}$
$q_{i H}^{I N 1^{*}}=\dfrac{3 A_{H}-u-2 w(1+R)}{6}$
$q_{e}^{I N 1^{*}}=\dfrac{u-w(1+R)}{3}$
Non-leakage equilibrium with bankruptcy $q_{i L}^{I N 2^{*}}=\dfrac{3 A_{L}-A_{H}-2 w(1+R)}{6}$
$q_{i H}^{I N 2^{*}}=\dfrac{A_{H}-w(1+R)}{3}$
$q_{e}^{I N 2^{*}}=\dfrac{A_{H}-w(1+R)}{3}$
Separating equilibrium $\phi^{I} \geq 3$ $q_{i L}^{I S 1^{*}}=\dfrac{A_{L}-w(1+R)}{2}$
$q_{e L}^{I S 1^{*}}=\dfrac{A_{L}-w(1+R)}{4}$
$q_{i H}^{I S 1^{*}}=\dfrac{A_{H}-w(1+R)}{2}$
$q_{e L}^{I S 1^{*}}=\dfrac{A_{H}-w(1+R)}{4}$
Separating equilibrium $\phi^{I}<3$ $\begin{array}{c} q_{i L}^{I S 2^{*}}=\dfrac{1}{2}\left(2 A_{H}-A_{L}-w-w R\right)\\ -\dfrac{1}{2} \sqrt{\left(A_{H}-A_{L}\right)\left(3 A_{H}-A_{L}-2 w-2 w R\right)} \end{array}$
$\begin{array}{c} q_{e L}^{I S 2^{*}}=\dfrac{1}{4}\left(3 A_{L}-2 A_{H}-w-R w\right)\\ +\dfrac{1}{4} \sqrt{\left(A_{H}-A_{L}\right)\left(3 A_{H}-A_{L}-2 w-2 w R\right)} \end{array}$
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