doi: 10.3934/jimo.2022005
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Dynamic pricing and sales effort in dual-channel retailing for seasonal products

School of Management, Hefei University of Technology, Hefei, 230009, China

*Corresponding author: Pengzhen Yin

Received  May 2021 Revised  September 2021 Early access February 2022

This paper studies the joint strategies of dynamic pricing and sales effort involving a seller and a third-party e-commerce platform (TPECP). The seller sells seasonal products through both offline and online channels, and the latter is opened on the TPECP. Considering a dual-channel setting, this paper fills a gap in the studies of joint strategies of dynamic pricing and sales effort. This paper formulates dynamic decision models and investigates them under both centralized and decentralized decision-making cases. Furthermore, a revenue-cost sharing contract is proposed to coordinate the two parties' decisions in the decentralized case, and it is validated theoretically and numerically. The key findings are as follows: For a given selling slot, the optimal online price decreases and the optimal sales effort level increases in the centralized case as the inventory level increases and remains below a threshold; however, when the inventory level exceeds the threshold, both the optimal online price and sales effort level remain fixed as the the inventory level increases in both cases. In addition, for a given inventory level below the threshold, the optimal online price decreases and the optimal sales effort level increases over time in both cases; however, when the inventory level exceeds the threshold, over time, the optimal online price increases and the optimal sales effort level decreases in the centralized case, and both the optimal online price and sales effort level increase in the decentralized case. Finally, a numerical experiment and some management implications are provided.

Citation: Wanting Hu, Jingjing Ding, Pengzhen Yin, Liang Liang. Dynamic pricing and sales effort in dual-channel retailing for seasonal products. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022005
References:
[1]

G. R. Bitran and S. V. Mondschein, Periodic pricing of seasonal products in retailing, Management Science, 43 (1997), 64-79.  doi: 10.1287/mnsc.43.1.64.

[2]

D. BoerV. Arnoud and B. Zwart, Dynamic pricing and learning with finite inventories, Oper. Res., 63 (2015), 965-979.  doi: 10.1287/opre.2015.1397.

[3]

G. P. Cachon, Supply chain coordination with contracts, Handbooks in Operations Research and Management Science, 11 (2003), 227-339. 

[4]

J. X. ChenL. LiangD. Yao and S. Sun, Price and quality decisions in dual-channel supply chains, European J. Oper. Res., 259 (2017), 935-948.  doi: 10.1016/j.ejor.2016.11.016.

[5]

X. ChenC. Liu and S. Li, The role of supply chain finance in improving the competitive advantage of online retailing enterprises, Electronic Commerce Research and Applications, 33 (2019), 100821. 

[6]

R. Y. ChenavazG. FeichtingerR. F. Hartl and P. M. Kort, Modeling the impact of product quality on dynamic pricing and advertising policies, European J. Oper. Res., 284 (2020), 990-1001.  doi: 10.1016/j.ejor.2020.01.035.

[7]

B. DanG. Xu and C. Liu, Pricing policies in a dual-channel supply chain with retail services, International Journal of Economics, 139 (2012), 312-320. 

[8]

Z. S. Dong, W. Chen, Q. Zhao and J. Li, Optimal pricing and inventory strategies for introducing a new product based on demand substitution effects, J. Ind. Manag. Optim., 16 (2020), 725-739, doi: 10.3934/jimo.2018175.

[9]

Y. Duan and J. Liu, Optimal dynamic pricing for perishable foods with quality and quantity deteriorating simultaneously under reference price effects, International Journal of Systems Science Operations & Logistics, 6 (2019), 346-355.  doi: 10.1080/23302674.2018.1465618.

[10]

C. Y. Dye and C. T. Yang, Optimal dynamic pricing and preservation technology investment for deteriorating products with reference price effects, Omega, 62 (2016), 52-67. 

[11]

G. Gallego and G. V. Ryzin, Optimal dynamic pricing of inventories with stochastic demand over finite horizons, Management Science, 40 (1994), 999-1020.  doi: 10.5555/2827835.2827840.

[12]

X. González, Chain heterogeneity and price-setting behavior: Evidence from e-grocery retailers, Electronic Commerce Research and Applications, 26 (2017), 62-72. 

[13]

H. G. Gurnani and Y. Xu, Resale price maintenance contracts with retailer sales effort: Effect of flexibility and competition, Naval Res. Logist., 53 (2006), 448-463.  doi: 10.1002/nav.20152.

[14]

W. Huang and J. M. Swaminathan, Introduction of a second channel: Implications for pricing and profits, European Journal of Operational Research, 194 (2009), 258-279. 

[15]

O. Kaya, Dynamic pricing of durable products with heterogeneous customers and demand interactions over time, Computers and Industrial Engineering, 65 (2013), 679-688. 

[16]

H. Ke and J. Liu, Dual-channel supply chain competition with channel preference and sales effort under uncertain environment, Journal of Ambient Intelligence & Humanized Computing, 8 (2017), 781-795.  doi: 10.1007/s12652-017-0502-8.

[17]

W. M. Kincaid and D. A. Darling, An inventory pricing problem, J. Math. Anal. Appl., 7 (1963), 183-208.  doi: 10.1016/0022-247X(63)90047-7.

[18]

T. C. Lee and M. Hersh, A model for dynamic airline seat inventory control with multiple seat bookings, Transportation Science, 27 (1993), 252-265.  doi: 10.1287/trsc.27.3.252.

[19]

B. LiP. W. HouP. Chen and Q. H. Li, Pricing strategy and coordination in a dual channel supply chain with a risk-averse retailer, International Journal of Production Economics, 178 (2016), 154-168. 

[20]

P. Li, Sale effort competition in dual-channel supply chain based on a Weakest online manufacturer, International Conference on Artificial Intelligence, Management Science and Electronic Commerce, (2011), 1491–1494, .

[21]

Q. Li and Z. Liu, Supply chain coordination via a two-part tariff contract with price and sales effort dependent demand, Decision Science Letters, 4 (2015), 27-34. 

[22]

S. LiJ. Zhang and W. Tang, Joint dynamic pricing and inventory control policy for a stochastic inventory system with perishable products, International Journal of Production Research, 53 (2014), 2937-2950.  doi: 10.1080/00207543.2014.961206.

[23]

X. LiY. Li and W. Cao, Cooperative advertising models in O2O supply chains, International Journal of Production Economics, 215 (2019), 144-152. 

[24]

X. LiG. Sun and Y. Li, A Multi-period ordering and clearance pricing model considering the competition between new and out-of-season products, Ann. Oper. Res., 242 (2016), 207-221.  doi: 10.1007/s10479-013-1498-x.

[25]

M. LiuE. Cao and C. K. Salifou, Pricing strategies of a dual-channel supply chain with risk aversion, Transportation Research Part E: Logistics and Transportation Review, 90 (2016), 108-120. 

[26]

S. Minsuk and A. Goker, Dynamic pricing of substitutable products with limited inventories under logit demand, IIE Transactions, 43 (2011), 323-331. 

[27]

X. PuL. Gong and X. Han, Consumer free riding: Coordinating sales effort in a dual-channel supply chain, Electronic Commerce Research and Applications, 22 (2017), 1-12. 

[28]

M. Radhi and G. Zhang, Pricing policies for a dual-channel retailer with cross-channel returns, Computers and Industrial Engineering, 119 (2018), 63-75. 

[29]

T. Taylor, Supply chain coordination under channel rebates with sales effort effects, Management Science, 48 (2002), 992-1007.  doi: 10.1287/mnsc.48.8.992.168.

[30]

Y. C. Tsao and G. J. Sheen, Dynamic pricing, promotion and replenishment policies for a deteriorating item under permissible delay in payments, Computers and Operations Research, 35 (2008), 3562-3580. 

[31]

A. Tsay and N. Agrawal, Modeling conflict and coordination in multi-channel distribution systems: A review, Handbook of Quantitative Supply Chain Analysis, 74 (2004), 557-606.  doi: 10.1007/978-1-4020-7953-5_13.

[32]

Y. Y. WangJ. C. Wang and B. Shou, Pricing and effort investment for a newsvendor-type product, European J. Oper. Res., 229 (2013), 422-432.  doi: 10.1016/j.ejor.2012.11.055.

[33]

S. WuQ. Liu and R. Q. Zhang, The reference effects on a retailer's dynamic pricing and inventory strategies with strategic consumers, Oper. Res., 63 (2015), 1320-1335.  doi: 10.1287/opre.2015.1440.

[34]

Y. B. XiaoF. Y. Chen and J. Chen, Optimal inventory and dynamic admission policies for a retailer of seasonal products with affiliate programs and drop-shipping, Naval Res. Logist., 56 (2009), 300-317.  doi: 10.1002/nav.20346.

[35]

R. Yan and Z. Pei, Retail services and firm profit in a dual-channel market, Journal of Retailing and Consumer Services, 16 (2009), 306-314. 

[36]

C. ZhangY. Wang and Y. Liu, Coordination contracts for a dual-channel supply chain under capital constraints, J. Ind. Manag. Optim., 17 (2021), 1485-1504.  doi: 10.3934/jimo.2020031.

[37]

F. Zhang and C. Wang, Dynamic pricing strategy and coordination in a dual-channel supply chain considering service value, Appl. Math. Model., 54 (2018), 722-742.  doi: 10.1016/j.apm.2017.10.006.

[38]

J. ZhangS. ZhaoT. C. E. Cheng and G. Hua, Optimisation of online retailer pricing and carrier capacity expansion during low-price promotions with coordination of a decentralised supply chain, International Journal of Production Research, 57 (2019), 2809-2827.  doi: 10.1080/00207543.2018.1516901.

[39]

S. X. Zhou and X. Chao, Dynamic pricing and inventory management with regular and expedited supplies, Production and Operations Management, 23 (2014), 65-80. 

show all references

References:
[1]

G. R. Bitran and S. V. Mondschein, Periodic pricing of seasonal products in retailing, Management Science, 43 (1997), 64-79.  doi: 10.1287/mnsc.43.1.64.

[2]

D. BoerV. Arnoud and B. Zwart, Dynamic pricing and learning with finite inventories, Oper. Res., 63 (2015), 965-979.  doi: 10.1287/opre.2015.1397.

[3]

G. P. Cachon, Supply chain coordination with contracts, Handbooks in Operations Research and Management Science, 11 (2003), 227-339. 

[4]

J. X. ChenL. LiangD. Yao and S. Sun, Price and quality decisions in dual-channel supply chains, European J. Oper. Res., 259 (2017), 935-948.  doi: 10.1016/j.ejor.2016.11.016.

[5]

X. ChenC. Liu and S. Li, The role of supply chain finance in improving the competitive advantage of online retailing enterprises, Electronic Commerce Research and Applications, 33 (2019), 100821. 

[6]

R. Y. ChenavazG. FeichtingerR. F. Hartl and P. M. Kort, Modeling the impact of product quality on dynamic pricing and advertising policies, European J. Oper. Res., 284 (2020), 990-1001.  doi: 10.1016/j.ejor.2020.01.035.

[7]

B. DanG. Xu and C. Liu, Pricing policies in a dual-channel supply chain with retail services, International Journal of Economics, 139 (2012), 312-320. 

[8]

Z. S. Dong, W. Chen, Q. Zhao and J. Li, Optimal pricing and inventory strategies for introducing a new product based on demand substitution effects, J. Ind. Manag. Optim., 16 (2020), 725-739, doi: 10.3934/jimo.2018175.

[9]

Y. Duan and J. Liu, Optimal dynamic pricing for perishable foods with quality and quantity deteriorating simultaneously under reference price effects, International Journal of Systems Science Operations & Logistics, 6 (2019), 346-355.  doi: 10.1080/23302674.2018.1465618.

[10]

C. Y. Dye and C. T. Yang, Optimal dynamic pricing and preservation technology investment for deteriorating products with reference price effects, Omega, 62 (2016), 52-67. 

[11]

G. Gallego and G. V. Ryzin, Optimal dynamic pricing of inventories with stochastic demand over finite horizons, Management Science, 40 (1994), 999-1020.  doi: 10.5555/2827835.2827840.

[12]

X. González, Chain heterogeneity and price-setting behavior: Evidence from e-grocery retailers, Electronic Commerce Research and Applications, 26 (2017), 62-72. 

[13]

H. G. Gurnani and Y. Xu, Resale price maintenance contracts with retailer sales effort: Effect of flexibility and competition, Naval Res. Logist., 53 (2006), 448-463.  doi: 10.1002/nav.20152.

[14]

W. Huang and J. M. Swaminathan, Introduction of a second channel: Implications for pricing and profits, European Journal of Operational Research, 194 (2009), 258-279. 

[15]

O. Kaya, Dynamic pricing of durable products with heterogeneous customers and demand interactions over time, Computers and Industrial Engineering, 65 (2013), 679-688. 

[16]

H. Ke and J. Liu, Dual-channel supply chain competition with channel preference and sales effort under uncertain environment, Journal of Ambient Intelligence & Humanized Computing, 8 (2017), 781-795.  doi: 10.1007/s12652-017-0502-8.

[17]

W. M. Kincaid and D. A. Darling, An inventory pricing problem, J. Math. Anal. Appl., 7 (1963), 183-208.  doi: 10.1016/0022-247X(63)90047-7.

[18]

T. C. Lee and M. Hersh, A model for dynamic airline seat inventory control with multiple seat bookings, Transportation Science, 27 (1993), 252-265.  doi: 10.1287/trsc.27.3.252.

[19]

B. LiP. W. HouP. Chen and Q. H. Li, Pricing strategy and coordination in a dual channel supply chain with a risk-averse retailer, International Journal of Production Economics, 178 (2016), 154-168. 

[20]

P. Li, Sale effort competition in dual-channel supply chain based on a Weakest online manufacturer, International Conference on Artificial Intelligence, Management Science and Electronic Commerce, (2011), 1491–1494, .

[21]

Q. Li and Z. Liu, Supply chain coordination via a two-part tariff contract with price and sales effort dependent demand, Decision Science Letters, 4 (2015), 27-34. 

[22]

S. LiJ. Zhang and W. Tang, Joint dynamic pricing and inventory control policy for a stochastic inventory system with perishable products, International Journal of Production Research, 53 (2014), 2937-2950.  doi: 10.1080/00207543.2014.961206.

[23]

X. LiY. Li and W. Cao, Cooperative advertising models in O2O supply chains, International Journal of Production Economics, 215 (2019), 144-152. 

[24]

X. LiG. Sun and Y. Li, A Multi-period ordering and clearance pricing model considering the competition between new and out-of-season products, Ann. Oper. Res., 242 (2016), 207-221.  doi: 10.1007/s10479-013-1498-x.

[25]

M. LiuE. Cao and C. K. Salifou, Pricing strategies of a dual-channel supply chain with risk aversion, Transportation Research Part E: Logistics and Transportation Review, 90 (2016), 108-120. 

[26]

S. Minsuk and A. Goker, Dynamic pricing of substitutable products with limited inventories under logit demand, IIE Transactions, 43 (2011), 323-331. 

[27]

X. PuL. Gong and X. Han, Consumer free riding: Coordinating sales effort in a dual-channel supply chain, Electronic Commerce Research and Applications, 22 (2017), 1-12. 

[28]

M. Radhi and G. Zhang, Pricing policies for a dual-channel retailer with cross-channel returns, Computers and Industrial Engineering, 119 (2018), 63-75. 

[29]

T. Taylor, Supply chain coordination under channel rebates with sales effort effects, Management Science, 48 (2002), 992-1007.  doi: 10.1287/mnsc.48.8.992.168.

[30]

Y. C. Tsao and G. J. Sheen, Dynamic pricing, promotion and replenishment policies for a deteriorating item under permissible delay in payments, Computers and Operations Research, 35 (2008), 3562-3580. 

[31]

A. Tsay and N. Agrawal, Modeling conflict and coordination in multi-channel distribution systems: A review, Handbook of Quantitative Supply Chain Analysis, 74 (2004), 557-606.  doi: 10.1007/978-1-4020-7953-5_13.

[32]

Y. Y. WangJ. C. Wang and B. Shou, Pricing and effort investment for a newsvendor-type product, European J. Oper. Res., 229 (2013), 422-432.  doi: 10.1016/j.ejor.2012.11.055.

[33]

S. WuQ. Liu and R. Q. Zhang, The reference effects on a retailer's dynamic pricing and inventory strategies with strategic consumers, Oper. Res., 63 (2015), 1320-1335.  doi: 10.1287/opre.2015.1440.

[34]

Y. B. XiaoF. Y. Chen and J. Chen, Optimal inventory and dynamic admission policies for a retailer of seasonal products with affiliate programs and drop-shipping, Naval Res. Logist., 56 (2009), 300-317.  doi: 10.1002/nav.20346.

[35]

R. Yan and Z. Pei, Retail services and firm profit in a dual-channel market, Journal of Retailing and Consumer Services, 16 (2009), 306-314. 

[36]

C. ZhangY. Wang and Y. Liu, Coordination contracts for a dual-channel supply chain under capital constraints, J. Ind. Manag. Optim., 17 (2021), 1485-1504.  doi: 10.3934/jimo.2020031.

[37]

F. Zhang and C. Wang, Dynamic pricing strategy and coordination in a dual-channel supply chain considering service value, Appl. Math. Model., 54 (2018), 722-742.  doi: 10.1016/j.apm.2017.10.006.

[38]

J. ZhangS. ZhaoT. C. E. Cheng and G. Hua, Optimisation of online retailer pricing and carrier capacity expansion during low-price promotions with coordination of a decentralised supply chain, International Journal of Production Research, 57 (2019), 2809-2827.  doi: 10.1080/00207543.2018.1516901.

[39]

S. X. Zhou and X. Chao, Dynamic pricing and inventory management with regular and expedited supplies, Production and Operations Management, 23 (2014), 65-80. 

Figure 1.  The schematic structure of the cooperation process between the seller and the TPECP
Figure 2.  Impact of selling slot on optimal strategies
Figure 3.  Impact of channel factors on optimal strategies
Figure 4.  The comparison of the total expected profits over whole selling season between two cases
Table 1.  Summary of the highly-related literature
Authors Dual channels Pricing strategy Sales effort / advertising Others
Li et al.[19]; Liu et al.[25] $ \checkmark $ Static pricing Risk aversion
Radhi & Zhang [28]; $ \checkmark $ Static pricing Channel coordination
Zhang et al.[36]
Gallego & Van Ryzin [11]; Dynamic pricing Stochastic demand
Kaya [15]; Li et al. [24]
Wu et al.[33]; Dye & Yang [10]; Duan & Liu[9] Dynamic pricing Reference price effect
Gurnani et al.[13] Static pricing $ \checkmark $ Competition
Li [20]; Ke and Liu [16] $ \checkmark $ $ \checkmark $ Competition
Li et al.[23] $ \checkmark $ $ \checkmark $ Online to offline
Chenavaz et al.[6] Dynamic pricing $ \checkmark $ Product quality
This paper $ \checkmark $ Dynamic pricing $ \checkmark $ revenue-cost sharing contract
Authors Dual channels Pricing strategy Sales effort / advertising Others
Li et al.[19]; Liu et al.[25] $ \checkmark $ Static pricing Risk aversion
Radhi & Zhang [28]; $ \checkmark $ Static pricing Channel coordination
Zhang et al.[36]
Gallego & Van Ryzin [11]; Dynamic pricing Stochastic demand
Kaya [15]; Li et al. [24]
Wu et al.[33]; Dye & Yang [10]; Duan & Liu[9] Dynamic pricing Reference price effect
Gurnani et al.[13] Static pricing $ \checkmark $ Competition
Li [20]; Ke and Liu [16] $ \checkmark $ $ \checkmark $ Competition
Li et al.[23] $ \checkmark $ $ \checkmark $ Online to offline
Chenavaz et al.[6] Dynamic pricing $ \checkmark $ Product quality
This paper $ \checkmark $ Dynamic pricing $ \checkmark $ revenue-cost sharing contract
Table 2.  Definitions of notations
Parameters
$ T $ The length of the whole selling season
$ N $ Initial total inventory
$ \beta $ The ratio of sales effort cost that borne by the seller
$ \phi $ The ratio of online store revenue that taken away by the TPECP
$ h $ The unit stock-holding cost per unit selling slot for the seller
$ \theta $ The customers channel preference coefficient for the online store
$ a $ The potential total demand rate for the product in dual-channel
$ b_1 $ The elasticity coefficient of online demand for online price
$ b_2 $ The elasticity coefficient of offline demand for offline price
$ k $ The elasticity coefficient of online demand for the sales effort exerted by the TPECP
$ c_1 $ The elasticity coefficient of online demand for offline price
$ c_2 $ The elasticity coefficient of offline demand for online price
$ p_2 $ The price of product that sold in the brick-and-mortar store, i.e., offline price
State variables
$ t $ The state variable of selling slot
$ n $ The state variable of inventory level
Other variables
$ \lambda _{1t}^{n} $ The demand rate of online product (or the probability of customer buying online product)
$ \lambda _{2t}^{n} $ The demand rate of offline product (or the probability of customer buying offline product)
Decision variables
$ p_{1t}^{n} $ The price of product that sold in the online store, i.e., online price, which decided by the seller
$ e_{t}^{n} $ Sales effort level, which exerted and decided by the TPECP, the sales effort cost is $ {{{(e_{t}^{n})}^{2}}}/{2}\; $
Parameters
$ T $ The length of the whole selling season
$ N $ Initial total inventory
$ \beta $ The ratio of sales effort cost that borne by the seller
$ \phi $ The ratio of online store revenue that taken away by the TPECP
$ h $ The unit stock-holding cost per unit selling slot for the seller
$ \theta $ The customers channel preference coefficient for the online store
$ a $ The potential total demand rate for the product in dual-channel
$ b_1 $ The elasticity coefficient of online demand for online price
$ b_2 $ The elasticity coefficient of offline demand for offline price
$ k $ The elasticity coefficient of online demand for the sales effort exerted by the TPECP
$ c_1 $ The elasticity coefficient of online demand for offline price
$ c_2 $ The elasticity coefficient of offline demand for online price
$ p_2 $ The price of product that sold in the brick-and-mortar store, i.e., offline price
State variables
$ t $ The state variable of selling slot
$ n $ The state variable of inventory level
Other variables
$ \lambda _{1t}^{n} $ The demand rate of online product (or the probability of customer buying online product)
$ \lambda _{2t}^{n} $ The demand rate of offline product (or the probability of customer buying offline product)
Decision variables
$ p_{1t}^{n} $ The price of product that sold in the online store, i.e., online price, which decided by the seller
$ e_{t}^{n} $ Sales effort level, which exerted and decided by the TPECP, the sales effort cost is $ {{{(e_{t}^{n})}^{2}}}/{2}\; $
Table 3.  The default values assigned to parameters
Parameter Value Parameter Value Parameter Value Parameter Value
$ p_2 $ 30 $ b_2 $ 0.016 k 0.015 h 0.06
$ \theta $ 0.5 $ c_1 $ 0.006 $ \beta $ 0.3 N 30
$ a $ 1.2 $ c_2 $ 0.008 $ \phi $ 0.2 T 40
$ b_1 $ 0.02
Parameter Value Parameter Value Parameter Value Parameter Value
$ p_2 $ 30 $ b_2 $ 0.016 k 0.015 h 0.06
$ \theta $ 0.5 $ c_1 $ 0.006 $ \beta $ 0.3 N 30
$ a $ 1.2 $ c_2 $ 0.008 $ \phi $ 0.2 T 40
$ b_1 $ 0.02
Table 4.  The numerical results given t = 30 in the centralized case
n $ U_{30}^{n} $ $ U_{30}^{n+1}-U_{30}^{n} $ $ p_{1}^{{}}{{_{30}}^{n\text{*}U}} $ $ e{{_{30}}}^{n\text{*}U} $ n $ U_{30}^{n} $ $ U_{30}^{n+1}-U_{30}^{n} $ $ p_{1}^{{}}{{_{30}}^{n\text{*}U}} $ $ e{{_{30}}}^{n\text{*}U} $
1 30.758 30.466 34.727 0.061 16 175.276 –0.660 25.449 0.392
2 61.224 29.584 34.603 0.065 17 174.616 –0.660 25.449 0.392
3 90.808 27.400 34.211 0.079 18 173.956 –0.660 25.449 0.392
4 118.208 23.413 33.308 0.111 19 173.296 –0.660 25.449 0.392
5 141.621 17.836 31.809 0.165 20 172.636 –0.660 25.449 0.392
6 159.458 11.607 29.916 0.232 21 171.976 –0.660 25.449 0.392
7 171.065 5.991 28.033 0.299 22 171.316 –0.660 25.449 0.392
8 177.056 2.041 26.579 0.351 23 170.656 –0.660 25.449 0.392
9 179.097 0.050 25.771 0.380 24 169.996 –0.660 25.449 0.392
10 179.147 –0.572 25.492 0.390 25 169.336 –0.660 25.449 0.392
11 178.576 –0.660 25.449 0.392 26 168.676 –0.660 25.449 0.392
12 177.916 –0.660 25.449 0.392 27 168.016 –0.660 25.449 0.392
13 177.256 –0.660 25.449 0.392 28 167.356 –0.660 25.449 0.392
14 176.596 –0.660 25.449 0.392 29 166.696 –0.660 25.449 0.392
15 175.936 –0.660 25.449 0.392 30 166.036 / 25.449 0.392
n $ U_{30}^{n} $ $ U_{30}^{n+1}-U_{30}^{n} $ $ p_{1}^{{}}{{_{30}}^{n\text{*}U}} $ $ e{{_{30}}}^{n\text{*}U} $ n $ U_{30}^{n} $ $ U_{30}^{n+1}-U_{30}^{n} $ $ p_{1}^{{}}{{_{30}}^{n\text{*}U}} $ $ e{{_{30}}}^{n\text{*}U} $
1 30.758 30.466 34.727 0.061 16 175.276 –0.660 25.449 0.392
2 61.224 29.584 34.603 0.065 17 174.616 –0.660 25.449 0.392
3 90.808 27.400 34.211 0.079 18 173.956 –0.660 25.449 0.392
4 118.208 23.413 33.308 0.111 19 173.296 –0.660 25.449 0.392
5 141.621 17.836 31.809 0.165 20 172.636 –0.660 25.449 0.392
6 159.458 11.607 29.916 0.232 21 171.976 –0.660 25.449 0.392
7 171.065 5.991 28.033 0.299 22 171.316 –0.660 25.449 0.392
8 177.056 2.041 26.579 0.351 23 170.656 –0.660 25.449 0.392
9 179.097 0.050 25.771 0.380 24 169.996 –0.660 25.449 0.392
10 179.147 –0.572 25.492 0.390 25 169.336 –0.660 25.449 0.392
11 178.576 –0.660 25.449 0.392 26 168.676 –0.660 25.449 0.392
12 177.916 –0.660 25.449 0.392 27 168.016 –0.660 25.449 0.392
13 177.256 –0.660 25.449 0.392 28 167.356 –0.660 25.449 0.392
14 176.596 –0.660 25.449 0.392 29 166.696 –0.660 25.449 0.392
15 175.936 –0.660 25.449 0.392 30 166.036 / 25.449 0.392
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