doi: 10.3934/jimo.2021227
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Optimal pricing and ordering strategies for dual-channel retailing with different shipping policies

School of Automation, Southeast University, Nanjing, Jiangsu 210096, China

* Corresponding author: Zheng Wang

Received  July 2021 Revised  October 2021 Early access January 2022

Fund Project: This research is supported by National Natural Science Foundation of China under grant No. 61673109

In this paper, considering dual-channel retailing (online channel and offline channel), we study the pricing and ordering problem under different shipping policies. In this research, we mainly consider three shipping policies: without shipping price (OSP), with shipping price (WSP) and conditional free shipping (CFP). Based on the principle of maximum utility, we firstly obtain the probability of demand for the online and offline channels and further model the pricing and ordering problem under the three shipping policies. Further, avoiding the curse of dimensionality, the deep deterministic policy gradient (DDPG) method is employed to solve the problem to obtain the optimal pricing and ordering policy. Finally, we conduct some numerical experiments to compare the optimal pricing and ordering quantity under the three different shipping policies and reveal some managerial insights. The results show that the conditional free shipping policy is better than the other two policies, and stimulates the increase of demand to gain more profit.

Citation: Ning Li, Zheng Wang. Optimal pricing and ordering strategies for dual-channel retailing with different shipping policies. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021227
References:
[1]

F. Alawneh and G. Q. Zhang, Dual-channel warehouse and inventory management with stochastic demand, Transportation Research Part E-Logistics and Transportation Review, 112 (2018), 84-106.  doi: 10.1016/j.tre.2017.12.012.

[2]

R. BatarfiM. Y. Jaber and C. H. Glock, Pricing and inventory decisions in a dual-channel supply chain with learning and forgetting, Computers and Industrial Engineering, 136 (2019), 397-420.  doi: 10.1016/j.cie.2019.07.034.

[3]

R. Becerril-ArreolaM. M. Leng and M. Parlar, Online retailers' promotional pricing, free-shipping threshold, and inventory decisions: A simulation-based analysis, European J. Oper. Res., 230 (2013), 272-283.  doi: 10.1016/j.ejor.2013.04.006.

[4]

J. X. ChenL. LiangD. Q. Yao and S. N. 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]

W. K. ChiangD. Chhajed and J. D. Hess, Direct marketing, indirect profits: A strategic analysis of dual-channel supply-chain design, Management Science, 49 (2003), 1-20.  doi: 10.1287/mnsc.49.1.1.12749.

[6]

L. DingX. Liu and Y. Xu, Competitive risk management for online Bahncard problem, J. Ind. Manag. Optim., 6 (2010), 1-14.  doi: 10.3934/jimo.2010.6.1.

[7]

C. FanY. M. LiuX. H. YangX. H. Chen and J. H. Hu, Online and offline cooperation under buy-online, pick-up-in-store: Pricing and inventory decisions, J. Ind. Manag. Optim., 15 (2019), 1455-1472.  doi: 10.3934/jimo.2018104.

[8]

F. Gao and X. M. Su, Omnichannel retail operations with buy-online-and-pick-up-in-store, Management Science, 63 (2017), 2478-2492.  doi: 10.1287/mnsc.2016.2473.

[9]

J. Geunes and Y. Q. Su, Single-period assortment and stock-level decisions for dual sales channels with capacity limits and uncertain demand, International Journal of Production Research, 58 (2020), 5579-5600.  doi: 10.1080/00207543.2019.1693648.

[10]

M. GümüsS. L. LiW. Oh and S. Ray, Shipping fees or shipping free? a tale of two price partitioning strategies in online retailing, Production and Operations Management, 22 (2013), 758-776.  doi: 10.1111/j.1937-5956.2012.01391.x.

[11]

Y. HeH. F. Huang and D. Li, Inventory and pricing decisions for a dual-channel supply chain with deteriorating products, Operational Research, 20 (2018), 1-43. 

[12]

G. W. HuaS. Y. Wang and T. C. E. Cheng, Price and lead time decisions in dual-channel supply chains, European Journal of Operational Research, 205 (2010), 113-126. 

[13]

G. HuangQ. DingC. Dong and Z. Pan, Joint optimization of pricing and inventory control for dual-channel problem under stochastic demand, Ann. Oper. Res., 298 (2021), 307-337.  doi: 10.1007/s10479-018-2863-6.

[14]

W. Huang and J. M. Swaminathan, Introduction of second channel: Implications for pricing and profits, European Journal of Operational Research, 194 (2009), 258-297.  doi: 10.1016/j.ejor.2007.11.041.

[15]

W. H. Huang and Y. C. Cheng, Threshold free shipping policies for internet shoppers, Transportation Research Part A, 82 (2015), 193-203.  doi: 10.1016/j.tra.2015.09.015.

[16]

R. X. KongL. LuoL. X. Chen and M. F. Keblis, The effects of BOPS implementation under different pricing strategies in omnichannel retailing, Transportation Research Part E, 141 (2020), 102014, 1-30.  doi: 10.1016/j.tre.2020.102014.

[17]

B. LiP. ChenQ. H. Li and W. G. Wang, Dual-channel supply chain pricing decisions with a risk-averse retailer, International Journal of Production Research, 52 (2014), 7132-7147.  doi: 10.1080/00207543.2014.939235.

[18]

G. LiL. LiS. P. Sethi and X. Guan, Return strategy and pricing in a dual-channel supply chain, International Journal of Production Economics, 215 (2019), 153-164.  doi: 10.1016/j.ijpe.2017.06.031.

[19]

G. LiL. Li and J. S. Sun, Pricing and service effort strategy in a dual-channel supply chain with showrooming effect, Transportation Research part E-Logistics and Transportation Review, 126 (2019), 32-48.  doi: 10.1016/j.tre.2019.03.019.

[20]

T. T. LiX. B. Zhao and J. X. Xie, Inventory management for dual sales channels with inventory-level-dependent demand, Journal of the Operational Research Society, 66 (2015), 488-499.  doi: 10.1057/jors.2014.15.

[21]

T. P. Lillicrap, J. J. Huant, A. Pritzel, N. Heess, T. Erez, Y. Tassa, D. Sliver and D. Wierstra, Continuous control with deep reinforcement learning, Computer Science, 529 (2016), 484–489, arXiv: 1509.02971.

[22]

B. LiuR. Zhang and M. D. Xiao, Joint decision on production and pricing for online dual channel supply chain system, Appl. Math. Model., 34 (2010), 4208-4218.  doi: 10.1016/j.apm.2010.04.018.

[23]

N. M. Modak and P. Kelle, Managing a dual-channel supply chain under price and delivery-time dependent stochastic demand, European J. Oper. Res., 272 (2019), 147-161.  doi: 10.1016/j.ejor.2018.05.067.

[24]

A. RoyS. S. Sana and K. Chaudhuri, Joint decision on eoq and pricing strategy of a dual channel of mixed retail and e-tail comprising of single manufacture and retail under stochastic demand, Computers & Industrial Engineering, 102 (2016), 423-434.  doi: 10.1016/j.cie.2016.05.002.

[25]

J. K. RyanD. Sun and X. Y. Zhao, Coordinating a supply chain with a manufacturer-owned online channel: A dual channel model under price competition, IEEE Transactions on Engineering Management, 60 (2013), 247-259.  doi: 10.1109/TEM.2012.2207903.

[26]

X. F. Shao, Free or calculated shipping: Impact of delivery cost on supply chains moving to online retailing, International Journal of Production Economics, 191 (2017), 267-277.  doi: 10.1016/j.ijpe.2017.06.022.

[27]

W. G. TangH. T. Li and K. Cai, Optimising the credit term decisions in a dual-channel supply chain, International Journal of Production Research, 59 (2021), 4324-4341.  doi: 10.1080/00207543.2020.1762018.

[28]

R. H. WangG. G. NanL. Chen and M. Q. Li, Channel integration choices and pricing strategies for competing dual-channel retailers, IEEE Transactions on Engineering Management, (2020), 1-15.  doi: 10.1109/TEM.2020.3007347.

[29]

J. Q. YangX. M. ZhangH. Y. Fu and C. Liu, Inventory competition in a dual-channel supply chain with delivery lead time consideration, Appl. Math. Model., 42 (2017), 675-692.  doi: 10.1016/j.apm.2016.10.050.

[30]

D. Q. YaoX. H. YueX. Y. Wang and J. J. Liu, The impact of information sharing on a returns policy with the addition of direct channel, International Journal of Production Economics, 97 (2005), 196-209.  doi: 10.1016/j.ijpe.2004.08.006.

[31]

Y. ZhangH. F. ZhongY. Liu and M. H. Huang, Online ordering strategy for the discrete newsvendor problem with order value-based free-shipping, J. Ind. Manag. Optim., 15 (2019), 1617-1630.  doi: 10.3934/jimo.2018114.

[32]

B. ZhouM. N. Katehakis and Y. Zhao, Managing stochastic inventory systems with free shipping option, European J. Oper. Res., 196 (2009), 186-197.  doi: 10.1016/j.ejor.2008.01.042.

show all references

References:
[1]

F. Alawneh and G. Q. Zhang, Dual-channel warehouse and inventory management with stochastic demand, Transportation Research Part E-Logistics and Transportation Review, 112 (2018), 84-106.  doi: 10.1016/j.tre.2017.12.012.

[2]

R. BatarfiM. Y. Jaber and C. H. Glock, Pricing and inventory decisions in a dual-channel supply chain with learning and forgetting, Computers and Industrial Engineering, 136 (2019), 397-420.  doi: 10.1016/j.cie.2019.07.034.

[3]

R. Becerril-ArreolaM. M. Leng and M. Parlar, Online retailers' promotional pricing, free-shipping threshold, and inventory decisions: A simulation-based analysis, European J. Oper. Res., 230 (2013), 272-283.  doi: 10.1016/j.ejor.2013.04.006.

[4]

J. X. ChenL. LiangD. Q. Yao and S. N. 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]

W. K. ChiangD. Chhajed and J. D. Hess, Direct marketing, indirect profits: A strategic analysis of dual-channel supply-chain design, Management Science, 49 (2003), 1-20.  doi: 10.1287/mnsc.49.1.1.12749.

[6]

L. DingX. Liu and Y. Xu, Competitive risk management for online Bahncard problem, J. Ind. Manag. Optim., 6 (2010), 1-14.  doi: 10.3934/jimo.2010.6.1.

[7]

C. FanY. M. LiuX. H. YangX. H. Chen and J. H. Hu, Online and offline cooperation under buy-online, pick-up-in-store: Pricing and inventory decisions, J. Ind. Manag. Optim., 15 (2019), 1455-1472.  doi: 10.3934/jimo.2018104.

[8]

F. Gao and X. M. Su, Omnichannel retail operations with buy-online-and-pick-up-in-store, Management Science, 63 (2017), 2478-2492.  doi: 10.1287/mnsc.2016.2473.

[9]

J. Geunes and Y. Q. Su, Single-period assortment and stock-level decisions for dual sales channels with capacity limits and uncertain demand, International Journal of Production Research, 58 (2020), 5579-5600.  doi: 10.1080/00207543.2019.1693648.

[10]

M. GümüsS. L. LiW. Oh and S. Ray, Shipping fees or shipping free? a tale of two price partitioning strategies in online retailing, Production and Operations Management, 22 (2013), 758-776.  doi: 10.1111/j.1937-5956.2012.01391.x.

[11]

Y. HeH. F. Huang and D. Li, Inventory and pricing decisions for a dual-channel supply chain with deteriorating products, Operational Research, 20 (2018), 1-43. 

[12]

G. W. HuaS. Y. Wang and T. C. E. Cheng, Price and lead time decisions in dual-channel supply chains, European Journal of Operational Research, 205 (2010), 113-126. 

[13]

G. HuangQ. DingC. Dong and Z. Pan, Joint optimization of pricing and inventory control for dual-channel problem under stochastic demand, Ann. Oper. Res., 298 (2021), 307-337.  doi: 10.1007/s10479-018-2863-6.

[14]

W. Huang and J. M. Swaminathan, Introduction of second channel: Implications for pricing and profits, European Journal of Operational Research, 194 (2009), 258-297.  doi: 10.1016/j.ejor.2007.11.041.

[15]

W. H. Huang and Y. C. Cheng, Threshold free shipping policies for internet shoppers, Transportation Research Part A, 82 (2015), 193-203.  doi: 10.1016/j.tra.2015.09.015.

[16]

R. X. KongL. LuoL. X. Chen and M. F. Keblis, The effects of BOPS implementation under different pricing strategies in omnichannel retailing, Transportation Research Part E, 141 (2020), 102014, 1-30.  doi: 10.1016/j.tre.2020.102014.

[17]

B. LiP. ChenQ. H. Li and W. G. Wang, Dual-channel supply chain pricing decisions with a risk-averse retailer, International Journal of Production Research, 52 (2014), 7132-7147.  doi: 10.1080/00207543.2014.939235.

[18]

G. LiL. LiS. P. Sethi and X. Guan, Return strategy and pricing in a dual-channel supply chain, International Journal of Production Economics, 215 (2019), 153-164.  doi: 10.1016/j.ijpe.2017.06.031.

[19]

G. LiL. Li and J. S. Sun, Pricing and service effort strategy in a dual-channel supply chain with showrooming effect, Transportation Research part E-Logistics and Transportation Review, 126 (2019), 32-48.  doi: 10.1016/j.tre.2019.03.019.

[20]

T. T. LiX. B. Zhao and J. X. Xie, Inventory management for dual sales channels with inventory-level-dependent demand, Journal of the Operational Research Society, 66 (2015), 488-499.  doi: 10.1057/jors.2014.15.

[21]

T. P. Lillicrap, J. J. Huant, A. Pritzel, N. Heess, T. Erez, Y. Tassa, D. Sliver and D. Wierstra, Continuous control with deep reinforcement learning, Computer Science, 529 (2016), 484–489, arXiv: 1509.02971.

[22]

B. LiuR. Zhang and M. D. Xiao, Joint decision on production and pricing for online dual channel supply chain system, Appl. Math. Model., 34 (2010), 4208-4218.  doi: 10.1016/j.apm.2010.04.018.

[23]

N. M. Modak and P. Kelle, Managing a dual-channel supply chain under price and delivery-time dependent stochastic demand, European J. Oper. Res., 272 (2019), 147-161.  doi: 10.1016/j.ejor.2018.05.067.

[24]

A. RoyS. S. Sana and K. Chaudhuri, Joint decision on eoq and pricing strategy of a dual channel of mixed retail and e-tail comprising of single manufacture and retail under stochastic demand, Computers & Industrial Engineering, 102 (2016), 423-434.  doi: 10.1016/j.cie.2016.05.002.

[25]

J. K. RyanD. Sun and X. Y. Zhao, Coordinating a supply chain with a manufacturer-owned online channel: A dual channel model under price competition, IEEE Transactions on Engineering Management, 60 (2013), 247-259.  doi: 10.1109/TEM.2012.2207903.

[26]

X. F. Shao, Free or calculated shipping: Impact of delivery cost on supply chains moving to online retailing, International Journal of Production Economics, 191 (2017), 267-277.  doi: 10.1016/j.ijpe.2017.06.022.

[27]

W. G. TangH. T. Li and K. Cai, Optimising the credit term decisions in a dual-channel supply chain, International Journal of Production Research, 59 (2021), 4324-4341.  doi: 10.1080/00207543.2020.1762018.

[28]

R. H. WangG. G. NanL. Chen and M. Q. Li, Channel integration choices and pricing strategies for competing dual-channel retailers, IEEE Transactions on Engineering Management, (2020), 1-15.  doi: 10.1109/TEM.2020.3007347.

[29]

J. Q. YangX. M. ZhangH. Y. Fu and C. Liu, Inventory competition in a dual-channel supply chain with delivery lead time consideration, Appl. Math. Model., 42 (2017), 675-692.  doi: 10.1016/j.apm.2016.10.050.

[30]

D. Q. YaoX. H. YueX. Y. Wang and J. J. Liu, The impact of information sharing on a returns policy with the addition of direct channel, International Journal of Production Economics, 97 (2005), 196-209.  doi: 10.1016/j.ijpe.2004.08.006.

[31]

Y. ZhangH. F. ZhongY. Liu and M. H. Huang, Online ordering strategy for the discrete newsvendor problem with order value-based free-shipping, J. Ind. Manag. Optim., 15 (2019), 1617-1630.  doi: 10.3934/jimo.2018114.

[32]

B. ZhouM. N. Katehakis and Y. Zhao, Managing stochastic inventory systems with free shipping option, European J. Oper. Res., 196 (2009), 186-197.  doi: 10.1016/j.ejor.2008.01.042.

Figure 1.  Dual-channel retailing for different shipping policies: (a) OSP; (b) WSP; (c) CFP
Figure 2.  The customer' choices for purchasing channels under the OSP policy
Figure 3.  The customer' choices for purchasing channels under the WSP policy
Figure 4.  The customer' choices for purchasing channels for $ 0<\theta<(p^k-p_s)/v $
Figure 5.  The customer' choices for purchasing channels for $ (p^k-p_s)/v<\theta<(v-2p_s)/v $
Figure 6.  The customer' choices for purchasing channels for $ (v-2p_s)/v<\theta<1 $
Table 1.  The procedure of DDPG strategy for the pricing and ordering problem
Algorithm 1 DDPG for the pricing and ordering problem
Initialization:
  Initialize the index of iteration $ l $ and the index of ordering period $ k $ to zero;
  Construct feedforward neural critic network and feedforward neural actor network with random parameters $ \theta^Q $ and $ \theta^\mu $;
  Initialize target critic network and target actor network with $ {\theta ^{\bar Q}} \leftarrow {\theta ^Q} $ and $ {\theta ^{\bar \mu }} \leftarrow {\theta ^\mu } $;
  Initialize the experience playback memory $ D $ to size $ W $;
While (iteration $ l $ < Max iteration)
  Initialize the state $ I^0 $ and the action exploration process $ N $;
  For $ k=1 $, $ n $ do
    Based on the state $ I^k $ to select action $ u(I^k)=\mu(I^k|\theta^\mu)+N^k $
    Based on the probability of demand (i.e., Eqs.(8), (14) and (32)) and the selected action, obtaining the random demand of product;
    Based on the Eq. (34) or Eq. (38) or Eq. (42), computing the reward function $ R^i $;
    Storing $ [I^k,u(I^k ),R^i(I^k,u(I^k),I^{k+1}] $ to experience playback memory $ D $;
    Randomly selecting a subset with fixed size $ G $ from memory $ D $;
    Based on the selected $ [I^j,u(I^j),R^i(I^j,u(I^j)),I^{j+1}] $, update the critic network by minimizing the loss function:
$ L = \frac{1}{G}\sum\limits_j {{{\left( {Q_{}^i\left( {\left. {{I^j},u({I^j})} \right|{\theta ^Q}} \right) - {y^j}} \right)}^2}} $
$ {y^j} = {R^i}\left[ {{I^j},u({I^j})} \right] + \gamma \bar Q_{}^i\left( {\left. {{I^{j + 1}},\bar \mu \left( {\left. {{I^{j + 1}}} \right|{\theta ^{\bar \mu }}} \right)} \right|{\theta ^{\bar Q}}} \right) $
    Update the actor network by using the policy gradient method:
$ {\nabla _{{\theta ^\mu }}}J = \frac{1}{G}\sum\limits_j {{{\left. {{\nabla _a}Q_{}^i\left( {\left. {I,a} \right|{\theta ^Q}} \right)} \right|}_{I = {I^j},a = \mu ({I^j})}}{{\left. {{\nabla _{{\theta ^u}}}\mu \left( {\left. I \right|{\theta ^\mu }} \right)} \right|}_{I = {I^j}}}} $
    Employ the Eq. (52) to update the target network;
  end for
end while
Algorithm 1 DDPG for the pricing and ordering problem
Initialization:
  Initialize the index of iteration $ l $ and the index of ordering period $ k $ to zero;
  Construct feedforward neural critic network and feedforward neural actor network with random parameters $ \theta^Q $ and $ \theta^\mu $;
  Initialize target critic network and target actor network with $ {\theta ^{\bar Q}} \leftarrow {\theta ^Q} $ and $ {\theta ^{\bar \mu }} \leftarrow {\theta ^\mu } $;
  Initialize the experience playback memory $ D $ to size $ W $;
While (iteration $ l $ < Max iteration)
  Initialize the state $ I^0 $ and the action exploration process $ N $;
  For $ k=1 $, $ n $ do
    Based on the state $ I^k $ to select action $ u(I^k)=\mu(I^k|\theta^\mu)+N^k $
    Based on the probability of demand (i.e., Eqs.(8), (14) and (32)) and the selected action, obtaining the random demand of product;
    Based on the Eq. (34) or Eq. (38) or Eq. (42), computing the reward function $ R^i $;
    Storing $ [I^k,u(I^k ),R^i(I^k,u(I^k),I^{k+1}] $ to experience playback memory $ D $;
    Randomly selecting a subset with fixed size $ G $ from memory $ D $;
    Based on the selected $ [I^j,u(I^j),R^i(I^j,u(I^j)),I^{j+1}] $, update the critic network by minimizing the loss function:
$ L = \frac{1}{G}\sum\limits_j {{{\left( {Q_{}^i\left( {\left. {{I^j},u({I^j})} \right|{\theta ^Q}} \right) - {y^j}} \right)}^2}} $
$ {y^j} = {R^i}\left[ {{I^j},u({I^j})} \right] + \gamma \bar Q_{}^i\left( {\left. {{I^{j + 1}},\bar \mu \left( {\left. {{I^{j + 1}}} \right|{\theta ^{\bar \mu }}} \right)} \right|{\theta ^{\bar Q}}} \right) $
    Update the actor network by using the policy gradient method:
$ {\nabla _{{\theta ^\mu }}}J = \frac{1}{G}\sum\limits_j {{{\left. {{\nabla _a}Q_{}^i\left( {\left. {I,a} \right|{\theta ^Q}} \right)} \right|}_{I = {I^j},a = \mu ({I^j})}}{{\left. {{\nabla _{{\theta ^u}}}\mu \left( {\left. I \right|{\theta ^\mu }} \right)} \right|}_{I = {I^j}}}} $
    Employ the Eq. (52) to update the target network;
  end for
end while
Table 2.  The price, ordering quantity and total profit under the OSP policy for different shipping price $ p_s $
$ k=1 $ $ k=2 $ $ k=3 $ $ k=4 $ $ k=5 $ $ {\bar p} $ or $ \bar{q} $
Optimal price $ p^k $
$ p_s=0.2 $ 0.953 0.965 0.951 0.916 0.931 0.943
$ p_s=0.4 $ 0.995 0.995 0.994 0.993 0.976 0.991
$ p_s=0.6 $ 1.068 1.060 1.069 1.066 1.059 1.064
Optimal ordering quantity $ q^k $
$ p_s=0.2 $ 54 56 54 50 53 53
$ p_s=0.4 $ 52 51 52 52 41 50
$ p_s=0.6 $ 47 41 46 51 40 45
Total profit
$ p_s=0.2 $ 50.12 96.95 137.11 183.86 228.88
$ p_s=0.4 $ 45.22 89.14 134.03 165.92 196.02
$ p_s=0.6 $ 36.85 68.45 110.48 143.63 185.43
$ k=1 $ $ k=2 $ $ k=3 $ $ k=4 $ $ k=5 $ $ {\bar p} $ or $ \bar{q} $
Optimal price $ p^k $
$ p_s=0.2 $ 0.953 0.965 0.951 0.916 0.931 0.943
$ p_s=0.4 $ 0.995 0.995 0.994 0.993 0.976 0.991
$ p_s=0.6 $ 1.068 1.060 1.069 1.066 1.059 1.064
Optimal ordering quantity $ q^k $
$ p_s=0.2 $ 54 56 54 50 53 53
$ p_s=0.4 $ 52 51 52 52 41 50
$ p_s=0.6 $ 47 41 46 51 40 45
Total profit
$ p_s=0.2 $ 50.12 96.95 137.11 183.86 228.88
$ p_s=0.4 $ 45.22 89.14 134.03 165.92 196.02
$ p_s=0.6 $ 36.85 68.45 110.48 143.63 185.43
Table 3.  The price, ordering quantity and total profit under the WSP policy for different shipping price $ p_s $
$ k=1 $ $ k=2 $ $ k=3 $ $ k=4 $ $ k=5 $ $ {\bar p} $ or $ \bar{q} $
Optimal price $ p^k $
$ p_s=0.2 $ 0.906 0.930 0.908 0.895 0.939 0.916
$ p_s=0.4 $ 0.833 0.809 0.806 0.820 0.812 0.816
$ p_s=0.6 $ 0.746 0.741 0.743 0.742 0.741 0.743
Optimal ordering quantity $ q^k $
$ p_s=0.2 $ 45 43 45 45 42 44
$ p_s=0.4 $ 50 46 45 48 46 47
$ p_s=0.6 $ 49 46 47 47 46 47
Total profit
$ p_s=0.2 $ 40.32 84.86 129.03 164.97 215.10
$ p_s=0.4 $ 39.45 77.69 119.06 159.37 194.45
$ p_s=0.6 $ 37.89 76.67 115.12 152.14 192.53
$ k=1 $ $ k=2 $ $ k=3 $ $ k=4 $ $ k=5 $ $ {\bar p} $ or $ \bar{q} $
Optimal price $ p^k $
$ p_s=0.2 $ 0.906 0.930 0.908 0.895 0.939 0.916
$ p_s=0.4 $ 0.833 0.809 0.806 0.820 0.812 0.816
$ p_s=0.6 $ 0.746 0.741 0.743 0.742 0.741 0.743
Optimal ordering quantity $ q^k $
$ p_s=0.2 $ 45 43 45 45 42 44
$ p_s=0.4 $ 50 46 45 48 46 47
$ p_s=0.6 $ 49 46 47 47 46 47
Total profit
$ p_s=0.2 $ 40.32 84.86 129.03 164.97 215.10
$ p_s=0.4 $ 39.45 77.69 119.06 159.37 194.45
$ p_s=0.6 $ 37.89 76.67 115.12 152.14 192.53
Table 4.  The price, ordering quantity and total profit under the CFP policy for different shipping price $ p_s $
$ k=1 $ $ k=2 $ $ k=3 $ $ k=4 $ $ k=5 $ $ {\bar p} $ or $ \bar{q} $
Optimal price $ p^k $
$ p_s=0.2 $ 0.873 0.857 0.867 0.873 0.869 0.868
$ p_s=0.4 $ 0.908 0.904 0.911 0.916 0.926 0.913
$ p_s=0.6 $ 0.990 0.980 0.970 0.990 0.980 0.982
Optimal ordering quantity $ q^k $
$ p_s=0.2 $ 79 71 75 79 79 77
$ p_s=0.4 $ 72 69 66 65 62 67
$ p_s=0.6 $ 61 64 67 70 58 64
Total profit
$ p_s=0.2 $ 52.07 110.86 176.08 241.16 292.13
$ p_s=0.4 $ 49.31 94.28 142.13 183.62 229.91
$ p_s=0.6 $ 40.07 81.73 130.55 174.87 212.56
$ k=1 $ $ k=2 $ $ k=3 $ $ k=4 $ $ k=5 $ $ {\bar p} $ or $ \bar{q} $
Optimal price $ p^k $
$ p_s=0.2 $ 0.873 0.857 0.867 0.873 0.869 0.868
$ p_s=0.4 $ 0.908 0.904 0.911 0.916 0.926 0.913
$ p_s=0.6 $ 0.990 0.980 0.970 0.990 0.980 0.982
Optimal ordering quantity $ q^k $
$ p_s=0.2 $ 79 71 75 79 79 77
$ p_s=0.4 $ 72 69 66 65 62 67
$ p_s=0.6 $ 61 64 67 70 58 64
Total profit
$ p_s=0.2 $ 52.07 110.86 176.08 241.16 292.13
$ p_s=0.4 $ 49.31 94.28 142.13 183.62 229.91
$ p_s=0.6 $ 40.07 81.73 130.55 174.87 212.56
Table 5.  The comparison results under the different shipping policies
$ p_s=0.2 $ $ p_s=0.4 $ $ p_s=0.6 $
Price difference
$ {{\bar p}^{WSP}} - {{\bar p}^{OSP}} $ -0.027 -0.175 -0.321
$ {{\bar p}^{CFP}} - {{\bar p}^{WSP}} $ -0.048 0.097 0.239
$ {{\bar p}^{CFP}} - {{\bar p}^{OSP}} $ -0.075 -0.078 -0.082
Ordering quantity difference
$ {{\bar q}^{WSP}} - {{\bar q}^{OSP}} $ -9 -3 2
$ {{\bar q}^{CFP}} - {{\bar q}^{WSP}} $ 33 20 17
$ {{\bar q}^{CFP}} - {{\bar q}^{OSP}} $ 24 17 19
Total profit difference
$ WSP-OSP $ -13.78 -1.75 7.1
$ CFP-WSP $ 77.03 35.46 20.03
$ CFP-OSP $ 63.25 33.89 27.13
$ p_s=0.2 $ $ p_s=0.4 $ $ p_s=0.6 $
Price difference
$ {{\bar p}^{WSP}} - {{\bar p}^{OSP}} $ -0.027 -0.175 -0.321
$ {{\bar p}^{CFP}} - {{\bar p}^{WSP}} $ -0.048 0.097 0.239
$ {{\bar p}^{CFP}} - {{\bar p}^{OSP}} $ -0.075 -0.078 -0.082
Ordering quantity difference
$ {{\bar q}^{WSP}} - {{\bar q}^{OSP}} $ -9 -3 2
$ {{\bar q}^{CFP}} - {{\bar q}^{WSP}} $ 33 20 17
$ {{\bar q}^{CFP}} - {{\bar q}^{OSP}} $ 24 17 19
Total profit difference
$ WSP-OSP $ -13.78 -1.75 7.1
$ CFP-WSP $ 77.03 35.46 20.03
$ CFP-OSP $ 63.25 33.89 27.13
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