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March  2022, 12(1): 109-120. doi: 10.3934/naco.2021054

Optimal pre-sale policy for deteriorating items

1. 

School of Management, Shanghai University, Shanghai 200444, China

2. 

School of Engineering, Open University of China, Beijing 100039, China

* Corresponding author: Qi An

Received  May 2020 Revised  July 2021 Published  March 2022 Early access  November 2021

Fund Project: This work is supported by Humanities and Social Sciences Foundation of the Chinese Ministry of Education (20YJAZH135)and National Natural Science Foundation of China (71502100)

Pre-sale policy is a frequently-used sales approach for deteriorating products, e.g, fruits, vegetables, seafood, etc. In this paper, we consider an EOQ inventory model under pre-sale policy for deteriorating products, in which the demand of pre-sale period depends on price and pre-sale horizon, and the demand of spot-sale period depends on the price and stock level. Optimal pricing decisions and economic order quantity are also provided. We compare pre-sale model with a benchmark inventory model in which all the products are sold in spot-sale period. Theoretical results are derived to show the existence and uniqueness of the optimal solution. Numerical experiments are carried out to to illustrate the theoretical results. And sensitivity analysis is conducted to identify conditions under which the pre-sale policy is better off than the spot-sale only policy.

Citation: Lianxia Zhao, Hui Qiao, Qi An. Optimal pre-sale policy for deteriorating items. Numerical Algebra, Control & Optimization, 2022, 12 (1) : 109-120. doi: 10.3934/naco.2021054
References:
[1]

T. AvinadavA. Herbon and U. Spiegel, Optimal inventory policy for a perishable item with demand function sensitive to price and time, International Journal of Production Economics, 144 (2013), 497-506.   Google Scholar

[2]

M. BakkerJ. Riezebos and R. H. Teunter, Review of inventory systems with deterioration since 2001, European Journal of Operational Research, 221 (2012), 275-284.  doi: 10.1016/j.ejor.2012.03.004.  Google Scholar

[3]

M. Cheng and G. Wang, A note on the inventory model for deteriorating items with trapezoidal type demand rate, Computers & Industrial Engineering, 56 (2009), 1296-1300.   Google Scholar

[4]

M. ChengB. Zhang and G. Wang, Optimal policy for deteriorating items with trapezoidal type demand and partial backlogging, Applied Mathematical Modelling, 35 (2011), 3552-3560.  doi: 10.1016/j.apm.2011.01.001.  Google Scholar

[5]

S.-H. Cho and C. S. Tang, Advance selling in a supply chain under uncertain supply and demand, Manufacturing & Service Operations Management, 15 (2013), 305-319.   Google Scholar

[6]

G. A. ChuaR. Mokhlesi and A. Sainathan, Optimal discounting and replenishment policies for perishable products, International Journal of Production Economics, 186 (2017), 8-20.   Google Scholar

[7]

A. DiabatA. A. Taleizadeh and M. Lashgari, A lot sizing model with partial downstream delayed payment, partial upstream advance payment, and partial backordering for deteriorating items, Journal of Manufacturing Systems, 45 (2017), 322-342.   Google Scholar

[8]

G. DobsonE. J. Pinker and O. Yildiz, An eoq model for perishable goods with age-dependent demand rate, European Journal of Operational Research, 257 (2017), 84-88.  doi: 10.1016/j.ejor.2016.06.073.  Google Scholar

[9]

R. M. Hill, Inventory models for increasing demand followed by level demand, Journal of the Operational Research Society, 46 (1995), 1250-1259.   Google Scholar

[10]

L. JanssenT. Claus and J. Sauer, Literature review of deteriorating inventory models by key topics from 2012 to 2015, International Journal of Production Economics, 182 (2016), 86-112.   Google Scholar

[11]

M. Khouja and J. Zhou, Channel and pricing decisions in a supply chain with advance selling of gift cards, European Journal of Operational Research, 244 (2015), 471-489.  doi: 10.1016/j.ejor.2015.01.045.  Google Scholar

[12]

S. Minner and S. Transchel, Order variability in perishable product supply chains, European Journal of Operational Research, 260 (2017), 93-107.  doi: 10.1016/j.ejor.2016.12.016.  Google Scholar

[13]

M. Möller and M. Watanabe, Advance purchase discounts versus clearance sales, The Economic Journal, 120(September), 120 (2010), 1125-1148.   Google Scholar

[14]

C. C. Murray, D. Talukdar and A. Gosavi, Joint optimization of product price, display orientation and shelf-space allocation in retail category management, Journal of Retailing, 86 (2010), 125–136, Special Issue: Modeling Retail Phenomena. Google Scholar

[15]

M. ÖnalA. Yenipazarli and O. E. Kundakcioglu, A mathematical model for perishable products with price- and displayed-stock-dependent demand, Computers & Industrial Engineering, 102 (2016), 246-258.   Google Scholar

[16]

S. PandaS. Senapati and M. Basu, Optimal replenishment policy for perishable seasonal products in a season with ramp-type time dependent demand, Computers & industrial engineering, 54 (2008), 301-314.   Google Scholar

[17]

A. PrasadK. E. Stecke and X. Zhao, Advance selling by a newsvendor retailer, Production and Operations Management, 20 (2011), 129-142.   Google Scholar

[18]

T. Roy and K. Chaudhuri, An inventory model for a deteriorating item with price-dependent demand and special sale, International Journal of Operational Research, 2 (2007), 173-187.   Google Scholar

[19]

S. S. Sana, An eoq model for perishable item with stock dependent demand and price discount rate, American Journal of Mathematical and Management Sciences, 30 (2010), 299-316.  doi: 10.2298/YJOR1002237P.  Google Scholar

[20]

B. SarkarB. Mandal and S. Sarkar, Quality improvement and backorder price discount under controllable lead time in an inventory model, Journal of Manufacturing Systems, 35 (2015), 26-36.   Google Scholar

[21]

Y.-C. Tsao, Retailer's optimal ordering and discounting policies under advance sales discount and trade credits, Computers & Industrial Engineering, 56 (2009), 208-215.   Google Scholar

[22]

Y.-C. TsaoJ.-C. LuN. AnF. Al-KhayyalR. W. Lu and G. Han, Retailer shelf-space management with trade allowance: A stackelberg game between retailer and manufacturers, International Journal of Production Economics, 148 (2014), 133-144.   Google Scholar

[23]

C. Wang and R. Huang, Pricing for seasonal deteriorating products with price- and ramp-type time-dependent demand, Computers & Industrial Engineering, 77 (2014), 29-34.  doi: 10.1051/ro/2015033.  Google Scholar

[24]

K.-S. Wu, An eoq inventory model for items with weibull distribution deterioration, ramp type demand rate and partial backlogging, Production Planning & Control, 12 (2001), 787-793.  doi: 10.1080/00207720110102575.  Google Scholar

[25]

P.-S. You, Optimal pricing for an advance sales system with price and waiting time dependent demands, Journal of the Operations Research Society of Japan, 50 (2007), 151-161.  doi: 10.15807/jorsj.50.151.  Google Scholar

[26]

C. Zeng, Optimal advance selling strategy under price commitment, Pacific Economic Review, 18 (2013), 233-258.   Google Scholar

[27]

L. Zhao, Optimal replenishment policy for weibull-distributed deteriorating items with trapezoidal demand rate and partial backlogging, Mathematical Problems in Engineering, 2016 (2016), 1-10.  doi: 10.1155/2016/1490712.  Google Scholar

[28]

L. Zhao and J. You, Optimal pricing and ordering policy for deteriorating items with stock-and-price dependent demand and presale rebate, Scientific Programming, 2016 (2016), 1-8.   Google Scholar

show all references

References:
[1]

T. AvinadavA. Herbon and U. Spiegel, Optimal inventory policy for a perishable item with demand function sensitive to price and time, International Journal of Production Economics, 144 (2013), 497-506.   Google Scholar

[2]

M. BakkerJ. Riezebos and R. H. Teunter, Review of inventory systems with deterioration since 2001, European Journal of Operational Research, 221 (2012), 275-284.  doi: 10.1016/j.ejor.2012.03.004.  Google Scholar

[3]

M. Cheng and G. Wang, A note on the inventory model for deteriorating items with trapezoidal type demand rate, Computers & Industrial Engineering, 56 (2009), 1296-1300.   Google Scholar

[4]

M. ChengB. Zhang and G. Wang, Optimal policy for deteriorating items with trapezoidal type demand and partial backlogging, Applied Mathematical Modelling, 35 (2011), 3552-3560.  doi: 10.1016/j.apm.2011.01.001.  Google Scholar

[5]

S.-H. Cho and C. S. Tang, Advance selling in a supply chain under uncertain supply and demand, Manufacturing & Service Operations Management, 15 (2013), 305-319.   Google Scholar

[6]

G. A. ChuaR. Mokhlesi and A. Sainathan, Optimal discounting and replenishment policies for perishable products, International Journal of Production Economics, 186 (2017), 8-20.   Google Scholar

[7]

A. DiabatA. A. Taleizadeh and M. Lashgari, A lot sizing model with partial downstream delayed payment, partial upstream advance payment, and partial backordering for deteriorating items, Journal of Manufacturing Systems, 45 (2017), 322-342.   Google Scholar

[8]

G. DobsonE. J. Pinker and O. Yildiz, An eoq model for perishable goods with age-dependent demand rate, European Journal of Operational Research, 257 (2017), 84-88.  doi: 10.1016/j.ejor.2016.06.073.  Google Scholar

[9]

R. M. Hill, Inventory models for increasing demand followed by level demand, Journal of the Operational Research Society, 46 (1995), 1250-1259.   Google Scholar

[10]

L. JanssenT. Claus and J. Sauer, Literature review of deteriorating inventory models by key topics from 2012 to 2015, International Journal of Production Economics, 182 (2016), 86-112.   Google Scholar

[11]

M. Khouja and J. Zhou, Channel and pricing decisions in a supply chain with advance selling of gift cards, European Journal of Operational Research, 244 (2015), 471-489.  doi: 10.1016/j.ejor.2015.01.045.  Google Scholar

[12]

S. Minner and S. Transchel, Order variability in perishable product supply chains, European Journal of Operational Research, 260 (2017), 93-107.  doi: 10.1016/j.ejor.2016.12.016.  Google Scholar

[13]

M. Möller and M. Watanabe, Advance purchase discounts versus clearance sales, The Economic Journal, 120(September), 120 (2010), 1125-1148.   Google Scholar

[14]

C. C. Murray, D. Talukdar and A. Gosavi, Joint optimization of product price, display orientation and shelf-space allocation in retail category management, Journal of Retailing, 86 (2010), 125–136, Special Issue: Modeling Retail Phenomena. Google Scholar

[15]

M. ÖnalA. Yenipazarli and O. E. Kundakcioglu, A mathematical model for perishable products with price- and displayed-stock-dependent demand, Computers & Industrial Engineering, 102 (2016), 246-258.   Google Scholar

[16]

S. PandaS. Senapati and M. Basu, Optimal replenishment policy for perishable seasonal products in a season with ramp-type time dependent demand, Computers & industrial engineering, 54 (2008), 301-314.   Google Scholar

[17]

A. PrasadK. E. Stecke and X. Zhao, Advance selling by a newsvendor retailer, Production and Operations Management, 20 (2011), 129-142.   Google Scholar

[18]

T. Roy and K. Chaudhuri, An inventory model for a deteriorating item with price-dependent demand and special sale, International Journal of Operational Research, 2 (2007), 173-187.   Google Scholar

[19]

S. S. Sana, An eoq model for perishable item with stock dependent demand and price discount rate, American Journal of Mathematical and Management Sciences, 30 (2010), 299-316.  doi: 10.2298/YJOR1002237P.  Google Scholar

[20]

B. SarkarB. Mandal and S. Sarkar, Quality improvement and backorder price discount under controllable lead time in an inventory model, Journal of Manufacturing Systems, 35 (2015), 26-36.   Google Scholar

[21]

Y.-C. Tsao, Retailer's optimal ordering and discounting policies under advance sales discount and trade credits, Computers & Industrial Engineering, 56 (2009), 208-215.   Google Scholar

[22]

Y.-C. TsaoJ.-C. LuN. AnF. Al-KhayyalR. W. Lu and G. Han, Retailer shelf-space management with trade allowance: A stackelberg game between retailer and manufacturers, International Journal of Production Economics, 148 (2014), 133-144.   Google Scholar

[23]

C. Wang and R. Huang, Pricing for seasonal deteriorating products with price- and ramp-type time-dependent demand, Computers & Industrial Engineering, 77 (2014), 29-34.  doi: 10.1051/ro/2015033.  Google Scholar

[24]

K.-S. Wu, An eoq inventory model for items with weibull distribution deterioration, ramp type demand rate and partial backlogging, Production Planning & Control, 12 (2001), 787-793.  doi: 10.1080/00207720110102575.  Google Scholar

[25]

P.-S. You, Optimal pricing for an advance sales system with price and waiting time dependent demands, Journal of the Operations Research Society of Japan, 50 (2007), 151-161.  doi: 10.15807/jorsj.50.151.  Google Scholar

[26]

C. Zeng, Optimal advance selling strategy under price commitment, Pacific Economic Review, 18 (2013), 233-258.   Google Scholar

[27]

L. Zhao, Optimal replenishment policy for weibull-distributed deteriorating items with trapezoidal demand rate and partial backlogging, Mathematical Problems in Engineering, 2016 (2016), 1-10.  doi: 10.1155/2016/1490712.  Google Scholar

[28]

L. Zhao and J. You, Optimal pricing and ordering policy for deteriorating items with stock-and-price dependent demand and presale rebate, Scientific Programming, 2016 (2016), 1-8.   Google Scholar

Figure 1.  The graphic of the sale model during a cycle
Figure 2.  Comparison of the profits and prices at different pre-sale market potential
Figure 3.  Comparison of the profits and prices at different deteriorating rate
Figure 4.  Comparison of the profits and prices at different price sensitivity
Figure 5.  Comparison of the profits and prices at different planning horizon
Table 1.  Summary of notations
Symbol Description
$ \epsilon $ The market demand in pre-sale period
$ \delta $ The price sensitivity of the demand
$ T $ The fixed spot-sale planning horizon
$ p_0 $ The selling price of a unit in pre-sale period under pre-sale policy
$ p $ The selling price of a unit in spot-sale period under pre-sale policy
$ p_b $ The selling price of a unit under spot-sale only policy
$ w $ The purchase price of a unit
$ \theta $ The deterioration rate of the item
$ c $ The sensitivity to the advertising policy, $ 0\leq\delta\leq1 $
$ c_h $ The holding cost per unit per unit time
$ c_d $ The deterioration cost per unit per unit time
$ L $ The length of the pre-sale horizon, and suppose $ L=\epsilon T (0\leq\epsilon\leq\frac{1}{2}) $
$ I(t) $ The instantaneous inventory level on hand at time $ [0, T]. $
Symbol Description
$ \epsilon $ The market demand in pre-sale period
$ \delta $ The price sensitivity of the demand
$ T $ The fixed spot-sale planning horizon
$ p_0 $ The selling price of a unit in pre-sale period under pre-sale policy
$ p $ The selling price of a unit in spot-sale period under pre-sale policy
$ p_b $ The selling price of a unit under spot-sale only policy
$ w $ The purchase price of a unit
$ \theta $ The deterioration rate of the item
$ c $ The sensitivity to the advertising policy, $ 0\leq\delta\leq1 $
$ c_h $ The holding cost per unit per unit time
$ c_d $ The deterioration cost per unit per unit time
$ L $ The length of the pre-sale horizon, and suppose $ L=\epsilon T (0\leq\epsilon\leq\frac{1}{2}) $
$ I(t) $ The instantaneous inventory level on hand at time $ [0, T]. $
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