American Institute of Mathematical Sciences

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April  2018, 14(2): 687-705. doi: 10.3934/jimo.2017069

A loss-averse two-product ordering model with information updating in two-echelon inventory system

Yanju Zhou 1, , Zhen Shen 1,, , Renren Ying 2, and  Xuanhua Xu 1,

1. 

School of Business, Central South University, Changsha 410083, China
2. 

School of Architecture Engineering, Jiangxi Modern Polytechnic College, Nanchang 330095, China

* Corresponding author: shenzhen@csu.edu.cn

The reviewing process was handled by Changjun Yu.

Received  December 2015 Revised  December 2016 Published  April 2018 Early access  June 2017

Fund Project: The Paper is supported by NNSF grants (No. 71221061, 71210003, 71431006, 71471178, 71171201, 71671189) and NCET grant( No. NCET-11-0524).

This paper integrates the prospect theory with two-product ordering problem and adopts Bayesian forecasting model under Brownian motion to propose a loss-averse two-product ordering model with demand information updating in a two-echelon inventory system. We also derive all psychological perceived revenue functions for sixteen supply-demand cases as well as the expected value functions and prospect value function for the loss-averse retailer. To solve this model, a Monte Carlo algorithm is presented to estimate the high dimensional integrals with curved polyhedral integral region of unknown volume. Numerical results show that the optimal order quantities of both high-risk product and low-risk product vary across different psychological reference points, which are also affected by information updating, and the loss-averse retailer benefits considerably from information updating. All results suggest that our model provides a better description of the retailer$'$s actual ordering behavior than existing models.

Keywords: Two-echelon inventory system, two-product ordering model, loss-averse, information updating, Monte Carlo algorithm.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Yanju Zhou, Zhen Shen, Renren Ying, Xuanhua Xu. A loss-averse two-product ordering model with information updating in two-echelon inventory system. Journal of Industrial and Management Optimization, 2018, 14 (2) : 687-705. doi: 10.3934/jimo.2017069
References:
[1]

L. Abdel-Malek, R. Montanari and L. C. Morales, Exact, approximate, and generic iterative models for the multi-product newsboy problem with budget constraint, International Journal of Production Economics, 2 (2004), 189-198. 

[2]

V. Agrawal and S. Seshadri, Impact of uncertainty and risk aversion on price and order quantity in the newsvendor problem, Manufacturing & Service Operations Management, 4 (2000), 410-423. 

[3]

S. Choi and A. Ruszczyński, A multi-product risk-averse newsvendor with exponential utility function, European Journal of Operational Research, 214 (2011), 78-84. 

[4]

A. Dvoretzky, J. Kiefer and J. Wolfowitz, The inventory problem: Ⅱ. Case of unknown distributions of demand, Econometrica: Journal of the Econometric Society, 20 (1952), 450-466. 

[5]

L. Eeckhoudt, C. Gollier and H. Schlesinger, The risk-averse (and prudent) newsboy, Management Science, 5 (1995), 786-794. 

[6]

G. Hadley and T. M. Whitin, Analysis of Inventory Systems, Prentice Hall, Upper Saddle River, 1994.

[7]

M. Joseph and K. Panos, On the complementary value of accurate demand information and production and supplier flexibility, Manufacturing & Service Operations Management, 2 (2002), 99-113. 

[8]

D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 2 (1979), 263-292. 

[9]

M. Khouja, The single-period (newsvendor) problem: Literature review and suggestions for future research, Omega, 5 (1999), 537-553. 

[10]

A. H. L. Lau and H. S. Lau, Decision models for single-period products with two ordering opportunities, International Journal of Production Economics, 5 (1998), 57-70. 

[11]

W. Liu, S. Song and C. Wu, Impact of loss aversion on the newsvendor game with product substitution, International Journal of Production Economics, 141 (2013), 352-359. 

[12]

X. Long and J. Nasiry, Prospect theory explains newsvendor behavior: The role of reference points, Management Science, 61 (2014), 3009-3012. 

[13]

L. Ma, Y. Zhao, W. Xue, T. Cheng and H. Yan, Loss-averse newsvendor model with two ordering opportunities and market information updating, International Journal of Production Economics, 140 (2012), 912-921. 

[14]

G. C. Mahata, A single period inventory model for incorporating two-ordering opportunities under imprecise demand information, International Journal of Industrial Engineering Computations, 2 (2011), 385-394. 

[15]

J. Miltenburg and C. Pong, Order quantities for style goods with two order opportunities and Bayesian updating of demand: Part 2-capacity constraints, International Journal of Production Research, 8 (2007), 1707-1723. 

[16]

J. V. Neuman and O. Morgenstern, Theory of Games and Economic Behavior, 2$^{nd}$ edition, Princeton university press, Princeton, 1994.

[17]

N. C. Petruzzi and M. Dada, Pricing and the newsvendor problem: A review with extensions, Operations Research, 2 (1999), 183-194. 

[18]

R. Pindyck, Irreversible investment, capacity choice, and the value of the firm, American Economic Review, 5 (1988), 969-985. 

[19]

Y. Qin, R. Wang, A. J. Vakharia, Y. Chen and M. M. H. Seref, The newsvendor problem: Review and directions for future research, European Journal of Operational Research, 213 (2011), 361-374. 

[20]

M. E. Schweitzer and G. P. Cachon, Decision bias in the newsvendor problem with a known demand distribution: experimental evidence, Management Science, 3 (2000), 404-420. 

[21]

G. H. Tannous, Capital budgeting for volume flexibility equipment, Decision Sciences, 2 (1996), 157-184. 

[22]

R. H. Thaler, A. Tversky, D. Kahneman and A Schwartz, The effect of myopia and loss aversion on risk taking: An experimental test, The Quarterly Journal of Economics, 112 (1997), 647-661. 

[23]

C. X. Wang and S. Webster, The loss-averse newsvendor problem, Omega, 37 (2009), 93-105. 

[24]

C. X. Wang, The loss-averse newsvendor game, International Journal of Production Economics, 124 (2010), 448-452. 

[25]

Q. Zhang, D. Zhang, Y. Tsao and J. Luo, Optimal ordering policy in a two-stage supply chain with advance payment for stable supply capacity, International Journal of Production Economics, 177 (2016), 34-43. 

[26]

Y. Zhou, X. Chen, X. Xu and C. Yu, A multi-product newsvendor problem with budget and loss constraints, International Journal of Information Technology & Decision Making, 5 (2005), 1093-1110. 

[27]

Y. Zhou, W. Qiu and Z. Wang, Product-portfolio Ordering Analysis with Update Information in the Two-echelon: Risk Decision-making Model, Systems Engineering-Theory & Practice, 28 (2008), 9-16. 

[28]

Y. Zhou, R. Ying, X. Chen and Z. Wang, Two-product newsboy problem based on prospect theory, Journal of Management Sciences in China, 11 (2013), 17-29. 

show all references

The reviewing process was handled by Changjun Yu.

References:
[1]

L. Abdel-Malek, R. Montanari and L. C. Morales, Exact, approximate, and generic iterative models for the multi-product newsboy problem with budget constraint, International Journal of Production Economics, 2 (2004), 189-198. 

[2]

V. Agrawal and S. Seshadri, Impact of uncertainty and risk aversion on price and order quantity in the newsvendor problem, Manufacturing & Service Operations Management, 4 (2000), 410-423. 

[3]

S. Choi and A. Ruszczyński, A multi-product risk-averse newsvendor with exponential utility function, European Journal of Operational Research, 214 (2011), 78-84. 

[4]

A. Dvoretzky, J. Kiefer and J. Wolfowitz, The inventory problem: Ⅱ. Case of unknown distributions of demand, Econometrica: Journal of the Econometric Society, 20 (1952), 450-466. 

[5]

L. Eeckhoudt, C. Gollier and H. Schlesinger, The risk-averse (and prudent) newsboy, Management Science, 5 (1995), 786-794. 

[6]

G. Hadley and T. M. Whitin, Analysis of Inventory Systems, Prentice Hall, Upper Saddle River, 1994.

[7]

M. Joseph and K. Panos, On the complementary value of accurate demand information and production and supplier flexibility, Manufacturing & Service Operations Management, 2 (2002), 99-113. 

[8]

D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 2 (1979), 263-292. 

[9]

M. Khouja, The single-period (newsvendor) problem: Literature review and suggestions for future research, Omega, 5 (1999), 537-553. 

[10]

A. H. L. Lau and H. S. Lau, Decision models for single-period products with two ordering opportunities, International Journal of Production Economics, 5 (1998), 57-70. 

[11]

W. Liu, S. Song and C. Wu, Impact of loss aversion on the newsvendor game with product substitution, International Journal of Production Economics, 141 (2013), 352-359. 

[12]

X. Long and J. Nasiry, Prospect theory explains newsvendor behavior: The role of reference points, Management Science, 61 (2014), 3009-3012. 

[13]

L. Ma, Y. Zhao, W. Xue, T. Cheng and H. Yan, Loss-averse newsvendor model with two ordering opportunities and market information updating, International Journal of Production Economics, 140 (2012), 912-921. 

[14]

G. C. Mahata, A single period inventory model for incorporating two-ordering opportunities under imprecise demand information, International Journal of Industrial Engineering Computations, 2 (2011), 385-394. 

[15]

J. Miltenburg and C. Pong, Order quantities for style goods with two order opportunities and Bayesian updating of demand: Part 2-capacity constraints, International Journal of Production Research, 8 (2007), 1707-1723. 

[16]

J. V. Neuman and O. Morgenstern, Theory of Games and Economic Behavior, 2$^{nd}$ edition, Princeton university press, Princeton, 1994.

[17]

N. C. Petruzzi and M. Dada, Pricing and the newsvendor problem: A review with extensions, Operations Research, 2 (1999), 183-194. 

[18]

R. Pindyck, Irreversible investment, capacity choice, and the value of the firm, American Economic Review, 5 (1988), 969-985. 

[19]

Y. Qin, R. Wang, A. J. Vakharia, Y. Chen and M. M. H. Seref, The newsvendor problem: Review and directions for future research, European Journal of Operational Research, 213 (2011), 361-374. 

[20]

M. E. Schweitzer and G. P. Cachon, Decision bias in the newsvendor problem with a known demand distribution: experimental evidence, Management Science, 3 (2000), 404-420. 

[21]

G. H. Tannous, Capital budgeting for volume flexibility equipment, Decision Sciences, 2 (1996), 157-184. 

[22]

R. H. Thaler, A. Tversky, D. Kahneman and A Schwartz, The effect of myopia and loss aversion on risk taking: An experimental test, The Quarterly Journal of Economics, 112 (1997), 647-661. 

[23]

C. X. Wang and S. Webster, The loss-averse newsvendor problem, Omega, 37 (2009), 93-105. 

[24]

C. X. Wang, The loss-averse newsvendor game, International Journal of Production Economics, 124 (2010), 448-452. 

[25]

Q. Zhang, D. Zhang, Y. Tsao and J. Luo, Optimal ordering policy in a two-stage supply chain with advance payment for stable supply capacity, International Journal of Production Economics, 177 (2016), 34-43. 

[26]

Y. Zhou, X. Chen, X. Xu and C. Yu, A multi-product newsvendor problem with budget and loss constraints, International Journal of Information Technology & Decision Making, 5 (2005), 1093-1110. 

[27]

Y. Zhou, W. Qiu and Z. Wang, Product-portfolio Ordering Analysis with Update Information in the Two-echelon: Risk Decision-making Model, Systems Engineering-Theory & Practice, 28 (2008), 9-16. 

[28]

Y. Zhou, R. Ying, X. Chen and Z. Wang, Two-product newsboy problem based on prospect theory, Journal of Management Sciences in China, 11 (2013), 17-29. 

Figure 1.  The Time Line of the Event
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Table 1.  Updated Demand Information Values of Two products
$\ u^{IU}_{A1}\ $ $\ \sigma^{IU2}_{A1}\ $ $\ u^{IU}_{A2}\ $ $\ \sigma^{IU2}_{A2}\ $ $\ u^{IU}_{B1}\ $ $\ \sigma^{IU2}_{B1}\ $ $\ u^{IU}_{B2}\ $ $\ \sigma^{IU2}_{B2}\ $
200123.6940063.7220057.4440059.79
Table Options

Download as excel

$\ u^{IU}_{A1}\ $ $\ \sigma^{IU2}_{A1}\ $ $\ u^{IU}_{A2}\ $ $\ \sigma^{IU2}_{A2}\ $ $\ u^{IU}_{B1}\ $ $\ \sigma^{IU2}_{B1}\ $ $\ u^{IU}_{B2}\ $ $\ \sigma^{IU2}_{B2}\ $
200123.6940063.7220057.4440059.79
Table 2.  Optimal Order Quantity with Different Psychological Reference Points and Information Updating
$\ \pi_0\ $ $\ x^{*}_{A1}\ $ $\ x^{*}_{B1}\ $ $\ x^{*}_{A2}\ $ $\ x^{*}_{B2}\ $ $\ U^*(\mathbf{x^*})\ $
02712814274293283.9
10002702784264282475.8
20002652734214271347.2
3000270268413427643.9
4000298280410403-235.7
5000315285460305-785.4
8000335290459303-1436.7
10000333285457302-2578.8
30000331283455301-3521.6
50000333288454300-4076.4
Table Options

Download as excel

$\ \pi_0\ $ $\ x^{*}_{A1}\ $ $\ x^{*}_{B1}\ $ $\ x^{*}_{A2}\ $ $\ x^{*}_{B2}\ $ $\ U^*(\mathbf{x^*})\ $
02712814274293283.9
10002702784264282475.8
20002652734214271347.2
3000270268413427643.9
4000298280410403-235.7
5000315285460305-785.4
8000335290459303-1436.7
10000333285457302-2578.8
30000331283455301-3521.6
50000333288454300-4076.4
Table 3.  Optimal Order Quantity with Different Psychological Reference Points and No Information Updating
$\ \pi_0\ $ $\ x^{*}_{A1}\ $ $\ x^{*}_{B1}\ $ $\ x^{*}_{A2}\ $ $\ x^{*}_{B2}\ $ $\ U^*(\mathbf{x^*})\ $
0402315258.5853
1000362214250.1422
200037231623-32.983
300039241821-421.655
400041252020-1674.67
500043252218-1975.9
800045282315-3452.9
1000043272513-3987.0
3000041252612-5436.9
5000043272711-6475.8
Table Options

Download as excel

$\ \pi_0\ $ $\ x^{*}_{A1}\ $ $\ x^{*}_{B1}\ $ $\ x^{*}_{A2}\ $ $\ x^{*}_{B2}\ $ $\ U^*(\mathbf{x^*})\ $
0402315258.5853
1000362214250.1422
200037231623-32.983
300039241821-421.655
400041252020-1674.67
500043252218-1975.9
800045282315-3452.9
1000043272513-3987.0
3000041252612-5436.9
5000043272711-6475.8
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