# American Institute of Mathematical Sciences

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

 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  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.

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 & 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

##### 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.
The Time Line of the Event
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}\$ 200 123.69 400 63.72 200 57.44 400 59.79
 $\ u^{IU}_{A1}\$ $\ \sigma^{IU2}_{A1}\$ $\ u^{IU}_{A2}\$ $\ \sigma^{IU2}_{A2}\$ $\ u^{IU}_{B1}\$ $\ \sigma^{IU2}_{B1}\$ $\ u^{IU}_{B2}\$ $\ \sigma^{IU2}_{B2}\$ 200 123.69 400 63.72 200 57.44 400 59.79
Optimal Order Quantity with Different Psychological Reference Points and Information Updating
 $\ \pi_0\$ $\ x^{*}_{A1}\$ $\ x^{*}_{B1}\$ $\ x^{*}_{A2}\$ $\ x^{*}_{B2}\$ $\ U^*(\mathbf{x^*})\$ 0 271 281 427 429 3283.9 1000 270 278 426 428 2475.8 2000 265 273 421 427 1347.2 3000 270 268 413 427 643.9 4000 298 280 410 403 -235.7 5000 315 285 460 305 -785.4 8000 335 290 459 303 -1436.7 10000 333 285 457 302 -2578.8 30000 331 283 455 301 -3521.6 50000 333 288 454 300 -4076.4
 $\ \pi_0\$ $\ x^{*}_{A1}\$ $\ x^{*}_{B1}\$ $\ x^{*}_{A2}\$ $\ x^{*}_{B2}\$ $\ U^*(\mathbf{x^*})\$ 0 271 281 427 429 3283.9 1000 270 278 426 428 2475.8 2000 265 273 421 427 1347.2 3000 270 268 413 427 643.9 4000 298 280 410 403 -235.7 5000 315 285 460 305 -785.4 8000 335 290 459 303 -1436.7 10000 333 285 457 302 -2578.8 30000 331 283 455 301 -3521.6 50000 333 288 454 300 -4076.4
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^*})\$ 0 40 23 15 25 8.5853 1000 36 22 14 25 0.1422 2000 37 23 16 23 -32.983 3000 39 24 18 21 -421.655 4000 41 25 20 20 -1674.67 5000 43 25 22 18 -1975.9 8000 45 28 23 15 -3452.9 10000 43 27 25 13 -3987.0 30000 41 25 26 12 -5436.9 50000 43 27 27 11 -6475.8
 $\ \pi_0\$ $\ x^{*}_{A1}\$ $\ x^{*}_{B1}\$ $\ x^{*}_{A2}\$ $\ x^{*}_{B2}\$ $\ U^*(\mathbf{x^*})\$ 0 40 23 15 25 8.5853 1000 36 22 14 25 0.1422 2000 37 23 16 23 -32.983 3000 39 24 18 21 -421.655 4000 41 25 20 20 -1674.67 5000 43 25 22 18 -1975.9 8000 45 28 23 15 -3452.9 10000 43 27 25 13 -3987.0 30000 41 25 26 12 -5436.9 50000 43 27 27 11 -6475.8
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