# American Institute of Mathematical Sciences

January  2017, 13(1): 329-347. doi: 10.3934/jimo.2016020

## Ordering policy for non-instantaneously deteriorating products under price adjustment and trade credits

 Department of Industrial Management, National Taiwan University of Science and Technology, Taipei, Taiwan

* Corresponding author: Yu-Chung Tsao

Received  January 2015 Published  March 2016

Non-instantaneously deteriorating products retain their quality for a certain period before beginning to deteriorate. Retailers commonly adjust their retail prices when products shift from a non-deteriorating state to a deteriorating state in order to stimulate demand. It is essential to consider this price adjustment for inventory models of non-instantaneously deteriorating products under trade credit, due to the fact that the calculation of earned interest is based on the retail price. This paper considers the problem of ordering non-instantaneously deteriorating products under price adjustment and trade credit. Our objective was to determine the optimal replenishment cycle time while minimizing total costs. The problem is formulated as three piecewise nonlinear functions, which are solved through optimization. Numerical simulation is used to illustrate the solution procedures and discuss how system parameters influence inventory decisions and total cost. We also show that a policy of price adjustment is superior to that of fixed pricing with regard to profit maximization.

Citation: Yu-Chung Tsao. Ordering policy for non-instantaneously deteriorating products under price adjustment and trade credits. Journal of Industrial and Management Optimization, 2017, 13 (1) : 329-347. doi: 10.3934/jimo.2016020
##### References:
 [1] C. T. Chang, J. T. Teng and M. S. Chern, Optimal manufacturer's replenishment policies for deteriorating items in a supply chain with up-stream and down-stream trade credits, International Journal of Production Economics, 127 (2010), 197-202. [2] M. C. Cheng, C. T. Chang and L. Y. Ouyang, The retailer's optimal ordering policy with trade credit in different financial environments, Applied Mathematics and Computations, 218 (2012), 9623-9634.  doi: 10.1016/j.amc.2012.02.066. [3] K. J. Chung, A complete proof on the solution procedure for non-instantaneous deteriorating items with permissible delay in payments, Computers and Industrial Engineering, 56 (2009), 267-273. [4] C. Y. Dye, L. Y. Ouyang and T. P. Hsieh, Inventory and pricing strategies for deteriorating items with shortages: A discounted cash flow approach, Computers and Industrial Engineering, 52 (2007), 29-40. [5] C. Y. Dye and T. P. Hsieh, A optimal replenishment policy for deteriorating items with effective investment in preservation technology, European Journal of Operational Research, 218 (2012), 106-112.  doi: 10.1016/j.ejor.2011.10.016. [6] C. Y. Dye and T. P. Hsieh, A particle swarm optimization for solving lot-sizing problem with fluctuating demand and preservation technology cost under trade credit, Journal of Global Optimization, 55 (2013), 655-679.  doi: 10.1007/s10898-012-9950-z. [7] J. J. Liao, K. N. Huang and P. S. Tinnng, Optimal strategy of deteriorating items with capacity constraints under two-levels of trade credit policy, Applied Mathematics and Computation, 233 (2014), 647-658.  doi: 10.1016/j.amc.2014.01.077. [8] Y. J. Lin, L. Y. Ouyang and Y. F. Dang, A joint optimal ordering and delivery policy for an integrated supplier-retailer inventory model with trade credit and defective items, Applied Mathematics and Computation, 218 (2012), 7498-7514.  doi: 10.1016/j.amc.2012.01.016. [9] L. Y. Ouyang, K. S. Wu and C. T. Yang, A study on an inventory model for non-instantaneous deteriorating items with permissible delay in payments, Computers & Industrial Engineering, 51 (2006), 637-651. [10] L. Y. Ouyang, J. T. Teng, S. K. Goyal and C. T. Yang, An economic order quantity model for deteriorating items with partially permissible delay in payments linked to order quantity, European Journal of Operational Research, 194 (2009), 418-431.  doi: 10.1016/j.ejor.2007.12.018. [11] L. Y. Ouyang and C. T. Chang, Optimal production lot with imperfect production process under permissible delay in payments and complete backlogging, International Journal of Production Economics, 144 (2013), 610-617. [12] L. Y. Ouyang, C. T. Yang, Y. L. Chan and L. E. Cárdenas-Barrón, A comprehensive extension of the optimal replenishment decisions under two levels of trade credit policy depending on the order quantity, Applied Mathematics and Computation, 224 (2013), 268-277.  doi: 10.1016/j.amc.2013.08.062. [13] Y. C. Tsao, Two-phase pricing and inventory management for deteriorating and fashion goods under trade credit, Mathematical Methods of Operations Research, 72 (2010), 107-127.  doi: 10.1007/s00186-010-0309-2. [14] Y. C. Tsao, Joint location, inventory and preservation decisions for non-instantaneous deterioration items under delay in payments, International Journal of Systems Science, 47 (2016), 572-585.  doi: 10.1080/00207721.2014.891672. [15] Y. C. Tsao, A piecewise nonlinear model for a production system under maintenance, trade credit and limited warehouse space, International Journal of Production Research, 52 (2014), 3052-3073. [16] J. T. Teng and T. C. Chang, Economic production quantity models for deteriorating items with price-and stock-dependent demand, Computers & Operations Research, 32 (2005), 297-308.  doi: 10.1016/S0305-0548(03)00237-5. [17] J. T. Teng and K. R. Lou, Seller's optimal credit period and replenishment time in a supply chain with up-stream and down-stream trade credits, Journal of Global Optimization, 53 (2012), 41-430.  doi: 10.1007/s10898-011-9720-3. [18] J. T. Teng, J. Min and Q. Pan, Economic order quantity model with trade credit financing for non-decreasing demand, Omega, 40 (2012), 328-335. [19] J. T. Teng, H. L. Yang and M. S. Chern, An inventory model for increasing demand under two levels of trade credit linked to order quantity, Applied Mathematical Modelling, 37 (2013), 7624-7632.  doi: 10.1016/j.apm.2013.02.009. [20] W. C. Wang, J. T. Teng and K. R. Lou, Seller's optimal credit period and cycle time in a supply chain for deteriorating items with maximum lifetime, European Journal of Operational Research, 232 (2014), 315-321.  doi: 10.1016/j.ejor.2013.06.027. [21] C. T. Yang, Q. H. Pan, L. Y. Ouyang and J. T. Teng, Retailer's optimal order and credit policies when a supplier offers either a cash discount or a delay payment linked to order quantity, European Journal of Industrial Engineering, 7-3 (2013), 370-392.

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##### References:
 [1] C. T. Chang, J. T. Teng and M. S. Chern, Optimal manufacturer's replenishment policies for deteriorating items in a supply chain with up-stream and down-stream trade credits, International Journal of Production Economics, 127 (2010), 197-202. [2] M. C. Cheng, C. T. Chang and L. Y. Ouyang, The retailer's optimal ordering policy with trade credit in different financial environments, Applied Mathematics and Computations, 218 (2012), 9623-9634.  doi: 10.1016/j.amc.2012.02.066. [3] K. J. Chung, A complete proof on the solution procedure for non-instantaneous deteriorating items with permissible delay in payments, Computers and Industrial Engineering, 56 (2009), 267-273. [4] C. Y. Dye, L. Y. Ouyang and T. P. Hsieh, Inventory and pricing strategies for deteriorating items with shortages: A discounted cash flow approach, Computers and Industrial Engineering, 52 (2007), 29-40. [5] C. Y. Dye and T. P. Hsieh, A optimal replenishment policy for deteriorating items with effective investment in preservation technology, European Journal of Operational Research, 218 (2012), 106-112.  doi: 10.1016/j.ejor.2011.10.016. [6] C. Y. Dye and T. P. Hsieh, A particle swarm optimization for solving lot-sizing problem with fluctuating demand and preservation technology cost under trade credit, Journal of Global Optimization, 55 (2013), 655-679.  doi: 10.1007/s10898-012-9950-z. [7] J. J. Liao, K. N. Huang and P. S. Tinnng, Optimal strategy of deteriorating items with capacity constraints under two-levels of trade credit policy, Applied Mathematics and Computation, 233 (2014), 647-658.  doi: 10.1016/j.amc.2014.01.077. [8] Y. J. Lin, L. Y. Ouyang and Y. F. Dang, A joint optimal ordering and delivery policy for an integrated supplier-retailer inventory model with trade credit and defective items, Applied Mathematics and Computation, 218 (2012), 7498-7514.  doi: 10.1016/j.amc.2012.01.016. [9] L. Y. Ouyang, K. S. Wu and C. T. Yang, A study on an inventory model for non-instantaneous deteriorating items with permissible delay in payments, Computers & Industrial Engineering, 51 (2006), 637-651. [10] L. Y. Ouyang, J. T. Teng, S. K. Goyal and C. T. Yang, An economic order quantity model for deteriorating items with partially permissible delay in payments linked to order quantity, European Journal of Operational Research, 194 (2009), 418-431.  doi: 10.1016/j.ejor.2007.12.018. [11] L. Y. Ouyang and C. T. Chang, Optimal production lot with imperfect production process under permissible delay in payments and complete backlogging, International Journal of Production Economics, 144 (2013), 610-617. [12] L. Y. Ouyang, C. T. Yang, Y. L. Chan and L. E. Cárdenas-Barrón, A comprehensive extension of the optimal replenishment decisions under two levels of trade credit policy depending on the order quantity, Applied Mathematics and Computation, 224 (2013), 268-277.  doi: 10.1016/j.amc.2013.08.062. [13] Y. C. Tsao, Two-phase pricing and inventory management for deteriorating and fashion goods under trade credit, Mathematical Methods of Operations Research, 72 (2010), 107-127.  doi: 10.1007/s00186-010-0309-2. [14] Y. C. Tsao, Joint location, inventory and preservation decisions for non-instantaneous deterioration items under delay in payments, International Journal of Systems Science, 47 (2016), 572-585.  doi: 10.1080/00207721.2014.891672. [15] Y. C. Tsao, A piecewise nonlinear model for a production system under maintenance, trade credit and limited warehouse space, International Journal of Production Research, 52 (2014), 3052-3073. [16] J. T. Teng and T. C. Chang, Economic production quantity models for deteriorating items with price-and stock-dependent demand, Computers & Operations Research, 32 (2005), 297-308.  doi: 10.1016/S0305-0548(03)00237-5. [17] J. T. Teng and K. R. Lou, Seller's optimal credit period and replenishment time in a supply chain with up-stream and down-stream trade credits, Journal of Global Optimization, 53 (2012), 41-430.  doi: 10.1007/s10898-011-9720-3. [18] J. T. Teng, J. Min and Q. Pan, Economic order quantity model with trade credit financing for non-decreasing demand, Omega, 40 (2012), 328-335. [19] J. T. Teng, H. L. Yang and M. S. Chern, An inventory model for increasing demand under two levels of trade credit linked to order quantity, Applied Mathematical Modelling, 37 (2013), 7624-7632.  doi: 10.1016/j.apm.2013.02.009. [20] W. C. Wang, J. T. Teng and K. R. Lou, Seller's optimal credit period and cycle time in a supply chain for deteriorating items with maximum lifetime, European Journal of Operational Research, 232 (2014), 315-321.  doi: 10.1016/j.ejor.2013.06.027. [21] C. T. Yang, Q. H. Pan, L. Y. Ouyang and J. T. Teng, Retailer's optimal order and credit policies when a supplier offers either a cash discount or a delay payment linked to order quantity, European Journal of Industrial Engineering, 7-3 (2013), 370-392.
The graphic illustrations of TC versus T

1. When $t_{d}$=0.5 and $t_{C}$=0.32. When $t_{d}$=0.3 and $t_{C}$=0.5 3. When $t_{d}$=0.5 and $t_{C}$=0.5

Effects of replenishment cycle time on total cost
 When ${\rm \; }t_{d}=0.5$ and $t_{C}=0.3$ When ${\rm \; }t_{d}=0.5$ and $t_{C}=0.3$ When ${\rm \; }t_{d}=0.5$ and $t_{C}=0.5$ $50\%T^{*}$ TC=481.46 $(9.64\%)$ ${}^{a}$ TC=448.79 $(8.75\%)$ TC=$471.03 (9.77\%)$ $75\%T^{*}$ TC=446.18$(1.61\%)$ TC=$418.69 (1.46\%)$ TC=436.08$(1.63\%)$ $T^{*}$ TC=439.13$(0\%)$ TC=412.67$(0\%)$ TC=429.09$(0\%)$ $125\%T^{*}$ TC=443.36 $(0.96\%)$ TC=416.28$(0.87\%)$ TC=433.28$(0.98\%)$ $150\%T^{*}$ TC=453.24$(3.21\%)$ TC=424.71$(2.92\%)$ TC=443.07$(3.26\%)$
 When ${\rm \; }t_{d}=0.5$ and $t_{C}=0.3$ When ${\rm \; }t_{d}=0.5$ and $t_{C}=0.3$ When ${\rm \; }t_{d}=0.5$ and $t_{C}=0.5$ $50\%T^{*}$ TC=481.46 $(9.64\%)$ ${}^{a}$ TC=448.79 $(8.75\%)$ TC=$471.03 (9.77\%)$ $75\%T^{*}$ TC=446.18$(1.61\%)$ TC=$418.69 (1.46\%)$ TC=436.08$(1.63\%)$ $T^{*}$ TC=439.13$(0\%)$ TC=412.67$(0\%)$ TC=429.09$(0\%)$ $125\%T^{*}$ TC=443.36 $(0.96\%)$ TC=416.28$(0.87\%)$ TC=433.28$(0.98\%)$ $150\%T^{*}$ TC=453.24$(3.21\%)$ TC=424.71$(2.92\%)$ TC=443.07$(3.26\%)$
Effects of replenishment cycle time on total cost (When ${\rm \; }t_{d}=0.5$ and $t_{C}=0.3$)
 Parameter $T^{*}$ $TC^{*}$ h =1 2.89 403.24 h=2 2.31 439.13 h=3 1.99 469.42 R =40 2.06 420.81 R =80 2.31 439.13 R =120 2.53 455.65 $\theta =0.05$ 2.63 425.04 $\theta=0.1$ 2.31 439.13 $\theta=0.15$ 2.10 450.08 $\lambda$=0.1 1.91 470.58 $\lambda$=0.2 2.31 439.13 $\lambda$=0.3 3.10 398.34 $I_{e}=0.05$ 2.32 440.10 $I_{e}=0.10$ 2.31 439.13 $I_{e}=0.15$ 2.30 438.15 $I_{P} =0.10$ 2.56 426.14 $I_{P}=0.15$ 2.31 439.13 $I_{P}=0.20$ 2.12 450.65 $D_{1}=25$ 1.33 367.97 $D_{1}=50$ 2.31 439.13 $D_{1}=75$ 2.98 488.22 $D_{2}=15$ 3.82 274.97 $D_{2}=30$ 2.31 439.13 $D_{2}=45$ 1.50 569.49
 Parameter $T^{*}$ $TC^{*}$ h =1 2.89 403.24 h=2 2.31 439.13 h=3 1.99 469.42 R =40 2.06 420.81 R =80 2.31 439.13 R =120 2.53 455.65 $\theta =0.05$ 2.63 425.04 $\theta=0.1$ 2.31 439.13 $\theta=0.15$ 2.10 450.08 $\lambda$=0.1 1.91 470.58 $\lambda$=0.2 2.31 439.13 $\lambda$=0.3 3.10 398.34 $I_{e}=0.05$ 2.32 440.10 $I_{e}=0.10$ 2.31 439.13 $I_{e}=0.15$ 2.30 438.15 $I_{P} =0.10$ 2.56 426.14 $I_{P}=0.15$ 2.31 439.13 $I_{P}=0.20$ 2.12 450.65 $D_{1}=25$ 1.33 367.97 $D_{1}=50$ 2.31 439.13 $D_{1}=75$ 2.98 488.22 $D_{2}=15$ 3.82 274.97 $D_{2}=30$ 2.31 439.13 $D_{2}=45$ 1.50 569.49
Effects of system parameters (when${\rm \; }t_{d}=0.5$ and $t_{C}=0.3$)
 Parameter $T^{*}$ $TC^{*}$ h =1 2.40 382.19 h=2 1.91 412.67 h=3 1.63 438.08 R =40 1.61 389.91 R =80 1.91 2.17 R =120 412.67 432.30 $\theta$=0.05 2.16 399.93 $\theta=0.1$ 1.91 412.67 $\theta=0.15$ 1.74 423.22 $\lambda$=0.1 1.60 438.45 $\lambda=0.2$ 1.91 412.67 $\lambda=0.3$ 2.49 380.27 $I_{e}$=0.05 1.95 416.21 $I_{e}=0.10$ 1.91 412.67 $I_{e}=0.15$ 1.86 409.05 $I_{P}$=0.10 2.10 404.72 $I_{P}=0.15$ 1.91 412.67 $I_{P}=0.20$ 1.76 419.51 $D_{1}$=25 1.32 368.04 $D_{1}=50$ 1.91 412.67 $D_{1}=75$ 2.35 446.46 $D_{2}=15$ 3.09 251.14 $D_{2}=30$ 1.91 412.67 $D_{2}=45$ 1.29 549.00
 Parameter $T^{*}$ $TC^{*}$ h =1 2.40 382.19 h=2 1.91 412.67 h=3 1.63 438.08 R =40 1.61 389.91 R =80 1.91 2.17 R =120 412.67 432.30 $\theta$=0.05 2.16 399.93 $\theta=0.1$ 1.91 412.67 $\theta=0.15$ 1.74 423.22 $\lambda$=0.1 1.60 438.45 $\lambda=0.2$ 1.91 412.67 $\lambda=0.3$ 2.49 380.27 $I_{e}$=0.05 1.95 416.21 $I_{e}=0.10$ 1.91 412.67 $I_{e}=0.15$ 1.86 409.05 $I_{P}$=0.10 2.10 404.72 $I_{P}=0.15$ 1.91 412.67 $I_{P}=0.20$ 1.76 419.51 $D_{1}$=25 1.32 368.04 $D_{1}=50$ 1.91 412.67 $D_{1}=75$ 2.35 446.46 $D_{2}=15$ 3.09 251.14 $D_{2}=30$ 1.91 412.67 $D_{2}=45$ 1.29 549.00
Effects of system parameters (when${\rm \; }t_{d}$=0.3 and $t_{C}$=0.5)
 Parameter $T^{*}$ $TC^{*}$ h =1 2.83 383.86 h=2 2.27 429.09 h=3 1.95 458.90 R =40 2.01 410.40 R =80 2.27 429.09 R =120 2.49 445.90 $\theta$=0.05 2.56 415.86 $\theta$=0.1 2.27 429.09 $\theta$=0.15 2.07 439.42 $\lambda$=0.1 1.88 459.17 $\lambda$=0.2 2.27 429.09 $\lambda$=0.3 3.00 390.41 $I_{e}$=0.05 2.30 431.83 $I_{e}$=0.10 2.27 429.09 $I_{e}$=0.15 2.23 426.31 $I_{P}$=0.10 2.51 418.60 $I_{P}$=0.15 2.27 429.09 $I_{P}$=0.20 2.09 438.18 $D_{1}$=25 1.33 359.50 $D_{1}$=50 2.27 429.09 $D_{1}$=75 2.92 477.32 $D_{2}$=15 3.75 269.41 $D_{2}$=30 2.27 429.09 $D_{2}$=45 1.47 555.33
 Parameter $T^{*}$ $TC^{*}$ h =1 2.83 383.86 h=2 2.27 429.09 h=3 1.95 458.90 R =40 2.01 410.40 R =80 2.27 429.09 R =120 2.49 445.90 $\theta$=0.05 2.56 415.86 $\theta$=0.1 2.27 429.09 $\theta$=0.15 2.07 439.42 $\lambda$=0.1 1.88 459.17 $\lambda$=0.2 2.27 429.09 $\lambda$=0.3 3.00 390.41 $I_{e}$=0.05 2.30 431.83 $I_{e}$=0.10 2.27 429.09 $I_{e}$=0.15 2.23 426.31 $I_{P}$=0.10 2.51 418.60 $I_{P}$=0.15 2.27 429.09 $I_{P}$=0.20 2.09 438.18 $D_{1}$=25 1.33 359.50 $D_{1}$=50 2.27 429.09 $D_{1}$=75 2.92 477.32 $D_{2}$=15 3.75 269.41 $D_{2}$=30 2.27 429.09 $D_{2}$=45 1.47 555.33
Effects of system parameters (when${\rm \; }t_{d}$=0.5 and $t_{C}$=0.5)
 Parameter $T^{*}$ $TC^{*}$ ${\rm \; }t_{d}=0.1$, $t_{C}=0.3$ 1.58 407.73 ${\rm \; }t_{d}=0.3$, $t_{C}=0.3$ 1.93 424.85 ${\rm \; }t_{d}=0.5$, $t_{C}=0.3$ 2.31 439.13 ${\rm \; }t_{d}=0.7$, $t_{C}=0.3$ 2.70 452.24 ${\rm \; }t_{d}=0.9$, $t_{C}=0.3$ 3.10 464.39 ${\rm \; }t_{d}=0.5$, $t_{C}=0.1$ 2.34 448.68 ${\rm \; }t_{d}=0.5$, $t_{C}=0.3$ 2.31 439.13 ${\rm \; }t_{d}=0.5$, $t_{C}=0.5$ 2.27 429.09 ${\rm \; }t_{d}=0.5$, $t_{C}=0.7$ 2.22 415.92 ${\rm \; }t_{d}=0.5$, $t_{C}=0.9$ 2.18 403.25
 Parameter $T^{*}$ $TC^{*}$ ${\rm \; }t_{d}=0.1$, $t_{C}=0.3$ 1.58 407.73 ${\rm \; }t_{d}=0.3$, $t_{C}=0.3$ 1.93 424.85 ${\rm \; }t_{d}=0.5$, $t_{C}=0.3$ 2.31 439.13 ${\rm \; }t_{d}=0.7$, $t_{C}=0.3$ 2.70 452.24 ${\rm \; }t_{d}=0.9$, $t_{C}=0.3$ 3.10 464.39 ${\rm \; }t_{d}=0.5$, $t_{C}=0.1$ 2.34 448.68 ${\rm \; }t_{d}=0.5$, $t_{C}=0.3$ 2.31 439.13 ${\rm \; }t_{d}=0.5$, $t_{C}=0.5$ 2.27 429.09 ${\rm \; }t_{d}=0.5$, $t_{C}=0.7$ 2.22 415.92 ${\rm \; }t_{d}=0.5$, $t_{C}=0.9$ 2.18 403.25
Comparison of two-phase pricing and one-phase pricing
 When ${\rm \; }t_{d}$=0.5 and $t_{C}$=0.3 When ${\rm \; }t_{d}$=0.3 and $t_{C}$=0.5 When ${\rm \; }t_{d}$=0.5 and $t_{C}$=0.5 Two-phase pricing $p_{1}$=39.22 $p_{1}$=38.28 $p_{1}$=38.81 $p_{2}$=23.77 $p_{2}$=28.48 $p_{2}$=23.25 T=1.19 T=2.33 T=1.10 TP=614.40 $(+14.05\%){}^{a}$ TP=415.09 $(+6.73\%){}^{ }$ TP=658.51 $(+14.13\%)$ One-phase pricing p=30.28 p=30.94 p=30.39 T=1.46 T=2.45 T=1.35 TP=538.69 TP=388.92 TP=576.50 a. the percentage of profit increasing
 When ${\rm \; }t_{d}$=0.5 and $t_{C}$=0.3 When ${\rm \; }t_{d}$=0.3 and $t_{C}$=0.5 When ${\rm \; }t_{d}$=0.5 and $t_{C}$=0.5 Two-phase pricing $p_{1}$=39.22 $p_{1}$=38.28 $p_{1}$=38.81 $p_{2}$=23.77 $p_{2}$=28.48 $p_{2}$=23.25 T=1.19 T=2.33 T=1.10 TP=614.40 $(+14.05\%){}^{a}$ TP=415.09 $(+6.73\%){}^{ }$ TP=658.51 $(+14.13\%)$ One-phase pricing p=30.28 p=30.94 p=30.39 T=1.46 T=2.45 T=1.35 TP=538.69 TP=388.92 TP=576.50 a. the percentage of profit increasing
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Using the algebraic approach to determine the replenishment optimal policy with defective products, backlog and delay of payments in the supply chain management. Journal of Industrial and Management Optimization, 2012, 8 (1) : 263-269. doi: 10.3934/jimo.2012.8.263 [19] Yu-Chung Tsao, Hanifa-Astofa Fauziah, Thuy-Linh Vu, Nur Aini Masruroh. Optimal pricing, ordering, and credit period policies for deteriorating products under order-linked trade credit. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021152 [20] 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

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