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 & Management Optimization, 2017, 13 (1) : 329-347. doi: 10.3934/jimo.2016020
References:
[1]

C. T. ChangJ. 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.   Google Scholar

[2]

M. C. ChengC. 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.  Google Scholar

[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.   Google Scholar

[4]

C. Y. DyeL. 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. J. LiaoK. 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.  Google Scholar

[8]

Y. J. LinL. 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.  Google Scholar

[9]

L. Y. OuyangK. 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.   Google Scholar

[10]

L. Y. OuyangJ. T. TengS. 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.  Google Scholar

[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.   Google Scholar

[12]

L. Y. OuyangC. T. YangY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

J. T. TengJ. Min and Q. Pan, Economic order quantity model with trade credit financing for non-decreasing demand, Omega, 40 (2012), 328-335.   Google Scholar

[19]

J. T. TengH. 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.  Google Scholar

[20]

W. C. WangJ. 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.  Google Scholar

[21]

C. T. YangQ. H. PanL. 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.   Google Scholar

show all references

References:
[1]

C. T. ChangJ. 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.   Google Scholar

[2]

M. C. ChengC. 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.  Google Scholar

[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.   Google Scholar

[4]

C. Y. DyeL. 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. J. LiaoK. 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.  Google Scholar

[8]

Y. J. LinL. 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.  Google Scholar

[9]

L. Y. OuyangK. 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.   Google Scholar

[10]

L. Y. OuyangJ. T. TengS. 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.  Google Scholar

[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.   Google Scholar

[12]

L. Y. OuyangC. T. YangY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

J. T. TengJ. Min and Q. Pan, Economic order quantity model with trade credit financing for non-decreasing demand, Omega, 40 (2012), 328-335.   Google Scholar

[19]

J. T. TengH. 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.  Google Scholar

[20]

W. C. WangJ. 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.  Google Scholar

[21]

C. T. YangQ. H. PanL. 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.   Google Scholar

Figure 1.  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

Table 1.  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\%)$
Table 2.  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=22.31 439.13
h=31.99469.42
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 $2.31439.13
$\theta=0.15 $ 2.10450.08
$\lambda $=0.1 1.91470.58
$\lambda $=0.22.31439.13
$\lambda $=0.33.10 398.34
$I_{e}=0.05 $ 2.32440.10
$I_{e}=0.10 $2.31439.13
$I_{e}=0.15 $ 2.30 438.15
$I_{P} =0.10$ 2.56426.14
$I_{P}=0.15 $ 2.31 439.13
$I_{P}=0.20 $2.12 450.65
$D_{1}=25 $1.33367.97
$D_{1}=50 $2.31439.13
$D_{1}=75 $2.98488.22
$D_{2}=15 $3.82274.97
$D_{2}=30 $2.31439.13
$D_{2}=45 $ 1.50569.49
Parameter $T^{*} $ $TC^{*} $
h =1 2.89 403.24
h=22.31 439.13
h=31.99469.42
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 $2.31439.13
$\theta=0.15 $ 2.10450.08
$\lambda $=0.1 1.91470.58
$\lambda $=0.22.31439.13
$\lambda $=0.33.10 398.34
$I_{e}=0.05 $ 2.32440.10
$I_{e}=0.10 $2.31439.13
$I_{e}=0.15 $ 2.30 438.15
$I_{P} =0.10$ 2.56426.14
$I_{P}=0.15 $ 2.31 439.13
$I_{P}=0.20 $2.12 450.65
$D_{1}=25 $1.33367.97
$D_{1}=50 $2.31439.13
$D_{1}=75 $2.98488.22
$D_{2}=15 $3.82274.97
$D_{2}=30 $2.31439.13
$D_{2}=45 $ 1.50569.49
Table 3.  Effects of system parameters (when${\rm \; }t_{d}=0.5 $ and $t_{C}=0.3 $)
Parameter $T^{*} $ $TC^{*} $
h =12.40382.19
h=21.91412.67
h=31.63 438.08
R =401.61389.91
R =801.912.17
R =120412.67432.30
$\theta $=0.052.16399.93
$\theta=0.1 $1.91412.67
$\theta=0.15 $1.74423.22
$\lambda $=0.11.60438.45
$\lambda=0.2 $1.91412.67
$\lambda=0.3 $2.49380.27
$I_{e} $=0.051.95416.21
$I_{e}=0.10 $1.91412.67
$I_{e}=0.15 $1.86409.05
$I_{P} $=0.102.10404.72
$I_{P}=0.15 $1.91412.67
$I_{P}=0.20 $1.76419.51
$D_{1} $=251.32368.04
$D_{1}=50 $1.91412.67
$D_{1}=75 $2.35446.46
$D_{2}=15 $3.09251.14
$D_{2}=30 $1.91412.67
$D_{2}=45 $1.29549.00
Parameter $T^{*} $ $TC^{*} $
h =12.40382.19
h=21.91412.67
h=31.63 438.08
R =401.61389.91
R =801.912.17
R =120412.67432.30
$\theta $=0.052.16399.93
$\theta=0.1 $1.91412.67
$\theta=0.15 $1.74423.22
$\lambda $=0.11.60438.45
$\lambda=0.2 $1.91412.67
$\lambda=0.3 $2.49380.27
$I_{e} $=0.051.95416.21
$I_{e}=0.10 $1.91412.67
$I_{e}=0.15 $1.86409.05
$I_{P} $=0.102.10404.72
$I_{P}=0.15 $1.91412.67
$I_{P}=0.20 $1.76419.51
$D_{1} $=251.32368.04
$D_{1}=50 $1.91412.67
$D_{1}=75 $2.35446.46
$D_{2}=15 $3.09251.14
$D_{2}=30 $1.91412.67
$D_{2}=45 $1.29549.00
Table 4.  Effects of system parameters (when${\rm \; }t_{d} $=0.3 and $t_{C} $=0.5)
Parameter $T^{*} $ $TC^{*} $
h =12.83383.86
h=22.27429.09
h=31.95458.90
R =402.01410.40
R =802.27429.09
R =1202.49445.90
$\theta $=0.052.56415.86
$\theta $=0.12.27429.09
$\theta $=0.152.07439.42
$\lambda $=0.11.88459.17
$\lambda $=0.22.27429.09
$\lambda $=0.33.00390.41
$I_{e} $=0.052.30431.83
$I_{e} $=0.102.27429.09
$I_{e} $=0.152.23426.31
$I_{P} $=0.102.51418.60
$I_{P} $=0.152.27429.09
$I_{P} $=0.202.09438.18
$D_{1} $=251.33359.50
$D_{1} $=502.27429.09
$D_{1} $=752.92477.32
$D_{2} $=153.75269.41
$D_{2} $=302.27429.09
$D_{2} $=451.47555.33
Parameter $T^{*} $ $TC^{*} $
h =12.83383.86
h=22.27429.09
h=31.95458.90
R =402.01410.40
R =802.27429.09
R =1202.49445.90
$\theta $=0.052.56415.86
$\theta $=0.12.27429.09
$\theta $=0.152.07439.42
$\lambda $=0.11.88459.17
$\lambda $=0.22.27429.09
$\lambda $=0.33.00390.41
$I_{e} $=0.052.30431.83
$I_{e} $=0.102.27429.09
$I_{e} $=0.152.23426.31
$I_{P} $=0.102.51418.60
$I_{P} $=0.152.27429.09
$I_{P} $=0.202.09438.18
$D_{1} $=251.33359.50
$D_{1} $=502.27429.09
$D_{1} $=752.92477.32
$D_{2} $=153.75269.41
$D_{2} $=302.27429.09
$D_{2} $=451.47555.33
Table 5.  Effects of system parameters (when${\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
Table 6.  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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