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