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Article Contents

# Optimal pricing and ordering policy for defective items under temporary price reduction with inspection errors and price sensitive demand

• * Corresponding author: Guiyang Zhu
• This paper studies the retailer's optimal promotional pricing, special order quantity and screening rate for defective items when a temporary price reduction (i.e., TPR) is offered. Although previous studies have examined the similar issue, they assume a constant demand and an error-free screening process. A subversion of these two assumptions differentiates our paper. First, using a price-sensitive demand, we analyze that the original screening rate may be insufficient, and propose the CPD (i.e., control the promotional demand) and the ISR (i.e., increase the screening rate through investment) strategy to handle it. Second, we incorporate both Type I and Type II inspection errors into our model. Then we establish an inventory model aiming to maximize the retailer's profit under CPD and ISR, respectively. Finally, numerical examples are conducted and several results are obtained: (1) a higher portion of defects makes ISR more profitable; (2) both a higher probability of a Type I error and a Type II error decrease the profit under CPD and ISR, but a Type I error has a more pronounced negative impact; and (3) comparing with the existing studies with a constant demand, our model generates a higher profit especially in markets with a higher price sensitivity.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

 Citation:

• Figure 1.  Retailer's regular order policy

Figure 2.  Retailer's special order policy with TPR at time 0

Figure 3.  Retailer's special order policy with TPR at time 0 (corresponding to ISR)

Figure 4.  Comparison of the solutions and profits under the CPD and ISR strategy as $p$ increases

Figure 5.  Comparison between the changing trends of the results as $m_1$ increases and those as $m_2$ increases

Figure 6.  Effects of an increase in $b$ on the profits of our model and the EOQ model in Hsu and Yu [22]

Table 1.  Brief literature on inventory models under limited-time price incentives and/or with defective items

Table 2.  Model parameters

 Parameters Value $A$ $＄500/order$ $r$ $0.1＄/＄/day$ $c$ $＄25/unit$ $x_0$ $242 units/day$ $d$ $＄0.5/unit$ $v$ $＄20/unit$ $s_0$ $＄50/unit$ $p$ $p$ varies between 0.001 and 0.025; for example, $p=0.001$ $m_1$ $m_1$ varies between 0.01 to 0.05; for example, $m_1=0.01$ $m_2$ $m_2$ varies between 0.01 to 0.05; for example, $m_2=0.01$ $p_e$ $p_e=(1-p)m_1+p(1-m_2)$; for example, when $p=0.001$, $m_1=0.01$, and $m_2=0.01$, we have $p_e=0.01098$ $c_a$ $＄500/unit$ $c_r$ $＄100/unit$ $N$ 365 $k$ $＄7.5/unit$

Table 3.  Effects of an increase in defect proportion $p$ on the solutions and profits under the CPD and ISR strategy ($m_1 = 0.02$, and $m_2 = 0.02$. $G^{(1)\star}$ represents $G^{(1)}(s^\star_1, Q^\star_1)$, and $G^{(2)\star}$ represents $G^{(2)}(s^\star_1, Q^\star_1, z^\star)$. Tables 4-8 follow this marking method)

 $p$ $Q^\star_0$ the CPD strategy the ISR strategy $s^\star_1$ $Q^\star_1$ $G^{(1)\star}$ $s^\star_1$ $Q^\star_1$ $z^\star$ $x^\star$ $G^{(2)\star}$ 0.001 283.81 46.92 1472.15 5221.87 46.02 1499.90 0.76 324.92 5089.82 0.003 283.89 46.96 1469.48 5207.56 46.03 1500.85 0.73 339.41 5084.93 0.005 283.97 47.00 1466.81 5193.30 46.04 1501.93 0.70 353.31 5080.96 0.007 284.04 47.04 1464.16 5179.07 46.05 1503.12 0.67 366.69 5077.82 0.009 284.12 47.08 1461.51 5164.89 46.06 1504.42 0.65 379.60 5075.43 0.011 284.19 47.12 1458.87 5150.74 46.07 1505.81 0.63 392.09 5073.71 0.013 284.26 47.16 1456.24 5136.64 46.08 1507.29 0.61 404.20 5072.60 0.015 284.33 47.19 1453.62 5122.58 46.09 1508.85 0.59 415.97 5072.06 0.017 284.41 47.23 1451.00 5108.56 46.10 1510.48 0.58 427.42 5072.05 0.019 284.48 47.27 1448.40 5094.59 46.11 1512.18 0.56 438.58 5072.51 0.021 284.54 47.31 1445.80 5080.66 46.13 1513.94 0.55 449.46 5073.43 0.023 284.61 47.35 1443.21 5066.78 46.14 1515.76 0.54 460.11 5074.77 0.025 284.68 47.39 1440.63 5052.96 46.15 1517.64 0.52 470.51 5076.52

Table 4.  Effects of an increase in the proportion of a Type I error $m_1$ on the solutions and profits under the CPD and ISR strategy ($p = 0.02$, and $m_2 = 0.02$)

 $m_1$ $Q^\star_0$ the CPD strategy the ISR strategy $s^\star_1$ $Q^\star_1$ $G^{(1)\star}$ $s^\star_1$ $Q^\star_1$ $z^\star$ $x^\star$ $G^{(2)\star}$ 0.010 284.12 47.09 1478.09 5353.91 45.62 1547.53 0.6340 398.43 5346.96 0.015 284.32 47.19 1462.26 5217.06 45.87 1529.80 0.5903 422.82 5203.96 0.020 284.51 47.30 1447.10 5087.62 46.12 1513.05 0.5552 444.05 5072.92 0.025 284.70 47.39 1432.59 4965.51 46.38 1497.22 0.5263 462.63 4953.37 0.030 284.88 47.49 1418.73 4850.64 46.64 1482.26 0.5018 478.93 4845.05 0.035 285.05 47.59 1405.54 4742.93 46.91 1468.16 0.4807 493.23 4747.85 0.040 285.22 47.69 1393.00 4642.31 47.18 1454.91 0.4623 505.74 4661.78 0.045 285.38 47.91 1377.04 4549.81 47.46 1442.51 0.4461 516.66 4586.92 0.050 285.53 48.23 1359.58 4469.73 47.74 1430.96 0.4316 526.13 4523.48

Table 5.  Effects of an increase in the proportion of a Type II error $m_2$ on the solutions and profits under the CPD and ISR strategy ($p = 0.02$, and $m_1 = 0.02$)

 $m_2$ $Q^\star_0$ the CPD strategy the ISR strategy $s^\star_1$ $Q^\star_1$ $G^{(1)\star}$ $s^\star_1$ $Q^\star_1$ $z^\star$ $x^\star$ $G^{(2)\star}$ 0.010 284.55 47.30 1449.00 5108.07 46.07 1521.57 0.5347 462.17 5126.40 0.015 284.53 47.30 1448.05 5097.85 46.10 1518.91 0.5359 460.61 5110.85 0.020 284.51 47.30 1447.10 5087.62 46.12 1513.05 0.5552 444.05 5072.92 0.025 284.49 47.29 1446.14 5077.39 46.15 1510.42 0.5564 442.55 5057.65 0.030 284.47 47.29 1445.19 5067.15 46.17 1507.80 0.5576 441.06 5042.48 0.035 284.46 47.29 1444.23 5056.90 46.20 1505.19 0.5588 439.56 5027.42 0.040 284.44 47.29 1443.28 5046.65 46.22 1502.59 0.5600 438.07 5012.46 0.045 284.42 47.29 1442.32 5036.39 46.25 1500.00 0.5612 436.58 4997.62 0.050 284.40 47.28 1441.36 5026.12 46.27 1497.42 0.5625 435.10 4982.86

Table 6.  Effects of an increase in the unit price discount $k$ on the solutions and profits under the CPD and ISR strategy ($p = 0.02$, $m_1 = 0.02$, and $m_2 = 0.02$)

 $k$ $Q^\star_0$ the CPD strategy the ISR strategy $s^\star_1$ $Q^\star_1$ $G^{(1)\star}$ $s^\star_1$ $Q^\star_1$ $z^\star$ $x^\star$ $G^{(2)\star}$ 2.5 284.51 47.68 603.27 531.03 - - - - - 5 284.51 47.30 975.23 2063.16 46.82 987.63 0.6429 370.45 1966.43 7.5 284.51 47.30 1447.10 5087.62 46.12 1513.05 0.5552 444.05 5072.92 10 284.51 47.30 2076.25 10343.92 45.41 2237.55 0.4932 517.20 10619.61 12.5 284.51 47.30 2957.06 19171.15 44.69 3280.39 0.4462 590.96 20142.16 15 284.51 47.30 4278.28 34247.53 43.97 4880.25 0.4090 665.97 36711.51

Table 7.  Effects of an increase in $M$ on the solutions and profits under the ISR strategy ($p = 0.02$, $m_1 = 0.02$, and $m_2 = 0.02$)

 $M$ $Q^\star_0$ the CPD strategy the ISR strategy $s^\star_1$ $Q^\star_1$ $G^{(1)\star}$ $s^\star_1$ $Q^\star_1$ $z^\star$ $x^\star$ $G^{(2)\star}$ 20 284.51 47.30 1447.10 5087.62 46.13 1534.35 0.4359 565.26 5222.67 25 284.51 47.30 1447.10 5087.62 46.13 1524.82 0.4889 504.16 5155.54 30 284.51 47.30 1447.10 5087.62 46.12 1516.26 0.5370 459.06 5095.41 35 284.51 47.30 1447.10 5087.62 46.12 1508.43 0.5816 424.01 5040.58 40 284.51 47.30 1447.10 5087.62 46.11 1501.18 0.6232 395.76 4989.93 45 284.51 47.30 1447.10 5087.62 46.11 1494.40 0.6625 372.35 4942.71

Table 8.  Effects of an increase in $b$ on the solutions and profits of our model and the EOQ model in Hsu and Yu [22] ($p = 0.01$, $m_1 = 0.02$, and $m_2 = 0.02$)

 $b$ Hsu and Yu [22] the CPD strategy the ISR strategy $Q^\star_1$ $D^{(1)}(Q^\star_1)$ $s^\star_1$ $Q^\star_1$ $G^{(1)\star}$ $s^\star_1$ $Q^\star_1$ $z^\star$ $G^{(2)\star}$ 7 1271.04 4221.78 50 1271.04 4221.78 45.02 1190.03 - 2491.92 8 1271.04 4221.78 50 1271.04 4221.78 45.65 1271.08 - 3234.62 9 1271.04 4221.78 49.09 1299.50 4265.23 46.13 1334.12 - 3846.09 10 1271.04 4221.78 47.95 1354.46 4468.04 46.52 1384.54 - 4356.57 11 1271.04 4221.78 47.04 1419.59 4790.88 46.81 1424.99 0.66 4633.18 12 1271.04 4221.78 47.10 1460.19 5157.81 46.06 1505.11 0.64 5074.49 13 1271.04 4221.78 47.32 1489.28 5477.32 45.44 1591.45 0.62 5585.67 14 1271.04 4221.78 47.51 1514.22 5756.21 44.92 1682.64 0.61 6153.24 15 1271.04 4221.78 47.68 1535.83 6001.65 44.48 1777.69 0.59 6767.56 16 1271.04 4221.78 47.82 1554.74 6219.27 44.09 1875.85 0.57 7421.47

Figures(6)

Tables(8)