doi: 10.3934/jimo.2021060

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

School of management, Xian Jiaotong university, Xi'an, Shaanxi 710049, China

* Corresponding author: Guiyang Zhu

Received  June 2020 Revised  February 2021 Published  March 2021

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.

Citation: Guiyang Zhu. Optimal pricing and ordering policy for defective items under temporary price reduction with inspection errors and price sensitive demand. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021060
References:
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[2]

P. L. Abad, Optimal price and lot size when the supplier offers a temporary price reduction over an interval, Computers & Operations Research, 30 (2003), 63-74.  doi: 10.1016/S0305-0548(01)00081-8.  Google Scholar

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show all references

References:
[1]

P. L. Abad, Optimal policy for a reseller when the supplier offers a temporary reduction in price, Decision Sciences, 28 (1997), 637-653.  doi: 10.1111/j.1540-5915.1997.tb01325.x.  Google Scholar

[2]

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[3]

A. A. AlamriI. Harris and A. A. Syntetos, Efficient inventory control for imperfect quality items, European J. Oper. Res., 254 (2016), 92-104.  doi: 10.1016/j.ejor.2016.03.058.  Google Scholar

[4]

F. J. ArcelusN. H. Shah and G. Srinivasan, Retailer's response to special sales: Price discount vs. trade credit, Omega, 29 (2001), 417-428.  doi: 10.1016/S0305-0483(01)00035-4.  Google Scholar

[5]

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[6]

A. Ardalan, A comparative analysis of approaches for determining optimal price and order quantity when a sale increases demand, European Journal of Operational Research, 84 (1995), 416-430.   Google Scholar

[7]

E. Babaee TirkolaeeA. GoliM. Pahlevan and R. M. Kordestanizadeh, A robust bi-objective multi-trip periodic capacitated arc routing problem for urban waste collection using a multi-objective invasive weed optimization, Waste Management & Research, 37 (2019), 1089-1101.  doi: 10.1177/0734242X19865340.  Google Scholar

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K. Ben, Xbox 360 failure rates worse than most consumer electronics, https://arstechnica.com/gaming/2008/02/xbox-360-failure-rates-worse-than-most-consumer-electornics/. Google Scholar

[9]

B. Caroline, Acceptable quality level (aql), https://www.investopedia.com/terms/a/acceptable-quality-level-aql.asp. Google Scholar

[10]

H.-C. Chang, An application of fuzzy sets theory to the EOQ model with imperfect quality items, Comput. Oper. Res., 31 (2004), 2079-2092.  doi: 10.1016/S0305-0548(03)00166-7.  Google Scholar

[11]

C.-T. ChangM.-C. Cheng and P.-Y. Soong, Impacts of inspection errors and trade credits on the economic order quantity model for items with imperfect quality, International Journal of Systems Science: Operations & Logistics, 3 (2016), 34-48.  doi: 10.1080/23302674.2015.1036473.  Google Scholar

[12]

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[13]

H. GaoD. WangE. D. R. Santibanez Gonzalez and Y. Ju, Optimal stocking strategies for inventory mechanism with a stochastic short-term price discount and partial backordering, International Journal of Production Research, 57 (2019), 7471-7500.  doi: 10.1080/00207543.2019.1567949.  Google Scholar

[14]

A. GoliE. Babaee Tirkolaee and M. Soltani, A robust just-in-time flow shop scheduling problem with outsourcing option on subcontractors, Production & Manufacturing Research, 7 (2019), 294-315.  doi: 10.1080/21693277.2019.1620651.  Google Scholar

[15]

A. Goli and S. M. R. Davoodi, Coordination policy for production and delivery scheduling in the closed loop supply chain, Production Engineering, 12 (2018), 621-631.  doi: 10.1007/s11740-018-0841-0.  Google Scholar

[16]

A. Goli, H. K. Zare, R. Tavakkoli-Moghaddam and A. Sadeghieh, Hybrid artificial intelligence and robust optimization for a multi-objective product portfolio problem case study: The dairy products industry, Computers & Industrial Engineering, 137 (2019), 106090. doi: 10.1016/j.cie.2019.106090.  Google Scholar

[17]

R. W. Grubbström and B. G. Kingsman, Ordering and inventory policies for step changes in the unit item cost: A discounted cash flow approach, Management Science, 50 (2004), 253-267.   Google Scholar

[18]

Z. Hauck and J. Vörös, Lot sizing in case of defective items with investments to increase the speed of quality control, Omega, 52 (2015), 180-189.  doi: 10.1016/j.omega.2014.04.004.  Google Scholar

[19]

L.-F. Hsu, A note on "an economic order quantity (EOQ) for items with imperfect quality and inspection errors", International Journal of Industrial Engineering Computations, 3 (2012), 695-702.  doi: 10.5267/j.ijiec.2012.03.008.  Google Scholar

[20]

J.-T. Hsu and L.-F. Hsu, An EOQ model with imperfect quality items, inspection errors, shortage backordering, and sales returns, International Journal of Production Economics, 143 (2013), 162-170.   Google Scholar

[21]

J.-T. Hsu and L.-F. Hsu, Two EPQ models with imperfect production processes, inspection errors, planned backorders, and sales returns, Computers & Industrial Engineering, 64 (2013), 389-402.  doi: 10.1016/j.cie.2012.10.005.  Google Scholar

[22]

W.-K. K. Hsu and H.-F. Yu, Eoq model for imperfective items under a one-time-only discount, Omega, 37 (2009), 1018-1026.   Google Scholar

[23]

M. Y. JaberS. K. Goyal and M. Imran, Economic production quantity model for items with imperfect quality subject to learning effects, International Journal of Production Economics, 115 (2008), 143-150.  doi: 10.1016/j.ijpe.2008.05.007.  Google Scholar

[24]

Jueves, Powerlocus history - the idea behind our company, https://powerlocus.com/es/blog1/powerlocus-history-the-idea-behind-our-company. Google Scholar

[25]

S. KhalilpourazariS. TeimooriA. MirzazadehS. H. R. Pasandideh and N. Ghanbar Tehrani, Robust fuzzy chance constraint programming for multi-item EOQ model with random disruption and partial backordering under uncertainty, Journal of Industrial and Production Engineering, 36 (2019), 276-285.  doi: 10.1080/21681015.2019.1646328.  Google Scholar

[26]

M. KhanM. Y. Jaber and A.-R. Ahmad, An integrated supply chain model with errors in quality inspection and learning in production, Omega, 42 (2014), 16-24.  doi: 10.1016/j.omega.2013.02.002.  Google Scholar

[27]

M. KhanM. Y. Jaber and M. Bonney, An economic order quantity (EOQ) for items with imperfect quality and inspection errors, International Journal of Production Economics, 133 (2011), 113-118.   Google Scholar

[28]

M. KhanM. JaberA. Guiffrida and S. Zolfaghari, A review of the extensions of a modified EOQ model for imperfect quality items, International Journal of Production Economics, 132 (2011), 1-12.  doi: 10.1016/j.ijpe.2011.03.009.  Google Scholar

[29]

C. KumarN. Naidu and K. Ravindranath, Performance improvement of manufacturing industry by reducing the defectives using six sigma methodologies, IOSR J. Eng, 1 (2012), 1-9.   Google Scholar

[30]

M. LengZ. Li and L. Liang, Implications for the role of retailers in quality assurance, Production and Operations Management, 25 (2016), 779-790.  doi: 10.1111/poms.12501.  Google Scholar

[31]

J. LiH. Feng and M. Wang, A replenishment policy with defective products, backlog and delay of payments, J. Ind. Manag. Optim., 5 (2009), 867-880.  doi: 10.3934/jimo.2009.5.867.  Google Scholar

[32]

Z. Lin, Price promotion with reference price effects in supply chain, Transportation Research Part E: Logistics and Transportation Review, 85 (2016), 52-68.  doi: 10.1016/j.tre.2015.11.002.  Google Scholar

[33]

T.-Y. LinB. R. Sarker and C.-J. Lin, An optimal setup cost reduction and lot size for economic production quantity model with imperfect quality and quantity discounts, J. Ind. Manag. Optim., 17 (2021), 467-484.  doi: 10.3934/jimo.2020043.  Google Scholar

[34]

R. Lotfi, Y. Z. Mehrjerdi, M. S. Pishvaee, A. Sadeghieh and G.-W. Weber, A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk, Numerical Algebra, Control & Optimization. Google Scholar

[35]

R. LotfiM. A. NayeriS. M. Sajadifar and N. Mardani, Determination of start times and ordering plans for two-period projects with interdependent demand in project-oriented organizations: A case study on molding industry, Journal of Project Management, 2 (2017), 119-142.  doi: 10.5267/j.jpm.2017.9.001.  Google Scholar

[36]

R. LotfiG.-W. WeberS. M. Sajadifar and N. Mardani, Interdependent demand in the two-period newsvendor problem, J. Ind. Manag. Optim., 16 (2020), 117-140.  doi: 10.3934/jimo.2018143.  Google Scholar

[37]

R. Lotfi, Z. Yadegari, S. H. Hosseini, A. H. Khameneh, E. B. Tirkolaee and G.-W. Weber, A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project, J. Ind. Manag. Optim., 13 (2017). Google Scholar

[38]

G. C. Mahata, Application of fuzzy sets theory in an EOQ model for items with imperfect quality and shortage backordering, International Journal of Services and Operations Management, 14 (2013), 466-490.  doi: 10.1504/IJSOM.2013.052839.  Google Scholar

[39]

Y. Meng and Y.Song, Optimal policy for competing retailers when the supplier offers a temporary price discount with uncertain demand, in Proceedings of the 6th International Asia Conference on Industrial Engineering and Management Innovation, Springer, (2016), 703–712. doi: 10.2991/978-94-6239-148-2_69.  Google Scholar

[40]

E. Naddor, Inventory Systems, Technical report, 1966. Google Scholar

[41]

H. Öztürk, Economic order quantity models for the shipment containing defective items with inspection errors and a sub-lot inspection policy, European Journal of Industrial Engineering, 14 (2020), 85-126.   Google Scholar

[42]

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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
22]">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 $
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
$ 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
$ 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
$ 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
$ 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
$ 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
$ 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
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