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October  2019, 15(4): 1617-1630. doi: 10.3934/jimo.2018114

Online ordering strategy for the discrete newsvendor problem with order value-based free-shipping

Yong Zhang , Huifen Zhong , , Yue Liu and  Menghu Huang

School of Management, Guangdong University of Technology, Guangzhou, Guangdong 510520, China

* Corresponding author: Huifen Zhong

Received  December 2016 Revised  May 2018 Published  August 2018

Fund Project: This research was supported by the National Natural Science Foundation of China (71501049, 71301029) and GDUPS(2016).

Suppliers always provide free-shipping for retailers whose total order value exceeds or equals an explicit promotion threshold. This paper incorporates a shipping fee in the discrete multi-period newsvendor problem and applies Weak Aggregating Algorithm (WAA) to offer explicit online ordering strategy. It further considers an extended case with salvage value and shortage cost. In particular, online ordering strategies are derived based on return loss function. Numerical examples serve to illustrate the competitive performance of the proposed ordering strategies. Results show that newsvendors' cumulative return losses are affected by the threshold of the order value-based free-shipping. Moreover, the introduction of salvage value and shortage cost greatly improves the competitive performance of online ordering strategies.

Keywords: Multi-period newsvendor problem, discrete, order value-based free-shipping, WAA, online ordering strategy.
Mathematics Subject Classification: Primary: 90B05, 68W27; Secondary: 68W40.
Citation: Yong Zhang, Huifen Zhong, Yue Liu, Menghu Huang. Online ordering strategy for the discrete newsvendor problem with order value-based free-shipping. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1617-1630. doi: 10.3934/jimo.2018114
References:
[1]

H. Alfares and H. Elmorra, The distribution-free newsboy problem: Extension to the shortage penalty case, International Journal of Production Economics, 93 (2005), 465-477.  doi: 10.1016/j.ijpe.2004.06.043.  Google Scholar

[2]

K. Arrow, T. Harris and J. Marshak, Optimal inventory policy, Econometrica, 19 (1951), 250-272.  doi: 10.2307/1906813.  Google Scholar

[3]

W. H. Huang, Y. C. Cheng and J. Rose, Threshold free shipping policies for internet shoppers, Transportation Research Part A, 82 (2015), 193-203.  doi: 10.1016/j.tra.2015.09.015.  Google Scholar

[4]

Y. Kalnishkan and M. V. Vyugin, The weak aggregating algorithm and weak mixability, Journal of Computer and System Sciences, 74 (2008), 1228-1244.  doi: 10.1016/j.jcss.2007.08.003.  Google Scholar

[5]

S. Karlin, Dynamic inventory policy with varying stochastic demands, Management Science, 6 (1960), 231-258.  doi: 10.1287/mnsc.6.3.231.  Google Scholar

[6]

M. Khouja, The single-period (news-vendor) problem: literature review and suggestions for future research, Omega: The International Journal of Management Science, 27 (1999), 537-553.  doi: 10.1016/S0305-0483(99)00017-1.  Google Scholar

[7]

K. Kwon and T. Cheong, A minimax distribution-free procedure for a newsvendor problem with free shipping, European Journal of Operational Research, 232 (2014), 234-240.  doi: 10.1016/j.ejor.2013.07.004.  Google Scholar

[8]

T. Levina, Y. Levin and J. McGill, Weak aggregating algorithm for the distribution-free perishable inventory problem, Operations Research Letters, 38 (2010), 516-521.  doi: 10.1016/j.orl.2010.09.006.  Google Scholar

[9]

M. Lewis, V. Singh and S. Fay, An empirical study of the impact of nonlinear shipping and handling fees on purchase incidence and expenditure decision, Marketing Science, 25 (2006), 51-64.  doi: 10.1287/mksc.1050.0150.  Google Scholar

[10]

W. Liu, S. Song, Y. Qiao and H. Zhao, The loss-averse newsvendor problem with random supply capacity, Journal of Industrial and Management Optimization, 13 (2017), 1417-1429.  doi: 10.3934/jimo.2016080.  Google Scholar

[11]

M. Morse and E. Kimball, Methods of operations research, Published jointly by the Teachnology Press of Massachusetts Institute of Technology, 4 (1951), 18-20.   Google Scholar

[12]

Y. Qin, The newsvendor problem: review and directions for future research, European Journal of Operational Research, 213 (2011), 361-374.  doi: 10.1016/j.ejor.2010.11.024.  Google Scholar

[13]

H. Scarf, Bayes solution to the statistical inventory problem, Annals of Mathematical Statistics, 30 (1959), 490-508.  doi: 10.1214/aoms/1177706264.  Google Scholar

[14]

H. Yu, J. Zhai and G. Y. Chen, Robust optimization for the loss-averse newsvendor problem, European Journal of Operational Research, 171 (2016), 1008-1032.  doi: 10.1007/s10957-016-0870-9.  Google Scholar

[15]

Y. Zhang, X. Yang and B. Li, Distribution-free solutions to the extended multi-period newsboy problem, Journal of Industrial and Management Optimization, 13 (2017), 633-647.  doi: 10.3934/jimo.2016037.  Google Scholar

[16]

Y. Zhang and X. Yang, Online ordering policies for a two-product, multi-period stationary newsvendor problem, Computers and Operations Research, 74 (2016), 143-151.  doi: 10.1016/j.cor.2016.04.031.  Google Scholar

show all references

References:
[1]

H. Alfares and H. Elmorra, The distribution-free newsboy problem: Extension to the shortage penalty case, International Journal of Production Economics, 93 (2005), 465-477.  doi: 10.1016/j.ijpe.2004.06.043.  Google Scholar

[2]

K. Arrow, T. Harris and J. Marshak, Optimal inventory policy, Econometrica, 19 (1951), 250-272.  doi: 10.2307/1906813.  Google Scholar

[3]

W. H. Huang, Y. C. Cheng and J. Rose, Threshold free shipping policies for internet shoppers, Transportation Research Part A, 82 (2015), 193-203.  doi: 10.1016/j.tra.2015.09.015.  Google Scholar

[4]

Y. Kalnishkan and M. V. Vyugin, The weak aggregating algorithm and weak mixability, Journal of Computer and System Sciences, 74 (2008), 1228-1244.  doi: 10.1016/j.jcss.2007.08.003.  Google Scholar

[5]

S. Karlin, Dynamic inventory policy with varying stochastic demands, Management Science, 6 (1960), 231-258.  doi: 10.1287/mnsc.6.3.231.  Google Scholar

[6]

M. Khouja, The single-period (news-vendor) problem: literature review and suggestions for future research, Omega: The International Journal of Management Science, 27 (1999), 537-553.  doi: 10.1016/S0305-0483(99)00017-1.  Google Scholar

[7]

K. Kwon and T. Cheong, A minimax distribution-free procedure for a newsvendor problem with free shipping, European Journal of Operational Research, 232 (2014), 234-240.  doi: 10.1016/j.ejor.2013.07.004.  Google Scholar

[8]

T. Levina, Y. Levin and J. McGill, Weak aggregating algorithm for the distribution-free perishable inventory problem, Operations Research Letters, 38 (2010), 516-521.  doi: 10.1016/j.orl.2010.09.006.  Google Scholar

[9]

M. Lewis, V. Singh and S. Fay, An empirical study of the impact of nonlinear shipping and handling fees on purchase incidence and expenditure decision, Marketing Science, 25 (2006), 51-64.  doi: 10.1287/mksc.1050.0150.  Google Scholar

[10]

W. Liu, S. Song, Y. Qiao and H. Zhao, The loss-averse newsvendor problem with random supply capacity, Journal of Industrial and Management Optimization, 13 (2017), 1417-1429.  doi: 10.3934/jimo.2016080.  Google Scholar

[11]

M. Morse and E. Kimball, Methods of operations research, Published jointly by the Teachnology Press of Massachusetts Institute of Technology, 4 (1951), 18-20.   Google Scholar

[12]

Y. Qin, The newsvendor problem: review and directions for future research, European Journal of Operational Research, 213 (2011), 361-374.  doi: 10.1016/j.ejor.2010.11.024.  Google Scholar

[13]

H. Scarf, Bayes solution to the statistical inventory problem, Annals of Mathematical Statistics, 30 (1959), 490-508.  doi: 10.1214/aoms/1177706264.  Google Scholar

[14]

H. Yu, J. Zhai and G. Y. Chen, Robust optimization for the loss-averse newsvendor problem, European Journal of Operational Research, 171 (2016), 1008-1032.  doi: 10.1007/s10957-016-0870-9.  Google Scholar

[15]

Y. Zhang, X. Yang and B. Li, Distribution-free solutions to the extended multi-period newsboy problem, Journal of Industrial and Management Optimization, 13 (2017), 633-647.  doi: 10.3934/jimo.2016037.  Google Scholar

[16]

Y. Zhang and X. Yang, Online ordering policies for a two-product, multi-period stationary newsvendor problem, Computers and Operations Research, 74 (2016), 143-151.  doi: 10.1016/j.cor.2016.04.031.  Google Scholar

Figure 1.  Daily cumulative return losses of $swaa$ and $best1$ when $V_0 = 99$
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Figure 2.  Daily cumulative return losses of $swaa$ and $best1$ when $V_0 = 110$
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Figure 3.  Cumulative return losses $swaa$ achieved under uniform and norm distribution
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Figure 4.  Cumulative return losses $twaa$ achieved under uniform and norm distribution
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Figure 5.  Daily cumulative return losses of $twaa$ and $best2$ when $V_0 = 99$
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Table 1.  Cumulative return losses of $swaa$ and $best1$ under different $V_0$
Trials$V_0=99$$V_0=110$
$swaa$$best1$$ratio1$$swaa$$best1$$ratio1$
11298.81107.41.17281406.81216.41.1565
21089.8964.601.12981119.8999.301.1206
31161.91049.21.10741209.91019.21.1871
4991.80962.501.03041081.81100.50.9830
51080.6999.701.06221182.61040.41.1367
61104.21000.81.10331224.21059.51.1555
71093.51026.31.06551099.51008.31.0904
8953.60924.401.0316983.60900.401.0924
91130.6990.301.14171124.61037.01.0845
10888.60867.301.0246900.60837.301.0756
111114.21005.21.10841126.2975.201.1548
12922.70922.701.00001072.71072.71.0000
13916.90857.201.06961030.91007.21.0235
141064.11005.51.05831118.11064.21.0506
15983.90928.201.06001079.9906.601.1912
161254.01155.31.08541368.01202.01.1381
17764.10742.901.0285860.10844.901.0180
181129.61035.01.09141141.61049.01.0883
191209.51113.31.08641215.51107.31.0977
20890.10879.701.0118890.10843.701.0763
211145.31053.31.08731157.31047.31.1050
221177.91061.91.10921213.91037.91.1696
23832.90801.101.0397898.90927.100.9696
241086.8989.701.09811098.8953.701.1521
251065.81010.21.05501125.81050.91.0713
26960.60861.901.11451092.61017.91.0734
271138.01057.21.07641162.01033.21.1247
28895.80832.801.0756985.80964.801.0218
291023.7981.301.04321131.71101.31.0276
301244.61107.41.12391352.61216.41.1120
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Trials$V_0=99$$V_0=110$
$swaa$$best1$$ratio1$$swaa$$best1$$ratio1$
11298.81107.41.17281406.81216.41.1565
21089.8964.601.12981119.8999.301.1206
31161.91049.21.10741209.91019.21.1871
4991.80962.501.03041081.81100.50.9830
51080.6999.701.06221182.61040.41.1367
61104.21000.81.10331224.21059.51.1555
71093.51026.31.06551099.51008.31.0904
8953.60924.401.0316983.60900.401.0924
91130.6990.301.14171124.61037.01.0845
10888.60867.301.0246900.60837.301.0756
111114.21005.21.10841126.2975.201.1548
12922.70922.701.00001072.71072.71.0000
13916.90857.201.06961030.91007.21.0235
141064.11005.51.05831118.11064.21.0506
15983.90928.201.06001079.9906.601.1912
161254.01155.31.08541368.01202.01.1381
17764.10742.901.0285860.10844.901.0180
181129.61035.01.09141141.61049.01.0883
191209.51113.31.08641215.51107.31.0977
20890.10879.701.0118890.10843.701.0763
211145.31053.31.08731157.31047.31.1050
221177.91061.91.10921213.91037.91.1696
23832.90801.101.0397898.90927.100.9696
241086.8989.701.09811098.8953.701.1521
251065.81010.21.05501125.81050.91.0713
26960.60861.901.11451092.61017.91.0734
271138.01057.21.07641162.01033.21.1247
28895.80832.801.0756985.80964.801.0218
291023.7981.301.04321131.71101.31.0276
301244.61107.41.12391352.61216.41.1120
Table 2.  Cumulative return losses of $twaa$ and $best2$ under different $V_0$
Trials$V_0=99$$V_0=110$
$twaa$$best2$$ratio2$$twaa$$best2$$ratio2$
11016.4935.601.08641058.4935.601.0827
21430.71360.81.05141454.71384.81.0505
3877.00832.001.0541895.00850.001.0529
4980.00943.601.03861010.0973.601.0374
5888.80814.801.0908924.80850.801.0870
61030.61081.60.95281048.61117.60.9383
7959.30929.601.0319977.30947.601.0313
81017.7969.21.05001041.71005.21.0363
9784.80722.401.0864808.80746.401.0836
101270.01157.61.09711312.01199.61.0937
111131.51127.21.00381143.51163.20.9831
12964.60924.801.04301000.6966.801.0350
131265.91140.81.10971295.91182.81.0956
14722.50663.601.0888746.50687.601.0857
151366.41272.81.07351390.41296.81.0722
161061.21012.01.04861121.21072.01.0459
171212.71146.81.05751230.71170.81.0512
18844.40810.401.0420856.40822.401.0413
191238.41163.61.06431262.41187.61.0630
201260.71230.41.02461278.71272.41.0050
211300.21200.01.08351330.21230.01.0815
221195.01102.81.08361219.01132.81.0761
231008.9947.101.06531044.91001.11.0438
241187.11096.81.08231211.11132.81.0691
251155.41087.91.06201173.41135.91.0330
26832.90792.601.0508844.90804.601.0501
271035.3942.001.09901035.3942.001.0990
28936.20861.701.0865942.20873.701.0784
29974.80907.501.07421016.8949.501.0709
30872.50825.001.0576896.50873.001.0269
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Trials$V_0=99$$V_0=110$
$twaa$$best2$$ratio2$$twaa$$best2$$ratio2$
11016.4935.601.08641058.4935.601.0827
21430.71360.81.05141454.71384.81.0505
3877.00832.001.0541895.00850.001.0529
4980.00943.601.03861010.0973.601.0374
5888.80814.801.0908924.80850.801.0870
61030.61081.60.95281048.61117.60.9383
7959.30929.601.0319977.30947.601.0313
81017.7969.21.05001041.71005.21.0363
9784.80722.401.0864808.80746.401.0836
101270.01157.61.09711312.01199.61.0937
111131.51127.21.00381143.51163.20.9831
12964.60924.801.04301000.6966.801.0350
131265.91140.81.10971295.91182.81.0956
14722.50663.601.0888746.50687.601.0857
151366.41272.81.07351390.41296.81.0722
161061.21012.01.04861121.21072.01.0459
171212.71146.81.05751230.71170.81.0512
18844.40810.401.0420856.40822.401.0413
191238.41163.61.06431262.41187.61.0630
201260.71230.41.02461278.71272.41.0050
211300.21200.01.08351330.21230.01.0815
221195.01102.81.08361219.01132.81.0761
231008.9947.101.06531044.91001.11.0438
241187.11096.81.08231211.11132.81.0691
251155.41087.91.06201173.41135.91.0330
26832.90792.601.0508844.90804.601.0501
271035.3942.001.09901035.3942.001.0990
28936.20861.701.0865942.20873.701.0784
29974.80907.501.07421016.8949.501.0709
30872.50825.001.0576896.50873.001.0269
Table 3.  $swaa$'s robustness in different computational days
TrialsDays
20406080100
11.21021.17281.10411.03271.0385
21.17381.08521.09331.06801.0718
31.15241.13621.15781.01601.0789
41.28681.16401.11581.06951.0328
51.12681.24411.16101.10561.0996
61.18011.06711.09951.03561.0640
71.32191.09251.13211.07911.0466
81.22621.11511.06611.05591.0424
91.19831.07361.07961.05631.0899
101.17831.06861.10731.06611.0602
$Avg1$1.20551.12221.11171.05851.0625
$SD1$0.003210.003020.000870.000590.00046
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TrialsDays
20406080100
11.21021.17281.10411.03271.0385
21.17381.08521.09331.06801.0718
31.15241.13621.15781.01601.0789
41.28681.16401.11581.06951.0328
51.12681.24411.16101.10561.0996
61.18011.06711.09951.03561.0640
71.32191.09251.13211.07911.0466
81.22621.11511.06611.05591.0424
91.19831.07361.07961.05631.0899
101.17831.06861.10731.06611.0602
$Avg1$1.20551.12221.11171.05851.0625
$SD1$0.003210.003020.000870.000590.00046
Table 4.  $twaa$'s robustness in different computational days
TrialsDays
20406080100
11.18451.10081.09931.03531.0300
21.24841.14901.04041.09761.0805
31.27521.10331.04571.08051.0503
41.24791.06721.06441.06741.0556
51.15461.10391.09841.06171.0298
61.22511.02571.06841.09381.0303
71.16171.11611.10261.02691.0380
81.10781.13471.09101.05051.0264
91.17721.05031.07761.06301.0696
101.27111.08921.04631.06521.0343
$Avg2$1.20531.09401.07341.06421.0445
$SD2$0.002850.001270.000520.000470.00032
Table Options

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TrialsDays
20406080100
11.18451.10081.09931.03531.0300
21.24841.14901.04041.09761.0805
31.27521.10331.04571.08051.0503
41.24791.06721.06441.06741.0556
51.15461.10391.09841.06171.0298
61.22511.02571.06841.09381.0303
71.16171.11611.10261.02691.0380
81.10781.13471.09101.05051.0264
91.17721.05031.07761.06301.0696
101.27111.08921.04631.06521.0343
$Avg2$1.20531.09401.07341.06421.0445
$SD2$0.002850.001270.000520.000470.00032
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