American Institute of Mathematical Sciences

Advanced
July  2021, 17(4): 1577-1591. doi: 10.3934/jimo.2020035

Design of differentiated warranty coverage that considers usage rate and service option of consumers under 2D warranty policy

Peng Tong 1,, and  Xiaogang Ma 2,,

1. 

School of Management, China University of Mining and Technology, Jiangsu, China
2. 

School of Management, Wuhan Textile University, Hubei, China

* Corresponding authors: flyingmantong@163.com; xgma@wtu.edu.cn

Received  June 2019 Revised  September 2019 Published  July 2021 Early access  February 2020

Fund Project: The first author is supported by NSF grant the National Natural Science Foundation of China (No. 71701200); the Postdoctoral Fund of China (No. 2016M590525); the Postdoctoral Fund of Jiangsu (No. 1601246C)

Warranty service providers usually provide homogeneous warranty service to improve consumer satisfaction and market share. Considering the difference of consumers, some scholars have carried out studies on maintenance strategies, service pricing, payment method, claim behaviour and warranty cost analysis in recent years. However, few scholars have focused on the differentiated coverage of warranty service that considers usage rate and service option of consumers. On the basis of previous classification criteria on usage rate, this paper divides consumers into heavy, medium and light usage rate groups with clear boundaries. To avoid discrimination in warranty service, this study divides 2D warranty coverage into disjoint sub-regions and adopts different maintenance modes in each sub-region. By formulating and calculating warranty cost model under warranty cost constraints, we can obtain the maximum warranty coverage under usage rate $ r $. Therefore, differentiated warranty scope for consumers in the three groups can be proposed, whilst consumers can choose the most suitable warranty service according to their usage rate. Evidently, the proposed warranty strategy can provide flexible warranty service for consumers, meet the requirements of the warranty cost constraints of warranty service providers and enable enterprises to occupy a favourable position in the market competition.

Keywords: Usage rate, warranty coverage, modelling, cost analysis, 2D warranty.
Mathematics Subject Classification: Primary: 90B50; Secondary: 90B25.
Citation: Peng Tong, Xiaogang Ma. Design of differentiated warranty coverage that considers usage rate and service option of consumers under 2D warranty policy. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1577-1591. doi: 10.3934/jimo.2020035
References:
[1]

A. Akbarov and S. Wu, Forecasting warranty claims considering dynamic over-dispersion, Int. J. Prod. Econ., 139 (2012), 615-622.  doi: 10.1016/j.ijpe.2012.06.001.

[2]

J. Baik, D. N. P. Murthy and N. Jack, Two-dimensional failure modeling with minimal repair, Naval Res. Logist., 51 (2004), 345-362.  doi: 10.1002/nav.10120.

[3]

W. L. Chang and J.-H. Lin, Optimal maintenance policy and length of extended warranty within the life cycle of products, Comput. Math. Appl., 63 (2012), 144-150.  doi: 10.1016/j.camwa.2011.11.001.

[4]

S. Chukova and M. R. Johnston, Two-dimensional warranty repair strategy based on minimal and complete repairs, Math. Comput. Modelling, 44 (2006), 1133-1143.  doi: 10.1016/j.mcm.2006.03.015.

[5]

G. Gallego, R. Wang, M. Hu, J. Ward and J. L. Beltran, No claim? Your gain: Design of residual value extended warranties under risk aversion and strategic claim behavior, Manufacturing Service Oper. Management, 17 (2015), 87-100.  doi: 10.1287/msom.2014.0501.

[6]

J. C. Hartman and K. Laksana, Designing and pricing menus of extended warranty contracts, Naval Res. Logist., 56 (2009), 199-214.  doi: 10.1002/nav.20333.

[7]

Y.-S. Huang, W.-Y. Gau and J.-W. Ho, Cost analysis of two-dimensional warranty for products with periodic preventive maintenance, Reliability Engineering System Safety, 134 (2015), 51-58.  doi: 10.1016/j.ress.2014.10.014.

[8]

Y.-S. Huang, C.-D. Huang and J.-W. Ho, A customized two-dimensional extended warranty with preventive maintenance, European J. Oper. Res., 257 (2017), 971-978.  doi: 10.1016/j.ejor.2016.07.034.

[9]

B. P. Iskandar and D. N. P. Murthy, Repair-replace strategies for two-dimensional warranty policies, Math. Comput. Modelling, 38 (2003), 1233-1241.  doi: 10.1016/S0895-7177(03)90125-7.

[10]

B. P. Iskandar, D. N. P. Murthy and N. Jack, A new repair-replace strategy for items sold with a two-dimensional warranty, Comput. Oper. Res., 32 (2005), 669–682. doi: 10.1016/j.cor.2003.08.011.

[11]

N. Jack, B. P. Iskandar and D. N. P. Murthy, A repair-replace strategy based on usage rate for items sold with a two-dimensional warranty, Reliability Engineering System Safety, 94 (2009), 611-617.  doi: 10.1016/j.ress.2008.06.019.

[12]

N. Jack and V. D. D. Schouten, Optimal repair-replace strategies for a warranted product, Int. J. Production Economics, 67 (2000), 95-100.  doi: 10.1016/S0925-5273(00)00012-8.

[13]

Z.-L. Lin and Y.-S. Huang, Nonperiodic preventive maintenance for repairable systems, Naval Res. Logist., 57 (2010), 615-625.  doi: 10.1002/nav.20418.

[14]

B. Liu, J. Wu and M. Xie, Cost analysis for multi-component system with failure interaction under renewing free-replacement warranty, European J. Oper. Res., 243 (2015), 874-882.  doi: 10.1016/j.ejor.2015.01.030.

[15]

D. T. Mai, T. Liu, M. D. S. Morris and S. Sun, Quality coordination with extended warranty for store-brand products, European J. Oper. Res., 256 (2017), 524-532.  doi: 10.1016/j.ejor.2016.06.042.

[16]

M. D. C. Moura, J. M. Santana, E. L. Droguett, I. D. Lins and B. N. Guedes, Analysis of extended warranties for medical equipment: A Stackelberg game model using priority queues, Reliability Engineering System Safety, 168 (2017), 338-354.  doi: 10.1016/j.ress.2017.05.040.

[17]

D. G. Nguyen and D. N. P. Murthy, An optimal policy for servicing warranty, J. Oper. Res. Soc., 37 (1986), 1081-1088.  doi: 10.1057/jors.1986.185.

[18]

D. G. Nguyen and D. N. P. Murthy, Optimal replace repair strategy for servicing products sold with warranty, European J. Oper. Res., 39 (1989), 206-212.  doi: 10.1016/0377-2217(89)90193-8.

[19]

M. Park, K. M. Jung and D. H. Park, Optimal warranty policies considering repair service and replacement service under the manufacturer's perspective, Ann. Oper. Res., 244 (2016), 117-132.  doi: 10.1007/s10479-014-1740-1.

[20]

M. Park and H. Pham, Cost models for age replacement policies and block replacement policies under warranty, Appl. Math. Model., 40 (2016), 5689-5702.  doi: 10.1016/j.apm.2016.01.022.

[21]

X. Qin, Q. Su and S. H. Huang, Extended warranty strategies for online shopping supply chain with competing suppliers considering component reliability, J. Systems Sci. Systems Engineering, 26 (2017), 753-773.  doi: 10.1007/s11518-017-5355-3.

[22]

M. Reimann and W. Zhang, Joint optimization of new production, warranty servicing strategy and secondary market supply under consumer returns, Pesquisa Operacional, 33 (2013), 325-342.  doi: 10.1590/S0101-74382013000300001.

[23]

M. Shafiee, M. Finkelstein and S. Chukova, Burn-in and imperfect preventive maintenance strategies for warranted products, Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, 225 (2011), 211-218.  doi: 10.1177/1748006X11398584.

[24]

K. Shahanaghi, R. Noorossana, S. G. Jalali-Naini and M. Heydari, Failure modeling and optimizing preventive maintenance strategy during two-dimensional extended warranty contracts, Engineering Failure Analysis, 28 (2013), 90-102.  doi: 10.1016/j.engfailanal.2012.09.006.

[25]

L. Shang, S. Si and Z. Cai, Optimal maintenance-replacement policy of products with competing failures after expiry of the warranty, Comput. Industrial Engineering, 98 (2016), 68-77.  doi: 10.1016/j.cie.2016.05.012.

[26]

C. Su and J. Shen, Analysis of extended warranty policies with different repair options, Engineering Failure Analysis, 25 (2012), 49-62.  doi: 10.1016/j.engfailanal.2012.04.002.

[27]

C. Su and X. Wang, A two-stage preventive maintenance optimization model incorporating two-dimensional extended warranty, Reliability Engineering System Safety, 155 (2016), 169-178.  doi: 10.1016/j.ress.2016.07.004.

[28]

C. Tom and P. Elmira, Maintenance policies with two-dimensional warranty, Reliability Engineering System Safety, 77 (2002), 61–69.

[29]

P. Tong, Z. Liu, F. Men and L. Cao, Designing and pricing of two-dimensional extended warranty contracts based on usage rate, Internat. J. Prod. Res., 52 (2014), 6362-6380.  doi: 10.1080/00207543.2014.940073.

[30]

P. Tong, X. Song and L. Zixian, A maintenance strategy for two-dimensional extended warranty based on dynamic usage rate, Internat. J. Prod. Res., 55 (2017), 5743-5759.  doi: 10.1080/00207543.2017.1330573.

[31]

H. Vahdani, H. Mahlooji and A. Eshraghnia Jahromi, Warranty servicing for discretely degrading items with non-zero repair time under renewing warranty, Comput. Industrial Engineering, 65 (2013), 176-185.  doi: 10.1016/j.cie.2011.08.012.

[32]

S. Varnosafaderani and S. Chukova, A two-dimensional warranty servicing strategy based on reduction in product failure intensity, Comput. Math. Appl., 63 (2012), 201-213.  doi: 10.1016/j.camwa.2011.11.011.

[33]

J. Wang, Z. Zhou and H. Peng, Flexible decision models for a two-dimensional warranty policy with periodic preventive maintenance, Reliability Engineering System Safety, 162 (2017), 14-27.  doi: 10.1016/j.ress.2017.01.012.

[34]

Y. Wang, Z. Liu and Y. Liu, Optimal preventive maintenance strategy for repairable items under two-dimensional warranty, Reliability Engineering System Safety, 142 (2015), 326-333.  doi: 10.1016/j.ress.2015.06.003.

[35]

W. Xie, Optimal pricing and two-dimensional warranty policies for a new product, Internat. J. Prod. Res., 55 (2017), 6857-6870.  doi: 10.1080/00207543.2017.1355578.

[36]

Z.-S. Ye and D. N. P. Murthy, Warranty menu design for a two-dimensional warranty, Reliability Engineering System Safety, 155 (2016), 21-29.  doi: 10.1016/j.ress.2016.05.013.

show all references

References:
[1]

A. Akbarov and S. Wu, Forecasting warranty claims considering dynamic over-dispersion, Int. J. Prod. Econ., 139 (2012), 615-622.  doi: 10.1016/j.ijpe.2012.06.001.

[2]

J. Baik, D. N. P. Murthy and N. Jack, Two-dimensional failure modeling with minimal repair, Naval Res. Logist., 51 (2004), 345-362.  doi: 10.1002/nav.10120.

[3]

W. L. Chang and J.-H. Lin, Optimal maintenance policy and length of extended warranty within the life cycle of products, Comput. Math. Appl., 63 (2012), 144-150.  doi: 10.1016/j.camwa.2011.11.001.

[4]

S. Chukova and M. R. Johnston, Two-dimensional warranty repair strategy based on minimal and complete repairs, Math. Comput. Modelling, 44 (2006), 1133-1143.  doi: 10.1016/j.mcm.2006.03.015.

[5]

G. Gallego, R. Wang, M. Hu, J. Ward and J. L. Beltran, No claim? Your gain: Design of residual value extended warranties under risk aversion and strategic claim behavior, Manufacturing Service Oper. Management, 17 (2015), 87-100.  doi: 10.1287/msom.2014.0501.

[6]

J. C. Hartman and K. Laksana, Designing and pricing menus of extended warranty contracts, Naval Res. Logist., 56 (2009), 199-214.  doi: 10.1002/nav.20333.

[7]

Y.-S. Huang, W.-Y. Gau and J.-W. Ho, Cost analysis of two-dimensional warranty for products with periodic preventive maintenance, Reliability Engineering System Safety, 134 (2015), 51-58.  doi: 10.1016/j.ress.2014.10.014.

[8]

Y.-S. Huang, C.-D. Huang and J.-W. Ho, A customized two-dimensional extended warranty with preventive maintenance, European J. Oper. Res., 257 (2017), 971-978.  doi: 10.1016/j.ejor.2016.07.034.

[9]

B. P. Iskandar and D. N. P. Murthy, Repair-replace strategies for two-dimensional warranty policies, Math. Comput. Modelling, 38 (2003), 1233-1241.  doi: 10.1016/S0895-7177(03)90125-7.

[10]

B. P. Iskandar, D. N. P. Murthy and N. Jack, A new repair-replace strategy for items sold with a two-dimensional warranty, Comput. Oper. Res., 32 (2005), 669–682. doi: 10.1016/j.cor.2003.08.011.

[11]

N. Jack, B. P. Iskandar and D. N. P. Murthy, A repair-replace strategy based on usage rate for items sold with a two-dimensional warranty, Reliability Engineering System Safety, 94 (2009), 611-617.  doi: 10.1016/j.ress.2008.06.019.

[12]

N. Jack and V. D. D. Schouten, Optimal repair-replace strategies for a warranted product, Int. J. Production Economics, 67 (2000), 95-100.  doi: 10.1016/S0925-5273(00)00012-8.

[13]

Z.-L. Lin and Y.-S. Huang, Nonperiodic preventive maintenance for repairable systems, Naval Res. Logist., 57 (2010), 615-625.  doi: 10.1002/nav.20418.

[14]

B. Liu, J. Wu and M. Xie, Cost analysis for multi-component system with failure interaction under renewing free-replacement warranty, European J. Oper. Res., 243 (2015), 874-882.  doi: 10.1016/j.ejor.2015.01.030.

[15]

D. T. Mai, T. Liu, M. D. S. Morris and S. Sun, Quality coordination with extended warranty for store-brand products, European J. Oper. Res., 256 (2017), 524-532.  doi: 10.1016/j.ejor.2016.06.042.

[16]

M. D. C. Moura, J. M. Santana, E. L. Droguett, I. D. Lins and B. N. Guedes, Analysis of extended warranties for medical equipment: A Stackelberg game model using priority queues, Reliability Engineering System Safety, 168 (2017), 338-354.  doi: 10.1016/j.ress.2017.05.040.

[17]

D. G. Nguyen and D. N. P. Murthy, An optimal policy for servicing warranty, J. Oper. Res. Soc., 37 (1986), 1081-1088.  doi: 10.1057/jors.1986.185.

[18]

D. G. Nguyen and D. N. P. Murthy, Optimal replace repair strategy for servicing products sold with warranty, European J. Oper. Res., 39 (1989), 206-212.  doi: 10.1016/0377-2217(89)90193-8.

[19]

M. Park, K. M. Jung and D. H. Park, Optimal warranty policies considering repair service and replacement service under the manufacturer's perspective, Ann. Oper. Res., 244 (2016), 117-132.  doi: 10.1007/s10479-014-1740-1.

[20]

M. Park and H. Pham, Cost models for age replacement policies and block replacement policies under warranty, Appl. Math. Model., 40 (2016), 5689-5702.  doi: 10.1016/j.apm.2016.01.022.

[21]

X. Qin, Q. Su and S. H. Huang, Extended warranty strategies for online shopping supply chain with competing suppliers considering component reliability, J. Systems Sci. Systems Engineering, 26 (2017), 753-773.  doi: 10.1007/s11518-017-5355-3.

[22]

M. Reimann and W. Zhang, Joint optimization of new production, warranty servicing strategy and secondary market supply under consumer returns, Pesquisa Operacional, 33 (2013), 325-342.  doi: 10.1590/S0101-74382013000300001.

[23]

M. Shafiee, M. Finkelstein and S. Chukova, Burn-in and imperfect preventive maintenance strategies for warranted products, Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, 225 (2011), 211-218.  doi: 10.1177/1748006X11398584.

[24]

K. Shahanaghi, R. Noorossana, S. G. Jalali-Naini and M. Heydari, Failure modeling and optimizing preventive maintenance strategy during two-dimensional extended warranty contracts, Engineering Failure Analysis, 28 (2013), 90-102.  doi: 10.1016/j.engfailanal.2012.09.006.

[25]

L. Shang, S. Si and Z. Cai, Optimal maintenance-replacement policy of products with competing failures after expiry of the warranty, Comput. Industrial Engineering, 98 (2016), 68-77.  doi: 10.1016/j.cie.2016.05.012.

[26]

C. Su and J. Shen, Analysis of extended warranty policies with different repair options, Engineering Failure Analysis, 25 (2012), 49-62.  doi: 10.1016/j.engfailanal.2012.04.002.

[27]

C. Su and X. Wang, A two-stage preventive maintenance optimization model incorporating two-dimensional extended warranty, Reliability Engineering System Safety, 155 (2016), 169-178.  doi: 10.1016/j.ress.2016.07.004.

[28]

C. Tom and P. Elmira, Maintenance policies with two-dimensional warranty, Reliability Engineering System Safety, 77 (2002), 61–69.

[29]

P. Tong, Z. Liu, F. Men and L. Cao, Designing and pricing of two-dimensional extended warranty contracts based on usage rate, Internat. J. Prod. Res., 52 (2014), 6362-6380.  doi: 10.1080/00207543.2014.940073.

[30]

P. Tong, X. Song and L. Zixian, A maintenance strategy for two-dimensional extended warranty based on dynamic usage rate, Internat. J. Prod. Res., 55 (2017), 5743-5759.  doi: 10.1080/00207543.2017.1330573.

[31]

H. Vahdani, H. Mahlooji and A. Eshraghnia Jahromi, Warranty servicing for discretely degrading items with non-zero repair time under renewing warranty, Comput. Industrial Engineering, 65 (2013), 176-185.  doi: 10.1016/j.cie.2011.08.012.

[32]

S. Varnosafaderani and S. Chukova, A two-dimensional warranty servicing strategy based on reduction in product failure intensity, Comput. Math. Appl., 63 (2012), 201-213.  doi: 10.1016/j.camwa.2011.11.011.

[33]

J. Wang, Z. Zhou and H. Peng, Flexible decision models for a two-dimensional warranty policy with periodic preventive maintenance, Reliability Engineering System Safety, 162 (2017), 14-27.  doi: 10.1016/j.ress.2017.01.012.

[34]

Y. Wang, Z. Liu and Y. Liu, Optimal preventive maintenance strategy for repairable items under two-dimensional warranty, Reliability Engineering System Safety, 142 (2015), 326-333.  doi: 10.1016/j.ress.2015.06.003.

[35]

W. Xie, Optimal pricing and two-dimensional warranty policies for a new product, Internat. J. Prod. Res., 55 (2017), 6857-6870.  doi: 10.1080/00207543.2017.1355578.

[36]

Z.-S. Ye and D. N. P. Murthy, Warranty menu design for a two-dimensional warranty, Reliability Engineering System Safety, 155 (2016), 21-29.  doi: 10.1016/j.ress.2016.05.013.

Figure 1.  Termination point of 2D warranty service
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Figure 2.  Schematic of the maintenance strategy under 2D warranty
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Figure 3.  Trend diagram of $ W_r-U_r $
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Figure 4.  Diagram of the differentiated warranty service strategy
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Figure 5.  Curve of $ W_r-U_r $ ($ ε = 0.9 $)
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Figure 6.  Curve of $ W_r-U_r $ ($ \varepsilon = 1.1 $)
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Table 1.  The interval of usage rate intensity
Usage intensity Low limit of interval Upper limit of interval
Light $ r_{l1} $ $ r_{l2} $
Medium $ r_{l2} $ $ r_{h1} $
Heavy $ r_{h1} $ $ r_{h2} $
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Usage intensity Low limit of interval Upper limit of interval
Light $ r_{l1} $ $ r_{l2} $
Medium $ r_{l2} $ $ r_{h1} $
Heavy $ r_{h1} $ $ r_{h2} $
Table 2.  The interval of usage rate intensity
Failure $ C_{mi}(Yuan) $ $ C_{ci} (Yuan) $ $ f_i(w,r_d) $ $ \lambda_{0} $ $ k $
A31 1000 5000 $ 6.32E-04 w^{1.06}{e^{-({w}/{4.03})}}^{2.06} $ 4.03 2.06
A18 3200 6400 $ 2.20E-02 w^{0.59}{e^{-({w}/{3.22})}}^{1.59} $ 3.22 1.59
A88 800 4800 $ 2.53E-03 w^{0.45}{e^{-({w}/{2.48})}}^{1.45} $ 2.48 1.45
A20 2600 7800 $ 5.67E-02 w^{0.49}{e^{-({w}/{2.90})}}^{1.49} $ 2.90 1.49
A30 4500 9000 $ 4.78E-03 w^{0.62}{e^{-({w}/{3.40})}}^{1.62} $ 3.40 1.62
A10 3000 12000 $ 5.82E-02 w^{0.51}{e^{-({w}/{2.94})}}^{1.51} $ 2.94 1.51
A16 2900 8700 $ 9.24E-03 w^{0.42}{e^{-({w}/{2.76})}}^{1.42} $ 2.76 1.42
A50 3500 10500 $ 6.98E-02 w^{0.85}{e^{-({w}/{3.28})}}^{1.85} $ 3.28 1.85
A15 1700 3400 $ 1.36E-02 w^{0.51}{e^{-({w}/{3.14})}}^{1.51} $ 3.14 1.51
A40 2800 5600 $ 1.33E-01 w^{0.49}{e^{-({w}/{2.69})}}^{1.49} $ 2.69 1.49
A17 4600 9200 $ 1.89E-02 w^{0.55}{e^{-({w}/{3.22})}}^{1.55} $ 3.22 1.55
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Failure $ C_{mi}(Yuan) $ $ C_{ci} (Yuan) $ $ f_i(w,r_d) $ $ \lambda_{0} $ $ k $
A31 1000 5000 $ 6.32E-04 w^{1.06}{e^{-({w}/{4.03})}}^{2.06} $ 4.03 2.06
A18 3200 6400 $ 2.20E-02 w^{0.59}{e^{-({w}/{3.22})}}^{1.59} $ 3.22 1.59
A88 800 4800 $ 2.53E-03 w^{0.45}{e^{-({w}/{2.48})}}^{1.45} $ 2.48 1.45
A20 2600 7800 $ 5.67E-02 w^{0.49}{e^{-({w}/{2.90})}}^{1.49} $ 2.90 1.49
A30 4500 9000 $ 4.78E-03 w^{0.62}{e^{-({w}/{3.40})}}^{1.62} $ 3.40 1.62
A10 3000 12000 $ 5.82E-02 w^{0.51}{e^{-({w}/{2.94})}}^{1.51} $ 2.94 1.51
A16 2900 8700 $ 9.24E-03 w^{0.42}{e^{-({w}/{2.76})}}^{1.42} $ 2.76 1.42
A50 3500 10500 $ 6.98E-02 w^{0.85}{e^{-({w}/{3.28})}}^{1.85} $ 3.28 1.85
A15 1700 3400 $ 1.36E-02 w^{0.51}{e^{-({w}/{3.14})}}^{1.51} $ 3.14 1.51
A40 2800 5600 $ 1.33E-01 w^{0.49}{e^{-({w}/{2.69})}}^{1.49} $ 2.69 1.49
A17 4600 9200 $ 1.89E-02 w^{0.55}{e^{-({w}/{3.22})}}^{1.55} $ 3.22 1.55
Table 3.  Age and usage parameters of the 2D warranty coverage ($ \varepsilon = 0.9 $)
$ w_n $ Value $ u_n $ Value
$ W_l $ 3.35 $ U_l $ 2.18
$ W_m $ 2.44 $ U_m $ 4.39
$ W_h $ 1.83 $ U_h $ 5.49
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$ w_n $ Value $ u_n $ Value
$ W_l $ 3.35 $ U_l $ 2.18
$ W_m $ 2.44 $ U_m $ 4.39
$ W_h $ 1.83 $ U_h $ 5.49
Table 4.  Age and usage parameters of the 2D warranty coverage ($ \varepsilon = 1.1 $)
$ w_n $ Value $ u_n $ Value
$ W_l $ 3.53 $ U_l $ 2.29
$ W_m $ 2.28 $ U_m $ 4.10
$ W_h $ 1.50 $ U_h $ 4.50
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$ w_n $ Value $ u_n $ Value
$ W_l $ 3.53 $ U_l $ 2.29
$ W_m $ 2.28 $ U_m $ 4.10
$ W_h $ 1.50 $ U_h $ 4.50
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