A collaborative EPQ inventory model for a three-echelon supply chain with multiple products considering the effect of marketing effort on demand

Katherinne Salas Navarro; Jaime Acevedo Chedid; Whady F. Florez; Holman Ospina Mateus; Leopoldo Eduardo Cárdenas-Barrón; Shib Sankar Sana

doi:10.3934/jimo.2019020

Article Contents

2020, Volume 16, Issue 4: 1613-1633. Doi: 10.3934/jimo.2019020

This issue Previous Article Deteriorating inventory with preservation technology under price- and stock-sensitive demand Next Article Model selection based on value-at-risk backtesting approach for GARCH-Type models

A collaborative EPQ inventory model for a three-echelon supply chain with multiple products considering the effect of marketing effort on demand

1.
Department of Industrial Management, Agroindustry and Operations, Universidad de la Costa, Colombia
2.
Department of Industrial Engineering, Universidad Tecnologica de Bolivar, Colombia
3.
Grupo de Energia y Termodinamica, Universidad Pontificia Bolivariana, Colombia
4.
Department of Industrial and Systems Engineering, School of Engineering and Sciences, Tecnológico de Monterrey, Monterrey, México
5.
Principal, Kishore Bharati Bhagini Nivedita College, 148 Ramkrishna Sarani, Behala, Kolkata-700060, India

^* Corresponding author: Shib Sankar Sana
^* Corresponding author: Shib Sankar Sana

Received: June 30, 2018

Revised: September 30, 2018

Early access: March 2019

Published: May 2020

Abstract Full Text(HTML) Figure(1) / Table(10) Related Papers Cited by

Abstract

This paper presents an inventory model for a three-echelon supply chain with multiple products and multiple members considering the demand as an increasing function of the marketing effort. In the proposed inventory model, a collaborative approach is studied and an analytical method is applied to obtain the optimal production lot size and the optimal marketing effort in order to achieve the maximum profits. Some numerical examples are illustrated to justify the model. Moreover, a sensitivity analysis is well done in order to analysis the effect of the changes of key parameters of inventory model on the the maximum benefits of all members of the chain.

Keywords:

Mathematics Subject Classification: Primary: 90B05; Secondary: 53C35.

Citation:

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Figure 1. Diagram of inventory level of the supply chain

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Table 1. Contributions of some authors related to EPQ model

Authors	Multiple	Three layer	Marketing effort	Defective	Collaborative	Different

	items	model	sensitive demand	items	approach	cycle time

Sana [44]		$\checkmark$		$\checkmark$	$\checkmark$
Pal et al. [29]		$\checkmark$		$\checkmark$
Ben-Daya et al.[4]		$\checkmark$				$\checkmark$
Sana et al.[43]	$\checkmark$	$\checkmark$		$\checkmark$	$\checkmark$
Roy et al. [39]		$\checkmark$			$\checkmark$
Abdelsalam and Elassal [1]		$\checkmark$
Giri and Bardhan [11]		$\checkmark$			$\checkmark$
Dai et al.[8]		$\checkmark$
Sana [45]			$\checkmark$
Cárdenas-Barrón and Sana [6]	$\checkmark$		$\checkmark$		$\checkmark$
Tsao and Sheen [48]			$\checkmark$		$\checkmark$
Roy et al.[38]			$\checkmark$		$\checkmark$
Cárdenas-Barrón and Sana [5]			$\checkmark$		$\checkmark$
Our present paper	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$

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Table 2. Values of parameters for manufacturers

Parameters of	Product 1	Product 2
Manufacturer 1	$p_{11} = 700$ units/year, $\alpha_{11} = 0.01$	$p_{12} = 740$ units/year, $\alpha_{12} = 0.03$,
	$c_{11} = 15$＄/units/year, $w_{11}^{M} = 55$＄/unit, $\epsilon = 1$	$c_{12} = 17$＄/ unit, $w_{12}^{M} = 75$＄/unit, $\epsilon = 1$
	$L_{11} = 2000$＄ /production cycle	$L_{12} = 1900$＄ /production,
	$AM_{11} = 200$＄/setup, $k_{11} = 5.1$＄/unit	$AM_{12} = 208$＄/setup, $k_{12} = 4.5$＄/unit
	$h_{11}^{P} = 2.75$＄/unit/year, $\beta_{11} = 15$%	$h_{12}^{P} = 3.75$＄/unit/year, $\beta_{12} = 15$%
Manufacturer 2	$p_{21} = 900$ units/year, $\alpha_{21} = 0.04$	$p_{22} = 1050$ units/year, $\alpha_{22} = 0.03$
	$c_{21} = 16$＄/units/year, $w_{21}^{M} = 65$＄/unit, $\epsilon = 1$	$c_{22} = 21$＄/ unit, $w_{22}^{M} = 80$＄/unit, $\epsilon = 1$
	$L_{21} = 1700$＄ /production cycle	$L_{22} = 1750$＄ /production cycle
	$AM_{21} = 205$＄/setup, $k_{21} = 6.0$＄/unit	$AM_{22} = 210$＄/setup, $k_{22} = 8.0$＄/unit
	$h_{21}^{P} = 3.25$＄/unit/year, $\beta_{21} = 10$%	$h_{22}^{P} = 4.0$＄/unit/year, $\beta_{22} = 10$%,

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Table 3. Values of parameters for distribution centers

Parameters of	Product 1	Product 2
Distribution Centre 1	$AD_{11} = 190$ ＄/setup, $w_{11}^{D} = 75$＄/units	$AD_{12} = 200$ ＄/setup, $w_{12}^{D} = 95$＄/units
	$DEM_{11}^{D} = 170 $ units/year	$DEM_{12}^{D} = 180 $ units/year
	$h_{11}^{D} = 3.75$＄/unit/year	$h_{12}^{D} = 4.75$＄/unit/year
	$k_{11} = 2.25$＄/unit, $\beta_{11} = 10%$	$k_{12} = 2.85$＄/unit, $\beta_{12} = 10%$
Distribution Centre 2	$AD_{21} = 180$ ＄/setup, $w_{21}^{D} = 85$＄/units	$AD_{22} = 195$ ＄/setup, $w_{22}^{D} = 100$＄/units
	$DEM_{21}^{D} = 250 $ units/year	$DEM_{22}^{D} = 290 $ units/year
	$h_{21}^{D} = 4.25$＄/unit/year	$h_{22}^{D} = 5.0$＄/unit/year
	$k_{21} = 2.55$＄/unit, $\beta_{21} = 5%$	$k_{22} = 3.0$＄/unit, $\beta_{22} = 5%$

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Table 4. Values of parameters for retailers

Parameters of	Product 1	Product 2
Retailer 1	$w_{11}^{R} = 115$ ＄/unit, $k_{11} = 6.6$, $\beta_{1} = 20$%	$w_{12}^{R} = 135$ ＄/unit, $k_{12} = 6.9$, $\beta_{1} = 20%$
	$AR_{11} = 130 $ ＄/setup, $h_{11 }^{R} = 5.75$＄/unit/year	$AR_{12} = 140 $ ＄/setup, $h_{12 }^{R} = 6.75$＄/unit/year
	$DEM_{11}^{R} = 220$units/year, $\tau_{11} = 44$units/year	$DEM_{12}^{R} = 230$units/year, $\tau_{12} = 46$units/year
Retailer 2	$w_{21}^{R} = 125$ ＄/unit, $k_{21} = 9.0$, $\beta_{2} = 25$%	$w_{22}^{R} = 140$ ＄/unit, $k_{22} = 10.2$, $\beta_{2} = 25$%
	$AR_{21} = 115 $ ＄/setup, $h_{21 }^{R} = 6.25$＄/unit/year	$AR_{22} = 118 $ ＄/setup, $h_{22 }^{R} = 7.0$＄/unit/year
	$DEM_{21}^{R} = 300$units/year, $\tau_{21} = 60$units/year	$DEM_{22}^{R} = 340$units/year, $\tau_{22} = 68$units/year

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Table 5. Optimal solution for Example 1

$\rho$	APM	APD	APR	$\mu_{j}$	$q_{11}^{M}$	$q_{12}^{M}$	$q_{21}^{M}$	$q_{22}^{M}$
0	60051.71	21632.00	43070.27	124753.98	264.86	246.98	304.93	305.42
1	59879.90	21860.12	57593.64	139333.66	320.20	292.53	366.26	359.89
2	59784.84	21956.74	62462.60	144204.18	347.51	314.00	396.29	385.45
3	59724.75	22010.88	64902.89	146638.52	363.97	326.60	414.30	400.41
4	59683.00	22045.53	66367.86	148096.40	375.01	334.90	426.33	410.25
5	59652.02	22069.56	67343.81	149065.39	382.93	340.79	434.95	417.22
6	59627.88	22087.13	68039.75	149754.76	388.90	345.19	441.43	422.42
7	59608.36	22100.49	68560.39	150269.24	393.56	348.60	446.48	426.45

45	59390.16	22159.61	71340.32	152890.10	424.16	370.47	479.48	452.19
46	59386.77	22159.26	71346.60	152892.63	424.31	370.57	479.65	452.32
47	59383.40	22158.88	71352.30	152894.58	424.46	370.68	479.80	452.44
48	59380.05	22158.49	71357.47	152896.01	424.60	370.78	479.95	452.56
49	59376.72	22158.08	71362.12	152896.92	424.74	370.87	480.10	452.67
50	59373.40	22157.66	71366.31	152897.37	424.87	370.96	480.24	452.77
51	59370.11	22157.23	71370.04	152897.37	424.99	371.05	480.37	452.88
52	59366.83	22156.78	71373.35	152896.96	425.11	371.13	480.50	452.98

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Table 6. Values of parameters for manufacturers

Parameters of	Product 1	Product 2
Manufacturer 1	$p_{11} = 500$ units/year, $\alpha_{11} = 0.02$	$p_{12} = 560$ units/year, $\alpha_{12} = 0.01$,
	$c_{11} = 12$＄/units/year, $w_{11}^{M} = 43$＄/unit, $\epsilon = 1$	$c_{12} = 10$＄/ unit, $w_{12}^{M} = 62$＄/unit, $\epsilon = 1$
	$L_{11} = 1500$＄ /production cycle	$L_{12} = 1450$＄ /production,
	$AM_{11} = 160$＄/setup, $k_{11} = 4.3$＄/unit	$AM_{12} = 176$＄/setup, $k_{12} = 3.9$＄/unit
	$h_{11}^{P} = 2.05$＄/unit/year, $\beta_{11} = 8$%	$h_{12}^{P} = 2.94$＄/unit/year, $\beta_{12} = 8$%
Manufacturer 2	$p_{21} = 870$ units/year, $\alpha_{21} = 0.022$	$p_{22} = 920$ units/year, $\alpha_{22} = 0.015$
	$c_{21} = 18$＄/units/year, $w_{21}^{M} = 53$＄/unit, $\epsilon = 1$	$c_{22} = 19$＄/ unit, $w_{22}^{M} = 66$＄/unit, $\epsilon = 1$
	$L_{21} = 1530$＄ /production cycle	$L_{22} = 1550$＄ /production cycle
	$AM_{21} = 185$＄/setup, $k_{21} = 4.9$＄/unit	$AM_{22} = 190$＄/setup, $k_{22} = 6.8$＄/unit
	$h_{21}^{P} = 3.05$＄/unit/year, $\beta_{21} = 5$%	$h_{22}^{P} = 3.2$＄/unit/year, $\beta_{22} = 5$%,

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Table 7. Values of parameters for distribution centers

Parameters of	Product 1	Product 2
Distribution Centre 1	$AD_{11} = 150$ ＄/setup, $w_{11}^{D} = 65$＄/units	$AD_{12} = 1650$ ＄/setup, $w_{12}^{D} = 84$＄/units
	$DEM_{11}^{D} = 150 $ units/year	$DEM_{12}^{D} = 164 $ units/year
	$h_{11}^{D} = 2.95$＄/unit/year	$h_{12}^{D} = 3.5$＄/unit/year
	$k_{11} = 1.83$＄/unit, $\beta_{11} = 5%$	$k_{12} = 2.05$＄/unit, $\beta_{12} = 5%$
Distribution Centre 2	$AD_{21} = 174$ ＄/setup, $w_{21}^{D} = 72$＄/units	$AD_{22} = 182$ ＄/setup, $w_{22}^{D} = 84$＄/units
	$DEM_{21}^{D} = 236 $ units/year	$DEM_{22}^{D} = 279 $ units/year
	$h_{21}^{D} = 3.10$＄/unit/year	$h_{22}^{D} = 3.90$＄/unit/year
	$k_{21} = 2.10$＄/unit, $\beta_{21} = 2%$	$k_{22} = 2.75$＄/unit, $\beta_{22} = 2%$

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Table 8. Values of parameters for retailers

Parameters of	Product 1	Product 2
Retailer 1	$w_{11}^{R} = 94$ ＄/unit, $k_{11} = 5.10$, $\beta_{1} = 10$%	$w_{12}^{R} = 119$ ＄/unit, $k_{12} = 5.5$, $\beta_{1} = 10$%
	$AR_{11} = 110 $ ＄/setup, $h_{11 }^{R} = 3.8$＄/unit/year	$AR_{12} = 115 $ ＄/setup, $h_{12 }^{R} = 5.5$＄/unit/year
	$DEM_{11}^{R} = 180$units/year, $\tau_{11} = 35$units/year	$DEM_{12}^{R} = 190$units/year, $\tau_{12} = 37$units/year
Retailer 2	$w_{21}^{R} = 107$ ＄/unit, $k_{21} = 7.9$, $\beta_{2} = 15$%	$w_{22}^{R} = 126$ ＄/unit, $k_{22} = 8.6$, $\beta_{2} = 15$%
	$AR_{21} = 102 $ ＄/setup, $h_{21 }^{R} = 5.2$＄/unit/year	$AR_{22} = 105 $ ＄/setup, $h_{22 }^{R} = 6.1$＄/unit/year
	$DEM_{21}^{R} = 265$units/year, $\tau_{21} = 49$units/year	$DEM_{22}^{R} = 310$units/year, $\tau_{22} = 56$units/year

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Table 9. Optimal solution for Example 2

$\rho$	APM	APD	APR	$\mu$	$q_{11}^{M}$	$q_{12}^{M}$	$q_{21}^{M}$	$q_{22}^{M}$
0	45832.63	18356.66	33700.57	97889.86	222.13	209.85	262.20	297.45
1	45763.73	18467.95	43950.02	108181.70	253.32	239.95	294.82	342.42
2	45727.88	18510.28	47378.86	111617.02	266.67	252.99	308.54	362.42
3	45706.07	18532.68	49095.31	113334.07	274.11	260.32	316.12	373.8
4	45691.26	18546.52	50125.18	114362.96	278.87	265.02	320.94	381.14
5	45680.40	18555.88	50811.22	115047.50	282.17	268.29	324.27	386.28
6	45671.99	18562.61	51300.58	115535.17	284.60	270.70	326.71	390.08
7	45665.20	18567.64	51666.90	115899.74	286.46	272.55	328.58	393.01

57	45566.23	18586.60	53725.82	117878.66	298.47	284.52	340.54	412.12
58	45564.88	18586.41	53728.36	117879.65	298.50	284.56	340.57	412.18
59	45563.53	18586.21	53730.69	117880.44	298.54	284.59	340.60	412.23
60	45562.19	18586.00	53732.84	117881.03	298.57	284.62	340.64	412.28
61	45560.85	18585.80	53734.80	117881.45	298.60	284.66	340.67	412.33
62	45559.51	18585.59	53736.58	117881.69	298.63	284.69	340.70	412.38
63	45558.18	18585.38	53738.20	117881.76	298.66	284.72	340.73	412.43
64	45556.85	18585.17	53739.66	117881.68	298.69	284.74	340.75	412.48

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Table 10. Sensitivity analysis of numerical example 1

Change	$\rho$	$\mu$	APM	APD	APR	$q_{11}^{M}$	$q_{12}^{M}$	$q_{21}^{M}$	$q_{22}^{M}$
-60%	51	139800.21	59899.63	22282.85	57617.73	320.20	292.53	366.26	1058.53
-40%	51	153314.43	58598.34	22828.59	71887.49	424.99	371.05	480.37	1758.29
-20%	51	153048.94	59343.71	22268.86	71436.37	424.99	371.05	480.37	620.22
$h_{22}^{P}$ = 4.0	51	152897.37	59370.11	22157.23	71370.04	424.99	371.05	480.37	452.88
20%	50	152778.80	59350.99	22092.75	71335.05	424.87	370.96	480.24	373.87
40%	50	152678.01	59316.37	22045.70	71315.95	424.87	370.96	480.24	325.64
60%	50	152588.82	59278.03	22008.07	71302.73	424.87	370.96	480.24	292.27
-60%	50	152984.89	59546.27	22100.25	71338.37	424.87	370.96	480.24	382.25
-40%	50	152954.03	59484.25	22121.56	71348.22	424.87	370.96	480.24	407.12
-20%	50	152924.95	59426.92	22140.53	71357.50	424.87	370.96	480.24	430.55
$AM_{22}$ = 210	51	152897.37	59370.11	22157.23	71370.04	424.99	371.05	480.37	452.88
20%	51	152871.09	59319.76	22172.90	71378.44	424.99	371.05	480.37	474.06
40%	51	152845.94	59272.08	22187.38	71386.47	424.99	371.05	480.37	494.34
60%	51	152821.78	59226.72	22200.86	71394.20	424.99	371.05	480.37	513.82

-60%	51	155297.37	45450.11	38477.23	71370.04	424.99	371.05	480.37	452.88
-40%	51	154497.37	50090.11	33037.23	71370.04	424.99	371.05	480.37	452.88
-20%	51	153697.37	54730.11	27597.23	71370.04	424.99	371.05	480.37	452.88
$w_{22}^{M}$ = 80	51	152897.37	59370.11	22157.23	71370.04	424.99	371.05	480.37	452.88
20%	51	152097.37	64010.11	16717.23	71370.04	424.99	371.05	480.37	452.88
40%	51	151297.37	68650.11	11277.23	71370.04	424.99	371.05	480.37	452.88
60%	51	150497.37	73290.11	5837.23	71370.04	424.99	371.05	480.37	452.88
-60%	52	152921.92	59391.79	22156.78	71373.35	425.11	371.13	480.50	452.98
-40%	51	152913.69	59386.43	22157.23	71370.04	424.99	371.05	480.37	452.88
-20%	51	152905.53	59378.27	22157.23	71370.04	424.99	371.05	480.37	452.88
$k_{22}$ = 8.0	51	152897.37	59370.11	22157.23	71370.04	424.99	371.05	480.37	452.88
20%	50	152889.37	59365.40	22157.66	71366.31	424.87	370.96	480.24	452.77
40%	50	152881.37	59357.40	22157.66	71366.31	424.87	370.96	480.24	452.77
60%	49	152873.40	59353.20	22158.08	71362.12	424.74	370.87	480.10	452.67
-60%	51	144762.14	59159.30	22857.50	62745.34	347.51	314.00	396.29	2147.77
-40%	51	153112.57	59319.01	22321.38	71472.18	424.99	371.05	480.37	710.55
-20%	51	152969.73	59374.85	22200.75	71394.13	424.99	371.05	480.37	513.66
$p_{22}$ = 1050	51	152897.37	59370.11	22157.23	71370.04	424.99	371.05	480.37	452.88
20%	50	152851.11	59362.66	22134.13	71354.32	424.87	370.96	480.24	422.51
40%	50	152817.83	59351.56	22119.19	71347.09	424.87	370.96	480.24	404.27
60%	50	152792.05	59341.01	22108.80	71342.25	424.87	370.96	480.24	392.04
-60%	50	152768.88	59461.52	21953.66	71353.71	424.87	288.55	480.24	452.77
-40%	50	152808.30	59440.42	22010.95	71356.93	424.87	309.65	480.24	452.77
-20%	50	152850.83	59412.31	22077.53	71360.99	424.87	336.18	480.24	452.77
$h_{12}^{D}$ = 4.75	51	152897.37	59370.11	22157.23	71370.04	424.99	371.05	480.37	452.88
20%	51	152949.33	59313.00	22258.88	71377.46	424.99	419.51	480.37	452.88
40%	51	153009.19	59220.99	22399.36	71388.84	424.99	493.82	480.37	452.88
60%	51	153082.24	59044.14	22628.41	71409.70	424.99	630.01	480.37	452.88
-60%	50	152979.22	59446.72	22176.51	71355.99	424.87	303.47	480.24	452.77
-40%	50	152950.06	59421.66	22168.73	71359.66	424.87	327.52	480.24	452.77
-20%	50	152922.90	59397.20	22162.61	71363.09	424.87	349.91	480.24	452.77
$AD_{12}$ = 200	51	152897.37	59370.11	22157.23	71370.04	424.99	371.05	480.37	452.88
20%	51	152873.23	59346.99	22153.15	71373.09	424.99	390.97	480.37	452.88
40%	51	152850.25	59324.52	22149.74	71375.99	424.99	409.92	480.37	452.88
60%	51	152828.3	59302.68	22146.85	71378.77	424.99	428.04	480.37	452.88
-60%	51	152897.37	59370.11	9047.23	84480.04	424.99	371.05	480.37	452.
-40%	51	152897.37	59370.11	13417.23	80110.04	424.99	371.05	480.37	452.88
-20%	51	152897.37	59370.11	17787.23	75740.04	424.99	371.05	480.37	452.88
$w_{12}^{D}$ = 95	51	152897.37	59370.11	22157.23	71370.04	424.99	371.05	480.37	452.88
20%	51	152897.37	59370.11	26527.23	67000.04	424.99	371.05	480.37	452.88
40%	51	152897.37	59370.11	30897.23	62630.04	424.99	371.05	480.37	452.88
60%	51	152897.37	59370.11	35267.23	58260.04	424.99	371.05	480.37	452.88
-60%	51	152906.10	59370.11	22165.95	71370.04	424.99	371.05	480.37	452.88
-40%	51	152903.19	59370.11	22163.04	71370.04	424.99	371.05	480.37	452.88
-20%	51	152900.28	59370.11	22160.13	71370.04	424.99	371.05	480.37	452.88
$k_{12}$ = 2.85	51	152897.37	59370.11	22157.23	71370.04	424.99	371.05	480.37	452.88
20%	50	152894.52	59373.40	22154.81	71366.31	424.87	370.96	480.24	452.77
40%	50	152891.67	59373.40	22151.96	71366.31	424.87	370.96	480.24	452.77
60%	50	152888.82	59373.40	22149.11	71366.31	424.87	370.96	480.24	452.77

-60%	50	152734.53	59481.70	22071.37	71181.46	424.87	370.96	480.24	351.04
-40%	50	152784.43	59456.07	22095.57	71232.80	424.87	370.96	480.24	377.00
-20%	50	152838.33	59421.60	22123.72	71293.02	424.87	370.96	480.24	409.72
$h_{22}^{R}$ = 7.0	51	152897.37	59370.11	22157.23	71370.04	424.99	371.05	480.37	452.88
20%	51	152963.44	59298.54	22200.46	71464.44	424.99	371.05	480.37	513.24
40%	51	153039.80	59181.53	22260.64	71597.62	424.99	371.05	480.37	606.70
60%	51	153133.59	58950.93	22360.44	71822.22	424.99	371.05	480.37	781.50
-60%	51	152982.69	59273.13	22214.42	71495.15	424.99	371.05	480.37	533.98
-40%	51	152952.65	59311.79	22192.94	71447.93	424.99	371.05	480.37	502.31
-20%	51	152924.30	59343.55	22174.07	71406.68	424.99	371.05	480.37	475.68
$AR_{22}$ = 118	51	152897.37	59370.11	22157.23	71370.04	424.99	371.05	480.37	452.88
20%	50	152871.68	59395.91	22142.43	71333.34	424.87	370.96	480.24	432.98
40%	50	152847.07	59415.22	22128.52	71303.34	424.87	370.96	480.24	415.57
60%	50	152823.41	59431.94	22115.69	71275.79	424.87	370.96	480.24	400.11
-60%	45	118742.28	59390.16	22159.61	37192.50	424.16	370.47	479.48	452.19
-40%	47	130125.92	59383.40	22158.88	48583.64	424.46	370.68	479.80	452.44
-20%	49	141511.00	59376.72	22158.08	59976.20	424.74	370.87	480.10	452.67
$w_{22}^{R}$ = 140	51	152897.37	59370.11	22157.23	71370.04	424.99	371.05	480.37	452.88
20%	52	164285.03	59366.83	22156.78	82761.42	425.11	371.13	480.50	452.98
40%	54	175673.72	59360.32	22155.84	94157.55	425.34	371.29	480.75	453.16
60%	55	187063.41	59357.09	22155.36	105550.96	425.45	371.37	480.87	453.25
-60%	54	152977.57	59360.32	22155.84	71461.41	425.34	371.29	480.75	453.16
-40%	53	152950.20	59363.57	22156.32	71430.32	425.23	371.21	480.63	453.07
-20%	52	152923.48	59366.83	22156.78	71399.87	425.11	371.13	480.50	452.98
$k_{22}$ = 10.2	51	152897.37	59370.11	22157.23	71370.04	424.99	371.05	480.37	452.88
20%	49	152871.93	59376.72	22158.08	71337.13	424.74	370.87	480.10	452.67
40%	48	152847.05	59380.05	22158.49	71308.51	424.60	370.78	479.95	452.56
60%	47	152822.67	59383.40	22158.88	71280.39	424.46	370.68	479.80	452.44

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