\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

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

  • * Corresponding author: Shib Sankar Sana

    * Corresponding author: Shib Sankar Sana
Abstract Full Text(HTML) Figure(1) / Table(10) Related Papers Cited by
  • 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.

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

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Diagram of inventory level of the supply chain

    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 $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
     | Show Table
    DownLoad: CSV

    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, $\beta_{22} = 10$%,
     | Show Table
    DownLoad: CSV

    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, $\beta_{22} = 5%$
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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, $\beta_{22} = 5$%,
     | Show Table
    DownLoad: CSV

    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, $\beta_{22} = 2%$
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
  • [1] H. M. Abdelsalam and M. M. Elassal, Joint economic lot sizing problem for a three-Layer supply chain with stochastic demand, International Journal of Production Economics, 155 (2014), 272-283. 
    [2] A. Ahmadi-Javid and P. Hoseinpour, A location-inventory-pricing model in a supply chain distribution network with price-sensitive demands and inventory-capacity constraints, Transportation Research Part E: Logistics and Transportation Review, 82 (2015), 238-255. 
    [3] F. Alawneh and G. Zhang, Dual-channel warehouse and inventory management with stochastic demand, Transportation Research Part E: Logistics and Transportation Review, 112 (2018), 84-106. 
    [4] M. Ben-DayaR. Asád and M. Seliaman, An integrated production inventory model with raw material replenishment considerations in a three layer supply chain, International Journal of Production Economics, 143 (2013), 53-61. 
    [5] L. E. Cárdenas-Barrón and S. S. Sana, A production-inventory model for a two-echelon supply chain when demand is dependent on sales teams' initiatives, International Journal of Production Economics, 155 (2014), 249-258. 
    [6] L. E. Cárdenas-Barrón and S. S. Sana, Multi-item EOQ inventory model in a two-layer supply chain while demand varies with promotional effort, Applied Mathematical Modelling, 39 (2015), 6725-6737.  doi: 10.1016/j.apm.2015.02.004.
    [7] L. E. Cárdenas-BarrónJ. T. TengG. Trevino-GarzaH. M. Wee and K. R. Lou, An improved algorithm and solution on an integrated production-inventory model in a three-layer supply chain, International Journal of Production Economics, 136 (2012), 384-388. 
    [8] Z. DaiF. Aqlan and K. Gao, Optimizing multi-echelon inventory with three types of demand in supply chain, Transportation Research Part E: Logistics and Transportation Review, 107 (2017), 141-177. 
    [9] S. K. De and S. S. Sana, The (p, q, r, l) mode l fo r stochastic demand unde r Intuitionistic fuzzy agg regation with Bonferroni mean, Journal of Intelligent Manufacturing, check (2016), 1-19. 
    [10] J. P. Dube, Multiple discreteness and product differentiation: Demand for carbonated soft drinks, Marketing Science, 23 (2004), 66-81. 
    [11] B. C. Giri and S. Bardhan, Sub-supply chain coordination in a three-layer chain under demand uncertainty and random yield in production, International, Journal of Production Economics, 191 (2017), 66-73. 
    [12] S. K. Goyal and C. T. Chang, Optimal ordering and transfer policy for an inventory with stock dependent demand, European Journal of Operational Research, 196 (2009), 177-185. 
    [13] S. K. Goyal and A. Gunasekaran, An integrated production-inventory-marketing model for deteriorating items, Computers & Industrial Engineering, 28 (1995), 755-762. 
    [14] K. L. Hou and L. C. Lin, An EOQ model for deteriorating items with price-and stock-dependent selling rate under inflation and time value of money, International Journal of Systems Science, 37 (2006), 1131-1139.  doi: 10.1080/00207720601014206.
    [15] M. JohariS. M. Hosseini-MotlaghM. NematollahiM. Goh and J. Ignatius, Bi-level credit period coordination for periodic review inventory system with price-credit dependent demand under time value of money, Transportation Research Part E: Logistics and Transportation Review, 114 (2018), 270-291. 
    [16] S. Karray, Periodicity of pricing and marketing efforts in a distribution channel, European Journal of Operational Research, 228 (2013), 635-647.  doi: 10.1016/j.ejor.2013.02.012.
    [17] S. Kim and D. I. Gilliland, Working more or working less? Contingent allocation of reseller effort in distribution channels, Industrial Marketing Management, 64 (2017), 44-56. 
    [18] H. KrishnanR. Kapusciniski and D. A. Butz, Coordinating contracts for decentralized supply chain with retailer promotional effect, Management Science, 50 (2004), 48-63. 
    [19] W. LeeS. P. Wang and W. C. Chen, Forward and backward stocking policies for a two-level supply chain with consignment stock agreement and stock-dependent demand, European Journal of Operational Research, 256 (2017), 830-840.  doi: 10.1016/j.ejor.2016.06.060.
    [20] P. MaH. Wnag and J. Shang, Supply chain chanel strategies with quality and marketing effort-dependent demand, International Journal of Production Economics, 144 (2013), 572-581. 
    [21] P. MaH. Wang and J. Shang, Contract design for two-stage supply chain coordination: Integrating manufacturer-quality and retailer-marketing efforts, International Journal of Production Economics, 146 (2013), 745-755. 
    [22] J. MinY. W. Zhou and J. Zhao, An inventory model for deteriorating items under stock-dependent demand and two-level trade credit, Applied Mathematical Modelling, 34 (2010), 3273-3285.  doi: 10.1016/j.apm.2010.02.019.
    [23] N. M. ModakS. Panda and S. S. Sana, Pricing policy and coordination for a two-layer supply chain of duopolistic retailers and socially responsible manufacturer, International Journal of Logistic: Research and Applications, 19 (2015), 487-508. 
    [24] N. M. ModakS. Panda and S. S. Sana, Two-echelon supply chain coordination among manufacturer and duopolies retailers with recycling facility, International Journal of Advanced Manufacturing Technology, 87 (2016), 1531-1546. 
    [25] P. A. NaikK. Raman and R. S. Winer, Planning marketing-mix strategies in the presence of interaction effects, Marketing Science, 24 (2005), 25-34. 
    [26] M. G. Nagler, An exploratory analysis of the determinants of cooperative advertising participation rates, Marketing Letters, 17 (2006), 91-102. 
    [27] T. PaksoyN. Y. Pehlivan and E. Ozceylan, A new tradeoff model for fuzzy supply chain network design and optimization, Human and Ecological Risk Assessment: An International Journal, 19 (2013), 492-514. 
    [28] T. PaksoyE. Ozceylan and G. W. Weber, Profit oriented supply chain network optimization, Central European Journal of Operations Research, 21 (2013), 455-478.  doi: 10.1007/s10100-012-0240-0.
    [29] B. PalS. S. Sana and K. Chaudhuri, Three-layer supply chain – A production-inventory model for reworkable items, Applied Mathematics and Computation, 219 (2012), 530-543.  doi: 10.1016/j.amc.2012.06.038.
    [30] S. PandaN. M. ModakM. Basu and S. K. Goyal, Channel coordination and profit distribution in a social responsible three-layer supply chain, International Journal of Production Economics, 168 (2015), 224-233. 
    [31] S. PandaS. Saha and S. K. Goyal, Dilema of rented warehouse and shelf for inventory systems with displayed stock level dependent demand, Economic Modelling, 32 (2013), 452-462. 
    [32] S. PandaN. M. ModakS. S. Sana and M. Basu, Pricing and replenishment policies in dual-channel supply chain under continuous unit cost decrease, Applied Mathematics and Computation, 256 (2015), 913-929.  doi: 10.1016/j.amc.2015.01.081.
    [33] M. PervinS. K. Roy and G. W. Weber, Analysis of inventory control model with shortage under time-dependent demand and time-varying holding cost including stochastic deterioration, Annals of Operations Research, 260 (2018), 437-460.  doi: 10.1007/s10479-016-2355-5.
    [34] M. PervinS. K. Roy and G. W. Weber, A Two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items, Numerical Algebra, Control & Optimization, 7 (2017), 21-50.  doi: 10.3934/naco.2017002.
    [35] M. PervinS. K. Roy and G. W. Weber, An integrated inventory model with variable holding cost under two levels of trade-credit policy, Numerical Algebra, Control & Optimization, 8 (2018), 169-191.  doi: 10.3934/naco.2018010.
    [36] M. Pervin, S. K. Roy and G. W. Weber, Multi-item deteriorating two-echelon inventory model with price-and stock-dependent demand: A trade-credit policy, Journal of Industrial & Management Optimization, (2018).
    [37] X. PuL. Gong and G. Han, A feasible incentive contract between a manufacturer and his fairness-sensitive retailer engaged in strategic marketing efforts, Journal of Intelligent Manufacturing, 30 (2019), 193-206.  doi: 10.1007/s10845-016-1239-5.
    [38] A. RoyS. S. Sana and K. Chaudhuri, A joint venturing of single supplier and single retailer under variable price, promotional effort and service level, Pacific Science Review B: Humanities and Social Sciences, 1 (2015), 8-14. 
    [39] A. RoyS. S. Sana and K. Chaudhuri, Optimal replenishment order for uncertain demand in three layer supply chain, Economic Modelling, 29 (2012), 2274-2282. 
    [40] K. Salas-NavarroJ. Acevedo-ChedidN. Mercado-Caruso and S. S. Sana, An inventory model of three-layer supply chain of wood and furniture industry in the Caribbean region of Colombia, Internarional Journal of Systems Science: Operations and Logistics, 5 (2018), 69-86. 
    [41] S. S. Sana, An EOQ model for salesmen's initiatives, stock and price sensitive demand of similar products – A dynamical system, Applied Mathematics and Computation, 218 (2011), 3277-3288.  doi: 10.1016/j.amc.2011.08.067.
    [42] S. S. Sana, Optimal contract strategies for two stage supply chain, Economic Modelling, 30 (2013), 253-260. 
    [43] S. S. SanaJ. Acevedo-Chedid and K. Salas-Navarro, A three layer supply chain model with multiple suppliers, manufacturers and retailers for multiple items, Applied Mathematics and Computation, 229 (2014), 139-150.  doi: 10.1016/j.amc.2013.12.006.
    [44] S. S. Sana, A production-inventory model of imperfect quality products in a three-layer supply chain, Decision Support Systems, 50 (2011), 539-547. 
    [45] S. S. Sana, Sales team's initiatives and stock sensitive demand – A production control policy, Economic Modelling, 31 (2013), 783-788. 
    [46] J. SongF. LiD. D. WuL. Liang and A. Dolgui, Supply chain coordination through integration of innovation effort and advertising support, Applied Mathematical Modelling, 49 (2017), 108-123.  doi: 10.1016/j.apm.2017.04.041.
    [47] H. Soni and N. H. Shah, Optimal ordering policy for stock-dependent demand under progressive payment scheme, European Journal of Operational Research, 184 (2008), 91-100.  doi: 10.1016/j.ejor.2006.10.048.
    [48] Y. C. Tsao and G. J. Sheen, Effects of promotion cost sharing policy with the sales learning curve on supply chain coordination, Computers & Operations Research, 39 (2012), 1872-1878.  doi: 10.1016/j.cor.2011.07.009.
    [49] S. Zhao and Q. Zhu, A risk-averse marketing strategy and its effect on coordination activities in a remanufacturing supply chain under market fluctuation, Journal of Cleaner Production, 171 (2018), 1290-1299. 
  • 加载中

Figures(1)

Tables(10)

SHARE

Article Metrics

HTML views(3198) PDF downloads(748) Cited by(0)

Access History

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return