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Optimum pricing strategy for complementary products with reservation price in a supply chain model

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  • This paper describes a two-echelon supply chain model with two manufacturers and one common retailer. Two types of complementary products are produced by two manufacturers, and the common retailer buys products separately using a reservation price and bundles them for sale. The demands of manufacturers and retailer are assumed to be stochastic in nature. When the retailer orders for products, any one of manufacturers agrees to allow those products, and the rest of the manufacturers have to provide the same amount. The profits of two manufacturers and the retailer are maximized by using Stackelberg game policy. By applying a game theoretical approach, several analytical solutions are obtained. For some cases, this model obtains quasi-closed-form solutions, for others, it finds closed-form solutions. Some numerical examples, sensitivity analysis, managerial insights, and graphical illustrations are given to illustrate the model.

    Mathematics Subject Classification: Primary: 90B05, 90B50; Secondary: 90B30.

    Citation:

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  • Figure 1.  Graphical representation for Case 1.1, total profit of manufacturer 1 versus selling-price and lot size

    Figure 2.  Graphical representation for Case 1.1, total profit of manufacturer 2 versus selling-price

    Figure 3.  Graphical representation for Case 1.1, total profit of retailer versus selling-price of bundle product

    Figure 4.  Graphical representation for Case 1.2, total profit of manufacturer 1 versus selling-price and lot size

    Figure 5.  Graphical representation for Case 1.2, total profit of manufacturer 2 and retailer versus selling-price and selling-price of bundle product

    Figure 6.  Graphical representation for Case 2.1, total profit of manufacturer 2 versus selling-price and lot size

    Figure 7.  Graphical representation for Case 2.1, total profit of manufacturer 1 versus sellingprice

    Figure 8.  Graphical representation for Case 2.1, total profit of retailer versus selling-price of bundle product

    Figure 9.  Graphical representation for Case 2.2, total profit of manufacturer 2 versus selling price and lot size

    Figure 10.  Graphical representation for Case 2.2, total profit of manufacturer 1 and retailer versus selling-price and selling-price of bundle product

    Figure 11.  Comparative studies of cooperation and non-cooperation for the selling-price of product 2 of manufacturer 2 in Case 1. Blue ink of the graphical representation indicates under cooperative strategy and the red ink of the graphical representation indicates under noncooperative strategy

    Figure 12.  Comparative studies of cooperation and non-cooperation for the selling-price of bundle product of retailer in Case 1. Blue ink of the graphical representation indicates under cooperative strategy and the red ink of the graphical representation indicates under noncooperative strategy

    Figure 13.  Comparative studies of cooperation and non-cooperation for the selling-price of product 1 of manufacturer 1 in Case 2. Blue ink of the graphical representation indicates under cooperative strategy and the red ink of the graphical representation indicates under noncooperative strategy

    Figure 14.  Comparative studies of cooperation and non-cooperation for the selling-price of bundle product of retailer in Case 2. Blue ink of the graphical representation indicates under cooperative strategy and the red ink of the graphical representation indicates under noncooperative strategy

    Table 1.  Comparison between the contributions of different authors

    Author (s)SCMCompetitive price studyReservation priceGame approachStochastic demand
    Choi [3]
    Yue et al. [22]
    Mukhopadhyay
    et al. [12]
    Wei et al. [18]
    Cárdenas-Barrón and Sana [2]
    Sarkar [14]
    McCardle et al. [10]
    This Model
     | Show Table
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    Decision variables
    $Q$ order quantity (units)
    $P_{i}$ selling-price of product j, j=1, 2 ($/unit)
    $P_{r}$ selling-price of the bundle product ($/unit)
    Random variables
    $D_{m_{i}}$ demand for product j, j=1, 2 (units)
    $D_{r}$ demand for the bundle product (units)
    Parameters
    $C_{i}$ manufacturing cost of product j, j=1, 2 ($/unit)
    $h_{m_{i}}$ holding cost of product j per unit per unit time, j=1, 2 ($/unit/unit time)
    $h_{r}$ holding cost of the bundle product per unit per unit time ($/unit/unit time)
    $S_{m_{i}}$ setup cost per setup of product j, j=1, 2 ($/unit)
    $K_{m_{i}}$ production rate of product j, j=1, 2 (units)
    $M$ known market size (units)
    $A$ ordering cost per order of the retailer ($/order)
    $I_{m_{i1}}$ inventory of manufacturer i at $ t \in[0, t_{m_{i}}]$, i=1, 2
    $I_{m_{i2}}$ inventory of manufacturer i at $t \in [t_{m_{i}}, T_{m_{i}}]$, i=1, 2
    $AP_{m_{i}}$ expected average profit of manufacturer i, i=1, 2
    $AP_{r}$ expected average profit of the retailer
    $t_{m_{i}}$ time required for maximum inventory of manufacturer i, i=1, 2
    $T_{m_{i}}$ cycle time of manufacturer i, i=1, 2
    $R_{i}^{a}$ lower limit of reservation price of manufacturer i, i=1, 2
    $R_{i}^{b}$ upper limit of reservation price of manufacturer i, i=1, 2
     | Show Table
    DownLoad: CSV

    Table 2.  Input data

    Player Market size (units) Manufacturer 1
    $M=1500$
    Manufacturer 2
    $M=1500$
    Retailer
    $M=1500$
    Setup cost
    ($/setup)
    $S_{m_{1}}=20$ $S_{m_{2}}=20$ $A=1$
    Holding cost
    ($/unit/year)
    $h_{m_{1}}=0.015$ $h_{m_{2}}=0.015$ $h_r=0.01$
    Production rate
    (units/year)
    $K_{m_{1}}=2000$ $K_{m_{2}}=2000$ -
    Purchasing cost
    ($/unit)
    $C_{1}=0.25$ $C_{2}=0.15$ -
    Reservation interval [0, 1] [0.1, 0.9] [0.1, 0.9]
      -indicates that the parameter is not available for this case.
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    Table 3.  Optimum results of Example 1

    Case $Q^*$ units $P_{1}^{*}$ $/unit $P_{2}^{*}$ $/unit $P_{r}^{*}$ $/unit $AP_{m_{1}}^{*}$ $/year $AP_{m_{2}}^{*}$ $/year $AP_{{r}}^{*}$ $/year $AP_{m_{1}r}^{*}$ $/year $AP_{m_{2}r}^{*}$ $/year
    1.1 1433.14 0.63 0.53 1.33 195.39 246.92 36.37 - -
    1.2 1433.14 0.63 0.44 1.29 195.39 - - - 294.18
    2.1 1690.88 0.63 0.53 1.33 195.18 247.15 35.90 - -
    2.2 1690.88 0.51 0.53 1.27 - 247.15 - 245.87 -
      -indicates that the average profit is not available for this case.
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    Table 4.  Input data from McCardle et al. [10]

    Player Market size (units) Manufacturer 1
    $M=100$
    Manufacturer 2
    $M=100$
    Retailer
    $M=100$
    Setup cost
    ($/setup)
    $S_{m_{1}}=0$ $S_{m_{2}}=0$ $A=0$
    Holding cost
    ($/unit/year)
    $h_{m_{1}}=0$ $h_{m_{2}}=0$ $h_r=0$
    Production rate
    (units/year)
    $K_{m_{1}}=0$ $K_{m_{2}}=0$ -
    Purchasing cost
    ($/unit)
    $C_{1}=0.25$ $C_{2}=0.25$ -
    Reservation interval [0, 1] [0.1, 0.9] [0.1, 0.9]
      -indicates that the parameter is not available for this case.
     | Show Table
    DownLoad: CSV

    Table 5.  Optimum results of Example 2

    Case $Q^* $ units $P_{1}^{*}$ $/unit $P_{2}^{*}$ $/unit $P_{r}^{*}$ $/unit $AP_{m_{1}}^{*}$ $/year $AP_{m_{2}}^{*}$ $/year $AP_{{r}}^{*}$ $/year $AP_{m_{1}r}^{*}$ $/year $AP_{m_{2}r}^{*}$ $/year
    1.1 $300$ $0.625$ $0.625$ $1.375$ $14.0625$ $14.0625$ $1.5625$ - -
    1.2 $300$ $0.625$ $0.541667$ $1.33$ $14.0625$ - - - $16.1458$
    2.1 $300$ $0.625$ $0.625$ $1.375$ $14.0625$ $14.0625$ $1.5625$ - -
    2.2 $300$ $0.625$ $0.541667$ $1.33333$ - $14.0625$ - 16.1458 -
      -indicates that the average profit is not available for this case.
     | Show Table
    DownLoad: CSV

    Table 6.  Sensitivity analysis for Case 1.1

    Parameter change(in %) $AP_{m_{1}}$
    (in %)
    $AP_{m_{2}}$
    (in %)
    $AP_{r}$
    (in %)
    $M$ -50% -52.76 -52.18 -59.85
    -25% -26.38 -26.09 -29.93
    +25% +26.38 +26.09 +29.93
    +50% +52.75 +52.18 +59.85
    Parameter change(in %) $AP_{m_{1}}$
    (in %)
    Parameter change(in %) $AP_{m_{2}}$
    (in %)
    $S_{m_{1}}$ -50% +1.99 $S_{m_{2}}$ -50% +1.97
    -25% +0.99 -25% +0.98
    +25% -0.99 +25% -0.98
    +50% -1.98 +50% -1.95
    $h_{m_{1}}$ -50% +2.33 $h_{m_{2}}$ -50% +1.42
    -25% +1.06 -25% +1.42
    +25% -0.94 +25% -0.71
    +50% -1.78 +50% -1.42
    $C_{1}$ -50% +38.63 $C_{2}$ -50% +22.18
    -25% +18.54 -25% +10.82
    +25% -17.04 +25% -10.29
    +50% -32.58 +50% -20.04
    Parameter change(in %) $AP_{r}$
    (in %)
    Parameter change(in %) $AP_{r}$
    (in %)
    $A$ -50% +0.25 $h_{r}$ -50% +9.85
    -25% +0.12 -25% +4.93
    +25% -0.12 +25% -4.93
    +50% -0.25 +50% -9.85
     | Show Table
    DownLoad: CSV

    Table 7.  Sensitivity analysis for Case 1.2

    Parameter change(in %) $AP_{m_{1}}$
    (in %)
    $AP_{m_{2}r}$
    (in %)
    $M$ -50% -52.15 -53.04
    -25% -26.24 -26.52
    +25% +26.49 +26.52
    +50% +53.18 +53.04
    Parameter change(in %) $AP_{m_{1}}$
    (in %)
    Parameter change(in %) $AP_{m_{2}r}$
    (in %)
    $S_{m_{1}}$ -50% +1.99 $S_{m_{2}}$ -50% +2.04
    -25% +0.99 -25% +1.02
    +25% -0.99 +25% -1.01
    +50% -1.98 +50% -2.02
    $h_{m_{1}}$ -50% +2.33 $h_{m_{2}}$ -50% +1.05
    -25% +1.06 -25% +0.52
    +25% -0.94 +25% -0.52
    +50% -1.78 +50% -1.04
    $C_{1}$ -50% +38.63 $C_{2}$ -50% +22.91
    -25% +18.55 -25% +11.18
    +25% -17.02 +25% -10.62
    +50% -32.52 +50% -20.67
     | Show Table
    DownLoad: CSV

    Table 8.  Sensitivity analysis for Case 2.1

    Parameters change(in %) $AP_{m_{1}}$
    (in %)
    $AP_{m_{2}}$
    (in %)
    $AP_{r}$
    (in %)
    $M$ -50% -53.25 -52.57 -61.77
    -25% -26.62 -26.28 -30.89
    +25% +26.62 +26.28 +30.89
    +50% +53.25 +52.57 +61.77
    Parameter change(in %) $AP_{r}$
    (in %)
    Parameter change(in %) $AP_{r}$
    (in %)
    $A$ -50% +0.21 $h_{r}$ -50% +11.77
    -25% +0.11 -25% +5.89
    +25% -0.11 +25% -5.89
    +50% -0.21 +50% -11.77
    Parameter change(in %) $AP_{m_{1}}$
    (in %)
    Parameter change(in %) $AP_{m_{2}}$
    (in %)
    $S_{m_{1}}$ -50% +1.70 $S_{m_{2}}$ -50% +1.96
    -25% +0.85 -25% +0.90
    +25% -0.84 +25% -0.79
    +50% -1.69 +50% -1.50
    $h_{m_{1}}$ -50% +2.34 $h_{m_{2}}$ -50% +1.96
    -25% +1.17 -25% +0.90
    +25% -1.17 +25% -0.79
    +50% -2.34 +50% -1.50
    $C_{1}$ -50% +38.76 $C_{2}$ -50% +22.27
    -25% +18.63 -25% +10.86
    +25% -17.13 +25% -10.32
    +50% -32.76 +50% -20.09
     | Show Table
    DownLoad: CSV

    Table 9.  Sensitivity analysis for Case 2.2

    Parameter change(in %) $AP_{m_{2}}$
    (in %)
    $AP_{m_{1}r}$
    (in %)
    $M$ -50% -52.57 -54.30
    -25% -26.28 -27.15
    +25% +26.28 +27.15
    +50% +52.57 +54.30
    Parameter change(in %) $AP_{m_{1r}}$
    (in %)
    Parameter change(in %) $AP_{m_{2}}$
    (in %)
    $S_{m_{1}}$ -50% +1.76 $S_{m_{2}}$ -50% +1.96
    -25% +0.88 -25% +0.90
    +25% -0.88 +25% -0.79
    +50% -1.75 +50% -1.50
    $h_{m_{1}}$ -50% +1.64 $h_{m_{2}}$ -50% +1.96
    -25% +0.82 -25% +0.90
    +25% -0.82 +25% -0.79
    +50% -1.64 +50% -1.50
    $C_{1}$ -50% +40.31 $C_{2}$ -50% +22.27
    -25% +19.36 -25% +10.86
    +25% -17.77 +25% -10.32
    +50% -33.95 +50% -20.09
    Parameter change(in %) $AP_{m_{1r}}$
    (in %)
    Parameter change(in %) $AP_{r}$
    (in %)
    $A$ -50% +0.04 $h_{r}$ -50% +1.72
    -25% +0.02 -25% +0.86
    +25% -0.02 +25% -0.86
    +50% -0.04 +50% -1.72
     | Show Table
    DownLoad: CSV
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