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Article Contents

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

• 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:

• 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) SCM Competitive price study Reservation price Game approach Stochastic 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 √ √ √ √ √
 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

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.

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.

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.

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.

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

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

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

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
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