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Optimal return and rebate mechanism based on demand sensitivity to reference price

  • * Corresponding author: Nengmin Wang

    * Corresponding author: Nengmin Wang 

The first author is supported by the Key Project of National Natural Science Foundation of China under Grant 71732006; the National Natural Science Foundation of China under Grants 71572138, 71390331, 71401132, and 71371150

Abstract Full Text(HTML) Figure(18) / Table(9) Related Papers Cited by
  • The acceleration of electronic products' upgrade affects consumers' purchase behaviour. How to encourage consumers to return old products in order to upgrade to new products and how to optimize such the closed-loop supply chain are important managerial topics. According to the theory of reference price, the closed-loop supply chain model with fixed rebate and variable rebate is established. The analysis results imply that, when consumers' willingness to return second-hand products depends on manufacturers' rebates and prices of new products, the profit of closed-loop supply chain decreases. In addition, when consumers are sensitive to price difference, enterprises can adopt low profit margin methods to increase new product demand. Furthermore, the profit of the manufacturer is closely related to whether consumers are loss-seeking or loss-averse. Finally, our analysis provides the insights of the relationship between the optimal return and rebate mechanism and the use time of the previous generation of products.

    Mathematics Subject Classification: Primary: 91A05, 91B42; Secondary: 91B99.

    Citation:

    \begin{equation} \\ \end{equation}
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  • Figure 1.  $ \ \pi _{1}^{*} $ when$ \ \beta $ varies

    Figure 2.  $ \ \pi _{R_1}^{*} $ when$ \ k $ varies

    Figure 3.  $ \ \pi _{M_1}^{*} $ when$ \ k $ varies

    Figure 4.  $ \ \pi _{R_1}^{*} $ when$ \ k $ varies

    Figure 5.  $ \ \pi _{M_1}^{*} $ when$ \ k $ varies

    Figure 6.  $ \ \pi _{1}^{*} $ when$ \ k $ varies

    Figure 7.  $ \ \pi _{2}^{*} $ when$ \ \beta $ varies

    Figure 8.  $ \ \pi _{R_2}^{*} $ when$ \ t $ varies

    Figure 9.  $ \ \pi _{M_2}^{*} $ when$ \ t $ varies

    Figure 10.  $ \ \pi _{R_2}^{*} $ when$ \ k $ varies

    Figure 11.  $ \ \pi _{M_2}^{*} $ when$ \ k $ varies

    Figure 12.  $ \ \pi _{2}^{*} $ when$ k $ varies

    Figure 13.  $ \ \pi _{R_2}^{*} $ when$ \ k $ varies

    Figure 14.  $ \ \pi _{M_2}^{*} $ when$ \ k $ varies

    Figure 15.  $ \ \pi _{2}^{*} $ when$ k $ varies

    Figure 16.  $ \ (\pi _{R_1}^{*}-\pi _{R_2}^{*}) $ when$ \ t $ varies

    Figure 17.  $ \ (\pi _{M_1}^{*}-\pi _{M_2}^{*}) $ when$ \ t $ varies

    Figure 18.  $ \ \pi _{1}^{*}-\pi _{2}^{*} $ when$ \ t $ varies

    Table 1.  The description of the symbols

    Notations Description
    $ \ p_0 $ The price at which the previous generation of products are sold
    $ \ p $ Sales price of new generation products
    $ \ \omega $ The unit wholesale price of the new product
    $ \ c $ Unit cost of new product
    $ \ T $ Product life cycle
    $ \ t $ Product usage time
    $ \ a $ Market demand scale
    $ \ b $ Price sensitivity coefficient
    $ \ \alpha $ The quantity of used products returned by consumers independent
    of price and rebate
    $ \ \beta $ Consumers' willingness to return used products
    $ \ \delta $ Discount factor according to product quality
    $ \ k $ Price difference (price difference between the new product and
    the previous one) sensitivity coefficient ($ \ k\geq0 $)
    $ \ g $ Cost of the unit of remanufactured product
    (including remanufacturing, transportation and detection costs)
    $ \ l $ Fixed rebate
    $ \ d $ The demand for new products
    $ \ r_i $ The quantity of previous generation products returned
    $ \ \pi_{M_i} $ Manufacturer's profit
    $ \ \pi_{R_i} $ Retailer's profit
    $ \ \pi_i $ Profits of closed loop supply chains
    $ \ i=1,2 $ Represent the corresponding parameters under fixed rebate and
    variable rebate respectively
     | Show Table
    DownLoad: CSV

    Table 2.  If consumers are loss-seeking

    $ \ \beta $ with fixed rebate mechanism with variable rebate mechanism
    $ \ p _{1}^{*} $ $ \ \omega _{1}^{*} $ $ \ \pi _{M_1}^{*} $ $ \ \pi _{R_1}^{*} $ $ \ \pi _{1}^{*} $ $ \ p _{2}^{*} $ $ \ \omega _{2}^{*} $ $ \ \pi _{M_2}^{*} $ $ \ \pi _{R_2}^{*} $ $ \ \pi _{2}^{*} $
    0 0.989 0.693 0.225 0.092 0.317 0.989 0.693 0.193 0.092 0.285
    0.2 0.987 0.688 0.215 0.094 0.309 0.989 0.692 0.191 0.093 0.284
    0.4 0.986 0.683 0.205 0.095 0.301 0.988 0.691 0.189 0.093 0.282
    0.6 0.982 0.679 0.196 0.097 0.293 0.988 0.690 0.187 0.093 0.280
    0.8 0.980 0.674 0.186 0.098 0.285 0.987 0.689 0.185 0.094 0.279
    1 0.977 0.669 0.177 0.100 0.277 0.987 0.688 0.183 0.094 0.277
     | Show Table
    DownLoad: CSV

    Table 3.  If consumers are loss-averse

    $ \ \beta $ with fixed rebate mechanism with variable rebate mechanism
    $ \ p _{1}^{*} $ $ \ \omega _{1}^{*} $ $ \ \pi _{M_1}^{*} $ $ \ \pi _{R_1}^{*} $ $ \ \pi _{1}^{*} $ $ \ p _{2}^{*} $ $ \ \omega _{2}^{*} $ $ \ \pi _{M_2}^{*} $ $ \ \pi _{R_2}^{*} $ $ \ \pi _{2}^{*} $
    0 0.989 0.860 0.445 0.018 0.463 0.989 0.860 0.471 0.018 0.489
    0.2 0.963 0.807 0.341 0.026 0.367 0.963 0.808 0.369 0.026 0.395
    0.4 0.937 0.755 0.240 0.035 0.275 0.937 0.756 0.269 0.035 0.304
    0.6 0.911 0.702 0.141 0.046 0.187 0.911 0.704 0.172 0.045 0.217
    0.8 0.885 0.650 0.046 0.058 0.104 0.886 0.652 0.078 0.057 0.135
    1 0.858 0.598 -0.047 0.071 0.024 0.860 0.600 -0.013 0.071 0.058
     | Show Table
    DownLoad: CSV

    Table 4.  If consumers are loss-seeking

    $ \ k $ with fixed rebate mechanism with variable rebate mechanism
    $ \ p _{1}^{*} $ $ \ \omega _{1}^{*} $ $ \ \pi _{M_1}^{*} $ $ \ \pi _{R_1}^{*} $ $ \ \pi _{1}^{*} $ $ \ p _{2}^{*} $ $ \ \omega _{2}^{*} $ $ \ \pi _{M_2}^{*} $ $ \ \pi _{R_2}^{*} $ $ \ \pi _{2}^{*} $
    0 1.088 0.746 0.168 0.081 0.249 1.095 0.761 0.157 0.078 0.235
    0.2 1.035 0.714 0.181 0.087 0.268 1.041 0.726 0.169 0.084 0.253
    0.4 0.998 0.690 0.194 0.093 0.287 1.003 0.701 0.182 0.090 0.272
    0.6 0.970 0.673 0.207 0.099 0.307 0.975 0.682 0.194 0.096 0.290
    0.8 0.949 0.659 0.221 0.106 0.327 0.953 0.667 0.208 0.103 0.311
    1 0.931 0.648 0.235 0.112 0.347 0.935 0.656 0.221 0.109 0.330
     | Show Table
    DownLoad: CSV

    Table 5.  If consumers are loss-averse

    $ \ k $ with fixed rebate mechanism with variable rebate mechanism
    $ \ p _{1}^{*} $ $ \ \omega _{1}^{*} $ $ \ \pi _{M_1}^{*} $ $ \ \pi _{R_1}^{*} $ $ \ \pi _{1}^{*} $ $ \ p _{2}^{*} $ $ \ \omega _{2}^{*} $ $ \ \pi _{M_2}^{*} $ $ \ \pi _{R_2}^{*} $ $ \ \pi _{2}^{*} $
    0 1.123 0.818 0.155 0.065 0.220 1.124 0.820 0.186 0.065 0.251
    0.2 1.024 0.773 0.173 0.053 0.226 1.024 0.775 0.203 0.052 0.255
    0.4 0.952 0.741 0.185 0.044 0.229 0.953 0.743 0.215 0.043 0.259
    0.6 0.899 0.717 0.194 0.037 0.231 0.899 0.719 0.225 0.037 0.262
    0.8 0.857 0.699 0.202 0.032 0.234 0.858 0.700 0.232 0.032 0.263
    1 0.824 0.684 0.208 0.028 0.236 0.825 0.685 0.237 0.027 0.265
     | Show Table
    DownLoad: CSV

    Table 6.  If consumers are loss-seeking

    $ \ t $ $ \ p_{2}^{*} $ $ \ \omega_{2}^{*} $ $ \ \pi _{M_2}^{*} $ $ \ \pi _{R_2}^{*} $ $ \ \pi _{2}^{*} $
    0 1.099 0.912 -0.531 0.037 -0.494
    0.2 1.075 0.864 -0.308 0.047 -0.261
    0.4 1.051 0.817 -0.122 0.058 -0.064
    0.6 1.027 0.769 0.026 0.070 0.096
    0.8 1.004 0.721 0.136 0.084 0.220
    1 0.980 0.674 0.209 0.098 0.307
     | Show Table
    DownLoad: CSV

    Table 7.  If consumers are loss-averse

    $ \ t $ $ \ p_{2}^{*} $ $ \ \omega_{2}^{*} $ $ \ \pi _{M_2}^{*} $ $ \ \pi _{R_2}^{*} $ $ \ \pi _{2}^{*} $
    0 0.980 0.841 0.080 0.020 0.100
    0.2 0.968 0.817 0.127 0.024 0.151
    0.4 0.956 0.793 0.165 0.028 0.193
    0.6 0.944 0.769 0.194 0.032 0.226
    0.8 0.932 0.745 0.213 0.037 0.250
    1 0.920 0.721 0.223 0.042 0.265
     | Show Table
    DownLoad: CSV

    Table 8.  If consumers are loss-seeking

    $ \ t $ $ \ p_{1}^{*}-p_{2}^{*} $ $ \ \omega_{1}^{*}-\omega_{2}^{*} $ $ \ \pi _{M_1}^{*}-\pi _{M_2}^{*} $ $ \ \pi _{R_1}^{*}-\pi _{R_2}^{*} $ $ \ \pi _{1}^{*}-\pi _{2}^{*} $
    $ t\in [0, 1-\frac{l}{p}) $ 0.1 -0.104 -0.207 0.615 0.055 0.670
    0.3 -0.080 -0.160 0.410 0.044 0.454
    0.5 -0.060 -0.112 0.243 0.032 0.275
    $ t\in [1-\frac{l}{p}, 1] $ 0.98 0.001 0.002 -0.004 -0.001 -0.005
    0.99 0.002 0.005 -0.006 -0.002 -0.008
    1 0.004 0.007 -0.009 -0.002 -0.011
     | Show Table
    DownLoad: CSV

    Table 9.  If consumers are loss-averse

    $ \ t $ $ \ p_{1}^{*}-p_{2}^{*} $ $ \ \omega_{1}^{*}-\omega_{2}^{*} $ {\mbox{$ \ \pi _{M_1}^{*}-\pi _{M_2}^{*} $ $ \ \pi _{R_1}^{*}-\pi _{R_2}^{*} $ $ \ \pi _{1}^{*}-\pi _{2}^{*} $
    $ t\in [0, 1-\frac{l}{p}) $ 0.1 -0.050 -0.100 0.085 0.018 0.103
    0.3 -0.038 -0.076 0.043 0.014 0.057
    0.5 -0.026 -0.052 0.009 0.010 0.019
    $ t\in [1-\frac{l}{p}, 1] $ 0.98 0.002 0.005 -0.032 -0.001 -0.033
    0.99 0.003 0.006 -0.032 -0.001 -0.033
    1 0.004 0.007 -0.033 -0.002 -0.035
     | Show Table
    DownLoad: CSV
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