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Pricing decisions for complementary products in a fuzzy dual-channel supply chain

The authors are supported by the National Natural Science Foundation of China 71571102 and the Key Scientific Research Projects of Colleges and Universities of Henan Province (No.18A630001)
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  • The paper considers the pricing problem of complementary products in a fuzzy dual-channel supply chain environment where there are two manufacturers and one retailer. Four different decision models are established to study this problem: the centralized decision model, MS-Bertrand model, RS-Bertrand model and Nash game model, where the consumer demand and manufacturing cost for each product are characterized as fuzzy variables. A closed form solution has been obtained for each model by using game theory and fuzzy theory. Numerical examples are presented to compare the maximal expected profits and optimal pricing decisions, and to provide additional managerial insights. The finding shows that the decision makers are more likely to choose industries with higher self-price elastic coefficient and lower complementarity in the retail channel to cooperate. We can obtain that consumers can benefit from the cooperation of the two manufacturers because of lower prices. We can also find that it might not be bad for retailer because it can expand demand and obtain more maximal expected profits.

    Mathematics Subject Classification: Primary: 91A25, 91B24; Secondary: 91A40.

    Citation:

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  • Figure 1.  The graphic method

    Figure 2.  The five kinds of CPs which selected randomly from the space$^*$

    Table 1.  Summary of the major literature review. (M: manufacturer; R: retailer)

    Paper Supply chain structure Product status Dual channel Fuzzy or not Decisions
    Chiang et al.[5] one M single Yes No pricing policy
    one R ordering policy
    Berger and Weinberg[1] one firm single Yes No pricing policy
    advertising
    Dan et al.[7] one M single Yes No pricing policy
    one R service decision
    Zhao et al.[32] one M substitutable No Yes pricing policy
    two Rs
    Zhao and Wang[33] one M single No Yes pricing policy
    two Rs service decision
    Leila and Mehdi[10] two Ms substitutable Yes No pricing policy
    one R
    Sang[19] two Ms substitutable No Yes pricing policy
    one R
    Wang et al.[23] two Ms complementary Yes No pricing policy
    one R service decision
     | Show Table
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    Table 2.  Relation between linguistic expression and triangular fuzzy variable.

    Linguistic expression Triangular fuzzy variable
    $\Phi_0$ Large (about 500) (350,500,700)
    Small (about 200) (150,200,300)
    $\Phi_1$ Large (about 800) (600,800,900)
    Small (about 300) (200,300,350)
    $\Phi_2$ Large (about 1200) (800,1200,1500)
    Small (about 600) (500,600,800)
    $ k_0 $ Very sensitive(about 5) (4, 5, 6)
    Sensitive (about 2) (1, 2, 2.5)
    $k_1$ Very sensitive(about 8) (5, 8, 10)
    Sensitive (about 4) (3, 4, 5)
    $k_2$ Very sensitive(about 30) (25, 30, 35)
    Sensitive (about 20) (10, 20, 25)
    $\beta$ Very sensitive(about 0.5) (0.3, 0.5, 0.8)
    Sensitive(about 0.2) (0.1, 0.2, 0.3)
    $\gamma_1$ Very sensitive(about 0.4) (0.3, 0.4, 0.6)
    Sensitive (about 0.2) (0.1, 0.2, 0.3)
    $\gamma_2$ Very sensitive(about 0.3) (0.1, 0.3, 0.5)
    Sensitive (about 0.1) (0.05, 0.1, 0.15)
    $c_1$ High(about 30) (25, 30, 40)
    Low(about 15) (10, 15, 20)
    $c_2$ High (about 5) (4, 5, 6)
    Low (about 2) (1, 2, 4)
     | Show Table
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    Table 3.  The optimal decisions of wholesale prices, direct sale price and retail prices

    $scenario$ $w_1^{*}$ $w_2^{*}$ $p_0^{*}$ $p_1^{*}$ $p_2^{*}$ $p_1^{*}-w_1^{*}$ $p_2^{*}-w_2^{*}$
    $CD$ $-$ $-$ 63.0097 60.8519 20.8009 $-$ $-$
    $MS-Bertrand$ 61.17 21.36 63.3476 82.0076 29.2987 20.8376 7.9387
    $RS-Bertrand$ 40.02 13.20 63.3317 82.0065 29.3002 41.9865 16.0902
    $NG$ 47.14 15.97 63.4526 74.9997 26.6042 27.8597 10.6342
     | Show Table
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    Table 4.  Maximal expected profits and demands in different models

    $scenario$ $E[D_0^{*}]$ $E[D_1^{*}]$ $E[D_2^{*}]$ $E[\Pi_{m_1}^{*}]$ $E[\Pi_{m_2}^{*}]$ $E[\Pi_r^{*}]$ $E[\Pi_c^{*}]$
    $CD$ 220.56 330.2375 505.94 $-$ $-$ $-$ 37076
    $MS-Bertrand$ 226.36 163.9087 244.52 19578 4173 5358 29109
    $RS-Bertrand$ 226.44 163.9089 244.48 16111 2180 10818 29109
    $NG$ 223.31 219.0836 327.41 18895 3761 9585 32240
     | Show Table
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    Table 5.  The corresponding triangular fuzzy variables of the CPs

    $\Phi_2$ $k_2$ $\gamma_1$
    $CP_1$ (1800,2000,2200) (3, 4, 5) (0.4, 0.5, 0.6)
    $CP_2$ (3201,3302,3403) (6, 8, 10) (0.1, 0.3, 0.5)
    $CP_3$ (800,1200,1500) (15, 20, 25) (0.1, 0.2, 0.3)
    $CP_4$ (500,600,700) (25, 30, 35) (0.3, 0.4, 0.6)
    $CP_5$ (1000,1156,1312) (15, 20, 25) (0.4, 0.5, 0.6)
     | Show Table
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    Table 6.  Comparison of the optimal decisions with and without CPs

    $CP_1$ $CP_2$ $CP_3$ $CP_4$ $CP_5$
    $p_0^{N*}-p_0^{Nocp}$ -36.3415 -28.0346 -19.6351 -20.7345 -19.284
    $p_1^{N*}-p_1^{Nocp}$ -22.3776 -18.0985 -2.4605 -2.5192 -0.827
    $p_2^{N*}$ 322.6005 271.3165 38.6035 37.2342 13.1502
    $w_1^{N*}-w_1^{Nocp}$ 12.2558 14.0035 18.0239 17.8967 18.3916
    $E[D_0^{N*}]-E[D_0^{Nocp}]$ 8.6757 49.2929 89.1796 83.7494 90.7422
    $E[D_1^{N*}]-E[D_1^{Nocp}]$ -26.3475 -23.7178 -12.0458 -11.9774 -10.7595
    $E[D_2^{NG*}]$ 640.6542 1070.9 342.9444 319.2612 129.3525
    $E[\Pi_{m_1}^{N*}]-E[\Pi_{m_1}^{Nocp}]$ -3141 0 5177 4690 5528
    $E[\Pi_{m_2}^{N*}]$ 102830 143660 6082 5501 611.5339
    $E[\Pi_r^{N*}]-E[\Pi_{r}^{Nocp}]$ 92082 133632 417 0 -4373
    $E[\Pi_{m_1+r}^{N*}]-E[\Pi_{c}^{Nocp}]$ 88941 133632 5594 4690 155
     | Show Table
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    Table 7.  The change of maximal expected demands and profits with the fuzzy degree of $\beta$

    $scenario$ $\beta$ $E[D_0^{*}]$ $E[D_1^{*}]$ $E[D_2^{*}]$ $E[\Pi_{m_1}^{*}]$ $E[\Pi_{m_2}^{*}]$ $E[\Pi_r^{*}]$ $E[\Pi_c^{*}]$
    $MS-Bertrand$ (0.2, 0.5, 0.8) 225.6026 163.7939 244.5593 19452 4175.0 5354.1 28981
    (0.3, 0.5, 0.7) 225.6886 163.8355 244.5621 19427 4175.1 5355.9 28958
    (0.4, 0.5, 0.6) 225.7746 163.8772 244.5649 19403 4175.2 5357.6 28936
    (0.45, 0.5, 0.55) 225.8176 163.8980 244.5663 19391 4175.3 5358.5 28925
    $RS-Bertrand$ (0.2, 0.5, 0.8) 225.6799 163.7941 244.5216 15990 2180.7 10810 28980
    (0.3, 0.5, 0.7) 225.7658 163.8357 244.5244 15964 2180.8 10813 28958
    (0.4, 0.5, 0.6) 225.8518 163.8774 244.5272 15938 2180.9 10817 28936
    (0.45, 0.5, 0.55) 225.8948 163.8982 244.5286 15925 2181.0 10818 28924
     | Show Table
    DownLoad: CSV

    Table 8.  The change of optimal prices with the fuzzy degree of $\beta$

    $scenario$ $\beta$ $w_1^{*}$ $w_2^{*}$ $p_0^{*}$ $p_1^{*}$ $p_2^{*}$ $p_1^{*}-w_1^{*}$ $p_2^{*}-w_2^{*}$
    $MS-Bertrand$ (0.2, 0.5, 0.8) 60.9727 21.3595 63.0687 81.7999 29.3032 20.8272 7.9437
    (0.3, 0.5, 0.7) 60.9608 21.3597 63.0508 81.7934 29.3034 20.8326 7.9437
    (0.4, 0.5, 0.6) 60.9489 21.3599 63.0329 81.7869 29.3037 20.8380 7.9438
    (0.45, 0.5, 0.55) 60.9430 21.3600 63.0240 81.7836 29.3038 20.8406 7.9438
    $RS-Bertrand$ (0.2, 0.5, 0.8) 39.8371 13.2063 63.0530 81.7988 29.3047 41.9617 16.0984
    (0.3, 0.5, 0.7) 39.8198 13.2064 63.0351 81.7923 29.3049 41.9725 16.0985
    (0.4, 0.5, 0.6) 39.8025 13.2065 63.0173 81.7858 29.3052 41.9833 16.0987
    (0.45, 0.5, 0.55) 39.7939 13.2066 63.0083 81.7825 29.3053 41.9886 16.0987
     | Show Table
    DownLoad: CSV

    Table 9.  The change of maximal expected demands and profits with the fuzzy degree of $k_2$

    $scenario$ $k_2$ $E[D_0^{*}]$ $E[D_1^{*}]$ $E[D_2^{*}]$ $E[\Pi_{m_1}^{*}]$ $E[\Pi_{m_2}^{*}]$ $E[\Pi_r^{*}]$ $E[\Pi_c^{*}]$
    $MS-Bertrand$ (10, 30, 50) 226.3554 163.9024 243.2656 19576 4320.2 5338.1 29234
    (15, 30, 45) 226.3583 163.9045 243.6821 19577 4271.3 5344.8 29192
    (20, 30, 40) 226.3613 163.9066 244.0987 19577 4222.4 5351.5 29151
    (25, 30, 35) 226.3642 163.9087 244.5152 19578 4173.6 5358.2 29109
    $RS-Bertrand$ (10, 30, 50) 226.4335 163.9026 243.2279 16110 2347.0 10777 29234
    (15, 30, 45) 226.4364 163.9047 243.6443 16110 2291.3 10791 29192
    (20, 30, 40) 226.4394 163.9068 244.0608 16111 2235.7 10804 29151
    (25, 30, 35) 226.4423 163.9089 244.4773 16111 2180.0 10818 29109
     | Show Table
    DownLoad: CSV

    Table 10.  The change of optimal prices with the fuzzy degree of $k_2$

    $scenario$ $k_2$ $w_1^{*}$ $w_2^{*}$ $p_0^{*}$ $p_1^{*}$ $p_2^{*}$ $p_1^{*}-w_1^{*}$ $p_2^{*}-w_2^{*}$
    $MS-Bertrand$ (10, 30, 50) 61.1638 21.4399 63.3457 82.0067 29.3404 20.8429 7.9005
    (15, 30, 45) 61.1644 21.4121 63.3463 82.0070 29.3265 20.8426 7.9144
    (20, 30, 40) 61.1650 21.3844 63.3469 82.0073 29.3126 20.8423 7.9282
    (25, 30, 35) 61.1656 21.3566 63.3476 82.0076 29.2987 20.8420 7.9421
    $RS-Bertrand$ (10, 30, 50) 40.0141 13.3298 63.3298 82.0055 29.3418 41.9914 16.0120
    (15, 30, 45) 40.0144 13.2881 63.3305 82.0059 29.3279 41.9915 16.0398
    (20, 30, 40) 40.0147 13.2465 63.3311 82.0062 29.3141 41.9915 16.0676
    (25, 30, 35) 40.0150 13.2048 63.3317 82.0065 29.3002 41.9915 16.0954
     | Show Table
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