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Analysis of the Newsboy Problem subject to price dependent demand and multiple discounts

  • * Corresponding author: Shouyu Ma

    * Corresponding author: Shouyu Ma 
The first author is supported by the China Scholarship Council.
Abstract Full Text(HTML) Figure(9) / Table(5) Related Papers Cited by
  • Existing papers on the Newsboy Problem that deal with price dependent demand and multiple discounts often analyze those two problems separately. This paper considers a setting where price dependence and multiple discounts are observed simultaneously, as is the case of the apparel industry. Henceforth, we analyze the optimal order quantity, initial selling price and discount scheme in the News-Vendor Problem context. The term of discount scheme is often used to specify the number of discounts as well as the discount percentages. We present a solution procedure of the problem with general demand distributions and two types of price-dependent demand: additive case and multiplicative case. We provide interesting insights based on a numerical study. An approximation method is proposed which confirms our numerical results.

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

    Citation:

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  • Figure 1.  sequence of events for a selling season

    Figure 2.  Expected profit $E(\pi(Q^{*}))$, as a function of the discount number, for normally distributed demand

    Figure 3.  Expected profit $E(\pi(Q^{*}))$, as a function of the intial price

    Figure 4.  discount schemes

    Figure 5.  The value of ($E(\pi(Q^{*}))-E_\sigma$), as a function of discount number, with normal distribution

    Figure 6.  The value of ($E(\pi(Q^{*}))-E_\sigma$), as a function of discount number, with uniform distribution

    Figure 7.  Expected profit as function of discount number n

    Figure 8.  Discount percentages at $v_0=6$ for different schemes

    Figure 9.  Expected profit as function of initial price

    Table 1.  Comparison with the work of Khouja(1995, 2000)

    parameterprice-demand relation demand distribution discount prices
    [6] fixed general known
    [8] additive uniform and normal linear
    our paper additive and multiplicative general all types
     | Show Table
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    Table 2.  The optimal order initial price, order quantity and expected profit for different combinations of n, b, $\sigma_0$ for normally distributed demand

    test n b $\sigma_0$ $v^*_{0}$ $Q^*$ $E(\pi(Q^*, v_0^*))$
    1 4 6 2 10.20 55.8 249.0
    2 4 6 4 10.18 55.9 246.9
    3 4 6 6 10.24 56.1 245.0
    4 4 6 8 10.23 56.9 243.4
    5 4 8 2 8.54 50.4 153.3
    6 4 8 4 8.58 49.8 151.6
    7 4 8 6 8.59 49.6 150.2
    8 4 8 8 8.57 50.0 148.6
    9 4 10 2 6.60 46.3 95.0
    10 4 10 4 6.64 44.5 94.3
    11 4 10 6 6.64 44.3 93.6
    12 4 10 8 6.61 44.6 92.2
    13 5 6 2 11.41 56.6 263.9
    14 5 6 4 11.51 56.4 262.0
    15 5 6 6 11.47 56.7 260.2
    16 5 6 8 11.54 57.4 258.2
    17 5 8 2 8.81 51.9 159.8
    18 5 8 4 8.71 50.9 158.6
    19 5 8 6 8.75 50.8 157.4
    20 5 8 8 8.81 51.2 155.8
    21 5 10 2 7.09 45.7 100.1
    22 5 10 4 7.06 45.0 99.8
    23 5 10 6 7.01 45.1 98.8
    24 5 10 8 7.09 45.3 97.6
    25 6 6 2 11.90 57.6 271.5
    26 6 6 4 11.90 57.2 270.0
    27 6 6 6 11.88 57.5 268.3
    28 6 6 8 12.0 58.2 266.3
    29 6 8 2 8.91 52.6 164.5
    30 6 8 4 8.91 51.5 163.7
    31 6 8 6 8.94 51.6 162.6
    32 6 8 8 8.91 52.1 161.0
    33 6 10 2 7.16 44.8 103.8
    34 6 10 4 7.18 45.7 103.3
    35 6 10 6 7.19 45.8 102.3
    36 6 10 8 7.18 46.1 100.0
     | Show Table
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    Table 3.  Optimal epected profit for different discount schemes

    scheme coe optimal expected profit
    linear 0 158.5
    1 -0.03 144.9
    2 -0.02 151.1
    3 -0.01 155.8
    4 0.01 159.1
    5 0.02 157.8
    6 0.03 153.4
     | Show Table
    DownLoad: CSV

    Table 4.  Expected profit function for uniform and normal distributions

    Distribution $U[\mu_0-\sigma_0, \mu_0+\sigma_0]$ $N(\mu_0, \sigma_0)$
    Condition for $\epsilon=0$ $\forall j, \sigma_0\leq \frac{\mu_{j}-\mu_{j-1}}{2}$ $\forall j, \sigma_0\leq \frac{\mu_{j}-\mu_{j-1}}{4}$
    $E(\pi(Q^*))$ $E_\sigma+E_v$ $E_\sigma+E_v$
    $E(\pi(Q^*))$ for linear case equation 4.11 equation 4.11
    $E_v$ equation 4.8 equation 4.8
    $E_\sigma$ equation 4.9 equation 4.10
     | Show Table
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    Table 5.  Expected profit function for uniform and normal distributions

    Distribution $U[\mu_0-\sigma_0, \mu_0+\sigma_0]$ $N(\mu_0, \sigma_0)$
    Condition that $\epsilon=0$ $\forall j, \sigma_0\leq\frac{\mu_{j}-\mu_{j-1}}{2}$ $\forall j, \sigma_0\leq\frac{\mu_{j}-\mu_{j-1}}{4}$
    $E(\pi(Q^*))$ $E_\sigma+E_v$ $E_\sigma+E_v$
    Exponential case equation 5.8 equation 5.8
    $E_v$ equation 5.5 equation 5.5
    $E_\sigma$ equation 5.6 equation 5.7
     | Show Table
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