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

• * Corresponding author: Shouyu Ma
The first author is supported by the China Scholarship Council.
• 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: • • 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)

 parameter price-demand relation demand distribution discount prices  fixed general known  additive uniform and normal linear our paper additive and multiplicative general all types

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

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

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

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
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Tables(5)

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