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.
Citation: |
Table 1. Comparison with the work of Khouja(1995, 2000)
Table 2.
The optimal order initial price, order quantity and expected profit for different combinations of n, b,
test | n | b | ||||
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 | ||
Condition for |
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equation 4.11 | equation 4.11 |
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equation 4.8 | equation 4.8 |
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equation 4.9 | equation 4.10 |
Table 5. Expected profit function for uniform and normal distributions
Distribution | ||
Condition that |
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Exponential case | equation 5.8 | equation 5.8 |
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equation 5.5 | equation 5.5 |
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equation 5.6 | equation 5.7 |
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