問12
頻出度★★★
ある資材を三つの工場 A，B，C に配分して，製品を生産している。資材の配分個数に応じた工場の利益は表のとおりである。合計4個の資材を工場に配分することによって得られる最大の利益は，何百万円か。 単位 百万円
| 資材の配分個数 | 0 個 | 1 個 | 2 個 | 3 個 | 4 個 |
|---|---|---|---|---|---|
| A 工場の利益 | 0 | 2 | 5 | 7 | 8 |
| B 工場の利益 | 0 | 1 | 3 | 5 | 7 |
| C 工場の利益 | 0 | 3 | 4 | 6 | 8 |
平成25年度 秋期 ITパスポート試験 問12
ストラテジ系企業活動 › 業務分析・データ利活用
This question's explanation and your answers so far.
解説
問 12
Incorrectア
8 is the value obtained when the 4 units are allocated all together to a single factory (4 units to A gives 8, 4 units to C gives 8), and similar cases. Looking closely at the table, both A and C gain little at the 4th unit (A goes 7→8, only +1; C goes 6→8, +2), so concentrating everything in one place is not a good strategy. Since combinations split among several factories can produce a total larger than 8, this is not the maximum profit.
Incorrectイ
9 is the total when 2 units go to A and 2 units to C (5+4 = 9), or 2 units to A, 1 unit to B, and 1 unit to C (5+1+3 = 9). It is certainly larger than 8, but it is not the maximum. The 3rd unit for A rises from 5 to 7, a gain of +2, so rather than stopping A at 2 units, giving A 3 units and sending the remaining 1 unit to C, that is (3,0,1) = 10, is higher. You need to compare all the patterns instead of stopping partway.
- ウ
正解 This question is an optimization problem of resource allocation, asking how to distribute a limited 4 units of material so that the total profit is maximized. The profit of each factory is not proportional to the number of units allocated; it is given individually in the table, so writing out every allocation pattern and comparing the totals is the sure way. The combinations (A,B,C) that distribute a total of 4 units among A, B, and C, and their total profits, are: (4,0,0)=8, (0,4,0)=7, (0,0,4)=8, (3,1,0)=7+1=8, (3,0,1)=7+3=10, (1,3,0)=2+5=7, (0,3,1)=5+3=8, (1,0,3)=2+6=8, (0,1,3)=1+6=7, (2,2,0)=5+3=8, (2,0,2)=5+4=9, (0,2,2)=3+4=7, (2,1,1)=5+1+3=9, (1,2,1)=2+3+3=8, (1,1,2)=2+1+4=7. The largest among these is the case of allocating 3 units to Factory A and 1 unit to Factory C, giving 7+3 = 10 million yen. The key point is that for Factory A the gain per additional unit is large up to the 3rd unit (0→2→5→7) but increases by only +1 at the 4th unit, while Factory C gains a large 3 from just its 1st unit; therefore the combination of 3 units to A and 1 unit to C builds up profit most efficiently. Hence the correct answer is ウ, 10.
Incorrectエ
11 is a value that cannot be produced by any combination in the table. Under the constraint of a total of 4 units, the total profits that can be formed are exactly the 15 listed above, and the maximum among them is 10. To obtain a value greater than 10 you would have to assume profits that are not in the table, so there is no basis for it.
ここがポイント
- Optimization of resource allocation: a problem of distributing a limited resource (here, 4 units of material) among several targets so as to maximize the overall effect. By listing every allocation pattern and comparing the totals, the maximum can be found for certain (this is the basis of the dynamic programming way of thinking).
- Profit is not necessarily proportional to the number of units allocated, so pay attention to the increment gained when adding one more unit (the marginal gain). Allocating first to where the gain is large, and avoiding allocations such as the 4th unit where the gain becomes small, is the knack for approaching the optimal solution.
用語