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+# Problem
+
+Exercises 15.3‐5<br>
+Suppose that in the rod-cutting problem, we also had limit l<sub>i</sub> on the number of pieces of length i that we are allowed to produce, for i = 1, 2, ..., n. Show that the **optimal-substructure** property described in Section 15.1 (unit05-演算法.pdf p28) no longer holds.
+
+# Solution
+
+## Example
+
+| length i | 1 | 2 | 3 | 4 | 5 |
+|:------:|:-:|:-:|:-:|:-:|:-:|
+| price p<sub>i</sub> | 10 | 15 | 24 | 28 | 35 |
+| limit l<sub>i</sub> | 3 | 1 | 1 | 1 | 1 |
+
+長度為 5 的 rod 可切為
+
+- [5] 35元
+- [1, 4] 34元
+- [2, 3] 39元
+- [1, 1, 3] 44元
+- [1, 2, 2] 違反limit
+- [1, 1, 1, 2] 45元
+- [1, 1, 1, 1, 1] 違反limit
+
+## Analysis
+
+i = 5 時，最佳解為 [1, 1, 1, 2] 45元。當我們再看 它的 sub-problem i = 2 的時候，解法有 [1, 1] 20元、[2] 15元，最佳解為 [1, 1]，**但是**我們卻不能將之套用回 i = 5的原問題，因為它違反了長度為 1 的 rod 數目限制 (l<sub>1</sub> = 3)。