题目
A restaurant sells two sizes of pizza, small and large. The restaurant owner knows that, each evening, she needs to make
- at least 85 pizzas in total
- at least twice as many large pizzas as small pizzas
In addition, at most 80% of the pizzas must be large.
Each small pizza costs £2 to make and each large pizza costs £3 to make.
The restaurant owner wants to minimise her costs.
Let represent the number of small pizzas made each evening and let represent the number of large pizzas made each evening.
Formulate the information above as a linear programming problem. State the objective and list the constraints as simplified inequalities with integer coefficients. You should not attempt to solve the problem.
(Total 5 marks)
题目中文翻译
一家餐厅出售两种尺寸的披萨，小号和大号。餐厅老板知道每天晚上她需要制作
- 总共至少 85 个披萨
- 大号披萨至少是小号披萨的两倍
此外，最多 80% 的披萨必须是大号。
每个小号披萨成本为 £2，每个大号披萨成本为 £3。
餐厅老板希望最小化成本。
设 表示每晚制作的小号披萨数量， 表示每晚制作的大号披萨数量。
将以上信息表述为线性规划问题。说明目标并列出简化为整数系数不等式的约束条件。不需要尝试解此问题。
解答
解法一
思路
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把每一句业务条件逐一翻译成关于 的不等式。80% 条件先写成“ 不超过总数 的 ”，再整理成整数系数形式。题目只要求建立模型，不需求最优解。
答题过程
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The total cost is
so the objective is to
At least pizzas are required, so
There must be at least twice as many large pizzas as small pizzas, so
At most of all pizzas may be large. Hence
Multiplying by and rearranging,
which gives
Therefore the linear programming model is