The board packing problem
提出板填充问题（BoPP），目标是在矩形板上放置矩形以最大化覆盖收益减去成本，证明其NP难并给出混合整数规划模型和进化算法。
We introduce the board packing problem (BoPP). In this problem we are given a rectangular board with a given number of rows and columns. Each position of the board has an integer value representing a gain, or revenue, that is obtained if the position is covered. A set of rectangles is also given, each with a given size and cost. The objective is to purchase some rectangles to place on the board so as to maximize the profit, which is the sum of the gain values of the covered cells minus the total cost of purchased rectangles. This problem subsumes several natural optimization problems that arise in practice. A mixed-integer programming model for the BoPP problem is provided, along with a proof that BoPP is NP-hard by reduction from the satisfiability problem. An evolutionary algorithm is also developed that can solve large instances of BoPP. We introduce benchmark instances and make extensive computer examinations.