# MATLAB 窗边框最优切割问题解决方案 针对窗边框生产中的最优切割问题，我将提供一个MATLAB实现方案，考虑原材料缺陷和锯口宽度的影响。 ## 问题分析 1. 需要为4种订单生产窗框，每种窗框需要2个宽度和2个高度组件 2. 原材料有3种规格，每种有已知缺陷区域 3. 切割时需要考虑0.005米的锯口宽度 4. 目标是最小化切割损失，最大化材料利用率 ## MATLAB实现代码 ```matlab function optimal_window_cutting() % 订单数据 orders = struct(... 'id', {1, 2, 3, 4}, ... 'name', {'学校教学楼', '酒店客房', '医院病房', '办公楼'}, ... 'quantity', {10, 20, 20, 15}, ... 'width', {1.6, 1.8, 1.7, 1.5}, ... 'height', {2.2, 2.4, 2.3, 2.0}, ... 'price', {480, 680, 550, 420}); % 原材料数据 materials = struct(... 'id', {1, 2, 3}, ... 'length', {5.5, 6.2, 7.8}, ... 'cost', {18, 22, 28}, ... 'defects', {{[1.0, 0.03; 2.5, 0.04], [0.5, 0.02; 1.8, 0.05], [3.0, 0.03]}}); kerf = 0.005; % 锯口宽度 tolerance = 0.01; % 尺寸允许误差 % 计算每种窗框需要的组件总数 total_pieces = struct(); for i = 1:length(orders) total_pieces(i).width = 2 * orders(i).quantity; total_pieces(i).height = 2 * orders(i).quantity; end % 生成所有可能的切割模式 cutting_patterns = generate_cutting_patterns(orders, materials, kerf, tolerance); % 使用整数线性规划求解最优切割方案 [solution, total_loss, utilization] = solve_ilp(cutting_patterns, orders, materials, total_pieces); % 输出结果 display_results(solution, cutting_patterns, orders, materials, total_loss, utilization); end function patterns = generate_cutting_patterns(orders, materials, kerf, tolerance) patterns = []; pattern_id = 1; % 为每种原材料生成切割模式 for mat_id = 1:length(materials) mat_length = materials(mat_id).length; defects = materials(mat_id).defects{:}; % 生成所有可能的有效切割段 valid_segments = generate_valid_segments(mat_length, defects, kerf); % 为每个有效段生成可能的窗框组件组合 for seg_idx = 1:size(valid_segments, 1) seg_start = valid_segments(seg_idx, 1); seg_end = valid_segments(seg_idx, 2); seg_length = seg_end - seg_start; % 尝试匹配所有订单的宽度和高度 for order_id = 1:length(orders) width = orders(order_id).width; height = orders(order_id).height; % 检查宽度组件 for w = [width - tolerance, width, width + tolerance] if w > 0 && (w + kerf) <= seg_length num_width = floor(seg_length / (w + kerf)); remaining = seg_length - num_width * (w + kerf); patterns = [patterns; struct(... 'id', pattern_id, ... 'material_id', mat_id, ... 'segment', [seg_start, seg_end], ... 'order_id', order_id, ... 'component_type', 'width', ... 'component_size', w, ... 'num_components', num_width, ... 'waste', remaining)]; pattern_id = pattern_id + 1; end end % 检查高度组件 for h = [height - tolerance, height, height + tolerance] if h > 0 && (h + kerf) <= seg_length num_height = floor(seg_length / (h + kerf)); remaining = seg_length - num_height * (h + kerf); patterns = [patterns; struct(... 'id', pattern_id, ... 'material_id', mat_id, ... 'segment', [seg_start, seg_end], ... 'order_id', order_id, ... 'component_type', 'height', ... 'component_size', h, ... 'num_components', num_height, ... 'waste', remaining)]; pattern_id = pattern_id + 1; end end end end end end function valid_segments = generate_valid_segments(mat_length, defects, kerf) % 初始化有效段为整个原材料 valid_segments = [0, mat_length]; % 根据缺陷分割有效段 for i = 1:size(defects, 1) defect_start = defects(i, 1); defect_end = defect_start + defects(i, 2); new_segments = []; for j = 1:size(valid_segments, 1) seg_start = valid_segments(j, 1); seg_end = valid_segments(j, 2); if defect_start >= seg_end || defect_end <= seg_start % 缺陷不与此段重叠 new_segments = [new_segments; seg_start, seg_end]; else % 缺陷与此段重叠，分割此段 if seg_start < defect_start - kerf new_segments = [new_segments; seg_start, defect_start - kerf]; end if defect_end + kerf < seg_end new_segments = [new_segments; defect_end + kerf, seg_end]; end end end valid_segments = new_segments; end end function [solution, total_loss, utilization] = solve_ilp(patterns, orders, materials, total_pieces) % 创建整数线性规划问题 num_patterns = length(patterns); num_orders = length(orders); % 目标函数：最小化总成本 + 浪费惩罚 f = zeros(num_patterns, 1); for i = 1:num_patterns mat_id = patterns(i).material_id; f(i) = materials(mat_id).cost + patterns(i).waste * 10; % 浪费惩罚系数 end % 约束矩阵：确保满足每个订单的宽度和高度需求 A_width = zeros(num_orders, num_patterns); A_height = zeros(num_orders, num_patterns); for i = 1:num_patterns order_id = patterns(i).order_id; if strcmp(patterns(i).component_type, 'width') A_width(order_id, i) = patterns(i).num_components; else A_height(order_id, i) = patterns(i).num_components; end end % 合并约束矩阵 A = [A_width; A_height]; b = [ [total_pieces.width]; [total_pieces.height] ]; % 变量下界 lb = zeros(num_patterns, 1); % 使用intlinprog求解 intcon = 1:num_patterns; options = optimoptions('intlinprog', 'Display', 'off'); [x, ~, exitflag] = intlinprog(f, intcon, -A, -b, [], [], lb, [], options); if exitflag <= 0 error('无法找到可行解'); end % 提取解 solution = x; % 计算总损失和利用率 total_material_cost = 0; total_waste = 0; total_material_length = 0; for i = 1:num_patterns if x(i) > 0 mat_id = patterns(i).material_id; total_material_cost = total_material_cost + x(i) * materials(mat_id).cost; total_waste = total_waste + x(i) * patterns(i).waste; total_material_length = total_material_length + x(i) * materials(mat_id).length; end end total_loss = total_waste / total_material_length * 100; utilization = (total_material_length - total_waste) / total_material_length * 100; end function display_results(solution, patterns, orders, materials, total_loss, utilization) fprintf('最优切割方案:\n'); fprintf('-------------------------------------------------\n'); fprintf('原材料使用情况:\n'); % 统计每种原材料的使用数量 material_usage = zeros(length(materials), 1); for i = 1:length(solution) if solution(i) > 0 mat_id = patterns(i).material_id; material_usage(mat_id) = material_usage(mat_id) + solution(i); end end for i = 1:length(materials) fprintf('原材料%d (长度%.1f米): %d根\n', ... materials(i).id, materials(i).length, material_usage(i)); end fprintf('\n订单完成情况:\n'); fprintf('-------------------------------------------------\n'); % 统计每个订单的组件完成情况 order_width = zeros(length(orders), 1); order_height = zeros(length(orders), 1); for i = 1:length(solution) if solution(i) > 0 order_id = patterns(i).order_id; if strcmp(patterns(i).component_type, 'width') order_width(order_id) = order_width(order_id) + solution(i) * patterns(i).num_components; else order_height(order_id) = order_height(order_id) + solution(i) * patterns(i).num_components; end end end for i = 1:length(orders) required_width = 2 * orders(i).quantity; required_height = 2 * orders(i).quantity; fprintf('订单%d (%s):\n', orders(i).id, orders(i).name); fprintf(' 需要宽度组件: %d个, 已生产: %d个\n', required_width, order_width(i)); fprintf(' 需要高度组件: %d个, 已生产: %d个\n', required_height, order_height(i)); if order_width(i) >= required_width && order_height(i) >= required_height fprintf(' 状态: 已完成\n'); else fprintf(' 状态: 未完成 (短缺: 宽度%d个, 高度%d个)\n', ... max(0, required_width - order_width(i)), max(0, required_height - order_height(i))); end end fprintf('\n性能指标:\n'); fprintf('-------------------------------------------------\n'); fprintf('总切割损失率: %.2f%%\n', total_loss); fprintf('材料利用率: %.2f%%\n', utilization); end ``` ## 数学模型 决策变量 设x_p为使用切割模式p的次数（非负整数） 目标函数 最小化总成本： ``` min Σ (c_p * x_p) + Σ (w_p * x_p * α) ``` 其中c_p是模式p使用的原材料成本，w_p是模式p产生的废料，α是废料惩罚系数 约束条件 1. 每个订单的宽度组件需求： ``` Σ (n_{p,o,w} * x_p) ≥ 2 * Q_o ∀o ``` 其中n_{p,o,w}是模式p为订单o生产的宽度组件数量，Q_o是订单o的数量 2. 每个订单的高度组件需求： ``` Σ (n_{p,o,h} * x_p) ≥ 2 * Q_o ∀o ``` 3. 切割模式必须避开缺陷区域： ``` 对于每个切割模式p，其切割区间不与任何缺陷区域重叠 ``` ## 结果分析 1. 该方案考虑了原材料缺陷和锯口宽度的影响 2. 使用整数线性规划确保得到最优解 3. 输出包括： - 每种原材料的使用数量 - 每个订单的完成情况 - 总切割损失率和材料利用率 ## 改进方向 1. 可以添加启发式算法预处理，减少切割模式数量 2. 考虑多目标优化，平衡成本和材料利用率 3. 添加可视化功能展示切割方案 要运行此代码，只需在MATLAB中调用`optimal_window_cutting()`函数即可得到最优切割方案。 [2025-08-09 09:46:40 | AI问答 | 2886点数解答]