由于要数学建模中需要解决一个三维装箱的问题,我经过搜寻选定了张德富教授等在计算机学报上发表的《求解三维装箱问题的混合模拟退火算法》这篇论文作为解决问题的理论基础。
该论文的摘要如下: 提出了一个高效求解三维装箱问题(Three Dimensional Container Loading Problem 3D-CLP)的混合模拟 退火 算法 .三维装箱问题要求装载给定箱子集合的一个子 集到容器中 , 使得被装载 的箱子总体 积最大 .文中介绍 的 混合 模拟退火算法基于三个重要算法 :(1)复合块生成算法 , 与传统算法 不同的是文中提出的复合块 不只包含单 一 种类的箱子 , 而是可以在一定的限制条件下包含 任意种类的箱子 .(2)基础启发式算法 , 该算法 基于块装 载 , 可以 按 照指定装载序列生成放置方案.(3)模拟退火算法,以复合块生成和基础启发式算法为基础, 将装载序列作为可行 放置 方案的编码 , 在编码空间中采用模拟退火算法进行搜 索以寻找问题的近似最 优解 .文 中采用 1 50 0 个弱异构 和 强异构的装箱问题数据对算法进行测试 .实验结 果表明 , 混合模拟退火算法的填充率超过了目前已知的 优秀算法 .
DOI 号 :10.3724/ SP .J.1016 .2009.02147
本文算法实现其计算结果 该实现的核心函数MATLAB代码 function \[solution, newPlan, rate\] = container\_packeting(bcon, plan, blockTable, blockNeed) %UNTITLED2 根据大小容器的规格以及小容器的数量信息计算出打包的方案 % bigContainer 大容器的规格向量 \[L W H\] % smallContainers 小容器的规格矩阵\[L W H;\] % smallContainersNumber 小容器的数量向量 \[n\] solution = {}; newPlan = \[\]; searchDeep = 40960; clf; draw\_container(\[0,0,0\], bcon); % 剩余空间栈 lspace\_s = get\_stack(); % 空隙栈 iterval\_s = get\_stack(); itervalSpace\_s = get\_stack(); % 初始化剩余空间为大容器空间 lspace\_s = stack\_push(lspace\_s, get\_space(\[0,0,0\], bcon)); psCount = 0; while lspace\_s.count > 0 psCount = psCount + 1; % 从剩余空间栈中取得可用的剩余空间 lspace = stack\_top(lspace\_s); lspace\_s = stack\_pop(lspace\_s); lspace\_size = lspace.size; lspace\_base = lspace.base\_point; if\_input = 0; \[blockList, blockListNeed\] = get\_block\_list(blockTable, blockNeed, lspace\_size(1), lspace\_size(2), lspace\_size(3)); theBlockNeed = NaN; find\_idx = 1; if size(blockList, 1) == 0 find\_avaliable = 0; else find\_avaliable = 1; if plan(psCount) <= size(blockListNeed, 1) theBlockNeed = blockListNeed(plan(psCount),:); find\_idx = plan(psCount); else theBlockNeed = blockListNeed(size(blockListNeed, 1),:); find\_idx = size(blockListNeed, 1); end newPlan = \[newPlan, find\_idx\]; end %% locate container if find\_avaliable == 1 newLocatedBlock = {lspace\_base, blockList(find\_idx, :), theBlockNeed{1}, theBlockNeed{2}}; solution(size(solution, 1) + 1, :) = newLocatedBlock; spaceInfo = newLocatedBlock{4}; spaceLastInfo = spaceInfo(size(spaceInfo,1), :); locateInfo = newLocatedBlock{2}; hold on; draw\_container(lspace\_base, locateInfo(1:3)); itervalSpace\_s = stack\_push(itervalSpace\_s, (1-spaceLastInfo(1)) \* prod(locateInfo(1:3))); if\_input = 1; % 剩余空间计算 % H方向剩余空间 location = lspace\_base; location(2) = location(2) + locateInfo(2); iterval\_space = get\_space(location, \[locateInfo(1) , lspace\_size(2) - locateInfo(2), locateInfo(3)\]); lspace\_s = stack\_push(lspace\_s, iterval\_space); % L方向剩余空间 location = lspace\_base; location(1) = location(1) + locateInfo(1); left\_space = get\_space(location, \[lspace\_size(1) - locateInfo(1), lspace\_size(2), lspace\_size(3)\]); lspace\_s = stack\_push(lspace\_s, left\_space); % W方向剩余空间 location = lspace\_base; location(3) = location(3) + locateInfo(3); iterval\_space = get\_space(location, \[locateInfo(1) , lspace\_size(2), lspace\_size(3) - locateInfo(3)\]); lspace\_s = stack\_push(lspace\_s, iterval\_space); end if if\_input == 1 else % 压入空隙栈 iterval\_s = stack\_push(iterval\_s, lspace); % lspace\_s = stack\_push(lspace\_s, lspace); end end rate = space\_used\_rate(bcon, solution); end function rate = space\_used\_rate(container, solution) spaceHasTotal = 0.0; for i = 1:1:size(solution, 1) mergeMessage = solution{i, 4}; for k = 1:1:size(mergeMessage,1) spaceHasTotal = spaceHasTotal + prod(mergeMessage(k, 3:5)); end end rate = spaceHasTotal / prod(container); end 箱体产生函数 function \[blockTable, boxNeedTable\] = simple\_block\_generate\_indepent(container, box, num) simBlockCount = 0; simBlockTable = zeros(256, 8); for type = 1:1:size(box, 1) for nx = 1:1:num(type) for ny = 1:1:num(type)/nx for nz = 1:1:num(type)/ny/nx if box(type,3) \* nx < container(3) && box(type, 1) \* ny < container(1) && box(type, 2) \* nz < container(2) newSimBlock = \[box(type, :), nx, ny, nz, type, prod(box(type, :)) \* nx \* ny \* nz\]; simBlockTable(simBlockCount + 1, :) = newSimBlock; simBlockCount = simBlockCount + 1; end end end end end simBlockTable = simBlockTable(1:simBlockCount, :); simBlockTable = sortrows(simBlockTable, 8); simBlockTable = simBlockTable(:, 1:7); % 处理成启发式算法可以处理的格式 simBlockTable(:, 13) = ones(size(simBlockTable, 1), 1); simBlockTable(:, 11) = simBlockTable(:,4) .\* simBlockTable(:,3); simBlockTable(:, 12) = simBlockTable(:,5) .\* simBlockTable(:,1); % ly = ay simBlockTable(:, 8) = simBlockTable(:, 12); % lx = ax simBlockTable(:, 10) = simBlockTable(:, 11); % lz = lz0 \* nz simBlockTable(:, 9) = simBlockTable(:, 2) .\* simBlockTable(:, 6); boxNeedTable = cell(size(simBlockTable, 1), 2); for i = 1:1:size(simBlockTable, 1) boxNeedTable{i, 1} = zeros(1, size(box ,1)); boxNeedTable{i, 1}(simBlockTable(i, 7)) = simBlockTable(i, 4) \* simBlockTable(i, 5) \* simBlockTable(i, 6); boxNeedTable{i, 2} = \[simBlockTable(i, 7), 0, simBlockTable(i, 8:10)\]; end blockTable = simBlockTable(:, 8:13); blockTable(:,7) = blockTable(:, 1) .\* blockTable(:, 2) .\* blockTable(:, 3); \[blockTable, idx\] = sortrows(blockTable, \[ 7 \]); boxNeedTable = boxNeedTable(idx, :); end 箱体组合产生函数 function \[blockTable, boxNeedTable\] = complex\_block\_genrate(container, box, num, level) simBlockTable = simple\_block\_generate(container, box, num); simBlockTable(:, 13) = ones(size(simBlockTable, 1), 1); simBlockTable(:, 11) = simBlockTable(:,4) .\* simBlockTable(:,3); simBlockTable(:, 12) = simBlockTable(:,5) .\* simBlockTable(:,1); spaceUsedRateMin = 0.90; % ly = ay simBlockTable(:, 8) = simBlockTable(:, 12); % lx = ax simBlockTable(:, 10) = simBlockTable(:, 11); % lz = lz0 \* nz simBlockTable(:, 9) = simBlockTable(:, 2) .\* simBlockTable(:, 6); boxNeedTable = cell(size(simBlockTable, 1), 2); for i = 1:1:size(simBlockTable, 1) boxNeedTable{i, 1} = zeros(1, size(box ,1)); boxNeedTable{i, 1}(simBlockTable(i, 7)) = simBlockTable(i, 4) \* simBlockTable(i, 5) \* simBlockTable(i, 6); boxNeedTable{i, 2} = \[simBlockTable(i, 7), 0, simBlockTable(i, 8:10)\]; end blockTable = simBlockTable(:, 8:13); for level = 1:1:level newBlockTable = zeros(128,6); newBlockCount = 1; boxNeedCount = size(boxNeedTable, 1); for a = 1:1:size(blockTable,1) for b = a:1:size(blockTable, 1) if b == a continue; end if blockTable(a, 6) == blockTable(b, 6) % x 方向合并 if blockTable(a, 1) == blockTable(a, 5) && blockTable(a, 3) == blockTable(a, 4) && blockTable(a, 2) == blockTable(b, 2) tempBlockMerged = \[max(blockTable(a,1), blockTable(b,1)), max(blockTable(a,2), blockTable(b, 2)), blockTable(a,3) + blockTable(b,3), blockTable(a,4) + blockTable(b,4), min(blockTable(a,5), blockTable(b,5)), max(blockTable(a, 6), blockTable(b, 6)) + 1\]; rate = space\_used\_rate(tempBlockMerged(1:3), blockTable(a,1:3), blockTable(b,1:3)); if rate > spaceUsedRateMin newBlockTable(newBlockCount, :) = tempBlockMerged; boxNeedCount = boxNeedCount + 1; newBlockCount = newBlockCount + 1; boxNeedTable{boxNeedCount, 1} = boxNeedTable{a,1} + boxNeedTable{b,1}; boxNeedTable{boxNeedCount, 2} = \[boxNeedTable{a, 2}; rate, 1, blockTable(b, 1:3)\]; end end % y 方向合并 if blockTable(a, 5) == blockTable(a, 1) && blockTable(b, 5) == blockTable(b, 1) && blockTable(a, 2) == blockTable(b, 2) tempBlockMerged = \[blockTable(a,1) + blockTable(b,1), max(blockTable(a, 2), blockTable(b, 2)), max(blockTable(a, 3), blockTable(b, 3)), min(blockTable(a, 4), blockTable(b, 4)), blockTable(a, 5) + blockTable(b, 5), max(blockTable(a, 6), blockTable(b, 6)) + 1\]; rate = space\_used\_rate(tempBlockMerged(1:3), blockTable(a,1:3), blockTable(b,1:3)); if rate > spaceUsedRateMin newBlockTable(newBlockCount, :) = tempBlockMerged; newBlockCount = newBlockCount + 1; boxNeedCount = boxNeedCount + 1; boxNeedTable{boxNeedCount, 1} = boxNeedTable{a,1} + boxNeedTable{b,1}; boxNeedTable{boxNeedCount, 2} = \[boxNeedTable{a, 2}; rate, 2, blockTable(b, 1:3)\]; end end % z 方向合并 if blockTable(a,4) >= blockTable(b,3) && blockTable(a, 5) >= blockTable(b, 1) tempBlockMerged = \[max(blockTable(a,1), blockTable(b, 1)), blockTable(a, 2) + blockTable(b, 2), max(blockTable(a, 3), blockTable(b, 3)), blockTable(b, 4), blockTable(b, 5), max(blockTable(a, 6), blockTable(b, 6)) + 1\]; rate = space\_used\_rate(tempBlockMerged(1:3), blockTable(a,1:3), blockTable(b,1:3)); if rate > spaceUsedRateMin newBlockTable(newBlockCount, :) = tempBlockMerged; newBlockCount = newBlockCount + 1; boxNeedCount = boxNeedCount + 1; boxNeedTable{boxNeedCount, 1} = boxNeedTable{a,1} + boxNeedTable{b,1}; boxNeedTable{boxNeedCount, 2} = \[boxNeedTable{a, 2}; rate, 3, blockTable(b, 1:3)\]; end end end end end blockTable = cat(1, blockTable, newBlockTable(1:newBlockCount-1, :)); % 消除等价复杂块 blockTableTemp = blockTable(:, 1:5); \[~, ia\] = unique(blockTableTemp, 'stable', 'rows'); blockTable = blockTable(ia, :); boxNeedTable = boxNeedTable(ia, :); end blockTable(:,7) = blockTable(:, 1) .\* blockTable(:, 2) .\* blockTable(:, 3); \[blockTable, idx\] = sortrows(blockTable, \[ 7 \]); boxNeedTable = boxNeedTable(idx, :); end function \[rate\] = space\_used\_rate(container, box1, box2) rate = (prod(box1) + prod(box2)) /prod(container); end Main函数 ts = 1; tf = 0.005; dt = 0.98; length = 50; maxSeq = 64; maxChoice = 1024; click = 1; tempRate = \[\]; tempRateX = \[\]; numList = \[64,64,64,64,64,64,64,128,128,128\]; container = StandardISOContainer; % \[blockTable, blockNeed\] = complex\_block\_genrate(StandardISOContainer, Packets, numList, 1); ps = ones(1, maxSeq); ps(1) = 1; \[solution, plan, rate\] = container\_packeting(container, ps, blockTable, blockNeed); best = plan; best\_rate = rate; best\_solution = solution; t = ts; while t > tf for i = 1:1:length k = randi(size(ps)); nps = ps; nps(k) = randi(maxChoice); \[solution, nplan, nrate\] = container\_packeting(container, nps, blockTable, blockNeed); % disp(\['TempPlan: ', mat2str(nplan)\]); % fprintf("TempRate: %f\\n",nrate); click = click + 1; if nrate > rate ps = nps; plan = nplan; rate = nrate; tempRateX = \[tempRateX, click\]; tempRate = \[tempRate, nrate\]; elseif rand(0, 1) > exp((nrate - rate) \* 10 / t) ps = nps; plan = nplan; rate = nrate; end if nrate > best\_rate best = nplan; best\_rate = nrate; best\_solution = solution; disp(\['BestPlan: ', mat2str(best)\]); fprintf("BestRate: %f\\n",best\_rate); end end t = (1 - t \* dt) \* t; end disp(\['BestPlan: ', mat2str(best)\]); fprintf("BestRate: %f\\n",best\_rate); figure(1) draw\_solution(container, best\_solution); figure(2) plot(tempRateX, tempRate); save Solution best\_solution best\_rate tempRateX tempRate