数学建模定义符号说明1.doc

2. 定义符号说明: 表示按照第i种模式切割I型号玻璃的块数（i=1,2,3.，20）； 表示按照第j种模式切割II型号玻璃的块数（j=1,2,3，.，20）； 3. 模型建立: 此问题为建筑公司下料问题，其与钢管易拉罐下料问题，自来水分派问题，奶制品的生产销售问题非常相似，都属于数学优化模型。所以我们可以用一些数学规划的相关知识来解决。按照假设把这其中规格的窗户分为两组，如表1： A1200x800 1200x650 1100x750 700x600 B1500x850 1200x750 800x600I,II型号玻璃分别以这两组的窗户的尺寸进行切割，例如I 型号玻璃按照A组切割，可以切3块1200x800,1块1200x650和1块700x600;II 玻璃按照A组切割，可以切3块1200x800,4块1200x650和2块700x600等等多种方案，这多种方案由表2,3表示： 表2模式尺寸1200x8001200x6501100x750700x600余量1460000001411202351111111179632530000099133.542424000094.5052421212193109.5624112323171108724002424358.561.5813550000391891344020234.512101323646421611131275755742121300808093193.5130250101093256.514023041241240.50150137606039541600282020120721700244040118.5136.5180012313173.531900106060112.50200000120120210 表3模式尺寸1500x8501200x750800x600 余量1340100142.5962241012132903230230126133.5413311279.5127.5513203473.5121.5612145067.581712037261.5758024204165699023146636310020098935711140208.512132202.513124196.514116190.515101088.5对A组建立的模型为：决策变量 它们为非负整数；决策目标 分别以切割后剩余的余料量和切割原料玻璃的总块数为最小目标，由表2,3可以得到目标函数 Min M1=141x1+117x+99x3+94.5x4+93x5+171x6+358.5x7+39x8+34.5x9+21x10+57x11+93x12+93x13+40.5x14+39x15+120x16+118.5x17+73.5x18+112.5x19+21x20+120y1+96y2+133.5y3+109.5y5+108.y6+61.5y7+18y8+12y9+6y10+42y11+193.5y12+256.5y13+54y15+72y16+136.5y17+3y18 ；（1）Min M2=+ ；（2）约束条件：4x1+3x2+2x3+2x4+2x5+2x6+2x7+x8+x9+x10+x11+x12+6y1+5y2+5y3+4y4+4y5+4y6+4y7+3y8+3y9+3y10+3y11+3y12+2y13+2y14+y15 =540 (3)x2+3x3+2x4+2x5+x6+5x8+4x9+2x10+x11+5x13+3x14+3x15+2x16+2x17+x18+x19+y2+4y4+y5+y6+5y8+4y9+3y10+2y11+7y15+8y16+4y17+2y18 =480 (4)x2+2x5+2x6+2x7+6x10+7x11+8x12+x13+4x14+6x15+2x16+4x17+3x18+6x19+12x20+y2+y5+3y6+4y7+2y9+4y10+5y11+12y14+y18 =480 (5)x2+2x5+2x6+2x7+6x10+7x11+8x12+x13+4x14+6x15+2x16+4x17+3x18+6x19+12x20+y2+y5+3y6+4y7+2y9+4y10+5y11+12y14+y18 =600 (6) 模型求解：将（1）（3）（4）（5）（6）构成的整数线性规划模型输入LINDO如下： min 141x1+117x2+99x3+94.5x4+93x5+171x6+358.5x7+39x8+34.5x9+21x10+57x11+93x12+93x13+40.5x14+39x15+120x16+118.5x17+73.5x18+112.5x19+21x20+120y1+96y2+133.5y3+109.5y5+108y6+61.5y7+18y8+12y9+6y10+42y11+193.5y12+256.5y13+54y15+72y16+136.5y17+3y18st4x1+3x2+2x3+2x4+2x5+2x6+2x7+x8+x9+x10+x11+x12+6y1+5y2+5y3+4y4+4y5+4y6+4y7+3y8+3y9+3y10+3y11+3y12+2y13+2y14+y15 =540x2+3x3+2x4+2x5+x6+5x8+4x9+2x10+x11+5x13+3x14+3x15+2x16+2x17+x18+x19+y2+4y4+y5+y6+5y8+4y9+3y10+2y11+7y15+8y16+4y17+2y18 =480x2+2x5+2x6+2x7+6x10+7x11+8x12+x13+4x14+6x15+2x16+4x17+3x18+6x19+12x20+y2+y5+3y6+4y7+2y9+4y10+5y11+12y14+y18 =480x2+2x5+2x6+2x7+6x10+7x11+8x12+x13+4x14+6x15+2x16+4x17+3x18+6x19+12x20+y2+y5+3y6+4y7+2y9+4y10+5y11+12y14+y18 =600end 求解可得到解如下：Global optimal solution found. Objective value: 0.000000 Total solver iterations: 6 Variable Value Reduced Cost X1 0.000000 141.0000 X2 0.000000 117.0000 X3 0.000000 99.00000 X4 0.000000 94.50000 X5 0.000000 93.00000 X6 0.000000 171.0000 X7 0.000000 358.5000 X8 0.000000 39.00000 X9 0.000000 34.50000 X10 0.000000 21.00000 X11 0.000000 57.00000 X12 0.000000 93.00000 X13 0.000000 93.00000 X14 0.000000 40.50000 X15 0.000000 39.00000 X16 0.000000 120.0000 X17 0.000000 118.5000 X18 0.000000 73.50000 X19 0.000000 112.5000 X20 0.000000 21.00000 Y1 0.000000 120.0000 Y2 0.000000 96.00000 Y3 0.000000 133.5000 Y5 0.000000 109.5000 Y6 0.000000 108.0000 Y7 0.000000 61.50000 Y8 0.000000 18.00000 Y9 0.000000 12.00000 Y10 0.000000 6.000000 Y11 0.000000 42.00000 Y12 0.000000 193.5000 Y13 0.000000 256.5000 Y15 0.000000 54.00000 Y16 0.000000 72.00000 Y17 0.000000 136.5000 Y18 0.000000 3.000000 Y4 120.0000 0.000000 Y14 50.00000 0.000000 Row Slack or Surplus Dual Price 1 0.000000 -1.000000 2 40.00000 0.000000 3 0.000000 0.000000 4 120.0000 0.000000 5 0.000000 0.000000 即按照模式4切割II号玻璃120块和模式14切割II号玻璃50块使得余量最小； 为0。将（1）（3）（4）（5）（6）构成的线性规划模型输入LINDO如下：min x1+x2+x3+x4+x+5x6+x7+x8+x9+x10+x11+x12+x13+x14+x15+x16+x17+x18+x19+x20+y1+y2+y3+y4+y5+y6+y7+y8+y9+y10+y11+y12+y13+y14+y15+y16+y17+y18+y19+y20st4x1+3x2+2x3+2x4+2x5+2x6+2x7+x8+x9+x10+x11+x12+6y1+5y2+5y3+4y4+4y5+4y6+4y7+3y8+3y9+3y10+3y11+3y12+2y13+2y14+y15 =540x2+3x3+2x4+2x5+x6+5x8+4x9+2x10+x11+5x13+3x14+3x15+2x16+2x17+x18+x19+y2+4y4+y5+y6+5y8+4y9+3y10+2y11+7y15+8y16+4y17+2y18 =480x2+2x5+2x6+2x7+6x10+7x11+8x12+x13+4x14+6x15+2x16+4x17+3x18+6x19+12x20+y2+y5+3y6+4y7+2y9+4y10+5y11+12y14+y18 =480x2+2x5+2x6+2x7+6x10+7x11+8x12+x13+4x14+6x15+2x16+4x17+3x18+6x19+12x20+y2+y5+3y6+4y7+2y9+4y10+5y11+12y14+y18 =600end 求解可得到下结果： Global optimal solution found. Objective value: 165.0000 Total solver iterations: 7 Variable Value Reduced Cost X1 0.000000 0.5000000 X2 0.000000 0.4375000 X3 0.000000 0.3750000 X4 0.000000 0.5000000 X5 0.000000 0.3750000 X6 0.000000 0.5000000 X7 0.000000 0.6250000 X8 0.000000 0.2500000 X9 0.000000 0.3750000 X10 0.000000 0.2500000 X11 0.000000 0.3125000 X12 0.000000 0.3750000 X13 0.000000 0.3125000 X14 0.000000 0.3750000 X15 0.000000 0.2500000 X16 0.000000 0.6250000 X17 0.000000 0.5000000 X18 0.000000 0.6875000 X19 0.000000 0.5000000 X20 0.000000 0.2500000 Y1 0.000000 0.2500000 Y2 0.000000 0.1875000 Y3 0.000000 0.3750000 Y4 75.00000 0.000000 Y5 0.000000 0.3125000 Y6 0.000000 0.1875000 Y7 0.000000 0.2500000 Y8 0.000000 0.000000 Y9 0.000000 0.000000 Y10 60.00000 0.000000 Y11 0.000000 0.6250000E-01 Y12 0.000000 0.6250000 Y13 0.000000 0.7500000 Y14 30.00000 0.000000 Y15 0.000000 0.000000 Y16 0.000000 0.000000 Y17 0.000000 0.5000000 Y18 0.000000 0.6875000 Y19 0.000000 1.000000 Y20 0.000000 1.000000 Row Slack or Surplus Dual Price 1 165.0000 -1.000000 2 0.000000 -0.1250000 3 0.000000 -0.1250000 4 120.0000 0.000000 5 0.000000 -0.6250000E-01即按照模式4购买II号玻璃75块，按照模式10购买II号玻璃60块和按照模式14购买II号玻璃35块使得总的购买量最小；为165块。对B组建立的模型为：决策变量 它们为非负整数；决策目标 分别以切割后剩余的余料量和切割原料玻璃的总块数为最小目标，由表2,3可以得到目标函数 Min N1=142.5x1+132x2+126x3+79.5x4+73.5x5+67.5X6+61.5x7+165x8+63x9+93x10+96y1+90y2+133.5y3+127.5y4+121.5y5+81y6+75y7+69y8+63y9+57y10+208.5y11+202.5y12+196.5y13+190.5y14+88.5y15 (1)Min N2=+ (2)约束条件为：3x1+2x2+2x3+x4+x5+x6+x7+4y1+4y2+3y3+3y4+3y5+2y6+2y7+2y8+2y9+2y10+y11+y12+y13+y14+y15 =600（3）x2+3x4+2x5+x6+4x8+3x9+y1+2y3+y4+4y6+3y7+2y8+y9+4y11+3y12+2y13+y14=960 （4）x2+3x3+x4+3x5+5x6+7x7+4x9+9x10+2y2+2y4+4y5+2y7+4y8 +6y9+8y10+2y12+4y13+6y14+10y15=1320（5）模型求解：将（1）（3）（4）（5）构成的整数线性规划模型输入LINDO如下： Min 142.5x1+132x2+126x3+79.5x4+73.5x5+67.5X6+61.5x7+165x8+63x9+93x10+96y1+90y2+133.5y3+127.5y4+121.5y5+81y6+75y7+69y8+63y9+57y10+208.5y11+202.5y12+196.5y13+190.5y14+88.5y15st3x1+2x2+2x3+x4+x5+x6+x7+4y1+4y2+3y3+3y4+3y5+2y6+2y7+2y8+2y9+2y10+y11+y12+y13+y14+y15 =600x2+3x4+2x5+x6+4x8+3x9+y1+2y3+y4+4y6+3y7+2y8+y9+4y11+3y12+2y13+y14=960x2+3x3+x4+3x5+5x6+7x7+4x9+9x10+2y2+2y4+4y5+2y7+4y8 +6y9+8y10+2y12+4y13+6y14+10y15=1320end可以求得解入下： Global optimal solution found. Objective value: 26640.00 Extended solver steps: 0 Total solver iterations: 6 Variable Value Reduced Cost X1 0.000000 57.00000 X2 0.000000 69.00000 X3 0.000000 69.00000 X4 0.000000 33.00000 X5 0.000000 33.00000 X6 0.000000 33.00000 X7 0.000000 33.00000 X8 0.000000 141.0000 X9 84.00000 45.00000 X10 0.000000 93.00000 Y1 0.000000 -24.00000 Y2 0.000000 -24.00000 Y3 0.000000 36.00000 Y4 0.000000 36.00000 Y5 0.000000 36.00000 Y6 0.000000 0.000000 Y7 204.0000 0.000000 Y8 0.000000 0.000000 Y9 96.00000 0.000000 Y10 0.000000 0.000000 Y11 0.000000 156.0000 Y12 0.000000 156.0000 Y13 0.000000 156.0000 Y14 0.000000 156.0000 Y15 0.000000 60.00000 Row Slack or Surplus Dual Price 1 26640.00 -1.000000 2 0.000000 -28.50000 3 0.000000 -6.000000 4 0.000000 0.000000即按照模式9分别切割I号玻璃84块和II号玻璃96块，按照模式7切割II号玻璃204块使得总的剩余量最小，为26640.00。将（2）（3）（4）（5）构成的整数线性规划模型输入LINDO如下：Min x1+x2+x3+x4+x5+x6+x7+x8+x9+x10+y1+y2+y3+y4+y5+y6+y7+y8+y9+y10+y11+y12+y13+y14+y15st4x1+3x2+2x3+2x4+2x5+2x6+2x7+x8+x9+x10+x11+x12+6y1+5y2+5y3+4y4+4y5+4y6+4y7+3y8+3y9+3y10+3y11+3y12+2y13+2y14+y15 =540x2+3x3+2x4+2x5+x6+5x8+4x9+2x10+x11+5x13+3x14+3x15+2x16+2x17+x18+x19+y2+4y4+y5+y6+5y8+4y9+3y10+2y11+7y15+8y16+4y17+2y18 =480x2+2x5+2x6+2x7+6x10+7x11+8x12+x13+4x14+6x15+2x16+4x17+3x18+6x19+12x20+y2+y5+3y6+4y7+2y9+4y10+5y11+12y14+y18 =480x2+2x5+2x6+2x7+6x10+7x11+8x12+x13+4x14+6x15+2x16+4x17+3x18+6x19+12x20+y2+y5+3y6+4y7+2y9+4y10+5y11+12y14+y18 =600end 求得结果为：Global optimal solution found. Objective value: 371.1000 Extended solver steps: 0 Total solver iterations: 1