设工厂生产A、B、C、D四种产品,每单位用甲、乙、丙、丁原料为(A)1,0.5 ,0.3,0; (B)0.7,2,0.5,0.8;(C)2,1,0.2,3;(D)0.5,1,2,2。售价分别是5,7,12,11,(1)如果各原料总计有1000,800,800,500单位,问各生产多少收入最大?(2)若原料价格分别是0.5,0.7,0.6,1,问各生产多少利润最大及原料使用情况?
数学模型:x记第i种产品产量,(1)为求 5x[1]+7x[2]+12x[3]+11x[4]满足x[1]+0.7x[2]+2x[3]+0.5x[4]<=1000,0.5x[1]+2x[2]+x[3]+x[4]<=800,0.3x[1]+0.5x[2]+0.2x[3]+2x[4]<=800,0.8x[2]+3x[3]+2x[4]<=500的最大值。
(2)只是目标函数不一样。 因为受到的约束与目标函数都是线性运算,故也称为线性最优化问题或线性规划问题。
>with(simplex): >maximize(5*x+7*y+12*z+11*w,{x+.7*y+2*z+.5*w<=1000,.5*x+2*y+z+w<=800,.3*x+.5*y+.2*z+2*w<=800,.8*y+3*z+2*w<=500},NONNEGATIVE); >maximize(5*x+7*y+12*z+11*w-.5*(x+.7*y+2*z+.5*w-1000)-.7*(.5*x+2*y+z+w-800)-.6*(.3*x+.5*y+.2*z+2*w-800)-(.8*y+3*z+2*w-500),{x+.7*y+2*z+.5*w<=1000,.5*x+2*y+z+w<=800,.3*x+.5*y+.2*z+2*w<=800,.8*y+3*z+2*w<=500},NONNEGATIVE); > assign(");#定义上述运算结果,即x,y,z,w值定义为求出的解
最优值及原料剩余: >R=5*x+7*y+12*z+11*w;l1=-(x+.7*y+2*z+.5*w-1000);l2=-(.5*x+2*y+z+w-800);l3=-(.3*x+.5*y+.2*z+2*w-800);l4=-(.8*y+3*z+2*w-500);
思考问题:如果有其他单位想购买该厂原料,应该如何对这些原料定价?
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