生产计划问题与投资问题
清华大学肖勇波梁湧老师翻译的Rardin教授的《运筹学》[1]已于今年年中出版，感谢机械工业出版社张有利老师的推荐和赠书。
问题
生产计划问题和投资问题见原书第二章的两个练习，习题2-1和习题2-2。
生产计划问题又叫混合生产问题（Mixed Production Problem),是经典的线型规划问题。
投资问题又叫投资组合问题，是金融背景上的一种背包问题。
+Leapms生成的两个问题的模型摘录
对任何问题，+Leapms都建议直接用+Leapms建模语言直接写出模型，并进行模型调试。
我们的经验是，模型调试能够发现模型的不足、促进建模的完美性，甚至可以促进对问题的更加深入的了解。当模型调试完毕，对模型的正确性有充分的信心后，+Leapms系统可生成模型摘录，包括数学概念模型（即使用标引符号表示的数学模型）和+Leapms源码供交流使用。
以下是两个问题的+Leapms模型摘录（pdf屏幕截图）, 其中 模型"02-1.leap" 是生产计划问题, 模型"02-02"是投资问题：
求解过程
+Leapms>load Current directory is "ROOT". ......... 02-01.leap 02-02.leap 02-03.leap 02-39.leap ......... please input the filename:02-01 ================================================================ 1: //2-1 The Notip Table Company sells two models of its patented 2: //five-leg tables. The basic version uses a wood top, requires 3: //0.6 hour to assemble, and sells for a profit of $200. The 4: //deluxe model takes 1.5 hours to assemble because of its glass 5: //top, and sells for a profit of $350. Over the next week the 6: //company has 300 legs, 50 wood tops, 35 glass tops, 7: //and 63 hours of assembly available. Notip wishes 8: //to determine a maximum profit production plan 9: //assuming that everything produced can be sold. 10: 11: max sum{j=1,..,n}c[j]x[j] 12: subject to 13: sum{j=1,..,n}a[i][j]x[j]<=b[i] | i=1,..,m 14: where 15: m,n are integers 16: b[i] is a number|i=1,..,m 17: c[j] is a number|j=1,..,n 18: a[i][j] is a number|i=1,..,m;j=1,..,n 19: x[j] is a variable of nonnegative number --> 20: |j=1,..,n 21: data 22: m=4 // number of kinds of resources 23: n=2 // number of product 24: c={200 350} // profits 25: b={63 300 50 35} //availabilities 26: a={ 27: 0.6 1.5 //hours 28: 5 5 //legs 29: 1 0 //wood tops 30: 0 1 //glass tops 31: } ================================================================ >>end of the file. Parsing model: 1D 2R 3V 4O 5C 6S 7End. .................................. number of variables=2 number of constraints=4 .................................. +Leapms>solve The LP is solved to optimal. 找到线性规划最优解.非零变量值和最优目标值如下： ......... x1*=30 x2*=30 ......... Objective*=16500 ......... +Leapms>load Current directory is "ROOT". ......... 02-01.leap 02-02.leap 02-03.leap 02-39.leap ......... please input the filename:02-02 ================================================================ 1: //2-2 Wiley Wiz is a mutual fund manager trying to 2: //decide how to divide up to $12 million between 3: //domestic and foreign stocks. Domestic stocks 4: //have been returning 11% per year and foreign 5: //17%. Naturally, Wiley would like to maximize 6: //the annual return from his investments. Still, he 7: //wants to exercise some caution. No more that $10 8: //million of the fund should go into domestic stocks 9: //and no more than $7 million into foreign. Also, at 10: //least half as much should be invested in foreign as 11: //domestic, and at least half as much in domestic as 12: //foreign to maintain some balance. 13: 14: max sum{i=1,..,m}R[i]x[i]/100 15: subject to 16: sum{i=1,..,m}x[i]<=T 17: x[i]<=B[i] | i=1,..,m 18: where 19: m is an integer 20: T is a number 21: R[i],B[i] are numbers|i=1,..,m 22: x[i] is a variable of nonnegative number--> 23: |i=1,..,m 24: data 25: m=2 26: T=12 27: R={11 17} 28: B={10 7} ================================================================ >>end of the file. Parsing model: 1D 2R 3V 4O 5C 6S 7End. .................................. number of variables=2 number of constraints=3 .................................. +Leapms>solve The LP is solved to optimal. 找到线性规划最优解.非零变量值和最优目标值如下： ......... x1*=5 x2*=7 ......... Objective*=1.74 ......... +Leapms>
参考文献
[1] Rardin R. L 著，肖勇波、梁湧译. 运筹学. 北京:机械工业出版社,2018