清華大學肖勇波樑湧老師的巨集篇譯著中的問題實踐之004-生產計劃問題與投資問題

清華大學肖勇波樑湧老師翻譯的Rardin教授的《運籌學》[1]已於今年年中出版，感謝機械工業出版社張有利老師的推薦和贈書。

問題

生產計劃問題和投資問題見原書第二章的兩個練習，習題2-1和習題2-2。

生產計劃問題又叫混合生產問題（Mixed Production Problem),是經典的線型規劃問題。

投資問題又叫投資組合問題，是金融背景上的一種揹包問題。

+Leapms生成的兩個問題的模型摘錄

對任何問題，+Leapms都建議直接用+Leapms建模語言直接寫出模型，並進行模型除錯。

我們的經驗是，模型除錯能夠發現模型的不足、促進建模的完美性，甚至可以促進對問題的更加深入的瞭解。當模型除錯完畢，對模型的正確性有充分的信心後，+Leapms系統可生成模型摘錄，包括數學概念模型（即使用標引符號表示的數學模型）和+Leapms原始碼供交流使用。

以下是兩個問題的+Leapms模型摘錄（pdf螢幕截圖）, 其中 模型"02-1.leap" 是生產計劃問題, 模型"02-02"是投資問題：

求解過程

+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.
找到線性規劃最優解.非零變數值和最優目標值如下：
    .........
    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.
找到線性規劃最優解.非零變數值和最優目標值如下：
    .........
    x1*=5
    x2*=7
    .........
    Objective*=1.74
    .........
+Leapms>

參考文獻

 [1] Rardin R. L 著，肖勇波、樑湧譯. 運籌學. 北京:機械工業出版社,2018