使用 线性规划 解决 数字 排序问题, +Leapms模型

问题

将如下一组数字从大到小排序。

{10, 20, -32, 177, 0, -11.5, 19, 7, 6.2, -6.28, -2.71, 44}

解决办法

建立数学模型，给出各个数字的次序值。

模型

设x[i]为第i个数的次序值。根据排序规则有如下约束：

x[i] <= x[j] -1 | i=1,...,n; j=1,...,n; i<>j; d[i] >d[j]

希望次序值从1开始，最大不超过数字的总数：

x[i] >= 1 | i=1,...,n

x[i] <= n | i=1,...,n

不需要目标：

max 0

最终模型为

max 0
subject to
	x[i] <= x[j] -1 | i=1,...,n; j=1,...,n; i<>j; d[i] >d[j]
	x[i] >= 1 | i=1,...,n
	x[i] <= n | i=1,...,n
where
	n is a number
	d is a set
	x[i] is a variable of nonnegative integer| i=1,...,n
data_relation
	n=_$(d)
data
	d={10, 20, -32, 177, 0, -11.5, 19, 7, 6.2, -6.28, -2.71, 44}

求解

+Leapms>load
 Current directory is "ROOT".
 .........
        sort.leap
 .........
please input the filename:sort
================================================================
1:  max 0
2:  subject to
3:      x[i] <= x[j] -1 | i=1,...,n; j=1,...,n; i<>j; d[i] >d[j]
4:      x[i] >= 1 | i=1,...,n
5:      x[i] <= n | i=1,...,n
6:  where
7:      n is a number
8:      d is a set
9:      x[i] is a variable of nonnegative integer| i=1,...,n
10:  data_relation
11:     n=_$(d)
12:  data
13:     d={10, 20, -32, 177, 0, -11.5, 19, 7, 6.2, -6.28, -2.71, 44}
================================================================
>>end of the file.
Parsing model:
1D
2R
3V
4O
5C
6S
7End.
..................................
number of variables=12
number of constraints=90
..................................
+Leapms>mip
relexed_solution=0; number_of_nodes_branched=0; memindex=(2,2)
The Problem is solved to optimal as an MIP.
找到整数规划的最优解.非零变量值和最优目标值如下：
  .........
    x1* =5
    x2* =3
    x3* =12
    x4* =1
    x5* =8
    x6* =11
    x7* =4
    x8* =6
    x9* =7
    x10* =10
    x11* =9
    x12* =2
  .........
    Objective*=0
  .........
+Leapms>

对上述结果进行解释，x1*=5即第一个数放在第5位， x2*=3即第2个数放在 第2位，或者说12数字的次序数分别为5，3，12，1，8，11，4，6，7，10，9，2。