lingo-运输问题的数学模型_垃圾运输问题数学建模lingo

建模

变量

ai表示第i个工厂的加工量
bj表示第j个门市部的销量
cij表示运价，从第i个加工厂到第j个门市部运送一吨糖果的价格
决策变量:xij 从第i个加工厂到第j个门市部的糖果数量

一般形式

m i n z = ∑ ∑ c i j ∗ x i j s . t . { ∑ j x i j = a i ∑ i x i j = b j x i j ≥ 0 min z = \sum \sum cij * xij\\ s.t. \left\{

$$\begin{matrix} \sum_j xij = ai\\ \sum_i xij = bj\\ xij \geq 0 \end{matrix}$$ \right. minz=∑∑cij∗xijs.t.⎩ ⎨ ⎧​∑j​xij=ai∑i​xij=bjxij≥0​

矩阵形式

A = [ 7 4 9 ] B = [ 3 6 5 6 ] C = [ 3 11 3 10 1 8 2 8 7 4 10 5 ] X = [ x 11 x 12 x 13 x 14 x 21 x 22 x 23 x 24 x 31 x 32 x 33 x 34 ] A = [

$$\begin{matrix}7&4&9\end{matrix}$$ ] \\ B= [ $$\begin{matrix}3&6&5&6\end{matrix}$$ ] \\ C=\left [ $$\begin{matrix} 3&11&3&10\\ 1&8&2&8\\ 7&4&10&5 \end{matrix}$$ \right]\\ X = \left [ $$\begin{matrix} x_{11}&x_{12}&x_{13}&x_{14}\\ x_{21}&x_{22}&x_{23}&x_{24}\\ x_{31}&x_{32}&x_{33}&x_{34}\\ \end{matrix}$$ \right] A=[7​4​9​]B=[3​6​5​6​]C= ​317​1184​3210​1085​ ​X= ​x11​x21​x31​​x12​x22​x32​​x13​x23​x33​​x14​x24​x34​​ ​

Lingo 建立模型

方法

定义集合
给常量赋值
目标函数的表达
约束条件的表达
模型求解

定义集合

! 建立加工厂集合及产量;
factory/1..3/:A；
! 建立门市部集合及产量；
shop/1..4/:B;
!建立联系集合及运价和运量,派生集合，双下标变量；
LINK(factory,shop):C,X;
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给常量赋值

DATA:
	A = 7 4 9;
	B = 3 6 5 6;
	C = 
	3 11 3 10
	1 9 2 8
	7 4 10 5;
ENDDATA
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目标函数的表达

$minz=\sum \sum cij\ast xij$

MIN = @SUM (LINK(i,j): C(i,j) * X(i,j));
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约束条件的表达

1.产量约束
${\sum }_{j}xij=ai$

@for(factory(i): @sum(shop(j): x(i,j)) = A(i));
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2.销量约束
${\sum }_{i}xij=bj$

@for(shop(j): @sum(factory(i): x(i,j)) = b(j));
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3.自身约束
$xij\ge 0$
lingo 默认非负

完整代码

MODEL:
	SETS:
	factory/1..3/:A;
	shop/1..4/:B;
	LINK(factory,shop):C,X;
	ENDSETS
	
	DATA:
	A = 7 4 9;
	B = 3 6 5 6;
	C = 
	3 11 3 10
	1 9 2 8
	7 4 10 5;
	ENDDATA
	
	MIN = @SUM (LINK(i,j): C(i,j) * X(i,j));
	@for(factory(i): @sum(shop(j): x(i,j)) = A(i));
	@for(shop(j): @sum(factory(i): x(i,j)) = b(j));
END
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