[mathematical modeling summer training] location of distribution center

Problem description

Jointly owned by a province 92 Cities , City Location 、 label , See the annex for the data of highway transportation network 1.
An enterprise is in front of the provincial label 20 Cities in have established direct sales centers , Each direct sales center is responsible for the sales of the city , See the attachment for the sales volume 1. The enterprise wants to set up a distribution center in the province to be responsible for distributing products to the direct sales center , The construction cost of the distribution center is 30 Ten thousand yuan . Freight per ton kilometer 2 element , The sales profit per ton of product is 300 element .
Try to establish a mathematical model to analyze and study the following problems ：

Symbol description

Symbol	explain
$dij$	Says from the first i City to city j City to distance
$sj$	It means the first one j Monthly sales of direct sales centers
pi={1,0}	1: In the i Distribution centers have been established in cities 0: Not in the i Distribution centers have been established in cities
qi={1,0}	1: In the j Distribution centers have been established in cities 0: Not in the j Distribution centers have been established in cities
aij={1,0}	1: Distribution Centre i To the distribution center j Delivery 0: Distribution Centre i Do not deliver to the distribution center
cij={1,0}	1: The first j The first direct selling center manages i Cities 0: The first j The first direct selling center does not manage i Cities

Question 1

In order to reduce the cost of transportation , Which city should the distribution center be located in ？
Model ：$$min\sum _{j=1}^{20}(2\ast d_{ij}\ast s_j)$$
Matlab Draw and solve

answer ：35

Question two

Please set a minimum cost for this enterprise 5 Annual product distribution plan ： Several distribution centers should be set up 、 Where are they located ？
Model ：$$min\sum _{i=1}^{92}(30\ast p_i+\sum _{j=1}^{20}0.012\ast a_{ij}\ast d_{ij}\ast s_j)$$

lingo The answer is ：

Question 3

If the enterprise considers to be 20 Location of direct sales centers , Please give the best 5 Annual product sales 、 Delivery plan .
$$max(\sum _{j=1}^{92}(1.8\ast q_j\ast s_j)-\sum _{i=1}^{92}(p_i\ast 30+\sum _{j=1}^{20}0.012\ast a_{ij}\ast d_{ij}\ast s_{ij}))$$

The above constraint should be qj, dial the wrong number .
lingo solve

Question 4

Suppose that customers in cities without direct sales centers buy products according to the principle of proximity , Please reconsider （3）.
$$max(\sum _{j=1}^{92}(1.8\ast q_j\ast s_j)-\sum _{i=1}^{92}(p_i\ast 30+\sum _{j=1}^{20}0.012\ast a_{ij}\ast d_{ij}\ast (s_j+c_{ij}\ast s_i)))$$

lingo solve

Code

Reference article
0-1 Planned lingo solve
Location problem of mathematical programming
Distribution location center problem