PHPSimplex

Location problem

A company has the exclusive for distribution of a product in 4 populations. Potential demand has been determined in a market research, according to he shows up in the following board:

Population 1	Population 2	Population 3	Population 4
3000 units	2000 units	2500 units	2700 units

Is knwon that the transport expenses are 0.02€ per Km and unit transported. The distance among towns is the shows in the board following:

	Population 1	Population 2	Population 3	Population 4
Population 1	-	25Km	35Km	40Km
Population 2	25Km	-	20Km	40Km
Population 3	35Km	20Km	-	30Km
Population 4	40Km	40Km	30Km	-

In order to cheap the transportation costs is decided to install in two of that 4 populations a store with capability for 6000 units. Determining in wich populations must install the stores.

 

Determining decision variables and expressing them algebraically. In this case:

• Xij: quantity sended from i store to j population
• Yi: store located at i population (0 points that there is not store, 1 yes)

Determining the restrictions and expressing them as equations or inequalities in function of the decision variables.

• The units that are sended to each population from each store, that must obey with the population demand:
  
  
  • X11 + X21 + X31 + X41 ≥ 3000
  • X12 + X22 + X32 + X42 ≥ 2000
  • X13 + X23 + X33 + X43 ≥ 2500
  • X14 + X24 + X34 + X44 ≥ 2700
  
  
• Only will be created two stores:
  
  
  • Y1 + Y2 + Y3 + Y4 = 2
  
  
• The units quantity that can send each store must be less or equal to the capability of this one:
  
  
  • X11 + X12 + X13 + X14 ≤ 6000·Y1
  • X21 + X22 + X23 + X24 ≤ 6000·Y2
  • X31 + X32 + X33 + X34 ≤ 6000·Y3
  • X41 + X42 + X43 + X44 ≤ 6000·Y4

Expressing all implicit conditions established by the origin of variables: negativeness, integer, only a few allowed values... . In this case, the restrictions are that the delivered units from each store can't be negatives and also the variable determine that determine if a store is created or not must be boolean (0 isn't created, 1 yes):

• Xij ≥ 0
• Yi are boolean

Determining objective function:

• Minimize Z = 0.5·X12 + 0.7·X13 + 0.8·X14 + 0.5·X21 + 0.4·X23 + 0.8·X24 + 0.7·X31 + 0.4·X32 + 0.6·X34 + 0.8·X41 + 0.8·X42 + 0.6·X43