# PHPSimplex

## Optimizing resources with Linear Programming

### Transporting goods' problem

For this type of problem, although can be solved by the Simplex Method, there is a specific method that make it easier: the transport method or Simplified Simplex Method for Transport. This method saves quite time and calculations against the traditional Simplex Method.

However the problem is modeled in the same way.

Example

A manufacturer wishes to dispatch several units of an article to three shops T1, T2, and T3. Has two stores from where accomplishing the shipment, A and B. In the first store, has 5 units of this article, and 10 in the second one. The request of every store is 8, 5, and 2 units respectively. The transport expenses of an article from store to each shop are shows below:

 T1 T2 T3 A 1 2 4 B 3 2 1

How must accomplish the transportation in order to be the more economic as be possible?

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

• Xi: number of units transported from each store to each shop

• X1: number of units transported from A store to T1 shop
• X2: number of units transported from A store to T2 shop
• X3: number of units transported from A store to T3 shop
• X4: number of units transported from B store to T1 shop
• X5: number of units transported from B store to T2 shop
• X6: number of units transported from B store to T3 shop

Determining the restrictions and expressing them as equations or inequalities in function of the decision variables. Such restrictions can be obteined from availability of units that there is at each store thus the request of each shop:

• Availability at A store: X1 + X2 + X3 = 5
• Availability at B store: X4 + X5 + X6 = 10
• Request of T1 shop: X1 + X4 = 8
• Request of T2 shop: X2 + X5 = 5
• Request of T3 shop: X3 + X6 = 2

Expressing all implicit conditions established by the origin of variables: negativeness, integer, only a few allowed values,... In this case, the restrictions are that quantity of units can't be negative and must be an integer number:

• Xi ≥ 0
• Xi are integers

Determining objective function:

• Minimize Z = X1 + 2·X2 + 4·X3 + 3·X4 + 2·X5 + X6
X

PHPSimplex
Version 0.81