Algebra - Simplex Algorithm

Simplex Algorithm

 

 

            Consider programming problem (2) and a basic program nondegenerated .  After some rearrangement and renumbering we can consider ; so, the variables Algebra Liniara are main and therefore Algebra Liniara secundary, and columns vectors form the base B of a basic programm . Let be

Writing:

 Algebra Liniara, Simplex, formula the problem (2) could be writed as:

Algebra Liniara, Simplexul

            Multiplying Algoritmul Simplex on the left with Algebra Liniarawe can obtain:

representing transcription system restrictions under B for if we write Algebra Liniara (expression vectors based on the vectors base column B) we have:


  Algoritmul Simplex 
According the programm Algebra Superioara, formula algoritm simplexthe problem (2) becomes:

So though Algoritmul Simplex are the vector component L based on B.

Algebra Liniara

So the relation (8) becomes:

Algoritmul Simplex from where

  and then

:  

or, more explicit:

Writing: Algoritmul Simplex then:

We notice that Algoritmul Simplex

         Now we can associate the problem PL-min following table:

 

Algoritmul Simplex

 c1     c2            cn

a1      a2            an

(S)

 

the vectors of the base

Unnull components of

.........................................

       

 

          Theorem II.4.1. Ifis a nondegenerate basic program for PL - min and in the associated table (S) have then is optimal program..

 

          Proof: We have the expression (13) and then:

for any  admissible program X. So is optimal.

 

      Theorem II.4.2. Ifis a nondegenerate basic program and in associated simplex table (S) we have a t, such as then PL - min doesn't have a finite optimal.

 

          Proof: Let be: where:

Such as we have

For we got:

       So though is an admissible solution. We have:

 

from the definition.

          Because then , i.e. the objective function has not an optimal finished.

 

      Theorem II.4.3. Ifis a nondegenerate basic program for PL - min and in the associated table (S)  and in the simplex table associated (S) exists  t, and at least one index i, , such as then if we choose with criteria:

may be substituted on the basis B the vector  with the vector , obtaining a base corresponding to the basic program which improves the objective function value.

 

          Proof. Because using Lemma substitution, replacing that with B and vectors system new obtained is a base. Corresponding to the basic solution it is given all the substitution lemma:

with all non-negative components (for if then

 , so the sum of non-negative numbers; and if we got and taking into account (14) means that   it is the product of two non-negative numbers).

So though is a basic solution. The value of the objectiv function for is:

 

          Now we can present a problem for the simplex algorithm PL - min in standard form.

- Step 10: There is a core(basic) program nondegenerated  having the base B; It is build the simplex table (S).

- Step 20: Check whether differences for any . If YES got to step 5; if NO, of all negative differences , it is choosen the smallest one. J index corresponding to denote by t. (If there are several t choose first from left to right). The vector will be in the base. It assesses whether for If YES, it goes to step 4, if NO, it goes to step 3.

- Step 30: It choose s, such as .

The vector will come out from the base. The item becomes pivot. It will be build a new simplex table using the rectangle rule:

a) The line of the pivot divides through the pivot

b) in the pivot's column the items are replaced by 0

c) the items are replaced by .

It will be obtained another core program with the base and a new value of the objectiv function.

You come back to step 20 with and

- Step 40 .Conclusion: “PL - min has not an finite optimal” and the algorithm stops.

- Step 50. Conclusion: “PL - min has an optimal and the minimum value is ". STOP.

          Example II.4.1. Let be the problem:

 

We choose . We got:

          The coordonates of the vector wit the base B are , and. To find the coordinates of proceed as follows: we put  , so:

which give us . So though, in base B, . The same way we can have:

So corresponding simplex table base B has the form:

 

 

 5      7      9      2      1

 

 

 

 3

 

 1       0      1       1      -1

 

 0       1     -1       1            

 0       0     11    -10    -15

       

          So comes in the base B, comes out from the base, z25 - pivot. Run and get:

 
 

 
 


4/3

 

 1    1/3    2/3             0

 

 0    1/3   -1/3   1/3     1

 0     5       6      -5      0

 

          Enter the base and comes out .

 

  15/4    25/4      17/2     0    0

 

          And we got So, the optimal program is .

         The algorithm applies problems PL - max in standard form with the observation that  . The algorithm applies if the objective function has the form , because of its extreme points are the same extreme points of the function:

 

Keywords: algebra, simplex, algorithm, superior maths, algebraic formula

 

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