Answer to Question #107859 in Operations Research for Mamabolo Letsatsi

Question #107859
Solve the following linear program using simplex algorithm
Minimize z= a+b+c
Subject to: 1. a - b -c<=0
2. a + b + c>=4
3. a +b-c=2
1
Expert's answer
2020-04-06T16:09:23-0400

problem formulating as,


Minimizez=a+b+csubject to,abc0a+b+c4a+bc=2Minimize\quad z=a+b+c\\ subject\ to,\\ a-b-c\le 0\\ a+b+c\ge4\\ a+b-c=2\\

After converting the simplex method(big M) form,

Minimizez=a+b+c+0s1+0s2+MA1+MA2subject to,abc+s1=0a+b+cs2+A1=4a+bc+A2=2Minimize\quad z=a+b+c+0s_1+0s_2+MA_1+MA_2\\ subject\ to,\\ a-b-c+s_1= 0\\ a+b+c-s_2+A_1=4\\ a+b-c+A_2=2\\

Steps of every table is described after all the tables..


Table 01:



S1S_1 has gone out from the basis and a has come in to basis.


Table 02:

Row operations:

Row#1=Row#1

Row#2=Row#2-Row#1

Row#2=Row#2-Row#1


A1A_1 has gone out from the basis and b has come in to basis.


Table 03:

Row operations:

Row#3=Row#3/2

Row#1=Row#1+Row#3

Row#2=Row#2-Row#3*2

A2A_2 has gone out from the basis and c has come in to basis.


Table 04:

Row operations:

Row#2=Row#2/2

Row#1=Row#1+Row#2

Row#3=Row#3


since all the ZCj0Z-C_j\le0 , optimal answer is occurred.

Answer to the minimization problem is,

a=2b=1c=1Zmin=4\red{a=2\\b=1\\c=1\\Z_{min}=4\\}

In all the tables,

  • blue numbers (Cj)(C_j) =corresponding coefficients in minimization function
  • ZCjZ-C_j = basis variable coefficients multiply by each column and add them -corresponding coefficient in minimization function(blue numbers)
  • Then find largest ZCjZ-C_j(green column),that variable in that column is incoming variable to basis.
  • Then b column is divide by that green column and find the smallest positive number. The variable is corresponding row(yellow row) is ​gone out from the basis.
  • Then next table is find by making row operations to make the corresponding basis variable matrix to identity matrix.
  • These steps are continue until all ZCj0.Z-C_j \le0.

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