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,


"Minimize\\quad z=a+b+c\\\\\nsubject\\ to,\\\\\na-b-c\\le 0\\\\\na+b+c\\ge4\\\\\na+b-c=2\\\\"

After converting the simplex method(big M) form,

"Minimize\\quad z=a+b+c+0s_1+0s_2+MA_1+MA_2\\\\\nsubject\\ to,\\\\\na-b-c+s_1= 0\\\\\na+b+c-s_2+A_1=4\\\\\na+b-c+A_2=2\\\\"

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


Table 01:



"S_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


"A_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

"A_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 "Z-C_j\\le0" , optimal answer is occurred.

Answer to the minimization problem is,

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

In all the tables,

  • blue numbers "(C_j)" =corresponding coefficients in minimization function
  • "Z-C_j" = basis variable coefficients multiply by each column and add them -corresponding coefficient in minimization function(blue numbers)
  • Then find largest "Z-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 "Z-C_j \\le0."

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