Chapter 6, Problem 34RP

Program Plan Intro

Optimal solution:

• Consider the linear programming problem given below:
• max z= $20x_1+10x_2$
• Subject to constraints:
  • $x_1+x_2=150$
  • $x_1\le 40$
  • $x_2\le 20$
  • $x_1,x_2\ge 0$
• Here,
• $x_i$= Fraction of investment I purchased by star oil.
• The Lindo output of the LP is given in the following figure:
• LP optimum found at step 5
• Objective function value: 57.44902.

Variable	Value	Reduced Cost
$x_1$	1.000000	0.000000
$x_2$	0.200860	0.000000
$x_3$	1.000000	0.000000
$x_4$	1.000000	0.000000
$x_5$	0.288084	0.000000

Row	Slack or Surplus	Dual prices
1	0.000000	0.190418
2	0.000000	0.984644
3	0.000000	7.951474
4	0.799140	0.000000
5	0.000000	10.124693
6	0.000000	12.063268
7	0.711916	0.000000

• Number of iterations: 5
• Ranges in which the basis is unchanged:

Obj coefficient ranges
Variable	Current coefficient	Allowable increase	Allowable Decrease
$x_1$	13.000000	Infinity	7.951474
$x_2$	16.000000	45.104527	9.117647
$x_3$	16.000000	Infinity	10.124693
$x_4$	14.000000	Infinity	12.063268
$x_5$	39.000000	51.666668	30.245285

Right-hand side ranges
Row	Current RHS	Allowable increase	Allowable Decrease
1	40.000000	38.264706	9.617647
2	20.000000	11.275863	8.849056
3	1.000000	1.139373	1.000000
4	1.000000	Infinity	0.799140
5	1.000000	1.995745	1.000000
6	1.000000	2.319149	1.000000
7	1.000000	Infinity	0.711916

Explanation of Solution

b.

• The given LP is called the primal. If the primal is a max problem, then the dual will be min problem. Define the variables for the max problem to be $z,x_1,x_2,\mathrm{...........},x_n$ and the variables for the min problem to be $w,y_1,y_2,\mathrm{........},y_n$.
• max z= $c_1{x}_1+c_2{x}_2+\mathrm{.......}+c_n{x}_n$
• subject to constraints
  • $a_{11}{x}_1+a_{12}{x}_2+\mathrm{.......}+a_{1n}{x}_n\le b_1$
  • $a_{21}{x}_1+a_{22}{x}_2+\mathrm{.......}+a_{2n}{x}_n\le b_2$
  • $a_{m1}{x}_1+a_{m2}{x}_2+\mathrm{.......}+a_{mn}{x}_n\le b_m$
• The dual of the normal max problem is defined below:
• min w=$b_1{y}_1+b_2{y}_2+\mathrm{.......}+b_2{y}_n$
• subject to constraints
  • $a_{11}{y}_1+a_{12}{y}_2+\mathrm{.......}+a_{1n}{y}_n\ge c_1$
  • $a_{21}{y}_1+a_{22}{y}_2+\mathrm{.......}+a_{2n}{y}_n\ge c_2$
  • $a_{m1}{y}_1+a_{m2}{y}_2+\mathrm{.......}+a_{mn}{y}_n\ge c_m$
• Given a max problem, the primal is read across the table and the dual is found by reading down.

min w		$\left(x_1\ge 0\right)$	$\left(x_2\ge 0\right)$	max z $\dots$	$\left(x_n\ge 0\right)$
		$x_1$	$x_2$		$x_n$
$\left(y_1\ge 0\right)$	$y_1$	$a_{11}$	$a_{12}$	$\dots$	$a_{1n}$	$\le b_1$
$\left(y_2\ge 0\right)$	$y_2$	$a_{21}$	$a_{22}$	$\dots$	$a_{2n}$	$\le b_2$
$⋮$	$⋮$	$⋮$	$⋮$		$⋮$	$⋮$
$\left(y_m\ge 0\right)$	$y_m$	$a_{m1}$	$a_{m2}$	$\dots$	$a_{mn}$	$\le b_m$
		$\ge c_1$	$\ge c_2$		$\ge c_n$

• The given problem is a non normal Max problem