Chapter 6, Problem 21RP

Program Plan Intro

Optimal solution:

• Consider the linear programming problem given below:
• max z= $4x_1+x_2$
• Subject to constraints:
  • $3x_1+x_2\le 6$
  • $5x_1+3x_2\le 15$
  • $x_1,x_2\ge 0$
• The graph of the LP is given below:

Explanation of Solution

b.

• In this case, $c_2$ represents the contribution of $x_2$ to the objective function.
• Here, $c_2=1$, the objective function or the isoprofit line has the form $4x_1+x_2=$ constant or $x_2=-4x_1+$ constant. Hence, each isoprofit line has a slope of -4.
• If the contribution of each $x_2$ is $c_2$ then the slope of each isoprofit line will be $-\frac{4}{c_2}$

Explanation of Solution

c.

• Change in $b_1$ shifts the black line parallel to its current position. The current optimal solution is point A where all the constraints bind.
• As long as the point where the two constraints bind remains feasible to any change in $b_1$ would not change the optimal solution and the optimal will still occur where all constraints intersect.
• In this case, the feasible region is the blue colored area.
• If $b_1\le 0$, the two constraints will be binding at an infeasible point having $x_1<0$ and the current basis will no longer be optimal

Explanation of Solution

d.

• Change in $b_2$ shifts the red line parallel to its current position. The current optimal solution is point A where all the constraints bind.
• As long as the point where the two constraints bind remains feasible to any change in $b_2$ would not change the optimal solution and the optimal will still occur where all constraints intersect.
• In this case, the feasible region is the blue colored area