Perform the exercise either by graphic method, branching and dimensioning or by cutting planes Max z = 2X1 + X2 5X1 + 2X2 ≤ 10 X1, x2 ≥ 0, integers

To solve this problem using graphical method, we first need to graph the constraint equation:

5X1 + 2X2 = 10

Now we find the intercepts by setting X1 = 0 :

2X2 = 10

X2 = 5

And by setting X2 = 0 :

5X1 = 10

X1 = 2

Plotting these points on the graph and drawing the line, we see that the feasible region is the triangle below the line and bounded by the axes.

Next, we identify the corner points of the feasible region. Since the region is bounded by the axes and the line, the corner points are the intersection of the line and the axes:

A(2,0), B(0,5), and C(0,0)

Now substitute these corner points into the objective function Z = 2X1 + X2 to find the maximum value:

Z_A = 2(2) + 0 = 4

Z_B = 2(0) + 5 = 5

Z_C = 2(0) + 0 = 0

The maximum value of Z is \boxed{5} when X2 = 5 and X1 = 0 .