Why is the objective function line running through points x2=10 and x1=5? Please

Question

Why is the objective function line running through points x2=10 and x1=5?

Please explain how to graph the objective function line.

Solve the following linear programming problem with the graphic method. Label each of the constraints and shade the feasible region. Draw at least one line for the objective function. Indicate the optimal solution on the graph. Find the values of x and x, as well as the value of the objective function z, at the optimal solution (20) min 9x +5x2 5x1 +7x2-35 8 x t3x2 2 24 2 x 9x2 2 18 5x +7x2 35 8x, + 3x2-24 s.t 15x, +21x2 -105 56x, +21x,-168 41x1 = 63 x 63 10 x, 160 41 z = 9(%)+5(1%1)-13%. -33.34 15 63 160 41,41) = (1 .5366, 3.9024) Optimal solution 4 Feasible region 4 10 8x1 +3x2 =24 9x1 +5x2 = 45 5x1 +7x2=35 2xi +9x2=18

Explanation / Answer

The objective function line is actually passing through the points (5,0) and (0,9) in the above graph. The objective function line has been represented by a dotted line to show one of the potentially infinite lines that can be drawn with the equation  Z = 9x1 + 5x2

Note that the slope of all those lines would be the same and would be equal to -9/5. For different values of Z, (in this case, taken as 45 arbitrarily) the objective function line would have the same slope but different intercept points on the Y and X-axes.

There is no single objective function line. We simply draw different parallel(same slope) iso-profit lines(lines with the same profit at each point on the line) and check which line yields the optimal value of the objective function.

In this linear programming case, the feasible region is a line segment that has been highlighted in the graph. All the points that lie on this line segment when supplied in the objective function equation would yield a distinct value. In this case, we need to find the point that yields the minimum value which is shown in the graph.

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