8Graphical Solution to a 2-Variable LPX1X2102040506080finishing constraintcarpentry constraintdemand constraintz = 60z = 100z = 180Feasible RegionGABCDEFHTo find the optimal solution, graph a line on which the points have the same z-value. In a max problem, such a line is called an isoprofit line while in a min problem, this is called the isocost line. The figure shows the isoprofit lines for z = 60, z = 100, and z = 180
9Graphical Solution to a 2-Variable LPThe last isoprofit intersecting (touching) the feasible region indicates the optimal solution for the LP. X1X2102040506080finishing constraintcarpentry constraintdemand constraintz = 60z = 100z = 180Feasible RegionGABCDEFH
10Graphical Solution to a 2-Variable LPThe optimal solution must lie somewhere on the boundary of the feasible region.The LP must have an extreme point that is optimal, because for any line segment on the boundary of the feasible area, the largest z value on that line segment must be assumed to be at one endpoint of the line segment.X1X2102040506080finishing constraintcarpentry constraintdemand constraintz = 60z = 100z = 180Feasible RegionGABCDEFH
Summary of the Graphical Solution Procedure for Maximization ProblemsPrepare a graph of the feasible solutions for each of the constraints.Determine the feasible region that satisfies all the constraints simultaneously.Draw an objective function line.Move parallel objective function lines toward larger objective function values without entirely leaving the feasible region.Any feasible solution on the objective function line with the largest value is an optimal solution.11