Decision Science
Optimization Methods: Linear Programming- Graphical Method Module – 3 Lecture Notes – 2 Graphical Method
1
Graphical method to solve Linear Programming problem (LPP) helps to visualize the procedure explicitly. It also helps to understand the different terminologies associated with the solution of LPP. In this class, these aspects will be discussed with the help of an example. However, this visualization is possible for a maximum of two decision variables. Thus, a LPP with two decision variables is opted for discussion. However, the basic principle remains the same for more than two decision variables also, even though the visualization beyond twodimensional case is not easily possible. Let us consider the same LPP (general form) discussed in previous class, stated here once again for convenience.
Maximize subject to Z = 6x + 5 y 2x − 3 y ≤ 5 x + 3 y ≤ 11 4 x + y ≤ 15 x, y ≥ 0 (C − 1) (C − 2) (C − 3) (C − 4) & (C − 5)
First step to solve above LPP by graphical method, is to plot the inequality constraints oneby-one on a graph paper. Fig. 1a shows one such plotted constraint. 5 4 3 2 1 0 -2 -1 -1 -2 0 1 2 3 4 5
2x − 3y ≤ 5
Fig. 1a Plot showing first constraint ( 2 x − 3 y ≤ 5 ) Fig. 1b shows all the constraints including the nonnegativity of the decision variables (i.e., x ≥ 0 and y ≥ 0 ).
D Nagesh Kumar, IISc, Bangalore
M3L2
Optimization Methods: Linear Programming- Graphical Method
2
5 4 3 2 1 0 -2 -1 -1 -2 0
x + 3 y ≤ 11
4 x + y ≤ 15 x≥0 y≥0
1
2
3
4
5
2x − 3y ≤ 5
Fig. 1b Plot of all the constraints Common region of all these constraints is known as feasible region (Fig. 1c). Feasible region implies that each and every point in this region satisfies all the constraints involved in the LPP.
5 4 3 2 1 0 -2 -1 -1 -2 0 1 2 3 4 5
Feasible region
Fig. 1c Feasible region
Once the feasible region is identified, objective function ( Z = 6 x + 5 y ) is to be plotted on it. As the (optimum) value of Z is
1
Graphical method to solve Linear Programming problem (LPP) helps to visualize the procedure explicitly. It also helps to understand the different terminologies associated with the solution of LPP. In this class, these aspects will be discussed with the help of an example. However, this visualization is possible for a maximum of two decision variables. Thus, a LPP with two decision variables is opted for discussion. However, the basic principle remains the same for more than two decision variables also, even though the visualization beyond twodimensional case is not easily possible. Let us consider the same LPP (general form) discussed in previous class, stated here once again for convenience.
Maximize subject to Z = 6x + 5 y 2x − 3 y ≤ 5 x + 3 y ≤ 11 4 x + y ≤ 15 x, y ≥ 0 (C − 1) (C − 2) (C − 3) (C − 4) & (C − 5)
First step to solve above LPP by graphical method, is to plot the inequality constraints oneby-one on a graph paper. Fig. 1a shows one such plotted constraint. 5 4 3 2 1 0 -2 -1 -1 -2 0 1 2 3 4 5
2x − 3y ≤ 5
Fig. 1a Plot showing first constraint ( 2 x − 3 y ≤ 5 ) Fig. 1b shows all the constraints including the nonnegativity of the decision variables (i.e., x ≥ 0 and y ≥ 0 ).
D Nagesh Kumar, IISc, Bangalore
M3L2
Optimization Methods: Linear Programming- Graphical Method
2
5 4 3 2 1 0 -2 -1 -1 -2 0
x + 3 y ≤ 11
4 x + y ≤ 15 x≥0 y≥0
1
2
3
4
5
2x − 3y ≤ 5
Fig. 1b Plot of all the constraints Common region of all these constraints is known as feasible region (Fig. 1c). Feasible region implies that each and every point in this region satisfies all the constraints involved in the LPP.
5 4 3 2 1 0 -2 -1 -1 -2 0 1 2 3 4 5
Feasible region
Fig. 1c Feasible region
Once the feasible region is identified, objective function ( Z = 6 x + 5 y ) is to be plotted on it. As the (optimum) value of Z is