Decision Science

Topics: Optimization, Operations research, Constraint Pages: 7 (998 words) Published: August 22, 2013
Optimization Methods: Linear Programming- Graphical Method Module – 3 Lecture Notes – 2 Graphical Method

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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 not known, objective function is plotted by considering any constant, k (Fig. 1d). The straight line, 6 x + 5 y = k (constant), is known as Z line (Fig. 1d). This line can be shifted in its perpendicular direction (as shown in the Fig. 1d) by changing the value of k. Note that, position of Z line shown in Fig. 1d, showing the intercept, c, on the

D Nagesh Kumar, IISc, Bangalore

M3L2

Optimization Methods: Linear Programming- Graphical Method
6 x + 5 y = k => 5 y = −6 x + k => y = k −6 x + , i.e., 5 5 m= −6 5

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y axis is 3. If,
c= k = 3 => k = 15 . 5

and

5 4 3 2 1 0 -2 -1 -1 -2
Z Line

0

1

2

3

4

5

Fig. 1d Plot of Z line and feasible region
5
Z Line

4 3 2 1 0 -2 -1 -1 -2 0 1 2 3

Optimal Point

4

5

Fig. 1e Location of Optimal Point Now it can be visually noticed that value of the objective function will be maximum when it passes through the intersection of x + 3 y = 11 and 4 x + y = 15 (straight lines associated with the second and third inequality constraints). This is known as optimal point (Fig. 1e). Thus the optimal point of the present problem is x * = 3.091 and y * = 2.636 . And the optimal solution is = 6 x * + 5 y * = 31.727

D Nagesh Kumar, IISc, Bangalore

M3L2

Optimization Methods: Linear Programming- Graphical Method
Visual representation of different cases of solution of LPP

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A linear programming problem may have i) a unique, finite solution, ii) an unbounded solution iii) multiple (or infinite) number of optimal solutions, iv) infeasible solution and v) a unique feasible point. In the context of graphical method it is easy to visually demonstrate the different situations which may result in different types of solutions. Unique, finite solution

The example demonstrated above is an example of LPP having a unique, finite solution. In such cases, optimum value occurs at an extreme point or vertex of the feasible region. Unbounded solution

If the feasible region is not bounded,...
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