LINEAR PROGRAMMING
4.2 GRAPHICAL SOLUTION OF LINEAR PROGRAMS IN TWO VARIABLES
In Section 4.1, some examples were presented to illustrate how practical prob-lems can be formulated mathematically as LP probprob-lems. The next step after formulation is to solve the problem mathematically to obtain the best possible solution. In this section, a graphical procedure to solve LP problems involving only two variables is discussed. Although in practice such small problems are not usually encountered, the graphical procedure is presented to illustrate some of the basic concepts used in solving large LP problems.
Example 4.3
Recall the inspection problem given by Example 4.1:
Minimize Z⫽ 40x1⫹ 36x2 Subject to x1 ⭐8 x2 ⭐ 10
5x1 ⫹3x2 ⭓ 45 x1 ⭓0 x2 ⭓ 0
In this problem, we are interested in determining the values of the variables x1 and x2 that will satisfy all the restrictions and give the least value for the objective function. As a first step in solving this problem, we want to identify all possible values of x1and x2that are nonnegative and satisfy the constraints.
For example, the point x1 ⫽ 8, x2 ⫽ 10 is positive and satisfies all the con-straints. Such a point is called a feasible solution. The set of all feasible solutions is called the feasible region. Solution of a linear program is nothing but finding the best feasible solution in the feasible region. The best feasible solution is called an optimal solution to the LP problem. In our example, an optimal solution is a feasible solution that minimizes the objective function 40x1⫹36x2. The value of the objective function corresponding to an optimal solution is called the optimal value of the linear program.
To represent the feasible region in a graph, every constraint is plotted, and all values of x1, x2 that will satisfy these constraints are identified. The non-negativity constraints imply that all feasible values of the two variables will lie in the first quadrant. The constraint 5x1 ⫹ 3x2 ⭓ 45 requires that any feasible solution (x1, x2) to the problem should be on one side of the straight line 5x1 ⫹ 3x2 ⫽ 45. The proper side is found by testing whether the origin satisfies the constraints. The line 5x1 ⫹ 3x2 ⫽ 45 is first plotted by taking two convenient points (e.g., x1 ⫽ 0, x2 ⫽ 15 and x1⫽ 9, x2 ⫽ 0).
Figure 4.3. Graphical solution of Example 4.3.
a straight line if the value of Z is fixed a priori. Changing the value of Z essentially translates the entire line to another straight line parallel to itself.
To determine an optimal solution, the objective function line is drawn for a convenient value of Z such that it passes though one or more points in the feasible region. Initially Z is chosen as 600. When this line is moved closer to the origin, the value of Z is further decreased (Figure 4.3). The only lim-itation on this decrease is that the straight line 40x1⫹ 36x2⫽Z must contain at least one point in the feasible region ABC. This clearly occurs at the corner
point A given by x1 ⫽ 8, x2 ⫽ –53. This is the best feasible point, giving the lowest value of Z as 380. Hence,
–5
x1 ⫽ 8 x2⫽ 3
is an optimal solution, and Z ⫽ 380 is the optimal value for the linear pro-gram.
Thus for the inspection problem the optimal utilization is achieved by using eight grade 1 inspectors and 1.6 grade 2 inspectors. The fractional value x2
⫽–53 suggests that one of the grade 2 inspectors is utilized for only 60 percent of the time. If this is not feasible, the normal practice is to round off the fractional values to get an approximate optimal integer solution as x1 ⫽8, x2
⫽ 2.
Unique Optimal Solution. In Example 4.3 the solution x1 ⫽ 8, x2 ⫽ –53 is the only feasible point with the lowest value of Z. In other words, the values of Z corresponding to the other feasible solutions in Figure 4.3 exceed the op-timal value of 380. Hence, for this problem, the solution x1 ⫽ 8, x2 ⫽ –53 is called a unique optimal solution.
Alternative Optimal Solutions. In some LP problems there may exist more than one feasible solution whose objective function values equal the optimal value of the linear program. In such cases, all these feasible solutions are optimal solutions, and the linear program is said to have alternative or mul-tiple optimal solutions.
The feasible region is shown in Figure 4.4. The objective function lines are drawn for Z⫽ 2, 6, 10. The optimal value for the linear program is 10, and the corresponding objective function line x1⫹ 2 x2 ⫽ 10 coincides with side BC of the feasible region. Thus, the corner-point feasible solutions x1 ⫽ 10, x2 ⫽ 0 (B), x1 ⫽ 2, x2 ⫽ 4 (C), and all other feasible points on the line BC are optimal solutions.
Figure 4.4. Graphical solution of Example 4.4.
Unbounded Optimum. It is possible for some LP problems not to have an optimal solution. In other words, it is possible to find better feasible solutions improving the objective function values continuously. This would have been the case if the constraint x1 ⫹ 2 x2 ⭐ 10 were not given in Example 4.4. In this case, moving farther away from the origin increases the objective function x1 ⫹ 2 x2, and maximum Z would be⫹⬁. In cases where no finite maximum exists, the linear program is said to have an unbounded optimum.
It is inconceivable for a practical problem to have an unbounded solution, since this implies that one can make infinite profit from a finite amount of resources! If such a solution is obtained in a practical problem, it usually means that one or more constraints have been omitted inadvertently during the initial formulation of the problem. These constraints would have prevented the objective function from assuming infinite values.
Conclusion. In Example 4.3 the optimal solution was unique and occurred at the corner point A in the feasible region. In Example 4.4 we had multiple optimal solutions to the problem, which included two corner points B and C.
In either case, one of the corner points of the feasible region was always an optimal solution. As a matter of fact, the following property is true for any LP problem.
Property 1
If there exists an optimal solution to a linear programming problem, then at least one of the corner points of the feasible region will always qualify to be an optimal solution.
This is the fundamental property on which the simplex method for solving LP problems is based. Even though the feasible region of an LP problem contains an infinite number of points, an optimal solution can be determined by merely examining the finite number of corner points in the feasible region.
For instance, in Example 4.4 the objective function was to maximize Z⫽ x1⫹ 2 x2
The corner points of the feasible region were A (1, 0), B (10, 0), C (2, 4), D (0, 4), and E (0, 1). Evaluating their Z values, we get Z(A)⫽ 1, Z(B)⫽ 10, Z(C)⫽ 10, Z(D) ⫽ 8, and Z(E)⫽ 2. Since the maximum value of Z occurs at corner points B and C, both are optimal solutions by Property 1.