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SIMPLEX METHOD
Basic Properties
The simplex method is based on the following fundamental properties:
Property 1: The collection of feasible solutions constitutes a convex set.
Property 2. If a feasible solution exists, a basic feasible solution exists where the basic feasible solution corresponds to the extreme points (corner points) of the set of feasible solutions.
Property 3. There exist only a finite number of basic feasible solutions.
Property 4. If the objective function possesses a finite maximum or minimum, then at least one optimal solution is a basic feasible solution.
These properties can be easily verified for their plausibility with reference to the graphical representation. The reason for these properties can be attributed to the complete linearity of the linear programming model. We shall clarify the hypothesis of properties 2 and 4. The linear programming problem need not necessarily have any feasible solution. This will be so when the constraints have inconsistencies or contradictions.
In the above example the third constraint x + y < 60 is replaced by, say, x + y > 100. Then we cannot have a single solution space to offer solution space to have an infinitely high value subject to the restrictions rather than some finite number. This will be the case when in the example the second and the third constraint have been deleted.
Then the solution space and the solution are unbounded. As per this model the objective function will have Z = + as the feasible solution. Suppose there exists a feasible solution and that the optimal value of Z is finite, properties 2 and 3 indicate that the number of basic feasible solutions is strictly positive and finite. From the property 4 we infer that only this finite number of solutions namely the basic feasible solutions or the extreme points are to be investigated to find an optimal solution. Hence even though there exists an infinite number of feasible solutions, it is enough to find the value of the objective function for the basic feasible solutions. Hence we limit to only a few of the feasible solutions. Therefore we examine the value of the objective function at each of the corner points or basic feasible solutions and select the one with the largest value of Z in a maximization problem and the smallest value of Z in a minimization problem.
If we apply the above logic to the first example the basic feasible solutions include the following with the value or Z as shown the table below:
( x, y ) (Basic feasible solution)
value of Z = 4 x + 7 y
We take the largest value of Z as the optimum denoted by Z* = 330 and the values of the decision variables as x = 30, y = 30.
One more interesting fact can be inferred from the property 4 that an optimal solution need not be a basic feasible solution. This can take place if a number of feasible solutions yield the same maximum feasible value of Z , since the property 4 guarantees only that at least one of these will be a basic feasible solution. To illustrate, suppose that the objective function in the above example is changed to Z = 4 x + 4 y. Then not only the two basic feasible solutions (30, 30) , (40, 20) but also all the non-basic feasible solutions that lie on the line segment between the two points, numbering to infinity, would have been optimal solutions.
Simplex Procedure
The simplex method is an algebraic procedure involving a well-defined iterative process, which leads progressively to an optimal solution in a few numbers of finite steps. Dantzig introduced the method in 1947 and even today this seems to be the most versatile and powerful tool to solve linear programming problems. For routine linear programming problems, computer solution packages have been developed to use in an electronic digital computer. In the next section we describe what a simplex method does and how it solves a linear programming problem.
Example
Consider Z = 3 x + 5 y
Subject to x < 40, y < 30, x + y < 60, & x, y > 0.
First step in the simplex method is to convert the linear programming model involving inequalities into strict equalities, by the use of "slack variables". To illustrate the above idea, suppose we have the inequality x < 40. How can one replace this in equation by an equivalent equation? This inequality x < 40 implies the meaning that x can take a value of 40 or less. If the value of x is exactly 40, then this is the required equation. But since it also tells that x can take a value less than 40, it is necessary to add some positive value to x to make it up to 40. This additional non-negative variable is called the slack variable denoted by S 1. We can then write
x + S 1 = 40
so that the above is an equation. This has been achieved by the addition of a slack variable S 1 to the left hand side of the in equation, which can take a value between 0 and 40 both inclusive. If S 1 = 0, then x = 40 and if S 1 = 40, then x = 0. Thus the slack variable is strictly non-negative.
So we conclude that the addition of a non-negative variable called slack variable converts the 'less than equal to' constraint into strict 'equality constraint'.
The second inequality, y < 30 can also be converted into an equation by adding another slack variable S 2 to the left hand side. Thus we have
y + S 2 = 30
Similarly the third constraint x + y < 60 is also converted with the addition of another slack variable S 3 so that we have,
