LINEAR PROGRAMMING:AN ALGEBRAIC APPROACH

Week 8

The Simplex Method

The method of corners is not suitable for solving linear programming problems when the number of variables or constraints is large. Its major shortcoming is that a knowledge of all the corner points of the feasible set S associated with the problem is required.

Thus we need to reduce the number of points to be inspected. One technique is the simplex method, which was developed in the late 1940s by George Dantzig and is based on the Gauss–Jordan elimination method.

The simplex method is readily adaptable to the computer, which makes it suitable for solving linear programming problems involving large numbers of variables and constraints.

An Iterative Method

The simplex method is an iterative procedure: it is repeated over and over again.

Beginning at some initial feasible solution (a corner point of the feasible set S, usually the origin), each iteration brings us to another corner point of S, usually with an improved (but certainly no worse) value of the objective function.

The iteration is terminated when the optimal solution is reached (if it exists).

4.1 The Simplex Method: Standard Maximization Problems

A Standard Maximization Problem

A standard maximization problem is one in which

1. The objective function is to be maximized.

2. All the variables involved in the problem are nonnegative.

3. All other linear constraints may be written so that the expression involving the variables is less than or equal to a nonnegative constant.

Example.

The optimal solution to the problem occurs at the corner point C(3, 2).

Using the Simplex Method

First replace the system of inequality constraints with a system of equality constraints.

2x + 3y 12 is converted into the equation 2x + 3y + u = 12

The variable u is called a slack variable.

2x + y 8 is converted into the equation 2x + y + v = 8

Finally, rewriting the objective function in the form -3x - 2y + P = 0.

We are led to the following system of linear equations:

From among all the solutions of the system for which x, y, u, and v are nonnegative (such solutions are called feasible solutions), determine the solution(s) that maximizes P.

The Augmented Matrix

Observe that each of the u-, v-, and P-columns of the augmented matrix (4) is a unit column. The variables associated with unit columns are called basic variables; all other variables are called non-basic variables.

The configuration of the augmented matrix suggests that we solve for the basic variables u, v, and P in terms of the non-basic variables x and y, obtaining

Basic Solution

A particular solution is obtained by letting x = 0 and y = 0. In fact, this solution is given by x = 0, y = 0, u = 12, v = 8, P = 0.Such a solution, obtained by setting the non-basic variables equal to zero, is called a basic solution of the system. This particular solution corresponds to the corner point A(0, 0) of the feasible set associated with the linear programming problem.If the value of P cannot be increased, we have found the optimal solution to the problem at hand.Continuing our quest for an optimal solution, our next task is to determine whether it is more profitable to increase the value of x or that of y (increasing x and y simultaneously is more difficult).Is it better to hold x = 0 or y = 0?

How Much Can x be Increased While Holding y = 0?Setting y = 0,

The first equation and the nonnegativity of u imply that x cannot exceed 12/2 or 6.

The second equation and the nonnegativity of v imply that x cannot exceed 8/2 or 4. Thus, we conclude that x can be increased by at most 4.

Now, if we set y = 0 and x = 4, we obtain the solution

x = 4, y = 0, u = 4, v = 0, P = 12

which is a basic solution to system, this time with y and v as non-basic variables.

We have to find an augmented matrix that is equivalent to the matrix and has a configuration in which the x-column is in the unit form

Pivot

Continuing

Consider the pivot column, use row operations to change the pivot element into 1and non-pivot elements into 0s.

Continue to find the next pivot column, pivot row, and pivot element with similar procedure as before.

Then, again, change the pivot element into 1 and non-pivot elements into 0s.

We stop the iteration when there are no negative entries in the last row, which means the solution is optimal and P cannot be increased further.

Simplex Method First Step: Set Up Initial Simplex Tableu

Example

Set up the initial simplex tableau for the linear programming problem

The Simplex Method

Example

Applied Example

Problems with Multiple Solutions and Problems with No SolutionsA linear programming problem will have infinitely many solutions if and only if the last row to the left of the vertical line of the final simplex tableau has a zero in a column that is not a unit column.

A linear programming problem will have no solution if the simplex method breaks down at some stage.

For example, if at some stage there are no nonnegative ratios in our computation, then the linear programming problem has no solution.

4.2 The Simplex Method: Standard Minimization Problems

Minimization with Constraints

By changing the problem to a standard maximization problem which satisfies three conditions:

1. The objective function is to be maximized.

2. All the variables involved are nonnegative.

3. Each linear constraint may be written so that the expression involving the variables is less than or equal to a nonnegative constant.

Example.

Minimizing the objective function C is equivalent to maximizing the objective function P = -C.

Maximize P = 2x + 3y subject to the given constraints.

Standard Minimization Problems

A class of linear programming problems which is characterized by the following conditions:

1. The objective function is to be minimized.

2. All the variables involved are nonnegative.

3. All other linear constraints may be written so that the expression involving the variables is greater than or equal to a constant.

Each maximization linear programming problem is associated with a minimization problem, and vice versa. For the purpose of identification, the given problem is called the primal problem; the problem related to it is called the dual problem.

Example

Write the dual problem associated with the following problem:

First, write the tableu for the primal problem:

Next, we interchange the columns and rows of the foregoing tableau and head the two columns of the resulting array with the two variables u and √, obtaining the tableau:

.

Primal Problem vs Dual Problem

Simplex Method for Minimization

Duality

Example

Assignment

Solve the following problems in a group of at most 2 persons.

1. Read Tan textbook page 228-231: solving linear programming with Excel. Then solve problems 1 and 4 in page 231 using Excel.

2. Read Tan textbook page 246-248: solving linear programming with Excel. Then solve problems 6 and 8 in page 248 using Excel.

Submit 4 excel files for each problem to ma1203itb@gmail.com.

The submission due on Sunday, 26 February 2017 at 23.00.