Where is simplex used

Connect with us. Sign up. Term of the Day. Best of Techopedia weekly. News and Special Offers occasional. Simplex Method. Techopedia Explains Simplex Method. What Does Simplex Method Mean? The simplex method, in mathematical optimization, is a well-known algorithm used for linear programming. The simplex method presents an organized strategy for evaluating a feasible region's vertices.

This helps to figure out the optimal value of the objective function. George Dantzig developed the simplex method in The method is also known as the simplex algorithm. Techopedia Explains Simplex Method The simplex method is used to eradicate the issues in linear programming. It examines the feasible set's adjacent vertices in sequence to ensure that, at every new vertex, the objective function increases or is unaffected.

In general, the simplex method is extremely powerful, which usually takes 2 m to 3 m iterations at the most here, m denotes the range of equality constraints , and it converges in anticipated polynomial time for specific distributions of random input. The simplex method uses a systematic strategy to generate and test candidate vertex solutions to a linear program. At every iteration, it chooses the variable that can make the biggest modification toward the minimum solution.

That variable then replaces one of its covariables, which is most drastically limiting it, thereby shifting the simplex method to another part of the solution set and toward the final solution. Furthermore, the simplex method is able to evaluate whether no solution actually exists. In real life situations, linear programming problems consist of literally thousands of variables and are solved by computers. We can solve these problems algebraically, but that will not be very efficient.

Suppose we were given a problem with, say, 5 variables and 10 constraints. By choosing all combinations of five equations with five unknowns, we could find all the corner points, test them for feasibility, and come up with the solution, if it exists.

But the trouble is that even for a problem with so few variables, we will get more than corner points, and testing each point will be very tedious.

So we need a method that has a systematic algorithm and can be programmed for a computer. The method has to be efficient enough so we wouldn't have to evaluate the objective function at each corner point. We have just such a method, and it is called the simplex method. The simplex method was developed during the Second World War by Dr. George Dantzig. His linear programming models helped the Allied forces with transportation and scheduling problems. In , a Soviet scientist named Leonid Khachian developed a method called the ellipsoid algorithm which was supposed to be revolutionary, but as it turned out it is not any better than the simplex method.

But the simplex method still works the best for most problems. The simplex method uses an approach that is very efficient. It does not compute the value of the objective function at every point; instead, it begins with a corner point of the feasibility region where all the main variables are zero and then systematically moves from corner point to corner point, while improving the value of the objective function at each stage.

The process continues until the optimal solution is found. To learn the simplex method, we try a rather unconventional approach. We first list the algorithm, and then work a problem. We justify the reasoning behind each step during the process. A thorough justification is beyond the scope of this course. We start out with an example we solved in the last chapter by the graphical method.

This will provide us with some insight into the simplex method and at the same time give us the chance to compare a few of the feasible solutions we obtained previously by the graphical method.

But first, we list the algorithm for the simplex method. Now, we use the simplex method to solve Example 3. She never wants to work more than a total of 12 hours a week. She has determined that for every hour she works at Job I, she needs 2 hours of preparation time, and for every hour she works at Job II, she needs one hour of preparation time, and she cannot spend more than 16 hours for preparation.

STEP 1. Set up the problem. Write the objective function and the constraints. STEP 2. Convert the inequalities into equations. This is done by adding one slack variable for each inequality. Later when we read off the final solution from the simplex table, the values of the slack variables will identify the unused amounts.

STEP 3. Construct the initial simplex tableau. Each inequality constraint appears in its own row. The non-negativity constraints do not appear as rows in the simplex tableau. Write the objective function as the bottom row. Now that the inequalities are converted into equations, we can represent the problem into an augmented matrix called the initial simplex tableau as follows.

Here the vertical line separates the left hand side of the equations from the right side. The horizontal line separates the constraints from the objective function. The right side of the equation is represented by the column C.

The reader needs to observe that the last four columns of this matrix look like the final matrix for the solution of a system of equations. The solution obtained by arbitrarily assigning values to some variables and then solving for the remaining variables is called the basic solution associated with the tableau.

So the above solution is the basic solution associated with the initial simplex tableau. We can label the basic solution variable in the right of the last column as shown in the table below. STEP 4. The most negative entry in the bottom row identifies the pivot column. The most negative entry in the bottom row is ; therefore the column 1 is identified.

Question Why do we choose the most negative entry in the bottom row? Answer The most negative entry in the bottom row represents the largest coefficient in the objective function - the coefficient whose entry will increase the value of the objective function the quickest.

It then moves from a corner point to the adjacent corner point always increasing the value of the objective function. STEP 5. Calculate the quotients. The smallest quotient identifies a row.