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1 Algebraic Simplex Method - Introduction To demonstrate the simplex method, consider the following linear programming model: This is the model for Leo Coco's problem presented in the demo, Graphical Method. That demo describes how to find the optimal solution graphically, as displayed on the right. Thus the optimal solution is,, and. We will now describe how the simplex method (an algebraic procedure) obtains this solution algebraically.

2 Algebraic Simplex Method - Formulation The Simplex Formulation To solve this model, the simplex method needs a system of equations instead of inequalities for the functional constraints. The demo, Interpretation of Slack Variables, describes how this system of equations is obtained by introducing nonnegative slack variables, and. The resulting equivalent form of the model is The simplex method begins by focusing on equations (1) and (2) above.

3 Algebraic Simplex Method - Initial Solution The Initial Solution Consider the initial system of equations exhibited above. Equations (1) and (2) include two more variables than equations. Therefore, two of the variables (the nonbasic variables) can be arbitrarily assigned a value of zero in order to obtain a specific solution (the basic solution) for the other two variables (the basic variables). This basic solution will be feasible if the value of each basic variable is nonnegative. The best of the basic feasible solutions is known to be an optimal solution, so the simplex method finds a sequence of better and better basic feasible solutions until it finds the best one. To begin the simplex method, choose the slack variables to be the basic variables, so and are the nonbasic variables to set equal to zero. The values of and now can be obtained from the system of equations. The resulting basic feasible solution is,,, and. Is this solution optimal?

4 Algebraic Simplex Method - Checking Optimality Checking for Optimality To test whether the solution,,, and is optimal, we rewrite equation (0) as Since both and have positive coefficients, can be increased by increasing either one of these variables. Therefore, the current basic feasible solution is not optimal, so we need to perform an iteration of the simplex method to obtain a better basic feasible solution. This begins by choosing the entering basic variable (the nonbasic variable chosen to become a basic variable for the next basic feasible solution).

5 Algebraic Simplex Method - Entering Basic Variable Selecting an Entering Basic Variable The entering basic variable is: Why? Again rewrite equation (0) as. The value of the entering basic variable will be increased from 0. Since has the largest positive coefficient, increasing will increase at the fastest rate. So select. This selection rule tends to minimize the number of iterations needed to reach an optimal solution. You'll see later that this particular problem is an exception where this rule does not minimize the number of iterations.

6 Algebraic Simplex Method - Leaving Basic Variable Selecting a Leaving Basic Variable The entering basic variable is: The leaving basic variable is: Why? Choose the basic variable that reaches zero first as the entering basic variable ( ) is increased (watch increase). when. What if we increase until?

7 Algebraic Simplex Method - Leaving Basic Variable Selecting a Leaving Basic Variable The entering basic variable is: The leaving basic variable is: Why? Choose the basic variable that reaches zero first as the entering basic variable ( ) is increased. when. What if we increase until (watch increase)? when. However is now negative, resulting in an infeasible solution. Therefore, cannot be the leaving basic variable.

8 Algebraic Simplex Method - Gaussian Elimination Scaling the Pivot Row In order to determine the new basic feasible solution, we need to convert the system of equations into proper form from Gaussian elimination. The coefficient of the entering basic variable ( ) in the equation of the leaving basic variable (equation (1)) must be 1. The current value of this coefficient is: 1 Therefore, nothing needs to be done to this equation.

9 Algebraic Simplex Method - Gaussian Elimination Eliminating from the Other Equations, we need to obtain a coefficient of zero for the entering basic variable ( ) in every other equation (equations (0) and (2)). The coefficient of in equation (0) is: -20 To obtain a coefficient of 0 we need to: Add 20 times equation (1) to equation (0). The coefficient of in equation (2) is: 3 Therefore, to obtain a coefficient of 0 we need to: Subtract 3 times equation (1) from equation (2).

10 Algebraic Simplex Method - Checking Optimality Checking for Optimality The new basic feasible solution is,,, and, which yields. This ends iteration 1. Is the current solution optimal? No. Why? Rewrite equation (0) as. Since has a positive coefficient, increasing from zero will increase. So the current basic feasible solution is not optimal.

11 Tabular Simplex Method - Introduction To demonstrate the simplex method in tabular form, consider the following linear programming model: This is the same problem used to demonstrate the simplex method in algebraic form (see the demo The Simplex Method - Algebraic Form), which yielded the optimal solution (, ) = (0, 7), as shown to the right.

12 Tabular Simplex Method - Initial Tableau Using the Tableau After introducing slack variables (, ), etc., the initial tableau is as shown above. Choose the slack variables to be basic variables, so and are the nonbasic variables to be set to zero. The values of and can now be obtained from the right-hand side column of the simplex tableau. The resulting basic feasible solution is,,, and.

13 Tabular Simplex Method - Entering Basic Variable Selecting an Entering Basic Variable Ahe entering basic variable is: Why? This is the variable that has the largest (in absolute value) negative coefficient in row 0 (the equation (0) row).

14 Tabular Simplex Method - Leaving Basic Variable Selecting a Leaving Basic Variable The entering basic variable is: The leaving basic variable is: Why? Apply the minimum ratio test as shown above.

15 Tabular Simplex Method - Gaussian Elimination Scaling the Pivot Row Although it is not needed in this case, the pivot row is normally divided by the pivot number.

16 Tabular Simplex Method - Gaussian Elimination Eliminating from the Other Rows Add 20 times row 1 to row 0. Subtract 3 times row 1 from row 2. Why? To obtain a new value of zero for the coefficient of the entering basic variable in every other row of the simplex tableau.

17 Tabular Simplex Method - Gaussian Elimination Eliminating from the Other Rows Add 20 times row 1 to row 0. Subtract 3 times row 1 from row 2. Why? To obtain a new value of zero for the coefficient of the entering basic variable in every other row of the simplex tableau.

18 Tabular Simplex Method - Checking Optimality Checking for Optimality This ends iteration 1. Is the current basic feasible solution optimal? No. Why? Because row 0 includes at least one negative coefficient.

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