Exercise 3 Work through the simplex method (in algebraic form) step by step to solve the following problem max 4x + 3y + 6z subject to 3x + y + 3z ≤ 30 2x + 2y + 3z ≤ 40 x, y, z ≥ 0.
linear programing question
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The simplex method is a widely used algorithm for solving linear programming problems. It involves transforming the problem into a canonical form, then iteratively improving the solution by pivoting through the vertices of the feasible region until the optimal solution is found. Here are the steps for solving the given problem using the simplex method in algebraic form:
Step 1: Convert the problem into standard form by introducing slack variables to convert the inequality constraints to equalities:
maximize 4x + 3y + 6z
subject to
3x + y + 3z + s1 = 30
2x + 2y + 3z + s2 = 40
x, y, z, s1, s2 >= 0
Step 2: Write the problem in tableau form by setting up a table with the coefficients of the variables and the constants in the objective function and the constraints, as well as the slack variables:
x y z s1 s2 b
-4 | -3 | -6 | 0 | 0 | 0
3 | 1 | 3 | 1 | 0 | 30
2 | 2 | 3 | 0 | 1 | 40
The columns represent the variables (including the slack variables), and the rows represent the constraints. The last column represents the right-hand side of each constraint.
Step 3: Choose the entering variable by selecting the most negative coefficient in the objective row. In this case, the most negative coefficient is -4, which corresponds to x. Therefore, x is the entering variable.
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