Final answer:
To convert the given minimization problem into linear programming standard form, additional variables representing the positive and negative parts of each variable's absolute value are introduced, and the objective function and constraints are rewritten accordingly. This allows for the application of linear programming techniques.
Step-by-step explanation:
The student's problem involves converting a minimization problem with absolute values into a linear programming problem in standard form. In the standard form, we need to minimize or maximize a linear objective function subject to a set of linear inequalities (constraints), and all the variables involved are non-negative. The original function to minimize is |x| + |y| + |z|, subject to the constraints x + y ≤ 1 and 2x + z = 3.
To convert the objective function to standard form, we introduce additional variables to represent the absolute values such that x1 = |x|, y1 = |y|, z1 = |z|, which we minimize instead. Since absolute values are always non-negative, we can split each into two cases, representing the positive and negative scenarios: x = x1 - x2, y = y1 - y2, z = z1 - z2 where x1, x2, y1, y2, z1, z2 ≥ 0.
The objective function thus can be rewritten as minimize x1 + x2 + y1 + y2 + z1 + z2. The constraints become x + y ≤ 1 and 2x + z = 3, rewritten using our new variables as (x1 - x2) + (y1 - y2) ≤ 1 and 2(x1 - x2) + (z1 - z2) = 3, with the additional constraints x1, x2, y1, y2, z1, z2 ≥ 0 to ensure non-negativity.
This transformation enables us to use linear programming techniques to find the values of x, y, z that minimize the original objective function while satisfying the given constraints.