Final answer:
To solve the min-max problem using linear programming, introduce a variable 'z' to represent the maximum of the absolute differences, transform the absolute values into linear inequalities, and minimize 'z' using these inequalities as constraints.
Step-by-step explanation:
Converting a Min-Max Problem into a Linear Programming Problem
To convert the given min-max problem into a linear programming (LP) problem, consider the original objective which is to minimize the maximum of the absolute differences, min max x - 7. Let's introduce a new variable, say 'z', that represents the maximum value among the three absolute differences. Hence, the original problem is equivalent to minimizing 'z' subject to a set of constraints that these absolute values are less than or equal to 'z'.
Step-by-step Transition to a Linear Programming Problem
- Introduce the variable 'z' where z ≥ |x - 1|, z ≥ |x - 3|, and z ≥ |x - 7|.
- Since the absolute value functions create nonlinearity, we rewrite them as linear inequalities. For example, |x - 1| ≤ z can be represented as two linear inequalities: x - 1 ≤ z and -x + 1 ≤ z.
- Apply this method to all three absolute values:
- x - 1 ≤ z
- -x + 1 ≤ z
- x - 3 ≤ z
- -x + 3 ≤ z
- x - 7 ≤ z
- -x + 7 ≤ z
Now, we have a set of linear inequalities that can be used as constraints in our LP model.The objective function is simply min z.To complete the LP problem, we need to specify the domain of 'x', which is given as x ∈ R (x is a real number), and the domain of 'z', which we can assume is also any real number hence, no additional constraints on 'z' are necessary besides being involved with the other constraints that tie it to x.
Final Linear Programming Formulation
The final LP model looks like this:
Minimize z
Subject to the constraints:
- x - 1 ≤ z
- -x + 1 ≤ z
- x - 3 ≤ z
- -x + 3 ≤ z
- x - 7 ≤ z
- -x + 7 ≤ z
- x, z ∈ R
This LP model can be solved using standard linear programming techniques.