Lecture 5: Inconsistent Information

Question 1

Over-determined system of equations

Answer 1

Total number of indifference statements > total number of objectives
-> Over-determined system of equations if no redundancies

Question 2

Redundancies

Answer 2

Two equations are redundant if they yield the same equation -> Test whether you can express one equation as a lines combination of the others

Question 3

Error Minimization Approach

Answer 3

1. Interpret the indifference statements as random draws from a distribution around the true value
2. Introduce an error term e for every equation such that f_i = e_i (before f_i = 0; E.g. 0,4 w1 - 0,5 w2 = 0) -> Leads to an under-determined system of equations
3. Search for the objective weights that lead to the lowest values of the absolute error terms (not a linear programming problem!)
4. Rewrite the constraint by replacing e_i with e_i(+) - e_i(-) (to make it a linear problem)
5. Solve for the weights that minimise the sum of absolute errors [constraints: sum of weights = 1, weights > 0, e_i(+) & e_i(-) > 0]

Question 4

Linear Programming

Answer 4

• Mathematical optimization model

- Describes the decision problem in terms of three elements: Decision variables, Objective function and Constraints

Question 5

Why is the problem not linear?

Answer 5

• Objective function and constraint have to be linear in the decision variables
• The problem can be transformed into a linear problem by writing e_i as the difference of two non-negative variables e_i = e_i(+) - e_i(-)
• The absolute terms can be simplified whenever either e_i(+) or e_i(-) equals zero -> |e_i(+) - e_i(-)| = |e_i(+)| + |e_i(-)| = e_i(+) + e_i(-)