Stands for Technique for Order of
Preference by Similarity to Ideal Solution, a
multi criteria analysis (MCA) method
based on the concept that best alternative is the one that is the shortest
"distance" from the positive ideal solution and the greatest
distance from the negative ideal solution. The positive ideal
solution is a hypothetical alternative that achieves the most desirable
levels with respect to each criterion across the options under
consideration. The ideal negative solution is defined in an analogous way,
it achieves the least desirable levels for any alternative with respect to
each criterion. TOPSIS ranks the alternatives based on a mathematical
concept of distance, the geometric distance in euclidian space from the
positive and negative ideal solutions.
TOPSIS assumes that all criteria are either monotonically increasing or
monotonically decreasing. In addition, TOPSIS assumes the criteria are
additive independent. TOPSIS also assumes that the performance of each
alternative with respect to each criterion has been established.
The steps for applying TOPSIS are as follows.
Step 1 is to construct the decision matrix. If there are M
alternatives and N criteria, each cell in the M by N matrix is the
estimated performance, x_{ij}, of the i'th alternative with
respect to the j'th criterion.
Step 2 is to create a normalized matrix
r_{ij} using a variation of the usual normalization approach.
Rather than convert the performance with respect to each
criterion to the fraction of the difference between the maximum and minimum
performances obtained for that criterion, TOPSIS normalizes each
alternative performance with respect to each criterion using the square
root of the sum of the squares of the performances achieved by any
alternative relative to that criterion. The normalization allows
performances with respect to the different criteria to be compared.
Step 3 is to construct the weighted normalized
decision matrix. Assume we have a set of weights for each criteria
w_{j} for j = 1,…N. Multiply each column of the
normalized decision matrix by its associated weight. An element of the new
matrix is: v_{ij} = w_{j} ×
r_{ij}
Step 4 is to determine the positive ideal and
negative ideal solutions. The positive ideal solution is denoted
where v_{1}^{+} is the maximum
normalized score achieved by any alternative for the first criterion (the
maximum value in the first column of the normalized decision matrix),
v_{2}^{+} is the maximum value achieved by any
alternative for the second criterion, and so forth. The negative ideal
solution,
where v_{1}^{} is the minimum
normalized score achieved by any alternative for the first criterion,
v_{2}^{} is the minimum value achieved by any
alternative for the second criterion, and so forth.
Step 5 is to calculate the separation distances of
each alternative from the positive ideal and negative ideal alternatives.
The distances for the i'th alternative are computed using a distance
concept referred to as Euclidian:
Step 6 is to measure the relative closeness of
each location to the positive and negative ideal alternatives, which is
computed as:
As shown, the measure of relative closeness is a
normalized score defined on the interval between the distance to the
positive ideal alternative and the distance to the negative ideal
alternative.
Step 7 is to rank the alternatives using the
relative closeness score, the higher the value of the relative closeness,
the higher the ranking.
The attractiveness of TOPSIS comes from its simplicity
and its ability to maintain the same steps regardless of problem size.
However, its use of just two reference points can create problems,
especially for problems for which the performance measures are nonlinear
problems.
