`"weight"_i = (1/ "n" ) sum_"i =1"^ "n" (1/(i))`

Enter a value for all fields

The Rank Ordered Centroid equation computes some number, n, which is a value that estimates the distance between adjacent ranks on a normalized scale e.g. running from 0 to 1. Moreover, this is used in modeling human decision logic.

INSTRUCTIONS: Enter the following:

• (n) Number of Criteria  the number of decision components being evaluated
• Rounding Mode - rounding mode for rounding to the specified precision
• Precision - number of decimal places to be used for resultant weights

Rank Ordered Centroid (ROC): The centroid weight is returned as a real number.

The Math / Science

This equation is often used to generate a modeled relative ranking between a set of criteria.  It is used when there is no data, time, or resources to do an exhaustive comparison of the criteria.  This method has been found to generate a distribution of criteria weightings that is statistically comparable to that generated by teams or panels of subject matter experts performing various algorithmic approaches to defining relative weights.

To use the equation you simply order the criteria from 1 through n with no regard for how much each criteria might differ in weight pairwise from any other criteria.  If you have seven criteria for example you just have to order them 1 through 7 in order of highest criteria importance.

Then you generate the Rank Order Centroid sequence of weights for the input number 7.  The resulting sequence of weights are then assigned to the ordered criteria.

Rank-Ordered Centroid (ROC) is a simple method from multi-criteria decision making (MCDM) used to convert a ranked list of criteria into a set of numerical weights, when you know the order of importance but not exact weights.

What problem does ROC solve?

Often, decision-makers can rank criteria (most important → least important) but cannot confidently assign precise weights. ROC provides a reasonable, theoretically grounded approximation of weights based only on the ranking.

How it works

Assume you have n criteria, ranked from:

• Rank 1 = most important
• Rank n = least important

The formula for Rank Ordered Centroid weight is:

`wk = 1/n * sum_(i=k)^n (1/i)`

where:

• wk = Ranked Ordered Centroid Weight
• n =​ Total number of criteria bing ranked.
• k = rank position of the criteria whose weight is being calculated.  (e.g., 1 being most important and n being the least important)
• i = sumation index (dummy variabl)

Example

Suppose you have 4 criteria ranked in importance:

1. Cost
2. Quality
3. Delivery time
4. Sustainability

Compute weights:

• `w1=1/4*(1+1/2+1/3+1/4)=0.521`
• `w2=1/4*(1/2+1/3+1/4)=0.27`
• `w3=1/4*(1/3+1/4)=0.146`
• `w4=1/4 * (1/4)=0.063`

NOTE: Weights sum to 1.

ROC represents the center (centroid) of all possible weight vectors that preserve the given ranking and sum to 1. It is the expected value of weights assuming no other information.

When to use ROC

• You only know rankings, not exact weights
• You want a defensible and unbiased weighting scheme
• Early-stage analysis or expert elicitation
• Common in:
  • Engineering design
  • Project selection
  • Policy analysis
  • Operations research

Rank-Ordered Centroid converts ordinal rankings into rational numerical weights without forcing subjective precision—making it a popular and robust choice in decision analysis.