Compute overall ranking

Question

You are given a table that represents the rankings of S subjects according to a number C of different criteria. The purpose is to

• compute an overall ranking according to a weighted sum of the C ranking criteria,
• using one of those criteria (i.e. columns), T, as tie-breaker.

The rankings are a S×C table of positive integers. The weights for computing the overall ranking are given as an array of non-negative numbers.

Entry (s,c) of the table contains the ranking of subject s according to criterion c; that is the position of subject s when all S subjects are sorted according to criterion c. (Another possible interpretation, not used here, would be: row s tells which user is ranked s-th according to criterion c).

Example

Consider the following table with S=6 and C=2:

1   4
4   5
6   1
3   3
5   2
2   6

Weights are

1.5 2.5

and column T=2 is used as tie-breaker.

The overall ranking for the first subject (first row) is 1*1.5+4*2.5 = 11.5. The overall ranking array is

11.5
18.5
11.5
12
12.5
18    

The resulting ranking, which is the output, is then

2
6
1
3
4
5

Note that the first-ranked row is the third, not the first, according to the tie-breaking specification.

Input and output

Use any reasonable format for input and output. The rankings may be a 2D array of any orientation, an array of arrays, or separate arrays. Inputs may be taken in any fixed order.

Input and output rankings should be 1-based (as in the example above), because that's the natural way to speak of rankings.

Output may be a string of numbers with a fixed separator character or an array.

You can write a function or a program.

Test cases

Table, weights, tie-breaking column, output.

--- (Example above) ---

1   4
4   5
6   1
3   3
5   2
2   6

1.5 2.5

2

2
6
1
3
4
5

--- (Three columns) ---

4     4     1
2     3     5
5     1     3
3     5     2
1     2     4

0.5   1.4   0.9

1

3
4
1
5
2

--- (Only one weight is nonzero) ---

1     5
3     3
4     2
5     1
2     4

1     0

2

1
3
4
5
2

--- (Only one column) ---

2
4
1
3

1.25

1

2
4
1
3

--- (Only one row) ---

1 1 1

0.8 1.5 2

3

1    

Answer 1

Jelly, 11 10 bytes

-1 byte by @Dennis

×⁵S€żị@€ỤỤ

Uses the fact that lists are lexicographically ordered.

This is a dyadic link that takes two arguments plus one input. The left argument is x, the rankings. The right argument is y, the tiebreaker. The input is the weights.

×⁵S€żị@€ỤỤ         Dyadic function. Inputs: x, y
×⁵                 Vectorized multiply x by the weights.

  S                Sum the rows.
                   This is the weighted ordering.
   ż               Zip with
    ị@€             x indexed at y, mapped over x
       ỤỤ          and compute the inverse of the permutation vector that sorts that.
                   A list is sorted by its inverse permutation, so
                   this is the inverse of the inverse; i.e. the original permutation.

Try it here.

Answer 2

JavaScript (ES6), 114 bytes

(a,w,t)=>a.map(s=>s.reduce((p,v,c)=>p+v*w[c],0)).map((n,i,b)=>b.map((v,j)=>r+=n>v|v==n&a[j][t-1]<a[i][t-1],r=1)|r)

Explanation

Fairly straight-forward. Expects an array of rows as arrays, an array of weights as numbers, and the tie-breaker column as a number. Returns an array.

var solution =

(a,w,t)=>
  a.map(s=>                        // for each row s in the criteria table
    s.reduce((p,v,c)=>p+v*w[c],0)  // create an array of the summed weights for each row
  )
  .map((n,i,b)=>                   // for each number n at i in the summed weights array b
    b.map((v,j)=>                  // compare it to each other summed weight
      r+=n>v                       // increment it's rank if n > v
        |v==n&a[j][t-1]<a[i][t-1], // or in case of tie, check the weights in column t
      r=1                          // initialise the rank to 1
    )|r                            // return the rank
  )

<textarea id="table" rows="5" cols="40">4     4     1
2     3     5
5     1     3
3     5     2
1     2     4</textarea><br />
Weights = <input type="text" id="weights" value="0.5   1.4   0.9" /><br />
Tie-Breaker = <input type="number" id="tie" value="1" /><br />
<button onclick="result.textContent=solution(table.value.split('\n').map(x=>x.split(/\s+/).map(n=>+n)),weights.value.split(/\s+/).map(n=>+n),+tie.value)">Go</button>
<pre id="result"></pre>

Answer 3

MATL, 21 20 bytes

2#2$XSwiY*4#S)tn:tb(

Takes input as a matrix for the table of rankings, then the tie breaking column as a number, then a column vector of weightings for each column. Attempted explanation:

2#2$XS        % specifies 2 inputs/2 outputs for sortrows, takes 2 implicit inputs, 
              % and sorts the matrix of ranks based on the second column. This means
              % ties are automatically broken in the correct way
wiY*          % swap the top 2 elements in the stack, take an input, and matrix multiply
              % the (sorted) ranking table by the weighting vector to give ranking array
4#S)         % sort the ranking array, taking only the permutation vector, and use it to
              % reorder the original sortrows permutation vector
tn:           % make a vector 1:(number of rows in ranking table)
tb(           % index 1:n vector using permutation vector to give final ranking
% implicit display

Try it Online!

(Written using MATL version 11.0.3 from a few weeks ago)

The equivalent Matlab code may be easier to understand:

[f,g]=sortrows(A,c); % 2#2$XS          (A=matrix of ranks, c=tie-break column)
[~,l]=sort(f*B);     % wiY*4#S        (B=weighting vector)
t=1:numel(g);        % 
t(g(l))=t            % )tn:tb(   

Answer 4

MATL, 16 bytes

Y*2GiY)v!4#XS4#S

Input is of the form: weights; table with each ranking on a different row; tie-breaker. Foe example,

[1.5 2.5]
[1 4 6 3 5 2; 4 5 1 3 2 6]
2

Try it online! First test case, second, third, fourth, fifth.

Y*         % implicitly input ranking table and weights. Matrix multiply, to produce
           % combined ranking as a row vector
2G         % push ranking table again
i          % input tie-breaker
Y)         % get that ranking from the table (as a row)
v!         % join combined ranking and tie-breaker column. Produce 2-col matrix
4#XS       % sort rows by first column (combined ranking) and then second
           % column (tie-breaker). Produce not the sorted matrix, but the indices
           % of the sorting, as a column vector
4#S        % sort that vector and get the indices of the sorting. Implicitly display