How can produce this matrix?

I want to produce a m×n matrix for example like this: x=[1 1 1 0 1 0 1 1; 0 0 0 1 0 1 0 0]  This matrix can give 0 or 1 for its elements. This 0 and 1 elements must be randomly distributed so that sum of each column is equal to one. I have a value like "k" that sum of the fisrt row can be equal or less than k. In defining the above matrix I also want to consider the following conditions: if the sum of the first row is less than k the all elements in the first row must be consider to 1 and all elements in the other rows must be zero. Otherwise, if the sum of the first row is equal to k the first row elements can be 0 or 1 but remind that the sum of each column must be 1.

[Dyuman Joshi]

What if the sum of the first row is greater than k?

[S AsZ]

The sum of first row can not be greater than k because this matrix is related to an optimization variable which changes between 0 to k

[Dyuman Joshi]

But what if it is? What should be the output then?

Should the function generate a new matrix in that case?

[S AsZ]

Acctually I want to code the below constraints in an optimization problem with the solver-based optimization approach in matlab. But I do not know how to code this constraints. In the following equations "U" denotes a binary variable, "Cd_k" is a determined parameter with one row and k columns (1×k), "Qdc_kl" is a continuous variable with k rows and l columns in which in each column just one element can be positive and the others are zero. Finally, "Y_k" is a binary variable wich is equal or less than to U. Actually, I am trying to code this constraints but I do not know how can define this constraints with solver-based optimization of matlab 😕 Can you help me?

[S AsZ]

So I tried to define "Qdc_kl" as a matrix in which sum of each column is equal to 1 to ensure that each column has only one positive value and the other values are equal to zero and also I wanted to ensure that the sum of each rows are less than or equal to "Cd_k"

Answer 1

[Dyuman Joshi]

M = 3; N = 7;

k=2;

arr = accumarray([randi(M,1,N); (1:N)].', 1, [M N])

arr = 3×7

1 1 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1 0 0

if sum(arr(1,:))<k
    arr = [ones(1,N);zeros(M-1,N)];
end
arr

arr = 3×7

1 1 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1 0 0

Answer 2

[Walter Roberson]

For the sum of each column to be 1, and the elements are each 0 or 1, then it follows that for every column, there must be exactly 1 row that is set to 1. You can use randi() to select the rows.

M = 5; N = 4;
matrix = zeros(M, N);
matrix(sub2ind([M,N], randi(M, N, 1), (1:N).' )) = 1;

matrix

matrix = 5×4

0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0