### 3.7.175. Pattern sequencing

A constraint allowing for expressing the pattern sequencing problem as one single global constraint. The pattern sequencing problem [FinkVoss99] can be described as follows: given a 0 -1 matrix in which each column $j$ $\left(1\le j\le p\right)$ corresponds to a product required by the customers and each row $i$ $\left(1\le i\le c\right)$ corresponds to the order of a particular customer (The entry ${c}_{ij}$ is equal to 1 if and only if customer $i$ has ordered some quantity of product $j$.), the objective is to find a permutation of the products such that the maximum number of open orders at any point in the sequence is minimised. Order $i$ is open at point $k$ in the production sequence if there is a product required in order $i$ that appears at or before position $k$ in the sequence and also a product that appears at or after position $k$ in the sequence.