A problem on the uniform assignment of jobs and its generalisation

V. A. Kropanov ; V. S. Rublev

Diskretnaya Matematika, Tome 13 (2001) no. 4, pp. 144-156

Résumé

We consider the following assignment problem. During $m$ days $n$ worker should fulfil some amount of jobs, which is given by the vector $S=\{s_1,\ldots,s_m\}$ of the numbers of jobs for each day. The possibilities of each worker for each day are given by a $(0, 1)$-matrix $R$ of size $n\times m$, the prescribed assignments are given by a matrix $A$, the cost of jobs is given by a matrix $C$, and the desirable numbers of jobs for each worker is given by a vector $H=\{h_1,\ldots,h_n\}$. It is required to construct a matrix $X$ of assignments of jobs to workers such that the prescribed assignments are fulfilled and the matrix minimises the functional of uniformity, that is, the least square deviation of the assigned numbers of jobs from the desirable numbers of jobs for all workers provided that it minimises the cost of jobs $\sum_{i,j}^{n,m}c_{ij}x_{ij}$. We find a characteristic property of the uniform assignment, and with the use of this property construct an algorithm for solving the problem.