Finding optimally balanced words for production planning and maintenance scheduling

Abstract

Balanced words are useful for scheduling mixed-model, just-in-time assembly lines, planning preventive maintenance, managing inventory, and controlling asynchronous transfer mode (ATM) networks. This paper considers the challenging problem of finding a balanced word (a periodic sequence) for a finite set of letters, when the desired densities of the letters in the alphabet are given.Two different measures of balance are considered. This paper presents a branch-and-bound approach for finding optimally balanced words and presents the results of computational experiments to show how problem characteristics affect the time required to find an optimal solution. The optimal solutions are also used to evaluate the performance of an aggregation approach that combines letters with the same density, constructs a word for the aggregated alphabet, and then disaggregates this word into a feasible word for the original alphabet. Computational experiments show that using aggregation with the heuristics not only finds more balanced words but also reduces computational effort for larger instances.

Do balanced words have a short period?

“…Indeed, for this example, the density of W A,e is 6 11 , 3 11 , 1 11 , 1 11 , with the last two densities (of a and b) being equal. By identifying the letters a and b, however, one obtains the density (of W ) 6 11 , 3 11 , 2 11 , and there is no balanced word with these densities.…”

“…If CAD is followed by A, then W = (CADA) * with density vector ( 8 14 , 4 14 , 1 14 , 1 14 ). If CAD is followed by C then W = (DCA) * with density vector ( 6 11 , 3 11 , 1 11 , 1 11 ). If W contains no CAD, then it contains DAC and the same reasonning applies.…”

“…Balanced sequences and balanced words have attracted the attention of many researchers in discrete mathematics or number theory, but also in application fields like scheduling, maintenance, queueing, or apportionment (see, e.g., [1,6,3,8,9]). Yet, in spite of this wealth of literature, balanced words and their properties are still poorly understood.…”