When difference in machine loads leadsto efficient scheduling in open shops

Abstract

We consider the open shop problem with n jobs, mmachines, and the minimum makespan criterion. Let l _i stand for the loadof the ith machine, l _max be the maximum machine load,and p _max be the maximum operation length. Suppose that the machines arenumbered in nonincreasing order of their loads and that p _max = 1, whileother processing times are numbers in the interval [0,1]. Then, given aninstance of the open shop problem, we define its vector of differences$VOD = \left( {\Delta \left( 1 \right), \ldots ,\Delta \left( m \right)} \right)$, where $\Delta \left( i \right) = l_{\max } - l_i$.An instance is called normal if its optimal schedule has length l _max.A vector Δ ∈ ℝ^m is called normalizing if every instancewith VOD = Δ is normal. A vector Δ ∈ ℝ^mis called efficiently normalizing if it is normalizing and there is a polynomial‐timealgorithm which for any instance with VOD = Δ constructs itsoptimal schedule. In this paper, a few nontrivial classes of efficiently normalizingvectors are found in ℝ^m. It is also shown that the vector$\left( {0,0,2} \right)$ is the unique minimal normalizingvector in ℝ^m, and that there are at least two minimal normalizingvectors in ℝ^m.