Nonstrict vector simulation in multi-operation scheduling

We consider several multi??operation scheduling problems with m machines and n jobs??, including fl??ow shop??, open shop,?? assembly line,?? and a few special cases of job shop with the makespan criterion. It is demonstrated that the problems in question can be effi??ciently solved by approximation algorithms with fairly good performance guaranteed in worst case. The algorithms constructed are based on a geometric technique called "??nonstrict vector summation".?? It consists of assigning an (m-1)-????dimensional vector to each job and then ??finding an order in which the resulting vectors should be summed so that all partial sums lied within half??-spaces of the best possible family (with respect to a certain objective function).?? The partial sums are allowed sometimes go out of this or that half-??space of the family,?? which explains the term "??nonstrict??" in the title of the paper. For the open shop problem this technique guarantees its polynomial??-time solution,?? provided that the maximum machine load (lmax)?? is large enough. In the case of three machines and lmax as large as at least 7 times the maxi??mum processing time of an operation,?? we can ??find the optimal schedule in O(n logn??) time. Keywords:?? open??-shop,?? fl??ow shop,?? job shop,?? scheduling,?? polynomial-??time approximation algorithms??, sequencing of vectors.