# An approximation algorithm for the two-machine flow shop with a single transporter

Soper, Alan and Strusevich, Vitaly A.
(2007)
*An approximation algorithm for the two-machine flow shop with a single transporter.*
International Journal of Foundations of Computer Science (ijfcs), 18 (3).
pp. 565-591.
ISSN 0129-0541
(doi:10.1142/S012905410700484X)

## Abstract

We study the two-machine flow shop problem with an uncapacitated interstage transporter. The jobs have to be split into batches, and upon completion on the first machine, each batch has to be shipped to the second machine by a transporter. The best known heuristic for the problem is a –approximation algorithm that outputs a two-shipment schedule. We design a –approximation algorithm that finds schedules with at most three shipments, and this ratio cannot be improved, unless schedules with more shipments are created. This improvement is achieved due to a thorough analysis of schedules with two and three shipments by means of linear programming. We formulate problems of finding an optimal schedule with two or three shipments as integer linear programs and develop strongly polynomial algorithms that find solutions to their continuous relaxations with a small number of fractional variables.

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