An Extension of Baranyai's Rounding Lemma Benjamin Doerr, Tobias Friedrich, Christian Klein, Ralf Osbild Max-Planck-Institut für Informatik, Saarbrücken, Germany

Baranyai (1975) Let X m n. Then there is a Y m n such that 1 (errors in rows) | )yx( | ]:m..1[i 1 (errors in columns) | )yx( | ]:n..1[j 1 (rounding)|yx|]:n..1[j],m..1[i n 1j ij m 1i

New Result 1 (errors in row intervals) | )yx( | ]:m..1[i],n..1[b 1 (errors in columns) | )yx( | ]:n..1[j 1 (rounding)|yx|]:n..1[j],m..1[i b 1j ij m 1i Such a rounding can be computed in time O(mn). Let X m n. Then there is a Y m n such that

Flexible transfer line scheduling Produce m different goods on a single machine in a balanced manner Machine can produce any good in one time unit No switch over costs... Product 1 Product 2 Product m Single machine (mixed-model assembly line)

Example time 3x Product 1 2x Product 2 1x Product 3 Time horizont: 6 time steps Total demand: 6 products Fractional demands per time step:

Steiner and Yeomans (1993) time Product 1 Product 2 Product 3 Can be solved with maximum error less than one

Our result allows… Non-constant demands ( not all columns identical) Multiple machines ( column sums larger than one) Thanks for your attention!