Maximizing total early work in a distributed two‐machine flow‐shop

The problem of maximizing total early work in a two‐machine flow‐shop, in which n jobs are to be scheduled subject to a common due date d, has been recently studied in the scheduling literature. An O(n2d4) time dynamic programming algorithm was presented first for the weighted case, and then for the unweighted case another O(n2d2) running time dynamic programming algorithm was proposed and converted into an On4ε2$$ O\left(\frac{n^4}{\varepsilon^2}\right) $$ time fully polynomial time approximation scheme (FPTAS). By establishing new problem properties, we present an O(nd2) time dynamic programming algorithm and an On3ε2$$ O\left(\frac{n^3}{\varepsilon^2}\right) $$ time FPTAS for the unweighted problem. We generalize the problem to a distributed setting of m parallel two‐machine flow‐shops, develop an O(nd3m) time dynamic programming algorithm, an On3m+1ε3m$$ O\left(\frac{n^{3m+1}}{\varepsilon^{3m}}\right) $$ time FPTAS, and three integer linear programming (ILP) formulations for it. Computational experiments are conducted to appraise the proposed ILP models.