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A. Strusevich and K. 1007/978-3-319-39574-6_3 37 38 3 Sequencing Under Precedence Constraints For some schedule S, the completion time of job j ∈ N is denoted by C j (S). If a schedule S is defined by a permutation π = (π(1), π(2), . . , π(n)) of jobs, we may denote the completion time of job j by C j (π). If no confusion arises, we may write C j , without making a reference to a particular schedule or sequence. The objective is to minimize a function (π), a non-decreasing function of C j , j ∈ N .

The link between problem Pm| | C j and the low-risk multiflight helicopter transportation problem is established by Qian et al. (2015). That paper presents a number of approximation algorithms for the capacitated version of the problem with pickup only. While problem 1| | w j C j is polynomially solvable, adding an additional machine changes the complexity dramatically. Indeed, problem P2| | w j C j is NP-hard even if p j = w j for all j ∈ N (see Bruno et al. (1974)). 36 2 Pairwise Interchange Argument and Priority Rules References Brucker P (2007) Scheduling algorithms, 5th edn.

N m , so that the jobs of set Ni and only those are assigned to be processed on machine Mi , 1 ≤ i ≤ m; • the sequence of jobs π [i] = π [i] (1), π [i] (2), . . , π [i] (h i ) on machine Mi , where h i = |Ni | and 1 ≤ i ≤ m. Take a machine Mi , 1 ≤ i ≤ m, and suppose that in some schedule S, the jobs π [i] (1), . . , π [i] (h i ) are processed on Mi in this order. We have that Cπ[i] (1) = pπ[i] (1) /si , Cπ[i] (2) = pπ[i] (1) /si + pπ[i] (2) /si , ... Cπ[i] (h i ) = pπ[i] (1) /si + . . + pπ[i] (h i ) /si .

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