文摘
We consider the problem of scheduling a set of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15005089&_mathId=si1.gif&_user=111111111&_pii=S0166218X15005089&_rdoc=1&_issn=0166218X&md5=88d44bd9b9f7be443caedc9ea6d5a2de" title="Click to view the MathML source">nclass="mathContainer hidden">class="mathCode"> jobs on a single machine with parallel batching and with rejection being allowed. Two bi-criteria problems are considered: (a) minimize the makespan subject to the constraint that the total rejection cost does not exceed a given threshold, and (b) minimize the total rejection cost subject to the constraint that the makespan does not exceed a given threshold. For the case of a batching machine with infinite capacity (i.e., the batch size allowed on the machine is larger than or equal to the number of jobs), we assume that the jobs have release dates. We present an class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15005089&_mathId=si2.gif&_user=111111111&_pii=S0166218X15005089&_rdoc=1&_issn=0166218X&md5=bc45b46dbb50661facc901c6d3152600" title="Click to view the MathML source">O(n2)class="mathContainer hidden">class="mathCode">-time 2-approximation algorithm for problem (a) and, in addition, we present dynamic programming algorithms and fully polynomial-time approximation schemes for both problems (a) and (b). For the case of a batching machine with finite capacity (i.e., the batch size allowed on the machine is less than the number of jobs), we assume that the jobs have identical release dates. We propose approximation algorithms for (a) and present dynamic programming algorithms and fully polynomial-time approximation schemes for both problems (a) and (b).