The bin packing problem with divisible item sizes and rejection penalties (the BP–DR problem, for short) is defined as follows. Given a lot of bins with same capacity limitation L and a set X={x1,…,xn}\documentclass[12pt]{minimal}
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\begin{document}$$X=\{x_{1},\ldots ,x_{n}\}$$\end{document} of items with a size function s:X→Z+\documentclass[12pt]{minimal}
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\begin{document}$$s: X\rightarrow Z^{+}$$\end{document} and a penalty function p:X→R+\documentclass[12pt]{minimal}
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\begin{document}$$p:X\rightarrow R^{+}$$\end{document}, where the item sizes are divisible, i.e., either s(xi)|s(xj)\documentclass[12pt]{minimal}
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\begin{document}$$s(x_{i})|s(x_{j})$$\end{document} or s(xj)|s(xi)\documentclass[12pt]{minimal}
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\begin{document}$$s(x_{j})|s(x_{i})$$\end{document} for any two items xi\documentclass[12pt]{minimal}
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\begin{document}$$x_{i}$$\end{document} and xj\documentclass[12pt]{minimal}
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\begin{document}$$x_{j}$$\end{document} with 1≤i<j≤n\documentclass[12pt]{minimal}
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\begin{document}$$1\le i < j\le n$$\end{document}, each of these n items must be either put into a bin under the constraint that the summation of sizes of items in that bin does not exceed L, or rejected with its penalty that we must pay for. No item can be spread into more than one bin. We consider the BP–DR problem and its important variant. (1) The BP–DR problem is asked to find a subset of items such that the items in that subset can be put in some bins to satisfy the constraint mentioned-above, the objective is to minimize the number of bins used plus the summation of penalties paid for the rejected items, and (2) Given a penalty budget B∈R+\documentclass[12pt]{minimal}
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\begin{document}$$B\in R^{+}$$\end{document}, the bin packing problem with divisible item sizes and bounded rejection penalties (the BP–DBR problem, for short) is asked to find a subset of items such that the items in that subset can be put in some bins to satisfy the constraint mentioned-above and that the summation of penalties paid for the rejected items does not exceed B, the objective is to minimize the number of bins used. In this paper, we design two exact combinatorial algorithms to solve the BP–DR problem and the BP–DBR problem, respectively.
