On the Hardness of the Maximum Lot Grouping Problem

We consider the problem of grouping orders into lots. The problem is modelled by a graph G=(V,E), where each node ${\nu}{\in}V$ denotes order specification and its weight ${\omega}(\nu)$ the orders on hand for the specification. We can construct a lot simply from orders of single specification. For a set of nodes (specifications) ${\theta}{\subseteq}V$, if the distance of any two nodes in $\theta$ is at most d, it is also possible to make a lot using orders on the nodes. The objective is to maximize the number of lots with size exactly $\lambda$. In this paper, we prove that our problem is NP-Complete when $d=2,{\lambda}=3$ and each weight is 0 or 1. Moreover, it is also shown to be NP-Complete when $d=1,{\lambda}=3$ and each weight is 1,2 or 3.

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