Preference-inspired co-evolutionary algorithms (PICEAs) are a new class of approaches which have been demonstrated to perform well on multi-objective problems (MOPs). The good performance of PICEAs is largely due to its clever fitness calculation method which is in a competitive co-evolutionary way. However, this fitness calculation method has a potential limitation. In this work, we analyze this limitation and propose to implement PICEAs within a local structure (LPICEAs). By using the local structure, the benefits of local operations are incorporated into PICEAs. Meanwhile, the limitation of the original fitness calculation method is solved. In details, the candidate solutions are firstly partitioned into several clusters according to a clustering technique. Then the evolutionary operations, i.e. selection-for-survival and genetic-variation are executed on each cluster, separately. To validate the performance of LPICEAs, LPICEAs are compared to PICEAs on some benchmarks functions. Experimental results indicate LPICEAs significantly outperform PICEAs on most of the benchmarks. Moreover, the influence of LPICEAs to the tuning of the parameter k, i.e. the number of clusters used in LPICEAs is studied. The results indicate that the performance of LPICEAs is sensitive to the parameter k.