Alzheimer's disease (AD) is a leading neurodegenerative condition and the primary cause of dementia, characterized by progressive cognitive decline and memory loss. Its progression, marked by shrinkage in the cerebral cortex, is irreversible. Numerous machine learning algorithms have been proposed for the early diagnosis of AD. However, they often struggle with the issues of noise, outliers, and class imbalance. To tackle the aforementioned limitations, in this article, we introduce a novel, robust, and flexible membership scheme called Flexi-Fuzz. This scheme integrates a novel flexible weighting mechanism, class probability, and imbalance ratio. The proposed flexible weighting mechanism assigns the maximum weight to samples within a specific proximity to the center, with a gradual decrease in weight beyond a certain threshold. This approach ensures that samples near the class boundary still receive significant weight, maintaining their influence in the classification process. Class probability is used to mitigate the impact of noisy samples, while the imbalance ratio addresses class imbalance. Leveraging this, we incorporate the proposed Flexi-Fuzz membership scheme into the least squares support vector machines (LSSVM) framework, resulting in a robust and flexible model termed Flexi-Fuzz-LSSVM. We determine the class-center using two methods: the conventional mean approach and an innovative median approach, leading to two model variants, Flexi-Fuzz-LSSVM-I and Flexi-Fuzz-LSSVM-II. To validate the effectiveness of the proposed Flexi-Fuzz-LSSVM models, we evaluated them on benchmark UCI and KEEL datasets, both with and without label noise. Additionally, we tested the models on the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset for AD diagnosis. Experimental results demonstrate the superiority of the Flexi-Fuzz-LSSVM models over baseline models.