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Domain adaptation aims to transfer knowledge from a domain with adequate labeled samples to a domain with scarce labeled samples. Prior research has introduced various open set domain adaptation settings in the literature to extend the applications of domain adaptation methods in real-world scenarios. This paper focuses on the type of open set domain adaptation setting where the target domain has both private ('unknown classes') label space and the shared ('known classes') label space. However, the source domain only has the 'known classes' label space. Prevalent distribution-matching domain adaptation methods are inadequate in such a setting that demands adaptation from a smaller source domain to a larger and diverse target domain with more classes. For addressing this specific open set domain adaptation setting, prior research introduces a domain adversarial model that uses a fixed threshold for distinguishing known from unknown target samples and lacks at handling negative transfers. We extend their adversarial model and propose a novel adversarial domain adaptation model with multiple auxiliary classifiers. The proposed multi-classifier structure introduces a weighting module that evaluates distinctive domain characteristics for assigning the target samples with weights which are more representative to whether they are likely to belong to the known and unknown classes to encourage positive transfers during adversarial training and simultaneously reduces the domain gap between the shared classes of the source and target domains. A thorough experimental investigation shows that our proposed method outperforms existing domain adaptation methods on a number of domain adaptation datasets.