On the one hand, the dehazing task is an ill-posedness problem, which means that no unique solution exists. On the other hand, the dehazing task should take into account the subjective factor, which is to give the user selectable dehazed images rather than a single result. Therefore, this paper proposes a multi-output dehazing network by introducing illumination controllable ability, called IC-Dehazing. The proposed IC-Dehazing can change the illumination intensity by adjusting the factor of the illumination controllable module, which is realized based on the interpretable Retinex model. Moreover, the backbone dehazing network of IC-Dehazing consists of a Transformer with double decoders for high-quality image restoration. Further, the prior-based loss function and unsupervised training strategy enable IC-Dehazing to complete the parameter learning process without the need for paired data. To demonstrate the effectiveness of the proposed IC-Dehazing, quantitative and qualitative experiments are conducted. Code is available at https://github.com/Xiaofeng-life/ICDehazing.