Waste classification is crucial as it brings economic benefits to cities and promotes environmental sustainability. Traditional waste classification studies focus on single-label classification tasks. However, single-label classification is inefficient when dealing with images containing multiple wastes. Supervised multi-label classification can manage more complex and realistic scenes, which contain a wider variety of objects in a single image. However, it necessitates pre-labeling all waste in each image, which is time-consuming. Therefore, we design a weakly supervised multi-label classification framework called Adaptive Weakly-supervised Waste Classification Framework (AWWCF). The AWWCF consists of the Target Preprocessing Module (TPM), the Prediction Module (PM), and the Computing Module (CM). Previous weakly-supervised classification works primarily exploit inner connections among observed labels while ignoring unannotated objects and unobserved labels. To properly utilize unobserved labels, we integrate the PM and AM of AWWCF with our newly proposed method, Adaptive Loss and Enhanced Class (ALEC) activation maps. In PM, ALEC dynamically enhances the attribution scores of the class activation maps to prevent predictions for positive observed labels from being affected by unobserved labels. In the CM, ALEC dynamically rejects and corrects unobserved labels with large loss values. Experimental results demonstrate the effectiveness of our framework in weakly supervised multi-label waste classification. ALEC can be integrated with different deep classification models to form an effective framework for a sustainable environment.