This paper introduces a new variant of the Multi-Compartment Waste Collection Problem (MC-WCP), which we call the MC-WCP with Bin Washer (MC-WCP-BW). The problem involves a fleet of compressed refuse collection trucks equipped with a bin washer, which compresses each waste type in the corresponding compartment of the vehicle. The vehicles can also wash the bins. Separation sites and water refilling stations are considered in the problem. A subset of the bins must be washed when emptied, which is not mandatory for the others. The problem is modeled using a mixed-integer linear programming formulation incorporating multiple visits to separation sites and water refilling stations. An efficient Hybrid Variable Neighborhood Search (HVNS) algorithm is developed and evaluated on a set of instances from the literature and newly generated instances. Computational experiments show that our proposed algorithm can identify solutions of better quality in a shorter computational time, as compared with the current state-of-the-art algorithms. The potential benefits of the compaction operation under several scenarios are also examined via extensive analyses.