In this paper, we study the pickup and delivery problem with time windows, multiple compartments, incompatibility constraints and cleaning operations (PDPTWMCI). In the PDPTWMCI, four types of item incompatibilities are defined and studied. We also allow cleaning operations to remove some existing incompatibilities. The following four types of incompatibilities are considered: incompatible items cannot be loaded 1) simultaneously in the same compartment, 2) simultaneously in the same vehicle, 3) simultaneously and subsequently in the same compartment, and 4) simultaneously and subsequently in the same vehicle. For the last two types of incompatibilities, if the items are loaded subsequently, i.e., the first item is loaded and unloaded before the second item is loaded, cleaning operations can remove existing incompatibilities on the route. Each cleaning operation is done for a subset of empty compartments in a vehicle, and allows to remove all existing incompatibilities from these compartments. A set of dedicated cleaning stations are available and the number of cleaning operations is not restricted. We propose a branch-price-and-cut (BPC) algorithm to solve the PDPTWMCI. In the proposed BPC, the pricing problem is solved using a forward labeling algorithm. We use acceleration techniques to speed up the algorithm. The resulting algorithm yields a variant of selective pricing. We generate benchmark instances for the PDPTWMCI and conduct an extensive numerical campaign to assess the performance of our algorithm and to derive relevant managerial insights for the PDPTWMCI.