A delivery processing apparatus comprising: a plurality of stackers configured to stack delivery objects; a conveyer configured to transport the delivery objects to a designated stacker out of the plurality of stackers; a detector configured to detect that a monitored stacker out of the plurality of stackers is in an empty state in which all delivery objects have been retrieved from the stacker; a main controller configured to specify a regular stacker that is a stacker out of the plurality of stackers and serving as a sort destination of the delivery object, based on address information that is obtained from the delivery object, and configured to control the conveyer such that the delivery object is transported to the regular stacker serving as the sort destination, out of the plurality of regular stackers, that is specified, characterized in that the delivery processing apparatus further comprises: a backup plate unit; a full sensor configured to detect that a monitored regular stacker out of the plurality of stackers is in an overflow state in which more than a predetermined amount of delivery objects have accumulated, and configured to output a full signal in response to coming in contact with the backup plate unit; a pre-full sensor configured to output a pre-full signal in response to coming in contact with the backup plate unit; wherein, if the regular stacker serving as the sort destination specified is in the overflow state, based on the full signal or the pre-full signal, then the main controller stops the transport of delivery objects to the regular stacker serving as the sort destination that has gone into the overflow state, and assigns a backup stacker that is a stacker out of the plurality of stackers not correlated with any address information, as a sort destination substituting for the sort destination that has gone into the overflow state, and after assigning the substitute sort destination, when the regular stacker that had gone into the overflow state goes into the empty state, the main controller stops transport of delivery objects to the backup stacker that has been assigned as the substitute sort destination, and resumes the transport of delivery objects to the regular stacker that has gone into the empty state. a plurality of stackers configured to stack delivery objects; a conveyer configured to transport the delivery objects to a designated stacker out of the plurality of stackers; a detector configured to detect that a monitored stacker out of the plurality of stackers is in an empty state in which all delivery objects have been retrieved from the stacker; a main controller configured to specify a regular stacker that is a stacker out of the plurality of stackers and serving as a sort destination of the delivery object, based on address information that is obtained from the delivery object, and configured to control the conveyer such that the delivery object is transported to the regular stacker serving as the sort destination, out of the plurality of regular stackers, that is specified, a backup plate unit; a full sensor configured to detect that a monitored regular stacker out of the plurality of stackers is in an overflow state in which more than a predetermined amount of delivery objects have accumulated, and configured to output a full signal in response to coming in contact with the backup plate unit; a pre-full sensor configured to output a pre-full signal in response to coming in contact with the backup plate unit; wherein, if the regular stacker serving as the sort destination specified is in the overflow state, based on the full signal or the pre-full signal, then the main controller stops the transport of delivery objects to the regular stacker serving as the sort destination that has gone into the overflow state, and assigns a backup stacker that is a stacker out of the plurality of stackers not correlated with any address information, as a sort destination substituting for the sort destination that has gone into the overflow state, and after assigning the substitute sort destination, when the regular stacker that had gone into the overflow state goes into the empty state, the main controller stops transport of delivery objects to the backup stacker that has been assigned as the substitute sort destination, and resumes the transport of delivery objects to the regular stacker that has gone into the empty state.