Delivery systems operated by autonomous robots use schedulers to allocate robots to the different orders. Such schedulers are often optimisation-based algorithms that aim to maximise the number of delivered goods. The oracle problem affects the testing of these schedulers, as it is not always possible to assess whether the schedule produced for a given scenario is the optimal one. In this work, we propose a framework, based on a novel use of metamorphic testing, to assess the optimality of the scheduling algorithm developed by Panasonic for the management of a fleet of autonomous delivery robots in the Fujisawa Sustainable Smart Town, Japan. In the framework, a metamorphic relation (MR) transforms a source test case in a followup test case in a predefined way, and compares the results of the execution of the two tests in a simulated environment: if the comparison violates the expected relation, we can claim that one of the two schedules produced by the scheduler is suboptimal. We propose 19 MRs that target different aspects of the delivery system. Experiments over more than 900,000 test cases show that the different MRs have different abilities in exposing suboptimal behaviour and that most of the MRs do not subsume each other. Moreover, they also show that MR violations can provide useful insights into the scheduler’s behaviour to Panasonic’s engineers.