This paper proposes a centralised model predictive control framework to address logistics management of supply chains of perishable goods. Meeting customer specific requirements is decisive to gain a competitive advantage in supply chain management. This fact motivates stakeholders to address solutions that continuously improve supply chain operations. The solution proposed in this work considers the supply chain as a dynamical system in a state-space representation where different categories of commodities, namely common goods and perishable goods, are included. Additionally, the dynamical model is able to store information of the complete supply chain regarding the quantity of commodities and the due time associated to the perishable goods. A centralised controller then collects the supply chain state information and optimises the commodity flow based on the model prediction over a fixed time horizon. The model predictive control solution assigns just-in-time commodity flows, schedules production according to customer demand (pull system) and monitors work-in-progress and in-transit commodities. The success of the proposed control approach is demonstrated in a numerical simulation of a three-tier supply chain following three distinct management policies.