Multi-evaporator vapor compression systems (ME-VCSs) are becoming widely used to meet the cooling needs for multiple thermal loads via a single system. As the number of evaporators increases, the size of these systems can make a centralized control approach impractical and computationally expensive; thus, motivating a decentralized control design. Linear gray-box modeling techniques show that ME-VCSs have a distinct underlying structure between the actuators and dynamic states known as a block arrow structure (BAS). This structure captures the high degree of coupling found in ME-VCSs, which can lead to poor decentralized control performance. This paper presents a partially decentralized model predictive control strategy which directly considers the coupling in the system when making control decisions by exploiting the BAS. The gray-box modeling approach and the decentralized nature of this BAS control strategy prove scalable to n-evaporator systems. Through simulated case studies, it is shown that this BAS control strategy can approximate the performance of a centralized control approach for ME-VCSs while significantly reducing computational costs.