Purpose:  The purpose of this study is the well-known Heterogeneous Fleet Vehicle Routing Problem (HVRP) is one of the developed problems of vehicle routing, which involves optimizing a set of routes for a fleet of vehicles with different costs and capacities. Methods: HVRP is usually modeled as a single objective that aims to minimize overall transportation costs (total fixed costs and costs commensurate with total distance). Results: These vehicles are located in the central depot and serve customers’ demands. Implications: In this case, the number of vehicles available (of any type) may be limited.

Let R be a commutative ring with identity which is not an integral domain. An ideal I of R is called an annihilating ideal if there exists $r\in R- \{0\}$ such that $Ir=(0)$ . The total graph of nonzero annihilating ideals of R is the graph $\Omega (R)$ whose vertices are the nonzero annihilating ideals of R and two distinct vertices $I,J$ are joined if and only if $I+J$ is also an annihilating ideal of R. We study the strong metric dimension of $\Omega (R)$ and evaluate it in several cases.

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