This paper explores the relationship between Shephard's input distance function and Luenberger's benefit function, which is a directional representation of preferences. The benefit function can be seen as a generalization of Shephard's input distance function. The paper discusses the properties of the directional input distance function, including its concavity, homogeneity, and the conditions under which it can represent the technology completely. It also examines the duality between the directional input distance function and the cost function, and derives shadow prices from these functions. Additionally, the paper extends McFadden's composition rules for input sets to the directional input distance function, providing a set of equivalent representations of the technology. The results are illustrated through various mathematical proofs and examples.This paper explores the relationship between Shephard's input distance function and Luenberger's benefit function, which is a directional representation of preferences. The benefit function can be seen as a generalization of Shephard's input distance function. The paper discusses the properties of the directional input distance function, including its concavity, homogeneity, and the conditions under which it can represent the technology completely. It also examines the duality between the directional input distance function and the cost function, and derives shadow prices from these functions. Additionally, the paper extends McFadden's composition rules for input sets to the directional input distance function, providing a set of equivalent representations of the technology. The results are illustrated through various mathematical proofs and examples.