In this article, we study the d-distance m-tuple (ℓ,r)-domination problem. Given a simple undirected graph G=(V,E), and positive integers d,m,ℓ and r, a subset V′⊆V is said to be a d-distance m-tuple (ℓ,r)-dominating set if it satisfies the following conditions: (i) each vertex v∈V is d-distance dominated by at least m vertices in V′, and (ii) each r size subset U of V is d-distance dominated by at least ℓ vertices in V′. Here, a vertex v is d-distance dominated by another vertex u means the shortest path distance between u and v is at most d in G. A set U is d-distance dominated by a set of ℓ vertices means size of the union of the d-distance neighborhood of all vertices of U in V′ is at least ℓ. The objective of the d-distance m-tuple (ℓ,r)-domination problem is to find a minimum size subset V′⊆V satisfying the above two conditions.We prove that the problem of deciding whether a graph G has (i) a 1-distance m-tuple (ℓ,r)-dominating set for each fixed value of m,ℓ, and r, and (ii) a d-distance m-tuple (ℓ,2)-dominating set for each fixed value of d(≥2),m, and ℓ of cardinality at most k (here k is a positive integer) are NP-complete. We also prove that for any ε>0, the 1-distance m-tuple (ℓ,r)-domination problem and the d-distance m-tuple (ℓ,2)-domination problem cannot be approximated within a factor of (12−ε)ln|V| and (14−ε)ln|V|, respectively, unless P=NP.