An orientation D of a graph G is a digraph obtained from G by replacing each edge by exactly one of the two possible arcs with the same ends. An orientation D of a graph G is a k-orientation if the in-degree of each vertex in D is at most k. An orientation D of G is proper if any two adjacent vertices have different in-degrees in D. The proper orientation number of a graph G, denoted by − → χ (G), is the minimum k such that G has a proper k-orientation. A weighted orientation of a graph G is a pair (D, w), where D is an orientation of G and w is an arc-weighting A(D) → N \ {0}. A semi-proper orientation of G is a weighted orientation (D, w) of G such that for every two adjacent vertices u and v in G, we have that S (D,w) (v) = S (D,w) (u), where S (D,w) (v) is the sum of the weights of the arcs in (D, w) with head v. For a positive integer k, a semi-proper k-orientation (D, w) of a graph G is a semi-proper orientation of G such that max v∈V (G) S (D,w) (v) ≤ k. The semi-proper orientation number of a graph G, denoted by − → χs(G), is the least k such that G has a semi-proper k-orientation. In this work, we first prove that − → χs(G) ∈ {ω(G) − 1, ω(G)} for every split graph G, and that, given a split graph G, deciding whether − → χs(G) = ω(G) − 1 is an NP-complete problem. We also show that, for every k, there exists a (chordal) graph G and a split subgraph H of G such that − → χ (G) ≤ k and − → χ (H) = 2k − 2. In the sequel, we show that, for every n ≥ p(p + 1), − → χs(P p n) = 3 2 p , where P p n is the p th power of the path on n vertices. We investigate further unit interval graphs with no big clique: we show that − → χ (G) ≤ 3 for any unit interval graph G with ω(G) = 3, and present a complete characterization of unit interval graphs with − → χ (G) = ω(G) = 3. Then, we show that deciding whether − → χs(G) = ω(G) can be solved in polynomial time in the class of co-bipartite graphs. Finally, we prove that computing − → χs(G) is FPT when parameterized by the minimum size of a vertex cover in G or by the treewidth of G. We also prove that not only computing − → χs(G), but also − → χ (G), admits a polynomial kernel when parameterized by the neighbourhood diversity plus the value of the solution. These results imply kernels of size 4 O(k 2) and O(2 k k 2), in chordal graphs and split graphs, respectively, for the problem of deciding whether − → χs(G) ≤ k parameterized by k. We also present exponential kernels for computing both − → χ (G) and − → χs(G) parameterized by the value of the solution when G is a cograph. On the other hand, we show that computing − → χs(G) does not admit a polynomial kernel parameterized by the value of the solution when G is a chordal graph, unless NP ⊆ coNP/poly.