You can use DijKstra's algorithm to solve your problem.

But in the process of Dijkstra, I do not see how it takes into account the difference between an AND node and a OR node

You can process the Or/And through your data structures:

B ---- x ------> A

C ---- x ------> A

D ----- y -----> A

Then there are two different paths from B or C to A using the same cost x

the path from D to A using the cost y. The algorithm will select the shorter one according to the values of x and y. 

suppose the following:

weight of edge B-->A = x

weight of edge C-->A = y

weight of edge D-->A = z

for example, suppose that x is the small value then the algorithm will select the edge B-->A

but to execute service A we need also service D !!!

In the resulting shorthest path, could we find service D in the path?

You need to construct the tree of all possible paths from node to another node. So you have all paths.