Solving less-than inequalities with positive integers
Pyth, 39 bytes

V&gt;1f.A&lt;MxMLTN.pS{s=Nm%2cd).zVNjd[H\==hZ


Try it online: Demonstration

Brute-forces through all possible permutations (and interpret them as sortings), check if they match the inequalities, and assign them the values 1, 2, ...., n. 

Explanation

f.A&lt;MxMLTN.pS{s=Nm%2cd).z  
                 m     .z  map each input line d to:
                    cd)       split d by spaces
                  %2          and remove the second element
               =N          save this list of pairs to N
              s            combine these pairs to a big list of variable names
             {             set (remove duplicates)
          .pS              generate all permutations
f                          filter for permutations T, which satisfy:
     xMLTN                    replace each variable in N by their index in T
 .A&lt;M                         check if each pair is ascending

V&gt;1...VNjd[H\==hZ          implicit: Z = 0
 &gt;1                        remove all but the last filtered permutation (if any)
V                          for each permutation N in ^ (runs zero times or once):
      VN                      for each variable H in N:
          [                      generate a list containing:
           H                        H
            \=                      "="
              =hZ                   Z incremented by 1 (and update Z)
        jd                       join this list by spaces and print