In this paper, we address three variations of the windy k-traveling salesman problem (the k-WTSP), which are related to the asymmetric traveling salesman problem. Concretely, given a weighted connected graph \(G=(V,E;w;r)\) of order n and an integer \(k\ge 1\) , where \(r\in V\) and a windy cost function \(w:A \rightarrow \mathbb {R^{+}}\) , where \(A=\{(v_{i},v_{j}),(v_{j},v_{i})~|~v_{i}v_{j}\in E\}\) , i.e., for each edge \(v_{i}v_{j}\in E\) , we denote \(w(v_{i},v_{j})\) to be the cost to traverse this edge \(v_{i}v_{j}\) from \(v_{i}\) to \(v_{j}\) and \(w(v_{j},v_{i})\) to be the cost to traverse the same edge \(v_{i}v_{j}\) from \(v_{j}\) to \(v_{i}\) , respectively, the k-WTSP is asked to find k circuits, starting and ending at r, such that each vertex of G is contained in at least one of these k circuits, the objective is to minimize the total costs of these k circuits. In addition, the minmax windy k-traveling salesman problem (the minmax k-WTSP) is asked to find k circuits as mentioned-above, the objective is to minimize the maximum cost of these k circuits. Finally, the minmax windy k-circuit cover problem (the minmax k-WCCP) is asked to find k circuits such that each vertex of G is contained in at least one of these k circuits, the objective is to minimize the maximum cost of these k circuits. We have three key contributions as follows. (1) We design a \(\frac{3}{2}(\alpha +1)\) -approximation algorithm to solve the k-WTSP in time \(O(n^{3})\) , where \(\alpha =\max \{\frac{w(v_{i},v_{j})}{w(v_{j},v_{i})}~|~v_{i}v_{j}\in E\}\) ; (2) We present a \((\frac{3}{2}(\alpha +1)+2(1-\frac{1}{k}))\) -approximation algorithm to solve the minmax k-WTSP in time \(O(n^{3})\) ; (3) We provide a \((\frac{3}{2}(\alpha +1)+k)\) -approximation algorithm to solve the minmax k-WCCP in time \(O(n^{3})\) .