This paper presents a new dual simplex algorithm for solving phase 1 linear programming problems. In the algorithm， first， the equation with the auxiliary objective function at the optimality as a new constraint is added to phase 1， and then a pivot is performed to generate one vertex on the associated objective hyperplane. This vertex may be feasible or not feasible. If it is feasible， the phase 1 simplex algorithm ends，otherwise， is proceed with the successive iterations fixed on the objective hyperplane. Next， taking the sum of the constraints with the righthandside negative as the objective， we use the dual pivot to make the value of the righthandside monotonically increased， and meanwhile keep the constraints with the righthandside positive or zero still feasible. Once a constraint with the righthandside negative is changed feasibly，it would be deleted from the sum until a feasible solution or the situation with no solution is achieved. Finally， computational study is done to test the efficiencies of the algorithm comparing to the ordinary simplex algorithm on some standard test problems from NETLIB and MIPLIB， showing that our algorithm uses fewer iterations and spends less executive time on most instances， thus is more efficient in computation.