Commercially viable automated picking in unstructured environments by a robot arm remains a difficult challenge. The problem of robot grasp planning has long been around but the existing solutions tend to be limited when it comes to deploy them in open-ended realistic scenarios. Practical picking systems are called for that can handle the different properties of the objects to be manipulated, as well as the problems arising from occlusions and constrained accessibility. This paper presents a practical solution to the problem of robot picking in an online shopping warehouse by means of a novel approach that integrates a carefully selected method with a new strategy, the centroid normal approach (CNA), on a cost-effective dual-arm robotic system with two grippers specifically designed for this purpose: a two-finger gripper and a vacuum gripper. Objects identified in the scene point cloud are matched to the grasping techniques and grippers to maximize success. Extensive experimentation provides clues as to what are the reasons for success and failure. We chose as benchmark the scenario proposed by the 2017 Amazon Robotics Challenge, since it represents a realistic description of a retail shopping warehouse case; it includes many challenging constraints, such as a wide variety of different product items with a diversity of properties, which are also presented with restricted visibility and accessibility.