A Fuzzy Gradient Method in Lagrangian Relaxation for Integer Programming Problems

A major issue in Lagrangian relaxation for integer programming problems is to maximize the dual function which is piece-wise linear, and consists of many facets. Available methods include the subgradient method, the bundle method, and the recently developed surrogate subgradient method. Each of the above methods, however, has its own limitations. Based on the insights obtained from these methods, this paper develops a fuzzy gradient method that makes use of all information available while solving the relaxed problem – not just the optimal solution, but also near optimal solutions. In this method, the fuzzy gradient direction is obtained by combining directions from all near minimum solutions following simple fuzzy rules. The resulting fuzzy gradient direction is continuous with respect to multipliers, thus zigzagging is significantly reduced. Furthermore, the direction can be obtained easily with small computation requirements. The convergence of the method is proved, and a general framework for maximizing the dual function is established. In fact, other methods mentioned above can be viewed as special cases. The fuzzy gradient method is then applied to job shop scheduling problems, and a fuzzy dynamic programming method is developed to effectively obtain the fuzzy gradient directions. Testing results show that the fuzzy gradient method leads to significant improvement over the frequently used subgradient method.