Motivated by applications in service systems, we consider queueing systems where each customer must be handled by a server with the right skill set. We focus on optimizing the routing of customers to servers in order to maximize the total payoff of customer--server matches. In addition, customer--server dependent payoff parameters are assumed to be unknown a priori. We construct a machine learning algorithm that adaptively learns the payoff parameters while maximizing the total payoff and prove that it achieves polylogarithmic regret. Moreover, we show that the algorithm is asymptotically optimal up to logarithmic terms by deriving a regret lower bound. The algorithm leverages the basic feasible solutions of a static linear program as the action space. The regret analysis overcomes the complex interplay between queueing and learning by analyzing the convergence of the queue length process to its stationary behavior. We also demonstrate the performance of the algorithm numerically, and have included an experiment with time-varying parameters highlighting the potential of the algorithm in non-static environments.

Transportation Problem is an important problem which has been widely studied in Operations Research domain. It has been often used to simulate different real life problems. In particular, application of this Problem in NP Hard Problems has a remarkable significance. In this Paper, we present the closed, bounded and non empty feasible region of the transportation problem using fuzzy trapezoidal numbers which ensures the existence of an optimal solution to the balanced transportation problem. The multivalued nature of Fuzzy Sets allows handling of uncertainty and vagueness involved in the cost values of each cells in the transportation table. For finding the initial solution of the transportation problem we use the Fuzzy Vogel Approximation Method and for determining the optimality of the obtained solution Fuzzy Modified Distribution Method is used. The fuzzification of the cost of the transportation problem is discussed with the help of a numerical example. Finally, we discuss the computational complexity involved in the problem. To the best of our knowledge, this is the first work on obtaining the solution of the transportation problem using fuzzy trapezoidal numbers.

Unfortunately, the required optimization problem is often intractable because there is a combinatorial increase in the number of local minima as the number of candidate basis vectors increases.

Unfortunately, the required optimization problem is often intractable because there is a combinatorial increase in the number of local minima as the number of candidate basis vectors increases.