NP [BSA], Ratio 10-19

Pachinko is a Japanese mechanical gambling game similar to pinball. Recently, several mathematical models of Pachinko have been proposed. A number of pins are spiked in a field. A ball drops from the top of the playfield and the ball falls down. In the 50-50 model, if the ball hits a pin, it moves to the left or right passage of the pin with an equal probability. An arrangement of pins generates a distribution of the drop probability for all of the columns. This problem was considered by generating uniform distributions. Previous studies have demonstrated that the (1/2a)-uniform distribution is possible for a is an element of {0, 1, 2, 3, 4} and is conjectured so that it is possible for any positive integer a. This study describes the constructive proof for this conjecture. This study also formalizes a natural decision problem yielded by this model while investigating its computational complexity. More precisely, given any drop-probability distribution A and any partial drop probability distribution B, this study uses non-deterministic polynomial-time (NP) hardness to determine if there exists a pin arrangement that transforms A into B. (C) 2020 Published by Elsevier B.V.

The green vehicle routing problem advised by the global sustainability conscience is a seminal work on the contemporary green research in supply chain management. Inspired by the replete, pollution routing literature, the present work acknowledges the significance of redesigning the logistic networks in a low emission perspective. The present research proposes three logistic models to address the vehicle routing problem of a dairy plant for collecting milk from chilling centres. The first model generates classical, distance centric and economically viable milk pickup plans. In the second model, the environmental friendly routes are generated. The third model is slated to bring a tradeoff between the first two models. The three models are formulated as integer linear programming models and are solved with branch-and-bound based exact algorithm. Owing to the complexity of the NP-hard problem for generating time-bound results, an ant colony optimisation algorithm is designed to solve the models. The performance of the algorithm is further enhanced by introducing a powerful variable neighbourhood search. The computational study on hybrid algorithm reports an average 0.25% improvement from a time interrupted solution of the exact algorithm. The study recommends decisive managerial insights for implementing green principles to the dairy industry.