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In this work, we study the δ\delta -chromatic number of a graph, which is the chromatic number of the δ\delta -complement of a graph. We give a structure of the δ\delta -complements and sharp bounds on the δ\delta -chromatic numbers of the Cartesian products of graphs. Furthermore, we compute the δ\delta -chromatic numbers of various classes of Cartesian product graphs, including the Cartesian products among cycles, paths, and stars.

This paper investigates manual order picking, where workers travel through the warehouse to retrieve requested items from shelves. To minimise the completion time of orders, researchers have developed various routing procedures that guide order pickers through the warehouse. The paper at hand contributes to this stream of research and proposes an optimal order picker routing policy for a conventional warehouse with two blocks and arbitrary starting and ending points of a tour. The procedure proposed in this paper extends an earlier work of Löffler et al. (2018. Picker routing in AGV-assisted order picking systems, Working Paper, DPO-01/2018, Deutsche Post Chair-Optimization of Distribution Networks, RWTH Aachen University, 2018) by applying the concepts of Ratliff and Rosenthal (1983. "Order-picking in a Rectangular Warehouse: a Solvable Case of the Traveling Salesman Problem." Operations Research 31 (3): 507–521) and Roodbergen and de Koster (2001a. "Routing Order Pickers in a Warehouse with a Middle Aisle." European Journal of Operational Research 133 (1): 32–43) that used graph theory and dynamic programming for finding an optimal picker route. We also propose a routing heuristic, denoted S*-shape, for conventional two-block warehouses with arbitrary starting and ending points of a tour. In computational experiments, we compare the average order picking tour length in a conventional warehouse with a single block to the case of a conventional warehouse with two blocks to assess the impact of the middle cross aisle on the performance of the warehouse. Furthermore, we evaluate the performance of the S*-shape heuristic by comparing it to the exact algorithm proposed in this study. [ABSTRACT FROM AUTHOR]

Order picking has often been considered as one of the most labor- and time-intensive tasks in warehouse operations. Among the various planning problems that have to be solved in manual picker-to-parts systems, the routing of the order picker usually accounts for the highest share of the total warehouse operating cost. To minimize the cost of order picking, researchers have developed various routing procedures that guide the order picker through the warehouse to complete given customer orders. For some warehouse layouts such as the chevron warehouse, an optimal routing algorithm has not yet been proposed. This article, therefore, contributes to filling this research gap by developing an optimal order picker routing policy for the chevron warehouse. The optimal routing algorithm proposed in this article is based on the concept of graph theory and utilizes a dynamic programming procedure. In addition, we propose various simple routing heuristics for the chevron warehouse. In computational experiments, the average order picking tour lengths resulting from optimal routing and from the simple heuristics are compared. Moreover, we compare the performance of the chevron warehouse to the conventional two-block warehouse under various conditions using the tour lengths obtained by the optimal algorithms. [ABSTRACT FROM AUTHOR]

We present an efficient numerical method for solving fractional pantograph differential equations by applying Taylor wavelets. We give an exact formula for the Riemann-Liouville fractional integral of the Taylor wavelets. We then apply the given formula to reduce our problem to the problem of solving a system of algebraic equations. Several examples are included to show the the effectiveness of our numerical method and in comparison with previous methods. [ABSTRACT FROM AUTHOR]