Markov properties of nonrecursive causal models

This paper aims to solve an often noted incompatibility between graphical chain models which elucidate the conditional independence structure of a set of random variables and simultaneous equations systems which focus on direct linear interactions and correlations between random variables. Various authors have argued that the incompatibility arises mainly from the fact that in a simultaneous equations system (e.g., a LISREL model) reciprocal causality is possible whereas this is not so in the case of graphical chain models. In this article it is shown that this view is not correct. In fact, the definition of the Markov property embodied in a graph can be generalized to a wider class of graphs which includes certain nonrecursive graphs. The resulting class of reciprocal graph probability models strictly includes the class of chain graph probability models. The class of lattice conditional independence probability models is also strictly included. It is shown that the resulting methodology is directly applicable to quite general simultaneous equations systems that are subject to mild restrictions only. Provided some adjustments are made, general simultaneous equations systems can be handled as well. In all cases, consistency with the LISREL methodology is maintained

Order batching in multi-server pick-and-sort warehouses.

In many warehouses, customer orders are batched to profit from a reduction in the order picking effort. This reduction has to be offset against an increase in sorting effort. This paper studies the impact of the order batching policy on average customer order throughput time, in warehouses where the picking and sorting functions are executed separately by either a single operator or multiple parallel operators. We present a throughput time estimation model based on Whitt's queuing network approach, assuming that the number of order lines per customer order follows a discrete probability distribution and that the warehouse uses a random storage strategy. We show that the model is adequate in approximating the optimal pick batch size, minimizing average customer order throughput time. Next, we use the model to explore the different factors influencing optimal batch size, the optimal allocation of workers to picking and sorting, and the impact of different order picking strategies such as sort-while-pick (SWP) versus pick-and-sort (PAS)Order batching; Order picking and sorting; Queueing; Warehousing;