Travel time estimation and order batching in a 2-block warehouse

Order batching problem (OBP) is the problem of determining the number of orders to be picked together in one picking tour. Although various objectives may arise in practice, minimizing the average throughput time of a random order is a common concern. In this paper, we consider the OBP for a 2-block rectangular warehouse with the assumptions that orders arrive according to a Poisson process and the method used for routing the order-pickers is the well-known S-shape heuristic. We first elaborate on the first and second moment of the order-picker’s travel time. Then we use these moments to estimate the average throughput time of a random order. This enables us to estimate the optimal picking batch size. Results from simulation show that the method provides a high accuracy level. Furthermore, the method is rather simple and can be easily applied in practice.     

Design and control of warehouse order picking: A literature review

Order picking has long been identified as the most labour-intensive and costly activity for almost every warehouse; the cost of order picking is estimated to be as much as 55% of the total warehouse operating expense. Any underperformance in order picking can lead to unsatisfactory service and high operational cost for the warehouse, and consequently for the whole supply chain. In order to operate efficiently, the order-picking process needs to be robustly designed and optimally controlled. This paper gives a literature overview on typical decision problems in design and control of manual order-picking processes. We focus on optimal (internal) layout design, storage assignment methods, routing methods, order batching and zoning. The research in this area has grown rapidly recently. Still, combinations of the above areas have hardly been explored. Order-picking system developments in practice lead to promising new research directions.     

Using a TSP heuristic for routing order pickers in warehouses

In this paper, we deal with the sequencing and routing problem of order pickers in conventional multi-parallel-aisle warehouse systems. For this NP-hard Steiner travelling salesman problem (TSP), exact algorithms only exist for warehouses with at most three cross aisles, while for other warehouse types literature provides a selection of dedicated construction heuristics. We evaluate to what extent reformulating and solving the problem as a classical TSP leads to performance improvements compared to existing dedicated heuristics. We report average savings in route distance of up to 47% when using the LKH (Lin–Kernighan–Helsgaun) TSP heuristic. Additionally, we examine if combining problem-specific solution concepts from dedicated heuristics with high-quality local search features could be useful. Lastly, we verify whether the sophistication of ‘state-of-the-art’ local search heuristics is necessary for routing order pickers in warehouses, or whether a subset of features suffices to generate high-quality solutions.     

A travel time estimation model for a high-level picker-to-part system with class-based storage policies

Most previous related studies on warehouse configurations and operations only investigated single-level storage rack systems where the height of storage racks and the vertical movement of the picking operations are both not considered. However, in order to utilize the space efficiently, high-level storage systems are often used in warehouses in practice. This paper presents a travel time estimation model for a high-level picker-to-part system with the considerations of class-based storage policy and various routing policies. The results indicate that the proposed model appears to be sufficiently accurate for practical purposes. Furthermore, the effects of storage and routing policies on the travel time and the optimal warehouse layout are discussed in the paper.     

A simulated annealing algorithm for transient optimization in gas networks

In this paper we present a simulated annealing approach for the gas network optimization problem. A gas network consists of a set of pipes to transport the gas from the sources to the sinks whereby gas pressure gets lost due to friction. Further on there are compressors, which increase gas pressure, and valves. The aim is to minimize fuel gas consumption of the compressors whereas demands of consumers have to be satisfied. The problem of transient (time-dependent) optimization of gas networks results in a highly complex mixed integer nonlinear program. We relax the equations describing the gas dynamic in pipes by adding these constraints combined with appropriate penalty factors to the objective function. A suitable neighborhood structure is developed for the relaxed problem where time steps as well as pressure and flow of the gas are decoupled. Our approach convinces with flexibility and very good computational results.     

Batch picking in narrow-aisle order picking systems with consideration for picker blocking

This paper develops strategies to control picker blocking that challenge the traditional assumptions regarding the tradeoffs between wide- and narrow-aisle order picking systems. We propose an integrated batching and sequencing procedure called the indexed batching model (IBM), with the objective of minimizing the total retrieval time (the sum of travel time, pick time and congestion delays). The IBM differs from traditional batching formulations by assigning orders to indexed batches, whereby each batch corresponds to a position in the batch release sequence. We develop a mixed integer programming solution for exact control, and demonstrate a simulated annealing procedure to solve large practical problems. Our results indicate that the proposed approach achieves a 5–15% reduction in the total retrieval time primarily by reducing picker blocking. We conclude that the IBM is particularly effective in narrow-aisle picking systems.     

A simulated annealing based hyperheuristic for determining shipper sizes for storage and transportation

The current drive to reduce packaging waste has led many companies to consider the use of multi-trip containers or shippers in which to transport their products in order to reduce packaging waste. The efficiency of such systems obviously depends on selecting shipper dimensions in such a way as to ensure high volumetric utilisation. As is the case with many practical problems the efficiency/solution quality can be improved if problem specific information is used to enhance the operation of a meta-heuristic solution approach. The problem can be modelled as a p-median problem but is too large to be solved in reasonable time without further modification. Four such modifications, all based on properties of the physical problem, are introduced and incorporated into a hyperheuristic driven simulated annealing solution approach.     

Mixed-width aisle configurations for order picking in distribution centers

Space required for the order picking area and labor required to perform the picking activity are two significant costs for a distribution center (DC). Traditionally, DCs employ either entirely wide or entirely narrow aisles in their picking systems. Wide aisles allow pickers to pass each other, which reduces blocking, and requires fewer pickers than their narrow-aisle counterpart for the same throughput. However, the amount of space required for wide-aisle configurations is high. Narrow aisles utilize less space than wide aisles, but are less efficient because of the increased likelihood of congestion experienced by pickers. We propose a variation to the traditional orthogonal aisle designs where both wide and narrow aisles are mixed within the configuration, with a view that mixed-width aisles may provide a compromise between space and labor. To analyze these new mixed-width aisle configurations, we develop analytical models for space and travel time considering randomized storage and traversal routing policies. Through a cost-based optimization model, we identify system parameters for which mixed-width aisle configurations are optimal. Experimental results indicate that annual cost savings of up to $48,000 can be realized over systems with pure wide or narrow aisle configurations.     

A priori policy evaluation and cyclic-order-based simulated annealing for the multi-compartment vehicle routing problem with stochastic demands

We develop methods to estimate and exactly calculate the expected cost of a priori policies for the multi-compartment vehicle routing problem with stochastic demands, an extension of the classical vehicle routing problem where customer demands are uncertain and products must be transported in separate partitions. We incorporate our estimation procedure into a cyclic-order-based simulated annealing algorithm, significantly improving the best-known solution values for a set of benchmark problems. We also extend the updating procedure for a cyclic order’s candidate route set to duration-constrained a priori policies.     

A mathematical model for the Multi-Levels Product Allocation Problem in a warehouse with compatibility constraints

The aim of this work is to address the products allocation problem in a multi-layers warehouse with compatibility constraints among the classes. The problem under study represents one of the most relevant topic in Logistics. The goal is to reduce, as much as possible, the delivery times; the inventories; the total logistic costs and to guarantee, at the same time, higher service levels (i.e., high customers satisfaction degree). In this work, a linear model to mathematically represent the problem is developed and its performance is evaluated on a set of instances, representing realistic situations. A sensitivity analysis is also carried out by considering the most relevant parameters of the model. Finally, an Iterated Local Search based heuristic is defined in order to solve large scale scenarios in a reasonable amount of time. Numerical results show that the proposed heuristic is able to find good quality solutions with a computational effort lower than that required to solve the proposed mathematical model.     

A simulated annealing approach for buffer allocation in reliable production lines

We describe a simulated annealing approach for solving the buffer allocation problem in reliable production lines. The problem entails the determination of near optimal buffer allocation plans in large production lines with the objective of maximizing their average throughput. The latter is calculated utilizing a decomposition method. The allocation plan is calculated subject to a given amount of total buffer slots in a computationally efficient way.     

A simulated annealing with a new neighborhood structure based algorithm for high school timetabling problems

This paper approximately solves the high school timetabling problem using a simulated annealing based algorithm with a newly-designed neighborhood structure. In search for the best neighbor, the heuristic performs a sequence of swaps between pairs of time slots, instead of swapping two assignments as in a standard simulated annealing. The computational results show that the proposed heuristic, which is tested on two sets of benchmark instances, performs better than existing approaches.     

A genetic algorithm for the problem of configuring a hub-and-spoke network for a LTL trucking company in Brazil

A heuristic based on genetic algorithms is proposed to the problem of configuring hub-and-spoke networks for trucking companies that operate less-than-truckload (LTL) services in Brazil. The problem consists of determining the number of consolidation terminals (also known as hubs), their locations and the assignment of the spokes to the hubs, aiming to minimize the total cost, which is composed of fixed and variable costs. The proposed formulation differs from similar formulations found in the literature in the sense that it allows variable scale-reduction factors for the transportation costs according to the total amount of freight between hub terminals, as occurs to less-than-truckload (LTL) freight carriers in Brazil. Our genetic algorithm approach incorporates an efficient local improvement procedure that is applied to each generated individual of the population. Computational results for benchmark problems are presented. A practical application to a real world problem involving one of the top-ten trucking companies in Brazil is also described.     

A Kantorovich-type analysis for a fast iterative method for solving nonlinear equations

We revisit a fast iterative method studied by us in [I.K. Argyros, On a two-point Newton-like method of convergent order two, Int. J. Comput. Math. 88 (2) (2005) 219-234] to approximate solutions of nonlinear operator equations. The method uses only divided differences of order one and two function evaluations per step. This time we use a simpler Kantorovich-type analysis to establish the quadratic convergence of the method in the local as well as the semilocal case. Moreover we show that in some cases our method compares favorably, and can be used in cases where other methods using similar information cannot [S. Amat, S. Busquier, V.F. Candela, A class of quasi-Newton generalized Steffensen's methods on Banach spaces, J. Comput. Appl. Math. 149 (2) (2002) 397-406; D. Chen, On the convergence of a class of generalized Steffensen's iterative procedures and error analysis, Int. J. Comput. Math. 31 (1989) 195-203]. Numerical examples are provided to justify the theoretical results.     

A fast collocation method for acoustic scattering in shallow oceans

In this paper, we investigate the numerical solution of the integral equation of the second kind reduced by acoustic scattering in shallow oceans with Dirichlet condition. Based on analyzing the singularity of the truncating kernel with a sum of infinite series, using our trigonometric interpolatory wavelets and collocation method, we obtain the numerical solution which possesses a fast convergence rate like o(2^−j). Moreover, the entries of the stiffness matrix can be obtained by FFT, which lead the computational complexity to decrease obviously.     

A fast numerical method for a natural boundary integral equation for the Helmholtz equation

A Neumann boundary value problem of the Helmholtz equation in the exterior circular domain is reduced into an equivalent natural boundary integral equation. Using our trigonometric wavelets and the Galerkin method, the obtained stiffness matrix is symmetrical and circulant, which lead us to a fast numerical method based on fast Fourier transform. Furthermore, we do not need to compute the entries of the stiffness matrix. Especially, our method is also efficient when the wave number k in the Helmholtz equation is very large.     

A Laplace transform method for order statistics from nonidentical random variables and its application in Phase-type distribution

In this paper, we derive a method for obtaining the Laplace transform of order statistics (o.s.) arising from general independent nonidentically distributed random variables (r.v.’s). A survey of the most important properties, applications and the o.s. of a Phase-type (PH) distribution are also presented. Two illustrative examples are provided.     

A fast method for discovering critical edge sequences in e-commerce catalogs

Web sites allow the collection of vast amounts of navigational data – clickstreams of user traversals through the site. These massive data stores offer the tantalizing possibility of uncovering interesting patterns within the dataset. For e-businesses, always looking for an edge in the hyper-competitive online marketplace, the discovery of critical edge sequences (CESs), which denote frequently traversed sequences in the catalog, is of significant interest. CESs can be used to improve site performance and site management, increase the effectiveness of advertising on the site, and gather additional knowledge of customer behavior patterns on the site.     

Defining order picking policy: A storage assignment model and a simulated annealing solution in AS/RS systems

Improving technology, variable customer demands, variety of products and increasing improvement of production systems has brought storage facilities, which has an important role on business operations, to a critical point. This circumstance has revealed the necessity on activating of storage systems and operations. In this study, mathematical model, which minimizes simultaneously storage allocation, storage retrieval and storage keeping costs, is generated for an AS/RS system of an enterprise. The mathematical model has run for small solution space and optimal solution is obtained. The problem in this study is an NP hard problem that is why optimal solution cannot be obtained for large solution spaces. So the problem in existing system is solved by using simulated annealing and an optimal order policy is proposed.