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Optimizing Warehouse Operations with Autonomous Mobile Robots
Abstract
Autonomous mobile robots (AMRs) can support human pickers in warehouse picking operations by reducing picker walking distance and increasing the warehouse’s throughput. AMR-assisted order picking is becoming popular as it can be conveniently implemented in conventional warehouses. This study proposes an integrated optimization model for scheduling the operations in AMR-assisted picker-to-parts warehouse systems. The model aims to minimize the makespan of all picking operations for a batch of orders by assigning batched orders to AMRs, selecting storage racks for AMRs and pickers to visit, and determining the routes of the AMRs and the pickers. A column- and row-generation algorithm is designed to solve the model using synchronization constraints between AMRs and pickers. Numerical experiments are conducted to validate the applicability of our proposed algorithm in a warehouse that handles 16,000 orders per day. Our algorithm can solve small-scale instances to optimality. Our algorithm can also obtain better solutions in less time than a column generation (CG)–based method. Extensive experiments are conducted to derive managerial insights.
Funding: This research was supported by the National Natural Science Foundation of China [Grants 72025103, 72394360, 72394362, 72401179, 72361137001, and 72371221], the Project of Science and Technology Commission of Shanghai Municipality China [Grant 23JC1402200], and the Research Grants Council of the Hong Kong Special Administrative Region, China (Project number HKSAR RGC TRS T32-707/22-N).
Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2024.0800 .
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Autonomous mobile robots (AMR) are currently being introduced in many intralogistics operations, like manufacturing, warehousing, cross-docks, terminals, and hospitals. Their advanced hardware and control software allow autonomous operations in dynamic environments. Compared to an automated guided vehicle (AGV) system in which a central unit takes control of scheduling, routing, and dispatching decisions for all AGVs, AMRs can communicate and negotiate independently with other resources like machines and systems and thus decentralize the decision-making process. Decentralized decision-making allows the system to react dynamically to changes in the system state and environment. These developments have influenced the traditional methods and decision-making processes for planning and control. This study identifies and classifies research related to the planning and control of AMRs in intralogistics. We provide an extended literature review that highlights how AMR technological advances affect planning and control decisions. We contribute to the literature by introducing an AMR planning and control framework to guide managers in the decision-making process, thereby supporting them to achieve optimal performance. Finally, we propose an agenda for future research within this field.
The efficiency of automated container terminals primarily depends on the synchronization of automated-guided vehicles (AGVs) and automated cranes. Accordingly, we study the integrated rail-mounted yard crane and AGV scheduling problem as a multi-robot coordination and scheduling problem in this paper. Based on a discretized virtualized network, we propose a multicommodity network flow model with two sets of flow balance constraints for cranes and AGVs. In addition, two side constraints are introduced to deal with inter-robot constraints to reflect the complex interactions among terminal agents accurately. The Alternating Direction Method of Multipliers (ADMM) method is adopted in this study as a market-driven approach to dualize the hard side constraints; therefore, the original problem is decomposed into a set of crane-specific and vehicle-specific subtasks. The cost-effective solutions can be obtained by iteratively adjusting both the primal and dual costs of each subtask. We also compare the computational performance of the proposed solution framework with that of the resource-constrained project scheduling problem (RCPSP) model using commercial solvers. Comparison results indicate that our proposed approach could efficiently find solutions within 2% optimality gaps. Illustrative and real-world instances show that the proposed approach effectively serves the accurate coordination of AGVs and cranes in automated terminals.
We present a multi-ob’jective two-echelon vehicle routing problem with vehicle synchronization and ‘grey zone’ customers arising in the context of urban freight deliveries. Inner-city center deliveries are performed by small vehicles due to access restrictions, while deliveries outside this area are carried out by conventional vehicles for economic reasons. Goods are transferred from the first to the second echelon by synchronized meetings between vehicles of the respective echelons. We investigate the assignment of customers to vehicles, i.e., to the first or second echelon, within a so-called ‘grey zone’ on the border of the inner city and the area around it. While doing this, the economic objective as well as negative external effects of transport, such as emissions and disturbance (negative impact on citizens due to noise and congestion), are taken into account to include objectives of companies as well as of citizens and municipal authorities. Our metaheuristic – a large neighborhood search embedded in a heuristic rectangle/cuboid splitting – addresses this problem efficiently. We investigate the impact of the free assignment of part of the customers (‘grey zone’) to echelons and of three different city layouts on the solution. Computational results show that the impact of a ‘grey zone’ and thus the assignment of these customers to echelons depend significantly on the layout of a city. Potentially pareto-optimal solutions for two and three objectives are illustrated to efficiently support decision makers in sustainable city logistics planning processes.
In a recent work, Muter, Birbil, and Bülbül (2013a) identified and characterized a general class of linear programming (LP) problems - known as problems with column-dependent-rows (CDR-problems). These LPs feature two sets of constraints with mutually exclusive groups of variables in addition to a set of structural linking constraints, in which variables from both groups appear together. In a typical CDR-problem, the number of linking constraints grows very quickly with the number of variables, which motivates generating both columns and their associated linking constraints simultaneously on-the-fly. In this paper, we expose the decomposable structure of CDR-problems via Benders decomposition. However, this approach brings on its own theoretical challenges. One group of variables is generated in the Benders master problem, while the generation of the linking constraints is relegated to the Benders subproblem along with the second group of variables. A fallout of this separation is that only a partial description of the dual of the Benders subproblem is available over the course of the algorithm. We demonstrate how the pricing subproblem for the column generation applied to the Benders master problem does also update the dual polyhedron and the existing Benders cuts in the master problem to ensure convergence. Ultimately, a novel integration of Benders cut generation and the simultaneous generation of columns and constraints yields a brand-new algorithm for solving large-scale CDR-problems. We illustrate the application of the proposed method on a time-constrained routing problem. Our numerical experiments confirm the outstanding performance of the new decomposition method.
Airline recovery presents very large and difficult problems requiring high-quality solutions within short time limits. To improve computational performance, various solution approaches have been employed, including decomposition methods and approximation techniques. There has been increasing interest in the development of efficient and accurate solution techniques to solve an integrated airline recovery problem. In this paper, an integrated airline recovery problem is developed, integrating the schedule, crew, and aircraft recovery stages, and it is solved using column-and-row generation. A general framework for column-and-row generation is presented as an extension of current generic methods. This extension considers multiple secondary variables and linking constraints and is proposed as an alternative solution approach to Benders' decomposition. The application of column-and-row generation to the integrated recovery problem demonstrates the improvement in the solution run times and quality compared to a standard column generation approach. Column-and-row generation improves solution run times by reducing the problem size and thereby achieving faster execution of each linear programming solve. As a result of this evaluation, a number of general enhancement techniques are identified to further reduce the run times of column-and-row generation. This paper also details the integration of the row generation procedure with branch and price, which is used to identify integer optimal solutions.
In this paper, we develop a simultaneous column-and-row generation algorithm that could be applied to a general class of large-scale linear programming problems. These problems typically arise in the context of linear programming formulations with exponentially many variables. The defining property for these formulations is a set of linking constraints, which are either too many to be included in the formulation directly, or the full set of linking constraints can only be identified, if all variables are generated explicitly. Due to this dependence between columns and rows, we refer to this class of linear programs as problems with column-dependent-rows. To solve these problems, we need to be able to generate both columns and rows on-the-fly within an efficient solution approach. We emphasize that the generated rows are structural constraints and distinguish our work from the branch-and-cut-and-price framework. We first characterize the underlying assumptions for the proposed column-and-row generation algorithm. These assumptions are general enough and cover all problems with column-dependent-rows studied in the literature up until now to the best of our knowledge. We then introduce in detail a set of pricing subproblems, which are used within the proposed column-and-row generation algorithm. This is followed by a formal discussion on the optimality of the algorithm. To illustrate the proposed approach, the paper is concluded by applying the proposed framework to the multi-stage cutting stock and the quadratic set covering problems.
This paper describes an exact branch-and-cut (B&C) algorithm for the split delivery vehicle routing problem. The underlying model is based on a previously proposed two-index vehicle flow formulation that models a relaxation of the problem. We dynamically separate two well-known classes of valid inequalities, namely capacity and connectivity cuts, and use an in-out algorithm to improve the convergence of the cutting phase. We generate no-good cuts from feasible integer solutions to the relaxation using a recently proposed single-commodity flow formulation in the literature. The exact methodology is complemented by a very effective adaptive large neighborhood search (ALNS) heuristic that provides high-quality upper bounds to initiate the B&C algorithm. Key ingredients in the design of the heuristic include the use of a tailored construction algorithm, which can exploit the situation in which the ratio of the number of customers to the minimum number of vehicles needed is low, and the use of a route-based formulation to improve the solutions found before, during, and after the ALNS procedure. An earlier version of this work was submitted to the DIMACS (Center for Discrete Mathematics and Theoretical Computer Science) implementation challenge, where it placed third. On sets of well-known benchmark instances for limited and unlimited fleet variants of the problem, we demonstrate that the heuristic provides very competitive solutions, with respective average gaps of 0.19% and 0.18% from best-known values. Furthermore, the exact B&C framework is also highly competitive with state-of-the-art methods, providing solutions with an average optimality gap of 1.82%.
History: This paper has been accepted for the Transportation Science Special Section on DIMACS Implementation Challenge: Vehicle Routing Problems.
Supplemental Material: The online appendices are available at https://doi.org/10.1287/trsc.2022.0353 .
During the last decade, several retailers have started to combine traditional store deliveries with the fulfillment of online sales to consumers from omni‐channel warehouses, which are increasingly being automated. A popular option is to use autonomous mobile robots (AMRs) in collaboration with human pickers. In this approach, the pickers' unproductive walking time can be reduced even further by zoning the storage system, where the pickers stay within their zone periphery and robots transport order totes between the zones. However, the robotic systems' optimal zoning strategy is unclear: few zones are particularly good for large store orders, while many zones are particularly good for small online orders. We study the effect of no zoning (NZ) and progressive zoning strategies on throughput capacity for balanced zone configurations with both fixed and dynamic order profiles. We first develop queuing network models to estimate pick throughput capacity that correspond to a given number of AMRs and picking with a fixed number of zones. We demonstrate that the throughput capacity is dependent on the chosen zoning strategy. However, the magnitude of the gains achieved is influenced by the size of the orders being processed. We also show that using a dynamic switching strategy has little effect on throughput performance. In contrast, a fixed switching strategy benefiting from changes in the order profile has the potential to increase throughput performance by 17% compared to the NZ strategy, albeit at a higher robot cost.
With the growing demand for speed and flexibility in order fulfillment, it is crucial to employ an efficient picking system that allows faster delivery of items from warehouse storage to a depot. The use of autonomous mobile robots (AMRs) for intra-logistics can improve picking productivity, while alleviating the strain on human workers. In this research, a collaborative human–robot order-picking system (CHR-OPS) is considered, where humans perform item retrieval tasks and AMRs handle item transportation to the depot. The delivery performance of a CHR-OPS for a given set of orders and their due dates depend on three subproblems: (i) order batching (how many items should be collected in each AMR tour?), (ii) batch assignment and sequencing (how to assign batches to AMRs, and in what order should they be processed?), and (iii) picker-robot routing (how should the AMR and picker be routed to coordinate the picking process?). Existing literature has not dealt with the three subproblems, and this work is the first to address them for a picker-to-parts system employing multiple pickers and AMRs. An optimization model is developed to jointly optimize the three problems with the objective of minimizing the total tardiness of all orders. A simulated annealing algorithm with adaptive neighborhood search and optimization-based restart strategy is proposed for handling large instances. The numerical experiments demonstrate the superior performance of the proposed solution approach compared to existing methods. Besides, our results also show that the picking efficiency is impacted by human–robot team composition, AMR speed, AMR capacity and warehouse layout.
In e-commerce warehouses, online retailers increase their efficiency by using a mixed-shelves (or scattered storage) concept, where unit loads are purposefully broken down into single items, which are individually stored in multiple locations. Irrespective of the stock keeping units a customer jointly orders, this storage strategy increases the likelihood that somewhere in the warehouse the items of the requested stock keeping units will be in close vicinity, which may significantly reduce an order picker’s unproductive walking time. This paper optimizes picker routing through such mixed-shelves warehouses. Specifically, we introduce a generic exact algorithmic framework that covers a multitude of picking policies, independently of the underlying picking zone layout, and is suitable for real-time applications. This framework embeds a bidirectional layered graph algorithm that provides the best known performance for the simple picking problem with a single depot and no further attributes. We compare three different real-world e-commerce warehouse settings that differ slightly in their application of scattered storage and in their picking policies. Based on these, we derive additional layouts and settings that yield further managerial insights. Our results reveal that the right combination of drop-off points, dynamic batching, the utilization of picking carts, and the picking zone layout can greatly improve the picking performance. In particular, some combinations of policies yield efficiency increases of more than 30% compared with standard policies currently used in practice.
This paper was accepted by Chung Piaw Teo, optimization.
As a new social e-commerce model, online community group-buying of perishable products has been under examined. This paper addresses a real-world delivery problem faced in common by an online community group-buying operators, in which operators may suffer revenue loss from product deterioration during delivery. Since delivery quantities of candidate orders may be beyond capacity resources, this paper investigates a new combined order selection and periodic vehicle routing problem with time windows for perishable products. An online community group-buying operator can design daily online community group-buying delivery plan by deciding to serve which customers, to deliver which products and delivery quantities to be transported. To solve this problem, we propose a branch-and-price algorithm that strongly relies on a new label setting algorithm with partial label dominance and a strong bounding strategy based on the definition of Pareto-optimal delivery patterns. Since en-route delivery quantities depend on the interval between two consecutive services, our label setting algorithm is also applicable to the pricing subproblem for the discrete split delivery vehicle routing problem. We conduct a case study on a real-world instance and propose management insights into the structure of delivery plan facing inadequate capacity resources. Numerical experiments on 64 randomly generated instances demonstrate the effectiveness of the proposed branch-and-price algorithm.
To reduce unproductive picker walking in traditional picker-to-parts warehousing systems, automated guided vehicles (AGVs) are used to support human order pickers. In an AGV-assisted order-picking system, each human order picker is accompanied by an AGV during the order-picking process. AGVs receive the picked items and, once a picking order is complete, autonomously bring the collected items to the shipping area. Meanwhile, a new AGV is requested to meet the picker at the first storage position of the next picking order. Thus, the picker does not have to return to a central depot and continuously picks order after order. This paper addresses both the routing of an AGV-assisted picker through a single-block, parallel-aisle warehouse and the sequencing of incoming orders. We present an exact polynomial time routing algorithm for the case of a given order sequence, which is an extension of the algorithm of Ratliff and Rosenthal [Ratliff HD, Rosenthal AS ( 1983 ) Order-picking in a rectangular warehouse: A solvable case of the traveling salesman problem. Oper. Res. 1(3):507–521], and a heuristic for the case in which order sequencing is part of the problem. In addition, we investigate the use of highly effective traveling salesman problem (TSP) solvers that can be applied after a transformation of both problem types into a standard TSP. The numerical studies address the performance of these methods and study the impact of AGV usage on picker travel: by using AGVs to avoid returns to the depot and by sequencing in (near-) optimal fashion, picker walking can be reduced by about 20% compared with a traditional setting. Sharing AGVs among the picker workforce enables a pooling effect so that, in larger warehouses, only about 1.5 AGVs per picker are required to avoid picker waiting.
Summary of Contribution: New technologies, such as automatic guided vehicles (AGVs) are currently considered as options to increase the efficiency of the order-picking process in warehouses, which is responsible for a large part of operational warehousing costs. In addition, picker-routing decisions are more and more often based on algorithmic decision support because of their relevance for decreasing unproductive picker walking time. This paper addresses both aspects and investigates routing algorithms for AGV-assisted order picking in parallel-aisle warehouses. We present a dynamic programming routine with polynomial runtime to solve the problem variant in which the sequence of picking orders is fixed. For the variant in which this sequence is a decision, we show that the problem becomes NP-hard, and we propose a greedy heuristic and investigate the use of state-of-the-art exact and heuristic traveling salesman problem solution methods to address the problem. The numerical studies demonstrate the effectiveness of the algorithms and indicate that AGV assistance promises strong improvements in the order-fulfillment process. Because of the practical relevance of AGV-assisted order picking and the presented algorithmic contributions, we believe that the paper is relevant for practitioners and researchers alike.
Efficient order fulfillment processes in warehouses are a key success factor in times of increasing global retail sales volumes and same-day deliveries. To streamline order fulfillment processes, warehouse managers frequently rely on autonomous mobile robots (AMRs). By supporting human order pickers with AMRs, unproductive picker walking time can be reduced, which is essential for increasing the performance of a traditional picker-to-parts system. We study an AMR-assisted picker-to-parts system in which the warehouse is partitioned into disjoint zones, each with one order picker assigned to it. A set of customer orders is given, each associated with a due date until which its items are to be collected. Each AMR of a given fleet is responsible for transporting the items of a single batch of customer orders from the picking area to the depot. An order picker collects those batch items that are stored in her zone and transports them to a handover location. The AMR visits the handover locations and returns to the depot after collecting all batch items. The objective is to minimize the total tardiness of all customer orders. We define the resulting problem as mixed integer program and provide an efficient heuristic solution approach. Furthermore, we investigate the influence of increasing the AMR fleet size and varying travel and walking speed ratios between AMRs and order pickers. We find that a slight increase in the speed ratio leads to a larger reduction of the total tardiness compared to increasing the AMR fleet size.
Intelligent part‐to‐picker systems are spreading across a broad range of industries as preferred solutions for agile order fulfillment, wherein mobile racks are carried by robots and moved to stations where human pickers can pick items from them. Such systems raise the challenge of designing good work schedules for human pickers; they also give rise to a new class of operational scheduling problems in human‐robot coordinated order picking systems. This work studies the problem of finding a suitable robot schedule that takes into account the schedule‐induced fluctuation of the working states of human pickers. A proposed model enables mobile racks with various workloads to be assigned to pickers, and schedule the racks that are assigned to every picker to minimize the expected total picking time. The problem is formulated as a stochastic dynamic program model. An approximate dynamic programming (ADP)‐based branch‐and‐price solution approach is used to solve this problem. The developed model is calibrated using data that were collected from a dominant e‐commerce company in China. Pickers’ working state transitions are modeled based on data obtained from this warehouse. Counter‐factual studies demonstrate that the proposed approach can solve a moderately sized problem with 50 racks in under two minutes. More importantly, the approach yields high‐quality solutions with picking times that are 10% shorter than the solutions that did not consider schedule‐induced fluctuations of pickers’ working states.
We consider a vehicle routing problem which seeks to minimize cost subject to time window and synchronization constraints. In this problem, the fleet of vehicles is categorized into regular and special vehicles. Some customers require both vehicles’ services, whose service start times at the customer are synchronized. Despite its important real-world application, this problem has rarely been studied in the literature. To solve the problem, we propose a Constraint Programming (CP) model and an Adaptive Large Neighborhood Search (ALNS) in which the design of insertion operators is based on solving linear programming (LP) models to check the insertion feasibility. A number of acceleration techniques is also proposed to significantly reduce the computational time. The computational experiments show that our new CP model finds better solutions than an existing CP-based ALNS, when used on small instances with 25 customers and with a much shorter running time. Our LP-based ALNS dominates the CP-based ALNS, in terms of solution quality, when it provides solutions with better objective values, on average, for all instance classes. This demonstrates the advantage of using linear programming instead of constraint programming when dealing with a variant of vehicle routing problems with relatively tight constraints, which is often considered to be more favorable for CP-based methods. We also adapt our algorithm to solve a well-studied variant of the problem, and the obtained results show that the algorithm provides good solutions as state-of-the-art approaches and improves four best known solutions.
Most search game models assume that the hiding locations are identical and the players’ objective is to optimize the search time. However, there are some cases in which the players may differentiate the hiding locations from each other and the objective is to optimize a weighted search time. In addition, the search may involve multiple search teams. To address these, we introduce a new network search game with consideration given to the weights at different locations. A hider can hide multiple objects and there may be multiple search teams. For a special case of the problem, we prove that the game has a closed-form Nash equilibrium. For the general case, we develop an algorithm based on column and row generation. We show that the Searcher’s subproblem is NP-hard and propose a branch and price algorithm to solve it. We also present a polynomial time algorithm for the Hider’s subproblem. Numerical experiments illustrate the performance of the approach and reveal insights on the properties of this game. The computational results also demonstrate the efficiency of the proposed algorithms.
This study addresses a vehicle routing problem (VRP) in which demands are discrete, split delivery is allowed, service time is proportional to the units of delivered products, multiple time windows are provided and the demand of each customer must be delivered in only one time window (this requirement is termed synchronization constraint). We formulate this problem into a three-index vehicle-flow model and a set-covering model. Then, we propose a branch-and-price-and-cut algorithm to solve the problem. We compare our branch-and-price-and-cut algorithm with CPLEX based on 252 randomly generated instances and the computational results demonstrate the effectiveness of our algorithm.
In this paper, we address a new variant of the cutting stock problem, in which skiving is allowed and setup costs are considered. Specifically, the output sheets are generally longer but narrower than the input coils. To satisfy the demand of each sheet, combining two or more coils is allowed. Moreover, because changing from one pattern to another involves considerable time and cost, minimizing the number of different patterns or setups is vital. Thus, the objective of our study is to minimize the material cost and the number of setups. We propose an integer programming (IP) formulation for the problem that contains an exponential number of binary variables and column-dependent constraints. The linear programming (LP) relaxation is solved using a column-and-row generation framework that involves a knapsack subproblem and a nonlinear IP subproblem. For the latter, we propose a decomposition-based exact solution method with pseudo-polynomial time. To obtain IP solutions, we develop a diving heuristic based on matching level. The computational experiments show that these algorithms are efficient and of high quality.
One of the main problems in production systems is the buffer sizing. Choosing the right buffer size, at each production stage, that allows to achieve some performance measure (usually throughput or waiting time) is known as Buffer Allocation Problem (BAP), and it has been widely studied in the literature. Due to its complexity, BAP is usually approached using decomposition methods, under very strict system assumptions, or using simulation-optimization techniques. In this paper, the approximated mathematical programming formulation of the BAP simulation-optimization based on the time buffer concept is used. Using this approximation, buffers are modeled as temporal lags (time buffers) and this allows to use Linear Programming (LP) instead of Mixed Integer Linear Programming (MILP) models. Although LP models are easier to solve than MILPs, the huge dimension and the complex solution space topology of the time buffer approximated BAP calls for ad hoc solution algorithms. To this purpose, a row-column generation algorithm is proposed, which exploits the theoretical properties of the time buffer approximation to reduce the solution time. The proposed algorithm has been compared with a standard LP solver (ILOG CPLEX) and with a state-of-the-art MILP solver and it proved to be better than the LP solver in most of the cases, and more robust than the MILP solver with respect to computation time. Moreover, the LP model (for flow lines) is able to solve the BAP also for assembly/disassembly lines.
Online restaurant aggregators with integrated meal delivery networks have become more common and more popular in the past few years. Meal delivery is arguably the ultimate challenge in last-mile logistics: a typical order is expected to be delivered within an hour (much less if possible) and within minutes of the food becoming ready. We introduce a novel formulation for a meal delivery routing problem (in which we assume perfect information about order arrivals) and develop a simultaneous column- and row-generation method for its solution. The analysis of the results of an extensive computational study, using instances derived from real-life data, demonstrates the efficacy of the solution approach, and provides valuable insights into, among others, the (potential) benefits of order bundling, courier-shift scheduling, and demand management.
Robotic handling systems are increasingly applied in distribution centers. They require little space, provide flexibility in managing varying demand requirements, and are able to work 24/7. This makes them particularly fit for e-commerce operations. This paper reviews new categories of automated and robotic handling systems, such as shuttle-based storage and retrieval systems, shuttle-based compact storage systems, and robotic mobile fulfillment systems. For each system, we categorize the literature in three groups: system analysis, design optimization, and operations planning and control. Our focus is to identify the research issue and operations research modeling methodology adopted to analyze the problem. We find that many new robotic systems and applications have hardly been studied in academic literature, despite their increasing use in practice. Because of unique system features (such as autonomous control, flexible layout, networked and dynamic operation), new models and methods are needed to address the design and operational control challenges for such systems, in particular, for the integration of subsystems. Integrated robotic warehouse systems will form the next category of warehouses. All vital warehouse design, planning, and control logic, such as methods to design layout, storage and order-picking system selection, storage slotting, order batching, picker routing, and picker to order assignment, will have to be revisited for new robotized warehouses.
The cost-effective and high-quality city logistics service is a key for the success of e-commerce enterprises. Synchronization (sync) is emerging as a typical yet challenging requirement, which asks for simultaneous deliveries of multiple products to the same customer. This paper is among the first to model the sync constraints with sliding time windows (STW). STW is a special type of time window of which only the window size is defined. Unlike traditional time windows, the start and end time of STW could be adjusted earlier or later, so long as their difference equals to the pre-defined window size. In this sense, the STW is a time constraint with partially unknown factors. Such flexibility of STW can greatly improve the efficiency of vehicle tours, because customers could be served in a more flexible sequence decided by the transporter. A novel divide and conquer based algorithm is developed to tackle the proposed problem with partially unknown time constraints. The values for STWs will dynamically be determined during the algorithm. Numerical studies show that by modelling sync requirements with STW, transportation cost could be saved as much as in half. We also carry out sensitivity analyses on the key factors such as promised sync level to customers and the complexity of online orders.
The robotic mobile fulfillment system (MFS) is widely used for automating storage pick and pack activities in e-commerce distribution centers. In this system, the items are stored on movable storage shelves, also known as inventory pods, and brought to the order pick stations by robotic drive units. We develop stylized performance evaluation models to analyze both order picking and replenishment processes in a mobile fulfillment system storage zone, based on multi-class closed queueing network models. To analyze robot assignment strategies for multiple storage zones, we develop a two-stage stochastic model. For a single storage zone, we compare dedicated and pooled robot systems for pod retrieval and replenishment. For multiple storage zones, we also analyze the effect of assigning robots to least congested zones on system throughput in comparison to random zone assignment. The models are validated using detailed simulations. For single zones, the expected throughput time for order picking reduces to one-third of its initial value by using pooled robots instead of dedicated robots; however, the expected replenishment time estimate increases up to three times. For multiple zones, we find that robots that are assigned to storage zones with dedicated and shortest queues provide a greater throughput than robots assigned at random to the zones.
Motivated by recent technological advances in mobile robotics, this paper explores a novel approach for warehouse order picking. In particular, this work considers two types of commercially available mobile robots – one that can grasp items from a shelf (a picker) and another (a transporter) that can quickly deliver all items from the pick list to the packing station. A new vehicle routing problem is defined which seeks to minimise the time to deliver all items from a pick list to the packing station, a problem termed the pick, place, and transport vehicle routing problem. A mixed integer linear programming formulation is developed to answer three related research questions. First, what combination of picker and transport robots is required to obtain performance exceeding traditional human-based picking operations? Second, how should the composition of the robot fleet be altered to affect the greatest performance improvements? Finally, what are the impacts of warehouse layout designs when coordinated mobile robots are deployed? An extensive numerical analysis reveals that, (1) increasing the number of cross aisles decreases system performance; (2) centrally located packing stations improve system performance; and (3) the average distance from each pick location to the packing station and the average distance between pick locations are effective metrics for identifying specific fleet modifications that are likely to yield system improvements.
E-commerce retailers face the challenge to assemble large numbers of time-critical picking orders each consisting of just a few order lines with low order quantities. Traditional picker-to-parts warehouses are often ill-suited for these prerequisites, so that automated warehousing systems (e.g., automated picking workstations, robots, and AGV-assisted order picking systems) are applied and organizational adaptions (e.g., mixed-shelves storage, dynamic order processing, and batching, zoning and sorting systems) are made in this branch of industry. This paper is dedicated to these warehousing systems especially suited for e-commerce retailers. We discuss suited systems, survey the relevant literature, and define future research needs.
This paper addresses a special vehicle routing problem, which extends the classical problem by considering the time window and synchronized-services constraints. A time window is associated with each client service and some services require simultaneous visits from different vehicles to be accomplished. This problem has many practical applications such as caregiver scheduling problem encountered in the home health care industry. The synchronization constraints in this problem interconnect various vehicles’ routes, making the problem more challenging than standard vehicle routing problem with time windows, especially in designing neighborhood search-based methods. A mixed-integer programming model is proposed for the problem. Motivated by the challenge of computational time, an efficient Adaptive Large Neighborhood Search heuristic is proposed to solve the problem. The approach is evaluated on benchmark instances acquired from the literature and new large-scale test instances first generated in this paper. The numerical results indicate that our solution method is able to outperform existing approaches.
Synchronization of workers and vehicles plays a major role in many industries such as logistics, healthcare or airport ground handling. In this paper, we focus on operational ground handling planning and model it as an archetype of vehicle routing problems with multiple synchronization constraints, coined as “abstract vehicle routing problem with worker and vehicle synchronization” (AVRPWVS). The AVRPWVS deals with routing workers to ground handling jobs such as unloading baggage or refuelling an aircraft, while meeting each job's time window. Moreover, each job can be performed by a variable number of workers. As airports span vast distances and due to security regulations, workers use vehicles to travel between locations. Furthermore, each vehicle, moved by a driver, can carry several workers. We propose two mathematical multi-commodity flow formulations based on time-space networks to efficiently model five synchronization types including movement and load synchronization. Moreover, we develop a branch-and-price heuristic that employs both conventional variable branching and a novel variable fixing strategy. We demonstrate that the procedure achieves results close to the optimal solution in short time when compared to the two integer models.
In this paper, we consider a two-stage extension of one-dimensional cutting stock problem which arises when technical requirements inhibit cutting large stock rolls to demanded widths of finished rolls directly. Therefore, demands on finished rolls are fulfilled through two subsequent cutting processes, in which rolls produced in the former are used as input for the latter, while the number of stock rolls used is minimized. We tackle the pattern-based formulation of this problem which typically has a very large number of columns and constraints. The special structure of this formulation induces both a column-wise and a row-wise increase when solved by column generation. We design an exact simultaneous column-and-row generation algorithm whose novel element is a row-generating subproblem that generates a set of columns and rows. For this subproblem, which is modeled as an unbounded knapsack problem, we propose three algorithms: implicit enumeration, column generation which renders the overall methodology nested column generation, and a hybrid algorithm. The latter two are integrated in a well-known knapsack algorithm which forges a novel branch-and-price algorithm for the row-generating subproblem. Extensive computational experiments are conducted, and performances of the three algorithms are compared.
We consider two types of disruptions arising in the multi-depot vehicle scheduling; the delays and the extra trips. These disruptions may or may not occur during operations, and hence they need to be indirectly incorporated into the planned schedule by anticipating their likely occurrence times. We present a unique recovery method to handle these potential disruptions. Our method is based on partially swapping two planned routes in such a way that the effect on the planned schedule is minimal, if these disruptions are actually realized. The mathematical programming model for the multi-depot vehicle scheduling problem, which incorporates these robustness considerations, possesses a special structure. This special structure causes the conventional column generation method fall short as the resulting problem grows also row-wise when columns are generated. We design an exact simultaneous column-and-row generation algorithm to find a valid lower-bound. The novel aspect of this algorithm is the pricing subproblem, which generates pairs of routes that form recovery solutions. Compromising on exactness, we modify this algorithm in order to enable it to solve practical-sized instances efficiently. This heuristic algorithm is shown to provide very tight bounds on the randomly generated instances in a short computation time.
This paper treats a special parts-to-picker based order processing system, where mobile robots hoist racks and bring them directly to stationary pickers. This technological innovation – known as the Kiva system – heavily influences all traditional planning problems to be solved when operating a warehouse. We, specifically, tackle the order processing in a picking station, i.e., the batching and sequencing of picking orders and the interdependent sequencing of the racks brought to a station. We formalize the resulting decision problem and provide suited solution procedures. In a comprehensive computational study we show that an optimized order picking allows to more than halve the fleet of robots compared to simple decision rules often applied in real-world warehouses.
This paper models Robotic Mobile Fulfillment Systems and analyzes their performance. A Robotic Mobile Fulfillment System is an automated, parts-to-picker storage system where robots bring pods with products to a workstation. It is especially suited for e-commerce distribution centers with large assortments of small products, and with strong demand fluctuations. Its most important feature is the ability to automatically sort inventory and to adapt the warehouse layout in a short period of time. Queueing network models are developed for both single-line and multi-line orders, to analytically estimate maximum order throughput, average order cycle time, and robot utilization. These models can be used to quickly evaluate different warehouse layouts, or robot zoning strategies. Two main contributions are that the models include accurate driving behavior of robots and multi-line orders. The results show that: 1) the analytical models accurately estimate robot utilization, workstation utilization, and order cycle time 2) maximum order throughput is quite insensitive to the length-to-width ratio of the storage area and 3) maximum order throughput is affected by the location of the workstations around the storage area.
In this article, we propose a branch-and-price-and-cut (BPC) algorithm to exactly solve the manpower routing problem with synchronization constraints (MRPSC). Compared with the classical vehicle routing problems (VRPs), the defining characteristic of the MRPSC is that multiple workers are required to work together and start at the same time to carry out a job, that is, the routes of the scheduling subjects are dependent. The incorporation of the synchronization constraints increases the difficulty of the MRPSC significantly and makes the existing VRP exact algorithm inapplicable. Although there are many types of valid inequalities for the VRP or its variants, so far we can only adapt the infeasible path elimination inequality and the weak clique inequality to handle the synchronization constraints in our BPC algorithm. The experimental results at the root node of the branch-and-bound tree show that the employed inequalities can effectively improve the lower bound of the problem. Compared with ILOG CPLEX, our BPC algorithm managed to find optimal solutions for more test instances within 1 hour. © 2016 Wiley Periodicals, Inc. Naval Research Logistics, 2016
We study a tactical level crew capacity planning problem in railways which determines the minimum required crew size in a region while both feasibility and connectivity of schedules are maintained. We present alternative mathematical formulations which depend on network representations of the problem. A path-based formulation in the form of a set-covering problem along with a column-and-row generation algorithm is proposed. An arc-based formulation of the problem is solved with a commercial linear programming solver. The computational study illustrates the effect of schedule connectivity on crew capacity decisions and shows that arc-based formulation is a viable approach.
This paper presents a branch-and-price-and-cut algorithm for the exact solution of the active-passive vehicle-routing problem (APVRP). The APVRP covers a range of logistics applications where pickup-and-delivery requests necessitate a joint operation of active vehicles (e.g., trucks) and passive vehicles (e.g., loading devices such as containers or swap bodies). The objective is to minimize a weighted sum of the total distance traveled, the total completion time of the routes, and the number of unserved requests. To this end, the problem supports a flexible coupling and decoupling of active and passive vehicles at customer locations. Accordingly, the operations of the vehicles have to be synchronized carefully in the planning. The contribution of the paper is twofold: First, we present an exact branch-and-price-and-cut algorithm for this class of routing problems with synchronization constraints. To our knowledge, this algorithm is the first such approach that considers explicitly the temporal interdependencies between active and passive vehicles. The algorithm is based on a nontrivial network representation that models the logical relationships between the different transport tasks necessary to fulfill a request as well as the synchronization of the movements of active and passive vehicles. Second, we contribute to the development of branch-and-price methods in general, in that we solve, for the first time, an ng-path relaxation of a pricing problem with linear vertex costs by means of a bidirectional labeling algorithm. Computational experiments show that the proposed algorithm delivers improved bounds and solutions for a number of APVRP benchmark instances. It is able to solve instances with up to 76 tasks, four active, and eight passive vehicles to optimality within two hours of CPU time.
The online appendix is available at https://doi.org/10.1287/trsc.2016.0730 .
This paper presents a survey of vehicle routing problems with multiple synchronization constraints. These problems exhibit, in addition to the usual task covering constraints, further synchronization requirements between the vehicles, concerning spatial, temporal, and load aspects. They constitute an emerging field in vehicle routing research and are becoming a "hot" topic. The contribution of the paper is threefold: (i) It presents a classification of different types of synchronization. (ii) It discusses the central issues related to the exact and heuristic solution of such problems. (iii) It comprehensively reviews pertinent literature with respect to applications as well as successful solution approaches, and it identifies promising algorithmic avenues.
This paper addresses a routing problem where the fulfillment of transport requests requires two types of transport resources, namely, passive and active means of transport. The passive means are used for holding the cargo that is to be shipped from pickup to delivery locations. The active means take up the passive means and carry them from one location to another. Compared to classical vehicle routing problems, the additional challenge in this combined routing problem is that the operations of both transport resources have to be synchronized. In this paper, we provide a modeling approach for the joint routing of passive and active means of transport. We solve the problem by large neighborhood search meta-heuristics that utilize various problem-specific components, for example local search techniques for the routes of active and passive means. A computational study on a large set of benchmark instances is used for assessing the performance of the meta-heuristics.
A general problem in health-care consists in allocating some scarce medical resource, such as operating
rooms or medical staff, to medical specialties in order to keep the queue of patients as short as possible. A major difficulty stems from the fact that such an allocation must be established several months in
advance, and the exact number of patients for each specialty is an uncertain parameter. Another problem
arises for cyclic schedules, where the allocation is defined over a short period, e.g. a week, and then
repeated during the time horizon. However, the demand typically varies from week to week: even if
we know in advance the exact demand for each week, the weekly schedule cannot be adapted accordingly.
We model both the uncertain and the cyclic allocation problem as adjustable robust scheduling
problems. We develop a row and column generation algorithm to solve this problem and show that it
corresponds to the implementor/adversary algorithm for robust optimization recently introduced by
Bienstock for portfolio selection. We apply our general model to compute master surgery schedules for
a real-life instance from a large hospital in Oslo.
In this study, we solve a robust version of the airline crew pairing problem. Our concept of robustness was partially shaped during our discussions with small local airlines in Turkey which may have to add a set of extra flights into their schedule at short notice during operation. Thus, robustness in this case is related to the ability of accommodating these extra flights at the time of operation by disrupting the original plans as minimally as possible. We focus on the crew pairing aspect of robustness and prescribe that the planned crew pairings incorporate a number of predefined recovery solutions for each potential extra flight. These solutions are implemented only if necessary for recovery purposes and involve either inserting an extra flight into an existing pairing or partially swapping the flights in two existing pairings in order to cover an extra flight. The resulting mathematical programming model follows the conventional set covering formulation of the airline crew pairing problem typically solved by column generation with an additional complication. The model includes constraints that depend on the columns due to the robustness consideration and grows not only column-wise but also row-wise as new columns are generated. To solve this difficult model, we propose a row and column generation approach. This approach requires a set of modifications to the multi-label shortest path problem for pricing out new columns (pairings) and various mechanisms to handle the simultaneous increase in the number of rows and columns in the restricted master problem during column generation. We conduct computational experiments on a set of real instances compiled from local airlines in Turkey.
The multistage cutting stock problem (CSP) generalizes the one-dimensional CSP when a lengthwise cutting process is distributed over two or more successive stages. At every stage of the cutting process incoming rolls are slit into smaller rolls by width. The problem is to minimize total trim loss occurring at all stages of technological process meeting customer demands for finished rolls. We propose a row and column generation technique for solving the multistage one-dimensional CSP. The technique is a generalization of the column generation method Suggested by Gilmore and Gomory for solving a classic CSP. The procedure generates only those intermediate rolls (rows) and cutting patterns (columns) that are needed. An auxiliary problem embedded into the frame of the revised simplex algorithm is a non-linear knapsack problem that can be solved efficiently. Computational results prove the overall method is a valuable addition to the toot set for modeling and solving the multistage CSP.