hierarchical

Development of a design methodology for warehousing systems:hierarchical frameworkMarc GoetschalckxLeon McGinnisGunter SharpDoug BodnerT. GovindarajKai HuangSchool of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlanta, GA 30332-0205AbstractWe will present a progress report on our effort to develop a science-based design methodology for warehousing systems. In particular, we will discuss the hierarchical structure of our design procedure. This includes the decisions and parameters used in the master model, the various sub-models, and the decision and data interaction between the various levels in the hierarchy. We will present several sub-models that are specific for a particular function or material handling equipment. Computational results of the application of the sub-model for modular storage drawers will also be shared.KeywordsWarehousing system, design methodology, modular storage drawers, bin shelving.1. IntroductionThe goal is the development of a systematic methodology to rapidly design warehousing systems. At the current time, most of warehousing design is either based on ad-hoc insight and experience of the warehouse designer or on detailed simulation of the equipment and material flows through the warehouse. However, the current business climate does not allow for the several weeks it takes to develop such a detailed simulation model. Third party logistics service providers (TPL) are routinely faced with a two-week cycle from the start of the project to the signing of a binding contract. With such tight deadlines, the highly detailed simulations by powerful contemporary simulation packages such as Witness, Arena, and AutoMod require too much time to develop, even with the built-in material handling constructs. There exists an urgent need for a design methodology that requires less detailed data, is less complex, and requires a shorter time-to-design. Typically the software programs that implement this type of design methodology are called rapid prototyping tools.The warehouse design has to satisfy a number of high-level objectives and constraints. Most high-level warehousing constraints are elastic, in other words, they can be violated with a certain penalty. We will develop a warehouse design formulation, whose overall structure is a multiperiod, multicommodity, capacitated network flow problem, where capacities are determined by binary configuration variables. There are costs associated both with the continuous flow and storage variables and the binary configuration variables. The formulation hence belongs to the class of mixed-integer linear programming problems. Such problems are known to be difficult to solve for large problem instances. While we realize that the data on which the design i s based is subject to a large degree of uncertainty and even the warehouse mission may change significantly over the lifetime of the warehouse, we assume at the current time that all data is deterministic and known with complete certainty.1.1 Science-Based Design MethodologyA scientific design methodology must satisfy certain necessary conditions. First it must be able to search over a large solution space that is not limited by the experience or intuition of the designer. In other words, it must be capable of generating “new” alternatives that were not known before the start of the design effort. A second requirement is that the result of the design effort must be repeatable. In other words, given an identical set of data, the same design should be g enerated. Given the large set of data and complex design calculations, the design procedure must also be programmable so that powerful computational tools can be developed to increase the designquality and reduce the design time.Due to the complexity of the design problem, a hierarchical design procedure appears to be a natural choice for a design framework. In any hierarchical design procedure, several questions must be addressed. The first decision determines at which level in the hierarchy different design decisions are made. The second decision determines how the decisions made in the different levels interact to generate a complete design specification. Often an iterative procedure is specified and for such procedures the convergence and optimality properties must be determined. Any design procedure must strike a balance between solution quality, solution resolution or level of detail, and solution time.Even though warehouse design is a very common activity during the configuration of logistics systems, there does not appear to exist a large literature focused on warehouse design. The warehouse design problem has been discussed by Frazelle (1996) and Ackerman (1999). An early review of models and design procedures by Ashayeri and Gelders (1985) advocate the combination of optimization and simulation design algorithms. Cormier and Gunn (1992) provide an early review of warehousing models, while Rouwenhorst et al (2000) and van den Berg and Zijm (1999) give a more recent reviews. Graves (1998) reports that there exist a large body of literature on the analysis and isolated design of warehousing components. Two hierarchical design procedures have been proposed by Gray et al. (1992) and Miller (2001). This manuscript is a progress report on a multi-year, multi-faculty effort in the Keck Virtual Factory Lab and some of the earlier results are presented several research proceedings such as Goetschalckx et al. (2001).2. Formulation Development – Warehouse System FormulationAn iterative hierarchical design procedure is proposed. At the top level the procedure iterates between a capacitated material flow model (CMF) and a warehouse block layout model (WBL). The CMF determines the major material flows through the warehouse and the technologies and equipment installed in the warehouse. The WBL determines the warehouse layout and thus the distances traveled by the various material flows. The CMF calls two types of sub models. The first type models all the activities inside a functional area and the second type models the transportation activities between various functional areas. Following is summary of the structure, constraints, and objectives of the formulation. A detailed formulation containing the mathematical equations is given in Goetschalckx and Huang (2001).2.1 Objective FunctionThe objective function is typically related to the economic performance of the warehousing system. The most common objective is the minimization of the sum of the time-discounted costs associated with operating t he warehouse. Both one-time investment costs, annual or major time-unit fixed costs, and variable costs can be included in the objective function.2.2 Functional AreasThe warehouse is modeled as a collection of components, also called departments or functional areas. These components are connected by a function flow network, which models the flow of materials from arrival at to departure from the warehousing system. The functional flow network is illustrated in Figure 1. The individual components are subject to constraints. The function flow network also requires conservation of flow constraints between the individual components. An example of a functional area would reserve storage.Material flow Function-to-space mapping2.3 Commodity flowsDifferent product or commodity flows arrive at the warehouse, pass through the functional areas of the warehouse, and finally depart the warehousing system. The number of commodity flows m odeled in the warehouse should be kept as small as possible to maintain control of the model size, data, and algorithm requirements, but it should belarge enough to capture the major activities in the warehouse. Pareto analysis may be used to group the l arge number of SKUs in a small number of commodities with similar characteristics, such as flow path, physical characteristics, and schedules. A different commodity flow is also defined for the three major unit load sizes being stored and handled in the w arehouse: pallet (also called unit load), case (also called box or carton), and item (also called piece).2.4 TechnologiesEach module or functional area that is implemented in the warehouse must use one or more technologies to execute its function. For example, the principal technologies for pallet-sized unit load storage are floor stacking, single deep rack, double deep rack, and deep lane storage. The main performance requirements for a pallet storage area are storage capacity and maximum input and output flows per time unit. The main resource requirements associated with a pallet storage area are floor space, cubic space, labor, material handling equipment acquisition and operating costs, and investment and operating capital. In the next paragraphs the variables and constraints to model these characteristics will be developed.The flow capacity of a technology has to be determined in advance. It is assumed that each technology is sufficiently configured before it is entered into the model so that the flow capacity, the storage capacity, and the cost coefficients can be determined. For example, an AS/RS system with four or fives aisles would be modeled as two different technologies, one with four aisles and one with five aisles, because of the significant impact of the number of aisles on the throughput capacity and cost of the AS/RS.2.5 ResourcesIf joint capacity restrictions exist that impact more than a single commodity, resources model these capacity restrictions. Typical resources are labor hours by labor grade, equipment hours by equipment type, space such as two-dimensional floor space and three-dimensional cubic space, and investment budget. By definition resources have a maximum availability during each time period.2.6 Generalized Conservation of FlowThere are three possible types of generalized conservation of flow constraints that must be enforced in order to yield a feasible warehouse design. The first type is the traditional balance of flow by commodity and time period of input and output flows in a functional area. The second type is the balance of flow across time periods, which incorporates the storage and inventory effects. The third type is the balance of flow among different commodities when commodities are transformed into another commodity. The prime example is a case picking functional area where all input flows are in pallets, but all output flows are in cartons or cases. The types are cumulative, in other words, type three constraints incorporate all effects of type one and type two. The goal is to keep the model as simple as possible, so if type one constraints suffice, type two and type three constraints will not be included in the model for that functional area.Type one constraints ensure balance of input and output flows and consistency with the throughput flow for a functional area. It should be noted that type one constraints do not incorporate any relationship between material flows in different time periods. Type two constraints incorporate in addition the initial and final inventory of the commodity in a functional area. Type two constraints model the relationships between material flows in different time periods.The conservation of flow constraints of type three model the transformation of one commodity into another commodity in a functional area. This type of conservation constraints is typically associated with manufacturing and supply chains, but it occurs also in warehousing. The prime example of such transformation operations are case picking operations, where the products enter the area on pallets and leave the functional area as individual cases. Here two different commodities are required to model the different physical characteristics of the material flows, even though all products in the warehouse may follow the same flow path and be otherwise identical.While each of the constraints or objective function components has a simple structure, the overall formulation represents a mixed-integer programming formulation. In addition, the costs and constraints associated with the various technologies may generate nonlinear models. It is very important to keep the number of commodities to a minimum to constrain the overall size of the formulation. The greatest challenge however is to determine and validate the data required to populate this formulation.Our next efforts will focus on constructing the formulation for a warehouse example out of the literature. This will be followed by computational experiment.3. Formulation Development – Small Parts Functional Area Formulation3.1 IntroductionIn this section we will describe the formulation developed for the storage of small parts. This type of functional area is present in repair parts warehouses and warehouse for small parts consumer and industrial products. There exist several alternative technologies for a small parts storage system, ranging from the low-investment bin shelving, through modular storage drawers, horizontal and vertical carousels, microload, and automated A-frame. To our knowledge, the only models for the design of such systems are given in a series of publications by the Material Handling Research Center in the late 1980’s and early 1990’s. Examples are Frazelle (1988, 1989), Frazelle and McGinnis (1989), Choe and Sharp (1992), Sharp et al (1991), Sharp and Yoon (1996), Herrera-Cuellar (1984), and Houmas (1986). The models presented there calculate the equipment cost, labor cost, and space requirements for technologies for small parts storage. The objective is minimize the aggregate sum of labor, equipment, and space costs, subject to technology constraints and a list of parts to be stored with a given maximum inventory per part. We will consider in this manuscript only two storage technologies: modular storage drawers and bin shelving.The formulations for the design of such small parts storage systems result in very large mixed-integer formulations. The integer variables correspond to the configuration and sizing of the technology. The orientation, assignment, and location of the parts can either be modeled with integer or continuous variables. These mixed-integer formulations cannot be solved to optimality in the framework of the iterative warehouse design algorithm because of their excessive computational requirements. There exist three possible approximations that can reduce the computational burden: 1) approximation of the comprehensiveness or completeness of the model by ignoring factors and cost categories, 2) approximation of modeling realism by aggregating individual parts or aggregating part and storage module dimensions, and 3) approximation of solution quality by accepting heuristic rather than optimal solutions. The extent of approximation in model and solution algorithm should be consistent. There exists very little benefit in spending additional computational resources to find an optimal solution to a very approximate formulation. It should be observed that each of the levels of the model completely answers all the design questions associated with its technology, albeit at different levels of detail. The models can thus be used independently or in parallel. The models can be used to validate each other and higher level models can be used to refine the parameter values for lower level models. Houmas (1986) d escribes a design procedure based on the sequential use of models, where each level answers a part of all the design questions and the models are used in a sequential manner.We developed three consistent models for the small parts storage systems design with increasing level of detail.3.2 Continuous Model (Level One)The continuous model computes the number of required shelving units or modular storage drawer units based on the aggregate cubic volume that has to be stored for all parts and the approximate net cubic volume that each shelving unit or drawer unit provides. The physical dimensions in the three axes for the parts and the storage units are ignored. The part storage inventory volume can be thought of as a deformable shape, such as water balloon, of the correct volume. Hence, this level of model is often called the fluid level or fluid model. The number of storage units is computed by rounding up the ratio of the total volume required divided by the volume capacity of each storage unit. The number of aisles is computed by rounding up the ratio of the total number of storage units and the number of units on both sides of an aisle. The area cost and equipment costs are computed directly in function of the number of storage units and footprint of the aisles. Labor cost is computed based on an average extraction time and replenishment time and then multiplied by the number of extractions (line items) and replenishments and the wages. This model does not include calculations of the number of drawers or shelves, only of drawer units and shelf units. The main advantage of this fluid model is that it can be solved extremely fast. Its level of accuracy may be sufficient to exclude a particular technology, e.g., bin shelving may be eliminated from further consideration because it requires too much space or modular drawer technology may be eliminated based on its investment cost. A more detailed description and development of the two fluid models for small parts storage based on bin shelving and modular drawer technologies are given in Goetschalckx et al. (2002).3.3 Vertical Dimension Model (Level Two)The vertical dimension model explicitly considers the vertical dimension of the individual parts, the number of layers of the parts stacked on top of each other in the storage drawer or shelf, and the vertical size of the modular drawers and the vertical distance between shelves in the shelving units. However, the other two dimensions of the part are still aggregated to yield a continuous area. Hence, the inventory of parts is considered to “fit” in a drawer if the sum of the areas of the parts in the drawer is less than or equal to the horizontal area of the drawer. The solution provides the assignment of individual parts to individual shelves, so that extraction times can include the vertical location of the parts in the shelving unit to model the “golden zone” for part extraction. The number of drawers or shelves iscomputed explicitly in the model. The aggregate cost calculations are performed i n a similar manner as for the continuous model, but the costs of drawers and shelves is included explicitly. The calculations in this model are much more realistic than in the continuous model, but the model can easily contain hundreds of thousands of variables and constraints. Since the technology sub-models have to be solved for repeatedly, heuristic solution methods are almost surely required. This level of model can be used to evaluate a limited number of alternative designs. A more detailed description of this level of model can be found in Goetschalckx and Huang (2001).3.3 Three-Dimensional Model (Level Three)The three dimensional model explicitly considers all three dimensions of the individual parts and the three dimensions of the storage drawer or shelf. The maximum number of layers can still be computed in advance for every combination of part and drawer or shelf. Computing the number of shelves thus involves solving a very large number of two-dimensional bin-packing problems. The number of constraints and variables is even larger than in the vertical dimension model, and the solution algorithms are much more demanding. This level of model can only be used to configure the final, selected technology for the small parts storage. It is the solution of this level of model than then can be used in digital simulation to verify the feasibility of the proposed warehouse design.4. Iterative Warehouse Design AlgorithmThe above formulation, encompassing both the system model and the various functional areas and technology sub-modes, are posed and solved before the layout of the warehouse is known, since at this time the required areas for the functional areas are unknown because their technologies have not yet been defined. This implies that the formulation cannot contain costs and resources used associated with inter-departmental moves and transportation. To determine the warehouse configuration and layout, we propose the following iterative algorithm.1. Solve the Capacitated Material Flow (CMF) f ormulation developed above. For each functional area thesolution determines the one or more technologies used, the required area for each functional area, and the material flows between the various functional areas.2. Based on the required areas for each f unctional department and the material flows solve the WarehouseBlock Layout (WBL) formulation. Traditional techniques for determining the block layout can be used. The resulting layout solution determines the location of each functional area and the dis tances between the various functional areas.3. Compute the required transportation resources and cost and add them to the objective function of CMF toobtain the overall cost of warehousing operations.4. If so desired, the cost parameters for the CMF formulation can be updated and CMF and WBL solvediteratively until both solutions converge.To solve the above formulation, the following technologies are currently used. All the data parameters and solution variables are stored in an object-oriented database. To solve the mixed-integer formulations, we use the MIP module of CPLEX, a commercial linear programming solver. A variety of custom heuristics for sizing and allocation of parts also has been developed and implemented. A custom program has been developed to extract the data from the database and populate the formulation and to extract the values of the decision variables of the solution and insert them back into the database. The current database is implemented in Microsoft Access.5. Future ResearchAt the current time, numerical experiments are being conducted to judge the required data parameters for the technology sub-models and the required running times for the various levels of solution algorithms. The iterative algorithms proposed in this manuscript may or may not converge to single design. Other decomposition schemes that have proven convergence are in their first stage of development.An obvious extension is to include more technology alternatives for the small parts storage and extraction processes. Models for more mechanized systems such as carousels, microloads, and A-frames have to be included to allow the system model to select the appropriate technology for this functional area.The data on which the design is based is subject to a large degree of uncertainty, because products and business conditions have to be forecasted. The warehouse mission itself may change significantly over the lifetime of the warehouse. The current formulation and solution procedure assume that all data is deterministic and known with complete certainty. An extension of the current formulation is to incorporate explicitly the uncertainty, which would result in a stochastic mixed-integer optimization problem.AcknowledgementsThis work has been funded jointly by the W. M. Keck Foundation, the Ford Motor Company and the National Science Foundation under grant no. DMI-0000051. 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