A novel framework for storage assignment optimization inspired by finite element method

Storage assignment literature

The literature suggests various methods for products storage location assignment. There are five basic storage assignment policies for AS/RSs (Roodbergen & Vis, 2009; Faber, De Koster & Smidts, 2013; Ekren et al., 2014). These rules are:

• Dedicated storage assignment

• Random storage assignment

• Open storage assignment in closest location

• Full-turnover-based storage assignment

• Class-based storage assignment

In dedicated storage method, for each product category a fixed location is considered. And at this location replenishments of that product are considered. The main disadvantage will be space requirements and low space utilization. This is due to limitation of this method in considering the out of stock products in the model and reserve locations for them. Also, to be able to fulfill the placement for probable maximum inventory level, the model should reserve enough spaces based on each product type (Heskett, 1963; Heskett, 1964; Zhang, Wu & Ma, 2017).

For random storage in all empty locations, equal probability exists for an incoming load assigned to it. The product is assigned to first free location which is encountered. The product demand frequency is considered for full-turnover storage policy for assigning storage locations near to the I/O-points. So, the easiest accessible locations are considered for high frequently demanding products. In addition, the products called slow-moving are located farther away from the I/O-point. For this rule, the product turnover frequencies will an important assumption which should be known first (Roodbergen & Vis, 2009).

The cube-per-order index (COI) rule presented by Heskett (Eynan & Rosenblatt, 1993; Moon & Kim, 2001; Roodbergen & Vis, 2009). Explains the full-turnover storage method. The COI is defined by the ratio of needed storage space for products to amount of product requests for a load in a specific period. The COI rule manages the assignment policy by placing loads with the lowest COI to the nearest locations to the I/O-point. The main challenge is that the demand frequencies and the product assortment are usually changing. The method will encounter challenge when the frequency of demands for products are changing dynamically or a new product is added to the system. It requires a huge number of calculations for reassignment of products and fulfilling the full turnover rule.

To reduce periodic repositioning and also fulfilling space requirements, class-based storage can be considered. The literature studies show that class-based storage method is the most consistent policy for storage assignment (Hausman, Schwarz & Graves, 1976; Eynan & Rosenblatt, 1994). Class based storage method splits the available space in the warehouse into several sectors. Then the demand frequency is considered for each item and items are subsequently assigned to the sectors. Inside a sector area, the assignment is completed randomly. So, the full turnover storage policy is a class-based policy which only consist one item per class. There is a so-called ABC storage which is a class-based storage with three classes (Graves, Hausman & Schwarz, 1977). The A-items are with the highest turnover and then in descending order of turnover frequency products are called B-items, and so on. The main merit of class-based storage is its efficiency for assigning high frequency products close to I/O-points and also benefiting from the flexibility of random storage method and being applicable for low storage space conditions.

There are three decision variables for fulfilling the class-based storage in an AS/RS:

• Zone division (i.e., in which the number of classes are determined),

• Zone sizing (i.e., the number of products, assigning to each zone),

• Zone positioning (i.e., defining the location each of the zones).

To include storage boundaries for above mentioned classes, several procedures have been found in the literature for zone sizing. According to the Eynan and Rosenblatt’s research studies (Guenov & Raeside, 1992; Eynan & Rosenblatt, 1994), a relatively small number of classes, less than 10, gain most of the probable savings in travel times as compared to full-turnover storage. However, there are limited number of classes to three in real case problems.

For zone positioning several strategies exist. Research studies (Graves, Hausman & Schwarz, 1977) have proven that when single command scheduling is considered in square-in-time frames, the optimal solution for classes A, B and C with square-in-time boundaries will be an L-shaped configuration. Also, in other research studies (Kulturel et al., 1999) it has been demonstrated by simulation there will be optimality for L-shaped configuration for dual command scheduling, in square-in-time frames. Another research (Aldenderfer & Blashfield, 1984) also studied dual command scheduling and compared three different zone shapes. Their conclusion is that the location of the I/O-point affects the performance of proposed shapes while none will be superior to the others.

Another research study (Kaylan & Medeiros, 1988) has discussed the application of single command scheduling in “n” classes in rectangular frames and combined “n-2” L-shaped zones in layout, i.e., a rectangular zone for class “n” and a transient region for class “n-1”.

Performance of storage assignment rules

An analytical approach for checking control rules can be travel time estimation in scheduling for different command types in different types of AS/RS configurations (Van Den Berg & Gademann, 2000). Using the simulation approach, the stochastic conditions were focused and more comprehensive studies have been accomplished (Goetschalckx & Ratliff, 1990; Jaikumar & Solomon, 1990; Muralidharan, Linn & Pandit, 1995; Malmborg, 1996; Malmborg, 1998; Kulturel et al., 1999). Before simulation study, a trade-off storage policy was developed for small systems and then based on that for unit load the rough comparisons of different policies were accomplished. Results from both simulation and analytical studies stated that class-based storage assignment and the full-turnover based outperform random storage policy. In Moon & Kim (2001), considering duration of stay policy, a comparison was made by a three class-based policy which was originally introduced informer studies for shared storage policies in Sadiq, Landers & Don Taylor (1996). Storage locations are assigned based on a policy which consider the nearest locations to the I/O-point for short staying products in the storage space while duration of stay policy was considered. In cases where the number of the product variety is fewer, the three-class-based policy overtakes the stay policy duration.

For today’s dynamic environment (Roodbergen & Vis, 2009), there is an essential need for proposing storage assignment policies in AS/RS system. This fulfills the requirements for having the required performance level for dynamic storage management (Heragu et al., 2005). The COI policy can be considered as an acceptable policy assignment when the logistics and handling frequency parameters for demand of products are static. (Moon and Kim) (Heragu, 2008) in a simulation study showed that when the production quantity of each storage items is fluctuating by time, there will requirements for item re-location. Sadiq (Roodbergen & Vis, 2006) proposed a dynamic storage assignment policy for reassignments of products in storage locations for short lifecycle products and also the mixtures which were dynamically changing. The research applied a prediction model for future product mixtures by using correlated demand of products as well as product demand. Then a dynamic policy for storage assignment was focused targeting to decrease the order processing times (relocating times and order picking times). The results showed that the proposed dynamic policy fulfilled the static COI shortcomings for dynamic conditions.

Considering the related research studies, by using simulation and analytical methods different storage assignment policies have been compared. Most research studies are limited by considering one-way passage AS/RSs which include only one I/O-point. There is considerable gap for storage assignment policies for other kinds of multiple I/O-points configurations or multiple shuttles equipped AS/RSs. By the increase of variables like number of I/O-points or number of classes in the warehouse, the complexity of forming optimal cost equation and solve parameters increases drastically (Barkanov, 2001). Besides the demands on storage assignment are changing rather than the level of automation, cost effectiveness and maximum throughput, other characteristics that are more important today are flexibility, configurability and high availability. Due to virtual development, the time needed to introduce a new model in the market has been shortened rapidly and the diversity of products is dramatically increasing, also shortening the product life cycle, it shortened planning horizons and smaller amortization periods result in smaller investments. Also, customer request for more individual products leads to mass customization, which in return it leads to smaller quantities of the product in almost every sector of the consumer market. To cope with the fast-changing demands, classic models and related algorithms are not effective and competent and so a new model has to be developed which needs to be dynamic, flexible and alterable.

Storage assignment models

As it was previously said, generally there are five major policies for storage assignment in an AS/RS. The most comprehensive and practical applications currently being used in warehouses are Dedicated storage assignment, Full-turnover-based storage assignment, Class-based storage assignment, or Integrated Optimal use of these policies.

In the first case, any commodity type always belongs to a particular location, but its main drawback is the necessity for large space for storage. Thus, little space left for storage equipment which is generally not mechanized. However, storage of large or heavy- volume goods is the most important advantage of this case.

In the second case, locating construction is based on the frequency of demand for goods, so the goods that are most needed are placed in the places close to entry/exit doors that provide better, simpler and faster access to those goods. In this method, as was explained before COI index rule for each entry is applied. This rule places lower loads with the less COI index in the nearest locations to entry or exit doors. The drawback here refers to any change in demand frequency or product definition (new load) which causes a large volume of displacements and possible layout changes in the warehouse.

In the third case, the space available in the warehouse is divided into several parts and the loads subsequently classified into each space based on demand frequency (each space is allocated only to a specific class), and the location of each class inside, can be randomly assigned. In this way in order to better managing space and time, mainly three classes of goods were considered (ABC Storage) that ranging from Class A (consuming) to Class C (low consumption). The advantage of this case is warehouse flexibility due to random storage of goods (in each class) and optimal use of space in warehouses. The cost model, developed by incorporating the benefits of these policies and currently being used in the literature and layout calculations, is set to minimize the cost of moving items to entry and exit points. The cost per pair of goods/storage space is calculated as the product of three factors (Ventsel & Krauthammer, 2001).

• Frequency of trips to each I/O point.

• The distance being traversed to any I/O point.

• Cost per unit distance (where cost per unit distance is different for different combinations of entry points and exit points for different products).

Dedicated storage policy model

In this model (which is called first model), a warehouse has “p” I/O points through which “m” items enter and leave the warehouse. The items are stored in one of “n” storage spaces or locations. Each location requires the same storage space, and it is known that item i requires s_i storage spaces so ideally (1)${\displaystyle \sum _{i=1}^m{s}_{i=n}}$

There are f_ik trips of item i through I/O point k, the cost of moving a unit load of item i a unit distance through I/O point k is c_ik and the distance of storage space j from I/O point k is d_kj, given a binary decision variable x_ij specifies whether or not item i is assigned to the storage space j, so to formulate a model to assign the items to storage spaces in the way that minimize the cost of moving the items in and out of the I/O points one would have: (2)${\displaystyle \text{minimize}\sum _{i=1}^m\sum _{j=1}^n\left(\sum _{k=1}^p{c}_{ik}{f}_{ik}{d}_{kj}∕s_i\right)\mathrm{}{x}_{ij}}$

Subjected to:

(3)${\displaystyle \sum _{j=1}^n{x}_{ij}=s_{i}i=1,2,\dots ,m}$ (4)${\displaystyle \sum _{i=1}^m{x}_{ij}=1j=1,2,\dots ,n}$ (5)${\displaystyle x_{ij}=0or1i=1,2,\dots ,m;j=1,2,\dots ,n.}$

Substituting $w_{ij}={\sum }_{k=1}^p{c}_{ik}{f}_{ik}{d}_{kj}∕s_i$

The objective function will be (6)${\displaystyle Minimize\sum _{i=1}^m\sum _{j=1}^n{w}_{ij}{x}_{ij}}$

COI policy model under certain conditions (Zhang & Kratzig, 1995)

In this model (second model), a special case of the design model for dedicated storage policy is considered.in this case all the items use the I/O point in the same proportion and the cost of moving a unit load of item i is independent of the I/O point. P_k isdefined as the percentage of trips through I/O point “k” where k = 1, 2, …, p (for any item because all the items use the I/O points in the same proportion). Due to the additional constraints included in this model, there is no need for the first subscript in f_ik or in C_ik. Therefor f_ik, C_ik can be replaced by f_i, C_i respectively, and the model may be formulated as follows: (7)${\displaystyle \text{minimize}\sum _{i=1}^m\sum _{j=1}^n\left(\sum _{k=1}^p{c}_i{f}_i{P}_k{d}_j∕s_i\right)\mathrm{}{x}_{ij}}$

Subjected to constraints Eqs. (3)–(5).

Substituting ${\displaystyle w_j=\sum _{K=1}^P{P}_k{d}_j}$

The objective function can write as: (8)${\displaystyle Minimize\sum _{j=1}^n\sum _{j=1}^n\frac{c_i{f}_i}{s_i}{w}_j{x}_{ij}}$

The model given by expression Eq. (8) and constraints Eqs. (3)–(5), is easier to be solved in comparison with the first model given in expression Eq. (6) and does not require the use of transportation algorithm. It involves rearranging the “cost” term $\frac{T_i}{s_i}$ (if c_if_i = Ti) for each item i (i = 1,2,…,m). Also the “distance” term w_j for each storage space j(j = 1, 2, …, n) in non-increasing and non-decreasing order, respectively, and matching the item i corresponding to the first element in the ordered “cost” list with the storage space corresponding to the first S_i, elements in the ordered “distance” list, the second item, j in the “cost” list with the storage space corresponding to the next S_i, elements in the ordered “distance” list and so on until all the items are assigned to all the storage spaces. This is exactly what the COI policy except it calculates the inverse of the “cost” term, COI, that is, the storage space requirement divided by the cost incurred as a result of handling item i (or the frequency of item i), and orders the elements in non-decreasing order of their COI values . Thereby producing the same result as the preceding algorithm.

As defined in the algorithm (Fig. 1), the objective function has two parts and the storage optimization on the layout is performed in two stages: The first stage involves partitioning of the storage space, the calculation of distance index (W_j) will be done on the storage area and the contribution of each unit in the storage division will be clear, the best location is the one that has the lowest index (Wj). It is evident that the index value will change according to the location of each block at the warehouse, the number of divisions in the warehouse level, the number of doors, the distance of the doors from the divisions, and the trip percentage of each good (Persson & Strang, 2004).

The second stage, is the calculations of (Ti/Si) index which is called cost index, any good that has higher cost per unit has higher priority for storage on nearest location. The computation has to be done in a way that the total cost of moving items is minimum and items with higher consumption factor are placed at optimum storage positions.

Example 1

A warehouse with 16,000 units area is available as follows: The area size 16,000 units, has two I/O points with 30% and

70% input and exit logs on one side and with 40 equal divisions on the area (20 × 20 = 400) (Fig. 2).

Three products of A and B and C are to be stored base on Table 1 specifications:

After calculating the first step of the classical storage algorithm, and implementation of the distance index on the warehouse area, the results are presented in Fig. 3.

According to Fig. 3, calculations of the spectrum of Fig. 4 shows the prior assignment locations are near to I/O point and as far as getting out of the I/O points, the storage assignment priority is reduced. The algorithm is then executed based on the computation of the Table 1 and outputs are placed in priority order on the partitions by the non–descending trend on the warehouse (Figs. 4–6).

Finithe element method (FEM)

Finite element method is a numerical method for solving the differential equations governing the real problem presented in the form of a mathematical model. To achieve accurate, strain and stress result of a structure, the equilibrium equations must be established, and in addition the boundary conditions of the problem have to be satisfied (NguyenVan, Mai-Duy & Tran-Cong , 2007) .

Solving equilibrium equations along with the establishment of boundary conditions and adaptation in non-complex problems is not hard work. However, if it is faced with a complex problem (from the point of view of geometry, material behavior, loading, etc.), the exact solution of the above equations is practically very difficult and, in some ways, it is almost impossible (Fig. 7) (Nadal et al., 2013).

Therefore, the finite element method is an accepted and efficient method for solving differential equations governing the behavior of the object (problem), whose efficiency and speed of operation have been practically proved and generally in accordance with the following steps (Zhang & Kratzig, 1995; Persson & Strang, 2004; NguyenVan, Mai-Duy & Tran-Cong , 2007; Nadal et al., 2013).

Stress in thin plates

Man-made objects are often made from stock plates of various materials by operations that do not change their essentially two-dimensional character, like cutting, drilling, gentle bending and welding along the edges. The stress description in thin plates is simplified by assuming the plate with two-dimension surfaces and neglecting the three-dimensional bodies. With this assumption, particles are considered to cover the plate’s surface, so that the boundary between adjacent particles will be tiny line. The surfaces of particles are located such that their normal vectors are straight through the plate. The Stress can be defined as the result of internal forces between two adjacent particles through their overlapping line element divided by the length of that line. Usually, some elements of the stress tensor are ignored, however, since particles are only considered through two-dimension, the third dimension cannot be neglected for the torque among particles positioned in neighbor location (Chandrupatla et al., 2002; Huston & Josephs, 2008; Smith, 2013).

Conceptual model

In this study, we try to design and develop mapping as an optimal layout allocation model based on natural stress distribution algorithm (considering the behavioral similarity of stress extension pattern), using finite element computational and analytical method. The potential energy minimum theory is used for the analysis and based on the new heuristic algorithm; the claimed capabilities of the new model are verified.

To use stress distribution analogy for storage assignment, first it is needed to define the conceptual mapping model of classical and finite element method, to match up some of the definitions, rules and activities, then an algorithm needed to be designed and implemented to prove that the new model is efficient and shows its new advantages as shown in Table 2 and Fig. 8.

To elaborate this mapping as illustrated in Table 2, first the properties of the classical model are defined as follows:

A-1) the warehouse is a rectangle.

A-2) The storage area of the warehouse is unobstructed, shelving and surface level.

A-3) the storage area has four walls (parallel to each other).

A-4) the storage area is divided into smaller, uniform areas (square, in this model) according to the requirements of the warehouse.

A-5) in this model, I/O doors are defined and installed on parallel (facing) walls.

A-6) It is possible to define two doors (I/O points) solely on parallel walls, and the doors are logically defined on the wall spacing most separately on the marginal points of the warehouse surface segmentation.

A-7) the storage area is loading or unloading at one period.

A-8) in this model, according to the number of the doors, the share of goods entering / leaving each door is defined as a percentage of the total supply capacity of the warehouse.

A-9) in a coordinated manner, over a period, the warehouse (all doors) is in the state of entry or exit.

Considering the abovementioned properties, the paper presents all the classical features in FEM perspective with the least variations. So, in the new proposed model the features are defined as:

B-1) a rectangular plate is considered as the surface of the storage area.

B-2) the plate is made of soft, homogeneous metal with a low thickness.

B-3) the boundary conditions are applied to the plate in such a way as the sides with no entry or exit doors (I/O points) restricted.

B-4) the warehouse surface is meshed with square elements according to the classical model classification.

B-5) in the new model, the entry/exit doors are defined on the pair of parallel sides with the greatest distance.

B-6) loading points on the same positions of classical I/O positions are defined on the nodes resulting from the meshing on non-restricted sides.

B-7) loading at all points over a period is merely compressive or tensile.

B-8) in the new model, the share of loading on I/O points are calculated as a percentage of unit load, like the proportion of entering/leaving of each door in the classical model.

simulation and analysis

Recalling the contents presented in the sections COI policy model, its numerical example and finite element method, to use the stress flux analogy for storage assignment problems, according the conceptual model, the area in example 1, considered as a plate with the same dimension and divisions, and the input/exit logs’ percentage of goods simulated as external forces in the same positions of the I/O points in the warehouse. The results show that, the stress distribution starts from the location with maximum stress concentration (load points), with maximum total potential energy towards the region which has the lowest stress with the minimum of potential energy (Fig. 9).

For more exact analysis, the average stress (calculated in the center of each element) shows that the elements with the maximum stress and highest potential energy are close to the loading points, and the lowest mean stress with lowest potential energy are exactly in counter (Fig. 10).

Comparing the result of the classical model (presented in Figs. 3 and 4 (‘Storage Assignment Example’)) with the new model (Fig. 10), clarify if the stress distribution calculations by finite element method relocate related to the I/O points, based on minimum stress and the least total energy, locations that have the preferences for storage assignment follow the same pattern as classical model and useable as a new model for storage optimization (Figs. 11 and 12). storage procedure diagrams in Figs. 13 and 14 show that both models have almost the same strategy up to step three but the procedure in step four and five are different.

SAO/FEM Algorithm - application

The SAO/FEM algorithm (Fig. 12) designed to check the similarity of the two analogies to prove that storage assignment with the new algorithm is possible and have more advantages. SAO/FEM application is designed for solving the same problem in both algorithms simultaneously, analysis the output data, compare the results in a specific computer and show the capabilities of the new proposed algorithm and its advantages over the classical model.

The graphical interface (GUI) shown in Fig. 15 designed for SAO/FEM application. It has three sections (Modeling, FEM and Classic) for data entry where the specifications data must be entered in order. In modeling part there are two sections, the first is the plate’s physical specifications and the second is dimensional information of the model in addition to its divisions biased on the unit of the length. In FEM part, boundary conditions apply to the area of the model, dimensions in no I/O point sections bounded. Then loads at the points that are equal to the I/O points proportional to the percentage of input and output of the goods apply to the model. In classical part, place of I/O points and each points’ entry and exit percentage of the goods define.

After data entry, the analysis phase starts, and software output shows the loads expansion and stress analysis (Since the elements with the maximum stress and the highest potential energy are close to the loading points, the lowest mean stress with lowest potential energy is exactly in counter).

After stress analysis, the application prepares a comparable model (Fig. 12) and calculate the average stress in the center of each element and then arrange the elements symmetrically. So, the elements with the minimum average stress and minimum potential energy, finally sort near the I/O point (virtual loading points) and display its graphs. Besides the application runs the classical calculations on the ideal assignment of goods in the warehouse performed and display its graph.

Then application calculate the index Ti/Si (Determination of each goods’ priority of storage) in both algorithms and arranges the prior goods on the elements from the lowest average stress to the highest in FEM model and from nearest distance to farthest distance in classical model. The next phase is comparison phase (Fig. 12). In this phase, SAO/FEM starts normalizing the result matrix of both algorithms and create the new matrixes with the elements calculate from 0 to 1 (independent of dimensions) for both. The graphical format of the results sample is shown in the Figs. 13 and 14, parts C, D.

Finally, SAO/FEM application compares the normal matrixes to calculate the conformity percentage of the performance of the storage assignment in both algorithms and calculating time. As observed in the SAO/FEM schematic diagram, the output of both methods must pass through data normalization filter to be able to be comparable with one another.

Numerical examples

In order to verify the capabilities of the proposed solution, some numerical examples have been solved by SAO/FEM application and the results are compered. SAO/FEM application solves each problem simultaneously in both algorithms, calculates the similarity of the performance along with the solution times.

It is notable that the verification started with the scenario 1, in this scenario SAO/FEM has solved the same problems discussed in ‘Simulation and analysis’. Which have been solved manually by classic and separately by FEM software (the storage area of this problem has divided in to 40 similar elements). The SAO/FEM results shows that storage assignment performance of the FEM model is 90% similar to the classical model and has the better performance in calculation time (Fig. 16).

In scenario 2, SAO/FEM application solved the storage problem with the storage area 4 times larger than the scenario 1 (the storage area has divided in to 160 similar elements).

The SAO/FEM results shows that storage assignment performance of the FEM model is 95.8% similar to the classical model and has the better performance in calculation time (Fig. 17).

Tables 3 and 4 show the SAO/FEM application’s different outputs of different problems and confirm the better performance of the FEM model. Figs. 18–22 demonstrate the storage assignment performance of the proposed FEM model and its similarity results to the classical model for scenarios in Tables 4 and 5. The results show that SAO/FEM method is successful for storage assignment optimization, and has more capabilities and advantages:

1. Higher computational speed

2. More computational accuracy

3. The ability to prioritize the arrangement without the restrictions of the classical model

4. Ability to apply any changes to storage priority in the shortest time

As a conclusion, in solving the storage assignment problems, the SAO / FEM method is efficient and more effective than the classic model.

Table 3:

SAO/FEM applicationresult for storage assignment with different order (example 2&3).

No	Mesh generation	I/O position	I/O frequency	FEM & Classic storage assignment similarity percentage	Classic time	FEM time
1	5*8 = 40	Node 4,6 (left side)	Both 50%	90%	0.36 s	0.16 s
2	10*16 = 160	Node 4,6 (left side)	Both 50%	95.8%	0.57 s	0.45 s

DOI: 10.7717/peerjcs.378/table-3

Table 4:

SAO/FEM application result for storage assignment with different order (index examples).

No	Mesh generation	I/O position	I/O frequency	FEM & Classic storage assignment similarity Percentage	Classic time	FEM time
1	5*8 = 40	Node 3,4 (left side)	30% 70%	91.7%	0.19 s	0.15 s
2	5*8 = 40	Node 2,5 (left side)	50% 50%	91.7%	0.19 s	0.15 s
3	5*8 = 40	Node 2,5 (left side)	30% 70%	95%	0.19 s	0.15 s
4	10*16 = 160	Node 3,7 (left side)	Both 50%	94.5%	0.49 s	0.43 s

DOI: 10.7717/peerjcs.378/table-4