Storage assignment under changing demand:
Periodically reslotting
Marij Hoekman
Master thesis Econometrics, Operations Research and Actuarial Studies Specialization: Operations Research
University of Groningen
Marij Hoekman (s1769286) Groningen, December 10, 2013
Supervisors:
Storage assignment under changing demand:
Periodically reslotting
Marij Hoekman
Abstract
In supply chain management, storage assignment is an important decision. Storage assignment assigns products to locations in a warehouse. It influences the order picking environment and thereby the travel time for retrieving orders. Many warehouses are dealing with fluctuations in demand, which results in an outdated storage assignment. Instead of completely rearranging a warehouse after large changes, we consider periodically reslotting in order to maintain a good storage assignment. Reslotting deals with changing an existing storage assignment. The focus is on maintaining or reaching a desired storage assignment, rather than finding the optimal storage assignment. The problem will be modelled as an Arc Orienteering Problem, an optimization problem in which the score of a route in a di-rected graph has to be optimized by visiting arcs, while the total travel time cannot exceed a predefined time limit. We will include precedence relations. This is where the paper con-tributes to current literature on the Orienteering Problem. A heuristic based on Ant Colony Optimization will be used. Ant Colony Optimization is a method based on the natural behaviour of ants. We show that periodically reslotting can successively reach and maintain a good storage assignment under changing demand.
Contents
1 Introduction 2
2 Problem Formulation 3
2.1 Mathematical Model . . . 5
3 Ant Colony Heuristic 6 3.1 Pheromone level . . . 7
3.2 Heuristic information . . . 8
3.3 Select next vertex . . . 8
3.4 Local improvements . . . 9
4 Experiments 9 4.1 Selecting vertices and rewards . . . 10
1
Introduction
In supply chain management storage assignment is an important decision. Storage assignment assigns products to locations in a warehouse. It influences the order picking environment and thereby the travel time for retrieving orders. Order picking, collecting products to satisfy cus-tomer orders, has been identified as the most labour-intensive and costly activity for most ware-houses, the performance of order picking can influence the entire supply chain [De Koster et al., 2007]. This is why a lot of research has been done on storage assignment (e.g. Hausman et al. [1976]). Three frequently used storage policies are random storage, volume based storage and class based storage. Random storage assigns products randomly to locations, volume-based stor-age assigns products to locations based on the order volume or demand and class-based storstor-age assigns products to classes, where each class has its own dedicated area. Volume and class based storage policies lead to less picker travel than random storage. However, random storage policies utilize the entire picking area and reduce congestion [Petersen and Aase, 2004]. Petersen and Schmenner [1999] show that the performance of routing policies for order picking are influenced by the storage assignment policy used.
The focus in research on storage assignment has been on filling a storage facility from scratch. However, nowadays product demand changes rapidly. For example, promotional actions can be a source of demand fluctuations. These changes in product demand cause the quality of the storage assignment to deteriorate rather quickly. One of the view articles about reslotting is Kofler et al. [2011]. Reslotting deals with changing an existing storage assignment. Kofler et al. [2011] show that performing a small number of clean-up tasks regularly leads to a good total warehouse assignment over the course of several months. They assume an initially random stor-age assignment and that a fixed number of clean-up tasks are performed every day. The focus is on changing the storage assignment policy, rather than maintaining it. Furthermore, they do not focus on which tasks should be performed. Christofides and Colloff [1973] find an effective way to rearrange items in a warehouse. They assume a fixed set of tasks to be executed and focus on minimizing the associated cost or time of rearranging the items. However, nowadays warehouses do not always have the time or resources to execute all these rearrangements. This is where the method developed in this paper can contribute to current literature. The focus is on maintaining the storage assignment in limited time.
In this thesis a method is developed for reslotting, aimed at improving the storage assignment of the facility as much as possible during a fixed time interval, by selecting which tasks to perform and in which order. The benefit of the time restriction is that the surpluses can be used, rather than needing additional staff or equipment. The reslotting can be performed at times that the workload is low and for short periods such that it does not interfere with the daily operations. We assume the desired storage assignment rule is known, and we attempt to reach and maintain this storage assignment by periodically reslotting. Based on the desired storage assignment, several products are selected to be eligible for reslotting. The tasks to be performed are selected from a list of possible tasks, where each task will improve the location of a product, but not necessarily move the product to its ideal location in the desired storage assignment. Without reslotting the storage assignment deteriorates and the time spend on order picking will increase. Completely rearranging the warehouse will affect the daily operations and will take additional resources, which many warehouses cannot afford. The goal of periodically reslotting is to keep the storage assignment up to date, without affecting the daily operations of the warehouse.
Problem. The Orienteering problem originates from the game of orienteering. In this game com-petitors start at the same point, can collect points at several control points, but have to reach the finish on time. The goal is to collect as much points as possible and return on time [Chao et al., 1996]. Usually it is not possible to reach all control points and reach the finish in time. Therefore players have to select a subset of control points to visit. Similarly the Orienteering Problem is an optimization problem in which the total collected score has to be optimized by visiting locations, while the total travel time cannot exceed a predefined time limit.
Vansteenwegen et al. [2011] provide an overview of recent developments and applications con-cerning the Orienteering Problem. They find that several real world problems can be modelled as Orienteering Problems. Many heuristic solution approaches are proposed for the Orienteering Problem. For example, Tsiligirides [1984] proposes two heuristic approaches, a deterministic and a stochastic heuristic. Chao et al. [1996] propose a five-step heuristic. Both Liang et al. [2002] and Schilde et al. [2009] construct a heuristic based on Ant Colony Optimization. They show that the Ant Colony heuristic outperforms the most competitive heuristics. Ant Colony Opti-mization has also been shown to be effective in several related problems such as the bi-objective Orienteering Problem [Schilde et al., 2009], in which locations have multiple scores, and the Clus-tered Orienteering Problem [Dijkstra, 2012], in which scores are given for visiting all locations in a group. Therefore, we will use an approach based on Ant Colony Optimization in this thesis.
The reslotting problem will be modelled as an Arc Orienteering Problem. The difference between the Orienteering Problem and the Arc Orienteering Problem is that in the Arc Orienteering Prob-lem, bonuses are not collected at locations but by travelling between locations. Souffriau et al. [2011] introduce the Arc Orienteering Problem for the planning of cycle trips. Since the Arc Orienteering Problem is quite similar to the Orienteering Problem, most heuristic solution ap-proaches for the Orienteering Problem can easily be adjusted for the Arc Orienteering Problem. This paper contributes to research on the (Arc) Orienteering Problem by including precedence constraints, constraints on the order in which locations can be visited.
The rest of the paper is outlined as follows. In the next section the problem is described in more detail, the construction of the graph is explained, and a mathematical model is given. The preceding paragraph outlines the approach based on the Ant Colony meta-heuristic. Next the experiments, results and conclusion are presented.
2
Problem Formulation
The problem of which reslottings tasks to perform in limited time can be modelled as an Arc Orienteering Problem. The Arc Orienteering problem is based on a graph G = (V, A), where V is a set of vertices and A a set of arcs. A graph is a collection of objects (vertices), where some (ordered) pairs of objects are connected. These unordered and ordered pairs of vertices are called edges or arcs respectively. In the Arc Orienteering Problem, the arcs have an associated reward and distance or travel time. The goal of the Arc Orienteering Problem is to find a route with a specified start and end vertex, that maximizes the total collected reward within a specified time (or distance) limit. Furthermore, each vertex is visited at most once.
possible to visit a location twice, once to pick-up a product (pick-up location) and once to drop-off a product (drop-drop-off location). If two vertices are associated with the same physical location, the vertex associated with the pick-up location should be visited before the vertex associated with the drop-off location. Otherwise, it would be possible to drop-off products at occupied locations. This restriction is incorporated through precedence relations, such that newly empty locations can be utilized. Furthermore, because each vertex can be visited at most once, we are able to allow multiple different reslottings tasks associated with the same product on the list of possible tasks. This allows the possibility to move one product to several locations and to move several products to one location. Because every pick-up and drop-off location can be visited at most once, no tasks that interfere with each other will be executed.
The depot is specified as the start and end vertex of the route and is denoted by vertex 0. There are several groups of arcs considered in the graph. First arcs associated with the possible reslottings tasks. These are arcs from a pick-up vertex to a drop-off vertex. Note that only the arcs associated with a task are included in the graph, not all arcs between the pick-up vertices and drop-off vertices. Second all possible arcs from drop-off vertices to pick-up vertices are in-cluded. These arcs depict selecting the next pick-up location. Finally all possible arcs from the depot to the pick-up vertices and the arcs from the drop-off vertices to the depot are included. No other arcs are considered, since this would be redundant. Travelling from pick-up location to pick-up location or from drop-off location to drop-off location will never give a better route than travelling directly from task to task, that is, alternating pick-up and drop-off locations where each subsequent pair of pick-up and drop-off locations is associated to a possible reslottings task.
The considered arcs have an associated travel time. For the arcs associated with possible reslot-tings tasks, the travel time may be increased by the time needed to pick-up the product and drop it off. These arcs also have an associated reward. This reward is based on the current and new location, and the difference in distance to the ideal location according to the desired storage assignment and on the demand of the product to be moved.
An example of possible tasks and part of the corresponding graph is given in Figure 1. On the left five locations are shown (location A, B, C, D and E) with corresponding reslottings tasks (denoted by arrows). On the right the corresponding arcs and vertices of these tasks are given. On top the pick up locations are shown (indicated by the added ”1” to the location) and below the drop off locations are shown (indicated by the added ”2” to the location). As can be seen from the graph, the locations that are used as pick up locations as well as drop off location have two associated vertices. Note that location C has two associated vertices, C1 and C2, while location A only has associated vertex, A1, since there is no task for which location A is the drop off point (no incoming arrows). For vertices C1 and C2 precedence relations hold. Vertex C1 should be visited before vertex C2 in the route, since location C should be cleared before a product can be dropped off there. This precedence relation will be captured with the parameter pC1,C2 = 1 and incorporated in the model with some additional constraints. The
Figure 1: Example tasks and corresponding network
The following notation is used:
n The number of vertices (excluding vertex 0). V The set of vertices (V = {0, 1, ..., n}). A The set of arcs.
tij The time it takes to traverse arc (i, j) ∈ A.
TmaxThe maximum allowed time to travel the route.
bij The bonus for travelling arc (i, j) ∈ A.
pij The binary parameter indicating whether vertex i should be visited before vertex j.
xij The binary decision variable indicating whether arc (i, j) ∈ A is traversed in the route.
ui The decision variable denoting the position of vertex i ∈ A in the route.
2.1
Mathematical Model
We use the mathematical model specified by Souffriau et al. [2011] for the Arc Orienteering Problem. We adjust this model to include the precedence relations:
The objective (1) is to maximize the total collected bonus. Constraint (2) ensures the tour starts and ends in vertex 0. Constraints (3) ensure that every vertex entered is also left (except vertex 0) and every vertex is visited at most once. This ensures that a product can only be picked-up once and at every location only one product can be dropped-off. The maximal travel time of the route is taken into account by constraint (4). Constraints (6) are sub-tour elimination constraints. The objective (1) and the constraints (2) till (5) are directly taken from Souffriau et al. [2011]. Constraints (6) actually consist of two separate constraints, sub-tour elimination constraints, ui− uj+ 1 ≤ n(1 − xij) (taken from Souffriau et al. [2011]) and constraints that
ensure that precedence relations are considered in the position variables, ui− uj+ 1 ≤ n(1 − pij).
Constraints (7) ensures that for every pair of vertices i and j with precedence relation pij = 1,
vertex j can only be visited if vertex i is visited. Hence constraints (6) and (7) ensure that the precedence relations are satisfied.
The model above will be used to find optimal routes, which will benchmark the performance of the Ant Colony Heuristic.
3
Ant Colony Heuristic
In the original (vertex) Orienteering Problem, Ant Colony Optimization was shown to be an effective heuristic. This heuristic is based on the way ants work together to find food. Ants leave a trail of pheromones when they go out to look for food. Other ants can use these pheromone trails to find food more quickly. The pheromones vaporize over time. The idea is that if an ants takes a long time to find food, his pheromones trail has already vaporized more. This way, routes with a high pheromone level are more likely to lead to food quickly. This way ants help each other discover the quickest routes.
The Ant Colony heuristic is based on the pheromone system of ants. It is used to construct several paths or routes. A path is a sequences of vertices that are connected through a sequence of arcs. If the first and last vertex in a path coincide, the path is called a route. The selection of the next vertex depends on the pheromone levels on the arcs. Arcs with a high pheromone level are more likely to be traversed. The local pheromone update ensures the diversification of the ant solutions, by lowering the pheromone levels on the arcs that are used in the ants paths [Liang et al., 2002]. This ensures that this arc is less likely to be used by the next ant. The global pheromone update increases the pheromone levels of the arcs used in the best found routes in a generation of ants. This way these arcs are more likely to be traversed again.
Stepwise the Ant Colony heuristic can be depicted as follows [Schilde et al., 2009]:
1. Generate an ant population, where each ant has initial path v0
2. Repeat until no (feasible) vertices can be added:
(a) Select next vertex and add to the path (b) Perform local pheromone update
3. Add vertex v0to each ants path, to finish the tour
4. Apply local improvements to the tours
In the warehouse, the selection of the next pick-up location from the current location and the selection of the corresponding drop-off point is guided by the artificial pheromones levels. The pheromone levels attempt to asses how likely it is that, picking a certain location to be the next location visited in the path, will lead to a good reslottings route.
The general heuristic can be adjusted for the reslotting problem. We will take advantage of the fact that alternating pick-up and drop-off locations will lead to the best solutions. Using this information leads to a smaller search area and therefore the heuristic will be more effective. Therefore we will add vertices in two phases, select a pick-up location and select a (correspond-ing) drop-off location.
The reslottings route will alternate pick-up and drop-off locations, or equivalently go from reslot-ting task to reslotreslot-ting task. Using this structure will reduce the non-value added time, since this keeps the route as short as possible, while collecting the same (or more) bonuses.
3.1
Pheromone level
The pheromone level of arc ij will be denoted by τij. The pheromone levels on all arcs are
initialized with τij= τ0for each arc ij. Here τ0is the parameter for the initial pheromone level.
Local pheromone updates are performed to ensure diversification in the constructed routes. This is done to avoid getting stuck in a local optimum. This update is performed directly after an arc ij is selected for the route:
τij= (1 − ρ)τij+ ρτ0,
where 0 ≤ ρ ≤ 1 denotes the evaporation rate [Liang et al., 2002].
The ants decide the next location to visit based on their current location. If an ant is at location A, i.e. a product is picked-up or dropped-off at location A, and decides to visit location B next, to drop-off the product picked-up at location A or to pick-up the next product respectively, the next ant that is in location A will be less likely to make the same move. This ensures that a lot of different routes are tested.
The global pheromones update is performed to make the arcs in good routes more attractive for the next generation ants. The idea is that it is likely that the arcs used in the best routes found in each generation will be used in other good routes. The global update is performed after a generation of ants has constructed their routes. To the arcs ij used in the best routes of the generation τu will be added to the pheromone level τij. To the arcs in the second best
route of the generation 0.5τu will be added to the pheromone level τij. Note that it could be
possible for arc ij to be included in the best route as well as the second best route, in this case a total of 1.5τuis added to the pheromone level. This is done in agreement with Schilde et al. [2009].
3.2
Heuristic information
The selection of the next vertex does not just depend on the pheromone levels but also on the heuristic information. The heuristic information is a measure for the attractiveness of visiting a vertex from the current position. In the reslotting problem, not all rewards associated with the arcs are strictly positive. This is where the graph slightly deviates from an ordinary (Arc) Orienteering Problem. The heuristic information of arc ij will be denoted by hij. Schilde et al.
[2009] suggest hij = btij
ij for the Orienteering Problem. We will adopt this heuristic information
measure for the arcs with a positive bonus, that is the arcs associated with selecting a corre-sponding drop off location.
The heuristic information used for selecting the corresponding drop-off location weighs the effort against the reward. It is used as a measure of attractiveness. Tasks that take little time but have a high reward will be preferred.
The suggestion of Schilde et al. [2009] does not work for arcs without bonuses, since these arcs will all have the same value, namely zero. This will lead to problems in selecting the next vertex to be visited. Dorigo and Colorni [1996] use the heuristic information hij = t1ij for the
Travelling Salesman Problem. Since the Travelling Salesman Problem does not use bonuses, this will work for the non-bonus arcs in our problem as well.
The selection of the next pick-up location is based on the distance from the current location. Se-lecting a location close by, reduces the non-value added time. The heuristic information ensures that this is taken into account by the heuristic.
The heuristic information reflects the fact that the reslottings problem does not immediately fit in the Arc Orienteering Problem, therefore some adjustments are needed. The heuristic in-formation of the reslottings problem is modelled using both the Arc Orienteering Problem and the Travelling Salesman Problem.
3.3
Select next vertex
The pheromone levels and the heuristic information are combined into an arc score. We denote the last vertex in the route by vertex i. For every vertex j which can be added to the route, such that the route remains feasible, the score of arc ij is given by:
sij= τijαh β
ij, (9)
where α and β are parameters to control the weight of the pheromone level and heuristic infor-mation [Dijkstra, 2012]. The score for all other arcs (which are not feasible for the current route) is set to zero. Then the selection of the next vertex (i.e. pick-up or drop-off location) to visit is done according to the following distribution [Dorigo et al., 2006]:
1. With probability q0, the vertex j = argmaxksikis selected
2. With probability 1 − q0, vertex j is selected with probability sij
P
ksik ∀j ∈ V
the pick-up locations that have not been visited are still occupied with the products that should be moved.
For the selection of the drop-off location, only the bonus arcs are considered. Also the precedence relations should be taken into account. It is possible to have selected a pick-up location that does not have any feasible drop-off locations at this point of the route (that can be added without violating the time constraint). In this case, the phase of selecting a drop-off point is skipped and the route is continued by adding a new pick-up location (if still possible). This means that the ant has made a detour, since he could have skipped visiting the previous pick-up location without loosing any bonuses. These detours will be removed in the local improvement phase.
3.4
Local improvements
It has been shown that applying local improvements in combination with the Ant Colony Al-gorithm can significantly improve the results. First the detours are eliminated from the routes. This is done by deleting the pick-up locations that are visited, without executing a task. This shortens the routes without lowering the total bonus collected. Since the route length was lowered and some vertices have been deleted from the route, it might be possible to add an additional task. We consider all feasible tasks that can be added to the end of the route (before returning to the depot) without violating the time constraint and we add the feasible task with the highest bonus. This is repeated if there are still feasible tasks left.
The number of empty locations and the corresponding precedence relations can limit the ef-fect of local improvements. If there is only one empty location, it is not possible to insert or delete a vertex or task, apply k-opt or exchange a vertex or task in the route with one not in the route, without making the route infeasible. Even when there are multiple empty locations, the precedence relations complicate the standard local search methods. Therefore we only consider 2-OPT. For the 2-OPT we use first-improvement, that is, we apply the first 2-OPT move found that shortens the route. For the 2-OPT we cluster the tasks, that is we only consider the order of tasks, we do not alter the tasks themselves. After the 2-OPT is applied, it is attempted to add tasks to the route, in the same way as above.
4
Experiments
The results of the Ant Colony heuristic are compared to optimal solutions and to results of simple heuristics and the effect of the periodically reslotting on the storage assignment is tested. The storage assignment is tested by randomly generating 1000 orders and finding the average route length for order picking. This is done to include enough orders to be representative, without spending extensive time on computations. This does however not eliminate all fluctuations. In the experiments in which the demand does not change, the same 1000 orders will be used to compute the average order pick route length throughout the experiment. This is done to reduce variation. The storage assignment with and without reslotting is compared.
For the experiments we will assume that the desired storage assignment policy is the nearest location policy. This policy orders the products based on the demand and the distance to the depot, with the products with the highest demand nearest to the depot. We use optimal order picking routes for the order picking. Ratliff and Rosenthal [1983] have found an algorithm to find optimal routes for order picking in warehouses without cross-aisles. However, for convenience we will find optimal order picking routes by solving the corresponding Linear Program. In practice it might not be desired to use optimal order picking. In this case the routing method used for order picking influences which storage assignment policy works well. We assume that there are 10 products per order. The number of products per order should not influence the heuristic, but rather the choice of storage assignment policy.
Most of the experiments are done on theoretically generated data. This ensures that the in-fluence of the number of aisles, the number of products and the number of blocks can be tested. We consider 8 different layouts to be able to determine the effect of the number of aisles, sub-aisles and the number of products. We assume all locations are of equal size. In the layouts the number of aisles, the number of blocks and the total number of locations are varied. The number of aisles considered are 12 or 24. The number of blocks are 1 or 2, or equivalently, the number of cross-aisles are 0 or 1. The number of pallet locations are 960, 1920 or 3840. Note that the number of locations are doubled in each step. We assume that 10% of the locations is empty. An overview of the layouts is given in Table 2. We assume that the depot is located at the front of the leftmost aisle. We assume the distance between adjacent pallet locations to be 1 meter, the distance between adjacent aisles to be 4 meter and the width of the cross-aisles to be 2 meter. We assume the average speed to be 3.3 km/h without load and 3 km/h while moving a pallet. We assume it takes an additional 10 seconds to switch aisles and an additional 10 seconds to lift and drop off a pallet.
The demand is generated using the analytical function given by Caron et al. [1998]. That is
F (x) =(1 + s)x
s + x , where F (x) ≥ 0, x ≤ 1, s ≥ 0 and s + x 6= 0. (10) Here x denotes the ratio of required storage space to total storage space, corresponding to the items whose order frequency represents a fraction F (x) of total warehouse activity. s denotes the demand skewness, we assume s = 0.15.
4.1
Selecting vertices and rewards
The choice of storage assignment influences the way products should be selected for reslotting and the rewards that are assigned to the reslottings tasks. We construct all possible reslottings tasks between the selected locations. We only allow tasks that move products closer to the ideal distance to the depot. The bonus for each of these tasks is constructed as follows:
bij = (|di− d∗i| − |dj− d∗i|) · |demandi− avdemand| , (11)
where di denotes the distance to the depot from location i, d∗i denotes the ideal distance to
the depot for the product currently in location i, demandi denotes the demand of the product
currently in location i and avdemand denotes the average demand.
difference between the demand of the product and the average demand. This is considered as a measure on how much the location of the product can be improved and how important a good location for this product is. In order to ensure that products are moved closer to the depot as well as further from the depot, we select the 20 products that score highest on this criteria and have higher than average demand and the 20 that score highest and have lower than average demand. This is done to ensure that the empty location remain spread out over the warehouse. This is necessary in order from keeping the heuristic from stagnating.
5
Results
5.1
Parameter Tuning
In order for the Ant Colony Heuristic to work properly, we need to find appropriate values for the parameters. We will adjust the parameters in steps. First we will adjust the relative weight of the pheromones and heuristic information, α and β respectively. These parameters strongly interact and should be adjusted simultaneously. We will use the settings of Liang et al. [2002] as initial settings for the other parameters. That is, τ0 = τu = 10, ρ = 0.1 and q0 = 0.2. We
use 20 iterations and 20 ants, since we would like the heuristic to reach a good solution quickly. We compare the average scores over 10 samples for 4 different layouts (Table 1) for a maximum route length of 20 minutes. We find that in general α ≥ β works best. This implies that there should be more weight on the pheromones than on the heuristic information. We select α = 5 and β = 4 since this results in the best found solution in 2 of the 4 layouts and scores relatively high on average.
Table 1: The different layouts used for parameter tuning.
Layout Number of locations Number of products Number of aisles Number of blocks
1 1400 1260 20 1
2 3000 2700 10 3
3 960 864 8 1
4 1960 1764 14 2
Next we select appropriate settings for τu and ρ. These parameters interact, since τudetermines
how much the pheromones of the arcs in the best routes found are increased and ρ determines how quickly the pheromones evaporate. We fix τ0 = 10, since only the relative difference with
τu should influence the effectiveness of the heuristic. We select τu= 10 and ρ = 0.05, since this
results in the best found solution in 2 of the 4 layouts and scores best on average. Finally we need to adjust q0, the probability of selecting the arc with the highest score. We find q0= 0.5
works best as it results in the best found solution in 2 of the 4 layouts and score the best on average.
5.2
Performance Heuristic
visited, and a heuristic that selects the next vertex based on the highest heuristic value (HH) as described in subsection 3.2.
We define eight different layouts, which are described in Table 2. We set the maximum com-putation time of the C-PLEX 12.5.1 solver to four hours. In case the optimum is not reached within four hours, the upper and lower bound are given. We test the heuristic for Tmax= 5 and
Tmax= 15, larger values are not considered because the optimal value cannot be attained within
reasonable time in these cases. We allow the Ant Colony Heuristic some additional time in order to compare the heuristic with the optimal values. We use 30 ants and 350 iterations.
The results are given in Table 3 and Table 4. Table 3 shows the bonus attained by the Ant Colony heuristic (AC), the nearest neighbour heuristic (NN) and the greedy heuristic that se-lects based on the heuristic value (HH) in the cases where the optimum value was attained by the C-PLEX 12.5.1. The optimal value was found in 110 out of 160 cases. In 72 of these cases the Ant Colony Heuristic attained the optimal value, while in the other cases the heuristic was at most 5.22% off. The Ant Colony heuristic outperforms both greedy heuristics in all cases. Furthermore, the highest heuristic value heuristic performs a lot better than the nearest neigh-bour heuristic.
For the cases that the optimum value was not obtained, we registered the the lower and up-per bound. The average scores are shown in Table 4. Notable is that in 3 situations, the Ant Colony score corresponds to the best found solution in C-PLEX 12.5.1 (lb), and in one situation, layout 8 with route length 5 minutes, the Ant Colony score even outperforms the best found solution in C-PLEX 12.5.1. In general, it appears that the cases which C-PLEX 12.5.1 cannot solve within four hours are also difficult for the Ant Colony heuristic, since the scores are rela-tively low and the gaps are larger.
As can be seen from Table 3, the Ant Colony Heuristic performs close to optimal for cases where few tasks can be performed within the time limit. This is the case with small maximum route lengths as well as larger warehouses. This can be explained by the number of feasible solutions. As more tasks can be executed, there are more feasible solutions. For the routes with fewer tasks, the Ant Colony Heuristic is able to find the optimal solution in almost all cases. For routes with more tasks the heuristic usually is close to the optimum route score.
Table 2: The different warehouse layouts.
Layout Number of locations Number of products Number of aisles Number of blocks
1 960 864 12 1 2 960 864 12 2 3 1920 1728 12 1 4 1920 1728 12 2 5 1920 1728 24 1 6 1920 1728 24 2 7 3840 3456 24 1 8 3840 3456 24 2
5.3
Performance Model
Table 3: Performance heuristic. Results reslotting routes over 10 samples for each layout and different maximum route lengths. The number of samples for which the optimal solution was found, and the corresponding average scores and the gap with the optimum.
Layout Tmax #opt opt AC NN HH
1 5 10 648 642 0.93% 58 91.14% 354 45.89% 15 8 1861 1808 2.86% 465 74.95% 1117 40.23% 2 5 9 881 875 0.76% 112 87.25% 566 35.49% 15 0 3 5 10 341 340 0.29% 175 95.08% 197 41.58% 15 10 1186 1163 1.96% 282 76.10% 787 33.59% 4 5 9 446 446 0% 59 86.65% 243 45.38% 15 2 1504 1487 1.10% 300 80% 1173 21.98% 5 5 10 346 346 0% 21 93.72% 214 38.76% 15 7 1220 1203 1.41% 371 67.94% 894 27.06% 6 5 8 437 436 0.15% 39 90.75% 268 38.97% 15 2 1499 1494 0.33% 480 67.94% 1160 22.57% 7 5 10 163 163 0% 50 72.23% 142 14.09% 15 6 648 643 0.78% 119 81.60% 463 28.53% 8 5 8 200 200 0% 3 97.58% 173 12.95% 15 1 776 776 0% 253 67.40% 575 25.90%
to test whether the storage assignment in the warehouse converges toward the desired storage assignment policy. We assume that the warehouse initially applied the random storage assign-ment policy. We use the Ant Colony Heuristic to reslot over several periods. We show the effect of reslotting by computing the average order pick route lengths. We use the same warehouse layouts as before (see Table 2).
The results are given in Table 5. As can be seen the average route length converges for the smaller warehouses. For the larger warehouses, there are still improvements made after 80 peri-ods. By allowing more reslotting time per period, the average order pick route length diminishes faster in general. However, especially for the smaller warehouses, there is little difference in 45 minutes reslotting or 60 minutes reslotting. This can be explained by the fact that there are only 40 products selected for reslotting together with 20 empty locations. In the smaller warehouses it is possible to execute the maximum number of reslotting tasks within 45 minutes. This implies that there is little difference in reslotting 45 minutes or 60 minutes. The results also suggest that less improvement is possible when warehouses have a cross aisle. This can be explained by the fact that the routes in a warehouse without a cross aisle are larger, which make a good storage assignment more important. The reslotting is more effective in warehouses with a cross aisle, which ensures a faster convergence.
Graph 2 shows the results for layout 3. There is little difference in using a reslottings time of 45 or 60 minutes, therefore the results for 60 minutes are left out. As can be seen from the graph, the improvements to the average order pick route are strongest in the first periods. After that, there are fewer reslottings tasks possible, which slows down the improvements.
Table 4: Performance heuristic. Results reslotting routes over 10 samples for each layout and different maximum route lengths. The number of samples for which the optimal solution was not found, and the corresponding average bounds and scores and the gap with the upper bound.
Layout Tmax # lb ub gap AC NN HH
1 5 0 15 2 1943 1968 1.27% 1895 5.52% 437 77.82% 904 54.09% 2 5 1 736 840 12.38% 736 12.38% 384 54.29% 375 55.36% 15 10 2440 2509 2.84% 2337 7.03% 744 70.44% 1450 42.48% 3 5 0 15 0 4 5 1 386 453 14.79% 386 14.79% 85 81.24% 279 38.41% 15 8 1443 1498 3.65% 1423 5.05% 360 75.85% 1081 27.93% 5 5 0 15 3 1137 1184 3.94% 1128 4.63% 443 62.56 % 786 33.53% 6 5 2 373 426 12.53% 373 12.53% 10 97.75% 186 56.73% 15 8 1338 1390 3.73% 1313 5.57% 406 71.32% 945 32.17% 7 5 0 15 4 598 624 4.24% 593 4.99% 113 81.71% 413 34.20% 8 5 2 65 229 71.74% 194 15.28% 0 100% 161 29.92% 15 9 735 781 5.94% 731 6.41% 195 75.05% 591 24.44%
Figure 2: Development of the average order pick route length when changing storage assignment policy from random to nearest location for layout 3 for different reslotting times.
from a random storage assignment to a nearest location storage assignment is a rather drastic change. Other changes, for example from random to class based, may converge more quickly.
Table 5: Development of the average order pick route length when changing storage assignment policy from random to nearest location for the different layouts and for different reslotting times.
Layout Tmax Periods Percentage
10 20 30 40 50 60 70 80 improvement 1 15 252.328 243.43 239.775 231.956 229.664 228.782 228.183 228.144 19.83% 30 243.066 233.829 229.338 225.85 227.406 226.176 224.011 224.436 21.14% 45 236.24 231.831 229.784 225.686 226.663 225.967 225.428 223.789 21.36% 60 234.769 227.334 224.96 223.358 225.754 225.431 223.503 223.393 21.50% 2 15 205.502 197.191 192.239 189.398 192.022 190.864 190.578 190.508 15.21% 30 197.696 189.484 187.548 185.982 185.709 185.104 184.916 185.439 17.46% 45 195.108 190.148 187.498 184.808 186.722 186.839 185.037 186.118 17.16% 60 194.78 188.238 187.966 186.057 186.917 185.506 184.46 186.584 16.95% 3 15 428.367 412.143 398.003 387.305 378.625 375.934 368.487 373.153 17.98% 30 414.45 390.378 374.88 372.082 366.691 365.597 360.286 363.57 20.09% 45 403.024 374.87 362.832 364.028 358.347 362.247 356.834 359.202 21.05% 60 393.156 374.688 362.998 361.487 358.852 359.181 356.224 360.64 20.73% 4 15 330.234 320.426 308.842 305.067 302.151 298.323 295.249 293.328 15.54% 30 317.807 306.175 298.896 292.244 294.239 290.272 286.96 285.304 17.85% 45 310.979 297.605 293.093 289.023 290.091 287.766 284.905 284.292 18.14% 60 307.158 293.666 292.829 286.406 287.725 286.669 285.539 284.088 18.20% 5 15 376.517 367.99 355.311 349.011 349.684 345.585 344.453 340.491 13.89% 30 363.156 353.496 347.916 339.474 339.012 335.69 336.98 335.489 15.15% 45 354.769 349.232 341.946 336.318 335.342 331.006 332.538 327.831 17.09% 60 351.443 344.442 338.503 334.847 331.29 330.44 330.03 326.538 17.41% 6 15 308.405 301.601 294.601 289.613 284.219 284.432 282.048 279.073 12.83% 30 299.792 292.94 287.414 280.621 277.015 279.946 276.785 274.707 14.20% 45 296.806 287.079 284.746 280.66 276.766 277.542 274.899 272.625 14.85% 60 293.219 286.645 283.03 278.26 274.862 276.941 273.643 273.096 14.70% 7 15 588.949 572.823 566.389 553.315 549.369 542.454 541.272 533.776 10.37% 30 574.388 556.054 547.932 531.485 528.821 524.232 520.111 513.086 13.85% 45 567.555 544.594 535.076 520.532 521.03 512.106 507.315 502.401 15.64% 60 558.248 535.884 530.219 512.818 513.782 508.246 507.15 499.545 16.12% 8 15 445.909 439.211 434.858 429.296 426.907 424.004 416.84 412.178 9.41% 30 439.745 428.633 418.794 412.571 410.27 407.688 402.528 399.076 12.29% 45 434.038 418.677 410.669 402.276 403.46 401.291 397.09 395.293 13.12% 60 428.269 410.692 405.517 398.928 399.523 398.878 391.258 393.174 13.59%
is slightly higher than the nearest location route length, however the difference remains small. If no reslotting is done the average route length for order picking increases quickly.
5.4
Case study
In addition to the theoretically generated data, the products in an actual warehouse are con-sidered. The products (and their corresponding demand) of this warehouse are used to fill a fictional warehouse, this is done since the actual layout uses unequally sized locations. The dataset contains 975 products, and the number of picks per week over 32 weeks. The fictional warehouse has 13 aisles, which is equal to the number of actual aisles. We adjust the number of locations per aisle such that the total number of locations is as close as possible to the actual number of locations. This results in 1300 pallet locations. This implies that approximately 10% of the locations are empty. We assume their are no cross-aisles. We use this data to test whether the Ant Colony Heuristic is able to maintain the nearest location storage assignment under the demand changes. We assume that the reslotting is performed five times per week.
Figure 3: The effect of demand changes on order picking. In each period, 10 reslottings routes are executed.
large changes in the first few periods. We note that even the average route length for perfect reslotting (nearest location storage assignment), increases strongly. This suggest major demand changes in these periods. It can be seen that the Ant Colony reslottings method struggles to maintain a good storage assignment in these periods. However, after these periods, the reslotting converges to the nearest location storage assignment again. If no reslotting is done, the route length quickly increase toward 325 meters, while with reslotting the average route length remains lower. However, the graph suggests that to be able to handle the major demand changes in the first periods, additional reslotting is necessary.
6
Conclusion
In this paper, an approach for periodically reslotting was proposed. The approach describes how to formulate the problem as an Arc Orienteering Problem with precedence constraints and to use an Ant Colony Heuristic to find reslottings routes. The results show that this method can be used to attain or maintain a certain storage assignment policy if the appropriate amount of reslotting is used. Attaining a desired storage assignment policy may be a lengthy process and therefore may not be desirable. However, as reslotting minimally effects the daily operations in a warehouse, it may still be preferred over completely rearranging a warehouse. The layout of a warehouse can effect the effectiveness of reslotting as well. A cross aisle causes reslotting to work more quickly. However, less improvements are possible.
The experiments have shown that the method described in this paper can successively obtain or maintain a storage assignment policy. It is important to use an appropriate amount of time on reslotting. This time may depend on the layout of the warehouse and its size, as well as on the severity of the demand changes. Furthermore, it was shown that even in cases were more reslotting is preferred in some periods, the method developed in this paper is able to catch up without additional reslotting in periods were less changes in demand occur.
In this paper, it was shown that, periodically reslotting can significantly reduce the time spend on order picking. Implementing periodically reslotting brings little or no additional costs, since it can be done in short intervals when the workload is low, using the surplus in staff. Furthermore, keeping up a good storage assignment reduces the time needed for order picking.
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