ANT algorithms for designing order picking systems

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(1)PRACE NAUKOWE POLITECHNIKI WARSZAWSKIEJ z. 97. Transport. 2013. Micha Kodawski, Roland Jachimowski Warsaw University of Technology, Faculty of Transport. ANT ALGORITHMS FOR DESIGNING ORDER PICKING SYSTEMS The manuscript delivered: May 2013. Summary: Paper presents suggestion of using ant algorithms for designing optimal order picking system variants. It discusses principles of operation and possibility of using ant algorithms for research problems. It presents how to map the order picking system structure to allow implementation of ant algorithms and shows a simplified algorithm for designing optimal order picking system variants with their use. Keywords: ant algorithms, designing order picking systems, order picking. 1. INTRODUCTION The order picking is one of the key steps in the warehouse process. It can absorb more than half of the time spent on all warehouse activities and generate about 55% of all operating costs of logistics facilities. In addition, it should be noted that the efficiency of order picking has direct impact on the level of customers’ service logistics facilities. However, maintaining a high level of customer satisfaction is now a priority objective of almost every business [8],[9],[10],[11],[12]. In order to minimize time-consuming of warehouse processes and operating costs generated by the storage facilities, as well as the need to improve the quality of services offered by these facilities, particular attention should be paid to the efficiency of the order picking systems used in them. For designing efficient order picking systems it is necessary to know both the "structure" of system components and relationships between these elements, ie organization of order picking system (OPS). Therefore, it is necessary to take into account a wide range of factors, including: order picking area layout, storage assignment method, routing order pickers method, replenishment method, picking method and order picking strategy, as well as order batching method, number of pickers and used technologies supporting order picking process. These factors determine the complexity of the order picking system and picking processes implemented in it, and consequently the need for researching them in a systemic way. It should be noted that omitting one of the above.

(2) 260. Micha Kodawski, Roland Jachimowski. . factors may result in impoverishment of designed order picking system, so consequently it may not fully match our expectations. Taking into account such a large number of factors and feasibility of their realization causes that in result we have considerable decision problem with high computational complexity. This problem concerns to such selection and interlinking of relevant variants of elements of order picking system, to get in result the efficient system from the view of accepted criteria. For the purpose of solving this problem in this article we used the ant algorithms.. 2. ANT ALGORITHMS Ant algorithms are processing operations, which is one of the so-called intelligent computing. These methods are based on the principle of swarm intelligence, and are most commonly used for routing in the graph [3]. Ant algorithms was originally proposed and implemented in computer application in 1991 by Marco Dorigo, Vittorio Maniezzo and Alberto Colorni [2],[4],[5],[6]. This solution is known as a ant system solving the TSP problems (traveling salesman problem). Initially, this technique was not particularly appreciated, because very often resulted in the achievement of local optimum. In the next years presented the modifications of the basic ant algorithm, which allowed for a wider application of this technique. An improved version of it, which envisages modifying the application of the pheromone trail, so as to minimize the cost of computation, is called the ant system. Modified versions of the ant algorithm quickly found applications in a larger number of problems, such as: scheduling, seeking partition and covering sets, detecting edges of the raster images and other optimization problems [1], [4], [6], [7], [13]. The studies showed that ant algorithms and ant systems can provide correct solutions (very often sub-optimal, but satisfactory from the practical point of view) where deterministic algorithms are unable to cope with, due to a considerable computational complexity. A heuristic of an ant algorithm is based on imitation of ants cooperate in the collection of food in the surroundings of anthill. These organisms create self-organizing systems, which are perfect not only in ideal conditions, but also in case of unforeseen changes in the surrounding (e.g. changes in ambient shape, appearing barriers on the road). In this case, the heuristic does not means the solution based on the proven mathematically algorithm, but on the imitation of a natural phenomenon. Ants are insects characterized by the fact that they live in colonies and are able to cooperate. Thus are creating a strong community. As a result of such an organization they are able to perform many tasks that could not be performed individually. An example of this is ability to find the shortest path from the anthill to a food source. Ants also can adapt to the environment changing. They do this without any visual signs - for example, may find a new road to food, if currently existing is unavailable. This process is described in the following paragraph and illustrated in Fig. 1. Suppose that the first ant leaving anthill and starts looking for food. She does not know where the food is located and any other ant did not leave her any information. So she starts.

(3) Ant Algorithms for Designing Order Picking Systems. 261. . the trip in a random direction. If there is any food in his "immediate surroundings" (within its senses), will take part of the food and come back with it to the anthill. Her way back she can find by following her own pheromone trail (ants are able to distinguish their personal pheromone), which left searching for food. Ants' road from the anthill to food source shows Fig. 1.. Anthill. Food. Anthill. Food. Anthill. Food. Fig. 1. Building routes by ants. After first ant the anthill is being left by others, who when looking for food can use pheromone trail left by predecessors. This increases the probability of finding food sources that have been already found by the other ant. As has already mentioned the environment of insects may change and may appear new difficulties, which were not there before. When ants encounter roadblock they have to change the direction of their motion (see Fig. 1). First ant randomly selects direction. The probability of choosing shorter path is equal to the probability of choosing the longer. Some ants that have chosen the shorter route will quickly find the lost pheromone trail. Therefore, on this path has been left larger amount of pheromone per unit time. This means that it will be more attractive for the next ants and more of them will choose this route. Ant systems (ant algorithms) are used to solve difficult problems, often unsolvable with deterministic algorithms. They are used to deal with, among others: Travelling Salesman Problems, Scheduling problems, Knapsack Problems, Connectionless Network Routing, Optical Network Routing, etc. Therefore they are well suited for solving NP-complete problems.. .

(4) 262. Micha Kodawski, Roland Jachimowski. . 3. ANT ALGORITHMS FOR DESIGNING ORDER PICKING SYSTEMS For implementation of an ant algorithm to the problem of designing order picking system we assume that the individual elements of the order picking system will be mapped by nodes. Among that nodes will be moving ants. Connections between elements will reflect the relationships between them. Ants' paths consisting individual connections present a complete variant of order system picking (see Fig. 2 and Fig. 3). 1. 2. 3. 4. .... .... n-1. n. Target. Source. No. of blocks. No. of aisles. No. of bays. No. of pick levels. .... SKU classification methods. .... etc.. Fig. 2. An example of mapping OPS structure for ant algorithms use. OPS structure is divided into subsets, which are grouped into n levels (Fig. 2). Each level gathers elements characterizing individual components of the OPS (for example: number of blocks, number of pick aisles, number of pick levels, pick location classification method, SKU classification method, number of pickers, routing method, batching or pick list dividing method). In addition, for purpose of mapping the order picking system structure added two points (source and outlet) in which ants will begin and end their trip. The number of levels of OPS structure (n) depends on assumed level of details mapped OPS. The more detailed mapping of the system structure, the more levels will be set out. A similar relationship exists in the number of components on each level (e(n)). The more detailed analysis of the system, the more characteristics of elements at each level is possible to choose. The formulation of the OPS variant is an appropriate connecting and arranging elements of OPS structure. Therefore, it is need to combine items selected for each of n levels. At the same time it is unacceptable to combine elements on the same level (Fig. 3). Formulated in this way problem of designing variants of order picking system allows to use ant algorithms. In this case, ants will set off from the node "source" (anthill) and moved between OPS structure elements from different levels. Their journey will continue until the end, that is, to find "target" (food). They can't move between the nodes located at.

(5) Ant Algorithms for Designing Order Picking Systems. 263. . the same level, or skip any of levels (then OPS variant would be incomplete). In addition, not always exist relationships between any pair of elements belonging to different levels. This is due to assumed constraints and will be illustrated by example of designing of order picking area. 1. 2. 3. 4. .... n-1. n. Target. Source. No. of blocks. No. of aisles. No. of bays. No. of pick levels. .... SKU classification methods. .... etc.. Fig. 3. Possible connections between elements of OPS structure. For study, it was assumed that it is need to design the OPS in which order picking area has 300 pick places. The maximum allowable area dimensions are: length - 70 m, width 40 m, height - 6 m. Other parameters were taken from Table 1. Table 1 Order picking area parameters Parameter name Width pick aisle Width cross aisle Width pick location Height pick location Length pick location Permitted reserve of pick locations. Value 2m 3m 1,2 m 1,3 m 1m 5%. In this case, there are 33 possible variants of the picking zone (except for the location of depot). Variants are listed in Table 2. Table 2However, further let's consider only systems with five blocks. In this case each ant starts from the source and always passes through node with five blocks (see Fig. 4). Then she can go to any node of the second level. However, if it will be chosen layout with 15 aisles, then in the next step may be visited only nodes with one or two columns. If it will be selected one corridor, his possible successors could be nodes of two or three levels of storage, and so on (Fig. 4). Therefore it should be noted that early ants elections have a significant impact, or even determine, course of the route in the following steps..

(6) 264. Micha Kodawski, Roland Jachimowski. . For the purpose of designing variants of order picking system will be used cyclic and density ant algorithms which are characterized by the fact that pheromone trail is not updated in each subsequent movement of ants, but only when all ants complete their route (variants of OPS will be constructed). Table 2 Variants of order picking area layout No. of variant. No. of blocks. No. of aisles. No. of bays. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1. 2 2 3 3 4 5 5 5 6 7 7 8 9 10 10 11 11. 19 25 17 25 19 10 15 30 25 11 22 19 17 5 15 7 14. No. of storage levels 4 3 3 2 2 3 2 1 1 2 1 1 1 3 1 2 1. No. of variant. No. of blocks. No. of aisles. No. of bays. 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 -. 1 1 1 2 2 2 2 2 3 3 5 5 5 5 5 5 -. 14 15 15 5 5 7 11 15 5 10 5 5 10 10 15 15 -. 11 5 10 5 15 11 7 5 5 5 2 3 1 3 1 2 -. No. of storage levels 1 2 1 3 1 1 1 1 2 1 3 2 3 1 2 1 -. To understand rules of using that type of algorithms for solving this problem, in the next section will be presented next steps of designing and evaluation order picking system variants and our essential assumptions. It was assumed, that to solve this problem we use M ants whose are looking for road from the source to the target in T iterations (trials). Connections between nodes of various levels are not described with any consistent characteristics (such as length). Therefore, choosing of connection is independent of its length, but only whether it connection exist and what amount of pheromone trial is on it. 1. 2. 5. 3. 4. 1 1. Source 5. 10. 2. 15. 2. 3 3. No. of blocks. No. of pick aisles. chosen order picking area layout (No. 32). No. of bays. No. of pick levels. Fig. 4. Example of constructing order picking area layout. Target.

(7) Ant Algorithms for Designing Order Picking Systems. 265. . So before "migration" of ants it should be chosen number of levels of order picking system structure (N) and number of elements at each level (e(n)). Then it is necessary to identify possible connections between different levels (ߟ௜௝ ), through which insects will be moving. Wherein the ߟ௜௝ ൌ ͳ if (i,j)-th connection exist, but ߟ௜௝ ൌ Ͳ otherwise. Additionally, to connections should be added initial pheromone trails, which in the first iteration are equally strong. In the first iteration the first ant starts searching for food. Standing in source she can choose link to any element of the first level of OPS structure. The selection is random. Ant doesn't know length of connections and at first iteration at any of connections is the same strength of pheromone trail. The probability that m-th ant will go through one of available ௠ connections (‫݌‬௜௝ ሺ‫ݐ‬ሻ), at the time when she is in the i-th element can be represented as follows: pijm t . . Iij t ¢¡ U ij t ¯±°. . s N im t. B. Iis t ¢ U is t ¯±. B. . (1). where: U ij t - strength of pheromone trail in t-th iteration and at connection between i-th and. j-th node, heuristic information related to the existence or not of a connection (i, j)-th nij t. between the i and j nodes in t-th iteration. N im t - set of successors of the i-th element is not visited by the m-th ant until the t-th. B. iteration, - parameter used to control the relative importance of intensity of the pheromone trail.. In this way, ant will choose next connections and construct her route. These steps are repeated N times, until she will visit one node from all levels of OPS structure. In the last step, ant reaches her destination, giving at the same time first variant of order picking system. This variant complies with constraints and is considered as acceptable solutions of analyzed problem. This process continues to the moment that all M ants construct her routes. Then current iteration ends and pheromone trail at all connections is updated Uij t

(8) N . It means that for each connection is performed strengthening and volatilization process of pheromone trail, by the following formula: M. Uij t

(9) N 1 S ¸ Uij t

(10) %U ijm t , t

(11) N. m1. where:. (2).

(12) 266. Micha Kodawski, Roland Jachimowski. M. %U m1. m ij. t , t

(13) N - intensity of pheromone trail at connection (i, j) - th after N passes of. all M ants. The intensity of pheromone trail at a connection (i, j) after t-th iteration is dependent on value that achieved function of OPS variant evaluation (criterion function), adopted for research. What can be presented as follows: x if criterion function (evaluation function of order picking system) tends to a minimum £¦ 1 ¦¦ m when m-th ant traverses the connection i, j. %U ijm t , t

(14) N ¤ H t

(15) N. ¦¦ 0 otherwise ¦¥¦. x. (3). if criterion function (evaluation function of order picking system) tends to a maximum £ ¦H m t

(16) N when m-th ant traverses the connection i, j. %U ijm t , t

(17) N ¦ ¤ ¦ 0 otherwise ¦ ¥. (4). where: H m t

(18) N -. value of evaluation function of order picking system constructed by m-th ant.. At the end of each t-th iteration is searched and saved route, for which criterion function is the lowest (or the highest - depending on form of the evaluation function). Then starts next iteration, and all ants again move from the source to the target. But this time, at individual connections is different amount of pheromone trail. Therefore, those with a higher intensity of pheromone will be more likely chosen by ants. Process of constructing new routes by ants is repeated T times until all of assumed iterations are executed. Then, is selected the best route in terms of adopted criterion among the best in the individual iterations. This route will be treated as rational (suboptimal) solution of the problem of designing order picking system. To evaluate different OPS variants (OPSV) time of order picking process could be used. We assumed that order picking process starts when op-th picker takes pl-th picking list and goes to first picking place. And it ends when op-th picker puts collected (according to pl-th picking list) mixed unit load. To determine duration of whole order picking process (time of picking all orders - picking lists) it is necessary to determine moments of starting msoppl OPSV and ending meoppl OPSV all pl-th picking lists by all op-th pickers, then placing them on the timeline. Next, it must be determined the earliest moment of starting and the latest moment of ending picking by each employee. The length of time period between these moments is time of whole order picking process. In analyzing current problem we are looking for that OPSV, which will give us minimum time of order picking process. It can be formulated as follows:.

(19) Ant Algorithms for Designing Order Picking Systems. 267. . F OPSV max \meoppl OPSV ^ min \msoppl OPSV ^ ¶¶¶ l min op , pl. op , pl. (5). For estimation value of criterion function was used method of researching order picking process and its computer implementation (application SymPick) presented in [12]. Simplified algorithm for designing optimal variants of order picking system using ant algorithms, step by step, presented as follows: Step 1. Select number of levels of order picking system structure (n); Step 2. Select number of elements (e(n)) at each n-th level; Step 3. Determine possible connections between elements at different levels ߟ௜௝. £ ¦1 Iij ¦ ¤ ¦ ¦ ¥0. if exist connection between element i -th and j -th otherwise. Step 4. Set initial values t:=0, m:=0; Step 5. Put the initial pheromone trail on connections U ij t ; Step 6. Start new iteration t:=t+1; Step 7. Start m-th ants trip (locate her in the source) m:=m+1, i:=0; Step 8. Defining by an m-th ant located in i-th element probability of choosing (i, j)-th connection ௠ - ‫݌‬௜௝ ሺ‫ݐ‬ሻ; Step 9. Selection by an m-th ant (i, j)-th connection and her passing to the j-th element, i:=j; Step 10. Verify that currently visited i-th element is at N-th level: YES - m-th ant constructed entire route in t-th iteration and goes to target (food) - go to step 11; NO - m-th ant continues construction of route in t-th iteration - go to step 8; Step 11. Verify that an m-th ant is the last ant in t-th iteration (m = M): YES - end t-th iteration - go to step 12; NO - start new ants trip - go to step 7; Step 12. Update pheromone trail at all routes; Step 13. Select and save the best one, in terms of assumed criteria, road of t-th iteration; Step 14. Verify that t-th iteration is the last one (t = T); YES - go to step 15; NO - start new iteration - go to step 6; Step 15. Select and save the best road among best roads from each t-th iteration; Step 16. Stop. 4. SUMMARY In paper has been taken thematics of order picking systems and processes. It drew attention to importance of these issues due to efficiency of logistics facilities and quality of their logistics services. And shows that due to the high complexity of order picking systems and processes their designing, research and analysis causes many difficulties. Therefore, in paper authors considered issue of designing variants of order picking systems, which enable a smooth and efficient order picking process. As an optimality.

(20) 268. Micha Kodawski, Roland Jachimowski. . criterion of OPS variants adopted time of order picking process, which is also basis for determining other characteristics of that process, such as efficiency or productivity. To solve a given decision problem proposed to use ant algorithms. Authors present how to map order picking system structure to allow implementation of ant algorithms and show a simplified algorithm for designing optimal order picking system variants with their use. Verifying correctness of developed algorithm was made with using SymPick application ([13]). This application is used to investigate efficiency of picking process so it helped to evaluate designed variants of order picking system. The initial implementation of the developed algorithm, as module of SymPick application, allowed to perform series of studies. As a result they have showed correctness of this algorithm. Research work carried out under a dean grant: "Algorytm ksztatowania optymalnych wariantów systemu komisjonowania z uwzgldnieniem problemu kongestii w strefie kompletacji" No.: 504/1160/0312. Bibliography 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.. Boryczka M.: Programowanie mrowiskowe w procesie aproksymacji funkcji. Wydawnictwo Uniwersytetu lskiego, Katowice 2006. Colorni A., Dorigo M., Maniezzo V.: Distributed optimization by ant colonies. Appeared in Proceedings of ECAL92, Paris 1992. Dorgio M., Maniezzo V., Colorni A.: Ant Colonies for the Travelling Salesman Problem. BioSystems, Vol.43 (1997), 73-81 Dorigo M., di Caro G.: The ant colony optimization meta-heuristic. New Ideas in Optimization. McGraw-Hill, London, 1999. Dorigo M., Gambardella L. M.: Ant Colony System: A Cooperative Learning Approach to the Traveling Salesman Problem. IEEE Transactions on Evolutionary Computation Vol. 1, No. 1, pp. 53-66, 1997. Dorigo M., Maniezzo V., Colorni A.: Positive feedback as a search strategy. Technical Report 91-016, 1991. Dorigo M., Socha K.: An introduction to ant colony optimization. Technical Report TR/IRIDIA/2006010, 2006. Drury J.: Towards more efficient order picking. The Institute of Materials Menagement, Cranfield, U.K., 1988 Kodawski M., Jacyna M.: Czas procesu kompletacji jako kryterium ksztatowania strefy komisjonowania. Logistyka 02/2011, artyku na CD, 2011 Kodawski M., Jacyna M.: Matematyczny model ksztatowania strefy komisjonowania, AUTOMATYKA 2011, z. 2, Tom 15, Kodawski M., Jacyna M: Wpyw ukadu strefy komisjonowania na dugo drogi kompletowania. Logistyka 04/2010, str. 18, artyku na CD, 2010 Kodawski M.: Metoda badania procesu komisjonowania w zalenoci od rozmieszczania artykuów w strefie kompletacji w obiektach logistycznych, rozprawa doktorska, WTPW, 2012 Yancang Li, Wanqing Li: Adaptive Ant Colony Optimization Algorithm Based on Information Entropy: Foundation and Application. Fundamenta Informaticae, IOS Press, 2007..

(21) Ant Algorithms for Designing Order Picking Systems. 269. WYKORZYSTANIE ALGORYTMÓW MRÓWKOWYCH DO KSZTATOWANIA SYSTEMÓW KOMISJONOWANIA Streszczenie: W artykule przedstawiono propozycj wykorzystania algorytmów mrówkowych do ksztatowania optymalnych wariantów systemu komisjonowania. Omówiono zasady dziaania i moliwoci wykorzystania algorytmów mrówkowych do problemów badawczych. Zaprezentowano sposób odwzorowania struktury systemu komisjonowania na potrzeby implementacji algorytmów mrówkowych oraz przedstawiono uproszczony algorytm ksztatowania optymalnych wariantów systemu komisjonowania z ich wykorzystaniem. Sowa kluczowe: algorytmy mrówkowe, ksztatowania systemów komisjonowania, kompletacja.

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