pheromone, increasing the concentration of pheromones in this trail. The ants return to the nest using always the same path, depositing in the way back another ...
OPTIMIZATION OF LOGISTIC PROCESSES USING ANT COLONIES∗ C. A. Silva§,† T.A. Runkler§ J.M. Sousa†,§ R. Palm§ §
Siemens AG Corporate Technology Information and Communications 81730 Munich, Germany
KEYWORDS
Ant colonies, logistic processes, scheduling. ABSTRACT
The scheduling of logistic processes modeled by birth-anddeath processes is a combinatorial problem. We consider here the problem of dynamic assignment of components to orders. In this paper, a distributed algorithm based on ant colonies is proposed to optimize this assignment. The ant-agents jointly test several different combinations and choose the solution that is able to deliver more orders at the correct delivery date, while keeping the delay variance small for the orders that are not delivered at the desired date. A simulation example is presented, comparing this algorithm with three other scheduling methods. Results show the effectiveness of the proposed algorithm. INTRODUCTION
In supply chains management, logistics can be defined as the subprocess of the supply chain process that deals with the planning, handling, and control of the storage of goods between the manufacturing point and the consumption point. In the past, goods were produced, stored and then delivered on demand. Nowadays, many companies do not work with stocks, using instead cross-docking centers [Jayashankar M. Swaminathan and Sadeh, 1998]. The goods are transported from the suppliers to these cross-docking centers, stored, and then shipped to the costumers. The lack of storage may increase the delivery time, but it considerably reduces the volume of invested capital and increases the flexibility of the supply chain. The key issue is to deliver the goods in time by minimizing the stocks. The goods should be delivered at the correct date (not earlier or later) in order to ensure the costumers satisfaction.
†
Technical University of Lisbon Instituto Superior T´ecnico Dept. Mech. Eng. GCAR-IDMEC 1049-001 Lisbon, Portugal
components are assigned to the orders at the moment when the components are ordered from the suppliers. They can therefore not be used by any other order. This static scheduling strategy revealed to be poor. Therefore, a dynamic assignment strategy, allowing to exchange components between the orders is necessary. Dynamic assignment consists of distributing the available components to orders. This can be done with a sorted list of orders. If the first order in the list can be delivered using the available components, the components are taken from the stock and delivery is triggered. Then, the dynamic assignment algorithm proceeds to the next orders in the list, an so on. The easiest way to sort this list is to use a First In First Served (FIFS) principle, where the orders are sorted by order date or a First Desired First Served (FDFS) principle where the orders are sorted by their desired date. Since both principles use a unique orders list, these scheduling methods are called centralized dynamic approaches. This paper proposes a dynamic assignment method using a distributed approach. The idea is to assign individual agents to the orders and let the population of agents interactively find an optimization scheduling solution [Palm and Runkler, 2002]. The interaction between the agents is realized by exchanging information about quantity, desired date and arriving date.
In a logistic scheduling problem, the number of agents involved and the quantity of information that has to be exchanged is very large. Multi-agent algorithms based on social insects can avoid this complexity. Social insects have captured the attention of scientists because of the high structuration level that the colonies can achieve, especially when compared to the relative simplicity of the individuals. Ants are one example of social insects. Even though ants are very simple animals, with no special abilities and almost blind, they are capable of establishing the shortest route paths from their colonies to feeding sources and back to the nests. Here, we propose an ant The scheduling algorithm has to decide, which goods are algorithm for logistic processes. delivered to which costumers. If the costumers’ orders are sets of different components, the most common method is to The paper is organized as follows. The next section presents pre-assign the components to the orders. In this strategy, all a global description of a logistic process. Then, some of ∗ This work is supported by the German Ministry of Education and Research (BMBF) under Contract no.13N7906 (project NIVELLI) and and by the Portuguese Fundation for Science and Technology (FCT) under Grant no. SFRH/BD/6366/2001.
the standard scheduling algorithms are briefly described. Further, the principles of the optimization algorithm using ant colonies are introduced, as well as the new framework for
an estimated delivery date, which is the predicted date for the order delivery.
Component request
When an order has been accepted by the logistic system, the different components must be requested from the suppliers. Each component is characterized by a certain quantity. Requesting the components consists of making a purchase list with the types of components and their quantities, so that the external suppliers can deliver the components.
Component arrival
Figure 1: General Representation of the Logistic Process Each component takes some time to be delivered to the logistic system. This time is called the supplier delay and accounts the application to scheduling logistic processes. Finally, it is for the necessary time for the supplier to deliver the demanded presented a simulation example and the analysis of the results. components. After this time, the component is delivered to The closing section concludes this paper and defines the future the so-called cross-docking places, e.g. airports. A component research work. stock list is built at these places, which contains the available components and their quantity. THE LOGISTIC PROCESS
Figure 1 presents a schematic representation of a logistic process, which can be described in probabilistic terms. In fact, the birth process of the system (arrival of new orders in a certain period of time) and the death process (delivery of orders per unit of time, or the time it took them to be processed), can be described by the classical theory of queuing processes [Wolff, 1989]. For the process being studied, this theory asserts the Poisson (and exponential depending on the perspective where the process is being looked at) distribution for the model of the birth process,
Component assignment
All the components have to be delivered to the docking places, but usually the components are not all available at the same time. For this reason, the orders have to wait for all required components to be available. This waiting list is called the order stock. Each order has its own desired delivery date. Since the components are available at different instants in time, the decision process has to decide which orders are going to be delivered, taking into account if their components are already available. This is usually done once per day. This process con(λt)x −λt sists of examining the component stock list, comparing it to the e (1) p(x, λt) = order stock, and making a decision based on some optimization x! routine. where x is the random variable ”number of orders” and λt is the parameter that indicates the probability of this event to occur in a certain time t. The death process is modeled by the STANDARD SCHEDULING ALGORITHMS exponential distribution (2) The component assignment is the key issue of a logistic process. The company can not influence the birth rate of the orwhere t is the random variable and µ is the death rate. The five ders, or the suppliers delay. The death rate of the orders are the only control variable, through the assignment of components steps of the process are described in the following sections. to the orders. The control goal is to generate particular death rates for each order by some decision process using external Order arrival information like desired times or internal information, such as The company delivers products, which can be seen as a the amount of stocks [Palm and Runkler, 2001]. collection of components. Whatever the type of product needed by the client is, it is called an order. An order is a set of The scheduling assigns components to orders at each day. one or more different items, called generally the components ci . This paper compares the pre-assignment method, the dynamic centralized approaches FIFS and FDFS based on sorting lists, When an order is submitted to the system, it must con- and a dynamic decentralized approach using the ant colonies. tain a desired delivery date, which is the date required by the The next sections describe in detail these different scheduling client for the order to be delivered. The system should offer policies. p(t, λ) = µe−µt
Pre-assignment (PA)
When the components arrive to the cross-docking center from the external suppliers, they are already assigned to specific orders. If the components do not arrive all at the same time at the docking center, they have to be stored there until all the missing components are arrived, so that all the components of the order can be delivered together. This strategy can not deal efficiently with disturbances, such as a delay in the component arrival or a change in the quantity of demanded components by the clients. Since the assignment is already defined as soon as an order enters the system, we call this a static scheduling method. First In First Served (FIFS)
The static approach cannot be influenced by the company, it totally depends on external variables. In contrast to that, the dynamic assignment is based on the exchange of components between the orders. All the components in the cross-docking center are gathered in a single stock, and the list of orders is sorted in such a way that the order on top of the list is the first one to be analyzed by the decision method. If all the components that it needs are available, the order is delivered. If not, this order stays in the list, and the next order is analyzed. The easiest way to sort this list is by arriving dates, so the first orders to get into the system are the ones that will be first served, thus it is called a FIFS. Notice again, that if the first order can not be delivered, the components are not assigned to that order,as with the pre-assignment scheduling method.
SCHEDULING USING ANT COLONIES
Ants are social insects. They live in colonies and all their actions are towards the survival of the colony as a whole, rather than the benefit of a single individual of the society. The individual ants have no special abilities. They communicate between each other using chemical substances, the pheromones. This indirect communication allows the entire colony to perform complex tasks, such as establishing the shortest route paths from their nests to feeding sources. In [Dorigo and Maniezzo, 1996] an optimization algorithm was proposed that tries to mimic the foraging behavior of real ants, i.e. the behavior of wandering in the search for food. This algorithm has already been successfully used to solve the Traveling Salesman problem [Gambardella and Dorigo, 1996], the Job-shop floor problem [Cicirello and Smith, 2001] and other NP hard optimization problems. The next subsections describe this algorithm and its application to scheduling in logistic processes.
General description of the ant colonies algorithm
When an ant is searching for the nearest food source and comes across with several possible trails, it tends to choose the trail with the largest concentration of pheromone τ , with a certain probability p. After choosing the trail, it deposits another pheromone, increasing the concentration of pheromones in this trail. The ants return to the nest using always the same path, depositing in the way back another portion of pheromone.
First Desired First Served (FDFS)
A disadvantage of FIFS is that orders that arrived early are also delivered early, even though their desired delivery dates might not be reached yet. On the other hand, orders with early desired dates might be delayed, just because they entered the system too late. This problem can be solved by sorting the orders by desired delivery date instead of orders arrival date. This sorting method can also be combined with a priority list [J.M.Sousa et al., 2002]. Distributed approach
The use of information about how many orders will be delivered or if the orders placed on the top of the list (thus having some kind of priority) are indeed the ones that can be delivered, can increase the performance of the system. None of the previous strategies uses this internal information. The idea is to use internal information in a distributed approach. The distributed approach also works with a global stock, but there is no sorting between the orders. The agents associated with orders and components, exchange information between each other. This information can be the desired delivery dates, the quantity of components in the stock or the number of orders that will be delivered. After exchanging all the information, the agents jointly decide which orders will be delivered. This approach is more flexible, because it allows to evaluate the scheduling result and possibly modify it before delivery.
Imagine then, that two ants at the same location choose two different trails at the same time. The pheromone concentration on the shortest way will increase faster than the other: the ant that chooses this way, will deposit more pheromones in a smaller period of time, because it returned earlier. If a whole colony of thousands of ants follows this behavior, soon the concentration of pheromone in the shortest path will be much higher than the concentration in other paths. Then the probability of choosing any other way will be very small, and only very few ants among the colony will fail to follow the shortest path. There are another phenomenon related with the pheromone concentration. Since it is a chemical substance, it tends to evaporate in the air, so the concentration of pheromones vanishes along the time. In this way, the concentration of the less used paths will be much lower than that on the most used ones, not only because the concentration increases in the other paths, but also because its own concentration decreases. In general, the ant colony behavior can be described formally using the following mathematical framework. Let the nest and the food source be connected by several different paths, connecting n intermediate nodes. The ant k in node i chooses one of the possible trails (i, j) connecting the actual node to one of other possible positions j ∈ {1, · · · , n}, with probability pkij = f (τij )
(3)
where τij is the pheromone concentration on the path connecting i to j, in the way to the food source. The pheromone in this trail will vary for in time according to: k τij (t + 1) = τij (t) × ρ + δij (4) k where δik is the pheromone released by the ant k on the trail (i, j) and ρ ∈ [0, 1] is the evaporation coefficient. The system is continuous, so the time acts as the performance index, since the shortest paths will have the pheromone concentration increased in a shorter period of time.
Initialization: Set for every pair (i, j): τij = τ0 Set N = 1 and define a Nmax Place the m ants While N 0 833 543 497 468 min d -32 -32 -34 -12 max d 7 11 6 6
orders delay. Figure 4 presents the histograms for the different scheduling methods. They show the number of orders with a specific delay d: if the orders were delivered earlier, d < 0; if they were delivered at the correct date, d = 0; and if they were delivered with some delay, d > 0. Table 1 quantifies this analysis in number of orders and also indicates the spread for each method in number of days. What can be interpreted from the results is that the ant colonies algorithm yields the maximum number of deliveries on the correct date (#d = 0), and the minimum number of delayed orders (#d > 0). It also yields the lowest spread between maximum and minimum delays (max d − min d). This means that, even if the orders are delivered before or after the correct date, it is more likely with this method that the delays, negative or positive, are not too different from the correct desired date. This shows clearly,the superiority of the ant algorithm to the other methods used in this application. CONCLUSIONS
tween this algorithm and three other scheduling methods (p.a., FIFS and FDFS) have shown that the distributed ant algorithm is better to solve the problem than the static or centralized dynamic approaches. The simplified example mimics a real-world logistic process at Fujitsu-Siemens Computers [Silva, 2001], and it shows that the ant colonies algorithm is a better alternative to the actual scheduling method. As the next step, we will apply the ant algorithm to the real full scale problem.
[Cicirello and Smith, 2001] Vicent A. Cicirello and Stephen F. Smith. Ant colony for autonomous decentralized shop floor routing. In Proceedings of ISADS-2001, Fifth International symposium on autonomous decentralized systems, 2001. [Dorigo and Maniezzo, 1996] Marco Dorigo and Vittorio Maniezzo. Ant system: Optimization by a colony of cooperating agents. IEEE Transactions on Systems, Man, and Cybernetics-Part B, 26(1):29–41, 1996. [Gambardella and Dorigo, 1996] Luca Maria Gambardella and Marco Dorigo. Solving symmetric and asymmetric tsps by ant colonies. In IEEE Conference on Evolutionary Computation (ICEC’96), 1996. [Jayashankar M. Swaminathan and Sadeh, 1998] Stephen F. Smith Jayashankar M. Swaminathan and Norman M. Sadeh. Modeling supply chain dynamics: A multiagent approach. Decision Sciences Journal, 29(3):607–632, 1998. [J.M.Sousa et al., 2002] J.M.Sousa, Rainer Palm, C.A.Silva, and Thomas Runkler. Fuzzy optimization of logistic processes. Accepted in WCCI-2002, Honolulu, Hawaii, 2002. [Lee et al., 1995] J. Lee, Hsing-Pei Kao, and Weiling Chiang. A tradeoff analysis of logistics performance based on relationships between criteria. In International Joint Conference of CFSA/IFIS/SOFT ’95 on Fuzzy Theory and Applications, pages 320–325, 1995. [Palm and Runkler, 2001] Rainer Palm and Thomas Runkler. Decentralized control of hybrid systems. In Proceedings of ISADS-2001, Fifth International symposium on autonomous decentralized systems, 2001. [Palm and Runkler, 2002] Rainer Palm and Thomas Runkler. Multi-agent control of queuing processes. Accepted in IFAC-2002, Barcelona, Spain, 2002.
[Silva, 2001] Carlos A. Silva. Arbeitspaket S1: Systemanalyse - Fujitsu Siemens Computers. Technical Report 1, Siemens This paper presents a modified ant colony algorithm for the AG, Corporate Technology, Department of Neural Compuoptimization of logistic problems. A dynamically distributed tation, CT IC-4, October 2001. assignment of components to orders is achieved by means of exchanging simple information between the agents. The [Wolff, 1989] R. W. Wolff. Stochastic Modeling and the Thepheromones used by the ants to determine the assignment repory of Queues. Prentice -Hall, 1989. resent distributed information about prior assignment attempts, as well as information about the desired delivery date and availability of components to fulfill the orders. The comparison be-
