Ant Colonies As Logistic Processes Optimizers

Outline Abstract Introduction The Logistic Process Scheduling Using Ant Colony Stimulation Results and Analysis Real work Example Conclusion References

Abstract The optimization of logistic processes using ant colonies. The analysis of the algorithm parameters is done in a simulation. It was applied to a real logistic process at Fujitsu-Siemens Computers. The results show that the ant colonies provide a better solution to logistic processes.

Introduction What is Logistics? Planning, handling, and control of the storage of goods between the manufacturing point and the consumption point. cross-docking centers instead of stocks.  The key issue is to deliver the goods in time by minimizing the stocks.  The scheduling algorithm has to decide which goods are delivered to which customers.  Centralized static scheduling vs. dynamic distributed scheduling

The Logistic Process Fig. 1. General representation of the logistic process

The Logistic Process The birth process (arrival of new orders). Poison distribution of the birth process: x : the random variable number of orders lambdaT : the probability of this event occur on a certain time T.

The Logistic Process The death process (delivery of orders) is modeled by the exponential distribution. T: the random variable μ: the death rate.

Process Description Order arrival. - order is a set of components ci and contain a desired delivery date Component request. - Each component has quantity. Component arrival. - supplier delay: time to be delivered to the logistic system.

Process Description Component assignment. The focus of this paper -A component stock contains the available components and their quantity. -A order stock is waiting list Order delivery -delay d is difference between the delivery date and the desired date.

Scheduling Policies Pre-assignment vs. dynamic decentralized approach Pre-assignment (p.a.). Components are assigned to specific orders. Not efficiently Distributed approach. The agents associated with orders and components exchange information between each other. More flexible than pre-assignment

Scheduling Using Ant Colonies The optimization of the scheduling process is a NP-hard problem.  The problem’s information can be translated into the pheromones, and used by all the interacting agents in order to achieve a good global solution.

Scheduling Using Ant Colonies Two different set of entities: component ≈ food source order ≈ nest. m ants, one per food source, distribute the food to the n nests.  In every iteration t of the algorithm, the ants have to choose with some probability p which is the nest to visit first. Then, they deposit a pheromone τ in the path from the food source to the nest.

Scheduling Using Ant Colonies Each ant delivers an amount qij from the total amount qi of component i  {1, . . . , m} to an order j  {1, . . . , n}. Since there are several nests to visit, the ant k chooses the path to a particular nest with a probability p

Scheduling Using Ant Colonies  ij is the amount of pheromone connecting i to j, nij is a visibility function Tk is the tabu list of the kth ant. This list contains all of the visited orders + the orders don’t need component type. α and β express the relative importance of trail pheromone (experience) with visibility (knowledge)

Scheduling Using Ant Colonies dj is the delay of order j  We try to have nij = 1. The local update of the pheromone concentration is then given by where  c is small constant.

Scheduling Using Ant Colonies At the end of a complete tour the change of pheromones in all paths is given by the solution can be evaluated using a performance measure n is the number of orders and dj is the delay of order j.

Scheduling Using Ant Colonies Fig. 2. Example of an ant colony applied to the logistic process with pheromone concentration level on the trails: High (—), Medium (- -) and Low (··)

Scheduling Using Ant Colonies At each tour N of the algorithm (where each tour has n × m iterations t), a z is computed and stored in the set Z = {z(1), · · · , z(N)} If z(N) is higher than the previous z  Z, then the actual solution has improved and the used pheromones should be increased. If it is worse, they should be reduced.

Scheduling Using Ant Colonies This is done by the global pheromone update

Algorithm Fig. 3. Ant colonies optimization algorithm for logistic processes

Simulation Results Let lambdaT = 10; each order can have at the most 7 different types of components ci; the quantity for each component varies randomly between 1 and 20; each type of component has a constant supplier delay, which are 1, 3, 2, 3, 1, 2, 6 days for components type c1, · · · , c7 respectively. For each order a desired date is generated using an exponential distribution with μ = 7. The simulation refers to an interval of 6 months.

Simulation Results The results are presented with the parameters α = 1, β = 10, p = 0.9 and Nmax = 20. Table 1. Comparison between the scheduling methods in number of orders 

Simulation Results Fig. 4. Histograms of the order delay d. Pre-assignment method (left) vs. the results for the ants (right).   Ants: higher number of delivery on time (d=0) lower spread between max and min delay

Tuning the Parameters The parameters α and β which are coupled between each other are changed at the same time, while others decoupled parameters remain constant. Varying α and β, using a fixed value p = 0.9.

Tuning the Parameters Fig. 5. Number of orders delivered for a fixed ρ and varying α and β.

Tuning the Parameters Number of orders delivered on the correct date is high, if β is small. It has an optimal value for α = 1  We can conclude that the α parameter tunes the number of orders in the right day and β controls the spread around that value

Tuning the Parameters Fig. 6. Number of orders in the right day, for different sets of fixed α and β and varying p  Evaporation coefficient (1-p)   p  0: the increment received by the new ants it will influence greatly the paths of the next ants p 1 the solution can rapidly stagnate.   As it can be seen, the value of evaporation should be around 0.1 (p = 0.9), in order to achieve a good solution

Tuning the Parameters Number of colonies per day Nmax. Fig. 7. Evolution of the solution for different number of colonies too few : not have enough iterations to find a good solution too many: increased severely the computational cost.

Real World Example The analysis presented an optimized solution with α = 1, β = 0.5, p = 0.9, and Nmax = 20.   In small data set of the data Fig. 8. Histograms of the orders delay d for the scheduling methods

Real World Example Ants: More orders are delivered on time. Less orders are delayed Good alternative to pre-assignment scheduling method !

Conclusions To apply in the ant colonies optimization algorithm to the optimization of logistic processes. It’s explored the correlations between the parameters and their role in the algorithm. The results show how the analysis is able to improve the algorithm performance, and explain the reasons for that improvement.  Finally, the algorithm was applied to a real data set, and the ant algorithm proved to be a better scheduling method than the pre-assignment. 

Conclusions Future work: The use of a different cost function z, The incorporation in the ants with some sort of prediction mechanism. There is still some information in the process, like priority of the orders, that was not used so far and can influence the scheduling of the logistic process.

References 1. Jayashankar M. Swaminathan, S.F.S., Sadeh, N.M.: Modeling supply chain dynamics: A multiagent approach. Decision Sciences Journal 29 (1998) 607–632 2. McKay, K., Pinedo, M., Webster, S.: A practice-focused agenda for production scheduling research. Production and Operations Management 10 (2001) 3. Palm, R., Runkler, T.: Multi-agent control of queuing processes. In: To appear in Proceedings ofWorld Conference on Automatic Controlo - IFAC’2002, Barcelona, Spain. (2002) 4. Silva, C.A., Runkler, T., Sousa, J.M., Palm, R.: Optimization of logistic processes using ant

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