Application Progress of Genetic Algorithm for Economic Lot Sizing Problem Jinhong ZHONG
Ling HUANG
school of management Hefei University of Technology, HFUT Hefei, China
[email protected]
302 Unit Electronic Engineering Institute, PLA Hefei, China
[email protected]
Abstract—Application progress of genetic algorithm is reviewed systemically from three aspects of algorithm parameters setting, local search capabilities and constraint handling techniques which have to be solved in the application. Then application status of genetic algorithm for economic lot sizing problem is surveyed. Finally, the future work is presented.
better positions in the search space. Too much Exploration will lead to premature convergence, while too much exploitation will make the search process into random search. To regulate the crossover and mutation rate may maintain a balance between exploitation and exploration, which directly affects the validity and efficiency of GA search.
Keywords-Genetic Algorithm; Economic Lot Sizing Problem; parametric self-adaptation
Static probability, a satisfactory parameter combination is first found out by trial and error, and then is applied to GA search. In literature, there are a lot of general guide rules which are insufficient and usually have good performance only for some problems. Essentially GA is a dynamic adaptive process which varies with considered problems. Thereby, parameter value should vary in search process to attain best performance at different stage of evolution. Evidently, this method violates the spirit for it uses rigid parameters. Dynamic probability, the algorithm change parameters dynamically by the predefined determinate rules. Dynamic self-adapting usually encodes parameters into another chromosome and evolves with the problem chromosome together. For instance, Grefensette[4] boiled parameters adjustment down to an optimization problem, which determines the optimal parameters combination by Meta-GA. Dynamic adaption, the parameter are adjusted using feedback information from the search, whose bottleneck is to develop an effective measurement of feedback information and adjustment policy.
I.
INTRODUCTION
Economic lot sizing problem is one of the most important and difficult problems in production planning. There have been many studies on this and related topics since the seminal work of Wagner and Whitin[1]. In these work, there are plenty of researches using metaheuristic approach, especially for the capacitated problems with setup time, joint start-up costs and general product structure. With economic globalization, the global-supply-chain-oriented production/procurement planning has become a focus in people’s eyes, which synthetically consider the optimization of production, procurement, inventory management and transportation in the whole supply chain. The type of problems is very complicated and hard to solve, so people often try to use meta-heuristic approach. Genetic algorithm(GA) is a random search technique and optimization algorithm which bases on natural selection and genetic mechanism[2,3]. It has good robustness, could explore huge and complicated searching space effectively. So it’s an ideal technology for solving optimization problem. There is a great deal of reports about successful application of GA for economic lot sizing problem. II.
THE ADAPTION OF GA’S PARAMETERS
The performance of GA highly depends on parameters. How to set these parameters is an art. Recently people take more attention to the parameters adaption. A. Crossover and Mutation Probability The searching ability of genetic algorithm depends on a balance between exploration and exploitation[3]. Exploration investigates new and unknown area in search space to maintain the diversity of population, and avoid premature convergence. Exploitation uses knowledge acquired by exploration to reach
It is the key to generate formal fuzzy sets and rules for using fuzzy logic controller successfully in GA. Herrera and Lozano[5] summarized the framework, term and design of fuzzy adaptive GA. All of conventional heuristic approaches have a common character that operator probability is adjusted adaptively according to the convergence of GA search. When the search tends to local convergence, the rate of mutation and crossover should be increased to prevent premature convergence. When individuals scatter in search space, operator probability should be decreased to restrain the introduction of new genome, and make the search tend to convergence. Srinivas and Patnaik[6] measured convergence by the distance(i.e. fitness difference) between individual and best one, and developed a adjustment scheme of genetic operator probability. In the method, the individual who is nearer to the best individual will gain more opportunity to keep itself, but there is a more latent risk of algorithm prematurity. Wu et al.[7]
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put forward a improvement scheme for enhancing mutation operator based on the work in [6]. Mark et al.[8] gave a heuristic scheme to adaptively adjust crossover and mutation probability. That is, the operator with good performance is encouraged to generate more effective offspring, while the opportunity that bad operator generates offspring will be reduced. A mutation probability adjustment based on fitness is provided in [9], which differentiates the most and least important bits on chromosome. Wong et al.[10] proposed a probability driven adaptive model of crossover and mutation probability. In [11] the probabilities of selection, crossover and mutation operator are adaptively adjusted based on the survival rate of offspring generated by the operator.
Genetic operators generate new solution, and promote the evolution exploration and exploitation of search space forward. It is very important for some special problems to adapt operation of multi-genetic operators. However, there are few researches about control of genetic operators. Zhang et al.[17] designed a multi-genetic operators adaptive scheme. That is, utilizing ratio of each operator is adjusted adaptively and dynamically by its contribution to the search performance. Hong and Wang[18] provided an adaptive control policy to balance the use proportion of multi-mutation operators, but without control of crossover operators.
When there are too many similar genomes in population, it is very difficult to avoid premature convergence even if the probability of crossover and mutation is increased. At this time, a feasible option is to prohibit new genomes generated by crossover operation, and add the novel genomes which are different from the individuals in current population but with similar high fitness. Mutation operator can satisfy this requirement, but its output is usually random so that it isn’t ideal method. Local search technology[12] can generate novel chromosomes with high fitness, is a possible substitute.
Genetic Algorithm boosts the search by changing the distribution of sub-structure with high-performance in a population, so individuals’ structures are not their concerning focus. That is to say, GA is more suitable for the global search, while more exact local search is not its strength. Therefore, when GA found the high-performance areas of search space, there is need to invoke a local search with high efficiency. Simulated Annealing (SA) is more suitable for local search, can be used to optimize each individual solution locally.
B. Other Parameters Population size is an important parameter affecting algorithm performance. If population size is too small with respect to the problem complexity, the search may get into local optimum. On the contrary, GA with large population size may turn into simple random search with low efficiency. Cavalieri and Gaiardelli[13] used dynamic population size controlled by dynamic birthrate and individual life. In each generation, the end-of-life individuals will be removed from the population, new individuals will also be added according to birthrate. Hinterding et al.[14] developed a genetic algorithm which could concurrently adapt Gaussian mutation intensity and population size. The choice of code length determines not only the resolution of the gene and the precision of its result, but also the size of the space of representation the GA has to search. As GA searches long place-value codes rather inefficiently, there is a balance between search efficiency and the precision of result. Dynamic parameter encoding[15] uses statistical information of population convergence to control encoding length adaptively. Its basic idea is that optimal solution can only be in some sub-region of search space, and usually comparatively coarse search resolution would be sufficient to identify such sub-region. Once the sub-region is identified, follow-up GA search will increase accuracy to the needed degree. The genotype-phenotype map is fixed in canonical GA. The design of map determines how large an area of solution space can be covered by the search, and the quality of solution. It is very difficult for designers of the algorithm to design suitable map when there is not priori knowledge of solution space. Murao et al.[16] developed a coevolutionary approach which the GA not only search the solution space defined by the map, but also adapt the map itself in search process.
III.
IMPROVING LOCAL SEARCH ABILITY
Combination of GA and SA can improve the search efficiency and the quality of solutions. Huntley and Brown [19], Grefenstette[20] first proposed the concept of GA-SA, and now there have been many researches using this method. The basic idea of this hybrid method is that enhancing the search of the solution space by means of ideal convergence property of SA, simultaneously preserving the population optimization and reorganization ability of GA. To add the annealing scheme into GA is helped to increase probability of escaping local optimum in the search, and improve the quality of solution and reduce computing time. Kazarlis et al.[21] proposed a hybrid genetic scheme, in which the main GA is in charge of global search while auxiliary GA with small population and short-period evolution is used as a local hill-climbing operator to amend the current solution. Marinakis and Migdalas[22] developed a new GA based on the Neighborhood expansion technology. IV.
CONSTRAINT-HANDLING METHODS
Most optimization problems in the real world have constraints, so handling constraints is a problem that we must face. Constraint-handling techniques[23] used commonly can be categorized as penalty functions, special presentations and operators, repair algorithms of infeasible solution, separation of objectives and constraints and other Hybrid Methods etc. A. Penalty Function Penalty Function is a mature method to solve constrained optimization problems by GA. It bases on the thought of trying to give a certain degree of punishment to the individual who violates constraint. By introducing penalty term into fitness function, a constrained optimization problem can be converted into an unconstrained one. Static penalty function, is a fixed constant in the whole evolutionary process, can not vary with the current search. The penalty strength of dynamic penalty function vary with the iteration, and smaller penalties are used to ensure the diversity
of population in the initial iteration while in the later period penalties will be increased gradually in order to assure or speed up the convergence of algorithm. Annealing penalty function is a dynamic penalty using the thought of simulated annealing, that is to say, that punishment will increase with iteration time (i.e. temperature falling). Adaptive penalty function can adaptively adjust the degree of penalty based on the feedback (i.e. feasible status) from the search. Absolute penalty function refuses infeasible individuals to participate in the reproduction process so that there are merely feasible individuals in population. Co-evolutionary penalty considering the degree and number of constraints violation, respectively evolves solutions and penalty factors by two GAs. With two penalty functions, segregated GA achieves a balance between moderate and heavy penalties through maintaining two subpopulations. Therefore, it can reduce the probability that the search is difficult to converge due to too small penalty and converges to local minimum because of too much penalty. B. Hybrid Method Ant Colony Optimization, whose inspiration comes from the principles of ants seeking food, optimizes search paths according to the distribution of pheromone. When a solution violates a constraint, it will be discarded no matter how better its fitness is, which is equivalent to the death penalty. As a result of the tightening of constraints, some previously accepted solutions will disappear with evolution. Culture is another kind of inheritance mechanism, which can be coded by symbols, and spread in and between groups. Computation model of cultural evolution, whose thought is to protect the accepted beliefs and discard or prune the ones that are not accepted by society, contain two levels of inheritance process: micro-evolution and macro-evolution. The accepted beliefs can be regarded as constraints instructing microevolution. Thus, constraints may affect the search process directly, so as to lead to an efficient process. To handle constraint, Artificial immune system(AIS)[24] utilizes two populations: antigen group that are comprised of feasible individuals or the best solutions, and antibodies group that are consisted of infeasible individuals or the worst solutions. Antibodies are evolved to be closer to antigens through embedded AIS or GA, in the process of which bit-level affinity and gene-level Hamming Distance are be used. V.
APPLICATION OF GA FOR ECONOMIC LOT SIZING
GA is the most attractive meta-heuristic method to different kinds of lot sizing problems such as multi-level problem with time-varying costs, general limited lot sizing problem, economic scheduling problem, and sequence-dependent economic scheduling problem. Many researchers have applied GAs to capacitated and uncapacitated economic lot sizing problems recently, and the numerical value shows that GA (or combined with other meta-heuristic algorithms) is an effective method to solve this type of problems. Dimopoulos and Zalzala[25], Chaudhry and Luo[26] and Aytug et al.[27] provided a general review on the application of GA in the optimization of manufacturing and operational
management. Dellaert and Jeunet[28] sum up different methods to solve multi-level lot sizing problem, and developed a new heuristic method. Jans and Degraeve[29] offered a more extensive overview on the application of meta-heuristic methods in a dynamic lot sizing problem. In [11] a single-item lot sizing problem is solved by GA. Gaafar [30] designed a new coding schemes called ‘012’ for the determinate time –varying lot sizing problem with batchordering and backlogging. In [31] a genetic algorithm is designed to solve multi-level lot sizing problem considering manufacturing partner. In [32] GA is used to solve the lot sizing problem is developed with fixed or rolling horizon, It is found that in the case of rolling horizon, GA performs better than commonly used heuristic algorithms. Xie and Dung [33], Li et al. [34] blended domain knowledge of problems into GA, and designed heuristic genetic algorithms, which are applied respectively to the general structure of capacitated multi-level lot sizing problem and capacitated dynamic lot sizing problem with remanufacturing. Dellaert and Jeunet [35] developed a hybrid GA for solving large-scale multi-level un-capacitated lot sizing problem, in which GA is used to improve initial solutions that generated by the methods of partial period, period ordering and WW. Ip et al.[36] studied production planning problem with lead time and backlogging, and took into account the balance between lot size of multi-product and the capacity of multi-processing. Basnet and Leung[37] addressed a multi-period inventory lot-sizing problem with multi-product and multi-vendor. Sarker and Newton[38] developed a GA with three different penalty functions to determine the optimal batch size of products and related raw materials procurement policy. Dellaert et al. [39] researched the general multi-level lot sizing problem with time-varying cost that is often encountered in MRP. Moreover, they showed that GA can give a better solution to large-scale problems in a reasonable time, so GA can be used as a useful component of the MRP system. VI.
CONCLUSION
GA is a powerful tool of solving complicated economic lot sizing problems, and to develop effective algorithms that can be used in real life is still the direction of future research in a very long time. It is vital to blend the problems’ domain knowledge into the design of GA for improving efficiency and validity of algorithms. In the application of GA, it is a prominent problem how to set suited algorithm parameters. At present, people mostly tend to use the parameters obtained by self-adaptation or adaptation complying with predefined rules. There have been many researches in this area, but to explore new adaptive mechanism of parameters that can better reflect the nature of evolution and then to design a more suitable self-adaptive scheme of parameters still need to continue efforts in the future. Especially when there are infeasible genomes in population, the adaptive mechanism of parameters is still open. Since the local search ability of GA is weaker, now people overcome the shortage mainly by combining GA with the other algorithms that are good at local search or developing a specific local search operator. It is the key for the design of the
hybrid algorithm to consider the specificity of the problem itself. It is worth noting that constraint-handling technique is based on Artificial Intelligence such as cultural algorithms, AIS etc. In this aspect, the further research is very necessary, especially cultural algorithms. REFERENCES [1] [2] [3] [4]
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