72 CHAPTER 3 RESEARCH METHODOLOGY Inventory management is considered to be an important field in Supply chain management. Once the efficient and effective management of inventory is carried out throughout the supply chain, service provided to the customer ultimately gets enhanced. Hence, to ensure minimal cost for the supply chain, the determination of the inventory to be held at various levels in a supply chain is unavoidable. Minimizing the total supply chain cost refers to the reduction of holding and shortage cost in the entire supply chain. Efficient inventory management is a complex process which entails the management of the inventory in the whole supply chain and getting the final solution as optimal, i.e. in the process of supply chain management, the stock level at each member of the supply chain should account to minimum total supply chain cost. The dynamic nature of the excess stock level and shortage level over all the periods is a serious issue when implementation is considered. In addition, consideration of multiple products leads to very complex inventory management process. The complexity of the problem increases when more distribution centers and agents are involved. In this present research, these issues of inventory management have been focused and novel models based on GA and PSO Meta heuristics have been proposed
73 in which the most probable excess stock level and shortage level required for inventory optimization in the supply chain is distinctively determined so as to achieve minimum total supply chain cost. 3.1 GENETIC ALGORITHM Genetic algorithm is a randomized search methodology having its roots in the natural selection process. Initially the neighborhood search operators (crossover and mutation) are applied to the preliminary set of solutions to acquire generation of new solutions. Solutions are chosen randomly from the existing set of solutions where the selection probability and the solution s objective function value are proportional to each other and eventually the aforesaid operators are applied on the chosen solutions. Genetic algorithms have aided in the successful implementation of solutions for a wide variety of combinatorial problems. The robustness of the Genetic algorithms as search techniques have been theoretically and empirically proved [95]. The artificial individual is the basic element of a GA. An artificial individual consists of a chromosome and a fitness value, similar to a natural individual. The individual's likelihood for survival and mating is determined by the fitness function [93]. In accordance with the Darwin s principle, individuals superior to their competitors, are more likely to promote their
74 genes to the next generations. In accordance with this concept, in Genetic Algorithms, a set of encoded parameters are mapped into a potential solution, named chromosome, to the optimization problem [94]. The population of candidate solutions is obtained through the process of selection, recombination, and mutation performed in an iterative manner. [96]. Chromosomes refer to the random population of encoded candidate solutions with which the Genetic algorithms initiate with. [95]. Then the set (called a population) of possible solutions (called chromosomes) are generated [99]. A function assigns a degree of fitness to each chromosome in every generation in order to use the best individual during the evolutionary process [98]. In accordance to the objective, the fitness function evaluates the individuals [96]. Each chromosome is evaluated using a fitness function and a fitness value is assigned. Then, three different operators- selection, crossover and mutation- are applied to update the population. A generation refers to an iteration of these three operators [97]. The promising areas of the search space are focused in the selection step. The selection process typically keeps solutions with high fitness values in the population and rejects individuals of low quality [96]. Hence, this provides a means for the chromosomes with better fitness to form the mating pool (MP) [99].
75 After the process of Selection, the Crossover is performed. In the crossover operation, two new children are formed by exchanging the genetic information between two parent chromosomes (say C1 and C2 which are selected from the selection process) [99]. A crossover point is chosen at random by the crossover operator. At this point, two parent chromosomes break and then exchange the chromosome parts after that point. Consequently, the partial features of two chromosomes are combined to generate two off springs. The chromosome cloning takes place when a pair of chromosomes does not cross over, thus creating off springs that are exact copies of each parent [98]. The ultimate step in each generation is the mutation of individuals through the alteration of parts of their genes [96]. Mutation alters a minute portion of a chromosome and thus institutes variability into the population of the subsequent generation [99]. Mutation, a rarity in nature, denotes the alteration in the gene and assists us in avoiding loss of genetic diversity [96]. Its chief intent is to ensure that the search algorithm is not bound on a local optimum [98]. Evolutionary algorithms like Genetic Algorithms offer practical advantages to several optimization problems. The advantages include: Simplicity of the approach Robust response to dynamic changes
76 Parallelism Hybridization Discovers global optimum Resistant to being trapped in local optima Handles large,complex search spaces easily Perform very well for large-scale optimization problems Used in a variety of optimization problems 3.2 INVENTORY OPTIMIZATION ANALYSIS USING GENETIC ALGORITHM The inventory control for more number of products along with different levels of supply chain is a complex task. To make the inventory control effective, the most primary objective is to predict where, why and how much of the control is required. Such a prediction is to be made here through the methodology proposed. To accomplish the same, Genetic algorithm is used and the optimal number of units of a particular product that needs to be kept in the level of control is determined on the basis of the knowledge of the past records. This leads to an easy estimation of the level of stocks of the respective products to be maintained in the upcoming periods. For instance, a three stage supply chain having seven members is depicted in Fig: 3.1.
77 Fig: 3.1 3 stage-7 member supply chain As illustrated in Fig: 3.1, a factory is the parent of the chain and it is having two distribution centers, Distribution center 1 and Distribution center 2. Distribution centers further comprise of two agents each. So, in aggregate there are four agents, Agent1 and Agent2 for Distribution center 1 and Agent 3 and Agent 4 for Distribution center 2. The factory manufactures different products that would be supplied to the distribution centers. From the distribution center, the stocks will be moved to the corresponding agents. The methodology adopted is intended to determine the specific product that needs to be concentrated on and the amount of stock levels of the product that has to be maintained by the different members of the supply chain. Also, the methodology analyses whether the stock level of the particular product at each member of the supply chain needs to be in abundance in order to avoid shortage of the product or needs to be held minimal in order to minimize the holding cost.
78 The steps involved in the methodology is illustrated in the Fig: 4.2 which would analyze the past records and facilitate efficient inventory management using Genetic Algorithm. Fig: 3.2 Genetic Algorithm flow for the proposed inventory management analysis The process depicted in Fig: 3.2 will be repeated along with the new chromosome obtained from the previous process. In other words, at the end of each of the iteration, a best chromosome will be obtained. This will be included along with a newly generated chromosome for the next iteration. Eventually, an individual which is the optimal one among
79 all the possible individuals is obtained. This best chromosome obtained has the information about excess/shortage of stock levels of a particular product corresponding to each member of the supply chain. From the information it can be concluded that the particular product and its corresponding stock levels play a significant role in the increase of supply chain cost. By controlling the stock level of that particular product at every member of the supply chain in the upcoming periods, the total supply chain cost can be minimized. 3.3 PARTICLE SWARM OPTIMIZATION In 1995, Kennedy and Eberhartin, inspired by the choreography of a bird flock, first proposed the Particle Swarm Optimization (PSO). In comparison with the evolutionary algorithm, PSO, relatively recently devised population-based stochastic global optimization algorithm, has many similarities and the robust performance of the proposed method over a variety of difficult optimization problems has been proved [111]. In accordance with PSO, either the best local or the best global individual affects the behavior of each individual in order to help it fly through a hyperspace [106]. Simulation of simplified social models has been employed to develop Particle Swarm Optimization techniques. The following are the features of the method [107]:
80 The researches on swarms such as fish schooling and bird flocking are the basis of the method. The computation time is short and it requires little memory as it is based on a simple concept. Nonlinear optimization problems with continuous variables were the initial focus of this method. Nevertheless, problems with discrete variables can be treated by easy expansion of the method. Hence, the mixed integer nonlinear optimization problems with both continuous and discrete variables can be treated with this method. In addition to PSO, several evolutionary paradigms exist which include Genetic algorithms (GA), Genetic programming (GP), Evolutionary strategies (ES) and Evolutionary programming (EP). Biological evolution is simulated by these approaches which are based on population [108]. Genetic algorithm and PSO are two widely used types of evolutionary computation techniques among the various types of Evolutionary Computing paradigms [109]. PSO and evolutionary computation techniques such as Genetic Algorithms (GA) have many similarities between them. A population of random solutions is used to initialize the system which updates
81 generations to search for optima. Nevertheless, PSO does not have evolution operators such as crossover and mutation that are available in GA. In PSO, the potential solutions, called particles follow the current optimum particles to fly through the problem space. Every particle represents a candidate solution to the optimization problem. The best position visited by the particle and the position of the best particle in the particle s neighborhood influences its position. Particles would retain part of their previous state using their memory. The particles would still remember the best positions they ever had even as there are no restrictions for particles to know the positions of other particles in the multidimensional spaces. An initial random velocity and two randomly weighted influences: individuality (the tendency to return to the particle s best previous position), and sociality (the tendency to move towards the neighborhood s best previous position) form each particle s movement [110]. When the neighborhood of a particle is the entire swarm, the global best particle refers to the best position in the neighborhood and in this case, gbest PSO refers the resulting algorithm. Generally, lbest PSO
82 refers the algorithm in cases when smaller neighborhoods are used [109]. A fitness function that is to be optimized evaluates the fitness values of all the particles [110]. PSO uses individual and group experiences to search the optimal solutions. Nevertheless, previous solutions may not provide the solution of the optimization problem. The optimal solution is changed by adjusting certain parameters and putting random variables. The ability of the particles to remember the best position that they have seen is an advantage of PSO [110]. 3.4 PREDICTION ANALYSIS BASED ON PARTICLE SWARM OPTIMIZATION ALGORITHM The supply chain cost increases because of the influence of lead times for supplying the stocks as well as the raw materials. Practically, the lead times will not be same through out all the periods. Maintaining abundant stocks in order to avoid the impact of high lead time increases the holding cost. Similarly, maintaining fewer stocks because of ballpark lead time may lead to shortage of stocks. This also happens in the case of lead time involved in supplying raw materials. A better optimization methodology would consider all these above mentioned factors in the prediction of the optimal stock levels to be maintained such that the total
83 supply chain cost can be minimized. Here, an optimization methodology that utilizes the Particle Swarm Optimization algorithm, one of the best optimization algorithms, is proposed to overcome the impasse in maintaining the optimal stock levels in each member of the supply chain. Taking into account the stock levels thus obtained from the proposed methodology, an appropriate stock levels to be maintained in the approaching periods that will minimize the supply chain inventory cost can be arrived at. The methodology proposed here will minimize the total supply chain cost by predicting optimal stock levels, not only by considering the past stock levels but also considering the lead time of the products to reach each supply chain member from its previous stage as well as the lead time involved in supplying the raw materials to the factory. Usually, shortage for a particular stock at a particular member, excess stock levels at a particular member, time required to transport stock from one supply chain member to another i.e. lead time of a stock in a member, time taken to supply raw materials to the factory to manufacture certain products i.e. lead time of raw materials in factory are some of the key factors that play a vital role in deciding the supply chain cost. A better optimization methodology should consider all these factors. In the proposed methodology, all the above mentioned key factors in predicting
84 the desired stock levels for the purpose of minimizing the supply chain inventory cost are considered. Also, different priorities are assigned to those above factors. As per the priority given, the corresponding factors will influence the prediction of optimal stock levels. Hence as per the desired requirement, the optimal stock level will be maintained by setting or changing the priority levels in the optimization procedure. Supply chain model is broadly divided into four stages in which the optimization is going to be performed. The supply chain model is illustrated in the Fig: 3.3. Fig: 3.3 Four stage supply chain model
85 Now, the particle Swarm Optimization (PSO) is utilized to predict the optimal stock levels to be maintained in the future so as to minimize the supply chain cost. The steps involved in determining the optimal stock levels are illustrated in Fig: 3.4 Fig: 3.4 Particle swarm optimization in optimizing the stock levels
86 3.5 LIMITATIONS Inventory at unit levels is considered instead of cost levels to maintain generalization of the approach. Usually the inventory cost of a unit is expressed as a percentage of the unit price which will vary from one product to another. Though there are many costs like transaction cost, transportation cost, facility cost, information cost etc., inventory cost is contributing maximum towards total supply chain cost. So by taking care of the emerging excess/shortage inventory levels in the forthcoming period, the inventory cost can be minimized and thus the total supply chain cost. The algorithms are not implemented by different meta heuristic techniques for comparison but instead implemented by popular techniques found suitable for the purpose. Similarly different types of cross over operators or mutation operators are not attempted in this study and again the popular operators of each one of them is chosen for implementation of GA. The algorithm and the technique is applied on the simulated data set to maintain generality of the approach and is not confined to any particular company/industry/product. These areas provide ample scope for prospective future researchers in the area of supply chain inventory optimization.
87 3.6 CONCLUDING REMARKS A brief overview of two popular Meta heuristics GA and PSO together with the advantages of the same is presented in this chapter. Also the methodology of these two methods to predict the optimal stock levels to facilitate optimal inventory to be maintained in the supply chain in the future to minimize the supply chain cost is illustrated. In addition, the limitations of the proposed research work have also been outlined.