Hybrid Techniques of Genetic Algorithm for Inventory of Auto Industry Model for Deteriorating Ttems with Two Warehouses

In this Paper Hybrid techniques are the integration of various soft computing techniques that synergizes the strength and weakness of an individual technique with the other therefore the overall performance of such techniques working in unison is increased. A deterministic inventory of Auto industry model has been developed for deteriorating items of Auto industry having a ramp type demand with the effects of inflation with two-storage of Auto industry facilities using Genetic Algorithms. Here, we assumed that the inventory of Auto industry holding cost in RW is higher than those in OW. Shortages in inventory of Auto industry are allowed and partially backlogged and it is assumed that the inventory of Auto industry deteriorates over time at a variable deterioration rate using Genetic Algorithms. The effect of inflation of Auto industry has also been considered for various costs associated with the inventory of Auto industry system. Cost minimization technique is using Genetic Algorithms to get the expressions for total cost and other parameters.


INTRODUCTION
Many researchers extended the EOQ model to time varying demand patterns. Some researchers discussed of inventory of Auto industry models with linear trend in demand. The main limitations in linear varying demand rate is that it implies a uniform change in the demand rate per unit time. This rarely happens in the case of any commodity in the market.
In this Paper Hybrid techniques are the integration of various soft computing techniques that synergizes the strength and weakness of an individual technique with the other therefore the overall performance of such techniques working in unison is increased. A deterministic inventory of Auto industry model has been developed for deteriorating items of Auto industry having a ramp type demand with the effects storage of Auto industry ng Genetic Algorithms. Here, we assumed that the inventory of Auto industry holding cost in RW is higher than those in OW. Shortages in inventory of Auto industry are allowed and partially backlogged and it is assumed that the inventory of teriorates over time at a variable deterioration rate using Genetic Algorithms. The effect of inflation of Auto industry has also been considered for various costs associated with the inventory of Auto industry system. Cost minimization Genetic Algorithms to get the expressions for total cost and other parameters.

warehouses, deterministic inventory, deteriorating items and Genetic Algorithms
Many researchers extended the EOQ model to timevarying demand patterns. Some researchers discussed of inventory of Auto industry models with linear trend in demand. The main limitations in linear-time varying demand rate is that it implies a uniform e in the demand rate per unit time. This rarely happens in the case of any commodity in the market.
In recent years, some models have been developed with a demand rate that changes exponentially with time. For seasonal products like clothes, Air conditions etc. at the end of their seasons the demand of these items is observed to be exponentially decreasing for some initial period. Afterwards, the demand for the products becomes steady rather than decreasing exponentially. It is believed that such type of demand is quite realistic. Such type situation can be represented by ramp type demand rate.
An important issue in the inventory theory is related to how to deal with the unfulfilled demands which occur during shortages or stock outs. In most of the developed models researchers assumed that the shortages are either completely backlogged or completely lost. The first case, known as backordered or backlogging case, represent a situation where the unfulfilled demand is completely back ordered. In the second case, also known as lost sale case, we assume that the unfulfilled demand is completely lost.
Furthermore, when the shortages occur, some customers are willing to wait for backorder and others would turn to buy from other sellers. In many cases customers are conditioned to a shipping delay and may be willing to wait for a short time in order to get their first choice. For instance, for fashionable commodities and high-tech products with short product life cycle, the willingness of a customer to wait for backlogging is diminishing with the length of the waiting time. Thus the length of the waiting time for the next replenishment would determine whether In recent years, some models have been developed with a demand rate that changes exponentially with time. For seasonal products like clothes, Air ons etc. at the end of their seasons the demand of these items is observed to be exponentially decreasing for some initial period. Afterwards, the demand for the products becomes steady rather than decreasing exponentially. It is believed that such type demand is quite realistic. Such type situation can be represented by ramp type demand rate.
An important issue in the inventory theory is related to how to deal with the unfulfilled demands which occur during shortages or stock outs. In most of the eloped models researchers assumed that the shortages are either completely backlogged or completely lost. The first case, known as backordered or backlogging case, represent a situation where the unfulfilled demand is completely back ordered. In the case, also known as lost sale case, we assume that the unfulfilled demand is completely lost.
Furthermore, when the shortages occur, some customers are willing to wait for backorder and others would turn to buy from other sellers. In many cases are conditioned to a shipping delay and may be willing to wait for a short time in order to get their first choice. For instance, for fashionable tech products with short product life cycle, the willingness of a customer to acklogging is diminishing with the length of the waiting time. Thus the length of the waiting time for the next replenishment would determine whether International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 @ IJTSRD | Available Online @ www.ijtsrd.com | Volume -2 | Issue -5 | Jul-Aug 2018 Page: 59 the backlogging would be accepted or not. In many real life situations, during a shortage period, the longer the waiting time is, the smaller is the backlogging rate would be. Therefore, for realistic business situations the backlogging rate should be variable and dependent on the waiting time for the next replenishment. Many researchers have modified inventory policies by considering the "time proportional partial backlogging rate".

Related Works
Zangwill (1966) developed a production multi period production scheduling model with backlogging. Inventory models with a mixture of backorders and lost sales were formulated by Montgomery et al. (1973). Economic production lot size model for deteriorating items with partial backordering was suggested by Wee (1993

Assumptions and Notations
In developing the mathematical model of the inventory system the following assumptions are being made:  A single item is considered over a prescribed period T units of time.  The demand rate of Auto industry D(t) at time t is deterministic and taken as a ramp type function of time i.e.
The replenishment rate is infinite and lead-time is zero.  When the demand for goods is more than the supply.  The variable rate of deterioration in both warehouse is taken as β(t) = βt. Where 0< β << 1 and only applied to on hand inventory.  No replacement or repair of deteriorated items is made during a given cycle.  The owned warehouse of Auto industry (OW) has a fixed capacity of W units; the rented warehouse of Auto industry (RW) has unlimited capacity.  The goods of OW are consumed only after consuming the goods kept in RW.
In addition, the following notations are used throughout this paper: I ow (t) The inventory of Auto industry level in OW at any time t. I rw (t) The inventory of Auto industry level in RW at any time t. C W The capacity of the own warehouse of Auto industry. The opportunity cost of Auto industry due to lost sales. A The replenishment cost of Auto industry per order.

Formulation and solution of the model
The inventory levels of Auto industry at OW of Auto industry are governed by the following differential equations: with the boundary conditions, I 0W (0) =C W and I(t 1 ) = 0 (4) The solutions of equations (1), (2) and (3) are given by respectively.
The inventory level of Auto industry at RW of Auto industry is governed by the following differential equations: With the boundary condition I rw (0)=0, the solution of the equation (8) is Due to continuity of I o (t) at point t = , it follows from equations (5) and (6) The total average cost consists of following elements: I.
Ordering cost of Auto industry per cycle = A (11) II.
Holding cost of Auto industry per cycle (H 1 ) in OW

III.
Holding cost of Auto industry per cycle (C HR ) in RW 2 0

IV.
Cost of deteriorated units of Auto industry per cycle (C D ) V. Shortage cost of Auto industry per cycle (C S )

VI.
Opportunity cost of Auto industry due to lost sales per cycle ( 0 C )

Genetic Algorithms
The basic concepts were developed by Holland, while the practicality of using the GA to solve complex problems was demonstrated. Genetic Algorithms (GAs) is a soft computing approach. GA is a maximization process. The problem it addresses usually has a very large search space with probable multiple local maxima inside it. The GA process has to ensure that it is not trapped at local maxima, so that at the end of the process it may find the global maxima even if the global maximum is not returned, we may expect a close approximation of it as the outcome of the GA process To achieve this, GA works on a set of solutions to the given problem instance, and evolves it through a number of generations. The evolution process stops when some predefined termination condition is satisfied. At each intermediate stage, the old generation is replaced by the new generation. The individuals of the population of a generation are processed with the help of a number of GA operators in such a way that the quality of the new generation, in general is improved in comparison with old generation. In this way we obtain better and better solution will be returned by the GA process.

Procedure Basic GA
Step 1 :-Initialize the population. Call this the current population.
Step 2 :-Repeat step 3 through step 5 till termination condition is satisfied.   Step 3 :-Apply selection operation on the current population to obtain the mating pool.
Step 4 :-Apply crossover and mutation operators n for the matting pool to generate the new Population.
Step 5 :-Replace the current population by the new population.
Step 6 :-Return the best solution of the current population. Therefore, numerical solution of these equations is obtained by using the software MATLAB 7.0.1.
Since the model is integer in nature, reaching an analytical solution (if any) to the problem is difficult (Gen and Cheng, 1997). So we need to use Meta heuristic algorithms. To solve the models under metaheuristic approach, four hybrid intelligent algorithms of harmony search (Taleizadeh et al., 2009), simulated annealing (Taleizadeh, 2008(Taleizadeh, , 2009 material product and occurrence of shortages in inventory are natural phenomenon in real situations. Inventory of Auto industry shortages are allowed in the model using Genetic Algorithms. In many cases customers are conditioned to a shipping delay, and may be willing to wait for a short time in order to get their first choice. Therefore, this concept is also taken in this model. From the numerical illustration of the model, it is observed that the period in which inventory holds of Auto industry increases with the increment in backlogging and ramp parameters while inventory period decreases with the increment in deterioration and inflation parameters of Auto industry using Genetic Algorithms. Initial inventory level of Auto industry decreases with the increment in deterioration, inflation and ramp parameters while inventory level increases of Auto industry with the increment in backlogging parameter using Genetic Algorithms. The total average cost of the system goes on increasing with the increment in the backlogging and deterioration parameters while it decreases with the increment in inflation and ramp parameters using Genetic Algorithms. The proposed model can be further extended in several ways. For example, we could extend this deterministic model in to stochastic model. Also, we could extend the model to incorporate some more realistic features, such as quantity discount or the unit purchase cost, the inventory holding cost and others can also taken fluctuating with time using Genetic Algorithms.