Partha Guchhait

Two storage inventory model of a deteriorating item with variable demand under partial credit period

In this paper, a two-warehouse inventory model for deteriorating item with stock and selling price dependent demand has been developed. Above a certain (fixed) ordered label, supplier provides full permissible delay in payment per order to attract more customers. But an interest is charged by the supplier if payment is made after the said delay period. The supplier also offers a partial permissible delay in payment even if the order quantity is less than the fixed ordered label. For display of goods, retailer has one warehouse of finite capacity at the heart of the market place and another warehouse of infinite capacity (that means capacity of second warehouse is sufficiently large) situated outside the market but near to first warehouse. Units are continuously transferred from second warehouse to first and sold from first warehouse. Combining the features of Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) a hybrid heuristic (named Particle Swarm-Genetic Algorithm (PSGA)) is developed and used to find solution of the proposed model. To test the efficiency of the proposed algorithm, models are also solved using another two established heuristic techniques and results are compared with those obtained using proposed PSGA. Here order quantity, refilling point at first warehouse and mark-up of selling price of fresh units are decision variables. Models are formulated for both crisp and fuzzy inventory parameters and illustrated with numerical examples.

A production inventory model with fuzzy production and demand using fuzzy differential equation: An interval compared genetic algorithm approach

In this paper, a production inventory model, specially for a newly launched product, is developed incorporating fuzzy production rate in an imperfect production process. Produced defective units are repaired and are sold as fresh units. It is assumed that demand coefficients and lifetime of the product are also fuzzy in nature. To boost the demand, manufacturer offers a fixed price discount period at the beginning of each cycle. Demand also depends on unit selling price. As production rate and demand are fuzzy, the model is formulated using fuzzy differential equation and the corresponding inventory costs and components are calculated using fuzzy Riemann-integration. @a-cut of total profit from the planning horizon is obtained. A modified Genetic Algorithm (GA) with varying population size is used to optimize the profit function. Fuzzy preference ordering (FPO) on intervals is used to compare the intervals in determining fitness of a solution. This algorithm is named as Interval Compared Genetic Algorithm (ICGA). The present model is also solved using real coded GA (RCGA) and Multi-objective GA (MOGA). Another approach of interval comparison-order relations of intervals (ORI) for maximization problems is also used with all the above heuristics to solve the model and results are compared with those are obtained using FPO on intervals. Numerical examples are used to illustrate the model as well as to compare the efficiency of different approaches for solving the model.

Multi-item inventory model of breakable items with stock-dependent demand under stock and time dependent breakability rate

Items made of glass, ceramic, etc. are normally stored in stacks and get damaged during the storage due to the accumulated stress of heaped stock. These items are known as breakable items. Here a multi-item inventory model of breakable items is developed, where demands of the items are stock dependent, breakability rates increase linearly with stock and nonlinearly with time. Due to non-linearity and complexity of the problem, the model is solved numerically and final decisions are made using Genetic Algorithm (GA). In a particular case, model is solved analytically as well as numerically and results are compared. Models are developed with both crisp and uncertain inventory costs. For uncertain inventory costs both fuzzy and stochastic parameters are considered. A chance constrained approach is followed to deal with simultaneous presence of stochastic and fuzzy parameters. Different numerical examples are used to illustrate the problem for different cases.

Inventory policy of a deteriorating item with variable demand under trade credit period

Stock, price and credit linked demand of a deteriorating item.Effect of customers credit period on demand.Effects of inflation and time value of money when demand depends on selling price.Imprecise planning horizon. In this paper, an inventory model of a deteriorating item with stock and selling price dependent demand under two-level credit period has been developed. Here, the retailer enjoys a price discount if he pays normal purchase cost on or before the first level of credit period, or an interest is charged for the delay of payments. In return, retailer also offers a fixed credit period to his customers to boost the demand. In this regard, the authors develop an EOQ model incorporating the effect of inflation and time value of money over all the costs. Keeping the business of seasonal products in mind, it is assumed that planning horizon of business is random and follows a normal distribution with a known mean and standard deviation. The model is formulated as retailers profit maximization problem for both crisp and fuzzy inventory costs and solved using a modified Genetic Algorithm (MGA). This algorithm is developed following fuzzy age based selection process for crossover and gradually reducing mutation parameter. For different values of MGA parameters, optimum results are obtained. Numerical experiments are performed to illustrate the model.

Inventory Policy with Stock, Price and Credit-Linked Demand: A Fuzzy Genetic Algorithm Approach

An inventory control problem under two-level trade-credit policy with fuzzy inventory costs is proposed where supplier provides not only a credit-period for settling account but also a cash discount to the retailers. Due to this advantage the retailer also offers a fixed credit period to all its customers to boost the demand. Demand also depends on stock and selling price. A Genetic Algorithm (GA) with chromosome’s life-time dependent varying population size is used to solve the model where, at the time of generation of initial population, diversity in the population is maintained using information entropy theory. In the algorithm crossover probability of a pair of parents is a function of their age-type (young, middle-aged, old, etc.) and is obtained using a fuzzy rule base and possibility theory. A fuzzy possibility/necessity based evolution process is proposed to deal with fuzzy objective function.

A production inventory model with price discounted fuzzy demand using an interval compared hybrid algorithm

Abstract An economic production quantity (EPQ) model in an imprecise environment is proposed, where the production rate, planning horizon and demand coefficients are fuzzy in nature. At the beginning of each cycle a price discount is offered for a period to boost the demand. During this period, demand increases with time depending upon the amount of discount. Here, demand also depends on the unit selling price. After withdrawal of the price discount, demand depends only on the unit selling price. The governing differential equation for the model is obviously fuzzy in nature as the production rate and demand are fuzzy. For this reason, the model is formulated using a fuzzy differential equation and the α -cut of the total profit from the planning horizon is obtained using fuzzy Riemann integration. To optimize the interval objective function, using a fuzzy preference relation on intervals and a fuzzy possibility/necessity measure, a hybrid algorithm with varying population size is developed by combining the features of particle swarm optimization (PSO) and a genetic algorithm (GA). This algorithm is named Interval Compared Hybrid Particle Swarm-Genetic Algorithm (ICHPSGA) and is used to find an optimal decision for the decision maker (DM) in different cases of the model. To test the efficiency of the algorithm, it is compared with two other established algorithms namely PSO and PSGA. Numerical experiments are performed to illustrate the model and some interesting observations are made.

A fuzzy lifetime-based particle swarm optimisation with varying swarm size to solve a production inventory model

Here, a modified particle swarm optimisation (MPSO) algorithm with varying swarm size for constrained optimisation problem is proposed. In this MPSO, a life time is assigned to each particle at the time of generation depending on its fitness. After completion of a generation, if no movement is made by the particle, its age is increased by unity. When age of a particle exceeds the lifetime, it is discarded from the swarm. Diversity in the swarm is maintained using information entropy theory. A fuzzy possibility/necessity-based fitness evolution is proposed to deal with fuzzy optimisation problems using this MPSO. Efficiency of the algorithm is tested against a list of crisp valued standard benchmark nonlinear test functions. This algorithm is used to solve a production inventory model with fuzzy costs, where lifetime of the product is random in nature. At the beginning of planning horizon price discount is offered to the customers for few cycles to boost the demand. Demand also depends on stock and selling price. The model is illustrated with numerical examples and some sensitivity analyses have been made.

An imperfect EPQ model for deteriorating items with promotional effort dependent demand

In this paper, a multi-item economic production quantity (EPQ) model for deteriorating seasonal products is developed with stock and promotional effort dependent demand in an imperfect production process. The promotional efforts are advertising, delivery facilities, better services, etc. The production process produces some imperfect quality units which are instantly reworked at a cost to bring back its quality to the perfect ones. Here, the rate of production is time dependent. Unit production cost is a function of production rate including the defective ones and the deterioration rate is considered as constant. The model is formulated as a profit maximization problem with space and budget constraints in the form of an optimal control problem. The total profit function with the effect of inflation and time-value of money is expressed as a finite integral over a finite planning horizon. The problem is solved using variational calculus to determine the minimum defective rates of the production process for which the total profit is maximum. Another three models are developed considering the constraints as uncertain (fuzzy, random and rough) in nature. For fuzzy model, three types of fuzzy numbers are considered. To deal with the fuzzy constraints, fuzzy possibility measure is used. Also, stochastic and rough constraints are reduced to the approximate crisp ones following chance constrained approach and rough expectation respectively. Numerical experiments are performed to illustrate the models. Also, some sensitivity analyses are performed and presented.

A Multi-item EPQ Model with Variable Demand in an Imperfect Production Process

Nowadays, to survive and promote the market competition, multi-item business strategy is more effective for any production/manufacturing sector. Here, an attempt has been made to develop a multi-item production inventory model, especially for seasonal products for a finite time horizon. Production process is not 100% reliable, and for this reason some imperfect quality items are produced, and these items are immediately remanufactured incurring some cost to get back its originality. Both demands and production rate of the items are time-sensitive. It is also assumed that production cost per unit varies with defective rates as well as production rates. Imposing two constraints, space and investment, profit maximization model is formulated. Incorporating inflation and time value of money into the model, total profit is represented as a definite integral with time period as its upper limit. The profit function becomes an optimal control problem. The defective rates of the production process are determined by variational calculus. As today’s competitive business transaction is full of uncertainty, another model is considered with fuzzy constraints and solved. Fuzzy constraints are converted to the crisp one, following fuzzy possibility. For the illustration of the developed models, numerical experiments and also some sensitivity analyses are performed and presented.

Imperfect production policy of a breakable item with variable breakability and demand in random planning horizon

Production inventory model of a breakable item, i.e., item made of glass, ceramic, etc. is developed where, above a certain stock level breakability increases with time and stock level. Demand of the item is stock and selling price dependent. Production process is not 100% perfect, i.e., not all produced units are of perfect quality. Defective units are sold at a reduced price. Duration of demand of the item in the market is assumed as stochastic in nature and follows normal distribution with a known mean and standard deviation. Here, the unit production cost depends on production rate and is derived from the particular production function under which it is being produced. Here selling price, reliability of production process, set-up cost and duration of each cycle are decision variables. Model is illustrated with numerical data and some sensitivity analyses have been presented.