Barun Das

A solid transportation problem for an item with fixed charge, vechicle cost and price discounted varying charge using genetic algorithm

In this paper, discount in transportation cost on the basis of transportated amount is extended to a solid transportation problem. In a transportation model, the available discount is normally offered on items/criteria, etc., in the form AUD (all unit discount) or IQD (incremental quantity discount) or combination of these two. Here transportation model is considered with fixed charges and vechicle costs where AUD, IQD or combination of AUD and IQD on the price depending upon the amount is offered and varies on the choice of origin, destination and conveyance. To solve the problem, genetic algorithm (GA) based on Roulette wheel selection, arithmetic crossover and uniform mutation has been suitably developed and applied. To illustrate the models, numerical examples have been presented. Here, different types of constraints are introduced and the corresponding results are obtained. To have better customer service, the entropy function is considered and it is displayed by a numerical example. To exhibit the efficiency of GA, another method-weighted average method for multi-objective is presented, executed on a multi-objective problem and the results of these two methods are compared.

Integrated supply chain model for a deteriorating item with procurement cost dependent credit period

This study develops an integrated production inventory model of supplier and retailer where a delay in payment is offered by supplier to retailer for a constant deteriorating item. Here shortages are not allowed. In this model retailers procurement cost linearly depends on the credit period and suppliers process cost also is a linear function of the amount of quantity purchased by retailer. In this model, the objective is to decide the position of the credit period and number of replenishment of retailer in finite time horizon in such a way that the integrated system gets the optimum cost. The model is explained with the help of numerical examples and the sensitivity analysis of some parameters are also carried out.

Multi-item partial backlogging inventory models over random planninghorizon in random fuzzy environment

Abstract In this paper, some multi-item inventory models for deteriorating items are developed in a random planning horizon under inflation and time value money with space and budget constraints. The proposed models allow stock dependent consumption rate and partially backlogged shortages. Here the time horizon is a random variable with exponential distribution. The inventory parameters other than planning horizon are deterministic in one model and in the other, the deterioration and net value of the money are fuzzy, available budget and space are fuzzy and random fuzzy respectively. Fuzzy and random fuzzy constraints have been defuzzified using possibility and possibility–probability chance constraint techniques. The fuzzy objective function also has been defuzzified using possibility chance constraint against a goal. Both deterministic optimization problems are formulated for maximization of profit and solved using genetic algorithm (GA) and fuzzy simulation based genetic algorithm (FAGA). The models are illustrated with some numerical data. Results for different achievement levels are obtained and sensitivity analysis on expected profit function is also presented. Scope and purpose The traditional inventory model considers the ideal case in which depletion of inventory is caused by a constant demand rate. However for more sale, inventory should be maintained at a higher level. Of course, this would result in higher holding or procurement cost, etc. Also, in many real situations, during a shortage period, the longer the waiting time is, the smaller the backlogging rate would be. For instance, for fashionable commodities and high-tech products with short product life cycle, the willingness for a customer to wait for backlogging diminishes with the length of the waiting time. Most of the classical inventory models did not take into account the effects of inflation and time value of money. But at present, the economic situation of most of the countries has been much deteriorated due to large scale inflation and consequent sharp decline in the purchasing power of money. So, it has not been possible to ignore the effects of inflation and time value of money any further. The purpose of this article is to maximize the expected profit of two inventory control systems in the random planning horizon.

An integrated production inventory model under interactive fuzzy credit period for deteriorating item with several markets

Graphical abstractDisplay Omitted HighlightsIt is an integrated production inventory model of a manufacturer and multiple markets which have different selling seasons.Raw-materials supplier offers a credit period to the manufacturer which may be crisp and fuzzy.The fuzziness of the credit period is analyzed in two different ways such as - triangular and trapezoidal fuzzy numbers.Raw-materials for production and finished item are both considered as deteriorating and nondeteriorating cases.Manufacturer allows a conditional part-payment to the markets and his selling price is assumed as an increasing function of production run time. The presence of multiple markets create profitable opportunities to the supply chain system. In this regard, this paper consists of the joint relationship between a manufacturer and multiple markets in which manufacturer offers part-payment to the markets due to their collection of finished products during the production run time. Here it is also considered that manufacturer is facilitated with credit period by raw material supplier where credit period has been presented as an interactive fuzzy fashion. In this paper, two types of deterioration have been assumed such as one for finished products and the other for raw materials. A solution algorithm is presented to get fuzzy optimal profit for the proposed integrated production inventory system optimizing production run time. A numerical example is used to illustrate the proposed model. Finally, sensitivity analysis has been carried out with respect to the major parameters to demonstrate the feasibility of the proposed model.

A fuzzy simulation via contractive mapping genetic algorithm approach to an imprecise production inventory model under volume flexibility

In this paper, an economic production quantity model for multi-item with storage space and budget constraints in a volume flexible manufacturing system is developed. Here it is assumed that the demand rate is constant up to a certain level of stock and after that it depends on stock itself. The unit production cost is taken to be a function of the finite production rate involving labour cost and wear and tear expenditure. Here, the inventory costs, selling price, storage space and available budget are defined imprecisely. Using necessary measure theory, the imprecise problem is reduced to deterministic problem. Here, necessity measure approach has been used for triangle fuzzy number and parabolic fuzzy number. Finally the crisp nonlinear optimization problem is solved by Fuzzy simulation, Contractive Mapping Genetic Algorithm and Generalized Reduced Gradient technique. The model is illustrated numerically and the results are compared.

An application of bi-level newsboy problem in two substitutable items under capital cost

A single period newsboy type inventory problem for two substitutable deteriorating items is studied with a resource constraint. In this system, a wholesaler and several retailers with their probabilistic demands and random lot sizes are considered. To make it more realistic, it is assumed that the wholesaler sells a certain portion of the stock instantaneously to the retailers as per their initial demands and later on, sells the rest on push-sale basis if a surplus exists but gets the sale proceed for this amount at the end of the time period for which an interest is charged and paid by the wholesaler. Following newsboy system, retailers also sell a portion of their stocks instantaneously and the rest amount, if it remains, also is instantaneously sold towards the end of the time period T. The deterioration of the left-over amount after the first sale of the retailers is considered for the time period T. As the items are substitutable, several cases of their mutual uses are considered as subsystem cases. Assuming that wholesaler and retailers under a single management, an integrated model is formulated to maximize a single objective profit function with the space constraint. It is solved through Genetic Algorithm. Taking wholesaler and retailers to be from different management houses, a multi-objective model has been developed, the expected profit functions are maximized and compromise optimum solutions are obtained via Multi-objective Genetic Algorithm based on the dominant criteria for each member of the system. Finally, numerical results have been presented for illustration.

A multi-item transportation problem with fuzzy tolerance

Abstract This paper presents the recently introduced modified subgradient method for optimization and its effectiveness in a fuzzy transportation model. Here a multi-item balanced transportation problem (MIBTP) is formulated where unit transportation costs are imprecise. Also available spaces and budgets at destinations are limited but imprecise. The objective is to find a shipment schedule for the items that minimizes the total cost subjected to imprecise warehouse and budget constraints at destinations. The proposed model is reduced to a multi-objective optimization problem using tolerances, then to a crisp single-objective one using fuzzy non-linear programming (FNLP) technique and Zimmermanns method. The above fuzzy MIBTP is also reduced to another form of deterministic one using modified sub-gradient method (MSM). These two crisp optimization problems are solved by Genetic Algorithm (GA). As an extension, fuzzy multi-item balanced solid transportation problems (STPs) with and without restrictions on some routes and items are formulated and reduced to deterministic ones following FNLP and Zimmermanns methods. These models are also solved by GA. Models are illustrated numerically, optimum results of fuzzy MIBTP from two deductions are compared. Results are also presented for different GA parameters.

A Partial Backlogging Inventory Model for Deteriorating Item under Fuzzy Inflation and Discounting over Random Planning Horizon: A Fuzzy Genetic Algorithm Approach

An inventory model for deteriorating item is considered in a random planning horizon under inflation and time value money. The model is described in two different environments: random and fuzzy random. The proposed model allows stock-dependent consumption rate and shortages with partial backlogging. In the fuzzy stochastic model, possibility chance constraints are used for defuzzification of imprecise expected total profit. Finally, genetic algorithm (GA) and fuzzy simulation-based genetic algorithm (FSGA) are used to make decisions for the above inventory models. The models are illustrated with some numerical data. Sensitivity analysis on expected profit function is also presented. Scope and Purpose. The traditional inventory model considers the ideal case in which depletion of inventory is caused by a constant demand rate. However, to keep sales higher, the inventory level would need to remain high. Of course, this would also result in higher holding or procurement cost. Also, in many real situations, during a longer-shortage period some of the customers may refuse the management. For instance, for fashionable commodities and high-tech products with short product life cycle, the willingness for a customer to wait for backlogging is diminishing with the length of the waiting time. Most of the classical inventory models did not take into account the effects of inflation and time value of money. But in the past, the economic situation of most of the countries has changed to such an extent due to large-scale inflation and consequent sharp decline in the purchasing power of money. So, it has not been possible to ignore the effects of inflation and time value of money any more. The purpose of this paper is to maximize the expected profit in the random planning horizon.

An EPQ model with promotional demand in random planning horizon: population varying genetic algorithm approach

One of the economic production quantity problems that have been of interest to researchers is the production with reworking of the imperfect items including waste most disposal form and vending the units. The available models in the literature assumed that the decay rate of the items is satisfied from three different points of view: (i) minimum demands of the customer’s requirement, (ii) demands to be enhanced for lower selling price and (iii) demands of the customers who are motivated by the advertisement. The model is developed over a finite random planning horizon, which is assumed to follow the exponential distribution with known parameters. The model has been illustrated with a numerical example, whose parametric inputs are estimated from market survey. Here the model is optimized by using a population varying genetic algorithm.

Fuzzy stochastic inequality and equality possibility constraints and their application in a production-inventory model via optimal control method

Abstract This paper deals with one equality constraint in fuzzy environment and other inequality constraint with both fuzzy and random parameter together. The purpose of this paper is to demonstrate the application of these type of constraints in a production inventory model solved as a Bang–Bang control problem in a finite time horizon. Finally numerical experiments have been performed for illustration.