Manoranjan Maiti

A fuzzy EOQ model with demand-dependent unit cost under limited storage capacity

Abstract A fuzzy EOQ model is developed with limited storage capacity where demand is related to the unit price and the setup cost varies with the quantity produced/purchased. Here fuzziness is introduced in both objective function and storage area. It is solved by both fuzzy nonlinear and geometric programming techniques for linear membership functions. The model is illustrated with a numerical example and a sensivity analysis is made. Generalisation to a multi-item problem is also presented and its numerical results are compared with those of the crisp model.

Multi-item fuzzy EOQ models using genetic algorithm

A soft computing approach is proposed to solve non-linear programming problems under fuzzy objective goal and resources with/without fuzzy parameters in the objective function. It uses genetic algorithms (GAs) with mutation and whole arithmetic crossover. Here, mutation is carried out along the weighted gradient direction using the random step lengths based on Erlang and Chi-square distributions. These methodologies have been applied to solve multi-item fuzzy EOQ models under fuzzy objective goal of cost minimization and imprecise constraints on warehouse space and number of production runs with crisp/imprecise inventory costs. The fuzzy inventory models have been formulated as fuzzy non-linear decision making problems and solved by both GAs and fuzzy non-linear programming (FNLP) method based on Zimmermanns approach. The models are illustrated numerically and the results from different methods are compared.

Two storage inventory problem with dynamic demand and interval valued lead-time over finite time horizon under inflation and time-value of money

A finite time horizon inventory problem for a deteriorating item having two separate warehouses, one is a own warehouse (OW) of finite dimension and other a rented warehouse (RW), is developed with interval-valued lead-time under inflation and time value of money. Due to different preserving facilities and storage environment, inventory holding cost is considered to be different in different warehouses. The demand rate of item is increasing with time at a decreasing rate. Shortages are allowed in each cycle and backlogged them partially. Shortages may or may not be allowed in the last cycle and under this circumstance, there may be three different types of model. Here it is assumed that the replenishment cycle lengths are of equal length and the stocks of RW are transported to OW in continuous release pattern. For each model, different scenarios are depicted depending upon the re-order point for the next lot. Representing the lead-time by an interval number and using the interval arithmetic, the single objective function for profit is changed to corresponding multi-objective functions. These functions are maximized and solved by Fast and Elitist Multi-objective Genetic Algorithm (FEMGA). The models are illustrated numerically and the results are presented in tabular form.

Fuzzy inventory model with two warehouses under possibility constraints

A multi-item inventory model with two-storage facilities is developed with advertisement, price and displayed inventory level-dependent demand in a fuzzy environment (purchase cost, investment amount and storehouse capacity are imprecise). The model is formulated as a single/multi-objective programming problem under fuzzy constraint. Constraints are satisfied with some pre-defined necessity and the problem is solved via the Goal Programming Method (GPM) when crisp equivalents of the constraints are available and by a fuzzy simulation-based single/multi-objective genetic algorithm (FSGA/FSMOGA) otherwise. The model is illustrated with some numerical examples and results from different methods are compared in some particular cases.

Deterministic inventory model with two levels of storage, a linear trend in demand and a fixed time horizon

Abstract A deterministic inventory model is developed for a single item having two separate storage facilities (owned and rented warehouses) due to limited capacity of the existing storage (owned warehouse) with linearly time-dependent demand (increasing) over a fixed finite time horizon. The model is formulated by assuming that the rate of replenishment is infinite and the successive replenishment cycle lengths are in arithmetic progression. Shortages are allowed and fully backlogged. As a particular case, the results for the model without shortages are derived. Results are illustrated with two numerical examples. Scope and purpose Throughout the world, the production of food grains is periodical. Normally, in countries where state control is less, the demand of essential food grains is lowest at the time of harvest and goes up to the highest level just before the next harvest. This phenomenon is very common in developing third world countries where most of the people are landless or marginal farmers. At the time of harvest, they share some grain/product with landowners and as soon as the small inventory is exhausted, they are forced to buy food grains from the open market. As a result, demand for food grains increases with time in a period along with the number of the people whose initial stock of food grains gets exhausted. In this paper, a two-storage inventory model with time-dependent demand and fixed time horizon is developed and solved by a mathematical programme based on gradient method. This methodology of model development and its solution are quite general and it can be applied to inventory models of any product whose production is periodical and demand increases linearly with time.

Multi-objective inventory models of deteriorating items with some constraints in a fuzzy environment

Abstract In multi-item, multi-objective constrained inventory models, objective goals, resource constraints, inventory costs and prices are assumed to be crisp and defined with certainty. In real-life, however, this is seldom the case. Due to the specific requirements and local conditions, the above goals and parameters are normally vague and imprecise, i.e. fuzzy in nature. Until now, a few people have attempted to solve such inventory problems. Impreciseness of objective goals and resource constraints have been expressed here by fuzzy membership functions and vagueness in inventory costs and prices by fuzzy numbers. Thus, the multi-item multi-objective constrained inventory problems reduce to fuzzy decision making problems which are solved by fuzzy non-linear programming (FNLP) and fuzzy additive goal programming (FAGP) methods. The exact fuzzy membership functions for goals and fuzzy number representations for inventory parameters can be obtained through past observations. Once these actual representations are available, the real-life inventory problems can be solved realistically which will be of much use for the management. In this paper, multi-item inventory models of deteriorating items with stock-dependent demand are developed in a fuzzy environment. Here, the objectives of maximizing the profit and minimizing the wastage cost are fuzzy in nature. Total average cost, warehouse space, inventory costs, purchasing and selling prices are also assumed to be vague and imprecise. The impreciseness in the above objective and constraint goals have been expressed by fuzzy linear membership functions and that in inventory costs and prices by triangular fuzzy numbers (TFN). Models have been solved by the fuzzy non-linear programming (FNLP) method based on Zimmermann [Zimmermann, H.-J., Fuzzy linear programming with several objective functions. Fuzzy Sets and Systems , 1978, 1 , 46–55] and Lee and Li [Lee, E. S. and Li, R. J., Fuzzy multiple objective programming and compromise programming with Pareto optima. Fuzzy Sets and Systems , 1993, 53 , 275–288]. These are illustrated with numerical examples and results of one model are compared with those obtained by the fuzzy additive goal programming (FAGP) [Tiwari, R. N., Dharmar, S. and Rao, J. R., Fuzzy goal programming: an additive model. Fuzzy Sets and Systems , 1987, 24 , 27–34] method.

A two warehouse inventory model for a deteriorating item with partially/fully backlogged shortage and fuzzy lead time

An optimization inventory policy for a deteriorating item with imprecise lead-time, partially/fully backlogged shortages and price dependent demand is developed under two-warehouse system. For display and storage, the retailer hires one warehouse of finite capacity at market place, treated as own warehouse (OW) and another warehouse of large capacity as it may be required at a distance place from the market, treated as rented warehouse (RW). Holding cost at RW decreases with the increase of distance from the market place. Units are transferred from RW to OW in bulk release pattern and sold from OW. Using the nearest interval approximation method the estimated fuzzy average profit function is defuzzified and transformed to multiple crisp objective functions which are solved by Global Criteria Method. The models are illustrated numerically. Sensitivity of the inventory costs on the location of RW has been depicted graphically. Also loss in profit due to deteriorations for both models have been presented.

Multi-objective fuzzy inventory model with three constraints: a geometric programming approach

A multi-item multi-objective inventory model with shortages and demand dependent unit cost has been formulated along with storage space, number of orders and production cost restrictions. In most of the real world situations, the cost parameters, the objective functions and constraints of the decision makers are imprecise in nature. Hence the cost parameters, the objective functions and constraints are imposed here in fuzzy environment. This model has been solved by geometric programming method. The results for the model without shortages are obtained as a particular case. The sensitivity analysis has been discussed for the change of the cost parameters. The models are illustrated with numerical examples.

Deterministic inventory model for deteriorating items with finite rate of replenishment dependent on inventory level

Abstract In industry, the replenishment rate of a manufacturing system at any time is constant and determined depending upon the demand of the production in the market and the present stock of the produced in the godown at that time. Until now, this practical aspect has not received the attraction of inventory researchers. In this paper, two models have been formulated and solved for deteriorating items with linearly time dependent demand where replenishment rate changes linearly with the change in demand and on-hand inventory amount. Different types of functional relations of replenishment rate with demand and on-hand inventory can be incorporated in this analysis. Using the present methodology, the above practical aspect can be incorporated to the problems of multi-item, multi-storage, discount, etc. In this paper, two deterministic inventory models for deteriorating items have been developed and solved with and without allowing shortages which are backlogged. The replenishment rate is finite and dependent on the instantaneous inventory level and also on demand. In both models, the rate of demand and deterioration at any moment of time is a linear increasing function of time. Two numerical examples are given to illustrate the results and the significant features of the results are discussed. Also, the effects of changes in the demand and deterioration parameters on the optimal average cost are graphically presented.

A two warehouse supply-chain model under possibility/ necessity/credibility measures

This article presents a joint performance of a supply chain (SC) with two warehouse facilities. A realistic two warehouse multi-collection-production-inventory model with constant/stock dependent demand, defective production system and fuzzy budget constraint has been formulated in an SC context. Here rate of defectiveness follows a probability distribution and fuzzy constraint is imposed in a necessity/possibility/credibility sense and defuzzied following fuzzy relations. The rates of collection for each raw material and production are assumed as control variables. Also the defective units are screened along with production and reworked with a known rework rate. The model is reduced to the equivalent deterministic one of minimizing the expected total cost with imprecise demand through possibility, necessity or convex combination of them (credibility). All these formulations are solved using Generalized Reduced Gradient (GRG) technique. The model is illustrated numerically for different types of demands and with respect to those demands, the optimal production and stock level are presented in both tabular and graphical forms. The sensitivity of the total operating cost on the variation of capacity of the market warehouse is also presented to illustrate the advantage of a two warehouse system in SC.