Tapan Kumar Roy

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.

Fuzzy goal programming approach to multilevel programming problems

The purpose of this paper is to propose a procedure for solving multilevel programming problems in a large hierarchical decentralized organization through linear fuzzy goal programming approach. Here, the tolerance membership functions for the fuzzily described objectives of all levels as well as the control vectors of the higher level decision makers are defined by determining individual optimal solution of each of the level decision makers. Since the objectives are potentially conflicting in nature, a possible relaxation of the higher level decision is considered for avoiding decision deadlock. Then fuzzy goal programming approach is used for achieving highest degree of each of the membership goals by minimizing negative deviational variables. Sensitivity analysis with variation of tolerance values on decision vectors is performed to present how the solution is sensitive to the change of tolerance values. The efficiency of our concept is ascertained by comparing results with other fuzzy programming approaches.

Fuzzy multi-objective mathematical programming on reliability optimization model

Abstract This paper considers multi-objective reliability optimization problem for system reliability where reliability enhancement is involved with several mutually conflicting objectives. It is conflicting to reduce the cost of the system and improve the reliability of the same system simultaneously. A new fuzzy multi-objective optimization method is introduced and it is used for the optimization decision making of the series and complex system reliability with two objectives. In general, cost of the components of the system’s objective goals has not been stated clearly. This imprecise reliability optimization model is solved through the fuzzy multi-objective optimization method. Examples are shown to illustrate the method.

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.

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.

A new fuzzy multi-objective programming: Entropy based geometric programming and its application of transportation problems

In this paper, we introduce a fuzzy mathematical programming with generalized fuzzy number as objective coefficients. We also examine a transportation problem with additional restriction. There is an additional entropy objective function in the transportation problem besides transportation cost objective function. Using new fuzzy mathematical programming, this multi-objective entropy transportation problem with generalized trapezoidal fuzzy number costs has been reduced to a primal geometric programming problem. Pareto optimal solution of the transportation model is found. Numerical examples have been provided to illustrate the problem.

Multiobjective Transportation Model with Fuzzy Parameters: Priority based Fuzzy Goal Programming Approach

Abstract This paper presents a priority based fuzzy goal programming approach for solving a multiobjective transportation problem with fuzzy coefficients. In the model formulation of the problem, first the membership functions for the fuzzy goals are defined. Subsequently, the membership functions are transformed into membership goals, by assigning the highest degree (unity) of a membership function as the aspiration level and introducing deviational variables to each of them. In the solution process, negative deviational variables are minimized to obtain the most satisficing solution. Sensitivity analysis of the solution, with a change in priorities of the fuzzy goals is performed. Next the Euclidean distance function is used to identify the appropriate priority structure of the goals, thereby obtaining the most satisficing decision for the decision-making unit, by minimizing their regrets of achieving the ideal point dependent decision in the decision-making context. A numerical example is solved to demonstrate the potential use of the proposed approach.

Multi-item stochastic and fuzzy-stochastic inventory models under two restrictions

Multi-item stochastic and fuzzy-stochastic inventory models are formulated under total budgetary and space constraints. Here, the inventory costs are directly proportional to the respective quantities, unit purchase/production cost is inversely related to the demand and replenishment/production rate is assumed to vary directly with demand. Shortages are allowed but fully backlogged. Here, for both models, demand and budgetary resource are assumed to be random. In fuzzy-stochastic model, in addition to the above assumptions, available storage space and total expenditure are imprecise in nature. Impreciseness in the parameters have been expressed with the help of linear membership functions. Assuming random variables to be independent and to follow normal distributions, the models have been formulated as stochastic and fuzzy-stochastic non-linear programming problems. The stochastic problem is first reduced to the equivalent single objective or multiple objectives problems following chance-constraint method. The problem with single objective is solved by a gradient-based technique whereas fuzzy technique is applied to the multi-objective one. In the same way, the fuzzy-stochastic programming problem is first reduced to a corresponding equivalent fuzzy non-linear programming problem and then it is solved by fuzzy non-linear programming (FNLP) following Zimmermann technique. The models are illustrated numerically and the results of different models are compared.

Inventory model of deteriorated items with a constraint : A geometric programming approach

An inventory model for deteriorating items is built-up with limited storage space. Here, demand rate for the items is finite, items deteriorate at constant rates and are replenished instantaneously. Following EOQ model, the problem is formulated with and without truncation on the deterioration term and ultimately is converted to the minimization of a signomial expression with a posynomial constraint. It is solved by modified geometric programming (MGP) method and non-linear programming (NLP) method. The problem is supported by numerical examples. The results from two versions of the model (with and without truncation) and two methods (i.e. MGP and NLP) are compared.

Multi-item integrated supply chain model for deteriorating items with stock dependent demand under fuzzy random and bifuzzy environments

Multi-item integrated supply chain models with deterioration has been developed.Supply chain models with budget and space constraints has been formulated.The objective and constraints under imprecise environments has been introduced.An example, with some sensitivity analysis has been provided to validate the model.Chance constraint techniques and CMGA have been used to solve the models. In this paper, we have investigated multi-item integrated production-inventory models of supplier and retailer with a constant rate of deterioration under stock dependent demand. Here we have considered suppliers production cost as nonlinear function depending on production rate, retailers procurement cost exponentially depends on the credit period and suppliers transportation cost as a non-linear function of the amount of quantity purchased by the retailer. The models are optimized to get the value of the credit periods and total time of the supply chain cycle under the space and budget constraints. The models are also formulated under fuzzy random and bifuzzy environments. The ordering cost, procurement cost, selling price of retailers and holding costs, production cost, transportation cost, setup cost of the suppliers and the total storage area and budget are taken in imprecise environments. To show the validity of the proposed models, few sensitivity analyses are also presented under the different rate of deterioration. The models are also discussed in non deteriorating items as a special case of the deteriorating items. The deterministic optimization models are formulated for minimizing the entire monetary value of the supply chain and solved using genetic algorithm (GA). A case study has been performed to illustrate those models numerically.