A multi-objective wholesaler–retailers inventory-distribution model with controllable lead-time based on probabilistic fuzzy set and triangular fuzzy number

Abstract

This paper develops a single wholesaler and multi retailers mixture inventory distribution model for a single item involving controllable lead-time with backorder and lost sales. The retailers purchase their items from the wholesaler in lots at some intervals throughout the year to meet the customers' demand. Not to loose the demands, the retailers offer a price discount to the customers on the stock-out items. Here, it is assumed that the lead-time demands of retailers are uncertain in both stochastic and fuzzy sense, i.e., these are simultaneously random and imprecise. To implement this behavior of the lead-time demands, at first, these demands are assumed to be random, say following a normal distribution. With these random demands, the expected total cost for each retailer is obtained. Now, the mean lead-time demands (which are crisp ones) of the retailers are fuzzified. This fuzzy nature of the lead-time demands implies that the annual average demands of the retailers must be fuzzy numbers, suppose these are triangular fuzzy numbers. Using signed distance technique for defuzzification, the estimate of total costs for each retailer is derived. Therefore, the problem is reduced to optimize the crisp annual costs of wholesaler and retailers separately. The multi-objective model is solved using Global Criteria method. Numerical illustrations have been made with the help of an example taking two retailers into consideration. Mathematical analyses have been made for global pareto-optimal solutions of the multi-objective optimization problem. Sensitivity analyses have been made on backorder ratio and pareto-optimal solutions for wholesaler and different retailers are compared graphically.

Evaluation and selection of emergency treatment technology based on dynamic fuzzy GRA method for chemical contingency spills

“…Since some of the evaluation criteria are subjective and qualitative in nature, it is difficult for decision makers to express the preferences using exact numerical values. In many practical cases, an overall triangular fuzzy numbers (TFNs) are used to effectively aggregate the assessment of the decision makers to each criterion [26][27][28][29]. Moreover, interval-valued triangular fuzzy numbers can provide a more accurate modeling to encounter with the problems and improve emergency response plans for chemical contingency spills [30,31].…”

Modeling and Optimization of the Multiobjective Stochastic Joint Replenishment and Delivery Problem under Supply Chain Environment

“…For the JRD policy, it is necessary to decrease the total cost while improving the service level. Although there are several papers that studied multiobjective inventory models (Roy and Maiti [ 16 ]; Rong et al [ 17 ]; Islam [ 18 ]; Wee et al [ 19 ]), no study on the multiobjective JRD can be found.…”

“…Many linear or nonlinear weighted methods (Rong et al [ 17 ]; Islam [ 18 ]; Wee et al [ 19 ]; Roy and Maiti [ 16 ]) were used to convert the multiobjective to a single one in the existing studies. These methods undoubtedly provide one easy way to deal with the multiobjective JRD model.…”

Variable demand model for periodically reviewing with allowing refunding parts of the orders

“…For example, Dey and Chakraborty [9] developed a fuzzy random periodic review model with variable lead-time. Rong et al [10] studied a multi-objective inventory model with controllable lead-time and triangular fuzzy numbers. Jauhari et al [11] introduced a fuzzy periodic review model involving stochastic demand.…”