مدل موجودی کمیت نظم فازی با مقدار کمبود فازی و شاخص تبلیغاتی فازی

The article deals with a backorder EOQ (Economic Order Quantity) model with promotional index for fuzzy decision variables. Here, a profit function is developed where the function itself is the function of m-th power of promotional index (PI) and the order quantity, shortage quantity and the PI are the decision variables. The demand rate is operationally related to PI variables and the model has been split into two types for the multiplication and addition operation. First the crisp profit function is optimized, letting it free from fuzzy decision variable. Yager (1981) ranking index method is utilized here to have a best inventory policy for the fuzzy model. Finally, a graphical presentation of numerical illustrations and sensitivity analysis are done to justify the general model.

The EOQ model is an essential methodology to overcome some bottlenecks of the supply chain (Cárdenas-Barrón, 2007, Cárdenas-Barrón et al., 2011, Cárdenas-Barrón et al., 2012a, Cárdenas-Barrón et al., 2012b and Cárdenas-Barrón et al., 2012c). Among many factors, promotional strategy is a more applicable issue in today's business strategy. The promotional effort is an important management strategy to introduce a new product to the customers when it is launched in the market. The promotional efforts are free gift, discount offer, delivery facilities, better services and advertising, etc. Nowadays these strategies are applied for all types of commodities, not only for new products. Goyal and Gunasekaran (1995) developed a production-inventory model while demand of the end customers is influenced by advertising. Krishnan et al. (2004) found out optimal promotional strategy to maximize the profit function. Szmerekovsky and Zhang (2009) developed two-layer supply chain model for retail price and advertising sensitive demand. Xie and Wei (2009) and Xie and Neyret (2009) investigated two-layer supply chain model to obtain optimal cooperative advertising strategies and equilibrium pricing of the chain. Recently, the works (Sana, 2010, Sana, 2011a, Sana, 2011b, Sana, 2012 and Sana and Chaudhuri, 2008) in this line are worth mentioning. The classical backorder inventory model lacks a new kind of variable named promotional Index (PI).This index usually enhance the demand of the customers of any kind of commodities. However, the demand rate is a function of the PI variable. Since the order quantity and shortage quantity are functionally related to demand rate. Hence, they are functionally related to the PI variable alone. Generally speaking, in the competitive world, no variable is fixed and hence they are flexible in nature. Several research papers have been published in fuzzy environment. Vojosevic et al. (1996) fuzzified the order cost into trapezoidal fuzzy number in the backorder model. Using this propositions other authors We and Yao (2003) studied a fuzzy inventory with backorder for fuzzy order quantity and fuzzy shortage quantity. With the help of fuzzy extension principle, an economic order quantity in fuzzy sense for inventory without backorder model has been developed by Lee and Yao (1999). Yao et al. (2000) analyzed a fuzzy model without backorder for fuzzy order quantity and fuzzy demand quantity. A lot-size reorder point inventory model with fuzzy demands has been developed by Kao and Hsu (2002) considering the α-cut of the fuzzy numbers and they have used ranking index to solve the model. De and Goswami (2001) developed an EPQ model for decaying items considering fuzzy deterioration and constant demand rate. Author like De et al. (2003) developed an EPQ model for fuzzy demand rate and fuzzy deterioration rate using the α-cut of the membership function of the fuzzy parameters. De et al. (2008) studied an economic ordering policy of deteriorated items with shortage and fuzzy cost coefficients for vendor and buyer. Recently Kumar et al. (2012) developed a fuzzy model with ramp type demand rate and partial backlogging. In this paper, we have used PI as the fuzzy variable. However, in practice, any decision maker may face lead time delay or uneven traffic situations for which an uncertain flexible shortage quantity occurs during backorder period that affects the total order quantity as a whole. Therefore instead of crisp rather we assume all the decision variables as fuzzy variables. As the best of our knowledge, no research papers were published along this direction. First we have optimized the profit function under crisp environment then we have constructed an optimized function in terms of PI variable in two types of operation namely, ‘×’ and ‘+’ naming Model-I and Model-II respectively. Here we have seen that the crisp optimal solution and the solution for optimized function of PI variable are same. The models have been solved for four different positions of the PI variables with respect to the crisp PI optimum. Using the lower and upper bounds of the α-cuts of the membership functions for the fuzzy variables, we have constructed ranking index for each of the variables and finally got a numerical result with some programming language. A sensitivity analysis and graphical illustrations have been done to justify the model. The rest of the paper is organized as follows: the notations and assumptions are given in 2 and 3 formulates the model, numerical examples are provided in Section 4, sensitivity analysis has been done in 5 and 6 concludes the achievements of our proposed model.

In this paper we have studied the nature of the average profit function for the classical backorder model under promotional effort where all the decision variables are considered as fuzzy variables. The total average profit is affected by the m -th power of the PI variable alone. Moreover, according to the nature of the demand rate, we have split the model into two types, and then we have solved the models in crisp environment. To construct a fuzzy problem we have used the optimized form of the crisp optimal solution in terms of PI variable. As the profit function is convex so we have selected the upper bound of the PI variable for finite m . Constructing the fuzzy membership functions for the different positions of the PI variable with respect to the PI optimal for crisp sense we have solved the model in four different cases for the model with demand rate in multiplication operator. The similar way may be adopted to solve the Model-II also. In the fuzzy optimal Table 3, Table 4, Table 5 and Table 6, we see that the average profit is decreased in all the cases. These results also show that the value of PI near zero may not give the maximum profit. From Table 4, we have at View the MathML sourceIρ˜=3.30 the ranking order quantity be View the MathML sourceq˜=59.90, that of shortage quantity be View the MathML sourceIs˜=11.33 and the corresponding ranking index of the average profit be View the MathML sourceIφ˜=$6707.752, but at View the MathML sourceIρ˜=1.30, the ranking index of the average profit be View the MathML sourceIφ˜=$4832.276. The numerical examples are provided for Model-1 only. Similar results also arise in Model-II. The new major contributions of the present article are: (i) new demand function of fuzzy advertising effort (ii) fuzzy ordering quantity and (iii) shortage quantity. This article may be extended further in many ways: one of them, the demand function may be constructed by selling price and advertising effort for perishable items immediately. Other limitations of deterministic values of parameters may be waived by considering fuzzy parameters for multi-item EOQ model.