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Figure from:A fuzzy inventory model with imperfect items and backorder with allowable proportionate discount

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The maximum inventory level is obtained by substituting the value of Eq.(3.1.7) and Eq.(3.1.8) in equation Eq. (3.1.9) we get the following:

Figure 4 The maximum inventory level is obtained by substituting the value of Eq.(3.1.7) and Eq.(3.1.8) in equation Eq. (3.1.9) we get the following:

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Figure 1. Behaviour of the inventory level over time Using the above assumption, the cycle wise total cost is given as follows: In this paper, we develop a fuzzy EOQ inventory model of imperfect quality items, present in each lot, with shortages by introducing allowable proportionate discount. To obtain the total profit, a 100% screening is conducted for the lot and an allowable proportionate discount is introduced for the defective items. In case of the fuzzy inventory model, the defective rates are taken to be fuzzy with the allowable proportionate discount for the defective items present in each lot, where the total profit per unit time is derived in fuzzy sense. The fuzzy EOQ models are derived by using the triangular fuzzy numbers for the defective rates and the fuzzy models are defuzzified by using the signed distance method. The solutions for maximizing the fuzzy total profits per unit time have been derived and numerical examples are given to illustrate the model. Finally, the sensitivity analysis is performed on the results to analyse the effect of the variations taken for the parameters involved in the model.
Figure 1. Behaviour of the inventory level over time Using the above assumption, the cycle wise total cost is given as follows: In this paper, we develop a fuzzy EOQ inventory model of imperfect quality items, present in each lot, with shortages by introducing allowable proportionate discount. To obtain the total profit, a 100% screening is conducted for the lot and an allowable proportionate discount is introduced for the defective items. In case of the fuzzy inventory model, the defective rates are taken to be fuzzy with the allowable proportionate discount for the defective items present in each lot, where the total profit per unit time is derived in fuzzy sense. The fuzzy EOQ models are derived by using the triangular fuzzy numbers for the defective rates and the fuzzy models are defuzzified by using the signed distance method. The solutions for maximizing the fuzzy total profits per unit time have been derived and numerical examples are given to illustrate the model. Finally, the sensitivity analysis is performed on the results to analyse the effect of the variations taken for the parameters involved in the model.
The cycle wise total profit m(z,Q) is the difference between the cycle wisetotal revenue and the cycle wise total cost per cycle. The total profit per unit time is denoted by 7,,(z, Q) and the formula used for the total profit is given as follows.
The cycle wise total profit m(z,Q) is the difference between the cycle wisetotal revenue and the cycle wise total cost per cycle. The total profit per unit time is denoted by 7,,(z, Q) and the formula used for the total profit is given as follows.
The optimality condition for the nonlinear problem the expected total profit per unit time given in Eq. (2.1.5), is demonstrated by finding the first and second partial derivatives of E7,,(z,Q) with respect to z,Q are obtained as follows:
The optimality condition for the nonlinear problem the expected total profit per unit time given in Eq. (2.1.5), is demonstrated by finding the first and second partial derivatives of E7,,(z,Q) with respect to z,Q are obtained as follows:
By setting (2.2.5(a)) and (2.2.5(b)) equal to zero, we obtain: In other hand by replacing (2.2.6) and (2.2.7) in equation (2.2.8) we have:
By setting (2.2.5(a)) and (2.2.5(b)) equal to zero, we obtain: In other hand by replacing (2.2.6) and (2.2.7) in equation (2.2.8) we have:
Taking A, = A, = A in Eq. (2.2.6), we get the following equation. In order to examine the second-order sufficient conditions (SOSC) for maximum value, we first obtain the Hessian matrix H which is given as follows:
Taking A, = A, = A in Eq. (2.2.6), we get the following equation. In order to examine the second-order sufficient conditions (SOSC) for maximum value, we first obtain the Hessian matrix H which is given as follows:
When g=landC,+C, = Sz, in Eq. (2.2.17), z reduces t the traditional EOQ backorder formula for the lot size i.e.,
When g=landC,+C, = Sz, in Eq. (2.2.17), z reduces t the traditional EOQ backorder formula for the lot size i.e.,