A Soft-Computing Approach to Fuzzy EOQ Model for Deteriorating Items with Partial Backlogging

Genetic Algorithm (GA) is an optimized method to find a perfect solution which is based on general genetic process of life cycle. In this article we discussed a crisp and a fuzzy inventory model keeping its demand rate constant for the imprecision and uncertainly deteriorating items with special reference to shortage and partially backlogging systems. The objective of this paper is to minimize the total cost of fuzzy inventory environment for which Graded mean representation, Signed distance and Centroid methods are used to defuzzify the total cost of the systems. Consequently, we are comparing the total average cost, obtained through these methods with the help of numerical example, and sensitively analysis is also given to show the effects of the values on these items. Moreover, Genetic Algorithm (GA) is also applied to the optimistic value of the total cost of the crisp model for the effective and fruitful results.


Introduction
Genetic algorithm (GA) is a set of tools to get the total inventory cost which is obtained by the fuzzy model. Fuzzification is a technique to transform the crisp set principles into fuzzy set principles, with the help of different sorts of membership functions. Zadeh [1] gives the principle sets of fuzzy, properties and features of membership functions. Economic order quantity prototype, to demonstrate the resonant charge with the help of the fuzzy number, is expressed by Park [2]. Kaufmann and Gupta [3] show the concept and real-time applications of fuzzy logic. Yao and Lee [4] instigate the fuzzy model for inventory control with various constraints such as backlog, fuzzy order amount, crisp principles, trapezoidal, triangular numbers as in fuzzy.
The interval mean value notion helps to resolve the issues of inventory control management, further down the state when the inventory price is less and the fixed order interval system is more explained by Gen et al. [5]. Chang et al. [6] modelled the subjects of backorder fuzzy with the triangular number of fuzzy. To explore a collection of computing schematics for the inventory control problems through fuzzy principles with and without backorder, this study is presented by Lee and Yao [7].
In this sequence Yao et al. [8] acknowledged the demand and request frequency is calculated by fuzzy sets without scarcity and accomplished the fuzzy model. Wu and Yao [9] explain a fuzzy model where order quantity is fuzzify. They also used fuzzy cost which is the fuzzy membership function of the fuzzy cost and to defuzzify they used centroid method. Yao and Chiang [10] express a fuzzy model to decrease the total inventory cost without backorders and to equate the result of the model they used signed distance and centroid defuzzification methods.
A fuzzy inventory model with the presence of demand fuzzy random variable and to validate the result of the fuzzy model they used graded the mean integration representation defuzzification method by Dutta et al. [11]. Chang et al. [12] study about the fuzzy model in the presence of backorders, misplaced sales where lead-time belongs to a variable and to validate the model they took help of the centroid method. Wee et al. [13] present a model of inventory for a substandard value of goods in the presence of backorders and scarcity. Lin [14] shows to control inventory for a recurrent study with parameters such as backorder, lead-time of variable and the conformist demand scarcity rate to verify the result they used the signed distance defuzzification method.
Roy and Samanta [15] described a model of fuzzy recurrent study in the absence of backorder and the sequence period and to validate the result they used the signed distance defuzzification method. Gani and Maheswari [16] present a study model of EOQ with the help of different properties of goods such as imperfect value, mandate, inventory costs, shortages and to verify the result of model they used the Graded mean integration method. Ameli et al. [17] shows a modified model of inventory to determine the technique of ordering items or goods for substandard belongings and in the presence of inflation to find the total revenue and validate the model with the signed distance defuzzification process.
Singh et al. [18] expressed a fuzzy two-warehouse model in the existence of interruption in amount is acceptable. Kumar et al. [19] expanded that a model with various parameters of demand rate is dependent on time, backlogged, deteriorating rate and to get the total revenue of inventory and for comparing the results they used defuzzification methods.
In another study of Kumar et al. [19] presented the two-warehouse inventory model with the effect of GA. Agarwal et al. [20] also present a fuzzy model with partially backlog, deteriorating goods and to make an operative and suitable procedure they use linear demand rate. Agarwal et al. [21] inflation should be especially measured for long period investment and forecast such as ice and cold storage industry and weather forecasts, which specifically help in the production industry. Agarwal et al. [22] discussing in which to deal with the actual situation of the ice and cold storage (Agra) and to find the solution to the problem. To get the solution of these problems they used GA.
Here, we present a study of fuzzy model for inventory control with the help of triangular fuzzy number of different inventory cost. To estimate the total inventory cost, we used three methods of defuzzification, i.e. Centroid, Signed distance and Graded Mean Representation methods and to optimised the result we took the help of Genetic Algorithm. To prove the above model, we compared our outputs with the help of mathematical illustration and sensitive examination by MATLAB. the total fuzzy inventory cost per unit per time. TIC(G) the total inventory cost by applying Graded mean integration method of defuzzification. TIC(S) the total inventory cost by applying Signed distance method of defuzzification. TIC(C) the total inventory cost by applying Centroid method of defuzzification.

Supposition
This Modified Model is Consequential Under the Subsequent Expectations (i) The period limit is fixed and Replacement is immediate with zero lead-time.
(ii) Shortages and partially backlogs are acceptable. When the cycle occurs no repair is done on deteriorated items. (iii) The demand is disgruntled backlog and shortages backlogged fraction where ϕ is a positive constant. And the demand rate is constant which can be presented as DR(T) = a. Where a ≥ 0.

Mathematical Formulation
Let us assume IL(T) be the level of inventory on T time with the primary inventory OQ. Moreover, based on the current market scenario and customer consummation, the inventory level regularly moderates on period (0, T 1 ) and afterwards the shortages period (T 1 , LC). At any random time, IL (T) is administered by the subsequent equations ( Figure 1):

Representation of the Crisp Model
Let the inventory level be IL(T) at any time, the equations are The solution is Let IL(T 1 ) = 0, so we obtain So, we can write the following equation as follows: (Ignoring the highest powers of φ).
The entire holding cost is H u on the period (0, LC) is represented by Deteriorated unit D u on the period of (0, LC) is specified by The shortage unit S u on the period of (0, LC) is represented by The total cost of inventory, TIC, is shown by

Representation of the Fuzzy Model
In the expansion of Economic Order Quantity models, Constant deterioration rate is supposed by most authors. But in this model, we let that the entire constraint is static and it can be assumed by certainty; but in the current bazaar state, It is problematic to specify all the parameters because of customer satisfaction, So, here we take the following constraints z which can be modified with some boundaries.
So, the triangular fuzzy numbers are

Total fuzzy inventory cost is
For defuzzification of the total fuzzy inventory cost, we used three methods.

Graded Mean Representation
The entire inventory price is represented by So, we obtain To decrease the entire inventory price per unit per time is TIC(G) and the optimum rate of T 1 and LC will be calculated by the following differential equations: Equation (17) is correspondent to Furthermore, the total inventory cost is TIC(G) to fulfil the result, the following differential equations have to be contented: Afterwards the next derivatives of the total inventory cost TIC(G) are complexes and it is hard to demonstrate mathematically.

Signed Distance
The entire inventory value is represented by To decrease the entire cost of inventory per unit time is TIC(S) and the optimum rate of T 1 and LC will be calculated by the following differential equations: Equation (25) is comparable to Furthermore, the total inventory cost is TIC(S), to fulfil the result, the following differential equations have to be contented: Afterwards the next derivatives of the total inventory cost TIC(S) are complexes and it is hard to demonstrate mathematically.

Centroid Method. Entire Cost is Given by
where the values of TIC(C 1 ), TIC(C 2 ), TIC(C 3 ) are To decrease the entire inventory price per unit per time is TIC(C) and the optimum rate of T 1 and LC will be calculated by the following differential equations: Furthermore, the total inventory cost is TIC(C), to fulfil the result, the following differential equations have to be contented: Afterwards the next derivatives of the total inventory cost TIC(C) are complexes and it is hard to demonstrate mathematically.

Mathematical Illustration
Consider the following parameters in the above model.

Sensitivity Examination
We can compare the values of parameters with changes in effects. And the outcomes are presented in tables. Observations: The observations from the above table are mentioned as follows.

1.
It is clearly visible if the parameter value of b will be increasing and also parametric values of T 1 and LC will be decreasing, so the total fuzzy inventory cost TIC(G) will also be increasing, as shown in Table 2 and Figure 3.

2.
Similarly, if the parameter value of ∅ will be increasing and at the time parametric values of T 1 and LC will be decreasing, so the total fuzzy inventory cost TIC(G) will also be increasing, as shown in Table 3 and Figure 4.

Execution of Genetic Algorithm
Here we are implementing (GA) Genetic algorithm in Table 2. So, we get the optimal values. Again, GA is applying in Table 3. So, changes have effects on parameter ∅ Observations: Here it is clearly visible that we are getting optimal result by using mutation and the crossover method of GA.

1.
It is clearly visible if the parameter value of b will be increasing and parametric values of T 1 and LC will be decreasing, so the total fuzzy inventory cost TIC(G) will also be increasing, as shown in Table 4 and Figure 5.

2.
Similarly, if the parameter value of ∅ will be increasing and similarly parametric values of T 1 and LC will be decreasing, so the total fuzzy inventory cost TIC(G) will also be increasing, as shown in Table 5 and Figure 6.

Conclusion
The presenting model for worsening objects in the situations of where shortage, partially backlog is acceptable. The main objective of this study is to abate the inventory value and increase the revenue. So, we used three defuzzification methods, i.e. Centroid, Signed Distance and Graded Mean Representation Method. And we can see that if parameter values of b and ∅ will be increasing and also parametric values of T 1 and LC will be decreasing, so the total fuzzy inventory cost TIC(G) will also be increasing. Furthermore, we used GA to get optimistic values of different cost factors and to conclude our result with the help of arithmetical illustrations and changes in the effects in parametric values of b and ∅. To show the graphical representation and to analyse the values of different cost factors MATLAB and Java are used.
This model can be modified with various factors such as stock-dependent demand rate, time-based demand and many more.