Multi-Source Backlogged Probabilistic Inventory Model for Crisp and Fuzzy Environment

Abstract This paper proposed a multi-item multi-source probabilistic periodic review inventory model under a varying holding cost constraint with zero lead time when: (1) the stock level decreases at a uniform rate over the cycle. (2) some costs are varying. (3) the demand is a random variable that follows some continuous distributions as (two-parameter exponential, Kumerswamy, Gamma, Beta, Rayleigh, Erlang distributions).


Introduction
In the practical situation, some costs are relating to some variables such as quantity ( )or length of the cycle

The mathematical model
This model developed the stock level decreases at a uniform rate over the cycle. Figure  The expected total cost of the cycle for multi-item multi source will be the sum of the expected purchase cost, the expected order cost, the expected varying holding cost, and the expected varying backlogged cost, -The expected purchase cost for the cycle is given by -The expected order cost for the cycle is given by -The expected varying holding cost for the cycle is given by where ̅ represents the expected average amount of inventory. The first situation in Figure (2). If ̅ ≤ . Then the average amount in inventory ̅ , is given by The second situation, the following relationships are evident: Thus, the expected average amount of inventory is given by -The varying expected varying holding cost for the cycle becomes -The expected varying backlogged cost for the cycle is given by where ̅ represents the expected average backlogged. In the first situation of Figure (2), the average backlogged is given by In the second situation, the following relationship is evident: Then, the expected total cost of the cycle for multi-item multi-source is the sum of Equations (1), (2), (3) and (4) There is a limitation on the available expected varying holding cost; The problem is to find the optimal maximum inventory level for a given which minimize the expected annual average total cost function (5) subject to the expected varying holding cost restriction. It may be written as Min ( (, , )) for all = 1,2, … , , = 1,2, … , subject to inequality constraint ( ( )) ≤ ℎ (7) To find the optimal values * for a given which minimize Equation (6) under the constraint (7), the Lagrange multipliers technique with the Kuhn-Tacker conditions is used, then the Lagrange function is given by:- (8) where ℎ , the Lagrange multiplier. The optimal values * can be calculated by setting the corresponding first partial derivatives of Equation (8) equal to zero, and then the following equations are obtained. Where the shortage cost per unit item. (Fabrycky W. J. and Banks Jerry [7]).

The Model when all parameters are fuzzy numbers
The inventory cost coefficients, elasticity parameters and other coefficients in the model are fuzzy in nature. Therefore, the decision variable and the objective function should be fuzzy as well, it should find the right and the left shape functions of the objective function and decision variable, by find the upper bound and the lower bound of the objective function, i.e. ̃( ∝) and̃(∝). Recall that ̃( ∝) and ̃( ∝) represents the largest and the smallest values (The left and right ∝ cuts) of the optimal objective function ̃( ∝). Using approximated value of TFN which observe in Figure 3 where ̃= + Likewise, the same steps as in crisp case will be applied here with replacing , , ℎ by ̃,̃,̃ℎ ̃. then the optimal value of * for a given which minimize expected annual total cost for fuzzy case can be calculated easily and the optimal value * for fuzzy case can be calculated by the following equations:

The model with some continuous distributions
Suppose that the demand for a particular item follows some continuous distribution such as: 5.1 The model with two-parameters exponential distribution: If the demand follows the two parameter exponential distribution then, continuous inverse scale parameter > 0, continuous location parameter. Hence, the optimal value * can be calculated by the following equation 1 , 2 continuous shape parameter 1 , 2 > 0, , continuous boundary parameters < . Hence, the optimal value * can be calculated by the following equation

Numerical example
To illustrate the above developed model, consider a hypothetical inventory system with the following parameter values which are given in Table 1. It is desired to determine to optimal value Q mrs * which minimize the expected total cost for β r between (0,1) and = 1 month (0.08333 year). Also, the optimal solutions of the crisp environment and triangular fuzzy number TFN are given in Tables 3, 5

Conclusion
This paper concerns with a multi-item multi-source (MIMS) constrained probabilistic periodic review inventory model. We determine the optimal maximum inventory level for a given that minimized the expected annual total cost under varying holding cost constraint using Lagrange multiplier technique for crisp and TFN environment. And we conclude that: the fuzzy environment is more closed to the practical situation than crisp number. Also, When β convergence to 1, the solution approaching to the optimal solution. Furthermore, under our assumption and from selected distributions: Raylieph distribution give the optimum expected annual total cost.