Determining the Efficiency of Fuzzy Logic EOQ Inventory Model with Varying Demand in Comparison with Lagrangian and Kuhn-Tucker Method Through Sensitivity Analysis

This paper considers an EOQ inventory model with varying demand and holding costs. It suggests minimizing the total cost in a fuzzy related environment. The optimal policy for the nonlinear problem is determined by both Lagrangian and Kuhn-tucker methods and compared with varying price-dependent coefficient. All the input parameters related to inventory are fuzzified by using trapezoidal numbers. In the end, a numerical example discussed with sensitivity analysis is done to justify the solution procedure. This paper primarily focuses on the aspect of Economic Order Quantity (EOQ) for variable demand using Lagrangian, Kuhn-Tucker and fuzzy logic analysis. Comparative analysis of there methods are evaluated in this paper and the results showed the efficiency of fuzzy logic over the conventional methods. Here in this research trapezoidal fuzzy numbers are incorporated to study the price dependent coefficients with variable demand and unit purchase cost over variable demand. The results are very close to the crisp output. Sensitivity analysis also done to validate the model. DOI: 10.14302/issn.2643-2811.jmbr-20-3465 Corresponding author: H. Mary Henrietta, Department of Mathematics, Saveetha Engineering College, Chennai, Mobile No. 9994991376, Email: henriettamaths@gmail.com


Introduction
In today's competitive scenario, organizations face immense challenges for meeting the transitional consumer's demand, and maintaining the inventory plays a major role. Indulging the betterment of various promotional activities that yield a reputation for the concern. The classical economic order quantity by Harris [1] and Wilson [2] was developed for problems where demand remains constant. In recent study of inexplicit demand rates by Chen [3] and few more resulting in the time-dependent demand rates and varying holding cost. In 1965, Zadeh [4] introduced the concept of fuzzy that carries vagueness or unclear in sense. Hence fuzzy sets came into prominence in describing the vagueness and uncertainty that impressed many researchers. In this paper, the total cost has been taken and the proposed model economic order quantity (EOQ) and the input parameters such as ordering cost, purchase cost, order size, holding cost, unit purchase cost, and unit selling price are fuzzified using trapezoidal numbers. The optimization is carried out by both Lagrangian and Kuhn-Tucker methods and graded mean integration for defuzzification of the total cost.
In the early 1990's the Economic order quantity (EOQ) and Economic Production Quantity (EPQ) played a vital role in the operations management area, contrastingly it failed in meeting with the real-world challenges. Since these models assumed that the items received or produced are of a perfect quality which is highly challenging. In inventory management, EOQ plays a vital role in minimizing holding and ordering costs.
Ford W. Harris (1915) developed the EOQ model but it was further extended and extensively applied by R.H.
Wilson. Wang [5] examined an EOQ model where a proportion of defective items were represented as fuzzy variables. In the year 2000, Salameh et al [6] came up with a classical EOQ that assumed a random proportion of defective items, and the recognized imperfect items are sold in a single batch at an economical rate.
Chen [3] in the year 2003, came up with an EOQ with a random demand that minimized the total cost. Firms to cope up with the current scenarios, have to adapt the diversity among inventory models due to the advancement of management strategies and varying production levels.
Briefly, the input parameters of inventory models are often taken as crisp values due to variability in nature.H.J. Zimmerman [7] studied the fuzziness in operational research. In a classical EOQ, Park [8] fuzzified the decision variables ordering cost and holding cost into trapezoidal fuzzy numbers that gave rise to study the fuzzy EOQ models, solving non-linear programming to obtain the optimal solution for economic order quantity. In 2002, Hsieh [9] initiated two production inventory models that applied the graded -mean representation method for the defuzzification process. In traditional inventory models when minimizing the annual costs, the demand rate was always assumed to be independent. Despite the promotional setups and the deterioration items, a wavering demand arises. In 2003, Sujit [10] came up with an inventory model that involves a fuzzy demand rate fuzzy deterioration rate. In 2013, Dutta and Pavan Kumar [11] proposed an inventory model without shortfall with fuzziness in demand, holding cost, and ordering cost. The study of inexplicit demand rates was doneby Chan [12] and few more resulting in the timedependent demand rates and varying holding cost. A detailed study of Lagrangian method to solve the nonlinear programming of the total cost was done by Kalaiarasi et al [13]. The Kuhn-Tucker optimization technique was for the NPP(non-linear programming) problem was carried out by Kalaiarasi et al [14].
Considering these inputs, the non-linear programming problem is solved for total cost function using Lagrangian and Kuhn-Tucker method and concluded with a sensitivity analysis, which exhibits the variations between the fuzzy and crisp values.

3) The subtraction of is
Where also x 1 , x 2 , x 3 , x 4 , y 1 , y 2 , y 3 and y 4 are any real numbers.
(4) The division of is where y 1 , y 2 , y 3 and y 4 are positive real numbers. Also x 1 , x 2 , x 3 , x 4 , y 1 , y 2 , y 3 and y 4 are nonzero positive numbers.
(5) For any ∝ ∈ R Extension of the Lagrangian Method.
Solving a nonlinear programming problem by obtaining the optimum solution was discussed by Taha [13] with equality constraints, also by solving those inequality constraints using Lagrangian method.
The constraints are non-negative say x ≥ 0 if included in the m constraints. Then the procedure of the extension of the Lagrangian method will involve the following steps.

Solve the unconstrained problem
Min y = f(x) If the resulting optimum satisfies all the constraints, then stop since all the constraints are inessential. Or else set K = 1 and move to step 2.
Step 2: Activate any K constraints (i.e., convert them into equalities) and optimize f(x) subject to the K active constraints by the Lagrangian method. If the resulting solution is feasible with respect to the remaining constraints, the steps have to be repeated. If all sets of active constraints taken K at a time are considered without confront a feasible solution, go to step 3.
Step 3: If K = m, stop; there's no feasible solution.

Numerical Examples
Let us consider an integrated inventory system having the following statistics with crisp parameters having following values [15], the ordering cost K=520 units, the optimal selling price P=36.52 units, unit purchasing cost c=4.75, g=0.2, constant demand rate coefficient a=100 units, price-dependent demand rate coefficient b=1.5.
As mentioned earlier, the trapezoidal numbers yielding the below results. Equation 1 was the optimal order quantity for crisp (equation 1) values and optimization to the EOQ is done by applying Lagrangian (equation 2) and Kuhn-Tucker (equation 3) methods under graded-mean defuzzification method. Fig. 1 represents the crisp and fuzzified output varying on demand.

Conclusion
The fuzzified output varies at a larger rate than crisp output while varying the demand. This shows that the optimization results can be obtained only by restricting the controlling parameters. Fig. 2 shows the variations in unit purchase cost for crisp and fuzzy output. An analysis of the total cost in a fuzzy environment is studied herewith by applying the Lagrangian and Kuhn-Tucker methods for optimization and using trapezoidal numbers. The value does not show much variation in optimal solutions. While varying the price-dependent demand rate coefficient among both the methods, fuzzified values decrease in Lagrangian and increases in the Kuhn-Tucker method. In Fig. 3, the curves of crisp and fuzzy EOQ remain closer.    optimization methods in operations research [13] Lagrangian and Kuhn-Tucker methods. The values are varied to an increase and decrease of 25% and 40% which clearly shows that the crisp values of EOQ stay perfectly aligned between both the methods. Fig. 4 collates the three parameters viz., price dependent coefficient, purchase cost, and varying demand which shows the variations in 3D model. Tab 2.