The authors have declared that no competing interests exist.

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.

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

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

Briefly, the input parameters of inventory models are often taken as crisp values due to variability in nature.H.J. Zimmerman

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.

Considering these inputs, non-linear programming is solved for total cost function using Lagrangian and Kuhn-Tucker method and done a sensitivity analysis between the slight variations between the fuzzy and crisp values.

Definition 1:

A fuzzy set Ã defined on R (∞,-∞), if the membership function of Ã is defined by

Definition 2:

A trapezoidal fuzzy number Ã=(a, b, c, d) with a membership function μÃ is defined by

Function principle

Suppose Ã=(x_{1}, x_{2}, x_{3}, x_{4}) and _{1}, y_{2}, y_{3}, y_{4}) be two trapezoidal fuzzy numbers. Then

The addition of

Ã ⊕ B̅ =(x_{1}+y_{1}, x_{2}+y_{2}, x_{3}+y_{3}, X_{4}+y_{4})

Where x_{1}, x_{2}, x_{3}, x_{4},y_{1}, y_{2}, y_{3}, y_{4} are any real numbers.

(2) The multiplication of

Ã ⊕ _{1}, c_{2}, c_{3}, c_{4})

Where Z_{1}={x_{1}y_{1}, x_{1}y_{4}, x_{4}y_{1}, x_{4}y_{4}}, Z_{2} = ={x_{2}y_{2}, x_{2}y_{3}, x_{3}y_{2}, x_{3}y_{4}}, C_{1} = min Z_{1}, C_{2} = min Z_{1}, C_{1} = max Z_{2}, C_{1} = max Z_{2}.

If x_{1}, x_{2}, x_{3}, x_{4},y_{1}, y_{2}, y_{3} and y_{4} are all zero positive real numbers then

Ã ⊗ _{1}y_{1}, x_{2}y_{2}, x_{3}y_{3}, x_{4}y_{4}).

(3) The subtraction of

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

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

Suppose if the problem is given as

Minimize y = f(x)

Sub to g_{i}(x) ≥ 0, i = 1, 2, · · ·, m.

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.

Step 1:

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.

Otherwise set K = K + 1 and go to step 2.

Graded mean Integration Representation Method was introduced by Hsieh et al _{1}, α_{2}, α_{3}, α_{4},)_{LR}. By graded mean integration are the inverses of L and R are L^{-1} and R^{-1} respectively. The graded mean h-level value of the generalized fuzzy number Ã-=(α_{1}, α_{2}, α_{3}, α_{4},)_{LR} is given by h/2[L^{-1}(h)+R^{-1}(h)]. Then the graded mean integration representation of P(Ã) with grade then

In this paper trapezoidal fuzzy numbers is used as fuzzy parameters for the production inventory model. Let

The input parameters for the corresponding model are

K - Ordering cost

a - constant demand rate coefficient

b - price-dependent demand rate coefficient

P - selling price

Q - Order size

c - unit purchasing cost

g - constant holding cost coefficient

Let’s consider the total cost

The total cost per cycle is given by

Partially differentiating w.r.t Q,

Equating

By using trapezoidal numbers, fuzzified input parameters are as follows

The optimal order quantity

Partially differentiating w.r.t ‘Q’ and equating to zero,

Hence the optimal economic order quantity for crisp values is derived,

Applying the graded mean representation

Now partially differentiating w.r.t Q_{1}, Q_{2}, Q_{3}, Q_{4} and equating to zero,

The above derived results depict that Q_{1} > Q_{2 }> Q_{3} > Q_{4} failing to satisfy the constraints 0 ≤ Q_{1} ≤ Q_{2 }≤ Q_{3} ≤ Q_{4} . So, converting the inequality constraint Q_{2 }- Q_{1} ≥ 0 into equality constraint Q_{2 }- Q_{1} = 0 Optimizing P(TC(Q,P) subject to Q_{2 }- Q_{1} = 0 by Lagrangian method.

From equations (1) and (2) the results are,

Since Q_{3} > Q_{4} which does not satisfy the constraint 0 ≤ Q_{1} ≤ Q_{2 }≤ Q_{3} ≤ Q_{4}. Now converting the inequality constraints Q_{2} - Q_{1} ≥ 0, Q_{3} - Q_{2} ≥ 0 into equality constraints Q_{2} - Q_{1} = 0 and Q_{3} - Q_{2} = 0. Optimizing,

From equations (1’) , (2’) and (3’),

In the above-mentioned results Since Q_{1} > Q_{4} which does not satisfy the constraint 0 ≤ Q_{1} ≤ Q_{2 }≤ Q_{3} ≤ Q_{4}. Converting the inequality constraints Q_{2} - Q_{1} ≥ 0, Q_{3} - Q_{2} ≥ 0 and Q_{4} - Q_{3} ≥ 0 into equality constraints Q_{2} - Q_{1} = 0, Q_{3} - Q_{2} = 0 and Q_{4} - Q_{3} = 0 . Optimizing,

After Partially differentiation (Appendix)

satisfies the required inequality constraints.

By applying Kuhn Tucker conditions, the total cost is minimized by finding the solution of Q_{1}, Q_{2}, Q_{3}, Q_{4} with Q_{1}, Q_{2}, Q_{3}, Q_{4}

The above conditions simplify to the following 𝞴_{1}, 𝞴_{2}, 𝞴_{3}, 𝞴_{4}.

It is known that, Q1 > 0 then in

𝞴_{1}Q_{1 }= 0 arrive at 𝞴_{4 }= 0 In a similar fashion, if 𝞴_{1}= 𝞴_{2} = 𝞴_{3 }= 0, Q_{4} ≤ Q_{3} ≤ Q_{2} ≤ Q_{1} does not satisfy 0 ≤ Q_{1} ≤ Q_{2} ≤ Q_{3} ≤ Q_{4.} Therefore the conclusion is Q_{2} = Q_{1}, Q_{3 }= Q_{2} and Q_{4} = Q_{3}

i.e., Q_{1} = Q_{2} = Q_{3} = Q_{4 }= Q^{*}

Let us consider an integrated inventory system having the following statistics with crisp parameters having following values

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.

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.

Price-dependentco-efficient | Lagrangian method | Kuhn-Tucker method |

b=1 | EOQ = 263.6169 | EOQ = 263.6169 |

Fuzzy = 260.2202 | Fuzzy = 321.2251 | |

b=1.5 | EOQ = 222.4949 | EOQ = 222.4949 |

Fuzzy = 207.7356 | Fuzzy = 258.333 | |

b=2 | EOQ = 171.7967 | EOQ = 171.7967 |

Fuzzy = 147.1273 | Fuzzy = 194.5951 |

Parameters | % change parameters | Graded-mean values | CrispEOQ | LagrangianMethod | Kuhn-Tucker Method |

K(520) | +40% | 728(708,718,738,748) | 263.2595 | 246.140 | 305.902 |

+25% | 650(630,640,660,670) | 248.7569 | 232.451 | 288.931 | |

-25% | 390(370,380,400,410) | 192.6863 | 179.430 | 223.231 | |

-40% | 312(292,302,322,332) | 172.3438 | 160.136 | 199.342 | |

P(36.52) | +40% | 51.128(31.128,41.128,61.128,71.128) | 159.7376 | 106.502 | 128.4246 |

+25% | 45.65(25.65,35.65,55.65,65.65) | 185.7729 | 147.128 | 178.9012 | |

-25% | 27.39(7.39,17.39,37.39,47.39) | 253.9615 | 236.628 | 289.3353 | |

-40% | 21.912(1.912,11.912,31.912,41.912) | 271.093 | 257.475 | 315.0044 | |

a(100) | +40% | 140(120,130,150,160) | 305.4398 | 301.807 | 374.9838 |

+25% | 125(105,115,135,145) | 277.2588 | 271.688 | 338.6781 | |

-25% | 75(55,65,85,95) | 148.7803 | 127.422 | 168.3332 | |

-40% | 60(40,50,70,80) | 75.5944 | 52.1485 | 49.2549 | |

b(1.5) | +40% | 2.1(1.6,1.8,2.2,3) | 159.7376 | 134.7605 | 187.338 |

+25% | 1.9(1.6,1.8,2,2.2) | 185.7729 | 180.8325 | 227.4581 | |

-25% | 1.1(0.5,0.8,1.4,1.7) | 253.9615 | 250.2417 | 310.0121 | |

-40% | 0.9(0.6,0.8,1,1.2) | 271.0938 | 271.8465 | 334.0561 | |

c(4.75) | +40% | 6.7(6.4,6.5,6.9,7) | 188.0425 | 169.2837 | 210.5158 |

+25% | 5.9(5.5,5.7,6,6.5) | 199.0055 | 179.1926 | 222.8382 | |

-25% | 3.6(3,3.2,3.8,4.6) | 256.9150 | 223.8899 | 278.4223 | |

-40% | 2.8(2.2,2.7,3,3.2) | 287.2397 | 258.3873 | 321.3222 | |

g(0.2) | +40% | 0.28(0.18,0.2,0.3,0.5) | 188.0425 | 170.5264 | 212.0612 |

+25% | 0.25(0.14,0.15,0.26,0.54) | 199.0055 | 176.1179 | 219.0145 | |

-25% | 0.15(0.1,0.14,0.16,0.2) | 256.9150 | 241.4086 | 300.208 | |

-40% | 0.12(0.1,0.11,0.13,0.14) | 287.2397 | 271.2692 | 337.3417 |