Fuzzy inventory model with quadratic demand, linear time dependent holding cost, constant deterioration rate and shortages (original) (raw)

Fuzzy inventory model with quadratic demand, linear time dependent holding cost, constant deterioration rate and shortages

S Balarama Murthy 1∗{ }^{1 *} and S Karthigeyan 2{ }^{2}

Abstract

Fuzzy inventory model with demand rate is quadratic, holding cost is linearly time dependent, constant deterioration rate and allowing shortages. Trapezoidal fuzzy number is used for holding cost, deterioration cost and shortage cost. For defuzzification purpose Graded Mean Representation Integration method is used. Sensitivity Analysis have done and the Total cost is minimized.

Keywords

Purchase inventory model, Quadratic demand, Shortage, Deterioration, Time dependent holding cost and Trapezoidal fuzzy number.

AMS Subject Classification

90B05, 62E86, 62P30.
1{ }^{1} Research Scholar, Bharathiar University, Coimbatore-641046, India.
2{ }^{2} P.G. and Research Department of Mathematics, Dr. Ambedkar Government Arts College, Chennai-600039, India.
3{ }^{3} Corresponding author: 1{ }^{1} balaramac@gmail.com; 2{ }^{2} karthigeyanshan@gmail.com
Article History: Received 01 January 2020; Accepted 12 February 2020
© 2020 MJM

Contents

1 Introduction … 157
2 Notations and Assumptions … 158
3 Fuzzy Concepts … 158
4 Formulation of the Model … 159
5 Analysis and Interpretation … 161
6 Conclusion … 162
References … 162

1. Introduction

Inventory is an essential resource needed for day-to-day operations and it deals with how much to keep on hand and how frequently reorder. Raw materials inventory, work-inprocess inventory and finished goods inventory are three basic types of inventory. The main purpose of inventory is to maintain trade-off between the minimization of the total cost and maximization of the customer satisfaction.

In the inventory process, the effect of deterioration is very important. While developing an optimal inventory policy for products such as fruits, vegetables, chemicals etc., deterioration cannot be ignored. In inventory models the demand is
either deterministic or probabilistic in nature. In reality both demand and supply are uncertain due to unpredictable events, like change of orders, random capacity of supplies, etc,. In the real situations Fuzzy concepts can be used in the inventory models.

Fuzzy set theory originally introduced by [12] Zadeh in 1965, provides a framework for considering parameters that are vaguely or unclearly defined or whole values are imprecise or determined based on subjective beliefs of individuals. In the development of inventory models, the quantities such as the demand rate, deterioration rate and all the costs related to inventory models are considered as fuzzy numbers.

Pande and Gowtham [8] have developed an inventory control model for fixed deterioration and logarithmic demand rates. They used cost minimization technique to obtain the optimal value of stock, time and total cost. They considered deterministic cases of demand by allowing shortage. They obtained an approximate expression for initial inventory, total number of deteriorated units and total minimum average cost.

Amutha and Chandrasekaran [1] have developed an EOQ model for deteriorating items with quadratic demand and time dependent holding cost. They used the exponential distribution for deterioration. Shortages are not allowed and holding cost is time dependent. Using numerical examples, the model

was solved to minimize the total inventory cost.
Mishra et al [7] have designed an inventory model for deteriorating items with time-dependent demand and time varying holding cost under partial backlogging. They considered a deterministic inventory model with time dependent demand and time varying holding cost where deterioration was time proportional. Shortages were allowed and the demand was partially backlogged. Using numerical examples, the model was solved analytically to minimize the inventory cost.

Venkateswarlu and Mohan [11] have developed an inventory model with quadratic demand, constant deterioration and salvage value. They considered two cases ( Retarded Decline Model and Accelerated Decline Model) to calculate the optimum total cost and total order quantity. They found that the retarded decline and accelerated decline have shown good results which will be useful to describe a realistic situation for any product. They also formulated inventory models for constant deterioration rate together with salvage value. They found the existence of retarded decline and accelerated decline models in this case.

Jaggiet al [5] have considered fuzzy inventory model for deteriorating items with time varying demand and shortages. They developed a fuzzy economic order quantity model for deteriorating items in which demand increases with time. Shortages were allowed and fully backlogged. They considered the demand, holding cost, unit cost, shortage constant deterioration cost as trapezoidal fuzzy numbers. They defuzzified the total cost function using graded mean integration representation method, signed distance method and centroid method. Using numerical examples, they found that the graded mean integration representation method gave the minimum cost as compared to signed distance method and centroid method.

Dutta and Kumar [4] have designed an inventory model without shortages have been considered in a fuzzy environment. They determined the optimal total cost and the optimal order quantity. Trapezoidal fuzzy number is used. The computation of economic order quantity (EOQ) is carried out through defuzzification process by using signed distance method. Sensitivity for this model is also studied, which shows a linear relation between demand, EOQ, and total cost.

Ranganathan and Thirunavekarasu [9] have developed an inventory control model for constant deterioration in fuzzy environment with demand rate. Shortages were allowed and fully backlogged. All inventory parameters were assumed to be trapezoidal fuzzy number. They defuzzified the fuzzy model using graded mean one level integration representation method. After analysing the result, they obtained the fuzzy optimal solution to minimize the total cost of the inventory system.

Karthigeyan et al [6] have expressed a fuzzy optimized production EOQ model with deterioration rate and allowing the shortages. In their continuous production inventory model the demand and production rates along with the holding cost, shortage cost and deteriorating cost are assumed as trapezoidal fuzzy numbers. The deterioration is assumed as exponential
distribution. They used graded mean integration method for defuzzification. They obtained the minimum total cost and optimal inventory quantity and optimal shortage quantity.

Balarama Murthy et al [2] have developed fuzzy inventory control problem with Weibull deterioration rate and logarithmic demand rate. They considered holding cost, shortage cost, and deterioration cost and demand rate as a fuzzy trapezoidal number. They defuzzified the fuzzy model using graded mean one level integration representation method. The total inventory cost was minimized.

2. Notations and Assumptions

1. Demand rate D(t)=(a+bt+ct2)D(t)=\left(a+b t+c t^{2}\right) where a>0,b>0,c>0a>0, b>0, c>0
2. A: Ordering cost per order
3. I(t)\mathrm{I}(\mathrm{t}) : Inventory level of the product
4. H(t)=(α+βt)\mathrm{H}(\mathrm{t})=(\alpha+\beta t), linearly time dependent holding cost. where α>0,β>0\alpha>0, \beta>0
5. CtC_{t} : shortage cost per unit per time period
6. CdC_{d} : deterioration cost per unit per time period
7. CbC_{b} : inventory holding cost per unit time
8. THC, TDC, TSC and TIC are the total holding cost, total deterioration cost, total shortage cost and total inventory cost respectively
9. Exponential distribution is used for deterioration rate of an item. The probability density function of exponential distribution is

f(t)={θe−θt, if t≥00, otherwise f(t)= \begin{cases}\theta e^{-\theta t}, & \text { if } t \geq 0 \\ 0, & \text { otherwise }\end{cases}

1. S is the initial inventory
2. D is deterioration units
3. At time t=t1t=t_{1} Inventory level reaches to zero
4. Shortages are in the time period (t1,T)\left(t_{1}, T\right). Shortages are fully backlogged

3. Fuzzy Concepts

3.1 Fuzzy Number

[10] If a fuzzy set is convex and normalized and its membership function is defined in R and piecewise continuous is called as fuzzy number. So fuzzy number represents a real number interval whose boundary is fuzzy.

3.2 Trapezoidal fuzzy number

A fuzzy set A~=(a1,a2,a3,a4)\tilde{A}=\left(a_{1}, a_{2}, a_{3}, a_{4}\right) where a1<a2<a3<a4a_{1}<a_{2}<a_{3}<a_{4} and defined on R, is called the trapezoidal fuzzy number, if the membership function of A is given by

μA~={0, if x≤a1x−a1a2−a1, if a1≤x≤a21, if a2≤x≤a3a4−xa4−a3, if a3≤x≤a40, if x≥a4\mu_{\tilde{A}}= \begin{cases}0, & \text { if } x \leq a_{1} \\ \frac{x-a_{1}}{a_{2}-a_{1}}, & \text { if } a_{1} \leq x \leq a_{2} \\ 1, & \text { if } a_{2} \leq x \leq a_{3} \\ \frac{a_{4}-x}{a_{4}-a_{3}}, \text { if } \quad a_{3} \leq x \leq a_{4} \\ 0, & \text { if } x \geq a_{4}\end{cases}

(or)
μAˉ=max⁡(min⁡(x−a1a1−a1,1,a4−xa4−a3),0)\mu_{\bar{A}}=\max \left(\min \left(\frac{x-a_{1}}{a_{1}-a_{1}}, 1, \frac{a_{4}-x}{a_{4}-a_{3}}\right), 0\right)
3.3 Fuzzy arithmetical operations :

If Aˉ=(a1,a2,a3,a4)\bar{A}=\left(a_{1}, a_{2}, a_{3}, a_{4}\right) and Bˉ=(b1,b2,b3,b4)\bar{B}=\left(b_{1}, b_{2}, b_{3}, b_{4}\right) are two trapezoidal fuzzy real numbers, Then
Aˉ+Bˉ=(a1+b1,a2+b2,a3+b3,a4+b4)\bar{A}+\bar{B}=\left(a_{1}+b_{1}, a_{2}+b_{2}, a_{3}+b_{3}, a_{4}+b_{4}\right)
Aˉ×Bˉ=(c1,c2,c3,c4)\bar{A} \times \bar{B}=\left(c_{1}, c_{2}, c_{3}, c_{4}\right) where T=(a1b1,a1b4,a4b1,a4b4)T=\left(a_{1} b_{1}, a_{1} b_{4}, a_{4} b_{1}, a_{4} b_{4}\right),
T1=(a2b2,a12b3,a3b2,a3b3)T 1=\left(a_{2} b_{2}, a_{1} 2 b_{3}, a_{3} b_{2}, a_{3} b_{3}\right),
c1=min⁡T,c2=min⁡T1,c3=max⁡T1c_{1}=\min T, c_{2}=\min T 1, c_{3}=\max T 1 and c4=max⁡Tc_{4}=\max T
If a1,a2,a3,a4,b1,b2,b3a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3} and b4b_{4} are all non-zero positive real numbers,

Then Aˉ∗Bˉ=(a1b1,a2b2,a3b3,a4b4)\bar{A} * \bar{B}=\left(a_{1} b_{1}, a_{2} b_{2}, a_{3} b_{3}, a_{4} b_{4}\right)
Aˉ=a1,a2,a3,a4=(a1,a2,a3,a4)\sqrt{\bar{A}}=\sqrt{a_{1}, a_{2}, a_{3}, a_{4}}=\left(\sqrt{a_{1}}, \sqrt{a_{2}}, \sqrt{a_{3}}, \sqrt{a_{4}}\right)
−Bˉ=(−b4,−b3,−b2,−b1)-\bar{B}=\left(-b_{4},-b_{3},-b_{2},-b_{1}\right) and Aˉ−Bˉ=(a1−b4,a2−\bar{A}-\bar{B}=\left(a_{1}-b_{4}, a_{2}-\right.
b3,a3−b2,a4−b1)\left.b_{3}, a_{3}-b_{2}, a_{4}-b_{1}\right)
1Bˉ=Bˉ2−1)=(1/b4,1/b3,1/b2,1/b1)\frac{1}{\bar{B}}=\bar{B}^{2}-1)=\left(1 / b_{4}, 1 / b_{3}, 1 / b_{2}, 1 / b_{1}\right)
Aˉ/Bˉ=(a1/b4,a2/b3,a3/b2,a4/b1)\bar{A} / \bar{B}=\left(a_{1} / b_{4}, a_{2} / b_{3}, a_{3} / b_{2}, a_{4} / b_{1}\right)

3.4 Graded mean Representation integration method

The method of defuzzification of a generalized trapezoidal fuzzy number Aˉ=(a1,a2,a3,a4)\bar{A}=\left(a_{1}, a_{2}, a_{3}, a_{4}\right) by its graded mean integration representation was proposed by [3] Chen and Hsieh and is defined by

GMRI⁡(Aˉ)=∫01h2[(a1+a4)+h(a2−a1−a4+a3)]dh∫01hdh=a1+2a2+2a3+a46\begin{aligned} \operatorname{GMRI}(\bar{A}) & =\frac{\int_{0}^{1} \frac{h}{2}\left[\left(a_{1}+a_{4}\right)+h\left(a_{2}-a_{1}-a_{4}+a_{3}\right)\right] d h}{\int_{0}^{1} h d h} \\ & =\frac{a_{1}+2 a_{2}+2 a_{3}+a_{4}}{6} \end{aligned}

4. Formulation of the Model

I(t)I(t) denote on-hand inventory at time (0,T)(0, T), then the linear first order differential equation which the on-hand inventory I(t)I(t) satisfies in two different parts of the cycle time TT are given by:

dI(t)dt+θ(t)I(t)=−(a+bt+ct2),0≤t≤t1dI(t)dt=−(a+bt+ct2),t1≤t≤T\begin{aligned} & \frac{d I(t)}{d t}+\theta(t) I(t)=-\left(a+b t+c t^{2}\right), 0 \leq t \leq t_{1} \\ & \frac{d I(t)}{d t}=-\left(a+b t+c t^{2}\right), \quad t_{1} \leq t \leq T \end{aligned}

With initial and boundary conditions

I(0)=S,I(t1)=0,I(T)=QI(0)=S, I\left(t_{1}\right)=0, I(T)=Q

Solving equations (4.1) and (4.2) with using equation (4.3)

I(t)eθt=−[at+(aθ+b)t22+(bθ+c)t33+cθt44]+C1\begin{aligned} I(t) e^{\theta t}=-[a t+(a \theta+b) \frac{t^{2}}{2} & +(b \theta+c) \frac{t^{3}}{3} \\ & +c \theta \frac{t^{4}}{4}]+C_{1} \end{aligned}

Put t=0,I(0)=S\mathrm{t}=0, \mathrm{I}(0)=\mathrm{S}, then S=C1\mathrm{S}=C_{1}

I(t)=−[at+(aθ+b)t22+(bθ+c)t33+cθt44]eθt+Seθt\begin{aligned} I(t)=-[a t+(a \theta+b) \frac{t^{2}}{2} & +(b \theta+c) \frac{t^{3}}{3} \\ +c \theta \frac{t^{4}}{4}] & e^{\theta t}+S e^{\theta t} \end{aligned}

Put t=t1,I(t1)=0t=t_{1}, I\left(t_{1}\right)=0

S=[at1+(aθ+b)t122+(bθ+c)t133+cθt144]S=\left[a t_{1}+(a \theta+b) \frac{t_{1}^{2}}{2}+(b \theta+c) \frac{t_{1}^{3}}{3}+c \theta \frac{t_{1}^{4}}{4}\right]

Substitute (4.6) in (4.5)

I(t1)=[at1+(aθ+b)t122+(bθ+c)t133+cθt144]−t(a(1+θt1)+bθt122+cθt133)−(b−aθ)t23−(2c−bθ)t36)+cθ120≤t≤t1\begin{aligned} I\left(t_{1}\right)=\left[a t_{1}+(a \theta+b) \frac{t_{1}^{2}}{2}+(b \theta+c) \frac{t_{1}^{3}}{3}+c \theta \frac{t_{1}^{4}}{4}\right] \\ -t\left(a\left(1+\theta t_{1}\right)+b \theta \frac{t_{1}^{2}}{2}+c \theta \frac{t_{1}^{3}}{3}\right) \\ -(b-a \theta) \frac{t^{2}}{3}-(2 c-b \theta) \frac{t^{3}}{6})+\frac{c \theta}{12} \\ 0 \leq t \leq t_{1} \end{aligned}

solving equation (4.2)

I(t)=−[at+bt22+ct33]+C2I(t)=-\left[a t+\frac{b t^{2}}{2}+\frac{c t^{3}}{3}\right]+C_{2}

Put I(t1)=0I\left(t_{1}\right)=0, then t=t1t=t_{1}
Where C2=at1+bt122+ct133C_{2}=a t_{1}+\frac{b t_{1}^{2}}{2}+\frac{c t_{1}^{3}}{3}

I(t)=a(t1−t)+b2(t12−t2)+c3(t13−t3)t1≤t≤T\begin{gathered} I(t)=a\left(t_{1}-t\right)+\frac{b}{2}\left(t_{1}^{2}-t^{2}\right)+\frac{c}{3}\left(t_{1}^{3}-t^{3}\right) \\ t_{1} \leq t \leq T \end{gathered}

Put I(t)=Q,t=TI(t)=Q, t=T
Q=a(t1−T)+b2(t12−T2)+c3(t13−T3)Q=a\left(t_{1}-T\right)+\frac{b}{2}\left(t_{1}^{2}-T^{2}\right)+\frac{c}{3}\left(t_{1}^{3}-T^{3}\right)
The deterioration units D=S−∫0t1(a+bt+ct2)dtD=S-\int_{0}^{t_{1}}\left(a+b t+c t^{2}\right) d t

D=aθt122+bθt133+cθt144D=\frac{a \theta t_{1}^{2}}{2}+\frac{b \theta t_{1}^{3}}{3}+\frac{c \theta t_{1}^{4}}{4}

Deterioration cost =Cd∗=C_{d} * (number of deterioration units)

=Cd∗[aθt122+bθt133+cθt144]=C_{d} *\left[\frac{a \theta t_{1}^{2}}{2}+\frac{b \theta t_{1}^{3}}{3}+\frac{c \theta t_{1}^{4}}{4}\right]

Shortage units =∫t1TI(t)dt=\int_{t_{1}}^{T} I(t) d t

=[at1T+bt12T2+ct13T3−aT22−bT36−cT412−at122−bt133−ct144]\begin{gathered} =\left[a t_{1} T+\frac{b t_{1}^{2} T}{2}+\frac{c t_{1}^{3} T}{3}-\frac{a T^{2}}{2}-\frac{b T^{3}}{6}-\frac{c T^{4}}{12}\right. \\ \left.-\frac{a t_{1}^{2}}{2}-\frac{b t_{1}^{3}}{3}-\frac{c t_{1}^{4}}{4}\right] \end{gathered}

Shortage cost =Cs∗=C_{s} * (no.of shortage units)

=Cs∗[at1T+bt12T2+ct13T3−aT22−bT36−cT412−at122−bt133−ct144\begin{aligned} =C_{s} *\left[a t_{1} T+\frac{b t_{1}^{2} T}{2}+\frac{c t_{1}^{3} T}{3}\right. & -\frac{a T^{2}}{2}-\frac{b T^{3}}{6}-\frac{c T^{4}}{12} \\ & -\frac{a t_{1}^{2}}{2}-\frac{b t_{1}^{3}}{3}-\frac{c t_{1}^{4}}{4} \end{aligned}

Holding units =∫0t1(α+β) dtt(t)dt=\int_{0}^{t_{1}}(\alpha+\beta) \mathrm{~d} t \mathrm{t}(\mathrm{t}) \mathrm{dt}

=[aαt122+αt136(aθ+2b)+αt148(bθ+2c)+αt1510(cθ)+aβt136+βt1424(aθ+3b))+βt1530(bθ+3c)+βt1636(cθ)]\begin{aligned} & =\left[\frac{a \alpha t_{1}^{2}}{2}+\frac{\alpha t_{1}^{3}}{6}(a \theta+2 b)\right. \\ & +\frac{\alpha t_{1}^{4}}{8}(b \theta+2 c)+\frac{\alpha t_{1}^{5}}{10}(c \theta) \\ & \left.+\frac{a \beta t_{1}^{3}}{6}+\frac{\beta t_{1}^{4}}{24}(a \theta+3 b)\right) \\ & +\frac{\beta t_{1}^{5}}{30}(b \theta+3 c)+\frac{\beta t_{1}^{6}}{36}(c \theta)] \end{aligned}

Holding cost =Cha=C_{h}{ }^{\mathrm{a}} (number of holding units)

=Ch∗[aαt122+αt136(aθ+2b)+αt148(bθ+2c)+αt1510(cθ)+aβt136+βt1424(aθ+3b))+βt1530(bθ+3c)+βt1636(cθ)]\begin{aligned} = & C_{h} *\left[\frac{a \alpha t_{1}^{2}}{2}+\frac{\alpha t_{1}^{3}}{6}(a \theta+2 b)\right. \\ & +\frac{\alpha t_{1}^{4}}{8}(b \theta+2 c)+\frac{\alpha t_{1}^{5}}{10}(c \theta) \\ & \left.+\frac{a \beta t_{1}^{3}}{6}+\frac{\beta t_{1}^{4}}{24}(a \theta+3 b)\right) \\ & \left.+\frac{\beta t_{1}^{5}}{30}(b \theta+3 c)+\frac{\beta t_{1}^{6}}{36}(c \theta)\right] \end{aligned}

Total Inventory Cost (TIC) during the cycle period [0, T]=[0, \mathrm{~T}]=
Tavg =T_{\text {avg }}= Ordering cost + Deterioration cost + shortage cost + holding cost

Tavg =A+CdT[aθt122+bθt133+cθt144]+CsT[at1T+bt12T2+ct13T3−aT22−bT36−cT412−at122−bt133−ct144]+ChT[aαt122+αt136(aθ+2b)+αt148(bθ+2c)+αt1510(cθ)+aβt136+βt1424(aθ+3b)+βt1530(bθ+3c)+βt1636(cθ)]\begin{aligned} T_{\text {avg }}= & A+\frac{C_{d}}{T}\left[\frac{a \theta t_{1}^{2}}{2}+\frac{b \theta t_{1}^{3}}{3}+\frac{c \theta t_{1}^{4}}{4}\right] \\ & +\frac{C_{s}}{T}\left[a t_{1} T+\frac{b t_{1}^{2} T}{2}+\frac{c t_{1}^{3} T}{3}-\frac{a T^{2}}{2}-\frac{b T^{3}}{6}\right. \\ & \left.-\frac{c T^{4}}{12}-\frac{a t_{1}^{2}}{2}-\frac{b t_{1}^{3}}{3}-\frac{c t_{1}^{4}}{4}\right] \\ & +\frac{C_{h}}{T}\left[\frac{a \alpha t_{1}^{2}}{2}+\frac{\alpha t_{1}^{3}}{6}(a \theta+2 b)\right. \\ & +\frac{\alpha t_{1}^{4}}{8}(b \theta+2 c)+\frac{\alpha t_{1}^{5}}{10}(c \theta) \\ & +\frac{a \beta t_{1}^{3}}{6}+\frac{\beta t_{1}^{4}}{24}(a \theta+3 b) \\ & \left.+\frac{\beta t_{1}^{5}}{30}(b \theta+3 c)+\frac{\beta t_{1}^{6}}{36}(c \theta)\right] \end{aligned}

The condition for optimality:

d(Tavg )dt1=CdT[aθt1+bθt12+cθt13]+CsT[aT+bt1T+ct12T−at1−bt12−ct13]+ChT[αat1+αt12aθ+2b2+αt13bθ+2c2+αt14cθ2+βat122+βt13aθ+3b6+βt14bθ+3c6+βt15cθ6]\begin{aligned} \frac{d\left(T_{\text {avg }}\right)}{d t_{1}}= & \frac{C_{d}}{T}\left[a \theta t_{1}+b \theta t_{1}^{2}+c \theta t_{1}^{3}\right] \\ & +\frac{C_{s}}{T}\left[a T+b t_{1} T+c t_{1}^{2} T-a t_{1}-b t_{1}^{2}-c t_{1}^{3}\right] \\ & +\frac{C_{h}}{T}\left[\alpha a t_{1}+\alpha t_{1}^{2} \frac{a \theta+2 b}{2}+\alpha t_{1}^{3} \frac{b \theta+2 c}{2}\right. \\ & +\alpha t_{1}^{4} \frac{c \theta}{2}+\frac{\beta a t_{1}^{2}}{2}+\beta t_{1}^{3} \frac{a \theta+3 b}{6} \\ & \left.+\beta t_{1}^{4} \frac{b \theta+3 c}{6}+\beta t_{1}^{5} \frac{c \theta}{6}\right] \end{aligned}

d2Tavg dt12=CdT[aθ+2bθt1+3cθt12]+CsT[bT+2ct1T−a−2bt1−3ct12]+ChT(αa+αt1(aθ+2b)+3αt12bθ+2c2+2αt13cθ+αβt1+βt12aθ+3b2+2βt14bθ+3c3+5βt13cθ6d2Tavg dt12>0\begin{aligned} \frac{d^{2} T_{\text {avg }}}{d t_{1}^{2}} & =\frac{C_{d}}{T}\left[a \theta+2 b \theta t_{1}+3 c \theta t_{1}^{2}\right] \\ & +\frac{C_{s}}{T}\left[b T+2 c t_{1} T-a-2 b t_{1}-3 c t_{1}^{2}\right] \\ & +\frac{C_{h}}{T}\left(\alpha a+\alpha t_{1}(a \theta+2 b)+3 \alpha t_{1}^{2} \frac{b \theta+2 c}{2}\right. \\ & +2 \alpha t_{1}^{3} c \theta+\alpha \beta t_{1}+\beta t_{1}^{2} \frac{a \theta+3 b}{2} \\ & +2 \beta t_{1}^{4} \frac{b \theta+3 c}{3}+5 \beta t_{1}^{3} \frac{c \theta}{6} \\ \frac{d^{2} T_{\text {avg }}}{d t_{1}^{2}} & >0 \end{aligned}

When d(Tavg )dt1=0\frac{d\left(T_{\text {avg }}\right)}{d t_{1}}=0 \quad gives the minimum point.

Fuzzy Model

Suppose Cd=(Cd1,Cd2,Cd3,Ch4)C_{d}=\left(C_{d_{1}}, C_{d_{2}}, C_{d_{3}}, C_{h_{4}}\right),
Cˉs=(Cˉs1,Cˉs2,Cˉs3,Cˉs4),Cˉh=(Cˉh1,Cˉh2,Cˉh3,Cˉh4)\bar{C}_{s}=\left(\bar{C}_{s_{1}}, \bar{C}_{s_{2}}, \bar{C}_{s_{3}}, \bar{C}_{s_{4}}\right), \bar{C}_{h}=\left(\bar{C}_{h_{1}}, \bar{C}_{h_{2}}, \bar{C}_{h_{3}}, \bar{C}_{h_{4}}\right),
θˉ=(θˉ1,θˉ2,θˉ3,θˉ4)\bar{\theta}=\left(\bar{\theta}_{1}, \bar{\theta}_{2}, \bar{\theta}_{3}, \bar{\theta}_{4}\right) are non-negative trapezoidal fuzzy numbers. From the equation (4.15) can be rewritten as

Tavg c=Cˉd⊗(θˉ⊗P)⊕Cˉs⊗(Q⊕R)⊕Cˉh⊗((θˉ⊗L)⊕M)T_{\text {avg }}^{c}=\bar{C}_{d} \otimes(\bar{\theta} \otimes P) \oplus \bar{C}_{s} \otimes(Q \oplus R) \oplus \bar{C}_{h} \otimes((\bar{\theta} \otimes L) \oplus M)

where

P=[at122+bt133+ct144]Q=[at1T+bt12T2+ct13T3−aT22−bT36−cT412]R=[−at122−bt133−ct144]L=[aαt136+bαt148+cαt1510+aβt1424+bβt1530+cβt1636]M=[aαt122+2bαt136+2cαt148+aβt136+3bβt1424+3cβt1530]Tavg c=P⊗(θˉ⊗Cˉd)⊕(Q⊗Cˉs)⊕(R⊗Cˉs)⊕L⊗(θˉ⊗Cˉh)⊕(M⊗Cˉh)Tavg c(t)=(Tavg 1(t1),Tavg 2(t2),Tavg 3(t3),Tavg 4(t4))Tavg 1(t1)=Pθˉ1Cd1c+QCs1c+RCs1c+Lθˉ1Ch1c+MCh1cTavg 1(t2)=Pθˉ2Cd2c+QCs2c+RCs2c+Lθˉ2Ch2c+MCh2cTavg 1(t3)=Pθˉ3Cd3c+QCs3c+RCs3c+Lθˉ3Ch3c+MCh3cTavg 1(t4)=Pθˉ4Cd4c+QCs4c+RCs4c+Lθˉ4Ch4c+MCh4c\begin{aligned} & P=\left[\frac{a t_{1}^{2}}{2}+\frac{b t_{1}^{3}}{3}+\frac{c t_{1}^{4}}{4}\right] \\ & Q=\left[a t_{1} T+\frac{b t_{1}^{2} T}{2}+\frac{c t_{1}^{3} T}{3}-\frac{a T^{2}}{2}-\frac{b T^{3}}{6}-\frac{c T^{4}}{12}\right] \\ & R=\left[-\frac{a t_{1}^{2}}{2}-\frac{b t_{1}^{3}}{3}-\frac{c t_{1}^{4}}{4}\right] \\ & L=\left[\frac{a \alpha t_{1}^{3}}{6}+\frac{b \alpha t_{1}^{4}}{8}+\frac{c \alpha t_{1}^{5}}{10}+\frac{a \beta t_{1}^{4}}{24}+\frac{b \beta t_{1}^{5}}{30}+\frac{c \beta t_{1}^{6}}{36}\right] \\ & M=\left[\frac{a \alpha t_{1}^{2}}{2}+\frac{2 b \alpha t_{1}^{3}}{6}+\frac{2 c \alpha t_{1}^{4}}{8}+\frac{a \beta t_{1}^{3}}{6}+\frac{3 b \beta t_{1}^{4}}{24}+\frac{3 c \beta t_{1}^{5}}{30}\right] \\ & T_{\text {avg }}^{c}=P \otimes\left(\bar{\theta} \otimes \bar{C}_{d}\right) \oplus\left(Q \otimes \bar{C}_{s}\right) \oplus\left(R \otimes \bar{C}_{s}\right) \oplus L \otimes\left(\bar{\theta} \otimes \bar{C}_{h}\right) \\ & \oplus\left(M \otimes \bar{C}_{h}\right) \\ & T_{\text {avg }}^{c}(t)=\left(T_{\text {avg } 1}\left(t_{1}\right), T_{\text {avg } 2}\left(t_{2}\right), T_{\text {avg } 3}\left(t_{3}\right), T_{\text {avg } 4}\left(t_{4}\right)\right) \\ & T_{\text {avg } 1}\left(t_{1}\right)=P \bar{\theta}_{1} C_{d_{1}}^{c}+Q C_{s_{1}}^{c}+R C_{s_{1}}^{c}+L \bar{\theta}_{1} C_{h_{1}}^{c}+M C_{h_{1}}^{c} \\ & T_{\text {avg } 1}\left(t_{2}\right)=P \bar{\theta}_{2} C_{d_{2}}^{c}+Q C_{s_{2}}^{c}+R C_{s_{2}}^{c}+L \bar{\theta}_{2} C_{h_{2}}^{c}+M C_{h_{2}}^{c} \\ & T_{\text {avg } 1}\left(t_{3}\right)=P \bar{\theta}_{3} C_{d_{3}}^{c}+Q C_{s_{3}}^{c}+R C_{s_{3}}^{c}+L \bar{\theta}_{3} C_{h_{3}}^{c}+M C_{h_{3}}^{c} \\ & T_{\text {avg } 1}\left(t_{4}\right)=P \bar{\theta}_{4} C_{d_{4}}^{c}+Q C_{s_{4}}^{c}+R C_{s_{4}}^{c}+L \bar{\theta}_{4} C_{h_{4}}^{c}+M C_{h_{4}}^{c} \end{aligned}

Defuzzifying the total inventory cost (Tavg (t))\left(T_{\text {avg }}(t)\right) by GMRI method

Tavg (t)=P6(θˉ1Cd1c+2θˉ2Cd2c+θˉ3Cd3c+θˉ4Cd4c)+Q6(Cˉs1+2Cˉs2+2Cˉs3+Cˉs4)+R6(Cˉs1c+2Cˉs2+2Cˉs3c+Cˉs4c)+L6(θˉ1Ch1c+2θˉ2Ch2c+θˉ3Ch3c+θˉ4Ch4c)+M6(Ch1c+2Ch2c+2Ch3c+Cˉh4c)\begin{aligned} & T_{\text {avg }}(t)=\frac{P}{6}\left(\bar{\theta}_{1} C_{d_{1}}^{c}+2 \bar{\theta}_{2} C_{d_{2}}^{c}+\bar{\theta}_{3} C_{d_{3}}^{c}+\bar{\theta}_{4} C_{d_{4}}^{c}\right) \\ + & \frac{Q}{6}\left(\bar{C}_{s_{1}}+2 \bar{C}_{s_{2}}+2 \bar{C}_{s_{3}}+\bar{C}_{s_{4}}\right) \\ + & \frac{R}{6}\left(\bar{C}_{s_{1}}^{c}+2 \bar{C}_{s_{2}}+2 \bar{C}_{s_{3}}^{c}+\bar{C}_{s_{4}}^{c}\right) \\ + & \frac{L}{6}\left(\bar{\theta}_{1} C_{h_{1}}^{c}+2 \bar{\theta}_{2} C_{h_{2}}^{c}+\bar{\theta}_{3} C_{h_{3}}^{c}+\bar{\theta}_{4} C_{h_{4}}^{c}\right) \\ + & \frac{M}{6}\left(C_{h_{1}}^{c}+2 C_{h_{2}}^{c}+2 C_{h_{3}}^{c}+\bar{C}_{h_{4}}^{c}\right) \end{aligned}

To get the optimal value of t1t_{1} can be obtained by solve the

following equation, from this t1t_{1} can obtain minimized total cost.

d(Tavg (t))dt1=P′6(θˉ1Cd1′+2θˉ2Cd2′+θˉ3Cd3′+θˉ4Cd4′)+Q′6(Cˉs1+2Cs2′+2Cs3′+Cs4′)+R′6(Cs1′+2Cs2′+2Cs3′+Cs4′)+L′6(θˉ1Ch1′+2θˉ2Ch2′+θˉ3Ch3′+θˉ4Ch4′)+M′6(Ch1′+2Ch2′+2Ch3′+Ch4′)=0d2Tavg (t)dt12=P′′6(θˉ1Cd1′+2θˉ2Cd2′+θˉ3Cd3′+θˉ4Cd4′)+Q′′6(Cs1′+2Cs2′+2Cs3′+Cs4′)+R′6(Cs1+2Cs2′+2Cs3′+Cs4′)+L′′6(θˉ1Ch1′+2θˉ2Ch2′+θˉ3Ch3′+θˉ4Ch4′)+M′′6(Ch1′+2Ch2′+2Ch3′+Ch4′)>0\begin{aligned} & \frac{d\left(T_{\text {avg }}(t)\right)}{d t_{1}}=\frac{P^{\prime}}{6}\left(\bar{\theta}_{1} C_{d_{1}}^{\prime}+2 \bar{\theta}_{2} C_{d_{2}}^{\prime}+\bar{\theta}_{3} C_{d_{3}}^{\prime}+\bar{\theta}_{4} C_{d_{4}}^{\prime}\right) \\ + & \frac{Q^{\prime}}{6}\left(\bar{C}_{s_{1}}+2 C_{s_{2}}^{\prime}+2 C_{s_{3}}^{\prime}+C_{s_{4}}^{\prime}\right) \\ + & \frac{R^{\prime}}{6}\left(C_{s_{1}}^{\prime}+2 C_{s_{2}}^{\prime}+2 C_{s_{3}}^{\prime}+C_{s_{4}}^{\prime}\right) \\ + & \frac{L^{\prime}}{6}\left(\bar{\theta}_{1} C_{h_{1}}^{\prime}+2 \bar{\theta}_{2} C_{h_{2}}^{\prime}+\bar{\theta}_{3} C_{h_{3}}^{\prime}+\bar{\theta}_{4} C_{h_{4}}^{\prime}\right) \\ + & \frac{M^{\prime}}{6}\left(C_{h_{1}}^{\prime}+2 C_{h_{2}}^{\prime}+2 C_{h_{3}}^{\prime}+C_{h_{4}}^{\prime}\right)=0 \\ & \frac{d^{2} T_{\text {avg }}(t)}{d t_{1}^{2}}=\frac{P^{\prime \prime}}{6}\left(\bar{\theta}_{1} C_{d_{1}}^{\prime}+2 \bar{\theta}_{2} C_{d_{2}}^{\prime}+\bar{\theta}_{3} C_{d_{3}}^{\prime}+\bar{\theta}_{4} C_{d_{4}}^{\prime}\right) \\ + & \frac{Q^{\prime \prime}}{6}\left(C_{s_{1}}^{\prime}+2 C_{s_{2}}^{\prime}+2 C_{s_{3}}^{\prime}+C_{s_{4}}^{\prime}\right) \\ + & \frac{R^{\prime}}{6}\left(C_{s_{1}}+2 C_{s_{2}}^{\prime}+2 C_{s_{3}}^{\prime}+C_{s_{4}}^{\prime}\right) \\ + & \frac{L^{\prime \prime}}{6}\left(\bar{\theta}_{1} C_{h_{1}}^{\prime}+2 \bar{\theta}_{2} C_{h_{2}}^{\prime}+\bar{\theta}_{3} C_{h_{3}}^{\prime}+\bar{\theta}_{4} C_{h_{4}}^{\prime}\right) \\ + & \frac{M^{\prime \prime}}{6}\left(C_{h_{1}}^{\prime}+2 C_{h_{2}}^{\prime}+2 C_{h_{3}}^{\prime}+C_{h_{4}}^{\prime}\right)>0 \end{aligned}

Optimal initial inventory after fulfilling backorder Sˉ\bar{S} denoted by Sˉ∗\bar{S}^{*} by GMRI method is

Sˉ∗=Sˉ1∗+2Sˉ2∗+2Sˉ3∗+Sˉ4∗6\bar{S}^{*}=\frac{\bar{S}_{1}^{*}+2 \bar{S}_{2}^{*}+2 \bar{S}_{3}^{*}+\bar{S}_{4}^{*}}{6}

where

Sˉ1∗=[at1+(aθ1+b)t122+(bθ1+c)t133+cθ1t144]Sˉ2∗=[at2+(aθ2+b)t222+(bθ2+c)t233+cθ2t244]Sˉ3∗=[at3+(aθ3+b)t322+(bθ3+c)t333+cθ3t344]Sˉ4∗=[at4+(aθ4+b)t422+(bθ4+c)t433+cθ4t444]\begin{aligned} & \bar{S}_{1}^{*}=\left[a t_{1}+\left(a \theta_{1}+b\right) \frac{t_{1}^{2}}{2}+\left(b \theta_{1}+c\right) \frac{t_{1}^{3}}{3}+c \theta_{1} \frac{t_{1}^{4}}{4}\right] \\ & \bar{S}_{2}^{*}=\left[a t_{2}+\left(a \theta_{2}+b\right) \frac{t_{2}^{2}}{2}+\left(b \theta_{2}+c\right) \frac{t_{2}^{3}}{3}+c \theta_{2} \frac{t_{2}^{4}}{4}\right] \\ & \bar{S}_{3}^{*}=\left[a t_{3}+\left(a \theta_{3}+b\right) \frac{t_{3}^{2}}{2}+\left(b \theta_{3}+c\right) \frac{t_{3}^{3}}{3}+c \theta_{3} \frac{t_{3}^{4}}{4}\right] \\ & \bar{S}_{4}^{*}=\left[a t_{4}+\left(a \theta_{4}+b\right) \frac{t_{4}^{2}}{2}+\left(b \theta_{4}+c\right) \frac{t_{4}^{3}}{3}+c \theta_{4} \frac{t_{4}^{4}}{4}\right] \end{aligned}

Optimal quantity of the unit deteriorated units Dˉ\bar{D} denoted by Dˉ∗\bar{D}^{*} by GMRI method is

Dˉ∗=Dˉ1∗+2Dˉ2∗+2Dˉ3∗+Dˉ4∗6\bar{D}^{*}=\frac{\bar{D}_{1}^{*}+2 \bar{D}_{2}^{*}+2 \bar{D}_{3}^{*}+\bar{D}_{4}^{*}}{6}

where

Dˉ1∗=[aθ1t122+bθ1t133+cθ1t144]Dˉ2∗=[aθ2t222+bθ2t233+cθ2t244]Dˉ3∗=[aθ3t322+bθ3t333+cθ3t344]Dˉ4∗=[aθ4t422+bθ4t433+cθ4t444]\begin{aligned} & \bar{D}_{1}^{*}=\left[\frac{a \theta_{1} t_{1}^{2}}{2}+\frac{b \theta_{1} t_{1}^{3}}{3}+\frac{c \theta_{1} t_{1}^{4}}{4}\right] \\ & \bar{D}_{2}^{*}=\left[\frac{a \theta_{2} t_{2}^{2}}{2}+\frac{b \theta_{2} t_{2}^{3}}{3}+\frac{c \theta_{2} t_{2}^{4}}{4}\right] \\ & \bar{D}_{3}^{*}=\left[\frac{a \theta_{3} t_{3}^{2}}{2}+\frac{b \theta_{3} t_{3}^{3}}{3}+\frac{c \theta_{3} t_{3}^{4}}{4}\right] \\ & \bar{D}_{4}^{*}=\left[\frac{a \theta_{4} t_{4}^{2}}{2}+\frac{b \theta_{4} t_{4}^{3}}{3}+\frac{c \theta_{4} t_{4}^{4}}{4}\right] \end{aligned}

Optimal value of the total cost Tavg (t)T_{\text {avg }}(t) denoted by Tavg (t)∗T_{\text {avg }}(t)^{*} by GMRI method is

Tavg (t)∗=A+16(Pθˉ1Cd1′+QCs1′+RCs1′+Lθˉ1Ch1′+MCh1′)+13(Pθˉ2Cd2′+QCs2′+RCs2′+Lθˉ2Ch2′+MCh2′)+13(Pθˉ3Cd3′+QCs3′+RCs3′+Lθˉ3Ch3′+MCh3′)+16(Pθˉ4Cd4′+QCs4′+RCs4′+Lθˉ4Ch4′+MCh4′)\begin{aligned} & T_{\text {avg }}(t)^{*} \\ & =A+\frac{1}{6}\left(P \bar{\theta}_{1} C_{d_{1}}^{\prime}+Q C_{s_{1}}^{\prime}+R C_{s_{1}}^{\prime}+L \bar{\theta}_{1} C_{h_{1}}^{\prime}+M C_{h_{1}}^{\prime}\right) \\ & +\frac{1}{3}\left(P \bar{\theta}_{2} C_{d_{2}}^{\prime}+Q C_{s_{2}}^{\prime}+R C_{s_{2}}^{\prime}+L \bar{\theta}_{2} C_{h_{2}}^{\prime}+M C_{h_{2}}^{\prime}\right) \\ & +\frac{1}{3}\left(P \bar{\theta}_{3} C_{d_{3}}^{\prime}+Q C_{s_{3}}^{\prime}+R C_{s_{3}}^{\prime}+L \bar{\theta}_{3} C_{h_{3}}^{\prime}+M C_{h_{3}}^{\prime}\right) \\ & +\frac{1}{6}\left(P \bar{\theta}_{4} C_{d_{4}}^{\prime}+Q C_{s_{4}}^{\prime}+R C_{s_{4}}^{\prime}+L \bar{\theta}_{4} C_{h_{4}}^{\prime}+M C_{h_{4}}^{\prime}\right) \end{aligned}

5. Analysis and Interpretation

All the cost related to inventory (holding cost, shortage cost and deterioration cost) and parameter are as a trapezoidal fuzzy number and graded mean representation integration method is used for defuzzification. MATHCAD is used for calculation.

Numerical Example

The given values are A=1000,Cd=1.5,Cs=2,Ch=α+\mathrm{A}=1000, C_{d}=1.5, C_{s}=2, C_{h}=\alpha+ βt=0.504t1=0.4,T=1,α=0.5\beta t=0.504 \quad t_{1}=0.4, T=1, \alpha=0.5 and β=0.011,θ=\beta=0.011, \theta= 0.08 are assumed.

For Crisp Model:> By substituting the above numerical values in equations (4.6), (4.9) and (4.17) results obtained as d2Tavg dt12>0\frac{d^{2} T_{\text {avg }}}{d t_{1}^{2}}>0

Sˉ∗=S=13.865D∗=D=0.236Tavg ∗=980.96\bar{S}^{*}=S=13.865 \quad D^{*}=D=0.236 \quad T_{\text {avg }}^{*}=980.96

For Fuzzy Model: -

θ=(0.077,0.079,0.081,0.083),Cd=(1.47,1.49,1.51,1.53),Cs=(1.7,1.95,2.1,2.3),Ch=(0.47,0.49,0.51,0.53),t=(0.37,0.39,0.41,0.43)T=1,α=0.5andβ=0.011\begin{aligned} & \theta=(0.077,0.079,0.081,0.083), C_{d}=(1.47,1.49,1.51,1.53), \\ & C_{s}=(1.7,1.95,2.1,2.3), C_{h}=(0.47,0.49,0.51,0.53), \\ & t=(0.37,0.39,0.41,0.43) T=1, \alpha=0.5 \mathrm{and} \beta=0.011 \end{aligned}

We have
Sˉ1∗=12.517,Sˉ2∗=13.409,Sˉ3∗=14.328,Sˉ4∗=15.272\bar{S}_{1}^{*}=12.517, \bar{S}_{2}^{*}=13.409, \bar{S}_{3}^{*}=14.328, \bar{S}_{4}^{*}=15.272
Sˉ∗=13.877\bar{S}^{*}=13.877

Dˉ1∗=0.191,Dˉ2∗=0.222,Dˉ3∗=0.256,Dˉ4∗=0.294Dˉ∗=Dˉ1∗+2Dˉ2∗+2Dˉ3∗+Dˉ4∗6=0.24\begin{aligned} & \bar{D}_{1}^{*}=0.191, \bar{D}_{2}^{*}=0.222, \bar{D}_{3}^{*}=0.256, \bar{D}_{4}^{*}=0.294 \\ & \bar{D}^{*}=\frac{\bar{D}_{1}^{*}+2 \bar{D}_{2}^{*}+2 \bar{D}_{3}^{*}+\bar{D}_{4}^{*}}{6}=0.24 \end{aligned}

Where P=2.981,R=−2.981,Q=−7.207, L=0.208,M=−3.69\mathrm{P}=2.981, \mathrm{R}=-2.981, \mathrm{Q}=-7.207, \mathrm{~L}=0.208, \mathrm{M}=-3.69

Tavg (t)∗=A+16(Pθˉ1Cd1′+QCs1′+RCs1′+Lθˉ1Ch1′+MCh1′)+13(Pθˉ2Cd2′+QCs2′+RCs2′+Lθˉ2Ch2′+MCh2′)+13(Pθˉ3Cd3′+QCs3′+RCs3′+Lθˉ3Ch3′+MCh3′)+16[Pθˉ4Cd4′+QCs4′+RCs4′+Lθˉ4Ch4′+MCh4′)=978.145\begin{aligned} & T_{\text {avg }}(t)^{*}=A+\frac{1}{6}\left(P \bar{\theta}_{1} C_{d_{1}}^{\prime}+Q C_{s_{1}}^{\prime}+R C_{s_{1}}^{\prime}+L \bar{\theta}_{1} C_{h_{1}}^{\prime}+M C_{h_{1}}^{\prime}\right) \\ + & \frac{1}{3}\left(P \bar{\theta}_{2} C_{d_{2}}^{\prime}+Q C_{s_{2}}^{\prime}+R C_{s_{2}}^{\prime}+L \bar{\theta}_{2} C_{h_{2}}^{\prime}+M C_{h_{2}}^{\prime}\right) \\ + & \frac{1}{3}\left(P \bar{\theta}_{3} C_{d_{3}}^{\prime}+Q C_{s_{3}}^{\prime}+R C_{s_{3}}^{\prime}+L \bar{\theta}_{3} C_{h_{3}}^{\prime}+M C_{h_{3}}^{\prime}\right) \\ + & \frac{1}{6}\left[P \bar{\theta}_{4} C_{d_{4}}^{\prime}+Q C_{s_{4}}^{\prime}+R C_{s_{4}}^{\prime}+L \bar{\theta}_{4} C_{h_{4}}^{\prime}+M C_{h_{4}}^{\prime}\right)=978.145 \end{aligned}

Sensitive analysis

Suppose Cˉd=(Cˉd1′,Cd2′,Cˉd3′,Cˉd4′)\bar{C}_{d}=\left(\bar{C}_{d_{1}}^{\prime}, C_{d_{2}}^{\prime}, \bar{C}_{d_{3}}^{\prime}, \bar{C}_{d_{4}}^{\prime}\right),
Cˉs=(Cˉs1,Cˉs2,Cˉs3,Cˉs4),Cˉh=(Cˉh1,Cˉh2,Cˉh3,Cˉh4′)\bar{C}_{s}=\left(\bar{C}_{s_{1}}, \bar{C}_{s_{2}}, \bar{C}_{s_{3}}, \bar{C}_{s_{4}}\right), \bar{C}_{h}=\left(\bar{C}_{h_{1}}, \bar{C}_{h_{2}}, \bar{C}_{h_{3}}, \bar{C}_{h_{4}}^{\prime}\right),
on the S, Q, TDC, TSC, THC and TIC. In the analysis one

parameter is changing by −50%,−25%,25%-50 \%,-25 \%, 25 \%, and 50%50 \% remaining parameters are constant(unchanged).

Table 1. Sensitivity analysis through CdC_{d}

TABLE 1 shows that: Percentage of changes in the TDC and TIC when changes in (Cd1,Cd2,Cd3,Cd4)\left(C_{d_{1}}, C_{d_{2}}, C_{d_{3}}, C_{d_{4}}\right)

Table 2. Sensitivity analysis through CsC_{s}

TABLE 2 shows that: Percentage of changes in the TSC and TIC when changes in (Cs1,Cs2,Cs3,Cs4)\left(C_{s_{1}}, C_{s_{2}}, C_{s_{3}}, C_{s_{4}}\right)

6. Conclusion

EOQ model for quadratic demand, time dependent holding cost with deteriorating items and allowing shortages can be summarized as follows:

1. Taking the deterioration cost, holding cost, shortage cost and parameter are considered as crisp and for a given cycle, deteriorated units, initial inventory after fulfilling back order and minimum total cost are determined.
2. Considering the deterioration rate, holding cost, deterioration cost and shortage cost are considered as trapezoidal fuzzy number. Defuzzyfied by Graded Mean Integration Representation method and for a given cycle and obtained deteriorated units, initial inventory after fulfilling back order. Total inventory cost is minimized.

References

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