# A Warehouse Imperfect Fuzzified Production Model with Shortages under Inflationary Conditions

Authors: S. R. Singh; Shalini Jain; Sarla Pareek
Journal: Advances in Decision Sciences
 (2012)

---

## Abstract

We develop a two-warehouse production model with imperfect items. Production rate is taken as the linear combination of on-hand inventory and demand, while demand rate is taken as function of time. Most of the researchers consider that the production rate is independent from the demand rate. In this paper we assume production rate as being dependent on the demand rate, and this assumption is more realistic. Shortages are allowed and partially backlogged with time-dependent backlogging rate. Due to different preservation facilities we consider that the deterioration rate is time dependent in own warehouse (OW) and Weibull distribution deterioration in rented warehouse (RW). Holding cost in RW is greater than in OW. We developed a fuzzy model with fuzzifying all the costs of the model as triangular fuzzy numbers. The present model is developed in both crisp and fuzzy senses. Finally, numerical example is shown, and sensitivity is also illustrated.


---

## Body

## 1. Introduction

One of the weaknesses of some production-inventory models is the unrealistic assumption that all items produced are of good quality. But production of defective units is a natural phenomenon in a production process. Defective items should be treated as a result of imperfect quality production. The effect of an imperfect process on production run time and EPQ was initially studied by Rosenblatt and Lee [1]. In their study, the elapsed time until the process shift was assumed to be exponentially distributed. The optimal production run was found to be shorter than that of classical EPQ model. In recent years, numerous research efforts have been undertaken to extend the work of Rosenblatt and Lee [1]. Kim and Hong [2] extended the work of Rosenblatt and Lee [1] by assuming that elapsed time until the process shift was arbitrarily distributed. However, neither of their models took into consideration of any allowable shortages. Chung and Hou [3], however, have generalized the work of Kim and Hong [2] by assuming that shortages were allowed. Nevertheless, the common assumption of all the above-mentioned models was that there were a fixed percentage of defective items produced during the out-of-control period. C. Singh and S. R. Singh [4] developed an imperfect production process with exponential demand rate, Weibull deterioration under inflation.In some cases, uncertainties can be defined as fuzziness or vagueness, which are characterized by fuzzy numbers of the fuzzy set theory. Zadeh [5] introduced fuzzy set theory to deal with quality-related problems with imprecise demand. Bellman and Zadeh [6] distinguished the difference between randomness and fuzziness by showing that the former deals with uncertainty regarding membership or nonmembership of an element in a set, while the later is concerned with the degree of uncertainty by which an element belongs to a set. Gen et al. [7] expressed the input data by fuzzy numbers, where they used interval mean value concept to solve an inventory problem. Yao and Chiang [8] considered an inventory model with total demand and storing cost as triangular fuzzy numbers. They performed the defuzzification by centroid and signed distance methods. Mondal and Maiti [9] applied genetic algorithms (GAs) to solve a multi-item fuzzy EOQ model. M. K. Maiti and M. Maiti [10] dealt with a fuzzy inventory model with two warehouses under possibility constraints. Mahapatra and Maiti [11] formulated a multi-item, multiobjective inventory model for deteriorating items with stock- and time-dependent demand rate over a finite time horizon in fuzzy stochastic environment. Halim et al. [12] developed a fuzzy inventory model for perishable items with stochastic demand, partial backlogging, and fuzzy deterioration rate. The model is further extended to consider fuzzy partial backlogging factor. Gani and Maheswari [13] discussed the retailer’s ordering policy under two levels of delay payments considering the demand and the selling price as triangular fuzzy numbers. They used graded mean integration representation method for defuzzification. Lee and Yao [14] developed an economic production quantity (EPQ) model in which the demand and the production quantity are assumed to be fuzzy. Halim et al. [15] addressed the lot sizing problem in an unreliable production system with stochastic machine breakdown and fuzzy repair time. They defuzzified the cost-per-unit time using the signed distance method. Production of defective items in any manufacturing industry is a natural phenomenon. The number of defectives may have a change from one lot to another that cannot be assessed by a crisp value. Chen and Chang [16] developed a fuzzy economic production quantity (EPQ) model with defective productions that cannot be repaired. In this model, they considered a fuzzy opportunity cost and trapezoidal fuzzy costs under crisp production quantity or fuzzy production.Since most decision makers think that the inflation does not have significant influence on the inventory policies, the effects of inflation are notconsidered in some inventory models. However, from a financial point of view, an inventory represents a capital investment and must compete with other assets for a firm’s limited capital funds. Thus, it is necessary to consider the effects of inflation on the inventory system. Buzacott [17], Bierman and Thomas [18], and Misra [19] investigated the inventory decisions under an inflationary condition for the EOQ model. Chandra and Bahner [20], Moon and Lee [21], Lo et al. [22], and Singh et al. [23, 24] are amongst the few who studied the concept of inflation with regard to inventory. Singh and Jain [25] explored a deterministic inventory model for a deteriorating item in an inflation-induced environment.In this paper, we assumed the following assumptions.(i)
Production rate is greater than the demand rate.(ii)
Production rate is the linear combination of on-hand inventory and demand rate.(iii)
Demand rate is exponentially an increasing function of time.(iv)
Model is considered for imperfect items.(v)
Deterioration is taken as time dependent for OW, while, Weibull distribution for RW.(vi)
Inflation is also taken in this model.(vii)
Costs are considered as a triangular fuzzy numbers.(viii)
Model is presented in both fuzzy and crisp senses.


## 2. Assumptions and Notations

### 2.1. Assumptions

The following assumptions have been used throughout the study.(i)
The demand rate is exponentially increasing and is represented byλ(t)=λ0eδt, where 0≤δ≤1 is a constant inflation rate, and λ0 is the initial demand rate.(ii)
Lead time is zero, and no replenishment or repair of deteriorated items is made during a given cycle.(iii)
Production rate is the linear combination of on-hand inventory and demand rate at any time subject to(2.1)P(t)=[I(t)+ηλ(t)](1-e-Δt),0≤Δ≤1,η>1,P(t)>λ(t).(iv)
Shortages are allowed and the backlogging rate is exponentially, a decreasing function of time when inventory is in shortages, the backlogging parameters is a positive constant s.t. 0<s<1.(v)
A single item is considered over the prescribed periodT units of time, which is subject to variable deterioration rate.(vi)
The owned warehouse (OW) has a fixed capacity ofW units, and the rented warehouse (RW) has unlimited capacity.(vii)
The goods of OW are consumed only after consuming the goods kept in RW.(viii)
The unit inventory costs (including holding cost) per unit time in RW are higher than those in OW.(ix)
Deterioration-rate of the items is considered to be different in different warehouses. In OW time-dependent deterioration rate isθ(t)=θt, while in RW Weibull distribution deterioration rate is αβtβ-1, where α,β>0, t>0.(x)
The model is developed in both fuzzy and crisp environment.(xi)
Finite planning horizon is considered.


### 2.2. Notations

The following notations have been used throughout the study.λ
: Demand rate,λ(t)=λ0eδt, increases with time, where λ0 is the initial demand rate and 0≤δ≤1.P(t): Production rate P(t)=[I(t)+ηλ(t)](1-e-Δt), which is the linear combination of on-hand inventory and demand rate at any time and P(t)>λ(t), where 0≤Δ≤1, η>1.B: Backlogging rate, B=e-st, where the backlogging parameter s is a positive constant s.t. 0<s<1.θ: Deterioration rate, where θ=θt for OW and θ=αβtβ-1 for RW, where α,β>0, t>0.Δ: Rate of imperfect production.r: Inflation rate.W: Fixed capacity level of OW.C1: Set up cost per production run.CRW: Carrying cost per inventory unit held in RW per unit time.COW: Carrying cost per inventory unit held in OW per unit time.C2: Deterioration cost per unit time.C3: Shortage cost for backlogged items.C4: Lost sale cost per unit time.C5: Rework cost per unit time.Ii1: Inventory level in OW at time t with t∈[0,t1].Ii2: Inventory level in RW at time t with t∈[t1,t2].Ii3: Inventory level in RW at time t with t∈[0,t3].Ii4: Inventory level in OW at time t with t∈[0,t4].Ii5: Inventory level in OW at time t with t∈[0,t2+t3].Ii6: Inventory level in OW at time t with t∈[0,t5].Ii7: Inventory level in OW at time t with t∈[0,T].C~1: Fuzzy set up cost per unit time.C~2: Fuzzy deterioration cost per unit time.C~3: Fuzzy shortage cost per unit time.C~4: Fuzzy lost sale cost per unit time.C~5: Fuzzy rework cost per unit time.C~OW: Fuzzy carrying cost per unit time FOR OW.C~RW: Fuzzy carrying cost per unit time for RW.t1,t2: The production periods for OW and RW.t3,t4: The non production periods.t5: The shortage period.T: Total cycle time.


## 2.1. Assumptions

The following assumptions have been used throughout the study.(i)
The demand rate is exponentially increasing and is represented byλ(t)=λ0eδt, where 0≤δ≤1 is a constant inflation rate, and λ0 is the initial demand rate.(ii)
Lead time is zero, and no replenishment or repair of deteriorated items is made during a given cycle.(iii)
Production rate is the linear combination of on-hand inventory and demand rate at any time subject to(2.1)P(t)=[I(t)+ηλ(t)](1-e-Δt),0≤Δ≤1,η>1,P(t)>λ(t).(iv)
Shortages are allowed and the backlogging rate is exponentially, a decreasing function of time when inventory is in shortages, the backlogging parameters is a positive constant s.t. 0<s<1.(v)
A single item is considered over the prescribed periodT units of time, which is subject to variable deterioration rate.(vi)
The owned warehouse (OW) has a fixed capacity ofW units, and the rented warehouse (RW) has unlimited capacity.(vii)
The goods of OW are consumed only after consuming the goods kept in RW.(viii)
The unit inventory costs (including holding cost) per unit time in RW are higher than those in OW.(ix)
Deterioration-rate of the items is considered to be different in different warehouses. In OW time-dependent deterioration rate isθ(t)=θt, while in RW Weibull distribution deterioration rate is αβtβ-1, where α,β>0, t>0.(x)
The model is developed in both fuzzy and crisp environment.(xi)
Finite planning horizon is considered.


## 2.2. Notations

The following notations have been used throughout the study.λ
: Demand rate,λ(t)=λ0eδt, increases with time, where λ0 is the initial demand rate and 0≤δ≤1.P(t): Production rate P(t)=[I(t)+ηλ(t)](1-e-Δt), which is the linear combination of on-hand inventory and demand rate at any time and P(t)>λ(t), where 0≤Δ≤1, η>1.B: Backlogging rate, B=e-st, where the backlogging parameter s is a positive constant s.t. 0<s<1.θ: Deterioration rate, where θ=θt for OW and θ=αβtβ-1 for RW, where α,β>0, t>0.Δ: Rate of imperfect production.r: Inflation rate.W: Fixed capacity level of OW.C1: Set up cost per production run.CRW: Carrying cost per inventory unit held in RW per unit time.COW: Carrying cost per inventory unit held in OW per unit time.C2: Deterioration cost per unit time.C3: Shortage cost for backlogged items.C4: Lost sale cost per unit time.C5: Rework cost per unit time.Ii1: Inventory level in OW at time t with t∈[0,t1].Ii2: Inventory level in RW at time t with t∈[t1,t2].Ii3: Inventory level in RW at time t with t∈[0,t3].Ii4: Inventory level in OW at time t with t∈[0,t4].Ii5: Inventory level in OW at time t with t∈[0,t2+t3].Ii6: Inventory level in OW at time t with t∈[0,t5].Ii7: Inventory level in OW at time t with t∈[0,T].C~1: Fuzzy set up cost per unit time.C~2: Fuzzy deterioration cost per unit time.C~3: Fuzzy shortage cost per unit time.C~4: Fuzzy lost sale cost per unit time.C~5: Fuzzy rework cost per unit time.C~OW: Fuzzy carrying cost per unit time FOR OW.C~RW: Fuzzy carrying cost per unit time for RW.t1,t2: The production periods for OW and RW.t3,t4: The non production periods.t5: The shortage period.T: Total cycle time.


## 3. Formulation and Solution of the Model

Figure1 shows the inventory level during a production cycle in which both OW and RW are used. Initially, the inventory level is zero. The production starts at time t=0, and items accumulate from 0 up to W units in OW and in t1 units of time. After time t1 any production quantity exceeding W will be stored in RW. After this production stopped and the inventory level in RW begins to decrease at t2 and will reach 0 units at t3 because of demand and deterioration. The inventory level in OW comes to decrease at t1 and then falls below W at t2+t3 due to deterioration. The remaining stocks in OW will be fully exhausted at t4 owing to demand and deterioration: the inventory becomes zero. At this time shortage starts developing, and at time t5 it reaches to maximum shortage level: at this time fresh production starts to clear the backlog by the time T.Figure 1
Inventory level in production system for deteriorating items with inflation-induced demand.The differential equations stating the inventory levels within the cycle are given as follows:(3.1)dIi1(t)dt+θIi1(t)=P(t)-λ(t),0≤t≤t1.(3.2)dIi2(t)dt+θIi2(t)=P(t)-λ(t),t1≤t≤t2.(3.3)dIi3(t)dt+θIi3(t)=-λ(t),t2≤t≤t3.(3.4)dIi4(t)dt+θIi4(t)=-λ(t),t3≤t≤t4.(3.5)dIi5(t)dt+θIi5(t)=0,t1≤t≤t3.(3.6)dIi6(t)dt=-Bλ(t),t4≤t≤t5.(3.7)dIi7(t)dt=P(t)-λ(t),t5≤t≤T.With the boundary conditionsIi1(0)=0,Ii2(t1)=0,Ii3(t3)=0,Ii4(t4)=0,Ii5(t1)=W,Ii6(t4)=0,andIi7(T)=0,  respectively, the above equations can be solved successively as follows:
(3.8)Ii1(t)=(η-1)λ0[Δt22+δΔt33-(θ-Δ)Δt44]-λ0[t+δt22-(θ-Δ)t32],(3.9)Ii2(t)=(η-1)λ0[Δ2(t2-t12)-Δαtβ2(t2-t12)+δΔ3(t3-t13)]-λ0[(t-t1)-αtβ(t-t1)+δ2(t2-t12)-δαtβ2(t2-t12)],(3.10)Ii3(t)=λ0[(t3-t)-αtβ(t3-t)+δ2(t32-t2)-δαtβ2(t32-t2)],(3.11)Ii4(t)=λ0[(t4-t)-θt22(t4-t)+δ2(t42-t2)-δθt24(t42-t2)],(3.12)Ii5(t)=We(-θ/2)(t12-t2),(3.13)Ii6(t)=λ0[(t4-t)+(δ-s2)(t42-t2)-δs3(t43-t3)],(3.14)I7(t)=(η-1)λ0[(t-T)+Δt22(t-T)+δ2(t2-T2)+δΔt24(t2-T2)]-ηλ0[(T-t)+Δt22(T-t)+(δ-Δ)2(T2-t2)].Therefore, we find out some costs such as the following.


### 3.1. Present Worth Ordering Cost

Consider the following:(3.15)Orderingcostpercycle=C1e-rt5.


### 3.2. Present Worth Inventory Level in RW

The total inventory level in RW can be derived as(3.16)IRW=∫t1t2I2(t)e-rtdt+∫t2t3I3(t)e-rtdt=λ0[(η-1)(Δ2(t233+2t133)-Δr3(t244+t142)+Δδ3(t244+3t144)-rΔδ3(t255+3t1510))]-[(t122+t322)-r(t136+t336)+δ2(2t133+2t333)-δr2(t144+t342)].


### 3.3. Present Worth Inventory Level in OW

The total inventory level in OW can be derived as(3.17)IOW=∫0t1I1(t)e-rtdt+∫t1t3I5(t)e-rtdt+∫t3t4I4(t)e-rtdt=(η-1)λ0[Δt136-rΔt148+δΔt1412]-λ0[t122-t422-t322-rt136-rt333-δt436]+W[(t3-t1)-r2(t32-t12)+θt132].


### 3.4. Present Worth Deteriorated Items

The total quantity of deteriorated items during the period(0,T) is given by(3.18)ID=∫0t1θtIi1(t)e-rtdt+∫t1t3θtIi5(t)e-rtdt+∫t3t4θtIi4(t)e-rtdt+∫t1t2αβtβ-1Ii2(t)e-rtdt+∫t2t3αβtβ-1Ii3(t)e-rtdt=[{∑∑∑∑}θλ0{(η-1)(Δt148+Δδt1515+Δrt1510)-(t133+δt148+rt144)}+Wθ{t322-t122+θt348-θt148-rt333+rt133}+θλ0{(-t336-t433)-r(-t3412-t444)+δr2(-2t3515-t455)}+λ0αβ{{(η-1)Δ(t2β+22(β+2)+t1β+22β(β+2))}-(t2β+1(β+1)+t1β+1β(β+1))}×λ0αβ{(t2β+1(β+1)+t3β+1β(β+1))-r(t2β+2(β+2)+t3β+2(β+2)(β+1))}{∑∑∑∑}].


### 3.5. Present Worth Shortage Cost

Total quantity of shortage units (IS) during the period (0,T) is given by
(3.19)IS=-∫t4t5Ii6(t)e-rtdt-∫t5TIi7(t)e-rtdt=s[λ0{(t422+t522)-r(t436+t533)-δs3(3t444+t544)}+(η-1)λ0{(T22+t522)-r(T36+t533)}+ηλ0{(T22+t522)-r(T36+t533)}].


### 3.6. Present Worth Lost Sales Quantity

The total amount of lost sales(IL) during the period (0,T) can be obtained as
(3.20)IL=∫t4t5{1-e-st}λ0eδtdt=sλ0[12(t52-t42)+δ3(t53-t43)].
Number of defective items is(3.21)N={0,X≥t2∫Xt2Pdt,X<t2},N=t2{0,X≥t2P(t2-X),X<t2}.Expected number of defective items in a production cycle is(3.22)E(N)=∫0t2P(t2-X)f(X)dX.f(X) is probability density function of X:
(3.23)f(X)=μe-μX.


### 3.7. Present Worth of Rework Cost

The total amount of rework cost (IR) during the period (0, T2) can be obtained as
(3.24)IR=E(N)e-rt2={∫0t2P(t2-X)f(X)dX}e-rt2=λ0μ[(μt277-t266)(η-1)Δ{Δ8(μt266-t255)+Δδ15(μt277-t266)}-Δ{13(μt255-t244)+δ8(μt266-t255)}+ηΔ{12(μt244-t233)+δ6(μt255-t244)}+(η-1)Δ{Δ2(t254-μt268)-Δ2α2(t2β+5β+4-μt2β+62(β+4))}-{(t243-μt256)-α(t2β+4β+3-μt2β+52(β+3))}+ηΔ(t232+δt243-μt244)](1-rt2).


### 3.8. Present Worth of Total Cost

The total average cost(TC) of the inventory system is given by
(3.25)TC=1T[C1+CRWIRW+COWIOW+C2ID+C3IS+C4IL+C5IR].


## 3.1. Present Worth Ordering Cost

Consider the following:(3.15)Orderingcostpercycle=C1e-rt5.


## 3.2. Present Worth Inventory Level in RW

The total inventory level in RW can be derived as(3.16)IRW=∫t1t2I2(t)e-rtdt+∫t2t3I3(t)e-rtdt=λ0[(η-1)(Δ2(t233+2t133)-Δr3(t244+t142)+Δδ3(t244+3t144)-rΔδ3(t255+3t1510))]-[(t122+t322)-r(t136+t336)+δ2(2t133+2t333)-δr2(t144+t342)].


## 3.3. Present Worth Inventory Level in OW

The total inventory level in OW can be derived as(3.17)IOW=∫0t1I1(t)e-rtdt+∫t1t3I5(t)e-rtdt+∫t3t4I4(t)e-rtdt=(η-1)λ0[Δt136-rΔt148+δΔt1412]-λ0[t122-t422-t322-rt136-rt333-δt436]+W[(t3-t1)-r2(t32-t12)+θt132].


## 3.4. Present Worth Deteriorated Items

The total quantity of deteriorated items during the period(0,T) is given by(3.18)ID=∫0t1θtIi1(t)e-rtdt+∫t1t3θtIi5(t)e-rtdt+∫t3t4θtIi4(t)e-rtdt+∫t1t2αβtβ-1Ii2(t)e-rtdt+∫t2t3αβtβ-1Ii3(t)e-rtdt=[{∑∑∑∑}θλ0{(η-1)(Δt148+Δδt1515+Δrt1510)-(t133+δt148+rt144)}+Wθ{t322-t122+θt348-θt148-rt333+rt133}+θλ0{(-t336-t433)-r(-t3412-t444)+δr2(-2t3515-t455)}+λ0αβ{{(η-1)Δ(t2β+22(β+2)+t1β+22β(β+2))}-(t2β+1(β+1)+t1β+1β(β+1))}×λ0αβ{(t2β+1(β+1)+t3β+1β(β+1))-r(t2β+2(β+2)+t3β+2(β+2)(β+1))}{∑∑∑∑}].


## 3.5. Present Worth Shortage Cost

Total quantity of shortage units (IS) during the period (0,T) is given by
(3.19)IS=-∫t4t5Ii6(t)e-rtdt-∫t5TIi7(t)e-rtdt=s[λ0{(t422+t522)-r(t436+t533)-δs3(3t444+t544)}+(η-1)λ0{(T22+t522)-r(T36+t533)}+ηλ0{(T22+t522)-r(T36+t533)}].


## 3.6. Present Worth Lost Sales Quantity

The total amount of lost sales(IL) during the period (0,T) can be obtained as
(3.20)IL=∫t4t5{1-e-st}λ0eδtdt=sλ0[12(t52-t42)+δ3(t53-t43)].
Number of defective items is(3.21)N={0,X≥t2∫Xt2Pdt,X<t2},N=t2{0,X≥t2P(t2-X),X<t2}.Expected number of defective items in a production cycle is(3.22)E(N)=∫0t2P(t2-X)f(X)dX.f(X) is probability density function of X:
(3.23)f(X)=μe-μX.


## 3.7. Present Worth of Rework Cost

The total amount of rework cost (IR) during the period (0, T2) can be obtained as
(3.24)IR=E(N)e-rt2={∫0t2P(t2-X)f(X)dX}e-rt2=λ0μ[(μt277-t266)(η-1)Δ{Δ8(μt266-t255)+Δδ15(μt277-t266)}-Δ{13(μt255-t244)+δ8(μt266-t255)}+ηΔ{12(μt244-t233)+δ6(μt255-t244)}+(η-1)Δ{Δ2(t254-μt268)-Δ2α2(t2β+5β+4-μt2β+62(β+4))}-{(t243-μt256)-α(t2β+4β+3-μt2β+52(β+3))}+ηΔ(t232+δt243-μt244)](1-rt2).


## 3.8. Present Worth of Total Cost

The total average cost(TC) of the inventory system is given by
(3.25)TC=1T[C1+CRWIRW+COWIOW+C2ID+C3IS+C4IL+C5IR].


## 4. Fuzzy Model

In order to develop the model in a fuzzy environment, we consider the costs as the triangular fuzzy numbers whereC~1=(C1-Δ1,C1,C1+Δ2), C~RW=(CRW-Δ3,CRW,CRW+Δ4),C~OW=(COW-Δ5,COW,COW+Δ6),  C~2=(C2-Δ7,C2,C2+Δ8),  C~3=(C3-Δ9,C3,C3+Δ10), C~4=(C4-Δ11,C4,C4+Δ12), and C~5=(C5-Δ13,C5,C5+Δ14) such that 0<Δ1<C1, 0<Δ2, 0<Δ3<CRW, 0<Δ4, 0<Δ5<COW, 0<Δ6, 0<Δ7<C2, 0<Δ8, 0<Δ9<C3, 0<Δ10, 0<Δ11<C4, 0<Δ12, 0<Δ13<C5, 0<Δ14, and Δ1, Δ2, Δ3, Δ4, Δ5, Δ6, Δ7, Δ8, Δ9, Δ10, Δ11, Δ12, Δ13, and Δ14 are determined by the decision maker based on the uncertainty of the problem. Thus, the costs are considered as the fuzzy numbers with membership function:
(4.1)TC=1T[C~1+C~RWIRW+C~OWIOW+C~2ID+C~3IS+C~4IL+C~5IR],C~1=(C1-Δ1,C1,C1+Δ2),C~RW=(CRW-Δ3,CRW,CRW+Δ4),C~OW=(COW-Δ5,COW,COW+Δ6),C~2=(C2-Δ7,C2,C2+Δ8),C~3=(C3-Δ9,C3,C3+Δ10),C~4=(C4-Δ11,C4,C4+Δ12),C~5=(C5-Δ13,C5,C5+Δ14).By Centroid Method consider(4.2)C~1=C1+13(Δ2-Δ1)C~RW=CRW+13(Δ4-Δ3)C~OW=COW+13(Δ6-Δ5)C~2=C2+13(Δ8-Δ7)C~3=C3+13(Δ10-Δ9)C~4=C4+13(Δ12-Δ11)C~5=C5+13(Δ14-Δ13),F1=1T[(C1+Δ2)+(CRW+Δ4)IRW+(COW+Δ6)IOW+(C2+Δ8)ID+(C3+Δ10)IS+(C4+Δ12)IL+(C5+Δ14)IR]F2=1T[C1+CRWIRW+COWIOW+C2ID+C3IS+C4IL+C5IR],F3=1T[(C1-Δ1)+(CRW-Δ3)IRW+(COW-Δ5)IOW+(C2-Δ7)ID+(C3-Δ9)IS+(C4-Δ11)IL+(C5-Δ13)IR].By defuzzification, we get(4.3)=1T[C1+14(Δ2-Δ1)+CRWIRW+14(Δ4-Δ3)+COWIOW+14(Δ6-Δ5)+C2ID+14(Δ8-Δ7)+C3IS+14(Δ10-Δ9)+C4IL+14(Δ12-Δ11)+C5IR+14(Δ14-Δ13)].


## 5. Numerical Example

To illustrate the theory of the model we consider the following data on the basis of the previous study.(5.1)δ=2,CRW=0.6,W=50,θ=0.03,C1=10,COW=1.2,η=5,C2=2,α=0.06,β=0.01,C3=5,μ=0.005,C4=6,C5=0.1,λ0=20,T=10
in appropriate units. Based on these input data, the findings are as follows So the optimum values of
(5.2)t1*=1.42655,t2*=2.08592,t3*=4.17695,t4*=5.72968,t5*=8.38866,TC=4007.4.


## 6. Fuzzy Numerical

(6.1)
C
~
1
=
(
9,10,11
)
,
C
~
OW
=
(
1.1,1.2,1.3
)
,
η
=
5
,
λ
0
=
20
,
δ
=
2
,
C
~
RW
=
(
0.5,0.6,0.7
)
,
W
=
50
,
θ
=
0.03
,
C
~
2
=
(
1.5,2
,
2.5
)
,
α
=
0.06
,
β
=
0.01
,
C
~
3
=
(
4,5
,
6
)
,
μ
=
0.005
,
C
~
4
=
(
5.5,6
,
6.5
)
,
C
~
5
=
(
0.09,0.1,0.2
)
,
T
=
10
(6.2)
t
1
*
=
(
1.41419,1.42655,1.44654
)
,
t
2
*
=
(
2.08595,2.08592,2.06724
)
,
t
3
*
=
(
4.16689,4.17695,4.23937
)
,
t
4
*
=
(
5.84037,5.72968,5.69694
)
,
t
5
*
=
(
8.90087,8.38666,8.08675
)
,
TC
=
(
3425.62,4007.4,4628.42
)
.


## 7. Sensitivity Analysis

See Table1.Table 1
Parameter
Change in parameter
t
1
*
t
2
*
t
3
*
t
4
*
t
5
*
TC
C
1
12
1.42651
2.08592
4.17679
5.72944
8.38843
4007.35
14
1.42648
2.08592
4.17664
5.7292
8.3882
4007.3
16
1.42644
2.08592
4.17648
5.72897
8.38797
4007.25
18
1.4264
2.08592
4.17633
5.72873
8.38774
4007.2
20
1.42637
2.08592
4.17617
5.7285
8.3875
4007.15
α
0.07
1.42891
2.08533
4.17599
5.72978
8.38871
4007.42
0.08
1.43164
2.08474
4.17505
5.72988
8.38877
4007.43
0.09
1.43472
2.08414
4.17413
5.72998
8.38833
4007.44
0.1
1.43816
2.08355
4.17321
5.73009
8.38889
4007.45
0.2
1.49168
2.07793
4.16471
5.73122
8.38951
4007.46
β
0.02
1.428
2.08244
4.17683
5.72969
8.38866
4007.41
0.03
1.42945
2.07897
4.17672
5.7297
8.38867
4007.42
0.04
1.4309
2.07552
4.17662
5.72971
8.38867
4007.43
0.05
1.43236
2.07209
4.1765
5.72972
8.38868
4007.44
0.06
1.43381
2.06868
4.17639
5.72973
8.38869
4007.45
λ
0
22
1.34214
2.08549
4.17764
5.72985
8.3888
4407.61
24
1.27294
2.08518
4.1764
5.7298
8.38881
4807.68
26
1.21496
2.08495
4.1742
5.72965
8.38876
5207.94
28
1.16551
2.08476
4.17155
5.72945
8.38868
5608.08
30
1.12271
2.08462
4.16873
5.72922
8.38857
6008.2
δ
2.1
1.44449
2.08501
4.22808
5.81669
8.46614
4048.1
2.2
1.46366
2.08411
4.27917
5.90235
8.5434
4089.6
2.3
1.48418
2.08325
4.33018
5.98681
8.62043
4131.9
2.4
1.50622
2.08242
4.38107
6.07019
8.69721
4175
2.5
1.52996
2.08161
4.43178
6.15258
8.77374
4218.9
Δ
0.32
1.34242
1.91621
4.18507
5.73049
8.38911
4007.8
0.34
1.27138
1.7726
4.19123
5.73111
8.38945
4008.13
0.36
1.2103
1.65176
4.19603
5.7316
8.38972
4008.4
0.38
1.15704
1.54628
4.19987
5.73199
8.38994
4008.63
0.4
1.11006
1.45395
4.20299
5.7323
8.39011
4008.83
C
OW
1.4
1.24759
2.08895
4.08125
5.72016
8.3834
4004.78
1.6
1.12142
2.09152
3.99301
5.70965
8.37761
4002.17
1.8
1.02623
2.09365
3.85523
5.69852
8.37149
3999.63
2.0
0.951127
2.09543
3.72869
5.68693
8.36512
3997.18
2.2
0.889949
2.09691
3.59326
5.67493
8.35856
3994.86
C
RW
0.7
1.56693
2.08682
4.31716
5.75917
8.40501
4013.99
0.8
1.70794
2.08788
4.4499
5.78821
8.42119
4020.78
0.9
1.85215
2.08912
4.57497
5.81672
8.43716
4027.78
1.0
2.00219
2.09058
4.69176
5.84457
8.45284
4034.98
1.1
2.16108
2.0923
4.79913
5.87156
8.46812
4042.38
C
3
5.2
1.42214
2.08589
4.15801
5.70089
8.25883
4101
5.4
1.4188
2.08586
4.14345
5.67877
8.14299
4198.27
5.6
1.41632
2.08584
4.13251
5.66213
8.03918
4298.68
5.8
1.41454
2.08583
4.12454
5.65002
7.94579
4401.77
6.0
1.41331
2.08582
4.11904
5.64165
7.86147
4507.17
C
4
6.1
1.42907
2.08594
4.18761
5.74588
8.44763
4036.48
6.2
1.43177
2.08596
4.19892
5.76306
8.50755
4066.33
6.3
1.43467
2.08598
4.21088
5.78123
8.5684
4096.98
6.4
1.43776
2.08601
4.22349
5.8004
8.63019
4128.45
6.5
1.44106
2.08603
4.23676
5.82055
8.69293
4160.76
C
5
0.12
1.42648
2.08134
4.17695
5.72968
8.38866
4007.41
0.14
1.42641
2.07681
4.17696
5.72968
8.38866
4007.41
0.16
1.42634
2.07232
4.17697
5.72968
8.38866
4007.42
0.18
1.42627
2.06786
4.17698
5.72968
8.38866
4007.42
0.2
1.4262
2.06345
4.17699
5.72968
8.38866
4007.43
r
0.16
1.3601
2.13665
3.84454
5.20265
7.86865
3662.23
0.17
1.31431
2.19421
3.55049
4.72879
7.40461
3348.43
0.18
1.28124
2.26059
3.29027
4.29964
6.98831
3059.69
0.19
1.25646
2.33895
3.0598
3.90794
6.61292
2791.16
0.2
1.23733
2.4348
2.85541
3.54734
6.27281
2539.13
s
0.6
1.51085
2.08657
4.48099
6.1919
8.39432
4033.45
0.7
1.5898
2.08718
4.6929
6.51616
8.39547
4051.61
0.8
1.66348
2.08777
4.8477
6.75601
8.39463
4064.91
0.9
1.73232
2.08835
4.96477
6.94047
8.3929
4075.05
1.0
1.7968
2.08891
5.05568
7.08662
8.39076
4083.01
W
55
1.52163
2.08646
4.1731
5.72929
8.38844
4007.93
60
1.61945
2.08709
4.16542
5.72852
8.38802
4008.42
65
1.72024
2.08781
4.15332
5.72731
8.38735
4008.87
70
1.82396
2.08864
4.13619
5.7256
8.3864
4009.26
75
1.9302
2.08957
4.11348
5.72336
8.38516
4009.62
η
6
1.14672
1.5716
4.00831
5.44543
8.11066
4627.9
7
0.988027
1.26651
3.87949
5.23885
7.90997
5256.31
8
0.881603
1.06271
3.7803
5.08204
7.75838
5889.69
9
0.803577
0.916472
3.70204
4.95889
7.63982
6526.39
10
0.743094
0.8063
3.63886
4.85959
7.54456
7165.4


## 8. Observations

(1)
As we increase the ordering cost, the optimum value oft1*, t3*, t4*, t5* decreases, and t2* remains same. By this effect, our total cost decreases.
(2)
As we increase the deterioration parameter for OW, the optimum value oft1*,t2* increases and that of  t3*,t4*,t5* decreases, so our total cost decreases.
(3)
As we increase the deterioration parameters for RW, the optimum value oft1*,t4*,t5*, increases and that of t2*,t3* decreases then total cost slightly increase decreases. So the total cost is minimum when the deterioration rate of own warehouse is less than the deterioration rate of rented warehouse.
(4)
As we increase the demand parameterλ0, the optimum value of t1*,t2*,t3*,t4*,t5* decreases, while total cost increases. More demand encourages more production which increases the total cost.
(5)
As we increase the demand parameterδ, the optimum value of t1*,t3*,t4*,t5* increases and t2* decreases, and then our total cost increases. In both the cases, we see that as we increase the demand, increase in total cost is then obvious.
(6)
On increasing the imperfect rate, the optimum value oft1*,t3*,t4*,t5* increases and t2* decreases; then our total cost slightly increases. This is very obvious since with an increment in the percentage of the defective items the total cost then increases.
(7)
As we increase the holding cost for OW, the optimum value oft1*,t3*,t4*,t5* decreases and t2* increases, and then our total cost decreases.
(8)
As we increase the holding cost for RW, the optimum value oft1*,t2*,t3*,t4*,t5* increases then our total cost increases.
(9)
On increasing the shortage cost, the optimum value oft1*,t2*,t3*,t4*,t5* decreases, and then our total cost increases.
(10)
As we increase the lost sale cost, the optimum value oft1*,t2*,t3*,t4*,t5* increases, and then our total cost increases.
(11)
As we increase the rework cost, the optimum value oft1*,t2* decreases, t3* increases, and t4*,t5* remains the same, and so our total cost slightly increases.
(12)
As we increase the inflation rate the optimum value oft1*,t3*,t4*,t5* decreases and t2* increases, while total cost decreases.
(13)
As we increase the backlogging parameter, the optimum value oft1*,t2*,t3*,t4*,t5* and total cost increase.
(14)
As we increase the warehouse capacity, the optimum value oft1*,t2* increases and t3*,t4*,t5* decreases, and so our total cost increases. As we increase the warehouse capacity, our expenses increases, and so total cost also increases.


## 9. Conclusion

This paper presented a warehouse imperfect production-inventory models for deteriorating items having time varying demand patterns with Weibull distribution deterioration and partial backlogging under the effect of inflation and time value of money. Inflation plays a very significant role in the inventory models. Demand, production, defective item production, and so forth are natural in any manufacturing system. In this paper, we have developed two models for a manufacturing system. In the first model, all costs were assumed to be stochastic whereas in the second model, the costs were assumed to be fuzzy. The production rate was proportional to the demand rate. Well-known triangular membership function was used for all the fuzzy numbers. Numerical examples are solved by the software MATHEMATICA 8.0. Convexity of the total average cost is also illustrated in Figure2. However, the advantage of the fuzzy approach is that it relaxes the assumptions such as defective rate, demand rate etc. Also, it eases the difficulties in searching for suitable probability distribution function to represent the random behavior of uncontrollable variables.Figure 2
Convexity of total cost with respect tot1* and t2*.



---

*Source: 638060-2012-12-18.xml*