# Imperfect Production System under Reverse Logistics in Stock-Out Situation: EPQ Model

Authors: Himani Dem; Leena Prasher
Journal: Advances in Decision Sciences
 (2013)

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## Abstract

This paper derives a reverse logistic inventory model with imperfect production, stock-dependent demand, flexible manufacturing, and shortages over infinite planning horizon. The objective is to determine the joint policy for optimal production, amount of remanufacturing, collection of reusable items, and collection as well as disposal of defective items which minimizes the total cost of the inventory system under consideration. To make the model more realistic, both of the cases of linear and nonlinear holding costs have been discussed. The results are discussed with a numerical example to illustrate the theory.


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## Body

## 1. Introduction

The assumption of perfect production is not ideal for practical production system. Even the best production system may produce defective items. The governmental guidelines clearly state reduction of wastages, conservation of precious resources, protection of environment, and prevention of environmental degradation as a guiding principles for the business organizations. The manufacturing organizations may reuse the defective items after suitably repairing and removal of defects in order to avoid waste of resources. The defective items which are either irreparable or cannot be repaired easily and cost effectively are disposed off. They also prefer to reuse or recycle the items procured from the customers and reconvert through the appropriate process to appear as new and useful.


## 2. Literature Review

The classical production inventory model assumes that all the items produced are of perfect quality. Such an assumption appears impractical in real system. Therefore researchers have attracted towards model formation in which some parts of the items produced are of imperfect quality and they can be reworked and repaired. Rosenblatt and Lee [1], Lee and Rosenblatt [2], Cheng [3], Das and Sarkar [4], and Chung and Hou [5] worked on the issue of imperfect quality items and proved that production inventory cost is affected by rework or repair. Cheng [3] developed an economic order quantity model with demand-dependent unit production cost and imperfect production processes. Hayek and Salameh [6] assumed that all of the defective items produced are repairable and derived an optimal operating policy for EPQ model under the effect of reworking of imperfect quality items. Chung and Hou [5] investigated the production inventory model with imperfect production processes and allowable shortages. Chiu et al. [7] derived an economic production quantity (EPQ) model with scrap, rework, and stochastic machine breakdowns and assumed some portion of the defective items to be scrapped and the other parts to be repairable. S. R. Singh and C. Singh [8] developed an imperfect production process with exponential demand rate and Weibull distribution deterioration under inflation.For the past few decades, reverse logistics have been receiving much attention. The problem of optimal lot sizes for production/procurement and recovery was first studied by Schrady [9]. For issues in the greening process, Nahmias and Rivera [10] studied an EPQ variant of Schrady’s model [9] with a finite recovery rate. Richter [11, 12] and Richter and Dobos [13] investigated a waste disposal model considering the returned rate as a decision variable. They gave the optimal number of production and reproduction batches depended on the returned rate. Dobos and Richter [14, 15] investigated a production/remanufacturing system with constant demand that is satisfied by noninstantaneous production and remanufacturing for single and multiple remanufacturing and production cycle. Dobos and Richter [16] extended their previous model and assumed that the quality of collected returned items is not always suitable for further repairing. El Saadany and Jaber [17] extended the models developed in Dobos and Richter [14, 15] by assuming that the collection rate of returned items is dependent on the purchasing price and the acceptance quality level of these returns. That is, the flow of used/returned items increases as the purchasing price increases and decreases as the corresponding acceptance quality level increases. A general reverse logistics inventory model was developed by Alamri [18]. Chung and Wee [19] developed an inventory model on short life-cycle deteriorating product remanufacturing in a green supply chain model. Singh and Saxena [20] derived an optimal returned policy for a reverse logistics inventory model with backorders.An increase in the shelf space can influence more customers. In this connection, the observations made by Levin et al. [21] and Silver and Peterson [22] may be noted. They observed that the presence of greater quantity of the same item tends to attract more customers. The reason behind this fact is a typical psychology of the customers. They may have the feeling of obtaining a wide range for selection when a large amount is stored/displayed. Gupta and Vrat [23] developed models for stock-dependent consumption rate. Mandal and Phaujdar [24] developed an inventory model for deteriorating items and stock-dependent consumption rate. Schweitzer and Seidmann [25] established optimizing processing rate for flexible manufacturing systems. Giri and Chaudhuri [26] developed deterministic model of perishable inventory with stock-dependent demand rate and nonlinear holding cost and proved that the nonlinear holding cost affected the total average cost. Sana et al. [27] established a production-inventory model for a deteriorating item with trended demand and shortages. Teng and Chang [28] proposed economic production model for deteriorating item with price and stock-dependent demand. Singh and Jain [29] worked on reserve money for an EOQ model in an inflationary environment under supplier credits. S. R. Singh and C. Singh [8] worked on supply chain model with stochastic lead time under imprecise partially backlogging for expiring items. Singh et al. [30] contributed on an inventory model for deteriorating items with shortages and stock-dependent demand under inflation for two shops under one management. Konstantaras and Skouri [31] presented a model by considering a general cycle pattern in which a variable number of reproduction lots of equal size were followed by a variable number of manufacturing lots of equal size. They also studied the case where shortages were allowed in each manufacturing and reproduction cycle and similar sufficient conditions, as the nonshortages case, were given. Yadav et al. [32] developed an inventory model of deteriorating items with stock-dependent demand using genetic algorithm in fuzzy environment. Dem and Singh [33] investigated an EPQ model for damageable items with multivariate demand and flexible manufacturing. Dem and Singh [34] developed a production model for imperfect production process under volume flexibility. Goyal et al. [35] explored an inventory system with variable demand as well as production under partially backordered shortages.


## 3. Assumptions and Notations

The following assumptions and notations are used throughout the model.


### 3.1. Assumptions

(1)
Production rate is linear function of demand and demand is a nonlinear function of on-hand inventory. Thus production is a nonlinear function of on-hand inventory.
(2)
Demand rate remains stock-dependent for a certain period after which a uniform demand rate follows as the stock comes down to zero level. The functional relationship between the demand ratef(q) and the inventory level q(t) is given by the following expression:
(1)f(q)=Dqβ,D>0,0<β<1,q≥0,
whereβdenotes the shape parameter and is a measure of responsiveness of the demand to changes in the level of on hand inventory and D denotes the scale parameter.
(3)
The repair work of defective items starts when the regular production work stops. After repair, they are as good as the new ones.
(4)
Deterioration rate is constant.
(5)
Items are returnable and are remanufactured. Remanufactured items are as good as new ones and they are used during the shortage period of forward manufacturing.
(6)
The time horizon of the inventory system is infinite. Only a typical planning schedule of lengthT is considered, and all remaining cycles are identical.
(7)
Shortages are allowed and completely backlogged.
(8)
The production time interval including repair time during forward production coincides with the collection time interval for reverse manufacturing. (This assumption is not applicable during the period of shortages.)


### 3.2. Notations

q
(
t
): On hand inventory level during regular production uptime
f
(
q
): Demand rate, f(q)=Dqβ, D>0, 0<β<1
P: Production rate during regular production uptime,P=lf(q), wherelis a scale parameter,P>f(q), l>1
P
1: Production rate of defective items repaired, P1=l1f(q), wherel1 is a scale parameter,l1>1
K: Setup cost
h
1: Holding cost per unit per unit time during regular production uptime, repair time, and reverse manufacturing
h
2: Holding cost per unit per unit time during the collecting and consuming process of defective items
h
3: Holding cost per unit per unit time during the collecting and consuming process for the reverse manufacturing
q
c
(
t
c
): Inventory level during collecting and consuming process of defective items
q
r
c
(
t
r
c
): Inventory level during the collecting process of reverse manufacturing
q
r
m
(
t
r
m
): Inventory level during the remanufacturing process for reverse manufacturing
ξ: Fraction of the production lot size in the interval[0,tp1]where0<ξ<1
Q: Maximum inventory level during regular production uptime
Q
c: Maximum inventory level of defective items collected
Q
h: Inventory level when repair work stops
t
p: Time when regular production stops and repair starts. It also represents the time when collection process of defective items stops
t
p
1: Time when repair work of defective items stops and remanufacturing starts. At this point of time collection process of reusable items also stops
t
s: Time when remanufacturing stops and also the time when accumulated inventory of forward manufacturing vanishes
t
s
1: Time when accumulated remanufactured inventory vanishes and shortages start
t
m: Time when production starts again during the period of shortage. At this time collection of defective items start again
t
m
1: Time when collection process of defective items stops and their repair work starts during the period of shortage
T: Length of a complete cycle
C
p: Production cost per unit of regular good quality items produced
C
r: Repair cost per unit of defective items repaired
C
s: Shortage cost per unit per unit time
s
r: Scrap cost per unit
S
′: Maximum shortages
S
1: Maximum inventory level of remanufactured items
S
c: Collection of defective items during period of shortage
S
h: Shortage level when repair work starts
R
c: Rate of collection of reusable items
R
m: Rate of remanufacturing of reusable items
θ
1: Rate of deterioration
HC: Total holding cost
SC: Total shortage cost
SR: Total scrap cost
RC: Total repair cost
PC: Total production cost.


## 3.1. Assumptions

(1)
Production rate is linear function of demand and demand is a nonlinear function of on-hand inventory. Thus production is a nonlinear function of on-hand inventory.
(2)
Demand rate remains stock-dependent for a certain period after which a uniform demand rate follows as the stock comes down to zero level. The functional relationship between the demand ratef(q) and the inventory level q(t) is given by the following expression:
(1)f(q)=Dqβ,D>0,0<β<1,q≥0,
whereβdenotes the shape parameter and is a measure of responsiveness of the demand to changes in the level of on hand inventory and D denotes the scale parameter.
(3)
The repair work of defective items starts when the regular production work stops. After repair, they are as good as the new ones.
(4)
Deterioration rate is constant.
(5)
Items are returnable and are remanufactured. Remanufactured items are as good as new ones and they are used during the shortage period of forward manufacturing.
(6)
The time horizon of the inventory system is infinite. Only a typical planning schedule of lengthT is considered, and all remaining cycles are identical.
(7)
Shortages are allowed and completely backlogged.
(8)
The production time interval including repair time during forward production coincides with the collection time interval for reverse manufacturing. (This assumption is not applicable during the period of shortages.)


## 3.2. Notations

q
(
t
): On hand inventory level during regular production uptime
f
(
q
): Demand rate, f(q)=Dqβ, D>0, 0<β<1
P: Production rate during regular production uptime,P=lf(q), wherelis a scale parameter,P>f(q), l>1
P
1: Production rate of defective items repaired, P1=l1f(q), wherel1 is a scale parameter,l1>1
K: Setup cost
h
1: Holding cost per unit per unit time during regular production uptime, repair time, and reverse manufacturing
h
2: Holding cost per unit per unit time during the collecting and consuming process of defective items
h
3: Holding cost per unit per unit time during the collecting and consuming process for the reverse manufacturing
q
c
(
t
c
): Inventory level during collecting and consuming process of defective items
q
r
c
(
t
r
c
): Inventory level during the collecting process of reverse manufacturing
q
r
m
(
t
r
m
): Inventory level during the remanufacturing process for reverse manufacturing
ξ: Fraction of the production lot size in the interval[0,tp1]where0<ξ<1
Q: Maximum inventory level during regular production uptime
Q
c: Maximum inventory level of defective items collected
Q
h: Inventory level when repair work stops
t
p: Time when regular production stops and repair starts. It also represents the time when collection process of defective items stops
t
p
1: Time when repair work of defective items stops and remanufacturing starts. At this point of time collection process of reusable items also stops
t
s: Time when remanufacturing stops and also the time when accumulated inventory of forward manufacturing vanishes
t
s
1: Time when accumulated remanufactured inventory vanishes and shortages start
t
m: Time when production starts again during the period of shortage. At this time collection of defective items start again
t
m
1: Time when collection process of defective items stops and their repair work starts during the period of shortage
T: Length of a complete cycle
C
p: Production cost per unit of regular good quality items produced
C
r: Repair cost per unit of defective items repaired
C
s: Shortage cost per unit per unit time
s
r: Scrap cost per unit
S
′: Maximum shortages
S
1: Maximum inventory level of remanufactured items
S
c: Collection of defective items during period of shortage
S
h: Shortage level when repair work starts
R
c: Rate of collection of reusable items
R
m: Rate of remanufacturing of reusable items
θ
1: Rate of deterioration
HC: Total holding cost
SC: Total shortage cost
SR: Total scrap cost
RC: Total repair cost
PC: Total production cost.


## 4. Formulation of Model

This model consists of two systems: forward manufacturing and reverse manufacturing. The manufacturing process is flexible but imperfect, and hence it can produce as per the demand rate, but it produces perfect as well as imperfect items. At the beginning of each cycle, the inventory is zero. The production starts at the very beginning of the cycle and system produces perfect as well as imperfect items. As production progresses, the inventory of perfect items piles up even after meeting the market demand and imperfect items are collected in a separate store. During the regular production uptime,x portion of produced items is assumed to be defective and is generated at a production rate d=Px where the regular production rate P is a function of demand. Among these defective items, θ portion is considered to be scrap and the other portions can be reworked and repaired. The rework or repair work starts when the regular production work stops. Along with the two stores of perfect quality items and imperfect quality items from the forward production system, there is one more store in which reusable items are collected from the customers, and therefore at the beginning of each cycle, the process of collecting returnable items in a separate store also begins and is continued till the forward production process stops. More precisely, at a point where the repair work during forward manufacturing system stops; the collection process of returnable items also stops at the same point (for simplicity, we assume that there is no collection of used items, once the remanufacturing of collected items starts). At this very point, the remanufacturing of reusable items begin at a constant rate. The accumulated inventory produced from the advanced manufacturing system in the meanwhile starts getting consumed and ultimately becomes nil. The accumulated inventory of remanufacturing products (which are assumed to be as good as the newly produced products) is consumed when the shortages from the forward manufacturing system begin to surface. Also at this stage, there is no production and inventory of remanufactured items is consumed till it becomes nil. When the inventory of remanufactured items is also nil, inventory shortages begin to accumulate for some time. Thereafter, production starts and collection of defective items also starts. After certain time, repair work starts and after clearing the shortages, the cycle ends with zero inventory. Graphical description is shown in Figure 1.Figure 1
Production inventory model with imperfect production and reverse manufacturing.As the production system is not perfect, the items produced during regular production are a mixture of perfect and imperfect quality. The production cycle begins with zero inventory and starts att=0. In the interval(0,tp), the inventory of perfect quality items piles up after meeting demand and removing imperfect items and imperfect quality items are collected in a separate store as shown by (2) and (5). Feasibility of this assumption implies that l-1-lx must be greater than zero. Regular production is stopped at t=tp, and at this very point repair work starts. The repairable portion of imperfect items collected in the interval (0,tp) is repaired at a repair rate P1=l1D(q) during the interval (tp,tp1). After repairing, these items are assumed to be as good to serve the demand as the perfect quality items and contribute in rising inventory of perfect quality items after meeting demand. As the inventory of perfect quality items rises, the inventory of defective items falls as shown by (3) and (6). The forward production process stops at t=tp1. The inventory of perfect quality items falls to zero level at t=ts.


### 4.1. Forward Manufacturing Process

The following equations depict the inventory level of perfect quality items in various intervals:(2)dqdt=P-Dqβ-Px,q(0)=0,0≤t≤tp,(3)dqdt=P1-Dqβ,q(tp)=Q,tp≤t≤tp1,(4)dqdt=-Dqβ,q(ts)=0,tp1≤t≤ts.The following equations depict the inventory level of imperfect quality items (collection and consumption process):(5)dqcdtc=Px(1-θ),qc(0)=0,0≤tc≤tp,(6)dqcdtc=-P1,qc(tp1)=0,tp≤tc≤tp1.Along with the two stores of perfect quality items and imperfect quality items from the forward production system, there is one more store in which reusable items are collected from the customers. At the beginning of each cycle, the process of collecting reusable items also starts at a rate of collectionRc. At time t=tp1, the collection process is stopped and remanufacturing of collected items starts at a rate Rm. At time t=ts, the inventory of reusable collected items falls to zero and the inventory of remanufactured items rises to S1. Also at this point, there is no inventory to serve the demand from the forward production system, so that the remanufactured items are used to serve the demand till its falling to zero.


### 4.2. Reverse Collection and Consumption Process

One has(7)dqrcdtrc=Rc-θ1qrc,qrc(0)=0,0≤trc≤tp1,dqrcdtrc=-Rm-θ1qrc,qrc(tp1)=Bξ,tp1≤trc≤ts,whereB=(l/(l-1-lx))Q+(l1/(l1-1))(Qh-Q)is the production lot size in the interval[0,tp1](see Appendix A).


### 4.3. Reverse Manufacturing Process

(8)
d
q
r
m
d
t
r
m
=
R
m
-
θ
1
q
r
m
,
q
r
m
(
t
p
1
)
=
0
,
t
p
1
≤
t
r
m
≤
t
s
,
d
q
d
t
=
-
D
q
β
-
θ
1
q
,
q
(
t
s
)
=
S
1
,
t
s
≤
t
≤
t
s
1
.


### 4.4. Shortage Phase

At timet=ts1, the inventory of remanufactured items is also nil and shortages accumulate in the interval (ts1,tm) and rise to S′ at t=tm. At time t=tm production restarts and shortages start clearing gradually. As the production starts, imperfect quality items also start to accumulate. The imperfect quality items collected in the interval (tm,tm1) are repaired and used to clear the demand in the interval (tm1,T).The following are the equations depicting the behavior of inventory during shortages as described above:(9)dqdt=-D1,q(ts1)=0,ts1≤t≤tm,dqdt=(l-1-lx)D1,q(tm)=S′,tm≤t≤tm1,dqdt=(l1-1)D1,q(tm1)=Sh,tm1≤t≤T.The following are the equations depicting the collection and consumption of imperfect quality items during shortage period:(10)dqcdtc=Px(1-θ),qc(tm)=0,tm≤tc≤tm1,dqcdtc=-P1,qc(T)=0,tm1≤tc≤T.Solving (2), (3), and (4) and using the associated boundary conditions, we get the inventory level of perfect quality items at various stages during forward manufacturing:
(11)qα=(l-1-lx)Dαt,0≤t≤tp,(12)qα=Qα+(l1-1)Dα(t-tp),tp≤t≤tp1,(13)qα=-αD(t-ts),tp1≤t≤ts.Usingq(tp)=Q, q(tp1)=Qh, and q(tp1)=Qh in (11), (12), and (13), respectively, we get the following relations:
(14)tp=Qα(l-1-lx)Dα,tp1=tp+Qhα-Qα(l1-1)Dα,ts=1αD(Qhα+αDtp1).For feasibility of the modeltp1 must be greater than tp; otherwise the remanufacturing process will start during the regular production time. Therefore tp1>tp, give rise to the constraint (Qhα-Qα)/(l1-1)Dα>0 must be satisfied which holds true (see Appendix B). The above constraint also implies Qhα>Qα with means that inventory level attp1is higher than the inventory level at tp which is shown in Figure 1. Also, ts>tp1 as the interval (tp1,ts) represents the interval of remanufacturing which implies that Qhα/αD>0 and which holds true.Consider(15)Holdingcostin[0,ts]=holdingcostin[0,tp]+holdingcostin[tp,tp1]+holdingcostin[tp1,ts]=h1(∫OQqndt+∫QQhqndt+∫Qh0qndt)=h1[Qn+α(l-1-lx)D(n+α)+Qhn+αl1(l1-1)D(n+α)iiiiiiiii-Qn+α(l1-1)D(n+α)].Solving (5) and (6) and using boundary conditions, we get inventory level of imperfect quality items during forward manufacturing process:
(16)qcα=lx(1-θ)Dαtc,0≤tc≤tp,qcα=-Dα(tc-tp1)l1,tp≤tc≤tp1.Usingqc(tp)=Qc in (16) and also using the values of tp and tp1 (already derived), we have
(17)Qcα=lx(1-θ)l-1-lxQα,Qhα-Qα=(l1-1)Dα(Qcαl1Dα).Asl-1-lx>0, l1-1>0⇒Qcα≥0 and Qhα>Qα which shows the existence of inventory levels at various stages.Consider(18)
Holding
cost
of
imperfect
quality
items
during
interval
[
0
,
t
p
1
]
=
holding
cost
in
[
0
,
t
p
]
iiiiiiiiiiiiiii
+
holding
cost
in
[
t
p
,
t
p
1
]
iiiiiiiiiii
=
h
2
[
Q
c
n
+
α
l
x
(
1
-
θ
)
D
(
n
+
α
)
+
Q
c
n
+
α
l
1
D
(
n
+
α
)
]
.Solving (7) and using boundary conditions, the inventory level of reusable items in various intervals is calculated as follows:
(19)qrc=Rcθ1(1-e-θ1trc),0≤trc≤tp1,(20)qrc=-Rmθ1+(Bξ+Rmθ1)eθ1(tp1-trc),tp1≤trc≤ts.Usingqrc(tp1)=Bξ in (19), we have Bξ=Rc(tp1-θ1tp12/2) which can be used to find the rate of collection of reusable items.Usingqrc(ts)=0, in (20), we have
(21)BξRm-B2ξ2θ12Rm2=(ts-tp1).The above relation gives the rate of remanufacturing.Consider(22)Holdingcostofreversecollectionininterval[0,ts]=holdingcostin[0,tp1]+holdingcostin[tp1,ts]=h2[∫0Bξqrcndt+∫tp1tsqrcndtrc]=h2[(Bξ)n+1Rc(n+1)+(Bξ)n+2θ1Rc2(n+2)+(Bξ)n+1Rm(n+1)iiiiiiiiiiiiiiiii-(Bξ)n+2θ1(n+2)Rm2],Deteriorationcost=θ1Cp[(Bξ)n+1Rc(n+1)+(Bξ)n+2θ1Rc2(n+2)+(Bξ)n+1Rm(n+1)iiiiiiiiiiiiiiiiiiiii-(Bξ)n+2θ1(n+2)Rm2].For reverse manufacturing process, after solving (8) and using boundary conditions, the inventory level of remanufactured items can be obtained as
(23)qrm=Rmθ1(1-eθ1(tp1-trm)),tp1≤trm≤ts,(24)qα=-Dθ1+(S1α+Dθ1)eθ1α(ts-t),ts≤t≤ts1,(25)Holdingcostin[tp1,ts1]=holdingcostin[tp1,ts]+holdingcostin[ts,ts1]=h1[∫0S1qrmn(1Rm+θ1qrmRm2)dqrm+∫S10qn(-qα-1D+θ1q2α-1D2)dq]=h1[S1n+1(n+1)Rm+θ1S1(n+2)Rm2(n+2)+(S1α+nD(α+n)-θ1S1n+2αD2(n+2α))],Deteriorationcost=θ1Cp[S1n+1(n+1)Rm+θ1S1(n+2)Rm2(n+2)+(S1α+nD(α+n)-θ1S1n+2αD2(n+2α))[S1n+1(n+1)Rm+θ1S1(n+2)Rm2(n+2)].Usingq(ts1)=0 in (24), we have ts1=ts+S1α/Dα-θ1S12α/2D2α, where ts=Qαl(1-xθ)/αD(l-1-lx).Solving (9), we have the shortage level in various intervals as given as follows:
(26)q=D1(ts1-t),ts1≤t≤tm,(27)q=(l-1-lx)D1(t-tm)+S′,tm≤t≤tm1,(28)q=(l1-1)D1(t-tm1)+Sh′,tm1≤t≤T.Usingq(tm)=S′, q(tm1)=Sh, and q(T)=0 in (26), (27), and (28), respectively, we have the following relations:
(29)tm=ts1-S′D1,tm1=tm+Sh-S′(l-1-lx)D1,-Sh(l1-1)D1+tm1=T.For feasibility of the practical situation,tm1must be greater than or equal totmas there cannot be any collection of imperfect items before production. Therefore (Sh-S′)/(l-1-lx)D1≥0 which implies Sh>S′ which holds true.Substituting all the relevant values, the cycle time is given as(30)T=[Qαl(1-xθ)αD(l-1-lx)+S1αDα-θ1S12α2D2α-S′D1(l(1-x)l-1-lx)+ShD1(l1-l+lx(l-1-lx)(l1-1))[Qαl(1-xθ)αD(l-1-lx)+S1αDα-θ1S12α2D2α-S′D1(l(1-x)l-1-lx)],whereSh=((l1-1)lx(1-θ)/(lxθ(1-l1)-lx+l1(l-1)))S′.Consider(31)Totalshortagecostin[ts1,T]=SC=Cs[×(Sh22-S′22)+Sh22(l1-1)D1]S′22D1-1D1(l-1-lx)iiiiiiiiiiiiiiiiiiiiiiiii×(Sh22-S′22)+Sh22(l1-1)D1].Solving (10), the inventory levels of imperfect quality items during period of shortage are
(32)qc=lx(1-θ)D1(tc-tm),tm≤tc≤tm1,qc=-l1D1(tc-T),tm1≤tc≤T.Usingqc(tm1)=Sc in (32), we get
(33)S′l-1-lx=Sh(1l-1-lx+l1(l1-1)lx(1-θ)).Holding cost of imperfect quality items during period of shortage(34)=h2[Sc(n+1)lx(1-θ)D1(n+1)-Sc(n+1)l1D1].Total holding cost in complete cycle[0,T] is as follows:
(35)HC=h1[-Qn+α(l1-1)D(n+α)]Qn+α(l-1-lx)D(n+α)+Qhn+α(l1-1)D(n+α)-Qn+α(l1-1)D(n+α)]+h2[Qcn+αlx(1-θ)D(n+α)+Qcn+αl1D(n+α)]+h3[(Bξ)n+1Rc(n+1)+(Bξ)n+2θ1Rc2(n+2)+(Bξ)n+1Rm(n+1)-(Bξ)n+2θ1(n+2)Rm2]+h1[S1n+1(n+1)Rm+θ1S1(n+2)Rm2(n+2)+(S1α+nD(α+n)-θ1S1n+2αD2(n+2α))[S1n+1(n+1)Rm+θ1S1(n+2)Rm2(n+2)]+h2[Sc(n+1)lx(1-θ)D1(n+1)-Sc(n+1)l1D1].Production cost is as follows:(36)PC=Cp[∫0tpPdt+∫tmtm1Pdt]=Cpll-1-lx[Q+(Sh-S′)].Repair cost is as follows:(37)RC=Cr[∫tptp1P1dt+∫tm1TP1dt]=Cr[l1(Qh-Q)l1-1-l1Shl1-1].Scrap cost is as follows:(38)SR=Srθx[lQl-1-lx+l(Sh-S′)l-1-lx].Total deterioration cost is as follows:(39)DC=θ1Cp×[-(Bξ)n+2θ1(n+2)Rm2+S1n+1(n+1)Rm+θ1S1(n+2)Rm2(n+2)](Bξ)n+1Rc(n+1)+(Bξ)n+2θ1Rc2(n+2)+(Bξ)n+1Rm(n+1)-(Bξ)n+2θ1(n+2)Rm2+S1n+1(n+1)Rm+θ1S1(n+2)Rm2(n+2)].The total inventory cost per unit time is given by(40)TAC(Q,S′)=K+HC+DC+SC+PC+SC+RCT.Our problem is to find the time to stop the production whenq takes optimum value Q and the time to again start the production when maximum shortages accumulate. For minimum value of total cost, we must have
(C1)∂∂Q(TAC)=0,∂∂S′(TAC)=0,T[∂HC∂S′+∂SC∂S′+∂PC∂S′+∂DC∂S′+∂RC∂S′+∂SR∂S′]=(K+HC+DC+SC+PC+RC+SR)∂T∂S′,T[∂HC∂Q′+∂SC∂Q′+∂PC∂Q′+∂DC∂Q′+∂RC∂Q′+∂SR∂Q′]=(K+HC+DC+SC+PC+RC+SR)∂T∂Q′.Provided the following condition of Hessian matrix,H is satisfied, where
(41)H=[∂2TAC∂Q2∂2TAC∂Q∂S∂2TAC∂S∂Q∂2TAC∂S2].The first principle minor determinant ofH, |H11|>0 and the second principle minor determinant of H,|H22|>0 (see Appendix C).Using(C1). We get the relation
(C2)(∂HC∂S′+∂SC∂S′+∂PC∂S′+∂DC∂S′+∂RC∂S′+∂SR∂S′)iiiiiiii×(∂HC∂Q+∂SC∂Q+∂PC∂Q+∂DC∂Q+∂RC∂Q+∂SR∂Q)-1iiii=∂T/∂S′∂T/∂Q.


### 4.5. Special Case

If we relax some conditions so as to approach the basic EOQ model with shortages.Asx→0, ξ→0, θ1→0, α→1, n→1, and D=D1, we have the following results: DC=0, RC=0, SR=0, PC=Cp(l/(l-1))(Q-S′), HC=h1lQ2/2(l-1)D, SC=CsS′2l/2(l-1)D1, and T=Ql/(l-1)D+(l/(1-l)D1)S′.Substituting all the above values in(C1) and (C2) and solving for Q, we have
(42)Q*=2k(l-1)DcSlh1(cs+h1).Aslincreases, production occurs at a more rapid rate. Hence for large l, the model should approach the instantaneous delivery situation of the EOQ model. For largel,1-1/l→1. Thus aslincreases towards infinity, the optimal run size for the model approaches the EOQ when shortages are allowed.


## 4.1. Forward Manufacturing Process

The following equations depict the inventory level of perfect quality items in various intervals:(2)dqdt=P-Dqβ-Px,q(0)=0,0≤t≤tp,(3)dqdt=P1-Dqβ,q(tp)=Q,tp≤t≤tp1,(4)dqdt=-Dqβ,q(ts)=0,tp1≤t≤ts.The following equations depict the inventory level of imperfect quality items (collection and consumption process):(5)dqcdtc=Px(1-θ),qc(0)=0,0≤tc≤tp,(6)dqcdtc=-P1,qc(tp1)=0,tp≤tc≤tp1.Along with the two stores of perfect quality items and imperfect quality items from the forward production system, there is one more store in which reusable items are collected from the customers. At the beginning of each cycle, the process of collecting reusable items also starts at a rate of collectionRc. At time t=tp1, the collection process is stopped and remanufacturing of collected items starts at a rate Rm. At time t=ts, the inventory of reusable collected items falls to zero and the inventory of remanufactured items rises to S1. Also at this point, there is no inventory to serve the demand from the forward production system, so that the remanufactured items are used to serve the demand till its falling to zero.


## 4.2. Reverse Collection and Consumption Process

One has(7)dqrcdtrc=Rc-θ1qrc,qrc(0)=0,0≤trc≤tp1,dqrcdtrc=-Rm-θ1qrc,qrc(tp1)=Bξ,tp1≤trc≤ts,whereB=(l/(l-1-lx))Q+(l1/(l1-1))(Qh-Q)is the production lot size in the interval[0,tp1](see Appendix A).


## 4.3. Reverse Manufacturing Process

(8)
d
q
r
m
d
t
r
m
=
R
m
-
θ
1
q
r
m
,
q
r
m
(
t
p
1
)
=
0
,
t
p
1
≤
t
r
m
≤
t
s
,
d
q
d
t
=
-
D
q
β
-
θ
1
q
,
q
(
t
s
)
=
S
1
,
t
s
≤
t
≤
t
s
1
.


## 4.4. Shortage Phase

At timet=ts1, the inventory of remanufactured items is also nil and shortages accumulate in the interval (ts1,tm) and rise to S′ at t=tm. At time t=tm production restarts and shortages start clearing gradually. As the production starts, imperfect quality items also start to accumulate. The imperfect quality items collected in the interval (tm,tm1) are repaired and used to clear the demand in the interval (tm1,T).The following are the equations depicting the behavior of inventory during shortages as described above:(9)dqdt=-D1,q(ts1)=0,ts1≤t≤tm,dqdt=(l-1-lx)D1,q(tm)=S′,tm≤t≤tm1,dqdt=(l1-1)D1,q(tm1)=Sh,tm1≤t≤T.The following are the equations depicting the collection and consumption of imperfect quality items during shortage period:(10)dqcdtc=Px(1-θ),qc(tm)=0,tm≤tc≤tm1,dqcdtc=-P1,qc(T)=0,tm1≤tc≤T.Solving (2), (3), and (4) and using the associated boundary conditions, we get the inventory level of perfect quality items at various stages during forward manufacturing:
(11)qα=(l-1-lx)Dαt,0≤t≤tp,(12)qα=Qα+(l1-1)Dα(t-tp),tp≤t≤tp1,(13)qα=-αD(t-ts),tp1≤t≤ts.Usingq(tp)=Q, q(tp1)=Qh, and q(tp1)=Qh in (11), (12), and (13), respectively, we get the following relations:
(14)tp=Qα(l-1-lx)Dα,tp1=tp+Qhα-Qα(l1-1)Dα,ts=1αD(Qhα+αDtp1).For feasibility of the modeltp1 must be greater than tp; otherwise the remanufacturing process will start during the regular production time. Therefore tp1>tp, give rise to the constraint (Qhα-Qα)/(l1-1)Dα>0 must be satisfied which holds true (see Appendix B). The above constraint also implies Qhα>Qα with means that inventory level attp1is higher than the inventory level at tp which is shown in Figure 1. Also, ts>tp1 as the interval (tp1,ts) represents the interval of remanufacturing which implies that Qhα/αD>0 and which holds true.Consider(15)Holdingcostin[0,ts]=holdingcostin[0,tp]+holdingcostin[tp,tp1]+holdingcostin[tp1,ts]=h1(∫OQqndt+∫QQhqndt+∫Qh0qndt)=h1[Qn+α(l-1-lx)D(n+α)+Qhn+αl1(l1-1)D(n+α)iiiiiiiii-Qn+α(l1-1)D(n+α)].Solving (5) and (6) and using boundary conditions, we get inventory level of imperfect quality items during forward manufacturing process:
(16)qcα=lx(1-θ)Dαtc,0≤tc≤tp,qcα=-Dα(tc-tp1)l1,tp≤tc≤tp1.Usingqc(tp)=Qc in (16) and also using the values of tp and tp1 (already derived), we have
(17)Qcα=lx(1-θ)l-1-lxQα,Qhα-Qα=(l1-1)Dα(Qcαl1Dα).Asl-1-lx>0, l1-1>0⇒Qcα≥0 and Qhα>Qα which shows the existence of inventory levels at various stages.Consider(18)
Holding
cost
of
imperfect
quality
items
during
interval
[
0
,
t
p
1
]
=
holding
cost
in
[
0
,
t
p
]
iiiiiiiiiiiiiii
+
holding
cost
in
[
t
p
,
t
p
1
]
iiiiiiiiiii
=
h
2
[
Q
c
n
+
α
l
x
(
1
-
θ
)
D
(
n
+
α
)
+
Q
c
n
+
α
l
1
D
(
n
+
α
)
]
.Solving (7) and using boundary conditions, the inventory level of reusable items in various intervals is calculated as follows:
(19)qrc=Rcθ1(1-e-θ1trc),0≤trc≤tp1,(20)qrc=-Rmθ1+(Bξ+Rmθ1)eθ1(tp1-trc),tp1≤trc≤ts.Usingqrc(tp1)=Bξ in (19), we have Bξ=Rc(tp1-θ1tp12/2) which can be used to find the rate of collection of reusable items.Usingqrc(ts)=0, in (20), we have
(21)BξRm-B2ξ2θ12Rm2=(ts-tp1).The above relation gives the rate of remanufacturing.Consider(22)Holdingcostofreversecollectionininterval[0,ts]=holdingcostin[0,tp1]+holdingcostin[tp1,ts]=h2[∫0Bξqrcndt+∫tp1tsqrcndtrc]=h2[(Bξ)n+1Rc(n+1)+(Bξ)n+2θ1Rc2(n+2)+(Bξ)n+1Rm(n+1)iiiiiiiiiiiiiiiii-(Bξ)n+2θ1(n+2)Rm2],Deteriorationcost=θ1Cp[(Bξ)n+1Rc(n+1)+(Bξ)n+2θ1Rc2(n+2)+(Bξ)n+1Rm(n+1)iiiiiiiiiiiiiiiiiiiii-(Bξ)n+2θ1(n+2)Rm2].For reverse manufacturing process, after solving (8) and using boundary conditions, the inventory level of remanufactured items can be obtained as
(23)qrm=Rmθ1(1-eθ1(tp1-trm)),tp1≤trm≤ts,(24)qα=-Dθ1+(S1α+Dθ1)eθ1α(ts-t),ts≤t≤ts1,(25)Holdingcostin[tp1,ts1]=holdingcostin[tp1,ts]+holdingcostin[ts,ts1]=h1[∫0S1qrmn(1Rm+θ1qrmRm2)dqrm+∫S10qn(-qα-1D+θ1q2α-1D2)dq]=h1[S1n+1(n+1)Rm+θ1S1(n+2)Rm2(n+2)+(S1α+nD(α+n)-θ1S1n+2αD2(n+2α))],Deteriorationcost=θ1Cp[S1n+1(n+1)Rm+θ1S1(n+2)Rm2(n+2)+(S1α+nD(α+n)-θ1S1n+2αD2(n+2α))[S1n+1(n+1)Rm+θ1S1(n+2)Rm2(n+2)].Usingq(ts1)=0 in (24), we have ts1=ts+S1α/Dα-θ1S12α/2D2α, where ts=Qαl(1-xθ)/αD(l-1-lx).Solving (9), we have the shortage level in various intervals as given as follows:
(26)q=D1(ts1-t),ts1≤t≤tm,(27)q=(l-1-lx)D1(t-tm)+S′,tm≤t≤tm1,(28)q=(l1-1)D1(t-tm1)+Sh′,tm1≤t≤T.Usingq(tm)=S′, q(tm1)=Sh, and q(T)=0 in (26), (27), and (28), respectively, we have the following relations:
(29)tm=ts1-S′D1,tm1=tm+Sh-S′(l-1-lx)D1,-Sh(l1-1)D1+tm1=T.For feasibility of the practical situation,tm1must be greater than or equal totmas there cannot be any collection of imperfect items before production. Therefore (Sh-S′)/(l-1-lx)D1≥0 which implies Sh>S′ which holds true.Substituting all the relevant values, the cycle time is given as(30)T=[Qαl(1-xθ)αD(l-1-lx)+S1αDα-θ1S12α2D2α-S′D1(l(1-x)l-1-lx)+ShD1(l1-l+lx(l-1-lx)(l1-1))[Qαl(1-xθ)αD(l-1-lx)+S1αDα-θ1S12α2D2α-S′D1(l(1-x)l-1-lx)],whereSh=((l1-1)lx(1-θ)/(lxθ(1-l1)-lx+l1(l-1)))S′.Consider(31)Totalshortagecostin[ts1,T]=SC=Cs[×(Sh22-S′22)+Sh22(l1-1)D1]S′22D1-1D1(l-1-lx)iiiiiiiiiiiiiiiiiiiiiiiii×(Sh22-S′22)+Sh22(l1-1)D1].Solving (10), the inventory levels of imperfect quality items during period of shortage are
(32)qc=lx(1-θ)D1(tc-tm),tm≤tc≤tm1,qc=-l1D1(tc-T),tm1≤tc≤T.Usingqc(tm1)=Sc in (32), we get
(33)S′l-1-lx=Sh(1l-1-lx+l1(l1-1)lx(1-θ)).Holding cost of imperfect quality items during period of shortage(34)=h2[Sc(n+1)lx(1-θ)D1(n+1)-Sc(n+1)l1D1].Total holding cost in complete cycle[0,T] is as follows:
(35)HC=h1[-Qn+α(l1-1)D(n+α)]Qn+α(l-1-lx)D(n+α)+Qhn+α(l1-1)D(n+α)-Qn+α(l1-1)D(n+α)]+h2[Qcn+αlx(1-θ)D(n+α)+Qcn+αl1D(n+α)]+h3[(Bξ)n+1Rc(n+1)+(Bξ)n+2θ1Rc2(n+2)+(Bξ)n+1Rm(n+1)-(Bξ)n+2θ1(n+2)Rm2]+h1[S1n+1(n+1)Rm+θ1S1(n+2)Rm2(n+2)+(S1α+nD(α+n)-θ1S1n+2αD2(n+2α))[S1n+1(n+1)Rm+θ1S1(n+2)Rm2(n+2)]+h2[Sc(n+1)lx(1-θ)D1(n+1)-Sc(n+1)l1D1].Production cost is as follows:(36)PC=Cp[∫0tpPdt+∫tmtm1Pdt]=Cpll-1-lx[Q+(Sh-S′)].Repair cost is as follows:(37)RC=Cr[∫tptp1P1dt+∫tm1TP1dt]=Cr[l1(Qh-Q)l1-1-l1Shl1-1].Scrap cost is as follows:(38)SR=Srθx[lQl-1-lx+l(Sh-S′)l-1-lx].Total deterioration cost is as follows:(39)DC=θ1Cp×[-(Bξ)n+2θ1(n+2)Rm2+S1n+1(n+1)Rm+θ1S1(n+2)Rm2(n+2)](Bξ)n+1Rc(n+1)+(Bξ)n+2θ1Rc2(n+2)+(Bξ)n+1Rm(n+1)-(Bξ)n+2θ1(n+2)Rm2+S1n+1(n+1)Rm+θ1S1(n+2)Rm2(n+2)].The total inventory cost per unit time is given by(40)TAC(Q,S′)=K+HC+DC+SC+PC+SC+RCT.Our problem is to find the time to stop the production whenq takes optimum value Q and the time to again start the production when maximum shortages accumulate. For minimum value of total cost, we must have
(C1)∂∂Q(TAC)=0,∂∂S′(TAC)=0,T[∂HC∂S′+∂SC∂S′+∂PC∂S′+∂DC∂S′+∂RC∂S′+∂SR∂S′]=(K+HC+DC+SC+PC+RC+SR)∂T∂S′,T[∂HC∂Q′+∂SC∂Q′+∂PC∂Q′+∂DC∂Q′+∂RC∂Q′+∂SR∂Q′]=(K+HC+DC+SC+PC+RC+SR)∂T∂Q′.Provided the following condition of Hessian matrix,H is satisfied, where
(41)H=[∂2TAC∂Q2∂2TAC∂Q∂S∂2TAC∂S∂Q∂2TAC∂S2].The first principle minor determinant ofH, |H11|>0 and the second principle minor determinant of H,|H22|>0 (see Appendix C).Using(C1). We get the relation
(C2)(∂HC∂S′+∂SC∂S′+∂PC∂S′+∂DC∂S′+∂RC∂S′+∂SR∂S′)iiiiiiii×(∂HC∂Q+∂SC∂Q+∂PC∂Q+∂DC∂Q+∂RC∂Q+∂SR∂Q)-1iiii=∂T/∂S′∂T/∂Q.


## 4.5. Special Case

If we relax some conditions so as to approach the basic EOQ model with shortages.Asx→0, ξ→0, θ1→0, α→1, n→1, and D=D1, we have the following results: DC=0, RC=0, SR=0, PC=Cp(l/(l-1))(Q-S′), HC=h1lQ2/2(l-1)D, SC=CsS′2l/2(l-1)D1, and T=Ql/(l-1)D+(l/(1-l)D1)S′.Substituting all the above values in(C1) and (C2) and solving for Q, we have
(42)Q*=2k(l-1)DcSlh1(cs+h1).Aslincreases, production occurs at a more rapid rate. Hence for large l, the model should approach the instantaneous delivery situation of the EOQ model. For largel,1-1/l→1. Thus aslincreases towards infinity, the optimal run size for the model approaches the EOQ when shortages are allowed.


## 5. Numerical Example

In this part, we have presented computational results obtained by using Mathematica 7.0 which give insight about the behavior of optimal run sizeQ*, production cycle timeT, and the effects of reverse manufacturing on the total average cost TAC. The parametric values in the models are taken as
(43)D=2,h1=h2=h3=0.5,K=200,l=2,l1=1.5,Cs=0.5,θ=0.1,Cp=0.6,Cr=0.4,θ1=0.0001,D1=1.5,x=0.1,Sr=0.3.The following observations are made from Table1.Table 1
Effect ofξ on optimal values of Q,T, and TAC when holding cost is linear.
α
=
1
n
=
1
ξ
0.2
0.4
0.6
0.8
1
Q
11.103
8.67645
7.12382
6.04315
5.24999
T
42.9282
42.02219
41.44897
37.75955
36.47535
HC
90.85782
87.58654
85.42628
83.89338
82.77261
SC
109.1283
112.4023
114.562
116.0942
117.2156
PC
40.43753
37.15183
35.05367
33.59508
32.52352
SR
0.202188
0.185759
0.175268
0.167975
0.162618
RC
1.950454
1.686772
1.61653
1.586091
1.569654
TAC
5.650744
5.700709
5.725624
6.243906
6.432465Case 1 (when holding cost is linear).
Consider the following.(i)
Asξ increases, production cost, scrap cost, and repair cost decrease. For higher values of ξ, the reusable items are produced more that result in less production from the forward manufacturing as some of the demand is adjusted by the reusable items. Consequently, less production leads to less production cost. The lower production also leads to lower defectives and hence decrease in scrap cost and repair cost.(ii)
Asξ increases, holding cost decreases and the cycle length also decreases. It is reasonable that lower holding cost and lower cycle length lead to more shortages and hence higher shortage cost. Consequently, there is slight increase in the total average cost.The following observations are made from Table2.Table 2
Effect ofξ on optimal values of Q,T, and TAC when holding cost is nonlinear.
α
=
1
n
=
2
ξ
0.2
0.4
0.6
0.8
1
Q
3.98022
2.86567
2.06824
1.55932
1.22518
T
31.06676
30.12545
29.4805
29.07657
28.81484
HC
50.06126
46.94452
44.62028
43.07698
42.03811
SC
100.0975
106.1109
110.7606
113.8455
115.9239
PC
28.74804
27.75043
27.06259
26.63059
26.35012
SR
0.14374
0.138752
0.135313
0.133153
0.131751
RC
1.890104
1.645637
1.592055
1.57176
1.561479
TAC
5.824253
6.060996
6.247208
6.371384
6.455194Case 2 (when holding cost is nonlinear).
In view of the reasons discussed in Case1, similar types of changes are observed in Case 2 also. (i)
For higher values ofξ, inventory level Q decreases and decrease in inventory level leads to lower inventory cost.(ii)
Asξ increases, inventory level of regular production falls and lower inventory level causes more shortages which results in higher shortage cost. While balancing between the inventory cost and shortage cost, there is slight increase in the total average cost.(iii)
Asξ increases, production cost, scrap cost, and repair cost decrease.The following are the comparative observations from Tables1 and 2.(i)
The optimal value ofQ is much higher in case of linear holding cost as compared to nonlinear holding cost. Generally, nonlinear holding cost leads to very high inventory cost as compared to linear holding cost. It is natural that higher inventory cost decreases the production rate to reduce the inventory cost. So it is very reasonable to have lower inventory level in case of nonlinear holding cost as compared to inventory level in case of linear holding cost.(ii)
The total average cost incurred is slightly less in case of linear holding cost as compared to nonlinear holding cost. Nonlinear holding cost causes high increase in the inventory cost. To avoid high increase, there is much decrease in the inventory level of Case2 as compared to Case 1. In spite of the lower inventory level in Case 2 as compared to Case 1, there is slight increase in the total average cost which shows the existence of nonlinear factor.The following observations are made from Table3.(i)
Asβ decreases, inventory level Q rises and holding cost also increases. Higher inventory leads to higher inventory cost.(ii)
Asβ decreases, scrap cost and production cost increase but repair cost slightly decreases.(iii)
Asβ decreases, total average cost slightly increases and cycle lengthTalso increases.Table 3
Effect ofα on optimal solution.
n
=
1
α
=
1
-
β
0.2
0.4
0.6
0.8
Q
3.55002
6.61844
8.76254
10.101
T
35.8966
36.316
37.6392
39.6777
HC
24.0557
33.5122
45.9702
64.2293
SC
98.64717
120.8368
123.672
119.2628
PC
27.93712
34.95404
38.46209
40.01435
SR
0.139686
0.17477
0.19231
0.200072
RC
1.880162
1.740152
1.673579
1.606051
TAC
4.252766
5.265391
5.578497
5.678569


## 6. Conclusion

This model addresses the various expected realistic features that usually arise while working on the optimal production policy that minimizes the associated inventory cost. It is very important to take the production system imperfect as no real system can be perfect in production. In view of the highly competitive situations in the real business problems, the production system cannot afford to be inflexible, therefore we have taken the production system flexible also. Along with the issues raised above, the issue of environmental protection by undertaking reverse manufacturing has also been incorporated and optimal production policy is derived which can reduce the inventory cost as much as possible and also can take care of the requirement of environmental protection. When remanufacturing is undertaken along with imperfect production, as the ratioξ increases, there is slight increase in the total average cost consisting of holding cost, shortage cost, deterioration cost, set-up cost, scrap cost, repair cost, and production cost. The total average cost incurred is slightly more in case of nonlinear holding cost as compared to linear holding cost. Optimal solution is derived for different values of the shape parameter. Further research can be extended to consider more realistic assumptions into the proposed model, for example, multiple production, stochastic nature of demand, machine breakdown, collection of used items during reverse manufacturing period, and so forth.



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*Source: 915675-2013-12-19.xml*