# An Optimal Returned Policy for a Reverse Logistics Inventory Model with Backorders

Authors: S. R. Singh; Neha Saxena
Journal: Advances in Decision Sciences
 (2012)

---

## Abstract

Environmental responsibility plays a significant role in the firm’s agendas nowadays. In this paper, we address the environmental operations of reverse logistics. Here we developed an integrated supply chain model with coordinated production and remanufacturing due to time-dependent rates. To study the problem we consider the demand to be satisfied with newly manufactured (produced) and the remanufactured products, so there is no difference between manufactured and remanufactured items. The shortages are allowed and excess demand is backlogged as well. The returned items are collected from the end user to be remanufactured. Optimal expression is obtained for the acceptable returned quantity, maximum inventory level, production and remanufacturing scheduling period, and the total average cost. Illustrative examples, which explain the application of the theoretical results as well as their numerical verifications, are given. The sensitivity of these solutions to change in underling parameter values is also discussed.


---

## Body

## 1. Introduction

Environmental issues are gaining reasonable attention among society, worldwide. Consumer demand for clean manufacturing and recycling is increasing. Consumers expect to trade in an old product when they buy a new one. Hence, for the past few decades the reverse flow of products from consumers to upstream businesses has received much interest. Due to the governmental regulations and consumer concerns regarding these environmental issues, an increasing number of companies have focused on reduction efforts in the amount of waste stream, diversion of the discarded products, and disposition of the retired products properly whereas reverse logistics is the process of retrieving the product from the end consumer for the purposes of proper disposal. To facilitate the reverse flow of used products from consumers to manufacturers in an efficient manner, the most appropriate approach is to create a reverse supply chain network. Wherein reverse logistics can take place through the original forward channel, through a separate reverse channel, or through combinations of the forward and the reverse channel. Generally, companies focus on setting up a reverse supply chain either because of environmental regulations or to reduce their operating cost by remanufactured products or components. For companies that utilise a reverse supply chain deals with handling and reprocess of repairable used products withdrawn from production and consumption process. Such a reuse is, for example, recycling or repair of spare parts. It has an advantage from economic point of view, as reduction of environmental load through return of used items in the manufacturing process.In the recent years researchers paid much attention to reverse logistics inventory models. There have been numerous studies and research on reverse logistics. In the past one approach adopted by many authors to the study of recovery systems is the use of Economic Ordering Quantity (EOQ) technique. The EOQ models are simple and they usually lead to closed-form solutions. The first reverse logistic model was investigated by Schrady [1]. He analyzed the problem in the EOQ model for repairable items which assumes that the production and repairing rates are instantaneous without disposal cost. Nahmias and Rivera [2] considered the model of Schrady [1] for the case of finite repair rate and limited storage in the repair and production shops. A very good review on quantitative models for recovery production planning and inventory control is given by Fleischmann et al. [3]. In this review they subdivided the field into three main areas, namely, distribution planning, inventory control, and production planning for each of these they discuss the implication of the emerging reuse efforts. There is a multiproduct generalization of EOQ-type reverse logistics models published by Mabini et al. [4]. They have extended the basic model of Schrady [1] with capital budget restriction. Richter [5, 6] investigated a modified version of the model of Schrady [1] by assuming multiple production and multiple repair cycles within a time interval. Most of the models investigated earlier are governed by two extreme (bang-bang) strategies, that is, “dispose all” or “recover all” [7]. In a similar work to Richter [5–7], Teunter [8] developed a deterministic EOQ inventory model with disposal option where recoverable and manufactured items have different holding costs and obtained a general finding similar to Richter [7]. Koh et al. [9] generalized the model of Nahmias and Rivera [2] by assuming a limited repair capacity. In a later study, several researchers have developed models along the same lines as Schrady and Richter, but with different assumptions, for example, Teunter [10], Inderfurth et al. [11], Dobos, and Richter [12, 13]. Dobos and Richter [14] explored their previous model by assuming that the quality of collected used items is not always suitable for further recycling. Konstantaras and Papachristos [15] have investigated an inventory model for stability. In his next work [16] he extended the work of Koh et al. [9] and followed a different analysis to obtain closed form expressions for both optimal number of set up in the recovery and the ordering processes. Konstantaras and Papachristos [17] extended the work of Teunter [10] by introducing the exact solution method for the same model. Jaber and El Saadany [18] extend the work of Richter [5, 6] by assuming the newly produced and remanufactured items are perceived differently by customers. Omar and Yeo [19] developed a production model that satisfies a continuous time varying demand for finished goods over a known and finite planning horizon by supplying either new products or repaired used products. The extended version of Dobos and Richter [12, 13] is made by El Saadany and Jaber [20] by assuming the returned rate of used items follows a demand like function depend on the purchasing price and acceptance quality level of returns. Alamri [21] derives a general reverse logistics inventory model for deteriorating items taking the returned rate as a decision variable. Another extension of Koh et al. [9] is made by Konstantaras et al. [22] by introducing the inspection and sorting of returned items. Hasanov et al. [23] extended the work Jaber and El Saadany [18] by assuming that unfulfilled demand for remanufacturing and produced items is either fully or partially backordered. This paper also considered the scenario of overlapping of one production and one remanufacturing cycle to minimize the effect of stock outs. A closed-loop supply chain inventory model is developed by Yang et al. [24], (in press). In this paper he considers price-sensitive demand and multiretailer and analysis of the problem with three optimization methods sequential optimization, a centralized optimization without benefit sharing, and a centralized optimization with benefit sharing. The comparative review is given in Table 1.Table 1
The comparative review in the tabular form.
References
Production rate
Remanufacturing/repairing rate
Demand rate
Returned rate
The quality of the remanufactured items
Shortages
Schrady [1]
Not considering
Constant
Constant
Constant
As-good-as new
Not allowed
Nahmias and Rivera [2]
Not considering
Constant
Constant
Constant
As-good-as new
Not allowed
Richter [5–7]
Constant
Constant
Constant
Constant
As-good-as new
Not allowed
Dobos and Richter [12–14]
Demand dependent
DemandDependent
Constant
Demand dependent
As-good-as new
Not allowed
Konstantaras and Papachristos [15]
Constant
Constant
Constant
Constant
As-good-as new
Not allowed
Konstantaras and Papachristos [16], Teunter [10]
Constant
Constant
Constant
Demand dependent
As-good-as new
Not allowed
Konstantaras and Papachristos  [17], Koh et al., [9]
Not considering
Constant
Constant
Constant
As-good-as new
Not allowed
Konstantaras and Skouri, [25]
Constant
Constant
Constant
Constant
As-good-as new
Allowed
Konstantaras et al. [22]
Not considering
Constant
Constant
Constant
Different from the new items
Not allowed
Jaber and  El Saadany, [18]
Demand dependent
DemandDependent
Constant
Demand dependent
Different from the newly produced
Not allowed
El Saadany and Jaber [20]
Demand dependent
DemandDependent
Constant
Price and quality   dependent
As-good-as new
Not allowed
Jaber and El Saadany, [26]
Demand dependent
DemandDependent
Constant
Demand dependent
As-good-as new
Not allowed
Alamri, [21]
Arbitrary function of time
Arbitrary function of time
Arbitrary function of time
Arbitrary function of time
As-good-as new
Not allowed
Hasanov et al. [23]
Constant
Constant
Constant
Constant
Different from the newly produced
Allowed
Yang et al., (in press) [24]
Constant
Constant
Price sensitive
Constant
As-good-as new
Not allowed
The present paper
Arbitrary function of time
Arbitrary function of time
Arbitrary function of time
Arbitrary function of time
As-good-as new
AllowedUsually in most of the models shortages are not permitted to occur. However, in many practical situations, stock out is unavoidable due to various uncertainties. Therefore, the occurrence of shortages in inventory is a natural phenomenon. In this paper we use this phenomenon. In the proposed model we determined the coordination of reverse manufacturing with the forward supply chain in the inventory management. The reverse logistics operations deal with the collection of returns, cleaning of the collected returns, and remanufacturing of the reusable collected items. The quality of the remanufactured items is assumed to be good as those of new products hence the demand is to be satisfied with newly manufactured (produced) and the remanufactured products. A general framework of the system is shown in Figure1. In Section 1, a comprehensive literature review and background of the model are presented. Section 2 is for assumption and notations. Section 3 demonstrates the model development. Section 4 presents the solution procedure to solve the optimization problem. Section 4.1 shows three numerical examples to illustrate the model and sensitivity analysis is presented in Section 4.2. A particular case of the given problem is given in Section 4.3. Concluding remarks are derived and future research topics are suggested in Section 5.Figure 1
The flow of material in the integrated inventory model.


## 2. Assumptions and Notations

In this paper the subscript “m” is used to indicate the quantity corresponds to the remanufactured stock, we will use the subscript “r” to indicate the quantity corresponds to the remanufactured stock and the subscript “R” to indicate the quantity corresponds to the returned stock.The model is developed with the following assumptions and notations.(i)
New products are produced at a rate ofPm(t).(ii)
Repairable used products are collected at a rate ofR(t) and then remanufactured at a rate of Pr(t). All the returned items are remanufactured.(iii)
Demand is satisfied from the newly produced and remanufactured items at a rate ofD(t).(iv)
Pm(t),R(t),Pr(t), and D(t) are assumed to be arbitrary functions of time.(v)
We will require that
(a)
Pr(t)>D(t),Pm(t)>D(t),D(t)>R(t),(b)
Pr(t)>R(t),D(t)≠0,R(t)≠0.(vi)
Shortage is allowed andcs is the unit shortage cost.(vii)
Cost parameters related to manufactured products are as follows.
Im(t)
is the inventory level at timet related to manufactured products.q2
is the maximum inventory level in the production process.cm
is the unit procurement cost.sm
is the unit production cost.hm
is the unit holding cost per unit time.km
is the setup cost per production cycle.(viii)
The cost parameters related to remanufactured products are as follows.
Ir(t)
is the inventory level at timet related to remanufactured products.q1
is the maximum inventory level in the remanufacturing process.sr
is the unit remanufacturing cost.hr
is the unit holding cost per unit time.kr
is the setup cost per remanufacturing cycle.(ix)
The cost parameters related to returns are as follows.
IR(t)
is the inventory level at timet related to returns.cR
is the unit acquisition cost.hR
is the unit holding cost per unit time.kR
is the setup cost per returned cycle.


## 3. Mathematical Modelling and Analysis

The cycle now starts at timet0 with backorders. At this instant of time, remanufacturing starts to clear the backlog by the time t1 and the inventory level Ir(t) increases at a rate Pr(t)-D(t) until the time t2 where stock level reaches its maximum value. Then the remanufacturing is stopped and hence the demand depletes the inventory level Ir(t) during the period  (t2,t3) and falls to zero at t=t3, thereafter shortages occur during the period (t3,t4) due to the absence of stock. At this instant of time, fresh production starts to clear the backlog by the time t5. Production raises the inventory level Im(t) at a rate Pm(t)-D(t) and reaches its maximum at time t=t6. Then the production is stopped and hence the demand depletes the inventory level Im(t) until the time t7 by which it becomes zero. Now shortages start developing and accumulate to their maximum (equal to the shortage level at time t=t0) at the time t=t8. However for each returned cycle the inventory level IR(t) is affected by the returned rate and the remanufacturing rate, as the remanufacturing process starts at t0, the stock level declines at a rate R(t)-Pr(t) and falls to zero at t=t2 by which the remanufacturing stops. Now the stock level increases at a rate R(t) by the time t=t8. This is depicted in the Figure 2.Figure 2
Inventory variation of an EPQ model for Reverse logistics system.The differential equations governing the stock level during the periodt0≤t≤t8 can be written as
(3.1)dIr(t)dt=Pr(t)-D(t)Ir(t1)=0,t0≤t≤t2,dIr(t)dt=-D(t)Ir(t3)=0,t2≤t≤t4,dIm(t)dt=Pm(t)-D(t)Im(t5)=0,t4≤t≤t6,dIm(t)dt=-D(t)Im(t7)=0,t6≤t≤t8,dIR(t)dt=R(t)-Pr(t)IR(t2)=0,t0≤t≤t2,dIR(t)dt=R(t)IR(t2)=0,t2≤t≤t8.
Solutions of the above differential equations using their boundary conditions are
(3.2)Ir(t)=-∫tt1(Pr(u)-D(u))du,t0≤t≤t1,Ir(t)=∫t1t(Pr(u)-D(u))du,t1≤t≤t2,Ir(t)=∫tt3D(u)du,t2≤t≤t3,Ir(t)=-∫t3tD(u)du,t3≤t≤t4,(3.3)Im(t)=-∫tt5(Pm(u)-D(u))du,t4≤t≤t5,(3.4)Im(t)=∫t5t(Pm(u)-D(u))du,t5≤t≤t6,Im(t)=∫tt7D(u)du,t6≤t≤t7,Im(t)=-∫t7tD(u)du,t7≤t≤t8,IR(t)=∫tt2(Pr(u)-R(u))du,t0≤t≤t2,IR(t)=∫t2tR(u)du,t2≤t≤t8.
Let I(x1,x2)=∫x1x2I(u)du, then from (3.3)–(3.6) we have
(3.5)Ir(t0,t1)=-∫t0t1u(Pr(u)-D(u))du,Ir(t1,t2)=∫t1t2(t2-u)(Pr(u)-D(u))du,Ir(t2,t3)=∫t2t3(u-t2)D(u)du,(3.6)Ir(t3,t4)=∫t3t4(u-t4)D(u)du,(3.7)Im(t4,t5)=∫t4t5(t4-u)(Pm(u)-D(u))du,Im(t5,t6)=∫t5t6(t6-u)(Pm(u)-D(u))du,Im(t6,t7)=∫t6t7(u-t6)D(u)du,Im(t7,t8)=∫t7t8(u-t8)D(u)du,IR(t0,t2)=∫t0t2u(Pr(u)-R(u))du,IR(t2,t8)=∫t2t8(t8-u)R(u)du.
Without loss of generality let us assume t0=0 then the per cycle cost components for the given inventory model are as follows:
(3.8)Procurementandacquisitioncost=cm∫t4t6Pm(u)du+cR∫0t8R(u)du,Productionandremanufacturingcost=sm∫t4t6Pm(u)du+sr∫0t2Pr(u)du,Holdingcost=hr[Ir(t1,t2)+Ir(t2,t3)]+hm[Im(t5,t6)+Im(t6,t7)]+hR[IR(0,t2)+IR(t2,t8)],Shortagecost=cs[-Ir(0,t1)-Ir(t3,t4)-Im(t4,t5)-Im(t7,t8)].
Hence the total cost per unit time of the given inventory model during the cycle [0,t8] as a function of t1,t2,t3,t4,t5,t6,t7, and t8 say Z(t1,t2,t3,t4,t5,t6,t7,t8) is given by
(3.9)Z(t1,t2,…,t8)=1t8{(cm+sm)∫t4t6Pm(u)du+cR∫0t8R(u)du+sr∫0t2Pr(u)du+hr[∫t1t2(t2-u)(Pr(u)-D(u))du+∫t2t3(t-t2)D(u)du]+hm[∫t5t6(t6-u)(Pm(u)-D(u))du+∫t6t7(u-t6)D(u)du]+hR[∫0t2u(Pr(u)-R(u))du+∫t2t8(t8-u)R(u)du]+cs[∫0t1u(Pr(u)-D(u))du-∫t3t4(u-t4)D(u)du-∫t4t5(t4-u)×(Pm(u)-D(u))du-∫t7t8(u-t8)D(u)du]+kr+km+kR}.
Here we have a cost function of the system in terms of t1,t2,t3,t4,t5,t6,t7, and t8. To find the optimum solution we have to find the optimum value of t1,t2,t3,t4,t5,t6,t7, and t8 that minimize Z(t1,t2,…,t8) but we have some relations between the variables as follows. 						(3.10)0≤t1≤t2≤t3≤t4≤t5≤t6≤t7≤t8,(3.11)∫t1t2(Pr(t)-D(t))dt=∫t2t3D(t)dt,(3.12)∫t3t4D(t)dt=∫t4t5(Pm(t)-D(t))dt,(3.13)∫t5t6(Pm(t)-D(t))dt=∫t6t7D(t)dt,(3.14)∫0t1(Pr(t)-D(t))dt=∫t7t8D(t)dt,(3.15)∫0t8R(t)dt=∫0t2Pr(t)dt.
Equation (3.10) is an essential condition for the existence of the model. Equation (3.11) show that the inventory levels Ir(t) are same at the time t=t2, similarly (3.12) and (3.13) show that the inventory level Im(t) and Ir(t) are same at the time t=t4 and t=t6, respectively, depicted in Figure 2. Equation (3.14) shows that the backorders at the time t=0 and t=t8 are alike as we have already assumed in the formulation of model. According to the presented model all the units buyback from the market will be remanufactured (no disposal) hence the total remanufacturing during the cycle is equal to the total returned items in the complete cycle which is demonstrated in (3.15).


## 4. Solution Procedure

LetQ be the acceptable returned quantity for used items in the interval [0,t8]:
(4.1)Q=∫0t8R(u)du.
And let we have the maximum inventory level q1 and q2 in remanufacturing and production process, respectively, so that from (3.11) and (3.13) we get
(4.2)∫t1t2(Pr(t)-D(t))dt=∫t2t3D(t)dt=q1,(4.3)∫t5t6(Pm(t)-D(t))dt=∫t6t7D(t)dt=q2.
Therefore from (3.11), (3.12),…, (4.2), and (4.3) the value of t1,t2,t3,t4,t5,t6,t7, and t8 can be determined as a function of Q,q1, and q2.Therefore the total variable cost function will be the function of three variablesQ,q1, and q2 which is Z(Q,q1,q2):
(4.4)Z(Q,q1,q2)=1t8{(cm+sm)∫t4t6Pm(u)du+cR∫0t8R(u)du+sr∫0t2Pr(u)du+hr[∫t1t2(-u)(Pr(u)-D(u))du+∫t2t3uD(u)du]+hm[∫t5t6(-u)(Pm(u)-D(u))du+∫t6t7uD(u)du]+hR[∫0t2u(Pr(u)-R(u))du+∫t2t8(f8-u)R(u)du]+cs[∫0t1u(Pr(u)-D(u))du-∫t3t4uD(u)du+∫t4t5u(Pm(u)-D(u))du-∫t7t8uD(u)du]+kr+km+kR},
where the values of t1,t2,t3,t4,t5,t6,t7, and t8 are given in the next section.To obtain the optimal solution of the proposed problem, we minimize the functionsZ(Q,q1,q2) with respect to Q,q1, and q2 Taking the first order derivatives of Z(Q,q1,q2) with respect to Q,q1, and q2. Thereafter setting it equal to zero gives
(4.5)∂Z(Q,q1,q2)∂Q=0,∂Z(Q,q1,q2)∂q1=0,∂Z(Q,q1,q2)∂q2=0.
The optimal value of Q,q1, and q2 can be derived from the above equations when the following conditions of Hessian matrix are satisfied. The Hessian matrix is
(4.6)H=(∂2Z∂Q2∂2Z∂Qq1∂2Z∂Qq2∂2Z∂q1Q∂2Z∂q12∂2Z∂q1q2∂2Z∂q2Q∂2Z∂q2q1∂2Z∂q22).
The first principal minor determinant of H, |H11|>0. The second principal minor determinant of H, |H22|>0 and the third principal minor determinant of H, |H33|>0.


### 4.1. Illustrative Examples with Numerical Analysis

Example 4.1.
The model is developed with linearly time-dependent demand, production, remanufacturing, and returned rates, in this example we consider(4.7)Pm(t)=am+bmt,Pr(t)=ar+brt,R(t)=c+dt,D(t)=α+βt.
On the basis of these above demand, production, remanufacturing, and returned rates we calculate the theoretical results and the total cost function as defined in the previous section.
From (4.1) we have
(4.8)Q=∫0t8R(u)du=∫0t8(c+du)du=dt822+ct8.
From which t8 is given by
(4.9)t8=-c+c2+2dQd.
From (3.15) and (4.1) we have
(4.10)Q=∫0t8R(u)du=∫0t2Pr(u)du=∫0t8(ar+bru)du.
Hence t2 is given by
(4.11)t2=-ar+ar2+2brQbr.
From (4.2) we get
(4.12)∫t1t2(Pr(t)-D(t))dt=q1(ar-α)(t2-t1)+(br-β)2(t22-t12)=q1.
By which we can find t1, say
(4.13)t1=-2ar+2α+(2ar-2α)2-4(br-β)(2q1-2art2-brt22+2t2α+t22β)2(br-β).
From (4.2) we have
(4.14)∫t2t3D(t)dt=q1α(t3-t2)+β2(t32-t22)=q1.
From which we can find t3, say
(4.15)t3=-α+α2+2q1β+2t2αβ+t22β2β.
From (3.14) we have
(4.16)∫0t1(Pr(t)-D(t))dt=∫t7t8D(t)dt(ar-α)t1+(br-β)2t12=α(t8-t7)+β2(t82-t72).
Hence the value of t7 is given by
(4.17)t7=-α+α2-2art1β-brt12β+2t1αβ+2t8αβ+t12β2+t82β2β.
From (4.3) we have
(4.18)∫t6t7D(t)dt=q2⇒α(t7-t6)+β2(t72-t62)=q2.
From which we can determine the value of t6, say
(4.19)t6=-α+α2-2q2β+2t7αβ+t72β2β.
Similarly from (4.3)
(4.20)∫t5t6(Pm(t)-D(t))dt=q2.
From which we can find t5(4.21)t5=-2am+2α+(2am-2α)2-4(bm-β)(2q2-2amt6-bmt62+2t6α+t62β)2(bm-β).
From (3.12) we have
(4.22)∫t3t4D(t)dt=∫t4t5(Pm(t)-D(t))dt(am-α)(t5-t4)+(bm-β)2(t52-t42)=α(t4-t3)+β2(t42-t32).
From which we can find t4(4.23)t4=-am+am2+2ambmt5+bm2t52+2bmt3α-2bmt5α+bmt32β-bmt52βbm.Example 4.2.
The above theoretical results are illustrated through the numerical verification. To illustrate the proposed model, we have considered the input parameters as given below.
c
m
=
8 $/unit, sm=3 $/unit, sr=2 $/unit, cR=2 $/unit, cs=5 $/unit, am=1500 unit/month, bm=30 unit/month2, α=800 unit/month, β=35 unit/month2, ar=1800 unit/month, br=30 unit/month2, hR=1 $/unit/month, hm=1.5 $/unit/month, km=1500 $/cycle, kR=1000 $/cycle, hr=1 $/unit/month, kr=800 $/cycle, c=600 unit/month, d=28 unit/month2.
Applying the solution procedure given in the last section we derive the optimal solution and results are presented in Table2.Table 2
The optimal results for the inventory model under the above parametric values as in Example4.2.
Q
*
q
1
*
q
2
*
t
1
*
t
2
*
t
3
*
t
4
*
t
5
*
t
6
*
t
7
*
t
8
*
Z
(
Q
*
,
q
1
*
,
q
2
*
)
1712
781
191
0.160528
0.94369
1.86346
1.92922
2.01177
2.28889
2.50498
2.6851
7616.1
units
units
units
Month
month
month
Month
month
Month
Month
month
$The convexity of the reverse logistics inventory model is shown in Figure3. The three dimensional graph shows that the integrated expected total annual cost is convex, and that there exists a unique solution minimizing the integrated expected total annual cost.Convexity of the reverse logistics inventory model is derived, and when the optimal value of acceptable returned quantityQ, maximum inventory level q1 and q2 is taken.
(a)
(b)
(c)


### 4.2. Sensitivity Analysis

To study the effects of the parameter changes on the optimal solutions derived by the proposed method, this investigation performs a sensitivity analysis by increasing or decreasing the parameters, one at a time.The main conclusions drawn from the sensitivity analysis given above are as follows.(i)
From Tables3 and 4 we have observed that as the production rate increases the optimum value of total acceptable returned quantity decreases and hence the total minimum cost slightly decreases. However we observe a little but unexpected increment in the total minimum average cost and this increment happened due to the reduction in total cycle length.(ii)
Similarly from Tables5 and 6 we have observed that as the remanufacturing rate increases the optimum value of total acceptable returned quantity decreases and hence the total minimum cost slightly decreases. But due to the reduction in total cycle length we observe a little increment in the total minimum average cost.(iii)
From Tables7 and 8 it is observed that as the return rate increases the optimum value of total acceptable returned quantity increases while the total time required to produce the optimum manufactured quantity decrease. Hence the total procurement and production cost will decrease. This reduction resulted in the decrease in the total minimum average cost.(iv)
Similarly from Table9 it is observed that as the constant demand rate increases the optimal values of total acceptable returned quantity and the total time required to produce the optimum manufactured quantity increase. Hence the total procurement, acquisition, and production cost increases. This increment resulted in the increase in the total minimum average cost per cycle.(v)
Now from Table10 it is observed that as the variable demand rate increases the optimal values of total acceptable returned quantity decrease and hence the total minimum cost slightly decreases. But due to the reduction in total cycle length we observe a little increment in the total minimum average cost.Table 3
For the different constant production rates optimal results for the same set of values as in Example4.2.
a
m
Q
*
q
1
*
q
2
*
t
1
*
t
2
*
t
3
*
t
4
*
t
5
*
t
6
*
t
7
*
t
8
*
Z
(
Q
*
,
q
1
*
,
q
2
*
)
1200
1719
785
134
0.1603
0.9475
1.8717
1.9177
2.0198
2.3642
2.5156
2.6954
7605.77
1350
1715
783
165
0.1601
0.9453
1.8673
1.9249
2.0173
2.3233
2.5098
2.6895
7611.48
1500
1712
781
191
0.1605
0.9436
1.8634
1.9292
2.0117
2.2888
2.5049
2.6851
7616.1
1650
1709
780
212
0.1598
0.9420
1.8607
1.9335
2.0086
2.2612
2.5012
2.6806
7619.9
1800
1707
779
229
0.1598
0.9409
1.8585
1.9379
2.0075
2.2389
2.49833
2.6777
7623.08Table 4
Optimal results for the same set of values as in Example4.2 for different variable production rates.
b
m
Q
*
q
1
*
q
2
*
t
1
*
t
2
*
t
3
*
t
4
*
t
5
*
t
6
*
t
7
*
t
8
*
Z
(
Q
*
,
q
1
*
,
q
2
*
)
28
1712
782
190
0.159527
0.94369
1.86462
1.93053
2.01376
2.29116
2.5061
2.6851
7615.98
29
1712
781
191
0.160528
0.94369
1.86346
1.92873
2.01091
2.28889
2.50498
2.6851
7616.04
30
1712
781
191
0.160528
0.94369
1.86346
1.92922
2.01177
2.28889
2.50498
2.6851
7616.1
31
1711
781
191
0.159983
0.943143
1.86293
1.92902
2.01175
2.288
2.5041
2.68362
7616.15
32
1711
781
191
0.159983
0.943143
1.86293
1.92951
2.01261
2.288
2.5041
2.68362
7616.21Table 5
Optimal results for the same set of values as in Example4.2 for different constant remanufacturing rates.
a
r
Q
*
q
1
*
q
2
*
t
1
*
t
2
*
t
3
*
t
4
*
t
5
*
t
6
*
t
7
*
t
8
*
Z
(
Q
*
,
q
1
*
,
q
2
*
)
1440
1823
661
202
0.2109
1.2497
2.0207
2.0903
2.1782
2.4717
2.69856
2.8489
7479.26
1620
1759
730
202
0.1814
1.0751
1.9312
1.9914
2.0671
2.3603
2.5882
2.7546
7556.35
1800
1712
781
191
0.1605
0.9436
1.8634
1.9292
2.0117
2.2888
2.5049
2.6851
7616.1
1980
1676
822
187
0.1430
0.8411
1.8122
1.8772
1.9586
2.2298
2.44188
2.6317
7663.79
2160
1648
855
184
0.1292
0.7589
1.7716
1.8359
1.9161
2.1829
2.39203
2.5901
7702.75Table 6
Optimal results for the same set of values as in Example4.2 for different variable remanufacturing rates.
b
r
Q
*
q
1
*
q
2
*
t
1
*
t
2
*
t
3
*
t
4
*
t
5
*
t
6
*
t
7
*
t
8
*
Z
(
Q
*
,
q
1
*
,
q
2
*
)
28
1712
781
191
0.160147
0.944177
1.86393
1.92969
2.01223
2.28935
2.50543
2.6851
7615.79
29
1712
781
191
0.160338
0.943934
1.86369
1.92945
2.012
2.28912
2.5052
2.6851
7615.95
30
1712
781
191
0.160528
0.94369
1.86346
1.92922
2.01177
2.28889
2.50498
2.6851
7616.1
31
1711
781
191
0.160173
0.9429
1.8627
1.92831
2.01066
2.28777
2.50387
2.68362
7616.25
32
1711
782
191
0.159362
0.942657
1.86362
1.92922
2.01156
2.28868
2.50477
2.68362
7616.4Table 7
Optimal results for the same set of values as in Example4.2 for different constant return rates.
c
Q
*
q
1
*
q
2
*
t
1
*
t
2
*
t
3
*
t
4
*
t
5
*
t
6
*
t
7
*
t
8
*
Z
(
Q
*
,
q
1
*
,
q
2
*
)
480
1460
668
315
0.1490
0.8769
1.7353
1.9548
2.2299
2.5963
2.87761
3.0425
8251.8
540
1590
726
252
0.1490
0.8769
1.7353
1.8226
1.9316
2.2971
2.58178
2.7485
7917.36
600
1712
781
191
0.1605
0.9436
1.8634
1.9292
2.0117
2.2888
2.5049
2.6851
7616.1
660
1826
833
133
0.1705
1.0060
1.9833
2.0284
2.0853
2.2783
2.42905
2.6209
7318.71
720
1934
882
77
0.1802
1.0649
2.0961
2.1226
2.1562
2.2680
2.35546
2.5588
7024.89Table 8
Optimal results for the same set of values as in Example4.2 for different variable return rates.
d
Q
*
q
1
*
q
2
*
t
1
*
t
2
*
t
3
*
t
4
*
t
5
*
t
6
*
t
7
*
t
8
*
Z
(
Q
*
,
q
1
*
,
q
2
*
)
26
1697
775
192
0.15836
0.935485
1.84862
1.9157
1.99985
2.2784
2.4957
2.67347
7629.03
27
1704
778
192
0.159172
0.939314
1.85578
1.92156
2.0041
2.28265
2.49992
2.67857
7622.57
28
1712
781
191
0.160528
0.94369
1.86346
1.92922
2.01177
2.28889
2.50498
2.6851
7616.1
29
1719
785
190
0.160338
0.947518
1.87177
1.93732
2.01965
2.29532
2.51023
2.69011
7609.62
30
1711
782
191
0.161149
0.951347
1.87892
1.94426
2.02635
2.30059
2.51432
2.69508
7603.14Table 9
Optimal results for the same set of values as in Example4.2 for different constant demand rates.
α
Q
*
q
1
*
q
2
*
t
1
*
t
2
*
t
3
*
t
4
*
t
5
*
t
6
*
t
7
*
t
8
*
Z
(
Q
*
,
q
1
*
,
q
2
*
)
640
1645
871
53
0.1544
0.9070
2.1625
2.1845
2.2030
2.2654
2.33903
2.5856
5949.82
720
1683
829
129
0.1582
0.9278
2.0026
2.0517
2.1022
2.2700
2.43081
2.6421
6774.6
800
1712
781
191
0.1605
0.9436
1.8634
1.9292
2.0117
2.2888
2.5049
2.6851
7616.1
880
1735
729
236
0.1614
0.9562
1.7424
1.8180
1.9347
2.3220
2.56647
2.7191
8467.36
960
1756
674
265
0.1626
0.9677
1.638
1.7164
1.8670
2.3675
2.62061
2.7501
9323.28Table 10
Optimal results for the same set of values as in Example4.2 for different variable demand rates.
β
Q
*
q
1
*
q
2
*
t
1
*
t
2
*
t
3
*
t
4
*
t
5
*
t
6
*
t
7
*
t
8
*
Z
(
Q
*
,
q
1
*
,
q
2
*
)
33
1732
792
192
0.1613
0.954628
1.88977
1.95572
2.03776
2.31463
2.53282
2.71471
7586.86
34
1722
787
191
0.160409
0.949159
1.87717
1.94361
2.02664
2.30292
2.51948
2.69991
7601.52
35
1712
781
191
0.160528
0.94369
1.86346
1.92922
2.01177
2.28889
2.50498
2.6851
7616.1
36
1702
776
190
0.159656
0.93822
1.85094
1.91713
2.00059
2.27709
2.49157
2.67029
7630.6
37
1692
772
190
0.157792
0.93275
1.83961
1.90504
1.98791
2.26524
2.47926
2.65547
7645.01


### 4.3. A Particular Case Whenq1 and q2 Are Constant

In a particular case let us haveq1 and q2 the maximum inventory level in the remanufacturing and production process, as a constant then from (4.9)–(4.23) the values of t1,t2,t3,t4,t5,t6,t7, and t8 can be found in the form of single variable Q, say
(4.24)ti=fi(Q),where1≤i≤8.
From (4.4) the total cost function will be the function of one variable, say Q and the problem will be converted into the following unconstrained problem with one variable Q:
(4.25)W(Q)=1f8{(cm+sm)∫f4f6Pm(u)du+cR∫0f8R(u)du+sr∫0f2Pr(u)du+hr[∫f1f2(-u)(Pr(u)-D(u))du+∫f2f3uD(u)du]+hm[∫f5f6(-u)(Pm(u)-D(u))du+∫f6f7uD(u)du]+hR[∫0f2u(Pr(u)-R(u))du+∫f2f8(f8-u)R(u)du]+cs[∫0f1u(Pr(u)-D(u))du-∫f3f4uD(u)du+∫f4f5u(Pm(u)-D(u))du-∫f7f8uD(u)du]+Kr+Km+KR}.
Now the necessary conditions for having a minimum for the problem
(4.26)dWdQ=0.
To find the solution of (4.26), let W=w/f8 then
(4.27)dWdQ=wQ′f8-f8.Q′wf82,
where, wQ′ and f8.Q′ are the derivatives of w and f8 (w.r.t.) Q, respectively. Hence, (4.26) is equivalent to
(4.28)wQ′f8=f8.Q′w.
Taking the first derivative of both sides of (3.11), (3.12), (3.13), (3.14), and (3.15) with respect to Q. We obtain
(4.29)f2.Q′(Pr(f2)-D(f2))-f1.Q′(Pr(f1)-D(f1))=f3.Q′D(f3)-f2.Q′D(f2),(4.30)f4.Q′D(f4)-f3.Q′D(f3)=f5.Q′(Pm(f5)-D(f5))-f4.Q′(Pm(f4)-D(f4)),(4.31)f6.Q′(Pm(f6)-D(f6))-f5.Q′(Pm(f5)-D(f5))=f7.Q′D(f7)-f6.Q′D(f6),f1.Q′(Pr(f1)-D(f1))=f8.Q′D(f8)-f7.Q′D(f7),f8.Q′R(f8)=f2.Q′Pr(f2).
As we have q1 and q2 are constant so that from (4.2) and (4.3) both sides of (4.29) and (4.30) will be equal to zero
(4.32)f2.Q′(Pr(f2)-D(f2))-f1.Q′(Pr(f1)-D(f1))=f3.Q′D(f3)-f2.Q′D(f2)=0(4.33)f4.Q′D(f4)-f3.Q′D(f3)=f5.Q′(Pm(f5)-D(f5))-f4.Q′(Pm(f4)-D(f4))=0.
Form the above equation we can find the values of f1.Q′,f2.Q′,f3.Q′,f4.Q′,f5.Q′,f6.Q′,f7.Q′, and f8.Q′.From (4.29)–(4.33), and (4.25) we have
(4.34)wQ′=(cm+sm){f6.Q′Pm(f6)-f4.Q′Pm(f4)}+cRf8.Q′R(f8)+srf2.Q′Pr(f2)+hr{f1.Q′(f1-f2)(Pr(f1)-D(f1))+f3.Q′(f3-f2)D(f3)}+hm{f5.Q′(f5-f6)(Pm(f5)-D(f5))-f7.Q′(f7-f6)D(f7)}+hR{f8.Q′f2R(f8)-f2.Q′f8R(f2)}+cs{f3.Q′(f3-f4)D(f3)+f5.Q′(f5-f4)(Pm(f5)-D(f5))+f8.Q′f1D(f8)+f7.Q′(f7-f8-f1)D(f7)}.From which and (4.28) we obtain
(4.35)w=f8f8.Q′[(cm+sm){f6.Q′Pm(f6)-f4.Q′Pm(f4)}+cRf8.Q′R(f8)+srf2.Q′Pr(f2)+hR{f8.Q′f2R(f8)-f2.Q′f8R(f2)}+hr{f1.Q′(f1-f2)(Pr(f1)-D(f1))+f3.Q′(f3-f2)D(f3)}+hm{f5.Q′(f5-f6)(Pm(f5)-D(f5))-f7.Q′(f7-f6)D(f7)}+cs{f3.Q′(f3-f4)D(f3)+f5.Q′(f5-f4)(Pm(f5)-D(f5))+f8.Q′f1D(f8)+f7.Q′(f7-f8-f1)D(f7)}],
where W is given by (4.25) and wQ′ is given by (4.34).Equation (4.35) can now, be used to determine the optimal value of Q. If we get more than one solution we choose the one for which the condition d2W/dQ2>0 is satisfied.Then the optimal values oft1,t2,t3,t4,t5,t6,t7, and t8 can be determined by using (4.9)–(4.23). From the above-mentioned results we can determine the value of wQ′ and w. Hence from (4.25) we can get the minimum total average cost.Example 4.3.
The same set of input data are considered as in the Example4.2 except that the constant values of q1 and q2 are as follows: q1=800 units, q2=200 units.
Applying the solution procedure given in last section we find that the optimal value of the acceptable returned quantityQ is 1737 and hence the corresponding optimal results are presented in Table 11.
The convexity of the total cost function has been established graphically in Figure4 it has also been verified numerically that the cost function is convex.Table 11
The optimal results for the inventory model under the parametric values given in Example4.2.
Q
*
t
1
*
t
2
*
t
3
*
t
4
*
t
5
*
t
6
*
t
7
*
t
8
*
Z
(
Q
*
,
q
1
*
,
q
2
*
)
1737
0.155131
0.957362
1.89856
1.95774
2.03215
2.32237
2.5483
2.7221
7616.6
units
Month
month
month
Month
Month
Month
Month
month
$Figure 4
Behavior of the inventory cost function, when the values ofq1 and q2 are taken to be constant.


## 4.1. Illustrative Examples with Numerical Analysis

Example 4.1.
The model is developed with linearly time-dependent demand, production, remanufacturing, and returned rates, in this example we consider(4.7)Pm(t)=am+bmt,Pr(t)=ar+brt,R(t)=c+dt,D(t)=α+βt.
On the basis of these above demand, production, remanufacturing, and returned rates we calculate the theoretical results and the total cost function as defined in the previous section.
From (4.1) we have
(4.8)Q=∫0t8R(u)du=∫0t8(c+du)du=dt822+ct8.
From which t8 is given by
(4.9)t8=-c+c2+2dQd.
From (3.15) and (4.1) we have
(4.10)Q=∫0t8R(u)du=∫0t2Pr(u)du=∫0t8(ar+bru)du.
Hence t2 is given by
(4.11)t2=-ar+ar2+2brQbr.
From (4.2) we get
(4.12)∫t1t2(Pr(t)-D(t))dt=q1(ar-α)(t2-t1)+(br-β)2(t22-t12)=q1.
By which we can find t1, say
(4.13)t1=-2ar+2α+(2ar-2α)2-4(br-β)(2q1-2art2-brt22+2t2α+t22β)2(br-β).
From (4.2) we have
(4.14)∫t2t3D(t)dt=q1α(t3-t2)+β2(t32-t22)=q1.
From which we can find t3, say
(4.15)t3=-α+α2+2q1β+2t2αβ+t22β2β.
From (3.14) we have
(4.16)∫0t1(Pr(t)-D(t))dt=∫t7t8D(t)dt(ar-α)t1+(br-β)2t12=α(t8-t7)+β2(t82-t72).
Hence the value of t7 is given by
(4.17)t7=-α+α2-2art1β-brt12β+2t1αβ+2t8αβ+t12β2+t82β2β.
From (4.3) we have
(4.18)∫t6t7D(t)dt=q2⇒α(t7-t6)+β2(t72-t62)=q2.
From which we can determine the value of t6, say
(4.19)t6=-α+α2-2q2β+2t7αβ+t72β2β.
Similarly from (4.3)
(4.20)∫t5t6(Pm(t)-D(t))dt=q2.
From which we can find t5(4.21)t5=-2am+2α+(2am-2α)2-4(bm-β)(2q2-2amt6-bmt62+2t6α+t62β)2(bm-β).
From (3.12) we have
(4.22)∫t3t4D(t)dt=∫t4t5(Pm(t)-D(t))dt(am-α)(t5-t4)+(bm-β)2(t52-t42)=α(t4-t3)+β2(t42-t32).
From which we can find t4(4.23)t4=-am+am2+2ambmt5+bm2t52+2bmt3α-2bmt5α+bmt32β-bmt52βbm.Example 4.2.
The above theoretical results are illustrated through the numerical verification. To illustrate the proposed model, we have considered the input parameters as given below.
c
m
=
8 $/unit, sm=3 $/unit, sr=2 $/unit, cR=2 $/unit, cs=5 $/unit, am=1500 unit/month, bm=30 unit/month2, α=800 unit/month, β=35 unit/month2, ar=1800 unit/month, br=30 unit/month2, hR=1 $/unit/month, hm=1.5 $/unit/month, km=1500 $/cycle, kR=1000 $/cycle, hr=1 $/unit/month, kr=800 $/cycle, c=600 unit/month, d=28 unit/month2.
Applying the solution procedure given in the last section we derive the optimal solution and results are presented in Table2.Table 2
The optimal results for the inventory model under the above parametric values as in Example4.2.
Q
*
q
1
*
q
2
*
t
1
*
t
2
*
t
3
*
t
4
*
t
5
*
t
6
*
t
7
*
t
8
*
Z
(
Q
*
,
q
1
*
,
q
2
*
)
1712
781
191
0.160528
0.94369
1.86346
1.92922
2.01177
2.28889
2.50498
2.6851
7616.1
units
units
units
Month
month
month
Month
month
Month
Month
month
$The convexity of the reverse logistics inventory model is shown in Figure3. The three dimensional graph shows that the integrated expected total annual cost is convex, and that there exists a unique solution minimizing the integrated expected total annual cost.Convexity of the reverse logistics inventory model is derived, and when the optimal value of acceptable returned quantityQ, maximum inventory level q1 and q2 is taken.
(a)
(b)
(c)


## 4.2. Sensitivity Analysis

To study the effects of the parameter changes on the optimal solutions derived by the proposed method, this investigation performs a sensitivity analysis by increasing or decreasing the parameters, one at a time.The main conclusions drawn from the sensitivity analysis given above are as follows.(i)
From Tables3 and 4 we have observed that as the production rate increases the optimum value of total acceptable returned quantity decreases and hence the total minimum cost slightly decreases. However we observe a little but unexpected increment in the total minimum average cost and this increment happened due to the reduction in total cycle length.(ii)
Similarly from Tables5 and 6 we have observed that as the remanufacturing rate increases the optimum value of total acceptable returned quantity decreases and hence the total minimum cost slightly decreases. But due to the reduction in total cycle length we observe a little increment in the total minimum average cost.(iii)
From Tables7 and 8 it is observed that as the return rate increases the optimum value of total acceptable returned quantity increases while the total time required to produce the optimum manufactured quantity decrease. Hence the total procurement and production cost will decrease. This reduction resulted in the decrease in the total minimum average cost.(iv)
Similarly from Table9 it is observed that as the constant demand rate increases the optimal values of total acceptable returned quantity and the total time required to produce the optimum manufactured quantity increase. Hence the total procurement, acquisition, and production cost increases. This increment resulted in the increase in the total minimum average cost per cycle.(v)
Now from Table10 it is observed that as the variable demand rate increases the optimal values of total acceptable returned quantity decrease and hence the total minimum cost slightly decreases. But due to the reduction in total cycle length we observe a little increment in the total minimum average cost.Table 3
For the different constant production rates optimal results for the same set of values as in Example4.2.
a
m
Q
*
q
1
*
q
2
*
t
1
*
t
2
*
t
3
*
t
4
*
t
5
*
t
6
*
t
7
*
t
8
*
Z
(
Q
*
,
q
1
*
,
q
2
*
)
1200
1719
785
134
0.1603
0.9475
1.8717
1.9177
2.0198
2.3642
2.5156
2.6954
7605.77
1350
1715
783
165
0.1601
0.9453
1.8673
1.9249
2.0173
2.3233
2.5098
2.6895
7611.48
1500
1712
781
191
0.1605
0.9436
1.8634
1.9292
2.0117
2.2888
2.5049
2.6851
7616.1
1650
1709
780
212
0.1598
0.9420
1.8607
1.9335
2.0086
2.2612
2.5012
2.6806
7619.9
1800
1707
779
229
0.1598
0.9409
1.8585
1.9379
2.0075
2.2389
2.49833
2.6777
7623.08Table 4
Optimal results for the same set of values as in Example4.2 for different variable production rates.
b
m
Q
*
q
1
*
q
2
*
t
1
*
t
2
*
t
3
*
t
4
*
t
5
*
t
6
*
t
7
*
t
8
*
Z
(
Q
*
,
q
1
*
,
q
2
*
)
28
1712
782
190
0.159527
0.94369
1.86462
1.93053
2.01376
2.29116
2.5061
2.6851
7615.98
29
1712
781
191
0.160528
0.94369
1.86346
1.92873
2.01091
2.28889
2.50498
2.6851
7616.04
30
1712
781
191
0.160528
0.94369
1.86346
1.92922
2.01177
2.28889
2.50498
2.6851
7616.1
31
1711
781
191
0.159983
0.943143
1.86293
1.92902
2.01175
2.288
2.5041
2.68362
7616.15
32
1711
781
191
0.159983
0.943143
1.86293
1.92951
2.01261
2.288
2.5041
2.68362
7616.21Table 5
Optimal results for the same set of values as in Example4.2 for different constant remanufacturing rates.
a
r
Q
*
q
1
*
q
2
*
t
1
*
t
2
*
t
3
*
t
4
*
t
5
*
t
6
*
t
7
*
t
8
*
Z
(
Q
*
,
q
1
*
,
q
2
*
)
1440
1823
661
202
0.2109
1.2497
2.0207
2.0903
2.1782
2.4717
2.69856
2.8489
7479.26
1620
1759
730
202
0.1814
1.0751
1.9312
1.9914
2.0671
2.3603
2.5882
2.7546
7556.35
1800
1712
781
191
0.1605
0.9436
1.8634
1.9292
2.0117
2.2888
2.5049
2.6851
7616.1
1980
1676
822
187
0.1430
0.8411
1.8122
1.8772
1.9586
2.2298
2.44188
2.6317
7663.79
2160
1648
855
184
0.1292
0.7589
1.7716
1.8359
1.9161
2.1829
2.39203
2.5901
7702.75Table 6
Optimal results for the same set of values as in Example4.2 for different variable remanufacturing rates.
b
r
Q
*
q
1
*
q
2
*
t
1
*
t
2
*
t
3
*
t
4
*
t
5
*
t
6
*
t
7
*
t
8
*
Z
(
Q
*
,
q
1
*
,
q
2
*
)
28
1712
781
191
0.160147
0.944177
1.86393
1.92969
2.01223
2.28935
2.50543
2.6851
7615.79
29
1712
781
191
0.160338
0.943934
1.86369
1.92945
2.012
2.28912
2.5052
2.6851
7615.95
30
1712
781
191
0.160528
0.94369
1.86346
1.92922
2.01177
2.28889
2.50498
2.6851
7616.1
31
1711
781
191
0.160173
0.9429
1.8627
1.92831
2.01066
2.28777
2.50387
2.68362
7616.25
32
1711
782
191
0.159362
0.942657
1.86362
1.92922
2.01156
2.28868
2.50477
2.68362
7616.4Table 7
Optimal results for the same set of values as in Example4.2 for different constant return rates.
c
Q
*
q
1
*
q
2
*
t
1
*
t
2
*
t
3
*
t
4
*
t
5
*
t
6
*
t
7
*
t
8
*
Z
(
Q
*
,
q
1
*
,
q
2
*
)
480
1460
668
315
0.1490
0.8769
1.7353
1.9548
2.2299
2.5963
2.87761
3.0425
8251.8
540
1590
726
252
0.1490
0.8769
1.7353
1.8226
1.9316
2.2971
2.58178
2.7485
7917.36
600
1712
781
191
0.1605
0.9436
1.8634
1.9292
2.0117
2.2888
2.5049
2.6851
7616.1
660
1826
833
133
0.1705
1.0060
1.9833
2.0284
2.0853
2.2783
2.42905
2.6209
7318.71
720
1934
882
77
0.1802
1.0649
2.0961
2.1226
2.1562
2.2680
2.35546
2.5588
7024.89Table 8
Optimal results for the same set of values as in Example4.2 for different variable return rates.
d
Q
*
q
1
*
q
2
*
t
1
*
t
2
*
t
3
*
t
4
*
t
5
*
t
6
*
t
7
*
t
8
*
Z
(
Q
*
,
q
1
*
,
q
2
*
)
26
1697
775
192
0.15836
0.935485
1.84862
1.9157
1.99985
2.2784
2.4957
2.67347
7629.03
27
1704
778
192
0.159172
0.939314
1.85578
1.92156
2.0041
2.28265
2.49992
2.67857
7622.57
28
1712
781
191
0.160528
0.94369
1.86346
1.92922
2.01177
2.28889
2.50498
2.6851
7616.1
29
1719
785
190
0.160338
0.947518
1.87177
1.93732
2.01965
2.29532
2.51023
2.69011
7609.62
30
1711
782
191
0.161149
0.951347
1.87892
1.94426
2.02635
2.30059
2.51432
2.69508
7603.14Table 9
Optimal results for the same set of values as in Example4.2 for different constant demand rates.
α
Q
*
q
1
*
q
2
*
t
1
*
t
2
*
t
3
*
t
4
*
t
5
*
t
6
*
t
7
*
t
8
*
Z
(
Q
*
,
q
1
*
,
q
2
*
)
640
1645
871
53
0.1544
0.9070
2.1625
2.1845
2.2030
2.2654
2.33903
2.5856
5949.82
720
1683
829
129
0.1582
0.9278
2.0026
2.0517
2.1022
2.2700
2.43081
2.6421
6774.6
800
1712
781
191
0.1605
0.9436
1.8634
1.9292
2.0117
2.2888
2.5049
2.6851
7616.1
880
1735
729
236
0.1614
0.9562
1.7424
1.8180
1.9347
2.3220
2.56647
2.7191
8467.36
960
1756
674
265
0.1626
0.9677
1.638
1.7164
1.8670
2.3675
2.62061
2.7501
9323.28Table 10
Optimal results for the same set of values as in Example4.2 for different variable demand rates.
β
Q
*
q
1
*
q
2
*
t
1
*
t
2
*
t
3
*
t
4
*
t
5
*
t
6
*
t
7
*
t
8
*
Z
(
Q
*
,
q
1
*
,
q
2
*
)
33
1732
792
192
0.1613
0.954628
1.88977
1.95572
2.03776
2.31463
2.53282
2.71471
7586.86
34
1722
787
191
0.160409
0.949159
1.87717
1.94361
2.02664
2.30292
2.51948
2.69991
7601.52
35
1712
781
191
0.160528
0.94369
1.86346
1.92922
2.01177
2.28889
2.50498
2.6851
7616.1
36
1702
776
190
0.159656
0.93822
1.85094
1.91713
2.00059
2.27709
2.49157
2.67029
7630.6
37
1692
772
190
0.157792
0.93275
1.83961
1.90504
1.98791
2.26524
2.47926
2.65547
7645.01


## 4.3. A Particular Case Whenq1 and q2 Are Constant

In a particular case let us haveq1 and q2 the maximum inventory level in the remanufacturing and production process, as a constant then from (4.9)–(4.23) the values of t1,t2,t3,t4,t5,t6,t7, and t8 can be found in the form of single variable Q, say
(4.24)ti=fi(Q),where1≤i≤8.
From (4.4) the total cost function will be the function of one variable, say Q and the problem will be converted into the following unconstrained problem with one variable Q:
(4.25)W(Q)=1f8{(cm+sm)∫f4f6Pm(u)du+cR∫0f8R(u)du+sr∫0f2Pr(u)du+hr[∫f1f2(-u)(Pr(u)-D(u))du+∫f2f3uD(u)du]+hm[∫f5f6(-u)(Pm(u)-D(u))du+∫f6f7uD(u)du]+hR[∫0f2u(Pr(u)-R(u))du+∫f2f8(f8-u)R(u)du]+cs[∫0f1u(Pr(u)-D(u))du-∫f3f4uD(u)du+∫f4f5u(Pm(u)-D(u))du-∫f7f8uD(u)du]+Kr+Km+KR}.
Now the necessary conditions for having a minimum for the problem
(4.26)dWdQ=0.
To find the solution of (4.26), let W=w/f8 then
(4.27)dWdQ=wQ′f8-f8.Q′wf82,
where, wQ′ and f8.Q′ are the derivatives of w and f8 (w.r.t.) Q, respectively. Hence, (4.26) is equivalent to
(4.28)wQ′f8=f8.Q′w.
Taking the first derivative of both sides of (3.11), (3.12), (3.13), (3.14), and (3.15) with respect to Q. We obtain
(4.29)f2.Q′(Pr(f2)-D(f2))-f1.Q′(Pr(f1)-D(f1))=f3.Q′D(f3)-f2.Q′D(f2),(4.30)f4.Q′D(f4)-f3.Q′D(f3)=f5.Q′(Pm(f5)-D(f5))-f4.Q′(Pm(f4)-D(f4)),(4.31)f6.Q′(Pm(f6)-D(f6))-f5.Q′(Pm(f5)-D(f5))=f7.Q′D(f7)-f6.Q′D(f6),f1.Q′(Pr(f1)-D(f1))=f8.Q′D(f8)-f7.Q′D(f7),f8.Q′R(f8)=f2.Q′Pr(f2).
As we have q1 and q2 are constant so that from (4.2) and (4.3) both sides of (4.29) and (4.30) will be equal to zero
(4.32)f2.Q′(Pr(f2)-D(f2))-f1.Q′(Pr(f1)-D(f1))=f3.Q′D(f3)-f2.Q′D(f2)=0(4.33)f4.Q′D(f4)-f3.Q′D(f3)=f5.Q′(Pm(f5)-D(f5))-f4.Q′(Pm(f4)-D(f4))=0.
Form the above equation we can find the values of f1.Q′,f2.Q′,f3.Q′,f4.Q′,f5.Q′,f6.Q′,f7.Q′, and f8.Q′.From (4.29)–(4.33), and (4.25) we have
(4.34)wQ′=(cm+sm){f6.Q′Pm(f6)-f4.Q′Pm(f4)}+cRf8.Q′R(f8)+srf2.Q′Pr(f2)+hr{f1.Q′(f1-f2)(Pr(f1)-D(f1))+f3.Q′(f3-f2)D(f3)}+hm{f5.Q′(f5-f6)(Pm(f5)-D(f5))-f7.Q′(f7-f6)D(f7)}+hR{f8.Q′f2R(f8)-f2.Q′f8R(f2)}+cs{f3.Q′(f3-f4)D(f3)+f5.Q′(f5-f4)(Pm(f5)-D(f5))+f8.Q′f1D(f8)+f7.Q′(f7-f8-f1)D(f7)}.From which and (4.28) we obtain
(4.35)w=f8f8.Q′[(cm+sm){f6.Q′Pm(f6)-f4.Q′Pm(f4)}+cRf8.Q′R(f8)+srf2.Q′Pr(f2)+hR{f8.Q′f2R(f8)-f2.Q′f8R(f2)}+hr{f1.Q′(f1-f2)(Pr(f1)-D(f1))+f3.Q′(f3-f2)D(f3)}+hm{f5.Q′(f5-f6)(Pm(f5)-D(f5))-f7.Q′(f7-f6)D(f7)}+cs{f3.Q′(f3-f4)D(f3)+f5.Q′(f5-f4)(Pm(f5)-D(f5))+f8.Q′f1D(f8)+f7.Q′(f7-f8-f1)D(f7)}],
where W is given by (4.25) and wQ′ is given by (4.34).Equation (4.35) can now, be used to determine the optimal value of Q. If we get more than one solution we choose the one for which the condition d2W/dQ2>0 is satisfied.Then the optimal values oft1,t2,t3,t4,t5,t6,t7, and t8 can be determined by using (4.9)–(4.23). From the above-mentioned results we can determine the value of wQ′ and w. Hence from (4.25) we can get the minimum total average cost.Example 4.3.
The same set of input data are considered as in the Example4.2 except that the constant values of q1 and q2 are as follows: q1=800 units, q2=200 units.
Applying the solution procedure given in last section we find that the optimal value of the acceptable returned quantityQ is 1737 and hence the corresponding optimal results are presented in Table 11.
The convexity of the total cost function has been established graphically in Figure4 it has also been verified numerically that the cost function is convex.Table 11
The optimal results for the inventory model under the parametric values given in Example4.2.
Q
*
t
1
*
t
2
*
t
3
*
t
4
*
t
5
*
t
6
*
t
7
*
t
8
*
Z
(
Q
*
,
q
1
*
,
q
2
*
)
1737
0.155131
0.957362
1.89856
1.95774
2.03215
2.32237
2.5483
2.7221
7616.6
units
Month
month
month
Month
Month
Month
Month
month
$Figure 4
Behavior of the inventory cost function, when the values ofq1 and q2 are taken to be constant.


## 5. Conclusion

In this paper designed model for a supply chain network with reverse flows is proposed. This paper generalises a reverse logistics inventory model for integrated production of new items and remanufacturing of returned items with shortages. In many practical situations, stock out is unavoidable due to various uncertainties. Therefore, the occurrence of shortages in inventory is a natural phenomenon. The developed model is solved by using the Hessian matrix. Results presented herein provide a valuable reference for decision makers in production, storage remanufacturing, and returning planning. A numerical example demonstrates that applying the proposed model can minimize the total average cost. In addition, sensitivity analysis is performed to examine the effect of parameters. According to those results, the presented model is least sensitive with respect to the production and remanufacturing rates while enough sensitive with respect to the returned and demand rates. It can be concluded from the obtained results that the higher returned rate can provide more profit. A future study should incorporate more realistic assumptions into the proposed model, for example, stochastic nature of demand, production, and remanufacturing rates follow learning and forgetting curves, and consider multiple production and remanufacturing batches per interval.



---

*Source: 386598-2012-11-11.xml*