# EPQ Model for Trended Demand with Rework and Random Preventive Machine Time

Authors: Nita H. Shah; Dushyantkumar G. Patel; Digeshkumar B. Shah
Journal: ISRN Operations Research
 (2013)

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## Abstract

Economic production quantity (EPQ) inventory model for trended demand has been analyzed with rework facility and stochastic preventive machine time. Due to the complexity of the model, search method is proposed to determine the best optimal solution. A numerical example and sensitivity analysis are carried out to validate the proposed model. From the sensitivity analysis, it is observed that the rate of change of demand has significant impact on the optimal inventory cost. The model is very sensitive to the production and demand rate.


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## Body

## 1. Introduction

An item that does not satisfy quality standards but can be attained after reprocess is termed as a recoverable item and the process is known as rework. It is observed that in an industrial sector, the rework reduced production cost and maintained quality standard of the item. Schrady [1] debated rework process. Khouja [2] formulated an economic lot-size and shipment policy by incorporating a fraction of defective items and direct rework. Koh et al. [3] and Dobos and Richter [4] discussed two production policies with options to order new products externally or recover old products. Chiu et al. [5] analyzed an imperfect rework process for EPQ model with repairable and scrapped items. Jamal et al. [6] advocated the policy for rework of defective items in the same cycle which was extended by Cárdenas-Barrón [7]. Widyadana and Wee [8] gave an analysis of these problems using an algebraic approach. Chiu [9] and Chiu et al. [10] discussed EPQ model by allowing shortages and considering service level constraint. Yoo et al. [11] discussed an EPQ model with imperfect production quality, imperfect inspection, and rework.Meller and Kim [12], Sheu and Chen [13] and Tsou and Chen [14] studied Variants of EPQ model with preventive maintenance. Abboud et al. [15] analyzed an economic order quantity model by considering machine unavailability owing to preventive maintenance and shortage. Chung et al. [16] extended the previous model to compute an economic production quantity for deteriorating inventory model with stochastic machine unavailable time and shortage. Wee and Widyadana [17] revisited the previous model incorporating rework.In this paper, we analyze an economic production quantity (EPQ) model with rework and random preventive maintenance time together when demand is increasing function of time. The consideration of random preventive maintenance time, rework, and trended demand in the model increases its applicability in the electronic and automobile industries. In this production system, produced items are inspected immediately. Defective items are stocked and reworked at the end of the production uptime. We will call these items as recoverable items. Out of these recoverable items, the fraction of the items will be labeled as “new” and rest will be scrapped. Preventive maintenance is performed at the end of the rework process, and the maintenance time is assumed to be random. When demand is increasing, shortages may occur which will be treated as lost sales in this study. It is observed that the rate of change of demand has significant impact on the optimal inventory cost. It is suggested that when demand is trended, preventive maintenance time should be controlled by recruiting well-qualified technicians. The uniform distribution and exponential distribution for preventive maintenance time are explored. The paper is organized as follows: Section2 is about the mathematical development of the proposed problem. In Section 3, example and sensitivity are given. Conclusions are highlighted in Section 4.


## 2. Mathematical Model

Assumptions.
(
1
) The inventory system under consideration deals with single item. (2) Standard quality items must be greater than the demand. (3) The production and rework rates are constant. (4) The demand rate, R(t)=a(1+bt), is increasing function of time, where a>0 is scale demand and 0<b<1 denotes the rate of change of demand. (5) Setup cost for rework process is zero or negligible. (6) Recoverable items are spawned during the production uptime, and scrapped items are produced during the rework uptime.The status of the serviceable inventory is depicted in Figure1. Production occurs during [0,T1a]. In phase x defective items per unit time are to be reworked. The rework process starts at the end of the predetermined production uptime. The rework time ends at T3r time period. The different production processes of the material and defective items result in different product rates. During the rework, some rejected and scrapped items will occur. LIFO policy is assumed for the production system. So, serviceable items during the rework uptime are utilized before the fresh items from the production in uptime. The new production run is started when the inventory level reaches zero at the end of T2a time period. It may happen that the production may not start at T2a time period because unavailability of the machine is randomly distributed with a probability density function f(t). The nonavailability of machine may result in shortage during T3 time period. The production will resume after the T3 time period.Figure 1
Inventory status of serviceable items with lost sales.From the above description, the inventory level in a production uptime period is governed by the differential equation(1)dI1a(t1a)dt1a=P-R(t1a)-x,0≤t1a≤T1a.
The inventory level in a rework uptime is
(2)dI3r(t3r)dt3r=P1-R(t3r)-x1,0≤t3r≤T3r.
The inventory level in a production downtime is
(3)dI2a(t2a)dt2a=-R(t1a),0≤t2a≤T2a.
The inventory level in a rework downtime is
(4)dI4r(t4r)dt4r=-R(t4r),0≤t4r≤T4r.Under the assumption of LIFO production system, the inventory level of good items depletes at a constant rate during rework uptime and downtime. The inventory level is governed by(5)dI3a(t3a)dt3a=0,0≤t3a≤T3r+T4r.
Using I1a(0)=0, the solution of (1) is
(6)I1a(t1a)=(P-a-x)t1a-ab2t1a2,0≤t1a≤T1a
which is the inventory level during [0,T1a]. Hence, the total inventory in a production uptime is
(7)TI1a=∫0T1aI1a(t1a)dt1a=(P-a-x)T1a22-ab6T1a3.
Using I3r(0)=0, I4r(0)=0, the total inventory of serviceable items for the rework uptime and rework downtime is
(8)TI3r=(P1-a-x1)T3r22-ab6T3r3,TI4r=a[T4r22+b3T4r3],
respectively.UsingI2a(I2a)=0, the total inventory level of a production downtime is
(9)TI2a=a[T2a22+b3T2a3].
The maximum inventory is
(10)Im=I1a(T1a)=(P-a-x)T1a-ab2T1a2
and hence, the total inventory in a rework uptime is
(11)TI3a=Im(T3r+T4r).Now, let us analyze the inventory level of recoverable items (Figure2).Figure 2
Inventory status of recoverable items.The inventory level of recoverable items in a production uptime is governed by the differential equation(12)dIr1(tr1)dtr1=x,0≤tr1≤T1a.
Since initially there are no recoverable items, that is, Ir1(0)=0, the solution of (12) is
(13)Ir1(tr1)=xtr1,0≤tr1≤T1a.
Hence, the total inventory of recoverable items in a production uptime is
(14)TTIr1=xt1a22
and the maximum recoverable inventory is
(15)IMr=Ir1(T1a)=xT1a.
The inventory level of recoverable item in the rework uptime is modeled as
(16)dIr3(tr3)dtr3=-P1,0≤tr3≤T3r.
UsingIr3(T3r)=0, the inventory level of recoverable item in rework uptime is
(17)Ir3(tr3)=P1(T3r-tr3),0≤tr3≤T3r.
Hence, the total inventory of recoverable item in the rework uptime is
(18)TTIr3=P1T3r22.
The number of recoverable inventories is
(19)IMr=Ir3(0)=P1T3r.
Hence,
(20)T3r=IMrP1.
Substituting IMr from (15), we get
(21)T3r=xT1aP1.
Hence, the total recoverable inventory is
(22)Iw=TTIr1+TTIr3=xT1a22(1+xP1).The inventory level at the beginning of the production downtime is equal to the inventory level at the end of the production uptime; that is,(23)I1a(T1a)=I2a(0).
Therefore,
(24)T2a≈1a[(P-a-x)T1a-ab2T1a2].
When t3r=T3r and t4r=0, the inventory level for serviceable item in rework process satisfies
(25)(P1-a-x1)T3r-ab2T3r2=a[T4r-b2T4r2].
Neglecting T4r2 (because 0<T4r<1), we get
(26)T4r≈1a(P1-a-x1)xP1T1a.
The total production inventory cost is the sum of the production set up cost, inventory cost of serviceable item, inventory cost of recoverable item, and scrap cost:
(27)TC=A+h[TI1a+TI3r+TI2a+TI4r+TI3a]+h1Iw+SCx1T3r
and the total cycle time is
(28)T=T1a+T3r+T2a+T4r.
Hence, the total cost per unit time without lost sales is given by
(29)TCTNL=TCT.The optimal production uptime for the EPQ system without lost sales can be obtained by setting(30)dTCTNL(T1a)dT1a=0.
When unavailability time of a machine is longer than the production downtime duration, lost sales will occur. So the total inventory cost is
(31)E(TC)=A+h[TI1a+TI3r+TI2a+TI4r+TI3a]+h1Iw+SCx1T3r+SL×∫t=T2a+T4r∞R(t)(t-(T2a+T4r))f(t)dt
and the total cycle time for lost sales is
(32)E(T)=T1a+T3r+T2a+T4r+∫t=T2a+T4r∞(t-(T2a+T4r))f(t)dt.
Hence, the total cost per unit time for lost sales is
(33)E(TCT)=E(TC)E(T).
We discuss lost sales scenario for two distributions, namely uniform distribution and exponential distribution.


### 2.1. Uniform Distribution

Define the probability distribution functionf(t), when the preventive maintenance time t is distributed uniformly as follows:
(34)f(t)={1τ,0≤t≤τ0,otherwise.
Substituting f(t) in (33) gives the total cost per unit time for uniform distribution as
(35)TCTU=((a(1+bt)τ)A+h[TI1a+TI3r+TI2a+TI4r+TI3a]+h1Iw+SCx1T3r+SL∫0τ(a(1+bt)τ)(t-(T2a+T4r))dt)×(∫t=T2a+T4rτ(t-(T2a+T4r))T1a+T3r+T2a+T4r+(1τ)×∫t=T2a+T4rτ(t-(T2a+T4r))dt)-1
substituting all the time variables in (35) in terms of T1a, the objective function; TCTu is a function of T1a only. The optimum value of T1a can be computed by setting
(36)dTCTU(T1a)dT1a=0.
To derive the best solution from nonlost sales and lost sales scenarios, we propose the following steps [17].Step 1.
Calculate (30), (24), and (26) and set Tsb=T2a+T4r.Step 2.
IfTsb<τ, then the obtained solution is not feasible, and go to Step 3; otherwise the       solution is obtained.Step 3.
SetTsb=τ. Find T1aub using (26) and (24). Calculate TCTNL(T1aub) using (29).Step 4.
Calculate (36), (24), and (26) and set Tsb=T2a+T4r.Step 5.
IfTsb≥τ, then T1a*=T1aub and the corresponding total cost is TCTNL(T1aub); otherwise,    calculate TCTU(T1a).Step 6.
IfTCTNL(T1aub)≤TCTU(T1a), then T1a*=T1aub: otherwise T1a*=T1a.


### 2.2. Exponential Distribution

Define the probability distribution functionf(t), when the preventive maintenance time t is distributed exponential with mean 1/λ as
(37)f(t)=λe-λt,λ>0.
Here, the total cost per unit time for the lost sale SL is
(38)TCTE=(∫t=T2a+T4r∞R(t)(t-(T2a+T4r))λe-λtA+h[TI1a+TI3r+TI2a+TI4r+TI3a]+h1Iw+SCx1T3r+SL∫t=T2a+T4r∞R(t)(t-(T2a+T4r))λe-λtdt)×(T1a+T3r+T2a+T4r+(1λ)e-λ(T2a+T4r))-1.Arguing as in (Section2.1), we can obtain optimum total cost. The high nonlinearity of the cost functions (29), (35), and (38) does not guarantee that the optimal solution is global. However, using parametric values, convexity of the objective function is established.


## 2.1. Uniform Distribution

Define the probability distribution functionf(t), when the preventive maintenance time t is distributed uniformly as follows:
(34)f(t)={1τ,0≤t≤τ0,otherwise.
Substituting f(t) in (33) gives the total cost per unit time for uniform distribution as
(35)TCTU=((a(1+bt)τ)A+h[TI1a+TI3r+TI2a+TI4r+TI3a]+h1Iw+SCx1T3r+SL∫0τ(a(1+bt)τ)(t-(T2a+T4r))dt)×(∫t=T2a+T4rτ(t-(T2a+T4r))T1a+T3r+T2a+T4r+(1τ)×∫t=T2a+T4rτ(t-(T2a+T4r))dt)-1
substituting all the time variables in (35) in terms of T1a, the objective function; TCTu is a function of T1a only. The optimum value of T1a can be computed by setting
(36)dTCTU(T1a)dT1a=0.
To derive the best solution from nonlost sales and lost sales scenarios, we propose the following steps [17].Step 1.
Calculate (30), (24), and (26) and set Tsb=T2a+T4r.Step 2.
IfTsb<τ, then the obtained solution is not feasible, and go to Step 3; otherwise the       solution is obtained.Step 3.
SetTsb=τ. Find T1aub using (26) and (24). Calculate TCTNL(T1aub) using (29).Step 4.
Calculate (36), (24), and (26) and set Tsb=T2a+T4r.Step 5.
IfTsb≥τ, then T1a*=T1aub and the corresponding total cost is TCTNL(T1aub); otherwise,    calculate TCTU(T1a).Step 6.
IfTCTNL(T1aub)≤TCTU(T1a), then T1a*=T1aub: otherwise T1a*=T1a.


## 2.2. Exponential Distribution

Define the probability distribution functionf(t), when the preventive maintenance time t is distributed exponential with mean 1/λ as
(37)f(t)=λe-λt,λ>0.
Here, the total cost per unit time for the lost sale SL is
(38)TCTE=(∫t=T2a+T4r∞R(t)(t-(T2a+T4r))λe-λtA+h[TI1a+TI3r+TI2a+TI4r+TI3a]+h1Iw+SCx1T3r+SL∫t=T2a+T4r∞R(t)(t-(T2a+T4r))λe-λtdt)×(T1a+T3r+T2a+T4r+(1λ)e-λ(T2a+T4r))-1.Arguing as in (Section2.1), we can obtain optimum total cost. The high nonlinearity of the cost functions (29), (35), and (38) does not guarantee that the optimal solution is global. However, using parametric values, convexity of the objective function is established.


## 3. Numerical Examples and Sensitivity Analysis

Consider, following parametric values to study the working of the proposed problem. LetA=$200 per production cycle, P = 10,000 units per unit time, a=5000 units per unit time, b=10%, x=500 units per unit time; x1=400 units per unit time, h=$5 per unit per unit time. h1=$3 per unit per unit time, SL=$10 per unit, SC=$12 per unit, and the preventive maintenance time is uniformly distributed over the interval [0,0.1] [17]. Using the solution procedure outlined, the optimal production uptime is T1a=41.5 days and the corresponding minimum total cost per unit time is TCTU=$2575. This establishes that some lost sales reduce the total cost per unit time. The convexity of TCTU is established in Figure 3.Figure 3
Convexity of total cost.The sensitivity analysis is carried out by changing each of the parameters by-40%,  -20%, +20%, and +40%. The optimal production uptime T1a and the optimal total cost per unit time for inventory parameters under consideration are shown in Table 1.Table 1
Sensitivity analysis ofT1a and total cost when preventive maintenance time is uniformly and exponentially distributed.
Parameter
Percentage change
Uniform distribution
Exponential distributionλ=20
T
1
a
TCT
T
1
a
TCT
A
−40%
0.1118836
2217.023407
0.165363168
3105.82179
−20%
0.1127598
2396.444313
0.168010377
3225.883231
0
0.1136351
2574.499976
0.170676999
3344.12915
20%
0.1145095
2751.208003
0.173362159
3460.599558
40%
0.1153831
2926.585673
0.176064945
3575.33395
P
−40%
0.119
0.224
0.873011973
3769.650118
−20%
0.116
0.220356527
0.280588937
3224.896639
0
0.1136351
2574.499976
0.170676999
3344.12915
20%
0.0784111
2671.944386
0.123903512
3421.121046
40%
0.0598742
2748.060997
0.09758743
3473.297234
P
1
−40%
0.1252239
2840.779326
0.179184671
3667.154126
−20%
0.117738
2670.650277
0.17374104
3462.440106
0
0.1136351
2574.499976
0.170676999
3344.12915
20%
0.1110459
2512.591082
0.168712009
3266.965218
40%
0.1093473
2469.539689
0.167344624
3212.637461
a
−40%
0.0468611
2285.690117
0.081365729
2483.982152
−20%
0.0736194
2419.311543
0.118822843
2928.274545
0
0.1136351
2574.499976
0.170676999
3344.12915
20%
0.1801641
2784.239588
0.248801609
3743.389279
40%
0.313401
3147.605249
0.385620567
4166.830717
b
−40%
0.1133496
2572.975288
0.170020919
3336.335153
−20%
0.113492
2573.73768
0.170347969
3340.233353
0
0.1136351
2574.499976
0.170676999
3344.12915
20%
0.1137789
2575.262185
0.171008033
3348.022553
40%
0.1139235
2576.024314
0.1713411
3351.913594
x
−40%
0.1075459
2440.729188
0.165578265
3180.040146
−20%
0.1105098
2506.778232
0.168074836
3261.520565
0
0.1136351
2574.499976
0.170676999
3344.12915
20%
0.1169347
2644.037111
0.173391457
3427.954424
40%
0.1204229
2715.549502
0.176225505
3513.094599
x
1
−40%
0.1131507
2451.633648
0.170247776
3220.242525
−20%
0.1133924
2513.000758
0.170462207
3282.119979
0
0.1136351
2574.499976
0.170676999
3344.12915
20%
0.1138787
2636.131758
0.17089215
3406.270464
40%
0.1141232
2697.89656
0.171107662
3468.544359
h
−40%
0.1163864
2035.802579
0.196138784
2497.162768
−20%
0.114993
2306.766649
0.18157314
2935.554059
0
0.1136351
2574.499976
0.170676999
3344.12915
20%
0.1123112
2839.068303
0.162020977
3729.597492
40%
0.11102
3100.535386
0.15486872
4096.262909
h
1
−40%
0.113733
2555.165605
0.171388527
3315.223296
−20%
0.113684
2564.8349
0.171031253
3329.691775
0
0.1136351
2574.499976
0.170676999
3344.12915
20%
0.1135862
2584.160837
0.170325715
3358.53567
40%
0.1135373
2593.817484
0.169977356
3372.911591
S
−40%
0.1119752
2570.475444
0.15263831
3147.25954
−20%
0.1129966
2572.958931
0.162618107
3255.743808
0
0.1136351
2574.499976
0.170676999
3344.12915
20%
0.1140719
2575.549497
0.1774544
3418.940914
40%
0.1143897
2576.310317
0.183312722
3483.929611
S
c
−40%
0.113643
2453.561123
0.170776446
3223.952673
−20%
0.113639
2514.030563
0.170726744
3284.041215
0
0.1136351
2574.499976
0.170676999
3344.12915
20%
0.1136311
2634.969362
0.17062721
3404.216467
40%
0.1136271
2695.438718
0.170577378
3464.303169Figures4 and 6 depict sensitivity analysis of production uptime, T1a, with respect to all the inventory parameters considered in the modeling when preventive maintenance time follows uniform distribution/exponential distribution. It is observed that production uptime is slightly sensitive to changes in P and a and moderately sensitive to changes in b and τ, with little impact due to changes in the other inventory parameters. T1a has negative impact with the increase in the production rate, P, and positive impact when scale demand, a, and rate of demand, b, increase.Figure 4
Sensitivity analysis of production uptime for uniform distribution.The optimal total cost per unit time is slightly sensitive to changes ina, P, x, and L and moderately sensitive to changes in A, b, τ, SC, x1, and SL. No change is observed in the optimal total cost per unit time for the remaining inventory parameters. The optimal total cost per unit time is inversely related to P and P1 and directly related to other inventory parameters (see Figures 5 and 7).Figure 5
Sensitivity analysis of total cost for uniform distribution.Figure 6
Sensitivity analysis of production uptime for exponential distribution.Figure 7
Sensitivity analysis of total cost for exponential distribution.


## 4. Conclusions

In this research, rework of imperfect quality and random preventive maintenance time are incorporated in economic production quantity model when demand increases with time. The random preventive maintenance time is distributed uniformly and exponentially. The models are validated by the example. The sensitivity analysis suggests that the optimal total cost per unit time is sensitive to changes in the production rate, the demand rate, and the product defect rate in both the uniform and the exponential distributed preventive maintenance time. To combat increasing demand, the management should adopt the latest machinery which decreases defective production rate, reducing rework, and as a consequence, the machine’s production uptime can be utilized to its utmost. Further research can be carried out to study the effect of deterioration of units.



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