Hindawi Publishing CorporationISRN Operations ResearchVolume 2013, Article ID 485172, 8 pageshttp://dx.doi.org/10.1155/2013/485172
Research ArticleEPQ Model for Trended Demand with Rework andRandom Preventive Machine Time
Nita H. Shah,1 Dushyantkumar G. Patel,2 and Digeshkumar B. Shah3
1 Department of Mathematics, Gujarat University, Ahmedabad, Gujarat 380009, India2Department of Mathematics, Government Polytechnic for Girls, Ahmedabad, Gujarat 380015, India3 Department of Mathematics, L.D. College of Engineering, Ahmedabad, Gujarat 380015, India
Correspondence should be addressed to Nita H. Shah; [email protected]
Received 20 April 2013; Accepted 9 June 2013
Academic Editors: I. Ahmad, P. Ekel, and Y. Yu
Copyright Β© 2013 Nita H. Shah et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Economic production quantity (EPQ) inventory model for trended demand has been analyzed with rework facility and stochasticpreventive 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 isobserved that the rate of change of demand has significant impact on the optimal inventory cost. The model is very sensitive to theproduction and demand rate.
1. Introduction
An item that does not satisfy quality standards but can beattained after reprocess is termed as a recoverable item andthe process is known as rework. It is observed that in anindustrial sector, the rework reduced production cost andmaintained quality standard of the item. Schrady [1] debatedrework process. Khouja [2] formulated an economic lot-sizeand shipment policy by incorporating a fraction of defectiveitems and direct rework. Koh et al. [3] andDobos and Richter[4] discussed two production policies with options to ordernew products externally or recover old products. Chiu et al.[5] analyzed an imperfect rework process for EPQ modelwith repairable and scrapped items. Jamal et al. [6] advocatedthe policy for rework of defective items in the same cyclewhich was extended by Cardenas-Barron [7]. Widyadanaand Wee [8] gave an analysis of these problems using analgebraic approach. Chiu [9] and Chiu et al. [10] discussedEPQ model by allowing shortages and considering servicelevel constraint. Yoo et al. [11] discussed an EPQ modelwith imperfect production quality, imperfect inspection, andrework.
Meller and Kim [12], Sheu and Chen [13] and Tsou andChen [14] studied Variants of EPQ model with preventive
maintenance. Abboud et al. [15] analyzed an economicorder quantity model by considering machine unavailabilityowing to preventive maintenance and shortage. Chung et al.[16] extended the previous model to compute an economicproduction quantity for deteriorating inventory model withstochastic machine unavailable time and shortage. Wee andWidyadana [17] revisited the previous model incorporatingrework.
In this paper, we analyze an economic production quan-tity (EPQ) model with rework and random preventive main-tenance time together when demand is increasing function oftime. The consideration of random preventive maintenancetime, rework, and trended demand in the model increasesits applicability in the electronic and automobile industries.In this production system, produced items are inspectedimmediately. Defective items are stocked and reworked at theend of the production uptime. We will call these items asrecoverable items. Out of these recoverable items, the fractionof the items will be labeled as βnewβ and rest will be scrapped.Preventivemaintenance is performed at the end of the reworkprocess, and the maintenance time is assumed to be random.When demand is increasing, shortages may occur which willbe treated as lost sales in this study. It is observed that the rateof change of demand has significant impact on the optimal
2 ISRN Operations Research
Table 1: Sensitivity analysis of π1πand total cost when preventive maintenance time is uniformly and exponentially distributed.
Parameter Percentage change Uniform distribution Exponential distribution π = 20π1π
ππΆπ π1π
ππΆπ
π΄
β40% 0.1118836 2217.023407 0.165363168 3105.82179β20% 0.1127598 2396.444313 0.168010377 3225.8832310 0.1136351 2574.499976 0.170676999 3344.12915
20% 0.1145095 2751.208003 0.173362159 3460.59955840% 0.1153831 2926.585673 0.176064945 3575.33395
π
β40% 0.119 0.224 0.873011973 3769.650118β20% 0.116 0.220356527 0.280588937 3224.8966390 0.1136351 2574.499976 0.170676999 3344.12915
20% 0.0784111 2671.944386 0.123903512 3421.12104640% 0.0598742 2748.060997 0.09758743 3473.297234
π1
β40% 0.1252239 2840.779326 0.179184671 3667.154126β20% 0.117738 2670.650277 0.17374104 3462.4401060 0.1136351 2574.499976 0.170676999 3344.12915
20% 0.1110459 2512.591082 0.168712009 3266.96521840% 0.1093473 2469.539689 0.167344624 3212.637461
a
β40% 0.0468611 2285.690117 0.081365729 2483.982152β20% 0.0736194 2419.311543 0.118822843 2928.2745450 0.1136351 2574.499976 0.170676999 3344.12915
20% 0.1801641 2784.239588 0.248801609 3743.38927940% 0.313401 3147.605249 0.385620567 4166.830717
b
β40% 0.1133496 2572.975288 0.170020919 3336.335153β20% 0.113492 2573.73768 0.170347969 3340.2333530 0.1136351 2574.499976 0.170676999 3344.12915
20% 0.1137789 2575.262185 0.171008033 3348.02255340% 0.1139235 2576.024314 0.1713411 3351.913594
π₯
β40% 0.1075459 2440.729188 0.165578265 3180.040146β20% 0.1105098 2506.778232 0.168074836 3261.5205650 0.1136351 2574.499976 0.170676999 3344.12915
20% 0.1169347 2644.037111 0.173391457 3427.95442440% 0.1204229 2715.549502 0.176225505 3513.094599
π₯1
β40% 0.1131507 2451.633648 0.170247776 3220.242525β20% 0.1133924 2513.000758 0.170462207 3282.1199790 0.1136351 2574.499976 0.170676999 3344.12915
20% 0.1138787 2636.131758 0.17089215 3406.27046440% 0.1141232 2697.89656 0.171107662 3468.544359
β
β40% 0.1163864 2035.802579 0.196138784 2497.162768β20% 0.114993 2306.766649 0.18157314 2935.5540590 0.1136351 2574.499976 0.170676999 3344.12915
20% 0.1123112 2839.068303 0.162020977 3729.59749240% 0.11102 3100.535386 0.15486872 4096.262909
β1
β40% 0.113733 2555.165605 0.171388527 3315.223296β20% 0.113684 2564.8349 0.171031253 3329.6917750 0.1136351 2574.499976 0.170676999 3344.12915
20% 0.1135862 2584.160837 0.170325715 3358.5356740% 0.1135373 2593.817484 0.169977356 3372.911591
ISRN Operations Research 3
Table 1: Continued.
Parameter Percentage change Uniform distribution Exponential distribution π = 20π1π
ππΆπ π1π
ππΆπ
π
β40% 0.1119752 2570.475444 0.15263831 3147.25954β20% 0.1129966 2572.958931 0.162618107 3255.7438080 0.1136351 2574.499976 0.170676999 3344.12915
20% 0.1140719 2575.549497 0.1774544 3418.94091440% 0.1143897 2576.310317 0.183312722 3483.929611
ππ
β40% 0.113643 2453.561123 0.170776446 3223.952673β20% 0.113639 2514.030563 0.170726744 3284.0412150 0.1136351 2574.499976 0.170676999 3344.12915
20% 0.1136311 2634.969362 0.17062721 3404.21646740% 0.1136271 2695.438718 0.170577378 3464.303169
Time
T1a T2aT4rT3r
T1 T2
T3
T
Inventory level
Im
Figure 1: Inventory status of serviceable items with lost sales.
inventory cost. It is suggested that when demand is trended,preventivemaintenance time should be controlled by recruit-ing well-qualified technicians. The uniform distribution andexponential distribution for preventive maintenance time areexplored.The paper is organized as follows: Section 2 is aboutthe mathematical development of the proposed problem. InSection 3, example and sensitivity are given. Conclusions arehighlighted in Section 4.
2. Mathematical Model
Assumptions. (1) The inventory system under considerationdeals with single item. (2) Standard quality items must begreater than the demand. (3) The production and reworkrates are constant. (4) The demand rate, π (π‘) = π(1 + ππ‘), isincreasing function of time, where π > 0 is scale demand and0 < π < 1 denotes the rate of change of demand. (5) Setup costfor rework process is zero or negligible. (6) Recoverable itemsare spawned during the production uptime, and scrappeditems are produced during the rework uptime.
The status of the serviceable inventory is depicted inFigure 1. Production occurs during [0, π
1π]. In phase π₯
defective items per unit time are to be reworked. The reworkprocess starts at the end of the predetermined productionuptime. The rework time ends at π
3πtime period. The
different production processes of the material and defectiveitems result in different product rates. During the rework,some rejected and scrapped items will occur. LIFO policyis assumed for the production system. So, serviceable itemsduring the rework uptime are utilized before the fresh itemsfrom the production in uptime. The new production run isstarted when the inventory level reaches zero at the end ofπ2πtime period. It may happen that the production may not
start at π2πtime period because unavailability of the machine
is randomly distributed with a probability density functionπ(π‘). The nonavailability of machine may result in shortageduring π
3time period. The production will resume after the
π3time period.From the above description, the inventory level in a
production uptime period is governed by the differentialequation
ππΌ1π(π‘1π)
ππ‘1π
= π β π (π‘1π) β π₯, 0 β€ π‘
1πβ€ π1π. (1)
The inventory level in a rework uptime is
ππΌ3π(π‘3π)
ππ‘3π
= π1β π (π‘
3π) β π₯1, 0 β€ π‘
3πβ€ π3π. (2)
The inventory level in a production downtime is
ππΌ2π(π‘2π)
ππ‘2π
= βπ (π‘1π) , 0 β€ π‘
2πβ€ π2π. (3)
The inventory level in a rework downtime is
ππΌ4π(π‘4π)
ππ‘4π
= βπ (π‘4π) , 0 β€ π‘
4πβ€ π4π. (4)
4 ISRN Operations Research
Under the assumption of LIFO production system, theinventory level of good items depletes at a constant rateduring rework uptime and downtime. The inventory level isgoverned by
ππΌ3π(π‘3π)
ππ‘3π
= 0, 0 β€ π‘3πβ€ π3π+ π4π. (5)
Using πΌ1π(0) = 0, the solution of (1) is
πΌ1π(π‘1π) = (π β π β π₯) π‘
1πβππ
2π‘2
1π, 0 β€ π‘
1πβ€ π1π
(6)
which is the inventory level during [0, π1π]. Hence, the total
inventory in a production uptime is
ππΌ1π= β«
π1π
0
πΌ1π(π‘1π) ππ‘1π
= (π β π β π₯)π2
1π
2βππ
6π3
1π.
(7)
Using πΌ3π(0) = 0, πΌ
4π(0) = 0, the total inventory of serviceable
items for the rework uptime and rework downtime is
ππΌ3π= (π1β π β π₯
1)π2
3π
2βππ
6π3
3π,
ππΌ4π= π[
π2
4π
2+π
3π3
4π] ,
(8)
respectively.Using πΌ
2π(πΌ2π) = 0, the total inventory level of a
production downtime is
ππΌ2π= π[
π2
2π
2+π
3π3
2π] . (9)
The maximum inventory is
πΌπ= πΌ1π(π1π) = (π β π β π₯) π
1πβππ
2π2
1π(10)
and hence, the total inventory in a rework uptime is
ππΌ3π= πΌπ(π3π+ π4π) . (11)
Now, let us analyze the inventory level of recoverableitems (Figure 2).
The inventory level of recoverable items in a productionuptime is governed by the differential equation
ππΌπ1(π‘π1)
ππ‘π1
= π₯, 0 β€ π‘π1β€ π1π. (12)
Since initially there are no recoverable items, that is, πΌπ1(0) =
0, the solution of (12) is
πΌπ1(π‘π1) = π₯π‘
π1, 0 β€ π‘
π1β€ π1π. (13)
x
Time
IMr
T1a T3r
Figure 2: Inventory status of recoverable items.
Hence, the total inventory of recoverable items in a produc-tion uptime is
πππΌπ1=π₯π‘2
1π
2
(14)
and the maximum recoverable inventory is
πΌππ= πΌπ1(π1π) = π₯π
1π. (15)
The inventory level of recoverable item in the rework uptimeis modeled as
ππΌπ3(π‘π3)
ππ‘π3
= βπ1, 0 β€ π‘
π3β€ π3π. (16)
Using πΌπ3(π3π) = 0, the inventory level of recoverable item in
rework uptime is
πΌπ3(π‘π3) = π1(π3πβ π‘π3) , 0 β€ π‘
π3β€ π3π. (17)
Hence, the total inventory of recoverable item in the reworkuptime is
πππΌπ3=π1π2
3π
2. (18)
The number of recoverable inventories is
πΌππ= πΌπ3(0) = π
1π3π. (19)
Hence,
π3π=πΌππ
π1
. (20)
Substituting πΌππ
from (15), we get
π3π=π₯π1π
π1
. (21)
Hence, the total recoverable inventory is
πΌπ€= πππΌ
π1+ πππΌ
π3=π₯π2
1π
2(1 +
π₯
π1
) . (22)
ISRN Operations Research 5
The inventory level at the beginning of the productiondowntime is equal to the inventory level at the end of theproduction uptime; that is,
πΌ1π(π1π) = πΌ2π(0) . (23)
Therefore,
π2πβ1
π[(π β π β π₯) π
1πβππ
2π2
1π] . (24)
When π‘3π= π3πand π‘4π= 0, the inventory level for serviceable
item in rework process satisfies
(π1β π β π₯
1) π3πβππ
2π2
3π= π [π
4πβπ
2π2
4π] . (25)
Neglecting π24π(because 0 < π
4π< 1), we get
π4πβ1
π(π1β π β π₯
1)π₯
π1
π1π. (26)
The total production inventory cost is the sum of theproduction set up cost, inventory cost of serviceable item,inventory cost of recoverable item, and scrap cost:
ππΆ = π΄ + β [ππΌ1π+ ππΌ3π+ ππΌ2π+ ππΌ4π+ ππΌ3π]
+ β1πΌπ€+ ππΆπ₯1π3π
(27)
and the total cycle time isπ = π
1π+ π3π+ π2π+ π4π. (28)
Hence, the total cost per unit time without lost sales is givenby
ππΆπNL =ππΆ
π. (29)
The optimal production uptime for the EPQ systemwithout lost sales can be obtained by setting
πππΆπNL (π1π)
ππ1π
= 0. (30)
When unavailability time of a machine is longer than theproduction downtime duration, lost sales will occur. So thetotal inventory cost is
πΈ (ππΆ) = π΄ + β [ππΌ1π+ ππΌ3π+ ππΌ2π+ ππΌ4π+ ππΌ3π]
+ β1πΌπ€+ ππΆπ₯1π3π+ ππΏ
Γ β«
β
π‘=π2π+π4π
π (π‘) (π‘ β (π2π+ π4π)) π (π‘) ππ‘
(31)
and the total cycle time for lost sales is
πΈ (π) = π1π+ π3π+ π2π+ π4π
+ β«
β
π‘=π2π+π4π
(π‘ β (π2π+ π4π)) π (π‘) ππ‘.
(32)
Hence, the total cost per unit time for lost sales is
πΈ (ππΆπ) =πΈ (ππΆ)
πΈ (π). (33)
We discuss lost sales scenario for two distributions, namelyuniform distribution and exponential distribution.
2.1. UniformDistribution. Define the probability distributionfunction π(π‘), when the preventive maintenance time π‘ isdistributed uniformly as follows:
π (π‘) =
{{
{{
{
1
π, 0 β€ π‘ β€ π
0, otherwise.(34)
Substituting π(π‘) in (33) gives the total cost per unit time foruniform distribution as
ππΆππ
= (π΄ + β [ππΌ1π+ ππΌ3π+ ππΌ2π+ ππΌ4π+ ππΌ3π] + β1πΌπ€
+ ππΆπ₯1π3π+ ππΏβ«
π
0
(π (1 + ππ‘)
π) (π‘ β (π
2π+ π4π)) ππ‘)
Γ (π1π+ π3π+ π2π+ π4π+ (1
π)
Γβ«
π
π‘=π2π+π4π
(π‘ β (π2π+ π4π)) ππ‘)
β1
(35)
substituting all the time variables in (35) in terms of π1π,
the objective function; ππΆππ’is a function of π
1πonly. The
optimum value of π1πcan be computed by setting
πππΆππ(π1π)
ππ1π
= 0. (36)
To derive the best solution from nonlost sales and lost salesscenarios, we propose the following steps [17].
Step 1. Calculate (30), (24), and (26) and set πsb = π2π + π4π.
Step 2. If πsb < π, then the obtained solution is not feasible,and go to Step 3; otherwise the solution is obtained.
Step 3. Set πsb = π. Find π1aub using (26) and (24). CalculateππΆπNL(π1aub) using (29).
Step 4. Calculate (36), (24), and (26) and set πsb = π2π + π4π.
Step 5. If πsb β₯ π, then πβ1π = π1aub and the correspondingtotal cost is ππΆπNL(π1aub); otherwise, calculate ππΆππ(π1π).
Step 6. If ππΆπNL(π1aub) β€ ππΆππ(π1π), then πβ1π = π1aub:otherwise πβ
1π= π1π.
2.2. Exponential Distribution. Define the probability distri-bution function π(π‘), when the preventive maintenance timeπ‘ is distributed exponential with mean 1/π as
π (π‘) = ππβππ‘, π > 0. (37)
6 ISRN Operations Research
0.11 0.112 0.114 0.116 0.118 0.1220.12 0.124T1a
2580
2590
2600
2610
2620
2630
2640
2650
2660To
tal c
ost
Figure 3: Convexity of total cost.
0.09
0.095
0.1
0.105
0.11
0.115
0.12
0.125
0 20 40
Prod
uctio
n up
time f
or u
nifo
rm d
istrib
utio
n
Change in inventory parameter (%)β40 β20
A
P
P1
a
b
x
x1
h1
h
S
Sc
Figure 4: Sensitivity analysis of production uptime for uniformdistribution.
Here, the total cost per unit time for the lost sale ππΏis
ππΆππΈ= (π΄ + β [ππΌ
1π+ ππΌ3π+ ππΌ2π+ ππΌ4π+ ππΌ3π]
+ β1πΌπ€+ ππΆπ₯1π3π
+ππΏβ«
β
π‘=π2π+π4π
π (π‘) (π‘ β (π2π+ π4π)) ππβππ‘ππ‘)
Γ (π1π+ π3π+ π2π+ π4π+ (1
π) πβπ(π2π+π4π))
β1
.
(38)
1800
2000
2200
2400
2600
2800
3000
3200
0 20 40Tota
l inv
ento
ry co
st fo
r uni
form
dist
ribut
ion
Change in inventory parameter (%)A
P
P1
a
b
x
x1
h
h1
S
β20β40
Figure 5: Sensitivity analysis of total cost for uniform distribution.
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
Change in inventory parameter (%)0 20 40β20β40
Prod
uctio
n up
time f
or ex
pone
ntia
ldi
strib
utio
n
A
P
P1
a
b
x
x1
h
h1
S
Figure 6: Sensitivity analysis of production uptime for exponentialdistribution.
Arguing as in (Section 2.1), we can obtain optimum totalcost. The high nonlinearity of the cost functions (29), (35),and (38) does not guarantee that the optimal solution isglobal. However, using parametric values, convexity of theobjective function is established.
3. Numerical Examples andSensitivity Analysis
Consider, following parametric values to study the workingof the proposed problem. Letπ΄ = $200 per production cycle,π = 10,000 units per unit time, π = 5000 units per unittime, π = 10%, π₯ = 500 units per unit time; π₯
1= 400
units per unit time, β = $5 per unit per unit time. β1=
ISRN Operations Research 7
3050
3100
3150
3200
3250
3300
3350
3400
3450
3500
Tota
l inv
ento
ry co
st fo
r exp
onen
tial d
istrib
utio
n
Change in inventory parameter (%)0 20 40β20β40
A
P
P1
a
b
x
x1
h
h1
S
Figure 7: Sensitivity analysis of total cost for exponential distribu-tion.
$3 per unit per unit time, ππΏ= $10 per unit, π
πΆ= $12
per unit, and the preventive maintenance time is uniformlydistributed over the interval [0, 0.1] [17]. Using the solutionprocedure outlined, the optimal production uptime is π
1π=
41.5 days and the correspondingminimum total cost per unittime is ππΆπ
π= $2575. This establishes that some lost sales
reduce the total cost per unit time. The convexity of ππΆππis
established in Figure 3.The sensitivity analysis is carried out by changing each
of the parameters by β40%, β20%, +20%, and +40%. Theoptimal production uptime π
1πand the optimal total cost per
unit time for inventory parameters under consideration areshown in Table 1.
Figures 4 and 6 depict sensitivity analysis of productionuptime, π
1π, with respect to all the inventory parameters
considered in the modeling when preventive maintenancetime follows uniform distribution/exponential distribution.It is observed that production uptime is slightly sensitive tochanges in π and π and moderately sensitive to changes in πand π, with little impact due to changes in the other inventoryparameters. π
1πhas negative impact with the increase in the
production rate, π, and positive impact when scale demand,π, and rate of demand, π, increase.
The optimal total cost per unit time is slightly sensitiveto changes in π, π, π₯, and πΏ and moderately sensitive tochanges inπ΄, π, π, π
πΆ, π₯1, and S
πΏ. No change is observed in the
optimal total cost per unit time for the remaining inventoryparameters. The optimal total cost per unit time is inverselyrelated to π and π
1and directly related to other inventory
parameters (see Figures 5 and 7).
4. Conclusions
In this research, rework of imperfect quality and randompreventive maintenance time are incorporated in economic
production quantity model when demand increases withtime.The randompreventivemaintenance time is distributeduniformly and exponentially.Themodels are validated by theexample. The sensitivity analysis suggests that the optimaltotal cost per unit time is sensitive to changes in theproduction rate, the demand rate, and the product defectrate in both the uniform and the exponential distributedpreventive maintenance time. To combat increasing demand,the management should adopt the latest machinery whichdecreases defective production rate, reducing rework, andas a consequence, the machineβs production uptime can beutilized to its utmost. Further research can be carried out tostudy the effect of deterioration of units.
Notations
πΌ1π: Serviceable inventory level in a production
uptimeπΌ2π: Serviceable inventory level in a production
downtimeπΌ3π: Serviceable inventory level in a rework
uptimeπΌ3π: Serviceable inventory level from rework
uptimeπΌ4π: Serviceable inventory level from rework
process in rework downtimeπΌπ1: Recoverable inventory level in a production
uptimeπΌπ3: Recoverable inventory level in a rework
uptimeππΌ1π: Total serviceable inventory in a production
uptimeππΌ2π: Total serviceable inventory in a production
downtimeππΌ3π: Total serviceable inventory in a rework
uptimeππΌ3π: Total serviceable inventory from a rework
uptimeππΌ4π: Total serviceable inventory from rework
process in a rework downtimeπππΌπ1: Total recoverable inventory level in aproduction uptime
πππΌπ3: Total recoverable inventory level in a reworkuptime
π1π: Production uptime
π2π: Production downtime
π3π: Rework uptime
π4π: Rework downtime
πsb: Total production downtimeπ1aub: Production uptime when the total
production downtime is equal to the upperbound of uniform distribution parameter
πΌπ: Inventory level of serviceable items at the
end of production uptimeπΌππ
: Maximum inventory level of recoverableitems in a production uptime
πΌπ€: Total recoverable inventory
8 ISRN Operations Research
π: Production rateπ1: Rework process rate
π = π (π‘): Demand rate; π(1+ππ‘), π > 0, 0 < π < 1π₯: Product defect rateπ₯1: Product scrap rate
π΄: Production setup costβ: Serviceable items holding costβ1: Recoverable items holding cost
ππΆ: Scrap cost
ππΏ: Lost sales cost
ππΆ: Total inventory costπ: Cycle timeππΆπ: Total inventory cost per unit time for
lost sales modelππΆπNL: Total inventory cost per unit time for
without lost sales modelππΆππ: Total inventory cost per unit time
for lost sales model with uniform dis-tribution preventive maintenance time
ππΆππΈ: Total inventory cost per unit time for
lost sales model with exponential dis-tribution preventive maintenance time.
References
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