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21
EPQ MODELS WITH GENERALIZED PARETO RATE OF
PRODUCTION AND WEIBULL DECAY HAVING DEMAND AS FUNCTION OF ON HAND INVENTORY
D.Madhulatha1, K. Srinivasa Rao2, B.Muniswamy3
Department of Statistics1, Andhra Loyola College, Vijayawada, India Department of Statistics 23, Andhra University, Visakhapatnam, India madhulatha.dasari@gmail.com1, ksraoau@yahoo.co.in2, munistats@gmail.com3
Abstract
Economic production quantity (EPQ) models are more important for scheduling production processes in particular batch production in which the production uptime and production downtime are decision variables. This paper addresses the development and analysis of an EPQ model with random production and Weibull decay having stock dependent demand. The random production is more appropriate in several production processes dealing with deteriorated items. The instantaneous state of on hand inventory is derived. With appropriate cost considerations the total cost function is derived and minimized for obtaining optimal production uptime, production downtime and production quantity. The model sensitivity with respect to changes in parameters and costs is also studied and observe that the production distribution parameters and deteriorating distribution parameters have significant influence on optimal operating policies of the model. This model is extended to the case of without shortages and observed that allowing shortages reduce total product cost. It is further observed that the demand being a function of on hand inventory can reduce inventory cost than other patterns of demand.
Keywords: Stochastic production, on hand inventory, Weibull decay, Generalized Pareto distribution, Production Schedules, Sensitivity analysis.
I. Introduction
In production scheduling problems the on hand inventory plays a dominant role. To have efficient decisions on when to start production and when to stop production the EPQ models provide the basic frame work. In developing the EPQ models the major string is on nature of the product. The product may be subjective deterioration or decay depending upon various random factors. Much work has been reported in literature regarding EPQ models for deteriorating items. The literature on inventory models for deteriorating items are reviewed by Pentico and Drake [1], Ruxian Lie et al [2], Goyal and Giri [3], Raafat [4] and Nahmias [5].
In addition to the nature of the commodity another important factor for developing EPQ models is demand. Several authors developed various models with different patterns of demand. Among them the inventory models with stock dependent demand gained importance due to this applicability in many places. Silver et al. [6] mentioned that the demand for many consumer items is directly proportional to the stock on hand. Gupta et al. [7] have pointed the inventory models with inventory models with stock dependent demand. Later Panda et al. [8], Roy et al. [9], Uma Maheswara Rao et al. [10] and others have developed inventory models for deteriorating items with stock dependent demand. Yang et al. [11], Srinivasa Rao et al. [12], Santanu Kumar Ghosh et al. [13] have developed an inventory model for deteriorating items with Weibull replenishment
and generalized Pareto decay. Brojeswar et al. [14], Srinivasa Rao et al[15], Lakshmana Rao et al [16], Srinivasa Rao et al [17], Ardak and Borade [18], Anindya Mandal, Brojeswar Pal and Kripasindhu Chaudhuri [19], Sunit Kumar, Sushil Kumar and Rachna Kumari [20]. and Jyothsna et al. [21] studied a production inventory system for deteriorating items. In all these papers the authors assumed that the production is either instantaneous or finite rate.
However, in many production processes the production is random due to various random factors such as availability of raw material, skill level of the manpower, tool wear, environmental conditions and availability of power (electricity). Very little work has been reported regarding EPQ models with random production for deteriorating items except the works of Sridevi et al. [22], Srinivasa Rao et al [23] who developed EPQ models with Weibull production and constant rate of deterioration. In reality, many products may not have constant rate of deterioration but will have a variable rate of deterioration can be well characterized by Weibull decay. The generalized Pareto rate of production can characterize the time dependent production. Hence, in this paper we develop and analyze an EPQ model with the assumption that the production process is characterized by generalized Pareto distribution and the lifetime of the commodity follows a Weibull distribution. It is further assumed that the demand is a linear function of on hand inventory. This type of model is much useful in textile industry where the lifetime of the government is random and may have decreasing or increasing or constant rates of deterioration and production is random.
Using the differential equations the instantaneous state of inventory at different states of production are derived. The total cost function is obtained with appropriate cost considerations. Assuming shortages are allowed and fully backlogged, the optimal operating policies of the production schedule such as production downtime and production uptime are derived. The optimal production quantity is also obtained. The effect of change in parameters on optimal production schedule and optimal production quantity are studied in sensitivity analysis. The case of without shortages is also discussed. The conclusions are given at the end.
II. Assumptions
For developing the model the following assumptions are made:
• The demand rate is a function of on hand inventory.
i.e. A(t) = 0i + <M(0 (1)
• The production is random and follows a Generalized Pareto distribution. The instantaneous rate of production is
K(t) = — ;0 < t < # (2)
v J a-yt y v '
• Lead time is zero.
• Cycle length is T. It is known and fixed.
• Shortages are allowed and fully backlogged.
• A deteriorated unit is lost.
• The life time of the item is random and follows a two parameter Weibull distribution with probability density function
f(t) = dijt'-1e-6t! ;6,-q>0, t > 0 Therefore the instantaneous rate of deterioration is
h(t)= 1£§) = Brlt '-1 ^>0, t>0 (3)
The following notations are used for developing the model. Q: Production quantity. A: Setup cost. C: Cost per unit.
h: Inventory holding cost per unit per unit time. n: Shortages cost per unit per unit time.
III. EPQ Model with Shortages
Consider a production system in which the stock level is zero at time t = 0. The stock level increases during the period (0, t1), due to production after fulfilling the demand and deterioration. The production stops at time tx when stock level reaches S. The inventory decreases gradually due to demand and deterioration in the interval (t1,t2). At time t2 the inventory reaches zero and backorders accumulate during the period (t2, t3). At time t3 the production again starts and fulfills the backlog after satisfying the demand. During (t3, T) the backorders are fulfilled and inventory level reaches zero at the end of cycle T. The Schematic diagram representing the inventory level is given in Figure 1.
Figure 1: Schematic Diagram representing the inventory level
Let I(t) be the inventory level of the system at time 't' (0 < t < T). The differential equations governing the instantaneous state of I(t) over the cycle of length are:
T i(t) + h(t)i(t) = -( fa + fai(t)) TI(t) + h(t)/(t) = -(fa + fa/(t))
dt d dt
d I(t) = -(fa + fa/(t)) T '(0= ^ -(01 +02/(0)
0 <t<ti
tl <t<t"
12 <t<t3
t3 <t<T
(4)
(5)
(6) (7)
Where, h(t) is as given in equation (3), with the initial conditions I(0) = 0, I(t1) = S, I(t2) = 0 and I(T) = 0. Substituting h(t) in equations (4) and (5) and solving the differential equations, the on hand inventory at time ' t ' is obtained as
I(t) = See/t & V - e-((t V f (— - fa ) e(du!+2#u) du ; 0 < t < t,
Jt \a-yu J 1
I(t) = Se8/t 1 v0+2#(-« - fae-(e t!+2# V f e(8uV+2#u) du ; t± <t<t2
I(t)= ^ (e2#(t V -l) ; t2 <t<t3
I(t) = e-2#t C ft - fa) e2#u du + f5 - fa) e2#u du\
LJt3 \a-yu J Jt3 \a-yu J J
Production quantity Q in the cycle of length T is
t- <t<T
(8) (9) (10) (11)
Q= £ K(t)dt + £ K(t)dt = % log((aa%#g^)
(12)
From equation (8) and using the initial condition I(0) = 0, we obtain the value of 'S' as
s = r&" (_1_ - ф\е{ви!+ф2и)du
J0 \а-%и J
(13)
When t = t3, then equations (10) and (11) become
I(t3)= g {еФ2(&2-&$) - i: and
I(t3) = е-ф2&3 ¡5 - 4>i) e2#и du respectively.
(14)
(15)
Equating the equations (14) & (15) and on simplification, one can get
t2 = t3 + 2- ln[l+ ф2 e-2#&$ £ (а$%4 -ф1)е22ийи]=х(13) (say)
(16)
Let K(t1, t2, t3) be the total production cost per unit time. Since the total production cost is the sum of the set up cost, cost of the units, the inventory holding cost. Hence the total production cost per unit time becomes
K(tv t2,t3)= 7+c-9+5 [jo£i i(t)dt + £ i(t)dt]+; [j* -i(t)dt + ¡5 -i(t)dt]
(17)
Substituting the values of I(t) and Q in equation (17), we can obtain K(t1, t2, t3) as
t"
A C I a (a — yt3) \ h K(ti,t2,t-)= = + -=7^
T ' - yt1)(a - yT)J + T
i 1
Jt
See(t!-t!)+22(t1-t)
-е-(в&! + Ф2&) V 1 (--ф)е((и! + ф2и) du
jt \a — yu J
t2 t
J See(t?-t!) + 22(ti-t) - ^e-(dt! + 22t) J е(ви1 + ф2и) du
+
n + T
dt
îl J(1 - e22(t2-t))dt - J e-ф2& J (^а1уи-ф1Хеф2иЛи
+
¡5 (— - ф1) еф2и du]
Jt3 \а-уи J J
dt
0n integration and simplification one can get
А С i a (a - yt3) \ h K(tv t2, t-)= r + ^log [-- , 3 ^ T + -
T yT \(a - yt1)(a - yT) J T
&!+ф2&") J Se^+^dt
t"t
J gHe^w) J" (-^--ф^е^^^аи
dt
(18)
fa J e~((&!l22t) J e(Su!+22u)du
&i 1
- & \e-*2t [£& du+ rnîïldu]dt + 21 (e*2(T-t) - 1)1 dtl
J&3 L lJt3 a-vu J&3 a-vu J 22 J J
n
dt
7
(t3 — t2
1 02
(19)
Substituting the values of S and t2 in equation (19), one can obtain
A C i a (a — yt3) K(tvt3)= - + —log' ' y 3
dt
T yT y (a — yti)(a — yT)
=(t3)
1
h + T
t"
V e-(dt!+22t) V 1 W--é!)e(du!+224)du
J J0 \a — yu J
j e-(et!+22&) J" W—^-—01Xe(0uV+*24)du
dt
— 0i
j e-(dt! + 22&) j e(du! + 22U) du
dt
+
n
Tfa
e-V2&3
a — yu
— fa ) e22u du — fa[t3 — T
—1 [1 — e22(T-&3) — in [i + ÈLe-<P2&3 f5 i-^. — e22u du]
"02 t e-*2& [& du + ft5!%ldu] dt]
2 Jt$ LJt3 a-vu Jt3 a-vu J J
(20)
IV. Optimal Production Schedules of the Model
In this section we obtain the optimal policies of the system under study. To find the optimal values of 1 and t3, we obtain the first order partial derivatives of ((¿forgiven in equation with respect to 1 and t3 and equate them to zero. The condition for minimization of (13) is Where D i s the Hessian matrix
d2K(t1,t3) d2K(t!,t3)
D =
dt-t2 dt1dt3 d2K(ti,t-) d2K(ti,t3)
dt1dt3
> 0
Differentiating K(tr, t3) given in equation (20) with respect to 1 and equating to zero, we get
a — yt1
.+ he/6t1l+22ti)
— fa" e-(6t!+*2t)
dt]+fa f=
=(t$) p-(0t!+<p2t)
dt] =
fc—W j <-"t,i'tMdt
(21)
Differentiating K(t1, t3) with respect to t3 and equating to zero, we get —C + _ _\a. . / 1
a — yt-
he->d/x(t$)0V+22=(t$)}y(t3) j 1 W—L- — fa) e(duVl*2u)du
=(t3)
i j
t1
n 02
3 — 2e22(t$-T) a — yt-
-01
1 _ e<Pi(T-t3) .
= 0
(a-Yt3)[<p1+<l,2e-<t>2t3 2" du] (22)
where, x(t3) = t2 = t3 + — In \l C (— - é,) du]
3 z 02 L 0! Jt3 \a-yu T x/ J
d 1 y (t3) = 3) =-f----7—--T
dts O - Yt3) 01 + J ^ - 0 ) e02U
- 1
Solving the equations (21) and (22) simultaneously, we obtain the optimal time at which production is stopped t{ of t1 and the optimal timetg of t3 at which the production is restarted after accumulation of backorders. The optimum production quantity Q* of Q in the cycle of length T is obtained by substituting the optimal values of t{, t3 in equation (12) as
= llogf )
(23)
V. Numerical Illustration
The numerical illustration is carried to explore the effect of changes in model parameters and costs on the optimal policies, by varying each parameter (-15%, -10%, -5%, 0%, 5%, 10%, 15%) at a time for the model under study. The results are presented in Table 1. The relationships between the parameters and the optimal values of the production schedule are shown in Figure 2.
Figure 2: Relationship betioeen parameters and optimal values with shortages
Table 1: Numerical Illustration of the Model - With Shortages
Variation Parameters Optimal Policies -15% -10% -5% 0% 5% 10% 15%
A t\ 5.8924 5.892 5.8916 5.8912 5.8908 5.8903 5.8899
tl 8.6156 8.6162 8.6169 8.6175 8.6181 8.6187 8.6193
Q* 27.2344 27.2313 27.2281 27.225 27.2219 27.2187 27.2156
K* 69.2564 72.1721 75.0877 78.0034 80.919 83.8347 86.7503
С t\ 5.8796 5.8835 5.8873 5.8912 5.895 5.8988 5.9026
tl 8.6648 8.6489 8.6332 8.6175 8.6018 8.5863 8.5708
Q* 27.0359 27.0993 27.1623 27.225 27.2872 27.3491 27.4106
K* 74.5927 75.7248 76.8617 78.0034 79.1498 80.3009 81.4566
h t\ 5.8891 5.8898 5.8905 5.8912 5.8919 5.8926 5.8933
tl 8.6181 8.6179 8.6177 8.6175 8.6173 8.6171 8.6169
Q* 27.218 27.2203 27.2226 27.225 27.2273 27.2297 27.232
K* 78.2736 78.1835 78.0934 78.0034 77.9133 77.8233 77.7334
n t\ 5.8909 5.891 5.8911 5.8912 5.8913 5.8914 5.8914
tl 8.5835 8.5948 8.6061 8.6175 8.6288 8.6402 8.6515
Q* 27.3403 27.3019 27.2635 27.225 27.1864 27.1477 27.109
K* 78.1961 78.1288 78.0646 78.0034 77.9451 77.8898 77.8375
a t\ 5.9436 5.9245 5.9067 5.8912 5.8776 5.8664 5.8564
¿3 8.2309 8.3944 8.5196 8.6175 8.6961 8.7571 8.8032
Q* 52.6288 39.1438 31.9239 27.225 23.8626 21.4632 19.8741
K* 102.832 89.4855 82.5151 78.0034 74.7664 72.4419 70.0115
Y t\ 5.8722 5.8809 5.8859 5.8912 5.8966 5.902 5.9072
¿3 8.7439 8.7112 8.6671 8.6175 8.5612 8.4968 8.423
Q* 21.5134 23.3696 25.0854 27.225 29.9954 33.7873 39.4837
K* 71.8131 74.203 75.8954 78.0034 80.7391 84.512 90.2701
в t\ 5.8921 5.8918 5.8915 5.8912 5.8909 5.8906 5.8904
¿3 8.6173 8.6174 8.6174 8.6175 8.6175 8.6176 8.6177
Q* 27.2277 27.2268 27.2259 27.225 27.2241 27.2233 27.2225
K* 77.9196 77.9484 77.9763 78.0034 78.0296 78.0549 78.0796
V t\ 5.8925 5.8921 5.8916 5.8912 5.8907 5.8903 5.8899
¿3 8.6172 8.6173 8.6174 8.6175 8.6176 8.6177 8.6178
Q* 27.2292 27.2278 27.2264 27.225 27.2236 27.2222 27.2208
K* 77.8796 77.9207 77.962 78.0034 78.0449 78.0864 78.128
0i t\ 5.8909 5.891 5.8911 5.8912 5.8913 5.8914 5.8915
¿3 8.5841 8.5951 8.6062 8.6175 8.6289 8.6405 8.6523
Q* 27.338 27.3009 27.2633 27.225 27.1861 27.1465 27.1062
K* 79.1187 78.7492 78.3773 78.0034 77.6275 77.25 76.8709
02 t\ 5.8922 5.8918 5.8915 5.8912 5.8909 5.8906 5.8903
¿3 8.592 8.6009 8.6094 8.6175 8.6253 8.6328 8.6402
Q* 27.3141 27.2828 27.2532 27.225 27.1978 27.1713 27.1455
K* 78.9696 78.6124 78.2924 78.0034 77.7405 77.4996 77.2776
VI. Observations The major observations drawn from the numerical study are:
• It is observed that the costs are having a significant influence on the optimal production quantity and production schedules.
• As the setup cost 'A' decreases, the optimal production downtime t** and the optimal production quantity Q* are increasing and the total production cost per unit time K* and the optimal production up time t- are decreasing.
• As the cost per unit 'C' decreases, the optimal production up time t- increases and the optimal production downtime i*, the optimal production quantity Q* and the total cost per unit time K* are decreasing.
• As the holding cost 'h' decreases, the optimal production up time t- and the total production cost per unit time K* are increasing, the optimal production downtime t** and the optimal production quantity Q* are decreasing.
• As shortage cost 'n' decreases the optimal production downtime i*, the optimal production quantity Q* and the total production cost per unit time K* are increasing and the optimal production up time t- decreases.
• As the production rate parameter 'y' decreases, the optimal production up time t-increases and the optimal production downtime i*, the optimal production quantity Q* and the total production cost per unit time K* are decreasing.
• Another production rate parameter 'a' decreases, the optimal production downtime i*, the optimal production quantity Q* and the total production cost per unit time K* are increasing and the optimal production up time t- decreases.
• As deteriorating parameter 0 decreases, the optimal production downtime t** and the total production cost per unit time K* are increasing and the optimal production up time t- and the optimal production quantity Q* are decreasing.
• Another deteriorating parameter n decreases the optimal production downtime t** and the optimal production quantity Q* are increasing and the optimal production up time t- and the total cost production per unit time K* are decreasing.
• As demand rate parameter 0* decreases, the optimal production quantity Q* and the total production cost per unit time K* are increasing and the optimal production downtime t** and the optimal production up time t- are decreasing.
• Another demand rate parameter increases, the optimal production quantity Q* and the total production cost per unit time K* are decreasing, the optimal production downtime t** and the optimal production up time t- are increasing.
Finally, from the numerical illustrations we can observe that the parameters are having tremendous influence on the optimal policies of the system.
VII. EPQ Model without Shortages
In this section the inventory model for deteriorating items without shortages is developed and analyzed. Here, it is assumed that shortages are not allowed and the stock level is zero at time t = 0. The stock level increases during the period (0, t*) due to excess production after fulfilling the demand and deterioration. The production stops at time t* when the stock level reaches S. The inventory decreases gradually due to demand and deterioration in the interval (t*, T) . At time T the inventory reaches zero. The schematic diagram representing the instantaneous state of inventory is given in Figure 3.
i(t)
Figure 3: Schematic diagram representing the inventory level
Let I(t))e the inventory level of the system at time 't' (0 < t < T). Then the differential equations governing the instantaneous state of I(t) over the cycle of length Tare:
. i(t) + h(t)i(t) = ^ -9 Ф1 + Ф"1(1))
0 <t<t,
(24)
.I(t) + h(t)I(t) = -{<pi + <P"i(t)) ; t± <t<T (25)
where, h(t) is as given in equation (3), with the initial conditions I(0) = 0, ¡(t^) = S and I(T) = 0. Substituting h(t) in equations (24) and (25) and solving the differential equations, the on hand inventory at time 't ' is obtained as
I(t) = Sed/tll-tV0+2#(t"-t) - e~(8tV+2#t) f1 - 01) e(du!+2#u) du ; 0 < t < t1 (26)
I(t) = See</t?-t,10+2#(ti-t) - (pie-(etV+2#t) f^ e(6u!+2#u) du ; t1 <t<T (27)
Production quantity Q in the cycle of length T is
Q= f1 = % (28)
From equation (26) and using the initial conditions I(0) = 0, we obtain the value of 'S' as
5 = e-(dt1+22ti) fti (_l_ - e(0u1+<p2u)du (29) J0 \a-yu J v '
Let K(t1) be the total production cost per unit time. Since the total production cost is the sum of the set up cost, cost of the units, the inventory holding cost. Therefore the total production cost per unit time becomes
K(ti) = 7 + f + 5 [ft1 №dt + ¡11(t)dt\ (30)
Substituting the values of I(t) and Q in equation (30), one can obtain K(ti) as
AC ( a \
K(ti) = - + —\og{--)
T vT \a- vt-i )
h
+ T
YT 1
J Se^i-^^"-^ - e-((&!+Ф2&) J 1 W^—^ - ф±Х е((и!+ф2и) du
О t ^
+ ¡5 [se^i-^^"-^ - ф^-^^) |* е((иГ1+ф2и) du ] dt]
dt
(31)
0n simplification, one can get
AC ( a \ h
K(t!) = - + —log{--) + -
T yT \a — ytJ T
(etT+fct") J Se~(8tV+2#t)dt
t
J e-(St!l2it) J" — e(Su!+<p2u) du
dt
~(8t!+22t) ^ e(8u!+22U)
du] dt]
Substituting the value of S in equation (32), one can obtain
AC ( a \
K(t±) = log(--)
T yT \a — Yh/
T
h + T
J e-((t!+22t) J" (—--fa)e((u!+22u)du
J J0 \a — yu J
— J e-((t! + 22t) J " — fa) e(6U!l*2U) du
0 t ^ ■ £ [e-((t!+22t) £ e(du!+22u) du] dt]
dt
dt
(32)
(33)
VIII. Optimal Production Schedules of the Model
In this section we obtain the optimal policies of the inventory system under study. To find the optimal values of ti, we equate the first order partial derivatives of K(ti) with respect to ti equate them to zero. The condition for minimum of K(ti) is
d"K(ti)
> 0
Differentiating K(t!) with respect to t1 and equating to zero, we get
T
a — Ytt
t—^ — 01) l
,-(et*+&2t) dt
J"'1 e-(#t*+&2t) dt] + f e-(#t* + &2t)
0 J $i
2 dt =0
(34)
Solving the equation (34), we obtain the optimal time t* of t1 at which the production is to be stopped.
The optimal production quantity Q* of Q in the cycle of length Tis obtained by substituting the optimal values of t1 in equation (28).
IX. Numerical Illustration
The numerical illustration is carried to explore the effect of changes in model parameters and costs on the optimal policies, by varying each parameter (-15%, -10%, -5%, 0%, 5%, 10%, 15%) at a time for the model under study. The results are presented in Table 2.
The relationship between the parameters and the optimal values of the production schedule is shown in Figure 4.
c o
>
\ 8,56
\ 8,53
\ 8,5
\ 8,47
8.3SV
8,29
—826
-15 -10
10 15
Percentage change in parameters
il
4>i
i/2
Percentage change in parameters
Percentage change in parameters
Figure 4: Relationship between parameters and optimal values without shortages
Table 2: Numerical illustration of the model - Without Shortages
Variation Parameters Optimal Policies -15% -10% -5% 0% 5% 10% 15%
A t\ 8.3703 8.3692 8.368 8.3669 8.3657 8.3645 8.3634
Q* 17.7291 17.7253 17.7215 17.7177 17.7139 17.7101 17.7062
K* 66.83 69.7424 72.6549 75.5674 78.4798 81.3923 84.3047
C t\ 8.3311 8.3431 8.355 8.3669 8.3787 8.3905 8.4023
Q* 17.6011 17.64 17.6789 17.7177 17.7565 17.7952 17.8339
K* 72.7991 73.718 74.6408 75.5674 76.4978 77.432 78.37
h t\ 8.3631 8.3644 8.3656 8.3669 8.3681 8.3694 8.3706
Q* 17.7055 17.7096 17.7136 17.7177 17.7218 17.7258 17.7299
K* 75.6268 75.6069 75.5871 75.5674 75.5476 75.528 75.5083
a t\ 8.5814 8.4906 8.4214 8.3669 8.3227 8.2863 8.2556
Q* 24.275 21.5265 19.4117 17.7177 16.3216 15.1462 14.1399
K* 82.7598 79.7446 77.4251 75.5674 74.0371 72.7491 71.6467
Y t\ 8.3039 8.3202 8.3423 8.3669 8.3944 8.4254 8.4607
Q* 16.1354 16.6849 17.1767 17.7177 18.3176 18.9892 19.7498
K* 73.7729 74.4309 74.9721 75.5674 76.2279 76.967 77.8039
0 t\ 8.3679 8.3676 8.3672 8.3669 8.3665 8.3662 8.3659
Q* 17.7212 17.72 17.7188 17.7177 17.7166 17.7155 17.7145
K* 75.55 75.5561 75.5619 75.5674 75.5726 75.5776 75.5823
V t\ 8.3685 8.368 8.3674 8.3669 8.3663 8.3657 8.3651
Q* 17.7231 17.7213 17.7195 17.7177 17.7158 17.7139 17.712
K* 75.5454 75.5529 75.5602 75.5674 75.5743 75.5811 75.5877
0i t\ 8.3667 8.3667 8.3668 8.3669 8.367 8.3671 8.3672
Q* 17.7173 17.7174 17.7176 17.7177 17.7178 17.718 17.7181
K* 75.888 75.7811 75.6742 75.5674 75.4605 75.3536 75.2467
02 t\ 8.3685 8.3679 8.3674 8.3669 8.3663 8.3658 8.3654
Q* 17.7231 17.7213 17.7194 17.7177 17.716 17.7144 17.7128
K* 75.5528 75.5578 75.5626 75.5674 75.572 75.5765 75.5809
X. Observations The major observations drawn from the numerical study are:
• It is observed that the costs are having a significant influence on the optimal production quantity and production schedules.
• As the setup cost 'A' decreases, the optimal production time t* and the optimal production quantity Q* are increasing and the total production cost per unit time K* decreases.
• As the cost per unit 'C' decreases, the optimal production time t*, the optimal production quantity Q* and the total production cost per unit time K* are decreasing.
• As the holding cost 'h' decreases, the total production cost per unit time K* increases and the optimal production time t* and the optimal production quantity Q* are decreasing.
• As the production rate parameter 'y' decreases, the optimal production time t*, the optimal production quantity Q* and the total production cost per unit time K* are decreasing.
• As the production rate parameter 'a' decreases, the optimal production time t*, the optimal production quantity Q* and the total production cost per unit time K* are increasing.
• As deteriorating rate parameter 6 decreases, the total production cost per unit time K* decreases and the optimal production time t* and the optimal production quantity Q* are increasing.
• As deteriorating parameter rç decreases, the total cost per unit time K* decreases and the optimal production time t* and the optimal production quantity Q* are increasing.
• As demand rate parameter decreases, the total production cost per unit time K* increases and the optimal production time t* and the optimal production quantity Q* are decreasing.
• Another demand rate parameter decreases, the total production cost per unit time K* decreases and the optimal production time t* and the optimal production quantity Q* are increasing.
Finally, from the numerical illustration we can observe that the parameters are having tremendous influence on the optimal policies of the system.
XI. Conclusions
This paper addresses the derivation of optimal ordering policies of an EPQ model with the assumption that the production process is random and follows a generalized Pareto distribution. Further it is assumed that the lifetime of the commodity is random and follows a Weibull distribution. The generalized Pareto distribution characterizes the production process more close to the reality. The Weibull rate deterioration can include increasing/decreasing/constant rates of deterioration for different values of parameters. The sensitivity analysis of the model reveals that the replenishment distribution parameters have significant influence on the optimal values of the production uptime, production downtime, production quantity and total cost per a unit time. The deterioration distribution parameters also influencing the optimal values of the model. The replenishment and deterioration distributions can be estimated by using historical data. With the distributional data the production and deterioration distributions parameters can be estimated and the analysis of the production process can obtain the optimal production downtime and uptime. This model also includes some of the earlier models as particular cases for specific or limiting values of the parameters. This model can be extended for the cases of changing money value (inflation) and multicommodity production systems which will be taken elsewhere.
Funding
No funding was provided for the research.
Declaration of Conflicting Interests
The Authors declare that there is no conflict of interest.
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