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 Authors : Senthil Kumar P
 Paper ID : IJERTV5IS010602
 Volume & Issue : Volume 05, Issue 01 (January 2016)
 DOI : http://dx.doi.org/10.17577/IJERTV5IS010602
 Published (First Online): 27012016
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
A Discrete Time Two Commodity Inventory System
P. Senthil Kumar
School of Mathematics Madurai Kamaraj University Madurai – 625021, India
Abstract In this article, we consider a twocommodity inventory system under discrete time review. The demands for each commodity – i(i 1,2) arrive according to a independent
Bernoulli process. The maximum inventory level for the
i
i
ith (i = 1,2) commodity is fixed as S , i = 1,2 and the
reorder level as si , i = 1,2 . The ordering policy is defined as, when both the inventory levels are less than or equal to their respective reorder levels, we place an order for Qi (= Si si ), i = 1,2 units . The lead time distribution is assumed to be geometric. The demands that occur during the stockout period are considered to be lost. Some system performance measures in the steady state are derived and the total expected cost rate under a suitable cost structure is calculated. The results are illustrated numerically.
Keywords D iscrete time; two commodity; inventory system; joint order policy;
I. INTRODUCTION
One of the factors that contribute the complexity of the present day inventory system is the multitude of items stocked and this necessitated the multicommodity systems. In dealing with such systems, in the earlier days models were proposed with independently established reorder points. But in situations were several product compete for limited storage space or share the same transport facility or are produced on (procured from) the same equipment (supplier) the above strategy overlooks the potential savings associated with joint ordering and, hence, will not be optimal. Thus, the coordinated, or what is known as joint replenishment, reduces the ordering and setup costs and allows the user to take advantage of quantity discounts.
Inventory system with multiple items have been subject matter for many investigators in the past. Such studies vary from simple extensions of EOQ analysis to sophisticated stochastic models. References may be found in [15, 2, 22, 24,
27, 17] and the references therein.
Kalpakam and Arivarignan [9] have introduced (s,
S) policy with a single reorder level s defined in terms of the total number of items in the stock. This policy avoids separate ordering for each commodity and hence a single processing of orders for both commodities has some advantages in situation where in procurement is made from the same supplies, items are produced on the same machine, or items have to be supplied by the same transport facility.
Krishnamoorthy and Varghese [10] have considered a two commodity inventory problem without lead time and with Markov shift in demand for the type of commodity namely commodity1, commodity2 or both commodity, using the direct Markov renewal theoretical results. Anbazhagan and Arivarignan [4, 5, 6, 7] have analyzed two commodity inventory system under various ordering policies. Yadavalli et al. [26] have analyzed a model with joint ordering policy and varying order quantities. Yadavalli et al. [27] have considered a twocommodity substitutable inventory system with Poisson demands and arbitrarily distributed lead time.
A twocommodity inventory system under continuous review is analyzed by Sivakumar [21]. They assumed that both the commodities are substitutable in the sense that at the time of zero stock, the other commodity is used to meet the demand. During the stockout of both commodities an arriving demand entered the orbit of infinite size. They assumed constant retrial policy with exponential retrial time.
In all the above models, the authors assumed that the time axis is continuous. But, the discrete time systems, are more appropriate than their continuous time counterparts for modelling diverse productive processes, since the basic units in these systems are digital. In discrete time setting, it is assumed that the time axis is calibrated into epochs by small units and that all the events are deemed to occur only at these epochs. With the advent of fast computing devices and efficient transaction reporting facilities, such epochs with small gaps can be conveniently assumed so that events can occur at these epochs.
The analysis of discrete time queueing models has received considerable attention in the literature over the past years, in view of its applicability in the study of many computer and communication systems in which time is slotted ([23], [25]). An important application stems from the secondary and tertiary sector since, for example the current production systems of numerous factories operate on a discrete time basis where events can only happen at regularly spaced epochs.
In the case of inventory modelling under discrete times, the first paper was by BarLev and Perry [8], who assumed that demands are nonnegative integer valued random variables and items have constant life times. Lian and Liu [12] developed a discrete time inventory model with geometrically distributed interdemand times, bulk demands
and constant life time for items. They assumed
(0, S)

The reorder level for the commodity – i is fixed as
ordering policy, with instantaneous supply which clears any
si (1 si < Si ) with an ordering quantity for the
backlog and restores the stock to the maximum capacity S.
commodity – i is Q (= S s > s 1)(i = 1,2) items
This assumption helped them to have fixed life time for all items. They derived the limiting distribution of inventory level through matrixanalytic method.
Lian et al. [13] developed a discrete time inventory system with discrete PHrenewal process for (batch) demand time points and assumed discretePHdistribution for life time of items. They also assumed zero lead time and that unsatisfied demand were completely backlogged.
Abboud [1] studied a discrete inventory model for production inventory systems with machine breakdowns. They assumed that the demand and production rates were constant and that the failure and repair times of each item were independently distributed as geometric.
In this paper we have considered a discrete time two commodity inventory system with independent reorder levels where a joint order for both commodities is placed only when the levels of both commodities are less than or equal to their respective reorder levels. The rest of the paper is organized as follows: In section 2, the mathematical model is described. Section 3 is the central one. The steady state analysis of the model is presented in section 4. Somekey system
i i i i
when both inventory levels are less than or equal to their respective reorder levels. The requirement Qi > si 1(i = 1,2) ensures that after the replenishment the inventory levels of both commodities are above their respective reorder levels; otherwise it may not be possible to reorder (according to this policy) which leads to perpetual shortage. The lead time is assumed to be distributed as geometric distribution with parameter b(> 0) .

The demands that occur during stockout periods are considered to be lost.
We assume that all the above activities are occurred in the slot. Therefore more than one event may occur in the same slot. For mathematical clarity, we need to define the order of the events to be occurred, here, first the replenishment of order, then the demand for the commodity – 1 and finally the demand for the commodity – 2 will be satisfied.
3 Analysis
Let X denote the inventory level of commodity – 1 and Y
performance measures are derived and the total expected cost tt
rate is calculated in section 5. In section 6, some numerical results are presented to illustrate the effect of the parameters on several performance characteristics.
II . Model Description
We consider a twocommodity inventory system where the time axis is divided into intervals of equal length, called slots. The end points of the slots are called slot boundaries. The
denote the inventory level of commodity – 2 at time t . From the assumptions made on the input and output processes, it can be shown that the stochastic process
{(Xt ,Yt ) : t = 0,1,2,,} is a discrete time Markov chain with state space given by
E = {(i, k) : i = 0,1,2,, S , k = 0,1,2,, S }.
system is monitored at the slot boundaries. The maximum 1 2
storage capacity for the commodity – i is Si (i = 1,2) which is used to meet the demands. The demand is for single item per customer. We assume the following:

The demand for the commodity – 1 arrives according to a
The transition probability function is defined as for
(i, k),( j,l) E ,
Bernoulli process with probability a1 . Thus a1
is the
p((i, k),( j,l)) = Pr[X
= j,Y = l  X = i,Y
= k].
probability that a demand occurs at a slot and
a (= 1 a ) ,
t 1
t 1 t t
1 1
is the probability that a demand does not occur in a slot.
The transition probability matrix P of this process,

The demand for the commodity – 2 arrives according to a
Bernoulli process with probability a2 . Thus a2
is the
P = (
p((i, k),( j,l)) ),
(i, k),( j,l) E
probability that a demand occurs at a slot and
a2 (= 1 a2 ) ,
is the probability that a demand does not occur in a slot.
Hence, we have
p((i, k),( j,l)) =
a1a2 ,
j = i, l = k, (or)
i = s1 1, s1 2,, S1 k = 1,2,, S2
ba1 ,
j = i 1,
l = k,
i = 1,2,, s1 k = 0
j = i,
l = k,
i = 1,2,, s1
k = s2 1, s2 2,, S2
ba2 ,
j = i, l = k,
i = 0
k = 1,2,, s2
a a ,
1 2
j = i,
l = k 1, (or)
j = i,
i = s1 1, s1 2,, S1 k = 1,2,, S2
i = 1,2,, s1
ba ,
2
2
ba1a2 ,
j = i,
l = k 1,
j = i,
i = 0
k = 1,2,, s2
i = 1,2,, s1
l = k 1,
k = s2
1, s2
2,, S2
l = k,
k = 1,2,, s2
1
1
2
2
1
1
a a , j = i 1,
i = s 1, s 2,, S
b
b
1
1
2
2
a a ,
j = i,
i = 1,2,, s
l = k,
(or)
j = i 1,
k = 1,2,, S2
i = 1,2,, s
a a ,
1
1
2
2
1
1
l = k 1,
j = i 1,
k = 1,2,, s2
i = 1,2,, s
l = k,
1
k = s2 1, s2 2,, S2
b 1 2
l = k,
1
k = 1,2,, s2
a a , j = i 1,
i = s 1, s 2,, S
a a , j = i 1, i = 1,2,, s
1 2
1 1 1
b 1 2 1
a1 ,
l = k 1, (or)
j = i 1,
l = k 1,
j = i,
k = 1,2,, S2
i = 1,2,, s1
k = s2 1, s2 2,, S2
i = s1 1, s1 2,, S1
ba a ,
1 2
ba1a2 ,
l = k 1,
j = i Q1 , l = k Q2 ,
j = i Q1 ,
k = 1,2,, s2
i = 0,1,2,, s1
k = 0,1,2,, s2
i = 0,1,2,, s1
a ,
l = k,
j = i 1,
k = 0
i = s 1, s 2,, S
a ,
l = k Q2 1,
j = i Q 1,
k = 0,1,2,, s2
i = 0,1,2,, s
1
l = k,
1 1 1
k = 0
ba1 2
1 2
1 2
1
l = k Q2 ,
1
k = 0,1,2,, s2
a2 , j = i,
i = 0
ba a ,
j = i Q1 1,
i = 0,1,2,, s1
We define
< i >= ((i,0),(i,1),,(i, S2 )),i = 0,1,2,, S1. By
ordering the set of states as (< 0 >,< 1 >,, < S1 >), the
transition probability matrix P of the discrete time Markov chain can be conveniently expressed in a block partitioned form with entries,
A1,
A ,
A ,
2
j = i 1
j = i
i = s1 1, s1 2,, S1 i = s1 1, s1 2,, S1
a2 ,
l = k
k = s2 1, s2 2,, S2
A3 ,
A ,
j = i 1
j = i
i = 1,2,3,, s1
i = 1,2,3,, s
a ,
a ,
2
l = k 1
k = s2 1, s2 2,, S2
[P] = 41 a2b ,
l = k
k = 1,2,3,, s2
<i>< j >
A ,
j = i
i = 0
[ A5 ]kl =5 a2b ,
l = k 1
k = 1,2,3,, s2
C1,
C2 ,
0,
j = i Q1 1 j = i Q1 otherwise.
i = 0,1,2,, s1
i = 0,1,2,, s1
b ,
0,
l = k otherwise.
k = 0
where
a1a2 ,
l = k
k = 1,2,3,, S2
[C ]a1a2b,
a a b,
=
=
l = k Q2
l = k Q 1
k = 0,1,2,, s2
k = 0,1,2,, s
a a ,
l = k 1
k = 1,2,3,, S
1 kl 1 2 2 2
[ A ]= 1 2
2 0,
otherwise.

kl
a ,
l = k
k = 0
1
0,
otherwise.
[C ]a1a2b,
a a b,
=
=
l = k Q2
l = k Q 1
k = 0,1,2,, s2
k = 0,1,2,, s
a a ,
l = k
k = 1,2,3,, S
2 kl
1 2 2 2
1 2 2
0,
otherwise.
a a , l = k 1 k = 1,2,3,, S
[ A ]= 1 2 2

kl
a ,
l = k
k = 0
It may be noted that the matrices
A , A , A , A , A ,C
and
1 1 2 3 4 5 1
0,
a1a2 ,
otherwise.
l = k
k = s2 1, s2 2,, S2
C2 are square matrices of size S2 1.
4 Steady State Analysis
It can be seen from the structure of the transition probability matrix P , that the discrete time Markov chain
a a ,
l = k 1
k = s 1, s 2,, S
{(Xt ,Yt ),t = 0,1,2,,} on the finite state space (E) is
1 2
2 2 2
irreducible. Hence, the limiting distribution, exits and it is
[ A3 ]kla a b ,
= 1 2
a a b ,
l = k
l = k 1
k = 1,2,3,, s2
k = 1,2,3,, s
2
2
defined as
1 2
= lim Pr[ X = i,Y = k  X , L ]
a1b ,
l = k
k = 0
(i,k ) t t 0 0
t
0,
a1a2 ,
otherwise.
l = k
k = s2 1, s2 2,, S2
exists and is independent of the initial state. We group the probabilities (i,k ) as follows:
1
1
(i) = (i,0),(i,1),,(i,S ) ,i = 0,1,, S1
a a ,
l = k 1
k = s 1, s 2,, S
2
1 2
2 2 2
and = (0) ,(1) ,,(S ) .
[ A4 ]kla a b ,
= 1 2
a a b ,
l = k
l = k 1
k = 1,2,3,, s2
k = 1,2,3,, s
Then, the limiting probability distribution satisfies the
1 2
2 following equations
a1b ,
0,
l = k
otherwise.
k = 0
P =
and e = 1.
1
1
The first equation of the above yields the following set of equations:
and(Q 1) can be obtained by solving
s1
2 1
2 1
s 1
(s j )
A
A = ,
i = 0
[(I A ) A
1] 1 D C A 1[(I (3A)
) A 1] 1
(i1) 3

5
(i)
(Q1 1)
j =0
j 1 1 2 1
(i1) A3 (i) A4 = (i) ,
i = 1,2,, s1 1
s1 1
( s 1k )
(4)

D C A 1[(I A ) A

1] 1
(I A )
A A = ,
i = s
k 2 1 2 1
(25) = 0
(i1) 1 (i) 4
(i) 1
k =0
1 s1 1
1
A
A = ,
i = s
1, s
2,,Q 2
[ A1(I A2 ) ] [ A1(I A4 )]C2
(6
(i1) 1
(i) 2
(i) 1 1 1
)
and
1
1
(i1) A1 (i) A2 (iQ 1)C1 = (i) ,
i = Q1 1
s1
Q1 2
(7)
1
1
Q 1i
A
A
C
C = ,
D0 Di
[ A1 (I A2 ) ] 1 I
(i 1) 1

2
(i Q1 1) 1
(i Q1 ) 2
(i)
(Q 1)
i =1
i = s1 1
i = Q , Q 1,, S 1
1
(8)
1
1
1
1 1 1
[(I A2 ) A1
] D0C1 A1
1
1
(i) A2 (iQ )C2 = (i) ,
i = S1
[(I A) A 1 ]i(Q11)
(9)
The equations (expect (7)) can be recursively solved to get
2 1
S1
S1
j =0
j =0
D C
D C
A 1[(I A
A 1[(I A
) A 1 ]i(Q1 j )
) A 1 ]i(Q1 j )
iQ1
=
=
,
,
i = 0,1,, S .
i = 0,1,, S .
(i) (Q11) i 1
j 1 1
2 1(10)
where
Di ,
i = 0,1,, s1,
i=Q11
A (I A )1 Q1 1i ,
iQ11
1 2

D C

A 1[(I A
) A 1 ]i(Q11 j ) e = 1
i = s
1, s
2,,Q 2
k 2 1 2 1
I ,
1
i = Q1
1 1
1,
k =0
5 System Performance Measures
= 2 1 0 1 1
= 2 1 0 1 1
[(I A ) A 1 ] D C A 1,
i
[(I A ) A 1 ]i (Q1 1)i = Q1
In this section, we derive some importance system performance measures.

Expected Inventory level
2 1
i Q1
i (Q j )
Let
denote the expected inventory level for the
D C A
1[(I A ) A 1 ] 1 I1
j = 0
j 1 1 2 1
commodity – 1 in the steady state.
i Q1 1
S1 S2
D C
A 1[(I A ) A 1 ]i (Q1 1 j )
I = i(i,k ).
k = 0
k 2 1 2 1 ,
1
i=1 k =0
i = Q
1,Q 2,, S .
Let
2
I
I
denote the expected inventory level for the
1 1 1
with
I
I
commodity – 2 in the steady state.
D = A (I A )1 s1 1A (I A )1 A (I A )1 s1 1
S1 S2
0 1 2
1 4 3 4
I = k(i,k ).
A (I A )1 and
2
i=0 k =1
3 5
D = A (I A )1 s1 1A (I A )1 A (I A )1 s1 i ,

Expected Reorder Rate
i 1 2 1 4 3 4
i = 1,2,, s1
Let R denote the expected reorder rate in the steady state.
s1 s2
function TC analytically. Hence a detailed computational study of the expected cost rate function is carried out in the
R = a2(i,s 1) a1( s 1,k ) a1a2( s 1,s 1).
next section.
(13)
i=0
2 1 1 2
k =0

6 Numerical Analysis


Expected Shortage Rate
1
1
Let SR denote the expected shortage rate of commodity – 1
To study the behaviour of the model developed in this work, several examples were performed and the set of representative results are shown here. Although we have not
in the steady state.
shown the convexity of TC(s1, s2 ), our experience with
S2 considerable numerical examples indicate that the function
1
1
SR = a1(0,k ).
TC(s , s ), is convex.
(14)
Let SR
k =0
denote the expected shortage rate of commodity – 2
1 2
A three dimensional plot of
TC(s1, s2 ) is
2 presented in Figure 1. We use simple numerical search
in the steady state.
S1
procedure to get the optimal values of TC, s1
and
s2 (say
SR 2 (i,0) 1 2
= a .
2
TC*, s* and s* respectively). The minimu(1m5) value of
i=0
TC = 0.829224 is obtained at (s*, s* ) = (12,4) .
1 2
2
2

Total Expected Cost Rate
The longrun total expected cost per unit time for this system in the steady state is given by
TC(S1, S2 , s1, s2 ) = ch I

ch I

csR cr SR

cr SR
where
1 1 2 2
1 1 2
ch
1
The inventory carrying cost of commodity – 1 per unit item per unit time.
ch
2
The inventory carrying cost of commodity – 2 per unit item per unit time.
cs
Setup cost per order.
cr
1
Shortage cost of commodity – 1 per unit item per unit time.
cr
2
Shortage cost of commodity – 2 per unit item per unit time.
By putting the values of s ' from the above measures of system performance, we obtain TC(S1, S2 , s1, s2 ) as
Figure 1: A three dimensional plot of total cost rate per unit time.
We have studied the effect of varying the costs and other system parameters on the optimal values and some of our
S1 S2 S1 S2
results are presented in Tables 1 to 12. The lower entry in
TC(S1, S2 , s1, s2 ) = ch i(i,k ) ch
k(i,k ) each cell corresponds to the total optimal cost value TC*
s1

c a
1
i=1 k =0
s2

a


a a
2
i=0 k =1
1
1
and the upper entries correspond to the local optima
S * respectively.
2
2
S * and
s 2
(i,s2
1)
1 ( s1 1,k )
1 2 ( s1 1,s2
1)
Example 1: We start by examining the effect of the system
i =0
k =0
parameters namely, demand for commodity1 with the probability, a1 ,demand for commodity 2 with the
S2

c a

c
S1
a
probability, a2 and lead time distributions success
r1
1 (0,k )
r2
2 (i,0)
probability, b , on the optimal values
(S*, S* )
and the
k =0
i=0 1 2
Due to the complex form of the limiting distribution, it is difficult to discuss the qualitative behaviour of the cost
corresponding optimal cost observe the following:
TC* . From tables 1 to 3, we

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Table 1: Effect of a1 and a2 on the optimal values
a
1
a2
0.6
0.62
0.64
0.66
0.68
0.7
0.5
36
32
36
31
37
31
38
31
39
31
40
31
0.596968
0.602735
0.608424
0.614051
0.619615
0.625116
0.52
35
32
36
32
37
32
38
32
38
31
39
31
0.601644
0.607352
0.613006
0.618606
0.624057
0.629384
0.54
35
33
36
33
36
32
37
32
38
32
39
32
0.606411
0.612075
0.617655
0.623039
0.628377
0.633667
0.56
35
34
35
33
36
33
37
33
38
33
38
32
0.611276
0.616877
0.62224
0.627569
0.63286
0.638019
0.58
35
35
35
34
36
34
37
34
37
33
38
33
0.616228
0.621672
0.626976
0.632251
0.63737
0.642355
0.6
35
35
35
34
36
34
37
34
37
34
38
34
0.620631
0.62609
0.631466
0.636824
0.641995
0.64691
Table 2: Effect of a1 and b on the optimal values
a
1
b
0.64
0.66
0.68
0.7
0.72
0.74
0.2 37
35
37
34
38
34
39
34
40
34
41
34
0.637157
0.642385
0.647428
0.652465
0.6575
0.662533
0.3
36
34
37
34
37
34
38
34
38
33
39
33
0.631466
0.636824
0.641995
0.64691
0.651736
0.656385
0.4
35
34
36
34
37
34
38
34
38
33
39
33
0.627004
0.632242
0.637487
0.642732
0.647956
0.653081
0.5
35
34
36
34
37
34
37
34
37
33
38
33
0.622057
0.62754
0.633048
0.638513
0.643441
0.648374
0.6
35
34
35
34
36
34
36
33
37
33
38
33
0.617222
0.622643
0.627774
0.632791
0.63789
0.643072
0.7
34
34
34
33
35
33
36
33
37
33
38
33
0.611996
0.617102
0.622171
0.627354
0.632645
0.63804
Table 3: Effect of a2 and b on the optimal values
a2
b
0.55
0.56
0.57
0.58
0.59
0.6
0.1
39
36
38
36
38
36
38
37
38
37
38
38
0.649383
0.652975
0.65673
0.660404
0.664113
0.667944
0.2
36
34
36
34
36
35
36
35
35
35
35
35
0.615055
0.617307
0.619912
0.621936
0.624798
0.626548
0.3
35
33
35
34
35
34
35
35
35
35
35
35
0.608965
0.611276
0.613486
0.616228
0.618129
0.620631
0.4
34
33
34
33
34
34
34
34
34
34
34
35
0.603423
0.605971
0.60882
0.610994
0.613707
0.616184
0.5
34
33
34
33
34
33
34
34
33
34
33
34
0.597483
0.600673
0.603083
0.605841
0.608789
0.611194
0.6
33
32
33
33
33
33
33
33
33
34
33
34
0.591711
0.594621
0.597293
0.600607
0.603158
0.60599
h
h
Table 4: Effect of c
1
and c
h
h
2
on the optimal values
c
p
c
p
0.01
0.011
0.012
0.013
0.014
0.015
0.01
39
36
38
35
38
35
36
34
36
34
36
34
0.658962
0.671693
0.684347
0.695961
0.706759
0.717556
0.011
38
35
38
35
37
34
36
34
35
33
35
33
0.681807
0.694461
0.706773
0.718256
0.728977
0.739367
0.012
37
34
37
34
37
34
35
33
35
33
35
33
0.704464
0.716751
0.729038
0.74038
0.750771
0.761162
0.013
37
34
36
33
36
33
35
33
35
33
34
32
0.726728
0.738965
0.750891
0.762175
0.772565
0.782552
0.014
36
33
36
33
36
33
34
32
34
32
34
32
0.7488
0.760726
0.772652
0.783876
0.793861
0.803847
0.015
36
33
35
32
35
32
34
32
34
32
33
31
0.770561
0.782485
0.794057
0.80517
0.815156
0.824974
h
h
Table 5: Effect of c
1
and cs on the optimal values
c
p
c
s
0.01
0.011
0.012
0.013
0.014
0.015
6
34
30
33
30
32
29
32
29
32
29
31
28
0.559854
0.572914
0.583524
0.594086
0.604649
0.615116
7
36
32
34
31
34
31
33
30
33
30
32
30
0.5878
0.600346
0.611571
0.622726
0.633614
0.644131
8
37
33
36
33
35
32
35
32
34
32
33
31
0.613436
0.625831
0.637443
0.649015
0.660361
0.670101
9
37
34
37
34
36
33
36
33
35
33
34
32
0.63714
0.649427
0.661682
0.673608
0.684192
0.694508
10
39
36
38
35
38
35
36
34
36
34
36
34
0.658962
0.671693
0.684347
0.695961
0.706759
0.717556
11
40
37
39
36
38
36
37
35
37
35
37
35
0.679541
0.692795
0.70546
0.717021
0.728226
0.739431
h
h
Table 6: Effect of c
1
and c
r
r
1
on the optimal values
c
p
c
r1
0.011
0.012
0.013
0.014
0.015
0.016
0.7
37
36
36
36
36
36
35
36
34
35
34
35
0.652792
0.663201
0.672823
0.682189
0.690589
0.698903
0.8
38
36
38
36
37
36
36
35
35
35
35
35
0.665074
0.676875
0.687695
0.697956
0.707688
0.716889
0.9
39
36
38
35
37
35
37
35
36
35
35
34
0.674766
0.687933
0.699703
0.711105
0.721956
0.732028
1.0
39
35
38
35
38
35
37
35
36
34
36
34
0.682774
0.696613
0.709516
0.722362
0.733689
0.744693
1.1
39
35
39
35
38
35
37
34
37
34
36
34
0.68885
0.703775
0.718196
0.730973
0.743522
0.755986
1.2
40
35
39
35
38
34
38
34
37
34
37
34
0.694616
0.70985
0.724622
0.739288
0.752122
0.764671
h
h
Table 7: Effect of c
1
and c
r
r
2
on the optimal values
c
p
c
r2
0.006
0.007
0.008
0.009
0.01
0.011
3
35
37
34
37
33
36
32
35
32
35
31
34
0.562557
0.577559
0.591813
0.605683
0.619037
0.632065
3.5
35
38
34
37
33
36
32
35
32
35
31
35
0.564629
0.579275
0.59347
0.607279
0.620633
0.632951
4
35
38
34
37
33
36
32
35
32
36
31
35
0.566403
0.58099
0.595126
0.608875
0.621751
0.633654
4.5
35
38
34
37
33
36
32
36
32
36
31
35
0.568176
0.582706
0.596783
0.610192
0.622483
0.634356
5
35
38
34
37
33
37
32
36
32
36
31
35
0.569949
0.584421
0.598219
0.610924
0.623215
0.635059
5.5
35
38
34
38
33
37
32
36
32
36
31
35
0.571723
0.585905
0.598981
0.611656
0.623946
0.635761
h
h
Table 8: Effect of c
2
and cs on the optimal values
c
p
c
s
0.01
0.011
0.012
0.013
0.014
0.015
6
34
30
33
29
33
29
32
28
32
28
31
27
0.559854
0.579678
0.598989
0.618152
0.636849
0.655391
7
36
32
35
31
34
30
34
30
33
29
33
29
0.5878
0.60832
0.6287
0.648599
0.668356
0.687667
8
37
33
36
32
36
32
35
31
35
31
34
30
0.613436
0.634927
0.655946
0.676762
0.697229
0.717445
9
37
34
37
34
36
33
37
33
36
32
36
32
0.63714
0.659405
0.681352
0.702948
0.724093
0.745113
10
39
36
38
35
37
34
37
34
36
33
36
33
0.658962
0.681807
0.704464
0.726728
0.7488
0.770561
11
40
37
39
36
39
36
38
35
37
34
37
34
0.679541
0.703039
0.72631
0.749101
0.771788
0.794052
h
h
Table 9: Effect of c
2
and c
r
r
1
on the optimal values
c
p
c
r1
0.012
0.013
0.014
0.015
0.016
0.017
0.75
35
33
34
32
33
31
33
31
33
31
32
30
0.655662
0.675952
0.696178
0.715965
0.735751
0.755042
0.8
35
33
34
32
33
31
33
31
33
31
32
30
0.661314
0.681624
0.701871
0.721658
0.741445
0.760759
0.85
35
32
35
32
34
31
34
31
33
30
33
30
0.666932
0.68717
0.707301
0.727034
0.746706
0.765932
0.9
35
32
35
32
34
31
34
31
33
30
33
30
0.671257
0.691494
0.711583
0.731316
0.750941
0.770167
0.95
35
32
35
32
34
31
34
31
34
30
34
30
0.675581
0.695819
0.715865
0.735598
0.755173
0.774125
1
36
32
36
32
35
31
35
31
34
30
34
30
0.679587
0.699618
0.719319
0.738814
0.757917
0.776868
h
h
Table 10: Effect of c
2
and c
r
r
2
on the optimal values
c
p
c
r2
0.012
0.013
0.014
0.015
0.016
0.017
2.5
36
32
36
32
35
31
35
31
34
30
34
30
0.679931
0.699962
0.71973
0.739225
0.758413
0.777365
3
35
32
35
32
35
31
35
31
34
30
34
30
0.680019
0.700256
0.720141
0.739636
0.75891
0.777861
3.5
35
32
35
32
34
31
35
31
34
30
34
30
0.680075
0.700313
0.720338
0.740046
0.759406
0.778358
4
35
32
35
32
34
31
34
31
33
30
34
30
0.680132
0.70037
0.720402
0.740135
0.7597
0.778854
4.5
35
32
35
32
34
31
34
31
33
30
33
30
0.680189
0.700427
0.720466
0.740199
0.759772
0.778999
5
35
32
35
32
34
31
34
31
33
30
33
30
0.680246
0.700484
0.72053
0.740263
0.759845
0.779071
1
1
Table 11: Effect of cs and cr
on the optimal values
cs
c
r1
6
7
8
9
10
11
0.5
32
31
33
33
34
35
35
36
35
37
36
38
0.520918
0.545697
0.568678
0.589853
0.609871
0.628745
0.55
32
31
34
33
35
35
36
36
36
37
37
38
0.528152
0.553657
0.577177
0.598884
0.61914
0.638388
0.6
33
31
35
33
35
34
36
36
37
37
38
38
0.534011
0.560708
0.584562
0.60691
0.627499
0.6472
0.65
34
31
35
33
36
34
37
36
38
37
38
38
0.539671
0.566374
0.591075
0.613847
0.635036
0.655015
0.7
34
31
35
32
36
34
37
35
38
37
39
38
0.543823
0.571555
0.596722
0.620002
0.641799
0.661972
0.75
34
31
35
32
37
34
38
35
39
37
40
38
0.547975
0.575761
0.601546
0.625613
0.647485
0.668296
2
2
Table 12: Effect of cs and cr
on the optimal values
cs
c
r2
9
10
11
12
13
14
2.5
34
29
35
30
36
31
37
32
38
33
38
34
0.587972
0.608885
0.628377
0.646635
0.663832
0.679977
3
34
29
35
30
36
31
37
32
38
33
38
34
0.588523
0.609261
0.628637
0.646818
0.663964
0.680007
3.5
34
29
35
30
36
31
37
32
37
33
38
34
0.589075
0.609637
0.628898
0.647002
0.664033
0.680036
4
34
29
35
30
36
31
37
32
37
33
38
34
0.589626
0.610012
0.629158
0.647186
0.664072
0.680065
4.5
34
29
35
30
36
31
36
32
37
33
38
34
0.590178
0.610388
0.629418
0.647304
0.664111
0.680095
5
33
29
34
30
35
31
36
32
37
33
38
34
0.590556
0.610735
0.629606
0.647357
0.66415
0.680124

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inventory model with Poisson demand, general lead
time and adjustable reorder size, in: G.V. Krishna Reddy et al.