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28
Impact of Negative Arrivals and Multiple Working Vacation on Dual Supplier Inventory Model with Finite
Lifetimes
M. L. Soujanya a*, P. Vijaya Laxmi b, E. Sirisha c
a Department of Mathematics, Vignan's Institute of Information Technology (Autonomous), Visakhapatnam, India. mailto:[email protected]@gmail.com department of Applied Mathematics, Andhra University, Visakhapatnam, India. mailto:[email protected]@gmail.com c Department of Computer Science and Engineering, Gayatri Vidya Parishad College of Engineering (Autonomous), Visakhapatnam, India. mailto:[email protected], [email protected]
Abstract
In this paper we analyzed an inventory model with two-suppliers, finite life times, multiple working vacations and customers who arrive according to RCE process. Perishable and replenishment rates of two-suppliers are exponentially distributed. The server takes exponential working vacations when the queue is empty. Arrival process follows Poisson distribution and the probability for an ordinary customer is p and for negative customer is q. Limiting distribution of the assumed model is obtained. Numerical results are presented for cost function and various system performance parameters. The impact of two-suppliers on the optimal reorder points will be useful in developing strategies for handling various perishable inventory problems with replenishment rates.
Keywords: (s,S) policy, Two-supply inventory, Lead time, RCE process, Multiple working vacations, Matrix analytic method.
1 Introduction
In an (s,S) inventory policy, the quantity Q(= S — s) is placed if inventory falls to s, so that the maximum inventory level is S. This policy has been widely discussed for almost a century. However in inventory models with more than one supplier we can improve the quality of service, develop strong relationship with the customers, reduce loss of sales due to stock shortages, enhanced profits, etc. In two-supply (s, S) inventory policy, two orders of quantities Qi and Q2 are placed whenever inventory level drops to r and s respectively. For literature on inventory models with two supplies one can refer Yang and Wei-Chung [12] and Vijayashree and Uthaykumar [8].
The life time of inventory items is indefinitely long in many classic inventory models like, vegetables, food items, medical products, etc., which become unusable after a certain period. That means there exists a real - life inventory system which consists of products having a finite lifetime.
M. L. Soujanya, P. Vijaya Laxmi, E. Sirisha RT&A, No 1 (61)
IMPACT OF NEGATIVE ARRIVALS AND MULTIPLE WORKING Volume 16, March 2021
VACATION ON DUAL SUPPLIER INVENTORY MODEL..._
These type of products are called as perishable products and the corresponding inventory system can be considered as a perishable inventory system. Yonguri et. al. [13] studied an inventory models for perishable items with and without backlogging. "A deterministic perishable inventory model with time dependent demands is developed by Sushil and Ravendra [7]. Dinesh and Roberto [1] discussed a perishable inventory model with style goals."
"Sivakumar and Arivarignan [6] introduced negative customers in inventories with finite queues". "For more literature on this concept, one may refer Manual et al. [3, 4]."
In the working vacation (WV) model, the server without stopping the service completely, he continuous with lower service rate. "A continuous review inventory system with a single and multiple server vacations is given by Jayaraman et al. [2]. Periyasamy [5] discussed an inventory system with finite life times, retrial demands and multiple server vacations." For more literature on working vacations on may refer the papers by Vijaya Laxmi and Soujanya, [9, 10] and [11].
In the present paper, an inventory policy with two supply chains for replacement in which one having a lesser lead time is considered. Demands occur according to Poisson distribution. The arriving person may join the system with possibility p or remove one customer from the queue with probability q. The perishable, service rates in busy and WV are exponentially distributed. Limiting distributions are found. Several system performance parameters of the assumed model are presented. Also the analysis of the cost function is also carried out using direct search method.
2 Description of the model
In two supply inventory model, when inventory shrinks to r(> -), a quantity Q1(= S — r) is placed
from the first supplier and is replenished with an exponential rate ■q1. If it falls to s(< Q1) a quantity Q2(= S — s > s + 1) is placed from another supplier and is replenished with an exponential rate > ill). Arrival process follows Poisson distribution with rate A. The arrived customer joins the queue with probability p and removes an existing customer from the end with probability q(= 1 — p). Consider exponential Vacation time, service times during regular period and WV with rates ft, nb and nv, respectively. We assume that, the server stays idle in any period at zero inventory level. Perishable rate follows exponential distribution with rate y.
Let N(t) be the length of the queue, L(t) be the quantity of inventory and <^(t) be the state of the server, which is defined as
f .. _ (0, server is active;
) {1, server is in WV. It is clear that the Markov process {(N(t), ((t),L(t)\t >0} is a state space model with
E = {{(i,j,k)\i > 0,j = 1,2,0 <k<S}
Describe the order sets as
C((< i,j,k >))j=hk=i,2.....s
< Ui >= i . . ,
[((< l,],k >))j=2,k=0,1.....s.
< i >= ((< i,j >))j = l,2.
Then E can be denoted as (< 0 >, < 1 >,...). Therefore, the transition rate matrix P is P =< 0 >< 1 >< 2 >< 3 > -<0> 40C00... <1> BAC0 ... <2> 0BAC ... <3> 00BA ...
where
A, = 121[A0]11[A0]122[A0]21[A0]22,
[¿o]1
— (pA + r]1 + r]2 + ly), m = I ,1 = 1,2,... ,s;
— (pA + TJi + ly),
— (pA + ly), lY,
V2, 0,
m = 1,1 = s + 1, ... , r; m = 1,1 = r + 1, ... , S; m = l — 1, 1 = 2, ... , S; m = l + Q1, 1 = 1, ... , r\ m = I + Q2, 1 = 1, ...,s; otherwise.
kp={0,
112=^ m = l — 1 ,l = 1; [. ~\2i = l0] {0, otherwise. , ["o] {0,
[Ao]
■0, m = I ,1 = 1,2,.,S; 0, otherwise. '
22
— (P^ + V1+V2),
m = 1,1 = 0;
— (pA + y1+y2 + p + ly), m = I , I = 1,2,..., s;
(pA + V1+p + ly), (pA + p + ly),
lY,
T)2. 0,
m = 1,1 = s + 1, ... , r; m = 1,1 = r + 1, ... , S; m = l — 1, 1 = 1, ... , S; , m = l + Q1, 1 = 0, ... , r; m = l + Q2, 1 = 0,...,s; otherwise.
C = 121[Co]11[Co]122[Co]21[Co]22,[Co]12 = [Co]21 = 0
[Co]11 = ^
m = I ,1 = 1, ...,S; otherwise.
; .[Co]22 =
m = I ,1 = 0,...,S; otherwise.
B = 121[B]11[B]122[B]21[B]22, [B]21 = 0, [B]12 =
m = 1 — 1,1 = 1; otherwise.
m = I — 1,1 = 2,... , S; mv, m = I — 1,1 = 1,... S;
qX, m = I ,l = 1,...,S; ,[B]22 =\qA, m = l,l = 0,.,S; 0, otherwise. v.0, otherwise.
A = 121[A1]11[A1]122[A1]21[A1]22,
— (pA +qA + -q1+-q2 + ly + ^b), m = I ,1 = 1,2,...,s;
— (pA + +qA + -q1 + ly + nb), m = 1,1 = s + 1, ... , r;
— (pA + qA + ly + ^b), m = 1,1 = r + 1, ... , S; ly, m = I — 1, I = 2, ... , S; , V1, m = I + Q1, I = 1, ... , r; y2, m = I + Q2, I = 1, ...,s;
0, otherwise.
[A]
11
M12={0,
y, m = 1 — 1 ,1 = 1; 0, otherwise.
P, m = l ,1 = 1,2,.,S; 0, otherwise.
(pA + qA + ■q1 + -q2 + ^v), m = 1,1 = 0;
(pA +qA + -q1+-q2++p + ly), m = I ,1 = 1,2, ...,s;
(pA + qA + ■q1 + p + ly + ^v), (pA + qA + p + ly + nv), lY,
V2, 0,
m = 1,1 = s + 1, ... , r; m = 1,1 = r + 1, ... , S; m = l — 1, 1 = 1, ... , S; ' m = l + Q1, 1 = 0, ... , r; m = l + Q2, 1 = 0,..., s; otherwise.
3 Analysis of the Model
Initially the stability condition of the defined model is determined and then the limiting probabilities are derived in this section.
3.1 Stability condition
For the stability condition, consider the matrix G = A + B + C as
G = 121[G]11[G]122[G]21[G]22,
№ =
(Vi +V2+ßb + ty), m = 1,1 = 1; (Vi +Vz + lY), (Vi + lY), lY,
lY, Vi.
0,
m = I ,1 = 2, ...,s; m = 1,1 = s + 1, ... , r; m = 1,1 = r + 1, ... , S; m = 1-1, 1 = 2, ... , S; m = l + Q1, 1 = 1, ... , r; m = l + Q2, 1 = 1, ...,s; otherwise.
y, m = 1 — 1 ,1 = 1; 0, otherwise.
ß, m = l ,1 = 1,2,.,S;
0, otherwise.
[G]22 =
— i
— i
— i
— i
m = 1,1 = 0; m = l ,1 = 1,2, ...,s;
■ r; . S; , S; ' .. , r; ,,s;
-(pA + qA + ^i+^2 +Vv),
-(pA + qA + Tji+Tj2+ Vv+ ft + lY), -(pA + qA + ■q1+ ft + ly + nv), -(pA + qA + ft + ly + nv), lY,
V2, 0,
Let n be the limiting distribution of G, i.e., nG = 0 and ne = 1, where n = (n1, n2). From nG = 0, we get
ni[G]11 + n2[G]21 = 0 ni[G]12 + n2[G]22 = 0
On solving the above two equations, one can
get n2 = —n1[G]12[G]22 1. Using n 2 value is ne = 1,
we get
n1 = [1 — [G]12[G]22-1]-1
m = 1,1 = s + 1, ... m = 1,1 = r + 1, ... m = 1 — 1, 1 = 1, . m = l + Qi, 1 = 0, m = l + Q2, 1 = 0,. otherwise.
n2 = —[1 — [G]i2[G]
22-i]-i[G]i2[G]2
3.2 Computation of Steady State Vectors
The limiting distribution for the defined model is
limPr[N(t) = i,Ç(t) = j,L(t) = k\N(0),Ç(0),L(0)]
where nis the probability of ith demand at jth state with k inventories. These probabilities are shortly represented as nt. "The limiting distribution is given by nt = n0Rl,i > 1, where R is the minimal non-negative solution of the matrix-quadratic equation R2B + RA + C = 0."
For finding n0 and n1, we have from nP = 0,
ni = —n0C(A + RB)
-i _
n0w,
where
w = —C(A + RB)~
Further, n0A0 + n1B = 0, i. e.,n0(A0 + wB) = 0.
First take n0 as the limiting distribution of A0 + wB. Then nt, for i > 1 can be found using
i
M. L. Soujanya, P. Vijaya Laxmi, E. Sirisha RT&A, No 1 (61)
IMPACT OF NEGATIVE ARRIVALS AND MULTIPLE WORKING Volume 16, March 2021
VACATION ON DUAL SUPPLIER INVENTORY MODEL ._
n1 = now,ni = n1Rl-1,i >2. Therefore the limiting distribution of the system is obtained as follows.
(no+n1 +n2 + -)e = no(1+w(I — R)-1)e.
4 System performance measures
1. Percentage of server busy period is
Pb = HT=1 Zl=1 n{1k) * 100.
2. Percentage of server working vacation period is
pv = HT=1 Yl=o * 100.
3. Average inventory level: The average inventory level (E1L) is defined as
c _ Vs m Vs tt(2,k)
nn = Li=o Lk=1 Kni + Li=1 Lk=o ni .
4. Average service rate: The average service rate is defined as
ESr = YT=o H=1 [Mb^ + wl2Jc)].
5. Average server vacation period: Average server vacation period (Esv) is
Esv = ZZo Tik=o Pnl '
6. Average negative arrivals: Average negative arrivals (ENA) is defined as
ENA = YT=o Yl^qA^ + nf'k)].
7. Average arrival rate: The average arrival rate (EAR) is defined as
EAR=YT=1 n=1 pA[n(1'k) + n(2'k)].
8. Average replenishment rate from 1st supplier: The average replenishment rate from 1st supplier (ERRl) is
Err1 = Y>i=o Yk=1 V1ni ' ^ + Y?=o Y!k=o V1ni ').
9. Average replenishment rate from 2nd supplier: The average replenishment rate from
RR2)
2nd supplier (ERR2) is
Err2 = YT=o Yk=1 V2ni ' ^ + Y= Yk=o V2n( ' ^
10. Average lifetime: The average lifetime (EFR) is defined as EFR=YZo YSk=1 krrt^+n^].
4.1 Cost analysis
Let
CSl = Setup cost from the 1st supplier, CS2 = Setup cost from the 2nd supplier, CH = Holding cost, CF = Failure cost,
CN = Loss due to negative arrivals,
Cv = Fixed cost when the server is on vacation,
CA = Fixed cost for arrivals,
CST = service cost.
Therefore, the total average cost is defined as
TC(s,r) =
Cs1Err1 + Cs2Err2 + Ch^il + ^f^pr + Cn^na + Cv^sv + Ca^ar + ^st^st.
5 Numerical analysis
For this section, let us fix the parameters as S = 14, p = 0.6, q = 0.4, A = 2.3,= 1.2, = 0.8, 0.4, r]2 = 4.7 y = 0.2, ft = 1.2.
Figure 1: Effect of (s, r) on Et
RR1
Figure 1 presents the effect of reordering points (s, r) on the Average replenishment rate of the first supplier (ERRl). Since ERRl is effected with r, it rises as r rises and drops as s rises.
Figure 2: Effect of (s, r) on E,
RR2
Figure 2 presents the effect of reordering points (s, r) on the Average replenishment rate of the second supplier (ERR2). Even though ERR2 is effected with s, the second replenishment order is placed after the first replenishment order is done. Due to this ERR2 increases with the increase in both s and r as show in Figure 2
Figure 3: Effect of (s, r) on the EIL Figure 4: Effect of (s, r) on the EP
The effect (s, r) on EIL and EPR are shown in Figure 3 and Figure 4 respectively. According to our assumption s < S — r, first order is placed if inventory falls to r. Also > replenishment time of the first supplier is greater than that of the second supplier. Due to this EIL and EPR increases up to r = 11(rvsEIL and rvsEPR) and from s = 4(svsElh and svsEPR) which is evident from figures 3 and 4.
Table 1: Values of s*,r* and TC(s*, r*)
Case 1 (s*,r*) (4,8) (3,9) (3,10) (5,11) (4,12)
TC(s*,r*) 684.852 690.974 696.967 703.395 705.422
Case 2 (s*,r*) (4,8) (3,9) (3,10) (5,11) (4,12)
TC(s*,r*) 686.896 693.546 700.301 707.848 711.938
Case 3 (s*,r*) (4,8) (3,9) (3,10) (5,11) (4,12)
TC(s*,r*) 684.948 691.060 697.054 703.518 705.543
Case 4 (s*,r*) (4,8) (3,9) (3,10) (5,11) (4,12)
TC(s*,r*) 903.71 918.745 933.036 947.758 949.734
Case 5 (s*,r*) (4,8) (3,9) (3,10) (5,11) (4,12)
TC(s*,r*) 700.172 706.918 713.492 720.500 722.524
Case 6 (s*,r*) (4,8) (3,9) (3,10) (5,11) (4,12)
TC(s*,r*) 586.104 592.269 598.219 604.647 606.674
Case 7 (s*,r*) (4,8) (3,9) (3,10) (5,11) (4,12)
TC(s*,r*) 684.575 690.697 696.690 703.118 705.114
Case 8 (s*,r*) (4,8) (3,9) (3,10) (5,11) (4,12)
TC(s*,r*) 388.608 394.730 400.723 407.151 409.178
Case 9 (s*,r*) (4,8) (3,9) (3,10) (5,11) (4,12)
TC(s*,r*) 678.421 684.543 690.536 696.964 698.991
Table 1 gives s* and r* that minimize TC(s,r), for different numerical examples which are defined as the following cases.
1. CSl = 10,CS2 = 14, CH = 10, CF = 13, CN = 150, Cv = 100, CA = 250, CST = 20
2. CS1 = 20, CS2 = 14, CH = 10, CF = 13, CN = 150, Cv = 100, CA = 250, CST = 20
M. L. Soujanya, P. Vijaya Laxmi, E. Sirisha RT&A, No 1 (61) IMPACT OF NEGATIVE ARRIVALS AND MULTIPLE WORKING Volume 16, March 2021 VACATION ON DUAL SUPPLIER INVENTORY MODEL ._
3. CSl = 10,CS2 = 20, CH = 10, CF = 13, CN = 150, Cv = 100, CA = 250, CST = 20
4. CS1 = 10,CS2 = 14, CH = 30, CF = 13, CN = 150, Cv = 100, CA = 250, CST = 20
5. Cs± = 10,CS2 = 14, CH = 10, CF = 20, CN = 150, Cv = 100, CA = 250, CST = 20
6. Cs± = 10,CS2 = 14, CH = 10, CF = 13, CN = 50, Cv = 100, CA = 250, CST = 20
7. Cs! = 10,CS2 = 14, CH = 10, CF = 13, CN = 150, Cv = 50, CA = 250, CST = 20
"S1 — J-0, uS2 = J-4, uH - J-0, uF - ^ uN = J-50, uV = 50, uA — uST
CSl — 10, CS2 — 14, CH — 10, CF — 13, CN — 150, Cv — 100, CA — 50, CST — CSi — 10,CS2 — 14, CH — 10, CF — 13, CN — 150, Cv — 100, CA — 250, CST
Table 1 gives s* and r* that minimize TC(s,r). One can observe that the optimum reorder points in all the cases is r = 8 for the first supplier and s = 3 for the second supplier. We know that the average inventory level is more when compared to other performance measures discussed in Section 4. However the total cost function increases with increase in holding cost which is evident form Case 4. Cost values are minimum in Case 8 due to decrease in the fixed cost per unit arrival. Also, cost value rises with the increase in the reordering point of the first supplier.
Conclusion
In this paper, some investigations are done on the impact of two suppliers on an inventory model having negative arrivals, finite lifetimes and multiple working vacations. The main motive of this paper is to place an order for inventory using two suppliers instead of a single supplier. The limiting distribution of the assumed model is derived. Various system performance parameters are discussed and analyzed assumed cost function to obtain s* and r*. Later, one can extend this paper with multi supply inventory models.
Funding
This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
Declaration of Conflicting Interests
The Authors declare that there is no conflict of interest.
References
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