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Analysis on Dual Supply Inventory Model having Negative Arrivals and Finite Life Time Inventory
M. L. Soujanya^*, P. Vijaya Laxmib
"Department of Mathematics, Vignan's Institute of Information Technology (Autonomous), Visakhapatnam, India. [email protected] bDepartment of Applied Mathematics, Andhra University, Visakhapatnam, India. [email protected]
b
Abstract
In this paper the impact of dual supply chain on a perishable inventory model with negative arrivals is evaluated. The perishable and replenishment rates of dual suppliers are distributed exponentially. Arrival process follows Poisson distribution and the probability for an ordinary customer is p and for the 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 dual suppliers on the optimal reorder points will be useful in developing strategies for handling various perishable inventory problems with replenishment rates.
Keywords: Matrix analytic method, Steady state distributions, Perishable inventory, Replenishment time, Negative arrivals, Dual supply inventory model, Cost function optimization.
In an (s, S) inventory policy, an order of quantity Q(= S — s) is placed if inventory drops 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 dual supply (s, S) inventory policy, two orders of quantities Q1 and Q2 are placed whenever inventory level drops to r and s respectively. For literature on inventory models with dual supply chains one can refer [9] and [8].
A review of the literature on fixed time perishable inventory models was given by 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 span. That means there exists a real - life inventory system which consists of products having a finite lifetime. These types of products are called as perishable products and the corresponding inventory system can be considered as a perishable inventory system. [10] studied inventory models for perishable items with and without backlogging. A deterministic inventory model for perishable items with time dependent arrivals is developed by [5]. [1] discussed a perishable inventory model with style goals.
Finite waiting hall inventory model with negraive arrivals is introduced by [4]. For more literature on negative arrivals, one may refer [2],[3] and [6, 7].
I. Introduction
In the present paper, an (s, S) inventory policy with dual supply chains for replenishment in which one having a shorter 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 and service rates 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.
II. Model description
In dual supply inventory model, when the inventory shrinks to a fixed level r(> |) an order of quantity Qi(= S — r) is placed from the first supplier and are replenished with an exponential rate n1. If it drops to a prefixed level s(< Q1) an order of quantity Q2(= S — s > s + 1) is placed from the second supplier and is replenished with an exponential rate 72(72 > ni). 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). Service time follows exponential distribution with rates p. Let us assume that, the server stays idle at empty queue. Perishable rate follows exponential distribution with rate 7.
Let N(t) be the queue length, L(t) be the quantity of inventory and Z(t) be the server state, which is defined as
Z( ) = J 1, server is idle at time t;
Z \ 2, server is busy at time t.
Therefore, the stochastic process {(n(t), Z(t), L(t) | t > 0} is a state space model with E = {(0,1,k) | 0 < k < S} U{(i,2,k) | i > 1, 1 < k < S}} and it is subdivided into level z defined as 0 = {(0,1,0), (0,1,1), ... , (0,1, S)} and 1 = {(i, 2,0), (i,2,1),..., (i,2, S)},for i > 1. Then the transition rate matrix P is
P
<0> <1> <2> <3>
<0> <1> <2> <3>
A0 C 0 0
B A C 0
0 B A C
0 0 B A
where
[ A0 ]
[ A]
— (pA + 71 + n2),
— (pA + 71 + 72 + n),
— (pA + 71 + n),
— ( pa + Уy), Уy,
0,
y
x = У,
x = У, x = ^
x = У,
x = y — 1, y
x = y + Ql, y x = y + Q2, y
otherwise.
0;
= 1,2,..., s; = s + 1,...,r; = r +1,..., S;
= 1.....S;
0, . . . , r; 0, . . . , r;
[C] , pA, x = ^ y = 0.....S; [B]
[ ] ' 0, otherwise. ' [ ]
— (pA + qA + 71 + 72),
— (pA + qA + 71 + 72 + yi + V-),
— (pA + +qA + 71 + yY + V-),
— (pA + qA + yY + p), УY,
71, 72, 0,
p, x = y — 1,y = 1,...,S; qA, x = y, y = 0,...,S; 0, otherwise.
y, y, y, y,
y— 1, y
y + Q^ y y + ^ y
x x x x x x x
otherwise.
= 0;
= 1,2,..., s; = s + 1,..., r; = r +1,..., S;
1.....S;
0, . . . , r; 0, . . . , r;
y
III. Analysis of the Model
Initially the stability condition of the defined model is determined and then the limiting probabilities are derived in this section.
I. Stability condition
For the stability condition, consider the matrix G = A + B + C as
— (ni + ), x = y, -(ni + n2 + Ub + yi), x = y,
— (ni + n2 + У7), x = y, -(ni + yrl
[G] = { -УY, УY, ni, n2, 0,
x = У, x = ^
y = 0; y = 1
y = 2,..., s; y = s + 1,...,r; y = r + 1,...,S;
x = y - 1, y = 2,...,S; x = y + Qi, y = 1,...,r; x = y + Q2, y = 1,..., r;
otherwise.
Let n be the limiting distribution of G,i.e., nG = 0and ne = 1 where n = (n(2,0), n(2,1),..., n(2, S)). From nG = 0, we get
—(m + m )n(2,0) + 7n(2,1) = 0
— (n1 + n2 + y?)n(2,l)+ y^n(2,l + 1) = 0,1 < y < s
— (m + y?)n(2, l) + y7n(2, l + 1) + n2n(2, l + Q2) = 0, s + 1 < y < r
— (m + y?)n(2,l) + y7n(2,l + 1) + n2n(2,l + Q1) = 0,r + 1 < y < S
On solving the above two equations and by using ne = 1, one can get the limiting probability values.
II. Computation of Steady State Vectors
The limiting distribution for the defined model is
rW _
K;
lim Pr
N(t) = i, Z(t)= j, L(t) = k |N(0), Z(0), L(0)
where n(i',k) is the probability of ith demand at jth state of the server with k inventories. These probabilities are shortly represented as ni. The limiting distribution is given by n = n0Rl, i > 1. For finding n0 and n1, we have from nP = 0,
n1 = -n0C( A + RB)-1
n0w,
where
w
-C( A + RB)-1.
Further, n0A0 + n\B = 0, i.e., n0(A0 + wB) = 0.
First take n0 as the limiting distribution of A0 + wB. Then n,, for i > 1 can be found using n\ = n0w, n = n\Rt—1, i > 2. Therefore the limiting distribution of the system is obtained as
follows.
(n + ni + K2 + .. .)e = K0(1 + w(I - R) 1)e.
IV. System performance measures (i) Average inventory level: The average inventory level (Ejl) is defined as
S
to S
eil = E k^0 + EE
n
k=0 i=1k=0
(ii) Average service rate: Let Esr denote the average service rate and is given by
Esr = E P
(2,k)
i=1
(iii) Average reorder rate: Average reorder rate (EOR) is given by
Efc=0 Y
E
OR =
n.
(1,s+1) _(1,r+1)
0
+ n.
0
+
E,=1 ES=0(p + y)
(2,s+1) . (2,r+1) ni + ni
(iv) Average negative arrivals: The average negative arrivals (ENA) is given by
S
ena = EE qAn(
(2,k)
i=1 k=1
(v) Average arrival rate: The average arrival rate (Ear) is given by
S
ear = EE Pa
n.
(2,k)
i=1 k=0
(vi) Average replenishment rate from 1st supplier: The average replenishment rate from 1
st
supplier (Err 1) is
err 1 = E 71 n01,k) + E EE mni2,k).
0
k=0 i=1k=0
(vii) Average replenishment rate from 2nd supplier: The average replenishment rate from 2nd
supplier (Err2) is
r ^ (1,kK ^ (2,k) ERR 2 = E 72 n0 ) + EE 72ni( ). k=0 i=0k=0
(viii) Average lifetime: The average lifetime (Efr) is defined as
S AkK f f _(2,k)
efr = E kYn01,k) + EE
n
k=1
i=1 k=1
I. Cost analysis
Let
CH = Carrying cost,
CS = Set up cost,
Cs1 = cost for the 1st supplier,
CS2 = cost for the 2nd supplier,
CF = Failure cost,
M. L. Soujanya and P. Vijaya Laxmi RT&A, No 3 (63)
Perishable inventory model with two suppliers Volume 16, September 2021
CN = Loss incurred due to negative customer, CA = Fixed cost for arrivals, Cst = service cost.
Therefore, the total average cost is defined as
TC(s r) = { CH EIL + CS EOR + CS1 ERR1 + CS2 ERR2 + I CFEPR + CN ENA + CA EAR + CST EST.
V. Numerical analysis
For this section, let us fix the parameters as S = 14, p = 0.6, q = 0.4, A = 2.3, ^ = 1.2, y1 2.4, n2 = 4.7, y = 0.2.
Figure 1: Impact of (s, r) on EOR Figure 2: Impact of (s, r) on EIL
Figure 1 and Figure 2 presents the influence of reordering points (s, r) on EOR and EIL respectively. As we know that, EOR and EIL increases with the increase of reorder points, which is evident from Figure 1 and Figure 2. However, one can observe that Eor decreases from r = 11 as reordering is done very near to S. From Figure 2, one can also observe that EIL is decreased till s = 4 since reordering is done very slowly.
Figure 3: Impact of s on EPR & EST Figure 4: Impact of r on EPR & EST
The influence of s and r on Epr and Est are shown in Figure 3 and Figure 4, respectively. According to our assumption s < S — r, first order is replenished when the inventory reaches to r. Also ^2 > the replenishment time of the first supplier is greater than that of the second supplier, due to this EST decreases with increase in s and r, EPR decreases with s and increases
with r.
Figure 5: Impact of (s, r) on ERR1 Figure 6: Impact of (s, r) on ERR2
Figure 5 and Figure 6 presents the influence of s and r on ERR1 and ERR2 respectively. Since ERR t is effected with r, it rises as r increases and falls as s increases as shown in Figure 5. Even though Err2 is effected with s, the second replenishment order is placed only after the first replenishment order is done. Due to this ERR2 rises with the increase in both s and r which is clearly evident from Figure 6.
Table 1: The values of s*, r* and TC(s*, r*)
Case 1 (s*,r*) TC(s*, r*) (4,8) 893.33 (5,9) 891.62 (7,10) 895.59 (7,11) 889.00 (4,12) 872.29 (7,13) 648.25
Case 2 (s*,r*) TC(s*, r*) (4,8) 1437.35 (5,9) 1436.10 (7,10) 1462.06 (6,11) 1455.07 (4,12) 1423.96 (7,13) 1373.02
Case 3 (s*,r*) TC(s*, r*) (4,8) 899.22 (5,9) 897.79 (7,10) 903.77 (7,11) 897.88 (4,12) 979.79 (7,13) 854.92
Case 4 (s*,r*) TC(s*, r*) (4,8) 893.88 (5,9) 892.27 (7,10) 896.37 (7,11) 890.19 (4,12) 874.56 (7,13) 851.97
Case 5 (s*,r*) TC(s*, r*) (4,8) 893.40 (5,9) 891.69 (7,10) 895.71 (7,11) 889.12 (4,12) 872.38 (7,13) 848.39
Table 1 gives s* and r* that minimize TC(s, r), for different numerical examples which are defined as the following cases.
1: Ch = 30, Cs = 100 Cs1 = 10, CS2 = 30, Cf = 13, CN = 150, Ca = 250, CST = 20
2: Ch = 80, Cs = 100 Cs1 = 10, CS2 = 30, Cf = 13, CN = 150, Ca = 250, CST = 20
3: CH = 30, Cs = 120 Cs 1 = 10, CS2 = 30, Cf = 13, CN = 150, Ca = 250, CST = 20
4: Ch = 30, Cs = 100 Cs 1 = 15, CS2 = 30, Cf = 13, CN = 150, Ca = 250, CST = 20
5: Ch = 30, CS = 100 Cs 1 = 10, CS2 = 35, CF = 13, CN = 150, CA = 250, CST = 20
One can observe that the optimum reorder point in all the cases is (s, r) = (5,9) as 5the optimum total cost exists. 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 2.
VI. Conclusion
In this paper, some investigations are done on the impact of two suppliers on an inventory model having negative arrivals and finite lifetimes. The limiting distribution of the assumed model is derived. Various system performance parameters are discussed and analyzed the assumed cost function to obtain s* and r*.
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.
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