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Sanjay Sharma, Vipin Kumar, Anand Tyagi RT&A, Special Issue No 2(64), A PRODUCTION INVENTORY MODEL WITH..._Volume 16, November 2021
A Production Inventory Model for Deteriorating Items with Effect of Price Discount under the Stock Dependent Demand
Sanjay Sharma1*, Vipin Kumar2, Anand Tyagi1
department of Applied Sciences & Humanities, Ajay Kumar Garg Engineering College, Ghaziabad, India. *[email protected]; [email protected] ^Department of Mathematics BKBIET, Pilani, India. [email protected]
Abstract
In most of the production-based model, the effect of deterioration at initial stage of the production is not assumed even if assumed than considered to be constant, in few productions-based model deteriorations is assumed time-dependent but during initial level of production its effect was not considered. For a product whose life is high this phenomenon is justifying but for those products whose life is very less this is not faring to not assume the effect of deterioration. In the present study, author have considered the effect of deterioration at the initial level of the model and for the rest cycle of the model as well. After a fixed time, a price discount is also offered for partially deteriorated items where a discount rate is offer at the original one, Shortages are not allowed, production is assumed as demand dependent, and demand is assumed as stock dependent where holding cost of the inventory is assumed as a function of time. A numerical example is also discussed.
Keywords: Stock Dependent Demand, Economic Production Quantity, Time dependent Holding Cost.
I. Introduction
Present study contains an inventory control model that decides the amount of a single commodity that satisfies the market requirement over an infinite planning horizon.
One of the important aspects of the production inventory model is deterioration. In previous research, so many researchers have been taken into an account on deterioration. The four major concerns of any manufacturing firm are production, planning, quality, and maintenance. In the present time, each business has a competition day by day for a superior quality item and provides better customer services. Because of globalization and new specialized advancement, the manufacturing infrastructure likewise changes quickly.
In this paper, the non-instantaneous deterioration model under production policy has been considered. Here the deterioration starts with the beginning time (from one day) because the proper storage condition is not good, so it will start to decline. In this circumstance to encourage the sales with more attractive offer on the decayed unit is known as the rebate rate. In the current paper, author
studied and derived a production-based inventory model under the effect of stock dependent demand rate and fixed deterioration rate. Also, author have taken a linear holding cost. The production inventory system has completed in four stages in each cycle.
Author assume that the production starts with zero inventory level in each cycle. After a certain time, demands will be fulfilled then break the production. The collected inventory is then quietly empty out and after some time it reached zero due to the effect of demand and deterioration. The decaying item which loss self-original value sells at a discounted rate. When the inventory falls down to zero then one cycle is complete, the same process will be repeated in the whole process.
Assuming constant rate of deterioration [1] analyzed the effect of Economic Production Quantity model of decaying items, [2] elaborate on the production model of deteriorating items that minimize the total cost. Also [3] focused on the inventory model for the exponential demand rate. An approach based on inventory model developed by [4] and explained a multi-lot size inventory-based system with constant demand and production rates. For different demand rates for different stage of inventory [5], [6], [7], [8] and [9] studied the Economic Production Quantity model under the consideration of partial backlogged.
[10] studied a deteriorating inventory model with exponentially decreasing demand, where author have assumed a finite planning horizon in the model. [11] developed an Economic Production Quantity model with time-varying demand and partially backlogged model. [12] explored the inventory production-based inventory model by considering shortages. [13], investigated and formulated an optimal returned policy-based model by considering reverse logistics with backorders where [14] extended these models with new criteria of limited storage facility under inflation. In all inventory models, author seen that the product almost all items either software and hardware have fixed shelf life. Because of the expiration, new technology, time-consuming, more efforts, non-auto start, etc. substantial decay of inventory system cannot be neglected, it was another major feature of the real world. In this field, [15] developed an Economic Production Quantity model with constant demand and exponentially decaying items.
The stock decreases due to demand which is a function of the on-hand inventory and deterioration which is constant. It is seen that items have a lifetime which cessation when advantages become zero of the on-hand inventories. It is noticed that products have a lifetime which cessation when the benefit becomes zero. Considering the concept of permissible delay in payment, [16] introduced a perishable inventory model with a parabolic rate of demand along with partial backlogging.
For the product of low cycle [17] presented a quality consideration deteriorating inventory model. [18] considered a multivariate demand model for decaying items having shortages. [19] presented a demand dependent production inventory model with price-sensitive demand and shortages. In the last few decades researcher pay attention to the deterioration-based models, they considered the different deterioration rates for different environment conditions. Now author discuss the non - instantaneous deteriorating items in our study so many researchers contribute to this field, but author have discussed some of them.
[20] developed an inventory control model under stock dependent consummation rate. [21] also did work on Gupta's model. [22] and considered different inventory models based on the time value of money and inflation. After that [23] developed the deterministic model where non-linear holding cost was considered for decay items with stock dependent consummation rate. [24] explored an inventory model with a stock dependent rate of demand and holding cost was time-dependent. After that [25] investigated the model in which demand and holding cost both are stock dependent.
[26] built up the echelon inventory model for decay products and assumed variable holding cost with stock dependent demand. In view of non-instantaneous deterioration, under the impact of inflation [27] analyzed and developed an inventory model with completely backlogged shortages
where price and stock dependent demand was taken. By considering trade-credit strategy and two warehouse [28] presented an Economic Production Quantity model for non-instantaneous deteriorating items. [29] analyzed and developed a two-shop based stock model for non-instantaneous and assumed variable holding cost with stock dependent demand. Cardenas Barron et al. (2020) built up an Economic Production Quantity model and examined a uniform inventory model with non- nonlinear stock dependent holding cost and demand. Based on Price discount facility [30] developed an EOQ model for deteriorating items with stock-dependent demand and partial backlogging. For multi-item deteriorating [31] presented a two-echelon inventory model with price-and stock-dependent demand.
For deteriorating items [32] described discount facility and discussed a study of inventory model for deteriorating items with price and stock dependent demand under-price discount facility. In some of the study a concept of overtime production was considered in the same way [33] presented an inventory model for deteriorating items under overtime production for deteriorating items with nonlinear price and stock dependent demand.
For the product with time varying holding cost by offering some quantity discounts [34] developed and discussed partial backlogging inventory model with price and stock level dependent demand.
In most of the inventory models, it is commonly seen that the ordering cost is assumed to be fixed but it is not certifiable in the present emulative market. Under this presumption, many attempts were made by various research scholars in the direction to developed inventory control model for decaying items under the effect of exponential demand where holding cost has been time-dependent. But in this study deterioration was considered at the initial level of the production. In the current paper, considering the more practical phenomena author have discussed the stock dependent rate of demand and time-varying holding cost with rebate rate in the production model and author have assumed the impact of decay at the initial level of production.
II. Assumptions and Notations
• The demand rate for the product is assumed to be stock-based.
• The produced unit of product is always available to face the demand of the market.
• Products start deteriorates at the initial level of the model but price discount on deteriorated
items is offered only when production is stopped.
• Shortages are not permitted in the model.
• There is no replacement or repair of perishable items.
• The holding cost of the inventory units is a function of time.
• p = k.D , where P is the production rate and k > 1, where D is the demand. The notations are as follows used in the present model
A = Setup cost
Ch = Holding cost per unit per unit time Cp = production cost per unit Cd = Deterioration cost per unit p = Production rate per unit time r = Price discount per unit cost ij = The time where production was stopped T = Duration of complete production cycle I (t) = Level of inventory at time t between the intervals 0 < t < tj
I (t) = Level of inventory at time t between the intervals tY < t < T
III. Mathematical Formulation of the Model
The stock level develops during the interval [0, tj. Thus, the stock level in a creation period is represented by the differential equation (1) where the inventory is increasing due to production and reached to the highest level at tt. The stock level exhausts along the [tt, T]. The stock level in a non-creation period is represented by the differential equation (2) also in the time slot due to combined effect of demand and deterioration the inventory level is decreasing and reached to the lowest point at T. This model is shown by the following Figure. 1
Inventory Level
t = 0
Figure 1 Graphical representation of the inventory with respect to time
+ 9tl1(t) = p- (a + bl1(t)) + 6tl2-t) = -(a + bl2(t))
dixit) dt dh(t) dt
Author have the following boundary conditions k(0) = 0 = I2(T)
By using the boundary conditions,
2W b+9 v
T-b + 9) _ et(b+6)\
Maximum inventory level
'Max=h(ti) = (-^f(e-ti(b+s)
I,
1)
Using the condition of continuity I1(t1) = 12(h)
-a-P) (cU(-b-8) - ^ =_^(eT(b+9)e-t1-b + 9) -b + 9 ^ ' b+9 ^
1)
0<t<t1 t1<t<T
0 < t <t1 t1<t<T
(1) (2)
(3)
(4)
(5)
(eti(-b-9) - !)(a -p) = a(e-b+9)-T-ti) - 1)
u =-t
Now various costs associated with the models are:
p
Production cost
PC = Cppt1
Setup cost
SC = A
Cost for Holding the Inventory
hc = Ch[Ç01 I±(t)dt + $Ttih(t)dt]
<2(\ + (T - t1)(b + e))k
(7)
(8)
a . -2h2-(b + e)[ \ " , "1'v" '
HC=T7T^T( 2 {+(2T + T2(b + Û)-t12(b + Û))h
2(b+8)3 I
\+2e(T-ti)(b+9)(h2 + (b + 9)(h1 + t1h2)) (a-P) f-2((b + 9)K + h2) + h(b + 9)2(2h1 + t^)
3 ^(b+e)fh2 + (b + g)(h1 + t1h2)
2
2(.b+e)3\+2e-ti(b+e)(h2 + (b + e)(h1 + t1h2))
(9)
Deterioration cost
1
DC = Cd [/01 eh(t)dt + IT eh(t)dt]
(8 + b)
((a-P) (j-^-^-(¿.((T-tMe + n + l- e<T-i«b+"))
(10)
Price Discount
PD = Cpr
= CP
Il^1( a + bl1(t))dt + lT(a + bl2(t))dt]
a<1 + ^+ i(T - <1)e2 + + (T - t1)e -1))
(11)
The average Total Cost per unit time
i
TC(t1t T)=- [PC + SC + HC + DC + PD]
Cppt1 + A +
-2h2 -(b + e)
2(1 + (T - t1)(b + e))h1
(a-p)
2(b+e)3 ( +2T + T2(b + e)- t12(b + e))h2
+2e(T-ti)(b+e)(h2 + (b + e)(h1 + th)) 2((b + e)h1 + h2) + ti (b + e)2(2h1 + t1h2) 2
2(b+9)3 y+2e-ti(b+e^(h2 + (b + e)(h1 + th))
+
Cd8 (8 + b) i
((a - P) K+P -'j-WT^-^e+V + l- e'T-"b+8'))
+Cpr [a^+iâ-l-t^+^iv- tlV2 + b(e'T-»b+8' + Te- t,e -1))
b+ 8
(12)
Substituting the value of t1=^T ,by (6) in (12),
A
Total Cost per unit time TC =--h aC
+
2 T( b+ 8)3
T
(a-p)\
f2((b + e)h1 + h2)
aT(b+6)
-2e ' v [h2 +
aT(b+ey
P
(2h1 + °-^)\ 1
(b + e)(ih + a-f-j)
- a
i 2(i + (T-f)(b + e))h1 + \\
2h2 + (b + e) v P J
\[2T + (b + e) (t2 - OpT-)) h2 j
\-2e(TJT>+8) (h-2 + (b + e) (h1 + Of)) )
1
v
y
+
cae
T-b + e)
aT(b+9)\
f aT 1 — e p
-a-p)\--+-——
1 p b + 9
\ -b + e)
(
1-
(r-f)(b+9)
+ (T — ^)-b + 9)
+ Cpr
b(a-p)\
Ï1 + .
P
I aT(b+e) \ 1-e V aT I
b+e V I
b+9
+
a\(r-^+b(e(T-aT)-b+e)+r9-ave-i))
(b+9)2
(13)
IV. Solution Procedure
Now the optimum value of T which minimize the total cost. The values of T for which
dTC(T) d2TC(T)
——— = 0, Satisfying the condition gr2 > 0
The optimal solution of the equation (13) is obtained by using Mathematica software. Above said process can also be seen through the following example.
V. Numerical Example
Considering A 20,b ^ 1.2, Cp ^ 15,h1 ^2,h2^ 0.15,p ^ 40,9 ^ .006, Cd ^3,Cp^2,r ^
.02 in appropriate units. The optimal value of T* = 2.20006,1* = 1.10003,7C* = 349.78825
Figure 2. Concavity of the profit function.
a
VI. Sensitivity Analysis
Based on the values used in above example in the model author have examined the sensitivity analysis by changing some parameters one at a time and keeping the rest fixed.
Table 1 Sensitivity Analysis w.r.t. Various Parameters
Parameter % Changed value ti T Auax TC
A +50% 75 1.34725 2.69451 169.0449 359.918
+25% 62.5 1.22741 2.40402 130.22971 354.026
0 50 1.10003 2.20006 114.77403 349.788
-25% 37.5 0.97816 1.98821 71.21354 341.498
-50% 25 0.87699 1.68724 32.21354 333.805
a +50 30 1.43291 1.09105 19.33721 489.125
+25 25 1.22886 1.10554 53.97692 431.226
0 20 1.10003 2.20006 114.77403 349.788
-25 15 0.92214 2.31269 148.29284 276.394
-50 10 0.78764 2.39991 189.76542 200.947
b +50 1.8 1.09371 2.18739 90.53786 354.985
+25 1.5 1.09685 2.19369 102.73451 352.063
0 1.2 1.10003 2.20006 114.77403 349.788
-25 0.9 1.10324 2.20647 133.95031 348.072
-50 0.6 1.11009 2.22018 149.24731 346.857
P +50 60 0.53185 1.59475 107.98073 361.688
+25 50 0.86818 1.82913 110.76742 357.296
0 40 1.10003 2.20006 114.77403 349.788
-25 30 1.33422 2.62627 143.91819 335.358
-50 20 1.67151 2.91326 201.21387 317.963
Author have calculated the sensitivity analysis based on different parameters. The outcome of the result is compared.
Figure 3. t1 v/s change in parameters
Total Cost w.r.t. % change of A,a,b and p
600 500 400 300 200 100 0
234 9 A »a • b ■ p
Figure 6. Total Cost v/s change in parameters
1
5
VII. Observations
In the present study, author have calculated the sensitivity analysis based on the parameters used in the study. Author have made changes in the parameters by (-50%, -25%, 0%, 25%, 50%). Some important inferences drawn from Table -1 and Figures 2 to 5 are as follows:
(i) Table No 1 shows that as the value of A goes from -50% to + 50%, the value of t1,T, Imax , and total cost also increases.
(ii) Table No 1 shows that as the value of a goes from -50% to + 50%, the value of T and Imax decreases while the value of ^ and total cost increases rapidly.
(iii) Table No 1 shows that as the value of b goes from -50% to + 50%, the value of T total cost increases while the value of h, T and Imax decrease rapidly.
(iv) Table No 1 shows that as the value of p goes from -50% to + 50%, the value of T total cost increases while the value of h, T ,and Imax decreases.
As per above analysis it is obvious that it becomes a new tool to take initial deterioration and it is also observed that we have maximize the average total profit.
VIII. Conclusions
In the present study, author analyzed some important facts related to maximization of total cost related to inventory models and developed an inventory control production-based model for deteriorating items to reduce total cost. Also, the model has been developed by assuming the rate of demand depends on in-hand stock and holding cost is time-dependent.
In this model, author have assumed that produced items that have been partially degraded, customers have been discounted on their selling price, and products that were completely damaged or deteriorated have been discarded which is a more realistic assumption and helps to improve the profit. Also, in the present model, author have assumed that for the items having low life deterioration take place at the very initial stage of the cycle and developed this model under the effect of deterioration when production is going on. To validate the optimality, a numerical example has been taken and explained in the model. Sensitivity between the ranges -50% to 50% of different parameters have been carried out to check the deviation. The assumption of taking initial time-based deterioration makes this study more effective. Managerial aspect of the study is that this paper provide a good platform for the research scholars to use this study to investigate various changes in the deterioration perimeter and formulate new study.
IX. Future scope of the study
Present study may be further expanded by making some more changes in the main parameters of the study like this model can also be developed under a shortage, trade credits, and some price discount on the in-hand inventory may also be assumed.
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