CC BY
58
An Inventory Model with Quantity Dependent Trade Credit for Stock and Price Dependent Demand, Variable Holding Cost and Partial Backlogging
Shilpy Tayal1*, S.R. Singh2, Chandni Katariya3, Nidhi Handa3
�
1*Department of Mathematics, Graphic Era Hill University, Dehradun, India
2Department of Mathematics, CCS University, Meerut, India
3Department of Mathematics and Statistics, KGC, Gurukul Kangri Vishwavidyalaya,
Haridwar, India
[email protected], [email protected]
Abstract
Here, the modelling of an inventory system for price and stock reliant demand with the combination of quantity discount and credit limit policy has been presented. Price and stock level are the key sources that always affect the demand of any product. In present study the cost of holding is considered as a time varying function. Vendors usually offer different policies or discounts to attract more customers. Different possible cases for offered trade credit period are discussed in the model. Shortages with partial backlogging are considered here in the development of the model. The different possible cases in this model is exemplified numerically with the help of software Mathematica 11.3 and a sensitivity analysis with respect to distinct system parameters is also presented.
Keywords: Inventory Model, Quantity Discount, Deterioration, Stock and Price Dependent, Demand, Trade Credit, Partial Backordering, Variable Holding Cost, Shortages.
I. Introduction
The aim of the present paper is to develop an inventory model for deteriorating items using price and stock dependent demand. Most of the previous inventory models were developed by assuming constant, time dependent, stock dependent and many different demand patterns. In general, there are more marketing schemes and strategies that influence the market demand. Available stock and selling price are the main factors that affect the demand of customers.
Deterioration is another important factor whose role in the construction of an inventory model is very useful. Deterioration can be defined as the reduction in the original quality and value of the product in any term. Mostly, all available products deteriorate, the only difference is that in some products the rate of deterioration is large and in some it is very low. So, for more accuracy in result, the deterioration should be considered in the modeling of inventory models.
Some researchers worked on inventory models without considering shortages. It cannot be predicted that the stocked units will always be enough to satisfy the demand of all the customers. So, shortage is also an important concept in inventory model that should be taken into account. In the construction of an inventory model the assumption that during stock out the occurring shortages are either completely lost or completely backlogged, is not realistic. So there will be a partial backlogging of the demand during stock out.
Further trade credit period is the useful incentive policy to attract more customers. In this time period vendor allows a certain time limit to retailer to pay all his dues. If the retailer pays all his dues before the credit limit then there will be no interest otherwise interest will be charged on unpaid amount. Retailer can increase his profit by earning interest on sales revenue.
I. ....... .... ........... ....... ....... ....... ..... ... ....... .. ........ .. ....... their business. In the present paper quantity discount policy is considered with trade credit period and both of these works as a promotional tool for the business. To make the study more realistic and to improve the efficiency of the model, holding cost is also taken as a linear function of time.
II. Literature Review
Skouri and Papakristos [1] proposed an inventory model with quantity discount policy using price dependent demand. Chang [2] introduced an inventory model based on price-dependent demand under quantity and freight discounts. Alfares [3] developed an inventory policy under stock level dependent demand, time varying holding cost and quantity discount. Tripathi et al [4] investigated a partial backordering inventory model for deteriorating items under quantity discount scheme.
Geetha and Udayakumar [5] proposed a non-instantaneous deteriorating model for price and advertisement dependent demand with partial backorder. Sanni and Chigbu [6] developed a three-parameter Weibull distribution deteriorating inventory model under stock level dependent demand with shortage backordering. Li and Teng [7] introduced pricing and lot-sizing strategies for perishable products when demand depends on stock level, selling price, product freshness and reference price. Rastogi et al. [8] developed an inventory model for non-instantaneous deteriorating items with price sensitive demand and partial backlogging. Shaikh et al. [9] studied an EOQ model for decaying products using time dependent demand under shortage backordering and trade credit. Tayal et al. [10] presented deteriorating inventory model for two level of shortage using stock dependent demand and fractional backlogging. Rani et al. [11] studied a green supply chain inventory model for decaying items with credit period dependent demand. Handa et al. [12] worked on an inventory model under trade credit policy and shortages in which stock level plays a major role for demand.
In present paper holding cost is taken as a linear function of time. Jaggi [13] proposed a non-instantaneous deteriorating inventory model with variable holding cost in which demand depends upon price, and holding cost is taken as a variable. Tayal et al. [14] introduced an EPQ model with exponential demand rate and time dependent holding cost. Rastogi et al. [15] developed a deteriorating inventory model for price sensitive demand, linear holding cost and trade credit period. Aggarwal et al. [16] proposed an inventory model for price dependent demand, linear holding cost and partial backlogging under inflation.
Skouri et al. [17] formulated an inventory model with Weibull distribution deterioration and ramp type demand rate. Dutta and Kumar [18] studied a deteriorating inventory model in which demand and holding cost is considered as a function of time and permitted shortages are partially backlogged. Mahapatra et al. [19] introduced a model for deteriorating items based on reliability dependent demand under partial backlogging. Singh et al. [20] worked on replenishment policy for decaying items with partial backordering under credit financing and inflation.
Patra [21] investigated effect of inflation and time value of money for two warehouse inventory model under shortages. Bhojak and Gothi [22] introduced Weibull distributed deteriorating inventory model for time reliant demand with deficiency and backordering. Singh and Sharma [23] proposed a reverse logistic supply chain inventory model for imperfect production/remanufacturing with partial backordering and inflation. Kumar et al. [24] studied the effect of preservation and learning on partial backordering inventory model for deteriorating items with the effect of Covid-19 pandemic.
Kumar et al. [25] worked on an inventory model for two-level storage under the effect of learning and inflation. Wang et al. [26] studied a supply chain inventory model for decaying ........ ..... ........ ....... .....t financing. Singh et al. [27] formulated an inventory model under preservation technology using stock dependent demand with credit financing. Shaikh [28] introduced a deteriorating inventory model based on price and advertisement dependent demand under shortage backordering and mixed type of trade credit. Sundararajan and Uthayakumar [29] formulated an optimal inventory policy with promotional efforts and backordering of shortages under trade credit period. Mishra and Talati [30] studied quantity discount inventory policy in which demand depends upon the frequency of advertisement and stock with preservation and backordering of shortages.
This study represents an inventory model considering variable demand, quantity discount and partial backlogging. To make the study more realistic, holding cost is taken as the function of time. Different cases for allowed trade credit period are also elaborated in the model. To improve the efficiency of the model numerical example for different cases and sensitivity analysis for distinct value of parameters have been discussed.
III. Assumptions and notations
level of inventory at any time t coefficients of demand
initial stock level
back order quantity during stock out
rate of backlogging
rate of deterioration
waiting time up to next arrival
cycle time
time at which level of inventory becomes zero
parameters of holding cost
per unit shortage cost
per unit deterioration cost
lost sale cost per unit
purchasing cost per unit
per order ordering cost
selling price per unit
unit time profit
allowed trade credit period
rate of interest charged
rate of interest earned
These following are the assumptions used here:
. Products considered in this model are of deteriorating nature.
. Demand rate is a function of price and stock and is given by
. No replacement policy is allowed for deteriorating products in whole cycle.
. The system allows shortages and partial backlogging.
. Deterioration rate is constant.
. In the model all-units quantity discounts and the length of credit periods are defined as
where denote the boundaries of quantity in units and .
. Backlogging rate is assumed as a function of waiting time. . Holding cost is considered as a linear function of time i.e., . . Trade credit is allowed for different time period.
.
4. Mathematical Modelling
Figure 1. Represents the behaviour of inventory system with respect to time. Here
denotes the initial inventory level at
. The level of inventory depletes in the interval
due to demand and deterioration. At
, inventory reaches to zero level and after that shortages occur. The depletion of the inventory is represented by the following Fig 1.
Inventory system can be represented by the following differential equations:
(1)
..... (2)
Boundary equations are given as follow: Solution of the equations (1) and (2) are given by (3) (4) (5)
V Associated Costs
V.I. Ordering Cost
Ordering cost per order of the system is as follow:
(6)
V.II. Purchasing Cost If c is the purchasing cost per unit and Q1, Q2 are the ordering quantity and backordered quantity respectively then purchasing cost of the system will be:
Where
(7)
Here I(0) denotes the initial inventory level at the starting of the cycle and Q2 is the backordered quantity during stock out of the inventory.
(8)
Hence, the purchasing cost of the system will be
(9)
V.III. Holding Cost Holding cost is considered in the duration of positive inventory. It is a linear function of time and is given by:
(10)
V.IV. Shortage Cost In the inventory system shortage occurs during the stock out of inventory. Shortage cost of the system will be as follow:
(11)
V.V. Lost Sale Cost
In the inventory system lost sale cost occurs when some customers fulfil their demand from other places, during the stock out conditions.
(12)
V.VI Deterioration cost
Deterioration cost is considered for those products that are deteriorated or decayed in the system.
The deterioration cost for the system is as follow:
V.VII Sales revenue Sales revenue:
Hence, the sale revenue of the system is given by
(14)
VVIII.Permissible delay
Two cases for allowed trade credit period are given as follow:
For this case retailer has an adequate amount of funds to clear up all his dues since the credit limit
period is more than the time of positive inventory.
Interest charged in this case will be:
=0
Interest earned in the duration of [0, M] is given by:
(15)
Case 2: When For this case the retailer has to settle all his payment before zero stock. On the basis of interest earned and interest charged, following two cases arise.
Case 2.1: When and
(16)
For this case retailer has enough money to settle all his payments.
Interest charged is given by
(17)
Interest earned in the duration [0, M] is given by
(18)
Case 2.2: When and
(19)
For this case retailer has not enough money to settle all his payments. Interest earned in the duration [0, M] is given by
(20)
Interest charged on unpaid amount is given by
(21)
where B is given by:
V.IX. Unit Time Profit
(22)
VI. Numerical Example
Case 1: When
A=500 per/order, c=35 Rs./unit, d=200 units, k=0.001, T=30 days, M=23 days,
=2.2, =0.1 Rs./unit, =18 Rs./unit,
=15 Rs./unit,
=0.8 Rs./unit,
=0.45 Rs./unit,
=18 Rs./unit
=0.02, After solving this model with the help of corresponding parameters optimal value of
The behavior of the system for
is given by the figure 4 with the help of Mathematica 11.3.
Case 2.1: When and
A=500 per/order, c=35 Rs./unit, d=200 units, k=0.001, T=30 days, M=20 days,
=2.2,
=0.1 =18 Rs./unit,
=15 Rs./unit,
=0.8 Rs./unit,
=0.45 Rs./unit,
=18 Rs./unit,
=0.02, After solving this model with the help of corresponding parameters optimal value of
=47.8656 Rs., =23.2287 days and
=8155.1 Rs. The behavior of the system for is given by the figure 5 with the help of Mathematica 11.3.
Fig. 5: Optimality of the system for case 2.1
Case 2.2: When and
A=500 per/order, c=35 Rs./unit, d=200, k=0.001, T=30, M=17 days,
=2.2,
=0.1, =18 Rs./unit, =15 Rs./unit,
=0.8 Rs./unit,
=0.45 Rs./unit,
=18 Rs./unit
=0.02, =0.03,
After solving this model with the help of corresponding parameters optimal value of
=48.2336,
=23.1396 days and
=8066.84 Rs
The behavior of the system for
is given by the figure 6 with the help of Mathematica 11.3.
The Algorithm
The solution procedure, to maximize the
for optimal ordering quantity is given as follows:
H... U.T.P. .. ... ........ .. ... ......... .... ... .... S.. .. ........ U.T.P. .. ... ... .......
derivatives and After solving these equations, system gives the optimal value of v and p
provided
Find the value of
and p for every credit period length.
Calculate
i.e., ordering quantity for every value of
and p.
Calculate a valid quantity
.
Find out the unit time profit for this
.
Evaluate
for all given credit period lengths and also for the value which is greater than
.
Quantity Discount Approach
Model is demonstrated numerically for the different trade credit period with the help of Mathematica 11.3.
With respect to these credit periods and the values of parameters discussed above, optimal ordering quantity i.e.,
5348.81 units,
47.8659 Rs. and =23.2287 days.
And U.T.Px = 8155.1 Rs. Also 8099.45 Rs., Clearly >
5348.81 is the optimal value of ordering quantity
8155.1 Rs. is the optimal value of unit time profit for the system. Also, optimal value of =23.2287 days and 47.8659 Rs.
VII. Sensitivity Analysis
Sensitivity analysis for distinct parameters is specified as follows. In this the effect of different system parameters on unit time profit is calculated to check the stability of the system.
Case 1: When M
Table 1: Variation in optimal solution for demand parameter (d):
% change in (d) (d)
-20% 160 21.3076 37.3959 5141.32
-15% 170 21.4102 39.7209 5826.7
-10% 180 21.5104 42.0471 6555.9
-5% 190 21.6084 44.3746 7329.19
0% 200 21.7042 46.7032 8146.47
5% 210 21.7979 49.0328 9007.97
10% 220 21.8896 51.3634 9913.83
15% 230 21.9793 53.6947 10864.2
20% 240 22.0672 56.0268 11859.1
Table 2: Variation in optimal solution for shortage parameter (
):
% change in ( ) ( )
-20% 1.76 22.1533 58.3596 10315.7
-15% 1.87 22.0261 54.9293 9675.04
-10% 1.98 21.9097 51.8814 9107.39
-5% 2.09 21.8028 49.1555 8600.98
0% 2.2 21.7042 46.7032 8146.47
5% 2.31 21.6130 44.4855 7736.29
10% 2.42 21.5284 42.4702 7364.30
15% 2.53 21.4497 40.6310 7025.42
20% 2.64 21.3762 38.9457 6715.43
Table 3: Variation in optimal solution for lost sale cost parameter (
):
% change in ( ) ( )
-20% 0.08 28.3126 37.3189 13184.9
-15% 0.085 26.1631 40.2893 11471.9
-10% 0.09 24.4238 42.7617 10127.1
-5% 0.095 22.9620 44.8712 9041.34
0% 0.1 21.7042 46.7032 8146.47
5% 0.105 20.6039 48.3159 7396.91
10% 0.11 19.6293 49.7510 6760.70
15% 0.115 18.7576 51.0395 6214.61
20% 0.12 17.9717 52.2053 5741.37
Table 4: Variation in optimal solution for deterioration cost parameter (
):
% change in ( ) ( )
-20% 14.4 21.6753 46.6497 8159.91
-15% 15.3 21.6827 46.6631 8156.54
-10% 16.2 21.6899 46.6765 8153.17
-5% 17.1 21.6970 46.6899 8149.82
0% 18 21.7042 46.7032 8146.47
5% 18.9 21.7113 46.7165 8143.12
10% 19.8 21.7184 46.7298 8139.79
15% 20.7 21.7255 46.7430 8136.43
20% 21.6 21.7326 46.7562 8133.14
Table 5: Variation in optimal solution for deterioration parameter (
):
% change in ( ) ( )
-20% 12 21.6131 46.4286 8227.84
-15% 12.75 21.6359 46.4977 8207.37
-10% 13.5 21.6588 46.5665 8186.98
-5% 14.25 21.6815 46.6350 8166.68
0% 15 21.7042 46.7032 8146.47
5% 15.75 21.7268 46.7712 8126.34
10% 16.5 21.7494 46.8389 8106.30
15% 17.25 21.7719 46.9063 8086.34
20% 18 21.7943 46.9735 8066.47
Table 6: Variation in optimal solution for interest earned parameter (
):
% change in ( ) ( )
-20% 0.016 21.9530 46.8795 8019.16
-15% 0.017 21.8887 46.8323 8050.30
-10% 0.018 21.8258 46.7873 8081.91
-5% 0.019 2.4643 46.7443 8113.97
0% 0.02 21.7042 46.7032 8146.47
5% 0.021 21.6453 46.6640 8179.39
10% 0.022 21.5877 46.6265 8212.72
15% 0.023 21.5306 46.5906 8246.49
20% 0.024 21.4760 46.5563 8280.55
Case 2: When M
Table 7: Variation in optimal solution for demand parameter (
):
% change in ( ) ( )
-20% 160 22.446 38.1208 5090.59
-15% 170 22.6416 40.5444 5784.96
-10% 180 22.8372 42.9764 6526.76
-5% 190 23.0329 45.4168 7316.64
0% 200 23.2287 47.8656 8155.1
5% 210 23.4247 50.3228 9042.74
10% 220 23.6209 52.7883 9980.14
15% 230 23.8175 55.2622 10967.9
20% 240 24.0146 57.7445 12006.8
Table 8: Variation in optimal solution for shortage parameter (
):
% change in ( ) ( )
-20% 1.76 24.2121 60.2352 10474.1
-15% 1.87 23.9218 56.5753 9782.19
-10% 1.98 23.6646 53.3374 9174.07
-5% 2.09 23.435 50.4524 8635.46
0% 2.2 23.2287 47.8656 8155.1
5% 2.31 23.0422 45.5332 7724.07
10% 2.42 23.8728 43.4195 7335.16
15% 2.53 22.7182 41.495 6982.49
20% 2.64 22.5764 39.7356 6661.24
Table 9: Variation in optimal solution for lost sale cost parameter ( ):
% change in ( ) ( )
-20% 0.08 31.417 40.0413 14157.7
-15% 0.085 28.5285 42.1984 12079.8
-10% 0.09 26.394 44.2577 10483.8
-5% 0.095 24.6733 46.1438 9206.48
0% 0.1 23.2287 47.8656 8155.1
5% 0.105 21.9865 49.4465 7270.26
10% 0.11 20.9008 50.9116 6511.38
15% 0.115 19.9409 52.2844 5849.5
20% 0.12 19.0845 53.5868 5263.38
):
Table 10: Variation in optimal solution for deterioration cost parameter (
% change in ( ) ( )
-20% 14.4 23.1993 47.8156 8163.83
-15% 15.3 23.2067 47.8282 8161.64
-10% 16.2 23.214 47.8407 8159.45
-5% 17.1 23.2214 47.8532 8157.28
0% 18 23.2287 47.8656 8155.1
5% 18.9 23.236 47.8780 8152.93
10% 19.8 23.2432 47.8904 8150.77
15% 20.7 23.2505 47.9028 8148.61
20% 21.6 23.2577 47.9151 8146.46
Table 11: Variation in optimal solution for deterioration parameter (
):
% change in ( ) ( )
-20% 12 23.1130 47.588 8219.98
-15% 12.75 23.1420 47.6579 8203.62
-10% 13.5 23.1710 47.7275 8187.68
-5% 14.25 23.1999 47.7968 8171.68
0% 15 23.2287 47.8656 8155.1
5% 15.75 23.2574 47.9341 8139.12
10% 16.5 23.2861 48.0023 8123.23
15% 17.25 23.3147 48.0701 8107.43
20% 18 23.3432 48.1376 8091.72
Table 12: Variation in optimal solution for interest earned parameter (
):
% change in ( ) ( )
-20% 0.016 23.2312 47.8977 8044.48
-15% 0.017 23.2306 47.8896 8072.14
-10% 0.018 23.230 47.8816 8099.79
-5% 0.019 23.2293 47.8736 8127.45
0% 0.02 23.2287 47.8656 8155.1
5% 0.021 23.2281 47.8577 8182.76
10% 0.022 23.2274 47.8499 8210.42
15% 0.023 23.2268 47.8421 8238.07
20% 0.024 23.2262 47.8344 8265.73
VIII. Observations
. Table 1 and 7 represent the effect of demand
on critical time , selling price
and on . It is observed that with an increment in , there is also an increment in , and in .
. Table 2 and 8 show the effect of on , and on , it is observed that after an increment in
, a pattern of decrement is observed in
, and in in both the tables.
. Table 3 and 9 list the variation in and its effect on , and , it is observed that as the value
of
increases the values of and decreases, while the value of p in both the tables increases.
. Table 4 and 10 represent the effect of
on , and on , it is observed that after an increment in , some increment in and is observed while some decrement in in both the tables is
detected.
. Table 5 and 11 list the variation in parameter on optimal value of , and on . It is
observed that after an increment in , some increment in and while some decrement in in
both the tables are detected.
. Table 6 and 12 represent the effect of on , and on , it is observed that after an increment
in , some decrement in and while some increment in in both the tables are detected.
IX. Conclusion
Present paper considers an inventory model for price and stock-dependent demand under some real-.... .......... .... ........ ....... .... ... ...... ......... ......... I. ....... ...... .... there is the high competition in the market, vendors usually offer new schemes or policies to customers to promote their business. In present study quantity discount policy is also applied because it works as an incentive and a promotional tool for any business. Shortages are allowed with partial backordering. To improve the efficiency of the model numerical examples for different cases and sensitivity analysis for distinct parameters have been discussed with the help of Mathematica 11.3. This Model further can be extended for different demand patterns, deterioration, backlogging rate and also for different realistic approaches like preservation, inflationary environment and green supply chain.
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