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29
An Economical Order Quantity Inventory Model for Time-
Dependent Deterioration Rate with Price Dependent
Demand under Permitted Delay
Mukesh Kumar1*, Satya Jeet Singh2, Sandeep Kumar3
�
1/ 2 Department of Mathematics/ Graphic Era (Deemed to be University) Dehradun/ India 3Department of Mathematics/ Graphic Era Hill University� Dehradun/ India
Abstract
In the current study, we develop an inventory model for the deteriorating items under the variable holding cost. The decline (or deterioration) rate depends on time; also, the demand rate depends on the price of the item. The shortage cost is linear in nature. The interest rate is a component of selling prices and shortages. Supplier offers a trade-credit period to the customer during which there is no interest charged, but upon the prescribed time limit expiry, the supplier will charge some interest. This study validates with a numerical example and explains the sensitivity analysis, and the optimal solution not only exists but also feasible.
Keywords: Inventory, permitted delay, deterioration and time varying holding cost.
I. Introduction
Generally, an inventory is typical of idle resources in organizations for upcoming use. Manufacturing organizations naturally have lists of raw materials, apparatuses, tools, equipment, semi-finished items, finished items, etc. In service organizations such as banks, financial institutions, hospitals, etc., the inventory consists of various items to be used in the multiple types of forms (for various banking operations), brochures, and pamphlets (for details of different banking policies, schemes, and instruments), etc. Banks also have inventories of currency notes and coins. Hospitals have medical equipment stocks such as a syringe, thermometer, drip, various one-time instruments used by medical professionals, etc. other accessories such as bandages, cotton, spirit, etc., to multiple medicines. Thus, no organization works without inventory. Due to the effects of deterioration, this assumption is not always applicable to certain commonly used physical commodities such as wheat, rice, or some other sort of grains, vegetables, organic products, and so on. A few parts of these products have been damaged, decayed, gasifier, or influenced by various elements. These degraded parts are not so much lost to the stock management office. The decrease in items is one of the severe variables in any stock and creation framework. The acknowledgment of this factor provoked modellers to consider the corruption factor as one of the displaying angles. The weakening stock issue has been widely concentrated by various analysts now and then. Covert and Philip [4] proposed a model of things under a consistent rate of weakening; Deb and Chaudhuri [5] finished the absolute most recent work here, Aggarwal et al. [2] broadened the Goyal model when the circumstance deteriorated. Goyal and Giri [9] projected a typical model affected by item weakening. Roy [17] proposed an inventory model for deteriorating items with price dependent demand and time-varying holding cost. Kumar et al. [14] built up a stock model that decayed after some time; Kingsman [11] developed the effect of payment rules on ordering and stockholding in purchasing. Aarya and Kumar [1] proposed a stock model with various restrictions at a constant decaying rate. An EOQ model accepts that the retailer must compensate the provider following getting the merchandise in practical situations. Providers may regularly permit retailers to direct advance financing to expand requests or diminish stock. This implies the vendor will support the purchaser with a credit period to settle the sum owed, and during this period, the sum owed won't acquire any intrigue. Goyal [8] built up a stock model affected by professional credit; additionally, Goh [7] presented an EOQ model with general demand and holding cost functions; Teng et al. [18] established an optimum pricing and assembling strategy under permitted delay, Mondal et al. [16] demonstrated a model of upgrading objects under demand rate depends on price, Kumar et al. [13] built up a model for the diverse interest rates under exchange credit and in [12] a stock model of the incremental holding cost with the admissible instalment delay proposed, Weiss [19] presented an economic order quantity model with non-linear holding cost. Giri and Chaudhuri [6[ presented a heuristic model for deteriorating items with shortages and time-varying demand and costs, and Yu
[20] proposed an inventory policy for products with price and time-dependent demands.
In the field of inventory management, many authors use various other types of needs and factors.
Some researchers believe that the demand rate is static, linear, price-related, and inventory-related.
The actual target demand may be related to time, inventory, and price. Haley [10] offered Inventory
policy and trade credit financing, Chapman et al. [3] developed Credit policy and inventory control,
Kumar et al. [15] presented a model on preservation technology with trade credits under demand
rate dependent on an advertisement, time and selling price.
In view of the above writing survey, we set up a degradation model. When the corruption rate is
relative to time, and the holding cost is variable, the interest rate is an element of selling price and
all-out deficiency, bringing about lack and total accumulation. On account of postponed instalment
authorization, the interest rate is an element of the business cost, and we utilize a numerical example
to check the model.
Assumptions and Notations:
Assumptions:
�
Deterioration rate changes with time.
�
Shortage is permitted and �completely backlogged�.
�
Demand function is a component of trade cost.
�
Cost of ownership is linear.
�
Renewal or �replacement� rate is immediate.
�
Leading-time is zero.
�
Trade credit is permitted.
Notations: 1) .. : Shortage cost per unit time. 2) .. : Cost of an item per unit. 3) .... . ..: Deterioration rate. 4) .............is the demand rate, where ...... 5) .... . ......, where .... ...is holding cost per unit time. 6) .: Order quantity per cycle. 7) .: Marketing charge per unit item. 8) .is the trade credit period. 9) .is the ordering cost.
10) .is the time period.
11) In period ...... the inventory is positive.
12) In time (..) the stock is drained because of the crumbling and request of the thing. At time
(..) the stock goes to zero and shortage starts.
13) ..: Interest received for each unit time.
14) .. : Interest paid for each unit time with .. . ...
II. Model Formulation and Solution
Modeling, and Solution of Proposed Model
.....
......... . ....., ...... (1)
.. .....
. ....., ...... (2)
..
with initial condition ..... . .
Solution of equations (1) and (2) are given as follows:
. ..
.. ....
.... . ................ . . .
.. .
... .
.. .... ....
....... .., ...... (3)
.... .. . and .... . ............. . ............, .. . . .. (4)
Holding cost
..
HC = . ..........
.
... ...
.. ... .... .. ... ....
........... ............ .. (5)
. .... . .....
Shortage cost
.
SC = .... ...............
..
......
. .. ....... (6)
.
Stock loss due to deterioration
.. ... ..
......... .... . ....... ..
..
.... .
....
. ...... .. . (7)
. ..
Order quantity
.
........ .........
.
..
... ....
................... (8)
. ..
Presently, there are two prospects with respect to the delay period . of allowable deferral in installments. Case I: ..... Case II: .... Case I: .... : In this case, the interest payable for each period of unsold stock after the maturity date (.) is
..
... . .... . ...... .
... ..
.. ... ....
............. . . ....... .
. .... .
. ..
... .. .... ... .. .... ....
............ . .. (9)
. .. . .... .. .. In the calculation, the interest earned (...) in each period is found as
..
... . .... . ........
.
...
. .... ...... (10)
.
Total profit function is
.
Z.T.T............ .O.......C....S.......C....H......C....O.............IP..IE..
T
..
. ........ .. ...
. ....... . ... ................ .
. . ...
. ...
.... .. ... ....
. .......... . . .
.. ......
.. ...
... .... .. ... ....
............ . .............. . . .
... . ....
.. .. ....
.... ...... .... . . .
. ....
.... .
... .... .... ............
...... . ... .
...... .. . Let .....; ..... Thus, total profit is
. ........ .... .....
............... ... ................. .
. . ... ...... .... ..... ...... ..... ...... . .......... . . ............. . .
.. ...... . ..
.... ..... ...... .. ..... .. ....
............. . . ....... .... . . .
. .. .. . ....
..... .. ...... .... ..............
...... ......
...... .. .
Presently, our goal is to optimize .... ... The essential situations for expanding the profit are
....... .......
.. and ..
.. ..
We obtain
. .............. .. ..... ......
.. . ......... . . .
.. . ....
... ..... ...... ......... . . .
. .. ..
.... ...... .......... . .
. .. .. ..... ....... .. .... .. ............. . . ... .... . .
.. . ........
..... .. ..... ............
.........= 0 (11)
........ .
and
. ................. .... ..... ......
.......... .. .............. . . .
. . ..... .... ..... ...... ..... ...... ............ . . ................ . .
...... . .. .... ..... ....... .. ................ . . ....... .
.... .
..... .. .... ..... .. ...... .... .................
.................= 0 (12)
.... ...... .. . Case II: .... For this situation, the interest unpaid for each cycle is zero, when ......on the grounds that the provider can be forked over the required funds at the time ., the allowable delay. In this manner, the premium received in each cycle is the premium received during the great stock time frame in addition to the premium earned from the money contributed during the timeframe .... ..after the
stock is depleted at time .., and it is given by .. ..
............................. ......
..
... . ........ ..................
. .............. ...... (13)
...
Total profit is
.
Z.T.T............ .!.S.......C....H......C......IP..IE..
T
.. .
. ........ .. ... ....
......... ... ................ . . .
. . ... ..
.. .
.. ... .... ..
............................... (14)
. .. ... .
Let ...... ...... Thus, the profit function is
. ........ .... ..... ......
............... ... ................. . . .
. . .....
.... ..... ...... ..
............................... (15)
. .. ... .
Presently, our goal is to optimize .... ... The essential conditions for expanding the profit are
....... .......
.. and ...
.. ..
. .............. .. ..... ......
... . ......... . . .
.. . ....
... ..... ......
.............................= 0 (16)
. .. .. . ................. .... ..... ......
And .......... .. .............. . . .
. . ... .. .... ..... ...... ..
....................................= 0 (17)
. .. ... . Solving equations (11) to (17), we find the .., and ... Also, optimize the function .......using the essential conditions for maximizing of ......are
........ ........ ........ ........ ........
... .. and . . .. at ........
... ... ... ... ....
III. Illustrative Example
Table-1: Sensitivity analysis table
Parameter Parameter Changing (%) Value of Parameter T p Z Changing in Z*(%)
A -50% 125 54 13759 413 -8%
-25% 187.5 61 16842 426 -5%
0% 250 69 24434 450 0%
25% 312.5 72 24444 449 -0.2%
50% 375 74 25061 450.77 0.17%
. -50% 50 72 14394 205.91 -54%
-25% 75 69 17424 319.85 -29%
0% 100 69 24434 450 0%
25% 125 64 20659 549.54 22.1%
50% 150 68 35908 683.68 51.93%
. -50% 0.425 30 182990 106120 23482%
-25% 0.6375 60 151030 7534 1574%
0% 0.85 69 24434 450 0%
25% 1.0625 58.8 1546.4 56 -87.6%
50% 1.275 62.48 580.2293 11.84 -97.37%
M -50% 0.0325 67 20658 438.97 -2%
-25% 0.04875 69 24434 450.5349 0%
0% 0.065 69 24434 450 0%
25% 0.08125 69 24434 450.5357 0.1%
50% 0.0975 69 24434 450.5361 0.12%
H -50% 0.25 69 24493 450.71 0%
-25% 0.375 72 32079 469.84 4%
0% 0.5 69 24434 450.5353 0%
25% 0.625 67 20659 438.97 -2.6%
50% 0.75 67 20658 438.96 -2.57%
C2 -50% 10 70 15247 419.29 -7%
-25% 15 67 46841 497.62 11%
0% 20 69 24434 450 0%
25% 25 66 24444 450.33 0.1%
50% 30 61 16914 425.0958 -5.53%
C1 -50% 0.65 71 28270 461.04 2%
-25% 0.975 69 24443 450.69 0%
0% 1.3 69 24434 450 0%
25% 1.625 66 20685 438.87 -2.5%
50% 1.95 69 28271 460.48 2.33%
a -50% 0.1 56.2 2230.5 306 -32%
-25% 0.15 67.8 6162.7 363.28 -19%
0% 0.2 69 24434 450 0%
25% 0.25 53 17454 426.89 -5.1%
50% 0.3 44 176669 606.61 34.80%
. -50% 0.01 0.75 12227 608.86 35%
-25% 0.015 69 1.5294 419.18 -7%
0% 0.02 69 24434 450 0%
25% 0.025 62 20659 4388.72 875.3%
50% 0.03 62 28875 461.83 2.63%
Ip -50% 0.08 70 15279 419.14 -7%
-25% 0.12 67 16844 425.39 -5%
0% 0.16 69 24434 450 0%
25% 0.2 69 3.2079 469.711 4.4%
50% 0.24 65 24434 450.33 0.07%
Ie -50% 0.065 70 28257 460.7 2%
-25% 0.0975 67 20659 438.96 -2%
0% 0.13 69 24434 450 0%
25% 0.1625 72 32103 469.9 4.4%
50% 0.195 67 20658 439.01 -2.44%
We applied our program to a major cosmetics retailer store in a mega-city to explain the proposed model. In advertising products, including TV/Internet, sunscreens, powders, lipsticks, and baby products, these products were initially promoted, but the products' sales declined slightly. For the validation of the model numerically, we consider the values of the parameters are as follows: A = 250/ . =100/ .= 0.85 M= 0.065/ C2 = 20,H = 0.5, C1 = 1.3/ a = 0.2/ . = 0.02/ Ip = 0.16/ Ie = 0.13. Using Mathematica software, we get the optimal values are as follows:
(Z*) = 450, p* = 24434, and T* = 69.
IV. Discussion
For the justification of the proposed model numerically, data for the numerical section, and using Mathematica software, we obtained the optimal values are as follows:
Profit (Z*) = 450, Price (p*) = 24434, and Time (T*) = 69. An affectability investigation is performed to contemplate the impacts of parameter values on the optimal solution. The Sensitivity analysis table shows the consequences of the model. The following decisions are as follows:
�
If the parameters ./ C2, and Ie are changed at the rate of 50%, 25%, -25%, and -50%, then the profit function Z decreases.
�
If the parameters A/ ./ M/ H/ C1, a, ., and Ip, are changes at the rate of 50%, 25%, -25%, and -50%, then profit function Z increases.
The following figures shows the affectability investigation as for parameters: A/ ./ ./ M/ H/ C2, C1, a, ./ Ip, Ie:
60%
Sensitivity analysis with A
-60% -40% -20% 0% 20% 40%
1 2 3 4 5
���1 -50% -25% 0% 25% 50%
���2 -8% -5% 0% -0,2% 0,17%
(Figure 1: w. r. to parameter A)
Sensitivity analysis with .
60%
-60% -40% -20% 0% 20% 40%
1 2 3 4 5
���1 -50% -25% 0% 25% 50%
���2 -54% -29% 0% 22,1% 51,93%
(Figure 2: w. r. to parameter .)
Sensitivity analysis with .
25000%
-5000% 0% 5000% 10000% 15000% 20000%
1 2 3 4 5
���1 -50% -25% 0% 25% 50%
���2 23482% 1574% 0% -87,6% -97,37
(Figure 3: w. r. to parameter .)
Sensitivity analysis with M
60%
-60% -40% -20% 0% 20% 40%
1 2 3 4 5
���1 -50% -25% 0% 25% 50%
���2 -2% 0% 0% 0,1% 0,12%
(Figure 4: w. r. to parameter M)
Sensitivity analysis with H
60%
-60% -40% -20% 0% 20% 40%
1 2 3 4 5
���1 -50% -25% 0% 25% 50%
���2 0% 4% 0% -2,6% -2,57%
(Figure 5: w. r. to parameter H) (Figure 6: w. r. to parameter C2)
(Figure 7: w. r. to parameter C1)
(Figure 8: w. r. to parameter a)
(Figure 9: w. r. to parameter .)
V. Conclusions
This study established an inventory model for the linear decline rate and the price-related demand rate under the holding cost. Shortages are permitted and are entirely reproduced and allowed a delayed payment period. Supplier offers a credit limit to the customer during which there is no interest charged, but the supplier will charge some interest upon the prescribed time limit expiry. However, the retailer has stored to make the installment, choosing to profit by as far as possible. Also, approve the model with the assistance of a mathematical model and study the sensitivity analysis. This model further developed with the inflation rate.
References
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