EOQ Pharmaceutical Inventory Model for Perishable
Products with Pre and Post Discounted Selling Price and Time Dependent Cubic
Demand
G. Santhi1, K. Karthikeyan2
1Research Scholar, Department of Mathematics, SAS, VIT
University, Vellore, India
2Associate Professor, Department of Mathematics, SAS,
VIT University, Vellore, India
*Corresponding Author E-mail:
kkarthikeyan67@yahoo.co.in
ABSTRACT:
In this article we propose pharmaceutical inventory
model for perishable products with pre and post discounted selling price and
time dependent cubic demand. Mostly the pharmaceutical company considered the
constant rate of deterioration. In the majority of the earlier studies, the
demand and holding cost has been considered to be constant function, which is
not true in most of the practical situations as the manufacturing
medicine/setting machine cost and patient record keeping costs or even cost of
keeping the pharmaceutical items in the cold storage increases with time. In
view of this, we develop a pharmaceutical inventory model in which the time
dependent demand is cubic and holding cost is linear function of time. Also we
introduce both pre and post-deterioration discounts on unit selling price which
determine discount to be given on unit selling price during deterioration so as
to maximize the total profit. Finally, numerical examples and sensitivity
analysis are given for illustration of the model.
KEYWORDS: Deteriorating items, Pharmaceutical products, Cubic
demand, holding cost, discounted selling price
1.
INTRODUCTION:
As we enter the new millennium, healthcare
organizations are facing new challenges and must continually improve their
services to provide the highest quality at the best cost. Everyday hospitals
deal with inventory complications, tracking materials, and patient validation.
Inventory control is a complex and time-consuming process that every healthcare
facility must deal with. Every pharmaceutical production process has unique,
proprietary requirements that must be satisfied to protect patient
outcomes while maximizing manufacturing quality and efficiency. Deterioration is defined as
decay, damage, spoilage evaporation and loss of utility of the product.
Deterioration in pharmaceutical inventory is a
realistic feature and need to consider it. Often we encounter the
pharmaceutical products such as generic inject able, tablets, caplets,
ophthalmic, ointments, creams, and liquids, drug etc., that have a
defined period of life time. Pharmaceutical products are more commonly
known as medicine or drugs. Due to defective items, pharmaceutical inventory
system faces the problem of short- ages and loss of good will or loss of
profit. Shortage is a fraction of those customers whose demand is not
satisfied in the current period reacts to this by not returning the next
period.
The authors5 were the first to address the inventory
model with deteriorating items. Under the assumption of constant demand and
constant deterioration rate, they developed a single economic order quantity
model. An economic order quantity model for constant rate of deterioration was
analyzed in detail12. A deterministic inventory system considered with
power-form dependent demand pattern1.The recent trends of the modeling in
deteriorating items inventory was introduced 4. Many authors are published the
research paper with inventory-level-dependent demand rate2, 3, 16, 11, 13.Panda
et al. 9 proposed a single item economic order quantity model in which stock
dependent demand with pre and post deterioration discounts are allowed. An
economic order quantity model for perishable item with demand rate depends on
stock level and price discount rate was analyzed14. The authors15 developed an
inventory model that integrates continuous review with production and
distribution for a supply chain involving a pharmaceutical company and a
hospital supply chain. An inventory model for constant deteriorating items in
which cubic demand rate without shortages and with salvage value was introduced
6. An inventory model for deterioration rate is constant under trapezoidal
fuzzy numbers when the supplier offered price discount to the retailer at the
time of replenishment studied by the author8.Optimal ordering, discounting, and
pricing in the single-period problem7.The author10 developed an optimal
inventory policy for instant deterioration of price discounts with constant
rate of demand and deterioration.
The proposed model can be considered as an extension
of 10model by investigating an inventory model for constant rate of
deteriorating items with time dependent cubic demand and holding cost as linear
function of time. The objective of this model is we find the optimal order
quantity, optimal time and maximize the total profit of inventory system.
2. ASSUMPTIONS AND
NOTATIONS:
2.1. Assumptions:
In this section we present the assumptions are used
throughout this paper:
1) Inventory system is included only one item.
2) The demand rate is time dependent cubic,
i.e., where and are the positive constants.
3) Deterioration rate 
is
constant and (
).
4) Shortages are not permitted.
5) Lead time is zero.
6) Replenishment rate is infinite.
7) The holding cost 
per
unit time is time dependent and it is assumed
8) is the percentage per-deterioration
discount offer on unit selling price. is the effect of pre-deterioration
discount on demand. is the percentage discount offer on unit selling price
during deterioration. (the set of real numbers), is the effect of discounted
selling price on demand during deterioration. is determined from priori
knowledge of the seller with cubic demand.
2.2 Notations:
The notations are used throughout this paper are given
as follows:
Demand rate is cubic function of time
Deterioration rate is constant
Holding cost per unit time,
Unit purchasing cost of the product
Unit constant selling price of the product,
where
Discount offer per unit before deterioration
Discount offer per unit after deterioration
Ordering cost per unit order is
constant
Inventory level at time 
where
Maximum inventory level for order
quantity
Cycle length of inventory
Total profit of an inventory system
3. MODEL FORMULATION:
Formulation of the model when pre and post
deterioration discounts on selling price are given by Figure-1. At the
beginning of the replenishment cycle the inventory level raises to
The
pharmaceutical inventory level decreases due to time dependent demand up to the
time. After deterioration starts and the pharmaceutical inventory level
decreases for deterioration and cubic demand. We assume that before the start
of deterioration from time 
discount
on unit selling price of the product is given in order to enhance the demand of
fresh items. As deterioration starts from 
discount
on unit selling price is provided to enhance the demand. Then, the behavior of
inventory level is governed by the following system of differential equations
Figure.1 The graphical representation of inventory
level
3.1 Model for pre-deterioration discount on unit
selling price:
In this case we consider the model post deterioration
discount does not come into account and only pre-deterioration discount on
selling price is given. So
(1)
The solution of above equation (1) is
(2)
With boundary conditions and,
The optimum order quantity is given by
(3)
Ordering cost in the cycle is given by 
Holding cost of inventories in the cycle is
Purchase cost in the cycle is given by 
Total sales revenue in the order cyclecan be found as
Thus total profit per unit time is given by
(4)
There is only pre deterioration discount on selling
price. Therefore, we have the maximization problem
Maximize 
Subject to,
3.2 Special case:
Model with only post deterioration discount on unit
selling price
If the product starts to deteriorate as soon as it is
received in the stock, then there is no option to provide pre-deterioration
discount. Only we may give post deterioration discount. In that case 
and 
(5)
The solution of above equation (5) is
(6)
with boundary conditions and,
The optimum order quantity is given by
(7)
Ordering cost in the cycle is given by
Holding cost of inventories in the cycle is
Purchase cost in the cycle is given by
Total sales revenue in the order cycle can be found as
Thus total profit per unit time is given by
(8)
Table 1: Sensitivity analysis of thepre-deterioration
discount model
|
Parameter
|
Parameter
value
|
|
|
|
|
|
100
|
2.6170
|
257.7038
|
266.9272
|
|
|
150
|
2.6576
|
267.2841
|
247.9710
|
|
|
200
|
2.6945
|
276.2307
|
229.2889
|
|
|
26
|
2.6170
|
257.7038
|
266.9272
|
|
|
27
|
2.6107
|
258.9053
|
270.9336
|
|
|
28
|
2.6043
|
260.0772
|
274.9468
|
|
|
21
|
2.6170
|
257.7038
|
266.9272
|
|
|
22
|
2.6146
|
260.6335
|
271.1082
|
|
|
23
|
2.6121
|
263.5276
|
275.2902
|
|
|
11
|
2.6170
|
257.7038
|
266.9272
|
|
|
12
|
2.6208
|
264.7112
|
273.2066
|
|
|
13
|
2.6243
|
271.6999
|
279.4882
|
|
|
4
|
2.6170
|
257.7038
|
266.9272
|
|
|
5
|
2.6375
|
274.8505
|
278.0197
|
|
|
6
|
2.6548
|
291.9558
|
289.1787
|
|
|
0.9
|
2.6170
|
257.7038
|
266.9272
|
|
|
1.0
|
2.5218
|
236.2899
|
250.4878
|
|
|
1.1
|
2.4313
|
217.2444
|
235.4910
|
|
|
0.7
|
2.6170
|
257.7038
|
266.9272
|
|
|
0.8
|
2.4884
|
229.1156
|
251.1585
|
|
|
0.9
|
2.3782
|
206.6393
|
237.8598
|
|
|
10.0
|
2.6170
|
257.7038
|
266.9272
|
|
|
12.0
|
3.1903
|
420.8437
|
494.0857
|
|
|
14.0
|
3.7152
|
632.6544
|
792.5609
|
|
|
4.0
|
2.6170
|
257.7038
|
266.9272
|
|
|
5.0
|
2.3150
|
194.5455
|
175.8312
|
|
|
6.0
|
2.0179
|
144.8108
|
98.1287
|
|
|
2.0
|
2.6170
|
257.7038
|
266.9272
|
|
|
3.0
|
2.6162
|
260.1190
|
270.0095
|
|
|
4.0
|
2.6153
|
262.5330
|
273.1230
|
|
|
0.01
|
2.6170
|
257.7038
|
266.9272
|
|
|
0.02
|
2.5852
|
255.5252
|
263.2242
|
|
|
0.03
|
2.5532
|
253.3251
|
259.4805
|
Table 2: Sensitivity analysis of the Post-deterioration
discount model
|
Parameter
|
Parameter
value
|
|
|
|
|
|
100
|
2.2006
|
203.8592
|
212.7776
|
|
|
150
|
2.2548
|
215.2951
|
190.3392
|
|
|
200
|
2.3022
|
225.7035
|
168.3997
|
|
|
26
|
2.2006
|
203.8592
|
212.7776
|
|
|
27
|
2.1932
|
204.9265
|
216.8263
|
|
|
28
|
2.1858
|
205.9845
|
220.8846
|
|
|
21
|
2.2006
|
203.8592
|
212.7776
|
|
|
22
|
2.1982
|
206.2586
|
216.3278
|
|
|
23
|
2.1959
|
208.6672
|
219.8790
|
|
|
11
|
2.2006
|
203.8592
|
212.7776
|
|
|
12
|
2.2037
|
208.8043
|
217.2623
|
|
|
13
|
2.2067
|
213.7661
|
221.7487
|
|
|
4
|
2.2006
|
203.8592
|
212.7776
|
|
|
5
|
2.2215
|
215.5347
|
220.1215
|
|
|
6
|
2.2400
|
227.2725
|
227.5391
|
|
|
0.04
|
2.2006
|
203.8592
|
212.7776
|
|
|
0.06
|
2.1015
|
188.8217
|
198.7281
|
|
|
0.08
|
2.0103
|
175.4469
|
186.3551
|
|
|
0.9
|
2.2006
|
203.8592
|
212.7776
|
|
|
1.0
|
2.1201
|
187.7571
|
200.0120
|
|
|
1.1
|
2.0444
|
173.5362
|
188.3140
|
|
|
0.7
|
2.2006
|
203.8592
|
212.7776
|
|
|
0.8
|
2.1075
|
185.3294
|
202.4410
|
|
|
0.9
|
2.0267
|
170.3352
|
193.4713
|
|
|
10.0
|
2.2006
|
203.8592
|
212.7776
|
|
|
12.0
|
2.6877
|
325.9383
|
401.0595
|
|
|
14.0
|
3.1380
|
484.7083
|
641.1138
|
|
|
4.0
|
2.2006
|
203.8592
|
212.7776
|
|
|
5.0
|
1.9187
|
151.7786
|
127.6998
|
|
|
6.0
|
1.6582
|
113.3491
|
54.5405
|
|
|
2.0
|
2.2006
|
203.8592
|
212.7776
|
|
|
3.0
|
2.1940
|
215.3215
|
229.3116
|
|
|
4.0
|
2.1876
|
227.4805
|
246.9091
|
|
|
0.06
|
2.2006
|
203.8592
|
212.7776
|
|
|
0.07
|
2.1718
|
202.2264
|
209.4880
|
|
|
0.08
|
2.1428
|
200.5769
|
206.1446
|
There is only post deterioration discount on selling
price. Therefore, we have the maximization problem
Maximize
Subject to,
4. NUMERICAL EXAMPLES:
Example 1: Here we consider the parameter values are and
get the optimal values are as and
Example 2: Here we consider the parameter values are and
get the optimal values are as and
5. SENSITIVITY
ANALYSIS:
Sensitivity analysis is performed by increasing the
parametersand keeping the remaining parameters at their original values. The
results are shown in Table 1 and Table 2.
The three dimensional graphs are shown in the
following Fig. 2 and Fig.3
Fig-2Effect of changing parameters and in Total
profit
Fig-3 Effect of changing parameters and in Total
profit
5. OBSERVATIONS:
From Table-1and Table-2, we observed the following
results
(1) When the ordering cost
increases,
the cycle time the ordering quantity increases and Total profitdecreases.
(2) When the parametersandare increases, the cycle timedecreases
and the ordering quantity Total profit are increases.
(3) When the parameters and increases, the cycle timethe
ordering quantity and increases.
(4) When the parameters and increases and the cycle time the
ordering quantity and decreases
6. CONCLUSION:
This article depicts a time dependent pharmaceutical
inventory model of perishable items where the demand rate is a cubic function
of time and deterioration occurs after a certain time, with time dependent
demand. Pre and post deterioration discounts are provided on unit selling price
of the product. The model is very practical for the healthcare industries in
which the demand rate and holding cost is depending upon the time. The numerical
examples shows that the order quantity, cycle time and total profit are
decreased during the post deterioration discount whereas the same are increased
during pre-deterioration discount and the sensitivity analysis are illustrated
to test the model In general, customers are satisfied when they get value for
their money. The frame work of the model presented in this paper guides a
business sector in addressing the reduction in the selling price and inventory.
This research work further can be extended in many directions like the
inflation and time value of money, price dependent demand, stock dependent
demand, etc.
7. REFERENCES:
1.
Baker RC and Urban TL. A
deterministic inventory system with an inventory-level dependent demand rate. Journal
of the Operational Research Society. 39; 1988: 823–831.
2.
Datta TK and Pal AK. A
note on an inventory model with inventory level dependent demand rate. Journal
of the Operational Research Society. 41; 1990: 971-975.
3.
Giri BC, Pal S, Goswami
A and Chaudhuri KS. An inventory model for deteriorating items with
stock-dependent demand rate. European Journal of Operational Research. 95;
1995: 604-610.
4.
Goyal SK and Giri BC.
Recent trends in modeling of deteriorating inventory. European Journal of
Operational Research. 134; 2001:1–16.
5.
Ghare PM and Schrader
GF. A model for exponentially decaying inventories. Journal of Industrial
Engineering. 14; 1963: 238-243.
6.
Karthikeyan K and
Santhi G. An Inventory Model for Constant Deteriorating Items with Cubic Demand
and Salvage Value. International Journal of Applied Engineering Research. 10
(55) ; 2015:3723- 3728.
7.
Khouja M. Optimal
ordering, discounting, and pricing in the single-period problem. Int J Prod
Econ. 65; 2000:201–216.
8.
Mohan R. An EOQ model
for Price Discount Linked to Order Quantity under Fuzzy Environment in
Quadratic Demand Pattern. IOSR Journal of Mathematics. 11(4); 2015: 20-26.
9.
Panda S, Saha S and Basu
M. An EOQ model for perishable products with discounted selling price and stock
dependent demand.CEJOR.17; 2009: 31–53.
10.
Pattnaik M. A Note on
Non Linear Optimal Inventory Policy Involving Instant Deterioration of
Perishable Items with Price Discounts. The Journal of Mathematics and Computer
Science. 3 (2); 2011: 145-155.
11.
Roy J and Chaudhuri KS.
An EOQ model with stock dependent demand, shortage, inflation and time
discounting. International Journal of Production Economics.53; 1997:171-180.
12.
Shah YK. and Jaiswal MC.
An order-level inventory model for a system with constant rate of
deterioration, Opsearch. 14; 1977: 174-184.
13.
Sana S and Chaudhuri
K.S. A stock-review EOQ model with stock-dependent demand, quadratic
deterioration rate. Advanced Modeling and Optimization. 6; 2004: 25-33.
14.
Sana S. An EOQ Model for
Perishable Item with Stock Dependent Demand and Price Discount Rate, American
Journal of Mathematical and Management Sciences.30(3-4); 2010: 299-316.
15.
Uthayakumar R and Priyan
S. Pharmaceutical supply chain and inventory management strategies: Optimization
for a pharmaceutical company and a hospital. Operations Research for Health
Care. 2; 2013:52-64.
16.
Wee HM and Yu J. A
deteriorating inventory model with a temporary price discount. Int. J.
Production Economics.53; 1997:81-90.