An Optimal Ordering Pharmaceutical Inventory Model for Time Varying Deteriorating Items with Ramp Type Demand
Manoj Kumar Sharma*, V. K. Srivastava
Department of Mathematics, Uttaranchal University, Dehradun248007, Uttarakhand, India
*Corresponding Author Email: manojsharmamath1984@gmail.com
ABSTRACT:
In this present paper we developed an economic order quantity (EOQ) inventory model for healthcare industry. We consider time varying deteriorating items with a constant and time dependent demand rate, constant holding cost without shortage. Here two numerical examples are used to explain the significance of the model. Sensitivity analysis with graphs of the optimal solution with respect to various parameters is carried out to see the effect of parameter changes on the solution.
KEYWORDS: EOQ, Ramp type demand, time varying deteriorating items, shortage, weibull distribution.
1. INTRODUCTION:
In the new era, healthcare or pharmaceutical industry are facing new challenges and must frequently improve their amenities to deliver the highest value at the best cost. Deterioration plays the major role in pharmaceutical industry because pharmaceutical products cannot be used after the expiry date and all pharmaceutical products are deteriorating after a time. It means pharmaceutical products have finite life time. Deterioration refers to the items that become spoiled, expired, undone, or devalued through time, such as fruits, vegetables, flowers, fish, meat, medicine, photographic films, etc. Products such as medicines, drugs and chemicals substances are also considered decaying products. Recently Benhadid et al. (2008) proposed an optimal control of production inventory systems with deteriorating items and dynamic costs. They dealt with two types of models, continuous and periodicreview. Patel and Patel (2013) developed a two warehouse inventory model with constant deteriorating rate under inflection and permissible delay in payments. ELGohary and Elsayed (2008) provide a new approach to study the problem of optimal control of twoitem inventory model with deteriorating items. Roy (2008) developed a deterministic inventory model with time relative deterioration rate selling price dependent demand rate and time dependent holding cost. Kumar et
al. (2017) considered with constant deteriorating item inventory model with variable holding cost. Skouri et.al. (2009) dealt an inventory model with general ramp type demand rate, time dependent deterioration rate with shortage that is partial backlogged. Karuppasamy and Uthayakumar (2018) derived a pharmaceutical inventory model for deteriorating items with time dependent deteriorating rate and timedependent demand, Shah and Shukla (2009), Taleizadeh (2014), Sarkar (2012a), Mishra and Singh (2010), Sharma, M. (2018), Sharma et.al. (2012) etc. have considerable remark in deteriorated inventory models.
In most of the of the pharmaceutical inventory model, the demand rate has been considered as a constant. But in reality demand vary according to time. Several papers on inventory systems consider time dependent or constant demand rate. Sugapriya and Jeyaraman (2008) developed an inventory model for deteriorating item in which production and demand rate are constant. Nahavandi and Haghighirad (2008) proposed an inventory model for products having fixed demand. Taleizadeh (2014) Silver and Meal (1973) studied an approximate approach for a deterministic time varying demand pattern. MaliK et al. (2017) consider Quadratic Demand based Inventory Model with Shortages. Sana and Chaudhuri (2003) consider an EOQ model with timedependent demand. Moon et al. (2005) analyzed EOQ models for deteriorating items with demand depending on time and considering inflation. Omar et al. (2008) considered an inventory replenishment model for deteriorating items with time varies demand. Sarkar et al. (2012) developed an economic order quantity model for finite production rate and deteriorating items with time dependent increasing demand. Mishra and Singh (2010) developed an inventory model with constant rate of deterioration and time dependent demand. Uthayakumar and Karuppasamy (2016) considered an inventory model for healthcare industries with quadratic demand, linear holding cost and shortages. Santhi and Karthikeyan (2018) proposed an EOQ Pharmaceutical Inventory Model for Perishable Products with time dependent demand. Some researcher like Nicholson (2002), Nicholson et al. (2004), Jarrett (2006), Safitri et al. (2017), David et al. (2017), Naveen et al. (2011) and Masoom et al. (2015) also studied inventory models in healthcare sector.
In this paper, we develop a pharmaceutical inventory model for healthcare industry with constant and time dependent demand rate. The pharmaceutical items approaching in the model are two parameter weibull distribution deterioration. Shortage are not consider in this model.
The rest of the paper is organized as follows. . In section 2, we define the notations and assumptions used throughout this paper. In Section 3, we establish the mathematical model and develop several theoretical results in and provide the decisionmaker with an algorithm for finding the optimal solution. A numerical example is provided in Section 4 to illustrate the solution procedure. In addition in section 5, a sensitivity analysis of the optimal solution with respect to all major parameters with graphically is also carried out. Finally, concludes our findings in Section 6 and provide some suggestions for future research
2. ASSUMPTIONS AND NOTATIONS:
To formulate the proposed mathematical model of the inventory system, the following assumptions are considered in this paper:
· The demand is deterministic and ramp type i. e. it has a two component form for the time horizon, i.e., it is constant for the part of the cycle and is a linear function of time in the second part of the cycle.
· The inventory system involves only Pharmaceuticals product
· There is no deterioration for the first part of the cycle and the second part it follows weibull deterioration. i.e.
· The inventory carrying cost, h per unit quantity per unit time and it is constant.
· The occurrence of replenishment is instantaneous and the delivery lead time is zero.
· The planning horizon is infinite. Only a typical planning schedule of length is considered and all the remaining cycles are identical.
· Shortages are not considered in this model.
· be the inventory level at any time t in first cycle.
· be the inventory level at any time t in second cycle.
· Deteriorated units are not replaced or repaired during the cycle period under consideration.
· The ordering cost and unit cost remain constant over time.
· d_{C }be the deterioration cost item per unit per unit of time.
· be the time point at which the demand increases with time as well as the deterioration starts.
· T be the length of the replenishment cycle.
· q be the number of items received at beginning of the inventory system.
· be the ordering cost per order.
· K be the average total cost per unit per unit time.
· T*be the optimal value of T.
· q* be the optimal value of q.
· TC* be the optimal average total cost per unit per unit time.
3. MATHEMATICAL MODEL:
Initially, the inventory cycle starts with maximum stocklevel q at t=0. The inventory level decreases during the time interval [0, ] due to demand. Finally, inventory level falls at zero level during the time interval [, T] due to both demand and deterioration. The total process repeats itself after a scheduling time T. The total inventory system is shown in Fig. 1.
Figure 1:Graphical representation of the inventory system: inventory versus time
From Fig. 1, it is shown that during the time interval [0,], the inventory level decreases owing to demand. Hence, to signify the inventory system at any time t, the governing differential equations are given by
(1)
(2)
With Boundary condition and
By equation (1)
(3)
Inventory level at starting of second period is
By equation (2)
(4)
At so by using equation (4)
(5)
The costs associate in this model
(i) Ordering cost : Ordering cost is given by OC= (6)
(ii) Deterioration Cost : the deterioration cost in time period (0,T) is given by
]
] (7)
(iii) Holding cost: Holding cost in the period (0,T) is HC
HC=holding cost in (0µ) + Holding Cost in (µT)
HC=
HC= (8)
Total cost TC =
TC= (9)
For maximum total cost
and
Now
i.e. (10)
4. NUMERICAL EXAMPLE:
Example1:To illustrate the model developed an example is considered based on the following values of parameters a=10, b=1.5, µ=2.5, β=3, h=10, α=0.015, C_{0}=$125 Per unit, d_{c}=$15 per unit per unit time . Solve equation (10) we get T*=0.7783. then by equation (5) and (9) get optimal solution
Example2 :Another example with following value of parameters a=6, b=1, µ=2.5, β=3, h=1.5, α=0.015, C_{0}=$185 per unit, d_{c}=$6 Per unit we get optimal solution .
5. SENSITIVITY ANALYSIS:
We now study the effect of changes in the values of the system parameters a,,, and h on the optimal length of the cycle (T*), the economic order quantity (q*) and the minimum total cost per unit time (TC*). The sensitivity analysis is performed by changing each of the parameters by 50%, 25%, −25%, −50%, and keeping the remaining parameters at their original values. The corresponding changes in the cycle time, total cost per unit and the economic order quantity are shown in Table 1.
Table1 Sensitivity analysis with respect to the parameters
Parameter 
% Change in T* 
% Change in q* 
% Change in TC* 

α 
50 
24.2 
127.2 
4.3 
25 
12.2 
68.7 
2.8 

10 
4.9 
28.9 
1.4 

0 
0.0 
0.0 
0.0 

10 
5.1 
31.1 
2.0 

25 
13.0 
82.5 
7.1 

50 
27.9 
Not Feasible 
27.1 

a 
50 
36.6 
16.5 
56.7 
25 
13.0 
12.5 
31.3 

10 
4.4 
5.6 
13.1 

0 
0.0 
0.0 
0.0 

10 
3.6 
6.1 
13.7 

25 
7.9 
15.8 
35.0 

50 
13.3 
33.3 
72.5 

b 
50 
2.1 
32.0 
17.2 
25 
1.1 
16.1 
8.8 

10 
0.4 
6.5 
3.6 

0 
0.0 
0.0 
0.0 

10 
0.4 
6.5 
3.6 

25 
1.1 
16.3 
9.2 

50 
2.4 
32.9 
18.8 

h 
50 
19.6 
40.4 
21.1 
25 
8.6 
18.0 
9.1 

10 
3.2 
6.8 
3.5 

0 
0.0 
0.0 
0.0 

10 
2.9 
6.2 
3.3 

25 
6.8 
14.7 
8.2 

50 
12.3 
26.8 
16.2 

C_{0} 
50 
24.6 
50.3 
59.8 
25 
11.6 
24.2 
22.3 

10 
4.5 
9.5 
7.7 

0 
0.0 
0.0 
0.0 

10 
4.3 
9.2 
6.4 

25 
10.5 
22.7 
14.0 

50 
20.1 
44.3 
22.9 

d_{c} 
50 
45.2 
103.2 
68.4 
25 
20.4 
44.8 
44.3 

10 
7.7 
16.6 
20.7 

0 
0.0 
0.0 
0.0 

10 
7.2 
15.1 
25.6 

25 
17.1 
35.5 
74.9 

50 
31.8 
64.2 
194.8 

Μ 
50 
15.4 
124.7 
4.7 
25 
15.7 
112.6 
4.0 

10 
8.9 
65.4 
1.3 

0 
0.0 
0.0 
0.0 

10 
91.4 
Not Feasible 
Not Feasible 

25 
65.2 
Not Feasible 
Not Feasible 

50 
Not Feasible 
Not Feasible 
Not Feasible 
”
Fig.2Sensitivity Analysis with “α”
Fig3:Sensitivity Analysis with “a”
Fig4: Sensitivity Analysis with “b”
Fig5:Sensitivity Analysis with “h”
Fig6: Sensitivity Analysis with “C_{0}”
Fig7: Sensitivity Analysis with “d_{c}”
Fig:9:Sensitivity Analysis with “μ
6. CONCLUSION:
In the present paper a deterministic inventory model have considered for deteriorating items in Pharmaceutical industry. The principle features of the model are as follows: The deterministic demand rate is assumed to be a ramp type function of time and the deterioration factor has been taken into attention in the present model. We have not considered shortage in this modeland the holding parameter is taken as constant in this model. We have given an analytic formulation of the problem on the framework described above and have presentedan optimal solution procedure to find optimal replenishment policy. Finally, the sensitivity of the solution to changes in the values of different parameters has been discussed. It is seen that changes in the parameter, and unit deterioration cost () lead to significant effects on the order quantity (q). The total cost is very sensitive to changes in the unit deterioration cost (), and the ordering cost (). In future, the present model can be extended by including shortage and inflation of money. We could study our model with stochastic types of competitive market demand. We also may extend the model by introducing one or two more members and then we may impose credit periods or different contracts among the members.
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Received on 26.07.2018 Modified on 13.09.2018
Accepted on 10.10.2018 © RJPT All right reserved
Research J. Pharm. and Tech 2018; 11(12): 52475252.
DOI: 10.5958/0974360X.2018.00957.5