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 Total Downloads : 396
 Authors : Kesavarao V.V.S., Srinivasarao.K., Srinivasarao.Y
 Paper ID : IJERTV1IS3188
 Volume & Issue : Volume 01, Issue 03 (May 2012)
 Published (First Online): 30052012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
On Optimal Production scheduling of an EPQ model with Stock dependent Production Rate having Selling Price Dependent Demand and Pareto decay
1Kesavarao V.V.S., 2Srinivasarao.K., 3Srinivasarao.Y.,
1Affiliation, Professor, Department of Mechanical Engineering, Andhra University, Visakhapatnam
2Affiliation, Professor, Department of Statistics, Andhra University, Visakhapatnam,
3Affiliation, St. Theressa Institute of Engg., & Tech., Garividi, Vizianagaram(Dist.),
Abstract
EPQ models play an important role in production and manufacturing units. Much work has been reported in literature regarding EPQ models with finite rate of production. But in many industries lik e agricultural products manufacturing units the production is dependent on stock on hand. Hence in this paper we develop and analyze an EPQ model for deteriorating items with stock dependent production rate having selling price dependent demand and Pareto rate of decay. Using the differential equations the instantaneous state of inventory is derived and with suitable cost considerations the optimal quantity, production uptime and production downtime are obtained for two cases of with and without shortages. The sensitivity analysis of the model revealed that the stock dependent production has a significant influence an optimal production schedule and can reduce total cost of production. This model also includes the finite rate of production inventory model with Pareto decay as a particular case.
Key words: EPQ model, Stock dependent production, Pareto decay.

Introduction
Much work has been reported in literature regard ing Economic Production Quantity (EPQ) mode ls during the last two decades. The EPQ mode ls are also a particular case of inventory models. The ma jor constituent components of the EPQ mode ls are 1) De mand 2) production (Replen ishment) and 3) Life
1
time of the commodity. Several EPQ mode ls have been developed and analyzed with various assumptions on demand pattern and life time of the commodity. In general it is customary to consider that the replenishment is either finite or infinite in production inventory models.
. Goel and Aggrawal (1980) , Teng, et al.(2005), Srinivasa Rao and Begum (2007), Maiti, et al. (2009), Srinivasa Rao and Patnaik (2010), Tripathy and Misra (2010), Sana (2011) and others have studied inventory models having selling price dependent demand. In all these papers they considered that the replenishment is infinite/fin ite and constant rate. Sridevi, et al. (2010) developed and analyzed an inventory model with the assumption that the rate of production is random and follows a we ibull distribution. However, in many practical situations arising at production processes the production (replenishment) rate is dependent on the stock on hand. The consideration of production rate being dependent on onhand inventory can significantly reduced wastage of resources and increase profitability.
Another important consideration for developing the EPQ models for deterio rating ite ms is the life time of the co mmodity. For ite ms like agricultural products, chemica ls etc., the life time of the commodity is random and follows a Pareto distribution. (Srinivasa Rao, et al. (2005), Srin ivasa Rao and Begum (2007), Srinivasa Rao and Eswara Rao (2011)). Very litt le work has been reported in the literature regard ing EPQ models for deteriorat ing items with Pareto decay having stock dependent production rate and selling price dependent demand, even though these models are more useful for deriv ing the optimal production schedules of many production processes. Hence, in this
paper we develop and analyze an economic production quantity model with stock dependent production having selling price dependent demand and Pareto decay. The Pareto distribution is capable of characterizing the life time of the commodit ies which have a min imu m period to start deterioration and the rate of deterio ration is inversely proportionate to time.
Using the differential equations the instantaneous state of inventory is derived. With suitable cost considerations the total cost function and profit rate function are derived. By ma ximizing the profit rate function the optima l production quantity, production up time , production down time are derived. A numerical illustration is also discussed. The sensitivity of the model with respect to the costs and parameters is also discussed.

Assumptions and notations of the model
The following assumptions are made for developing the inventory model under study.

Life t ime of the commod ity is random and follo ws
a pareto distribution having probability density function of the form
The instantaneous rate of deteriorat ion is .

The demand is a function of selling price and is of the form where, a and d are constant, a > 0, d 0, s is the unit selling price.
If d = 0 then the demand rate will be constant

The rate of production is dependent on stock on hand and is of the form
, such that R (t ) 0.
where, I (t ) is the stock on hand at time t, > 0,
S2 ma ximu m shortage level
R (t) rate of production at any time t Sh total shortage cost in a cycle t ime t1 time point at wh ich production
stops (production down time)
t2 time point at wh ich shortage begins t3 time point at wh ich production
resu mes (production uptime)
3 EPQ model without shortages
3.1 Model formulation
Consider a production system in which the production starts at time t = 0 and inventory level gradually increases with the passage of time due to production and demand during the time interval (0, t1). At time t1 the production is stopped and let S1 be the inventory level at that time . Du ring the time interval (t1, T) the inventory decreases partly due to demand and partly due to deterioration of items. The cycle continues when inventory reaches zero at time t = T. The schematic diagra m rep resenting the model is shown in fig.1.
Inventory level I (t)
S1
0 k 1.
When k=0, this production rate reduces to constant rate of production.
0 t1
T Time (t)

There is no repair or replace ment of deteriorated ite ms.

The planning horizon is fin ite. Each cycle will have length T.

Lead time is ze ro.

The inventory holding cost per unit time (h), the shortage cost per unit per unit time (), the unit production cost per unit time (c) and set up cost(A) per cycle are fixed and known.
H total inventory holding cost in a cycle t ime I (t) inventory level at any time t
Q production quantity
S1 ma ximu m inventory level
Fig.1.The schemat ic diagra m representing the inventory level of the system without shortages.
The differentia l equations governing the system in the cycle time (0, T) a re;
.
(2)
With the boundary conditions I (0) = 0 and I (T) = 0. Solving the equations (1) and (2), the instantaneous
2
state of inventory at any time t during the interval (0, t 1) is obtained as
where, (4)
The instantaneous state of inventory at any time t during the interval (t1, T) is obtained as
The total inventory in the time period 0 t t1 is
purchasing cost per unit time and holding cost per unit time i.e.
The total holding cost in a cycle time T is
.By substituting the values for I (t) and Q fro m the equations (3), (5) and (11) in TC(t1,T,s) equation one can get
where, g (t, b, k) is as defned as in equation (4) The total inventory in the time period t1 t T is
The ma ximu m inventory level I (t1) = S1 is
(6)
where ,g (t, b, k) is as defined as in equation (4) Let TR (t1, T, s) be the total revenue per unit time.
where, (9)
The stock loss due to deterioration in the interval (0, T) is given by
Let TP (t1, T, s) be the profit rate function. Then,
The total profit per unit time = total revenue per unit time total cost per unit time,
This imp lies
This imp lies
where, g (t,b,k) is as defined as in equation (4) The total production in the cycle time T is
This imp lies
(14)
where ,TC (t1,T, s) is as defined as in equation (12)
3.2. Optimal Operating Policies of the model
In this section, we obtain the optima l pricing and ordering polic ies of the inventory model developed in section.3.1. The proble m is to find the optima l values of t1 and s that ma ximize the profit rate function TP ) over (0, T). To obtain these values, diffe rentiate TP ( given in equation (14) with respect to and s and equate them to zero. The condition for the solutions to be optimal (min imu m) is that the determinant of the Hessian matrix is negative definite i.e.
where ,g(t,b,k) is as defined as in equation (4)
(11)
Let TC (t1, T, s) is the total cost per unit time. Then, TC(t1,T,s) sum of the set up cost per unit time,
Diffe rentiating TP (t1, T, s) with respect to t1 and equating to zero one can get
3
This imp lies
where , g(t1,b,k) is as defined as in equation (9)
where, g(t,b,k) is as defined as in equation (4)
(15)
(16)
months, decreases the unit selling price s* from Rs.17.343 to Rs .17.073, increases the production quantity Q* from 184.141 to 222.324 units and decrease the total profit TP* fro m Rs. 71.908 to Rs.
61.484. The increase in the parameter a 25 to 45 increase the production down time , the unit selling price s*, the production quantity Q* and the total profit TP*. Whereas the increase in the parameter d 0.8 to 1.2 decrease in the production down time , the unit selling price s*, the production quantity Q* and the total profit TP*.
The increase in unit cost c from Rs. 5 to Rs. 9 has a decreasing effect on , Q* and TP* and increasing effect on s* viz. Production down time fro m 3.876 to.2.173 months, production quantity Q* fro m 184.141to 112.513units and total profit TP* fro m Rs. 71.908 to Rs.23.562 and unit selling price fro m Rs.
17.343 to Rs. 19.33 respectively. The increase in holding cost h fro m Rs. 1 to Rs. 1.8 results increase in optima l values of , s* and Q* and decrease in TP* i.e. production down time fro m3.876 to 5.114 months, unit selling price s* fro m Rs. 17.343 to Rs. 17.591, production quantity Q* from 184.141 to 228.62 units
Solving the nonlinear equations (15) and (16) simu ltaneously using numerica l methods and verifying the determinant of Hessian matrix to be negative semi definite for concavity one can get the optima l values fo r t1 and s. Substituting the optima l values of t1 and s in the equations (11) and (14) the optimal values of production quantity Q and total profit TP can be obtained.

Numerical illustration
To e xpound the model developed, consider the case of deriving an economic production quantity and production down time for an edible oil manufacturing unit. He re, the product is deteriorating type and has random life t ime and assumed to follow a Pa reto distribution. Based on the discussions held with the personnel connected with the production and market ing of the plant and the records, the values of different parameters are considered as T = 12 months, A = Rs. 50, b = 1.2, a = 30, d = 1, h = Rs. 1, c = Rs. 5, k = 0.4
and = 60.By substituting these values of the parameters and costs in the equations (15) and (16) then solving numerically, the optimal values for production down time t1, unit selling price s, production quantity Q and total profit TP are obtained and are presented in Table.1.
Fro m Table 1, It is observed that the increase in deterioration para meter b fro m 1.2 to 1.6 increases the production down time fro m 3.876 to 4.678
4
and total profit TP* fro m Rs.71.908 to 26.078.
The increase in production rate parameter k fro m 0.4 to 0.8 results an increase in optimal values of , Q* and TP* and decreasing in s* i.e. production down time fro m 3.876to 5.053 months, production quantity 184.141to 189.103 units and total profit TP* fro m Rs. 71.908 to Rs 88.524. and fro m Rs. 17.343 to Rs. 16.384 Whereas the increase in production rate parameter fro m 60 to 80 results a decrease in optima l values of production down time fro m 3.876 to 2.912 months, total profit Rs.71.908 to Rs.47.218, increase in optima l values of unit selling price s*from Rs. 17.343 to Rs. 18.245 and production quantity from 184.141to 190.872 units respectively.

Sensitivity Analysis
To study the effects of changes in the parameters on the optima l values of production down time and production quantity, sensitivity analysis is performed taking the values of the parameters as b = 1.2, c = Rs. 5, h = Rs. 1, k = 0.4, = 60, a = 30, d = 1,T = 12 months and A = Rs. 50.
Sensitivity analysis is performed by changing the parameter values by 15% , 10%, 5% , 0%, 5%, 10% and 15%. First changing the value of one para meter at a time wh ile keeping a ll the rest at fixed values and then changing the values of all the parameters simu ltaneously, the optima l values of production down time, production quantity, selling price and total
Table 1
OPTIMAL VA LUES OF
t1, s, Q, TP for different values of the para meters for model without shortages
PARAM ETERS 
OPTIMAL VA LUES 

b 
a 
D 
c 
h 
k 
A 

1.2 
30 
1.0 
5 
1.0 
0.4 
60 
50 
3.876 
17.343 
184.141 
71.908 
1.3 
4.102 
17.265 
194.657 
69.069 

1.4 
4.309 
17.195 
204.450 
66.398 

1.5 
4.500 
17.135 
213.310 
63.875 

1.6 
4.678 
17.073 
222.324 
61.484 

25 
3.081 
15.559 
150.366 
16.476 

35 
4.495 
19.424 
211.029 
142.568 

40 
5.001 
21.647 
234.136 
227.433 

45 
5.426 
23.950 
254.872 
325.966 

0.8 
3.998 
21.003 
189.365 
127.103 

0.9 
3.939 
18.963 
<>186.834 
96.370 

1.1 
3.809 
16.031 
181.281 
52.014 

1.2 
3.737 
14.952 
178.215 
35.556 

6 
3.332 
17.839 
162.406 
57.486 

7 
2.875 
18.338 
143.366 
44.763 

8 
2.493 
18.838 
126.836 
33.520 

9 
2.173 
19.335 
112.513 
23.562 

1.2 
4.290 
17.412 
119.600 
59.293 

1.4 
4.563 
17.474 
211.603 
47.564 

1.6 
4.893 
17.532 
221.063 
36.581 

1.8 
5.114 
17.591 
228.620 
26.078 

0.5 
4.197 
17.163 
187.093 
76.282 

0.6 
4.506 
17.02 
188.804 
80.559 

0.7 
4.794 
16.912 
189.426 
84.658 

0.8 
5.053 
16.834 
189.103 
88.524 

65 
3.599 
17.566 
186.712 
65.125 

70 
3.347 
17.792 
188.609 
58.772 

75 
3.119 
18.018 
189.981 
52.814 

80 
2.912 
18.245 
190.872 
47.218 

40 
3.876 
17.343 
184.141 
72.741 

45 
3.876 
17.343 
184.141 
72.325 

55 
3.876 
17.343 
184.141 
71.491 

60 
3.876 
17.343 
184.141 
71.075 

Cycle length T = 12 months 
profit are co mputed. The results are presented in Table
2. The relationships between parameters, costs and the optima l values are shown in Fig.2.
Fro m Table 2, It is observed that variation in the deterioration para meters b has considerable effect on production down time , unit selling price s*, optima l production quantity Q* and total profit TP*.Simila rly variation in de mand para meters a and d has slight effect on production down time , unit selling price s*,
5
production quantity Q* and significant effect on total profit TP*.
The decrease in unit cost c results an increase in production down time , optima l production quantity Q*, total profit TP* and decrease in unit selling price s*. The increase in production rate parameter k result variat ion in production down time ,
slight increase in production quantity Q* and total profit TP*.Whereas the increase in production rate parameter result decrease in production down time ,
The differentia l equations describing the instantaneous states of I(t) in the interval (0, T) are given by
total profit TP* and slight increase in production quantity Q*.The increase in holding cost h has
significant effect on optima l values of production down
time , production quantity Q* and total profit TP*. When all the parameters change at a time it has a
significant effect on optima l values of production down time , unit selling price s*, production quantity Q* and total profit TP*.

EPQ Model with Sho rtages

Model Formulation
Let I (t ) denote the inventory level of the system at t ime
t. (0 t T)
(18)
(19)
Consider an inventory system for deteriorat ing ite ms in which the life time of the commod ity is random and
(20)
follows a pareto distribution. He re, it is assumed that
shortages are allo wed and fully bac klogged. In this model the stock level for the ite m is init ially zero. Production starts at time t=0 and continues adding ite ms to stock until the on hand inventory reaches its
with the boundary conditions I (0) =0, I (t2) =0 and I
(T) =0.Solving the equations (18) to (21) ,the instantaneous state of inventory at any time t, during the interval (0,t1) is obtained as
ma ximu m level S1 at time t = t1. During the time (0, t1) stock is depleted by demand and deterioration while
(21)
production is continuously adding to it. At t = t 1 the production is stopped and stock will be depleted by deterioration and demand until it reaches zero at t ime t
= t2. As demand is assumed to occur continuously, at this point shortages begin to accumulate until the backlog reaches its ma ximu m level of S2 at t = t3. At this point production resumes meeting the current demand and clearing the backlog. Fina lly shortages will
where , g (t,b,k) is as defined as in equation (4)
The instantaneous state of inventory at any time t, during the interval (t1, t2) is obtained as
The instantaneous state of inventory at any time t, during the interval (t2, t3) is obtained as
be cleared at time t = T. Then the cycle will be repeated
identically. Thes e types of production systems are
, t2 t t3 (23)
common in production process dealing agricultural products, where production rate is stock dependent. The
The instantaneous state of inventory at any time t during the interval (t3, T) is obtained as
schematic diagra m representing the invento ry system is
shown in figure 3
, t3 t T (24)
Inventory level I (t)
Using the equations (21) and (22) the total volume of inventory for the respective time periods are obtained as follows
The total inventory in the time period 0 t t1 is
S1
Time(t )
where ,g(t,b,k) is as defined as in equation (4) The total inventory in the time period t1 t t2 is
0 t1 t2 t3 T
S2
Fig 3; Sche matic diagra m representing the inventory level of the system for the modelwith shortags
6
(26)
Table 2
Sensitivity analysis of the model without shortages
Variation Para meters
Optima l Policies
Change in para meters (T = 12 Months)
15%
10%
5%
0%
5%
10%
15%
b(1.2)
3.414
3.577
3.731
3.876
4.014
4.144
4.269
17.508
17.499
17.394
17.343
17.296
17.251
17.209
163.100
170.461
177.473
184.141
190.541
196.637
202.544
77.525
75.571
73.702
71.908
70.183
68.522
66.920
a(30)
3.303
3.508
3.698
3.876
4.044
4.202
4.352
15.959
16.395
16.859
17.343
17.845
18.36
18.887
159.798
168.489
176.555
184.141
191.341
198.172
204.095
28.813
42.188
56.563
71.908
88.198
105.415
116.112
d(1)
3.969
3.939
3.908
3.876
3.843
3.809
3.774
19.921
18.963
18.109
17.343
16.654
16.031
15.466
188.120
186.834
185.508
184.141
183.994
181.281
179.789
110.815
96.370
83.478
71.908
61.467
52.014
43.411
c(5)
4.347
4.184
4.027
3.876
3.732
3.593
3.460
16.977
17.098
17.220
17.343
17.466
17.59
17.714
202.236
196.045
190.011
184.141
178.479
172.952
167.606
83.975
79.826
75.805
71.908
68.131
64.471
60.924
h(1)
3.500
3.632
3.758
3.876
3.988
4.095
4.195
17.281
17.303
17.324
17.343
17.361
17.379
17.396
169.558
174.739
179.621
184.141
188.383
192.394
196.106
82.401
78.786
75.293
71.908
68.619
65.415
62.289
k(0.4)
3.685
3.748
3.812
3.876
3.941
4.005
4.069
17.468
17.425
17.383
17.343
17.304
17.267
17.231
181.863
182.650
183.422
184.141
184.848
185.47
186.050
69.276
70.152
71.030
71.908
72.786
73.663
74.539
(60)
4.451
4.248
4.057
3.876
3.706
3.546
3.396
16.951
17.080
17.211
17.343
17.477
17.611
17.747
177.715
180.152
182.303
184.141
185.741
187.114
188.301
85.324
80.669
76.200
71.908
67.785
63.821
60.010
A(50)
3.876
3.876
3.876
3.876
3.876
3.876
3.876
17.343
17.343
17.343
17.343
17.343
17.343
17.343
184.741
184.741
184.741
184.741
184.741
184.741
184.741
72.533
72.325
72.116
71.908
71.7
71.491
71.283
All para meters
3.44
3.590
3.735
3.876
4.013
4.144
4.271
17.534
17.457
17.394
17.343
17.302
17.270
17.246
144.276
157.362
170.646
184.141
197.822
211.588
225.495
90.993
85.331
78.970
71.908
64.146
55.683
46.522
7
Fig.2: Relat ionship between optimal va lues and parameters
Since I (t) is continuous at t2 equating (22) and (23) one can get
(27)
This equation can be used to establish the relationship between t3 and t2.
The ma ximu m inventory level I (t1) = S1 obtained as
(28)
The stock loss due to deterioration in the interval (0, T) is
This imp lies
where, g(t1,b,k) is as defined as in equation (.9).
Similarly the ma ximu m shortage level
I (t3) = S2 obtained as
(29)
Backlogged demand at time t is
8
This imp lies
(31)
By substituting the values of I(t) and Q fro m the equations (21) to (24) and (32) in TC(t1,t3,T,s) equation, one can get
The total production in the cycle time T is
(32)
On integrating and simplify ing the above equation one can get
where, g(t,b,k) is as defined as in equation (4)
The total cost per unit time TC (t1, t3,T, s) is the sum of the setup cost per unit time, purchasing cost per unit
time, hold ing cost per unit time and the shortage cost
per unit time i.e.
The total holding cost in a cycle time is
The total shortage cost in a cycle time is
Therefore
where, g(t,b,k) is as defined as in equation (4)
Let TR (t1, t3, T, s) be the total revenue per unit time .
9
Also let TP (t1, t3, T, s) be the profit rate function. Then,
Total profit per unit time = Total Revenue per unit time
Total cost per unit time.
This imp lies
(35)
where ,TC (t1, t3, T,s) is as given in equation (33)

Optimal operating policies of the model
In this section, the optima l policies of the invento ry system developed in section 4.1 are derived. To find
the optimal va lues of production down time (t1) and
production up time (t3) and optima l selling price (s)
,one has to ma ximize the total profit TP (t1, t3,T,s) in
equation (35) with respect to t1, t3 and s and equate the resulting equations to zero. The condition for the
solutions to be optima l (minimu m) is that the determinant of the Hessian matrix is negative definite i.e.
The necessary conditions which ma ximize TP (t1, t3, T, s) is
2 11 3 + 1+ + 121
where, g(t1,b,k) is as defined as in equation (9)
(36)
where , g(t,b,k) is as defined as in equation (4)
Solving the nonlinear equations (36) to (38) by using MathCAD one can obtain the optima l production down and up times , and selling price .Substituting in equation (27) is obtained. The optima l production quantity Q* is obtained by substituting and in equation (32).

NUMERICAL ILLUS TRATION
To e xpound the model developed, consider the case of deriving and economic production quantity, production down time, production up time and selling price for an edible oil plant. He re the product is of a deteriorating
10
type and has a random life t ime which is assumed to follow pareto distribution. Form the records and discussions held with the production and market ing personnel the values of various parameters are considered. For different values of the parameters and costs, the optima l va lues of production down time, production up time, selling price, optima l production quantity and total profit are co mputed and presented in Table3.
Fro m Table 3, it is observed that the when b
increases from 1.2 to 1.6 units the production down time is decreasing, production quantity Q* is increasing and the total profit TP* is decreasing i.e. decreases fro m 1.989 to 1.860 months, Q* increases fro m 162.212 to 173.697 units and total profit TP* decreases from Rs. 114.092 to Rs.112.809. There is a decrease in production up time fro m 11.038 to 10.870 months and slight increase in selling price
fro m Rs. 13.275 to Rs. 13.330.
When the demand parameter a increases 25 to 29 then the optima l production down time is increases, production up time is decreasing, optima l values of selling price, production quantity and total profit a re increasing i.e . fro m 1.989 to 2.001 months, fro m 11.038 to 10.765 months, fro m Rs. 13.275 to Rs.15.166, Q* fro m 162.212 to 179.802 units and TP* fro m Rs. 114.092 to Rs. 164.702. Similarly when the demand parameter d increases 0.8 to 1.2 results , increase production up time fro m 11.038 to 11.059 months, decrease in production down time fro m 1.989 to 1.976 months, selling price fro m Rs. 13.275 to Rs. 11.205, p roduction quantity Q* fro m 162.212 to
160.299 units and total profit TP* fro m Rs. 114.092 to
Rs. 88.241.
The increase in holding cost h from Rs. 0.2 to Rs. 0.6 results decrease in production down time
fro m 1.996 to 1.979 months, production up time , fro m 11.280 to 10.733 months, increase in selling price
fro m Rs. 13.170 to Rs. 13.457, production quantity
Q* fro m 148.048 to 179.995 units and decrease in total profit TP* fro m Rs. 117.352 to Rs.109.026. The increase in unit cost c from Rs. 1 to Rs. 5 results slight increase in production down time fro m 1.986 to
2.005 months, production up time, fro m 10.680 to 11.730 months, selling price fro m Rs. 12.858 to Rs. 13.854, decrease in production quantity Q* from 183.681 to 121.249 units and total profit TP* fro m Rs. 127.722 to Rs. 77.587.
The increase in shortage cost from Rs. 0.2 to
Rs. 0.6 has effect on all optimal va lues of fro m 1.990 to 1.899 months, fro m 11.034 to 11.081 months, selling price from Rs.13.316 to Rs.13.227,
production quantity Q* from 162.486 to 155.468 units and total profit TP* fro m Rs. 115.048 to Rs. 111.855.The increase in production rate parameter k
0.3 to 0.7 results decrease in production down time fro m 1.989 to 1.988 months, production up time , fro m11.075 to 10.945 months, selling price s * fro m Rs.
13.308 to Rs. 13.188, production quantity Q* fro m
163.272 to 159.272 units and total profit TP* increase fro m Rs. 113.514 to Rs. 115.446.Simila rly the increase in production rate parameter 50 to 70 results increase in production down time fro m 1.986 to 1.990 months, production up time, fro m10.719 to
11.259 months, selling price s * fro m Rs. 13.160 to Rs. 13.376, production quantity Q* fro m 151.646 to
173.199 units and total profit TP* decrease from Rs.
117.071 to Rs. 110.952.
4.4 S ENSITIVITY ANALYS IS
To study the effect of changes in the parameters and costs on the optima l values of production down time, production up time, unit selling price and production quantity, sensitivity analysis is performed taking the values A = Rs. 50, c =Rs. 2, h = Rs. 0.3, T = 12 months, = Rs. 0.3, a = 25, d = 1, k = 0.4, b = 1.2 and = 60.
Sensitivity analysis is performed by changing the parameters by 15% , 10% , 5% , 0% , 5% , 10% and 15%. First changing the value of one para meter at a time wh ile keeping a ll the rest at fixed values and then changing the values of all the parameters simu ltaneously, the optimal values t1,t3,s,Q and TP are computed and the results are presented in Table 4. The relationships between parameters, costs and the optima l values are shown in figure4.
Fro m Table 4, it is observed that the deteriorating parameter b has less effect on production down time, production up time, unit selling price and significant effect on production quantity and total profit. Decrease in unit cost c results decrease in production down time, production up time, selling price, increase in production quantity Q* and total profit TP*. The increase in production rate parameter has less effect on production down time, production up time, unit selling price, moderate effect on production quantity Q* and total profit TP* respectively.Increase in holding cost h results significant variation in production quantity Q* and decrease in total profit TP*. The increase in shortage cost results less effect on production quantity Q* and total profit TP*.
11
Table .3
OPTIMAL VA LUES OF
t1, t3, s, Q and TP for different values of the para meters and costs for the model with shortages
PARAM ETERS(T = 12 Months)
OPTIMAL POLICIES
b
a
d
c
h
k
A
1.2
25
1.0
2
0.3
0.4
60
0.2
50
1.989
11.038
13.275
162.212
114.092
1.3
1.988
10.994
13.276
165.385
113.677
1.4
1.987
10.955
<>13.277 168.212
113.282
1.5
1.986
10.918
13.280
170.878
112.905
1.6
1.860
10.870
13.330
173.697
112.809
26
1.990
10.973
13.749
166.32
126.021
27
1.995
10.906
14.320
170.704
138.411
28
1.996
10.833
14.692
175.325
151.320
29
2.001
10.765
15.166
179.802
164.702
0.8
2.009
11.018
16.383
164.387
152.932
0.9
1.997
11.028
14.656
163.207
131.352
1.1
1.983
11.049
12.145
161.249
99.981
1.2
1.976
11.059
11.205
160.299
88.241
1
1.986
10.680
12.858
183.681
127.722
3
1.991
11.317
13.562
145.476
101.125
4
1.999
11.537
13.740
132.589
88.945
5
2.005
11.730
13.854
121.249
77.587
0.2
1.996
11.280
13.170
148.048
117.352
0.4
1.991
10.899
13.350
170.621
111.893
0.5
1.975
10.803
13.412
175.624
110.341
0.6
1.979
10.733
13.457
179.995
109.026
0.3
1.989
11.075
13.308
163.272
113.514
0.5
1.988
11.005
13.243
160.98
114.617
0.6
1.988
10.974
13.215
160.154
115.051
0.7
1.988
10.945
13.188
159.275
115.446
50
1.986
10.719
13.160
151.646
117.071
55
1.987
10.895
13.220
156.792
115.605
65
1.990
11.159
13.328
167.598
112.541
70
1.990
11.259
13.376
173.199
110.952
0.2
1.990
11.034
13.316
162.486
115.048
0.4
1.986
11.044
13.238
161.725
113.170
0.5
1.988
11.054
13.204
161.228
112.266
0.6
1.899
11.081
13.227
155.468
111.855
40
1.989
11.038
13.275
162.212
114.926
45
1.989
11.038
13.275
162.212
114.509
55
1.989
11.038
13.275
162.212
113.676
60
1.989
11.038
13.275
162.212
113.259
12
Tab le 4; sensitivity analysis of the model with shortages
Variation
Para meters
Optima l
Policies
Change in para meters(T = 12 Months)
15%
10%`
5%
0%
+5%
+10%
+15%
b
1.990
1.989
1.989
1.989
1.989
1.988
1.987
11.133
11.100
11.071
11.038
11.012
10.986
10.963
s*
13.277
13.276
13.276
13.275
13.274
13.274
13.274
Q*
155.38
157.728
159.86
162.212
164.126
165.976
167.629
TP*
114.893
114.623
114.356
114.092
113.839
113.595
113.361
a
1.984
1.984
1.985
1.989
1.992
1.994
1.996
11.272
11.202
11.122
11.038
10.954
10.861
10.782
s*
11.450
12.097
12.687
13.275
13.864
14.561
15.048
Q*
147.352
151.736
156.785
162.212
167.594
173.438
178.507
TP*
73.738
86.426
99.885
114.092
129.069
144.79
161.327
d
1.996
1.996
1.993
1.989
1.985
1.983
1.980
11.021
11.028
11.033
11.038
11.044
11.049
11.054
s*
15.468
14.656
13.928
13.275
12.683
12.145
11.654
Q*
163.595
163.161
162.710
162.212
161.654
161.249
160.798
TP*
141.534
131.356
122.265
114.092
106.703
99.981
93.852
c
1.988
1.988
1.989
1.989
1.989
1.989
1.99
10.941
10.975
11.007
11.038
11.069
11.099
11.128
s*
13.165
13.203
13.239
13.275
13.308
13.341
13.372
Q*
168.020
165.968
164.083
162.212
160.342
158.532
156.828
TP*
118.126
116.777
115.429
114.092
112.763
111.441
110.12
h
1.991
1.989
1.989
1.989
1.988
1.987
1.987
11.130
11.096
11.069
11.038
11.013
10.989
10.967
s*
13.234
13.247
13.265
13.275
13.287
13.300
13.310
Q*
156.797
158.741
160.355
162.212
163.662
165.02
166.369
TP*
115.375
114.924
114.497
114.092
113.718
113.363
113.023
k
1.989
1.989
1.989
1.989
1.989
1.989
1.989
11.060
11.053
11.045
11.038
11.031
11.024
11.018
s*
13.294
13.288
13.281
13.275
13.268
13.262
13.256
Q*
162.829
162.593
162.428
162.212
162.006
161.810
161.562
TP*
113.755
113.871
113.983
114.092
114.199
114.303
114.405
1.987
1.988
1.988
1.989
1.989
1.99
1.992
10.759
10.87
10.956
11.038
11.119
11.180
11.246
s*
13.173
13.216
13.242
13.275
13.315
13.337
13.348
Q*
152.602
155.4
158.922
162.212
165.027
168.732
171.786
TP*
116.782
115.909
115.005
114.092
113.169
112.226
111.252
1.989
1.989
1.989
1.989
1.989
1.989
1.988
11.036
11.037
11.038
11.038
11.039
11.04
11.041
s*
13.293
13.287
13.28
13.275
13.269
13.263
13.257
Q*
162.327
162.268
162.211
162.212
162.154
162.096
161.991
TP*
114.522
114.378
114.235
114.092
113.950
113.809
113.673
All Para meters
1.999
1.991
1.989
1.989
1.988
1.986
1.982
11.139
11.102
11.073
11.038
11.01
10.983
10.98
s*
13.165
13.203
13.243
13.275
13.31
13.342
13.352
Q*
133.922
143.16
152.381
162.212
171.752
181.332
189.304
TP*
101.411
105.866
110.092
114.092
117.887
121.473
124.883
13
Fig 4. Re lationship between optima l values and parameters
14
International Journal of Engineering Research & Technology (IJERT)
ISSN: 22780181
Vol. 1 Issue 3, May – 2012
A comparative study of with and without shortages revealed that allowing shortages has significant influence in optima l production schedule and total profit. This model includes some of the earlier inventory models for deteriorating ite ms with Pareto decay as particular cases for specific values of the parameters. When k = 0 this model inc ludes EPQ model for deteriorating ite ms with Pareto decay and selling price dependent demand and finite rate of replenishment. When b = 0 this model beco mes EPQ model with stock dependent production and selling price dependent demand. When d=0 this model includes EPQ model for deterio rating ite ms with pareto decay and constant demand.


Conclusions
In this paper, production level inventory models for deteriorating ite ms with selling price dependent demand and Pareto deterioration for both without and with shortages are developed and analyzed. By ma ximizing the total profit function the optimal values of the production quantity, production down time, production uptime and unit selling price are derived. The sensitivity model with respect to the parameters and costs revealed that the change in production rate parameters and deteriorating parameters have significant influence on optimal production schedule. By suitably estimating the parameters and costs the production manager can optima lly derive the production schedule and reduce waste and variation of resources. This model is having potential applications in manufacturing and production industries like edib le oil mills, sugar factories, etc., where the deterioration of the commodity is random and follo ws Pareto distribution and having selling price dependent demand .

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