On Optimal Production scheduling of an EPQ model with Stock dependent Production Rate having Selling Price Dependent Demand and Pareto decay

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.

1. 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 on-hand 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.

2. Assumptions and notations of the model

The following assumptions are made for developing the inventory model under study.

1. 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 .

2. 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

3. 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)

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

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

6. Lead time is ze ro.

7. 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 non-linear 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.

1. 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

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   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.

2. 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

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*.

1. EPQ Model with Sho rtages

   1. 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)

   2. 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 non-linear 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).

   3. 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: 2278-0181

      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.

2. 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 .

3. References

1. Goel, V.P. and A ggarwal, S.P.Pricing and ordering policy with general Weibull rate of deteriorating inventory , Indian Journ.Pure Appl.Math., Vol.11 (5),1980, 618-627.

2. M aiti, A.K., M ahathi, M .K. and M aiti, M . , Inventory model with Stochastic lead time and price dependent demand incorporating advance payment, Applied Mathematical Modeling, Vol.33,2009, No.5, 2433-2443.

3. Sana, S.S Price sensitive demand for perishable items- An EOQ model applied mathematics and computation Vol.217, 2011, 6248-6259.

4. Sridevi G, Nirupama Devi K and Srinivasa Rao K., Inventory model for deteriorating items with weibull rate of replenishment and selling price dependent demand International journal of operational research,2010, Vol. 9(3) 329-349.

5. Srinivasa Rao K and begum K.J Inventory models with generalized pareto decay and finite rate of production, Stochastic modeling and application, Vol. 10(1 and 2),2007, 13-27.

6. Srinivasa Rao K. Begum K.J and Vivekananda M urty M Optimal ordering policies of inventory model for deteriorating items having generalized pareto lifetime, Current science, 2007, Vol. 93, No.10,1407-1411.

7. Srinivasa Rao K, Vivekananda murthy and Eswara Rao S Optimal ordering and pricing policies of inventory models for deteriorating items with generalized pareto lifetime, Journal of stochastic processes and its applications Vol. 8(1) 59-72,2005.

8. Teng, J.T and Chang, C.T.Economic production quantity models for deteriorating items with price and stock dependent demand, Computers operations research, Vol. 32,2005, No.2, 297-308.

9. Tripathy, C.K. and M ishra, U An inventory model for weibull deteriorating items with price dependent demand and time-varying holding cost, International journal of commutating and applied mathematics, Vol. 4,2010, No. 2, 2171-2179.

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