# An EOQ Model with Time Dependent Demand and Weibull Distributed Deterioration

DOI : 10.17577/IJERTV4IS090195

Text Only Version

#### An EOQ Model with Time Dependent Demand and Weibull Distributed Deterioration

Ankit Bhojak

Department of Statistics

GLS Institute of Commerce, GLS University Ahmedabad, Gujarat, India

1. B. Gothi

Head, Department of Statistics St. Xaviers College (Autonomous)

Abstract In this paper, we developed an inventory model for deteriorating items under time dependent demand function. Inventory holding cost is a linear function of time. For deterioration of units we considered two parameters Weibull distribution. In this study, a more realistic scenario is assumed where the part of shortages was backordered and the rest was lost with time. The backordering rate is variable and it is considered exponential function depending on waiting time for the next replenishment. Solution procedure of the developed model is presented with numerical example and its sensitivity analysis.

Keywords EOQ model; deterioration; time dependent demand; weibull distribution; shortages; inventory

1. INTRODUCTION

Deterioration of produced items is very important factor in inventory management and in any production. In the recent time more research work has been carried out in the inventory management to develop inventory models with deterioration of items and shortages are permitted. Commodities like vegetables, fruits and food items from depletion by direct spoilage while kept in store. It has been observed that the failure of many items may be expressed by Weibull distribution.

Ghare and Scharder [5] first formulated a mathematical model with a constant deterioration rate. Khemlnitsky and Gerchak [7] developed optimal control approach to production systems with inventory level dependent demand. Ruxian , Hongjie Lan and John Mawhinney [12] reviewed on deteriorating inventory study. Vipin Kumar, Singh and Sanjay Sharma [21] developed inventory model for profit maximization production with time dependent demand and partial backlogging. Papachristos and Skouri

[10] gave an inventory model with deteriorating items, quantity discount, pricing and time-dependent partial backlogging. Samanta and Ajanta Roy [14] presented a production inventory model with deteriorating items and shortages. Inventory model with price dependent demand and time varying holding cost was given by Ajantha Roy [1]. Vinod Kumar Mishra and Lal Sahab Singh [20] have given deteriorating inventory model for time dependent demand and holding cost with partial backlogging. Kirtan Parmar, Indu Aggarwal and Gothi [9] have formulated an order level inventory model for deteriorating items under varying demand condition.

Analysis of inventory-production systems with weibull distributed deterioration is given by Azizul Baten and Anton Abdulbasah Kamil [3]. Chen and Lin [4] developed optimal replenishment scheduling for inventory items with weibull distributed deterioration and time varying demand. Sanni [13] studied an economic order quantity inventory model with time dependent weibull deterioration and trended demand.

An EOQ Model for varying items with weibull distribution deterioration and price-dependent demand was given by Begum, Sahoo, Sahu and Mishra [11]. One more condition was imposed in the form of shortage by Wu and Lee [22] in their work, an EOQ inventory model for items with weibull distributed deterioration, shortages and time varying demand. Vikas Sharma and Rekha Rani Chaudhary [19] have also developed an inventory model for deteriorating items with Weibull deterioration with time dependent demand and shortages. Tripathy & Pradhan [17] formulated an integrated partial backlogging inventory model having weibull demand and variable deterioration rate with the effect of trade credit. Amutha and Dr. Chandrasekaran [2] developed an inventory model for deteriorating products with weibull distributed deterioration, time varying demand and partial backlogging. Tripathy and Pradhan [15] have given an EOQ model for weibull deteriorating items with power demand and partial backlogging.

Some more conditions were applied on inventory holding cost by Ghosh and Chaudhuri [6] in their research in an order level inventory model for a deteriorating item with weibull distribution deterioration, time-quadratic demand and shortages. Tripathy and Mishra [16] developed an inventory model for Weibull deteriorating items with price dependent demand and time varying holding cost. Kirtan Parmar & U. B. Gothi [8] developed an EOQ model of deteriorating items using three parameter weibull distribution with constant production rate and time varying holding cost.

Vashistha & Gupta [18] has developed an inventory model with weibull distribution deterioration and time dependent demand. We have modified the same inventory model by considering inventory holding cost as a linear function of time and also including production cost to the total cost function to make the model more realistic. The

sensitivity analysis is also carried out by changing the values of all the parameters one by one. We have used different formulae to find costs like IHC, DC, SC and LSC.

2. NOTATIONS

The following notations are used to develop the mathematical model:

• Q t : Inventory level of the product at time t ( 0).

1. Shortages are allowed and unsatisfied demand is

backlogged at a rate e T t , where the backlogging parameter is a positive constant.

2. Total inventory cost is a real and continuous function which is convex to the origin.

1. MATHEMATICAL MODEL AND ANALYSIS

The initial stock is S at time t = 0, then inventory level decreases mainly due to meet up demand with rate

• R t : Demand rate varying over time.

• a + b t + c t2 and partly from deterioration with rate

• t : Deterioration rate

• A : Ordering cost per order during the cycle period.

• Ch : Inventory holding cost per unit.

• Cd : Deterioration cost per unit.

t 1 and reaches to S at t = Âµ. The stock reduces to zero at t = t1 due to demand with rate a + m t where m b + c and the deterioration with rate t 1 . Thereafter, shortages are allowed to occur during the time

1

interval t1 , T and the demand during the period t1 , T

• Cs : Shortage cost per unit.

• pc : Production cost per unit.

is partially backlogged.

• l : Opportunity cost due to lost sale per unit.

• S : Initial stock level at the beginning of every

Inventory Level

t t 1

cycle.

• S1 : Inventory level at t = Âµ.

• S2 : The maximum inventory level during shortage

R t a + b t + c t2

period.

• T : Duration of a cycle.

• TCt1 , T : Total cost per unit time.

III. ASSUMPTIONS

R t a + m t

S

S1 where m b c

T

Time

The following assumptions are considered to develop this model.

1. Replenishment rate is infinite.

3. A single item is considered over the prescribed period of time.

4. No repair or replacement of the deteriorated items

O t1 S2

Fig. Graphical representation of inventory system

The rate of change of inventory during the

takes place during a given cycle.

5. We consider finite time horizon period.

6. Demand rate R(t) is assumed to be a funcion of time such that

intervals 0 , , , t1 and t1 , T

following differential equations

d Qt

is governed by the

R(t) =a +b t +c{ t t H t }t , where H

t is the Heavisides function defined as

+ t 1 Qt

d t

a + b t + c t2

0 t

(1)

H t 1,

0,

if t

and a is the initial rate of

if t

d Qt + t 1 Qt d t

a + m t where m

b + c

demand, b is the rate with which the demand rate increases. The rate of change in demand rate itself increases at a rate c. a, b and c are positive constants.

7. Holding cost is linear function of time and it is

d Qt

T t

t t1

(2)

Ch = h + r t (h, r > 0).

8. t t 1 is the two parameter Weibull distributed deterioration rate, where is scale

a + m t e t1

d t

t T

(3)

Now solving equation (1), (2) and (3) with boundary

parameter and is shape parameter and

0 < < < 1 .

condition Q(0) = S and Qt1 0 we get,

Q t = S 1 t

b t2 c t3 a t + 1

a t + + +

The total cost can be obtained by adding the following costs.

2 3 + 1

b t + 2

+ 2 + 2

c t + 3

+ 3 + 3

1

2

m a

0 t

(4)

Ordering cost OC = A

The deterioration cost during the period

0 , t1

(10)

Q t =

a t t1

t2 t2 +

+ 1

t + 1 t + 1

1

DC C

t 1 Q t dt +

t1

t 1 Q t dt

m

d

1 1

+ + 2

t + 2 t + 2 a t + 1 t t

0

S

a + 1

b + 2

c + 3

m t + 2 t2 t

t t

Cd

+ 1

+ 2 + 2 + 3 + 3

2 1

1

(5)

m t2 t

+ a t + 1 1

t2 t2 1 2

Q t =

a 1 Tt t a + m 1 T 1

1

2

t 1 1 2

2

t3 t3

a 1 m t1

+ 1

+ 2

+ m 1

t t T

3

1

(6)

The shortage cost during the period t1 , T

(11)

In equation (5) , Q S1 and so T

S = a t m t2 2 + a

t 1 1

SC Cs Q t dt

t

2

1 1 1

+ 1 1

1

T t 2

m 2

2

= C a 1 T 1

+ + 2 t1

a t1

s 2

2

• m

t2 2

a + m 1 T T t 3

1

1 t T t 2

2 3

1 1

In equation (4), Q S1 which gives

(7)

m T t 4 3 t2

+ 1 t T t 3 1 T t 2

1

b 2

c 3

a + 1

3 4 1 1 2

1

S = 1

S1 a + 2 + 3

+ 1

(12)

b + 2

2 + 2

c + 3

3 + 3

The opportunity cost due to lost sales during the period

t1 , T

In equation (6), QT S2

and hence

(8)

T2 t2

LSC

T

l

t1

a + m t

1 e

T t

dt

S = a 1 T T t a m 1 T 1

m T a 2 2

2 1 2

l

a T T t1 T t1

m T3 t3

2

m 3 3

3 1

(9)

3 T t1

(13)

The holding cost during the period 0 , t1

S 1 a 2

b 3

c 4

t1

A h S

1

2 + 6 +

12

IHC

h r t Q t dt h r t Q t dt

0

a

2

+ b

3

S 1 a 2

b 3

c 4

1

2

2

3

h S

+ +

1

2 6 12

+

c 4

3

4

a 2

b 3

c 4

+ +

m t2

a t + 1

m t + 2

1

2

2

3

3

4

a t

1 1

1 t

1 2 + 1 + 2 1

m t2

a t + 1

m t + 2

a t1

1 1

1 t

1

a 2 t2 m 3 t3 a 2 t 2

2 + 1

+ 2

2 1 6

1

1

2 1

a 2 t2 m 3 t3 a 2 t 2

m 3 t 3

2 1 6

1

1

2 1

2

2

3 1

m

3

3

m t2 1 t 1

• t

a t 1 1

2

2

3 1

1

2

+ 1

m t2 1 t 1

S 2 S 2 a 3 b 4 c 5

a t 1 1

1

+ r 2

2

3

+ 8 + 15

2

+ 1

2 2 3 4 5

a 3

b 4

r S

S a + b + c

+

2 2 3 8 15

1 3 2 4

c 5

a 3

+

b 4

+ c

5

+

3

5

1 3 2 4 3 5

2 + 1

+ 2 2 2

m t a t m t t

a t

1

1 1 1 1

m t2

a t + 1

m t + 2 2 t2

1 2

+ 1

+ 2 2

a t1

1 1

1 1

TC t , T

2 + 1

+ 2 2

1 T a

3 3

m 4 4

a 3 3

3

t1

8

t1

1

3

t1

a 3 t3 m 4 t4 a 3 t 3

3 1 8

1

1

3 1

m 4 t 4

2

2

4 1

m

4

4

t1

m t2 2 t 2

2 2 4

a t 1 1

1

2

+ 2

m t2 2 t 2

a t 1 1

1

T t 2

a + m 1 T

2

+ 2

C a 1 T 1

(14)

s

2 2

Production cost

T t 3

1 t T t 2

PC = pc S S2

(15)

3

1 1

4

m T t1 3 3 t2

2

The total cost per unit time is given by

+ t T t 1 T t

1 3 4 1 1 2

1

TCt1 , T

OC IHC SC DC LSC PC

T

S

a + 1

b + 2

c + 3

Cd

+ 1 + 2 + 2 + 3 + 3

m t2 t

+ a t + 1 1

1

2

t 1 1 t 2 2

a 1 m 1

+ 1

+ 2

m T a 2 2 m

3 3

l a T T t1 T t1 T

t1

2

3

p S S

c 2

1

 Parameter % change T t1 TCt1, T % changes in TCt1, T 20 7.445406199 5.29016578 9759.947791 0.107044875 10 7.446835483 5.290354243 9765.177951 0.053514251 A 10 7.449691919 5.290730829 9775.633473 0.053497897 20 7.451119074 5.290918953 9780.858839 0.106979459 20 7.342798114 5.300616593 9325.835862 4.550175556 10 7.394243966 5.295717103 9542.005314 2.337683668 10 7.505103902 5.285035598 10012.36947 2.476488128 20 7.565040232 5.279127689 10269.41498 5.107346055 20 5.048259926 4.210974646 4023.445977 58.82007598 10 6.14066536 4.752816003 6253.2881 35.99766711 10 9.176093149 5.811514762 15991.71499 63.67502185 20 12.12255666 6.206800713 31100.4103 218.3123472 20 7.442946559 5.289841412 9730.31566 0.410329403 10 7.445608265 5.290192425 9750.362325 0.205152018 a 10 7.450913956 5.290891916 9790.44823 0.205126771 20 7.453557987 5.291240402 9810.487494 0.410228412 20 7.39193315 5.287053749 9554.925649 2.205444184 10 7.42019919 5.288813518 9662.84006 1.100941408 b 10 7.476130652 5.292241495 9877.627337 1.097403945 20 7.503801812 5.29391074 9984.504834 2.191293912 20 7.046694019 5.270130313 8255.748149 15.50251118 10 7.252724663 5.281124747 9021.998138 7.659951222 c 10 7.634206651 5.298550067 10501.30021 7.480688706 20 7.811326283 5.305288724 11215.01677 14.78556963 20 7.448155361 5.290556183 9770.554284 0.001512462 10 7.448209705 5.290549388 9770.480401 0.000756263 10 7.448318413 5.290535795 9770.332615 0.000756326 20 7.448372776 5.290528998 9770.258712 0.001512716

Our objective is to determine optimum values

t and T

TABLE – I SENSITIVITY ANALYSIS

such that

TCt1 , T is minimum. Note that

t and T are

1

the solutions of the equations

TCt1 , T 0 &

t1

such that

TCt1 , T 0

T

(17)

2TCt , T 2TC t , T 2TC t , T 2

1

t2

1

T2

1

t T

0

1 1

t t , T = T

1 1

2TCt , T

1

1

t2

0

1 1

t t , T = T

(18)

The optimal solution of the equations in (17) can be obtained by using the software. This has been illustrated by the following numerical example.

1. NUMERICAL EXAMPLE

Let us consider the following example to illustrate the above developed model. We consider the following values of the parameters A = 300, h = 2, r = 16, = 5, Cd = 18, l

= 10,

= 0.0001, = 5, a = 2, b = 3, c = 5, m = 28, Cs = 17,

= 0.0002 & pc = 15 (with appropriate units of

measurement). We obtain the optimal values

1

T = 7.448264056 units, t = 5.2905425919 units and

optimal total cost

TCt1 , T

= 9770.406510 units by

using appropriate software.

2. SENSITIVITY ANALYSIS

Sensitivity analysis helps in identifying the effect of optimal solution of the model by the changes in its parameter values. In this section, we study the sensitivity

of total cost per time unit TCt1 , T with respect to the

changes in the values of the parameters A, , Âµ, a, b, c, , h, pc, Cs, Cd, l , m and r.

The sensitivity analysis is performed by considering 10% and 20% increase and decrease in each one of the above parameters keeping all other remaining parameter as fixed. The results are presented in Table 1. The last column of

table shows the % change in TCt1 , T as compared to

the original solution corresponding to the change in parameters values.

td>

0.474813625

 Parameter % change T t1 TCt1, T % changes in TCt1, T 20 7.423514172 5.288219977 9677.559835 0.950284678 10 7.435909423 5.289386344 9724.01529 h 10 7.460578348 5.291688823 9816.733642 0.474157667 20 7.472852573 5.292825141 9862.996829 0.947660856 20 7.610933659 5.326173188 9690.57666 0.817057616 10 7.529209709 5.308272694 9731.07186 0.402589707 pc 10 7.368092895 5.272982062 9808.5799 0.390704208 20 7.288692191 5.255590224 9845.591279 0.769515246 20 7.70899727 5.260068445 9378.29687 4.013237731 10 7.56889612 5.276149679 9584.207555 1.905744205 Cs 10 7.343048417 5.303527912 9939.95053 1.735281115 20 7.250290241 5.315322778 10095.23773 3.324643822 20 7.446451526 5.293180369 9756.435631 0.142991795 10 7.447344685 5.291852949 9763.390681 0.071806942 Cd 10 7.449209062 5.289248958 9777.48175 0.072414993 20 7.450179142 5.287971718 9784.615065 0.145424388 20 7.448290504 5.290539379 9770.364749 0.000427436 10 7.44827728 5.290540986 9770.38563 0.000213717 l 10 7.448250832 5.290544198 9770.427392 0.000213715 20 7.448237609 5.290545805 9770.448272 0.000427427 20 7.966953092 5.312540139 9504.831551 2.718156703 10 7.687810269 5.30157837 9652.993137 1.201724553 m 10 7.240214212 5.279778227 9864.625505 0.964330334 20 7.057659597 5.269453198 9940.922426 1.745228462 20 7.065941013 5.277231694 8311.034745 14.9366535 10 7.263975282 5.28540387 9055.049867 7.321667149 r 10 7.62108883 5.293563685 10460.48512 7.062946719 20 7.784171635 5.295077033 11128.07187 13.89568957

3. CONCLUSION

From the above sensitivity analysis we may conclude that the total cost per time unit TCt1 , T is highly sensitive to

changes in the values of the parameters Âµ, c and r moderately sensitive to changes in the values of the parameters , b, Cs, m and less sensitive to changes in the values of the parameters A, a, , h, pc, Cd, and l .

REFERENCES

1. Ajantha, R. (2008). An inventory model for deteriorating items with price dependent demand andtime-varying holding cost. AMO- Advanced Modeling and Optimization, Volume 10, Number 1.

2. Amutha R. & Dr. E. Chandrasekaran. (2012). An Inventory Model for Deteriorating Products with Weibull Distribution Deterioration, Time-Varying Demand and Partial Backlogging. International Journal of Scientific & Engineering Research, Volume 3, Issue 10.

3. Baten Azizul & Kamil Anton Abdulbasah. (2009). Analysis of inventory-production systems with Weibull distributed deterioration. International Journal of physical Sciences Vol.4(11), 676-682.

4. Chen J.M. & Lin S. C. . (2003). Optimal replenishment scheduling for inventory items with Weibull distributed deterioration and time- varying demand. Journal of Information and Optimization Sciences, 24 , 1-21.

5. Ghare P. M. & Schrader G. F. (1963). A model for an exponentially decaying inventory. Journal of Industrial Engineering, 238 – 243.

6. Ghosh S. K. & Chaudhuri K. S. (2004). An order-level inventory model for a deteriorating item with Weibull distribution deterioration, time-quadratic demand and shortages. Adv. Model.

Optim. 6(1), 21-35.

7. Khemlnitsky E. & Gerchak Y. (2002). Optimal Control Approach to Production Systems with Inventory-Level-Dependent Demand. IIE Transactions on Automatic Control 47 , 289- 292.

8. Kirtan Parmar & U. B. Gothi . (2015). An EPQ model of deteriorating items using three parameter Weibull distribution with constant production rate and time varying holding cost. International Journal of Science, Engineering and Technology Research, Vol. 4,

409 416.

9. Kirtan Parmar, Indu Aggarwal & U. B. Gothi. (2015). Order level inventory model for deteriorating item under varying demand condition. Sankhya Vignan,(NSV11),20 – 30

10. Papachristos S. & Skouri K. . (2003). An inventory model with deteriorating items, quantity discount, pricing and time-dependent partial backlogging. International Journal of Production Economics, 83 , 247-256.

11. R. Begum, R. R. Sahoo, S. K. Sahu & M. Mishra. (2010). An EOQ Model for Varying Items with Weibull Distribution Deterioration and Price-dependent Demand. Journal of scientific research, Res. 2 (1), 24-36.

12. Ruxian Li, Hongjie Lan & John R. Mawhinney. (2010). A Review on Deteriorating Inventory Study. J.Service Science & Management, 3, 117-129.

13. Sanni S. S. (2012). An economic order quantity inventory model with time dependent Weibull deterioration and trended demand. M. Sc. Thesis, University of Nigeria.

14. Samanta G.P. & Roy Ajanta. (2004). A Production Inventory Model with Deteriorating Items and Shortages. Yugoslav Journal of Operations Research 14, No 2, 219-230.

15. Tripathy C.K. & Pradha L.M. (2010). An EOQ Model for Weibull Deteriorating Items with Power Demand and Partial Backlogging. Int.J.Contemp, Math.Sciences, Vol 5, No.38, 1895 – 1904.

16. Tripathy C.K. & Mishra U. (2010). An Inventory Model for Weibull Deteriorating Items with Price Dependent Demand and Time- varying Holding Cost. Applied Mathematical Sciences, Vol.4, no.44, 2171-2179.

17. Tripathy P.K. & Pradhan S. (2011). An Integrated Partial Backlogging Iventory Model having Weibull Demand and Variable Deterioration rate with the Effect of Trade Credit. International Journal Scientific & Engineering Research Volume 2, Issue 4.

18. Vashistha P.K. & Gupta V.S. (2011). An Inventory Model with Weibull distribution deterioration and Time dependent Demand. International journal of advances in engineering research Vol. No. 2, Issue No. I.

19. Vikas Sharma & Rekha Rani Chaudhary. (2013). An inventory model for deteriorating items with Weibull deterioration with time dependent demand and shoratages. Research Journal of Management Sciences Vol. 2(3), 28-30.

20. Vinod Kumar Mishra & Lal Sahab Singh. (2011). Deteriorating inventory model for time dependent demand and holding cost with partial backlogging. International Journal of Management Science and Engineering Management, 6(4), 267-271.

21. Vipin Kumar, S R Singh & Sanjay Sharma. (2010). Profit Maximization production inventory models with time dependent demand and partial backlogging. International Journal of Operations Research and Optimization, Volume 1, No.2, 367-375.

22. Wu J. W. & Lee W. C. (2003). An EOQ inventory model for items with Weibull distributed deterioration, shortages and time-varying demand. Information and Optimization Science, 24 , 103-122.