Optimal J.I.T Epq Model with Items of Imperfect Quality, Exponential Declining Demand and Regular Preventive Maintenance.

Optimal J.I.T Epq Model with Items of Imperfect Quality, Exponential Declining Demand and Regular Preventive Maintenance.

Sudipta Sinha Departments of Mathematics Burdwan Raj College,Burdwan

West-Bengal

Abstract:

This paper attempts to establish an Economic Production Quantity (EPQ) model where items produced are not considered to be of good quality. Defective items are sold at a low price after the items are being screened.

It is highly necessary to start preventive maintenance as soon as the production stops to ensure smooth running and long lasting of machine. Idea of JIT is also considered to avoid shortages. The effectiveness and utility of this model is illustrated with numerical example and sensitivity analysis has been performed.

Key words: Preventive maintenance, Inventory, Just-in-time, Imperfect, Declining demand.

Introduction: In EOQ or EPQ models of traditional method, it often happens that some of the inventories are either excess or short resulting a penalty cost. To overcome this problem, use of just-in-time production method of Japanese has proved fruitful and effective to minimize the cost. Salameh etal [14] proposed a JIT buffer inventory model in order to provide regular preventive maintenance. A comparative cost function between inventory ordering under an

economic quantity (EOQ) system or a just in time system was presented by Fazal et al [5]. Schonberger [16] and Wantuck [19] have clearly proved that the conversion to JIT will reduce space in plants and factories. In this paper conception of JIT is assumed to avoid shortages.

Olorunniwo et al [12] pointed out that preventive maintenance is based on one of the two extreme assumptions. The production unit is restored to either a bad as old or good as new condition .Actually due to preventive maintenance the condition of equipment may be enhanced at a level between those of two extremes. In developing the mathematical model of this paper it is assumed that regular preventive maintenance will keep the production unit to such condition that prevents sudden failure as will as maintains the quality as before

.Groenevelt et al [6] showed the effects of the machine breakdown and corrective maintenance on the economic lot size decisions. An approach for preventive maintenance scheduling in light of production plan is discussed by Balasubramanian [2].Valdez-Flores [18] and Nakagawa [11] show in their survey papers that the preventive maintenance improves to a certain extent the health condition of the equipment reducing the effective ages of the machine. Monga et al [10] presented optimal system which incorporated design model the effects of preventive maintenance and minimal repair. An algorithm for preventive maintenance policy is presented by Lie et al [7].Schniederjans et al

From the above discussions it may be pointed that none of them has taken into consideration JIT and items of imperfect quality practically it is seen that all the items produced in a machine are not of the some quality. Therefore, it is necessary to separate the defective items. Naturally after being screened the

defective items are sold at a low price Zhang and Gerchak [20] proposed an EOQ model where defective items are replaced by nondefective ones. Rosenblatt et al [13] come to a conclusion that presence of defective products leads to the motivation of smaller lot sizes. The effect of defective items on the lot sizing policy is amply illustrated by chakraborty et al [3] ; Moinzadheh et al [9] , urban et al [17] , Anily et al [1] Salameh et al [15] considered an EPQ model for items with imperfect quality .

The demand for an item naturally deelines over time due to various reason such as introduction of better products, change in trend or perception about the product and so on. An EOQ model with demand dependent unit cost is developed by Cheng [4].

In the present paper an EPQ model is suggested when all the items are not of same quality. The items having some defects are sold at a low price. Here we would like to emphasize on the regular preventive maintenance for maintaining the same quantity just-in-time production quantity is also solved here .The demand declines exponentially with time in this model.

.Lau et al [8] developed a multi-item newsboy problem with capacity constraints. Storage capacity of items in this model is limited .Now a numerical example is taken and is solved by penalty function method and sensitivity analysis has been performed.

Notations:

The following notations are used S=number of produced units in each cycle. p=rate of defective items among S.

N(S,p)=S-pS=(1-p)S=Number of good items among the produced items. V= unit selling price of defective items.

d= unit screening cost.

h= inventory holding cost per unit per unit time. c2= shortage cost per unit per unit time.

= a constant (>0)

T0= preventive maintenance time per cycle.

Q= number of produced unit+ the amount of shortage t1= time of screening.

t2= time of production

p/= defective items/unit. x= production /unit time.

Assumptions:

The following assumption are made

1. The items produced are not of the same quality, some of those are defectives.

2. Defective items are sold at a low price.

3. The regular preventive maintenance of machine guarantees that the probability of a break down of the production unit during T is approximately zero.

4. The just-in-time inventory is also determined.

5. Exponential declining demand with time is considered.

6. Limiting storage capacity is also considered here

   S

   T0 T1 T

   Case I (T1<T-T0)

   Time

   S

   T0 T T1

   Case II (T1>T-T0)

   S

   T0 T

   Case(III) (T1=T-T0)

   Time

   Inventory Inventory

   dq e t

   dt

   (1)

   q(0) (1 p)S …………..(1a)

   where

   q(T ) 0

   …………………..(1b)

   Integrating and using the boundary condition (1a) we get from (1)

   D e t / D

   q(t) 0 (1 p )S 0

   (2)

   The boundary condition (1b) gives

   T 1 ln[

   D0

   D0 ]

   (1 p/ )S

   (3)

   Let T1=(T-T0) = t1+t2

   Where t1=Time of screening & t2=Time of production.

   Now p.t1=S and x.t2=S, Therefore T1= t1+t2 S S (p x)S

   (4)

   p x px

   The total variable cost is comprised of the sum of the holding cost, stock out cost and screening cost. In case JIT stock-out cost does not occur.

   The defective items among S =p.S

   Case I

   In this case since T1<T-T0, inventories are accumulated during the period (T-T1) which will be used for the next cycle. Therefore an extra holding cost is added to each cycle. The holding cost is

   T T T e t

   1 T D et D

   HC hq(t)dt q(t)dt h{

   (1 p/ )S }dt h{ 0 (1 p/ )S 0 }dt

   0 T1 0

   T1

   1 e T /

   T D0

   T

   T1

   / D0

   h[ 2

   (1 p )ST

   2 (e

   e ) (T

   T1 )(S

   p S )]

   The screening cost is SRC= d.S

   The selling price of defective items DP= v.p/.S The production cost PC = C*.S

   The total variable cost function is defined as TVC= HC+SRC + PC S.P. of defective items

   D (1 p/ )S

   D (1 p/ )S D

   D ( p x) S

   (1 p/ )S

   TVC h[ 0 ln[ 0 ]( 0 ) 0 (e px 1 )

   D0 (1p )S D

   / 2 2

   0

   0

   0

   0

   { 1 ln[ D0 ] ( p x)S }{S p/ S D0 }]

   D0 (1 p )S px

   /

   • d.S C*S vp/ S

   .

   …………………………………………………………………………………………………..(6)

   The variable cost per unit time TCU1

   TVC1 T

7. The problem is Min TCU(S)

   s.t. aS A

   Case II

   In this case since T1>(T-T0),shortages occur during the period (T1-T).Therefore a stock out cost is added to each cycle.

   Stock out time = T1-T

   Therefore, the number of unit short per preventive maintenance cycle can be

   expressed as

   W (t) = e-t (T1-T)

   Stock-out cost SHC is

   T1

   2 1 1

   C (T T)e tdt; (T T)

   T

8. C2 (T1 T) (e T e T1 )

   The holding cost

   T

   HC2 q(t) dt

   0

   T

   h{

   D et

   (1 p/ )S

   D0 }dt

   0

   0

   0

   (1 p/ )SD (1 p/ )S D D D

9. h{ 0 ln[ 0 ] 0 ln[ 0 ]}

   D0 (1 p )S D (1 p )S

   / 2 /

   0

   0

   0

   0

   The screening cost is SRC= d.S

   The selling price of defective items DP= v.p/.S The production cost PC = C*.S

   The total variable cost function is defined as

   TVC= HC+SRC + PC+SHC S.P. of defective items

   D (1 p/ )S D (1 p/ )S D

   0

   0

   TVC h{ 0 ln[ 0 ]( 0 )}

   C (p x)S D

   D (1 p/ )S

   D

   2

   (1 p/ )S

   • (p x)S

10. 2 {

    0 ln[ 0 ]}{1 e

    px }

    px

    D0 (1 p )S D

    /

    0

    0

    • d.S C*S vp/S

    .

    The variable cost per unit time TCU1

    TVC1

    T

11. The problem is

    Min TCU(S)

    s.t. aS A

    Case III

    In case of JIT since T1=T-T0 i.e. the perfect inventories are accumulated just at the end of the cycle period.Therefore neither an extra holding cost nor the stockout cost is added to each cycle.

    The holding cost is

    T

    HC3 hq(t)dt

    0

    T

    0

    0

    h{

    D et

    (1 p/ )S

    D0 }dt

12. 0

    D (1 p/ )S

    D (1 p/ ) D

    h{ 0 ln[ 0 ]( 0 )}

    D0 (1 p )S

    / 2

    The screening cost is SRC= d.S

    The selling price of defective items DP= v.p/.S The production cost PC = C*.S

    The total variable cost function is defined as TVC= HC+SRC + PC S.P. of defective items In this case the total cost is

    D (1 p/ )S D (1 p/ )S D * /

    TVC h[ 0 ln[ 0 ]( 0 ) d.S C S vp S

    D0 (1 p )S

    / 2

    .

13. TCU(S)=

    h[ (1 P)s (1 p)s ln[1 (1 p)s]

    1 ln[1 (1 p/ )S] d.S v.p/ .S.

    1 ln[

    2

    1 ]

    1 (1 p)s

s.t. aS A The problem is

Min TCU(S)

s.t. aS A

5. Numerical example & Sensitivity Analysis

In this paper the optimal production quantity is determined under the situation of exponential declining demand, quantities with defective items, regular preventive maintenance of machine. A numerical example is considered to illustrate the developed

The following parametric values =0.15

p=$ 0.25/unit

h=$ 5.00/unit/time d=$ 0.5/unit

c2=$ 1.00

v=$ 20.00/unit/time A=20

a=0.5

Case I (T1<T-T0)

Case II (In Case Of Stock Out Occurs(T1>T-T0)

CASE-III (In Case Of JIT) (T1=T-T0)

Conclusions:

In this study we consider are JIT EPQ model with defective items. Regular maintenances of machine are also considered to avoid break down and to produce the same quality. The demand declines with time exponentially. The assumptions are very realistic as it will help producer to determine the optimal product and total cost under capacity constraint. In this paper the optimum JIT inventory is determined by trading off the HC, SRC, and SHC.

The results of the sensitivity analysis are also consistent.

References:

1. Anily.S, Single machine lot-sizing with uniform yields and rigid demands . Rebustness of optimal solution.IIE Transactions 27(5)(1995)633-635

2. Balasubramanian .R, Preventive maintenance scheduling in presence of a production plan production Inventory Management 1(1987) 80-87. [3]Chakraborty.A.K, Stub. A, Strategic allocation of inspection effect in a serial , multi product production systems IIE Transactions 19(1)(1987)18-22 [4]Cheng.T.C.E, An economic order quantity model with demand dependent unit cost, European Journal Of Operational Research (1989),Vol. 24,pp 252-256.

1. Fazal F, Fischer.P.K, Gilbert.E.W, JIT purchasing VS EOQ with a price discount; An analytical comparison of inventory costs. Internationd journal of production Economics 54(1998) 101-109.

2. Groenevelt.H..Pintelon.L, Scidmann.A, Production batching with machine break downs and safety stock. Operations Research 40 (1992) 959-971.

3. Lie.C.H and Chun.Y.H, An algorithm for preventive maintenance policy IIE Trans Reliabil (1986) R-35: 71-75

4. Lau.H andLau.A,The newsboy problem: A capacitted multiple product single period inventory problem,European Journal Of Operational Research,vol.94,(1996),pp.29-42.

5. Moinzadeh.K H,.Lee.L, A continuous review inventory model with constant re supply time and defective items .Naval Rescarch Logistics 37(4)(1987)457- 467

6. Monga .A, Duo.M.G and Toogood .R, Reliability based design of systems considering preventive maintenance and minimal repair . Int.J.Reliabil quality and safety Engng (1997) 4:55-71

7. Nakagawa.T ,Sequential imperfect preventive maintenance policies .IIE Trans Reliabil .(1988) 37: 295-298

8. Olorunniwo.F, Izuchukwu.A , Preventive maintenance scheduling in presence of a production plan. production inventory management 1(1987) 67-79 [13]Rosenblatt.M.J, Lee.H.L, Economic production cycles with imperfect production processes IIE Transaction 18(1986)48-55

1. Salameh.M.K, Ghattars.R.E, Optimal just-in-time buffer inventory for regular preventive maintenance. International Journal of Production Economics. 74(2001) 157-161.

2. Salameh.M.K, Jaber.M.Y , Economic production quantity model for items with imperfect quality International journal of production Economics 64(2000)59-64

3. Schniederjans.J.M. and Cao.Q ,a note on JIT purchasing Vs.EOQ with a price discount :An expansion of inventory costs.International Journal of Production Economics,65(2000)289-294.

4. Urban T.L, Deterministic inventory models in corporating marketing decisions, Computers and Industrial Engineering22(1)(1992)85-93

5. Voldez.C-F and Feldman.R.M, A survey of preventive maintenance models for stochastically deferiorating single unit systems . Naval Res Light (1989) 36:419-446.

6. Wantuck.K.A,Just in-time for America.The Forum, Milwaukee.WI,(1989). [20]Zhang.X,Gerchak.Y,Joint lot-sizing andinspection policy in an EOQ model

with random yield.IIE Transaction 22(1)(1991)41.

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