Economic Order Model With Quadratic Backorder Costs

Transcription

1 International Journal of scientific research and management (IJSRM) Volume Issue 1 Pages Website: ISSN (e): Economic Order Model With Quadratic Bacorder Costs Dr. Martin Osawaru Omorodion, Mr. Ayinde, Oladunmi Moses Joseph Ayo Babalola University IejiAraeji, Osun State, Nigeria odionmart@yahoo.com ABSTRACT Joseph Ayo Babalola University IejiAraeji Osun State, Nigeria yahoo.com The paper considers bacorder costs as a quadratic cost depending on the length of time of the bacorder. The quantity ordered is Q and the reorder level is R. We assume that the bacorder cost C β (t) is b 1 b t b 3 t. Where b 1, b and b 3 are the coefficients with respect to t, t 1 e and t, The equations for the linear cost is derived first in section 1 by deriving the expected bacorder cost per cycle using the assumption that demand follows the normal distribution. We extend the method of analysis to derive the bacorder costs when the bacorder costs is a quadratic function of the length of time of a bacorder. The additional costs to the incurred by the linear costs to the costs that is not time dependent is.economic ORDER MODEL WITH QUADRATIC BACKORDER COSTS. INTRODUCTION In this paper, the cost depending upon the length of time for which the bacorder exists is taen as a quadratic cost. Without inventories customers would have to wait until their orders were filled from a source or were manufactured. The time lag could be a quadratic cost. The longer it taes to meet the order the more severe the bacorder costs. The paper considers the economic model (Q,R). LITERATURE REVIEW The simple models of economic bacorder inventory control were extensively dealt with by Hatdley and Whitin (1) Uthayaumar and Parvathi () investigates a continuous review inventory model to reduce lead time, yield variability and set up costs simultaneously through capital investments. The bacorder rate is depending on the lead time through the amount of shortage. Zhang G.U. and B.E. Dathwu (3) develops a hybrid inventory system with a time limit in bacorders. BASIC MATHEMATICS ( Well dealt with in Ref (1) (Hatley and Whitin) Let x Ɲ(0, 1) and g( be the probability density function. and g ( Integrating by parts If n 0 If n 1 x1 x F( n 1 n1 x g( dx n 1 F ( dx F( ) g( ) xf( dx B. 1. LINEAR BACKORDER COST g( ) F ( ) B.1 1 F( ) g ( ). 1 Let be the cost of a bacorder which has been outstanding for time t. Dr. Martin Osawaru Omorodion, IJSRM volume issue 1 Jan 014 [ Page 517

2 If a bacorder is incurred at time z, z< L then L z is the time for which a bacorder exists. Let Ry, 0< y<q be the inventory level at time 0, then if the system is out of stoc in the time interval z, to Zdz after the reorder point R is reached then Ry was demanded in time Z and a demand occurred in time dz. This probability is 1.1 exp Hence the probability that t L Z, Length of time of a bacorder Giving an inventory level R y at time 0 Expected cost of bacorder 0 < z < L Hence the expected bacorder cost per cycle Expected bacorder costs 1.5 Integrating and noting that R Vg( V ) g( R) We have the expected bacorder costs equal to D 1. For the linear bacorder cost function When the cost function is linear Assume that F 1.6 Then the expected bacorder costs equals dy and substituting into 1., expected bacorder cost per cycle Noting that Then expected bacorder costs equals Expected bacorder costs per cycle 1.4 Let V Then expected bacorder equals Letting v Dr. Martin Osawaru Omorodion, IJSRM volume issue 1 Jan 014 [ Page 518

3 1.8 using B1 and B. equations and integrating, the expected bacorder costs equals ( ) ( b1 bl b3l ) R Y LV D g(v)dvdy 1.9, the expected bacorder costs B (Q, R). Integrating with respect to v And applying B1 and B ( ) )*. QUADRATIC BACKORDER COST We shall extend this method of analysis to derive the bacorder costs when the bacorder cost is a function of the length of time of a bacorder.1 Expected bacorder costs per cycle and applying equation Assuming that F, expected bacorder cost per cycle *F 3 Let v Then we have Dr. Martin Osawaru Omorodion, IJSRM volume issue 1 Jan 014 [ Page 519

4 .4 Then we have rearranging.7 Integrating and applying B1 and B. we have.5 Noting that And noting that Then substituting into 5 We have.8.6 Dr. Martin Osawaru Omorodion, IJSRM volume issue 1 Jan 014 [ Page 50

5 .13 Substituting.9 letting L ( V ) 1 v F v vg v.10 Then we have expected bacorders per cycle to be.11 CONCLUSION With the only additional cost to the linear cost C β b 1 b t being the b factor: b Q / L DQ And the additional cost of the linear to the cost without length of time of shortage is Db Q 1 Q L b Db1 Since, the additional costs of the quadratic DQ Q cost to the linear cost is far less than the additional cost of the linear cost to the cost that is not time dependent. Hence expected bacorder costs per year.1 Hence from.6.8 the inventory costs for model (Q, R) with quadratic bacorder cost terms and C REFERENCES 1.Hadley G and Whitin T.M. Analysis of Inventory Systems, PrenticeHall International, Inc, Englewood Cliffs, N.T Uthayaumar R and Parvathi P. Continuous Review Inventory Model International Journal of System Science Vol 10 Issue 3, 7, Zhang G.P, Patuno B.E.(A Hybrid Inventory System with a Time Limit or Bacorder), Transactions of Industrial Engineers, 003. Q L Dr. Martin Osawaru Omorodion, IJSRM volume issue 1 Jan 014 [ Page 51