Developing an Inventory Mathematical Model with Deterioration Variables for Discounted Stochastic Goods

Transcription

1 Indian Jornal of Science and Tecnology, Vol 7(11), , November 2014 ISSN (Print) : ISSN (Online) : Developing an Inventory Matematical Model wit Deterioration Variables for Disconted Stocastic Goods Moammad Reza Saraki *, Alireza Saraki and Navid Javdan Department of Indstrial Engineering, University of Sistan and Balcestan; mr.saraki@eng.sb.ac.ir, sarakiar@amoon.sb.ac.ir, javdan.navid@pgs.sb.ac.ir Abstract Experts believe tat a large percentage of total corporate capitals lie in organizational inventories. Terefore, a more torog and intelligent review of te inventory isse empowers organizations to optimally make se of all teir resorces to acieve better efficiency and effectiveness. Tis migt, in trn, elp organizations in reacing teir goals. In tis stdy, an inventory model nder general discont and stocastic deteriorating conditions is developed. Deterioration was stdied as a Stocastic variable wile oter variables were deterministic. Te objective of tis stdy was to redce te expected cost of te wole system. First, te reorder point was determined by te EOQ model, and ten oter price ct points (de to discont or inventory Deterioration) were determined. Finally, te total cost of te system was compted for all points, by wic te reorder point ad te lowest cost. Te developed model was frter analyzed trog a nmerical example. Keywords: Inventory; Deterioration; Discont; Stocastic Model; EOQ 1. Introdction Timely provision of demands for goods as trned logistics and inventory operations into delicate ndertakings for every indstrial organizations. Te capital invested in inventory stocks olds a great sare in every economic sector. Tis ge capital also amonts for a great sare of GDP. Inventory management activities necessitate considering inventory provision and maintenance processes as an integrated system and not as separate operational components 2. Te main objective of inventory planning and control is to adopt te best reordering and inventory maintenance policies by analyzing costs and conditions 10. Inventory control is an operation tat ensres reqired organizational items are delivered to te respective operational segment (manfactring, distribtion, sales, engineering, etc.) wit regards to time, place, qantity, qality and cost 9. Te inventory incldes raw materials, semi-finised materials, spare parts and prodcts, te correct management of wic wold bring abot an efficient work flow. Moreover, inventories play a determining role in creating more reliability, more robst planning and more endrance against canges 14. In tis stdy, stocks are stdied nder te conditions tat more disconts wold be given for more prcased qantities. Additionally, it is assmed tat te system sold not face inventory sortage since it wold incr ig costs on te system. Tis model is intended for Deteriorateable prodcts, i.e. migt become nfasionable or wasted and sable. 2. Literatre Review Prodction and inventory control incldes planning and controlling for material flow and prodction operations. Materials are tose items wic are system inpts (raw materials and parts), are flowing in te prodction line (semi-finised prodcts), and are system otpts (finised prodcts). In oter words, prodction and *Ator for correspondence

2 Moammad Reza Saraki, Alireza Saraki and Navid Javdan inventory control is defined as maintaining items in desirable conditions to meet cstomer needs in a timely fasion wit minimized system costs 10. Wen an organization is ot of stock and fails to meet cstomer demands, it will face stock sortage costs 3. If available inventory is lower tan cstomer demands, ten tere will be cstomer attrition 9. Usally large prcase orders are offered wit a discont, terefore, encoraging more prcases. Accordingly, organizations sold seek sales qantities tat wold minimize total costs of prcase, maintenance and ordering, despite disconts 11. Tere are sally two kinds of disconts: general and special. Te former is only stdied in tis paper. Dring stock ordering and procrement nder general discont conditions, stores qote teir prices in different qantity ranges, were te price of eac range incldes te wole order 8. Te iger te economic vale of a reorder, te iger te maintenance costs. Besides te maintenance costs, tere is anoter cost called deterioration costs. deterioration can be defined as spoiling, damages, or becoming nfasionable. For example, in a prodction line, if te prodction eqipment or te final prodct is canged, te wareose stocks wold become seless, terefore te prcase cost of te crrent inventory will be regarded as te deterioration cost 8. Gcait 7 Mlti-item inventory model of breakable items wit stock-dependent nder stock and time dependent breakability rate. Here a mlti-item inventory model of breakable items is developed, were demands of te items are stock dependent, breakability rates increase linearly wit stock and nonlinearly wit time 7. Lin 12 in an article A stocastic periodic review integrated inventory model involving defective items, backorder price discont, and variable lead time. Te prpose of tis article is to investigate a stocastic integrated spplier-retailer inventory problem. Te model analyzed in tis article explores te problem of te protection interval, te backorder price discont, te lead time, and te nmbers of sipments from te spplier to te retailer in one prodction rn as control variables to widen applications for an integrated periodic review inventory model 13. Glock 6 in an article Lead time redction strategies in a single-vendor-singlebyer integrated inventory model wit lot size-dependent lead times and stocastic demand. Tis paper stdies alternative metods for redcing lead time and teir impact on te safety stock and te expected total costs of a (Q,s) continos review inventory control system 6. Cakrabortty 4 in an article Intitionistic fzzy optimization tecniqe for Pareto optimal soltion of manfactring inventory models wit sortages. Tis paper discsses a manfactring inventory model wit sortages were carrying cost, sortage cost, setp cost and demand qantity are considered as fzzy nmbers 4. Giami 5 in an article A two-ecelon inventory model for a deteriorating item wit stock-dependent demand, partial backlogging and capacity constraints. Tis stdy investigates a two-ecelon spply cain model for deteriorating inventory in wic te retailer s wareose as a limited capacity 5. Sazver 15 a replenisment policy for perisable prodcts wit nonlinear olding cost nder stocastic spply lead time. In tis stdy, we develop an inventory model for a main class of deteriorating items, namely perisable prodcts, nder stocastic lead time assmption 15. Lee 12 in an article An optimal policy for a single-vendor single-byer integrated prodction-distribtion model wit bot deteriorating and defective items. Tis article develops an integrated prodction distribtion model to determine an optimal policy wit bot deteriorating and defective items nder a single-vendor single-byer system. Te objective of tis article is to maximize te spply cain profit and to find te optimal nmbers of delivery after incorporating deterioration and defectiveness into one model 12. Tayal 16 in an article A mlti item inventory model for deteriorating items wit expiration date and allowable sortages. Tis article presents a mlti item inventory model to optimize te nit time profit of inventory management for te prodcts aving an expiration date after wic te prodct can not be sold Statement of te Problem Spply cain management, i.e. controlling over te raw materials to final cstomer domain, is essential for more or less every organization. Senior managers today appreciate te strategic significance of spply cain management. In indstrial factories, te capital invested in te inventory is ig. Moreover, controlling te capital pt into raw materials, semi-finised prodcts and final prodcts can offer a great potential for improvement. Using scientific metods for inventory control can generate competitive advantages for companies 1. In te classic EOQ model, all parameters are considered deterministic and wit a fixedvale. However, in or proposed model, prcase and maintenance costs are inclded along wit disconts and probable deterioration costs. Indian Jornal of Science and Tecnology 1751

3 Developing an Inventory Matematical Model wit Deterioration Variables for Disconted Stocastic Goods 4. Problem Modelling To propose te inventory model, parameters and variables are first introdced and te proposed model is ten presented. C : Cost of eac order D : Rate of demand (consmption rate per nit time) : Cost of maintenance (cost of maintaining one item per nit time) : nit price of an item Q = Q w : Qantity of eac reorder (Wilson s point) Q s : Negative inventory (sortage) s : Cost of exposre to inventory sortage (cost indced by sortage of eac nit item per nit time) d : Amont of ndeterioration items P(d) : stocastic of deterioration TOC : Total Ordering Cost (total cost of ordering per nit time) THC : Total Hosing Cost (total cost of osing or storing per nit time) TMC : Total Paid Cost for prcasing items per nit time TDC : Total deterioration Cost (total deterioration, becoming nfasionable, per nit time) TSC : Total Sortage Cost (total cost of exposre to sortage per nit time) TIC : Total Inventory Cost (total cost of inventories per nit time) 4.1 Modeling Indstrial factories are faced wit varios inventory costs and tey try to implement inventory control policies in one way or anoter to minimize total costs. Costs incrred to a system by inventory inclde ordering cost (Eqation 2), osing cost (Eqation 3), and procrement cost (Eqation 4) (7). CD Qw = Q = 2 (1) TOC= C D (2) Q THC Q = 2 (3) TMC= D (4) TIC= TOC + THC+ TMC (5) TIC= C D + Q Q + D 2 (6) In tis stdy, an inventory model was stdied nder no sortage costs, becase te sortage cost is infinitely ig (Eqation 6). In te first assmption, tere is a discont id offered on larger orders. Te discont type in tis case is general; tat is, by prcasing a certain qantity, a discont is received on all items. Terefore, te prcase cost is calclated based on te ordered qantity. Te administrative affairs and general costs of wareosing (like air conditioning, ligting, etc.) are common for all items, accordingly, te wareosing cost is also redced, wic we call it a discont as well 8. Te second assmption of te stdied problem is deterioration, for wic a new cost is defined as deterioration cost. In tis stdy, deterioration is stocastic (Eqation 7). Since, in market predictions, it is possible to assert tat a certain amont of materials wold not peris ntil reordering, te mean amont of deteriorate items is considered. Te total cost can be determined sing Eqation 8 and Eqation 9. Q- dˆ ˆ TDC= P( d) 2 (7) TIC= TOC + THC+ TMC + TDC (8) TIC C D ˆ Q Q d = D P d Q + ˆ - ˆ ˆ + ( )+ ( ) 2 2 (9) Wit regards to Figre 1, in order to determine te economical reorder point, first, te TIC at Wilson s point and Q points on te rigt of Q w, Q vales on te left of Q w, and te starting point of deteriorating (d) sold be examined. Ten te vale of Q for wic TIC wold be minimm can be selected. 5. Nmerical Example Te following prices and terms are qoted by a seller for a given commodity: Te total consmption of tis item is estimated at Te cost of eac ordering is 7500 nit crrency and 1752 Indian Jornal of Science and Tecnology

4 Moammad Reza Saraki, Alireza Saraki and Navid Javdan Annal costs Qantity per order Figre 1. Annal costs for te qantity of eac reorder (7). Table 1. wareosing cost is 56 nit crrency, wic is disconted by 3 nit crrency per nit item for eac 5000 items or more. According to te predictions, 1700 non-perised items wold remain in te wareose and more tan 1700 items wold peris according to Poison s distribtion. Te economical qantity for eac order is: D = 23000, C = 7500 At first, we determined Wilson s point, provided tat te wareosing cost is variable. Terefore, we examined Wilson s point for eac price range: Q = = = Unacceptable Item s nit price for different qantities Ordered qantity (nit item) Price of nit item for total ordered qantity 0 to to to and more 4125 Q - = = = Acceptable Q - = = = Unacceptable Q> = = = Unacceptable Since only te order qantity of 2482 nits is in te acceptable range, te economical point is Next, te stocastic of deterioration is determined. Since te deterioration fnction is Poison s, its stocastic is determined as follows: P(d = 1700) = 0 P(1700 < d 2000) = 0.05 P(1700 < d 2482) = 0.1 Indian Jornal of Science and Tecnology 1753

5 Developing an Inventory Matematical Model wit Deterioration Variables for Disconted Stocastic Goods P(1700 < d 5000) = 0.3 P(1700 < d 10000) = 0.6 For te tird step, te vale of TIC from Eqation 8 was examined for eac Wilson s vale and vales on te rigt of Wilson s. Ten, te vales on te left of Wilson s point to te point deterioration were also examined. TIC= TOC + THC+ TMC + TDC TIC C D ˆ Q Q d = D P d Q + ˆ - ˆ ˆ + ( )+ ( ) 2 2 TICw = ˆ ˆ + ( ) - + P ( 1700 < d ˆ ) ˆ TIC w = TIC was ten examined for 5000 nit items: TIC5000 = ˆ ˆ 2 + ( ) P 1700 < ˆ ( ) ˆ d TIC 5000 = Ten for nit items: TIC = ˆ ˆ 2 + ( ) P 1700 < d ˆ ( ) ˆ TIC = Finally, TIC was determined for 1700 nit items: TIC1700 = ˆ ˆ 2 + ( ) P d= ˆ ( ) ˆ TIC 1700 = Comparing te determined TICs sowed tat te total cost wold be minimized for 5000 nit items. 6. Discssion and Conclsion Te most important and fndamental responsibilities of every indstrial nit are inventory planning and control. In tis regard, te responsibility of prodction and inventory control personnel is to adopt te best policies tat minimize costs, wile considering all factors. In tis stdy, an inventory model nder probable deterioration and discont conditions was developed, trog wic te economical qantity of orders can be determined. Reslts revealed tat, nder stdy conditions, te amont of deteriorative items and terefore te final costs increase wit larger order qantities. However, on te oter and, iger disconts are offered for larger order qantities. Accordingly, selecting te best reorder point can redce incrred costs to an organization. In addition to stocastic deterioration, ftre researc may inclde stocastic consmption and/or receiving rates in te inventory model, or examine tis model wile all tree are stocastic. Frtermore, te inventory model can be investigated wit mltiple prodcts. 7. References 1. Axsäter S. Inventory Control: Springer; Banks J, Fabrycky WJ. Procrement and inventory systems analysis: Prentice-Hall; Bezadian M. Inventory planning and control (1). 1, editor. Amol: nasr somal paydar; p. 4. Cakrabortty S, Pal M, Nayak PK. Intitionistic fzzy optimization tecniqe for Pareto optimal soltion of manfactring inventory models wit sortages. Eropean Jornal of Operational Researc Giami Y, Williams T, W Y. A two-ecelon inventory model for a deteriorating item wit stock-dependent demand, partial backlogging and capacity constraints. Eropean Jornal of Operational Researc Glock CH. Lead time redction strategies in a singlevendor single-byer integrated inventory model wit lot size-dependent lead times and stocastic demand. International Jornal of Prodction Economics. 2012; 136(1): Gcait P, Maiti MK, Maiti M. Mlti-item inventory model of breakable items wit stock-dependent demand nder stock and time dependent breakability rate. Compters & Indstrial Engineering. 2010; 59(4): Haj sirmoammadi A. principles of prodction and inventory planning and control. Isfaan: arkan danes; p Indian Jornal of Science and Tecnology

6 Moammad Reza Saraki, Alireza Saraki and Navid Javdan 9. Hosseini baaranci R. Order planning and control of prodction and inventory. 1, editor. Teran: Jaan jam jam; p. 10. Javadian n, Naij agigi J. Inventory planning and control. 1, editor. Babol: Olomrayane; p. 11. Kademi Zarea H. Inventory control. 1, editor. teran: Soros danes; Lee S, Kim D. An optimal policy for a single-vendor singlebyer integrated prodction distribtion model wit bot deteriorating and defective items. International Jornal of Prodction Economics. 2014; 147: Lin Y-J. A stocastic periodic review integrated inventory model involving defective items, backorder price discont, and variable lead time. 4OR. 2010; 8(3): Mller M. Essentials of Inventory Management: AMACOM; Sazvar Z, Baboli A, Jokar MRA. A replenisment policy for perisable prodcts wit non-linear olding cost nder stocastic spply lead time. Te International Jornal of Advanced Manfactring Tecnology. 2013; 64(5 8): Tayal S, Sing S, Sarma R. Mlti Item Inventory Model for Deteriorating Items wit Expiration Date and Allowable Sortages. Indian Jornal of Science and Tecnology. 2014; 7(4): Indian Jornal of Science and Tecnology 1755