Optimal Management of the Spare Parts Stock at Their Regular Distribution

Transcription

1 Joural of Evirometal Siee ad Egieerig 7 (018) doi: /16-598/ D DVID PUBLISHING Optimal Maagemet of the Spare Parts Stok at Their Regular Distributio Svetozar Madzhov Forest Researh Istitute, Bulgaria ademy of Siees, Sofia 1756, Bulgaria bstrat: The artile presets a model for optimal maagemet of the SP (Spare Parts) stok at their regular distributio. The preset researh disussio is a multi-omelature model with a fixed limit of the stok s value. study was arried out o the optimal value of storage ost for a uit of stok at a uit of time (h ), optimal value of the demad itesity (λ ) ad the optimal prie of a stok uit ( ), subet to the restritio for the upper limit K of the total value of stoks. Key words: SP, optimum maagemet, stok, forestry. 1. Itrodutio The questio of the optimal stok maagemet has bee disussed i oe way or aother i may publiatios [1-6]. Most ofte, moo omelature determiat models are disussed. They are of iterest, beause they eable us to study the ideology of researh methods used i more sophistiated systems. Pratially, a large umber of differet types of stoks are stored i the warehouses, whih imposes that differet kids of omelature models for stok maagemet of SP (Spare Parts) are developed. other importat feature of the SP maagemet system i the pratie is the presee of restritios, more total umber of deliveries durig the year, the maximal storage spae, maximal volume of eah separate delivery, miimal quatity of the ordered produtio, et. The ompaies i the real eoomy usually do ot dispose of ulimited fiaial resoure, to be ivested i stok, but ust the opposite they dispose of a limited workig apital, by whih they maitai the stok. The preset review studies a multi-omelature model with a prelimiary defied limitatio of the stoks value. The oduted researh of the optimum Correspodig author: Svetozar Madzhov, ssistat Professor, Ph.D., researh field: Forest researh. ost values for a uit storage for a uit of time (h ), the optimum value of the demad itesity (λ ) ad the optimum prie for a stok uit (с ), upo keepig the upper limit (K) of the total stoks value is preseted i Figs. 1-3 respetively. The output data are olleted i the led i the tow of Eli Peli researh for λ, ad the other data are borrowed by Ref. [7], with updated pries.. Materials ad Methods Let us assume that kids of stoks are stored i the warehouse ad there is always a upper limit K of their total value. If we assume ad desigate the deliveries kid volume by, the the limitatio may be writte dow by the iequality: K (1) Whereas α is the revaluatio fator due to the irregularity of some deliveries types, 0,1., с the prie of a stok. Let us assume that λ is the itesity of demad of produts of the kid (the demad is determiat ad with a regular itesity withi the time), А is the ostat otet of deliveries osts, h the storage osts for a stok uit for a time uit. The model does ot allow usatisfied demad of either of the stoks types. Therefore the average osts for a time uit for stoks of the type

2 56 Optimal Maagemet of the Spare Parts Stok at Their Regular Distributio shall be illustrated by the equatio of the Wilso model [-5]: h d the total osts of all stoks shall be expressed by the Eq. (): R () It is eessary to fid suh umbers 1 0,..., 0, whih substituted i Eq. () to give the maximal value of the total osts R. The latest sum i Eq. () does ot deped o the maageable parameters 1,, ad will ot ifluee the solutio of the task. The sum of the positive added will be miimal, if eah of the addeds reahes its maximal value. But for the added this value will be reahed below i Ref. [3]: If these umbers satisfy the limitatio (1), the they are the optimal oes. I this ase the limitatio (1) is isigifiat, i.e. the apital ivestmets are eough ad their irease will ot lead to derease of the total osts. If o the other had, do ot satisfy Eq. (1), the limitatio of the stok value is sigifiat ad the optimal others. are I the last ase, for gettig the optimal the Lagrage method of the udefied multiplier shall be applied. uthors are formig the Lagrage futio: h K 1 1 where is Lagrage s udefied multiplier. The set of umbers, = 1,,, at whih the futio R has a absolute miimum ad satisfies the limitatio Eq. (1), is the solutio of the equatio system. h 1, 1 / h, 1,,..., System (3) has oe sigle solutio: ; 1,,...,. (3) (4) h 1 1 ad to fulfill also the last oditio (3), it is eessary to be seleted i suh a way that: 3. Results ad Disussio h 0, K 0. 1,..., K K (5) 1 h whih is ew ad allows optimizig the SP stok maagemet. The oduted researh of the optimum ost values for a uit storage for a uit of time (h ), the optimum value of the demad itesity (λ ) ad the optimum prie for a stok uit (с ), upo keepig the upper limit (K) of the total stoks value is preseted i Figs. 1-3 respetively. The allowed values of the preseted parameters belog to the multipliity, ad the ot allowed of the multipliity B. The optimum value is loated o the ross poit of the graphi of the futios of the upper limit (K) of the total value of the storages ad the ostat ompoet of delivery osts () (Fig. 1), the value of the demad itesity (λ ) (Fig..) ad the prie of a stok uit (с ) (Fig. 3). s the left had side of eq. (4) is a mootoously dereasig futio of ω ad К(0) > αк, upo the kow α ad К, Eq. (4) has oe sigle solutio ω i regards to the parameter ω. The maagemet strategy upo the presee of the limitatio (1) is defied by the quatities of =, alulated by equality (3). The umerial algorithm for the determiatio of ω turs out uder the followig algorithm:

3 Optimal Maagemet of the Spare Parts Stok at Their Regular Distributio K max B А h, BGN h opt h, BGN. Fig. 1 The optimum osts value for storage of a stok uit for a uit of time (h ), upo keepig the upper limit (K) of the total stoks value; the multipliity of allowable storage values for a storage of a stok uit for a uit of time (h ); B the multipliity of uaeptable values of storage of a stok uit for a uit of time (h ). K max B А λ λ opt λ Fig. The optimum value of the demad itesity (λ ), upo keepig the upper limit (K) of the total stoks value; multipliity of the allowed values of demad itesity (λ ); B the multipliity of uaeptable values of demad itesity (λ ).

4 58 Optimal Maagemet of the Spare Parts Stok at Their Regular Distributio K max B А с, BGN opt, BGN Fig. 3 The optimum value of prie for a uit of stok (с ), upo keepig the upper limit (K) of the total stoks value; multipliity of the allowed values of the prie for a uit of stok (с ), B the multipliity of uaeptable values of the prie for a uit of stok (с ). uthors hoose oe radom value ω > 0 ad alulate К(ω 1 ). We ompare К(ω 1 ) with αк (1) If К(ω 1 ) = αк, the ω = ω 1 ad the searhed are =, ad the alulatios ed. () If К(ω 1 ) > αк, authors hoose suh a value ω > ω 1 to alulate К(ω ) ad proeed to a ompariso of К(ω ) with αк, i.e. to the preedig poit of the ewly hose value of ω. (3) If К(ω 1 ) αк, authors hoose ω ω 1, to alulate К(ω ) ad proeed to poit for the ew ω = ω. The optimizatio proedure may ed, whe the urret solutio ω is osidered satisfatory, i.e. whe К(ω ) αк ε (ε > is a aeptably small value). I order to use this model i the pratie we should kow the flutuatio of its followig parameters: upper limit of the stok value (Кw), stok deliveries volume by the -kid ( ) ad the total osts of all stoks (R). The researh obet is the flutuatio of the model parameters, i relatio to the maagemet fators: demad itesity of produts of the -kid (λ ), ostat osts ompoet ( ), storage osts (h ), stok uit s prie ( ) ad the idefiite multiplier of Lagrage (w ). Figs. 4-6 show the flutuatio of the total osts of all stoks (R) depedig o the flutuatio of deliveries volume of sok of -kid ( ) ad the itesity of demad of produts of -kid (λ ). The other parameters, amely the ostat otet of osts ( ) = 0 BGN, storage osts (h ) = 0 BGN ad the prie of a uit of stok ( ) = 30 BGN are assumed ostat [8]. It is obtaied by regressio aalysis ad desribed i Fig. 4 that with the irease of the deliveries volume of the stok -( ) the osts R are ireased liearly, ad the urves of R = f{} are shifted i the diretio irease of total osts of all stoks (R). There is a irease i R with a irease i the itesity of demad of produts of -kid (λ ). Fig. 5 shows that their hage is also liear, but the irease of the volume of deliveries of stok of -kid ( ) ifluees more sigifiatly o R = f{λ}. greater iterest represets the flutuatio of the total osts (R), whe the two parameters hage: stok deliveries, volume of -type ( ) ad the itesity of the produt demad of -type (λ ). s a rule, the total

5 Optimal Maagemet of the Spare Parts Stok at Their Regular Distributio 59 osts for all stoks (R) irease liearly (Fig. 6), but with the irease i the stok volume of -type ( ) of the osts R irease faster tha i the ase of irease of the itesity of demad of -type produts (λ ). ordigly, the total osts for all stoks (R) irease slower with the irease of the itesity of demad of -type produts (λ ), the total osts for all stoks (R) have a maximal value of the volume of the stok delivery of the -type ( ) ad the itesity of demad of produts of the -type (λ ), tha i the opposite ase, osequetly the volume of supplies of -type stok ( ) ifluees more over the total osts of all stoks (R), tha the demad itesity of produts of the -type (λ ). The ext study desribes the hage of the total osts for all stoks (R ) i relatio to the ostat ompoet of osts ( ) ad the prie of a uit stok ( ). The flutuatio of the deliveries volume of stok of -type ( ), storage osts (h ) ad produt of the -type demad itesity (λ ) are osidered ostats. Their values are: =400 items, h =5 BGN ad λ =30 items [[8,9]. 4х х10 5 3х10 5.5х10 5 х х λ=00 λ =400 λ =600 λ =800 λ =1000 Fig. 4 Flutuatio of the total stok osts (R ) i relatio to the deliveries volume of the stok of the type ( ) at a ostat ompoet of osts ( ) = 0, storage osts (h ) = 0 ad prie of a stok uit ( ) = 30. 4х х10 5 3х10 5.5х10 5 х х10 5 =500 =600 =700 =800 =900 = =1100 λ, Fig. 5 Flutuatio of the total stok osts (R ) i relatio to itesity of produts of the type (λ ) at a ostat ost ompoet ( ) = 0, storage osts (h ) = 0 ad prie of a stok uit ( ) = 30.

6 60 Optimal Maagemet of the Spare Parts Stok at Their Regular Distributio R, лв , бр. λ, бр. Fig. 6 Flutuatio of the total stok osts (R ) i relatio to the flutuatio of the volume of stok deliveries of the type ( ) ad the demad itesity of produts of the type (λ ) at a ostat ( ) = 0, storage osts (h ) = 0 ad prie of a stok uit ( ) = 30. The flutuatio of the total osts of all stoks (R) depedig o the prie for a uit of stok ( ) has bee studied. The aalysis of results (Fig. 4) idiates that the stok uit prie irease ( ) leads to a liear irease of osts R, ad the flutuatio of the ostat ompoet of osts do ot ifluee o R = f() (Fig. 5). The study, oduted at a simultaeous flutuatio of the ostat ompoet of osts ( ) ad the prie of a stok uit ( ) (Fig. 6) over the total osts of al stoks (R ) shows that with the irease of prie of a stok uit ( ) the osts R irease quikly, but with the irease of the ostat ompoet of osts ( ) the osts R irease isigifiatly. The total flutuatio of the total osts of all stoks (R) is show as a surfae, whih is ilied at a ertai agle to a level of three-dimesioal width А с. 4. Colusios mathematial model for SP stok optimizatio at their regular distributio for maiteae of mahie effiiey has bee elaborated. The ifluee of the mai parameters of the model over the flutuatio of the total osts has bee studied. Referees [1] Barlow, R., Prosha, F Statistial Theory of Reliability ad Reliability Tests.M Holt,Riehart & Wisto of Caada Ltd,. p. 38. [] Dimitrov, B Sietifi Ivetory MaagemetSofia. (i Bulgaria) [3] Kostati, G., leksadrov, K., et al. 00. Survey of OperatiosSofia. (i Bulgaria) [4] Ryzhikov, Y Ivetory MaagemetMoskow. (i Russia) [5] Sakovih, V Models of Ivetory Maagemet. Misk. (i Russia) [6] Spiridoov, G., ad Tasev, G Colletio of Methodial ad Normative Materials o the Maiteae of the griultural Mahiery. Rousse. (i Bulgaria) [7] Spiridoov, G., ad Tasev, G Justifiatio of the Parameters of the System for Distributio of Spare Parts betwee the Uits of ROSB. Rousse, Report o the subet No. 8530/1 (ot published). (i Bulgaria) [8] Tasev, G., ad toov, B Basi Issues i Regulatig ad Maagig Stoks of Spare Parts, FTS, 7, pp Rousse (i Bulgaria) [9] Spyridoov, G., ad Tasev, G Some Theoretial ad pplied spets of the Repair ad Maiteae of griultural Mahiery. Rousse. (i Bulgaria)