MAHEMAICAL DESCIPION OF HEOEICAL MEHODS OF ESEVE ECONOMY OF CONSIGNMEN SOES Péer elek, József Cselényi, György Demeer Universiy of Miskolc, Deparmen of Maerials Handling and Logisics Absrac: Opimizaion of he operaion of a consignmen sore in generally is no an easy ask. he main problem is he sochasic characerizaion of he consumpion of he producs which makes he using of deerisic mehods impossible. In our research projec we developed a heoreical mehod which uses hisoric daa of a previous erm o deere he opimal values for he parameers of he consignmen soring process. Main advanage of he researched mehod is o give possibiliy for he designers and operaors of he consignmen soring process o find (or a leas approach) he opimal parameer values which is very difficul by any oher way. he new mehod helps o deere he parameers of he soring process previously as here can be seen in an example. Keywords: Logisics, socking, supplying, consignmen sore Consignmen sores have special siuaion because of he operaion specialies which require seing he sock level near he opimal value. his opimal value is enforced by he operaor company (which would like o increase he sored quaniy o avoid he lack of producs) and also by he manufacurer company (which would like o decrease he sored quaniy o reduce he amoun of money blocked by he sock). 1. Consignmen sores he calculaion, acualizaion and susaining of he opimal sock value are in generally no easy asks and require exac descripion of he main characerisics of consignmen sores and heir relaions. he mos imporan characerisics of consignmen sores (Figure 1.): uploading level:, imal sock limi:, J exhausion of he sock: Q, F ime of he exhausion of he sock:, angle of he exhausion funcion:, which can be calculaed in case of linear exhausion: g F Q, (1) - 3 -
difference in exhausion of he angle: ±, o by he highes exhausion (if he exhausion ime is he shores):, (2) o by he lowes exhausion (if he exhausion ime is he longes): erm of he produc order:. max +, (3) Q M max J max max Figure 1. Changing of he angle of he exhausion a socks of consignmen sores - imal value of he imal sock levels, max - maximal value of he imal sock levels, - imal value of he ime of he exhausion, - maximal value of he ime of he exhausion. max 2. Analyzing of he sock levels of consignmen sores A he analysis of sock levels of consignmen sores if deerisic or prognosic daa can no be available, he only one soluion is he mahemaical-saisical processing of hisoric daa of a previous producion erm. In his case, a firs we have o analyze periodically he values of he angle of he exhausion funcion, he difference beween changings of he angle. - 4 -
hen based on he analysis of he angle of he exhausion funcion he nex parameers can be deered: uploading level, imal sock limi, erm of he order. 2.1. Uploading quaniies If we use consan uploading quaniies o fill he sock of he consignmen sore (Figure 1.), he required uploading quaniy can be calculaed by: Q J + M J + - average of he imal sock values, M - average of he exhausions below he imal sock limi. g, (4) As he differen values of are no consan so he uploaded level ( ) can no be consan, is value will change beween he wo limi values (imal and maximal) depend on he imal sock values. Limi values of he uploaded level ( ) are: imal: + Q <, (5) maximal: 2.2. ime of he exhausion of he sock + Q >. (6) max max In he aspec of he ime of he exhausion of he sock we can also analyze limi values which can be based on he nex expression: g. (7) ransforg equaion (7) and subsiue he average value of he angle of he exhausion: g. (8) o analyze he average ime of he exhausion i have o be deered A a imal value ( ) and a maximal value ( ) for he ime of he exhausion. F Depend on he relaion beween he average and he limi values of he ime of he exhausion we have o deere wo differen cases in he aspec of uploading levels: - 5 -
if > F, hen he loading level requires reducion because in any cases he average sock level can be oo high which will cause sufficien increasing in cos and soring capaciy, if < A, hen he loading level requires increasing because in any cases he inensiy of he uploading process can be oo high which will cause sufficien increasing in logisic asks. A F he imal ( ) and maximal ( ) limis of he ime of he exhausion have o be deered o fi o he available soring and logisic capaciies. 2.3. Correcion of he loading quaniies If he uploading levels can no be changed here is anoher possibiliy o correc he uploading quaniy. In his case ( cons. ) he correced uploading quaniies can be calculaed by : Qmax ( J) + M1 J +, (9) g g, (1) M 1 by : max Q ( J) +, () g max g max. () M 2 If cons., he required imal sock limi: and, (13) J. J Based on Figure 2. and g J, () J g. () - 6 -
M1 max M2 J max Figure 2. elaions for he limi values of he angle of he exhausion and he imal sock limi Based on he equaions (2) and (3), if he value of is high a consan uploading quaniy we have o ake very low uploading levels ( < ) or very high uploading levels ( > ) ino consideraion. If we use an uploading sraegy wih changing quaniies i is no enough o ake only he imal sock limi ino accoun, bu we have o ake care on he angle of exhausion of he sock and is changing. 2.4. Analysis of he sock level-ime funcions of elemen sores A consan uploading erms he sock levels of he elemen sore have o be suied o he disribuion requiremens of he finished producs. Deeraion of he required sock levels based on Figure 3. Q K Q j + Q j - Q B2 Q Bj Q B1 QJ1 Q Q B(j+1) B Q J2 Figure 3. Example for an elemen supply funcion ( ) Based on Figure 3. he prognosic value of he uploading when reach he imal sock limi in erm j can be calculaed as: Q Q. () he real exhausion unil erm j can be if Q Q, hen he real exhausion is equal o he prognosic value, Fj Bj - 7 -
if if Q Q Q Q Q, hen he real exhausion is higher hen he Fj > j Fj prognosic value, Q Q Q Q Q, hen he real exhausion is lower hen he + Fj < j Fj prognosic value. he safey sock level can be calculaed wih wo differen mehod: a) 1 { B Fj} Max Q, () b) B 2 QFj + k δ. QF () Q Fj he average value, δ QF he disribuion value, k an ineger value (k1, 2, 3, ec.) which akes he effec of he disribuion ino consideraion. equired uploading level required by he supplier a consignmen sores or { Fj} 1 Max Q Q k 2 F + δfj, (). (2) Q + B(j-k) Q B(jk-1) Q B(jk+1) Q Bj Q B(j+1) Q B Q - B(j-k) Figure 4. Example for he sock level of an elemen sore ( > ) - 8 -
o conrol he uploading process he uploading parameers have o be checked in deered erms, which are Q he required sock level, depend on he acual producion orders, J ( jk) Q F ( jk) he real exhausion (suied o he producion orders), Q ( ) Q ( ) Q he overexhausion, j k F j k J( jk) + Q( jk) QJ( jk) Q he underexhausion. F( jk) he quaniy of he elemens o order can be calculaed if QF ( jk) Q, hen J( jk) 1 ( ) j Q νεq, j k Bε () ε j k+ 1 ν coefficien for aking he uncerainy of he order ino accoun (has o be ε calculaed by he hisoric daa), is value can be if he real exhausion is lower han he prognosic: ν < 1, if he real exhausion is higher han he prognosic: if QF ( jk) > Q, hen J( jk) ε ν > 1. ε + 1 ( ) j Q ν εqbε + Q, j k ( jk ) () ε j k+ 1 if QF ( jk) < Q, hen J( jk) he safey sock level Bmax + 1 + ( ) j Q νεqb ε Q, j k ( jk ) () ε j k+ 1 { } B 1 Max Qj + Q, Bmax () Q he faul of he evaluaion during erm k. A consignmen sores in generally he fulfillmen of he saemen > is no required (Figure 4.). 3. Example for analysis of sock levels of a consignmen sore he heoreical mehod was esed a a given produc ype of he consignmen sore of a Hungarian company named Schefenacker Auomoive Pars Ungarn. - 9 -
he sock level-ime funcion of he given produc can be seen on Figure 5. he Figure shows ha he sock level is no consan and he changing is no periodic so he funcion can no be exaed by mehods designed for deerisic funcions. If we creae he angles of exhausion from his funcion and compare hem o he heoreical values calculaed by he above menioned mehod for he analyzed erm hen we ge a new diagram can be seen on Figure 6. 1 Sock level [ps] 8 6 4 2 nap 3 31 3 4 567 8 9 1 13 2 3 31 3 4 567 8 9 1 13 2 3 3 4 567 8 9 1 2 3 31 hó777777788888888888888888888888888888889999999999999999999999999999991 ime [day] Figure 5. Sock level of he consignmen sore of he seleced produc ype Angle of exhausion 1 9 8 7 6 5 4 3 2 1 3 32 3 45 6 7 89 1 13 2 3 32 3 45 6 78 9 1 13 2 3 3 45 6 78 9 1 2 3 3 7777778888888888888888888888888888888999999999999999999999999999999 ime [day] real exhausion heoreical exhausion Figure 6. Angles of exhausion of he produc in he consignmen sore By he analysis of he real exhausion we ge real angles of he exhausion for he process which are suiable o deere a heoreical angle of he exhausion o approach he real values. If we use his heoreical value o check he real exhausion hen funcions on Figure 7. can be creaed heoreical resuls are based on hisoric daa. Quaniy [ps] 1 8 6 4 2 1 5 9 13 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 ime [day] consan imal sock limi consan uploading level calculaed sock Figure 7. Effec of he heoreical angle of he exhausion - -
Figure 8. shows he sock level of he seleced produc ype in he consignmen sore using he heoreical angles and he real angles of exhausion of he produc. 2 Quaniy [ps] 1-1 1 7 13 31 37 43 49 55 61 67 73 79 85 91 97 ime [day] ' Figure 8. Comparison of he heoreical and he real sock level ( - heoreical, - real) o exae he effecs of he heoreical exhausion he nex parameers have o be used average value of he sock levels, fulfillabiliy of he coninuous supplying, frequency of he supplying process. Comparison of he average value of he sock levels in he heoreical and he real process can be seen on Figure 9. Comparison says ha he amoun of pieces sored in sock is higher abou 2% in he heoreical siuaion han in he real process. Average value [ps] 1 9 8 7 6 5 4 3 2 1 calculaed sock level real sock level Figure 9. Average sock levels In he aspec of he coninuiy of he supplying process here is only one case when he sock level of heoreical process goes under zero, in he oher cases he mehod gives suiable resuls. In he aspec of he inensiy of he supplying process here is no significan difference beween he real and he heoreical processes. Summarizing he experiences in our example we can see ha some parameer of he socking process worse using he evaluaed exhausion han a he real siuaion, bu his mehod le us - 1 -
o design he socking process of consignmen sores giving suiable resuls. As in our example we used he average value of he angles of exhausion o calculae he evaluaed exhausion, o increase he precisiy of he resuls (approach he opimal value) we can use he disribuion of he angles of exhausion in he furher process. 4. Summary Opimizaion of he operaion of a consignmen sore is no an easy ask in general. he main problem is he sochasic characerizaion of he exhausion of he producs which makes he using of deerisic mehods impossible. In our research projec we developed a heoreical mehod which uses hisoric daa of a previous erm o deere he opimal values for he parameers of he consignmen soring process. Main advanage of he researched mehod is o give possibiliy for he designers and operaors of he consignmen soring process o find (or a leas approach) he opimal parameer values which is very difficul by any oher way. As here can be seen in our example some resuls of he mehod conains misakes. However his problem can be reduced by addiional analysis which is he par of our furher research aciviy. eferences [1] Auóipari beszállíók szerelősorai üemezésének és készleezésének logiszikai szemponú opimalizálása, ME-AKKK research repor, Miskolci Egyeem,. [2] D. CSELÉNYI J. D. ILLÉS B. (szerk.): Logiszikai rendszerek I., Miskolci Egyeemi Kiadó,. [3] D. CSELÉNYI J. D. ILLÉS B. (szerk.): Anyagáramlási rendszerek ervezése I., Miskolci Egyeemi Kiadó,. ISBN 963 661 672 8-2 -