Uncertain Supply Chain Management

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1 Ucerta Supply Cha Maagemet 3 (5) Cotets lsts avalable at GrowgScece Ucerta Supply Cha Maagemet homepage: Modelg of a vetory system wth mult varate demad uder volume flexblty ad learg Surbh Sghal a* ad S.R. Sgh b a Departmet of Statstcs, Vardhma College, Bjor, Ida b Departmet of Mathematcs, D.N. College, Meerut, Ida C H R O N I C L E A B S T R A C T Artcle hstory: Receved July 8, 4 Accepted December 8 4 Avalable ole December 4 Keywords: Learg Volume flexblty Stochastc backorder Mult-varate demad I ths study, a volume flexble vetory system for deteroratg tems wth stock & tme depedet demad has bee developed over a fte plag horzo. Shortages are permtted wth partal backorder. Ucertates are heret real vetory problems due to complextes of market stuato. Ths ucertaty ca be hadled by the cocept of radomess. As a result, backorder rate s take as radom ad follows a probablty dstrbuto. All the costs are flueced by the learg effect. The optmal umber of producto cycles that mmze the total cost s cosdered. Numercal llustratos together wth sestvty aalyss are gve to elucdate the model. Furthermore, the umercal results of the fte plag horzo model have bee plotted graphcally. 5 Growg Scece Ltd. All rghts reserved.. Itroducto Producto s a orgazed actvty of covertg raw materals to useful products. Ths actvty takes place a wde rage of maufacturg ad servce sectors. Producto system requres the optmal utlzato of atural resources lke labor, moey, mache, materals ad tme. Thus, t s essetal that before startg the work of actual producto, producto plag s doe order to atcpate possble dffcultes ad to decde advace as to how the producto should be carred out a best ad ecoomcal way. I geeral, the Ecoomc Producto Quatty models are formulated wth costat producto. I real lfe, t may ot be so. I the chagg market scearo, flexblty s recogzed as a mportat feature maufacturg ad volume flexblty s gettg pheomeal mportace amogst the researchers. Volume flexblty permts a maufacturg system to adjust producto upwards or dowwards wth wde lmts pror to the start of producto of a lot. The effect of learg from repettve process caot be gored whle developg the vetory model. Learg suggests that the performace of a perso or a orgazato egaged a repettve task mproves wth tme. Ths mprovemet s represeted as a decrease the cost of the product, but f the * Correspodg author E-mal address: sghalsurbh6@gmal.com (S. Sghal) 5 Growg Scece Ltd. All rghts reserved. do:.567/j.uscm.4..6

2 48 savgs due to learg are sgfcat, the effect o producto tme ad hece vetory should also be sgfcat. Factors cotrbutg to ths mproved performace clude more effectve use of tools ad maches, creased famlarty wth operatoal tasks, the work evromet ad ehaced maagemet effcecy. There s almost uamous agreemet amog practtoers ad academcas that the learg curve s best descrbed by a power as suggested by Wrght (936). It s worth otg that the learg curve practce s a S -shaped curve (Jordo, 958; Carlso, 973). The theory ts most popular form states that as the total quatty of uts produced becomes double, the cost per ut decles by some costat percetage (e.g., Yelle, 979; Jaber, 6). The form of the learg curve has bee debate by Jaber (6). Zagwll (966) dscussed a producto schedulg model wth partal backloggg. They have take the costat demad ad backloggg rate. Holler ad Mak (983) developed vetory repleshmet polces for deteroratg tems wth egatve expoetally demad ad costat rate of deterorato. Wee (999) cosdered a vetory model for deteroratg tems wth quatty dscout, prcg ad partal backordered. Wu () formulated a order level vetory model for decayg tems wth tme depedet demad ad shortages were allowed wth partal backloggg. Teg et al. () developed a optmal repleshmet polcy for costat deteroratg tems wth tme-varyg demad ad partal backloggg. Akse et al. (3) cosdered the sgle tem lot szg vetory model wth the effect of lost sales. Saa et al. (4) cosdered a producto vetory model for deteroratg tems wth treded demad. They allowed shortages wth complete backloggg ad producto rate was take as costat. Ouyag et al. (5) preseted a order level vetory model for deteroratg tems wth expoetally decreasg demad ad partally backlogged. The backloggg rate was take as tme depedet ther model. Ouyag et al. (6) developed a optmal orderg polcy for deteroratg tems wth partal backloggg. Uthayakumar ad Parvath (6) dscussed a determstc vetory model for deteroratg tems wth stock ad tme depedet demad ad partally backlogged. Dye (7) proposed jot prcg ad orderg polcy for deteroratg tems wth costat partal backloggg rate. Sgh ad Sgh (8) developed a optmal orderg polcy for decayg tem wth stock depedet demad. Sgh et al. (8) developed a vetory model for deteroratg tems havg stock depedet demad. They were allowed shortages wth partal backloggg ther study. Arya et al. (9) dscussed a vetory system for pershable tems wth stock depedet demad ad tme depedet partal backloggg. I ther model, costat holdg cost has bee take. Sgh et al. () developed a volume flexble vetory model for defectve Items wth mult-varate demad ad partal backloggg. Sgh et al. () studed a ecoomc producto lot-sze (EPLS) model wth rework ad flexblty uder allowable shortages. Sgh et al. (3) developed a supply cha vetory model for shortages wth varable demad rate. Kumar et al. (3) preseted two-warehouse vetory model wth K-release rule ad learg effect. Sghal ad Sgh (3) developed volume flexble mult tems vetory system wth mprecse evromet. I ths model, volume flexble system for decayg tems wth stock ad tme depedet demad over a fte plag horzo has bee developed. Our study cludes the stuato of shortages wth partal backloggg, where backloggg rate depeds upo stochastc evromet. The combato of more tha oe parameter grats more geueess to the formulato of the model ad makes t more close to realty. We have dscussed the learg effect o all cost. Numercal examples are preseted to llustrate the theoretcal results. The sestvty of the optmal solutos wth respect to system parameters s examed. Graphcal aalyss also has bee dscussed. The proposed model has a broad area of applcablty.

3 . Assumptos ad Notatos S. Sghal ad S.R. Sgh / Ucerta Supply Cha Maagemet 3 (5) 49 The proposed vetory model s developed uder the followg assumptos ad otatos: Assumptos The followg assumptos are as follows:. Demad rate s take as both tme ad stock depedet.. The ut producto cost s a fucto of producto rate. 3. The rate of producto s cosdered to be decso varable. 4. Shortages are allowed wth partal backloggg. 5. Backloggg rate s radom varable ad follows a beta dstrbuto of frst kd. 6. Deterorato rate s take as costat. 7. All the costs are take wth the effect of learg. 8. Tme horzo s fte. 9. The fte tme horzo s dvded to a fte umber of repleshmet cycles, each of equal durato. Notatos The followg otatos are used our study: abt ci() t Tme ad stock depedet demad, a, b>, <c< P Producto rate Deterorato rate, C d Deterorato cost per ut per ut tme H Fte tme horzo ( P) Ut producto cost of a tem ad ( ) G, P N P where P N s materal cost, s tool or de cost ad G s eergy ad labor cost The fracto of the demad durg the stock-out perod that wll be backordered ad a radom varable,. M g( ) d : g( ) The probablty desty fucto (p.d.f.) of ad follows Beta dstrbuto of frst kd ( ),,, g( ) B(, ), otherwse Number of cycles [,H] The learg effect s very much mportat; therefore ths model we studed the effect of learg. The earlest learg curve represetato s a geometrc progresso that expresses the decreasg cost requred to accomplsh ay repettve operato. Several learg curve models were ftted to the collected data ad the S-shaped logstc learg curve was foud to ft well ad t s of the form g R ( ) m, where m ad g > are the model parameters, s the cumulatve umber of shpmets, ad R() s the percetage defectve per shpmet.

4 5 From the Fg. : the frst phase (cpet) s the phase durg whch the worker s gettg acquated wth the set-up, the toolg, structo, blueprts, the workplace arragemet ad the codtos of the process. I ths phase mprovemet s slow. The secod phase (learg) s where most of the mprovemet, e.g., reducto errors, chages the dstace moved takes place. The thrd ad last phase (maturty) represets the learg of the curve. Hours per Ut Output Icpet Learg Maturty Phase- Phase- Phase-3 Uts Fg.. The three phases of learg curve The vetory carryg cost, backorder cost, lost sale cost, set up cost also follows the learg effect C C s C LS C SP ad fucto of these cost are C, Cs, CLS, CSP. 3. Model Formulato I ths model, volume flexble vetory system wth the effect of learg has bee developed. Ths vetory system cosdered four phases each cycle. I th cycle (=,,., ), the tal vetory s zero ad producto starts at the very begg of the cycle. As producto cotues, vetory begs to ple up cotuously after meetg demad ad deterorato. At tme t, producto stops. The accumulated vetory s just suffcet eough to accout for demad ad deterorato over the terval[ t, t ]. After tme t, shortage starts wth partal backloggg ad reach to maxmum shortage level at tme S. Producto restarts after S to fulfll the backlog demad ad the cycle eds wth zero vetory. Ivetory S T T- t t Fg.. Tme versus vetory of th cycle Tme horzo The vetory level I(t) of the system at ay tme t [ T, T] s descrbed by the followg equatos:

5 S. Sghal ad S.R. Sgh / Ucerta Supply Cha Maagemet 3 (5) 5 I () t I() t R( ) P( abt ci()) t T t t I() t It () ( abt cit ()) t t t () I () t ( a bt) t t S (3) I () t R( ) P( a bt) S t T (4) Wth boudary codtos () IT ( ), It ( ), IT ( ) (5) Soluto of Eq. () s gve by: ( RP ( ) a) ( c)( T t) b ( c)( T t) It () [ e ] [{( ct ) } {( ct ) } e ] ( c) ( c) Soluto of Eq. () s gve by: T t t ( c) ( ct ) a ( ct ) ( c) t b ( ct ) ( c) t Ite () It ( ) e [ e e ] [{( ct ) } {( ) } ] e ct e ( c) ( c) From Eq. (6) substtute the value of ( ) It Eq. (7), ths relato becomes: (6) (7) RP ( ) a b It () [ ( e e ) e (( ct ) ) e ] e ( c) ( c) ( c) a b(( c) t) [ ] t t t ( c) ( c) ( ct ) ( ct ) ( ct ) ( ct ) ( ct ) (8) Usg the codtos It ( ) Eq. (8), oe ca have ( crh ) bh a K l[ { a ( r )} e ( crh ) RP ( ) ( ch ) RP ( ) bh ( )] R( ) P ( c) H (9) Soluto of Eq. (3) s gve by: b It () [( tt){ a ( t t)}] t t S () Soluto of Eq. (4) s gve by: b It () IS ( ) ( RP ( ) a)( ts) ( t S ) S t T Substtutg the value of IS ( ) from Eq. (), ths relato becomes: b b It () ( RP ( ) a)( ts) ( t S ) a ( S t) ( S t ) S t T Usg the codtos IT ( ) Eq. (), oe ca have () ()

6 5 a bh( r) d R( P ) RP ( ) Usg all the values of t, t, S, T from Appedx, the followg costs are as follows: Holdg cost occurs durg the terval [ T, t] s gve by: t C C ( C ) I( t) dt h C b( ) rh K brh K( rk ) b( ) rh ( K) brh( K) T ( C )[ ( c ) ( c ) ( c ) brh ( K)( r rk ) brh ( K) ] ( c) ( c) Deterorato cost occurs durg the terval [ T, t ] s gve by: t CD Cd I() t dt T b( ) rh K brh K( rk ) b( ) rh ( K) brh ( K) Cd[ ( c) ( c) ( c) brh ( K)( r rk ) brh ( K) ] ( c) ( c) Shortage cost occurs durg the terval [ t, T ] s gve by: C C C I t dt S T s ( )[ ( )] t s C ah d ( r) bh { ( d )( r)} bh { ( d )( r)}( r) bh ( r) H { ( d )( )} { ( )( )} 3 ] ( )[ r H d r H Pa ] b H H { ( d)( r)} H { ( d)( r)} [ ] H { ( d)( r)} H { ( d)( r)} H ( r) a [ 3 3 H { ( d)( r)}( r) b H { ( d)( r)} H ( r) ] [ H { ( d)( r)} H { ( d)( r)}( r) 3 3 ]] Lost sale cost occurs durg the terval [ t, S ] s gve by: s ( Cs )[ [ 3 3 S CLS CLS ( CLS )[( )( abt)] dt t (7) CLS ahd ( ) ( )( ) r bh d r d r rd ( CLS )( )[ ] Producto cost occurs durg the terval [ T, t ]& [ S, T ] s gve by: G t T PC ( N RP ( ) )[ RPdt ( ) RPdt ( ) ] RP ( ) T S (8) H r ( NRP ( ) G RP ( ) ) [ ( d )( r)] Set up cost occurs durg the terval [ T, T] s gve by: (3) (4) (5) (6)

7 S. Sghal ad S.R. Sgh / Ucerta Supply Cha Maagemet 3 (5) 53 T CSP CSP H (9) AS [ ( CSP ) dt] ( C ) T SP The total cost s the sum of holdg cost, deterorato cost, shortage cost, lost sale cost, producto cost ad set up cost. The average total cost durg the tme horzo (, H) usg the equatos from Eq. (4) to Eq. (9) s gve by: () Avc() r [ C C C C P A ] H h D S LS C S Sce the backorder rate s a radom varable wth p.d.f. g( ), the expected backorder rate s M g( ) d. Thus, the expected average total cost durg the tme horzo (,H) s gve by: EAP() r E[ Avc()] r () Ths s the objectve fucto whch eeds to be mmzed. It s a fucto of servce level r. For optmzg the expected average total cost EAP(r) () r The Eq. () s solved for dfferet values of servce level r ad the equato () s solved to fd the values of total cost. These both equatos are solved usg the software for a fxed plag horzo H. 4. Numercal Illustratos The umercal examples are gve below to llustrate the above soluto procedure. O the bass of prevous studes, let us cosdered the followg data proper uts: C S =5.5, d =., C SP =5, SP C LS =8.5, =.5, a=5, b=6, c=.7, H=, =, G=35, N=, =., M =.75, d =.8, K =.8, K =., C =5.5, =.5, P=3, =.5, C =7.5, C S =5.5, C LS =6.7, m=6, g=4, C = Table Varato Cycles No. of Cycles Servce level r Total cost Ivetory cost Producto cost The optmum values are: Servce level r =.99593, Total cost=444 ad Producto cost=3573. The graphcal represetato of the optmum values for = has bee show by Fg. 3. The graphcal represetato of the servce level r ad o. of cycles s show by Fg. 4 ad the graphcal represetato of the total cost ad o. of cycles s show by Fg. 5.

8 54 Total cost P..4 8 r Fg. 3. Graphcal represetato of the system Varato Servce level r w.r.t. No. of cycles Varato Total cost w.r.t. No. of cycles Servce level r Total cost No. of cycles 9 No. of cycles 9 Fg. 4. Graphcal represetato of Servce level r w.r.t. No. of cycles Fg. 5. Graphcal represetato of Total cost w.r.t. No. of cycles 5. Sestvty Aalyss I ths secto, the effects of chages the system parameters a, b, c, H, P ad M o the values of r, total cost ad the producto cost has bee studed. The results are preseted Table. Table Effect of percetage chage system parameters of the vetory model Parameter % Chage -5% -% -5% 5% % 5% Servce level r a Total cost Producto cost b c H P M Servce level r Total cost Producto cost Servce level r Total cost Producto cost Servce level r Total cost Producto cost Servce level r Total cost Producto cost Servce level r Total cost Producto cost......

9 S. Sghal ad S.R. Sgh / Ucerta Supply Cha Maagemet 3 (5) Observatos Servce level r s very slghtly sestve to chage the parameters of demad ( a, b ad c ). The total cost s somewhat sestve to chage the demad parameters ( a, b ad c ). 3 The total cost s decreases wth the creases of the values of demad parameters ( b ad c ). The chage values of demad parameters (a, b ad c) do t have ay effect o the producto cost. 4 The servce level r ad total cost are fatly sestve to chage the parameter of plag horzo. 5 Producto cost s hghly sestve to chage the parameter of plag horzo. 6 The total cost ad producto cost are extremely sestve ad r s slghtly sestve to chage the parameter of producto rate. 7 Servce level r ad total cost are lttle sestve to chage the backloggg parameter. The values of backloggg rate do t gve the effect o the producto cost. All the varatos cted above have bee show graphcally Fgs. (6-)..5.5 % % % % % % %. % % % r Total cost r Total cost Fg. 6. Graphcal represetato of sestvty of the r ad total cost w.r.t. a Fg. 7. Graphcal represetato of sestvty of the r ad total cost w.r.t. b.6.4. % % % % % % % % 5 % % 5 r Total cost r Total cost Producto cost Fg. 8. Graphcal represetato of sestvty of the r ad total cost w.r.t. c Fg. 9. Graphcal represetato of sestvty of the r, total cost ad producto cost w.r.t. H

10 % % % 5 % % % % % % %.5.5 r Total cost Producto cost r Total cost Fg.. Graphcal represetato of sestvty of the r, total cost ad producto cost w.r.t. P Fg.. Graphcal represetato of sestvty of the r ad total cost w.r.t. M 7. Cocluso I ths paper, a vetory model for deteroratg tems wth volume flexblty ad stock & tme depedet demad has bee developed. Large quattes of goods dsplayed market accordg to seasos lure the customer to buy more. I fact a customer s demad s flueced by more tha oe parameter. It s very realstc to cosder the practcal demad rate, whch depeds upo both tme ad stock. It s a well kow fact that there are very much ucertates real lfe busess scearos wth respect to lost sales. Therefore, t s worthwhle to cosder the backorder rate stochastc ature. The proposed model s very useful the preset market stuato as almost every tem havg a demad rate varyg accordg to tme ad stock avalable ca be detfed. Ths whole setup s very practcal ad ca be appled to may commodtes today s market. All these facts together make ths study very uque ad matter-of-fact. Appedx Here t, t, S ad T are coected by the followg relatos: t Kt ( K) T, t rt ( r) T H S dt ( d) t, T ad H( rk ) H( r) t, t H rh S { ( d)( r)}, T t Hd ( r) S t, S t H [ d r rd] r, K,d ad =,

11 S. Sghal ad S.R. Sgh / Ucerta Supply Cha Maagemet 3 (5) 57 Refereces Akse, D., Altıkemer, K., & Chad, S. (3). The sgle-tem lot-szg problem wth mmedate lost sales. Europea Joural of Operatoal Research,47(3), Arya, R. K., Sgh, S. R., & Shakya, S. K. (9). A order level vetory model for pershable tems wth stock depedet demad ad partal backloggg. Iteratoal Joural of Computatoal ad Appled Mathematcs, 4(), 9 8. Carlso, J.G.H. (973). Cubc learg curve: Precesso tool for labour estmatg. Maufacturg Egeerg ad Maagemet, 7(5), -5. Dye, C.Y. (7). Jot prcg ad orderg polcy for a deteroratg vetory wth partal backloggg. Omega, 35(), Holler, R.H., & Mak, K.L. (983). Ivetory repleshmet polces for deteroratg tems a declg market. The Iteratoal Joural of Producto Research,, Jaber, M.Y. (6). Learg ad forgettg models ad ther applcatos. I: Badru, A.B. (Ed.). Hadbook of Idustral ad Systems Egeerg, CRC Press, Boca Rato, FL, pp Jorda, R.B. (958). Learg how to use the learg curve. N.A.A Bullet, 39(5), Kumar, N., Sgh, S. R., & Kumar, R. (3). Two-warehouse vetory model wth K-release rule ad learg effect. Iteratoal Joural of Procuremet Maagemet, 6(), Ouyag, L.Y., Wu, K.S., & Cheg, M.C. (5). A vetory model for deteroratg tems wth expoetal declg demad ad partal backloggg. Yugoslav Joural of Operatos Research, 5(), Ouyag, L.Y., Teg, J.T., & Che, L.H. (6). Optmal orderg polcy for deteroratg tems wth partal backloggg uder permssble delay paymets. Joural of Global Optmzato, 34(), Saa, S., Goyal, S. K., & Chaudhur, K. S. (4). A producto vetory model for a deteroratg tem wth treded demad ad shortages. Europea Joural of Operatoal Research, 57(), Sgh, S.R., Sgh, C. (8). Optmal orderg polcy for decayg tem wth stock depedet demad uder flato a supply cha. Iteratoal Revew of Pure ad Appled Mathematcs,, Sgh, S.R. et al. (8). A orderg polcy for pershable tems havg stock depedet demad wth partal backloggg ad flato. Iteratoal Joural of Mathematcs, Computer Scece ad Techology, (-), Sgh, S. R., Sghal, S., & Gupta, P. K. (). A volume flexble vetory model for defectve tems wth mult-varate demad ad partal backloggg. Iteratoal Joural of Operatos Research ad Optmzato, (4), Sgh, N., Vash, B., & Sgh, S. R. (). A ecoomc producto lot-sze (EPLS) model wth rework ad flexblty uder allowable shortages. Iteratoal Joural of Procuremet Maagemet, 5(), 4-. Sgh, S., Gupta, V., & Gupta, P. (3). Three stage supply cha model wth two warehouse, mperfect producto, varable demad rate ad flato. Iteratoal Joural of Idustral Egeerg Computatos, 4(), 8-9. Sghal, S., & Sgh, S.R. (3). Volume flexble mult tems vetory system wth mprecse evromet. Iteratoal Joural of Idustral Egeerg Computatos, 4(4), Teg, J. T., Chag, H. J., Dye, C. Y., & Hug, C. H. (). A optmal repleshmet polcy for deteroratg tems wth tme-varyg demad ad partal backloggg. Operatos Research Letters, 3(6), Uthayakumar, R. ad Parvath, P. (6). A determstc vetory model for deteroratg tems wth partally backlogged ad stock ad tme depedet demad uder trade credt. Iteratoal Joural of Soft computg, (3) Wee, H. M. (999). Deteroratg vetory model wth quatty dscout, prcg ad partal backorderg. Iteratoal Joural of Producto Ecoomcs, 59(), Wrght, T. (936). Factors affectg the cost of arplaes. Joural of Aeroautcal Scece, 3, -8.

12 58 Wu, K. S. (). A EOQ vetory model for tems wth Webull dstrbuto deterorato, ramp type demad rate ad partal backloggg. Producto Plag & Cotrol, (8), Yelle, L.E. (979). The learg curve: Hstorcal revew ad comprehesve survey. Decso Sceces, (), Zagwll, W.I. (966). A determstc mult-perod producto schedulg model wth backloggg. Maagemet Scece, 3, 5-9.