MAGNT Researc Reort ISSN. 444-8939 Vo.3 4. PP: 47-54 Mut-Objectve Otzaton or Inventory ontro In Sera-Producton Systes wt eendent eand and Uncertanty o ead Tes Habatoa Sadeg Industra Engneerng et. Iran Unversty o Scence and Tecnoogy Teran Iran P.O. Bo.684634 Aad Maku Industra Engneerng et. Iran Unversty o Scence and Tecnoogy Teran Iran P.O. Bo.684634 Med Heydar Industra Engneerng et. Iran Unversty o Scence and Tecnoogy Teran Iran P.O. Bo.684634 Receved Feb. 5 & Pubsed onne Ar. 5 ssue Abstract: Suy annng or ut-erod sera roducton systes under ead-te uncertantes s consdered n ts aer. Te order ocy s erodc order quantty POQ and objectve s to nd te anned ead-te and erodcty or te tota tes n order to nze te eected ed orderng odng and arta backoggng costs and to aze te custoer servce eve or te nsed roduct. In ts aer assued a ercentage o tes at eac stage are scras. For ts new robe a new ateatca ode s suggested and ten ndng te ota souton or ts ode. Keywords: Sera roducton systes; anned ead te; Perodc order quantty; Uncertanty; Suy can arta backoggng. Introducton In an ndustra contet data are oten recse or uncertan. In roducton anageent t ay or nstance be te case or te deand te ead tes te resources requred ter caactes te transortaton tes te nventory or roducton costs etc. Wen anayzng te state o te art on ts subject t can be see tat te uncertanty on te deand s a great ocus o te terature we te uncertanty on costs and caactes s aso oten consdered. By otzng tes sues enterrses can save oney and ncrease te custoer satsacton. In te terature o roducton annng and nventory contro assued tat ead-te s equa to zero or constant. In reaty ead-tes are rarey constant; unredctabe events can cause rando deays. For derent reasons acne breakdowns transort deays or quaty robes etc. te te ead tes te o te devery ro an eterna suer or rocessng te or te se-nsed roduct at te revous eve are oten rando. To reduce te eectve o rando actors rs eent saety ead tes but saety ead tes are eensve. In contrast te saety ead tes are not enoug we ave stock-out and ten backoggng cost. Tere are varous aers wc consder to uncertanty ead te but ost o te used ot-or-ot orderng ocy and gnore te custoer s servce eve and n ts aer s assued ercentage o te ack s OI: d.do.org/4.983/444-8939.5/3-4/magnt.5
MAGNT Researc Reort ISSN. 444-8939 Vo.3 4. PP: 47-54 backogs and te ercentage reanng s te ost ses. Te a o ts aer s deternng annng ead-tes orderng quantty and erodc te wc su o setu cost odng cost o a tes odng cost o nsed roduct and backoggng costs nzed. Tere are severa ubcatons one eve wt ut tes asseby syste. Researcer usng derence etod or ndng ota souton suc as nze su o tota costs. In [] consder to ts ode by usng ot-or-ot order oce. Te objectve s to deterne saety ead-te suc as nze su o odng costs. In [] used Markovan ode or a dynaca ut-erod annng and one eve asseby syste and te a was ndng saety ead te suc as nze te su o odng and backoggng costs and [3] were usng suated anneang or ndng saety ead te. In [4] consder to one eve assebe syste agan and used a Branc and Bound etod or ndng ota anned ead te wt ot-or-ot order oce. Te a was to nd te ota MRP osettng suc as nze te su o te setu cost and average odng costs or te tes we satsyng a desred servce eve. Te recent researc or ts robe s dong by [5] wc consder to a uterod sera roducton syste wen ead tes or a tes are uncertantes but don t consder to arta back-order and custoer servce eve.. Notaton and assutons In ts researc we consder ut-stage Pannng suy tes wt deendent deand wt uncertanty ead-te and arta back-order wt POQ order ocy. Te assuton o ts ode s as oow: A ercentage o tes at eac stage are west Te deand at eac erod s constant Orderng ocy s POQ ead te or eac stage s uncertanty Sortages s aow and t s wt arta backoggng Te raw ateras are reeased at stage N te se-nsed roducts are rocessed at stage n n and nay te nsed roduct s roduced. Te objectve s to nd te te anned ead te at eac stage and erodcty P or nzng te su o te odng costs or te tes o eac eve and back-orderng cost ost sae cost odng cost or te nsed roduct and aze custoer servce eve. I. Notaton t : Inde o erod's t= 3 A : orderng cost er order : eand or nsed roduct : Te erods or eac orderng : Per unt odng cost er te unt : Backoggng costs or eac stage n eac erod : ost Saes or eac te n eac erod s : Mau unt Backorderng b : Mau unt backoggng : Percentage o Backorderng wc s Backoggng : resent o west or stage. : Rando ead te or stage. : Servce eve N : Tota ead-te o te syste. : Panned ead te or stage. N : Su o anned ead-te. : strbuton o ead-te n stage. : onvouton o ead-te. : Producton cost er te at stage. Varabes P: erodcty X: anned ead te 3. Mode deveoent Te ead-te s assued robabstc. Te anned ead-te s or stage. Te orders OI: d.do.org/4.983/444-8939.5/3-4/magnt.5
MAGNT Researc Reort ISSN. 444-8939 Vo.3 4. PP: 47-54 or roducts are ade at te begnnng o te erods + + and tere s no order ade n oter erods. Order quantty s constant and equa to P P s a decson varabe Te objectve n ts syste s to nd te anned ead-te and erodcty or te tota tes n order to nze te eected costs and aze te custoer servce. osts are odng costs or te a tes o eac stage and back-orderng cost ost sae cost odng cost or nsed roduct and roducton/assebe cost. 3.. Te syste costs I. Producton/Assebe costs In ts syste at eac stage a ercentage o tes are scra tereore accordng to te Fg. te tota nuber o te n eac stage s equa to: Q j j Ten roducton/assebe costs s Q j j j j State : Ts state as not back-order cost and te costs or ts stat s equa to: A A Were t s te convouton o ead-te. 3 State : I at a stage a job s not receved as anned a deay s ncurred at devery o roducton and t s aybe ncurred backorderng at nsed roduct. In ts case t s assued ercentage o te ack s backogs and te ercentage reanng s te ost ses. A 3 P State 3: 4 4 A P 5 II. Fna roduct costs Because o uncertany ead-te tere are tree states n acton. One o te s anned ead te equa to actua ead te Fg. and te second state s anned ead-te s saer tan actua ead-te Fg. 3 and te trd state s anned ead te s bgger tan te actua ead te Fg. 4 Te cost or tese stats are as oow: OI: d.do.org/4.983/444-8939.5/3-4/magnt.5
MAGNT Researc Reort ISSN. 444-8939 Vo.3 4. PP: 47-54 Raw Matera Q - Q3 - Q- Fna roduct custoer waste waste waste waste Stage Stage - Stage - Stage Fg. : Te -stage anuacturng syste na P- INVENTORY P- P- P- Fg. : anned ead te equa to actua ead te PERIO INVENTORY P- P- ß b P- PERIO -ß b oss se oss se Fg. 3: An ustraton o te annng robe wen anned ead-te s saer tan actua ead-te OI: d.do.org/4.983/444-8939.5/3-4/magnt.5
MAGNT Researc Reort ISSN. 444-8939 Vo.3 4. PP: 47-54 INVENTORY P- P- P- k k PERIO Fg. 4: An ustraton o te annng robe wen anned ead-te s bgger tan te actua ead-te Wt suarze te Eq Eq3 Eq4 and Eq5 ave: 3 4 A j j P P P P Ect ors o te tota cost as been resented as oow: A j j P P P P 6 7 An ect or or te stock-out robabty s s te oowng: s 8 N N Were and. Te robe s to nze te tota costs gven by 7 we azng te servce eve.e. nzng te stock out robabty gven by 8 Mnze Mnze s For sovng te ode we soud deterne te dstrbuton o. We assue tat eadtes dstrbuton are ndeendent. Tereore dstrbuton o w be cacuated sy by usng convouton concet. For ore detas ease reer to [5]. 3.. Ota souton roertes Te na ode can be wrtten as oows: OI: d.do.org/4.983/444-8939.5/3-4/magnt.5
MAGNT Researc Reort ISSN. 444-8939 Vo.3 4. PP: 47-54 OI: d.do.org/4.983/444-8939.5/3-4/magnt.5 P P P P A j j Mn Mn 9 Subject to Were N and N. Sovng te ode Te uncton o s conve. Tereore we can nd te ota souton or t. By reacng ts souton on te constrant saer or equa to ten ts answer s ota souton but s not true ten we reace wt on and nd te ota souton. Teore : Te objectve uncton n Eq. 9 s conve. d d d d Eq s ostve ten objectve uncton s strcty conve. Te tota cost equaton s conve tereore or ndng e ota souton used..5 d And te strbuton o ead-te n stage s dscrete we can use and or ota anned ead-te. By used we ave: E E E 3 By sy Eq3 we ave: F F F. 5 4 3. 4 3 4 3 3 F F 4 For ndng erodc order quantty araeter we resectvey reace te vaues
MAGNT Researc Reort ISSN. 444-8939 Vo.3 4. PP: 47-54 nstead o at te cost uncton and obtan tota cost by Eq. 9 and ota vaue o. Noted tat s actua ead-te or one erod but we consder to ut erod odeng ten anned ead te s dened or tes. Tereore anned ead te or one erod s equa to erod. wc 4. Nuerca eae s anned ead-te or onsder ve stages Pannng suy tes wt deendent deand wt oowng araeters 5 A 5.8 3 4 5.5 and. strbuton o ead-te at eac stage s as oow: Tabe : dstrbuton o ead-te at eac stage.5..5.4..3..3.. 3..3...3 Souton: Accordng to Eq. 4 ave: 4.5..5.5.5 F F 5.5...5.5 Accordng to strbuton o ead-tes te nu o N s equa to 5 and te au o t equa to 5. Tereore convouton o ead-te wt dscrete dstrbuton s as oow: F5.6; F8.47; F.6955; F4.63467; F6.4; F9.8855; F.38565; F5.748945; F7.95356; F8.9499; F.98953; F.99598695; F.998673; F3.999637575; F4.9999635; Ste : =; F7.547; F.63575; F3.58; F6.83893; F9.9753488; F5 6 54.4848 F F * 6 Ste : =; F F * 4 4 89.494 Ste 3: =3; F F.5 * 3 33 76.67 Ste 4: =4; F F * 3 34 7.699 Ste 5: =5; F F.5 * 5 68.668 Ste 6: =6; F F 3 * 6 67.5983 Ste 6: =7; F F 3.5 * 6 7.58 ost reduce or = unt =5 and ten cost s ncreased Tereore ota souton wtout consderng to custoer servce eve s equa to * * 5 * 5 6 * 5 55.7 And te custoer s servce eve s equa to: F F.38565 Tereore custoer s servce eve s saer tan.38 ts answer s te ota but custoer s servce eve s bgger tan. 38 ten we reace wt on and ndng te ota souton. Tabe reresent te ota souton or varous custoer servce eve. OI: d.do.org/4.983/444-8939.5/3-4/magnt.5
MAGNT Researc Reort ISSN. 444-8939 Vo.3 4. PP: 47-54 Tabe : Te ota souton or varous custoer servce eve. Servce eve Ota anned ead te * stock-out robabty Tota cost % 6.69435 67.5983 3% 6.69435 67.5983 38% 6.69435 67.5983 4% 3 5.48999 7.6784 5% 3 5.48999 7.6784 6% 4 5.36533 77.9776 7% 5 5.555 84.4968 8% 6 5.6869 9.75 85% 7 5.946494.598 9% 7 5.946494.598 95% 9 5.465 9.37 98% 5.7468 9.6 % 5 5 78.8778 5. oncuson In ts aer we consder to ut-stage annng suy tes wt deendent deand wt stocastc te s ead-tes. We use new etod or odeng ts ode. Te an goa n ts ode s nze su o nventory odng cost back-order cost and setu cost n order to nd te otze souton o te anned ead tes and order erodcty or asseby systes. A MRP aroac wt erodc order quantty POQ ocy s used or te suy annng o tes. We consder a new aroac or odeng ut-stage Pannng suy tes wt deendent deand under uncertanty te s ead-tes. We roved tat te cost uncton s conve and ten usng te conve concets or ndng ota soutons. By usng o ts equaton we can nd te ota souton or ts ode. Econocs vo. no.. 4 47 9. [4]. A. A.-A. A. Oud ouy M.A. "Ota MRP osettng or asseby systes wt stocastc ead tes: POQ ocy and servce eve constrant" Journa o Integent Manuacturng vo. 3 no. 6. 485-495 eceber. [5] H. Sadeg A. Maku and M. Heydar "eternng te erodcty and anned ead te n sera-roducton systes wt deendent deand and uncertan ead te" Uncertan Suy an Manageent vo. no.. 87-98 3. Reerences []. P. J. a. X. X. u "Suy anageent n asseby systes" Nava Researc ogstcs vo. 4 no. 7. 933 949 993. [] M. a.. A. Oud-oudy "Te MPS araeterzaton under ead-te uncertanty" Internatona Journa o Producton Econocs vo. 9 no. 3. 369 376 4. [3]. A. P. J. auan S.S. "A contnuous ode or suy annng o asseby systes wt stocastc te rocureent tes" Internatona Journa o Producton OI: d.do.org/4.983/444-8939.5/3-4/magnt.5