A HEURISTIC SOLUTION OF MULTI-ITEM SINGLE LEVEL CAPACITATED DYNAMIC LOT-SIZING PROBLEM

Transcription

1 A eurstc Soluton of Mult-Item Sngle Level Capactated Dynamc Lot-Szng Problem A EUISTIC SOLUTIO OF MULTI-ITEM SIGLE LEVEL CAPACITATED DYAMIC LOT-SIZIG POBLEM Sultana Parveen Department of Industral and Producton Engneerng Bangladesh Unversty of Engneerng and Technology, Dhaka 000, Bangladesh AFM Anwarul aque ashah Unversty of Engneerng and Technology, ashah, Bangladesh Abstract: The mult-tem sngle level capactated dynamc lot-szng problem conssts of schedulng tems over a horzon of T perods The obectve s to mnmze the sum of setup and nventory holdng costs over the horzon subect to a constrant on total capacty n each perod o backloggng s allowed Only one machne s avalable wth a fxed capacty n each perod In case of a sngle tem producton, an optmal soluton algorthm exsts But for mult-tem problems, optmal soluton algorthms are not avalable It has been proved that even the two-tem problem wth constant capacty s P (nondetermnstc polynomal)-hard That s, t s n a class of problems that are extremely dffcult to solve n a reasonable amount of tme Ths has called for searchng good heurstc solutons For a mult-tem problem, t would be more realstc to consder an upper lmt on the lot-sze per setup for each tem and ths could be a very mportant parameter from practcal pont of vew The current research work has been drected toward the development of a model for mult-tem problem consderng ths parameter Based on the model a program has been executed and feasble solutons have been obtaned Keywords: eurstcs, nventory, lot-szng, mult-tem, schedulng ITODUCTIO The mult-tem sngle level capactated dynamc lotszng problem conssts of schedulng tems over a horzon of T perods Demands are gven and should be satsfed wthout backloggng The obectve s to mnmze the sum of setup costs and nventory holdng costs over the horzon subect to a constrant on total capacty n each perod Mathematcally, the problem can be stated as: Mnmze Subect to I = I, = k x C, 0 x I Z ( X ) = ( S δ ( x + x D = = ) + h I ) =, 2,, and =, 2,, =, 2,, =, 2,, and =, 2,, where, = the number of tems, = the tme horzon, D = the gven demand for tem n perod, I = the nventory of tem at the end of perod, x = the lot-sze of tem n perod, S = the setup cost for tem, h = the unt holdng cost for tem, k = the capacty absorpton rate for tem, C = the capacty n perod and δ ( x ) s a bnary setup varable ndcatng whether a setup cost must be ncurred for tem n perod or not δ ( x ) f x > 0 and 0 f x = 0 A mult-tem, mult-echelon nventory problem, wth stochastc varables s extremely dffcult to solve n a realstc tme perod, whch leads to P (nondetermnstc polynomal) -hardness, qute smlar to schedulng problem ence, t appears hghly unlkely that an effcent optmal algorthm wll ever be developed So the search for a good heurstc method s defntely warranted As a consequence, many heurstcs were developed for ths problem Esenhut s procedure 2 could be called perod-byperod heurstc s procedure was later extended by many, ncludng Dxon and Slver 3 Basc assumptons of = the Dxon-Slver model are: () the requrements for each product are known perod by perod, () for each product there s a fxed setup cost ncurred each tme producton takes place, () unt producton and holdng costs are lnear, (v) the tme requred to set up the machne s neglgble, (v) all costs and producton rates can vary from product to product but not wth respect to tme, and (v) n each perod there s a fnte amount of machne tme avalable that can vary from perod to perod The obectve s to determne lot-szes so that () costs are mnmzed, () no backloggng occurs, and () capacty s not exceeded It would be more realstc to assume an upper lmt, a maxmum value of the lot-sze from a machne Ths restrcton may be mposed per setup and ths could be a very mportant parameter from practcal pont of vew for several reasons Stuatons lke () machne s nablty to run contnuously, and (2) machne may not be avalable for ndefnte perod for a partcular product, (3) there may be storage lmtaton for WIP nventory can be consdered n ths regard The current research work has thus been drected toward an extenson of the Dxon-Slver model consderng the above mentoned stuaton It s to be noted that Dxon-Slver heurstc allows only one setup for each tem n each perod But the lmtaton on lot-sze may need more than one setup n a partcular perod So should ths lmtaton be ncorporated nto Dxon-Slver heurstc, each tme an tem processed n a new setup s to be consdered a new tem Ths may call for splttng an tem nto several new tems n a partcular perod owever, the maxmum number of the new spltted tems wll be restrcted by the maxmum perodcal demand of the tem Let the maxmum perodc demand and the lmted lot-sze for the th tem be d max and x max, respectvely Then the number of new tems for the th tem wll be n = dmax / xmax Thus the total number of new tems wll be = n So after meetng the lot-sze lmtaton, the total number of tems to be consdered n the model should be = n + = Journal of Mechancal Engneerng, vol ME38, Dec 2007 Transacton of the Mech Eng Dv, The Insttuton of Engneers, Bangladesh

2 A eurstc Soluton of Mult-Item Sngle Level Capactated Dynamc Lot-Szng Problem 2 In vew of the above dscussons, the model may now be presented as follows Mathematcal Model: Mnmze Subect to I I + x =, I 0 = I = 0 Z ( X ) = D = = ( S δ ( x ) + h I ) =, 2,, and =, 2,, =, 2,, k x C =,, = 0 x x max =, 2,, and =, 2,, I 0 =, 2,, and =, 2,, The unt producton cost s assumed to be constant for each tem Therefore, the total producton cost (excludng setup costs) wll be a constant and hence s not ncluded n the model If ntal nventory exsts, or f postve endng nventory s desred, then the net requrements should be determned That s, use the ntal nventory to satsfy as much demand as possble n the frst few perods The net requrements, wll be that demand not satsfed by the ntal nventory ence, an equvalent problem s created wth zero startng nventory ow ncrease the demand n the last perod,, by the desred endng nventory ow the equvalent problem satsfes the startng and endng nventory constrants TE STUCTUE OF TE EUISTIC For a detaled statement of the algorthm the reader s referred to the orgnal publcaton by Dxon and Slver 3 Several other mathematcal models have been developed to solve these types of P-hard nventory problem, those are computatonally harder and thus requre more tme n nformaton processng Often, they become near P-hard problem, wth global search optons 4-6 Ths research thus concentrates on basc Dxon-slver heurstc The purpose of ths secton s to outlne the structure of the heurstc The Lot-Szng Technque Dxon-Slver heurstc s perod-by-perod heurstc whch s undrectonal n that they proceed by constructng a schedule perod by perod, startng wth perod To determne whch producton lots should be scheduled n each perod a prorty ndces s used Consder a perod n the process: one certanly has to produce max{0, d I,- } for all n order to avod stock outs n the current perod The remanng capacty (f any) can be used to produce future demands for some future setup costs may be saved at the expense of some added nventory holdng costs Consder tems whch need a setup n the current perod (e, d > I,- ) Prorty ndces whch ndcate the vablty of producng future demands for these tems n the current perod are then computed A very smple prorty ndex for the next perod s demand would be (S h d, ) The actual prorty ndces (U ) used by the heurstc are more sophstcated n that they try to capture potental savngs per tme perod In fact they are derved from wellknown heurstcs for the sngle level uncapactated dynamc lot-szng problem, eg, the Slver-Meal crteron 7 In any case, future demands are ncluded nto the current producton lot based on the prorty ndex n a greedy fashon untl ether no lots wth a postve ndex reman or untl the capacty constrant s ht The heurstcs then proceed to the next perod and the process s repeated Ensurng Feasblty If the total capacty demanded exceeds the capacty avalable n some perod, then some or all of the requrements of that perod must be satsfed by producton n precedng perods and by such pre-producton the nfeasblty can be removed Consder the determnaton of lot-szes n perod Let AP be the amount of producton (n capacty unts) n perod that wll be used n future perod If I, s the nventory at the end of perod for tem whch s resulted from only the currently scheduled producton n perod, then AP = k ( I, t I, ) () = Let C be the total demand (n capacty unts) n perod Then C = k d (2) = The producton plan for perod s feasble f and only f the followng condton s satsfed for t = 2,, t t AP ( C C ), (3) = = where C s the capacty n perod That s, the producton n perod for perods to t must exceed the total amount that demand exceeds capacty n those perods, and ths must be the case for all t Ths set of constrants can be used to gude the selecton of whch tme supples to ncrease It s now the case though that a lot-sze may be forced to be ncreased when U < 0 Furthermore, t may be necessary to schedule lots whch do not exactly satsfy an nteger number of perods requrements A smple approach to rectfyng ths dffculty s to ncrease the lotszes untl the feasblty condtons are satsfed, whle mnmzng the addtonal costs ncurred Implementaton of the eurstc The orgnal mult-tem problem wth constant capacty s P-hard In the present work a new constrant on upper lmt of the lot-sze s consdered Wth ths new constrant the problem s also P-hard Therefore, a smple heurstc has been developed whch guarantees a feasble soluton Step Create an equvalent demand matrx Convert the ntal demand matrx nto equvalent demand matrx wth the use of ntal nventory, endng nventory and safety stock Use the ntal nventory to satsfy as much demand as possble n the frst few perods The net requrements wll be that demand not satsfed by the ntal nventory Durng the calculaton of the net demands, the amount of the safety stock should be mantaned Let In = ntal nventory for tem, Iend = endng nventory for tem, Irem = remanng ntal nventory for tem, SS = safety stock for tem, and d = equvalent demand for product n perod Intally set Irem = In - SS and perod = 0 f Irem > D Then set d = D Irem f Irem D Compute Irem = Irem D Set = + and recycle tll Irem > 0 Journal of Mechancal Engneerng, vol ME38, Dec 2007 Transacton of the Mech Eng Dv, The Insttuton of Engneers, Bangladesh

3 A eurstc Soluton of Mult-Item Sngle Level Capactated Dynamc Lot-Szng Problem 3 Snce the amount of the safety stock s always mantaned, the demand n the last perod would be partally satsfed by the safety stock of the perod - If endng nventory s desred, then the requrements n perod should be ncreased by the desred endng nventory Then d = D + Iend SS Compute the net demands for all =, 2,, Step 2 Check the feasblty of the problem Feasblty Condton: = C If the feasblty condton s not satsfed, the problem s nfeasble, e, all demands cannot be met wth the avalable capacty Step 3 Convert the mult-setup problem nto sngle setup problem Step 3 Fnd the maxmum demand d max for each tem by d max = max {d =, 2,, } Fnd the number of new tems n to be consdered to satsfy demand d max usng the formula n = dmax / xmax Then the number of total tems after lmtng the lot-sze s = + = n Item s spltted nto n + tems Let the new tems be 0,, n Intally set d rem = d and l = 0 d rem f d rem xmax Then set d l, = xmax f d rem > xmax 0 f d rem xmax Compute d rem = d rem xmax f d rem > xmax Set l = l + and recycle up to l = n ow the equvalent demand matrx s converted nto a new demand matrx Step 32 Intalze the values of setup cost, holdng cost and capacty absorpton rate for the new tems from that of the tems by usng the formulas S 0 = S = = S = S, = = = = h and h n 0 h h = = = = k n k k 0 k n = C C = C k d = Intalze I wth zero, e, set I = 0, =, 2,, and =, 2,, Step 45 Calculate AP and C by the followng formulas AP = k = ( I I ),, and C = k d Determne the earlest perod t c at whch the feasblty constrant (3) s not satsfed, e, set + t + t t c = mn { t AP < ( C C )} = + = + If there s no nfeasblty, set t c = + Step 46 Consder only tems wth () T < t, (2) x can > 0, c where xcan = mn{ d, T, x } and (3) C s suffcent to rem produce x can Among these fnd the tem that has the largest U, where AC ( T ) AC ( T + ) U = kd, T + and T AC( T ) = S + h + ( ) d / T = Step 47 (a) If U > 0, then t s economc to produce x can n perod Increase the value of lot-sze, x nventory I for =,, T, and x rem,t by x can Decrease the value of lot-sze x, + T, demand d, + T, remanng capacty C and x rem by x can Set T = T + and contnue from Step 45 (b) If U 0, then t s not economc to ncrease T of any tem (total cost ncreases) Check the value of t c () If t c >, then no nfeasbltes left and lot-szng of the current perod s complete Go to Step 42 () If t c <, there are nfeasbltes and producton of one or more tem s to be ncreased and t s done through Steps 48 to 4 = Step 4 Apply the heurstc wth ncluson of the lmted lot-sze per setup [through Steps 4 to 42] Step 4 Start at perod, e, set = Step 42 Intalze lot-sze x by equalzng to demand d, e, set =, 2,, x = d and =, 2,, Calculate remanng allowable amount x rem that can be produced f x s produced at perod by x rem = x max x =, 2,, and =, 2,, Step 43 Intally set the value of tme supply to one e T =, where =, 2,, Tme supply T denote the nteger number of perods requrements that ths lot wll exactly satsfy Step 44 For each tem, =, 2,,, produce d (> 0) n the lot-szng perod After producng d calculate remanng capacty n perod, denoted by, C by Step 48 Calculate the amount of producton Q stll needed n the current perod to elmnate nfeasbltes n the later perod by the followng formula t Q = max ( C C AP ) tc t = Step 49 Consder only tems wth () T <, t c (2) x can > 0, where xcan = mn{ d, T, x } and (3) C s suffcent to rem produce x can To decde the best tem (from a cost standpont) to be produced n perod, calculate the prorty ndex for all of these tems, where AC( T + ) AC( T ) = k d, T + Among these fnd the one, denoted by, that has the smallest Steps 40 Let W = k x can If Q > W then Increase the value of lot-sze x, nventory I for =,, T, and x rem, T by x can Decrease the value of lot-sze,demand,remanng capacty C x, + T d, + T Journal of Mechancal Engneerng, vol ME38, Dec 2007 Transacton of the Mech Eng Dv, The Insttuton of Engneers, Bangladesh

4 A eurstc Soluton of Mult-Item Sngle Level Capactated Dynamc Lot-Szng Problem 4 and x rem by x can Set Q = Q W and T = T +, and contnue from Step 49 else Set IQ = [Q/K ] Increase the value of lot-sze, x nventory I for =,, T, and x rem, T by IQ Decrease the value of lot-sze x, + T, demand d, + T and x rem by IQ Step 4 Set = + (a) If <, then contnue from Step 43 (b) If >, lot-szng s complete up to perod for tems Step 42 Convert the lot-szng matrx nto lotszng matrx by applyng the formula n x = =, x 0 l, l Step 5 Calculate the values of Forecasted machne tme requred/perod Total expected setup cost Total expected nventory holdng cost v Total expected safety stock cost Stop ESULTS WIT TE LIMITED LOT-SIZE PE SETUP The algorthm developed to generate feasble soluton for mult-tem sngle level capactated lot-szng problem wth lmted lot-sze was tested n PC verson wth of a Item o () oldng Cost (h ) Table : elevant product data for the hypothetcal machne Setup Maxmum Producton Safety Cost Lot-Sze ate Stock (S ) (x max ) (/k ) (SS ) programmng language A near optmal soluton was obtaned Ths secton presents the results obtaned from the modfed model The algorthm has been tested wth hypothetcal data It s assumed that entre producton to meet demands s done n the plant and no subcontractng s permssble Moreover, a further assumpton s made that plant capacty could not be ncreased Product Data The relevant product data (eg, holdng cost, setup cost, producton rate, safety stock, ntal nventory and endng nventory) has been depcted n Table The problem sze has been restrcted at 2 products and 2 tme perods; each tme perod corresponds to a month Product Demand and Plant Capacty Product demands are qute seasonal and the same seasonal ndces are used for all the products Forecasted demand and the capacty of the machne are shown n Table 2 It has been assumed that the capacty per month s the total number of hours avalable per month Two percent of the capacty s reserved as a buffer to guard aganst uncertanty n the actual producton rate In ths hypothetcal problem, Perod corresponds to the month of June, Perod 2 corresponds to the month of July Thus the machne capacty n Perod s 98% of the total hours n June, e, = 706 hours To be n the safe sde, t has been assumed that the number of days n February s 28 Then the machne capacty n Perod 9 s = 660 hours Smlarly the machne capacty for the other perods has been calculated Intal Inventory (In ) Journal of Mechancal Engneerng, vol ME38, Dec 2007 Transacton of the Mech Eng Dv, The Insttuton of Engneers, Bangladesh Endng Inventory (Iend ) Table 2: Forecasted demand and capacty of the hypothetcal machne Item Perod o Avalable Machne ours

5 A eurstc Soluton of Mult-Item Sngle Level Capactated Dynamc Lot-Szng Problem 5 Table 3: Equvalent demand wth the use of ntal nventory, endng nventory and safety stock Item o Perod Table 4: Fnal lot-szes and forecasted machne tme requrements for the heurstc wth the lmted lot sze per setup Item Perod o Forecasted Machne equrements (hours) Table 5: Inventores at the end of each perod for all tems Item Perod o Equvalent Demand Schedule Table 3 depcts the equvalent demand after consderng ntal nventory, endng nventory and safety stock esults of the eurstc Table 4 shows the fnal lot-szes and forecasted machne hour requrements for each perod, and Table 5 shows the nventores at the end of each perod for all tems The followng results have also been found after applyng the heurstc wth the lmted lot-sze per setup = Total avalable machne tme ( C ) : hour t t Total setup tme : 0 hour Total forecasted machne tme : 8398 hour Total nventory holdng cost, C nv = = = t : $ Total expected safety-stock cost, C ss = = SS : $ Total expected setup cost, C set = n S = : $ Total expected cost (C nv + C ss + C set ) : $ Effect of the lmtaton on the lot-sze s dependent on the extent of reducton of the lot-sze It s obvous that the smaller the allowable lot-sze, the greater wll be the number of setup whch wll eventually lead to more spltted tems Thus when the lot-sze was reduced by 9, the Journal of Mechancal Engneerng, vol ME38, Dec 2007 Transacton of the Mech Eng Dv, The Insttuton of Engneers, Bangladesh ( I t SS )

6 A eurstc Soluton of Mult-Item Sngle Level Capactated Dynamc Lot-Szng Problem o of tems, Lmted Lot-sze Fgure : The growth rate of number of tems wth the lmted lot-sze Setup Cost Lmted Lot-sze Fgure 2: The varaton of setup cost wth the lmted lot-sze model yelded the total number of spltted tems of 95 from the orgnal twelve tems Ths n turn led to the ncrease number of requred setups Costs due to mplementaton of ths restrcton on lotsze went up qute sgnfcantly- the extent of whch was found to be more than 23% Further decrease n lot-sze would obvously result n hgher costs But at the lower range of allowable lot-sze, there has been a trend of slght ncrease n setup costs To see the effect of the lmted lot-sze to dfferent parameters, the frst value of the lmted lot-sze of each tem has been chosen as shown below These values have been chosen so that the number of total tems after lmtng the lot-sze remans unchanged and a lttle decrease n these values wll ncrease the number of total tems Item o Maxmum Lot-Sze ext the value of the lmted lot-sze of each tem s reduced step by step Wth the varaton of the lmted lotsze, the change of the values of the number of total tems, the machne utlzaton tme, total nventory cost, total setup cost, total safety stock cost and total cost has been shown n the followng fgures Fgure shows the growth rate of number of tems as a functon of the lmted lot-sze Ths growth rate s ncreasng wth the decrease of the lmted lot-sze The decrease n the lmted lot-sze decreases the amount of producton quantty per setup of an tem Ths decrease n producton quantty results n an ncrease n the number of tems Fgure 2 shows the varaton of setup cost wth the lmted lot-sze Wth the decrease of the lmted lot-sze, the setup cost ncreases sgnfcantly If the lmted lot-sze per setup s decreased, then the number of setup needed s ncreased accordngly Therefore the setup cost s also ncreased Fgure 3 shows the varaton of total nventory holdng cost wth the lmted lot-sze Wth the decrease of the lmted lot-sze, the varaton of the total nventory holdng cost s fluctuatng Ths nature of the varaton needs to be more nvestgaton Fgure 4 shows the varaton of total cost wth the lmted lot-sze Wth the decrease of the lmted lot-sze, total cost ncreases, snce the setup cost ncreases sgnfcantly, the nventory holdng cost s fluctuatng and safety stock cost remans almost unchanged COCLUSIO Lot-szng problem has been recognzed to be one of the most mportant functons n ndustral unts Thus efforts have been gven to develop usable optmzng routnes but wthn lmted boundary condtons Varous models have been developed wth restrcted applcatons n real-lfe settngs because of ther demandng computatonal enormsty Thus heurstc models have been evolved These heurstcs produce optmal and near optmal Journal of Mechancal Engneerng, vol ME36, Dec 2006 Transacton of the Mech Eng Dv, The Insttuton of Engneers, Bangladesh

7 A eurstc Soluton of Mult-Item Sngle Level Capactated Dynamc Lot-Szng Problem Total Inventory oldng Cost Lmted Lot-sze Fgure 3: The varaton of total nventory holdng cost wth the lmted lot-sze Total cost Lmted Lot-sze Fgure 4: The varaton of total cost wth the lmted lot-sze solutons In the present work the Dxon-Slver heurstc was extended to nclude a very mportant parameter, maxmum lmt of producton lot-sze from a machne From analyss and results, the present work has demonstrated that feasble solutons could be obtaned wth compettve computer usage to a realstc lot-szng problem The heurstc s based on a lot-szng technque and a set of feasblty condtons whch should be ntutvely appealng to managers Ths paper has been concerned wth a sngle stage process Extenson of the heurstc for multple producton stages could be a sgnfcant contrbuton EFEECES [] Shaker,, Mult-echelon Inventory Problem n Constant Usage ate Stuaton: Mult-modal Dspatchng, Journal of Producton esearch, Vol 4, o 2, 2006, pp [2] Esenhut, P S, A dynamc lot-szng algorthm wth capacty constrants, AIIE Transactons, Vol 7, o 2, 975, pp [3] Dxon, P S and Slver, E A, A heurstc soluton procedure for the mult-tem, sngle-level, lmted capacty, lot-szng problem, Journal of Operatons Management, Vol, 98, pp [4] arrs, C, An algorthm for Solvng Stochastc Mult-echelon Inventory Problem, Journal of Producton esearch, Vol 5, o 3, 2007, pp 90-0 [5] Brunett, M P, A Dsaggregaton Model for Solvng Computatonal Complexty In Mult-tem Inventory Problem, Internatonal Journal of Operatons Management, Vol 5, o 2/3, 2007, pp 0-2 [6] Adam, J, Global Search for Economc Lot Szng for Mult-tem Orderng Polcy, Internatonal Journal of Operatons Management, Vol 5, o, 2007, pp 0-8 [7] Slver, E A and Meal,, A eurstc for selectng lot-sze quanttes for the case of a determnstc tme varyng demand rate and dscrete opportuntes for replenshments, Producton and Inventory Management, Vol 2, o 2, 973, pp Journal of Mechancal Engneerng, vol ME36, Dec 2006 Transacton of the Mech Eng Dv, The Insttuton of Engneers, Bangladesh