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1 University of Groningen Time bucket size and lot-splitting approac Riezebos, Jan IMPORTANT NOTE: You are advised to consult te publiser's version (publiser's PDF) if you wis to cite from it. Please ceck te document version below. Document Version Publiser's PDF, also known as Version of record Publication date: 2002 Link to publication in University of Groningen/UMCG researc database Citation for publised version (APA): Riezebos, J. (2002). Time bucket size and lot-splitting approac. s.n. Copyrigt Oter tan for strictly personal use, it is not permitted to download or to forward/distribute te text or part of it witout te consent of te autor(s) and/or copyrigt older(s), unless te work is under an open content license (like Creative Commons). Take-down policy If you believe tat tis document breaces copyrigt please contact us providing details, and we will remove access to te work immediately and investigate your claim. Downloaded from te University of Groningen/UMCG researc database (Pure): ttp:// For tecnical reasons te number of autors sown on tis cover page is limited to 10 maximum. Download date:

2 TIME BUCKET SIZE AND LOT-SPLITTING APPROACH Jan Riezebos Faculty of Management and Organization, Production Systems Design Group, University of Groningen, Te Neterlands Som-teme A Primary processes witin firms ABSTRACT We address te problem of lot splitting for various time bucket lengts in MRP systems. Two approaces for lot splitting can be applied: eiter use te same (equal) or a variable number of subbatces. Equal subbatcing strategies ave logistical and computational advantages. Literature states tat variable batcing strategies are only marginal better. However, tese results do not take into account te sensitivity for canges in time bucket lengt. Managers ave reduced time bucket lengts in planning systems. We examine te sensitivity of lot splitting for tese canges. Our study reveals tat it is not cost-effective to disregard time bucket lengt wen deciding on te number of subbatces. Using te same number of subbatces per time bucket for all products results in substantial cost-differences, were te magnitude is affected by te discontinuity of te total cost curve. For a given time bucket lengt, a cost difference wit a variable number of subbatces per operation of only 2.1% can be obtained if an appropriate, equal number of subbatces for eac product can be found. Oter equal subbatcing strategies sow muc larger cost differences on average, ranging from 4-11%. In order to obtain tese results, a new variable subbatc euristic as been designed. KEYWORDS: Lot sizing, Lot splitting, Time bucket size, Planning and Control 1

3 1 INTRODUCTION OF THE PROBLEM MRP planning systems use time buckets for time-pasing te co-ordination of several stages of production, during eac of wic one or more operations are being performed. Te lengt of te time bucket is an important design parameter, as it constrains te replanning frequency of te planning system and affects te lengt of internal work order lead times. Planned lead times in an MRP system are always a multiple of te lengt of te time bucket P. We would terefore expect determination of te time bucket lengt to ave received muc attention in literature. However, important and influential studies on understanding lead times in MRP systems - suc as Kanet 1986, Karmarkar 1987, and Krajewski et al do not pay attention to te lengt of te time bucket at all. More recent studies by Matsuura et al. 1996, Portioli Staudacer 2000, Enns 2001, and Enns 2002 devote attention to te problem of setting MRP system design parameters. Tese studies consider te effect of time bucket size on lead time performance and on te costs of system operation. Several oter papers ave studied te effect of time bucket lengt variation on system performance, e.g. Luss 1989, Rees, Huang and Taylor 1989, Yang and Jacobs 1992, and Steele et al Tese studies ave sown tat te lengt of a time bucket as a substantial effect on system performance. Many firms are aware of te benefits of time bucket lengt reduction and managers try to improve cycle times by reducing time buckets in te planning systems used (Burbidge 1996). Software developers for ERP systems ave made it possible to exploit tese benefits troug te introduction of so-called bucketless systems. Beind te scenes, tese systems still use discrete time units, but tey allow various ordering intervals and make it easier to sorten te actual time bucket lengt tat is used in operating te system. Setting te time bucket lengt is an important medium-term decision in planning system design. A design parameter tat as received muc more attention in literature on planning system design is te lot size, see e.g. Wemmerlöv 1979, Karmarkar 1987, Wemmerlöv 1989, Stub 1990, and Yeung, Wong and Ma Te lot sizes used witin an MRP system are eiter static or dynamic. Dynamic lot sizes cover demand during one or more time buckets. Te minimum number of periods a lot size as to 2

4 cover demand is one, in wic case it is denoted as Lot for Lot ordering. Te size of te lot may terefore not be smaller tan te demand for tis product during one time bucket, as te time bucket sets te minimum time between ordering moments. Tis condition olds true bot for dynamic and static lot sizes. Terefore, lot sizes in an MRP system depend on te time bucket lengt P. Te size of te lot influences te amount of time necessary to produce te lot (see e.g. Kanet 1986). A decrease of te lot size results in a smaller production time and trougput time of te lot in te system. Lot size reduction can be obtained by decreasing te minimum time between ordering moments, te time bucket lengt P. Anoter way to sorten trougput times is by applying lot splitting. We define lot splitting as te division of te lot into nb subbatces tat can be transferred to te next operation as soon as te former operation as been performed for all items in te subbatc. Lot splitting may ave a substantial effect on te minimal required trougput time of te complete lot. Lot splitting affects te required number of stages N and ence te total amount of inventory in te system at a certain point in time. Te popularity of lot splitting as grown in practice since te success of cellular manufacturing. Olorunniwo 1996 reports tat 45.6% of te firms moving towards cellular manufacturing apply lot splitting. About 90% of tese firms indicate tat suc a cange in planning and controlling te system significantly contributes to teir success. Several types of lot splitting ave been described in te literature, for example, repetitive lots (Jacobs and Bragg 1988), overlapping operations (Graves and Kostreva 1986), lot streaming (Baker and Pyke 1990), transfer lot sizes (Trietsc 1989) and powered nested batcing policies (Muckstadt and Roundy 1993). A long-standing debate in tis literature concerns te size of te subbatces. Size can be expressed eiter in te number of items in te subbatc or in te amount of time necessary to process te subbatc. Some autors (e.g., Veral 1995) focus on utilization of te equipment and argue tat te number of items in te subbatces may cange between operations if suc a cange elps maintain a constant amount of time 3

5 necessary for te subbatc at te operations. Oters (e.g., Muckstadt and Roundy 1993) use consistent subbatces but allow different subbatc combinations at te various operations. A related point of discussion is te effectiveness of unequally sized subbatces. Some autors (e.g., Kropp and Smunt 1990) argue tat te logistical complexity of various subbatces in te system sould be avoided as muc as possible. Usage of unequally sized subbatces for te products in te system decreases te transparency of te planning system. Oter autors are more positive. Bogascewsky, Buscer and Lindner 2001 state tat for geometrically canging subbatc sizes at subsequent operations, total costs may be sligtly lower. An important study by Baker and Pyke 1990 states tat te advantage of unequally sized subbatces for flow time performance is only marginal (on average 4.6% reduction of make span in teir test cases). Tey conclude tat decision makers sould trade off te complexity increase of unequally sized subbatces wit tese small savings. However, in drawing tese conclusions concerning te effectiveness of equally sized subbatces, te lengt of te time bucket as not been considered as an experimental design factor. In situations were managers reduce time buckets, tey ave also to decide on te appropriateness of te lot splitting approac used. Our study supports tis decision and examines te effectiveness and appropriateness of an equal number of subbatces approac if time bucket lengt varies. We expect tat te effectiveness and near optimality of a strategy involving an equal number of subbatces depends on te lengt of te time bucket in te planning system and on te type of equal subbatc strategy applied. Te objective of tis study is terefore to explore te effect of time bucket size on te use of an equal subbatc strategy. We know tat te lengt of te time bucket affects te total lot size and te amount of material transfers per year. An increase in te number of subbatces nb increases te number of material transfers witin a time bucket P. Tis leads to larger transfer costs and possibly smaller olding costs. Terefore, a cost perspective will be useful wen analysing te effectiveness of an equal number of subbatces strategy. 4

6 Tis paper is organised into six sections. Section 2 specifies te type of planning system explored and provides a matematical model for te relations between te design parameters of tis planning system. It develops an enumerative searc euristic for finding a variable subbatc solution. Section 3 presents te experimental design for evaluating several equal subbatc strategies, using a simulation approac. Section 4 presents te performance of te equal subbatc strategies for successive values of te time bucket lengt. Section 5 presents our conclusions. 2 MODELLING THE PLANNING SYSTEM Te planning system studied in tis paper may be caracterized according to two basic principles. First, for all products and parts tis system uses an identical amount of time between two order releases (te reorder interval), wic equals one time bucket P. Second, it is a lot for lot ordering system, i.e., te amount ordered equals te demand during te next period of lengt P. Lots consist of one type of end product; ence, te total trougput time of a lot equals te total trougput time of te products contained in tis lot. Te number of stages N in te planning system is determined by te number of periods of lengt P a lot remains in te production system in various levels of completion. Te trougput time of te system is te product of N and P. Te basic idea is to determine te effect of time bucket lengt P, number of stages N, and number of subbatces nb i on tree relevant cost factors: olding costs HC in te system, ordering or set-up costs SC i, and transfer costs TC i. Te number of stages N is a function of P and nb i. Tese factors directly influence te total trougput time TT needed for a batc q of product. We assume for simplicity tat te operations ave to be performed in sequence (a linear product structure), so we can use as a lowerbound for TT : n max n q TT si + pi q + pt i= 1 t=+ i 1 nb i + (1) 5

7 wit s i being te set-up time and p i te unit processing time for operation i of product (i=1..n ). Te number of stages N is an integer. We assume tat te planning system uses N stages for all products. Formula (1) models a variable subbatc strategy, as it allows variation of te number and size of te subbatces between operations i and i+1. An equal subbatc strategy uses te same number of subbatces for any operation. It can be modelled using an extra constraint wit respect to nb i, as sown in Formula (2) and (3). Equal subbatc strategies of type A use te same number of subbatces for all products. Type B strategies allow te number of subbatces per product to vary, but use te same number of subbatces for all operations of a product. nbi = nbi 1, i > 1 (equal subbatc strategy type A) (2) nbi = nbi 1 i > 1 (equal subbatc strategy type B) (3) Holding costs are incurred from te moment tat te required raw material is introduced into te system until te moment te finised product leaves te system. We assume tat all required material is introduced during te first stage of te planning system. On average, te raw material will be introduced alfway troug te first production stage and te final product will be sold or delivered alfway troug stage N+1. Te material worked into a lot q is terefore, on average, N stages of lengt P present in te system, irrespective of te actual progress in making te product. Te average total inventory in te system is ten N q = N P D. Holding one item of product in stock during a standard time unit costs HC, so te total cost of olding tis inventory during a standard time unit is N P HC D. Te second cost factor is costs tat will be involved every time a new cycle starts irrespective of te number of subbatces or te lengt of te time bucket. Examples of tese costs are ordering and set-up costs. Te number of cycles is 1/P per standard time unit. Te set-up costs for an operation are te product of te set-up time for tis operation s i and its set-up costs per standard time unit SC i. Eac operation i of product is performed once per cycle, unless te lot size in one of te cycles is zero. We terefore sum up te set-up times for all n operations of product. Te total set- 6

8 up costs per cycle of lengt P are =1..H i=1..n s i SC i. Total set-up costs per standard time unit can be obtained troug dividing tis by te lengt of P. Finally, we consider te costs of transferring subbatces to te next operation. Tese costs depend on te number of subbatces of a product : nb. TC i reflects te cost of transportation and administration effort required at te successive operations. If te lot of product is split at operation i into several subbatces (nb i >1), te total transfer costs increase. Te transfer costs TC i may vary per product and operation. We assume tat te transfer cost is linear wit respect to te number of subbatces. Terefore, te transfer costs per standard time unit are [ =1..H i=1..n nb i TC i ] / P. We obtain te following non-linear model for finding P and a variable subbatc strategy: Minimize P, nb i (=1..H, i= 1.. n ) = 1 { } H n H N P D HC + + = 1i= 1 { s SC + nb TC } i i i i H max TT s.t. N wit TT as defined in Formula =1 P (1) (5) Te model searces for te optimal combination of a time bucket lengt P and a variable subbatc strategy nb i wit respect to te sum of olding costs, set-up costs, and costs for te transfer of subbatces between operations. Te model gives us insigt into te combined effect of time bucket size and splitting a lot into nb i subbatces. However, no solution metods are available to optimally solve tis model witin polynomial time. We can solve a simplified model tat fixes P and tries to find a suitable subbatcing strategy at tis time bucket lengt. Repeatedly applying tis model for successive values of P will yield an idea of te cost curves as a function of te time bucket lengt P. For fixed P, te problem of finding a suitable subbatcing strategy can be reformulated as: P (4) 7

9 H n { nb TC } H i i = 1i= 1 Minimize N { q HC } + st nb i (=1..H, i= 1.. n ) = 1 P..(1) and (5) (6) For tis problem, we determine te cost of various subbatc strategies at succeeding values of te time bucket lengt P. We develop an approximate solution approac to find a variable number of subbatces strategy and evaluate te effect of equal versus varying numbers of subbatces at different time bucket sizes. 2.1 Finding a variable subbatc solution Graves and Kostreva (1986) discuss te problem of finding a suitable number of subbatces in a two-macine flow sop wit equal processing times p and suggest a value equal to HC Q p wit Q = * * TCi i+ 1 ( ) 2 D s SC t t t= i A rounding procedure is proposed in order to find an integer valued number of subbatces for te batc of size Q *. For te case of more tan two operations, unequal processing times p i and fixed period lengt P, nb i as to reflect te number of subbatces per period of lengt P. We terefore transform te function to: [ ] nb % = P D min p, p i i i+ 1 HC TC i 2 nbi = nb % i if nb % i nb % i nb % i ( nbi is te largest integer nb % i ) (8) 2 = nb % i if nb % i nb % i < nb % i ( nbi is te smallest integer nb % i ) Te reduction in trougput time due to te increase of te number of subbatces at operation i is a function of te minimum of te processing times at operations i and i+1: HC (7) 8

10 TT q q min p, p [ ] i i+ 1 nbi + 1 nbi (9) We terefore propose to use te minimum of bot processing times in Formula (8). Note tat Formula (8) does not explicitly model te effect of trougput time reduction. Furtermore, it does not consider te interdependency between te decisions on te number of subbatces at te various operations. For te cost structure of our model, bot factors ave a strong impact. However, no solution approaces are known tat solve tis problem witin polynomial time. We develop a euristic tat takes te effect of an increase in te number of subbatces at an operation on te expected trougput time into account. 2.2 Enumerative searc euristic Discontinuous total cost curve Continuous Total Cost Curve Discontinuous Total Cost Curve Minimum Cost Slope Curve Holding costs N=6 N=5 N=4 N=3 N=2 N= Transfer costs Figure 1 Discontinuous total cost curve for fixed P 9

11 Te enumerative searc euristic tries to find te most cost effective variable batcing strategy for a specific value of te time bucket lengt P. Tis batcing strategy as to balance transfer costs (nb i ) and olding costs (N). It takes into account tat te actual total cost curve is discontinuous, as illustrated in Figure 1. Measures to increase te number of subbatces do terefore not directly pay off in a reduction of system trougput time, as would ave been te case if te total cost curve ad been continuous. Trougput time reduction is only acieved if te number of stages N decreases. If te associated olding cost reduction is still wortwile depends on te increase in transfer costs necessary to acieve te trougput time reduction. Te law of diminising marginal returns olds ere, illustrated by te sape of te total cost curve. Te intersection of te minimum cost slope curve and te discontinuous total cost curve gives us te optimal balance between olding costs and transfer costs. Total costs can be read from te point were tis minimum cost slope curve intersects wit one of te axes. Te enumerative searc euristic subsequently evaluates suitable variable batcing strategies for decreasing values of N. Te euristic begins wit an initial (large) number of stages N, and a subbatc strategy in wic all products ave only one subbatc (nb i =1). Te euristic directs its attention to finding breakpoints in te cost curve - a function of N and nb i - compares te solutions for tese breakpoints, and selects te batcing strategy tat results in te lowest total cost. Te euristic considers an increase in te number of subbatces per operation up to a user defined maximum number of subbatces at an operation. Te decision to increase te number of subbatces at an operation is based on information related to te breakpoints locations and te estimated reduction in te total trougput time. We use te lowerbound of Formula (9) for estimating tis reduction. All products for wic te trougput times ave to be sortened are considered. Te increase in te number of subbatces for an operation causes extra transfer costs, but a trougput time reduction will eventually reduce olding costs. Te ratio between te two cost canges, Cost i, drives te selection of a suitable batcing strategy. For Cost i, we use a ratio of olding costs and transfer costs similar to te one used in Formula (8). 10

12 1 Set nb 1, i i 2 ComputeTT ( lengt of pattt is trougput timeof product ) maxtt N P Total cost N = N q HC + nb tc [ ] i i i 3 Consider if a solution wit sorter trougput time( less stages) and more subbatces as lower total cos ts : * i ( ) FOR ALL wittt > N 1 P q q 4 TTi = min [ p, 1] max i pi+ max nb, nbi 1 nb i + ( TT is alowerbound ontetrougput time reduction) 5 Cost i i 1 TCi = TT HC i 6 IF Cost = min Cost = :STOP, return solution wit N stages ELSE nb nb + 1 [ ] ( ) i * * i i [ ] [ ] i ( ) 7 REPEAT steps4 6 UNTIL TT N 1 P 8 Total cost N 1 = N 1 q HC + IFTotal cost N 1 Total cos t N :STOP, return solution wit N stages ELSE N N 1 Goto step 3 updatett i nb P i TC i Tis enumerative searc euristic as been compared wit te result of Formula (8) (Graves and Kostreva modified) for finding a variable subbatc strategy. Te enumerative searc euristic outperformed Formula (8) in 89.4% of te cases, wile 11

13 in only 2% of te cases te modified Graves and Kostreva formula performed better. If te enumerative euristic was better, te average cost difference was 5.3%. If it performed worse, cost difference was only marginal (less tan 1%). We terefore use te result of te enumerative searc euristic wen evaluating te effect of equal versus varying numbers of subbatces at different time bucket sizes. 2.3 Discontinuous total cost curve as a function of P Costs as a function of P Total cost Holding cost Setup cost Transfer cost Cost ,005 0,01 0,015 0,02 0,025 0,03 0,035 0,04 0,045 0,05 Period lengt (years) Figure 2 Partial and total cost curves as a function of P Tis subsection discusses te cost beaviour for successive values of te time bucket lengt. Total costs are a function of P as sown in Formula (4). Te contribution of 12

14 set-up and transfer costs to te total costs diminises if P increases. Tis is a rater familiar pattern, well known from inventory researc (see e.g. Silver et al. 1998). Te reason is tat te number of set-ups and transfers are a linear function of te number of cycles per year, and tis number is inversely proportional to te lengt of te time bucket. Holding costs sow a less familiar pattern. Usually, olding costs are a linear function of te batc size and ence of P as q =P D. However, olding costs also depend on te total time TT tese units stay in te system. If P increases, possibly less stages are needed to cover te total trougput time of te batc. If a reduction in te number of stages occurs, tis reduces te amount of items in te system and ence te olding costs. A tendency exists to require a lower number of stages for increasing values of P. Eac reduction in te number of stages causes a breakpoint in te olding costs and total costs curves. Figure 2 illustrates te various partial cost curves as well as te total cost curve as a function of P. We used te same number of subbatces for all values of P. Hence, te discontinuity of te olding cost curve is not caused by canges in te number of subbatces. It can only be addressed to canges in P. From our analysis we expect tat te lengt of te time bucket P as a strong effect on te total cost of a subbatc policy. IF P canges, a different number of stages may be required. Accordingly, a different number of subbatces may be appropriate. Te decision on te number of subbatces is strongly interrelated wit te coice of te time bucket lengt. Bot coices ave direct consequences for te required number of stages and resulting total costs. We sould terefore examine te performance of subbatc strategies for various values of te time bucket lengt. 3 EXPERIMENTAL DESIGN In order to investigate te cost differences between several types of equal subbatc strategies and a variable subbatc policy as a function of te time bucket lengt, a simulation study as been performed. Te problem set contained 40 randomly generated problems for wic several subbatc strategies ave been evaluated at 80 13

15 different time bucket lengts. A problem consisted of tree product types wit eac 12 operations tat ad to be performed successively. We used a fixed demand rate tat remained constant over te experiments. We assumed tat eac operation ad to be performed at a different macine, because scarce capacity was not an issue in our study. Te time needed at an operation was terefore influenced only by te randomly generated set-up and processing time, te time bucket lengt (affecting q ), and te subbatc policy applied. Operation processing and set-up times were normally distributed wit mean as described in Table 1 and standard deviation equal to 1/3 of its mean. Table 1 Set-up and processing time data used in experiments (years) Operation Process time Process time Process time Set-up time number product 1 product 2 product 3 1 Mean Sum Table 2 gives details on te experimental design settings. Product 2 as a iger demand but a smaller mean processing time tan product 3. It terefore depends on te randomly generated set-up and processing times and on te actual lengt of te time bucket wic product will ave te longest trougput time. 14

16 Table 2 Experimental design data Demand (year) Time bucket Costs Product 1 Product 2 Product Minimum value (year) Costs for olding one unit for one year Step value (year) Set-up costs per operation Number of steps 80 Costs per transfer of a subbatc Types of equal subbatc strategies Te first equal subbatc policies tat we consider do not consider te effect of canges in te time bucket size. No matter wat time bucket lengt, te same number of subbatces is used. Tese policies are denoted by AI-nb=1, AI-nb=2, AI-nb=3 and AI-nb=4, indicating te number of subbatces tat are used. Tese strategies are of type A, indicating tat tey use te same number of subbatces for all products, resulting in a batcing policy tat is very transparent and easy to use. However, type A policies may result in unnecessary ig costs, as tey do not distinguis between products for wic trougput time reduction sould or sould not be realised troug lot splitting. Type B policies consider equal subbatces tat differ per product. Second, an intuitively attractive strategy is introduced tat considers te effect of time bucket size. It uses at te specified time bucket size te equal subbatc strategy tat performs on average best. We denote te strategy as AII-MinAvg. Te results of te AI strategies for all randomly generated problems were used to determine te on average best policy at a specific time bucket lengt. Tis policy presumes te existence of knowledge on wat is in general optimal at a specific time bucket lengt, independent of te caracteristics of te product to be made. Example of suc knowledge is to use no more tan two subbatces if te time bucket lengt is less tan a week. Equal subbatc strategies like tis are very often applied in practice. 15

17 Table 3 Subbatc strategies Subbatc Strategy Time bucket dependent Problem dependent Identical per product (A/B) Identical per operation AI-nb=x (x=1,2,3,4) No No (standard) Yes (A) Yes (Equal) AII-MinAvg Yes No (average) Yes (A) Yes (Equal) AII-AlmostMin Yes No (average) Yes (A) Yes (Equal) AIII-Min Yes Yes Yes (A) Yes (Equal) BIII-OptEqual Yes Yes No (B) Yes (Equal) Variable Subbatc Yes Yes No (B) No (Variable) Te inerent danger of te above mentioned strategy is tat unnoticed te wrong equal subbatc strategy is used. In practice, tis may be caused wen canges in product caracteristics or time bucket lengt occur witout reconsidering te subbatc strategy used. Tis results in cost differences, wic we measure by considering te use of te on average second best equal subbatc strategy AII- AlmostMin. Te tird type of equal subbatc strategy searces te appropriate number of subbatces dependent on te product caracteristics. For eac problem situation, instead of a general policy a specific equal batcing strategy is being determined and applied to problems tat belong to tis category. We denote te results of tis equal subbatc strategy as type III results. Two variants are distinguised: AIII-Min and BIII-OptEqual. AIII-Min uses te same number of subbatces for all products, wile BIII-OptEqual allows te number of subbatces per product to vary. Note tat a variable subbatc policy allows variation between te operations of te same product, contrary to BIII-OptEqual. An optimal variable subbatc policy will lead to te lowest cost solution possible. Unfortunately, suc a solution is bot 16

18 difficult and time-consuming to find, and we need to use a euristic procedure to find an approximation of tat solution. Te enumerative searc euristic tat we developed in order to find tis approximation gives us a lowerbound on te cost increase if an equal subbatc strategy is used. We used it to obtain variable subbatc solutions for successive values of te time bucket lengt. 4 RESULTS First, te performance of te four equal subbatc strategies AI-nb=1, AI-nb=2,.., AInb=4 was compared wit te solution found wit te euristic for te problems generated. Figure 3 and Table 4 sow for all four equal subbatc strategies of type AI, at eac time bucket size, te average distance to te cost of te variable subbatc strategy. For very small lengts of te time bucket (< year) tere is no difference in te effect of equal and variable subbatc strategies. Tis is not surprising, as te batc size in suc a sort time bucket is also very small and often cannot be split. Te difference begins increasing at a time bucket lengt of years. Te variable subbatc strategy increases te number of subbatces at certain operations, wic results in less costs tan obtained wen applying an equal subbatc strategy wit nb i =1. Te cost difference can only be attained by reducing te number of stages (total trougput time), as a consequence of te increased number of subbatces at tese operations. Applying tis increased number of subbatces for all operations (nb i =2) results in large cost differences (10-25%) for tese small time bucket lengts. Hence, te flexibility of a variable subbatc strategy at small time bucket sizes pays. 17

19 Sensitivity of performance equal subbatc strategies for cange in time bucket size Average distance to cost variable subbatc solution 50% 45% 40% 35% 30% 25% 20% 15% 10% 5% AI-nb=1 AI-nb=2 AI-nb=3 AI-nb=4 0% 0 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 Lengt of time bucket (years) Figure 3 Cost effectiveness of equal subbatc strategies independent of P For time bucket sizes of almost a week (P=0.015 year), lot splitting becomes for tese products almost necessary in order to avoid uge cost differences. Te use of an equal subbatc strategy of nb i =2, instead of a variable subbatc strategy, results in a cost difference of between 5 and 10%. For P=0.025 and larger, tree equal subbatces result in still iger cost differences (>10%). However, if no cange were made towards an increase in te number of subbatces (nb i still equal to 2), te cost difference would increase muc faster. For large time bucket lengts, te cost differences wit an equal subbatc strategy of nb i =4 tend to decrease. Te variable 18

20 subbatc policies at tese ig time bucket lengts do not cange anymore, but te relative contribution of te transfer costs to te total costs steadily decreases. Table 4 Sensitivity of equal subbatc strategies AI to time bucket size AI-nb=1 AI-nb=2 AI-nb=3 AI-nb=4 From- P -To Mean St.Dev Mean St.Dev Mean St.Dev Mean St.Dev % 0.05% 23.87% 1.90% 48.44% 3.23% 73.26% 4.68% % 0.94% 16.75% 2.54% 36.35% 4.40% 57.48% 6.08% % 2.68% 9.95% 2.33% 23.14% 3.78% 39.43% 5.46% % 4.43% 6.88% 2.92% 14.92% 3.17% 26.64% 4.03% % 6.32% 8.47% 4.11% 11.68% 3.39% 19.80% 3.85% % 7.02% 12.28% 5.16% 10.72% 4.10% 16.29% 3.76% % 6.98% 16.62% 6.11% 11.46% 5.50% 13.82% 3.14% % 8.21% 19.49% 7.29% 11.32% 6.91% 11.87% 2.50% % 8.95% 22.59% 8.29% 11.61% 7.94% 10.19% 2.06% % 10.11% 25.58% 9.41% 12.41% 9.03% 8.76% 1.74% % 11.31% 27.34% 10.27% 12.41% 9.86% 7.68% 1.44% % 12.30% 29.15% 11.20% 12.64% 10.65% 6.73% 1.19% % 13.35% 30.83% 12.00% 13.52% 11.31% 5.91% 1.02% % 14.24% 32.38% 12.61% 13.34% 11.82% 5.23% 0.89% % 15.05% 33.46% 13.26% 13.39% 12.30% 4.66% 0.76% % 15.80% 34.30% 13.79% 13.73% 12.69% 4.21% 0.67% We conclude tat te lengt of te time bucket as a strong impact on te performance of te distinct equal subbatcing strategies. Te use of equal subbatces independent of te time bucket lengt results in substantial cost differences and is not effective. We conclude furter tat te size of te cost difference wit a variable subbatc strategy also depends on te lengt of te time bucket. 19

21 Table 5 Results of equal subbatc strategies dependent on time bucket size AII-MinAvg AII-AlmostMin AIII-Min BIII-OptEqual From-P -To Mean St.Dev Mean St.Dev Mean St.Dev Mean St.Dev % 0.05% 23.87% 1.90% 0.02% 0.05% 0.02% 0.05% % 0.94% 16.75% 2.54% 0.88% 0.94% 0.87% 0.93% % 2.68% 9.95% 2.33% 4.63% 2.61% 3.27% 1.67% % 2.92% 11.81% 3.42% 6.74% 2.85% 3.28% 2.23% % 4.11% 11.68% 3.39% 8.14% 3.78% 3.34% 1.97% % 4.10% 12.28% 5.16% 9.51% 3.77% 3.23% 2.06% % 5.50% 13.82% 3.14% 8.76% 3.21% 2.95% 1.85% % 6.03% 12.25% 4.11% 7.49% 2.95% 2.29% 1.62% % 2.06% 11.61% 7.94% 6.50% 2.53% 2.05% 1.34% % 1.74% 12.41% 9.03% 5.75% 2.26% 1.72% 1.18% % 1.44% 12.41% 9.86% 4.99% 1.96% 1.53% 1.04% % 1.19% 12.64% 10.65% 4.39% 1.78% 1.35% 0.89% % 1.02% 13.52% 11.31% 3.96% 1.55% 1.23% 0.81% % 0.89% 13.34% 11.82% 3.46% 1.38% 1.06% 0.69% % 0.76% 13.39% 12.30% 3.07% 1.23% 0.96% 0.62% % 0.67% 13.73% 12.69% 2.80% 1.13% 0.85% 0.56% Te sensitivity to time bucket size canges does also occur for AII type policies. Te mean and standard deviation of te distance of te AII-MinAvg solution to te best solution found is sown in te first column of Table 5. Canges in te number of subbatces took place at 0.015, 0.025, and years. Te results sow clearly tat te mean cost difference is not constant over a broad range of time bucket sizes. Te average cost difference varies between 4 and 11%. Note also tat variation increases if te average cost distance is iger. Te largest cost difference we found in our experiments was 22%. If product caracteristics or time bucket ave canged witout notice, te cance exists tat te wrong equal subbatc strategy is being used. Te strategy AII- AlmostMin indicates te cost of not using te best equal subbatc strategy. Table 5 sows tat te cost difference wit te best variable subbatc solution is generally around 12%, but is muc iger for very small time bucket lengts (< P=0.010). Te 20

22 average cost difference does not fluctuate strongly wit P, but its standard deviation does increase as a function of P. For specific product configurations, AII-AlmostMin proved to be more cost effective tan te AII-MinAvg policy. Our experiments sowed tat te magnitude of te cost difference between te AII-MinAvg solution and te AII-AlmostMin solution was 34.85% in favour of te AII-AlmostMin solution (see Figure 4). Due to te fact tat at very small time bucket sizes (P<0.03) AII-MinAvg almost always outperforms AII-AlmostMin, te 34.85% sare will be realised in te mid and end range of time bucket sizes. Witin tis range, te inerent danger of applying AII-MinAvg is terefore substantial. Cost difference between AII-ALmostMin and AII-MinAvg 10% 9% AII-AlmostMin - AII-MinAvg 8% 7% 6% 5% 4% 3% 2% 1% 0% -12% -9% -6% -3% 0% 3% 6% 9% 12% 15% 18% 21% 24% 27% Frequency Percentage cost difference Figure 4 Cost difference between AII-MinAvg and AII-AlmostMin strategies Finally, we performed an analysis using equal subbatc strategies tat take product caracteristics into account wen tey are used to determine te appropriate number of subbatces. We compared strategy AIII-Min wit BIII-OptEqual. Note tat B allows te number of subbatces per product to vary, wic results in lower costs. Table 5 presents te results. Bot strategies realise a significant cost difference in relation to te oter equal subbatc strategies. If we compare AIII wit AII-MinAvg, te cost difference is an average of approximatley 2%. Te difference wit BIII is 21

23 iger, more tan 5.5%. Taking product caracteristics into account results, terefore, in lower costs. However, note tat te efforts to acieve suc a solution are also iger. Te cost advantage of varying te number of equal subbatces per product (strategy B instead of A) is substantial. Te largest average cost difference wit te best known solution is around 4%, wit a mean of 2%. It also as a positive effect on te standard deviation of te cost difference wit te best known solution, indicating tat te results it produces are more stable. Compared to te efforts needed for obtaining a variable subbatc strategy, equal subbatc strategies tat allow different numbers of subbatces per product seem to be a good alternative. 5 CONCLUSION Te lengt of te time bucket as an important effect on te performance of equal subbatc strategies. Te effectiveness of using te same equal subbatc is igly sensitive wit respect to an increase in te time bucket lengt. Hence, from a cost perspective, it would be inappropriate to use equal subbatc strategies tat specify te number of subbatces independent of te time bucket lengt. If one canges to anoter equal subbatc strategy at appropriate lengts of te time bucket (i.e., equal subbatc strategies tat depend on te time bucket lengt), large differences wit a variable subbatc strategy can be avoided. In our experiments, canges at 0.015, and years were required for strategy AII-MinAvg. In tat case, te average cost differences wit te best solution found were still between 4 and 11 %. Te problem is to determine at wat lengt of te time bucket a cange in te number of subbatces sould be considered. If te cange takes place too late, cost differences grow rapidly, as sown troug strategy AII-AlmostMin. Tus, te inerent danger of tis intuitively attractive strategy is tat product caracteristics or time bucket may ave canged, wile consequences for te number of equal subbatces ave not been considered. Equal subbatc strategies tat take notice of te product caracteristics at a specific time bucket size perform even better (on average 2.0%) tan AII-MinAvg. We compared an equal subbatc strategy tat used te same number of subbatces for all 22

24 products wit a strategy tat allowed tis number to vary. Te cost difference between bot strategies was substantial, on average 3.8%. Te average cost advantage of a variable subbatc strategy, wic also allows te number of subbatces per operation to vary, is 2.1%. We obtained te variable subbatc solution using te euristic searc procedure, as described in Section 2.2. Te presented cost difference is terefore a lowerbound for te cost difference wit te unknown optimal solution. Te euristic outperformes te modified formula of Graves and Kostreva 1986 for finding a variable subbatc solution in almost 90% of te cases wit an average cost difference of 4.7%. We conclude tat determination of te number of subbatces sould receive careful attention of senior operations management in relation to te decision on te time bucket lengt. Te cost difference wit a variable subbatc solution depends strongly on te caracteristics of te equal subbatc strategy used. Scientific studies comparing te performance of equal versus variable subbatc strategies sould clearly delineate te assumptions beind te evaluated strategies. Te performance differences tat we found between te various equal subbatc strategies make general conclusions on te effectiveness of equal subbatc strategies wortless. Summarizing our conclusion, our study reveals tat: Using te same equal number of subbatces for various time bucket lengts results in substantial cost inefficiencies. Selecting te in general most cost effective equal subbatc strategy at a specific time bucket lengt interval still results in an average cost difference of between 4 and 11%, as compared to te result of our variable subbatc searc euristic. Equal subbatc strategies tat determine te number of subbatces based on product caracteristics and time bucket lengt perform 2.0% better, on average, tan AII-MinAvg. Equal subbatc strategies tat allow te number of subbatces per product to vary are te most cost effective. Mean cost difference wit te best solution known is only 2.1%. Cost improvements compared to equal subbatc strategies tat use te same number of subbatces for all products were, on average, 3.8%. 23

25 REFERENCES Baker, K.R., D.F. Pyke Solution procedures for te lot-streaming problem. Decision Sci Bogascewsky, R.W., U.D. Buscer, G. Lindner Optimizing multi-stage production wit constant lot size and varying number of unequal sized batces. Omega Burbidge, J.L Period Batc Control, Oxford University Press, Clarendon Press, Oxford. Enns, S.T MRP performance effects due to lot size and planned lead time settings. Internat. Production Res. 39(3) Enns, S.T MRP performance effects due to forecast bias and demand uncertainty. Eur. J. Oper. Res Graves, S.C., M.M. Kostreva Overlapping operations in material requirements planning. J. Oper. Management 6(3) Jacobs, F.R., D.J. Bragg Repetitive lots: flow-time reduction troug sequencing and dynamic batc sizing. Decision Sci Kanet, J.J Toward a better understanding of lead times in MRP systems. J. Oper. Management 6(3) Karmarkar, U Lot sizes, lead times and in-process inventories. Management Sci. 33(3) Krajewski, L.J., B.E. King, L.P. Ritzman, D.S. Wong Kanban, MRP and saping te manufacturing environment. Management Sci. 33(1), Kropp, D.H., T.L. Smunt Optimal and euristic models for lot splitting in a flow sop. Decision Sci Luss, H Syncronized manufacturing at final assembly and feeder sops. Internat. Production Res. 27(8) Matsuura, H., H. Tsubone, M. Kanezasi Setting planned lead times for multioperation jobs. Eur. J. Oper. Res Muckstadt, J.A., R.O. Roundy Analysis of multistage production systems in: Graves, S.C., A.H.G. Rinnooy Kan, P.H. Zipkin (eds) Logistics 24

26 of production and inventory. Handbooks in operations researc and management science, vol. 4, Nort Holland, Amsterdam. Olorunniwo, F.O Canges in production planning and control systems wit implementation of cellular manufacturing. Prod. Inventory Management J. 37(1) Portiolo Staudacer, A ISTS: a new procedure for te integrated sceduling of multi-stage systems. Internat. Production Res. 38(11) Rees, L.P., P.Y. Huang, B.W. Taylor A comparative analysis of an MRP lotfor-lot system and a Kanban system for a multistage production operation. Internat. Production Res. 27(8) Stub, A Lot sizing in MRP/GT systems. Production Planning & Control 1(1) Silver, E.A., D.F. Pyke, R. Peterson Inventory management and production planning and sceduling, 3 t edition. Jon Wiley & Sons, New York. Steele, D.C., W.L. Berry, S.N. Capman Planning and control in multi-cell manufacturing. Decision Sci. 26(1) Trietsc, D Polynomial transfer lot sizing tecniques for batc processing on consecutive macines. Tecnical report NPS Naval Postgraduate Scool, Monterey, California, USA. Veral, E Using production and transfer batces in flow sops under MRP control. J. Oper. Management Wemmerlöv, U Design factors in MRP systems: a limited survey. Prod. Inventory Management J. 20 (4) Wemmerlöv, U Te beavior of lot-sizing procedures in te presence of forecast errors. J. Oper. Management 8(1) Yang, K.K., F.R. Jacobs Comparison of make-to-order job sops wit different macine layouts and production control systems. Internat. Production Res. 30(6) Yeung, J.H.Y., W.C.K. Wong, L. Ma Parameters affecting te effectiveness of MRP systems: a review. Internat. Production Res. 36(2)

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