Mathematical Sciences. Bristol BS16 1QY. UK

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1 This article was downloaded by:[university of the West of England] [University of the West of England] On: 9 May 27 Access Details: [subscrition number ] Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Production Planning & Control The Management of Oerations Publication details, including instructions for authors and subscrition information: htt:// Rolling horizon heuristics for roduction lanning and set-u scheduling with backlogs and error-rone demand forecasts Alistair R. Clark a a University of the West of England, Faculty of Comuting, Engineering and Mathematical Sciences. Bristol BS16 1QY. UK To cite this Article: Alistair R. Clark, 'Rolling horizon heuristics for roduction lanning and set-u scheduling with backlogs and error-rone demand forecasts', Production Planning & Control, 16:1, To link to this article: DOI: 1.18/ URL: htt://dx.doi.org/1.18/ PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: htt:// This article maybe used for research, teaching and rivate study uroses. Any substantial or systematic reroduction, re-distribution, re-selling, loan or sub-licensing, systematic suly or distribution in any form to anyone is exressly forbidden. The ublisher does not give any warranty exress or imlied or make any reresentation that the contents will be comlete or accurate or u to date. The accuracy of any instructions, formulae and drug doses should be indeendently verified with rimary sources. The ublisher shall not be liable for any loss, actions, claims, roceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. Taylor and Francis 27

2 Production Planning & Control, Vol. 16, No. 1, January 25, Rolling horizon heuristics for roduction lanning and set-u scheduling with backlogs and error-rone demand forecasts ALISTAIR R. CLARK University of the West of England, Faculty of Comuting, Engineering and Mathematical Sciences, Bristol BS16 1QY, UK Three families of models and fast heuristic methods are develoed for identifying a roduction lan and immediate set-u schedule for a manufacturing line with changeover times. The initial model is exact, but is otimally solvable only for very short lanning horizons. A second modelling and solution method otimizes roduction one eriod at a time, first in a backward ass to identify target inventory levels, and then in a forward ass to build u these target inventories. Finally, a third method lans set-us and lots on a eriod-by-eriod basis, estimating the caacity usage of future set-us. All three methods are first tested under static conditions and then on a rolling horizon basis with differing degrees of demand forecast accuracy, tightness of caacity and length of horizon. Comutational exeriments confirm that even under great forecasting uncertainty the lanning horizon should extend beyond the time at which the horizon is rolled forward and the forecasts udated. Tests also show that the degree of caacity tightness and horizon length affects which aroximate models and methods are most successful. The degree of forecast error aears to have limited imact on the lanning horizon to be used and relative erformance of the models. Keywords: Production lanning; Set-us; Demand forecasts; Rolling horizons; Heuristics 1. Introduction The motivation for this aer arose from a roduction lanning and scheduling roblem encountered in the canning of liquid roducts at a drinks manufacturer. The challenge was to otimize the fulfilment of customer orders, taking into account forecast demand, the caacity of the canning line and the imact of roduct changeover (set-u) times. As the forecasts of individual customer orders and roduct-secific demand at the drinks manufacturer were rone to error, it was not clear that it was worthwhile to carry out scheduling for more than a week in advance. However, it was useful to identify a rovisional Alistair.Clark@uwe.ac.uk roduction lan for several weeks ahead in order to have advance warning of ossible roduction backlogs and be able to act accordingly. Consequently, the roosed lanning and scheduling system had the following oututs: (1) A firm roduction schedule for week 1, detailing which roducts should be roduced on the canning line, their lot sizes and in what sequence. (2) A rovisional lan showing the lot size of roduct set u in each of the subsequent weeks of the lanning horizon. Many roducts would not be roduced every week if scarce caacity was to be used efficiently. If a large number of roducts had to be sequenced in week 1, then a mathematical algorithm would almost certainly identify a more efficient roduction sequence Production Planning & Control ISSN rint/issn online # 24 Taylor & Francis Ltd htt:// DOI: 1.18/

3 82 A. R. Clark than a human scheduler. However, the straightforward nature of roduct changeover times meant that the sequencing of roducts was well handled by the manufacturer s exerienced scheduler. Consequently the mathematical lanning and scheduling model that was develoed to decide oututs (1) and (2) above did not seek to sequence the set-us scheduled for week 1. A simlified version of the model develoed for the drinks manufacturer is exlained in section 3 and then used to test the fast solution aroaches roosed in sections 4 and 5. A schedule that is otimal for forecast demand over a given horizon will almost certainly be sub-otimal when imlemented for the actually occurring demand. Furthermore, only the schedule s roduction decisions relating to the first lanning eriod would be imlemented, after which the horizon would be rolled forward and a new schedule calculated once more with udated demand, inventory and caacity information. This raises the question as to how far ahead does roduction need to be lanned in order to otimize, over the long term, the actually imlemented schedules? The answer is not obvious. If forecasts are generally reliable, then it could well ay to schedule medium-term roduction in detail in order to rovide more accurate roduction targets for the immediate eriod to be scheduled. However, if the forecasts are of oor quality, then medium-term scheduling might be unnecessary. It could be a waste of comutational resources and effort in eriod 1 to decide otimal set-u allocations in, say, eriod 3 if the demand forecasts will have changed by then. So to what extent and how should set-us, roduction quantities and their imact in eriod 3 be modelled so that this imact is taken account of in eriod 1, the (only) actually imlemented eriod of the model outut? Too crude a modelling reresentation in eriod 3 could lead to a bad eriod 1 decision if a forecast surge in demand in eriod 3 beyond immediately available caacity necessitates roduction ahead of time. If the surge fails to occur, then the eriod 1 decision of the crude model might be quite reasonable. the crude model could be just as effective as an exact model, but much more efficient (or even comutationally viable) in terms of the comuting effort invested. Accordingly, the objective of this aer is to exlore and comare the quality and comuting times of several alternative models and associated solution aroaches, articularly when imlemented on a rolling horizon basis with forecasts of demand, and then assess the tradeoffs between solution comuting effort (i.e. inut) and the imact of lanning and scheduling decisions (outut). The aer will use a simlified (but widely encountered) roduction lanning and scheduling roblem, distilled from that encountered at the drinks manufacturer, to gain insights into the research question osed above and offer guidance through comutational testing. 2. Review of revious research Closely related to the forecasting of demand is the nervousness that generally occurs when lot-sizing models are alied using a rolling horizon. The concern here is that the demand in the new eriod(s) added as the horizon rolls forward can cause changes to already lanned (but unimlemented) roduction quantities, thereby necessitating more set-us than would have occurred with a static solution calculated over the entirety of the rolled horizons. Such nervousness occurs even if demand is erfectly forecast. A further comlication is the different behaviour of models in static and rolling horizon environments. Kazan et al. (2) observed that simle heuristic aroaches, for examle that by Silver and Meal (1973), often outerform statically otimal aroaches such as the Wagner Whitin algorithm Wagner and Whitin (1958) when alied on a rolling horizon basis. The issue of how to minimize the imact of nervousness has been extensively researched by many investigators (Kro et al. 1983, Blackburn et al. 1986, Maesand Van Wassenhove 1986, Baker 1993, Kadiasaoglu and Sridharan 1997, Kazan et al. 2). A comrehensive study of dynamic lot-sizing rules, including nervousness and lanning horizon lengths, but excluding demand forecast error, was recently carried out by Simson (21). The ability of certain simle lot-sizing rules to damen nervousness was also researched by Ho (22). Generally, as one would exect, researchers have found that the longer the lanning horizon the better the rolling horizon erformance of static models (Baker and Peterson 1979). When demand is erfectly known in advance, nervousness is reorted to be accentuated by short lanning horizons (Sridharan and Berry 199). Almost all of the above studies do not consider the imact of nervousness-induced set-us on available roduction caacity through, for examle, increased total set-u time. Maes and Van Wassenhove (1986) comared several caacitated lot-sizing heuristics under rolling horizon conditions, warning that tight caacity may cause extra roblems as small lot-sizes may be lanned just to satisfy strict caacity constraints whereas in reality a certain flexibility (e.g. overtime, backlogging) exists. The model resented below in section 3 ermits backlogging so as to allow the batching of small lot sizes if this decreases backlogs overall.

4 Rolling horizon heuristics for roduction lanning and set-u scheduling 83 Kimms (1998) suggested several stability measures for caacitated rolling-horizon roduction lanning and develoed an effective single-item interactive method for reducing nervousness. He commented that if caacities are scarce, we do exect that finding a robust lan is easier than in uncaacitated cases, because the degree of freedom for rescheduling is smaller. This may turn out to be false, if even finding a feasible lan is a non-trivial task. This for instance is the case with ositive setu times. As the model below in section 3 ermits backlogging, finding a feasible lan is generally not roblematic, but the degree of freedom for rescheduling will be greater than if backlogs were rohibited. Drexl and Kimms (1997) noted that little research has been carried out into caacitated lot-sizing on a rolling horizon basis. Excetions include Dimitriadis et al. (1997) who, on a different roblem to that treated in this aer, develoed aggregate models to rovide estimates of future caacity requirements and demonstrated their effectiveness. In this aer, we will evaluate and comare both static and rolling horizon erformance of several finite caacity lot-sizing solution methods that take set-u times into account. The relative effect of nervousness will be evaluated by the imact of additional set-us in decreasing caacity available for lot roduction. Scarcity of such caacity will be felt through the inability to meet demand and hence in backlogs, the rincial criterion in the objective function of the model in section 3 below. Forecasts of demand are rarely free of error. Not surrisingly, and in line with intuition, research confirms the negative imact of forecast errors. De Bodt and Van Wassenhove (1983) showed that even small errors have a large effect on costs. Wemmerlo v (1985) found that as forecast errors increase, so do inventory and nervousness costs while the average order cycle decreases. Zhao and Lee (1993) confirmed that forecasting errors significantly increase total costs and schedule instability, reducing service levels and, in later research (Zhao et al. 1995), found that they often have a large imact on the relative erformance of lot-sizing rules. Xie et al. (23) concluded that tightness of caacity and length of lanning horizon, both of which are exerimental factors in the tests of section 6, significantly affected the imact of master roduction schedule (MPS) freezing arameters on cost, stability and service in multi-item caacitated systems under demand uncertainty. Significantly, both De Bodt and Van Wassenhove (1983) and Fildes and Kingsman (1997) confirm that imroved forecasting of demand is more imortant for erformance than the choice of lot-sizing rule, an issue that is exlored in the comutational tests of section 6. Interestingly, Zhao and Lee (1993) discovered that rolongation of the lanning horizon can actually worsen material requirements lanning (MRP) erformance when demand is uncertain but result in better erformance when demand is free of forecast error. Similarly, Sridharan and Berry (199) also found that while a large lanning horizon decreases costs when demand is deterministic, costs tend to rise as demand forecast error increases. This imortant effect of forecast error is investigated in section 6. This aer builds uon revious research by the author. Motivated by industrial roblems encountered as a consultant, Clark (1998) develoed a fast myoic rule-based heuristic for rolling-horizon lot-sizing and scheduling on a set of arallel machines with sequence-deendent set-u times. The heuristic roduced schedules that were robust to demand forecast errors. However, the model assumed just a single set-u at the beginning of each lanning time eriod. Clark and Clark (2) extended this work, modelling multile set-us er eriod and demonstrated the otential of using fast aroximate models on a rolling horizon basis. The oor quality of myoic models was artly avoided by considering forecasts of future demand, but assumed they were error-free. The current aer further develos similar models and their solution methods, and tests them under a wider variety of caacity and forecast accuracies. Clark (23) develoed aroximate static MIP models and heuristic methods to assist in identifying a caacity feasible MPS for large roduct structures in MRP systems with sequencedeendent set-u times. However, that aer used a narrower range of solution methods than the current aer, did not investigate rolling horizon use of the models and worked with erfectly-known demand. The model develoed in section 3 below is an extension of the general lotsizing and scheduling roblem (GLSP) of Fleischman and Meyr (1997). The GLSP schedules and sizes lots of multile roducts on a single finite caacity machine, ermitting multile set-us in each single large-bucket time eriod. This aer extends the GLSP to include roduct-grou set-u times and backlogs. The review on caacitated lot sizing by Karimia et al. (23) notes that there is very little literature on roblems with backlogging and multigrou joint set-us, both features of the model in section 3. The review s authors highlight the develoment of heuristics with reasonable seed and solution quality for this kind of model as an interesting research area. Thecurrent article s solution methods in subsections 4.2 and 4.3 can be seen as a contribution in that direction, although their eriod-by-eriod MIP aroach is generally alicable to a wide variety of lot-sizing models.

5 84 A. R. Clark 3. The lanning and scheduling model Consider a manufacturing line that rocesses P distinct roducts from G grous g. Production is lanned on a rolling horizon basis for each of T consecutive lanning eriods t, but demand often overshoots caacity and so backlogs must be allowed for. A generally alicable mixed integer rogramming (MIP) formulation of the roblem is: Model PPS ( roduction lanning and scheduling) min X h It þ þ I t ð1þ, t such that I, þ t 1 I, t 1 þ x t It þ þ It ¼ d t 8, t ð2þ X ax t þ s X! y t 1 þ e X l z gt 1! B t 8 t ð3þ y t z gt 8 g, 2 PðgÞ, t ð4þ x t M t y t 8, t ð5þ x t, I þ t, I t 8, t ð6þ y t ¼ or1 8, t ð7þ z gt ¼ or1 8 g, t ð8þ where the symbol 8 means for all. The decision variables are: x t Quantity (lot size) of roduct roduced in eriod t (). It þ Inventory of roduct at the end of eriod t (). It Backlog of demand for roduct at the end of eriod t (). y t Binary variable taking value 1 if roduct is roduced in eriod t, and otherwise. z gt Binary variable taking value 1 if a roduct from grou g is roduced in eriod t, and otherwise. The arameter and data inuts are: h Penalty for holding one unit of inventory of roduct from one eriod to the next. d t Forecast demand for roduct at the end of week t. I þ Current inventory of roduct at the start of eriod 1. I Current backlog of demand for roduct at the start of eriod 1. B t Available roduction time on the manufacturing line in eriod t. a Manufacturing line time required to roduce one unit of any roduct, excluding changeovers. s Manufacturing line set-u time needed to changeover between roducts if no change of grou is involved. e Extra manufacturing line set-u time needed in the changeover between roduct grous if a change of grou is involved. P(g) Set of roducts in grou g. M t Uer bound on x t, calculated as minðb t =a, P T ¼1 d Þ. In model PPS, the lanning horizon is T eriods while the scheduling horizon is 1 eriod, the length of time before demand forecasts are udated on a rolling horizon basis. The objective function minimizes a weighted sum of backlogsr and inventory quantities, the former generally having much more weight than the latter (i.e. < h 1) so as to allow, but strongly discourage, lanned backlogs rather than outrightly rohibiting them. The dimension of the backlog and inventory enalties can be secific costs, but are more likely to be cost-roxies that reflect the relative undesirability of backlogs and inventory of each roduct. If desired, backlogs can be strictly forbidden by fixing all the It values to be zero, but this is not helful when demand exceeds caacity as model PPS will not be able to identify even a feasible lan. A scheduling erson needs to know by how much and in which lanning eriods demand will be backlogged, so as to be forewarned and take aroriate action. Note that the objective function (1) is able to measure the caacity imact of nervousness through its effect rincially on backlogs and, to a lesser extent, on inventories. The first constraints (2) balance roduction, inventories and backlogs with demand. The inventory It þ and backlog It cannot both be ositive for a given air (, t), an occurrence that is revented by both having ositive coefficients in the objective function (1). Many roducts have current inventory I þ (or backlog I ) which, in a data rearation hase, is discounted from (added to) demand in the first eriod(s), resulting in zero current inventory and what is known as effective demand, calculated as follows: For t ¼ 1 to T do f Let d t ¼ maxðd t I,Þ; Let I ¼ maxði d t,þ; If I ¼ then sto; gðend of t looþ If I > then do not lan for roduct :

6 where I ¼ I þ I is the current inventory (ositive or negative) of roduct. Constraints (3) ensure that roduction and set-us take lace within the available time. The use of the different s and e set-u times reflect the common reality that a changeover between roducts from different grous takes longer than one within the same grou. For examle, on the canning line that motivated this research, consecutive roducts may be filled from the same liquid (i.e. grou), making ossible a half-hour set-u, or from different liquids, necessitating a onehour set-u. The 1 terms in (3) reflect the fact that in many lants, and certainly in the one that insired this research, the first roduct of a eriod is either a continuation from the revious eriod or a new set-u carried out in the idle time between consecutive eriods. If this is not the case, then the 1 terms may be taken out of exression (3) without altering the generality of the methods develoed and evaluated in this aer. Constraints (4) and (5) ensure that if a roduct is rocessed then both it and its grou need to be set u. Safety stocks reduce roduction backlogs of demand in the resence of demand forecast errors (Bertrand et al. 199), but are excluded from the model so that the imact of the alternative heuristics methods in reducing backlogs can be strictly evaluated. Generally seaking, the smaller the backlogs, the less the level of safety stocks that is needed. Secifically, future research, outside the scoe of the current aer, can investigate which of the heuristic methods erforms best in reducing backlogs at various levels of safety stock. Maes and Van Wassenhove (1986) oint out that caacitated lot-sizing models such as PPS are owerful and very flexible (i.e. one can add additional constraints such as maximum inventory or overtime availability), but slow (or imossible) to solve if the roblem instance is very large. In fact, model PPS is NP-hard (Paadimitriou 1994) and so the time needed to solve it otimally exlodes exonentially as the roblem size increases. In addition, as already discussed, exact otimization is illusory when demand forecasts are often revised and usets such as machine failure are common, necessitating frequent relanning. For such oerational use, the model needs to be solved heuristically if an otimal solution is not quickly available. Bearing this in mind, the next section develos several fast solution aroaches. 4. Static heuristic solution aroaches Rolling horizon heuristics for roduction lanning and set-u scheduling 85 Maes and Van Wassenhove (1986) argue from a ractical oint of view for simle but very fast algorithms, a ersective with which this aer wholly agrees in develoing the methods in subsections 4.2 and 4.3 below. They also advise the use of a routine that looks ahead in time to facilitate future caacity feasibility, an aroach we also follow. In this sirit, several alternative heuristic solution aroaches are now develoed and evaluated, first for a static fixed-horizon alication in this section, and then in a rolling horizon environment in section 5 with both erfect and imerfect forecasts of demand. 4.1 Default branch-and-cut search The first solution aroach to assess must be the lazy default simly let Clex 7.1, as an industrial strength branch-and-cut MIP solver, try to find a good solution within a re-secified amount of search time. The tests in subsection 4.4 below showed that this aroach could obtain good results within a minute or less, but often did not make the best use of limited comuting time. As exected, exlicitly rohibiting backlogs resulted in infeasible solutions for the test instances where demand was too much for the cumulative caacity available, creating unavoidable backlogs. This method was tested with search time limits of 1 seconds and 1 minute, denoted Exact1secs and Exact1min resectively. 4.2 Backward Forward method The T eriods in model PPS could each be otimized searately were it not for the inventories It þ and backlogs It sanning more than one eriod in constraints (2). If we knew or could intelligently estimate their values beforehand, then model PPS would decomose into T single-eriod models that can each be solved very quickly for the size of roblem under consideration. Thus the second solution aroach otimizes roduction one eriod at a time, first in a backward ass to identify the target inventory levels S t that are necessary to fulfil demand after eriod t, and then in a forward eriod-by-eriod ass that lans roduction to build u these target inventories. First, the Backward ass is carried out as follows: Take out the back-order variable It from the model: Let S T ¼ 8 : For t ¼ T down to 1 do f Fix It þ ¼ S t as data 8, and free the I, þ t 1 as variables 8 : Solve the model just for the I, þ t 1 and eriod t variables: Let S, t 1 ¼ I, þ t 1 8 : gðend of t looþ

7 86 A. R. Clark The target inventory levels S for eriod zero rovide indicative minimum levels for the current inventories I þ needed to satisfy demand over the T-week lanning horizon without backlogs. Assuming that effective demand has already been calculated so that current inventories and back-orders have been zeroed, then an indicator of insufficient caacity to meet overall demand will be the value P S. Then the Forward ass is executed: Restore the backorder variables It to the model: For t ¼ 1toT do f Inflate demand by the target: Let d t ¼ d t þ S t 8 : Solve the model for just the eriod t variables: Restore demand: Let d t ¼ d t S t 8 : Recalculate and fix inventories I,t 1 þ and back orders I,t 1as data 8 : gðend of t looþ: This simle Backward Forward method (denoted BF) is fast as the MIPs are small, and can result in a good lan as shown in the comutational tests in subsection 4.4 below. 4.3 Forward ass with linear set-u aroximations now aly only to eriods 1 to, with the values of the set-u variables y t and z gt having been fixed for t ¼ 1,..., 1 by revious alications of the model in the forward ass. The constraints for eriod þ1 onwards are: I þ, t 1 I, t 1 þ x t I þ t þ I t PPS B : PPS a : ¼ d t ðunchangedþ 8, t þ1 ð9þ X ax t B t or X a x t B t 8 t þ1 ð1þ The number of binary variables in model PPS B or is the same as that for each of the 2T MIPs in PPS a the BF method, but there are more continuous variables, so the solution time of each of the T constituent MIPs will be longer. However, only a single (forward) ass is needed. How much should the value of the reduced caacity B t or the inflated roduction time a* be determined? One ossibility is to do nothing and simly let B t ¼ B t for all t (equivalently let a* ¼ a). This method is denoted B* ¼ B. Three other ossibilities are also considered and tested: The third and final aroach also determines setus eriod by eriod, but in a single forward ass. Initially, the binary y and z variables reresenting the set-us for eriod 2 onwards are eliminated and comensated for by (1) either subtracting from the caacity B t the estimated total time sent on set-us, resulting in reduced caacity B * t in eriod 2 onwards, denoted model PPS B, (2) or increasing the value of the unit roduction time from a to a* in those eriods, denoted model PPS a : After solving and fixing the set-us in eriod 1, each model is reformulated so that now the set-us for eriod 2 are to be determined, with either the reduced caacity B t or the increased unit roduction time a* comensating for the lack of exlicit reresentation of set-u times in eriod 3 onwards. The model is solved afresh, the set-us in eriod 2 are fixed, and so on, with T 2 further models being solved, until the set-us for eriods 1,..., T have all been decided. Used in this manner, each model determines the set-us in eriod and has the same objective function (1) as model PPS. However, constraints (3) to (5) Use Backward Forward solution. By constraint (3), B t and a* should satisfy B t B t ¼ s X! y t 1 þ e X! z gt 1 8 t g ð11þ and X a x t ¼ X ax t þ s X! y t 1 þ e X! z gt 1 8 t g ð12þ resectively. Thus reasonable values of B t could be calculated from the quickly-obtained results x BF, y BF and z BF of the Backward-then-Forward ass: B t ¼ B t s X t! y BF t =T 1 e X gt z BF gt =T 1! 8 t ð13þ

8 averaging set-u usage over all eriods. Similarly a* could be estimated as s P a, t ybf t T þ e P g, t zbf gt T ¼ a þ ð14þ P, t xbf t These two methods are denoted B*BF and a*bf resectively Assume demands and caacity are roughly matched. Alternatively, assuming that (1) the total roduction P t x t of a roduct equals its total effective demand P t d t, which is a reasonable assumtion unless caacity is very tight and not enough to meet overall demand, (2) the canning line is using all available caacity time on set-us and roduction, which is also a reasonable assumtion unless caacity is loose, so that the right-hand side of constraint (3) is equal to B t, then the following formula for B t can be derived: P B t t ¼ B t B t a P, t d t in model PPS B T and similarly for a : P a ¼ P t B t Rolling horizon heuristics for roduction lanning and set-u scheduling 87, t d t in model PPS a ð15þ ð16þ These two methods are denoted B*Formula and a*formula resectively. By assumtions 1 and 2 above, we a riori exect them to erform best when caacity is moderately tight, as verified by the tests in subsections 4.4 and 6.2 below. X a x þ s X! y 1 þ e X l! z g 1 B ð19þ y z g 8 g, 2 PðgÞ ð2þ x M y 8 ð21þ where the decision variables are: ( 1 if roduct is roduced; y ¼ otherwise: ( 1 if a roduct from grou g is roduced; z g ¼ otherwise: x ¼ Mean quantity of roduct roduced: I ¼ Resulting mean back orders of roduct : The data rovided are d, the mean demand for roduct over eriods 1,..., T after discounting initial stocks I þ and back orders I and B, the mean available roduction time er eriod. The uer bound M on x is calculated as min( B=a, d ). Model EBA has PþG binary variables and, like the one-eriod models in the Backward-then-Forward ass, is quickly solvable to otimality. Thus values of B t could be estimated from the x, y and z of the model solution: B t ¼ B t s X! y 1 e X! z g 1 8 t ð22þ g Similarly a* could be estimated as s P a y 1 þ e P g z g 1 ¼ a þ P x ð23þ These two methods are denoted B*EBA and a*eba resectively Solve a fast MIP to estimate Bt and a*. A third alternative is to formulate a reresentative singleeriod model that catures the general relationshi between the tightness B t of caacity and the frequency ( y, z) and size (s, e) of set-us. Mean er-eriod values of demand and caacity are inuts to the following MIP whose solution is then used to estimate B t and a*: Model EBA (Estimate B t and a*): such that min X x þ I g I ð17þ ¼ d 8 ð18þ 4.4 Static comutational tests The methods of subsections 4.1 to 4.3 are now tested to comare their solution quality and comutational run times. All tests were carried out on a Sun Enterrise 45 workstation with a 4 MHz Ultrasarc rocessor and 1.5 Gb of RAM, using the mathematical rogramming language AMPL (Fourer et al. 23) for all modelling and basic calculations, calling Clex 7.1 (ILOG 21) to solve the MIPs by a branch-and-cut algorithm. The test data were based on a generalization of the P ¼ 41 roducts from G ¼ 14 grous on a single manufacturing line that normally oerated for B t ¼ B ¼ 76 hours er eriod (week) t. The demand rofiles

9 88 A. R. Clark d t were based on actual data sulied by the manufacturer, suitably modified for exerimental uroses in order to assess how well the solution aroaches develoed in this aer erformed under different conditions (such as tightness of caacity, and varying accuracy of demand forecasts). The roduction rate, excluding changeovers, was 1, units an hour, i.e. a ¼.1 hours/unit. The set-u time to change between roducts in the same grou was s ¼.5 hours while the extra setu time involving a change of grou was e ¼.5 hours also. These values are aroximate and simlified for research uroses. However, it must be ointed out that the roduction line was not erfectly under control and that in ractice the actual values of a, s and e varied over a range of 2%. This aer will not exlore the imlications when roduction arameter values are not well known and also vary, but it should be clear that under such conditions, even with erfect demand forecasts, exact otimization of model PPS is again illusory, generally resulting in sub-otimal schedules. The inventory enalty was set to h ¼.1 so as to give overwhelming riority to the minimization of backlogs over inventory. For situations where this riority is not aramount and tradeoffs between backlogs and inventories are interesting, the model PPS could be bi-objective and the efficient (Pareto) frontier of non-dominated solutions exlored to rovide the human scheduler with a choice of tradeoffs (Ergott and Gandibleux 22, Good win Wright 24). The mean demand d er roduct er eriod was set to B nðp 1Þs nðg 1Þe Pa so that caacity was either very tight (i.e. n ¼ ) d ¼ 1, 854, allowing for no set-us at all, inevitably causing backlogs), moderately tight (n ¼ :5 ) d ¼ 1, 53, allowing for a given roduct to be set-u every second eriod, robably causing some backlogs), or loose (n ¼ 1 ) d ¼ 1, 27, allowing for a given roduct to be set-u in every eriod, so that any backlog will be due to the variability of demand around the roduct s mean). The comutational tests below will show that tightness of caacity has a strong effect on the relative erformance of the solution methods. The mean demand d of a roduct was in roortion to that encountered at the manufacturer where some roducts had high demand while certain others were secialist low-volume brands. The demand rofile d t for each roduct varied normally around its mean d with coefficient of variation of.25 so that 95.5% of demands d t are within the interval [.5d, 1.5d ] with none negative. The initial inventories I þ and backlogs I were set to zero, noting that this would not be the case in a steady state under tight caacity. One hundred different demand sets {d t j ¼ 1,...,41; t ¼ 1,..., 1} were randomly generated, each of which could then be roortionally scaled for tightness of caacity as defined above. Each demand set was solved and evaluated for all 3 combinations of solution method (1 levels), tightness of caacity (three levels) and lanning horizon (T ¼ 1, 2,..., 1). The emhasis of the comutational tests was on the relative quality of fast solution methods, i.e. that would execute in a minute or less. All methods gave identical results for T ¼ 1, of course. Figure 1 summarizes the mean solution quality, as given by objective function (1), of all 1 methods over T ¼ 1, 2,..., 1 for each of the three levels of tightness of caacity. For moderately and very tight caacity, the solution values increase as the horizon T increases because there are more eriods over which caacity struggles to coe with demand. For loose caacity, the solution values for the best-erforming methods generally level off as T increases because caacity is easily able to meet demand in the later eriods. For loose caacity, most of the backlog values were zero, but some were ositive due to the absence of initial inventory to meet any above-average demand in the first eriod. For moderately tight caacity, some backlogs and stock were unavoidable. For very tight caacity, the backlog values were inevitably very large. For each of the 27 combinations of caacity tightness and horizon T ¼ 2, 3,..., 1, the Friedman nonarametric chi-squared statistical test (Lowry 23) or evaluating multile treatments was alied on the same grou of individuals (i.e. the 1 random demand sets). The null hyothesis was H : the median values of the solution objective (1) are identical for the 1 methods. In all 27 tests the solution method treatment was highly significant with -value <.1, resulting in accetance of the alternative hyothesis H 1 : the median objective values of the 1 solution methods are not all identical. Observe from figure 1 that, for loose caacity, the default branch-and-cut Exact method of subsection 4.1 was cometitive for all values of the lanning horizon T when 1 minute of comuting time was allowed. If only 1 seconds is allowed, then for T ¼ 4 onwards it was bettered by other methods, articularly for larger values of T. In fact, for T ¼ 5 onwards, method Backward Forward (BF) was not only the most cometitive, but also the fastest (taking from 1.5 seconds elased time at T ¼ 2 to just 4.2 seconds at T ¼ 1). Method B* ¼ B erformed almost as well, but took about twice as long when T ¼ 2, rising to nine times as long when

10 Solution Value Solution Value Rolling horizon heuristics for roduction lanning and set-u scheduling Loose Caacity with Static Use Moderately Tight Caacity with Static Use Exact 1 secs Exact 1 min BF B*Formula B*BF B*EBA B*=B a*formula a*bf a*eba Exact 1 secs Exact 1 min BF B*BF B*Formula B*EBA B*=B a*bf a*formula a*eba Solution Value Very Tight Caacity with Static Use Figure 1. Static solution quality. Exact 1 secs Exact 1 min BF B*BF B*Formula B*EBA B*=B a*bf a*formula a*eba T ¼ 1. In the best two methods (Exact1min and BF), 55 of the 1 instances had zero backlogs. Note the reasonable erformance of the Formula method for estimating a* and B*, and then the oor erformance of the BF and EBA methods for doing so. Observe also that each B* method generally did better than the equivalent a* method. In contrast, when caacity was moderately tight, method B* ¼ B (followed by BF) was the least cometitive for all values of T. However, methods B*BF and a*bf erformed the best for T 7, but took from 4 to 75 seconds. The exact method was the most cometitive for T 6 allowing 1 minute of comuting time, but only for T 4 if just 1 seconds was ermitted. In fact, both exact methods were unable to find feasible solutions for T 8 and 6 resectively. The Exact1secs solution value for T ¼ 5 was so large that it was lotted as way off the to of the grah in figure 1. Overall the B*Formula and a*formula methods erformed cometitively for all values of T, taking from 6 to 5 seconds of elased comuting time. Observe also that each B* method generally did the same as the equivalent a* method. When caacity was very tight, all methods were overwhelmed by the lack of caacity, building u cumulative backlogs. However, note that the exact methods were

11 9 A. R. Clark the best for small T, but once more were unable to identify feasible solutions in less than 1 seconds (1 minute) from T ¼ 6(T ¼ 9) uwards. Observe that the BF solution was noticeably the worst for all values of T. Generally, the B* and a* methods (including B* ¼ B) all erformed similarly and well, and certainly so for T 7 taking from 7 to just 11 seconds of elased comuting time. Again, each B* method generally did the same as the equivalent a* method. To conclude overall, the Exact 1 minute method is the best when caacity is loose, but otherwise only for T 6 or 7, However, for longer lanning horizons, the Exact methods are unable to even identify feasible solutions when caacity is moderately or very tight, recisely the circumstances when a lanning model is most needed. In this situation, the a =B BF/Formula methods are the best. The very fast B. method erforms reasonably well when caacity is loose or moderately tight, but is the worst when very tight. Since each B* method generally erformed the same as or somewhat better than the equivalent a* method, results from now on in this aer will be resented only for the B* methods in order to avoid over-cluttered figures. The rolling horizon tests below did include all the a* methods, but their erformance was again the generally same or a little worse than the equivalent B* method. 5. Solution aroaches for rolling horizons Only the first eriod s schedule is actually imlemented the other eriods are resent in the model just to take account of forecast future demand. If the forecasts are error-rone, then it may be effective to weight earlier eriods (articularly the first) more heavily in the objective function, for examle with a geometric decay: min X t It þ h I þ t ð24þ, t where is a measure of the error of the demand forecasts (as exlained below in section 6) and is a time-eriod weighting exonent. The less reliable the forecasts, then the greater the value of and so the less future eriods are relatively weighted in objective function (24). If ¼ then all eriods are equally weighted, but if ¼ 2 and ¼.5 or.1 (values used in the tests of section 6), then the weighting is non-trivial, as shown in figure The backward-then-forward ass Only the first eriod s lan is actually imlemented, and so the forward ass of subsection 4.2 can be curtailed after eriod 1, enabling some economy of calculation. As each model otimizes over just a single eriod, it makes no difference to weight the objective function as in exression (24) above. If demand forecasts remain unchanged between one rolling horizon and the next, then the roduction lan for the first horizon s eriods 2 to T should change only a little by the realication of model PPS, becoming the lan for the next horizon s eriods 1 to T 1. The addition of eriod T will cause some changes as fresh needs or oortunities for set-us are imlemented by the model. If demand forecasts are substantially modified, then the new horizon s lan for eriods 1 to T 1 should be exected to change significantly. How can the methods develoed for static use in section 4 be adated for rolling horizons? Several ossibilities are now considered. 5.1 The exact model PPS Model PPS, if used on a rolling horizon basis, still has to be solved in its entirety, but, as in its static use, each Clex branch-and-cut search will need to be terminated after a limited amount of comuting time, say, 1 seconds or 1 minute, as in subsection Forward ass with linear set-u aroximations The case made in subsection 5.1 for weighting the objective function (24) also alies here. The first of the T single-eriod otimizations of model PPS B rovides a eriod 1 solution that has imlicitly Weighting Alha =.5 Alha =.1 Period t Figure 2. Time eriod weights when ¼ 2 and ¼.5 or.1.

12 taken into account and fedback future caacity requirements through the use of the reduced caacity B t. This eriod 1 solution could be used on a rolling horizon basis, thus saving aroximately a T-fold reduction of comuting effort by not solving model PPS B T times. However, although the eriod 1 set-us ð y 1, z 1 Þ are fixed in the subsequent T 1 single-eriod otimizations of PPS B, the eriod 1 lot-sizes x 1 are not fixed. It is ossible that they could change as a consequence of exlicitly taking into account the caacity requirements of future set-us in the remaining T 1 single-eriod otimizations of PPS B. Thus all of the T single-eriod set-u otimizations of model PPS B in subsection 4.3 should be carried out, as was done in the comutational tests of section 6 below. 6. Rolling horizon comutational tests 6.1 Exerimental design The quality of a set of imlemented rolling-horizon decisions is evaluated by comaring their outcome to that resulting from an alternative set of decisions. In the tests below, the outcome is the eventual result, i.e. the value of the objective exression (25): X Rolling horizon heuristics for roduction lanning and set-u scheduling 91 X 1 ¼1 h I þ þ I ð25þ calculated from the imlemented roduction decisions and the actual demand that occurs as the horizon rolls forward a total of 1 times, large enough to enable valid comarisons to be made between the exerimental factors. The final eriods of the last T 1 rolling horizon models lie beyond eriod 1 and so use forecasts of demand whose actual values are never eventually known to the model and user. Five factors were taken into account in the statistical exerimental design to comare the rolling horizon outcomes, namely: (1) the length T of the model lanning horizon; (2) the solution method; (3) the degree of demand forecast error; (4) the use, or not, of weights t in objective function (24); (5) the tightness of roduction caacity. We assume that as the lead-time from the forecast to the actual demand event decreases, accuracy will imrove with the incororation of new market and customer information into the forecast. As mentioned in subsection 5.1, a arameter is used to quantify the degree of error in the demand forecasts, where ¼ corresonds to erfect demand forecasts. The base value V T for first forecast F T of demand for a given roduct at the end of the lanning horizon T eriods ahead is simulated as the eventual actual demand value V, multilied by unity lus a random element that is roortional to T, and the standardized normally distributed random variable r: V T ¼ maxf, V ð1 þ TrÞg ð26þ Thus, the larger the value of, the further the first forecast s base value V T is likely to be from the eventual value V. The values V T and V are the same when ¼. As the lanning horizon rolls forward, the udated forecast F t of demand for the same roduct in eriod t (¼ T, T 1,..., 1, ) is simulated as the interolated base value V t ¼ V þðt=tþðv T V Þ at fraction t / T of the distance from V to V T which is then, similarly to exression (26), multilied by unity lus a random element that is roortional to the eriod t, and the standard normal variable r: F t ¼ maxf, V t ð1 þ trþg t ¼ T, T 1,...,1, ð27þ The value V t assures the convergence of the forecast F t to the eventual actual demand V, as illustrated in figure 3. The larger the value of, the further the forecast F t will be from V t. When ¼, the forecast is erfect: F t ¼ V t ¼ V. This method of simulating forecasts is simle and intuitively sensible it not only incororates a random element, but also recognizes that accuracy tends to imrove as the lead-time between the forecast and the actual demand diminishes. It enables us to test how the methods in section 5 erform under varying degrees of forecast accuracy. It should be emhasized, however, that exressions (26) and (27) are not intended to reflect how eole actually make forecasts, an active subject of ongoing research (Goodwin 22). In addition, note that confirmed orders can imlicitly be considered as that art of the demand quantity which is taken as almost certain and below which the forecast is highly unlikely to fall. For examle, in figure 3 demand never falls below 5 units. The forecast uncertainty is within a range from 5 units uwards. Similarly to the static tests of subsection 4.4, 25 different eventual demand sets {F t j ¼ 1,...,41; t ¼ 1,..., 19} were randomly generated and roortionally scaled for tightness of caacity. Each of these 25 demand sets was solved on a rolling horizon basis for 1 eriods, with demand forecasts udated as described above, and evaluated for all ¼ 81 combinations of lanning horizon (T ¼ 2,..., 1), solution method (1 levels), tightness of caacity (three levels), and forecast error (¼,.5 and.1). To evaluate the imact of a non-trivial weight ¼ 2 in the objective function (24), all methods were then

13 92 A. R. Clark 1 tested again on the same 25 demand and forecast sets, for ¼.5 and.1 and all three levels of tightness of caacity. The one-eriod horizon T ¼ 1, whose solution for a given caacity tightness does not deend on the values of and nor on the solution method used, was also evaluated as a benchmark for the T ¼ 2,..., 1 tests. 6.2 Exerimental results 9 Tyical Evolution of a Demand Forecast F t to Actual Demand V = 1 when Alha =.5 Figure 4 to 6 summarize the mean quality of the solutions, i.e. the value of the objective exression (25): P P 1 ¼1 h I þ þ I calculated from the imlemented roduction decisions and the actual demand that occurs as the horizon rolls forward a total of 1 times. As such, the values can only be concetually comared with figure 1 s static T ¼ 1 values to which they are broadly similar in the equivalent case of erfect forecasts. Although some of the detail is not visible, figure 4 to 6 nicely illustrate the range of results and major atterns. Relevant details for secific solution methods are brought out in the discussion below. To answer the question osed in section 1, looking at the T ¼ 1 results for the better erforming methods, it is clear overall that roduction does need to be scheduled ahead of eriod 1 in order to otimize, over the long term, the actually imlemented schedules. In other words, on a rolling horizon basis, it is worthwhile to schedule ahead, with the ossible excetion of tight caacity under very oor forecasts of demand. The Friedman test was again alied on the 25 random demand sets for each of the combinations of caacity tightness, horizon T, forecasts error and weight secified just above in subsection V t F t Periods t before the Actual Demand V Figure 3. Tyical evolution of a demand forecast The null hyothesis was H : the median values of objective function (1), calculated from the imlemented roduction decisions and the actual demand as the horizon rolls forward a total of 1 times, are identical for the 1 methods. In all tests for t ¼ 2, 3,..., 1 the solution method treatment was highly significant with -value <.1, resulting in accetance of the alternative hyothesis H 1 : the median solution values of all 1 methods are not all identical. For examle, in the last grah in figure 5 (moderately tight caacity for ¼.1 and ¼ 2) where the solutions for different methods at T ¼ 3 aear to be very close to each other, the Friedman test gave a -value less than.1. Figure 4 shows that, with erfect forecasts, the Exact1min method continues to be the best when caacity is loose, closely followed by the BF and then B* ¼ B methods, as in the static case. Note that for these three methods the length of the lanning horizon T makes little difference to the quality of the solution when forecasts are erfect, but certainly does for the other methods. In fact, counter-intuitively, the longer the lanning horizon, the worse the quality of the already inferior B* BF/Formula/EBA solutions. As forecasts worsen, the erformance of all methods deteriorates. However, the Exact1min method still continues to be the best, closely followed by the B* ¼ B and BF methods. Observe how all but one of the other methods, including Exact1secs, erform very badly (going u and off the grah) when the forecasts are very oor ( ¼.1). Note that all methods erform relatively well at T ¼ 3 when forecasts are oor, i.e. scheduling three eriods ahead is worthwhile, even when only the immediate eriod s schedule is actually imlemented. Figure 5 shows that for moderately tight caacity, the Exact1min method also continues to be the best with Demand

14 Solution Rolling horizon heuristics for roduction lanning and set-u scheduling 93 Solution Loose Caacity for Alha= Loose Caacity for Alha=.5 Exact 1 secs Exact 1 min BF B*Formula B*BF B*EBA B*=B Loose Caacity for Alha=.5 and Beta=2 Loose Caacity for Alha=.1 Loose Caacity for Alha=.1 and Beta=2 Solution Figure 4. Rolling horizon solutions for loose caacity. erfect forecasts for T 7, closely followed by the B*Formula method, as in the static case. The next best are the B* ¼ B and BF methods, somewhat in contrast to their worst erformance in the static case. Observe again that for these five methods the length of the lanning horizon T makes little difference to the quality of the solution, but certainly does for the B*EBA method, getting worse as the horizon lengthens. As forecasts become more error-rone, the erformance of all methods deteriorates overall, although the Exact1min method still continues to be the best, closely followed broadly identically by the B*Formula, B* ¼ B and B. methods, i.e. the quality ranking is roughly maintained as forecasts worsen. Note how the best solutions are obtained with a lanning horizon T between 3 and 5 and when T ¼ 9, i.e. scheduling three to five eriods ahead is worthwhile when forecasts are oor, even when just imlementing eriod 1 s schedule. When caacity is very tight with erfect forecasts, figure 6 shows that the Exact1min method also continues to be the best for T 8, closely followed by the B* Formula/EBA and B* ¼ B methods, as in the static case. The next best is the B. method and by far the worst is the B*BF method. Again, as forecast error

15 94 A. R. Clark Solution Solution Moderately Tight Caacity for Alha= 4, Exact 1 secs 3, Exact 1 min BF 2, B*Formula B*BF 1, B*EBA B*=B Moderately Tight Caacity for Alha=.5 Moderately Tight Caacity for Alha=.5 & Beta=2 4, 3, 2, 1, Moderately Tight Caacity for Alha=.1 Moderately Tight Caacity for Alha=.1 and Beta=2 4, 3, Solution 2, 1, Figure 5. Rolling horizon solutions for moderately tight caacity. increases, the erformance of all methods deteriorates overall, with the Exact1min, B* Formula/EBA and B* ¼ B methods continuing to be the best. Note again how the best solutions are obtained with a lanning horizon T between 3 and 6 and when T ¼ 9, i.e. scheduling three to six eriods ahead is worthwhile with oor forecasts, even when only the eriod 1 s schedule is imlemented. For all tightnesses of caacity, the value ¼ 2 of the time-eriod weighting exonent in exression (24) has a slightly worsening effect on the best methods at the well-erforming small values of the horizon T. However, as can be seen in figure 4 to 6, it has a dramatic imroving imact on the methods whose ¼ solution quality deteriorated with the lengthening of the horizon T. The value ¼ 2 enables these methods to heavily discount unreliable demand forecasts far in the future, ermitting a radical imrovement in their erformance. Nevertheless, their solutions are still inferior to those of the best-erforming methods identified in figures 4 to Conclusions At the beginning of this aer, the question was raised as to how far ahead roduction needs to be scheduled in order to otimize, over the long term,

16 Solution Rolling horizon heuristics for roduction lanning and set-u scheduling 95 Solution Very Tight Caacity for Alha= 1,4, 1,2, Exact 1 secs 1,, Exact 1 min 8, BF 6, B*Formula B*BF 4, B*EBA 2, B*=B Very Tight Caacity for Alha=.5 Very Tight Caacity for Alha=.5 and Beta=2 1,4, 1,2, 1,, 8, 6, 4, 2, Very Tight Caacity for Alha=.1 Very Tight Caacity for Alha=.1 and Beta=2 Solution 1,4, 1,2, 1,, 8, 6, 4, 2, Figure 6. Rolling horizon solutions for very tight caacity. the actually imlemented schedules. The test results analysed above in subsection 6.2 show that, irresective of the quality of demand forecasts and tightness of caacity, the scheduling horizon should extend to between 3 and 5 eriods ahead. This is a useful result that, while strictly valid only for the system and data in hand, can be extraolated from the canning line, a tyical roduction system, as a guiding indicator for other systems. Turning to the exact and heuristic methods used to solve the model, the tests show that the relative static erformance of the methods generally tends to be reroduced in their rolling horizon use, which is good news for those researchers who have carried out tests only under static conditions. The one excetion is the case of moderately tight caacity where the bad static erformance of the B* ¼ B and BF methods contrasts with their relative sueriority over the very oor erformance of the B* BF/EBA methods. Interestingly, the comutational results do not mirror the MRP findings of Lee (1993) that rolongation of the lanning horizon worsens erformance when demand is uncertain, but roduces better erformance when demand is free of forecast error. However, the test results do suort the finding of Sridharan and Berry (199) that costs tend to rise as demand forecast error increases. But their other finding, namely, that a large lanning horizon decreases costs when demand is deterministic, is not suorted by the results for the data used.

17 96 A. R. Clark In addition, as Zhao and Lee (1993) found, forecasting errors do significantly reduce service levels (as measured by backlogs). However, contrary to the results of Zhao et al. (1995) with various lot-sizing rules, figures 4 to 6 show that the degree of forecast error aears to have limited imact on the relative erformance of the solution methods tested. Overall the Exact 1 minute method reigns sureme in terms of quality and robustness to forecast error, but is not viable for the longer lanning horizons T when caacity is moderately or very tight. This is not really a disadvantage (at least for data arameters similar to those used in the tests) as the rolling horizon results show that best Exact1min solutions occur over shorter horizons when 3 T 6. Nevertheless, a faster method that erforms nearly as well on a rolling horizon basis under conditions of moderately or very tight caacity, and takes a fraction of the time, may well be referable and more racticable if numerous schedules need to be generated and comared under many varying scenarios. If there are more than the 41 roducts of this study, then the Exact 1 minute method could well be imracticble under loose caacity. In this case, the test results suggest that the quicker BF and B* ¼ B methods (but not B*Formula) would rovide a good alternative, even with very oor forecasts of demand. Under moderately tight caacity, the results suggest the use of method B* ¼ BF, BF or B*Formula. If caacity is very tight, then method B* ¼ B or B*Formula (but not BF) is indicated by the test results. All three methods will be relatively fast as their MIPs cover only one lanning eriod. If the number of roducts is so large that the MIPs cannot be otimally solved in reasonable time, then limits can be imosed on each MIP s solution time, as in Clark (23). Some indication of caacity tightness, such as that in subsection 4.4, needs to enter into the choice of fast solution method. However, in the absence of such caacity information, method B* ¼ B would aear to be a robust choice that avoids bad erformance. Simlicity wins, it seems. Acknowledgements I would like to thank the referees for their valuable and constructive feedback, resulting in a better aer. References Baker, K.R., Requirements lanning. In Handbooks in Oerations Research and Management Science, edited by S.C. Graves; Vol. 4, , 1993 (Elsevier Science: New York). Baker, K.R. and Peterson, D.W., An analytical framework for evaluating rolling horizon schedules. 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18 Rolling horizon heuristics for roduction lanning and set-u scheduling 97 Kro, D.H., Carlson, R.C. and Jucker, J.V., Heuristic lot-sizing aroaches for dealing with MRP systems nervousness. Decision Sciences, 1983, 14, Lowry, R., Concets and Alications of Inferential Statistics, 23 (htt://faculty.vassar.edu/lowry/webtext.html). Maes, J. and Van Wassenhove, L.N., Multi item single level caacitated dynamic lotsizing heuristics: a comutational comarison. Part ii: Rolling horizon. IIE Transactions, 1986, 18, Paadimitriou, C.H., Comutational Comlexity, 1994 (Addison-Wesley: Reading MA). Silver, E.A. and Meal, H.C., A heuristic for selecting lot size quantities for the case of a deterministic time-varying demand rate and discrete oortunities for relenishment. Production and Inventory Management, 1973, 14, Simson, N. C., Questioning the relative virtues of dynamic lot sizing rules. Comuters and Oerations Research, 21, 28, Sridharan, V. and Berry, W.L., Freezing the master roduction schedule for make-to-stock roducts under demand uncertainty in a rolling horizon lanning environment. Decision Sciences, 199, 21(1), Wagner, H.M. and Whitin, T.M., Dynamic version of the economic lot size model. Management Science, 1958, 5, Wemmerlo v, U., Comments on Cost increases due to demand uncertainty in MRP lot-sizing : a verification of ordering robabilities. Decision Sciences, 1985, 16(4), Xie, J., Zhao, X., Lee, T.S. and Goodale, J., Freezing the master roduction schedule under single resource constraint and demand uncertainty. International Journal of Production Economics, 23, 83, Zhao, X. and Lee, T.S., Freezing the master roduction schedule in multilevel material requirements lanning systems under demand uncertainty. Journal of Oerations Management, 1993, 11, Zhao, X., Lee, T.S. and Goodale, J., Lot-sizing rules and freezing the master roduction schedule in MRP systems under demand uncertainty. International Journal of Production Research, 1995, 33(8), Alistair Clark is a Princial Lecturer in Oerational Research (OR) in the Faculty of Comuting, Engineering and Mathematical Sciences at the University of the West of England, Bristol. He was formerly a Production Systems Secialist at UniSoma (an OR consulting and systems develoment comany in Brazil), an academic adviser at IBM s Latin American Institute of Technology, an analyst in the UK government OR Service, a lecturer at several UK universities, and a VSO volunteer teacher in Wa, Ghana. Alistair has led roduction lanning and scheduling consulting and systems develoment assignments for Polyenka, Tilibra and Thomson Consumer Electronics in Brazil, and Matthew Clark Brands in the UK, among other comanies. He belongs to the UK Oerational Research Society (ORS) on whose Council he reresents the Western region of England. He is also the UK reresentative on the Council of EURO the Association of Euroean Oerational Research Societies.