The Division of Labour under Uncertainty. Nigel Wadeson *

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1 Te Division of Labour under Uncertainty By Nigel Wadeson * Date of First Submission: 19 t May, 011 Date of Second Submission: 9t May, 01 How to cite tis paper: Wadeson, Nigel (013), "Te Division of Labour under Uncertainty," Journal of Institutional and Teoretical Economics (JITE), Publised Online First April, DOI: / X Abstract Reductions in te division of labour are a significant feature of modern developments in work organisation. It as been recognised tat a reduced division of labour can ave te advantages of job enricment and lower coordination costs. In tis paper it is sown ow advantages from a lesser division of labour can stem from te flow of work between different sets of resources were te work rates of individual production stages are subject to uncertainties. Bot process and projectbased work are considered. Implications for te boundaries of te firm and for innovation processes are noted. Keywords: Coordination, Division of Labour, Uncertainty, Production, Project JEL: D0 * Department of Economics, University of Reading 1

2 1. Introduction Tis paper explores implications for te division of labour tat result from uncertainties in production processes. Te firm as to balance te need for workers and oter resources to be available to carry out tasks wen needed against keeping tem from idleness and from carrying out tasks wic do not make full use of teir skills and capabilities. Due to work rate uncertainties, availability as to be traded off against lower utilization rates if a rigid division of labour is maintained. Literature on te division of labour seems so far to ave given suc uncertainties little attention. In fact, wit te notable exception of inframarginal analysis (Yang and Ng, 1998), tere as been a relative deart of modern teoretical work on te division of labour (Stigler, 1976, p. 109; Ceng and Yang, 004). Empirical work by management researcers and sociologists also suffered a substantial decline from te early 1970s (Carter and Keon, 1986). Te division of labour as been central to our understanding of te organisation of production and of economic progress since Adam Smit s (1776) Wealt of Nations. According to Smit, te division of labour is a powerful metod of increasing productivity: it improves dexterity, eliminates time spent in switcing between tasks, and leads to improvements in tools and macinery. Later, Babbage (183) pointed out tat te division of labour allows ig-skilled workers to concentrate on ig skill tasks rater tan spending some of teir time on tasks wic do not require teir skill level, wile lower paid, lower-skilled workers perform te low skill tasks. Tis is known as te Babbage Principle and can be considered as a key part of Taylorism. It as also been argued tat te division of labour is advantageous to te firm in allowing it to exercise more control over workers and tat a greater division of labour will develop were worker power to oppose management decisions is low (Reinstaller, 007). According to Adam Smit, te division of labour is limited by te extent of te market as, in order to be more specialised, workers need to face large enoug markets for teir specialist outputs. Te division of labour is terefore increased wen barriers to trade are reduced so tat markets become less fragmented. Improvements in productivity generated troug an increased division of labour are traded off against increases in bot coordination and transportation costs (Houtakker, 1956). Becker and Murpy (199) claimed tat increasing te division of labour leads to greater agency costs and old-up problems, te communication of misleading information, and breakdowns in production caused by poor coordination, also stressing tat a growt in knowledge leads to increases in specialisation. A reduction in te division of labour is sometimes termed job enlargement. Tis refers to widening te number of tasks undertaken by a worker and as often been described in terms of te motivational advantages of job enricment (Parker et al., 001). It as also been sown tat job enlargement can reduce non-productive time, wic includes balance delays (Conant and Kilbridge, 1965, 383-5; Kilbridge and Wester, 1961). Balance delays occur due to bottlenecks in

3 production systems (Kilbridge, 1960) were one stage processes units at a lower rate tan oter stages (Matanacat and Yano, 001). Georgescu-Roegen (1970) saw te elimination of delays as central to explaining te success of te factory system of production (see also Morroni, 1999), as idleness can be eliminated were production is sufficiently large and a number of processes are arranged in a production line. Note, owever, tat work rates are normally subject to uncertainties. Delays are terefore caused by variances in work rates and not just by differences in average or constant work rates. Te term variance delays can be used to differentiate delays caused by bottlenecks resulting from variances in te production rates of different stages of production from tose caused by differences in average rates. Variance delays ave been recognised in operations researc (Scultz et al., 1998), in particular tat greater inventory olding between stages reduces tem, and tat aving a greater number of stations working in parallel in a stage of production increases te predictability of its output (Buxey, 1974). Te term task consolidation refers to combining multiple tasks into one so tat tey are undertaken by te same workers or resource set. Task consolidation often accompanies a decentralization of decision making (Seidmann and Sundarajan, 1997). Rummel et al. (005) ave considered its advantages in eliminating andoff delays between activities including delays in transferring knowledge and materials and in waiting for te resources to undertake te next stage to become available to start teir work. Note tat literature on project planning normally takes te activities as fixed, not considering te possibility of task consolidations. In addition, most literature on resource constrained project sceduling does not consider te effects of uncertainty (Ballestín and Leus, 009). Employee and macine flexibility and team working are stressed under lean production wic involves teams of multi-skilled workers (Alony and Jones, 008; Womack et al., 1990). One advantage of flexible equipment and workers is tat tey facilitate te rapid switcing of a production line between different products wic would oterwise be more disruptive to te production process. Te importance of limits to te divisions of labour are also reflected in te stress tat employers ave placed more generally in industrial relations on increased employee flexibility and in te central part tat task consolidation plays in business process re-engineering (Rummel et al., 005). In addition, under bucket brigade manufacturing workers move from station to station wit te product, wic makes te line to some extent self-balancing (Bartoldi and Eisenstein, 1996). As modern production tecniques often involve reductions in te division of labour it is important tat we are clear about te circumstances under wic a greater or lesser division of labour is advantageous. Matters ave canged significantly since te days of Adam Smit. For instance, IT systems, including te use of flexible, programmable production macinery (Milgrom et al, 1991), 3

4 can significantly affect te breadt of tasks tat workers undertake (Lindbeck and Snower, 000; Borgans and ter Weel, 006). Also, te skills needed to operate different types of macines can be similar. In fact, taking Adam Smit s famous example of a pin factory, Pratten (1980) commented tat, not only ad pin making macines replaced many separate operations, but tat eac operative also controlled multiple macines. Modern production metods also often involve little or no in-process stock between production stages, so tat te balance of work rates becomes of muc greater importance (Piore, 1986: p. 7). Hig quality standards also make workflow more vulnerable to quality variations at different stages of production. Advanced manufacturing tecnologies increase interdependencies between different parts of firms (Zammuto and O Connor, 199: ). Additionally, services make up a large part of many modern economies and delaying te completion of a service by queuing units of output during te production process can amount to a serious reduction in service quality. Te division of labour is also clearly important in understanding different forms of project-based organisation. Some projects involve a succession of different trades. In oters te same team largely sees tings troug from start to end. Te sceduling of resources is an important form of coordination, bot for te firm s internal resources and in sceduling te work of oter firms resources involved in a project, wic elps to ensure tat tey will be available at te times needed. So, for instance, a firm contracted to do work on te project will want to know wen it will be able to start. It will need to scedule its resources to do oter work before and after wen tey will be needed on te project. However, project scedules ave to be repeatedly revised due to te uncertainties tat exist wen tey are made. Suc revisions disrupt resource coordination. Te paper will proceed by first considering te implications of task consolidations for process-based work, suc as a manufacturing line. It will ten go on to consider project-based work. Te term process-based is used to refer to production tat is on-going wit eac stage of production processing different units of output at any time. Wit project-based work it is assumed tat tere is only one unit of output. Here, any particular resource may only be needed for a single task or a restricted set of tasks witin te project. Te commencement of te work of furter resources may ten depend on a task done by te resource in question aving been completed or aving reaced a certain stage. Te initial consideration of project-based work is followed by a furter section giving a more detailed model demonstrating te benefits of task consolidation.. Process-Based Production Consider process-based production. Ideally eac stage of production will work consistently at te same rate as te oters. Once one stage as finised working on a unit of output, te next stage will ave just become ready to work on it and te previous stage will ave just finised working on te next unit of output. Hence, te production system is balanced across its different stages. However, assume instead tat tere is variation in te productivity of eac stage of production. If, at 4

5 any time, some stages are working faster tan oters ten te system is out of balance. Te stages completing work at a lower rate ten act as bottlenecks in te system, resulting in variance delays. Sort-lived imbalances can be buffered wit stocks of part finised outputs, were applicable. A stage may ten still be forced into idleness once stocks from te previous stage ave run out. It may also be forced into idleness wen available space in wic to store stocks of its own output run out. Suc space is itself costly. Buffering wit inventory can be costly. It involves not only te costs of olding stock, but also, were units of output sit idle between stages of production, te costs of delaying supply in te case were individual units are produced to order. In te case of some services, te customer may be kept idle in passing from one stage of a service to anoter, suc as wen being passed from one telepone extension to anoter and being eld in a queue of calls. Inventory buffering can also postpone te identification of a fault in te output of a stage, creating a delay before te next stage starts to work on eac unit. Te consolidation of two or more consecutive stages of production, reducing te division of labour so tat te same set of resources carries out eac of tem, is one metod of addressing tese problems. Workers can continue working on eac unit of output until tey ave finised all of teir stages of production. Balance across te stages concerned is tus in-built. Say tat tere are two production stages tat could be consolidated. Were tere are multiple units of resources in a stage assume tat tey are working in parallel wit eac simultaneously carrying out te full stage on different units of output. Wen te production stages are unconsolidated, te rate of output of resource unit i in te first stage of production is r 1di. In te second stage te rate of output of resource unit j is r dj. Assume tat te production rate of eac resource unit as a random component,, so tat r1 di r1d 1 di and r dj r d dj. Tere are q 1d units of resources in te first stage and q d in te second. For simplicity, assume tat tere is no inventory between te stages. Eac unit of output passes directly from one stage of production to te next. Te overall rate of production of te non-consolidated stages at any time is ten: q1d qd 1d 1di d i1 j1 (1) R Min( q r1d, q r d ) d Hence, even in te case were te average rates of production are balanced, if te random components of te two stages ave different signs or magnitudes ten one stage of production is constrained to operate below its potential rate of output. Te resource quantities, q 1d and q d, are, of course, endogenous. For instance, if te resources used in one of te stages of production are very costly, ten tey can be kept operating more fully to teir potential by employing extra resources in te 5 dj

6 oter stage. Tis is at te cost of incurring increased expected idleness in te oter stage. Hence, production balancing based on average rates of output is not necessarily efficient. Now consider te consolidated case. Te total number of resource units used in te two stages is now q c. Te rate of output of eac unit of resources, k, per unit of time spent in eac stage is now r 1ck in Stage 1 and r ck in Stage, were r1 ck r1c 1 ck and r ck r c ck. Assume tat no resource time is now wasted. Eac unit of output continues to be worked on by te same resource unit until it is finised. Te overall rate of output of eac resource unit, k, now averages out te rates of te two stages being te inverse of te sum of te times taken to complete eac of te two stages: r ck r r 1ck ck 1ck r r ck If one of te stages slows down ten te workers increase te output from te slowed-down stage by using some of te time tey would normally use in te oter stage of production. Hence, te rate of output across te consolidated stages as a wole is not as badly affected as if te stages ad teir own dedicated workers. Te extent of te bottleneck in te wider production system is reduced. In addition, te overall rate of output of te consolidated stages is likely to be more tigtly distributed around te mean, relative to te mean, tan for te unconsolidated tasks. Tis is inerent in te fact tat te time taken to process a single unit involves te sum of te random elements involved in te time taken for a resource unit to complete eac stage, so long as tey are less tan perfectly correlated. If work in one of te consolidated stages of production slows down, te oter may progress at its normal rate or speed up. An extreme value of te sum of a number of random terms tat are not igly correlated as a low probability as it requires tat every term is of large magnitude and as te same sign. So, for example, te sum of two identical and independent uniform distributions as a triangular distribution. Furter, if we define r c as te expected value of r ck, ck as te random component of r ck, and R c as total production summed over every resource unit, k, ten. () Rc qc r c q c k1 ck A furter advantage of consolidation is tat it can lead to a greater degree of parallelism of production as q c will often be greater tan eiter q 1d or q d. Tis 6

7 again reduces te variability of production rates due to te summation of te random terms in Equation. For instance, say tat tere are two identical macines plus teir operators in eac of two separate stages of production. If one macine breaks down ten te capacity of its stage of production is alved, so creating a substantial bottleneck in te overall production process. Now say tat te two stages are consolidated and tat tere are now four identical macines and teir operators wic eac carry out bot stages. Now if one macine breaks down capacity falls by only a quarter. Te bottleneck is muc reduced. Consolidation could be described in terms of making te cain of production stages sorter and fatter. Tere are not only less interfaces between stages of production; tere is also less dependence on individual workers and macines. Tere can terefore be significant advantage in macines and workers being capable of undertaking multiple production tasks. Te production rates of individual resources can also vary if te nature of te resources used depends on weter or not tasks are consolidated. For instance, a macine for consolidated production migt be more complex and more likely to break down tan a more specialised one. On te oter and, it migt be a simpler, more general purpose item of equipment. Note also a source of economies of scale inerent in te summations of te random terms in Equation 1. If larger scale results in a greater parallelism of activities in a stage of production ten tis reduces te variability of te overall output of tat stage of production, so facilitating te division of labour. However, increased scale will also involve a discontinuity were a different production tecnology is employed tat results in a muc iger rate of production per unit of resources, or if te new tecnology is more automated and makes te production rate less susceptible to worker related variations. Te above discussion assumes tat resources cannot be added or sed instantly and witout cost. Te firm cannot count on being able to buy on spot markets as any one of tem may lack available supply. Searcing and matcing also takes time. So does te induction of new workers into te firm. Similarly, workers cannot be sure of being able to find alternative work at sort notice. Workers wo are not employed by te firm, and terefore given some degree of forward security of demand for teir services, may contract wit oter firms and so become unavailable. Indeed, tey ave an incentive to seek forward contracts in order to avoid being unemployed in future time periods. However, under certain circumstances, te firm can instead use te flexibility tat it as over te coordination of its internal resources. For instance, it could use resources tat are capable of working across multiple stages of production and redeploy some of tem between stages as and wen imbalances occur. If one stage is processing units of output at a iger rate tan anoter, ten resources will be transferred in order to rebalance te work rates. Evidence on te benefits of tis is cited by Daniels, Mazzola and Si (004). Note tat it may be possible to deal 7

8 wit a significant proportion of te variance by only making a fraction of te resources flexible in tis way. Suc redirection could be acieved troug ierarcical management control or troug a degree of self-management of production teams working under appropriate incentives. Te latter could facilitate quick decisions based on knowledge of current conditions eld witin te teams. 3. Project-Based Production Now consider a project suc as a new product development or construction project. Assume tat different specialist resources could be used in eac of its stages. Eac resource as to be brougt into te project to perform its tasks and, once tey are complete, is no longer needed. Te project scedule as uncertain timings. Here, te uncertainty is not about te number of units of output processed in eac time period. Rater, it is ow long eac task witin te project will take to be completed, or at least to reac a point wic allows oters to start work. As above, te firm cannot rely simply on spot contracting for resources. Wole sets of resources need to be available simultaneously and in sequence so tat coordinated availability across multiple markets is needed. Te firm can instead forward book te resources. However, given te uncertainty in te project s scedule, te firm does not know exactly wen tey will actually be needed. If te divisions of labour are rigidly maintained, terefore, ten were a delay in a resource starting and completing its work will be costly, tere is an incentive to insert extra time buffers into te scedule (te use of buffers is common in projects), so tat a resource is sceduled to be available earlier tan it migt actually be needed and is sceduled to finis later tan its work migt actually be completed. Widening te time periods of te resource bookings increases te cances tat tey will be available wen it will turn out tat tey will actually be needed. In addition, tere may be uncertainty as to weter te resources will actually become available wen booked, as tey may not complete previous work to scedule and so be late in starting work on te project. Tese problems apply weter te firm is contracting for resources in te market or weter it is using internal resources; tere are competing demands for resources witin te firm and teir use needs to be coordinated troug sceduling. Te costs can be split into four components wic are traded off against eac oter in determining te optimal start time and duration of te resource booking. Assume tat te task in question can only be started once te preceding project task as been completed. Firstly, z s1 is te expected cost due to te possibility tat te preceding task ends after te resources for te task ave been sceduled to start teir work. Assume tat tis keeps tem idle wile waiting for te preceding task to be finised. Note tat anoter possibility in reality is tat tey would move to anoter project wic could ten keep tem unavailable for some time. Secondly, z s is te expected cost due to te possibility tat te preceding project task is completed before te resources for te task become available to start teir work. Tese are costs of te project being unnecessarily delayed wile awaiting te resources. z s1 and z s depend bot on te uncertainty over te duration of te 8

9 preceding task and on te endogenously determined time, S, at wic te resources are sceduled to start work on te task. Te minimum possible value of S is at te earliest time tat te preceding task migt be completed and its maximum value is at te latest possible time tat te preceding task migt be completed. A later start-time of te resource booking, S, decreases z s1 wile increasing z s : δz s1 /δs<0 and δz s /δs>0. As te start time becomes later it becomes progressively less likely tat te resources will be left idle wile waiting for te previous task to end and more likely tat tere will be a delay between te end of te previous task and te start of te resource booking. Hence, as S increases, z s1 falls at a decreasing rate and z s rises at an increasing rate: δ z s1 /δs >0, δ z s /δs >0. Te marginal benefit of a later start time, in terms of a reduction in z s1, is terefore downward sloping and te marginal cost in terms of an increase in z s is upward sloping. Te optimal start time is earlier te iger are te costs of delay and te lower are te costs of te resources per time period. Were a task s duration is fairly long it may be quite likely tat it will be possible to lengten te booked resource time after a delayed start. Were tere is some likeliood of not being able to do tis ten for a given duration of te resource booking, due to te increased expectation of some idleness before starting work, an earlier start time increases z e1 and decreases z e, were z e1 is te expected cost resulting from te possibility tat te sceduled resource time ends before te task is finised and z e is te expected cost of te resource booking ending after te task as been completed. Assume tat te latter involves expected costs of idleness. In reality, owever, workers migt actually slow down to fill te time available. However, te effects of an earlier start time on tese two expected costs can be countered by an increase in te duration, T, of te sceduled resource time, δt/δs <0, in order to make up for te expected idleness before starting work. Te task end time is made more uncertain by an earlier start time, S. If tere were furter tasks tat could only start wen te task ad been finised ten tis would ave knock-on effects on teir sceduling. Tis terefore gives an increased incentive to scedule later start times for te resources undertaking eac task, so delaying te expected project completion date. Te longer te duration of te resource booking, te lower is z e1 and te greater is z e : δz e1 /δt <0 and δz e /δt >0. Increases in te duration are progressively more likely to result in extra idle time rater tan avoided delays: δ z e1 /δt >0 and δ z e /δt >0. Te marginal benefit of a longer duration, in terms of a lower z e1, is terefore downward sloping and te marginal cost in terms of a iger z e is upward sloping. Te optimal duration will be longer te more costly are delays per time period, resulting in greater expected idleness. It will be sorter te iger are te costs of te resources per time period, resulting in iger expected delays. Te total of te expected costs, over and above tose tat would be borne if te resources could be sceduled wit certainty over ow long eac task will take, or 9

10 if tey could be obtained instantly as an wen needed, is: z z z z z s1 s e1 e Sceduling uncertainty terefore results in bot expected costs of idleness and of resources not being available wen needed. Te expected cost, z, will be ig wen uncertainty over te end time of te previous task and over te duration of te task itself are ig and were te costs of bot te delays (and disruptions) caused by non-availability and te costs of resources per time period are ig. Non-availability will be more likely were te resource type is fairly scarce in te firm and in te market, suc as may be te case were specific knowledge is involved. Note tat te duration, T, will be set close to te latest possible completion time for te task and te start time close to its earliest possible value if delay costs dominate resource costs. However, if tis reflects very ig delay costs ten te expected costs of idle resources may still be large in teir own rigt. Similarly, if resource costs dominate delay costs ten te start time will be set at near te latest possible time and te duration near its sortest possible value, so resulting in significant expected delays. However, te firm does not simply ave to accept tese expected costs, even after minimising tem in terms of selecting te sceduled start times and durations. Instead tey provide an incentive for task consolidations wic reduce tem in te following ways. Firstly, if te same resources undertake multiple tasks ten teir overall work duration is made more predictable relative to te mean tan te durations of te individual tasks. Hence, te expected costs due to te possibilities tat a sceduled resource duration will turn out to be insufficient or tat it will be too long (represented by z e1 and z e above) are bot reduced. Te overall sceduled resource time can terefore be bot sorter and more effective. Secondly, some tasks will be very unlikely to run out of forward booked resource time. For instance, if a number of tasks are consolidated, undertaken one after anoter by te same resources, ten te earlier ones will be very unlikely to take so long as to exaust te total booked resource time. Hence, te risk of disruption costs being incurred wen tose tasks run out of sceduled resource time (represented by z e1 above) is reduced or eliminated by consolidation. In addition, if te work begins to slip beind scedule ten managers ave more time to react in order to gain extra resource time before te sceduled time runs out. Tirdly, resources used to carry out a consolidated sequence of tasks are available to undertake teir next task once tey ave completed teir current task. Hence, te expected costs z s1 and z s are bot eliminated if te task is consolidated wit te previous task. Tere will often also be advantages in workers being able to utilise project specific knowledge gained in earlier tasks wile undertaking later tasks and also in avoiding te costs of transferring materials. Finally, sometimes projects will require some reworking of tasks. Tis can result 10

11 from information gained during te performance of later tasks. If a worker is used across a range of project tasks ten it is more likely tat tey will still be working on te project wen te need for reworking is discovered. Tis is particularly advantageous wen te individual concerned as significant project specific knowledge. A furter strategy, as wit process-based work, is to move resources between tasks witin a project, or from oter activities witin te firm, as and wen required. Again, resources must be available tat are capable of undertaking te task types in question. Te more tat te firm's resources are capable of working across multiple task types, te greater te flexibility tere will be. However, tere will also be clases of resource needs witin te firm. A resource will not always be available to transfer immediately into te project even wen it is internal to te firm. Tere will be more freedom to move resources out of oter tasks if tose tasks are relatively non time-critical and if te disruption costs involved are low. It is significant tat different tasks witin a project ave different levels of time criticality. For instance, if te laying of te foundations of a ouse is completed too late ten oter tasks, suc as building te walls, will be delayed wile if turf laying in te garden is delayed somewat ten it may ave no effect on oter tasks. One strategy would be to consolidate critical pat tasks, giving te advantages set out above. However, critical pat tasks migt also be consolidated wit non-critical pat tasks. A critical pat task could ten take sceduled resource time from non-critical pat tasks as and wen needed. It could also release resource time to tem once completed. Hence, overall resource bookings could be made wic would ensure tat te critical pat task would not become sort of resource time. Note tat a similar strategy could also be applied to process-based work. Wile work on te production line itself will often be timecritical, if some of a worker's time is allocated to oter tasks tat are not ten te worker can be switced into tem wen not needed on te production line and also back from tem wen necessary. Masten et al. (1991, p. 1) found evidence of skilled workers in a large naval construction project being kept busy in tasks tat were not time-critical in order to utilise tem for more significant periods of time, keeping tem occupied outside times wen tey were needed to carry out teir primary tasks. Love (010, pp ) reports evidence of internalisation due to time criticality and a ig cost of delay even in te absence of opportunistic old-up. Hameri and Heikkila (00) give case study evidence sowing major delays between a project task being completed and te commencement of te subsequent task. Tey also present evidence demonstrating te importance of good communications on task progress and reallocations of resources in leading to te improved interfacing between different tasks. Hammer and Campy (1993) provide furter evidence of delays between tasks. Serpell et al. (1997) give evidence of labour and equipment lying idle in Cilean construction projects observing c. 53% of work time being spent on non-productive activities. Eden et al. (000) discuss ow small delays and 11

12 disruptions can ave serious knock-on effects on a project. 4. A Model of Project Task Consolidation Te model tat follows demonstrates ow task consolidations, under wic te same resources are made to carry out more tan one task one after anoter, can ease te effects of a combination of project scedule uncertainty and resource constraints. Te model demonstrates te first two of te above sources of advantage from task consolidation. Firstly, task duration uncertainties are consolidated, so resulting in a more favourable probability distribution. As explained above, te summation of independent random variables, in tis case task durations, results in a variable more tigtly distributed around te mean, relative to te mean. Secondly, were tasks are consolidated, an earlier task is less likely to be disrupted troug te resource booking not being long enoug to complete it. Indeed, te overall resource booking may well be longer tan te maximum time tat migt be taken to complete te first task. Te model involves two optimisations. Firstly, te optimal resource booking for a non-consolidated task is derived. Ten te same is done for consolidated tasks. Tis ten allows te expected costs of consolidated and non-consolidated tasks to be compared. First consider te case were resources are to be allocated to a single task in isolation. Tey are to be booked to carry out only tat task and ten leave te project. Forward booking reserves resources to work on te project for a specified number of time periods. Te duration of te task, t, is uniformly distributed (0, ]. Hence, te probability density of t is 1/. Te initial resource booking is for a duration of T (T ) time periods. Te cost of te resources per time period is w (w>0). For simplicity, assume tat te resources actually do become available on te date for wic tey are booked to start work and tat te state of te project is suc tat tey can start work on tat date rater tan sitting idle. Tis places te focus of te model on te uncertainty over ow long te task itself will take. If te initial resource booking turns out not to be long enoug for te task to be completed ten an extra expected cost of D (D>0) is incurred due to disruption of te work. Tis migt actually involve eiter a delay to te next task tat te resources will work on so tat tey can complete teir current task before moving on (in wic case D migt be a penalty carge) or a delay wile waiting for furter resource time. Tis as to be weiged against te cance tat te initial resource booking turns out to be longer tan required to complete te task, in wic case, for simplicity, it is assumed tat te resources stand idle from te time of completion of te task until te end of te booking (rater tan being expected to be used in some less tan ideal way). Te expected extra costs, Y, resulting from possible disruption or resource idleness over and above te costs of te resource time tat will actually turn out to be needed to carry out te task, tw, are te disruption cost, D, multiplied by te probability tat te task takes longer tan te duration of te resource booking, T, plus te per period resource cost multiplied by te expected value of te duration 1

13 of booked resource time tat migt be left over following te completion of te task: (3) Y dt ( T T D T 0 w TD T w D t) dt Differentiating tis wit respect to T gives: dy dt Tw D Setting tis equal to zero gives te optimal resource booking: T * 01 D w Te corresponding value of Y, obtained by substituting te value of T 01 * into Equation 3, is: (4) * Y1 D D w Tis illustrates te trade-off between saving on te costs of a longer resource booking and avoiding te disruption costs tat result wen te booking turns out to be too sort. If resource time is ceap relative to te potential disruption costs ten te optimal booking is relatively long. Tese values apply so long as D w, oterwise T as reaced its maximum value (i.e. te maximum time tat te task could possibly take): * T0 Te value of Y corresponding to T=T 0 * is: (5) * w Y Now consider te case were tere are two suc tasks, A and B, and a strategy of task consolidation is applied so tat Task B is carried out immediately after te completion of Task A by te same resources. Tis means tat a single resource booking is made to cover bot tasks. Tis consolidates te uncertainties over te completion times of te tasks. If one of te tasks is completed slowly compared to 13

14 expectations ten te oter may be completed relatively quickly. Te initial resource booking for te consolidated tasks is again T (T ). First consider te case were te resource booking is at least as great as te maximum time taken to undertake Task A, T. Tis is equivalent to assuming tat D w/, as is sown later. Te expected extra costs due to te possibility tat te tasks will be completed before te end of te initial resource booking are as follows. Note tat tis expression allows bot for te case were Task A is completed early enoug for tere to be more tan te maximum time to complete Task B left, t A <T-, and te case were it does not, t A T-. In te latter case, for bot tasks to be completed before te end of te resource booking requires tat t B T- t A. z 11 w 6 T w ( T t A tb ) dt Bdt A 0 0 T 3 6T 6 T T 3 T t A 0 w ( T t Tis result can be interpreted by noting tat it can also be obtained by utilising te fact tat te sum of te two durations wit independent uniform distributions as a duration, t, wit a triangular distribution. Note tat te apex of te triangle of te distribution is at t=, te probability density to te left of te apex (0<t ) is t/, and to te rigt of te apex up until te maximum possible value of t is /-t/, giving te following expression: t T t z 11 ( T t) wdt ( T t) wdt 0 Te expected cost resulting from te possibility tat te initial booking of resources will turn out not to be long enoug to complete Task B, wic requires tat tere are less tan time periods of te booking left after te completion of Task A (t A >T-) and tat Task B takes more tan te remaining duration of te booking (t B >T- t A ), is: z1 T T t A D D T dt B dt A Te overall expected extra costs, over and above tose tat would be incurred if te resources were booked for te exact actual durations of Tasks A and B, is Z 1 : A t B ) dt B dt A (6) Z 1 = z 11 + z 1 14

15 Differentiating Z 1 wit respect to te duration, T, gives: dz dt 1 1 4wT DT wt w 4D Te optimal value of T is tus: T 1 w * 1 D w D w Note tat tis value is increasing in D and approaces as D rises to values tat are very ig relative to w. Hence, were disruption costs are ig relative to resource costs te duration of te resource booking for te consolidated tasks, being close to te maximum duration of te tasks in order to largely eliminate te possibility of disruption costs being incurred, is close in value to te sum of te resource bookings made wen tey are not consolidated. Substituting tis value into Equation 6 gives te associated value of Z 1 : * 1 3 3/ 3 3 Z1 D ( w D ) 3D w D 6 w 6 w ( D w D 6 w For convenience, define r as a measure of te strengt of potential disruption costs relative to resource time costs, suc tat r=d/w. Now consider te superiority of consolidation. Tis is measured in terms of te focus of te model wic is te sceduling problem. It sould terefore be interpreted relative to oter factors not included in te model, particularly weter using resources specialised in individual tasks as advantages over te same resources undertaking bot tasks. Oter factors tat migt favour consolidation sould also be noted, suc as project specific knowledge gained in te first task being useful in te second and any incentive advantages tat migt result. Te superiority, S 1, of consolidation over te two tasks being carried out by separate resource sets, were r 1 (i.e. D w) so tat te value of Y is Y 1 * as defined in Equation 4, is: ) * 1 S1 Y Z * 1 Expanding tis gives: S ( r ) r w r r

16 Were r 1, so tat te value of Y is Y * as defined in Equation 5, te superiority of consolidation is instead given by S : S Y * Z * 1 ( r w ) r 3 r Now consider te case were te duration of te booked resource time, T, is for less tan te maximum time tat it will take to complete Task A, T<. Note tat tis puts T to te left of te apex of te triangular distribution of te duration of te consolidated tasks. Assume now tat eac task as a minimum duration, m, and tat T, t A, t B, and are terefore measured as times over and above m. Hence te overall booking is m+t, and te overall duration of a task is m+t. Te significance of m comes from te furter assumption, for simplicity, tat m>, so tat Task A is never interrupted wen te overall resource booking is exausted. It is always Task B tat is interrupted wen tis appens. Hence, we do not need to consider te case were Task A runs out of resource time and ten later Task B does as well. Note tat, even if it were assumed tat m<, te time booked for Task B would still give extra security tat Task A could be completed witin te overall booked resource time, so reducing expected disruption costs. Te expected cost of resources becoming idle due to te two tasks being completed witin T is now: z 1 T T t A w ( T t t ) dt dt 3 T w 6 A B B A 0 0 Te expected cost resulting from Task B not being completed witin te initial resource booking, allowing bot for te case were Task A takes less tan T (t A <T) and were it does not (T t A ) is now: z D D D T dt Bdt A dt A ( T ) 0 T t A T Te overall extra expected cost is Z : Z z1 z Differentiating wit respect to T gives: 16

17 dz dt 1 ( T w TD) Setting tis equal to zero gives te optimal value of T, wic is now te same as te optimal overall resource booking if te tasks are not consolidated (T 01 * ): T * 3 D w Hence, te condition T is equivalent to: w D (or, equivalently, r 0.5) Te corresponding value of Z is: Z * 3 D D 3 w Te superiority of consolidation over te two tasks being carried out by separate sets of resources (Y 1 * - Z * ) is now: S 3 w r r r 3 3 Te figure below plots te superiority of consolidation (S 1, S, and S 3 ) for te case were w=1. S 3 is applicable in te range 0 r 0.5. S 3 meets S 1 at r=0.5, and S 1 is ten applicable in te range 0.5 r 1. S 1 meets S at r=1, wit S being applicable to te rigt of tis point. [Insert Figure around ere] A point illustrated by te figure is tat te superiority of consolidation must decline after some point as te optimal resource booking, T, under consolidation becomes closer to its maximum of wit increasing values of r. As te disruption cost increases te duration of te resource booking for an unconsolidated task climbs more quickly towards its maximum value tan does te duration of te booking made if te tasks are consolidated. In fact, for an unconsolidated task, te resource booking reaces its maximum duration of at r=1 (i.e. D=w). Once it as reaced tis value any furter increases in te disruption cost ave no impact as te task will ten never run out of booked resource time before it is finised, te booking aving been made to cover te maximum time tat te task can take. However, for consolidated tasks increases in 17

18 te disruption cost still continue to increase te expected costs and so te superiority of consolidation declines. It is at values of r close to unity, were D and w are close in value, tat te superiority of task consolidation is greatest. Were, on te oter and, one tese costs dominates te oter ten te overall resource bookings for consolidated and non-consolidated tasks are similar and consolidation is less advantageous. Note tat tis result is based on te consolidation of two similar tasks. However, consolidation is clearly also valuable if te disruption and resource costs are bot ig for te first task but te disruption cost is significantly lower for te second task, so long as te resources are not replacing ones in te second task tat are so muc less costly tat te benefits are wiped out. Te trade-off involved in sceduling te first task wen not consolidated is eased by consolidation, under wic te disruption cost becomes te lower cost involved wit te second task. However, consolidation is still valuable if one of te two types of cost is significantly iger tan te oter if te task can be consolidated wit a second wit a lower disruption cost. For instance, if disruption costs dominate resource costs in te first task, so tat te resource booking for te task if not consolidated would be towards te top end of te possible value of te task duration, ten consolidation wit a second task wit a low disruption cost can lead to a significant saving in resource costs. Hig disruption costs are particularly associated wit critical pat tasks and wit resources tat are difficult to replace or to secure extra time for quickly. An example of a situation were te disruption cost and resource cost would bot be ig would terefore be a critical pat task needing ig-cost specialists using expensive equipment wo gain significant specific knowledge during teir work. If tey are not booked for long enoug ten tey are difficult and costly to replace. Note tat disruption costs can potentially be reduced somewat by prompt information saring so tat managers can react earlier wen tasks are not running to scedule. 5. Conclusion Te effects of production uncertainties on te division of labour ave been considered. Dividing labour into narrower sets of tasks as te consequence of making te production process more vulnerable to te performance of individual workers and macines. In addition, it creates extra interfaces between production stages. A unit of production can be delayed at an interface between production stages, awaiting te attention of te resources tat will undertake te next stage of production. Knowledge may ave to be transferred across it in order to process eac unit of output or suc knowledge may be lost at it. A production stage may be forced into idleness wile waiting for te subsequent stage to catc up or wile awaiting te output of te previous stage. Dividing tasks can make production rates more uncertain. Tis increases expected costs of bot idleness and delays for a given level of resources. Altoug it is not a focus of te paper, it sould also be 18

19 recognised tat te division of labour is important for incentives witin firms and for contracting issues between firms. For instance, if two tasks are interdependent ten it is likely to be easier to identify wo is responsible for performance if tey are performed witin te same firm and by te same team. A model of project-based work was presented in order to sow formally ow a consolidation of tasks, so tat tey are undertaken one after anoter by te same resources, can reduce bot resource costs and te expected value of disruption costs tat result wen a resource booking turns out not to be long enoug to complete its tasks. Te model demonstrated two advantages to a reduced division of labour. Firstly, te consolidation of tasks results in a task duration probability tat is more tigtly distributed around its mean, relative to te mean. For instance, if one task takes longer tan expected ten anoter may be completed more quickly tan expected. Te consolidation of tasks terefore eases te problem of sceduling resources to undertake tasks wit uncertain durations. Secondly, te model demonstrated ow te consolidation of tasks reduces expected disruption costs. Tis is because te earlier of a set of consolidated tasks is less likely to run out of booked resource time as te resource booking is made to cover a full sequence of tasks rater tan just te first task alone. Wile te model focused on advantages of task consolidations, tese ave to be weiged against disadvantages suc as losing advantages of more specialised knowledge. Te model furter demonstrated tat te advantages of consolidating similar tasks are greatest wen bot te disruption cost and resource costs are ig but neiter type of cost is so great tat one dominates te oter. At very low values of te disruption cost relative to resource costs very sort resource bookings are made weter te tasks are consolidated or not. Hence, tere is very little expected idle resource time but a ig expectation tat te very low disruption cost will be incurred. Te overall resource costs incurred in bot cases are ten similar and te disruption cost as little impact on overall expected costs, being very low in value. Hence, te expected costs are similar weter or not te tasks are consolidated. As te disruption cost rises from low values consolidation becomes increasingly superior. As it rises furter to intermediate values relative to resource costs, te resource bookings for unconsolidated tasks rise more quickly towards teir maximum durations tan is te case for consolidated tasks. Because of tis, furter rises in te disruption cost, aving no furter impact on unconsolidated tasks, ten start to cause a decline in te superiority of consolidation. At ig values of te disruption cost relative to resource costs te resource bookings are at or near te maximum times tat te tasks migt take, weter tey are consolidated or not, and tere is little or no cance of te disruption cost being incurred. In tis case, terefore, te superiority of consolidation is again low. In addition to te gains tat can be made by consolidating similar tasks, significant gains can also be made from consolidating a pair of tasks were te first task performed as ig values of bot disruption and resource costs and te second task as a significantly lower disruption cost. An example could be were 19

20 a critical pat task is consolidated wit a second task tat is not time critical. Note tat te model did not make te assumption tat tere is a dependency between te tasks involved suc tat one must always be done after te oter, weter or not tey are undertaken by te same resources. Tere migt be suc a dependency or it migt be tat te tasks can be arranged in tat way in order to gain te advantages of task consolidation. However, for te latter case it sould be recognised tat interdependencies between tasks mean tat tey cannot always be resequenced witout costs (Simon, 00; Langlois, 00). Hence, it is more likely tat some tasks will be cosen for consolidation wit any particular task tan oters. A task wit only weak interdependencies wit oter tasks could be useful for task consolidation, if its sequencing in te work scedule could be easily canged to allow it, or alternatively two tasks wit strong interdependencies migt be consolidated wit eac oter. Te consolidation of two tasks tat in any case ave to be completed one after te oter as additional advantages to tose explored in te model because, as explained in Section 3, consolidation avoids a possible delay between te first task ending and resources becoming available to start te second task and also a possible overlap were resources booked to undertake te second task are left idle wile waiting for te first task to be finised. Wile tasks may be fully consolidated so tat tey are always undertaken by te same workers, tere is often, in reality, also te alternative of moving resources between tasks as te need arises. For instance, if one stage of process-based production is moving slowly ten it may be possible to speed it up by moving in resources from oter stages. Suc dynamic transfers of resources can also reduce idleness by redirecting oterwise idle (or underutilised) resources into oter tasks, among wic could be quality assurance activities and macine maintenance. Te arguments made in tis paper may seem less relevant to production on a larger scale. For instance, if tere are many workers and macines carrying out te same stage of production in parallel ten tis will make te overall output of tat stage more predictable, altoug sometimes large scale production will involve individual macines wit very ig capacities. Also, worker related uncertainties can be reduced troug automation. However, it sould be remembered in respect to process-based production tat, firstly, modern production metods often involve te olding of little stock between different stages of production. Secondly, ig quality standards can make te system more vulnerable to disruptions caused by variations in output quality tat occur in individual stages. Tirdly, te introduction of advanced manufacturing tecnologies tends to result in close integration and ence interdependencies between different parts of te firm. Fourtly, te flexibility of some modern production tecnologies allows for easier switcing between products. However, tis can result in muc more frequent switcing wic can increase te frequency of disruptions to te production system. Variations in work flow can terefore be more important tan migt oterwise be expected. Instead of an ever iger division of labour, multi- 0