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1 Available online at ScienceDirect Procedia CIRP 17 (2014 ) Variety Management in Manufacturing. Proceedings of the 47th CIRP Conference on Manufacturing Systems Evaluation Of Capacity Control And Planned Lead Time Control In A Control-Theoretic Model Mathias Knollmann a *, Katja Windt a, Neil Duffie b a School of Engineering and Science, Jacobs University Bremen, Bremen, Germany b Department of Mechanical Engineering, University of Wisconsin-Madison, Madison, WI, USA * Corresponding author. Tel.: ; address: M.Knollmann@jacobs-university.de Abstract Reducing lead time variability of production systems has been shown to be advantageous, enabling constant flows and improved on time production. Nevertheless, fluctuating order inflows, disturbances and other factors are known to induce varying workloads, and hence produce fluctuating actual lead times. Adjusting planned lead times as a countermeasure against low due date reliability could lead into the drawbacks of the Lead Time Syndrome. The update frequency of planned lead time adjustments as well as the delay until changes take effect in a production system can significantly influence the occurrence of the Lead Time Syndrome. Nevertheless, taking these effects into account, production control via planned lead time adjustments remains a suitable means for increasing due date reliability. Another production planning and control approach is to avoid fluctuating actual lead times using capacity adjustments, which can be implemented by either lead time regulation or work in process regulation. These strategies have been integrated into a control theoretic simulation model that enables comparisons to be made of resulting performance; thus, preferable strategies can be identified for different settings of inflow fluctuations, update frequencies and delay The Elsevier Authors. B.V. Published This is an by open Elsevier access B.V. article under the CC BY-NC-ND license Selection ( and peer-review under responsibility of the International Scientific Committee of The 47th CIRP Conference on Manufacturing Systems Selection in and the peer-review person of the under Conference responsibility Chair Professor of the International Hoda ElMaraghy. Scientific Committee of The 47th CIRP Conference on Manufacturing Systems in the person of the Conference Chair Professor Hoda ElMaraghy Keywords:Capacity Control; Order Release Control; Control Theory; Lead Time Syndrome 1. Introduction The aim of production planning and control is to maintain a high logistic target achievement. Targets include short lead times, low work in process, high capacity utilization, and high due date reliability, with due date reliability as the most important target from customer s point of view [1]. It has been shown that low lead time fluctuation is a key factor to decreasing safety stocks or safety times [2] and improving the performance of production systems [3]. Previous research showed the strong influence of the lead time variability on due date reliability [4]. Reducing lead time length and variability in production systems enables constant flows and a higher due date reliability. Nevertheless, various influences are known to increase lead time fluctuations; these include input fluctuations, output disturbances and inappropriately high work in process levels. Addition of safety lead times is a common strategy used by production planners to increase due date reliability, but this can lead to the chain reaction of the Lead Time Syndrome (LTS), that was firstly described by Mather and Plossl [5] and is summarized in Fig. 1. Safety lead times are added because, apparently, prior planned lead times were set too short to produce in time [6], which results in earlier order releases. This reaction directly increases the process workload. Consequently, the WIP level rises and lead times get longer and more erratic [7]. Finally, this circle of mistakes leads to an even lower due date reliability - although the aim was to improve it - thus demanding further measures to be undertaken [1,8]. Knollmann & Windt [4,9] formally analyzed LTS using the logistic operating curve theory of Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( Selection and peer-review under responsibility of the International Scientific Committee of The 47th CIRP Conference on Manufacturing Systems in the person of the Conference Chair Professor Hoda ElMaraghy doi: /j.procir

2 Mathias Knollmann et al. / Procedia CIRP 17 ( 2014 ) Nyhuis & Wiendahl [1], and reasons for lead time fluctuations within the scope of the LTS cycle were evaluated [7]. It was shown that the LTS has strong similarities to a positive feedback loop where, as known from control theory, even small system disturbances can lead to an increasing magnitude of perturbation. In feedback loops the output value influences itself, as a fraction of the output value is fed back to the input value [10]. A positive feedback loop occurs, if the output value is amplified by the own feedback, thus causing an oscillatory response [10,11]. This behavior could be observed at sound systems that amplify the signal of a microphone that receives the output signal of the speakers. The signal amplifies itself, causing the known deafening sound. In a positive feedback loop even small perturbations can lead to an almost uncontrollable system behavior, as the amplification of the input signal increases exponentially with each loop [12]. Hence, the occurrence of the LTS strongly depends on the magnitude of response, the update frequency of planned lead time adjustments as well as the delay until changes take effect in a production system. actual lead times get longer queues get longer actual lead times get more erratic work center loads are increased due dates are missed production system orders are released earlier planners reaction planned lead times are increased Fig. 1. Lead time syndrome of production control (based on [5,8]) Selçuk et al. [13,14] used queuing theory to investigate the influence of the planned lead time update frequency and the capacity utilization level on the occurrence of the LTS. They found that the LTS triggers uncontrolled production system states with a high mean and standard deviation of lead times. However, this research did not take into account delay and the system s transient response, although Deif et al. found that system s responsiveness is inversely proportional to delay [15]. According to the manufacturing control model of Lödding, lead time and WIP can be influenced directly by capacity control or indirectly by a planned lead time control [16]. A study by Duffie et al. compared two possible strategies controlling lead time by adjusting the actual manufacturing capacity [17]: lead time control and work output control. Both of these capacity control strategies are influenced by the length of time between capacity adjustments and the delay until these adjustments take effect. The aim of this paper is to compare control of lead time using adjustments in capacity to control of lead times using adjustments in release times; the latter will be referred to as planned lead time control. The influence of adjustment period and delay on their performance will be investigated, as will the effect on performance of higher input fluctuations. This will enable the identification of preferable strategies for certain environmental conditions, by considering the benefits or drawbacks that are linked to each strategy. Moreover, a central issue is to clarify if planned lead time control is a good choice when the drawbacks of LTS are considered. The next section of this paper describes the control theoretic model that has been developed, and defines the control strategies. Then, the control theoretic model is used to evaluate the effects of different inflow fluctuations, adjustment periods and delays. Finally, the control strategies are evaluated, and conclusions are presented regarding which strategy obtains the best performance under which circumstances and restrictions. 2. Control Theoretic Model and Control Strategies A control-theoretic model was developed to compare the planned lead time control and work output control strategies. The model was programmed in Simulink (MathWorks 2012) and is shown in simplified form in Fig. 2. The input and output control structure is adapted from a closed-loop production planning and control system proposed by Duffie & Falu [18] and a control theoretic model of production control proposed by Petermann [19]. Capacity Adjustments c(z) Delay Planned Capacity c p(z) inflow/input Work Input Rate IR a(z) Capacity Disturbance c d(z) - Capacity c a(z) outflow/output Integrator Work Total Disturbance Work w d(z) In w i(z) Total Work Out w o(z) WIP w a(z) Due Date Reliability DR(z) f(x) dx Work Adjustments Δw(z) Planned Lead Time tl p(z) Lead Time tl a(z) - Lateness l a(z) l sd(z) l Lateness m(z) Lower&Upper Due Date Tolerance Fig. 2. Control-theoretic simulation model of the production system Analysis of In Fig. 2, the total work in w i (kt) is the sum of the integrated input rate, any work disturbances such as rush orders or order cancellations, and any work input deviations applied as a result of planned lead time adjustments. T=1 shop calendar day [scd] represents the smallest time unit, k is a positive integer, and kt is a discrete instant in time (w i (z) is the z transformation that represents the sequence w i (kt), k=0,1,2 ). The actual capacity is the sum of the planned capacity and any capacity adjustments, minus any capacity disturbances such as equipment failures and worker illness. The actual capacity cannot be negative, and is zero if work in progress (WIP) is zero. For comparability reasons of the different control strategies, capacity disturbance and work disturbance are zero in the simulation. The total work out w o (kt) is the integrated actual capacity, and work in progress is the difference between total work in and total work out. The actual lead time tl a (kt) is calculated by finding the value that satisfies the relationship w i (kt-tl a (kt))=w o (kt), assuming First-In-First-Out as sequencing rule. The relative actual lateness l a (kt) is the difference between actual lead time and the planned lead time [1,7]. Eq. 1 describes Due date reliability DR as a cumulative distribution function of mean and standard deviation of lateness, which was evaluated by Knollmann & Windt [4]. Thereby, a company specific - tla

3 394 Mathias Knollmann et al. / Procedia CIRP 17 ( 2014 ) tolerance period defines which orders are considered to be produced on time [20]. This results in a value DR(kT), which represents the due date reliability at time kt. Finally, to compare the simulation results of the different control strategies, the average of all DR(kT) was calculated for each simulation setting. The values planned lead time, work adjustments, planned capacity and capacity adjustments in Fig. 2 relate to planned lead time control and work output control, which are described next. 1 x lm ( kt) upper due date tol lsd ( kt) DR( kt ) = e dx l kt 2π x= lower due date tol. sd ( ) 1 l m(kt ) = tl i k 4 a it tlp it 5 = k ( ) ( ) DR due date reliability [%] l lateness [scd] m mean sd standard deviation k current period a actual tl lead time [scd] p planned a) Capacity Adjustment Using Work Output Control The lead time control strategy described by Duffie et al. [17] considers a work system that adjusts the actual capacity with the goal of reducing the difference between actual and planned lead time. They also described the work output control strategy in which work output deviation is controlled. In this case, a work system is considered that periodically calculates and adjusts the actual capacity to minimize the difference between desired work out and actual work out, using the accumulated values of actual work in and actual work out acc. to Fig. 2. ( ) c(kt) a = c p (k d)t (k d)t tl p (k d)t k c w(x) x 0 i w x 0 o(x) = = k d d = T d 1 c d 1 ( ) w i work input [h] w o work output [h] k c magnitude of response c p planned capacity [h/scd] tl p planned lead time [scd] 2 The calculation is performed at instants in time separated by the time period nt, where n is a positive integer. Adjustments are delayed by d (time delay dt [scd]). The magnitude of response amplifies or attenuates the calculated capacity value, thus accelerating or decelerating the response to fluctuations. Eq. 5 calculates a simulation setting dependent to avoid both slow and oscillatory response [21]. The work output control strategy is summarized in the block diagram in Fig. 3. Duffie et al. [17] showed that the performance of lead time control and work output control are similar to each other for low work input and capacity fluctuations, but work output control is preferred to lead time control because it produces more consistent and stable behavior for higher fluctuations. Therefore, only work output control was considered in the present work and compared with planned lead time control. The planned lead time and planned capacity were assumed to be constant. Total Work Out w Capacity Planned Capacity c o(z) p(z) Adjustment Total Work In w i(z) z -tlp Fig. 3. Work output control b) Work Adjustment Using Planned Lead Time Control In this control strategy capacities are not adjusted, and a work system is considered that periodically adjusts both planned lead times and consequentially order releases based on due date reliability DR(kT) [9]. To model the LTS cycle shown in Fig.1, it is assumed that production planners monitor the due date reliability and the mean lead time and adjust release times at the beginning of each time period T. Two control modes are defined in this control strategy, planned lead time adjustment and no lead time adjustment as shown in Fig. 4. For (kt) due dates are met, hence no planned lead time adjustments are necessary: tlp( kt ) = tlp( ( k- d ) T ) ΔwkT ( ) =0 For DR(kT)<10, the planners reaction is to adapt planned lead times every n th period (time period between adjustments nt [scd]), following the LTS cycle in Fig. 1. In order to maximize DR(kT), the planned lead time is adjusted by noting in Eqs. 1 and 2 that the maximum due date reliability can be achieved by setting tl p (kt) to the latest mean value of actual lead time tl m (kt). More specifically, tl p (kt) is only adjusted if DR(kT)<10 and mod(k,n)=0. As a result, the value of mean and standard deviation of lateness will decrease, thus improve due date reliability. Analog to the k c of work output control, a control variable k pl is added to amplify or dampen the calculated planned lead time adjustment. Also, the calculation and decision-making minimally takes one period (dt=1scd) to implement and can take even longer. This is represented by the time delay dt [scd]. Eq. 8 defines the production planning logic of Fig. 4 for this case: ( ) (( ) ) (( ) ) (( ) ) tlp kt = tlp k- d T k pl tlm k- d T tlp k- d T Planned Lead Time tl p(z) DR=10 or mod(k,n)=0 Mean Lead Planned Planned Lead Time tl m(z) - Lead Time Time Adjustment k pl tl p(z) tl DR<10 p(z) and mod(k,n)=0 Fig. 4. Planned lead time calculation Capacity c a(z) Planned Lead Time tl p(z) - k c c(z) WIP w a(z) Fig. 5. Work adjustment calculation - Work Adjustment Δw(z) Capacity c a(z)

4 Mathias Knollmann et al. / Procedia CIRP 17 ( 2014 ) If planned lead times are changed in a production system, orders are released earlier or later. This work adjustment is calculated in Eq. 9: ( ) a( ( ) ) p( ( ) ) a( ( ) ) Δw kt =c k-d T tl k-d T w k-d T Once again, the calculation and implementation of order release adjustments is subject to delay in practice. For simplification the delay of planned lead time adjustments and the delay of order release adjustments have the same length. The production control logic is shown in the block diagram in Fig. 5. This control logic represents the planners reaction to due dates being missed in the LTS cycle shown in Fig.1: To increase due date reliability planned lead times are adjusted, which directly leads to order release adjustments. The impacts of the anticipated LTS drawbacks caused by these adjustments on the work center load, WIP level, lead time and finally on the due date reliability were evaluated in the simulation for different settings of adjustment periods, information delay and input fluctuation. If adjustments are implemented too often in proportion to the delay or the magnitude of response to disturbances is too high, system`s performance might decrease significantly due to the LTS in which the control s own short term adjustments are amplified before the system reaches a steady state [4]. 3. Influence of Delay and Frequency of Adjustments Fig. 6 shows the work input rate IR(kT) that was used in the simulations. These data are from a supplier to the automotive industry [17]. The mean input rate is standard deviation of Because some simulation runs had long adaption periods and long delays, this input time series was replicated 10 times (IR(kT)=IR((k74j)T), with jt=1..10). Amplifying the same input sequence might not be different enough to draw conclusions for both strategies and has to be validated in further research under use of a normally distributed input fluctuation. h/scd 10 5 IR(kT) scd 70 Fig. 6. Work input rate in simulated production system Depending on the chosen control strategy, either capacities or planned lead times are adjusted. The results obtained for the different control strategies are highly dependent upon how often these values are adjusted and whether there is time delay in making the adjustments; therefore, the influence of update frequency and delay on the performance of each control strategy was analyzed. Fig. 7 shows the resulting mean due date reliability for work output control with delays dt=1..9scd and for either short and long adjustment periods (nt=1 and nt=9scd, respectively). The adjustment periods nt=2..8scd are not shown, but lie in ascending order between the curves. According to Eq. 5, for each setting of dt and nt a specific k c is defined to obtain the best results for work output control. Work and capacity disturbances were assumed to be zero. Planned capacity c p =5h/scd, planned lead time tl p =3scd, and the upper/lower due date tolerance ±0.5scd. As expected, the best performance is achieved when actual capacities are updated more rapidly and there is less delay. With increasing delay, the performance of work output control decreases approximately exponentially, and the influence of the adjustment frequency decreases. For longer adjustment periods the influence of delay decreases and only for very low delays an increasing performance could be observed. Thus, if planners are able to maintain a high adjustment frequency and decrease delay in adjustment implementation, performance as measured by due date reliability can be expected to significantly improve under work output control. nt=1 nt= delay 7dT 8 9 Fig. 7 Mean due date reliability under work output control For planned lead time control the resulting mean due date reliability does not deteriorate with longer adjustment periods as is the case under work output control. Fig. 8 shows that the mean due date reliability is lowest, even for a low delay, if planned lead times are adjusted daily (nt=1scd). Once again, work and capacity disturbances were assumed to be zero. Planned capacity c p =5h/scd, magnitude of response k pl =1, and the upper/lower due date tolerance ±0.5scd. This unexpected behavior is most significant for a delay dt=1scd, for which the best results were obtained for longer adjustment periods. Thus, of particular interest in this context is the distribution of the obtained due date reliability for the investigated adjustment periods at dt=1scd, therefore which adjustment period length enables the best performance at minimal delay. nt=2 nt=9 nt= delay 7 dt 8 9 Fig. 8 Mean due date reliability under planned lead time control Fig. 9 shows the discussed distribution revealing three main characteristics: nt=1scd produces the worst performance; nt=2scd and nt=9scd produce the best performance; and the performance decreases around nt=5scd and for longer adjustment periods (nt 10scd).

5 396 Mathias Knollmann et al. / Procedia CIRP 17 ( 2014 ) adj. period nt 11 Fig. 9 Planned lead time control with various adjustment periods and dt=1scd The LTS cycle can be triggered if the adjustment period is too short or the magnitude of response is too high [4]. For example, a poor due date reliability in one adjustment period with nt=1scd and dt=1scd leads to a planned lead time adjustment in the next adjustment period and an order release adjustment in the subsequent adjustment period (see Eqs. 8 and 9). However, before the order release adjustment takes, the ongoing due date reliability monitoring calls for new planned lead time adjustments. This helps to the poor performance for nt=1scd with k pl =1. With a direct impact on work inflow, work output control tends to produce better results for shorter adjustment periods. With a direct impact on due date reliability, planned lead time control produces a better performance for longer adjustment periods. In order to dampen the drawback of the LTS for frequent planned lead time adjustments, the control variable k pl was added to the LTS control logic. Setting k pl =0.5, the magnitude of each planned lead time adaption is halved. Compared to Fig.9 a significant performance increase could be observed for the results shown in. Fig.10. Especially the mean due date reliability of nt=1scd rose from 17% to 56% for the given input sample. changes for higher input fluctuations, and this was simulated by amplifying the input rate by a factor of four. Fig.12 shows the obtained mean due date reliability for the same adjustment period and delay settings as in Fig. 11, but for higher input fluctuations with a planned capacity c p =20h/scd and a planned lead time tl p =10scd. In contrast to work output control, planned lead time control obtains even better results at higher fluctuations for a low delay and short adjustment periods or one or two periods. Work output control obtains very stable results for either low or high input fluctuations, reaching nearly the same due date levels for all simulation settings as under low fluctuations. Nevertheless, planned lead time control produced much better results for long delays (dt=9scd) for all analyzed adjustment periods. Moreover, planned lead time control obtained a significantly higher performance than work output control when adjustments were performed less frequently for either high and low delays. Only for high adjustment frequencies work output control is able to obtain similar performances as planned lead time control. delay of 1scd delay of 2scd delay of 9scd nt=1scd; planned lead time control nt=1scd; work output control nt=2scd; planned lead time control nt=2scd; work output control nt=9scd; planned lead time control nt=9scd; work output control Fig. 11 Comparison of work output control and planned lead time control for different adjustment periods and delays with k c acc. to Eq. 5 and k pl= adj. period nt 11 Fig. 10 Planned lead time control with various adjustment periods for dt=1scd and k pl= Control Strategy Comparison Fig. 11 shows that even if the influence of LTS is taken into account and dampened by setting k pl =0.5, the performance of work output control is significantly better for short adjustment periods for both high and low delays. In contrast to this, the performance of planned lead time control is significantly better for increasing adjustment period lengths. The performance decrease for an increasing delay is very significant for both control strategies, except for planned lead time control with long adjustment periods. This behavior can also be observed for longer delays dt=9scd, for which planned lead time control (DR m =34%) outperforms work output control (DR m =2%). The preceding results were obtained for the input fluctuation shown in Fig. 6. It is likely, that the performance delay of 1scd delay of 2scd delay of 9scd nt=1scd; planned lead time control nt=1scd; work output control nt=2scd; planned lead time control nt=2scd; work output control nt=9scd; planned lead time control nt=9scd; work output control Fig.12 Comparison of work output control and planned lead time control with a four times amplified input rate and different adjustment periods and delays with k c acc. to Eq. 5 and k pl=0.75 Depending on system characteristics such as capacity flexibility, frequency of adaptions, and information delay the most suitable control strategy can be selected. The differences in strategy performance are primarily explained by the different approaches: Work output control monitors the work deviation, which is a past-oriented value [17]. In contrast to this planned lead time control monitors the actual lead time, but as a consequence of a possible planned lead time adaption it also controls the order release. Therefore, this strategy combines the reactive aspect of lead time control and the proactive aspect of planning. A shift from reactive to proactive controlling should obtain better results (except for

6 Mathias Knollmann et al. / Procedia CIRP 17 ( 2014 ) negative effects of the LTS) when delays and adjustment periods become long. The obtained performance for both high and low input fluctuations for an adjustment period nt=9scd is shown in Fig. 13. The results support the conclusion that when adjustment periods get longer planned lead time control is able to perform better than work output control for either short and long delays. For the selection of one control strategy it is also important to consider that, on one hand, work output control requires flexible capacities while, on the other hand, planned lead time control could lead to the LTS drawbacks. Thereby, simulation results imply that three strategies are feasible to dampen LTS: A suitable magnitude of response k pl, which should be lower for small fluctuations. Secondly, an update frequency that is lower than the expected delay; and finally a delay which is as short as possible. Fig. 13 Comparison of work output control and planned lead time control at the longest analysed adjustment period nt=9scd with k c acc. to Eq. 5 and k pl =0.75 for high input fluctuations, and k pl=0.5 for low input fluctuations 5. Conclusion planned lead time control with low and high input fluctuation work output control with low and high input fluctuation delay dt 8 9 The aim of this paper was to compare the control strategies capacity control and planned lead time control in order to define preferable strategies for certain environmental conditions. Therefore, the influence of the adjustment period and the information delay on their performance was investigated for both high and low input fluctuations. It was shown that planned lead time control triggers the LTS drawbacks if the adjustment period length and delay are low. Anticipating this effect by including a damping factor k pl significantly increased the performance as measured by due date reliability. However, even with this anticipation of the LTS, the reactive work output control produced better performances than the proactive planned lead time control for shorter adjustment periods and delays at low input fluctuations. However, if delays get longer and adjustments are less frequent, planned lead time control was superior to work output control at both high and low input fluctuations. In order to further validate the conclusions regarding strategies to avoid or dampen LTS and to find the optimal variable settings for planned lead time control, further research is necessary. Additional input sequences such as normally distributed input fluctuations should be studied. Moreover, a more detailed investigation of system s transient response after planned lead time adjustments would improve understanding how the LTS manifests itself and influences results. Acknowledgements The research of Prof. Dr.-Ing. Katja Windt is supported by the Alfried Krupp Prize for Young University Teachers of the Alfried Krupp von Bohlen und Halbach-Foundation. References [1] Nyhuis, P., H.-P. Wiendahl, 2009, Fundamentals of Production Logistics: Theory, Tools and Applications, Springer. [2] Zipkin, P.H., 1986, Models for Design and Control of Stochastic, Multi- Item Batch Production Systems, Oper. Res. 34: [3] Erlebacher, S.J., M.R. Singh, 1999, Optimal Variance Structures and Performance Improvement of Synchronous Assembly Lines, Oper. 47: [4] Knollmann, M., K. Windt, 2013, Control-Theoretic Analysis of the Lead Time Syndrome and its Impact on the Logistic Target Achievement, in: Procedia CIRP, Vol. 7, 46th CIRP Conf. Manuf. Syst. (CMS 2013): [5] Mather, H., G. Plossl, 1977, Priority fixation versus throughput planning, Prod. Invent. Manag. J. Am. Prod. Invent. Control Soc. APICS. 3rd Q.: [6] Lindau, R.A., K.R. 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