CONTINUOUSLY IMPROVE THE PERFORMANCE OF PLANNING AND SCHEDULING MODELS WITH PARAMETER FEEDBACK
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CONTINUOUSLY IMPROVE THE PERFORMANCE OF PLANNING AND SCHEDULING MODELS WITH PARAMETER FEEDBACK

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Continuously improving the accuracy and precision of planning and scheduling models is not new; unfortunately it is not institutionalized in practice. The intent of this paper is to highlight a ...

Continuously improving the accuracy and precision of planning and scheduling models is not new; unfortunately it is not institutionalized in practice. The intent of this paper is to highlight a relatively simple approach to historize or memorize past and present actual planning and scheduling data collected into what we call the past rolling horizon (PRH). The PRH is identical to the future rolling horizon (FRH) used in hierarchical production planning and model predictive control to manage omnipresent uncertainty in the model and data. Instead of optimizing future decisions such as throughputs, operating-modes and conditions we now optimize or estimate key model parameters. Although bias-updating using a single time-sample of data is common practice in advanced process control and optimization to incorporate “parameter” feedback, this is only realizable for real-time applications with comprehensive measurement systems. Proposed in this paper is the use of multiple synchronous or asynchronous time-samples in the past in conjunction with simultaneous reconciliation and regression to update a subset of the model parameters on a past rolling horizon basis to improve the performance of planning and scheduling models.

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CONTINUOUSLY IMPROVE THE PERFORMANCE OF PLANNING AND SCHEDULING MODELS WITH PARAMETER FEEDBACK CONTINUOUSLY IMPROVE THE PERFORMANCE OF PLANNING AND SCHEDULING MODELS WITH PARAMETER FEEDBACK Presentation Transcript

  • CONTINUOUSLY IMPROVE THE PERFORMANCE OF PLANNING AND SCHEDULING MODELS WITH PARAMETER FEEDBACK Jeffrey D. Kelly* and Danielle Zyngier Abstract Continuously improving the accuracy and precision of planning and scheduling models is not new; unfortunately it is not institutionalized in practice. The intent of this paper is to highlight a relatively simple approach to historize or memorize past and present actual planning and scheduling data collected into what we call the past rolling horizon (PRH). The PRH is identical to the future rolling horizon (FRH) used in hierarchical production planning and model predictive control to manage omnipresent uncertainty in the model and data. Instead of optimizing future decisions such as throughputs, operating- modes and conditions we now optimize or estimate key model parameters. Although bias-updating using a single time-sample of data is common practice in advanced process control and optimization to incorporate “parameter” feedback, this is only realizable for real-time applications with comprehensive measurement systems. Proposed in this paper is the use of multiple synchronous or asynchronous time- samples in the past in conjunction with simultaneous reconciliation and regression to update a subset of the model parameters on a past rolling horizon basis to improve the performance of planning and scheduling models. Keywords Rolling horizons, reconciliation, parameter estimation, error-in-variables, closed-loop, feedback. Introduction * To whom all correspondence should be addressed. jdkelly@industrialgorithms.ca (Industrial Algorithms LLC.) Planning and scheduling decision-making is traditionally based on simplified models that can, with any luck, accurately and precisely interpolate and extrapolate the dominate behavior of the underlying production in terms of its processes, operations and maintenance. Related to this is the well known notion that "all models are wrong, but some are useful" (G.E.P. Box) which implies that even detailed models do not guarantee their accuracy and precision in being able to predict and optimize production (Forbes and Marlin, 1994, Zyngier and Marlin, 2006). The focus of this paper is to suggest a symmetrical methodology to the future rolling horizon (FRH, Baker and Peterson, 1979) we call the past rolling horizon (PRH) which implements a continuous improvement strategy similar to the Deming Wheel, Shewhart Cycle, Kaizen or the Plan-Perform-Perfect-Loop (Kelly, 2005). The use of a past rolling horizon for planning and scheduling is a similar concept to moving-window estimators (Robertson et al., 1996, Zyngier et al., 2001, Yip and Marlin, 2002). In the PRH, “parameter” optimization is performed with essentially fixed variables looking backwards in time whereas in the FRH of planning and scheduling we use “variable” optimization with fixed parameters looking forwards in time where fixed implies exogenously defined as opposed to endogenously determined via the optimization process.
  • The structure of paper is to first illuminate the different aspects of a model, second to highlight the issues with typically passive data collection, third, updating and estimating techniques are discussed and fourth a motivating example is overviewed to demonstrate the need for what we call “parameter” feedback and not just “variable” feedback used in existing planning and scheduling implementations. Model Morphology For decision-making found in the process industries, models can usually be segregated into three different types. (1) Macro or lumped models are the ones that usually consider an entire plant (or a large section thereof) taking into account only the core or critical unit-operations as well as aggregations of units. Macro models are commonly encountered in planning, scheduling and yield accounting applications; (2) Micro or distributed models are more restricted in scope than macro models in that they usually consider a smaller number of unit-operations and are usually distributed across several spatial dimensions including time. The modeling of these unit-operations comprise more detailed mass, energy and momentum balances including vapor-liquid equilibrium and reaction kinetic relationships. Micro models are widely applied in advanced process simulation and optimization; and finally (3) Molecular models address the relationships that occur at an atomic or elemental level of granularity within a small section of a unit-operation with very detailed thermodynamics and transport phenomena. Whether a model is macro, micro or molecular, there are three relevant aspects of the model morphology. A model may be classified by its structural form such as if it is linear, piece-wise linear, polynomial, rational, multi- linear or non-linear. On the other hand, its functional form relates to its parameters, coefficients and/or factors. Its syntactical form relates to how the model functions, formulae or formulations are expressed. Syntactically, models can be explicit or implicit (i.e., use closed- or open- form1 equations respectively) of which the latter is the more general form i.e., comprises explicit models as a subset. Other aspects of a model such as whether it is static or dynamic (steady or unsteady), continuous or discrete and deterministic or non-deterministic (stochastic or chaotic) is also worth mentioning. Planning and scheduling models are usually dynamic in the sense of having multiple time- periods built-up from essentially static models, have a mix of continuous and discrete variables to represent the quantity, logic and quality dimensions and are mostly deterministic. Another important aspect of a model is related to its fidelity and size. Bigger and more detailed models do not guarantee its precision or accuracy as shown 1 There can also be “pried-open” models which break-apart the internal convergence loops inside closed-form models. by Forbes and Marlin (1994), i.e., smaller and simpler models can be just as “useful” if they meet certain point- wise model accuracy criteria. Therefore, we can class models into being either rigorous or rough. Rough models are related to meta-models or surrogate models where a blend of rigorous and rough sub-models is termed hybrid modeling. Rigorous models are also known as first- principle models and rough models are empirical models. The types of models used in planning and scheduling are mostly rough models where it is common practice to linearize available rigorous models into first-order Taylor- series expansions. These linearized models are called base plus delta, fixed and variable, absolute and relative, slope with intercept and shift models, (Bodington, 1995). Data Issues As is well known in the mathematical programming community, any decision-making problem can be decomposed into its model, data and solution. Therefore, how to collect, clean and compile data for the purposes of what we call “parameter” feedback merits some discussion. As mentioned, the focus of this work is to establish a past rolling horizon (PRH) for planning and scheduling problems which is symmetrical to the future rolling horizon (FRH) that exists at the heart of hierarchical production planning (HPP) (Bitran and Hax, 1977) and model predictive control (MPC) (Richalet et. al., 1978). Ideally the data used to perform data reconciliation and parameter estimation (DRPE), error-in-variables method (EVM) (Reilly and Patino-Leal, 1981) or instrumental variables regression (IVR) (Young, 1970) should be independent and identically normally distributed, else systemic or gross-errors in the data may exist hence skewing the results. Unfortunately the data collected after a plan or schedule has been completed is most often passive and not perturbed, happenstance and not holistic and degenerate and not designed. This means that the calibration or training-data used to fit the key2 model parameters in the PRH may not be representative of the production or operating regions or ranges seen in the control or testing-data found in the FRH. After all, the sole purpose of planning and scheduling decision-making using optimization is to push/pull the production to new and more profitable/performant regimes perhaps not implemented hitherto. Along this line, the quality of the data can be classified into three main characteristics: (1) diversity or richness of the data i.e., all sampled points span different regions of the control-data, (2) consistency of the data i.e., all sampled points in both the calibration- and control-data are taken from the same system and (3) statistical homogeneity of data i.e., all sampled points in the calibration- and control-data have the same noise, error, random shock/perturbation or uncertainty 2 See Krishnan et. al. (1992) or Zyngier (2006) to determine key model parameters.
  • distributions including their non-linear correlation structure (Rooney and Biegler, 2001). To compound the issue, planning and scheduling also forms a closed-loop feedback control circuit similar to that found in MPC. The issues with structural analysis and parameter estimation when closed-loop data is used were first addressed by Box and MacGregor (1974) when fitting linear and rational time-series transfer function models. These issues also exist for planning and scheduling models. Perhaps one of the main results of their work is to introduce a small but persistent and uncorrelated dithering signal or excitation to either the manipulated variables or the set-points which continuously stimulates the process. Additionally, closed-loop identification can be implemented similarly to the approach of Koung and MacGregor (1993). These same techniques can also be applied to planning and scheduling optimization systems. Finally, potential sources of error that exist in the data arise from several diverse sources as enumerated by Kelly (2000). They are forecast-errors, measurement-errors, execution-errors (processing, operating and maintaining), model structural- and functional-errors (including decomposition- and aggregation-errors) and last but not least, solution-errors due to the non-convexity of the problems (existence of local optima). Updating and Estimating Methods The fundamental objective of any model updating and estimating technique is to find the “best” functional form which balances the tradeoffs between: (1) the best fit of the calibration-data i.e., interpolation and (2) the most accurate and precise parameter estimates when noise exists. There is a third requirement which is the best prediction of the control-data i.e., extrapolation, which is the overriding goal of design-of-experiments and control-relevant identification. Obviously the quality of the functional form will depend on both the quality of the structural form and the quality of the data discussed previously. And, in advanced process or real-time optimization (RTO) applications, recognition of the fidelity of the models must be understood in order to increase the performance of the models in terms of minimizing the offsets (accuracy) between the true-plant’s response and the noise-free model prediction and the variability (precision) of the predictions due to disturbances (Yip and Marlin, 2004). Although simple bias-updating is the standard technique used in both MPC and RTO for “parameter” feedback, it utilizes a single data point for one time-point or period in the past and updates the bias, base, intercept or fixed value of the model formula or equation only from the measured “variable” feedback. Albeit this is sufficient to asymptotically remove the offset between the actual and assumed value of the plant, it is not particularly suited to planning and scheduling systems. The reason is that in MPC/RTO, real-time electronic and digitized measurements of temperatures, pressures, flows, levels, concentrations and properties are readily available. Unfortunately, in planning and scheduling applications it can take days, weeks or months to obtain measured “variable” feedback given field/control laboratory, accounting, billing and invoicing system delays. Instead, a more sophisticated approach is necessary which continuously collates data over the PRH and performs a robust method of parameter estimation whilst respecting errors in the variables. For our purposes, we choose the method which incorporates simultaneously both reconciliation and regression from Kelly (1998). This method is identical to EVM but has useful diagnostics tailored to the estimability and variability of the both the reconciled and regressed estimates (Kelly and Zyngier, 2008). More specifically, it reliably computes the observability of the parameters, the redundancy of the measurements and the precision of the parameters and adjusted measurements. In addition, it can also provide necessary missing-data capability when some data sources are not available usually intermittently. Motivating Example In order to illustrate the importance of “parameter” feedback in addition to “variable” feedback, a simple example is presented. In Figure 1, a system is shown that consists of receiving a supply of material, processing it in a reactor, storing it in a tank and shipping it to some demand point. The reactor has a yield of product (Y) which is the only uncertain parameter in this system. The demands are exogenously defined by the customer, i.e., it is not a degree-of-freedom when determining the plan or schedule. Supply Reactor Tank Demand True Plant Tank Holdup (Variable) Supply (Solution) Reactor Yield (Parameter) Demand (Parameter) Figure. 1. Closed-Loop System. At each planning and scheduling cycle, the supply profile is dispatched to the true plant for implementation after the solution is calculated. In terms of the feedback mechanism, two strategies were compared: (i) “variable” feedback only, i.e., inventory information is available at the start of each cycle, and (ii) “variable” and “parameter” feedback, i.e., both inventory and updated yield information is available. For illustrative purposes, it is assumed that there is no noise in the measurements and therefore only regression is necessary to update Ymodel. The equations used to determine the supply solution at time-period t (St), given the demand (Dt) and assuming that
  • the inventory in the tank (It) must remain at a constant target value (Itarget) of 2.0 is provided below. tY/)DII(S elmodttetargtt  (1) The inventory It in equation (1) is obtained through “variable” feedback in that the value of the inventory at the start of the cycle is measured and is used in the model for the next cycle. The “true” inventory value is determined after the supply profile is calculated by using the “true” plant yield in equation (2): tDYSII tplantttt   -1 (2) Initially, Ymodel was assumed to be 0.7 whereas the true plant yield Yplant has a value of 0.6. The results are shown in Figure 2 with the demand profile the same for both scenarios or situations. For the case where the yield is not updated, i.e., there is “variable” feedback only, the dotted line inventory profile shows an offset or bias from the target inventory value of 2.0. By updating the yield at every cycle using the PRH data, the offset from the inventory target is quickly corrected (cycle 2) by the time the Ymodel has been updated to the true value of 0.6. 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 9 Supply (true) Inventory (true) Supply (fixed y) Inventory (fixed y) Supply (updated y) Inventory (updated y) Figure 2. Supply & Inventory Responses. Therefore, it is evident that with “variable” feedback only, it is impossible to remove the persistent offset or inaccuracy in terms of meeting the planned/scheduled inventory target of 2.0. Consequently, plan/schedule versus actual reporting, common place in planning and scheduling stewardship, will always display a non-zero bias when significant parameter uncertainty exists of which “variable” feedback alone will not correct. Conclusions Shown in this paper is the limitation of “variable” feedback when moving from one planning and scheduling cycle to another. Without both “variable” and “parameter” feedback, offsets to planning and scheduling targets, set- points and/or upper/lower bounds will exist similar to the persistent offset found in proportional-only control and those observed in real-time process optimization. By employing reconciliation and regression technology on a past rolling horizon (PRH) basis, it is possible to reduce these offsets or inaccuracies asymptotically or evolutionary over the life-time of the models. References Baker, K. R., Peterson, D. W. (1979). An Analytic Framework for Evaluating Rolling Schedules. Mgmt. Sci., 25, 341. Bitran, G.R., Hax, A. C. (1977). One the Design of Hierarchical Production Planning Systems. Decision Sciences, 8, 28. Bodington, C.E. (1995), Planning, Scheduling and Control Integration in the Process Industries, McGraw-Hill Inc. Box, G.E.P. and MacGregor, J.F. (1974). The Analysis of Closed Dynamic-Stochastic Systems, Technometrics, 16, 391. Forbes, J. F., Marlin, T. E. (1994). 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  • Zyngier, D., Marlin, T.E. (2006) Monitoring and improving LP optimization with uncertain parameters. In Proc. of ESCAPE-16,Garmisch-Partenkirchen,Germany, 427.