Control Theory Applications for Dynamic Production Systems. Neil A. Duffie

point that was used to design lead-time regulation decision rules. An option⁶ in this case could be to

       calculate the parameters for a linearized model for each of several capacity operating points

       design lead time regulation decision rules for each operating point using the model for that operating point

       switch between decision rules as operating conditions vary.

      2.4.2 Linearization Using Taylor Series Expansion – Multiple Independent Variables

      A nonlinear function f(x,y,…) of several variables x, y, … can be expanded into an infinite sum of terms of that function’s derivatives evaluated at operating point xo, yo, …:

       (2.4)

      Over some range of (x – x_o), (y – yo), … higher-order terms can be neglected and a linear model is a sufficiently good approximation of the nonlinear model in the vicinity of the operating point:

(2.5)

      where

(2.6)

      Example 2.10 Production System Lead Time when WIP and Capacity are Variable

      In the case where the production work system illustrated in Figure 2.15 has variable work in progress (WIP) w(t) hours and variable production capacity r(t) hours/day, the lead time is

      For work in progress operating point wo and capacity operating point ro, an approximating linear function for lead time in the vicinity of operating point wo,ro, can be obtained using Equations 2.5 and 2.6:

      where

      2.4.3 Piecewise Approximation

      In practice, variables in models of production systems may have a limited range of values. Maximum values of variables such as work in progress and production capacity cannot be exceeded, and these variables cannot have negative values. In many cases, operating conditions where limits have been reached may not be of primary interest when analyzing and designing the dynamic behavior of production systems. On the other hand, models can be developed that represent important combinations of operating conditions, each of which represents dynamic behavior under those specific conditions. A set of piecewise linear approximations then can be used to represent non-linear relationships between variables.

      Example 2.11 Piecewise Approximation of a Logistic Operating Curve

The relationship between work in progress (WIP) and actual capacity shown in Figure 2.17 is another example of nonlinear behavior. When WIP w(t) is relatively low in a production work system such as that shown in Figure 2.15, production capacity may not be fully utilized and actual capacity can be less than full capacity because of the work content of individual orders and the timing of arriving orders. Conversely, when WIP is relatively high, work is nearly always waiting and the work system is nearly fully utilized. The actual capacity ra(t) of the work system therefore may depend on both its full capacity rf and the work in progress w (t).

      Figure 2.17 Actual production capacity function and a piecewise linear approximation.

      As shown in Figure 2.17, the actual capacity function can be approximated in a piecewise manner by two segments, delineated by a WIP transition point wt hours, where for w(t) ≥ wt

      and for w(t) < wt

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