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The percent error in lead time calculated using the linear approximation due to deviation of actual capacity r(t) from the chosen capacity operating point ro is shown in Figure 2.16 and calculated using
Figure 2.16 Percent error in lead time due to deviation of actual capacity from capacity operating point chosen for linear approximation.
Clearly, capacity should not deviate significantly from the operating point if this approximation is used in a model. If, for example, lead time is to be regulated by adjusting capacity, capacity might vary significantly from the operating 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, …:
Over some range of (x – xo), (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:
where
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
