rel="nofollow" href="#fb3_img_img_5657a54c-c424-5146-95cc-0cab4f2132a0.png" alt="r Subscript p Baseline left-parenthesis left-parenthesis k plus d right-parenthesis upper T right-parenthesis equals r Subscript f Baseline left-parenthesis k upper T right-parenthesis"/>
where dT days is the delay in implementing permanent worker capacity adjustments. Hence, the portions of fluctuating order input that are addressed by permanent worker capacity rp(kT) orders/day and cross-trained capacity rc(kT) orders/day are
2.4 Model Linearization
A component behaves in a linear manner if input x1 produces output y1, input x2 produces output y2, and input x1 + x2 produces output y1 + y2. The following are examples of linear relationships:
The following are examples of nonlinear relationships:
In reality, most production system components have nonlinear behavior, but often the extent of this nonlinearity is insignificant and can be ignored, with care, when a model is formulated. On the other hand, behavior that is significantly nonlinear often can be modeled in a simpler but sufficiently accurate manner using approximate linear models obtained using approaches such as those described in the following subsections.5
2.4.1 Linearization Using Taylor Series Expansion – One Independent Variable
A nonlinear function f(x) of one variable x can be expanded into an infinite sum of terms of that function’s derivatives evaluated at operating point xo:
xo is the operating point about which the expansion made. Over some range of (x – xo) higher-order terms can be neglected, and the following linear model in the vicinity of the operating point is a sufficiently good approximation of the function:
where
Such an approximation is illustrated in Figure 2.14.
Figure 2.14 Linear approximation of function f(x) at operating point xo.
Example 2.9 Production System Lead Time when WIP Is Constant and Capacity Is Variable
A production work system such as that illustrated in Figure 2.15 has constant work in progress (WIP) w hours and variable production capacity r(t) hours/day. The lead time l(t) hours then is approximately
Figure 2.15 Production work system with variable capacity.
The relationship between lead time and capacity is nonlinear; however, a linear approximation of this relationship in the vicinity of operating point ro can be obtained using Equations 2.2 and 2.3:
