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Economic Control Chart Models with Cycle Duration Constraints

Table 1. Cost Minimization Techniques with Respect to Continuous

Control Chart Parameters
	X -Charts (Cycle over values of n)	p-charts (Cycle over values of n and R)
Unconstrained Optimization	Minimize cost over L and h using Nelder-Mead algorithm or IMSL® routine UMINF	Minimize cost over h using Nelder-Mead algorithm or IMSL® routine UVMIF
Constrained Optimization	2-D Search Approach: Minimize cost over L and h, subject to cycle duration constraint, using IMSL routine NCONF	Find h to satisfy Pq 95 = Tmax using Newton’s method or IMSL®routine ZREAL
	1-D Search Approach: Minimize cost (via Nelder-Mead or UVMIF) over L. Given L, h is chosen to satisfy p0 95 = Tmax (solved via Newton’s method or ZREAL)

subroutine based for example on Newton's method or using IMSL subroutine ZREAL. For constrained X -chart examples, two altematives were considered for minimizing over L and h (given a fixed value n). The constrained optimization problem represents a nonlinear programming problem with a

single constraint and can be solved using IMSL routine NCONF. Altematively, we can assume the constraint holds with equality and proceed as follows: (1) Use a single variable minimization routine such as UVMIF or the Nelder-Mead based subroutine to minimize the expected cost function with respect to the parameter L; and (2) for each value of L at which the