, Volume 28, Issue 1, pp 211–236 | Cite as

Kostenoptimale Prüfpläne für die laufende Kontrolle eines normalverteilten Merkmals

  • E. v. Collani


The process under consideration is assumed to start in control with mean μ and standard deviation σ. There is a single assignable cause of variation, which takes the form of a shift in the process mean from μ to μ+δσ or to μ−δσ. The time until the assignable cause occurs has an exponential distribution with parameter λ. To maintain current control 3 different strategies are compared.
  1. 1.

    no control at all,

  2. 2.

    in constant time intervals the process is stopped with a given probability and a search for the assignable cause and if necessary a repair is undertaken,

  3. 3.

    samples of sizen are drawn after constant time intervalls; if the sample mean\(\bar x\) exeeds the action limits\( \pm k\sigma /\sqrt n \) the process is stopped and a search for the assignable cause, and if necessary, a repair is undertaken.

When operating with \gm, the process derives a profit ofGI per item and when operating with mean \gm + \gd\gs or \gm\t-\gd\gs it derives a profit ofGII<GI per item. An expected time oft D hours and an average search cost ofK D are required if the assignable cause does not exist. If the assignable cause actually exists, it can always be discovered and eliminated but takes further average revision time oft R hours and a further average revision cost ofK R to restore the process. The process starts a-new with mean \gm after adjustment. The costs for maintaining the repair workshop are considered to bed R per time unit. Similar the costs for taking a sample are divided in current costsd P per time unit and the actual sampling costs, which are assumed to be proportional to the sample size. Generalized control plans are defined including all 3 strategies, and for the mean loss per item in the long run a loss function is derived. It is shown that there exists an optimal control plan and, suitable defined, that it is unique.


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  4. Duncan, A.J.: The economic design ofX-charts used to maintain current control of a process. J. Amer. Statist. Ass.51, 1956, 228–242.MATHGoogle Scholar

Copyright information

© Phydica-Verlag 1981

Authors and Affiliations

  • E. v. Collani
    • 1
  1. 1.Institut für Angewandte Mathematik und StatistikWürzburg

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