Design of Optimal Reliability Acceptance Sampling Plans for Exponential Distribution

In this paper, an attempt is made to derive the most efficient economic reliability sampling plans for accepting a lot containing identical units having exponentially distributed lifetime with parameter θ. We consider two types of sampling plans, namely, (a) sequential sampling plan <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>t</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>t</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:math> (t₁,t₂) and (b) repetitive group sampling plan <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>t</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>t</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:math> (n,t₁,t₂) . Under plan (a), the lot is rejected when the time between successive failures <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>Y</m:mi> <m:mi>r</m:mi> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:math> (Yᵣ) is less than t 1 , and accepted when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>Y</m:mi> <m:mi>r</m:mi> </m:msub> <m:mo>≥</m:mo> <m:msub> <m:mi>t</m:mi> <m:mn>2</m:mn> </m:msub> </m:mrow> </m:math> Yᵣ≥ t₂ . The testing will continue for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>t</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>≤</m:mo> <m:msub> <m:mi>Y</m:mi> <m:mi>r</m:mi> </m:msub> <m:mo><</m:mo> <m:msub> <m:mi>t</m:mi> <m:mn>2</m:mn> </m:msub> </m:mrow> </m:math> t₁≤ Yᵣ< t₂ . Also we formulate an optimization problem that minimizes the total expected testing cost. Under plan (b), four different criteria are used to derive the sampling plan. The optimization problem formulated under each criterion is solved using a genetic algorithm to obtain the plan parameters <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>t</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>t</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:math> (n,t₁,t₂) . Several numerical examples are discussed to illustrate our plans. In addition, a real example is also considered to demonstrate our plan. Finally, we compare the cost of our plan with that of an existing plan in the literature. Our plan has significant potential to reduce the testing cost by about 50%.