One of the main criteria for judging about the power of control charts is their ability in fast detection of deviations and shifts in the process. Average time to signal (ATS) or adjusted average time to signal (AATS) are among such criteria calculated under a certain state and assumption. Several studies have shown that using the idea of variable design for control charts, by separating their limits to the safe and the warning regions, can allow quick discovery of shifts and increase sensitivities to small changes. In this paper, a new variable sampling scheme with three sample sizes and two different sampling intervals, called SVSSI, is developed to increase the efficiency of the np control chart. Through various numerical examples, the performance of this scheme is evaluated by calculating ATS and AATS values using Markov chain method. Monte Carlo simulation method is used to validate the results of Markov chain method of SVSSI sampling scheme. In comparison with other schemes, better performance of applying SVSSI is proved in all conditions.

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Due to the imperfection of processes, the quality of some products may be unsatisfactory. Moreover, equipment failure can stop production for a while. Therefore, integrating the triple concepts of quality, maintenance, and inventory control has attracted attention. Triple concepts are the constituents of the pre-sale costs. Selling price and warranty are generally considered to maintain market share and maximize the producer's profit. Quality and maintenance in production should be considered to reduce the post-sale costs of the warranty. Despite interactions, integrating warranty with triple concepts has been neglected. We integrate the quadruple concepts in a biobjective model to maximize the profit and minimize the pre-sale and post-sale costs under the free minimal repair warranty policy. A non-central chi-square (NCS) control chart monitors the mean and variance, simultaneously. The technology level is also considered for increasing product quality and reducing failures during the warranty period. Due to its high complexity, the model is solved by the particle swarm optimization algorithm. The proposed model is applied through a numerical example and three comparative studies. The results indicate the better performance of the NCS chart, the superiority of bi-objective optimization versus single-objective optimization, and the importance of integrating presale and postsale costs.