The Principle of Mathematical Induction Applied to the Generalized Model for the Economic Design of x-Control Charts

Rahim and Banerjee [1] developed a general model for the optimal design of x -control charts. The model minimizes the expected cost per unit time. The heart of the model is a theorem that derives the expected total cost and the expected cycle length. In this paper an alternative simple proof for the theorem is provided based on mathematical induction.


Introduction
Quality control charts are graphical statistical tools used for process control.The first control chart was developed by Walter A. Shewart [2].Since then, these charts have been widely applied in industry and also received intensive attention from researchers.The x -control charts are the most used statistical control charts when continuous variables are used to measure quality characteristics [3].Generally, control charts are designed using four approaches: heuristics, statistical design, economic design and economic statistical design [4].For more information on the design of control charts, interested readers are referred to [1][2][3][4][5][6][7].
The economic design of x -control charts was originated by Duncan [8].In his model, Duncan considers a production process that is subject to an occurrence of an assignable cause that will drive the process out of control.The output of the process is measurable on a certain scale and is normally distributed with mean 0  and standard deviation 0  .The assignable cause is assumed to occur according to a Possion process with intensity  and causes a shift in the process mean to 0    , where  is a positive parameter.The control limits of the x -control chart are set at 0 , where 0  is the standard deviation of the process.A sample of size n is taken from the output of the process every h hour, and used to decide whether the process is in control or not.Banerjee and Rahim [9] generalized Duncan's model [8] for the economic design of x -control charts by relaxing Duncan's assumption that the in-control period follows an exponential distribution.Instead, they assumed it follows a general probability distribution having an increasing hazard rate function.The increasing hazard rate assumption resulted in the modification of the fixed sampling interval to a variable sampling interval, dependent on the process age.In both models, if the sample falls outside the control limits, a search is initiated to locate the assignable cause.If the search indicates that there is a false alarm, the process continues.On the other hand, if the alarm is true, repair or replacement is undertaken to bring the process in control.Later, Rahim and Banerjee [1] extended the model in Banerjee and Rahim [9] by assuming a general distribution of in-control periods with increasing failure rate and considering an age-dependent repair before failure.The objective of Rahim and Banerjee's model [1] is to find the parameters n, h j , and k.Under the assumptions described for this generalized model, they derived expressions for the expected cycle length and the total expected cost per cycle.They also presented proofs for these expressions using a recursive argument.In this paper a simpler and shorter mathematical induction proof of these results is presented.Thus, our alternative proof may make it easier for students and professors interested in the topic within graduate coursework or further research on extending the model.In this paper, we adopt the same notations and model descriptions as defined in Rahim and Banerjee [1].
In order to make the paper self-contained, we describe the main elements involved in the design of control charts, as modeled in [1].

Model Development
In [1], the following assumptions are made: 1) The duration of the in-control period is assumed to follow an arbitrary probability density function, f(t), having an increasing hazard rate, r(t), and F(t) as its cumulative density function.
2) The process is monitored by drawing random samples of size n at times 3) A production cycle begins with a new machine and ends either with a true alarm or at time m  , whichever occur first.In other words, if no true alarm is observed by the time 1 m   , then the cycle is allowed to continue for an additional time m ; at time m h  , the old component is replacement by a new one.Thus, there is no cost of sampling and charting during the m th -sampling interval.4) For mathematical simplicity, it is assumed that the production ceases during search and repair.
In their paper, Rahim and Banerjee [1] state the following theorem and provide a proof for it based on recursive relationships.Below, we first present their result and then provide a simpler mathematical induction proof.
Theorem 1 Under the assumptions (1)-( 3) described above the following is true:

The Mathematical Induction Proof
or Theon inducdel with one interval.Thus, E(T) = h 1 + Z 1 in Equation ( 1) (the basis st vals less than m.Then we show that it is true for a m ach possible st ing data.
In this section, an alternative proof is provided f rem 1.The proof consists of the two well-know tion steps: the basis step and the inductive step.For this purpose, we consider the case after the first sampling interval as a new model with a smaller number of intervals and as having a new density function for the incontrol period.
1) Expected cycle length: a) Let M be a mo , obtained by letting m = 1 ep).b) Assume that ( 1) is true for a model with a number of inter odel with m intervals (the inductive step).
As in [7], let us view the possible states of the system at the end of the first sampling interval.For e ate of the system, the expected residual times in the cycle and the associated probabilities are presented in Table 1.
E(T * ) is the expected cycle length for a model M * with the follow 1) M * has m − 1 intervals with *

 Expected resi
In control and no alarm (1 ) In control and false alarm Out of control but no al

Out of control and true alar
Copyright © 2012 SciRes.OJAppS Thus, Thus, according to the induction hypothesis along with the relations ( 3) and ( 4) and Substituting ( 6) and ( 7) into (5), we obtain hich gives the proof for Equation (1) in the theorem.he ex costs for M * and M ** respectively, from the above table m w 2) The Expected Total Cost ost E(C), we consider t we obtain To obtain the expected total c pected residual cost beyond time h 1 as the expected total cost for a model with less than m intervals.For each possible state of the system at the end of the first sampling interval, the expected residual costs in the cycle (8) Thus, employing the induction hypothesis along with th and the associated probabilities are presented in Table 2.
Where E(C * ) and E(C ** ) are the two expected total e relations (3) and ( 4), Substituting ( 9) and ( 10) into (8), we obtain In control and false alarm 0 h 1 Out of control but no alarm This completes the proof.

Acknowledgements
cknowledge the support for subtracting the term p 1 D 0 h 1 to the right hand side of (11) and substituting