Open Journal of Applied Sciences, 2012, 2, 236-240

doi:10.4236/ojapps.2012.24035 Published Online December 2012 (http://www.SciRP.org/journal/ojapps)

The Principle of Mathematical Induction Applied to the

Generalized Model for the Economic Design of

-Control Charts

Mohamed E. Seliaman1, Salih O. Duffuaa2

1Department of Information Systems, King Faisal University, Al-Ahsa, Saudi Arabia

2Department of Systems Engineering, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia

Email: mseliaman@gmail.com

Received August 4, 2012; revised September 6, 2012; accepted September 25, 2012

ABSTRACT

Rahim and Banerjee [1] developed a general model for the optimal design of

-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.

Keywords: Discrete Mathematics; Mathematical Induction; Statistical Quality Control; Design of

-Control Charts;

Hazard Rate Function

1. 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 inten-

sive attention from researchers. The

-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 ap-

proaches: heuristics, statistical design, economic design

and economic statistical design [4]. For more information

on the design of control charts, interested readers are re-

ferred to [1-7].

The economic design of

-control charts was origi-

nated 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

-control chart are set at 00

k

, 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

-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 assump-

tion 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] ex-

tended 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 be-

fore failure. The objective of Rahim and Banerjee’s

model [1] is to find the parameters n, hj, 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 re-

search on extending the model. In this paper, we adopt the

same notations and model descriptions as defined in Ra-

him and Banerjee [1].

In order to make the paper self-contained, we describe

the main elements involved in the design of control charts,

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