1.
ECONOMIC DESIGN OF
AN X CHART FOR
SHORT-RUN
PRODUCTION
LINDA LEEHO A,
, ANDERSONLAE ´ CIO GALINDOTRINDADE
* 2009 ELSEVIERB .V. ALL RIGHTS RESERVED.
NAME :-VAIBHAV .V. KADU
REG NO:- 2015BPR031

2.
ABSTRACT
• The aim of this paper is to present an economical design of an X chart for
a short-run production. The process mean starts equal to x0 (in-control ,
State I) and in a random time it shifts to x1>x0 (out-of-control , State II).
The monitoring procedure consists of inspecting a single item at every m
produced ones . If the measurement of the quality characteristic does not
meet the control limits , the process is stopped, adjusted, and additional r-
1 items are inspected retrospectively . The probabilistic model was
developed considering only shifts in the process mean . A direct search
technique is applied to find the optimum parameters which minimizes the
expected cost function.

3.
WHAT LEAD TO FOLLOW THIS PAPER
• The aim is to answer some questions faced by the manufacturers of short-run production (since they want
to produce high quality products) as:
1. How to design an economical on-line control procedure for this kind of production?
2. Which is the best sampling strategy? Inspecting all items or is it less costly not to use any kind of process
control?
3. What are the consequences if the long-run production parameters (on-line) are used to control a process
in a short-run production?

4.
TYPES OF CONTROL CHART
• It is graphical representation of collected information .It detect the variation in
processing and warn if there is any departure from specified departure limit
• CONTROL CHART FOR VARIABLES (measurable quality characteristics )
1. X chart
2. R chart
3. Sigma (o-) chart
• CONTROL CHART FOR ATTRIBUTES ( rigid quality characterstics)
1. P chart
2. U chart
3. Np chart
4. C chart

5.
INTRODUCTION
• According to Hillier (1969) short-run control charts are necessary in the initiation of a new process or during the start up of a
process just brought into statistical control again and for processes whose total outputs are not large enough to use the
conventional control charts constants.
• They classify short-run production in two types: repetitive and non-repetitive.
• Repetitive:- The production is repetitive and many small lot sizes of similar parts are manufactured on the same machine
without major setup operations . Ex:- shirts, shoes (orders of small quantities)
• Non-repetitive:- It require completely different setups of the manufacturing equipment to produce different lots
Ex:- Shipbuilding , product for aerospace shuttle
• Various paper were presented by HILLIER (1964,1967,1969), HILLIER & YANG (1970) for controlling quality for small number
of sample. Quesenberry (1991) has also proposed Q charts to address the problems encountered in short-run control charts

6.
ENROUTING THE ECONOMIC DESIGN
• After efforts of many researcher in order to stabalise the x chart for short run production the ship sailed to
economic analysis
• A new approach to get robust economic design of control chart is proposed by Vommi and Seetala
(2007), where each cost and process parameter can be expressed in a range such that it covers the true
parameter and the best control chart parameters are selected.
• In this context, the purpose of this paper is to present a quality on-line monitoring model for variables in a
short run production that minimize the expected average cost per produced item, through a set of
parameters: sampling interval; control limits, and sample size (to be taken in a retrospective inspection)

7.
NOTATIONS AND ASSUMPTIONS
• To develop the probabilistic model, the following assumptions and notations were considered:
1. The process starts in State I (in-control, process mean xo=0)
2. At random time i, the process mean changes to x1=e (out-of-control, State II) with probability p, 0<p<1. Once out-of-control,
the process remains out-of-control until stoppage for investigation, which means Pðyi ¼ ejyi1 ¼ eÞ ¼ 1. There are
supposedly no shifts in the process variability.
3. At every m produced items, one item is inspected, so is the value of the observable variable at i-th
inspection; is the sampling interval.
4. The process stoppage follows immediately (no lag between the signal and the stoppage) and additional (r-1) items are
inspected retrospectively; the process is adjusted and restarts in-control.

8.
THE PRODUCTION AND MONITORING CYCLE CEASE AFTER PRODUCTION
OF N ITEMS (SEE FIG.). A NEW PRODUCTION CYCLE BEGINS AFTER STATING
THE SIZE N FOR A NEW ORDER, WITH OPTIMUM VALUES OF SAMPLING
INTERVAL M, CONTROL LIMITS D, AND A TOTAL SAMPLE SIZE R.

9.
INVESTIGATION OF PARABOLISTIC MODEL
• Consider items produced in a process and a control system that consist of inspecting a single item at
every m items produced. If /Zkm/>D, the production is stopped and additional (R-1) items are inspected
retrospectively. With the process in-control, let
be the probabilities to manufacture an item with the characteristic of interest, respectively, lower and
higher than the control limits.
• In that case a signal led an erroneous stoppage of the process (a false alarm) and the probability of such
event (called error type I) is given by

10.
PROBABILITIES ASSOCIATED WITH CONTROL LIMIT

11.
COST FUNCTION
• Basically the cost function is mainly categorized in 3 parts
1. total fixed cost
2. Total variable cost
3. Total cost
Cost associated with strategy costs of non-conforming
items

12.
A BRIEF REVIEW OF COST FUNCTION

13.
CONCLUSION
• In this paper the authors propose an
economic design of an X chart for short-
run production considering the
possibility of a retrospective inspection
when the examined item is non-
conforming

14.
VAIBHAV KADU
2015BPR031
3RD YEAR PRODUCTI
SGGSIE&T