BITS Pilani presentationBITS Pilani, Pilani Campus
Control charts for Range
BITS Pilani, Pilani Campus
The choice between the mean and R-chart is a managerial problem. Mean chart is used to control the quality averages of the samples drawn from a given process, where as R-Chart is used to control the quality variability's of the samples.
Also it is better to construct R-Chart first. If it indicates that the dispersion of the quality by the process is out of control, it is better not to construct Mean chart, until the quality dispersion is brought under the control.
Mean chart reveals undesirable variations between samples while R-Chart reveals any undesirable variations within the samples
INTERPRETATION OF MEAN CHART AND R-CHART
BITS Pilani, Pilani Campus
1. A drilling machine bores holes with a mean diameter of 0.5230cm and a standard deviation of 0.0032cm. Calculate the 2-sigma and 3-sigma upper and lower control limits for means of samples of 4 and prepare a control charts. [Ans: LCL=0.5198cm,UCL=0.5262cm
LCL=0.5182cm,UCL=0.5278CM]
BITS Pilani, Pilani Campus
Suppose random samples of size n are taken from the product output at some time intervals. If d be the number of defectives in a sample, then the fraction defective in that sample is given by p=d/n.
We may suppose that the number of defectives d=np is binomial with E(np)=nP and V(np)=
Where P is the population fraction defective.
The process will be in statistical control if P is the same for all the samples.This will be so if the statistic np lies within the 3-sigma limits nP±3
Control charts for number of defectives (or np-Chart)
BITS Pilani, Pilani Campus
BITS Pilani, Pilani Campus
When the sample size is constant, it is immaterial whether one uses np-chart or p-chart. However, when the sample size varies, all the three control lines will vary with n, in the case of an np-chart. The resulting chart will be highly confusing.
On the other hand in a p-chart the central line remains invariant. Therefore in case the sample size varies, it is simpler and preferable to use the p-chart.
Choice between p-chart and np-chart
BITS Pilani, Pilani Campus
Defective : It is an item that fails to satisfy some given specification(s)
Defect : It is an event of the item’s lack of conformity to given specifications.
Every defectve item contains one or more defects.
Examples : defective rivets in an aircraft , exposed wires in a refrigerator, surface defects in a roll of a paper, Crumpled pages in a book, loose screws in a bicycle etc.
Control charts for the number of defects (or c-chart)
BITS Pilani, Pilani Campus
BITS Pilani, Pilani Campus
The c-chart relates to samples of constant size. In case of varying sample sizes, it is more convenient to study the control charts for the number of defects per unit, u=c/n.
Control charts for per unit Defects ( or u-chart)
BITS Pilani, Pilani Campus
Control limits for u-chart
BITS Pilani, Pilani Campus
An u-chart is applied to control the number of defects per unit in the case of fairly complex assembled units as T.V.sets, aircraft engines, refregirators etc.
Applications of u-chart
In Decision making we deal with devising Future Plans :
For instance, in Financial Planning like share markets, Investments in Fixed and Recurring deposits, we need to predict the pattern of cash-flow over time.
FORECASTING AND REGRESSION MODELS
BITS Pilani, Pilani Campus
BITS Pilani, Pilani Campus
The value for a period of time t is given by
Where the time series is stable, follows a constant process
b = Unknown constant parameter estimated from historical data
= Random component for period t with zero mean and constant variance
The data for different periods are not correlated
Moving Average Technique
A Constructive process for selecting the best alternatives.
The goodness of a selected alternative depends on the quality of the data used in describing the decision situation
DECISION THEORY(Sequential Decision Models)
BITS Pilani, Pilani Campus
I Decision making under certainty in which the data are known deterministically (Data well defined)
II Decision making under risk in which the data can be described by probability distribution (Data are ambiguous)
III Decision making under uncertainty in which the data cannot be assigned weights that represent their degree of relevance in the decision process. (Data middle-of-the-road situation)
THREE CATEGORIES OF DECISION MAKING
BITS Pilani, Pilani Campus
EXPECTED MONETARY VALUE (EMV) :
S-1 : List the conditional profit for each act event combination, along with the corresponding event probabilities.
S-2 : For each act, determine the expected conditional profits.
S-3 : Determine EMV for each act.
S-4 : Choose the act which corresponds to the optimal EMV.
Decision making under risk in which the data can be described by probability distribution (Data are ambiguous)
BITS Pilani, Pilani Campus
S-1 : List the conditional profit table for each act-event combination , along with corresponding event probabilities.
S-2 : For each event, determine the Conditional Opportunity Loss (COL=Highest Pay off-other act loss) values by first locating the most favorable act (maximum pay-off) for that event and then taking the difference between that conditional profit value and each conditional profit value for that event.
S-3 : For each act, determine the expected COL values and sum these values to get the expected opportunity loss (EOL) for that act.
S-4 : Choose that act which corresponds to the minimum COL value.
EXPECTED OPPORTUNITY LOSS(EOL)
BITS Pilani, Pilani Campus
1.A manager has a choice between (i) A risky contract promising Rs.7 lakhs with probability 0.6 and Rs.4 lakhs with probability 0.4 , and (ii) A diversified portfolio consisting of two contracts with independent outcomes each promising Rs.3.5 lakhs with probability 0.6 and Rs.2 lakhs with probability 0.4. Construct a decision tree for using EMV criteria. Can you arrive at the decision using EMV criteria ?
problems
BITS Pilani, Pilani Campus
Using the EMV criteria , the manager must go for the risky contract which will yield him a higher expected monetary value of Rs.5.8 lakhs
Event
Probability
Posterior (or Bayes’) probabilities: The probabilities obtained using sampling or experimentation.
Prior probabilities are obtained from the raw data.
The Bayesian Decision -Making
BITS Pilani, Pilani Campus
Analytic Hierarchy process(AHP) :
Designed for situation in which ideas , feelings and emotions are quantified based on subjective judgment to provide a numeric scale for prioritizing decision alternatives
Decision making under certainty in which the data are known deterministically
BITS Pilani, Pilani Campus
1.Mr X has received full academic scholarships from three Institutions : U of A , U of B and U of C. To select a university Mr X specifies two main criteria : location and academic reputation. Being excellent student he judges academic reputation to be five times as important as location, which gives a weight of approximately 17% to location and 83% to reputation. Analyze the best choice of the university by ranking the three universities from the standpoint of location and reputation.
problems
BITS Pilani, Pilani Campus
Given the table of ranks based on two criteria for the three universities :
Percent
BITS Pilani, Pilani Campus
It reflects the decision makers judgment of the relative importance of the different criteria.
Aij = criteria in the row i is ranked relative to each of the criteria represented by the n columns.
Also Aij = 1 signifies that i and j are equally important
Aij = 5 indicates that i is strongly more important than j
Aij = 9 indicates that I is extremely more important than j
Other intermediary values between 1 and 9 are interpreted correspondingly
For consistency Aij = 1 automatically implies that Aij=1/k
Also all the diagonal elements Aii of A must equal 1 because they rank a criteria against itself.
Determination of relative weights :
Then the correspnding matrix is given by
L R
L R Row Average
R(.83 .83) WR= .83
The columns are identical implies the charecteristic that occurs only when decision maker exhibits perfect consistency in specifying the entries of the comparison matrix A.
Consider the previous problem