STAT702 Notes Compilation PDF

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Lecture 1Quality - degree of fitness for a specific purpose, helps with current and future needsQuality in - 1. Baked beans - the beans themselves, the ingredients used in the sauce, process, how it is to eat, shelf life, the can 2. Bus system - ease of use, route, ability to get through traffic, co...

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Lecture 1 Quality - degree of fitness for a specific purpose, helps with current and future needs Quality in 1. Baked beans - the beans themselves, the ingredients used in the sauce, process, how it is to eat, shelf life, the can 2. Bus system - ease of use, route, ability to get through traffic, comfort of seats 3. Motor car - engine, comfort, design, durability, extra features such as A/C, WOF Use 1. 2. 3.

statistics in quality - this makes a quality management system Describe process Solve problem Make decision

Quality Assurance - management system, control and maintain product quality, need different checkpoints of quality Quality Control - process to review quality of all factors involved in production 1. 2. 3. 4. 5.

Failure Failure Prevention Appraisal Prevention (can also be appraisal)

Quality Criteria Practice : 1. Can of vegetable soup - taste, actual quantity in mL (weight), proportion of dented cans, appearance defects 2. Delivery of AUT paper - student satisfaction, class grade averages 3. Carton of yoghurt - weight, expiry date, taste Kaizen philosophy - improve constantly and forever the system of production and service, to improve quality and productivity and thus decrease costs PDCA - plan, do, check, act Elegant Eight - 8 tools for quality improvement 1. Flow chart

2. 3. 4. 5. 6. 7. 8.

e.g. Plan process, pick fruit, transport to factory, sort fruit into categories, examine fruit based on quality criteria, pack good fruit, export fruit Time chart Cause and effect chart (fishbone, ishikawa) Control chart Pareto chart Dot plot/histogram Scatter plot Factorial Design Experiment

EXAMPLE : CANNED WHOLE TOMATOES Materials



Alumini um



Supplie rs

Measuremen  ts Personnel

Inspect ors

 

Rulers Employ ee training



Operat ors

Environment 

Moistur e

Methods

  

Machines

heat Stop/st art of machine transpo rt - Conveyor belts - pulleys

Lecture 2 Acceptance Sampling Sample raw materials for quality at beginning of process AND sample at end of process to make sure finished product is ideal for customer Start ----------> End ^ | | | Control charts Accept or reject a lot based on a sample we take. There may be a maximum number of defectives allowed before rejection for things like export.

Defective VS. Defect Peaches example - defects include bruising, discolouration, size, shape, rot Vinyl example - defects include scratching, wear and tear, Item that contains at least ONE defect = classified as defective Sampling by attribute - each item selected is classified in 2 categories i.e. Defective or not, within specifications or not, minor or major defects, local or overseas markets, acceptable or not. 3 sampling plans : 1. Single sampling - one sample taken for quality 2. Double sampling - take another sample because unsure 3. Sequential sampling - take many samples before judging Poisson approximation to binomial - large number of trials with small probability of success then approximate binomial with Poisson. Mean = n * p (greater N is and the less P is = better approximation. N at least 30, p less than 0.1) EXAMPLE Find proability that sample of no more than 2 defectives is accepted N * p = 200 * 0.04 P(acceptance) = p(x = 0) + p(x = 1) + p(x = 2) Operating Characteristic Curve -

Basic curve code

Producers risk - risk that good quality lot is rejected by sampling plan. Quality of this lot is called acceptable quality level (AQL). This is a type 1 error in hypothesis testing.

Consumers risk - risk that poor quality lot is accepted by sampling plan, gets to consumer, quality of such lot is called rejectable quality level or Lot tolerant percent defective. This is a type 2 error in hypothesis testing. EXAMPLE :

Basically the plan is to take samples of 198, and accept samples that have no more than 4 defectives. Assess Command in R :

Put in the rejection and acceptance points and this is the output :

To find a better plan :

Rectification - replace defective items with good ones, schemes usually restricted to small lots of specialized product. Average outgoing quality - measure of proportion of defectives in lot after rectification scheme has been applied. Calculated by Proportion defectives * probability of acceptance . Max value of AOQ is called AOQL (acceptable outgoing quality limit)

To create AOQ Curve :

Max(aoq) is the code to find the maximum AOQ.

Lecture 3 SETTING TARGET MEANS 1. We want to set a target mean based on the minimum value to obtain and a measure of standard deviation (how much it is allowed to vary from that target mean) 2. To do this we need to find a Z score, one that when used in calculation, will result in z = 0.

3. This shows that -3.9 is the best solution for obtaining a score that is 0 (so z = 0). FORMULA FOR TARGET MEAN : Minimum value - (z x standard deviation) EXAMPLE : Minimum value = 400, standard deviation is 1.5 and we know z = -3.9 SO 400 - (-3.9 * 1.5) = 405.85

USING AN N OR NP CHART N = sample size P = desired mean proportion Q = (1 - P) Mean/centre line = N * P Sigma / standard deviation = sqrt(N*P*Q) FORMULAS FOR NP CHART LIMITS :

EXAMPLE :

Steps : 1. For each day, record how many rooms are actually ready (for the last 25 days) 2. Use quality tools in R (qcc library) 3. Input data in R For the NP chart for the unready rooms, we need to subtract each of number of ready rooms from 80 (the total number of rooms) 4. Np = number of rooms not ready pers day with n = 80 5. Use R Code to construct the chart : 6. Which creates

QUESTIONS : Say we want a 98% room readiness. Are we achieving this? To calculate :

(N - Centre line) / N For this example : ((80 - 3.64) / 80) * 100 = 95.45 %. This is the room readiness we are currently achieving. This is < 98%, and therefore we are not achieving our goals. UCL = 3.64 + (3 * 1.863939) = 9.231817 To improve this, we may add more staff, change rosters to include training, reviewing procedure more for room readiness, adjusting the times of cleaning so that staff start earlier

CUSUM CHARTS 1. Short for cumulative sum 2. Can detect sudden and huge changes as well as small and persistent departures from centre line 3. This is a plot of cumulative differences between successive values and target, use for both proportions and means/ranges Has 3 slopes :

V MASK - helps decide when action needs to be taken when using CUSUM chart, helps you see whether the line is going upwards or down as a general trend. When the graph cuts the "upper arm" - this is when we need to take action. That point is an aberration.

V mask angle = theta = gives u the angle between the end of the V = can calculate angle based on the scales given. EXAMPLE :

TO CONSTRUCT THE CHART : 1. Library (qcc) for quality control tools 2. Observations are a sequence from 1 to 12 in steps of 1 i.e. Observation = seq(1, 12, 1) 3. Input the number of defectives for each sample i.e. Defectives = c(1, 1, 1, 0, 0, 0, 4, 4, 4, 3, 3, 3) 4. Now do the cusum(defectives, 1, 2). This does the cumulative sum of all the defectives, tells it that each number is 1 sample, and that each sample is allowed 2 defectives. 5. This creates the chart

If we change it to 1 defective per sample -

Doing a CUSUM for the Room Readiness example Room readiness target of 95% is being met at 2 periods - days 1-7 and 14 onwards On day 14 there is a distinct change in the % of rooms that are ready

EWMA CHARTS 1. These charts take into account the current observation and the observation that was previous 2. Current observation = alpha, previous = (1 - alpha) 3. This chart acknowledges the fact that current observations are affected by those previous 4. The weighting factors: α, α(1-α), α(1 – α)**2, α(1 – α)**3, … are generally used. These factors form a geometric sequence with a sum to infinity of 1.The weighting factors: α, α(1-α), α(1 – α)**2, α(1 – α)**3, … are generally used. These factors form a geometric sequence with a sum to infinity of 1.

EXAMPLE : 1. Install library(qcc) for quality control tools 2. Input dataset, in this case lemonade 3. Create a matrix for the data 4. Make a variable that stores the ewma chart with lambda and nsigmas.

20 groups of 3, no points along the limits, we have the smoothing aspect - the solid line which is the smooth graph. The UCL and LCL depend on the smoothing aspect and when the standard deviation settle down. First we have actual values - our 20 means of 3 with a smooth average. We use our z value as 2.7, this is the highest z score that gives 0.99 as a value (99% confidence interval). Look at previous calculations for z score to see how this is obtained.

Lecture 6 PROCESS CONTROL :  

Evaluate how stable a process is over time Use control charts in conjunction w this



Evaluates how well a process meets specifications

The more spread out a time plot looks, the more we see that there are special causes of variation present

If the process goes outside of its spec limits, it is not capable, though it may be stable. Capable processes has almost all measurements fall inside specification limits. DEFINITION : PROCESS CAPABILITY Compares output of in control process to specification limits using

capable indices. Comparison is made by formation of ratio of spread between specifications (width) to spread of actual process values (6 process standard deviation units / process width). PROCESS CAPABILITY :  Compare natural variability to spec limits of process  Evaluates performance of stable process  Reflect variation in individual observations  Standard calculations of the processes capability assume that the process itself has underlying normality  The analysis is always an estimate o Always variation of sampling o No process is always fully in control o Exact distribution unknown PROCESS / NATURAL VARIABILITY :

Width of the process = 6 standard deviations = This amount contains 99.7% of the data generated by the process. SPECIFICATION WIDTH : Always between the USL and LSL (lower and upper specification limits) CAPABILITY INDICES :  Must be used on a large enough sample i.e. 50 or more  They provide a summary of the processes performance (relating to customer requirements)  They are commonly used by many professions  They are sensible  PURPOSE - the distribution of the capability indices can be monitored as an indication of overall performance. They can be used to estimate the % out of spec PROCESS CAPABILITY FORMULA :

EXAMPLE : USL = 20 , LSL = 8, SD = 2. What Cp = (20 - 8) / (6 * 2) Cp = 1 SO the process is capable as long midpoint m = (20 + 8) / 2 = 14 PROCESS CABILITY INDEX FORMULA :

For the process to be capable,

is Cp? as the

Cpk should be at least 1. Anything under is not considered A "good" process EXAMPLE : USL = 20 , LSL = 8, SD = 2, AVERAGE = 16

Not a good process or capable, as it is under 1 CP AND CPK :  Cp is ratio of allowable spread : actual process spread  Cp unaffected by relative position of process mean compared to limits  Cpk is ratio of spread from process mean to nearest spec limit : actual spread / 2  WHEN PROCESS IS CENTRED, CP = CPK  Cp in this case represents the highest possible value of Cpk ONE SPEC LIMIT :  If given only 1 limit then Cp is not defined  We need 2 limits  Cpk still calculated though  Uses indices : 

Cases w 2 limits like this are known as bilateral. If only 1 then unilateral

PROCESS CABILITY INDEX WHEN T DIFFERS FROM MIDPOINT :

PROCESS PERFORMANCE INDICES :

These are the same formula for Cp and Cpk BUT these always use total variation (overall std.dev) When using Cp and Cpk we use within spec variation to assess best case performance - this is only ok when the process is STABLE Otherwise :

When process is not stable, Pp and Ppk are a more realistic assessment of capability because they use the total variation % CAPABILITY PERFORMANCE : Specify upper and lower spec limits - you have a mean and standard deviation based on the process too. You can now use the normal distribution to calculate the % of output that is BETWEEN the spec limits :

3 COMPONENTS OF PROCESS CAPABILITY : 1. Design specs or customer expectations 2. Centre of natural process (mean) 3. Spread of process (variance) 6 steps of capability study (determines if process is stable and capable and consistently making good products: 1. Select critical parameters 2. Collect data 3. Check assumptions 4. Calculate required stats 5. Interpret Cp and Cpk 6. Establish monitoring system

INTERPRET CP :  If above 1.33 it is acceptable  Process is capable INTERPRET CPK :  Jointly measures centring and variation

NOTE - Centring the data increases the Cpk

Lecture 7 EOQ MODEL : Economic Order Quantity

Quantity ordered -> needs to minimize cost function to balance total cost from over/understock Inventory cycle -> period between max stock levels Set up cost -> fixed charge for order placement or with initial prep of production site. Is independent of quantity ordered. Time between orders = Q (time taken for order quantity to be sold)/ A(amount ordered in one year to meet annual demand) Annual inventory cost (Tc) = annual order cost(k x (A/Q)) + annual holding cost (h x (Q/2)) Annual order cost = k(cost of single order) x (A / Q)  A/ Q = number of orders placed in a year Holding cost = Q / 2(average inventory level) x h (annual holding cost for one item) Optimum order quantity : Q* :

Which makes :

EXAMPLE :

EXAMPLE 2 :

Q* = sqrt((2 * 10 * 5200)/0.4) = 509.9019513592785 T = 509.9 / 5200 = 0.0981 years 0.0981 * 365 = 35.8065 days So optimum is 510 bottles produced every 36 days.

Annual Inventory Cost = (10 * (5200 / 510)) + (0.4 * (510 /2)) = 203.9608 This rounded up is $204 Annual order cost and annual holding cost are always equal.

OPTIMUM BACKORDER MODEL :

In this model, demand does not have to be met from the stock immediately. Demand is accepted when there is no stock because it is understood that these orders will be met when stock becomes available.

Holding cost only associated with first section of each cycle, when there is stock. Proportion of year when there is stock = S/Q Average stock during first phase of cycle = S/2

Average backorder level during out of stock section of inventory : (Q - S) / 2 Proportion of year for which

If these are set then inventory costs will be miminized. If p becomes very very large then cost is too large to be viable so model becomes EOQ EXAMPLE :

EPQ Model - Economic production quality  Product produced for 

inventory Goods produced internally

  

We assume goods are produced at a continuous rate for a period of time and then stops Production runs begin when 0 is reached This model determines best order quantity

Lecture 8 Single Period Inventory Model :  One order placed for product  At the end of this period the product is either sold out or the company has a surplus of items left over  Used to solve inventory problem in quantitative way  Used for o Seasonal items o Perishables  Examples include newspapers  Single product because product can't be sold later or at salvage value  No initial inventory  1 decision is number of units to order  Unknown distribution EXAMPLE :

Suppose uniform probability - range of demand is from 350 to 650 with average demand of 500. USE INCREMENTAL ANALYSIS : this compares cost/loss of ordering 1 additional unit with the cost of not ordering 1 additional unit.

Exact level of demand is unknown, therefore we consider probability of demand / probability of obtaining those costs/losses

From this we compute the E[X] of loss or the expected loss (EL).

However, we need to find the specific amount to order i.e. The optimal quantity Q*. To find this we can use the equation :

This then becomes :

Which simplifies to :

We can now use this equation to solve for the optimal order size P(demand...