om ii - class 5
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Simple linear regression takes the formy = a + bx + e
y is the dependent variable
x is the independent or explanatory variable
Constants „a‟ and „b‟ represent the intercept and slope,
respectively, of the regression line
Term „e‟ represents an “error” term or “residual”
In other words, we may think of the relationship between xand y as if it follows the straight line formed by the function y
= a + bx, subject to some unexplained factors captured in
error term
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Regression is a means to find the line that most closely
matches the observed relationship between x and y.
Most common approach is to minimize the sum of
squared differences between the observed values and
the model values
Sum of squared differences SS: 22
1 1
n n
i i i
i i
SS e y a bx
1 1 1
2
2
1 1
n n n
i i i i
i i i
n n
i i
i i
n x y x y
b and a y bx
n x x
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Much of the information in the regression tables is quite
detailed and beyond the scope of our coverage
Focus on five measures of how well the regression model is
supported by the data: R2, Standard error, F-statistic, p-
statistics, and confidence intervals
First three of the measures apply to the regression model
as a whole
Last two apply to individual regression coefficients
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Having found the values of the constants a and b that give
the best fit of the model to the observed data, we might
wonder, how good a fit is the best fit?
Goodness can be interpreted as the extent to which all of the variation in y-values is completely accounted for (or
explained) by their dependence on x-values.
In order to quantify the outcomes and to provide a measureof how well the regression equation fits the data, we
introduce the coefficient of determination, known as R2
Its square root is the coefficient of correlation, „r‟
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Values of R2 must lie between zero and one.
Closer R2 to one, the better the fit
When R2 is relatively large, then the model explains much of
the variation in observed y-values
When R2 is equal to one, the regression equation is perfect – it
explains all of the observed variation
When R2 is zero, the regression equation explains none of the
observed variation
Unexplained variation could be because of “noise” or other
systematic factors that were not considered in the regression
equation
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If the standard error of estimate (Se) is small, the fit isexcellent and the linear model should be used for
forecasting
If it is large, the model is poor But, what is small and what is large?
Judge the value of Se by comparing it with the sample mean
of the dependent variable In this example, Se is 32.165 and sample mean is 783.9
Hence, in this case, the linear regression model developed
is good
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The significance level of F answers this question: Howlikely is it that we would get the R2 we observe (or
higher) if, in fact, all the true regression coefficients
were zero? In other words, if our model really had no explanatory
power at all, how likely is it that, sampling at random, we
would encounter the explained variation we observed?
Precisely, what is the probability of getting a coefficient
of determination of R2 by fluke or by chance?
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p-statistic or p-value is somewhat akin to thesignificance level of the F-statistic.
It answers this question: How likely is it that we would
get an estimate of the regression coefficient at least thislarge (either positive or negative) if, in fact, the true
value of the regression coefficient were zero?
In other words, if there really was no influence of this
particular explanatory variable on the response variable,
how likely is it that we would encounter the estimated
coefficient we observed?
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The most useful way to determine the range of uncertainty in a regression coefficient is to form a
confidence interval
A confidence interval for the coefficient will tell uswhether the actual parameter is likely to lie between two
numbers
This information, along with the point estimate of the
coefficient itself and our judgment, will help us to select
a final parameter value for the coefficient in our model
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Value of R2
is about 0.745, which suggests that themodel fits the data rather well
Probability under Significance of F is about 6 percent,
which suggests that it is unlikely that this value R2
couldhave arisen by chance
a = 708 and b = -27.7, indicating the average y value
drops by almost 28 points each round
p-value for the intercept is very small, suggesting that
this estimate would be unlikely by chance
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However, the p-value for X variable 1 is 6 percent, whichsuggests that we cannot be so sure that this estimate
could not have arisen by chance
The 95% confidence interval for X variable is extremelywide, from -57 to +2. This suggests that our point
estimate, from which we concluded that the average y
value drops by almost 28 points each year, is ratherimprecisely estimated
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Demand Forecasting
Capacity Planning
Aggregate Planning
Inventory Control
Scheduling
Quality Control & Maintenance
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Demand Forecasting
Capacity Planning
Aggregate Planning
Inventory Control
Scheduling
Quality Control & Maintenance
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Capacity refers to a system‟s potential for producing goods or
delivering services over a specified time interval
Goal of capacity planning is to achieve a match between long-
term supply capabilities and the predicted level of long-term
demand
Basic questions in capacity planning:
- What kind of capacity is needed?
- How much is needed?
- When is it needed?
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Capacity often refers to an upper limit on the rate of
output
In selecting a measure of capacity, it is important to
choose one that does not require updating (capacity of
$30 million a year)
When only one product or service is involved, capacity
may be expressed in terms of that item When multiple products are involved, capacity is often
expressed in terms of inputs
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Type of BusinessInput Measures of
Capacity
Output Measures
of Capacity
Car manufacturer Labor hours Cars per shift
Hospital Available beds Patients per month
Pizza parlor Labor hours Pizzas per day
Retail storeFloor space in
square feet Revenue per foot
No single measure of capacity will be appropriate in everysituation. Rather it must be tailored to the situation
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Design capacity:
◦ Maximum output rate under ideal conditions
Effective capacity:
◦
Design capacity minus allowances such as personaltime, maintenance, and scrap
Actual output cannot exceed effective capacity and is
often less because of machine breakdowns,absenteeism, shortage of materials, and quality
problems
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Actual outputEfficiency =
Effective Capacity
Actual outputUtilization =
Design Capacity
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Facilities – Location, Size, Provision for expansion etc.,
Product and service factors – Design similarities
Process & human factors – Quality considerations
Policy factors – Overtime, second or third shifts
Operational factors – Scheduling, inventory, purchasing
Supply chain factors – Suppliers, warehousing, logistics
External factors – Safety, Unions, Pollution control etc.,
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1. Estimate future capacity requirements
2. Evaluate existing capacity
3. Identify alternatives
4. Conduct financial analysis
5. Assess key qualitative issues
6. Select one alternative7. Implement alternative chosen
8. Monitor results
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1. Design flexibility into systems (capacity cushions)
2. Take stage of life cycle into account
3. Take a “big picture” approach to capacity changes
4. Prepare to deal with capacity “chunks”
5. Attempt to smooth out capacity requirements
6. Identify the optimal operating level
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100% –
80% –
60% –
40% –
20% –
0 –
N i s s a n
C h r y s l e
r
H o n d a
G M
T o y o t a
F o r d
Percent of North American Vehicles Made on Flexible Assembly Lines
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(a) Leading demand withincremental expansion
D e m a n d
Expecteddemand
Newcapacity
(b) Leading demand withone-step expansion
D e m a n d
Newcapacity
Expecteddemand
(c) Capacity lags demand withincremental expansion
D e m a n d
Newcapacity
Expecteddemand
(d) Attempts to have an averagecapacity with incrementalexpansion
D e m a n d
Newcapacity Expected
demand
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Demand exceeds capacity Curtail demand by raising prices, scheduling
longer lead time
Long term solution is to increase capacity
Capacity exceeds demand
Stimulate market
Product changes
Adjusting to seasonal demands
Produce products with complementary demandpatterns
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4,000 –
3,000 –
2,000 –
1,000 –
J F M A M J J A S O N D J F M A M J J A S O N D J
S a l e s i n
u n i t s
Time (months)
Air-conditioners
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4,000 –
3,000 –
2,000 –
1,000 –
J F M A M J J A S O N D J F M A M J J A S O N D J
S a l e s i n u n i t s
Time (months)
Air-conditioners
Room heaters
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4,000 –
3,000 –
2,000 –
1,000 –
J F M A M J J A S O N D J F M A M J J A S O N D J
S a l e s i n u n i t s
Time (months)
Air-conditioners
Room heaters
Combining both demandpatterns reduces the variation
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BOLis the output that results in the lowest average unit
cost
Economies of Scale:
◦ Where the cost per unit of output drops as volume of output
increases
◦ Spread the fixed costs of buildings & equipment over multiple
units, allow bulk purchasing & handling of material
Diseconomies of Scale:
◦ Where the cost per unit rises as volume increases
◦ Often caused by congestion (overwhelming the process with too
much work-in-process) and scheduling complexity
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Cost-volume analysis
◦ Break-even point
Financial analysis
◦ Cash flows
◦ Present value
Decision theory
Waiting-line analysis
Simulation
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-$90,000Market unfavorable (.6)
Market favorable (.4)$100,000
Market favorable (.4)
Market unfavorable (.6)
$60,000
-$10,000
Medium plant
Market favorable (.4)
Market unfavorable (.6)
$40,000
-$5,000
$0
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-$90,000Market unfavorable (.6)
Market favorable (.4)$100,000
Market favorable (.4)
Market unfavorable (.6)
$60,000
-$10,000
Medium plant
Market favorable (.4)
Market unfavorable (.6)
$40,000
-$5,000
$0
EMV = (.4)($100,000)+ (.6)(-$90,000)
Large Plant
EMV = -$14,000
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-$14,000
$13,000
$18,000
-$90,000Market unfavorable (.6)
Market favorable (.4)$100,000
Market favorable (.4)
Market unfavorable (.6)
$60,000
-$10,000
Medium plant
Market favorable (.4)
Market unfavorable (.6)
$40,000
-$5,000
$0