Quantitative Review I

Report
Quantitative Review I
Spring 2013
Vicky Gu
Key Concepts:
1. Productivity
2. Productivity Change
Ch.1, p.14
Productivity is the ratio of outputs (goods and services) divided by
the inputs (resources, such as labor and capital)
Productivity (P) =
=
Productivity Change (Productivity index) is used to compare a process’
productivity at a given time (P2) to the same process’ productivity at an
earlier time (P1)
P2  P1
Growth Rate 
P1
Example
– Last week a company produced 150 units using 200
hours of labor, and found to have 10 defective units
– This week, the same company produced 180 units with
3 defective units using 230 hours of labor
What is the change in productivity?
(150  10) units
P1 
 0.70 units / hour
200 hours
(180  3) units
P2 
 0.77 units / hour
230 hours
P2  P1 0.77  0.70
Growth Rate 

 0. 1
P1
0.70
A 10% increase in productivity
Productivity of last week
Productivity of this week
Productivity change
If inputs increase by 30% and outputs decrease by 15%, what
is the percentage change in productivity?
P1= outputs/inputs = 1/1
P2 = (1- 0.15)/ (1+0.3) =0.654
Productivity change = (P2-P1)/ P1
= 0.654-1 = -0.3462
1
Key Concept: Multifactor Productivity
It measures productivity using ratio of outputs to several
inputs such as labor, material, energy…… Ch 1. p.15
• Convert all inputs & outputs to $ value
• Example:
– 200 units produced sell for $12.00 each
– Materials cost $6.50 per unit
– 40 hours of labor were required at $10 an hour
Calculate the multifactor productivity
200 units $12 / unit
$2400

 1.41
200units $6.50 / unit  40 hours $10 / hour $1700
Revenue Management Systems
(also called Yield Management)
Ch.2
•Airline booking
Overbooking –accepting more reservations than capacity
available, assuming that a certain percentage of customers
will not show up or will cancel prior to using the service
Example: A regional airline that operates a 50-seat jet prices the ticket
for one popular business flight at $250. If the airline overbooks the
reservations, overbooked passengers receive a $450 travel business
flight voucher. The airline is considering overbooking by up to 2 seats,
and the demand for the flight always exceeds the number of
reservations it might accept. The probabilities of the number of
passengers who show up is given for each booking scenario in the
following table:
Number of passengers showing up
45
Number of
reservations
46
47
48
49
50
0.1
51
50
0.18 0.25 0.15 0.22
0.1
51
0.06 0.13 0.13
0.28 0.28 0.02
52
0.06 0.125 0.175 0.2
0.1
How many passengers should they book?
52
0.35 0.05 0.02 0.02
# of passengers actually showed up
45
# of seats booked
Reservations
46
47
48
49
50
0.1
51
50
0.18 0.25 0.15 0.22
0.1
51
0.06 0.13 0.13
0.28 0.28 0.02
52
0.06 0.125 0.175 0.2
0.1
52
0.35 0.05 0.02 0.02
Expected Profit
50
=250*(45*0.18+46*0.25+47*0.15+48*0.22+49*0.1+50*0.1)
=$11777.5
51
=250*(45*0.06+46*0.13+47*0.13+48*0.1+49*0.28+50*0.28) -450*0.02
=$11818.5
52
=250*(45*0.06+46*0.125+47*0.175+48*0.2+49*0.35+50*0.05)-450*(0.02+0.02)
=$11463.2
They should book 51 passengers
•Hotel Management
-Contribution to profit and overhead
-Hotel Management Effectiveness
Your first job is in hotel management and recently you were promoted
to Hotel Manager for a large convention hotel in downtown New
Orleans. Answer the following questions given the information below
for one day. What is the total contribution to profit and overhead?
What is your hotel effectiveness percentage?
Characteristic/Variable
Business Hotel Customers
(B)
Convention Association
Hotel Customers (C)
260 room nights rented (DB)
400 room nights rented
(DC)
$125 (PB)
$85 (PC)
Variable cost/room night
(VC)
$25
$25
Maximum price/room night
(called the rack rate)
$150
$110
300 room nights available
700 room nights
available
Customers for this day (D)
Average price/room night(P)
Maximum number rooms
available for sale this day
Contribution to profit and overhead ($)
= (PB - VC)*DB+(PC -VC)*DC
= ($125 - $25)*260 + ($85- $25)*400
= $50000
Hotel Management
Effectiveness (%) =
Actual hotel revenue
Maximum possible hotel revenue
(Actual prices for each room night)*(Actual number of room nights rented)
=
Maximum price for each room night)*(Maximum number of room nights available
150 *300+110* 700
=
125 * 260 +85* 400
54.5%
=
Key Concept: Forecasting
Forecasting is the art and science of predicting future events.
Quantitative forecasting involves taking historical data and project them
Into the future with mathematical models. Ch. 4. p.104
Important forecasting methods to project the demand
1)
2)
3)
4)
5)
Moving Average (Simple vs. Weighted)
Exponential Smoothing
Seasonality forecasting
Linear Regression
Tracking signal
Time Series Models
Casual Model
Used to monitor forecast
accuracy
Simple Moving Average – Uses an average of the n most
recent periods of data to forecast the next period (Ch 4. p.109)
(when we assume that market demands will stay fairly steady over time)
Example: Lauren's Beauty Boutique has experienced the
following weekly sales. Calculate a 3 period moving average
for Week 6.
Week Sales
415 + 458 +460 = 444.3
1
432
3
2
396
3
415
4
458
5
460
6
Weighted Moving Average – use weights to place more
emphasis on recent values (Ch 4. p. 110)
(This is used when a detectable trend or pattern is present)
Example: A firm has the following order history over the last 6
months. What would be a 3-month weighted moving average
forecast for July, using weights of 40% for the most recent
month, 30% for the month preceding the most recent month, and
30% for the month preceding that one?
January
120
February 95
March
100
April
75
May
100
June
50
50*40% +100*30%+75*30% = 72.5
Exponential Smoothing – Uses a weighted average of past
time-series values to forecast the value of the time series in the
next period (Ch 4. p. 112)
Ft 1  At  1   Ft
–
–
–
–
Last period’s forecast (Ft)
Last periods actual value (At)
Select value of smoothing coefficient α, between 0 and 1.0
The forecast “smoothes out” the irregular fluctuations in the time
series
– Forecast quality is dependent on selection of alpha
(Typical values for α are in the range of 0.1-0.5, larger values of α place
more emphasis on recent data, if the time series is very volatile and
contains substantial random variability, a small value of the smoothing
constant is preferred.)
Example: The manager of a small health clinic would like to use
exponential smoothing to forecast demand for emergency
services in their facility. If she uses an alpha value of 0.2, what is
the mean absolute deviation of her forecasts from Weeks 2
Through 6? (Assume that the forecast for Week 1 is 430).
Week
Actual Demand
in Patients
Exponential
Smoothing
Forecast
1
430
430
2
234
3
506
4
470
5
468
6
365
Absolute Deviation
Week
Actual
Demand in
Patients
Exponential
Smoothing
Forecast
1
430
430
2
234
430
234-430 = 196
3
506
391
506-391 =115
4
470
414
470-414 = 56
5
468
425
468-425 = 43
6
365
434
365- 434 = 69
Mean absolute
deviation(MAD)
for wk 2~6
Absolute Deviation
430-430 =0
=(196+115+56+43+69)/5
= 95.8
Ft+1 = αAt+(1-α)Ft
Week 2 forecast
F2 = .2 (430)+.8(430) = 430
Week 3 forecast
F3 = .2 (234)+.8(430) = 391
Week 4 forecast
F4 = .2 (506)+.8(391) = 414
Week 5 forecast
F5 = .2 (470)+.8(414) = 425
Week 6 forecast
F6 = .2 (468)+.8(425) = 434
• Mean absolute deviation (MAD) – A measure of
the overall forecast error for a model (Ch 4. p. 113)
MAD =
N: number of periods of data
Tracking Signal – It is used to measure of how well a
(TS)
forecast is predicting actual values
Ch. 4, p. 132
• Mean Absolute Deviation
(MAD):
1 n
MAD =  Ai  Fi
n i =1
– A good measure of the
actual error in a forecast
(See the previous exponential smoothing example)
• Tracking Signal (TS)
- Exposes forecast bias
(positive or negative)
- Positive tracking signal
=under-forecasting
- Negative = over-forecasting
Cumulative error
actual- forecast

TS 
MAD
Example: Given the actual demand and forecast from Jan.
to Apr. what will be the MAD and TS?
Month
Actual Demand (A)
Forecast (F)
Jan
60
68
Feb
50
52
Mar
65
55
Apr
35
40
A-F
Absolute
Deviation
Total
-8
8
-2
2
10
10
-5
5
-5
25
MAD =25/4 =6.25
TS 
 actual- forecast
MAD
TS  5/ 6.25  .8
Seasonal Forecasting –forecast method used to project
seasonal demand based on seasonal variation in historical
data (regular up-and-down movements in a time series that
relate to recurring events such as weather or holidays)
(Ch.4, p. 121)
Example: Joe’s Equipment Distributors sells “Raider Power” brand
lawn mowers. The demand forecast for 2002 is 2000 units. Given the
historical sales figures listed below derive a forecast for each quarter in
2002.
Historical Data
1999
2000
2001
90
110
200
120
420
500
300
600
650
380
450
510
Historical Data
The given data
1999
2000
2001
Spring
90
110
200
Summer
Fall
120
420
500
300
600
650
Winter
380
890
450
1580
510
1860
2000
890/4=222.5
1580/4=395
1860/4= 465
2000/4=500
Total
1. Calculate the average for each year
Current Year
Average
2002
Seasonal Index
1999
2. Calculate the seasonal index
for each quarter in each year
3-year spring average index
3. Calculate the average index
for each season, then calculate the
forecast of each season
3-year winter average index
2000
2001
90/222.5= .40
110/395 = .28
200/465 = .43
120/222.5=.54
420/395 =1.06
500/465 = 1.08
300/222.5=1.35
600/395 = 1.52
650/465 =1.40
380/222.5=1.71
450/395 =1.14
510/465 =1.10
Average index
Forecast
(1999-2001)
2002
(.40+.28+.43)/3 = .37
.37*500= 185
(.54+1.06+1.08)/3 = .89
.89 *500=446
(1.35+1.52+1.40)/3=1.42
1.42*500=711
(1.71+1.14+1.10).3=1.31
1.31*500=657
Regression analysis – A method for building a statistical model
that defines a relationship between a single dependent variable
and one or more independent variables (Ch 4. p.126)
The Regression Equation or Trend Forecast
Tx  y  a  bX
Tx = trend forecast or y variable
a = estimate of Y-axis intercept where x = 0
b = estimate of slope of the demand line
X = period number or independent variable
Linear Regression
• Identify dependent (y) and independent (x) variables
• Solve for the slope of the line
XY  n X Y

b
 X  n((X) )
2
2
• Solve for the y intercept
a  Y  bX
• Develop your equation for the trend line
Tx or y =a + bX
Example:
Cover Me, Inc. sells umbrellas in three cities. Management
assumes that annual rainfall is the primary determinant of
umbrella sales, and it wants to generate a linear regression
equation to estimate potential sales in other cities. Given
the data, what is the estimated amount of sales for 40
inches of rain utilizing a linear regression equation?
XY  n XY

b
 X  n((X) )
2
a  Y  bX
2
Rainfall "X" Sales "Y"
X*Y
X2
City A
35
$2800
98000
1225
City B
30
$2000
60000
900
City C
15
$800
12000
225
Total
80
$5600
170000
2350
26.67
$1866.67
Average
b  (170000  3 * 26.67 * 1866.67) /[(1225  900  225)  3 * (26.67 ^ 2)]  95.38
a  1866.67  95.38 * 26.67  677
Y = a +bX = -677 +95.4*40= $3138
Key Concept: Break-Even Analysis
A way of finding the point, in dollars and units, at which
costs equal revenues (Supplement 7 p. 292)
Total cost = FC +VC*Q
Total revenue = SP *Q
At break-even point
FC +VC*Q= SP*Q
Solve for Q:
Q (SP-VC) =FC
FC
Q
SP  VC
FC : Fixed Cost
VC: Variable Cost
SP: Selling Price
Q: Number of units produced
Example: Blaster Radio Company is trying to decide
whether or not to introduce a new model. If they introduce
it, there will be additional fixed costs of $400,000 per year.
The variable costs have been estimated to be $20 per
radio. If Blaster sells the new radio model for $30 per radio,
how many must they sell to break even?
FC
Q
SP  VC
Q = $400,000/ ($30-$20)
Q = 40,000
The company has to sell 40,000 radios to break even
Example: If Blast radio company can’t sell 40,000 radios in the first year,
instead, their sales forecast is as follows:
Year 1: Sell 25,000
Year 2: Sell 42,000
Year 3: Sell 60,000
At which year will the company achieve break even?
Answer: To achieve break even in each year (i.e. to cover both the FC & VC),
Sales need to reach 40,000 unit per year from what we just found out
Year 1: 25,000 – 40,000 = -15,000 (short of 15,000 radios)
Year 2: 42,000 – 40,000 = 2,000 (over 2000 radios)
Year 3: Need 40,000 + (15000-2000)= 53,000 to break even
53,000/60,000 =0.88 0.88*12 months = 10.6,
10.6 months in year 3 or by November the BE will be reached
Key Concept:
Manufacture capacity utilization and efficiency
Supplement 7, p. 283
Capacity - The maximum output rate of production or service
facility or units of resource availability
Theoretical capacity - Also called ideal capacity, designed
capacity, (best operating level)
Maximum output rate under idea conditions
e.g. A bakery can make 30 custom cakes per day when
pushed at holiday time
Effective capacity - Also called realistic capacity
It is the maximum output rate under normal conditions
e.g. On the average this bakery can make 20 custom
cakes per day
Capacity Utilization - measures how much of the available
capacity is actually being used
actual output
(100%)
Utilization effective =
effective capacity
Utilization design =
Example: A bakery can make 30 custom cakes per day when
pushed at holiday time (or the design capacity is 30 custom cakes
per day), but under normal condition, it makes 20 custom cakes per
day on average. Currently the bakery is producing 28 cakes per
day. What is the bakery’s capacity utilization relative to both
theoretical and effective capacity?
Utilization effective
Utilization design 
actual output
28
(100%) 
(100%)  140%
effective capacity
20
actual output
28
(100%) 
(100%)  93%
theoretical capacity
30
• The current utilization is only slightly below its theoretical capacity
and considerably above its effective capacity
• The bakery can only operate at this level for a short period of time
Example: A clinic has been set up to give flu shots to the elderly in a
large city. The theoretical capacity is 50 seniors per hour, and the
effective capacity is 44 seniors per hour. Yesterday the clinic was open
for ten hours and gave flu shots to 330 seniors.
(a) What is the theoretical utilization?
(b) What is the effective utilization?
Yesterday the clinic was open for ten hours and gave flu shots to 330
seniors
So the actual output is 330 senior / ten hours  33 senior / hour
We know the theoretical capacity is 50 senior / hour
We also know the effective capacity is 44 senior / hour
Utilization theoretical = 33/50 =66%
Utilization effective = 33/44 = 75%
Example: A manufacturer of printed circuit boards has a
theoretical capacity of 900 boards per day. The theoretical
capacity utilization is 83% currently, what is the current
production?
Utilization design 
Solve for X:
actual output
X
(100%)
(100%) 83%
theoretical capacity
900
83% * 900 =74 7
Key Concept: Decision Trees for Capacity Planning Decisions
• Build from the present to the future:
– Distinguish between decisions (under your
control) & chance events (out of your control,
but can be estimated to a given probability)
• Solve from the future to the present:
– Generate an expected value for each decision
point based on probable outcomes of
subsequent events
Example: The owners of Sweet-Tooth Bakery have determined that they
need to expand their facility in order to meet their increased demand for
baked goods. The decision is whether to expand now with a large facility or
expand small with the possibility of having to expand again in 5 years. The
owners have estimated the following chances for demand:
The likelihood of demand being high is 0.65. ·
The likelihood of demand being low is 0.35.
for each alternative have been estimated as follows:
•Large expansion has an estimated profitability of either $110,000 or
$40,000, depending on whether demand turns out to be high or low.
•Small expansion has a profitability of $40,000, assuming demand is low.
•Small expansion with an occurrence of high demand would require
considering whether to expand further. If the bakery expands at this point,
the profitability is to be $60,000, if not, $20,000.
What decision should the bakery make, and what is the expected value of
that decision?
Step 1. We start by drawing the decision trees
Low demand
Expand small
Expand large
High demand
Expand
Don’t expand
Low demand
High demand
Step 2. Add our possible states of probabilities, and
potential revenue
$40,000
0.35
Expand small
Expand
0.65
1
$60,000
2
Don’t expand
Expand large
0.35
$40,000
$20,000 X
0.65
$110,000
It is obvious that not to expand
is not a good choice
Step 3. Determine the expected value of each decision
0.35
Expand small
$40,000
Expand
0.65
1
$60,000
2
Do nothing
Expand large
0.35
0.65
$40,000
$20,000
$110,000
EVsmall = (0.35)*40,000 +0.65*60,000 = $53000
EVlarge = (0.35)*40,000+(0.65)*110,000 = $85500
Expanding large generates the greatest expected profit, so our choice is
to expand large, and the expected value for this decision is $85500
Interpretation
• At decision point 2, we chose to expand to maximize
profits ($60,000 > $20,000)
• Calculate expected value of small expansion:
– EVsmall = 0.35($40,000) + 0.65($60,000) = $53000
• Calculate expected value of large expansion:
– EVlarge = 0.35($40,000) + 0.65($110,000) = $85500
• At decision point 1, compare alternatives & choose
the large expansion to maximize the expected profit:
– $85500 > $53000
• Choose large expansion despite the fact that there is
a 35% chance it’s the worst decision:
– Take the calculated risk!
Key Concepts:
Bottleneck
- The limiting factor or constraint in a system.
Process time of a station
-The time to produce units at a single workstation.
Process time of a system
-The time of the longest (slowest) process; the
bottleneck.
Process cycle time
- The time it takes for a product to go through the
production process with no waiting.
Three-Station Assembly Line
There are 60 minutes in each hour
Capacity: (60 min/hr) /2 (min/unit) = 30 units/hr
B
Capacity: 60 (min/hr) /3 (min/unit) = 20 units/hr
A
C
4 min/unit
2 min/unit
Capacity: 60 (min/hr) /4 (min/unit) = 15 units/hr
3 min/unit
Process time for each station: 2 minutes, 4 minutes, 3 minutes
Process time for the system: 4 minutes (the bottleneck)
Process cycle time: 2+4+3 =9 minutes
(the time to produce one finished product)
Capacity Analysis with Simultaneous Process
order
Make
patties
Cook
burgers
30 sec/unit
60 sec/unit
Add Veggie
& cheese
10 sec/unit
Wrap
20 sec/unit
45 sec/unit
Make
patties
30 sec/unit
Cook
burgers
Add Veggie
& cheese
60 sec/unit
10 sec/unit
The process time of each assembly line is 60 second
The process time of the combined assembly line operations is 60 sec per two burgers, or
30 sec per burger. Thus, the wrapping becomes the bottleneck for the entire operation
which is 45 sec per burger.
Capacity: within each hour which is 3600 second, 80 burgers are made (3600 /45=80)
If productivity needs to be increased, then the bottleneck station should be the first to start

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