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Quantitative Review I Spring 2013
Vicky Gu
Key Concepts: 1. Productivity2. Productivity
Change
Ch.1, p.14
1
12
P
PPRateGrowth
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)
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?
typroductiviin increase 10%A
1.070.0
70.077.0
1
12
/77.0230
)3180(2
/70.0200
)10150(1
P
PPRateGrowth
hourunitshours
unitsP
hourunitshours
unitsP 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
• 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 $10 an hour
Calculate the multifactor productivity
It measures productivity using ratio of outputs to several inputs such as labor, material, energy…… Ch 1. p.15
41.11700$
2400$
/10$40/50.6$200
/12$200
hourhoursunitunits
unitunits
Revenue Management Systems (also called Yield Management) Ch.2
•Airline bookingOverbooking –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:
45 46 47 48 49 50 51 52
50 0.18 0.25 0.15 0.22 0.1 0.1
51 0.06 0.13 0.13 0.1 0.28 0.28 0.02
52 0.06 0.125 0.175 0.2 0.35 0.05 0.02 0.02
Number of passengers showing up
Number of reservations
How many passengers should they book?
Reservations 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
45 46 47 48 49 50 51 52
50 0.18 0.25 0.15 0.22 0.1 0.1
51 0.06 0.13 0.13 0.1 0.28 0.28 0.02
52 0.06 0.125 0.175 0.2 0.35 0.05 0.02 0.02
# of seats booked
# of passengers actually showed up
•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)
Customers for this day (D)
260 room nights rented (DB)
400 room nights rented
(DC) Average price/room night(P)
$125 (PB)
$85 (PC)
Variable cost/room night (VC)
$25
$25
Maximum price/room night (called the rack rate)
$150
$110
Maximum number rooms available for sale this day
300 room nights available
700 room nights
available
Contribution to profit and overhead ($)
= (PB - VC)*DB+(PC -VC)*DC = ($125 - $25)*260 + ($85- $25)*400= $50000
Hotel Management Actual hotel revenueEffectiveness (%) 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=
=
= 125 * 260 +85* 400
150 *300+110* 700
=
54.5%
Important forecasting methods to project the demand1) Moving Average (Simple vs. Weighted)
2) Exponential Smoothing
3) Seasonality forecasting
4) Linear Regression
5) Tracking signal
Key Concept: Forecasting
Forecasting is the art and science of predicting future events. Quantitative forecasting involves taking historical data and project themInto the future with mathematical models. Ch. 4. p.104
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 Sales123456
432396415458460
415 + 458 +460 = 444.3 3
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 120February 95March 100April 75
May 100June 50
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)
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)
– 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.)
ttt FAF 11
Example: The manager of a small health clinic would like to use exponential smoothing to forecast demand for emergencyservices in their facility. If she uses an alpha value of 0.2, what isthe mean absolute deviation of her forecasts from Weeks 2Through 6? (Assume that the forecast for Week 1 is 430).
Week Actual Demandin Patients
ExponentialSmoothingForecast
Absolute Deviation
1 430 430
2 234
3 506
4 470
5 468
6 365
WeekActual
Demand inPatients
ExponentialSmoothing
ForecastAbsolute Deviation
1 430 430 430-430 =0
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 absolutedeviation(MAD)
for wk 2~6
=(196+115+56+43+69)/5 = 95.8
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
Ft+1 = αAt+(1-α)Ft
• 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
• Mean Absolute Deviation (MAD):– A good measure of the
actual error in a forecast
• Tracking Signal (TS)
- Exposes forecast bias (positive or negative)
- Positive tracking signal =under-forecasting
- Negative = over-forecasting
MAD
TS forecast - actual
n
1=iii FA
n
1=MAD
(See the previous exponential smoothing example)
Cumulative error
Ch. 4, p. 132
Month Actual Demand (A) Forecast (F)
Jan 60 68
Feb 50 52
Mar 65 55
Apr 35 40
MAD
TS forecast - actual
A-F AbsoluteDeviation
-8 8 -2 2 10 10 -5 5
Total -5 258.25.6/5 TS
MAD =25/4 =6.25
Example: Given the actual demand and forecast from Jan. to Apr. what will be the MAD and TS?
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
Seasonal Index
1999 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
1. Calculate the average for each year
2. Calculate the seasonal index for each quarter in each year
3. Calculate the average index for each season, then calculate theforecast of each season
The given data Historical Data Current Year
1999 2000 2001 2002
90 110 200
120 420 500
300 600 650
380 450 510
Total 890 1580 1860 2000
Average 890/4=222.5 1580/4=395 1860/4= 465 2000/4=500
Spring
FallSummer
Winter
3-year spring average index
3-year winter average index
The Regression Equation or Trend Forecast
bXayTx xT = 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
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)
Linear Regression• Identify dependent (y) and independent (x) variables• Solve for the slope of the line
• Solve for the y intercept
• Develop your equation for the trend line
Tx or y =a + bX
XbYa
)(X)n(X
YXnXYb 22
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?
Example:
)(X)n(X
YXnXYb 22
Rainfall "X" Sales "Y" X*Y X2
City A 35 $2800 98000 1225City B 30 $2000 60000 900City C 15 $800 12000 225Total 80 $5600 170000 2350Average 26.67 $1866.67
38.95)]2^67.26(*3)2259001225/[()67.1866*67.26*3170000( b
67767.26*38.9567.1866 a
XbYa
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
VCSP
FCQ
FC : Fixed CostVC: Variable CostSP: Selling PriceQ: Number of units produced
(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
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?
VCSP
FCQ
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 42,000Year 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
Utilization effective =
Utilization design =
(100%)capacity effective
output actual
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?
93%(100%)30
28(100%)
capacity ltheoretica
output actualn Utilizatio
140%(100%)20
28(100%)
capacity effective
output actualn Utilizatio
design
effective
• 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 / hourWe 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?
83%(100%)900
X(100%)
capacity ltheoretica
output actualn Utilizatio design
Solve for X: 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
Don’t expand
Expand small
Expand large
High demand
Low demand
Expand
Low demand
High demand
1Don’t expand
Expand small
Expand large
20.65
0.35
Expand
0.35
0.65
Step 2. Add our possible states of probabilities, and potential revenue
$40,000
$40,000
$110,000
$60,000
$20,000 X
It is obvious that not to expand is not a good choice
Step 3. Determine the expected value of each decision
1Do nothing
Expand small
Expand large
20.65
0.35
Expand
0.35
0.65
$40,000
$110,000
$40,000
$60,000
$20,000
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.
A
B
C
Three-Station Assembly Line
2 min/unit
4 min/unit
3 min/unit
Process time for each station: 2 minutes, 4 minutes, 3 minutesProcess time for the system: 4 minutes (the bottleneck)Process cycle time: 2+4+3 =9 minutes (the time to produce one finished product)
Capacity: (60 min/hr) /2 (min/unit) = 30 units/hr
Capacity: 60 (min/hr) /4 (min/unit) = 15 units/hr
Capacity: 60 (min/hr) /3 (min/unit) = 20 units/hr
There are 60 minutes in each hour
Capacity Analysis with Simultaneous Process
order
Make patties
Cook burgers
Add Veggie & cheese
Wrap
20 sec/unit
30 sec/unit 60 sec/unit 10 sec/unit
45 sec/unit
30 sec/unit 60 sec/unit 10 sec/unit
Make patties
Cook burgers
Add Veggie & cheese
The process time of each assembly line is 60 secondThe 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