Spring 2013
Vicky Gu

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)
Growth Rate

P
2

P
1
P
1

– 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?

P
1

P 2

( 150

10 ) units

0 .
70 units / hour
200 hours
( 180

3 ) units

0 .
77 units / hour
230 hours
Growth Rate

P
2

P
1
P
1 
0 .
77

0 .
70
0 .
70

0 .
1
A 10% increase in productivi ty
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?
P
1
= outputs/inputs = 1/1
P
2
= (1- 0.15)/ (1+0.3) =0.654
Productivity change = (P
2
-P
1
)/ P
1
= 0.654-1 = -0.3462
1

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

200
$ 6 .
50 /

$ 12
 
40
/

$ 10 /


$ 2400
$ 1700

1 .
41

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
Number of reservations
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
How many passengers should they book?

# of passengers actually showed up
# of seats booked
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
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

•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
Customers for this day (D)
Average price/room night(P)
Variable cost/room night
(VC)
Maximum price/room night
(called the rack rate)
Maximum number rooms available for sale this day
Business Hotel Customers
(B)
260 room nights rented (D
B
)
$125 (P
B
)
Convention Association
Hotel Customers (C)
400 room nights rented
(D
C
)
$85 (P
C
)
$25
$150
300 room nights available
$25
$110
700 room nights available

Contribution to profit and overhead ($)
= (P
B
- VC)*D
B
+(P
C
-VC)*D
C
= ($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
125 * 260 +85* 400
=
54.5%
150 *300+110* 700

: 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) Moving Average (Simple vs. Weighted)
2) Exponential Smoothing
3) Seasonality forecasting
4) Linear Regression
5) 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.
415 + 458 +460 = 444.3
Week Sales
3
4
1
2
5
6
432
396
415
458
460
3

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)
F t

1
 
A t


1
  
F t
– 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
5
6
3
4
1
2
Actual Demand in Patients
430
234
506
470
468
365
Exponential
Smoothing
Forecast
430
Absolute Deviation

Week
3
4
1
2
5
6
Actual
Demand in
Patients
430
234
506
470
468
365
Week 2 forecast
Week 3 forecast
Week 4 forecast
Week 5 forecast
Week 6 forecast
Exponential
Smoothing
Forecast
430
430
391
414
425
434
Mean absolute deviation(MAD) for wk 2~6
Absolute Deviation
430-430 =0
234-430 = 196
506-391 =115
470-414 = 56
468-425 = 43
365- 434 = 69
=(196+115+56+43+69)/5
= 95.8
F t+1
= αA t
+(1-α)F t
F
2
= .2 (430)+.8(430) = 430
F
3
= .2 (234)+.8(430) = 391
F
4
= .2 (506)+.8(391) = 414
F
5
= .2 (470)+.8(414) = 425
F
6
= .2 (468)+.8(425) = 434

• Mean absolute deviation (MAD) – A measure of the overall forecast error for a model
(Ch 4. p. 113)
N: number of periods of data

– It is used to measure of how well a
(TS) forecast is predicting actual values
Ch. 4, p. 132
• Mean Absolute Deviation
(MAD):
– A good measure of the actual error in a forecast
MAD =
1 n n 
A i
 i = 1
F i
(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


Jan
Feb
Mar
Apr
Example: Given the actual demand and forecast from Jan.
to Apr. what will be the MAD and TS?
Month Actual Demand (A) Forecast (F)
60
50
65
35
68
52
55
40
Total
-8
-2
10
-5
-5
A-F Absolute
Deviation
10
5
25
8
2
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.
1999
Historical Data
2000 2001
90
120
110
420
200
500
300
380
600
450
650
510

The given data
Spring
Summer
Fall
Winter
Total
1. Calculate the average for each year
1999
90
120
300
380
890
Historical Data
2000
110
420
600
450
1580
2001
200
500
650
510
1860
Current Year
2002
2000
Average 890/4=222.5
1580/4=395 1860/4= 465 2000/4=500
2. Calculate the seasonal index for each quarter in each year
1999
90 / 222.5= .40
120 / 222.5=.54
300 / 222.5=1.35
380 / 222.5=1.71
Seasonal Index
2000
110 / 395 = .28
420 / 395 =1.06
600 / 395 = 1.52
450 / 395 =1.14
2001
200 / 465 = .43
500 / 465 = 1.08
650 / 465 =1.40
510 / 465 =1.10
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
Average index
(1999-2001)
( .40
+ .28
+ .43)/ 3 = .37
(.54+1.06+1.08)/3 = .89
(1.35+1.52+1.40)/3=1.42
(1.71+1.14+1.10).3=1.31
Forecast
2002
.37*500= 185
.89 *500=446
1.42*500=711
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
T x = 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 b



X
XY
2 
 n( n X Y
(X)
2
)
• Solve for the y intercept a

Y
 b X
• Develop your equation for the trend line
Tx or y =a + bX

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?

b



X
XY
2 
 n X Y n( (X)
2
) a

Y
 b X
City A
City B
City C
Total
Average
Rainfall "X" Sales "Y"
35 $2800
30 $2000
15
80
26.67
$800
$5600
$1866.67
X*Y
98000
60000
12000
170000
X 2
1225
900
225
2350 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

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
Q

FC
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?
Q

FC
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

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
= actual output effective capacity
(100%)
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?
Ut ilizat io n effective
 act ual out put effect ive capacit y
(100%)

28
(100%)

140%
20
Ut ilizat io n design
 act ual out put t heoret ica capacit y
(100%)

28
(100%)

30
93%
• 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?
Utilizatio n design
 actual output theoretica l capacity
(100%)

X
900
(100%)

83%
Solve for X: 83% * 900 =74 7

Key Concept : Decision Trees for Capacity Planning Decisions
• Build from the present to the future:
– Distinguish between
(under your control) &
(out of your control, but can be estimated to a given probability)
• Solve from the future to the present:
– Generate an
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
Expand small
Expand large
Low demand
High demand
Low demand
High demand
Expand
Don’t expand

Step 2. Add our possible states of probabilities, and potential revenue
$40,000
0.35
$60,000
1
Expand small
Expand large
0.65
0.35
Expand
2
$40,000
Don’t expand
$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
1
Expand small
Expand large
0.35
0.65
0.35
$40,000
Expand
2
Do nothing
$40,000
$60,000
$20,000
0.65
$110,000
EV small
= (0.35)*40,000 +0.65*60,000 = $53000
EV large
= (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:
– EV small
= 0.35($40,000) + 0.65($60,000) = $53000
• Calculate expected value of large expansion:
– EV large
= 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!

:
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
A
4 min/unit
Capacity: 60 (min/hr) /4 (min/unit) = 15 units/hr 2 min/unit
Capacity: 60 (min/hr) /3 (min/unit) = 20 units/hr
C
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
Make patties
30 sec/unit
Cook burgers
60 sec/unit
Add Veggie
& cheese
10 sec/unit order
20 sec/unit
Wrap
45 sec/unit
Make patties
30 sec/unit
Cook burgers
60 sec/unit
Add Veggie
& cheese
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