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