ISQS 3344 Introduction to Production and

Operations Management

Spring 2014

Quantitative Review I

SUPPLEMENT 7Capacity and Constraint Management

(Break-Even Analysis)

Portland Radio Company (PRC) 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 $30 per radio. If PRC sells the new radio model for $40 per radio, how many must they sell to break even?

VCSP

FQ

unitsQ 000,4030$40$

000,400$

Which Year to Break Even (BE)Year BE

Quantity(At)

Available for Sale

(Bt=At+Dt-1)

Demand(Ct)

Left Over /Inventory(Dt=Bt-Ct)

1 40,000 40,000 30,000 10,000

2 40,000 50,000 35,000 15,000

3 40,000 55,000 60,000 -5,000

Which Month of Year 3 to Break EvenYearly Demand = 60,000Monthly Demand = 60,000/12 = 5,000BE = 5,000 x 11 months = 55,000 or Month

11 of Year 3OR 55,000/60,000 = 0.9167

0.9167 x 12 = 11

CHAPTER 1Operations and Productivity

Productivity

Inputs

OutputsP

#

#

Labor Productivity

hourtablehours

tables

Inputs

OutputsP /1

24

24

Three workers paint twenty-four tables in eight hours

Inputs: 24 hours of labor (3 workers x 8 hours)Outputs: 24 painted tables

Multifactor Productivity

• Convert all inputs & outputs to $ value

50.1600,1$

400,2$

/10$40/6$200

/12$200

P

hourhoursunitunits

unitunitsP

Example:– 200 units produced sell for $12 each– materials cost $6 per unit– 40 hours of labor were required at

$10 an hour

Productivity Index

• Can be used to compare a process’ productivity at a given time (P2) to the same process’ productivity at an earlier time (P1)

1

12

P

PPRateGrowth

Productivity Growth RateExample:– Last week, a company produced 150 units

using 200 hours of labor.– This week, the same company produced 170

units using 240 hours of labor.

hourunitshours

unitsP

hourunitshours

unitsP

/71.0240

170

/75.0200

150

2

1

rategrowthnegativeaor

P

PPRateGrowth

%5

05.075.0

75.071.0

1

12

If inputs increase by 25% and outputs decrease by 10%, what is the percentage change in productivity? Assume P1 = 1/1.

decreaseorChange

Change

P

P

%2828.%1

172.0%

72.025.1

90.0

11

1

2

1

Production increased to 75 from 65 pieces per day. Defective items have dropped from 12 to 5 pieces per day. Production facility operates eight hours per day. Seven people work daily in the plant. What is the change in productivity?Period 1 Output = 65 Defects = 12 Net Output = 53Period 2 Output = 75 Defects = 5 Net Output = 70Period 1 Input = 8 hours x 7 workers = 56Period 2 Input = 8 hours x 7 workers = 56

P1 = 53/56 = 0.95P2 = 70/56 = 1.25Change = (1.25-0.95)/0.95 = 0.32

CHAPTER 2Revenue Management Systems

Revenue Management Systems (Hotel)Contribution to profit and overhead ($)

= [(Selling Price – Variable Cost) x Demand]1 + [(Selling Price – Variable Cost) x Demand]2 +… + [(Selling Price – Variable Cost) x Demand]n

Hotel management effectiveness (%)= Actual hotel revenueMaximum possible hotel revenue

= Actual prices for each room night x Actual number of room nights rented Maximum legal price for each room night x Maximum number of room nights available in hotel

Hotel ManagementCharacteristic/Variable Business

Hotel Customers Convention Association

Hotel Customers Customers for this day (Demand)

400 room nights rented

800 room nights rented

Average price/room night (Selling Price) $200 $120

Variable cost/room night (Variable cost)

$50 $50

Maximum price/room night $250 $150

Maximum number rooms available for sale this day

500 room nights available

900 room nights available

Hotel ManagementContribution to profit and overhead ($)

= [(Selling Price – Variable Cost) x Demand]= [($200 - $50) x 400] + [($120 - $50) x 800]= ($150 x 400) + ($70 x 800) = $116,000

Hotel management effectiveness (%)= Actual prices for each room night x Actual number of room nights rented

Maximum legal price for each room night x Maximum number of room nights available in hotel= ($200 x 400 rooms) + ($120 x 800 rooms) ($250 x 500 rooms) + ($150 x 900 rooms)= ($80,000 + $96,000) / ($125,000 + $135,000) = 67.69%

Revenue Management Systems (Airlines)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 $300 travel voucher. The airline is considering overbooking by up to 5 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 in the following table:

Number of reservations 45 46 47 48 49 50 51 52 53 54 5550 0.100 0.150 0.150 0.200 0.300 0.10051 0.080 0.130 0.180 0.150 0.250 0.110 0.10052 0.060 0.125 0.175 0.200 0.250 0.100 0.050 0.04053 0.040 0.050 0.070 0.200 0.250 0.150 0.100 0.080 0.06054 0.020 0.040 0.050 0.090 0.120 0.210 0.180 0.140 0.100 0.05055 0.010 0.030 0.040 0.060 0.100 0.120 0.200 0.180 0.150 0.090 0.020

Number of passengers showing up

Overbooking Strategies and AnalysisNumber of reservations 45 46 47 48 49 50 51 52 53 54 55

50 0.100 0.150 0.150 0.200 0.300 0.10051 0.080 0.130 0.180 0.150 0.250 0.110 0.10052 0.060 0.125 0.175 0.200 0.250 0.100 0.050 0.04053 0.040 0.050 0.070 0.200 0.250 0.150 0.100 0.080 0.06054 0.020 0.040 0.050 0.090 0.120 0.210 0.180 0.140 0.100 0.05055 0.010 0.030 0.040 0.060 0.100 0.120 0.200 0.180 0.150 0.090 0.020

Number of passengers showing up

Expected revenue for 50 reservations = $250 x (45*.1 + 46*.15 + 47*.15 + 48*.2 + 49*.3 + 50*.1) = $11,937.50Expected re venue for 51 reservations = [$250 x (45*.08 + 46*.13 + 47*.18 + 48*.15 + 49*.25 + 50*.11)] – ($300 x .1) = 10,717.50Expected re venue for 52 reservations = [$250 x (45*.06 + 46*.125 + 47*.175 + 48*.2 + 49*.25 + 50*.1)] – [$300 x (.05+.04)] = $10,854.25Expected re venue for 53 reservations = [$250 x (45*.04 + 46*.05 + 47*.07 + 48*.2 + 49*.25 + 50*.15)] – [$300 x (.1+.08+.06)] = $9,113Expected revenue for 54 reservations = $6,306.50Expected revenue for 55 reservations = $4,180.50

CHAPTER 4Forecasting

Lauren's Beauty Boutique has experienced the following weekly sales. Calculate a 3 period moving average for Week 6.

Week Sales123456

432396415478460451

4513

4604784156

SalesWeek

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 60% for the most recent month, 20% for the month preceding the most recent month, and 20% for the month preceding that one?

January 120February 95March 100April 75May 100June 50July

65)75)(2.0()100)(2.0()50)(6.0( JulyOrder65

Exponential Smoothing

– Last period’s actual value (At)– Last period’s forecast (Ft)– Select value of smoothing coefficient, between 0

and 1.0

ttt FAF 11

Summary of Single Exponential Smoothing Milk-Sales Forecasts with α = 0.2

F2 = .2(172) + .8(172) =172

F3 = .2(217) + .8(172) = 181

F4 = .2(190) + .8(181) = 182.8

F5 = .2(233) + .8(182.8) = 192.84

F6 = .2(179) + .8(192.84) = 190.07

F7……….

You start with past data and calculate forecasts working forward.

Determine forecast for periods 7 and 8exponential smoothing with alpha = 0.2 and the period 6 forecast being 375.

Period Actual

1 300

2 315

3 290

4 345

5 320

6 360

7 375

8 372.6372.0

ExponentialSmoothing

Period 7 = 0.2(360) + 0.8(375) = 72 + 300 = 372.0

Period 8 = 0.2(375) + 0.8(372) = 75 + 297.6 = 372.6

375.0

Quarterly ForecastingExpected total demand in 2012 is 3,000 units. Given the historical sales figures below, derive a forecast for each quarter in 2012.

2009250500700900

0.430.851.191.53

2010270530800970

0.420.821.251.51

20113106008501000

0.450.871.231.45

Q1Q2Q3Q4

0.430.851.221.50

2350 2570 2760Total

Quarter 587.5 642.5 690

20123246369171123

3000

750

250/587.5

(1.53+1.51+1.45)/3

750 x 0.43

The Regression Equation orTrend 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

Linear Regression

XXX

YXXYb

2

• 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

A maker of golf shirts has been tracking the relationship between sales and advertising dollars. Use linear regression to find out what sales might be if the company invested $60,000 in advertising next year.

)X)n((X

YXnXYb 22

Sales$ (Y)

Advertising$ (X)

XY X^2

1 130 48 6240 2304

2 151 52 7852 2704

3 150 50 7500 2500

4 158 55 8690 3025

5

Total 589 205 30282 10533

Average 147.25 51.25 2633.25

XbYa

bXayxT

Y = a + bXY = -36.17 + 3.579XY = -36.17 + 3.579(60)Y = 178.57 or $178,570 in sales

)X)n((X

YXnXYb 22

XbYa

a = 147.25-3.579(51.25) = -36.17

XbYa

579.375.26

75.95

)25.51(410533

)25.147)(25.51(430282b

2

Tracking Forecast Error Over Time

• Mean Absolute Deviation (MAD)– A good measure of the actual error in

a forecast

• Tracking Signal (TS)

– Exposes bias (positive or negative)Positive TS = under-forecastingNegative TS = over-forecasting

MAD

TS forecast - actual

n

1=iii FA

n

1=MAD

Mean Absolute Deviation• MAD sums only absolute values errors, both

positive and negative errors add to the sum and the average size of the error (whether positive or negative) is determined.

n = number of periods of dataF = forecast of demand in period iA = actual demand in period i

n

1=iii FA

n

1=MAD

A company is comparing the accuracy of two forecasting methods. Forecasts using both methods are shown below along with the actual values for January through May. The company also uses a tracking signal with ±4 limits to decide when a forecast should be reviewed. Which forecasting method is best?

Month Actual sales

Method A Method B

Forecast Error Abs. Value Forecast Error Abs. Value

Jan. 30 28 2 2 27 3 3

Feb. 26 25 1 1 25 1 1

Mar. 32 32 0 0 29 3 3

Apr. 29 30 -1 1 27 2 2

May 31 30 1 1 29 2 2

MAD = 5/5 = 1TS = 3/1 = 3

MAD = 11/5 = 2.2TS = 11/2.2 = 5

Supplement 7Capacity and Constraint

Management

Capacity Utilization

Theoretical Capacity: Maximum output rate under ideal conditions

Effective Capacity: Maximum output rate under normal (realistic) conditions

(100%)capacity

rateoutput actualn Utilizatio

Computing Capacity Utilization

(100%)capacity ltheoretica

output actualn Utilizatio

(100%)capacity effective

output actualn Utilizatio

ltheoretica

effective

In the bakery example, the design capacity is 60 custom cakes per day. On average, this bakery can make 40 custom cake per day. Currently, the bakery is producing 56 cakes per day. What is the bakery’s capacity utilization relative to both theoretical and effective capacity?

Computing Capacity UtilizationIn the bakery example, the design capacity is 60 custom cakes per day. On average, this bakery can make 40 custom cake per day. Currently, the bakery is producing 56 cakes per day. What is the bakery’s capacity utilization relative to both theoretical and effective capacity?

93%(100%)60

56(100%)

capacity ltheoretica

output actualn Utilizatio

140%(100%)40

56(100%)

capacity effective

output actualn Utilizatio

ltheoretica

effective

A clinic has been set up to give flu shots to the elderly in a large city. The theoretical capacity is 80 seniors per hour, and the effective capacity is 55 seniors per hour. Yesterday the clinic was open for ten hours and gave flu shots to 350 seniors.

What is the effective utilization?What is the theoretical utilization?

%75.34(100%)80

350/10(100%)

capacity ltheoretica

output actualn Utilizatio

%64.36(100%)55

350/10(100%)

capacity effective

output actualn Utilizatio

ltheoretica

effective

Decision Trees• 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

Should you expand large or small? Low Demand (0.40) $60,000 Exp Rev Expand Large High Demand (0.60) $100,000 Exp Rev Low Demand (0.40) $50,000 Exp Rev

Expand Small Expand $70,000 Exp Rev High Demand (0.60) Do not Expand $45,000 Exp Rev

$70,000 > $45,000.60 x

$70,000= $42,000

.40 x $50,000 = $20,000

$20,000 + $42,000

= $62,000

.40 x $60,000 = $24,000

.60 x $100,000 = $60,000

$24,000 + $60,000

= $84,000

Capacity AnalysisCapacity analysis determines the

throughput capacity of workstations in a system.

A bottleneck is a limiting factor or constraint.

A bottleneck has the lowest effective capacity in a system or takes the most time.

What is the bottleneck?

What is the maximum production per hour?

Station A (50 seconds)

hourUnitsunit

hour

Bottleneck

imeAvailableTputMaximumOut /72

sec/50

sec/3600

That’s all, folks!

Thanks for today!