Chapter 2. Forecasting

Chapter 2: Quantitatve MChapter 2: Quantitatve Methods in Health Care Maethods in Health Care Managementnagement

Yasar A. OzcanYasar A. Ozcan 11

Chapter 2. Chapter 2. ForecastingForecasting



OutlineOutline Why Forecast?Why Forecast? Steps in the Forecasting ProcessSteps in the Forecasting Process Forecasting ApproachesForecasting Approaches

– JudgmentalJudgmental– Time Series-- Historical DataTime Series-- Historical Data– Techniques for AveragingTechniques for Averaging– Techniques for TrendTechniques for Trend– Techniques for SeasonalityTechniques for Seasonality– AssociativeAssociative

Accuracy and Control of ForecastsAccuracy and Control of Forecasts Choosing a Forecasting TechniqueChoosing a Forecasting Technique



Why forecasting is important?Why forecasting is important?

Forecasts serve as a basis for planningForecasts serve as a basis for planning Enable health care managers to anticipate the Enable health care managers to anticipate the

future to plan the system and plan the use of future to plan the system and plan the use of that systemthat system

Forecasting is more than predicting demandForecasting is more than predicting demand It is not an exact science; one must blendIt is not an exact science; one must blend experience, judgment, and technical expertiseexperience, judgment, and technical expertise



All forecasts have common elementsAll forecasts have common elements

Assumption that past continues into futureAssumption that past continues into future Errors occur-- actual differs from Errors occur-- actual differs from

predicted; presence of randomnesspredicted; presence of randomness Forecasts of group of items (aggregate) Forecasts of group of items (aggregate)

tends to be more accurate than individual tends to be more accurate than individual items (i.e., departmental vs. whole items (i.e., departmental vs. whole hospital)hospital)

Forecast accuracy decreases as time Forecast accuracy decreases as time horizon increaseshorizon increases



Characteristics of a Good ForecastCharacteristics of a Good Forecast

Timely

Reliable

Accurate

Meaningful units ($$’s, visits, discharges, patient days, etc.)

Easy to use



Step 1 Identify the goal of the forecast

Step 2 Establish a time horizon

Step 3 Select a forecasting technique

Step 4 Conduct the forecast (analyze data)

Step 5 Determine its accuracy

Step 6 Monitor the forecast

Steps in the Forecasting ProcessSteps in the Forecasting Process



What approaches can we use?What approaches can we use?

Judgmental– Delphi method– Executive opinions– Contracts/insurance/HMO/PPO/POS

estimates – Consumer surveys– Outside opinions– Opinions of managers/staff



The Delphi MethodThe Delphi Method Method of obtaining opinions of Method of obtaining opinions of

managers and staffmanagers and staff Involves circulating a series of Involves circulating a series of

questionnaires, each developed from the questionnaires, each developed from the previous one, to achieve a consensus on previous one, to achieve a consensus on an issue (in this case, a forecast)an issue (in this case, a forecast)

Useful for forecasting technological Useful for forecasting technological changes and their impactschanges and their impacts



The Delphi Approach, cont.The Delphi Approach, cont. AdvantagesAdvantages

– More individuals may be engaged than can effectively More individuals may be engaged than can effectively interact face-to-faceinteract face-to-face

– It is important to avoid bandwagon effectIt is important to avoid bandwagon effect– Preserves anonymity of participantsPreserves anonymity of participants

WeaknessesWeaknesses– Questions may be ambiguous leading to false Questions may be ambiguous leading to false

consensusconsensus– Panel members may changePanel members may change– Studies do not prove that Delphi forecasts are highly Studies do not prove that Delphi forecasts are highly

accurateaccurate– Preserving anonymity removes accountabilityPreserving anonymity removes accountability



Forecasting Approaches, cont.Forecasting Approaches, cont. Time series-- Time series-- identify the behavior of the series by using identify the behavior of the series by using

factors such as trend, seasonality, cycles, irregular factors such as trend, seasonality, cycles, irregular variations, and random variationsvariations, and random variations

Techniques for Techniques for averagingaveraging Naive forecastsNaive forecasts Moving averages (MA)Moving averages (MA) Exponential smoothingExponential smoothing

– Techniques for Techniques for trendtrend Linear equations using regression (yLinear equations using regression (ytt = a + bx = a + bxtt)) Trend adjusted exponential smoothing Trend adjusted exponential smoothing

– Techniques for Techniques for seasonalityseasonality Seasonal Variations Seasonal Variations Indices TechniqueIndices Technique



Forecasting Approaches, cont.Forecasting Approaches, cont.

Associative TechniquesAssociative Techniques– Simple linear regression (y = a + bx)Simple linear regression (y = a + bx)– Scatter diagram-- plot dataScatter diagram-- plot data– CorrelationsCorrelations



Jan Mar May Jul Sep Nov

SeasonalVariation

2005

2004

2003

SeasonalVariation

CycleRandom Variation

Trend

Figure 2.1 Variation Characteristics



Averaging TechniquesAveraging Techniques Smooth out fluctuations in time serious Smooth out fluctuations in time serious

because individual highs and lows cancel because individual highs and lows cancel each other outeach other out

So, would forecasts based on averages So, would forecasts based on averages exhibit more or less variability?exhibit more or less variability?



Naive Forecasts A naive forecast for any period equals

the previous period’s actual value Low cost, easy to prepare, easy to

understand, but less accurate forecasts Can be applied to seasonal or trend

dataExamples: If last week’s demand was 50 units, the naive forecastfor the coming week is 50 units.

If seasonal pattern exists, the naive forecast for next January would equal the actual demand for January of this year.



Moving AveragesMoving Averages

where, i = “Age” of data (i=1,2,3. . .) n = number of periods in moving average Ai = actual value with age i

n

AMAF i

nt

Forecast uses a number of the most recent actual data values in generating a forecast



Moving AveragesMoving AveragesExample 2.1:

An OB/GYN clinic has the following yearly patient visits, and would like to predict the volume of business for the next year for budgeting purposes.

Period (t) Age Visits

1 5 15908

2 4 15504

3 3 14272

4 2 13174

5 1 10022



Moving Averages, cont.Moving Averages, cont.Solution:The three-period moving average (MA3) for period 6 is F6 = MA3 = (14272+13174+10022) ÷ 3 = 12489.3

Period (t) Age Visits Forecast

1 5 15908

2 4 15504

3 3 14272

4 2 13174 15228

5 1 10022 14317

6 12489



Moving Averages, cont.Moving Averages, cont.

1 2 3 4 5 6 7 8 9

Data

MA5

MA3

The greater the number of periods in a moving average, the greater the forecast will lag with changes in the data



Moving Averages, cont.Moving Averages, cont.

Easy to compute and understand, but data storage requirements can be high and all values are weighted equally (i.e., in a ten year moving average, each value is given a weight of 1/10, adding up to 1).

A weighted average assigns more weight to recent values



Using Weighted ValuesUsing Weighted ValuesExample:Example: Continuing Continuing

with Example 2.1; since with Example 2.1; since there is a downward there is a downward trend in visits and in trend in visits and in period 5 there is a sharp period 5 there is a sharp decline, a weight of .5 or decline, a weight of .5 or even higher is justified even higher is justified by the healthcare by the healthcare manager to calculate a manager to calculate a weighted average for weighted average for period 6 period 6

Period (t) Age Visits Weights

1 5 15908

2 4 15504

3 3 14272 0.2

4 2 13174 0.3

5 1 10022 0.5

6

iint AwMAF



Using Weighted ValuesUsing Weighted ValuesSolution: Solution:

In this analysis, a weighted average, using In this analysis, a weighted average, using formula [2.2], for the OB/GYN clinic for the formula [2.2], for the OB/GYN clinic for the period 6 would be: period 6 would be: FF66 = 14272*.2+13174*.3+10022*.5 = 14272*.2+13174*.3+10022*.5FF66 = 11818. = 11818.

Period (t) Age Visits Weights Forecast

1 5 15908

2 4 15504

3 3 14272 0.2

4 2 13174 0.3

5 1 10022 0.5

6 11818



Simple Exponential SmoothingSimple Exponential Smoothing Each new forecast is based on the previous forecast plus a Each new forecast is based on the previous forecast plus a

percentage of the difference between that forecast the actual percentage of the difference between that forecast the actual value of the series at that pointvalue of the series at that point

New forecast = Old forecast + New forecast = Old forecast + αα(Actual-Old forecast), where (Actual-Old forecast), where αα is a percentage or is a percentage or

FFtt = F = Ft-1 t-1 + + αα(A(At-1 t-1 - F- Ft-1t-1),),

where, Fwhere, Ftt = Forecast for period t = Forecast for period t FFt-1t-1 = Forecast for period t-1 = Forecast for period t-1 αα = Smoothing constant = Smoothing constant AAt-1 t-1 = Actual demand or sales in period t-1= Actual demand or sales in period t-1



Exponential Smoothing, cont.Exponential Smoothing, cont.Example 2.4:Example 2.4: Using the data Using the data

from Example 2.1, build from Example 2.1, build forecasts with smoothing forecasts with smoothing constant α = 0.3constant α = 0.3

Solution: Solution: Following the previous Following the previous example and formula [2.3], example and formula [2.3], we can build forecasts for we can build forecasts for periods as data become periods as data become available. available.

FF33 = 15908 + .30(15504- = 15908 + .30(15504-15908)15908)

FF33 = 15786.8 = 15786.8

Smoothing constant α = 0.3 Error

Period (t) Actual (Visits) Forecast (Actual – Forecast)

1 15908 --

2 15504 15908 -404.0

3 14272 15786.8 -1514.8

4 13174 15332.4 -2158.4

5 10022 14684.9 -4662.9



Smoothing constant Smoothing constant αα = 0.5 = 0.5 ErrorError

Period(t)Period(t) VisitsVisits ForecastForecast (Actual – Forecast)(Actual – Forecast)

11 1590815908 ----

22 1550415504 1590815908 -404.0-404.0

33 1427214272 15706.015706.0 -1434.0-1434.0

44 1317413174 14989.014989.0 -1815.0-1815.0

55 1002210022 14081.514081.5 -4059.5-4059.5

Example 2.5:Example 2.5: Using the Using the data from data from Example 2.1Example 2.1, build , build forecasts with smoothing forecasts with smoothing constantconstant

αα = 0.5. = 0.5.

Solution:Solution:

Exponential Smoothing, cont.Exponential Smoothing, cont.



Example 2.6:Example 2.6: Using the data Using the data from from Example 2.1Example 2.1, build forecasts , build forecasts with smoothing constants with smoothing constants

αα = 0.0 and = 0.0 and αα = 1.0. = 1.0. Solution:Solution:

Exponential Smoothing, cont.Exponential Smoothing, cont.

Period(t)

α = 0.0 Error α = 1.0 Error

Visits Forecast (Actual – Forecast) Visits Forecast (Actual – Forecast)

1 15908 -- 15908 --

2 15504 15908 -404.0 15504 15908 -404.0

3 14272 15908.0 -1636.0 14272 15504.0 -1232.0

4 13174 15908.0 -2734.0 13174 14272.0 -1098.0

5 10022 15908.0 -5886.0 10022 13174.0 -3152.0

6 15908.0 10022.0



Techniques for TrendsTechniques for Trends

Least squares regressionLeast squares regression-- minimizes the sum -- minimizes the sum of the squared errorsof the squared errors

Least squares lineLeast squares line::y = a + bxy = a + bx, y = predicted (dependent) variable, y = predicted (dependent) variable

x = predictor (independent) x = predictor (independent) variablevariable

b = slope of data lineb = slope of data linea = value of y when x = 0a = value of y when x = 0

n(xy) - (x)(y)n(x2) - (x)2b = a = y - bx

n



Figure 2.9 Linear RegressionFigure 2.9 Linear Regression

y

x

y = a + bx

a

errorerror

Δy

Δx b =(Δy/Δx) , where b>0




Multi Hospital System Revenues and Profits Data

Hospital Revenue (x) Profit (y) x*y x2

1 7 0.15 1.05 49

2 2 0.10 0.2 4

3 6 0.13 0.78 36

4 4 0.15 0.6 16

5 14 0.25 3.5 196

6 15 0.27 4.05 225

7 16 0.24 3.84 256

8 12 0.20 2.4 144

9 14 0.27 3.78 196

10 20 0.44 8.8 400

11 15 0.34 5.1 225

12 7 0.17 1.19 49

Total 132 2.71 35.29 1796

Example 2.7: A multi-hospital system (MHS) owns 12 hospitals. Revenues (x, or the independent variable) and profits (y, or the dependent variable) for each hospital are given below. Obtain a regression line for the data, and predict profits for a hospital with $10 million in revenues. All figures are in millions of dollars.



Solution:Solution:After calculating

x y xy x, , , 2substitute into the equations [2.5] for a and [2.6] for b, respectively.

Hence, the regression line is: yx = 0.0506 + 0.01593x. To predict the profits for a hospital with $10 million in revenue, simply plug 10 in as the value of x in the regression equation: Profit = 0.0506 + 0.01593(10) = .209903 Multiplying this value by one million, the profit level with $10 million in revenue is found to be $209,903.

.01593.0)132(132)1796(12)71.2(132)29.35(12

)()(

))(()(22

xxn

yxxynb

.0506.012

)132(01593.071.2

nxby

a




Linear Regression as a Trend Line

y = a + b*t y = predicted (dependent) variablet = predictor (time) variableb = slope of data linea = value of y when x = 0

n(ty) - (t)(y)n(t2) - (t)2b = a = y - bt

n



Example 2.8:Example 2.8: Referring back to the OB/GYN example, the health care manager can Referring back to the OB/GYN example, the health care manager can estimate the trend line using regression analysis. estimate the trend line using regression analysis.

Solution:Solution:



Techniques for SeasonalityTechniques for SeasonalitySeasonal variationsSeasonal variations in a data set consistently repeat upward in a data set consistently repeat upward

or downward movements of the data values that can be or downward movements of the data values that can be traced to recurrent events. traced to recurrent events.

In the In the additiveadditive model model, seasonality is expressed as a quantity , seasonality is expressed as a quantity (example: 5 units), which is added or subtracted from the (example: 5 units), which is added or subtracted from the series average in order to incorporate seasonality. series average in order to incorporate seasonality.

In the In the multiplicativemultiplicative model model, seasonality is expressed as a , seasonality is expressed as a percentage of the average amount (example: 1.15) percentage of the average amount (example: 1.15)

Quarterly, Monthly, Daily Indices TechniqueQuarterly, Monthly, Daily Indices Technique



Techniques for SeasonalityTechniques for SeasonalityEmploying Seasonal Indices in ForecastsExample 2.10: A forecast based on linear regression yields the following

trend equationDemand (Yt) = 511.06 + 1.259 t.

The forecast of demand for periods 29 through 31 would be:Y29 = 511.06 + 1.259 (29) = 547.6.Y30 = 511.06 +1.259 (30) = 548.8.Y31 = 511.06 + 1.259 (31) = 550.1.

Having forecast the next three months, the healthcare manager needs to incorporate seasonality back into those forecasts. The periods t = 29, 30 and 31 represent the months of November, December and January, respectively, with corresponding monthly indices 0.984, 0.973, and 1.036. Monthly adjustments to those forecasts are calculated

Monthly Adjusted Forecast (t): Forecast * Monthly IndexPeriod 29 (November): 547.6 (0.984) = 538.8.Period 30 (December): 548.9 (0.973) = 534.0.Period 31 (January) : 550.1 (1.036) = 569.9.



Techniques for SeasonalityTechniques for SeasonalityEmploying Seasonal Indices in ForecastsEmploying Seasonal Indices in Forecasts

The next step in adjustment of the forecasted The next step in adjustment of the forecasted demand would be for daily fluctuations. As was demand would be for daily fluctuations. As was shown in Table 2.4, Heal Me Hospital experiences shown in Table 2.4, Heal Me Hospital experiences daily variation in demand. Thus, the monthly index daily variation in demand. Thus, the monthly index adjusted forecasts should be further adjusted for adjusted forecasts should be further adjusted for daily variations.daily variations.

Daily Adjusted Forecast = Monthly Adjusted Forecast (t) * Daily Adjusted Forecast = Monthly Adjusted Forecast (t) * Daily IndexDaily Index

For example, for November (period 29), the For example, for November (period 29), the adjusted forecasts for Monday and Tuesday are:adjusted forecasts for Monday and Tuesday are:Monday, November: 538.8 * (0.972) = 523.7.Monday, November: 538.8 * (0.972) = 523.7.Tuesday, November: 538.8 * (1.023) = 551.2.Tuesday, November: 538.8 * (1.023) = 551.2.



How accurate are we?How accurate are we?

Errors may be caused by:Errors may be caused by:– An inadequate forecasting modelAn inadequate forecasting model– Irregular variations due to severe weather, Irregular variations due to severe weather,

shortages or breakdowns, catastrophes, etc.shortages or breakdowns, catastrophes, etc.– Forecasting technique may be used Forecasting technique may be used

improperlyimproperly– There may be random variations in the dataThere may be random variations in the data

Forecast Error equals the actual value minus the forecasted value.Error = Actual – Forecast



Is your forecast accurate?Is your forecast accurate?

• Mean Absolute Deviation (MAD)

Seek lowest of MAD or MAPE for given set of data; also examine historical performance versus responsiveness to current situation.

MAD weights allerrors evenly.

MAPE avoids the problem of interpreting the measure of accuracy relative to the magnitudes of the actual and the forecast values.

nForecastActual

MAD

||

ActualForecastActual

MAPE||

• Mean Absolute Percent Error (MAPE)



Periodt

Smoothing constant α = .3 Error Absolute Error

Actual Forecast (Actual – Forecast) |Actual – Forecast|

1 15908 --

2 15504 15908 -404 404

3 14272 15786.8 -1514.8 1515

4 13174 15332.4 -2158.4 2158

5 10022 14684.9 -4662.9 4662.9

6 13286

Sum Σ 52972 8740.1

Is your forecast accurate?Is your forecast accurate?Using data from Example 2.4, SES with α = 0.3, we observe the necessary error calculations in Table below.

Hence,MAD = 8740.1 ÷ 4 = 2185.03, andMAPE = 8740.1 ÷ 52972 = 0.165 or 16.5%.



Is your forecast accurate?Is your forecast accurate? Controlling forecasts-- set predetermined upper/lower Controlling forecasts-- set predetermined upper/lower

limits for forecast errorslimits for forecast errors Forecasts can be monitored using either a Forecasts can be monitored using either a tracking tracking

signal signal or or control chartcontrol chart..– Tracking signals show cumulative errorsTracking signals show cumulative errors- Control Charts-- set upper and lower limits for - Control Charts-- set upper and lower limits for

individual forecast errorsindividual forecast errors

MADForecastActual

signalTracking

)(



Control Chart for Tracking SignalControl Chart for Tracking Signal

Range ofAcceptableVariation

Need for Corrective Action

-6

-4

-2

0

2

4

6

1 3 5 7 9 11 13 15 17 19 21 23 25 27

Trac

king

sig

nal

-During periods 12 through 15 the tracking signal went beyond the acceptable control limits (down to -5.51), but recovered at period 16 and stayed within acceptable limits after that. -Until period 8 the predicted values were below the actual. That changed from period 9 to period 20, when forecasts were higher than actual data. -At the period 21 a return to under-forecast occurred.



So what technique should we use?So what technique should we use?

Factors of importance:Factors of importance:– FrequencyFrequency– Level of aggregationLevel of aggregation– Type of Model- Errors [MAD, MAPE]Type of Model- Errors [MAD, MAPE]– Degree of managerial involvementDegree of managerial involvement– Cost per forecastCost per forecast

Time horizon considerations-- short, Time horizon considerations-- short, intermediate, or longintermediate, or long



What makes a forecast a good one?What makes a forecast a good one?

TimelinessTimeliness AccuracyAccuracy Meaningful Units ($$’s, visits, etc.)Meaningful Units ($$’s, visits, etc.) In writingIn writing Simple to understand and useSimple to understand and use



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Chapter 2. Forecasting

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