HMP 654 Operations Research and Control Systems in Health Care Spring-Summer 2013

HMP654 -1-

HMP 654

Operations Research and Control Systems

in Health Care

Fall 2015

HMP654 -2-

Forecasting - Introduction

• Forecasting in Health Care• Forecasting Models

– Structural Models– Time Series Models– Expert Judgment

• Time Series Models:– Demand has exhibited some

measurable structure in the past.– The structure will continue into

the future.

HMP654 -3-

Forecasting - Time Series

• Signal vs. Noise• Extrapolation Models

• Accuracy of Forecasts

MADY Y

n

MSEY Y

n

i i

i

i

i

2

, , ,.....Y f Y Y Yt t t t 1 1 2

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Forecasting - Stationary Models

• Stationary Time-Series

• Moving Averages

.....Y Y Y Ykt

t t t k

11 1

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Forecasting - Moving Avgs.

Time Period Demand1 332 383 314 355 306 367 348 399 39

10 3611 4012 3813 3714 3915 3216 3817 3718 3919 3720 3521 3722 3423 3524 36

28

30

32

34

36

38

40

42

1 2 3 4 5 6 7 8 9 1011 12 13 14 15 1617 18 19 20 2122 23 24 25

Time Period

Un

its

So

ld

Demand

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2-Month 4-MonthTime Period Demand Moving Avg. Moving Avg.

1 33 -- --2 38 -- --3 31 35.50 --4 35 34.50 --5 30 33.00 34.256 36 32.50 33.507 34 33.00 33.008 39 35.00 33.759 39 36.50 34.75

10 36 39.00 37.0011 40 37.50 37.0012 38 38.00 38.5013 37 39.00 38.2514 39 37.50 37.7515 32 38.00 38.5016 38 35.50 36.5017 37 35.00 36.5018 39 37.50 36.5019 37 38.00 36.5020 35 38.00 37.7521 37 36.00 37.0022 34 36.00 37.0023 35 35.50 35.7524 36 34.50 35.25

MSE: 6.93 7.66

050

0 50

33 + 38 2

38 + 31 2

33 + 38 + 31 + 35 4

30 34 25 36 33 50

20

2 2. . . . . . . .

SUMXMY2(B7:B26,D7:D26)/COUNT(D7:D26)

HMP654 -7-


28

30

32

34

36

38

40

42

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Time Period

Unit

s So

ld

Demand

2-Month Moving Avg.

4-Month Moving Avg.

HMP654 -8-

Forecasting - Weighted M.A.

• Weighted Moving Averages

......Y w Y w Y w Yw

w

t t t k t k

i

ii

k

1 1 2 1 1

1

0 1

1

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2-Month WeightedTime Period Demand Moving Average ----Weights----

1 33 -- w2 0.3002 38 -- w1 0.7003 31 34.50 sum 1.0004 35 35.905 30 32.206 36 33.507 34 31.808 39 35.409 39 35.50

10 36 39.0011 40 38.1012 38 37.2013 37 39.4014 39 37.7015 32 37.6016 38 36.9017 37 33.8018 39 37.7019 37 37.6020 35 38.4021 37 36.4022 34 35.6023 35 36.1024 36 34.30

MSE: 6.29

050

0 500.3 x 33 + 0.7 x 38

0.3 x 38 + 0.7 x 31

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Finding the Optimal WeightsForecasting - Weighted M.A

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Finding the Optimal Weights

MSE vs W2

W2

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Forecasting - Weighted M.A.Finding the Optimal Weights

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Finding the Optimal Weights


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Forecasting - Exp. Smoothing

• Exponential Smoothing

Y Y Y Y

Y Y Y

t t t t

t t t

1

1

0 1

1

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Exp. SmoothingTime Period Demand Prediction

1 33 33.00 alpha 0.7002 38 33.003 31 36.504 35 32.655 30 34.306 36 31.297 34 34.598 39 34.189 39 37.55

10 36 38.5711 40 36.7712 38 39.0313 37 38.3114 39 37.3915 32 38.5216 38 33.9617 37 36.7918 39 36.9419 37 38.3820 35 37.4121 37 35.7222 34 36.6223 35 34.7924 36 34.94

MSE: 9.59

0.7 x 33 + 0.3 x 33

0.7 x 38 + 0.3 x 33

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Exp. SmoothingTime Period Demand Prediction

1 33 33.00 alpha 0.7002 38 33.003 31 36.504 35 32.655 30 34.306 36 31.297 34 34.598 39 34.189 39 37.55

10 36 38.5711 40 36.7712 38 39.0313 37 38.3114 39 37.3915 32 38.5216 38 33.9617 37 36.7918 39 36.9419 37 38.3820 35 37.4121 37 35.7222 34 36.6223 35 34.7924 36 34.94

MSE: 9.59

050

0 502830323436384042

12345678910111213141516171819202122232425

Time

Uni

t28

30

32

34

36

38

40

42

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

Time Period

Units

Sol

d

Demand

Exp. Smoothing Prediction

HMP654 -17-

Forecasting - Trend Models

Ti me Ac t ua lYe a r Qt r Pe r i o d De ma n d1990 1 1 $684 . 2

2 2 $584 . 13 3 $765 . 44 4 $892 . 3

1991 1 5 $885 . 42 6 $677 . 03 7 $1 , 00 6 . 64 8 $1 , 12 2 . 1

1992 1 9 $1 , 16 3 . 42 10 $993 . 23 11 $1 , 31 2 . 54 12 $1 , 54 5 . 3

1993 1 13 $1 , 59 6 . 22 14 $1 , 26 0 . 43 15 $1 , 73 5 . 24 16 $2 , 02 9 . 7

1994 1 17 $2 , 10 7 . 82 18 $1 , 65 0 . 33 19 $2 , 30 4 . 44 20 $2 , 63 9 . 4

$0.0

$500.0

$1,000.0

$1,500.0

$2,000.0

$2,500.0

$3,000.0

1 3 5 7 9 11 13 15 17 19 21

Time Period

Sale

s (i

n $1

,000

s)

Actual Demand

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Forecasting - Holt’s Method

(11.5)

(11.6)

(11.7)

Y E kT

E Y E T

T E E T

t k t t

t t t t

t t t t

1

1

1 1

1 1

• Compute the base level Et for time period t using equation 11.6

• Compute expected trend value Tt for time period t using equation 11.7

• Compute the final forecast Y^t+k for time period t+k

using equation 11.5

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Forecasting - Holt’s Method

Ti me Ac t u a l Ba s e Pr e di c t e dYe a r Qt r Pe r i o d De ma n d Le v e l Tr e nd De ma n d1990 1 1 $684 . 2 684. 2 0 . 0 - - a l p ha 0 . 500

2 2 $584 . 1 634. 2 - 25 . 0 $684 . 2 b e t a 0 . 5003 3 $765 . 4 687. 3 14 . 0 $609 . 14 4 $892 . 3 796. 8 61 . 8 $701 . 3

1991 1 5 $885 . 4 872. 0 68 . 5 $858 . 62 6 $677 . 0 808. 7 2 . 6 $940 . 53 7 $1 , 00 6 . 6 909. 0 51 . 4 $811 . 44 8 $1 , 12 2 . 1 1041 . 2 91 . 8 $960 . 4

1992 1 9 $1 , 16 3 . 4 11 48 . 2 99 . 4 $1 , 133 . 12 10 $993 . 2 11 20 . 4 35 . 8 $1 , 247 . 73 11 $1 , 31 2 . 5 1234 . 3 74 . 9 $1 , 156 . 24 12 $1 , 54 5 . 3 1427 . 3 133 . 9 $1 , 309 . 2

1993 1 13 $1 , 59 6 . 2 1578 . 7 142 . 6 $1 , 561 . 12 14 $1 , 26 0 . 4 1490 . 9 27 . 4 $1 , 721 . 33 15 $1 , 73 5 . 2 1626 . 7 81 . 6 $1 , 518 . 34 16 $2 , 02 9 . 7 1869 . 0 162 . 0 $1 , 708 . 4

1994 1 17 $2 , 10 7 . 8 2069 . 4 181 . 2 $2 , 031 . 02 18 $1 , 65 0 . 3 1950 . 4 31 . 1 $2 , 250 . 53 19 $2 , 30 4 . 4 2143 . 0 111 . 8 $1 , 981 . 54 20 $2 , 63 9 . 4 2447 . 1 208 . 0 $2 , 254 . 8

MS E 70179 . 3

Set initial trend to 0Initial base level = first demand value

Forecast for Qtr. 3, 1990:634.2= 0.5 x 584.1 + (1 - 0.5) x (684.2 + 0) -25 = 0.5 x (634.2 - 684.2) + (1 - 0.5) x 0609.1 = 634.2 + 1 x (- 25)

HMP654 -20-

Forecasting - Regression

• Linear Trend Model

Ti me Ac t u a l Li n e a rYe a r Qt r Pe r i o d De ma n d Tr e n d1990 1 1 $684 . 2 $467 . 8

2 2 $584 . 1 $560 . 43 3 $765 . 4 $653 . 04 4 $892 . 3 $745 . 7

1991 1 5 $885 . 4 $838 . 32 6 $677 . 0 $930 . 93 7 $1, 00 6 . 6 $1 , 02 3. 54 8 $1, 12 2 . 1 $1 , 11 6. 2

1992 1 9 $1, 16 3 . 4 $1 , 20 8. 82 10 $993 . 2 $1 , 30 1. 43 11 $1, 31 2 . 5 $1 , 39 4. 14 12 $1, 54 5 . 3 $1 , 48 6. 7

1993 1 13 $1, 59 6 . 2 $1 , 57 9. 32 14 $1, 26 0 . 4 $1 , 67 1. 93 15 $1, 73 5 . 2 $1 , 76 4. 64 16 $2, 02 9 . 7 $1 , 85 7. 2

1994 1 17 $2, 10 7 . 8 $1 , 94 9. 82 18 $1, 65 0 . 3 $2 , 04 2. 43 19 $2, 30 4 . 4 $2 , 13 5. 14 20 $2, 63 9 . 4 $2 , 22 7. 7

$0.0

$500.0

$1,000.0

$1,500.0

$2,000.0

$2,500.0

$3,000.0

1 3 5 7 9 11 13 15 17 19 21

Time Period

Sale

s (i

n $1

,000

s)

Actual Demand

Linear Trend

Y tt t 0 1

0

0

37517 92 6255

. , .

1

1Y tt

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Blank Slide

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Blank Slide

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• Linear Trend ModelSUMMARY OUTPUT

Re g r e s s i on S t a t i s t i c sMul t i p l e R0. 9 3414R Sq ua r e 0 . 8 7262Adj u s t e d R Squa r e0 . 8 6554St a n da r d Er r or215 . 102Obs e r va t i on s 20

ANOVAd f SS MS F Si g n i f i c anc e F

Re gr e s s i on 1 5705 356 570 5 356 123 . 309 1. 7E- 09Re s i dua l 18 832 837 462 6 8. 7To t a l 19 6538 193

Coe f f i c i e n t sS t a ndar d Er r ort S t a t P- v a l ue Lowe r 95%Upp e r 95%Lowe r 95 . 000 %Upp e r 95 . 000 %I n t e r c e p t 375 . 17 99 . 9 215 3. 7 5 465 0. 00 145 165 . 243 585 . 097 165 . 2 43 585 . 097X Va r i a b l e 192. 6255 8. 34 128 11 . 1 045 1. 7E- 09 75. 1 011 11 0 . 1 5 75. 10 11 11 0 . 15

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• Quadratic Trend Model Y t tt t 0 1 2

2

Ti me Ac t u a l Qu a d r a t i cYe a r Qt r Pe r i o d Ti me ^2 De ma nd Tr e n d1990 1 1 1 $684. 2 $674. 0

2 2 4 $584. 1 $701. 53 3 9 $765. 4 $736. 24 4 16 $892. 3 $778. 2

1991 1 5 25 $885. 4 $827. 52 6 36 $677. 0 $883. 93 7 49 $1, 006 . 6 $947. 64 8 64 $1, 122 . 1 $1 , 018 . 5

1992 1 9 81 $1, 163 . 4 $1 , 096 . 72 10 100 $993. 2 $1 , 182 . 13 11 121 $1, 312 . 5 $1 , 274 . 74 12 144 $1, 545 . 3 $1 , 374 . 6

1993 1 13 169 $1, 596 . 2 $1 , 481 . 72 14 196 $1, 260 . 4 $1 , 596 . 03 15 225 $1, 735 . 2 $1 , 717 . 64 16 256 $2, 029 . 7 $1 , 846 . 4

1994 1 17 289 $2, 107 . 8 $1 , 982 . 42 18 324 $1, 650 . 3 $2 , 125 . 73 19 361 $2, 304 . 4 $2 , 276 . 24 20 400 $2, 639 . 4 $2 , 433 . 9

$0.0

$500.0

$1,000.0

$1,500.0

$2,000.0

$2,500.0

$3,000.0

1 3 5 7 9 11 13 15 17 19 21

Time Period

Sale

s (i

n $1

,000

s)

Actual Demand

Quadratic Trend

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• Quadratic Trend Model

SUMMARY OUTPUT

Re gr e s s i on S t a t i s t i c sMul t i p l e R0. 95 27 6R Sq ua r e 0 . 90 77 5Adj u s t e d R Squa r e0 . 89 68 9St a n da r d Er r or188. 36 3Obs e r va t i on s 20

ANOVAd f SS MS F Si gn i f i c anc e F

Re gr e s s i on 2 5935 02 0 2967 510 83 . 6 372 1. 6E- 09Re s i dua l 17 6031 73 3548 0. 7Tot a l 19 6538 19 3

Coe f f i c i e n t sS t an dar d Er r ort S t a t P- v a l ue Lowe r 95 %Uppe r 95 %I n t e r c e pt 653 . 67 140. 13 9 4. 66 444 0. 00 022 358. 002 949. 337X Va r i a bl e 116 . 6 71 30 . 7 34 6 0. 54 242 0. 59 457 - 48. 173 81 . 5 155X Va r i a bl e 23 . 61 68 8 1. 42 16 2 2. 54 42 0. 02 096 0. 61 752 6. 61 624

HMP654 -28-

Forecasting - Seasonality

• Adjusting trend predictions with seasonal indices

Ti me Ac t u a l Qu a d r a t i c Ac t u a l a s a Se a s o n a lYe a r Qt r Pe r i o d Ti me ^2 De ma n d Tr e n d % o f Tr e n d Fo r e c a s t1990 1 1 1 $684 . 2 $674 . 0 102% $712 . 6

2 2 4 $584 . 1 $701 . 5 83% $561 . 93 3 9 $765 . 4 $736 . 2 104% $758 . 94 4 16 $892 . 3 $778 . 2 11 5% $864 . 8

1991 1 5 25 $885 . 4 $827 . 5 107% $874 . 92 6 36 $677 . 0 $883 . 9 77% $708 . 03 7 49 $ 1 , 0 06 . 6 $947 . 6 106% $976 . 84 8 64 $ 1 , 1 22 . 1 $1 , 018 . 5 11 0% $1, 1 31 . 8

1992 1 9 81 $ 1 , 1 63 . 4 $1 , 096 . 7 106% $1, 1 59 . 62 10 100 $993 . 2 $1 , 182 . 1 84% $946 . 83 11 121 $ 1 , 3 12 . 5 $1 , 274 . 7 103% $1, 3 14 . 04 12 144 $ 1 , 5 45 . 3 $1 , 374 . 6 11 2% $1, 5 27 . 5

1993 1 13 169 $ 1 , 5 96 . 2 $1 , 481 . 7 108% $1, 5 66 . 62 14 196 $ 1 , 2 60 . 4 $1 , 596 . 0 79% $1, 2 78 . 43 15 225 $ 1 , 7 35 . 2 $1 , 717 . 6 101% $1, 7 70 . 54 16 256 $ 2 , 0 29 . 7 $1 , 846 . 4 11 0% $2, 0 51 . 7

1994 1 17 289 $ 2 , 1 07 . 8 $1 , 982 . 4 106% $2, 0 96 . 02 18 324 $ 1 , 6 50 . 3 $2 , 125 . 7 78% $1, 7 02 . 63 19 361 $ 2 , 3 04 . 4 $2 , 276 . 2 101% $2, 3 46 . 34 20 400 $ 2 , 6 39 . 4 $2 , 433 . 9 108% $2, 7 04 . 6

Se a s o n a lQt r I n de x

1 105. 7%2 80. 1%3 103. 1%4 111 . 1%

102 + 107 + 106 + 108 + 106 5

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$0.0

$500.0

$1,000.0

$1,500.0

$2,000.0

$2,500.0

$3,000.0

1 3 5 7 9 11 13 15 17 19 21Time Period

Dem

and

(in

$1,

000s

)

HMP654 -30-


• Use of Seasonal Indices1 Create a trend model and calculate the

estimated value for each observation in the sample.

2 For each observation, calculate the ratio of the actual value to the predicted trend value

3 For each season, compute the average of the ratios calculated in step 2. These are the seasonal indices.

4 Multiply any forecast produced by the trend model by the appropriate seasonal index calculated in step 3.

HMP654 -31-

Forecasting - Seasonal Regression Models

Y t t Q Q Qt t 0 1 22

3 1 4 2 5 3

Ti me I n d i c a t o r f o r Qt r : Ac t u a l S e a s o n a lYe a r Qt r Pe r i o d Ti me ^2 1 2 3 De ma n d Mo d e l1990 1 1 1 1 0 0 $684. 2 $758. 5

2 2 4 0 1 0 $584. 1 $448. 43 3 9 0 0 1 $765. 4 $784. 34 4 16 0 0 0 $892. 3 $949. 5

1991 1 5 25 1 0 0 $885. 4 $911 . 42 6 36 0 1 0 $677. 0 $629. 23 7 49 0 0 1 $1 , 00 6. 6 $993. 04 8 64 0 0 0 $1 , 12 2. 1 $1 , 18 6. 1

1992 1 9 81 1 0 0 $1 , 16 3. 4 $1 , 17 5. 92 10 100 0 1 0 $993. 2 $921. 63 11 121 0 0 1 $1 , 31 2. 5 $1 , 31 3. 34 12 144 0 0 0 $1 , 54 5. 3 $1 , 53 4. 3

1993 1 13 169 1 0 0 $1 , 59 6. 2 $1 , 55 1. 92 14 196 0 1 0 $1 , 26 0. 4 $1 , 32 5. 53 15 225 0 0 1 $1 , 73 5. 2 $1 , 74 5. 14 16 256 0 0 0 $2 , 02 9. 7 $1 , 99 4. 0

1994 1 17 289 1 0 0 $2 , 10 7. 8 $2 , 03 9. 52 18 324 0 1 0 $1 , 65 0. 3 $1 , 84 1. 03 19 361 0 0 1 $2 , 30 4. 4 $2 , 28 8. 54 20 400 0 0 0 $2 , 63 9. 4 $2 , 56 5. 3

HMP654 -32-

Forecasting - Seasonal Regression Models

SUMMARY OUTPUT

Re g r e s s i on S t a t i s t i c sMul t i p l e R0. 9 9274R Sq ua r e 0 . 9 8553Adj u s t e d R Squa r e0 . 9 8037St a n da r d Er r or82. 1926Obs e r va t i on s 20

ANOVAd f SS MS F Si g n i f i c anc e F

Re gr e s s i on 5 6443 614 128 8 723 190 . 763 2. 3E- 12Re s i dua l 14 9457 8. 8 675 5 . 63Tot a l 19 6538 193

Coe f f i c i e n t sS t a ndar d Er r ort S t a t P- v a l ue Lowe r 95%Upp e r 95%Lowe r 95 . 000 %Upp e r 95 . 000 %I n t e r c e p t 824 . 473 71 . 3 884 11 . 5 491 1. 5E- 08 671 . 3 6 977 . 586 671 . 36 977 . 586X Va r i a b l e 117. 3189 13 . 4 331 1. 2 8 927 0. 2 1 82 - 11 . 492 46 . 1 3 - 11 . 4 92 46 . 1 3X Va r i a b l e 23 . 4 8548 0. 62 068 5. 6 1 558 6. 4E- 05 2. 15 425 4. 816 7 2. 1 54 25 4. 81 67X Va r i a b l e 3- 86 . 805 52 . 8 891 - 1 . 6 413 0. 12 301 - 200 . 24 26. 6 309 - 20 0 . 24 26 . 6 309X Va r i a b l e 4- 42 4 . 74 52 . 4 024 - 8 . 1 053 1. 2E- 06 - 537 . 13 - 31 2 . 34 - 53 7 . 13 - 312 . 34X Va r i a b l e 5- 12 3 . 45 52 . 0 994 - 2 . 3 696 0. 03 272 - 235 . 2 - 11 . 711 - 235 . 2 - 11 . 711

HMP 654 Operations Research and Control Systems in Health Care Spring-Summer 2013

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Transcript of HMP 654 Operations Research and Control Systems in Health Care Spring-Summer 2013