P=Pt-1+U(At-1-Pt-1) U=

mes ventas (miles)(a)(D) p D-P

Enero 190 190 0Febrero 200 190 10Marzo 210 191 19Abril 200 192.9 7.1Mayo 220 193.61 26.39Junio 210 196.249 13.751Julio 230 197.6241 32.3759Agosto 220 200.86169 19.13831

202.775521promedio

mes ventas (miles)(a)(D) p D-P

Enero 190 190 0Febrero 200 190 10Marzo 210 192 18Abril 200 195.6 4.4Mayo 220 196.48 23.52Junio 210 201.184 8.816Julio 230 202.9472 27.0528Agosto 220 208.35776 11.64224

210.686208Promedio

mes ventas (miles)(a)(D) p D-P

Enero 190 190 0Febrero 200 190 10Marzo 210 193 17Abril 200 198.1 1.9Mayo 220 198.67 21.33Junio 210 205.069 4.931Julio 230 206.5483 23.4517Agosto 220 213.58381 6.41619

215.508667

promedio

0.1 0.2 0.3

U(D-P) Pt=P+U(D-P) (d-p)^2

0 190 01 191 100

1.9 192.9 3610.71 193.61 50.41

2.639 196.249 696.43211.3751 197.6241 189.090001

3.23759 200.86169 1048.19891.913831 202.775521 366.27491

suma 2811.40591351.4257389

U(D-P) Pt=P+U(D-P) (d-p)^2

0 190 02 192 100

3.6 195.6 3240.88 196.48 19.36

4.704 201.184 553.19041.7632 202.9472 77.721856

5.41056 208.35776 731.8539882.328448 210.686208 135.541752

suma 1941.668242.7084995

U(D-P) Pt=P+U(D-P) (d-p)^2

0 190 03 193 100

5.1 198.1 2890.57 198.67 3.61

6.399 205.069 454.96891.4793 206.5483 24.314761

7.03551 213.58381 549.9822331.924857 215.508667 41.1674941

suma 1463.04339

182.8804235

P=Pt-1+U(At-1-Pt-1) U= 0.1

mes ventas p D-P U(D-P)

enero 500 500 0 0febrero 850 500 350 35marzo 315 535 -220 -22abril 620 513 107 10.7mayo 700 523.7 176.3 17.63junio 514 541.33 -27.33 -2.733julio 816 538.597 277.403 27.7403agosto 500 566.3373 -66.3373 -6.63373septiembre 550 559.70357 -9.70357 -0.970357octubre 900 558.733213 341.266787 34.1266787noviembre 780 592.8598917 187.140108 18.7140108diciembre 800 611.57390253 188.426097 18.8426097

promedio

enero 500 500 0 0febrero 850 500 350 70marzo 315 570 -255 -51abril 620 519 101 20.2mayo 700 539.2 160.8 32.16junio 514 571.36 -57.36 -11.472julio 816 559.888 256.112 51.2224agosto 500 611.1104 -111.1104 -22.22208septiembre 550 588.88832 -38.88832 -7.777664octubre 900 581.110656 318.889344 63.7778688noviembre 780 644.8885248 135.111475 27.022295diciembre 800 671.91081984 128.08918 25.617836

promedio

enero 500 500 0 0febrero 850 500 350 105marzo 315 605 -290 -87abril 620 518 102 30.6mayo 700 548.6 151.4 45.42junio 514 594.02 -80.02 -24.006julio 816 570.014 245.986 73.7958agosto 500 643.8098 -143.8098 -43.14294septiembre 550 600.66686 -50.66686 -15.200058octubre 900 585.466802 314.533198 94.3599594

noviembre 780 679.8267614 100.173239 30.0519716diciembre 800 709.87873298 90.121267 27.0363801

promedio

0.2 0.3

Pt=P+U(D-P) (d-p)^2

500 0535 122500513 48400

523.7 11449541.33 31081.69

538.597 746.9289566.3373 76952.424409

559.70357 4400.6373713558.733213 94.159270745

592.8598917 116463.01991611.5739025 35021.420135630.4165123 35504.394208

suma 482613.674240217.80618

500 0570 122500519 65025

539.2 10201571.36 25856.64

559.888 3290.1696611.1104 65593.356544

588.88832 12345.520988581.110656 1512.3014324

644.8885248 101690.41372671.9108198 18255.110731697.5286559 16406.838074

suma 442676.3510936889.69592

500 0605 122500518 84100

548.6 10404594.02 22921.96

570.014 6403.2004643.8098 60509.112196

600.66686 20681.258576585.466802 2567.1307023

679.8267614 98931.132644

709.878733 10034.677732736.9151131 8121.8427693

suma 447174.3150237264.52625

P=Pt-1+U(At-1-Pt-1) U= 0.1

periodo ventas ( en miles) P (D-P) U(D-P)

2006 120 120 0 02007 121 120 1 0.12008 117 120.1 -3.1 -0.312009 118 119.79 -1.79 -0.1792010 124 119.611 4.389 0.43892011 125 120.0499 4.9501 0.495012012 120 120.54491 -0.54491 -0.0544912013 118 120.490419 -2.490419 -0.24904192014 113 120.2413771 -7.2413771 -0.72413771

PROMEDIO 12.9463963

periodo ventas ( en miles) P (D-P) U(D-P)

2006 120 120 0 02007 121 120 1 0.22008 117 120.2 -3.2 -0.642009 118 119.56 -1.56 -0.3122010 124 119.248 4.752 0.95042011 125 120.1984 4.8016 0.960322012 120 121.15872 -1.15872 -0.2317442013 118 120.926976 -2.926976 -0.58539522014 113 120.3415808 -7.3415808 -1.46831616

PROMEDIO 13.6798995

periodo ventas ( en miles) P (D-P) U(D-P)

2006 120 120 0 02007 121 120 1 0.32008 117 120.3 -3.3 -0.992009 118 119.31 -1.31 -0.3932010 124 118.917 5.083 1.52492011 125 120.4419 4.5581 1.367432012 120 121.80933 -1.80933 -0.5427992013 118 121.266531 -3.266531 -0.97995932014 113 120.2865717 -7.2865717 -2.18597151

PROMEDIO 14.1396991

0.2 0.3 EL ERROR MINIMO PROMEDIO ES:u=0.1

Pt=P+U(D-P) (d-p)^2 APROXIMACION EXPONENCIAL DOBLE

120 0 Periodo Pt P' Pt-P'120.1 1

119.79 9.61 enero 120 120 0119.611 3.2041 febrero 120.1 120 -0.1

120.0499 19.263321 marzo 119.79 120.01 0.22120.54491 24.50349 abril 119.611 119.988 0.377

120.490419 0.29692691 mayo 120.0499 119.9503 -0.0996120.241377 6.2021868 junio 120.54491 119.96026 -0.58465119.517239 52.4375423 julio 120.490419 120.018725 -0.471694

SUMA 116.517567 agosto 120.241377 120.065894 -0.1754827septiembre 119.517239 120.083443 0.56620328

A=2AES-AED 118.894416Pt=P+U(D-P) (d-p)^2

B=(U/1-U)(AES-AED) -0.06920262120 0

120.2 1 AÑOS119.56 10.24 1 2015 118.825213

119.248 2.4336 2 2016 118.756011120.1984 22.581504 3 2017 118.686808

121.15872 23.0553626 4 2018 118.617605120.926976 1.34263204 5 2019 118.548403120.341581 8.5671885 6 2020 118.4792118.873265 53.8988086

SUMA 123.119096

Pt=P+U(D-P) (d-p)^2

120 0120.3 1

119.31 10.89118.917 1.7161

120.4419 25.836889121.80933 20.7762756

121.266531 3.27367505120.286572 10.6702248

118.1006 53.0941271SUMA 127.257292

EL ERROR MINIMO PROMEDIO ES:

U(Pt-P') P't=P'+U(Pt-P')

0 120-0.01 119.990.022 120.032

0.0377 120.0257-0.00996 119.94034

-0.058465 119.901795-0.0471694 119.9715556

-0.01754827 120.048346130.05662033 120.140062998

Y=A+BX

P=Pt-1+U(At-1U= 0.1 0.2

PERIODO VENTAS (MILES) P (D-P) U(D-P) Pt=P+U(D-P)

2008 85 85 0 0 852009 89 85 4 0.4 85.42010 92 85.4 6.6 0.66 86.062011 95 86.06 8.94 0.894 86.9542012 93 86.954 6.046 0.6046 87.55862013 98 87.5586 10.4414 1.04414 88.60274

SUMAPROMEDIO 47.5100917

PERIODO VENTAS (MILES) P (D-P) U(D-P) Pt=P+U(D-P)

2008 85 85 0 0 852009 89 85 4 0.8 85.42010 92 85.8 6.2 1.24 86.422011 95 87.04 7.96 1.592 87.8362012 93 88.632 4.368 0.8736 89.06882013 98 89.5056 8.4944 1.69888 90.35504

SUMAPROMEDIO 34.8393092

PERIODO VENTAS (MILES) P (D-P) U(D-P) Pt=P+U(D-P)

2008 85 85 0 0 852009 89 85 4 1.2 85.42010 92 86.2 5.8 1.74 86.782011 95 87.94 7.06 2.118 88.6462012 93 90.058 2.942 0.8826 90.35222013 98 90.9406 7.0594 2.11782 91.64654

SUMAPROMEDIO 26.3290154

0.3 EL ERROR MINIMO PROMEDIO ES:u=0.1

(d-p)^2 APROXIMACION EXPONENCIAL DOBLE

0 Periodo Pt P' Pt-P' U(Pt-P')16

43.56 2008 85 85 0 079.9236 2009 85.4 85 -0.4 -0.12

36.554116 2010 86.06 85.12 -0.94 -0.282109.022834 2011 86.954 85.402 -1.552 -0.4656

285.06055 2012 87.5586 85.8676 -1.691 -0.50732013 88.60274 86.3749 -2.22784 -0.668352

A=2AES-AED 91.498932(d-p)^2

B=(U/1-U)(AES-AED) 1.241225140

16 AÑOS38.44 5 2018 97.7050577

63.3616 6 2019 98.946282919.079424 7 2020 100.187508

72.1548314209.035855

(d-p)^2

016

33.6449.8436

8.65536449.8351284157.974092

P't=P'+U(Pt-P')

8584.88

84.83884.936485.3603

85.706548

Y=A+BX