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Sensitivity analysis of the response characteristics of pattern search techniques applied to exponentially… Bitz, Brent William John 1972

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CI  SENSITIVITY ANALYSIS OF THE RESPONSE CHARACTERISTICS OF PATTERN SEARCH TECHNIQUES APPLIED TO EXPONENTIALLY SMOOTHED FORECASTING MODELS  by BRENT WILLIAM JOHN BITZ B. Comm., University of B r i t i s h Columbia, 1970  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF BUSINESS ADMINISTRATION i n the Faculty of Commerce and Business Administration  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA June, 19?2  In p r e s e n t i n g t h i s  thesis  an advanced degree at  further  fulfilment  the U n i v e r s i t y of  the L i b r a r y s h a l l make i t I  in p a r t i a l  freely  of  the  requirements  B r i t i s h Columbia, I agree  available  for  agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f  of this  representatives. thesis for  It  financial  this  thesis  of  g a i n s h a l l not  Commerce and Business  The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada  Date  June 8, 1972  or  i s understood that copying o r p u b l i c a t i o n be allowed without my  written permission.  Department  that  r e f e r e n c e and s t u d y .  f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department by h i s  for  Administration  i  ABSTRACT  The  purpose o f t h i s study was to undertake a s e n s i t i v i t y  a n a l y s i s o f s e l e c t e d input parameters o f the p a t t e r n s e a r c h e x p o n e n t i a l smoothing f o r e c a s t i n g system.  The i n p u t s s u b j e c t e d  to the a n a l y s i s werei 1) maximum number o f p a t t e r n moves, 2) minimum step s i z e , 3) p a t t e r n search step  size,  4) step s i z e r e d u c t i o n f a c t o r , 5) e x p o n e n t i a l smoothing constants  (A, B and C ) .  As the v a l u e s o f these i n p u t parameters were changed d u r i n g the course  o f the a n a l y s i s the r e s u l t a n t changes i n c e r t a i n  v a r i a b l e s o f the system were noted.  criterion  These v a r i a b l e s were:  1) f o r e c a s t e r r o r standard d e v i a t i o n , 2) number o f i t e r a t i o n s  ( o r p a t t e r n moves),  3) e x p o n e n t i a l smoothing constants The  three separate  (A, B and C ) .  time s e r i e s that were used i n t h i s study were  f u r n i s h e d by the F r a z e r V a l l e y M i l k Producer's  Association.  data s e r i e s are composed o f u n i t s a l e s o f f l u i d m i l k  The  segregated  a c c o r d i n g to c o n t a i n e r s i z e , b u t t e r f a t content and channel o f distribution.  Each of the time s e r i e s analysed  d i f f e r e n t type o f trend f a c t o r . and s t a b l e trend f a c t o r s .  represents a  One each f o r r i s i n g ,  falling  The three time s e r i e s were subjected  to i d e n t i c a l a n a l y t i c a l procedures.  The r e s u l t s were then  ii compared a c r o s s the three time s e r i e s i n order t o determine i f the response  p a t t e r n s o f the p a t t e r n s e a r c h system were s e n s i t i v e  to changes i n the s e r i e s t r e n d . As measured hy the response  p a t t e r n s o f the c r i t e r i o n  v a r i a b l e s , the accuracy o f the system i s not i n f l u e n c e d s i g n i f i c a n t l y hy changes i n the input parameters. the s e n s i t i v i t y a n a l y s i s there developed  Throughout  a consistent pattern  of minimal change i n the f o r e c a s t e r r o r standard d e v i a t i o n and the e x p o n e n t i a l smoothing c o n s t a n t s .  The search process was  able to c o n s i s t e n t l y reach v e r y s i m i l a r f o r e c a s t e r r o r standard d e v i a t i o n values and e x p o n e n t i a l smoothing constant v a l u e s , g i v e n the range o f input v a l u e s t e s t e d . The  o n l y dependent v a r i a b l e t h a t experienced any marked  change was the number o f i t e r a t i o n s .  There does appear t o be  c e r t a i n input v a l u e s that minimize the number o f i t e r a t i o n s that the p a t t e r n s e a r c h system needs, to a r r i v e a t s o l u t i o n v a l u e s . N e i t h e r the maximum number o f p a t t e r n moves nor the minimum step s i z e exerted much of an e f f e c t on the s i z e o f the f o r e c a s t e r r o r standard d e v i a t i o n or the "optimum v a l u e s " f o r the e x p o n e n t i a l smoothing c o n s t a n t s .  However, changes i n the  minimum step s i z e do a f f e c t the number o f i t e r a t i o n s the p a t t e r n search system makes before r e a c h i n g a minimum f o r e c a s t e r r o r standard d e v i a t i o n .  I f the minimum s t e p s i z e i s decreased the  number o f i t e r a t i o n s i s i n c r e a s e d . and  The opposite i s a l s o t r u e ,  i f the minimum s t e p s i z e i s i n c r e a s e d , the number o f  i t e r a t i o n s i s decreased.  Changes i n the maximum number o f  iii pattern moves have no effect on the number of i t e r a t i o n s . The pattern search system also appears to be unresponsive to changes i n the pattern search step size.  Neither the  forecast error standard deviation nor the expotential smoothing, constant values can be improved through the use of different pattern search step sizes. what more responsive.  The number of iterations i s some-  Both large and small pattern search step  size y i e l d l a r g e r numbers of i t e r a t i o n s than do middle values i.e.  .10  -  .20.  Like the other inputs, the step size reduction factor, also does not e l i c i t  change i n the results of the search process.  Movements i n the forecast error standard deviation and the exponential smoothing constants are small enough to be considered insignificant. •500  Step size reduction factor values from . 1 0 0 to  minimize the number of i t e r a t i o n s , although within this  interval there i s l i t t l e  change.  Larger values of the step size  reduction factor tend to increase the number of i t e r a t i o n s . There i s l i t t l e responsiveness i n the pattern search system to changes i n the i n i t i a l smoothing constants.  values for the exponential  Between the three time series used, there  is l i t t l e consistency with regards to the effects of changes in the i n i t i a l  constant values on the number of i t e r a t i o n s .  r i s i n g series benefits most from small values i . e . . 2 5 0 .  The The  f a l l i n g series benefited most with a middle value i . e . . 5 0 0 .  The  stable series reacted opposite to the f a l l i n g one and benefited most with values at the extremes i . e . . 2 5 0 and . 7 5 0 . One  important finding i s that most of the responsiveness  of the p a t t e r n s e a r c h system takes p l a c e before the f i r s t size reduction.  step  The bulk o f a l l improvement i n the f o r e c a s t  e r r o r standard d e v i a t i o n and the m a j o r i t y o f a l l change i n the e x p o n e n t i a l smoothing constants occurs p a t t e r n moves. insensitivity  T h i s i s an important  i n this f i r s t set of  r e s u l t as i t e x p l a i n s the  o f the search system to changes i n the maximum  number o f p a t t e r n moves, the minimum step s i z e and the step size reduction factor.  V  TABLE OF CONTENTS Page 1  FORWARD CHAPTER I. General Discussion of Forecasting Introduction Forecasting Systems Descriptive Models Time Series Analysis Factor L i s t i n g Causal Techniques Leading Series Econometrics Summary CHAPTER I I . Exponential Smoothing Introduction Characteristics of Exponential Exponential Smoothing Method Measures of Forecast Accuracy Mean Forecast Error Mean Absolute Error Error Variance Seasonal Adjustments Trend Adjustment Weighting Constants  •  Smoothing .  3 3 6 6 7 10 10 11 13 14 15 15 15 18 22 22 23 23 24 26 28  CHAPTER III.Pattern Search Methodology Introduction Pattern Search Process Exploratory and Pattern Moves Step Size Reduction Direction Change General Discussion of Pattern Search  31 31 32 36 42 43 45  CHAPTER IV. Data» Source and Selection Data Source Dairy Industry Data Selection  48 48 49 55  CHAPTER V.  58 58 60 6l 63  Analytical Methodology Selection of I n i t i a l Values Error Measurement Standard Run Values Methodology  vi  Page CHAPTER VI. Data Analysis And Conclusions 68 Maximum Number of Pattern Moves and Minimum Step Size 68 Pattern Search Step Size 73 Step Size Reduction Factor 77 Exponential Constants - A, B and C .... 79 Conclusions 83 CHAPTER VII. Recommendations  88  BIBLIOGRAPHY  90  APPENDIX  vii  LIST OF TABLES  Page TABLE I  Shipping Volume D i s t r i b u t i o n , 1 9 6 6  51  TABLE II  Per Capita and Total Consumption of Dairy Product 1 9 6 5 * 1 9 6 ? and Projections for 1975 and 1 9 8 0 , Assuming Constant Real Prices  5^  TABLE III  Cow Numbers, Yield Levels and Milk Sales f o r 1 9 6 5 and Projections f o r 1 9 7 5 and 1 9 8 0  TABLE IV  Input Values on an Experimental Run Basis  TABLE V  L i s t of Experimental Run Values f o r  55  66  a l l Time Series  69  TABLE VI  Results from Changes i n DELO  7^  TABLE VII  Results from Changes i n RHO  78  TABLE VIII  Results from Changes i n Exponential Constants (A, B and C)  81  viii  LIST OF FIGURES Page Figure 1  Different Values of the Smoothing Constant Give Different Weights to Past Data Items  21  Figure 2  Path of a Pattern Search Application  35  Figure 3  Exploratory and Pattern Moves  37  Figure 4  Pattern Search Flow Chart  40  Figure 5  Exploratory Move Flow Chart  4l  Figure 6A  Pattern Search: Change of Direction  44  Figure 6B  Pattern Searchs Change of Direction  44  Figure 7  Pattern Search Step Size and Associated Error Value  72  Pattern Search Step Size and Associated Number of Iterations  76  Step Size Reduction Factor and Associated Number of Iterations  80  Constant Value and Associated Number of Iterations  84  Figure 8 Figure 9 Figure 10  ix  ACKNOWLEDGEMENT  The author o f t h i s paper wishes to thank Dr. Doyle Weiss f o r h i s i n v a l u a b l e a s s i s t a n c e .  Dr. Weiss'  criticisms  and comments were extremely u s e f u l i n p r e p a r i n g t h i s paper. I would a l s o l i k e  to extend my thanks to the F r a z e r V a l l e y  M i l k Producer's A s s o c i a t i o n f o r t h e i r c o - o p e r a t i o n i n s u p p l y i n g the data necessary to the study.  1 FORWARD The focus of this study i s e s s e n t i a l l y s e n s i t i v i t y analysis with regards to some of the inputs of the pattern search technique.  This analysis i s performed within the context  of an exponential smoothing application.  What I did was to  change, i n a systematic way, the i n i t i a l values of the smoothing constants, and also the values of the maximum number of pattern moves (MAX), the pattern search step size (DELO), the step size reduction factor (RHO) and the minimum step size (D). While doing this I noted the e f f e c t of these changes on the dependent variables, the standard deviation of the forecast error, the number of i t e r a t i o n s or pattern moves that the search process takes i n a r r i v i n g at a p a r t i c u l a r solution and at the smoothing constants.  The purpose i s to see what effect the changes had  and to determine i f there are any values or range of i n i t i a l values that are "better" than the r e s t .  By "better" I mean any  values that consistently minimize the standard deviation of the forecast error and the number of i t e r a t i o n s .  By determining this  I hoped to develop some generalizable information about the response c h a r a c t e r i s t i c s of the pattern search process as i t i s r e f l e c t e d i n the values of the dependent variables. Chapter I gives an overview of forecasting methodology in general and t r i e s to place exponential smoothing i n an overa l l context.  I t also discusses the r e l a t i v e advantages and dis-  advantages of the various approaches.  Chapter II i s an i n depth  2 d i s c u s s i o n o f the e x p o n e n t i a l smoothing p r o c e s s .  Chapter I I I  d i s c u s s e s the p a t t e r n search technique and r e l a t e s i t to t h i s a p p l i c a t i o n i n e x p o n e n t i a l smoothing.  Chapter IV d e a l s w i t h  the c h a r a c t e r i s t i c s o f the data used i n t h i s study and g i v e s an overview o f the i n d u s t r y from which the data was taken.  Chapter  IV a l s o goes i n t o the s e l e c t i o n o f the-data s e r i e s , that were a c t u a l l y used, from the l a r g e r number that were a v a i l a b l e to the study.  Chapter V r e l a t e s the s p e c i f i c s o f the methodology  used i n the s e n s i t i v i t y a n a l y s i s .  Chapter VI d i s c u s s e s the  r e s u l t s o f the a n a l y s i s and s t a t e s the c o n c l u s i o n s o f the study. Chapter V I I makes some recommendations as to areas o f f u r t h e r research.  3 CHAPTER I General D i s c u s s i o n o f F o r e c a s t i n g Introduction The  purpose o f t h i s chapter i s to d e s c r i b e i n an  o v e r a l l sense,  s e v e r a l approaches to f o r e c a s t i n g s a l e s .  doing so, e x p o n e n t i a l smoothing, the technique  By-  t h a t i s the  focus o f t h i s study, w i l l be p l a c e d w i t h i n a broader  context.  I t i s intended t h a t t h i s w i l l help the reader understand the advantages and disadvantages  o f e x p o n e n t i a l smoothing and g a i n  a better feeling f o r i t s applicability. Any  form o f p l a n n i n g , be i t b u s i n e s s , p e r s o n a l o r  governmental, i n v o l v e s some e x p e c t a t i o n s o f the f u t u r e .  Such  e x p e c t a t i o n may be o p t i m i s t i c , p e s s i m i s t i c , based on i n t u i t i o n , or simply a s t r o n g assumption-that to the p a s t .  the f u t u r e w i l l be s i m i l a r  In some cases, the accuracy o f f u t u r e e x p e c t a t i o n s  i s not o f c r i t i c a l  importance.  I f the f o r e c a s t proves  wrong,plans can be e a s i l y and q u i c k l y a d j u s t e d .  to be  However, on  o t h e r o c c a s i o n s , when r e s o u r c e s have to i r r e t r i e v a b l y  committed,  a c c u r a t e f o r e c a s t i n g i s e s s e n t i a l to the f u t u r e s u r v i v a l and success o f the e n t e r p r i s e . that a technique  I t i s under c o n d i t i o n s such as these  such as e x p o n e n t i a l smoothing becomes o f  value. As mentioned e a r l i e r , f o r e c a s t i n g i s concerned  w i t h the  f u t u r e but i s based i m p e r f e c t l y on i n f o r m a t i o n from the past and present.  The f u t u r e i s u n c e r t a i n and u n t i l the c r y s t a l b a l l i s  p e r f e c t e d must remain so.  No matter what f o r e c a s t i n g methodology  4 or  technique w i l l he developed  will  continue to be u n c e r t a i n .  i n the years to come the f u t u r e As a r e s u l t o f t h i s ,  forecasts  are estimates o f f u t u r e s t a t e s about which no one can be s u r e . How f a r the f u t u r e i s capable o f b e i n g p r e d i c t e d depends on the extent to which i t i s r e l a t e d to the past and whether the r e l a t i o n s h i p s between the past and f u t u r e can be d i s c o v e r e d . J u s t because apples have f a l l e n from t r e e s f o r as l o n g as man can remember (Newton not excepted), t h i s i s no proof that the phenomenon w i l l  continue tomorrow.  Moreover, I may be w i l l i n g  to r i s k my f u t u r e on the continuence o f t h i s f a c t but I am s t i l l t a k i n g a r i s k because i t i s u n c e r t a i n . The view t h a t economic events can be f o r e c a s t depends upon the seemingly reasonable assumption have some c o n t i n u i t y ( l ) .  that economic events  I f there were no r e l a t i o n between the  past and the f u t u r e then there would be no r a t i o n a l b a s i s on which f o r e c a s t s c o u l d be made o r on which p l a n n i n g could take place.  The purpose  o f d e v e l o p i n g f o r e c a s t i n g techniques i s to  e x p l o i t the r e l a t i o n s h i p s that are thought  to e x i s t between past  and f u t u r e s t a t e s . Although f o r e c a s t s a r e sometimes made as though the v a r i a b l e to be f o r e c a s t w i l l , o r w i l l not, a t t a i n a p a r t i c u l a r v a l u e , complete  accuracy i s not to be expected.  Ideally, a  f o r e c a s t should take the form o f a d i s t r i b u t i o n o f the v a l u e s that might be assumed by the v a r i a b l e to be f o r e c a s t a l o n g w i t h an estimate o f the p r o b a b i l i t y that each w i l l  occur.  Throughout the study, I have taken a pragmatic view.  point of  I have assumed that the value o f a f o r e c a s t i n g technique  5 lies i n its applicability.  Given t h i s general  l i n e of reasoning,  then the c r i t e r i o n to be used i n judging a f o r e c a s t i s whether i t enables b e t t e r d e c i s i o n s to be made. business's  I f t h i s i s true then a  d e c i s i o n s w i t h regards to f o r e c a s t i n g should  the f o l l o w i n g c o n s i d e r a t i o n s technique i n c r e a s e s  include  ( l ) . I f the use o f a f o r e c a s t i n g  the accuracy o f assumptions made about the  f u t u r e , and as a consequence improves the q u a l i t y o f managerial d e c i s i o n making, then t h e use o f f o r e c a s t i n g i n such circumstances may be s a i d to be worthwhile. o n l y i f the i n c r e a s e  Of course, i t w i l l be worthwile  i n p r o f i t s t h a t r e s u l t s from improved  f o r e s i g h t exceeds the costs o f making the f o r e c a s t s . the other  I f , on  hand, the f o r e c a s t s r e s u l t i n p r o f i t s no b e t t e r than  would have been r e a l i z e d without them then there  i s no point i n  forecasting. The exponential  reason t h a t I make the above p o i n t s  smoothing i s an e a s i l y a p p l i e d technique.  be apparent, when the v a r i o u s exponential  i s that As w i l l  f o r e c a s t i n g approaches a r e d i s c u s s e d ,  smoothing does have some weaknesses.  These weaknesses,  however, do not d e t r a c t from the f a c t that the method i s q u i t e accurate  f o r a wide range o f commercial s i t u a t i o n s and i s q u i t e  inexpensive  t o use.  I f e e l t h a t these advantages outweigh the  t h e o r e t i c a l disadvantages that the system possesses. I would now l i k e to d i s c u s s some o f the problems that a f o r e c a s t e r might face en route self.  to p r o d u c i n g the f o r e c a s t i t -  As the f o r e c a s t e r seeks to determine t h e i n t e r r e l a t i o n -  s h i p s that are o f importance to h i s problem, he w i l l l i k e l y  find  6  that they change over time.  He i s also l i k e l y to find that  for each f l u c t u a t i o n there are a number of possible a l l of which may may  be p a r t i a l l y or wholly correct.  change from time to time.  explanations,  The cause  More than one cause may  work and t h e i r r e l a t i v e strengths may  be at  vary.  The forecaster i s often faced with a great volume of data.  Some of i t may  questionable  validity.  be l a t e , some incomplete and some of In order to permit prompt  conclusions  the data must be organized i n some l o g i c a l sequence. i t i o n , i f one i s to have a f a i t h i n the conclusions  In addor i n t e r -  pretations, the r e l i a b i l i t y and meaning of the data must also be determined.  This organization and  interpretation i s , of  course, only a necessary preliminary to the actual forecasting itself. Forecasting Systems I would now  l i k e to discuss a number of methods and  techniques commonly used i n forecasting.  Although I have used  the c l a s s i f i c a t i o n system delineated by Spencer et a l . (9) i t should be remembered that the i d e n t i f i c a t i o n of s p e c i f i c methods does not mean that they must or even should be used alone.  In  f a c t , i t i s sometimes d i f f i c u l t to separate one method from another, and there are numerous conceivable  classifications.  Descriptive Models This approach can also be c a l l e d naive methods.  The  basic idea i s to project the current s i t u a t i o n into the future.  7 Such methods are u s u a l l y d i s t i n g u i s h e d from o t h e r f o r e c a s t i n g methods i n t h a t they are mechanical  and are not c l o s e l y  grated w i t h surrounding economic or business theory.  inte-  If a  f o r e c a s t e r decides to use a d e s c r i p t i v e approach the number of problems t h a t he f a c e s w i t h regards to understanding reduced. ships.  He does not have to know the cause-and-effect  relation-  He simply l o o k s f o r a method t h a t w i l l work without  e n q u i r i n g as to why have any  are  i t works.  judgement about why  The f o r e c a s t e r may  not know or  the method worked i n the past  and  consequently he must assume t h a t i t w i l l work a g a i n i n the future. Time S e r i e s A n a l y s i s - I t i s to t h i s c l a s s i f i c a t i o n that e x p o n e n t i a l smoothing belongs.  At t h i s p o i n t however I w i l l  d e s c r i b e the c h a r a c t e r i s t i c s of the system as a whole l e a v i n g to the next chapter more i n depth d i s c u s s i o n o f e x p o n e n t i a l smoothing i t s e l f .  A time s e r i e s i s a sequence of v a l u e s  c o r r e s p o n d i n g to p a r t i c u l a r p e r i o d s of time.  Data such as  s a l e s , p r o d u c t i o n volume or p r i c e s when ordered are r e f e r r e d to as time s e r i e s .  chronologically  The b a s i c assumption f o r  f o r e c a s t i n g these s e r i e s i s t h a t there w i l l be a  continuous  development of the v a r i a b l e i n q u e s t i o n and t h a t the s i z e , r a t e and nature o f h i s t o r i c a l change w i l l continue i n t o the f u t u r e . As a quick l o o k at most time s e r i e s w i l l show, they are s u b j e c t to c e r t a i n p a t t e r n s o f f l u c t u a t i o n .  W i t h i n economic  time s e r i e s these sources of v a r i a t i o n can be c l a s s i f i e d f o u r typesJ t r e n d , s e a s o n a l , c y c l i c a l and random f o r c e s .  into Trend  8 represents  the l o n g - r u n growth or d e c l i n e of the s e r i e s .  Seasonal v a r i a t i o n s , that are due  to weather and  custom,  m a n i f e s t themselves d u r i n g the same approximate time each year e.g.  summer or Christmas.  periods  Seasonal v a r i a t i o n s could  even be reduced to a weekly b a s i s i f the p a t t e r n of change i n the  time s e r i e s moved i n a seven day  cover s e v e r a l years at a time and  cycle.  Cyclical variations  are a r e f l e c t i o n of  the  p o s i t i o n of the economy w i t h i n the b u s i n e s s c y c l e e.g. F i n a l l y , random f o r c e s such as s t r i k e s , wars and  recession.  competitive  i n f l u e n c e are e r r a t i c i n t h e i r i n f l u e n c e on p a r t i c u l a r s e r i e s . Of these f o u r f o r c e s a f f e c t i n g economic time s e r i e s , the seasonal All  v a r i a t i o n i s f a i r l y easy to measure and  predict.  that i s needed i s a good set of data that covers the  in question  f o r a number o f y e a r s .  name i m p l i e s i s u n p r e d i c t a b l e  but  The  season  random f a c t o r , as  can be m i t i g a t e d  by  the  such  smoothing out processes as moving average. The  method o f trend p r o j e c t i o n i s o f t e n used as a  f o r e c a s t i n g procedure i n i t s e l f .  T h i s method assumes that  r e c e n t r a t e of change of the v a r i a b l e w i l l future.  On  past trends  t h i s b a s i s expectations i n t o the f u t u r e .  method used by b u s i n e s s f i r m s .  continue i n t o  the  the  are e s t a b l i s h e d by p r o j e c t i n g  T h i s i s perhaps the most common T h i s method i s used because  economic s e r i e s e x h i b i t a p e r s i s t e n t and  c h a r a c t e r i s t i c rate  growth which can be approximated by a mathematical trend The  trend may  may  be weighted by a t t a c h i n g g r e a t e s t  (9).  c o n s i s t of a simple unweighted p r o j e c t i o n or i t importance to the most  of  9  recent period and successively l e s s importance to periods i n the more distant past. Because economic time series show a persistent tendency to move i n the same d i r e c t i o n for a period of time a forecaster using the method of trend projection w i l l be r i g h t as to the d i r e c t i o n of change more times than he w i l l be wrong.  In fact,  he w i l l he r i g h t i n every forecast except those at turning points.  This points out some major weaknesses of trend projec-  t i o n and indeed of a l l time series methods.  Because forecasts  are based e n t i r e l y on data within the time series under consideration, factors outside the data series w i l l not be looked at.  In  many cases factors outside the data series can be used to help predict the rate and d i r e c t i o n of movement of the s e r i e s .  For  example, business analysts have been using leading indicators (see Leading Series) f o r years.  In these cases changes i n one  factor consistently preceed changes i n another and hence can be used as a predictive t o o l .  Forecasting with time series method-  ology alone would preclude the use of such other tools. Another weakness of time series methods exists because they assume the future to be a systematic extension of the past. In many cases, new  changes  i n the environment that the time series  operates i n , w i l l cause changes i n the rate and d i r e c t i o n of movement of the series. these new  I f only h i s t o r i c a l data i s used,  changes w i l l not be picked up and any forecasts made  a f t e r that point w i l l be i n error. On the other hand, time series methods do have some  10 d i s t i n c t advantages.  To begin with, because the method uses  only h i s t o r i c a l data l i t t l e new data c o l l e c t i o n has to be done for each forecast.  I f the time series i s one of sales  then a firm would only have to keep a record of i t s sales i n order to s a t i s f y the requirements of the method. discussed  As w i l l be  l a t e r many other techniques are not nearly so simple  in t h e i r data requirements.  Another advantage i s that time  series forecasts usually involve the use of r e p e t i t i v e mathematical  computations.  use i n conjunction  Because of t h i s , the system lends i t s e l f to with electronic computers.  I f the number  of forecasts being made i s large this i s a great asset.  In  addition because of this repetitiveness costs per forecast can usually be kept low r e l a t i v e to other methods. Factor L i s t i n g - This i s a descriptive method of forecasting whereby the analyst simply enumerates the factors that he believes are influencing the dependent variable.  From t h i s l i s t the  forecaster draws a conclusion as to the l i k e l i h o o d of various events occurring.  In i t s most basic form the l i s t makes no  provision for the quantitative evaluation of each of the factors and t h e i r r o l e i n influencing the variable under consideration. Causal Techniques If the forecaster decides to employ a causal approach, he i s faced with a series of decisions.  These decisions occur  in the s e l e c t i o n of the causal explanation business movements.  of, for example,  The decisions also occur i n the selection,  11 organization and  interpretation of data.  Whereas descriptive  methods of forecasting, p a r t i c u l a r l y time series analysis, imply that the future i s an extension of the past, the use  of  causal techniques i s "based on the idea that the knowledge of the interrelationships between independent and dependent variables enables accurate forecasts to be made.  In many cases  these interrelationships are based on happenings i n the present. To be more s p e c i f i c , barometric methods of which leading series analysis i s one,  involve the use of s t a t i s t i c a l  usually selected time s e r i e s .  indicators,  When these indicators are used  i n conjunction with one another or when combined i n certain ways, they provide an i n d i c a t i o n of the d i r e c t i o n i n which the dependent variable i s moving. barometers of change. econometrics.  The  Thus the time series serve as  second technique to be discussed  is  This method works on the assumption that changes  i n a dependent variable, usually an economic one,  can be  explained by a set of mathematical relationships. Leading Series - Within the f i e l d of forecasting, this method has received by f a r the most attention. rather simple.  The basic idea i s  I f a series or index could be discovered that  showed leads of, say, six months with substantial r e g u l a r i t y i t would successfully indicate the turns i n the dependent variable. This being the case, the problems of forecasting would be largely  solved. In the early 1950's Geoffrey Moore and his  at the National  associates  Bureau of Economic Research did some extensive  12 examination o f time s e r i e s ( 7 ) . series exhibited leading  the  q u i t e a degree of c o n s i s t e n c y  "reference  turning points.  At f i r s t  First,  the  i n d i c a t o r s are not  t h e i r tendency to l e a d . s i g n a l what turns out  always  indices  to be a true change i n the  or e l s e s i g n a l too l a t e to be  purposes.  consistent  F r e q u e n t l y , some of the  v a r i a b l e , w h i l e the remaining i n d i c e s e i t h e r f a i l all  sight  U n f o r t u n a t e l y the method s u f f e r s from a number of  limitations.  will  w i t h i t or  i n d i c a t o r s would seem to p r o v i d e a u s e f u l guide f o r  prediction.  in  either i n  c y c l e , " running coincident  l a g g i n g behind i t at the leading  They found t h a t a number of  to s i g n a l at  of much v a l u e f o r p r e d i c t i v e  Secondly, i t i s not always p o s s i b l e  index i s s i g n a l l i n g an a c t u a l  dependent  change i n the  to t e l l  i f an  dependent v a r i a b l e  or whether i t i s merely e x h i b i t i n g a random f l u c t u a t i o n o f real significance. s i g n a l the t r u e still  only  F i n a l l y even i f the  turning points  of l e a d i n g  i t can  e a s i l y be  s e r i e s i s somewhat l i m i t e d .  reason b e i n g t h a t the l e a d i n g s e r i e s are not in  change, w h i l e d i s -  or n o t h i n g about the magnitude of the  Since both f a c t o r s are d e s i r e d  consistently  the economy, they would  i n d i c a t e the d i r e c t i o n of f u t u r e  closing l i t t l e  use  of, say,  i n d i c e s could  change.  seen t h a t  Perhaps the  Nor  the  chief  causally related,  a f u n c t i o n a l sense, to the b a s i c f a c t o r s r e s p o n s i b l e  them.  no  for  are they weighted a c c o r d i n g to t h e i r i n t r i n s i c  importance r e l a t i v e to the dependent v a r i a b l e . i n d i c a t o r s are  Instead,  s e l e c t e d because of t h e i r h i s t o r i c a l  of performance and  are  the  uniformity  u s u a l l y g i v e n equal weights of performance.  Econometrics - T h i s approach i s based on the i d e a that changes i n a c t i v i t y of a dependent v a r i a b l e can be e x p l a i n e d of i n t e r r e l a t i o n s h i p s between v a r i a b l e s .  by a s e t  For example, i t  attempts to e x p l a i n past economic a c t i v i t y and  predict future  economic a c t i v i t y by d e r i v i n g mathematical equations that  will  express the most probable i n t e r r e l a t i o n s h i p between a set of economic v a r i a b l e s .  By  combining the r e l e v a n t v a r i a b l e s i n t o  what seems to be the best mathematical arrangement econometricians attempt to p r e d i c t the f u t u r e course of one  or more of these  v a r i a b l e s on the b a s i s o f the e s t a b l i s h e d r e l a t i o n s h i p s .  The  "best mathematical arrangement" i s thus a model which takes the form of an e q u a t i o n or system of equations that seems best describe  the past s e t of r e l a t i o n s h i p s .  to  In other words, the  model i s a s i m p l i f i e d a b s t r a c t i o n o f a r e a l s i t u a t i o n , expressed i n e q u a t i o n form and  employed as a p r e d i c t i o n system t h a t w i l l  y i e l d numerical r e s u l t s .  In a c t u a l i t y econometrics i s q u i t e  h e a v i l y based on e x i s t i n g economic theory. attempt to apply  that theory  in real situations.  With regards to a p p l i c a t i o n s two considered.  The  first  has  w e l l the model d e s c r i b e s i z a t i o n one  E s s e n t i a l l y i t i s an  f a c t o r s must  be  to do w i t h accuracy of " f i t "  reality.  As a r a t h e r simple  or  general-  can say that the g r e a t e r degree of complexity  elegance t h a t the model possesses the b e t t e r w i l l be the However o f f s e t t i n g t h i s i s the problem of weighing the i n research value  and  construction costs against  of g r e a t e r accuracy, i n order  how  and "fit".  increment  the increment i n  to decide on  j u s t how  complete  14 a model to c o n s t r u c t .  The second p o i n t has to do w i t h the  f a c t t h a t s i n c e the model i s a r e p l i c a o f a dynamic  situation  i t must he r e v i s e d p e r i o d i c a l l y to allow f o r the changing weights  o f the constants o r parameters i n the equations.  Summary As a summary and c o n c l u s i o n to the chapter i t should be noted t h a t the major s t r e n g t h o f time s e r i e s a n a l y s i s i . e . e x p o n e n t i a l smoothing, r e l a t i v e to o t h e r f o r e c a s t i n g methods is i t s applicability.  As mentioned p r e v i o u s l y i t i s f a i r l y  simple to use, q u i t e cheap and q u i t e a c c u r a t e f o r a broad of uses.  I t does not possess  the t h e o r e t i c a l  and completeness o f say econometrics,  range  sophistication  but i t a l s o does not  possess the disadvantages w i t h regards to data c o l l e c t i o n and the r e q u i r e d understanding o f the complex i n t e r r e l a t i o n s h i p s o f the system b e i n g l o o k e d a t . casting.  I t i s a users approach to f o r e -  15 CHAPTER II Exponential Smoothing Introduction As was noted e a r l i e r the purpose of this study i s to investigate the response c h a r a c t e r i s t i c s of the pattern search method as i t applies to exponential smoothing.  Consequently,  the intent of this chapter i s to provide the reader with a description of the exponential smoothing method of forecasting. The f i r s t section of this chapter w i l l deal with the general c h a r a c t e r i s t i c s and conceptual foundations of exponential smoothing? while the second section w i l l give a detailed description of the technique i t s e l f .  It should he noted here  that the content of this chapter r e l i e s heavily on the material presented i n the a r t i c l e by Peter R. Winters, "Forecasting Sales by Exponentially Weighted Moving Average" (11).  This  chapter integrates Dr. Winters methodology and a number of ideas from other sources which have also been referenced where appropriate. Characteristics of Exponential Smoothing As a generalization the forecasting method discussed here deals with products whose " l i v e s " may indefinitely.  be assumed to continue  Therefore, the forecasting procedure i s concerned  with generating routine forecasts that take into account random fluctuations, trends and the recurrent seasonal movements of sales.  The needs of t h i s type of routine forecast imply certain  16 c h a r a c t e r i s t i c s t h a t should be present  i n the techniques  used.  Such a f o r e c a s t must be made q u i c k l y , i n e x p e n s i v e l y and To accomplish  t h i s the technique  easily.  used must be c l e a r l y s p e l l e d  out, so t h a t i t can be f o l l o w e d r o u t i n e l y , e i t h e r manually or by an e l e c t r o n i c computer.  The  number o f p i e c e s of  information  r e q u i r e d to make a s i n g l e f o r e c a s t must be kept a t a minimum, or e l s e the t o t a l amount of i n f o r m a t i o n r e q u i r e d f o r a l l products w i l l be expensive to s t o r e and the technique  must be able to i n t r o d u c e  i n f o r m a t i o n e a s i l y and Exponential these requirements. simple  expensive to m a i n t a i n .  equations.  cheaply  i n t o new  the l a t e s t  computational  These equations  which meets a l l  format c o n s i s t s of a  and  Even i n i t s  t r e n d adjustments are  i n c l u d e d , the amount of i n f o r m a t i o n r e q u i r e d f o r an forecast i s quite small. e a s i l y i n c o r p o r a t e s new Quite a few  By i t s very nature  the  individual  technique  i n f o r m a t i o n i n t o the system.  f o r e c a s t i n g techniques,  or systems, are  a v a i l a b l e or could be developed f o r p r e d i c t i n g item s a l e s . one  few  are e a s i l y programmed i n t o  an e l e c t r o n i c computer or manipulated manually. most complete form, when seasonal  sales  forecasts.  smoothing i s a technique The  Finally,  d i s c u s s e d here does not  The  " p r e d i c t " w i t h a b e h a v i o u r a l model  o f s a l e s , but uses an a n a l y s i s of the s a l e s time s e r i e s taken out of c ontext.  That i s , the only input to the f o r e c a s t i n g system  i s the past h i s t o r y of item s a l e s .  The  model does not  consider  such i n f o r m a t i o n as the market, the i n d u s t r y or the economy.  17 Various  forms of the e x p o n e n t i a l  model have been used  f o r a wide v a r i e t y of f o r e c a s t i n g a p p l i c a t i o n s .  Most a p p l i c a -  t i o n s have r e q u i r e d l i t t l e m o d i f i c a t i o n of the b a s i c model. For example, monthly f o r e c a s t s of cooking u t e n s i l s a l e s f o r the Aluminum Company of America, and  bi-monthly p r o j e c t i o n s of p a i n t  s a l e s f o r P i t t s b u r g h P l a t e Glass are  two  characteristic applica-  tions. In i t s s i m p l e s t  form, the e x p o n e n t i a l  system makes a  f o r e c a s t o f expected s a l e s i n the next p e r i o d from a weighted average of a c t u a l s a l e s i n the c u r r e n t p e r i o d and s a l e s f o r the c u r r e n t p e r i o d In the same way, constructed previous  (made d u r i n g the previous  the f o r e c a s t f o r the c u r r e n t p e r i o d  p e r i o d and  was  the f o r e c a s t of s a l e s f o r t h a t p e r i o d made it.  T h i s r e c u r s i v e process continues  to the b e g i n n i n g of the s a l e s data f o r the item. t i o n made i n any  cast i s constructed  s a l e s data f o r the i n such a way  item.  information  However the  t h a t only one  back  Thus a p r e d i c -  p e r i o d i s based on c u r r e n t s a l e s  a l l the p r e v i o u s  most r e c e n t  period).  from a weighted average of a c t u a l s a l e s f o r the  i n the p e r i o d before  and  forecasted  fore-  number (the  f o r e c a s t f o r the c u r r e n t p e r i o d ) must be r e t a i n e d to  produce the next f o r e c a s t . T h i s scheme o b v i o u s l y a f o r e c a s t i n g method. introduced.  The  Current  has  desirable characteristics for  sales information  is easily  c a l c u l a t i o n of an i n d i v i d u a l f o r e c a s t i s f a s t .  Only a l i m i t e d amount of i n f o r m a t i o n must be kept and  maintained.  18 For products w i t h s t a b l e s a l e s r a t e s and l i t t l e seasonal  i n f l u e n c e , the simple  proves q u i t e u s e f u l .  e x p o n e n t i a l smoothing model  Many products,  however, have a marked  trend i n t h e i r s a l e s , p a r t i c u l a r l y when they a r e f i r s t duced o r when competing products a r e i n t r o d u c e d . f o r many products to s a l e s .  In a d d i t i o n ,  there i s a l s o a s u b s t a n t i a l seasonal  Because o f t h i s ,  intro-  pattern  i t i s u s u a l l y worthwhile to extend  the e x p o n e n t i a l system to account f o r l o n g - r u n trends and seasonal e f f e c t s .  These two f a c t o r s are handled  same way as the simple  e x p o n e n t i a l system.  i n much the  More i n f o r m a t i o n i s  r e q u i r e d f o r t h i s more complete model but, f o r most the accuracy  products,  o f the f o r e c a s t i s s u b s t a n t i a l l y i n c r e a s e d .  next s e c t i o n w i l l  The  give a more in-depth d e s c r i p t i o n o f the  system. E x p o n e n t i a l Smoothing Method The  advantages o f e x p o n e n t i a l weighted  d i s c u s s e d i n the p r e v i o u s s e c t i o n , are obvious  averages, from i t s simple  formulation: Expected Sales f o r the Coming P e r i o d = A ( R e a l i z e d Sales d u r i n g the L a s t P e r i o d ) + ( l - A ) (Expected Sales f o r the L a s t P e r i o d ) . From t h i s i t can be seen t h a t j u s t three b i t s o f i n f o r m a t i o n a r e called  fori 1) expected s a l e s f o r the p e r i o d j u s t completed, 2) r e a l i z e d s a l e s f o r the p e r i o d j u s t completed, 3) a w e i g h t i n g f a c t o r A, whose value l i e s between zero and one.  19  The  r o l e played  by the w e i g h t i n g f a c t o r and  the  and  f o r e c a s t data can be demonstrated by r e a r r a n g i n g  and  expressing  them i n the n o t a t i o n  introduced  sales  the  terms  by Winters  (11)  Expected s a l e s becomes: S  t  = S_ t  x  + A  -  S _ t  l ) f  where S^ = s a l e s d u r i n g  period  &t = s a l e s f o r e c a s t made i n p e r i o d -t f o r the coming p e r i o d , S^-i  = s a l e s f o r e c a s t made i n p e r i o d -t-1 f o r p e r i o d  A = a number between one This formulation the l a s t  and  zero i n c l u s i v e .  makes i t c l e a r that a f o r e c a s t  sales forecast  ±,  ( S ^ - i ) adjusted  (S^)  i s equal to  hy some f r a c t i o n (A)  of the d i f f e r e n c e between the s a l e s a c t u a l l y achieved d u r i n g l a s t period Therefore, w i l l be  (S^) and  the s a l e s f o r e c a s t f o r t h a t p e r i o d  i f the l a s t  f o r e c a s t was  equal to the l a s t  cast e r r o r .  I f the l a s t  be equal to the l a s t  high,  the new  sales  f o r e c a s t l e s s a f r a c t i o n of the f o r e c a s t was  low,  the new  (S^-i). forecast fore-  forecast  f o r e c a s t plus some f r a c t i o n of the  the  will  forecast  error. Using the same n o t a t i o n ,  the new  f o r e c a s t may  be  i n t e r p r e t e d as a simple weighted average of the s a l e s r e a l i z e d and  the s a l e s t h a t were f o r e c a s t . S  The past  t  = AS*  + (1-A)  That i s :  St-1.  smoothing constant A determines how data w i l l have on the R.G.  much of an e f f e c t the  estimate.  Brown (3) presented a formula f o r f i n d i n g the  20 number of past time i n g system. N =  u n i t s of data t h a t are b i n g used i n a smooth-  T h i s number i s d i r e c t l y i n f l u e n c e d by the v a l u e of A.  2 - A A  where N i s the number of time u n i t s of data r e p r e s e n t e d . example, a low v a l u e of A, e.g. present data and  0.2  For  g i v e s l i t t l e weight to the  c o n s i d e r a b l e weight to past d a t a .  If this  i s used, N = 9 time u n i t s of data are used i n t h i s case. 0.5»  where l a r g e r emphasis i s p l a c e d on c u r r e n t data and  l y l e s s emphasis on past data, N = 3. e a s i l y be seen how  value  If A = consequent-  From t h i s example i t can  the value of A e f f e c t i v e l y c o n t r o l s the  t r i b u t i o n of emphasis p l a c e d on c u r r e n t and past data.  dis-  Figure 1  g r a p h i c a l l y demonstrates the e x p o n e n t i a l l y d i s t r i b u t e d weight g i v e n to each data pexce i n a time s e r i e s f o r two  different  values  of A. Choosing  an a p p r o p r i a t e value f o r the smoothing constant  i s a n o n - t r i v i a l problem.  The value of the constant should be  enough to give the system s t a b i l i t y . the emphasis on the c u r r e n t data and  low  I t does t h i s by d e c r e a s i n g consequently  f l u e n c e of c u r r e n t random f l u c t u a t i o n s .  r e d u c i n g the i n -  On the other hand the  value must be l a r g e enough to give the system s e n s i t i v i t y to s y s tematic changes i n the time s e r i e s .  I t does t h i s by having a value  that p l a c e s c o n s i d e r a b l e emphasis on c u r r e n t d a t a . One  approach to choosing a proper value f o r the smoothing  constant uses a " g r i d of v a l u e s . "  Each value on the g r i d i s  t r i e d i n the smoothing system and t h a t value which r e s u l t s i n the minimum f o r e c a s t e r r o r i s used i n f u t u r e c a l c u l a t i o n s . *  Forecast  For d i s c u s s i o n of f o r e c a s t e r r o r seet Measures of Accuracy.  21  +3  cd P -p  weighting of data with smoothing constant = 0.5  CO  o <U  o <H  98.4$ of forecasted data i s made up of data 5 periods old or l e s s  o +» c  0>  o u  weighting of data with smoothing constant = 0.2  0)  PH  a.  S  «•  5  w  1  91.6$ of forecasted data i s made up of data 10 periods old or les  Age of Data (months)  Figure 1  Different Values of the Smoothing Constant Give Different Weights to Past Data Items D'Amico (1971)  22 O b v i o u s l y t h i s i s an i n e f f i c i e n t and cumbersome method. pattern  search method, which w i l l be d i s c u s s e d  The  i n the next  chapter, i s an attempt to take a d i f f e r e n t and more  efficient  route f o r choosing the v a l u e of the smoothing constant. Measures o f F o r e c a s t  Accuracy  Roberts and Whybark  (8) present a number of ways o f  measuring f o r e c a s t accuracy. e r r o r i n each f o r e c a s t p e r i o d .  Each measure i s a f u n c t i o n o f The f o r e c a s t e r r o r i n any  p e r i o d ±, g i v e n by e^, f o r which the f o r e c a s t was F^, can be written: e where  t  = F  ±  - X^,  i s the a c t u a l s e r i e s value f o r p e r i o d  f  With the measure o f e r r o r , f o r e c a s t accuracy may then be assessed by three measures: A) mean f o r e c a s t e r r o r , B) mean absolute e r r o r (MAD), C) e r r o r v a r i a n c e . Mean Forecast The  Error  mean f o r e c a s t  c a s t i n g technique. forecast  e r r o r a s s e s s e s the b i a s of the f o r e -  I f the system i s l e a d i n g o r l a g g i n g  in its  the d e v i a t i o n o f e r r o r w i l l be i n d i c a t e d by the mean  f o r e c a s t e r r o r , given by: e =  e. / N, t=l  X  where N denotes the number of f o r e c a s t p e r i o d s under i n v e s t i g a tion.  23 Mean Absolute  Error  Another common and f r e q u e n t l y  used measure o f f o r e c a s t  accuracy i s the mean a b s o l u t e e r r o r o r mean a b s o l u t e (MAD).  deviation  The MAD i s d e f i n e d as:  MAD i s an i n d i c a t i o n o f the v a r i a b i l i t y i n the f o r e c a s t  error.  E r r o r Variance An a l t e r n a t i v e t o MAD i s the f o r e c a s t  error  variance.  2 T h i s measure, denoted by S , i s d e f i n e d a s : e  Se = ^  N  (e +  e) /N-l. 2  T h i s measure i s u s e f u l because many d e c i s i o n s knowledge o f f o r e c a s t r e l i a b i l i t y .  are i n f l u e n c e d by  I f the v a r i a n c e  i s high t h i s  means that the i n t e r v a l w i t h i n which the f o r e c a s t i s l i k e l y to fall  i s q u i t e wide.  Put i n o t h e r words, i t means t h a t  of e r r o r s i z e i s q u i t e wide.  the range  The d e c i s i o n then, must r e f l e c t  t h i s knowledge o f the v a r i a n c e ,  i n order to absorb the u n c e r t a i n -  t y o f the f o r e c a s t . I d e a l l y one would l i k e a f o r e c a s t i n g system which, i n terms o f f o r e c a s t accuracy, y i e l d s a l l measures c l o s e to zero. Unfortunately, i t i s very d i f f i c u l t  to determine a technique  which s i m u l t a n e o u s l y s a t i s f i e s a l l c r i t e r i a because these measures f r e q u e n t l y out, the  conflict.  As Roberts and Whybark (8) p o i n t  one may be able t o reduce to a minimum the v a r i a b i l i t y o f f o r e c a s t e r r o r by m i n i m i z i n g MAD.  may i n c r e a s e  In doing so, however, he  the mean e r r o r i n the f o r e c a s t .  The q u e s t i o n o f  24 which type of c r i t e r i o n to minimize must be evaluated  in light  of the expected b e n e f i t s from improved f o r e c a s t s f o r each individual application. Seasonal Adjustments Seasonal adjustments of s a l e s f o r e c a s t s are u s e f u l  and  p r a c t i c a l i n those cases where r e c u r r i n g c y c l e s i n s a l e s behaviour can be  identified.  To s e a s o n a l l y a d j u s t an exponen-  t i a l l y smoothed s a l e s f o r e c a s t r e q u i r e s an i n c r e a s e computations performed and from one  p e r i o d to another.  the procedure and  the amount of data c a r r i e d  monthly s a l e s  forward  By c o n s i d e r i n g the f o r m u l a t i o n  In doing  convenient to use an annual s a l e s c y c l e  and  data.  To c a l c u l a t e a s a l e s f o r e c a s t i n one  p e r i o d f o r some  f u t u r e p e r i o d , i t i s necessary to form the product of an of the smoothed deseasonalized seasonal  of  the steps performed i n o b t a i n i n g a f o r e c a s t ,  the workings o f the added f a c t o r s w i l l be apparent. so, i t w i l l be  i n both the  s a l e s r a t e and  f a c t o r f o r the f o r e c a s t p e r i o d .  estimate  an estimate of  the  Again u s i n g Winters'  n o t a t i o n t h i s estimate i s i s+,i  =  S ^ V L + I ,  where S^.,1  = smoothed f o r e c a s t of expected s a l e s made i n p e r i o d -t f o r p e r i o d «t + 1.  S^  = estimate o f the smoothed expected deseasona l i z e d r a t e of s a l e s f o r p e r i o d = expected r a t i o of smoothed seasonal s a l e s to smoothed d e s e a s o n a l i z e d s a l e s i n the p e r i o d  25 for which the forecast i s being made. In making a forecast for the coming month, the r a t i o calculated one year ago f o r that month is used. The subscript bears this out since L represents the number of periods i n the cycle or season. I f smoothed seasonal  sales (S^.) are expected to be  greater than expected smoothed deseasonalized  sales (S^), the  seasonal r a t i o w i l l be greater than one and i f the reverse i s true, the r a t i o w i l l be l e s s than The  one.  formula that y i e l d s an estimate of the smoothed  expected deseasonalized  sales i s :  Its s i m i l a r i t y to the formula for simple exponential i s e a s i l y observed.  The  smoothing  current sales are deseasonalized  d i v i d i n g by an appropriate  seasonal  structure of the r e l a t i o n s h i p s are  factor.  by  Otherwise the  identical.  To compute the expected deseasonalized  rate of sales  (S4,l) each month i t i s necessary to have available the following: A) B) C) D)  unit sales f o r the month (S±) value of the seasonal factor applicable to the same month l a s t cycle (F-t-L) weighting constant A expected deseasonalized rate of sales computed l a s t period (§t-l)  To complete this part of the procedure only one b i t of information  (the applicable seasonal  extra  sales r a t i o ) and  only one additional c a l c u l a t i o n (a d i v i s i o n ) are needed.  Both  of these additions being r e l a t i v e to what would have been needed i n the case where a correction f o r the seasonal  sales  26 behaviour was  not r e q u i r e d .  There i s  however, the added problem of computing and  f  s t o r i n g values f o r the seasonal H  Again i t may form.  -tr  = B  The  +  ( l  -  be observed  new  B )  F  ratios,  t-L  t h a t the formula i s i n a f a m i l i a r  w e i g h t i n g constant B (a number between one  and  zero i n c l u s i v e ) a s s i g n s the r e l a t i v e i n f l u e n c e between the most r e c e n t seasonal r a t i o the year b e f o r e .  (S-t)/(S+,) and  I t should be noted  the r e l e v a n t r a t i o that the seasonal  calculated ratio  f o r each p e r i o d i s a new  a d d i t i o n to the r e q u i r e d data  computing requirements.  Since an annual monthly c y c l e has been  used to i l l u s t r a t e t h i s method, twelve s t o r e d and Trend  and  such f i g u r e s would be  each employed once d u r i n g the course  of the year.  Adjustment So f a r , the problems of h a n d l i n g normal random and  seasonal behaviour  have been c o n s i d e r e d .  The  final  adjustment  i n the e x p o n e n t i a l smoothing method of s a l e s f o r e c a s t i n g i s to a d j u s t data to r e f l e c t t r e n d s .  In the l i g h t of the  preceding  d i s c u s s i o n i t w i l l not be s u r p r i s i n g to f i n d t h a t the amount of computation i n c r e a s e s but t h a t the g e n e r a l format technique  i s not r a d i c a l l y  The  of the  altered.  e x p o n e n t i a l l y smoothed t r e n d and s e a s o n a l l y  a d j u s t e d s a l e s f o r e c a s t f o r T p e r i o d s i n the f u t u r e can be written, S  t t  T = (S  t  + TR^)  F -L+T. t  27  Two new terms are introduced here, a revised estimate  of the  trend (R-fc) and the number of periods i n the future for which the forecast i s being made (T). The above formulation i s a straightforward of the simpler procedure. period's estimated  extension  By adding the product of the current  trend (Rt) and the number of periods i n the  future f o r which the forecast i s being made (T) to the smoothed and deseasonalized  current rate of sales (S+)i an  estimate of  the expected l e v e l of smoothed and trend adjusted sales i s obtained f o r the forecast period.  This result i s then multi-  p l i e d by the appropriate seasonal factor (F^._Tj m) i n order to +  obtain the desired sales forecast. The expression used to obtain the seasonal r a t i o i s unchanged i n the process of adding a trend adjustment. computation of the deseasonalized ever, i s s l i g h t l y modified. ^  =  A  {^i)  +  (  1  "  A  )  The  expected rate of sales, how-  I t becomes (  ^ - i  +R  t-D.  The e f f e c t of this to update the previous deseasonalized  sales  forecast by adding to i t the previous period's trend f a c t o r . Although t h i s i s a simple adjustment, i t necessitates s t o r i n g and carrying forward additional data. To compute the value of the trend f a c t o r , a weighting constant  C (again, a number between one and zero i n c l u s i v e )  must also be added to the l i s t of required data.  This i s  necessary since the trend factor must also be smoothed. The  28 formula f o r smoothing the t r e n d f a c t o r i s Rt = C(§t " §t-l)  +  (1-C) R-t-1.  T h i s e x p r e s s i o n smooths the v a r i a t i o n i n the t r e n d from one p e r i o d t o the next.  Again, the usual form o f the smoothing  r e l a t i o n s h i p i s preserved and the c l o s e r the w e i g h t i n g  constant  C i s t o one, the g r e a t e r the r e l a t i v e importance attached to the most r e c e n t d e v i a t i o n s . Weighting  Constants  The a f o r e g o i n g d i s c u s s i o n has h i g h l i g h t e d many aspects of  the f l e x i b i l i t y  i n h e r e n t i n the e x p o n e n t i a l smoothing method.  Two o t h e r f e a t u r e s t h a t enhance f l e x i b i l i t y a r e s i g n i f i c a n t . first  o f these i s the assignment o f i n i t i a l  The  v a l u e s w i t h i n the  smoothing system and the second i s the a v a i l a b i l i t y of separate weighting  constants.  I n i t i a l v a l u e s must be g i v e n to the f o r e c a s t o f expected s a l e s , the seasonal r a t i o f o r each p e r i o d d u r i n g the season and the trend f a c t o r .  I d e a l l y , these values would be based on  h i s t o r i c a l s a l e s data f o r the item i t s e l f o r from an item whose s a l e s behaviour  i s expected  to be s i m i l a r to the one under  consideration.  Lacking information of this character,  informed  judgements must be made on the f a c t s at hand, no matter how fragmentary  these f a c t s may be.  accuracy o f the i n i t i a l method permits  D i s r e g a r d i n g the a b s o l u t e  c o n d i t i o n s employed, t h i s f o r e c a s t i n g  the f o r e c a s t e r to i n c o r p o r a t e i n the f o r e c a s t i n g  mechanism a l l o f the i n f o r m a t i o n a t h i s command.  Chapter V  d i s c u s s e s t h i s area o f i n t i a l v a l u e s i n g r e a t e r d e t a i l i s presented here.  than  The purpose o f these few l i n e s i s to give  the reader a f e e l i n g f o r the system under c o n s i d e r a t i o n . The  second  f e a t u r e i s the a v a i l a b i l i t y o f three  separate w e i g h t i n g c o n s t a n t s .  Each o f these constants may be  a d j u s t e d to allow f o r an items expected  s a l e s behaviour.  was mentioned e a r l i e r the constant A determines  the r e l a t i v e  i n f l u e n c e t h a t the most r e c e n t s a l e s r e s u l t s w i l l have. ly  As  Relative  high v a l u e s o f t h i s w e i g h t i n g constant w i l l make a system  responsive to c u r r e n t changes and w i l l when a system i s f i r s t available. reduced  f r e q u e n t l y be a p p r o p r i a t e  i n t r o d u c e d and l i t t l e  h i s t o r i c a l data i s  When t h i s c o n d i t i o n p r e v a i l s , the v a l u e of A may be  to a "normal" l e v e l f o r the item.  T h i s can take p l a c e  a f t e r a s u f f i c i e n t time has passed t o permit the system to e s t a b l i s h a r e a s o n a b l y s t a b l e p a t t e r n o f response. The  second w e i g h t i n g constant performs  the same  smoothing f u n c t i o n f o r the seasonal r a t i o t h a t A performs unadjusted  forecast.  f o r the  I f B i s h i g h , the seasonal r a t i o s are  s e n s i t i v e to the seasonal behaviour d i s p l a y e d i n the p r e v i o u s cycle.  I f B i s low, l i t t l e weight i s g i v e n to the seasonal  behaviour  t h a t o c c u r r e d l a s t c y c l e and as a consequence the  i n f l u e n c e o f seasonal behaviour extending f u r t h e r back i n time becomes the p r i n c i p a l determinant employed f o r f o r e c a s t i n g purposes.  o f the seasonal  factor  In the case o f the w e i g h t i n g  constant C used to smooth the t r e n d f a c t o r s , the same i s t r u e . High v a l u e s p l a c e more weight on r e c e n t t r e n d s i z e than do low  30 values.  Again,  as i n the d i s c u s s i o n of the o t h e r  initial  v a l u e s , I have r a i s e d the matter i n order to p l a c e i t i n i t s context.  Chapter VI d i s c u s s e s i n i t i a l  much g r e a t e r depth.  constant v a l u e s i n  31 CHAPTER I I I Pattern Search Methodology The purpose of t h i s chapter i s to present a detailed discussion of pattern search procedures as they apply to f i t t i n g smoothing constants to the exponential forecasting system.  The  chapter starts o f f with a short general introduction to the pattern search process.  Following t h i s there i s an extensive  description of the workings of the process, including a discussion of the two basic search moves: exploratory and pattern.  Next i s  a discussion of the stopping rules as they relate to the accuracy of the procedure, while the l a s t part of the description looks at the systems a b i l i t y to change the d i r e c t i o n of the search. F i n a l l y , the l a s t section of the chapter, discusses some of the strengths and weaknesses of such a system. Introduction While the pattern search technique i n i t s e l f i s a method with enough generality to be able to be applied i n a number of problem situations I w i l l only deal with i t as i t relates to the selection of parameters f o r the exponential smoothing model. The parameters under consideration are the smoothing constants; A, B, and C, (see Chapter I I ) . Berry and Bliemel (2) discuss three approaches to the problem of s e l e c t i n g the exponential weights (A, B, and C). F i r s t , there are some general guidelines for s e l e c t i n g smoothing  32 constant values. These guidelines require an assessment of the extent to which the sales series average i s subject to random variations and major s h i f t s i n i t s l e v e l . dynamic techniques^mentioned  Second, there are  but not discussed,for the continuously  adjusting smoothing constant values.  These adaptive  techniques  are concerned with detecting important changes i n the sales average and specifying the proper smoothing constant value to be applied.  A t h i r d method of selecting the smoothing constant  values conducts a simulation analysis, using past sales data, to evaluate the forecast error associated with alternative smoothing constant values.  The quality of the smoothing constant  values derived from t h i s analysis depends on the sales history being a r e l i a b l e description of future sales.  I t i s this t h i r d  method with which Berry and Bliemel were concerned. the one which t h i s chapter w i l l discuss i n d e t a i l .  I t i s also While the  i n i t i a l work i n this area was done by Hook and Jeeves (6) I have used the Berry and Bliemel paper (2) as my major reference source because of i t s t o p i c a l relevance. Pattern Search Process The pattern search process u t i l i z e s a simple search strategy to maximize or minimize the value of a function. The strategy enables the procedure  to move progressively towards  "better" parameter values by building on what i t has already learned from previous functional evaluations.  Pattern search i s  applied to problems where a minimum (or maximum) value f o r an  33  expression i s sought, i . e . an expression of the formj y = f (X]_, X£, X^, ...» X ) . As Van Wormer and Weiss (10) point out, n  the search technique views the function to be minimized or maximized as a black box. The box allows certain inputs to be set  ( t r i a l values of the functions arguments) and responds  with a single output value (the value of the function at that point). The search strategy i t s e l f consists of conducting a series of t r a i l evaluations, where the expression i s evaluated successively with p a r t i c u l a r sets of X values.  The strategy  then uses these t r i a l results to decide what to do next.  The  basic notion underlying this strategy involves moving from one set  of t r i a l s  to another, and when desirable results are  obtained the moves towards "better" values are made i n increasingly l a r g e r step sizes.  Thus, the search f o r an improved solu-  t i o n i s guided by the successes (or f a i l u r e s ) obtained i n previous function evaluations.  In e f f e c t , the procedure i s  capable of crude learning. In order to describe and i l l u s t r a t e the pattern search techniques, I have l i f t e d out of context, the example used by Berry and Bliemel.  They used the method i n an application to  the problem of s e l e c t i n g smoothing constant values (A and C) f o r the trend adjusted expotential forecasting model.  Insofar as  the model does not deal with the weighting constant, B, that performs the smoothing function for the seasonal r a t i o s , the description i s incomplete.  Fortunately, t h i s does not detract  3^ from the g e n e r a l i t y o f the search procedure i t s e l f . he noted  that i n my a n a l y s i s o f the technique  I t should  I used the  complete e x p o n e n t i a l smoothing model and considered a l l o f the A, B and C parameters. In s e l e c t i n g smoothing constant v a l u e s , the e x p r e s s i o n to  be evaluated  involves a simulation using a sales history  sequence and a s p e c i f i c s e t o f smoothing constant v a l u e s . are used to o b t a i n the estimate  o f the f o r e c a s t e r r o r standard  d e v i a t i o n ( o r an analogous c r i t e r i o n ) a s s o c i a t e d w i t h values.  These  these  F o r the trend adjusted model the e x p r e s s i o n t o be  evaluated i s represented technique  as; S = f ( A , C ) .  begins by e v a l u a t i n g an i n i t i a l  constant v a l u e s  (eg. A =.5  and C = .5).  The p a t t e r n search s e t of smoothing From t h i s i n i t i a l  position  subsequent moves are made towards parameter values which produce a s m a l l e r estimate  o f the f o r e c a s t e r r o r s standard d e v i a t i o n .  To i l l u s t r a t e t h i s procedure,  F i g u r e 2 shows the path  taken by the search procedure i n s u c c e s s i v e e v a l u a t i o n s s o l u t i o n s ) f o r v a r i o u s smoothing constant v a l u e s . at  the p o i n t (A = . 5»  C = .5)  (trial  I t begins  and moves r a p i d l y " d o w n h i l l " to  smoothing constant v a l u e s with s u c c e s s i v e l y s m a l l e r f o r e c a s t e r r o r standard d e v i a t i o n s and stops t e m p o r a r i l y at the p o i n t (A = . 0 0 , C = . 0 0 ) . tracks s l i g h t l y  At t h i s p o i n t the s e a r c h procedure back-  ( t o A = .07, C = .07) and changes d i r e c t i o n ,  moving towards the t e r m i n a l p o i n t (A = . 0 2 , C = . 7 8 ) . procedure terminates  a t t h i s p o i n t because another  The s e a r c h  set of  35  Figure 2  Path o f a P a t t e r n Search A p p l i c a t i o n  r&R -mi's.  36  smoothing constants having a smaller forecast error standard deviation cannot he located. There are several important  features of the path  followed by the pattern search technique considered.  i n Figure 2 to be  F i r s t , as the path moves from the s t a r t i n g point  (A = . 5 , C = . 5 ) and more information i s obtained about the response surface, longer steps are taken between successive evaluations.  Second, when the t r i a l evaluations reveal that the  search i s producing i n f e r i o r solutions, e.g. at (A = .00, C = .00), the next step i n the procedure i s a series of l o c a l explorations that r e s u l t i n establishing a new pattern i n a new d i r e c t i o n . F i n a l l y , the new pattern progresses simultaneously  i n an a r c - l i k e manner, moving  i n two directions towards the f i n a l point (A = .02,  C = . 7 8 ) . These points,as well as the search l o g i c , are  illustrat-  ed i n Figures 3» 6A and 6B where smaller portions of the search path are shown i n more d e t a i l . Exploratory and Pattern Moves Pattern search employs two types of moves i n looking f o r improved parameter values:  exploratory and pattern moves.  The  search procedure begins by making a series of exploratory moves to examine the nature of the response surface around an a r b i t r a r i l y selected s t a r t i n g point, e.g. (A= . 5 , C= - 5 ) .  1  The moves, which  are i l l u s t r a t e d i n Figure 3 , include:  The s e l e c t i o n of t h i s s t a r t i n g point w i l l be discussed i n d e t a i l l a t e r on as i t i s one of the factors under consideration i n this research.  37  X  *  o  *  ,50 . .  .Mb -•  Exploratory Move Pattern Move Forecast Error Standard Deviation  •oa... —\—t—A/ .0 .03.  .«W  -Hb  M  .5o  ,5H  ^  Weighted Average Smoothing Constant (A)  Figure 3  Exploratory and Pattern Moves  38  A" Value  C" Value  I n i t i a l Evaluation .5 Exploratory Move No.l . 5 • - 0 1 Exploratory Move No.2 . 5 - . 0 1 Exploratory Move No.3 . 4 9 Exploratory Move No.3 - 4 9  •5 •5 •5 .5 + .5 -  .01 .01  Forecast E r r o r Standard Deviation 758.5 763.1 753.9 756.4 751.4  The exploratory moves provide an evaluation of the alternative weighted average smoothing constant values (A) a short distance (.01)  from the s t a r t i n g point.  Although an increase i n A to . 5 1  r e s u l t s i n a larger forecast error standard deviation, the smaller value of . 4 9 leads to a reduction i n this measure.  Noting  t h i s , the procedure continues by evaluating the effect of small changes i n C, with the new value of A = . 4 9 .  The changes i n C  r e s u l t i n reducing the forecast error standard deviation to 751.4  and the selection of a d i r e c t i o n f o r future moves, i . e .  from (A= . 5 , C - . 5 ) to (A = . 4 9 , C = . 4 9 ) .  In e f f e c t , the  exploratory moves establish an estimate of the slope of the contour map around a given point.  I f one wanted to add the  seasonal r a t i o constant value B to the smoothing formula then the procedure would be extended to holding the A and C values constant while sweeping B through the same series of exploratory moves. The next step i n the search procedure involves making what i s c a l l e d a pattern move. moves i s shown i n Figure 3 . distance than i t s predecessor  The i n i t i a l series of pattern  Each of these moves covers a greater as long as the successive  trial  evaluations meet with continued success ( i . e . progressively  39  smaller forecast standard deviations).  The length of a pattern  move i s determined by multiplying the distance covered by the immediately preceeding exploratory and/or pattern moves by a factor (K).  To i l l u s t r a t e t h i s , (X^, X ) are defined to be the 2  best exploratory move co-ordinate i n the l a s t set of exploratory moves and (Y^» Yg) are the i n i t i a l co-ordinates of the previous pattern move.  As an example, f o r pattern moves one, two and  three i n Figure 3 , the co-ordinates are: Pattern Move Number  Best-Exploratory Move Coordinate (X-pXp)  Previous Pattern Move I n i t i a l Coordinate ( Y j ^ )  1  (.49,.49)  ( I n i t i a l Point A= . 5 , C= . 5 )  2  (.4?,  .47)  ( . 4 9 ,  .49)  3  (.44,  .44)  ( . 4 7 ,  .47)  Next the length of the pattern move i s computed using the following equation: New Co-Ordinate ( Z  l f  Z ) = (X 2  X ) + K [(X  l p  2  lP  X ) - (Y 2  l f  Y 7| . 2  In this example the pattern move m u l t i p l i e r (K) i s set equal to 1 . 0 and the r e s u l t i n g pattern moves are: Pattern Move Number  New Coordinate (Zi ,Zp)  (X ,X ) 1  2  (Yi ,Y ) ?  KJJX-I ,X?)-(Yi ,Yp  1  (.48,.48)  (.49,-49)  (.50,.50)  (-.01,-.01)  2  ( . 4 5 , - 4 5 )  ( . 4 7 , - 4 7 )  ( . 4 9 , - 4 9 )  (-.02,-.02)  3  ( . 4 1 , . 4 1 )  ( . 4 4 , . 4 4 )  ( . 4 7 , - 4 7 )  (-.03,-.03)  Thus as long as the search procedure continues to encounter successful t r i a l evaluations, the distance covered by each of the pattern moves increases rapidly.  In the smoothing model that  40 (w  me  GO  EX*>LOfcftvoR.y EQUALS  ex9uoR.moKy 100  expuo«.fK  oR.y S"  WO  VS  UHS5  C4>  (2L SETT  S  yes  KEY Ai Ci  Si  Starting Value f o r A Starting Value f o r C Forecast E r r o r Standard Deviation f o r Point (A,C) Best Exploratory Move Value f o r A Best Exploratory Move for C Forecast E r r o r Standard Deviation f o r Point ( A ' , C )  Figure 4 SOURCEi  Pattern Search Flow Chart  Berry and Bliemel  CO  MO  41  IAOVE  yes  (\  Success ?  \s ^  wove  ves  success ?  f\WQ ft N5u>  too  COOR.t>\W«TE  EXIT  Figure 5  SOURCE:  +  Exploratory Move Flow Chart (The route shown i s carried out f o r each co-ordinate separately)  Hook and Jeeves  42 was  used i n this study the pattern move m u l t i p l i e r was  value of 2.0.  given a  This had the e f f e c t of doubling the distance  between the best exploratory move co-ordinate and the previous pattern move i n i t i a l coordinate andassigning that value to the next pattern move step s i z e . Berry and Bliemel (2) experimented with m u l t i p l i e r values i n the interval between 1.0 and 4.0 and found that the computing time and forecast error obtained were r e l a t i v e l y insensitive to such changes.  A value of 2.0,  however, was  found  to be marginally superior with regards to minimizing the number of exploratory moves needed to obtain an optimum f o r the function. The exploratory and pattern moves are used i n a r e p e t i t i v e manner to guide the search toward improved smoothing constant values.  A flow chart summarizing the pattern search procedure  i s presented  i n Figure 4 and a flow chart describing the  exploratory moves i s presented Step Size  i n Figure 5«  Reduction  As long as the exploratory moves continue to provide improvement i n the forecast error standard deviation (S*) pattern moves continue to be made.  This process i s represented  blocks 4 to 7 i n Figure 4.  The pattern moves continue  by until  the d i r e c t i o n of the search leads to unsatisfactory r e s u l t s . In t h i s case, either the d i r e c t i o n of the search i s changed through a series of exploratory moves (block 2) or i f a better d i r e c t i o n cannot be found, the step size for the exploratory move  43  (.01  i n the example shown i n F i g u r e 2) i s reduced.  s i z e r e d u c t i o n i s accomplished 9 i n Figure 4.  step  i n the l o o p c o n n e c t i n g b l o c k s  2,  3,  to  produce u n s a t i s f a c t o r y r e s u l t s  8 and  The  Should  the reduced  step s i z e  (a higher f o r e c a s t e r r o r  standard d e v i a t i o n i n t h i s a p p l i c a t i o n ) , the search In F i g u r e 2,  finally  i s terminated  (block 1 0 ) .  C = .78.  i n f l u e n c e of the s t e p s i z e r e d u c t i o n f a c t o r  The  continue  t h i s occurs a t A =  .02, on  the s e l e c t i o n of the smoothing constants i s an area t h a t w i l l considered  later.  G e n e r a l l y speaking,  the s t e p s i z e f o r t h i s  application  can be any number i n the open i n t e r v a l between 0 and reasons  be  1.  The  f o r t h i s i s that a f t e r u n s a t i s f a c t o r y r e s u l t s follow  from a p a r t i c u l a r p a t t e r n move and  i f subsequent e x p l o r a t o r y  moves do not y i e l d b e t t e r r e s u l t s then the step s i z e r e d u c t i o n f a c t o r i s m u l t i p l i e d by the p a t t e r n s e a r c h step s i z e to y i e l d to a new  p a t t e r n search step s i z e  (a s m a l l e r one).  s i z e r e d u c t i o n f a c t o r were 0 then the new s i z e would a l s o be 0 and e r a t e and  stop.  consequently  I f the s t e p  p a t t e r n s e a r c h step  the process would degen-  S i m i l a r l y , i f the value were 1 then the  new  p a t t e r n s e a r c h step s i z e would be the same as the o l d one and process would enter i n t o an i n f i n i t e ing  the  s e r i e s of i t e r a t i o n s attempt-  to reduce the c u r r e n t step s i z e .  D i r e c t i o n Change The illustration  example shown i n F i g u r e s „6A and  6B p r o v i d e s a good  of the p a t t e r n search procedures  a b i l i t y to change  44  •v, r  •a +  F i g u r e 6A Trend Smoothing Constant (C)  .OS +  P a t t e r n Search: Change o f Direction  .ow 4.  I  .ta. .tn .oi, .0% ,ia .is ,\k> Weighted Average Smoothing Constant (A)  AM  F i g u r e 6B  (650.?)  *  Trend * Smoothing Constant (C) .  x Cssa.H)  , J>oim at .  w  P a t t e r n Search: Change of Direction  x<l  •DM  4  •02-  .OS  .OL  .08  .ft  ,U  ,\H  Weighted Average Smoothing Constant (A) -—x X (  P a t t e r n Moves E x p l o r a t o r y Moves ) Forecast E r r o r Standard D e v i a t i o n  45 the d i r e c t i o n o f the search. left  These f i g u r e s present  hand p o r t i o n o f F i g u r e 2 i n more d e t a i l .  the lower  In t h i s  case,  the l a s t p a t t e r n move l e a d s the search to the p o i n t (A = . 0 0 , C = .00).  The e x p l o r a t o r y moves about t h i s p o i n t ( p o i n t #4 i n  F i g u r e 6A) produce r e s u l t s that are i n f e r i o r to those by the l a s t s e t o f e x p l o r a t o r y moves ( p o i n t #1-3) s e a r c h r e t u r n s to the best previous i n F i g u r e s 6A and 6B).  •  I t then conducts a s e r i e s o f e x p l o r a t o r y  ending a t p o i n t #5 i n F i g u r e 6 B .  important  a t p o i n t #4  T h i s new s e t o f p a t t e r n  moves l e a d s r a p i d l y away from p o i n t #3, smaller A co-ordinates  Thus the  e x p l o r a t o r y move ( p o i n t #3  moves and e s t a b l i s h e s a new p a t t e r n move b e g i n n i n g and  provided  t e n d i n g towards s l i g h t l y  and much l a r g e r C c o - o r d i n a t e s .  to note that because a p a t t e r n move begins  It i s  from the  best e x p l o r a t o r y move, the d i r e c t i o n of the moves need not be i n a s t r a i g h t l i n e , but can be a l t e r e d s l i g h t l y to allow f o r d i r e c t i o n a l changes.  As an example, the d i r e c t i o n of the p a t t e r n move  s h i f t s s l i g h t l y a t p o i n t #7,  t a k i n g advantage o f the i n f o r m a t i o n  p r o v i d e d by the l a s t p a t t e r n move. General  D i s c u s s i o n o f P a t t e r n Search Hook and Jeeves ( 6 ) r e p o r t that i n a c t u a l p r a c t i c e , p a t t e r n  search has proved p a r t i c u l a r l y s u c c e s s f u l i n l o c a t i n g minima on hypersurfaces  which c o n t a i n "sharp v a l l e y s . "  On such s u r f a c e s  c a l procedures behave badly and can only be induced  classi-  to reach the  In cases where e x p l o r a t o r y moves i n v o l v e A and C values o u t s i d e o f the i n t e r v a l from 0 to 1, e.g. A= - . 0 1 and C= .00 then an a r b i t r a r i l y high f o r e c a s t e r r o r standard d e v i a t i o n value o f 99999 i s a s s o c i a t e d w i t h such c o - o r d i n a t e s to d r i v e the s e a r c h back i n t o the c o r r e c t l e v e l .  46 minimum s l o w l y .  D i r e c t search procedures u s i n g only  simple  moves are f o r c e d i n t o s m a l l step s i z e s i n order to keep from moving out o f the v a l l e y on each s t e p . f a s t e r than c l a s s i c a l techniques, are n o t o v e r l y f a s t .  Consequently, though  such d i r e c t search procedures  P a t t e r n search has the inherent  potential-  i t y o f making p a t t e r n moves d i r e c t l y down the v a l l e y , and hence r a p i d l y approaching the minimum (or maximum).  Another advantage  i s t h a t p a t t e r n search i s w e l l adapted to use on e l e c t r o n i c computers, s i n c e i t uses repeated  identical arithmetic  operations.  C l a s s i c a l methods, developed f o r human use, o f t e n s t r e s s minimizat i o n o f a r i t h m e t i c by i n c r e a s e d s o p h i s t i c a t i o n o f l o g i c , a goal which may not he d e s i r a b l e when a computer i s used.  Pattern  search a l s o p r o v i d e s an approximate s o l u t i o n , improving w h i l e , a t a l l stages o f c a l c u l a t i o n .  a l l the  T h i s f e a t u r e can be  when a t e n t a t i v e s o l u t i o n i s needed before  important  the c a l c u l a t i o n s are  completed. On the other hand, as Van Wormer and Weiss (10) p o i n t out, the technique  does have some l i m i t a t i o n s .  I t can o n l y be  used i n d e t e r m i n i s t i c r a t h e r than s t o c h a s t i c search problems. Other l i m i t a t i o n s a r e that a s o l u t i o n i s not guaranteed i n a finite  time and, any s o l u t i o n s d i s c o v e r e d by the procedure are  not guaranteed to be the minimum (or maximum).  The nature o f  the search procedure i s such that i t may a r r i v e and stop a t a l o c a l minimum ( o r maximum) p o i n t . T h i s concludes p a t t e r n search method.  the chapter  on the workings of the  As was mentioned e a r l i e r ,  this material  has been presented to provide an a i d to understanding the search system. will  In t u r n i t i s hoped that t h i s  understanding  a i d the r e a d e r i n understanding and e v a l u a t i n g the mean-  i n g and consequences o f the a n a l y s i s presented i n t h i s The  study.  f o l l o w i n g chapter i s intended to p r o v i d e a more s p e c i f i c  type o f i n f o r m a t i o n . used i n the study.  I t deals w i t h the data that was a c t u a l l y  48 CHAPTER IV Data:  Source and Selection  The purpose of t h i s chapter i s to give a description of the industry from which the data series used i n this study's analysis were taken.  By doing so i t i s hoped that the reader  w i l l gain a better understanding of the context within which the pattern search and exponential smoothing techniques were applied.  I w i l l also discuss the rationale behind the s e l e c t i o n  of the s p e c i f i c data series from the much l a r g e r set available to t h i s study. Data Source The three data series that form the basis for the present study were selected from among a t o t a l of t h i r t y - e i g h t series made available by the Frazer Valley Milk Producer's Association.  In an aggregate sense the series are composed  of units sales of f l u i d milk segregated according to container size, butterfat content and by d i s t r i b u t i o n channel.  The data  represents sales of f l u i d milk f o r the period January 1966 to December 1971 i n c l u s i v e .  Unit sales were used as opposed to  d o l l a r sales so as to eliminate the problem of price changes over time.  I f d o l l a r sales had been used and not adjusted to  account f o r price increases there would be an upward bias i n the data that would tend to d i s t o r t the trend factor i n the various series.  49  Dairy  Industry This section draws heavily on a paper prepared by the  Federal Task Force on Agriculture ( 5 ) '  While discussion deals  with the dairy industry on a national l e v e l , the authors of that paper f e l t that many of the trends are general be applicable to s p e c i f i c provinces.  enough to  I w i l l deal i n added  depth with the processing sector of the industry as t h i s i s the sector within which the Frazer Valley Milk Producer's Association operates. The Canadian dairy industry has at l e a s t as many problems as most industries i n the a g r i c u l t u r a l sector. i c conditions f o r milk production than i n most countries.  i n Canada  The domestic per capita consumption of  milk i n a l l forms i s f a l l i n g . population,  are l e s s favourable  Climat-  Even with a rapid increase i n  t o t a l Canadian consumption of milk i n a l l forms i s  increasing only s l i g h t l y .  In addition, substitutes  threaten  to continue to erode the f l u i d milk producers' markets. ments which should have occurred  Adjust-  on farms and i n processing  plants have been slow i n coming, p a r t l y because of government p o l i c i e s which have attempted to protect and maintain a type of a g r i c u l t u r a l production  i n which Canada has a marked disadvantage.  Throughout the post-war years (1946-1972) federal dairy p o l i c i e s have supported manufacturing milk and cream prices by o f f e r s - t o purchase programs, by embargoes on v i r t u a l l y a l l dairy imports except s p e c i a l t y cheeses, and by other forms of subsidization. Support programs have provided  seasonally stable prices, but  50 the year-to-year changes i n dairy programs have created uncertainties f o r investment  i n the entire industry.  These  economic conditions have retarded the rate of structural adjustment  both i n the producing and processing sectors.  At present there are about 110,000 manufacturing milk and cream shippers of whom about 78,000 ship l e s s than 100,000 pounds of milk (equivalents) annually, and about 21,000 f l u i d milk shippers of whom nearly a l l ship over 100,000 pounds of milk.  Except f o r those small scale producers who have l i t t l e  alternative use f o r the few resources they devote to dairying, and the largest scale producers (predominantly f l u i d shippers) who  have obtained substantial economies of scale, primary dairy  production i n Canada i s characterized by high costs.  There i s  a s i m i l a r s i t u a t i o n i n the composition of the processingd i s t r i b u t i n g sector.  Of the close to 1,300 dairy f a c t o r i e s ,  about two-fifths are r e l a t i v e l y small with annual sales of l e s s than $250,000.  Both primary and secondary sectors are  characterized by wide d i s p a r i t i e s i n l e v e l s of technology and average costs.  The smaller processors are expected to continue  facing serious f i n a n c i a l problems. The composition of the B r i t i s h Columbia dairy industry can be seen i n Table I. T r a d i t i o n a l l y B.C.  has had a dairy  industry heavily oriented to the f l u i d milk market. 70 percent of B.C.'s milk production i s u t i l i z e d  i n f l u i d sales  as compared to the national average of 11 percent. hand only 10 percent i s u t i l i z e d  Approximately  On the other  f o r manufactured dairy products  51  compared to a n a t i o n a l average o f  36 percent.  p a t t e r n i s expected to continue through the  This  utilization  1970's.  TABLE I Shipping.Volume D i s t r i b u t i o n , 1 9 6 6  British Columbia  S h i p p i n g Volume l b s . per annum  Cream Mfg. F l u i d percent of a l l d a i r y farms  under 4 8 , 0 0 0 48,000 - 95,999 9 6 , 0 0 0 and over  18  Total Canada  under 4 8 , 0 0 0 48,000-95.999 9 6 , 0 0 0 and over  Total * l e s s than . 5  22 20 58  (-)* 15 55  20  10  70  43 8 3  12 10  14  (-)* 2 9  54  36  11  100 54 20 26 100  percent  Sources The  4 3 3  2 (_)*  Total  see  Bibliography  No.  5  p r o c e s s i n g - d i s t r i b u t i n g s e c t o r o f the Canadian d a i r y  i n d u s t r y comprises c l o s e to 1 , 3 0 0 f a c t o r i e s or p l a n t s owned by n e a r l y h a l f as many companies.  A l a r g e p a r t of the s e c t o r i s  s t i l l made up of companies p r o c e s s i n g single plants.  Large s c a l e and m u l t i p r o d u c t  by major c o r p o r a t i o n s and  have t h e i r own  product l i n e s .  b u t t e r or cheese i n small p l a n t s , operated  which s e l l a wide range of d a i r y products  brand names, are  Through mergers and  i n t e g r a t i n g the s e c t o r consolidations  across  the degree of  52  c o n c e n t r a t i o n i n the s e c t o r i s i n c r e a s i n g markedly. Changes i n technology and favoured l a r g e volume p l a n t s . merchandising  New  forms of packaging  as w e l l as changes i n the s t r u c t u r e of  have had a d i r e c t and and  i n d u s t r i a l s t r u c t u r e have  e q u a l l y important  s i z e of these p r o c e s s i n g f i r m s .  impact  and competition  on the number  Condenseries,  process  cheese p l a n t s , and the l a r g e r i c e cream p l a n t s , which  typically  have been operated by major c o r p o r a t i o n s , are faced w i t h c o u n t e r v a i l i n g power of the r e t a i l retail  chains.  The  impact  the  of the  chains on d a i r i e s , has had f a r g r e a t e r consequences.  The  retail  chains have o f f e r e d consumers lower p r i c e s f o r milk  and  other d a i r y items, and a g r e a t e r choice of c o n t a i n e r s i z e s .  More r e c e n t l y c o m p e t i t i o n at the r e t a i l l e v e l has been  heightened  by the emergence of m i l k s p e c i a l t y s t o r e s i n major c i t i e s , by means of h i g h volume s a l e s and l o n g e r s t o r e hours, milk  i n 3 quart  requirements  jugs at a lower p r i c e .  which  offer  The l a r g e c a p i t a l  f o r modern p a s t e u r i z i n g and b o t t l i n g p l a n t s and  the need to meet the demand f o r d i v e r s i f i e d s i z e s and types of c o n t a i n e r s and types of products have put d a i r i e s i n a weaker competitive p o s i t i o n .  In a d d i t i o n , the b a r g a i n i n g s t r e n g t h of  the supermarkets, which are a c c o u n t i n g f o r an i n c r e a s e d  propor-  t i o n o f the d a i r i e s s a l e s , has put great pressure on the to  expand t h e i r businesses or to s e l l  out to other  distributors.  A c c o r d i n g to the F e d e r a l Task Force paper ( 5 ) trends i n milk p a s t e u r i z i n g p l a n t s show an average  dairies  the  annual  c l o s i n g of 2 5 p l a n t s or about a 3 percent per year drop i n the  53  number of operating plants.  Over the past eight years there  has been no s i g n i f i c a n t change i n t o t a l employment.  The growth  i n sales per plant has only r i s e n by about 5 percent annually. This change i n structure i s most pronounced i n urban centers. The degree of industry concentration has increased s i g n i f i c a n t l y i n the post-war period.  The implications of these trends are  that i n the long run the processing-distributing sector w i l l be completely integrated across product l i n e s , and operated by a small number of large corporations and co-operatives. At t h i s stage i n the process and i n the foreseeable future the degree of competition w i l l be high and margins w i l l be r e l a t i v e l y low. As i n the case of B r i t i s h Columbia where a provincial milk board administers r e t a i l prices, the board  effectively  determines the marketing margin for f l u i d milk products. Evidence suggests that these margins are set to cover the costs of the l e a s t e f f i c i e n t d i s t r i b u t o r s and thus serve to reduce price competition.  The existence of fixed margins for processing  d i s t r i b u t i o n provides considerable incentive f o r backward inte^ gration by chain stores into the processing f i e l d . Projections as to future consumption of milk i n a l l forms depend upon the assumptions made and type of analysis used.  The Task Force paper notes two projections, one made by  the Task Force i t s e l f and the other by Department of Agriculture. The Task Force study forecast a 9 percent growth i n t o t a l Canadian consumption i n the 1 5 years 1 9 6 5 to 1 9 8 0 , the Depart-  54  ment of Agriculture forecast a 14 percent growth i n the same period.  The main source of discrepancy between the two arises  from differences i n the treatment of 2 percent f l u i d milk. The Task Force estimates appear i n Table I I . They indicate that between 1 9 6 5 and 1980 per capita consumption of milk i n a l l forms w i l l decline by 18 percent but that t o t a l consumption w i l l r i s e by 9 percent because population growth.  While cheese  consumption i n 1980 i s expected to be more than double that of 1 9 6 5 . t o t a l consumption of other milk products w i l l  fall.  TABLE II Per Capita and Total Consumption of Dairy Products 1 9 6 5 . 1 9 6 7 and Projections f o r 1 9 7 5 and 1 9 8 0 , Assuming Constant Real Prices.  Per Capita Consumption i n Pounds of Products  1965  F l u i d Milk 275.0 Butter 18.5 Cheese 9.0 Other Milk Products 114.4  1967  267.5  16,9 9.9 114.5  1975  --pounds246. 0 14. 0 12.8 102.8  1980  233.0 13-1  14.4  1980  as a  % of  1965  84.7 70.8  161.1  99.6  89.4  Total Consumption i n Milk Equivalents 000,000  •pounds— F l u i d Milk Butter Cheese Other Milk Products Total  5,263 8,372  5,325  1,714 2,178  7,933 1,971 2,337  5,703 7,773 2,949 2,438  5,943 7,973 3,653 2,594  115-3 95-2 213.1 119.1  17,230  17,149  ,17.809  18,831  109.3  55  On the supply s i d e Table  I I I represents a p o r t i o n of  the Task Force p r o j e c t i o n s f o r the changes i n cow numbers, y i e l d l e v e l s , and milk s a l e s f o r the s a l e time p e r i o d . TABLE I I I Cow Numbers, Y i e l d L e v e l s and M i l k Sales f o r 1 9 6 5 and P r o j e c t i o n s f o r 1975 and 1980  Cow Numbers 1965 "  B.C. Canada  1975  Sales per Cow  1980  1965  1975  (000's)  Milk  1980  1965"  (lbs)  86 77 82 2,822 2 , 2 1 3 2,048  9,536 5,909  11,100 7,909  As can be seen i n the Table cows, both i n B.C. and Canada, w i l l  1975  1980  (millions 11,700 8,722  lbs)  =805 855 1 6 , 6 7 2 17,502  '.-842 17,863  the absolute number o f decline.  T h i s w i l l be  o f f s e t by marked i n c r e a s e s i n p r o d u c t i v i t y per cow. s a l e s w i l l be up over  Sales  While milk  the p e r i o d i n q u e s t i o n they w i l l  only  i n c r e a s e by a l i t t l e over 7 percent. Data S e l e c t i o n The  purpose o f t h i s s e c t i o n i s to d e s c r i b e the r a t i o n a l e  that was used i n s e l e c t i n g the three data s e r i e s that were used from the l a r g e r number that were a v a i l a b l e . to  s e l e c t from the o v e r a l l s e t those  complete.  By t h i s I mean those  The f i r s t  s t e p was  data s e r i e s which were  s e r i e s i n which data was  avail-  able f o r every monthly p e r i o d from January 1 9 6 6 to December 1 9 7 1 . Out  o f the 3 8 s e r i e s i n the i n i t i a l  requirement.  s e t only 2 0 met t h i s b a s i c  The purpose o f t h i s procedure was to i n s u r e  56 c o n s i s t e n c y between the s e r i e s and thereby to a l l o w comparisons to be made.  By making sure t h a t each data s e r i e s used  was  composed of 72 p e r i o d s the problem o f v a r i a b i l i t y i n the number of periods used to determine  the constant v a l u e was e l i m i n a t e d .  A l s o by doing t h i s each of the data s e r i e s r e p r e s e n t e d the same time p e r i o d . The second  s t e p was to p l o t each s e r i e s on a very  coarse g r i d i n order to get a rough f e e l i n g f o r the s i z e and d i r e c t i o n o f the t r e n d and the degree of random between i n d i v i d u a l p e r i o d s .  fluctuation  The purpose o f t h i s was to t r y  to s e l e c t from the s e t a data s e r i e s f o r each of the trend types r i s i n g , f a l l i n g and s t a b l e .  There was an attempt  to s e l e c t a r e l a t i v e l y c l e a n s e r i e s f o r each, low.'random f l u c t u a t i o n s .  i n doing t h i s  i . e . one w i t h  fairly  The i d e a behind u s i n g three separate  s e r i e s f o r the v a r i o u s t r i a l  runs on the p a t t e r n search process  was to see i f there was any d i f f e r e n c e  i n the techniques  a b i l i t y to s e l e c t constant v a l u e s f o r v a r i o u s types of t r e n d s . The attempt  to minimize  the random f a c t o r was to i n c r e a s e the  c o m p a r a b i l i t y of the three s e r i e s by r e d u c i n g influences. visual basis. accomplished  In eachcase  extraneous  the s e l e c t i o n was done on a p u r e l y  Once the i n i t i a l  s e l e c t i o n of 9 series  was  the next step was to more a c c u r a t e l y p l o t the s e r i e s  i n order to make a f i n a l s e l e c t i o n .  At t h i s stage the PLOT  r o u t i n e s compiled by the U.B.C. Computing Center were  used.  Again on a v i s u a l b a s i s the same c r i t e r i o n of t r e n d and degree  57 o f random f l u c t u a t i o n were used. was  The r e s u l t  of this  the s e l e c t i o n o f the t h r e e data s e r i e s l i s t e d  R i s i n g Trend - ( 2 Quart Wholesale 600  108 486 1,240 2,227 2,343 2,965 4,075 4,158  64 2 04 534 1,279 2,301 2,851 2,515 4,129 4,289  738 192 491 1,341 2,2 09 2,551 3,726 4,i4o 5,525  667 141 -591 1,702 2,244 2,843 3,733 3,974 4,443  F a l l i n g Trend - ( l Quart 574769 460373 372176 327378 247858 245087 221610 284984 257794  510639 470456 348348 308466 257293 220530 362808 302933 235652  111352 132229 123528 124197 110619 119697 120634 114781 32428  F o r a graph  594 226 556 1,511 1,947 3,103 3,330 3,580 4,423  Retail  526489 482295 330660 308882 242758 214127 351441 288486 19444  123339 129546 124755 117636 122181 83750 107147 124755 31671  122281 138445 120333  122422 119546 95055  H636O 98810 34430  459 251 1,118 1,610 1,698 1,994 3.985 3,937 4,818  519697 466049 346101 310063  265143 221125  324770 285604 21180  474330  365637 342330  280988 244302 228845 301804 259205 2 0904  117136 128749 131781 114896 126026 IO8323 114096 35044 35046  Carton  Shipping)  297 306 1,057 2,450 2,4l4 2,854 3.756 3,860 4,l6l  Carton) 421067 391404 341494 262749 258999 213942 2 82703 285006 21843  3.895 B u t t e r f a t  116527 134002 132671 116971 116882 98567 116125 43383 33146  below.  471 2 88 1,075 2,044 2,106 2,238 3.870 4,524 5,141  3 . 2 $ Butterfat  - ( 2 Quart Wholesale  S t a b l e Trend 121085 129930 124665 124569 129979 112291 118079 II6361 36350  558940 470608 317454 317440 251992 220600 374502 277750 90051  2.0% B u t t e r f a t  process  Carton)  117326 120016 131747 109114 111753 113988 108220 32472 35473  o f these t h r e e s e r i e s see Appendix A.  435102 358698 332260 234490 24862 8 226993 219985 265072 24114  127116 136481 131102 117715 116678 110252 117774 36151 27902  58 CHAPTER V A n a l y t i c a l Methodology The  purpose o f t h i s chapter i s to provide the reader  w i t h a d i s c u s s i o n o f the r a t i o n a l e and procedure a n a l y z i n g the response  used i n  c h a r a c t e r i s t i c s o f the p a t t e r n search -  e x p o n e n t i a l smoothing f o r e c a s t i n g system.  I will  start o f f  w i t h a d i s c u s s i o n o f the approach used i n s e t t i n g the i n i t i a l v a l u e s f o r the s a l e s average (BEGAVG), the t r e n d estimate (BEGTND) and the seasonal f a c t o r s (BSEAS(N)). present the r e a s o n i n g behind used i n t h i s study.  Next I w i l l  the s e l e c t i o n o f the e r r o r measure  T h i r d l y w i l l be a d i s c u s s i o n o f the  standard v a l u e s used as a comparison f o r the experimental r u n results.  Lastly I will  present the methodology t h a t was used  i n the a n a l y s i s . Selection of I n i t i a l  Values  The method used to determine the i n i t i a l s a l e s average,  v a l u e s o f the  the t r e n d estimate and the seasonal  factors  remained the same f o r a l l three o f the s e r i e s t e s t e d . a d d i t i o n , once the values were determined, throughout  a l l the t r i a l s .  three i n p u t s to a n a l y s i s .  In  they remained  constant  I t was decided not to s u b j e c t these Due to the v e r y nature of a weighted  moving average method such as e x p o n e n t i a l smoothing, i n p u t s such as these l o s e t h e i r impact By the time the system reaches and  as the time p e r i o d s move  foreward.  the f i f t e e n t h or t w e n t i e t h p e r i o d  i s f o r e c a s t i n g i n t o the f u t u r e any changes i n these  initial  59  v a l u e s w i l l have a n e g l i g i b l e  effect.  To begin w i t h , the i n i t i a l simply by determining the average data s e r i e s .  T h i s was  value f o r the f i r s t  accomplished  twelve monthly p e r i o d s of the f i r s t sum  by twelve.  s a l e s average, was  computed  year i n the  by summing the values of the year and  then d i v i d i n g  this  A s l i g h t b i a s i s i n t r o d u c e d by t h i s procedure, i f  there i s a t r e n d i n the data s e r i e s . then the value of the i n i t i a l  I f the trend i s p o s i t i v e  s a l e s average  w i l l be b i a s e d upward.  The degree of b i a s depends on the s i z e of the t r e n d .  Of  course,  the opposite holds true f o r a negative t r e n d . I c a l c u l a t e d the i n i t i a l the f i r s t  trend estimate by s u b t r a c t i n g  value of the data s e r i e s from the l a s t value and  then  d i v i d i n g t h i s d i f f e r e n c e by the number of p e r i o d s i n the data series.  T h i s gave me a mean value f o r the t r e n d of the  T h i s mean v a l u e was  then used as an estimate of the i n i t i a l  By the time the s e r i e s reached initial  series. trend.  i t s l a t e r v a l u e s the e f f e c t of the  estimate w i l l have been d i l u t e d by the smoothing  procedure  so as not to n o t i c e a b l y i n f l u e n c e the r e s u l t s of the process. i s because of t h i s that an estimate c o u l d be used f o r the  It  initial  value. The  t h i r d i n p u t determined  was  the i n i t i a l  seasonal  for  each month of the seasonal c y c l e .  was  a year and hence a seasonal f a c t o r had to be developed  month.  factor  In t h i s study the seasonal  The method used to develop a s e t of i n i t i a l  cycl  f o r each  values was  to  sum  6o values of the months i n a yearly cycle, then divide this sum hy 12.  This yields an average value f o r each year.  divided each monthly value hy the yearly average. the r a t i o that represents the seasonal factor. process f o r the f i r s t  I then This gave me  I repeated t h i s  two years to the data series, summed the  two values f o r the individual months, then divided hy two to determine  the mean value f o r the two years.  The twelve values,  realized from this procedure, were then normalized so that they summed to twelve.  The reason that I chose to use two years  rather than three or more, was simply one of convenience. as i n the other two values being determined,  Again,  the size and  d i r e c t i o n of the trend of the series influences the c a l c u l a t i o n of  i n i t i a l trend estimates.  I f the values of the series r i s e  then the seasonal factors representing the months at the l a t t e r end of the cycle w i l l be biased upwards r e l a t i v e to those months at  the beginning of the cycle.  Error Measurement As I mentioned e a r l i e r i n my discussion of exponential smoothing, any of a number of measures can be used to track the accuracy of the smoothing process.  The reason the standard  deviation of the forecast error was chosen i s that i f a forecast system i s to be used i n a real l i f e application then the method of  error measurement should be linked to p r o f i t .  I f , as i s true  in many business situations, a large mistake i s more than proport i o n a l l y "costly" than a small one then the error measurement of  61 the forecast system should r e f l e c t t h i s non-linearity.  As  an example, the mean absolute error, which i s a l i n e a r measurement scale, can only appropriately be used where a 10 point i n accuracy i n a forecast i s only twice as "bad" error.  The  as a 5 point  standard deviation of the forecast error i s not  l i n e a r and can appropriately be applied to situations where bigger errors are more than proportionally costly than small ones.  Because most p r o f i t situations are not l i n e a r i n t h e i r  reaction to error I f e l t that the use of this measure would be more appropriate. to l i n k my  It should be noted that I made no attempt  error measure with the a c t u a l i t i e s of the dairy  industry. Standard Run  Values  As part of my methodology I developed a set of standard values which when used as inputs into the pattern search process yielded a series of benchmark values against which other experimental runs could be evaluated. standard value sets.  In a c t u a l i t y there were two  The f i r s t was  used i n analyzing changes  in the pattern search parameters, e.g. etc.  The  pattern search step s i z e ,  second, which resulted from my  meter changes, was  analysis of the para-  used i n the analysis of the changes i n the  beginning values f o r the smoothing constants. The values that I used for the f i r s t set of standards can appropriately be described  as " t r a d i t i o n a l " ones.  They are  values which are t y p i c a l l y used i n a pattern search application  62  where more " e f f i c i e n t " values are unknown. listed  The values are  belowt Maximum number of pattern moves (MAX) Pattern search step size (DELO) Step size reduction factor (RMO) Minimum step size (D) Starting Sales Average Smoothing Constant Starting Trend Smoothing Constant (B) Starting Seasonal Factor Smoothing Constant (C)  50 .05 .500 .01 (A).500 .500 .500  When applied to the three sets of data used i n this study these values yielded! 1)  minimum error value  2)  t o t a l number of i t e r a t i o n s used i n a r r i v i n g at the minimum error value  3)  value f o r each of the smoothing constants -A, B and C.  Subsequent changes were then compared against these figures to determine  i f the change had resulted i n improved r e s u l t s .  As I have just mentioned the second set of standard values were derived from the analysis of changes i n MAX, DELO, D, and RH0.  After I had completed  the analysis I determined  which values had yielded the best results and used these values i n my analysis of changes i n A, B and C.  The values I used  were: Maximum number of pattern moves (MAX) Pattern search step size (DELO) Step size reduction factor (RHO) Minimum Step Size (D) Again as before, when these were applied to the data computations  100 .1 .5 .001 used,the  resulted i n a number of figures which I subsequently  63  compared a l l changes against. Methodology I intend to discuss i n t h i s section the approach taken for  the s e n s i t i v i t y analysis of the smoothing systems parameters.  E s s e n t i a l l y the method involved a t r i a l and error approach to the s e l e c t i o n of the input values. It should he noted here that a l l of the analysis performed was  f a c i l i t a t e d by the use of a previously developed pattern  search-exponential  smoothing computer program.  The computer  program "evolved" i n several stages over a period of three years.  People associated with i t s development were* W. Berry - Purdue University J. Wilcox - Indiana University D. Weiss - University of B r i t i s h Columbia Given that I had developed a base l e v e l against which  to make comparisons then I proceeded to change the input values one at a time.  I used a "one at a time" approach so that I  could e f f e c t i v e l y isolate the r e s u l t s of any changes made and hence determine the cause-and-effect  r e l a t i o n s h i p . The f i r s t  change I made was to extend the length of the pattern search process.  I did this by increasing the value of MAX (maximum  number of pattern moves) to 100 and by decreasing D (minimum step size) to .001.  Together these two parameters form the  boundary that terminates  the pattern search process.  The purpose  of t h i s move was to get a more complete f e e l i n g for the response surface of the search process as i t applied to my data s e r i e s .  6k I also wanted to find out i f I was prematurely terminating the process before i t arrived at the optimum constant values.  I  subsequently incorporated these changes into the rest of ray trials. The second series of changes that made were changes i n DELO (pattern search step s i z e ) .  The values weres . 0 1 , . 1 ,  each with a different value. .2,  .3.  I made f i v e experimental runs, .15,  The purpose of t h i s series of changes was to see i f  either the smoothing constant values, minimum error value or number of  iterations  were sensitive to search step sizes.  As a result of these t r i a l s I selected a value of DELO = .1 as the most e f f i c i e n t value f o r use i n a l l subsequent experiment a l runs. The next series of changes made were on RHO reduction f a c t o r ) . .250,  was  .400,  .600,  (step size  Again I t r i e d f i v e different values .750).  (.100,  The reason for these experimental runs  the same as those above for DELO.  I wanted to see the  e f f e c t of these changes on the smoothing constant values, the minimum error value and the number of i t e r a t i o n s .  The results  of these changes l e d me to select the value of RHO  that had  been used i n the standard run remainder of my  (.500)  and use t h i s value f o r the  analysis.  Up to t h i s point I had been keeping the smoothing constant values unchanged.  Now  that I had developed what  appeared to be the best set of pattern search parameter values. I applied the analysis to the smoothing constants.  The previous  65  analysis had given me values that I used as a standard f o r t h i s second part of the analysis. MAX =  DELO = RHO = D=  A = B =  C=  These values were;  100 .100 .500 .001 .500 .500 .500  Because I was working with three smoothing constants (A, B and C) I took a two-pronged approach.  F i r s t l y , I made two changes  that were consistent f o r a l l three constants. were i n i t i a l l y  A, B and C  a l l given values of . 2 5 0 and then were a l l  given values of . 7 5 0 .  I used t h i s consistent approach because  i n most cases of p r a c t i c a l application the user i s not aware of the range within which the best constant values might l i e and consequently usually w i l l give a l l three constants the same value.  The second set of changes that were made were  based on the consistency of the constant value results over a l l of the previous analysis.  Because the A, B and C values  remained almost constant over a l l the previous experimental  runs  that were made i t was decided to see what would happen i f the i n i t i a l constant values were put near the solution constant values.  Given that the three data series each yielded d i f f e r e n t  r e s u l t s I gave them d i f f e r e n t i n i t i a l values.  For the r i s i n g  series I assigned the values A = . 2 0 0 , B = . 6 0 0 , C = . 2 0 0 while for both the f a l l i n g and stable series I assigned the values A = .800, B = .800 and G = . 2 0 0 as i n i t i a l values.  66  Table IV gives a l i s t i n g of the t r i a l runs made and the  values associated with each t r i a l run.  As can be seen both  TABLE IV Input Values on an Experimental Run Basis Run Standard 2nd Run 3rd Run 4th Run 5th Run 6th Run 7th Run 8th Run 9th Run 10th Run 11th Run 12th Run 13th Run 14th Run *15th Run 116th Run  * -V  MAX  DELO  50  .05 .05  100 100 100 100 100 100 100 100 100 100 100 100 1*00 100 100  .500 .500 .500 .500 .500 .500 .500 •750 .250  -.1 .01  .15  .2  ±1 .1 .1 .1 .1 .1 .1 .1 .1 .1  \00  D  RH0  .4oo  .01 .001 .001 .001 .001 .001 .001 .001 .001 .001 .001  .500 .500 .500 .500  .001 .001 .001 .001  .100  .001  .600  .500 .500 .500 .500 .500 .500 .500. .500 .500 .500 .500 .500 .250 .750 .800 .200  . 001  • 5oo  .\  A  B  .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .250  •750 .800  .600  C  .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .500 .250 .750 00 .200  .2  These values were applied only to the atahlie aa4 f a l l i n g serie These value s were applied only to the r i s i n g series  N.B .  11  11  n  11  H  i>  The underlined figures represent the parameter input under consideration during an experimental run.  from the Table and my description of the a n a l y t i c a l process I t r i e d to base each change on the degree of success or f a i l u r e that I had noted i n the previous moves.  Because I followed this  rule-of-thumb procedure I only analyzed a rather narrow range of values when compared to the p o s s i b i l i t i e s open to me. on purpose.  I did this  Given the usual constraints on time, computer  resources, etc. I f e l t that the analysis had to be approached i n  67 some s o r t of l o g i c a l manner.  By f o l l o w i n g a l o n g what l o o k e d  l i k e good l e a d s or d i r e c t i o n s I attempted to maximize the  results  of the e f f o r t i n v e s t e d and minimize p r o d u c t i o n of r e s u l t s w i t h l i t t l e value.  By u s i n g a more e x t e n s i v e "blanket" approach I  would have covered areas t h a t would not have been  worthwhile.  68 CHAPTER VI Data A n a l y s i s and  Conclusions  As has been mentioned s e v e r a l times before the purpose of  t h i s study was  to undertake a s e n s i t i v i t y a n a l y s i s of a  number of i n p u t s i n t o the p a t t e r n search technique a p p l i e s to e x p o n e n t i a l smoothing.  Consequently  as i t  the focus of  t h i s chapter w i l l be on an a n a l y s i s of the data generated the study.  Each of the inputs t h a t were changed are d e a l t w i t h  on a separate b a s i s . and  The maximum number of p a t t e r n moves  the minimum step s i z e  (D) are d i s c u s s e d f i r s t .  t h i s i s the p a t t e r n search s t e p s i z e (DELO). size reduction factor  (RHO)  will  be d i s c u s s e d .  Following  As i n the standard v a l u e s .  Next, based on these standard v a l u e s the experimental i n v o l v i n g the e x p o n e n t i a l constants w i l l t h i s the f i n a l  (MAX)  Then the step  methodology t h i s a n a l y s i s w i l l l e a d to some new  ing  by  runs  be d i s c u s s e d .  Follow-  c o n c l u s i o n s o f the study w i l l be drawn.  At  each stage of the a n a l y s i s the three separate time s e r i e s , t h a t were used, w i l l be compared f o r d i f f e r e n c e i n r e s u l t s . Maximum Number of P a t t e r n Moves and Minimum Step S i z e Given that i n the i n i t i a l D = .01 to ing of  100  the f i r s t  experimental  standard r u n MAX  = 50  and  r u n c o n s i s t e d of i n c r e a s i n g MAX  and d e c r e a s i n g D to .001.  T h i s had  the e f f e c t of  the t e r m i n a l l i m i t s on the p a t t e r n s e a r c h p r o c e s s .  extendIn each  the three time s e r i e s that were used the e f f e c t of the  initial  69 TABLE V L i s t o f Experimental Run Values f o r a l l Time S e r i e s * Experimental Run  MAX  PELO  RHO  Standard  50 100 100 100 100 100 100 100 100 100 100 100  • 05 .05 .10 .01 .15 .20 •30 .10 .10 .10 .10 .10  .500 .500 .500 .500 .500 .500 .500  2nd 3rd 4th 5th 6 th  ?th  8th 9th 10th 11th 12th  For  .750 .250 .100  .4oo  .600  D .01 .001 .001 .001 .001 .001 . 001 .001 . 001 .001 .001 .001  the next s e r i e s o f experimental runs the f o l l o w i n g v a l u e s  were h e l d as standards MAX = 1 0 0 DELO = . 1 0 RHO = . 5 0 0 D = . 0 0 1  .250 .750  13th  14th +I5th il6th **17th  .800 .200 .500  + =fe ** *  .250 .750 .800 .600 .500  .250 .750 .200 .200 .200  Values used f o r vj^nj^g time s e r i e s o n l y Values used f o r f a l l i n g time s e r i e s only Values used f o r s t a b l e time s e r i e s o n l y The u n d e r l i n e d value i n each r u n was the change made f o r that r u n .  v a l u e s o f MAX and D was to have the s e a r c h process terminate when the minimum step s i z e was reached.  MAX never  effectively  a c t e d as an upper bound on the number o f p a t t e r n moves p e r m i t t e d . On the f i r s t  experimental r u n the same h e l d true a g a i n .  In each  of the three cases the search was terminated by reaching the minimum step s i z e . Given that the computer printed the results of the search process i n such a format that every pattern move and i t s associated error and smoothing constant values were a v a i l able i t was decided to use the new values (MAX = 100, for the remainder of the study. move.  .001)  D =  This was an e f f i c i e n c y oriented  It enabled a l l subsequent moves to be compared both to  the standard run and the second experimental run.  For a complete  l i s t i n g of a l l of the experimental runs r e f e r to Table V. The effects of the changes i n MAX great.  and D are not very  In the 2nd experimental run of the r i s i n g series this  extension of the search process only reduced the error factor from a value of 43.611 i n the standard run to 43.6l?. v i r t u a l l y inconsequential change.  This i s a  In order to achieve this  reduction the search process had to increase the number of pattern moves made from 16 i n the standard run to 24 i n the experimental run.  This i s a rather large increase given the  small decrease i n error. changed very l i t t l e also. values of A = .187,  The values of the exponential constants The standard run yielded solution  B = . 6 l 8 and C = .012.  run solution values were A = .190,  The experimental  B = 6l9 and C = .011.  Again,  this i s a v i r t u a l l y inconsequential change. This above pattern of results held true f o r every other experimental run and was consistent across the time series.  71 Without going into the actual results of the remaining 16 runs i t can be said, without exception, that error value and exponent i a l constant values benefited very l i t t l e from extending the search process to a minimum step size of .001. Figure 7 i s i l l u s t r a t i v e of why the extentions resulted' in such minor improvement.  By f a r the largest portion of error  value reduction and associated constant value change takes place within the pattern moves that are linked to the f i r s t two step sizes i n the search process.  In other words, the pattern moves  associated with the i n i t i a l step size value and the f i r s t  reduc-  t i o n of i t , account f o r most of the changes i n the r e s u l t s . Indeed, the bulk of this improvement i n error and constant values occurs before any step size reduction takes place. l i t t l e change takes place after these i n i t i a l moves.  Very  As Figure  7 shows the 8 pattern moves that are associated with a DELO of .05,  reduced the error value from 97.404 to 44.304.  The  remaining 16 pattern moves that account f o r the rest of the step size reduction only further reduce the error value to 43.617. An obvious conclusion i s that an extension of the pattern search process i s not a worthwhile move.  Of course, such a  statement must be taken within the context of the time series used, the importance of forecast accuracy and the costs incurred in extending the search process.  The existence of t h i s  response  c h a r a c t e r i s t i c of the pattern search process i s quite important as i t had a marked impact on many other changes that were made In the system during the various experimental runs.  This area  w i l l be discussed i n greater d e t a i l i n the appropriate sections.  72  100  % PATTERN MOVE"}  90 -•  Error Value Io0  1 ?rVn£R|0  e\ove i  x  40  . c>?>  l  .oa«o  .ora&  x  -+-  -+-  H  .oowa  .oo'iv .M>\U  \-  .cor/a  Pattern Search Step Size (DELO)  Figure 7  Pattern Search Step Size and Associated Error Value (Rising Time Series - 2nd Experimental Run)  73 Pattern Search Step Size This section deals with those changes that were made i n the pattern search step size (DELO).  As Table V shows DELO  was given a series of values ranging from .01 to .30. The purpose of making these changes was to see i f any consistent patterns of response developed.  One point that should be  mentioned here i s the i n t e r r e l a t i o n s h i p of the pattern search step size (DELO) and the step size reduction factor (RHO) • As the search process proceeds DELO i s multiplied by RHO to y i e l d a new DELO value f o r the following pattern and exploratory moves. This m u l t i p l i c a t i o n process occurs each time a step size reduction i s required. product of the two.  Changes i n either value w i l l affect the  This i n t e r r e l a t i o n s h i p was not explored  in this study because i t did not appear f r u i t f u l to do so. This w i l l be clearer as the results of the changes i n DELO and RHO are studied. To start o f f with, changes i n DELO produced very l i t t l e change i n either the minimum error value or the exponential constant values.  As Table VI shows the values f o r the exponen-  t i a l constants and the error factor remained very stable throughout the experimental '.runs.  The pattern search process seems  capable of consistently reaching the same (or very similar) end values no matter what step size was used. The major difference observed between the various values  TABLE VI Results From Changes i n DELO RISING SERIES Experi No. of Minimum mental Itera- Error Run tions Value  FALLING SERIES Constant Values A B C  No. of Minimum Itera- Error tions Value  STABLE SERIES  Constant Values A B C  No.of Minimum Itera- Error tions Value  Constant Values A B C  2nd  24  43.617  .190 .619 . Oil  18  3406.812  .822.  999 . 001  20  709.172  .532 .474 .025  3rd  16  43.617  .190 .617 .011  8  3405.810  .821 .999 .000  17  709.173  .532 .473 .025  4th  32  43.617  .191  31  3406.617  .822 .999 .000  23  709.172  • 533 .477 .025  5th  15  43.617  .190 .618 .012  16  3406.075  .822 .999 .000  18  709.170  .532 .474 .025  6th  14  43.617  .190 .617 .012  9  3405.807  .822 .999 .000  13  709.172  .532 .474 .015  7th  17  43.617  .191  .617 .012  14  3406.076  .822 .999 .000  18  709.170  .532 .474 .025  .618 . Oil  75  -\  •o\  \  \  .05  .10  \  as  1  \  .2o  Pattern Search Step Size (DELO)  Figure 8  Pattern Search Step Size and Associated Number of Iterations  r .^0  76  of DELO was the number of i t e r a t i o n s or pattern moves needed to arrive at the f i n a l values.  This i s a f a i r l y  important  point because, as a generalization, the lower the number of necessary pattern moves the greater the e f f i c i e n c y of the system.  Here e f f i c i e n c y refers primarily to computer time  costs. There i s not, however, a one-to-one r e l a t i o n s h i p between number of i t e r a t i o n s and computer time costs.  This  i s primarily due to the existence of base l e v e l s of compilation, input and output costs associated with computer usage. As Figure 8 shows there i s quite a b i t of v a r i a t i o n i n the number of i t e r a t i o n s , as a r e s u l t of changes i n DELO. For example, the number of i t e r a t i o n s f o r the f a l l i n g time series ranges from 8 when DELO = .10 to 32 when DELO = .01. I t i s i n t e r e s t i n g to note the consistency between the time series. The fact that the three of them move i n a f a i r l y consistent pattern strengthens  the conclusions with regards to generality.  Although the data presented here only represents three d i f f e r e n t time series and as such i s a very small sample, tentative conclusions can be drawn.  I t seems that the best values f o r  DELO l i e i n the middle ground of the range that was looked a t . Between .10 and .20 seems to be an area within which the number of i t e r a t i o n s i s minimized, given any p a r t i c u l a r l e v e l of error factor.  Given these results a value of DELO = .10 was chosen  to be used for the remaining experimental  runs.  77 Step Size Reduction Factor The focus of t h i s section i s on the results that were generated from changes i n the step size reduction factor (RHO). As with the other experimental runs the purpose of subjecting RHO  to a series of orderly changes was to see what the effects  of these changes would be on the error and exponential constant values and on the number of i t e r a t i o n s . As Table V shows RMO 'was given a series of s i x separate values. •750.  A l l of these values f a l l i n the range from .100  to  Even though the spread of this range of values i s quite  large the r e s u l t i n g changes i n the minimum error value and the exponential constant values are very small.  Indeed, the changes  i n these values are inconsequential. For an example, the minimum error value f o r the r i s i n g time series at 43.617 remained constant throughout the changes.  Table VII presents  a complete l i s t i n g of the results of these experimental runs. As that table shows, the exponential constant values also remained very consistent.  In v i r t u a l l y every case, the changes  that did occur took place at the third decimal place.  This i s  a pattern which turned out to be consistent across time series. The reason f o r these r e s u l t s can be traced to the nature of the responsiveness of the pattern search system.  As  was discussed e a r l i e r i n this chapter and as Figure 7 shows, most of the changes i n the error and constant values takes place before RHO  has a chance to have an e f f e c t .  Most of the change  takes place before the f i r s t step size reduction i s implemented.  TABLE VII Results From Changes i n RHo Rising Series Experi- No. of Minimum mental IteraError Run tions Value  F a l l i n g Series Constant Values A B C  3rd  16  43.617  .190 .617 . O i l  8th  33  43.617  .191  .619 .011  9th  18  43.617  .191  10th  13  11th 12th  No. of Minimum Itera- Error tions Value  8  Stable Series  Constant No. of Minimum Values Itera- Error A B C tions Value  Constant Values A B C  3405.810  .821  • 999 .000 17  709.173  • 532 .^73 .025  21  3405.808  .823  .999  .000 39  709.172  • 532 .474  .024  .617 .011  10  3405.808  .822  • 999 .000 14  709.170  .532  .025  43.617  .190 .619 . 012  13  3405.807  .822  • 999 .000 19  709.173  • 531 .473  .025  19  43.617  .190 .617 .011  10  3405.808  .822  • 999 . 000 19  709.170  • 532 .474  .o?5  22  43.61?  .191  .62 0 .011  19  3405.807  .822  .999  . 000 22  709.171  • 532 .^75 .025  -S3  79  The  r e s u l t of t h i s i s t h a t the p a t t e r n s e a r c h system w i t h  regards to the e r r o r and to  changes i n  constant value, i s v e r y  insensitive  .R'HO.  F i g u r e 9 d e p i c t s g r a p h i c a l l y the changes that occur in  the number o f i t e r a t i o n s when the step s i z e r e d u c t i o n f a c t o r  is altered.  The number of i t e r a t i o n s i s the only f a c t o r that  i s responsive  to changes i n RH9.  marked changes occur when RHO .600  or . 7 5 0 *  A S the graph shows the most  i s assigned l a r g e v a l u e s i . e .  When these v a l u e s are used the number of  i t e r a t i o n s r i s e s sharply.  The  s t a b l e s e r i e s almost  doubled i t s  = .750.  towards  normal number of i t e r a t i o n s when RHO  Looking  the other end of the graph i t can be seen t h a t the r e s p o n s i v e ness to s m a l l e r values i s more l i m i t e d . and  . 5 0 0 the s t a b l e s e r i e s f a l l s  For values between . 1 0 0  between 14 and 19  iterations.  Here too, the three time s e r i e s demonstrate a c l o s e c o n s i s t e n c y . The  c o n c l u s i o n to be drawn i s t h a t the i n i t i a l  not of p a r t i c u l a r importance. avoided  value of RHO  is  As l o n g as l a r g e values are  the system w i l l perform w e l l .  mental runs a value of . 5 0 0 was  For the remaining e x p e r i -  used f o r RHO.  This value  was  chosen as i t r e p r e s e n t s both a reasonable value based on the f i n d i n g s o f t h i s study and a value that i s commonly used i . e . "traditional." E x p o n e n t i a l Constants The of  - A, B and  C.  focus of t h i s s e c t i o n i s to determine the  changes i n the i n i t i a l  values of the e x p o n e n t i a l  effects  constants.  \  • 100  1  1  -35v  A&)  V  .500  Step Size Reduction Factor  Figure 9  1  -\JX>  1  r&\>  (RHO)  Step Size Reduction Factor and Associated Number of Iterations  TABLE VIII Results From Changes In Exponential Constants (A, B, C) Rising Series Experimental Run  No. of Minimum IteraError tions Value  F a l l i n g Series  Constant Values A B C  Constant Values A B C  Series  No. of Minimum Itera- Error tions Value  Constant Values A B C  8  3405.810  .821  .999  .000  17  709.173  .532  .473  .025  . Oil  12  3405.808  .822  .999  . 0 0 0 15  709.172  .532  .473  .025  .619  .011  15  3406.819  .822  .999  . 0 0 0 12  709.171  .533  .476  .024  -  -  -  6  3405.808  .822  .999  . 000  43.617  .190  .618  .011  -  -  -  -  -  -  -  -  -  -  -  -  16  709.172  .531  .474  .025  16  43.617  .190  .617  .011  13th  14  43.617  .190  .617  14th  23  43.617  .190  15th  -  -  16th  13  17th  -  3rd  No. of Minimum Itera- Error tions Value  Stable  -  —  CO  82 A two pronged approach was made i n t h i s s e c t i o n o f the a n a l y s i s . The  first  p a r t c o n s i s t e d o f g i v i n g each o f the three c o n s t a n t s ,  A, B and G the same v a l u e .  T h i s was done f o r three  v a l u e s , i . e . . 2 5 0 , . 5 0 0 and . 7 5 0 . of  approaching  Based on i n f o r m a t i o n a l r e a d y  the study each constant was g i v e n an i n i t i a l  c l o s e to the s o l u t i o n v a l u e . these changes. of  The second p a r t c o n s i s t e d  each time s e r i e s and each constant value i n t h a t  series individually. in  different  generated  value that was  Table V presents the l i s t o f  I t a l s o should be noted t h a t a standard s e t  v a l u e s were used f o r the other i n p u t s i n t o the system.  These  v a l u e s aret MAX D DELO RN0  = = = =  100 .001  .10  .500  T h i s s e t o f v a l u e s c o n s i s t s o f the "best" v a l u e s t h a t the a n a l y s i s has generated  up to t h i s p o i n t .  With regards t o the f i r s t  s e t o f changes, T a b l e VIII  shows very l i t t l e movement e i t h e r i n the minimum e r r o r value o r the s o l u t i o n constant v a l u e s , as a f u n c t i o n o f the i n i t i a l constant v a l u e s . C -  .500).  #13  Experimental  (A =  .250, B =  runs #3 (A = . 5 0 0 , B = . 5 0 0 , .250, C =  . 2 5 0 ) and  B = . 7 5 0 , C = . 7 5 0 ) present a complete l i s t  #14  (A =  .750,  o f the r e s u l t s .  V i r t u a l l y the only changes that take place do so a t the t h i r d decimal p o i n t .  T h i s p a t t e r n o f n e g l i g i b l e change i s a g a i n  present i n runs # 1 5 B =  .600, C =  (A = .800, B = .800, C =  . 2 0 0 ) and  #17  (A =  .500, B =  .200),  .500, C =  #16  (A = . 2 0 0  .200).  8-3 S e t t i n g the i n i t i a l  constant value near the s o l u t i o n value  has very l i t t l e e f f e c t on e i t h e r the s o l u t i o n value or the minimum e r r o r v a l u e . with regards constant to  As a c o n c l u s i o n , i t can he s a i d that  to the minimum e r r o r value and  v a l u e s , the p a t t e r n search process  changes i n the i n i t i a l With regards  constant  exponential  is  unresponsive  values.  to number of i t e r a t i o n s , F i g u r e  g r a p h i c a l l y d e p i c t s the responsiveness system f o r t h i s a r e a .  the  The  first  of the p a t t e r n  The  there  T h i s l a c k of  p a t t e r n weakens the g e n e r a l i t y of any  that might be drawn.  search  t h i n g to n o t i c e i s that  i s not very much c o n s i s t e n c y between the s e r i e s . a systematic  10  conclusions  r i s i n g s e r i e s appears to be more  a p p r o p r i a t e f o r a p p l i c a t i o n of a set o f low v a l u e s .  The  falling  s e r i e s i s r a t h e r i n c o n s i s t e n t with a middle value i . e . . 5 0 0 seeming to work b e t t e r than e i t h e r l a r g e r or s m a l l e r v a l u e s . The  opposite i s true f o r the s t a b l e s e r i e s .  In t h i s case both  small i . e . . 2 5 0 or l a r g e i . e . . 7 5 0 seem to work b e t t e r than middle v a l u e s .  Because these  statements r e p r e s e n t  only  one  time s e r i e s each the reader should r e f r a i n from making broad i n f e r e n c e s based on t h i s  data.  Conclusions T h i s s e c t i o n i s e s s e n t i a l l y a c o m p i l a t i o n of a l l the c o n c l u s i o n s that have been made so f a r .  To begin with,  the response p a t t e r n s of the search p r o c e s s ,  given  the system i s  not i n f l u e n c e d very much by changes i n the input parameters.  84  -i  -^50  <  .500  Constant Value  Figure 10  Constant Value and Associated Number of Iterations  s  .ISO  8<5 Throughout the experimental runs there developed p a t t e r n of minimal constants.  a consistent  change i n the e r r o r value and the e x p o n e n t i a l  The s e a r c h process appears  to be a b l e to c o n s i s t e n t l y  r e a c h the same minimum e r r o r value and s o l u t i o n constant v a l u e s no matter what the i n p u t s .  T h i s statement  does not encompass  the use o f such extreme values as 0 o r 1, however.  The o n l y  dependent v a r i a b l e t h a t changes a t a l l i s the number o f i t e r a t i o n s o r p a t t e r n moves. input v a l u e s that minimize  There does appear to be c e r t a i n  the number o f i t e r a t i o n s that the  p a t t e r n s e a r c h system needs to a r r i v e a t the s o l u t i o n v a l u e s . F o r the maximum number o f p a t t e r n moves (MAX) 50 i s a r b i t r a r i l y chosen as a s a t i s f a c t o r y number.  In no case i n the study was  the search process terminated by running over MAX.  Because o f  t h i s i t seems to be a p p r o p r i a t e to s e t MAX a t such a value as not t o h i n d e r the s e a r c h process yet s t i l l  be able to stop i t  i n cases where s u c c e s s i v e p a t t e r n moves are p r o d u c i n g improvement.  little  The b e s t way to do t h i s i s to s e t i t a t a value  j u s t beyond that number o f i t e r a t i o n s which one would the search process t o r u n t o .  In the study, even under the  most unfavourable v a l u e assignments, never reached beyond 39 •  expect  the number of i t e r a t i o n s  Hence an a p p r o p r i a t e v a l u e f o r MAX  would be 50. With regards to minimum step s i z e  (D) the s m a l l s i z e  (.001) t h a t was used throughout the study was never needed. I t c o u l d e a s i l y have been i n c r e a s e d to .01 w i t h o n l y a n e g l i g i b l e e f f e c t on the dependent v a r i a b l e s .  At v a l u e s s m a l l e r than  .01  8'6 the step s i z e becomes too small to y i e l d much improvement. At t h i s stage i t becomes i n e f f i c i e n t w i t h regards to computation time, g i v e n the r e s u l t s r e c e i v e d , to continue the search. The  p a t t e r n search system a l s o appears to be  s i v e to changes i n the p a t t e r n search step s i z e  unrespon-  (DELO).  Neither  the e r r o r value nor the constant v a l u e s can be improved or changed through  the use of d i f f e r e n t DELO v a l u e s .  i t e r a t i o n s i s somewhat more r e s p o n s i v e .  The number of  Both l a r g e and  values y i e l d l a r g e r numbers of i t e r a t i o n s than do middle i.e.  .10-.20.  approximately  small values  Hence i t would be a d v i s a b l e to use a value of .10 when s e t t i n g up a s e a r c h system.  L i k e the o t h e r i n p u t s the s t e p s i z e r e d u c t i o n f a c t o r (RHO)  a l s o does not e l i c i t  search system.  much change i n the r e s u l t s of the  Movements i n the e r r o r f a c t o r and  values are small enough to be i n s i g n i f i c a n t . the number of i t e r a t i o n s RHO minimize the number. t h i s range.  the  constant  With regards  values of from .100  to  to  .500  There i s q u i t e a b i t of c o n s i s t e n c y w i t h  L a r g e r values than these should be avoided as  they  tend to i n c r e a s e the number of i t e r a t i o n s . There i s l i t t l e in  responsiveness  the e x p o n e n t i a l constant v a l u e s .  i n the system to changes  Even when the  initial  values are placed c l o s e to the s o l u t i o n value there i s i n s i g n i f i cant change i n e i t h e r the e r r o r or s o l u t i o n constant v a l u e s . Between the time s e r i e s there i s l i t t l e to  consistency with  regards  the e f f e c t s o f constant changes on the number of i t e r a t i o n s .  87  The r i s i n g series benefited most from a small set of values i.e.  . 2 5 0 . The f a l l i n g series was most e f f i c i e n t with a middle  value i . e . . 5 0 0 . The stable series reacted opposite to the f a l l i n g one and reacted best to values at the extremes i . e . .250  and  .750.  Perhaps the most important f i n d i n g of the study i s that of the responsiveness of the search system i n the series of exploratory and pattern moves that occur before the f i r s t step size reduction.  By f a r the bulk of a l l improvement in the error  value and the bulk of a l l change i n the constant values takes place before the f i r s t step size reduction.  This i s important  because i t accounts f o r the i n s e n s i t i v i t y of the search system to changes i n MAX,  D and RHO .  8'8 CHAPTER V I I Recommendations Whereas the l a s t  chapter d e a l t w i t h the a n a l y s i s  and c o n c l u s i o n s o f the study t h i s chapter b r i e f l y d e a l s w i t h some recommendations based on the f i n d i n g s .  F i r s t l y , I would  l i k e to d e a l w i t h the q u e s t i o n o f the g e n e r a l i t y o f the f i n d i n g s . Although  the r e s u l t s o f the study showed some c o n s i s t e n t response  p a t t e r n s w i t h i n the p a t t e r n search system, o v e r - g e n e r a l i z a t i o n should be avoided.  Due t o the small sample s i z e used i . e .  three time ...series, i t might be more a p p r o p r i a t e to c a l l the findings "tentative conclusions." I do, however, f e e l that some s i g n i f i c a n t p a t t e r n s have been brought  to l i g h t by the study.  response Further  i n v e s t i g a t i o n i n t o some o f these p a t t e r n s might prove  worthwhile.  One a r e a o f recommendation i s the r e p l i c a t i o n of t h i s  study,  u s i n g d i f f e r e n t time s e r i e s .  By doing t h i s ,  i n f o r m a t i o n necessary  to the d e t e r m i n a t i o n o f the r e l i a b i l i t y of the r e s u l t s o f t h i s study, would be generated.  As an important aspect o f t h i s  t i o n the amount o f the e r r o r r e d u c t i o n before the f i r s t  replica-  step s i z e  r e d u c t i o n move should be looked a t w i t h p a r t i c u l a r a t t e n t i o n . I t would be v e r y u s e f u l to know i f o t h e r time s e r i e s d i s p l a y p a t t e r n o f response. A second  area that would y i e l d f r u i t f u l  this  s  f i n d i n g would  be a c o r r e l a t i o n study between number o f i t e r a t i o n s and computer costs.  While  i t i s r e a l i z e d that a study o f t h i s nature would  be h i g h l y machine dependent w i t h regards to r e s u l t s , the f i n d i n g s  8:9 would y i e l d v a l u a b l e i n f o r m a t i o n w i t h regards to the c o s t s i n v o l v e d i n o p e r a t i n g a p a t t e r n s e a r c h - e x p o n e n t i a l smoothing forecasting  system.  A t h i r d area of p o t e n t i a l l y v a l u a b l e r e s e a r c h would be to r e p l i c a t e the study but u s i n g time s e r i e s of v a r y i n g length.  P a r t of the reason f o r the i n s e n s i t i v i t y of the  p a t t e r n search technique i n t h i s study may  have been due  f a c t that a l a r g e number of p e r i o d s per time s e r i e s was i.e.  .72.  By f o r c i n g the i n p u t s of the search system  to the used  through  such a l a r g e number o f time p e r i o d s the responsiveness of the system may  have been smoothed out.  search system appears  As i t stands now  the p a t t e r n  capable of producing good s o l u t i o n r e s u l t s  no matter what the v a l u e s of the i n p u t s a r e .  I f the number of  data s e r i e s used were s m a l l e r then perhaps more responsiveness would be observed as the r e s u l t o f a reduced In  smoothing e f f e c t .  c l o s i n g , I would l i k e to emphasize, once more, the  a p p l i e d nature of the approach use of a more complete  taken i n t h i s study.  While  the  range of experimental runs would have  been more s a t i s f y i n g i n a t h e o r e t i c a l sense, i t a l s o would have been l e s s e f f i c i e n t .  The b a s i s f o r the use of the e x p o n e n t i a l  smoothing technique i s t h a t , f o r c e r t a i n a p p l i c a t i o n s , a c c u r a t e f o r e c a s t i n g method.  T h i s study t r i e d  i t i s an  to extend  that  f e e l i n g of u t i l i t y by proceeding i n t o areas where r e s u l t s seemed f r u i t f u l and by a v o i d i n g areas where r e s u l t s seemed r e p e t i t i v e or i n c o n s e q u e n t i a l .  90  BIBLIOGRAPHY  1.  Bates, James and Parkinson J.R., Business Economics, Oxfordt Basil Blackwell, 1969.  2.  Berry, William L. and Bliemel, Friedhelm W., "Selecting Exponential Smoothing Model Parameters: An Application of Pattern Search."  3.  Brown, R.G., S t a t i s t i c a l Forecasting for Inventory Control, New York: McGraw-Hill, 1959.  4.  D'Amico, Peter, "Forecasting System Uses Modified Smoothing," Industrial Engineering, June 1971.  5.  Federal Task Force on Agriculture, Dairy,- Ottawa: Canadian Agricultural Congress, 1969.  6.  Hook, R. and Jeeves, T.A. , "'Direct Search',..Solution of Numerical and. S t a t i s t i c a l Problems," Journal of the Association of Computing Machinery, V o l . 8 , A p r i l 1961.  7.  More, Geoffrey, S t a t i s t i c a l Indicators of C y c l i c a l Revivals and Recessions, New York: National Bureau of Economic Research, 1950.  8.  Roberts, Stephen D. and Whybark, D. Clay, "Adaptive Forecasting Techniques," Paper No. 3 3 5 , Herman C. Krannert Graduate School of Industrial Administration, Purdue University, 1971.  9.  Spencer, M.H. , Mack,. C.G. and Hoguet, P.W. , Business and Economic F o r e c a s t i n g - An Econometric Approach, New York: Richard D. Irwin Inc., 1961.  10.  Van Wormer, T.A. and Weiss, Doyle L., " F i t t i n g Parametersto Complex Models by Direct Search," Journal of Marketing Research, Vol. VII, Nov. 1970.  11.  Winters, Peter R., "Forecasting Sales by Exponentially Weighted Moving Averages," Management Science, Vol. VI,  I960.  APPENDIX A Data S e r i e s P l o  000.00-  200.00-  400.00  600.00  800 .00-  Y AXIS, 000.00-  200 .00-  5|C  400 .00-  'i*  -V-  -r>  "3|C  5jc  T  PLOT OF RISING TIME SERIES 600.00-  *J* ^  ^  X AXIS -^Time period i n months Y AXIS - Time series data values  800.00-  4 00  • * * o « o e e «  .00  | »  9  8 .00  0  •^ * 0  0  4  0  0  P  )  O  O  16 .00  O  O O  O  O  O  O  l  O  O  24.00  O  O  O  32. 00  40.00  ,..| ... 48.00  . . I. .. 56.00  64.00  72.00  80.00  O  0 0 0 . 0 0 -  000,00'  0 0 0  -  -  -  -  -  • -  -  -  —  -  -  -  -  -  -  -  . 0 0 ...  •  . „  _  „  _  -  -  --  - -  -  -  *  0 0 0 . 0 0 -  * 0 0 0 .  Y 0 0 0  co-  A X I S . . 0 0 * * *  * *  oco.oo-  *  *  * -v- '1^  "c-  *  * *  * *  *  f  * *r  t~  T  j{c  • s}: j[e  —  -• -  -  -  ' — • - -  -  -  *  * * *  *  * * *  $ * * *  *  *v* 5):  0 0 0 . 0 0 -  ^  -  * * *  *  *  ;{e #  >)c #  -*r-  *r #  • jjc  *  *  PLOT OF FALLING TIME SERIES  * * 0 0 0  *  . 0 0 -  X AXIS - Time period i n mont!  * * *  Y AXIS - Time s e r i e s data values  * * 0 0 0 . 0 0 -  -  *  -  J . 0 0 -  . . . . 0 0  1  8 . 0 0  . . I 1 6 . 0 0  1 2 4 . 0 0  I 3 2 . 0 0  .  I 4 0 . 0 0  I 4 8 . 0 0  I 5 6 . 0 0  I 6 4 . 0 0  I  !  7 2 . 0 0  8 0 . 0 0  i.J  000.00  000.00  000.00  *  000.00  *  * * *  *  000.00-  **  * •*  Y A X I S ,* * * * 000.00--* * * •* * * ****  *  *  ****  * ** * * *  *  *  * *  *** **  *  * *  *  * * *  * * *  . * 000 .00  *  ** * *  *  * Hp  >  ** * ** ** * ** * * *  * *  *  * * * * * * *  * * ** * * * * ** * * * * ** * * * * ** ** ** * * * PLOT OF STABLE TIME SERIES  000 .00 X AXIS - Time period i n months Y AXIS - Time s e r i e s d a t a v a l u e s 000.00  000.00  Of 0.00-. . . .00  8. 00  16. 00  24.00  3 2.00  40.00  48.00  56.00  I  6 4 . 00  72.00  80. 00  

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