A MATHEMATICAL MODEL FOR THE SELECTION OF HAYING MACHINERY by JOHN PERCIVAL WELDON JEFFERS B.S.A.(Hons) University of British Columbia, 1959 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN AGRICULTURE i n the Department of Agricultural Mechanics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1966 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y avail able f o r reference studyo 1 further agree that permission-for thesis f o r s c h o l a r l y purposes may extensive copying of t h i s be granted by the Head of Department or by his representatives. and my I t i s understood that copying or p u b l i c a t i o n of" t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of A g r i c u l t u r a l Mechanics. The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada ABSTRACT Surveys of hay harvesting machinery in the Lower Fraser Valley of British Columbia indicated wide variation in sizes of machines employed on farms of comparative size. Quality hay depends on 1. the types of forages grown of growth at which the crop i s harvested and i t effects curing. 3. 2. the stage the influence of weather as Bleaching, leaf shatter, and leaching of soluble nutrients are the worst hazards to which the crop i s exposed. The system of harvesting w i l l affect the time the crop is exposed to the effects of the weather. Methods that reduce the time needed for curing in the field tend to produce a better quality hay. Factors affecting the selection of least cost combinations of machines used i n sequence are 1. capacity of the machines, the cost which bears a linear relationship to the 2. the area they have to service and available for the performance of the operations. 3. the time The time available for a sequence of operations to be performed in this case i s a function of the weather. A study of the weather of the Lower Fraser Valley indicated that in any given ten day period during the months of June, July and August, the probabilities of two or more, three or moreor four or more open days for field curing hay are constant. Using this linear cost capacity relationship, acreages from ten to one hundred acres, and the time available obtained from weather probability data, a mathematical model is derived to select least cost machinery combinations for hay harvesting in the Lower Fraser Valley. A computer programme for the I.B.M. 7040 digital computer is also developed. iii TABLE OF CONTENTS Page Abstract ii Table of Contents iii List of Tables iv List of Figures V Acknowledgements vi JJfTHODUCTION 1 LITERATURE REVIEW 3 THEORY 6 Cost and Capacity Relationships 7 Weather 10 Mathematical Model 13 RESULTS 19 Field Capacity 19 Weather 21 Mathematical Model DISCUSSION OF RESULTS 23 25 CONCLUSION 32 LITERATURE CITED 33 APPENDIX 35 I II III IV V VI Graphs Showing Linear Relationship Between Costs and Capacities of Machines 35 Periods and Duration of Periods (Days) of Dry Weather During June, July and August 40 Frequency of Dry Spells at Abbot sfor d, 42 IBM 7040 Digital Computer Program 45 Machine Sizes i n Feet Selected on Time not Affected by Weather Probability 46 Machine Sizes i n Feet as Affected by Weather Probability 47 iv LIST OF TABLES Table I II III Page Nomenclature Calculated and Observed Field Capacities V VI 20 Probability of 2 or more, 3 or more, 4 or more "open" days i n June, July and August at Abbotsford, B.C. IV vi 22 Machine Hours Available June 1 to August 29 23 Theoretical and Observed Machinery Selections 28 Cost Comparison of Theoretical and Other Selections of Equal Capacities 30 v. LIST OF FIGURES Figure P a g e 1 Mower Cost vs Capacity 35 2 Conditioner Cost vs Capacity 36 3 Mower-Conditioner Cost vs Capacity 37 4 Side Delivery Rake Cost vs Capacity 38 5 P.T.O. Baler, Cost vs Capacity 39 vi. NOMENCLATURE A TASK SIZE IN ACRES B A PROPORTIONALITY CONSTANT C PURCHASE PRICE IN DOLLARS D ANNUAL DEPRECIATION E FIELD EFFICIENCY IN PERCENT E g ENERGY SUPPLIED F FIXED CHARGE IN DOLLARS L SERVICE LIFE IN YEARS N OBSERVED NUMBER OF 2 OR MORE "OPEN" DAYS PTO POWER TAKE OFF R SUM OF THE SERIES OF OBSERVED "OPEN" DAYS S SPEED OF TRAVEL IN MILES PER HOUR S SALVAGE OR TRADE-IN VALUE T TIME AVAILABLE FOR COMPLETING TASK IN HOURS W RATED WIDTH OF MACHINE IN FEET W WORK ACCOMPLISHED Y EFFECTIVE FIELD CAPACITY IN ACRES PER HOUR "H EFFICIENCY OF ENERGY CONVERSION X LAGRANGES MULTIPLIER v k ACKNOWLEDGEMENTS The writer i s grateful for the assistance and guidance of Professor L.M. Staley during this study. Thanks are extended to Professor E.L. Watson, Dr. M.J. Dorling and Dr. V.C. Brink for serving as members of the committee directing this study. Gratitude i s also expressed to Mr. A.G. Fowler for his assistance with the Digital Computer program and to Dr. F.V. McHardy for the use of his unpublished Ph. D. Thesis. 1. INTRODUCTION Of the 4384 commercial farms in the Lower Fraser Valley of British Columbia i n 1961, 2095 were dairy farms (18).*- Consequently the growing, harvesting and storage of forage crops i s of prime importance to approximately 50$ of the commercial farmers. In this area of the province, the weather during the months of June, July and August i s very variable. Generally the f i r s t ten days of June are rainy, the second ten day period less rainy, and the third, much like the f i r s t . The average r a i n f a l l for these three months i s 5.70 inches, but the distribution i s not uniform. This has a profound effect on the harvesting and subsequent handling of forage stored as hay. The selection of minimum cost combinations of f i e l d machinery for harvesting the hay crop i s therefore of v i t a l economic importance to the dairy farmers. A study by Lange (13) i n 1959 indicated that there was wide variation in capital invested i n forage harvesting machinery on farms of comparable size. A more detailed study in 1961 by Crossfield and Woodward (10) further emphasized Lange's findings. From the information i n these reports, i t would appear that either some farms were undercapitalised and others overcapitalised or that * (Numbers i n parenthesis refer to appended references) 2. seemingly undercapitalised farms were dependant on custom operators, and the seemingly overcapitalised farms'-were doing custom work for others. A third alternative i s that some farmers were purchasing larger machines with a greater capacity as a form of insurance against the uncertainty of the weather. In this connection i t must be realised that combination of resources by farmers i s not always i n a manner to minimise costs for a given output and hence to maximise profits. This behavior pattern can be explained partly by the fact that profit maximisation i s not always the ultimate end of farming. Farmers may use labour saving equipment, though i t cannot be justified i n terms of resource prices, in order to lessen the drudgery of farm work and to increase the satisfactions of living. With these alternatives i n mind, a c r i t i c a l appraisal of haymaking i n the Lower Fraser Valley was undertaken during the summer of 1965 with a view to establishing a mathematical model capable of determining least cost combinations of f i e l d machinery for haying in this area. The model would combine the inter- relationships of acreage, management methods and r a i n f a l l probabilities. 3. LITERATURE REVIEW Kranick (12) has defined grassland farming as a general plan of farm operations calculated to combine quick return advantages i n forage, l i v e stock production and land use efficiency with long range advantages i n s o i l conservation. High protein legumes and other forage crops must be harvested to provide feed for the livestock which can be carried during months of good pasture. This implies that forage crop production, harvesting, handling, storage and feeding operations should be done economically and must be implemented by power machinery and other modern f a c i l i t i e s . High quality hay has been defined as hay of bright green colour, pleasant aroma, pliable texture, leafiness, high nutrient value and palatability, (4). The original feeding value of a hay crop can best be conserved by the retention of a l l the leaves and by a rate of curing that quickly stops respiration, enzyme action and fermentation. Hay cut at an early stage of growth i s richer i n vitamins, proteins and minerals (3). Leaching and bleaching as well as leaf shatter are some of the chief sources of nutrient loss affecting hay. It has been found that that 20% to U0% of the nutrients i n hay can be extracted by cold water. These soluble nutrients are most easily digested and are biologically some of the most important nutrients i n hay (16). Damage by rain i s less severe soon after cutting, but increases i n severity with time. Bohstedt (4) quoted Kisselbach and Anderson who concluded that the most practical way of curing hay was to leave i t for three hours i n the swath and then put i t i n windrows. It thus reached 'J0% moisture content i n twenty-seven hours with a loss of only 1.7% dry matter. Casselman and Flncham (8) have shown i n Iowa that a practical method of reducing f i e l d curing time i s by use of conditioners. Using these implements, the f i e l d curing time may be reduced by as much as eighteen hours, when compared with the usual method. From the considerations on hay quality, i t w i l l be noted that time i s an important variable, and therefore that any method that would reduce the period between mowing and storage would result i n a better quality product. While this may be achieved by using larger machines, a balance must be achieved between the cost-price, size, operating cost and the magnitude of the haying operation. Due to the complexities of the problem and the d i f f i c u l t i e s involved i n solving them by manipulation i n the f i e l d , a mathematical approach to the problem was deemed most practicable. Viewed generally, a model i s a representation of some subject such as events, processes or systems, and i s used for purposes of prediction and, or control. It i s intended to make possible or to facilitate determinations of how changes i n one or more aspects of the modelled entity may affect other aspects of the whole. In the employment of models, these determina- tions are made by manipulating the model rather than by imposing changes on the modelled entity i t s e l f . 5 Three types of models are recognised: (9). (a) Analogue models which employ one set of properties to represent some other set of properties possessed by the system being studied. (b) Iconic models which pictorially or visually represent certain aspects of a system. (c) Symbolic models which employ symbols to designate the properties of a system. After reviewing the types of models and their potential, i t was decided that a symbolic model was the one most appropriate to the problem under consideration. 6. THEORY A study of the haying systems in the Lower Fraser Valley showed two approaches and four systems of making hay. The f i r s t approach, which i s on the wane, i s that the entire hay acreage should be mowed and receive the subsequent treatments. The second approach, which i s becoming increasingly popular, i s that the hay should be cut and treated in lots of ten to twelve acres. This ensures that i n the event of a wet spell immediately after cutting, the entire crop would not be lost. The four systems i n common uses are as follows :(a) The use of a sickle bar mower to cut the crop, followed by a side delivery rake to put i t into windrows. After curing for a day or two, depending on weather conditions a baler i s used to bale the hay. (b) The use of a sickle bar mower to cut the crop followed immediately by a conditioner. A side delivery rake i s then used to put the crop into windrows. After curing for a day, a baler i s used to bale the hay. (c) The sickle bar mower and the conditioner i s used in tandem to combine the operations of cutting and conditioning the crop. This i s followed by the side 7. delivery rake to put the crop into windrows. After curing for a day, the crop is baled. (d) The fourth system, which i s very promising i s the use of mow drying to complete the drying of the baled hay to a moisture content which w i l l prevent spontaneous combustion or spoilage when stored. In this system, lots of ten to twelve acres are cut, conditioned and raked on one day and baled on the afternoon of the second day. Immediately after baling, the hay i s transported to the mow for further drying by heated or unheated a i r . For the purposes of this study, the above combinations were examined with a view to selecting the least cost combinations for each system using a mathematical model. Cost and Capacity Relationships In keeping with the use of a symbolic model, the f i r s t variable considered was that of machine capacity and cost. To this end, prices of machines f.o.b. Vancouver were obtained from manufacturers representatives, and using the American Society of Agricultural Engineers specifications concerning speed of operation and field efficiency and their formula for determining capacity, the capabilities of the machines .were calculated (1). The formula i s :- „ _ S.W.E. where Y = effective f i e l d capacity i n acres per hour. S = speed of travel, i n miles per hour. W = rated width of machine, i n feet. E = f i e l d efficiency i n percent. \ 8. The effective f i e l d capacities so calculated were compared with the costs of the machines. Costs i n this instance were taken as the purchase price, and the associated fixed and operating costs were omitted for the following reasons :1. To rationally estimate the operating cost of farm machines, i t i s useful to recall the energy balance concept as applied to farm work (14) HE where T) '•' E W^. g = W ,-; (2) k = efficiency of energy conversion, = energy supplied, = work accomplished, and power P, i s defined as the rate of doing work during time T, then "HPT = \ (3) t and for a particular quantity of work accomplished , Tl PT i s a constant. Where T) i s a constant, P and T may vary inversely while maintaining a constant product. The quantity of energy that must be applied varies directly with the work done and i s independent of the rate at which the work i s performed. Within limits, the quantity of fuel required to perform a particular operation remains"the same whether a large tractor and machine perform the task or a smaller tractor and machine are used over a longer time (2). Fuel costs may therefore be considered as a linear function of task size alone. 2. Similarly, the operators pay covers a particular number of hours per day, and i s independent of the time spent on each operation. This too i s a constant 9. expense and may therefore be considered as a l i n e a r f u n c t i o n of the task s i z e . In l i k e manner, r e p a i r costs which tend to be "lumpy", but which are p r e d i c t a b l e , can be prorated as l i n e a r functions of work done (14). 3. In the case of f i x e d costs such as depreciation, i n t e r e s t on investment, housing e t c . , the reason f o r not i n c l u d i n g them i n the costs of the machines concurred i s as f o l l o w s :Each c l a s s of machine has a c e r t a i n f i e l d e f f i c i e n c y , l i f e expectancy, recommended speed of operation e t c . , regardless of the make ( l ) . I f then the l i f e of two machines of d i f f e r e n t s i z e s i s the same, then i t f o l l o w s that the l a r g e r or more expensive machine w i l l have the higher depreciation value per hour. But while the depreciation value per hour i s higher, i t s capacity i s also higher than the smaller l e s s expensive machine. Therefore, i f a task of a p a r t i c u l a r s i z e i s to be performed, the smaller machine w i l l take longer than the l a r g e r machine and the d i f f e r e n c e i n time w i l l o f f s e t the d i f f e r e n c e i n the hourly depreciation r a t e . For example, consider two mowers, one 5' and the other 7' wide, each with an expected l i f e of 12 years and average annual use of 167 hours. I f the purchase p r i c e s are $385.00 and $460.00, the e f f e c t i v e f i e l d c a p a c i t i e s 2.4 and 3.4 acres per hour and the salvage value i s zero, then the annual d e p r e c i a t i o n C — (h.) can be c a l c u l a t e d from D = —-jr '' * when D = annual depreciation. C = purchase p r i c e . S= r salvage or trade i n value. L = service l i f e . Applying t h i s formula, the depreciation i s $32.08 and $38.33 per year or $0.19 and $0.23 per hour r e s p e c t i v e l y . But the f i e l d c a p a c i t i e s are 2.4 and 10 3.4 acres per hour, therefore, the depreciation per acre i s $0.07 and $0.06 respectively. As the depreciation per hour i s almost equal, i t may be considered a constant coefficient of the enterprise size. It follows then that the other fixed costs can be treated i n the same way, The linear nature of the price capacity relationship i s apparent as shown i n Appendix I, i n every class of machinery considered. This relationship may be expressed for each class of machine as : C = F + BY where (5) C = purchase price in dollars. F = fixed price in dollars. B = a proportionality constant. Y = effective f i e l d capacity i n acres per hour. The relationship between the task size and the time available for performing the task may be expressed as : A = TY ,,... where (6) A = task size in acres. T = time available for completing the task, i n hours. Y = effective field capacity of the machine in acres per hour. Weather As stated earlier, the weather has a marked effect on the time available for making hay and also on the quality of the end product. Prolonged rainy spells prevent f i e l d operations, and, i f the crop has already been cut, cause serious leaching of the digestible nutrients. Another effect of prolonged 11. rainy weather i s that the forage crop may have passed the stage of optimum nutritive value by the time the weather i s once more suitable for f i e l d operations. The f i r s t stage in the weather investigations was to define the conditions that are considered suitable for haying operations. Borgman and Booker (6) working in Missouri define an "open" haying day as follows :1. Less than 0.1 inch of rain f e l l on that day. 2. The sun had to shine more than 70% of the time between sun rise and sun set on that day. 3. Less than 1.0 inch of rain f e l l the day before. These criteria were examined to determine whether they were applicable to the Lower Fraser Valley. It was not possible to determine the effect of duration of sunshine due to the absence of appropriate meteorological data for the area. Examination of the weather data for Kansas City and Springfield Missouri disclosed that the vapour pressure there was lower than the vapour pressure i n the Lower Fraser Valley. This in conjunction with higher temperatures in Missouri indicates a lower relative humidity and a faster rate of drying i n Missouri than in the Lower Fraser Valley. Due to these factors i t i s suggested that the r a i n f a l l criteria for the Lower Fraser Valley may be as follows :1. Less than 0.05 inch of rain f e l l on that day. 2. Less than 0.50 inch of rain f e l l on the day before. These criteria were further substantiated by observations of f i e l d conditions after r a i n f a l l during the summers of 1965 and 1966 when average temperatures were lower than normal. Having established the criteria, a preliminary study of the available meteorological data for the area (15) showed that the number of periods of three 12. days or more without rain during the months of June, July and August were relatively uniform as shown by the small standard deviation (18). of the periods however showed wide variability. The durations Appendix II shows the number and duration of dry spells during June, July and August based on meteorological data for the last twenty years. Brooks and Carruthers (7) investigated the persistance of runs of wet or dry days i n some areas of England. When dealing with events which either can or cannot occur, the term "run" i s taken to mean an unbroken succession of occurrences or non-occurrences. To appreciate the effect of persistence, i t i s necessary to consider the number of runs of different lengths expected i n a series i n which there i s no persistence. In such a series, both the probability p that an event w i l l occur and the probability q •* (1 - p) that an event w i l l not occur are independant of what has gone before. In attempting to f i t a theoretical structure to observed sequences of "open" days at Abbotsford, B.C., Brooks and Carruthers (7) procedure for f i t t i n g a theoretical structure to observed sequen«es of rain days at Kew was adopted. The three possibilities to be considered are :1. The probability of an "open" day following an "open" day i s constant (i.e. p^ .** p for a l l values of k where k i s the number of "open" days). 2. The probability of an "open" day after two "open" days i s constant, (i.e. p^ = Pg for a l l values of k ^ 1 or more open days). 3. p^ continually increases as k increases. 2 The f i r s t possibility when tested by the X test does not give a very go«d f i t and the n u l l hypothesis i s tentatively rejected and the second possibility examined. 13 The second possibility, that frequencies of two or more, three or more, four or more N(l + P 2 + + ?2 n or more "open" days i s given by the series + p n ~ Equating the sum (R) of the series to the sum of the cumulative frequencies from two or more "open" days onwards, a probability for two or more "open" days can be calculated, and a set of calculated frequencies obtained as discussed under Results. The observed and calculated frequencies when tested by the \)i test give a better f i t than the f i r s t possib i l i t y and i s therefore adopted. From this, we may conclude that the probability of having two or more "open" days i s constant. N(l + P 2 + P 2 + P^ + P ~ From the series ) i t can be seen that the probability of having 2 three or more or four or more "open" days i s 3 and respectively. The third possibility i s rejected because no simple law of increase of p^ with k i s l i k e l y to improve appreciably on the value of X: when allowance i s made for the decrease in the number of degrees of freedom required by each additional assumption. Mathematical Model. Problems of maximising and minimising may often be solved by linear programming. This technique requires that B i n equation (5) be a constant coefficient relating to Y and also that at least one of the three terms of equation (6) equation (5) be a constant. or As A and T are normally system variables, the requirements for linear programming are not directly met (14). The Monte Carlo technique as applied to queuing problems was also investigated ( 9 ) . The theory of queuing provides techniques for determining such things as the average queue length and average waiting time when the arrival and service rates are known. In applying this technique to the haying problem, i t was assumed that the acres of hay to be harvested were "queuing" and that they 14 were to be serviced by the implements concerned. These implements each had a mean service rate and associated standard deviation. It was further assumed that time was one of the input variables with a mean and standard deviation. When the. mean and standard deviations calculated i n appendix II was applied, i t was found that i n too many instances the time available for the servicing of the acres to be harvested was unrealistic. By this technique numbers taken from the table of random numbers are applied to the mean service and delivery rate. When this factor was applied to the time, minus quantities or excessively long periods were often obtained. The Monte Carlo technique was found on these grounds mainly to be unsuitable. It was decided to attempt Langranges method to determine the least cost combinations of machines. Langranges multipliers may be used to determine the maximum or minimum value of a function when the variables are connected by some relation (16, 17). Thus consider a function, (7) u = f ( x , y, z) x where one of the variables, say z, i s connected with x and y by some restraining relation 0 (x, y, z) = 0 (8) : Regarding x and y as the independent variables, the necessary conditions for a minimum give ^ /dx = 0 and ^"/dy = 0 or, U du dx df df dz dx . dz dx = 0 du Sy df Sy i df dz dz Sy = 0 • 15. The total differential du ^ , du dx + ^ df , df , dy = ^ dx ^ + , dy + ^ df /dz , dx + ^ dz , \ dy) = 0 „ and since the expression in brackets i s precisely dz, i t follows that Sf dx ^— , dx + of dy df dz , dy + ^J , . dz=0. . (Q) The total differential of the restraining condition i s dx dy dz J Let this equation be multiplied by some undetermined multiplier equation 9 . ^ and added to The result i s (f • x M • (f . x;.f) • (f • x | | ) d . o • )dx dy If the ^ i s so chosen that dx dx dz dz = 0 >• = 0 > 0 ( x , y, z)=0-, Z 16. then the necessary condition for a mi nimum or maximum i n equation 7 w i l l be satisfied. Thus i n order to determine the minimal values of equation 7» i t i s necessary to obtain the solution of the system of equation 11 for the four unknowns x, y, z and A., The multiplier X i s termed a Lagrangian multiplier. Equations 5 and 6 express the relationship between cost and capacity of the machine and between task size and the time available for performing the task, A working sequence of operations may then be expressed as, ? A - TY. ,± (12) i»=l subject to the constraints S \ (13) - T i-l and - § BY. i«=l j Applying the above concept, the auxiliary equation may be written, n A " 2 Y. i=l 1 where A i s constant^ 17. therefore n j = ^ 1=1 A ^ i X or n iw> Langranges m u l t i p l i e r i s then applied to equations 14 and 15 to y i e l d the following r e s u l t z~ 2 Y + X 2 |1 B i=l 1 i=l 1 1 Taking the p a r t i a l derivative B-, - B 0 * A. - ^2 * 0 we obtain , =0 , (17) 2 B -i2 n n * - 0 , n E i=l therefore . ^ A t = (16) /Bj ^ T 1 » (18) A Substituting f o r ^ i n the p a r t i a l derivatives, equation 17, y i e l d s 18. n 2 /B_. 1=1 TBT where the subscript j designates a particular value for i . (19) A minimum value for Z i n equation 16 subject to the conditions imposed by equation 15 may then be found by the use of LagraMglian multipliers. (14). RESULTS Field Capacities The effective f i e l d capacities as calculated by equation 1 were checked by observing and timing the relevant operations on several farms i n the area without the knowledge of the operators. In a l l the cases observed the actual f i e l d capacities of the machines were very close to their calculated values as seen i n Table II. In the case of the balers, the observations were further substantiated by recent test reports issued by the National Institute of Agricultural Engineers (11). Table II Machine CALCULATED AND OBSERVED FIELD CAPACITIES Size Calculated Field Capacity Ac/Hr. Observed Field Capacity Ac/Hr. 6' 2.9 3.0 7' 3.39 3.4 9' 4.36 4.3 6« 2.9 2.8 7' 3.39 3.3 7' 3.5 3.6 Side Delivery 8' 4.0 4.0 Rake 9' 4.5 4.6 10» 5.0 5.2 4.5' 1.62- 1.6 5' 1.85 1.8 Mowers Conditioner P.T.O. Baler 21. Weather Using the Brooks and Carruthers technique (7) the period June 1 to August 29 was divided into discrete periods of ten days. The probability of two, three, four or up to ten "open" days for each ten day period was calculated as follows : The daily r a i n f a l l data for Abbotsford, B.C., for the months of June, July and August during the years 1944 to 1964 was tabulated. The period June to August was divided into discrete ten day intervals starting with June 1» In each period, the total number of discrete two day, three day, four day and up to ten day "open" days were counted and recorded i n column (2) of Appendix III. Starting with the last figure i n column (2) the figures were successively summed and recorded i n column (3) to give an observed cumulative frequency. The sum of column 2 gives the total number, N, of spells of two or more "open" days. Given N spells of two or more "open" days, the assumption of a constant probability, P, implies that the theoretical frequency of spells of two or more, three or more, or four or more "open" days i s given by the successive terms of the series :— N(l + P + P 2 + P 3 + P 11 ~ ) , where P, P , P , etc. are 1 2 3 the probabilities of getting two or more, three or more, four or more "open" days etc, The sum of the series i s T andnultiplying both sides by P and subtracting we get R ~ PR - N(l - P ) . n Solving for R, we have R *» N(l - P ) . n 1-P As n approaches i n f i n i t y , P n approaches zero for a l l P < 1 whence 1 «p and P - 1- N R 22. The sum of column(3) i s the sum of the series K from which the probability P of having two or more "open" days i s calculated. Column(4) i s constructed by use of the observed frequency for two or more "open" days and multiplying i t success2 3 ively by P, P , P , etc. Successive subtraction of the figures in column(4)gives the computed frequency i n column(5). The computed frequencies i n column(5)are compared with the observed 2 frequencies i n column(2)by an X: test. hypothesis. In no case was there reason to reject the The probabilities calculated are shown i n Table III. It can be seen that the f i r s t and third periods of June are least suited to haymaking and that July i s the month most likely to have spells of "open" days. Table III THE PROBABILITY OF HAVING 2 OR MORE, 3 OR MORE OR 4 OR MORE "OPEN" DAYS IN JUNE, JULY AND AUGUST AT ABBOTSFORD B.C. JUNE 11-20 21-30 1-10 11-20 21-30 31.9 10-19 0.50 0.64 0.58 0.62 0.66 0.68 0.63 0.64 0.65 0.25 0.41 0.34 0.38 0.44 0.46 0.40 0 . 4 1 0.42 0.125 0 . 2 6 0.20 0.24 0.29 0.31 0.25 0 . 2 6 0.27 ..1-J.O Period P Probability 2 P P 2 2 3 2 AUGUST JULY 20-29 The probabilities were used as follows to calculate the machine times available for making hay. These times are based on the assumption that . 3 6 of a day (24 hours) i s used for machine operations and the rest of the time i s f i e l d curing time and night time. For example, i f i t were 100$ certain that no rain would f a l l during a given two day period, then 1 7 . 2 8 hours of machine time would be available for completing a sequence of operations. If however the probability 23. of obtaining two or more "open" days i s .50, then the machine time available is decreased by the probability factor to allow for additional drying time to compensate for the uncertainty of the weather. Or another way of considering the influence of weather probability on machine size would be to equate the machine size ratio?, to the ratio of probabilities. is assumed to have a P = 1.0, Since the i n i t i a l condition then the theoretical machine size i s determined from the product of the i n i t i a l machine size and the probability for the number of "open" days. required On this basis, the machine times available for making hay during the period June 1 to August 29 are shown in Table IV. Table IV MACHINE HOURS AVAILABLE AT ABB0TSFORD B.C. JUNE 1 TO AUGUST 29 Machine Time - JUNE ' JULY , AUGUST Hours 1-10 11-20 21-30 1-10 11-20 21-30 -31-9 10-19 20-29 2 17.28 8.64 11.10 10.02 10.71 11.40 11.75 10.89 11.10 11.23 3 25.92 6.48 10.63 8.91 9.85 11.40 11.92 10.37 10.63 10.89 •4 34.56 4.32 8.99 6;91 8.29 10.71 8.64 Days 10.02 8.99 9.33 Mathematical Model On examination of equation 19 i t i s seen that each Y varies directly with A and inversely with T. The ratio of optimum machine size i s obtained by dividing i n turn the square root of each B value into the sum of the square roots of the B values. The solution of equation 18 w i l l therefore give a solution to the selection of sizes for machinery combinations operating in sequence. It should be noted that the machines with the lowest B. coefficient by being divided into the J sum of the B^ values w i l l always be selected i n the largest sizes compatible with the acreage and time available. A programme for evaluating equation 19 by Computer was developed and appears i n Appendix IV. the I.B.M. 7040 Digital The results for three different machine combinations appear in Appendix V. 25. DISCUSSION OF RESULTS The solution of the mathematical model as given i n Appendices V and VI clearly demonstrates the relationship between machine sizes, acreage and the time available for the performance of operations carried out in sequence. The weather directly affects the time available for the operation of field machines as already reported i n Tables III and IV. From the probabilities reported in Table IV, i t would appear that on the whole, the month of June i s least suited to making hay. Furthermore, during the month of June, to reduce the possibility of the hay crop being rained on larger machines w i l l be required especially i f a three day or four day harvesting cycle i s employed. The uncertainty of obtaining long spells of "open" days i n June i s the chief factor influencing the increase of mow drying i n the Abbotsford area. The crop i s cut, conditioned, windrowed and baled i n two days and the drying i s completed i n the barn using heated or unheated air. In this way less time i s spent i n the f i e l d and the probability of damage by rain i s appreciably reduced. July and August are both more suitable for haying operations. When Appendices V and VI are compared the effect of the weather probability on machine size i s very marked. 26. As previously stated when the probability factor i s applied, the available machine time i s decreased to allow for additional drying time and to compensate for the weather uncertainty. This results in relatively short periods being available especially for the three and four day cycles. The result i s that multiple units are required i n each case, and the impression i s given that hay cannot be made in the area without the use of multiple units. Another way of viewing the situation i s as follows :During the f i r s t ten days i n June, i t i s probable that i n five years out of ten, two or more "open" days w i l l be obtained and i n the other five years less than two. During the same period i t i s probable that in only one year in four w i l l three or more "open" days occur and in only one year i n eight w i l l four or more "open" days occur. The effect of the probability factor on the machine size i s clearly demonstrated i n .'.Appendix VI. It i s also noticeable that as the l o t size increases, that multiple units become necessary. It would indicate that where haying operations are done by one operator, that lot sizes should be ten acres or smaller regardless of which cycle i s practised. Tillage machines are often used i n tandem to operate on large areas. Hay harvesting machines because they are usually P.T.O. operated cannot be similarly utilised. Where multiple units are indicated, additional tractors and operators are required. As the mathematical model described applies specifically to a single operator carrying out the sequence of operations, i t may not furnish a least cost solution for multiple units. In trying to verify the results of the model some farmers were interviewed to compare the theoretical selections with farmers selections in the f i e l d . 27, Even when as many as 60 acres of forage were being harvested for hay, farmers were cutting i n lots of 10 to 15 acres. This method i s being used to insure against loss of the entire acreage i n case of rain. As quality hay must be cut before i t i s too mature, careful planning i s needed to ensure that the entire crop i s cut before i t has passed the optimum stage of growth. The start of the harvest may have to be advanced by a week or more, but the quality of the hay crop would not be as adversely affected i f the crop were a week or more past i t s optimum stage of growth. The effect of acreage on machine size i s quite a simple one. Each machine size varies directly with the acreage. An attempt was made to evaluate the mathematical model by comparing the theoretical selections with selections from some farms i n the Abbotsford area. Tw» groups of ten farmers were selected with the assistance of the British Columbia Department of Agriculture Extension staff i n order to reflect a wide spectrum of management techniques and efficiencies; group A^being the more efficient farmers. Eventually, only the information from six farmers i n each group was used, as the others were either unavailable for interviewing or even largely dependent on custom operators. Table I V shows the theoretical selections compared-with the selections found on the farms. In comparing the theoretical selections with farmers selections i n the f i e l d , several interesting selections were observed. using 6 1 In a l l cases, farmers were mowers. In some cases this size selection was justified, and i n others, this size had been selected on the basis of personal preference. Most farmers however stated that i n most years the forage crop was so dense that larger mowers became tangled and valuable time was wasted i n clearing the obstruction. Conditioners were only found i n 6 f t , sizes to match the flowers. Farmers complain that there i s a greater tendency for conditioners to become "plugged up" when they are 28. Table V:y * THEORETICAL AND OBSERVED MACHINERY SELECTIONS FQR A 10 ACRE BLOCK GROUP A Theoretical Selections GROUP B Observed Selections Theoretical Selections Observed Selections Mower 7' 6' 6' 6' Rake 7' 7' 8' Baler 4.5' 4.5' 7' 4.5' Mower 9' 6' 6' 6' Conditioner 7' 6» - - Rake 9' 9' 10' Baler 4.5' 4.5' 4.5' 4.5' Mower 9' 6' 6' 6' 10' 10' 7' 7' Rake 4.5' 9' Baler 4.5' 4.5' 4.5« 4,5' Mower 7' 6' 6' 6' Rake 7' 10' 7' 8' Baler 4.5' 4,5' 4.5' 4.5' Mower 6» 6« 6' 6' Rake 9' 8< 7' 8' Baler 4.5' 4.5' 4.5' 4.5' Mower 6' 6' 6' 6» Rake 9' 10' 10' 9' Baler 4.5' 4.5' 4.5' 4.5' ^Theoretical Selections Calculated From P =."1.0 29 not the same width as the mower. Side Delivery Rakes size were found to compare closely with the theoretical selection without weather considerations. cases the sizes selected were larger. In five In one case the farmer had bought the larger size i n an effort to insure against weather damage. In two cases, farmers depended on custom operators for their baling. A l l balers seen were of the 4«5i pick-up width and had been selected i n each case on farmers preference. In two cases, the farmers selections were far removed from any of the theoretical selections. It would appear that their selections were haphazard and were not based on past experience or analysis of the conditions under which they were farming. On examination of Table V, the observed selections for group B appear to take into account to some extent the variability of the weather. cases, rakes are larger than the theoretical selection. In three Group A, the more efficient group do not appear to have considered the weather variability as shown by their machine size selections. It would appear that i t i s i n some other aspect of management that Group B i s less efficient than group A. The total cost of a theoretical machine combination was compared with a combination of other machines having an overall, equal capacity, but made up of different sizes. The costs were calculated on the basis of an estimated l i f e of 12 years for mowers and rakes and 10 years for balers; repairs and maintenance at the rate of 12$, 7$ and 3.i$ for mowers, rakes and balers respectively. Interest on investment, housing and insurance were calculated on the basis of 6$, 1«6;$ and 0.4$ respectively for a l l implements. These costs were calculated to verify that the theoretical combination was a least cost combination of machines. 30. From the r e s u l t s set out i n Table V I I i t w i l l be seen that the c a p i t a l expended t o purchase the t h e o r e t i c a l combination of machines i s $2930 compared w i t h $3425 f o r the other combination. are S i m i l a r l y , the operating and f i x e d costs $655.13 f o r the t h e o r e t i c a l combination and $784.44 f o r the other combination. Table VII' COST COMPARISON OF THEORETICAL AND OTHER COMBINATION OF EQUAL CAPACITIES T h e o r e t i c a l Combination 6'MOWER 9'RAKE 4.5' BALER Purchase Price $ 480.00 650.00 1800.00 Repairs and M/tc. $/Year 57.60 45.50 55.80 Depreciation $/Year 40.00 Interest $/Year Other Combination TOTAL 7' MOWER 7' RAKE ..2930.00 520.00 5' BALER TOTAL 605.00 2300.00 3425.00 158.90 62.40 42.70 81.30 186.40 54.16 180.00 274.16 43.33 50.41 230.00 323.74 28.80 39.00 108.00 175.80 31.20 36.60 138.00 205.80 Insurance $/Year 1.92 2.60 7.20 11.72 2.08 2.42 9.20 13.70 Housing $/Year 5.88 10.05 18.80 34.73 9.68 36.80 .:. 54.80 TOTAL COST $ 655.31 8.32 784.44 31. This i s due to the fact that i n making the theoretical selection, the machine with the smallest B value i s always obtained i n the largest size compatible with the time available and the acreage. The machines with the larger B values are always selected i n smaller sizes as can be seen from equation 19. combination does i n fact represent a least cost combination. The theoretical 32 CONCLUSION The mathematical model described i s suitable for least cost machineryselections as advocated by McHardy (14). An attempt has been made to improve this method by the inclusion of a weather probability factor. Machine size selections made by farmers were similar to the theoretical selections when the probability p •» 1.0 was used, and l o t sizes were from 10 to 15 acres. When the lot sizes were increased to 30 or 40 acres, even with a probability factor of 1,0, the use of multiple units was indicated. When a probability factor of less than 1,0 was applied, the need for multiple units was indicated for l o t sizes as small as 10 acres. In cases where labour i s available on the farm, the use of multiple units presumes that this labour would be directed to assist with haying operations. No further expenditure for labour would be incurred, but rather there would be a temporary re-allocation of resources to assist i n the hay harvesting. Where smaller farms are concerned the multiple unit situation must be regarded i n a different light. If an operator and the associated equipment cannot perform a task of a given size i n a specified time, and i f labour i s not available to be diverted to assist, then one of three decisions have to be made. 1. Reduce the l o t size so that i t can be performed by one operator with the necessary equipment. 2. The owner or manager has to decide whether extra expenditure i n labour and machinery i s economically feasible. In other words, are the marginal returns equal to greater or less than the marginal expenditure. 3. The other possible solution i s to use the existing machines, but employ a different system for preserving the harvested forage during the period that the probability of having runs of "open" days i s low. Later i n the season 32a. when the probability of runs of "open" days increases, then haying operations may be performed without resorting to multiple units. The low probability of obtaining runs of "open days i n the early a and latter part of June indicate that: 1, Supplementary hay drying i s essential i f the crop i s to be harvested without the risk of damage by rain, 2. The crop must be harvested i n small lots, 3» Alternate methods of preservation and storage may have to be employed for the early cut« In July and August the chance of obtaining spells of "open" days is greater, but the crop s t i l l has to be harvested i n small lots or extra expenditure for additional labour and machines w i l l be incurred. It would appear that present machine selections used on Abbotsford dairy farms do not take into account the influence of the weather to the same extent as has been done by the application of the weather probabilities calculated in this study. 33. Literature Cited 1. Agricultural Engineers Yearbook 1964. pp. 230-234. 2. Barger, E. L., W. M. Carlton, E. G. McKibben, Roy Bainer, Tractors and their Power Units. John Wiley & Sons, New York, 1955. 3. Backenbach, Edwin, F. Editor Applied Mathematics for the Engineer. McGraw H i l l Book Co. Inc. New York, Toronto, London, 1956. 4. Bohstedt, Gustav. Nutritional Value of Hay and Silage as Affected by Harvesting, Processing and Storage. Agricultural Engineering: Vol. 25, No. 9, September 1944 pp. 337-88. 5. Bohstedt, Gustav. What is Quality Hay? FORAGES: Editors Hughes Heath and Metcalf. Iowa State College Press, Ames-Iowa 1953. 6. Borgman, Earl and D. B. Brooker. The Weather and Haymaking in Missouri. Agricultural Experiment Station, University of Missouri, 8777, 1961. 7. Brooks, C. E. P., and H. Carruthers. Handbook of Statistical Methods in Meteorology. London, Her Majesties Stationary Office 1953. 8. Casselman, T. W., and Robert C. Fincham. How Effective are Hay Conditioners? Iowa Farm Service Nol. 15 Nos. 5-6 1960. 9. Churchman, C. West, Russell, L. Ackoff, E. Leonard Arnoff. Introduction to operations Research. John Wiley and Sons Inc., New York. 1964. 10. Crossfield, D. C , and E. D. Woodward. Dairy Farm Organization in the Fraser Valley of British Columbia, 1961. Dominion Economics BranchCanada Department of Agriculture. 11. Farm Mechanization. Users Test Reports. Vol. 17. No. 191 July 1965. 12. Kronick, Frank N. G. Equipment for grassland farming. Engineering: Vol. 32 No. 1 January 1951. 13. Lange, Dierk. A Cost Comparison of Forage Harvesting Machine Operation on Lower Fraser Valley Dairy Farms, 1959. Undergraduate Thesis Agricultural Mechanics Dept. University of British Columbia. 14. McHardy, F. V. An Investigation of the Application of Programming Techniques To Farm Management Problems, 1964. Unpublished Ph.D. Thesis. University of Edinburgh. 15. Monthly Records. Meteorological Records in Canada. 1944-64. Dept. of Transport. Meteorological Branch, 315 Bloor St. West, Toronto 5, Ontario. Seven Pick-up Balers. Compared. Agricultural 34. 16. S o k o l n i k o f f , I . S . and E . S . S o k o l n i k o f f . Mathematics f o r Engineers and P h y s i c i s t s . McGraw H i l l Book Co. I n c . , London and New Y o r k , 1 9 4 1 . 17. W o l f f , E. Farm F o o d s . English E d i t i o n . Gurney and J a c k s o n , London 18. T r a n s l a t i o n b y H.B. C o u s i n s , 1895. 1961 Census. Canada. Dominion Bureau o f S t a t i s t i c s . B u l l e t i n 5. 3-4. V o l . 5, P a r t 3. Census D i v i s i o n . - _ H H 1 1 1 1 1 i r~ J D • 1 1 I A.TM D •I-X- -: - VI ( &y. 1 I - 1 cn I 3 uU — i - 1 - i i Ji ' •', n'n i T } - 1 — I — 1 ' in 1 - o A A - - r J (JU - 1 ... - r 1 /• / J•J - 1 "^ i 1 1 1 )• 1 _C.ap a c. 1 — ict •es ic3 U E - — J' i g ure !. 1 1 IC 1 t i 31u l -B. X ie 1 i ,d - 0 1 -7„ L u c uu i -5 C 3!r t M ..] V s -c •p"ad 1:.. a L-l r .8OK. it * I c i t »ri r V —t - - -7r5f 1 - - 1 i j j 1 I i — A RET,mTiV. Lfir: 0 :-, 1 • -) - I — rI— - 1 - 1 - - - - -- - - - - 1 If n <£lJUU 1 1 "r 11 )U (0 - - T r u - .... 1 "T A -- o 1 - 1 1 - J 1 If 1 in ! - 1 -- ... - r J) 3 2 Gi yc - Fi 1 0weir ii e_3. P ConeIi t i_ iesr. I a s 2<J- en d = LUUU - 1 — t-]- k r ( A t ?£ V5 m il - - - i c apacit y • j 1 41.» i -i ,nd - f- "J.c.i e - >C0 •1 i 1 i 1 •8• AJP:?i - - 0 rv t, ) X - ... - - •- - - n r\ OUU - - •f T 1 - n r 0 00 - 1 n j I ^ - Li » o a - - Lnn -j - - - onn u £. 1 /if I J/ \ :•£ p ac L - ,- _ ( ur j s TI 11 -- . 1 1 II , De.1:..\ :er V. J Lake• 0 Vj n.- x | t<:i •es/ f:.cst. _5 i ;»/_ a Hi3- 2 * c 5" -i ir Ca ?-« U :1t - • •e-ff-L< ;enG - ] - - i Y 1 •- 1 ! -- 1 ! I < 1 1 9 - -I I-:{- k]>I>E ( it •) 1 - ... - - - - - - - 1 ri n ri J UU J - - - - 1 n 4-1 cn -- n j~i -- - 1^ Z 00J 1 T 1 A | u - A t < > — t l n nn KJ - i I 1 - ; 1 1 >- i- •- es/. Hotir Pa C. it\ fie ur.2..5..- i ? 1 f B, J 0.. i«tl r •- .C.cst j »u—\J 1- 1 1 -5- J .2L 90 ~ M >_ .-r ;-. a if ;_ Ca xac:i t y- -7A »• | •f-f-U — n •6301! - - 1 40. APPENDIX II Period and Duration of Periods (Days) of Dry Weather During June, July and August at Abbotsford, B. C. June July No. o£ Duration of No. of Duration of No. of Duration of Periods (Days) Periods Periods(Days Year Eeriods Periods (Days) Periods UgU 10 9 3 3 12 10 13 4 14 19 4 7 16 3 46 5 3 5 5 16 4 21 47 3 3 6 10 8 4 6 21 9 10 3 7 3 4 12 6 6 3 7 14 5 3 3 19 20 7 3 7 8 8 1944 45 48 49 50 51 4 3 8 7 52 3 3 6 53 3 6 4 4 7 5 5 18 25 3 10 5 7 22 5 7 3 16 3 3 14 4 41. APPENDIX II (cont'd) i Year No. of Periods June Duration of Periods(Days) No. of Periods 54 1 5 2 55 2 2 56 1 10 3 4 10 5 4 14 4 3 13 4 8 3 5 5 20 3 12 3 3 9 4 57 58 3 4 59 2 60 4 61 2 62 3 63 3 64 2 4 7 21 55 2.619 .29 350 X a 6.363 4.23 2 2 1 July Duration of Periods(Days) 6 21 12 9 3 15 4 10 31 August No. of Duration Periods Periods(Days) 1 10 2 17 H 14 9 9 8 2 2 2 2 17 8 2 1 31 2 2 4 15 9 14 2 2 4 2 50 2.38 . 307 4 4 3 9 3 7 461 9.220 6.66 3 3 3 47 2.238 . 28 6 11 12 3 15 12 3 4 5 10 10 6 3 4 5 441 9.382 6.18 42. APPENDIX III Frequency of Dry Spells at Abbotsford Period Length of dry spell June 1-10 (days) Observed Frequency (0) 2 3 4 5 6 7 8 9 10 45 21 11 6 3 ff =. 172, N = 86, 2 3 4 5 6 7 8 9 10 50 25 14 10 7 6 4 4 2 Observed Cumulative Frequency 86.0 43.0 21.5 10.8 5.4 2.7 1.4 0.7 0.4 86 41 21 11 5 2 2 2 2 p = 0.50, X Computed Computed Comulative Frequency Frequency (C) 2 43.0 21.5 10.7 5.4 2.7 1.3 0.7 0.3 0.2 (0-C)' C .093 .011 .008 .066 .033 1.300 .700 .300 .200 = 2.711 for 8 degrees of freedom 122 72 47 33 23 16 10 6 2 122.0 78.1 50.0 32.0 20.5 13.1 8.4 5.4 3.5 43.9 28.1 18.0 11.5 7.4 4.7 3.0 1.9 1.2 .847 .341 .888 .021 .021 .359 •333 2.321 .533 R = 331, N « 122, p = 0.64, x = 5.691 for 8 degrees of freedom 2 2 3 4 5 6 7 8 9 10 51 26 17 12 6 3 2 1 1 R = 281, m 2 3 4 5 6 7 8 9 10 R = 391, 119 68 42 25 13 7 4 2 1 119.0 69.0 40.0 23.2 13.5 7.8 4.5 2.6 1.5 50.0 29.0 16.8 9.7 5.7 3.3 1.9 1 .1 0.64 .020 .310 .002 .545 .015 .027 .005 .005 .130 119, p = 0.58, X- = 1.059 for 8 degrees of freedom 2 63 31 21 15 7 6 3 3 3 N = 152, 152 89 58 37 22 15 9 6 3 p = .62, X = 152.0 94.2 58.4 36.2 22.4 13.9 8.6 5.3 3.3 2.704 57.8 35.8 22.2 13.8 8.5 5.3 3.3 2.0 1.2 .467 .643 .064 .064 .264 .092 .027 .500 .583 for 8 degrees of freedom. 43. APPENDIX I I I (cont'd) Period July Length of dry s p e l l (days) 11-20 Observed Frequency (0) Observed Cumulative Frequency 74 43 27 19 14 11 8 5 4 205, 79 47 29 24 15 13 9 7 5 205 131 88 61 42 28 17 9 4 2 3 4 5 6 7 8 9 10 R = 585, N = J u l y 21-30 2 3 4 5. 6 7 8 9 10 p = 0.66, * 2 3 4 5 6 7 8 9 10 August 10-19 R- = 471, 172 105 65 43 29 19 11 6 3 p = 0.63, 66 39 24 16 8 7 6 5 3 = 174, 174 108 69 p = 45 29 21 14 8 3 9.64, 69.7 46.0 30.4 20.1 13.2 8.7 5.7 3.9 2.5 (o-c) c .207 . 195 .380 .060 .048 .608 .928 .310 .900 = 3.636 f o r 8 degrees o f freedom X = N = 172, N 11.2 7.3 5 40 22 15 10 8 5 3 3 2 3 4 5 6 7 8 9 10 38.8 25.6 16.9 149 102 73 49 34 21 12 67 RC = 453, 205.0 135.3 89.3 58.9 228 Rj = 673, N = 228, p = 0.68, J u l y 31 August 9 2 Computed Computed Cumulative Frequency Frequency (c) x 2 ' 228.0 73.0 49.6 33.7 22.9 15.6 10.6 7.2 5.4 3.2 .493 . 136 .655 .052 .023 .543 .450 .474 1.012 174.0 62.6 .184 .071 155.0 105.4 71.7 48.8 33.2 22.6 15,4 10.0 3.838 f o r 8 degrees of freedom .181 172.0 63.6 108.4 40.1 . 002 .430 68.3 25.3 43.0 15.9 .227 27.1 10.0 .000 .458 17.1 6.3 .250 10.8 4.0 6.8 2.5 .100 4.3 1.6 1.225 = 2.873f o r 8 degress of freedom 111.4 71.3 45.6 29.2 18.7 12.0 7.7 4.9 X2 40.7 25.7 16.4 10.5 6.7 4.3 2.8 1.8 .112 .009 . 595 .013 .672 1.728 .800 4.184 f o r 8 degrees of freedom. 44. APPENDIX III (cont'd) Period Aug.20-29 Length of dry spell (days) Observed Frequency (0) 2 3 4 54 31 18 12 12 8 5 6 7 8 145 91 10 18 10 55 2 453, N = 145, P » 0.6j>, 33.0 16.8 9.1 5.9 3.8 10.9 7.1 4.6 X - 50.7 94.3 61.3 39.8 25.9 42 30 2 Computed Computed Cumulative Frequency Frequency (C) 145.0 60 5 3 9 R = Observed Cumulative Frequency 3.364 21.5 13.9 2.5 1.7 ^ (0-C) -C .214 .121 .569 .259 .924 .747 .378 .100 .052 for 8 degrees of freedom. 45. APPENDIX IV I.B.M. 7040 Digital Computer Program. To Calculate Least Cost Machinery Combinations . $ JOB $ FORTRAN 1. DIMENSION TIME (100), B (100), LA (10), BJ (100), FINAL (10) 2. DATA IA 3. READ, N.M. (N = No. of pieces of equipment; M = times equipment available). 4. READ, (B), 1 = 1 , N) 5. SUM = 0.0 6. DO 100 3 = 1, N 7. BJ (J) = SQRT (B (J) ) 10. 100 SUM = SUM + BJ (J) 11. READ ( TIME ( l ) , L = 1, M) 12. DO 110 I = 1, Mo 13. WRITE ( t, 1) I, TIME ( I), IA 14. 1 FORMAT ( t HO TIME, 13, 1H =, F8. 2, 30X, 5H,ACRES/4X, 1HB, 2 x 10 I 12/) 15. ST = S/TIME (I) 16. D0 110 J = 1, N 17. P = ST/BJ (J) 20. D0 105 K = 1, 10 21. ACRES - K * 10 22. 105 FINAL K = P * ACRES 23. WRITE ( t, 2 ) B (J), FINAL 24. 2 FORMAT ( IX, F6.1, 10G 12.4) 25. 110 CONTINUE 26. STOP 27. END $ENTRY. 46. APPENDIX V MACHINE SIZES IN FEET SELECTED ON TIME NOT AFFECTED BY WEATHER PROBABILITY Activity I - Use of mower, rake, baler in sequence. Activity II - Use of mower, conditioner, rake, baler in sequence Activity III - Use of mower-conditioner in tandem, rake, baler in sequence. Days 2 10 3 4 2 Acreages 20 3 4 2 30 3 4 Activity 6 <D* 6 (1) 4.5(1) I II III * 7 (1) 7 (1) 4.5(1) 6 (1) 6 (1) 7 (1) 4.5(1) 9 7 9 5 (2) (2) (2) (1) 6 (1) 6 (1) 9 (1) 7 (1) 4.5(1) 4.5(1) 7 9 5 (1) 6 (1) (2) 10 (1) (1) 4.5(1) 9 (1) 7 (1) 9 (1) 4.5(1) 6 (2) 6 (2) 7 (2) 4.5(1) 9 (1) 11 (1) 4.5(1) 9 (3) 7 (3) 9 (3) 4.5(2) 9 7 9 5 (3) (3) (2) (1) 6 (2) 7 (2) 9 (3) 9 (3) 4.5(2) 4.5(2) Numbers in parentheses refer to the numbers of units of the size indicated. 4Z*> APPENDIX VI MACHINE SIZES IN FEET SELECTED ON TIME AFFECTED BY WEATHER PROBABILITY Activity I - Use of mower, rake, baler in sequence Activity II - Use of mower, conditioner, rake, baler in sequence Activity III - Use of mower-conditioner in tandem, rake, baler in sequence JUNE 1 - 10 Days P Activity 2 0.50 10 3 0.25 III * 4 0.125 9 (3)* 9 (3) 4.5(2) I II Acreages 20 2 3 0.50 0..25 4 0..125 9 9 5 2 0.50 30 3 0.25 4 0.125 9 (8) 9 (8) 5 (5) (6) (6) (3) 9 7 9 5 (2) (2) (2) (1) 7 (3) 7 (3) 8 (3) 4.5(2) 9 6 9 5 (4) 9 (4) 7 (4) 10 (2) 5 (5) (5) (5) (3) 9 9 9 5 (6) 9 (6) 7 (6) 11 (3) 5 (8) (7) (7) (4) 7 9 5 (1) 9 (1) (2) 11 (2) (1) 4.5(2) 7 11 5 (3) 7 (4) 11 (3) 5 (4) (7) (4) 7 11 5 (4) 7 (7) 11 (4) 5 (7) (9) (6) Numbers in parentheses refer to the number of units of the size indicated. JUNE 11 - 20 AUGUST 10 - 19 Days P Activity 2 0.64 10 3 0.41 7 (2) 7 (2) 4.5(1) I II Ill 4 0.26 Acreages 20 2 3 0.64 0.41 4 0.26 2 0.64 30 3 0.41 9 10 5 9 (3) 10 (3) 4.5(2) 7 (2) 7 (2) 7 (2) 4.5(1) 9 (3) 9 (3) 7 (3) 7 (3) 10 (3) 10 (3) 4.5(2) 4.5(2) 9 7 11 5 7 (1) 7, (1) 7 (2) 8 (2) 4.5(1) 5 (1) 7 (2) 7 (2) 10 (3) 11 (3) 4.5(2) 5 (2) 7 (3) 7 (4) 11 (4) 11 (5) 4.5(3) 5 (3) 6 (2) 6 (2) 7 (2) 4.5(1) 4 0.,26 (4) 9 (4) (4) 7 (5) (4) (5) (3) 48'. APPENDIX VI (Cont'd) JUNE 21 - 30 Days P Activity 2 0.58 10 3 0 .34 9 9 5 I II III 4 0.20 2 0.58 Acreages 20 3 0,.34 4 0.20 2 0.58 30 3 0.34 9 11 5 9 (4) 9 (4) 5 (2) (2) (2) (1) (2) 7 (2) 7 (2) 9 (1) 5 (2) (2) (2) (1) 9 7 10 5 (3) 9 (3) 7 (3) 9 (2) 5 (4) (4) (4) (2) 9 7 11 5 (5) 9 (5) 7 (4) 11 (3) 5 (5) (5) (5) (3) 7 (1) 7 8 (2) 9 4.5(1) 5 (1) (2) (1) 7 (2) 7 10 (3) 9 4.5(2) 5 (2) (4) (2) 7 11 5 (3) 7 (4) 11 (3) 5 (3) (5) (3) 7 6 7 5 4 0..20 (6) (5) (3) JULY 1 - 1 0 Days P Activity 2 0.62 10 3 0.38 III 2 0.62 7 (2) 8 (2) 5 (1) I II 4 0.24 Acreages 20 3 0.38 9 10 5 7 7 8 5 (2) (2) (2) (1) 9 (3) 7 (4) 10 (3) 4.5(2) 6 (1) 7 7 (2) 8 4.5(1) 5 (1) (2) (1) 6 (2) 7 (2) 10 (3) 11 (3) 4.5(2) 5 (2) 7 (2) 6 (2) 7 (2) 4.5(1) 4 0.24 9 7 11 5 (3) (3) (3) (2) 2 0.62 30 3 0.38 4 0.24 9 (5) 10 (5) 4.5(3) (3) (3) (2) 9 (4) 7 (5) 11 (4) 4.5(3) 9 7 10 5 (5) (5) (5) (3) 7 (3) 7 11 (4) 11 4.5(3) 5 (3) (4) (3) APPENDIX VI (Cont'd) JULY 11 - 20 Days P Activity 10 2 0.66 0.44 6 (2) 8 (2) 4.5(1) I II III 4 0.29 Acreages 20 2 3 0.66 0.44 7 (2) 6 (2) 7 (2) 4.5(1) 7 (2) 6 (2) 7 (2) 4.5(1) 4 0.29 30 3 0.44 7 (3) 9 (3) 4.5(2) 9 (3) 7 (3) 9 (3) 4.5(2) 9 (3) 7 (3) 9 (3) 4.5(2) 4 0.29 9 (4) 10 (4) 4.5(3) 9 (4) 7 (4) 11 (4) 4.5(3) 6 (2) 6 (2) 9 (3) 9 (3) 4.5(2) 4.5(2) 6 (1) 6 (1) 7 (2) 7 (2) 4.5(1) 4.5(1) 2 0.66 9 (4) 7 (4) 11 (4) 4.5(3) 6 (3) 6 (3) 10 (4) 10 (4) 4.5(3) 4.5(3) JULY 21 : - 30 Days P Activity 2 0.68 10 3 0.46 6 (2) 7 (2) 4.5(1) I II III 4 0.31 Acreages 20 3 2 0.68 0.46 6 (2) 6 (2) 7 (2) 4.5(1) 6 (2) 6 (2) 7 (2) 4.5(1) 6 (1) 6 (1) 7 (2) 7 (2) 4.5(1) 4.5(1) 4 0.31 2 0.68 30 3 0.46 9 9 5 7 (3) 8 (3) 4.5(2) 7 (3) 7 (3) 9 (3) 4.5(2) 7 (3) 7 (3) 9 (3) 4.5(2) 6 (2) 6 (2) 9 (3) 9 (3) 4.5(2) 4.5(2) 4 0.31 9 (4) 7 (4) 10 (4) 4.5(3) 9 (4) 7 (4) 10 (4) 4.5(3) 6 (3) 6 (3) 10 (4) 10 (4) 4.5(3) 4.5(3) (4) (4) (2) so;.. APPENDIX VI (Cont'd) JULY 31 - AUGUST 9 Days P Activity 2 0.63 10 3 0.40 III 2 0.63 7 (2) 7 (2) 4.5(1) I II 4 0.25 Acreages 20 3 0.40 7 (2) 6 (2) 7 (2) 4.5(1) (3) (3) (3) (2) 30 3 0.40 4 0.25 9 (4) 11 (4) 5 (3) 9 (4) 7 (4) U (4) 4.5(3) 6 (2) 7 (2) 10 (3) 10 (3) 4.5(2) 5 (2) 6 (1) 7 (1) 7 (2) 8 (2) 4.5(1) 4.5(1) 2 0.63 9 (3) 10 (3) 4.5(2) 9 (3) 9 7 (3) 7 10 (3) 10 4.5(2) 5 7 (2) 6 (2) 8 (2) 4.5(1) 4 0.25 9 7 11 5 (4) (4) (4) (3) 6 (3) 7 (3) U (4) 11 (4) 4.5(3) 5 (3) AUGUST 20 - 29 Days P Activity 2 0.65 10 3 0.42 7 (2) 7 (2) 4.5(1) I II III 4 0.27 Acreages 20 2 3 0.65 0.42 4 0.27 2 0.65 30 3 0.42 9 (3) 9 (3) 4.5(2) 9 (4) 10 (4) 4.5(3) 7 (2) 6 (2) 7 (2) 4.5(1) 9 (3) 9 (3) 7 (3) 7 (3) 9 (3) 10 (3) 4.5(2) 4.5(2) 9 (4) 7 (4) 10 (4) 4.5(3) 6 (1) 7 (1) 7 (2) 8 (2) 4.5(1) 4.5(1) 6 (2) 7 (2) 9 (3) 10 (3) 4.5(2) 5 (2) 6 (3) 7 (3) 10 (4) 11 (4) 4.5(3) 5 (3) 6 (2) 6 (2) 7 (2) 4.5(1) 4 0.27 9 (4) 7 (4) U (4) 4.5(3)
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Mathematical model for the selection of haying machinery Jeffers, John Percival Weldon 1966
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Title | Mathematical model for the selection of haying machinery |
Creator |
Jeffers, John Percival Weldon |
Publisher | University of British Columbia |
Date Issued | 1966 |
Description | Surveys of hay harvesting machinery in the Lower Fraser Valley of British Columbia indicated wide variation in sizes of machines employed on farms of comparative size. Quality hay depends on 1. the types of forages grown 2. the stage of growth at which the crop is harvested and 3. the influence of weather as it effects curing. Bleaching, leaf shatter, and leaching of soluble nutrients are the worst hazards to which the crop is exposed. The system of harvesting will affect the time the crop is exposed to the effects of the weather. Methods that reduce the time needed for curing in the field tend to produce a better quality hay. Factors affecting the selection of least cost combinations of machines used in sequence are 1. the cost which bears a linear relationship to the capacity of the machines, 2. the area they have to service and 3. the time available for the performance of the operations. The time available for a sequence of operations to be performed in this case is a function of the weather. A study of the weather of the Lower Fraser Valley indicated that in any given ten day period during the months of June, July and August, the probabilities of two or more, three or more or four or more open days for field curing hay are constant. Using this linear cost capacity relationship, acreages from ten to one hundred acres, and the time available obtained from weather probability data, a mathematical model is derived to select least cost machinery combinations for hay harvesting in the Lower Fraser Valley. A computer programme for the I.B.M. 7040 digital computer is also developed. |
Subject |
Hay-Harvesting |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-07-19 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0059210 |
URI | http://hdl.handle.net/2429/36171 |
Degree |
Master of Science - MSc |
Program |
Chemical and Biological Engineering |
Affiliation |
Applied Science, Faculty of Chemical and Biological Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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