J. agric. Engng Res. (1986) 33, 13-22

Combine Speed Strategies in Cereal Harvesting.Part 2: Adjustment for Weather Variability

M. B. McGEcHAN*; C. A. GLASBEYt

Existing models of cereal harvesting have derived a common optimum combine speed for everyharvesting day on a particular farm. This study is an assessment of the economic benefits ofselecting a different speed for each day of the harvest, taking into account the history of previousweather. Both a simple simulation approach, and a dynamic-stochastic programming approachwhich incorporates a forecast of future work-days, have been developed.

Overall, the results showed small benefits from a strategy based on a selected daily speedcompared with single constant speed operation, when averaged out over a number of years.However, substantial benefits were shown in the occasional very wet harvests, when combiningcould take place on only a small number ofdays, and in some other exceptional situations.

1. Introduction

Operational Research (OR) models ofcereal harvesting, such as those developed by Audsley andBoyce' and by Philips and O'Callaghan," determine the optimum values for combine size andspeed for the crops and conditions on a particular farm. This two-part paper describes adap­tations of the models to assess the value of adjusting the combine speed for changes in circum­stances on the farm. Part 13 considered adjustments for the crop parameters (such as straw yield)in different harvest years, or for different crops, varieties or fields in the same year. Part 2 assessesthe benefits of a daily adjustment of speed to allow for weather variations. The adjustmentmakes use of information about the weather history and the amount of crop remaining to be cuton each day throughout the harvest. This study indicates the potential value of a program whicha farmer can run himself, on his own microcomputer, to recalculate the optimum combine speeddaily as the harvest progresses.

Existing models include unvalidated equations for determining work-days from weather data.So far, in work with adaptations of these models,3-g the criterion suggested by Audsley andBoyce using daily rainfall data has been assumed. By comparing survey data on commercialcombine working periods (Mcffechant") with weather records, Glasbey and McGechan"developed a new criterion, which they considered to be an improvement on the arbitrary Audsleyand Boyce criterion. In the current study, parallel assessments of benefits ofa selected dailyspeed strategy were carried out using both Audsley and Boyce's and Glasbey and McGechan'scriteria for determining work-days from weather. Since the effect of vatriations in other modelparameters has already been examined thoroughly.t:" a single set of values was assumed here.Front end (i.e. shedding and cutter bar) losses were determined from the equations suggested by

Audsley and Boyce; a farm with 200 ha ofcereals with mean yield of both grain and straw 5 t/ha,and grain price £120/t, was assumed. As in Part I, a range of alternative sizes of combine wasassumed, each with a set of costs and values of the parameters of the exponential threshing lossequation suggested by Philips and O'Callaghan" (Part I, Table I).

'Scottish Institute of Agricultural Engineering. Bush Estate. Penicuik, Midlothian EH26 OPI!. Scotland

tAgricultural and Food Research Council Unit of Statistics. University of Edinburgh. King's Buildings. Edinburgh EII9 3JZ. Scotland

Received 20 February 1984; accepted in revised form 19 December 1984

13

0021-8634/86/010013+ 10 $03.00/0 © 1986The British Society for Research in Agricultural Engineering

14

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COMBINE SPEED STRATEGIES

Notationarea ofcrop cut on day i, haexpected cost with optimal strategyday length, hfield efficiency, % (assumed = 75%)grain cost, £jthourly cost of harvesting, £jh (costs of labour, fuel and repairs")front-end loss, tjhathreshing loss, tjha

} minimum value of bracketed expression with respect to variation in S only

probability of weather state I following weather state koverall work-rate, hajhcombine forward speed, kmjhcutter bar width, myield ofgrain, tjha

£+1iHIjj-AkI

Subscriptsday following finish day (assumed to be 8th October)day of harvest (integer)day following day iarea ofcrop remaining to be cut at beginning ofday, ha (integer in range 0-200)area ofcrop remaining to be cut at end of day, ha (integer in range 0-200)weather state on day i (integer in range 1-8, as in Table 1)weather state on day i+ 1 (integer in range 1-8, as in Table 1)

Table 1Conditional probabilities

Weather Ability to combine Probability ofbeing able tostatus on last three days combine on day i+ I

numberon day i i-2 i-I i Audsley and Boyce Glasbey and McGeehan

I 0 0 0 0·30 0·472 I 0 0 0·25 0·503 0 I 0 0·23 0·454 I I 0 0·34 0'535 0 0 I 0·62 0·606 I 0 I 0·65 0·647 0 I I 0·68 0·698 I I I 0·78 0·77

2. Derivation of combining work-days from weather data

2.1, Audsley and Boyce's combining work-days criterionAudsley and Boyce' assumed that no combining would take place if the discounted sum of pastrainfall exceeded the arbitrary value of 1·27 mm. The discounted sum was the rainfall in the past24 hours plus 20% of the previous day's discounted sum of past rainfall.

Glasbey and McGeehan" showed that combining work-days derived from daily rainfall usingthe Audsley and Boyce criterion were not in complete agreement with those observed in a surveyduring six harvests on up to six commercial farms to the south of Edinburgh."?

M. n. MCGECHAN; C. A. GLASBEY 15

2.2. Glasbey and M ctiechan's combine work-days criterionGlasbey and McGeehan" investigated alternative criteria for deriving combining work-daysfrom daily rainfall data. They proposed a new criterion which states that combining can takeplace when rainfall in the previous 24 h is less than 1·4 mm. This value gave the best fit to thesurvey data; no significant improvement in fit was obtained by including proportions of rainfallfrom earlier days.

2.3. Start andfinish datesWhen running their model, Audsley and Boyce specified the day when the crop reached 30% m.c.(w.b.) and the moisture content at which combining could start; from this, the combining startday was computed from a sixth order polynomial for the moisture content curve, dependent ontime but not weather. Grain was regarded as a total loss if it remained uncut 70 days after 1stAugust, or ifits moisture content rose above 30% m.c. (w.b.) towards the end of the season.

For this study, combining was assumed to start with amoisture content of21 % m.c, (w.b.) on20th August, an average date observed in the survey. No information about the grain moisturecontent throughout each harvest was recorded in the survey, so the last available date forcombining was calculated as 7th October by assuming the Audsley and Boyce moisture contentcurve with 10th August as the 30% m.c. (w.b.) day [which gave 21% m.c. (w.b.) on 20th August].

2.4. Length ofwork-days recordsfor simulationAudsley and Boyce' determined costs and optimum combine speeds by simulation over combinework-days derived from 10 years' rainfall datafor six different areas of England. Mcfiechan"has shown that simulation over longer periods gives results very different from those using 10years' data, indicating that 10 years' rainfall is inadequate to obtain a reliable indication ofoptimum speed.

For this study, a set of 38 years' daily rainfall was available for a site at Penicuik, under 8 kmdistant from the furthest farm. Simulations were carried out over work-days derived from all 38years of this data set.

3. Variable speed strategies

3.1. Simulation approachThe method used to derive the optimal single constant speed, used extensively in the authors'earlier work,3-9 was to calculate the mean total annual cost of combining the total area of cropfor a range of typical constant speeds, by a simulation over a number of years of work-daysderived from rainfall data," and then to select the speed with the least cost. The main costsdependent on combine speed are those associated with threshing losses, front-end losses andgrain which remains uncut if the harvest is not completed in the time available." Since the first ofthese losses increases with increasing speed, and the others decrease with increasing speed, theselected speed is a compromise between these losses.

The simulation approach to deriving an optimal daily speed was a natural extension of theapproach in the single constant speed case. For each day of harvesting in each of 38 years, theoptimum combine speed was selected which minimized an estimated mean total annual cost ofharvesting the area ofcrop remaining uncut at the beginning of that day. This estimated cost wasderived by a simulation over the work-days derived from all 38 year's daily rainfall data; thework-days considered in each year were only those from the current day to the last availabledate for combining. If rainfall on the current day in the current year prevented combiningtaking place, no cost would be incurred; otherwise, the actual cost of harvesting on that day wascalculated at this optimum speed. The total annual cost was then calculated by repeating theprocedure and summing for all the days of the harvest. Finally, the mean total annual cost wasobtained for the 38 year period.

16 COMBINE SPEED STRATEGIES

3.2. Dynamic-stochastic programming approach3.2.1. Weather forecastingThe simulation approach made use of the weather history from previous years only. Theeffectiveness of a selected daily speed strategy should be capable of improvement from a currentweather forecast based on weather earlier that season, so this was considered.

Glasbey and McGeehan11 examined the time dependence of days when combining could takeplace (denoted by one), and days when it could not (denoted by zero). Eight "weather states"were used to represent the possible combinations of ability to combine on each of the three pre­vious days. For each weather state, they determined conditional probabilities, i.e. the probabilityof being able to combine on the current day, using both Audsley and Boyce's and Glasbey andMcGeehan's criteria for deriving work-days from daily rainfall data (Table 1). The proportion ofdays when combining could take place was 0·51 using the Audsley and Boyce work-days criterionand 0·59 using Glasbey and McGeehan's criteria.

3.2.2. Dynamic-stochastic programmingDynamic-stochastic programming (DSP) is a method of mathematical optimization in an uncer­tain environment, when a choice has to be made at a range of points in time. The optimal choiceis that which minimizes the expected cost from each point to the final time. Working backwardsin time, expected costs are determined recursively, to take account of the probability of beingin each of a number of possible states at the next time point (WhittleP}, An important simpli­fication is when the number of states is finite. Values of certain parameters must be integers toachieve this, so the range and increment of parameter values must be chosen carefully. Thismakes the problem manageable in computer time and workspace.

3.2.3. Formulation ofDSP modelFor the current problem, using the conditional probabilities for each weather state (Table 1), themodel was formulated to estimate the optimum speed on each day, which minimizes the expectedoverall cost from the beginning of that day to the end of the season, Ck,i,i as follows:

(1) on days when combining was possible (i.e. k= 5-8)

8

Ck,i,i=mjn {(Lf+L,)AG+AH/R+ 'L,Pk,IC"j-A)+ I},1=1

whereA=RDifj~RD or A=jifj<RD, and R=O'OOI SFW;(2) on days when combining was not possible (i.e. k= 1-4)

8

C, .. ="Pk ,C, "+1' and,),1 l..J , ,1.' ,1=1

(3) at the end of the harvesting period the value of the uncut grain

C',i,E+l =j YG.

For each of the eight weather states (Table 1), for all integer areas of crop between zero and200 ha, and for each harvesting day recursively from the last to the first within the permissiblerange, the expected cost Ck,i,i was calculated for each of a range of speeds, so that the optimum(least cost) speed could be selected. A three-dimensional array was thus created, containing theoptimum speed for each weather state, area of crop, and harvest day. Speeds considered werethose at which a whole number of hectares was cut in a day, i.e. multiples of 1000/(FWD).

1\1. B. MCGECHAN; C. A. GLASBEY 17

4. Assessment of the benefits of variable speed strategies

4.1. General method ojassessmentMean costs, according to both Audsley and Boyce's and Glasbey and McGechan's criteria forderiving work-days from weather, were calculated for each strategy considered. The benefits ofthe selected daily speed strategy were assumed to be the difference in the costs derived from the 38year simulation and those derived from operating at the single optimum constant speed. Thus,any assumptions in the strategies which were inaccurate or unvalidated would affect the costs ofboth similarly, so their difference would be largely unaffected. For the nspapproach, the costswere obtained by carrying out a 38 year simulation, using the optimal daily speeds selected fromthe array previously created and the set of work-days derived from daily rainfall data.

4.2. A/eon benefits orer an extendedperiodThe annual costs, averaged over 38 years, of harvesting a 200 ha crop using all the speedstrategies considered, based on both work-days criteria, are listed in Table 2. The savings fromthe selected daily speed strategy compared with single constant speed operation were greatest forthe smaller combine sizes, and greater using the Audsley and Boyce work-days criteria; this gavea smaller number of available days than Glasbey and McGechan's criterion. Thus the savingswere greatest when the combine was stretched in terms of the size of the task relative to the size ofthe combine. For a size 4 combine, the optimum size for this crop, the annual saving accordingto the Audsley and Boyce work-days criterion was £300 when derived by simulation, rising to£400 when derived by the nspapproach: However, using Glasbey and McGechan's work-dayscriterion, the benefits were less than £100 for either approach. The effect of the selected dailyspeed strategy was not sufficiently great in any situation to change the optimum size ofcombine.

4.3 Benefits in individualyearsThe year by year costs of harvesting using either an optimum single constant speed, or theselected daily speed strategy are listed in Table 3. By far the most substantial benefits from the

Table 2Mean annual costs. Crop size 200 ha, grain and straw yleld 5tfha

Audsley and Boyce Glasbey and Mctlechan

Combining work-days criterion AverageTotal Annual cost So ring

AverageTotal Annual cost Savingspeed speed

(kmjh} loss (t) (£100) (£100) {kmlh} loss (I) (£100) (£/OO)

Size] Single constant speed 5·50 97·6 176 5·00 66·9 141Daily speed. simulation 4·24 93·0 171 5 4·49 60·8 136 5Daily speed. DSP 4·10 90'7 169 7 4·43 60·2 136 6

Size 3 Single constant speed 4·75 89·7 172 4·25 61·8 141Daily speed. simulation 4·39 85'7 167 5 3·95 57·2 136 4Daily speed. DSP 4·25 83·9 166 6 3-92 56·9 136 4

Size 4 Single constant speed 5·25 69·7 150 4·75 51·5 130Daily speed. simulation 5·12 66·7 147 3 4·32 46·7 129 IDaily speed. DSP 4·94 65-4 146 4 4·46 49·9 129 I

Size 5 Single constant speed 5·75 61·3 155 5·00 48·1 141Daily speed. simulation 5'73 58·3 152 3 4·71 47·3 141 0Variable speed. DSP 5·49 57·4 152 3 4·83 47·5 141 0

Size 6 Single constant speed 5·75 56·2 154 5·00 46·7 145Daily speed. simulation 6·08 53·3 152 2 4·75 45·9 145 0Daily speed. DSP 5·94 52·9 152 2 4·92 46·3 145 0

Table 3Year by year costs for alternative speed strategies. Size 4 combine 38 year simulation

Audsley and Boyce combiningwork-dayscriterion Glasbeyand AIctlechan combilling work-dayscriterion

Ami/able Single Selecteddaily speed. Selected daily speed.Available Single Selecteddaily speed. Selecteddaily speed..

Y('ar combining constantspeed simulation DSP combining constantspeed simulation DSPwork-days work-days

Cost Ullclll COSI Uncut Saving Cost UIICIII Saving Cost Ullclll COSI Uncut Saving COSI Uncut Saving(£100) grain (I) grain (£100) (£100) grain tstoo) (£fOO) grain (I) grain (£100) (£/00) grain (£fOO)

(I) (I) (I) (I) (I) (I)

1943 20 143 0 149 0 -5 146 0 -3 29 131 0 131 0 I 131 0 I1944 24 133 0 132 0 I 131 0 2 29 130 0 128 0 0 128 0 11945 26 134 0 133 0 I 132 0 2 31 127 ·0 127 0 1 127 0 11946 20 140 0 154 0 -14 151 0 -II 25 134 0 135 0 -1 133 0 01947 34 127 0 124 0 3 124 0 3 37 124 0 123 0 I 123 0 I\948 23 142 0 149 0 -7 147 0 4 29 134 0 134 0 0 133 0 I1949 38 128 0 126 0 2 126 0 3 41 124 0 124 0 0 124 0 \1950 II 459 289 386 203 73 401 210 58 17 158 18·5 144 0 14 147 0 II1952 26 129 0 127 0 2 126 0 3 31 126 0 125 0 0 125 0 I1953 27 133 0 131 0 0 131 0 3 29 128 0 128 0 1 128 0 11954 16 139 0 143 0 -4 140 0 -I 24 131 0 131 0 0 130 0 01955 28 131 0 130 0 2 129 0 2 33 128 0 127 0 0 127 0 11956 22 143 0 147 0 -4 146 0 -3 31 133 0 133 0 I 133 0 I\957 28 135 0 133 0 2 133 0 2 32 13\ 0 131 0 0 130 0 0

-00

("J

o=::e:c-Zmen-emttl

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1958 21 136 0 134 0 I 134 0 2 31 128 0 127 0 I 127 0 I1959 42 131 0 129 0 1 128 0 3 42 128 0 127 0 I 127 0 11961 23 136 0 135 0 1 134 0 2 28 131 0 131 0 0 130 0 11962 18 146 0 161 0 -15 158 0 -12 30 135 0 134 0 1 134 0 I1963 23 139 0 139 0 0 138 0 1 32 129 0 128 0 I 128 0 I1964 22 137 0 139 0 -2 137 0 1 25 133 0 133 0 0 133 0 01965 12 391 226 '307 140 84 323 155 68 21 135 0 135 0 0 135 0 01966 25 135 0 133 0 2 132 0 2 33 128 0 128 0 0 128 0 01967 25 132 0 130 0 2 129 0 3 29 128 0 127 0 0 127 0 I1968 22 132 0 130 0 2 129 0 3 28 127 0 127 0 0 127 0 01969 27 133 0 132 0 I 131 0 2 34 129 0 128 0 0 128 0 I1970 23 136 0 134 0 2 134 0 2 29 129 0 129 0 0 129 0 01971 33 133 0 131 0 2 131 0 3 38 129 0 129 0 1 129 0 11972 47 127 0 124 0 3 124 0 3 47 124 0 123 0 1 123 0 11973 34 130 0 128 0 2 127 0 3 38 126 0 125 0 0 125 0 01974 22 135 0 134 0 1 133 0 2 27 129 0 128 0 0 128 0 01975 20 132 0 131 0 1 130 0 2 28 126 0 126 0 0 126 0 01976 27 128 0 126 0 2 125 0 3 34 124 0 124 0 1 124 0 11977 20 141 0 145 0 -4 142 0 -1 27 132 0 131 0 1 131 0 11978 19 139 0 140 0 -I 138 0 I 28 132 0 131 0 0 131 0 11979 27 132 0 131 0 0 130 0 3 34 127 0 127 0 1 127 0 I1980 25 132 0 130 0 2 130 0 2 31 127 0 127 0 0 127 0 01981 24 127 0 125 0 2 124 0 3 31 124 0 123 0 1 123 0 11982 22 138 0 137 0 0 136 0 2 26 132 0 132 0 1 132 0 1

overall 150 IH 147 % 3 146 % 4 130 0-49 129 0 I 129 0 1

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-\0

20 C O MB I N E SPEED STRA T EG IE S

3 5 7 9 \I 13 15 17 19 21 23 25 27 29 I 3 5 7September Octob er

1950

--r""'l- ,i --

~-- --, r'----,.,., ......;.;' Ft-:' r~=;-,,,

IIIII

•III

o20 22 24 26 28 30 1

AugU5t

4·0

2·0

4,0

8·0

o20 22 24 26 28 30 1 3 5 7 9 II 13

AugU5t September1973

6·0

.c 2·0<,

E'"~..0.

'"..c

:0E8 6·0

Fig. J. Daily combine speeds throughout harvest in tll"O sample years, Audsley and Boyce work-days criterion. -.dynamic- stochastic programming approach:---, simulation approach.

selected daily speed strategy accrued in the very wet har vests, 1950 and 1965 using the Audsleyand Boyce work-days criterion, and 1950 alone using Glasbey and McG eehan's criterion. Inthese years, a substantial quantity of grain was not cut in the available time when operating at theoptimal single constant speed; the selected daily speed strategy gave a cons iderable reduction inthe quantity of uncut grain using the Audsley and Boyce work-days criterion, and reduced it tozero using Glasbey and McGeehan 's criterion. In practice, even if he were otherwise adopting aconstant speed strategy, a farmer would probably take drastic measures 'in a very wet year toreduce the quantity of uncut grain, such as hiring another combine (at additional cost) or increas­ing his combining speed. Hence, the benefits of the selected daily speed strategy relative to such amodified constant speed strategy may not be quite as great as have been calculated. Nevertheless,since the optimum speed is critical , there would still be a benefit from operating at a correctlycalculated rather than an arbitrarily increased speed, although it would be difficult to calculatethe value of this benefit.

Th e daily selected speeds for both approaches to derivation in two sample years are illustratedin Fig. 1. This shows a progressive increase in speed throughout the harvest in a very wet season(1950), and a decrease in speed in a fairly dry season (1973), with slight differences between thedaily speed derived by each of the two approaches in both years.

4.4. Benefits ill other circumstancesThe benefits of the DSP strategy relative to constant speed operations were investigated for thr ee

M. B. MCGECHAN; C. A. GLASBEY

Table 4Annual costs of combining, in critical years and mean of 38 years, where work-days

differ from expectation

Annual costs (£100) Annual

Actual work-daysCritical Arai/able savingyears work-days Constant DSP {romDSP

speed strategy ([JOO)

(I) Rainfall in 1950 16 221 155 66previous 24hours 1965 16 223 159 64less than 0·88 mm

Mean of38 years 135 131 4

(3) Audsley and Boyce 1950 II 532 476 56work-days formula 1954 16 212 181 31

1965 12 470 392 78

Mean of38 years 153 149 4

21

cases where the relationship between work-days and weather was not as expected. For a size 4combine, apparent optimum speeds were derived from historical rainfall data according to theGlasbey and McGechan criterion, but in fact the criterion was incorrect as follows:

(1) and (2) Combining could take place when rainfall in the previous 24 hours was less thaneither 0·88 mm or 1·96 mm; these values represent the 95% confidence limits in Glasbey andMcGechan's" estimate of the threshold rainfall of 1·4mm.

(3) Combining work-days were related to daily rainfall using the Audsley and Boyce formula.In each case, costs were higher at the apparent optimum speed than they would have been at thetrue optimum speed, but to a much greater degree at a single constant speed than with the dailyselected speed strategy. In cases 1 and 3, average annual benefits of the daily selected speedstrategy were about £400, compared with single constant speed operation, with benefits of up to£7800 in individual years (Table 4). This demonstrates the capability of the selected daily speedstrategy if the work-days criterion is incorrect and predicts more work-days than are actuallyavailable.

5. Conclusions

A selected daily speed strategy derived by dynamic-stochastic programming, which incorporatessome weather forecasting, performed slightly better than a selected daily speed strategyderived by simulation with no forecasting, and better still than a single constant speed strategy.However, the average benefits over a number of years were very small.

Glasbey and McGeehan's criterion for deriving combining work-days from weather data gavemore available work-days than Audsley and Boyce's criterion, with a consequent reduction in thebenefits from a selected daily speed strategy. Although the mean annual saving from the selecteddaily speed strategy to take account of weather variability was small (about £400 using theAudsley and Boyce work-days formula), it was greater than that from an automatic speed con­trol system under the same conditions." It was also greater than that from a strategy to takeaccount of crop variability if the crop variability was typical, but not in the minority of instanceswhen the crop variability was high for a particular reason." A selected daily speed strategy to

22 COMBINE SPEED STRATEGIES

take account of weather variability would be easier to implement than one to take account ofcrop variability, because it is easier to record whether combining has taken place each day andthe amount of crop cut than to make accurate measurement of crop parameters such as strawyield.

Despite the small benefits from the selected daily speed strategy when averaged over a numberof years, there appeared to be large benefits in the occasional very wet harvests, when the com­bine could work on only a small number of days. Also, if the relationship between work-daysand weather was not as expected, very substantial benefits were shown in occasional criticalyears.

A farmer may wish to employ a selected daily speed strategy, even for a relatively small averagebenefit, if it makes use of computer equipment which he already has and so incurs little or noadditional expense. He may also favour this strategy because it gives substantial benefits indisastrous years, for which any small additional cost in the remaining years can be regarded as aninsurance premium. Large benefits of this type would be obtained even if the program were runonly in a crisis situation, rather than every day.

References1 Audsley, E.; Boyce, D. S. A method of minimising the cost of harvesting and high temperature grain

drying. Journal ofAgricultural Engineering Research 1974, 19 (2): 173-189:2 Philips, P. R.; O'Callaghan, J. R. Cereal harvesiing-a mathematical mode!' Journal of Agricultural

Engineering Research 1974, 19 (4): 415-433 .3 McGeehan, M. B. Combine speed strategies in cereal harvesting. l. Adjustment for long term crop

variability. Journal ofAgricultural Engineering Research 1985;31 (3): 243-2544 ·MeGeehan, M. B.; Glasbey, C. A. The benefits of different speed control systems for combine harvesters.

Journal ofAgricultural Engineering Research 1982,27 (6): 537-5525 Glasbey, C. A.; McGeehan, M. B. Threshing loss stochastic variability on combine harvesters. Journal of

Agricultural Engineering Research 1983,28 (2): 163-1746 McGeehan, M. B. A preliminary investigation into the benefits of some alternative speed strategies in

cereal harvesting. Dep. Note SINj369 Scottish Institute of Agricultural Engineering, Penicuik 1983(unpub!.)

7 McGeehan, M. B. A parametric study of cereal harvesting models. l. Critical assessment of measureddata on parameter variability. Journal ofAgricultural Engineering Research, 1985,31 (2): 149-158

8 McGeehan, M. B.. A parametric study of cereal harvesting models. 2. Analysis of sensitivity toparameter variability. Journal ofAgricultural Engineering Research 1985,31 (2): 159-170

9 McGeehan, M. B. Details of a parametric study of cereal harvesting models. Dep. Note SINj376,Scottish Institute ofAgricultural Engineering, Penicuik 1983(unpub!.)

10 McGeehan, 1\1. B. Information used for testing combining work-days criteria. Dep. Note SINj396,Scottish Institute ofAgricultural Engineering, Penicuik, 1984(unpubl.)

11 Glasbey, C. A.; Mcflechan, M. B. The assessment of combining work-days criteria and forecastingmodels. Journal ofAgricultural Engineering Research 1986,33(I): 23-31

12 Whittle, P. Optimization over time, dynamic programming and stochastic control, Volume l. Chichester:John Wiley and Sons, 1982