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Mathematical Models Can Make Farmers More Efficient
Maurice A. Vodounon *
Many of the problems faced by farmers in Sub-Saharan Africa can be solved
analytically. Mathematics modeling, mathematical theory, operations research,
and practical applications are useful tools for predicting crop yield or for
planning how much land should be cultivated. Some of these methods could be
included as practical examples in mathematics curricula, allowing students to
gain an interest in these topics.
A case study is used here to illustrate how a real-life farming problem can
be modeled as a set of mathematical equations and used to advise farmers on how
best to cultivate their land. There are many possible formulations of this
problem based on the assumptions made, and readers are encouraged to try out
some solutions of their own.
New Applications for Mathematical Modeling
In developing countries, local daily life problems are rarely mentioned in
the mathematics classroom. Mathematics modeling is often restricted to large
scale research projects. Egypt, India, Benin, and Niger use mathematical
modeling for national and regional economic long-term planning. South American
countries use it for water control and analysis of water reservoirs.
Mathematical modeling can also be used as an aid for long-term planning. These
studies are carried out by international organizations or consulting agencies
in cooperation with government departments in developing countries. However,
the results of these large scale projects often reflect the point of view of
the organization financing the research, which may differ from the farmers'
point of view. Very often, there is little understanding of the problems that
poor farmers struggle with in the village. Village organizations, farmers
cooperatives, and district planners do not have access to the organizations that
decide which research projects receive funding. Yet there are many urgent
problem areas in the everyday life of farmers that could benefit from such
research.
Agriculture employs the largest percentage of the population in most
developing countries. Some of the basic problems that farmers face include:
• What crops should be grown?
• What area should be sown to minimize the risk of producing too little food
in the event of insufficient rainfall?
• Should the emphasis be on subsistence or cash crops?
• Should a village or district use ox-drawn plows or tractors?
• Is small-scale irrigation possible? What are the economic advantages?
• What size of plot should be used for communal farming?
Some of these questions could be answered by using mathematics models and
operations research methods. Staff and students in universities are in a
position to develop models and present their findings to farmers in village
organizations. The mathematical analysis could bring up many important points
for discussion. For example:
• What is best for a village may not be best for the government.
• The selling price of the produce is not always in reasonable proportion to
the amount of agricultural labor used to produce it.
• Can the farmers carry the risks of fluctuating world market prices
themselves, or would it be better to accept a lower income that was stabilized
by the government?
Case Study: How Many Acres Should Be Plowed?
Seboto is a community of Porto-Novo in the Republic of Benin, where about
one thousand families live. They grow maize as a major crop for subsistence.
During the growing season, farmers use ox-plows, but they have been discussing
whether they should hire a tractor to plow and whether they should use
herbicides to avoid weeding. The purpose of this investigation is to help them
make that decision. In quantitative terms, how many acres have to be plowed by
a tractor to make it cost-effective, and for how many acres should they buy
herbicides?
The study will examine the farming activities of an average family in the
village with 5 people. Studying a family unit instead of the village as a whole
is acceptable because all families use the same methods to grow maize. It is
assumed that each family owns an ox-plow. The main quantities and constraints
are first laid out as variables to analyze this problem.
The land available to the family can be plowed by tractor or by ox. The
farmers may also choose to seed the land manually or apply herbicide. The
number of acres assigned to each of these choices is shown in Table 1.
The total acreage for maize can be expressed as the sum of land cultivated
using each of the above methods.
Xtotal = Xtm + Xth + Xom + Xoh
We assume there is enough land available to the farming family, so there is
no land constraint. To formulate the labor constraint, the timing of plowing,
planting, weeding, and harvesting must be considered due to the seasonal nature
of the farming activities.
Farming Schedule and Labor Constraints
Plowing and planting are done for three months between October and December.
Plowing season = Planting season = 3 months/yr.
Constraints when using a tractor to plow one acre: A tractor needs 1 hour to
plow an acre.
Tractor plow rate = 1 hr/acre (1)
The tractor can be hired for 8 hours per day.
Tractor time = 8 hr./day (2)
Ten person-hours of family labor are needed for clearing, preparation, and
planting of each acre.
Clearing rate = 10 pers. hr. /acre (3)
The plowing is done twice, the first time for clearing, the second time for
planting.
Number of plow passes = 2 passes/acre (4)
Constraints when using an ox-plow to plow one acre: An ox-plow needs 12
hours to plow an acre.
Ox plow rate = 12 hr./acre (5)
An ox-plow can be used 4 hours per day.
Ox time = 4 hr./day (6)
Forty person-hours of family labor are needed for clearing, preparation, and
planting of each acre.
Clearing rate = 40 pers. hr. /acre (7)
Plowing is done twice.
Number of plow passes = 2 passes/acre (8)
Weeding is done for two months in January and February.
Weeding season = 2 months/yr.
Constraints when weeding is done manually: The weeding takes 100
person-hours.
Weeding labor rate = 100 pers. hr. /acre (9)
Constraints when weeding is done using a herbicide: It takes 5 hours to
spray one acre during the planting period.
Spraying rate = 5 hr. /acre (10)
Harvesting is done for two months in April and May.
Harvesting season = 2 months/yr.
Harvesting is done manually, and it takes 30 person-hours to harvest an acre
of maize.
Harvesting labor rate = 30 pers. hr. /acre (11)
The above information will help formulate a set of labor constraints, taking
into account that one month has 25 working days.
Plowing
The tractor must plow the acreage (Xtm+Xth) twice at a rate of one acre per
hour. The use of the tractor is limited to 8 hours a day for the 3-month (75
day) planting season. This gives the following constraint for the use of the
tractor:
(2 passes) x (Xtm + Xth) + 75days x 8hrs/day (12)
The ox-plow takes 12 hours to plow each acre, and can only be used 4 hours
per day. Therefore the constraint for use of the ox-plow is:
(2 passes) x (12hr.) (Xom + Xoh) + 75days x 4hr./day
(13)
Clearing
The human labor constraints are based on 5 family members working 10 hours
per day during planting (Oct.-Dec.), weeding, and harvesting. During each
three-month season, they work the land twice; clearing it after each plowing
pass.
(5 pers.)[2 x 3 x (Xtm + Xth) + 24(Xom + Xoh)
+ (Xth + Xoh)] + 75days x 10hr./day x 5pers.
30(Xtm +Xth)+120(Xom + Xoh)+5(Xth
+ Xoh) + 3750 (14)
The first term in Equation 14 is the time needed to clear and plant when
using a tractor. The second term is the corresponding time when using an
ox-plow. The final term represents the time taken to apply herbicide.
Planting
When a tractor is used, 10 person-hours are needed for planting each acre
(Equation 3). With an ox-plow, 40 person-hours are needed for planting
(Equation 7). The constraint for human labor during the planting season shown
in Equation 15 is based on Equations 3, 7 and 10.
10(Xtm + Xth) + 40(Xom + Xoh) +
5(Xth + Xoh)
+ 75days x 10hr./day x 5pers. (15)
Weeding
The weeding takes 100 person-hours during the weeding period in January and
February. The human labor constraints for weeding will be:
100(Xtm + Xom) + 50days x 10hr./day x 5pers. (16)
Harvesting
Based on the number of hours needed to harvest maize (Equation 11), The
constraint for the harvesting period in April and May is represented in
Equation 17.
30(Xtm+Xth+Xom+X oh) +
50daysx10hr./dayx5pers. (17)
Maize Yields, Production, and Costs
The yield of maize per acre is 1,000kg, independent of the method of plowing
or weeding. The consumption on maize by the family is estimated as 50kg per
month, or 600kg annually. Therefore, the food requirement constraint can be
described as:
1000(Xtm + Xth + Xom + Xoh) +
600kg (18)
The objective is to maximize the total production total production of maize:
MAX: 1000(Xtm + Xth + Xom + Xoh)
(19)
Net profit is the difference between the revenue function and the cost
function from selling the surplus maize at 60 CFA per kilogram. The cost of
hiring a tractor to plow one acre is 300 CFA, and the cost of the herbicide is
also 300 CFA. The cost of the ox-plow is very small and can be omitted.
The maximum profit function is given by:
MAX: 60{1000(Xtm + Xth + Xom + Xoh)
_ 600}
_ 300(Xtm + Xth) _ 300(Xth + Xoh)
(20)
We have now defined a solvable set of equations that can be solved using
linear programming methods. While it is possible to calculate a solution by
hand using variables and graphs, there are software packages readily available
to solve such sets of equations. In this case a software package called EUREKA
was used to solve the constraints in Equations 12-20 to provide us with the
maximum revenue or the minimum cost.
Results
Results from the analysis are shown in Table 2.
The focus of the study was to determine whether or not a tractor should be
hired and herbicides used. The results suggest it would be beneficial to hire a
tractor (Xtm = 50.27 acres), and apply herbicides as well (Xth
= 22.63 acres). These numbers represent two dominant figures in the profit
function:
MAX: 59700Xtm + 59400Xth + 60000Xom
+ 59700Xoh _ 36000 (21)
The herbicides should be bought and used on about 32 acres (22.63 acres +
9.75 acres) when weeding is done manually. The result shows the significance of
manual work in relation to the machine (tractor). As a matter of fact, manual
labor is involved during the planting period (October, November, December), the
weeding period (January, February), and the harvesting period (April and May).
Using these values of cultivated land, production of maize is found to be
much higher than consumption. With a total cost of 36,000 CFA, the estimated
profit can be found from Equation 21.
Profit = 59700x50.27 + 59400x22.63 + 60000x0.62 +
59700x9.75 _ 36000 = 4,928,616 CFA
The results from a model like this one could be disseminated through
training and mass media. It is also worthwhile to refine the model by
investigating some other alternatives. For instance, what would be the impact
on the revenue obtained if tractors were not hired? What would happen if only
half as much herbicide was bought? Does it make much difference to maximize the
yield or the net revenue? What would be the income per family? This case study
illustrates that it is possible to use some straightforward equations to
provide guidance to small farmers.
_______________
Dr. Maurice A. Vodounon is a professor of mathematics and Computer
Science at John Jay College of Criminal Justice, The City University of New
York, 445 W 59th Street, New York, NY 10019. Phone:(212)237-8860, FAX:(212)
237-8742, E-mail: VAMJJ@CUNYVM.CUNY.EDU.
References
1 Kolawole', S. A. & Boko, M. "Le Benin" EDICEF., Paris 1983.
2 Hitch, C. "Operations Research and National planning, A
Dissent," Operations Research, 5, 718-723, 1957.
3 Schweigman, S. "Doing Mathematics in a Developing Country," by
Tanzania Publishing House p. 63-67, 1979.
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