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.

X_{total} = X_{tm} +
X_{th} + X_{om} + X_{oh}

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 (X_{tm} + X_{th}) + 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.) (X_{om} +
X_{oh}) + 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 (X_{tm} +
X_{th}) + 24(X_{om }+
X_{oh}) + (X_{th} + X_{oh})] + 75days x 10hr./day x 5pers.

30(X_{tm} +X_{th})+120(X_{om
}+ X_{oh})+5(X_{th} +
X_{oh}) + 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(X_{tm} + X_{th}) +
40(X_{om} + X_{oh}) + 5(Xth +
X_{oh})

+ 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(X_{tm} + X_{om}) + 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(X_{tm}+X_{th}+X_{om}+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(X_{tm} + X_{th} +
X_{om} + X_{oh}) + 600kg (18)

The objective is to maximize the total production total production of maize:

MAX: 1000(X_{tm} + X_{th} +
X_{om} + X_{oh}) (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(X_{tm} + X_{th} +
X_{om} + X_{oh}) _ 600}

_ 300(X_{tm} + X_{th}) _
300(X_{th} + X_{oh}) (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
(X_{tm} = 50.27 acres), and apply herbicides as well
(X_{th} = 22.63 acres). These numbers represent two dominant figures in the profit function:

MAX: 59700X_{tm }+ 59400X_{th
}+ 60000X_{om}

+ 59700X_{oh }_ 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.

*_______________
*

**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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