Maximizing profit in the poultry farming sector : An application of the robust linear programming technique

Managing the limited resources of money, time and space poses a real problem in the livestock business in general and the poultry sector in particular. The stochastic linear programming with recourse will permit the formulation of a mathematical model that could easily be solved in order to attain the objective of maximizing profit but this will be hindered by the presence of some parameters like demand and supply which have no predetermined probability distributions. In an attempt to address the problem of unavailable probability distribution functions, this research is aimed at proposing a robust linear programming model which can handle problems and others in similar circumstances with the need of probability distributions.The robust model thus constructed is based on a modification of the stochastic model by Soyster. An application of this model on a real life data produced results showing an increase in profit made by a local poultry farmer from 241,485 FCFA to 362,580FCFA representing an increase in profit of more than 50% in over that obtained when using the ordinary Linear Programming Technique. The belief therefore is that, if models like this are implemented, not only would the livelihood of the poultry farmers be improved, but it will go a long way to better the economy and satisfy the ever increasing need of poultry products by the communities.


INTRODUCTION
Livestock systems occupy about 30% of the planet's icefree terrestrial surface area (Steinfeld et al., 2006) and are a significant global asset with a value of at least $1.4 trillion.The livestock sector is increasingly organized in long market chains that employ at least 1.3 billion people globally and directly support the livelihoods of 600 million poor smallholder farmers in the developing world (Thornton et al., 2006).Keeping livestock is an important risk reduction strategy for vulnerable communities, and livestock are important providers of nutrients and traction for growing crops in smallholder systems.Livestock products contribute 17% to kilocalorie consumption and 33% to protein consumption globally (Rosegrant et al., 2009).From the World Bank 2009 report, livestock *Corresponding author.E-mail: mathakong@yahoo.fr.
Author(s) agree that this article remain permanently open access under the terms of the Creative Commons Attribution License 4.0 International License systems have both positive and negative effects on the natural resource base, public health, social equity and economic growth.Despite this, livestock is one of the fastest growing agricultural subsectors in developing countries.Its share of agricultural GDP is already 33% and is quickly increasing.This growth is driven by the rapidly increasing demand for livestock products; this demand being driven by population growth, urbanization and increasing incomes in developing countries (Delgado, 2005) while traditional livestock systems contribute to the livelihoods of 70% of the world's rural poor, increasingly the emerging large-scale operations with sophisticated technology and international trade cater for the rapidly growing markets for meat, milk and eggs.The AGA's program on animal production focuses primarily on smallscale dairying, small-medium scale poultry and, to a lesser extent, on small ruminant systems which can make a significant contribution to improved livelihoods and local economic development.This will be achieved through the provision of topical information, guidance and technical support to farmers.

Problem statement
In the developing world where more than 600 million small holder farmers and in Cameroon where about 80% of the population depends on agriculture for livelihood (Nji, 2004), with millions relying on small-medium scale poultry, there is need to provide technical support to these farmers in order to boost production.There has been rapid global expansion of consumption of poultry products and which is expected to continue to grow.Increasing productivity, especially in the small to medium scale production systems, is currently constrained by lack of skills, knowledge and appropriate technologies compounded by the inability to manage limited resources of money, space and time, especially as demand and supply cannot be determined in advance.
The production and commercialization of livestock like every economic activity requires the rational use of limited resources of money, space and time available at any given time with the objective of maximizing profits.The Simplex methods of Dantzig (1955) would have been a better tool to solve the problem if the parameters were static and could be modelled by Linear Programming.This is not the case with the livestock sector especially with the small-medium scale poultry farming where most parameters such as demand and supply cannot be determined ahead of time.The significance of uncertainty has prompted a number of works addressing random parameters in tactical level supply chain planning involving production and distribution of products in different sector of businesses; examples can be seen in Alidaee and Kochenberger (2005), Cooper and LeBlanc (1997), Powell and Topaloglu (2003), Yu and Li (2000), and Van Landeghem and Vanmaele (2002).On the other hand, Mustafa (2005) and Ziemba and Vickson (1975) suggested that, under some assumed probability distribution of the uncertain parameters, a stochastic model could handle the optimization.What happens in the absence of any assumed probability distribution of these parameters?This is the concern of these researchers.
Therefore, the objective of this research was to propose a robust linear programming (RLP) model that could be used to handle optimization problems in the livestock business sector in general and in poultry farming in particular in the absence of probability distribution of the uncertain parameters involved.In the case of the small-medium scale poultry farming, the main concern shall be to determine how many species of each type and age of birds to be kept at a particular period in order to maximize the benefit without violating the constraints of money, time, and available space while using the robust linear programming model.

BACKGROUND AND RELATED WORK
Most often in a linear program problem, the parameters are random variables whose probability distributions cannot be determined ahead of time within each production period.This has thus left most decision makers in the field Stochastic programming (SP) and Robust Optimization (RO) are the main alternative techniques to deal with uncertain data both in a single period and in a multiperiod decision making process.The main difficulty associated with the former is the need to provide the probability distribution functions of the underlying stochastic parameters.This requirement creates a heavy burden on the user because in many real world situations, such information is unavailable or hard to obtain (Birge and Louveaux, 2011;Ruszczinski and Shapiro, 2003).On the other hand, RO addresses the uncertain nature of the problem without making specific assumptions on probability distributions.The goal is to seek solutions which are insensitive to variations of certain data of the problem without destroying all or part of the potential optimum.This is made possible by using the technique of Soyster (1973) or Minoux (2009Minoux ( , 2010)).Other reflections on treating problems on uncertainty can be seen in Dupacova (1998).Soyster (1973) introduced uncertainty as a column in the linear program.It showed that a linear programming problem can be written in the form: Soyster (1973) equally showed that, assuming ̃ , the optimal robust solution boils down to the solution of the initial linear programming problem by assigning values to the coefficients of the random variables in the matrix A as follows: Minoux (2009) the uncertainty on the second member follows the robust formulation according to Soyster (1973) by integrating the vector b in the matrix of constraints and adding an (n+1) th variable of decision .Thus, the problem becomes: ∑ Therefore, in order to build a Robust Linear Programming Model, it suffices to developed the stochastic model and then modify it following the proposal of Soyster (1973).

Description of the Problem
The objective here is to use the stochastic linear programming to maximize profits in a smallmedium size poultry farming project while respecting customers' demands.The stochastic linear programming is useful here since most of the parameters are random, especially commands and productions.Other parameters shall be included and treated during modelling.Since it is a poultry farming project, the main objective shall be to find the number of birds of each species and age that the farmer must have at a period t to maximize the benefit while respecting the orders from customers.
It is worth noting that several other constraints could come into consideration out of that of command and production.These could include without restriction the constraints of money, time, and space.The farmer may have to rent more space if demand is high or may even be forced to buy from elsewhere if there is not enough to meet up with the request of the customers.

Modelling
The intension here is to build a stochastic/robust linear programming model that will maximize the profits while respecting the constraints mentioned in the problem description.Assuming that orders are made by period that is the duration of an order is a period, assume also that the species of the same age have the same weight.Suppose now that: is the variable that represents the number of species of type and of age a to be raised in the owned space during the period t to meet the command of the period .
is the variable that represents the number of species of type and of age a to be raised in the rented space during the period to meet the command of the period .
is the variable that represents the number of species of type and age a to be raised by a subcontractor during the period to meet the command of the period .
represents the money available to the farmer during the period t.
represents the space available to the farmer during the period t.
represents the maximum time available to the farmer during the period t.
is the money needed to care for a species of type and of age raised during the period to satisfy orders of the period .
is the money needed to care for a species of type and of age raised during the period and stored during the period .is the space needed to contain a species of type i and of age a during the period t.
is the time needed to take care of a species of type and of age during the period t.
is the number of species of type and of age produced by a species of type and of age during the period t.This parameter turns random because it is not always possible to know the number of young ones a bird can produce, thus the association with the effect of a random variable ̅ .and of age commanded during the period t.This parameter becomes random because production may not meet the command or customers may not respect their commands, hence the association with the effect of a random variable ̅ . is the number of species of type and of age commanded during the period but which will be delivered during the period .Since this parameter is random, a random effect variable ̅ is associated with it.
. During the period t, it could happen that customers may either still be in possession of the produce that was delivered during the period t-1 or that are to be delivered in subsequent periods.
In such a case, let say: and of age remaining from the period which must be kept in the farmers space in order to maximise profit of the period as well as subsequent periods.
and of age remaining from the period which must be kept in the rented space in order to maximise profit of the period as well as subsequent periods.
and of age remaining from the period which must be sub contracted in order to maximise profit of the period as well as subsequent periods.
and of age remaining from the period which must be continued with during the period t.Since have been declared as random variables, and it happens that production does not meet the commands for the period t, then let be the number of species of type i and of age a that has to be bought to satisfy the command and let be the profit on a species bought to satisfy the command, then this two quantities shall be used to construct the demand function.
After describing the possible but not all different variables and parameters needed for the construction of the stochastic linear programming model, the construction of the constraints is required.

Constraint on money
The maximum money available to the farmer during the period t is .For a species of type i and age a, one could spend in some cases and in others, thus, for , one would spend Therefore for n species of type i with ages ranging from 1 to T, the expected spending should be: during the period t using the farmers owned space.
Similarly, let a unit of space be rented at , thus to raise in the rented space, the expected spending is ( for the rent.Analogous to the reasoning in Equation (1), one should spend in the rented space for n species of type i with ages ranging from 1 to T ∑ ∑ (2) It follows that, the money to be spend by the farmer during the period t is the total amount of money in Equations ( 1) and (2).Expenditures should not exceed the maximum money available during the period t.Thus the constraint on money can be written as:

Constraint on the space
The maximum space available to the farmer during the period t is .To raise species of type i with age a requires units of the owned space, thus , will require [( ) ) , and accordingly for n species of type i with ages ranging from 1 to T, the farmer would need amount of space during the period t using the owned space.The use of the owned space should not exceed the maximum space available to the farmer during the period t.The constraint on space can therefore be written as follows: (4)

Constraint on the time
To care for a species of type i and of age a, unit of time is needed.The assumption is that, the farmer takes care of the stocks both in the owned and rented spaces.This means that it will require time to take care of a bird.Hence for n species of type i with ages ranging from 1 to T, will require a total time given as follows: Since the time to be used should not be more than the maximum available time of the period t, it implies the constraint on time can be written as:

Constraint on the commands
Here the constraints, are in two folds; a constraint on the orders placed during the previous periods and the period t to satisfy periods t and later and a constraint to satisfy orders placed on the species of age zero, for periods t and later.
In the first case, the quantity must equal the sum of orders placed during the previous periods, the period t and later and can be evaluated by ∑ ̅ .This constraint is then written as: For the second constraint, + ̅ is the number of species produced by one species of type i and of age during the period t, then for which represents the sum of species of type i and of age raised in the owned and rented spaces, those subcontracted and the stock of period t-1.The total production would be given as + ̅ .Sometimes, this production might not meet the command and will therefore require making up for the deficit.Since the production has to be equal to the orders of species of type i and that will be of age j at the period t+j, this constraint can be written as:

Constraint on the stock
The stocks in owned, rented spaces plus the stock sub contracted during the period t-1, should equal the total stock for the same period, hence the constraint is written as: The objective function The objective function is that which will maximise profits after the model has been solved taking into consideration the identified constraints.Therefore if: : is the profit obtained on a species of type i and age a raised in owned space during the period t.
: is the profit obtained on species of type i and age a of the stock of the period t-1elevated in owned space during the period t.
: is the profit obtained on one species of type i and age a raised in rented space during the period t.
: is the profit obtained on species of type i and age a of the stock of the period t-1elevated in the rented space during the period t.
:is the profit obtained on species of type i and age a raised Onabid and Tchoffo 91 by a subcontractor during the period t.
: is the profit obtained on species of type i and age a of the stock of period t-1 raised by a subcontractor during the period t.
Then the total profit would be written as: It could happen that the production does not satisfy the demand, which means other birds have to be bought from the market to make up the deficit.
As the number of species of type i and age a to pay for the deficit is a random variable, stochastic linear programming and particularly the L-Shaped method requires calculating the mathematical expectation of profit for the recourse function.As is the profit achieved on a species of type i and age a needed to make up for the deficit, the recourse function would then be given as: Therefore the objective function would be written as:

The Stochastic model
With the definition of the objective function, the Stochastic Linear Programming model for solving this problem can now be written as:

The Robust model
The robust equivalence of this problem as proposed by Soyster (1973) would be written as: Table 1.Represents unit cost of production for each specie in the owned and rented spaces for each period.

REAL LIFE APPLICATION
The proposed model was then tested with data collected from a poultry farm in the West Region of Cameroon.In this farm, two species of birds were raised; table birds (broilers) and traditional birds (Gallus Gallus) and this is carried out twice a year thus giving two periods for the year.Data collected was pre-processed to give the figures in Tables 1 to 7. The breeder did not give values per species, but in a global form of which it was divided by the number of birds in each species to get unit values.

Construction of the real life model
For simplicity, the model was applied to the periods of one time step.The variable designate broilers of the same age of first period raised in owned space, rented space, subcontracted and bought respectively.
The variable designate broilers raised in the second period respectively in owned space, rented space, subcontract and bought.
The variable designate Gallus gallus raised in the first period in owned space, rented space, subcontract and bought respectively.
The variable designate Gallus gallus of the second period raised respectively in owned space, rented space, subcontracted and bought.
The linear program of this period can be written as follows:

Solving the problem using Matlab
The function ' linprog' of Matlab was used to solve the problem in order to come out with the best decision that will maximise the profit of the poultry farmer.In other words, results returned by this function will allow the farmer to maximise profit.
After having entered the values of the various variables in the LINPROG function of MATLAB, the following results were obtained in order to maximise profit: 1) The farmer should raise in the first period 100 broilers in owned space, 250 broilers of same age should be subcontracted in the second period, while 350 Gallus gallus should be raised in owned space in the first period, 227 Gallus gallus of the same age village should be raised in the second period in owned space and 73 Gallus gallus of the same age should be sub contracted in the second period in order to make maximal profit.
This model yielded a profit of 362,580 FCFA, which is far much greater that the profit of 241,485 FCFA that is usually made when he uses his local model made up of the following: 1) 100 broilers raised in the first period in owned space; 2) 200 broilers of the same age in the second period in owned space, 50 subcontracted; 3) 105 Gallus gallus raised in owned space in the first period and 195 subcontracted; 4) 350 Gallus gallus of the same age subcontracted in the second period.
It is obvious that while using the Linear Programming model, there will be an increase in profit of 121,095FCFA which represent a more than 50% increase in profit.This figure is highly significant and would contribute greatly in poverty alleviation especially in most African countries where majority livelihood depends on agriculture.

CONCLUSION
Stochastic Linear Programming and Robust Linear Programming are little known in the livestock sector despite their potential of optimising even in the presence of uncertainty in data.This uncertainty is a typical characteristic of most of the parameters involved in this livestock sector in particular and the poultry industry in particular.The main difficulty associated with the stochastic model is the need to provide the probability distribution functions of the underlying stochastic parameters whereas the robust model addresses the uncertain nature of the problem without making specific assumptions on probability distributions.In this article, we have been able to construct a simple stochastic model for the poultry farming problem and then convert it to a robust model by using the proposal of Soyster (1973).
The model thus constructed was tested with data obtained from a poultry farm in the Bambutus Division of the Western Region of Cameroon.Since the farmer could only give global amounts for each of the factors studied, the unit prices were obtained by dividing these global figures by the number of birds of each species being raised.The Linprog function of Matlab was used to run the program.
The results showed a greater than 50% increase in profit if the values obtained from our model were implemented as compared to the result obtained when using the locally used model.In fact our model yielded a profit of 362,580 FCFA, which is far much greater that the profit of 241,485 FCFA made when implementing the local model.
It is therefore hoped that if farmers are educated on the use of such models in the day to day running of their poultry activities, the result will not only be the improvement of livelihood on the part of the poultry farmers, but it will go a long way to satisfy the daily growing needs of poultry products by the entire community.

Future work
To the best of our knowledge, this technique has not been applied to this type of activity before, thus it is very possible that we might not have been able to exploit all the possible factors that could come into action in the attempt to obtain maximum profit in this economic activity.Therefore, many other factors could be included depending on the circumstances and the volume of the activity.

Table 2 .
Represents unit profit from each bird and specie in FCFA from all the sources.

Table 3 .
Represents the quantity of space required by each species for its production.

Table 4 .
Represents the time required by each species for its production.

Table 5 .
Represents number of chicks that can be produced by each species in a given period.

Table 6 .
Represents orders made for the different species per period.

Table 7 .
Summary of the available resources for the first period.