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### The Overview of Linear Programming

Linear Programming is a mathematical device developed by the mathematician George Dantzig in 1947 for planning the diversified activities of the US Air Force connected with the problem of supplies to the forces. Linear or mathematical programming, also known as activity analysis, has been further developed in its application to the firm, managerial economics and finally to development planning. It is a mathematical technique for the analysis of optimum decisions, subject to certain constraints in the form of linear inequalities.

Mathematically speaking, it applies to those problems which require the solution of maximization or minimization problems subject to a system of linear inequalities stated in terms of certain variables.
The problems of maximization and minimization are also called optimization problems. When cost and price per unit change with the size of output, the problem is non – linear and if they do not change with output, the problem is linear. Linear programming may thus be defined as a method to decide the optimum combination of factors to produce a given output or the optimum combination of products to be produced by given plan and equipment. It is also used to decide between a variety of techniques to produce a commodity. The technique involved in linear programming is similar to the one adopted in input – output analysis for the industry.
Conditions And Generalisations Of Linear Programming
You should know that for any principle to hold in any situation, certain conditions and generalizations has to be satisfied. The application of linear programming (LP) technique to any problem rests on certain conditions and generalizations.
First, there is a definite objective. It may be the maximization of profits or national income or employment of the minimization of costs. It is known as the objective function or the criterion function. If a quantity is maximized, its negative quantity is minimized. Every maximization problem has its dual problem, that of minimization. The original problem is the primal problem which always has its dual. If the primal problem pertains to maximization, the dual involves minimization and vice versa.
Secondly there should be alternative production processes for achieving the objective. The concept of process or activity is the most important in linear programming. A process is a specific method of performing an economic task. It is some physical operation, e.g consuming something, storing something, selling something, throwing something away, as well as manufacturing something in a particular manner. The LP technique enables the planning authority to choose the most efficient and economical process in attaining the objectives.
Thirdly, there must be certain constraints or restraints of the problem. They are the limitations or restrictions pertaining to certain conditions of the problems, as to what cannot be done and what has to be done. They are also known as inequalities. They may be limitations of resources such as land, labour or capital.
Fourthly, there are the choice variables, the various production processes or activities so as to maximize or minimize the objective function and to satify all the restraints.
Lastly, there are the feasible and optimal solutions. Given the income of the consumer and the prices of goods, feasible solutions are all possible combinations of the goods he can feasibly buy. Feasible solutions of two goods for the consumer are all combinations that lie on and to the left of the budget line. Whereas, on an isocost line, they are the combinations that lie on and to right of it. We may put it differently that a feasible solution is one which satisfies all the restraints.
The optimal solution is the best of the feasible solutions. If a feasible solution maximizes or minimizes the objective function, it is an optimal solution. The best available procedure for finding out the optimal solution out of the possible feasible solutions is the simplex method. It is a highly mathematical and technical method involved in linear programming.
However, the main aim of linear programming is to find out optimal solutions and study their characteristics.
Assumptions Of Linear Programming Technique
The linear programming analysis is based upon the following assumptions;
i. The decision-making body is faced with certain constraints or resource restrictions. They may be credit, raw material and space constraints on its activities. The type of constraints in fact depend upon the nature of problem. Mostly, they are fixed factors in they production process.
ii. It assumes a limited number of alternative production processes.
iii. It assumes linear relations among the different variables which implies constant proportionality between inputs and outputs within a process.
iv. Input – output prices and coefficients are given and constant. They are known with certainty.
v. The assumption of additivity also underlies linear programming techniques which means that the total resources used by all firms must equal the sum of resources used by each individual firm.
vi. The LP technique assumes continuity and divisibility in produces and factors.
vii. Institutional factors are also assumed to be constan

Limitations Of Linear Programming
Linear programming has turned out to be a highly useful tool of analysis in development planning. But it has its limitations. As a matter of fact, actual planning problems cannot be solved directly by the LP technique due to a number of restraints.
Firstly, it is not easy to define a specific objective function.
Secondly, even if a specific objective function is laid down, it may not be so easy to find out the various social institutional, financial and other constraints which may be operative in pursuing the given objective.
Thirdly, given a specific objective and a set of constraints, it is possible that the constraints may not be directly expressible as linear inequalities.
Fourthly, even if the above problems are surmounted, a major problem is one of estimating relevant value of the various constant co-efficients that enter into an LP problem, i.e. population, prices, etc.
Fifthly, one of the defects of this technique is that it is based on the assumption of linear relations between inputs and outputs. This implies that inputs and outputs are additive, multiplicative and divisible. But the relations between inputs and outputs are not always
linear. In real life, most of the relations are non-linear.
Sixth, this technique assumes perfect competition in product and factor markets. But perfect competition is not a reality.
Seventh, the LP technique is based on the assumption of constant returns in the economy. In reality, there are either diminishing or increasing returns.
To further buttress this point, it is a highly mathematical and complicated technique. The solution of a problem with linear programming requires the maximization or minimization of a clearly specified variable. The solution of a linear programming problem is also arrived at with the Simplex method which involves a large number of mathematical calculations.
It requires a special computational technique, an electric computer or desk calculator. Such computers are not only costly, but also require experts to operate them. Mostly, the LP models present trial-and-error solutions and it is difficult to find out really optimal solutions to the various economic problems.
Uses Of Linear Programming In Planning
Linear programming as a tool of economic development is more realistic than the input￾output approach. In input-output analysis only one method is adopted to produce a commodity. It does not take into consideration the bottlenecks (constraints) which a development project has to face in underdeveloped countries. But in linear programming a definite objective is set to maximize income or minimize costs.
All possible processes or techniques are taken into account for achieving the desired objective. This necessitates even the substitution of one factor for another till the most efficient and economical process is evolved. So projects and techniques which are too uneconomical to implement are not undertaken.
By assuming certain constraints, linear programming as a tool of development planning is superior to the input-output technique. In underdeveloped countries, the planning agencies are faced with such constraints as the lack of sufficient capital and machinery, growing populations, etc. Resources exist that cannot be used properly for want of the co operant factors. Linear programming takes to due note of
these constraints and helps in evolving an optimum plan for attaining the objectives within a specified period of time. Thus the LP technique has been used for constructing theoretical multi- sector planning models for countries like India. Such models extend the consistency models of the input-output type to optimization of income or employment or any other quantifiable plan objective under the constraints of limited resources and technological conditions of production.
In practice, however, the LP technique is being used in solving a limited number of economic problems in developing countries. This is due to the lack of proper personnel for working out mathematical equations and for operating highly mechanical computers.
Mostly the LP technique has been found to be extremely useful for sectoral planning in developing countries, for example, in selecting optimum alternatives in respect of location and technologies in industries, transport, and power or in farm management.This technique is being used in farm management for determining the optimum combination of different crops e.g Livestock and crops. The objective function used in such studies is either the minimization of costs or the maximization of income.
The constraints are set by pre-determined levels of demand or the availability of resources such as raw materials or capacity. Besides, this technique is being used for the solution of diet problem where the aim is to minimize costs, given the values of minimum nutrients of the diet and the prices of products as constraints. It is also with the LP technique that the transport problems is being solved by the railways, airways and transport companies with regard to the selection of routes, transportation of goods, allocation of the means of transport ( i.e. railway, wagons, aircrafts,trucks etc.depending on the type of transport under study).Again, this technique is used to assign jobs to the work force for maximum effectiveness and optimum results subject to constraints of wages and other costs.
Similarly, purchasing, assembling, production and marketing problems are being solved through the LP technique in order to minimize costs and maximize profits, given the various constraints in the case of each problem. However, for an extensive use of this technique for development planning, developing economies will have to depend upon larger resources of trained personnel, and finance.

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Summary and Conclusion
From our discussion so far on the introduction to linear programming as a concept , we can deduce the following facts:
That Linear or mathematical programming, also known as activity analysis, has been further developed in its application to the firm, managerial economics and finally to development planning for the analysis of optimum decisions, subject to certain constraints in the form of linear inequalities. It also applies to those problems which require the solution of maximization or minimization problems subject to a system of linear inequalities stated in terms of certain variables which is also regarded as Optimisation problems. Mostly the LP technique has been found to be extremely useful for sectoral planning in developing countries, for example, in selecting optimum alternatives in respect of location and technologies in industries, transport, and power or in farm management.
Also, from the point of view of our discussion, you have learnt that the main aim of linear programming is to find out optimal solutions and study their characteristics.  Production capacity technique of LP, Limitations of linear programming technique and the uses of linear programming technique in planning. Also, from the point of view of our discussion, you have learnt that the main aim of linear programming is to find out optimal solutions to available constraints posed by available resources.

References

• Jhingan M.L. (2007):- The Economics of development and planning, Vrinda publications India. (39th Edition).
• Michael P. Todaro and Stephen C Smith (2011).Economic development, pearson education ltd, Edinburgh gate harlow, Essex, England.
• Michael P.Todaro: Development planning, models and methods, Chapter 2-3.
• Akosile I.O, Adesanya A.S Ajani A.O (2012):- Management of development ( A Nigeria per spective ) Olas Ventures, Mushin, Lagos.
• Olajide O.T (2004):_ Theories of Economics development and planning, Lagos, Nigeria, Pumark Nigeria Ltd.
• Otokiti S.O (1999):- Issnes and strategies in Economic Planning, Bitico publishers,Ibadan