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Limitations Of Linear Programming

 Linear Programming

Meaning

            The mathematical technique LP is used for analysing optimum decisions subject to certain constraints in the form of linear inequalities. Mathematically it applies to those problems which require the solution of maximisation problems subject to a system of linear inequalities stated in terms of certain variables. If and b, the two variables are the function of c, the value of c is maximised when any movement from that point results in a decreased value of c. The value of c is minimised when even a small movement results in an increased value of c.

An illustration could explain us the model of linear programming.

Let us consider a linear programming problem and solve it by algebraic method. An important thing that has to be understood is to ascertain the given problem as linear programming, is to write the objective function and the constraints in the form of equations or inequalities.

Illustration 1

            Presume an industry manufactures two commodities M1 and M2. Each unit of commodity M1 supplies $30 to profit and each unit of commodity M2 donates $40 to profits. The manufacture of these commodities requires inputs X, Y and Z and their available volume are 14, 10 and 4 relatively. Assumed that the manufacture of one commodity M1 procures 2 units of input X, 1 unit of input Y and does not require input Z and the manufacture of one unit of commodity M2 requires 2 units of input X, 2 units of Y and 4 units of Z.

Derivate the above problem into linear programming and solve it with algebraic method.

Solution

First we shall convert the above write-up as linear equations in order to determine solution.

  1. Objective Function –> π = 30M1 + 40M2
  1. Input X Constraint –> 2M1 + 2M2 ≤ 14
  1. Input Constraint Y –> M1+ 2M2 ≤ 10
  1. Input Constraint Z  –> M2 ≤ 4
  1. Non- Negative Constraint –> M1, M2 ≥ 0

With respect to solve these linear equation problems by algebraic method we first ascertain possible area and its intense points. This is represented in the below diagram which could give us a graphical solution to the equation.

Three constraint lines XY, ZI and JL denotes input constraints have been constructed to obtain the region JKPY as the possible area. There are three J, K, P and Y corner points or intense points of this possible area.

The intense point J is ascertained by only one constraint of input Z and according to it 4 units of commodity M2 and no amount of M1 are produced. Substituting this in profit function we get profits at intense point J.

Thus profits at intense point J = 30 x 0 + 40 x 4 = $160

                                                                                    Hence J =160

At intense point Y, 7 units of commodity M1 and no unit of commodity M2 are manufactured.

Thus π at intense point Y = 30 x 7 + 40 x 0 = $210

                                                                                    Hence Y = 210

The intense point K is ascertained by the intersections constraints of inputs Y and Z. As will be noted from the possible area, according to the intense point K, 4 units of commodity M2 are manufactured. To get the quantity manufactured of commodity M1, we substitute M2 = 4 in the constraint equation of input Y.

Thus M1 + 2 x 4 = 10
M1 = 10 – 8 = 2

Now with M1 = 2, and M2 = 4, at the intense point K, profits are

π = 30 M1 + 40 M2
 = 30 x 2 + 40 x 4 = 60 +160 = 220                            Hence K = 220

Let us take the intense point P, which is ascertained by intersection of the constraints of inputs X and Y. The productivity of two commodities at intense point be got by solving the constraints equation of input X and Y.

Thus we get,

2M1 + 2M2 = 14                                 …………….Equation (1)

1M1 + 2M2 = 10                                 …………….Equation (2)

If we deduct the Equation (2) from Equation (1), we get the following,

M1 = 4

Now we have to substitute the value of M1 in the Equation (2), we obtain

4 + 2M2 = 10

2M2 = 6

M2 = 3

Thus at intense point P, 4 units of commodity M1 and 3 units of commodity M2 are manufactured. With these productivities of M1 and M2 profits are:

π = 30 M1 + 40 M2

= 30 x 4 + 40 x 3

= 120 + 120 = 240;                                                                 Hence, P = 240

Now after ascertaining the profits earned in the intense points, we are going to present the tablet containing productivities at intense points with their respective profits.

Productivities and Revenue Earned at Different Intense points of the Possible Area

Intense Points

Productivities of Commodities M1 and M2

Revenue earned
Value in $

P

M1 = 4; M2 = 3

240

Y

M1 = 7; M2 = 0

210

K

M1 = 2; M2 = 4

220

J

M1 = 0; M2 = 4

160

Intense P represents the optimum or profit maximising productivities of two commodities, such as 4 units of commodity M1 and 3 units of Commodity M2.

Intense Y represents the optimum or profit maximising productivities of two commodities, such as 7 units of commodity M1 and 0 units of Commodity M2.

Intense K represents the optimum or profit maximising productivities of two commodities, such as 2 units of commodity M1 and 4 units of Commodity M2.

Intense J represents the optimum or profit maximising productivities of two commodities, such as 0 units of commodity M1 and 4 units of Commodity M2.

Limitations of Linear Programming

Linear programming has turned out to be a highly useful tool of analysis for the business executives. It is being increasingly made use of in theory of the firm, in managerial economics, in inter regional trade, in general equilibrium analysis, in welfare economics and in development planning.

However, there are limitations and they are discussed below.

  1. It is complex to determine the particular objective function
  1. Even if a particular objective function is laid down, it may not be so easy to find out various technological, financial and other constraints which may be operative in pursuing the given objective.
  1. Given a Specified objective and a set of constraints it is feasible that the constraints may not be directly expressible as linear inequalities.
  1. Even if the above problems are surmounted, a major problem is one of estimating relevant values of the various constant co-efficient that enter into a linear programming mode, i.e. prices etc.
  1. This technique is based on the hypothesis of linear relations between inputs and outputs. This means that inputs and outputs can be added, multiplied and divided. But the relations between inputs and outputs are not always clear. In real life, most of the relations are non-linear.
  1. This technique presumes perfect competition in product and factor markets. But perfect competition is not a reality.
  1. The LP technique is based on the hypothesis of constant returns. In reality, there are either diminishing or increasing returns which a firm experiences in production.
  1. It is a highly mathematical and complicated technique. The solution of a problem with linear programming requires the maximisation or minimisation of a clearly specified variable. The solution of a linear programming problem is also arrived at with such complicated method as the simplex method which comprises of a huge number of mathematical calculations.
  1. Mostly, linear programming models present trial and error solutions and it is difficult to find out really optimal solutions to the various economic complexities.

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