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Linearity Assumptions And Choice Of Product And Process

  Linearity Assumptions and Choice of Product and Process

Illustration 61

The linear programming function is given that:

Maximise:        π          =          20A + 12B and this is the objective function.

Subject to:                               4A + 6B <= 24

The restraint Equation             4A + 2B = 16

Now, A is zero and B is 8 at one corner point Say X while in another, A is 8 and B is zero, say Y. Solve the Equation through linear programming.

Solution

(1) Non negativity requirements are A > 0, B > 0.

             Thus,               π          =          20 (0) + 12 (8)

                                                =          96                                ….. (1)

(2) Now A being 8 and B being zero, at corner point Y, the profit would become,

                                    π          =          20 (8) + 12 (0)

                                                =          160                              ….. (2)

(3) Now, at the corner point say Z, the two restraint equation curves crosses. That is, point Z contends the two restraint equations. The values of variables i.e. productivities of two products relating to this can be procured by resolving them simultaneously. The two restraint equations are:

                                    4A + 6B = 24              …..(i)
                                    4A + 2B = 16              …..(ii)

Subtracting Equation (ii) from (i), we procure

                                    4B       =          8

                                    B         =          2

Substituting the value of B into equation (i) we have

                                    4A + 6 (2)       =          24
                                   
                                                4A       =          24 – 12

                                                A         =          12 / 4   =          3

Therefore, at the corner point Z, the productivities of A and B are 2 and 3 correspondingly. Now substituting these values in the profit equation we can obtain the profits at the corner point Z.

                                    π          =          20 A + 12 B

Substituting the values of A = 3 and B = 2, we get,

                                    π          =          10 * 3 + 12 * 2

                                                =          54                                ….. (3)

Hence the corner points X, Y and Z would posses profits of 96, 160 and 54 respectively. Therefore, provided the restraints, profits at corner point Z are the optimum and according to this the firm produces 2 units of B and 3 units of A.

It is therefore clear in linear programming solution; maximum solution takes place at a corner of the possible region. This is for the reason that both the objective function and restraints are linear.

Basic Aspects And Terms Of Linear Programming

  1. The linear programming as practical to the thesis of the industry describes decision making by an industry about productivity and processes it will select, provided the prices of both inputs and products.
  1. Therefore, in the analysis of the industry conducted through the linear programming it is presumed that prices for the products as well as the inputs utilised by them will stay invariable.
  1. With the provided invariable prices of raw materials and productivities, their linear programming offers numerical resolution to the complexity of making maximum choice by the industry when it has to work within specified restraints.
  1. The maximisation can denote to either maximisation or minimisation. For instance, objective may be optimisation of profits for a provided for a provided degree of expenditure or reduction in cost for a provided degree of productivity.
  1. Maximisation and selection are therefore central aspects of linear programming difficulties.

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