(a) Describe two applications of linear programming to management problems.

           What are the main disadvantages of the technique?

When managers relate to the decision-making processes they face a problem of allocating several scarce resources. The opportunity cost of these scarce resources can be determined by the use of linear programming techniques. Linear programming was developed by George B. Dantzig in 1947 as a technique for planning the diversified activities of the U.S Air Force. Linear programming is a powerful mathematical technique that can be applied to the problem of rationing limited facilities and resources among many alternative uses in such a way that the optimum benefits can be derived from their utilization. The main objective of the linear programming problem is maximizing profit or minimizing cost. Applications of the linear programming are numerous in a variety of problem situations such as the blending problem and the product-mix problem.

One of the most important applications of linear programming is the formulation of blends which meet certain requirements at minimum cost. Blending problems occur whenever managers decide how to produce a blend out of specified commodities or constituents whose characteristics and costs are given.

     3X+Y=9

                     IF                X=0        therefore      Y=9                Point:(0,9)

                     IF                Y=0        therefore      X=3                Point:(3,0

Series 2

     X+Y=4

              IF                   X=0       therefore      Y=4                Point:(0,4)

              IF                   Y=0       therefore      X=4                Point:(4,0)

Series 3

     X=4

                     IF                  Y=0       therefore     X=4                 Point:(4,0)

               IF                  Y=10     therefore     X=4                  Point:(4,0)

 Third equation is a parallel straight line to the y-axis

Series 4

Y=6

               IF                  X=0      therefore     Y=6

               IF                  X=5      therefore     Y=6

 Fourth equation is a parallel straight line to the x-axis

First

Graphical representation:

Suppose that the maximum profit is £200

then the objective function would be :

               O.F. 40X+20Y=200

                    When X=0      then       Y=10                  Point (0,10)

                    When Y=0      then       X=5                    Point (5, 0)

When the profit is equal to £200 the iso-profit curve (40X+20Y=200) is displayed above all 4 budget constraints. Therefore the iso profit curve is shifted south-west, parallel to itself, until it hits the highest point of an intersection between two budget constraints in the feasible region. This point is the number three in the figure above. The optimum point is      (1, 6) .That means 1 cow and 6 pigs.

Therefore: if we increase the number of bushels by one at the right-hand side of the first   constraint                  then            3X+Y=10

                                                             and            Y      =6

               equation 4 into 1:              

                                                      3X+6=10

                                                       X=4/3   ,    Y=6

Plugging into the O.F:  40*(4/3) +20*(6) = £173⅓                                             Dual price for increase or decrease of bushel by 1 is £13⅓                                        

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Now increase of 1 pig at the right-hand side of the fourth constraint                                                                                                            

                                                                                                Y=7

and   3X+Y=9

                        equation 4 into 1

3X+7=9

                                                      X=⅔    , Y=7

Plugging into the O.F:   40*(⅔) + 20*(7) = 166⅔

Dual price for an increase or decrease of a pig is £ 6⅔

Shadow price and dual price are exactly the same for all maximization linear programs. The economic meaning of shadow prices is very important for the managers. Shadow price is the changes in profit (positive or negative) of a marginal increase of scarce resources in any of the constraints in the linear programming procedure. Therefore the managers can gather information from the performance of each constraint and give emphasis to an increase of those resources with the highest opportunity cost or shadow price. In our case it would be beneficial, and more profitable for the farmer to increase the number of the bushels rather than the number of pigs to achieve a greater profit.