Linear programming

In the field of business and management, linear programming is a method for solving complex problems in the two main areas of (i) product mix and (ii) distribution of goods. It is a tool within the field of operational research.

In product mix the technique may be used where it is difficult to decide just how much of each variable to use in order to satisfy certain criteria such as maximizing profits or minimizing costs. The situation may be subject to certain constraints that may be present. A simplified example will illustrate this concept.

A large sub-contractor machines special parts to order. For one order he has two machines (X and Y) available. Unfortunately, the machines perform more effectively on some jobs than others. Machine Y can do 10 units per hour on contracts from Alpha Ltd., 12 per hour on components from Beater & Co. and 26 per hour on those from Chester Inc., while Machine X can produce 16, 9 and 10 units per hour respectively. However, this is complicated by certain constraints which are:

  1. maxima per month of units from Alpha, Beater and Chester are 6,500, 4,440 and 800 per month respectively
  2. the machines X and Y are restricted to working a maximum of 260 and 350 hours per month respectively.

The problem is: how many hours should each machine work to maximize profits? This is answered by plotting machine Y's hours (y) against machine X's hours (x) using mathematical models for the three suppliers and then solving these using linear programming.

The model for Alpha can be generated using the above data as follows: machine Y can produce 10 per hour, so if it works y hours the total is 10y hours.
Similarly, X can produce 16 per hour so if it works x hours the total is 16x hours.
So the model for Alpha, which cannot exceed 6,500 is: 10y + 16x x 6500
and the model for Beater which cannot exceed 4,400 is: 12y + 9x x 4400.
and the model for Chester which cannot exceed 8,000 is: 26y + 10x x 8000.

A profit model is necessary in order to find the conditions for maximum profit i.e. the optimum number of hours for X and for Y. In this case, the profits on each machine's output are: X = £40 per hour and Y = £24, so the total profit is P = 40x + 24y.

Because of the existence of only two variables (x and y) this can be solved by plotting the models on a graph for the three suppliers and moving the profit model until it reaches the highest point of intersection of the three supplier lines.


A quicker way is to use a linear programming "Simplex" computer model and is the only feasible way where there are more than two variables. Another linear programming tool is useful for solving transportation and distribution problems. These can cover questions such as the most efficient and profitable pattern of distribution from warehouses to customers or the determination of the most efficient areas over which bus routes should operate.

One way to tackle such a problem is by iteration (trial-and-error) using a matrix or, again, having access to the relevant computer programme.

A third example is to be found in generating computer models for assessing the work in terms of time for carrying out projects. This can be done using Linear Programming but a far superior method is using Multiple Regression Analysis.

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