Linear programming


Linear programming has become nowadays, the mathematical technique most used in solving a variety of problems related with management, from scheduling, media selection, financial planning to capital budgeting, transportation and many others, with the special characteristic that linear programming expect always to maximise or minimise some quantity.

Two possibilities appear at the solution of a linear-programming problem:

1-Simplex method, developed by Dr.Dantzing, this method is remarkable due to its efficiency and calculating facility. The simplex method can be used where distributions method cannot. Therefore the field of application is quite broad. In complex cases this method saves time and effort by taking us to the optimal solution in a finite number of steps.

2-Graphically, this option will be developed in the later example.

By means of one example we are going to gradually check the advantages and disadvantages of linear programming as a management aid.

First of all it is known that one of the main advantages of linear programming is that it fits strictly with reality, as we will see, the example reflects this property.


I have designed this example searching for a high grade of reality, suppose we are running a football club and launching a new merchandising campaign and we have to decide the quantity of scarves and shirts produced, considering current constraints.

The sale prices of shirts and scarves are £35 and £10 respectively, also we know the maximum annual manufacture capacity is 2000 units, secondly four times more time is needed to sew a shirt compared with a scarf having at the most 2300 hours a year and finally space is limited up to 2500 square inches, requiring shirts and scarves, 3 and 2 square inches respectively.

The first advantage is the calculation facility, as can be checked in the first step where we have to model or formulate the problem. This is a process where verbal statement is translated to mathematical statement; here this example is quite simple.

The incomes must be maximised knowing the different prices of scarves and shirts but some limitations have been set which are called constraints, in this case limitations are related with capacity, time and sales space.

Objective function: 35X+10Y


1-capacity x+y<=2000

2-time 4x+y<=2300

3-sales space 3x+2y<=2500

4-nonnegativity constraints x,y>=0

The next step can vary depending of the selected method; two options can be chosen such as graphical or simplex algorithm. In this occasion we have selected the first option due to the fact that only two variables are being studied, but this method sometimes cause objections.

The maximum profit is in the point where the second and third constraints intersect each other. As a result is known that X=420 and Y=620 .

To draw the objective function we have suppose that we want to make the maximum profit equal to 3000, but it means nothing, because we can choose any number, the slope of the line is the matter that concerns us. If we move parallel the function toward larger objective function values, the maximum profit point will be found when the line will be completely outside of the feasible region. Then the largest value is the optimal solution.

It can be ascertained that in frequent occasions some variables are ignored in this sort of problems (In our example trends and fashions have been ignored but maybe people prefer to buy a shirt than a scarf for certain reasons).

Hence the problem is less rigorous and it lose accuracy and certainly, that become a disadvantage, furthermore the model is static which means that it does not consider the changes and the evolution of variables as time goes by. Another obstacle arises in the formulation process which should be taken into account, values must be known with certainty.

Having seen the example it is obviously that talking in disadvantages terms functions lineality is an important bound, it means that each decision variable appears in a separate term and has an exponent of 1, so that non-lineal function cannot be used.

This example has been resolved by graphical method because it has only two variables, if it would have had more than two variables we would have used the simplex method, due to the fact of the impossibility of solving problems having more than two variables, with the graphical method. Therefore this is another disadvantage of linear programming; graphical method can be used only under determined conditions. There may be another two problems consisting of numerous optimal solutions, this is not a simple matter despite the fact could seem a minor concern, and the other problem could be infeasibility. When no solution to the linear programming problem satisfies all the constraints, feasible region does not exist and therefore any solution cannot be reached.

In this case the problem we have focused on is related with the portfolio selection area, one of the varieties of applications where linear programming can be used, helping managers in inventory design (in the example, number of scarves and shirts), in control and in capacity planning. But this is not the only linear programming advantage, also it can be used in much more different areas such as marketing (helping marketing managers in allocating a fixed advertising budget to various advertising media), finance (resolving situations of capital budgeting, financial planning an so on), product mix problems (problems based on blending resources to produce 1 item, management must decide the quantity of each product), multiple plant location studies (the new location will be where the total production and distribution cost will be minimised), maximizing material utilization (determine the combination of cuts that will meet requirements for the amounts of the different sizes with a minimum trim loss) and finally data envelopment analysis (has been used in efficiency measure)

But there still are more advantages, linear programming analysis can help both with determining whether management's plans are feasible and in unbounded cases where the value of the solution is infinitely large, without violating any of the constraints, warning us that the problem is improperly formulated.

As we can see, in the example, it does not occur, consequently there are finite solutions and therefore we can ascertain that the problem is properly formulated.

It must be mentioned, another remarkable characteristic of linear programming problems such as, the adapting facility to reality, which allows solving, by computer programs, problems with thousands of variables and constraints.

The next step to analyse is what happens when we change the values of the objective function or in the constraints. This is another advantage of using linear programming, when can check easily how the results vary when we change the old coefficients for others. This is called sensitivity analysis, which determines how changes affect the optimal solution to the original linear programming problem. The range of values over which the current solution will remain optimal, despite the change of the coefficients, is called range of optimality.

For example we suppose that the space for sales in the third constraint is reduced, now the available space is 2499 square inches, therefore we have changed in one unit the right hand side of the third constraint and as a result the solution has varied. The new results are X=420,2 and Y=619,2 therefore it can be seen that the solutions have changed in 0.2 and in 0,8 respectively these variations are called dual prices as well as implicit cost and shadow prices.

Dual prices can be used what will happen to the value of the objective function when we make a one unit change in the right-hand side of the constraints. We have to take into account that dual price is only operative for small variations.

In conclusion if we evaluate the pros and cons it can be ascertained that is not coincidence that linear programming is the most used program in the management area ,despite it has several arguments against, there are sound reasons which take us to select this method solving management problems owing to the complexity of the problems that can be handled.


-Anderson, Sweeney and Williams. An introduction to management science.2003

-Elwood S.Buffa. Operations Management: Problems and Models. University of California. Los Angeles.1963

-Lev.B and Weiss.H . Introduction to Mathematical Programming; Quantitative Tools for Decision Making.

Enviado por:Froilan
Idioma: inglés
País: Reino Unido

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