Linear Programming as an Approach in Math

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Integer Programming is a mathematical approach which maintains that the solution of any mathematical problems should be in terms of whole numbers or integers. The purpose of integer programming is to find a nearest whole number solution to the Linear Programming (LP) problem within the constraints imposed. For example, the decimal solutions like 30. 5 tables, 3. 96 cars, 9. 25 chairs or 2. 66 persons may be realistic without violating the constraints of problem; however, simply rounding off the values to the nearest integer would not produce a feasible solution.

Integer Programming can be considered as part of LPM or Linear Programming Models.

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LPMs seek to minimize or maximize the variable which is subjected to constraints or limitations. The value of the variable, obtained by solving the LPM can be in decimals while in case of integer programming, it has to be in integers (Taylor, 2009).

Identify the three basic types of integer programming models mentioned in the Taylor text (total integer model, 0-1 integer model, mixed integer model).

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In your own words, distinguish these

Many combinatorial problems can be stated as Linear programming problems with the requirement which is over and above the existing as that all or some of the variables can only have integral values. As per Taylor in his book of Management Science, the Integer Linear Programming (ILP) can be divided into three categories as given below:

Total Integer Model – In this type of ILP, all the variables are integers. The problem does not have any relaxation for decimal results. The model is designed to result in whole numbers or the output is either rounded up or down by applying the logic provided in the problem requirements itself.
0-1 Integer Model – This type of Linear Programming model is also known as Boolean as the linear program variables in this case can take values 0 or 1.
Mixed Integer linear program – This is another category of linear programming in which some of the only some of the variables are allowed to be integers.

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This criterion in this case is little relaxed in terms of results being obtained in decimals or whole numbers (Taylor, 2009).

Explain the statement: Sensitivity analysis can be much more critical in integer programming that it may be in traditional LPMs

Integer programs for managerial analysis under conditions of uncertainty must be used with great care. The difficulty of performing systematic sensitivity analyses on integer models further limits the advantages of the model. Rounding continuous linear programming solutions to integers instead of using an integer programming routine may lead to non- optimal integer solution. But such conditions do not necessarily warrant the use of an integer rather than a rounded linear programming solution for purposes of managerial analysis.

Integer solution as an ideal it suggests that all relevant variables in the decision situation have been properly incorporated in the model and the values of the coefficients are sufficiently accurate. The uncertainties of demand, prices, costs etc make such type of assumptions impractical and sensitivity analysis is used as a management tool to analyze the effect of these uncertainties. The limitation in the use of integer programming is that in “in performing sensitivity analysis on an integer linear programming model, it may be necessary to rely on intuition and ingenuity rather than on systematic procedures.” Further, the difficulty associated with measuring and interpreting dual variables is still another problem connected with the integer programming (Karlof, 2006).

Generally speaking, why is integer (linear) programming preferable to simply rounding the solution determined by traditional LPMs? Are there any exceptions where simply rounding the linear programming solution is adequate? If yes, explain in business content

During the formulation of any Linear Programming it is found that certain variables can be regarded as taking integer values. However, for the convenience sake they are regarded to be taking fractional values as they can be ignored because of high integer values. Sometimes, this could be possible but there are cases when finding a numeric solution in which the variables take integer values is necessary. As discussed, Integer Programming is advantageous, in situations where solution need not be integer, when the fractions can be ignored, the LP with rounding off can be the best approach. For example, if the solution to a problem comes to be 3. 55 chairs, the next best number i. e. 3 can be chosen but its not the right decision as fraction is of higher value. Hence, an optimized number needs to be arrived by using integer linear programming. However, if the answer to a problem is 30000. 55 chairs then rounding it off to 30000 is fine as . 55 is a small fraction compared to the integer, 30000 (Karlof, 2006).

References

Taylor, Bernard W (2008). Introduction to Management Science, Prentice Hall, ISBN 0136064361, 9780136064367
Karlof, John K, (2006). Integer Programming – Theory & Practice, CRC Press, ISBN 0849319145, 9780849319143