Agree/Disagree and Why?

Integer linear programs involve a class of problems that are modeled as linear programs with the additional requirement that one or more variables must be integer. If all variables must be integer, we have an all-integer linear program. As some, but not all, variables must be integer of a mixed-integer linear program. The cost of the added modeling flexibility provided by integer programming is that problems involving integer variables are often much more difficult to solve. (Anderson)

As discussed Bradley, Hax, and Magnati, The linear-programming models that have been discussed thus far all have been continuous, in the sense that decision variables are allowed to be fractional. Often this is a realistic assumption. At other times, however, fractional solutions are not realistic, and we must consider the optimization problem. This problem is called the (linear) integer-programming problem. It is said to be a mixed integer program when some, but not all, variables are restricted to be integer, and is called a pure integer program when all decision variables must be integers. If the constraints are of a network nature, then an integer solution can be obtained by ignoring the integrality restrictions and solving the resulting linear program. In general, though, variables will be fractional in the linear-programming solution, and further measures must be taken to determine the integer-programming solution.

If we drop the phrase and integer from the last line of this model, we have the familiar two variable linear program. The linear program that results from dropping the integer requirements is called the LP relaxation of the integer linear program. When analyzing the LP Relaxation model, it is possible use a graphical solution just as accomplished with the familiar two variable linear program. In many cases, a non-integer solution can be rounded to obtain an acceptable integer solution. It should be recognized however that rounding may not always be a good strategy. When the decision variables take on small values that have a major impact on the value of the objective function, an optimal integer solution is needed. Rounding to an integer solution is a trial-and-error approach. Another aspect of integer linear program is a result of the need to use 0-1 variables. In many applications, 0-1 variables provide selections or choices if the value of the variable equal to 1 corresponds to activities undertaken, and equal to 0 if the corresponding activity is not undertaken. (Anderson) In this application of integer linear programming, the story involving the wisdom of King Solomon comes to mind. In the story, two women came to him with one baby with each woman claiming that she was the mother of the baby. King Solomon, without knowing which woman was truly the mother of the baby, asked for a sword. Because neither one of the women would confess that she was not the mother, he ordered the baby to be cut in half and give each of the women half of the baby. The real mother who truly loved the child requested that King Solomon give the baby to the other woman so the child would not be injured. The other woman who was not the real mother said go ahead and cut the baby in half, whereupon, in his wisdom inspired by God, King Solomon realized the first woman was the true mother. In a simple example maintaining the constraint of 0 or 1, representing a whole baby or half of a baby, King Solomon was able to determine the true identity of the babys mother.

Integer linear program can be applied to many real-world situations such as distribution system design for shipping, business center location problems for optimum customer service, product design and market share optimization, and determining number of weapon systems for the DoD (Anderson)