Integer Programming
Integer Programming
9
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. For instance, we might 3 easily produce 102 4 gallons of a divisible good such as wine. It also might be reasonable to accept a solution 1 giving an hourly production of automobiles at 58 2 if the model were based upon average hourly production, and the production had the interpretation of production rates. At other times, however, fractional solutions are not realistic, and we must consider the optimization problem: n Maximize j=1 cjxj,
subject to: n j=1
ai j x j = bi xj ≥ 0 x j integer
(i = 1, 2, . . . , m), ( j = 1, 2, . . . , n), (for some or all j = 1, 2, . . . , n).
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. As we saw in the preceding chapter, 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. The purpose of this chapter is twofold. First, we will discuss integer-programming formulations. This should provide insight into the scope of integer-programming applications and give some indication of why many practitioners feel that the integer-programming model is one of the most important models in management science. Second, we consider basic approaches that have been developed for solving integer and mixed-integer programming problems.
9.1 SOME INTEGER-PROGRAMMING MODELS
Integer-programming models arise in practically every area of
9
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. For instance, we might 3 easily produce 102 4 gallons of a divisible good such as wine. It also might be reasonable to accept a solution 1 giving an hourly production of automobiles at 58 2 if the model were based upon average hourly production, and the production had the interpretation of production rates. At other times, however, fractional solutions are not realistic, and we must consider the optimization problem: n Maximize j=1 cjxj,
subject to: n j=1
ai j x j = bi xj ≥ 0 x j integer
(i = 1, 2, . . . , m), ( j = 1, 2, . . . , n), (for some or all j = 1, 2, . . . , n).
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. As we saw in the preceding chapter, 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. The purpose of this chapter is twofold. First, we will discuss integer-programming formulations. This should provide insight into the scope of integer-programming applications and give some indication of why many practitioners feel that the integer-programming model is one of the most important models in management science. Second, we consider basic approaches that have been developed for solving integer and mixed-integer programming problems.
9.1 SOME INTEGER-PROGRAMMING MODELS
Integer-programming models arise in practically every area of