Combinatorial Optimization Permutation Problem

Several problems are solved by finding an optimum permutation of the first n integers. As an example consider the simple assignment problem. There are n tasks and n machines. A cost is specified for completing each task with each machine. The problem is to assign one task to each machine so that all tasks are completed with the smallest total cost. This is a combinatorial problem whose solution is described by a permutation of n integers. For a seven task problem one solution is given by the permutation: (1, 2, 3, 4, 5, 6, 7). This implies assigning task 1 to machine 1, task 2 to machine 2 and so on. Every permutation describes an assignment.

The linear assignment problem just described is one of the simplest of the combinatorial problems and can be solved with linear programming as well as with several special purpose algorithms. Here we formulate and solve the problem as a combinatorial search problem. Although this is a poor choice from a computational point of view, it serves to illustrate the features of permutation problems. There are a number of problems whose solution is a permutation that that are not so easy to solve including the quadratic assignment problem and the facility location problem. These problems are solved in the Combinatorial add-in and the Layout add-in with procedures that call the permutation option of this Optimize add-in.

We state the general permutation combinatorial model in terms of assigning n machines to n tasks.

 Permutation Problem: Permute-COP The solution x is a permutation if all of the components of x have different values.

The general model allows any objective function or constraint set. For the linear assignment problem a cost is given for each machine-task assignment and the objective is to find the permutation that minimizes the sum of the assignment costs. In the following we assume all assignments are feasible.

Excel Model

The general Permute-COP is implemented in the Optimize add-in. The figure below shows the model form for the linear assignment problem. Exhaustive enumeration was used to obtain the optimum solution for this seven task case. The matrix of assignment costs is in range B12:H19. The solution vector is in range B7:H7. The costs of the selected assignments are indicated in B9:H9. The objective in D3 is the sum of the numbers in B9:H9.

In terms of the model notation the solution is:

It is interesting to observe how the integer programming model for the assignment problem is different from the combinatorial model. The assignment problem is written as an IP below. It also has a pure network flow model.

When the integrality constraints are neglected and the model is solved with an LP algorithm, the solution turns out to be integer. This fortunate event occurs because the constraint matrix has the total unimodularity property. This property assures that every extreme point of the feasible region of the LP, and hence the optimal extreme point, has all integer values. The math programming models for the transportation, maximum flow, shortest path and pure min-cost flow problems also have this characteristic. These are combinatorial problems that are easy to solve. Why create a COP model for them?

One reason is that the COP model is simpler. The IP has 0-1 variables and 2n constraints where the combinatorial model only has n variables and no constraints. A solution described by a permutation automatically satisfies the requirements that all machines are assigned and all tasks are performed.

Another reason for the COP model is that it can handle a broader range of assumptions. If one adds logical constraints to the IP model, the solution of the LP relaxation may no longer be integer and an IP algorithm such as branch and bound is necessary. A nonlinear objective function results in an integer-nonlinear model. Mild changes in the situation change the problem from easy to hard. This is illustrated in the next section.

The quadratic assignment problem (QAP) illustrates a combinatorial problem with a nonlinear objective function. The mathematical model has the form of a linear assignment problem, but the objective function is a quadratic function of the variables. We describe the problem in terms of assigning n departments to n locations. Data describes the flow between every pair of departments with f(i, j) the flow from i to j. The distance between locations k and l is given as d(k, l) for all location pairs k and l.

The QAP combinatorial model for assigning n locations to n departments is the same as the Permute-COP with some terms redefined.

 Quadratic Assignment Problem: QAP-COP The solution x is a permutation if none of the components of x have the same value.

The instance of the QAP that represents the location problem has the objective defined below. Notice that the variables appear as indices of the distance matrix.

The mathematical programming model for the QAP has the same constraints as the linear assignment problem, but a much more complicated objective. This is a very difficult problem to solve and there are a number of papers about solving the QAP. Although there are still variables, there are nonlinear terms in the objective function.

The Combinatorics add-in builds a model for the QAP and solves the problem as a permutation COP. The Layout add-in has an option to solve certain facility layout problems as QAP.

QAP Excel Model

The QAP model constructed by the Combinatorics add-in for a seven department problem is shown below. The assignment is in D9:J9. The distance matrix is in D14:J20 and the Flow matrix is in D23:J29. A critical part of the Excel model is the Sorted Distance matrix in D32:J38. This matrix is created with Excel equations, and the matrix changes with the assignment. The objective function value is computed in F5 with a simple SUMPRODUCT function.

Although the exhaustive enumeration approach used to solve this example is not practical for larger problems, partial enumeration methods are available that offer a good chance of finding a satisfactory, if not optimum, solution.

The combinatorial description described on this page is very general and the methods can be used for a variety of problems. Excel is primarily an evaluation tool and very complex models can be built to evaluate specific solutions. Unless great care is take by a skilled analyst an Excel model is unlikely to have the form necessary for solution by an algorithm specialized to a particular set of problems. The combinatorial approach is available for most models.

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by Paul A. Jensen