The Element Constraint

By -
The Element Constraint

The element global constraint is most commonly used to look up a cost or profit associated with an integer variable. In particular, suppose that a variable y has domain {1, 2, . . . , n} and that the cost associated with each of these values is c1, c2, . . . , cn, respectively. Then the constraint

element (y, [c1, c2, . . . , cn], z)

constrains the variable z to equal the yth constant in the list [c1, c2, . . . , cn]. In other words, z = cy. This variable z can now be included in the objective function to provide the cost associated with y.

To illustrate the use of the element constraint, consider the assignment problem again and let

cij = cost of assigning assignee i to task j

for i, j, = 1, 2, . . . , n. The complete constraint programming model (including the ob- jective function for this problem is

Minimize Z = Izi,

i=1

subject to

element (yi, [ci1, ci2, . . . , cin], zi) for i = 1, 2, . . . , n, all-different (y1, y2, . . . , yn),

yi ∈ {1, 2, . . . , n} for i = 1, 2, . . . , n.

This complete model now has 2n variables and (n + 1) constraints (plus the one domain for all the variables), which still is far smaller than the traditional integer programming formulation presented in Sec. 9.3. For example, when n = 100, this model has 200 variables

and 101 constraints whereas the traditional integer programming model has 10,000 variables and 200 functional constraints.

As an additional example, reconsider Example 2 (Violating Proportionality) presented in Sec. 12.4. In this case, the original decision variables are

xj = number of TV spots allocated to product j

for j = 1, 2, 3, where a total of five TV spots are to be allocated to the three products. However, because the profits given in Table 12.3 for different values of each xj are not proportional to xj, Sec. 12.4 formulates two alternative integer programming models with auxiliary binary variables for this problem. Both models are fairly complicated.

A constraint programming model that uses the element constraint is much more straightforward. For example, the profit for Product 1 given in Table 12.3 is 0, 1, 3, and 3 for x1 = 0, 1, 2, and 3, respectively. Therefore, this profit is simply z1 when the value of z1 is given by the constraint

INTRODUCTION TO OPERATIONS RESEARCH-0161

Now compare this model to the two integer programming models for the same prob- lem in Sec. 12.4. Note how the use of element constraints provides a considerably more compact and transparent model.

The all-different and element constraints are but two of the various available global constraints (Selected Reference 5 describes nearly 40), but they nicely illustrate the power of constraint programming to provide a compact and readable model of a complex problem.

Comments

Post a Comment

Popular posts from this blog

DUALITY THEORY:THE ESSENCE OF DUALITY THEORY

By -
Image
THE ESSENCE OF DUALITY THEORY Given our standard form for the primal problem at the left (perhaps after conversion from another form), its dual problem has the form shown to the right. Thus, with the primal problem in maximizatio n form, the dual problem is in minimization form instead. Furthermore, the dual problem uses exactly the same pa r amete r s as the primal problem, but in different locations, as summarized below. 1. The coefficients in the objective function of the primal problem are the right-hand sides of the functional constraints in the dual problem. 2. The right-hand sides of the functional constraints in the primal problem are the coef- ficients in the objective function of the dual problem. 3. The coefficients of a variable in the functional constraints of the primal problem are the coefficients in a functional constraint of the dual problem. To highlight the comparison, now look at these same two problems in matrix notation (as introduced at the beginn...
Read more

NETWORK OPTIMIZATION MODELS:THE MINIMUM SPANNING TREE PROBLEM

By -
Image
THE MINIMUM SPANNING TREE PROBLEM The minimum spanning tree problem bears some similarities to the main version of the shortest-path problem presented in the preceding section. In both cases, an undi r ecte d and connected network is being considered, where the given information includes some mea- sure of the positive length (distance, cost, time, etc.) associated with each link. Both problems also involve choosing a set of links that have the shortest total length among all sets of links that satisfy a certain property. For the shortest-path problem, this property is that the chosen links must provide a path between the origin and the destination. For the minimum spanning tree problem, the required property is that the chosen links must provide a path between ea c h pair of nodes. The minimum spanning tree problem can be summarized as follows: 1. You are given the nodes of a network but not the links. Instead, you are given the po- tential links and the positive length for each ...
Read more

INTEGER PROGRAMMING:THE BRANCH-AND-CUT APPROACH TO SOLVING BIP PROBLEMS

By -
TH E BRANCH-AND-CUT APPROACH TO SOLVING BIP PROBLEMS Integer programming has been an especially exciting area of OR since the mid-1980s be- cause of the dramatic progress being made in its solution methodology. Background To place this progress into perspective, consider the historical background. One big break- through had come in the 1960s and early 1970s with the development and refinement of the branch-and-bound approach. But then the state of the art seemed to hit a plateau. Rel- atively small problems (well under 100 variables) could be solved very efficiently, but even a modest increase in problem size might cause an explosion in computation time be- yond feasible limits. Little progress was being made in overcoming this exponential growth in computation time as the problem size was increased. Many important problems arising in practice could not be solved. Then came the next breakthrough in the mid-1980s, with the introduction of the b r an c h- and-cu t approach to sol...
Read more