INTEGER PROGRAMMING:The All-Different Constraint

By -
The All-Different Constraint

The all-different global constraint simply specifies that all the variables in a given set must have different values. If x1, x2, . . . , xn are the variables involved, the constraint can be written succinctly as all-different (x1, x2, . . . , xn)

while also specifying the domains of the individual variables in the model. (These domains collectively need to include at least n different values in order to enforce the all-different constraint.)

To illustrate this constraint, consider the classical assignment problem presented in Sec. 9.3. Recall that this problem involves assigning n assignees to n tasks on a one- to-one basis so as to minimize the total cost of these assignments. Although the as- signment problem is a particularly easy one to solve (as described in Sec. 9.4), it nicely illustrates how the all-different constraint can greatly simplify the formulation of the model.

With the traditional formulation presented in Sec. 9.3, the decision variables are the binary variables, for i, j = 1, 2, . . . , n. Ignoring the objective function for now, the functional constraints are the following.

Each assignee i is to be assigned to exactly one task:

Thus, there are n2 variables and 2n functional constraints.

Now let us look at the much smaller model that constraint programming can provide.

In this case, the variables are yi = task to which assignee i is assigned for i = 1, 2, . . . , n. There are n tasks and they are numbered 1, 2, . . . , n, so each of the yi variables has the domain {1, 2, . . . , n}. Since all the assignees must be assigned different tasks, this restriction on the variables is precisely described by the single global constraint, all-different (y1, y2, . . . , yn).

Therefore, rather than n2 variables and 2n functional constraints, this complete constraint programming model (excluding the objective function) has only n variables and a single constraint (plus one domain for all the variables).

Now let us see how the next global constraint enables incorporating the objective function into this tiny model as well.

Comments

Post a Comment

Popular posts from this blog

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

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

DECISION SUPPORT SYSTEMS:DIALOG GENERATION AND MANAGEMENT SYSTEMS

By -
Image
DIALOG GENERATION AND MANAGEMENT SYSTEMS In our efforts in this chapter, we envision a basic DSS structure of the form shown in Figure 12. This figure also shows many of the operational functions of the database management system (DBMS) and the model base management system (MBMS). The primary purpose of the dialog generation and management system (DGMS) is to enhance the propensity and ability of the system user to use and benefit from the DSS. There are doubtless few users of a DSS who use it because of necessity. Most uses of a DSS are optional, and the decision maker must have some motivation and desire to use a DSS or it will likely remain unused. DSS use can occur in a variety of ways. In all uses of a DGMS, it is the DGMS that the user interacts with. In an early seminal text, Bennett (1983) posed three questions to indicate this centrality: 1. What presentations is the user able to see at the DSS display terminal? 2. What must the user know about what is seen at the display
Read more