Variable binary

Binary variables usually have values of 0 (for attribute absent) or 1 (for attribute present). The simplest type of similarity measure is the matching coefficient. For two objects i and i and attribute j  

This means that one counts the number of attributes for which i and i have the same value and divides this by the number of attributes. 

The Jaccord similarity coefficient is slightly more complex. It considers that the simultaneous presence of an attribute in objects i and i indicates similarity, but that the absence of the attribute has no meaning. Therefore  

The Jaccard similarity coefficient is then computed with eq. (30.13), where m is now the number of attributes for which one of the two objects has a value of 1. This similarity measure is sometimes called the Tanimoto similarity. The Tanimoto similarity has been used in combinatorial chemistry to describe the similarity of compounds, e.g. based on the functional groups they have in common [9]. Unfortunately, the names of similarity coefficients are not standard, so that it can happen that the same name is given to different similarity measures or more than one name is given to a certain similarity measure. This is the case for the Tanimoto coefficient (see further). 

It can be shown [5] that the Hamming distance is a binary version of the city block distance (Section 30.2.3.2). 

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Now, we define the binary variable Ei j, , which takes the values of 0 when there is no match between streams / and j in SN , and takes the value of 1 when there exists a match between streams i and j (and hence an exchanger) in SN . Based on Eq. (6.13), one can write... 

In the case of binary variables a G 0,1, we can let density p t) at time t represent the probability of finding value a = 1 at any site at time t. A simple counting of all possible configurations in an arbitrary neighborhood Af gives us the general mean-field expression for the time-dependence of p ... 

Kauffman ([kauffSO], [kauffOOa]) has introduced a class of parametrizable fitness landscapes called NK-landscapes, that provide a formalism for studying the efficacy of GA evolution as a function of certain statistical properties of the landscape. Given N binary variables Xi = 1, so that x = (xi, X2, , Xjv) represents a vertex of an A -dimensional hypercube, an NK-landscape is defined by a fitness function, JF, of the form... 

For the ZW (zero wait) policy idle times (slacks) between consecutively produced batches may appear in all stages including the one that defines the bottleneck. It should be noted that the slacks are only a function of consecutive pairs of batches. Therefore, the slacks for each pair of batches can easily be calculated a priori with the binary variable for any two consecutive batches ... 

Xik - binary variable, product produced in campaign k yes or no dimensionless... 

Zjk - binary variable for selection of equipment size dimensionless... 

The methods discussed for linear and nonlinear programming can be adapted to deal with structural optimization by introducing integer (binary) variables that identify whether... 

A binary variable can be used to set a continuous variable to 0. If a binary variable y is 0, the associated continuous variable x must also be 0 if a constraint is applied such that ... 

When a linear programming problem is extended to include integer (binary) variables, it becomes a mixed integer linear programming problem (MILP). Correspondingly,... 

Example 3.5 A problem involving three binary variables yu y2 and >>3 has an objective function to be maximized1. 

Figure 3.16 Setting the binary variables to 0 or 1 creates a tree structure. (Reproduced from Floudas CA, 1995, Nonlinear and Mixed-Integer Optimization, by permission of Oxford University Press).

Thus, the solution of the MILP problem is started by solving the first relaxed LP problem. If integer values are obtained for the binary variables, the problem has been solved. However, if integer values are not obtained, the use of bounds is examined to avoid parts of the tree that are known to be suboptimal. The node with the best noninteger solution provides a lower bound for minimization problems and the node with the best feasible... 

Needless to mention, the exact capturing of time presents further challenges in the analysis. Fundamentally, a decision has to be made on how the time horizon has to be represented. Early methods relied on even discretization of the time horizon (Kondili et al., 1993), although there are still methods published to date that still employ this concept. The first drawback of even time discretization is that it inherently results in a very large number of binary variables, particularly when the granularity of the problem is too small compared to the time horizon of interest. The second drawback is that accurate representation of time might necessitate even smaller time intervals with more binary variables. Even discretization of time is depicted in Fig. 1.8a. 

Recent approaches tend to adopt the uneven discretization of the time horizon of interest wherein each time point along the time horizon coincides with either the start or the end of a task (Schilling and Pantelides, 1996). In addition to accurate representation of time this approach results in much smaller number of time points, hence fewer binary variables, as shown in Fig. 1.8b. 

In their formulation, Ierapetritou and Floudas (1998), separated task and unit events by assigning corresponding binary variables to tasks, wv (i,p), and units, yv (i,p), respectively. This led to an overall number of binary variables of P(Nt+Nj), where P is the number of time points, whilst N, and Nj are the numbers of tasks... 

The results in the second and third columns were obtained using GAMS 2.5/OSL in a 600 MHz Pentium III processor, while those in the fourth and fifth columns were taken directly from Ierapetritou and Floudas (1998). The approach based on the SSN representation gives an objective value of 71.473 and requires only 15 binary variables, compared to 48 and 46 binary variables required in approaches proposed by Zhang, and Schilling and Pantelides, respectively. The formulation by Ierapetritou and Floudas (1998) initially consisted of 30 binary variables that were later reduced to 15 by exploiting one to one correspondence of units and tasks. It... 

NTP = number of time points NC = number of constraints NV = total number of variables NB = number of binary variables... 

Table 2.3 Values of the binary variables at different time points for the literature example... 

Since there are 8 effective states for the overall problem, the resulting number of binary variables is 8 xf. Note that this number of effective states emanates from the fact that each of states s. 2, s6 and, v8 can be fed to any of the 2 reactors. The first set of effective states was chosen throughout the formulation. 

Table 2.6 shows the values of the binary variables at different time points. It is worthy of note that all the binary variables are zero at the end of the time horizon, i.e. ps, implying that no state can be used at this point. The Gantt chart corresponding to these values of binary variables is shown in Fig. 2.8. 

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