Optimization integer variables

Mixed-integer programming contains integer variables with the values of either 0 or 1. These variables represent a stmcture or substmcture. A special constraint about the stmctures states that of a set of (stmcture) integer variables only one of them can have a value of 1 expressed in a statement the sum of the values of (alternate) variables is equal to 1. In this manner, the arbitrary relations between stmctures can be expressed mathematically and then the optimal solution is found with the help of a computer program. (52). 

All of these variables must be varied in order to minimize the total cost or maximize the economic potential (see Chapter 2). This is a complex optimization problem involving both continuous variables (e.g. batch size) and integer variables (e.g. number of units in parallel) and is outside the scope of the present text9. 

An optimization model which considers all these constraints - especially those which can only be modeled using binary or general integer variables - can be highly complex. 

The integer solution property is particularly important in assignment problems. These are transportation problems (like the problem just described) with n supply nodes and n demand nodes, where each supply and demand is equal to 1.0, and all constraints are equalities. Then the model in Equations (7.41) through (7.44) has the following interpretation Each supply node corresponds to a job, and each demand node to a person. The problem is to assign each job to a person so that some measure of benefit or cost is optimized. The variables xi are 1 if job i is assigned to person ./, and zero otherwise. 

Liquid-liquid extraction is carried out either (1) in a series of well-mixed vessels or stages (well-mixed tanks or in plate column), or (2) in a continuous process, such as a spray column, packed column, or rotating disk column. If the process model is to be represented with integer variables, as in a staged process, MILNP (Glanz and Stichlmair, 1997) or one of the methods described in Chapters 9 and 10 can be employed. This example focuses on optimization in which the model is composed of two first-order, steady-state differential equations (a plug flow model). A similar treatment can be applied to an axial dispersion model. 

As explained in Chapter 9, a branch-and-bound enumeration is nothing more than a search organized so that certain portions of the possible solution set are deleted from consideration. A tree is formed of nodes and branches (arcs). Each branch in the tree represents an added or modified inequality constraint to the problem defined for the prior node. Each node of the tree itself represents a nonlinear optimization problem without integer variables. 

In sub-problem 4 the process model constraints (function of both integer and continuous variables) are considered along with the objective function. The optimal solvent is identified by either solving a smaller MINLP problem (if the number of feasible solutions is large) or a set of NLP problems (if the number of feasible solutions is small) by fixing the values of integer variables. 

The above problem becomes an NLP problem when we fix the integer variables and since we have only 10 feasible compounds, 10 NLP problems were solved by fixing the binary variables representing the 10 compounds. Sequential quadratic programming algorithm was used to solve the NLP problems. The molecular structure and design results of the optimal solvent and 2-ethoxy ethyl acetate are shown in Table 1. 

As reported [see problem 7, Chem, Eng. ScL, 45, 595-614 (1990)], the attack on this problem used 2077 continuous variables, 204 integer variables, 2108 constraints, and gave as an optimal solution the design shown in Fig. PlO.ll. 

A large number of optimization models have continuous and integer variables which appear linearly, and hence separably, in the objective function and constraints. These mathematical models are denoted as Mixed-Integer Linear Programming MILP problems. In many applications of MILP models the integer variables are 0 - 1 variables (i.e., binary variables), and in this chapter we will focus on this sub-class of MILP problems. 

Nonlinear and Mixed-Integer Optimization addresses the problem of optimizing an objective function subject to equality and inequality constraints in the presence of continuous and integer variables. These optimization models have many applications in engineering and applied science problems and this is the primary motivation for the plethora of theoretical and algorithmic developments that we have been experiencing during the last two decades. 

Integer variables assigned under the control of a clock edge are also inferred as flip-flops. Here is an example where an integer variable is assigned under clock control. Four flip-flops are inferred for the variable IntState) the other high-order bits of the variable are optimized away (since they are not used). 

For the first problem, one will usually write a mathematical model of how insulation of varying thicknesses restricts the loss of heat from a pipe. Evaluation requires that one develop a cost model for the insulation (a capital cost in dollars) and the heat that is lost (an operating cost in dollars/year). Some method is required to permit these two costs to be compared, such as a present worth analysis. Finally, if the model is simple enough, the method one can use is to set the derivative of the evaluation function to zero with respect to wall thickness to find candidate points for its optimal thickness. For the second problem, selecting a best operating schedule involves discrete decisions, which will generally require models that have integer variables. 

Binary integer variables can be used to formulate optimization problems that choose between flowsheet options. For example, consider the problem of selecting a reactor. We can set up a unit cell consisting of a well-mixed reactor, a plug-flow reactor and a... 

The molecular structure of the unknown chemical could be found by inverting these three relationships. However, an explicit inversion is not analytic (the molecular structure is described by integer variables denoting the presence or absence of specific atoms and bonds), and it accepts multiple solutions (there may be several molecules satisfying the constraints). Implicit inversion of Eqs. (1) is possible through the formulation of appropriate optimization problems. However, in such cases the complexity and nonlinear character of the functional relationships used to estimate the values of physical properties in conjunction with the integer variables description of molecular structures, yield very complex mixed-integer optimization formulations. 

Both LP and IP are optimization tools used especially by operations research people. These tools find optimal solutions to problems that include an objective and some constraints. LP variables are defined in jxrsitive real numbers domain while IP variables are defined in positive integer numbers domain. IP problems are hard to solve. The models reviewed in this chapter are linear integer programming model (ILP). If a model employs both continuous and integer variables, it is called mixed integer programming (MIP). 

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