Multiobjective Optimization Problem ---多目标优化问题介绍

1. Introduction

1.1 Multiobjective Optimization Problem

In engineering, it is often a problem to formulate a design in which there are several criteria or design objectives. If the objectives are opposing, then the problem becomes finding the best possible design which still satisfies the opposing objectives. An optimum design problem must then be solved, with multiple objectives and constraints taken into consideration. This type of problem is known as either a multiobjective, multicriteria, or a vector optimization problem.

---

1.2 Multiobjective Optimization Examples

As an example, in the design of an automobile an engineer may wish to maximize crash resistance for safety and minimize weight for fuel economy. This is a multiobjective problem with two opposing objectives, that is, a step towards improving one of the objectives, increasing crash resistance, is a step away from improving the other, increasing weight.

As a second example, an engineer is given the task to design a beam with minimum deformation and weight. This is a multiobjective problem, again with two opposing objectives. That is, an increase in weight would cause a reduction in deformation.

A third example is the design of a lathe for maximum metal removal rate and also maximum tool life. In order to increase the tool life, it is necessary to decrease the metal removal rate.

---

1.3 History (Eschenauer et al 1986)

G.W. Leibniz (1646-1716) and L. Euler (1707-1783) used infinitesimal calculus to find the extreme values of functions. This made it possible for pioneers to study various new fields of mechanics. J. Bernoulli (1655-1705), D. Bernoulli (1700-1782), and Sir Isaac Newton (1643-1727) used these methods to lead them to their findings; Newton in minimizing the resistance of a revolving body and the Bernoulli's in isoperimetric problems.

J.L. de Lagrange (1736-1813) and W.R. Hamilton (1805-1865) developed the several theorems which serve as the basis for the solution of all optimum design problems.

Later, function approximations were developed by Lord Rayleigh (1842-1919), W. Ritz (1878-1909), B.G. Galerkin (1871-1945) and others to solve complicated time-consuming functions, because they could be approximated relatively accurately.

A French-Italian economist named V. Pareto (1848-1923) first developed the principle of multiobjective optimization for use in economics. His theories became collectively known as Pareto's optimality concept.

---

1.4 Problem Solution

A Multiple-Objective (MO) optimum design problem is solved in a manner similar to the Single- Objective (SO) problem . In a SO problem, the idea is to find a set of values for the design variables that, when subject to a number of constraints, yield an optimum value of the objective (or cost) function.

Ideal Solution

In MO problems, the designer tries to find the values for the design variables which optimize the objective functions simultaneously, in this manner the solution is chosen from a so-called Pareto optimal set. In general, for multiobjective problems the optimal solutions obtained by individual optimization of the objectives (i.e., SO optimization) is not a feasible solution to the multiobjective problem.

Ideal solution,(f, f), obtained via individual minimization of each function is not within the feasible region.

2. Definitions

The first step in the optimization process is the formulation of the problem. The designer will need to develop a mathematical model that will closely mimic the behavior of the physical system, for all possible situations.

---

2.1 Design Variables

Design variables will be identified as . For example, a tool manufacturing company wants to optimize the production of allen wrenches and bolt cutters. Then the design variables would be as follows:

x = the number of allen wrenches to manufacture,
x = the number of bolt cutters to manufacture.

Sometimes, the designer may choose to hold one or more of the design variables fixed in order to simplify the problem. These fixed quantities will be classified as parameters. In general, the vector of n design variables will be represented as

---

2.2 Constraints

The next step in the formulation of the problem is identifying the constraints. Constraints are conditions which must occur, in order for the design to function as intended.

For example, a beam is required to sustain a certain load without failing. Constraints are expressed as inequalities and/or equalities.

Inequality Constraints

Inequalities are usually specified by (where is a vector representing the constraints g, j = 1,..., J). The standard form of an inequality constraint is shown below

Equality Constraints

Equality constraints are shown as . In a scalar form they are written as

---

2.3 Objective Functions

The final step in the problem statement is the definition of the objective functions. These are the quantities that the designer wishes to optimize. These functions are designated by

The functions may be defined so that they are all minimized. This can be done by multiplying any objective function which is to be maximized by -1, i.e.

---

2.4 Standard Form

The problem, when written in what is termed the standard form, will appear as follows

The above notation reads as follows: find the real values of the design variables (i.e., they belong to R) which will result in the smallest values of the objective functions subject to both equality and inequality constraints.

---

2.5 Problem Formulation Examples

Example

Design a simply supported beam with a solid rectangular cross section that is 4 m long and can sustain a centrally loaded 50 kN force while minimizing both mass and deformation of the beam. The beam will be constructed of 2014-T6 Aluminum which has the following properties

The desired factor of safety ( ) of the beam is 2.

Figure 1: Shear and Bending Moment Diagrams

Solution:

The box beam will be described by the dimensions of the cross section. The letter h will represent the height, while b will represent the width. These are the design variables.

First the maximum allowable stresses must be calculated. From the maximum distortion energy theory (Fenster and Ugural, 1987), the following two equations can be written:

Now, the two stresses must be defined in terms of the design variables.

The allowable normal stress in a beam is defined as):

There is no need to worry about the shear stress here, because when is maximum, = 0. In relation, when is maximum, = 0. The only stress that we need to consider is.

The first constraint (g) can now be defined.

The second and third constraints are rather obvious; both the base and the height must be non-negative.

The last step in the problem definition is to define the two objective functions. The first function will be the mass of the beam.

The second objective function is the deformation of the beam. The maximum deformation will occur at the center of the beam (L = 2m), where the force is applied.

It is now possible to define the problem in the standard notation.

Example (Osyczka, 1984)

A tool manufacturing company wants to make two products, allen wrenches and bolt cutters. What number of each products should be produced in order to maximize profit? Both tools require construction in two different departments. Allen wrenches require 1 hour in the first department and 1.25 hours in the second department. Bolt cutters requires 1 hour in the first department and 0.75 hours in the second department. Each department may work 200 hours per month. The maximum market demand for bolt cutters is 150 units. The company makes a $4 profit on allen wrenches and a $5 profit on bolt cutters. In addition, the company's best client wants as many bolt cutters as possible.

Solution:

The design variables are the number of units of allen wrenches and bolt cutters to produce.

x = number of allen wrenches
x = number of bolt cutters

The constraints are then related to the working time between the two departments and the maximum market for the bolt cutters. They will be written as follows

Finally, there are two objective functions to define, the first is the maximization of the profit and the second is the production maximization of product A.

---

2.6 Determining Feasible Space

If the feasible space of decision variables is graphed, i.e. the constrained region, then it is then possible to go from a plot of the design variables (Figure 2) to a plot of the objective functions (Figure 3) and map the feasible range of the objective functions. This is known as mapping the space of the objective functions from the space of the decision variables.

Figure 2: Space of Design Variables

Figure 3: Space of Objective Functions

Example (Osyczka 1984)

The Production Planning problem is used to illustrate the feasible space of design variables and objective functions. The feasible space of design variables is shown on the first slide, while the feasible space of the objective functions is shown on the second slide.

2. Definitions

The first step in the optimization process is the formulation of the problem. The designer will need to develop a mathematical model that will closely mimic the behavior of the physical system, for all possible situations.

---

2.1 Design Variables

Design variables will be identified as . For example, a tool manufacturing company wants to optimize the production of allen wrenches and bolt cutters. Then the design variables would be as follows:

x = the number of allen wrenches to manufacture,
x = the number of bolt cutters to manufacture.

Sometimes, the designer may choose to hold one or more of the design variables fixed in order to simplify the problem. These fixed quantities will be classified as parameters. In general, the vector of n design variables will be represented as

---

2.2 Constraints

The next step in the formulation of the problem is identifying the constraints. Constraints are conditions which must occur, in order for the design to function as intended.

For example, a beam is required to sustain a certain load without failing. Constraints are expressed as inequalities and/or equalities.

Inequality Constraints

Inequalities are usually specified by (where is a vector representing the constraints g, j = 1,..., J). The standard form of an inequality constraint is shown below

Equality Constraints

Equality constraints are shown as . In a scalar form they are written as

---

2.3 Objective Functions

The final step in the problem statement is the definition of the objective functions. These are the quantities that the designer wishes to optimize. These functions are designated by

The functions may be defined so that they are all minimized. This can be done by multiplying any objective function which is to be maximized by -1, i.e.

---

2.4 Standard Form

The problem, when written in what is termed the standard form, will appear as follows

The above notation reads as follows: find the real values of the design variables (i.e., they belong to R) which will result in the smallest values of the objective functions subject to both equality and inequality constraints.

---

2.5 Problem Formulation Examples

Example

Design a simply supported beam with a solid rectangular cross section that is 4 m long and can sustain a centrally loaded 50 kN force while minimizing both mass and deformation of the beam. The beam will be constructed of 2014-T6 Aluminum which has the following properties

The desired factor of safety ( ) of the beam is 2.

Figure 1: Shear and Bending Moment Diagrams

Solution:

The box beam will be described by the dimensions of the cross section. The letter h will represent the height, while b will represent the width. These are the design variables.

First the maximum allowable stresses must be calculated. From the maximum distortion energy theory (Fenster and Ugural, 1987), the following two equations can be written:

Now, the two stresses must be defined in terms of the design variables.

The allowable normal stress in a beam is defined as):

There is no need to worry about the shear stress here, because when is maximum, = 0. In relation, when is maximum, = 0. The only stress that we need to consider is.

The first constraint (g) can now be defined.

The second and third constraints are rather obvious; both the base and the height must be non-negative.

The last step in the problem definition is to define the two objective functions. The first function will be the mass of the beam.

The second objective function is the deformation of the beam. The maximum deformation will occur at the center of the beam (L = 2m), where the force is applied.

It is now possible to define the problem in the standard notation.

Example (Osyczka, 1984)

A tool manufacturing company wants to make two products, allen wrenches and bolt cutters. What number of each products should be produced in order to maximize profit? Both tools require construction in two different departments. Allen wrenches require 1 hour in the first department and 1.25 hours in the second department. Bolt cutters requires 1 hour in the first department and 0.75 hours in the second department. Each department may work 200 hours per month. The maximum market demand for bolt cutters is 150 units. The company makes a $4 profit on allen wrenches and a $5 profit on bolt cutters. In addition, the company's best client wants as many bolt cutters as possible.

Solution:

The design variables are the number of units of allen wrenches and bolt cutters to produce.

x = number of allen wrenches
x = number of bolt cutters

The constraints are then related to the working time between the two departments and the maximum market for the bolt cutters. They will be written as follows

Finally, there are two objective functions to define, the first is the maximization of the profit and the second is the production maximization of product A.

---

2.6 Determining Feasible Space

If the feasible space of decision variables is graphed, i.e. the constrained region, then it is then possible to go from a plot of the design variables (Figure 2) to a plot of the objective functions (Figure 3) and map the feasible range of the objective functions. This is known as mapping the space of the objective functions from the space of the decision variables.

Figure 2: Space of Design Variables

Figure 3: Space of Objective Functions

Example (Osyczka 1984)

The Production Planning problem is used to illustrate the feasible space of design variables and objective functions. The feasible space of design variables is shown on the first slide, while the feasible space of the objective functions is shown on the second slide.