Multiobjective presentation
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Transcript of Multiobjective presentation
The engineer is often confronted with “simultaneously”
minimizing (or maximizing) different criteria. The
structural engineer would like to minimize weight and
also maximize stiffness; in manufacturing, one would
like to maximize production output and quality while
minimizing cost and production time.
These practical applications are we refer to them as
multiobjective optimization problems.
Introduction
Mathematically, the problem with multiple objectives may be stated as
where the weights wi are ≥ 0, ∑wi = 1. The weights are chosen from experience.
Example:Each unit of product Y requires 2 hours of machining in the first cell and 1 hour in the second cell. Each unit of product Z requires 3 hours of machining in the first cell and 4 hours in the second cell. Available machining hours in each cell = 12 hours. Each unit of Y yields a profit of $0.80, and each unit of Z yields $2. It is desired todetermine the number of units of Y and Z to be manufactured to maximize:(i) total profit(ii) consumer satisfaction, by producing as many units of the superior quality product Y. If x1 and x2 denote the number of units of Y and Z, respectively, then the problem is:maximize f1 = 0.8 x1 + 2 x2 and f2 = x1subject to 2x1 + 3 x2 ≤ 12x1 + 4 x2 ≤ 12x1, x2 ≥ 0
Concept of Pareto Optimality
The preceding problem is shown graphically in Fig.1. Point A is the solution if only f1 is to be maximized, while point B is the solution if only f2 is to be maximized. For every point in x1 − x2 space, there is a point ( f ((x1), f (x2)) in criterion space.
Referring to Fig. 2, we observe the following :•There is interesting aspect of points lying on the line A − B: no point on the line is “better” than any other point on the line with respect to both objectives. •A point closer to A will have a higher value of f1 than a point closer to B but at the cost of having a lower value of f2. •In other words, no point on the line “dominates” the other.•Furthermore, a point P in the interior is dominated by all points within the triangle as shown in Fig. 2. •the line segment A − B represents the set of “nondominated” points or Pareto points in Ω.•We refer to the line A − B as the Pareto curve in criterion space.•This curve is also referred to as the Pareto efficient frontier or the nondominated frontier.
• general, no solution vector X exists that maximize all the objective
functions simultaneously..
• A feasible solution X is called Pareto optimal if there exists no
other feasible solution Y such that
fj(Y) ≤ fi(X) for i = 1, 2, . . . , k
with fj(Y) < fi(X) for at least one j.
• In other words, a feasible vector X is called Pareto optimal if there
is no other feasible solution Y that would maximize some objective
function without causing a simultaneous decrease in at least one
other objective function.
Definition of Pareto Optimality
Example 2Referring to Fig. 3-a, consider the feasible region and the problem of maximizing f1 and f2. The (disjointe) Pareto curve is identified and shown as dotted lines in the figure. If f1 and f2 were to be minimized, then the Pareto curve is as shown in Fig. 3-b
• Several methods have been developed for solving a
multiobjective optimization problem.
• Some of these methods will be briefly described in the
following slides.
• Most of these methods basically generate a set of Pareto
optimal solutions and use some additional criterion or rule
to select one particular Pareto optimal solution as the
solution of the multiobjective optimization problem.
Solving a multiobjective optimization problem
Weighted Sum Method
• Weight of an objective is chosen in proportion to the
relative importance of the objective.
• Scalarize a set of objectives into a single objective by
adding each objective pre-multiplied by a user supplied
weight .
Advantage
•Simple
Disadvantage
•It is difficult to set the weight vectors to obtain a
Pareto-optimal solution in a desired region in the
objective space
•It cannot find certain Pareto-optimal solutions in the
case of a nonconvex objective space
• Keep just one of the objective and restricting the
rest of the objectives within user-specific values .
ε-Constraint Method
• Keep f2 as an objective Minimize f2(x)• Treat f1 as a constraint f1(x) ≤ ε1
Advantage•Applicable to either convex or non-convex problemsDisadvantage•The ε vector has to be chosen carefully so that it is within the minimum or maximum values of the individual objective function
Lexicographic MethodWith the lexicographic method, preferences are imposed by ordering the objectives according to their importance or significance, rather than by assigning weights. The objective functions are arranged in the order of their importance. Then, the following optimization problems are solved one at a time:
• Here, i represents a function’s position in the preferred sequence, and fj(x*j ) represents the minimum value for the jth objective function, found in the jth optimization problem.
Advantages :
•it is a unique approach to specifying preferences.
•it does not require that the objective functions be
normalized.
•it always provides a Pareto optimal solution.
Disadvantages :
•it can require the solution of many single objective problems
to obtain just one solution point.
•it needs additional constraints to be imposed.