In the previous sections, our focus was on solving MCDM problems with afinite number of alternatives, where each alternative is measured by several conflicting criteria. These MCDM problems were called multiple criteria selection problems (MCSP). The methods we discussed earlier helped in identifying the best alternative or rank order all the alternatives from the best to the worst.
In this and the subsequent sections, we will focus on MCDM problems with aninfinite number of alternatives. In other words, the feasible alternatives are not known a priori but are represented by a set of mathematical (linear/nonlinear) constraints. These MCDM problems are calledmulticriteria mathematical programming(MCMP) problems.
MCMP Problem
Max F(x)={f1(x), f2(x), . . ., fk(x)}
Subject to gj(x)≤0 forj= 1, . . ., m
(5.46)
wherexis ann-vector ofdecision variablesandfi(x),i= 1, . . ., kare thekcriteria/objective functions.
LetS={x/gj(x)≤0, for all “j”}
Y={y/F(x)=yfor somex∈S}
S is called thedecision spaceandY is called thecriteria or objective spacein MCMP.
A solution to MCMP is called a superior solutionif it is feasible and maximizes all the objectives simultaneously. In most MCMP problems, superior solutions do not exist as the objectives conflict with one another.
5.6.1 Definitions
Efficient, Non-Dominated, or Pareto Optimal Solution: A solution xo∈S to MCMP is said to beefficientiffk(x)> fk(xo) for somex∈Simplies thatfj(x)<
fj(xo) for at least one other indexj. More simply stated, an efficient solution has the property that an improvement in any one objective is possible only at the expense of at least one other objective.
A Dominated Solutionis a feasible solution that is not efficient.
Efficient Set:The set of all efficient solutions is called theefficient set orefficient frontier.
Note: Even though the solution of MCMP reduces to finding the efficient set, it is not practical because there could be an infinite number of efficient solutions.
Example 5.1
Consider the following bi-criteria linear program:
Max Z1= 5x1+x2
MaxZ2=x1+ 4x2
Subject to: x1≤5 x2≤3 x1+x2≤6 x1, x2≥0
The decision space and the objective space are given in Figures 5.1 and 5.2, respectively.
Corner Points C and D are efficient solutions whereas corner points A, B, and E are domi- nated. The set of all efficient solutions is given by the line segment CD in both figures.
An ideal solution is the vector of individual optima obtained by optimizing each objective function separately ignoring all other objectives.
In Example 5.1, the maximum value of Z1, ignoring Z2, is 26 and occurs at point D.
Similarly, maximumZ2 of 15 is obtained at point C. Thus the ideal solution is (26,15) but isnot feasible or achievable.
Note:One of the popular approaches to solving MCMP problems is to find an efficient solu- tion that comes “as close as possible” to the ideal solution. We will discuss these approaches
later.
B(0,3) C(3,3)
Optimal for Z2 Optimal for Z1 x2
x1 D(5,1)
E(5,0) Feasible
decision space
A(0,0)
FIGURE 5.1 Decision space (Example 5.1).
B(3,12) Z2
C(18,15)
D(26,9)
E(25,5) (26,15)
A(0,0) Z1
Achievable objective
values
FIGURE 5.2 Objective space (Example 5.1).
5.6.2 Determining an Efficient Solution [12]
For the MCMP problem given in Equation 5.46, consider the following single objective optimization problem, called thePλ problem.
MaxZ= k i=1
λifi(x)
Subject to:x∈S k i=1
λi= 1 λi ≥0
(5.47)
THEOREM 5.1(Sufficiency) Letλi>0for allibe specified. Ifxois an optimal solution for the Pλ problem (Equation 5.47), thenxois an efficient solution to the MCMP problem.
In Example 5.1, if we set λ1=λ2=0.5 and solve the Pλ problem, the optimal solution will be at D, which is an efficient solution.
The Pλ problem (Equation 5.47) is also known as the weighted objective problem.
Warning: Theorem 5.1 is only a sufficient condition and is not necessary. For example, there could be efficient solutions to MCMP that could not be obtained as optimal solutions to the Pλ problem. Such situations occur when the objective space is not a convex set. However, for MCMP problems, where the objective functions, and constraints are linear, Theorem 5.1 is both necessary and sufficient.
5.6.3 Test for Efficiency
Given a feasible solutionx∈Sfor MCMP, we can test whether it is efficient by solving the following single objective problem.
Max W =
k i=1
di
Subject to: fi(x)≥fi(x) +di fori= 1,2, . . ., k x∈S
di≥0
THEOREM 5.2
1. If MaxW >0, thenx is a dominated solution.
2. If MaxW= 0, thenx is an efficient solution.
Note: If MaxW >0, then at least one of the di’s is positive. This implies that at least one objective can be improved without sacrificing on the other objectives.
5.6.4 Classification of MCMP Methods
In MCMP problems, often there are an infinite number of efficient solutions and they are not comparable without the input from the DM. Hence, it is generally assumed that the DM has a real-valuedpreference function defined on the values of the objectives, but it is not known explicitly. With this assumption, the primary objective of the MCMP solution methods is to find thebest compromise solution,which is an efficient solution that maximizes the DM’s preference function.
In the last two decades, most MCDM research have been concerned with developing solu- tion methods based on different assumptions and approaches to measure or derive the DM’s preference function. Thus, the MCMP methods can be categorized by the basic assumptions made with respect to the DM’s preference function as follows:
1. Whencompleteinformation about the preference function is available from the DM.
2. Whenno information is available.
3. Wherepartial information is obtainable progressively from the DM.
In the following sections we will discuss MCMP methods such as goal programming, com- promise programming and interactive methods as examples of categories 1, 2, and 3 type approaches.