Multiple criteria decision making (MCDM) is one of operations research disci- plines that evaluates a set of alternatives or solutions with respect to a set of decision criteria or objectives. One class of MCDM problems is multi-objective mathematical programming (MOMP) problem where the alternatives are not explicitly known. In MOMP, an alternative (solution) can be found by solving a mathematical model. The number of solutions can be infinite (in case of continuous variables) or very large to count (in case of discrete variables).
The mathematical model of the desired order picking problem introduced in the previous chapter, falls into the category of MOMP problems. These types of problems are solved using multi-objective optimization methods. For a general introduction to multi-objective optimization, refer to Ehrgott 2006 and Gandibleux 2006.
Before continuing any further, some of the standard terminology and defini- tions in multi-objective optimization are described.
A multi-objective optimization problem can be stated as follows: min
x∈X f(x) := {f1(x), . . . , fn(x)} (18)
where n > 1 and X ⊆ Rn is the set of constraints or the feasible set in the decision space(the space of feasible solutions). The image Y of X under vector-valued function f = f1, . . . , fn represents the feasible set in the criterion space (the space of objective
function values). Mathematically speaking:
Y := f(X ) := {y ∈ Rn: y = f(x) for some x ∈ X }
To distinguish between the two spaces, the elements of the decision space are called
solutions and elements of the criterion space are referred to as points.
Typically, in multi-objective optimization problem, there does not exist a sin- gle solution that simultaneously optimizes all objective functions, as the objective functions are conflicting (note that if some objective functions need to be maximized, its negative can be minimized). In this case, the notion of Pareto optimality is in- troduced. A solution is called Pareto optimal or efficient when none of the objective function values can be improved without degrading at least one other objective value. The relation between Pareto optimal and feasible solutions can be described using
dominance relation.
Definition III.1. Solution x1 ∈ X is said to (weakly) dominate solution x2 ∈ X, and
is shown as x1 x2, if and only if it is not worse than x2 in all the objectives and is
strictly better than x2 in at least one objective. Mathematically speaking:
x1 x2 ⇐⇒ fi(x1) ≤ fi(x2) ∀i ∈ {1, 2, . . . , n} fj(x1) < fj(x2) ∃j ∈ {1, 2, . . . , n} (19)
Also, for x1 to strongly dominate x2:
x1 ≺ x2 ⇐⇒ fi(x1) < fi(x2) ∀i ∈ {1, 2, . . . , n} (20)
Some properties of dominance relation are as follows:
Reflexive: The dominance relation is not reflexive since by definition of dominance,
Symmetric: The dominance relation is not symmetric because x1 x2 does not
mean x2 x1. But the opposite is true, i.e. if x1 x2 then x2 x1.
Transitive: The dominance relation is transitive. So, if x1 x2 and x2 x3, then x1 x3.
Note that if x1 x2, it does not imply that x2 x1.
Definition III.2. A feasible solution x ∈ X is called weakly efficient, if there is no
other x0 ∈ X such that f
i(x0) < fi(x) for i = 1, . . . , n. If x is weakly efficient, then f(x) is called a weakly nondominated point.
Definition III.3. A feasible solution x ∈ X is called efficient or Pareto optimal, if
there is no other x0 ∈ X such that f
i(x0) ≤ fi(x) for i = 1, . . . , n and f(x0) 6= f(x).
If x is efficient, then f(x) is called a nondominated point. The set of all efficient solutions x ∈ X is denoted by XE. The set of all nondominated points is denoted by
YN.
Definition III.4. A feasible solution x ∈ X is called ideal if it minimizes all the
objectives simultaneously. In other words, if x ∈ arg minx∈Xfi(x) ∀i = 1, . . . , n.
Definition III.5. The point fI ∈ Rn is the ideal point if
fiI = min
x∈X fi(x) ∀i ∈ {1, . . . , n}
.
Definition III.6. The point fN ∈ Rn is the nadir point if
fiN = max x∈XE
fi(x) ∀i ∈ {1, . . . , n}
The number of Pareto optimal solutions can be infinite or very large to count. The set of Pareto optimal solutions is called Pareto optimal set, Pareto frontier,
Pareto front, or efficient front. As mentioned in Definition III.3, if x is an efficient
solution in the decision space, its image in criterion space is a nondominated point. The set of all nondominated points is called nondominated frontier or nondominated
front.
As defined in Definitions III.5 and III.6, nondominated front is bounded by ideal point as an array with the lowerbound of all the objective functions in the entire search space and nadir point as the upperbound of each objective function in the Pareto optimal set.
Definition III.7. A set ˜YN ⊆ Y is called an approximation of the set YN if no point
in ˜YN is dominated by any other point in ˜YN. In other words, all the points in ˜YN
are nondominated points.
Note that ˜YN is not necessarily a subset of Y and there is no guarantee on
quality of the points or how good of an approximation ˜YN is. But still, having an
approximation or approximate nondominated front is very important. The reason is although the goal of multi-objective optimization is to find the set of nondomi- nated points, typically this is not easy (they are NP-hard to compute). Because, there may not exist an algorithm to find the entire front. Moreover, it may be very time-consuming to compute the exact nondominated front, as usually the number of nondominated points grows exponentially with the size of the problem (Boland, Charkhgard, and Savelsbergh 2016). There were lots of effort in finding the entire nondominated frontier such as the work of Boland, Charkhgard, and Savelsbergh 2015 on bi-objective mixed integer linear programs. For further information about exact methods on multi-objective integer linear programming, refer to the dissertation of Charkhgard 2016.
terms of criteria such as closeness to the nondominated frontier, diversity, etc., it is possible to compare the quality of one approximate nondominated front with another (Zitzler et al. 2003). To do so, there are different approaches which one of them, namely hypervolume metric (Zitzler and Thiele 1999), will be discussed with more detail in Section III.D.