2.3 Advanced Pareto Optimality
2.3.3 Advanced Pareto Dominance
In this section we will discuss how to combine the concepts of relaxation and tightening, which will result in what we call advanced Pareto dominance. We will first look at an example. Suppose we want to minimize the following criteria: travel time, number of interchanges, ticket cost, and an additional criterion (add) measured in some unit U. The last criterion acts as a wildcard for the further criteria (a measure for the reliability of interchanges and the sleeping time in night trains among others) that we will introduce in Chapter6.
Using Pareto optimality, all connections in Table2.2except connectionF are optimal.
Now we define our desired goal as
• the travel time should be a relaxed Pareto criterion,
• the number of interchanges should be a Pareto criterion,
• we want an hourly wage of at least 5ein ticket cost, and
• we want an hourly wage of at least 10U in the additional criterion.
We relax the travel time using our time difference formula
reltime(AB) =α(A, B)·min{|dA−dB|,|aA−aB|, ω(A, B)} (2.5) with the third argument to min(·)
ω(A, B) =
½ 0 ifA overtakesB 100,000 otherwise
2.3 Advanced Pareto Optimality 17
Name Departure Criteria Dominance Rules
time time ic add cost Pareto I II III
A 8:00 120 1 50U 90e √ √ √ √
B 8:00 180 1 50U 84e √ √ √
C 8:00 180 1 39U 90e √ √ √
D 8:00 180 1 42U 86e √ √ √
E 8:00 240 1 49U 89e √
F 8:45 125 1 50U 90e √ √ √
Table 2.2: Example connections for advanced Pareto dominance and different sets of rules. Hourly Wages of 5efor ticket cost or 10U in our additional criterion are assumed.
to void the relaxation if Aovertakes B andα(A, B) = 12timeA/timeB as introduced on Page 13. To use tightening on ticket cost and our additional criterion we define
ΛAB(c) := max{cB−cA,0}
for criterion c. Let the relation symbol 4describe the concept of “less or equal in all dimensions and less in at least one of the dimensions.”
Ruleset (I) Our first set of rules (I) consists of these inequalities:
timeA+reltime(AB) 4 timeB
icA 4 icB costA−ΛAB(time)·δcost 4 costB
addA−ΛAB(time)·δadd 4 addB
with the parametersδcost= 5e/h andδadd= 10E/h. With this set of rules we relax the time criterion, have interchanges as a Pareto criterion and tighten the additional and cost criteria requiring hourly wages.
Now all but connectionsD and E are optimal. The time difference formula ensures that connection F is not dominated by connectionA. ConnectionsB andCachieve the desired decrease in either ticket cost or our additional criterion for the additional hour travel time. ConnectionD is dominated by connectionA as it fails to reach the desired hourly wages.
Ruleset (II) However, connectionD obtains an hourly wage of 4eand 8U simultane-ously and therefore could be considered attractive as well. We might even want connection D to dominate connectionsB andC (and not be dominated itself). To this end we may use ∆AB(c) :=cB−cA instead of adding or subtracting ΛAB(c) := max{cB−cA,0}. In doing so, we are able to reward and penalize for a single criterion at the same time. As we replace the maximum term by a simple difference, we automatically gain transitivity,
asymmetry, and irreflexivity (see Page19). This leads to our second, alternative set of rules (II):
timeA+reltime(AB) 4 timeB
icA 4 icB
costA−∆AB(t)·δcost−∆AB(add)·δadd·δcost 4 costB (2.6) addA−∆AB(t)·δadd−∆AB(cost)·δcost·δadd 4 addB (2.7) timeA−∆AB(add)·δadd−∆AB(cost)·δcost 4 timeB (2.8) whereδi= δ1
i.
In the formulae the terms ∆AB(c)·δc are used to determine the tradeoff in time from the difference in criterionc. By multiplying it withδc0, we obtain the tradeoff in criterion c0, e.g. a difference of 15U is equivalent to 7.50e, because we have
∆AB(add)·δadd·δcost= 15E· 1h 10E ·5e
1h = 3h 2 ·5e
1h = 7.5e.
The Inequalities2.6, 2.7, and 2.8 convert time, ticket cost and the additional criterion into only one of them. Either of them is suitable to make our connections A and D optimal and allow connectionD to dominate connectionsBandC, but only one of them is needed. So, although we have four criteria, only three equations are necessary.
Ruleset (III) If we do not want to lose connectionsB and C, we only need to keep separate inequalities for the ticket cost and the additional criterion. Note that we will again use ∆AB(c) instead of ΛAB(c). Our rule set (III) is:
timeA+reltime(AB) 4 timeB
icA 4 icB
costA−∆AB(time)·δcost 4 costB addA−∆AB(time)·δadd 4 addB
timeA−∆AB(add)·δadd−∆AB(cost)·δcost 4 timeB (2.9) This is essentially the rule set (I) plus Equation2.8. Thus we keep the tightening for ticket cost and the additional criterion (protecting connectionsBandC) as well as the weighted sum that protects connections reaching a “combined” hourly wage, like connectionD.
We could also incorporate our trade-off for the number of interchanges from Sec-tion2.3.2. For example by adding
−∆AB(ic)· δic0
100·timeA
on the left hand side of Formula2.9.
Reformulation by sorting If we sort the terms on the left and right hand side in Formula2.9appropriately, we obtain
timeA−addA·δadd−costA·δcost4timeB−addB·δadd−costB·δcost.
2.3 Advanced Pareto Optimality 19 i aitime aiic aiadd aicost reli(AB)
1 1 0 0 0 reltime(AB)
2 0 1 0 0 0
3 δcost 0 0 1 0
4 δadd 0 1 0 0
5 1 0 δadd δcost 0
Table 2.3: The coefficientsaic and relaxation termsreli(A, B) in Formula2.11for rule set (III).
That is, we only compare two weighted sums as our fifth criterion. Similarly, we may sort the whole rule set (III) to look as follows:
timeA+reltime(AB) 4 timeB (2.10)
icA 4 icB
timeA·δcost+costA 4 timeB·δcost+costB
timeA·δadd+addA 4 timeB·δadd+addB
timeA−addA·δadd−costA·δcost 4 timeB−addB·δadd−costB·δcost
In fact, this leads to our final formulation for advanced Pareto Dominance.
Formulation for Advanced Pareto Dominance
Given k criteria and r inequalities, we can formulate each of our inequalities (i) for i∈ {1, . . . , r}as
(i)
k
X
j=1
αicjcj A+reli(A, B)4
k
X
j=1
αicjcj B (2.11) with cj A and cj B denoting the value of criterion cj for connections Aand B. Function reli(A, B) is our relaxation for criterion ci, e.g. a constant or the time difference rule (Formula2.5).
The coefficients for the rule set (III) are shown in Table2.3. For example, we have α4time = δadd, α4add = 1, rel4(A, B) = 0, for the fourth formula, and all α4c
j = 0 for all other criteriacj.
Transitivity, Antisymmetry, and Irreflexivity
We compare only weighted sums of the criteria in an extension of the fundamental Pareto formulation, which is of course transitive, antisymmetric, and irreflexive. Applying relax-ation using functions of the type discussed in Section 2.3.1does not violate the desired properties of our smaller relations.
Expressiveness of Our Formulation Our formulation for advanced Pareto domi-nance can model all introduced variants of multi-criteria domidomi-nance. Classical Pareto dominance and relaxed Pareto dominance are obtained, withk= 4,r= 4 andaic
j = 1 for i=j, andaicj = 0, otherwise. With relaxation functionsreli(AB) we get relaxed Pareto dominance. Without relaxation functions we have Pareto dominance. We will use these
formulations in the computational study in Section9.5. There, they are also given in a less condensed form including coefficient tables like Table2.3.
Using only one equation (r= 1) and weights a1cj 6= 0 for each of the criteria cj, we have a simple weighted sum. If all but one of the a1cj are zero with r = 1, we model dominance on a single criterion.