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The interactive analysis of the multicriteria shortest path problem by the reference point method
European Journal of Operational Research 151 (2003) 103–118 www.elsevier.com/locate/dsw
Decision Aiding
The interactive analysis of the multicriteria shortest path problem by the reference point method Janusz Granat a, Francesca Guerriero a
b
b,*
Institute of Control and Computation Engineering, Warsaw University of Technology, Warsaw and National Institute of Telecommunications, Szachowa 1, Warsaw, Poland Dipartimento di Elettronica, Informatica e Sistemistica Via Bucci, Universit a della Calabria, Rende (CS) 87030, Italy Received 19 March 2001; accepted 26 June 2002
Abstract The multicriteria shortest path problem is considered. The paper presents the interactive method of analyzing this problem by the reference point approach. The reference point guided labeling algorithm was developed. This algorithm finds the Pareto-optimal shortest path which is best attuned to the specified preferences. 2002 Elsevier B.V. All rights reserved. Keywords: Multiple criteria analysis; Pareto-optimal paths; Reference point method
1. Introduction Traditionally, the problem of finding the shortest path from a specified origin node to another node has been considered in the framework of the single objective optimization. More specifically, it is assumed that some value is associated to each arc (for example, the length or the travel time), and the goal is to determine the feasible path for which either the total distance or the total travel time is minimized. In many real applications it is often found that a single objective function is not sufficient to characterize adequately the problem. For instance, in the routing of hazardous materials, it is important *
Corresponding author. Tel.: +39-984-494620; fax: +39-984494713. E-mail address: [email protected] (F. Guerriero).
to consider not only the total distance from a source to a destination, but also the number of people brought into contact with the hazardous materials along a route (Batta and Chiu, 1988). In transportation networks, a typical situation that can be adequately represented only considering more objectives is related to highway construction, where time, cost, and ecological factors must be taken into account at the same time (Current et al., 1987). Another application, in which it is important to deal with several factors, is represented by path planning, where the goal is to find a navigation path for a mobile robot (Fujimura, 1996). In this case, the navigation path can be considered acceptable only if it satisfies multiple objectives, such as safety, time and energy consumption. It is important to observe that the list of the potential applications of the multicriteria shortest
0377-2217/$ - see front matter 2002 Elsevier B.V. All rights reserved. doi:10.1016/S0377-2217(02)00594-5
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path problem (in the sequel, referred to as MSPP) is very extensive, too long to be cited in detail. The reader is referred to (Current and Min, 1986) and (Current and Marsh, 1993) for a more detailed review of the application areas. In the context of the multicriteria optimization, the concept of Pareto-optimality, nondominance or efficient solutions plays a crucial role since it is usually assumed that the criteria are in conflict and thus, in general, it is not possible to find a single optimal solution, but rather a set of Pareto-optimal solutions. In this case, it is necessary to choose from the Pareto-optimal solution set a reasonable solution, not the best for all criteria, but the most satisfactory. In the case of the MSPP, the Pareto-optimal solutions represent those paths such that it is not possible to find another feasible path with a better value in at least one criterion without worsening the value of at least one other criterion. It is important to point out that, while the shortest path problem has a polynomial complexity bound, there exist problem instances for which the computational effort required to solve the MSPP increases exponentially with the size of the problem. In other words, as shown in (Garey and Johnson, 1979), the MSPP is NP-complete. This theoretical aspect has been the cause of a slowing of the research activity in this area for several years. However, recently, a significant number of papers, in which different instances of the MSPP are formulated and solved, have been published. The different approaches proposed in the literature for solving the MSPP can be classified on the basis of the strategy used for exploring the Paretooptimal solution set. More specifically, we can distinguish the following three categories: • Generating methods; • Methods based on utility functions; and • Interactive methods. The methods belonging to the first group can be used either to generate the whole set of nondominated solutions or to evaluate the Pareto-optimal solution set approximately. Approaches for determining an approximation of the Pareto-optimal paths are proposed in (Tung
and Chew, 1992). In addition, Warburton (1987) presents scaling procedures for obtaining all feasible paths that are -nondominated (where is a predetermined degree of accuracy) and demonstrates that the computational effort of these procedures and the number of approximately Pareto-optimal paths obtained are bounded by a polynomial function in the problemÕs size and . In the case in which the aim is to determine exactly all nondominated solutions, it is possible to use multiple labelling approaches described in (Vincke, 1974; Brumbaugh-Smith and Shier, 1989; Hansen, 1980; Skriver and Andersen, 2000; Martins, 1984), ranking methods presented in (Climaco and Martins, 1982; Azvedo and Marins, 1991) or the parametric approach proposed by Mote et al. (1991). Solution approaches in which the decision makerÕs preferences are represented by using an utility function are described in (Carraway et al., 1990; Modesti and Sciomachen, 1996). It is worth noting that in these methods the multicriteria optimization becomes a single-objective optimization problem. The methods, briefly described above, cannot be used for efficiently solving real problems. First of all, generating the whole Pareto-optimal solution set may be computationally intractable, even in the case when a small number of criteria is considered. Furthermore, supposing the Paretooptimal solution set has been determined, it may not be easy for the decision maker to choose from a very large set the nondominated path that he/she likes best, since the number of Pareto-optimal solutions may grow exponentially with the number of nodes. Secondly, it is difficult for the decision maker to define a utility function representing his/ her preferences. In order to overcome the drawbacks mentioned, it is possible to consider interactive methods that are based on a direct interaction with the decision maker. During the process of interaction the decision maker indicates his/her preferences in various forms. The algorithms find the nondominated solutions that best correspond to the decision makerÕs preferences. The interactive methods are very efficient from a computational point of view, since the search for
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the Pareto-optimal solution is done by considering a limited portion of the nondominated solution set, defined on the basis of the decision makerÕs preferences. On the basis of the previous considerations, it is apparent that the use of interactive procedures is an attractive way of dealing with the MSPP. An interactive approach for solving the MSPP with only two criteria has been proposed by Current et al. (1990). This method is characterized by two phases. The main aim of the first phase is to give to the decision maker an approximation of the possible tradeoffs. In the second phase, the search inside a specific portion of the Pareto-optimal solution set, defined on the basis of the decision makerÕs preferences, is carried out by solving a constrained shortest path problem. Similar approaches, in which the search of Pareto-optimal paths inside the duality gap is done by using a k-shortest path algorithm, have been proposed by Climaco and Coutinho-Rodrigues (1988) and Coutinho-Rodrigues et al. (1994). As shown in (Coutinho-Rodrigues et al., 1999), interactive methods based on the k-shortest path algorithm are more efficient than the method proposed by Current et al. (1990). Murthy and Olson (1999) proposed an interactive procedure for solving the bicriterion shortest path problems by using the concept of domination cones. This procedure assumes the decision makerÕs implicit utility function to be quasi-concave and nonincreasing. From the decision makerÕs pairwise comparisons, domination cones are developed which help in reducing the number of Paretooptimal solutions. Antunes et al. (1999) considers MSPP with more than two objectives for solving the routing problem in an integrated communication network. Keeping the above considerations in mind, in this paper, we propose an interactive procedure for the MSPP that is based on the reference point methodology of Wierzbicki (1977), see also (Wierzbicki et al. (2000)). The paper is organized as follows. In Section 2 the definitions related to Pareto-optimal paths with respect to domination cones are introduced. An interactive method of analyzing the MSPP by the reference point approach is presented in Section 3. Section 4 is
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devoted to a description of a label-correcting method for finding the shortest path with respect to the preferences specified by the decision maker. Finally, in Section 5 an example of interactive selection of the Pareto-optimal path is presented.
2. The Pareto-optimal paths with respect to domination cones Pareto-optimality can be generalized by using partial ordering implied by a negative cone. In this section we introduce some definitions related to the Pareto-optimal paths. Let GðN; AÞ be a directed network graph, where N ¼ f1; . . . ; ng is a finite set of nodes and A N N is a finite set of m arcs. Each arc is denoted by an ordered pair (i; j), where i 2 N and j 2 N. A path pij between the two distinct nodes (i 2 N and j 2 N) is defined as the sequence pij ¼ fi ¼ i1 ; ði1 ; i2 Þ; i2 ; . . . ; il 1 ; ðil 1 ; il Þ; il ¼ jg of alternating nodes and arcs. The path is called elementary when no node is repeated in the sequence. When the nodes are repeated in the sequence we call it a non-elementary path. Associated with any arc ði; jÞ 2 A, there is a pði;jÞ dimensional vector of criteria qði;jÞ ¼ ðq1 ; ði;jÞ ði;jÞ q2 ; . . . ; qp Þ. A path pst from the origin node s 2 N to the destination node t 2 N is evaluated by the pdimensional vector of criteria qpst ¼ ðqp1 st ; qp2 st ; . . . ; qpp st Þ. The value of any criterion k 2 f1; . . . ; pg for the given path s 2 N to t 2 N is defined as P pst from ði;jÞ qpk st ¼ ði;jÞ2pst qk . We can generally express this in the form of vector valued linear function q : P0 7! Rp ; where P0 represents the set of all paths from the origin node s to the destination node t in the graph GðN; AÞ. On the basis of the definitions and notation introduced above, the MSPP can be stated as follows: min qpst :
pst 2P0
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The partial ordering in the objective space is implied by a cone D, given by D ¼ fq 2 Rp : qk 6 0; k ¼ 1; . . . ; pg: A strictly negative cone can be defined as e ¼ fq 2 Rp : qk 6 0; D
8k ¼ 1; . . . ; p;
9k ¼ 1; . . . ; p : qk < 0g: Definition 2.1. The path p^st 2 P0 such that ^ q ¼ qð^ pst Þ is called Pareto-optimal if it belongs to the set: b 0 ¼ f^ e Þ \ qðP0 Þ ¼ ;g: P pst 2 P0 : ðqð^ pst Þ þ D A strongly negative cone can be defined as int D ¼ fq 2 Rp : qk < 0;
8 k ¼ 1; . . . ; pg:
Definition 2.2. The path p^st 2 P0 such that ^ q ¼ qð^ pst Þ is called weakly Pareto-optimal if it belongs to the set: pst 2 P0 : ðqð^ pst Þ þ int DÞ \ qðP0 Þ ¼ ;g: p^w0 ¼ f^
3. Interactive selection of the properly Paretooptimal path In this section, we show how the reference point methodology can be applied to the MSPP. The key problem here is a selection of a particular properly Pareto-optimal path out of typically large set of such paths. This selection is implicitly determined by a conversion of a multiobjective problem into a parametric single-objective problem whose solution provides a properly Pareto-optimal path. Different multicriteria optimization methods apply different conversions, see, e.g. (Haimes and Hall, 1974; Sawaragi et al., 1985; Steuer, 1986). We will consider the achievement scalarizing function (ASF). The concept of ASF has been introduced by Wierzbicki (1977). In this approach, the set A Rp , of controlling parameters (reference points) q is considered. A parametric scalarizing function is a function S : qðP0 Þ A 7! R1 . Such a function should desirably have the following two basic properties: The sufficiency property: b : arg max Sðqðpst Þ; q; wÞ P 0
The most interesting from the practical computations are properly Pareto-optimal paths. Let us define the -neighborhood of the negative cone D int D ¼ fq 2 Rp : distðq; DÞ < kqkg;
ð1Þ
pst 2P0
This condition means that for each reference point q the solution path belongs to the set of properly efficient paths. The necessity property:
p
where we can choose any norm in R and a concept of distance between the point q and the cone D ( is a given small number).
For each p^st 2 P0 exists q p^st 2 arg max Sðqðpst Þ; q; wÞ:
ð2Þ
pst 2P0
^ ¼ Definition 2.3. The path p^st 2 P0 such that q qð^ pst Þ is called properly Pareto-optimal if it belongs to the set: b ¼ f^ P pst 2 P0 : ðqð^ pst Þ þ D Þ \ qðP0 Þ ¼ ;g: 0 It can be shown that Pareto-optimal solutions for a specific form of the cone D are properly Pareto-optimal with a prior bound 1 þ 1= ¼ M on trade-off coefficients, see Wierzbicki (Wierzbicki, 1986). If we have a priori bound M on tradeoff coefficients between objectives then ¼ 1= ðM 1Þ.
This condition means that each of the properly Pareto-optimal path can be generated by selection of q. The selection of a particular properly Paretooptimal path is determined by the definition of the reference point (aspiration point). Most of those methods use the maximization of an ASF in the form: Sðq; q; wÞ ¼ min fwk ð qk qk Þg þ 16k6p
p X
wk ð qk qk Þ
k¼1
ð3Þ
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where q 2 Rp is a vector of criteria, q 2 Rp is an aspiration point (reference point), wk > 0, k ¼ 1; . . . ; p, are scaling coefficients (see the comment below) and is a given small positive number. Maximization is over all feasible paths from the origin node s to the destination node t. Maximization of (3) for all paths from s to t generates a properly Pareto-optimal path with the trade-off coefficients smaller than (1 þ 1=). For a nonattainable q, the vector of criteria of the resulting properly Pareto-optimal path is the nearest––in the sense of a Chebyshev weighted norm––to the specified aspiration point q. If q is attainable, then the vector of criteria of the properly Pareto-optimal path is uniformly better. The selection of properly Pareto-optimal paths is controlled by the vector q. There is a common agreement that the aspiration point is a very good controlling parameter for examining a properly Pareto-optimal paths. The scaling coefficients wk should not be confused with the weights used by some methods for conversion of a multicriteria problem into a single-criterion problem with a weighted sum of original criteria. In the weighted sum methods, the weights are the main controlling parameters. In the reference point approaches, the scaling coefficients are needed to provide for uniform scaling of all criteria and are not used as controlling parameters. The information provided by the user could be extended by specification of the reservation levels. These levels determine the values which the user would not like to achieve. In this case the ASF usually takes the form: Sðq; q; qÞ ¼ min fuk ðqk ; qk ; qk Þg 16k6p
þ
p X
uk ðqk ; qk ; qk Þ;
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(4) over the set of feasible paths provides a properly Pareto-optimal solution with the properties discussed above for the function (3). CAFs uk ðÞ are strictly monotone––decreasing for minimized––functions of the objective vector component qk with values uk ðUk ; Þ ¼ 1 þ c; uk ðqk; Þ ¼ 1; uk ðqk ; Þ ¼ 0; uk ðNk ; Þ ¼ c
ð5Þ
where c and c are given positive parameters, Uk represents the utopia point and it is composed of best values out of the set of all properly Paretooptimal paths for each criterion, whereas Nk is the nadir point composed of worst values out of the same set. The piecewise linear component achievement functions uk ðÞ proposed in (Wierzbicki, 1986) are defined by: 8 < fk wk ðqk qk Þ þ 1; if qk < qk if qk 6 qk 6 qk uk ðq; q; qÞ ¼ wk ðqk qk Þ þ 1; : g w ðq q Þ if qk < qk k k k k ð6Þ where wk ¼ 1=ðqk qk Þ, and fk , gk (k ¼ 1; 2; . . . ; p) are given parameters that are set in such a way that uk ðÞ takes the values defined by (5). The aspiration point and the reservation point is selected arbitrary by the user. There are various methods that supports the selection of these points in the process of interaction, see (Wierzbicki et al., 2000). We can distinguish methods based on specialized graphical user interfaces. One of them was proposed by Granat and Wierzbicki (1996) and Granat and Makowski (2000). This approach will be applied in the example presented in the Section 5.
ð4Þ
k¼1
k ; qk, rewhere q; q are vectors (composed of q spectively) of aspiration and reservation levels respectively, and uk ðqk ; qk ; qk Þ are the corresponding component achievement functions (CAF), which can be simply interpreted as nonlinear monotone transformations of qk taking into account the information represented by qk and qk > 0 is the same as the one used when defining properly Pareto-optimal paths. Maximization of the function
4. The reference point guided labeling algorithm 4.1. The label vector based optimality conditions In order to apply efficiently the reference point methodology to the MSPP, it is important to develop a specific approach for the maximization of the ASF over all feasible paths from the origin node s to the destination node t.
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In what follows, we refer to this problem as the reference point guided efficient (RPGE) path problem. In this section, we formally describe the optimality conditions for the RPGE path problem, which allow to assess when a label vector is optimal, representing, in this case, the value of the path for which ASF is maximized. We assume that the paths are elementary paths. On the basis of the optimality conditions, we define in the next section a prototype method and demonstrate the power of these conditions in guiding the design of solution algorithms. For the sake of simplicity, it is assumed that the ASF takes the form (3) and that wk ¼ 1, 8k ¼ 1; . . . ; p. However, these assumptions can be done without loss of generality, since the results stated below and the algorithms to be described in the following can be easily adapted to solve the RPGE path problem in the case in which the ASF takes the form (4) or other forms that fulfill the conditions (1) and (2). Before the statement of necessary and sufficient optimality conditions, let us introduce the following definition. Definition 4.1. Given a path psj from s to j, we define the value of the path as the following quantity: p
min fwk ð qk qk sj Þg þ
16k6p
p X
p wk qk qk sj :
ð7Þ
k¼1
The method described in the next section is based on a labeling procedure. More specifically, for each node j 2 N, it maintains and updates a label yj which represents either the value of the RPGE path from node s to node j (at the end of computation) or a lower bound for this value (at an intermediate stage of computation). The use of labels is motivated by simple optimality conditions, which are given in the following proposition. Proposition 4.1 (RPGE path optimality conditions). For every node j 2 N, let yj denote the value of some path psj from the origin node s to node j.
Then the labels yj , 8j 2 N, represent the value of the RPGE path from s to j if and only if they satisfy the following optimality conditions: ys ¼ 0;
ð8Þ
n o ðijÞ yj P min qk qpk si þ qk 16k6p
þ
p n o X ðijÞ qk qpk si þ qk ;
ð9Þ
k¼1
8ði; jÞ 2 A, and for all paths psi from s to i such that: j 62 psi :
ð10Þ
Proof. It is worth noting that condition (8) states that loops containing node s are not allowed, whereas condition (10) ensures that the RPGE path from node s to node j does not include loops, that is it is an elementary path. We use a contradiction argument to establish that an optimal label yi must satisfy conditions (9) and (10). Let us suppose that a node i exists such that: n o ðijÞ yj < min qk qpk si þ qk 16k6p
þ
p n o X ðijÞ qk qpk si þ qk ; k¼1
where psi is a path from s to i that satisfies condition (10). In this case, we could improve the value of the RPGE path to node j by passing through node i, thereby contradicting the optimality of the labels yj . Conditions (9) and (10) are also sufficient for optimality, in the sense that if each label yj ; j 6¼ s represents the value of some directed path from node s to node j and it satisfies conditions (9) and (10), then it must be optimal. This result can be proved by contradiction. More specifically, we suppose that the label yj associated to a node j 6¼ s satisfies the conditions (9) and (10) and let psj ¼ ðs ¼ i1 ; ði1 ; i2 Þ; i2 ; . . . ; ðil 1 ; il Þ; il ¼ jÞ be the corresponding path. We assume that psj is not optimal (i.e., it is not the RPGE path from the origin node s to the node j).
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This means that it is possible to find another path, from node s to node j, for example, p^sj ¼ ðs ¼ ^i1 ; ð^i1 ; ^i2 Þ; ^i2 ; . . . ; ð^il 1 ; ^il Þ; ^il ¼ jÞ whose value is greater than yj . Thus, on the basis of the Definition 4.1, we have p X p^ p^ min qk qk sj þ qk qk sj
16k6p
¼ min
16k6p
þ
k¼1 p^s^i
qk qk
p X
l 1
ð^i
^i Þ
þ qk l 1 l
p^s^i ð^i ^i Þ qk q^k l 1 þ qk l 1 l > yj ;
k¼1
thereby contradicting the hypothesis that yj satisfies the conditions (9) and (10). 4.2. A class of label correcting methods for the RPGE path problem The class of methods presented in this section can be viewed as an extension of the generic label correcting methods proposed in (Bertsekas, 1993), for solving the classical single-origin single-destination shortest path problem. In order to describe the proposed method, we refer to the following algorithm, for which we assume that yj , 8j 2 N, defines a lower bound for the value of the RPGE path from node s to node t and V represents the set of candidate nodes, whose incident arcs might violate the optimality conditions given in Proposition 4.1. Initially, we have ys ¼ 0; yj ¼ 1;
8j 2 N; j 6¼ s;
V ¼ fsg: Assuming V is nonempty, a typical iteration of the algorithm is as follows: 1. Remove from V a node i; 8j 2 N : ði; jÞ 2 A and j 6¼ s, Pp ði;jÞ If yj < min16k 6p f qk ðqpk si þqk Þgþ k¼1 ði;jÞ f qk ðqpk st þqk Þg and j 62 psi then ði;jÞ • set f qk ðqpk si þ qk Þg þ Pp yj ¼ minp1st6 k 6 p ði;jÞ k¼1 f qk ðqk þ qk Þg • add j to V if j does not already belong to it.
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The algorithm terminates when V is empty. On the basis of the operations executed by the prototype algorithm stated above, we can demonstrate the following proposition. Proposition 4.2 (correctness of the prototype method). 1. At the end of each iteration, the following conditions hold: (a) ys ¼ 0; (b) 8j, if yj 6¼ 1 and j 6¼ s, then yj is the value of some path from node s to node j; (c) if i 62 V , then either yi ¼ 1 or else a path from node s to node i exists (whose value is equal to yi ) such that 8j : ði; jÞ 2 A if j 62 psi then yj P min1 6 k 6 p fqk ðqpk si þ P ðijÞ ðijÞ qk Þg þ pk¼1 fqk ðqpk si þ qk Þg: 2. When the algorithm terminates, for all yj 6¼ 1, j 6¼ s the following condition holds: (a) n o ðijÞ yj P min qk qpk si þ qk 16k 6p
þ
p n o X ðijÞ qk qpk si þ qk ;
8ði; jÞ 2 A
k¼1
and for all path psi from node s to node i that does not contain the node j.
Proof (1) Condition (1a) holds because, initially, ys ¼ 0 and, by the rules of the algorithm, ys cannot change. We prove condition (1b) by induction on the iteration count. In fact, initially, condition (1b) holds, since node s is the only node with a finite label value. Suppose that (1b) holds at the beginning of some iteration. Let i be the node removed from V. If i ¼ s, it happens only at the first iteration. At the end of this iteration, we have ðsjÞ
yj ¼ min fqk qk g þ 16k6p
p X
ðsjÞ
fqk qk g;
k¼1
for all outward neighbors j of s, yj ¼ 1 for all other j 6¼ s. Thus, the label vector has the required
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property. If i 6¼ s then yi > 1, which is true for all nodes of V by the rules of the algorithm, and yi is the value of some path psi starting from s and ending at i (by the induction hypothesis). When a label yj changes as a result of the iteration, yj is set to ði;jÞ
min f qk ðqpk si þ qk Þg
16k6p
þ
p X
ði;jÞ
f qk ðqpk st þ qk Þg;
k¼1
which represents the value of the path p^sj from node s to node j. Finally, note that the new path p^sj does not contain loops, since the label yj is updated only if psi does not include the node j. This completes the induction proof of (1b). To prove (1c), note that if node i is removed from V and the path psi from node s to node i does not contain the node j the condition ði;jÞ
qk ðqpk si þ qk Þg yj P min f 16k6p
þ
p X
ðijÞ
f qk ðqpk si þ qk Þg
k¼1
is satisfied by the rules of the algorithm. Up to the next entrance of i into the list V the path from node s to node i remains the same, while the label yj ; 8j : ði; jÞ 2 A cannot decrease. Therefore, the condition is preserved. (2) At this point we demonstrate that for all yj 6¼ 1 condition (2a) is satisfied upon termination. Indeed, condition (1b) implies that upon termination we have, for all nodes j such that yj 6¼ 1, yj represents the value of some path from node s to node j. Furthermore, condition (1c) implies that for all nodes j such that yj 6¼ 1 upon termination we have ðijÞ
qk ðqpk si þ qk Þg yi P min f 16k6p
þ
p X
ðijÞ
f qk ðqpk si þ qk Þg;
8ði; jÞ 2 A
to be removed from the candidate list V. More specifically, the selection strategies used in the case of the classical single-origin single-destination shortest path problem can be generalized to the case of the problem considered here. In this case, since the goal is to maximize the ASF, the performance of the algorithm could get better if, at each iteration, the node exiting the candidate list is the node with relatively large label. It is worth noting, that in the case in which the criteria associated to each arc are nonnegative, it is not possible to develop a method for the solution of the RPGE path problem that guarantees that a node once removed from the candidate list V has a permanent label value. Different methods have been developed and they are basically based on the following node selection strategies, that can be viewed as a generalization of the approaches proposed for the classical shortest path problem (Bertsekas, 1993): • The simplest method of Bellman–Ford (BF, for short), for which the candidate list is maintained in a FIFO queue; • The large label first (LLF, for short) that can be viewed as a generalization of the small label first proposed in (Bertsekas, 1993). In this method, the node removed is always the top node of the candidate list V, while the entering node i is added at the top or the bottom of it, depending whether the label of i is larger or equal, or smaller than the label of current top node, respectively; • The small label last (SLL, for short) that is an adaptation of the method proposed in (Bertsekas et al., 1996). It is characterized by a sophisticated node removal strategy. When the node at the top of the candidate list V has a smaller label than the average node label in the candidate list, this node is not removed, but rather it is repositioned at the bottom of the list.
k¼1
and for all path psi from node s to node i that does not contain the node j. There are many implementations of the generic algorithm. They differ in how they select the node
As in the case of the classical shortest path problem, it is simple to combine the LLF queue insertion and the SLL node selection strategies, obtaining in this way combined methods, whose performances are improved.
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4.3. Computational experiments and discussion The proposed label-correcting method has been implemented and tested by using a single processor of an Origin 2000, a multiprocessor consisting of four nodes, each of them having a shared memory of 128 MB. Each node consists of two processors R10000 at 195 MHz, with a 4 MB cache memory and a hub device, which carries out duties similarly to a bus in a bus-based system. The nodes are connected by two routers. The operating system used is IRIX 6.4, whereas the compiler used is Fortran 77. With the aim to evaluate numerically the efficiency of the methods described in the preceding section, we have tested and compared the following codes: 1. BF: Bellman–Ford algorithm; 2. LLF: Large label first method; 3. LLF SLL: Large label first method, using, in combination, the small label last strategy for node removal. The computational results have been carried out on the basis of different choices of the number of criteria. More specifically, p has been set to four and eight, respectively. In all the computational experiments, we have chosen the origin node s equal to node 1 and the destination node t equal to jNj. The codes have been tested on two classes of networks: random networks and grid networks. These two topologies are well recognized and have been used extensively to test algorithms for the classical shortest path problem. For all test problems, the criteria have been chosen according to a uniform distribution in the range from ½1; 1000. • Random networks We have considered a set of eight random networks of varying size. All the problems have been generated using a modified version of the public domain program NETGEN (Klingman et al., 1974). The characteristics of the random networks are reported in Table 1, where for each test problem,
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Table 1 Characteristics of the random networks Problem
Nodes
Arcs
R1 R2 R3 R4 R5 R6 R7 R8
31,622 15,811 11,952 10,000 40,000 5000 20,000 25,000
1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000
the number of nodes and the number of arcs are provided. • Grid networks These problems, whose characteristics are reported in Table 2, have been generated by using a modified version of the Gridgen generator of Bertsekas (Bertsekas, 1993). We have considered both square (G1–G4) and rectangular (G5–G8) test problems in which the nodes are arranged in a planar grid. Each pair of adjacent nodes is connected in both directions. We can also have additional arcs with random starting and ending nodes. The number of arcs is 1,000,000 for all problems and the number of nodes is selected so that the total number of additional arcs is approximately 2, 3, 4 and 5 times the number of grid arcs. The details of the computational results are given in Tables 3–6, where, for each test problem and for each value of p, we indicate the solution time and the total number of iterations required by each algorithm. Table 2 Characteristics of the grid networks Problem
Dimension
Nodes
Arcs
G1 G2 G3 G4 G5 G6 G7 G8
289 289 250 250 224 224 204 204 204 408 177 354 158 316 144 288
83,521 62,500 50,176 41,616 83,232 62,658 49,928 41,472
1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000
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Table 3 Time (in seconds) required to solve the problems on random networks Problem
p
BF
LLF
LLF SLL
R1
4 8
4.00 5.71
3.88 5.63
3.32 4.95
R2
4 8
3.58 5.22
3.46 4.66
2.95 4.24
R3
4 8
3.56 4.65
3.29 4.41
2.78 4.08
R4
4 8
3.75 4.86
3.41 4.54
2.80 4.05
R5
4 8
4.35 5.82
3.94 5.71
3.44 5.27
R6
4 8
3.56 4.39
3.40 4.08
2.73 3.78
R7
4 8
3.64 4.79
3.43 4.71
2.98 4.38
R8
4 8
3.88 5.06
3.72 5.00
3.16 4.55
Table 4 Time (in seconds) required to solve the problems on grid networks Problem
p
BF
LLF
LLF SLL
G1
4 8
99.47 199.44
83.22 173.57
70.52 151.74
G2
4 8
81.27 174.71
67.18 150.32
57.35 139.03
G3
4 8
66.40 144.67
61.36 127.45
47.29 108.40
G4
4 8
74.89 139.04
63.43 119.17
51.57 98.85
G5
4 8
99.08 203.14
85.41 164.75
76.32 138.67
G6
4 8
88.75 177.36
74.66 142.71
59.36 122.38
G7
4 8
81.12 136.14
63.77 123.50
51.30 101.35
G8
4 8
72.16 119.65
62.04 111.52
47.29 89.93
The computational results clearly demonstrate that, in all the test problems, LLF SLL is much faster than the other methods. For example,
for the test problem G7 (see Table 4) the execution time of BF is equal to 81.12 for p ¼ 4 and equal to 136.14 for p ¼ 8, the execution time of LLF is
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Table 5 Number of iterations required to solve the problems on grid networks Problem
p
BF
LLF
LLF SLL
G1
4 8
1,065,003 1,269,252
794,782 1,022,322
696,154 897,515
G2
4 8
681,760 821,713
533,540 681,911
453,996 617,091
G3
4 8
460,660 548,353
403,972 466,995
306,088 399,974
G4
4 8
425,170 457,734
345,776 385,679
277,520 314,668
G5
4 8
1,052,572 1,216,845
834,823 948,652
747,585 782,279
G6
4 8
745,948 854,359
571,202 647,615
462,785 539,537
G7
4 8
541,875 542,272
398,462 460,415
315,944 380,063
G8
4 8
409,358 401,872
332,635 361,151
253,889 285,906
Table 6 Number of iterations required to solve the problems on random networks Problem
p
BF
LLF
LLF SLL
R1
4 8
43,245 37,802
42,317 37,720
35,308 33,274
R2
4 8
21,557 19,453
21,132 17,759
17,826 16,210
R3
4 8
16,649 13,514
15,666 13,232
13,081 12,277
R4
4 8
14,683 11,975
13,602 11,554
11,119 10,345
R5
4 8
56,114 45,503
50,902 45,074
43,752 41,648
R6
4 8
7185 5592
7071 5417
5654 5042
R7
4 8
27,192 22,183
25,872 22,137
22,256 20,624
R8
4 8
35,122 28,354
33,650 28,357
27,797 25,697
equal to 63.77 for p ¼ 4 and equal to 123.50 for p ¼ 8, whereas the execution time of LLF
SLL is of 51.30 for p ¼ 4 and of 101.35 for p ¼ 8.
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A similar trend has also been observed for the random networks. For example, for the test problem R4 (see Table 3) the execution time of BF is equal to 3.75 for p ¼ 4 and 4.86 for p ¼ 8, the execution time of LLF is of 3.41 for p ¼ 4 and 4.54 for p ¼ 8, whereas the execution time of LLF SLL is equal to 2.80 for p ¼ 4 and 4.05 for p ¼ 8. This behavior can be explained by comparing the number of the iterations executed by the methods (see Tables 5 and 6): it is more for BF and LLF than LLF SLL. For example, for the test problem G7 and p ¼ 4, LLF SLL performs 315,944 iterations, BF executes 541875 iterations (41% more), whereas the number of iterations of LLF is equal to 398,462 (20% more). In addition, for the test problem G7 and for p ¼ 8, the number of iterations executed by BF is more than 24% of the LLF SLL iterations, while LLF iterations are more than 18% of the LLF SLL iterations. In conclusion, the LLF and SLL strategies are very helpful in reducing the number of iterations, with a significant reduction in the execution time. 4.4. An algorithm for finding the utopia point and the nadir point In this section, we describe a label-correcting method for finding nadir and utopia criterion values for the multiple objective shortest path problem. As mentioned in Section 4, the utopia point Uk is composed of best values out of the set of all properly Pareto-optimal path for each criterion, whereas the nadir point Nk is composed of worst values out of the same set. An utopia point can be easily computed as a result of p single criterion optimization with each criterion at a time serving as an objective function. In other words, it is necessary to solve p shortest path problems from node s to node t by considering one criterion at a time. In order to solve the problem of finding utopia criterion values, it is possible to consider two different strategies.
The easiest way is to apply p times a classical algorithm for solving the shortest path problem, by considering each time as a cost one of the p criteria. However, this method is not very efficient, since during the computation of the shortest path with respect to the ith criterion it is possible to generate information that can be used to determine the shortest path with respect to the i þ 1th criterion. For this reason, a more efficient method can be defined if the p shortest paths are obtained in one run. Following this idea, in what follows, we present a method that is characterized by p different sweeps. During each sweep k, we iteratively improve the current estimates of the path values with respect to the k; k þ 1; . . . ; p criterion. At the end of the first sweep, we compute the shortest path with respect to the first criterion, at the end of the second sweep, the shortest path with respect to the second criterion, and so on. Finding a nadir point is typically difficult and different methods have been proposed in the literature. Exact algorithms for computing nadir criterion values have been proposed in (Benson, 1992; Benson, 1993a), whereas heuristic approaches for obtaining estimates of the nadir point have been presented in (Benson, 1993b; Korhonen et al., 1997). The method, described in what follows, allows one to compute in one run the utopia point and an approximation of the nadir point, defined as the worst value obtained for each criterion during the determination of the utopia point. This method is divided in two main phases. In the first phase, the utopia point is determined, and in the second phase, the information found in the previous phase is used to compute an estimate of the nadir point. In order to describe the proposed method, we assume that for each node t 2 N and for each criterion k, k ¼ 1; . . . ; p, Ukt defines an upper bound on the value of the shortest path from node s to node t with respect to the kth criterion. At the end of the algorithm Ukt , 8t 2 N and 8k ¼ 1; . . . ; p will be the optimal label, representing the utopia point. In order to compute the nadir point, the algorithm considers a set of p label vectors, Nk ¼ ðNk1 ;
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Nk2 ; . . . ; Nkn Þ k ¼ 1; . . . ; p, defined in such way that for each node t 2 N and for each criterion k, k ¼ 1; . . . ; p Ntk , represents the worst value obtained during the computation of the utopia point and it uses a double-linked list Pstn for storing the sequence of nodes defining the optimal path from node s to node t with respect to the nth criterion. In addition, the algorithm maintains a set of p lists of candidate nodes V ðkÞ , k ¼ 1; . . . ; p, in which the nodes with the potential to improve the label of other nodes are stored. Initially, we have V ð1Þ ¼ fsg; Ukt
¼ 1;
V ðkÞ ¼ ;; t 6¼ s;
k ¼ 1; . . . ; p;
Uks ¼ 0;
k ¼ 1; . . . ; p;
Nkt
8t 2 N;
¼ 0;
Psjk ¼ ftg;
8k 6¼ 1;
8t 2 N;
k ¼ 1; . . . ; p; k ¼ 1; . . . ; p:
The sketch of the proposed method is the following: 1. First phase (computation of the utopia point) For all k ¼ 1; . . . ; p: (a) remove a node i from V ðkÞ ; 8j 2 N : ði; jÞ 2 A; j 6¼ s For all n ¼ k; . . . ; p ðijÞ
IF Unj > Uni þ qn THEN ðijÞ
Unj Uni þ qn update Psjn add j to V ðnÞ (if j does not already belong to it). (b) if V ðkÞ 6¼ ;; go to step a. 2. Second phase (computation of the nadir point) For all k ¼ 1; . . . ; p: For all t 2 N For all n ¼P 1; . . . ; p ði;jÞ IF Nkt < ði;jÞ2Pst qk THEN n P ði;jÞ Nkt ði;jÞ2Pst qk
5. An example of interactive selection of the Paretooptimal paths In order to better clarify the basic operations of the interactive selection of the Pareto-optimal path, we refer to the simple network of Fig. 1 (modified example considered in (Martins, 1984)). There are four criteria and the goal is to find the path which corresponds to specified preferences. For specification of the preferences, we have applied the software ISAAP, developed by Granat and Makowski (2000). This software supports specification of user preferences in terms of both aspiration/reservation levels, and it also provides the user with other means of control over the problem analysis by allowing to change in criteria status, to select the displayed solutions, etc. At the beginning (i.e., initialization phase), the ISAAP module computes the utopia point, the nadir point, and it determines the so-called compromise Pareto-optimal path, which corresponds to the optimal solution of the RPGE path problem for which the aspiration and reservation levels, used to select the RPGE path, are set to the utopia and to an approximation of the nadir points, respectively (see Eq. (4)). The related solution is presented in Fig. 2 and the corresponding RPGE path is reported in Fig. 3. The graphs of the component achievement functions (for 0 6 uk 6 1 (see Eq. (6))) are presented to the user on the screen. The solution, marked by circle, is projected into two dimensional space, in which, for each
n
On the basis of the operations executed by the method stated above, it is easy to verify that the method determines correctly the utopia/nadir point.
115
Fig. 1. Example of the network.
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Fig. 2. A compromise solution.
achievement functions (see Eq. (4)) which are used to select a Pareto-optimal path. In our example, the user changes reservation point for criterion q1 to value equal to 14 and the reservation point for criterion q2 to the value equal to 6. The new solution is presented in Fig. 4 and the corresponding path is reported in Fig. 5. The user can continue the interactive process in order to find the optimal RPGE path. Fig. 3. The compromise Pareto-optimal path.
6. Conclusions criterion, its values (the x-axis) and the degree of satisfaction (the y-axis) of meeting preferences, expressed by aspiration and reservation levels, are reported. In the next step, an interactive procedure is used for helping the user in selecting an Pareto-optimal path that best corresponds to his/her preferences. During such a procedure the user specifies preferences, including values of criteria that he/she wants to achieve and to avoid. Those values are called aspiration and reservation levels, respectively. Such a specification defines component
In this paper we have focused on the solution of the MSPP. In particular, we have presented a new approach to the interactive selection of the Paretooptimal path, in which the selection process is supported by a graphical user interface. The proposed method does not require the generation of all the properly Pareto-optimal paths, but it allows the user to focus his/her attention only on the path that appears the most promising, that is, on the reference point guided Pareto-optimal path (i.e., RPGE path).
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117
Fig. 4. Setting of preferences in the first iteration.
Fig. 5. The selected path for the first iteration.
An algorithm to find the RPGE path has been designed and implemented. The code has been tested extensively on a different variety of random and grid networks with a number of criteria chosen up to eight. The computational results obtained are very encouraging.
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International Teletraffic Congress––ITC 16 Volume 3b Teletraffic Science and Engineering. Elsevier, pp. 1291– 1300. Azvedo, J., Marins, E.Q.V., 1991. An algorithm for the multiobjective shortest path problem on acyclic networks. Investigacao Operational 11, 52–69. Batta, R., Chiu, S.S., 1988. Optimal obnoxious paths on a network: Transportation of hazardous materials. Operations Research 36, 84–92. Benson, H., 1992. A finite, nonadjacent extreme-point search algorithm over the efficient set. Journal of Optimization Theory and Application 73, 47–64. Benson, H., 1993a. A bisection-extreme point search algorithm for optimizing over the efficient set. Journal of Global Optimization 3, 95–111. Benson, H., 1993b. A fase search heuristic algorithm for optimizing over the efficient set. Naval Research Logistics 40, 103–116. Bertsekas, D.P., 1993. A simple and fast label correcting algorithm for shortest paths. Networks 23, 703–709. Bertsekas, D.P., Guerriero, F., Musmanno, R., 1996. Parallel asynchronous label correcting methods for shortest paths. Journal of Optimization Theory and Applications 88, 297– 320. Brumbaugh-Smith, J., Shier, D., 1989. An empirical investigation of some bicriterion shortest path algorithms. European Journal of Operational Research 43, 216–224. Carraway, R., Morin, L., Moskowitz, H., 1990. Generalized dynamic programming for multicriteria optimization. European Journal of Operational Research 44, 95–104.
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Decision Aiding
The interactive analysis of the multicriteria shortest path problem by the reference point method Janusz Granat a, Francesca Guerriero a
b
b,*
Institute of Control and Computation Engineering, Warsaw University of Technology, Warsaw and National Institute of Telecommunications, Szachowa 1, Warsaw, Poland Dipartimento di Elettronica, Informatica e Sistemistica Via Bucci, Universit a della Calabria, Rende (CS) 87030, Italy Received 19 March 2001; accepted 26 June 2002
Abstract The multicriteria shortest path problem is considered. The paper presents the interactive method of analyzing this problem by the reference point approach. The reference point guided labeling algorithm was developed. This algorithm finds the Pareto-optimal shortest path which is best attuned to the specified preferences. 2002 Elsevier B.V. All rights reserved. Keywords: Multiple criteria analysis; Pareto-optimal paths; Reference point method
1. Introduction Traditionally, the problem of finding the shortest path from a specified origin node to another node has been considered in the framework of the single objective optimization. More specifically, it is assumed that some value is associated to each arc (for example, the length or the travel time), and the goal is to determine the feasible path for which either the total distance or the total travel time is minimized. In many real applications it is often found that a single objective function is not sufficient to characterize adequately the problem. For instance, in the routing of hazardous materials, it is important *
Corresponding author. Tel.: +39-984-494620; fax: +39-984494713. E-mail address: [email protected] (F. Guerriero).
to consider not only the total distance from a source to a destination, but also the number of people brought into contact with the hazardous materials along a route (Batta and Chiu, 1988). In transportation networks, a typical situation that can be adequately represented only considering more objectives is related to highway construction, where time, cost, and ecological factors must be taken into account at the same time (Current et al., 1987). Another application, in which it is important to deal with several factors, is represented by path planning, where the goal is to find a navigation path for a mobile robot (Fujimura, 1996). In this case, the navigation path can be considered acceptable only if it satisfies multiple objectives, such as safety, time and energy consumption. It is important to observe that the list of the potential applications of the multicriteria shortest
0377-2217/$ - see front matter 2002 Elsevier B.V. All rights reserved. doi:10.1016/S0377-2217(02)00594-5
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path problem (in the sequel, referred to as MSPP) is very extensive, too long to be cited in detail. The reader is referred to (Current and Min, 1986) and (Current and Marsh, 1993) for a more detailed review of the application areas. In the context of the multicriteria optimization, the concept of Pareto-optimality, nondominance or efficient solutions plays a crucial role since it is usually assumed that the criteria are in conflict and thus, in general, it is not possible to find a single optimal solution, but rather a set of Pareto-optimal solutions. In this case, it is necessary to choose from the Pareto-optimal solution set a reasonable solution, not the best for all criteria, but the most satisfactory. In the case of the MSPP, the Pareto-optimal solutions represent those paths such that it is not possible to find another feasible path with a better value in at least one criterion without worsening the value of at least one other criterion. It is important to point out that, while the shortest path problem has a polynomial complexity bound, there exist problem instances for which the computational effort required to solve the MSPP increases exponentially with the size of the problem. In other words, as shown in (Garey and Johnson, 1979), the MSPP is NP-complete. This theoretical aspect has been the cause of a slowing of the research activity in this area for several years. However, recently, a significant number of papers, in which different instances of the MSPP are formulated and solved, have been published. The different approaches proposed in the literature for solving the MSPP can be classified on the basis of the strategy used for exploring the Paretooptimal solution set. More specifically, we can distinguish the following three categories: • Generating methods; • Methods based on utility functions; and • Interactive methods. The methods belonging to the first group can be used either to generate the whole set of nondominated solutions or to evaluate the Pareto-optimal solution set approximately. Approaches for determining an approximation of the Pareto-optimal paths are proposed in (Tung
and Chew, 1992). In addition, Warburton (1987) presents scaling procedures for obtaining all feasible paths that are -nondominated (where is a predetermined degree of accuracy) and demonstrates that the computational effort of these procedures and the number of approximately Pareto-optimal paths obtained are bounded by a polynomial function in the problemÕs size and . In the case in which the aim is to determine exactly all nondominated solutions, it is possible to use multiple labelling approaches described in (Vincke, 1974; Brumbaugh-Smith and Shier, 1989; Hansen, 1980; Skriver and Andersen, 2000; Martins, 1984), ranking methods presented in (Climaco and Martins, 1982; Azvedo and Marins, 1991) or the parametric approach proposed by Mote et al. (1991). Solution approaches in which the decision makerÕs preferences are represented by using an utility function are described in (Carraway et al., 1990; Modesti and Sciomachen, 1996). It is worth noting that in these methods the multicriteria optimization becomes a single-objective optimization problem. The methods, briefly described above, cannot be used for efficiently solving real problems. First of all, generating the whole Pareto-optimal solution set may be computationally intractable, even in the case when a small number of criteria is considered. Furthermore, supposing the Paretooptimal solution set has been determined, it may not be easy for the decision maker to choose from a very large set the nondominated path that he/she likes best, since the number of Pareto-optimal solutions may grow exponentially with the number of nodes. Secondly, it is difficult for the decision maker to define a utility function representing his/ her preferences. In order to overcome the drawbacks mentioned, it is possible to consider interactive methods that are based on a direct interaction with the decision maker. During the process of interaction the decision maker indicates his/her preferences in various forms. The algorithms find the nondominated solutions that best correspond to the decision makerÕs preferences. The interactive methods are very efficient from a computational point of view, since the search for
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the Pareto-optimal solution is done by considering a limited portion of the nondominated solution set, defined on the basis of the decision makerÕs preferences. On the basis of the previous considerations, it is apparent that the use of interactive procedures is an attractive way of dealing with the MSPP. An interactive approach for solving the MSPP with only two criteria has been proposed by Current et al. (1990). This method is characterized by two phases. The main aim of the first phase is to give to the decision maker an approximation of the possible tradeoffs. In the second phase, the search inside a specific portion of the Pareto-optimal solution set, defined on the basis of the decision makerÕs preferences, is carried out by solving a constrained shortest path problem. Similar approaches, in which the search of Pareto-optimal paths inside the duality gap is done by using a k-shortest path algorithm, have been proposed by Climaco and Coutinho-Rodrigues (1988) and Coutinho-Rodrigues et al. (1994). As shown in (Coutinho-Rodrigues et al., 1999), interactive methods based on the k-shortest path algorithm are more efficient than the method proposed by Current et al. (1990). Murthy and Olson (1999) proposed an interactive procedure for solving the bicriterion shortest path problems by using the concept of domination cones. This procedure assumes the decision makerÕs implicit utility function to be quasi-concave and nonincreasing. From the decision makerÕs pairwise comparisons, domination cones are developed which help in reducing the number of Paretooptimal solutions. Antunes et al. (1999) considers MSPP with more than two objectives for solving the routing problem in an integrated communication network. Keeping the above considerations in mind, in this paper, we propose an interactive procedure for the MSPP that is based on the reference point methodology of Wierzbicki (1977), see also (Wierzbicki et al. (2000)). The paper is organized as follows. In Section 2 the definitions related to Pareto-optimal paths with respect to domination cones are introduced. An interactive method of analyzing the MSPP by the reference point approach is presented in Section 3. Section 4 is
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devoted to a description of a label-correcting method for finding the shortest path with respect to the preferences specified by the decision maker. Finally, in Section 5 an example of interactive selection of the Pareto-optimal path is presented.
2. The Pareto-optimal paths with respect to domination cones Pareto-optimality can be generalized by using partial ordering implied by a negative cone. In this section we introduce some definitions related to the Pareto-optimal paths. Let GðN; AÞ be a directed network graph, where N ¼ f1; . . . ; ng is a finite set of nodes and A N N is a finite set of m arcs. Each arc is denoted by an ordered pair (i; j), where i 2 N and j 2 N. A path pij between the two distinct nodes (i 2 N and j 2 N) is defined as the sequence pij ¼ fi ¼ i1 ; ði1 ; i2 Þ; i2 ; . . . ; il 1 ; ðil 1 ; il Þ; il ¼ jg of alternating nodes and arcs. The path is called elementary when no node is repeated in the sequence. When the nodes are repeated in the sequence we call it a non-elementary path. Associated with any arc ði; jÞ 2 A, there is a pði;jÞ dimensional vector of criteria qði;jÞ ¼ ðq1 ; ði;jÞ ði;jÞ q2 ; . . . ; qp Þ. A path pst from the origin node s 2 N to the destination node t 2 N is evaluated by the pdimensional vector of criteria qpst ¼ ðqp1 st ; qp2 st ; . . . ; qpp st Þ. The value of any criterion k 2 f1; . . . ; pg for the given path s 2 N to t 2 N is defined as P pst from ði;jÞ qpk st ¼ ði;jÞ2pst qk . We can generally express this in the form of vector valued linear function q : P0 7! Rp ; where P0 represents the set of all paths from the origin node s to the destination node t in the graph GðN; AÞ. On the basis of the definitions and notation introduced above, the MSPP can be stated as follows: min qpst :
pst 2P0
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The partial ordering in the objective space is implied by a cone D, given by D ¼ fq 2 Rp : qk 6 0; k ¼ 1; . . . ; pg: A strictly negative cone can be defined as e ¼ fq 2 Rp : qk 6 0; D
8k ¼ 1; . . . ; p;
9k ¼ 1; . . . ; p : qk < 0g: Definition 2.1. The path p^st 2 P0 such that ^ q ¼ qð^ pst Þ is called Pareto-optimal if it belongs to the set: b 0 ¼ f^ e Þ \ qðP0 Þ ¼ ;g: P pst 2 P0 : ðqð^ pst Þ þ D A strongly negative cone can be defined as int D ¼ fq 2 Rp : qk < 0;
8 k ¼ 1; . . . ; pg:
Definition 2.2. The path p^st 2 P0 such that ^ q ¼ qð^ pst Þ is called weakly Pareto-optimal if it belongs to the set: pst 2 P0 : ðqð^ pst Þ þ int DÞ \ qðP0 Þ ¼ ;g: p^w0 ¼ f^
3. Interactive selection of the properly Paretooptimal path In this section, we show how the reference point methodology can be applied to the MSPP. The key problem here is a selection of a particular properly Pareto-optimal path out of typically large set of such paths. This selection is implicitly determined by a conversion of a multiobjective problem into a parametric single-objective problem whose solution provides a properly Pareto-optimal path. Different multicriteria optimization methods apply different conversions, see, e.g. (Haimes and Hall, 1974; Sawaragi et al., 1985; Steuer, 1986). We will consider the achievement scalarizing function (ASF). The concept of ASF has been introduced by Wierzbicki (1977). In this approach, the set A Rp , of controlling parameters (reference points) q is considered. A parametric scalarizing function is a function S : qðP0 Þ A 7! R1 . Such a function should desirably have the following two basic properties: The sufficiency property: b : arg max Sðqðpst Þ; q; wÞ P 0
The most interesting from the practical computations are properly Pareto-optimal paths. Let us define the -neighborhood of the negative cone D int D ¼ fq 2 Rp : distðq; DÞ < kqkg;
ð1Þ
pst 2P0
This condition means that for each reference point q the solution path belongs to the set of properly efficient paths. The necessity property:
p
where we can choose any norm in R and a concept of distance between the point q and the cone D ( is a given small number).
For each p^st 2 P0 exists q p^st 2 arg max Sðqðpst Þ; q; wÞ:
ð2Þ
pst 2P0
^ ¼ Definition 2.3. The path p^st 2 P0 such that q qð^ pst Þ is called properly Pareto-optimal if it belongs to the set: b ¼ f^ P pst 2 P0 : ðqð^ pst Þ þ D Þ \ qðP0 Þ ¼ ;g: 0 It can be shown that Pareto-optimal solutions for a specific form of the cone D are properly Pareto-optimal with a prior bound 1 þ 1= ¼ M on trade-off coefficients, see Wierzbicki (Wierzbicki, 1986). If we have a priori bound M on tradeoff coefficients between objectives then ¼ 1= ðM 1Þ.
This condition means that each of the properly Pareto-optimal path can be generated by selection of q. The selection of a particular properly Paretooptimal path is determined by the definition of the reference point (aspiration point). Most of those methods use the maximization of an ASF in the form: Sðq; q; wÞ ¼ min fwk ð qk qk Þg þ 16k6p
p X
wk ð qk qk Þ
k¼1
ð3Þ
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where q 2 Rp is a vector of criteria, q 2 Rp is an aspiration point (reference point), wk > 0, k ¼ 1; . . . ; p, are scaling coefficients (see the comment below) and is a given small positive number. Maximization is over all feasible paths from the origin node s to the destination node t. Maximization of (3) for all paths from s to t generates a properly Pareto-optimal path with the trade-off coefficients smaller than (1 þ 1=). For a nonattainable q, the vector of criteria of the resulting properly Pareto-optimal path is the nearest––in the sense of a Chebyshev weighted norm––to the specified aspiration point q. If q is attainable, then the vector of criteria of the properly Pareto-optimal path is uniformly better. The selection of properly Pareto-optimal paths is controlled by the vector q. There is a common agreement that the aspiration point is a very good controlling parameter for examining a properly Pareto-optimal paths. The scaling coefficients wk should not be confused with the weights used by some methods for conversion of a multicriteria problem into a single-criterion problem with a weighted sum of original criteria. In the weighted sum methods, the weights are the main controlling parameters. In the reference point approaches, the scaling coefficients are needed to provide for uniform scaling of all criteria and are not used as controlling parameters. The information provided by the user could be extended by specification of the reservation levels. These levels determine the values which the user would not like to achieve. In this case the ASF usually takes the form: Sðq; q; qÞ ¼ min fuk ðqk ; qk ; qk Þg 16k6p
þ
p X
uk ðqk ; qk ; qk Þ;
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(4) over the set of feasible paths provides a properly Pareto-optimal solution with the properties discussed above for the function (3). CAFs uk ðÞ are strictly monotone––decreasing for minimized––functions of the objective vector component qk with values uk ðUk ; Þ ¼ 1 þ c; uk ðqk; Þ ¼ 1; uk ðqk ; Þ ¼ 0; uk ðNk ; Þ ¼ c
ð5Þ
where c and c are given positive parameters, Uk represents the utopia point and it is composed of best values out of the set of all properly Paretooptimal paths for each criterion, whereas Nk is the nadir point composed of worst values out of the same set. The piecewise linear component achievement functions uk ðÞ proposed in (Wierzbicki, 1986) are defined by: 8 < fk wk ðqk qk Þ þ 1; if qk < qk if qk 6 qk 6 qk uk ðq; q; qÞ ¼ wk ðqk qk Þ þ 1; : g w ðq q Þ if qk < qk k k k k ð6Þ where wk ¼ 1=ðqk qk Þ, and fk , gk (k ¼ 1; 2; . . . ; p) are given parameters that are set in such a way that uk ðÞ takes the values defined by (5). The aspiration point and the reservation point is selected arbitrary by the user. There are various methods that supports the selection of these points in the process of interaction, see (Wierzbicki et al., 2000). We can distinguish methods based on specialized graphical user interfaces. One of them was proposed by Granat and Wierzbicki (1996) and Granat and Makowski (2000). This approach will be applied in the example presented in the Section 5.
ð4Þ
k¼1
k ; qk, rewhere q; q are vectors (composed of q spectively) of aspiration and reservation levels respectively, and uk ðqk ; qk ; qk Þ are the corresponding component achievement functions (CAF), which can be simply interpreted as nonlinear monotone transformations of qk taking into account the information represented by qk and qk > 0 is the same as the one used when defining properly Pareto-optimal paths. Maximization of the function
4. The reference point guided labeling algorithm 4.1. The label vector based optimality conditions In order to apply efficiently the reference point methodology to the MSPP, it is important to develop a specific approach for the maximization of the ASF over all feasible paths from the origin node s to the destination node t.
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In what follows, we refer to this problem as the reference point guided efficient (RPGE) path problem. In this section, we formally describe the optimality conditions for the RPGE path problem, which allow to assess when a label vector is optimal, representing, in this case, the value of the path for which ASF is maximized. We assume that the paths are elementary paths. On the basis of the optimality conditions, we define in the next section a prototype method and demonstrate the power of these conditions in guiding the design of solution algorithms. For the sake of simplicity, it is assumed that the ASF takes the form (3) and that wk ¼ 1, 8k ¼ 1; . . . ; p. However, these assumptions can be done without loss of generality, since the results stated below and the algorithms to be described in the following can be easily adapted to solve the RPGE path problem in the case in which the ASF takes the form (4) or other forms that fulfill the conditions (1) and (2). Before the statement of necessary and sufficient optimality conditions, let us introduce the following definition. Definition 4.1. Given a path psj from s to j, we define the value of the path as the following quantity: p
min fwk ð qk qk sj Þg þ
16k6p
p X
p wk qk qk sj :
ð7Þ
k¼1
The method described in the next section is based on a labeling procedure. More specifically, for each node j 2 N, it maintains and updates a label yj which represents either the value of the RPGE path from node s to node j (at the end of computation) or a lower bound for this value (at an intermediate stage of computation). The use of labels is motivated by simple optimality conditions, which are given in the following proposition. Proposition 4.1 (RPGE path optimality conditions). For every node j 2 N, let yj denote the value of some path psj from the origin node s to node j.
Then the labels yj , 8j 2 N, represent the value of the RPGE path from s to j if and only if they satisfy the following optimality conditions: ys ¼ 0;
ð8Þ
n o ðijÞ yj P min qk qpk si þ qk 16k6p
þ
p n o X ðijÞ qk qpk si þ qk ;
ð9Þ
k¼1
8ði; jÞ 2 A, and for all paths psi from s to i such that: j 62 psi :
ð10Þ
Proof. It is worth noting that condition (8) states that loops containing node s are not allowed, whereas condition (10) ensures that the RPGE path from node s to node j does not include loops, that is it is an elementary path. We use a contradiction argument to establish that an optimal label yi must satisfy conditions (9) and (10). Let us suppose that a node i exists such that: n o ðijÞ yj < min qk qpk si þ qk 16k6p
þ
p n o X ðijÞ qk qpk si þ qk ; k¼1
where psi is a path from s to i that satisfies condition (10). In this case, we could improve the value of the RPGE path to node j by passing through node i, thereby contradicting the optimality of the labels yj . Conditions (9) and (10) are also sufficient for optimality, in the sense that if each label yj ; j 6¼ s represents the value of some directed path from node s to node j and it satisfies conditions (9) and (10), then it must be optimal. This result can be proved by contradiction. More specifically, we suppose that the label yj associated to a node j 6¼ s satisfies the conditions (9) and (10) and let psj ¼ ðs ¼ i1 ; ði1 ; i2 Þ; i2 ; . . . ; ðil 1 ; il Þ; il ¼ jÞ be the corresponding path. We assume that psj is not optimal (i.e., it is not the RPGE path from the origin node s to the node j).
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This means that it is possible to find another path, from node s to node j, for example, p^sj ¼ ðs ¼ ^i1 ; ð^i1 ; ^i2 Þ; ^i2 ; . . . ; ð^il 1 ; ^il Þ; ^il ¼ jÞ whose value is greater than yj . Thus, on the basis of the Definition 4.1, we have p X p^ p^ min qk qk sj þ qk qk sj
16k6p
¼ min
16k6p
þ
k¼1 p^s^i
qk qk
p X
l 1
ð^i
^i Þ
þ qk l 1 l
p^s^i ð^i ^i Þ qk q^k l 1 þ qk l 1 l > yj ;
k¼1
thereby contradicting the hypothesis that yj satisfies the conditions (9) and (10). 4.2. A class of label correcting methods for the RPGE path problem The class of methods presented in this section can be viewed as an extension of the generic label correcting methods proposed in (Bertsekas, 1993), for solving the classical single-origin single-destination shortest path problem. In order to describe the proposed method, we refer to the following algorithm, for which we assume that yj , 8j 2 N, defines a lower bound for the value of the RPGE path from node s to node t and V represents the set of candidate nodes, whose incident arcs might violate the optimality conditions given in Proposition 4.1. Initially, we have ys ¼ 0; yj ¼ 1;
8j 2 N; j 6¼ s;
V ¼ fsg: Assuming V is nonempty, a typical iteration of the algorithm is as follows: 1. Remove from V a node i; 8j 2 N : ði; jÞ 2 A and j 6¼ s, Pp ði;jÞ If yj < min16k 6p f qk ðqpk si þqk Þgþ k¼1 ði;jÞ f qk ðqpk st þqk Þg and j 62 psi then ði;jÞ • set f qk ðqpk si þ qk Þg þ Pp yj ¼ minp1st6 k 6 p ði;jÞ k¼1 f qk ðqk þ qk Þg • add j to V if j does not already belong to it.
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The algorithm terminates when V is empty. On the basis of the operations executed by the prototype algorithm stated above, we can demonstrate the following proposition. Proposition 4.2 (correctness of the prototype method). 1. At the end of each iteration, the following conditions hold: (a) ys ¼ 0; (b) 8j, if yj 6¼ 1 and j 6¼ s, then yj is the value of some path from node s to node j; (c) if i 62 V , then either yi ¼ 1 or else a path from node s to node i exists (whose value is equal to yi ) such that 8j : ði; jÞ 2 A if j 62 psi then yj P min1 6 k 6 p fqk ðqpk si þ P ðijÞ ðijÞ qk Þg þ pk¼1 fqk ðqpk si þ qk Þg: 2. When the algorithm terminates, for all yj 6¼ 1, j 6¼ s the following condition holds: (a) n o ðijÞ yj P min qk qpk si þ qk 16k 6p
þ
p n o X ðijÞ qk qpk si þ qk ;
8ði; jÞ 2 A
k¼1
and for all path psi from node s to node i that does not contain the node j.
Proof (1) Condition (1a) holds because, initially, ys ¼ 0 and, by the rules of the algorithm, ys cannot change. We prove condition (1b) by induction on the iteration count. In fact, initially, condition (1b) holds, since node s is the only node with a finite label value. Suppose that (1b) holds at the beginning of some iteration. Let i be the node removed from V. If i ¼ s, it happens only at the first iteration. At the end of this iteration, we have ðsjÞ
yj ¼ min fqk qk g þ 16k6p
p X
ðsjÞ
fqk qk g;
k¼1
for all outward neighbors j of s, yj ¼ 1 for all other j 6¼ s. Thus, the label vector has the required
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property. If i 6¼ s then yi > 1, which is true for all nodes of V by the rules of the algorithm, and yi is the value of some path psi starting from s and ending at i (by the induction hypothesis). When a label yj changes as a result of the iteration, yj is set to ði;jÞ
min f qk ðqpk si þ qk Þg
16k6p
þ
p X
ði;jÞ
f qk ðqpk st þ qk Þg;
k¼1
which represents the value of the path p^sj from node s to node j. Finally, note that the new path p^sj does not contain loops, since the label yj is updated only if psi does not include the node j. This completes the induction proof of (1b). To prove (1c), note that if node i is removed from V and the path psi from node s to node i does not contain the node j the condition ði;jÞ
qk ðqpk si þ qk Þg yj P min f 16k6p
þ
p X
ðijÞ
f qk ðqpk si þ qk Þg
k¼1
is satisfied by the rules of the algorithm. Up to the next entrance of i into the list V the path from node s to node i remains the same, while the label yj ; 8j : ði; jÞ 2 A cannot decrease. Therefore, the condition is preserved. (2) At this point we demonstrate that for all yj 6¼ 1 condition (2a) is satisfied upon termination. Indeed, condition (1b) implies that upon termination we have, for all nodes j such that yj 6¼ 1, yj represents the value of some path from node s to node j. Furthermore, condition (1c) implies that for all nodes j such that yj 6¼ 1 upon termination we have ðijÞ
qk ðqpk si þ qk Þg yi P min f 16k6p
þ
p X
ðijÞ
f qk ðqpk si þ qk Þg;
8ði; jÞ 2 A
to be removed from the candidate list V. More specifically, the selection strategies used in the case of the classical single-origin single-destination shortest path problem can be generalized to the case of the problem considered here. In this case, since the goal is to maximize the ASF, the performance of the algorithm could get better if, at each iteration, the node exiting the candidate list is the node with relatively large label. It is worth noting, that in the case in which the criteria associated to each arc are nonnegative, it is not possible to develop a method for the solution of the RPGE path problem that guarantees that a node once removed from the candidate list V has a permanent label value. Different methods have been developed and they are basically based on the following node selection strategies, that can be viewed as a generalization of the approaches proposed for the classical shortest path problem (Bertsekas, 1993): • The simplest method of Bellman–Ford (BF, for short), for which the candidate list is maintained in a FIFO queue; • The large label first (LLF, for short) that can be viewed as a generalization of the small label first proposed in (Bertsekas, 1993). In this method, the node removed is always the top node of the candidate list V, while the entering node i is added at the top or the bottom of it, depending whether the label of i is larger or equal, or smaller than the label of current top node, respectively; • The small label last (SLL, for short) that is an adaptation of the method proposed in (Bertsekas et al., 1996). It is characterized by a sophisticated node removal strategy. When the node at the top of the candidate list V has a smaller label than the average node label in the candidate list, this node is not removed, but rather it is repositioned at the bottom of the list.
k¼1
and for all path psi from node s to node i that does not contain the node j. There are many implementations of the generic algorithm. They differ in how they select the node
As in the case of the classical shortest path problem, it is simple to combine the LLF queue insertion and the SLL node selection strategies, obtaining in this way combined methods, whose performances are improved.
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4.3. Computational experiments and discussion The proposed label-correcting method has been implemented and tested by using a single processor of an Origin 2000, a multiprocessor consisting of four nodes, each of them having a shared memory of 128 MB. Each node consists of two processors R10000 at 195 MHz, with a 4 MB cache memory and a hub device, which carries out duties similarly to a bus in a bus-based system. The nodes are connected by two routers. The operating system used is IRIX 6.4, whereas the compiler used is Fortran 77. With the aim to evaluate numerically the efficiency of the methods described in the preceding section, we have tested and compared the following codes: 1. BF: Bellman–Ford algorithm; 2. LLF: Large label first method; 3. LLF SLL: Large label first method, using, in combination, the small label last strategy for node removal. The computational results have been carried out on the basis of different choices of the number of criteria. More specifically, p has been set to four and eight, respectively. In all the computational experiments, we have chosen the origin node s equal to node 1 and the destination node t equal to jNj. The codes have been tested on two classes of networks: random networks and grid networks. These two topologies are well recognized and have been used extensively to test algorithms for the classical shortest path problem. For all test problems, the criteria have been chosen according to a uniform distribution in the range from ½1; 1000. • Random networks We have considered a set of eight random networks of varying size. All the problems have been generated using a modified version of the public domain program NETGEN (Klingman et al., 1974). The characteristics of the random networks are reported in Table 1, where for each test problem,
111
Table 1 Characteristics of the random networks Problem
Nodes
Arcs
R1 R2 R3 R4 R5 R6 R7 R8
31,622 15,811 11,952 10,000 40,000 5000 20,000 25,000
1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000
the number of nodes and the number of arcs are provided. • Grid networks These problems, whose characteristics are reported in Table 2, have been generated by using a modified version of the Gridgen generator of Bertsekas (Bertsekas, 1993). We have considered both square (G1–G4) and rectangular (G5–G8) test problems in which the nodes are arranged in a planar grid. Each pair of adjacent nodes is connected in both directions. We can also have additional arcs with random starting and ending nodes. The number of arcs is 1,000,000 for all problems and the number of nodes is selected so that the total number of additional arcs is approximately 2, 3, 4 and 5 times the number of grid arcs. The details of the computational results are given in Tables 3–6, where, for each test problem and for each value of p, we indicate the solution time and the total number of iterations required by each algorithm. Table 2 Characteristics of the grid networks Problem
Dimension
Nodes
Arcs
G1 G2 G3 G4 G5 G6 G7 G8
289 289 250 250 224 224 204 204 204 408 177 354 158 316 144 288
83,521 62,500 50,176 41,616 83,232 62,658 49,928 41,472
1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000 1,000,000
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Table 3 Time (in seconds) required to solve the problems on random networks Problem
p
BF
LLF
LLF SLL
R1
4 8
4.00 5.71
3.88 5.63
3.32 4.95
R2
4 8
3.58 5.22
3.46 4.66
2.95 4.24
R3
4 8
3.56 4.65
3.29 4.41
2.78 4.08
R4
4 8
3.75 4.86
3.41 4.54
2.80 4.05
R5
4 8
4.35 5.82
3.94 5.71
3.44 5.27
R6
4 8
3.56 4.39
3.40 4.08
2.73 3.78
R7
4 8
3.64 4.79
3.43 4.71
2.98 4.38
R8
4 8
3.88 5.06
3.72 5.00
3.16 4.55
Table 4 Time (in seconds) required to solve the problems on grid networks Problem
p
BF
LLF
LLF SLL
G1
4 8
99.47 199.44
83.22 173.57
70.52 151.74
G2
4 8
81.27 174.71
67.18 150.32
57.35 139.03
G3
4 8
66.40 144.67
61.36 127.45
47.29 108.40
G4
4 8
74.89 139.04
63.43 119.17
51.57 98.85
G5
4 8
99.08 203.14
85.41 164.75
76.32 138.67
G6
4 8
88.75 177.36
74.66 142.71
59.36 122.38
G7
4 8
81.12 136.14
63.77 123.50
51.30 101.35
G8
4 8
72.16 119.65
62.04 111.52
47.29 89.93
The computational results clearly demonstrate that, in all the test problems, LLF SLL is much faster than the other methods. For example,
for the test problem G7 (see Table 4) the execution time of BF is equal to 81.12 for p ¼ 4 and equal to 136.14 for p ¼ 8, the execution time of LLF is
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Table 5 Number of iterations required to solve the problems on grid networks Problem
p
BF
LLF
LLF SLL
G1
4 8
1,065,003 1,269,252
794,782 1,022,322
696,154 897,515
G2
4 8
681,760 821,713
533,540 681,911
453,996 617,091
G3
4 8
460,660 548,353
403,972 466,995
306,088 399,974
G4
4 8
425,170 457,734
345,776 385,679
277,520 314,668
G5
4 8
1,052,572 1,216,845
834,823 948,652
747,585 782,279
G6
4 8
745,948 854,359
571,202 647,615
462,785 539,537
G7
4 8
541,875 542,272
398,462 460,415
315,944 380,063
G8
4 8
409,358 401,872
332,635 361,151
253,889 285,906
Table 6 Number of iterations required to solve the problems on random networks Problem
p
BF
LLF
LLF SLL
R1
4 8
43,245 37,802
42,317 37,720
35,308 33,274
R2
4 8
21,557 19,453
21,132 17,759
17,826 16,210
R3
4 8
16,649 13,514
15,666 13,232
13,081 12,277
R4
4 8
14,683 11,975
13,602 11,554
11,119 10,345
R5
4 8
56,114 45,503
50,902 45,074
43,752 41,648
R6
4 8
7185 5592
7071 5417
5654 5042
R7
4 8
27,192 22,183
25,872 22,137
22,256 20,624
R8
4 8
35,122 28,354
33,650 28,357
27,797 25,697
equal to 63.77 for p ¼ 4 and equal to 123.50 for p ¼ 8, whereas the execution time of LLF
SLL is of 51.30 for p ¼ 4 and of 101.35 for p ¼ 8.
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A similar trend has also been observed for the random networks. For example, for the test problem R4 (see Table 3) the execution time of BF is equal to 3.75 for p ¼ 4 and 4.86 for p ¼ 8, the execution time of LLF is of 3.41 for p ¼ 4 and 4.54 for p ¼ 8, whereas the execution time of LLF SLL is equal to 2.80 for p ¼ 4 and 4.05 for p ¼ 8. This behavior can be explained by comparing the number of the iterations executed by the methods (see Tables 5 and 6): it is more for BF and LLF than LLF SLL. For example, for the test problem G7 and p ¼ 4, LLF SLL performs 315,944 iterations, BF executes 541875 iterations (41% more), whereas the number of iterations of LLF is equal to 398,462 (20% more). In addition, for the test problem G7 and for p ¼ 8, the number of iterations executed by BF is more than 24% of the LLF SLL iterations, while LLF iterations are more than 18% of the LLF SLL iterations. In conclusion, the LLF and SLL strategies are very helpful in reducing the number of iterations, with a significant reduction in the execution time. 4.4. An algorithm for finding the utopia point and the nadir point In this section, we describe a label-correcting method for finding nadir and utopia criterion values for the multiple objective shortest path problem. As mentioned in Section 4, the utopia point Uk is composed of best values out of the set of all properly Pareto-optimal path for each criterion, whereas the nadir point Nk is composed of worst values out of the same set. An utopia point can be easily computed as a result of p single criterion optimization with each criterion at a time serving as an objective function. In other words, it is necessary to solve p shortest path problems from node s to node t by considering one criterion at a time. In order to solve the problem of finding utopia criterion values, it is possible to consider two different strategies.
The easiest way is to apply p times a classical algorithm for solving the shortest path problem, by considering each time as a cost one of the p criteria. However, this method is not very efficient, since during the computation of the shortest path with respect to the ith criterion it is possible to generate information that can be used to determine the shortest path with respect to the i þ 1th criterion. For this reason, a more efficient method can be defined if the p shortest paths are obtained in one run. Following this idea, in what follows, we present a method that is characterized by p different sweeps. During each sweep k, we iteratively improve the current estimates of the path values with respect to the k; k þ 1; . . . ; p criterion. At the end of the first sweep, we compute the shortest path with respect to the first criterion, at the end of the second sweep, the shortest path with respect to the second criterion, and so on. Finding a nadir point is typically difficult and different methods have been proposed in the literature. Exact algorithms for computing nadir criterion values have been proposed in (Benson, 1992; Benson, 1993a), whereas heuristic approaches for obtaining estimates of the nadir point have been presented in (Benson, 1993b; Korhonen et al., 1997). The method, described in what follows, allows one to compute in one run the utopia point and an approximation of the nadir point, defined as the worst value obtained for each criterion during the determination of the utopia point. This method is divided in two main phases. In the first phase, the utopia point is determined, and in the second phase, the information found in the previous phase is used to compute an estimate of the nadir point. In order to describe the proposed method, we assume that for each node t 2 N and for each criterion k, k ¼ 1; . . . ; p, Ukt defines an upper bound on the value of the shortest path from node s to node t with respect to the kth criterion. At the end of the algorithm Ukt , 8t 2 N and 8k ¼ 1; . . . ; p will be the optimal label, representing the utopia point. In order to compute the nadir point, the algorithm considers a set of p label vectors, Nk ¼ ðNk1 ;
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Nk2 ; . . . ; Nkn Þ k ¼ 1; . . . ; p, defined in such way that for each node t 2 N and for each criterion k, k ¼ 1; . . . ; p Ntk , represents the worst value obtained during the computation of the utopia point and it uses a double-linked list Pstn for storing the sequence of nodes defining the optimal path from node s to node t with respect to the nth criterion. In addition, the algorithm maintains a set of p lists of candidate nodes V ðkÞ , k ¼ 1; . . . ; p, in which the nodes with the potential to improve the label of other nodes are stored. Initially, we have V ð1Þ ¼ fsg; Ukt
¼ 1;
V ðkÞ ¼ ;; t 6¼ s;
k ¼ 1; . . . ; p;
Uks ¼ 0;
k ¼ 1; . . . ; p;
Nkt
8t 2 N;
¼ 0;
Psjk ¼ ftg;
8k 6¼ 1;
8t 2 N;
k ¼ 1; . . . ; p; k ¼ 1; . . . ; p:
The sketch of the proposed method is the following: 1. First phase (computation of the utopia point) For all k ¼ 1; . . . ; p: (a) remove a node i from V ðkÞ ; 8j 2 N : ði; jÞ 2 A; j 6¼ s For all n ¼ k; . . . ; p ðijÞ
IF Unj > Uni þ qn THEN ðijÞ
Unj Uni þ qn update Psjn add j to V ðnÞ (if j does not already belong to it). (b) if V ðkÞ 6¼ ;; go to step a. 2. Second phase (computation of the nadir point) For all k ¼ 1; . . . ; p: For all t 2 N For all n ¼P 1; . . . ; p ði;jÞ IF Nkt < ði;jÞ2Pst qk THEN n P ði;jÞ Nkt ði;jÞ2Pst qk
5. An example of interactive selection of the Paretooptimal paths In order to better clarify the basic operations of the interactive selection of the Pareto-optimal path, we refer to the simple network of Fig. 1 (modified example considered in (Martins, 1984)). There are four criteria and the goal is to find the path which corresponds to specified preferences. For specification of the preferences, we have applied the software ISAAP, developed by Granat and Makowski (2000). This software supports specification of user preferences in terms of both aspiration/reservation levels, and it also provides the user with other means of control over the problem analysis by allowing to change in criteria status, to select the displayed solutions, etc. At the beginning (i.e., initialization phase), the ISAAP module computes the utopia point, the nadir point, and it determines the so-called compromise Pareto-optimal path, which corresponds to the optimal solution of the RPGE path problem for which the aspiration and reservation levels, used to select the RPGE path, are set to the utopia and to an approximation of the nadir points, respectively (see Eq. (4)). The related solution is presented in Fig. 2 and the corresponding RPGE path is reported in Fig. 3. The graphs of the component achievement functions (for 0 6 uk 6 1 (see Eq. (6))) are presented to the user on the screen. The solution, marked by circle, is projected into two dimensional space, in which, for each
n
On the basis of the operations executed by the method stated above, it is easy to verify that the method determines correctly the utopia/nadir point.
115
Fig. 1. Example of the network.
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Fig. 2. A compromise solution.
achievement functions (see Eq. (4)) which are used to select a Pareto-optimal path. In our example, the user changes reservation point for criterion q1 to value equal to 14 and the reservation point for criterion q2 to the value equal to 6. The new solution is presented in Fig. 4 and the corresponding path is reported in Fig. 5. The user can continue the interactive process in order to find the optimal RPGE path. Fig. 3. The compromise Pareto-optimal path.
6. Conclusions criterion, its values (the x-axis) and the degree of satisfaction (the y-axis) of meeting preferences, expressed by aspiration and reservation levels, are reported. In the next step, an interactive procedure is used for helping the user in selecting an Pareto-optimal path that best corresponds to his/her preferences. During such a procedure the user specifies preferences, including values of criteria that he/she wants to achieve and to avoid. Those values are called aspiration and reservation levels, respectively. Such a specification defines component
In this paper we have focused on the solution of the MSPP. In particular, we have presented a new approach to the interactive selection of the Paretooptimal path, in which the selection process is supported by a graphical user interface. The proposed method does not require the generation of all the properly Pareto-optimal paths, but it allows the user to focus his/her attention only on the path that appears the most promising, that is, on the reference point guided Pareto-optimal path (i.e., RPGE path).
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Fig. 4. Setting of preferences in the first iteration.
Fig. 5. The selected path for the first iteration.
An algorithm to find the RPGE path has been designed and implemented. The code has been tested extensively on a different variety of random and grid networks with a number of criteria chosen up to eight. The computational results obtained are very encouraging.
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