Multiobjective transportation network design and ... - ScienceDirect

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European Journal of Operational Research 65 (1993) 4-19 North-Holland

Multiobjective transportation network design and routing problems: Taxonomy and annotation John Current The Ohio State University, Columbus, OH 43210, USA

Michael Marsh Shippensburg University, Shippensburg, PA 17257, USA Received May 1991; revised February 1992

Abstract: In this paper we update an earlier review of multiobjective network design and routing problems. Forty-one articles are annotated. The diversity of objectives presented demonstrates the multiobjective nature of such problems and the difficulty of measuring these objectives in commensurate units. The variety of problems addressed reflects the importance and complexity of transportation network analysis.

Introduction

The purpose of this paper is to update an earlier review on the multiobjective design of transportation networks (Current and Min, 1986). The title has been expanded to include 'routing problems' to more accurately reflect the intention of both articles. We annotate the articles in a manner identical to those in the earlier review; however, several changes have been made to the taxonomy. These changes are discussed in the next section. It is expected that the interested reader will also read Current and Min (1986). Consequently, we do not provide a historical overview of multiobjective transportation network design problems as one is given in the earlier article. As is the case with any literature review, the decision rules regarding inclusion are somewhat

Correspondence to: Dr. John Current, The Ohio State University, Columbus, OH 43210, USA.

arbitrary. For the most part, we have followed those used by Current and Min (1986). Specifically, we do not include articles on the evaluation of a given set of planning alternatives as the focus of the review is on the use of multiobjective techniques for the generation of planning alternatives. Although clearly related mathematically (Magnanti and Wong, 1984) and practically, facility location articles are also excluded as reviews of this literature already exist (ReVelle et al., 1981; Current, Min and Schilling, 1990). The review does not include multiobjective resource allocation articles even when directly related to transportation planning, unless the underlying transportation network is considered directly. Similarly, transportation articles which do not present specific mathematical models are excluded as well. Finally, this paper only covers articles published in major refereed journals published in English. As a consequence, books, book chapters, working papers, conference proceedings, Master's and Doctoral Theses and non-English journal articles are not examined.

0377-2217/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

J. Current, M. Marsh / Taxonomy and annotation

5

I

I ShortestPath Problem

ILransportation Probem

Ill

( T~°bi:~tiNetw ork/ , ~

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//

Assignment Problem

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IVTransshipment Problem

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v;timNaletwOrk \ ~ I'snPr°°

Vehicle Routing Probem

Figure 1. First level of the taxonomy

We have tried to be as objective, exhaustive and extensive as possible and apologize for any oversights. Most of the articles annotated here have

Certainly, the selection of articles for any literature review is a subjective process and reflects the biases and special interests of the reviewers.

I

I Shortest Path Problem

i ~ ~xacA~or,h~ I

I 1 GeneratingTechnique II 2 P~e~re~°~° Technique • • • • • •

White (1982) Martins (1984) Corley and Moon (1985) Henig (1985) Current et al (1987) Current et al (1988) • Brumbaugh-Smith and Shier (1989) • Carrawy et al (1990) • Moteet al (1991) • Wijeratne el ai (1993)

• Henig (1985) • Current et al (1990b)

I ~ ~eor c~0or'h~

II

1 GeneratingTechnique • Warburton (1987)

Figure 2. Shortest path problem branch of taxonomy

I I

2 PreferenceBased Technique

6

J. Current, M. Marsh / Taxonomy and annotation

I

I

II. Transportation Problem

A. Exact Algorithm

B. Heuristic Algorithm

I

Technique I1. GeneratingTechnique II 2++°++++ • Egly and Wright (1987)

I

I

Technique I1. GeneratingTechnique II 2++°°+° • Ringuest and Rinks (1987)

• Klingman and Phillips (1988) • Climaco et al (1993)

Figure 3. Transportationproblembranchof the taxonomy appeared since 1985. Several earlier articles, however, which were inadvertently omitted from Current and Min (1986) have been included.

Taxonomy As stated earlier, the taxonomy used in this article is similar to the one developed in Current

I

III. Assignment Problem

I ~. ~xao,~,oori,~ I

I1. GeneratingTechnique I I

and Min (1986). The first level of classification in the taxonomy is based upon the underlying mathematical structure of the model a n d / o r the underlying purpose of the formulation. This level of the taxomony is presented in Figure 1. The differences to the taxonomy presented in Current and Min (1986) appear at this level. Specifically, we have made the following changes. First, we have made the Traveling Salesman Problem (TSP) cat-

Technique

~ +°+°++++°

I

+. +u+t+++~ I

Technique I1. GeneratingTechnique 112++++°1 • Rosenblatt (1979) • Ignizio et al. (1982) • Dutta and Sahu (1982) • Murphy and Ignizio (1984) • Fortenbery and Cox (1985) • Malakooti and D'souza • Urban (1987) (1987) • Malakooti (1989)

Figure 4. Assignmentproblembranchof the taxonomy

J. Current, M. Marsh / Taxonomy and annotation

I

IV. Transshipment Problem

] ~ ~xa~,~0or,~ ]

7

I

I ~ .~0r~,c~0or~ I

I 1 GeneratingTechnique I I ~ ~e'e~nce~ase~ I 1. GeneratingTechnique !1 ~ ~'~a~° Technique Technique • Ogryczak et al. (1989) Figure 5. Transshipmentproblembranch of the taxonomy

egory more general by making it the Vehicle Routing Problem (VRP), as the TSP is a specific instance of the VRP. Secondly, we have eliminated the Generalized Network Problem category. This was done for two reasons: 1) the 'problem' primarily refers to the transformation of non-network problems into a network formulation (Bazaraa et al., 1990) and consequently includes many non-network applications; and 2) the category included only one article in Current and Min (1986) and none in this review.

I

A. Exact Algorithm

[

1. GeneratingTechnique

V. Vehicle Routing Problem

I

I

B. HeuristicAIgorithm

I1 ~ ~ ' ~Technique °°~°

• Sutcliffe and Board (1990)

Thirdly, we have included two new categories: the Spanning Tree Problem and Network Flow Problems. Clearly, the Network Flow Problems category could include categories I-IV (Shortest Path Problem, Transportation Problem, Assignment Problem, and Transshipment Problem) as special cases (e.g., Hillier and Lieberman, 1990). The category was added because articles addressing multiobjective minimum cost network flow problems and multicommodity network flow problem have appeared since the earlier review.

1. GeneratingTechnique • Keller (1989)

Figure 6. Vehicleroutingproblem branch of the taxonomy

[

Technique • Park and Koelling (1986) • Park and Koelling (1989)

8

J. Current, M. Marsh / Taxonomy and annotation

I VI. Optimal Network Design Problem ]

] A Exac,A0orthm I

l ~ "eurs*cA0or*hmI

Technique Technique I 1 GeneratingTechnique I I ~e*~°~e~ase° I I 1. GeneratingTechnique Ir ~ ~*~~ • Friesz et al. (1980) • Perl (1980)

• Friesz et al. (1980)

• Friesz and Harker (1983) • Jain and Dutta (1986) • Friesz et al. (1993) Figure 7. Optimal networkdesign problem branch of the taxonomy

Categories I-IV were not generalized and included in this new category because, as special cases of the minimum cost network flow problem, they have received considerable attention in the operations research and transportation network design and routing literature. Finally, we have modified the article inclusion criteria for category III (The Assignment Problem) by not including most non-transportation related articles (e.g., personnel assignment,

scheduling). This rule would have led to the exclusion of all six entries in this category in Current and Min (1986). Most of these articles, however, are formulation oriented and are not particularly applicable to transportation network planning. Our search uncovered fourteen such articles (Chen and Bulfin, 1990; Crum and Namazi, 1987; Deckro and Rangachari, 1990; Franz et al., 1989; Geetha and Vartak, 1989a,b; Geoffrion et al., 1972; Klingman and Phillips, 1984; Musa and

VII. Spanning Tree Problem I

i ~. ~xac,~.oo~,,~ I

l 1 Generating T Technique e c h ~J CI l q u

r ~. ~,,r,~,,~,~or,,~ I

Based

• Hutson and ReVelle (1989) • Hutson and ReVelle (199_) Figure 8. Spanning tree problem branch of the taxonomy

J. Current, M. Marsh / Taxonomy and annotation

saxena, 1982; Liang and Lee, 1985; Phillips, 1987; Saatcioglu, 1987; Wilamowsky et al., 1990; Zanakis, 1983); however, they are not included in the annotation. We did, however, include facility layout and computer network design articles in category III. As the annotation demonstrates, most of these articles were based on the quadratic assignment problem. The rationale for inclusion includes the importance of inter-nodal flows in such problems, and the relevance of the quadratic assignment problem to transportation networks (e.g., airline hub networks: O'Kelly, 1987). Readers are referred to Scott (1969), Dionne and Florian (1979), Phillips and Garcia-Diaz (1981), Magnanti and Wong (1984), Golden and Assad (1988), Bazaraa et al. (1990) and Hillier and Lieberman (1990) for formulations and descriptions of the underlying network problems. The second level of the taxonomy is based upon solution technique. This level is divided into two broad categories: exact solution techniques and heuristic techniques. Heuristics constitute an important part of the network literature because of the computational complexity involved in solving many large scale (50-100 nodes) network design problems (Magnanti and Wong, 1984). The third tier of the taxonomy is based upon the multiobjective analysis technique employed.

I

I

I

9

This classification is an adaptation of the scheme proposed by Cohon (1978) where multiobjective methods are classified as either being generating techniques or preference based techniques. Generating techniques are those which generate an exact representation or an approximation of the noninferior solution set. The most commonly employed of these are the weighting method and the constraint method. Preference based techniques elicit preferences from the decision maker(s) regarding the relative importance of the various objectives. These preferences are then incorporated into the mathematical formulation of the problem. The preferences may be stated a priori (e.g. lexicographic GP) or may evolve progressively as subsets of the underlying noninferior solution set are generated and presented to the decision maker. These techniques generally identify only a single solution or a small subset of the noninferior solution set. Figures 2 through 9 list the articles in each sub-category for the 8 problem categories.

Annotation

In the following annotation each entry includes author(s), year of publication and title of the article. This is followed by three subentries.

VIII. Network Flow Problems ]

A. Exact Algorithm

I

I B. Heuristic Algorithm I

tl

Technique

[

1. GeneratingTechnique • Martins (1987) • Ruhe (1988) • Lee and Pulat (1991)

• Rees et al. (1987)

• Fruhwirth et al. (1989)

Figure 9. Networkflowproblemsbranch of the taxonomy

10

J. Current, M. Marsh / Taxonomyand annotation

The first identifies certain model characteristics and which of the multiobjective analysis techniques is employed. The second categorizes the article as having either a problem formulation orientation or a problem solution orientation. This categorization of articles is undoubtedly the most subjective of the annotation, because many articles include both formulation and solution aspects. Decisions were based upon what we considered to be the primary intent of the article. Also, where appropriate, the problem setting is described. Finally, the objectives of the problem are summarized in the third entry. Please note that the full citations are given in the Reference section at the end of this article. The entries in the annotation are grouped by their respective node in the taxonomy. Within each nodal grouping, the articles are listed in chronological order. The following is the general format of an annotation entry: Author(s) (year of publication), title: • deterministic or stochastic, static or dynamic, multiobjective technique employed, • formulation or solution oriented; problem setting if appropriate, • objectives.

L A. 1. Shortest path problem, exact algorithm, generating technique

White (1982), "The set of efficient solutions for multiple objective shortest path problems": • deterministic, static, weighted MOLP, • solution oriented, • non-specific; suggests minimize total cost, time, number of operations in steel strip rolling; maximize quality. Martins (1984), "On a special class of bicriterion path problems": • deterministic, static, specialized algorithms, • solution oriented, • MAXMIN; MINSUM, MINRATIO arc costs and lengths. Corley and Moon (1985), "Shortest paths in networks with vector weights": • deterministic, static, labeling algorithm, • solution oriented, • non-specific. Henig (1985), "The shortest path problem with two objective functions":

• deterministic, static, specialized algorithms, • solution oriented, • non-specific; suggests minimize total cost, travel time. Current, ReVelle and Cohon (1987), "The median shortest path problem: A multiobjective approach to analyze cost vs. accessibility in the design of transportation networks": • deterministic, static, specialized algorithm based on K-shortest path procedure, • formulation oriented, • minimize total path length; maximize user accessibility. Current, ReVeUe and Cohon (1988), "The minimum-covering/shortest-path problem": • deterministic, static, weighted shortest path algorithm, • formulation oriented; routing of hazardous materials, • minimize total population at risk, total path length or cost. Brumbaugh-Smith and Shier (1989), "An empirical investigation of some bicriterion shortest path algorithms": • deterministic, static, label correcting shortest path algorithm, • solution oriented, • non-specific. Carraway, Morin and Moskowitz (1990), "Generalized dynamic programming for multicriteria optimization": • stochastic, static, dynamic programming, • solution oriented, • minimize total distance, maximize probability of reaching destination. Mote, Murthy and Olson (1991), "A parametric approach to solving bicriterion shortest path problems": • deterministic, static, parametric MOLP, • solution oriented, • nonspecific, two generic costs to be minimized. Wijeratne, Turnquist, and Mirchandani (1993), "Multiobjective routing of hazardous materials in stochastic networks": • stochastic, static, weighted MOLP, • solution oriented; routing of hazardous materials; develops methodology to evaluate stochastic dominance, • Minimize total travel time, accident ral~e; also suggests minimize operating costs.

J. Current,M. Marsh / Taxonomyand annotation L A.2. Shortest path problem, exact algorithm, preference based technique Henig (1985), "The shortest path problem with two objective functions": • deterministic, static, exact algorithm for quasiconcave and quasiconvex utility functions, • solution oriented, • non-specific; suggests minimize total cost, travel time. Current, ReVelle and Cohon (1990), "An interactive approach to identify the best compromise solution for two objective shortest path problems": • deterministic, static, weighted and constrained MOLP, interactive, • solution oriented, • non-specific. LB.1. Shortest path problem, heuristic algorithm, generating technique Warburton (1987), "Approximation of Pareto optima in multiple-objective, shortest-path problems": • deterministic, static, E-bounded approximation of non-inferior set, • solution oriented, • non-specific. LB.2. Shortest path problem, heuristic algorithm, preference based technique No articles. II. A.1. Transportation problem, exact algorithm, generating technique Egly and Wright (1987), "Microcomputer-based multiobjective personnel management model": • deterministic, static, constrained MOLP, • solution oriented; personnel reassignment to satisfy seasonal demand for highway maintenance, • minimize total distance, maximum distance. II.A.2. Transportation problem, exact algorithm, preference based technique Klingman and Phillips (1988), "Equitable demand adjustment for infeasible transportation problems": • deterministic, static, weighted MOLP, • formulation oriented; determination of eq-

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uitable sharing of demand shortages for infeasible transportation problems, • minimize infeasibilities, total transportation costs, maximum deviation between undersupply and demand. Cllmaco, Antunes and Alves (1993), "Interactive decision support for multiobjective transportation problems": • deterministic, static, weighted MOLP, interactive, • solution oriented, • non-specific; suggests minimize total cost, delay.

ll.B.i. Transportation problem, heuristic algorithm, generating technique No articles. H.B.2. Transportation problem, heuristic algorithm, preference based technique Ringuest and Rinks (1987), "Interactive solutions for the linear multiobjective transportation problem": • deterministic, static, weighted MOLP, interactive, • solution oriented, • non-specific; suggests minimize total cost, unfilled demand, underused capacity, product deterioration; maximize quantity of goods delivered, reliability of delivery, safety of delivery. IlL A. 1. Assignment problem, exact algorithm, generating technique No articles. III. A.2. Assignment problems exact algorithm, preference based technique No articles. III.B.1. Assignment problem, heuristic algorithm, generating technique Rosenblatt (1979), "The facilities layout problem: A multi-goal approach": • deterministic, static, weighted quadratic assignment problem, graphic solution approach, • formulation oriented; facility layout, • minimize total work-flow volume; maximize closeness rating.

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J. Current, M. Marsh / Taxonomy and annotation

Dutta and Sahu (1982), "A multigoal heuristic for facilities design problems: MUGHAL": • deterministic, static, weighted quadratic assignment problem, exchange heuristic, • solution oriented; facility layout, • minimize total materials handling costs; maximize closeness rating. Fortenberry and Cox (1985), "Multiple criteria approach to the facilities layout problem": • deterministic, static, weighted quadratic assignment problem, exchange heuristic, • formulation oriented; facility layout, • minimize total work-flow volume; maximize closeness rating. Urban (1987), "A multiple criteria model for the facilities layout problem": • deterministic, static, weighted quadratic assignment problem, exchange heuristic, • formulation oriented; facility layout, • minimize total work-flow volume; maximize closeness rating. Malakooti (1989), "Multiple objective facility layout: A heuristic to generate efficient alternatives": • deterministic, static, weighted quadratic assignment problem, exchange heuristic, • solution oriented; facility layout, • non-specific, suggests minimize total material handling costs; maximize flexibility, production rate.

III.B.2. Assignment problem, heuristic algorithm, preference based technique Ignizio, Palmer and Murphy (1982), "A multicriteria approach to supersystem architecture definition": • stochastic, static, non-linear goal program, interactive, • solution oriented; design of distributed computer system architecture, • minimize system cost, data spread, technological dispersion; maximize reliability, maintainability, testability. Murphy and Ignizio (1984), "A methodology for multicriteria network partitioning": • deterministic, static, weighted quadratic assignment problem; exchange-, penalty-, goal programming-based heuristic, • solution oriented; design of airport terminals, distributed computing networks.

• non-specific; suggests; minimize cost, time delay, vulnerability to message interception and destruction; maximize total network intracommunication, technological dispersion and diversity, system reliability, file availability, connectivity, node similarity, maintainability. Malakooti and D'souza (1987), "Multiple objective programming for the quadratic assignment problem": • deterministic, static, weighted quadratic assignment problem, exchange heuristic, interactive, • solution oriented; facility layout, • minimize total cost, material movement time; maximize flexibility.

IIZ.A.1. Transshipment problem, exact algorithm, generating technique No articles.

II~.A.2. Transshipment problem, exact algorithm, preference based technique Ogryczak, Studzifiski and Zorychta (1989), "A solver for the multi-objective transshipment problem with facility location": • deterministic, static, goal programs, interactive, • solution oriented; sugar beet distribution network, • minimize total cost, flow through depots, rail deliveries; maximize sugar production.

1E.B.1. Transshipment problem, heuristic algorithm, generating technique No articles.

IE.B.2. Transshipment problem, heuristic algorithm, preference based technique No articles.

V.A.1. Vehicle routing problem, exact algorithm, generating technique Sutcliffe and Board (1990), "Optimal solution of a vehicle-routing problem: Transporting mentally handicapped adults to an adult training centre": • deterministic, static, constrained MOLP,

J. Current, M. Marsh / Taxonomy and annotation

• formulation oriented, transporting handicapped adults to training centers, • minimize total distance, time; maximize equalization of vehicle trip times, capacity utilization.

V.A.2. Vehicle routing problem, exact algorithm, preference based technique No articles. V.B.1. Vehicle routing problem, heuristic algorithm, generating technique Keller (1989), "Algorithms to solve the orienteering problem: A comparison": • deterministic, static, constrained MOLP, exchange/insertion heuristic, • solution oriented, • minimize total penalties; maximize total rewards. V.B.2. Vehicle routing problem, heuristic algorithm, preference based technique Park and Koelling (1986), "A solution of vehicle routing problems in a multiobjective environment": • deterministic, static, weighted GP, interactive, • solution oriented, • minimize total travel distance, deterioration of goods; maximize fulfillment of urgent demand, conditional dependencies of stations. Park and Koelling (1989), "An interactive computerized algorithm for multicriteria vehicle routing problems": • deterministic, static, weighted GP, interactive, • solution oriented, • minimize total travel distance, deterioration of goods; maximize fulfillment of urgent demand, conditional dependencies of stations. VI. A. 1 Optimal network design problem, exact algorithm, generating technique Friesz, Tourreilles, Han and Fernandez (1980), "Comparison of multicriteria optimization methods in transport project evaluation": • deterministic, static, weighted and constrained MOLP,

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• solution oriented; rural transportation network investment, • maximize agricultural production in various regions. Perl (1980), "Goal-programming approach to multiobjective highway design model": • deterministic static, weighted GP, • solution oriented; network improvement, • minimize total investment, household relocation, vehicle miles traveled; maximize level of service.

VI.A.2. Optimal network design problem, exact algorithm, preference based technique Friesz, Tourreilles, Han and Fernandez (1980), "Comparison of multicriteria optimization methods in transport project evaluation": • deterministic, static, weighted and constrained MOLP, • solution oriented; rural transportation network investment, • maximize agricultural production in various regions. VI.B.1. Optimal network design problem, heuristic algorithm, generating technique Friesz and Harker (1983), "Multicriteria spatial price equilibrium network design: Theory and computational results": • deterministic, static, spatial price equilibrium, iterative optimization-equilibrium algorithm and nonlinear direct search technique, • solution oriented, rail and freight network design, • minimize total investment; maximize consumers' surplus, producers' surplus. Jain and Dutta (1986), "Distributed computer system design: A multicriteria decision-making methodology": • deterministic, static, exchange heuristic, interactive, • formulation oriented; distributed computer network design, • minimize total response time, system cost; maximize system availability for queries/transactions. Friesz, Anandalingam, Mehta, Nam, Shah and Tobin (1993), "The multiobjective equilibrium network design problem revisited: A simulated annealing approach":

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J. Current, M. Marsh / Taxonomy and annotation

• deterministic, static, weighted simulated annealing, • solution oriented, network investment, • minimize total user transportation costs, construction costs, miles traveled, dwelling units taken for right-of-way.

VI.B.2. Optimal network design problem, heuristic algorithm, preference based technique No articles.

VIL A.1. Spanning tree problem, exact algorithm, generating technique Hutson and ReVelle (1989), "Maximal direct covering tree problems": • deterministic, static, weighted MOIP, • formulation oriented, • minimize total cost; maximize total demand satisfied (covered). Hutson, and ReVelle (1993), "Indirect covering tree problems on spanning tree networks": • deterministic, static, weighted MOIP, • formulation oriented, • minimize total cost; maximize total demand satisfied (covered).

VII.A.2. Spanning tree problem, exact algorithm, preference based technique No articles.

• non-specific; suggests minimize total time, cost. Ruhe (1988), "Complexity results for multicriteria and parametric network flows using a pathological graph of Zadeh": • deterministic, static, maximal flow problem, min-cost flow problem, • solution oriented; demonstrates an exponential number of breakpoints in the optimal value function of the maximal flow problem in generalized networks with parametric capacities and shows that, in the worst case, the bicriteria mincost flow problem has 2 n efficient extreme points, • non-specific. Lee and Pulat (1991), "Bicriteria network flow problems: Continuous case": • deterministic, static, min-cost flow problem, out-of-kilter method, • solution oriented, • non-specific.

VIIL A.2. Network flow problems exact algorithm, preference based technique Rees, Clayton and Taylor (1987), "A linear goal programming model of a multi-period, multicommodity network flow problem": • deterministic, multi-commodity network flow problem, lexicographic GP, • formulation oriented; carpet distribution system, • minimize total transportation costs, inventory; maximize satisfaction of customer demands, maintenance of shipping arrangements.

VII.B.1. Spanning tree problem, heuristic algorithm, generating technique No articles.

VII.B.2. Spanning tree problem, heuristic algorithm, preference based technique No articles.

VIII. A. 1. Network flow problems, exact algorithm, generating technique Martins (1987), "On a particular quadratic network problem": • deterministic, static, quadratic min-cost flow problem, • solution oriented,

VIII.B.1. Network flow problems, heuristic algorithm, generating technique Fruhwirth, Burkard and Rote (1989), "Approximation of convex curves with application to the bicriterial minimum cost flow problem": • deterministic, static, rain-cost flow problem, e-approximation of efficient set, • solution oriented, • non-specific; suggests minimization of total time, cost.

VIII.B.2. Network flow problems, heuristic algorithm, preference based technique No articles.

J. Current, M. Marsh / Taxonomy and annotation

Summary and conclusions The primary purpose of this paper is to update the review of multiobjective research on transportation network design and routing problems presented in Current and Min (1986). A total of 41 articles are annotated. Twelve of these articles appeared prior to 1986 but are included because they did not appear in the earlier review. We have made several changes to the taxonomy presented in the earlier review. Specifically, we have generalized the 'transportation problem' category to that of the 'vehicle routing problem'; eliminated the 'generalized network problem' category; and added two additional categories: the 'spanning tree problem' and 'network flow problems'. In addition, we have restricted articles in the 'assignment problem' category to those which directly address transportation network analysis or quadratic assignment problems. Had Current and Min (1986) employed this more restrictive inclusion rule for the assignment problem category, all six of their entries would not have been included. This would have reduced the total number of articles in that review from 42 to 36. Our goal for the taxonomy was to keep it relatively simple and to follow that proposed by Current and Min (1986) as closely as possible. Undoubtedly, important similarities and differTable 1 Years of publication Year

Number of articles

1969 •973 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1993

1 1 3 2 2 7 5 4 8 7 7 7 2 8 3 7 3 2 4

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ences among articles are not revealed by this taxonomy. For example, Deo and Pang (1984) present five separate taxonomies for shortest path algorithms alone and Hwang and Masud (1979) divide multiobjective decision methods into 30 categories. It is our hope, however, that the basic structure provided here will help analysts to identify existing research which is similar to their problems at hand. Table 1 demonstrates the continuing interest in the multiobjective design of transportation networks since 1973. This table includes the articles annotated in Current and Min (1986) (excluding the six in the assignment problem category) as well as those annotated in this paper. Undoubtedly, we have inadvertently missed some articles in our search. It is expected that, as was the case with Current and Min (1986), many of these omissions will be from the most recent years reviewed because of journal availability problems and the lack of citations for recent articles. The variety of conflicting objectives in transportation planning and the difficulty of measuring these objectives is clearly demonstrated by the articles covered by this review. For example, one broad categorization of objectives might include cost, profit, demand satisfaction, and quality of service oriented objectives. Obviously, the boundaries among these are fuzzy. The difficulty of creating an objective function taxonomy increases as one looks at these broad categories in detail. Specific objectives one might include in the cost category are: construction a n d / o r transportation costs (26 articles), distance (12), travel time (8), deterioration of goods (3), households relocated (2) inventory (1), accident rate (1), and population at risk (1). The difficulty of measuring costs in a single unit is further demonstrated by the fact that four of these articles include both a construction/transportation cost and a distance objective and 7 of the 8 articles with a travel time objective also include a c o n s t r u c t i o n / transportation cost objective. An additional factor complicating cost measurement occurs when the costs are borne by different parties (e.g., private sector vs. public sector, providers vs. users, various political jurisdictions). Six articles address demand satisfaction directly and 8 include an objective related to accessibility of demand. This latter group includes 3 articles which consider accessibility of the de-

16

J. Current, M. Marsh / Taxonomy and annotation

Table 2 Journal outlets

Table 3 Problem type

Journal

Number of articles

European Journal of Operational Research International Journal of Production Research Computers & Operations Research Decision Sciences Journal of the Operational Research Society Transportation Research Record Transportation Science Computers and Industrial Engineering Engineering Costs and Production Economics IEEE Transactions on Computers Journal of Business Logistics Journal of Computing in Ovil Engineering Journal of Optimization Theory and Applications Operations Research Transportation Research B Zeitschrift fiir Operations Research

15 6 3 2 2 2 2 1 1 1 1 1 1 1 1 1

mand not on the network to the network, and 5 consider accessibility among specific demands on the network. In certain circumstances, these demand satisfaction objectives may be viewed as cost or profit objectives. Quality oriented objectives include those related to quality (2), flexibility (2), reliability (2), and safety (1) of service. Certainly, in given circumstances these objectives might more appropriately be categorized as demand satisfaction or profit oriented objectives. Profit oriented objectives include the maximization of production (2), producers' surplus (1), and rewards (1). Objectives presented which do not fit clearly into one of these four general categories include equalization of vehicle trip miles (1), maintenance of shipping arrangements (1), and facility capacity utilization (2). The 41 articles annotated in this paper appeared in 16 different journals. The number of articles cited per journal is given in Table 2. Including the articles reviewed in Current and Min (1986), multiobjective transportation network design and routing articles have appeared in 31 journals representing the disciplines of economics, engineering, logistics, management science/operations research, production, and regional science. This diversity of journals reflects the multiobjective nature of transportation planning and suggests the importance of increased interdisciplinary research.

Problem type

Year of publication

I. Shortest Path Problem II. Transportation Problem III. Assignment Problem IV. Transshipment Problem V. Vehicle Routing Problem VI. Optimal Network Design Problem VII. Spanning Tree Problem VIII. Network Flow Problems

19731985 a

19861992

17 14 5 4 2

8 4 3 1 4

8 0 0

2 2 5

a Includes articles reviewed in Current and Min (1986).

Tables 3 and 4 compare the research activity at the various levels of the taxonomy between the years 1973-1985 and 1986-1992. Table 3 lists the number of publications for each of the 8 categories in the top level of the taxonomy for these two periods. The percentage of articles included in the shortest path, transportation, transshipment and optimal network design problem are lower for the later time period than they were for the earlier time period, while the percentage of articles addressing the vehicle routing, spanning tree and network flow problem categories are higher in the 1986-1992 period. Table 4 lists the number of publications for each of the four categories in the second level of the taxonomy for these two time periods. The ratio of entries proposing heuristic solution techniques to entries proposing exact solution procedures increased from the first to the second time period. This trend reflects an increased interest in analyzing more difficult problems. The ratio of generating based entries to preference based entries also increased in the second time period.

Table 4 Multiobjective solution approach Solution Approach

Exact Heuristic Generating Preference based

Year of publication 1973-1985

1986-1992

46 12 36 22

17 12 21 8

J. Current, M. Marsh / Taxonomy and annotation

This may reflect decision makers' desire to be presented with alternative options rather than a single, 'best' solution (Hall, 1985); the difficulty of determining preferences prior to knowledge of the actual tradeoffs involved (Zionts and Wallenius, 1976); and the increased speed by which noninferior solutions can be generated as better heuristics are developed. Several of the preference based articles proposed techniques in which the decision makers progressively state their preferences among objectives as they learn more about the actual tradeoffs involved as various noninferior solutions are generated and presented to them. These interactive techniques generally require less computational effort than is necessary to generate the entire set of noninferior solutions but do not require the decision maker to state preferences among the objectives without some knowledge of the actual tradeoffs involved. As the articles in this review demonstrate, transportation analysis must frequently consider multiple objectives and it is often difficult or impossible to measure these objectives in commensurate units. Multiobjective analysis permits the measurement of criteria in their natural units and provides decision makers with efficient options which demonstrate the inherent tradeoffs among the objectives. Given the multiobjective nature, complexity, diversity, and practical importance of transportation network design and routing problems, it is expected that the multiobjective analysis of such problems will continue to be a fertile area of research.

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