# Goal-Programming Approach to Multiobjective Highway GOAL-PROGRAMMING MODEL A discussion of linear...

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Transportation Research Record 751

CONCLUSION

We have shown through a hypothetical problem of in- vestment in rural roads that multicriteria optimization methods may be used to assist the decision maker in evaluating transport projects. The methods con- sidered-the generating technique known as the con- straint method and the yes-no, iterative preference- incorporation method of Zionts and Wallenius-differ substantially in the types of information required by the decision maker and the degree of interaction be- tween the decision maker and the analyst. The con- straint and weighting methods, prototypical of the methods usually classified as generating methods, strive to approximate the noninferior (or efficient) set, under the implicit assumption that knowledge of this set will allow the decision maker to select a best com- promise solution. The iterative techniques of Zionts and Wallenius and Geoffrion and others are preference- incorporation techniques and they seek to identify the best compromise solution without generating the entire noninferior set by soliciting preference information from the decision maker. The Zionts and Wallenius method does not lead to a definitive statement of the best compromise solution unless the overall utility measure is assumed to be a linear additive function of the multiple objectives. The method of Geoffrion and others by contrast always leads to a definitive best com- promise solution. This precision is at the expense of requiring more sophisticated preference information from the decision maker.

Abridgment

41

An ideal multicriteria optimization algorithm would accommodate yes-no preference information and be able to treat discrete alternatives (i.e., integer vari- ables). This last capability can only be introduced in the methods discussed here with a severe loss of com- putational efficiency.

REFERENCES

1. J. L. Cohon. Multiobjective Programming and Planning. Academic Press, New York, 1978.

2. S. Marglin. Public Investment Criteria. MIT Press, Cambridge, MA, 1967, 103 pp.

3. S. Zionts and J. Wallenius. An Interactive Pro- gramming Method for Solving the Multiple Criteria Problem. Management Science, Vol. 22, 1976, 652 pp.

4. A. Geoffrion, J. Dyer, and A. Feinberg. An Inter- active Approach for Multi-Criterion Optimization with an Application to the Operation of an Academic Department. Management Science, Vol. 19, 1972, 357 pp.

5. T. Friesz, F. Tourreilles, and A. Han. Multicriteria Optimization Methods in Transport Project Evalua- tion: The Case of Rural Roads in Developing Coun- tries. Proc., Transportation Research Forum, Vol. 20, Oct. 1979.

Publication of this paper sponsored by Committee on Transportation Systems Design.

Goal-Programming Approach to Multiobjective Highway Network Design Model Jossef Perl

A new approach to the highway network design model is presented that allows comparisons of networks on the basis of multiple incommensur- able objectives with different degrees of importance. The goal-program- ming approach is not only capable of solving the multiobjective network design problem in a relatively efficient manner, but it can also be used to generate the multidimensional trade-off curve that provides additional important information to that provided in the two-dimensional curve derived by using the linear programming model. An example illustrates the application of the linear goal-programming model with four objec- tives.

Decision problems in general, and particularly those associated with transportation systems, are made in the context of multiple conflicting objectives. Decisions about transportation systems should be weighted against the social, economic, environmental, and aesthetic needs of the community. Among the most important decisions in the transportation planning process are those regard- ing the structure of transportation networks and the level of service to be offered on them.

In recent years there has been interest in the applica- tion of advanced analytic techniques to the search for

good alternative transportation networks. A class of models known as network design models has been de- veloped to solve the following problem: Given an existing network, a list of improvement options for various links, and projected increase in demand between various origin and destination pairs, select the optimal set of links to be improved or added to the existing network. The models perform two tasks simultaneously: (a) they choose the optimal set of links to be improved or added and (b) they assign the projected traffic to the new net- work.

This paper deals with the extension of a continuous Highway Network Design Model (HNDM) developed by Agarwal (1-3). The approach adopted by Agarwal follows that used by -Hay and others ( 4) in an urban context and Morlok and others (5) for the -Northeast Corridor Trans- portation Project. The HNDM is a linear programming model developed as a sketch planning tool.

There are two ways to incorporate multiple objectives in linear programming-as elements of the objective function or as constraints. In the first method, various objectives are collapsed into a single objective by using

42

Table 1. Link characteristics for example network. Average

Link Daily Segment Travel

Time, ij m c;,{h)

12 1 0.121 43 2 0.805 80

14 1 0.040 0 2 0.246 73

23 1 0.064 29 2 0.396 27

34 1 0.057 14 2 0.376 67

45 1 0.100 00 2 0.077 87

35 1 0.021 43 2 0.139 27

Table 2. Demand matrix for example network.

To Node

From Node ?. 4

1 15 000 4516 5 222 7000 2 7685 6218 10 414 1122 3 5517 3 229 711 2217 4 6224 3 429 1704 858 5 1333 2 129 3209 5 817

some kind of weighting scheme (6). The difficulty with this approach is that it is not possible to collapse objec- tives that are significantly different to a common unit without a great deal of arbitrary judgment. The prob- lems with the second approach are both conceptual and practical. In linear programming terminology, an ob- jective represents a target and a constraint represents a restriction on feasibility. In practice, if all the con- straints cannot be satisfied, the problem is termed in- feasible. Such a conclusion is obviously misleading when objectives are formulated as constraints . The inflexibility of linear programming does not allow one to consider some , if not all, of the constraints as not absolutely binding.

The purpose of this paper is to demonstrate an ap- proach to the multiobjective HNDM. The proposed goal- programming approach has significant advantages over the interactive programming techniques for solving multiobjective problems previously applied by Agarwal (5). The goal-programming approach can generate trade- off curves in their full dimensionality. The interactive approaches are very time consuming because they re- quire a continuous interaction with the decision maker. Goal programming can solve multiobjective problems while still employing the simplex algorithm (on a modi- fied basis). This allows exploration of a much larger number of alternatives in a given time period than does an interactive programming technique.

In the process of using the HNDM, information that describes the trade-offs between the achievement of various objectives is perhaps more valuable than the actual point solution. This information is presented by a trade-off curve . A serious issue in the multiobjective trade-off analysis is the display of the multidimensional trade-off curve in two-dimensional space. This paper will demonstrate an approach to the presentation of multidimensional trade-off curves.

GOAL-PROGRAMMING MODEL

A discussion of linear goal programming is beyond the scope of this paper and can be found in Ignizio ('.!_). Lin-

Transportation Research Record 751

Dwelling Units to Be

Average Link Relocated Dally Capac tlles , Total Cosl, Length, per Capacity k~, (vehleles/day) B,, [{$1000/ vehicle) / day) v,,{miles) Unit (OOOs)

15 000 600 400 2.0 2 5 000

40 500 252 800 1.0 3 12 500 15 000 376 200 1.4 4

5 000 15 000 347 600 1.4 4

5 000 15 000 126 400 1.7 10

5 000 15 000 505 600 0.6

5 000

ear goal programming can be defined as (7) "a sys- tematic methodology for solving linear, multiple objec- tive problems wherein preemptive priorities and weights are associated with the objectives."

The linear goal-programming model presented here includes the following objectives:

Gi = flow conservation objective, G.! = flow definition objective, Gi = link capacity objective, G4 = budget objective, Gs = level-of-service objective, Gs = household relocation objective, and G, = vehicle-miles-of-travel objective.

Since Gi, G.!, and G3 can be legitimately viewed as absolute objectives (objectives that must be satisfied), they are assigned a top priority. In the following formu- lation of the linea r goal-progr amming model, the house- hold relocation objective is assigned to priority level two, the level of service objective to priority three, the budget objective to priority four, and the vehicle-miles- of-travel objective to priority five. The linear goal program for a network wit.h N nodes, a set of Lundi- rected links , and a set of S origin and destination nodes can be written as follows :

Find: 'x1l, x~ l• k 1l so as to minimize

a = [(n, +p, +n2 + p2 + .. . nH +pH+ ... nq +pq + .. . No +Po + Po+1 + · ·. P, + . . . PR), (Pd), (P,) , (Pb), CPvl l (I )

such that for Gi,

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