The Solutions

Although the projects are presented as a sequence, in reality they are grouped by year, with some being spread across adjoining years. A solution to the road project construction timetable problem is a collection of information on the various aspects of the road projects in question, between which the relationships are complicated. Solutions are presented in the following order:

Construction sequence and net present value for the ten best solutions ( Allocation of investment amounts according to the GA sequence so as to satisfy the annual budget constraints, the limits to annual expenditure on any one A detailed construction timetable, containing all relevant information (

)

project and the preferred investment profiles ( ) )

3.4.2.1 The Ten Best Project Sequences

The project sequence and net present value for each of the best ten project schedules across all experiments are shown in

.

The solution ranked first, which is in Experiment 1, is probably near-optimal. The other highly ranked GA individuals differ considerably in sequence, as well as in objective function values.

Although the entire ordered string of projects is presented, only those which could be completed within budget are actually implemented. All projects remain in the string but those which are not implemented have no influence on the solution or the calculated net present value. The projects to be implemented are shown in bold in each of the solutions in

.

In some cases, the final project is only partially completed in Year 10, the final construction year of the analysis.

The benefit from a road project in a sequence to be implemented in a network depends on the other projects in the sequence and the benefits and benefit-cost ratios for individual projects are unknown. The marginal benefit-cost ratio of an individual project could be calculated by deleting it from the optimal schedule, re- calculating total benefits of the remaining project group and expressing the lost benefits as a ratio to the cost saved on the particular project.

Table 3.5

Table 3.6

Table 3.7

Table 3.5

Table 3.5

Table 3.5 Summary of the best ten investment sequences

Ranking of Road Project Construction Order' Net Present Value

Solution

c€J

The projects in bold are those to be wholly or partly implemented within the program period which contribute to the objective function. In some cases, an oullier in bold is a 'predecessor' project lo he implemented as part of its more highly ranked 'successor.'

3.4.2.2 Project Sequence Converted to Annual Investment

shows the best project sequence converted to amounts invested by years. It shows how the computation procedure maps the sequence of projects in bold in the first row of into annual expenditures. If there are insufficient funds to complete a project in a year, then further amounts are allocated to it in subsequent years. If a benefit indivisible project cannot be completed within a ten-year program period, nominal rather than actual amounts of investment are allocated to it during the program period. This leads to total investment in later years of the program (the 8th, 9th and 10th) being less than the annual budget of Table 3.6

Table 3.5

$27 million. This treatment of such benefit indivisible projects is justified on two grounds :

The firmness of the schedule of road projects decreases with time because funding projcctions and travcl dcmand forccasts bccomc lcss rcliablc.

Road project programming is a rolling process. which should be repeated every year or every few years. In the next programming round, those benefit indivisible projects which cannot be completed in the current program period will have an extended time span for completion, and the nominal amounts of investment in the projects will become actual amounts of investment in the new program period.

Projects 29 and 26 are the last two to enter the investment program of

thus exhausting the budget. Project 12, which came between them in the GA sequence (first case in ). was a predecessor of Project 13, which enters the program in Year 7 and therefore absorbs Project 12. Such cases of absorption of predecessor by successor has been taken into account i n the assessment of network effects and benefits for every GA individual in every generation. Other cases of I)redecessor-successor relationships in the best solution are considered in the next section.

It can also be seen in that some projects are spread over more than two succcssivc ycars. This is duc to financial constraints, cithcr thc absolutc limit on annual expenditure on an individual project, as in Project 22, or the effect of a preferred investment profile, as in the case of Project 14 (which is also constrained by the absolute limit in its third construction year). In some cases, spreading over only two years is also due to a financial constraint specific to the project, as in Project 6 where the preferred investment profile has been imposed.

3.4.2.3 Full Statement of the Best Project Schedule

Details of the implementation plan for the best solution are shown in

Presentation in the table of the predecessor project numbers, divisibility of benefits. preferred investment profile by years, and budget provides a reminder that the solution had to satisfy a series of constraints.

As alrcady noted, thc proccdurc for stagcd construction has a substantial cffcct. In , cases where a predecessor project has been absorbed by its successor, because the siiccessor is higher in the GA ranking, are:

Table 3.6 Table 3.5

Table 3.6

Table 3.7

Table 3.7

122 Optimise the Selection and Scheduling of Road Projects

Predecessor Proiect No.

1 12 21 27

Successor Project No.

2 13 22 28

Project 1 is very low in the ranking but the combination of 1 and 2 ranks high enough to be implemented in Year 7. An important case of two stages being constructed separately is provided by predecessor Project 25 followed by successor Project 26, with a two-year gap between.

3.4.3 Similarity and Dissimilarity of Solutions: Euclidean Distance

If the GA individuals with large objective function values cluster together in the search space, it is likely that there is a single peak in the vicinity of these individuals; otherwise, there may be multiple peaks. The shape of the search space is one of the factors affecting the ability of a genetic algorithm to find the optimal solution.

Three different types of outcome are observed in the actual genetic algorithm results:

l Similar project sequences with similar objective function values

l Significantly different project sequences but similar objective function values

l Similar project sequences but significantly different objective function values These phenomena are associated with the interdependence of road project benefits. Because it is extremely difficult to investigate algebraically the shape of the search space of the road project problem, a numerical investigation is carried out. In each of the ten experiments, the best ten GA individuals represent ten good solutions obtained by running the genetic algorithm repeatedly. Pooling the best ten GA individuals in each of the ten experiments forms a set of 100 fair to good solutions for the problem and provides the basis for investigating the shape of the search space and the relationships between solutions.

The Euclidean distance for any two vectors

X = ( x l 9

^{x 2 3}

... x n )

and

y

=

(Y

^{I 3} Y 2 7

.

^{‘ ’}

_Y

_n

)

in a Rn space can algebraically be written as:

where d ~ ? ; is the Euclidean distance between vector X and vector Y .

‘Dimension 2

/

Dimension 1

Figure 3.5 Euclidean distance between two vectors in a R3 space

It is difficult to comprehend differences between road project construction timetables because the investment in each construction timetable has 340 elements or dimensions (i.e., 34 projects x 10 years). Consequently, it is not feasible to investigate the distribution of solutions by plotting objective values against the corresponding space dimensions, and a summary measurement for the space dimensions is necessary. Euclidean distance has been used to measure distance between vectors in the real number space (i.e., R,,, where

“R’

stands for a real number space and “n” for the number of dimensions of the space), as shown in for two vectors in a R , Figure 3.5 space.

Taplin and Qiu 127

Two sets of Euclidean distances are calculated:

l Distances between the very best solution and other solutions in the set of 100 good solutions

l Distances between the second best solution and other solutions in the set of 100 good solutions

Table 3.8 Euclidean distances between the best ten solutions

Euclidean distances between solutions’

1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th

1st 0.00 4.99 5.23 4.66 5.09 4.86 5.09 5.00 4.87 4.96

2nd 0.00 4.96 4.44 4.99 4.98 4.96 4.81 4.92 5.45

3rd 0.00 4.14 4.98 4.71 4.78 5.16 4.55 5.61

4th 0.00 4.76 4.40 4.37 4.48 4.35 4.97

5th 0.00 4.62 4.67 5.36 4.95 5.49

6th 0.00 2.77 5.23 4.76 5.48

7th 0.00 5.18 4.54 5.14

8th 0.00 5.34 5.26

9th 0.00 5.36

10th 0.00

NPV 9.566 9.083 8.826 8.826 8.825 8.825 8.825 8.729 8.615 8.543 CM

The matrix of Euclidean distances between vectors in a real number space is a symmetric matrix with values of zero for the diagonal elements. In this case, the possible Euclidean distances range between 0 and 18.44.

The distances are calculated by using the amounts of investment in projects scaled by the corresponding project costs, i.e., the proportions of road projects constructed over years. The reason for this is that each of the 340 elements in the proportion matrix has a uniform domain range of [O,l], and the Euclidean distance accordingly has a domain range of

[&zm = 0, &ii% = 18.441

The first set of distances (i.e., the Euclidean distances between the best solution and other solutions in the set of the 100 good solutions) has a minimum value of 4.3072 and a maximum value of 5.6512. The second set of distances (i.e., the Euclidean distances between the second best solution and other solutions in the set of the 100 good solutions) has a minimum value of 2.1539 and a maximum value of 5.6738. The Euclidean distance between the best solution and the second

best solution is 4.9857. Euclidean distances between the best ten solutions are shown in

Another way of illustrating differences and the practical meaning of Euclidean distance is presented in . The last two columns are not surprising in indicating that where the Euclidean distance is small, there is a high proportion of projects common to both solutions, with many of these being implemented over identical periods. The striking result is the dissimilarity of the two best solutions.

Thus, it is of practical importance to take note of more than one GA solution because it may turn out that, for other reasons, the radically different plan given by the second best solution is more acceptable than the best and yet its implcmcntation would mean a ncgligiblc sacrifice in terms of total payoff.

Table 3.9 Differences between solutions: Euclidean distance and program similarities

Best and Second Best Solution Sixth and Second Best and the Second Seventh Best

Solutions Closest to it in Solutions Euclidean Distance

Number of oroiects (and percentage of total1

Total projects to be 28 (100%) 26 (1007c) 24 (1007c)

implemented by either solution

Projects common to both 12 (43%) 20 (77%) 23 (96%)

solutions

- Identical implementation I (47r) 14 (54%) 12 (50%) ? periods and construction

proportions

- Different implementation 1 I (39%) 6 (23%) 11 (46%) ? periods or construction

proportions

Projects not common to both 16 (57%) 6 (23%) 1 (4%)

solutions

Euclidean distance between

solutions 4.99 3.03 2.77

3.4.3.1 Similar Solutions with Close Objective Function Values

The Euclidean distance between the sixth and seventh best solutions is 2.77 ( ), meaning that the road project construction plans from the two solutions are very similar. Differences between them are merely in slight divergences in construction periods and project proportions. Yet another feature

Table 3.9

Table 3.8 Table 3.8

of the two solutions is that their objective function values are virtually the same, the objective function value being 8,824,545 for the sixth best solution and 8,824,536 for the seventh.

3.4.3.2 Different Solutions with Similar Objective Function Values

As shown in ^, the Euclidean distances between the best ten solutions, except for that between the sixth and seventh best solutions, are all greater than 4.0. If this is taken as the borderline between similarity and dissimilarity of solutions then all of the top ten, except for the sixth and seventh best, can be regarded as dissimilar from one another. On the other hand, as shown in

, objective function values for these solutions are very close to each other. The largest objective function value is 9,566,049 and the smallest in the top ten is 8,542,897, a difference of 11%. If the results are expressed in cost-benefit terms, then the largest and smallest benefit-cost ratios differ by about 1%.

In order to capture differences between solutions in the context of the road network in question, the road project construction plans derived from the best and second-best solutions are shown in

.

These solutions are shown to differ not only in projects to be implemented but also in their construction years and the proportions of projects to be implemented annually.

3.4.3.3 Similar Solutions with Dissimilar Payoffs: The Shape of the Search Space It is apparent from that a solution with a large objective function value may not be close, in terms of Euclidean distance, to the solution with the next highest objective function value. The best and second best solutions illustrate this point ( ). For any two superior solutions with large objective function values, some inferior solutions with small objective function values may exist between the two superior solutions, as illustrated in three-dimension space R’ in

There are a number of examples in the top 100 solutions (some being shown in ) of both spatial separation between solutions which have similar payoffs and also the spatial affinity in terms of Euclidean distance between solutions with very different payoffs. This phenomenon implies that around very good solutions to the road construction timetable problem, there may exist some inferior solutions ( ). If the inferior ones are developed from the good solutions through the evolutionary process in the genetic algorithm, they are likely to be similar in construction sequence but with significantly different objective function values.

130 Optimise the Selection and Scheduling of Road Projects

Table 3.10 Comparison of project implementation in the best and second best solutions (Euclidean distance = 4.99)

100.0

Table 3.10 (cont’d)

31 32 33 34

100.0

Best 4.0 62.0 34.0 2nd Best 33.0 33.0 34.0 Best

2nd Best 50.0 Best 100.0

2nd Best 100.0 Best 100.0

A good example of similar road project solutions with radically different payoffs is provided by the second-best solution and its second-closest solution which are separated by a Euclidean distance of only 3.03 (second last column of

). Twenty out of a total of 26 projects in the two solutions are the same and 14 have identical implementation timetables and construction proportions.

However, the objective function value for the second-best solution is 9,083,365, whereas it is a very poor -1,038,558 for the other very similar solution. This suggests that in some situations, small differences between road project construction timetables may lead to a significant disparity between the Table 3.10

132 Optimise the Selection and Scheduling of Road Projects

corresponding net present values, one being economically desirable and the other highly undesirable on the net present value criterion.

Dimension 3 A

O l oo 0 0 0 0 0

Dimension 1

0 0 0

”

)Dimension 2

Legend:

l a superior solution with a large objective function value

0

an inferior solution with a small objective function value

Figure 3.6 Hypothetical superior solutions and surrounding inferior solutions

Thus, it is concluded that the search space of the road project problem has multiple peaks in the vicinity of the good solutions, with some of these peaks being surrounded by inferior solutions. This finding is consistent with the result of an early investigation into the shape of solution space for the transport network design problem (Pearman 1979), which is similar to the road project construction timetable problem in this study.

3.5 Conclusions: Scheduling Interactive Road Projects by GA

The results of this study to optimise the selection and scheduling of road projects are based only on road user and supplier effects, but other environmental and social impacts could be incorporated in the model. The narrow focus is appropriate in this rural area where the effects on road users and suppliers would dominate any comprehensive evaluation.

Not only does the genetic algorithm for the project scheduling problem generate an apparently best solution, but it also generates a set of very good solutions which make it easy for the decision-maker to select between alternatives without reducing the payoff to any substantial extent. The GA also discloses that traffic responses make the fitness of a program of road projects sensitive to small changes in project sequence. Two contrasting cases have emerged from the

Taplin and Qiu 133

analysis, both having significant implications for practical decisions and the formulation of policy.

3.5.1 Dissimilar Construction Schedules with High and Almost Equal Payoffs The presence of a number of good solutions makes the search for the optimal solution to the road project problem difficult and may even indicate that the optimum has not been found. However, the multiplicity of good but dissimilar solutions may be valuable for decision-makers. They are interested not only in the efficiency of resource allocation but also in other issues, such as environment protection and the development of the local economy. These considerations could influence a marginal decision. From a group of almost equally good solutions, in terms of narrowly specified resource allocation efficiency, decision-makers can choose the most effective by other criteria. In other words, they can choose a solution that is still good in terms of efficiency but better on other grounds.

3.5.2 Similar Construction Schedules with Dissimilar Payoffs

The presence of similar construction timetables with different economic payoffs, of which some are positive and high and the others are negative, has an important implication for practical decision-making on road investment programs. When a road project construction timetable with high economic payoff needs to be modified to a seemingly small degree, caution is required to avoid a substantial decrease in the total economic payoff which may be caused by the adjustment.

The genetic algorithm search has generated a number of examples of programs with very poor payoff which differ only slightly from those giving excellent results. These show that small modifications may cause large and damaging results.

References

Dial R. B. (1971) A Probabilistic Multipath Traffic Assignment Model which Obviates Path Enumeration, Transportation Research, Vol. 5, 83-l 11.

Han R. L. (1999) Optimising Road Maintenance Projects on a Rural Network Using Genetic Algorithms, Proceedings of the Australian Transport Research Forum 1999, Perth, Australia, Vol. 23,477-491.

Michalewicz, Z. (1992) Genetic Algorithms + Data Structures = Evolution Programs, Springer-Verlag.

Pearman, A.D. (1979) The Structure of the Solution Set to Network Optimisation Problems, Transportation Research, Vol. 13B, 81-90.

134 Optimise the Selection and Scheduling of Road Projects

Qiu, M. (1995) Prioritising and Scheduling Road Projects by Genetic Algorithm, Proceedings of the International Congress on Modelling and Simulation, Newcastle, Australia, Vol. 1,290-295

Chapter 4 Decoupled Optimization of Power

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