simulation-based scheduling of mutually exclusive projects ... · consider several complicating...

Shiaau-Lir Wang, Paul Schonfeld 11/14/2011

1

Simulation-Based Scheduling of Mutually Exclusive Projects with Precedence and Regional Budget Constraints by Shiaaulir Wang (Corresponding Author) Research Scientist University of Maryland College Park, MD 20742 Tel: (301) 405-3160 Fax: (301) 405-2585 Email: [email protected] and Paul Schonfeld Professor University of Maryland College Park, MD 20742 Tel: (301) 405-1954 Fax: (301) 405-2585 Email: [email protected] November 2011 Word Count: 4907 + (3 Tables + 7 Figures) * 250 = 7407

TRB 2012 Annual Meeting Paper revised from original submittal.


2

Abstract An improved scheduling model for waterway projects is presented which can

consider several complicating factors: (1) multiple project alternatives at each location, of which only one per location may be selected; (2) multiple budget sources or regional funding constraints and (3) constrained precedence relations among projects. A simulation-based optimization model is developed to solve the problem, which uses simulation to evaluate alternative project schedules. A genetic algorithm is developed in order to efficiently solve this large investment optimization problem, using some prescreening rules to reduce the number of simulated alternatives. The mutually exclusive alternatives at each location allow us jointly optimize the sizing and timing of improvements. The multiple budget constraints realistically reflect actual funding practices but considerably complicate the problem because project sequencing no longer uniquely determines the schedules and projects may now be funded concurrently. The numerical example shows how the additional factors considered here can be properly incorporated in the analysis and how the quality and reliability of results from such a relatively complex model can be verified.



3

Introduction Scheduling waterway improvement projects is a complex combinatorial

optimization problem with numerous local optima, which is difficult to solve with conventional optimization approaches. The objective function may minimize the system costs or maximize the net benefits over a multi-year period. There may be several constraints regarding budgets (possibly by region or type of expense), precedence, mutually exclusivity, minimum improvement steps, construction times, capacities, service quality, and geographic distributions. It is difficult to model the probabilistic features of a waterway system analytically. In addition, the solution space increases quickly with problem size, i.e., with the number of projects considered. Furthermore, in transportation networks, some capacity improvement projects may mostly shift elsewhere the bottlenecks and delays. Project benefits and/or costs may depend on which other projects are implemented, thus greatly complicating the solution of project scheduling problems.

The relevant literature includes various methods for evaluating schedules of interdependent projects, such as integer programming [1, 2], dynamic programming [3], queuing metamodels [4], artificial neural networks [5], equilibrium traffic assignment [6], and microscopic simulation models [7]. Among those evaluation techniques, simulation models provide more precise estimation of system performance than other estimation models by considering detailed system characteristics and operations. Various optimization approaches have also been explored in previous studies, including swapping algorithms [8], Lagrange relaxation [6], simultaneous perturbation stochastic approximation (SPSA) [9], and genetic algorithms (GA) [10, 11]. With metaheuristic approaches such as simulated annealing (SA) and GA, the optimization performance is based on specified search parameters, and is also affected by the stochastic aspects of the search. Therefore, a simulation-based optimization process inherits its stochastic nature from two stochastic processes.

None of the published studies found on the selection and scheduling of interdependent projects (including non-waterway applications) consider the following three complicating factors arising in realistic waterway planning problems: (1) multiple project alternatives at each location of which only one per location may be selected; (2) multiple budget sources or regional funding constraints and (3) constrained precedence relations among projects. The model presented in this paper aims to overcome these three important limitations.

At any specific lock site, several expansion alternatives with discretely specified capacities may be considered. Project multiplicity of two types should be considered: multiple alternatives might be selected to be implemented at different times over the analysis period [12], or only one project among those alternatives can be selected. If at most one project is selected among the alternatives at each site, the problem becomes that of determining project sizes and local implementation times with independent construction costs. Based on political or geographical considerations, there may be precedence relations among projects [12] or locations with pre-specified requirements for scheduling some projects ahead of others. Since construction period overlaps may occur, the precedence relations considered restrict the order of project sequence as well as project implementation times. That is, they constrain the order in which projects are



4

funded rather than finished. Additional budget constraints may be based on regional boundaries (e.g., states, basins, or divisions). With limited budgets, not all projects in one region can be fully funded during the analysis period. Thus, if multiple alternatives are considered at one lock location, the available regional budget might not suffice for all project combinations in that region.

Problem Formulation For years, there has been considerable interest in integrating simulation into

optimization [13, 14, 15]. With a number of controllable decision variables and an objective function to be maximized or minimized, the optimization model runs the simulation model to evaluate alternative solutions and eventually determines a combination of the decision variables that produces an optimal or near optimal solution. In order to consider the project interdependencies, a simulation model developed by Wang and Schonfeld [7, 11] is employed to evaluate the performance of interdependent improvement projects. This simulation model considers probabilistic aspects of waterway traffic, lockage times, travel times, demand variability to service levels, as well as and operational lock control alternatives such as different control strategies, chamber preference and chamber assignment for multiple-chamber locks. With growing interest in evolution algorithms, various network problems have recently been analyzed with genetic algorithms [16, 17]. Among network combinatorial optimization problems, scheduling problems have been promisingly solved with GA’s [18, 19]. Wang and Schonfeld [17] also demonstrate a simulation-based GA approach in scheduling waterway interdependent projects. In this study, mutual exclusivity among projects as well as constraints on precedence and regional budgets are embedded in the proposed simulation-based optimization model. A prescreening technique is also provided to save the simulation/evaluation time during the search process.

Mutual Exclusivity At any specific lock site, several improvement projects or expansion alternatives

with discretely specified capacities may be considered. Those projects might be independent of each other, but might also be dependent with interrelated costs. With non-exclusive multiple projects, several alternatives could be selected for one site but implemented at different times over the planning period [12]. If there are mutually exclusive projects at the same location, i.e. if only one can be selected, we may consider the inclusion of sizing decisions in the project scheduling problem.

When combining the project sizing and scheduling problem, the solution space of fully permutated sequences is further enlarged through the inclusion of all project alternatives at each lock. Thus, if there are N lock locations and im ( i = 1,…, N) project

alternatives for each lock, the total number of solution including all possible combinations and permutations would be

iimN! . The project constraints must ensure

that only one project at each location is selected among all available alternatives. Let jX

be a binary variable. If 1jX , the project is selected; otherwise 0jX . If j denotes the



5

project alternatives, then the project constraints for any location can be formulated as 1

jjX .

In order to consider project multiplicity, the chromosome should be defined to include variables representing project ID (identification number) and project size. In this study we encode these two variables together by using the same path representation but applying project ID instead of lock ID in a sequence, as shown in Figure 2. With the original representation, the proposed GA operators [17] could still be applied in the mutation and crossover processes to produce offspring. However, if we consider only one alternative for each location, the sequences with full lists of projects are not feasible solutions, in the sense that all alternatives will be implemented at different times (as shown in the middle part of Figure 2). Therefore, it is necessary to have a “refining” scheme embedded to create the feasible solutions for simulation evaluation. Instead of sequences with full lists of projects, a shorter sequence with only one project at each lock should be formed after the refining procedure, as shown in the lower part of Figure 2.

The simplest way is to discard, for every location, all projects beyond the first one in the full-list sequence. As shown in Figure 3, whenever a project alternative at one lock is selected, a refining technique will automatically discard the other project alternatives at the same lock. As noted, all the mutation and crossover operators are applied on the full-list chromosomes, but not the refined chromosomes. Before starting any evaluation through simulation, the chromosome refining processes are performed on all offspring from any mutation or crossover operations.

Prescreening of Solutions It is noted that the comparison of sequences is straightforward if there are no

mutually exclusive projects within individual locks, since the full list of projects is the same as the full list of lock locations. However, with mutually exclusive projects, the comparison results could be different. Two types of sequences are created when considering mutually exclusive projects. Full sequences of project alternatives are generated from the offspring production process. Partial sequences with only one project per lock are refined for evaluation by simulation. To avoid duplication in the evaluation process, we should compare the refined partial sequences, rather than the full sequences. That is, as shown in Figure 4, after the refining process (performed in the case of mutually exclusive projects), three different full sequences of project alternatives could become the same partial sequence with only one project per lock. Since the simulation model evaluates the refined sequence rather than the original sequence, the same fitness value is calculated for those three original sequences. Therefore, in order to efficiently reduce simulation time, it is preferable to record the “refined” partial sequences rather than the “original” full sequences in the solution list.

Precedence Relations Based on technical, political or geographical considerations, some precedence

relations among projects or locations may be imposed on the scheduling process. As in resource-constrained project scheduling problems (RCPS), it may be necessary or preferable to schedule some particular projects ahead of some others.



6

In order to determine the sequence of predecessor/successor projects, precedence constraints define precedence relations among various projects and are represented by an arrow between any two projects with precedence relations. If two projects iP and jP are

related by a precedence constraint ji PP , project jP can only be funded after iP is

fully funded, or later. Given an array of integers ix where i = 1, 2, 3, …, n, n is the

number of projects, each array element represents the place of one project in the sequence. The precedence constraint can be formulated as ji xx . Similarly, the precedence

constraints can be further applied for the cases in which projects are considered at different lock locations (that is, location precedence constraints). If projects at two locks

iL and jL are related by a precedence constraint ji LL , any selected project at lock

iL can only be funded after any selected project at lock jL is fully funded.

With precedence constraints, some solutions (i.e., project sequences) will be infeasible and should be prescreened and discarded before being simulated. To impose the precedence constraints and preserve population diversity, infeasible solutions which violate any of the precedence relations should be very unlikely to be selected to reproduce offspring in the next generation. Thus, if a sequence violates the precedence constraints, instead of simulating it, we assign its fitness value a large number (i.e., 1015), which represents the penalty in a minimization problem. In a maximization problem, a number close to 0 (i.e., 10-15) is assigned as the fitness value for a sequence violating the precedence constraints. Let a binary variable ip denote the relevant precedence

constraints, i = 1,2,…, k, if 1kp , the kth precedence constraint is satisfied; if 0kp ,

the kth precedence constraint is violated. Since k denotes any given precedence constraint, then the objective function is multiplied by a factor of

kkp . In a minimization problem,

when 0k

kp , the fitness value ends with a large number, i.e., 1015. Otherwise, when

1k

kp , the fitness value is the simulated total system cost.

Regional Budget Constraints A large waterway system is typically operated by multiple geographic divisions

which may have separate budgets. The formulation of regional budget constraints is similar to that in Figure 1 and the equation for regional budget constraints becomes

oit

k

i

j k

oj dttbc

01)( , where

k

ojc is the capital cost of the jth project to be

implemented, and )(tbk is the annual budget in region k. If regional budgets are

independent, the budget constraint for the problem is easily divided into several regional budget constraints. That is, projects are funded one by one in each region and funds from one region cannot be used elsewhere. Therefore, projects from different regions may be funded at the same time. The overall implementation sequence then combines the implementation sequences of each region.

As shown in Figure 5, a sequence containing only one project at each lock location is refined from the constraint of mutually exclusive projects. If there are different



7

regions and each region has its own improvement projects and budget constraints, three regional implementation sequences are then generated with their own implementation schedules. With regional budget constraints, an overall implementation sequence to be evaluated by the simulation model is rearranged chronologically.

Model Test

Numerical Example A simple test network shown in Figure 6 is used here to test the proposed

simulation-based optimization model. There are 3 rivers, 5 ports, and 7 locks (4 single-chamber locks and 3 double-chamber locks). Locks are numbered with ID 0, 1, 2, 3, 4, 6, 7. Locks #5 and #8 are dummy locks. (Lock definitions are borrowed from [7]) Not all locks require improvement projects, but all improvement projects are located at real locks. We assume here that a project is completed and starts affecting traffic operations as soon as its funding is fully spent. First, the current lock congestion level is determined from the baseline simulation run of current conditions without any improvement projects. Ranked from the highest V/C (volume capacity ratio) to lowest V/C, the intuitive project implementation sequence is 7160243.

Simulation inputs include network statistics (O/D trip generation rates, tow size distributions, chamber service time distributions and speed distributions), lock operation (FIFO control, towboats priority, lockage cuts, chamber assignment and chamber bias), demand variables (baseline O/D travel time, annual growth rates), and system variables (simulation period, warm-up period, number of replications) [11].

The project-relevant inputs include budget rate, project IDs, locations, costs, capacity expansion ratios, regional budget, and precedence relations. The precedence constraints limit the sequence of locks receiving improvement projects. For example, the alternative projects at lock #6 should be funded before the alternative projects at locks #2 and #3. The regional budget constraints limit the project annual funds to $40 ×106, $70 ×106, $40 ×106 for regions 1, 2, and 3, respectively. For example, the alternative projects at locks #7, #2, and #6 are funded annually by the 2nd regional budget, 70 ×106, which is uniformly distributed throughout each year. The termination rule for GA search is set at 20 generations without further improvement. Mutation and crossover rates are 0.07 and 0.3, respectively. All the tests are run on a Pentium III processor with 3.6 GHz CPU and 1GB memory.

In this test, multiple projects are considered at some lock locations. However, at most one of the alternative projects for each location is selected in any implementation sequence. There are 18 projects: 4 alternatives at lock #7, 3 alternatives at lock #1, 2 alternatives at lock #6, 3 alternatives at lock #0, 2 alternatives at lock #2, 3 alternatives at lock #4, and one alternative at lock #3. With these mutual exclusivity constraints for projects at some locks, the solution space is 7! × (4 × 3 × 2 × 3 × 2 × 3 × 1) = 2,177,280. That is much less than 18! = 6,402,373,705,728,000. Additionally, with 2 precedence constraints, the solution space is further reduced to ( 7

4C × 4! × 2!) × (4 × 3 × 2 × 3 × 2 × 3 × 1) = 725,760.

In order to accelerate the analysis, a fast budget flow is applied for simulating three years after a one-year warming up period. The average results from 10 simulation



8

replications with different random number seeds are used to evaluate each candidate solution (i.e., generated project sequence and resulting schedule). The population size in this test is set at 20. A 4% interest rate is used to discount the present value of cost, and tow time is assumed to cost $450/tow-hour. In the evolution process, if the generated sequence violates any constraint, its fitness value is assigned a large cost and has nearly no chance of being selected for next generation. The search process is stopped here after 20 unchanged solutions.

Since the proposed optimization search is probabilistic and requires random numbers in the evolution process, 20 GA search processes for the same problem but with different random seeds are presented in Table 1. Based on 20 search processes, the results show that most searches (15 of 20 replications) converge to the same optimized solutions, with optimized total cost of $594,187,215. Searches #2 and #9 converge to even lower optimized solutions ($593,965,950). The best solution in 15 of the 20 searches has project sequence of 26816111318. On average, approximately 566 solutions are generated in each of the 20 search processes, of which 447 are new. (The other 119 are screened out if they exactly match previously generated solutions.) As the number of search generations increases, the discrepancy between the number of evaluated solutions and number of generated sequences also increases. Among the 447 new solutions 23 (=5%) are infeasible (i.e., they violate some constraints) and are screened out before being simulated. Thus, on average only 424 solutions are simulated for this problem before an optimized solution is identified and the search stops.

In this test example, approximately 43 seconds are required to evaluate one solution, which is averaged from 10 simulation replications. The prescreening process, however, takes less than 1 second to determine if the generated sequence has been evaluated. Therefore, it is worth prescreening any new solution against the recorded solutions whenever a sequence is generated to avoid the duplication of simulation. As can be seen in this example, approximately one fourth of generated sequences result in duplicate solutions. We thus save one fourth of simulation runs by avoiding duplicated evaluations.

Table 2 shows the optimized solutions and four project implementation schedules for four of the searches which produce four different optimized solutions in 20 search replications. This indicates that the objective function for this combinatorial problem does have several local optima. As can be seen in the results, with independent regional budget constraints, more than one project during different regions may be funded at the same time (e.g., project #11 at lock #0 in region #1 and project #13 at lock #2 in region #2). The evolution of objective values from those searches is plotted in Error! Reference source not found.. The optimized solutions decrease relatively quickly in early generations and converge at the end of genetic search with the best solution in the previous generation always being saved in the current generation. In this test example, the optimized solution (with an objective value of $594,187,215) located in most search processes (15 out of 20) is slightly inferior to the best one (with an objective value of $593,965,950) found in two other search processes. The discrepancy between these two final objective values is due to the sequence rather than choice of projects. That is, most local optimal solutions have the same projects (projects #2, #6, #8, #11, #13, #16, #18), but different implementation sequences (26816111318, or 86216111318). Increasing the mutation



9

rate or the number of search generations might be helpful in finding the globally optimal solution.

Using the 3rd, 10th, 11th, 12th, 15th, 16th search processes as an example, Table 3 shows how the parameters specifying mutation rate or the number of search generations affect the optimization results. Scenarios with “-2”, “-3” and “-4”are presented with different GA parameters, mutation rates (0.07 or 0.7) and termination conditions (20 or 50 unchanged solutions). Scenarios “-1” are taken from the original searches in Table 1, with 0.07 for mutation rate and 20 unchanged solutions before stopping.

As can be seen, the scenarios with increasing mutation rate do increase the number of offspring, i.e., the number of generated sequences. Convergence in fewer generations is also shown in those scenarios. When increasing the search time, i.e., by allowing 50 rather than 20 generations without any improvement, more generations are required in the GA search but better solutions may be found. Changing either or both of those GA parameters does help in finding slightly lower optimized solutions in most cases (except for scenarios 3-2 and 3-4).

GA Solution Quality In such a complex combinatorial problem with numerous local optima, it is

difficult to find the exact globally optimal solution. No existing methods can guarantee finding that global optimum when the problem is realistically large. The solution space for this test case contains 7! × (4 × 3 × 2 × 3 × 2 × 3 × 1) = 2,177,280 candidate solutions. An experiment is designed to evaluate 25,000 randomly generated solutions to the problem (which cover approximately 1.15% of the solution space) with a sampling process. 12,454 out of our 25,000 randomly generated solutions are infeasible ones which violate precedence constraints. Among the 12,546 feasible solutions, the best fitness value in this sample is 593,965,950 (in present value $), while the worst is 769,730,300. The sample mean is 678,628,766 and the standard deviation is 31,650,572.

Since the sample is randomly generated, the fitted distribution should approximate the actual distribution of fitness values for all possible solutions in the search space. Excluding the infeasible solutions, the distribution for the remaining 12,546 sampled solutions is shown in Figure 7. Based on the plotted histograms, the best distributions may be approximated as normal, but also as gamma or lognormal distributions with slightly skewed bell shapes which cover the domain from 0 to infinity. Figure 7 also shows how similar are the normal, gamma and lognormal distributions that best fit the 12,546 feasible solutions.

Given that similarity, we can choose to use the fitted normal distribution, with a mean of $678,629,000 and standard deviation of $31,650,600, to assess the goodness of solutions obtained by the genetic algorithm. Among the 25,000 randomly generated solutions, only one (.004%) has a fitness value of 593,965,950 and six (.024%) have a fitness value of 594,187,215. Those are the values found in search #2 and #15, respectively, off our twenty GA searches. These GA optimized solutions are located at the extreme low end of the distribution. Thus, very conservatively, we can estimate the probability of finding a solution better than 593,965,950 at .0035. In other words, the solution found by the GA dominates 99.65% of the solutions in this distribution. In practice, further investigation of the finite lower bounds of the distribution of random solutions may confirm that no solutions significantly better than 593,965,950 can be



10

obtained for this problem. It should also be remembered that estimation errors in the input information regarding demand, project costs, lock reliability and tow characteristics, limit the usefulness of further searching for perfect mathematical solutions that are globally optimal but limited in accuracy by the input data.

Hence, the solutions optimized through GA searches, although not necessarily globally optimal, are still extremely good when compared with a large sample of random solutions and leave only a very small probability that significant improvements might still be found by letting the GA search further. That practically shows the reliability and validity of the proposed search algorithm. Besides, the above GA search, after evaluating 447 solutions, only expends 1.8% of the effort of evaluating 25,000 random solutions to locate a near-optimal solution. We expect that such relative advantage of genetic search would increase as the problem gets larger (i.e., as the number of project permutations considered increases).

Conclusions An improved simulation-based optimization model is proposed for evaluating

waterway systems and optimizing the selection and scheduling of interdependent projects. Unlike previous models, this one can consider (1) multiple mutually-exclusive projects at each lock site, which allow us to jointly optimize project sizing and timing decisions, (2) multiple budget sources and regional funding constraints, and (3) constrained precedence relations among projects. When considering mutually exclusive projects, the GA chromosome contains a full list of mutually exclusive projects. However, solutions with full lists of projects are not feasible when we allow at most one project per lock. Therefore, a refining technique is applied to create feasible solutions with lists of projects having at most one project per lock. Introducing project precedence relations does reduce the solution space of this problem. Given the penalty costs assigned to solutions which violate constraints, such infeasible solutions are almost never produced beyond the earliest generations. With regional budget constraints, projects in different regions are funded independently by their own regional budgets in different regions and it becomes possible to have projects funded concurrently.

Although the feasibility of combining simulation and optimization is demonstrated, the computation time is a crucial factor. To reduce running time in a simulation-based optimization model, any newly evaluated solution is recorded in a solution list. Whenever a new sequence is produced from mutation or crossover operations, a pre-screening process is first performed to check throughout the solution list. If that solution is also found in the list, its simulation is waived and its fitness value is directly assigned from the saved records. About one quarter of the evaluations can be omitted simply by avoiding duplicated simulation runs according to our test results. A validation process is also conducted to show the quality and reliability of the solutions optimized with the proposed GA algorithm. It demonstrates that the GA search can efficiently identify solutions that leave no room for significant improvements.

Since the optimization method can be fully separated from the simulation model, the development of these two processes can proceed concurrently. With parallel computing, the search time for the simulation-based GA can be greatly reduced by evaluating different solutions within each generation or multiple replications of those



11

solutions on parallel processors. Future research may achieve further improvements in search efficiency through additional problem-specific genetic operators, better mutation/crossover rules, and migration of solutions between generations. More importantly, the basic approach developed here for evaluating, selecting and scheduling waterway projects is also applicable to the analysis of interdependent projects in other kinds of transportation networks as well as general investment planning and capital budgeting applications in many industries. Future research should extend it to such new applications.



12

References

1. Weigartner, H. M. (1966), “Capital Budgeting of Interrelated Projects: Survey and Syntheses”, Management Science, 12(7), 485-516.

2. Janson, B. N. (1988), “Method of Programming Regional Highway Projects”, Journal of Transportation Engineering, ASCE, 114(5), 584-605.

3. Nemhauser, G. L. and Ullman, Z (1969), “Discrete Dynamic Programming and Capital Allocation”, Management Science, 15(9), 494-505.

4. Dai, M. and Schonfeld, P. (1998), “Metamodels for Estimating Waterway Delays through Series of Queues”, Transportation Research B, Vol. 32, No. 1, pp.1-19

5. Zhu, L. and Schonfeld, P. (1999), “Queuing Network Analysis for Waterways with Artificial Neural Networks”, Artificial Intelligence for Engineering Design, Analysis and Manufacturing, 13, 365-375.

6. Tao, X. and Schonfeld P. (2005), “A Lagrangian Relaxation Heuristic for Selecting Interdependent Transportation Projects Under Cost Uncertainty”, Annual TRB Meeting Jan. 2005 (05-0565 on CD-ROM)

7. Wang, S. and Schonfeld, P. (2002) “Development of Generalized Waterway Simulation Model for Waterway Operations”, Transportation Research Board 2002 Annual Meeting (CD-ROM 02-2194).

8. Martinelli, D., Dai, M., Schonfeld, P., and Antle, G. (1993), “Methodology for planning efficient investments on inland waterways”, Transportation Research Record, 1383: 49-57

9. Ting and Schonfeld (1998), “Optimization through Simulation of Waterway Transportation Investments”, Transportation Research Record 1620, 11-16.

10. Jong, J.C. and Schonfeld, P. (2001), “Genetic Algorithm for Selecting and Scheduling Interdependent Projects”, Journal of Waterway, Port, Coastal and Ocean Engineering, ASCE, 127(1), 45-52.

11. Wang, S. and Schonfeld, P. (2005), “Scheduling Interdependent Waterway Projects through Simulation and Genetic Optimization,” Journal of Waterway, Port, Coastal and Ocean Engineering, ASCE, 131(3), May/June, 89-97.

12. Wang, S. and Schonfeld, P. (2008), “Scheduling Waterway Projects with Complex Interrelations,” Transportation Research Record, No. 2062, 2008, pp. 59-65.

13. Fu, M. C. (1994), “Optimization via Simulation: A Review”, Annals of Operations Research, 53, 199-248.

14. Fu, M. C., Andradóttir, S., Carson, J. S., Clover, F., Harrell, C. R., Ho, Y. C., Kelly, J. P., and Robinson, S. M. (2000), “Integrating Optimization and Simulation: Research and Practice”, Proceedings of the 2000 Winter Simulation Conference, 610-616.

15. Fu, M. C. (2002), “Optimization for Simulation: Theory vs. Practice” (Feature Article), INCORMS Journal on Computing, 14(3), 192-215.

16. B. Golden, S. Raghavan, and D. Stanojevic (2005), “Heuristic Search for the Generalized Minimum Spanning Tree Problem”, INFORM Journal on Computing, 17(3), 290-304



13

17. Xiong Y., Golden B., and Wasil E. (2005), “A One-Parameter Genetic Algorithm for the Minimum Labeling Spanning Tree Problem”, IEEE Transactions on Evolutionary Computation, 9(1), February, 55-60.

18. Michalewicz, Z. (1995), Genetic Algorithms + Data Structures = Evolution Programs, 3rd Edition, Springer.

19. Gen, M, and Cheng, R. (1997), Genetic Algorithms & Engineering Design, Wiley Interscience.



14

Table 1 Optimized Results and Optimized Solutions

GA Search

# of Gen.

# of Generated Sequences

# of Infeasible Solutions / # of Evaluated Solutions

(% of Infeasible Solutions)

Optimal Total Cost ($)

1 44 659 25 / 524 (4.8) 594,187,215 2 34 529 19 / 417 (4.6) 593,965,950 3 25 384 12 / 313 (3.8) 594,187,215 4 47 673 37 / 543 (6.8) 594,187,215 5 48 728 21 / 525 (4.0) 594,187,215 6 41 626 28 / 492 (5.7) 594,187,215 7 38 598 8 / 471 (1.7) 594,187,215 8 33 501 39 / 410 (9.5) 601,896,915 9 27 440 14 / 368 (3.8) 593,965,950

10 32 487 16 / 388 (4.1) 594,187,215 11 33 502 25 / 411 (6.1) 594,187,215 12 31 481 17 / 391 (4.3) 594,187,215 13 40 586 28 / 445 (6.3) 594,187,215 14 29 456 19 / 356 (5.3) 601,550,790 15 29 448 22 / 365 (6.0) 594,187,215 16 29 466 20 / 363 (5.5) 594,187,215 17 39 576 21 / 465 (4.5) 601,833,750 18 41 625 22 / 484 (4.5) 594,187,215 19 47 730 28 / 580 (4.8) 594,187,215 20 55 829 41 / 637 (6.4) 594,187,215

Avg. 37 566 23 / 447 (5.1) -



15

Table 2 Examples of Optimized Solutions GA Search 1

Total Cost ($594,187,215) GA Search 2

Total Cost ($593,965,950) Project # Lock # Region # Time Project # Lock # Region # Time

2 7 2 0.29 8 6 2 0.39 6 1 1 0.50 6 1 1 0.50 8 6 2 0.67 2 7 2 0.67

16 4 3 0.68 16 4 3 0.68 11 0 1 1.13 11 0 1 1.13 13 2 2 1.13 13 2 2 1.13 18 3 3 1.55 18 3 3 1.55

GA Search 8 Total Cost ($601,896,915)

GA Search 14 Total Cost ($601,550,790)

Project # Lock # Region # Time Project # Lock # Region # Time 2 7 2 0.29 2 7 2 0.29 6 1 1 0.50 11 0 1 0.63 8 6 2 0.67 8 6 2 0.67

16 4 3 0.68 16 4 3 0.68 11 0 1 1.13 6 1 1 1.13 14 2 2 1.17 13 2 2 1.13 18 3 3 1.55 18 3 3 1.55



16

Table 3 Search Scenarios with Different GA Parameters

Scenario

GA Parameters Search Results

Mutation Rate

# of Unchanged Solutions

# of Gen.

# of Generated Sequences

# of Evaluated Solutions

Optimal Total Cost ($)

3-1 0.07 20 25 384 313 594,187,215 15-1 0.07 20 29 448 365 594,187,215 16-1 0.07 20 29 466 363 594,187,215 12-1 0.07 20 31 481 391 594,187,215 10-1 0.07 20 32 487 388 594,187,215 11-1 0.07 20 33 502 411 594,187,215

3-2 0.7 20 21 609 357 594,187,215

15-2 0.7 20 32 905 515 593,965,950 16-2 0.7 20 44 1191 667 593,965,950 12-2 0.7 20 38 1073 622 593,965,950 10-2 0.7 20 54 1491 793 593,965,950 11-2 0.7 20 41 1149 662 593,965,950

3-3 0.07 50 92 1393 932 593,965,950

15-3 0.07 50 95 1393 969 593,965,950 16-3 0.07 50 85 1271 895 593,965,950 12-3 0.07 50 99 1474 973 593,965,950 10-3 0.07 50 92 1378 932 593,965,950 11-3 0.07 50 88 1318 947 593,965,950

3-4 0.7 50 51 1408 706 594,187,215

15-4 0.7 50 62 1721 869 593,965,950 16-4 0.7 50 74 1991 1018 593,965,950 12-4 0.7 50 68 1916 936 593,965,950 10-4 0.7 50 84 2282 1099 593,965,950 11-4 0.7 50 71 1987 991 593,965,950



17

1t 2t 3t 4t 5t 6t

321 ccc 21 cc

1c

54321 ccccc 4321 cccc

654321 cccccc

Figure 1 Relations of Budget Flow, Cumulative Cost, Project Sequence, and Project Schedule



18

Chromosome containing all project alternatives

81371116 32

Project Alternatives at specific lock

Lock ID

Project ID 1 2

0

3 4 5

1

8

3

6 7

2

9 10

4

11 12 13

6 7

14 15 16

Chromosome containing one project alternative at each lock location

71849 1613

11515316 214 86101 137

154109 1613 1171423 18

4

12 5

9

6

12

Figure 2 Path Representation of Chromosome for Mutually Exclusive Projects



19

81371116 32

11515316 214 98101 137 4 12

Select 2(Discard 1)

Select 3(Discard 4, 5)

Select 7(Discard 6)

Select 13(Discard 12)

Select 8


6


Figure 3 Proposed Refining Technique to Create Feasible Solutions for Mutually Exclusive Projects



20

81371116 32

11515316 214 98101 137 4 126

11314116 152 12695 410 7 813

41131416 215 98101 137 12 56

Refining

Figure 4 Refined Sequence



21

Re-arrange Sequence

Regional Budget Constraints

1373811 1620.2 0.3 0.7 0.9 1.2 1.5 1.8

Reshuffled Project Sequencein Chronological Order

2 3

16 7

11 8

13

0.3 1.2

0.7 1.5 1.8

0.2 0.9

Region 1

Region 2

Region 3

Project ID

Schedule

Project ID

Schedule

Project ID

Schedule

81371116 32

Refined Project Sequence from Constraintof Mutually Exclusive Projects

Figure 5 Implementation Sequence with Regional Budget Constraints



22

Port

Two-Chamber Lock

One-Chamber Lock

Junction

0 1 2 3 4

6

7

5

8

Figure 6 Test Network and Project Information



23

5.8 6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8

x 108

0

200

400

600

800

1000

1200

1400

1600

Data Histogram

Fitted GammaFitted Lognormal

Fitted Normal

Figure 7 Fitted Gamma, Lognormal, and Normal Distributions


simulation-based scheduling of mutually exclusive projects ... · consider several complicating...

Documents