Optimization of Complex Systems with OptQuest

Transcription

1 Optimization of Complex Systems with OptQuest MANUEL LAGUNA Graduate School of Business Administration University of Colorado, Boulder, CO Latest revision: April 8, ABSTRACT Optimization of complex systems has been for many years limited to problems that could be formulated as mathematical programming models of linear, nonlinear and integer types. Problemspecific heuristics that do not require a mathematical formulation of the problem have been also used for optimizing complex systems, however, they generally must be developed in a case-by-case basis. Recent research in the area of metaheuristics has placed the ambitious goal of building a general-purpose optimizer within reach. Specifically, population-based metaheuristics have made possible the development of general-purpose optimizers for which the solution process is context independent. Two of the best known population-based metaheuristics are genetic algorithms and scatter search. In this paper we described the implementation and functionality of OptQuest, a general-purpose optimizer that was developed using the scatter search methodology. We illustrate the system s features with applications in stochastic, nonlinear and combinatorial optimization.

2 2 Optimization with OptQuest 1. Introduction The best-known optimization tool is without a doubt linear programming. Linear programming is considered a general-purpose tool because the only requirement is to represent the optimization model as a linear objective function subject to a set of linear constraints. The state-of-the-art linear programming solvers are quite powerful and can successfully solve models with thousands and even millions of variables employing reasonable amounts of computer effort. Evidently, however, not all business and industrial problems can be expressed by means of a linear objective and linear equalities or inequalities. Many complex systems may not even have a convenient mathematical representation, linear or nonlinear. Techniques such as linear programming and its cousins (nonlinear programming and integer programming) generally require a number of simplifying assumptions about the real system to be able to properly frame the problem. Linear programming solvers are design to exploit the structure of a well defined and carefully studied problem. The disadvantage to the user is that in order to formulate the problem as linear program, simplifying assumptions and abstractions may be necessary. This leads to the popular dilemma of whether to find the optimal solution of models that do not necessarily represent the real system or to create a model that is a good abstraction of the real system but for which only very inferior solutions can be obtained. When dealing with the optimization of complex systems, a course of action taking for many years has been to develop specialized heuristic procedures which, in general, do not require a mathematical formulation of the problem. These procedures were appealing from the standpoint of simplicity, but generally lacked the power to provide high quality solutions to complex problems. Metaheuristics provided a way of considerably improving the performance of simple heuristic procedures. The search strategies proposed by metaheuristic methodologies result in iterative-procedures with the ability to escape local optimal points. Genetic algorithms (GAs) and scatter search, for example, are metaheuristics designed to operate on a set of solutions that is maintained from iteration to iteration. On the other hand, metaheuristics such as simulated annealing and tabu search typically maintain only one solution by applying mechanisms to change this solution from one iteration to the next. Metaheuristics have been developed to solve complex optimization problems in many areas, with combinatorial optimization being one of the most fruitful. Generally, the most efficient procedures achieve their efficiencies by relying on context information. The solution method can be viewed as the result of adapting metaheuristic strategies to specific optimization problems. In these cases, there is no separation between the solution procedure and the model that represents the complex system. Metaheuristics can also be used to create solution procedures that are context independent. The original genetic algorithmic designs were based on this model. The advantage of this design is that the same solver can be used to solve a wide variety of problems. The disadvantage, of course, is that the solutions found by a context-independent solvers might be inferior to those of specialized procedures given the same amount of computer effort (e.g., total search time). We will refer to solvers that do not use context information as general-purpose optimizers. One of the main design considerations when developing a general-purpose optimizer is the solution representation to be employed. The solution representation is used to establish the communication between the optimizer and the solution evaluator (which is in fact the abstraction of the complex system). Classical genetic algorithms, for example, used binary strings to represent solutions. This representation is not particularly convenient in some instances, for example when natural solution representation is a sequence of numbers, as in the case of permutation problems. Once again the tradeoff is between solution representations that are generic and those that are specific. General-purpose optimizers can incorporate more than one representation but at least one should allow for maximum flexibility. The most flexible

3 Laguna 3 representations is possibly one in which a solution is represented by an n-dimensional vector X, where each x i may be a continuous or integer bounded variable (for i = 1, K, n ). This representation can be used in a wide rage of applications, which include all those problems that can be formulated as mathematical programs. Other applications might not be immediately obvious. For example, consider a situation where one wishes to solve a traveling salesperson problem (TSP) employing a general-purpose optimizer that represents solutions as X. We first assume that a tour-evaluation routine exists that uses the following tour representation: { (), 1 L, ( n )} Π= π π where π(i) is the index of the i th city visited, for i = 1,, n. So, the tour evaluator takes Π as an input and produces TourLength(Π) as an output. Also assume that an n-dimensional vector X is created., where the visiting priority for city i is given by the x i variable. Furthermore, assume that the variables are continuous and bounded between 0 and 1. Therefore, as far as the general-purpose optimizer is concerned, the problem to be solved consists of assigning a value between 0 and 1 to n variables. Since { } X= x 1, L, x n does not represent a solution to the TSP, the mapping X Π must be performed before the length of the tour can be calculated by the tour evaluator. One possible X Π mapping assigns π(i) for i = 1,, n in such a way that x π(i) > x π(j) if i > j. This mapping is equivalent to ordering the x variables according to their value and using this order as the tour Π. The main disadvantage of this process is that it does not result in a one-to-one mapping of X into Π. For instance, the solutions X 1 = {0.378, 0.456, 0.123, 0.965} and X 2 = {0.786, 0.836, 0.623, 0.885} represent the same 4-city tour Π = {3, 1, 2, 4}. The lack of a one-to-one mapping creates inefficiencies when solving this problem with a general-purpose optimizer, since there is nothing to prevent the search from constructing solutions X that map to tours Π that have already been considered. Obviously since highly efficient exact and heuristic methods are available for the solution of the TSP, the use of a general-purpose optimizer is not justified. However, it is possible that a sequence may be the natural representation for an optimization problem in a complex system. For example, a sequence may represent the job priority in a production scheduling problem, for which the performance of a given sequence is determined by simulating a job shop. Since the job shop simulation represents a computationally expensive black box evaluator of the objective function, the use of a general-purpose optimizer is therefore justified. The rest of this paper describes the implementation and functionality of the general-purpose optimizer known as OptQuest, which uses X as a generic solution representation. This optimizer employs the scatter search framework coupled with tabu search strategies to obtain high quality solutions for optimization problems defined in complex settings. OptQuest was developed by F. Glover, J. P. Kelly and M. Laguna at the University of Colorado. The first version of OptQuest was developed to optimize discrete event simulations modeled with Micro Saint 2.0 (Glover, Kelly and Laguna, 1996). (Micro Saint is a registered trademark of Micro Analysis and Design, Inc.) The descriptions provided in this paper, however, correspond to a new version of OptQuest that operates both as an Add-in function to Excel and as an optimizer for the risk analysis and forecasting software Crystal Ball. (Excel is a registered trademark of Microsoft Corp. and Crystal Ball is a registered trademark of Decisioneering, Inc.) 2. Scatter Search In this section we provide a general description of scatter search as a foundation to discuss the solution algorithm embedded in OptQuest. For a more detailed description of the scatter search methodology see Glover and Laguna (1997). Scatter search is a methodology that operates on a population of solution

4 4 Optimization with OptQuest vectors X. The scatter search process is organized to (1) capture information not contained separately in the original vectors, (2) take advantage of auxiliary heuristic solution methods (to evaluate the combinations produced and to actively generate new vectors), and (3) make dedicated use of strategy instead of randomization to carry out component steps. Figure 1 sketches the scatter search approach in its original form. Extensions are typically created that additionally take advantage of the memory-based designs of tabu search. Two particular features of the scatter search proposal deserve mention. The use of clustering strategies is suggested for selecting subsets of points in Step 2, which allows different blends of intensification and diversification by generating new points within clusters and across clusters. Also, solutions selected in Step 3 as starting points for reapplying heuristic processes are not required to be feasible, since heuristics proposed to accompany scatter search include those capable of starting from an infeasible solution. Fig. 1 Scatter search procedure. 1) Apply heuristic processes to generate a starting set of solution vectors (trial points). Designate a subset of the best vectors to be reference points. (Subsequent iterations of this step, transferring from Step 3 below, incorporate advanced starting solutions and best solutions from previous history as candidates for the reference points.) 2) Form linear combinations of subsets of the current reference points to create new points. The linear combinations are: chosen to produce points both inside and outside the convex region spanned by the reference points. modified by generalized rounding processes to yield integer values for integer-constrained vector components. 3) Extract a collection of the best points generated in Step 2 to be used as starting points for a new application of the heuristic processes of Step 1. Repeat these steps until reaching a specified iteration limit. In sum, scatter search is founded on the following premises. (P1) Useful information about the form or location of optimal solutions is typically contained in a (sufficiently diverse) collection of elite solutions. Useful information may also be contained in bad solutions. However, such solutions are usually much more numerous and varied than good ones, and consequently there is less advantage in trying to make use of them. On the other hand, valuable information can be contained in trajectories from bad solutions to good solutions (or from good solutions to other good solutions), and such trajectory-based information is one of the elements that tabu search seeks to exploit. (P2) When solutions are combined as a strategy for exploiting such information, it is important to provide for combinations that can extrapolate beyond the regions spanned by the solutions considered, and further to incorporate heuristic processes to map combined solutions into new points. (P3) Taking account of multiple solutions simultaneously, as a foundation for creating combinations, enhances the opportunity to exploit information contained in the union of elite solutions.

5 Laguna 5 The outline in Figure 1 reveals significant differences between classical GA implementations and scatter search. While classical GAs heavily rely on randomization and a somewhat limiting operation to create new solutions (e.g., one-point crossover on binary strings), scatter search employs strategic choices and memory along with (convex and nonconvex) linear combinations of solutions to create new solutions. Scatter search explicitly encourage the use of additional heuristics to process selected reference points in search for improved solutions. This is specially advantageous in settings where heuristics that exploit the problem structure can either be developed or are already available. The adaptation of scatter search for OptQuest does not incorporate heuristics to process reference points. The general-purpose nature of OptQuest prevents the use of such heuristics unless specific knowledge about the problem context is communicated to the optimizer. For instance, if instead of using a set of continuous variables to represent a visiting priority that maps into a sequence in the TSP illustration of Section 1, a sequence representation can be directly used to link the optimizer with the solution evaluator, then specialized heuristics could be employed to find improved tours. 3. The OptQuest Optimizer The discussion in this section addresses implementation details as well the system s functionality. OptQuest seeks to find an optimal solution to a problem defined on a set X of bounded variables. The scatter search procedure in OptQuest starts by generating an initial population of reference points (i.e., a population of X vectors). The initial population may include an initial point suggested by the user, and it always includes the following midpoint: x i = l i + (u i - l i )/2, for i = 1,, n, where L = {l i : i = 1,, n} is the set of lower bound values and U = {u i : i = 1,, n} is the set of upper bound values for all x i X. Additional points are generated with the goal of creating a diverse population. A population is considered diverse if its elements are significantly different from one another. The system uses a Euclidean distance measure to determine how close a potential new point is from the points already in the population, in order to decide whether the point is included or discarded. Since the system allows for linear constraints to be imposed on a solution X, newly created reference points X are subjected to a feasibility test before they are evaluated (i.e., before the objective function value f(x) is calculated). Note that the evaluation of the objective function may entail the execution of a simulation, as in the case of discrete event simulation models created with Micro Saint or risk analysis simulation models created with Crystal Ball. Although the system allows for equality as well as general inequality constraints, we will represent the set of constraints as AX B. The feasibility test consists of checking (one by one) whether the linear constraints imposed by the user are satisfied. An infeasible point X is made feasible by formulating and solving a linear programming (LP) problem. The LP (or mixedinteger program, when X contains integer variables) has the goal of finding a feasible X * that minimizes the absolute deviation between X and X *. Mathematically, the problem is given by: Minimize d - + d + subject to A X * B X - X * + d - + d + = 0 L X * U where d - and d + are negative and positive deviations from the feasible point X * to the infeasible reference point X. When constraints are not specified, infeasible points are made feasible by simply adjusting

6 6 Optimization with OptQuest variable values to their closest bound. That is, if x i > u i, then xi * = ui, for all i = 1,.., n. Similarly, if x i < l i, then x * = l, for i = 1,.., n. i i The population size is automatically adjusted by the system considering the time that is required to complete one evaluation of f(x) and the time limit the user has allowed the system to search. This is particularly relevant for problems in which the evaluation of f(x) is computationally expensive. For fast objective function evaluations, the population size defaults to a value of 100. Once the population is generated, the procedure iterates in search of improved outcomes. At each iteration two reference points are selected to create four offspring. Let the parent-reference points be X 1 and X 2, then the offspring X 3 to X 6 are created as follows: X 3 = X 1 + d X 4 = X 1 - d X 5 = X 2 + d X 6 = X 2 - d where d = (X 1 - X 2 )/3. The selection of X 1 and X 2 is biased by the values f(x 1 ) and f(x 2 ) as well as the search history. In particular, tabu search memory functions are used to keep track of those reference points that have been recently used to create linear combinations. An iteration ends by replacing the worst parent with the best offspring, and giving the surviving parent a tabu-active status for a given number of iterations (i.e., the tabu tenure). During its tabu tenure, a tabu-active reference point is not chosen as parent. This is a very simple form of recency-based memory within the tabu search methodology. More sophisticated uses of memory can be found in Glover and Laguna (1997). 3.1 Restarting Strategy In the course of searching for a global optimum, the population may contain many reference points with similar characteristics. That is, in the process of generating offspring from a mixture of high-quality reference points and ordinary reference points member of the current population, the diversity of the population may tend to decrease. A strategy that remedies this situation considers the creation of a new population. The restarting mechanism within OptQuest has the goal of creating a population that is a blend of highquality points found in earlier explorations (referred to as elite points) complemented with points generated in the same way as during the initialization phase. The restarting procedure, therefore, injects diversity through newly generated points and preserves quality through the inclusion of elite points. 3.2 Adaptive Memory and the Age Strategy Some of the points in the initial population may have poor objective function values. Therefore, with a high probability, they may never be chosen to play the role of a parent and would remain in the population until restarting. To additionally diversify the search, the system increases the attractiveness of these unused points over time. This is done by using a form of long-term memory that is different from the conventional frequency-based implementation. In particular, OptQuest introduces the notion of age and defines a measure of attractiveness based on the age and the objective function value of a particular point. The idea is to use search history to make reference points that have not been used as parents attractive, by modifying their objective function values according to their age. At the start of the search process, all the reference points X in a population of size p have an age of zero. At the end of the first iteration, there will be p-1 reference points from the

7 Laguna 7 original population and one new offspring. The ages of the p-1 reference points are set to one and that of the new offspring is set to zero. The same logic is followed in subsequent iterations, and therefore, the age of every reference point increases by one in each iteration except for the age of the new population member which is initialized to zero. (A variant of the above procedure sets the surviving parent s age also to 0, however, this is not absolutely necessary given that the selection of this reference point is forbidden in the short term during its tabu tenure.) Each reference point in the population has an associated age and an objective function value. These two values are used to define a function of attractiveness that makes an old high-quality point the most attractive. Low-quality points become more attractive as their age increases. 3.3 Neural Network Accelerator This strategy is designed to increase the power of the system s search engine. The motivation behind embedding a neural network is to screen out reference points that are likely to have inferior objective function values as compared to the best known objective function value. The neural network is used as a prediction model to help the system accelerate the search by avoiding the need for evaluating f(x) for a newly created reference point X, in situations where the f(x) value can be predicted to be of low quality. Engaging the neural network accelerator is an option to the user. When the neural network is used, the values of X and f(x) are collected for a number of iterations. These data points are then used to train the neural network during the search. The system automatically determines how many data points to collect and how much training is done, based once again on both the time to perform a simulation and the optimization time limit provided by the user. When the complex system is represented by a simulation model, an additional piece of information becomes relevant to improve the prediction power of the neural network. In most simulations, the value of f(x) is calculated after every trial. The f(x) value that becomes the output of the simulation is the one found after a user-specified number of trials. If we define f α (X) as the objective function value found by performing α% of the total number of trials specified by the user, then f α (X) is an estimate of f(x). By this definition f(x) = f 100 (X). In this situations, the neural network accelerator uses not only X but also f α (X) to predict f(x). Therefore, the predicted value of the objective function for X determined by the neural network is given by f $ ( X, fα ( X ) ), where values for α in the range of 10 and 20 have been found quite effective. The neural network is trained on the historical data collected during the search and an error value is calculated during each training round. This error reflects the accuracy of the network as a prediction model. That is, if the network is used to predict f(x) for a newly created X, then the error indicates how good the prediction is expected to be. The error term can be calculated by computing the differences between the known f(x) and the predicted f $ ( X, fα ( X ) ) objective function values. The training continues until the error falls within a specified maximum value. For problems where f(x) can be rapidly evaluated, the use of the neural network accelerator is obviously not recommended. 3.4 Objective Versus Goals So far, it has been assumed that the complex system to be optimized can be treated by OptQuest as a black box which takes X as an input to produce f(x) as an output. It was also assumed that for reference point X to be feasible, the point must be within a given set of bounds and, when applicable, also satisfy a set of linear constraints. It is assumed that the bound values and the coefficient matrix are both known. However, there are situations where the feasibility of X is not known prior to performing the process that determines f(x), i.e., prior to executing the black box system evaluator. In other words, the feasibility

8 8 Optimization with OptQuest test for X cannot be performed in the input side of the black box but instead has to be performed on the output side. This situation is depicted in Figure 2. Fig. 2 Solution evaluation process. X X * f(x * ) Complex System LP Map NN Filter (Evaluator) G(X * ) Penalized Function P(X * ) X * discarded Figure 2 shows the process of evaluating a reference point X. First the mapping X X * is performed ( LP Map ), using either the linear program at the beginning of section 3, a MIP equivalent, or the simple bounding process. Then the neural network is applied, and those solutions that are predicted to be inferior are discarded ( NN Filter ). If a reference point is not discarded, then the solution-evaluator is used to determine the value of the objective function f(x) and of other measures used to determine the feasibility of X. These other measures are denoted by G(X) and are referred to as goals. To illustrate the use of goals, consider an investment problem for which X represents the allocation of funds to n investment instruments. The objective is to maximize the expected return represented by R µ (X). A Monte Carlo simulation is performed to determine the expected return R µ (X) for a given fund allocation X. Restrictions on the fund allocations, which establish relationships among the x i variables, can be clearly handled by the LP Filter. Thus, a restriction of the type the combined investment in instruments 2 and 3 should not exceed the total investment in instrument 7, results in the simple linear constraint x 2 + x 3 x 7. However, a restriction that limits the variability of the returns (as measured by the standard deviation) to be no more than a critical value c, establishes that relationship R σ (X) c. Clearly, this simple bound constraint cannot be enforced in the input of the Complex System Evaluator and therefore is treated as a goal. Instead of discarding a solution that violates a goal (or a set of goals), the system creates the penalized function P(X). The value used to penalize goal infeasibility is not static, because it depends on whether a goal feasible solution is known. The system assumes that the user is interested in finding a goalfeasible solution if one exists. Therefore, goal-infeasible solutions are penalized more heavily when no goal-feasible solution has been found than when one is available. The current version of OptQuest can only handled simple bound goals, i.e., a goal G(X) can be bounded from below, from above, or both. 4. OptQuest Applications This section provides three applications of OptQuest. The first application illustrates the use of the system to solve a stochastic optimization problem. The stochastic nature of the problem is capture by a Crystal Ball model. Hence, the objective function evaluation consists of performing a simulation for a set number of trials. The last two applications consider nonlinear optimization problems for which the objective function can be quickly evaluated. Overall the goal of this section is to illustrate the flexibility and general-purpose nature of the system.

9 Laguna Project Selection Under Uncertainty This problem may be described as follows. A set of n projects are divided into m categories or subsets N i (for i = 1,..., m). It is desired to select at least l i projects but no more than u i projects from each subset N i, in such a way as to maximize total returns without violating a single budget constraint. The cost c j and the investment return r j are associated with the selection of project j. The problem can be characterized as follows: Maximize n j= 1 rx j j subject to c x b n j= 1 j j li x j ui for i = 1,, m j N x j = { 01, } i The problem formulation corresponds to a 0-1 knapsack problem with side constraints that enforce the bounds on project selection for each category. We assume that the return values are random variables with an underlying probability distribution function that depends on the project. A Crystal Ball model is created such that for a given selection of projects (i.e., a set of x values), a simulation is run to predict total returns. An OptQuest search uses the Crystal Ball model as the function evaluator and incorporates the set of constraints on the x values. Consider the instance of this problem given in Table 1. Assume that projects X01 to X03 belong to category A, projects X04 to X07 belong to category B, and projects X08 to X10 belong to category C. Also assume that a total budget of 100 is available. Table 1. Project selection problem instance Return Project Cost Mean Std. Deviation X X X X X X X X X X For each category there is a lower bound on the number of projects that should be selected ( At least column in Table 2) and a corresponding upper bound ( At most column in Table 2). The Distribution column on Table 2 shows the probability distribution function associated with the return values of projects in each category.

10 10 Optimization with OptQuest Table 2. Project category information. Category At least No more Distribution A 1 2 Normal B 2 4 Log Normal C 0 2 Exponential The information in Table 2 along with the budget restriction results in the following set of constraints that must be declared in OptQuest: 12*X *X *X03 + 5*X *X *X *X *X *X09 + 9*X10 <= 100 X01 + X02 + X03 >= 1 X01 + X02 <= 2 X04 + X05 + X06 + X07 >= 2 X04 + X05 + X06 + X07 <= 4 X08 + X09 + X10 <= 2 Finally, assume that the decision maker would like to select the projects that maximize expected return, however, she would like for the total return to have a reasonable variability value. Specifically, the decision maker wants a portfolio of projects that maximizes the expected total return subject to the standard deviation of the returns being less than 120. Since the standard deviation calculation is an output to the simulation, this restriction must be handled as a goal. Final settings before the search can begin include the number of trials to be performed during the simulation and the activation or deactivation of the neural network accelerator. Figure 3 shows the Performance Graph that results from an OptQuest run of 10 minutes on Dell OptiPlex GXpro 200 Mhz. with the Windows NT 4.0. Fig. 3 Performance graph. The graph in Figure 3 reveals that the search began from a solution with an expected total return of approximately 700, which happened to be goal-feasible solution. Starting the search from a goal-feasible solution is not a system requirement, goal-infeasible solutions may be part of the search trajectory until the first goal-feasible solution is found. The best solution found (on the 16 th simulation) has a total expected return of 1, Details of this OptQuest run are given in Table 3.

11 Laguna 11 Table 3. Best solutions found. Simulation Mean Std. Dev. X01 X02 X03 X04 X05 X06 X07 X08 X09 X The best solution is to select projects X01, X05, X06 and X09. These selections meet the category bounds specified in Table 2 and result in a total cost of 100. The project selection problem under uncertainty can be formulated using the robust optimization framework (Laguna 1995). Robust optimization relies on the use of scenarios to model uncertainty in key data. In this case, a number of scenarios would be necessary to account for the variability on the return values related to project. Robust optimization models tend to be large (because the number of variables typically depends on the number of scenarios), and specialized techniques are often necessary to solve the related formulations (Mulvey, et al. 1995). The advantage of using a general-purpose optimizer such as OptQuest is that the uncertainty can be directly modeled via simulation and the optimization model does not increase in size (although, the problem remains combinatorially difficult). 4.2 Nonlinear Optimization OptQuest can be used to find solutions to optimization problems with nonlinear objective functions, linear constraints and bounded continuous and/or integer variables. An instance of such a problem is: Maximize Z = x 1 sin(x 2 ) + x x 4 subject to 2x 1 + x 2-3.5x 4 = 1 x 2 + x 3 3 x 1-2x x x x x For this problem, we create an Excel worksheet which allocates a cell for each variable and designates one cell for the objective function calculation. The linear constraints and the bounds are entered into the appropriate OptQuest windows and the neural network accelerator is deactivated. A minute OptQuest search results in the solution given in Table 4. (The solution was actually found during the first few seconds of the search.) Table 4. OptQuest solution. Variable Value Z x 1 1 x x x

12 12 Optimization with OptQuest The solution in Table 4 is identical to the one found by the Excel Solver (using the default settings). However, if the problem is changed to restrict the values of x 1 and x 2 to be integer, then the solutions found by OptQuest and Solver are different (as indicated in Table 5). Table 5. Solution comparison. Variable OptQuest Solver Z x x x x As shown in Table 5, OptQuest is able to find a better solution than the one provided by Solver. Although it may be possible to change the Solver options to be able to match the solution found by OptQuest, this result shows the robustness of the scatter search approach embedded in OptQuest. 4.3 Pipe Network Design The New York City water supply tunnel problem was first addressed by Schaake and Lai (1969), and has since been used to test several optimization approaches to the design of water distribution systems. The problem consists of finding a least cost capacity expansion plan that satisfies the water demand associated with an increase in the population. The pipe network consists of 20 nodes, some of which have less than the required minimum pressure. The objective is then to add parallel pipes to improve the hydraulic performance of the system in order to meet the minimum pressure requirements at all nodes while minimizing total construction costs. The problem has been addressed considering discrete pipe diameters (Dandy, et al. 1996) and continuous pipe diameters (Loganathan, et al. 1995). Once a set of pipes is selected, the pressures at each node are determined by solving a system of nonlinear equations that represent the hydraulics of the system. In this problem, therefore, the objective function (i.e., the total construction cost) is immediately known once the pipes are selected, but the feasibility of the design must be determined after a black box evaluation. In a recent study, Lippai and Heaney (1996) developed an Excel model for this problem. The model consists of a worksheet that performs the hydraulic computations for every proposed solution (i.e., a pipe selection). The authors used a genetic algorithm to obtain the best solution known for this problem considering discrete pipe diameters, which has a total construction cost of $38.13 million. Using the worksheet developed by Lippai and Heaney, we formulated an OptQuest model for this optimization problem. The model consists of 21 integer variables (one for each potential parallel pipe) bounded between 0 and 15. If the lower bound is assigned to a particular pipe, this means that the pipe is not selected. The values between 1 and 15 correspond to discrete pipe diameters. The model also considers 5 goals, one for each node that violates minimum pressure requirements in the existing network. Therefore, feasible network designs correspond to goal-feasible solutions. The OptQuest model consistently finds the best solution known within 10,000 calls to the function evaluator. This compares well with the 18,053 function evaluations that the genetic algorithm employed to find the same solution, as reported by Lippai and Heaney (1996).

13 Laguna Conclusions In this paper we have described the general-purpose optimizer OptQuest. The system was developed adapting the scatter search methodology with the purpose of building an optimizer that can effectively deal with problems arising in complex settings. OptQuest has been linked to the discrete event simulation package Micro Saint 2.0 and is currently under testing to add optimization capabilities to Crystal Ball, a risk analysis and forecasting software. The OptQuest applications presented here have shown the flexibility of the system. Although, these applications were modeled using a spreadsheet software, the system is also capable of tackling problems beyond those that can be effectively represented in Excel. We have in fact successfully linked OptQuest to black box evaluators that are stand-alone executable programs. References Dandy, G. C., A. R. Simpson and L. J. Murphy (1996) An Improved Genetic Algorithm for Pipe Network Optimization, Water Resource Research, Vol. 32, No. 2, pp Glover, F., J. P. Kelly and M. Laguna (1996) New Advances and Applications of Combining Simulation and Optimization, Proceedings of the 1996 Winter Simulation Conference, J. M. Charnes, D. J. Morrice, D. T. Brunner, and J. J. Swain (eds.), pp Glover, F. and M. Laguna (1997) Tabu Search, Kluwer Academic Publishers, Boston. Loganathan, G. V., J. J. Greene and T. J. Ahn (1995) Design Heuristic for Globally Minimum Cost Water-Distribution Systems, Journal of Water Resource Planing and Management., Vol. 121, No. 2, pp Laguna, M. (1995) Methods and Strategies for Robust Combinatorial Optimization, Operations Research Proceedings 1994, U. Derigs, A. Bachem, and A. Drexl (eds.), Springer-Verlag, Berlin Heidelberg, pp Lippai, I. and J. P. Heaney (1996) Robust Pipe Network Design Optimization Using Genetic Algorithms, Department of Civil and Environmental Engineering, University of Colorado. Mulvey, J. M., R. J. Vanderbei, and S. A. Zenios (1995) Robust Optimization of Large-Scale Systems, Operations Research, Vol. 43, No. 2, pp Schaake, J. C. and D. Lai (1969) Linear Programming and Dynamic Programming Application to Water Distribution Network Design, Rep. 116, Hydrodynamics Lab., Department of Civil Engineering, MIT.