Optimizing Transportation Problems with Multiple Objectives

Virtually all models developed for transportation problems have focused upon the optimization of a single objective criterion, namely the minimization of total transportation costs. They have generally neglected or often ignored the multiple conflicting objectives involved in the problem, the priority structure of these objectives, various environmental constraints, unique organizational values of the firm, and bureaucratic decision structures. However, in reality these are important factors which greatly influence the decision process of transportation problems. In this study the goal programming approach is utilized in order to allow for the optimization of multiple conflicting goals while permitting an explicit consideration of the existing decision environment.     

Developing a fuzzy proportional–derivative controller optimization engine for engineering design optimization problems

In real world engineering design problems, decisions for design modifications are often based on engineering heuristics and knowledge. However, when solving an engineering design optimization problem using a numerical optimization algorithm, the engineering problem is basically viewed as purely mathematical. Design modifications in the iterative optimization process rely on numerical information. Engineering heuristics and knowledge are not utilized at all. In this article, the optimization process is analogous to a closed-loop control system, and a fuzzy proportional–derivative (PD) controller optimization engine is developed for engineering design optimization problems with monotonicity and implicit constraints. Monotonicity between design variables and the objective and constraint functions prevails in engineering design optimization problems. In this research, monotonicity of the design variables and activities of the constraints determined by the theory of monotonicity analysis are modelled in the fuzzy PD controller optimization engine using generic fuzzy rules. The designer only needs to define the initial values and move limits of the design variables to determine the parameters in the fuzzy PD controller optimization engine. In the optimization process using the fuzzy PD controller optimization engine, the function value of each constraint is evaluated once in each iteration. No sensitivity information is required. The fuzzy PD controller optimization engine appears to be robust in the various design examples tested.     

Applicability of surrogates to improve efficiency of particle swarm optimization for simulation-based problems

Particle swarm optimization (PSO) is a population-based, heuristic technique based on social behaviour that performs well on a variety of problems including those with non-convex, non-smooth objective functions with multiple minima. However, the method can be computationally expensive in that a large number of function calls is required. This is a drawback when evaluations depend on an off-the-shelf simulation program, which is often the case in engineering applications. An algorithm is proposed which incorporates surrogates as a stand-in for the expensive objective function, within the PSO framework. Numerical results are presented on standard benchmarking problems and a simulation-based hydrology application to show that this hybrid can improve efficiency. A comparison is made between the application of a global PSO and a standard PSO to the same formulations with surrogates. Finally, data profiles, probability of success, and a measure of the signal-to-noise ratio of the the objective function are used to assess the use of a surrogate.     

GENERALIZED CENTER METHOD FOR MULTIOBJECTIVE ENGINEERING OPTIMIZATION

This paper presents a new approach to multiobjective engineering optimization: the generalized center method (GCM). A multiobjective problem is solved by calculating the centers of a sequence of level sets. These sets comprise intersections of the original constraints and level constraints imposed on objective functions. In view of the different dimensions and conflicting nature of multiple objectives, some scaling and trade-off procedures are implemented. Several engineering optimization examples are given to demonstrate the effectiveness of the proposed method.     

Multi-objective scheduling problems: Determination of pruned Pareto sets

There are often multiple competing objectives for industrial scheduling and production planning problems. Two practical methods are presented to efficiently identify promising solutions from among a Pareto optimal set for multi-objective scheduling problems. Generally, multi-objective optimization problems can be solved by combining the objectives into a single objective using equivalent cost conversions, utility theory, etc., or by determination of a Pareto optimal set. Pareto optimal sets or representative subsets can be found by using a multi-objective genetic algorithm or by other means. Then, in practice, the decision maker ultimately has to select one solution from this set for system implementation. However, the Pareto optimal set is often large and cumbersome, making the post-Pareto analysis phase potentially difficult, especially as the number of objectives increase. Our research involves the post Pareto analysis phase, and two methods are presented to filter the Pareto optimal set to determine a subset of promising or desirable solutions. The first method is pruning using non-numerical objective function ranking preferences. The second approach involves pruning by using data clustering. The k-means algorithm is used to find clusters of similar solutions in the Pareto optimal set. The clustered data allows the decision maker to have just k general solutions from which to choose. These methods are general, and they are demonstrated using two multi-objective problems involving the scheduling of the bottleneck operation of a printed wiring board manufacturing line and a more general scheduling problem.     

A practical sequential lexicographic approach for derivative-free black-box constrained optimization

Many engineering optimization problems involve models that might not exhibit the necessary smoothness to warrant efficient use of gradient algorithms. Many of these problems are also subject to constraints that might be simulation-based and as costly to compute as the objective function. Traditionally, such problems are solved using either penalty methods or lexicographic ordering that evaluates aggregate constraints prior to computing objective values. This study describes a cost-effective approach to performing such optimizations. After classifying all constraints depending on their computational cost, points not satisfying linear constraints are feasibilized, and a suitable penalty term constructed. A sequential lexicographic ordering is then applied in which inexpensive nonlinear constraints take precedence over expensive ones, which in turn take precedence over objective function values. The performance advantage of the proposed method over traditional ones is demonstrated with a set of analytical test problems, and with oilfield-production optimization examples that use ‘black-box’ simulators.     

Time-response-based evolutionary optimization

Solutions to engineering problems are often evaluated by considering their time responses; thus, each solution is associated with a function. To avoid optimizing the functions, such optimization is usually carried out by setting auxiliary objectives (e.g. minimal overshoot). Therefore, in order to find different optimal solutions, alternative auxiliary optimization objectives may have to be defined prior to optimization. In the current study, a new approach is suggested that avoids the need to define auxiliary objectives. An algorithm is suggested that enables the optimization of solutions according to their transient behaviours. For this optimization, the functions are sampled and the problem is posed as a multi-objective problem. The recently introduced algorithm NSGA-II-PSA is adopted and tailored to solve it. Mathematical as well as engineering problems are utilized to explain and demonstrate the approach and its applicability to real life problems. The results highlight the advantages of avoiding the definition of artificial objectives.     

An adaptive direct multisearch method for black-box multi-objective optimization

Optimization and Engineering - At present, black-box and simulation-based optimization problems with multiple objective functions are becoming increasingly common in the engineering context. In...     

A splitting algorithm for simulation-based optimization problems with categorical variables

In the design of complex products, some product components can only be chosen from a finite set of options. Each option then corresponds to a multidimensional point representing the specifications of the chosen components. A splitting algorithm that explores the resulting discrete search space and is suitable for optimization problems with simulation-based objective functions is presented. The splitting rule is based on the representation of a convex relaxation of the search space in terms of a minimum spanning tree and adopts ideas from multilevel coordinate search. The objective function is underestimated on its domain by a convex quadratic function. The main motivation is the aim to find—for a vehicle and environment specification—a configuration of the tyres such that the energy losses caused by them are minimized. Numerical tests on a set of optimization problems are presented to compare the performance of the algorithm developed with that of other existing algorithms.     

The benefits of adaptive parametrization in multi-objective Tabu Search optimization

In real-world optimization problems, large design spaces and conflicting objectives are often combined with a large number of constraints, resulting in a highly multi-modal, challenging, fragmented landscape. The local search at the heart of Tabu Search, while being one of its strengths in highly constrained optimization problems, requires a large number of evaluations per optimization step. In this work, a modification of the pattern search algorithm is proposed: this modification, based on a Principal Components’ Analysis of the approximation set, allows both a re-alignment of the search directions, thereby creating a more effective parametrization, and also an informed reduction of the size of the design space itself. These changes make the optimization process more computationally efficient and more effective – higher quality solutions are identified in fewer iterations. These advantages are demonstrated on a number of standard analytical test functions (from the ZDT and DTLZ families) and on a real-world problem (the optimization of an axial compressor preliminary design).     

A mixed-integer simulation-based optimization approach with surrogate functions in water resources management

Efficient and powerful methods are needed to overcome the inherent difficulties in the numerical solution of many simulation-based engineering design problems. Typically, expensive simulation codes are included as black-box function generators; therefore, gradient information that is required by mathematical optimization methods is entirely unavailable. Furthermore, the simulation code may contain iterative or heuristic methods, low-order approximations of tabular data, or other numerical methods which contribute noise to the objective function. This further rules out the application of Newton-type or other gradient-based methods that use traditional finite difference approximations. In addition, if the optimization formulation includes integer variables the complexity grows even further. In this paper we consider three different modeling approaches for a mixed-integer nonlinear optimization problem taken from a set of water resources benchmarking problems. Within this context, we compare the performance of a genetic algorithm, the implicit filtering algorithm, and a branch-and-bound approach that uses sequential surrogate functions. We show that the surrogate approach can greatly improve computational efficiency while locating a comparable, sometimes better, design point than the other approaches.     

A TABOO SEARCH BASED APPROACH TO FIND THE PARETO OPTIMAL SET IN MULTIPLE OBJECTIVE OPTIMIZATION

Taboo search is a heuristic optimization technique which works with a neighbourhood of solutions to optimize a given objective function. It is generally applied to single objective optimization problems. Taboo search has the potential for solving multiple objective optimization (MOO) problems, because it works with more than one solution at a time, and this gives it the opportunity to evaluate multiple objective functions simultaneously. In this paper, a taboo search based algorithm is developed to find Pareto optimal solutions in multiple objective optimization problems. The developed algorithm has been tested with a number of problems and compared with other techniques. Results obtained from this work have proved that a taboo search based algorithm can find Pareto optimal solutions in MOO effectively.     

Multi-objective optimization of fuzzy structural systems

It is recognized that there exists a vast amount of fuzzy information in both the objective and constraint functions of the optimum design of structures. Since most practical structural design problems involve several, often conflicting, objectives to be considered, a multi-objective fuzzy programming method is outlined in this work. The fuzzy constraints define a fuzzy feasible domain in the design space and each of the fuzzy objective functions defines the optimum solution by a fuzzy set of points. A method of solving a fuzzy multi-objective structural optimization problem using ordinary single-objective programming techniques is presented. The computational approach is illustrated with two numerical examples.     

Aerodynamic optimization of turbomachinery blades using evolutionary methods and ANN-based surrogate models

A fast, flexible, and robust simulation-based optimization scheme using an ANN-surrogate model was developed, implemented, and validated. The optimization method uses Genetic Algorithm (GA), which is coupled with an Artificial Neural Network (ANN) that uses a back propagation algorithm. The developed optimization scheme was successfully applied to single-point aerodynamic optimization of a transonic turbine stator and multi-point optimization of a NACA65 subsonic compressor rotor in two-dimensional flow, both were represented by 2D linear cascades. High fidelity CFD flow simulations, which solve the Reynolds-Averaged Navier-Stokes equations, were used in generating the data base used in building the ANN low fidelity model. The optimization objective is a weighted sum of the performance objectives and is penalized with the constraints; it was constructed so as to achieve a better aerodynamic performance at the design point or over the full operating range by reshaping the blade profile. The latter is represented using NURBS functions, whose coefficients are used as the design variables. Parallelizing the CFD flow simulations reduced the turn-around computation time at close to 100% efficiency. The ANN model was able to approximate the objective function rather accurately and to reduce the optimization computing time by ten folds. The chosen objective function and optimization methodology result in a significant and consistent improvement in blade performance.     

Redundancy allocation for multi-state systems using physical programming and genetic algorithms

This paper proposes a multi-objective optimization model for redundancy allocation for multi-state series–parallel systems. This model seeks to maximize system performance utility while minimizing system cost and system weight simultaneously. We use physical programming as an effective approach to optimize the system structure within this multi-objective optimization framework. The physical programming approach offers a flexible and effective way to address the conflicting nature of these different objectives. Genetic algorithm (GA) is used to solve the proposed physical programming-based optimization model due to the following three reasons: (1) the design variables, the number of components of each subsystems, are integer variables; (2) the objective functions in the physical programming-based optimization model do not have nice mathematical properties, and thus traditional optimization approaches are not suitable in this case; (3) GA has good global optimization performance. An example is used to illustrate the flexibility and effectiveness of the proposed physical programming approach over the single-objective method and the fuzzy optimization method.     

Balancing multiple criteria in formulation of weighted,single-objective genetic algorithm optimization for CNC machining problems

This paper presents results obtained from the implementation of a genetic algorithm (GA) to a simplified multi-objective machining optimization problem. The major goal is to examine the effect of crucial machining parameters imparted to computer numerical control machining operations when properly balanced conflicting criteria referring to part quality and process productivity are treated as a single optimization objective. Thus the different combinations of weight coefficient values were examined in terms of their significance to the problem's response. Under this concept, a genetic algorithm was applied to optimize the process parameters exist in typical; commercially available CAM systems with significantly low computation cost. The algorithm handles the simplified linear weighted criteria expression as its objective function. It was found that optimization results vary noticeably under the influence of different weighing coefficients. Thus, the obtained optima differentiate, since balancing values strongly affect optimization objective functions.     

AN OPTIMIZATION METHOD FOR THE DESIGN OF LARGE,HIGHLY CONSTRAINED COMPLEX SYSTEMS

A practical and efficient optimization method for the rational design of large, highly constrained complex systems is presented. The design of such systems is iterative and requires the repeated formulation and solution of an analysis model, followed by the formulation and solution of a redesign model. The latter constitutes an optimization problem. The versatility and efficiency of the method for solving the optimization problem is of fundamental importance for a successful implementation of any rational design procedure.

In this paper, a method is presented for solving optimization problems formulated in terms of continuous design variables. The objective function may be linear or non-linear, single or multiple. The constraints may be any mix of linear or non-linear functions, and these may be any mix of inequalities and equalities. These features permit the solution of a wide spectrum of optimization problems, ranging from the standard linear and non-linear problems to a non-linear problem with multiple objective functions (goal programming). The algorithm for implementing the method is presented in sufficient detail so that a computer program, in any computing language, can be written.     

Optimal power system generation scheduling by multi-objective genetic algorithms with preferences

Power system generation scheduling is an important issue both from the economical and environmental safety viewpoints. The scheduling involves decisions with regards to the units start-up and shut-down times and to the assignment of the load demands to the committed generating units for minimizing the system operation costs and the emission of atmospheric pollutants.As many other real-world engineering problems, power system generation scheduling involves multiple, conflicting optimization criteria for which there exists no single best solution with respect to all criteria considered. Multi-objective optimization algorithms, based on the principle of Pareto optimality, can then be designed to search for the set of nondominated scheduling solutions from which the decision-maker (DM) must a posteriori choose the preferred alternative. On the other hand, often, information is available a priori regarding the preference values of the DM with respect to the objectives. When possible, it is important to exploit this information during the search so as to focus it on the region of preference of the Pareto-optimal set.In this paper, ways are explored to use this preference information for driving a multi-objective genetic algorithm towards the preferential region of the Pareto-optimal front. Two methods are considered: the first one extends the concept of Pareto dominance by biasing the chromosome replacement step of the algorithm by means of numerical weights that express the DM’ s preferences; the second one drives the search algorithm by changing the shape of the dominance region according to linear trade-off functions specified by the DM.The effectiveness of the proposed approaches is first compared on a case study of literature. Then, a nonlinear, constrained, two-objective power generation scheduling problem is effectively tackled.     

Simulation optimization of logistics systems through the use of variance reduction techniques and criterion models

Simulation is a useful tool for modelling logistics systems. However, simulation itself is not an optimization tool. Therefore attempts have been made to combine simulation and optimization. Optimization of a logistics system through the use of simulation is difficult for several reasons. Because of the size and complexity of logistics systems, it is often necessary to consider the trade-offs between multiple conflicting performance measures for the system. One major drawback associated with commercially available tools is that users are not able to consider these trade-offs easily. To overcome this drawback, a simulation model can be developed to employ multi-objective decision analysis techniques such as criterion models which can then be optimized. This article illustrates how criterion models can be interfaced with simulation models of logistics systems. In addition, this article includes the programming and implementation of the variance reduction techniques of common random numbers and antithetic variates.     

An alternative desirability function for achieving ‘six sigma’ quality

To achieve high process yields or ‘six sigma’ quality, engineers often need to evaluate and optimize processes that are characterized by multiple quality characteristics. Existing desirability functions weigh together multiple objectives but they have a number of limitations. Most importantly, available desirability functions do not explicitly account for the combined effect of the mean and the dispersion of the quality characteristic. Therefore, it is easy to incur excessive expenditures or unknowingly to fail to achieve targeted yields. In this paper, a desirability function is proposed that addresses these limitations. This function conservatively estimates the ‘effective yield’ under assumptions described in the ‘six sigma’ literature. We use an arc‐welding application to illustrate how the proposed desirability function can yield a substantially higher level of quality as well as a more accurate assessment of the process capability than available alternatives. We suggest that the proposed desirability function should be used to facilitate multicriterion optimization when dispersion data are available. Copyright © 2003 John Wiley & Sons, Ltd.