Optimization problems subject to noise are a challenging type of problem because the evaluation of the solutions will rarely (if ever) reflect their true objective values. Instead, the effect of noise will lead to underestimations or overestimations that will deteriorate the performance of metaheuris- tics in general. This uncertainty is usually modeled in controlled environ- ments as sampling noise from a Gaussian distribution [69], and the sever- ity of noise is determined by its effect on the objective values and by the standard deviation of the noise distribution. For example, common types of noise are modeled as follows,
ˆ f+(x) = f (x) + N 0, σ2 (2.6) ˆ f×(x) = f (x) × N 1, σ2 (2.7) where ˆf+ and ˆf× refer to additive and multiplicative noise (respectively),
2.5. OPTIMIZATION PROBLEMS SUBJECT TO NOISE 29 value sampled from a Gaussian distribution with mean µ and standard de- viation σ. Hereinafter, the true objective value of solution x is represented as f (x), a single noisy evaluation of solution x is represented as ˆf (x), and the estimated objective value of solution x is represented as ˜f (x). Thus, in the absence of noise, f (x) = ˆf (x) = ˜f (x).
2.5.1
Classification according to Uncertainty
The classification proposed in [69] divides optimization problems accord- ing to the type of uncertainty into four classes, namely noise, robustness, objective approximation and time-varying objective functions. The category of noise covers the problems whose objective space is subject to noise. The category of robustness covers the problems whose solution space is subject to noise. The category of objective approximation avoids evaluating costly objective functions and instead creates and updates a model which esti- mates the objective values of the solutions from previously collected data. The category of time-varying objective functions covers the dynamic opti- mization problems whose objective space changes in time and yet remains deterministic at any given instant.
In the category of noisy optimization problems, the noise mitigation mechanisms are classified into explicit averaging when resampling methods are involved, implicit averaging when methods other than resampling are involved (e.g. increasing the population size), and as modifying selection when the individuals are discarded from being selected unless they satisfy certain requirements regarding the effect of noise [69].
2.5.2
Direction of the Optimization Problem
The direction of the optimization problem [111, 112] refers to the challenge for algorithms when the objective space is subject to levels of noise that vary depending on the location of the solutions. The direction of an opti- mization problem is backwards when the global optimum can be found by
directing the search towards the low-noise regions. Conversely, the direc- tion is forwards when the global optimum can only be located by travers- ing the regions subject to high levels of noise. According to these defi- nitions [111, 112], multiplicative noise on the objective values will define a direction for the optimization problem given that the severity of noise will change according to the objective values of the solutions. Thus, the additional challenge of multiplicative noise over additive noise is a larger corruption of the objective values whose magnitude changes across the search space proportionally to the objective values of the solutions.
In optimization problems subject to multiplicative noise, the direction of the problem will depend on the objective of the optimization and on the range of the objective space. Specifically, on minimization problems whose objective space is only positive, the objective values of better solu- tions will be affected by a smaller severity of noise. Conversely, in maxi- mization problems, the objective values of better solutions will be affected by a larger severity of noise. Hence, the direction of these two instances of optimization problems are backwards and forwards, respectively.
2.5.3
Simulation-Based Optimization
In the discipline of Operations Research, optimization problems subject to noise are comprised within the general field of simulation-based optimiza- tion [2, 3, 4, 36, 56, 57, 58, 103], which is defined as the process of finding the best parameter values for a system whose performance is evaluated from a stochastic simulation model [139]. The formulation of a general stochastic simulation model is commonly found as Equation (2.8), where
xis a solution in the search space Θ, ˜f (x)is the objective function to op-
timize, ω represents a simulation replication, L is the sample performance measure, and E is the expected objective value.
min
x∈Θ
˜
2.5. OPTIMIZATION PROBLEMS SUBJECT TO NOISE 31 Some examples of real-world problems that are modeled as stochas- tic simulation models are the design and operations of call centers [57], inventory policies for orders in inventory control systems [57], schedul- ing of manufacturing cells for the production of aircraft and gas turbine engines [139], and scheduling of camshaft machining lines [139]. In all these cases, the optimization is performed on stochastic simulation mod- els whose samples could be obtained, for example, over a period of time on an online system or from already known probability distributions.
Design and Operations of a Call Centre
A call centre provides support to customers via telephone. The design of a call centre depends on many variables that will define its operation towards the handling of the calls. The variables can be obtained from Customer Relationship Management (CRM) systems that store informa- tion about customers such as inquiries, requests, complaints, and prior- ities [57]. Based on these variables, different policies determine the as- signment of available operators to the calls depending on other variables such as skills of the operator, routing algorithms and types of queues [57]. The optimization of the design and operations of a call centre consists of finding potential solutions with optimal settings for the variables in- volved. The objective values of these solutions may consist of metrics rel- ative to the customers or operators such as reducing waiting times, oper- ational costs, network usage, abandonment rates of calls, and operators usage [57].
Inventory Control System
An inventory control system manages the availability of items in stock by automatically placing new orders when the stock falls below a certain level. A simple example considers the optimization of two parameters, namely the re-order level and the order-up-to level [57]. The re-order level
refers to the minimum availability of an item in stock before a new or- der is placed to increase it. The order-up-to level refers to the amount of items that the new order will place to stock up with. Solutions to this problem will find the optimal values to both levels such that a cost objec- tive function is minimized. Examples of the objective functions involve reducing ordering and holding costs, and lost sales components, amongst others [57].
Scheduling of Manufacturing Cells
The scheduling of a real-world manufacturing cell of components for air- craft and gas turbine engines was addressed in [139]. The cell consists of five machines and five burring stations that operate differently according to the components that arrive at the cell. If multiple components arrive simultaneously, a priority function p determines their precedence based on their due times and that results in a critical ratio value p < 1.0 if the component is behind schedule, p = 1.0 if it is on schedule, and p > 1.0 if it is ahead of schedule, where those components with smaller ratio values take precedence. The due time for each type of component is determined by an inter-arrival time that specifies the frequency at which the type of component enters the system. The optimization problem is to find the op- timal values of inter-arrival times for the different types of components such that the utilization of the cell is maximized and the tardiness is mini- mized.
Scheduling of Camshaft Machining Lines
The scheduling of a real-world camshaft machining line was addressed in [139]. The machining line produces 15 different variants of camshafts, and the tasks to be performed on each camshaft are allocated between 14 specialized groups containing 34 machines in total that operate in parallel. Each machine has a processing time, physical capability and limitations,
2.6. PSO FOR NOISY OPTIMIZATION PROBLEMS 33