Optimization in Lot-Sizing Problems

Optimization is the process which is executed iteratively for finding the value of variables for which objective function can be either minimized or maximized by satisfying some constraints. For a given problem domain, the main goal is to provide the mode of obtaining the best value of objective function (Gupta & Jain, 2015).

The range of techniques that have been applied to tackle combinatorial optimization problems can be classified into two general categories, the exact methods and the approximate (heuristic) methods. Exact methods seek to solve a problem to guaranteed optimality but their execution on large real world problems usually require too much computation time. Consequently, resolution by exact methods is not realistic for large- sized problems, justifying the use of powerful heuristic and metaheuristics methods (Dhingra, 2006).

A heuristic is a problem-dependent algorithm that exploits problem dependent information to find a sufficiently good solution (not necessarily optimal) to a specific problem (Saka et al., 2013). As such, they usually are adapted to the problem at hand and try to take full advantage of the particularities of the problem. However, because they are often too greedy, they usually get trapped in a local optimum and thus fail, in general, to obtain the global optimum solution.

Metaheuristics are a class of heuristic techniques that have been successfully applied to solve a wide range of combinatorial optimization problems over the years as they provide ways to escape the local optimum solutions (Osman & Laporte, 1996; Voß et al., 2012). They are also often claimed to be able to solve larger instances of a problem and/or to obtain faster results than pure enumerative exact approaches. Moreover, metaheuristics are general purpose algorithms that can be applied to almost any type of optimization problem (Boussaïd et al., 2013). They do not take advantage of any

specificity of the problem, and generally they are not greedy. In fact, they may even accept a temporary worsening of the solution (for example, simulated annealing technique), which allows them to explore more thoroughly the solution space and thus to get a better solution (that sometimes will coincide with the global optimum). Although a metaheuristic is a problem-independent technique, it is nonetheless necessary to do some fine-tuning of its intrinsic parameters in order to adapt the technique to the problem at hand.

The drawbacks (efficiency and accuracy) of existing numerical methods have encouraged researchers to rely on metaheuristic algorithms based on the simulations and nature inspired methods to solve engineering optimization problems. Metaheuristic algorithms commonly operate by combining rules and randomness to imitate natural phenomena (Lee & Geem, 2005). These phenomena may include the biological evolutionary process such as genetic algorithm (GA) proposed by Holland (1975), animal behavior such as particle swarm optimization (PSO) proposed by Kennedy and Eberhart (1995), or the physical annealing which is generally known as simulated annealing (SA) proposed by Kirkpatrick et al. (1983).

There are several advantages of using metaheuristic algorithms such as (Madić et al.,

2013):

1. Broad applicability: they can be applied to any problems that can be formulated as function optimization problems. The problem can be continues or discrete. 2. Hybridization: they can be combined with more traditional optimization

techniques.

3. Ease of implementation: typically easier to understand and implement.

4. Efficiency and flexibility: they can solve large-sized problems faster. Moreover, they are simple to design and implement, and are very flexible.

The use of metaheuristics can be justified due to: (i) complexity of the internal problem that prevents the application of exact techniques, and (ii) a very large quantity of possible solutions that prevent the use of exhaustive algorithms (Gendreau & Potvin, 2005; Talbi, 2009).

It is known that the decision making associated with the lot-sizing and scheduling problem belongs to the category of combinatorial optimization problems. The difficulty to find a general approach for the lot-sizing and scheduling problem is considered in complexity theory as a NP-hard problem (França et al., 1997). Therefore, metaheuristic solution methods must be developed in order to find near optimal solution by exploring the search space efficiently. Metaheuristics has become a great choice for solving NP- hard problems because of their multi-solution and strong neighborhood search capabilities in a reasonable computational time.

As it has been reported in the literature, three types of metaheuristic-based search algorithms namely GA, SA and PSO have been mostly applied in the domain of the lot- sizing and scheduling optimization problems. However, in recent years there is also an increasing trend in the application of newly developed metaheuristic algorithms such as artificial bee colony (ABC) and imperialist competitive algorithm (ICA) for solving lot- sizing and scheduling problems. Therefore, these metaheuristic algorithms are selected as they are tested vastly in plenty of combinatorial optimization problems.

Wolpert and Macready (1997) introduced “No Free Lunch Theory” and concluded that every metaheuristic algorithm has different searching abilities and has its own advantage to deal with the problem domain. So no single algorithm is able to offer satisfactorily results for all problems. In other words, a specific algorithm may show very promising results on a set of problems, but may show poor performance on a different set of problems (Gupta & Jain, 2015).

A classification of different solution methods is shown in Figure 1.1.

Optimization

Continuous Combinatorial

Linear Quadratic Nonlinear

Approximate Method Exact Method Local Method Global Method Classical

Method Metaheuristic Heuristic

Population Based Neighborhood Based Tabu Search Simulated Annealing Swarm Intelligence Evolutionary Computation Particle Swarm Optimization Artificial Bee Colony Imperialist Competitive Algorithm Evolutionary Programming Evolution Strategies Genetic Algorithm

Figure 1.1: Classification of common search methodologies and common