Solution Representation and Chromosome Encoding

The solution representation for an optimization problem encompasses a solution structure and its possible encoding system for the problem elements [123]. The solution structure can be represented by a one (1D, or vector), two (2D), or a multi-dimensional (ND) matrix. The basic solution structure in GA is a vector chromosome. Each element in the chromosome structure is called gene, and the content of this gene is called an allele [86]. The values of a chromosome can be encoded using different encoding systems. The encoding of a chromosome can be in a binary, permutations, value, or tree structure of genotype [86].

The solution representation of the first team allocation method named “Static Teams with Queue Simulator” (STQS) is illustrated from the approaches in [20-22, 94]. This method uses two vector chromosomes. The first chromosome represents the resources and their distribution into teams, and the second one represents the tasks and their order in a single queue system. The representations of both chromosomes having n number of resources, and m number of tasks are depicted in the following Figure 21.

Figure 21: STQS method Chromosomes

The first chromosome - in Figure 21(a) – represents the resources distribution solution, where each gene represents a resource, and each allele represents a team. That is, the value for each resource (gene) holds the team number that this resource is assigned to. The encoding system used for this chromosome is represented by an integer value encoding. The distribution of resources into teams should be performed in this chromosome according to the pigeonhole principle, which as a key aspect allows more than one resource to have the same team number in the solution (multiple genes have the same allele value) [124]. For instance, resource 1 and n in the figure both work in team 2.

The second chromosome depicted in Figure 21(b) provides the ordering solution representation that show the execution order for each project task i.e. to where each task should be sorted in the queue. Each gene in this chromosome represents a task, and each allele represents the task order in the queue. Unlike the previous chromosome, the ordering representation do not allow similarity between allele values. Therefore, the encoding system used in this solution representation to

comply with this restriction is a permutation encoding. For example, if we have three elements representing the project tasks, the encoding of these tasks for their order within the queue using the permutation encoding can be 1, 2, 3, or 2, 1, 3, or 3, 1, 2, etc.

The second resource allocation method defined in this chapter is “Static Teams with Time Simulator” (STTS). The solution representation for this method is depicted in Figure 22 by two chromosomes. The structure of those chromosomes is a vector. The first chromosome similar to the one in STQS method represents the distribution of resources into teams. Unlike STQS method representation, the second chromosome in this method solution represents the task allocation by providing the team number that is responsible to work on each task. This representation of STTS method solution having n number of resources, and m number of tasks is depicted as follow.

Figure 22: STTS Method Chromosomes

From the previous Figure 22, it can be seen that the pigeonhole principle again is the one that forms the solution for both chromosomes. For the first chromosome representation (a), the resources are distributed into teams, where each gene represents a resource, and each allele represents a team. This representation provides to which team each resource is assigned to.

Representation (b) on the other hand, depicts the distribution of teams to tasks. From this representation, it can be seen that which task should be done by which team. For example, from the Figure 22(b), we can see that task 2 is assigned to team 1, and task 3 is assigned to team 3.

The third method of “Dynamic Teams with Binary Participation” (DTBP) depicted in the following Figure 23 represents the solution by a 2-D matrix structure. The vertical dimension (columns) represents the resources, and the horizontal one (rows) represents project tasks. This representation having n number of resources, and m number of tasks is depicted as follow.

Figure 23: DTBP Method Chromosome 1 2 3 … n

1 1 0 1 … 1

2 0 0 1 … 0

3 0 1 0 … 1

: 0 1 1 … 1

m 1 1 0 … 1

From Figure 23, it can be seen that this representation assigns each resource with a high probability to serve in different teams during project time. For instance, it can be seen in Figure 23 that resource 1 assigned with resources 3, and n to work on tasks 1, where on task m this resource works with resources 2, and n. We have defined this representation as an arbitrary assignment of resources to tasks. Each gene in this chromosome has two positioning points (v and h) that define the resource (v) and the task (h) that (s)he assigned to. In addition, the allele of each gene should be encoded using the binary system, which implies whether resource (v) is assigned to task (h) by value of one, or zero otherwise. This representation moreover, requires important assignment constraint in order to gain realistic and reasonable solution. For instance, this representation can provide a solution where all resources are assigned to all tasks. For this reason, a constraint is implemented with this representation to make sure that any resource works on more than one task at a time, its productivity will be normalized to the number of these simultaneous tasks. By doing so, the solution then of having all the resources works on all the tasks will provide low quality solution of project time span.

The fourth method of “Dynamic Teams with Participation Rate” (DTPR) represents the solution by 2-D structure similar to DTBP. This representation having n number of resources, and m number of tasks is depicted in the following Figure 24.

Figure 24: DTPR Method Chromosome

From Figure 24, it can be seen that this assignment representation is similar to the one in DTBP method. However, this method enforces the resources to partially dedicate a percentage of their working time to each task to which they are assigned. With five different values described in the problem formalization Section 5.2 of this method, each allele in the chromosome representing the gene (v, h) of resource (v) and task (h) can hold a real number value encoding from the range {0, 0.25, 0.5, 0.75, 1}. It is noteworthy that the same constraint described for the pervious method DTBP is adopted in this method implementation too. This is to make sure that a full dedication of all the resources to all tasks, implying overtime work assignment is not considered.

1 2 3 … n

1 0.25 0.5 0.75 … 0

2 0 0 1 … 0.5

3 0 0.5 0.25 … 1

: 0.75 1 1 … 0.25

m 1 0.25 0 … 0.5