A multi-objective military system of systems architecting problem with inflexible and flexible systems: formulation and solution methods

Abstract

System of systems (SoS) architecting is the process of bringing together and connecting a set of systems so that the collection of the systems, i.e., the SoS is equipped with a set of required capabilities. A system is defined as inflexible in case it contributes to the SoS with all of the capabilities it can provide. On the other hand, a flexible system can collaborate with the SoS architect in the capabilities it will provide. In this study, we formulate and analyze a SoS architecting problem representing a military mission planning problem with inflexible and flexible systems as a multi-objective mixed-integer-linear optimization model. We discuss applications of an exact and an evolutionary method for generating and approximating the Pareto front of this model, respectively. Furthermore, we propose a decomposition approach, which decomposes the problem into smaller sub-problems by adding equality constraints, to improve both the exact and the evolutionary methods. Results from a set of numerical studies suggest that the proposed decomposition approach reduces the computational time for generating the exact Pareto front as well as it reduces the computational time for approximating the Pareto front while not resulting in a worse approximated Pareto front. The proposed decomposition approach can be easily used for different problems with different exact and heuristic methods and thus is a promising tool to improve the computational time of solving multi-objective combinatorial problems. Furthermore, a sample scenario is presented to illustrate the effects of system flexibility.

Appendix

1.1 Determining Pareto efficient solutions in a given set

Let $P^o$, $D^o$, and $C^o$ denote the objective function values of a given solution $\mathbf U ^o$ in a set of solutions $\Phi$ such that $1\le o\le |\Phi |$. The following algorithm determines the set of Pareto efficient solutions within $\Phi$, denoted by ${PE}(\Phi )$.

1.2 Details of the numerical studies

Given n, $|J_1|$, and $|J_2|$, we randomly generate ten problem instances where each problem instance is generated as follows: First, we generate an $n\times (|J_1|+|J_2|)$-matrix where each entry is uniformly distributed between 0 and 1, and then, we round the entries to the nearest integer and construct a binary $n\times (|J_1|+|J_2|)$-matrix (that is, an entry is 1 with probability 0.5 and 0 with probability 0.5). After that, we check if there is at least one 1 in each row. If there is at least one 1 in each row, it is accepted as a feasible A for the problem instance because it means that there is at least one system that can provide each capability; otherwise, for those rows without a 1, we randomly select a column and make the entry of that row in the randomly selected column equal to 1. After that, we generate $\mathbf P$, $\mathbf C$, $\mathbf D$, $\mathbf E$, and $\mathbf H$ matrices such that $p_{ij}\sim U[10,20]$, $d_{ij}\sim U[5,10]$, $c_{ij}\sim U[20,40]$, and $h_{j^1j^1}\sim U[1,5]$, where U[a, b] denotes a continuous uniform distribution with the range [a, b]. Without loss of generality, we round $\mathbf P$, $\mathbf C$, $\mathbf D$, $\mathbf E$, and $\mathbf H$ to the nearest integers (given that these parameters are not in the constraints except $\mathbf D$ and $\mathbf D$ is only in the constraints that define the completion time of a SoS, this generalization does not change the model).

In the evolutionary methods, we randomly generate $\alpha =n$ chromosomes initially, we randomly generate $\gamma =n$ chromosomes to be added to each population, and set $\beta =n$ as the termination criterion. Furthermore, in the exact methods, we set $\epsilon =1$ as $\mathbf C$, $\mathbf D$, $\mathbf E$, and $\mathbf H$ are integers. We set $M_c$ and $M_d$ equal to the total cost and total time of the solution defined by $\Upsilon$, respectively (see Observation 1).

All of the methods are coded in Matlab 2014a (8.3.0.352) and executed on a personal computer with 3 GHz dual-core processor and 16 GB RAM. For solving the mixed-integer-linear problems in the form of $\widehat{\mathbf{SP }}$, we use the mixed-integer-linear solver of IBM-ILOG’s CPLEX version 12.6.1.

1.3 Tables of Sect. 5

See Tables 3, 4, 5 and 6.

Table 3 Quantitative comparison of EM-1, DM-1, EM-2, and DM-2 for {3, 4, 5} combinations

Table 4 Qualitative comparison of EM-2 and DM-2 for {3, 4, 5} combinations

Table 5 Quantitative comparison of EM-2 and DM-2 for {5, 6, 7} combinations

Table 6 Quantitative comparison of EM-2 and DM-2 using $\widehat{{PE}}$ for {5, 6, 7} combinations