Keywords

1 Introduction

Industry 4.0 involves high-tech systems and requires reliable subsystems to meet the requirements of the companies. Reliability of systems belongs to dependability studies. By definition, the reliability is the ability of an item to perform given functions during a given period time and under given conditions. A system with high-level reliability should be investigated at the design stage by resorting to various methods, notably adding identical and/or different redundant components that perform the same functions, increasing the component reliability, or both options a mixture. The problem is described by a nonlinear optimization problem [1]. These problems are hard to solve due to the complexity, nonlinearity, high computational time, and finding the optimal solutions. Therefore, various methods of artificial intelligence (IA), notably nature-inspired algorithms, have been proposed to solve these problems. During the last decades these algorithms have been widely used and proven their effectiveness in solving various problems.

The paper aims to implement a MO optimization algorithm (namely the NSGA-II) to deal with the reliability optimization problem to reach the highest reliability level at the lowest cost under the design constraints of space, weight, and cost.

2 Problem Description

The MO reliability optimization problems are mainly described as [2, 3]:

2.1 Reliability Allocation

$$ \begin{array}{*{20}c} {{\text{Maximize}}\;R_{S} (r) = R_{S} (r_{1} r_{2} , \ldots ,r_{m} )} \\ {{\text{Minimize}}\;C_{S} (r) = C_{S} (r_{1} r_{2} , \ldots ,r_{m} )} \\ \end{array} $$
(1)

Subject to

$$ \begin{array}{*{20}c} {g_{j} (r_{1} ,r_{2} , \ldots ,r_{m} ) \le {\text{b}}} \\ {0 \le r_{i} \le 1;\;i = 1,2, \ldots ,m} \\ {r \subset {\mathbb{R}}^{ + } } \\ \end{array} $$
(2)

where \({R}_{S}\)(·) and \({C}_{S}\)(·) are the system reliability and cost, \(g\)(·) is the set of constraints, \({r}_{i}\) is the component reliability, \(m\) is the number of subsystems, and \(b\) is the vector of limitations. This problem involves real design variables only.

2.2 Redundancy Allocation

$$ \begin{array}{*{20}c} {{\text{Maximize}}\;R_{S} (n) = R_{S} (n_{1} n_{2} , \ldots ,n_{m} )} \\ {{\text{Minimize}}\;C_{S} (n) = C_{S} (n_{1} n_{2} , \ldots ,n_{m} )} \\ \end{array} $$
(3)

Subject to

$$ \begin{array}{*{20}c} {g_{j} (n_{1} ,n_{2} , \ldots ,n_{m} ) \le {\text{b}}} \\ {0 \le n_{i} \le n_{i\;max} ;\;i = 1,2, \ldots ,m} \\ {n_{i} \in {\mathbb{Z}}^{ + } } \\ \end{array} $$
(4)

where \({n}_{i}\) is the number of redundant components. This problem involves integer design variables only.

2.3 Reliability-Redundancy Allocation (RRAP)

$$ \begin{array}{*{20}c} {{\text{Maximize}}\;R_{S} (r,n) = R_{S} (r_{1} r_{2} , \ldots ,r_{m} ;\;n_{1} n_{2} , \ldots ,n_{m} )} \\ {{\text{Minimize}}\;C_{S} (r,n) = C_{S} (r_{1} r_{2} , \ldots ,r_{m} ;\;n_{1} n_{2} , \ldots ,n_{m} )} \\ \end{array} $$
(5)

Subject to

$$ \begin{array}{*{20}c} {g_{j} (r_{1} ,r_{2} , \ldots ,r_{m} ;\;n_{1} ,n_{2} , \ldots ,n_{m} ) \le {\text{b}}} \\ {0 \le r_{i} \le 1;\;0 \le n_{i} \le n_{i\;max} ;\;i = 1,2, \ldots ,m} \\ {r \subset {\mathbb{R}}^{ + } ,n \in {\mathbb{Z}}^{ + } } \\ \end{array} $$
(6)

The values of Rs and Cs are given in the Pareto front [4].

3 NSGA-II

The NSGA-II has been proposed in [4]. It is the MO version of the genetic algorithms which is inspired by nature evolution. It has been successfully implemented to solve many problems, such as design optimization, energy management, and layout problems. Algorithm 1 illustrates the pseudo-code of the NSGA-II implemented in the present paper.

figure a

Constraint Handling

In the literature, many techniques were developed to deal with the constraints. To handle the design constraints (resource limitation), the penalty function method is adopted in the present paper [5]. The constraints are introduced to the objective function using penalty terms. Therefore, the MO RRAP becomes as follows:

$$Fitnes{s}_{1}=-{R}_{S}\left({r}_{1},{r}_{2},\dots ,{r}_{m,}{n}_{1},{n}_{2},\dots ,{n}_{m}\right)+\psi \left({r}_{1},{r}_{2},\dots ,{r}_{m,}{n}_{1},{n}_{2},\dots ,{n}_{m}\right)$$
(7)
$$Fitness\_2={C}_{S}\left({r}_{1},{r}_{2},\dots ,{r}_{m,}{n}_{1},{n}_{2},\dots ,{n}_{m}\right)+\psi \left({r}_{1},{r}_{2},\dots ,{r}_{m,}{n}_{1},{n}_{2},\dots ,{n}_{m}\right)$$
(8)

where ψ \(\left({r}_{1},{r}_{2},\dots ,{r}_{m,}{n}_{1},{n}_{2},\dots ,{n}_{m}\right)\) is the penalty function, calculated as follows:

$$\psi \left({r}_{1},{r}_{2},\dots ,{r}_{m,}{n}_{1},{n}_{2},\dots ,{n}_{m}\right)={\sum }_{j=1}^{M}{\phi }_{j}.\mathrm{max}{(0,{g}_{j}\left({r}_{1},{r}_{2},\dots ,{r}_{m,}{n}_{1},{n}_{2},\dots ,{n}_{m}\right))}^{2}$$
(9)

where \({\phi }_{j}\) are the penalty factors (constant values). The values of these factors are fixed after several tests.

4 Numerical Case Study

The investigated case study consists of a pharmaceutical plant (see Fig. 1). The NSGA-II including the constraint handling described in Sect. 3 is used to solve this problem.

Fig. 1.
figure 1

Pharmaceutical plant

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This pharmaceutical plant involves ten subsystems connected in series [6]. The raw material is transferred from a subsystem to another one till the end of the production line, chronologically.

The MO RRAP of this pharmaceutical plant is given as follows:

$$ \begin{array}{*{20}c} {{\text{Maximize}}\;R_{S} = \prod\limits_{i = 1}^{10} {[1 - (1 - r_{i} )^{{n_{i} }} ]} } \\ {{\text{Minimize}}\;C_{S} = \sum\limits_{i = 1}^{10} {C(r_{i} )(n_{i} + exp\left( {\frac{{n_{i} }}{4}} \right))} } \\ \end{array} $$
(10)

Subject to

$$ \begin{array}{*{20}c} {g_{1} (r,n) = \sum\nolimits_{i = 1}^{5} {C(r_{i} )(n_{i} + exp\left( {\frac{{n_{i} }}{4}} \right)) \le {\text{C}}} } \\ {g_{2} (r,n) = \sum\limits_{i = 1}^{10} {v_{i} n_{i}^{2} \le V} } \\ \end{array} $$
(11)
$$ \begin{array}{*{20}c} {g_{3} (r,n) = \sum\limits_{i = 1}^{10} {w_{i} (n_{i} *exp\left( {\frac{{n_{i} }}{4}} \right)) \le W} } \\ {0.5 \le r_{i} \le 1 - 10^{ - 6} ,\;r \subset {\mathbb{R}}^{ + } } \\ {1 \le n_{i} \le 10,\;n \subset {\mathbb{Z}}^{ + } } \\ {0.5 \le R_{S} \le 1 - 10^{ - 6} } \\ \end{array} $$

where \(C\left({r}_{i}\right)={{\alpha }_{i}(-T/\mathrm{ln}{r}_{i})}^{{\beta }_{i}}\) is the cost of the component at subsystem i, T is the mission time, wi is the weight of the component at subsystem i. C, V, and W are the limits of cost, volume, and weight, respectively.

In [5, 7], the problem has been investigated as a single-objective problem by taking the overall reliability as a target. Data of this system are given in Table 1.

Table 1. Data of the system [5, 7].
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5 Results and Discussion

The implemented NSGA-II with the constraint handling was implemented using MATLAB and run on a PC with Intel Core I7 (6 GB of RAM and 2.20 GHz) under Windows 7 of 64 bits. The parameters of the implemented NSGA-II are given in Table 2. These parameters were carefully fixed after several simulations.

Table 2. Parameters of the implemented NSGA-II.
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Fig. 2.
figure 2

Pareto front

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Figure 2 shows the obtained Pareto front for the tradeoff between the system reliability and system cost. It can be observed that the redundancy and reliability of the components which give high reliability increases the cost, i.e., highest system reliability is more expensive. Each point corresponds to an optimal number of redundant components and the corresponding reliabilities. The solutions of the Pareto front are optimal and the decision-maker can choose a specific solution after deep further investigations based on the main target.

6 Conclusions

MO optimization problems are complex problems that need strong solution approaches. Artificial intelligence has contributed by proposing nature-inspired optimization algorithms which can tackle these problems. This paper addressed the MO RRAP through a pharmaceutical plant as a case study. The NSGA-II has been implemented to deal with the problem and the penalty function has been used to handle the constraints. The results obtained have been given in a Pareto front that helps the decision-maker choosing an adequate solution. Future works will focus on an approach allowing to consider the constraints as other objectives.