A Reconfigurable Cellular Remanufacturing Architecture: a multi-objective design approach

Abstract

Remanufacturing is a practice that postpones the product ‘end-of-life’ by returning the properties or features of a new product to a used product. Such a process represents an efficient circular economy strategy to extend product life, reducing its environmental footprint. However, remanufacturing systems must overcome distinct challenges related to information uncertainties about quantity and conditions of used products. Current strategies to address these issues include smart approaches towards smart remanufacturing systems. However, remanufacturing architectures must manage the negative effects on operations derived from the used product’s deterioration and return rate variability, as well as remanufactured products demand fluctuations. This study contributes to address this issue by a Reconfigurable Cellular Remanufacturing Architecture that is integrated in a business strategy towards smart sustainable remanufacturing. The design process is based on a multi-objective optimization model that minimizes the grouping cost, the workload balancing cost, the investment cost, the makespan cost and the reconfigurable cost. A customized version of the well-known multi-objective evolutionary non-dominated sorting genetic algorithm II (NSGA2), a mono-objective genetic algorithm (GA-mo) and a GAMS model were implemented to obtain the potential architecture’ configurations for an explanatory case study based on real-world industrial and random data, respectively. Two procedures to identify the best architecture were also considered. A sensitivity analysis is presented to illustrate the robustness of the proposed model. Managerial insights illustrate about solution methods, and best architecture selection. Practical implications also are provided to offer useful options for practitioners.

Appendix

Genetic algorithms NSGA2 and GA-mo

This study implements in Matlab 2021a two genetic algorithms to solve the proposed multi-objective model, both share the same solutions coding and operators. The first is a customized version of NSGA2 [20], and the second is a mono-objective genetic algorithm that uses weighted factors as goal programming.

NSGA2 starts randomly generating the initial population, and a set of non-dominated solutions was obtained. These solutions fed the next iteration step as part of the initial population, which was then completed randomly. Following this incremental solution process, the final iteration provided the best last non-dominated solutions. Here, the NSGA2 and GA-mo were run with the following parameters: number of individuals = 100, generations = 100, iterations = 1000, crossover probability = 0.9, and the mutation probability = 0.1.

Coding solutions

The customization of NSGA2 starts by defining a hybrid data structure for the solutions codification. Similarly, the generation of the initial population, crossover, and mutation processes were customized.

The decision variables ${X}_{ik, }{Y}_{jk}, {Z}_{ijl}$ and ${W}_{pjj{\prime}k}$, described in the mathematical model, were coded in variable Var as a set of four vectors:

$$V\; ar=({Var}^{X},{Var}^{Y},{Var}^{Z},{Var}^{W})$$

(23)

The first vector, Var^X, is defined by a binary structure that contains the rows of the ${X}_{ik,}$ matrix.

$${Var}^{X} = [0,1,1,0,1,1,0,1,0,1,1,1,0,1]$$

(24)

The second vector, Var^Y, contains the number of the product family or cell k to which each product j was assigned, according to the solution codification based on the cell number exposed by Gen et al. [52] This data structure allows to directly manage Eq. (13), which ensures that each product j is only assigned to a single family or cell k.

$${V\; ar}^{Y} = [1,2,2,1,2,1]$$

(25)

The third vector, Var^Z, contains the number of type i workplaces required to manufacture product i.

$${V \;ar}^{Z}=[3,4,7,1,2,1,3,5,4,2,3,7,3,4,1,2,3,4,1,2,.....,2,5,1,2,3,1,1,7]$$

(26)

The fourth vector, Var^W, contains numbers between 0 and 1, which represent the relative position among the products to determine the production sequences ${W}_{pjj{\prime}k}$, as is described next.

$${V\; ar}^{W}= [\mathrm{0.34,0.25,0.16,0.07,0.14,0.95,0.86,0.77,0.17,0.27,0.49,0.63}]$$

(27)

The decodification of the variable ${W}_{pjj{\prime}k}$, requires more steps than the previously described ones. This process starts with the vector Var^W that store the rows of the matrix ${d}_{pj}^{W}$, where each row presents a period p and each column presents a product j. In a next step, for each period of d^W_pj, the values which belong to each product family were identified using variable Y_jk, hence defining d^temp. Then, in a following step, the obtained values were classified in ascending order to identify the production sequence for each product family in each period. Finally, using this information, the full W_jj′p matrix was defined.

The model constrains were managed using data structure, and penalization and solution correction operators as described in the following. The constrains described as the model Eqs. (12), (13) and (14) were managed using a penalization strategy. Here, each model constraint is verified in a prior step to calculate the objective function for each solution. If a solution is not in agreement with a constraint, a penalization factor is increased. Finally, the objective related to this constraint is then multiplied by the penalization factor.

In addition, two solution correction operators were implemented to fulfil the constraints described in Eqs. (12) to (14). After the solution decoding process, if a solution did not fulfil the constraints Eqs. (12) and (13), workplace i was assigned to the cell or family belonging to product j.

Initial population

The initial population brings the seeds from it starts optimization process to NSGA2-RCRSA, based on solutions coded as Var = (Var^X,Var^Y ,Var^Z,Var^W), which was implemented according to two strategies. The first strategy defined the population randomly, and the second one followed a heuristic process. For each Var^X vector position, a random number a was generated, if a ≤ 0.5 V ar^X = 1, otherwise Var^X = 0. For each position of Var^Y and Var^Z a random integer ranging from 1 to mc was generated. Finally, Var^W was generated as a random vector. This strategy allowed to explore the search space; however, it could not be optimized. Hence, for half of the population, the second strategy was implemented. First, a base V ar^Y was defined, where a product j that is processed in a workplace i was assigned to the same cell or product family. Second, to balance the workload and to reduce the makespan in scheduling, Var^Z adopts the values of the balanced workplaces $=\lceil\frac{{t}_{ijl}^{op}}{{t}_{j}^{c}}\rceil$, which approximates the ideal number of workplaces i to balance t^op_ijl. Finally, the random values are assigned to Var^W trying to be near to the ideal production sequences.

Reproduction

The algorithms use sexual reproduction as a mechanism to exchange information among individuals for improving the population. In this way, this process allows the population’s evolution, providing the features of the best adapted individuals of the current generation to the next one [20]. This process starts with parents’ selection, which in this study was realized through a binary tournament.

A crossover process was conducted after parent selection. During this process, the information of the parents was exchanged. For Var^X, Var^Y, and Var^Z a crossover with a single cross point was implemented, as shown in the following example:

$$\begin{array}{cc}{Parent}^1=\left[1,2,3,4,5\right]&{Parent}^2=\lbrack6,7,8,9,10\rbrack\end{array}$$

(28)

For this example, the cross point is the third position, hence as a result two children are obtained:

$${Child}^{1} = [\mathrm{1,2},\mathrm{3,9},10] {Child}^{2} = [\mathrm{6,7},\mathrm{8,4},5]$$

(29)

Given that V ar^W is a real vector, its crossover is implemented as a random weighted average of the two parents to generate a child:

$$Child=\alpha *P{arent}^{1}+(1 - \alpha )*{Parent}^{2}$$

(30)

where α is a random weighted factor.

The last reproduction process is mutation. It allows exploring the search space to generate tiny random changes to some individuals. This mutation is an essential feature of evolutionary algorithms which helps to avoid local optimal solutions. The mutation process was applied to randomly selected individuals (GA-mo) or dominated by others (NSGA2). For each element of an individual vector, a random number was generated, and if less than the mutation probability the element was mutated. In the case of Var^X, if the element is ”1”, its value is changed to ”0” and vice-versa. Var^Y and Var^Z were mutated increasing or decreasing the selected element by one unit. If the element value was ”1”, it can only be increased. Finally, Var^W is mutated multiplying the element by a random factor that can increase or decrease its value.