Using Electromagnetic Algorithm for Total Costs of Sub-contractor Optimization in the Cellular Manufacturing Problem

In this paper, we present a non-linear binary programing for optimizing a specific cost in cellular manufacturing system in a controlled production condition. The system parameters are determined by the continuous distribution functions. The aim of the presented model is to optimize the total cost of imposed sub-contractors to the manufacturing system by determining how to allocate the machines and parts to each seller. In this system, DM could control the occupation level of each machine in the system. For solving the presented model, we used the electromagnetic meta-heuristic algorithm and Taguchi method for determining the optimal algorithm parameters.


Introduction
Group technology (GT) is a production philosophy that improves the proactivity of production by grouping the parts and productions based on their similarities in design and production process [1].Cellular manufacturing system (CMS) framework is one of the applications of GT.Some of the constraints of this system in real world condition are as follows: the maximum number of cells limitation, minimum cost of available parts between the machines in a cell and efficient usage of the machines in a cell [2].For drawing the problem near to real conditions, some system parameters like the time between two consecutive arrivals of the parts, the parts processing time, machines setup time, and the system demand in deferent periods can be considered as probabilistic and/or fuzzy [3].Queuing theory and stochastic programming will be used to model the problem, if the parameters are not considered as deterministic [4].In a real condition, some parts need the process in more than one cell that is called exceptional elements (EEs).One of the assumptions of CMS is that the parts cannot move between the cells, so for completing the production process of these parts using the sub-contractors is a useful policy [5].In this regard, some more detailed research has been conducted in this area [6 and 7].Some of the costs of this model are as follows: the cost of allocating the machines to the cells, imposed costs of EEs, cost of losses, t cost of machines failures, and so on.When a part needs to process by a specific demand that is not allocated to the cell, the part must process by a sub-contractor and this part impose the penalty cost to system [8,9].Some of the investigations that have been made on CMS with stochastic parameters and EEs after 1970 are presented in Table 1.In this paper, we identify the EEs I system and the sub-contractor that has been allocated to these parts.Also we used the continuous probability functions for system parameters so that we have a probabilistic CMS with EEs.We used queuing theory by introducing each part as a customer and each machine as a server in the system.This paper has five sections.In the second section, we present the mathematical model of the system.The third part deals with the solving method.A numerical example is presented in section four and the last part is devoted to conclusion and further studies.

Mathematical model
In the presented model in CMS, each machines is considered as a server with a queue of the customers (parts) waiting for service.The objective function is to minimize the imposed cost of sub-contractors.The processing time of each machine and the time between two consecutive arrivals have exponential p.d.f.So, we can consider an M/M/1 queuing model for this system and the service method is FIFO.

System assumptions
There is no opportunity for moving a part between the cells, The system policy is using sub-contractors for EES, All the cost parameters are deterministic, The operational requirements of the part are determined by the net requirement matrix, The DM expected occupation rate of each machine is pre-defined.

Nomenclatures
: P Total number of the parts,

Mathematical model
Considering the presented assumptions, the mathematical model of the system is as follows: . . .

Solving algorithm
As the CMS belongs to Np-Hard problems, we used the electromagnetic meta-heuristic problem for solving the presented model.Electromagnetic algorithm is a population-based problem that was presented by Birbil and Fang in 2003 [20].This algorithm has been used for solving periodic production planning [21], rout finding [22], single machine scheduling [23], limited non-linear optimization [24], facility location of re-adjustable production systems [25], and many other problems.This algorithm is based on attraction-repulsion mechanism to find the appropriate solution.This algorithm has four basic processes:  Algorithm initialization,  Local search for finding local optimal solution,  Calculation of the total power entered to each particle,  Moving in the direction of the entered power.
The pseudo-code of electromagnetic algorithm is presented as follows: F= CalcF() 6.

Parameter tuning
For parameter tuning of this algorithm, we used Taguchi method.The results of using this method are presented in Table 2.

Numerical example
In this section, a numerical example is solved.In this problem, we considered a system with 3 cells, 5 machines, and 10 different parts.The maximum number of machines that can be allocated to each cell is considered 2 and the available budget is 200.In Table 3, the matrix of part-machines requirement is presented.Table 4 contains the processing cost without using sub-contractor and the processing cost using sub-contractors is presented in Table 5.The costs of allocating machines to the cells are presented in Table 6.Table 7 contains the minimum occupation rate and processing rate of the machines.Finally Table 8 contains the input and output rate of the parts.The presented example was solved with electromagnetic algorithm and tuned parameters.The convergence trend is presented in Figure 1.Also the final results are presented in Tables 9 and 10.


The value of demand matrix using uniform p.d.f between 0, 1.


Other parameters:

Conclusion and further studies
In this paper a non-linear binary programing for optimizing total cost of sub-contractors in a CMS in a stochastic area has been developed.Because this problem belongs to Np-Hard problems we used electromagnetic meta-heuristic algorithm for solving the problem and tuned the algorithm parameters with Taguchi algorithm.Many different problems with different sizes have been solved and the results compared with each other's.For further studies adding some assumptions for changing in state-space diagram and the queuing model can be consider.Also this problem can be solved with other methods.

Figure 1 :
Figure 1: Convergence diagram of Electromagnetic algorithm

Table 1 :
Research on CMS with EEs after 1970

Table 2 :
The electromagnetic parameter levels in Taguchi method

Table 4 :
Processing cost without using sub-contractor

Table 5 :
Processing cost using sub-contractors

Table 6 :
The cost of allocating the machines to the cells

Table 7 :
Minimum occupation rate and processing rate of the machines

Table 8 :
Input and output rate of the parts

Table 11 .
Results for the examples with different sizes