Optimal Components Assignment Problem for Stochastic-Flow Networks 1

Corresponding Author: Abdellah Aissou IGEE, Université de M'hamed Bougara Boumerdes, Algeria Email: aaissou1969@gmail.com Abstract: Components assignment problem to maximize the network reliability in the case of each component has both an assignment cost and lead-time is never discussed. Therefore, this paper focuses on solving this problem under the above mentioned constraints. The presented problem is called an Optimal Components Assignment Problem (OCAP) and it is formulated based on three constraints namely total assignment cost, total lead-time and system reliability. Also, an approach based on a Random Weighted Genetic Algorithm (RWGA) is presented to solve the OCAP. The results revealed that an optimal components assignment leads to the maximum reliability, minimum assignment cost and minimum total leadtime using the proposed approach.


Introduction
Assignment Problem (AP) is one of the most important issues in manufacturing and service systems; it has also received significant attention among researchers.AP deals with the allocation of the various resources to the various activities on one to one basis.The generalized assignment problem is defined as the problem of finding the minimum cost assignment of n jobs to m agents such that each job is assigned to exactly one agent, subject to an agent's capacity Chu and Beasley (1997).They used a Genetic Algorithm (GA) based heuristic for solving such problems.As a special case of a generalized assignment problem, Wang (2002) has used a GA to solve the teacher assignments problem.The obtained results showed that the proposed technique can save a significant time spent on teacher assignments problem.Furthermore, Harper et al. (2005) used GA to solve the project assignment problem.Also, Lin and Yeh (2015) used GA to find the optimal two-class allocation of a computer network subject to a budget.
The CAP for a Stochastic-Flow Network (SFN) found by Lin and Yeh (2010), they proposed an efficient approach based on a GA to solve the resource assignment problem by finding the optimal resource assignment, which leads to maximal system reliability.The Component Assignment Problem (CAP) aims to find the optimal arrangement of n available components to m positions in a system to maximize the system reliability Lin and Yeh (2011a).In addition, Lin and Yeh (2011b) have solved the above problem as a multiobjective CAP.They proposed a two-stage approach to solving the multi-objective CAP subject to reliability and assignment cost for SFN.Furthermore, Chen (2014) addressed the optimal double resource assignment for the robust design problem, a minimum capacity assignment for each link and node is searched to keep the network working even both links and nodes are subject to failures.Lin and Yeh (2013) presented a new algorithm based on GA to find the optimal two class allocation subject to a budget leading to maximal system reliability of a computer network.Hassan (2015) discussed each component possessing a lead-time and proposed a GA to search the set of components that maximize the system reliability such that the total lead-time cannot exceed a specified amount.
The system reliability evaluation in the case of solving CAP under cost constraint is based on flow assignment for the assigned components given the demand Lin and Yeh (2013;Lin et al., 1995;Lin, 2001;2002).
In the case of considering each link has a lead-time and capacity, the transmission time of network paths can be evaluated and determined the quickest one (de Queiros et al., 1997;Chen and Hung, 1993;Chen and Chin, 1990).The system reliability of an SFN network is evaluated according to a given demand under the time constraints (Sedeno-Noda and Gonzalez-Barrera, 2014;Lin, 2003;2009a;2009b;2010;2011a;2011b;El Khadiri and Yeh, 2016).
GA is a heuristic search method used in artificial intelligence and computing, it uses techniques inspired from evolutionary biology such as selection, mutation, inheritance and recombination to solve a problem.Classical GA was implemented to solve reliability optimization design problems and the idea behind reliability optimization is to find the best way to increase the system reliability (Coit David and Smith, 1994;Berna et al., 1995;1997a;1997b;Fulya et al., 1998;Berna and Abbas, 2001).Some other researchers extended the idea of GA to evaluate the system reliability of a stochastic-flow network such as Younes and Hassan (2011) and Yeh et al. (2018).Also, GA is used to solve multiple objective multi-state reliability optimization design problems Taboada et al. (2008).Coello and Christiansen (2000) used GA as a tool to solve multiobjective optimization problems in structures and Deb et al. (2002) developed a new algorithm based on a GA called non-dominated sorting GA II which leads to better convergence near the true Pareto-optimal front.Azaron et al. (2009) have used the GA in solving a discrete reliability optimization problem.In addition, Lin (2016) proposed a new approach incorporating GA with fuzzy control and neural network to evaluate network reliability with the number of network nodes increases exponentially.
The CAP subject to reliability and the total cost studied by Lin and Yeh (2010;2011a;Yeh et al., 2018).The purpose was to maximize the reliability and minimize the total assignment cost.While, in Hassan (2015), the aim was to maximize the reliability and reduce the total lead-time of the assigned components.
While the CAP subject to system reliability, total assignment cost and total lead-time is never discussed in the previous literature.Therefore, this paper examines a multi-objective CAP problem for the case where each component has three attributes: An assignment cost, lead-time and system reliability.Such a problem is called OCAP.The main goal is to search the optimal components that maximize system reliability and minimize both the total assignment cost and the total lead-time.Furthermore, a multi-objective GA based on a Random Weighted GA (RWGA) is proposed to solve the presented problem.
The rest of this paper is organized as follows: Section 2 deals with the needed notations and section 3 presents the problem formulation.Next, Section 4 discusses the reliability evaluation under a time constraint.Section 5 explains the proposed multiobjective GA-based on RWGA.To demonstrate the usability of the proposed approach, several examples included in Section 6.Finally, Section 7 draws conclusions and future work.

L j
The lead-time of mp j. c(cp k ) The cost of the component cp k R d,T The system reliability to the demand d under time limit T.  Therefore, the mathematical programming formulation of the OCAP is given by: ( )

Problem Formulation
(2) , 1,2, , .= ∑ be the total lead-time of a MP j .Constraints (4) and ( 5) assert that each link should be given one component and each component can be assigned to at most one link.All feasible component assignments are generated using constraints (4) and ( 5).Constraint (6) states that the lead-time of the path MP j (L j ) is less than the time limit (T) Hassan (2015).
Since, the multi-objective components assignments problem in the case of a maximal and minimal objective transformed into either a multi-objective minimization problem or a multi-objective maximization problem (Konak et al., 2006;Murata and Ishibuchi, 1995;Murata et al., 1996).Therefore, the original problem formulation can be modified to be of the minimal type: , 1,2, , .

Reliability Evaluation
Each component cp k has a maximum capacity M k and capacity values of cp k vary from 0-M k .Also, the leadtime of component cp k is l(cp k ) and the system reliability of the candidate chromosome evaluated as follows: Step 1. Check if the candidate chromosome satisfies constraint (12).
Step 2. Use the procedure described by Lin, (2003), to generate the capacity vector X j corresponding to the path mp j .
Step 3. Calculate the network reliability of the chromosome: ∪ using the inclusion-exclusion rule given by Xue (1985).

Proposed Approach Based on RWGA
In this section, we develop an approach to solve the multi-objective optimization problem based on RWGA, which is used to determine the highest ranking solution to the problem.The initial inputs include data related to the components such as the lead-time, assignment cost, probability, the parameters of RWGA and the network topology.

Crossover, Mutation and Selection Operations
We use the modified uniform crossover and mutation presented in Hassan (2015) to generate new offspring.The crossover operation is described as follows: Given two parents, the new offspring is filled randomly by selecting genes from them.This crossover process showed in Fig. 1.We note that swap mutation is used to avoid duplicated genes in a chromosome.The mutation process showed in Fig. 2.

Determine Fitness
Let R d,T (i), C(B) and S l (i) be the corresponding values for the solution i, i = 1,2, …, ps: Step 1. Find the normalized values of R d,T , C(B) and S l as follows: Step 1.1.Normalized value for R d,T (i): Step 1.2.Normalized value for S i (i) : Step 2. Calculate the Fitness value for each solution as follows: Step 2.1.Generate a random number u k in [0,1] for each objective k, k = 1, 2 and 3.
Step 2.2.Calculate the random weight of each objective k as Step 2.3.Calculate the fitness of the solution as Step 3. Calculate the selection probability of each solution: The Entire Algorithm Based on RWGA Step 1. Set the population size (ps), the crossover rate (pc), the mutation rate (pm) and the number of generations (gn).

Illustrative Examples
In this section, we demonstrate the effectiveness of our proposed approach using examples taken from literature (six nodes network and Taiwan Academic Network (TANET)).The genetic parameters used in the proposed multi objective GA are ps = 10, Maxgen = 100, P c = 0.95 and P m = 0.05.The algorithm was iterated 10 times.Also, N E equals to 3, i.e., for each objective the algorithm search the best solution and stores it as a member of E.

Six-Node Network Example
The network has six nodes and 10 links (Fig. 3), as studied by Hassan (2015).The mps are as follows: , , , ,

Discussion and Comparison
This study is presented and solved the optimal components assignment in the case of coexistence of three conflicting objectives: System reliability, Cost and Total lead-time.Previous studies focused on studying two objectives such as Lin and Yeh, (2013); solved the CAP to maximize the system reliability under cost constraint.While in Hassan (2015) the CAP is studied subject to reliability and lead-time constraints.So, to assert the efficiency of our algorithm we will apply it to the CAP with two objectives; reliability and lead time.Accordingly, the optimization problem is formulated in the following minimization form: Minimize S S B = , 1,2, , Table 7 and 8 show the results of applying the proposed algorithm in comparison with Hassan (2015).The components information for network given in Fig. 3 and 4 are shown in Table 1 and 4 respectively.While the components information for the network of Fig. 5 is given in Table 6, Hassan (2015).

Conclusion
This paper explored how to determine the optimal assignment component for an SFN with maximal system reliability under a minimum of both assignment cost and total lead-time.A multi-objective components assignments problem subject to system reliability, assignment cost and total lead-time is presented and formulated as a multi-objective minimization problem.Furthermore, a multi-objective GA-based on RWGA approach is proposed to solve the presented problem.Using the proposed approach, the optimal solution is obtained i.e., the system reliability is maximized and both the total lead-time and the assignment cost are minimized.Future work may consider the OCAP for an SFN with multiple sources subject to multiple constraints.
Many real-life systems, such as computer network systems, manufacturing systems, electrical power systems and logistics systems are modelled as SFN to evaluate the reliability taking into account one or more constraints.However, this study presented a solution approach to maximize the reliability subject to assignment cost and total lead-time constraints and helping the decision makers to determine the optimal solution.

Let
CP = {cp k |1≤k≤cn} be a set of available components and B = (b 1 , b 2 ⋅⋅⋅,b m ) be the components assignment in which cp k is assigned to the arc a e if b e = k.The total lead-time and the total cost associated with a specified components assignment B are, respectively,

Fig. 1 :
Fig. 1: Modified uniform crossover Step 2. Generate the initial population including successful individuals B 1 , B 1 ,⋅⋅⋅ B ps .Step 3.For each individual, evaluate the network unreliability S 1 = 1-R d,T (B), total lead time S 2 = S l (B) and the total assignment cost S 3 = C(B) Step 4. Determine the set of dominated solutions E and the number of non-dominated solutions NE.Step 5. Calculate the Fitness value and the selection probability for each individual B in the current population as presented in section 5.2.Step 6. Select parents using the selection probabilities calculated in Step 5. Apply the GA operations (described in section 5.1) to generate new populations.Apply crossover on the selected parent pairs to create N offspring.Mutate offspring with a predefined mutation rate.Copy all offspring to Pt +1 and update E if necessary.Step 7. Randomly remove NE solutions from the new population and add the same number of solutions from E to NE. Step 8.If the stopping condition is not reached, set gn = gn +1 and go to Step 5. Otherwise, return to E. After obtaining a Pareto set, change the network unreliability to be the network reliability for each Pareto solution.

Fig. 5 :
Fig. 5: Computer network with 4 nodes and 6 arcs Table 1 lists the 20 components and associated information.Table 2 lists the candidate solutions found by the proposed approach for this network given d = 6 and T = 7. While, Table 3 concisely lists the best candidate solutions for given different values for d, T. i.e., Maximum value for R d , C(B) and minimum values for S l (B) and C(B).

Table 1 :
Arc capacity, probability, lead-time and cost for the 20 available components Capacity

Table 2 :
The candidate solutions to the network of Fig.3

Table 3 :
The best candidate solutions for different values for d,T.

Table 4 :
Component information Capacity

Table 5 :
The results of applying the proposed approach to the network given in Fig.4

Table 6 :
Components capacities, probabilities and lead-time for the network of Fig.5Capacity

Table 7 :
Comparison results for the Four-node network example Hassan's approach Proposed approach -

Table 8 :
Comparison results for the Six-node network example Hassan's approach Proposed approach -