COMPONENT ALLOCATION COST MINIMIZATION

JOURNAL OF INDUSTRIAL AND MANAGEMENT OPTIMIZATION Volume 12, Number 1, January 2016

doi:10.3934/jimo.2016.12.141 pp. 141–167

COMPONENT ALLOCATION COST MINIMIZATION FOR A MULTISTATE COMPUTER NETWORK SUBJECT TO A RELIABILITY THRESHOLD USING TABU SEARCH

Cheng-Ta Yeh Department of Business Administration Shih Hsin University Taipei 106, Taiwan

Yi-Kuei Lin Department of Industrial Management National Taiwan University of Science and Technology Taipei 106, Taiwan

Abstract. From the perspective of business management, system supervisors are usually more concerned with the cost of a system rather than its reliability. This study determines the optimal component allocation based on the cost criterion for a computer system subject to a reliability threshold in which the computer system is represented as a network composed of a set of links and a set of vertices. The component allocation means allocating some from the set of components to the network’s links, where the cost of allocating a component is counted in terms of the length. Any computer network associated with a component allocation is called a multistate computer network (MCN) because each component has multiple states with a probability distribution. Associated with a component allocation, the system reliability is the probability that the specific units of data are successfully transmitted through the MCN. An optimization algorithm, which integrates tabu search and minimal paths, is proposed to solve the problem under consideration. Several benchmark computer networks are utilized to demonstrate the computational efficiency of the proposed algorithm compared with several popular meta-heuristic algorithms.

1. Introduction. In a modern society, when a computer system fails, a business activity may not operate efficiently and the company involved may suffer a great loss in revenue. Hence, for computer systems, optimization problems related to system reliability are highly relevant to many system supervisors. The problems are usually solved via modeling systems networks, each of which consists of several links and vertices. Lisnianski and Levitin [28] categorized the optimization problems related to system reliability into two types, namely achieving the maximal system reliability subject to various constraints (e.g. the cost constraint) and minimizing the resources needed for providing a specific system reliability level. Many researchers 2010 Mathematics Subject Classification. Primary: 90B15, 68T20; Secondary: 60K10. Key words and phrases. Multistate computer network, cost minimization, reliability threshold, tabu search, minimal paths. This work was supported in part by the National Science Council, Taiwan, Republic of China, under grant no. NSC 101-2811-E-011-013-MY3. The reviewing process of the paper was handled by Ryan Loxton as a Guest Editor.

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[7, 8, 31, 32, 33, 34] have studied the network topology design for maximizing the reliability or minimizing the cost under various constraints. Some of them [7, 8, 31, 34] regarded the systems as binary-state networks, that is, the links/vertices are either working or nor working. In these studies, the system reliability is the probability that the network can successfully connect the source vertex/vertices and sink vertex/vertices. The others [32, 33] modeled the systems as capacity-related networks, where each vertex/link owns two levels, namely zero and a positive integer. The system reliability means the probability that the network can successfully satisfy the given demand at the sink vertex/vertices. Many real-life systems should be regarded as multistate flow networks since each link/vertex has several states with a probability distribution. For instance, in a computer network, a link denotes a transmission line which combines with several physical lines such as optical fibers or coaxial cables. Each physical line has a capacity and may fail. In the multistate flow network, the system reliability is defined as the probability that a specified unit for a given demand can be successfully transmitted from a source vertex to a sink vertex. The studies [12, 13, 14, 15] focused on how to allocate the various commodities at multiple source vertices to maximize the system reliability. Liu et al. [29] determined the optimal flow assignment with maximal system reliability and minimal transmission cost. Xu et al. [35] designed a transportation flow assignment model with stochastic chanceconstraint programming based on the reliability of transportation time, and then developed a hybrid intelligent algorithm that integrates a genetic algorithm (GA), stochastic simulation, and neural network. Lin and Yeh [24, 25, 26] addressed the system reliability maximization problem about finding the optimal component allocation to the networks including computer networks, logistics networks, and electronic power networks in which the components, such as transmission lines or carrier’s available containers, are multistate, and the network topologies are fixed. Furthermore, Lin and Yeh [27] considered an extended problem about the tradeoff between the system reliability maximization and the allocation cost minimization for the multistate computer network (MCN) which is a typical multistate flow network, and proposed a multi-objective GA to solve it in which the allocation cost is the sum of the costs of allocating the components to the network. Although Lin and Yeh [24, 25, 26, 27] have solved the component problem for maximizing the system reliability under a budget and the extended multi-objective problem, the component allocation problem of minimizing the allocation cost subject to a reliability threshold for a multistate flow network has yet to be addressed. From the viewpoint of business management, cost is the significant factor that usually concerns supervisors. For instance, most components (e.g. transmission lines or devices) meet the high reliability requirement in the modern life, and thus supervisors emphasize the importance of the allocation cost rather than the reliability. Thus, they tend to minimize the allocation cost to improve the system reliability in order to achieve a specified level. In this paper, we focus on solving the allocation cost minimization problem related to an MCN where a component represents a transmission line. The proposed problem is also NP-hard because it involves both NP-hard problems of determining the optimal allocation and evaluating the system reliability [5, 6] Hence, the meta-heuristic algorithms, such as GA, ant colony optimization (ACO), particle swarm optimization (PSO), tabu search (TS), and simulated annealing (SA), are suitable to solve the proposed problem. Our study proposes a TS-based approach in which the system reliability is evaluated in terms

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of minimal paths (MPs). The proposed TS is based on the structure from Glover and Laguna [11]. Until now, many researchers [1, 2, 3, 20] have adopted the Glover and Laguna’s structure to solve different kinds of allocation problems. The major advantage of Glover and Laguna’s TS is that it has powerful local search capability [11, 37]. Therefore, this study focuses on strengthening the global search of TS by adopting the intensification and diversification strategies and driving them according to the current convergent times in which the global search is used to explore different solution areas. Such a mechanism can improve the efficiency of TS in searching for the best solution. In the proposed problem, an infeasible solution means that it does not satisfy the reliability threshold. Since the reliability constraint is very difficult to consider in solution encoding, the infeasible solution may be generated while executing the proposed TS. Hence, a penalty function based on the dead penalty approach is proposed to deal with the infeasible solutions, where the death penalty approach is based on the rejection of infeasible solutions and is also an easy-to-use and efficient approach [36]. The proposed penalty function can worsen an infeasible solution’s objective value so that the infeasible solution has worse objective value than any feasible solution. In order to show the computational efficiency for the proposed TS, four benchmark computer networks are utilized to compare it with several popular metaheuristic algorithms. In addition, the best solution found by the TS is also compared with the exact solution. The remainder of this paper is organized as follows. The assumptions and problem formulation are described in Section 2. The MCN model and reliability evaluation algorithm (REA) are developed in Section 3. The proposed TS for solving the proposed problem is illustrated in Section 4. Numerical experiments of several simple and benchmark networks are performed to compare TS, GA, PSO, ACO, and SA in Section 5. Conclusions and discussion are finally drawn in Section 6. Acronyms and notations. MCN MP REA TS GA PSO ACO SA RSDP IEA TANET NSFNET NTD A; n V (A, V) O, D ai B; z

multistate computer network minimal path reliability evaluation algorithm tabu search genetic algorithm particle swarm optimization ant colony optimization simulation annealing recursive sum of disjoint products implicit enumeration approach Taiwan academic network National Science Foundation Network new Taiwan dollar set of links connecting a pair of vertices; number of links set of vertices computer network unique source vertex; unique sink vertex link #i for i = 1, 2, . . . , n set of components; number of components

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bt Mt Ktw ´t K ct yi Y CY xi X MY UY V (X) D Rd (Y ) XY RT r Pj fj F FM Y FX ΩY Ψ Ψmin τ ϕ π λ0 λ Γ Γ0 ε

CHENG-TA YEH AND YI-KUEI LIN

component #t for t = 1, 2, . . . , z number of states that component bt owns, t = 1, 2, . . . , z capacity #w of component bt , w = 1, 2, . . . , Mt where Kt1 = 0 maximal capacity of component bt cost per unit of length of component bt the index of the component allocated to link ai , i = 1, 2, . . . , n (y1 , y2 , . . . , yn ): a component allocation or a solution allocation cost associated with Y current capacity of ai , i = 1, 2, . . . , n (x1 , x2 , . . . , xn ): (current) capacity vector ´y , K ´y , . . . , K ´ y ): maximal capacity vector associated with Y (K 1 2 n set of all feasible X associated with the component allocation Y maximal flow of (A, V) under X demand at the destination vertex D system reliability according to the component allocation Y set of X ∈ UY satisfying the demand d reliability threshold number of MPs MP #j, j = 1, 2, . . . , r amount of flow through Pj , j = 1, 2, . . . , r (f1 , f2 , . . . , fr ): flow vector set of all feasible F under MY set of all feasible F under X set of all F ∈ FMY meeting exact demand d set of all capacity vectors transformed from all F ∈ ΩY set of all minimal elements in Ψ iteration #τ candidate list size tabu tenure current convergent times coefficient of determination which an integer value to judge whether the intensification and diversification strategies should be executed terminal time executional time penalty value

2. Assumptions and problem formulation. In this section, with a fixed computer network topology, we introduce notations and assumptions, and then define the allocation cost, system reliability, and problem formulation. 2.1. Notations and basic assumptions. Let A = {ai |1 ≤ i ≤ n} be a set of links and V be a set of vertices, where ai denotes the ith link and V includes the origin vertex O and destination vertex D. Thus, a computer network is represented as (A, V). The length of ai is denoted by li for i = 1, 2, . . . , n. Let B = {bt |t = 1, 2, . . . , z} be the set of z components, where each component bt has multiple states, 1, 2, . . . , Mt , with corresponding capacities, 0 = Kt1 < Kt2 < . . . < KtMt = ´ t according to a given probability distribution. In particular, Ktw is the wth K capacity of bt for w = 1, 2, . . . , Mt . The cost per unit of length of component bt is denoted by ct for t = 1, 2, . . . , z. A component allocation is represented as a

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vector Y = (y1 , y2 , . . . , yn ) where yi = t when component bt is allocated to ai for i = 1, 2, . . . , n. The computer network (A, V) associated with the allocation Y is thus an MCN. Several assumptions and all notations are proposed as follows: I. Flow in (A, V) must satisfy the flow-conservation law [10]. That is, no flow will reduce or be increased during the transmission. II. Each link must contain exactly one component. III. No component can be allocated to more than one link. IV. The capacities of different components are statistically independent. 2.2. Problem formulation. The cost for allocating component bt to link ai is counted in terms of length li , i.e., allocating cost is cyi · li if yi = t. Therefore, the allocation cost associated with component allocation Y , denoted by CY , is CY =

n X

(cyi · li ).

(1)

i=1

A capacity vector denoted by X = (x1 , x2 , . . . , xn ) shows the current state of computer network (A, V), where xi represents the current capacity of link ai for i = 1, 2, . . . , n. The capacity vector X is not deterministic because xi is a ran´ yi for i = 1, 2, . . . , n. Let dom variable taking the values Kyi 1 , Kyi 2 , . . ., and K ´ ´ ´ MY = (Ky1 , Ky2 , . . . , Kyn ) be the maximal capacity vector associated with Y , where ´y = K ´ ρ is the maximal capacity of aπ if the component bρ is allocated to aπ . K π Any capacity vector X associated with Y must satisfy the following constraint: X ≤ MY .

(2)

Constraint (2) indicates that the current capacity xi cannot exceed the maximal ´ y for each ai . Let UY be the set of X satisfying constraint (2) associated capacity K i with Y and V (X) be the maximal flow of (A, V) under X, where the maximal flow means the largest amount of data that the computer network (A, V) can transmit from O to D under X. The system reliability associated with Y is the probability that the maximal flow of the network is not less than d and is denoted by Rd (Y ) = Pr{X|V (X) ≥ d, X ∈ UY }. For convenience, the set {X|V (X) ≥ d, X ∈ UY } is designated as XY and thus Rd (Y ) = Pr{X|X ∈ XY }. A reliability threshold is denoted by RT. The mathematical programming formulation of the proposed problem is thus given as follows: Minimize CY =

n X

(cyi · li )

(3)

i=1

Subject to yi = t yi 6= yj

t ∈ {1, 2, . . . , z} ∀i = 1, 2, . . . , n, ∀i 6= j, and

Rd (Y ) ≥ RT.

(4) (5) (6)

Constraints (4) and (5) depend on assumptions II and III. Constraint (6) expresses the condition that the system reliability associated with Y should be larger than or equal to the reliability threshold RT. All feasible component allocations are generated through constraints (4)–(6). The optimal component allocation Y with minimal allocation cost CY is obtained via minimizing objective function (3).

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3. Multistate computer network model. The system reliability has been defined in the previous section. Subsequently, the MCN model and an algorithm to evaluate the system reliability Rd (Y ) are illustrated in this section. Definition 1. A path is an ordered set of links that connects O and D. Definition 2. An MP is a path without any cycle. 3.1. Relationship between capacity and flow. Suppose there are a total of r MPs in (A, V): P1 , P2 , . . . , Pr . A flow vector F = (f1 , f2 , . . . , fr ) represents the flow transmission through the network, where f j is the flow traveling through Pj . Any F is feasible under MY if and only if X ´ y for each i = 1, 2, . . . , n, fj ≤ K (7) i j:ai ∈Pj

where

P

fj is the total flow traveling through ai , and constraint (7) says that

j:ai ∈Pj

the total flow cannot exceed the maximal capacity of ai . For convenience, let FM Y be the set of F feasible under constraint (7). Similarly, any F satisfying the following constraint is said to be feasible under X ∈ UY (i.e., F ∈ FX ), X fj ≤ xi i = 1, 2, . . . , n. (8) j:ai ∈Pj

Therefore, the maximal flow of (A, V) under X is represented as V (X) = ( ) r P max fj |F ∈ FX . j=1

3.2. System reliability evaluation. For demand d, the system reliability associated with Y is denoted by Rd (Y ) = Pr{X|X ∈ XY }. However, it is inefficient to evaluate Rd (Y ) by enumerating all X ∈ XY and then summing up their probabilities [21, 22]. Instead, a minimal capacity vector for demand d, called a d-MV, is utilized to increase the computational efficiency for the system reliability evaluation and defined as follows: Rule 3.1. X ≤ H: (x1 , x2 , . . . , xn ) ≤ (h1 , h2 , . . . , hn ) if and only if xi ≤ hi for each i. Rule 3.2. X < H: (x1 , x2 , . . . , xn ) < (h1 , h2 , . . . , hn ) if and only if X ≤ H and xi < hi for at least one i. Definition 3. The d-MV is a minimal capacity vector X in XY such that H ∈ / XY for any capacity vector H with H < X. Suppose there are q d-MVs: X1 , X2 , . . . , Xq. Then, the set XY can be  repreq S sented as a union of sets in terms of d-MVs, i.e., {X|X ≥ Xi , X ∈ UY } . Thus, i=1   q S the system reliability is equal to the probability Pr {X|X ≥ Xi , X ∈ UY } . i=1

Such a probability can be calculated by using the inclusion-exclusion principle [22], the state-space decomposition [15], or the recursive sum of disjoint products (RSDP) [25, 27, 38]. Since the study [38] has shown that the RSDP approach has better computational efficiency than  q the others, especially for  larger networks, this paper S utilizes it to calculate Pr {X|X ≥ Xi , X ∈ UY } . i=1

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3.3. Search for all d-MVs. Any feasible F is said to satisfy the exact demand if F ∈ FMY meets the following constraint: r X

fj = d.

(9)

j=1

For convenience, let ΩY be the set of F ∈ FMY satisfying constraint (9). The following lemma shows a necessary condition for a d-MV. Lemma 1. If X ∈ UY is a d-MV, then there exists an F ∈ ΩY such that  P  if fj = 0 0 j:ai ∈Pj xi = , P  fj > Kyi (w−1) , w ∈ {2, 3, . . . , Myi } Kyi w if Kyi w ≥ j:ai ∈Pj

for i = 1, 2, . . . , n.

(10)

Proof. To explain this necessary condition of d-MV, we consider a d-MV X and an P F ∈ FX satisfying d, and illustrate case (i) xi = 0 if fj = 0 and case (ii) j:ai ∈Pj P xi = Kyi w if Kiw ≥ fj > Kyi (w−1) , w ∈ {2, 3, . . . , Myi }, respectively. j:ai ∈Pj

(i) Suppose that there exists a link aµ such that xµ >

P

fj = 0. Set H =

j:aµ ∈Pj

(h1 , h2 , . . . , hn ) where hµ = 0Pand hi = xi ∀i 6= µ. Thus, H < X and F is feasible under H due to fj ≤ hi for i = 1, 2, . . . , n. This indicates j:ai ∈Pj

H ∈ XY and contradicts that X is a d-MV. (ii) Suppose that there exists a link aτ such that xτ > Kyτ ε ≥

P

fj >

j:ai ∈Pj

Kyτ (ε−1) , ε 6= 1. Set H = (h1 , h2 , . . . , hn ) where hτ = Kyτ ε and Phi = xi ∀i 6= τ . That is, H < X and F must be feasible under F due to fj ≤ hi for j:ai ∈Pj

i = 1, 2, . . . , n. This indicates H ∈ XY and contradicts that X is a d-MV. In accordance with Lemma 1, any capacity vector X transformed from F ∈ ΩY through Eq. (10) is guaranteed to satisfy d, i.e., V (X) ≥ d, and can be treated as a d-MV candidate. For convenience, let Ψ be the set of such candidates. Furthermore, let Ψmin be the set of all minimal elements in Ψ. That is, there exists no capacity vector H ∈ Ψ such thatX ∈ Ψmin and H < X. The following lemma further presents that the set of d-MVX is equivalent to the set Ψmin . Lemma 2. The set of d-MV X is equivalent to the set Ψmin . Proof. Suppose X ∈ Ψ is a d-MV but X ∈ / Ψmin . Then, there exists an H ∈ Ψmin such that H < X. It is given that V (H) ≥ d (i.e., H ∈ XY ) which contradicts X is a d-MV. Hence, a d-MV X is an element in Ψmin . Conversely, suppose X ∈ Ψmin (that means V (X) ≥ d according to Lemma 1) but not a d-MV (that is, it is not a minimal element in XY ). Then, there exists a d-MV H (i.e., H is a minimal element in XY ) such that H < X. This implies that H ∈ Ψ which contradicts X ∈ Ψmin . Thus, X ∈ Ψmin is a d-MV. Based on Lemma 2, a comparison approach shown as follows is adopted to find out all d-MVs from the set Ψ.

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The comparison approach. //Find out all d-MVs from the set of the d-MV candidates. Step 1. S = ∅ (S is the stack which stores the index of each non-d-MV. Initially, S is empty.) Step 2. For i = 1 to v&i ∈ / S. // there are v d-MV candidates. Step 3. For j = i + 1 to v, &j ∈ / S. Step 4. If Xj < Xi , then Xi is not a d-MV, S = S ∪ {i}, and go to step 7. Else if Xj ≥ Xi , thenXj is not a d-MV, S = S ∪ {j}. Step 5. Next j. Step 6. Xi is a d-MV. Step 7. Next i. Step 8. End 3.4. The reliability evaluation algorithm. Based on the proposed MCN model, an algorithm is developed to evaluate the system reliability as follows. Reliability evaluation algorithm (REA). //calculate the system reliability for Rd (Y ). Step 1. Find all F satisfying the following constraints: X ´ yi for each i = 1, 2, . . . , n, and fj ≤ K (11) j:ai ∈Pj r X

fj = d.

(12)

j=1

If no feasible F exists, then Rd (Y ) = 0 and terminate the algorithm. Step 2. Transform each F into X via the following equation: X   if fj = 0 0  j:ai ∈Pj X , xi =  fj > Kyi (w−1) , w ∈ {2, 3, . . . , Myi }  Kyi w if Kyi w ≥ j:ai ∈Pj

for i = 1, 2, . . . , n.

(13)

All X transformed from this step are d-MV candidates. Step 3. Utilize the comparison approach to obtain all d-MVs from the set of d-MV candidates. Step 4. Suppose X1 , X2 , . . . , Xq are  qq d-MVs from Step 3. Use  the RSDP approach S to calculate Rd (Y ) = Pr {X|X ≥ Xi , X ∈ UY } . i=1

4. Tabu search development for finding the optimal component allocation. The proposed TS, based on Glover and Laguna’s version [11], utilizes flexible memory structures including short-term-memory and long-term-memory. The short-term-memory mechanism consisting of tabu list and aspiration criterion prevents the proposed TS from falling into local optima, and the long-term-memory mechanism including intensification strategy and diversification strategy reinforces the attractive solution in short-term-memory and then diversifies the search direction to drive it into new regions. Thus the proposed TS can avoid searching for the recently-visited solutions, and then guide the search towards new and promising areas.

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4.1. Solution encoding. The solution encoding needs to consider the constraints of the proposed problem as far as possible. Since a solution for the proposed problem means a component allocation, the notation Y continues to be utilized in the proposed TS but is modified to be Y τ = (y1τ , y2τ , . . . , ynτ ) by adding a superscript τ , where Y τ is the current solution at the τ th iteration. For example, Figure 1 shows a solution for the network with 6 links. The term, y110 = 2, signifies that the component b2 is allocated to link a1 at the 10th iteration.

Figure 1. The solution encoding 4.2. Search space and neighborhood structure. Search space or solution space is the space consisting of all potential solutions to the problem. In the search space, the way of finding another solution through a specified solution is called ‘move’. Thus, a neighbor of the current solution is obtained from performing a move, and the set of all neighbor solutions from the current solution is called neighborhood. When a current solution Y τ is created, the proposed TS will explore the possible moves to obtain the neighbor solutions of the current one. Glover and Laguna [11] proposed several move operators: swap, insert, and add/drop. This study adopts the swap operator to generate the neighbor solutions of the current solution because it is the most common and flexible operator [16, 18, 19]. Figure 2 illustrates the swap operator. The swap operator randomly selects two links and exchanges their allocated components with each other.

Figure 2. The swap operator A candidate list is utilized to store the possible neighbor solutions from the current one, whose size means the maximal number of neighbor solutions in the candidate list and is denoted by ϕ. Hence, the swap operator is continuously executed until ϕ possible neighbor solutions are created. Subsequently, the best neighbor solution with the minimal cost in the candidate list is chosen to be the new current solution Y τ +1 to re-initiate the search procedure at the next iteration τ + 1. Such a move resulting in the best neighbor solution is the best move for the current solution Y τ. 4.3. Tabu list and tabu tenure. A tabu list is established to forbid the recent moves for the next specified number of iterations for preventing a cycling or reversed move. The best move of Y τ will be forbidden for each τ . If the best move of Y τ has been recorded in the tabu list, the second best neighbor solution whose corresponding move is not recorded in the tabu list will be regarded as the new current solution Y τ +1 . Moreover, the size of a tabu list, called tabu tenure, is the number of moves recorded in the tabu list and is denoted by π. At each iteration, a

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new move is added to the tabu list, and the oldest move in the list is dropped while the list is full. 4.4. Aspiration criterion. The aspiration criterion is an override condition that may allow the tabu restriction to be violated. In the proposed TS, a move value is defined as the allocation cost of a neighbor solution from the current solution. Suppose the best move of the current solution has been recorded in the tabu list, but the corresponding move value is better than the other move values whose corresponding moves are in the tabu list. Then, such a move will be recorded as a new forbidden one and its corresponding neighbor solution will be the next current solution. 4.5. Intensification and diversification strategies. In this study, the intensification strategy is firstly executed to store an elite solution, where the elite solution is equivalent to the best solution at the current iteration. Subsequently, the diversification strategy is executed to change the elite solution. Figure 3 illustrates how to change the elite solution. One position of the elite solution will be chosen randomly to change its value into another value which is different with the other positions’. Then, the revised elite solution will replace the current solution to re-initiate at the next iteration.

Figure 3. The process of diversification Whether the intensification and diversification strategies should be executed is according to the current convergent times (denoted by λ0 ). If the best solution doesn’t get any improvement while compared with the previous one, set λ0 = λ0 +1. Otherwise, set λ0 = 0. When the current convergent time reaches the value λ, both strategies are executed. 4.6. Penalty function to deal with infeasible solutions. A penalty function is proposed to increase an infeasible solution’s allocation cost to be larger than that of any feasible solution. The penalty function proposed to deal with an infeasible solution Y is shown as follows: CˆY = [CY × (RT − Rd (Y ))] × ε, (14) where the term (RT − Rd (Y )) is the difference between the reliability threshold RT and the infeasible solution’s system reliability Rd (Y ), and is utilized to control the level for each infeasible solution. The term ε denotes a penalty value to make the output CˆY as large as possible, and thus must be large enough (e.g. ε = 108 ) such that the output CˆY is always larger than the cost CY as well as any feasible solution’s allocation cost. Furthermore, the proposed penalty function has the capability to rank the infeasible solutions’ quality levels according to their allocation costs and system reliabilities. According to the penalty function, the infeasible solution with smaller system reliability has the greater CˆY than the one with larger system reliability if the two solutions’ costs have no significant difference. For instance, Y1 and Y2 are the total solutions in the candidate list and both are infeasible. If CY1 =

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100, CY2 = 95, Rd (Y1 ) = 0.8, Rd (Y2 ) = 0.6, ε = 1, 000, and RT = 0.9, then CˆY1 = [100 × (0.9 − 0.8)] × 1, 000 = 10, 000 and CˆY2 = [95 × (0.9 − 0.6)] × 1, 000 = 28, 500. Thus, the infeasible solution Y1 will be adopted to find the neighbor solutions.

Figure 4. The TS Process

4.7. The process of tabu search. The following process describes how to use the proposed TS to solve the proposed problem (see Figure 4) in which the TS is repeated until the executional time |itΓ0 reaches the terminal time Γ .

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Step 0. (Initialization) Input the problem parameters: A, {Pj |1 ≤ j ≤ r}, d, RT, and B, and TS parameters: ϕ, π, λ, ε, and Γ . Set τ = 1 and λ0 = 0. Then, start the proposed TS. Step 1. Create an initial solution Y 1 randomly. 1.1) Calculate the allocation cost via equation (1) and use the REA to evaluate the system reliability. 1.2) If the system reliability Rd (Y 1 ) is less than RT, adopt the penalty function to re-calculate the allocation cost. 1.3) Set the initial solution as the best solution. Step 2. Use the current solution Y τ to start (or re-initiate) the neighborhood search. 2.1) Use the swap operator to generate the neighbor solutions of the current solutions for producing a candidate list. 2.2) Calculate the allocation cost via equation (1) and use the REA to evaluate the system reliability for each neighbor solution. 2.3) For each neighbor solution, if the system reliability is less than RT, then adopt the penalty function to re-calculate the allocation cost. 2.4) Find the best neighbor solution with minimal allocation cost from the set of the neighbor solutions. 2.5) Update the tabu list. (If the best move is one of the moves in the tabu list, then apply the aspiration criterion.) Step 3. If the best neighbor solution is better than the current best solution, then set the best neighbor solution as the best solution, λ0 = 0. Otherwise, set λ0 = λ0 + 1. Step 4. If the executional time Γ0 is more than or equal to the terminal time Γ , then stop this algorithm and output the best solution (the best component allocation with minimal allocation cost). Otherwise, set τ = τ + 1 and continue this algorithm. Step 5. If λ0 = λ, then execute the intensification and diversification strategies to create an elite solution and then set it as the current solution. Otherwise, set the best neighbor solution as the current solution. Then, go to Step 2. 5. Numerical experiments. In this section, the proposed TS is applied to five illustrative networks including a simple computer network, ARPA, OCT, TANET, and NSFNET. For the simple computer network, there are 10 components ready to be allocated, and the TS is compared with the implicit enumeration approach (IEA) to show that it has the capability to find the exact solution. In other cases, there are 80 components ready to be allocated. We will discuss the setting of the TS parameters and compare TS with GA, PSO, ACO, and SA through 4 benchmark networks. In all cases, each component ready to be allocated is composed of several OC36 (Optical Carrier 36) or OC-18 (Optical Carrier 18) lines. Each OC-36 (resp. OC-18) line owns both capacities of approximate 2 (resp. 1) Gbps (giga bits per second) as it operates and 0 Gbps as it fails. For example, a component combining two OC-36 lines has three capacities, 0 Gbps, 2 Gbps, and 4 Gbps, and follows a binomial distribution where each OC-36 line operates with a probability of 0.9. The component provides 0 Gbps when both two OC-36 lines are failed, and the corresponding probability of the state is C02 (0.9)0 (0.1)2 = 0.01. Similarly, the component provides 2 Gbps with probability C12 (0.9)1 (0.1)1 = 0.18, and 4 Gbps with probability C22 (0.9)2 (0.1)0 = 0.81. Generally, each component has a different probability

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Table 1. Length data of Figure 5 Link: ai Length (kilometer): li

a1 15

a2 25

a3 35

a4 10

a5 20

Table 2. Probability distributions of capacities for 10 components Component Cost (NTD) bt ct 0 1 100 0.005929 2 80 0.003 3 65 0.000729 4 45 0.000857375 5 60 0.0001 6 30 0.001 7 15 0.097 8 105 0.000001 9 85 0.000000343 10 70 0.001 a. The component does not provide this

Capacity (Gbps) 1 2 3 0a 0.142142 0 0 0.997 0 0.022113 0.223587 0.753571 0.024503 0.233422 0.741218 0.0198 0.9801 0 0.027 0.243 0.729 0 0.903 0 0.000297 0.029403 0.970299 0.000146 0.020707 0.979147 0 0.999 0 capacity.

4 0.851929 0 0 0 0 0 0 0 0 0

distribution due to the different manufacturers. For the operating environment, all algorithms are programmed in MATLAB programming language and operated on a personal computer with Intel Core i7-2600 CPU 3.4GHz and 4GB RAM.

Figure 5. Simple computer network 5.1. Simple computer network. A simple computer network with 5 links is presented in Figure 5. There are 4 MPs: P1 = {a1 , a2 }, P2 = {a1 , a3 , a5 }, P3 = {a2 , a3 , a4 } and P4 = {a4 , a5 }. The length of each link is listed in Table 1. The data of 10 components are shown in Table 2. At first, IEA is executed to find the optimal component allocation with minimal allocation cost subject to several combinations of d and RT : (i) d = 4 and RT = 0.98, (ii) d = 5 and RT = 0.95, and (iii) d = 6 and RT = 0.7. Subsequently, the TS is executed and will stop until the obtained solution is the same with the exact solution from IEA. Table 3 shows

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Table 3. The comparison among the meta-heuristic algorithms through Figure 5 Demand,

Minimal Optimal allocationa

Reliability Rd (Y )b

IEA

TS

Reliability allocation CPU timec CPU time threshold costa d = 4Gb, 3925 (5, 6, 7, 9, 4) 0.982381 27.614 0.404 TR = 0.98 d = 5Gb, 5525 (1, 6, 7, 8, 9) 0.956207 20.709 0.582 TR = 0.95 d = 6Gb, 5600 (9, 6, 7, 8, 1) 0.767647 19.631 0.468 TR = 0.7 a. Each optimal component allocation is obtained by IEA. b. Each value is the system reliability of the corresponding optimal component allocation. c. Unit: second the TS can find the exact solution in a shorter period of time (within 1 sec) than IEA. Through this simple case, the proposed TS has the capability to find the exact solution for the proposed problem.

Figure 6. ARPA network [27] 5.2. Discussion on the setting of TS parameters using four benchmark networks. The major influence on the proposed TS’s computational efficiency is the setting of several parameters including candidate list size (ϕ), tabu tenure (π), and coefficient of determination (λ). Therefore, this subsection focuses on discussing the setting of them through 4 computer networks including TANET, NSFNET, ARPA, and OCT. (a) The ARPA network is composed of 9 links, 4 vertices, and 13 MPs (see Figure 6) [27] where its link data are listed in Table 4(a). The settings of demand level and reliability threshold for the ARPA network are given as d = 5 GB and RT = 0.95. (b) The OCT network is composed of 29 links, 24 vertices, and 9 MPs (see Figure 7) [2] where its link data are listed in Table 4(b). The settings of demand level and reliability threshold for the OCT network are given as d = 5 GB and RT = 0.9.

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Table 4. The length data of ARPA, OCT, TANET, and NSFNET (a) The length of each link in Figure 6 link: ai 1 2 3

Length (km): li 24 18 20

link: ai 4 5 6

Length (km): li 26 37 26

link: ai 7 8 9

Length (km): li 24 18 20

(b) The length of each link in Figure 7 link: ai 1 2 3 4 5 6 7 8 9 10

Length (km): li 30 8 10 11 12 12 15 13 14 27

link: ai 11 12 13 14 15 16 17 18 19 20

Length (km): li 19 26 16 17 17 16 16 17 17 16

link: ai 21 22 23 24 25 26 27 28 29

Length (km): li 31 27 14 25 10 16 20 32 23

(c) The length of each link in Figure 8 link: ai 1 2 3 4 5 6 7 8 9 10

Length (km): li 41 47 58 52 33 35 43 36 53 22

link: ai 11 12 13 14 15 16 17 18 19 20

Length (km): li 20 37 14 93 97 59 78 26 72 24

link: ai 21 22 23 24 25 26 27 28 29 30

Length (km): li 13 54 33 54 46 113 102 91 35 53

(d) The length of each link in Figure 9 link: ai 1 2 3 4 5 6 7 8

Length (km): li 996 1547 1123 825 780 796 633 782

link: ai 9 10 11 12 13 14 15 16

Length (km): li 248 876 1985 286 1297 1683 1184 307

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Table 5. Probability distribution and allocation cost of capacities for 80 components Comp- Cost onent (NTD) bt ct 0 1 100 0.0004 2 50 0.000512 3 65 0.000343 4 80 0.015 5 70 0.0016 6 135 0.005929 7 60 0.003 8 35 0.007225 9 35 0.005929 10 80 0.003 11 55 0.034 12 40 0.0036 13 110 0.000001 14 65 0.000784 15 70 0.000225 16 15 0.095 17 35 0.005776 18 75 0.000625 19 40 0.000729 20 35 0.001 21 45 0.000512 22 30 0.004225 23 85 0.005929 24 70 0.003 25 55 0.000216 26 30 0.034 27 55 0.000512 28 60 0.000343 29 35 0.001 30 85 0.0009 31 60 0.002809 32 70 0.000166375 33 80 0.000125 34 140 0.0001 35 10 0.025 36 60 0.024 37 75 0.000125 38 85 0.000110592 39 100 0.0001 40 60 0.001849

Capacity (Gbps) 1 2 0.0392 0.9604 0.017664 0.203136 0.013671 0.181629 0.985 0 0.0768 0.9216 0 0.142142 0 0.997 0 0.15555 0 0.142142 0.997 0 0.966 0 0.1128 0.8836 0.000297 0.029403 0.054432 0.944784 0.02955 0.970225 0.905 0 0.140448 0.853776 0.04875 0.950625 0.022113 0.223587 0.027 0.243 0.017664 0.203136 0.12155 0.874225 0 0.142142 0 0.997 0.010152 0.159048 0.966 0 0.017664 0.203136 0.013671 0.181629 0.027 0.243 0.0582 0.9409 0.100382 0.896809 0.008575875 0.147349125 0.007125 0.135375 0.0198 0.9801 0.975 0 0.976 0 0.007125 0.135375 0.006580224 0.130507776 0 0.0198 0 0.082302

3 0 0.778688 0.804357 0 0 0 0 0 0 0 0 0 0.970299 0 0 0 0 0 0.753571 0.729 0.778688 0 0 0 0.830584 0 0.778688 0.804357 0.729 0 0 0.843908625 0.857375 0 0 0 0.857375 0.862801408 0 0

4 0 0 0 0 0 0.851929 0 0.837225 0.851929 0 0 0 0 0 0 0 0 0 0 0 0 0 0.851929 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.9801 0.915849

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Table 5. (Continued). Probability distribution and allocation cost of capacities for 80 components Comp- Cost onent (NTD) bt ct 0 41 60 0.001024 42 65 0.000676 43 35 0.007921 44 25 0.000512 45 20 0.001 46 40 0.097 47 135 0.000001 48 70 0.022 49 145 0.000256 50 70 0.001225 51 70 0.025 52 65 0.000274625 53 120 0.000529 54 110 0.000144 55 70 0.000216 56 60 0.000117649 57 50 0.046 58 40 0.083 59 105 0.000015625 60 60 0.000274625 61 85 0.001369 62 125 0.000001 63 50 0.000512 64 40 0.006084 65 45 0.004096 66 50 0.003481 67 60 0.035 68 70 0.022 69 85 0.000166375 70 95 0.000042875 71 100 0.000024389 72 95 0.000324 73 145 0.000000343 74 30 0.004356 75 15 0.055 76 55 0.001936 77 85 0.000035937 78 115 0.000484 79 100 0.000121 80 100 0.001

1 0.061952 0.050648 0.162158 0.017664 0.027 0 0.000297 0.978 0 0 0.975 0.011851125 0 0 0.010152 0.006850053 0 0 0.001828125 0.011851125 0.071262 0.000297 0.017664 0.143832 0.119808 0.111038 0.965 0 0.008575875 0.003546375 0.002449833 0 0.000145971 0.123288 0.945 0.084128 0.003159189 0 0 0.999

Capacity (Gbps) 2 0.937024 0.948676 0.829921 0.203136 0.243 0.903 0.029403 0 0.031488 0.06755 0 0.170473875 0.044942 0.023712 0.159048 0.132946947 0.954 0.917 0.071296875 0.170473875 0.927369 0.029403 0.203136 0.850084 0.876096 0.885481 0 0.978 0.147349125 0.097778625 0.082027167 0.035352 0.020707029 0.872356 0 0.913936 0.092573811 0.043032 0.021758 0

3 0 0 0 0.778688 0.729 0 0.970299 0 0 0 0 0.817400375 0 0 0.830584 0.860085351 0 0 0.926859375 0.817400375 0 0.970299 0.778688 0 0 0 0 0 0.843908625 0.898632125 0.915498611 0 0.979146657 0 0 0 0.904231063 0 0 0

4 0 0 0 0 0 0 0 0 0.968256 0.931225 0 0 0.954529 0.976144 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.964324 0 0 0 0 0 0.956484 0.978121 0

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Figure 7. OCT network [2] (c) The TANET connects all academic and research organizations (universities, high schools, and elementary schools, research centers) in Taiwan, with each vertex denoting an organization (see Figure 8) [27]. For the TANET, we focus on the data transmission from NTU to NSYSU where there are 30 links, 27 vertices, and 6MPs. The link data are listed in Table 4(c). The settings of demand level and reliability threshold for the TANET are given as d = 5 GB and RT = 0.9. (d) The NSFNET is a network backbone built by a team from the National Center for Supercomputing Applications at the University of Illinois and Cornell Theory Center (see Figure 9) [23]. For the NSFNET, we set Seattle, WA and Ithaca, NY as the origin vertex and the destination vertex, respectively where there 16 links, 14 vertices, and 6 MPs. The link data are listed in Table 4(d). The settings of demand level and reliability threshold for the NSFNET are given as d = 5 GB and RT = 0.9. The data of 80 components ready to be allocated are shown in Table 5. The candidate list size (ϕ) is commonly discussed in the interval [5, 17, 20, 30]. Several studies [17, 30] considered the tabu tenure (π) in the interval [4, 15]. Bilgin and Azizo˘ glu [4] pointed that a high π may make the TS difficult to obtain a good move, and oppositely, a low π may easily result in that the same solutions may be obtained in the next few iterations. Referring to the previous parameter settings, we initially discuss the combination of candidate list size ϕ from {10, 15, 20} and tabu tenure π from {4, 7, 14}, and fix the coefficient of determination with λ = 5 and the terminal time with Γ = 1800 secs. For each combination, the proposed TS is executed for 10 trials to obtain the average minimal allocation cost. The experimental results shown in Table 6 indicate the average minimal allocation cost increases as the candidate list size increases. The combination of π = 7 and ϕ = 10 is the best setting for each network while comparing it with the others. Subsequently, we discuss the setting for the coefficient of determination (λ). This parameter is a newly-proposed parameter and must be a non-negative integer. Thus, we focus on finding the best setting for the coefficient of determination with λ ∈ [0, 10]. For each setting, the proposed TS is executed for 10 trials with π = 7, ϕ = 10, and Γ = 1800 sec. The experimental

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Figure 8. TANET [27] results shown in Table 7 suggest that the coefficient of determination should not be too high or too low. The best value is 5 for Figures 6 and 7 (resp. 4 for Figures 8 and 9). 5.3. Comparison among the meta-heuristic algorithms. Following the discussion for the four benchmark networks in Section 5.2, this subsection focuses on comparing the proposed TS with 4 meta-heuristic algorithms. The best experimental results from the TS in Section 5.2 continue to be used in this section. For each network, each algorithm is executed for 10 trails to obtain the smallest minimal allocation cost and average minimal allocation cost where each algorithm runs for 1800 secs. The experimental results are summarized in Tables 8–11. Apparently, the proposed TS can find the better solutions than the others from the viewpoint of minimal allocation cost and average minimal allocation cost. Figures 10–13 depict each algorithm’s search process based on the smallest minimal allocation cost for the four networks, respectively. Although the ACO can find a great solution within 500 seconds, it seems to converge quickly and drop into local optimum because the

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Table 6. Analysis for tabu tenure (π) and candidate list size (ϕ) in Figures 6–9 Figure 6 (ARPA) Average minimal allocation cost (Unit: NTD) Candidate list size (ϕ)

4 10 7482 15 7810 20 7865 Figure 7 (OCT)

Tabu tenure (π) 7 7206 7993 8672

14 7876 7877 8275

Average minimal allocation cost (Unit: NTD)

Tabu tenure (π) 4 7 14 10 20002 19027 19,949 15 20642 20568 20,692 Candidate list size (ϕ) 20 22123 22438 22,350 Figure 8 (TANET) Average minimal allocation Tabu tenure (π) cost (Unit: NTD) 4 7 14 10 59222 58128 59561 15 60224 59803 59652 Candidate list size (ϕ) 20 61488 62294 60722 Figure 9 (NSFNET) Average minimal allocation Tabu tenure (π) cost (Unit: NTD) 4 7 14 10 641705 640026 642250 15 669652 678588 679023 Candidate list size (ϕ) 20 705352 729889 718703 a. The value marked in boldface is the smallest average minimal allocation cost in all combinations for the network. Table 7. Analysis for coefficient of determination (λ) in Figure 6–9 Average minimal allocation cost (Unit: NTD) 0 1 2 3 4 Coefficient of determination (λ) 5 6 7 8 9 10

Figure 6 Figure 7 Figure 8 Figure 9 (ARPA) (OCT) (TANET) (NSFNET) 7722 19407 58567 645370 7872 19302 58557 642698 7560 19264 58368 636146 7451 19262 58204 638674 7386 19225 58044 630951 7206 19027 58128 640026 7845 19295 58169 636073 7748 19260 58249 637047 7965 19262 58223 637967 7994 19296 58736 642580 7897 19377 58474 640828

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Figure 9. NSFNET [23] Table 8. The experimental results of Figure 6 via five metaheuristic algorithms Terminal constrain = 1800 secs Smallest Metaheuristic minimal algorithm allocation Cost ACO 6830 GA 7050 PSO 10185 SA 8265 TS 6790

Best component allocation

Corresponding Reliability

(40,8,44,16,35,75,45,29,79) (79,20,8,16,45,35,44,29,9) (39,17,25,16,20,76,45,24,40) (40,2,20,29,44,45,8,9,50) (40,9,20,16,45,35,44,8,50)

0.95266 0.95374 0.967485 0.95745 0.95616

Average minimal Allocation Cost 7400 7533 10954 9659 7206

solution does not significantly improve after the 500th second. On the contrary, the proposed TS performs worse initially but its powerful local search and global search make it more efficient in improving the best solution, and can obtain the better solutions within 1800 secs. That is, the proposed problem is solved in a reasonable time by the proposed TS. In the experiments, the proposed TS provides the system supervisors with a great alternative for each network. For instance, the supervisor can allocate Y = (56, 8, 74, 21, 20, 29, 50, 40, 79, 39, 45, 17, 35, 9, 44, 72) to the NSFNET. 6. Conclusion and discussion. From the perspective of business management, cost is usually given more priority over reliability when enterprisers try to design or improve a system, especially for computer systems. One of the challenging problems is to find the optimal component allocation such that the allocation cost is

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Table 9. The experimental results of Figure 7 via five metaheuristic algorithms Terminal constrain = 1800 secs Soft Smallest computing minimal algorithm allocation Cost ACO

19325

GA

19350

PSO

28810

SA

21155

TS

18805

Best component allocation

(8,76,21,66,27,65,58,26,75,22, 20,44,64,16,43,35,19,56,25,40, 45,9,2,29,28,63,12,74,17) (22,41,66,25,64,17,12,65,58,45, 27,9,26,75,16,35,2,21,19,60, 44,8,63,29,3,7,43,74,20) (56,68,54,47,15,27,45,24,52,69, 21,9,75,32,46,26,42,12,39,61,44, 19,59,60,40,29,20,65,74) (22,65,20,32,41,56,43,75,27,45, 8,60,17,46,16,35,63, ,40,29,74, 19,24,9,18,28,21,44,12) (35,64,66,7,65,6,17,58,46,9,20, 44,43,16,75,26,21,29,19,27,45, 8,25,2,56,40, 12,22,74)

CorresAverage ponding minimal Reliability Allocation Cost 0.90803

20660

0.90205

19733

0.90269

29939

0.90279

22104

0.90272

19207

Table 10. The experimental results of Figure 8 via five metaheuristic algorithms Terminal constrain = 1800 secs Soft com- Smallest minimal puting algorithm allocation Cost ACO

61095

GA

59510

PSO

86400

SA

62945

TS

57165

Best component allocation

(17,12,35,22,46,64,43,65,74,21, 66,63,27,40,9,72,50,56,25,60,26, 75,2,20,8,44,45,29,19,16) (64,17,22,43,2,12,20,7,74,52,27, 65,50,8,9,63,19,37,21,60,16,35, 40,25,56,44,45,29,66,26) (28,21,8,9,17,25,42,37,75,74,14, 43,70,45,20,31,64,38,62,49,65, 35,27,71,47,55,50,44,18,76) (21,43,74,66,14,50,25,60,22,41, 18,56,15,44,29,40,20,33, ,63, 57,16,55,19,27,45,9,2,17,35) (26,43,35,22,12,65,17,64,74,60, 7,20,2,8,9,50,40,27,19,28,66,75, 56,21,25,44,45,29,63,16)

Corresponding Reliability

Average minimal Allocation Cost

0.90095

66971

0.90028

61105

0.90421

91956

0.90163

64978

0.90078

58044

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Table 11. The experimental results of Figure 9 via five metaheuristic algorithms Terminal constrain = 1800 secs Soft comSmallest puting minimal algorithm allocation Cost ACO

642740

GA

639610

PSO

937100

SA

686855

TS

623860

Best component allocation

(40,44,20,9,19,21,79,25, 54,72,45,74,35,8,29,39) (50,9,20,22,19,29,25,8, 28,39,44,17,35,40,45,79) (72,45,38,23,43,52,50,59, 54,39,44,56,31,40,22,49) (27,9,12,57,40,8,79,72, 39,50,45,24,75,44,20,71) (56,8,74,21,20,29,50,40, 79,39,45,17,35,9,44,72)

Corresponding Reliability

Average minimal Allocation Cost

0.90147

659686

0.90127

668210

0.91947

1066807

0.90015

727642

0.90075

630951

Figure 10. The comparison results of soft computing algorithm for Figure 6 minimized under a reliability threshold. Since each component is multistate, the computer network associated with a component allocation is regarded as an MCN. A TS-based algorithm is proposed in this paper to solve the proposed problem where the intensification and diversification strategies and the coefficient of determination related to the convergent times are proposed to improve the TS’s global search capability. In addition, the penalty function is proposed to deal with the infeasible solutions where the infeasible solutions are ranked according to their allocation costs and system reliabilities. The major contributions are listed as follows: 1. Different from the problems mentioned in the previous studies [24, 25, 26, 27], this paper discusses the problem of minimizing the allocation cost for multistate computer networks with the viewpoint of business management, and such a problem has yet to be addressed in the literature.

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Figure 11. The comparison results of soft computing algorithm for Figure 7

Figure 12. The comparison results of meta-heuristic algorithm for Figure 8

2. An improved TS-based algorithm is proposed to solve the proposed problem, and its utility and computational efficiency are verified via several benchmark computer networks. 3. Via simple computer networks and four benchmark computer networks, the experimental results show that the proposed TS has better computational efficiency than the other popular meta-heuristic algorithms such as GA, ACO, PSO, and SA. 4. The analysis of TS parameters is also discussed, in which the suggested setting of TS parameters is π = 7, ϕ = 10, and λ ∈ {4, 5}. It seems meaningful to compare the solution of the cost minimization subject to a reliability threshold with the one of the reliability maximization subject to a budget. However, the study [27] indicates that the cost minimization and the reliability maximization are two conflicting objectives. Hence, their optimal solutions should

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Figure 13. The comparison results of meta-heuristic algorithm for Figure 9

be different. The consideration of determining the criterion for a system usually depends on the situation. For instance, if there are many component allocations with high reliability, the system supervisors may choose the cost minimization as the objective. Furthermore, through the observation of Figures 10–13, the ACO can obtain better solutions than the TS before 300 secs. The reason may be that the ACO can produce a better initial solution for determining a good search direction [9] and thus it performs better before 300 secs. In other words, if the TS is improved to find a better initial solution, it can determine a good search direction and has the chance to find a better best solution than the current best one. Thus, future research may focus on strengthening the search capability of the proposed TS from the perspective of the initial solution improvement. REFERENCES [1] H. Ahonen, A. G. de Alvarenga and A. R. S. Amaral, Simulated annealing and tabu search approaches for the corridor allocation problem, European Journal of Operational Research, 232 (2014), 221–233. [2] A. Amiri and H. Pirkul, Routing and capacity assignment in backbone communication networks, Computers and Operations Research, 24 (1997), 275–287. [3] D. Berend, E. Korach and S. Zucker, Tabu search for the BWC problem, Journal of Global Optimization, 54 (2012), 649–667. [4] S. Bilgin and M. Azizo˘ glu, Operation assignment and capacity allocation problem in automated manufacturing systems, Computers and Industrial Engineering, 56 (2009), 662–676. [5] P. C. Chu and J. E. Beasley, A genetic algorithm for the generalized assignment problem, Computers and Operations Research, 24 (1997), 17–23. [6] C. J. Colbourn, The Combinatorics of Network Reliability, Oxford University Press, New York, 1987. [7] M. Dasgupta and G. P. Biswas, Design of multi-path data routing algorithm based on network reliability, Computers and Electrical Engineering, 38 (2012), 1433–1443. [8] R. K. Dash, N. K. Barpanda, P. K. Tripathy and C. R. Tripathy, Network reliability optimization problem of interconnection network under node-edge failure model, Applied Soft Computing, 12 (2012), 2322–2328. [9] M. Dorigo, V. Maniezzo and A. Colorni, Ant system: Optimization by a colony of cooperating agents, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 26 (1996), 29–41. [10] L. R. Ford and D. R. Fulkerson, Flows in Networks, Princeton University Press, NJ, 1962.

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Received February 2014; 1st revision May 2014; final revision November 2014. E-mail address: [email protected] E-mail address: [email protected]