Ant Colony Optimization for Constrained Combinatorial ... - CiteSeerX

Proceedings of The 5th Annual International Conference on Industrial Engineering -Theory, Applications and Practice December 13-15, 2000, Hsinchu, Taiwan

Ant Colony Optimization for Constrained Combinatorial Problems Yun-Chia Liang and Alice E. Smith Department of Industrial and Systems Engineering Auburn University Auburn, AL 36849, USA [email protected] and [email protected]

Abstract This paper develops and demonstrate Ant Colony Optimization (ACO) methods for constrained combinatorial problems. ACO is one of the meta-heuristic optimization methods and is inspired by the behavior of real ants. Two problem-specific ACO algorithms are deployed for two NP-hard problems - the redundancy allocation problem (RAP) of a series-parallel system and the single machine total weighted tardiness (SMTWT) problem. The objective of the RAP is to select components and redundancy-levels to maximize system reliability, given system-level constraints on cost and weight. The SMTWT problem is to find a processing order of the jobs on a single machine with minimum total weighted tardiness, given processing time, weight, and due date of each job. The ACO algorithm for the RAP presented in this paper is combined with an adaptive penalty method to deal with the highly constrained problem. Elitist strategy, mutation strategy, and local search are introduced to our ACO algorithms. Experiments were conducted on a well-known set of sample problems from the literature.

1. Introduction Ant Colony Optimization (ACO) is one of the adaptive meta-heuristic optimization methods which include simulated annealing (SA), genetic algorithm (GA), evolutionary strategies (ES), and tabu search (TS). ACO is distinctly different from these methods in that it is a constructive, rather than an improvement, algorithm. ACO methods have been successfully applied to different optimization problems including continuous and combinatorial problems such as the travelling salesperson problem (TSP), the quadratic assignment problem (QAP), the vehicle routing problem (VRP), telecommunication networks, graph coloring, scheduling, etc. The main focus of this paper is to examine the robustness and versatility of the ACO algorithms as applied to diverse problems that have not previously been explored by ACO. A further emphasis is to borrow techniques from other methods to improve ACO algorithms. Two different classes of problems, the Redundancy Allocation Problem (RAP) and the Single Machine Total Weighted Tardiness problem (SMTWT), both difficult combinatorial problems, are chosen as the focus of this research. For the computational complexity of these combinatorial problems, Chern [1] proves that reliability redundancy optimization problems are NP-hard; Karp [12, 13], Lenstra, Rinnooy Kan and Brucker [14], and Pinedo [19] prove that the SMTWT problem is NP-hard. ACO has been inspired by the behavior of real ants. Ethologists studied how blind animals, such as ants, could establish shortest paths from their nest to food sources [6, 7, 11]. The medium that is used to communicate information among individual ants regarding paths consists of pheromone trails. A moving ant lays some pheromone on the ground, thus marking the path by a pheromone trail. If an isolated ant moves randomly, it will detect a previously laid trail and decide where to go. A trail with more pheromone has a higher probability to be chosen by subsequent ants. The process is thus characterized by a positive feedback loop, where the probability with which an ant chooses a path increases with the number of ants that previously chose that same path. Marco Dorigo first introduced the Ant System (AS), the earliest version of the ACO methods, in his dissertation in 1992 [8]. The artificial ants behave in a similar way as real ants. However, they differ in two aspects. First, the artificial ants are not blind, i.e. they can “see” information regarding their environment; second, they have a memory, such as a tabu list in the TSP application. Moreover, the characteristics of an artificial ant colony include positive feedback, negative feedback, and the use of a constructive greedy heuristic. Positive feedback based on the pheromone-lying and trail-following behavior accounts for rapid discovery of good solutions. Negative feedback

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implemented through pheromone evaporation avoids premature convergence of the algorithm, and the greedy heuristic helps find acceptable solutions in the early stages of the search process. The reliable performance of a system for a predefined mission time under various conditions is very important in many industrial and military applications. The series-parallel system with k-out-of-n:G subsystems redundancy is a common representation for many system design problems. In the formulation of a series-parallel system problem, for each subsystem, multiple component choices (assuming an unlimited supply of each component) are used in parallel. For those systems designed using off-the-shelf component types, with known cost, reliability, and weight, system design and component selection becomes a combinatorial optimization problem. The problem is to select the optimal combination of parts and redundancy levels to meet cost and weight constraints collectively while maximizing system reliability. Scheduling is a form of decision-making which plays a crucial role in manufacturing as well as in service industries. In the modern competitive environment, effective scheduling has become a necessity for survival in the marketplace. Thus, meeting due dates and avoiding delay penalties are the most typical goals of scheduling. The costs of tardy deliveries, such as company’s goodwill, future sales loss, and rush shipping costs, vary significantly over customers and orders, and this implied “strategic weight” should be reflected in job priority. Therefore, minimizing total weighted tardiness for the jobs awaiting completion is a useful and important measure in practice. This static sequencing problem has a fixed number of jobs in a queue; each with a processing time, a due date, and a corresponding weight. Moreover, the static problem permits direct comparison of sequencing procedures without regard to either the random effects of job arrivals, or the due date setting methodology. In section 2, detailed problem statements of the redundancy allocation problem for the series-parallel system and the single machine total weighted tardiness problem are shown, respectively. All notation is problem-specific. Section 2 also presents methodologies and computational results. For the RAP and SMTWT problems, benchmark test problems are used to examine the efficiency and practicality of the ACO algorithm. Finally, section 3 discusses conclusions.

2. Methodology 2.1 Redundancy Allocation Problem (RAP) 2.1.1 Problem definition Nomenclature R overall reliability of the series-parallel system C cost constraint W weight constraint s number of subsystems ai number of available component choices for subsystem i rij , cij , wij

reliability, cost, and weight of component j available for subsystem i

yij

quantity of component j used in subsystem i

yi

( yi1 ,...,yiai )

pi

=

ai

∑ yij , total number of components used in subsystem i j =1

p max

maximum number of components in parallel (user assigned)

ki

minimum number of components in parallel required for subsystem i to function

Ri ( y i | k i )

reliability of subsystem i , given k i

Ci ( y i )

total cost of subsystem i

Wi ( y i )

total weight of subsystem i

Rk

unpenalized objective function value of solution k

Rkp

penalized objective function value of solution k

2


Rall

unpenalized value of the best solution found

R feas

value of the best feasible solution found

NFT

Near-Feasibility Threshold, the threshold distance from the feasible region

NFTc

Near-Feasibility Threshold for the cost constraint

NFTw

Near-Feasibility Threshold for the weight constraint

∆c k

magnitude of cost constraint violation for solution k

∆w k γ

magnitude of weight constraint violation for solution k preset severity parameter

Assumptions • The states of components and the system have only two options - good or bad. • If the number of good components is less than k i in a subsystem i , then subsystem i fails. • • • • • •

The failure of any subsystem will cause system failure. Failures of components are independent events. Failed components do not damage the system, and are not repaired. The failure rates of components when not in use are the same as when in use (i.e. active redundancy). Component reliabilities are known and deterministic. The supply of components is unlimited (i.e. off-the-shelf). 1

2

s

1

1

1

2

2

2

3

3

:

:

:

p1

p2

ps

k1

k2

ks

...

3

Figure 1 Series-parallel system configuration The redundancy allocation method of system reliability discussed here is designed to select the optimal solution in the context of a trade-off analysis. Given overall restrictions on system cost of C and weight of W , the problem is to determine which components to select and how many redundant components to use in order to achieve the maximum reliability. This formulation of the redundancy allocation problem for series-parallel system, such as depicted in Figure 1, leads to the maximization of system reliability, R , given by the product of subsystem reliabilities s

R = Π Ri ( y i | k i )

(1)

i =1

Subject to the constraints s

∑ Ci ( y i ) ≤ C ,

(2)

∑ Wi ( y i ) ≤ W ,

(3)

i =1 s i =1

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ki ≤

ai

∑ yij ≤ p max ∀ i = 1,2 ,..., s

(4)

j =1

2.1.2 Ant systems for the RAP An adaptive penalty function has been successfully employed in the redundancy allocation problem with GA [2, 3, 4, 5]. Similarly, it is also incorporated into the ACO as the tool of measuring the violation of constraints in order to ensure sufficient search over the feasible and infeasible regions. Following discussion of the adaptive penalty function, the algorithms of the Ant System (AS), which is the first version of ACO, will be discussed and followed by the computational results of a well-known test problem in section 2.1.3. The penalty function used employs the notion of a “Near-Feasibility Threshold” (NFT) for each constraint [2, 3, 4, 5]. The NFT is the threshold distance from the feasible region that is considered as being close to feasibility. The redundancy allocation problem is usually formulated with a max objective function with two independent constraints (cost and weight) so the penalty function is a linear summation as follows: Rkp = Rk − ( R all − R feas )((

∆w k γ ∆c k γ ) +( ) ) NFTw NFT c

(5)

s

Rk is the unpenalized objective function value of solution k ; i.e. Rk = Π Ri ( y i | ki ) subject to k i ≤ i =1

ai

∑ y ij ≤ p max j =1

∀ i = 1,2 ,..., s . The exponent γ is a preset severity parameter and its value is set to 2 for problems in this section. NFTw and NFTc are the “Near-Feasible Thresholds” for the weight and cost constraints, respectively, and take dynamic formations which monotonically decrease during research [4] NFT =

NFT0 1 + λ × NC

(6)

where NFT0 is a starting point for NFT , NC represents the number of iterations, and λ denotes a constant which assures that the entire region between NFT0 and zero is searched. Each possible solution to the redundancy allocation problem is a collection of p i parts in parallel ( ki ≤ pi ≤ p max ) for s different subsystems. The p i parts can be chosen in any combination from the a i available components. The a i components are indexed in descending order in accordance with their reliability; i.e. 1 represents the most reliable component, etc. An index of a i + 1 is assigned to a position where an additional component was not used (left blank). Each of the s subsystems is represented by p max positions with each component listed according to its reliability index. The parameters considered in the Ant System algorithm are shown below: • •

α : the relative importance of the pheromone trail, α ≥ 0; ρ : trail persistence, 0 ≤ ρ < 1;

•

Q : a constant related to the magnitude of the pheromone trail laid;

•

c : initial quantity of pheromone substance for the “blank”, 0 ≤ c < 1.

In an Ant System, each ant represents one design of the entire system. In the beginning of the process, an initialization phase takes place during which ants select components in each subsystem according to the transition probability Pij =

[τij ]α ai

(7)

∑ [τil ]

α

l =1

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which is based on the trail intensity (τij ). After all of the components for the ants are selected, the unpenalized reliability Rk and the penalized reliability Rkp for the system are calculated. Thereafter, according to which mechanisms are applied, the AS algorithm is divided into two versions – AS-EM [15], which stands for Elitist and Mutated ants, and AS-EL, which denotes Elitist ants and Local search. In the AS-EM algorithm, the trail is updated by τijnew = ρτijold +

NA

∑ ∆τijk

(8)

k =1

where ∆τijk is the quantity of trail substance laid for combination (i, j) by the k th ant if the k th ant’s objective function is greater than zero. The magnitude of a new pheromone trail is updated by vaporizing (1 − ρ ) percentage of pheromone from the previous iteration and accumulating pheromone laid by the current ant population. Also, the best feasible solution found is saved. Starting from the second iteration, a certain number of mutated ants are generated by following the initial transition probability, and a certain number of the best feasible ant (elitist ants) found are duplicated. The use of mutated ants can help prevent the search from stagnating in a local optimum. The purpose of duplicating the best feasible solution is to enhance the magnitude of trail for “good” combinations. This process continues until the maximum (user-defined) number of iterations, NCmax , is reached. Formally the AS-EM algorithm is: Set all parameters and initialize the pheromone trails Loop Generate colony based on the transition probability ( Pij ) and/or the original transition probability (mutated ants), and duplicate the best feasible solution (elitist ants) Evaluate all solutions Update the best solution so far Update the pheromone trail ( τij ) Continue until the stopping criterion is reached In the AS-EL algorithm, after each colony is generated, the neighborhood of the best feasible ant in each iteration is explored by local search. The 2-position Opt, which exchanges the 1st and last ( p max ) position components with all other available choices, provides an exhaustive search of the chosen neighborhood. If the best neighboring ant is feasible and superior to the original ant, then the related information will be updated. Thereafter, the trail is updated by τijnew = ρτijold +

NA

∑ ∆τijk + e ⋅ ∆τije

(9)

k =1

where ∆τijk is as in AS-EM method, e denotes the number of elitist ants, and ∆τije represents the quantity of trail substance laid for combination (i, j) by the best feasible ant found so far. The procedure continues until the best feasible ant has not changed for a certain (user-defined) number of iterations. Formally the AS-EL algorithm is: Set all parameters and initialize the pheromone trails Loop Generate colony based on the transition probability ( Pij ) Evaluate all solutions Apply 2-position Opt and update the best solution so far Update the pheromone trail ( τij )

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Continue until the stopping criterion is reached 2.1.3 Test problems and computational results The best known problem of a series-parallel system, Fyffe, Hines, and Lee [10], specifies 130 units of system cost, 170 units of system weight and k i =1; i.e., 1-out-of- n :G subsystems. In the optimal solution, the weight constraint is active and the cost of system is 119. Both of the Ant Systems are implemented on the 33 variations of the Fyffe, Hines, and Lee problem [10], which were revised by Nakagawa and Miyazaki [18]. In the set of problems [18], C = 130 and W is increased incrementally from 159 to 191. In both papers, the approach required is that only identical components can be placed in redundancy. Different component types will be allowed to reside in parallel assuming that the value of k i is 1 and the value of p max is 8 for all subsystems, as was done by Coit and Smith [2, 3, 4, 5]. In the AS-EM algorithm, for the decreasing NFT implementation, two different λ values (0.04, 0.4) were used, W and NFTc 0 was set to 100 and NFTw 0 was set to . The number of ants, NA , was set equal to 1000. An 1. 3 NA

alternative trail updating method τijnew = ρτ ijold + (1 − ρ) ∑ ∆τijk was also tested but not shown to influence the k =1

results. In order to show the effect of duplicating the best feasible solution, two different values of elitist ants, 1 (i.e. no duplication) and 20, were examined. Three different Q values (1, 100, 10000) were tested but were not shown to influence the algorithm. Three different α values (0, 1, 2) were tested and the results of α = 1 were superior. Also, four different c values (0.5, 0.65, 0.7, 0.85), five different ρ values (0.3, 0.5, 0.7, 0.85, 0.9), and three different number of mutated ants (20, 50, 100) were tested. The value of parameters, (c, ρ , λ ) equal to (0.5, 0.7, 0.4) with 20 duplicated best feasible ants and 50 mutated ants produced better results. Ten different random number seeds were used for each of the 33 problem variations, and 1500 iterations were used in each run. In the AS-EL algorithm, for the decreasing NFT implementation, λ was set to 0.04, NFTc 0 was set to 100 and W . Four different NA values (20, 50, 100, 150), the number of ants, were used corresponding 1. 3 to four different number of elitist ants (5, 10, 20, 30), respectively. Q was set to 1 since the variety of Q values did not show obvious influence to the algorithm. α was set to 1 since α = 1 was superior to other values. Also, four different c values (0.6, 0.7, 0.8, 0.9) and four different ρ values (0.7, 0.8, 0.9, 0.99) were tested. When the numbers of ants were 100 and 150, the results defeated the ones with 20 or 50 ants. However, the results with 100 and 150 did not show obvious difference, so a colony size of 100 was set. The value of parameters, (c, ρ , λ ) equal to (0.6, 0.8, 0.04) with 20 elitist ants and 100 ants produced better results. Ten different random number seeds were used for each of the 33 problem variations. The procedure was stopped if after 500 iterations the best feasible solution did not change for the next 300 iterations. NFTw 0 was set to

The best feasible results of the 33 variations are summarized in Table 1 and also compared with the results from Nakagawa and Miyazaki [18] and from Coit and Smith [4]. The percentage of deviation between C&S and the two AS algorithms are calculated. The average percentage of deviation of the AS-EM algorithm over 33 variations is 0.44% and that of AS-EL algorithm is 0.59%. Both of the AS algorithms provided good alternatives to the test problems. When comparing the performance of the two AS algorithms, among 33 variations of the test problem, AS-EM results are superior to AS-EL results in 26 of them. It can be concluded that the addition of mutated ants makes the difference. However, the performance of the AS-EL algorithm seems more stable in the higher constrained problems, from problem 26 to problem 33.

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Table 1 Comparison of N&M [18], C&S [4], AS-EM [15], and AS-EL performance AS-EM AS-EL No C W N&M C&S best %deviation best %deviation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130

191 190 189 188 187 186 185 184 183 182 181 180 179 178 177 176 175 174 173 172 171 170 169 168 167 166 165 164 163 162 161 160 159

0.9864 0.98675 0.98340 0.9854 0.98603 0.97985 0.9850 0.98556 0.98157 0.9847 0.98503 0.98157 0.9840 0.98429 0.98151 0.9831 0.98362 0.98189 0.9829 0.98311 0.97972 0.9822 0.98239 0.97972 0.9815 0.98190 0.97793 0.9815 0.98102 0.97757 0.9800 0.98006 0.97483 0.9796 0.97942 0.97744 0.9792 0.97906 0.97332 0.9772 0.97810 0.97458 0.9772 0.97715 0.97458 0.9764 0.97642 0.96683 0.9744 0.97552 0.96683 0.9744 0.97435 0.96825 0.9723 0.97362 0.97070 0.9720 0.97266 0.97070 0.9700 0.97186 0.96650 0.9700 0.97076 0.96351 0.9675 0.96922 0.96606 0.9666 0.96813 0.96357 0.9656 0.96634 0.96357 0.9646 0.96504 0.96086 0.9621 0.96371 0.95090 0.9609 0.96242 0.95894 0.9602 0.96064 0.95894 0.9589 0.95912 0.95894 0.9565 0.95803 0.94668 0.9546 0.95567 0.95456 0.9546 0.95432 0.95456 Avg. %deviation

0.33950 0.62676 0.40485 0.35126 0.28244 0.17588 0.34482 0.27179 0.40432 0.35167 0.53364 0.20216 0.58628 0.35988 0.26301 0.98216 0.89081 0.62606 0.29991 0.20151 0.55152 0.74684 0.32604 0.47101 0.28665 0.43314 1.32924 0.36159 0.17697 0.01877 1.18472 0.11615 -0.02515 0.43867

0.98373 0.98170 0.97913 0.97914 0.97875 0.97678 0.97608 0.97375 0.97726 0.97443 0.97480 0.97441 0.97273 0.97297 0.96894 0.97252 0.96749 0.96782 0.96843 0.96854 0.96602 0.96815 0.96124 0.96086 0.95971 0.95818 0.96189 0.95635 0.95444 0.95277 0.95299 0.94916 0.95184

0.30606 0.43913 0.65242 0.59795 0.56284 0.69539 0.71508 0.87949 0.47255 0.67175 0.53670 0.51153 0.64654 0.52449 0.84020 0.39942 0.82315 0.67019 0.53306 0.42358 0.60091 0.26886 0.82334 0.75093 0.68609 0.71085 0.18885 0.63070 0.64540 0.66207 0.52608 0.68120 0.25987 0.58596

2.2 Single Machine Total Weighted Tardiness Problem (SMTWT) 2.2.1 Problem definition Nomenclature wj weight of job j pj

processing time of job j

dj

due date of job j

n Cj

total number of jobs completion time of job j

Tj

= max{ 0, C j − d j } , tardiness of job j

P

=

p

average processing time of the unscheduled jobs a look-ahead parameter related to the number of competing critical and near-critical jobs

k

n

∑ p j , total processing time of all jobs j =1

7


The single-machine total weighted tardiness problem can be stated as follows. Each of n jobs (numbered 1,..., n ) is to be processed without preemption on a single machine that can handle no more than one job at a time. The processing and set-up requirement of any job is independent of its position in the sequence. The release time of all jobs is zero. Thus, job j ( j = 1,...,n ) becomes available for processing at time zero, requires an uninterrupted positive processing time p j , which includes set-up and knock-down times, on the machine, has a positive weight w j , and has a due date d j by which it should ideally be finished. For a given processing order of the jobs, the completion time C j and the tardiness T j = max{ 0, C j − d j } of job j ( j = 1,...,n ) can be computed. Then, the n

problem is to find a processing order of the jobs with minimum total weighted tardiness

∑wjT j . j =1

2.2.2 Ant colony optimization for the SMTWT problem Each ant generates a complete solution, i.e., schedule sequence. To construct a feasible solution the artificial ants successively choose jobs to be appended to the actual subsequence, until all jobs are scheduled. For this selection process, the ants use problem-specific heuristic information, denoted by ηij , as well as pheromone trails, denoted by τij . The former is an indicator of how good the choice of that job seems to be, and the latter indicates how good the choice of the job was. The transition probability Pij that job j is selected to be processed on position i in the sequence is formally given by: (τij ) α (ηij ) β Pij =

If j ∈US

∑ (τil )α (ηil ) β

(10)

l∈US

0

Otherwise

where α and β are two parameters that control the relative weight of pheromone trail and local heuristic value, and US is the set of unscheduled jobs. Two heuristic methods were used in this algorithm. Morton, Rachamadugu and Vepsalainen [17] developed an exponential function, called MRV here, to foresee job clashes.

MRV

η ij = w j ⋅ e

−

max( d j −t − p j , 0 ) kp

(11)

where t is the current time, k is a “look-ahead parameter” which relates to the number of competing critical and nearcritical jobs, and p is the average processing time of the unscheduled jobs which replaces the position of the average processing time of all jobs in the original MRV rule. Another heuristic used is the well-known Apparent Tardiness Cost (ATC) rule [19, 21].

ATC

ηij =

wj pj

⋅e

−

max( d j −t − p j , 0 ) kp

(12)

A fixed value of k = 2 that has been used in static flow shops including a single-machine study [16, 21], was used for both heuristics and both rules are used in descending order. The job yielding the highest ranking is appended to the sub-sequence generated so far. After all ants have followed the selection process described above and thus constructed a complete solution to the problem, the pheromone trails are globally updated. Depending on the solution quality, i.e., depending on total weighted tardiness, the corresponding pheromone trails are increased. At the same time evaporation reduces the pheromone trails. This global trail update can formally be expressed as follows:

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τijnew = ρ ⋅ τijold + (1 − ρ) ⋅ ∆τij

(13)

where ρ ∈ [0,1] is a parameter that controls the pheromone persistence, i.e., 1 − ρ represents the proportion of the pheromone evaporated, and ∆τij is the amount of pheromone trail added to τij by the ants. As it was the case for the heuristic information to calculate ηij , there are several possibilities regarding the quantity ∆τij in ACO algorithms. Here we followed the Ant Colony System (ACS) idea [9], i.e., only the best ant contributes to the 1 pheromone trail update. Thus, we have ∆τij = for all combinations (i,j) belonging to the best solution found so TB 1 far, where TB is the total weighted tardiness of that best solution. All initial trail intensities ( τij ) are set to n ⋅ TWSPT where n is the number of total jobs, and TWSPT is the total weighted tardiness for a processing sequence generated according to the Weighted Shortest Processing Time (WSPT) rule. Ant Colony Optimization algorithms have been shown to work very well if combined with a local search procedure [9]. In such a case, a local search procedure is applied to the solutions generated by the ants before updating the pheromone trails. For our ant approach to the SMTWT problem, the Adjacent Pairwise Interchange strategy (API) is considered, i.e., only adjacent jobs which have a predecessor-successor relation will be swapped. Also, considering the computational expense, we only apply API to the best solution of the current iteration instead of applying it to all ants generated during the iteration. Formally the ACO algorithm is: Set all parameters and initialize the pheromone trails Loop Generate the sequence based on the transition probability ( Pij ) Evaluate all solutions during the iteration and record the best one Apply the API rule and update the best solution so far Update the pheromone trail ( τij ) Continue until the stopping criterion is reached 2.2.3 Test problems and computational results The test problems used in this project were the 125 benchmark problems from Potts and Van Wassenhove [20]. The instances were randomly generated as follows: for each job j ( j = 1,...,n ), an integer processing time p j was generated from the uniform distribution [1,100] and an integer processing weight w j was generated from the uniform distribution [1,10]. Using different uniform distributions for generating the due dates generated instance classes of varying hardness. For a given relative range of due dates RDD (RDD = 0.2, 0.4, 0.6, 0.8, 1.0) and a given average tardiness factor TF (TF = 0.2, 0.4, 0.6, 0.8, 1.0), the integer due date d j for each job j was randomly n RDD RDD ), P ⋅ (1 − TF + )] , where P = ∑ p j . Five 2 2 j =1 instances were generated for each of the 25 pairs of the values of RDD and TF, yielding 125 instances for both values of n = 40 and n = 50. The test problem files can be downloaded from OR-Library (URL: http://mscmga.ms.ic.ac.uk/jeb/orlib/wtinfo.html).

generated from the uniform distribution [ P ⋅ (1 − TF −

In our tests we applied MRV or ATC for the local heuristic information ηij . Three different numbers of ants (20, 40, 60) were used; four different ρ values (0.1, 0.5, 0.9, 0.95) were tested; three different number of iterations (10, 50, 100) were examined as well. Among those settings, the value of parameters - 40 ants, ρ = 0.9 , and 100 iterations produced the best results. Also, the following parameter settings, α = 1 and β = 2 , were used. Each instance was run five times according to different random number seeds. To compare with the other simple heuristic methods such as WSPT, MRV and ATC, C codes for each method were generated. The optimal solutions (OPTIMUM) were downloaded from the OR-Library for comparison.

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In order to prove that local search can improve the search method, ACO algorithms with API and without API were implemented. Thus, there are four categories of ACO algorithms based on different local heuristics and local search: ATC with API, ATC without API, MRV with API, and MRV without API.

2 0 0 0 0

1 5 0 0 0 A T C M R V - A P I M R V

Solutions

Difference of Total Weighted Tardiness between ATC-API and Other ACO

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Figure 2 Comparison of four categories of ACO when n = 40 In Figures 2 and 3, the difference of the best runs of each instance between ATC-API and other three categories are shown for both n = 40 and n = 50. When considering the effect of local search, the results showed that, compared to ATC without API, ATC-API was even or superior in all of 125 instances for n = 40 and n = 50. Compared to MRV without API, MRV-API was even or superior in all of 125 instances for n = 40 and n = 50. Therefore, it can be seen that local search did help find better solutions regardless of the ηij heuristic used. Furthermore, compared to MRV with API, ATC-API was even or superior in 122 out of 125 instances for n = 40 and 115 out of 125 instances for n = 50, and ATC-API was even or superior in all of 125 instances for n = 40 and in 122 out of 125 instances for n = 50 compared to MRV without API. Therefore, comparisons between these four categories show that the ATC heuristic is better than the MRV heuristic, and ATC-API dominates others. 4 0 0 0 0

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Difference of Total Weighted Tardiness between ATC-API and Other ACO

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Figure 3 Comparison of four categories of ACO when n = 50 In Figures 4 and 5, the best runs from the ACO with ATC and API (ATC-API) are compared with results from the optimum (OPTIMUM), WSPT, Pure ATC, and Pure MRV methods, i.e. the differences between each heuristic method and the OPTIMUM are shown in the figures. Among 125 instances, ATC-API reached optimum in 48 instances when n = 40 and in 32 instances when n = 50. WSPT never reached the optimum in any instance. Pure ATC reached optimum in 16 out of 125 instances for n = 40 and 15 out of 125 instances for n = 50, and Pure MRV reached optimum in 17 of 125 instances for n = 40 and 16 of 125 instances for n = 50. ATC-API showed the highest percentage of optimum among all heuristic methods examined here. The relative comparison between the heuristics is summarized in Table 2. When the values of the tardiness factor (TF) is small, ACO performs very well, i.e., the results of ACO are the same or very close to the OPTIMUM and outperform other heuristics. When the values of tardiness factor (TF) becomes larger, the performance of ACO is either close to or better than other heuristics.

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Table 2 Relative comparison between the heuristics (number of times reaching OPTIMUM out of 125 instances) Heuristics n = 40 n = 50

WSPT 0 0

Pure ATC 16 15

Pure MRV 17 16

ATC-API 48 32

5 0 0 0 0

4 5 0 0 0

Difference between OPTIMUM and Other Methods

4 0 0 0 0

3 5 0 0 0

3 0 0 0 0

2 5 0 0 0

W S P P u r e P u r e A T C -

T A T C M R V A P I

W S P P u r e P u r e A T C -

T A T C M R V A P I

2 0 0 0 0

1 5 0 0 0

1 0 0 0 0

5 0 0 0

121

116

111

106

96

101

91

86

81

76

71

66

61

56

51

46

41

36

31

26

21

16

6

11

1

0


Figure 4 Comparison of ACO with other heuristic methods when n = 40 7 0 0 0 0

Difference between OPTIMUM and Other Methods

6 0 0 0 0

5 0 0 0 0

4 0 0 0 0

3 0 0 0 0

2 0 0 0 0

1 0 0 0 0

121

116

111

106

101

96

91

86

81

76

71

66

61

56

51

46

41

36

31

26

21

16

11

6

1

0


Figure 5 Comparison of ACO with other heuristic methods when n = 50

3. Conclusions The Ant Colony Optimization (ACO) algorithm is a promising heuristic method for solving combinatorial optimization problems. Unlike most of meta-heuristic methods, ACO is a constructive algorithm. The efficiency of this algorithm will depend on the selection of parameters, trail update method, and other supplemental devices such as local search. In this paper, we show how to apply the ACO algorithms to the redundancy allocation problem and the single machine total weighted tardiness problem. Unlike the original Ant System, we introduce an elitist strategy, mutation, and a local search method to our AS algorithm for the RAP. The elitist strategy enhances the magnitude of the trails of good selections of components. The mutated ants can help explore new search areas. For the SMTWT problem, we introduce a local search (API) to improve the performance of the ACO algorithm.

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