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Simulated annealing heuristics for the average flow-time and the number of tardy jobs bi-criteria identical parallel machine problem
Computers ind. Engng VoI. 33, Nos 1-2, pp. 257-260, 1997 © 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0360-8352/97 $17.00 + 0.00
Pergamon Plh S0360.8352(97)00087-9
Simulated Annealing Heuristics for the Average Mow-Time and the Number of Tardy Jobs Bi.Criteria Identical Parallel Machine Problem Alex J. Ruiz-Torres', E. Emory Enscore h, and Russell R. Bartonh dDepartment of Management College of Business Florida Gulf Coast University, Ft. Myers, FL "Department of Industrial and Manufacturing Engineering College of Engineering, 207 Hammond Building Pennsylvania State University, University Park, PA
ABSTRACT This paper presents the first part of a study of a scheduling problem where the objective is to minimize both the number of jobs late and the average flow-time. Unlike previous approaches, a weighted function of the two criteria is not utilized. Heuristics based on simulated annealing and neighborhood search are proposed with the objective of generating schedules that produce a compromise across both criteria. Simulation experiments showed that the heuristics perform well when compared to the optimal solution set for a small problem instance. © 1997 Elsevier Science Ltd KEYWORDS Scheduling; Multiple Criteria; Parallel Machines; Average Flow-Time; Number of Tardy Jobs; Simulated Annealing.
1
INTRODUCTION
The relevance of parallel machine scheduling has been well documented and includes applications in manufacturing, service, and data processing settings. Most of the work in parallel machines has concentrated on single criteria problems. However, several researchers have recognized the importance of considering multiple measures of performance as a better representation of today's decision making (Nagar et al. 1995). This is because single criteria optimal solutions may prove too expensive in relation to other important criteria. This research utilizes the efficient frontier approach which has a stronger appeal in many settings, as it allows users to pick the best tradeoff between the criteria based on other dynamic conditions (Bemardo and Lin 1994). This paper presents the initial results on the performance of four heuristics to optimally solve the problem of scheduling N jobs on M identical parallel machines with the objective of generating tradeoff schedules for the average flowtime (F) and number of tardy jobs (Nt) criteria. As per the notation used in Blazewicz et al. (1993), the problem is defined as P/IF, Nt. The heuristics are based on neighborhood search and simulated annealing, methodologies that have been widely used in scheduling problems (Zegordi et al. 1995). The remainder of this paper includes a problem definition and its complexity, a description of the four heuristics, and finally, computational results and conclusions.
2
PROBLEM DEFINITION AND COMPLEXITY
In general there are M identical machines to process N one operation jobs. Each job j has a deterministic processing time pj and due date dj. All jobs are ready at time zero. Preempuon, alvtston or can257
258
21st International Conference on Computers and Industrial Engineering
cellations of jobs are not allowed. Finally, M and N are fixed. The objective is to generate the nondominated (efficient) schedules/solutions which minimize the average flow-time (F) and the number of tardy jobs (Nt). A schedule S is efficient if there is no schedule S' such that F >_.F' and Nt > Nt', where at least one inequality holds strictly. These two criteria can seldom be optimized simultaneously, therefore tradeoffs exist. This is shown in Figure 1 (each job is identified by its processing time and due date). The problem P / / F , Nt is NPHard given one of its 'component' single criteria problems is NP-Hard (P//Nt is NP-Hard: Ho and Chang 1995). The bi-criteria problem is NP-Hard given the single criteria Nt solution is part of the nondominated set, and if finding one of the solutions in the set is NP-Hard, finding the total set is NPHard. This was shown in general by Chen and Bulfin (1993).
ml m2
3,3
I F=2.3 Nt = 1
ml m2
F=2.7 Nt=0
Fig. 1. Tradeoffs between F and Nt.
3
FOUR HEURISTIC ALGORITHMS
This section describes four heuristics to solve the P / I F , Nt with the objective of generating a set of efficient alternative schedules (nondominated solution set). These heuristics are based on neighborhood search methods. The basic elements in this methodology are (Morton and Pentico 1995): the starting solution (original seed), the solutions close to the original solution (neighbor of the seed), a method for selecting the new seed (selection criterion), and a method for terminating the procedure (termination criterion). The difference among the four heuristics is the speed of the search and the use of simulated annealing, both reflected in the selection criterion step. First the similarities are described, and then the differences between all four heuristics are described. Original seed:: Two seeds are used for every problem (therefore two searches): 1) The schedule generated by the well known parallel machine SPT algorithm, and 2) The best (minimum Nt) schedule generated from the four Ho and Chang (1995) algorithms to minimize Nt in parallel machines. Neighborhood of the seed:: Two methods - pairwise interchanges, where each job j (in SPT order), is swapped with all other jobs N-I, and single job insertions, where each jobj is inserted in all positions of all M machine sequences. When the complete neighborhood has been evaluated for job j, then the best move (selection criterion) is accepted. The best move is different for heuristics GNS/SA and Q-NS/SA. An efficiency check (Section 2) is made after each move. Termination criterion:: When no reduction of the Y criteria is made, or when the number of cycles through the search, Z, is equal to the number of jobs N. These four heuristics are based on the assumption that each of the two original seeds has the best single criteria solution (criteria X); the schedule generated by Ho and Chang (1995) has the lowest number of tardy jobs criteria possible, and the SPT schedule has the lowest average flow-time criteria possible. In that context, the search is made in the direction of the other criteria (criteria I/). Heuristic Q-NS (Quick-Neighborhood Search) selects from the neighborhood, the schedule with the lowest Y' (if there are ties, the lowest X'). Heuristic G-NS (Gradual-NS) selects from the neighborhood, the schedule that has the smallest X' < X of those where Y' < Y. The proposed simulated annealing heuristics use a similar framework as their NS counterparts. The major difference is that during each cycle, a move can be accepted as the best even if it has a higher Y
21st International Conference on Computers and Industrial Engineering
259
criteria. Heuristics Q-SA and G-SA accept moves in the same manner as Q-NS and G-NS, respectively, but also accept a move when Y' > Y if r, a random number in the interval (0,1), is less than exp(-(Y'-Y)/BZ). B is defined as the cooling parameter and it is an initial input. Given that Y'-Y is always positive in this case (Y'>Y), then exp(-(Y'-Y)/Bz) always falls within the (0,1) range. Note that as the number of cycles, Z, increases, the probability of accepting of a move is reduced (cooling effect). Finally, all four heuristics have the same complexity level. For each job j, there are N + M + N - 1 insertions and exchanges. There are N jobs and the maximum number of cycles, Z, is equal to N. Including all the steps, the heuristics have a computational complexity of O(N3).
4
COMPUTATIONAL EXPERIMENTS
The heuristics are compared to full enumeration for a problem instance where N=10 and M=2. Processing times and due dates are generated by a uniform distribution pj =U(1,99) and d =U(l,500/q) + pj, where q is defined as the traffic congestion ratio (Ho and Chang 1995). The higher the value of q, the higher the number of late jobs. The congestion ratio is set at three levels 3, 4, and 5. One hundred replications are taken at each experimental point. The cooling parameter, B, is set at 0.85 based on pilot experimental runs. The number of optimal solutions for a problem is defined as n g, while the number of optimal solutions found by a heuristic is nk (k = G-SA, G-NS, Q-NS, Q-SA). For example, in the problem depicted in Figure 1, the optimal solution is the set of schedules with criteria {(1,2.3), (0,2.7)}, therefore the value of n,~, = 2. The statistic of interest is the proportion of solutions found by each heuristic over the total number of optimal solutions; w~ = n k / nm (k = G-SA, G-NS, Q-NS, Q-SA). If in the example of Figure 1, the Q-SA heuristic had an efficient solution set equal to {(1,2.3), (0,3.0)}, then %s^ = 1, and WO.SA= 0.5. Table 1 shows the resulting average and standard deviation for the proportion w~ of the four heuristics under the three levels of q. This table also shows the best and worst case scenarios; percentage of time that a heuristic found all the optimal solutions for a problem (wk = 1), and the percentage of time it found none of the optimal solutions (w~ = 0). In general, the four heuristics performed well, finding more than 60% of the optimal solutions. The results show that there is a slight difference between the NS and SA heuristics, where in all cases, the SA heuristics performed slightly better than their NS counterparts. Also, as q increased, the difference between the SA and NS heuristics decreased. The
avg. (w~) std. dev. (w~)
Wk~ 1 Wk~0 avg. (w~) std. dev. (w~)
Wk~ 1 wk~O avg. (w,) std. dev. (w,)
Wb.~ l Wk'~--0
G-NS 62.9 30.8 35% 4% 70.5 32.6 47% 6% 77.5 27.4 57% 0%
Q-NS 65.9 29.1 36% 2% 76.8 23.0 46% 0% 71.2 26.1 42% 0%
G-SA 67.0 28.4 37% 2% 74.0 29.1 48% 5% 81.3 23.7 60% 0%
Table 1. Experiment Results.
Q.SA 68.2 29.4 40% 2% 78.2 23.4 50% 0% 72.2 25.3 42% 0%
260
21st International Conference on Computersand Industrial Engineering
performance of the G heuristics increases as q increases, while the performance of the Q heuristics is not affected by changes in q. The performance of the heuristics in regards to the worst and best case scenarios, follows a similar pattern to that of the proportions average; it improves in the G heuristics as q increases, and it is not affected in the Q heuristics as q increases. In relation to the optimal solution set, it has found that the number of optimal solutions per problem, n~, is 2.74, 2.48, and 2.24, when q is 3, 4, and 5, respectively. This shows that as q increases the number of tradeoff solutions decreases. This is intuitive since as q increases, there is more congestion, and therefore less flexibility. 5
CONCLUSIONS AND FUTURE WORK
This paper addresses the multi-criteria problem of scheduling N jobs on M identical parallel machines with the objective of minimizing the number of tardy jobs and the average flow-time simultaneously. The four heuristics presented generate a nondominated solution set based on simulated annealing and neighborhood search. When compared to the optimal set, the four heuristics perform well, finding more than sixty percent of all the optimal nondominated solution points. Future work includes the analysis of larger problems based on an estimated benchmark. Finally, pilot experiments have shown that even in larger problems (80 jobs) the number of optimal/efficient solutions is small ( < 7). This result is very interesting in relation to the applicability of the problem and methodology to industrial settings. Users can pick from the small set of efficient schedules with relative ease (given the small number of alternatives) based on other qualitative criteria or a secondary quantitative criteria. REFERENCES Bemardo, J. J., and K. Lin (1994). An interactive Procedure for B-criteria scheduling with Flow Time and Earliness Penalties. Journal of Global Optimization 3, 289-309. Blazewicz, J., K. Ecker, G. Schmidt, and J. Weglarz (1993). Scheduling in Computer and Manufacturing Systems. Springer-Verlag, Berlin. Chen, C., and R. L. Bulfin (1993). Complexity of single machine, multi-criteria scheduling problems. European Journal of Operational Research 70, 115-125. Morton, T. E., and D. W. Pentico (1993). Heuristic Scheduling Systems. Wiley, New York. Nagar, A., J. Haddock, and S. Heragu (1995). Multiple and bicriteria scheduling: A literature survey. European Journal of Operational Research 81, 88-104. Zegordi, S. H., I. Itoh, and T. Enkawa (1995). Minimizing makespan for flow shop scheduling by combining simulated annealing with sequencing knowledge. European Journal of Operational Research 85, 515-531.
Pergamon Plh S0360.8352(97)00087-9
Simulated Annealing Heuristics for the Average Mow-Time and the Number of Tardy Jobs Bi.Criteria Identical Parallel Machine Problem Alex J. Ruiz-Torres', E. Emory Enscore h, and Russell R. Bartonh dDepartment of Management College of Business Florida Gulf Coast University, Ft. Myers, FL "Department of Industrial and Manufacturing Engineering College of Engineering, 207 Hammond Building Pennsylvania State University, University Park, PA
ABSTRACT This paper presents the first part of a study of a scheduling problem where the objective is to minimize both the number of jobs late and the average flow-time. Unlike previous approaches, a weighted function of the two criteria is not utilized. Heuristics based on simulated annealing and neighborhood search are proposed with the objective of generating schedules that produce a compromise across both criteria. Simulation experiments showed that the heuristics perform well when compared to the optimal solution set for a small problem instance. © 1997 Elsevier Science Ltd KEYWORDS Scheduling; Multiple Criteria; Parallel Machines; Average Flow-Time; Number of Tardy Jobs; Simulated Annealing.
1
INTRODUCTION
The relevance of parallel machine scheduling has been well documented and includes applications in manufacturing, service, and data processing settings. Most of the work in parallel machines has concentrated on single criteria problems. However, several researchers have recognized the importance of considering multiple measures of performance as a better representation of today's decision making (Nagar et al. 1995). This is because single criteria optimal solutions may prove too expensive in relation to other important criteria. This research utilizes the efficient frontier approach which has a stronger appeal in many settings, as it allows users to pick the best tradeoff between the criteria based on other dynamic conditions (Bemardo and Lin 1994). This paper presents the initial results on the performance of four heuristics to optimally solve the problem of scheduling N jobs on M identical parallel machines with the objective of generating tradeoff schedules for the average flowtime (F) and number of tardy jobs (Nt) criteria. As per the notation used in Blazewicz et al. (1993), the problem is defined as P/IF, Nt. The heuristics are based on neighborhood search and simulated annealing, methodologies that have been widely used in scheduling problems (Zegordi et al. 1995). The remainder of this paper includes a problem definition and its complexity, a description of the four heuristics, and finally, computational results and conclusions.
2
PROBLEM DEFINITION AND COMPLEXITY
In general there are M identical machines to process N one operation jobs. Each job j has a deterministic processing time pj and due date dj. All jobs are ready at time zero. Preempuon, alvtston or can257
258
21st International Conference on Computers and Industrial Engineering
cellations of jobs are not allowed. Finally, M and N are fixed. The objective is to generate the nondominated (efficient) schedules/solutions which minimize the average flow-time (F) and the number of tardy jobs (Nt). A schedule S is efficient if there is no schedule S' such that F >_.F' and Nt > Nt', where at least one inequality holds strictly. These two criteria can seldom be optimized simultaneously, therefore tradeoffs exist. This is shown in Figure 1 (each job is identified by its processing time and due date). The problem P / / F , Nt is NPHard given one of its 'component' single criteria problems is NP-Hard (P//Nt is NP-Hard: Ho and Chang 1995). The bi-criteria problem is NP-Hard given the single criteria Nt solution is part of the nondominated set, and if finding one of the solutions in the set is NP-Hard, finding the total set is NPHard. This was shown in general by Chen and Bulfin (1993).
ml m2
3,3
I F=2.3 Nt = 1
ml m2
F=2.7 Nt=0
Fig. 1. Tradeoffs between F and Nt.
3
FOUR HEURISTIC ALGORITHMS
This section describes four heuristics to solve the P / I F , Nt with the objective of generating a set of efficient alternative schedules (nondominated solution set). These heuristics are based on neighborhood search methods. The basic elements in this methodology are (Morton and Pentico 1995): the starting solution (original seed), the solutions close to the original solution (neighbor of the seed), a method for selecting the new seed (selection criterion), and a method for terminating the procedure (termination criterion). The difference among the four heuristics is the speed of the search and the use of simulated annealing, both reflected in the selection criterion step. First the similarities are described, and then the differences between all four heuristics are described. Original seed:: Two seeds are used for every problem (therefore two searches): 1) The schedule generated by the well known parallel machine SPT algorithm, and 2) The best (minimum Nt) schedule generated from the four Ho and Chang (1995) algorithms to minimize Nt in parallel machines. Neighborhood of the seed:: Two methods - pairwise interchanges, where each job j (in SPT order), is swapped with all other jobs N-I, and single job insertions, where each jobj is inserted in all positions of all M machine sequences. When the complete neighborhood has been evaluated for job j, then the best move (selection criterion) is accepted. The best move is different for heuristics GNS/SA and Q-NS/SA. An efficiency check (Section 2) is made after each move. Termination criterion:: When no reduction of the Y criteria is made, or when the number of cycles through the search, Z, is equal to the number of jobs N. These four heuristics are based on the assumption that each of the two original seeds has the best single criteria solution (criteria X); the schedule generated by Ho and Chang (1995) has the lowest number of tardy jobs criteria possible, and the SPT schedule has the lowest average flow-time criteria possible. In that context, the search is made in the direction of the other criteria (criteria I/). Heuristic Q-NS (Quick-Neighborhood Search) selects from the neighborhood, the schedule with the lowest Y' (if there are ties, the lowest X'). Heuristic G-NS (Gradual-NS) selects from the neighborhood, the schedule that has the smallest X' < X of those where Y' < Y. The proposed simulated annealing heuristics use a similar framework as their NS counterparts. The major difference is that during each cycle, a move can be accepted as the best even if it has a higher Y
21st International Conference on Computers and Industrial Engineering
259
criteria. Heuristics Q-SA and G-SA accept moves in the same manner as Q-NS and G-NS, respectively, but also accept a move when Y' > Y if r, a random number in the interval (0,1), is less than exp(-(Y'-Y)/BZ). B is defined as the cooling parameter and it is an initial input. Given that Y'-Y is always positive in this case (Y'>Y), then exp(-(Y'-Y)/Bz) always falls within the (0,1) range. Note that as the number of cycles, Z, increases, the probability of accepting of a move is reduced (cooling effect). Finally, all four heuristics have the same complexity level. For each job j, there are N + M + N - 1 insertions and exchanges. There are N jobs and the maximum number of cycles, Z, is equal to N. Including all the steps, the heuristics have a computational complexity of O(N3).
4
COMPUTATIONAL EXPERIMENTS
The heuristics are compared to full enumeration for a problem instance where N=10 and M=2. Processing times and due dates are generated by a uniform distribution pj =U(1,99) and d =U(l,500/q) + pj, where q is defined as the traffic congestion ratio (Ho and Chang 1995). The higher the value of q, the higher the number of late jobs. The congestion ratio is set at three levels 3, 4, and 5. One hundred replications are taken at each experimental point. The cooling parameter, B, is set at 0.85 based on pilot experimental runs. The number of optimal solutions for a problem is defined as n g, while the number of optimal solutions found by a heuristic is nk (k = G-SA, G-NS, Q-NS, Q-SA). For example, in the problem depicted in Figure 1, the optimal solution is the set of schedules with criteria {(1,2.3), (0,2.7)}, therefore the value of n,~, = 2. The statistic of interest is the proportion of solutions found by each heuristic over the total number of optimal solutions; w~ = n k / nm (k = G-SA, G-NS, Q-NS, Q-SA). If in the example of Figure 1, the Q-SA heuristic had an efficient solution set equal to {(1,2.3), (0,3.0)}, then %s^ = 1, and WO.SA= 0.5. Table 1 shows the resulting average and standard deviation for the proportion w~ of the four heuristics under the three levels of q. This table also shows the best and worst case scenarios; percentage of time that a heuristic found all the optimal solutions for a problem (wk = 1), and the percentage of time it found none of the optimal solutions (w~ = 0). In general, the four heuristics performed well, finding more than 60% of the optimal solutions. The results show that there is a slight difference between the NS and SA heuristics, where in all cases, the SA heuristics performed slightly better than their NS counterparts. Also, as q increased, the difference between the SA and NS heuristics decreased. The
avg. (w~) std. dev. (w~)
Wk~ 1 Wk~0 avg. (w~) std. dev. (w~)
Wk~ 1 wk~O avg. (w,) std. dev. (w,)
Wb.~ l Wk'~--0
G-NS 62.9 30.8 35% 4% 70.5 32.6 47% 6% 77.5 27.4 57% 0%
Q-NS 65.9 29.1 36% 2% 76.8 23.0 46% 0% 71.2 26.1 42% 0%
G-SA 67.0 28.4 37% 2% 74.0 29.1 48% 5% 81.3 23.7 60% 0%
Table 1. Experiment Results.
Q.SA 68.2 29.4 40% 2% 78.2 23.4 50% 0% 72.2 25.3 42% 0%
260
21st International Conference on Computersand Industrial Engineering
performance of the G heuristics increases as q increases, while the performance of the Q heuristics is not affected by changes in q. The performance of the heuristics in regards to the worst and best case scenarios, follows a similar pattern to that of the proportions average; it improves in the G heuristics as q increases, and it is not affected in the Q heuristics as q increases. In relation to the optimal solution set, it has found that the number of optimal solutions per problem, n~, is 2.74, 2.48, and 2.24, when q is 3, 4, and 5, respectively. This shows that as q increases the number of tradeoff solutions decreases. This is intuitive since as q increases, there is more congestion, and therefore less flexibility. 5
CONCLUSIONS AND FUTURE WORK
This paper addresses the multi-criteria problem of scheduling N jobs on M identical parallel machines with the objective of minimizing the number of tardy jobs and the average flow-time simultaneously. The four heuristics presented generate a nondominated solution set based on simulated annealing and neighborhood search. When compared to the optimal set, the four heuristics perform well, finding more than sixty percent of all the optimal nondominated solution points. Future work includes the analysis of larger problems based on an estimated benchmark. Finally, pilot experiments have shown that even in larger problems (80 jobs) the number of optimal/efficient solutions is small ( < 7). This result is very interesting in relation to the applicability of the problem and methodology to industrial settings. Users can pick from the small set of efficient schedules with relative ease (given the small number of alternatives) based on other qualitative criteria or a secondary quantitative criteria. REFERENCES Bemardo, J. J., and K. Lin (1994). An interactive Procedure for B-criteria scheduling with Flow Time and Earliness Penalties. Journal of Global Optimization 3, 289-309. Blazewicz, J., K. Ecker, G. Schmidt, and J. Weglarz (1993). Scheduling in Computer and Manufacturing Systems. Springer-Verlag, Berlin. Chen, C., and R. L. Bulfin (1993). Complexity of single machine, multi-criteria scheduling problems. European Journal of Operational Research 70, 115-125. Morton, T. E., and D. W. Pentico (1993). Heuristic Scheduling Systems. Wiley, New York. Nagar, A., J. Haddock, and S. Heragu (1995). Multiple and bicriteria scheduling: A literature survey. European Journal of Operational Research 81, 88-104. Zegordi, S. H., I. Itoh, and T. Enkawa (1995). Minimizing makespan for flow shop scheduling by combining simulated annealing with sequencing knowledge. European Journal of Operational Research 85, 515-531.