The 11th Asia Pacific Industrial Engineering and Management Systems Conference The 14th Asia Pacific Regional Meeting of International Foundation for Production Research Melaka, 7 – 10 December 2010
A Heuristic Method in Nurse Rerostering Following a Sudden Absence of Nurses Manabu Kitada† 1and Kazuko Morizawa2 Dept. of Electrical and Information Systems, Graduate School of Engineering Osaka Prefecture University 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka, 599-8531, JAPAN Email:
[email protected] [email protected] Hiroyuki Nagasawa Osaka Prefectural College of Technology 26-12 Saiwai-cho, Neyagawa, Osaka, 572-8572, JAPAN Email:
[email protected]
Abstract - This paper considers nurse rerostering problems triggered by a sudden incident that at least one nurse cannot fulfill tasks assigned in a predetermined schedule. Since every nursing-shift should satisfy required numbers and skill levels of nurses at every shift, the scheduler should find a nurse who can fill up the vacancy of the absentee and modify some task assignments for the related nurses on and after the day in the schedule quickly. Furthermore, the modification of the schedule in rerostering should be as small as possible to avoid confusion in implementing tasks according to the predetermined schedule. Therefore, this paper proposes a heuristic nurse rerostering method for finding an optimal schedule with the minimum number of re-assigned tasks by incorporating a tree-search algorithm into a recursive search algorithm to generate feasible schedules in rerostering. The proposed method first generates a new feasible schedule by using the recursive search algorithm. Setting the feasible schedule as an incumbent, backtracking to promising nodes in the search tree constructed in the recursive search process is implemented according to ‘depth-first search’ strategy for obtaining better feasible schedules. Some numerical examples in the 3-shift nurse rerostering problems with real data demonstrate that the proposed method can generate acceptable schedules for practical use efficiently. Keywords: Nurse scheduling, rerostering, heuristic, search tree, backtracking.
1. INTRODUCTION Nurse scheduling problem (or sometimes called nurse rostering problem) is to generate a schedule by assigning nurses to every-day-shifts in a scheduling horizon so as to satisfy a lot of conditions such as assigning the required number of nurses with appropriate skill at each shift; prohibiting any violation of both monthly limits for evening- and night-shifts according to the related regulation law; balancing workloads among nurses every month; maintaining standard shift patterns for each nurse as often as possible; meeting requests from each nurse for taking holiday or a special assignment of an evening- or nightshift, and considering a good combination of personnel based on personal relationships among nurses. Since most
________________________________________ † : Corresponding Author
nurse scheduling problems are over-constrained combinatorial optimization problems, it is difficult to generate such a schedule satisfying all these conditions completely. Therefore, schedulers, usually head/chief nurses in Japan, take a long time and much effort to make a schedule acceptable for all nurses and allowable in various rules and law. Aiming at generating better schedules automatically and reducing the head/chief nurses‟ effort, various kinds of approaches have been developed for the problems (Ikegami and Niwa 1998/2003, Nonobe and Ibaraki 1998, Nonobe and Ibaraki 1995, Gutjahr and Rauner 2007, Morizawa and Nagasawa 2007, Burcker et al. 2009 etc.). Most of the existing nurse scheduling methods are “static” in the sense that they generate a schedule for a
The 11th Asia Pacific Industrial Engineering and Management Systems Conference The 14th Asia Pacific Regional Meeting of International Foundation for Production Research Melaka, 7 – 10 December 2010 given scheduling horizon under given conditions, but there exists another type of nurse scheduling problem called a “nurse rerostering problem.” Nurse rerostering problems arise in usual hospitals almost every week when at least one nurse becomes in no condition to fulfill tasks assigned in a predetermined schedule. Since the absence is announced early in the morning on any day, the scheduler should find a nurse who can fill up the vacancy of the absentee from among nurses in his/her module unit and generate a new schedule satisfying all important conditions by modifying some task assignments for the related nurses on and after the day in the current schedule as quickly as possible to prepare the start of the day-shift on that day. Unfortunately, it is not a good idea to apply promising static nurse scheduling methods for rerostering under the new condition, because the modification of the schedule in rerostering should be as small as possible to avoid any confusion in implementing the tasks according to the predetermined schedule. Therefore, it is crucial to develop a nurse rerostering method for its own purpose. For nurse rerostering, Moz and Pato (2004) formulated it as an integer multicommodity flow problem and solved it using a standard software package CPLEX 7.0. Pato and Moz (2007) also proposed some kinds of genetic algorithms for the single-objective nurse rerostering problem, and Pato and Moz (2008) expanded it into a biobjective version. On the other hand, Kitada et al.(2009) proposed a heuristic recursive search method to find a new schedule by repeating two kinds of task assignment procedures: (1) a task re-assignment for filling up the absent position with an appropriate nurse, and (2) a „dayoff‟ task assignment for eliminating any violation of the feasible conditions in the schedule in a recursive way. This paper extends the nurse rerostering method proposed by Kitada et al.(2009) and develops a heuristic method for obtaining an optimal schedule with the minimum number of re-assigned tasks by incorporating a tree-search algorithm into the recursive search method. Some numerical examples in the 3-shift nurse rerostering problems with real data demonstrate the effectiveness of the proposed method.
2. THE NURSE REROSTERING PROBLEM This paper considers the nurse rerostering problems with the same assumptions as those given by Kitada et al. (2009) as follows. General assumptions (1) All nurses work in three shifts: night-shift (0:00-8:00), day-shift (8:00-16:00) and evening-shift (16:00-24:00). (2) Each nurse is assigned to only one task every day from among a day-shift, an evening-shift, a night-shift, a meeting and a day-off. Every meeting is held during
8:00-16:00(day-shift hours). (3) Every nurse belongs to one of the nursing modules (teams) and any temporary change of module is not allowed. (4) In each shift and in each module, the number of nurses required to work together is prespecified individually for workday and for Sunday or holiday. (5) Nurses are classified into five ranks according to their own skill and experience, that is, „module leader (leader, in short),‟ „highly experienced,‟ „experienced,‟ „second year‟ and „newcomer.‟ (6) The required mean skill level of assigned nurses is prespecified for each shift on each day. (7) A skillful nurse should be assigned together with each newcomer so that any newcomer can get on-the-job training. (8) Scheduling horizon for static scheduling is one month (30 days). (9) Lower and upper bounds for the number of shift assigned to each nurse during each scheduling horizon are prespecified. (10) Prohibited shift patterns to be avoided and standard shift patterns to be maintained are given in advance. (11) Requests with their priority for the task assignment such as days-off, meeting or special combination of shifts are submitted to a head/chief nurse by each nurse in advance. Requests for meeting have a priority higher than those for the other tasks or day-off. (12) Personal relationships among nurses are known as „good,‟ „neutral‟ and „bad.‟ Nurses with bad relationship should not be assigned to a night- or evening-shift if possible. (13) The following historical information on task assignment to each nurse for the last two months is available, and should be considered in assigning tasks for the next scheduling horizon. - acceptance ratio of the requests for the task assignment during the last two months - number of evening-shifts assigned for the last two months - number of night-shifts assigned for the last two months - date of the latest evening-shift in the last month - date of the latest night-shift in the last month (14) A schedule for the next scheduling horizon is presented to nurses by the end of the current scheduling horizon. The schedule satisfies the following four conditions: (a) In each shift on each day, the number of nurses assigned to the shift is larger than or equal to the prespecified number both in total and for each module. (b) In each shift on each day, the required mean skill level is satisfied by nurses assigned to the shift.
The 11th Asia Pacific Industrial Engineering and Management Systems Conference The 14th Asia Pacific Regional Meeting of International Foundation for Production Research Melaka, 7 – 10 December 2010 (c) Numbers of the day-, evening-, and night-shift assigned to each nurse in each scheduling horizon are within the corresponding range. (d) No prohibited shift pattern is assigned to any nurse. Moreover, requests for task assignment posed by nurses should be satisfied as well as possible, and standard shift patterns for each nurse should be realized as frequently as possible in the schedule. Nurse rerostering will occur after the next scheduling horizon starts, and then the scheduling horizon is called the “current” scheduling horizon in discussing nurse rerostering. Additional assumptions in rerostering (15) Nurses usually fulfill their tasks according to a predetermined schedule for the current scheduling horizon. (16) An absence of a nurse caused by an unexpected event/accident suddenly happens and is informed to a scheduler (or a head/chief nurse) before 8:00 a.m. on any day during the scheduling horizon. (17) When an absence is announced, the scheduler immediately starts generating new schedules for filling up the absence by modifying some task assignments in the current schedule. The modification process is called „rerostering.‟ (18) Except the night-shift and meeting on that day, any task assigned on and after that day in the predetermined schedule can be changed in rerostering. (19) The new schedules obtained through the rerostering process should satisfy all conditions (a) to (d) in assumption (14).
3. PROPOSED METHOD Under the above assumptions, this paper proposes a heuristic method for the nurse rerostering problem with a sudden absence of any nurse. In rerostering, it is very hard (often impossible) to generate a schedule satisfying all conditions completely. Therefore, as described in assumption (19), the proposed method aims at generating schedules which should satisfy at least some important conditions such as the required number and skill levels of nurses for each shift, the prohibited shift patterns and the limits of workload for individual nurse. A schedule is called “feasible” when it satisfies all of these important conditions. Kitada et al. (2009) proposed a recursive search method which can generate a “feasible” schedule efficiently. Though the method is effective enough, but it only aims at finding a feasible schedule in rerostering. To enhance its performance in terms of minimization of number of reassigned tasks, this paper proposes a heuristic tree-search
method. The proposed method first finds a feasible schedule by using the recursive search method. In this process, an initial search tree is constructed by generating nodes corresponding to all candidate nurses who can fill up the absent position at each recursive level. Setting the feasible schedule as an incumbent, backtracking to promising nodes along the search tree is implemented according to „depth-first search‟ strategy for obtaining better feasible schedules. 3.1 Recursive Search Method The basic procedure of the recursive search method is summarized in the following three steps, where a nurse assigned to task k on the t-th day in the predetermined schedule is assumed to become absent. Step 1: Find a nurse i who is assignable to task k on the t-th day without violating the condition of the required skill level of the shift (called an „assignable nurse,‟ in short) and assign the nurse to the absent position. Step 2: If the assignment in Step 1 results in any violation of the other important conditions for the nurse i, then change a task assigned to the nurse i on and after the (t+1)-th day in the current schedule into a „day-off‟ task so as to eliminate the violation. Let the t’-th day be the day on which a „day-off‟ task is newly assigned, and task k’ be the task pre-assigned to nurse i on the t’-th day. Step 3: If the day-off assigned in Step 2 results in any violation of the other important conditions, then deal with the „day-off‟ as a new absence of nurse, set t =:t’ and k =:k’, and go back to Step 1. Otherwise, terminate the procedures. In Step 1, nurse i is selected from among assignable nurses according to a priority rule based on the importance among conditions. The above procedures are implemented recursively until the schedule has no more violated condition, resulting in the “initial schedule” with an “initial search tree” to be used in the succeeding tree-search algorithm. The initial search tree is constructed by generating nodes corresponding to all candidate nurses in Step 1 at each recursive level. Note that the “height” of the search tree means the number of re-assigned tasks in rerostering. 3.2 Tree Search Algorithm If the height of the initial search tree is larger than two, the following tree-search algorithm is implemented in order to find better schedules with less number of task reassignments. Let h be number of re-assigned tasks for the current best schedule (incumbent) and its initial value is set as the height of the initial search tree. Step 1: Find an active node i with the deepest search level
The 11th Asia Pacific Industrial Engineering and Management Systems Conference The 14th Asia Pacific Regional Meeting of International Foundation for Production Research Melaka, 7 – 10 December 2010 in the current search tree. If there exists more than two active nodes with the deepest search level, select a node according to the priority of assignable nurses in the same way as used in Step 1 in the recursive search method. Step 2: Backtrack to the node i, and implement the recursive search procedures given in the previous section provided that nurse i is set as the nurse corresponding to node i and that the related task k and date t are also set as a new task and date in node i, respectively. If either the height of the current search tree reaches h or the search fails to find a feasible schedule, then terminate the recursive search and set h’=h+1. Otherwise, h’ is set as the height of the current search tree. Step 3: If h’< h, then set the current schedule as the incumbent schedule with h:=h’. Step 4: If there exists no active nodes in the current search tree, then terminate. Otherwise, go to Step 1.
4. NUMERICAL EXPERIMENTS To evaluate the effectiveness of the proposed method, it is applied to some instances of a three-shift nurse scheduling problem with a sudden absence under real conditions. Experimental design is given as follows: • Number of nurses is 24. • Number of nursing module is 3. • Composition of nursing modules and skill levels of nurses are given in Table 1. • Required numbers of nurses for each shift are 10 in total and at least 3 for each module every day-shift, 3 in total and at least 1 for each module every evening-shift, and 3 in total and at least 1 for each module every night-shift. • Required mean skill level of assigned nurses for each
shift every day is 1.3. • Lower and upper bounds (LB, UB) for the monthly number of shift assigned to each nurse are (5, 15) for each day-shift, (2, 8) for each evening-shift, (2, 8) for each night-shift and (4, 9) for the sum of evening- and night-shifts. • Prohibited shift patterns with the first priority are six days consecutive shifts without day-off (prohibited in law); three consecutive night-shifts; day-shift, meeting or night shift succeeding immediately after the eveningshift. • Standard shift pattern is a sequence of tasks composed of up to three successive day-shifts, night-shift, eveningshift and finally day-off. • Personal relationships between any pair of nurses and historical information on task assignment to each nurse for the last two months are given but omitted to save space. Eight schedules are obtained first by solving the static nurse scheduling problem randomly generated under the above conditions by using a heuristic method proposed by Morizawa and Nagasawa (2007). For these schedules, 3840 instances of nurse rerostering problems are generated by inserting an absence to any task (except „meeting‟ and „dayoff‟) assigned to any nurse on the t-th day in the predetermined static schedule. The algorithm of the proposed method is coded in C-language and is run on a personal computer (Pentium IV, 2.8 GHz). Tables 2 and 3 show the results for one of instances obtained by the original recursive search method by Kitada et al. (2009) and by the proposed method, respectively. In this instance, an absence is supposed to happen to No.6 nurse on the 9th day, and green cells in these tables represent the tasks re-assigned in rerostering.
Table 1: Nursing module and skill level of each nurse. Module 1 Nurse No.
Rank
Module 2 Skill
Nurse
level
No.
Rank
Module 3 Skill
Nurse
level
No.
Rank
Skill level
1
leader
5
2
leader
5
3
leader
5
4
highly-experienced
3
5
highly-experienced
3
6
highly-experienced
3
7
highly-experienced
3
8
experienced
1
9
experienced
1
10
experienced
1
11
experienced
1
12
experienced
1
13
experienced
1
14
experienced
1
15
experienced
1
16
second-year
0.8
17
second-year
0.8
18
second-year
0.8
19
second-year
0.8
20
second-year
0.8
21
newcomer
0.5
22
newcomer
0.5
23
newcomer
0.5
24
newcomer
0.5
The 11th Asia Pacific Industrial Engineering and Management Systems Conference The 14th Asia Pacific Regional Meeting of International Foundation for Production Research Melaka, 7 – 10 December 2010 Table 2: An example of new schedules obtained by the recursive search algorithm (Kitada et al. 2009)
module 2
skill level
module 3
nurses (24 persons)
module 1
1 1 4 7 10 13 16 19 22 2 5 8 11 14 17 20 23 3 6 9 12 15 18 21 24
D N M D E D D E D N D D D N D E D
2
3
E N D D D D D E N D D N D E D D
D E N D D N D E D D D E D D D N
4 D D E D N E D D D N D N D D D E
5 N D D E D D D D D E N E D D D N -
6 E N D D D D D M D N D E D D E M N D
7 E N D D D D D E D N D D D N E D
8 D E D D N D N D D E D D D E N D
9 D D N E D D E D N D N N D D E D D
scheduling period (30 days) 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 D D D N E D N D E D D E D D D N -
D D D E D N E D N D D N D D E D
N M D D E D M E D D D N E D D N D D
D D N D E D N D D E E D D D N D -
E N D D D D D N E D D D D E N D -
E D D D N N E D D D D D D E D N
N D D E D E D N D D D D N E D D
D D E D D N M D D D E N N E D D D
D E N D D D D N D D E N D D D E
D D D N D E N D E D D E N D D D M -
D N D E M D E D D N D D D E N D D -
N E D D D D D E N D D D D E N D
E D D N D D N D E D D D D E D N
N D D E D D E D N D D D N D D E
D E D N D D D E D D N N E D D D -
M E D D D N N D D D E D D D N E D
D E D D N D D D E N D N D D D E -
D N E D D N D D E D M D E N D D D -
D D D E N E D D D N D N E D D D
N D D D E D D D E N E D N D D D
E N D D D D D N D D E D E D N D
D
1.44 1.44 1.46 1.91 1.73 1.69 1.41 1.66 2.13 1.86 1.73 1.63 1.73 1.37 1.89 1.31 1.39 1.89 2.10 1.41 1.46 1.47 1.74 1.89 1.64 1.93 1.39 2.11 1.88 1.46
E
1.60 1.50 1.60 2.10 1.50 2.27 2.33 1.53 1.50 2.17 1.60 2.10 1.67 1.43 1.67 2.17 2.33 1.53 1.67 2.20 1.67 1.43 1.50 2.33 2.00 1.67 2.20 1.50 2.33 1.43
N
1.60 1.60 2.10 1.50 2.27 2.33 1.53 1.50 2.27 1.60 2.17 1.67 1.43 1.67 2.17 1.43 1.53 1.67 2.10 1.67 2.10 2.17 2.17 2.17 1.67 2.20 1.50 1.67 1.43 2.17
(symbols „D,‟ „N,‟ ‟E,‟ „M‟ and „-‟ denote day-shift, night-shift, evening-shift, scheduled meeting and day-off, respectively)
module 2
skill level
module 3
nurses (24 persons)
module 1
Table 3: An example of new schedules obtained by the proposed method 1 4 7 10 13 16 19 22 2 5 8 11 14 17 20 23 3 6 9 12 15 18 21 24
1
2
3
4
5
6
7
8
9
D － N M D － E D D E D N D － D － － D － N － D E D
E － － N D D － D D － D E N D － － D N D E D － － D
－ D － E N D － D N D － － E － D D D E － － D D D N
D D － － E D N － E － D D － D N D N － － D D D － E
N D D E － － D － － D D D － D E N E D D D N － － －
E N D － D － D － D D M D N D － E － － D D E M N D
－ E N D D D － D － D E － D N D － D － D － － N E D
D － E － D D N － D N － － D D E － D D － D E N － D
－ D － D － N E D D E D － N － － D N N D D － E D D
scheduling period (30 days) 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 D D D － N E － D N － － D E D D － － E D D D － N －
D － D D E － D N E D － N － － D D N － － － D D E D
N M D D － E D M － D D E D D － N － E D N D － － D
D D － N D － － E － N D － D D － E E － D D N D D －
E N D D － D D － D － N E D － D － － D D － E N D －
－ E D － D D N － － N E － D D D D D － D － － E D N
－ － N D － D E D － E － D － N D D D D N E － － D D
D D E D D － － N M － D D D E － N － N － E D － D D
－ D E N D － D D D － N D D － － E － N D － D D － E
D D － D N D － E N D E － D － D － － E N D D D M －
D N D － E M D － E D － D N D D － D － E N － D D －
N D D E － D D － － D － D E N － D E D － D － N － D
E － D － － D N D D N － D － E D D － － D － D E D N
－ － N D D － E D D E D N － － D D D － N － D － D E
D － E － D N － D D － D E D D N － － N E － D D D －
M E － D － D D N N D D － － D E － D D － D N － E D
D E D D － N D － － D D － E N － D N D D D E － － －
－ － － D N E D D N D D － － E D M D E N D － D D －
D － － D D E － N E D － D D － N D N － E － － D D D
N D D D － － － E － D － D D － E N E D － N D － D D
E N D － D D D － D － N D D － － E － D － E － D N D
D
1.44 1.44 1.46 1.91 1.73 1.69 1.41 1.66 2.13 1.86 1.73 1.63 1.73 1.37 1.89 1.31 1.39 1.89 2.10 1.41 1.46 1.47 1.74 1.89 1.64 1.93 1.39 2.11 1.88 1.46
E
1.60 1.50 1.60 2.10 1.50 2.27 2.33 1.53 1.50 2.17 1.60 2.10 1.67 1.43 1.67 2.17 2.33 1.53 1.67 2.20 1.67 1.43 1.50 2.33 2.00 1.67 2.20 1.50 2.33 1.43
N
1.60 1.60 2.10 1.50 2.27 2.33 1.53 1.50 2.27 1.60 2.17 1.67 1.43 1.67 2.17 1.43 1.53 1.67 2.10 1.67 2.10 2.17 2.17 2.17 1.67 2.20 1.50 1.67 1.43 2.17
The 11th Asia Pacific Industrial Engineering and Management Systems Conference The 14th Asia Pacific Regional Meeting of International Foundation for Production Research Melaka, 7 – 10 December 2010 From Tables 2 and 3, it is obvious that the number of reassigned tasks to get a feasible schedule after rerostering processes is only 12 for the proposed method and much smaller than 21 for the original recursive search method.
5. CONCLUSION A nurse rerostering problem with a sudden absence of nurses was formulated on the basis of real data in a Japanese hospital, and a heuristic method based on the recursive search method proposed by Kitada et al. (2009) was proposed in this paper. The proposed method is developed by incorporating a tree-search algorithm into the recursive search algorithm for obtaining an optimal schedule to minimize the number of re-assigned tasks. The results of numerical experiments showed that the method can obtain feasible solutions for almost all instances and the tree-search algorithm is effective for finding better solutions in terms of the number of re-assigned tasks. The computation time of the proposed method is sufficiently small within ten seconds. We can adopt some other criteria such as the number of days affected by rerostering or the number of nurses affected by rerostering in the proposed method.
Pacific Industrial Engineering & Management Systems Conference, Kitakyusyu, Japan, 9 pages (CD-ROM). Morizawa, K. and Nagasawa, H. (2007) A heuristic method in static nurse scheduling, Proceedings of the 19th International Conference on Production Research, Valparaiso, Chile, 6 pages (CD-ROM). Moz, M. and Pato, M. V. (2004) Solving the problem of rerostering nurse schedules with hard constraints: new multicommodity flow model, Annals of Operations Research, 128, 179-197. Nonobe, K. and Ibaraki, T. (1998) A tabu search approach to the constraint satisfaction problem as a general problem solver, European Journal of Operational Research, 106, 599-623. Pato, M. V. and Moz, M. (2007) A genetic algorithm approach to a nurse rerostering problem, Computers & Operations Research, 34, 667-691. Pato, M. V. and Moz, M. (2008) Solving a biobjective nurse rerostering problem by using a utopic pareto genetic heuristic, Journal of Heuristics, 14, 359-374. AUTHOR BIOGRAPHIES
ACKNOWLEDGMENT This work was supported by Grant-in-Aid for Scientific Research (C) 22510157.
REFERENCES Brucker, P., Burke, E. K., Curtois, T., Qu, R. and Vanden Berghe, G. (2009) A shift sequence based approach for nurse scheduling and a new benchmark dataset, Journal of Heuristics (Online first). Gutjahr, W.J. and Rauner, M.S. (2007) A ACO algorithm for a dynamic regional nurse-scheduling problem in Austria, Computers & Operations Research, 34, 642-666. Ikegami, A. and Niwa, A. (1998) An efficient approach to nurse scheduling – implementation for a 2-shift case, Journal of Operations Research Society in Japan, 41, 572-587(in Japanese). Ikegami, A. and Niwa, A. (2003) A subproblemcentric model and approach to the nurse scheduling problem, Mathematical Programming Series B, 97, 517541. Kitada, M., Morizawa, K. and Nagasawa, H. (2009) Recursive search approach in nurse rerostering following a sudden absence of nurses, Proceedings of the 10th Asia
Manabu Kitada is a graduate student for master‟s course in Department of Electrical and Information Systems, Division of Electrical Engineering and Information Science, Graduate School of Engineering, Osaka Prefecture University, Japan. His research interests include production management and staff scheduling. His e-mail address is Kazuko Morizawa is an Associate Professor in Department of Electrical and Information Systems, Division of Electrical Engineering and Information Science, Graduate School of Engineering, Osaka Prefecture University, Japan. She received a Doctoral Degree from the Graduate School of Engineering at Osaka Prefecture University, Japan in 1994. Her teaching and research interests include production management, production scheduling and staff scheduling. Her e-mail address is Hiroyuki Nagasawa is a President in Osaka Prefectural College of Technology and an Emeritus Professor at Osaka Prefecture University, Japan. He received a Doctoral Degree from the Graduate School of Engineering at Kyoto University, Japan in 1985. His research interests include production management, flexible manufacturing systems, multiobjective planning and scheduling under uncertainty. His e-mail address is
