Part I. Scheduling optimization for minimizing single machine with weighted earliness and tardiness in paper industry
5. Computational experiments
To verify the performance of the VNS/TS algorithm, we tested the same set of problems as Biskup and Feldmann (2001), Feldmann and Biskup (2003) and Hino et al. (2005). The algorithm was code in C++ and implemented on a Pentium 4 3.2GHz with 512MB memory. Each instance was run 10 times and the best solution was selected.
The benchmark problem instances were provided by Biskup and Feldmann (2001), which can be obtained at http://people.brunel.ac.uk/~mastjjb/jeb/orlib/ schinfo.html (Beasley, 2005). There are seven different numbers of jobs (n=10, 20,50,100, 200,500,1000) with four restrictive due-date factors (h=0.2,0.4,0.6,0.8), where the common due date was determined by the expression
[ nj 1 j]
d = ×h
∑
= p . For each combination, 10 different problem instances were generated, resulting in a total of 280 problem instances.Before conducting a formal experiment for comparing with existing algorithms, the following four preliminary experiments will be carried out:
1. Determine if the insertion move can improve the algorithm. If yes, determine the best ratio of the two neighborhoods.
2. Determine the best number of generated schedules in a neighborhood.
3. Evaluate the algorithm with and without the use of the B/F local search.
4. Evaluate the algorithm with and without the combination of TS.
All of the four experiments were conducted for two job sizes n=200 and 500. To have a fair comparison, 1 second of computation time is set as the stopping criterion for all the instances with
200
n= , and 7 seconds for all the instances with n=500. To measure the effectiveness of algorithms, we compute the percentage improvement (PI) of the solution value obtained (F ) with a respect to the benchmark value provided by Biskup and Feldmann (2001; F ) as follows: BF
PI 100 a BF
BF
F F
F
= × − (2)
The benchmark values from Biskup and Feldmann (2001) are upper bounds on the optimal objective function values, except for the instances with 10 jobs which were solved optimally with LINDO software.
The result of the first experiment is for all the eight combinations of n and h, ratio 1:10 always gives the best solution, and hence it is chosen as the insertion/swap ratio in our algorithm.
The result of the second experiment shows that generating 5 points in a neighborhood is superior to generating 1 point or 10 points.
The third experiment evaluates the algorithm with and without the use of the B/F local search, which is implemented only for the loose due date (i.e., h=0.6,0.8). The results show that the B/F local search always results in an obvious improvement.
The fourth experiment evaluates the algorithm with and without the combination of TS, i.e., to compare the hybrid VNS/TS with the pure VNS. The results show that the hybrid VNS/TS indeed has a better performance.
In the formal experiment, we compare our VNS/TS algorithm with the best algorithms for the problem in the literature. The following parameters, determined by a series of pilot experiments, are used in our algorithm: TL1 = , 7 TL2 = (for 7 n=10, 20,50), 14 (for n=100, 200), 21 (for
500,1000
n= ); MaxIter=600 for h=0.2,0.4 and MaxIter=150 for h=0.6,0.8 , where MaxIter is the maximum number of iterations.
We compare our results with current best methods presented by Hino et al. (2005). They developed two metaheuristics and two hybrid algorithms: GA, TS, HTG (TS+GA), and HGT (GA+TS). The hybrid algorithms simply apply the metaheuristics in a sequential form. Among the
four algorithms, GA and HGT are the two best ones, where GA obtained an average percentage improvement (API) of −2.12% against those benchmark values with a mean processing time of 21.5 seconds; HGT obtained a −2.06% API with only 7.8 seconds.
The comparative results of our VNS/TS with the better one of GA and HGT is given in Table 1.
It can be observed that VNS/TS achieves better results in 14 and worst in 2 of 28 suites of problem instances and gains average 0.09% and 0.03% against HGT and GA respectively. Note that the results of VNS/TS for all 10-job problems are equal to the benchmark values which had been proved by Biskup and Feldmann (2001) to be optimal. As for the computation time, our VNS/TS requires only 4.7 seconds for each run (on a Pentium 4 3.2GHz with 512MB memory), HGT takes 7.8 seconds and GA takes 21.5 seconds (both on a Pentium 4 1.7GHz with 512MB memory). By considering the different computer environments, VNS/TS and HGT have relatively the same efficiency, but VNS/TS is much more efficient than GA.
The solution values for all the 280 instances yielded by our VNS/TS are provided in Table 2.
Because we cannot obtain the results from Hino et al. (2005), the current best known solutions for the benchmark problems are unknown except for the instances with 10 jobs. Nevertheless, based on the above comparative results, we believe that most of the solutions in Table 6 constitute the best solutions known for the benchmark problems as to date.
6. Conclusions
Pulp and paper production have many features in common with the continuous production (Farla et al., 1997; Yin et al., 2003). In this paper, we consider the process in a single type of paper in the paper industry. Under ignoring the missing operations at the coating stage, its process can be regarded as the continuous production. Such a continuous production is classified as the single machine problem in the literature. The objective is to minimize total weighted earliness and tardiness. This paper presents a metaheuristic algorithm for the scheduling problem, the total weighted earliness and tardiness with the common due date on a single machine. There are several distinctive features in the developed VNS/TS algorithm, including different ratio of the two neighborhoods, generating five points simultaneously in a neighborhood, implementation of the B/F local search, and combination of TS with VNS. These features make the algorithm very efficient and effective. By examining the 280 benchmark problem instances, the algorithm shows an excellent performance in not only the solution quality but also the computation time. The results obtained are better than those reported previously in the literature and constitute the best solutions known for the benchmark problems as to date.ġ
Extension of the VNS/TS algorithm with its simple, efficient and effective characteristics to other machine environment problems involving paper industry is a possible direction of more research. Consideration of other classes of objective functions is also interesting. Finally, development of an exact solution method (e.g., branch and bound method) for solving relatively larger sized problem instances, which could be used to better evaluate the effectiveness of metaheuristic, is a challenging area for further research.ġ
Table 1.
Comparison with HGT and GA of Hino et al. (2005)
n h HGT GA VNS/TS
10 0.2 0.12 0.12 0.00 + 0.4 0.19 0.19 0.00 + 0.6 0.01 0.03 0.00 + 0.8 0.00 0.00 0.00 20 0.2 −3.84 −3.84 −3.84
0.4 −1.62 −1.62 −1.63 + 0.6 −0.71 −0.68 −0.72 + 0.8 −0.41 −0.28 −0.41 50 0.2 −5.70 −5.68 −5.70 0.4 −4.66 −4.60 −4.66 0.6 −0.31 −0.31 −0.34 + 0.8 −0.23 −0.19 −0.24 + 100 0.2 −6.19 −6.17 −6.19
0.4 −4.93 −4.91 −4.94 + 0.6 0.04 −0.12 −0.15 + 0.8 −0.11 −0.12 −0.18 + 200 0.2 −5.76 −5.74 −5.78 + 0.4 −3.75 −3.75 −3.75 0.6 0.07 −0.13 −0.15 + 0.8 0.07 −0.14 −0.15 + 500 0.2 −6.41 −6.41 −6.42 + 0.4 −3.58 −3.58 −3.56 − 0.6 0.15 −0.11 −0.11 0.8 0.13 −0.11 −0.11 1000 0.2 −6.74 −6.75 −6.75 0.4 −4.39 −4.40 −4.37 − 0.6 0.42 −0.05 −0.05 0.8 0.40 −0.05 −0.05 Mean −2.06 −2.12 −2.15
+ VNS/TS is the best one.
− Either HGT or GA is the best one.
Table 2. The solution values of VNS/TS for the 280 instances 50 1 40697 23792 17969 17934 100 1 145516 85884 72017 72017 2 30613 17907 14050 14040 2 124916 72981 59230 59230 10 32958 19167 14366 14363 10 118911 71850 61361 61361 200 1 498654 295697 254259 254259 500 1 2955335 1787979 1579140 1579109 2 541181 319205 266002 266002 2 3365952 1995161 1712429 1712466 3 488665 293888 254488 254476 3 3102930 1864852 1641706 1641718 4 586257 353034 297109 297109 4 3221423 1887902 1640785 1640784 5 513240 304678 260278 260278 5 3114982 1807378 1468256 1468263 6 478022 279940 235702 235702 6 2792362 1610445 1411867 1411841 7 454757 275030 246330 246313 7 3173069 1902767 1634330 1634330 8 494286 279177 225215 225215 8 3122442 1819891 1540458 1540470 9 529292 310418 254659 254637 9 3364709 1973920 1680486 1680647 10 538354 323109 268353 268354 10 3120925 1837705 1519215 1519205 1000 1 14057673 8112258 6411260 6411352 10 12430428 7278460 6146054 6145999
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行政院國家科學委員會補助國內專家學者出席國際學術會議報告 98 年 5 月 10 日
報告人姓名
廖慶榮 服務機構
及職稱 台灣科技大學工管系教授
會議 時間 地 點
自 2009 年 5 月 1 日至 2009 年 5 月 4 日止 地 點：美國，佛羅里達 州，奧蘭多 (Orlando)
核定 補助
補助編號:
NSC 95-2221-E-011-098-MY3
會議名稱 (中文) 生產與作業管理學會 2009 年之全球性挑戰和機會
(英文) POMS 2009: GLOBEL CHALLENGES AND OPPORTUNITIES
發表論文題目 (中文) 應用變動鄰域搜尋法求解育嬰配方推廣之醫院拜訪排程 (英文) An application of variable neighborhood search to hospital call scheduling of infant formula promotion
一、參加會議經過及其學術地位、重要性：
POMS (Production and Operations Management Society) 2009 會議係國際作業 管理與應用領域十分重要的國際會議，該學會所出版的學術期刊 Production and Operations Management 是 MS/OR 領域排名第一的刊物，且已是美國著名刊物 Business Week 的 20 個頂尖期刊之一（該期刊以此 20 個期刊作為全美商學院
MBA 的排名）。本研討會對我國作業管理與應用的研究發展助益良多，研討會
議程共 4 天，發表 200 多篇論文，每篇均經過多位專家審查推薦才被接受。參與 論文發表的國家約 10 多國; 包括 USA、Netherlands、Australia、Vietnam、Japan、
Taiwan、Hong Kong、China、Switzerland、Austria、Belgium、China 等。
本屆 POMS 年度研討會副標題訂為全球的挑戰與機會「Global Challenges and
Opportunities」，來自全球 45 個國家約 800 位學者以及研討會論文摘要參與了此
次盛會。研討會除了召開 POMS Board Meeting 外，分別以各 College 舉辦 Panels、
Workshops 及 Tutorials；Colleges 包括：Service Operations、Sustainable Operations、
Production Innovation and Technology Management、Human Behavior in Operations Management、Healthcare Operations Management、Supply Chain Management 等。
然而，本屆研討會之主軸顯然是在供應鏈與作業管理 (Operations and Supply Chain Management)。至於，本人和學生所發表的研討會論文則歸屬於 Healthcare Operations Management，被安排於 5 月 1 日上午 8 點 30 分第一場次第一順位發 表。
同行參加此次研討會者，除了本人外，尚有本校工管系王孔政教授（發表 3 篇論文）及其指導博士生林鈺祥同學、中央大學企管系的張東生教授、朝陽科技 大學企管系的楊文華副教授、朝陽科技大學工管系的曾兆堂助理教授，以及龍華
科技大學林湘沅講師（本人目前的博士學生），總共有十餘位來自台灣各地的學
者與會。
二、與會心得
此次大會的內容非常豐富，並邀請多位資深學術界代表對作業管理與應用領 域做專題演講。本會議的特色之一是場次安排細緻。除了論文發表之外，大會並 安排了多場主題資深學者專題演講。各領域的與會學者們將研究所得發表並和與 會者熱烈討論。本人與他國教授建立溝通管道、論文合作及心得交換。此外也從 他人發表的論文中，激發新的研究構想。從不同國家的研究成果中了解作業管理 與應用發展的現狀與未來趨勢。
三、建議
本項會議為國際生產與作業管理（Production and Operations Management）領 域十分重要的國際會議，建議國內學者應多參與。與會學者以美國人為主，亞洲 方面則以韓國各大學教師參與人數最多，各國人員幾乎全是博士身份；盼往後有 更多機會參與相同的國際性研討會以增廣見聞，增進國際觀。
四、攜回資料名稱與內容
會議論文摘要集及會議 Programs and Information。