Procedia Engineering 29 (2012) 1137 – 1141 1877-7058 © 2011 Published by Elsevier Ltd. doi:10.1016/j.proeng.2012.01.101 Procedia Engineering 00 (2011) 000–000
Procedia
Engineering
www.elsevier.com/locate/procedia2012 International Workshop on Information and Electronics Engineering (IWIEE)
A Note on “A MAX-MIN Ant System for Unconstrained
Multi-Level Lot-Sizing Problems”
Yi Han
a, Jianhu Cai
a,*, Ikou Kaku
b, Yingsong Li
c, Huazhen Lin
a, Xuhong Ye
aaCollege of Economics and Management, Zhejiang University of Technology, Hangzhou 310023, China bDepartment of Management Science and Engineering, Akita Prefectural University, Yulihonjo 015-0055, Japan cCollege of Information and Communications Engineering, Harbin Engineering University, Harbin 150001, China
Abstract
Pitakaso et al.[1] presented an ant systems based on random cumulative Wagner-Whitin (RCWW) (RCWW-STVS) for uncapacitated multilevel lot-sizing (MLLS) problems and gave out a computational result for Dellaert’s instance [2] with a random variable r = 0.43. The result is not quite right. This paper highlighted the error and presented a revision to the result.
© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Harbin University of Science and Technology
Keywords: Multilevel lot-sizing, RCWW; ant systems, material requirements planning, metaheuristic
1. Introduction
Pitakaso et al. [1] presented a RCWW-STVS algorithm to solve multilevel lot-sizing problems without resource constraints. To prove their algorithm is superior to Dellaert et al. [2]’s RCWW algorithm, they presented a result reached by RCWW-STVS algorithm for reference [2]’s instance with product order of {2, 5, 1, 3, 4, 6, 8, 7, 9}, setup cost accumulative parameters ri ={0.43, 0.43, 0.43, 0.43, 0.43, 0.43, 0.43, 0.43, 0.43} and the inventory costs for material 1 to 9 is {13, 8, 4, 4, 3, 3, 2, 1, 1}.
The product structure is shown in Fig.1. The underlined number is lead time for each material and the right side number is setup cost for each material. The inner demand relation between materials is 1 to 1.
The result from [1] is presented in Table 1. There are no errors for parameters but the result for
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reference [2]’s instance. Here we note in Table 1, the demand for material 1 is {0, 0, 0, 0, 10, 15, 10, 12, 20}. Material 1’s production lot, according to [1], is {0, 0, 0, 0, 10, 15, 10, 12, 20}. Then for material 3, taking into consideration of material 1’s lead time, the demand is {0, 0, 0, 10, 15, 10, 12, 20, 0}. As we seen from Table 1, the revised setup cost of material 3 is {0, 0, 0, 41.5, 30, 30, 30, 30, 30}.
1
2
3
4
5
6
7
9
8
1 50 2 60 1 30 90 80 20 10 80 40 1 0 1 2 0 0Fig.1 Product Structure.
The production lot of material 3 reached by [1] is {0, 0, 0, 10, 25, 0, 12, 20, 0}. However, we can find that the setup cost in period 6 of material 3 is 30 and the inventory cost in period 5 of material 3 is 4. If producing 10 products in period 6, the cost is 30. However, according to Pitakaso et al.[1]’s result, 10 units of material 3 are kept in inventory for 1 period. Then the cost is 10*4=40. So we can easily tell a production decision {0, 0, 0, 10, 15, 10, 12, 20, 0} is better than another one {0, 0, 0, 10, 25, 0, 12, 20, 0}. After a practical verification with C program language, we got a result, which is a revision to reference [1]’s result, for Dellaert et al.[2]’s instance and presented it in Table 2. The material presentation of MLLS problem is shown in Fig.2. Fig.3 shows the output interface of VC++ program.
Fig.2 Struct Definition in C Language for Materials.
By the way, the results of Pitakaso et al. [1]’s are based on different product sequence. So, detailed presentations of each sequence to each solution are strongly needed. Only by doing so, the other scholars can reproduce those benchmark results.
Table 1. Result from Reference [1] i si ri 1 2 3 4 5 6 7 8 9 2 60 0.84 S2,t 195.7 195.7 195.7 195.7 (0.43) (129.4) (129.4) (129.4) (129.4) X2t 50 60 30 (50) (40) (20) (30) 5 80 0.74 S5,t 91.1 91.1 91.1 91.1 91.1 91.1 (0.43) (86.5) (86.5) (86.5) (86.5) (86.5) (86.5) X5t 50 60 30 (50) (60) (30) 1 50 0.64 S1,t 128.6 128.6 170.6 128.6 170.6 (0.43) (103.2) (103.2) (103.2) (103.2) (131.6) X1t 10 25 12 20 (10) (15) (10) (12) (20) 3 30 0.53 S3,t 44.3 30 30 44.3 30 30 (0.43) (41.5) (30) (30) (30) (30) (30) X3t 10 25 12 20 (10) (25) (12) (20) 4 90 0.43 S4,t 96.5 90 90 96.5 90 96.5 (0.43) (96.5) (90) (90) (90) (90) (96.5) X4t 60 85 62 (60) (55) (30) (62) 6 20 0.33 S6,t 39.7 39.7 26.6 26.6 39.7 26.6 26.6 (0.43) (45.9) (45.9) (28.6) (28.6) (28.6) (28.6) (28.6) X6t 70 120 25 74 32 20 (70) (90) (45) (84) (32) (20) 8 80 0.22 S8,t 80 80 80 80 80 80 80 80 (0.43) (80) (80) (80) (80) (80) (80) (80) (80) X8t 70 155 131 52 (70) (145) (141) (52) 7 10 0.12 S7,t 12.4 10 10 10 10 10 10 (0.43) (18.6) (10) (10) (10) (10) (10) (10) X7t 60 135 60 62 30 (60) (105) (90) (62) (30) 9 40 0.02 S9,t 40 40 40 40 40 40 40 40 40 (0.43) (40) (40) (40) (40) (40) (40) (40) (40) (40) X9t 60 205 180 87 104 52 (60) (175) (180) (107) (114) (52)
Table 2. Result Reached by Authors i si ri 1 2 3 4 5 6 7 8 9 2 60 0.43 S2,t (129.4) (129.4) (129.4) (129.4) X2t 50 40 20 30 5 80 0.43 S5,t (86.5) (86.5) (86.5) (86.5) (86.5) (86.5) X5t 50 60 30 1 50 0.43 S1,t (103.2) (103.2) (103.2) (103.2) (131.6) X1t 10 15 10 12 20 3 30 0.43 S3,t (41.5) (30) (30) (30) (30) (30) X3t 10 15 10 12 20 4 90 0.43 S4,t (96.5) (90) (90) (90) (90) (96.5) X4t 60 55 30 62 6 20 0.43 S6,t (45.9) (45.9) (28.6) (28.6) (28.6) (28.6) (28.6) X6t 70 80 55 84 32 20 8 80 0.43 S8,t (80) (80) (80) (80) (80) (80) (80) (80) X8t 70 145 141 52 7 10 0.43 S7,t (18.6) (10) (10) (10) (10) (10) (10) X7t 60 105 90 62 30 9 40 0.43 S9,t (40) (40) (40) (40) (40) (40) (40) (40) (40) X9t 60 175 170 117 114 52 2. Conclusion
In this paper, we pointed out an error in literature [2] and gave out the revision for this error. The significance of this paper is twofold: 1) the reason why Pitakaso et al.[1]’s algorithm have produced questionable results is pointed out; 2) all those results from reference [1] are not fully optimized.
Acknowledgment
This research is supported by National Natural Science Foundation of China under grant numbers (70971017), Natural Science Foundation of Zhejiang Province of China under grant number (Y1100854), Philosophy and Social Science Foundation of Zhejiang Province of China under grant number
(10CGGL21YBQ), Foundation of Zhejiang Educational Bureau under grant number (Y201016979), Science and Technology Research Program of Zhejiang Province of China under grant number (2009C35007) and Foundation of Ministry of Education of China under grant number (10YJC630009). References
[1] Pitakaso, R., Almeder, C., Doerner, K.F., Hartl, R.F., 2007. A MAX-MIN ant system for unconstrained multi-level lot-sizing problems. Computers & Operations Research, 34(9): 2533-2552.
[2] Dellaert, N., Jeunet, J., 2003. Randomized multi-level lot-sizing heuristics for general product structures. European Journal of Operational Research, 148(1): 211-228.
