An efficient genetic algorithm for decentralized multi-project scheduling with resource transfers

Jingwen Zhang; Wanjun Liu; Wanlin Liu

doi:10.3934/jimo.2020140

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2022, Volume 18, Issue 1: 1-24. Doi: 10.3934/jimo.2020140

This issue Previous Article Next Article Statistical mechanics approach for steady-state analysis in M/M/s queueing system with balking

An efficient genetic algorithm for decentralized multi-project scheduling with resource transfers

School of Management, Northwestern Polytechnical University, Xi’an 710072, Shaanxi, China

^* Corresponding author: Jingwen Zhang
^* Corresponding author: Jingwen Zhang

Received: December 31, 2019

Revised: June 30, 2020

Early access: September 2020

Published: January 2022

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Abstract

This paper investigates the decentralized resource-constrained multi-project scheduling problem with transfer times (DRCMPSPTT) in which the transfer times of the global resources among different projects are assumed to be sequence-independent, while transfers of local resources take no time within a project. First, two decision variables ($ {y_{ijg}} $ and $ {w_{ijg}} $) are adopted to express the transition state of global resources between projects. $ {y_{ijg}} $ (takes a value of 0 or 1) represents whether activity i transfers global resource g to activity j; accordingly, the transferred quantity is denoted as $ {w_{ijg}} $. Then, we construct an integer linear model with the goal of minimizing the average project delay for the DRCMPSPTT. Second, an adaptive genetic algorithm (GA) is developed to solve the DRCMPSPTT. To gain the schedules for the DRCMPSPTT, the traditional serial and parallel scheduling generation schemes (SGSs) are modified to combine with different resource transfer rules and to design multiple decoding schemes. Third, the numerical experiments are implemented to analyse the effects of eight decoding schemes, and we found that the scheme comprising the parallel SGS and maxRS rule can make the GA work the best; furthermore, the effectiveness of the GA_maxRS (GA embedded with the best scheme) is demonstrated by solving some instances with different sizes.

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Mathematics Subject Classification: Primary: 90B35, 90C11.

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Figure 1. An example of a DRCMPSPTT

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Figure 2. Renumbered network of the example

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Figure 3. Individual representation for the DRCMPSPTT

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Figure 4. A precedence feasible AL of the multi-project example

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Figure 5. The adaptive serial SGS with transfer times

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Figure 6. The adaptive parallel SGS with transfer times

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Figure 7. Two-point crossover operator

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Figure 8. Average APDs obtained by different schemes

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Figure 9. (a) APDs obtained by the two schemes, and (b) the deviation of the APDs obtained by the two schemes varies as $ UF_{AV} $ increases

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Figure 10. (a) The three types PREs, and (b) computation times of the solution methods

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Figure 11. (a) mean value of the APD obtained by different methods, and (b) The three types PREs, (c) The $ PRE_{_GA} $ of the 12 problem sets, and (d) The change trend of $ PRE_{_GA} $ as $ UF_{AV} $ increases

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Table 1. Literature review of the project scheduling with transfer 09:49:34

Authors	Project			Objective		Transfer times		Algorithm
Authors	Single	Multi-	Decentralized multi-	Makespan	Others	sequence-dependent	sequence-independent
Kolisch (1995)	$ \surd $			$ \surd $			$ \surd $		Parallel scheme based heuristic
Neumann et al (2003)	$ \surd $			$ \surd $		$ \surd $		Branch-and-bound
Vanhoucke(2008)	$ \surd $			$ \surd $			$ \surd $	Branch-and-bound
Mika (2008)	$ \surd $			$ \surd $		$ \surd $		Tabu search
Afshar-Nadjafi et al (2014)	$ \surd $			$ \surd $			$ \surd $	GA
Poppenborg(2016)	$ \surd $			$ \surd $		$ \surd $		Tabu search
Kadri and Boctor (2017)	$ \surd $			$ \surd $		$ \surd $		GA
Yang and Sum(1993, 1997)		$ \surd $		$ \surd $			$ \surd $	Computational experiment
Mittal and Kanda (2009)		$ \surd $		$ \surd $	Cost		$ \surd $	Heuristics
Kruger and Scholl(2009, 2010)	$ \surd $	$ \surd $		$ \surd $	Cost	$ \surd $		Priority-rule based heuristic
Cai and Li(2012)		$ \surd $			APD	$ \surd $		Hybrid GA
Adhau and Mittal(2013)			$ \surd $		APD		$ \surd $	DMAS/RIA
This research			$ \surd $		APD		$ \surd $	GA

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Table 2. Resource transfer rules

Transfer rules	Priority value $ \pi_i $	Extremum
minTT	$ \pi_i = \Delta_{ijg} $	min
minGAP	$ \pi_i = gap_{ijg} = t - (F_i + \Delta_{ijg}) $	min
maxRS	$ \pi_i = v_{ig} = u_{ig} - \sum\limits_{l \in PS} {w_{ilg}} $	max
minRS	$ \pi_i = v_{ig} = u_{ig} - \sum\limits_{l \in PS} {w_{ilg}} $	min

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Table 3. The instances for the DRCMPSP

Problem subset	NOI	Characterization per instance			UF$ _{AV} $
Problem subset	NOI	$ \left\| {P} \right\| $	$ \left\| J_{p} \right\| $	$ (\left\| G \right\| $; $ \left\| L_{p} \right\|) $	UF$ _{AV} $
MP30_2	5	2	30	(1;3)/(2;2)/(3;1)	0.84
MP90_2	5	2	90	(1;3)/(2;2)/(3;1)	0.57
MP120_2	5	2	120	(1;3)/(2;2)/(3;1)	1.31
MP30_5	5	5	30	(1;3)/(2;2)/(3;1)	0.82
MP90_5	5	5	90	(1;3)/(2;2)/(3;1)	0.61
MP120_5	5	5	120	(1;3)/(2;2)/(3;1)	1.32
MP30_10	5	10	30	(1;3)/(2;2)/(3;1)	2.38
MP90_10	5	10	90	(1;3)/(2;2)/(3;1)	1.14
MP120_10	5	10	120	(1;3)/(2;2)/(3;1)	1.91
MP30_20	5	20	30	(1;3)/(2;2)/(3;1)	3.37
MP90_20	5	20	90	(1;3)/(2;2)/(3;1)	0.9
MP120_20	5	20	120	(1;3)/(2;2)/(3;1)	0.87

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Table 4. APDs obtained by the adaptive GA with different schemes

Problem subsets	Serial				parallel
Problem subsets	minTT	minGAP	maxRS	minRS	minTT	minGAP	maxRS	minRS
MP30_2	14.8	13.7	14.8	16.1	13.9	13.3	13.7	14
MP90_2	10	8.8	10.1	11.3	6.9	6.8	7.1	7.3
MP120_2	86.3	83.6	85.1	92.3	57.2	56.7	56.6	57.2
MP30_5	34.24	34.2	36.44	39.04	23.16	23.28	22.48	23.36
MP90_5	23.96	21.76	22.92	25.44	12.4	12.4	11.76	13
MP120_5	104.04	104.44	106.56	112.64	67.72	68.32	67.56	67.8
MP30_10	131.78	126.4	134.3	145.34	85.1	85.22	83.48	87.62
MP90_10	104.14	106.72	108.96	114.3	61.34	60.4	61.12	62.1
MP120_10	252.43	254.44	256.73	262.54	164.38	166.78	166.84	171.96
MP30_20	333.61	340.69	338.15	352.6	205.06	208.3	203.74	213.16
MP90_20	65.24	65.04	66.76	72.84	44.16	43.95	44.06	44.49
MP120_20	71.28	70.49	71.48	77.56	47.48	48.28	47.24	47.52
Average	102.65	102.52	104.36	110.17	65.73	66.14	65.47	67.46

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Table 5. Parameter configurations to generate test problems

Parameter	J	P	K	$ r_j $	$ d_j $	NC	RF	RS
value	5, 8, 10 or 14	2 or 3	4	$ [1,8] $	$ [1,5] $	1.5	0.25	0.2

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Table 6. Results of the two methods

instances		CPLEX		GA_maxRS (100 solutions)				$ PRE_w $ (%)	$ PRE_b $ (%)	$ PRE_m $ (%)
instances		opt	$ ct_{ave} (s) $	worst	best	mean	$ ct_{ave} (s) $	$ PRE_w $ (%)	$ PRE_b $ (%)	$ PRE_m $ (%)
J5_2	1	3	0.08	3	3	3	0.11	0	0	0
	2	1.5	0.03	1.5	1.5	1.5	0.14	0	0	0
	3	2	0.05	2	2	2	0.1	0	0	0
	4	5	0.08	5	5	5	0.15	0	0	0
	5	2	0.05	2	2	2	0.13	0	0	0
J8_2	1	0.5	0.06	0.5	0.5	0.5	0.11	0	0	0
	2	1	0.09	1	1	1	0.17	0	0	0
	3	2.5	0.08	2.5	2.5	2.5	0.2	0	0	0
	4	5	0.3	5	5	5	0.24	0	0	0
	5	3	0.2	3	3	3	0.31	0	0	0
J10_2	1	2.5	2.11	2.5	2.5	2.5	0.94	0	0	0
	2	9	3.72	9.5	9	9.1	0.87	6	0	1
	3	10.5	2.23	12	10.5	11	1.06	14	0	5
	4	7.5	3.2	7.5	7.5	7.5	0.56	0	0	0
	5	7	4.83	7	7	7	1.42	0	0	0
J14_2	1	2	0.94	2.5	2	2.2	0.89	25	0	10
	2	6	25.23	6	6	6	0.99	0	0	0
	3	4.5	1.91	5.5	4.5	4.7	1.18	22	0	4
	4	7.5	30.42	7.5	7.5	7.5	1.5	0	0	0
	5	4	15.2	6	4	5.5	1.45	50	0	38
J5_3	1	4.33	0.23	4.33	4.33	4.33	0.44	0	0	0
	2	2.67	0.63	3.67	3.33	3.53	0.82	37	25	32
	3	4	0.86	5.33	4	4.67	0.61	33	0	17
	4	3.33	0.19	4.33	3.33	3.94	0.45	30	0	18
	5	3.33	0.7	3.33	3.33	3.33	0.63	0	0	0
J8_3	1	2	0.5	2	2	2	0.75	0	0	0
	2	7.67	15.69	8.67	7.67	7.94	1.18	13	0	4
	3	2	0.22	2	2	2	0.58	0	0	0
	4	6	152.2	8.33	6	6.47	1.85	39	0	8
	5	1.33	0.41	1.33	1.33	1.33	0.66	0	0	0
J10_3	1	8.67	157.94	9.33	8.76	9.01	1.86	8	1	4
	2	10.67	249.08	11	10.67	10.79	1.67	3	0	1
	3	4.33	320.17	5.67	4.33	4.79	1.27	31	0	11
	4	8	824.63	8	8	8	1.28	0	0	0
	5	6	1832.06	6	6	6	1.06	0	0	0
J14_3	1	6.33	131.98	6.33	6.33	6.33	2.14	0	0	0
	2	3.33	411.58	5	3.33	3.79	2.93	50	0	14
	3	5.67	3600.44	6.67	5.67	5.79	3.03	18	0	2
	4	11	3600.13	13.67	11	12.1	7.33	24	0	10
	5	5	450.97	7	5	5.4	3.01	40	0	8

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Table 7. Results of GA_maxRS and DMAS/RIA

Problem subsets	GA_maxRS			DMAS/RIA			$ PRE_{_A} $ (%)
Problem subsets	with TT	without TT	$ PRE_{_GA} $ (%)	with TT	without TT	$ PRE_{_D} $ (%)	$ PRE_{_A} $ (%)
MP30_2	with TT	without TT	13	with TT	without TT	10.9	19	23.3	18	29	-39
MP90_2	6.9	6.7	3	17.1	11.8	45	-43
MP120_2	56.1	55.6	1	89.6	80.3	12	-31
MP30_5	22.2	18.08	23	35.28	21.64	63	-16
MP90_5	11.68	9.36	25	22	13.32	65	-30
MP120_5	66.84	65.48	2	86.12	77.68	11	-16
MP30_10	81.58	78.16	4	92.68	83	12	-6
MP90_10	59.34	50.36	18	62.66	61.52	2	-18
MP120_10	164.28	140.3	17	162.6	144.1	13	-3
MP30_20	199.63	189.6	5	206.9	193	7	-2
MP90_20	43.38	34.92	24	42.88	43.13	-1	-19
MP120_20	46.42	39.71	17	48.09	42.11	14	-6
Average	64.28	58.26	10	74.1	65.8	13	-11

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References

[1]	S. Adhau, M. L. Mittal and A. Mittal, A multi-agent system for decentralized multi project scheduling with resource transfers, International Journal of Production Economics, 146 (2013), 646-661. doi: 10.1016/j.ijpe.2013.08.013.
[2]	S. Adhau, M. L. Mittal and A. Mittal, A multi-agent system for distributed multi-project scheduling: An auction-based negotiation approach, Engineering Applications of Artificial Intelligence, 25 (2012), 1738-1751. doi: 10.1016/j.engappai.2011.12.003.
[3]	B. Afshar-Nadjafi and M. Majlesi, Resource constrained project scheduling problem with setup times after preemptive processes, Computers and Chemical Engineering, 69 (2014), 16-25. doi: 10.1016/j.compchemeng.2014.06.012.
[4]	C. Artigues, P. Michelon and S. Reusser, Insertion techniques for static and dynamic resource-constrained project scheduling, European Journal of Operational Research, 149 (2003), 249-267. doi: 10.1016/S0377-2217(02)00758-0.
[5]	D. Bedworth and J. Bailey, Integrated Production Control Systems Management, Analysis, Design, 2$^nd$ edition, John Wiley & Sons, Inc., New York, 1999.
[6]	C. Bierwirth, D. Mattfeld and H. Kopfer, On permutation representations for scheduling problems, in Parallel Problem Solving from Nature, Lecture Notes in Computer Science, 1996,310–318. doi: 10.1007/3-540-61723-X_995.
[7]	P. Brucker, S. Knust, A. Schoo and O. Thiele, A branch and bound algorithm for the resource-constrained project scheduling problem, European Journal of Operational Research, 107 (1998), 272-288. doi: 10.1016/S0377-2217(97)00335-4.
[8]	Z. Cai and X. Li, A hybrid genetic algorithm for resource-constrained multi-project scheduling problem with resource transfer time, IEEE International Conference on Automation Science and Engineering, Seoul, 2012,569–574. doi: 10.1109/CoASE.2012.6386457.
[9]	G. Confessore, S. Giordani and S. Rismondo, A market-based multi-agent system model for decentralized multi–project scheduling, Annals of Operations Research, 150 (2007), 115-135. doi: 10.1007/s10479-006-0158-9.
[10]	E. W. Davis, Project network summary measures constrained resource scheduling, AIIE Transactions, 7 (1975), 132-142. doi: 10.1080/05695557508974995.
[11]	E. L. Demeulemeester and W. S. Herroelen, An efficient optimal solution procedure for the preemptive resource-constrained project scheduling problem, European Journal of Operational Research, 90 (1996), 334-348. doi: 10.1016/0377-2217(95)00358-4.
[12]	L. Djerid, M. C. Portman and P. Villon, Performance analysis of permutation cross-over genetic operators, Journal of Decision Systems, 5 (1996), 131-140. doi: 10.1080/12460125.1996.10511679.
[13]	S. Hartmann and R. Kolisch, Experimental evaluation of state-of-the-art heuristics for resource-constrained project scheduling problem, European Journal of Operational Research, 127 (2000), 394-407. doi: 10.1016/S0377-2217(99)00485-3.
[14]	J. B. Holland and J. H. Holland, Adaption in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, Mich., 1975.
[15]	J. Homberger, A multi-agent system for the decentralized resource constrained multi-project scheduling problem, International Transactions in Operational Research, 14 (2007), 565-589. doi: 10.1111/j.1475-3995.2007.00614.x.
[16]	J. Homberger, A (${\mu}$, ${\lambda}$)-coordination mechanism for agent-based multi-project scheduling, OR Spectrum, 34 (2012), 107-132. doi: 10.1007/s00291-009-0178-3.
[17]	R. Jans and Z. Degraeve, Modeling industrial lot sizing problems: A review, International Journal of Production Research, 46 (2008), 1619-1643. doi: 10.1080/00207540600902262.
[18]	R. L. Kadri and F. F. Boctor, An efficient genetic algorithm to solve the resource-constrained project scheduling problem with transfer times: The single mode case, European Journal of Operational Research, 265 (2018), 454-–462. doi: 10.1016/j.ejor.2017.07.027.
[19]	J. E. Kelley, The critical-path method: Resources planning and scheduling, Industrial Scheduling, 13 (1963), 347365.
[20]	S. Khalilpourazari, B. Naderi and S. Khalilpourazary, Multi-objective stochastic fractal search: A powerful algorithm for solving complex multi-objective optimization problems, Soft Computing, 24 (2020), 3037–-3066. doi: 10.1007/s00500-019-04080-6.
[21]	S. Khalilpourazari and S. H. R. Pasandideh, Modeling and optimization of multi-item multi-constrained EOQ model for growing items, Knowledge-Based Systems, 164 (2019), 150-162. doi: 10.1016/j.knosys.2018.10.032.
[22]	S. Khalilpourazari and S. H. R. Pasandideh, Sine-cosine crow search algorithm: Theory and applications, Neural Computing and Applications, 32 (2019), 7725–-7742. doi: 10.1007/s00521-019-04530-0.
[23]	R. Kolisch, Project scheduling with setup times, in Project Scheduling Under Resource Constraints, Production and Logistics, Physica, Heidelberg, 1995,177–185. doi: 10.1007/978-3-642-50296-5_8.
[24]	R. Kolisch and S. Hartmann, Heuristic algorithms for solving the resource-constrained project scheduling problem: Classification and computational analysis, in Project Scheduling, International Series in Operations Research & Management Science, Springer, Boston, 1999,147–178. doi: 10.1007/978-1-4615-5533-9_7.
[25]	R. Kolisch and A. Sprecher, PSPLIB–A project scheduling library, European Journal of Operation Research, 96 (1996), 205-216.
[26]	D. Kruger and A. Scholl, A heuristic solution framework for the resource constrained (multi-)project scheduling problem with sequence-dependent transfer times, European Journal of Operation Research, 197 (2009), 492-508. doi: 10.1016/j.ejor.2008.07.036.
[27]	D. Kruger and A. Scholl, Managing and modelling general resource transfers in (multi-)project scheduling, OR Spectrum, 32 (2010), 369-394. doi: 10.1007/s00291-008-0144-5.
[28]	C. J. Liao, C. W. Chao and L. C. Chen, An improved heuristic for parallel machine weighted flowtime scheduling with family set-up times, Computers and Mathematics with Applications, 63 (2012), 110-117. doi: 10.1016/j.camwa.2011.10.077.
[29]	A. Lova and P. Tormos, Analysis of scheduling schemes and heuristic rules performance in resource-constrained multiproject scheduling, Annals of Operations Research, 102 (2001), 263-286. doi: 10.1023/A:1010966401888.
[30]	A. Lova, P. Tormos, M. Cervantes and F. Barber, An efficient hybrid genetic algorithm for scheduling projects with resource constraints and multiple execution modes, International Journal of Production Economics, 117 (2009), 302-316. doi: 10.1016/j.ijpe.2008.11.002.
[31]	D. Merkle, M. Middendorf and H. Schmeck, Ant colony optimization for resource-constrained project scheduling, IEEE Transactions on Evolutionary Computation, 6 (2002), 333-346. doi: 10.1109/TEVC.2002.802450.
[32]	M. Mika, G. Waligra and J. Weglarz, Tabu search for multi-mode resource-constrained project scheduling with schedule-dependent setup times, European Journal of Operational Research, 187 (2008), 1238-1250. doi: 10.1016/j.ejor.2006.06.069.
[33]	M. L. Mittal and A. Kanda, Scheduling of multiple projects with resource transfers, International Journal of Mathematics in Operational Research, 1 (2009), 303-325. doi: 10.1504/IJMOR.2009.024288.
[34]	K. Moumene and J. A. Ferland, Activity list representation for a generalization of the resource-constrained project scheduling problem, European Journal of Operational Research, 199 (2009), 46-54. doi: 10.1016/j.ejor.2008.10.030.
[35]	M. S. Nagano, A. A. Silva and L. A. N. Lorena, A new evolutionary clustering search for a no-wait flow shop problem with set-up times, Engineering Applications of Artificial Intelligence, 25 (2012), 1114-1120. doi: 10.1016/j.engappai.2012.05.017.
[36]	K. Neumann, C. Schwindt and J. Zimmermann, Project Scheduling with Time Windows and Scarce Resources, Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-540-24800-2.
[37]	S. H. R. Pasandideh and S. Khalilpourazari, Sine cosine crow search algorithm: A powerful hybrid meta heuristic for global optimization, preprint, arXiv: 1801.08485.
[38]	J. Poppenborg and S. Knust, A flow-based tabu search algorithm for the RCPSP with transfer times, OR Spectrum, 38 (2016), 305-334. doi: 10.1007/s00291-015-0402-2.
[39]	V. Roshanaei, B. Naderi, F. Jolai and M. Khalili, A variable neighborhood search for job shop scheduling with set-up times to minimize makespan, Future Generation Computer Systems, 25 (2009), 654-661. doi: 10.1016/j.future.2009.01.004.
[40]	M. Rostami, M. Bagherpour, M. M. Mazdeh and A. Makui, Resource Pool Location for Periodic Services in Decentralized Multi-Project Scheduling Problems, Journal of Computing in Civil Engineering, 31 (2017), 04017022. doi: 10.1061/(ASCE)CP.1943-5487.0000671.
[41]	V. Valls, F. Ballestin and M. S. Quintanilla, A hybrid genetic algorithm for the RCPSP, Technical report, Department of Statistics and Operations Research, University of Valencia, 2003.
[42]	V. Valls, F. Ballestin and M. S. Quintanilla, Justification and RCPSP: A technique that pays, European Journal of Operational Research, 165 (2005), 375-386. doi: 10.1016/j.ejor.2004.04.008.
[43]	M. Vanhoucke, Setup times and fast tracking in resource-constrained project scheduling, Computers & Industrial Engineering, 54 (2008), 1062-1070. doi: 10.1016/j.cie.2007.11.008.
[44]	K. K. Yang and C. C. Sum, A comparison of resource allocation and activity scheduling rules in a dynamic multi-project environment, Journal of Operation Management, 11 (1993), 207-218. doi: 10.1016/0272-6963(93)90023-I.
[45]	K. K. Yang and C. C. Sum, An evaluation of due date, resource allocation, project release, and activity scheduling rules in a multi project environment, European Journal of Operational Research, 103 (1997), 139-154.