A multi-objective approach to the scheduling problem with workers allocation

Pantuza Júnior, Guido

doi:10.1590/0104-530X1432-14

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Articles • Gest. Prod. 23 (1) • Jan-Mar 2016 • https://doi.org/10.1590/0104-530X1432-14 copy

A multi-objective approach to the scheduling problem with workers allocation

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

This paper addresses the scheduling problem with workers allocation (SPWA). In SPWA, the objective is to minimize the number of workers and the total time taken to perform all tasks (makespan). To this end, we propose the use of two different mathematical programming models and a VNS-based multi-objective heuristic. As the objectives are conflicting, the proposed methods generate a set of efficient solutions, and the manager chooses which solution should be adopted. The proposed methods achieved satisfactory results for the SPWA resolution, showing that it is possible to use a limited staff and finish all tasks in a timely manner, thereby utilizing the resources of a firm optimally.

Keywords:
Multi-objective optimization. Epsilon-restricted. Method of weighted sums. VNS-Multi-objective. SPWA.

Procedure VNS (f(.), N(.), r, s);
1	Being s_o a starting solution;
2	Being r the number of different neighborhood structures;
3	s ← s_o ;
4	While ( non-satisfied stopping criterion )do
5		k ← 1;
6		While (k ≤ r) do
7			Generate any neighbor s’ N⁽^k⁾(s);
8			s” ← LocalSearch (s’);
9 10 11			if (f(s”) < f(s)) them s ←s”; k ← 1;
12			else
13				k ← k + 1;
14			End-if
15		End-while;
16	End-while;
17	Return s;
End VNS;

Procedure VND (f(.), N(.), r, s);
1	Being r the number of different neighborhood structures;
2	k ← 1 ;
3 4	While (k ≤ r) do Find a neighbor s’ N⁽^k⁾(s);
5 6 7	if (f(s’) < f(s)) them s ←s’; k ← 1;
8	else
9	k ← k + 1;
10		End-if
11	End-while;
12	Return s;
End VND;

Procedure VNSM (f(.), N(.), r, S₀)
1	Being S₀ a set of initial solutions;
2	Being r the number of different neighborhood structures;
3	for (l < NSol) do
4	s ← s_lS ₀ ;
5	it ←0;
6	While (it < It_SM) do
7	k ← 1;
8	While (k ≤ r) do
9	Generates any neighbor s_l’N⁽^k⁾(s_l);
10	s” ← LocalSearch (s’);
11	if (f(s_l”) < f(s_l)) them s_l ←s_l”; k ← 1;
12
13
14	else
15	k ← k + 1;
16	it ← it+1;
17	End-if
18	End-while;
19	End-while;
20	s_lS;
21	End-for
22	Return S;
End VNSM

		Useful	Jobs
Worker	1	1	1	3	4
	2	0	0	0	0
	3	0	0	0	0
	4	1	2	5	0

		Useful	Jobs
Worker	1	1	1	3	4	0	0
	2	0	0	0	0	0	0
	3	0	0	0	0	0	0
	4	1	2	5	0	0	0
		Useful	Jobs
Worker	1	1	1	3	0	0	0
	2	0	0	0	0	0	0
	3	0	0	0	0	0	0
	4	1	2	5	4	0	0

		Useful	Jobs
Worker	1	1	1	3	0	0	0
	2	1	4	0	0	0	0
	3	0	0	0	0	0	0
	4	1	2	5	0	0	0
		Useful	Jobs
Worker	1	0	0	0	0	0	0
	2	1	4	1	0	0	0
	3	0	0	0	0	0	0
	4	1	2	5	3	0	0

	*#Worker*	*#Job*
Problem 1	5	10
Problem 2	10	50
Problem 3	15	100
Problem 4	25	100

		Epsilon-Restricted			Weighted sum
Problem 1	Solution	Epsilon - ε	#Worker	makespan (min)	#Worker	makespan (min)
	1	0	5	8	5	8
	2	1	4	10	4	10
	3	2	3	13	3	13
	4	3	2	20	2	20
	5	4	1	43	1	43

		Epsilon-Restricted			Weighted sum
Instance 2	Solution	Epsilon - ε	#Worker	makespan (min)	#Worker	makespan (min)
	1	0	10	20	10	20
	2	1	9	22	9	22
	3	2	8	25	8	25
	4	3	7	28	7	28
	5	4	6	33	6	33
	6	5	5	39	5	39
	7	6	4	49	4	49
	8	7	3	65	3	65
	9	8	2	98	2	98
	10	9	1	215	1	215

		Epsilon-Restricted			Weighted sum
Instance 3	Solution	*Epsilon - ε*	*#Worker*	makespan (min)	*#Worker*	makespan (min)
	1	0	15	27	15	27
	2	2	13	31	13	31
	3	5	10	39	8	49
	4	7	8	49	7	58
	5	9	6	65	5	81
	6	12	3	130	4	107
	7	14	1	430	1	430

		Epsilon-Restricted			Weighted sum
Instance 4	Solution	*Epsilon - ε*	*#Worker*	makespan (min)	*#Worker*	makespan (min)
	1	0	25	16	25	16
	2	3	22	18	22	18
	3	6	19	22	19	22
	4	9	16	25	15	28
	5	12	13	31	11	37
	6	15	10	40	10	40
	8	18	7	57	8	49
	9	21	4	100	3	135
	10	24	1	433	1	433

Method		Problem 1	Problem 2	Problem 3	Problem 4
Epsilon-restricted		5	149	1972	6365
Weighted Sum		5	136	1861	5992
VNSM	Smaller	5.5	78.5	61.4	146.7
	Medium	5.8	101.8	146.8	211.7
	Bigger	6.8	136.5	265.7	277.3

Universidade Federal de São Carlos Departamento de Engenharia de Produção , Caixa Postal 676 , 13.565-905 São Carlos SP Brazil, Tel.: +55 16 3351 8471 - São Carlos - SP - Brazil
E-mail: gp@dep.ufscar.br

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