A HYBRIDIZED MULTI-OBJECTIVE MEMETIC ALGORITHM FOR THE MULTI-OBJECTIVE STOCHASTIC QUADRATIC KNAPSACK PROBLEM

Guerrouma, Amina; Aïder, Méziane

doi:10.1590/0101-7438.2022.042.00257386

Amina Guerrouma

LaROMaD, Fac. Maths, USTHB, PB 32 Bab Ezzouar, 16111 Algiers, Algeria - E-mail: amina.g2008@hotmail.com

https://orcid.org/0000-0003-2625-3973

Méziane Aïder ^**Corresponding author

LaROMaD, Fac. Maths, USTHB, PB 32 Bab Ezzouar, 16111 Algiers, Algeria - E-mail: meziane.aider@usthb.edu.dz

https://orcid.org/0000-0001-7195-810X

*Corresponding author

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Greedy Constructive Heuristic

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Subpopulation construction

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Algorithm 1
Non-dominated Sort Algorithm

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Figure 1
Computation of the Crowding distance.

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Algorithm 2
Crowding-Distance Computation Algorithm

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Algorithm 3
Selection Neighborhood Pareto Local Search SNPLS Algorithm

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Algorithm 4
Gradient algorithm

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Algorithm 5
Memetic Algorithm With Selection Neighborhood Pareto Local Search (MASNPL)

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Figure 2
Comparison between Pareto Fronts of an exact algorithm and MASNPLfor 50 objects.

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Figure 3
Comparison between Pareto Fronts of an exact algorithm and MASNPL for 20 objects.

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Figure 4
Comparison between Pareto Fronts of an exact algorithm and MASNPLfor 15 objects.

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Table 1
Numbers of non-dominated solutions obtained by MASNPL and NSGA-II.

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Table 2
Numbers of non-dominated solutions obtained by MASNPL and NSGA-II.

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Table 3
Numbers of non-dominated solutions obtained by MASNPL and NSGA-II.

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Table 4
Numbers of non-dominated solutions obtained by MASNPL and NSGA-II.

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Table 5
Numbers of non-dominated solutions obtained by MASNPL and NSGA-II.

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Table 6
Numbers of non-dominated solutions obtained by MASNPL and NSGA-II.

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	Number of solutions of Rank 1		CPU-time (s)		Ratio %
	MASNPL	NSGA-II	MASNPL	NSGA-II	MASNPL	NSGA-II
1	10	0	1713, 51311	3419, 76764	100	0
2	6	0	1609, 18844	3376, 87798	100	0
3	10	0	1516, 1261	3250, 44791	100	0
4	20	5	1347, 10586	2879, 00458	80	20
5	2	0	1400, 02226	2831, 42838	100	0
6	5	0	1319, 84907	2909, 47836	100	0
7	5	0	1328, 41828	2930, 77582	100	0
8	6	1	1418, 68944	2882, 2089	85, 71	14, 28
9	5	1	1357, 27806	2911, 37711	83, 33	16, 67
10	16	2	1422, 045	2757, 8156	88, 89	11, 11

	Number of solutions of Rank 1		CPU-time (s)		Ratio %
	(MASNPL)	NSGA-II	MASNPL	NSGA-II	MASNPL	NSGA-II
1	4	0	2555, 20082	4160, 13939	100	0
2	1	1	2485, 20063	4235, 54793	50	50
3	3	0	2393, 36046	4176, 23395	100	0
4	8	0	2481, 62	3965, 26625	100	0
5	1	0	2501, 27197	4039, 40369	100	0
6	1	0	2578, 42903	4144, 85823	100	0
7	8	0	2270, 357	4106, 02298	100	0
8	1	0	2590, 46484	8054, 85057	100	0
9	1	0	2504, 63287	4205, 24089	100	0
10	16	0	2092, 41251	3490, 4851	100	0

	Number of solutions of Rank 1		CPU-time (s)		Ratio %
	(MASNPL)	NSGA-II	MASNPL	NSGA-II	MASNPL	NSGA-II
1	5	0	4298, 38137	4627, 86451	100	0
2	5	0	4329, 33696	4577, 69908	100	0
3	2	1	4286, 61502	4587, 7257	66, 67	33, 33
4	1	0	4457, 91234	4567, 6765	100	0
5	5	0	4040, 51267	4342, 15335	100	0
6	1	0	4341, 55699	4465, 75378	100	0
7	3	0	4451, 06075	4527, 91025	100	0
8	3	0	4537, 64983	4554, 9791	100	0
9	5	0	4267, 92233	4471, 04282	100	0
10	1	0	4479, 37789	4662, 25679	100	0

	Number of solutions of Rank 1		CPU-time (s)		Ratio %
	(MASNPL)	NSGA-II	MASNPL	NSGA-II	MASNPL	NSGA-II
1	66	26	1851, 74066	2445, 45638	71, 74	28, 26
2	31	2	1905, 95985	2648, 07941	93, 94	6, 06
3	71	22	1905, 95985	2648, 07941	76, 34	23, 65
4	69	19	1930, 86798	2560, 80004	78, 41	21, 59
5	29	9	1886, 09592	2627, 31683	76, 31	23, 68
6	14	31	1951, 97476	2582, 64445	36, 84	68, 69
7	97	10	1946, 74633	2408, 61901	90, 65	9, 34
8	13	5	1938, 47225	2592, 48961	72, 22	27, 78
9	1	13	2003, 46515	2616, 81209	7, 14	92, 86
10	9	11	1965, 69301	2734, 79297	45	55

	Number of solutions of Rank 1		CPU-time (s)		Ratio %
	(MASNPL)	NSGA-II	MASNPL	NSGA-II	MASNPL	NSGA-II
1	18	6	3338, 65774	4686, 83084	75	25
2	1	0	3255, 72489	4753, 71708	100	0
3	17	1	3316, 3073	4848, 68752	94, 44	5, 55
4	10	1	3164, 97535	4761, 14919	90, 91	9, 09
5	0	2	3261, 58644	4751, 8666	0	100
6	18	6	3251, 30331	4100, 82273	75	25
7	1	0	3204, 16986	4137, 24102	100	0
8	17	1	3233, 53408	4234, 42546	94, 44	5, 55
9	10	1	3138, 91231	4260, 57573	90, 91	9, 09
10	1	0	3101, 46999	4057, 35701	100	0

	Number of solutions of Rank 1		CPU-time (s)		Ratio %
	(MASNPL)	NSGA-II	MASNPL	NSGA-II	MASNPL	NSGA-II
1	14	15	4994, 0142	7138, 69308	48, 27	51, 72
2	6	11	5149, 28687	7544, 84421	35, 29	64, 7
3	2	3	5273, 99488	7636, 07286	40	60
4	30	21	5119, 07871	7084, 38893	58, 82	41, 18
5	12	2	5262, 87698	7011, 5114	85, 71	14, 28
6	3	1	4743, 11459	6919, 07195	75	25
7	4	0	4885, 45103	7210, 53789	100	0
8	3	1	4791, 12316	7187, 80439	75	25
9	5	1	5134, 35748	7932, 27036	83, 33	16, 67
10	3	1	5063, 31278	7366, 53774	75	25

\max_{x \in {0,1}^{n}} E [\sum_{i = 1}^{n - 1} \sum_{j = i + 1}^{n} r_{i j}^{(k)} x_{i} x_{j} (χ_{i} + χ_{j})] + E [\sum_{i = 1}^{n} r_{i}^{(k)} x_{i} χ_{i}] - d E {[\sum_{i = 1}^{n} x_{i} χ_{i} - c]}^{+}, k = 1,..., m .

{\begin{matrix} \max_{x \in {0,1}^{n}} E [\sum_{i = 1}^{n - 1} \sum_{j = i + 1}^{n} r_{i j}^{k} x_{i} x_{j} (χ_{i} + χ_{j})], k = 1, \dots, m & (3.1) \\ s . t . E [1_{ℝ^{+}} (c-g (x, χ))] \geq p . & (3.2) \end{matrix}

{\begin{matrix} \max_{x \in {0,1}^{n}} E [\sum_{i = 1}^{n - 1} \sum_{j = i + 1}^{n} r_{i j}^{k} x_{i} x_{j} (χ_{i} + χ_{j})], k = 1, \dots, m & (4.1) \\ s . t . P [g (x, χ) \leq c] \geq p, & (4.2) \end{matrix}

J^{(k)} (x, χ) = E [\sum_{i = 1}^{n - 1} \sum_{j = i + 1}^{n} r_{i j}^{(k)} x_{i} x_{j} (χ_{i} + χ_{j})] + E [\sum_{i = 1}^{n} r_{i}^{(k)} x_{i} χ_{i}] - d E {[\sum_{i = 1}^{n} x_{i} χ_{i} - c]}^{+}, k = 1, \dots, m .

\begin{matrix} a) h h a s a u n i q u e m a x i m u m a t x = 0, \\ b) \forall x \in ℝ, h (x) \geq 0, \\ c) \forall x \in ℝ, h (x) = h (- x), \\ d) \int_{- \infty}^{+ \infty} h (x) d x = 1. \end{matrix}

\nabla (J_{t}^{(k)}) (x, χ) = [{(r_{1}^{(k)} χ_{1}, \dots, r_{n}^{(k)} χ_{n})}^{T} + {(\sum_{\begin{matrix} j = 1 \\ j \neq 1 \end{matrix}}^{n} r_{1 j}^{(k)} (χ_{1} + χ_{j}) x_{j}, \dots, \sum_{\begin{matrix} j = 1 \\ j \neq n \end{matrix}}^{n} r_{n j}^{(k)} (χ_{n} + χ_{j}) x_{j})}^{T}] - d (- \frac{1}{t} h (\frac{g (x, χ) - c}{t}) . χ . (g (x, χ) - c) + 1_{ℝ^{+}} (g (x, χ) - c) . χ), k = 1, \dots, m .

\nabla (J_{t}^{(k)}) (x, χ) = [{(r_{1}^{(k)} χ_{1}, \dots, r_{n}^{(k)} χ_{n})}^{T} + {(\sum_{\begin{matrix} j = 1 \\ j \neq 1 \end{matrix}}^{n} r_{1 j}^{(k)} (χ_{1} + χ_{j}) x_{j}, \dots, \sum_{\begin{matrix} j = 1 \\ j \neq n \end{matrix}}^{n} r_{n j}^{(k)} (χ_{n} + χ_{j}) x_{j})}^{T}] - d . (- \frac{3}{4 t} (1 - {(\frac{g (x, χ) - c}{t})}^{2}) \cdot 1_{1} \cdot (\frac{g (x, χ) - c}{t}) . χ . (g (x, χ) - c) - 1_{ℝ^{+}} (g (x, χ) - c) . χ), k = 1, \dots, m .

E [{[X - c]}^{+}] = \int_{- \infty}^{\infty} {[X - c]}^{+} φ (X) d X = \int_{c_{\infty}}^{\infty} (X - c) φ (X) d X = \int_{c}^{\infty} X φ (X) d X - c \int_{c}^{\infty} φ (X) d X = \hat{μ} \int_{c}^{\infty} φ (X) d X - {\hat{σ}}^{2} \int_{c}^{\infty} φ' (X) d X - c \int_{c}^{\infty} φ (X) d X = {\hat{σ}}^{2} φ {[(X)]}_{c}^{\infty} + (\hat{μ} - c) {[Φ (x)]}_{c}^{\infty} = {\hat{σ}}^{2} φ (c) + (\hat{μ} - c) [1 - Φ (c)] = \hat{σ} f (\frac{c - \hat{μ}}{\hat{σ}}) + (\hat{μ} - c) \cdot [1 - F (\frac{c - \hat{μ}}{\hat{σ}})]

J_{det}^{k} (x) = \sum_{i = 1}^{n - 1} \sum_{j = i + 1}^{n} r_{i j}^{(k)} x_{i} x_{j} (μ_{i} + μ_{j}) + \sum_{i = 1}^{n} r_{i}^{(k)} μ_{i} x_{i} - d [\hat{σ} f (\frac{c - \hat{μ}}{\hat{σ}}) - (c - \hat{μ}) [1 - Ϝ (\frac{c - \hat{μ}}{\hat{σ}})]], k = 1, \dots, m .

J^{k} (x, χ) = E [\sum_{i = 1}^{n - 1} \sum_{j = i + 1}^{n} r_{i j}^{(k)} x_{i} x_{j} (χ_{i} + χ_{j})] + E [\sum_{i = 1}^{n} r_{i}^{(k)} x_{i} χ_{i}] - d . E {[\sum_{i = 1}^{n} x_{i} χ_{i} - c]}^{+}, k = 1, \dots, m .

{\begin{matrix} J (x, χ) = \sum_{k = 1}^{m} λ^{(k)} f^{(k)} (x) & (31.1) \\ \sum_{k = 1}^{m} λ^{(k)} = 1 & (31.2) \\ λ^{(k)} \geq 0, k = 1, \dots, m & (31.3) \end{matrix}

\begin{matrix} J^{(k)} (x, χ) = \sum_{k = 1}^{m} λ^{(k)} . [E [\sum_{i = 1}^{n - 1} \sum_{j = i + 1}^{n} r_{i j}^{(k)} x_{i} x_{j} (χ_{i} + χ_{j})] + E [\sum_{i = 1}^{n} r_{i}^{(k)} x_{i} χ_{i}] - d E {[\sum_{i = 1}^{n} x_{i} χ_{i} - c]}^{+}], \end{matrix}

\nabla (J_{t}^{(k)}) (x, χ) = λ^{(k)} [[{(r_{1}^{(k)} χ_{1}, \dots, r_{n}^{(k)} χ_{n})}^{T} + {(\sum_{\begin{matrix} j = 1 \\ j \neq 1 \end{matrix}}^{n} r_{1 j}^{(k)} (χ_{1} + χ_{j}) x_{j}, \dots, \sum_{\begin{matrix} j = 1 \\ j \neq n \end{matrix}}^{n} r_{n j}^{(k)} (χ_{n} + χ_{j}) x_{j})}^{T}] - d . (- \frac{3}{4 t} (1 - {(\frac{g (x, χ) - c}{t})}^{2}) \cdot 1_{1} \cdot (\frac{g (x, χ) - c}{t}) . χ . (g (x, χ) - c) - 1_{ℝ^{+}} (g (x, χ) - c) . χ)], k = 1, \dots, m .

[1] *Corresponding author

Brasil

Brasil

A HYBRIDIZED MULTI-OBJECTIVE MEMETIC ALGORITHM FOR THE MULTI-OBJECTIVE STOCHASTIC QUADRATIC KNAPSACK PROBLEM

ABSTRACT