BIOBJECTIVE INTEGER STOCHASTIC OPTIMIZATION OVER THE INTEGER STOCHASTIC EFFICIENT SET

Badaoui, Ilias; Moulaï, Mustapha; Chaiblaine, Yacine; Chaabane, Djamal

doi:10.1590/0101-7438.2023.043.00281853

Ilias Badaoui ^**Corresponding author

USTHB, LaROMaD, Department of Operational Research, Bp 32 El Alia, 16111 Algiers, Algeria - E-mail: iliesbadi@hotmail.fr

https://orcid.org/0000-0003-0189-327X

Mustapha Moulaï

USTHB, LaROMaD, Department of Operational Research, Bp 32 El Alia, 16111 Algiers, Algeria - E-mail: mmoulai@usthb.dz

https://orcid.org/0000-0002-2013-356X

Yacine Chaiblaine

USTHB, LaROMaD, Department of Operational Research, Bp 32 El Alia, 16111 Algiers, Algeria - E-mail: ychaiblaine@usthb.dz

https://orcid.org/0000-0002-8935-1020

Djamal Chaabane

USTHB, AMCD&RO, Department of Operational Research, Bp 32 El Alia, 16111 Algiers, Algeria - E-mail: chaabane_dj@yahoo.fr

https://orcid.org/0000-0002-7722-311X

*Corresponding author

B	Rhs	x ₉	x10
x ₈	$\frac{1}{3}$	$\frac{- 1}{3}$	$\frac{- 5}{3}$
x ₃	12	−1	−4
x ₄	6	−1	3
x ₅	1	−2	1
x ₆	1	−1	0
x ₂	3	0	−1
x ₇	2	0	−1
θ	$\frac{668}{27}$	$\frac{32}{27}$	$\frac{295}{27}$
x ₁	3	1	0
${\bar{Z}}_{1}$	−9	5	2
${\bar{Z}}_{2}$	3	−4	−3
${\bar{Z}}_{3}$	−9	−1	−4
${\bar{d}}_{1}$	9	1	4
${\bar{d}}_{2}$	9	−2	1

		CPU (Sec)			Nbr of Nodes			ρ
R	n × m × k	Min	Mean	Max	Min	Mean	Max	Mean
2	5 × 3 × 3	0.41	2.47	6.43	10	42	88	0.40
	10 × 5 × 3	2.08	4.49	6.80	40	79	116	0.39
	20 × 10 × 5	5.39	8.51	12.58	134	197.20	338	0.09
	30 × 15 × 5	21.15	30.11	39.15	322	490	652	0.10
	40 × 20 × 5	55.80	82.15	164.64	608	721.20	954	0.10
	50 × 25 × 10	76.74	113.55	135.27	754	918	1044	0.27
	60 × 30 × 10	138.55	181.98	243.79	916	1112.80	1478	0.12
	70 × 35 × 10	346.84	405.40	489.74	1580	1822.40	2344	0.09
	80 × 40 × 20	513.05	565.49	649.62	1446	1770	2206	0.11
	90 × 45 × 20	783.20	1070.44	1395.43	2546	2667.60	2978	0.10
	100 × 50 × 20	1003.98	1407.96	1897.15	2288	2816	3438	0.12
Glob	Avrg(R = 2)	267.93	352.05	458.24	967.64	930.49	1421.45	0.17
3	5 × 3 × 3	0.97	3.51	11.65	8	33.20	98	0.57
	10 × 5 × 3	1.06	4.23	7.50	18	66.40	118	0.37
	20 × 10 × 5	3.54	11.78	18.64	106	190.80	256	0.29
	30 × 15 × 5	18.64	23.62	26.31	322	398.80	454	0.13
	40 × 20 × 5	38.28	60.72	88.62	576	693.20	904	0.30
	50 × 25 × 10	71.32	104.45	187.82	538	862.40	1424	0.19
	60 × 30 × 10	110.91	191.95	358.30	832	1111.60	1634	0.06
	70 × 35 × 10	205.65	340.08	492.87	864	1425.20	1798	0.07
	80 × 40 × 20	349.53	612.98	876.32	1180	2019.20	2844	0.08
	90 × 45 × 20	689.63	954.20	1466.57	1706	2415.20	3040	0.06
	100 × 50 × 20	273.65	1402.61	2272.41	746	2535.60	4096	0.12
Glob	Avrg(R = 3)	160.24	337.29	527.91	337.29	626.91	1515.09	0.20
5	5 × 3 × 3	1.84	3.94	6.70	10	26.40	46	0.29
	10 × 5 × 3	5.65	7.89	14.33	58	84.40	154	0.40
	20 × 10 × 5	7.60	11.75	20.91	152	183.20	238	0.27
	30 × 15 × 5	22.31	27.64	33.54	302	361.20	450	0.13
	40 × 20 × 5	36.35	72.61	118.72	384	648.80	992	0.08
	50 × 25 × 10	101.96	143.70	213.36	826	1036.80	1434	0.08
	60 × 30 × 10	154.29	267.86	360.80	1048	1419.60	1816	0.07
	70 × 35 × 10	244.93	350.09	471.33	1324	1523.20	1804	0.05
	80 × 40 × 20	545.48	812.29	993.73	1860	2357.20	2882	0.04
	90 × 45 × 20	913.20	1033.18	1240.23	1944	2503.60	2966	0.04
	100 × 50 × 20	873.47	1282.77	1674.22	1744	2548.80	3398	0.07
Glob	Avrg(R = 5)	264.28	364.88	467.98	877.45	1153.93	1470.91	0.14
10	5 × 3 × 3	1.40	5.12	8.36	8	21.20	34	0.33
	10 × 5 × 3	3.44	9.40	16.91	18	56	104	0.27
	20 × 10 × 5	8.59	23.96	44.38	68	231.20	434	0.07
	30 × 15 × 5	18.74	41.13	63.64	172	419.60	552	0.11
	40 × 20 × 5	67.11	86.19	127.58	598	762.80	994	0.16
	50 × 25 × 10	68.85	135.97	170.81	570	1040	1288	0.07
	60 × 30 × 10	167.50	261.94	304.53	904	1348	1844	0.07
	70 × 35 × 10	290.50	373.66	485.72	1404	1499.20	1752	0.07
	80 × 40 × 20	427.21	596.96	759.83	1462	1771.20	2274	0.05
	90 × 45 × 20	621.69	1032.79	1626.56	1934	2383.20	3042	0.06
	100 × 50 × 20	477.19	1263.33	1814.29	1180	2535.60	3724	0.07
Glob	Avrg(R = 10)	195.66	348.22	492.96	756.18	1097.09	1458.36	0.12

(M O S I L P_{R}) \{\begin{cases} m i n Z_{i}^{r} = C_{i} (ξ^{r}) x, i \in \{1, 2, \dots, K\}, r \in \{1, 2, \dots, R\}, \\ s . t ., \\ A x = b, \\ T (ξ^{r}) x = h (ξ^{r}), r \in \{1, 2, \dots, R\}, \\ x \in ℕ . \end{cases}

\{\begin{cases} {\tilde{Z}}_{i} = E [Z_{i}^{r}] = \sum_{r = 1}^{R} P^{r} Z_{i}^{r} = \sum_{r = 1}^{R} P^{r} C_{i} (ξ^{r}) x = {\tilde{C}}_{i} x, i \in \{1, 2, \dots, K\}, \\ Q (x) = E [Q (x, ξ)] = \sum_{r = 1}^{R} P^{r} {(q^{r})}^{T} y^{r} . \end{cases}

\{\begin{cases} {\tilde{ϕ}}_{1} = E [ϕ_{1}^{r}] = E [d_{1} (ξ) x] = \sum_{r = 1}^{R} P^{r} d_{1} (ξ^{r}) x = {\tilde{d}}_{1} x, \\ {\tilde{ϕ}}_{2} = E [ϕ_{2}^{r}] = E [d_{2} (ξ) x] = \sum_{r = 1}^{R} P^{r} d_{2} (ξ^{r}) x = {\tilde{d}}_{2} x, \\ Q (x) = E [Q (x, ξ)] = \sum_{r = 1}^{R} P^{r} {(q^{r})}^{T} y^{r} . \end{cases}

(P_{R}) \{\begin{cases} m i n {\tilde{ϕ}}_{1} = {\tilde{d}}_{1} x + E [Q (x, ξ)], \\ m i n {\tilde{ϕ}}_{2} = {\tilde{d}}_{2} x + E [Q (x, ξ)], \\ s . t ., \\ x \in S, \end{cases}

(E K (x^{l})) \{\begin{cases} m a x Ψ = \sum_{i = 1}^{K} ψ_{i}, \\ s . t ., \{\begin{cases} {\tilde{C}}_{i} x + ψ_{i} = {\tilde{C}}_{i} x^{l}, i \in \{1, 2, \dots K\}, \\ x \in S, \\ ψ_{i} \geq 0, i \in \{1, 2, \dots, K\} . \end{cases} \end{cases}

(E K^{1} (x^{l})) \{\begin{cases} m a x \sum_{i = 1}^{2} v_{i}, \\ s . t ., \\ {\tilde{d}}_{i} x - v_{i} = {\tilde{d}}_{i} x^{l}, i \in \{1, 2\}, \\ x \in S_{l}, \\ v_{i} \geq 0, i \in \{1, 2\} . \end{cases}

(E K^{2} (x^{l})) \{\begin{cases} m a x \sum_{i = 1}^{K} w_{i}, \\ s . t ., \\ {\tilde{C}}_{i} x - w_{i} = {\tilde{C}}_{i} x^{l}, i \in \{1, 2, \dots, K\}, \\ x \in S_{l}, \\ w_{i} \geq 0, i \in \{1, 2, \dots, K\} . \end{cases}

\{\begin{cases} S_{l + 1}^{1} \{x \in S_{l} | \sum_{j \in H_{l}} x_{j} \geq 1\}, \\ S_{l + 1}^{2} \{x \in S_{l} | \sum_{j \in H_{l}^{'}} x_{j} \geq 1\}, \end{cases}

{\tilde{Z}}_{i} (x) - {\tilde{Z}}_{i} (x^{i}) = \sum_{j \in N_{l}} {\bar{C}}_{i}^{j} x_{j}, = \sum_{j \in H_{l}} {\bar{C}}_{i}^{j} x_{j} + \sum_{j \in N_{l} | H_{l}} {\bar{C}}_{i}^{j} x_{j}, = \sum_{j \in N_{l} | H_{l}} {\bar{C}}_{i}^{j} x_{j} .

(E K (x^{l})) \{\begin{cases} m a x \sum_{i = 1}^{p} v_{i}, \\ s . t ., \\ {\tilde{d}}_{i} x - v_{i} = {\tilde{d}}_{i} x^{l}, i \in \{1, 2, \dots, p\}, \\ x \in S_{l}, \\ v_{i} \geq 0, i \in \{1, 2, \dots, p\} . \end{cases}

S c e n a r i o 1 S c e n a r i o 2 M O S I L P (ξ^{l}) \{\begin{array}{l} m i n Z_{1}^{1} = - 9 x_{1} + 4 x_{2}, \\ m i n Z_{2}^{1} = 3 x_{1} - 5 x_{2}, \\ m i n Z_{3}^{1} = 5 x_{1} - 11 x_{2}, \\ s . t ., \\ x_{1} - 4 x_{2} \leq 3, \\ x_{1} + 3 x_{2} \leq 18, \\ 2 x_{1} + x_{2} \leq 10, \\ x_{1} + 2 x_{2} \leq 3, \\ - 3 x_{1} + x_{2} \leq 5, \\ x_{1}, x_{2} \in ℕ . \end{array} M O S I L P (ξ^{2}) \{\begin{cases} m i n Z_{1}^{2} = 3 x_{1} - 2 x_{2}, \\ m i n Z_{2}^{2} = 6 x_{1} + x_{2}, \\ m i n Z_{3}^{2} = - 7 x_{1} + 10 x_{2}, \\ s . t ., \\ x_{1} - 4 x_{2} \leq 3, \\ x_{1} + 3 x_{2} \leq 18, \\ 2 x_{1} + x_{2} \leq 10, \\ 5 x_{1} + x_{2} \leq 4, \\ - 3 x_{1} + 2 x_{2} \leq 1, \\ x_{1}, x_{2} \in ℕ . \end{cases}

{\tilde{Z}}_{1} = E [C_{1} (ξ) x] = ℙ (ξ^{1}) C_{1} (ξ^{1}) x + ℙ (ξ^{2}) C_{1} (ξ^{2}) x, {\tilde{Z}}_{2} = E [C_{2} (ξ) x] = ℙ (ξ^{1}) C_{2} (ξ^{1}) x + ℙ (ξ^{2}) C_{2} (ξ^{2}) x, {\tilde{Z}}_{3} = E [C_{3} (ξ) x] = ℙ (ξ^{1}) C_{3} (ξ^{1}) x + ℙ (ξ^{2}) C_{3} (ξ^{2}) x, (M O S I L P_{D}) \{\begin{cases} m i n {\tilde{Z}}_{1} = - 5 x_{1} + 2 x_{2}, \\ m i n {\tilde{Z}}_{2} = 4 x_{1} - 3 x_{2}, \\ m i n {\tilde{Z}}_{3} = x_{1} - 4 x_{2}, \\ s . t ., \\ x_{1} - 4 x_{2} \leq 3, \\ x_{1} + 3 x_{2} \leq 18, \\ 2 x_{1} + x_{2} \leq 10, \\ x_{1}, x_{2} \in ℕ . \end{cases}

S c e n a r i o 1 S c e n a r i o 2 P (ξ^{1}) \{\begin{cases} m i n ϕ_{1}^{1} = - 2 x_{1} + 4 x_{2}, \\ m i n ϕ_{2}^{1} = - 2 x_{1} - 1 x_{2}, \\ s . t ., \\ x_{1}, x_{2} \in X_{E D} . \end{cases} P (ξ^{2}) \{\begin{cases} m i n ϕ_{1}^{2} = x_{1} + 4 x_{2}, \\ m i n ϕ_{2}^{2} = 2 x_{1} + 5 x_{2}, \\ s . t ., \\ x_{1}, x_{2} \in X_{E D} . \end{cases}

{\tilde{ϕ}}_{1} = E [d_{1} (ξ) x] = ℙ (ξ^{1}) d_{1} (ξ^{1}) x + ℙ (ξ^{2}) d_{1} (ξ^{2}) x, {\tilde{ϕ}}_{2} = E [d_{2} (ξ) x] = ℙ (ξ^{1}) d_{2} (ξ^{1}) x + ℙ (ξ^{2}) d_{2} (ξ^{2}) x, (P_{D R}) \{\begin{cases} m i n {\tilde{ϕ}}_{1} = - x_{1} + 4 x_{2}, \\ m i n {\tilde{ϕ}}_{2} = 2 x_{1} + x_{2}, \\ s . t ., \\ x_{1}, x_{2} \in S, \end{cases}

h (ξ^{1}) - T (ξ^{1}) x^{2} = (\begin{matrix} 3 \\ 5 \end{matrix}) - (\begin{matrix} 1 & 2 \\ - 3 & 1 \end{matrix}) (\begin{matrix} 4 \\ 1 \end{matrix}) = (\begin{matrix} - 3 \\ 16 \end{matrix}) . h (ξ^{2}) - T (ξ^{2}) x^{2} = (\begin{matrix} 4 \\ 1 \end{matrix}) - (\begin{matrix} 5 & 1 \\ - 3 & 2 \end{matrix}) (\begin{matrix} 4 \\ 1 \end{matrix}) = (\begin{matrix} - 6 \\ 11 \end{matrix}) . S c e n a r i o 1 S c e n a r i o 2 σ (ξ^{1}) \{\begin{array}{l} m a x + - 3 σ_{1}^{1} + 16 σ_{1}^{2}, \\ - 2 σ_{1}^{1} + 3 σ_{1}^{2} \leq 0, \\ - σ_{1}^{1} + 2 σ_{1}^{2} \leq 0, \\ 2 σ_{1}^{1} - 5 σ_{1}^{2} \leq 0, \\ σ_{1}^{1} - 6 σ_{1}^{2} \leq 0, \\ σ_{1}^{1} + σ_{1}^{2} \leq 1 . \end{array} σ_{1}^{T} = (\frac{2}{3}, \frac{1}{3}), σ (ξ^{2}) \{\begin{array}{l} m a x + - 6 σ_{2}^{1} + 11 σ_{2}^{2}, \\ - 2 σ_{2}^{1} + 3 σ_{2}^{2} \leq 0, \\ - σ_{2}^{1} + 2 σ_{2}^{2} \leq 0, \\ 2 σ_{2}^{1} - 5 σ_{2}^{2} \leq 0, \\ σ_{2}^{1} - 6 σ_{2}^{2} \leq 0, \\ σ_{2}^{1} + σ_{2}^{2} \leq 1 . \end{array} σ_{2}^{T} = (0, 0) .

S c e n a r i o 1 S c e n a r i o 2 σ (ξ^{1}) \{\begin{array}{l} m a x - 6 σ_{1}^{1} + 11 σ_{1}^{2}, \\ - 2 σ_{1}^{1} + 3 σ_{1}^{2} \leq 0, \\ - σ_{1}^{1} + 2 σ_{1}^{2} \leq 0, \\ 2 σ_{1}^{1} - 5 σ_{1}^{2} \leq 0, \\ σ_{1}^{1} - 6 σ_{1}^{2} \leq 0, \\ σ_{1}^{1} + σ_{1}^{2} \leq 1 . \end{array} σ_{1}^{T} = (0, 0), σ (ξ^{2}) \{\begin{array}{l} m a x - 14 σ_{2}^{1} + 4 σ_{2}^{2}, \\ - 2 σ_{2}^{1} + 3 σ_{2}^{2} \leq 0, \\ - σ_{2}^{1} + 2 σ_{2}^{2} \leq 0, \\ 2 σ_{2}^{1} - 5 σ_{2}^{2} \leq 0, \\ σ_{2}^{1} - 6 σ_{2}^{2} \leq 0, \\ σ_{2}^{1} + σ_{2}^{2} \leq 1 . \end{array} σ_{2}^{T} = (0, 0) .

S c e n a r i o 1 S c e n a r i o 2 π (ξ^{1}) \{\begin{cases} m a x - 6 π_{1}^{1} + 11 π_{1}^{2}, \\ - 2 π_{1}^{1} + 3 π_{1}^{2} \leq 1, \\ - π_{1}^{1} + 2 π_{1}^{2} \leq 3, \\ 2 π_{1}^{1} - 5 π_{1}^{2} \leq 6, \\ π_{1}^{1} - 6 π_{1}^{2} \leq 2 . \end{cases} π_{1}^{T} = (7, 5), π (ξ^{2}) \{\begin{cases} m a x - 14 π_{1}^{1} + 4 π_{1}^{2}, \\ - 2 π_{1}^{1} + 3 π_{1}^{2} \leq 5, \\ - π_{1}^{1} + 2 π_{1}^{2} \leq 3, \\ 2 π_{1}^{1} - 5 π_{1}^{2} \leq 2, \\ π_{1}^{1} - 6 π_{1}^{2} \leq 1 . \end{cases} π_{2}^{T} = (- \frac{11}{3}, - \frac{7}{9}) .

(E K^{1} (x^{5})) \{\begin{cases} m a x ω = ω_{1} + ω_{2}, \\ s . t ., \\ x \in S_{8}, \\ θ \geq \frac{689}{27} + \frac{32}{3} x_{1} - \frac{295}{27} x_{2}, \\ - x_{1} + 4 x_{2} + ω_{1} = 9, \\ 2 x_{1} + x_{2} + ω_{2} = 9, \\ x_{1}, x_{2} \in ℕ, θ, ω_{1}, ω_{2} \in ℝ^{+} . \end{cases}

(E K^{2} (x^{5})) \{\begin{cases} m a x Ψ = ψ_{1} + ψ_{2} + ψ_{3}, \\ s . t ., \\ x \in S_{8}, \\ θ \geq \frac{689}{27} + \frac{32}{3} x_{1} - \frac{295}{27} x_{2}, \\ - 5 x_{1} + 2 x_{2} + ψ_{1} = - 9, \\ 4 x_{1} - 3 x_{2} + ψ_{2} = 3, \\ x_{1} - 4 x_{2} + ψ_{3} = - 9, \\ x_{1}, x_{2} \in ℕ, θ, ψ_{1}, ψ_{2}, ψ_{3} \in ℝ^{+} . \end{cases}

[1] *Corresponding author

Brasil

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BIOBJECTIVE INTEGER STOCHASTIC OPTIMIZATION OVER THE INTEGER STOCHASTIC EFFICIENT SET

ABSTRACT