STOCHASTIC KNAPSACK PROBLEM: APPLICATION TO TRANSPORTATION PROBLEMS

Kosuch, Stefanie; Letournel, Marc; Lisser, Abdel

doi:10.1590/0101-7438.2017.037.03.0597

Stefanie Kosuch

Laboratoire de Recherche en Informatique (LRI), Université Paris Sud - XI, Bât. 650, 91405 Orsay Cedex, France. E-mails: stefanie@kosuch.eu; marc.letournel@lri.fr; lisser@lri.fr

Marc Letournel

Laboratoire de Recherche en Informatique (LRI), Université Paris Sud - XI, Bât. 650, 91405 Orsay Cedex, France. E-mails: stefanie@kosuch.eu; marc.letournel@lri.fr; lisser@lri.fr

Abdel Lisser ^**Corresponding author.

Laboratoire de Recherche en Informatique (LRI), Université Paris Sud - XI, Bât. 650, 91405 Orsay Cedex, France. E-mails: stefanie@kosuch.eu; marc.letournel@lri.fr; lisser@lri.fr

*Corresponding author.

	Arrow-Hurwicz & FD-method			Arrow-Hurwicz & IP-method			SOCP
n	Optimum	CPU-time (msec)	Gap	Optimum	CPU-time (msec)	Gap	Optimum	CPU-time (msec)
C./B.	4695.525	3	0.02%	4667.790	30	0.75%	4696.413	4
15	4954.314	5	0.01%	4910.434	30	0.89%	4954.804	4
20	6712.219	8	0.03%	6653.941	40	0.89%	6713.987	6
30	10308.293	14	0.02%	10174.878	53	1.31%	10310.45	18
50	16992.444	28	0.01%	16743.260	92	1.47%	16993.514	65
75	25565.525	52	0.01%	25155.023	138	1.63%	25569.379	213
100	33902.119	85	0%	33208.278	181	2.05%	33903.672	503
150	50677.120	174	0%	49609.044	271	2.10%	50678.312	1802
250	85186.119	438	0%	83785.657	431	1.64%	**	**
500	170226.568	1638		167342.988	868		**	**
1000				334984.436	1720		**	**
5000				1676670.246	8640		**	**
20000				6731686.102	36031		**	**

	Arrow-Hurwicz & FD-method		Arrow-Hurwicz & IP-method		MOSEK
n	CPU (msec)	Gap	CPU (msec)	Gap	CPU (msec)
C./B.	1	0.03%	2	0.03%	25
15	1	0.02%	2	0.02%	22
20	1	0.02%	3	0.02%	22
30	1	0.02%	4	0.02%	22
50	3	0.01%	6	0.02%	24
75	4	0.02%	9	0.02%	26
100	5	0.01%	12	0.01%	28
150	8	0.01%	17	0.01%	31
250	13	0.01%	28	0.01%	37
500	25	0.01%	56	0.01%	52
1000	51	0.01%	111	0.01%	89

	Arrow-Hurwicz & FD-method				Arrow-Hurwicz & IP-method				SOCP
n	Optimum	Number of nodes	CPU-time B-and-B	Gap	Optimum	Number of nodes	CPU-time B-and-B	Gap	Optimum	Number of nodes	CPU-time B-and-B
C./B.	4595	122	0.09	0%	4595	122	0.21	0%	4595	122	0.406
15	4840	31	0.02	0%	4840	34	0.06	0%	4840	34	0.082
20	6634	69	0.07	0%	6634	63	0.13	0%	6634	66	0.236
30	10272	271	0.35	0%	10272	436	1.22	0%	10272	350	1.801
50	16975	3341	6.43	0%	16975	7051	31.34	0%	16975	7406	70.914
75	25548	6187	18.43	0%	25548	23911	161.47	0%	25548	62175	1535.520
100	33895	12093	45.87	-	33894	98479	1049.18	-	*	*	*
150	50672	37076	244.11	-	*	*	*	*	*	*	*
250	85189	65890	623.53	-	*	*	*	*	*	*	*
500	*	*	*	*	*	*	*	*	*	*	*

Θ (x) = [u (x, χ) v (x, χ)]_{- \infty}^{\infty} - \int_{- \infty}^{\infty} u (x, χ) v'_{χ_{κ}} (x, χ) d χ = - \int_{- \infty}^{\infty} u (x, χ) v'_{χ_{κ}} (x, χ) d χ = - \int_{- \infty}^{\infty} Y_{ℝ^{+}} (c - g (x, χ)) v'_{χ_{κ}} (x, χ) d χ

φ'_{χ_{κ}} (χ) = \prod_{i \neq κ} φ^{i} (χ_{i}) \cdot (φ^{κ})'_{χ_{κ}} (χ_{κ}) = \prod_{i \neq κ} φ^{i} (χ_{i}) \cdot (- \frac{(χ_{κ} - μ_{κ})}{σ_{κ}^{2}} φ^{κ} (χ_{κ})) = - \frac{(χ_{κ} - μ_{κ})}{σ_{κ}^{2}} φ (χ)

Θ (x) = - \int_{- \infty}^{\infty} Y_{ℝ^{+}} (c - g (x, χ)) \frac{(χ_{κ} - μ_{κ})}{x_{κ} σ_{κ}^{2}} φ (χ) d χ = E [- Y_{ℝ^{+}} (c - g (x, χ)) \frac{(χ_{κ} - μ_{κ})}{x_{κ} σ_{κ}^{2}}] □

{(\nabla_{x} (- Y_{ℝ^{+}} (c - g (x, χ)) \frac{(χ_{κ} - μ_{κ})}{x_{κ} σ_{κ}^{2}}))}_{k} = ℍ_{ℝ^{+}} (c - g (x, χ)) \frac{(χ_{κ} - μ_{κ})}{x_{κ} σ_{κ}^{2}} χ_{κ} + [c - g (x, χ)]^{+} \frac{(χ_{κ} - μ_{κ})}{x_{κ}^{2} σ_{κ}^{2}} = ℍ_{ℝ^{+}} (c - g (x, χ)) (χ_{κ} - μ_{κ}) (\frac{χ_{κ}}{x_{κ} σ_{κ}^{2}} + \frac{(c - g (x, χ))}{x_{κ}^{2} σ_{κ}^{2}})

0 \leq (\nabla_{x} (- Y_{ℝ^{+}} (c - g (x, χ)) \frac{(χ_{κ} - μ_{κ})}{x_{κ} σ_{κ}^{2}}))_{κ} \leq (χ_{κ} - μ_{κ}) (n \cdot \frac{χ_{κ}}{σ_{κ}^{2}} + n^{2} \cdot \frac{(c - g (x, χ))}{σ_{κ}^{2}})

{(\nabla_{x} \tilde{l} (x, λ, χ))}_{h} = r_{h} χ_{h} + λ (ℍ_{ℝ^{+}} (c - g (x, χ)) \frac{(χ_{κ} - μ_{κ})}{x_{κ} σ_{κ}^{2}} (χ_{h} + \frac{(c - g (x, χ))}{x_{κ}} ν_{h}^{κ}))

{(\nabla_{x} \tilde{l} (x, λ, χ))}_{h} = r_{h} χ_{h} - λ (ℍ_{ℝ^{+}} (g (x, χ) - c) \frac{(χ_{κ} - μ_{κ})}{x_{κ} σ_{κ}^{2}} (χ_{h} - \frac{(g (x, χ) - c)}{x_{κ}} ν_{h}^{κ}))

{(\nabla_{x} \tilde{l} (x, λ, χ))}_{h} = r_{h} χ_{h} - λ (ℍ_{ℝ^{+}} (g (x, χ) - c) \frac{(χ_{κ} - μ_{κ})}{x_{κ} σ_{κ}^{2}} (χ_{h} - \frac{(g (x, χ) - c)}{x_{κ}} ν_{h}^{κ}))

[1] *Corresponding author.

Brasil

Brasil

STOCHASTIC KNAPSACK PROBLEM: APPLICATION TO TRANSPORTATION PROBLEMS

ABSTRACT