BRANCH-AND-CUT METHOD FOR SOLVING THE INTEGER LINEAR MULTIPLICATIVE BILEVEL PROBLEM

Maachou, Nacera; Moulaï, Mustapha

doi:10.1590/0101-7438.2023.043.00278191

Nacera Maachou ^** Corresponding author

LaROMaD Laboratory, Department of Operations Research, USTHB, Faculty of Mathematics, BP 32, Bab Ezzouar 16111, Algiers, Algeria - E-mail: maachou.nacera.72@gmail.com

https://orcid.org/0000-0001-6710-5622

Mustapha Moulaï

LaROMaD Laboratory, Department of Operations Research, USTHB, Faculty of Mathematics, BP 32, Bab Ezzouar 16111, Algiers, Algeria - E-mail: mmoulai@usthb.dz

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

* Corresponding author

Step 1: Initializationl = 0, 𝒳 ₀ = 𝒳 , 𝒟 _E = Ø, ${\bar{D}}_{E} = \emptyset$ .
Step 2: General Step .
	While there is a non-fathomed node in the tree, choose the node l not yet fathomed and solve the corresponding linear problem (16).
		• If the problem (16) is unfeasible, then the corresponding node l is fathomed.
		• Otherwise, let ( ẑ ( l )) be an optimal solution of the problem (16).
			1. If ẑ ( l ) is integer, goto Step 2a.
			2. Otherwise, goto Step 2b.
Step 2a:
		• If the vector ( g ₁( ẑ ( l )) , ..., g _k ( ẑ ( l ))) is not dominated by the vector ( g ₁( z ) ,..., g _k ( z )) for any z ∈ 𝒟 _E then $D_{E} = D_{E} \cup {{\hat{z}}^{(l)}}$ and ${\bar{D}}_{E} = {\bar{D}}_{E}$ .
		• If there is a solution z ∈ 𝒟 _E such that ( g ₁( ẑ ( l )) , ..., g _k ( ẑ ( l ))) dominates ( g ₁( z ) ,... ; g _k ( z )) then $D_{E} = D_{E} \ {z} \cup {{\hat{z}}^{(l)}}$ and ${\bar{D}}_{E} = {\bar{D}}_{E} \cup {z}$ .
	Determine sets N _l , H _l ;
		1. If H _l = Ø then the node l is fathomed and goto Step 2;
		2. Otherwise, add the efficient cut $\sum_{j \in H_{l}} z_{j} \geq 1$ to problem (16) and goto Step 2, a new node is created, l = l + 1.
Step 2b: Branching Process Since there is at least one non-integer value among the values ẑ ( l ), choose one of them such that ${\hat{z_{j}}}^{(l)} = ν_{j}$ , where ν _j is a fraction. Partition the feasible set 𝒳 _l into two parts ( 𝒳 _l1 and 𝒳 _l2 ) ( l ₁ > l + 1 , l ₂ > l + 1 , l ₁ ≠ l ₂) by adding to 𝒳 _l new branching constraints ${\hat{z_{j}}}^{(l)} \leq ⌊ ν_{j} ⌋$ and ${\hat{z_{j}}}^{(l)} \geq ⌊ ν_{j} ⌋ + 1$ , respectively, and goto Step 2.

Step 1: Initializationp = 0, F _opt = − ∞ and f _opt = − ∞, z _opt = Ø.
	Solve the upper level problem ULILMP₀. Using the above Algorithm 2 , the sets of efficient and dominated solutions ?? _E and ${\bar{D}}_{E}$ respectively are determined,
	$D_{E}^{0} = D_{E}, {\bar{D}}_{E}^{0} = {\bar{D}}_{E} .$
		• If ULILMP₀ is unfeasible, then the bilevel integer optimal solution of the main problem (8) does not exist, and the algorithm stops.
		• Otherwise, let z ^∗(0) = ( x ^∗(0) , y ^∗(0)), the optimal solution of the upper level ( ULILMP ₀), F _opt = F ( z ^∗(0)) and goto step 3.
Step 2: General Step. Use the above Algorithm 2 to solve the following upper level problem at node p .
${ULILMP}_{p} \{\begin{cases} \max_{(z)} F (z) = \prod_{i = 1}^{k} (c^{i} z + α_{i}) \\ s . t z \in S_{p} \end{cases}$ (28)
	The sets of efficient and dominated solutions, 𝒟 _E and ${\bar{D}}_{E}$ respectively, are then updated, $D_{E}^{p} = D_{E}$ , ${\bar{D}}_{E}^{p} = {\bar{D}}_{E}$ . Let z ^(p) = ( x ^∗(p) , y* ^∗(p) ) be the optimal solution of the upper level ULILMP_p .
		• If ULILMP_p is unfeasible or F ( z ^∗(p) ) > F _opt (for p ≥ 1), then the corresponding node p is fathomed.
		• Otherwise, F _opt = F ( z ^∗(p) ) and goto step 3.
Step 3: Put x = x ^∗(p) and solve the lower level problem
$LLILMP \{\begin{cases} \max_{y} f (x, y) = \prod_{j = 1}^{l} (p_{1}^{j} x^{* (p)} + p_{2}^{j} y + β_{j}) \\ s . t (x^{* (p)}, y) \in {\hat{S}}_{p} \end{cases}$ (29)
	using the same technique as in Algorithm 2 above.
		• Let $\tilde{y}$ be the integer optimal solution of the lower level problem (29).
			1. If $y^{* (p)} = \tilde{y}$ then z ^∗(p) = ( x ^∗(p) , y ^∗(p) ) is the bilevel integer optimal solution of the main problem (8), F _opt = F ( z ^∗(p) ), f _opt = f ( z ^∗(p) ), z _opt = z ^∗(p) and the algorithm stops.
			2. Otherwise, goto step 4.
Step 4: Update the set $D_{E}^{p}$ and the set ${\bar{D}}_{E}^{p}$ . $D_{E}^{p} = D_{E}^{p} \ {z^{* (p)}}$ and ${\bar{D}}_{E}^{p} = {\bar{D}}_{E}^{p}$ .
	• If there is a solution $\bar{z} \in {\bar{D}}_{E}^{p}$ such that the vector $(F_{1} (\bar{z}), . . ., F_{k} (\bar{z}))$ is not dominated by the vector ( F ₁( z ),…, F _k ( z )) for any $z \in D_{E}^{p}$ , then $D_{E}^{p} = D_{E}^{p} \cup {\bar{z}}$ , ${\bar{D}}_{E}^{p} = {\bar{D}}_{E}^{p} \ {\bar{z}}$ , goto step 5.
Step 5: Determine the set N _p corresponding to the integer solution z ^∗(p) obtained by Algorithm 2 . A Dantzig cut (27) is added to the corresponding domain to the integer solution z ^∗(p) . $D_{E} = D_{E}^{p}$ , ${\bar{D}}_{E} = {\bar{D}}_{E}^{p}$ , p = p + 1, and goto step 2.

B ₀	x ₁	x ₂	x ₆	Rhs
x ₄	1	-3	-2	1
x ₅	-1	-1	-1	1
x ₃	1	2	1	2
x ₇	-2	-1	-2	2
$c_{1 j}^{1} + c_{2 j}^{1} - z_{j}^{1 (1)}$	-1	-4	-2	-7
$d_{1 j}^{1} + d_{2 j}^{1} - z_{j}^{1 (2)}$	0	3	0	-2
$c_{1 j}^{2} + c_{2 j}^{2} - z_{j}^{2 (1)}$	2	1	0	-2
$d_{1 j}^{2} + d_{2 j}^{2} - z_{j}^{2 (2)}$	-1	-2	-1	-3

B ₁	x ₂	x ₆	x ₈	Rhs
x ₄	-4	-2	1	0
x ₅	0	-1	-1	2
x ₃	1	1	1	1
x ₇	1	-2	-2	4
x ₁	1	0	-1	1
$c_{1 j}^{1} + c_{2 j}^{1} - z_{j}^{1 (1)}$	-3	-2	-1	-6
$d_{1 j}^{1} + d_{2 j}^{1} - z_{j}^{1 (2)}$	3	0	0	-2
$c_{1 j}^{2} + c_{2 j}^{2} - z_{j}^{2 (1)}$	-1	0	2	-4
$d_{1 j}^{2} + d_{2 j}^{2} - z_{j}^{2 (2)}$	-1	-1	-1	-2

B ₂	x ₄	x ₆	x ₉	Rhs
x ₂	-1/5	2/5	-1/5	1/5
x ₅	1/5	-7/5	-4/5	14/5
x ₃	0	1	1	0
x ₇	3/5	-16/5	-7/5	27/5
x ₁	2/5	-4/5	-3/5	8/5
x ₈	1/5	-2/5	-4/5	4/5
$c_{1 j}^{1} + c_{2 j}^{1} - z_{j}^{1 (1)}$	-2/5	-6/5	-7/5	-23/5
$d_{1 j}^{1} + d_{2 j}^{1} - z_{j}^{1 (2)}$	3/5	-6/5	3/5	13/5
$c_{1 j}^{2} + c_{2 j}^{2} - z_{j}^{2 (1)}$	-3/5	6/5	7/5	-27/5
$d_{1 j}^{2} + d_{2 j}^{2} - z_{j}^{2 (2)}$	0	-1	-1	-1

B ₃	x ₆	x ₉	x ₁₀	Rhs
x ₂	0	-1/2	-1/2	1/2
x ₅	-1	-1/2	1/2	5/2
x ₃	1	1	0	0
x ₇	-2	-1/2	3/2	9/2
x ₁	0	0	1	1
x ₈	0	-1/2	1/2	1/2
x ₄	-2	-3/2	-5/2	3/2
$c_{1 j}^{1} + c_{2 j}^{1} - z_{j}^{1 (1)}$	-1	-2	-2	-4
$d_{1 j}^{1} + d_{2 j}^{1} - z_{j}^{1 (2)}$	3/2	0	3/2	-7/2
$c_{1 j}^{2} + c_{2 j}^{2} - z_{j}^{2 (1)}$	-3/2	0	1/2	-9/2
$d_{1 j}^{2} + d_{2 j}^{2} - z_{j}^{2 (2)}$	0	-1	-1	-1

B ₄	x ₆	x ₉	x ₁₁	Rhs
x ₂	0	0	-1	1
x ₅	-1	-1	1	2
x ₃	1	1	0	0
x ₇	-2	-2	3	3
x ₁	0	-1	2	0
x ₈	0	-1	1	0
x ₄	-2	1	-5	4
x ₁₀	0	1	-2	1
$c_{1 j}^{1} + c_{2 j}^{1} - z_{j}^{1 (1)}$	-2	-1	-2	-3
$d_{1 j}^{1} + d_{2 j}^{1} - z_{j}^{1 (2)}$	0	0	3	-5
$c_{1 j}^{2} + c_{2 j}^{2} - z_{j}^{2 (1)}$	0	2	-3	-3
$d_{1 j}^{2} + d_{2 j}^{2} - z_{j}^{2 (2)}$	-1	-1	0	-1

B ₅	x ₂	x ₉	x ₈	Rhs
x ₄	-1	-3	4	3
x ₅	0	0	-1	2
x ₃	1	0	1	1
x ₇	1	0	-2	4
x ₁	0	1	-2	0
x ₆	1	-1	1	1
$c_{1 j}^{1} + c_{2 j}^{1} - z_{j}^{1 (1)}$	-2	0	-1	-5
$d_{1 j}^{1} + d_{2 j}^{1} - z_{j}^{1 (2)}$	3	0	0	-2
$c_{1 j}^{2} + c_{2 j}^{2} - z_{j}^{2 (1)}$	1	4	-2	-2
$d_{1 j}^{2} + d_{2 j}^{2} - z_{j}^{2 (2)}$	-1	-1	0	-2

B ₆	x ₂	x ₁₀	x ₄	Rhs
x ₉	5/7	-4/7	-1/7	1/7
x ₅	2/7	-3/7	1/7	20/7
x ₃	5/7	3/7	-1/7	1/7
x ₇	11/7	-6/7	2/7	40/7
x ₁	-1/7	-2/7	3/7	11/7
x ₆	10/7	-1/7	-2/7	2/7
x ₈	2/7	-3/7	1/7	6/7
$c_{1 j}^{1} + c_{2 j}^{1} - z_{j}^{1 (1)}$	-9/7	-1/7	-4/7	-34/7
$d_{1 j}^{1} + d_{2 j}^{1} - z_{j}^{1 (2)}$	3	0	0	-2
$c_{1 j}^{2} + c_{2 j}^{2} - z_{j}^{2 (1)}$	9/7	-6/7	4/7	-36/7
$d_{1 j}^{2} + d_{2 j}^{2} - z_{j}^{2 (2)}$	-5/7	1/7	-3/7	-8/7

B ₇	x ₂	x ₁₁	x ₁₀	Rhs
x ₉	2/3	-1/3	-2/3	1/3
x ₅	1/3	1/3	-1/3	8/3
x ₃	2/3	-1/3	1/3	1/3
x ₇	5/3	2/3	-2/3	16/3
x ₁	0	1	0	1
x ₆	4/3	-2/3	-1/3	2/3
x ₈	1/3	1/3	-1/3	2/3
x ₄	-1/3	-7/3	-2/3	4/3
$c_{1 j}^{1} + c_{2 j}^{1} - z_{j}^{1 (1)}$	-4/3	-2/3	-1/3	-14/3
$d_{1 j}^{1} + d_{2 j}^{1} - z_{j}^{1 (2)}$	3	0	0	-2
$c_{1 j}^{2} + c_{2 j}^{2} - z_{j}^{2 (1)}$	1	0	-2	-4
$d_{1 j}^{2} + d_{2 j}^{2} - z_{j}^{2 (2)}$	-2/3	-1/3	1/3	-4/3

B ₈	x ₁₂	x ₁₀	x ₁₁	Rhs
x ₉	1	-1	0	0
x ₅	1/2	-1/2	1/2	5/2
x ₃	1	0	0	0
x ₇	5/2	-3/2	3/2	9/2
x ₁	0	0	1	1
x ₆	2	-1	0	0
x ₈	1/2	-1/2	1/2	1/2
x ₄	-1/2	-1/2	-5/2	3/2
x ₂	-3/2	1/2	-1/2	1/2
$c_{1 j}^{1} + c_{2 j}^{1} - z_{j}^{1 (1)}$	0	-1	-2	-4
$d_{1 j}^{1} + d_{2 j}^{1} - z_{j}^{1 (2)}$	-3/2	3/2	9/2	-7/2
$c_{1 j}^{2} + c_{2 j}^{2} - z_{j}^{2 (1)}$	-1/2	-3/2	3/2	-9/2
$d_{1 j}^{2} + d_{2 j}^{2} - z_{j}^{2 (2)}$	0	0	-1	-1

B ₉	x ₁₃	x ₁₁	x ₁₂	Rhs
x ₉	-2	-1	-2	1
x ₅	-1	0	-1	3
x ₃	0	0	1	0
x ₇	-3	0	-2	6
x ₁	0	1	0	1
x ₆	-2	-1	-1	1
x ₈	-1	0	-1	1
x ₄	-1	-3	-2	2
x ₂	1	0	0	0
x ₁₀	-2	-1	-3	1
$c_{1 j}^{1} + c_{2 j}^{1} - z_{j}^{1 (1)}$	-1	-2	0	-4
$d_{1 j}^{1} + d_{2 j}^{1} - z_{j}^{1 (2)}$	0	0	-3	-2
$c_{1 j}^{2} + c_{2 j}^{2} - z_{j}^{2 (1)}$	-2	0	-1	-4
$d_{1 j}^{2} + d_{2 j}^{2} - z_{j}^{2 (2)}$	0	-1	0	-1

B ₁₀	x ₁₀	x ₆	x ₁₃	Rhs
x ₉	-1/2	-1/2	0	0
x ₅	-1/2	1/2	1	2
x ₃	1/2	-1/2	0	0
x ₇	-1	1	3	3
x ₁	-1/2	3/2	2	0
x ₁₁	1/2	-3/2	-2	1
x ₈	-1/2	1/2	1	0
x ₄	1/2	-7/2	-5	4
x ₂	0	0	-1	1
x ₁₂	-1/2	1/2	0	0
$c_{1 j}^{1} + c_{2 j}^{1} - z_{j}^{1 (1)}$	-1/2	-1/2	-2	-3
$d_{1 j}^{1} + d_{2 j}^{1} - z_{j}^{1 (2)}$	0	0	-3	-5
$c_{1 j}^{2} + c_{2 j}^{2} - z_{j}^{2 (1)}$	-3	1	-3	-3
$d_{1 j}^{2} + d_{2 j}^{2} - z_{j}^{2 (2)}$	1/2	-1/2	0	-1

B ₁₁	x ₂	x ₆	x ₈	Rhs
x ₄	-4	-3	1	0
x ₅	0	0	-1	2
x ₃	1	0	1	1
x ₇	1	0	-2	4
x ₁	1	1	-1	1
$c_{1 j}^{1} + c_{2 j}^{1} - z_{j}^{1 (1)}$	-3	-1	-1	-6
$d_{1 j}^{1} + d_{2 j}^{1} - z_{j}^{1 (2)}$	3	0	0	-2
$c_{1 j}^{2} + c_{2 j}^{2} - z_{j}^{2 (1)}$	-1	-2	2	-4
$d_{1 j}^{2} + d_{2 j}^{2} - z_{j}^{2 (2)}$	-1	0	-1	-2

k	l	n ₁	m	n ₂	CPU(s)
					Mean	Max
5	5	5	10	5	1.65	2.82
		10	10	10	13.72	18.7
		10	20	10	8.13	20.21
		20	20	10	38.30	56.23
		20	30	10	21.28	28.84
		20	30	20	47.06	72.66
		20	40	20	55.16	89.02
		20	40	30	128.91	220.98
		20	50	30	131.98	241.93
		30	50	30	249.84	279.76
		30	60	30	255.7	297.81
		40	60	30	559.91	738.83
		40	70	30	580.31	662.88
		50	70	30	1113.62	1228.86
		40	80	40	1217.02	1506.73
		40	80	50	2304.6	2526.94
		40	90	50	2416.11	2575.7
		40	90	60	4220.07	4419.8
		40	100	60	4368.74	4782.02
10	10	5	10	5	1.83	3.34
		10	10	10	13.1	25.37
		10	20	10	10.73	20.56
		20	20	10	35.88	62.69
		20	30	10	20.41	25.15
		20	30	20	81.75	153.84
		20	40	20	74.71	188.1
		20	40	30	190.87	297.93
		20	50	30	142.79	188.83
		30	50	30	318.63	398.29
		30	60	30	365.57	666.59
		40	60	30	705.42	838.83
		40	70	30	780.43	1139.49
		50	70	30	1375.11	1609.09
		40	80	40	1473.51	1694.83
		40	80	50	2917.5	3345.03
		40	90	50	2826.23	3234.92
		40	90	60	4833.1	5158.01
		40	100	60	5331.13	5656.49

ILMBP \{\begin{cases} \max_{(x, y)} F (x, y) = \prod_{i = 1}^{k} (c_{1}^{i} x + c_{2}^{i} y + α_{i}) \\ where y solves \\ \{\begin{cases} \max_{y} f (x, y) = \prod_{j = 1}^{l} (p_{1}^{j} x + p_{2}^{j} y + β_{j}) \\ s . t \\ (x, y) \in D = \tilde{S} \cap ℤ^{n} \end{cases} \end{cases}

LLILMP \{\begin{cases} \max_{y} f (x, y) = \prod_{j = 1}^{l} (p_{1}^{j} x^{* (p)} + p_{2}^{j} y + β_{j}) \\ s . t (x^{* (p)}, y) \in {\hat{S}}_{p} \end{cases}

(ILMBP) \{\begin{cases} \max_{x_{1}} F (x_{1}, y_{1}, y_{2}) = (x_{1} + 2 y_{2} + 3) (3 y_{1} + 2) (2 x_{1} + y_{1} + 2) (y_{2} + 1) \\ where (y_{1}, y_{2}) solves \\ \{\begin{cases} \max f (x_{1}, y_{1}, y_{2}) = (y_{1} + 1) (x_{1} + y_{1} - y_{2} + 3) \\ s . t \\ 3 x_{1} + y_{1} + 2 y_{2} \leq 5 \\ y_{1} + y_{2} \leq 3 \\ x_{1} + 2 y_{1} + y_{2} \leq 2 \\ 3 y_{1} + 2 y_{2} \leq 6 \\ x_{1} \geq 0, y_{1} \geq 0, y_{2} \geq 0, i n t e g e r s \end{cases} \end{cases}

({ULILMP}_{0}) \{\begin{cases} \max F (x_{1}, x_{2}, x_{3}) = (x_{1} + 2 x_{3} + 3) (3 x_{2} + 2) (2 x_{1} + x_{2} + 2) (x_{3} + 1) \\ s . t \\ 3 x_{1} + x_{2} + 2 x_{3} \leq 5 \\ x_{2} + x_{3} \leq 3 \\ x_{1} + 2 x_{2} + x_{3} \leq 2 \\ 3 x_{2} + 2 x_{3} \leq 6 \\ x_{1} \geq 0, x_{2} \geq 0, x_{3} \geq 0, i n t e g e r s \end{cases}

(IMLP) \{\begin{cases} \max g_{1} (x_{1}, x_{2}, x_{3}) = (x_{1} + 2 x_{3} + 3) \\ \max g_{2} (x_{1}, x_{2}, x_{3}) = (3 x_{2} + 2) \\ \max g_{3} (x_{1}, x_{2}, x_{3}) = (2 x_{1} + x_{2} + 2) \\ \max g_{4} (x_{1}, x_{2}, x_{3}) = (x_{3} + 1) \\ s . t \\ 3 x_{1} + x_{2} + 2 x_{3} \leq 5 \\ x_{2} + x_{3} \leq 3 \\ x_{1} + 2 x_{2} + x_{3} \leq 2 \\ 3 x_{2} + 2 x_{3} \leq 6 \\ x_{1} \geq 0, x_{2} \geq 0, x_{3} \geq 0, i n t e g e r s \end{cases}

(LLILMP) \{\begin{cases} \max f (1, y_{1}, y_{2}) = (y_{1} + 1) (y_{1} - y_{2} + 4) \\ s . t \\ s . t \\ y_{1} + 2 y_{2} \leq 2 \\ y_{1} + y_{2} \leq 3 \\ 2 y_{1} + y_{2} \leq 1 \\ 3 y_{1} + 2 y_{2} \leq 6 \\ y_{1} \geq 0, y_{2} \geq 0, i n t e g e r s \end{cases}

(LLILMP) \{\begin{cases} \max f (0, y_{1}, y_{2}) = (y_{1} + 1) (y_{1} - y_{2} + 3) \\ s . t \\ s . t \\ y_{1} + 2 y_{2} \leq 5 \\ y_{1} + y_{2} \leq 3 \\ 2 y_{1} + y_{2} \leq 2 \\ 3 y_{1} + 2 y_{2} \leq 6 \\ y_{1} \geq 0, y_{2} \geq 0, i n t e g e r s \end{cases}

(LLILMP) \{\begin{cases} \max f (0, y_{1}, y_{2}) = (y_{1} + 1) (y_{1} - y_{2} + 3) \\ s . t \\ s . t \\ y_{1} + 2 y_{2} \leq 5 \\ y_{1} + y_{2} \leq 3 \\ 2 y_{1} + y_{2} \leq 2 \\ 3 y_{1} + 2 y_{2} \leq 6 \\ y_{1} \geq 0, y_{2} \geq 0, i n t e g e r s \end{cases}

[1] * Corresponding author

Brasil

Brasil

BRANCH-AND-CUT METHOD FOR SOLVING THE INTEGER LINEAR MULTIPLICATIVE BILEVEL PROBLEM

ABSTRACT