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INTEGER LINEAR PROGRAMMING
ANTONIO J. CONEJO
2002
CONTENTS
1) REFERENCES
2) INTEGER LINEAR PROGRAMING• PURE INTEGER LINEAR PROGRAMMING
⇒ BRANCH AND BOUND• MIXED INTEGER LINEAR PROGRAMMING
⇒ BRANCH AND BOUND⇒ GOMORY CUTS
3) AVAILABLE SOFTWARE
REFERENCES
1. G.L. NEMHAUSER, L.A. WOLSEY. “INTEGER AND COMBINATORIAL OPTIMIZATION”. JOHN WILEY AND SONS. NEW YORK, 1988
2. S.P. BRADLEY, A.C. HAX, T.L. MAGNANTI. “APPLIED MATHEMATICAL PROGRAMMING”. ADDISON-WESLEY PUBLISHING COMPANY. READING, MASSACHUSETTES, 1977
Antonio Conejo 4
INTEGER LINEAR PROGRAMMING (1/2)
c xj jj
n
=∑
1
MINIMIZE
SUBJECT TO a b mij jj
n
i x i=∑ = =
112, , ,K
x nj ≥ =0 12 j , , ,K
n,1,2,j some or all for Ix j K=∈
INTEGER LINEAR PROGRAMMING (2/2)
POSSIBILITIES
• PURE INTEGER LINEAR PROGRAMMING• MIXED INTEGER LINEAR PROGRAMMING• 0/1 MIXED INTEGER LINEAR PROGRAMMING
SOLUTION METHODS
• BRANCH AND BOUND• GOMORY CUTS• BRANCH & CUT
PURE INTEGER LINEAR PROGRAMMING
VIABRANCH AND BOUND
PROCEDURE
1. SET UP OF BOUNDS
2. BRANCHING STRATEGY
3. SYSTEMATIC SOLUTION OF FURTHER CONSTRAINED LP PROBLEMS
4. TITHING OF BOUNDS
z z z< <∗
OBSERVATIONS
1. AN UPPER BOUND OF THE SOLUTION (OBJECTIVE FUNCTION VALUE) OF AN INTEGER LINEAR PROGRAMMING PROBLEM IS THE LOWEST VALUE OF ANY FEASIBLE INTEGER SOLUTION ENCOUNTERED
2. A LOWER BOUND OF THE SOLUTION OF AN INTEGER LINEAR PROGRAMMING PROBLEM IS THE OPTIMAL SOLUTION OF THE ASSOCIATED LINEAR PROGRAMMING PROBLEM
BRANCHING (1/3)IF IS NOT INTEGER BUT SHOULD BE
⇓BRANCHING
xj
x a bj = .
x aj ≤ x aj ≥ + 1
xxxj
j
j= ⇒
≤≥
3 7534.
BRANCHING (2/3)
LP0
x = a.b j
LP 1 LP 2
xj ≤ a xj ≥ a+1
BRANCHING (3/3)PROCEDURE
1. BY SYSTEMATIC BRANCHING TIGHTER BOUNDS OF THE FORM
z z z< <∗
ARE PROGRESSIVELY GENERATED2. THE BRANCHING REQUIRES THE SOLUTION OF MULTIPLE
LP'S WITH ADDITIONAL CONSTRAINTS, THEREFORE THE DUAL SIMPLEX ALGORITHM SHOULD BE USED
THE PRIMAL PROBLEMWITH AN ADDITIONAL CONSTRAINT
MINIMIZE
SUBJECT TO
[ ]− −
3 5
2 x1
x
−−
− −
−
≥
−−−
−
1 00 23 2
0 1
41218
4
1
2L L L x
x
xx
1
2
26
∗
∗
=
INFEASIBLE
THE DUAL PROBLEMWITH AN ADDITIONAL CONSTRAINT
MAXIMIZE
SUBJECT TO [ ]
[ ]λ λ λ λ1 4, ,
41218
42 3
---
-L
[ ]λ λ λ λ1 4
1 00 23 2
0 1
3, , , - 52 3
−−
− −
−
≤ −L L
[ ] [ ]λ λ λ λ1 4 0, , , 0, 0 02 3 ≥λλλ
λ
1
2
3
4
03 2
1
0
∗
∗
∗
∗
=
L
L
/
FEASIBLE BUT NOT OPTIMAL
CUTTING RULESIT IS NOT REQUIRED TO FURTHER SUBDIVIDE BRANCH Lj IF
1. THE LP OVER Lj IS INFEASIBLE2. THE OPTIMAL LP SOLUTION OVER Lj IS INTEGER3. THE OPTIMAL LP SOLUTION zj IS SUCH THAT
(A VALUE LARGER THAN THE CURRENT UPPER BOUND)
IT IS SAID THAT Lj IS "FATHOMED" BY1. INFEASIBILITY2. INTEGRALITY3. BOUNDS
z zj >
OBSERVATION
IF ALL OBJETIVE FUNCTION COEFICIENTS ARE INTEGER AND ALL VARIABLES ARE INTEGER, THE BOUND
z* ≥ - a.b
IS EQUIVALENT TO THE BOUNDz* ≥ - a
AND THE BOUNDz* ≥ a.b
IS EQUIVALENT TO THE BOUNDz* ≥ a + 1
EXAMPLE (1/2)
MINIMIZE -5 x1 -8 x2 = z
SUBJECT TO x1 + x2 ≤ 6
5 x1 +9 x2 ≤ 45
x1 ≥ 0 & INTEGER
x2 ≥ 0 & INTEGER
EXAMPLE (2/2) L 0
x = 2.251
x ≥ 4 2 x ≤ 3 2
x = 3.752 z = - 41.25
z* ≥ - 41.25
L 1x = 1.8 1x = 4 2z = - 41
L 2x = 3 1x = 3 2z = - 39
x ≥ 2 1
L 3 L 4x = 1 1x = 4.44 2z = - 40.55
INFEASIBLE
* z* ≥ - 40.55
*
L 5x = 1 1x = 4 2z = - 37
L 6x = 0 1x = 5 2z = - 40
x ≤ 4 2 x ≥ 5 2
z* ≤ - 37
z* ≤ - 40 * *
z* ≤ - 39x ≤ 1 1
z* ≥ - 41
z* = - 40x*1 = 0x*2 = 5
MIXED INTEGER LINEAR PROGRAMMING
VIABRANCH AND BOUND
MIXED INTEGER LINEAR PROGRAMMING
JUST THE SAME THING THAT PURE INTEGER LINEAR PROGRAMMING BUT BRANCHING ONLY THOSE VARIABLES WHICH HAVE TO BE INTEGER
EXAMPLE (1/2)MINIMIZE 3x1 +2x2 -10 = z
SUBJECT TO x1 -2x2 +x3 = 5/2
2x1 +x2 +x4 = 3/2
x1 ≥ 0
x2 ≥ 0
x3 ≥ 0
x4 ≥ 0x2, x3 INTEGER
EXAMPLE (2/2) L 0
x = 0 ,1
x ≤ 2 3 x ≥ 3 3
x = 0 2
z = -10
z* ≥ -10
x ≤ 0 2 x ≥ 1 2
x = 2.5 , 3 x = 1.5 4
L 1x = 0 .5, 1 x = 0 2
z = -8.5
x = 2 ,3 x = 0.5 4
L 2x = 0 ,1 x = 0.252
z = -9.5
x = 3 ,3 x = 1.254z* ≤ - 8.5
L 3 L 4x = 0 ,1 x = 1 2
z = -8
x = 4.5 , 3 x = 0.5 4INFEASIBLE
*
* *
z* ≥ -9.5
(z* ≥ -8)
z* = - 8.5
x*1 = 0.5x*2 = 0x*3 = 2x*4 = 0.5
MIXED INTEGER LINEAR PROGRAMMING
VIAGOMORY CUTSANTONIO J. CONEJO
2002
GOMORY ALGORITHM (1/2)CONSTRAINTS
Ax b=
[ ]B NM L x
x b
B
N
=
Bx Nx bB N+ =
x B Nx B bB N+ =− −1 1
[ ]I B Nx
xb
B
N
M L− −
=1 1 B
[ ]I Yx
xb
B
N
M L
= ~
x Yx bB N+ = ~
GOMORY ALGORITHM (2/2)
• SELECT xBi NONINTEGRAL, THEN(1)
BECAUSE xNj ARE NON BASIC, IS NONINTEGRAL
• EACH Yij AND IS WRITTEN AS THE SUM OF AN INTEGER AND A NONNEGATIVE FRACTION LESS THAN 1, SO
x Y x bBi ij Nj i+ =∑ ~
Y I Fij ij= +ij
~ ~ ~b i fi = +i i
~bi
~bi
GOMORY ALGORITHM (2/2)
SOME Fij MAY BE 0, BUT IS GUARANTEED TO BE POSITIVE
• EQUATION (1) BECOMES
OR(2)
( )x I F x i fBi ij Nj i i+ + = +∑ ij~ ~
x I x i f F xBi Nj i i ij Nj+ − = −∑ ∑ij~ ~
~fi
GOMORY OBSERVATION• IF EACH VARIABLE IS REQUIRED TO BE INTEGRAL, THE
LEFT-HAND SIDE OF (2) SHOULD BE INTEGRAL, AND AS A RESULT, THE RIGHT-HAND SIDE TOO
• EACH AND IS NONNEGATIVE, SO TOO
• THEN, THE RIGHT-HAND SIDE OF (2) IS AN INTEGER WHICH IS SMALLER THAN A POSITIVE FRACTION LESS THAN ONE; THAT IS, A NONPOSITIVE INTEGER
OR(3)
• (3) IS THE SO CALLED GOMORY CUT
F x fij Nj − ≥∑ i 0
Fij xNjF xij Nj∑
~f F xij Nj i − ≤∑ 0
~
GOMORY ALGORITHM (1/2)
SELECT ONE (ANY) NONINTEGRAL BASIC VARIABLE, AND WITHOUT ASSIGNING ZERO VALUES TO THE NONBASIC VARIABLES, CONSIDER THE CONSTRAINT EQUATION REPRESENTED BY THE ROW OF THE SELECTED VARIABLE
REWRITE EACH FRACTIONAL COEFFICIENT AND CONSTANT IN THE CONSTRAINT EQUATION OBTAINED FROM STEP 1 AS THE SUM OF AN INTEGER AND A POSITIVE FRACTION BETWEEN 0 AND 1
STEP 1
STEP 2
GOMORY ALGORITHM (2/2)
THEN, REWRITE THE EQUATION SO THAT THE LEFT-HAND SIDE CONTAINS ONLY TERMS WITH FRACTIONAL COEFFICIENTS AND A FRACTIONAL CONSTANT, WHILE THE RIGHT-HAND SIDE CONTAINS ONLY TERMS WITH INTEGRAL COEFFICIENTS AND AN INTEGRAL CONSTANT
REQUIRE THE LEFT-HAND SIDE OF THE REWRITTEN EQUATION TO BE NONNEGATIVE. THE RESULTING INEQUALITY IS THE NEW CONSTRAINT
STEP 3
STEP 4
GOMORY CUTS EXAMPLE 1/14
CONSIDER THE PROBLEM:
RELAXED AND IN STANDARD FORM:
IN x IN, x 0 x 0 x 28 8x 7x 6 x 2x TO SUBJECT
80x120xZ MAXIMIZE
21
2
1
21
21
21
∈∈≥≥≤+≤+
+=
0x ,x ,x ,x 28 x 8x 7x 6 x x 2x TO SUBJECT
80x120xZ MAXIMIZE
4321
421
321
21
≥=++=++
+=
GOMORY CUTSEXAMPLE 2/14
THE SOLUTION IS
AND
THE EQUATION ASSOCIATED TO x2 IS USED TO GENERATE A CUT:
391.11z* =
.
92
97
91
98
Y;
914
920
x
xBxb~ *
*2
*1*
−
−
=
=
==
914x
92x
97x 432 =+−
GOMORY CUTSEXAMPLE 3/14
THEREFORE:
AND THE CUT
OR(1)2
5xx 43 ≥+
95f~
951
914f~i~b~
92f
920
92fiy
92f
921
97fiy
2222
22222222
21212121
=⇒+=⇔+=
=⇒+=⇔+=
=⇒+−=−⇔+=
95x
92x
920
xx
92
92
95
434
3 ≥+⇔≤
−
GOMORY CUTSEXAMPLE 4/14
THE CUT CAN BE EXPRESSED AS A FUNCTION OF x1 AND x2
TAKING INTO ACCOUNT THAT
THE CUT BECOMES
(1)
THE FEASIBILITY REGION IS REDUCED WITHOUT ELIMINATING INTEGER SOLUTIONS
2xx 21 ≤+
214
213
8x7x28xx2x6x−−=
−−=
7
GOMORY CUTSEXAMPLE 5/14
NEXT PROBLEM TO SOLVE IS
0x,x,x,x,x2
5xxx
28x8x7x6xx2xTOSUBJECT
80x120xZMAXIMIZE
54321
543
421
321
21
≥
=−+
=++
=++
+=
THE GOMORY CUT HAS BEEN INCLUDED USING SLACK VARIABLE x5
GOMORY CUTSEXAMPLE 6/14
THE SOLUTION IS
380z* =
−
−
−
=
=
=
921
11911
Y;
1
2525
x
x
x
Bx; *
*3
*2
*1
*
GOMORY CUTSEXAMPLE 7/14
x1 IS USED TO GENERATE A NEW CUT. TAKING INTO ACCOUNT THAT 25*xb~ 11 ==
25x9
1xx 541 =−+
THEREFORE
21f~
212
25f~i~b~
98f
981
91fiy
0f011fiy
1111
12121212
11111111
=⇒+=⇔+=
=⇒+−=−⇔+=
=⇒+=⇔+=
GOMORY CUTSEXAMPLE 8/14
THE CUT BECOMES
OR(2)
EXPRESSING THIS CUT AT A FUNCTION OF VARIABLES x1AND x2
21x
980
x
x)
98(0
21
55
4≥⇔≤
−
169x5 ≥
1655xx 21 ≤+ (2)
GOMORY CUTSEXAMPLE 9/14
NEX PROBLEM TO BE SOLVED
0x,x,x,x,x,x9/16xx5/2xxx28x8x7x6xx2xTOSUBJET
80x120xz MAXIMIZE
654321
65
543
421
321
21
≥
=−
=−+
=++
=++
+=
GOMORY CUTSEXAMPLE 10/14
THE SOLUTION IS
−
−
−
−
=
=
==
921
10
11911
Y;
87
169
1649
1641
x
x
x
x
x377.5;z *
*2
*5
*4
*1
**B
GOMORY CUTSEXAMPLE 11/14
USING x2 TO GENERATE A NEW CUT AND TAKING INTO ACCOUNT THAT 87/b~ 2 =
87x
92xx 632 =+−
THEN
87f~
870
87f~i~b~
92f
920
92fiy
0f011fiy
4444
42424242
41414141
=⇒+=⇔+=
=⇒+=⇔+=
=⇒+−=−⇔+=
GOMORY CUTSEXAMPLE 12/14
THE CUT BECOMES
( )87x
920
xx
2/9087
66
3 ≥⇔≤
−
OR
1663x 6 ≥ (3)
GOMORY CUTSEXAMPLE 13/14
THE LAST CUT AS A FUNCTION OF THE ORIGINAL VARIABLES BECOMES
3xx 21 ≤+
NEXT PROBLEM IS
0x,x,x,x,x,x,x63/16xx9/16xx5/2xxx28x8x7x6xx2x TO SUBJET
80x120xz MAXIMIZE
7654321
76
65
543
421
321
21
≥
=−
=−
=−+
=++
=++
+=
GOMORY CUTSEXAMPLE 14/14
THE SOLUTION IS
=
==
31663
29
7
0
x
xx
x
x
x360;z
*1
*6
*5
*4
*3
**B
THIS SOLUTION IS INTEGER AND THEREFORE IS THE OPTIMAL SOLUTION OF THE ORIGINAL PROBLEM
360z0;x3;x **2
*1 ===
Antonio Conejo 43
MODELINGUSING 0/1 VARIABLES
ANTONIO J. CONEJO
2002
MODELING (1/6)ALTERNATIVE CONSTRAINTS
AT LEAST ONE OF THE CONSTRAINTS BELOW SHOULD BE MET
THIS IS EXPRESSED AS
a x ba x b
T
T1 1
2 2
≤
≤
{ }0,1y,y1yy
byBxabyBxa
21
21
222T2
111T1
∈≤+
≤−
≤−
MODELING (2/6)ALTERNATIVE CONSTRAINTS
AND
22T22
11T11
bBxa THAT SO BbBxa THAT SO B
≤−
≤−
MODELING (3/6)ALTERNATIVE CONSTRAINTS
POSIBILITIES:
cba100y
010y
2
1
a. Ambas restricciones han de cumplirseb. Restricción 1 desactivada, ha de cumplirse 2c. Restricción 2 desactivada, ha de cumplirse 1
MODELING (4/6)DISCONTINUITY
THE FUNCTION
IS EXPRESSED AS
AND
f xx
k cx x( ) =
=+ >
0 00
{ }
f x ky cxx Byxy
( )
,
= +≤≥
∈
001
B SO THAT x B≤
MODELING (5/6)DISCONTINUITY
1)
( )
0x,Bx
cxkxf
1y
≥≤
+=
=
2)
( ) 0)x(f
0x,0x
cxxf
0y
=
≥≤
=
=
MODELING (6/6)DISCONTINUITY
f(x)
c
k
x
MODELING (1/4)PIECE-WISE LINEAR NONCONVEX FUNCTION
THE FUNCTION
WHERE
f xx x aa x a a x ba b a x b b x c
( ) ( )( ) ( )
=≤ ≤
+ − < ≤+ − + − < ≤
αα βα β γ
0
cba0
0
<<<
γ<α<β<
MODELING (2/4)PIECE-WISE LINEAR NONCONVEX FUNCTION
IS EXPRESSED AS
{ }
f x x x x
x x x x
aw x aw b a x b a w
x c b w
w w
w w
( )
( ) ( )( )
, ,
= + +
= + +
≤ ≤− ≤ ≤ −
≤ ≤ −
≥
∈
α β γ1 2 3
1 2 3
1 1
2 2 1
3 2
1 2
1 2
0
0 1
MODELING (3/4)PIECE-WISE LINEAR NONCONVEX FUNCTION
cbaCase
100w2
110w1
) ( ) 1321 xxf;0x,0x,ax0a α===≤≤
) ( ) 2321 xaxf;0x,abx0,axb β+α==−≤≤=
) ( ) ( ) 3321 xabaxf,bcx0;abx,axc γ+−β+α=−≤≤−==
MODELING (4/4)PIECE-WISE LINEAR NONCONVEX FUNCTION
f(x)
x1 x2 x3 x
a b c
α
β
γ
MODELING (1/4)PIECE-WISE LINEAR NONCONVEX FUNCTION WITH
INITIAL DISCONTINUITY
THE FUNCTION
WHERE
≤<−γ+−β+−α+≤<−β+−α+≤<−α+
=
=
dxc)cx()bc()ab(fcxb)bx()ab(fbxa)ax(f
0x0
)x(f
0
0
0
dcba0
0
<<<<
β<α<β<
MODELING (2/4)PIECE-WISE LINEAR NONCONVEX FUNCTION WITH
INITIAL DISCONTINUITYIS EXPRESSED AS
{ }
f x vf x x x
x va x x x
w b a x b a vw c b x c b w
x d c w
v ww w
v w w
( )
( ) ( )( ) ( )
( )
, , ,
= + + +
= + + +
− ≤ ≤ −− ≤ ≤ −
≤ ≤ −
≥≥
∈
0 1 2 3
1 2 3
1 1
2 2 1
3 2
1
1 2
1 2
0
0 1
α β γ
MODELING (3/4)PIECE-WISE LINEAR NONCONVEX FUNCTION WITH
INITIAL DISCONTINUITY
b0
01 110V
dcaCase100w2
110w1
) ( ) 0xf;0x;0x,0xa 321 ====
) ( ) 10321 xfxf;0x,0x,abx0b α+===−≤≤
) ( ) ( ) 20321 xabfxf;0x,bcx0,abxc β+−α+==−≤≤−=
) ( ) ( ) ( ) 30321 xbcabfxf,cdx0;bcx,abxd γ+−β+−α+=−≤≤−=−=
MODELING (4/4)PIECE-WISE LINEAR NONCONVEX FUNCTION WITH
INITIAL DISCONTINUITY f(x)
x1 x2 x3 x
a b c
α
β
γ
f0
d
AVAILABLE SOFTWARE
• LINDO• GAMS (CPLEX)