Intro
Review
Methodology
Computations
Conclusions
Two-Period Relaxations on Big-Bucket Production
Planning Problems
Kerem Akartunalı
Department of Mathematics and Statistics
The University of Melbourne
Joint work with Andrew J. Miller
4 August 2008
K. Akartunalı
Big-Bucket Production Planning: Two-Period Relaxations
Intro
Review
Methodology
Computations
Conclusions
Problem Description
Multiple items and levels (BOM structure)
Assembly (a) or general (b) structures
4
5
5
3
1
2
2
3
1
2
1
6
1
4
3
1
4
1
1
(a)
2
2
(b)
Demands
Big-bucket capacities (items share resources)
Extensions possible, e.g. overtime and backlogging
Production plan minimizing total cost to be determined
K. Akartunalı
Big-Bucket Production Planning: Two-Period Relaxations
Intro
Review
Methodology
Computations
Conclusions
Basic Formulation
min
s.t.
NT X
NI
X
fti yti +
t=1 i=1
i
xti + st−1
−
NT X
NI
X
hti sti
t=1 i=1
i
st = dti
i
xti + st−1
− sti =
X
r ij xtj
(1)
t ∈ [1, NT ], i ∈ endp
(2)
t ∈ [1, NT ], i ∈
/ endp
(3)
t ∈ [1, NT ], k ∈ [1, NK ]
(4)
t ∈ [1, NT ], i ∈ [1, NI ]
(5)
j∈δ(i)
NI
X
(aki xti + STki yti ) ≤ Ctk
i=1
xti ≤ Mti yti
y ∈ {0, 1}NTxNI
(6)
x ≥0
(7)
s≥0
(8)
K. Akartunalı
Big-Bucket Production Planning: Two-Period Relaxations
Intro
Review
Methodology
Computations
Conclusions
What do we know?
Many of the test problems remain challenging
We do not have an adequate approximation of the convex hull
of the multi-item, single-machine, single-level capacitated
problems!
Item 1
Item 2
Item 3
Period
t
Period
t+1
Generalizes the “bottleneck flow” model of Atamtürk and
Muñoz [2003]
K. Akartunalı
Big-Bucket Production Planning: Two-Period Relaxations
Intro
Review
Methodology
Computations
Conclusions
The Model to Study
e i 0 y i0
xti 0 ≤ M
t t
i
i i
e
x 0 ≤ d 0y 0 + si
t
x1i
x1i
t
t
+ x2i ≤ de1i y1i + de2i y2i + s i
+ x i ≤ dei + s i
NI
X
2
1
et 0
(ai xti 0 + ST i yti 0 ) ≤ C
i = [1, ..., NI ], t 0 = 1, 2
(9)
i = [1, ..., NI ], t 0 = 1, 2
(10)
i = [1, ..., NI ]
(11)
i = [1, ..., NI ]
(12)
t 0 = 1, 2
(13)
i=1
x, s ≥ 0, y ∈ {0, 1}2xNI
(14)
Let X 2PL = {(x, y , s)|(9) − (14)}
K. Akartunalı
Big-Bucket Production Planning: Two-Period Relaxations
Intro
Review
Methodology
Computations
Conclusions
Separation Over the 2-Period Convex Hull: Details
From the LPR of the original problem, we obtain (x̄, ȳ , s̄)
min z =
X
i
[(∆−
s ) +
2
X
i
− i
+ i
− i
(∆+
x )t 0 + (∆x )t 0 + (∆y )t 0 + (∆y )t 0 ]
t 0 =1
i
s.t. x̄ti 0 =
X
ȳti 0
X
i
− i
λk (xk )it 0 + (∆+
x )t 0 − (∆x )t 0
∀i, t 0 = 1, 2 (αti 0 )
i
− i
λk (yk )it 0 + (∆+
y )t 0 − (∆y )t 0
∀i, t 0 = 1, 2
k
=
(βti 0 )
k
i
s̄ ≥
X
i
λk (sk )i − (∆−
s )
∀i
(γ i )
k
X
λk ≤ 1
(η)
k
λk ≥ 0, ∆ ≥ 0
K. Akartunalı
Big-Bucket Production Planning: Two-Period Relaxations
Intro
Review
Methodology
Computations
Conclusions
Separation Over the 2-Period Convex Hull: Details (cont’d)
The dual of the distance problem:
max
NI X
2
X
i=1
s.t.
t 0 =1
NI X
2
X
i=1
(x̄ti 0 αti 0 + ȳti 0 βti 0 ) +
NI
X
((xk )it 0 αti 0 + (yk )it 0 βti 0 ) +
NI
X
(sk )i γ i + η ≤ 0
∀k
(16)
i=1
− 1 ≤ αti 0 ≤ 1
βti 0
i
(15)
i=1
t 0 =1
−1≤
s¯k i γ i + η
≤1
∀i, t 0
(17)
0
(18)
∀i, t
−1≤γ ≤0
∀i
η≤0
(19)
(20)
K. Akartunalı
Big-Bucket Production Planning: Two-Period Relaxations
Intro
Review
Methodology
Computations
Conclusions
Separation Over the 2-Period Convex Hull: Details (cont’d)
Theorem
Let z > 0 for (x̄, ȳ , s̄), and (ᾱ, β̄, γ̄, η̄) optimal dual values. Then,
NI X
2
X
(ᾱti 0 xti 0 + β̄ti 0 yti 0 ) +
i=1 t 0 =1
X
γ̄ i s i + η̄ ≤ 0
(21)
i
is a valid inequality for conv (X 2PL ) that cuts off (x̄, ȳ , s̄).
Proof.
Validity: Using (16), γ̄ ≤ 0 and λ ≥ 0.
Violation for (x̄, ȳ , s̄): Using (15)
K. Akartunalı
Big-Bucket Production Planning: Two-Period Relaxations
Intro
Review
Methodology
Computations
Conclusions
Separation Over the 2-Period Convex Hull: Extreme Points
How to generate (xk , yk , sk )?
Using column generation
Solve the minimum reduced cost problem using the optimal
dual values (ᾱ, β̄, γ̄, η̄):
max zP =
x,y ,s
2
XX
i
(ᾱti 0 xti 0 + β̄ti 0 yti 0 ) +
t 0 =1
X
γ̄ i s i + η̄
i
s.t. (x, y , s) ∈ X 2PL
If zP > 0, then the solution is an extreme point of X 2PL ;
otherwise, generating extreme points is done
K. Akartunalı
Big-Bucket Production Planning: Two-Period Relaxations
Intro
Review
Methodology
Computations
Conclusions
Separation Algorithm
repeat
Solve the distance problem for conv (X 2PL )
if z = 0 then break
else Solve column generation problem
if zP ≤ 0 then break
else Add new extreme point
until z = 0 or zP ≤ 0
if z=0 then (x̄, ȳ , s̄) ∈ conv (X 2PL )
else Add the violated cut (21)
Note: The same is valid for L∞
K. Akartunalı
Big-Bucket Production Planning: Two-Period Relaxations
Intro
Review
Methodology
Computations
Conclusions
Using Euclidean Distance (L2 )
We can also apply the same framework with Euclidean distance
"
#
2
X
X
i 2
i 2
i 2
min z =
[(∆s ) ] +
[(∆x )t 0 ] + [(∆y )t 0 ]
∆,λ
t 0 =1
i
s.t. x̄ti 0 =
X
ȳti 0
X
λk (xk )it 0 + (∆x )it 0
∀i, t 0 = 1, 2
(αti 0 )
λk (yk )it 0 + (∆y )it 0
∀i, t 0 = 1, 2
(βti 0 )
k
=
k
i
s̄ ≥
X
λk (sk )i − (∆s )i
∀i
(γ i )
k
X
λk ≤ 1
(η)
k
λk ≥ 0, ∆s ≥ 0, ∆x , ∆y free
K. Akartunalı
Big-Bucket Production Planning: Two-Period Relaxations
Intro
Review
Methodology
Computations
Conclusions
Using Euclidean Distance: Dual
"
max zD = −
∆,α,β,γ
−
s.t.
NI
X
NI
X
X
i
2
X
[(∆s )i ]2 +
2
X
#
[(∆x )it 0 ]2 + [(∆y )it 0 ]2
t 0 =1
(x̄ti 0 αti 0 + ȳti 0 βti 0 ) +
i=1 t 0 =1
2
X
NI
X
!
s̄ i γ i + η
i=1
NI
X
((xk )it 0 αti 0 + (yk )it 0 βti 0 ) +
i=1 t 0 =1
αti 0 = −2(∆x )it 0
βti 0 = −2(∆y )it 0
− γ i ≥ −2(∆s )i
(sk )i γ i + η ≥ 0 ∀k
i=1
∀i, t 0
∀i, t 0
∀i
γ ≥ 0, η ≥ 0, ∆s ≥ 0, α, β, ∆x , ∆y free
K. Akartunalı
Big-Bucket Production Planning: Two-Period Relaxations
Intro
Review
Methodology
Computations
Conclusions
Using Euclidean Distance: Theory
Theorem
¯ x, ∆
¯ y, ∆
¯ s , λ̄),
Let z > 0 for (x̄, ȳ , s̄), with optimal primal values (∆
and (ᾱ, β̄, γ̄, η̄) be the associated optimal dual values. Then,
2
XX
i
(ᾱti 0 xti 0 + β̄ti 0 yti 0 ) +
t 0 =1
X
γ̄ i s i + η̄ ≥ 0
(22)
i
is a valid inequality for conv (X 2PL ) that cuts off (x̄, ȳ , s̄).
Proof.
Using a similar approach to the previous proof and also using the
strong duality theorem for QP.
K. Akartunalı
Big-Bucket Production Planning: Two-Period Relaxations
Intro
Review
Methodology
Computations
Conclusions
Defining 2-Period Subproblems
Question 1: On which two periods to run the separation?
We can look at all the 2-period problems (NT − 1 of them)
Question 2: Which period’s stock is represented by s i ?
Let φ(i) ∈ [t + 1, .., NT ] be the horizon parameter for each i
i
Obvious choice: t + 1, i.e., s i = st+1
Then, parameters are defined as follows (∀i, t 0 = 1, 2):
e i0 = M i 0 , C
e i0 = C i 0
M
t
t+t −1
t
t+t −1
i
i
ei
ei 0 = d i 0
ei
d
t
t+t −1, t+1 , i.e., d1 = d12 and d2 = d2 .
Observation 1: If a number of periods following t + 1 have
no setups, their demands should be incorporated
Observation 2: If a setup occurs after t + 1, (`, S)
inequalities will be weakened if that period’s demand is in
K. Akartunalı
Big-Bucket Production Planning: Two-Period Relaxations
Intro
Review
Methodology
Computations
Conclusions
2-Period Convex Hull Closure Framework
Following Miller, Nemhauser, Savelsbergh (2000)
0
0
0
φ(i) = max{t |t ≥ t + 1,
t
X
i
yti 00 ≤ yt+1
+ Θ}
t 00 =t+1
where Θ ∈ (0, 1] is a random number
Let Xt2PL be Xt2PL (φ(1), φ(2), ..., φ(NI ))
Solve LPR of the original problem
→ (x̄, ȳ , s̄)
for t=1 to NT-1
Define Xt2PL
Apply 2-period convex hull separation algorithm
K. Akartunalı
Big-Bucket Production Planning: Two-Period Relaxations
Intro
Review
Methodology
Computations
Conclusions
Computational Results: 2-Period Problems
2PCLS instances: 20 problems with two periods only and with
two to six items
cdd might provide the full description of the convex hull
Generate all the extreme points and rays of the LPR
Eliminate all the fractional extreme points
Using these integral extreme points, generate all the facets of
the integral polyhedron
The more items share a resource, the more the structure tends
to resemble that of an uncapacitated problem
The separation algorithm implemented in Mosel (Mosel
version 2.0.0, Xpress 2007 package)
K. Akartunalı
Big-Bucket Production Planning: Two-Period Relaxations
Intro
Review
Methodology
Computations
Conclusions
Computational Results: 2-Period Problems (cont’d)
Inst.
NI
XLP
IP
2pcls01
2pcls02
2pcls03
2pcls04
2pcls05
2pcls06
2pcls07
2pcls08
2pcls09
2pcls10
3
3
3
2
3
3
2
2
2
3
17.033
12.6253
76.5345
14.7674
38.39
117.375
36.5
21.45
129
17.6539
25
19
104
19
52
173
43
26
153
24
K. Akartunalı
# cuts
(L1 )
11
13
5
4
8
5
2
7
2
1
# cuts
(L∞ )
8
7
3
2
6
6
1
2
3
3
# cuts
(L2 )
19
7
1
1
4
5
1
2
3
1
Big-Bucket Production Planning: Two-Period Relaxations
Intro
Review
Methodology
Computations
Conclusions
Computational Results: 2-Period Problems (cont’d)
Inst.
NI
XLP
IP
2pcls11
2pcls12
2pcls13
2pcls14
2pcls15
2pcls16
2pcls17
2pcls18
2pcls19
2pcls20
3
3
4
4
4
4
5
5
6
6
71.7209
46.68
85.6256
70.2961
54.1848
34.0844
164.858
57.0825
115.131
59.2412
102
69
113
81
74
39
211
97
150
89
K. Akartunalı
# cuts
(L1 )
4
4
7
6
6
6
39
34
11
34
# cuts
(L∞ )
1
1
7
8
3
7
19
10
6
11
# cuts
(L2 )
1
2
9
5
1
4
14
6
1
5
Big-Bucket Production Planning: Two-Period Relaxations
Intro
Review
Methodology
Computations
Conclusions
Computational Results: Multi-Period Problems
tr6-15 detailed results (30 iterations):
L1
2PL
lim=100
φ(i)
37,234.6
lim=150
φ(i)
37,298.8
lim=100
t +1
37,306.7
L2
lim=150
t +1
37,306.8
lim=150
φ(i)
37,331.2
Trigeiro instances results with Euclidean approach:
XLP
2PL
OPT
tr6-15
37,201
37,344
37,721
tr6-30
60,946
61,090
61,746
K. Akartunalı
tr12-15
73,848
73,932
74,634
tr12-30
130,177
130,215
130,596
Big-Bucket Production Planning: Two-Period Relaxations
Intro
Review
Methodology
Computations
Conclusions
Computational Results: Multi-Period Problems (cont’d)
TDS instances results with L∞ approach:
LB (l,S)
LB (2PL)
Best Sol.
Gap was
Gap now
AK501131
96,968
114,788
123,366
27.22%
7.47%
AK501132
101,699
108,846
123,473
21.41%
13.44%
AK501141
134,805
150,954
170,897
26.77%
13.21%
AK502131
93,369
107,477
115,819
24.04%
7.76%
AK502132
96,312
110,542
118,319
22.85%
7.04%
LB (l,S)
LB (2PL)
Best Sol.
Gap was
Gap now
BK511131
92,602
101,565
120,303
29.91%
18.45%
BK511141
125,307
134,997
172,762
37.87%
27.97%
BK521131
92,350
106,302
118,217
28.01%
11.21%
BK521142
124,988
139,090
154,258
23.42%
10.91%
BK522142
119,559
131,552
148,471
24.18%
12.86%
K. Akartunalı
Big-Bucket Production Planning: Two-Period Relaxations
Intro
Review
Methodology
Computations
Conclusions
Conclusions
Study of a stronger relaxation
A new framework independent from defining families of valid
inequalities or reformulations a priori, although expected
output is to define new valid inequalities using the results from
the framework
To our knowledge, this is an original approach in production
planning literature
Different norms useful to generate cuts and improve lower
bounds significantly
Euclidean and L∞ approaches computationally much more
efficient than Manhattan approach
Observed both on the efficiency of cuts and on the number of
extreme points generated in column generation before
termination
K. Akartunalı
Big-Bucket Production Planning: Two-Period Relaxations
Intro
Review
Methodology
Computations
Conclusions
Future Directions
Immediate directions
Achieving more computational results on realistic size problems
Applying lifting on the generated inequalities
Possibility to obtain facets instead of faces
Near-future directions
Careful analysis of the inequalities generated by the framework
and the facets from cdd
Significant potential to identify new families of valid
inequalities
Possibilities for extending results to other MIP techniques
Example: Disjunctive cuts
Normalization or Manhattan distance used by researchers
Euclidean has potential to provide more efficient cuts
Extension to other MIP problems possible
“Local Cuts” of Cook, Espinoza and Chvátal [2006]
K. Akartunalı
Big-Bucket Production Planning: Two-Period Relaxations