The Box Assignment Problem in Log YardsThe Box Assignment Problem in Log Yards Maria Anna Huka...
Transcript of The Box Assignment Problem in Log YardsThe Box Assignment Problem in Log Yards Maria Anna Huka...
The Box Assignment Problem in Log Yards
Maria Anna Huka
Institute for Production and LogisticsBOKU - University of Natural Resources and Life Sciences, Vienna
11/07/2012
Overview
1 Problem statement
2 Log yard and material flow
3 Spadework
4 Notation
5 Real world solution
6 Mathematical models
7 Solution
8 Perspective
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Overview
1 Problem statement
2 Log yard and material flow
3 Spadework
4 Notation
5 Real world solution
6 Mathematical models
7 Solution
8 Perspective
Maria Anna Huka The Box Assignment Problem in Log Yards 3
Problem statement
• European middle size hardwood sawmill
• Annual production capacity of 30,000 m3 processed withband saws
• 75% of the total annual production is beech
• Crane schedule plan
• 15 assortments (diameters)
• 18 ejector boxes
• 42 storage boxes reduced to 28
• Since 2008 the log supply declined by 5.9%
⇒ Logistic optimization, minimizing transportation time⇒ New arrangement of the log yard
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Overview
1 Problem statement
2 Log yard and material flow
3 Spadework
4 Notation
5 Real world solution
6 Mathematical models
7 Solution
8 Perspective
Maria Anna Huka The Box Assignment Problem in Log Yards 5
Log yard and material flow
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Overview
1 Problem statement
2 Log yard and material flow
3 Spadework
4 Notation
5 Real world solution
6 Mathematical models
7 Solution
8 Perspective
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Spadework
• Estimate the amount of used logs with differentdiameters
• Compute the required storage space per assortment pershift per day
• Rate the given storage capacity at the log yard
• Calculate the distances between ejection boxes to storageboxes and material charge
• Reckon the amount of crane movements for eachassortment
• Assess the corresponding transportation time for theassortments
• Investigate given constraints/space on the log yard
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Spadework
Minimum MaximumVola 5.46 12.20Na 21 68
Cap28 4.1 12.3Cap42 4.0 10.6
• Minimal volume exceeds the capacity of the smalleststorage box
• Largest volume tops the capacity of the biggest storagebox in the case of the original setting
• Numbers of moves per assortment vary from 21 to morethan three times as much
⇒ Reduction and the amalgamation of the storage boxes wasnecessary to implement a model without fragmentation of theassortments
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Overview
1 Problem statement
2 Log yard and material flow
3 Spadework
4 Notation
5 Real world solution
6 Mathematical models
7 Solution
8 Perspective
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NotationA Set of assortments (a = 1, ..., 15)E Set of ejection boxes (e = 1, ..., 18)S Set of storage boxes (s = 1, ..., 28/42)
Na Number of trips per assortment a, due to the diameter of theassortment and the demand per set assortment
TTTes Total transportation time for every assortment from ejection boxe to storage box s and to the material charge
TTes Transportation time from ejection box e to storage box sTTs Transportation time from storage box s to the material chargeCAPs Capacity of storage box sVOLa Maximal volume of assortment aM Large number
xas Binary assignment variable, 1 if assortment a is assigned to sto-rage box s, 0 otherwise
yae Binary assignment variable, 1 if assortment a is assigned to ejec-tion box e, 0 otherwise
wes Binary assignment variable, 1 if the storage box s is filled withmaterial from the ejection box e, 0 otherwise
fillaes Filling factor of assortment a in ejection box e and storage box s
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Overview
1 Problem statement
2 Log yard and material flow
3 Spadework
4 Notation
5 Real world solution
6 Mathematical models
7 Solution
8 Perspective
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Real world solution
TTas matrix with the corresponding transportationtimes for each assortment
A0 = Awhile A0 6= {} do
for i ∈ A0 dofind minimal TTis
move (pick s ∈ S) is forbiddenassign corresponding assortment i to box sremove i from A0
end forend while
Providing an easy to handle logistic optimization approach,unproblematic applicable with a commercial available computersystem like Excel.
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Real world solution
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Overview
1 Problem statement
2 Log yard and material flow
3 Spadework
4 Notation
5 Real world solution
6 Mathematical models
7 Solution
8 Perspective
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Double stage - Storage box allocationThe transportation time from storage box s to thematerial charge is minimized, taking into accountthe numbers of trips per assortment a.
min∑
a∈A,s∈S
Na × TTs × xas
VOLa × xas ≤ CAPs ∀a ∈ A, s ∈ S
∑
s∈S
xas = 1 ∀a ∈ A
∑
a∈A
xas ≤ 1 ∀s ∈ S
xas ∈ {0, 1} ∀a ∈ A, s ∈ S
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Double stage - Ejection box assignmentWith the result of stage 1 the next instance is goingto be solved.Therefore, the best assignment of assortment a toejection box e is calculated.
min∑
a∈A,e∈E ,s∈S
Na × yae × x̃as × TTes
∑
e∈E
yae = 1 ∀a ∈ A
∑
a∈A
yae ≤ 1 ∀e ∈ E
yae ∈ {0, 1} ∀a ∈ A, e ∈ E
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Partition approach
• To simplify matters reduced number of boxes and nofragmentation of an assortment
• Expansion of the double-stage model is approach wherethe assortment can be divided into several storage boxes
• Confirming the solution of the approach with fewer storageboxes original numbers of boxes used
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Partition approach
min∑
a∈A,e∈E,s∈S
Na
VOLa× fillaes × TTTes
∑
a∈A
yae ≤ 1 ∀e ∈ E
∑
e∈E
yae = 1 ∀a ∈ A
∑
e∈E
wes ≤ 1 ∀s ∈ S
∑
e∈E,s∈S
fillaes = VOLa ∀a ∈ A
∑
s∈S
fillaes ≤ yae × VOLa ∀a ∈ A, e ∈ E
∑
a∈A
fillaes ≤ wes × M ∀e ∈ E , s ∈ S
∑
a∈A,e∈E
fillaes ≤ CAPs ∀s ∈ S
yae ∈ {0, 1} ∀a ∈ A, e ∈ E
wes ∈ {0, 1} ∀e ∈ E , s ∈ S
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Overview
1 Problem statement
2 Log yard and material flow
3 Spadework
4 Notation
5 Real world solution
6 Mathematical models
7 Solution
8 Perspective
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Comparable solutions of the log yard and boxassignment problem
Assortment Original Exceldouble-stage
Xpressdouble-stage
Xpresspartition
Xpress partition(original layout)
1 1/- 1/- 1/- 1/- 1/-2 2/- 2/- 2/- 2/- 2/-3 3/- 3/- 3/- 3/- 3/-4 12/12 a 17/16 c 18/18 a 18/18 a 18/18 a, 18c5 8/8 a 11/18 c 14/18 c 17/17 b, 18 c 17/17 b, 18d6 2/2 a 9/17 b 17/17 b 15/17 a 15/17 a7 16/16 a 18/18 a 16/16 a 16/16 a 16/16 a, 18 b8 10/10 a 10/14 c 12/16 c 11/13 a 12/14 b, 16 c9 4/4 a 8/12 c 10/14 c 12/16 c 13/4 a, 15 a10 17/17 a, b 16/16 a 11/14 a 14/14 a 14/14 a, 16 b11 9/10 c 7/10 a 8/12 c 10/14 c 9/12 a, 16 d12 6/6 a 6/10 c 6/10 c 6/10 c 11/8 c, 13 a13 13/13 a 13/15 a 13/15 a 13/15 a 10/12 b, 14 c, 14 d14 18/18 a 14/14 a 9/12 a 9/12 a 7/10 a, 10 b15 11/12 c 5/8 a 7/10 a 8/12 c 8/12 c, 12 d16 15/15 a 15/13 a 15/13 a 7/5 a, 10 a 6/8 b, 10 c17 14/14 a 12/12 a 5/8 a 5/8 a 5/5 a, 8 a18 7/8 c 4/8 c 4/8 c 4/6 a 4/6 a, 6 b
# storage boxes 42 28 28 28 42Transportation
time (min)342.821 326.311 301.453 288.117 293.659
Saving - 4.8 % 12.1 % 16.0 % 14.3 %
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Overview
1 Problem statement
2 Log yard and material flow
3 Spadework
4 Notation
5 Real world solution
6 Mathematical models
7 Solution
8 Perspective
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Perspective
• Apply model on a standardized log yard◮ No crane but stacker◮ More than one stacker
• Enhance the model by different wood types
• Consider frequency distribution of diameters
• Here model is based on feeding data and log storagecapacities not considering production data
• Optimization of the lumber yard
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Thank you for your attention!
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University of Natural Resourcesand Life Sciences, Vienna
Department of Economics and Social SciencesInstitute for Production and Logistics
Maria Anna Huka
Feistmantelstrasse 4, A-1180 ViennaTel.: +43 1 47654-4425, Fax: +43 1 [email protected], www.boku.ac.at
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