The Box Assignment Problem in Log YardsThe Box Assignment Problem in Log Yards Maria Anna Huka Institute for Production and Logistics BOKU - University of Natural Resources and Life

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

Maria Anna Huka The Box Assignment Problem in Log Yards 2

Overview

1 Problem statement


3 Spadework

4 Notation



7 Solution

8 Perspective


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


Overview

1 Problem statement


3 Spadework

4 Notation



7 Solution

8 Perspective


Log yard and material flow


Overview

1 Problem statement


3 Spadework

4 Notation



7 Solution

8 Perspective


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


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


Overview

1 Problem statement


3 Spadework

4 Notation



7 Solution

8 Perspective


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


Overview

1 Problem statement


3 Spadework

4 Notation



7 Solution

8 Perspective


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.


Real world solution


Overview

1 Problem statement


3 Spadework

4 Notation



7 Solution

8 Perspective


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


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


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


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


Overview

1 Problem statement


3 Spadework

4 Notation



7 Solution

8 Perspective


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 %


Overview

1 Problem statement


3 Spadework

4 Notation



7 Solution

8 Perspective


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


Thank you for your attention!


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


www.boku.ac.at

Documents

The Box Assignment Problem in Log YardsThe Box Assignment Problem in Log Yards Maria Anna Huka Institute for Production and Logistics BOKU - University of Natural Resources and Life