optimization of a mix bed plant - lunds tekniska högskola · optimization of a mix bed plant 3–7...

Optimization of a Mix Bed Plant

Kalle Oskari SaastamoinenDepartment of Information Technology,Lappeenranta University of Technology,

Lappeenranta,Finland.

Eskil HansenDepartment of Mathematics,

Lund University,Lund, Sweden.

Mari Susanna KahkonenDepartment of Mathematics,

University of Joensuu,Joensuu, Finland

Peter SteinhorstDepartment of Mathematics,

University of Chemnitz,Chemnitz, Germany

Maciej NiedzielaDepartement of Mathematics,University of Zielona Gora,

Zielona Gora, Poland

Instructor: Dirk KehrwaldDepartment of Mathematics,University of Kaiserslautern,

Kaiserslautern, Germany

Abstract

The mix bed plants are used to mix several different materials, e.g. in acement production. In this report we will present a simple model which showsthe right shape of the mix bed plant if we know the plants length and two angles.We also provide four algorithms which lets a company to optimize amount ofingredients to achive as homogenuous mix bed plant as possible.

3–1

3–2 14th ECMI modelling week Lund 2000

Contents

1 Introduction - Optimization of a Mix Bed Plant 3–3

2 Optimal Shape 3–4

3 Controlling Flow of the Material - General 3–53.1 Company Wishes . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–53.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3–63.3 General Notations and Conditions . . . . . . . . . . . . . . . . . . 3–6

4 Modelling 3–74.1 First Approach - Minimum Number of Oscillations . . . . . . . . . 3–7

4.1.1 Minimizing the overload . . . . . . . . . . . . . . . . . . . 3–84.1.2 Minimazing the Number of Layers . . . . . . . . . . . . . . 3–9

4.2 Second Approach - Minimizing the Local Error . . . . . . . . . . . 3–94.3 Model Four - Compromize . . . . . . . . . . . . . . . . . . . . . . 3–10

5 Conclusions 3–11

Optimization of a Mix Bed Plant 3–3

1 Introduction - Optimization of a Mix Bed Plant

Figure 1a: Scetch of the mix bed

Figure 1b: The longitudinal mix bed plant in practice

This report deals with a problem brought to the European Consortium of Math-ematics in Industry 14th Modelling Week that took place at the Lund University inLund from July 1st to July 10th,

��. The problem instructor was Dirk Kehrwald

from University of Kaiserslautern, Germany and the problem was originated fromthe machine company Maschinen- und Verfahrenstechnik. According to MVT homepages [1] it is a company that has specialised in providing complete solutions andindividual machines for a longitudinal (see figure 1b) and circular mix bed plants aswell as bulk material storages of a different design. The mix bed plants are used toconstruct a homogenious stockpiles, e.g. in a cement production. At first ingredientsare pre-mixed, after this one load of pre-mixed material will be put through a stacker(which is continuosly moving longitudinal direction to and for). This one load ofpre-mixed material forms a layer during a one sweep of a stacker. Typically the mixbeds are around 100 meters long and consists of around 240 layers, thus achieving avolume of 15 000 �� . The basic principle in how they are made can also be found


from figure 2 below.

Figure2: Scetch of a mix bed plant

Our problem had two goals to find an optimal shape of the mix bed and to find acontrolling algorithm which minimizes the amount of lacking ingredient to minimalamount of layers.

2 Optimal Shape

The optimal shape is important to find because heaping up the mix bed in a wrongway, one might waste a lot of space and thus reduce the amount of the material thatcan be mixed (and thus sold) in a one period. So we are trying to make the lengthof the base as near as possible the length of the top. In practise this means that weminimize a cross-sectional area of the mix bed, so the optimal shape of this area is asnear of a rectangle as possible. This was relatively easy to achieve because we know


how the optimal shape looks like (below).

Figure 3. Scetch of the optimal shape.

We also know the angles of stability in a case that we know what materials have beenused, which is a normal situation in practice. We also know the volume of the heap.We managed to formulate a relatively simple non-linear optimization problem of theform:

Goal ��

�

Subject to:

��

��

�� !#"%$'&)("%$'&�* �,+.-"%$'&)(When tested with the different angles resulting figures appeared to be very near

of the estimated shape.

3 Controlling Flow of the Material - General

For a technical reasons (e.g. pre-mixing is ineffective) one or more material mightbe missing in a given layer. This of cource causes a bad quality of the end productso this failure must be compensated elsewhere. We tried to do this by controlling theflow of the material and this we will do by a control algorithm.

3.1 Company Wishes

The company had many wishes. Main items have been listed here (below).

/ No layer contains only one material.

/ Few changes in conveyors speed (no oscillations).

/ Small global error.


/ Small local error.

The first wish is obvious, cause for example cement is totally something else if itcontains only one material. Other wishes caused our main problems. One can expectthat a small global vs. a small local error is somewhat contradictionary. There arealso two other main problems which arise from these three last wishes and they arefollowing.

Conflict 1 A little changes in a conveyors speed causes a big local error.

Conflict 2 Small local error (small overload) causes a constant long period standingerror in a many layers or chronical changes in a conveyors speed.

So we came to the conclusion that these wishes are incompatible, this is the rea-son why we finally had four algorithms which all had their benefits.

3.2 Assumptions

At first we made a few assumptions.

Assumtion 1 Homogeniuous material after pre-mixing. This is resonable, but not infact totally correct.

Assumtion 2 ’Input’ is always of the same volume.

Assumtion 3 One input gives always a one layer. From this assumtion and the pre-vious one it follows that all the separate layers have the same volume.

Assumtion 4 We always run out of material in whole layers. This is artificial butactually it is impossible to correct lacking of the material in the middle of thelayer.

Assumtion 5 When running out of a one ingredient, we have to correct this mistakebefore the next ingredient runs out. This means that we should reach the initialstable situation before we have the next lack of an ingredient.

Like one can see we have made quite a few assumtions which is the normal casein the real life modelling. All of these assumtions are anyhow a very reasonable ones.

3.3 General Notations and Conditions

We have also a few general notations and conditions. These are general cause theyare valid in all of our four models.

Notation 1 �� , where�

��

��

�� These are expected percentages of the differentmaterials

�.

Notation 2 ��'�� , where�

��

��

�� These are functions defining percentage ofmaterial

�in an arbitrary layer .

Condition 1� � ��'��

, where�

��

��

�� . This is due that the functionsdefined above are giving values in percentages.


Condition 2��

�and

��

�where � is the number of materials

used in an arbitrary layer . So sums of percentages can not be more than�.

Condition 3 �� . This is the case whenmaterial

�is missing from the layers . This ratio fixes the amount of missing

ingredient�

to the arbitrary layers .

Condition 4

��

�� "!#�$��%�

� �'�� '& �)( �

, where *� is the number of the

layer where the error happens and ,+��- is a number of the layer where error

has been fixed. So by this we are expecting global error to be as near of zeroas possible.

4 Modelling

In our first three models we took into account only some of the company wishescause like noted earlier they were a bit contradictionary. In our fourth model we weretrying to find a some kind of compromize between a contradictionary wishes.

Remark 1 Note that we are now expecting that we have three ingredients and miss-ing ingredient is marked as an index

�.

Remark 2 We also expect the total amount of layers to be 240.

4.1 First Approach - Minimum Number of Oscillations

At first we tried to minimize the number of oscillations and there has to obviouslybe at least one oscillation after an error, which comes when we try to compensate anerror which has occured. We got two models with different kinds of conditions. Inboth cases we have only one oscillation which means only three control point.

Figure 4. First approach.


4.1.1 Minimizing the overload

We used here a simple optimization algorithm (below):

Remark 3 Constraint 1 is valid in all of our models.

Goal ��

, where�

is the number of layers which are used to compensate

the error.

Subject to:

Constraint 1� �

� �

and� � � ��

, where is the overload of the missingingredient and is the number of the layer. So this means just that the morethan hundred percent overload is not allowed.

Constraint 2� � � �� + , where + is the number of the layer where we start

to fix an error and �+� - is the number of the layser where the error has been

fixed. This tries to quarantee that we have enough layers to fix an error.

Under conditions:

Condition 1 � � �� , when � +

�,��

��

. So after error has beenfixed the amount of ingredients has been balanced.

Condition 2

��

�� + ��

� � � �� & � , where + is the number of the layer

where where we start to fix an error and��

��

. So the difference be-tween total overload during the fixing period and expected percentages of thematerials

�can not be more than constant

& � while we are fixing the error.So if

& � is small then we can put only a little extra to all layers which arecompensating the error.

When we calculated this with Matlab we got following result (below).

Figure 5. Illustration of the model 1.


Remark 4 One can see that the constant overload for several layers is the maindisadvantage of this model. Here it takes more 150 layers to compensate an error.This is due that in condition 2 the constant

& � is selected to be too small. Next modeltries to solve this problem.

4.1.2 Minimazing the Number of Layers

Algorithm (below):

Goal �� , where

�is the number of layers which are used to compensate an error.

Subject to:

Condition 1 � � ��

, when +� � +

�, + is the number of the layer

where we start to fix the error and + �

� +

��- this is number of the layerwhere the error has been fixed, so tells us that how many layers it takes to fixthe error.

Condition 2 � �� , �� is the maximal overload for � � , which is allowed.

Condition 3��

+� � � , where

�

��, this is the minimum value for � + and �

� .

Remark 5 Conditions 2 ja 3 allow company to set limits for concentration.

When we calculated this with the Matlab we got following the result (below).


4.2 Second Approach - Minimizing the Local Error

Here should be noted that in the following model we are no longer trying to minimizethe number of the oscillations at all. Instead of that we are now trying to minimizemaximal local error in the following way:


Goal �� +��-

, where is the maximal overload.

Subject to:

Condition1

��

��

� � � �� & , � ��

�� +

��- � � � , � ��

and � �� ! � � . So the local error can not be more than constant

& while weare fixing the error.

We implemented this with the Matlab and got the following result:


Remark 6 Very high number of oscillations is the main disadvantage of this model.

4.3 Model Four - Compromize

This model is the same as the previous one except that we are also minimizing thenumber of oscillation by the following condition:

Condition2 � � � �� & � , � � + �� +

��- � ��,� � �

��

. Thisfixes the number of oscillations to the minimum when we take the constant

& �low enough.

We implemented this with the Matlab and got the following result:



Remark 7 This is also hard to control, because amount of concentration is changingall the time.

5 Conclusions

We did the following test-runs with models 4, 2 and 1, where we compared a localerror which they generate. Model three is not here because it is the only one whichproduces a lot of oscillations so it is not compatible with other models. Note that

& is the period of layers where an error occurs, which means actually layers which aremissing some ingredient and

� is the average amount of overload during the fixingprocess, where number of the index refers to the model.

Test � � � + ��

� & �

+ �

1 0.50 0.30 0.20 20 10 0.02 0.25 0.172 0.33 0.33 0.33 20 25 0.04 0.23 0.153 0.70 0.20 0.10 20 10 0.03 0.15 0.124 0.10 0.80 0.10 20 30 0.04 0.09 0.08

Table 1: Test-run of models 4, 2 and 1.

Model one This model is obviously easy to control, because it has only three controlpoints. It has only one oscillation. This model takes anyway long time tostabilize.

Model two This model is also easy to control, because it has only three controlpoints. It also has only one oscillation. It stabilizes itself quite fast and mainprofit is perhaps that company can set limits for concentration.


Model three From model three we found a lot of disadvantages like a lot of oscil-lations, very hard to control, in some layers probably only one ingredient andcan take a long time to stabilize.

Model four From table 1 we can easily see that the model four gives the minimumoverload that means a minimal local error it also stabilizes itself a quite fastand there is only one oscillation. Bad thing is that it seems to be a quite hardto control.

In future perhaps becoming better controlling systems like ones which are basedon fuzzy logic, neural networks, genetic algorithms or some of their combinationperhaps allow mathematicians to use more oscillations in their models. It can alsobe that run-outs of the other materials can be available and then we can fix errorsstraight when they happens. One can also easily improve our models and includemore ingredients.

References

[1] Homepage of Maschinen- und Verfahrenstechnic: http://www.mvt.de

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