multivariate optimization

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MULTIVARIATE OPTIMIZATION CONSIDERING QUALITY AND

MANUFACTURING COSTS: A CASE STUDY IN A DRYING PROCESS

Carla Schwengber ten CatenPPGEP/UFRGS – BRAZIL

[email protected]

Carlos Eduardo Appollo UnterleiderPPGEP/UFRGS - [email protected]

José Luis Duarte RibeiroPPGEP/UFRGS - BRAZIL

[email protected]

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MULTIVARIATE OPTIMIZATION

This paper describes a multivariate optimization case study performed in five steps:

Problem characterization;

Experiment planning;

Experiment execution;

Individual analysis of response variable;

Multivariate Optimization.

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Generally, the optimization process involves multiple quality characteristics. It is also necessary to satisfy multiple goals to achieve quality.

Typical goals are: (i) minimize deviations from targets and (ii) maximize robustness to noise.

Any deviation from target represents a quality loss and this loss leads to higher costs.


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Another important goal, which could be included in the optimization study, is the minimization of the cost of raw material and energy, spent during the manufacturing process.

Thus, the optimum setting is the one which minimizes the global costs, including costs due to poor quality as well as manufacturing costs.


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PROBLEM CHARACTERIZATION

The Drying Process Studied:

Is carried out in a period from 4.5 to 6 hours;

Reduces humidity from 50% to 13%;

Produces 48 units per second.

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The response variable used to quantify the quality characteristics of the drying process are:

Specification Response variable Type Target Min Max

RI

Y1: Productivity (units/s) Larger-is-better 60 30 - 2,0 Y2: AI Loss (mg) Smaller-is-better 5,5 - 20 2,0 Y3: Humidity (%) Nominal-is-better 8 2 13 1,0

PROBLEM CHARACTERIZATION

In order to capture their relative importance (RI), weights were assigned to each response variables.

The type and specification limits of each response variables were also identified.

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The research purpose was to find out the optimum setting for the process parameters:

Maximum temperature of dry chamber;

Vertical disposition of the trays;

Horizontal disposition of the trays;

Dry chamber used;

Ventilator inversion interval, and

Ventilator velocity.

THE PROBLEM CHARACTERIZATION

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The process parameters were prioritized and the following control factors were chosen, i.e., the subset process factors that would be essayed in the experiment

Control Factors Investigation Interval Current Setting

A: Vertical disposition of the trays A1: 50% A2: 100% 100% B: Horizontal disposition of the trays B1: 50% B2: 100% 100% C:Maximum temperature of the dry chamber C1: 70ºC C2: 95ºC 70ºC D: Dry Chamber used D1: nº1 D2: nº2 nº1

EXPERIMENT PLANNING

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The experiment plan considered four control factors in 2 levels.

A 2k factorial project was adopted, with one repetition, totalizing 2 x 2 x 2 x 2 = 16 essays.

To test the linear model adjustment, two central points were added.

EXPERIMENT PLANNING

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The trials were randomly executed and the data were collected in four weeks.

For each one of the 16 trials, three response variables were measured:

Productivity;

Active Ingredient (AI) Loss;

Humidity.

EXPERIMENT EXECUTION

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INDIVIDUAL ANALYSIS OF RESPONSE VARIABLE

Initially, the analysis was carried out individually for each response variable;

Using multiple regression, an individual model for each response variable mean and variability was built, including linear and interaction terms;

The coefficients with p-value less than =0.05= 5% were considered significant in the drying process.

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The adopted objective function was the multivariate quadratic loss, in which will be considered poor quality and also manufacturing costs;

Thus, the optimum setting of the control factors is the one that minimizes the global costs (GC), including costs due to poor quality (QC) as well as manufacturing costs (MC).


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MULTIVARIATE QUADRATIC LOSS FUNCTION

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2

1

ˆˆ)(ˆjYjj

J

jj TYwiZ

( )Z i

w j

is the objective function to be minimized; i refers to a certain treatment, i.e., certain factors setting;

are weights that take into account the units and therelative importance of each response variable;

2 ˆ,ˆjYjY are estimations of mean and standard

deviation of the response variable j;

jT is the target value of the response variable j;

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CONSTANT wj

The weights wj were defined taking into account the

relative importance (RI) of each response variable and the semi-amplitude of the specification interval (E).

wIR

Ej

j

j2

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LOW QUALITY COST (QC)

In its original form, the Multivariate Quadratic Loss Function provides values which are proportional to the financial loss due to low quality.

In order to obtain the financial loss in monetary values, it is necessary to know the proportionality constant K.

Once the K value is found, it is possible to transform the loss value Z into monetary units, it means, into quality costs.

)(ˆ)( iZKiQC

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The company offers two product categories classified according to quality (high/low) and with different selling prices.

It was calculated the loss value Z for a category A (high quality) product and its value was ZA = 2.60 loss units.

It was calculated the loss value Z (using the same target values considered for category A) for a category B (low quality) product and its value was ZB = 4.71 loss units .

CONSTANT K

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The category A product selling price is U$A 9.65/box and it is bigger than category B product selling price that is U$B 7.15/box;

The K constant is calculated as followed:

2.372.60)(4.71

7.15)(9.65

ZZ

)U$(U$

AB

AB

K

CONSTANT K

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GLOBAL COST (GC)

With this information, it is possible to compute the Global Cost (GC):

This way, the optimum setting found is a compromise among quality costs and raw material and energy spent in a product manufacture.

)()()( iMCiQCiGC

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A regression analysis for the manufacturing cost (MC) was carried out considering the control factors:

MC = 19.3 + 2.49 x horizontal disposition + 1.62 x maximum temperature of dry chamber – 0.877 x horizontal disposition x maximum temperature of dry chamber

MANUFACTURING COST (MC)

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Once the objective function was defined, linear programming routines were used to determine the control factors setting that minimizes the objective function.

The setting that minimizes the objective function is the one that best attend simultaneously the Global Cost, considering manufacture and quality costs.


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Control Factors Current Setting

Optimum Setting

A: Vertical disposition of the trays 100% 100% B: Horizontal disposition of the trays 100% 50% C: Maximum temperature of the dry chamber 70ºC 70ºC D: Dry chamber used nº1 nº1 ou nº2

Global Cost U$ 15.30 U$ 10.24


The current setting leads to a global cost of U$ 15.30

The optimum setting leads to a global cost of U$10.24

These are the current and optimum settings:

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The optimum setting provides the response variable estimations that are shown on the table below

Specification Response variable Current Optimum Type Target Min Max

RI

Y1: Productivity (units/s) 48 33 Larger-the-better

60 30 - 2,0

Y2: AI Loss (mg) 20,23 12,01 Smaller-the-better

5,5 - 20 2,0

Y3: Humidity (%) 2,40 3,69 Nominal-the better

8 2 13 1,0


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Controllable Current Setting

Optimum Setting

A: Vertical disposition of the trays 100% 100% B: Horizontal disposition of the trays 100% 50% C: Maximum temperature of the dry chamber 70ºC 70ºC D: Dry chamber used nº1 nº1 ou nº2

Total Costs Z* U$ 15.30 U$ 10.24

The cost difference represents a gain of U$ 222,640.00 per year (220 days), considering a 200 box/day production.


00.640,222$200220)24.1030.15( UG

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CONCLUSIONS

The optimum setting for the drying process control factors was identified considering quality costs and manufacturing costs;

The responses variable of the process were: productivity, loss of active ingredient (LA) and

final humidity.

The response variable humidity was close to the target and the LA loss was reduced in 40.63%;

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The productivity in the optimum setting is lower compared to the current setting, but an investment to build 5 new dry chambers would maintain the same value of the current setting and the invested value would quickly be amortized;

With the results found with the multivariate quadratic loss function, an economy of U$222,640.00 per year was possible.

CONCLUSIONS