MULTIVARIATE OPTIMIZATION
CONSIDERING QUALITY AND
MANUFACTURING COSTS: A CASE
STUDY IN A DRYING PROCESS
Carla Schwengber ten Caten
PPGEP/UFRGS – BRAZIL
[email protected]
Carlos Eduardo Appollo Unterleider
PPGEP/UFRGS - BRAZIL
[email protected]
José Luis Duarte Ribeiro
PPGEP/UFRGS - BRAZIL
[email protected]
1
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.
2
MULTIVARIATE OPTIMIZATION

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.
3
MULTIVARIATE OPTIMIZATION

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.
4
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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PROBLEM CHARACTERIZATION
 The response variable used to quantify the quality
characteristics of the drying process are:
Response variable
Y1: Productivity (units/s)
Y2: AI Loss (mg)
Y3: Humidity (%)


Type
Target
Larger-is-better
Smaller-is-better
Nominal-is-better
60
5,5
8
Specification
Min
Max
30
20
2
13
RI
2,0
2,0
1,0
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.
6
THE PROBLEM CHARACTERIZATION

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.
7
EXPERIMENT PLANNING

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
A: Vertical disposition of the trays
B: Horizontal disposition of the trays
C:Maximum temperature of the dry chamber
D: Dry Chamber used
Investigation Interval
A1: 50%
B1: 50%
C1: 70ºC
D1: nº1
A2: 100%
B2: 100%
C2: 95ºC
D2: nº2
Current
Setting
100%
100%
70ºC
nº1
8
EXPERIMENT PLANNING



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.
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EXPERIMENT EXECUTION
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.
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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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MULTIVARIATE OPTIMIZATION

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
J

Zˆ (i)   w j  Yˆj  T j

j 1
Z (i )
2
ˆ
 Y 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 the
relative importance of each response variable;
wj
Tj
Yˆj , ˆ 2Y

2
is the target value of the response variable j;
j
are estimations of mean and standard
deviation 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).
wj 
IR j
E 2j
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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.
QC (i )  K  Zˆ (i )
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CONSTANT K



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 .
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CONSTANT K

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:
(U$ B  U$ A ) (9.65  7.15)
K

 2.37
ZB  ZA
(4.71  2.60)
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GLOBAL COST (GC)

With this information, it is possible to compute the
Global Cost (GC):
GC(i )  QC (i )  MC (i)

This way, the optimum setting found is a compromise
among quality costs and raw material and energy
spent in a product manufacture.
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MANUFACTURING COST (MC)

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
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MULTIVARIATE OPTIMIZATION
 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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MULTIVARIATE OPTIMIZATION
 These are the current and optimum settings:
Control Factors
A: Vertical disposition of the trays
B: Horizontal disposition of the trays
C: Maximum temperature of the dry chamber
D: Dry chamber used
Global Cost
Current
Setting
100%
100%
70ºC
nº1
U$ 15.30
Optimum
Setting
100%
50%
70ºC
nº1 ou nº2
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$10.24
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MULTIVARIATE OPTIMIZATION

The optimum setting provides the response variable
estimations that are shown on the table below
Response variable
Y1: Productivity (units/s)
Current Optimum
48
33
Y2: AI Loss (mg)
20,23
12,01
Y3: Humidity (%)
2,40
3,69
Type
Target
Larger-thebetter
Smaller-thebetter
Nominal-the
better
60
Specification
Min
Max
30
-
RI
2,0
5,5
-
20
2,0
8
2
13
1,0
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MULTIVARIATE OPTIMIZATION
Controllable
A: Vertical disposition of the trays
B: Horizontal disposition of the trays
C: Maximum temperature of the dry chamber
D: Dry chamber used
Total Costs Z*
Current
Setting
100%
100%
70ºC
nº1
U$ 15.30
Optimum
Setting
100%
50%
70ºC
nº1 ou nº2
U$ 10.24
The cost difference represents a gain of U$ 222,640.00 per
year (220 days), considering a 200 box/day production.
G  (15.30  10.24)  220  200  U $222,640.00
23
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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CONCLUSIONS
 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.
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