2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

Application of Multi-objective Optimization in Food Refrigeration Processes

T.T.H. Luong, F.J. Trujillo and Q.T. PhamUniversity of New South Wales

Sydney 2052, Australia

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

PART I: MULTI-OBJECTIVE

OPTIMISATION CONCEPTS

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

What is Multi-objective Optimisation (MO)

• MO is an optimisation problem which has several contradictory objectives.

• ALL real-life problems have several contradictory objectives!

– Big house vs big boat

– More comfort vs more energy consumption

– Product quality vs cost of production

– Safety vs capital cost

etc.

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

Conventional approach to MO

• The conventional or economist’s approach: Use weighted objective function (Assign a unit cost or weight to each objective and add up).

F = c1f1 + c2f2 + c3f3 + ...

This transform a MO problem into a single objective optimisation.

• Problem with this approach:• What values should the unit costs ci be?• User should have a range of alternatives to choose from, i.e.

make final choice on a subjective basis.

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

“True” Multi-objective Optimization

• True MO aims to obtain a range of solutions, each being “optimal in its own way”, i.e. is at least as good as each of the others in at least ONE respect.

• Such solutions are called Pareto-optimal solutions or non-dominated solutions.

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

Illustration of MO Optimisation

• Suppose we want to minimise two conflicting objectives A and B and have found 4 possible solutions.

(Plot of Objective function B

vs Objective function A)

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

Illustration of MO Optimisation

• Suppose we want to minimise two conflicting objectives A and B and have found 4 possible solutions.

• Solution 1 is dominated by 4: it is worse than 4 in both objectives.

Region

dominated by 4

(Plot of Objective function B

vs Objective function A)

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

Illustration of MO Optimisation

• Suppose we want to minimise two conflicting objectives A and B and have found 4 possible solutions.

• Solution 1 is dominated by 4: it is worse than 4 in both objectives.

• Similarly solutions 1 and 2 are dominated by solution 3.

Region dominated by 3

(Plot of Objective function B

vs Objective function A)

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

Illustration of MO Optimisation

• Suppose we want to minimise two conflicting objectives A and B and have found 4 possible solutions.

• Solution 1 is dominated by 4: it is worse than 4 in both objectives.

• Similarly solutions 1 and 2 are dominated by solution 3.

• But neither 3 and 4 dominate each other. They are non-dominated (at least, among these 4).

(Plot of Objective function B

vs Objective function A)

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

Levels of domination

Actually the solutions can be classified into several “levels of dominance”, by successively removing the more dominant solutions:

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

The Pareto Front• When all possible solutions are plotted on the objective

function graph, the non-dominated solutions form a smooth Pareto front. Ideally, we would like to find as many solutions lying on the Pareto front as possible.

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

The Pareto Front

• We would like also that the solutions are nicely spread along the front…

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

The Pareto Front

• We would like also that the solutions are nicely spread along the front…

and not clumped up like this...

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

PART II:MO OPTIMISATION

BY GENETIC ALGORITHM

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

Genetic Algorithm (GA) - General principles

• GA aims to optimise a function by evolving a population of solutions (instead of a single solution)

• Solutions combine their features in a directed but randomised way to produce the next generation.

• A randomised selection process cause the best solutions to survive and produce offsprings while the others die off.

• The use of multiple solutions and randomisation ensures that the search escapes from local optima and is not affected by small errors.

• The use of multiple solutions are ideal to give a range of Pareto-optimal solutions in multi-objective optimisation.

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

Search direction

Genetic Algorithm - graphical illustration(for a single objective problem)

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

GA: Pseudocode

Initialize random population of solutions

Loop

Select “parents” from present population (*)

Create “children” (new solutions) (†)

Select next generation from existing population (*)

Until maximum number of generation is reached

(*) Selection is randomised (throwing dices), but better solutions have more chance of being selected.

(†) Create a new solution from two existing solution by extrapolation, interpolation or mutation.

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

How do we rank the solutions when there are several objectives?

• Non-dominated solutions are always better than 1st-level dominated solutions, which are always better than 2nd-level dominated solutions, etc.

• Within the same level of dominance, solutions which are isolated are better than solutions that are clumped together (we must define how close is close!)

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

How do we rank the solutions when there are several objectives? (cont)

By using the above criteria, we favour dominant solutions that are spread out over a large range.

(numbers

represent

“fitness value”)

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

PART III:CASE STUDIES

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

Problem 1: OBJECTIVES

Design a temperature regime to chill a beef carcass while • maximising the tenderness of the meat in the loin, and• minimising the weight loss.

Constraints: • Chilling time = 16 hours• Final temperature of the leg must not be greater than

7oC.

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

Details of model

• A multi-region finite difference model is used to represent the carcass (Davey & Pham , 1999)

• A second, finer FD grid is superimposed near the surface to calculate moisture diffusion (Pham and Karuri,1999)

• Surface water activity obtained experimentally and correlated by Lewicky (1998) model.

• Microbial growth obeys the equation by Ross (1999).

• Tenderness evolves according to Arrhenius law (Graafhuis et al.,1992).

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

Results

Pareto fronts at some generations

0.38

0.39

0.4

0.41

0.42

0.43

0.44

0.45

0.46

0.47

0.48

1.85 1.9 1.95 2 2.05 2.1 2.15Weight loss (%)

Tend

erne

ss Gen.20

Gen.30

Gen.40

Gen.50

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

Some temperature regimes

• 1, 2: low weight loss, high toughness.• 4, 5: high weight loss, low toughness.• 3: intermediate.

-15

-10

-5

0

5

10

15

20

0 4 8 12 16

Time, h

Tem

per

atu

re,

C 1

2

3

4

5

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

Weight loss curves for different regimes

00.20.40.60.81

1.21.41.61.82

2.2

60 180 300 420 540 660 780 900 1020

Time (minute)

Wei

ght l

oss

(%)

S1

S3

S5

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

Tenderness change for different regimes

0.2

0.30.4

0.50.6

0.70.8

0.91

1.1

0 120 240 360 480 600 720 840 960 1080

Time (minute)

Te

nd

ern

es

s S1

S3

S5

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

Changes in surface water activity (Regime 1)

0.975

0.98

0.985

0.99

0.995

0 120 240 360 480 600 720 840 960 1080Time (minute)

Wat

er a

ctiv

ity

Leg

Rump

Loin

Ribs

Shoulder

Foreleg

Neck

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

Problem 2: OBJECTIVES

Design a temperature regime to • Chill a beef carcass within 16 hours, while• maximising the tenderness of the meat in the loin,

and• minimising the microbial growth.• (Constraint) Final temperature of the leg must not

be greater than 7oC.

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

Some solutions

-10

-5

0

5

10

15

20

0 4 8 12 16

Time, h

Tem

per

atu

re,

C

1

2

3

4

5

1: least tender

5: most tender

2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003

CONCLUSIONS

• Multi-objective optimisation is a powerful tool for decision making in industry.

• Problems with more than two objectives can be solved: product quality aspects, economics, etc.

• Unlike classical optimisation methods, GA is very robust and never gets “stuck”by numerical errors in numerical models.