Application of Multiobjective Optimization in Food Refrigeration Processes

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Application of Multi-objective Optimization in
Food Refrigeration Processes

• T.T.H. Luong, F.J. Trujillo and Q.T. Pham
• University of New South Wales
• Sydney 2052, Australia

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PART I MULTI-OBJECTIVE OPTIMISATION CONCEPTS
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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.

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Conventional approach to MO

• The conventional or economists 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.

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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.

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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)
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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)
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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)
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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)
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Levels of domination

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

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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.

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The Pareto Front

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

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The Pareto Front

• We would like also that the solutions are nicely
  spread along the front
• and not clumped up like this...

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PART IIMO OPTIMISATION BY GENETIC ALGORITHM
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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.

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

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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!)

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How do we rank the solutions when there are
several objectives? (cont)
(numbers represent fitness value)

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

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PART IIICASE STUDIES
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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.

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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).

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Results

• Pareto fronts at some generations

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Some temperature regimes

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

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Weight loss curves for different regimes
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Tenderness change for different regimes
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Changes in surface water activity (Regime 1)
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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.

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Some solutions
1 least tender 5 most tender
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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 stuckby numerical errors
  in numerical models.