1. 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. Pham University of New South Wales Sydney 2052, Australia

2. 2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 PART I: MULTI-OBJECTIVE OPTIMISATION CONCEPTS

3. 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.

4. 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 = c 1 f 1 + c 2 f 2 + c 3 f 3 +... This transform a MO problem into a single objective optimisation. Problem with this approach: What values should the unit costs c i be? User should have a range of alternatives to choose from, i.e. make final choice on a subjective basis.

5. 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.

6. 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)

7. 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)

8. 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)

9. 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)

10. 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:

11. 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.

12. 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…

13. 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...

14. 2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 PART II: MO OPTIMISATION BY GENETIC ALGORITHM

15. 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.

16. 2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 Search direction Genetic Algorithm - graphical illustration (for a single objective problem)

17. 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.

18. 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!)

19. 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”)

20. 2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 PART III: CASE STUDIES

21. 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 7 o C.

22. 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).

23. 2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 Results Pareto fronts at some generations

24. 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.

25. 2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 Weight loss curves for different regimes

26. 2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 Tenderness change for different regimes

27. 2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 Changes in surface water activity (Regime 1)

28. 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 7 o C.

29. 2003 International Congress of Refrigeration, Washington, D.C., August 17-22, 2003 Some solutions 1: least tender 5: most tender

30. 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.