Through the analysis of the heat distribution along the edges of pans
affected by different pan shapes, the article finds the reason for the
overcooking of the product at the edges when using a rectangular pans.
Furthermore, according to the requirements given by the problem, the a
uthors propose the optimization model for designing the optimal baking pan.
The paper propose two models and one algorithm. Model one is established
to investigate the heat distribution at the edges of pans in different
shapes, while model two and the algorithm are used to design the ultimate pan.
Model one describes the heat distribution at the edges of polygons. Based
on the equation of heat conduction while taking the convection effects
into account, the authors design a model that can describe the thermal
conduction inside the pan. Then, given certain Neumann boundary condition,
Finite-Element Analysis derives the approximate result of heat distribution,
where we can find the phenomena of overcooking at the edges, which matches
the practical phenomena described in the problem. Through curve fitting, the
authors also find that, for polygons, the standard deviation of temperature of
the edges is in inversely proportional to the square of the number of edges.
Model two can be used to find the ultimate solution of baking pan. In the analysis
of this model, according to the requirements given by the problem, the author
proposed two factors: area utilization rate and the degree of uneven of temperature.
Then, the paper analyzed the changing speed of the two factors as the number of
edges increases, adding correction factors R and weight p in order to construct
proper objective function. After modeling, the authors proposed three evaluation
methods of the model in response to the three conditions given by the problem.
For the first condition, integer programming is adopted to give the proper arrangement
of rectangular pans; for the last two conditions, the authors use a new algorithm to
give the proper arrangement of circular and regular polygon pans.
The new algorithm proposed in this paper is a polygon arrangement model based on Center
Dot Matrix Statistics, which is used to test the maximum number of a certain kind of polygon
or circles that can be put in a certain area. Then, based on this algorithm, the models
are tested with several examples, the shapes and numbers of pan for different W/L and p
are calculated.
Finally, the sensitivity analysis and model evaluation are performed.
During sensitivity analysis, the sudden change phenomenon that emerged
when W/L and p change are found. During model evaluation, strengths and
weaknesses are analyzed for different models, and suggestion of using rounded
corners and normal polygons are proposed for model two.