(Use MATLAB Optimization Commands (linprog, quadprog, or fmincon).) What mixture

Question

(Use MATLAB Optimization Commands (linprog, quadprog, or fmincon).)

What mixture of metals should we use to minimize the cost of the alloy? P.S: Put commands at the beginning of your Matlab code that explains what are the design variables and how you found the cost function and constraints. Also explain how you decide the category of the optimization problem.

Question 4 Use MATLAB Optimization Commands (linprog, quadprog, or fmincon).) There are four metals with the following properties Metal | Density | % Carbon | % Phosphor | Price($/kg) 6500 5800 6200 5900 0.2 0.35 0.15 0.11 0.05 0.015 0.065 2.0 2.5 1.5 2.0 We want to make an alloy with properties in the following range: Range | Density | %Carbon | %Phosphor 0.1 0.3 0.045 0.055 Minimum5950 Maximum6050 What mixture of metals should we use to minimize the cost of the alloy? P.S: Put commands at the beginning of your Matlab code that explains what are the design variables and how you found the cost function and constraints. Also explain how you decide the category of the optimization problem.

Explanation / Answer

% this is the classic example of operation research and resource allocation suite of problems
% similar map coloring problems - you can use only 3 colors - Red Green, Blue - neighboring regions must have different colors
% like wise, we will have to minimize the cost here and also meet out alloy mixing requirements

% density between 5950 and 6050
% the metalic bonding character of alloys:
% the base metal contribution to the strength of materials
% for example, when we mix approximately 1.5 % of Carbon by weight to iron
% galvenize it, make it as a Galvenized Iron pipe, we get a GI Steel pipe as an alloyed product
% steel ropes are also made by mixing iron and carbon in the ratio of 1.9 % carbon by weight to iron
% in plumbing, gate valves are made of alloys
% the underlying physics detail is that the Allotropes of iron comprising of alpha and ghama
% has varied type of arrangements in their atomic structure
% rapid cooling and slow cooling will create different types of alloys
% as the carbon gets locked during rapid cooliing and produces stress inside the alloy

% the economic scaling is maintained by gold coins where gold and silver alloyed together to yield Electrum
% to get a density of 5950, we are looking at Metal D and A
% D has 5900, A has 6500
% if you mix metals A and D that will have the density of ( 6500 + 5900 ) / 2 = 12400 / 2 = 6200
% which exceeds the maximum range - hence skip this choice
% let us mix metal D and metal C
% 5900 + 6200 = 12100 / 2 = 6050
% it is inside the range
hence we can go for this combination of metals C and D mixing

% Matlab command description:
% optimization options can be made by utilizing the optimoptions function
% we can also use optimset with fminbnd, fminsearch, fzero
% linprog
% Linear Programming model
% quadprog
% this is the quadratic program solver in matlab
% fmincon
optimoptions(@fmincon)
% the above command will return an array of choices for the fmincon algorithm - in which the the default values
% for interior - points will also be returned
%
% the algorithm option can be set explicitly for another fmincon algorithm by making use of the algorithm set option
choices = optimoptions(@fmincon, 'NewRoutines', 'khg')

% alloy is made up of mixing of few or many metals

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