How can I find the vector that simultaneously maximizes my two functions?

Question

Guglielmo 2024-2-23

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2085773-how-can-i-find-the-vector-that-simultaneously-maximizes-my-two-functions

移动：Torsten 2024-2-23

Hi everyone, I am new in using matlab, I am using it for my own work, this is the Code:

fa=0:1; %[1:100] ho cambiato per uguagliare dimensione vettori
fh=0:4;
% Frequenze di viaggio
%Fa=1:4;   
Fa=1;
%Fh=1:4;  %[1:10] ho cambiato per uguagliare dimensione vettori
Fh=1;
%Tempi di viaggio
Ta=2;
Th=1;
diffTemp=Ta-Th;
%VOT e costi
lVOT=1;
uVOT=0;
diffVOT=uVOT-lVOT;
ca=1;
Ca=1;
ch=0;
Ch=0;
%for i=2:length(fa);
%    j=2:length(fh);
%   Pa(i,j) = (fa(i)./diffVOT)*((fh(j)-fa(i))/diffTemp)-lVOT-Fa.*ca-Ca
%end
% Calcolo di Pa senza l'uso di cicli
%Pa = (fa./diffVOT).*((fh - fa)./ diffTemp) - lVOT - (Fa.*ca) - Ca
for i=1:length(fa)
    for j=1:length(fh)
        Pa(i,j) = (i/diffVOT).*(((j-i)/diffTemp) - lVOT) - (Fa*ca) - Ca
        Ph(j,i) = ((j/diffVOT).*(uVOT - ((j-i)/diffTemp))) - (Fh*ch) - Ch        
    end 
end

In this way i'm able to calculate the value of the function for the different combinations of (fa,fh), but my problem consists in the following: I have two players "a" and "h", each of them has its own function that expresses the profit "Pa , Ph"; all values in the function are constants, except for "fa" and "fh" which are expressed as vectors of different sizes.

I need to find an x(fa*,fh*) vector made from the value of "fa" and "fh" which simultaneously maximizes the "Pa" and "Ph" profit functions of both players, but I don’t know how to implement this on matlab.

Can anyone help me?

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Walter Roberson 2024-2-23

I suggest you use Problem Based Optimization https://www.mathworks.com/help/optim/problem-based-basics.html

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Answer 1

Hassaan 2024-2-23

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2085773-how-can-i-find-the-vector-that-simultaneously-maximizes-my-two-functions#answer_1415023

编辑：Hassaan 2024-2-23

在 MATLAB Online 中打开

% Clear workspace, close all figures, and clear command window
clear; close all; clc;
% Define constants
diffTemp = 2 - 1; % Ta - Th, example values
lVOT = 1;
uVOT = 0;
diffVOT = uVOT - lVOT;
Fa = 1;
Fh = 1;
ca = 1;
Ca = 1;
ch = 0;
Ch = 0;
% Define optimization variables
fa = optimvar('fa', 'LowerBound', 0, 'UpperBound', 1); % Adjust bounds as necessary
fh = optimvar('fh', 'LowerBound', 0, 'UpperBound', 4); % Adjust bounds as necessary
% Define the objective functions using anonymous functions
objPa = @(fa, fh) (fa./diffVOT).*((fh - fa)./ diffTemp) - lVOT - (Fa.*ca) - Ca;
objPh = @(fa, fh) ((fh./diffVOT).*(uVOT - ((fh - fa)./ diffTemp))) - (Fh.*ch) - Ch;
% Combine the objectives into a single function (for demonstration)
totalObjective = objPa(fa, fh) + objPh(fa, fh);
% Create the optimization problem
prob = optimproblem('Objective', totalObjective, 'ObjectiveSense', 'maximize');
% Solve the optimization problem
[sol, fval, exitflag, output] = solve(prob);
% Display the solution
disp('Solution:');
disp(sol);
fprintf('Combined Profit: %f\n', fval);
fprintf('Exit Flag: %d\n', exitflag);
fprintf('Solver Output:\n');
disp(output);

The interaction between two players into a single optimization problem for illustrative purposes. In real-world scenarios, especially in game theory, interactions might require more sophisticated approaches to find equilibria or optimal strategies.
The effectiveness and applicability of combining objectives like this depend on the nature of the interaction between fa and fh. In cases where their interests are opposed, a more nuanced approach to find a Nash equilibrium or Pareto optimality might be necessary.
The solve function automatically selects an appropriate solver based on the problem type. For more control or for large-scale problems, consider specifying solver options.

Note:

It's an initial attempt and may need adjustment as per your requirements.

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It's important to note that the advice and code are based on limited information and meant for educational purposes. Users should verify and adapt the code to their specific needs, ensuring compatibility and adherence to ethical standards.

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Guglielmo 2024-2-23

Thx for the help, i'm doing this work for my Master Degree Thesys, i need to find the Generalized Nash Eq (GNE) in this game, where the competition is between 2 different mode of transport (a=air, h=high speed rail) for a common path.

I'm quite new to Game Theory and Matlab, and for that reason i'm finding problems, so thanks a lot for ur time

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