How can I write an objective function with its constrain for optimization?

The problem formulation for which I need optimization is as follow:

D- BUYER AGENT

S-SELLER AGENT

p-price

pj – price per unit

aj- amount of energy available

gj- total generation energy

ui- buyer utilities

vj- seller utilities

bi - bid of each buyer

maximize w.r.t 𝑑𝑖, 𝑠𝑗,

𝛩(𝑑𝑖, 𝑠𝑗)=Σ𝑢 𝑖(𝑑𝑖) +Σ𝑣𝑗(𝑔𝑗 − 𝑠𝑗)

𝑖∈𝒟 𝑗∈𝒮

subject to,

𝑑𝑖 ≤ 𝑏𝑖; ∀𝑖 ∈ 𝒟

𝑠𝑗 ≤ 𝑎𝑗; ∀𝑗 ∈ 𝒮

Σ𝑑𝑖 =Σ𝑠𝑗

𝑖∈𝒟 𝑗∈𝒮

Auction happens in iterative manner

Answer 1

[Torsten]

1. Put your unknowns in a vector x.
2. Write your function to be maximized as f*x where f is to be determined from your above problem formulation.
3. Write your inequality constraints as A*x <= b where A and b are to be determined by your above problem formulation.
4. Write your equality constraint as Aeq*x = beq where Aeq and beq is to be determined by your above problem formulation.
5. Determine lower and upper bounds on your unknowns and put them in the arrays lb and ub.
6. Call linprog.

[Jatinder Azrot]

Thanks for your response @Torsten, but as I'm new to MATLAB, can you please elaborate how to put unkowns in vectorx, and further steps also.

Thanks in advance

[Torsten]

So let's try linprog for your problem:

https://de.mathworks.com/help/optim/ug/linprog.html

Start with the vector of unknowns. These are d1,...,dD,s1,...,sS.

Since you want to maximize theta, you want to minimize -theta.

So your vector f that has to be passed to linprog is

f = [-u,v]

where u and v are vectors of length D and S, respectively (buyer and seller utilities).

Now your first and second constraints are bound constraints that have to be defined with the help of lb and ub:

lb = [zeros(size(u)),zeros(size(v))]
ub = [b,a];

a = amount of energy available, b = bid of each buyer.

The third constraint has to be put in Aeq and beq:

Aeq = [ones(size(u)),-ones(size(v))];
beq = 0;

Now you can call linprog:

sol = linprog(f,[],[],Aeq,beq,lb,ub)

That's all.

Of course, you must give values to the vectors a, b, u and v first.

[Jatinder Azrot]

Thanks a lot @Torsten, can I assign the values to a,b,u and v like:

u = randperm(5,1);

v = randperm(4,1);

a = randperm(50,1);

b = randperm (30,1);

and do I not need to mention g and s

[Torsten]

You want u,v,a,b to have one single element ?

[Jatinder Azrot]

I want them to have a range of values

[Torsten]

rng('default')
u = rand(1,6);
b = rand(1,6);
v = rand(1,9);
a = rand(1,9);
f = [-u,v];
lb = [zeros(size(u)),zeros(size(v))];
ub = [b,a];
Aeq = [ones(size(u)),-ones(size(v))];
beq = 0;
sol = linprog(f,[],[],Aeq,beq,lb,ub)

Optimal solution found.

sol = 15×1

0.2785 0.5469 0 0.9649 0.1576 0 0 0.5114 0 0.6787

[Jatinder Azrot]

Thank you @Torsten, I wanted to ask you one more thing, I'm working on NSGA II and I've two function and defined functions like below, is it correct or I need to do as you described earlier:

function z = MOP41(x)

u = randperm(5,1); % buyer utility

d = randperm(10,1); %amount of energy delevered to each buyer

c = randperm(4,1); % seller utility

s = randperm(15,1); % amount of energy each seller can deliver

g = randperm(30,1); % total energy generated

a = randperm(50,1);

b = randperm (30,1);

s1 = 0;

s2 = u*d;

s3 = 0;

s4 = c*(g-s);

n= 7;

for i=1:n

s1= s1+ s2;

end

for i= 1:n

s3 = s3+s4;

end

d<=b;

s<= a;

% Edit the lines below with your calculations.

benergy=randperm(50,1);

bhighest= randperm(40,1);

d=randperm(30,1);

t=5;

g=randperm(20,1);

T = 7;

K= 100;

I= 50;

for i=1:K

d=d;

end

for i= 1:I

g=g;

end

g<d;

for i= 1:T

x = d-g;

end

y= x*t*benergy;

y<= pi;

bhighest < benergy;

% Edit the lines below with your calculations.

%objective = b* log(d) - z;

% objective = max (objective);

%F(1) = s1 + s3;

%F(2) = x*t*benergy;

%x = optimInput(1);

%y = optimInput(2);

%end

z1 = s1 + s3;

z2 = x*t*benergy;

z = [z1 z2]';

end

[Torsten]

I gave you the code to solve the optimization problem you posted.

I'm completely lost what you try to do with the function from above since it is full of errors and settings I do not understand.

I suggest you first try to improve your MATLAB skills here

https://de.mathworks.com/learn/tutorials/matlab-onramp.html

and then work through some examples for "linprog" included here

https://de.mathworks.com/help/optim/ug/linprog.html

[Jatinder Azrot]

Thanks for the help with the code, I'm sorry I was trying other objective function and by mistake posted that, but the code you gave did not mentioned 'g' and 's' in the function. you only mentioned funtion of u and v, so I wanted to ask will it effect anyway on the performance

[Jatinder Azrot]

Also can you tell me is the below written code correct for the two objective function as mentioned below:

ist objective

min(g,d,benergy): Σ(Σ𝑑 - Σ𝑔) *t * benergy

subject to:

g<d, bhighest<benergy, Σ(Σ𝑑𝑖 - Σ𝑔𝑗) *t * benergy <= pi

and 2nd objective

max(g,d,benergy): Σ(Σg - Σd) *t * benergy - Σcczrbon

subject to:

d<g, bhighest<benergy, 0<ccarbon<=pi

code for both objective function:

function z = MOP41(x)

benergy1 = randperm(50,1);

bhighest1 = randperm(40,1);

d1 = randperm (30,1);

g1 = randperm(20,1);

t1 = 5;

ccarbon = randperm(30,1);

Tseller = 7;

K1 = 100;

I1 = 50;

for i = 1:I1

g1=g1;

end

for i = 1:K1

d1 =d1;

end

for i=1:Tseller

u = g1-d1;

end

v= u*t1*bhighest1 + ccarbon;

d1 < g1;

bhighest1<benergy1;

0<ccarbon<pi;

% Edit the lines below with your calculations.

benergy=randperm(50,1);

bhighest= randperm(40,1);

d=randperm(30,1);

t=5;

g=randperm(20,1);

T = 7;

K= 100;

I= 50;

for i=1:K

d=d;

end

for i= 1:I

g=g;

end

g<d;

for i= 1:T

x = d-g;

end

y= x*t*benergy;

y<= pi;

bhighest < benergy;

z1 = (u*t1*bhighest1 + ccarbon);

z2 = x*t*benergy;

z = [z1 z2]';

end

[Torsten]

d = sol(1:numel(u))

and

s = sol(numel(u)+1:end)

They have to be put in one common solution vector "sol" for linprog to work.

sum(v.*g) is constant and does not need to appear in the objective function.

rng('default')
u = rand(1,6);
b = rand(1,6);
v = rand(1,9);
a = rand(1,9);
f = [-u,v];
lb = [zeros(size(u)),zeros(size(v))];
ub = [b,a];
Aeq = [ones(size(u)),-ones(size(v))];
beq = 0;
sol = linprog(f,[],[],Aeq,beq,lb,ub);

Optimal solution found.

d = sol(1:numel(u))

d = 6×1

0.2785 0.5469 0 0.9649 0.1576 0

s = sol(numel(u)+1:end)

s = 9×1

0 0.5114 0 0.6787 0.7577 0 0 0 0