How to optimize the run time in my optimization problem.

Hi guys. I have an optimization problem. The code is as follows:

Objective=@MassTransferErrors;
kL=0.5;
kH=0.5;
p0=[kL,kH];
A = [];
b = [];
Aeq = [];
beq = [];
lb=[0 ; 0];
ub=[100;100];

k = fmincon(Objective, p0, A, b, Aeq, beq, lb, ub);
disp(k)

function MTE=MassTransferErrors(p)
kL = p(1);
kH = p(2);
%% Constants
tMax=18000;                                 % reaction duration (s)
Q0=[100 200 350 400 400 400 500];            % Q_G etylene inflow (ml/min)
T_C =[230 230 230 180 200 230 230];                %T for different cases
MTE_j=zeros(1,7);
Experiments = {[ 0.2985 0.6498 0.6147 0.43917 0.40398],[0.68662 1.6373 1.4260 1.4437 1.53169],[2.90493 5.68662 5.75704 2.65845 1.00352],[3.50352 11.3908 6.77817 3.46831 2.2007],[4.73592 10.8979 4.48944 3.01056 2.76408],[4.80634 9.45423 6.60211 4.03169 2.83451],[4.41901 10.4754 7.09507 4.13732 2.27113]};
for i=1:7
    Q1=Q0(i)*1e-6/60;     % Q_G ethylene inflow (m3/s)
    Q2=0;              % Q_G butene inflow
    Q3=0;              % Q_G hexene inflow
    Q4=0;              % Q_G octene inflow
    Q5=0;              % Q_G decene inflow
    Q6=0;              % Q_G dodecene inflow
    Q7=0;              % Q_G undecane inflow
    P1=36e5;           % ethylene inflow pressure [Pa]

    T1=T_C(i)+273.15;     % T_Ethylene [K]
    T2=230+273.15;     % T_ref  [K]
    R=8.314;           % gas constant [J/(mol.K)]
    C1=P1/(R*T1);      % ethylene inlet gas concentration [mol/m^3]
    
    VR=300e-6;         % reactor volume [m^3]
    VG=250e-6;         % gas volume [m^3]
    VL=50e-6;          % liquid volume [m^3]
    K=[3.24;2.23;1.72;0.2;0.1;0.08;0.09];       % solubility  [nondim]
    moleWt=[28;56;84;112;140;168;156];              % mole weight C2,C4,...,C12,C11 [g/mol]
    wc=(0.3+0.25)*1e-3;                             % catalyst weight [kg]
    kref=[2.224e-4;1.533e-4;3.988e-5;1.914e-7;4.328e-5;...
        2.506e-7;4.036e-5;1.062e-6;6.055e-7;];    % rate at Tref=230C [mol/(s.g_cat)]
    Eact=[109.5; 15.23; 7.88; 44.45; 9.438; 8.426; 10.91; 12.54; 7.127]; % activation energy [J/mol];
    k=kref.*exp(-Eact*(1/T1-1/T2)/R);               % rate at T=T2 [mol/(s.g)]
    tauOF=5;                                        % outflow time constant (s)
    % Specify initial conditions
    xinit=zeros(15,1);                  % initial state vector
    xinit(1)=C1*VR;                     % initial ethylene in gas (mol)
    xinit(14)=36.63/156;                % initial undecane in liquid (mol)
    xinit(7) = xinit(14)*VG*K(7)/VL;    % initial undecane in gas (mol)
    xinit(8) = xinit(1)*VL/(K(1)*VG);   % initial ethylene in liquid (mol)
    xinit(15)=Q1*C1;                    % initial outflow rate (mol/s)
    nToti=sum(xinit(1:7));              % initial moles in gas (mol)
    dNdt=@(t,x) [Q1*C1-x(15)*x(1)/sum(x(1:7))-VR*kL*(x(1)/VG-K(1)*x(8)/VL);   % gas phase ethylene (mol/s)
                    Q2-x(15)*x(2)/sum(x(1:7))-VR*kL*(x(2)/VG-K(2)*x(9)/VL);    % gas phase butene (mol/s)
                    Q3-x(15)*x(3)/sum(x(1:7))-VR*kL*(x(3)/VG-K(3)*x(10)/VL);   % gas phase hexene (mol/s)
                    Q4-x(15)*x(4)/sum(x(1:7))-VR*kH*(x(4)/VG-K(4)*x(11)/VL);   % gas phase octene (mol/s)
                    Q5-x(15)*x(5)/sum(x(1:7))-VR*kH*(x(5)/VG-K(5)*x(12)/VL);     % gas phase decene (mol/s)
                    Q6-x(15)*x(6)/sum(x(1:7))-VR*kH*(x(6)/VG-K(6)*x(13)/VL);     % gas phase dodecene (mol/s)
                    Q7-x(15)*x(7)/sum(x(1:7))-VR*kH*(x(7)/VG-K(7)*x(14)/VL);     % gas phase undecane (mol/s)
                    VR*kL*(x(1)/VG-K(1)*x(8)/VL)+wc*(-2*k(1)*x(8)^2/VL^2-k(2)*x(8)*x(9)/VL^2-k(3)*x(8)*x(10)/VL^2-k(5)*x(8)*x(11)/VL^2-k(7)*x(8)*x(12)/VL^2);
                    VR*kL*(x(2)/VG-K(2)*x(9)/VL)+wc*(k(1)*x(8)^2/VL^2-k(2)*x(8)*x(9)/VL^2-2*k(4)*x(9)^2/VL.^2-k(6)*x(9)*x(10)/VL^2-k(8)*x(9)*x(11)/VL^2);
                    VR*kL*(x(3)/VG-K(3)*x(10)/VL)+wc*(k(2)*x(8)*x(9)/VL^2-k(3)*x(8)*x(10)/VL^2-k(6)*x(9)*x(10)/VL.^2-2*k(9)*x(10)^2/VL^2);
                    VR*kH*(x(4)/VG-K(4)*x(11)/VL)+wc*(k(3)*x(8)*x(10)/VL^2+k(4)*x(9)^2/VL^2-k(5)*x(8)*x(11)/VL^2-k(8)*x(9)*x(11)/VL^2);
                    VR*kH*(x(5)/VG-K(5)*x(12)/VL)+wc*(k(5)*x(8)*x(11)/VL^2+k(6)*x(9)*x(10)/VL^2-k(7)*x(8)*x(12)/VL^2);
                    VR*kH*(x(6)/VG-K(6)*x(13)/VL)+wc*(k(7)*x(8)*x(12)/VL^2+k(8)*x(9)*x(11)/VL^2+k(9)*x(10)^2/VL^2);
                    VR*kH*(x(7)/VG-K(7)*x(14)/VL);
                    (sum(x(1:7))-nToti)/tauOF];        % d(outflow rate)/dt (mol/s^2)
    [t,x]=ode45(dNdt,[0,tMax],xinit);
    molGend=x(end,1:7);
    molLend=x(end,8:14);
    massGend=molGend'.*moleWt;
    massLend=molLend'.*moleWt;
%Total Product
    TotalProduct = zeros(1,7);
    for j=1:7
    TotalProduct(j) = massGend(j) + massLend(j);        %Sum of the liquid and gas phase products(g)
    end
    Experiment_i = cell2mat(Experiments(i));            %Converting Experiments set to matrix 
    MTE_i = (TotalProduct(2)-Experiment_i(1))^2+(TotalProduct(3)-Experiment_i(2))^2+(TotalProduct(4)-Experiment_i(3))^2+(TotalProduct(5)-Experiment_i(4))^2+(TotalProduct(6)-Experiment_i(5))^5;   %Defining an Mass Transfer Error relation
    MTE_j(i) = MTE_i;                                                                                                                                                                              %Defines a Mass Transfer Error Vector(1*7) that contains the error for each case
end
MTE = sum(MTE_j(1:7));      %Objective function(Sum of the all arrays in MTE_j Vector) what I need to minimize is each array that is on the MTE_j Vector but since I can't return a Vector as an objective function I sum all the arrays as my objective function.                                                                                                                                                                       
end

What I need to minimize in this problem are all the seven values in MTE_j vector but since objective function can't return a vector. I have used the sum of the all values. I doubt this is the correct way to minimize all the seven values in MTE_j vector also my code took an extremely long run time(6 hours last time I have checked with no answers yet) . I know the objective function is complicated but I guess I'm doing something wrong. I also test my objective function with an assumption for kL and kH values. my objective function code seems to work OK. Here's the test code:

%%Test Objective function
kL = 0.1; %Assumption for kL
kH = 0.1; %Assumption for kH
%% Constants
tMax=18000;                                  % reaction duration (s)
Q0=[100 200 350 400 400 400 500];            % Q_G etylene inflow (ml/min)
T_C =[230 230 230 180 200 230 230];                %T for different cases
MTE_j=zeros(1,7);
Experiments = {[ 0.2985 0.6498 0.6147 0.43917 0.40398],[0.68662 1.6373 1.4260 1.4437 1.53169],[2.90493 5.68662 5.75704 2.65845 1.00352],[3.50352 11.3908 6.77817 3.46831 2.2007],[4.73592 10.8979 4.48944 3.01056 2.76408],[4.80634 9.45423 6.60211 4.03169 2.83451],[4.41901 10.4754 7.09507 4.13732 2.27113]};
for i=1:7
    Q1=Q0(i)*1e-6/60;     % Q_G ethylene inflow (m3/s)
    Q2=0;              % Q_G butene inflow
    Q3=0;              % Q_G hexene inflow
    Q4=0;              % Q_G octene inflow
    Q5=0;              % Q_G decene inflow
    Q6=0;              % Q_G dodecene inflow
    Q7=0;              % Q_G undecane inflow
    P1=36e5;           % ethylene inflow pressure [Pa]

    T1=T_C(i)+273.15;     % T_Ethylene [K]
    T2=230+273.15;     % T_ref  [K]
    R=8.314;           % gas constant [J/(mol.K)]
    C1=P1/(R*T1);      % ethylene inlet gas concentration [mol/m^3]
    
    VR=300e-6;         % reactor volume [m^3]
    VG=250e-6;         % gas volume [m^3]
    VL=50e-6;          % liquid volume [m^3]
    K=[3.24;2.23;1.72;0.2;0.1;0.08;0.09];       % solubility  [nondim]
    moleWt=[28;56;84;112;140;168;156];              % mole weight C2,C4,...,C12,C11 [g/mol]
    wc=(0.3+0.25)*1e-3;                             % catalyst weight [kg]
    kref=[2.224e-4;1.533e-4;3.988e-5;1.914e-7;4.328e-5;...
        2.506e-7;4.036e-5;1.062e-6;6.055e-7;];    % rate at Tref=230C [mol/(s.g_cat)]
    Eact=[109.5; 15.23; 7.88; 44.45; 9.438; 8.426; 10.91; 12.54; 7.127]; % activation energy [J/mol];
    k=kref.*exp(-Eact*(1/T1-1/T2)/R);               % rate at T=T2 [mol/(s.g)]
    tauOF=5;                                        % outflow time constant (s)
    % Specify initial conditions
    xinit=zeros(15,1);                  % initial state vector
    xinit(1)=C1*VR;                     % initial ethylene in gas (mol)
    xinit(14)=36.63/156;                % initial undecane in liquid (mol)
    xinit(7) = xinit(14)*VG*K(7)/VL;    % initial undecane in gas (mol)
    xinit(8) = xinit(1)*VL/(K(1)*VG);   % initial ethylene in liquid (mol)
    xinit(15)=Q1*C1;                    % initial outflow rate (mol/s)
    nToti=sum(xinit(1:7));              % initial moles in gas (mol)
    dNdt=@(t,x) [Q1*C1-x(15)*x(1)/sum(x(1:7))-VR*kL*(x(1)/VG-K(1)*x(8)/VL);   % gas phase ethylene (mol/s)
                    Q2-x(15)*x(2)/sum(x(1:7))-VR*kL*(x(2)/VG-K(2)*x(9)/VL);    % gas phase butene (mol/s)
                    Q3-x(15)*x(3)/sum(x(1:7))-VR*kL*(x(3)/VG-K(3)*x(10)/VL);   % gas phase hexene (mol/s)
                    Q4-x(15)*x(4)/sum(x(1:7))-VR*kH*(x(4)/VG-K(4)*x(11)/VL);   % gas phase octene (mol/s)
                    Q5-x(15)*x(5)/sum(x(1:7))-VR*kH*(x(5)/VG-K(5)*x(12)/VL);     % gas phase decene (mol/s)
                    Q6-x(15)*x(6)/sum(x(1:7))-VR*kH*(x(6)/VG-K(6)*x(13)/VL);     % gas phase dodecene (mol/s)
                    Q7-x(15)*x(7)/sum(x(1:7))-VR*kH*(x(7)/VG-K(7)*x(14)/VL);     % gas phase undecane (mol/s)
                    VR*kL*(x(1)/VG-K(1)*x(8)/VL)+wc*(-2*k(1)*x(8)^2/VL^2-k(2)*x(8)*x(9)/VL^2-k(3)*x(8)*x(10)/VL^2-k(5)*x(8)*x(11)/VL^2-k(7)*x(8)*x(12)/VL^2);
                    VR*kL*(x(2)/VG-K(2)*x(9)/VL)+wc*(k(1)*x(8)^2/VL^2-k(2)*x(8)*x(9)/VL^2-2*k(4)*x(9)^2/VL.^2-k(6)*x(9)*x(10)/VL^2-k(8)*x(9)*x(11)/VL^2);
                    VR*kL*(x(3)/VG-K(3)*x(10)/VL)+wc*(k(2)*x(8)*x(9)/VL^2-k(3)*x(8)*x(10)/VL^2-k(6)*x(9)*x(10)/VL.^2-2*k(9)*x(10)^2/VL^2);
                    VR*kH*(x(4)/VG-K(4)*x(11)/VL)+wc*(k(3)*x(8)*x(10)/VL^2+k(4)*x(9)^2/VL^2-k(5)*x(8)*x(11)/VL^2-k(8)*x(9)*x(11)/VL^2);
                    VR*kH*(x(5)/VG-K(5)*x(12)/VL)+wc*(k(5)*x(8)*x(11)/VL^2+k(6)*x(9)*x(10)/VL^2-k(7)*x(8)*x(12)/VL^2);
                    VR*kH*(x(6)/VG-K(6)*x(13)/VL)+wc*(k(7)*x(8)*x(12)/VL^2+k(8)*x(9)*x(11)/VL^2+k(9)*x(10)^2/VL^2);
                    VR*kH*(x(7)/VG-K(7)*x(14)/VL);
                    (sum(x(1:7))-nToti)/tauOF];        % d(outflow rate)/dt (mol/s^2)
    [t,x]=ode45(dNdt,[0,tMax],xinit);
    molGend=x(end,1:7);
    molLend=x(end,8:14);
    massGend=molGend'.*moleWt;
    massLend=molLend'.*moleWt;
%Total Product
    TotalProduct = zeros(1,7);
    for j=1:7
    TotalProduct(j) = massGend(j) + massLend(j);        %Sum of the liquid and gas phase products(g)
    end
    Experiment_i = cell2mat(Experiments(i));
    MTE_i = (TotalProduct(2)-Experiment_i(1))^2+(TotalProduct(3)-Experiment_i(2))^2+(TotalProduct(4)-Experiment_i(3))^2+(TotalProduct(5)-Experiment_i(4))^2+(TotalProduct(6)-Experiment_i(5))^2;
    MTE_j(i) = MTE_i;
end
MTE = sum(MTE_j(1:7));

I just want the kL and kH values that minimize each value on MTE_j vector and I want a code to actually give me these values. My code take an extremely long run time with no answers.

Answer 1

[Torsten]

I don't see anything obviously wrong in your coding - except for the last term in MTE_i which should be

(TotalProduct(6)-Experiment_i(5))^2

instead of

(TotalProduct(6)-Experiment_i(5))^5

Try whether switching to ode15s from ode45 will reduce the runtime.

[naiva saeedia]

@Torsten

Hi. thanks for your comment. Thanks for noticing the last term error. I have a question though can we return the vector results as multiple objective functions? I want all the seven values in MTE_j to be minimum but as I said the objective function in fmincon should be a single value hence I summed all the values in MTE_j . Can we use anything other than fmincon to solve a problem for multiple objective functions at the same time?? Can this enhance the run time in your opinion? Or it's pointless ?

[Torsten]

I want all the seven values in MTE_j to be minimum but as I said the objective function in fmincon should be a single value hence I summed all the values in MTE_j .

You can run 7 independent optimizations for the 7 experiments, but you will get 7 different parameter pairs for kL and kH. I don't think that makes much sense.

Also multiobjective optimization doesn't optimize all objectives simultaneously, but finds Pareto optimal solutions.

[naiva saeedia]

@Torsten

I have already done optimization for each case but the thing is as you said different values for different cases just don't make sense cause kL and kH should be constant on every case. What are Pareto optimal solutions? From my search when we use multi objective optimization these solutions don't minimize all the objective functions in short minimizing for example first obejective function cause some problem to minimize second obejective function. If that's the case I don't mind it actually cause with my optimization code I couldn't find any solutions due to long run time.

[Torsten]

Your approach to find kH and kL by minimizing the sum of squared differences over all measurements is the correct approach.

If you trust in some measurements more than in others, you can introduce weighting factors for the MTE_j and minimize sum wi*MTE_j(i) instead of sum MTE_j(i) with sum wi = 1.

If the computation doesn't finish, choose smaller values than the predefined ones for "maximum number of iterations" or "maximum number of function evaluations" .

Maybe the result doesn't make sense, but with "octave" - choosing ode15s instead of ode45 as integrator - I get kL = 0, kH = 2.492 within 2 minutes.

[naiva saeedia]

@Torsten

Thanks for the great suggestion but unfortunately I don't trust measurements differently. They all are same. Choosing different values for ""Maximum number of itteratierations"" cause some problems in convergaence I guess. What is "'Octave"'? kL shouldn't be 0 cause it doesn't make sense.

[Torsten]

Thanks for the great suggestion but unfortunately I don't trust measurements differently.

Then leave your code as it is. It gives the same weight to all your masurement data.

Choosing different values for ""Maximum number of itteratierations"" cause some problems in convergaence I guess.

It was a suggestion to inspect intermediate results to see whether they point in the "correct" direction - assuming you can say what is physically senseful.

What is "'Octave"'?

Do you have an internet connection ? Then googling "octave" solves this problem.

kL shouldn't be 0 cause it doesn't make sense.

Since you set the lower bound for the parameters to be 0, you admit kL = 0. The result from octave (kL = 0, kH = 2.492) gives a value of MTE in the order of 1e2 while your guess values give an MTE in the order of 1e11. So the direction kL -> 0 gives a huge decrease in MTE.

[naiva saeedia]

@Torsten

Yep I should leave it as it is.

Yeah for investigating the answers it's a good idea.

I asked my question in a wrong way.When you mentioned ""Octave"" I thought ""Octave"" was an option for ODE15s() in matlab but it was a programming language so I asked ""What is Octave?"" to completely understand what's your mean by that. I have found this link about solving differential equations by ""Octave"" using MATLAB

https://docs.octave.org/v4.4.0/Matlab_002dcompatible-solvers.html

but the problem is I'm torn right now. Is Octave pre-installed in MATLAB or should I add it to MATLAB? Also how you used ""Octave"" in my code?

kL and kH can be really small but they shoudn't be zero even If they improve MTE they shouldn't be zero cause it just doesn't make sense by the physics of the system. I didn't change the lower bound since it was never a problem when I optimized MTE for each case but here I should change it to a small number(order of 10^-5).

[Torsten]

"Gnu Octave" is a scientific programming language that can be used independent of MATLAB and is able to execute many MATLAB scripts (e.g. yours). Octave is free and has its own solvers.

https://octave.org/download

[naiva saeedia]

@Torsten

Thanks a lot for your answers. I really appreciate your help.

[naiva saeedia]

@Torsten

I have tried to run my code in Octave but I ended up getting my initial values for kL and kH. Should I change my code syntax completely in order to run it in Octave? Also changing ODE45 to ODE15s was a great idea. I ran the code in matlab in 1 min so thanks a lot for that.

[Torsten]

Load it into the editor and click the "Run" button under "Octave".

But now with 1 min runtime, I'd prefer MATLAB ...

    
function main
  pkg load "optim"
Objective=@MassTransferErrors;
kL=0.5;
kH=0.5;
p0=[kL,kH];
A = [];
b = [];
Aeq = [];
beq = [];
lb=[0 ; 0];
ub=[100;100];
k = fmincon(Objective, p0, A, b, Aeq, beq, lb, ub);
disp(k)
end
function MTE=MassTransferErrors(p)
kL = p(1)
kH = p(2)
%% Constants
tMax=18000;                                 % reaction duration (s)
Q0=[100 200 350 400 400 400 500];            % Q_G etylene inflow (ml/min)
T_C =[230 230 230 180 200 230 230];                %T for different cases
MTE_j=zeros(1,7);
Experiments = {[ 0.2985 0.6498 0.6147 0.43917 0.40398],[0.68662 1.6373 1.4260 1.4437 1.53169],[2.90493 5.68662 5.75704 2.65845 1.00352],[3.50352 11.3908 6.77817 3.46831 2.2007],[4.73592 10.8979 4.48944 3.01056 2.76408],[4.80634 9.45423 6.60211 4.03169 2.83451],[4.41901 10.4754 7.09507 4.13732 2.27113]};
for i=1:7
    Q1=Q0(i)*1e-6/60;     % Q_G ethylene inflow (m3/s)
    Q2=0;              % Q_G butene inflow
    Q3=0;              % Q_G hexene inflow
    Q4=0;              % Q_G octene inflow
    Q5=0;              % Q_G decene inflow
    Q6=0;              % Q_G dodecene inflow
    Q7=0;              % Q_G undecane inflow
    P1=36e5;           % ethylene inflow pressure [Pa]
    T1=T_C(i)+273.15;     % T_Ethylene [K]
    T2=230+273.15;     % T_ref  [K]
    R=8.314;           % gas constant [J/(mol.K)]
    C1=P1/(R*T1);      % ethylene inlet gas concentration [mol/m^3]

    VR=300e-6;         % reactor volume [m^3]
    VG=250e-6;         % gas volume [m^3]
    VL=50e-6;          % liquid volume [m^3]
    K=[3.24;2.23;1.72;0.2;0.1;0.08;0.09];       % solubility  [nondim]
    moleWt=[28;56;84;112;140;168;156];              % mole weight C2,C4,...,C12,C11 [g/mol]
    wc=(0.3+0.25)*1e-3;                             % catalyst weight [kg]
    kref=[2.224e-4;1.533e-4;3.988e-5;1.914e-7;4.328e-5;...
        2.506e-7;4.036e-5;1.062e-6;6.055e-7;];    % rate at Tref=230C [mol/(s.g_cat)]
    Eact=[109.5; 15.23; 7.88; 44.45; 9.438; 8.426; 10.91; 12.54; 7.127]; % activation energy [J/mol];
    k=kref.*exp(-Eact*(1/T1-1/T2)/R);               % rate at T=T2 [mol/(s.g)]
    tauOF=5;                                        % outflow time constant (s)
    % Specify initial conditions
    xinit=zeros(15,1);                  % initial state vector
    xinit(1)=C1*VR;                     % initial ethylene in gas (mol)
    xinit(14)=36.63/156;                % initial undecane in liquid (mol)
    xinit(7) = xinit(14)*VG*K(7)/VL;    % initial undecane in gas (mol)
    xinit(8) = xinit(1)*VL/(K(1)*VG);   % initial ethylene in liquid (mol)
    xinit(15)=Q1*C1;                    % initial outflow rate (mol/s)
    nToti=sum(xinit(1:7));              % initial moles in gas (mol)
    dNdt=@(t,x) [Q1*C1-x(15)*x(1)/sum(x(1:7))-VR*kL*(x(1)/VG-K(1)*x(8)/VL);   % gas phase ethylene (mol/s)
                    Q2-x(15)*x(2)/sum(x(1:7))-VR*kL*(x(2)/VG-K(2)*x(9)/VL);    % gas phase butene (mol/s)
                    Q3-x(15)*x(3)/sum(x(1:7))-VR*kL*(x(3)/VG-K(3)*x(10)/VL);   % gas phase hexene (mol/s)
                    Q4-x(15)*x(4)/sum(x(1:7))-VR*kH*(x(4)/VG-K(4)*x(11)/VL);   % gas phase octene (mol/s)
                    Q5-x(15)*x(5)/sum(x(1:7))-VR*kH*(x(5)/VG-K(5)*x(12)/VL);     % gas phase decene (mol/s)
                    Q6-x(15)*x(6)/sum(x(1:7))-VR*kH*(x(6)/VG-K(6)*x(13)/VL);     % gas phase dodecene (mol/s)
                    Q7-x(15)*x(7)/sum(x(1:7))-VR*kH*(x(7)/VG-K(7)*x(14)/VL);     % gas phase undecane (mol/s)
                    VR*kL*(x(1)/VG-K(1)*x(8)/VL)+wc*(-2*k(1)*x(8)^2/VL^2-k(2)*x(8)*x(9)/VL^2-k(3)*x(8)*x(10)/VL^2-k(5)*x(8)*x(11)/VL^2-k(7)*x(8)*x(12)/VL^2);
                    VR*kL*(x(2)/VG-K(2)*x(9)/VL)+wc*(k(1)*x(8)^2/VL^2-k(2)*x(8)*x(9)/VL^2-2*k(4)*x(9)^2/VL.^2-k(6)*x(9)*x(10)/VL^2-k(8)*x(9)*x(11)/VL^2);
                    VR*kL*(x(3)/VG-K(3)*x(10)/VL)+wc*(k(2)*x(8)*x(9)/VL^2-k(3)*x(8)*x(10)/VL^2-k(6)*x(9)*x(10)/VL.^2-2*k(9)*x(10)^2/VL^2);
                    VR*kH*(x(4)/VG-K(4)*x(11)/VL)+wc*(k(3)*x(8)*x(10)/VL^2+k(4)*x(9)^2/VL^2-k(5)*x(8)*x(11)/VL^2-k(8)*x(9)*x(11)/VL^2);
                    VR*kH*(x(5)/VG-K(5)*x(12)/VL)+wc*(k(5)*x(8)*x(11)/VL^2+k(6)*x(9)*x(10)/VL^2-k(7)*x(8)*x(12)/VL^2);
                    VR*kH*(x(6)/VG-K(6)*x(13)/VL)+wc*(k(7)*x(8)*x(12)/VL^2+k(8)*x(9)*x(11)/VL^2+k(9)*x(10)^2/VL^2);
                    VR*kH*(x(7)/VG-K(7)*x(14)/VL);
                    (sum(x(1:7))-nToti)/tauOF];        % d(outflow rate)/dt (mol/s^2)
    [t,x]=ode15s(dNdt,[0,tMax],xinit);
    molGend=x(end,1:7);
    molLend=x(end,8:14);
    massGend=molGend'.*moleWt;
    massLend=molLend'.*moleWt;
%Total Product
    TotalProduct = zeros(1,7);
    for j=1:7
    TotalProduct(j) = massGend(j) + massLend(j);        %Sum of the liquid and gas phase products(g)
    end
    Experiment_i = cell2mat(Experiments(i));            %Converting Experiments set to matrix
    MTE_i = (TotalProduct(2)-Experiment_i(1))^2+(TotalProduct(3)-Experiment_i(2))^2+(TotalProduct(4)-Experiment_i(3))^2+(TotalProduct(5)-Experiment_i(4))^2+(TotalProduct(6)-Experiment_i(5))^5;   %Defining an Mass Transfer Error relation
    MTE_j(i) = MTE_i;                                                                                                                                                                              %Defines a Mass Transfer Error Vector(1*7) that contains the error for each case
end
MTE = sum(MTE_j(1:7))      %Objective function(Sum of the all arrays in MTE_j Vector) what I need to minimize is each array that is on the MTE_j Vector but since I can't return a Vector as an objective function I sum all the arrays as my objective function.
end

[naiva saeedia]

@Torsten

Thank you so much.