Problems Minimizing a Function

Question

I need to minimize a function, but it takes a long time. So I don't know if it's normal or if there's something wrong with my model.

u=(y0-μ)/(σ+r);
v=(y0-μ)/(σ-r);

a=PDF[NormalDistribution[μ,σ],u];
b=PDF[NormalDistribution[μ,σ],v];
q=CDF[NormalDistribution[μ,σ],u]-CDF[NormalDistribution[μ,σ],v];

c=μ+σ/q(b-a);
d=σ^2(1+1/q(v*b-u*a)-1/q^2(b-a)^2);
L=k((c-y0)^2+d^2);
γ=1-∫(y0-rσ)^(y0+rσn)(1/q)(1/(√2π σ) e^(((-1)/2)(yn-μ)^2/σ^2 ) dyn
γe=(1-γ)*m1+γ*(1-m2)
Cr=(ICr*(1-γ)*m1)/(1-γe)
Ca=(ICa*γ*m2)/(1-γe )
TC=L+(1-q)SC+Ca+Cr+Exp[1-Abs[1-μ/y0]]MC+(Exp[(σ1-σ)/(σ1-σ2)]-1)DC;

n=D[TC,{{σ,μ}}];

Solve[n==0,{σ,μ}]

The model is so complicated, so I will attach the wolfram mathematica file and related paper.

Thank you

Answer 1

After correcting all typos, Mathematica 12 finds a solution.

k = 8;
DC = 1;
MC = 1;
SC = 1;
IC = 1/10;
Subscript[y, 0] = 20;
r = 3;
Subscript[\[Sigma], 1] = 7256/10000;
Subscript[\[Sigma], 2] = 5/100;
u = (Subscript[y, 0] - \[Mu])/(\[Sigma] + r
  );
v = (Subscript[y, 0] - \[Mu])/(\[Sigma] - r);
a = PDF[NormalDistribution[\[Sigma], \[Mu]], u];
b = PDF[NormalDistribution[\[Sigma], \[Mu]], v];
q = CDF[NormalDistribution[\[Sigma], \[Mu]], u] - 
   CDF[NormalDistribution[\[Sigma], \[Mu]], v];
Subscript[\[Mu], t] = \[Mu] + \[Sigma]/q (b - a);
Subscript[\[Sigma], t] = 
  Sqrt[\[Sigma]^2 (1 + (1/q (v*b) - (u*a)) - (1/q^2 (b - a)))];
L = k ((Subscript[\[Mu], t] - Subscript[y, 0])^2 + 
\!\(\*SubsuperscriptBox[\(\[Sigma]\), \(t\), \(2\)]\));
TC = L + (1 - q)*SC + IC + 
   Exp[\[Mu]/Subscript[y, 0]]*
    MC + (Exp[(Subscript[\[Sigma], 1] - \[Sigma])/(
       Subscript[\[Sigma], 1] - Subscript[\[Sigma], 2])] - 1)*DC;
n = D[TC, {{\[Sigma], \[Mu]}}];

FindRoot[n == {0, 0}, {{\[Sigma], .19}, {\[Mu], 19.9}}]

(*Out[]= {\[Sigma] -> 0.19285, \[Mu] -> 19.9092}*)

To make sure that this is solution, we plot TC

ContourPlot[TC, {\[Sigma], .1, 1}, {\[Mu], 19, 20}, Contours -> 150, 
 PlotLegends -> Automatic, ColorFunction -> "Rainbow", 
 PlotRange -> {0, 10}]