TheSimpliFire
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We can work from scratch. Rearrange $\beta=1-n(S)/\mu$ to get $$-\frac{\mu\sqrt{2\pi}}\sigma(\beta-1)=e^{-z^2/2}-z\sqrt{2\pi}+z\int_{-\infty}^ze^{-t^2/2}\,dt$$ on using {\cal L}(z)=\phi(z)-z(1-\Phi(z)... View answer 2 answers 7 votes 304 views 4 votes What you are looking for is the term domination analysis. For instance, see: Punnen, A., Margot, F., Kabadi, S. (2003) TSP heuristics: domination analysis and complexity. Algorithmica 35(2):111-127. ... View answer 1 answers 5 votes 130 views 8 votes In Pelánek (2011)1, Sudoku difficulty evaluation was investigated across four existing metrics. These are based on incidences of various logic techniques (see constant folding). Results based on ... View answer 1 answers 3 votes 105 views 3 votes \begin{align}\text{diff}^2&amp;=c^2+4\left(\left(\sum s_jx_j\right)^2-c\sum s_jx_j\right)\\&amp;=c^2+4\left(\sum s_j^2x_j^2+\sum_{\rm cyc}s_ks_\ell x_kx_\ell-c\sum s_jx_j\right)\\&amp;=c^2+4\left(\sum ... View answer 1 answers 6 votes 135 views Accepted answer 5 votes Interesting historical question. In Section 8.7, Chapter 8 of Algorithms (2019)1, Erickson notes that The simplest implementation of Ford’s generic shortest-path algorithm was first sketched by ... View answer 5 answers 8 votes 682 views 3 votes Blot et al. (2018)1 did an extensive literature review on multi-objective LS algorithms. You may wish to take a look at Ishibuchi and Murata (2004)2 which focuses on issues in evolutionary algorithms. ... View answer 3 answers 0 votes 182 views 2 votes Let(x_i)_{i\le n}$and$(\hat x_i)_{i\le n}$be sequences of observed data and forecast values respectively. Suppose we wish to split each$x_i$equally over$m$elements so that$(x_i)_{i\le n}$... View answer 3 answers 2 votes 341 views Accepted answer 4 votes As$x_i\in[0,1]$we just need to compare the values of the endpoints since$(x_i-c_i)^2$is minimised at$x_i=c_i$. It is easy to see that$x_i=0$gives the maximum whenever$c_i\ge1/2$and$x_i=1$... View answer 1 answers 3 votes 140 views Accepted answer 4 votes The notation$C^1$means$f'$is continuous (on$\Bbb R$as the interval is not stated). In general$C^k(a,b]$means that all of$f',f'',\cdots,f^{(k)}$are continuous on$(a,b]$. You are correct ... View answer 2 answers 12 votes 189 views 6 votes Is operations research really research?1 discusses how the Design Research paradigm can be used to evaluate OR problems. Reference [1] Manson, N. J. (2006). Is operations research really research?.... View answer 2 answers 1 votes 122 views Accepted answer 3 votes Dyer and Proll (1977)1 showed that for an M/M/c queue, the mean waiting time is a strictly decreasing and convex function of c. Reference [1] Dyer, M. E., Proll, L. G. (1977). On the Validity of ... View answer 1 answers 4 votes 115 views Accepted answer 5 votes Note that$\phi$is a function of$q$so$d\phi(q)$is interpreted as w.r.t.$\phi(q)$, which is the same as$d\phi$. This is just shorthand for$\phi'(q)\,dq$. View answer 1 answers 4 votes 66 views 2 votes Note that$V(O)$is simply of the form$\sf Q_1/L_1$where$\sf Q_1$is a quadratic and$\sf L_1$is a linear function of$p$. This can be written as${\sf{L_2}}+c/\sf{L_1}$where$\sf L_2$is also ... View answer 7 answers 16 votes 2k views 10 votes In terms of conferences, the Egon Balas Symposium in 2019 contained numerous talks on mathematical optimization. For integer programming specifically, it includes the Geometric Firefighter Routing ... View answer 4 answers 6 votes 166 views 3 votes In the book Nonlinear Optimization (2010) by Bomze, Demyanov, Fletcher and Telarky, there is the chapter The Sequential Quadratic Programming Method written by Fletcher. View answer 3 answers 13 votes 737 views 4 votes I think the first three are OK, more so the first two. I agree with what Larry said about defining all variables first before usage and in this case it means defining$t\in[T]$first. However, the ... View answer 2 answers 11 votes 3k views 6 votes I would not say that knowing such concepts is necessary unless you wish to not only apply certain results to solve OR problems, but also to understand why/how the results work mathematically. For the ... View answer 1 answers 4 votes 218 views 7 votes No. The constraint$a\ge\gamma b$sets no upper bound on$a$so it cannot be bounded above by$b$as your second formulation suggests. There are a few posts here on the linearisation of the product ... View answer 2 answers 5 votes 1k views 6 votes The package netgen (v1.3) can be used to generate networks and benchmark instances of VRP and TSP. As this article shows it is also possible to solve a variant of VRP (CVRP) in R Shiny, which is ... View answer 4 answers 8 votes 381 views 6 votes This is possible purely under DCP. As you are interested in the interval$[0,1]$, rewrite your function as $$x\sqrt{1-x}=\exp\left(\ln x+\frac12\ln(1-x)\right),\quad x\in[0,1].$$ Then the following ... View answer 1 answers 5 votes 823 views Accepted answer 2 votes I do not understand why the formula for$C_{0,4}$is as such. A simple check reveals that the unit on the RHS are pounds / (pounds / inch2) = inch2 whereas the unit for the cost is pounds. We have ... View answer 1 answers 6 votes 208 views 2 votes You may wish to take a look at the paper by Cozman (2000)1; the following is taken from the Introduction. One of the advantages of junction tree algorithms is that it is possible to efficiently ... View answer 1 answers 11 votes 344 views 7 votes Since$\hat q_{N'}(\hat x)\approx\Bbb E[Q(\hat x,\xi)]$and$\Bbb V[\hat q_{N'}(\hat x)]=\Bbb V[Q(\hat x,\xi)]/N', we have \begin{align}\hat\sigma_{N'}^2(\hat x)=\Bbb V[\hat q_{N'}(\hat x)]&amp;=\... View answer 2 answers 8 votes 170 views 6 votes It is possible to apply the notions of stationary points and the second derivative of a function to functionals. For|\varepsilon|\ll1$and and a differentiable function$h$, we can write, using ... View answer 4 answers 5 votes 557 views 3 votes Goerigk et al. (2014) may be a good start to your literature search. It uses mixed-integer programming to propose a genetic algorithm that can be used for urban evacuation planning. Examples of ... View answer 3 answers 9 votes 491 views 5 votes The constant car price means that $$c_{i,j}=12\,000-t_j+\sum_i^{j-1}m_i$$ where$t_j$denotes the trade-in price on year$j$and$m_i$the maintenance cost on year$i$. Thus $$c_{i,i+1}=12\,000-t_{i+1}... View answer 6 answers 13 votes 213 views 1 votes If B_i is the set whose elements are connected by a_i, then for the "at least" condition we want$$\sum_{b_k\in B_i\cap B_j\\\quad i\ne j}|b_k|&gt;0\qquad\forall a_i,a_j\in A$$where k=1,\cdots,K... View answer 2 answers 6 votes 173 views Accepted answer 5 votes The utility function u(x)=a-be^{-x/20\,000} with the conditions u(0)=0 and u(100\,000)=1 gives$$u(0)=a-be^{-0/20\,000}\implies 0=a-be^0=a-b\implies a=b\tag1$$and$$u(100\,000)=a-be^{-100\,000/... View answer 4 answers 12 votes 2k views 2 votes If you are thinking about working in the UK, indeed.co.uk might be useful for preliminary searching. As you ask for PhD level jobs, this and this (both Operational Research Scientists) are good ... View answer 2 answers 6 votes 792 views Accepted answer 3 votes If$u(d)=c$then$d=u^{-1}(c)$since$u\circ u^{-1}$forms the identity. Thus in general, under suitable constraints for$a,b,c\$,\begin{align}a(1-e^{-d/b})=c&amp;\implies1-e^{-d/b}=\frac ca\\&amp;\...