You're out of luck. k=x+e*sin(k) doesn't have a closed form solution in terms of k. If you supply values for e and x, you can find k numerically using fsolve.

@Kitonum and Tom

@mehdibaghaee do you expect someone to copy source code pasted as image? upload a worksheet!

@vv Happy new year to you. I have changed the question slightly to reflect the depot being node 1 not 0. and the customers are 2 to 8.

and a slightly different methodology....But as Carl Love points out, the distance matrix provided violates the triangle inequality: c[1,2]+c[2,8]<c[1,8], so it may not be salvagable.

VRP_IPv2.mw

@Carl Love when the facts spoil a good story. I found the matrix after trawling the internet for VRP questions (and answers)

@tomleslie why it works.......magic? But then i encountered a problem with a imported distance matrix...

revEngVRP.mw

@Markiyan Hirnyk a forum about maths, not English. Why don't you lookup "pedantry".

@vv thanks for your erudite solution. I like the way you dealt with the Y[ i<>j] constraint. I'll try and apply your method to a VRP....

@Christopher2222 what I wanted, thanks. The link you found contains some good information. It's a pity such a lot of interesting material is "buried" beneath an ambiguous title (virtually unsearchable). Hopefully this polemic will inspire improvement of the search engine by the powers that be.

@Kitonum Thx.  I swore I'd seen the formula before, so I looked through my repository of worksheets and found a worksheet from my Maple 15 folder:

poly_area.mw

and a link to similar work.

http://www.mapleprimes.com/questions/200154-Find-The-Area-Of-A-Polygon-Using-Maple

I didn't move it or remove it. Acer turned it into a post.
http://www.mapleprimes.com/posts/207664-Mapleprimes-Enquiry-Confusion

@Carl Love 

Looks like homework to me. Draw 3 lines from the vertexes of the triangle meeting in the centre of the circle. What angle is subtended at the centre? Get the radius by applying a little trigonometry.

@Carl Love Thanks for your effort. And Tom. And Markiyan.

Carl, on my Maple 2015, I don't get the output you got.

I believe the equation to optimize should be z := add(add(dist[i, j]*d[i]*Y[i, j], j = 1 .. n), i = 1 .. m);

try it and you will get different sites {compared to d[j] }