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I am new to zimpl and I am currently trying to modell the GTSP. The setting is that we have nodes which are grouped into clusters. My problem is i dont know how to implement in zimpl which node belongs to which cluster.

What I did so far:

set V:= {1..6}; set A:= { in V*V with i < j};
set C:= {1,2,3};
set W:= { in C*C with p < q};
set P[]:= powerset(C); set K:= indexset(P);

I am guessing something is missing because i want to group node 1,2 in cluster 1, 3,4 in cluster 2 and 5,6 in cluster 3.

Some background Information:

Let G = (V, A) be a graph where V=1,2,...,n is the set of nodes and A = {(i, j): i, j ∈ V, i ≠ j} is the set of directed arcs (or edges), and let c_ij be the travel distance (or cost or time) from node i to node j. Let V1, V2, ... , Vk be disjoint subsets of V such that union of these subsets equals to V. These subsets are called clusters. The GTSP is to find the tour that (i) starts from a node and visits exactly one node from each cluster and turns back to the starting node (ii) never visit a node more than once and (iii) has the minimum total tour length. Associated with each arc, let x_ij be a binary variable equal to “1” if the traveler goes from node i to node j, and “0” otherwise.

Thats the mathematicl model I want to model: min∑i∈V ∑j∈V\{i} cijxij subject to: ∑i∈Vp ∑j∈V\Vp xij = 1 (p= 1, ..., k) ∑i∈V\Vp ∑j∈Vp xij = 1 (p= 1, ..., k) ∑j∈V\{i} xji − ∑j∈V\{i} xij = 0 (∀i∈V) xij∈{0,1} ∀(i, j)A up−uq+k ∑i∈Vp ∑j∈Vq xij+(k−2)∑i∈Vq ∑j∈Vp xij ≤ k−1 (p≠q;p,q=2,...,k) up≥0 (p=2, ..., k) (Thats the link for the paper: http://www.wseas.us/e-library/conferences/2012/Vouliagmeni/MMAS/MMAS-09.pdf)

Maybe someone can help! thanks

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