Abstract In this short note, the dual problem for the traveling salesman problem is constructed through the classic Lagrangian. The existence of optimality conditions is expressed as a corresponding inverse problem. A general 4-cities instance is given, and the numerical experiment shows that the classic Lagrangian may not be applicable to the traveling salesman problem.

Different manufacturing enterprises use regularly scheduling algorithms in order to help meeting demands over time and reducing operational costs. Nowadays, for a better useofresources and manufacturingin accordance withcustomer needs and given the level ofcompetitionbetweencompanies, employing asuitablescheduling programhasa double importance. Conventional productionmethods are constantly subs...

This paper applies the multi-time window traveling salesman problem to not only optimize logistics cost, but also effectively endow users with multiple discrete idle optional time periods meet requirements of just-in-time production. In process solving, firstly, dynamic penalty factor is introduced into objective function and added relax constraints in order construct relaxation model. Secondly...

in press. 17 H and L chain preparations in 1 M propionic acid were mixed in the ratio of 3: 1 based on O.D. and then dialyzed against cold water followed by borate buffer. 18 This arc was not apparent in the plate of the original y-globulin nor in any of the other plates. It did not appear when the horse anti-rabbit globulin was absorbed with normal rabbit 72-globulin prior to diffusion. Since ...

Traveling Salesman Problems with Profits (TSPs with Profits) are a generalization of the Traveling Salesman Problem (TSP) where it is not necessary to visit all vertices. With each vertex is associated a profit. The objective is to find a route with a satisfying collected profit (maximized) and travel cost (minimized). Applications of these problems arise in contexts such as traveling salesman ...