Examples: Family or General Practice/Primary Care Nursing,
Chapter 7 Getting ready: education, role preparation and credentialing 7.1 Introduction
7.2 Programme development and curriculum design (2001 2010)
7.2.2 Curriculum Development
Chapter 8
Conclusions & Recommendations
The theme of the first part of this thesis has been the Travelling Salesman Prob-lem (TSP). A novel way of describing the probProb-lem mathematically has been pro-posed, which resulted in reducing the binary degrees of freedom, for the first time, to O(ndlog2(n)e). Three mathematical programming formulations have been in-troduced and a series of computational studies has been conducted in order to evaluate their computational performance in practice.
The second part of this dissertation reported the utilisation of Mixed-Integer Programming (MIP) for scheduling the cleaning actions for heat exchanger net-works subject to fouling. It described the extension of a novel work presented for isolated units to networks of exchangers. The concept of choice between alterna-tive cleaning methods was explored, with respect to the state of the deposit for networks subject to chemical reaction fouling and the conditioning of the heat transfer surface for networks subject to biological fouling.
8. Conclusions & Recommendations
that respect, the original problem can be viewed as a data storage problem on a binary tree structure.
The new contribution of this work is that it reduces the binary degrees of freedom for the Travelling Salesman Problem to O(ndlog2(n)e). To the author’s knowledge, up to date, no other mathematical description of the problem, among those found in the literature, succeeded in reducing the required number of bi-nary variables below O(n2). The current work takes an incremental step towards decreasing the formulation complexity of the Travelling Salesman Problem.
Two Mixed-Integer Linear Programming (MILP) formulations are developed for the general case of the Asymmetric Travelling Salesman Problem (ATSP), where the length of an arc connecting two cities depends on the direction in which it is travelled. These are the Tree-1 and Tree-2 formulations. Another MILP formulation is introduced for the special case where the distance between two cities is calculated on the basis of the rectilinear metric. This special case is referred to as the Manhattan Travelling Salesman Problem.
8.1.1 Asymmetric formulations
The two asymmetric formulations utilise a set of logical checks which force an arc to be present in the optimal tour if two cities are placed on neighbouring leaves of the binary tree. These adjacency constraints are the only difference between formulations Tree-1 and Tree-2. The set of adjacency constraints proposed by Millar and Cyrus [2000] is used in the Tree-1 formulation, while for Tree-2 the logical checks are derived by Theorem 3.1. The Tree-1 and Tree-2 formulations include O(n3) and O(n2dlog2(n)e) adjacency constraints, respectively.
The proposed formulations were implemented for small instances of the Trav-elling Salesman Problem (n ≤ 12). It emerged from the results that the Tree-1 formulation is superior in computational performance to the Tree-2 formulation.
In particular, the branch-and-cut solver visits considerably fewer nodes of the solution tree for Tree-1 than for Tree-2. In fact, the exact algorithm failed to converge after one day of execution, when applying Tree-2 to a problem involving 12 cities. For the same problem, the optimal solution was obtained after 363 CPU seconds when Tree-1 was applied. The dissimilar computational performance is
8. Conclusions & Recommendations
due to the different adjacency constraints included in each formulation.
Nevertheless, neither Tree-2 nor Tree-1 were successful when applied to prob-lems involving 12 < n ≤ 20 cities. In all cases the branch-and-cut algorithm failed to obtain the optimal solution after one day of execution. For all intents and purposes, the solution of such small problems should be trivial.
The computational efficiency of the two formulations was found to be worse than that of the well-known formulations proposed by Wong [1980] and Miller et al. [1960]. The basis for the comparison was the strength of the Linear Pro-gramming (LP) relaxation of each formulation. The formulation suggested by Miller et al. [1960] is proven to be one of the weakest in comparison to others existing in the literature [ ¨Oncan et al., 2009]. Therefore, the Tree-1 and Tree-2 are also placed among the weakest formulations.
The key finding that emerged from the computational studies is that the Tree-1 and Tree-2 formulations are not tightly constrained, which means that the resulting mixed-integer models have a large feasible region. It is due to this that the proposed formulations can only be applied to very small instances of the Travelling Salesman Problem.
8.1.2 Manhattan formulation
The computational performance of the Manhattan formulation in practice is found to be worse that of Tree-1. For the solution of two small problems (n= 8 and n = 10), noticeably more computational effort is required when implement-ing the former rather than the latter formulation. Therefore, it is concluded that the Manhattan formulation is also loosely constrained.
8.1.3 Recommendations for future work
Future attempts for continuation of this work should focus on improving the Tree-1 formulation. It is strongly recommended that additional constraints are added to the formulation in order to reduce the size of the feasible region. Nonetheless, the search for appropriate tightening constraints is not a trivial task and it might not have fruitful results.
8. Conclusions & Recommendations
At the same time, future work should be directed towards developing pertinent heuristic procedures in order to exploit the hierarchical structure of the asym-metric formulations. Such a heuristic procedure might be successful in generating initial tours that are close to the optimal solution.