H. Paul Williams

ANALYSIS OF MATHEMATICAL PROGRAMMING PROBLEMS PRIOR TO APPLYING THE SIMPLEX ALGORITHM

Large practical linear and integer programming problems are not always presented in a form which is the most compact representation of the problem. Such problems are likely to posses generalized upper bound(GUB) and related structures which may be exploited by algorithms designed to solve them efficiently.The steps of an algorithm which by repeated application reduces the rows, columns, and bounds in a problem matrix and leads to the freeing of some variables are first presented. The ‘unbounded solution’ and ‘no feasible solution’ conditions may also be detected by this. Computational results of applying this algorithm are presented and discussed.An algorithm to detect structure is then described. This algorithm identifies sets of variables and the corresponding constraint relationships so that the total number of GUB-type constraints is maximized. Comparisons of computational results of applying different heuristics in this algorithm are presented and discussed.

The Progressive Party Problem: Integer Linear Programming and Constraint Programming Compared

Many discrete optimization problems can be formulated as either integer linear programming problems or constraint satisfaction problems. Although ILP methods appear to be more powerful, sometimes constraint programming can solve these problems more quickly. This paper describes a problem in which the difference in performance between the two approaches was particularly marked, since a solution could not be found using ILP.The problem arose in the context of organizing a “progressive party” at a yachting rally. Some yachts were to be designated hosts; the crews of the remaining yachts would then visit the hosts for six successive half-hour periods. A guest crew could not revisit the same host, and two guest crews could not meet more than once. Additional constraints were imposed by the capacities of the host yachts and the crew sizes of the guests.Integer linear programming formulations which included all the constraints resulted in very large models, and despite trying several different strategies, all attempts to find a solution failed. Constraint programming was tried instead and solved the problem very quickly, with a little manual assistance. Reasons for the success of constraint programming in this problem are identified and discussed.

The reformulation of two mixed integer programming problems

Two practical problems are described, each of which can be formulated in more than one way as a mixed integer programming problem. The computational experience with two formulations of each problem is given. It is pointed out how in each case a reformulation results in the associated linear programming problem being more constrained. As a result the reformulated mixed integer problem is easier to solve. The problems are a multi-period blending problem and a mining investment problem.

Handbook on modelling for discrete optimization

Methods.- The Formulation and Solution of Discrete Optimisation Models.- Continuous Approaches for Solving Discrete Optimization Problems.- Logic-Based Modeling.- Modelling for Feasibility - the Case of Mutually Orthogonal Latin Squares Problem.- Network Modelling.- Modeling and Optimization of Vehicle Routing and Arc Routing Problems.- Applications.- Radio Resource Management.- Strategic and Tactical Planning Models for Supply Chain: An Application of Stochastic Mixed Integer Programming.- Logic Inference and a Decomposition Algorithm for the Resource-Constrained Scheduling of Testing Tasks in the Development of New Pharmaceutical and Agrochemical Products.- A Mixed-Integer Nonlinear Programming Approach to the Optimal Planning of Offshore Oilfield Infrastructures.- Radiation Treatment Planning: Mixed Integer Programming Formulations and Approaches.- Multiple Hypothesis Correlation in Track-to-Track Fusion Management.- Computational Molecular Biology.

The Two-Period Travelling Salesman Problem Appliedto Milk Collection in Ireland

We describe a new extension to the Symmetric Travelling Salesman Problem (STSP) in which some nodes are visited inboth of 2 periods and the remaining nodes are visited in either 1 ofthe periods. A number of possible Integer Programming Formulationsare given. Valid cutting plane inequalities are defined for thisproblem which result in an, otherwise prohibitively difficult, modelof 42 nodes becoming easily solvable by a combination of cuts andBranch-and-Bound. Some of the cuts are entered in a “pool” andonly used when it is automatically verified that they are violated.Other constraints which are generalisations of the subtour and combinequalities for the single period STSP, are identified manuallywhen needed. Full computational details of solution process aregiven.

Connections Between Integer Linear Programming and Constraint Logic Programming-An Overview and Introduction to the Cluster of Articles

The fields of integer linear programming and constraint logic programming have developed from different standpoints in mathematics and operations research, but offer synergy. This article will briefly review some aspects of their separate development that give rise to connections between the two fields and will describe some of these connections which appear to have potential for harnessing and for future research. The article will then introduce the other four articles in this cluster, which further explore and develop these interconnections.

A new framework for the solution of DEA models

We provide an alternative framework for solving data envelopment analysis (DEA) models which, in comparison with the standard linear programming (LP) based approach that solves one LP for each decision making unit (DMU), delivers much more information. By projecting out all the variables which are common to all LP runs, we obtain a formula into which we can substitute the inputs and outputs of each DMU in turn in order to obtain its efficiency number and all possible primal and dual optimal solutions. The method of projection, which we use, is Fourier–Motzkin (F–M) elimination. This provides us with the finite number of extreme rays of the elimination cone. These rays give the dual multipliers which can be interpreted as weights which will apply to the inputs and outputs for particular DMUs. As the approach provides all the extreme rays of the cone, multiple sets of weights, when they exist, are explicitly provided. Several applications are presented. It is shown that the output from the F–M method improves on existing methods of (i) establishing the returns to scale status of each DMU, (ii) calculating cross-efficiencies and (iii) dealing with weight flexibility. The method also demonstrates that the same weightings will apply to all DMUs having the same comparators. In addition it is possible to construct the skeleton of the efficient frontier of efficient DMUs. Finally, our experiments clearly indicate that the extra computational burden is not excessive for most practical problems.

The allocation of shared fixed costs

We consider the problem of sharing the fixed costs of facilities among a number of users. Typically the users have a benefit or revenue from the use of the facilities. Although the problem can be formulated and solved as an integer programme this provides limited accounting information. Such information is often needed in order to (i) decide on which facilities are viable and (ii) to charge the users. It is shown that it is impossible to meet both these needs in a satisfactory way. We examine different ways of partially meeting them. In addition, we consider the issue of fairness among different possible cost allocations and how such ‘fair’ costs may be derived.

Model Building in Linear and Integer Programming

This paper surveys the topic of model building in mathematical programming discussing, (i) the systematisation of model building, including the use of Matrix Generating Languages, (ii) the use of Boolean Algebra for formulating 0–1 integer programming models and the efficient formulation of integer programming models considering both their facial structure and the desirability of creating meaningful dichotemies for the branch-and-bound tree search, (iii) the desirability and possibility of converting models to network flow models, (iv) the building of stable models.

A reduction procedure for linear and integer programming models

A procedure is described for simplifying linear and integer programming models. The procedure performs tests which may: i Detect infeasibility or unboundedness; ii Detect and remove weakly and strongly redundant constraints; iii Detect strongly binding constraints and remove them by a suitable adjustment of the objective function; iv Fix variables and remove them; v Replace constraints by simple bounds; vi Replace columns by bounds on shadow prices; vii Tighten (or relax) bounds on variables and shadow prices.