Developments in linear and integer programming

In this review we describe recent developments in linear and integer (linear) programming. For over 50 years Operational Research practitioners have made use of linear optimisation models to aid decision making and over this period the size of problems that can be solved has increased dramatically, the time required to solve problems has decreased substantially and the flexibility of modelling and solving systems has increased steadily. Large models are no longer confined to large computers, and the flexibility of optimisation systems embedded in other decision support tools has made on-line decision making using linear programming a reality (and using integer programming a possibility). The review focuses on recent developments in algorithms, software and applications and investigates some connections between linear optimisation and other technologies

Decision-Stage Method: Convergence Proof, Special Application, and Computation Experience

This paper presents a new method for obtaining exact optimal solutions for a class of discrete-variable non-linear resource-allocation problems. The new method is called the decision-state method because, unlike the conventional dynamic programming method which works only in the state space, the new method works in the state space and the decision space. It generates and retains only a fraction of the points in the state space at which the state functions are discontinuous; and thus overcomes to some extent the curse of dimensionality. It carries the cumulative decision-strongs associated with these points, and thus avoids the backtracking entailed by the conventional dynamic programming method for recovering the optimal decisions. A concise and complete statement of the method is given in Algorithm 2 and it is proved that the algorithm finds all exact optimal solutions. In addition the method is adapted for solving some problems with special structures such as block-angular or split-block-angular constraints and the resultant substantial advantages are demonstrated. The performance of Algorithm 2 on many resource-allocations problems is reported, along with investigations on many tactical decisions which have substantial impact on the performance. The performance of the computer implementation of Algorithm 2 is compared with that of the MMDP algorithm and it showed that for the class of problems at which the two are aimed, the decision-state Algorithm 2 performed better than MMDP algorithm both in terms of storage requirement and solution time. In fact, it achieved an order of magnitude saving in storage requirement.

Fuzzy linear programming problems : models and solutions

We investigate various types of fuzzy linear programming problems based on models and solution methods. First, we review fuzzy linear programming problems with fuzzy decision variables and fuzzy linear programming problems with fuzzy parameters (fuzzy numbers in the definition of the objective function or constraints) along with the associated duality results. Then, we review the fully fuzzy linear programming problems with all variables and parameters being allowed to be fuzzy. Most methods used for solving such problems are based on ranking functions, alpha-cuts, using duality results or penalty functions. In these methods, authors deal with crisp formulations of the fuzzy problems. Recently, some heuristic algorithms have also been proposed. In these methods, some authors solve the fuzzy problem directly, while others solve the crisp problems approximately

Sensitivity Analysis as a Managerial Decision Making Tool

Decision making is an integral part of operations management. It may be useful to a decision maker to have some indication of how sensitive an alternative choice might be to the changes in one or more of those values. Unfortunately, it is not possible to explore all the possible combinations of all the variables in a typical problem. In spite of this, there are some elements that a decision maker can use to assess the sensitivity of assumption probabilities. One of the tools useful for the analysis in some decision making problems is sensitivity analysis. It provides a range of feasibility over which the choice of alternative remains the same. Successful decision making consists of several steps, the first and most important being carefully defining the problem. Given that linear problems can be extensive and complex, they are solved by using sophisticated computer methods. This paper will present software solutions available for personal computers (Lindo, POM). For a manager taking the decision, however, a solution model is only part of the answer. Sensitivity analysis offers a better understanding of the problem, different effects of limitations and “what if“ questions. The insights obtained are frequently much more valuable that a specific numerical answer. One of the advantages of linear programming lies in the fact that it provides rich information on sensitivity analysis as a direct part of the solution.feasibility range, linear programming, Lindo, POM, optimum solution, optimum range, sensitivity analysis.

Does choice of programming language affect student understanding of programming concepts in a first year engineering course?

Most undergraduate engineering curricula include computer programming to some degree,introducing a structured language such as C, or a computational system such as MATLAB, or both. Many of these curricula include programming in first year engineering courses, integrating the solution of simple engineering problems with an introduction to programming concepts. In line with this practice, Roger Williams University has included an introduction to programming as a part of the first year engineering curriculum for many years. However, recent industry and pedagogical trends have motivated the switch from a structured language (VBA) to a computational system (MATLAB). As a part of the pilot run of this change,the course instructors felt that it would be worthwhile to verify that changing the programming language did not negatively affect students’ ability to understand key programming concepts. In particular it was appropriate to explore students’ ability to translate word problems into computer programs containing inputs, decision statements, computational processes, and outputs. To test the hypothesis that programming language does not affect students’ ability to understand programming concepts, students from consecutive years were given the same homework assignment, with the first cohort using VBA and the second using MATLAB to solve the assignment. A rubric was developed which allowed the investigators to rate assignments independent of programming language. Results from this study indicate that there is not a significant impact of the change in programming language. These results suggest that the choice of programming language likely does not matter for student understanding of programming concepts. Course instructors should feel free to select programming language based on other factors, such as market demand, cost, or the availability of pedagogical resources

Geometric Ideas in Nonlinear and Multicriteria Optimization

Some geometric properties of the solution set for nonlinear and multicriteria programming problems and the related numeric algorithms are considered. The author deals with necessary and sufficient conditions for nonlinear programming problem stability (in the nonconvex case), with Pareto set stability, Pareto set connectedness conditions, with weak efficiency, efficiency and proper efficiency criteria. A study of numerical algorithms based on geometric properties of the so-called convolutions function is also considered. Necessary and sufficient convergence conditions for large classes of algorithms are presented and easy to check sufficient conditions are given. Further results deal with problems of using local unconstrained minimization algorithms to solve quasi-convex problems and the problem of using some convolution functions for constructing decision making procedures. New classes of inverse nonlinear programming problems are discussed and software implementations of DISO/PC-MCNLP are presented

Towards the implementation of a preference-and uncertain-aware solver using answer set programming

Logic programs with possibilistic ordered disjunction (or LPPODs) are a recently defined logic-programming framework based on logic programs with ordered disjunction and possibilistic logic. The framework inherits the properties of such formalisms and merging them, it supports a reasoning which is nonmonotonic, preference-and uncertain-aware. The LPPODs syntax allows to specify 1) preferences in a qualitative way, and 2) necessity values about the certainty of program clauses. As a result at semantic level, preferences and necessity values can be used to specify an order among program solutions. This class of program therefore fits well in the representation of decision problems where a best option has to be chosen taking into account both preferences and necessity measures about information. In this paper we study the computation and the complexity of the LPPODs semantics and we describe the algorithm for its implementation following on Answer Set Programming approach. We describe some decision scenarios where the solver can be used to choose the best solutions by checking whether an outcome is possibilistically preferred over another considering preferences and uncertainty at the same time.Postprint (published version

Problems of Dynamic Linear Programming

Dynamic linear programming (DLP) can be considered as a new stage of linear programming (LP) development. Nowadays it becomes difficult, maybe even impossible, to make decisions in large systems and not take into account the consequences of the decision over a long-range period. Thus, almost all problems of optimal decision making become dynamic, multi-stage ones. New problems require new approaches. With DLP it is difficult to exploit only LP ideas and methods: even having ,found the optimal program, we often do not know how to use it. This paper represents in some sense the statement of the problem; although it contains a brief survey of DLP, it is focused on the things to be done, rather than on those already being tackled