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Transformation of propositional calculus statements into integer and mixed integer programs: An approach towards automatic reformulation
A systematic procedure for transforming a set of logical statements or logical conditions imposed on a model into an Integer Linear Progamming (ILP) formulation Mixed Integer Programming (MIP) formulation is presented. An ILP stated as a system of linear constraints involving integer variables and an objective function, provides a powerful representation of decision problems through a tightly interrelated closed system of choices. It supports direct representation of logical (Boolean or prepositional calculus) expressions. Binary variables (hereafter called logical variables) are first introduced and methods of logically connecting these to other variables are then presented. Simple constraints can be combined to construct logical relationships and the methods of formulating these are discussed. A reformulation procedure which uses the extended reverse polish representation of a compound logical form is then described. These reformulation procedures are illustrated by two examples. A scheme of implementation.ithin an LP modelling system is outlined
Optimized shunting with mixed-usage tracks
We consider the planning of railway freight classification at hump yards, where the problem
involves the formation of departing freight train blocks from arriving trains subject to
scheduling and capacity constraints. The hump yard layout considered consists of arrival
tracks of sufficient length at an arrival yard, a hump, classification tracks of non-uniform
and possibly non-sufficient length at a classification yard, and departure tracks of sufficient
length. To increase yard capacity, freight cars arriving early can be stored temporarily
on specific mixed-usage tracks. The entire hump yard planning process is covered in this
paper, and heuristics for arrival and departure track assignment, as well as hump scheduling,
have been included to provide the neccessary input data. However, the central problem
considered is the classification track allocation problem. This problem has previously
been modeled using direct mixed integer programming models, but this approach did not
yield lower bounds of sufficient quality to prove optimality. Later attempts focused on
a column generation approach based on branch-and-price that could solve problem instances
of industrial size. Building upon the column generation approach we introduce
a direct arc-based integer programming model, where the arcs are precedence relations
between blocks on the same classification track. Further, the most promising models
are adapted for rolling-horizon planning. We evaluate the methods on historical data
from the Hallsberg shunting yard in Sweden. The results show that the new arc-based
model performs as well as the column generation approach. It returns an optimal schedule
within the execution time limit for all instances but from one, and executes as fast
as the column generation approach. Further, the short execution times of the column
generation approach and the arc-indexed model make them suitable for rolling-horizon
planning, while the direct mixed integer program proved to be too slow for this.
Extended analysis of the results shows that mixing was only required if the maximum
number of concurrent trains on the classification yard exceeds 29 (there are 32 available
tracks), and that after this point the number of extra car roll-ins increases heavily
Water Supply Planning under Interdependence of Actions: Theory and Application
An ongoing water supply planning problem in the Regional Municipality of Waterloo, Ontario, Canada, is studied to select the best water supply combination, within a multiple-objective framework, when actions are interdependent. The interdependencies in the problem are described and shown to be essential features. The problem is formulated as a multiple-criteria integer program with interdependent actions. Because of the large number of potential actions and the nonconvexity of the decision space, it is quite difficult to find nondominated subsets of actions. Instead, a modified goal programming technique is suggested to identify promising subsets. The appropriateness of this technique is explained, and the lessons learned in applying it to the Waterloo water supply planning problem are described
Time-optimal Coordination of Mobile Robots along Specified Paths
In this paper, we address the problem of time-optimal coordination of mobile
robots under kinodynamic constraints along specified paths. We propose a novel
approach based on time discretization that leads to a mixed-integer linear
programming (MILP) formulation. This problem can be solved using
general-purpose MILP solvers in a reasonable time, resulting in a
resolution-optimal solution. Moreover, unlike previous work found in the
literature, our formulation allows an exact linear modeling (up to the
discretization resolution) of second-order dynamic constraints. Extensive
simulations are performed to demonstrate the effectiveness of our approach.Comment: Published in 2016 IEEE/RSJ International Conference on Intelligent
Robots and Systems (IROS
Stochastic Constraint Programming
To model combinatorial decision problems involving uncertainty and
probability, we introduce stochastic constraint programming. Stochastic
constraint programs contain both decision variables (which we can set) and
stochastic variables (which follow a probability distribution). They combine
together the best features of traditional constraint satisfaction, stochastic
integer programming, and stochastic satisfiability. We give a semantics for
stochastic constraint programs, and propose a number of complete algorithms and
approximation procedures. Finally, we discuss a number of extensions of
stochastic constraint programming to relax various assumptions like the
independence between stochastic variables, and compare with other approaches
for decision making under uncertainty.Comment: Proceedings of the 15th Eureopean Conference on Artificial
Intelligenc
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