By Pascal van Hentenryck, Michela Milano
This quantity makes a speciality of the mixing of man-made intelligence and constraint programming to unravel difficulties in operations learn and combinatorial optimization. This quantity collects the contributions of specialists from numerous examine components together with choice idea, structures engineering, propositional satisfiability, mathematical optimization, and synthetic intelligence. those invited students describe and show probably the most very important themes and effects from the final ten years of study on the topic of hybrid optimization. Key positive aspects: - contains either good verified learn effects, and instructions for destiny study. - offers a number of answer tools for universal combinatorial optimization and selection difficulties. - provides either theoretical thoughts and real-world functions in synthetic intelligence and operations examine. Hybrid Optimization can function a important source for graduate scholars, researchers and practitioners learning man made intelligence or operations examine who're attracted to investigating or utilising constraint programming techniques.
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Additional info for Hybrid Optimization: The Ten Years of CPAIOR
If m < n, then dummy task nodes are created so that supply balances demand. The cost cij is set to zero for a dummy task i . A unit flow yij D 1 indicates that worker i is assigned task j . N. Hooker Fig. 7 A flow model for an assignment problem This model can be solved very rapidly with specialized algorithms. Obviously, the solution is meaningful only if each yij 2 f0; 1g, but this is assured by total unimodularity. 10 Circuit Problems Circuit problems have been given several MILP formulations , but by far the most popular is the subtour elimination formulation.
A unit flow yij D 1 indicates that worker i is assigned task j . N. Hooker Fig. 7 A flow model for an assignment problem This model can be solved very rapidly with specialized algorithms. Obviously, the solution is meaningful only if each yij 2 f0; 1g, but this is assured by total unimodularity. 10 Circuit Problems Circuit problems have been given several MILP formulations , but by far the most popular is the subtour elimination formulation. i; j / for which i 2 S and j 62 S . The assignment constraints (line 2) ensure that exactly one vertex precedes, and exactly one vertex follows, each vertex in the tour.
Hooker X is a set of variables, S a set of values (not necessarily numerical), and ` and u are lower and upper bounds. The constraint requires that at least ` and at most u variables in X have a value that is among those in S . Constraints of this kind can enforce the capacity constraints by applying them to every subsequence of five variables for air conditioners and every subsequence of three variables for sun roofs: among ftj ; : : : ; tj C4 g; fb; d g; 0; 3 ; j D 1; : : : ; 46 among ftj ; tj C1 ; tj C2 g; fc; d g; 0; 1 ; j D 1; : : : ; 48 Although this is a correct formulation, it fails to recognize that the among constraints are closely related.
Hybrid Optimization: The Ten Years of CPAIOR by Pascal van Hentenryck, Michela Milano