Some Basics on Tolerances. Gerold Jäger

Transcription

1 Some Basics on Tolerances Gerold Jäger University Halle, Germany joint work with Boris Goldengorin and Paul Molitor June 21, 2006

2 Acknowledgement This paper is dedicated to Jop Sibeyn, who is missed since a snow-hike in spring He has contributed to the results presented here by lively and inspiring discussions.

3 Overview 1 Introduction Sensitivity Analysis Combinatorial Minimization Problems 2 Tolerances Upper Tolerances Lower Tolerances Example

4 Overview 3 Results Upper Tolerances Lower Tolerances Relationship between Lower and Upper Tolerances 4 Future Work

5 Introduction Sensitivity Analysis What is Sensitivity Analysis? Is applied after an optimal solution to a combinatorial optimization problem has been determined. Purpose: Determine how the optimality of the given optimal solution depends on the input data. Also denoted by Post-Optimality Analysis or What-If Analysis.

6 Introduction Sensitivity Analysis Why Sensitivity Analysis? The data used are inexact or uncertain. Determine the credibility of the optimal solution and conclusions based on that solution. Rather significant considerations have not been built into the model due to the difficulty of formulating them. Having solved the simplified model, we want to know how well the optimal solution fits in with the other considerations.

7 Introduction Combinatorial Minimization Problems A combinatorial minimization problem (CMP) depends on the following parameters: finite ground set of elements E set of the feasible solutions D 2 E cost function c : E R objective (cost) function f c : 2 E R (we consider only cost functions of type,, or MAX) Find a feasible solution D with minimum objective cost function. Essential Condition: Set D of feasible solutions is independent of cost function c.

8 Tolerances Upper Tolerances Definition of upper tolerance Let e be an element in a fixed optimal solution S. Upper tolerance: u(e) := maximum by which the costs of e can be increased such that S remains optimal, if the costs of all other elements e E \ {e} remain unchanged.

9 Tolerances Lower Tolerances Definition of lower tolerance Let e be an element not in a fixed optimal solution S. Lower tolerance: l(e) := maximum by which the costs of e can be decreased such that S remains optimal, if the costs of all other elements e E \ {e} remain unchanged.

10 Tolerances Example Consider the asymmetric TSP of a complete graph with the following cost matrix: CMP with objective cost function.

11 Tolerances Example Feasible solutions (tours): (1,2,3,4): Length: = 38 (1,2,4,3): Length: = 38 (1,3,2,4): Length: = 58 (1,3,4,2): Length: = 85 (1,4,2,3): Length: = 49 (1,4,3,2): Length: = 64 Upper tolerances: u(1,2) = 11 Lower tolerances: l(1,3) = 20 l(2,1) = 26 Upper and lower tolerances: u(2,3) = 0 l(2,3) = 0

12 Results Purpose of paper: Overview over the terms of upper and lower tolerances for types,, MAX.

13 Results Upper Tolerances A ground set element e is contained in every feasible solution, iff u(e) =. The upper tolerance of a ground set element e doesn t depend on a particular optimal solution. Let z be the value of an optimal solution S, and v(e) the value of a solution which is optimal under the condition that e / S. Then u(e) = v(e) z, if the cost function is of type. u(e) = v(e) z z c(e), if the cost function is of type. u(e) = v(e) c(e), if the cost function is of type MAX.

14 Results Upper Tolerances Computation of v(e): Set c(e) =. Solve the new CMP and receive value v(e). For a cost function either of type or of type it holds: The optimal solution of the CMP is unique iff u(e) > 0 for all e.

15 Results Lower Tolerances For a cost function either of type or of type it holds: A ground set element e isn t contained in any feasible solution, iff l(e) =. The lower tolerance of a ground set element e doesn t depend on a particular optimal solution. Let z be the value of an optimal solution S, and w(e) the value of a solution which is optimal under the condition that e S. Then l(e) = w(e) z, if the cost function is of type. l(e) = w(e) z w(e) c(e), if the cost function is of type.

16 Results Lower Tolerances For a cost function either of type or of type it holds: e isn t contained in any optimal solution iff l(e) > 0.

17 Results Relationship between Lower and Upper Tolerances For a cost function of type it holds: Provided that no feasible solution is a subset of another feasible solution, then the minimum upper tolerance equals the minimum lower tolerance. For a cost function of type it holds: Provided a connectivity condition, the maximum upper tolerance equals the maximum lower tolerance.

18 Future Work Generalization of tolerances for more than one element of the ground set. Upper tolerance: Let e 1, e 2 be two elements in a fixed optimal solution. How much larger is the optimal solution containing neither e 1 nor e 2? Lower tolerance: Let e 1, e 2 be two elements not in a fixed optimal solution. How much larger is the optimal solution containing e 1 and e 2?

19 Future Work Application of tolerances for speeding up algorithms for CMPs. For example: Traveling Salesman Problem (TSP) Three-dimensional Assignment Problem (3-AP) Capacitated Vehicle Routing Problem (CVRP) Can be used for heuristics as well as for exact algorithms (for example branch-and-bound algorithms).

20 Future Work Thanks for your attention!